TOPICS IN CLASSICAL KINETIC TRANSPORT THEORY

WITH APPLICATIONS TO THE SPUTTERING AND SPUTTER-INDUCED

MASS FRACTIONATION OF SOLID SURFACES

AND PLANETARY ATMOSPHERES

Charles Coffey Watson

A DissertationPresented to the Faculty of the Graduate School of Yale University

In Candidacy for the Degree of Doctor of Philosophy

1980

ABSTRACT

We have investigated several problems pertaining to the sputtering, or ejection, of atomic particles from material bodies following the impact of ions whose energy places them in the nuclear stopping regime.We have addressed both the physical mechanisms underlying these phenomena and their consequences. We begin in Chapter II by considering how the collisional transfer of momentum in a hard sphere gas is affected by the spatial correlations of particles which can not be avoided at high densi­ties. The mutual shielding of nearest neighbors is found to enhance head-on encounters, producing a collisional chaining effect.

In Chapter III we lay the ground work for our subsequent discussion of energy transport by formally reducing the Boltzmann equation to energy variables alone. Then, adopting a variable-radius hard-sphere model for atomic collisions, we derive an expression for the energy spectrum of atoms ejected from a solid surface which agrees satisfactorily with measurements on the sputtering of uranium metal.

The problem of nonstoichiometric sputter-erosion in multicomponent media is addressed in Chapter IV. We first inquire whether such preferen­tial ejection arises as a consequence of the inherent nature of energy sharing in collisional cascades as they occur in an isotropic medium.Both binary targets and an illustrative polyatomic mineral are considered. The negative results lead us to propose that the emerging flux may actually be dominated by those atoms which couple more efficiently, in terms of collisional energy transfer, to the bulk of the medium. This concept results in estimates for Ca isotopic fractionation in the minerals plagioclase and fluorite which are consistent with the available experi­mental data. Since the sputtering process is largely independent of the target's density, a gravitationally bound gas may also be subject to such erosion. In this context, the exobase of an atmosphere plays the role of its surface. But the question arises of how the pairwise binding of atoms into diatomic molecules may affect collisional energy transport in such a system. We find that the mutual shielding of the molecular partners tends to offset dissociational energy loss.

We then proceed in Chapter V to a discussion of the sputtering of a planetary atmosphere by an energetic ion flux, e.g. the solar wind. We utilize both an analytical transport theoretic approach and independent Monte Carlo simulations to estimate atomic and molecular yields, both for cascade-type sputtering in the exobase region and for the direct ejection of primary recoils from the upper exosphere. Under favorable, but not improbable, conditions the atmospheric mass sink generated by solar wind sputtering may be substantial. The sputter-erosion of the Venusian atmosphere is also considered, but only the loss of its helium component is potentially significant. Finally, we point out that certain consti­tuents of an atmosphere may be preferentially sputtered, both because of the diffusive enrichment of the lighter species at higher altitudes and because of their lower gravitational escape energies. Thus, calculations on a model Martian atmosphere suggest that a large depletion of N2 with respect to CO2 could have evolved over a geological period of time. Over a similar period, sputter-induced fractionations of C and N isotopes on the order of those observed by the Viking landers could have been produced.

ACKNOWLEDGMENTS

This research has been supported in part by the United States

Department of Energy at Yale University and by the National Aeronautics

and Space Administration and the National Science Foundation at the

California Institute of Technology.

In many ways I have felt and appreciated the support of Professor

D. A. Bromley and others associated with the A. W. Wright Nuclear

Structure Laboratory at Yale. I am equally grateful to the personnel

of the W. K. Kellogg Radiation Laboratory at Caltech, and particularly^

Professor T. A. Tombrello, for the encouragement they have offered.

My largest debt is that which I owe to Dr. P. K. Haff. He has

not only taught me a great deal of physics; he has also taught me how

one does physics.

My best friend, and wife, Nancy Griffith, has shared many of the

burdens, but few of the rewards which have accompanied the writing of

this paper. She has my deepest gratitude.

Through their example, my parents imparted to me the fundamental

respect for intellectual endeavor which is essential in any undertaking

such as this. I dedicate this work to my mother, and to the memory of

my father.

iTABLE OF CONTENTS

Page

ACKNOWLEDGMENTS

I. INTRODUCTION ............................................... 1

II. MOMENTUM TRANSPORT IN DENSE GASES ........................ 16

A. Impact Parameter Distribution ........................ 16

B. Collision Chains ...................................... 32

III. SPUTTERING OF A MONATOMIC SYSTEM.......................... 42

A. Boltzmann Equation .................................... 42

B. Energy S p e c t r u m ...................................... 53

IV. SPUTTERING OE POLYATOMIC SYSTEMS: FRACTIONATION ........ 69

A. Introduction........................................... 69

B. Binary M e d i a ........................................... 72

C. Polyatomic Media 94 -

D. Surface Flux M o d e l .................................... 100

E. Diatomic Molecular G a s ................................ 118

V. SPUTTERING OF PLANETARY ATMOSPHERES ...................... 131

A. Introduction........................................... 131

B. M a r s ................................................... 135

1. Cascade Sputtering ................................ 135

2. Direct E j e c t i o n .................................. 155

C. V e n u s ................................................. 162

D. Fractionation......................................... 168

1. Exobase Region.................................... 168

2. Upper E x o sphere.................................. 172

VI. S U M M A R Y ................................................... 175

REFERENCES....................................................... 186

TABLES........................................................... 194

F I G U R E S ......................................................... 197

ii

1

I. INTRODUCTION

The phenomenon known as sputtering may occur whenever a material

body is subjected to the impact of atomic particles having energies

much greater than that energy, U, with which atoms are bound to the

bombarded body. Typically, U is on the order of a few electron volts.

The energy deposited in the target medium by the incident particle,

most commonly an ion, may be initially manifested both in the form of

electronic excitations, and as recoil motions of the atoms as a whole.

Before this energy is dissipated to the bulk of the medium, some of

the atoms recoiling in the vicinity of the surface may escape the target.

These latter particles, most of which have energies on the order of U,

are said to have been sputtered. The number of atoms of a given species

ejected per incident ion defines the sputtering yield of that component.

The magnitudes of these sputtering yields vary widely, but to a large

extent predictably, with the characteristics of the incident ions, and

with the structure of the target. The more important general trends

will be outlined below, but anticipating our conclusion, we point out

here the single most important fact which emerges, namely, that under

conditions which are easily attainable in the laboratory, are of

technological interest, and which also prevail in a variety of natural

environments, sputter-related effects can be large. In many instances

(see below) the sputtering yield itself may be large, i.e., on the

order of unity or greater. In other cases a smaller yield may combine

with a sizable incident particle flux or lengthy exposure time to

produce a substantial net result. For these reasons, the sputter-induced

erosion and compositional evolution of surfaces can be significant,

useful or even harmful phenomena in many contexts. Two illustrative

examples which spring to mind are the utility of sputter-etching in

the production of microcircuits, and the deleterious consequences of

the erosion of the walls of a controlled thermonuclear reactor. Our

goal in undertaking the work described in this dissertation was to

contribute to a better understanding of these sputtering processes

with regard both to the underlying physical mechanisms, and to the

consequences of sputter-induced erosion in certain naturally occurring

systems.

The mechanism underlying the sputtering process may be resolved

into a sequence of three steps; one may imagine an ion beam from a

laboratory accelerator impinging the surface of a solid target:

(1) The incident ion deposits some of its energy in the surface region

of the target medium, either through collisions with electrons, which

is the dominant mechanism at higher energies; or else through screened

Coulomb collisions with atomic nuclei, which are more important at

lower energies. (2) The energy deposited in these primary recoils is

shared among other particles in the target via secondary collisions.

(3) Some of these secondary electrons or atoms may then escape the

body if they approach the surface with sufficient energy. The first

two steps cited are involved in effects of ion penetration through

matter other than sputtering, e.g., displacement damage. The work to

be described here, however, deals principally with the latter two

aspects of the process, and more specifically, with the low-energy

regime in which atomic recoils are predominant.

The second stage listed above is not quite so clear cut as it

might appear, for three reasons. If the ion beam is not normally

2

incident, but approaches the surface obliquely the direct ejection of

primary recoils is possible so that (2) is occasionally bypassed.

Similarly, if the target is thin enough so that the ions pass through it,

sputtering may occur from both penetrated surfaces. Sputtering from the

second, or back, surface, known as transmission sputtering, may also be

contributed to by primary recoils. But there can be no primary contri­

bution to the backward sputtering of the first surface penetrated when

the beam is normally incident. We have this latter situation in mind

for most of our subsequent work, but much of the discussion is equally

applicable to the other configurations.

Step (2) may also be circumvented in a peculiar fashion when

certain dielectric and track registering materials are bombarded by ions

sufficiently fast to generate mainly energetic electrons. It then appears

that secondary atomic recoils may arise as the core of positively

charged ions left along the track of the penetrating ion blows apart due

to Coulombic repulsions (Haff, 1976a; Griffith, 1979). The nature

of the ensuing energy sharing process (Seiberling e_t al. , 1980) differs

in several essential respects from that which occurs in the low energy

atomic sputtering case, though, and we shall not pursue the topic further

here.

Among the three steps listed above, the first is most sensitive

to the characteristics of the individual ions and atoms, and least

sensitive to the structure of the target. The manner in which the

impinging ions deposit their energy depends critically on their velocity.

When their speed is comparable to the orbital velocities of the atomic

electrons, or greater, they may transfer energy efficiently to these

electrons. For target atoms of medium mass, this electronic energy loss

3

is greatest for ions having energies on the order of a MeV per nucleon.

At these velocities, however, the Rutherford scattering from the atomic

nuclei is small so that most of the energy loss goes into thfe production

of ionization along the path of the incoming particle. These circum­

stances characterize what is known as the electronic stopping energy

regime. Since it is difficult for the electrons to couple any substantial

amount of their kinetic energy into motion of atomic recoils, because of

the mass difference, ions in the electronic stopping regime generally

produce little sputtering of atoms (except when the ion explosion mechan­

ism is operative). As the energy of the incident particle decreases, its

coupling to the electrons becomes less effective and consequently its

energy deposition per unit path length diminishes. At the same time,

though, the nuclear Coulomb cross section is growing. At some point the

energy transfer to the atomic nuclei begins to dominate the electronic

loss. This point marks the beginning of the nuclear stopping energy

regime.

Ion-atom collisions in the nuclear stopping regime are well described

in terms of a screened Coulomb interaction potential. Such a potential

has the general form

ZZ e2V(r) = — 2— $(i) ,

where r is the internuclear separation, and Z are the nuclear charges

of the incident and target particles respectively, and a is some charac-

teristic screening length. The screening function <!>(— ) approaches unitydi

for r<<a and goes to zero for r>>a. Low energy collisions involve only

radii much greater than a and hence are highly screened, while at higher

energies most of the energy transfer in a collision occurs at smaller

4

radii where screening effects are reduced. A boundary between these two

regions might be defined in terms of the distance of closest approach of2 2the particles in an (unscreened) head-on collision, b = 2ZZ. e /pv ,,b rel

where p = nra / (m+m^) is the reduced mass of the two particles and vre^

is their relative velocity. When b>>a the cross section for the highly

screened collision is relatively insensitive to the energy of the

incident ion and thus this particle's energy transfer in a collision

tends to fall off in proportion to its initial energy. When b<<a on the

other hand, the cross section tends toward the Rutherford type and thus

decreases with increasing ion energy, which also results in a decreasing

energy deposition. Thus the condition b^a provides an estimate of

those conditions for which the energy deposited directly into atomic

recoils achieves a maximum.

A widely used value for a, which derives from the static Thomas-2/3 2/3 -1/2Fermi model of the atom is a = 0.8853 a (Z + Z, ) , where a iso b o

the first Bohr radius. With this, the condition that a'V/b becomes

c . , .v . = v , (a=b) ^ rel rel7 772e b w „2/3 . „ 2/3.1/2

a mm,. o b

From this expression it is seen that the velocity characteristic of the

maximal energy deposition is not especially sensitive to the masses or

charges of the particles. For ions and atoms of intermediate mass, the

energy corresponding to this relative velocity is on the order of 1 keV

per nucleon.

One can understand how the properties of the incident ion andctarget atom influence the sputtering yield in the neighborhood of vre^

on the basis of some simple dimensional arguments. If the energy of the

incoming ion is E^, then the maximum energy which it can transfer to a_2target atom is yE^, where y = 4 mm^(m-Hn^) . This primary recoil will go

on to share its energy among ^ y E^/U secondary recoiling atoms having suf­

ficient energy to escape the target. These recoils are spread out over a

depth of A from the surface on the average, where A, is the beam ion's D bmean collision free path. But only those secondaries within a distance of

A from the surface are likely to be ejected, with A being the low energy

atomic mean free path. That is, of all the recoiling atoms generated by

one primary collision, one would expect a fraction on the order of A/A,bto escape. Thus one might estimate that the sputtering yield,

y e k x

s ' ~ r r • <1-1)b

Since A - (no) and A, = (no, ) \ where n is the density of the targetb band o and o, are the cross sections for the low energy secondary and bhigher energy primary collisions respectively, it is apparent that the

yield is not expected to depend greatly on the target's density. This

fact suggests that sputtering effects may be significant in other than

solid media, a point which we shall later discuss at length.

If one estimates the cross section for screened ion-atom Coulomb

collisions when a ^ b by ^ irab, the essential features of the

sputtering yield may be exhibited:

2

u°” <z2/3 + h m i ' ~ (m4v

a e ZZ, m,s ~ — ------r-rr— ---^ 7 1/2 / ° . . (1.2)

We have dropped the numerical factors here. A more rigorous calculation,

the results of which are given in Eqs. (3.20) and (3.21) of Chapter III

leads, in fact, to a form identical to that of Eq. (1.2). Rather fortui­

tously, expression (1 .2) also agrees quantitatively with the more

accurate derivation, to within about 10%.

An important point to note is that S is independent of the ion's

incoming energy, E^. This is indicative of the fact that the yield

achieves a maximum in the region of energy under consideration here.

We also observe that S is more sensitive to the charges than to the

masses of the ions and atoms. Finally, the yield varies inversely with

the target atom’s binding energy and with its low energy cross section.

For intermediate mass particles, e.g., Ar ions bombarding a Cu target,

the yield given by Eq. (1.2) is on the order of unity, as indicated

earlier.

It should be evident from our discussion that a reasonable estimate

for the sputtering yield may be obtained without recourse to detailed

calculations. Conversely, the comparison of empirical observation with

calculated yield values can provide only a limited amount of information

on the mechanism underlying the sputtering process. For this reason,

we have sought to explore certain other aspects of the sputtering phenom­

enon which depend more sensitively on the two remaining and less well

understood steps in the process.

The problem of the sharing of the primary recoil's energy among

recoiling secondary particles has at least two facets which are accessible

to experimental examination. For instance, the number of secondary

recoils in a monatomic target undergoing sputtering, as a function of

their kinetic energy, characterizes the dissipative process in a

straightforward way. Measurements of the energy spectra of ejected

particles thus provide a useful window on some of the details of the

transport mechanism. Furthermore, it is observed that the various

components in a polyatomic target medium are not always sputtered exactly

in proportion to their abundances therein. The explanation of this

nonstoichiometric partitioning of recoil energy among the constituent

species is another challenge to our understanding of the energy transport

process.

Our analysis of these questions is based on the concept of a cascade

of binary elastic collisions between the recoiling atoms. We shall treat

these cascades through the formalism of classical kinetic transport

theory. However, certain questions concerning the validity of such an

approach arise when the density of the medium is that appropriate to a

solid target. It is clear, for instance, that the assumption of molecular

chaos fundamental to Boltzmann's transport theory can not be retained in

a solid target. This problem is circumvented in the present circumstance

by the fact that we do not need to consider correlated collisions if we

assume that the density of recoils is so low that collisions between

energetic particles remain negligible, as is usually the case. A more

persistent problem arises from the fact that at high densities, the

mutual shielding of nearest neighbors can restrict the range of impact

parameters available to the collision partners. Thus the transferal of

energy between two colliding atoms does not proceed precisely as it would

in a rarefied gas. In Chapter II we investigate this latter problem

through a calculation of the impact parameter distribution as a function

of density in a model hard sphere gas. We find that as the density of

the system increases, collisions tend to be more head-on. A pulse of

8

momentum may be propagated through a chain of such collisions with

inhibited dissipation, even in a spatially random system. The proper­

ties of such collision chains will be discussed. An attempt to relate

this work to the interactions of atoms in a solid suggests that such

correlation effects may be significant in the energy sharing processes

leading to sputtering. But a final answer is postponed until a transport

theory framework for the analysis of this dissipation is developed. This

latter is the task undertaken in Chapter III.

We commence Chapter III with a formal reduction of the Boltzmann

transport equation to energy variables. The time dependence is removed

by restricting ourselves to the case in which primary recoils are

generated by the incident ions at a constant rate. The aim is to

explicitly map out the context of, and provide the basic tools for, our

subsequent discussion of transport. For the purposes of this derivation,

we refer to an amorphous, isotropic system of identical particles. Real,

solid materials generally have some structure, however. One worries,

for example, about the fact that in a covalently bound solid, the atoms

within certain sets are bound more strongly to each other than to atoms

in other sets. At the recoil energies relevant for sputtering (> 10 eV),

though, this asymmetry is probably not very important. There is also

the possibility that the interaction between two atoms at low energies

in an ionically bound solid might depend somewhat on their electron

affinities. On the whole it appears that metallic targets are the

simplest materials to model. One may take the view that the atomic

cores in a metal are bound together principally through their inter­

actions with the electron sea in which they are embedded. For this

reason, metals tend to crystallize in close packed structures. Such

9

crystallization, not just in metals, but in any material, introduces

another new element into the transport process in that collision chains

may propagate along lattice rows in a highly cooperative fashion. Such

effects may lead to enhanced sputtering in directions correlated with

the crystal's axes of symmetry. We shall limit our considerations to

systems in which such structural effects can presumably be safely

neglected.

The second section of Chapter III is devoted to a discussion of

the energy spectra of sputtered atoms. The canonical hard sphere gas3

model predicts an energy dependence of the form E/(E+U) . But recent,

precise experiments on uranium metal (Weller, 1978) demonstrate signifi­

cant departures from this rule. The solution we propose to this problem

is based on the concept that real atoms are better treated within the

hard sphere model if their radii are allowed to vary with the energy of

their collision. Quantitatively, the results of this model compare

favorably with the observations.

The second piece to the energy sharing problem mentioned above,

namely, the partitioning of recoils among the species in a multicomponent

medium, is taken up in Chapter IV. We were motivated to pursue this

topic by the availability of extensive, accurate data on sputter-induced

fractionation of Ca isotopes in several minerals (W. A. Russell, 1979).

The term fractionation refers simply to the difference in stoichiometry

between the sputtered and bulk materials. The challenge, and opportunity,

offered here is to determine whether the origin of the observed preferen­

tial sputtering of lighter isotopes can be found within our transport

theoretic approach.

The first possibility examined, in Section IV.B, is that the

differences in mass and size between two species might lead to a

10

substantial asymmetry in the generation of recoil fluxes in the interior

of the target as collisional cascades develop. (One knows fortiori

that the fluxes emanating from the surface are fractionated.) Perhaps

surprisingly, this turns out not to be the case. That asymmetry which

exists is due to differences in the couplings of the atoms to the

impinging ions, and fades away at lower energies. The net fractionations

one derives from this mechanism are an order of magnitude smaller than

those measured. These calculations are extended to an illustrative

polyatomic medium in Section IV.C, with the conclusions remaining

unchanged.

The negative results of Sections IV.B and C lead us in the following

section to rethink the physics pertaining to the escape of atoms from

the surface of a target. We propose that at a solid-vacuum interface,

the recoil flux is substantially reduced from its value at greater depths,

due to the loss of the escaping recoils from the cascade. The remaining

flux is thought to consist primarily of those atoms which have suffered

only one energetic collision in the course of the cascade in which they

are produced. This "surface" flux is found to be naturally fractionated,

with those species which are more efficiently coupled to the bulk of the

medium, in terms of energy transfer, being favored. The quantitative

comparison between this model and measurements made on the minerals

plagioclase and fluorite, is encouraging. Closer connection between

experiment and theory is attained through calculations of the time

evolution of a surface undergoing sputtering, and the results are again

satisfactory.

There are only two basic ingredients in the sputtering process,

or, at least, two requirements for the applicability of the discussion

11

of Chapters II, III and IV. There must first be an ensemble of atoms

bound together with a separation energy U per atom. Then, there must

exist a source of atomic particles having energy much greater than U.

Such source-target systems can and do exist naturally. The most promi-

nant source of energetic atomic particles in the solar system is the

sun itself. Hydrogen and helium nuclei (and trace amounts of other

particles) stream outward from the sun through almost all of interplan­

etary space. This particle flux is known as the solar wind. The

typical kinetic energy of the protons and a-particles is 1 keV/nucleon,

which suggests that they would be effective in sputtering. Indeed, any

object not shielded from this solar wind flux will be subjected to

sputter-induced erosion. Such objects include interplanetary dust

grains, the surface of our moon and, perhaps, the tails of comets. A

slightly less obvious target is considered in Chapter V, namely, the

atmosphere of a nonmagnetic (and thus unshielded) planet.

We have already pointed out that the sputtering yield of a target

should not depend significantly on its density. On the other hand, it

would seem that an atmosphere does not possess a well defined surface.

The essential characteristic of a surface as far as sputtering is

concerned is that an atom moving within the surface of a body is likely

to suffer a collision, where as one moving above the surface is not.

But this statement is a good description of the exobase of an atmosphere.

Moreover, the gravitational binding energy of a molecule to a planet

such as Mars is comparable to the chemical binding of an atom at a

solid's surface. To a large extent, then, the exobase plays the role

of the surface of an atmosphere in the context of sputtering phenomena.

12

There are, of course, several novel features of an atmospheric

target. One of these is the binding of the target atoms into small

clusters, i.e., molecules. Since we have previously treated solid

targets as if they were comprised of a monatomic gas, the question arises

as to what consequences this molecular binding may hold for the energy

sharing process as we have outlined it. In Section IV.E we find that

two important effects accompany this binding. In the first place, the

molecular association of the atoms results in a mutual shielding which

tends to reduce the average low energy collision cross section per atom

and hence increase the yield (recall that S a ^). Secondly, the energy

lost as molecules are dissociated in collisions tends to reduce the

yield. These two trends nearly compensate each other for typical

molecular targets, so that the calculational tools we have developed may

be applied with only slight modification to the problem at hand.

In Chapter V therefore, we proceed to an analysis of sputter-

induced mass loss from the atmospheres of Mars and Venus. Section V.A

contains a brief outline of the nature of solar wind flow past a nonmag­

netic planet. In Section V.B we delineate the atmospheric sputtering

process with specific reference to Mars. Two distinct mechanisms are

pointed out. When the solar wind penetrates to the exobase of an

atmosphere, cascade-type sputtering ensues. We present calculations for

this mechanism based on the transport theory developed in Chapter IV.

Support for this model is offered in the form of an independent Monte

Carlo computer simulation of atmospheric sputtering. The general

features of the process appear to be well understood. When the solar

wind does not penetrate to the atmosphere's exobase, sputtering occurs

primarily via the direct ejection of the primary recoils. Numerical

13

estimates for mass loss rates to be expected from both mechanisms

are presented in a format in which the poorly known details of the

actual solar wind flow are separated out into multiplicative structure

factors. We conclude that under favorable, though not unreasonable

circumstances, sputter induced mass loss may have played a significant

role in the history of the Martian atmosphere.

Turning to Venus in Section V.C, we note first that because of

the massiveness of its atmosphere, sputter-erosion is of little conse­

quence in terms of total mass loss. On the other hand, helium, which is

diffusively enriched in the upper Venusian atmosphere, may be sputtered

to a significant extent. Of course, the solar wind may also deposit He.

The balance bewteen these two processes, as well as other sources and

sinks of He, is explored.

Both because the lighter components of an atmosphere tend to be

enriched in its upper reaches, and because they have lower gravitational

binding energies, these lighter species may be preferentially sputtered.

Substantial sputter-induced mass loss would thus lead to both the

chemical and isotopic fractionation of an atmosphere. Section V.D

presents an analysis of such possible fractionation effects in the

Martian atmosphere, based on a two component model (CO^ and

Prolonged sputtering can lead to a large depletion of ^ with respect15 14to CC^. At the same time, a large Increase in the N/ N isotopic

ratio would occur. The calculated enrichment is similar to that

observed by the Viking landers.

On the whole, the investigations of Chapter V establish that the

exospheric mass sink generated by the collisional ejection of neutral

species is a factor which must be reckoned with in the discussion of

14

the evolution of any atmosphere which is subjected to direct impact

by an energetic particle flux such as the solar wind.

We conclude this dissertation by a brief summary of the more

important concepts which we have developed in the course of our

research.

15

16

II. MOMENTUM TRANSPORT IN DENSE GASES

A. Impact Parameter Distribution

An energetic ion passing through a material body may suffer a

collision with one of the atoms comprising that body, imparting to it

a certain amount of kinetic energy. The central task of any model of

sputtering is to describe how this primary recoil energy is distributed

among the other atoms of the medium in the course of time. The usual

approach is to discuss this problem in terms of a cascade of two-body

secondary collisions. But, of course, the concept of true binary

collisions in a many-body system is physically well founded only when

the average spatial separation between nearest neighbors is very much

greater than the range of the interatomic potential. Nevertheless,

even in the case of a solid target the program of resolving the energy

dissipation process into a cascade of two-body collisions provides a

reasonable approximation; the reason being that the kinetic recoil

energies in which one is typically interested are much greater than

the interatomic potential energy between nearest neighbors at the

average separation distance. Thus, in a close encounter between two

atoms in which a large amount of energy is transferred, the perturba­

tion introduced by other neighboring atoms may be neglected. The

collisional cascade picture is the one which we have adopted for our

analysis of sputtering.

The fundamental problem which remains, therefore, is to determine

the probability that the first collision suffered by an energetic recoil

with an essentially stationary atom (i.e., one of thermal energy) will

result in the transfer of a given amount of kinetic energy. In terms

of classical mechanics, this question actually has two parts. In the

first place, it is necessary to know the atomic interaction potential.

Secondly, one must specify the probability that the collision will

involve an impact parameter between, say, b and b+db. It is the

density dependence of this latter distribution on which we wish to

focus here. We choose to discuss this problem with reference to a gas

of particles interacting as hard spheres. We do this in part because

it allows many of the essential features of the physics involved to be

expressed in relatively simple mathematics, in part because the true

atomic potentials are not well known, and finally because the hard

sphere model for low energy atomic collisions is used extensively in

our discussions of energy sharing in later chapters.

The probability that a given hard sphere, moving some distance

through a uniform gas of similar particles, will collide only with a

particle in the impact parameter interval (b,db) is the product of the

probability that this incident sphere will find a target sphere within

(b,db), times the probability that there are no other target particles

in the volume swept out by the first sphere. The first probability

here depends only on the two-body spatial correlation function (this is

the radial distribution function referred to below). The second, however,

involves three-body, four-body and all higher order correlations of

position. In a rarefied system such many-body correlations may be

neglected, but at high densities they become essential to the calculation

of the impact parameter distribution. In this section we offer a simple,

intuitive scheme by which one may handle these correlations. Although

elaborate techniques for the calculation of two-body correlations in

fluids have been developed in recent years (useful surveys may be found

in McQuarrie [1976] and Goodstein [1975]), we shall not make use of

these results here. Instead, we adopt an approximation for the radial

17

distribution of nearest neighbors in a gas, the use of which allows

us, in effect, to incorporate all higher order position correlations

in a self-consistent, tractable fashion. These ideas will be clarified

in the course of the following discussion.

Consider then a homogeneous, isotropic gas of hard, completely

elastic spheres, each of diameter s and mass m. (We shall later

generalize our results to include the case of inelastic collisions.)

Assume that these particles are randomly distributed in space and that

they are all at rest, save for one which has an initial momentum ’p’po2 °

and energy Eq = • We ask for the probability P(b;x)db that the first

collision of this sphere involves an impact parameter between b and b+db.

The variable x used here parameterizes the density of the gas; s+x is

defined as the average separation between nearest neighbors, as in

Fig. 2.1. The dashed circle in Fig. 2.1 represents the so-called "sphere

of influence" of the moving, or incident, sphere inside of which the

center of no target particle may enter. A collision occurs in this

picture when the moving sphere of influence contacts a stationary scat­

tering center, i.e., the center of a target sphere. Of course, no two

scattering centers may lie closer together than a distance s; their

average separation distance is s+x. The problem of calculating P(b;x)

as a function of the density is reduced here to determining the effects

of the mutual shielding of the scattering centers due to their position

correlations. We do not propose a rigorous treatment of this problem but

offer an approximate solution based on plausible arguments.

Consider first the problem of relating the separation parameter x

to the number density of the gas, n(x). As x->-0 we wish n(x) to approach

the maximum possible density for a collection of hard spheres, viz., that

18

which obtains for a hexagonally close packed (hep) crystal. In such a

crystal the hard spheres occupy 74% of the total volume (Kittel, 1976),

so that the density is

0.74 / T

19

n (x=0) =

Now suppose that the lattice parameter is increased from s to s+x, as

shown in cross section in Fig. 2.2, while maintaining the hep structure.

Obviously, the average separation distance of nearest neighbors is well

defined and equal to s+x. We now argue that a gas of constant density

may be modeled by displacing each particle in Fig. 2.2 in some random

manner about its lattice site, but that this randomizing will not alter

the average separation between a given hard sphere and its 12 (in 3

dimensions) nearest neighbors. In effect, this amounts to adopting a

specific function for the radial distribution of particles about some

given central particle. Such radial distribution functions form the

central element of the modern microscopic theory of liquids since,

assuming only two body interactions, they completely characterize all

the thermodynamic properties of the liquid (McQuarrie, 1976). With

this assumption, then,

n(x) = ° - 7,4 = ^ ■- . (2 .1 )j i r ( ^ ) 3 (s+ x)3

This picture, relating the random gas to an hep lattice, is

also useful for analyzing the position correlations of the scattering

centers in a collision. In Fig. 2.3 the incident particle is shown

moving a distance dr. Its sphere of influence sweeps out a volume 2dV = 4tts dr. If the gas is quite dilute, x>>s, then one could argue

that the density of scattering centers is uniform over dV so that

P(b;x)db is simply proportional to 2Tibdb. For x<<s, however, the

scattering centers are restricted to lie in a more or less diffuse

spherical shell of (mean) radius s+x about the initial position of the

incident sphere. In this case the distribution of scattering centers

is certainly not uniform over dV and consequently P(b;x)db will be

modified. By confining the nearest neighboring scattering centers to

a spherical shell about the incident particle we have, in an approximate

manner, accounted for correlations in radial positions of the spheres.

In what follows we shall treat the scattering centers as if they all

lay on the "scattering sphere" of radius s+x. But a knowledge of this

average radial distribution is not sufficient to determine where the

first collision will occur. Thus we make the additional assumption1 2that any area element on the scattering sphere of magnitude yy 4tt(s+x )

and of more or less circular shape will contain one and only one scatter­

ing center, and that this center is equally likely to be found anywhere

within this area. Although this assumption is well justified as x-*-0,

for large x it is true only in an average sense. Nevertheless, it

provides an approximation for the angular position correlations of the

target particles.

The sphere of influence of the incident particle is thus to be

thought of as passing through the scattering sphere, sweeping out a

maximum of yy the area of the latter. The probability that the impact

parameter of the ensuing collision is in (b,db) is just the area dA of the1 2target sphere included between b and b+db, normalized to — Air (s+x) .

If B is the angle between the direction of motion of the incident sphere

and the line of centers connecting the incident and target spheres, as

20

21

shown in Fig. 2.3, then

so that

P(b;x)db = — -Y 2 4*(s+x)

12 2trbdb2 4n(s+x)2 cos B

But

so

P(b;x)db = (2 .2)

This function is plotted versus b in Fig. 2.4.

A maximum allowed impact parameter exists in this model, determined

The fact that glancing collisions, i.e., those with b>b , aremaxforbidden here is a consequence of the mutual screening of the target

particles. In other words, before the incident sphere can travel far

enough to collide with a particle at large b it must with virtual

certainty collide with another, shielding, target sphere at smaller b.

The discontinuity in P(b;x) at b=bmav is a result of our approximations

and would presumably, in a more elaborate theory, be somewhat smoothed

by

bmaxP(b;x)db = 1.

0

With P(b;x) given by Eq. (2.2), one easily finds

(0.55)

out though still sharp for x<<s.

Equation (2.2) may be expressed more concisely in terms of the angle

(Fig. 2.3). Since sin 6 = - j 4 . we find(s+x)

P(g;x)dg = 6 sin g dg, cos g> •

Introducing the azimuthal angle 4>, over which the probability of

collision is uniformly distributed,

P(fi;x)dft = — , cos g> |n (4lT)

where dft = sin3dgd<J>. In other words, the probability per unit solid

angle that the first collision of the incident sphere will occur at

angles (B,) is constant, where the angles (S,<t>) are measured from the

initial position of the sphere and referenced to the direction of its

velocity.

As it stands, Eq. (2.2) is incomplete for physically we know

that b cannot exceed s. The condition that b = s corresponds to a max maxcritical value of x,

xc = 8 ( 0 3 5 - 1) = (0.81)s .

Evidently, for x>xc> the sphere of influence may pass completely

through the scattering sphere without intercepting one of the nearest

neighboring scattering centers. We must then consider the distribution

of target particles at larger distances. If we choose one particle in

the hep structure of Fig. 2.2 and proceed radially outward, we note

that the fluctuations in the density of scattering centers as a function

of r decreases rapidly for r>s+x. This must be even more the case when

x>xc in our randomized lattice picture of the dense gas. Indeed, we

recall that the scattering sphere itself is more correctly a diffuse

spherical shell whose diffuseness increases with increasing x. Hence

we shall assume that once the incident sphere has penetrated the

scattering sphere it finds additional scattering centers essentially

uniformly distributed in both angle and radius. This is to say that

we shall neglect all position correlations beyond those described

by the scattering sphere of nearest neighbors. This approximation

is deemed to be sufficient since we are primarily concerned with

collisions in dense systems. Our main interest here is simply to

connect the model expressed by Eq. (2.2) with the corresponding

situation in a rarefied gas, in a consistent manner.

Although it is true that both molecular dynamics calculations and

more elaborate analytical models have been used extensively to investi­

gate the radial distribution of particles in dense fluids (Goodstein,

1975), these results do not bear directly on the problem at hand since

they involve only two-body, and not higher order spatial correlations.

We shall therefore be content with our present approximation.

The contribution to P(b;x)db from those target particles located

in the scattering shell is, for x>xc> still given by Eq. (2.2), but since

23

6b db .1172 = -1’/ bb db

(s+x) [<s+*)2 - b2]J

there is an additional contribution, uniformly distributed over the

cross section of the incident sphere, given by

u - uits

24

Thus the total P(b;x) for any x is

1

6b [(s+x)2 - s2] 2 - 5b\ \ H(x - x ), (2.3)^(s+x) “ j ^ 2

where H is the Heaviside step function defined by

H(x - y) =0 x < y

1 x > y

It should not be thought that the derivation of Eqs. (2.2) and

(2.3) applies only to some special particle which happens to be

symmetrically located at the center of a scattering sphere. Rather,

this sphere is taken to represent the average correlations in position

of the nearest neighbors of any particle in the dense gas. Thus P(b;x)

applies not only to the initially incident particle A in Fig. 2.1 but

also to the subsequent collision of the target particle B. The second

collision of A however is not well described by P(b; x) since the fact

that A has followed a certain collision free trajectory is not taken

into account in determining the distribution of scattering centers.

Of course this last point is obviated as x-*00 since then the distribu­

tion of scattering centers is spatially uniform. Another point of

possible confusion is that we have apparently assumed for x<x£ that

the incident sphere definitely has a collision with the scattering

sphere. In fact, for x>0 there is a finite probability that the

moving particle will pass through the scattering sphere without

collision. We have merely assumed that P(b;x) may be determined

from the characteristics of an "average" collision, namely, one in

which the incident particle sweeps out a volume occupied on the average

by one target particle. This concept is analogous to the situation in

a dilute gas wherein the average distance, X, travelled by a particle

of cross section a between collisions through a stationary background,

is such that

Xo = 1/n .

Before proceeding with a discussion of the above expressions for

P(b;x) we digress briefly to mention that Sanders and Roosendaal (1976)

have also considered the problem of calculating the impact parameter

distribution as a function of the density. However, their work is

predicated on certain hypotheses to which we must take exception.

Considering atoms which interact via an inverse power-law screened

Couloumb potential they assume that the moving particle interacts only

with its nearest neighbor in the forward directed spherical half-space.

This collision is treated as if it proceeded unhindered by other

neighboring atoms. Now interatomic potentials are well described by a

given inverse power law only within a restricted range of interatomic

separations. Beyond some maximum radius, Rq , the atomic potential is

essentially zero. As long as the separation of two atoms exceeds Rq

they will not interact significantly, whether or not they are nearest

neighbors. Thus, when the density of the gas decreases to the point

that the average nearest neighbor separation distance exceeds Rq, the

nearest neighbor interaction hypothesis is not adequate. On the other

hand, it is difficult to see how one may isolate two-body collision

processes for such interatomic potentials when the average nearest

neighbor separation is less than Rq , i.e., at high densities.

25

A second hypothesis of Sanders and Roosendaal (1976) is that

the distribution of atoms in the gas is given bv the Poisson law,

according to which the probability of finding k atoms in a volume V is

(nV)k -nV~ k T 6

where n is the average number density. Clearly it is necessary that 3V>>kRQ. In addition, this distribution is based on the assumption that

the probability of finding any given atom of the gas in V is small and

independent of the position of the other atoms in the gas. In order3

for this assumption to be well justified it is necessary that nRQ<<l.

Neither of these conditions is met when one is discussing the distribu­

tion of nearest neighbors in a dense system. The Poisson distribution

is completely inadequate for the description of the position correlations

which, as we have seen, are so characteristic of a dense gas.

Returning to our model of a hard sphere gas, we note that in some

sense x = x determines a critical density, n(x ). When n>>n(x ), c c ccollisions involve a great deal of coordination and screening among the

target particles, while for n<<n(x^) the correlations are negligible

and the dilute gas picture is appropriate. Substituting x£ = (0.81)s

into Eq. (2.1), we find

n(x ) = 1

26

c 4 3y its

Thus at the critical density the average volume per particle is just the

volume of its sphere of influence. In terms of the maximal density n(0),

n(x ) = (0.17)n(0) . c

A more quantitative expression of the criticality of nCx^) is

given in Fig. 2.4, wherein P(b;x) is plotted versus b (in units of s)

for various densities. The straight line labeled “ is the well known

distribution of impact parameters for a dilute gas, P(b) = • For

x<xc , P(b;x) is seen to increase more rapidly with increasing b than

does the geometric cross section 2irbdb, until b = b , at which pointmaxP(b;x) goes to zero. As x increases from zero toward x , P(b;x)crapidly approaches the limiting distribution P(b;°°); indeed, this approach

might be characterized as being linear in x. For x>X£ the approach to

P(b;°°) is asymptotic, but as a practical matter P(b;xc> is already well

approximated by P(b;°°). The critical density n(xc) thus does in fact

mark a quite sharp transition in the nature of the impact parameter

distribution. One might make the analogy that as n increases through

n(xc), a disordered hard sphere gas "freezes" into a highly ordered,

or position correlated, state.

The concept of a critical density is not limited to a hard sphere

gas. In order for the preceding arguments to go through it is not

essential that two particles interact at a point in the radial dimension

but only that their region of interaction be small on the scale of

their average separation length. Such a situation might arise, for

example, in the low energy, 10eV<E<lkeV, highly screened collisions

between atoms in a solid matrix. Physically, such collisions are

important in the collisional cascade model for the sputtering of solid

surfaces by low energy (^l kev/amu) ion beams, which is developed at

some length in Chapters III and IV. The question naturally arises,

then, as to what restrictions on atomic collisions may be implied by

the present discussion of a hard sphere gas. We can illuminate this

27

connection through the example of a typical metallic target medium.3

Consider then an Au target of density 19.3 gm/cm . According toO

Eq. (2.1), s+x = 2.88 A, which is precisely the experimental metallic

diameter of a gold atom (Samsonov, 1968). This agreement is a result

of the fact that metals tend to crystalize in hep and other close-

packed structures. Now in the energy range under consideration the

best available analytic approximation to the screened Coulomb interaction

potential between two atoms is a Born-Mayer potential of the form

(Sigmund, 1972)

V(r) = A e"r/aBM , (2.4)

3/4 °where A = 52 (Z,Z.) eV, a_w = 0.219 A, and r is the internuclear i I BMseparation. If we identify the screening length a with the regionBMgof interaction then, since ^10 , we may sensibly discuss the

present system in terms of our hard sphere formalism. Thus, let us

define the "diameter" of an Au atom as the distance of closest approach

in a head-on collision between two atoms having a total relative

(center-of-mass) energy E . That is,cm

E = V(s) = A e~S^aBM cm

or,

s ( E ) = a £n ( A / E ) . cm BM cm

On the basis of this definition, we may estimate that energy below

which the solid medium begins to look dense to a moving Au atom of

laboratory energy Ec = 2E^m impinging upon stationary atoms, which is

the case of most importance in models of sputtering. We require that

28

energy at which s+x£ = 2.88 A. Recalling that x^ = (0.81)s, we find

Ec (Au+Au) = 51eV. Applying a similar analysis to a Cu target, one

finds E (Cu+Cu) = 25eV. Likewise, for uranium metal E (U+U) = 39eV. c cThe significance of these values is that for energies E>EC it

should be reasonable to treat a solid essentially as if it were a

dilute gas in a discussion of collisional mass and energy transport.

Such a situation arises in the context of sputtering when the energy

deposited by the incident beam is degraded through a series of nuclear

collisions in which kinetic energy is shared between atoms as a whole

instead of going into electronic modes of excitation. The quantitatively

most successful theory of sputtering in this regime to date (Sigmund, 1969)

is indeed based on the linearized Boltzmann transport equation as it

applies to rarefied gaseous systems. A prominent feature of such a model,

confirmed by experiment, is that a large majority of the atomic recoils

participating in a cascade have energies on the average less than a few

tens of eV, and therefore on the order of E . But the above analysiscsuggests that the position correlations of the target atoms might have a

noticeable effect on the energy sharing process at these energies. We

shall return to this question in the next chapter. The more immediate

problem, however, is to elucidate the general features which distinguish

energy transport in dense systems from that in rarefied ones. To this

end we continue our discussion of the hard sphere gas model.

For the purpose of comparison, consider first the limiting form

of P(b;x) for x -*■ °°. From Eq. (2.3)

29

, ... 2bdbP(b ;°°)db = — - j - .s

This can be expressed in terms of the conventional hard sphere scattering

Of more interest for the energy sharing problem is the probability that

a particle of initial lab energy E will, in a collision with a stationary

particle, transfer an energy between T and T+dT to that particle. Theb2

kinematics of such a collision imply that T = E (1 - so thats

/P(T;°°)dT = 41 (2.5)E

Consequently, for the collision of two equally massive hard spheres in

a rarefied system any energy transfer, 0<T<E, is equally probable, as

is well known. This means that the energy sharing in a collisional

cascade is quite efficient, in that the energy of the initiating particle

tends to be efficiently distributed among the recoils which it generates.

The consequences of this form for P(T;°°)dT are discussed in detail in

Chapter III.

In the opposite extreme, when x-*-0 and the spheres are just

touching, P(b;x) reduces to

P(b;0)db = — , 1 , 0<b<bs , 2 ,2,-r- max(s - b )2

This is more simply expressed in terms of the probability for a given

momentum transfer p. If pQ is the momentum of the incident particle 2 2 1 / 2then p = p (1 - b /s ) , so that

31

P<P:0>dP=T (2 .6)

Now, we have seen that for x<x there exists a b <s such thatc max-collisions with ^> )inax are forbidden. Consequently, there is a non­

zero minimum allowed momentum transfer, p , = p cos a , where a ismin o maxthe angle between the velocity of the incident particle and the line

of centers of the two spheres on impact (Fig. 2.3). For x->-0,2 2 1/2 5cos a = ( l - b /s) =-r. Expression (2.6) may thus be written:max max 6

P(p;0 )dp = 7=-----^ -----rr- = 7--- ^ 7 , p , <p<p . (2.7)r ^ _ cos a )p (p - P . ) min v *0max o vto *inin

In contrast to the uniform energy sharing which occurs in the low density

case, at maximum density it is momentum sharing which is uniform, with

any allowed momentum transfer, Pmin-P-P0 ’ being equally probable.

Eq. (2.7) may be expressed in terms of the energy transfer T as

o «T o

P(T;0)dT = -z-rz------ — --- .-.^.1/2 , E cos a <T<E .2(1 - cos a )(ET) ' ’ max-max

2Although energy transfers less than E cos a are forbidden, withinmaxthe allowed range of T lower energy transfers are favored. Energy

sharing in this case is not efficient.

An implicit assumption embodied in Eqs. (2.5) - (2.7) is that

the collision between the spheres is perfectly elastic. The case of

inelastic collisions may also be treated, with only a slight modifi­

cation of the above formulae. We shall assume that the inelasticity

of the particles may be described by a coefficient of restitution, e,

which is defined by

where vre and vrg are the magnitudes of the initial and final relative

velocities of the two spheres along their line of centers at impact.

Given this, it follows that the initial momentum and the momentum

transfer in a collision are related by

(1+e)p = — ~— cos a p 2 ro

Defining f = (l+e)/2 for convenience, Eq. (2.5) thus becomes

P(T;°°)dT = , (2.5a)f E

while for x->-0, the expression corresponding to Eq. (2.7) is

P(p;0)dp = -ri------ — -----— fp cos a <Plfp • (2.7a)r (1 - cos a )fp ’ *o max- romax o

Note that f = 1 when the collision is perfectly elastic as before,

while f = y (e = 0) is appropriate for a completely inelastic collision.

For simplicity the following arguments will be developed for f = 1.

The extension to f 4 0 is straightforward; only the results will be

given.

B. Collision Chains

The fact that, in a close packed hard sphere system, a particle

must transfer at least of its momentum to its first collision partnerosuggests the concept of a collision chain. The (n+l)tk particle in

this chain is just the first collision partner of the n ^ particle. One

may picture a momentum pulse propagating down this chain with a certain

amount of leakage occurring at each step. The concept is useful since

the amount of momentum lost to recoil in each collision will be only a

small fraction of the momentum in the pulse at that point. Let us

suppose the chain commences with a particle of momentum pQ . The second2particle will have cpo<p<pQ ; and similarly for the third, c po<p<pQ ,

where c = cos a . In general after n collisions the pulse momentum maxmust lie between c°p and p . This pulse is characterized by a functionro ro r

P(p,n) such that P(p,n)dp is the probability that after n collisions the

last particle in the chain will have momentum between p and p+dp. A

recursion relation for this probability distribution derives from Eq. (2.7)

Po

P(P,n+l) = J dp' Y l^-' cfp' H P " Cp H P " " P ' 2 '8^nC Po

The specification of P(p,n) is completed by assuming

33

p(p,l> - (I , cp0£P5P0 •

which according to Eq. (2,7) is the distribution following the first

collision of a particle of initial momentum p^.

The integration of Eq. (2.8) is simplified if we make the change

of variables:

Yp = c po ,

= £nc dy ,P

and define

34

P(Y »n) = P(c PQ »n)

Then

o

and Eq. (2.8) becomes

nP(Y,n+l) = £(ni --1/cC)) / * y ' P(Y',n) H(Y " - (y - D ) H(y - y'> • (2.9)

Taking n=l, we find

P(Y >2) = ^ V - 'cV f dY' P ( V M ) , 0<Y<1 ,0

/Y-l

P(v,2) = (} f~ / dY ' P ( y M ) , 1<Y<2 ,

and thus,

P(Y,2) l n (1^C) j "Y_ (2.10)(1 - c) po ( (2 - y) » 1<Y^2

The succeeding P(Y,n) may be similarly calculated; however, the direct

integration of Eq. (2.9) becomes increasingly tedious for larger n and

has not been pursued for n>5. Indeed, it appears that P(y,n) takes on

a different functional form in each of the n regions [0 ,1], ..., [n-l,n],

Figure 2.5 is a plot of the functions P(y,n) (and hence P(p,n)) for

35

l<n<5, showing graphically the propagation of a momentum pulse down the

collision chain.

The lack of a closed expression for P(y,n) for arbitrary n is of

little consequence, since essentially all of the interesting properties

of P(p,n) and P(y,n) may be determined from the general forms of Eqs. (2.8)

and (2.9). For instance, Fig. 2.5 suggests that P(y,n) is symmetric

about That this is in fact true may be proven by induction on n.

Suppose that P(y,n) = P(n-y,n) — this is certainly the case for P(y,2)

as exhibited in Eq. (2.10). Then, from Eq. (2.9),

symmetry of P(y,n) for arbitrary n. In terms of the momentum variable,

this result can be expressed as

n

dy' P(n -y' ,n) H(y' - (y - 1)) H(y - y')

0

Letting ys = n-y', then dy4 = -dy', andn

0so that

and thus P(y,n+1) is symmetric about from which follows the

It is also apparent from Fig. 2.5 that P(y,n) achieves its maximum

value at y = y and strictly decreases as y increases to n from this value.

Again, this property may be proven by induction. Consider the derivative

of Eq. (2.9) with respect to y. Making use of the fact that

36

where 6 (x - y) is the Dirac delta-function, we find

3P(y ,n+l) _ £n(l/9y (1

0

nJ dY' P(Y'»n) |-5 ((y'+1) -y) + 6 ( Y - Y ' ) J >

or

— [P ^,n) - P(y - l ,n)J . (2.11)

Now, assuming that P(y,n) is strictly decreasing for y<n, and

recalling its symmetry around y, it is clear that

P(y1,n) < P(y 2 ,n)

if

I f “ Yx I > I § " Y2 I •

Letting y = + c(e>0) and y2 = Y^-l = + c ’ f ° H ows

P(y,n) < P(y-l,n) for Y>^ ^ 1 • From Eq. (2.11) then, iZlLi-SjlH < 0

for Y^^p1* and so P(y,n+1) is strictly decreasing on the inverval

2The significance of this last result is that P(p,n) must attain

/ oits maximum value for p = c p^; in other words, this is the most

probable momentum of the (n+l)1"*1 particle in the collision chain. It

is interesting to note that this is the geometric mean of the highest/ 0

and lowest possible momenta at this step. On the other hand c p is notocoincident with the arithmetic mean, or average value, of the momentum

in the pulse since P(p,n) is not symmetric on a linear momentum scale.

This average value and, indeed, all the moments of the probability

distribution P(p,n) may be calculated from Eq. (2.8). Consider first

the average momentum of the last particle in a chain of n collisions,

<p>n , defined by

37

?on = y* dp' p' P(p',n) . ( 2 . 12)

nC Po

Multiplying Eq. (2.8) by pdp and integrating, we have

r ° f °J dp p J dp' H(p - cp') H(p* - p)<P>n+l = + .n+1 nc p c po o

Interchanging the integration order,

po P

<p>n+l ' / dp‘ (f- cf c f PdP ’n ac p cpoor

Po<p>n+1 = - Q jr ^ J " dp'p' P(p',n) = <p>n . (2.13)

nC Po

But since P(p,l) = [pQ(1 - c)] , <P> = PQ (l+c)/2, from Eq. (2.12).

Therefore Eq. (2.13) yields the result that

(2.14)

38

<P>n/P£) is graphed versus n in Fig. 2.6 from which we see that it

takes 8 collisions for the average momentum in the pulse to be reduced

to half its original value, and that <P>n decreases to PQ/10 only after

26 collisions. Certainly this justifies the concept of a collisional

"chain."

In a similar manner, moments of P(p,n) of any order may be calcu­

lated from Eq. (2.8). One finds

,, m+1,m \ x “ c / m i c \‘P n+l ‘ fa+1)(1 - c) <>> >„ 1 <2'15)

so that

r m+1, -|nm (1 - c ) m<p > = T- TTTTi t \ P • (2-16)n [_(m+l)(l - c)J o

The characterization of P(p,n) is essentially completed by Eq. (2.16).2It is of particular interest to consider <p > since the average energyn

t h 2of the pulse at the n step is <E> = <p > /2m, where m is the massn nof one of the hard spheres. Thus,

3,1 n<E> . r i i ^ i i ] n E ,

n |_3 (1 — c)J o

owith E = p /2m. A plot of <E> versus n is shown in Fig. 2.6. Oneo o nsees that <E> is reduced to half its initial value after 4 collisions,nand to one tenth of Eq at n = 13. In general, by considering only the

even moments of P(p,n) one has all the moments of the energy distribu­

tion function P(E,n)dE, which is the probability that the (n+l)

particle in the chain will have energy between E and E+dE:

39

n<E-> = U - I

n |_(2m+l)(l E . (2.17)o

The sharpness of the P(p,n) distribution may be characterized by its2 2 2relative width, 2o/<p> , where a = <p > - <p> . It follows fromn n n

Eq. (2.16) that

-Jo- . 2 i| 4(1 - c3) ,1 ” - J 2<P>„ 11 3(1 - c) (1 + c)2jn

For c = , this reduces to

^ n20 = 2 [(1.002755)° - l]

Therefore the relative width increases with n, although of course the

absolute width, 2o, must decrease. After 8 collisions = 0.30, and2o <P>8for n= 26> ----- = 0.54. The fluctuations in the pulse momentum26

therefore remain small relative to the mean, <P>n » until a quite large 9

fraction CV/ q) o f the initial momentum has leaked away to recoils.

These results may be extended to cover the possibility of

inelastic collisions, as mentioned in Section II.A. It follows from

Eq. (2.7a) that instead of the recursion relation (2.8) we should write

fPo

/P(p,n+1) = I dp' P(p' ,n) H(p - cfp') H(fp' - p) . (2.8a)_ „ (1 ~ c)fp'n _n c f p ro

Proceeding from this equation as in the derivation of Eq. (2.16), one

also has that

Thus, as one might have guessed, the m power of the average momentum

in the pulse after n collisions is reduced by a factor of fmn.

The discussion of this section has been based on the momentum

transfer probability function given by Eq. (2.7) which is rigorously

valid only as the average interparticle separation x goes to zero. In

this limit, however, the hard sphere system must be crystallized into

a definite regular structure (either hexagonal close packed or face

centered cubic), and it must be acknowledged that this fact is not

entirely consistent with the assumption, implicit in the above, that

any particle in the collision chain will have its nearest neighbors

randomly oriented with respect to its momentum vector. But as a

practical matter this does not pose a serious problem because Eq. (2.7)

provides an adequate approximation for the exact P(p;x) when x> 0, as

long as cos a is properly chosen. Normalization of P(b;x) requiresmax r b2 -il/2

that c = cos a = 1 - m^X , with now b =0.55 (s+x). Notemax I s I max

that for x>0 , c<-r ; consequently the momentum chain in this case will 6be more diffuse, that is, the average momentum transfers will be less,

more momentum will leak out at each step, and the chain effect will

persist for fewer collisions. The dashed curves in Fig. 2.4 represent

the above approximation, in terms of the impact parameter, for x = xc/2

and x = xc> Clearly, the approximation is poor for x = xc> and in

fact diverges as b-*s. On the other hand, for x = xc/2 the approximation

is already quite reasonable. We conclude then that the properties of

momentum transfer chains as we have discussed them provide a useful

characterization of hard sphere gases so long as x<x£/2 = (0.4)s.

In summary, we have shown in this chapter that it is possible

to incorporate the spatial correlations of a dense many-body system

in a relatively simple model for the distribution of the impact param­

eters in the first collision of an energetic particle. When the

average separation between nearest neighbors in such a system is less

than '^1.81 s, then mutual shielding by these neighbors tends to result

in more head-on collisions than would be the case in a rarefied gas;

with the result that a momentum pulse may be propagated through a chain

of collisions with significantly inhibited dissipation. Aside from

their intrinsic interest, these results may bear upon the low-energy

collisions of atoms in solids, in the context of sputter-related

phenomena. In particular, one might anticipate that the increased

inefficiency of collisional energy transfer which results from the

restrictions on impact parameters would influence the energy spectra

of sputtered particles at low energies. Actually, it turns out that

such effects appear not to be large, for reasons that will be made

clear by the conclusion of the next chapter.

41

42

III. SPUTTERING OF A MONATOMIC SYSTEM

A. Boltzmann Equation

The erosion of solid surfaces as a result of bombardment by

energetic ions provides perhaps the most familiar example of the

sputtering phenomenon. For ions whose energy places them in the

nuclear stopping regime, generally 1 keV < E < 100 keV, it seems well

established that the dominant mechanism for the partitioning of the

ion-deposited energy is a cascade of essentially elastic collisions

among the target atoms (Thompson, 1968; Sigmund, 1969). Three charac­

teristic energies should be singled out in this process. Incident

ions promote primary recoils in the target from the ambient thermal

energies up to some energy £Emax* Secondary recoils produced in the

ensuing cascade may escape the target if they approach the surface

with some energy >U. In general, only a very small fraction of the

energy imparted by the ions is carried off by such sputtered particles.

Instead, most of the energy of the cascade remains localized in the

surface region of the solid until the average recoil energy begins to

approach the thermal value kT. This heat is then conducted away from

the surface region and into the bulk of the material.

In this chapter we shall be concerned with modeling this colli-

sional transport of kinetic energy in a monatomic many-body system for

the case ^<<U<<Emax. Anticipating application to sputtering experi­

ments performed with constant beam flux, we focus on the nonequilibrium

steady state which arises when there is a constant flow of energy

through such a system. This energy flow may be characterized by the

distribution function f(E)dE, known as the recoil density, which is

the number of particles per unit volume with energy in the interval dE

about E. A master equation governing f(E) may be derived from the

Boltzmann transport equation as Williams has discussed in some detail

(Williams, 1976b). In what follows, we first offer our own derivation

in order to elucidate the specific assumptions and approximations

employed in our model. We then proceed to a discussion of the energy

spectra of sputtered atoms.

Neglecting external body forces, the Boltzmann equation for a

monatomic medium is commonly written as

ftrVv^t) = r (+) (rYv^t) “ R^Cr.v^.t) , (3.1)

3 3 3where f(r,v^,t) d r d v^ is the number of atoms in the interval d r

about "r, and about v^ at time t. R^+ and R^ are, respectively,

the rates at which atoms enter and leave this interval due to collisions.

Rigorously, our use of this equation already restricts us to systems

in which the range of interaction between two atoms is much less than

their average separation.

As a result of Boltzmann's H theorem, we know that a closed,

isolated system governed by Eq. (3.1) must eventually approach the time

independent state in which R ^ = R^ \ Our interest, however, is in

a closed system which is not isolated but exchanges energy with its

environment. One way to approach this problem is to introduce a source

of particles having some distribution 4> (rYv^, t) on the right hand

side of Eq. (3.1). We shall consider a time independent source such (+) 3 3that 4> (r,v^)d rd v^ is the number of atoms introduced into the

43

44

interval d r d v^ at (rYv^) per second. We similarly define a sink

of particles <5> (iT.v- ) which gives the rate at which atoms are removed 3 3from d r d v^ about (r,v^). Equation (3.1) thus becomes

( ’ T? ) f ^ v ' ) ■

R(+)(r,vr t) - R("}(?,Vi,t) + 0 (+)(?,Vi) - 4»(_}(r,Vi) . (3.2)

Although we do not yet introduce an explicit form for it should

be considered to be fixed, and chosen so as to model the physical

situation of interest. Likewise, we might specify $ to model certain

boundary conditions on the system. For purposes of the following discus­

sion, however, it will not be necessary to quantitatively characterize

As a model for the following development consider a box of volume V

containing N atoms each of mass m, whose walls are in contact with a heat

bath at temperature T. We assume that in the interior of the box the

source of energy is localized in a small region whose volume is much less

than V as shown in Fig. 3.1(a). One might for instance imagine there to

be a source of fast ions in this region. then characterizes the

atoms recoiling from collisions with these energetic particles. These

of thermal energy, and is the rate at which such particles are promoted

from thermal to higher energies through collisions with the source ions.

$ \ since we shall be concerned primarily with those regions of phase

space in which it is appropriate to assume that 4> is negligibly small.

recoils have energies up to Emax >>kT. will serve to provide

conservation of particle number in the source region

Physically, is intended to represent particles

Now, we are interested in the steady state of the system, so that

3f(r»v^,t)/3t = 0. If the power introduced by the source is sufficiently

small, then we expect that f(r‘,v^) will have approximately the form of

a Maxwell-Boltzmann distribution at a temperature = T throughout the box,

with the addition of a small high energy tail in the vicinity of the

source region. Thus, we seek a solution of Eq. (3.2) having the form

f (r, vL) = f ^ ^ r , ^ ) + ftrVv^) , (3.3)

where f^0^ ? , ^ ) describes the nearly thermal distribution of atoms at1/2velocities near = VT = (kT/m) but drops off rapidly with increasing

velocity so that above some energy E (kT<<E <<E ), f^°^(r“,v,) <<f(r‘,'v1).c c max 1 1

45

Furthermore, we assume that at all points in thee box/d"^v^ f(r“,v^) << j

(-) ,(+) ....3 (o)d v^ f (r,v^). According to this picture, $ v ' and <*>'■ ' describe the

promotion of atoms from thermal energies to much higher recoil veloci­

ties in the source region, hence $ is appreciable only for v^ = v^.

Outside the source region both $ and are zero except at the

walls of the box, where they provide appropriate boundary conditions.

Substituting Eq. (3.3) into Eq. (3.2) and writing out and

R^ we have,

Vf • |f (r.v^ + f^.v^j

= y d 3v2 da(ft, Iv-j-v^j | V ° ^ ( r , v ' ) + f ( r , v ' ) J ^ f ^ ^ r . v ' ) + f ( r , v 2)j

- ^ ° ^ ( r , v 1) + f { v yv 1 ^ J d 3v 2J d o ^ l y \\1- ^ 2 \]j l ^ - v ^ j^°^(r,v2) + f(r,v2)j

+ 4>(+) (r,v^) - t(-)(r,v1) . (3.4)

This equation refers to the collision of two atoms having initial

velocities (v^,v2), final velocities (v^/v^), and center of mass

scattering angles ft = (0 ,<J>) which are the angles between and

v'-v'. The differential cross section da = (do/dft)dft is assumed to

depend only on the scattering angles and the relative velocities.

Aside from the spatial coordinates, eight independent variables appear

in Eq. (3.4): v^, v^ and ft. v' and v^ may be considered functions of

these quantities. Alternatively, v' and v^ and ft may be taken as the3 3 3 . 3independent variables. We shall use the facts that d v^d v^ = d v^d v',

Iv^ - v2 | = |v' - v'|, and da(ft,|v^ - v 2 | ) = d a ' ^ . l ^ - v2|) where

da' is the differential cross section for the inverse collision

"v + v' -*■ "v + v2 (see, e.g., Huang, 1963).

We are primarily interested in Eq. (3.4) at high energies.

According to the assumptions we have made, for v j>>v j Eq. (3.4) reduces to

vx . I f ^ )

46

’ / d V2 / d° 1 W [£i <0) f2 <0) + ££(o ') ^ ^ ( o ) ^ ^f ' + f' f + f ' { ' 2 1 2 1 2

J ' d 3v 2 J ' d a 1^ - v2 | ^ 2°^ + f2^ + » (3.5)

where da e do(ft,|v^-v2 1) and f^°^ = f^°^(r,v^), etc. Now the f'^°^f'^°^

term drops out, since f^°^ is large only at thermal velocities, but two

thermal atoms cannot collide to produce a fast recoil. We may further*Vf *\j *\j t

neglect the f'f' and f f terras when compared with the remaining ones.

Thus the equation is linearized:

47

=J d3v2 ^ (0 ) ( r , v ' ) f ( r , v ' ) + f ( r , v ' ) f ( o ) ( r , v ' ) j

(?’V f * \ f da 1^1 " ^2 I f(0)(? »v2) + * (+)Cr,v1) (3'6)%

- f

In the third term on the right hand side (RHS) the integral over do may

be immediately performed, with J *d o ^ , ! ^ - v2 | ) = o (1 - v2 |) = a .

We note that |v^ - v2| - v^, and we finally assume that the spatial

variation of f d3v2 f ^ ( r \ v 2) may be neglected. ‘That is, we takeion of I c

/■'d^v2 fw '(r,v"2) = N/V = n. Therefore, the third term is simply

-nov^f (r^v^).

A further simplification may be obtained by integrating Eq. (3.6)

over some spatial volume V' which contains the source but is large

enough so that f(r',v1 ) goes to zero on its boundaries. It follows that

on the left hand side of Eq. (3.6)

J " d r v- . V_ f C?,v^) = J " d3r V . .

= J dA . v1 iE(r,v1o -___

v f (r .v^

S0 ,

v /* 3 ^where S is the surface of V'. Letting f(v^) = I d rf(r",v^)

♦ ( v ^ ) = £d3r $^(r,v^), Eq. (3.6) becomes v

and

^ f ( V i ) = J J d o l ^ - v ^nov, t (v, ; = f d rfd'v, |do |v,-v„| |f^°\r,v') f (r ,v')+f Cr, v') f ^ ( r \ v 2

+ * (+)(v1) (3.7)

48

The energy spectrum of recoils f(E^)dE^, which is the total number%

of atoms with energy between E^ and E^ + dE^, is related to f(v^) by

or

f(E1)dE1 = v12dv1 f <m1 f(v1 ,n1)/■

V1 f. — / dn, ! m J 1

2E„m • ai)

(e i > - v / dni ' “ i),

1 2where = (v^,fij) and E^ = -j mv^ . A similar relation holds between

<5>+ (v^)d2v^ and $^ (E^)dE^, the total number of recoils introduced into

the system per unit time in (E^,dE^). In order to express Eq. (3.7) in3terms of energy spectra, let us multiply by 6 (E^-E)d and integrate:

navf (E) = J d3r J 'd^v \ J " J da^ l _ ^ fi(Ej“E)

[ \ (o)(r,v^ ) f(r,v:) + f (?, v *) f (0)(r,v:)

+ $ (+^(E). (3.8)

Consider the first term on the right. By changing variables from v ^»v 2 t0

v',v', this term may be written as

(o)do' (fi,|v^-l^ I ) |vj-v'| 6 (Ej-E) f (r,v') f(r,v')

Since f^Cr.v'p contributes to this integral only for v' = vT> while

it is necessary that v^>>vj> we may take |v^-v'| = v^ and neglect the

49

dependence of on v'. The integral over v' may thus be carried out.

producing

n "d^r J" d i^ v ^ J " do'(Sl,v p v' dfE^F.) fOr.v')

Next, the volume integral is performed, yielding

n j" d ? v ' 2 £ d o ' ( ! ? , ,v p v' d ^ - E ) f(v')

Now if we assume that do' does not depend on the azimuthal angle,

4), then the kinematics of a collision in which a particle of velocity v'

strikes an atom of equal mass essentially at rest are such that

2 0E1 = E' sin (— ) , E2 - Ei

Therefore the angles of v' may be integrated over, giving

/ o C da'(e.v')dv' v' I d0 sin0 ---— ---- d(E^-E) f(E^)

Finally, we change variables from v' and 0 to E^ and E^. It follows

from the above relation that

2dEsin0 d0 = • v

2

which prompts us to define the differential cross section daCE'jE^),

that an atom of energy E^ will transfer an energy between E^ and

Ej+dE^ to a stationary collision partner, by

50

do(E',E1) 4ir do(0,v')dE^ " E^ dfi

With this, the collision term becomes

OO GO/ /* do(E',E )dE! J dE2 v2 — dEl 6 (Er

0 E,

E) f(E') ,

1w

/■or n I dE' dq(Ee,E) v ' f(E') .

E

A similar reduction may be carried out for the second term on the

RHS of Eq. (3.8). In this case, however,

E' - E1 = E' sin2(|) , E' > Ex ,

so that the term becomes

OO

n / dE'do (E' , E'-E)

d(E'-E) V f(E }

Therefore, we have finally for Eq. (3.8),

OO

■ " / drdo(E',E)no(E) v f (E) = n / dE' d'E v ' f ( -E

ECD

n/dE

+ » #dE' d°d(E:-E)E> V ' f(E'>

+ $ (+)(E) (3.9)

We reiterate that Eq. (3.9) applies to those atoms with energies E>>kT

in a system in which most of the atoms have thermal energies, and in which

the total particle number density is essentially constant.

Equation (3.9) has a simple physical interpretation in terms of

the balance between the rate at which atoms enter and leave the energy

bin (E,dE). The left side is clearly the rate at which atoms are

removed from a unit energy interval at E through collisions. The first

term on the RHS is the rate at which atoms are promoted from thermal

energies to this interval, and the second term on the RHS gives the- (+)rate at which atoms fall down to E from higher energies. $ (E)dE,

we recall, is the number of atoms introduced into (E,dE) per unit time

by the source.

The distribution function f(E) appearing in Eq. (3.9) is distinct

from the recoil density f(E), which is the number of atoms per unit

energy per unit volume at E. Actually, f(E) is in general a function'b ^

of r. It might be calculated by solving Eq. (3.6)for f(r,v^) and then-=“■ ^ _ a .performing the integration over the angles of v" . But if f(r,v^) - 0

in some part of the system, for instance in the center of the source

region, then one would obtain an equation for f("r,E) in that region

identical in form to Eq. (3.9). We have avoided assuming that

V^f fr ,v^) = 0 by integrating f(r,v^) over the volume V' outside of which

it is negligible. f(E) is thus related to f("r,E) by

= ^ y * d 3r f(r,E) , (3.10)V'

that is, f(E)/V' is the average of the recoil density over the volume V ' .

In practice we are interested in modeling experiments which involve

an ion beam of particle current i impacting a target's surface, as

51

52

illustrated in Fig. 3.1(b) (it is not essential that the beam be normal

for z within Az of the surface, then we may choose V' to be the volume

of depth Az and cross sectional area A ' , taken sufficiently large so that

f(r-,E) vanishes on the lateral sides of V'. With this choice, the

derivation of Eq. (3.9) goes through essentially as before.

The energy spectrum of sputtered particles, S(E)dE, is defined

as the ratio of the number of particles ejected from the surface per

unit time in (E,dE) to the total number of beam ions entering the target

per unit time. Since vf(r",E) is the recoil flux, clearly

Apparently then, this problem is entirely equivalent to one in which a

beam of uniform flux of magnitude i/A' gives rise to a uniform recoil

density of magnitude f(E)/V". If we therefore identify f(E) with

to the surface). If we make the assumption

#out „---- 'V.sec

Thus

S (E) ^

- (+)f(E)/V' and let 4>(E) 5 4>v (E)/V' be the number of recoils of energy E

produced per unit volume per unit time by a beam of flux = i/A',

Eq. (3.9) can be written

o o

no(E) v f(E) = n/dE' o(E',E) v' f(E')

E °°

E+ *(E) , (3.11)

where for brevity o(E,T) = -d°

53

B. Energy Spectrum

Equation (3.11) has been investigated extensively by various

authors (Robinson, 1965; Kostin, 1966) for the purpose of modeling

sputtering energy spectra. It is well known, for instance, that

Eq. (3.11) may be solved analytically for a system composed of hard

spheres, since in this case the differential cross section has the

particularly simple form

where s is the sphere diameter. We shall consider here the somewhat

more general case in which we assume only that o(E,T) does not depend

on the energy transfer T. The total cross section is then

This o(E,T) is similar to the hard core cross section in that, for a

given incident particle energy E, any energy transfer, 0<T<E, to the

stationary stuck particle is equally probable. The difference is, in

effect, that the hard sphere diameter of the particles is allowed to

vary with E. It is reasonable that this might provide an improvement

over the true hard sphere approximation for atomic collisions.

2

E

0

so that

(3.12)

54

Substituting (3.12) into Eq. (3.11) and defining the collision

density k(E), which is the number of collisions per unit energy per

volume per second, by

k(E) = no(E) v f(E) ,

we have

0 0

■ / '

dE'k(E) = 2 1 k(E') ~ + 0(E) . (3.13)

Differentiation with respect to E and multiplication by E gives

d(E k(E)) = 2 dO(E)dE dE

2 2 Integrating over E and assuming lim E 0(E) = 0, and lim E k(E) = 0,E -► °o

we find

o o

/ •k(E) = 0(E) + — /O(E') E' dE' ,

E E

and thus the recoil flux g(E) = vf(E) is given by

g(E) = 0(E) no (E)

2no(E)E^

OO

/ O(E') E' dE' (3.14)

More precisely, g(E) is directly related to the average recoil

flux. It follows from the definition of f(r,v) that

^ 3cos 0 v f(r,v) d v = cos 0 — f (?,m2_Em , ft) dEdft

55

is the flux of particles with energies in dE about E and velocities

directed into df2 about v through a surface whose normal makes an angle

0 with v. Averaging this quantity over the volume V' and over all

directions of v ,

Our use of this average flux distribution in the calculations of

sputtering fluxes which follow neglects deviations which may arise at

the surface of the target (Thompson, 1968; Williams, 1976a). Such

variations have two sources: The presence of the solid-vacuum interface

may lead to a distortion of g(E) in the immediate vicinity of the surface,

since particles which once cross this interface cannot return to partici­

pate in the cascade development. Furthermore, the collimation of the

incident ion beam leads to an anisotropic source of primary recoils.

Indeed, for a normally incident beam no primaries can escape a sufficiently

thick target without further collisions. We shall limit the following

discussion to this latter configuration. In Eq. (3.14) it is clear that

the term 4>(E)/no(E) represents the primary flux distribution, so following

Weller (1978) we shall drop this term and consider only the flux of

secondary recoils:

V 1cos 9 v dE dll - —m 4tt1 % dQd r f(r,v,ft) = cos 0 g(E) dE - f - .

OO(3.15)

E00

The factor $(E')E'dE' here will be recognized as the rate at £

which the beam deposits energy per unit volume of the target in the

form of primary recoils having energy greater than E. Only such recoils

can contribute to the flux at E. We consider a normally incident,

monoenergetic beam of ions whose energy, E^, places them in the nuclear

stopping regime. That is, such ions lose energy primarily through

binary elastic collisions which are governed by a screened Coulomb

potential. For intermediate mass ions and atoms, generally 1 keV<E^<100 keV.

At these energies, an ion deposits only a small fraction of its initial

energy in the surface region, so to a good approximation

$>(E)dE = ndo (E, ,E) <f> , E<E , (3.16)b b max

where n is the number density in the target, $ is the (average) beam flux,✓

do, (E, ,E) is the differential cross section for an ion of energy E to b b btransfer an energy between E and E+dE to a stationary atom, and E = yE,t max d

2is the maximum kinematically allowed energy transfer (y= (4mm^)/(m+m^) ,

where m^ is the mass of a beam ion). With this, Eq. (3.15) becomes

Emax

ss (E) - 7< f i 7 / E' dW E'> • <3-17)E

The conversion of this internal recoil flux into an external,

sputtered flux may be accomplished by assuming that an atom approaching

the target's surface sees a planar potential barrier of height U.

Typically, U ^ 5 eV, which easily satisfies our criterion ^<<U<<Emax,

for targets at room temperature or lower. If internal to the target the

atoms have energy E' and their velocities make an angle 6 ' to the normal,

then one finds (Thompson, 1968)

56

where the unprimed quantities are the corresponding external values

(E = E' -U). The differential sputtering yield S(E,fi)dEdf2 is therefore

S(E,fi)dEdO = x g (E+U) -rTTT cos 6 -j—- dE .<{> s E+U 4tt

This is the number of atoms per beam ion ejected with energy in (E,dE) and

velocity directed into dO at an angle 0 to the outward normal. Integrating

over angles and substituting Eq. (3.17), we find

S(E)dE = 42 o(E+U)(E+U)3

Emax

/ E'dob (Eb ,E')

L-E+U

dE . (3.18)

S(E) is the energy spectrum of sputtered atoms. For energies E<<E ,max3

the behavior of S(E) is dominated by the factor E/(E+U) , so thatEmax

/S(E) * J - I E'do (E ,E') , E « E , (3.19)(E^U) max

0

where o is some constant low-energy cross section. Integrating over E,

one then obtains an approximate expression for the sputtering yield:

(3.20)

E -U Emax max

’ j s<E>dE ’ z i I E' d»b (Eb>E' )S =

0 0 Thus the yield is expected to be inversely proportional to the surface

binding energy and the interatomic cross section, while it is directly

proportional to the nuclear stopping power of the beam ions

defined by

dE \dx) ’' n

EmaxE'dab (Eb ,E')

0

Up to a factor on the order of unity, expression (3.20) is identical

to the results of Sigmund's more detailed calculation of sputtering

yields (Sigmund, 1969), which in turn are well confirmed experimentally

(Sigmund, 1972) for a wide variety of targets and incident ions.

The proportionality between S and dE/dx which we have derived

in Eq. (3.20) is a consequence of our assumption (3.16) for the distri­

bution of energy deposited by the beam. Actually, the deposited energy

function may be expected to depart from the stopping power for two

reasons. First an ion's path through the surface region is not a

straight line, so that its energy loss per unit depth generally exceeds

its energy loss per unit path length, which therefore tends to increase

the energy deposition. On the other hand, the energetic recoils created

near the surface tend to have forward directed velocities, so that a

certain fraction may leave the surface region with the result that less

energy is available for the production of collisional cascades. Sigmund

(1969) has shown that these effects may be accounted for by introducing

a multiplicative factor a(m/m ) in Eq. (3.20) which is a slowly varyingD

function of the target atom to beam-ion mass ratio. But it should be

clear from the form of Eq. (3.15), and from the discussion immediately

thereafter, that a change in the magnitude of the energy deposition would

not alter the shape of the energy spectrum of sputtered particles.

That is, the introduction of such an a factor would change only the over­

all scale of the secondary recoil flux; contingent, of course, upon our

fundamental assumption that the energy deposition is not so great as to

invalidate the linearization of the transport equation performed in the

last section. Since a is on the order of unity in all cases, its

precise value is not critical with regard to this last point. We do

not discuss this correction further since we are not concerned here with

absolute sputtering yields, but rather with the form of the energy

spectrum.

58

According to Eq. (3.19), S(E) at low energies is proportional 3

to E/(E+U) . This is the well known result for energy sharing in a

system of hard spheres (Thompson, 1968; Robinson, 1965; Kostin, 1966).

Not surprisingly, experimentally measurable departures of the energy

spectrum from this form occur (Thompson, 1968; Chapman e£ al., 1972;

Weller, 1978; Sigmund, 1972). Near E , S(E) must of course fallmaxrapidly to zero. This behavior is accounted for in Eq. (3.18) by the

bracketed energy deposition factor. The extent to which this factor

affects the spectrum at lower energies depends on the precise form of

do^(E^,T). The spectrum we shall model below is that of uranium atoms

sputtered from a metal target by 80 keV ^ A r + ions as measured2 35 40by Weller (1978). At this energy collisions between U and Ar atoms

are well approximated analytically by a cross section of the form

(Lindhard jet _al. , 1968)

dob<Eb'T> s - (3-21% 1

59

where

, Q.w«2 ... V / M 2c2 6.93 (A eV) ^ 2/3 + z2/3}l/2 \ m /

and Z(Z, ) is the target atom's (beam ion's) nuclear charge. This b-2expression for do,(E, ,T) derives from the r approximation to the b b

Thomas-Fermi interatomic potential. More generally, for a potential— g

V(r) <* r the differential cross section may be accurately represented

by (Lindhard et al., 1968)

60

dV V T)

E, dT b (3.22)c 1+1 Iss<EbT)

This formula is exact for s = 1, which is the case of unscreened,

Rutherford scattering. The Thomas-Fermi interaction can be well

appropriate choices of s>l, while s very large adequately models the

Born-Mayer potential (Sigmund, 1969). For a large range of target and

projectile masses and energies, namely, near the peak of the nuclear

stopping power, where in fact most experimental work is done, s = 2 is

the preferred value. Consequently, extensive use will be made of

Eq. (3.21) in our further discussion of sputtering.

At energies much less than E , S(E) is found to fall off moremax_2slowly with increasing energy than E . Several mechanisms have been

proposed to explain this trend. Chapman et _al. (1972) have measured

the energy spectra for 10, 20, and 41 keV Ar on a polycrystalline Au3target at room temperature. They found that between 10 and 10 eV the

spectra are approximately proportional to E They propose that

since they detect the ejected atoms at an angle nearly normal to the

incident beam direction there may be a sizable contribution from

deflected surface primary recoils. Such a contribution would tend to*

result in a flatter energy spectrum both because deflected primaries

would be of higher energy on the average, and because they would not

contribute to the generation of lower energy secondary recoils.

Weller (1978), in his study of the sputtering of uranium metal, has-2also found an energy spectrum which falls off more slowly than E ,

namely, he finds S(E) «E The deflected primary recoil mechanism

approximated in given ranges of the energy parameter t “ (E,T)b1 / 2 by

would not help explain this latter data though, since Weller measures

the distribution of U atoms emitted back along the beam direction,

normal to the target's surface.-2A modification to the E form may also arise from a more rigorous

treatment of the surface discontinuity in the application of the Boltzmann

equation (Williams, 1976a) for the reasons mentioned previously. Indeed,

Thompson (1968) pointed out some time ago that the loss of energetic

atoms through the surface should lead to fewer low-energy recoils than

one would otherwise expect. Unfortunately, there have been few quantita­

tive calculations of this effect.

A third factor affecting the spectrum at lower energies is the form

of the interaction potential which governs the secondary collisions.

Robinson (1965), however, has found that the spectrum is relatively

insensitive to whether one assumes this interaction is governed by the

Born-Mayer, the Thomas-Fermi, or some more general potential function.

More specifically, Sigmund (1972) has shown that if low energy recoils—sinteract according to a power law potential proportional to r then

for E<<E , max

S(E) « E2/S_2 .

But for energies on the order of a few tens of eV, s must be chosen

quite large. In fact, (Sigmund, 1969) the best model of the Born-Mayer

potential at these energies is obtained by taking 1/s - 0._2

The modification to the E spectrum we propose also arises from

the form assumed for the secondary collisional cross section, Eq. (3.12).

According to Eq. (3.18), S(E) is inversely proportional to a(E), the

61

total atom-atom cross section. Physically, this result may be interpreted

as follows: By adopting Eq. (3.12) we have assumed in essence that the

dynamics of the individual atomic collision is not critical as long as

energy sharing is fairly efficient, that is, as long as the average energy

transfer in a collision is not far from half the initial energy. This has

the consequence that the collision density k(E) is completely independent

of the atomic cross section but depends only on the source conditions.

The magnitude of the collision density is fixed by the requirement that

a certain amount of energy flow through the system. Ek(E)dE is just the

power per unit volume dissipated by atoms in the interval (E,dE) to

particles of lower energy. The recoil flux, g(E), must be sufficient to

maintain this collision rate. If the total atomic cross section, a(E),

decreases, each atom has less frequent collisions so that the density of

recoils in (E,dE) must increase' in proportion. Thus g(E) a l/a(E). It

is interesting to note that while on one hand Sigmund (1972) finds that

the energy dependence of the recoil flux is a function of the exponent

of the assumed power law potential, on the other hand a quantity which-2he terms the recoil density, F(EQ ,E)dE, has an exact E form for E<<Eq.

This latter quantity is closely related to our k(E) and is defined as

the average number of secondary recoils having initial energy (E,dE) in

a cascade for which the primary atom had energy Eq . These results

are seen to be consistent with an analysis based on the cross section of

Eq. (3.12).

The energy spectrum of sputtered atoms is typically influenced

least by the surface boundary or source conditions in the range

10 eV<E<l keV so that our primary interest is in obtaining a good model

62

for S(E) at these energies. Therefore we take a(E) = u[s(E)] where

s(E) is the distance of closest approach in a head-on collision between

a stationary atom and one of energy E, as determined by the Born-Mayer

potential. As in Chapter II (E = E/2),cm

632

s(E) = aBM in(^) (3.23)

p 3 / 2and we again take a = 0.219A and A = 52 Z eV. For uranium,d M2A = 9.2 x 10^ eV. With this,

o(E) = Tra2M £n2C^r) . (3.24)

Substituting (3.21) and (3.24) into Eq. (3.18), the energy spectrum is

1 / 2 . 1 Y cS (E) =

Tra2M £n2(2A/(E+U)) (E+U)3 3 - ( f x ) 2\ max/(3.25)

Recall that E = yE. . For 80 keV ^ A r on 233U, y = 0.50, so that m a x d

E = 4 0 keV. Thus the second term in the brackets is ^ 0.05 for maxE - 100 eV. Consequently, at low energies our particular choice for

the source function 4>(E) is not critical and

1/2

S(E) = — — =------ =• , E « E . (3.26)Tra2M £n2(2A/(E+U))(E+U)3 “aX

3Compared to the canonical form, E/(E+U) , for a true hard sphere gas,

the S(E) of Eq. (3.26) does not drop off as rapidly with increasing E.

It is also noteworthy that the variable radius spectrum peaks at a

slightly higher energy than U/2, which is the peak of the fixed radius

spectrum.

It is conventional in reporting sputtering data to fit the measured

energy spectrum with a power law of the form

S (E) a ? (3.27)6XP (E+U)a

235Weller (1978) finds that the U spectrum at low energies is well

fitted with U = 5.4 eV and a = 2.77. It is of considerable interest

then to ask what power a best approximates the S(E) of Eq. (3.26) in a

given range of E. For the purpose of determining this power, it is

useful to consider the following derivative of £n S(E):

d &n S (E) = / ________ 2 \ Ed £n E I £n(2A/(E+U))I E+U ’

whereas the power law of Eq. (3.27) yields

64

d £n S (E)--------- = l - c cd £n E E+U

In the neighborhood of any given E, then, S(E) falls off as E/(E+U)a

with

“ 3 ' £n(2A/(E+U)) * (3.28)

235At E = 100 eV, for instance, a = 2.70 for a U target with U = 5.4 eV,

which compares favorably with the experimental value a = 2.77. This

exponent is not a strong function of the mass of the target atoms. For

a gold target with U = 3.8 eV (Gschneidner, 1964) we find nearly the

same value, a = 2.69, while the empirical value quoted by Chapman et al.

(1972) is a = 2.6.

In Fig. 3.2 we plot the S(E) of Eq. (3.25) labeled C, together235with the measured spectrum for U (Weller, 1978), Eq. (3.27), labeled

B. All curves in this figure are normalized so that they pass through

the same point at E = U/2 =2.7 eV. Curves B and C are seen to agree

quite well at all energies below 1 keV, the highest energy at which

data were reported. In fact, they are indistinguishable on the scale

of the graph at energies below 100 eV. The curve labeled A is the

spectrum which arises from the fixed radius hard sphere model. It

follows from Eqs. (3.18) and (3.21) with o(E+U) held constant; thus it3

falls off approximately as E/(E+U) in the energy range shown. Clearly,

the slope of curve A is much too steep. Spectrum D results from the

variable radius model, with the assumption that the source of primaries

is monoenergetic, i.e., 4>(E) = N6(E-E ), where N is some strength ° maxconstant. Use of this ^(E) provides an upper limit for the energy

spectrum within the framework of our model. To see this, consider00Eq. (3.15). As long as 5>(E) is positive definite,^*4>(E")E'dE' must

Ebe a monotonically decreasing function of E, so it can only increase

the rate at which the recoil flux falls off. But curve D has no such

contribution from a distributed source; instead, it has the functional

form of Eq. (3.26).

Although it follows from Eqs. ’(3.22) and (3.18) that S(E) should

fall off somewhat more rapidly the smaller the value of s, or in other

words, the less highly screened the Coulomb collisions, it turns out

65

that this variation is small, as a comparison of curves C and D suggests.

Thus we again conclude that the low-energy spectrum resulting from our

model is insensitive to the details of the beam-target interaction.

There is one clear reason to expect our model curve C to over­

estimate the spectrum at energies > 1 keV, which derives from our use

of the Born-Mayer potential. The potential of Eq. (2.4) is valid only

over a limited range of r. Because this exponential form does not

possess a hard core, it does not increase rapidly enough as r decreases

below some critical radius r which is on the order of the distance of

closest approach of two atoms in a head-on collision at E = 1 keV. The

effect is that the hard sphere diameter of Eq. (3.23) decreases too

rapidly with E for E> 1 keV. A better approximation to the interatomic

potential at these energies is the power law form:

where v is some constant. Using this, the hard sphere diameter would

c

vj3s S > 1 ,

r

sbe

(2v /E)1/S s

so that

tt(2v /E) s2/s

Let us compare o_T (E) and a (E), the cross section of Eq. (3.24), at PL BM

67

if

s > £n(2A/E ) = 4.5o

for uranium. This value of s is quite reasonable for the energy range

under consideration (Lindhard £t £l. , 1968; Sigmund, 1972). Hence we

conclude that the use of a more accurate interatomic potential in our

model would result in a calculated energy spectrum which fell off

somewhat more rapidly with E than curve C of Fig. 3.2 for E ^ 1 keV,

with the result, perhaps, of improving the agreement with the data.

The success of our rather simple model in reproducing the

empirical curve at energies on the order of U might seem surprising

at first glance considering the discussion of Chapter II on correlation

effects in dense systems. It should be emphasized though that curve B

of Fig. 3.2 is only a fit to the data. The relative uncertainties in

the measured values are largest at energies < U/2, where they are on the

order of a few times 10%. Hence we can only conclude that position

correlation effects do not produce a substantial deviation from the

dilute gas model. A possible explanation for this result follows.

Recall that in Chapter II we found that the interatomic separation x,

in a uranium target is equal to the critical value for incident

energies E - 39 eV. However, x does not decrease to xc/2 until

E = 4 eV. We may estimate from Eq. (2.7) that the average energy

transfer T at these energies falls in the range .5E<T<.7E. But this

model is based on the hard sphere approximation. If we consider

instead the 1/s = 0 power law cross section (Eq.(3.22)) we see that

atoms modeled by the more realistic Born-Mayer potential prefer to

transfer less than half their initial energy. These two tendencies are

in opposition, so the net effect is that the energy transfer probability

at low E could be surprisingly uniform for atoms in a solid matrix. In

other words, position correlations tend to exaggerate the importance of

small impact parameters, while the "bare" atomic potential favors larger

ones. The result is that a dense gas of atoms may behave much as a

dilute gas of hard spheres, as far as the efficiency of collisional

energy transfer is concerned, which would justify the choice of cross

section made in Eq. (3.12). This picture is at least consistent with

the data in hand.

68

69

IV. SPUTTERING OF POLYATOMIC SYSTEMS: FRACTIONATION

A. Introduction

In a number of polyatomic solids, sputtering in the nuclear

stopping regime is found to be accompanied by substantial changes in

both the elemental (Liau _et aj.. , 1977; Kelly, 1978) and isotopic

(W. A. Russell, 1979) composition of the exposed surficial layer.

These fractionation effects are a result of the fact that the partial

sputtering yields of the component species, S^, generally do not exhi­

bit the stoichiometry of the surface region. Within the framework of

the collision-cascade model as we have developed it in Chapter III,

such preferential sputtering may arise from the generation of nonstoich-

iometric recoil fluxes as the energy of the beam ion is shared among

the various target components, or from a species dependent boundary

potential which governs the escape of energetic secondaries from the

surface, or both. The latter mechanism is expected to contribute

significantly only when the species concerned differ chemically, that

is, in electron configuration.

In the particular case of a multicomponent ideal gas which is in

thermal equilibrium at a temperature T, a differentiation of particle

flux occurs due to the equipartition of energy among the constituent

species. It follows from the Maxwell-Boltzmann distribution that the

number of particles per unit volume of species i, having energy in the

interval dE about E, is

fi (E)dE = n± 4n(2nkT)_3/2 (2E)1/2 e_E/kTdE ,

70

where ru is the number density of species i. Consequently, =

n^/n^ for any two components, and thus the energy distributions are

stoichiometric. The fluxes however are given by

1/2g.(E) = (2E/ny) f±(E)

with ny being the mass of an i-type particle. The fluxes of two

species therefore are not in the ratio of their respective abundances,1/2since g./g. = (n./n.)(m./m.) . According to this one would expecti J i J J i

material evaporating from a surface to be fractionated by the factor1/2(ny/ny) if the surface binding energies IL of the species were equal

Such a picture might be relevent to so-called spike phenomena (Thompson

1978) in which either the ion's energy deposition or the ambient

target temperature is sufficiently high so that a local thermodynamic

equilibrium develops about the ion's track.

Our interest, though, is in the collision cascade regime. The

central problem undertaken in this chapter, therefore, is the descrip­

tion of the partitioning of recoil flux for a system in the steady

state which is established as the energy deposited by beam ion cascades

down through energetic recoils and is dissipated in the form of heat

to the bulk of a multicomponent target. We first develop a model for

this mechanism in the case of a binary medium. The structure of this

model serves not only to set limits on the relative contribution of

the mechanism to the fractionation process, but also illuminates some

of the general trends which have emerged from the data, such as the

insensitivity of the surface enrichment to the mass or energy of the

incident ion (Liau et al., 1977) and its nonlinear dependence on the

species' abundances (Winters and Coburn, 1976). Quantitative

estimates are derived in the case of isotopic fractionation under

the assumption that surface binding effects are negligible.

Isotopic fractionation data provide the greatest insight into

the mechanics of collision cascades since in this case obscuring

effects related to surface chemistry are minimal. The most incisive

set of experiments pertaining to this phenomenon is that due to

W. A. Russell (1979) who measured the sputter-induced calcium isotopic

fractionation of several minerals for the purpose, in part, of defining

the possible contribution of sputter-related effects in producing the

isotopic fractionations observed in lunar soils. In order to compare

the model developed in Section IV.B with W. A. Russell's data, we

shall extend our analysis in a perturbative manner in Section IV.C so

that it may be applied to the common mineral plagioclase, which was

one of the experimental targets. The particular sample examined is a

mixture of albite (Na A1 Si_ 0o) and anorthite (Ca Al„ Si„ 0 ) inJ o Z Z o

equal parts.

The result of this comparison between experiment and model

strongly suggests that any nonstoichiometry inherent in the energy

sharing process, as we have pictured it for cascades developing in

the interior of the target, is quite insufficient to explain the

observed fractionations. We are consequently led to rethink the

physical consequences of the surface discontinuity for the cascade

mechanism. The solution which we propose to the problem, in Section IV.D,

involves the concept of the generation of a recoil flux in the surficial

layer of the target medium which does not fully participate in the

collisional cascades. The fractionations calculated in this model are

71

in satisfactory agreement with the observed effects.

The sputtering of the surface of a solid offers only the most

familiar example of ion-induced erosion. An equally important problem

in the context of astrophysics concerns the sputter-induced mass loss

from planetary atmospheres due to the impact of energetic particle

fluxes such as the solar wind. In order to provide the background

analysis necessary to our detailed discussion of this phenomenon in

Chapter V., we are led in Section IV.E to consider the physics of the

sputtering of a diatomic molecular gas. In this case there are two

components: molecules and dissociated atoms. The distinctive feature

of this system is that inelastic collisions occur which remove kinetic

energy from the cascade and sometimes result in the conversion of one

species into the other. We shall make use of the analytical techniques

developed in the earlier sections of this chapter to investigate the

consequences of such molecular association and energy loss on the

cascade mechanism. Perhaps surprisingly, we find that the mass loss is

quite insensitive to the details of the molecular binding.

B. Binary Media

We consider a homogeneous, amorphous solid target whose two

components have number densities n^ and n^. The calculations of the

recoil fluxes g^(E) herein are based on the assumption that a steady

state of energy flow exists in the surface region. This implies that

we shall neglect variations in the n ^ /n ^ ratio. Actually, fractiona­

tion is a nonlinear process because each incremental change in the

surficial composition acts to alter the stoichiometry of the next

material removed. The prolonged sputtering of a surface eventually

72

results in the attainment of some equilibrium composition such that

the material being removed exhibits the bulk stoichiometry. But

because the time scale for the occurrence of a substantial change in

concentration is much longer than the time required for a typical

cascade to run its course, the calcirlational procedure adopted in this

section and the next is appropriate for a description either of the

initial fractionation of an undisturbed surface with respect to the

bulk— that is, prior to the removal of more than one or two monolayers—

or of the instantaneous fractionation of the surface region with respect

to its own concurrent composition. In Section IV.D we shall extend our

analysis to the determination of cumulative time integrated partial

yields, in order to provide a closer comparison with experimental

observations.

We shall again adopt a hard sphere model for the target atom —

target atom collisions. Since we are not interested in the energy

spectra of the sputtered particles, but only in the relative total

yields, it suffices to use the mathematically simpler fixed radii model,

as opposed to the variable radii model of Chapter III. The differential

cross section for an i-atom of energy E to transfer an energy between

T and T+dT to a stationary j-atom in such a collision is

.toyCE.T) ■ »i3 ^ 7 1 ’ TiT« E- (4'1)

where o „ is the total (energy independent) cross section and

is the coefficient of maximum energy transfer. We take o^. = us.. ,

where the collision diameter s.. is determined from the distance ofij

closest approach of the two atoms when their relative kinetic energy

in the center of mass frame has the typical cascade value of ^5 eV.

Again adopting the Born-Mayer potential of Eq. (2.4), we have

74

2

For example, the cross section for two Ca atoms is 7.0 A .

Aside from the physical transparency and mathematical simplicity

obtained from the hard sphere approximation, we can offer several

heuristic arguments to justify its use in the following calculations.

As we have indicated elsewhere, we believe that the characteristics of

a statistical steady state such as is involved here should be governed

more by the kinematics than by the dynamics of the atomic interactions.

Thus, the Maxwell-Boltzmann velocity distribution for a gas in thermal

equilibrium is independent of the form of the cross section assumed for

the molecular collisions. This point is supported by the work of

Robinson (1965), mentioned previously, on collision densities in mono-

elemental targets. Furthermore, one must question the utility of

adopting a more realistic interatomic potential without simultaneously

taking account of the density related correlation effects discussed in

Chapters II and III. But probably the strongest evidence for the

sufficiency of the hard sphere model is the study by Andersen and

Sigmund (1974) of the consequences of anisotropic, power-law scattering

for recoil energy sharing in a binary medium. They find that the recoil

fluxes are quite insensitive to the interaction potential assumed,

except perhaps in the case of an extreme mass ratio for the two species.

Our primary interest, however, is in systems whose constituents are of

rather similar mass.

The integral equations governing the g^(E) may be derived most

simply by balancing the rates at which particles enter and leave an

energy bin (E,dE). As usual, we neglect thermal energies and collisions

between energetic recoils. We assume a normally incident beam of

constant flux <f> whose ions have laboratory energy E^, nuclear charge ,

and mass m^. The maximum primary recoil energy of species i is then

Ei ■ max [''biEf Yij’'b3Eb

2where y = 4m^m^/ (m^+m^) .

Let us now focus on species 1. Type 1 atoms in (E,dE) may leave

this bin either through collisions with stationary atoms of type 1, in

which case the loss rate is

Ej n1do11(E,E>)g1(E)dE = n^g-^EjdE ,

E'=0

or they may collide with type 2 , with a loss rate

75

j n2do12(E,E')g;L(E)dE = n ^ ^ g ^ d E .

E"=0

Type 1 atoms may enter (E,dE) in a number of ways. They may be

promoted from rest by other energetic type 1 atoms, for which the

appropriate term is

Ei Ei

/ n1do1 1 (E ',E )g1 (E ')d E ' = J %1 (E') dE ;

E'=E

or they may be promoted from rest by energetic type 2 atoms:

E2 E2

j njdo^E'.DgjCE'OdE' - ^ J g2 (E')

E'- Eh n E/y12

The lower integration limit here derives from Eq. (4.1) since we must

have Species 1 atoms may also fall into (E,dE) from higher

energies. If they collide with stationary type 1 atoms, the rate of

gain is

E1 E1

j n 1do1 1 ( E ; E ' - E ) g 1 (E ' ) d E " = n ^ JE'=E E

If they collide with type 2 atoms, the term is

(E') dE

min[E/(l-Y12),E1] min[E/(1~Y12),E1 ^

The upper integration limit is a consequence of the two requirements

that E'- E<y12E', and E'<E1-

Finally, we shall let <5> (E)dE be the number of primary recoils

of type i created per unit volume per second by the beam. As in

Eq. (3.16) we take

0. (E)dE = n.<j>da . (E , E) , E<y,.E, . (4.4)l l bi b bi b

77

For the beam-target combinations and energies (E.^100 keV) to beb-2considered, the r approximation to the Thomas-Fermi screened Coulomb

potential is again the appropriate choice for da, .(E, ,E):b l b1

Z, 7-. I m \2 .b l / b I 1 dEd°bi(Eb ,E) 6,93 (A eV) j z 2/3 + z 2/3U/2 (m.) £ 1/2 r3/2 * ^ b i ^

l b 1 IE'

(4.5)

-3/2Consolidating all the factors multiplying E into a single source

strength parameter, s^, we write

sM E ) = 7 7 2 , ES1,blEb , (4.6)

E"

witn

1

Z Z ' 2°2 b isi = 6-93(A eV) / 2/3 2/3\l/2 \i». / 1/2/ M

/zb2/3 + z ^ y 1 \ ”i /

We digress momentarily to note that according to Eq. (3.20) and

the discussion following it we should expect that the sputtering yields

will be approximately proportional to the stopping power of the beam,

which in the case at hand is

But with dab^(Eb ,E) given by Eq. (4.5) both terms in dE/dx are indepen­

dent of E^. This suggests that neither the total amount of energy

deposited in the surface region nor the way this energy is initially

divided between the two species will depend strongly on E, in the presentbmodel. Thus we may anticipate that the consequent fractionation effects

will be insensitive to E^.

Returning to the above expressions, we equate the total loss rate

of type 1 atoms in (E,dE) to the total rate of gain to find

Ei n o ^(nl°ll+n2°12>®l<E) " 2nl°ll f S1(E J ~F~ + yu j 82<E * ~F~

e/Y 12

min|E/(l-Y12) .Ej.

+ - J 1 1 j + V E) (4'8)

The corresponding transport equation for g2 (E) may be obtained from

Eq. (4.8) by interchanging all subscripts. Hereafter we use the symbol

l-«-*-2 to indicate additional equations obtained in this manner.

Due to the nature of the integration limits, the g^(E) which

solves Eq. (4.8) will exhibit some transient structure in the vicinity

of the boundary at E^. By this we mean that g^(E) has slightly varying

functional forms in the various energy regions defined by Ei»E2 ,Yi2El’

^12^2 ’ 12^T’ ■“ ' ^or ^nstance» the second term on the right does not

contribute to g^(E) for E ^ y ^ ^ * This structure is clearly a mathemati­

cal artifact resulting from our assumption of an infinitely narrow beam

energy width. Furthermore, the hard sphere collision model incorporated

in Eq. (4.8) is not well justified at energies near E^ when E^>1 keV.

But fortunately we need not be concerned with the exact form of g^(E)

near E^, for we wish only to obtain a good estimate of the relative total

yields. In light of our results for the mono-elemental case, we must_2expect g^(E) to be roughly proportional to E at low energy and to fall

off more rapidly near E^. It then follows from the form of Eq. (4.8)

that as long as E^ is sufficiently larger than the surface binding

energy IK, the energy dependence of g^(E) near IK will be quite insensi­

tive to the precise value of E^. This will be the case, for instance,14 40for 50 keV N incident upon a Ca target, an example to which we

shall later return. Here E^ = 38.4 keV and = 1.83 eV. We shall

therefore determine the low energy behavior of g^(E) by solving Eq. (4.8)

with E /-*■<*>, while s^ is held fixed. The boundary conditions at E^ which

are removed by this operation will be reinstituted later through energy

flux considerations. Such a procedure provides a more accurate normali­

zation for g^(E), and hence the sputtering yields, than would a direct

attack on Eq. (4.8), since it deals primarily with the fluxes at low

energy. The equation to be considered is thus

80

+ n2°12 / gx(E') dE,3/2 (4.9)

and 1-^2 , with y 5 y = y .

This equation is solved by functions of the form

A. B.8i(E) = “i ‘ 1 7 2 (4,10)E E

The energy dependence of this expression is the same as that which we

found in the mono-elemental case, as can be seen from Eqs. (3.17) and

(3.21). Through other considerations, Thompson (1968) has also obtained

an expression having the structure of Eq. (4.10) for a single species_2target. As expected, g^(E)aE at low energies.

Substituting (4.10) into Eq. (4.9) and noting that = Q2i’ we

find that the A. must be related byl

A 1 A 2 (4.11)nl n 2

-2The significance of this result is that the dominant E terms in the

flux distribution must be in the stoichiometric ratio. The magnitudes

of the A^ will be fixed below through the imposition of an auxiliary

condition. The B^ coefficients, on the other hand, are completely-3/2determined from the E terms of Eq. (4.9):

81

[3„i‘B1 , y1 / 2 i 3 s 2 + |3nl ° 2 1 ' n2 °2 2 ~ <1”V)

2 12

-B„

- 2 (<

and l«->-2. Thus the are not stoichiometric. Their relative magnitude

is determined by the source strengths s^, concentrations n^, and cross

sections but since the y factor is symmetric, it does not depend

explicitly on the atomic masses. Any contribution to fractionation due

to the bulk energy sharing process must originate in these terms, the

strength of this contribution depending on the ratio of B^ to A^.

To fix the normalization of the g^(E) we balance the power intro­

duced by the source in the form of atoms having energy greater than some

Eo<<E^ against the rate at which such atoms dissipate their energy

through collisions (Kostin, 1966), for in the steady state the total

energy carried by such recoils must be constant. Since there are -3/2s^E dE i-type primaries produced per volume per second in (E,dE),

TbiEb yb2Eb/ EdE . f EdE771 2 J E3/2Power In _ I ue, , / / ,Volume '"I 1 + S~ 1 • (4'

E Eo o

The power outflow from type 1 particles in the interval (Eo»E)

arises from the transfer of energy to stationary atoms. For collisions

with other type 1 particles,

E1 oPower Out (1+1) , n ^ t | ^ E j p l + J d|lj

Volume 1 11

E 0 E-Eo o

• 1 2 )

13)

(4.14)

The first term on the right applies to interactions in which the energy

transferred to the struck particle is less than E . The second termodescribes transfers such that the energy retained by the incident

particle is less than Eq . The corresponding expression for collisions

between energetic type 1 and thermal type 2 atoms is

82

E, min[E ,yE]1 oPower Out (1+2) = j ^ dEg f E"_dE_*

Volume 2 12I

j dEgl(E) jYE

Eo

min[Eo/(1-y) ,E^] yE

j dEgl(E) jE E-Eo o

+ I dEg, (E) I (E~EyE)dE ) • (4.15)

The power lost by type 2 particles in (Eq ,E2) is given by Eqs. (4.14)

and (4.15) with l-<->-2.

If the exact g^(E) were known, the normalization determined by

equating Eq. (4.13) to the sum of Eqs. (4.14) and (4.15) would necessarily

be independent of Eq. We cannot be assured of this in the present case,

however. Consider Eq. (4.14), which may be reduced to

EiPower Out (1+1) E 2 f g,(E> fVolume 1 11 o “1'“' E/

Eo

This suggests that the fluxes in the immediate vicinity of Eo are the

most significant for the normalization procedure. It is consistent

therefore to take Eo<<E^, for not only is this the region in which we

may have the most confidence in the g^(E) of Eq. (4.10), but it is also

the energy range in which we most desire an accurate evaluation of the

A coefficients. In particular we shall take E <E_,(1~y ) (but note thato iwe do not set E^ = °°). As a matter of fact, it eventuates that the A^

thus determined are independent of E .oThe equation obtained from Eqs. (4.13) - (4.15) may be reduced

through the use of Eqs. (4.8), (4.11), and (4.12). After some algebraic

manipulation we find

83

Ai . 4 Sl<Yb A )1/2 + S2<Vb2Eb)‘/2 (4.16)^ 2 2 1 nl°ll + 2nln2°12r(Y) + n2°22

where

T (y) = 1 + Y (1-Y)

It is useful to note that lim r ( y ) = 1 and lim T(y ) = 0.Y+l Y^O

From the symmetry of Eq. (4.16) it is clear that A^ do not

depend on the way in which the source energy is initially partitioned

between the species, hence partially accounting for the insensitivity

of the enrichment effect to the beam characteristics. Furthermore,- 1/2for the beam-target interaction considered here, s^ « E^ so that A^

is independent of the beam energy. Both of these facts are consistent

with the observation made following Eq. (4.7) that the stopping power

of the beam, as we have modeled it, is independent of E^. On the other

hand, the B^ do exhibit a dependence on E^; the reason for this being

that the energy available for the production of recoils at energy E

is not strictly proportional to dE/dx, but rather to (see the

discussion following Eq. [3.15])

84

E' dcbl(Eb .E')

Again we see that the expected characteristics of the fractionation

effects depend critically on the relative magnitude of the two terms

in g^(E). This quantity may be estimated by considering the limits y =1»

0ll*°12=a22’ sl/nl"s2/n2 and Ybl=Yb2 = V WhiGh yiGld

Consequently, the flux distribution is dominated at low energies by the

sputtering yields. This conversion can be effected by introducing the

surface as a planar potential step of height for species i. As in

the monatomic case (Chapter III), the external flux of i-atoms per

incident ion, with (external) energy in the interval (E,dE) and directed

into the solid angle element dfi is

-1 1(4.17)

The significance of this last result is more readily assimilated

when the internal fluxes g^(E) are converted to the corresponding

S1 (E,Q)dEdn ? [ m e+v -

(nlall+n2°12') (£+IVr n 2 12}3/2j (E+l^) 4tt

EdE cos6df;

(4.18)

85

and l-«-»-2 , with 0 = 0 ° taken along the outer normal to the surface of

the target. The second term in the brackets, which gives the primary

somewhat for the anisotropy of the beam source, since obviously these

recoils cannot contribute directly to the ejected flux in the case of

backward sputtering at normal incidence. Alternatively, one may take

the view that this step accounts, in a first approximation, for the

recoil implantation of surface atoms into the bulk of the material.

On the other hand, both the transmission sputtering and backward sput­

tering at oblique incidence could be contributed to by such primary

recoils, but these are not problems we shall address here.

Integrating Eq. (4.18) over E and fi, the partial sputtering yield

of species i is

and l-<-*2. We nave chosen to indicate explicitly the dependence of the

yield on the species' concentration. In view of relation (4.17) and

the fact that the four smallest terms here tend to cancel pairwise,

we may safely retain only the two dominant terms, writing

recoil contribution to g^(E), is subtracted in order to compensate

• (4.19)

The first term here is greater in magnitude than the second by1/2 2 about a factor of (y E./U) , which is typically on the order of 10 forb b

beam energies such that the cross section of Eq. (4.5) is appropriate.

Thus the importance of the surface binding energies in determining

preferential sputtering is evident, for if the U differ by more thani

a few parts per thousand this difference will dominate any nonstoich-

iometric contribution from the B^. In this case, recalling Eq. (4.11),

we find

In other words, intracascade energy sharing should generally not be an

important factor in elemental fractionation, due to the nearly stoichio­

metric ratio of the flux distributions. This result offers a quantitative

explanation of the observed fact that fractionation phenomena do not

depend significantly upon the mass and energy of the beam ion (Liau et L.,

1977), at least when

Equation (4.20) also predicts that surface binding effects, and

not the partitioning of energy, are responsible for the observed

enrichment of surface regions in certain heavier elemental components.

A particular instance in which surface binding effects are large, and

one in which the present kinetic theory approach is most fully justified,

is the sputtering of planetary atmospheres by energetic ion bombardment.

The binding energy here is just the gravitational escape energy of a

molecule, so that Eq. (4.20) becomes

The consequences of Eq. (4.21) for the evolution of the atmospheres of

Mars and Venus due to solar wind sputtering will be fully explored in

Chapter V.

Before proceeding further with our discussion of multicomponent

media, we should point out that the model developed here does in fact

result in reasonable estimates for absolute sputtering yield magnitudes.20As an example let us consider a 100 keV Ne beam incident upon a pure

63Cu target. Previous expressions for sputtering of a binary medium may

be applied to a single species target simply by assuming the two compo­

nents to be identical, with the result that

87

s = - L4$ i t - (B no-) 3U,1/2 (4.22)

or

, 1/2 ,Determining sE, /nfy from Eq. (4.6), o from Eq. (4.3), and taking bU = 3.5 eV for copper (Gschneidner, 1964), we find S = 4.4 Cu atoms

sputtered per incoming Ne ion. This yield value does not include the

energy deposition correction factor discussed in Chapter III. For

the beam-target combination under consideration, a = 0.5, so that

S = 2.2 is in fact our best estimate of the yield. The sputtering

theory of Sigmund (1969) applied to this system predicts S = 2.7,

which is also the observed value (Dupp and Scharmann, 1966). These

figures inspire some confidence in the present model even for the

determination of absolute sputtering yields. The problem of relative

88

yields for two components of the same target is even less sensitive

to the various approximations entering our formalism and hence, we

contend, may be much more accurately represented within its framework.

Another problem of some formal interest is that of a delta

function source of primary recoils. Instead of Eq. (4.4), we might

assume

$.(E) = s' 6 (E' - E) , (4.23)

and for purposes of comparison

si ■ ni * j dobi(Eb ’E) -

T .min

-3/2with T . : U.. For such a source there will be no E term in themin i-3/?solution of Eq. (4.8). It bears reiterating here that the E "term

in Eq. (4.10) originated basically as a consequence of the fact that

the only energy available for the production of recoils in the bin

(E,dE) is that which is deposited by the incident ion in the form of

primary recoils having energy greater than E. Since our assumed-3/2source distribution of primaries was proportional to E , it was to

be expected that the resulting recoil flux would be reduced by a

term with this same functional form. In the present example, however,

all of the deposited energy appears in primaries whose energy exceeds

E, so long as E<E'. There is thus no modification of the recoil flux

89

due to the cut off of the primary spectrum at E.

Proceeding otherwise as for the distributed source one has

8i(E) = ~2 • E<<E:E 1

with

a ; 2CSJEJ + S-E')n . 2---------------— ------- — ------- j ----- ‘ (A,24)i r1!0!! + 2n1n2o12r(-Y) + n2<J22

Note the similarity to our previous expression for A^/n^. If we take

s£/nl = s2^n2 E s'* sl^nl ~ s2^n2 H S’ ^°bi aS ®^ven *n andalso E' r E' = E', then in order of magnitude

^ s 2s V b )1/2 s « W 172A'± s'E' E'

Our conclusion is that a source which produces n.<J>o (E ) primariesi bi b1/2of species i per volume per second, each at energy E' = (Uy E ) results

in roughly the same low energy fluxes of recoils as does a beam of-3/2energy E^ and flux 4* which deposits energy according to the E law.

-3/2Of course we are neglecting here the B^E terms which reduce the

g^(E) in the latter instance. The lack of these terms in the delta

function source solutions results in their being in the stoichiometric

ratio.

Models for the compositional changes in the surface region of a

target which accrue from prolonged sputtering have frequently been

based on the assumption of a linear dependence of the partial yields on

the concentrations, i.e., S^(n^) = n^S^(l). As Winters and Coburn (1976)

have pointed out, however, this can at best be a zeroth order approximation.

This problem may be addressed within our model in a particularly straight-_2forward manner by considering only the dominant E term in Eq. (4.19).

The sputtering yields for pure targets S .(1) follow from Eqs. (4.16) and

(4.19) by alternately taking n^ = 1 and n£ = 1. Letting U^(n^) be the

surface potential step for species i when this species comprises a

fraction n^ of the target, we have

[nl°llUl(1)Sl(1) + V 22U2 (1)52 (1)] i i U.(n.) 2 2 ( • 5)

l l ni ° n + ln2°12 n2 22

Recognizing that the denominator of the second factor in this expression

does not vary strongly with n^, we find that, unless U^(1)S^(1) is very

similar to U2 (1)S2 (1), the general trends of the concentration dependence

may be adequately represented by

(4.26)

90

n .S.(n.) = 1i'“i' U±(n1) nlUl(1)Sl(1) + n2U2(1)S2(1)

Experimental support for this relation has been noted by Haff and

Switkowski (1976), who originally proposed it as a model for partial

sputtering yields under the assumption that the recoil flux is stoichio­

metric. According to these authors (see also Haff, 1976b) the factor

[nlUi(l)Sf(1) + n2U2 (l)S2 (l)] is proportional to the total number of

low-energy secondaries, while the additional factor n^ is just the

fraction of those recoils represented by atoms of species i. In light

of this interpretation, the denominator of Eq. (4.25) would seem to

incorporate the effects of energy loss from the cascade due to the

imperfect coupling of the species.

91

The present microscopic analysis indicates the extent to which

elemental surface binding effects do in fact dominate the preferential

partitioning of energy within the bulk material. On the other hand,

when the species concerned are isotopically related, thus exhibiting

essentially identical chemical properties, they must experience nearly

the same average surface potential. Mass dependent effects are possible

of course, but do not seem to be large. Consider, for example, two40 44calcium isotopes, Ca and Ca, bound with an energy on the order of

2 eV to the surface of a solid at room temperature. Such a system is

relevant to the discussion of the sputtering of plagioclase pursued in

the next section. To a suitable approximation, the low energy vibra­

tional states of such an atom are those which obtain in a three-dimensional,

isotropic harmonic oscillator potential. Assuming identical chemical

properties, the force constant k for this potential is the same for both°2species. We may estimate k : 2 eV/A . Differences in the binding

energies of the two atoms then result from the mass dependence of the

average thermal energy of a harmonic oscillator. The average energy of

a particle of mass nu in such a potential, at temperature T, is

E3h

(4.27)i

3 3 2where y h u> = y ti(k/mj is the ground state energy.

isotopes we find

For the two Ca

E'40 7.96 x 10-2 eV

and

E 7.94 x 10-2 eV.

Consequently, the heavier atom is more tightly bound because it resides

lower in the potential well on the average. The surface binding

energies are thus expected to be in the ratio:

> V U«, ‘ 1 + 1 0 ' 4 •

Since we shall be concerned with fractionation effects on the order of

one part per thousand or greater, this variation in the boundary poten­

tials is quite negligible.

Assuming the equality of the binding energies in the isotopic case,

it becomes necessary to evaluate both terms of the flux distribution in

order to account for fractionation effects. For illustrative purposes14we consider the action of a 50 keV N beam on a pure calcium target

40 44composed hypothetically of 98% Ca and 2% Ca. Since, according to

our prescription, the hard-sphere cross sections involved are equal, the

determining factor in the calculation is the relative magnitude of the

source strengths (Eq. [4.6]). Labeling the isotopes by their mass

numbers, the s. are found to be related byl

92

S4Q _ /m44\2 n40 = 1>05 ^40f - Y\m40/s44 \40/ n44 n44

Then, from Eqs. (4.12) and (4.16) one calculates

A40 - 1931/2 B40

where we have set U = 1.83 eV (Gschneidner, 1964). This last ratio

is in accord with the estimate of Eq. (4.17). Also from Eq. (4.12) we

find

93

^ , o.9544 n44

Considering these numbers, it is clear that the partial yields will be3

stoichiometric to within the order of one part in 10 .

Now the fractionation of species 1 with respect to species 2 in

the first material eroded from a target having an initially uniform

composition given by n£ and is defined as

S (n )/nV 1;2) ' s ^ ) 7 ^ - 1 • <4 -28>

This is the fractionation of the sputtered material, as opposed to the

target's surface, in that a positive value of 6£(1 :2) indicates an

enrichment of the effluent atoms in species 1 when compared to the bulk

stoichiometry. Actually, since sputtering involves only the surface

region of the target medium, Eq. (4.28) may be understood to be the

fractionation with respect to the initial composition of this surficial

layer whether or not this is identical to the bulk inventory.

For the current example, Eqs. (4.19) and (4.28) yield

6f(40:44) = 5 x 10_4

Within the framework of our model, the preferential sputtering of the

lighter species can be most directly attributed to its more efficient

coupling to the beam ions as is evident from the ratio of the source

strengths above. On the other hand, this nonstoichiometric effect is

greatly diluted by the stoichiometric nature of the intracascade

energy sharing process. The fractionation thus predicted is to be

compared with effects on the order of 6 ^ 10 which have been observed

for these two calcium isotopes in fluorite (CaF2) and plagioclase

(W. A. Russell, 1979). The implication is that the bulk energy sharing

mechanism is not the controlling factor in fractionation, even in the

isotopic case. But in order to examine this result in the light of the

most precise measurements available, we shall develop in the next section

an extension of the present model applicable to certain multicomponent

targets. Following this, we turn to the question of alternative mechanisms

for preferential sputtering.

C. Polyatomic Media

We focus our attention in this section on compounds in which the

isotopic constituents of interest account for only a small fraction of

the total mass, enabling us to adopt a perturbative approach to the14calculation. To be specific, we consider the sputtering, by a 50 keV N

beam, of the mineral plagioclase, with composition 50% CaAl2Si20g and

50% NaAlSi-0o. We shall again be concerned with the preferentialJ o40 44sputtering of Ca with respect to Ca. This target, although complex,

is of special interest because sputter-induced isotopic fractionation

effects in plagioclase have been measured very carefully by W. A. Russell

(1979), whose purpose, in part, was the elucidation of the contribution

of solar wind sputtering to mass fractionation effects in lunar soils

(Switkowski et al., 1977).

It should be noted that the experiment was actually performed with

100 keV ^ molecules, as opposed to a 50 keV elemental beam. Since the

molecules must dissociate immediately upon impact, one might expect their

stopping power, and hence their sputtering yield, to be just twice that 14for a 50 keV N atom. On the other hand, the simultaneous juxtaposition

94

-2

of two particle tracks might increase the recoil density in the cascades

to the point that nonlinear effects, i.e., collisions between energetic

recoils, become important. Under these circumstances, the energy in

the cascade would be more evenly distributed among a greater number of

recoils and thus an enhancement of the sputtering yield would eventuate.

Such effects are indeed observed when relatively heavy targets, e.g.,

Ag, Au, and Pt, are bombarded by heavy molecular beams such as Se2 , Sb2>

and Te2 (Andersen and Bay, 1974; Thompson and Johar, 1979). However,

the nonlinearity appears to diminish rapidly with decreasing target and/or

beam ion mass; in particular, Thompson and Johar (1979) have found that,

for an Ag target, no enhanced sputtering effects occur when the stoppingO

power of the incident beam is less than ^100 eV/A. If the same criterion

is applicable to plagioclase, then the induced cascades are well

within the linear region, as the stopping power of 100 keV N2 on plagioclaseO

is ^27 eV/A. Consequently, the linearized transport theory adopted here

is deemed to be sufficient.

We shall model plagioclase as a three-component medium composed of40 44Ca (n^ = .0377), Ca (n^ = .0008) and a bulk species (n^ = .9615), where

n^ should now be interpreted as the atomic concentration of species i.

The atomic mass and nuclear charge of this last component are to be chosen

so as to represent in an average manner the interactions of the Ca atoms

with the remaining species. The two parameters which characterize these

interactions are the hard-sphere cross section and the coefficient of

maximum energy transfer The cross section for collisions between40Ca atoms and the bulk species is determined by the average

40 44That is, the sums extend over all components except Ca and Ca. Note

that 0^2 = ®2i = q13 = °3l’ w^ere the last two equalities are due to the

fact that Z = Z_. As usual, our estimates of the a , , are based on 2 3 ijEq. (4.3). Once a ^ is found, an effective Z^ follows from Eq. (4.3).

We find Z^ = 9.62. Now in order to fix y consider a ^ C a atom of

energy E travelling a small distance dx. The probability that this atom

will collide with species i is n Furthermore, the probability40that the Ca will transfer an energy (T,dT) in this collision is

d^ Yi2^’ ^°r ^“Yi2 " ^ ^a r aS e n e r §y -transfer properties are concerned,

it therefore seems appropriate to define

y'n.o ../y1 “ l i2 i2

, i ¥ 2,3 ,

96

Mi2

and 2-<-»-3. The value obtained for y implies m^ = 18.6 amu, while y ^

gives m^ = 18.7 amu. It is immaterial which value of m^ is chosen for

the following calculation because this quantity is directly involved

only in the interaction of the beam ions with the bulk species. The

coupling of the latter to the two isotopes is determined solely by the

gamma factors. We note that y ^ = 0.866, while y ^ = 0.837, which

leads us to anticipate that somewhat larger fractionation effects may

be present in this model as compared to the binary case, where Y23 = 0.998.

Two further approximations are embodied in the model. First, we

assume that, except for the purpose of overall normalization, the small

Ca component may be neglected in calculating the flux distribution of

the bulk species, g^(E). Again adopting a source of the form of Eq. (4.6),

it follows from Eq. (3.14) that

97

M E ) = 4 - BlVW E2 E3/2 ’

with

A ’ J <TblEb /2Band

B ’ 3 si/on

(The n^ appearing in s£ is now the concentration of the bulk species.)

In effect, this bulk flux acts as a second source of energetic Ca

recoils, in addition to the direct collisions with beam ions.

Our second assumption is that, due to the dilution of the two Ca

isotopes in the target, it suffices for the purpose of calculating their

interaction with each other (only) to approximate = and 82^n2 =

g^/n^. The flux equations for g2 (E) and g^(E) are thereby uncoupled,

and so we have

[nl°21 + (n2+n3)°22J 82 (E) = 2(n2+n3)a22 f g2(E'} f 7

e/(i-y 12)

e3/2

(4.29)

and 2-*-»-3. As in Eq. (4.9), we have here taken E-*- “ in the integra­

tion limits.

The solution of Eq. (4.29) has the form

a 2 ®2 ^2^ - p ? 2 + T 7 < 4 - 3 0 >

The constants a2 , B2» and X a r e fixed by Eq. (4.29). In particular,_2

Q2 = n 2^^ni so bbat the E term is stoichiometric, as in the binary

case. We also have X2 ~ 1; but 62 must be found by imposing some

boundary condition on the problem. If we do so, however, we find thatX ?the 62/E term is smaller in magnitude than the second term by a factor/ P 2of (yblEb/E) . The first two terms are related by the same factor, as

in Eq. (4.17). We shall therefore be safe in neglecting the contribution

of the 5^/E 2 term to g2 (E). The boundary condition may be disposed of

in this manner because the dominant source for the Ca recoil flux is its

interaction with the bulk flux g^(E), and not the beam ions. Fractiona­

tion effects arise from the different couplings of the two isotopes to-3/2the bulk species, as is manifested in the E terms of their fluxes.

Upon substituting Eq. (4.30) into Eq. (4.29) one finds

98

B2

c By ( 2y ^2 — k )2 12 12 V (4.31)

with

and

| (3-4c23)y12 - 2cl[3 -(1-y12)3 ]

k2 S2all/n2Sl°12 ,

C23 (n2+n3)022/c

c = n i a i 2 ^ c ’ 1 = 1 ’2 ’3 »

where

The corresponding expression for 6 may be obtained by replacing c ^ , Y^2>

and k2 in Eq. (4.31) with c^, Y ^ ’ and k^, respectively.

Assuming a planar surface potential of height U = 1.83 eV

(W. A. Russell, 1979; Gschneidner, 1964) the partial isotopic sputtering

yields may be found after the manner of Eqs. (4.18) and (4.19) including

the subtraction of the appropriate primary fluxes. The resulting expres-40 44sion for the fractionation of Ca with respect to Ca is

1/2(y E /U)' - 2c.. (B~/c B + k /3)6,(40:44) =■ ■ bl b L ? _ J ----- 1--- . j , (4.32)

(YblEb/U) ” 2ci(B3/c3B + k3/3>

which yields the value

6f(40:44) = 8 x 10 4

A comparison of this result to that for the binary target indicates

that the intermediation of a third species of dissimilar mass tends to

enhance the nonstoichiometric partitioning of recoil flux between the

isotopic species. Yet, again, this value is much smaller than observed-2effects (W. A. Russell, 1979) which are on the order of 10 for plagio­

clase. Thus it appears that the bulk recoil generation process does not

play a major role in the enrichment effect. The alternative, within the

framework of the present analysis, is that preferential isotopic loss

occurs through the differentiation of the flux distributions as they

exit the target, notwithstanding the equality of the boundary potentials.

One such possible alternative mechanism is discussed in the following

section.

D. Surface Flux Model

Previously, the surface of the target has been treated as an

imaginary plane in the interior of an infinite solid, with the sputtering

yields being calculated from that half of the flux crossing this plane

which has a positive component of velocity normal to it. As such, the

presence of the surface did not alter the form of the spectra of the

cascade secondaries in its neighborhood. A more careful physical exami­

nation reveals, however, that certain conditions unique to the surface

region do in fact bring about a modification of the flux distributions.

In the first place, there exists in most solids a displacement threshold,

E^, usually on the order of 25 eV, such that atoms recoiling with smaller

energies are not permanently displaced from their lattice site, but

instead lose their energy through collisions with their nearest neighbors.

These low energy (IKE^E^) secondaries typically compose 80-90% of that

portion of the internal recoil fluxes which contribute to sputtering.

The implication is that the great majority of sputtered atoms must origi­

nate within the one or two monolayers nearest the surface. Higher energy

recoils could, of course, escape from deeper within the solid; however

the anisotropy of the flux increases with energy, with the velocities

being directed preferentially away from the surface.

On the other hand, atoms in the topmost monolayer can acquire

velocities directed toward the outer hemisphere only, except for

infrequent collisions with beam ions. Then, due to the proximity of

the free surface and the location of their nearest neighboring atoms in

the surface plane and lower hemisphere, it is difficult for such atoms

to return much of their kinetic energy to the bulk of the target before

they escape. This is especially the case for those particles escaping

100

at small angles with respect to the outward normal to the surface. At

larger angles, interaction with other atoms in the surficial layer

becomes more probable.

The flux distributions, g^(E), we have examined heretofore contain,

in any energy interval (E,dE), roughly equal contributions from particles

being promoted from rest and from those falling down from higher energies.

The above considerations, however, lead us to propose that the sputtered

flux is actually dominated by those atoms which have been promoted from

rest immediately prior to ejection and escape the target without further

collision. Let us refer to this flux, composed of atoms in the surface

region which have participated in only one energetic collision, as

G^+^(E). Admittedly, G^+^(E) will represent a limiting form for the

surface flux distribution, our thesis being simply that it provides

a more realistic model for recoil flux at the surface than does the

bulk flux g^(E).

One approach to the calculation of G ^ + (E) is to assume that the

particles described by the various g^(E) do not escape the target directly

but instead are incident on stationary atoms in the surface which then

recoil to form G^+ ^(E). This is the view developed in Watson and Haff

(1980). A qualitatively very similar mechanism has also been discussed

by Winters and Sigmund (1974) as a possible contribution to the sputter­

ing of nitrogen atoms chemisorbed on a tungsten substrate. In the

alternative model described below, we choose to retain the concept that

the surface recoils are participants in collisional cascades which

extend up to, but are terminated at, the solid-vacuum interface. In

essence, the loss of energetic particles at this boundary is viewed as

being responsible for the reduction of the internal flux g^(E) to the

101

surficial flux ^(E). Numerically, the results of this model do not

differ appreciably from those of Watson and Haff (1980).

We consider a target medium composed of N species. In accordance

with our conclusion that the internal recoil fluxes are very nearly in

the stoichiometric ratio, we assume that the bulk flux distribution for

each type of atom is adequately represented by

A.8i(E) = ~2 ’ 1 = (4.33)E

with

A./n. = A./n.3 3

In this approximation, the energy source, i.e., the primary recoil

distribution, is accounted for only in the overall normalization of the

A^. A function having the form of Eq. (4.33) is a solution of a homoge­

neous energy sharing integral equation, that is, Eq. (4.9) without the

source term. For any reasonable beam-target interaction, a hard sphere_2system is dominated at low energy by this E behavior, and thus so will

be the relative yields of the various species present. The only

significant exception to this statement occurs when the source function_2is also proportional to E , as obtains when the beam ions are suffi­

ciently energetic that the primary interaction is through Rutherford2collisions. In this case g(E)=£n(Eb/E)/E (see Eq. [3.14]). We shall

not be concerned with this energy regime, however.

Neglecting then the source terms, the various internal flux distri­

butions are related by

102

In line with our other approximations, we have here assumed the beam

energy to be arbitrarily large. We thereby remove any dependence of

the fractionation on the source characteristics. Introducing Eq. (4.33)

into Eq. (4.34) we find that the flux of species k may be written as the

sum of two terms:

gk (E) G^+)(E) + g£_)(E) , (4.35)

where

G^+) (E)

and

with

i°ikY ik(4.36)

l

Note that y is just a weighted average of the gamma factors for the

various possible interactions of k-type atoms. Thus The

quantity G^+ ^(E) represents that fraction of gk (E) arising from atoms

which are promoted to energy E from rest, while G^ ^(E) is that portion

of the flux falling down to E from higher energies. For species of not

too dissimilar mass we see that, as mentioned earlier, g, (E) containskroughly equal contributions from each type of recoil.

In the interior of the target both the G^+\ e ) and G^~^ (E) fluxes

are present. In the one or two monolayers nearest the surface, however,

recoils promoted from rest escape the region to a large extent before

they have a chance to cascade to lower energies. Thus we propose that

only the G^+^(E) type flux will be prevalent in the surficial layer, or

more precisely, that G^+^(E) dominates the sputtered flux. One conse­

quence of this picture is that the recoil flux emanating from the target's

surface is reduced in magnitude by about a factor of one-half from the

flux crossing some plane in the interior of the medium in a single sense.

Such a reduction in flux at an interface with the vacuum is familiar in

the context of diffusion theory; for instance, the flux of thermal

neutrons in a moderating material decreases sharply at the boundary of

the moderator. Thompson (1968) has also pointed out that treating the

surface as if it were embedded in an infinite medium should lead to an

overestimation of the ejection rate in sputtering experiments, particu­

larly of the low energy recoils. This conjecture has been confirmed

for certain circumstances by the calculations of Williams (1976a) in a

hard-sphere model of a monatomic medium. Unfortunately, Williams does

not discuss the problem of interest to us— that of a collimated,

normally incident beam.

In order to calculate the sputtering yields from the G^+^(E), it

remains to introduce the surface binding potential. We again adopt a

planar potential step of height U^ for species k. Furthermore, since

the boundary between the surficial layer and the bulk of the target

104

exhibits random irregularities on the scale of an atomic diameter,

we shall assume that G^+ ^(E) is isotropic over the outer hemisphere,

neglecting possible edge effects at angles near 90° from the outward

normal. These assumptions permit the evaluation of the sputtering

yields in the usual manner, save that it is not necessary here to

subtract a primary recoil contribution. The result for the partial

yield of k-type atoms is

\ Y k\ ■ ■ <4-37)

We therefore derive that the expected relative fractionation of the

constituent species in the first material removed from the target, as

compared to the composition of the surface region, is

Yl U26 (1:2) = = ± - f - - 1 . (4.38)

Y2

In the isotopic case so that type 1 atoms are preferen­

tially sputtered if Y 1>Y 9- But Y, is a direct measure of the coupling1 / Kof species k to the bulk of the medium, as far as energy sharing is

concerned. The larger Y^ is, the greater is the average energy transfer

in collisions involving k-type atoms. Loosely speaking, a given

component is preferentially sputtered because it more easily acquires

energy in the collisional cascades. Since, according to our prescription,

the hard sphere cross sections do not depend on the atomic masses, it

follows from Eq. (4.36) that, between two isotopes, Y^ typically will be

larger for that one whose mass is closer to the average mass per atom in

the target. Deviations from this rule may arise due to the nonlinear

dependence of °n ny and m_., but the important point is that the

105

106

lighter isotope is not necessarily preferentially ejected.

For a binary isotopic medium, where we may equate all cross

sections, Eq. (4.38) reduces to

nl + n2Y 126 (1 :2) = -± - 1 . (4 .39)f V l 2 + n2

According to this expression, and in agreement with the above remark, the

species which is more abundant in the surface region is expected to be

positively fractionated, irrespective of its mass. This result has

been anticipated by Kelly (1978) in his model for the preferential

sputtering of a binary compound. This model was based on the concept

of substrate fluxes impinging upon the surface layer proposed by Winters

and Sigmund (1974), and made use of the findings of Andersen and Sigmund

(1974) that these fluxes should be essentially in the stoichiometric

ratio. Since, in standard sputtering theory, the low-energy secondary

collision cross section is independent of the participating species

(Sigmund, 1969), this cross section would not, in the model due to Kelly,

enter Eq. (4.38) even when the two species are not isotopically related.

The present derivation has the advantage of indicating how differences

in atomic sizes may be treated. For the pure calcium target considered

in Section IV.B, n^ = .98, = .02, and y^2 = *99773, so that the

predicted initial fractionation is

6f(40:44) = 2.2 x 10_3

which is substantially larger than the effect which derives from a consi­

deration of the internal flux distributions g^(E), alone.

The distinction between the present model and that of Sections IV.B

and IV.C is even more pronounced when the target medium contains other

species of dissimilar mass, for this tends to enhance the difference in

the energy sharing properties of the two isotopes. The simplicity of

Eq. (4.38) now allows us to treat each constituent of plagioclase indivi­

dually instead of approximating it as a three-component medium through an

averaging process. The predicted fractionation of that material removed

from a plagioclase target before the composition of the surface region

changes significantly is then

6f(40:44) = 3.2 x 10_2 .

This value represents a very substantial enrichment effect. We reiterate,

though, that within the framework of the present model this value must

rigorously be considered an upper limit to the expected effect, since we

have assumed a purely (E) type surface flux.40 44W. A. Russell (1979) has measured the Ca/ Ca fractionation in

material sputtered from a plagioclase surface by a 100 keV N2 beam,_2finding (40:44) = (2.12 ±.02) x 10 . Unfortunately, this result is

not directly comparable to the value quoted above, since the latter

refers to an initial fractionation, whereas in the experiment theOsurface was eroded to a depth of 140A, which was ^27% of the range of

a beam ion. In the course of such extensive sputtering, the surface

region of the target becomes increasingly enriched in the heavier

isotope. This enhancement tends to counterbalance the preferential loss

of the lighter species so that the fractionation, with respect to the

bulk composition, of the material being removed from the surface

107

continually decreases. Eventually an equilibrium must be reached in

which the stoichiometry of the sputtered mass is identical to that of

the bulk, that is, 6 = 0. Consequently, the initial fractionation

is expected to be somewhat larger than that observed after prolonged

sputtering.

If, in making a sputtering yield measurement, one commences with

an undisturbed target of uniform composition, exposes the surface to a

constant ion flux 4> for a time T, and then measures the total fractiona­

tion of two components in the material sputtered, with respect to the

initial composition of the target, i.e., its bulk composition, the result

will be: T

— N I S (t)dt1 ( 0 , Z j J 1

6“eaS (1:2,T) =--- \ ------- - 1 • (4.40)

— f S (t)dtn2 (0 ,z) J 2

0

Here n^,(t,z) represents the density of species i at a depth z from the

surface of the target at time t, which for t = 0 is assumed to be

independent of z. We develop a model below for the calculation of ID63S6 which is patterned after the discussion of surface compositional

evolution under sputtering found in Liau et al. (1978). We shall thus40 44be able to compare the results of our model for Ca/ Ca fractionation

directly to the effects observed by W. A. Russell.

It is well known that sputter-induced stoichiometric alterations

in a polyatomic medium are not confined to the immediate surface region

of the target, but extend up to depths approaching the range of the

impinging ions, R (Liau £t al_., 1977; Liau ftl., 1978; Poate et al. ,

1976). Such subsurface manifestations of preferential sputtering may

108

109

be understood in part in terms of radiation enhanced diffusion driven

by atomic mixing in collisional cascades (Haff and Switkowski, 1977).

A consideration of mass conservation therefore implies that the

following relation must hold at any time (we refer explicitly to the40 44Ca/ Ca system):

k t

j n40 (t’z)dz + J *s40(t' )dt0 _ n40(° ' z)t n44(°*2)

(4.41)

J n^(t,z)dz + J <}>S^(t')dt'0 0

Now Liau et a l . (1978) have found experimental evidence that the

species densities in a PtSi target undergoing fractionation tend to

vary linearly with z over the range 0<z<R, with n^(t,R) = n^(0,z).

This may be understood theoretically in the diffusion picture if the

subsurface atomic displacement fluxes are much greater in magnitude

than the flux differential which arises at the surface due to nonstoich-

iometric loss. In this case the compositional changes of the surface

are rapidly communicated to the interior and one has a situation

analogous to an infinite slab of a good heat conductor, one side of

which is held at constant temperature while the other is slowly cooled.

The temperature profile will be nearly linear with depth in such a

plate. We shall therefore assume that

R

/ n40(t’z)dz = 2 [ n40(t,R) + n40(t’0) ] *

and similarly for n^(t,z). With this, Eq. (4.41) becomes

(4.42)

Let S = S^g(t) + S^(t) be the total Ca sputtering yield (we neglect

other isotopes). We shall assume that S is constant in time. This

implies that the total Ca component is not being factionated with respect

to the other species present. If actually Ca is being enriched in the

surface region, as is probable, the value of 6 calculated below is

somewhat of an overestimate because the lower concentration of other

constituents would have the effect of increasing the parity between the

average gamma factors, 7^* The effect should not be large, however.

It is consistent with this last approximation to assume that the total

Ca density, n = n /r.(t,z) + n (((t,z) is independent of t and z.4U 44Equation (4.43) may then be reduced to

t

I*S40( O d t ' = c40(0)S4>t + ¥ [C40(0) " C40(t)] ’ (4‘AA)

where we have introduced the concentration of species i at the surface

at time t, c.(t) = n^(t,0)/n. It is now possible to express 6™*

terms of the single unknown quantity c^q(T):

„meas .in

2S<f>T/Rn +6meas (40;44jT) = 1-C40(T)/C40(0) - 12S<J>T/Rn - j-cA0 (0)/c4 4 (0)jj-l-c4 0(T)/c40(0)j

(4.45)

As a further consequence of our assumption that the Ca concentration

in the surface region is constant, it follows that the initial value of

6 we have previously calculated applies to the fractionation of the

sputtered fluxes with respect to the surface composition at all times,

that is,

Ill

S40^t C4 0 ^-o rn = r Y m (6f + 1} • (4*46)4 4' ' c44(t) f

40 44We neglect here the minor effect on 6 of the changing Ca/ Ca ratio.

Using Eq. (4.46) and differentiating Eq. (4.44) with respect to t, one

obtains a differential equation for c^g(t):

Rn dc40(t) ..... S*(«£+ D c 40(t) (4 4?)T — ' c40 ' --1+6,c,.(t) 'f 40

Separating variables in Eq. (4.47) and integrating up to time T we arrive

at the relation

<l+6f) In* + 62 c40(0)c44(0> (l - i) - [l+«fc44(0>]2 ,

(4.48)

where

6fc40(0)c44(0)x = 7 i+6rc,,(0)T c40(T) - c4 0(0)[1+!£c44(0)]

Note that x ■+ 1 when T -*■ 0. In order that Eq. (4.48) be satisfied when

T -► °°, we must have x -*■ « in this limit. This implies that the equilib-40rium concentration of Ca in the surface is

which is quite a small change in the case under consideration. This

last result also derives from Eq. (4.46) if we demand that

lim S/n(t)/S.. (t) = c. (0)/c..(0), as it must, due to mass conservation. 4(J 44 40 44With c^q(T) derived from Eq. (4.48), 6^eas is determined. It

remains only to note that the quantity S<f>T/Rn is just the fraction of

an ion's range which is eroded from the target in a sputtering run of

duration T. In the experiment of W. A. Russell on plagioclase, this

quantity is 0.27, as we have mentioned. Actually, this value was

determined from the total amount of Ca collected, the calculated ion

range, and the density of Ca in the undisturbed target. It thus

provides a measure of the amount of Ca processed, whether or not

elemental fractionation enriches Ca in the surface region. This circum­

stance tends to mitigate our assumption that the elemental Ca sputtering

yield, S, is constant.

Recalling that = 3.2 x 10 2 and taking c^q(0) = 0.98 and

c. . (0 ) = 0 .02, we find 44

^meas (40:44,T) = 2.5 x 10_2 .

This number is in much better agreement with the observed value of _2

2.1 x 10 , but it is consistent with our assertion that our calcula­

tion provides an upper limit for the expected effect. The conclusion

to be drawn here is that the sputtered flux appears to be quite well

described by the G^+^(E) distribution, with perhaps only a small

G^ ^(E) component.

Another medium for which calcium isotopic fractionation data

are available is polycrystalline fluorite, CaF2 (W. A. Russell, 1979).

Since for fluorine, Z = 9 and m = 19 amu, a major distinction between

this case and that of plagioclase considered above is the much higher

concentration of calcium in CaF^. (Recall that in the approximation

of Section IV.C the bulk plagioclase material had average charge

Z = 9.62 and mass m = 18.6 amu.) It is also true that the incident14 + 14 +ions in this experiment were 130 keV N as opposed to 100 keV ( N)2*

But according to the model presented here, the fractionation process

should be insensitive to this difference in beam characteristics.

Equation (4.38) applied to this system yields

6f(40:44) = 2.2 x 10~ 2 .

_2The relevant experimental value is 6 = (1.27 ± 0.07) x 10 , subsequentO

to the removal of ^4.2% of a beam ion's range (70A). Again we must

conclude that the predicted value is not incompatible with observation,_2both because 6 = 2.2 x 10 represents an upper limit to the initial

fractionation and because the measured quantity is not truly an initial

value. The latter qualification may be circumvented through the appli­

cation of Eq. (4.45). Since here only 4.2% of the ion's range was

eroded, we should expect the correction to be much snaller than for

plagioclase. Confirming this, the predicted value for CaF2 is

6meas (40:44jT) = 2 .1 x 10-2 •

This value does not compare as favorably with the observed effect as did

the previous calculation on plagioclase. It is possible that the

113

polycrystalline structure of CaF2 might partially account for the

difference, but neglecting this, the apparent explanation within the

context of the surface flux model is that the CaF2 sputtered flux

contains a larger ^(E) component. Although we might offer some

heuristic arguments as to why this should be, it is clear that more

work, both experimental and theoretical, is needed to clarify the

situation. Perhaps the clearest, though not the easiest, experimental

test of the surface flux mechanism would come from measurements of

fractionation in a binary isotopic target at varying relative concen­

trations. The magnitude of the predicted effects are considerably

greater than the resolution limits which have already been achieved.

It would be most interesting to establish whether such a surface may

become enriched in its lighter component when initially this species is

the less abundant.

We expect the model advanced in this section to be most applicable

when thermal (or electronic) sputtering mechanisms are not operative.

At room temperatures and for beam energies in the nuclear stopping regime

these mechanisms are not dominant in the type of targets considered here,

but they may be able to explain the sputtering of some volatile substances,

such as ice (Brown et al., 1978) and UF^, uranium tetrafluoride (Griffith

1979; Seiberling et al., 1980). In the thermal spike picture of sputtering

it is assumed that the energy deposited by the beam ion is sufficiently

great and remains localized in the vicinity of the ion track for a suffi­

ciently long period that a small region of the surface achieves a near-

thermal equilibrium at an elevated temperature. Sputtering then occurs

as a result of the evaporation of atoms from this region. To the extent

that this model is valid, it is probably more appropriate to assume an

114

equipartition of energy among the constituent species, instead of the

equipartition of flux we have derived._ A model along these lines has

been proposed by Haff (1977), who finds that such equipartition can

lead to substantial enrichment effects.

As another possible contribution to fractionation, Liau et £tl. (1977;

1978) have proposed that a less massive recoiling atom has a greater

range in the target than a heavier atom of equal energy, so that,

assuming an equipartition of energy, the lighter atoms may escape from

a greater depth below the surface, thus enriching the surface region

in heavier components. This would offer an explanation for the preferen­

tial loss of lighter constituents which is frequently observed in

elemental fractionation. But even though such variations in the ranges

of low energy recoils may exist, we would argue that they are not in

themselves sufficient for the production of fractionation effects because,

if the flux of recoils approaching the surface region from the interior

of the target is stoichiometric in the first place, it is immaterial

whether these particles originated close by or further away from the

surface, for they will be replenished to the same depths by recoils

moving from the surface region toward the interior and the composition

of the surficial layer will remain unchanged. From our point of view,

one must first offer a mechanism for the differentiation of the fluxes

of atomic recoils. Ironically, such a mechanism exists in the model

of Liau et al. (1978), for as was shown in the introduction to this

chapter, the equipartitioning of energy implies a nonstoichiometric

recoil flux. Thus it would not seem necessary to invoke variations in

range in order to explain fractionation if in fact an equipartition of

energy were to exist. But it was precisely our failure to find any

115

substantial nonstoichiometry in the bulk fluxes that led us to conclude

that fractionation is essentially a surface related phenomenon.

Although the predictions of the surface flux model have been based

on a flux distribution which was purely of the (E) type, in reality

it is likely that an exact solution of the Boltzmann equation in the

half-space would be intermediate between the (E) and the original

g^(E) distributions, that is, it would involve some G^ ^(E) component.

Attention is called to the fact that if the G^+ \ e ) fluxes are, so to

speak, positively fractionated with respect to a particular pair of

constituents, the G^ ^(E) fluxes will be negatively fractionated in this

same pair (see Eq. [4.35]). We would like to suggest that there may be

considerable utility in characterizing the sputtered flux by a mixture

of contributions from such oppositely biased distributions. It seems

intuitively reasonable, for instance, that an energetic recoil created

in the topmost monolayer of the target's surface will have a better

chance of escaping without further collision the more nearly normal to

the surface its velocity vector is directed. Thus one might expect

the distribution of those particles escaping at small angles to the

outward normal to exhibit substantially more of the (E) character

than that of those escaping at more oblique angles, while the latter

would be closer to the gk (E) or perhaps the G^ ^(E) type. For an

isotopic target this would imply that one should expect larger frac­

tionations in the material ejected normally from the target than in

that ejected obliquely. Such an angular dependence has indeed been

observed by Wehner (1977) in the low-energy sputtering of Cu and Mo.

He finds that the lighter isotopes are enriched in the normally

collected samples as compared to the obliquely sputtered material.

116

However, these experiments were carried out at the very low energy of

100 eV, so that well developed cascades as assumed in our model are

not present. Thus his results cannot be compared with the predictions

of this chapter.

The isotope effect has been investigated in more detail by

W. A. Russell (1979) for energetic bombardment of calcium fluorite.

He finds that the fractionation of ^ C a with respect to ^Ca, as

compared to the bulk composition, tends in general to decrease substan­

tially at larger angles. In fact it was observed that after extensive

sputtering the material ejected to larger angles actually showed a

negative fractionation with respect to the bulk while the normal

fractionation remained positive, although its magnitude decreased. This

trend might be explained in terms of the surface flux mechanism as

follows: As sputtering proceeds, the surface region becomes enriched44 (+)in Ca. However, the surface flux, (E), which escapes more or less

40normally from the target, is enriched in Ca compared to the composition

of the surface region. This enrichment may be great enough to give the

material carried off by G^+ ^(E) a positive fractionation with respect to

the bulk. At the same time, the flux escaping obliquely reflects more

closely the composition of the surface region, which, due to the enrich- 44ment in Ca, will exhibit a negative fractionation with respect to the

bulk. Clearly, a more detailed examination of this picture is warranted.

3ut any quantitative analysis must deal consistently with several other

aspects of the problem. For example, after extensive bombardment the

target's surface develops roughness on a scale of hundreds of angstroms.

Why does this roughness not obliterate any dependence on the angle with

respect to the normal, which at a microscopic level is not well defined?

117

118

Also, it is true that the variation of the fractionation with angle is

not necessarily monotonic, but in one case showed a peak at intermediate

angles. Finally, it seems likely that the angular dependence of the

fractionation may be closely related to the general over cosine behavior

of the total Ca sputtering yield observed by W. A. Russell. By this it

meant that relatively more Ca was sputtered to small angles, and less

to large ones, than predicted by the canonical cos 0 distribution. We

suspect that the answers to these questions will depend in large part

on the details of the binding of atoms to the surface. In this connec­

tion, the fundamental distinction to be made, according to the present

point of view, is not between the isotopic species per se, but between

those ejected particles (of every species) which have suffered only

E. Diatomic Molecular Gas

So far in our discussion of sputtering we have neglected the

attractive interactions between the atoms comprising the target, aside

from their implicit role in providing the cohesive energy of the medium

which is manifested in the surface binding potential, U. That is to

say, the examples considered heretofore have been treated as systems

of noninteracting particles confined to some spatially uniform potential

well. In the next chapter, however, we shall be concerned with the

sputtering of planetary atmospheres. A distinctive feature of such a

medium is that the constituent atoms tend to be bound together into

diatomic or triatomic molecules. Two important consequences of such

a molecular association should be singled out. First, since ion-molecule

one energetic collision and those escaping particles which have

undergone more than one energetic collision

and molecule-molecule collisions may lead to excitation or dissociation

of these molecules, it is clear that some of the kinetic energy which

would otherwise be present in a collisional cascade will be absorbed in

inelastic processes. Of course, a loss of energy in the form of recoil

motion also occurs in the case of a monatomic target in the sense that

recoils produced with energy less than U are thermalized without

further possibility of escape. The distinction between such thermal

loss and the loss to molecular excitation is not sharp and should not

be critical at the quantitative level adopted in the present work. The

dissociative energy loss in a molecular target, on the other hand, is

an essentially novel feature.

The other significant consequence of the molecular association of

the target medium's atoms is that each atom is partially screened from

collisions by its molecular partner(s). That is, the effective cross

section per atom is less in the molecular than in the monatomic case.

Such a mutual screening effect is reminiscent of the position correla­

tions we investigated in Chapter II with regard to a dense hard sphere

gas. Indeed, in the latter system one might mentally associate groups

of nearest neighbors into "molecular" clusters, but this would not

change the physics of the gas. A typical recoil would move the same

average distance between collisions. The screening effects arise simply

due to the (uniform) atomic density. On the other hand, one might

imagine that a true molecular gas is formed from an originally monatomic

system by causing the atoms to bind together. This would involve an

actual physical displacement of the atoms, resulting in a strong spatial

correlation distinct from that considered in Chapter II. That is to

say, the resulting system would no longer be uniform on the atomic scale

119

(although it would be on the molecular scale). An alternative way to

describe this situation is to note that a typical recoil would have a

longer mean collision free path in a molecular gas than it would in

a monatomic system having the same atomic number density.

In this section we shall examine these aspects of the sputtering

of a molecular target in some detail. The example chosen for quantita­

tive analysis is a diatomic oxygen gas bombarded by 1 keV protons. The

original interest in this system arose from speculation that the Jovian satel­

lite Ganymede might possess an appreciable 02 atmosphere due to photo­

dissociation of H20 vapor (Yung and McElroy, 1977). Although this has

apparently turned out not to be the case (Broadfoot et al., 1979), the

calculations on C>2 nevertheless provide a convenient basis for the

discussion of those aspects of atmospheric sputtering which are directly

related to the molecular nature of the target. Other features of a

real atmosphere, such as its density variation, and the existence of

multiple components, will not be considered here. Consequently, the

following examples are not intended to provide a complete model of any

planetary atmosphere, although it is true that protons of about 1 keV

energy form the major part of the interplanetary solar wind (Brandt,

1970). The macroscopic characteristics distinguishing an atmosphere

will be incorporated more realistically in the models employed in

Chapter V.

For the moment then we shall consider a hypothetical, spatially

uniform gaseous target with a well defined surface at which the binding

energy of each oxygen atom is 2 eV, and of each molecule, 4 eV. These

values are the gravitational potential energies for the two species on

Mars. Our analysis proceeds through the examination of two limiting

120

models for the sputtering of the 02 gas. In the first, referred to as

the primary dissociation model, it is assumed that the incident protons

break up a molecule with certainty if they transfer an energy greater

than the kinematically allowed minimum, T . , to one of the atoms. Them mquantity T . = (m/p)D, where D = 5 eV is the dissociation energy of them mmolecule, m is the atomic oxygen mass, and u = m /2 is the reduced mass

of the two atoms in the molecule. But it is further assumed in this

model that no additional dissociation occurs in the collisional cascade

generated by these primary atomic recoils. The alternative, secondary

dissociation model is predicated on the assumption that any primary or

secondary collision which results in an energy transfer to an atom which

exceeds T . will result in the break-up of the molecule, m m r

A rigorous calculation of collisional dissociation cross sections

is a complex problem and one which we shall not address. Nevertheless,

it is not difficult to see that at least the assumption of primary

dissociation is probably quite reasonable. In the first place, the total

cross section for a 1 keV proton to transfer an energy greater than Tmin°2to a free oxygen atom, calculated from Eq. (4.5) is 0.1 A . Therefore,

it is safe to say that the proton interacts directly with only a single

atom in the molecule. Furthermore, since the collision time T£ ^ aQ/vjj+

10 sec, where a is the first Bohr radius and v ^ is the velocity of

a 1 keV proton, is much shorter than a typical molecular vibration -14period T^ ^ 10 sec, the impluse approximation for the collision is

valid. Thus one would expect the uncertainty in the energy transfer to

be on the order of the ground state energy of the molecule. This

expectation is confirmed by the quasiclassical calculation of Gerasimenko

and Oksyuk (1965) who find that the dissociation probability in the

Impulse approximation, as a function of the energy transfer, exhibits

121

122

quite a sharp cutoff at T . provided that D is much larger than theminground state energy. The situation with respect to subsequent low-

energy atom-molecule collisions is necessarily more ambiguous. However,

the consideration of the two limits described above should provide an

adequate estimate of the sensitivity of our sputtering calculations to

the details of molecular binding.

For purposes of comparison it is appropriate to first consider

the sputtering yield for a hypothetical monatomic oxygen target, i.e.,

one in which the effects of the pairwise association of the particles

are completely absent. The flux of secondary recoils in this case is

given by Eq. (3.15). Since we are interested in the total yield, we

shall assume an energy independent hard sphere cross section for the

atom-atom collisions. In all the following calculations, as well as in

Chapter V, we have chosen to define atomic hard sphere radii in terms

of the distance of closest approach of two particles having a relative

energy of 10 eV in the Thomas-Fermi screened Coulomb potential, as

opposed to the Born-Mayer potential. Thus the collision diameter,

s ^ , is determined by

where $(x) is the Thomas-Fermi screening function and

is the screening radius. The cross sections obtained here are somewhat

larger than those given by Eq. (4.3); for instance, the 0 + 0 cross

0.8853 aoa

123°2 o2section is o_„ = 10.6 A as compared to o_„ = 4.4 A . Consequently, If BM

the calculated sputtering yields both here and in Chapter V will be

somewhat lower than they would be otherwise, since S<xo 3. Although we

cannot point to a cogent physical reason for the preference of one

over the other, the Thomas-Fermi cross section at least provides a

conservative estimate of sputter-induced mass loss.

Again, the primary source distribution 5>(E) of Eq. (3.15) is well

where 4> is the incident proton flux and n is the density of the oxygen

target. Therefore the secondary atomic oxygen flux distribution is

In converting g(E) into a sputtering yield, it is not appropriate

in the case of an atmosphere to assume a planar surface boundary as we

have for solid targets. Instead, we note that any recoiling particle

whose kinetic energy exceeds its gravitational potential energy, U,

will escape the planet if it is moving in the upward direction and

suffers no further collisions. This leads to a spherical boundary

condition according to which the differential sputtering yield is given

modeled by Eq. (4.6). We find <I>(E) = (0.196 A2 eV^2)<!>n/E3 2 = s/E3 2

g(E) (4.51)

E

with y E, = 221 eV being the maximum primary recoil energy, b b

by

S(E,fi)dEdQ = x g(E+U) cos 9 7 - dE .9 H 7T

(4.52)

124

The polar angle 0 is still restricted to the interval O<0<n/2.

Substituting Eq. (4.51) and carrying out the integrations, we find

a total yield,

For the current example, S = 0.11 oxygen atoms sputtered per incident

proton.

In order to avoid confusion over the interpretation of Eq. (4.52)

as applied to atmospheres, it should be pointed out that an ejected

particle does not lose its binding energy U until it is infinitely far

removed from the planetary body. Equation (4.52) is thus an idealized

expression which is not physically realized. The flux of recoils near

the top of the atmosphere is closer to g(E) than to g(E+U). Equation

(4.53) gives the yield of atoms which are promoted into open orbits.

This brings up the interesting point that even recoils which receive

less than the nominal escape energy U may travel to great distances

from the planet and be subject to other interactions which could prevent

them from returning. One might picture a large cloud of weakly bound

particles surrounding the planet and being swept away by other gravita­

ting bodies or perhaps by the solar wind. In this regard, the sputtering

yield estimate of Eq. (4.53) is almost certainly a lower bound to the

total charged particle induced loss rate.

Turning now to the primary dissociation scheme, we first observe

that there will be two species of recoils, viz. , 0^ molecules with

density and free 0 atoms. The latter are assumed to be produced

S s1

U

only in collision between molecules and the incident protons, thus

the source function for energetic atomic recoils will be taken to be

°2 1/2 2(0.196 A eV )n <p«(E) = ----------- w . (4.54)

<E+D)

Here each struck atom is assumed to lose an energy D in the dissociation

process— an approximation which becomes more accurate for higher energy

transfers. The spectator atom of the dissociated molecule will be

neglected.

As a practical matter, the density of free 0 atoms is sufficiently

low that we need consider only collisions between atoms and molecules.

Thus there is no multiplication of atomic recoils in the cascades. We

shall further treat energy sharing in the secondary collisions as if

each incident atom interacted with a single molecular atom. Although

this last assumption is not entirely consistent with the no-break-up

approximation, it has the advantage of simplifying the calculations

without significantly altering the results, as we have determined through

numerical integration of a more accurate model.

The equation governing the energetic atomic recoil flux, g^(E), can

now be written as

Em

n2°12gl ^ = n2°12 j gl^E + ’ (4-55)

E

with E = y. E, - D = 216 eV. The 0 + CL cross section here, = m b b 2 12°213.6 A , is determined from the sum of the atomic and molecular hard

sphere radii. The molecular radius in turn is taken equal to the radius

125

126

of a sphere having twice the atomic volume. This procedure provides

in some sense an average over the possible orientations of the mole­

cule. A similar procedure, which should result in conservative

estimates for molecular cross sections, is followed in all subsequent

calculations. The fact that the molecular cross section is less than

twice the atomic cross section accounts for the mutual shielding of

the atoms in the molecule.

Equation (4.55) may be integrated with the result that

Em

n2°12 Jg (E) = „ I *(E') dE' . (4.56)1 2 12

This expression may be compared to Eq. (3.14) which applies in the

monatomic case. Subtracting the primary recoil contribution and intro­

ducing a spherical boundary condition, the atomic sputtering yield is

E Em m

S1 = 4 4 / - I ^ f $(E'} dE' ' (4’5?)/ f /U1 E

EThe presence of the factorj ™ O(E') dE' here indicates the importance

Eof the number of recoils produced by the incident ions as opposed to

their total energy (see the remarks following Eq. [3.15]), as we

would expect when no multiplication of atomic recoils occurs. We-3find = 9.5 x 10 atoms ejected per proton.

The atomic flux given by Eq. (4.56) acts as the source of

energetic molecular recoils. The molecular flux distribution is then

127

described by

Yi ? Em E1/ m mn.o,n2°2282 (E) 2n2°22 j g2 (E^ E' + j gi(E^ d£E-

E/Y12

(4.58)

o2with o - 16.9 A . Note that g0 (E) must vanish at E = y , , E , and that ^ 2 12 mthere will be no primary molecular flux, since all primary recoils are

atomic. Integration of Eq. (4.58) leads to the following simple

expression:

Y12°12

Em

/g2 {E) = ~ ' 2 / g.(E')E'dE' . (4.59)a 22E

E/y12

The integration for g2 (E) may be performed analytically, but that for

the sputtering yield,

y12E*

■ifS2 ’ U / 82 (E') dE' ’ (4'60)

U2

-2must be carried out numerically. The result is S2 = 4.3 x 10 02

sputtered per proton.

The total sputtering yield, which we may define to be the total

number of atoms lost per proton is S = 2S„ + S. = 0.096. Thistot 2 1

result is only 13% less than the atomic yield for our monatomic target

example. Apparently the energy loss due to dissociation is not critical

128

in the present model. But what if an amount of kinetic energy on the

order of D were lost in each secondary collision with a resulting mole­

cular break-up? This assumption defines our secondary dissociation

model.

If dissociation occurs as a result of each sufficiently energetic

collision, the recoil flux is entirely atomic. We shall neglect any

contribution from the spectating dissociation products, and shall again

consider only atora-molecule collisions. The atomic recoil flux may be

estimated from the following expression:

In this case E = y .E, - otD and the source strength, s, is identical m b bto the numerator of Eq. (4.54). Implicit in this relation is the

assumption that energy transfers occur as if between free atoms. Then,

according to the second term on the right hand side we assume that

each atom being promoted from rest has its kinetic energy reduced by

an amount aD. Of course, there is somewhat of an inconsistency at low

E here, in that the energy transfer to the bound atom must actually

exceed T . = 2D in order that dissociation may occur. But, on the

other hand, we are free to consider values of a larger than unity.

Equation (4.61) may be rewritten as

E Em m

/ 81 (e') - f r + n

E E+aD

(4.61)

min

129

g1(E) = 2m

/jr> s/n*o 0

gl ( E ' ) f - ♦ 2 12(E+aD

E+aD

1° ' /

(4.62)

It is clear that the third term on the right hand side gives the

reduction in g^(E) due to the energy lost in secondary dissociation. We

have obtained an approximate solution to Eq. (4.62) through iteration.

That is, we first solve Eq. (4.62) with a -*■ 0 and then reintroduce this

function into the complete equation in order to obtain a correction to

g^(E). Proceeding in this manner we derive the following expression

for the yield

*n2°12

1/2m 1

2( l + - M - 2 / 1 | 2 1 + p - \ 2 1\ U-hxD/ \Ej \WhH) J

m(l + - 2 - ) \ E +aD/ ' m / (4.63)

This result should be compared to Eq. (4.53) which gives the yield in

the monatomic model. We see that because of the difference in the low

energy secondary collision cross sections, the atomic yield in the

secondary dissociation model, S , tends to exceed the monatomic yield,bl/S„, by a factor of n-a.,/n_o._ = 2a../a.-. On the other hand, S M 1 11 I Iz XI lztends to be reduced by the factor ^ [1 + U/(U+aD)] due to the energy lost

in inelastic processes. Thus

SDSM

11’l2 ( - U ^ )

(4.64)

For a = 2 this estimated ratio is 0.91. It would appear then, that

the screening effect of molecular association nearly compensates for

the dissociative energy loss in the present model. The smaller effec­

tive cross section per atom which results from screening in the

secondary dissociation scheme has the consequence that a higher flux

must be maintained in order to achieve approximately the same collision

rate, and thus energy flow, as in the monatomic model. Essentially

this same point was made in Chapter III in connection with the variable

radius hard sphere model.

The yield given by Eq. (4.63) has been calculated for several

values of a. These are presented in Table 4.1 along with the results

of the monatomic and primary dissociation examples. In particular,_2for a = 2 we have = 9.5 x 10 which agrees well with the estimate

of Eq. (4.64).

The conclusion to be drawn from the analysis presented in this

section is that the molecular associations of the atomic constituents

of the target are not particularly influential in the determination of

the sputtering yields at the level of accuracy with which we shall be

concerned in our practical applications. That is to say, the uncertain­

ties inherent in our knowledge of the characteristics of the solar

wind flow about planetary atmospheres, as well as in the structure of

the atmospheres themselves, make it unprofitable for the present to

pursue the collisional mechanics of dissociative molecular sputtering

in greater detail. Consequently, the primary dissociation model has

been employed for most of the numerical calculations of Chapter V.

130

131

V. SPUTTERING OF PLANETARY ATMOSPHERES

A. Introduction

One might infer from the generality of the formalism developed in

Chapters III and IV that the phenomena of sputtering should not be

associated exclusively with the surfaces of condensed bodies. Indeed,

in the present chapter we wish to point out that a gaseous system is

equally susceptible to such erosion. The most obvious such system,

and the one to be considered here, is a planetary atmosphere. Our aim

in part is to demonstrate the basic similarities in behavior under

bombardment by energetic atomic particles which exist between an atmos­

phere and a solid body, despite the immense disparity in physical scale

Hopefully, this similarity will not only emphasize the universality of

the sputtering process, but will also serve to widen the scope, and

thus the utility of our previous discussions. But our primary goal is

the investigation of the practical consequences of a sputter-induced

atmospheric mass sink, for although this is a problem of considerable

astrophysical import, it has nevertheless received scant attention in

the literature (see, however, Haff et al., 1978). However, as we shall

see, the quantity of data relating to this phenomenon is not sufficient

to justify a detailed formal analysis at the present time. Our discus­

sion therefore will stress general estimation procedures which should

quite adequately reveal the magnitude of the effects to be expected

under various circumstances.

It is safe to say that every planet and planetary satellite in

the solar system is embedded in an energetic particle flux of some sort

whether it be the solar wind (SW) or a magnetospheric plasma. In the

former case, the sputtering mechanism is feasible when the intrinsic

planetary magnetic field is not strong enough to exclude the SW from

the atmospheric volume. This condition is satisfied when the magnetic2 2induction is such that B /8ir<pv , where p is the mass density and v is

the bulk velocity of the SW. For typical p and v in the vicinity of

the Earth, the critical value of B is on the order of a milligauss.

Measured by this criterion, both Venus and Mars possess at most only

weak magnetic fields. We shall address specifically the problem of

the SW-atmospheric interaction for these two planets.

But a planetary body which does not possess an appreciable magnetic

field may nevertheless at least partially divert the streaming solar wind

plasma round itself if it possesses a sufficiently dense ionosphere

(Spreiter et aJ., 1970). The Mariner, Viking, and Pioneer spacecraft

probes have obtained substantial evidence for this latter type of inter­

action for Venus, and, with less certainty, for Mars. Assuming a

nonabsorptive flow pattern, the location of the ionopause, which is the

boundary between the SW and the ionosphere, is fixed by the pressure

balance betwaen the flowing and static plasmas. The maximum, or stagna-2tion, SW pressure (^pv ) occurs at the nose of the ionopause, i.e., the

subsolar point. Unless there is absorption, the SW does not penetrate

below the altitude of the nose.

Upstream from the ionopause, a standing bow shock wave is formed

which acts to deflect the supersonic flow. Such a bow shock has been

observed for both Venus and Mars. A basic difference between the bow

shocks of nonmagnetic planets and magnetic planets such as the Earth is

that in the former case the shock surface lies much closer to the planet

132

due to the much smaller apparent obstacle size offered by the ionosphere,

compared to an Earth-like magnetosphere.

Between the bow wave and the ionopause, the shocked SW forms a

plasma sheath of ions flowing with reduced bulk velocity but greatly

increased temperature and dayside density. This circumferential plasma

flow about the ionosphere will interact directly with that portion of

the neutral atmosphere which extends above the ionopause.

The preceding remarks describe a completely nonabsorptive SW flow.

However, there is growing evidence, particularly in the case of Venus,

that the SW is absorbed through the ionopause to a significant extent in

the vicinity of the subsolar point (C. T. Russell, 1977; C. T. Russell

al. , 1979; Wolfe et al. , 1979; Taylor et a K , 1979). The mechanism(s)

responsible for this absorption are at present poorly known. Substantial

fluctuations of the SW flux above its mean value may frequently result

in pressures sufficient to overcome the ionospheric shielding (Spreiter

et al., 1970). A favorable alignment of the interplanetary magnetic

field with the SW flow might also act to reduce the effectiveness of

this shielding (Taylor et al., 1979). Mass loading due to photo-ion

pickup and the production of fast neutrals via change exchange can also

lead to enhanced penetration of the ionopause (C. T. Russell, 1977).

Whatever the mechanisms involved, data gathered by Mariners 5 and

10, Venera 4, 6 , and 9, and Pioneer Venus at Venus indicate that the

bow shock wave lies much closer to the planetary surface than would be

anticipated simply from scaling the (nonabsorptive) flow pattern around

the Earth's magnetosphere. C. T. Russell (1977) has interpreted this

to mean that a substantial fraction of the incident SW is directly

absorbed in the subsolar region. Indeed, the absorption may on occasion

133

be strong enough to result in a bow shock attached to the atmosphere. On

the average, C. T. Russell (1977) estimates that as much as 29% of the

SW flux incident on the planetary cross section may be absorbed.

To the extent that SW ions of sufficient energy impinge upon a

planetary atmosphere, sputtering of its neutral components will occur.

There are two distinct mechanisms involved, depending upon the depth of

the plasma's penetration. The critical altitude in this regard is the

base of the exosphere, for it is at this height that the collisional

interaction of atmospheric molecules begins to dominate their ballistic

motion. Thus an ion passing through the exobase region may generate a

recoil cascade, much as at the surface of a solid. This may or may not

require that the SW penetrate the ionopause, depending on its altitude

relative to the exobase. If, on the other hand, the SW does not pene­

trate to the bottom of the exosphere, but flows around the planet at

some higher altitude, sputtering will still occur since any collision

between a SW ion and an atmospheric atom which transfers an energy to the

atom greater than its gravitational potential energy has essentially a

probability of 1/2 of removing that atom from the atmosphere.

In Section B of this chapter, we develop estimates of the mass loss

to be expected from both types of sputtering mechanisms with explicit

reference to the Martian atmosphere. A combination of Monte Carlo model

calculations and analytical techniques based in part on the results of

the last chapter will be employed. The discussion of Section V.C

extends this analysis to the atmosphere of Venus. Under favorable,

though not unreasonable circumstances we find that these sputtering

mechanisms can result in significant atmospheric mass loss.

134

But perhaps more important than the total mass loss rate is the

fact that the stoichiometry of the sputtered material may differ

substantially from that of the bulk atmosphere, due primarily to the

diffusive separation of the lighter components above the turbopause and

to their lower gravitational binding energy. We shall explore in some

detail the implications of such a mass fractionation effect for the

compositional evolution of an atmosphere in Section V.D. We focus15 14particularly on the anomalous N/ N isotopic ratio observed in the

Martian atmosphere.

Much of the material presented in this chapter appears also in

Watson ^t a^. (1980).

B. Mars

On the basis of data from Mariners 4, 6 , and 7, together with their

model for nonabsorptive SW flow about Mars, Spreiter et aJL. (1970) have

estimated the altitude of the nose of the Martian ionopause to lie between

155 and 175 km. The neutral Martian atmosphere as seen by the Viking 1

lander has its exobase at 176 km. These figures imply a substantial

direct interaction between the SW and the neutral Martian atmosphere. We

shall consider individually the two distinct sputtering processes mentioned

above. The first model addresses the possibility of SW flow down through

the exobase region. The second model will examine the consequences of a

mainly tangential (circular) flow above the exobase.

1. Cascade Sputtering

The number density of a unimolecular atmosphere which is in thermal

equilibrium above some reference altitude z (> the turbopause) varies asR

135

136

I \ - (z-z„) /Hn(z) = nRe R

where H = kT/mg is the scale height, T is the absolute temperature, m

is the molecular mass, and g is the gravitational acceleration. (We

shall neglect the variation of g with altitude.) The exobase, or criti­

cal height h^, of such an atmosphere is defined as that altitude at

which the mean free path of an atmospheric molecule in the horizontal

direction equals the scale height. Thus

h = H £n(Hon ) ,c o

2 / Hwhere o is the molecular cross section, and n = n e R . A fast SW iono Rpassing down through the exosphere may collide with an atmospheric mole­

cule. Recoiling atoms or molecules moving at altitudes z<<hc suffer

frequent collisions and are quickly thermalized with little chance for

escape. A primary collision occurring in the vicinity of h^ however may

generate a cascade of energetic secondary recoils, each of which has a

substantial probability for escape if its velocity exceeds the escape

velocity in magnitude and is directed into the upper hemisphere. Thus,

for normally incident ions, we expect sputtering to occur in a "critical

layer" extending a few scale heights on either side of h£.

There are marked similarities between this picture and our model

for the sputtering of a solid surface. Evidently, the exobase plays

the role of the surface of the atmosphere in the context of sputtering.

The surface binding energy U has as its parallel the gravitational

potential of the molecule, which is also typically a few electron volts.

The closest point of analogy, though, derives from our use of a

linearized Boltzmann-type equation in discussing mass transport in a

solid. Surely this approach is more tenable in the case at hand, where

the system considered actually is a dilute gas.

There are distinctions to be made between the two situations,

however. We have already examined the consequences of molecular associ­

ation for sputtering yields at some length in Section IV.E. It was also

pointed out there that gravitational binding does not provide quite as

well defined a work function for molecular ejection as does (presumably)

the electronic interaction at a solid surface. But the most salient

feature differentiating these two targets is their density. The atomic9 -3number density at the exobase on Mars or Venus is '\'10 cm , compared to

23 -3^10 cm in a solid. We recall, however, that one of the more interest­

ing results of our previous analysis was that the recoil flux generated

by, and hence the sputtering yield of an energetic ion should be

independent of the target's density. For although the production rate

of energetic recoils, both through primary and secondary collisions,

increases in proportion to n, the mean time between the collisions of

these recoils varies inversely with n. The product of these two quanti­

ties is proportional to the net recoil density (f(E)), and hence the

flux (g(E) = vf(E)), which are therefore independent of n.

There remains one further apparent complication in the atmospheric

case though; namely, that the target density is not uniform but instead

varies exponentially with altitude. We would argue, however, that such

nonuniformity is immaterial in the context of sputtering. Consider any

small but macroscopic atmospheric volume element in the exobase region

whose linear dimensions are much less than the scale of the density

137

variation (e.g., H). Within such a volume, our discussion of transport

in a spatially uniform system is applicable. Thus the recoil flux

generated within this volume, both by the SW source and by the fluxes

entering the given element from neighboring volumes, will not depend

on the background density of the gas therein. The same remarks apply

equally well to each such volume element in the exobase region.

Therefore, it is evident that the spatial distribution of recoil flux

in the exobase region must be the same as it would be if this region were

of uniform density; that is, it will depend only on the incident SW flux

and the conditions at the boundaries of the critical layer, which is

precisely the situation encountered in the surface region of a solid

target. This argument is, of course, subject to the qualification that

the background density at a given point be great enough to support the

recoil flux in the linear transport regime discussed in Chapter III.

This criterion is well satisfied for sputtering in the exobase region for

typical incident SW fluxes.

These considerations lend support to our proposal that the models

we have developed with regard to the sputtering of a solid surface may

equally well be applied to cascade type sputtering in an atmosphere. In

order to substantiate this connection, we have performed Monte Carlo

calculations of the sputtering due to normally incident ions in single

component, isothermal atmospheres with plane isobaric surfaces. Since the

Martian atmosphere is composed predominantly of C02 (95.32%) and N2 (2.7%)

(Owen et al_. » 1977), we have chosen to examine pure C02 and N2 atmospheres

whose densities in the critical layer were respectively taken to be n _ (z)V / U •

. „ in16 -3. -z/10 km . 12 -3. -z/15.35 km(1.2 x 10 cm )e and n^ (z) = (4.6 x 10 cm ) x e ,

with reference to the initial Viking 1 data (Nier and McElroy, 1976).

138

(But note that according to the above remarks, our results should

not be sensitive to the choice of n(z).) A rather complete description

of the calculational details may be found in Watson and Haff (1979). We

shall concentrate here more on the interpretation and application of the

results.

We have based our model of the solar wind on its observed quiescent

properties in the vicinity of the Earth (Brandt, 1970). This yields a7 -2 -11 keV proton flux of 9.5 x 10 cm sec at Mars and a corresponding 4 keV

6 -2 — 1a-particle flux of 5 x 10 cm sec . Of course, the energy of SW ions

approaching the ionopause may be considerably reduced from these undis­

turbed values. Indeed, Cloutier £t a^. (1969) have estimated that the

maximum interpenetration velocity of the SW at the stagnation point does

not exceed 1 km/sec. According to the model calculations of Spreiter

et a] . (1970), however, the SW velocity increases rapidly away from the

nose of the ionopause. In addition, it is expected that the shocked

plasma would exhibit a large temperature increase so that over a large

fraction of the dayside exosphere mean proton energies could possibly

exceed 50 eV. This is roughly the minimum incident energy which will

result in sputtering in our model. It has furthermore been postulated

(Wallis and Ong, 1975) that charge exchange reactions may play an important

role in the interaction between SW ions and atmospheric molecules. Such

reactions would lead to the production of fast neutral H and He particles

penetrating the exosphere.

It seems reasonable then that a substantial fraction of the SW

crossing the bow shock could pass through the exosphere with sufficient

energy to produce sputtering. In this energy range (50 eVsEsl keV for

H+ , 20 eVsE$4 keV for a) the nuclear stopping powers for H+ and a on (X>2

139

140

and N2 exhibit broad peaks. Recall that in Chapter III (Eq. [3.20]) we

found the sputtering yield for a monatomic target to be approximately

proportional to the stopping power of the incident ion when the surface

binding energy U is much smaller than the maximum primary recoil energy,

Em - At lower ion energies, however, we must take account of the fact

that only those primaries having energy greater than U can be effective

in sputtering, although the stopping power itself may still be quite

large. There is in effect some cutoff value for the ion's energy below

which the sputtering yield drops rapidly to zero, but above which the

yield should be fairly insensitive to the incident energy due to its

proportionality to the stopping power. An estimate of this cutoff may

be obtained by integrating the spectrum of Eq. (3.25) from U to E^.

Neglecting the energy dependence of the low energy cross section a, we

find that the yield varies with the ratio x = U/E according to

As a rule of thumb, we define the critical value of x by the condition

S(x ) = (1/10) S(0). This results in x = 0.32. In the case of a c c

Thus efficient cascade type sputtering by protons and a-particles remains

viable as long as the condition

m

S(x) (5.1)

molecular target, we let Ug be the gravitational binding energy of the

species of interest and use the E^ appropriate for the dominant primary

atomic recoil, taking account of its molecular dissociation energy D^.

(5.2)

is satisfied. As usual, is the coefficient of maximum energy

transfer to the primary. For C02 sputtering on Mars we find ^+>100 eV

and E > 35 eV, while for N„, £,_+> 80 eV and E £ 30 eV.a. l IT ~ a

Within these rough limits, the energies of the impinging ions are

not of primary importance in the estimation of the cascade type sputtering

process. Thus the basic quantity we shall consider in order to character­

ize the SW interaction with the critical layer is the sputtering yield

for ions with vi keV/amu energy passing normally through the exobase. We

shall outline the calculational procedure with specific reference to the

C02 target. The N2 computation is similar in all essential respects.

First, the ion-atom interactions are modeled by the usual screened

Coulomb potential (Lindhard et al. , 1968). Dissociation of the CC>2

molecule is assumed if the energy transfer to the C or 0 atom exceeds

some minimum value taken equal to the sum of its binding energy in the

molecule and its gravitational potential energy. These energies are,>

respectively, 5.5 eV and 2.0 eV for 0, and 11 eV and 1.5 eV for C atoms.

Smaller energy transfers cannot contribute to sputtering. For incident

protons the most energetic atomic recoils are 216 eV, 0, and 273 eV, C.

These primary particles generate a random cascade of recoiling molecules

through subsequent collisions which are modeled in terms of a hard-sphere

type interaction. It is assumed that no molecular breakup occurs in

these secondary collisions, and we furthermore neglect ionization and

excitation processes.

Note that the picture described here is just the primary dissocia­

tion scheme. The adequacy of this model was commented on in Section IV.E.

But further experimental support for this nondissociative low energy

molecular collision picture may be found in the recent study by Sheridan

141

et al. (1979) of the dissociation of 6 to 12 keV N2 ions in collision

with 02- These authors found that the dissociation cross section is

only 12-18% of the total cross section at these energies, and appears

to be falling toward lower energies. In addition, the measured magni­

tude of the total molecular cross section is commensurate with our°2assumed C02+CC>2 hard-sphere cross section, namely 22.5 A . Thus the

latter vaLue is surely not an underestimate.

Each recoiling secondary particle is followed until either its

energy falls below its escape energy (5.5 eV for C02) or until it

moves above an altitude of 300 km. In the latter case it is recorded as

a sputtered particle if its kinetic energy exceeds its escape energy.

The proton and cx-particle sputtering yields as determined by these

calculations are given in Table 5.1.

We see immediately that these yields are of the order of magnitude

we would expect on the basis of our calculations on 02 sputtering. The

analysis given in Section IV.E may in fact be applied directly to the H+

and a sputtering of N2- Such a computation results in the values given

in the second column of Table 5.1. The calculation of the sputtering

of a polyatomic molecule, p, composed of two or more atomic species, i,

may equally well be addressed if Eqs. (4.56) and (4.59) for the recoil

flux distributions are modified to read,

E .mi

g (E) = (a E2) £ o . y < ( g.(E')E'dE' (5.3)y pm Y ip ip I i

142

E/y .Ip

and

Recall that

*a (E) d°b(Eb ’E+Di) (5.5)

where the subscript b now refers to the SW ions. As a practical matter,

Eq. (5.3) may be reduced to

should. The sputtering yields may be determined from these fluxes after

the manner of Eq. (4.60). The values we obtain for protons and

a-particles on are also given.in Table 5.1.

An alternative method for the estimation of atmospheric sputtering

yields may be found in Haff and Watson (1980). These authors adopt a

primary dissociation scheme in which the primary atomic recoil flux is

treated as the sputtering flux incident on the molecular target. The

molecular yields are then calculated after the manner of Sigmund (1969);

they are proportional to the stopping powers of the recoiling atoms

(integrated over the energy distribution of the latter). There is a

E . E .mi mi

From Eqs. (5.4) and (5.6) one can see that if i = p and = 0, then

g^ + g^ reduces to the monatomic flux of Eq. (3.14), as of course it

great deal of similarity between this approach and that taken in the

present work— which derives not only from the fact that we have likewise

treated the primary flux as a source function for molecular recoils

(Section IV.E), but also from the connection between the recoil flux

and the stopping power of the incident particle discussed in Chapter III.

It is not surprising then that the molecular yield results of Haff and

Watson (1980) for H and a-particles (1 keV/amu) incident on the same

pure C02 atmosphere considered here are quite close to the values given

by Eq. (5.6). Their results are listed in Table 5.1 for comparison. The

technique of Haff and Watson (1980) has the advantage that in general it

is considerably easier to carry through computationally.

Haff and Watson (1980) estimated the atomic sputtering yields shown

in Table 5.1 on the basis of a picture in which these primaries execute

a stochastic random walk through the critical layer. The description of

the diffusion process provided (by this model is evidently quite similar

to that given by Eq. (5.4).

The estimates made by Haff and Watson (1980) and in the present

work are predicated on the same basic conceptualization of the physical

situation; only the mathematical approaches differ. The agreement

between the two resulting sets of numbers strongly attests to their

accuracy, within the overall framework of the primary dissociation

scheme. The agreement between these yields and those calculated by the

Monte Carlo technique is also quite satisfactory. At most, the Monte

Carlo molecular yields are lower by a factor of two, while the atomic

yields are more similar. One might expect the Monte Carlo procedure

to give somewhat smaller values for two reasons. In the first place the

computer simulation incorporates the anisotropy of the incident ion flux,

144

whereas the analytical calculations do not. The effect of this

discrepancy should be minimal, though, since most escaping atoms and

molecules are several collisions removed from the primary event. More im­

portantly, the Monte Carlo approach allows an exact treatment of the

surface boundary condition. In Section IV.D it was pointed out that

the actual flux escaping a surface should be expected to be somewhat

lower than that crossing an imaginary plane interior to the target. In

fact, according to the surface flux model described there, the escaping

flux should be lower by about a factor of two. The suggestion then is

that a large part of the difference between the Monte Carlo and analyti­

cal molecular yields listed in Table 5.1 may be due to the neglect of

the boundary condition in the latter case. The atomic yields are not so

strongly affected presumably because atomic recoils do not multiply in a

cascade process. In light of the foregoing interpretation, the a-particle

Monte Carlo yield of molecular CC>2 appears to be relatively large. But

since the one standard deviation statistical error for this value is

^ 15%, we do not attach great significance to its departure from the

general trend.

Figures 5.1 - 5.4 present some results of the proton sputtering

calculation on the C02 gas. Figure 5.1 shows the lateral position of

the last collision suffered by a sputtered particle prior to leaving

the atmosphere. The protons are all incident at the point (0,0).

Nearly all sputtered atoms and molecules originate within about 50 km

of the initial H+-molecule impact, in the lateral dimension, and the

full width of a typical cascade is perhaps 60 km. The density of points

shown in this figure should very accurately reflect the average recoil

collision density distribution (number of collisions per unit volume) in

145

a cascade, since the probability of escape following a collision does

not depend strongly on the position coordinates transverse to the

normal. The reason for the observed width of this distribution may be

understood along the following lines.

The primary atomic recoils tend to have initial velocity vectors

making fairly large angles with respect to the SW ion trajectory, due

to the nature of our assumed scattering cross section. Thus a character­

istic radius for the distribution of Fig. 5.1 is the path length an atom

of typical initial energy Eq would follow until its remaining energy is

about the escape energy of a molecule, U. Neglecting differences in

mass, the stopping power of such a primary recoil is

II ~ _ JLdx 2A

where X is the mean free path in the critical layer, X- 10 km. Thus,

neglecting straggling, the estimated radius is

r = 2X £n(E /U).o

Setting E = 4 0 eV, which is the average of the mean C and 0 primary oenergies, and taking U = 5.5 eV, we find r = 40 km which is consistent

with the Monte Carlo result.

Figure 5.2 provides similar information on the vertical extent

of the cascade, but in histogram form. The distribution is centered

around the critical height h = 171.1 km as expected, with a fullcwidth of perhaps 40 km. This distribution is not directly proportional

to the collision density in the cascade as was the case in Fig. 5.1.

146

Since we have argued that the recoil flux distribution g(E) should be

independent of the altitude, the collision density k(E) = n(z)og(E)

should follow the atmospheric number density. But there is an addi­

tional factor reflected in the distribution of Fig. 5.2, namely, the

probability that a recoiling molecule headed upward from an initial

altitude z will escape without further collision. For a normally

directed molecule, this probability is

P(z) = exp [-aHn(z) ]

Therefore the histogram of Fig. 5.2 is expected to follow

n(z)P(z) = n(z) exp[-aHn(z)]

This function is plotted in Fig. 5.2 as the dashed line. The normaliza­

tion is estimated, but the maximum occurs precisely at hc. The two

distributions are obviously quite similar. If anything, it would seem

that the above expression underestimates the histogram slightly at

lower altitudes. This would be consistent with our expectation,

discussed earlier, that the recoil flux falls off in the surface

region, compared to its bulk value, due to nature of the boundary

condition. Thus the position, width, and even the asymmetry of the

Monte Carlo distribution appear to be explained. Most importantly,

we now understand that as far as sputtering is concerned, the exobase

region does indeed provide a reasonably sharp, surface-like boundary.

It is of further interest to compare the energy and angle distri­

butions of Figs. 5.3 and 5.4 with those predicted by cascade theory for

147

a uniform target (Eq. 4.52). Retaining only the dominant term, the

differential sputtering yield is

148

S(E.fi)dEdQ - ■ cos 9 dEdfl .(E+U)

Here 0 is the polar angle with respect to the normal to the target's2surface. We find that the spectrum of Fig. 5.3 follows 1/(E+U)

fairly well, with U between 3.5 and 4.0 eV. The average gravitational

binding energy for all particles considered here is 3.7 eV.

The angular distribution of Fig. 5.4 indicates that there is a

depletion of the flux at small angles when compared to the above cos 0

dependence. 0 = 0 is upward from the planetary surface, or backward

relative to the solar wind flux. This indicates that, in fact, the

flux of secondaries in the cascades is not completely isotropic.

The overall congruence between the Monte Carlo results and the

expectations which derive from our physical picture of the sputtering

process, as expressed in our analytical model, strongly suggests that

we have attained a fairly complete understanding of the fundamental

processes involved in cascade-type atmospheric sputtering. We

therefore proceed to estimate loss rates due to this mechanism for the

Martian atmosphere.

Sputtering will remove any component present in the critical layer.

As a model for the Martian upper atmosphere we adopt the composition

and structure as determined by Viking 1 (Nier and McElroy, 1976; 1977)

given in Fig. 5.5. We shall consider only the four largest components.

The elemental oxygen component was not measured directly by VL1. Its

structure has been estimated on the basis of its concentration at 130 km

and an exospheric temperature of 169.2°K (Nier and McElroy, 1977),

assuming diffusive equilibrium. The critical height for a multicomponent

atmosphere may be defined as that height at which the probability for a

suitably energetic molecule, traveling radially outward, to escape the

atmosphere without collision, averaged over all species present at that

height, equals e For the atmosphere of Fig. 5.5 we find h^ = 175.6 km.

At this height, the atmosphere is 69.3% CC>2, 16.5% 0, 10.7% N2> and

3.6% CO.

According to our analysis in Chapter IV of energy sharing mediated

by a collisional cascade, we must expect the sputtering yields of the

various molecular constituents of this multicomponent atmosphere to be

proportional to their abundances in the critical layer n^. Thus the

relation between the yields of a two molecular species is that of

Eq. (4.21), viz.,

149

S./S. = U.n^/U.n? = m.n^/m.n? . (5.7)i j J i i J j i i j

On the other hand, we have seen that the atomic yields are more directly

related to the ion-atom dissociation cross section. But since this .

cross section does not vary widely between C, 0, and N in the above

molecules, and since we are not interested here in the individual atomic

yields per se, it is sufficient for the purpose of order of magnitude

estimation to scale the molecular equivalent yields (MEY) of the various

components from our Monte Carlo C02, C and 0 yields according to

Eq. (5.7). The MEY of a species is defined as the total mass of that

species lost per incident ion, whether in atomic or molecular form,

divided by the molecular mass. In other words, we treat the mass loss

as if it occurred entirely in molecular form, using Eq. (5.7) and the

CO2 MEY which derives from Table 5.1, corrected for its abundance, 0.693, in

the Martian exosphere. Averaging over a 95% proton and 5% a-particle SW

flux, we adopt the sputtering yields given in Table 5.2.

Even though more accurate estimates could be made through the use

of Eqs. (5.4) and (5.6), coupled to an energy sharing analysis such as

in Chapter IV, such a calculation would be intractable, and the evidence

suggests that the results would differ from the present values by at most

a factor of two, with most of the difference being attributable to the

failure of the analytic calculation to fully account for the physical

boundary conditions. Such precision is uncalled for here.

The magnitude of the exospheric mass sink generated by the sputter­

ing process must be determined by coupling the above results with an

analysis of the structure of the SW flow about Mars. The paucity of data

relating to the details of this flow limit us at present to offering some

general estimates which will be subject to refinement as the actual

character of the SW-planetary interaction emerges.

Let the phase space density of the incident plasma ions be denoted

by fswCr,v,t). We define a particle current density:

TCr,v,t) = vf (r,v,t)SW

so that the flux of ions through the exobase at a point r and a time t,3with velocities in d v about v is

, „ . 3Ir . j (r,v,t)|d v

150

where r is the unit radius vector. Now because the mean free path of

a solar wind ion in the exobase region is much greater than the width

of the critical layer, the i1”*1 partial sputtering yield of such a

particle, which traverses the exobase region at an angle 6 with respect

to r is expected to be enhanced by a factor of |cos e| E over its value

for normal incidence, for angles 0 not too near 90°. This results

simply from the ion's increased path length through, and energy deposi­

tion in, the critical layer. Thus the partial yield of an ion impinging

on the exobase at point r" is

151

S.si ( ?’ c) = n — z n m — r r r r z — r • <5- 8)I r . j(r,v,t)|/|j(r,v,t) |

with being the normal yield given in Table 5.2. We neglect here the

dependence of the yield upon the ion's kinetic energy as long as it is

within the limits of effectiveness noted above, e.g., 100 eV<Esl keV

for a proton.

Assuming the exobase to have a more or less constant planetocentric

radius r , the total loss rate of species i per unit area, averaged exover its entire surface may be written as

Ri (t) = — f d2r f d3v [r . YCr.v.t)! Si(T,t) (5.9)4*re* J J

exobase

As a practical matter, the areal integration extends only over the

exposed hemisphere. The integration over the ion velocities should

include only those consistent with the above energy limits.

With Eq. (5.8),

152

Ri ( t ) - i s i | £ j dar | d3y

vexobase

|?(r,v,t)| [ . (5.10)

. exobase

If there were no planetary ionosphere and the SW flowed unperturbed

directly into the atmosphere, then at all points on the dayside

|j(r,v,t) | = <j>(t), where <f>(t) is the magnitude of the interplanetary

SW flux, and consequently

Ri ( t ) = } s . <j>(t)

In the general case, R^(t) is proportional to the average magnitude

of the SW particle current density intercepting the exobase, as indicated

by the bracketed factor in Eq. (5.10). We are thus led to introduce a

structure factor a(t) which directly measures the ionospheric deviation

of the SW. Recalling our definition of j‘('r,v,t) we define

“ (t) ■ 2 n M I d0r I d3v v£swfF>y't) • (5'n )

exobase

so that

Ri(t) = 2 Si • (5,12)

The details of the SW plasma flow are therefore entirely incorporated in

the multiplicative factor a(t). It is important to note that a(t) may

be greater than unity even if all the flux incident on the planetary

cross section does not penetrate the critical layer. Nor is it necessary

that the flow maintain a nonzero mean velocity; a sufficiently high

plasma temperature (^10^ °K) can also result in efficient sputtering.

Two distinct factors influencing the loss rates are isolated in

expression (5.12). R^(t) scales linearly with the magnitude of the

interplanetary flux <t>(t) , while its dependence upon the distribution of

this flux in the exosphere is reflected in a(t). A third factor,

namely, variation in the composition of the critical layer itself, will

be discussed in Section V.D below. As far as loss rates today are

concerned, the greatest uncertainty must be attached to the value of a.

But until such time as j“(r",v,t) can be accurately specified, the most1 8 ~2 — 1 useful estimate is R^ = j S<J>, with 4 = 10 cm sec , its average

contemporary value.

Summing over the molecular yields of Table 5.2, we find the

following erosion rates for the various elemental species:

6 2Rq = 3 x 10 0 atoms/cm sec ,6 2R = 1 x 10 C atoms/cm sec ,5 2R^ = 5 x 10 N atoms/cm sec

These rates would imply substantial mass loss from the planet over

geological periods if we neglect the possible time variations discussed9

above. Integrated over a period of 4.5 x 10 yr, the total loss is on

the order of the present mass of the Martian atmosphere.

Although there is no information available which would allow us

to establish the time dependence of a(t), the magnitude of the inter­

planetary flux <f>(t) is a more tractable quantity. A recent study of

SW nitrogen deposition in the lunar surface has suggested that the

153

154

present-day SW flux is atypically low (Clayton and Thiemens, 1980).

It is inferred that the average solar wind intensity over the entire

lunar history has been greater by at least a factor of three than it

is in the present epoch. Because such a sizable increase in the SW

intensity could substantially enhance the penetration of the SW ions

into the neutral atmosphere, a(t) might be expected to be greater

under such conditions if other factors, such as the structure of the

ionosphere, remain the same. Consequently, on a geological time scale,

sputter-induced mass loss may have on the average considerably exceeded

its present rate. The estimates given above would be amplified by a

factor of three even if a(t) has remained equal to unity over such

periods.

An important measure of the significance of the above figures

is to compare them with the loss rates estimated for other mechanisms.

McElroy and others (McElroy, 1972; McElroy £t a^., 1977) have calculated

loss rates for C, 0, and N atoms due to chemical and photochemical

processes in the Martian atmosphere. We list below the important

reactions and associated loss rates:

Oxygen, CO* + 0 -► 0* + CO

0 + e -*■ 0 + 0

CO* + e -► CO + 0

7 -2 -1Rq = 6 x 10 cm sec

Carbon C0+ + e -»• C + 0 RC5 - 2 - 1 1.5 x 10 cm sec

C02 +hv -► C + 0 + 0

CO +hv -*■ C + 0

C02 + e-*-C + 0 + 0 + e

CO + e -► C + 0 + e

RC-.n5 "26 x 10 cm sec

CC>2 + e-*-C + 02 R , s 4 x 103 cm 2sec 3

Nitrogen, N2 + e -► N + N R ^ = 3 x 10^ cm 2sec 3

According to these numbers, the dominant mass loss mechanism is chemical

ejection of oxygen atoms. The total mass loss due to sputtering of the

critical layer with a structure factor equal to unity would amount to

about 7% of the loss due to these other mechanisms. However, the single

most important exospheric sink for carbon and nitrogen would be solar

wind sputtering, especially in light of a long-term enhancement of the

SW flux.

2. Direct Ejection

The preceding analysis is valid to the extent that the SW does in

fact penetrate the exobase region. If, on the other hand, the ionopause

lies above the exobase, a substantial fraction of the SW may flow around

the planet above the critical layer. In this event there can still be

considerable interaction between the SW plasma and the lighter components

of the atmosphere, which dominate at these higher altitudes. For the

atmosphere of Fig. 5.5, the most important component in this connection

is elemental oxygen.

A solar wind ion passing horizontally through the exosphere may

collide with one of the ambient atoms, transferring to it an energy

greater than its escape energy. If this primary atomic recoil is

scattered into the upper hemisphere it will almost certainly escape

the planet. The recoiling ion, scattered downward, may lead to additional

sputtering in the critical layer as discussed for normal incidence. Based

on the results of our C02 calculation, the mass loss due to this secondary

155

ionic sputtering process would be on the order of 10% of that due to

the direct ejection of the primary recoil. If, on the other hand, it

is the primary atomic recoil, instead of the incident ion, which is

scattered downward to the exobase, sputtering may still occur via the

collisional cascade mechanism. A Monte Carlo calculation for protons

and a-particles on both N2 and He atmospheres (cf. our subsequent

discussion of Venus) indicates that the mass loss due to this type of

secondary atomic sputtering should be roughly comparable to that

resulting from direct ejection— for SW ions of interplanetary energy.

At lower energies, such secondary sputtering effects are reduced. To

the extent that the sputtering yield follows the stopping power of the

atom, one would expect this latter type of sputtering to fall off

linearly with the average energy of the primary recoil. It should also

be borne in mind that both the ionic and the atomic secondary sputtering

mechanisms remove material in accordance with the composition of the

critical layer, although there is a non-negligible probability in the

latter case that the primary atomic particle will be reflected from the

atmosphere and eventually escape.

Since these secondary sputtering processes can be treated by the

formalism developed in the preceding section, we shall focus here on

the single step, direct sputtering mechanism which is the primary source

of mass loss generated by a SW flow in the upper exosphere. The loss

rate estimates arrived at in this manner provide only a lower bound to

the actual mass loss, but they may, if necessary, be combined with those

derived for sputtering in the critical layer, contingent of course upon

a reliable determination of the SW particle current density "jC?,v,t).

156

For ion-molecule collisions occurring above the exobase, the

escape of the energetic atomic recoil depends essentially on its velocity

vector having a positive radial component. We shall let f d o . denote1

the cross section for such a favorable collision of a SW ion with an

i-type atom. The differential cross section do will depend on the

ion's kinetic energy and the binding of the atom in its molecule, while

the angular range should be a function only of the orientation of

the SW ion’s velocity vector with respect to the radial vector r. If

n^Cr,t) is the number density of species i, then the total loss of3i-atoms in a time dt from an exospheric volume element d r is

157

ni(r,t) d3rdt / d"v ( j do .\ |j(r,v,t)

where now the velocity integration is to extend over all ions sufficiently

fast that they can transfer to the struck atom its escape energy. The

escape energy includes both the gravitational binding energy and the

molecular dissociation energy, if any, of the atom. For H+ + 0 on Mars,

the proton's energy must exceed 9 eV.

Supposing now that the SW flow extends downward to some more or less

well defined and constant radius on the dayside, r . , the average lossminrate per unit area can be written as

00

R^t) = — J r2dr f d^r ni(r‘,t) j d3v ^ d 0 ^ lT(r.v,t)| ,min r omin “+

(5.13)

with no constraint relating the radial and angular integrations. In

the exosphere, the density n^^OT,t) typically changes exponentially with

radius on a scale of <10 km. If the radial variation of 'j‘(r\v,t) is

much less rapid than this, as we shall assume (except of course at r . ),minthen Eq. (5.13) may be well approximated by replacing IXdv.t)! with

| j (rmin ,^r »'> t) I • Assuming furthermore that the variation of n^(r*,t)

with angle is negligible, the radial and angular integrations may be

separated with the result that

158

00

Ei(t> “ rj j j ^ J * o A | .Br , v , t) I )

rmin +

(5.14)

Even if there is no sharply defined inner boundary to the SW flow,

as would occur when there is substantial penetration through the

ionopause, Eq. (5.14) may be applied to calculate that sputtering which

occurs at radii exceeding some arbitrary rm£n- But let us concentrate

here on the case in which the SW is effectively excluded from below the

ionopause. It then follows that the mean velocity of the plasma flow

can have no radial component at rm^n * This implies that a thermal ion

velocity distribution will be, symmetric with respect to a plane tangent

to the ionopause at "r = *s also clear from our definition

of the direct ejection cross section defined above that the quantity

J do. - a . 1 2 , considered as a function of the ion's direction of motion,0+ 1 1will be antisymmetric with respect to this tangent plane. Here is

the total cross section for the transfer of an energy in excess of the

i-atom's escape energy. As a consequence of these symmetries, it follows

that

159

Although is in fact a function of the incident ion's energy E, such

10 eV<E<100 eV (for protons; cf. Eq. [3.21]), we shall neglect this

the value appropriate for the undisturbed SW. As with our other approxi­

mations, the precision lost here in favor of simplicity may be retrieved

when our knowledge of ^(r^Vjt) is more complete.

Carrying out the radial integral over the exponentially decaying

species density, we find that Eq. (5.14) reduces to

- 1/2that E at higher energies, with a peak roughly in the range

rather moderate variation and conservatively evaluate o. at E = 1 keV/amu,

R.(t) ,0 Hioi<J)(t)a(t) , (5.15)

with being the atmospheric scale height of the itk component. The

structure factor a(t) appearing here,

(5.16)

is, aside from the different value of the radius parameter, the same as

that defined by Eq. (5.11) for normal sputtering. Its ubiquity points

to its fundamental importance in characterizing the SW planetary inter­

action.

There are two marked formal differences distinguishing the expres­

sion for R^(t) in the collisional cascade model from that of Eq. (5.15).

The loss rate in the case of the direct ejection mechanism is not

proportional to the ion's stopping power (its energy loss per unit path

length), which is approximately the case for cascade type sputtering;

but instead to the ion-atom cross section. The second important

distinction is that the loss rate derived for sputtering in the upper

exosphere depends not on the concentration of species i, but on its

absolute density. Consequently, the total mass loss due to tangential

sputtering depends critically on the value of r . .minEquation (5.15) may be extended to a situation in which i-atoms are

present in the atmosphere in several different molecular species, which

we denote by the subscript j. The factor which varies most significantly

with j is the scale height H „ . The cross section should also depend

somewhat on the molecular binding. Therefore we may write

R^t) - ± »(t) «(t) £ Hij "ijfrmin’0 ' (5'17)

The results of our application of Eq. (5.17) to the Martian atmos­

phere, as modeled in Fig. 5.5, are shown in Fig. 5.6. The contemporary

loss rate, R(t), is given as a function of the altitude z corresponding

to the minimum SW approach radius r . . In these calculations a(t) wasrain

160

set equal to unity. The cross sections a., have been estimated on theij

-2basis of an r screened Coulomb potential and averaged over a solar

wind composition of 95% protons and 5% a-particles. We have again8 ~2 —1adopted a value of 10 cm sec for the SW flux intensity <HO> but

we reiterate that the various loss rates may be substantially greater

when averaged over geological periods. Note that the direct sputtering

mechanism is not operative at altitudes less than h = 176 km.cLeaving aside the question of the proper value for the structure

factor, we can see from Fig. 5.6 that the mass sink generated by

sputtering of the upper exosphere is likely to be quite small compared

to that arising from either cascade type sputtering of the exobase

region or photochemical loss processes. It is also instructive to

compare these loss rates with those due to the SW sweeping of photo-ions

in the upper atmosphere (Cloutier et a^., 1969; Michel, 1971). According

to this mechanism, photo-ions produced in the upper atmosphere are

carried away by their drift motion in the magnetized SW plasma. Such

mass loading of the SW can proceed up to some critical mass addition

rate before a substantial modification of the flow pattern must occur.

It is not clear to what altitude the SW must penetrate in order for this

optimal loss to be realized. However, maximum loss rates for C02 and N

have been estimated to be (McElroy, 1972):

5 2R = 5 x 10 molecules/cm sec2

6 2R^ = 1 x 10 atoms/cm sec

This quite large nitrogen loss rate would apparently dominate both

chemical ejection and the direct sputtering mechanism at high altitudes.

161

It has been suggested (Vaisberg e£ aJL. , 1976) that the nose of

the Martian ionopause lies at an altitude of about 400 km, instead of

in the 155-175 km interval estimated by Spreiter et_ al.(1970). This

value was obtained simply by scaling near-Earth data on the basis of

the bow shock wave altitude at the subsolar point of 1500 km, as observed

by the Mars 2, 3, and 5 spacecraft. In this event and if there is no

absorption of the SW through the ionopause, we would conclude that

sputtering does not play a substantial role in overall atmospheric mass

loss from Mars. On the other hand, the penetration of the SW to even

moderate altitudes could result in a significant preferential ejection

of those components which are diffusively enriched in upper atmosphere,

even if the total mass loss rate is small. We shall examine this

situation in Section D below.

C. Venus

The recent Pioneer Venus mission has revealed a dynamic, highly

variable interaction between the SW and the ionosphere of Venus. A

well-defined standing bow shock wave has been observed (Wolfe et al.,

1979), the nose of which lies at a radius of 1.23 R^ (altitude = 1400 km)

(C. T. Russell et al., 1979). Relative to the planetary radius, the

Venusian bow shock thus lies much closer to Venus than the Martian shock

does to Mars. A distinct ionopause is seen whose altitude varies widely

on a time scale of 24 hours, in apparent response to varying SW pressure

(Brace ^t al. , 1979; Taylor £t al. , 1979; Kliore e t al. , 1979; Knudsen

et al., 1979). The dayside ionopause has been observed at altitudes

varying from 250 km to 1950 km (Taylor ejt al. , 1979; Brace et al. , 1979;

Wolfe et al., 1979). These observations, however, were made at fairly

162

large solar zenith angles (£60°). It is to be expected that the altitude

of the ionopause at the subsolar point would be substantially lower.

Between the ionopause and the bow wave the shocked solar wind forms a

plasma sheath. The mean, or bulk velocity of this plasma is expected to

vary from near zero at the subsolar point on the ionopause to almost

its undisturbed value near the planet's limbs (Spreiter, 1976). One

must also expect that for a largely nondissipative flow, the energy

loss represented by this reduction in bulk velocity should be compensated

for by the compression and heating of the SW as it crosses the bow shock.

The Pioneer Venus data (Taylor et al . , 1979; Wolfe et al_. , 1979) offer

support for this view. Indeed, it appears that ionosheath temperature

may frequently approach 10^°K (kT = 86 eV). But in order for a proton

to eject, for instance, a helium atom from the Venusian atmosphere, its

required minimum energy is only about 3 eV. Thus the sputtering mechanism

appears energetically quite tenable, at least for that portion of the flow

above the exobase region.

The position of the ionopause with respect to the exobase in the

subsolar region is still a matter of conjecture. But even if the iono­

pause lies above the critical layer, it may nevertheless be substantially

penetrated by the SW. C. T. Russell (1977) has suggested that the

relatively low position of the Venus bow shock indicates that on average

perhaps 29% of the SW flux impinging Venus is absorbed through the iono­

pause. (For a less optimistic analysis, however, see (Cloutier, 1976)).

If this flux is of sufficient energy, efficient sputtering of the critical

layer can occur. The requisite ion energies are somewhat different here

than those estimates given for Mars, due to the difference in gravitational

binding energies. The criteria for the sputtering of elemental oxygen,

163

164

which is the dominant constituent of the exobase region (see below)

are i 120 eV and Eq> 41 eV. The atmospheric component in which

we shall have the most interest, however, is helium. Due to its low

* 30 eV

would

:ion to

mass, cascade type sputtering of helium may be important for Eh+

and E^> 10 eV. Of course, the energy available to these SW ions

depend critically on the mechanism responsible for their penetra

the exobase. But in any case, the quite low altitudes frequent!

attained by the ionopause indicate a substantial interaction between the

SW and the neutral atmosphere.

Our discussion of sputtering in the Martian atmosphere is <|ualita-

tively equally valid for Venus. The total atmospheric mass of Venus,4however, is about a factor of 10 greater than that of Mars. This means

that unless the SW structure factor is exceedingly large, the total mass

loss due to sputtering would have little consequence for the evolution

of the atmosphere as a whole. On the other hand, the selective sputtering

of a minor component of the atmosphere could be of significance for the

evolution of that component. Specifically, this is the case for helium.

Although He dominates the neutral atmosphere of Venus at high altitudes,

its bulk mixing ratio is probably at the 130 ppm level or less (von Zahn

et al. , 1979).

In order to make our analysis quantitative, we adopt the model

atmosphere of Fig. 5.7 which derives from Pioneer Venus observations at a

solar zenith angle of 88°, as the best available approximation to the

average dayside composition (Niemann et al., 1979. The critical height of

this atmosphere lies at about 160 km. The sputter-induced mass loss due

to passage of the SW through the exobase region may be estimated on the

basis of Eq. (5.7) and the Monte Carlo yield for our model C02 Martian

atmosphere, with adjustment for the difference in gravitational binding

energy. Recall that the atmospheric density is immaterial in a cascade

type sputtering process. We shall again set a(t) = 1, and assume a9 -2 -1contemporary SW flux in the vicinity of Venus equal to 10 ions cm sec

This value for <b (t) is consistent with the Pioneer Venus observations of

the quiescent interplanetary flux (Wolfe et al., 1979). As such, it

likely underestimates the mean flux intensity. In this respect, our

results should be considered conservative.

From Eq. (5.12) we derive a total mass loss of

p _ - 7 in-16 -2 -1 R = 2.7 x 10 g cm sec

9Over a period of T = 4.5 x 10 yr, with a SW flux enhanced by a factor- 2of three, this rate would result in a total erosion of only 12 g cm

This is to be compared with the present mass of the Venusian atmosphere,4 -29.3 x 10 g cm . Thus it is not likely that the sputtering process,

at least as we have outlined it, has produced substantial erosion.

The situation with respect to the He component is quite different.

Sputtering of the critical layer with a unitary structure factor is

estimated to result in a contemporary loss rate of

5 -2 -1R„ = 1.2 x 10 atoms cm secHe

9Were we to integrate this value over 4.5 x 10 yr, and allow for a

factor of three enhancement in the SW flux over this period, the implied22 -2total loss would be 4.9 x 10 atoms cm . If the value of 130 ppm for

the mixing ratio of He in the atmosphere of Venus is adopted, the present

165

23 -2abundance of He Is 1.7 x 10 a t o m s cm . By this measure, SW sputtering

of the exobase region would provide a significant sink for He.

The direct sputtering mechanism may also be operative in removing

He, particularly since, according to Fig. 5.7, He dominates the Venusian

atmosphere above ^250 km. Applying Eq. (5.15) with a(t) = 1 and <*> (t) =9 -2 -110 cm sec , the present He loss rate is given by

u fQ c. ,n5 -2 -1. -z , /52.9 kmR^e - (3.6 x 10 cm sec )e m m

Only for z . near h = 160 km does this direct ejection loss rate begin m m cto approach the above estimate for cascade type sputtering. But we

stress the fact that all of the values presented here are rather tentative.

More accurate estimations are contingent, among other things, upon a

reliable evaluation of the appropriate a(t), and more complete data on the

structure of the atmosphere itself.

Another measure of the significance of these loss rates is to

compare them with possible sources of He in the atmosphere. One such

source is outgassing. If we assume with Knudsen and Anderson (1969) that

the production rate of He on Venus is the same as for the Earth, then the6 — 2 “ 1 outgassing rate on Venus is 2 x 10 He atoms cm sec . A second source

of He is the SW itself. Assuming the SW ion flux is 5% He, and that an

average of 29% of the flux incident on Venus is absorbed (C. T. Russell,

1977), one finds a present deposition rate of R^ = 3.6 x 10B He atoms -2 -1cm sec . Thus one would expect SW deposition to be at least as impor­

tant a source of He in the Venusian atmosphere as is outgassing.

A comparison between these source strengths and the sputtering

loss rates given above would imply that unless the SW structure factor

166

Is fairly large, it is likely that He accumulates in the atmosphere

more rapidly than it is sputtered. But if there is in fact a substantial

SW-planetary interaction, such an imbalance between the deposition and

loss rates could not continue indefinitely. Since the deposition of He

by the above two mechanisms is independent of the abundance of He in the

atmosphere, while the sputter erosion rate increases roughly linearly

with that abundance, the sputtering process tends to drive the concentra­

tion of He to some stable value. This value would not depend on the

initial abundance of He, but only on the efficiency of the sputtering

mechanism relative to the He sources.

There is at least one other possible exospheric sink for He,

namely, the entrainment of photo-ions by the SW. Michel (1971) estimates

that the upper limit for the present He loss rate which could result5 -2 -1from this sweeping process is 7.5 x 10 He atoms cm sec . This value

is comparable to our exobase sputtering estimate, but in contrast to the

latter, the SW sweeping rate would not increase above the quoted value

in proportion to the concentration of He in the exosphere. Thus the

conclusions reached in the last paragraph are not altered.

There is further evidence that the imbalance between the sources

and sinks for He suggested by the above numbers could not have persisted

over the entire history of the planet. If the outgassing and SW sources9

of He had been operative at these strengths for 4.5 x 10 yr, then the24 -2total deposition of He would have been 1.8 x 10 atoms cm . This

would imply a mixing ratio of He to C02 in the present atmosphere of_3

1.4 x 10 , which is an order of magnitude greater than the observed

limit on this ratio of about 130 ppm. It is clear, then, that either

the above deposition rate estimates are considerably too large when

167

applied over, geological periods, or else there is in fact an efficient

sink for atmospheric He. In either case it is quite possible that

sputtering due to solar wind impact could have played a substantial

role in the evolution of the Venusian atmosphere.

D. Fractionation

1. Exobase Region

To the extent that solar wind sputtering occurs in the critical

layer of the upper atmosphere, the loss rates for the various species

should reflect their concentrations in this region and not their total

abundances. For instance, since the Viking data (Nier and McElroy,

1976, 1977; Nier .et £l. , 1976a; Owen and Biemann, 1976; Owen el al.,

1977) show that the mixing ratio of N2 to C02 is much greater at alti­

tudes near 176 km than for the bulk Martian atmosphere, we should expect

the N2 to be sputtered preferentially. Furthermore, an additional

fractionation effect arises from the difference in the gravitational

potential energies of the various species, as given in Eq. (5.7). These

considerations apply equally well to the case of isotopes.

The implications of these effects for the history of the Martian

atmosphere have been investigated by considering a simple model atmos­

phere with a present composition of 2.5% N2 and 97.5% CC>2, assumed to9have been rapidly outgassed 4.5 x 10 years ago. We assume a subsequent

passive role for the surface and consider only that mass loss which is-2due to sputtering. Let n^(t) be the column density (molecules/cm ) of

species i for the bulk atmosphere and n^(t) be its number density in

the critical layer at a time t (the atmosphere being formed at t = 0).

The corresponding mixing ratios of N2 to C02 are denoted by

168

169

f(t)

and

The parameter R = fc(t)/f(t) (McElroy and Yung, 1976) is then a measure

of the enrichment of the critical layer in the lighter species due to

diffusive separation. The dynamics of the atmosphere are assumed to be

such as to maintain R essentially constant in time.

The sputter-induced fractionation of the atmosphere is determined

by the relative magnitude of the time-integrated molecular loss rates.

In addition to their obvious dependence on the intensity and structure of

the SW flow, these rates will vary in response to the changing composition

of the critical layer. In accordance with our proposal that energy

sharing effects result in only small departures of the secondary recoil

fluxes from the stoichiometry of the target medium, the C0 2 sputtering

yield of a normally incident ion in the exobase region of our model

atmosphere may be written as

instance, by our Monte Carlo calculation, and is constant in time. Using

Eq. (5.7), the N2 yield is then given by

SCO2

(5.18)

Here S is the yield for a unimolecular C02 atmosphere as determined, for

(5.19)

170

The equations governing the evolution of the sputtered atmosphere

are:

Integrating these with the use of Eqs. (5.18) and (5.19) one finds

The reference time T is the age of the atmosphere. Given the present

abundances of N2 and C02> Eqs. (5.21) and (5.22) may be solved to deter­

mine their abundances at any previous time t as a function of (S/2)x

J ' i i (t')a (t')dt' . The latter quantity is a measure of the total amount

of material lost to SW sputtering in the interval (t,T).

The results of such a calculation for n (t) and n (t) are givenl«u 2in Figs. 5.8 and 5.9, respectively, for three different values of the

diffusive separation parameter R. The value R = 5.52 is in the best

dn„ (t)

dt - SN (t) *(t)a(t)/2 ,2

(5.20a)

and

= " sco(t) *(t)a(t)/2dt (5.20b)

(5.21)

and

T

t

(5.22)

T

agreement with the Viking data for the present Martian atmosphere. In

each figure both scales are in units of the present total column density

ntot(T) = nC0 (T) + n^ (T) = 2.25 x 1023cm 2. The indicated point on

the abscissa marks our nominally best estimate for the erosion parameter atQ _ O — 1

t = 0. It results from taking a(t) = 1, <f>(t) = 3 x 10 cm sec” and

S = 0.031. At this point, with R = 5.52, the model implies that the

initial Martian atmosphere was 67% N2 and 33% C02. Over a period of9

4.5 x 10 yr, 75% of the initial atmospheric mass could have been removed

by sputtering, with 43% of the C02 and 99% of the h'2 being lost. An

N2 depletion of 99% is as great or greater than mass loss estimates

based on other mechanisms, e.g., photochemical reactions (Nier et al.,

1976b; McElroy et a] . , 1976; Brinkman, 1971) so it would appear that

sputtering of the critical layer could be significant for atmospheric

evolution.

Given a value for n (0), we may make an analysis similar to the2

above to determine the enrichment expected among nitrogen isotopes if15 14the initial ratio of N to N equalled its present terrestrial value,

15 14N/ N = .00368. Bearing in mind that nitrogen is lost mainly in

molecular form, we have

[ n ^ w ( ® T Q ^ n o )

n 2 9 ( T ) / n 2 9 ( 0 ) l ( 5 , 2 3 )

which is analogous to Eq. (5.21). Assuming that the turbopause of the

Martian atmosphere is well defined and lies at an altitude of 125 km28 29(Nier and McElroy, 1977), the N2/ N2 diffusive separation parameter

R^ may be determined from the observed N2 scale height (Fig. 5.5), with

the result that R j = 1 .12.

171

172

Figure 5.10 shows the behavior of the enrichment parameter

EN (T) 5 n15(T) nl4(0)/[ni4(T) nl5(0)] <5-2A>

as a function of the integrated C02 equivalent molecular loss over the

planet's history under the assumption that the N2/C0 2 diffusive enrich­

ment parameter R = 5.52. The indicated value is again our best

estimation, viz., = 1.97. This is to be compared to the enrichment

observed by the Viking landers, eN = 1.62 ±0.2 (Nier and McElroy, 1977).

Clearly, fractionation due to efficient sputtering is a viable explana­

tion for the observed isotopic ratio.

Other sets of atmospheric isotopes may also suffer fractionation13 1°as a result of preferential mass sinks. One such possibility is C/ “C.

The carbon isotopic enrichment in the Martian atmosphere relative to the

terrestrial ratio is less than 5% (Nier and McElroy, 1977). The smallness

of this value when compared to the nitrogen isotopic enrichment suggests

that much less C02 than N2 has been processed through the atmosphere,

relative to their present abundances. This situation does in fact occur

in the sputtering model developed here. Applying the above analysis to12 13the relative ejection of CC>2 and CC>2 , we obtain the second curve

shown in Fig. 5.10. The indicated value of e, is about 1.07. Thus the

effects expected from SW sputtering of the exobase region are consistent

with the observed scale of isotopic enrichments.

2. Upper Exosphere

Fractionation may also result when sputtering occurs at higher

altitudes. Indeed, the higher the altitude, the greater the diffusive

separation which will exist between components. But, on the other

hand, the amount of material processed by sputtering decreases with

increasing height. We shall again consider the fractionation of 15 14N/ N in the Martian atmosphere, neglecting any elemental oxygen

component.15 14In speaking of the diffusive separation of N/ N, we are actually

29 28referring to the separation of N2/ N2. However, if we neglect the14 29small amount of N which is carried in , then Eq. (5.15) implies:

173

dnu (t)= - <J>(t)a(t)cr H„0n..(r . ,t)/4 (5.25a)dt 28 14 min

and

dn (t)— --- = - 4>(t)a(t)o H29n15(rn.n ,t)/4 , (5.25b)

where o = ~ Now because these loss rates depend on the exospher­

ic densities, not the concentrations, of the species, it is necessary to

relate directly the exospheric and bulk abundances of a given isotope.

We shall adopt the simplest possible assumption, namely

n..(r . ,t)/n .(t) = kiy(r . ) ,14 min 14 14 min

"l5(rmin-t)/n15<t> ‘ k15<rmin> • (5'26)

with k ^ and k ^ constant in time. The diffusive separation parameter

R^, evaluated at is then (approximately) equal to With

this assumption, the above equations may be solved for n^(t) and

with the result that

174

/■en (T) = exp j *(rnin) / <Kt')a(t')dtV«0T } , (5.27)

where

X(r . ) = ("ho k1 . (r . ) - H„.kic(r . )la* T/4 . (5.28)min [28 14 min 29 15 min J o

8 —2 —1For the purposes of normalization we shall set <p = 3 x 10 cm seco9

and T = 4.5 x 10 yr. The constants k^(r ) may evaluated by assuming

a sharp turbopause at some.altitude z . We have again chosen z = 125 km.R RFigure 5.11 shows the enrichment c x ( T) as a function of theN x

integrated SW particle current density at z . , / <t> (t')oi(t')dt'/$ T.min J o0The lower curve is for z . = 200 km, while the second one, correspondingmin > r o

to = 176 km, probably provides an upper limit to mass fractionation

effects generated by the direct sputtering process. Although under

favorable conditions sputtering of the upper exosphere could by itself

produce significant isotopic enrichments, on the whole it seems that the

direct ejection mechanism is not as effective in this regard as is

sputtering of the exobase region.

175

We have been impelled to the consideration of the various problems

addressed in the body of this dissertation by our desire to understand

the phenomenon of sputtering, which is the ejection of atomic particles

from a material system following impact by an energetic ion. We have

particularly focused on the fundamental mass and energy transport

processes which are operative in the erosion of solid surfaces and

planetary atmospheres by ions in the nuclear stopping energy regime. At

these energies, this transport proceeds predominantly through the

motions and elastic collisions of atomic recoils. The average number of

such energetic atoms which escape the target body per incident ion, the

sputtering yield, is the most readily observable quantity characterizing

the sputtering effect. Consequently, most of the theoretical work on

the problem to date has been aimed at calculating this yield. Although

we have been interested in the determination of the absolute sputtering

yield in the novel case of an atmospheric target, we have principally

dealt with certain other aspects of the problem which probe more deeply

the nature of the underlying energy sharing process. Thus, among other

things, we have developed models for the description of the energy

spectra of the ejectiles, and of the relative yields of the various

species in a multicomponent medium.

In these calculations, the transport theory of Boltzmann provided

the central element of our approach. But our desire to treat energy

transport in such a dense system as a solid led us to first consider

the question of how position correlations may restrict the collisional

VI. SUMMARY

interaction of particles. In Chapter II, therefore, we proposed a

model for what amounted to the radial distribution of particles in a

dense hard sphere gas. From this model we derived the distribution of

impact parameters in a collision as a function of the density of the

system. It was shown that at high densities, spatial correlations

place important limits on the transfer of momentum. For instance, at

densities approaching the maximal, the incident sphere most probably

transfers at least 83% of its initial momentum to its first collision

partner; but within this limit, any momentum transfer is equally likely.

At very low densities, of course, it is the energy transfer probability

which is uniformly distributed.

Another important implication of the model of Chapter II was that

there exists a critical interparticle separation distance x^, equal to

about 8/10 of the sphere diameter, above which scattering proceeds much

as in the limit of a very dilute gas, but below which the mutual screening

of nearest neighbors at large impact parameters rapidly becomes important.

It proved possible to extend the idea of a critical separation distance

to the low energy collisions of atoms in a solid, with the role of xc

being taken over by a critical energy E , typically on the order of a fewctens of electron volts. It appeared that at energies less than Ec> the

density related modifications of the scattering probabilities begin to

become significant. But we postponed until Chapter III considering the

full implications of this effect for the sputtering process.

The concept of a collision chain in an amorphous target was also

introduced in Chapter II. Such a chain occurs in a dense system as a

result of the enhanced tendency toward head-on collisions when compared

176

to a rarefied gas. Although both elastic and inelastic collisions may

be treated by the same formalism, we concentrated on the former case.

At maximum density, for example, it requires 26 collisions in order to

reduce the average momentum of the last particle in the chain to 1/10

of the initial value. When the chaining of collisions in this manner

is operative, the efficiency with which an energetic particle shares

its energy among others is substantially reduced. In other words, the

time scale for the dissipation of kinetic energy in the system is longer

than one might expect simply from scaling all times according to t ^ x/v,

where x is the average interparticle separation distance and v is the

particle velocity characteristic of the system. It is possible, for

example, that due to this effect the equilibration time for a dense gas

approaching thermal equilibrium might tend to be somewhat greater than

one would estimate by scaling down a rarefied system.

Bearing in mind the implications of the results of Chapter II

with reference to the sputtering of a solid surface, we proceeded in

Chapter III to develop a framework of transport theory within which we

could analyze the collisional cascade process. Chapter III commenced

with a formal reduction of the Boltzmann equation to energy space, which

served to delineate the assumptions and qualifications incorporated in

the model from which all our subsequent calculations derived. A generali­

zation of this equation appropriate to a discussion of a dissipative state

was effected via the introduction of energy source and sink terms. It

was proposed that perhaps the most useful solution to the resulting

equation could be obtained by treating the low-energy atomic collisions

in terms of a hard sphere cross section (as for a dilute gas), but with

the modification that the hard sphere radius should vary with the energy

177

of the collision. The model at which we thus arrived quite successfully

described the observed energy spectrum of atoms sputtered from a uranium40 +metal target by a beam of 80 keV Ar ions (Weller, 1978).

This achievement brought to light two fundamental characteristics

of the state of a system in which energy is dissipated through collisions

at a constant rate. In the first place it would appear that the precise

form of the energy transfer probability in individual collisions is not

critical so long as the sharing of energy is efficient, by which we mean

that a given primary atomic recoil tends to generate a maximum number of

coexisting secondaries having energies greater than some minimum value

on the order of a few electron volts. In this sense, the collisions of

hard spheres are quite efficient since they result on the average in the

transfer of one half the initial energy. The screened Coulomb inter­

actions between atoms are less efficient in that they tend to favor

smaller energy transfers. In solid media however, we believe this

tendency is partially compensated for by the mutual shielding effects

discussed in Chapter II. In any case, one would expect that the sensi­

tivity of the statistical distribution of atomic recoils to the dynamics

of the individual collisions should decrease at lower energies as the

average number of collisions which intervene between these secondaries

and their parent primary recoils increases. It is suggestive to note

in this context that for an ideal gas in thermal equilibrium, where one

may imagine that each particle has suffered an indefinitely large

number of collisions, the form of the energy distribution is completely

independent of the collision cross section.

On the other hand, it is true that the magnitude of the total

low-energy cross section is important in determining the recoil flux

178

g(E), and thus the energy spectrum of the ejected atoms. A second

conclusion to be drawn from Chapter III is that a given source condition

tends to maintain a fixed recoil collision density, irrespective of the

atomic cross section. The corresponding recoil flux, however, depends

inversely on the atomic size since the smaller the atom, the greater

the average time between its collisions, and thus the greater the flux

needed to maintain a given collision rate. The relevant point in

connection with the energy spectrum of sputtered atoms is that the

distance of closest approach, and hence the effective size, of two

particles interacting via a screened Coulomb potential varies inversely

with their relative velocity. Since higher energy collisions involve a

smaller atomic core, the requisite recoil flux is greater, with the

result that the energy spectrum falls off more slowly with increasing

energy than would be the case for an ideal hard sphere gas.

It is generally observed experimentally that the sputtering of

media comprised of two or more species of distinct masses and sizes

produces changes in the stoichiometry of their surface regions, or

fractionation. When the species under consideration differ chemically,

such fractionation effects are probably dominated by differences in

the binding of the atoms to the target's surface. But when these

constituents are isotopically related, the explanation of the preferen­

tial sputtering phenomena places important additional constraints on

any model for the underlying kinetic transport mechanism. The central

problem addressed in Chapter IV, then, was the partitioning of recoil

kinetic energy among the components of a polyatomic medium.

From our calculations on a binary system it became evident that,

neglecting possible surface related effects, the departure of recoil

179

fluxes from stoichiometry was to be attributed almost exclusively to

the differences in the couplings of the species to the impinging ions.

It emerged that the sharing of energy between two species in a collisional

cascade process tends to generate a recoil flux which is partitioned

between the components in proportion to their concentrations. At higher

energies the asymmetry of the source dominates the flux distribution,

but at lower energies the overwhelming majority of recoils are well

separated from the source characteristics by several generations of

collisions. The net result is that the total flux, integrated over

energy, is quite stoichiometric. The equipartition of flux derived here

is to be contrasted with the equipartition of energy which obtains for a

gas in thermal equilibrium. In the latter case it is the energy distri­

butions f^(E) which are proportional to the corresponding concentrations,

while in the dissipative state it is the g^(E) = vf^(E) which exhibit the

stoichiometry of the target medium.

The tendency toward the equipartition of flux was confirmed for

polyatomic media through calculations on the mineral plagioclase, for

which extensive empirical data were available (W. A. Russell, 1979). The

conclusion to which we were forced by this comparison was that the

differentiation of recoil flux resulting from the cascade energy sharing

model as it applies in the interior of the target cannot explain the

magnitude of the measured preferential mass loss. An immediate corollary

to this result is that fractionation effects should be insensitive to

the characteristics (charge and mass) of the impinging ions, which is

indeed observed to be the case.

Since the internal recoil fluxes were found not to be significantly

differentiated according to species, we proposed that fractionation arises

180

as a consequence of a modification of these fluxes in the vicinity of

the surface of the target body. A closer consideration of the nature of

the boundary conditions led to the concept that the flux emerging from

the surface is composed primarily of those atoms which have suffered

only one energetic collision in the course of a cascade. Within this

set of recoils, those species are favored whose mass is closer in a

suitably defined average sense to the mean mass per atom of the bulk

material. The greater the similarity in mass between two species, the

greater will be the efficiency of energy transfer between them in a

collision. Thus, those species are expected to be preferentially

sputtered whose averaged coefficients of maximum energy transfer, Y>

are closer to unity. The predictions deriving from this model were found

to be in acceptable quantitative agreement with the sputter-induced Ca

isotopic fractionation, on the order of 6(40:44) = 10-20 parts per thou­

sand, observed in plagioclase and in fluorite.

The proposed explanation of preferential sputtering exhibits the

novel feature that the magnitude, and even the sense of the fractionation

produced is expected to depend not only upon the masses of the species

considered but also upon their abundances in the target, and to a lesser

extent, their sizes. In a binary isotopic medium for instance, the more

abundant species would be preferentially removed from the surface whether

or not it was the less massive. More generally, the fractionation

induced between two species would depend on the matrix in which they

were embedded. Furthermore, one finds that, within a given set of

isotopes, the fractionation should vary with the mass of the species in

a significantly nonlinear manner. For these reasons, the ptesent model

of sputter-induced isotopic fractionation, if substantiated by further

181

experimentation, would offer a fresh basis for the interpretation of

some of the natural variations in isotopic compositions occurring, for

example, in lunar soils and certain meteoritic inclusions (Haff ^t a^. ,

1980).

A final problem to which we applied the techniques for the analysis

of collisional energy dissipation developed in Chapter IV was the sputter­

ing of a diatomic molecular gas. The aim here was to provide the ground­

work for our subsequent discussion of the erosion of planetary atmospheres

due to energetic ion impact. The particular aspect of the problem

addressed in Section IV.E was to determine what effect the binding of

target particles into small clusters, which might or might not be broken

up in the course of a cascade, would have on the sputtering yield. Two

important consequences of such molecular binding transpired. In the

first place, the dissociation of molecules upon impact results in the

loss of kinetic energy from the fragments with a resulting tendency

toward the reduction of the recoil flux. On the other hand, the

clustering of atoms into molecules produces a mutual shielding of the

atoms from collisions in excess of that which occurs when the atoms

are more uniformly distributed. In effect, the average cross section

per atom is reduced in the molecular case, with the result that the

recoil flux is increased from what it would be in a monatomic gas,

all other factors remaining the same. It was found that these two

tendencies very nearly compensated each other in the example considered.

The conclusion was that the sputtering yield, in terms of atoms ejected

per ion, is quite insensitive to the details of the assumptions which

one makes concerning molecular binding. This result enhanced the

credibility of the quantitative estimates made in Chapter V.

182

Desiring to extend the purview of the theory of sputtering as

well as to address specifically a potentially significant astrophysical

phenomenon, we devoted Chapter V to a consideration of the sputtering

of a gravitationally bound gaseous body. Our analysis there showed that,

to the extent which an energetic particle flux, such as the solar wind,

impinges upon a planetary atmosphere, the collisional ejection of neutral

species will be a potent force driving that atmosphere's evolution. The

total mass loss to be expected for a given intensity of the incident

plasma increases with the depth of penetration of the flow in proportion

to the atmospheric density. A limiting sputtering rate is reached at the

exobase at which point the direct ejection of primary recoils which

dominates at higher altitudes is superseded by a cascade type process

similar in many respects to the mechanism which we have described in the

context of the sputter erosion of solid surfaces. Both because this

cascade mechanism is independent of the target density, and because in

any case the densities in the exobase regions of most atmospheres are

similar, the importance of the total sputter-generated mass loss will be

greater in less massive atmospheres. We saw this to be the case in the

comparison between Mars and Venus.

On the other hand, sputtering can lead to a significant loss of the

less massive atmospheric components even when the total erosion is small.

Such preferential ejection is due in part to the diffusive enrichment of

the upper atmosphere in the lighter species. Although this enrichment

increases with altitude, it is in competition with a decreasing sputtering

rate so that the largest fractionation effects are experienced when loss

from the exobase region is dominant. Needless to say, this diffusive

separation factor will be operative for practically any exospheric mass

183

sink. Yet sputter-induced loss rates involve an additional departure

from stoichiometry, being inversely proportional to the masses of the

sputtered species. In the case of cascade type sputtering this is a

consequence of the inverse proportionality of the yield to the gravita­

tional binding energy, while for the direct ejection mechanism we have

shown that the loss rate of a species depends directly on the scale

height.

Preferential sputtering can lead to both elemental and isotopic

mass fractionation of an atmosphere. The latter effect may be exempli-15 14fied by the anomalous N/ N ratio on Mars. Not only can the sputtering

mechanism account for a large part of the effect observed for this

pair, but it is also consistent with the quite small fractionations found 13 12 18 16for C/ C and 0/ 0. On Venus, the elemental fractionation of He is

of primary importance. The extent to which the present abundance of

He in the Venusian atmosphere has been controlled by interaction with

the solar wind, as opposed to outgassing, is a question of fundamental

significance in the context of those models for planetary formation

which postulate a hot primordial solar nebula followed by condensation

and grain accretion (Pollack and Black, 1979).

Our discussion of sputtering phenomena was necessarily phrased in

terms of contingencies due to the uncertainty in the details of the

SW-planetary interaction. It seems probable that the SW does, in fact,

impact the Venusian atmosphere to a considerable extent, but the nature

of the interaction at Mars is more equivocal. Fortunately, the informa­

tion needed in order to evaluate the collisional processes explored here

is actually rather limited. Basically what is desired is the average of

the ion particle current density magnitude over either the exobase, or

184

else over the inner boundary of the ionosheath. Even if a determination

of this quantity implies only a marginal importance for SW induced

sputtering in the present epoch, the fact that the SW wind intensity

has probably been much greater on the average in the past could signify

a substantial geological role for the phenomena we have delineated. It

is our further hope that the utility of the discussion of Chapter V will

extend to other astrophysical environments not directly addressed here;

for instance, to the interaction of the Jovian magnetosphere with the

Galilean satellite Io.

185

186

REFERENCES

Andersen H. H. and Bay H. L. (1974) Nonlinear effects in heavy-ion

sputtering. J. Appl. Phys. 45, 953-954.

Andersen N. and Sigmund P. (1974) Energy dissipation by heavy ions

in compound targets. Kgl. Danske Vidensk. Selskab Mat.-fys.

Medd. 39, no. 3.

Brace L. H., Theis R. F. , Krehbiel J. P., Nagy A. F., Donahue T. M.,

McElroy M. B. and Pedersen A. (1979) Electron temperatures and

densities in the Venus ionosphere: Pioneer Venus orbiter electron

temperature probe results. Science 203, 763-765.

Brandt J. C. (1970) Introduction to the Solar Wind. W. H. Freeman and

Co., San Francisco.

Brinkmann R. T. (1971) Mars: has Nitrogen escaped? Science 174, 944-945.

Broadfoot A. L. , Belton M. J. S., Takacs P. Z., Sandel B. R., Shemansky

D. E., Holberg J. B., Ajello J. M., Atreya S. K., Donahue T. M.,

Moos H. W., Bertaux J. L., Blamont J. E., Strobel D. F., McConnell

J. C., Dalgamo A., Goody R. and McElroy M. B. (1979) Extreme

ultraviolet observations from Voyager 1 encounter with Jupiter.

Science 204, 979-982.

Brown W. L., Lanzerotti L. J., Poate J. M. and Augustyniak W. M. (1978)

"Sputtering" of ice by MeV light ions. Phys. Rev. Lett. 40,

1027-1030.

Chapman G. E., Farmery B. W., Thompson M. W. and Wilson I. H. (1972)

The energy distribution of sputtered atoms from gold. Rad. Eff.

13, 121-129.

Clayton R. N. and Thiemens M. H. (1980) Lunar Nitrogen: evidence for

secular change in the solar wind. Submitted to Proceedings of

the Conference on the Ancient Sun: Fossil Record in the Earth,

Moon and Meteorites.

Cloutier P. A., McElroy M. B. and Michel F. C. (1969) Modification of

the Martian ionosphere by the solar wind. J. Geophys. Res. 74,

6215-6228.

Cloutier P. A. (1976) Solar-wind interaction with planetary ionospheres.

In NASA SP-397 (Ness N. F., ed), 111-119.

Dupp G. and Scharmann A. (1966) Die Zerstaubung von Kupfer durch Ne+-,+ + +Ar -, Kr - und Xe - Ionen im Energiebereich von 75 keV bis 1 MeV.

Z. Physik 192, 284-298.

Gerasimenko V. I. and Oksyuk Yu. D. (1965) Dissociation of diatomic

molecules in fast collisions. Sov. Phys. JETP 21, 333-336.

Goodstein D. L. (1975) States of Matter. Prentice-Hall, Inc., Englewood

Cliffs, New Jersey.

Griffith J. E. (1979) Sputtering of Insulators in the Electronic Stopping

Region. Thesis, California Institute of Technology (unpublished).

Gschneidner K. A . , Jr. (1964) Physical properties and interrelationships

of metallic and semimetallic elements. Solid State Phys. 16,

275-426.

Haff P. K. (1976a) Possible new sputtering mechanism in track registering

materials. Appl. Phys. Lett. 29, 473-475.

Haff P. K. (1976b) Notes and comments on sputtering. Band Aid Preprint -7,

California Institute of Technology.

Haff P. K. and Switkowski Z. E. (1976) On the sputtering of binary

compounds. Appl. Phys. Lett. 29, 549-551.

187

Haff P. K. (1977) A model for surface layer composition changes in

sputtered alloys and compounds. Appl. Phys. Lett. 31, 259-260.

Haff P. K., Switkowski Z. E. and Tombrello T. A. (1978) Solar-wind

sputtering of the Martian atmosphere. Nature 272? 803-804.

Haff P. K. and Watson C. C. (1980) The erosion of planetary and satellite

atmospheres by energetic atomic particles. J. Geophys. Res. 82,

8436-8442.

Haff P. K., Watson C. C. and Tombrello T. A. (1980) Possible isotopic

fractionation effects in sputtered minerals. Unpublished.

Huang K. (1963) Statistical Mechanics. John Wiley & Sons, Inc.,

New York.

Kelly R. (1978) An attempt to understand preferential sputtering. Nucl.

Inst. Meth. 149, 553-558.

Kittel C. (1976) Introduction to Solid State Physics, 5th ed. John Wiley

& Sons, Inc., New York.

Kliore A. J., Woo R., Armstrong J. W., Patel I. R., and Croft T. A. (1979)

The polar ionosphere of Venus near the terminator from early

Pioneer Venus orbiter radio occultations. Science 203, 765-768.4

Knudsen W. C. and Anderson A. D. (1969) Estimate of radiogenic He and 40Ar concentration in the Cytherean atmosphere. J. Geophys. Res.

74, 5629-5632.

Knudsen W. C., Spenner K., Whitten R. C. Spreiter J. R., Miller K. L.

and Novak V. (1979) Thermal structure and major ion composition

of the Venus ionosphere: first RPA results from Venus orbiter.

Science 203, 756-763.

Kostin M. D. (1965) Energy distribution of energetic atoms in an irradi­

ated medium. J. Appl. Phys. 36, 850-851.

188

Kostin M. D. (1966) Energy distribution of energetic atoms in an

irradiated medium. III. Several species case. J. Appl. Phys.

37, 3801-3804.

Liau, Z. L., Brown W. L., Homer R. and Poate J. M. (1977) Surface-layer

composition changes in sputtered alloys and compounds. Appl. Phys.

Lett. 30, 626-628.

Liau Z. L., Mayer J. W. , Brown W. L. and Poate J. M. (1978) Sputtering

of PtSi. J. Appl. Phys. 49, 5295-5305.

Lindhard J., Nielsen V. and Scharff M. (1968) Approximation methods in

classical scattering by screened Coulomb fields. Kgl. Danske

Vidensk. Selskab, Mat.-fys. Medd. 36, no. 10.

McElroy M. B. (1972) Mars, an evolving atmosphere. Science 175, 443-445.

McElroy M. B. and Yung Y. L. (1976) Oxygen isotopes in the Martian

atmosphere: implications for the evolution of volatiles. Planet.

Space Sci. 24, 1107-1113.

McElroy M. B., Yung Y. L. and Nier A. 0. (1976) Isotopic composition of

nitrogen: implications for the past history of Mars' atmosphere.

Science 194, 70-72.

McElroy M. B., Kong T. Y. and Yung Y. L. (1977) Photochemistry and

evolution of Mars' atmosphere: a Viking perspective. J. Geophys.

Res. 82, 4379-4388.

McQuarrie D. A. (1976) Statistical Mechanics. Harper and Row, New York.

Michel F. C. (1971) Solar-wind-induced mass loss from magnetic field-free

planets. Planet. Space Sci. 19, 1580-1583.

Niemann H. B., Hartle R. E., Kasprzak W. T., Spencer N. W., Hunten D. M.

and Carignan G. R. (1979) Venus upper atmosphere neutral composition

preliminary results from the Pioneer Venus orbiter. Science 203,

770-772.

189

Nier A. 0., Hanson W. B., Seiff A., McElroy M. B., Spencer N. W.,

Duckett, R. J. , Knight T. C. D. and Cook W. S. (1976a) Composition

and structure of the Martian atmosphere; preliminary results from

Viking 1. Science 193, 786-788.

Nier A. 0., McElroy M. B. and Yung Y. L. (1976b) Isotopic composition of

the Martian atmosphere. Science 194, 68-70.

Nier A. 0. and McElroy M. B. (1976) Structure of the neutral upper

atmosphere of Mars: results from Viking 1 and Viking 2. Science

194, 1298-1300.

Nier A. 0. and McElroy M. B. (1977) Composition and structure of Mars'

upper atmosphere: results from the neutral mass spectrometers

on Viking 1 and 2. J. Geophys. Res. 82, 4341-4349.

Owen T. and Biemann K. (1976) Composition of the atmosphere at the

surface of Mars: detection of argon-36 and preliminary analysis.

Science 193, 801-803.

Owen T., Biemann K., Rushneck D. R., Biller J. E., Howarth D. W. and

Lafleur A. L. (1977) The composition of the atmosphere at the

surface of Mars. J. Geophys. Res. 82, 4635-4639.

Poate J. M., Brown W. L., Homer R., Augustyniak W. M., Mayer J. W.,

Tu K. N. and van der Weg W. F. (1976) The sputtering of PtSi and

NiSi. Nucl. Inst. Meth. 132, 345-349.

Pollack J. B. and Black D. C. (1979) Implications of the gas compositional

measurements of Pioneer Venus for the origin of planetary atmos­

pheres. Science 205, 56-59.

Robinson M. T. (1965) The energy spectra of atoms slowing down in

structureless media. Phil. Mag. 12, 145-156.

190

Russell C. T. (1977) The Venus bow shock: detached or attached?

J. Geophys. Res. 82, 625-631.

Russell C. T., Elphic R. C. and Slavin J. A. (1979) Initial Pioneer Venus

magnetic field results: dayside observations. Science 203, 745-748.

Russell W. A. (1979) Studies of calcium isotope fractionation found in

nature and produced during ion sputtering. Thesis, California

Institute of Technology (unpublished).

Samsonov G. V., ed. (1968) Handbook of the Physicochemical Properties

of the Elements. IFI/Plenum, New York.

Sanders J. B. and Roosendaal H. E. (1976) Non-linear dependence on density

of elastic stopping power. Nucl. Inst. Meth. 132, 267-271.

Seiberling L. E., Griffith J. E. and Tombrello T. A. (1980) Enhanced

sputtering of dielectric materials and its relationship to track

registration. Lunar and Planetary Science XI, 1021-1023.

Sheridan J. R. , Merla T. and Enzweiler J. A. (1979) Cross sections for

asymmetric charge transfer and collisional dissociation reactions

of atmospheric ions at keV energies. J. Geophys. Res. 84, 7302-7306.

Sigmund P. (1969) Theory of sputtering. I. Sputtering yield of amorphous

and polycrystalline targets. Phys. Rev. 184, 383-416.

Sigmund P. (1972) Collision theory of displacement damage, ion ranges,

and sputtering. Rev. Roum. Phys. 17, 823-870; 969-1000; 1079-1106.

Spreiter J. R. , Summers A. L. and Rizzi A. W. (1970) Solar wind flow past

nonmagnetic planets - Venus and Mars. Planet Space Sci. 18,

1281-1299.

Spreiter J. R. (1976) Magnetohydrodynamic and gasdynamic aspects of

solar-wind flow around terrestrail planets a critical review.

In NASA SP-397 (Ness N. F., ed), 135-149.

191

Switkowski Z. E., Haff P. K., Tombrello T. A. and Burnett D. S. (1977)

Mass fractionation of the Lunar surface by solar wind sputtering.

J. Geophys. Res. 82, 3797-3804.

Taylor H. A., Brinton H. C. , Bauer S. J., Hartle R. E., Cloutier P. A.,

Michel F. C., Daniell R. E., Jr., Donahue T. M. and Maehl R. C.

(1979) Ionosphere of Venus: first observations of the effects of

dynamics on the dayside ion composition. Science 203, 755-757.

Thompson D. A. and Johar S. S. (1979) Nonlinear sputtering effects in

thin metal films. Appl. Phys. Lett. 34, 342-345.

Thompson M. W. (1968) II. The energy spectrum of ejected atoms during

the high energy sputtering of gold. Phil. Mag. 18, 377-414.

Thompson M. W. (1978) Mechanisms of sputtering. In The Physics of

Ionized Gases (Janev R. K., ed), 289-334. Institute of Physics,

Beograd, Yugoslavia.

Vaisberg 0. L., Bagdanov A. V., Smirnov V. N. and Ramanov S. A. (1976)

On the nature of the solar-wind-Mars interaction. In NASA SP-397

(Ness N. V., ed), 21-40.

von Zahn U., Krankowsky D., Mauersberger K., Nier A. 0. and Hunten D. M.

(1979) Venus thermosphere: in situ composition measurements, the

temperature profile, and the homopause altitude. Science 203,

768-770.

Wallis M. K. and Ong R. S. B. (1975) Strongly-cooled ionizing plasma

flows with application to Venus. Planet. Space Sci. 23, 713-721.

Watson C. C. and Haff P. K. (1979) Solar wind sputtering effects in the

Martian atmosphere. Lime Aid Preprint - 28, California Institute

of Technology.

Watson C. C. and Haff P. K. (1980) Sputter-induced isotopic fractionation

at solid surfaces. J. Appl. Phys. 51, 691-699.

Watson C. C., Haff P. K. and Tombrello T. A. (1980) Solar wind sputtering

effects in the atmospheres of Mars and Venus. Submitted to Proc.

Lunar and Planet. Sci. Conf. XI.

Wehner G. K. (1976) Isotope enrichment in sputter deposits. Appl. Phys.

Lett. 30, 185-187.

Weller R. A., Jr. (1978) The Energy Spectra of Uranium Atoms Sputtered

from Uranium Metal and Uranium Dioxide Targets. Thesis, California

Institute of Technology (unpublished).

Williams M. M. R. (1976a) The energy spectrum of sputtered atoms. Phi 1.

Mag. 34, 669-683.

Williams M. M. R. (1976b) The Boltzmann equation for fast atoms. J. Phys.

A: Math. Gen. j), 771-783.

Winters H. F. and Coburn J. W. (1976) Influence of the altered layer on

depth profiling measurements. Appl. Phys. Lett. 28, 176-179.

Winters H. F. and Sigmund P. (1974) Sputtering of chemisorbed gas

(nitrogen on tungsten) by low-energy ions. J. Appl. Phys. 45,

4760-4766.

Wolfe J., Intriligator D. S., Mihalov J., Collard H., McKibben D.,

Whitten R. and Barnes A. (1979) Initial observations of the Pioneer

Venus orbiter solar wind plasma experiment. Science 203, 750-752.

Yung Y. L. and McElroy M. B. (1977) Stability of an oxygen atmosphere on

Ganymede. Icarus 30, 97-103.

193

Table 4.1. Sputtering yields for monatomic and diatomic oxygen, in

various models

194

Monatomic:

Primary Dissoc.:

Secondary Dissoc.:

so = 0.1125

so = 0.0095

V 0.0432

a so

1.0 0.1068

1.5 0.0997

2.0 0.0954

2.5 0.0924

3.0 0.0902

3.5 0.0884

4.0 0.0869

Table 5.1. H+ and a sputtering yields for pure CC^ or pure atmospheres,

exobase region

195

Monte Carlo Eqs. (5.4) and (5.6) Haff and Watson (1980)

H+: s c o 2 = ° - 014 s co 2 = ° - 027 SC02 = °-029

Sc = 0.0064 Sc = 0.0049 Sc = 0.0050

SQ = 0.0068 SQ = 0.0088 SQ = 0.0085

SXT = 0.022 S„ = 0.053N_ Nr2 2S„ = 0.0056 S„ = 0.0090N N

“ = sc o 2 ' 0 - 26 sc o 2 - ° - 28 s c o 2 - ° - 21

S - 0.010 S - 0.011 S - 0.015v w L

SQ = 0.025 SQ = 0.020 SQ = 0.019

S.. = 0.22 S„ = 0.532 2

S„ = 0.018 S„ = 0.025 ______N N

196

Table 5.2. Molecular equivalent yields, exobase region, Mars

Species Yield

C02 0.021

0.014

N2 0.0051

CO 0.0017

Totals: 0 0.058

C 0.023

N 0 .0 1 0

197

Two hard spheres, each of diameter s, are shown in the process of

collision, but in their initial positions. Sphere A, with momentum i?

is incident on stationary sphere B at an impact parameter b. One may

imagine that p" has just been imparted to A in a collision with another

particle. The dashed curve of diameter 2s about A represents the sphere

inside of which the center of no other particle may enter. The parameter

x is the average separation between nearest neighbors in a hard sphere

gas. The collision angle B is measured with reference to the initial

positions.

Figure 2.1

Figure 2.1

sphere of influence

199

The definitions of the sphere diameter s and the separation

distance x are again illustrated. Our model for a hard sphere gas

at less than maximal density is arrived at by first expanding a close

packed array, shown here in only two dimensions, until the lattice

parameter is s+x. One may then imagine each sphere to be displaced

in some random fashion about the lattice position shown.

Figure 2.2

ooooo o p o o ooooo o o o o

Figure 2.2

Figure 2.3

201

In this figure we illustrate the model from which we derive the

distribution of impact parameters for collisions in a dense hard sphere

gas, Eq. (2.2). The incident sphere (central solid circle) moves a

distance dr. Its sphere of influence (dashed circles) sweeps out a

volume dV. Eventually this latter sphere intercepts the scattering

sphere (outer solid circle) on which we assume that the centers of the

first particle's nearest neighbors are localized, each occupying ~

of its area. The probability that the collision involves an impact

parameter between b and b+db is then the ratio of the area dA of the1 2scattering sphere included between b and b+db to yy 4u(s+x) . In three

dimensions, dA is an annular region. As shown, the incident sphere has

already swept out the maximum area of the scattering sphere; that is,

it must have had a collision by this point. When x>xc however, the

incident sphere may penetrate the scattering sphere without collision,

in our model. The parameters s, x, and B are as in Fig. 2.1. The

scattering angle amax is the largest allowed angle between the direction

of motion of the incident sphere and the lines of centers of the two

spheres upon contact.

scatteringsphere

Figure 2.3

203

The impact parameter distribution, P(b;x) of Eq. (2.3) is plotted

in units of s 3 versus the impact parameter b, which is in units of s (solid

curves). The curves are labeled by their value of the separation

parameter x. The curve for x = 0 corresponds to the maximum density

n(0) = /2/s . The density for x = x /2 is n(x /2) = 0.36 n(0); andc cfor x = x , n(x ) = 0.17 n(0). The straight line (x = °°) is the canon- c cical hard sphere impact parameter distribution in a rarefied gas (Eq. [2.3]),

sP(b;°°) = 2b/s. When x<X£ collisions at large b are restricted; the dis­

tributions vanish outside the ranges for which they are shown. The

dashed curves are approximations to the distributions obtained from the

x = 0 distribution simply by extending its range and adjusting its

normalization.

Figure 2.4

sP(b

;x)

20U

Figure 2.U

205

We show here the propagation of a momentum pulse through a chain

of up to 5 collisions in a close packed hard sphere array. The variablevy is defined by p = c p where p is the momentum of the particleo o

initiating the chain, p is momentum of the last particle in the chain,

and c = cos a (Fig. 2.3) which here is set equal to the limiting max5 Yvalue, c = — . The momentum distributions, P(y,n) = P(c p ,n) = P(p,n), o o

of the last particle in a chain of n collisions are shown in units of

Pq \ as functions of both y (lower scale) and the momentum p (upper

scale). Note that each curve is symmetrical in the y variable andn / 2achieves its maximum value for y = n/2, or p = c pQ , which is the most

probable (but not the mean) momentum after n collisions.

Figure 2.5

Figure 2.5

0 I 2 y 3 4 5

207

The first and second moments of the momentum distributions

P(p,n) of the last sphere in a collision chain are plotted versus the

collision number n. <p> is the average momentum and <E> is the average

energy of the (n+l)tb hard sphere in a chain of n collisions, initiated2by a particle of energy = PQ/2m and mass m. These curves are appro­

priate to the limiting close packed case, i.e., c = \ (Fig. 2.5). Note

that it requires 26 collisions for the average momentum in the pulse to

be reduced to ~ of its initial value. <E> = E /10 after 13 collisions. 10 n o

Figure 2.6

208

0 10 20 30 40 50n

Figure 2.6

Figure 3.1

209

a)

This figure provides a picture of the context in which our

reduction of the Boltzmann equation in Chapter III is carried out.

We imagine a box of volume V, shown here in cross section, immersed

in a heat bath at temperature T. The phase space distribution of

particles inside the box may be written in the form fC?,v^) =

f^ ^fr,^) + f(r\v^) where f^°^ is an essentially Maxwellian distribu-

tion and f is a high energy tail which is appreciable only in the

region V'. Within V', a source of energy promotes atoms from thermal

to much higher energies in such a manner that particle number is

conserved. The loss of atoms at thermal energies is described by

^(r",v^) while their introduction at higher energies is given by

4>(+)(r,v1).

b)

A more realistic situation to which we wish to apply our energy

transport formalism is depicted here. An ion beam penetrates the

surface of a solid, shown cut away in the y-z plane. The discussion of

Chapter III is applicable to this situation if we assume that the

energetic recoil density fC?,E) vanishes on the lateral faces of the

arbitrarily large volume V', and that J ' J ' fOr,E)dxdy = 0 within a depth

Az of the surface. The area A' is the cross section of the volume V'.

In the discussion of Chapter III, we neglect any modification of fCV.E)

which occurs in the immediate vicinity of the solid-vacuum interface due

to the escape of energetic recoils.

Figure 3.1

211

The dashed curve, labeled B, is a fit to the energy spectrum of235 40 +uranium atoms sputtered from a U metal foil by an 80 keV Ar beam

as measured by Weller (1978). The normalization is arbitrary, but the-2.77functional form is E(E+U) * , where U = 5.4 eV is thought to be the

surface binding energy of the atoms. Curve A derives from the fixed

radius hard sphere model of energy sharing and has the approximate3form E(E+U) . A better fit to the observed spectrum, curve C, is

obtained from the variable radius hard sphere model developed in

Chapter III, Eq. (3.25). A monoenergetic, as opposed to a distributed,

source of primary recoils would result in the spectrum labeled D.

Curve D provides an upper limit to the energy distribution within the

variable radius hard sphere model. All curves here are adjusted so

that they pass through the same point at E = U/2.

Figure 3.2

S(E)

(a

rbitr

ary

norm

aliz

atio

n)

212

E (e V )

Figure 3.2

213

Each point in this plot marks the lateral coordinates of the last

collision of a particle escaping a model Martian CC>2 atmosphere under­

going sputtering, as calculated in the Monte Carlo simulation discussed

in Chapter V. One keV protons are normally incident on the atmosphere

at the origin in this figure. The full width of a typical collisional

cascade indicated by this distribution is perhaps 60 km.

Figure 5.1

KIL

OM

ETE

RS

2U»

i 1 t— 1— i— •— i— 1— i— •— i— >— r 6 0 l . P O S IT IO N O F L A S T

C O L L IS IO N

4 0 -

2 0 -

0 -

- 2 0 -

- 4 0 -

- 6 0 -

• •• •

• •

• • •

• •• •% •• • • • •• - ••

. •• v i . • *• •• • ••

• •

• •

• •

I___ , l__ , i «■ l___, I___ , I___ i I_- 6 0 - 4 0 -2 0 0 2 0 4 0 6 0

K I L O M E T E R S

Figure 5.1

215

This figure presents information similar to that of Fig. 5.1 on

the Monte Carlo sputtering simulation of Chapter V. The histogram

gives the number of escaping particles having their last collision in

each 5 km interval. The point marked h on the abscissa is the criticalcheight, or exobase, of the model CC^ atmosphere. Note that the zero

of this scale is suppressed. The dashed curve is a model for this

altitude distribution based on the proposal that the flux of energetic

recoils is insensitive to the density variations of the atmosphere. It

has the form n(z) exp[-aHn(z)], where z is the altitude, o is the low--z/Henergy molecular cross section, and n(z) = n e is the density ofo

the atmosphere. The normalization of the curve is the only free

parameter in this fit.

Figure 5.2

NUMB

ER

SPU

TTER

ED

A L T IT U D E (k m )

217

This histrogram is the energy spectrum of sputtered particles

calculated in the Monte Carlo simulation of Chapter V. We plot the

number of particles whose energies at large distances from the planet

lie within each 2.5 eV interval. The distribution is similar to that

one expects from a model of a hard sphere gas of uniform density with- 1/2a spherical boundary potential (see text), i.e., (E+U) , where

3.5 eV<U<4.0 eV.

Figure 5.3

Figure 5.3

80

S 6 0c rLxJ

8) 4 001UJCD5 3 2 20

00 5 10 15 20 25

E N E R G Y OF S P U T T E R E D P A R T IC L E (e V )

1 T 1 1

—E N E R G Y S P E C T R U M

— -

1 I i 1------------------ -

Figure 5.4

219

The number of particles sputtered per solid angle for each 5°

interval of the polar angle is shown. This emission angle is measured

with respect to the normal to the surface, i.e., with respect to the

local outward radius vector, thus 6 = 0 ° corresponds to upward emission.

The angular distribution calculated by this Monte Carlo simulation is

depleted at small angles with respect to the cos 0 dependence expected

in the isotropic hard sphere gas model.

NUMB

ER

SP

UTT

ER

ED

/STE

RA

DIA

N

E M I S S I O N ANGLE (d e g )

220

221

Our estimates of solar wind induced sputtering effects in the

Martian atmosphere are based on this model for its structure. We

derive this model from data gathered by the Viking 1 lander, as pub­

lished in Nier and McElroy (1977). The less abundant atmospheric

components (not shown here) are neglected. The calculated exobase

altitude, hc = 176 km, is indicated. The elemental oxygen component

is calculated from its density at 130 km and an assumed exospheric

temperature of 169.2°K.

Figure 5.5

ALTITUDE

(km)

222

Figure 5.6

223

The loss rates for the three dominant elemental components of

the Martian atmosphere due to the direct ejection mechanism have been

estimated from Eq. (5.17) and the model atmosphere of Fig. 5.5. The8 —2 ’ —1assumed solar wind flux is 10 ions cm sec , and the structure

factor a is set to one. The abscissa is the minimum altitude to which

the solar wind penetrates. The scale begins at the exobase, z = 176 km,

below which the direct ejection process is not operative.

LOSS

R

ATE

, R

(T)

(cm

^sec

'1)

22U

2 min ( k m )

Figure 5.6

225

This model for the dominant components of the Venusian atmosphere,

from which the estimates of sputtering effects in Chapter V derive, is

based on Pioneer Venus observations at a solar zenith angle of 88°

(Niemann et al., 1979). The exobase is estimated to lie at an altitude

of about 160 km.

Figure 5.7

ALTITUDE

(km)

Figure 5.8

227

These curves describe the depletion of C02 due to prolonged

cascade-type sputtering in the exobase region of a model Martian

atmosphere, composed of 97.5% CC>2 and 2.5% N2 [Eqs. (5.21) and (5.22)].9

This atmosphere is formed at time t = 0. The period T = 4.5 x 10

years. The ordinate is the ratio of the column density of C02 at a

time t after formation to the assumed present total column density of23 -2the atmosphere, ntot(T) = 2.25 x 10 molecules cm . The quantity

parameterizing the abscissa is the total sputter-induced loss (molecules -2cm in units of n (T)) which one would expect from a pure C0„tot 2

atmosphere characterized by a constant sputtering yield S, a structure

factor a(t), and subjected to an incident solar wind flux <J>(t). The

abscissa may be viewed as a time scale with earlier times (t 0)

toward the right. The indicated point represents our best estimate for8 —2 —1t = 0: a(t) = 1, <J>(t) = 3 x 10 cm sec , and S = 0.031, based on our

c cMonte Carlo calculation. The parameter R = n„ n „ /n„ n_. , wheren 2 c o2 n 2 c o2n^ is the total column density of species i, and n£ is its density in

the critical layer. The value R = 5.52 is in best agreement with the

Viking data.

ncOg

^/ntot

CT)

228

T!feSJ ) d t ' / n , o t (T )

t

Figure 5.8

Figure 5.9

229

The time evolution of the N2 component of the model Martian

atmosphere of Fig. 5.8 is displayed. This figure is similar in all

respects to Fig. 5.8 save that the vertical scale is now logarithmic.

Note that the N2 depletion indicated by the point marked on the

abscissa is much greater than the corresponding C02 depletion.

(t) / n t0t

(T)230

(VIZ

C

IJ <#>( t') <x( t')dt'/nfot(T)

Figure 5.9

Figure 5.10

231

The preferential sputtering of the lighter components of the model

atmosphere of Fig. 5.8 and 5.9 will lead to the isotopic enrichment of

with respect to 34N (left) and 33C with respect to *2C (right). The

enrichment parameter is defined by e^(T) = a15(T) n^O)/[n.^(T)n15(0) ],

and similarly for carbon, where n^(t) is the species' total column

density at time t. We assume that the isotopic ratios when the atmosphere

was formed (t = 0) were equal to their present terrestrial values. The

horizontal scale is the same as those of Figs. 5.8 and 5.9, except that

we have explicitly set t = 0. The results given here correspond to the

curves labeled R = 5.52 in Figs. 5.8 and 5.9.

€n(T

)

232

T

/2 s J V ( t ' ) a ( t ' ) d t 7 n t0t(T ) o

Figure 5*10

(1)°

3

233

The direct ejection mechanism can also lead to isotopic

fractionation in the two component model for the atmosphere of Mars

which was discussed in Figs. 5.8 - 5.10. Equation (5.27) for the 15 14N/ N isotopic enrichment is graphed here for two values of the

minimum solar wind penetration altitude. Again it is assumed that

the initial (t = 0) isotopic ratio was terrestrial. The abscissa

is the solar wind particle current density at z integrated over

the history of the atmosphere, normalized to the quantity 4>qT. We8 -2 —1have assumed the solar wind flux d> = 3 x 10 cm sec and set theo

9time T = 4.5 x 10 years.

Figure 5.11

€n

(T)

23U

T

0

Figure 5*11