Theory of Ion Neutralization at Metal Surfaces

Damascus University Journal for BASIC SCIENCES Vol. ١٩, No ٢٠٠٣ ,١

٢١

Theory of Ion Neutralization at Metal Surfaces

Abdalaziz A. Almulhem

Department of Physics, King Faisal University, Saudi Arbia

Received ١٠/١١/٢٠٠١ Accepted ١٨/٠١/٢٠٠٣

ABSTRACT

This work will concentrate on the mechanisms that produce neutralized atoms upon scattering of ions from metal surfaces. Three mechanisms are usually considered for ion neutralization at metal surfaces. Those are: resonance tunneling; Auger neutralization and surface plasmon-mediated ion neutralization. The third mechanism, namely surface plasmon-mediated ion neutralization, was suggested and calculated by us in an earlier work, where a unitary transformation was introduced to the second quantized Hamiltonian. The atoms are described by a state orthogonal to all conduction band states of the metal. The transformation gave rise to an additional term in the matrix elements ''the orthogonalization term''. This theory of applying a unitary transformation to the second quantized Hamiltonian is applied to the resonance and Auger neutralization mechanisms. The orthogonalization correction to the neutralization rate is found to be important at small distances from the surface when, applied to the scattering of protons from aluminum surface.

Key words: Ion neutralization; Orthogonalization, Ion-solid interaction; Auger neutralization; resonance neutralization; Surface plasmon - mediated neutralization.

Almulhem - Theory of Ion Neutralization at Metal Surfaces

٢٢

نظرية تعادل الأيونات عند سطوح المعادن

ن الملحمعبد العزيز بن عبد المحس المملكة العربية السعودية ـ الملك فيصلجامعة ـ الفيزياءقسم

١٠/١١/٢٠٠١تاريـخ الإيداع

١٨/٠١/٢٠٠٣قبل للنشـر في

ملخصال

الأيونـات ركز هذا البحث على الطرق الإجرائية التي يتم عن طريقها إنتاج ذرات متعادلة بعد تشتت يق مقترحة يتم بواسطتها تعادل الأيونات عند سطوح ائلك ثلاثة طر عادة ما يكون هنا . عند سطوح المعادن

التعادل عن طريـق تحفيـز بلازمـون تعادل أوجيه و التعادل بالرنين النفقي و : ق هي ائهذه الطر . المعادنتحفيز بلازمون سـطحي بواسـطة هي التعادل عن طريق لقد تم سابقا اقتراح الطريقة الثالثة و . سطحي

يـتم وصـف حيـث . و ذلك باستخدام تحويل وحدوي للهاميلتونين المكمم ثنائياً ،قالمؤلف في بحث ساب فينتج عن التحويل حدود . التوصيل للمعدن )عصابة (الذرات بواسطة حالة متعامدة مع جميع حالات حزمة

ذه النظرية لاسـتخدام التحويـل تم تطبيق ه . »حدود التعامد «التي تدعى إضافية في عناصر المصفوفة و قد تم اكتشاف أن التصحيحات على معدل التحـول و. هتعادل اوجي حدوي على إجراءات الرنين النفقي و الو

ذلك بالتطبيق على حالـة لهذه الإجراءات يكون مهما خصوصا عند المسافات القصيرة من سطح المعدن و . تصادم البروتون مع سطح الألمنيوم

تعادل أوجيه، تحفيز تعادل الأيونات، حدود التعامد، الرنين النفقي، :لكلمات المفتاحية ا

.بلازمون سطحي


٢٣

INTRODUCION Inelastic ion-surface collision has affected a great deal of

theoretical and experimental work [٢٥-١]. When ions are scattered in-elastically from a surface, the ions will interact with the surface resulting in a specific distribution of the charge states. This is due to the fact that some of the ions will pick up electrons to be neutralized. The great interest in this problem arose due to the desire to use the ion-surface scattering as a surface sensitive analytical tool. This field of work has grown up to cover all ranges of incident ion energies [٨, ١٢, and ١٧] and to include molecules and lately to also include negative ions. Also scattering from insulators is being discussed in the literature.

This work will concentrate on the mechanisms that produce neutralized atoms upon scattering of ions from metal surfaces. Four mechanisms are usually considered for ion neutralization at metal surfaces. Those are: resonance tunneling [٢،٥،١١،١٣،٢٤], Auger neutralization [٢٥ ,٤،١٢،١٤،١٦ ,٣], radiative recombination, and surface plasmon-mediated ion neutralization [٢٠ ,٦]. The radiative recombination is usually neglected due to the low probability [٥]. The fourth mechanism namely ion neutralization by surface plasmon excitation was suggested by us on an earlier work and argued to be of comparable in importance as the two main mechanisms namely resonance tunneling and Auger neutralization [٦].

The understanding of the process of neutralization of ions scattered from the surface is of great importance for many reasons: Ion neutralization spectroscopy is used as an analytical technique to study surfaces [٣٣-٣١]. In heterogeneous catalysts, adhesion and segregation processes a powerful tool to study solid surfaces composition, is considered low energy ion scattering [٣٤]. The advantage of low energy ion scattering is the informative depth reduction (practically outermost atomic layers), which is due to the high neutralization probability [٣٥]; and since only charged scattered particles are detected in the experiment, a good understanding of the process of neutralization is of great importance for the understanding


٢٤

of the results. For structure analysis the need for reliable theories is more pressing [٣٦]. In secondary ion mass spectroscopy, it is necessary to know the final charge state of the sputtered particles in order to draw conclusions about the surface concentration of the sputtered species [٣٩-٣٧]. This makes the understanding of ion neutralization theories more pressing.

In an earlier work [٦], to introduce the mechanism of surface plasmon-mediated ion neutralization, a theory that uses a unitary transformation applied on the modeled Hamiltonian was implemented. It gave rise to a new term ''the orthogonalization term'' in the matrix elements. The effect of nonorthogonality was also discussed for resonant tunneling neutralization by Wille [١١]. The orthogonalization is similar to that appearing in the theory of repulsive potential in atom-surface interaction due to electron kinetic energy gain. This is required as the decaying Bloch state of the metal electron orthogonal to the incident atomic states [٤٢-٤٠]. The orthogonalization comes out due to the use of a pseudopotential with metallic states that are explicitly orthogonal to the atomic states.

The unitary transformation used rotates the Fock space ''the physical space'' by / 2π into a new space called ''the ideal space''. The transformation being unitary, preserves the matrix elements and the Hermiticity of the Hamiltonian. An important characteristic of the transformation is that the transformed Hamiltonian will simultaneously describe all possible scattering and reactive channels [٢٦]. The problem of the final atomic state being not orthogonal to the initial band function is solved automatically in this formalism.

In this work analysis and comparison of the theory and result for the mechanisms of neutralization namely: resonance tunneling, Auger, and surface plasmon-mediated neutralization will be considered.

MODEL


٢٥

The metal will be simulated by a jellium (electron gas embedded in a constant neutralizing positive background) filling the half space 0z< . To make the calculation simpler, the ion will be taken to be a proton and its final bound atomic state to be the ground state of hydrogen. The shift of the atomic energy levels due to the interaction with the surface will be neglected since it is small down to distances close to 02a [٢٧] where 0a is the Bohr radius. The unperturbed hydrogen ١s state function will be used. Also the fixed ion approximation, where the ion is taken to be at rest during the evaluation of the transition rate, will be implemented.

The second-quantized Hamiltonian of our model is

† 1

† †

††

( ) [ ( ) ( )] ( )

1 ( ) ( ) ( , ) ( ) ( )2

( ) ( ) ( )

b

el el

q qs

H dr r T r r s V r r

dr dr r r V r r r r

c c dr r r r

ψ ψ

ψ ψ ψ ψ

ω ψ ψ

−

−

= − − +

′ ′ ′ ′ ′+

+ + Φ

∫

∫

∑ ∫

v v v v v v v

v v v v v v v v

v v v v$ $

(١)

In the last term the second-quantized potential of the surface plasmons is given by

†

( ) ( )q z i q R i q Rq qr g e e c e c− −Φ = +∑

v vv vv $ $ (٢) and the coupling constant g is given by

1/ 2[ ]sgq Aπ ω

=

Here qc$ and †qc$ are the surface plasmon annihilation and

creation operators. The prime on the summation implies that cq q< where cq is the plasmon cutoff wave vector (maximum

plasmon q ). No volume plasmons are assumed, since at low ion energies considered herein, the ion does not penetrate the metal surface and hence couples mainly to surface modes. In the Hamiltonian (١), ( )T rv is the electron kinetic energy, bV is the potential of the positive background, ( , )el elV r r−′ ′v v the electron-electron


٢٦

interaction, and ( )rψ v and †( )rψ v are the electron annihilation and

creation operators. The z -axis is perpendicular to the metal surface, the half plane 0z< constitutes the jellium metal and 0z> constitutes the exterior region. The electron position vector rv in (١) may be either inside or outside the metal. This allows for tunneling of the electron. Throughout this paper atomic units will be used.

The electron field operator will be expanded in terms of the complete orthonormal set of orbitals and corresponding annihilation operators kψ :

( )k krψ φ ψ=∑ v (٣) The orbitals kφ will be chosen to be eigenstates in a potential

( )V z which is constant inside and outside the metal with a step of height 0V at the surface ( 0)z =

0( ) ( )V z V zϑ= Here 0V F W= + with F the Fermi energy and W the work

function; and energies are measured from the bottom of the conduction band. It has been argued [٢٨ ,١٣] that this potential is a reasonable approximation for the surface potential of a metal in the presence of an ion. The corresponding eigenfunctions ( )k rφ v are

[( )exp( ) ( )exp( )1/ 2

1/ 2

1( ) [ ] 0( )

1 [2 0( )

z z z z z z

z

ik r k ik ik z k ik ik zk

k zik rz

r e zk V

k e e zk V

ν

ν

φ ′ ′ ′ ′+ + − −−

−−

= <

= >

v v

v v

v

(٤)

where V is the volume of the metal and zk′ , zk′ and kν are defined by

2 2 2 20 0( ) 2 ; 2( ) ; ( ) 2z k z k zk E k V E k k Vν′ ′′ ′= = − + =

with 2(1/ 2)k kE E K′ = −v


٢٧

kE is the eigenvalue of kφ and Kv

is the component of kv

parallel to the surface. Within the conduction band kE F< one has

00 k kE E F V′< < < < so that the eigenfunctions are oscillatory inside the metal and decay in

z+ direction outside.

The wave functions kφ in (٣) constitute a complete set, so the bound atomic wave atφ function in the final state of the process we are considering can be expanded in terms of them. This can cause problems in calculating the matrix elements of the neutralization process if one uses a simple-minded Born approximation [٢٨]. The problem is that the final atomic state atφ is not orthogonal to the initial band function kφ . This implies that the matrix elements

at kHφ φ (٥) is not invariant under addition of a constant to the perturbation H . However, a constant potential generates no force and therefore cannot cause a transition. The right form of matrix elements should be [٢٩]

(1 )at i kHφ π φ− (٦)

Here iπ is the projector onto iφ

i i jπ φ φ= (٧)

where iφ is the initial state. On his work on atom-metal interaction

Gadzuk [٢٨] has argued that this problem can be overcomed if one chooses the right initial state and interaction potential. Using this result Hentschke et. al. [١٣] calculated the neutral rate of the process of resonance tunneling.


٢٨

The Transformed Hamiltonian

A unitary transformation U applied on the Hamiltonian. This transformation proved very useful in tackling problems of reactive scattering that included composite particles (the atom). This transformation is

2( ) FU eπ

= ; † †

( )F A Aν νν νψ ψ= −∑ (٨)

Acting on the Hamiltonian in (١), a transformed Hamiltonian in which the matrix elements of the possible reactive channels are orthogonalized will be produced. Each matrix element will contain two parts, the first being the usual matrix element of the process, and the second is the orthogonalization term. In (٨) †

A ν is the creation operator for an electron in a bound hydrogen orbital ( )r sνφ −v v centered on the proton (position sv ) and ν stands for the atomic quantum numbers ( )n l mν =

† †( ) ( )A dr r s rν ν νφ ψ= −∫ v v v v

(٩) The physical states on which the transformed Hamiltonian

acts, are of the form )1

.... ....U−

=

where .... is any standard Fock state represented in terms of electron

creation operators †( )rψ v acting on the vacuum state 0

Using (٨) and (٩) and making use of the commutation rules of

νψ and ( )rψ v , the electron field operators transform as follows 1

( ) ( ) ( , ) ( ) ( )U r U r dr r s r s r r sν νψ ψ ψ φ ψ−

′ ′ ′= − ∆ − − + −∑∫v v v v v v v v v v (١٠)

where ( , )r s r s′∆ − −v v v v is the hydrogen bound state kernel *( , ) ( ) ( )r s r s r s r sν νφ φ′ ′∆ − − = − −∑v v v v v v v v (١١)


٢٩

Transforming the Hamiltonian in (١) using the unitary

transformation in (٨) we can see that all channels of scattering are represented, including those of reactive scattering that we are after. The Auger neutralization channel can only come from transforming the fourth term in the Hamiltonian.

† †1 ( ) ( ) ( , ) ( ) ( )2 el eldr dr r r V r r r rψ ψ ψ ψ−′ ′ ′ ′ ′∫

v v v v v v v v (١٢)

The transformed term takes the form † †11 ( ) ( ) ( , ) ( ) ( )

2 el eldr dr U r r V r r r r Uψ ψ ψ ψ−

−′ ′ ′ ′ ′∫v v v v v v v v (١٣)

This can be manipulated by inserting an identity operator

U1

U−

in between the operators in the following way

† †1 1

1 1

1 ( ) ( )2

( , ) ( ) ( )el el

dr dr U r UU r

UV r r U r UU r U

ψ ψ

ψ ψ

− −

− −

−

′ ′

′ ′ ′

∫v v v v

v v v v (١٤)

This transformation can now be carried out using equation (١٠) to get eighty one terms. This will result from multiplying three terms by three terms and the result by another three terms and then finally by three new terms. All two-electron scattering channels that might occur will be represented in those eighty one terms. The terms of interest here which are for the neutralization via Auger process are

† †*11 ( ) ( ) ( ) ( )el elT dr dr r s r V r rµ µφ ψ ψ ψ ψ−′ ′= −∑∫

v v v v v v v (١٥)

† †*

12 ( ) ( , ) ( ) ( ) ( )el elT drdr dr r s r s r s r V r rµ µφ ψ ψ ψ ψ−′ ′′ ′′ ′′ ′= − − ∆ − −∑∫v v v v v v v v v v v v

(١٦) The physical interpretation of these two terms is clear. The first term 11T corresponds to an Auger process in which the annihilation field operators ( ) ( )r rψ ψ′v v annihilate two electrons at


٣٠

rv and r′v as a result the creation operators † †( )r µψ ψ′′v create an Auger

electron and an electron bound to the ion. The second term 12T corresponds to the same process as the first except that it involves the bound state kernel (∆ is the kernel of the integral projection operator onto the bound atomic states). This guarantees orthogonalization of the metal orbitals to all bound atomic orbitals. Therefore no spurious contributions representing Auger atomic bound-bound electrons enter into the matrix elements representing the scattering studied. Similarly resonance tunneling will come from transforming the second term and giving the transformed terms

† 1*21 ( ) ( )T dr r s r rµ µφ ψ ψ−= −∑∫

v v v v v (١٧)

† 1*

22 ( ) ( , ) ( )T drdr r s r s r s r rµ µφ ψ ψ−′ ′= − − ∆ − −∑∫v v v v v v v v v v (١٨)

Here an electron tunnels from the metal surface to the proton and gets bounded to it through resonance tunneling. Transforming the sixth term in the Hamiltonian, the surface plasmon-mediated ion neutralization channel would arise. The terms take the form

†*31 ( ) ( ) ( )T dr r s r rµ µφ ψ ψ= − Φ∑∫

v v v v v (١٩) †*

32 ( ) ( , ) ( ) ( )T drdr r s r s r s r rµ µφ ψ ψ′ ′=− − ∆ − − Φ∑∫v v v v v v v v v v (٢٠)

Inserting (٣) in the different terms 11T , 12T , 21T , 22T , 31T and 32T one finds the following expressions for the perturbation inH . leading to the different channels of neutralization. For Auger neutralization one finds the expression for the perturbation 1H

† †1 1 1 1 2 1 2 3( , , ) k k kH k H k k µµ ψ ψ ψ ψ=∑ (٢١)

where the matrix element is given by

* *1 1 2 3 1 2 3

* * *1 2 3

( , , ) ( ) ( ) ( ) ( )

( ) ( , ) ( ) ( ) ( )

k el el k k

k el el k k

k H k k drdr r s r V r r

drdr dr r s r s r s r V r r

µ

µ

µ φ φ φ φ

φ φ φ φ

−

−

′ ′ ′= −

′ ′′ ′ ′′ ′− − ∆ − −

∫∫

v v v v v v v

v v v v v v v v v v v v (٢٢)

For resonance tunneling


٣١

†2 2( ) kH H k µµ ψ ψ=∑ (٢٣)

where the matrix elements are given by

1*

1

1* *

( ) ( ) ( )

( ) ( , ) ( )

k

k

H k dr r s r r

drdr r s r s r s r r

µ

µ

µ φ φ

φ φ

−

−

′= −

′ ′− − ∆ − −

∫∫

v v v v v

v v v v v v v v v v (٢٤)

For surface plasmon-mediated neutralization

† †

3 3( , ) q kH q H k cµµ ψ ψ=∑ $ (٢٥) where the matrix elements are given by

*

3

* *

( , ) ( ) ( )

( ) ( , ) ( )

q z iq rk

q z iq rk

q H k g dr r s r e e

drdr r s r s r s r e e

µ

µ

µ φ φ

φ φ

− −

− −

= −

′ ′ ′− − ∆ − −

∫∫

v v

v v

v v v v

v v v v v v v v v (٢٦)

This form of the matrix elements shows clearly the advantage of using the unitary transformation. It includes (the second term) an orthogonalization term that comes out automatically with the theory. The orthogonalization term takes care of orthogonalizing the metal orbitals to all bound atomic orbitals. The matrix elements in (٢٢), (٢٤)

and (٢٦) are corrected forms of the Born approximation to the exact T -matrix elements for the scattering process, and due to the inclusion of the orthogonalization term it is a better approximation. This term was found to be important in the ion neutralization at surfaces [٢٤ ,٦,

and ٢٥].

Calculations The evaluation of the matrix elements will be carried out using (٢٢), (٢٤) and (٢٦). The final atomic state will be assumed to be the unperturbed ground state of hydrogen

11 2

r ss eφ − −=

v v

(٢٧) The Transition rate of the scattering P is given by


٣٢

2 2122 (1 ) ( (1 ))inP s H k k E sπ δ= ∑ (٢٨)

where the final hydrogen state has been taken to be the 1s state. The prime on the summation sign indicates the restriction that the k sum is over the interior of the filled Fermi sea only. The matrix elements M for each channel are now taken from the actual evaluations [٢٤ ,٦,

and ٢٥]. This gives for the transition rate the simple form

2 2122 ( (1 ))P M k E sπ δ= −∑

(٢٩) Changing the sum into an integral we get

2 2122 ( (1 ))P M k E sπ δ= −∑

(٣٠)

Here 21

2i kε = is the metal electron energy and (1 )f E sε = is the atomic electron energy, V is the volume and 2 is for the double spin of the electron.

22( ) ( )

2 z i fVP s dKdk K M δ ε επ

= −∫∫v v

(٣١)

In the last equation Kv

is the component of kv

in the direction parallel to the surface. The delta function is used now to evaluate the integral over K

vwhere K

vis the component of k

vparallel to the surface.

22( )

2 zVP s dk Mπ

= ∫ (٣١)

with the matrix element evaluated at 2 2 21 1 1

02 2 2(1 )f zK k E s kε= − = −v

(٣٢) It should be stated that the atomic energies are shifted upward

by V since energies are measured from the bottom of the conduction band. Hence the value of (1 )E s should be given by

0(1 ) 1/ 2 1/ 2E s V F W= − = + − (٢٩) To calculate the neutral fraction we will assume that the proton follows a classical trajectory. This is an acceptable approximation


٣٣

since we are considering protons of low energies for which the de Broglie wave length is small compared to the atomic dimensions [٣٠]. As in all earlier calculations of the neutral fraction of low energy ions scattered from surfaces the following assumptions will be made. First it is assumed that the particle follows a straight classical path such that the perpendicular distance from the surface s is simply given by s v t⊥= . Secondly it will be assumed that we have specular reflection and that the perpendicular velocity v⊥ is constant up to the point of reflection (point of closest approach to the surface) where it changes its direction. The total neutralization probability can then be obtained by integrating over this orbit. Following previous work [١٦ ,١٤ ,١١] we calculate the number, ( )N s , of un-neutralized incident protons at distance s from the surface using the rate equation

( ) ( ) ( )( )

dN s P s N sds v s⊥

=− (٣٥)

where v⊥ is the perpendicular component of the ion velocity; and ( )P s is the neutralizing rate . We will assume constant v and specular

reflection to get 1

0( ) exp[ ( ) ]vN s N P s ds⊥

= − ∫ (٣٦)

where 0 ( )N N= ∞ . The neutral fraction will be given by

0 2

0

(0)1 1 exp[ ( ) ]vNf P s dsN ⊥

= − = − − ∫ (٣٧)

The 2 in the exponential comes from the assumption of specular reflection.

Results and Discussion Apply the theory developed here we assume the scattering system ( ) (1 )H e Al metal H s−+ → (٣٨) The aluminum is chosen because it best satisfies the assumptions made in the theory. First it can be well approximated by a jellium


٣٤

model. Second its Fermi surface is very close to the free electron surface for a face centered cubic monatomic Bravais lattice with three conduction electrons per atom. Third surface plasmons are well defined for aluminum and their existence has been demonstrated experimentally. Fourth the existence of experimental work on this system [١٥] in addition to the theoretical work where

orthogonalization is not taken into account [١٥-١٢].

The parameters used for aluminum are: ٠,٩٢٦١ for the Fermi

wave vector k , ٠,٥٨٦٢ for the surface potential V , ٠,٤١٠٦ for the

surface plasmon energy ω and ٠,٦٢٥٧ for the surface plasmon cutoff wave vector. The ground state in the hydrogen atom (1 )H s lies energetically within the conduction band of aluminum. All other states lie above the Fermi level. This makes Auger and resonant tunneling possible neutralization mechanisms for this system. Also, experimental and theoretical (not including orthogonalization term in the matrix elements) calculations on this system is available for comparison with the theory of this work [٩ ,٣].

Figure 1a

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+000 1 2 3 4 5 6 7 8

Distance

Tran

sitio

n R

ate


٣٥

Figure 1b

1.00E-06

1.00E-04

1.00E-02

1.00E+00

1.00E+02

0 2 4 6 8

Distance

Tran

sitio

n R

ate

Figure 1c

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+000 1 2 3 4 5 6 7 8

Distance

Tran

sitio

n R

ate

Figure.١.Transition rate ( )P s as a function of the distance s from the metal

surface for a proton scattered from aluminum surface (solid curve - the orthogonalization term is included,. Dashed curve - the orthogonalization term is not included). Figure ١-a for resonance

tunneling. Figure ١-b for Auger. Figure ١-c for surface plasmon-mediated neutralization.

Figure 2a

1.00E-071.00E-051.00E-031.00E-011.00E+011.00E+03 0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5 6 6 5 7 7 5 8

Distance

Tran

sitio

n R

ate


٣٦

Figure 2b

1.00E-07

1.00E-05

1.00E-03

1.00E-01

1.00E+01

0 1 2 3 4 5 6 7 8

Distance

Tran

sitio

n R

ate

Figure. ٢. Transition rate ( )P s as a function of the distance s from the metal surface for a proton scattered from aluminum surface for the three different mechanisms. Dashed curve - surface plasmon-mediated. Solid curve - Auger neutralization. Thick curve - resonance tunneling, Figure ٢-a for nonorthogonalized case and figure ٢-b for orthogonalized case.

Figure 3a

00.5

11.5

0 .01 0 .04 0 .07 0 .1 0 .13 0 .16 0 .19 0 .22 0 .25 0 .28 0 .31 0 .34

Perpendicular Velocity

Neu

tral

Fra

ctio

n


٣٧

Figure 3b

00.5

11.5

0.01

0.04

0.07 0.1 0.1

30.1

60.1

90.2

20.2

50.2

80.3

10.3

4


Neu

tral

Fra

ctio

n

Figure 3c

00.5

11.5

0.01

0.04

0.07 0.1 0.1

30.1

60.1

90.2

20.2

50.2

80.3

10.3

4


Neu

tral

Fra

ctio

n

Figure. ٣. The neutral fraction 0f for a proton scattered from aluminum surface

as a function of the perpendicular velocity v⊥ of the proton. (Dashed curve-the orthogonalization term is included,. solid curve - the orthogonalization term is not included). Figure ٣-a is for resonance

tunneling, Figure ٣-b is for Auger. Figure ٣-c for surface plasmon-mediated neutralization.


٣٨

orthogonalized case.

Figure. ٤. The neutral fraction 0f for a proton scattered from aluminum

surface as a function of the perpendicular velocity v⊥ for the three different mechanisms.Dashed curve-surface plasmon-mediated. Solid curve-Auger neutralization. Thick curve- resonance tunneling. Figure ٤- a for nonorthogonalized case and figure ٤-b for orthogonalized case.

Figure 4b

0

0.2

0.4

0.6

0.8

1

1.2

P V

0.02

0.04

0.06

0.08 0.

1

0.12

0.14

0.16

0.18 0.

2

0.22

0.24

0.26

0.28 0.

3

0.32

0.34


Neu

tral

Fra

ctio

nFigure 4a

0

0.2

0.4

0.6

0.8

1

1.2

P V0.0

20.0

40.0

60.0

8 0.1 0.12

0.14

0.16

0.18 0.2 0.2

20.2

40.2

60.2

8 0.3 0.32

0.34


Neu

tral

Fra

ctio

n


٣٩

Equation (٣٠) is used to calculate the transition rate P as a

function of the distance $s$ of the proton from the surface. The integration over k is calculated numerically. All other calculations are done analytically using the inverse Fourier transform integrals and making use of the calculus of residues [٢٥ ,٢٤ ,٦]. Figure (١) shows the neutralization rate P as a function of the distance s from the surface (solid curve for the case when orthogonalization is included while the dashed curve represents the nonorthogonalized case). The three different mechanisms are given in figures (١-a, ١-b, ١-c). From the figures it is evidently clear that the orthogonalization term in the matrix elements affects the transition rate especially at small distances. The inclusion of the orthogonalization tends to lower the transition rate. This lowering is attributed to the subtraction of terms that represent bound-bound atomic transitions, which are physically distinct channels that should not be included in the recombination rate. The transition rate is greatly affected in the important region for neutralization namely small distances from the surface. In this region the repulsive potential due to the positive background of the metal has large values. This decreases the total potential (attractive and repulsive) and consequently the transition rate. The effect of lowering the transition rate is accomplished in our theory by the orthogonalization term in the matrix element, which is subtracted from the original matrix element. The attractive potential (ion charge with the electron density) also increases at small distances from the surface. The two effects add up to a potential that give rise to the turnover region. The incident ion will penetrate into the turnover region when the electron density on the surface is such that the attractive potential rate of increase is greater than that of the repulsive potential. The transition rate attains a local maximum value at 1.3s= and a local minimum at 0.7s= in the case of resonance tunneling. For Auger neutralization a local maximum is attained at a greater distance s = ٢,


٤٠

while in the case of surface plasmon-mediated neutralization the maximum occurs at 0.5s= . At high distances 3s> the effect of neutralization is diminishing. It should not be forgotten however that figures are semi-logarithmic. The region 3s> affects the neutral fraction appreciably. Also it is important to notice that for all different mechanisms the orthogonalization corrections are very important, decreasing the transition rates by one to two orders of magnitude.

Figure (٢-a) gives the transition rate as a function of the ion

distance for the three different mechanisms for the nonorthogonalized case. Figure (٢-b) shows the result with the orthogonalization included. In the region of large distances from the surface the transition rate ( )P s is exponentially decaying for the different mechanisms. At small distances where ( )P s value is important, the relationship is somewhat ambiguous between s and ( )P s . This is because the approximations might affect the results at these distances considerably. The three different mechanisms are of equal importance as can be concluded from the overall curves. This makes it necessary not to neglect any of them in any complete theory.

Figures (٣-a, ٣-b, ٣-c) gives the neutral fraction 0f as a function of the perpendicular velocity v of the incident proton. The neutral fraction 0f is computed by numerical integration using eq. (٣٧). From the figures, the effect of the orthogonalization is clearly observed. The neutral fraction is greatly decreased for all velocities considered and consistent with the assumptions of the theory (low energy protons). The turning point of the ion ms (position as smallest s value) is taken to be zero. In other words it is assumed that the ion reaches the surface. However, if ms takes greater values it is seen that the neutral fraction 0f is decreased due to the small values of the matrix


٤١

elements. Also it is noticed that curves ٢a and ٢b begin to merge into one another. The reason of this is the fact that as the ion position is taken further outside the surface the effect of orthogonalization becomes negligible. At 7ms = the two curves almost coincide. As a conclusion we can see that proper calculation of the matrix elements in the neutralization process is important.

Figure (٤-a) gives the neutral fraction 0f as a function of the

perpendicular velocity v⊥ of the ion for the three different mechanisms for the nonorthogonalized case. Figure (٤-b) shows the result with the orthogonalization included. It is clear that surface plasmon-mediated neutralization would account for most of the neutralization processes if assumptions in our theory apply. The other two mechanisms namely Auger and resonance are of equal importance. Of course a more comprehensive theory that considers the different channels of neutralization simultaneously would be very valuable.


٤٢

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Theory of Ion Neutralization at Metal Surfaces

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