Pergamon Adv. Space Res. Vol. 14, No. 10, pp. (10)841-(10)852, 1994

1994 COSPAR Printed in Great Britain. All rights reserved.

0273-1177/94 $7.00 + 0.00

GALACTIC COSMIC RAY TRANSPORT METHODS: PAST, PRESENT, AND FUTURE

J. W. Wilson,* L. W. Townsend,* J. L. Shinn,* F. A. Cucinotta,* R. C. Costen,* F. F. Badavi** and S. L. Lamkin***

* NASA Langley Research Center, Hampton, VA 23681, U.S.A. ** Christopher Newport College, Newport News, VA 23601, U.S.A. *** Old Dominion University, Norfolk, VA 23508, U.S.A.

ABSTRACT

The development of the theory of high charge and energy (HZE) ion transport is reviewed. The basic solution behavior and approximation techniques will be described. An overview of the HZE transport codes currently available at the Langley Research Center will be given. The near term goal of the Langley program is to produce a complete set of one-dimensional transport codes. The ultimate goal is to produce a set of complete three-dimensional codes which have been validated in the laboratory and can be applied in the engineering design environment. Recent progress toward completing these goals is discussed.

~TRODUCTION

Propagation of galactic ions through matter has been studied for the past 40 years as a means of determining the origin of these ions/1,2,3/. The "solution" to the steady-state equations is given as a Volterra equation by Gloeclder and Jokipii/4/, which is solved to first order in the fragmentation cross sections by ignoring energy loss. They provide an approximation to the first- order solution with ionization energy loss included that is only valid at relativistic energies. Lezniak /5/ gives an overview of cosmic-ray propagation and derives a Volterra equation including the ionization energy loss which he refers to as a solution "only in the iterative sense" and evaluates only the unperturbed term. The main interest among cosmic-ray physicists has been in first-order solutions in the fragmentation cross sections, since path lengths in interstellar space are on the order of 3-4 g/cm 2. Clearly, higher order terms cannot be ignored in accelerator or space shielding transport problems/6-10/.

Several approaches to the solution of high-energy heavy ion propagation including the ionization energy loss have been developed over the last 20 years /7-19/. All but one have assumed the straight ahead approximation and velocity conserving fragmentation interactions/7/. Only two have incorporated energy-dependent nuclear cross sections/7,10/. The approach by Curtis, Doherty, and Wilkinson/15/for a primary ion beam represented the first-generation secondary fragments as a quadrature over the collision density of the primary beam. Allkofer and Heinrich /16/used an energy multigroup method in which an energy-independent fragmentation transport approximation was applied within each energy group after which the energy group boundaries were moved according to continuous slowing down theory (-dE/dx). Chatterjee, Tobias, and Lyman /17/solved the energy-independent fragment transport equation with primary collision density as a source and neglected higher order fragmentation. The primary source term extended only to the

(10)841

(10)842 J.W. Wilson et al.

primary ion range from the boundary. The energy-independent transport solution was modified to account for the finite range of the secondary fragment ions.

Wilson/8/der ived an expression for the ion transport problem to first order (first collision term) and gave an analytic solution for the depth-dose relation. Wilson/7/examined the more common approximations used in solving the heavy ion transport problem. Errors generated by assuming conservation of velocity on fragmentation and the straight ahead approximation are found to be negligible for cosmic-ray applications. Methods of solution for the energy-dependent nuclear cross sections have been developed/7/. Letaw, Tsao, and Silberberg /18/ approximated the energy loss term and ion spectra by simple forms for which energy derivatives more evaluated explicitly (even if approximately). This approximation results in a decoupling of motion in space and a change in energy. In Letaw's formalism, the energy shift was replaced by an effective attenuation factor. Wilson added the next higher order (second collision) t e rm/9 / . This term was found to be very important in describing 20Ne beams at 670 MeV/nucleon. The three-term expansion of Wilson/9/ was modified to include the effects of energy variation of the nuclear cross sections/10/. The integral form of the transport equa t ion /7 /was further used to derive a numerical marching procedure to solve the cosmic-ray transport problem/11/. This method can easily include the energy-dependent nuclear cross sections within the numerical procedure. Comparison of the numerical procedure /11/ with an analytic solution to a simplified problem/12/val idates the solution technique to about 1 percent accuracy. Several solution techniques and analytic methods have been developed for testing future numerical solutions to the transport equation/19/. More recently, an analytic solution for the laboratory ion beam transport problem has been derived assuming a straight ahead approximation, velocity conservation at the interaction site, and energy-independent nuclear cross sections/13/. These analytic techniques were used to derive the Green's function to be used for space or laboratory exposure/20/.

In the previous overview of past developments, the applications generally split into two separate categories according to a single ion species with a single energy at the boundary versus a broad host of elemental types with a broad, continuous energy spectrum. Techniques requiring a representation of the spectrum over an array of energy values require vast computer storage and computation speed for the laboratory beam problem to maintain sufficient energy resolution. On the other hand, analytic methods/7 ,8 ,13/are probably best applied in a marching procedure/11/, which again has within it a similar energy resolution problem. This is a serious limitation because we require a final High Charge and Energy (HZE) Code for cosmic-ray shielding that has been validated by laboratory experiments.

TRANSPORT THEORY

The massive particle transport equations are derived by balancing the change in particle flux as it crosses a small volume of material with the gains and losses caused by nuclear collision. The resulting equations for a homogeneous material are given b y / 2 1 /

I N. V 1 (E)] Cj (x, n, E) A---~ :E Sj(E) +

= ~ / dEI dflrajk (E, EI, n, fll) Ck

where Cj (x, N, E) is the flux of ions of type j with atomic mass energy E in units of MeV/nucleon, aj(E) is the corresponding macroscopic cross section, Sj(E) is the linear energy transfer (LET), and ajk (E, E I, fl, f~l) is the production cross section for type j particles with energy E and direction f / b y the collision of a type k particle of energy E I and direction f~t. The term on the left side of equation (1) containing Sj (E) is a result of the continuous slowing-down approximation, whereas the remaining terms of equation (1) are seen to be the usual Boltzmann terms. The solutions to equation (1) exist and are unique in any convex region for which the inbound flux of each particle type is specified everywhere on the bounding surface. If the boundary is given as the loci of the two-parameter vector function ~/(s, t) for which a generic point

(x, f~', E') (1)

Aj at x with motion along N and

GCR Transport Methods (10)843

on the boundary is given by r, then the boundary condition is specified by requiring the solution of equation (1) to meet

~jcr, a , E) = c j ( r , rJ , E) (2)

for each value of f /such that r l . n(r) < 0 (3)

where n ( r ) is the outward-directed unit normal vector to the boundary surface at the point r and Cj is a specified boundary function.

The fragmentation of the projectile and target nuclei is represented by the quantities ajk (E, E r, f/, f/i), which are composed of three functions:

a jk ( E, E r, N, N r) = ak ( Er)vjk ( Er) f jk ( E, E', N, N r) (4)

where vflc(E' ) is the average number (which we loosely refer to as multiplicity) of type j particles being produced by a collision of a type k of energy E r, and fjk (E, E r, f~, f/r) is the probability density distribution for producing particles of type j of energy Einto direction N from the collision of a type k particle with energy E r moving in direction f/r. For an unpolarized source of projectiles and unpolarized targets, the energy-angie distribution of reaction products is a function of the energies and cosine of the production angle relative to the incident projectile direction. The secondary multiplicities ujk(E r) and secondary energy-angle distributions are the major unknowns in ion transport theory.

The spectral distribution function is found to consist of two terms that describe the fragmentation of the projectile and the fragmentation of the struck nucleus as follows/22,23/:

F__ __

ajk(E, E t, ~, N') = ak(E' ) /v~ (E, E t, N, N t)

U T [E I~tT [E ] + :jk~ JJ/k~ ,E ' , f I , N') /

(B)

where v~ and fPjk depend only weakly on the target and v~ and f ~ depend only weakly on

the projectile. Although the average secondary velocities associated with fP are nearly equal to the projectile velocity, the average velocities associated with fT are near zero. Experimentally, Heckman /22/ observed for massive fragment (A _> 4) that

F i "+ 271" ( a ~ ) 2 /

+Ira '2m- (0-~)

3/2

v ~ exp

3/2

(p - pl)21

V / ~ exp - - -p -~ '

(6)

where p and p/axe the momenta per unit mass of j and k ions, respectively, and

f ~ (E, E', N, N')

i (-:,, v ~ exp P'I]

7 2 2

(7)

where a ~ and a ~ are related to the root-mean-square (rms) momentum spread of secondary products. These parameters depend only on the fragmenting nucleus. Feshbach and Huang/24/ suggested that the parameters c ~ and c ~ depend on the average square momentum of the nuclear

JASR 14:IO-CCC

(10)844 J.W. Wilson et al.

fragments as allowed by Fermi motion. A precise formulation of these ideas in terms of a statistical model was obtained by Ooldhaber/25/.

The notation is simplified by introducing a vector of flux fields as

= [ms(x, n , E)] (8)

the linear Boltzmann operator

[ 1 0 S,(E)+~,(E)] (9) B = a - v Aj

and the integral collision operator

Each component of the field vector ~b corresponds to a given particle type and by convention we place the most massive particle to the top of the vector and least massive to the bottom. The Boltzmann operator B representing field drift and collisional losses (atomic and nuclear) is diagonal and the collisional operator ~ tends to be lower triangular. There exists an integrating factor for B; we will refer to its inverse as the Boltzmann propagator Go and it has been found using the method of characteristic/6,7/as a solution of

BGo = 0 (11)

The general solution to the Boltzm~nn equation is then/6 ,7 /

~b = G o ~bB + B - l ~ b (12)

and satisfies the boundary conditions (2) provided Go reduces to the identity operators at the boundary (note, we choose the constants of integration for equation (11) so this is true). A number of approximate methods have been developed based on equation (12).

A Newman series/6,7,20/may be developed for equation (12) as

q5 = Go ~bs + B - I ~ Go ~bB + B - 1 Z B - I ~ Go ~bs + ... (13)

which we rewrite in terms of the complete propagator G a s / 2 0 /

= a (14)

It is clear from equation (13) that the complete propagator is given by

G = Go + B -1 ~ G (15)

Clearly G depends on the bounding surface and the physical properties of the media/26/. There are two streams of development in solving the transport problem, the first is to establish solutions to equation (12) according to some procedure /6-11/ and the second is to develop methods for evaluation of the complete propagator of equation (15) for application to specific input spectra 120,261.

APPROXIMATION PROCEDUI~S

In the remainder of this report, we will discuss a progression of development towards increasing levels of sophistication in evaluation methods of particle fields within complex geometric objects. We will not discuss methods for which the relation to the aforediscussed formalism is at best tenuous

GCR Transport Methods (10)845

/11/. Nor shall we dwell on strictly finite difference procedures or Monte Carlo simulation although they shall at times provide insight into the accuracy of the final methodology/6,14,19,26/.

Decoupling Of Target Fragments. The separation of the interaction cross sections into projectile and target fragment contributions as in equations (5) to (7) provides a basis of simplifying the computational procedure. We may separate the fields as ¢ = Cp + ¢T in which

BCp = ~ p (¢p + ¢T) (16)

BeT = ~T (¢P "{- ~T) (17)

In that the second term on the right-hand side of equations (16) and (17) are negligible since the range of the multiple charged target fragments is small compared to the nuclear mean free paths, we may take

, , o , = o , + o (o ; : / ,m)] :10>

where Rj(E) is the range energy relation for ion type j . Equation (18) must be yet evaluated for which equation (19) becomes a simple quadrature/7,27/. The reminder of this report will focus

the solution of equation (18)neglecting terms to the order of o" k Rj (a~2/2m/ ~ 10 -3. In the o n

remainder, we will drop the subscript P from equation (18) to simplify notation so that ¢ will refer to the projectile fields only and ~ will refer to specifically target fragments.

Conservative Field Estimates. A guiding principle in radiation protection practice is that if errors are committed in risk estimates they should be overestimates. The presence of strong scattering terms in the collision terms in equation (10) provides lateral diffusion along a given ray. Such diffusive processes result in leakage at near boundaries /26/. If c r ( r ) is the solution of the Boltzmann equation for a source of particles on the boundary surface F then the solution for the surface source on F within a region enclosed by r I denoted by Cr I (F) has the property

Or' (r) = r(r) + er' (20)

where er ~ is positive definite provided 1 ~ completely encloses r . The most strongly scattered component is the neutron fields for which ~r ~ ~ 0.2 percent for an infinite media for most practical problems/26/. Standard practice in space radiation protection replaces G as required at some point on the boundary and along a given ray by the corresponding GN evaluated for normal incidence on a semi-infiuite slab. The errors in this approximation are second order in the ratio of beam divergence and radius of curvature of the object and rarely exceeds a few percent and is always conservative/26/.

Straight Ahead Approximation. The adequacy of the straight ahead approximation in shielding from space protons was demonstrated by Alsmiller and eoworkers many years ago /28/. The straight ahead approximation for multiple charged ions is accomplished by approximating equation (6) as

6(#. a ' - 1) (21)

The error term gene ra t ed /7 /by the replacement of equation (21) is

~,5 ~ ~2 /2~4'-~E' (22)

and is quite small provided the angular distributions of the fields at the boundary are relatively uniform /7 / since the width a ~ of the fragment momentum spectrum is small compared to the projectile momentum. ~hrthermore, the straight ahead approximation overestimates the transmitted flux and is therefore conservative in most space shielding applications. The success of the straight ahead approximation results in part from the small increase in attenuation for lateral diffusion through angles as large as 30 ° /21 / .

(10)846 J.W. Wilson et al.

Velocity Conserving Interactions. The multiple charged fragments formed by nuclear interaction are mainly the spectators of the collision process which conceptually lead Goldhaber /25/ to suggest that the momentum spread a ~ in the fragment spectrum is related to the spectators random Fermi motion at the time of collision. The final fragment velocity is then the collective spectator velocity prior to collision and is nearly equal to the velocity of the projectile. The velocity conserving interaction is affected by replacing

l iP(E, E') ~" 6(E - E') (23)

in equation (21). The error term genera ted /7 /by the replacement is

(24)

Although this error is small when energy variation in the fields is modest as for space radiations, the velocity conserving interaction is an inferior approximation to the straight ahead approximation for space radiations as seen by comparing equations (22) and (24).

ONE-DIMENSIONAL THEORY

This section will deal with evaluation of the particle fields under approximations given by equations (18) and (21). There is no lateral spread so that the surviving spatial variable is the depth of penetration and the integral operator ~ is reduced to a simple integral operator over the energy variable only. The transmitted flux in this approximation is always conservative but the degree of error is small for space radiation exposure estimates/28/. We now consider methods by which equation (18) can be solved under the approximation given by equation (21).

Perturbation Theory. The integral form of equation (18) is given as

¢ = Go ¢B + B-1IE ¢ (25)

and has the Neuman series given by equation (13). The first two terms have been used by various workers to implement an approximate solution for low penetration depths /4,5,8,15,17/. An iterative procedure was developed by Lamldn and Wilson/6,7,29/which is continued until convergence. The charged particle fields were found to converge rapidly while the neutral neutron component required a greater number of terms/30/ . Although these methods showed promise as a very efficient shielding code as compared to Monte Carlo procedures, the computational demands were considered excessive compared to marching procedures.

Numerical Marching Procedures. As a consequence of the straight ahead approximation, the integral equation (25) is a Volterra equation and may be solved using marching procedures. Consider any point on the boundary, the solution can be propagated from the boundary F0 to an interior surface r 1 using equation (13) as

¢(ri) = Go ¢(r0) + B-IS Go ¢(r0) + o (Ir - r0[ 2) (26)

where the error term is on the order of the square of the distance between r 0 and 1~1 which can be made arbitrarily small. Equation (26) may be used repeatedly to cover the solution domain as

¢(r.+1) = Go + B - i S Go ¢(r . ) (27)

The propagated error at the nth step is

¢(h) [1 - exp(-anh)] (28)

where h is the distance between rn+1 and rn and ¢(h) is the maximum error committed on any step. The truncated error of equation (26) is on the order of (ah) 2 giving reasonable error propagation in equation (28). This method is the basis of the BRYNTRN and HZETRN codes/11,14,21/.

GCR Transport Methods (10)847

For convenience of notation and to simplify the computational procedures, we scale the flux vector by multiplying by the proton stopping power as

¢( r .+~) = s e ¢(r .+~) = go v ( r . ) + s -~ ~ ' 9o ¢ ( r . ) (29)

where B , ~ r, ~o are new operators corresponding to B, ~, Go. The component equations of equation (29) are written along a given ray as

x/0 h /? Cj (x + h, r) = e-ai(r)h Cj (x, r + vjh) + dz e-at (r)z -fjk (r + vj z , / ) +vjz

x e -%(¢)(h-z) Ck [x, r' + vk(h - z)] dr'

(30)

where E has been replaced by proton residual range and vj the ion range scale parameter Z2/Aj. It was shown by Lamldn et al. / 29 / t ha t the integrals of equation (30) may be evaluated as (for zj , zk _< 2)

j•OO k +vjh/2

×¢k x, +vk~ d / F-jk )

(31)

where

j•0 h

F-~k (r, vjh, r t) = -f jk (r + v j z , / ) dz (32)

Equations (31) and (32) are the bases for the BRYNTRN code for nucleon transport. The "Fjk (r, vjh, r t) is related to the integral spectrum of particles produced by the nuclear collision.

The ions with Z > 2 can be written as

Cj(x + h,r) ~ e-ai (r)h ¢j(x ,r + vjh)

+ dz e-aJ (r)z-%(r)(h-z) Ck[x, r + vjz + vk(h - z)] (33)

and may be reduced to

Cj(x + h,r) = e-aJ (r)h Cj(x,r + vjh)

e - a j ( r ) h _ e - a k ( r ) h "

$

Ck(z, r + v~h) (34)

Note that this formula is similar to the prescription of AUkofer and Heinrich/16/. Equation (34) for Z _> 2 coupled to equation (31) provides the source for the HZETRN code.

As a method of validation, we show in figure 1 a comparison of BRYNTRN with a three-dimensional Monte Carlo simulation (HETC) for a rather thick aluminum shield (20 g/cm 2) in front of a 30-cm tissue slab (phantom). The HZE propagation of equation (34) compares to a converged numerical so lu t ion /19 /o f equation (18) under approximations (21) and (23) to within 2 percent. Further attributes of these codes are described elsewhere/21/.

(10)848 J.W. Wilson e/02.

10-1

Dose, G y

1 0 - 2

10-3

10 -1

10 -2

Dose, Gy

10-3

10-1

Dose e q u i v a l e n t ,

~ V

10 -2

l l i i , f 1 0 "3

1 0 - 4

0 Tissue depth, g/cm2

, , J i l l

- - Monte Carlo method • BRYNTRN

10"1

10 -2 ~ Dose

equivalent,

Sv 10 -3 - - Total secondaries (Monte , Carlo method)

• o - Total secondaries (BRYNTRN) ° • ° HZE recoils (BRYNTRN)

, , , , ,° - 1 0 - 4 i I i l i , 1 0 20 30 0 1 0 20 30

Tissue depth, g/cm2

Fig. 1. Comparison of the BRYNTRN code with Monte Carlo calculations.

Green's Functions. Although the numerical procedures discussed above are adequate when the primary particles have broad continuous spectra, the problem of code validation would be limited to space flight experiments in which the primary particle environmental models are only approximately known, the spacecraft geometry is to a degree uncertain, and detector response is only partly understood. Code validation is ultimately to be achieved in particle accelerator experiments where the primary particle type is know with certainty, its energy is well defined and the highest quality detection systems can be employed under optimal configuration design to measure the reaction products transmitted through shield materials. We now discuss methods which are efficient tools for space shield design and may be validated in a laboratory environment.

The content of the Green's function method is when ¢( r ) defined on a closed boundary r is related to ¢ in the interior region as

¢ = Gr ¢ ( r ) (35)

where G r is the Green's function which reduces to the identity on the boundary and satisfies

G r = G o r + B -1 2; G r (36)

We noted in connection with equation (20) that Gr could be replaced at each point on the boundary by the Green's function for a semi-infinite slab value Gpr and that a conservative estimate of Gpr within the interior is found by using the straight ahead approximation of equation (21). We therefore consider a conservative approximate solution of equations (35) and (36) by using Gpr in place of Gr but must yet develop Gpr for the semi-infinite slab.

The propagator Gp relates solutions in a semi-infinite slab to any arbitrary flux ¢(p) at the planar boundary p as

Cp = Gp ¢(p) (37)

Suppose we evaluate Cp(/) at a plane / parallel to p which is chosen such that

Gp ~ Gap + B-lID Gap + B -1 Z B-I~ Gop (38)

GCR Transport Methods (10)849

Then the solution beyond pl is given as

¢~ = G d Cp(pl) = G// %(p:) ~b(p) (39)

If we denote Gp(p t) as the propagator from p --, pt and G~ as the propagator beyond p~(p~ - , c¢), while Gp is the propagator from p --* p~ --* c¢ then equations (37) and (39) yield

% = G f %(p') (4o)

Since Gid and Gp differ only by a translation they are functionally equal and equation (38) can be used to cover a restricted region of the space while equation (40) is a nonperturbative relation which can be used to cover the entire space.

Approximate Green's Functions. The scaled Green's functions in residual range space is given by

gjm(~, rj, r ' ) = ~j(e) v~m(~, E, S') (41)

t where r j , r m are the residual ranges. This Green's function may be approximated by

(42)

where

The function gjm (x) is a solution to the energy independent problem and is approximated by

(43)

g~m(~) = ~m e -~j~ + ~jm "e-O'jx _ e-fire, x"

am - a j +.. . (44)

where higher order terms are discussed elsewhere/13,20/. The perturbation series may be used to cover a portion of the space and the nonperturbative equation to cover the remaining space is

(45)

These nonperturbative techniques hold great promise for accurate and efficient computational methods for evaluation of the HZE particle fields in space or laboratory problems. They are yet to be extended to light ion and especially neutron fields.

Values for the collision related terms of g j m ( x , r j , rim) are shown in figure 2. The x is depth in a water medium, Zp is the charge of the incident projectile and specifically produced species are noted in the figure label. Clearly the production of any given species is dominated by the projectiles of nearly the same but greater charge. The multiply collision terms are mostly important for those projectiles whose charge is fax removed from the specific species. These Green's functions are used to evaluate the composition of a 600 MeV/nucleon iron beam in a water column at several depths with results in figure 3. These will be compared to experiments in the near future.

(10)850 J.W. Wilson et rd.

Cr fragment Ar fragment .24 .24

.16 G .16 G

.08 .08

0

:. cm -- 40.~n----~%"~'~ 21 7. : m 3 0 ~ 1 4 Z , '" bU 28 Zp X, p X, cm

O:

X, cm

O fragment .24

,.16 G .16 G

.08 0 t 0 2 0 " ~ ~ ~ "08

28 Zp X, : ~ 3 0 ~ 1 4 ;p

Fig. 2. Scaled Greens function for specific species produced in water shield.

I'0004 5 ¢m .0004.0003 ~ 10 cm I "0003 ¢

I ooo2 !.ooo2 .0001

" 2 0 0 ~ ~ 1 8 _. 2 0 0 ~ ~ / 1 8 _ - - - 4 0 ~ 8 " - Zf - - - 4 ( ~ / o a "- Z~

E, (MeV/amu) " ' ' E, (MeV/amu) 600 ,:.o - -

[.0004 [.0004

15 ¢m 1.0003 20 ¢m 1.0003 t ooo ° t _ _ _ j:oo

o ~ - - ~ M ~ , , , ~ ~ 1 o o ,o 400 _~_-----~_ "~a L~ 600-o ' - - - ,oo6-~o--~8- zf

E, (MeV/amu) E, (MeV/amu)

Fig. 3. Composition of a 600 MeV/nucleon Fe beam in water shield. The Z / > 2 flux is scaled by 1/Z/. The Z/--- 1, 2 flux is scaled by 1/10.

GCR Transport Methods (10)851

REFERENCES

1. B. Peters, The Nature of Primary Cosmic Radiation. Progress in Cosmic Ray Physics, J. G. Wilson, ed., Interscience Publ., Inc., pp. 191-242 (1958).

2. Leverett Davis, Jr., On the Diffusion of Cosmic Rays in the Galaxy. Proceedings of the Moscow Cosmic Ray Conference, International Union of Pure and Applied Physics (Moscow), pp. 220-225 (1960).

3. V. L. Ginzburg and S. I. Syrovatskii (H. S. H. Massey, transl., and D. Ter Haar, ed.), The Origin of Cosmic Rays. Macmillan Co. (1964).

4. G. Gloeckler and J. R. Jokipii, 1969: Physical Basis of the Transport and Composition of Cosmic Rays in the Galaxy. Phys. Review Leg., vol. 22, no. 26, pp. 1448-1453.

5. J. A. Leznlak, The Extension of the Concept of the Cosmic-Ray Path-Length Distribution to Nonrelativistic Energies. Astrophys. & Space Sci., vol. 63, no. 2, pp. 279-293 (1979).

6. John W. Wilson and Stanley L. Lamkin, Perturbation Theory for Charged-Particle Transport in One Dimension. Nucl. Sci. & Eng., vol. 57, no. 4, pp. 292-299 (1975).

7. John W. Wilson, Analysis of the Theory of High-Energy Ion Transport. NASA TN D-8381 (1977).

8. J. W. Wilson, Depth-Dose Relations for Heavy Ion Beams. VirginiaJ. Sci., vol. 28, no. 3, pp. 136- 138 (1977).

9. John W. Wilson, Heavy Ion Transport in the Straight Ahead Approximation. NASA TP-2178 (1983).

10. John W. Wilson, L. W. Townsend, H. B. Bidasaria, Walter Schimmerling, Mervyn Wong, and Jerry Howard, 20Ne Depth-Dose Relations in Water. Health Phys., Vol. 46, no. 5, pp. 1101-1111 (1984).

11. John W. Wilson and F. F. Badavi, Methods of Galactic Heavy Ion Transport. Radiat. Res., vol. 108, pp. 231-237 (1986).

12. John W. Wilson and L. W. Townsend, A Benchmark for Galactic Cosmic-Ray Transport Codes. Radiat. Res., voL 114, no. 2, pp. 201-206 (1988).

13. John W. Wilson, Stanley L. Lamkin, Hamidullah Farhat, Barry D. Ganapol, and Lawrence W. Townsend, A Hierarchy of Transport Approximations for High Energy Heavy (HZE) Ions. NASA TM-4118 (1989).

14. John W. Wilson, Lawrence W. Townsend, John E. Nealy, Sang Y. Chtm, B. S. Hong, Warren W. Buck, S. L. Lamkin, Barry D. Ganapol, Ferdous Khan, and Francis A. Cucinotta, BRYNTRN: A Baryon Transport Model. NASA TP-2887 (1989).

15. S. B. Curtis, W. R. Doherty, and M. C. Wilkinson, Study of Radiation Hazards to Man on Extended Near Earth Missions. NASA CR-1469 (1969).

16. O. C. AUkofer and W. Heinrich, Attenuation of Cosmic Ray Heavy Nuclei Fluxes in the Upper Atmosphere by Fragmentation. Nucl. Phys. B, vol. BT1, no. 3, pp. 429-438 (1974).

17. A. Chatterjee, C. A. Tobias, and J. T. Lyman, Nuclear Fragmentation in Therapeutic and Diagnostic Studies With Heavy Ions. Spallation Nuclear Reactions and Their Applications, B. S. P. Shen and M. Merker, eds, D. Reidel Publ. Co., pp. 169-191 (1976).

18. John Letaw, C. H. Tsao, and R. Silberberg, Matrix Methods of Cosmic Ray Propagation. Composition and Origin of Cosmic Rays, Maurice M. Shapiro, ed., D. Reidel Publ. Co., pp. 337-342 (1983).

(10)852 J.W. Wilson et aL

19. Barry D. Ganapol, Lawrence W. Townsend, Stanley L. Lamkin, and John W. Wil- son, Benchmark Solutions for the Galactic Heavy-Ion Transport Equations With Energy and Spatial Coupling. NASA TP-3112 (1991).

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