-_- .j

- NASATechnical

_Paper

_October 1991

Benchmark Solutionsfor the Galactic

Heavy-Ion TransportEquations With Energyand Spatial Coupling

Barry D. Ganap@ __

Lawrence W. Townsend,

Stanley L. Lamkin,

and John W. Wilson

[ (_'-A _,A-T _- 311))_ C, FI',ICHMA_K SflLIJTION3 FOR THE

.... GALACTIC HEAVY-TON TR_NSPOKT E(_UATIONS WITH

ENERGY ANO qPATIAL CGUPLING (NASA) 58 p

CSCL 2OA

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N92-13750

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NASATechnical

Paper3112

1991

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

Benchmark Solutionsfor the Galactic

Heavy-Ion TransportEquations With Energyand Spatial Coupling

Barry D. Ganapol

University of Arizona

Tucson, Arizona

Lawrence W. Townsend

Langley Research Center

Hampton, Virginia

Stanley L. Lamkin

Old Dominion University

Norfolk, Virginia

John W. Wilson

Langley Research Center

Hampton, Virginia

The use of trademarks or names of manufacturers in this

report is for accurate reporting and does not constitute an

official endorsement, either expressed or implied, of such

products or manufacturers by the National Aeronautics and

Space Administration.

Abstract

Nontrivial benchmark solutions are developed for the galactic heavy-

ion transport equations in the straight-ahead approximation with energy

and spatial coupling. Analytical representations of the ion fluxes are

obtained for a variety of sources with the assumption that the nuclear

interaction parameters are energy independent. The method utilizes

an analytical Laplace transform inversion to yield a closed-form repre-

sentation that is computationaUy efficient. The flux profiles are then

used to predict ion dose profiles, which are important for shield-designstudies.

1. Introduction

With our ever-increasing interest in establishing

mankind's presence in space, the protection of per-

sonnel from energetic radiation in the space envi-ronment has become a relevant concern. As high-

energy radiation interacts with target nuclei, the ions

undergo nuclear fragmentation and energy degrada-tion. This radiation is composed of heavy ions, called

galactic cosmic rays (GCR), originating in deep space

and/or protons from the Sun. The nuclear frag-

ments generated by direct nuclear impact or electro-

magnetic dissociation form a secondary radiationfield that again interacts with the target nuclei and

fragment to add to the biological radiation hazard

represented by the incident ions. Thus, to ensure

that the habitat in the space environment is properly

shielded from energetic radiation, a knowledge of the

changing nature of the incident radiation field as it

penetrates protective spacecraft shielding is imper-ative. To anticipate future space shielding require-

ments, NASA has initiated an effort to formulate

computational methods that simulate radiation ef-

fects in space. Deterministic transport algorithms

have been developed for the Boltzmann equation thatdescribe GCR interactions with nuclei and cause the

inevitable changing composition of the incident ra-diation field while traversing bulk shielding mate-

rial. An important component of the NASA pro-

gram is the assessment of the accuracy of proposed

deterministic algorithms. For this reason, analyti-

cal benchmark solutions to mathematically tractable,

galactic cosmic ray problems have recently been de-

veloped (refs. 1 and 2). Even though these prob-

lems involve some simplifying assumptions of the as-

sociated GCR physics, they still contain the essential

features of the transport processes. These solutions,

when compared with the corresponding numerical re-

sults from algorithms, provide assurance of proper

programming and a measure of the accuracy of thenumerical methods used in the algorithm. The moti-

vation for such comparison is that algorithms devel-

oped for realistic problems must also yield reliableresults for the less complicated problems.

In this report, the analytical solution to the galac-

tic ion transport (GIT) cquations that describe the

straight-ahead motion of energetic heavy ions (HZE)

is developed for the first time. An analytical rep-resentation of the ion fluxes for a variety of sources

is obtained with the assumption that the nuclear in-

teraction parameters are energy independent. The

method utilizes an analytical Laplace transform in-

version which yields a closed-form representationthat is also computationally efficient. From the flux

profiles, it is then possible to obtain ion dose profiles,

which are important for shield design.

In section 2, the analytical solution technique is

presented in detail, along with solutions for several

source distributions. In section 3, the numerical

implementation of the flux and dose profiles is dis-cussed. Results for selected case studies and com-

parisons are presented in section 4, with special em-

phasis on understanding the behavior of the ion flux

from both the physical and mathematical points ofview. The report concludes with a summary and rec-

ommendations for future development.

2. Theory

2.1. Galactic Ion Transport Equations

Because of the high energy of the galactic ions

(refs. 3 to 5), the straight-ahead approximation canbe introduced into the Boltzmann equation with a

high degree of confidence. In this approximation,

ions are assumed to move without angular deflection;

and, as the colliding ions break up in nuclear frag-

mentation, the fragments continue in the incident iondirection. Thus, for ions of charge numbcr j, the

appropriatetransportequationis (ref.6)

0 st (E) + (E)/CJ0 ]

E)ffz OE J

J

Ek=j+l

Mtk(E ) ak(E ) Ck(x, E) (la)

where Cj is the flux of the jth ion flux at position xwith an energy per nucleon of E. The macroscopic

absorption cross section is aj, Mjk is the multiplicity

of ion j produced in collision with ion k, and Sj isthe absolute value of the change in energy E perunit distance traveled. The added assumptions of

continuous slowing-down theory and that target ion

fragmentation can be neglected have been made in

equation (la). Source ions are assumed incident on asemi-infinite shield with an initial energy distributionof

Cj(0, E) = fj(E) (lb)

Because of the success of the analytical investigations

applied to previous benchmarks (refs. 1 and 2), the

analytical approach is again followed here: To make

equation (la) mathematically tractable, the crosssections and multiplicities are assumed to be energy

independent. This is a reasonable assumption at high

energy. Also, the well-known scaling law for stopping

powers at high energies (ref. 6), or

St(E) = .j S (E) (2a)

is introduced into equation (la), where

2vj - Zj /Aj (2b)

and where Sp is the proton stopping power and Ajis the atomic mass of thc jth ion. With this substi-

tution, and with the change of variable from energy

to proton path length at a given energy s(E, Eo),

equations (1) become

(9 (9 ] JCt(x,s) = E Ck(x,8)k=j+ 1

(3a)and

Cj(0, 8) = fj(s) (3b)

where

9jk = Mtk (3c)

The path length is given by

s(E, EO) = dE'/Sp(E') (3d)

2

which, from the definition of the flux distribution,

implies

¢j(x, s) = Sp(E) Cj(x, E) (3e)

and

It(s) = Sp(E) :j(E) (3f)

Some readers may be accustomed to seeing the en-

ergy variable in equations (la) and (lb) transformed

by using the residual range Rp(E) (refs. 7 and 8),where

E dE'Rp(E) = S(E')

The variables Rp(E) and s(E, Eo) are directlyrelated. The variable s(E, Eo) is the penetration

distance for a proton of initial energy E0 to reach

present energy E; R(E) is the remaining distance

the proton will travel before stopping.

The major difficulty in applying the path-length

transformation is in the determination of s(E, Eo)

from the energy E by using equation (3d), since anintegral must be inverted to return to the energy-

dependent flux given by equation (3e). This added

inconvenience, however, is readily acceptable be:

cause of the enormous simplification offered by theenergy path-length transformation. Indeed, equa-

tions (3a) and (3b) now closely resemble the energy-

independent case considered in reference 1. This sim-

ilarity suggests that similar solution methods should

therefore be adopted.

While the emphasis here is on generating highly

accurate benchmark solutions, the only limiting as-

sumption made to obtain equations (3) is that of

constant cross sections and multiplicities. Thus, the

analytical results to be presented can be of use as

a predictive as well as a benchmarking tool to giveinsight into realistic shield design.

2.2. Analytical Solution for

Monoenergetic Source

hfitially, the solution will be obtained for a mono-

energetic beam source of ions of charge J and en-

ergy Eo(s = 0) incident on the semi-infinite shield

[fj(E) = 6(E - Eo) 6iJ]. For this case, Cj corre-sponds mathematically to the Green's function, sincethe source is of the form

¢j(0, s) = 6(s) 6ij (3g)

which translates into the following volume source:

• = 6,j

The Green's function

generate solutions for

ing beams composed

will be used subsequently tomore general sources, includ-

of several different ion typcs

(compositebeams).Forthe incidention J and for

ion J- 1, produced by target fragmentation, equa-

tions (3) can be solved analytically. It can be shownthat the solutions for ions Y and J- 1 are as follows:

Cj(x, s) = exp(- jx) e(s - (4a)

Cj_l(X, s) = _J-1,J {exp(-ajx)al

x exp[-a t (s - ujx)/bl]O(s - ujx) - exp(-ctj_ ix)

×exp[-o,(s-vj_lx)/b ]0(s j_lx)}(4b)

where al = crj-1-a j, bl =- uj_i-u j, and 0(a) is theHeaviside step function. The increased complexity ofthese solutions relative to tile benchmarks considered

in reference 2 is clearly evident. Because of the

interaction of x and s through the transport operator,

a wave of particles in x,s space is induced. The

delta fimction in the expression for Cj represents thatwave for the incident ions. The delta fimction is a

mathematical statement required by the continuous

slowing-down approximation. For ions of type J to

be at a given position x, measured from the surface,

they must have traversed a path length s/uj exactly.

The leading exponential represents the attenuation

(loss) of these ions as they fragment into lighter ions

along their trajectories. Ion J- 1, produced by the

fragmentation of ion J, cannot exist before its source

(at x = s/uj). Also, because of the finite range ofa charged particle, ion J- 1 cannot travel farther

than x = s/uj_l. Thus, ¢J-1 is nonzero only intile interval s/Uj _ ;C < 8/uj 1, as indicated by

equation (4t)). This technique is better illustrated

in the (x,s) space diagram in figure 1, where the

region of nonzero solution is indicated for each ion.

The above analysis holds for all ions j producedby fragmentation, and the farthest nonzero solution

boundary approaches s = x as j goes to one. The

region x > s is forbidden, since no ion has yet

traversed a path long enough to get there.

At first glance, one wouht not attempt to carry on

the analytical manipulation necessary to determine

the flux for the lower vahles of j. Fortunately, the

analytical results already derived in reference 2 arc

very useful in pursuing the solution here.

Taking a Laplace transform of s in equations (3a)

and (3b) gives

(0 )-_x + o'j + ujp Cj(x,p) = E _qjk -Ok(x,P) (5a)

k=j+l

Cj (0, p) = 5ia (5b)

where the transform is defined by

-¢j(x,p) - ds exp(-sp) _b(x, s)

Treating equations (5a) and (Sb) as a set of ordinary

differential equations yields the general solution

g

Cj_g(x,p) = Z Cti, J-t(p) exp [--(crj_ i + t.,j_ip)x ]

i=0

(6a)where the coefficients O_i, J g depend on the trans-formed variable p and are given recursively by (ref. 2)

1

ai'J-t(P) = (aj_g - aj_i) p(uj_ i - uj_i)

g-1

E flJ t,J-k' °_i,J u(P)

Id=i

(0<i<g-1)

(6b)and

g-1

c_e,j_e(p) = - _ c%]_t(p) (6c)i=0

The last expression is necessary to satisfy the bound-

ary condition Cj(0,p) = 0 for 1 < j <_ J - 1.

At this point, a numerical inversion procedure

could be attempted, Such an attempt, however,

would most likely fail, since it is known a priori that

the expression for Cj__. contains step flmctions that

characterize the spatially discontinuous ion density

waves; and in general, numerical inversions fail nearsuch discontinuities. Thus, a numerical inversion will

not work well in this case, even for the heavy ions.

Therefore, the only other option is to perform the

inversion analytically, as was done for ions J and J- 1

(eqs. (4a) and (4b)). To do so, however, requires the

explicit knowledge of i_he dependence of c_i,j_ t on p,which will now be determined.

For ion J - 1, we have

o,J(p) = 1

O%j_ 1 (p) _-

c_l,j_l (p) =

y J-1 /

0,1

aJ-1,j + pbj-1,j

y J-11,1

aj-1,J + pbj-1,j

(7a)

wherey J-1

0,1 = _J-1,J

V J-11,1 ---- _Y-l,J

ag m _ Off -- O-m

btm =- u t - Um

For _ = 2 and i = 0,

(7b)

a2,J-2(P) = Y_,/1-2A1 (P) + Y_2A2(p) + Y2J32A3(p) (10b)

where

A3(p) -=aJ-2,J 1 + PbJ-2,J-1

It should be noted that AI is common to both the

g-- 1 and g = 2 a-coefficients. A condensed expres-

sion for ai,j_ 2 is therefore

ao;-2(p) =aj-2,j + Pbj-2,j

× ( 't_J-2'J-1]3J-l'J '1-]_J-2,J_ (8)/

Upon using the partial fraction expansion

1 1

a j_2, J+pbJ 2,jaj-1,J+Pbj-1,J

bJ-2,J

aj-2,J bj-1,J - aj_l, J bj-2,jAI(p)

bJ-1,J+ A2(p)

aj-2,J bJ-1,J- a j_l, J bj-2,J

ai,J-2(p)= E Y/Jr-2Ar (p) (11)r=l

with the Y coefficients set to zero as appropriate.

When the last equation is introduced into equa-

tions (65) and (6c) with j = J - 3, the following

form for ai,j_ 3 is found:

6

= (o< i < e)r----1

where the rule for expansion in partial fractions

where

and

we obtain

A1 (p)

A2(p) -=

aj-1,j + pbj-1,j

aj-2,J + Pbj-2,j

Am(p)An(p) = XlmnAm(p) + X2mnAn(p)

has been used. Apparently, with each unit increase in

g, g, new terms appear in the expression for o_i,j_g(p),with the appropriate Y coefficients again set to zero.

Thus, the general expression for oq,j_g(p) is

_o,y-2(p) : Yi(f2hl(P)+ ']_2A2(P) (9a)

with

y J-2 bj_ 1, J _3J-2,J- 1 _,I- 1,.I1,1 =

a j_2, J bj_l, J-aJ 1,J bj-2,J(9b)

and

yj _ bj-2,d ZJ-2,J-1 /D-t,y (9c)1,2 = flJ-2,J -- aj_2,j bj_l,j _ aj_l,j bj_2,j

The coefficients of At(p) are independent of p. Simi-

larly, for al,J_2(p) and a2,J_2(p) ,

al,J_2(p) = )_(1-2Al(p)+ Y1J,3-2A3(p) (10a)

and

ai,j-e(p) = Y_ Y/J(eh_(p) (Ne - e(e + 1)/2)r=l

(12)

and the explicit dependence of e%j_e(p) on p hasbcen cstablished. Table I shows how, given an

index r, the appropriate values of am,n and bm,n forthe term

Ar(p) =am,n + bm,nP

are dctcrminbd. A general recurrence relation for

y:J-e is obtained by introducing equation (12) into

the relation for ai,j_g(p ) (eqs. (65) and (6c)) to yield(after much algebra)

=

J-i

yJ-e ^J-gi,r --" Xl,i,r

k=J-f+l

(I <__ ___N,_ 1, o<i<e-1) (13a)

J-i Nj-k

y J-_ei,r = _ /_J-g,k _ Ay_gx2,i,ry.k2,r +/3d_f,j 6i,o

k=J-t+l r=l

(r*=N__l+i+l , 0<i<f-1) (13b)

(Ne_l+l<r<_Ne_t+i, Ne_l+i+2<r<N_) (13c)

yJ-tg,r = - _ Y'J-gz,r

i=0

(1 _< r < N,, i = g) (13d)

where

AJ-_ ( bJ-g,J-i )Xl,i,r =- -- -at bj_,,d-i + ad_g,j_ i br

^j__ br

X2,i,r = -ar bj_e,j_ i + aj_e,j_ i br

ar -- a,n(r),,_(r)

br - bm(r),n(r )

and

I m_>0h(m)=- 0 m<0

Interchanging the summations in equation (13b)yields the following further simplification:

NI

y.J_g = _ _J-tz,r Xl,i,rr=l

J-i

k=J-g+l

( + 6,,0r=]

(13e)

The final expression for the transform is obtained

by substituting equation (12) into equation (6a) asfollows:

, .,,v, ]s.,_,/:.p)= ox,/-o.,_,:/Zi=0 r=l Lr L ar + brp j

(14)

The inversion is easily performed analytically, since

]= 1

to give the following solution for 1 < g < J - 1:

N_ _J-g

CJ_?.(Xt8) = _ exp(-aj_ix)O(s- vj_ix ) _ ;;;

i=0 r=l

x exp L[- arbr(s - vj_ix)] (15)

Thus, a closed-form analytical solution to the GIT

equation for constant properties has been derived.A formulation more fundamental and therefore more

appropriate for computational purposes is obtained

from equation (15).

As mentioned previously, some of the coefficients,

y_J-e are identically zero in order to write o_i,j_g(p )2,r '

as the simple summation of equation (12). This pro-

cedure dramatically increases the required computer

storage and also increases the potential for the ac-cumulation of roundoff error. These zero coefficients

can be conveniently eliminated by noting the physicalrequirement that

Cj_t(z, s) = 0 (16)

for x < s/vj and by interchanging the summationsin equation (15) to give

Ne

brr=l i=0

ar _,r:,<exp [-(aj-i- :_'j-i)z] YJ-' O(s- uj_iz) (17)

Equation (16) comes about because ion fragmentscannot be produced before the incident ion has trav-

eled to its position (x = s/vj) as mandated by the

continuous slowing-down theory. Since in most cases

of interest, vj > pal_ 1 for 1 < i < g, we have¢(x,s) = 0, where s > xuj > xuj_i. Also from

equation (17),

0=_ exp(-_s) ___r=l br i=0

x exp [-- (crj-i -- _rVJ-i) x] YJ-ei,r (18)

which must hold for all values of x and s that

satisfy the above inequality. Equation (18) is trivially

satisfied for x = 0 as a result of equation (13d) of the

5

recurrencerelationsfor YJ-¢ For equation (18) toi,r "

be satisfied in general, the following must hold ateach value of r and g, independently of x:

g

_[2 exp -(crj_i- _ruJ_i)x EJ-ei,r = 0 (19)i=0

One way for equation (19) to be a true statement is

for the coefficient of x to be identically zero. This

requirement, however, is obviously too restrictive. A

second possibility is that the coefficients vanish at

several values of i taken in pairs; when this is not the

case, YJ.-f should be zero. It can be observed thatr#

ar aro'J_il _]J_il -_ _rj_i2 tlJ_i2 (20)

br br

for two values of i(il, i2) at each value of r. Also, as

required by equation (19),

y J-e _y.J-t (21)ihr _ z2,r

values of it and i2 for 1 < r < 15 arc given in table II,

from which a recognizable pattern emerges.

By introducing equations (20) and (21) into equa-

tion (17), the expression for Cj-t(x, s) then becomes

r-----1

x (22a)

where

g-il,rJ-g =- YiJl,r g /br

The recurrence relation for YJ-g becomesil,r

(22b)

J-il_j_g AJ_g _kh,r = Xl,fi,r E /3J-g,k Yil,r h (N j_ k - r)

k=J-g+l

(23a)

for l<r<Nt_ 1 and

Nf-1

(vj-e,s- ) er, J-,eil,r = E k, br, ] it,r + flJ-g'd 8i'°

r_=l

(23b)

for N__ 1 + 1 < r < N_, where the sum in equa-

tion 23(b) is over the r t, for which f'J-,g gives all_r

nonzero contribution. A sample of these particular

values of r I and the parities Pr' are given in table III.

Equation (22a) is the expression for the analyticalsolution of the GIT equations to be evaluated. One

6

advantage of the analytical solution (eq. (22a)) in

the path-length variable s is that it is independent of

the energy range relation. Once the proton stopping

power is specified, then Cj_t(x, E) is obtained from

equation (3@ Additional quantities of interest, as

well as solutions to more comprehensive benchmarks

to be considered in the following sections, arc ob-

tained from the analytical form of Green's function.

2.3. Dose Profile

The dose profile (in x) for the ion J - g (ref. 2)

is given by integrating the total track length ofion j over all energy in a differential volume element

multiplied by the energy loss per track length a.sfollows:

f0 CX)Dj_e(x) = Aj_e dE Sj_g Cj_g(x, E) (24)

Recall that Aj_g is the atomic mass of the J - g

ion. Upon substituting the stopping-power scaling

law (eq. (2a)) and equation (3e) into equation (24),

Rpo(co)Dj_g(x) = Aj_t uj_g ds Sp(E(s)) ¢(x, s)

(25)where Rpo is the proton range in the shield material

for incident ions of energy Eo. When equation (22a)is introduced into the integral, the following closed-

form representation for the dose results:

N_

Dj_e(x ) = Aj_g _J-e _ exp(-_x) _<J-F,I,_l_(z) (26a)r=l

where

(265)1_ - a_/b_

[r (x) -- O(ro - vJ_il x) frjvdoq x ds Sp(E) cxp (-_rS)

rio- O(ro - uj_i.2x ) ds Sp(E)exp( f3rS)

t]J-i2X

(26c)

ro = min(Rpo, ujx) (26d)

To perform the integration either numerically or ana-

lytically, we are now faced with the task of specifyingthe range energy relation for protons.

Initially, a simplified range energy relation is spec-

ified so that the integrals Ir can be performed analyt-

ically to guarantee reliable benchmark results. This

relationtakesthegeneralform(ref.9)

s = Rpo - no ln(1 + al En°) (27)

The simplest case is for no taken as unity, which

yields a simplified stopping power

sp = exp[(s - Rpo)Z] (28)

where1

oqct o

1

O_o

The following prescription is used to determine the

parameters a and _:

(a) The more accurate proton stopping-power (called

the Wilson stopping power) relation (-dE/ds)given by equation (27) with (ref. 9)

no = 1.79 /

no = 500

al -- 3.66 x 10 _6

(29a)

is used to specify the proton range as follows:

Rpo = no ln(1 + al En°) (29b)

(b) Since, in the energy variable, the simplified stop-

ping power is

sp( ) = + BE (29c)

can be extracted as the slope

= [SpiEl) --Sp(E2)I/(E1 - E2) (29d)

given two values of stopping power Sp(E1) and

Sp(E2) at energies E1 and E2, respectively. Thesestopping powers are obtained from the more

comprehensive stopping power referred to in (a)above.

(c) If E = 0 when s = Rpo and E = Eo when s = 0,then

Eo

= exp (Rpo_) - 1 (29e)

gives a consistent value of c_ with fl and the proton

range.

When the stopping power, as represented by

equation (28), is introduced into the integrand of Ir

(eq. (26c)), the integrals can be performed analyti-

cally to give

a exp (Rpo,3)

It(x) - fir + fl e(ro - vJ_il x)

For the more comprehensive stopping-power formula

with no, C_l, and no specified by equation (29a),

the integrals must be performed numerically. This

is most conveniently done by transforming to the

energy variable. Then the integrals in equation (26d)are of the form

= fE(.j_i=)Ir'i JE(ro) dE exp [-3rs(E)] (31a)

where s(E) is given by equation (27) and

E(s) { exp [(Rp° - s)/c_°] - l } 1/n°= -- (31b)

c_ 1

Finally, Ir, i is evaluated by using a reliable quadra-ture scheme. The dose for the uneollided ions,

g = 0, is obtained by substituting equation (4a) intoequation (24)

_0 Rp°Dj(x) = Ajuj ds Sp(S) ¢j(x, s)

= Ajuj exp (-ajx) Sp(b'jx) O(Rpo - ujx)

(32)

for any proton stopping power Sp.

2.4. Source With an Energy Distribution

Since the solution for a beam source is actually

Green's function in the energy variable, a source for

ions of charge J impinging on the shield surface ofthe form

¢j (0, E) = 5ij f(E) O(Eo - E) (33)

can be accommodated. An energy cutoff Eo has

been imposed so that the path length given by equa-

tion (3d) will again have meaning.

Thejth ion flux for this source is therefore

/?¢_(x, E) = dE' I(E') Cj(x, E') (34)

where the explicit dependence of Green's function

Cj on the beam energy E' has been included. Theappropriate change of variable yields

¢;(x, E) = ¢;(z, s)/Sp(E) (35a)

and

where

f(E) = f(s)/Sp(E) (35b)

fO¢_(x, s) -- ds' f(s - s') Cj(x, s') (35c)

Then, by substituting equation (22a) into equa-

tion (35c), the closed-form representation of the fluxfrom a distributed source is

Nt

¢3_e(x,_) : _ exp (-_rZ) U -e Jr(x, _)1,r

r=l

where

(36a)

roJr(x, s) _ O(r o - uj_qx) dJ f(s - st)exp (-13rs')

a xPj_ii

f_ TM- O(_o- .j__) ds' f(s - s')exp (-_')

,vj _ i2

(365)and

ro = rain(s, xuj) (36c)

Finally, for the incident ions (g = 0) from equa-

tion (4a),

¢_](x, s) = exp(-ajx) f(s - ujx) O(s - udx) (36d)

To generate benchmark results, we consider, as anexample, a source of the form

f(s) = Qo exp(-Mos) (37a)

or in the energy variable

f(E) = Qo exp [-Mos(E)] (3Zb)S_(E)

yields

8

Jr(X,_) Qo exp (-Mos) (O(ro - ug_it x)Mo - fl_

× {exp[(Mo-_r)ro}-exp[(Mo-_r)XVj_il]}

- O(_o- _j_,_) {¢xp[(Mo- _)_o1

-_p [(Mo - _)z_j_,_] }) (37c)

For the source ions,

¢_(x,s) = Qo exp[-(aj - Mouj)x] exp(-Mos) O(s- ujx)

(37d)

A more general class of analytical sources can also

provide benchmark flux profiles. If a source f(s) pos-

sesses an explicit Laplace transform f(p), then since

the integral in equation (35c) is of the convolutiontype, we find after application of a Laplacc trans-form that

-_(x,p) = 7(p) _j(x,p)

The transform of Cj(x,s) is easily obtained fromequation (22a) as

Nt

r=l

l a_+P _+P J(38)

Then, upon inversion of equation (38),

_;(x,_)-- _ _xp(-_) Vs-eil,r

r=l

× [exp(-]3rb'J_ilX)Lr(s - uJ_ilX ) O(s - uJ_ilX )

-exp(-_ruj_i2z)Lr(s - uj_i2x) O(s - uj_i2x)]

(39a)where

{ ] (39b)

If the inversion £:x I can be performed analyti-

cally, then accurate benchmarks can be generated.

In most cases, however, the transform cannot be

performed analytically. For a general f(p), it is

possible to perform the inversion numerically by

using a recently developed numerical-inversion algo-rithm (ref. I0). Since numerous numerical inversions

would have to be performed for a heavy incident ion

(J > 10), the required computational time would

be c_nsiderab'le. For lighter incident ions (J < 10),

the above method could be applied for a reasonable

computational effort. Of course, an even more gen-

eral class of sources could be accommodated by using

aquadratureschemeto evaluateJr(x, s) as was donefor the dose.

2.5. Composite Beam With DistributedSources

Another generalization of the beam source is the

composite beam. In this case, several ions of different

charge are allowed to impinge on the shield surface

with the same source incident energy per nucleon of

Eo (same velocities). The GIT equations (eqs. (3a

and 3b)) then become

J

LjCj(x,S) = E f3jk Ck(x's) (1 _< j _< J) (40a)k=j+l

and

where

J

¢,(0,s) = E dy 6,J' _(_)jl=l

0 0

Lj :-- _x T t/j-_s T crj

and dj is the abundance of ion j in the beam.

(40b)

Since the GIT equations are linear, the composite beam can be decomposed into its individual ion

components with a source at x = 0. Thus,

k=j+l

J-1

LJ-I¢J -l(x'8) ---- E _jk sJ-l(x, 8)

k=j+l

2

k=j+l

L 1 ¢_(x, s) = 0

¢](o, _)

CJ-l(o,s)

¢_(0,_)

,_(o,_)

= dj 5iJ 5(s)

= da-1 5j,j-1 _(_)

=d2 552_(_)

= dl _a_(_)

(41)

where the superscript indicates the incident ion

charge. When equations (41) are summed, weobtain

and

J-1 J-I J-1

L3 E ¢_-e' (x,s)= E _jk _ ePJ-e' (x,s) (42a)tl=O k=j+l _l=O

J-1 J-1

g_=O t _=0

J

= _ # % _(_)j'=l

(42b)

where use was made of the requirement that

CJ-e' (x, s) = 0

for j _> J - g'. Therefore, because of the uniqueness

of the GIT equations, a comparison of equations (40)

and (42) yields

d-1

Cj(X,8) = E CJ -t'(x's) (43)e'=O

Since CJ-t'(x,s) is Green's function, the sum in

equation (41) is obtained from equation (22a). Sim-

ilarly, for a distributed source, the sum in equa-

tion (43) is obtained from equation (36a).

2.6. Distinction BetweenNeutrons andProtons

Ascurrentlywritten,theGIT equationstreatallthe ionswithamassnumberofoneasprotons.Theyall haveanassociatedstoppingpowerandrange.Tomakethe GIT equationsmorerealistic,a neutroncomponenthasbeenadded.Neutronsaregiventhechargenumberzero(j = 0), andthe equationthatdescribestheir straight-aheadneutronmotionis

J

(0)_x + _rO ¢O(X' s) = E MO, kakCk(x' s) (44)k=2

where v0 = 0 and where it is assumed that protonscannot act as a source of neutrons. The absorptioncross section is taken to be the same as for protons,

and the nmltiplicities are assumed to be a fraction

fn of those for protons as follows:

M0,1 -- 0

JM0,k = AMI,k (k >__2)(45)

The proton multiplicities are then modified by a

factor 1 - fn.

The formulation also assumes that the neutrons

are monoenergetic and do not have scattering inter-actions. The inclusion of neutrons here is done only

for completeness; a more substantial model is cur-

rcntly under development.

3. Numerical Implementation

3.1. Nuclear Fragmentation Model and

Stopping Power

To numerically evaluate the ion flux and dose

for bcnchmarking purposes, the following simplifiednuclear model is assumed:

2

aj = _ j_ (46a)

2 (k > j)

g--z--r

Mjk=

o (k < j)

(46b)

The choice of aj is based on nuclear liquid drop modelconsiderations. The multiplicities are chosen so that

charge is conserved in each interaction. The cross-section normalization is an input parameter that is

taken to be representative of an air shield as follows:

= 0.01247 cm2/g

10

Unless otherwise stated, the simplified stopping

power is used in the computations of this report. The

values of energy for the determination of/3 from equa-

tion (29d) were taken as E1 = 10 MeV/nucleon andE2 = 10 "t MeV/nucleon, which provided a represen-

tation over the energy interval of interest.

3.2. Round-off Error

Because the solution for the ion flux that results

from an incident beam is an analytical representa-

tion, discretization and truncation errors arc not rel-

evant to the numerical evaluation. However, a veryreal concern is the error that results from round off.

The central numerical procedure used to evaluatethe flux is the recurrence relation for the coefficients

y J-e (eqs. (23)), which is rather involved. It is well-'flit

known that significant round-off error can accumu-

late when using recurrence relations. For this rea-

son, the simplifications described in section 2.1 play

an important role in round-off error mitigation.

The major cause of round-off error can bc tracedto the r summation in equation (22a). Table IVshows each term calculated in Control Data Corpora-

tion (CDC) double-precision arithmetic (_29 digits)and the corresponding partial sums of the summa-

tion over r in equation (22a) for x = 16.004 g/cm 2,

E = 100 MeV/nucleon, and j = 16. The incident

ion is nickel (J = 28) at Eo = 5 GcV/nucleon. Thesummation is remarkable, in that the partial sum os-cillates between ±1011 with a final result of _10 -3.

Thus, a precision of at least 14 digits is required for

this computation to have one correct digit. The cor-

responding single-precision calculation fails to give a

positive flux. The failure of single-precision arith-

metic is expected because of the significant loss of

precision. All computations are performed in double-

precision arithmetic. Even though round-off errorwill cause failure of the r sum for the light ions pro-

duced from the heavy incident ions for J > 30, there

is no apparent ill effect for J _< 30. If necessary

and available, quadruple-precision routines could be

employed. Round-off error is also expected to limitthe dose and distributed source evaluations to only

heavy ions when the integrals Ir and Jr cannot be

performed analytically.

4. Results

The flux and dose expressions determined in sec-

tion 2 are numerically evaluated for air shields in

this section by using the nuclear model and stopping

power discussed in section 3.

4.1. Flux Profile From a MonoenergeticBeam Source

According to the continuous slowing-down ap-

proximation, the incident ion that impacts the shield

surface with energy Eo has lost a known amount of

energy after having traversed a given path length s.Thus, at a particular energy El, a fraction of the in-

cident ions given by exp(-ajx) have survived to the

distance xj = sl(E1)/uj, where Sl is determinedfrom equation (27). At this point, o'jCj(x, 8) dx in-

cident ions/sec per cross-sectional area fragment intolighter ions (0 _< j < J - 1). Because of their finite

velocities, the secondary ions of charge J - 1 that

originate at xj can then propagate to a maximumrange of x j-1 = sl(E1)/uj-1 at this energy (fig. 2).

The flux profile decays with x because there is no

source of secondary ions between the creation and

allowable range. The fluxes Cg(x, s) for several ener-

gies E1 < E2 < ... < E5 are displayed in figure 2(b).

In keeping with the range energy relation, the region

for which Cg_l(x, E) is nonzero moves farther into

the shield with decreasing energy (increasing path

length). Also, the flux decreases with decreasing en-

ergy, because the penetration distances are greater

at lower energies; these greater distances allow moreof the J - 1 ions to fragment into lighter ions. Fig-

ures 3(a) and 3(b) show the variation of Cj-l(x, E)with E at a fixed positron x and at several positions

Xl < x2 < .. • < x5. The decrease in the flux at lower

energies is clearly evident.

In the region Sl(E1)/r'j __ x <_ Sl(E1)/r'y_l,

the secondary ions (J- 1) continuously generate

lighter ions. Figure 4(a) shows both Cj_l(x,E)

and Cj_2(x,E) at energy El. In the region where

ions J- 2 are produced (Sl/Uj < x < Sl/Uj_l),the flux increases as it should. When the source

abates, the flux decreases until the ions reach their

allowable range x i_ 2 = Sl/r_j_ 2. At Xj_l, the slope

of ¢J-2 changes dramatically (being infinite), which

indicates the reduction of the source abruptly to zero.

Figure 4(b) shows the secondary and tertiary ionfluxes for several energies (El < E2 < ... < Es).

Considering the lighter ions (J - 3, J - 4, J - 5)

with representative fluxes shown in figures 5(a) and

5(b), we observe similar behavior. In the region

Sl/r,j <_ x _ 8liP j-l, where ions J - 1 and J - 2produce ion J - 3, the flux increases as a result of

the increasing source. The slope again changes when

the source from Cj_l becomes zero. Because of the

production of ion J - 2 in the region Sl/Vj_ 1 "_ x <

81/vJ_2, and because of the attendant competition

with loss from fragmentation, the flux for ion J - 3

initially increases but then decreases. In the region

of no J - 3 ion source (8/lzj_ 2 "_ x < 8lib'J_3), the

flux is monotonically decreasing. The flux profiles

become increasingly smooth for the lighter ions. Thisincrease is a consequence of the smallei _ fraction

that the discontinuous source represents of the total

source at xj_i, 1 _< i < g - 2 for ion J - e. In

addition, at xj_g+l, where the only source is from

the preceding ion, attenuation has reduced the sourcecontribution significantly. However, a kink in the

¢J-5 flux curve can still be observed at x j_ 4.

When neon ions of energy 1 GeV/nucleon are

incident on the shield surface, the resulting fluxes

of the ions produced by fragmentation are shown

in figures 6(a) to 6(e) for E = 995 MeV/nucleon,

950 MeV/nucleon, and 100 MeV/nucleon. For the

energy nearest the source energy, E = 995 MeV/

nucleon, the fluxes are a series of waves of almost

uniform density that originate at x _ 0.6 and extendto the maximum range of the proton (j = 1) at this

energy. Since the ions have penetrated only a shortdistance into the shield, significant fragmentation has

not yet occurred. For this reason, the maximumflux for each successive ion is less than that of the

preceding ion until j = 1. Because all ions for

which j > 1 can produce protons, the proton source

starts to become significant and results in an increase

in the flux near the end of the proton trajectory.

This increase in fragmentation is clearly evident at

950 MeV/nueleon; these are indicated by significant

increases in the fluxes of light ions j = 1, 2, 3. At

100 MeV/nueleon, the ions have almost penetrated to

their maximum ranges (309.75/vj), and the light ionsnow have the largest fluxes. Figure 6(d) displays the

three energies and clearly shows the changing natureof the particle flux with penetration that results from

fragmentation. Figures 7(a) and 7(b) show the fluxenergy dependence at several positions and show that

the energy spectrum broadens with penetration.

The incident ion energy influences the flux profiles

significantly, as shown in figures 8(a) and 8(b), where

profiles at E = 100 MeV/nucleon for neon ions of in-

cident energies 500 MeV/nueleon and 2 GeV/nucleonare shown. The heavy-ion flux is still present for an

incident energy of 500 MeV/nucleon; however, for

2 GeV/nucleon, so much penetration has occurredthat the heavy ions have almost all fragmented and

left a negligible heavy-ion flux.

The cross-section normalization has a major effect

on the profiles. Figures 9(a) and 9(b) show the

flux profiles at 995 MeV/nucleon for neon incident

at 1 GeV/nucleon with cross-section normalizationsof 10 .4 cm2/g and 1.0 cm2/g. The normalization

is proportional to the shield density. Thus, for

the rarefied shield (_ = 10-4), the ions behave

as expected with only a small amount of light ion

11

productionbecauseof therelativelyfewinteractionsthat causefragmentation.However,for the denseshield(_ = 1.0),a significantflux of light ionshasalreadybeengeneratedafter a penetrationof only3g/cm2.

To demonstratethe generalapplicabilityof theanalyticalsolution,the flux profilesfor E -- 4.995

and 2 GeV/nucleon for nickel incident on a shield

at 5 GeV/nucleon are presented in figure 10. To

obtain better resolution for the 2-GeV case, ¢j for8 _< j _< 27 and 1 _< j < 7 are plotted on separate

plates. With CDC double precision, incident ions of

charge up to _30 can be considered. For J > 30,

the round-off error becomes a problem and higherprecision routines must be used at the expense of

storage and computational time.

4.2. Dose Profiles From a MonoenergeticBeam Source

The dose profiles given by equations (26a) to

(26d) for the stopping power of equation (27) are

evaluated here. Figures 11 to 13 show the dose

profiles for incident neon beams of energies 670 MeV/nucleon, 2 GeV/nucleon, and 10 GeV/nucleon; part a

of each figure presents the individual ion doses and

part b the normalized total dose D(x)/D(O). In each

case, the Bragg profile for the incident ion is clearly

seen. As the incident energy increases, the maximum

flux for the lighter ions moves farther into the shield

and away from the Bragg cutoff at Rpo/V J. Thismovement causes the total dose to extend past the

Bragg cutoff.

Unlike in the previous cases, when 5-GeV/nucleon

cobalt ions axe the incident ions (figs. 14(a) and14(b)), a significant amount Of the dose occurs after

the Bragg cutoff (_105 g/cm2). This is a result of the

larger number of fragments produced and the deeperpenetration of these fragments at a given energy per

nucleon. Also, the larger number of ions gives riseto an increase in the dose near x = 0, rather than adecrease as seen for incident neon.

The following dose calculations were performed

with the more realistic stopping power given by

equations (27) and (29a). The integrals in equa-

tion (26c) were performed numerically by using a

reliable Romberg integration routine (ref. 6). For in-

cident neon of 670 MeV/nucleon, figure 15(a) shows

the familiar Bragg peak and cutoff, and the contribu-

tion of the lighter ions is shown in figure 15(b). The

profile of ion j = 9 seems to follow the dose profile of

the incident ion near the Bragg peak. This aspect ofcourse results from the simple fact that the incident

ion is the sole producer of ion j = 9. As j decreases,

12

the maximum dose for each value of j moves far-

ther into the medium away from the Bragg peak as

the production of each subsequent ion becomes less

dependent on the source ion. The similarity of thedose profiles for ions j = 9 and 10 is even more strik-

ing for an incident energy of 1 OeV/nucleon. (See

figs. 16(a) and (b).) For higher incident energies

(figs. 17(a) and (b)), the Bragg peak dissipates asa result of the increased fragmentation at larger pen-

etration distances. Figurc 18 presents the total dose

for the previous three cases. The Bragg peaks again

stand out for the lower incident energies. Figures 19

and 20 show the doses for nickel incident at energies

of 1 GeV/nueleon and 5 GeV/nucleon, respectively.

The curves represent only ions 18 _< j _< 28, because,

for j _< 18, catastrophic round-off error accumulationoccurs and limits the evaluation.

As an example of the use of the analytical solu-

tion as a predictive tool, the experimental dose from a

neon beam incident on water (ref. 11) was compared

with that obtained from the analytical solution. The

comparison of the normalized doses is shown in fig-ure 21. The curves are normalized to the dose at

x = 0, and cross-section normalization _ was ad-

justed slightly to obtain a reasonable fit to the data.

If desired, the fragmentation parameter could also be

adjusted to obtain an even better fit.

4.3. Distributed Sources

Numerical results are now presented for analytical

and solar distributed sources. An analytic source

is one that allows the integral value of Jr to be

expressed in closed form; the solar source requires

a numerical quadrature evaluation.

4.3.1. Analytic source. Figures 22 and 23show the flux profilcs for neon ions that are inci-

dent with a distributed source in s (or E), given

by equation (37a) or 37(b), and a source cutofftaken at 104 MeV/nucleon. For small values of Mo

(0.01) and an energy very near the cutoff (E =

9999.9 MeV/nucleon), the incident ion source dom-

inates all other sources from fragmentation. Thus,

in the region where the incident ion has a nonzero

flUX (0 < X < 8�u j), all lighter ion flux profiles are

essentially identical. For this energy, the ions haveonly penetrated _0.01 g/em 2, and significant frag-

mentation has not occurred. As the energy decreases

(s increases) to 1 GeV/nucleon, the lighter ions build

up similar to the beam source.

Similar behavior is observed in the flux profiles

for a source that is more peaked at s = 0 (Mo =

1.0). As shown in figure 23(a), the profiles are

! m

virtually identical to the preceding case, since over

the very short penetrations both sources look alike.

For an energy of 1 GeV/nucleon (fig. 23b) the sourceresembles a delta function because of its rapid decay

in s, and the profiles look like those for a beam source.

4.3.2. Solar source. A source of the form

f(E) = Qo/(bl + E) n

is typically taken to represent solar ions from the

Sun. The Jr integral in equation (37c) cannot be

performed analytically, so the numerical Romberg

integration must be used. (See ref. 12.) Becauseof the excessive computer time required for these

calculations, only a few ions (<10) can be considered.

Figures 24(a) and (b) show the resulting ion fluxes

(to j = 19) as a function of E for incident nickel with

Q0 = 28446, bl = 1000, and n = 2.5 at positions

x = 1 g/cm 2 and 10 g/cm 2. As the distance into the

shield increases, the lighter ion fluxes again increasebecause of fragmentation. All ion fluxes seem to have

an energy variation similar to the incident ion source;

this similarity suggests that a scaling law may apply.

4.4. Composite Beam

Figures 25(a) to 25(c) show the flux profiles using

the Wilson stopping power for an artificial cosmic ray

beam composed of the elements neon through ironwith the assumed abundances given in table V. The

jagged appearance of the flux profiles near the source

energy 4.9 GeV/nucleon is a result of the relativelyshort penetration distances and the associated lack of

fragmentation so close to the source energy. For this

reason, the profiles for the ions with a charge nearthat of the incident ion still reflect the source distri-

butions. The lighter ions, however, have smoother

flux profiles as more fragmentation occurs at lower

energies.

4.5. Generation of Neutrons

The parameter fn in the cross-section model

(eq. (45)) has been introduced to make the distinc-

tion between the neutron and proton components.

(See section 2.6.) In figures 26(a) and (b), the re-

sulting ion fluxes at E = 500 MeV/nucleon for an in-

cident neon beam of Eo = 1 GeV/nucleon are shownfor fn = 0 and 0.5. In addition, the Wilson stop-

ping power has been used. As expected when fn is

increased from zero to one, the neutron component

also increases with an attendant decrease in the pro-

ton component. The neutron component does not

go to zero at a finite distance as for charged par-

ticles. After a distance equal to the proton range

(_200 g/cm2), the neutron flux decays exponentially

(not shown), since there is no longer a neutron source

in this region--only absorption.

4.6. Comparisons With NumericalSolutions

4.6.1. Finite-difference/power series ex-

pansion solution. For G intervals, if the GIT

equations are discretizcd in the path-length variable,

equations (3a) and (3b) become

3

+ Z Z_ O_(x) (47a)k=j+l

¢gj(o) = fj _jj (9 - 1,2 ..... G) (47b)

The advantage of the discretized model is that

energy-dependent cross sections and multiplicities

can now readily bc included. The solution to equa-

tions (47a) and (47b) is most easily obtained by ap-

plying the power series method, where the expansion

g,rhu'J (x - ar-1) (48a)

r_=0

is introduced into equations (47) to yield the follow-ing recursion relation for h g'r"

n_ 3 •

h g'r. =- aj,g+ +

J

+Z _gk=j+ 1

= ¢n,j (ar-1)

(4Sb)

The particular form of equation (48a) is valid for

x in the interval [ar-l,ar]. For incident neon at

Eo -- 670 MeV/neutron, a flux comparison of the

above interval solution with the analytical solution(eq. (22a)) is given in figure 27. The flux energy

spectrum is shown at a position of 10 g/cm 2 from the

surface. Even for relatively large values of As, the

proton flux (j = 1) is in relatively good agreement;

there is poor agreement for the heavy ions. This

poor agreement is probably a result of the highly

singular nature of the incident beam (eq. (4a)), which

leads to discontinuous distributions for ions 9 and 10;these ions are poorly simulated with finite differences.

However, as As is reduced, the general agreement

13

improves.Anotherareaof disagreementis at tilebeginningandendof the ion travel,wherethefluxfallssharplyto zero. Because of numerical diffusion,the finite-difference solution does not exhibit step

function behavior, but the fluxes do become steeper

with increasing As. The finite-difference solution

does, however, give the correct flux average at thesediscontinuities as As decreases.

4.6.2. Integral transport finite-difference

solution. Our final comparison is with the GCR

code (ref. 11) currently in use at Langley ResearchCenter and Johnson Space Center. The GCR code

solves the GIT equations by integration along the ion

trajectories on a spatial and energy grid. From pre-vious benchmark comparisons, the code is expected

to be accurate to better than 10 percent.

Figure 28 shows the flux profiles for the GCR codeand benchmark for a distributed source of neon of the

form

f(s) -- exp(O.Ols)O(lO 5 - E)

impinging on an air shield. Tim nuclear fragmen-tation model specified by equations (46) was used

along with the simplified stopping power given by

equations (29). To provide a contrast, the bench-

mark calculation begins at E = 20 MeV, whereas

the GCR calculation begins at E -- 100 MeV. Thetwo evaluations are in excellent agreement even at

100 g/cm 2. Figure 29 shows the flux profiles for

incident nickel with the same source; these profiles

again indicate virtually complete agrccmcnt. Thesecomparisons provide added confidence that the GCR

code employs a very accurate transport algorithm.

5. Summary and Recommendations

Based on its success in generating previous bench-mark solutions, a straightforward analytical ap-

proach has been adopted to solve the full (lIT

equations. As a result, a closed-form analyticalrepresentation of the ion fluxes that result from a

monoenergctic incident ion source has been derived.While there are no truncation errors to contend with,

round-off error does present a problem. The prob-

lem becomes serious only when the incident ion is

very heavy (J >_ 28) and when the energies are nearthe source energy; otherwise, 64-bit, double-precision

arithmetic adequately allows the evaluation of the

flux representation. The dose profiles can also be

obtained from the analytical representation with the

additional evaluation of an integral.

Since tile flux for the beam source is actually a

Green's flmction, the flux profiles for sources dis-

tributed in energy can bc obtained by integration

over the source energy. This integration, unless it

is performed analytically, introduces round-off errorcontamination that may limit the total number of

isotope flux profiles that can be evahmted. A more

general beam source composed of several ions withthe same velocity can also be accommodated.

Finally, a distinction between neutrons and pro-

tons is made to more correctly take into account the

attenuation of neutrons. The neutrons, however, are

still monoenergetie, and therefore do not contributeto the overall dose. It is this last point that deservesfurther attention.

An improved neutron flux model, including scat-tering interactions, should be the first step of a fu-

ture effort. By using tile ions where j >_ 3 to specify

a neutron sourcc, it is possible to formulate a more

realistic Boltzmann equation for the neutron compo-

nent. A multigroup treatment can then be appliedto this equation both with and without the straight-

ahead assumption. In this way, the neutron energyspectrum can be obtained either by a discretization

in the spatial variable or a discrete-ordinates formu-

lation in the angular variable. Once angular depen-dence has been introduced into the _ormulation, two-and three-dimensional shields can be considered.

In addition to continuing to develop benchmarks

for more realistic problems, such as including en-

ergy dependence in the cross sections or angular de-

pendence, the benchmark solutions developed thus

far should bc compared with existing transport al-

gorithms to establish a common basis of compari-son for these algorithms. The benchmark solutions

should be considered as a package that includes the

energy-independent, spatially independent and cou-pled energy-space solutions. Each solution should be

checked to verify each specific part of a transport

algorithm.

NASA Langley Research CenterHampton, VA 23665-5225August 26, 1991

I

!

|i

!

_=

14

6. References

I. Wilson, John W.; and Townsend, L. W.: A Benchmark

for (_alactic Cosmic-Ray Transport Codes. Radiat. Res.,

vol. 114, no. 2, May 1988, pp. 20i 206.

Ganapol, Barry D.; Townsend, Lawrence W.; and Wil-

son, John W.: Benchmark Solutions for the Galactic

Ion Transport Equations: Energy and Spatially Dependent

Problems. NASA TP-2878, 1989.

Wilson, John W.: Analysis of the Theory of High-Energy

Ion Transport. NASA TN D-8381, 1977.

Alsmiller, R. G., Jr.; Irving, D. C.; Kinney, W. E.;

and Moran, H. S.: The Validity of the Straightahead

Approximation in Space Vehicle Shielding Studies. Second

Symposium on Protection Against Radiations in Space,

Arthur Reetz, Jr., ed., NASA SP-71, 1965, pp. 177 181.

Alsmiller, R. G., Jr.; Irving, D. C.; and Moran, H. S.:

Validity of the Straightahead Approximation in Space-

Vehicle Shielding Studies, Part II. NucI. Sci. 8,_ Eng.,

vol. 32, no. 1, Apr. 1968, pp. 56 61.

Wilson, John W.: Heavy Ion Transport in the Straight

Ahead Approximation. NASA TP-2178, 1983.

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11.

12.

Wilson, John W.; Townsend, Lawrence W.; Nealy, John

E.; Chun, Sang Y.; Hong, B. S.; Buck, Warren W.;

Lamkin, S. L.; Ganapol, Barry D.; Khan, Ferdous; and

Cucinotta, Francis A.: BRYNTRN: A Baryon Transport

Model. NASA TP-2887, 1989.

Wilson, John W.; and Lamkin, Stanley L.: Perturbation

Theory for Charged-Particle Transport in One Dimension.

Nucl. Sci. _ Eng., vol. 57, no. 4, Aug. 1975, pp. 292 299.

Wilson, John W.; and Badavi, F. F.: Methods of Galac-

tic Heavy Ion Transport. Radiat. Res., vol. 108, 1986,

pp. 231 237.

Ganapol, B. D.: The Solution of the Time-Independent

Neutron Slowing Down Equation Using a Numerical

Laplace Transform Inversion. Transp. Theory _ Star.

Phys., vol. 19, no. 1, 1990, pp. 1 27.

Wilson, John W.; Townsend, L. W.; Bidasaria, H. B.;

Schimmcrling, Walter; W'ong, Mervyn; and Howard,

Jerry: 2°Ne Depth-Dose Relations in Water. Health Phys.,

vol. 46, no. 5, May 1984, pp. 1101 1111.

Miller, Alan R.: FORTRAN Programs for Scientists and

Engineers, 1st ed. SYBEX, Inc., 1982.

15

o

Aj

ai

agm

al

b_rn

bl

Dj

a_E

f_

L

g_r

hn,j

Ini

J

Jr

J

Lj

Lr

Mjk

N_

N.

Q(x,s)

Qo,Mo

R

Symbols

mass number of jth ion

position of ith interval boundary,

g/cm 2

---- (_ -- (7 m

-------if J-1 - ffJ

-- vt - v m

_d-1 --Ud

dose due to jth ion, MeV/g

abundance of jth beam ion

energy per nucleon, MeV/nucleon

monoenergetic beam source term

Laplace transform of fj

power series coefficient

defined by equation (26c)

defined by equation (31a)

charge number of incident ion

defined by equation (36b)

charge number of jth ion

defined by equation (40a)

defined by equation (39b)

multiplicity of ion j produced byion k

defined by equation (12)

number of intervals

volume source coefficient

parameters in equation (37a)

range, g/cm 2

ro

Sp

_(E, Eo)

x

y.kzJ

Ot

aij

O_o,o_1, Tto

As

5(s)

5ij

Ar

defined in equation (26d)

stopping power of jth ion,

MeV-cm2/g

proton stopping power,

MeV-cm2/g

proton path length, g/cm 2

position, g/cm 2

defined by equation (Tb)

defined by equation (22b)

._. ° 1

-- OtlO o

recursion coefficient

parameters used in equation (27)

=J_-- O_O

= Mjk_k (eq. (3c))

finite-diffcrencc interval, g/cm 2

Dirac 5-function

Kronecker delta

Heaviside step function

defined by equation on page 4

z?

-- _ (eq. (2b))

cross-section parameter, cm2/g

macroscopic absorption cross

section for jth ion, cm2/g

flux of jth ion

Laplace transform of Cj

defined by equation on page 5

16 =

i

Table I. Relation of r Index to g

r g m n Ne

1234

56789

10

J-1J-2J-2J-3J-3J-3J-4J-4J-4J-4

JJ

J-1J

J-1J-2

JJ-1J-2J-3

13

10

Table II. Variation of il and i2 To Satisfy Equation (20) With r

g i2

4

r il

1 02 - 03 14 05 16 27 08 19 2

10 311 012 113 214 315 4

12233344'44

55555

17

Table Ill. Nonzero Contributions to Sum in Equation (23b)

18

i=0

i=2

i=4

r !

01

2

4

7

11

2

36

9

13

18

7

8

9

10

15

20

PT !

-1

-1

1

1

1

1

-1

-1

-1

-1

11

i=1

i=3

i=5

T !

35

8

12

17

4

5

6

1014

19

8

9

1011

12

16

PF

-1

1

1

1

1

1

-1

-1-1

1

1

1

-1

-1

-1

-1

-1

1

TableIV. PartialSumsin Evaluationof Equation(22a)

[r=l

r % Sm r % S,n

37

38

3940

41

42

43

44

45

46

47

48

49

50

51

5253

54

56

57

58

59

6O61

62

63

64

67

68

697O

71

72

73

74

75

3.9228E+07

9.7776E+081.I020E+10

-1.5280E+06

8.7956E+06

-1.5549E+02

1.3975E+02

1.2963E+01

1.7750E+01

- 1.7809E+03

-3.3768E+04

1.6875E+04

2.2094E+03

-1.1694E+05-5.3195E+01

1.3282E+05

6.4344E+02

-3.7529E-02

-3.8850E+06

-5.6258E+09

2.7668E+08

-1.1491E+11-2.0332E+10

-1.7720E+05

1.9891E+07

6.6318E+10

-5.9047E+03

- 1.3597E+10

-2.5725E+081.8921E+08

1.0403E+07

-3.9645E+07

-5.2003E+02

1.1232E+04

-7.1222E+047.3132E+07

3.9228E+07

1.0170E+09

1.2037E+10

1.2035E+10

1.2044E+ 10

1.2044E+ 10

1.2044E+101.2044E+ 10

1.2044E+10

1.2044E+ 10

1.2044E+10

1.2044E+10

1.2044E+10

1.2044E+10

1.2044E+10

1.2044E+10

1.2044E+101.2044E+10

1.2040E+10

6.4145E+09

6.6911E+09

-1.0821E+11

- 1.2855E+ 11- 1.2855E+ 11

- 1.2853E+ 11

-6.2209E+10

-6.2209E+10

-7.5806E+10

-7.6063E+10

-7.5874E+ 10

-7.5864E+10-7.5903E+ 10

-7.5903E+10

-7.5903E+10

-7.5903E+ 10

-7.5830E+ 10

79

80

81

82

83

84

85

8687

92

93

94

95

96

97

98

99

100

106107

108

109

110

111112

113

114

121

122

123

124

125126

127

128

129

-1.1811E+08

-8.1489E+09

1.2471E+10

1.6089E+ 111.6572E+10

3.8245E+05

2.2172E+08

- 1.1062E+ 11

1.2098E+06

-7.5948E+10

-8.4097E+10

-7.1626E+10

8.9259E+10

1.0583E+11

1.0583E+11

1.0605E+11

-4.5652E+09-4.5640E+09

2.2978E+ 10

- 1.8297E+09

-7.1510E+09

- 1.6057E+08

3.5538E+08

8.8145E+04

- 1.7218E+062.4350E+07

- 1.4887E+08

1.3371E+08

1.2474E+10

-1.1917E+10

8.1157E+08

5.9305E+092.4703E+04

-4.6283E+06

-2.2498E+09

- 1.3697E+07

-9.4346E+09

2.3988E+09

-4.8884E÷096.2213E+09

-2.4306E+09

- 3.1752E+05

-2.3539E+08

-6.3879E+09

9.0967E+07

1.8414E+10

1.6584E+10

9.4334E+09

9.2729E+09

9.6282E+09

9.6283E+09

9.6266E+09

9.6510E+099.5021E+09

9.6358E+09

2.2110E+10

1.0192E+10

1.1004E+10

1.6934E+10

1.6934E+10

1.6930E+101.4680E+ 10

1.4666E+10

5.2317E+09

7.6304E+09

2.7420E+09

8.9633E+09

6.5326E+09

6.5323E+09

6.2969E+09-9.0967E+07

9.7023E-04

19

TableV. Composite Cosmic Ray Beam*

Element

Fe

Mn

Cr

VTi

Sc

Ca

K

Ar

C1

S

P

SiAI

MgNa

Ne

Fet

Abundance, percent

1.3

0.61.7

1.4

2.0

2.0

1.8

2.5

1.82.6

2.5

5.1

7.2

11.5

16.314.7

23.1

1.9

• Portion of cosmic ray spectrum from neonto iron.

t Added to make 100 percent.

2O

S

X

Figure 1. The x, s representation of nonzero region of jth ion flux Cj(x, s).

21

22

¢J-1 (x,E)

Cj_l(X, E)

xj_ 1 = Sl/Vj_ 1

x

(a) Energy Ei.

i'xii• |

iii |

i iEs

II| !! !

!!II

I!iii i

i

i

IIi |

'iI' 1!

I I | I

E3

I- -I E2I , ! .... ',

(b) Energies E1 < E2 < ... < E5,

Figure 2. Variation of flux for ion J - 1 with x.

E1| !

r

_J-1 (x,E)

E

(a) Position x].

¢J-1 (x,E)

x 2

x4 x3 _], r

E

(b) Positions x] < x2 < ... < xs.

Figure 3. Variation of flux for ion J - 1 with E.

x 1

23

_j(x,E)

J=J'17 . .--"

\

\

\

I!

I!

I!

I!

_-- xj = Sl/V J xJ-I = Sl/VJ-1 -_ xj_ 2 = Sl/Vj_2- _

(a) Energy El.

24

_j(x,E)

P,

I,l

I'iI

I!

I!

I!

I!

I!

I!

II

E4

F...-..--.,_,.,..\ \

(b) Energies E] < E2 < ... < ES,.

Figure 4. Variation of flux for ions J - 1 and J - 2 with x.

Cj(x,E)

/ \ \

i.Ii l \

/" J- , k

/ / \ ',i \/ \ \

// \ \

I

/ I1

I!

IiiI

I

\\

\\

X

(a) e=l, 2, 3.

\\

\

Cj(x,E)

/

,4'J "",,-/, ,,. _ ',

k, /I,'1_ _ '\, /, ,- , • \

X' /' ,,'1_ \ '

I: / _/, ! -,/ \

'/ \, \

II

, I

\

\\

\

\I

\ \

Ii \

I I _,

3',,,,f J-4!

I

!

!

\

\

(b) e=_, 2,..., 5.

Figure 5. Variation of flux for ions J - g at energy El.

5

, I

25

.03 -

Cj(x,E),arbitrary .02 -

units

.01 -

o.5

_.._j =7

__J...cJ=G= ,, fj=5 .

--I---,---z'__, /"J =4 j= 1b..._.-....J....._..:...-..--,.--._---_z-....._..._............... ..,_-_j ............ .i-I-,

I ' II I

I '._ I I _ I I,,

1.0 1.5 2.0 2.5 3.0x, g/cm 2

(a) E = 995 MeV/nucleon.

.025

.020 -fj =9

I

.015 I

Cj(x,E), I : 7 /-J_.. .............. "-.arbitrary .4-,,_j = i.. - ...

units.olo- _Fj =.s.... /-j-2 .....

"II "IL . L. "-

. ' I i I I .. I I' i05 ' I I ,10 15 2O 25 3O

x, g/cm 2

(b) E = 950 MeV/nucleon.

Figure 6. Variation of Cj(x, E) with x for incident neon (J = 10) ions at 1 GeV/nucleon.

26

Cj(x,E),arbitrary

.940

.752

.564

#• _'•

• %

## •%

/ ,¢---j = 1i1 %%

%

F

I %

I %

units " J'=6 7 ",.376 - /" '%

J 9 "II I %%2 _

/ "//// ,,.188 - ,, , ",.... ',/ s /, /// • ^ "_.///.,;4, r-/--:--,_J = _ -..,. ",

o _'f_: __ ......._ ' - --_ .... '50 100 150 200 250 300

x, g/cm 2

(c) E = 100 MeV/nucleon.

.400 -

.320

.240

Cj(x,E),arbitrary

units.160

.080

0

E = 995 MeV/nucleon

E = 900 MeV/nucleon

x, g/cm 2

(d) Composite overlay of all three energies.

Figure 6. Concluded.

27

.175

.140

.105Cj(x,E),arbitrary

units.070

.035i

077(

• I

• "4

• j = 1---_/-;=9 ,,," j =2;.___, ,,

• I I/" ." __._ i -, \. "... \ _,

i J : I i It

E, MeV/nucleon

(a) x ----=20 g/era 2.

I

975

.040

.O3O

¢j(x,E),arbitrary .020

units

.010

0100

x = 1 g/cm 2

x = 60 g/cm 2 x = 20 g/cm 2

1E, MeV/nucleon

)00

(b) x = 1 g/cm 2, 20 g/cm 2, and 60 gu/cm 2.

Figure 7. Variation of Cj(x, E) with E for incident neon (J = 10) ions at 1 GeV/neutron.

28

.490

.392

.294Cj(x,E),arbitrary

units.196

.098

j

j = 8-,,

j = 7-,

.

),

e,I

7

,!I2O

• %

i • %•

# •# •

_.--j= 1 ',

j =6.-._j = 2

,// \_

%

\

4O 60x, g/cm 2

(a) Eo = 500 MeV/nucleon.

I

80I

100

.220

Cj(x,E),arbitrary

units

#

i #

II

/I

#

I

I

//!

#l

.176 -

.132

.088 -

•l

•0441 ' ",,,// j = 3 " ",

, -...,300 400 500 600 700

J = x, g/cm 2

(b) Eo = 2 GeV/nucleon.

Figure 8. Sensitivity of flux at 100 MeV/nucleon to incident neon (J = 10) energy.

29

.3-

.2

Cj(x,E),arbitrary

units

.1

0.5

--L/-J=9

i,

i- i--

I

i_/,- j =7

I /-j=6--I- --(,

t ,, /-j=5 ,

-_---4,=-;-----".............-E'J=_-...." /-j=.--3 '7__ _!i, !,! , | , ,:,

1.0 1.5 22.0 2.5 3.0x, g/cm

(a) _ = 10 -4 cm2/g (rarefied shield).

i i̧ :::

£bj(x,E),

arbitraryunits

2.70 -

2.16

1.62

1.08

.54

0

j-

i •

I

i •

16

i I

•#

I

I

11

/!

%%

'v--J = 1

•t

I•

•t

!

J=9-- 7 ,' ..._j = 21 /-'j = 3 "_.. ••,

/ / . "_ •

:..-............... .,,,

_/___.o ,5 _.o _.5 3.0t',, "",- _-'" ...... _........_ "" 7" ..... ',,

d8 7 j= = : _-; : = :

- x, g/cm 2

(b) _ = 1 cm2/g (dense shield).

Figure 9. Sensitivi_ty of flux at 995 MeV/nucleon to incident neon (J = 10) ions at Eo = 1 GeV/nucleon withcross-section normalization.

3O

|

!

I

||!!

i=

!

d|

!|_=

i||

=E

|

1E

E

i!

12.5 -

10.0

7.5

_j(x,E),arbitrary

units5.0

2.5

/--j =27

I "llS#SjS_'° o • ,,,, •• ••%%

g ,,'..-".......... .../-.J=. -,,,,•,-.... .. .../ , v ..

ii11_iiIi i I I ii I I • %_. "*. _ '*._ t

'l_l_i[ll II • I i l, I l l l l " _i " " . " -- . l

,_'_,._.. ,l ', , , I I I I _ I _' I1.0 2.0 3.0 4.0 520

x, g/cm 2

(a) E = 4.995 GeV/nucleon.

.180

.144

.108

Cj(x,E),arbitrary

units.072

.036

085

i I ••

I •

I •

I •

, __1" = 9 '

/,,'" ",, ',Fi;,, _-j= lO',, (

# I • *"'_ I

iii • • _ 1

;,,,,_:1_,, ,,, -,

x, g/cm 2

I

325

(b) E = 2 GeV/nucleon for J _> 8.

Figure 10. Flux variation with z for incident nickel (J = 28) ions at 5 GeV/nucleon.

31

.220

.176

32

.132

Cj(x,E),arbitrary

units.088

.044 j=2

=4,-j = 5j = 6

I

x, g/cm 2

(c) E = 2 GeV/nucleon for J < 7.

Figure 10. Concluded.

1000

150

100

Dj(x),arbitrary

units j = 9

50-

0 50 100 150x, g/cm2

(a) Individual ions.

d

20O

.6

D(x),arbitrary

units.4

I I I

0 50 I00 150 200x, g/cm 2

(b) Sum of all ions.

Figure 11. Dose profiles (no = 1) for incident neon (3 = 10) ions at Eo = 670 MeV/nucleon.

33

Dj(x), 3

arbitrary II J-J= 10

units 2 II

0 200 400 6O0

x, g/cm 2

(a) Individual ions.

I

800

.8

.6

D(x),arbitrary

units.4

Figure 12.

.2

l i

0 200 400 600 800x, g/cm 2

(b) Sum of all ions.

Dose profiles (no = 1) for incident neon (J = 10) ions at Eo = 2 GeV/nucieon.

34

1.0

.8

.6

Dj(x),arbitrary

units.4

.2 j =3

0×, g/cm 2

(a) Individual ions.

I900

.6

D(x),arbitrary

units.4

0I 1 I

x, g/cm 2

I I900

(b) Sum of all ions.

Figure 13. Dose profiles (no = 1) for incident neon (J = 10) ions at Eo = 10 GeV.

35

.6

Dj(x),arbitrary

units.4

= 27

1.25

x, g/cm 2

(a) Individual ions.

750

!

ii

i36

1.00

.75D(x),

arbitraryunits

.50

I -_.-_ i j I0 200 400 600 800

x, g/cm 2

(b) Sum of all ions.

Figure 14. Dose profiles (no = 1) for incident cobalt (J = 27) ions at Eo = 5 GeV.

6O0

arDl[raryunits

50O

400

3O0

200

100

0I I I

x, g/cm 2

(a) J =- 10.

! I I

150

42.0 ,-.

j=a

arO,,ra units 16.8 = 4

j=3

84

0 150x, g/cm 2

(b) j < 10.

Figure 15. Dose profiles (no = 1.79) for incident neon (J = 10) ions at Eo = 670 MeV/nucleon.

37

25O

a?bJl_r_ry

units

200

150

100

50

0

i

5OI

100I I I I

150 200 250 300

x, g/cm 2

(a) J = 10.

I

35O

L

38

D.(x),arbitrary

units

24.0

19.2

14.4

9.6

4.8

0

.j =9j=8

.j=6

j=2 =

50 100 150 200 250 300x, g/cm 2

I

35O

(b) j _< 10.

Figure 16. Dose profiles (no = 1.79) for incident neon (J = 10) ions at Eo= 1 GeV/nucieon.

400

300

D(x),arbitrary 200

units

100

0

V/'--J = 10

I|

|

\

\_ , I I

x, g/cm 2

(a) J = 10.

I !

400.

Dj(x),arbitrary

units

3O

24

18

=10

98

=7=6

=5"=4

=3=2

6

x, g/cm 2

(b) j _< 10.

Figure 17. Dose profiles (no = 1.79) for incident neon (J = 10) ions at Eo = 3 OeV/nucleon.

39

3.20 -

2.56

1.92

D(x),, arbitrary

units

1.28

.64

tIIII EoIIfl 670 MeV/nucleonII 1 GeV/nucleon

3 GeV/nucleon

-/i _ i _

50 100 150 200 250x, g/cm 2

Figure 18. Total dose for previous cases.

I I300 350

4O

2000

1500

D./_),ar_trary 1000

units

500

0

x, g/cm 2

(a) 3 = 28.

110,.

/as H ] = 28

iB6 _'J = 27&L / / _,-?_

x, g/crn 2 50

(b) 28 < j < is.

Figure 19. Dose profiles (no -_ 1.79) t'or incident nickel (j = 28) ions at Eo = 1 CeV/nucIeon.

50'

120

96

72D(x),

arbitraryunits

48

' = 27= 26

=25= 24

= 23' = 22

j_21I = 20

=19=18

= 28

24

Figure 20.

I , I I I I

0 x, g/cm 2 50

Dose profiles (no = 1.79) for incident nickel (J = 28) ions at Eo = 5 GeV/nucleon.

1.4

1.2

1.0

D(x),arbitrary

units .8

.6

Calculated• Experimental

\

"40 1tO 2}:) 310 gO

x, g/cm 2

Figure 21. Experimental dose comparison for incident neon ions (J = 10) on water (_ _ 1.5 x 10-2 cm2/g).

42

100

10-2

_j(x,E),arbitrary 10-4

units

10-6

10-80

fj=lO

.002

1

.004 .006 .008x, g/cm 2

(a) E = 9999.9 MeV/nucleon.

I

.010

ej(x,E),arbitrary

units

10 -4

10-6

10-8

10-10

10-12

10-140

j=l

250 500 750 1000 1250 1500

x, g/cm 2

I

750

(b) E = 1 GeV/nucleon.

Figure 22. Flux profiles for analytic distributed source of incident neon (J = 10) ions with Mo = 0.01.

43

100

10-2

Cj(x,E),arbitrary 10 -4

units

10-6

10-80

fj=lO

/

.002

n

.004 .006 .008x, g/cm 2

(a) E = 9999.9 MeV/nucleon.

I

.010

10-6

44

10-8

Cj(x,E),ai'bitrary 1010

units

10-12

10-140

j=4

j=l

=10__

j=6

i=7

j--8j= )

!

250 500 750 1000 1250 1500 750

x, g/cm 2

(b) E = 1 GeV/nucleon.

Figure 23. for analytic distributed source of incident neon (J 10) ions with Mo = 1.0.

10-4

10-6

Cj(x,E),arbitrary 10-8

units

10-10

10-12

102

,,,,,I , , , ,,, ,,i56/ 89 2 3 4 56/89

103 104

E, MeV/nucleon

(a) x---- 1 g/cm 2.

10-4

10-5

Cj(x,E),arbitrary 10-6

units

10-7

10-8

i 19_-_._" _._'. *_-....

102

I I I I l l II_ I I I | | I I I [

2 3 4 56789 2 3 4 56789103 104

E, MeV/nucleon

(b) x = tO g/cm 2.

Figure 24. Flux profiles for solar source for incident nickel (J = 28) ions.

45

.630

.504

.378_j(x,E),arbitrary

units.252

.126

0

7,8, ..125

2.5 5.0 7.5 10.0 12.5 15.0 17.5

x, g/cm 2

(a) E = 4.9 GeV/nucleon.

46

.120

.096

_j(x,E), .072arbitrary

units .048

.024

I, |

0 20 40 60 80 1O0 120 140x, g/cm 2

(b) E = 4.3 GeV/nucleon.

Figurc 25. Flux profiles for a composite beam of neon through iron ions at Eo = 5 GeV/nucleon.

" i

.980

.784

.588¢j(x,E),arbitrary

units.392

.196

0 100 200 300

x, g/cm 2

(c) E = 2.5 GeV/nucleon.

Figure 25. Concluded.

400 50O 6OO

47

Cj(x,E),arbitrary

units

.680

.544

.408

.272

.136

ol25

Js •

I •

gl 8 '_NNNI

i ¢ \

/ \

/

I

i I

/I

i I

/l

s / ; 3_.,/ /-J=

50 75 1O0 150

I

',f J= 1I

\

•l

I

125

x, g/cm 2

t

t

ll

- I

175 200

(a) fn = 0 (no neutrons).

I

225

.680

t_j(x,E),arbitrary

units

.544

.408

.272

"136 I

0 l25

_.. _......_ ./- Neutronf

/

/

/ _.

./L ../j =2 -,..I

z/ . _,__.. /,'- Proton

_'4"_"1"',-",'--. .........,......."'r--_, ".,-.--.___.,50 75 100 125 150 175 200

x, g/cm 2

I

225

(b) /,_= o.5.

Figure 26. Distinction between neutrons and protons at E = 500 MeV/nucleon for incident neon (J = 10) ionsatEo=l GeV.

48

¢j(x,E),arbitrary

units

.4O

.32

.24

.16

.O8

0520

It,!

I

' !• I

!" , :II i-1I .._''" ,".. J. _ _

I .."" , ..-r'--'--t,,...-" ,")! .'° I I "_ i...

f /_' , ,,.,_. _ ............ .

v --:.[.-..'_- , _ ".4 - •II / -J sr i i • •

s sins : I • •

[_.-:._1"9:8 17 n6 15 , ".|- .-.- , ! I , ',

, ! I

' i ' J ',! , , ! _I, !, _ ,

E, MeV/nucleon

(a) Exact.

I

Ii l i

650

.40

Cj(x,E),arbitrary

units

.32

.24

Np= 10

.16 .- ....... j = 1

I .,,-',,.-:.._..T ..............3 _..-,/ .. _/-.'"_ - _.._ ""........ _ _ • •

.OSl- ___ ".._,.. _ ",,',

0520 650

E, MeV/nucleon

(b) ,'ks = 6.414.

Figure 27. Flux profile comparison with finite-difference calculation for an incident (J = 10) neon beam of

Eo = 670 MeV/nucleon.

49

.4O

Cj(x,E),arbitrary

units

.32

.24

.16

.O8

Np = 2O

0520

. .. ....... .<j=l,_ -,L

,,;',....:-------, ,,,,.._- -:__.....,.,__ ",,,,,,

__-,, _--_.... ° \ K

I I I I _-J.. _ _--,--z-1

650E, MeV/nucleon

(c) As = 3.207.

.4O

.. k

5O

.32

.24

Cj(x,E),arbitrary

units .16

.08

Np = 4O

052O

s

J _' '_"-'" _ %" _l

,, - --,---.;.,._-- _..............._ \

! I I

650E, MeV/nucleon

(d) As = 1.604.

Figure 27. Continued.

.40

.32

.24

(i)j(x,E),arbitrary

units.16

.08

0520

Np = 80

• 1....----<--.!, =

,_.._......:-- . \ "\"///,,'/._-_'\', \ ", ,,, \.,.\ \ ',

,650

E, MaY/nucleon

(e) /ks = 0.802.

Np = 150

Cj(x,E),arbitrary

units

E, MeV/nucleon

(f) As = 0.428.

Figure 27. Concludcd,

650

51

1018 E

1017

4)(x,s), 016arbitrary 1

units

1015

1014101

,,/-j = 10

.............. .._..j=2,3.. .... 9

................... ..._.._...%_ _- %,,_---_

1 • ! 11111| ..1 I I lallil i I_1 lllill I I I liittl

102 103 104 105

E, MeV/nucleon

(a) 1 g/cm 2.

1018

1017

¢(x,s), 016arbitrary 1

units

1015

=

j=lO

............ ..zc_.J_-2

1014 , i ...... I ........ I ........ I , , , ,,,,,I101 102 103 104 105

E, MeV/nucleon

(b) 10 g/cm 2.

Figure 28. Flux profile comparison with finite-difference solution for incident neon (J = 10) ions at177o= 100 GeV/nucleon.

52

1017 .-

! '=2

1016 i '

_o,_! - " -'_--\o_x,_, F -- " 4..,.\

arbitrary / -- " "_"_'_

un,,s_o"_- -- " -Z...'-,..... • 8

.....

1012

101 102 103 104 105

E, MeV/nucleon

(e) 100 g/cm 2.

1017

¢(x,s),arbitrary 1016

units

1015101

I I I IIII1[ ! I I IIIII]

102 103

j=3

104 105E, MeV/nucleon

(d) j = 2, 3, 4, 5 at 100 g/cm 2.

Figure 28. Concluded.

53

1018 _

1015

1012

_(x,s),arbitrary

units

=

10 9 -

10 6 -

103 I.3 .4

j = 28

I I I I I.5 .6 .7 .8 .9 1.0 x 105

E, MeV/nucleon

Figure 29. Flux profile comparison with finite-difference solution for incident nickel (J = 28) ions at

Eo = 100 GeV/nuclcon and 100 g/cm 2.

54

Form Approved

REPORT DOCUMENTATION PAGE OMBNo. 0704-0188

Pubfic reporting burden for this collection of _nformat;on is estimated to average 1 hour per response, includlngthe time for reviewing instructions, searching existing data sources.gathering and malntain;ng the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of thlscollection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports. t21S JeffersonDavis Highway, Suite 1204, Arlington. VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, OC 20SQ3

I 1. AGENCY USE ONLY(Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

October 1991 Technical Paper

4. TITLE AND SUBTITLE ,5. FUNDING NUMBERS

Benchmark Solutions for the Galactic Heavy-Ion 2_ansport

Equations With Energy and Spatial Coupling WU 199-04-16-11

6. AUTHOR(S)

Barry D. Ganapol, Lawrence W. Townsend, Stanley L. Lamkin, andJohn W. Wilson

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23665-5225

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-16909

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TP-3112

11. SUPPLEMENTARY NOTES

Barry D. Ganapol: University of Arizona, Tucson, AZ; Lawrence W. Townsend and John W. Wilson: LangleyResearch Center, Hampton, VA; Stanley L. Lamkin: Old Dominion University, Norfolk, VA.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -Unlimited

Subject Category 70

13. ABSTRACT (Maximum 200 words)

12b. DISTRIBUTION CODE

Nontrivial benchmark solutions are developed for the galactic heavy-ion transport equations in the straight-

ahead approximation with energy and spatial coupling. Analytical representations of the ion fluxes are obtainedfor a variety of sources with the assumption that the nuclear interaction parameters are energy independent.

The method utilizes an analytical Laplace transform inversion to yield a clbsed-form representation that iscomputationally efficient. The flux profiles are then used to predict ion dose profiles, which are important for

shield-design studies.

14. SUBJECT TERMS

Benchmark solutions; Space radiation; Heavy-ion transport

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified

_ISN 7540-01-280-5500

1S. NUMBER OF PAGES

55

16, PRICE CODE

aQ420. LIMITATION

OF ABSTRACT

Standard Form 298(Rev. 2-89)Prescribed by ANSI Std z3g-18298-102

NASA-Langley, 19gi