An Improved Elastic and Nonelastic Neutron Transport ... · PDF fileAn Improved Elastic and Nonelastic Neutron Transport Algorithm for Space ... The NASA STI Program ... An Improved

NASA/TP-2000-210299

An Improved Elastic and Nonelastic

Neutron Transport Algorithm for SpaceRadiation

Martha S. Clowdsley and John W. Wilson

Langley Research Center, Hampton, Virginia

]ohn H. Heinbockel

Old Dominion University, Norfolk, Virginia

R. K. Tripathi, Robert C. Singleterry, Jr., and Judy L. Shinn


July 2000

https://ntrs.nasa.gov/search.jsp?R=20000073842 2018-05-26T00:47:06+00:00Z

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NASA/TP-2000-210299

An Improved Elastic and Nonelastic

Neutron Transport Algorithm for SpaceRadiation

Martha S. Clowdsley and John W. Wilson


John H. Heinbockel

Old Dominion University, Norfolk, Virginia

R. K. Tripathi, Robert C. Singleterry, Jr., and Judy L. Shinn


National Aeronautics and

Space Administration

Langley Research CenterHampton, Virginia 23681-2199

July 2000

Available from:

NASA Center for AeroSpace Information (CASI)7121 Standard Drive

Hanover, MD 21076-1320

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171(703) 605-6000

Symbols and Abbreviations

A

AT_

C3H 10T 1/2

Ei, Ei,E,

fJ_,3

fj/,.,_3

Itij, Kij

HETC

tI Z E T ILN

h

L,i, It,i, _i

LAHET

MCNP

-_N rM( : PX

SI)E

x j, yj, x

# = col{o0, ol ..... 0X-l}

0

o,,o:,ol, o_

g = colK0,_l .... ,_x-1 }

P,_

crj(E)

crjk(E,E')

0"_ ,3, O'r 3

O"3

o, O, %( F)

coefficient matrix

atomic weight

Boltzlnann differential operator

mouse embryo cell culture

energy, MeV

direct, knockout redistribution, MeV-1

eval)oration spectral redkstributioll, MeV-1

elastic spectral redistribution, M eV- 1

integral operators

Itigh Energy Transport Code

High Charge and Energy Transport, (',ode

step size for numerical integration

integral operators

integrals from Ei to Ei+l of terms on right-hand side of

Boll, zniann equation

Los Alalnos High Energy Tratrsport Code

Monte Carlo N-Partiche TransI)ort Code

L A H ET/M C N P Code merger

stopl)ing power jth particle

solar particle event

depth of penet.ration of neutron radiation, g/cm 2

cohlnm vector of Oi terms

paralneter values

scattering angle

mean value fl'actions

column vector of source terms

nmnber density of/3 type atoms per mlil. llla-SS, g particles

tota.1 macroscopic cross section per unit mass, cm2/g

macroscopic differential cross section for particle k with energy

F_r producing particle j with energy E, cm2/g-MeV particles

scattering terms, cm2/g-MeV

microscopic cross sect, ion. cm 2

average macroscopic cross _ction, cm2/g

integral of fluenee for ith energy group, neutrolt_/cm :_

particle fluence, particles/cm2-MeV

°°.Ill

Abstract

A neutron transport algorithm including both elastic and nonelas-

tic particle interaction processes for use in space radiation pwtection

for arbitrary shield material is developed. The algorithm is based

upon a multiple ene_yy grouping and analysis of the straight-ahead

Boltzmann equation by using a mean value theorem for integrals.

The algorithm i.s then coupled to the Langley HZETRN code through

a bidi_,ctionalneutron evaporation source term. Evaluation of the

neutron fluence generated by the .solar particle event of l,_bruary 23,

1956, for an aluminum-water shield-targel configuration is then com-

pa,ed with MCNPX and LAHET Monte Carlo calculations for thesam_ shield-target configuration. With the Monte Carlo calculation

as a benchmark, the algorithm developed in this paper showed a great

improvement in r_sults over the unmodifiat HZETRN solution. In

addition, a h_gh-energy bidirectional neutron source based on a for-

mula by Ranfl showed even further inq_rov_ment of the fluenee re-

sults over p_vious results near the fTmtt of the' water ta_yet wherv

diffusion out tb e fivnt surface is important. Effects of imp_vved in-teraction cross sections are modest compared with the addition of the

high-_ ne rgy bidtrectio na I souwe Ie rm.s.

Introduction

This paper presents an improved algorithm for the analysis of the transport of secondary neu-

trons arising in space radiation protection studies. The design and simulation of the operational

processes in space radiation shielding and prol.ection require highly efficient COlnputatiollM pro-

cedures to adequately characterize time-dependent envh'onments and time-dependent geometric

factors and t,o address shield evaluation issues in a multidisciplinary integrated engineering design

environment. One example is the recent, study of the biological response in exposures to space

solar particle events (SPE's) in which the changing quMity of the radiation fields at specific tissue

sites are followed over 50 hours of satellite data to evMuate time-dependent factors in biological

response of the hematopoietic system (ref. 1). Similarly, the study of cellular repair dependent

effects on the neoplastic cell t.ransfornlation of a C3H10T½ mouse embryo cell culture popula-

tion in low Earth orbit, where trapped radiations and galactic cosmic rays vary continuously in

intensity and spectral content about the orbital path (ref. 2), requires computationMly efficient

codes to simlflate time-dependent boundary conditions around the orbital path. But even in

a steady-state environment which is holnogeneous and isotropic, the radiation fields within a

spacecraft have large spatial gradients and highly anisotropic factors so that the mapping ofthe radiation fields within the astronaut's tissues depends on the astronaut timeline of location

and orientation within the spacecraft interior where large differences in exposure patterns that

depend on the activity of the astronaut have been found (ref. 3). Obv ious cases exist, where rapid

evaluation of exposure fields of specific tkssues is required to describe the effects of variations in

the time-dependent exterior environment or changing geometric arrangement. A recenl study

of the time-dependent respo_Lse factors for 50 hours of exposure to the SPE of August 4, 1972,

required 18 CPU hours on a VAX 4000/500 computer by using the nucleon light, ion _ct, ion of

the deterministic high charge and energy trattsport code HZETRN (ref. 4). In comparison, it, isestimated that the related calculation with a standard Monte Carlo code such a.s HETC (ref. 5)

or LAItET (ref. 6), which are restricted to only neutrol_s, protons, piolts, and alphas, would have

required approximately 2 years of computer time on a VAX 4000/500 compnter. The spacecraft

design envu'onment also requires rapid evaluation of the radiation fiehks to adequately determine

effects of multiparameter design changes on system performance (refs. 7 and 8). These effects

are tile driving factors in the development and use of deterministic codes and in particular the

HZETRN code system which simulates 56 naturally occurring atomic ions and neutrons.

The basic philosophy for the development of the deterministic HZETRN code began with thestudy by Alsmiller et al. (ref. 5) using an early version of HETC, wherein they demonstrated

that the straight-ahead approximation for broad beam exposures was adequate for evaluation of

exposure quantities. Wilson and Khandelwal (ref. 9) examined the effects of beam divergenceoll tile estimation of exposure in arbitrary convex geometries and demonstrated that. the errors

hi the straight-ahead approximation are proportional to the square of the ratio of the beamdivergence (lateral spread) to the radius of curvature of the shield material. Tiffs ratio is small

hi typical space applications. From a shielding perspective, the straight-ahead approximation

overestimates the transmitted flux and the error is found to be small in space radiation exposure

quantities. Langley Research Center's first implementation of a numerical procedure was

performed by Wilson and Lamkin (ref. 10) as a numerical iterative procedure of the chargedcomponents perturbation series expansion of the Boltzmann transport equation and showed

good agreement with Monte Carlo calculations for modest penetrations to where neutrons pl_v

an important role. The neutron component was added by Lamkin (ref. 11) and closed the gap

between the deterministic code and the Monte Carlo code. The resulting code was fast compared

with the Monte Carlo codes but still lacked efficiency in generating and operating with largedata arrays which would he solved in the next generation of codes.

The transport of high-energy ions ks well adapted to the straight, ahead approximation. In

fact, the usual assumption that secondary ion fragments are produced with the same velocity as

the primary initial ion (ref 12) is less accurate than the straight-ahead approximation contrary

to intuition (ref. 13). The Boltzmann transport, equation for the particle fields ¢_j(x, E) is givenfor tile straight, ahead and continuous slowing down approximations as

[00 ] j?0. dE' (1)t,

where x is the depth of penetration, E is the particle kinetic energy, Sj(E) is the particle stopping0" t "power, crj(E) is the macroscopic interaction cross section, and ja.(E,E) is the macroscopac

cross section for particle j of energy E produced as a result of the interactioil with particle k

of energy E _. It has been customary in codes developed at Langley to invert, the differential

operator and implement it exactly as a marching procedure (ref. 14). The remaining issue hasbeen to approximate the integral term on the right-hand side of equation (1). The formulation

of the code to approximate heavy fragments was facilitated by assuming that. their fragmentvelocity is identical to that of the primary ion velocity. This assumption is inadequate for the

description of the coupled nucleolfic and light ion components. A computationally compatiblenucleonic transport procedure was developed by Wilson et al. (ref. 15) and agreed well with

exposure quantities evaluated by Monte Carlo transport procedures (refi 16). The trausport

of the nucleonic component was developed by assumhlg the midpohlt energy, within the stepsize h, was tile appropriate energy to evaluate the integral term. Thus, the residual range of

the proton will reduce by" h/2 before the interaction, and the secondary" proton residual rangewill reduce by h/2 before arriving at the next marching step. Neutrons show no loss in residualrange as their stopping power is zero. Tiffs choice was shown to mhlimize the second-order

corrections to the marching procedure (ref. 17). Although reasonable agreement on exposurequantities from Monte Carlo calculations was obtained, the resultant neutron flux at the lowest

energies was substantially below the Monte Carlo result, in the range of 0.01 to several MeV

and required improvement (ref. 18). Analysis revealed that the problem was in the rescatterhlgterms hi which the nutnber of elastic scattered neutrons was underestimated numerically, which

must be addressed as suggested by Shinn el. al. (ref. 18).

The issue of evaluation of the integral term of the Boltzmann equation for the elastic

scattering was developed ill a prior report (ref. 19). In reference 19. a nmltiple energy

group (multigroup) method based upon a mean value approximation to tile integral terms for

trartsporting evat)oration neutrolts showed vast. improvement in the low-energy neutron spectra.

Now that the elastic scattering events are adequately repre_nted, consider the addition of

unproved estimates of nonelastic processes on the neutron tra.ltsport solution. In the present

paper, nonelastic proces_s are added to the algorithm developed by tteinbockel, Clowdsley, and

Wilson (ref. 19). The code is then modified t.o a.ccount for the high-energy neutron production

at, backward angles. Improved neut.roll interaction cross sections with less dramatic changes on

the neutron spectrum are introduced.

Formulation of Transport Equations

Define the linear differenlial operator as

a s)(L') + ,,j(E)] O(x Lc)OE

aO(x,E)_-@-[,'gj(E) O(x,E)] + crj(E) ¢(x,E) (2)

and consider the following one-dimensional Boltzmann equation from reference 20

_0 'DCB[oj] : c9a.(E,tg') O_,(a,,E')dE' (3)

where Oj is the differential flux spectrum for the type j particles .... j(E) is the slopping power

and crj(E) is the total macroscopic cross section. The term crj_.(E,E r),of the type J particles,

a macroscopic differential energy cross section for redistribution of particle type and energy, is

written as

(rjh.(E,E') = _ P3 °'3(E') Ija,,,#(E,E') (4)3

where fj&3(E,E I) is the spectral redistribution, o-,,_ is a micro_opic cross section, and p2 is the

number density of/3 type atoms per unit mass of material. The spectral terms are expressed as

fit',3 el _ d, = f]k,,3 + fjk,J + f)k,3 (5)

where f]_,.3 represents the elastic redistrit)ution in energy, f]k 3 represents evaporation terms,d

and f)t.,3 represents direct, knockout terms. The elastic term is generally' limited to a small

energy range near that of the primary particle. The evaporation process dominates over the

low-energy range (E < 25 MeV), and the direct cascading effect dominates over lhe high-energy

range (E > 25 MeV), a.s illustrated in figure 1 with data. from reference 20.

Equation (3) is then written for j = n as

j? ( )= f,;t.,:_+ fnl" d + f,,t,,cl 0_,(x,E') dE'

k 3

(_)

which is expanded t,o the fore1

B[¢,,] = p,d cr,_(E') f,,,fl + f,,,,,,j + fn,,,,L_ _nJ

(7)

Define the integral operators as

= O(x,#) dE' (8),,3

= O(x,#)dE'E ,,'3

(9)

Id(_')[4)] = _ PJ ad(E') f,,dk,,3 _(a",E') dE' (10)E '

d

where k = n denotes coupling to neutron collisions, k = 1.,denotes the neutron source from

proton collisions, and similarly for other ions. When considering only neutrons and protons,equation (7) can be written in the linear operator form as

(") , _ i(") /v) l}p)[_/,] + (/') ,B[0,,] =/el [qmJ + II")[0,,] + [q_"] + "el [_P] + I'j [Op] (11)

(p) ,Note that /el [0el does not. contribute to the neutron field because protons cannot, produce

neutrons through elastic scattering and therefore equation (11), with ¢_, replaced by 0, is writtenas

(") I_I,)B[O] = /el [0] + [0] + I_{")[01 + I_V)[Op] + i(p)[Op] (12)

In reference 19, we assumed the evaporation source to be isotropic and evaluated tile transport in

forward and backward directions by using the straight-ahead approximation for elastic scattering

and found improved agreement, with Monte Carlo calculations. The first, step in this study is toadd effects of nonelastic events into the tralrsport, process.

Assume a solution to equation (12) of the form ¢ = _, + Od, where 0e is the solution forevaporation sources and contributes over the low-energy range and 0d is the solution for tile

direct knockout, sources and contributes mainly over the high-energy range as suggested byfigure 1. Substitute this assumed solution into equation (12) and find

+C'I,<+ + +/7'I ,l (13)

hi reference 19, the terms I_'n)[0e,] and ld(n)[0_] were set to zero to consider only elastic scattering.This allows estimates of the elastic scattering effects on the transport of evaporation neutrons.

In contrast, these terms are retained and they demoustrate nonelastic effects on the transport

of evaporated neutrons. This change al_ allows flexibility in fiirther improving the HZETRN

code ms shown later in this report. As in reference 19, we assume that 0d was calculated by the

4

HZETRN program because for the direct cascade neutrons the straight-ahead apwoximation is

valid. Consequently, Od is a solution of the equation

Tt,,)r _ 1 1{"B[Od] = 'el {OdJ + "d )[Od] + ll/')[Op] (14)

Tins assumption simplifies equation (13) t,o the form

+ + ,]+ + I?)[M

Tile elastic and reaction scattering terms are defined as

°-._,d P,:3 crd(L't) ,,1 HI= . f,,t.d(E, )

, , d E,E' )]_,.,_= p_ <_(E') [f,h.,_dE,E )+ f,,t,,_(

(15)

with units of cm2/g-MeV, and for neutrons the stopping power @(E) is a_umed zero, and

therefore, equation (15) reduces to the integro-differential trausport equation with source term

a.s

[0 ] ]?_+_,(L:) ,_(_,,E)= _ [,,_,,_(F,_')+ _-,.,e(E,#)] 0,(x,#) d#+ .q(E,_.) (1(5)3

Equation (16) repre_mt.s the steady state evaporation neul.ron fluence O_ (x,E) at. depl.h x and

energy /2. The various terms in equation (16) are energy E with units of MeV, depth in medium

is x with unit, s of g/cm 2, O,(x,E)(in particles/cn,2-MeV) ks the evaporation neutron fluence,

and g( E, x) = l}")[Od] + I, 0') [0p] (in partic-les/g-MeV) ks a volun,e source tern, to be evaluated by

the HZETRN algorithm. Equation (16) is further reduced by considering the neutron energies

before and after an elastic collision. The neutron energy E after an elastic collision with a

nucleus of mass number ATi#, initially al rest, is from reference 21

E-- E t

A , +2AT, #cos0+l

, 7,,3 (17)

where U ks the neutron energy before the collision, AT: _ is the atomic weight of the ;hh type of

atom being bombarded, and 0 is the angle of scatter. Note that for forward scattering 0 = 0,

E = E r, and for backward _al.t.ering 0 = rr, E = Ero3, where o d is the ratio

2

= \A<,,7(18)

wlfich ks a constant less than 1. Therefore, change lhe limits of integration for the elastic

scattering t.erm in equation (16) 1o [E,E/a.3], which represents the kinetically allowed energies

for the scattered neutron to result in an energy E. Equation (16) then is written as

b-i+_(E) ¢,(_-,E)=}--] E <'J( )O_(_,#)dE'

E+ E cr,.,3(E,E') O_ (x,E') dE' + g(E,x)

,]

(19)

5

Tile quantity _r(E) has units of cm2/g and is a macroscopic cross section given by

,3 ,3

(20)

where p a is the number of atoms per gram, a_l(E) is a. microscopic elastic cross section in units of

cm2/atom, and a_(E) is the corresponding reaction cross section. Other units for equation (16)

are obtahled from the previous mills by using the scale factor representing the density of ttlematerial in units of g/era a.

Mean Value Theorem

Throughout the remaining discu_ious tile following mean value theorem is used for integrals.

Mean Value Theorem: For 9(x,E) and f(E) contimmus over an interval a _< E _< b

such that. (1) ¢(x,E) does not change sign over the hlterval (a, b), (2) O(x,E) is integrable

over the interval (a, b), and (3) f(E) is bomlded over the hlterval (a,b), then there existsat. least one point ¢ such that.

f(E) ¢(x,E) dE = f(¢) O(x,E) dE (a < ( < b) (21)

In particle transport this mean value approach is not commonly used. In reactor neutron

calculations, an assumed spectral depen&nce for O(x,E) is used to approximate the hlt.egralover energy groups. The present use of the mean value theorem is free of this assumption; thus,

more flexibility is allowed in the HZETRN code, and the result, is a fast and efficient algorithmfor low-energy neutron analysis.

Multigroup Method

To solve equation (19), partition the energy domain into a set. of energies {E0,E1 .....Ei, Ei+l,. • .}. Consider first the case where there ksonly one value of/_ which represents neutron

penetration into a single elelnent material and let, (p, be denoted by 0. Equation (19) is integrated

from Ei to Ei+I with respect, to the energy E to obtain

where

[Ei+ 1Ei+l O0(x,E) dE+ o(E) 0(x,E) dE= I_.,i + I,.i + _i (22)Ei Ox j Ei

The quantity

fEi+ /E/aI_. i = as(E,#) O(x,E') dE' dEd E i dE

/Ei+I/E_'1,. ,: : a,.(E,E') _(x,E') dE' dE" Ei

[L)+I_i = g(E,x) dEJ Ei

Ei+lqbi(x) = 0(x,E) dEEi

(23)

(24)

(25)

(26)

6

is associated with the ith energy group (El,El+l), so that. 1 d_i(x ) represents an averageEi+ 1 -E i

fluence for the ith energy group. Then equation (22) can be m'itten as an ordinary differential

equation in terms of (Pi(x) as follows. In the first term in equation (22), interchange the order

of integration and differentiation to obtain

EEi+I O_(x,E) d_i(x)i cgx dE - dx(27)

By using the previously stated mean value theorem for integrals, the second terln ill equal, ioll (22)

can be expressed as

Ei+I cr O(x,E) dE = _ ¢bi( x ) (28)i

whe re _- = cr[Ei + O(Ei +1 - El)] is a mean value associated with some vat ue of 0 between 0 and 1.

For the term l._,i in equation (23), interchange the order of integration as illustrated in

figure 2. The inlegration of equation (23) depends upon the energy partition _qected. For

example, figure 2(b) illustrates an energy partition where Ei+l < El�O; for this ca._ _ we can

write equation (23) as

jftEi+l/E _' Ei/{_f Ei+l dE_I_ i = H,. dE dE _+ / tt._ dE

• ' U=Ei =El JE_=Ei+I E=Ei

m ,lE tF' (2(.)1q- aEt=Ei/( t JE=oE t "

where H, = o'.,(E,E I) 0(x,U). Figure 2(c) depicts the case where El+ 1 = Ella exactly for all i.

In this special caw, equal, ion (23) reduces to

= [Ei+I fU [Ei+l/t, /Ei+lI_ i H,. dE dE' + Hs dE dE' (30)"' jEt=EiJE=Ei JEI=Ei+I JE=(_E t

The selection of an energy partition can lead to two or more distinct energy groups associated

with each interchange in the order of integration. For example, see figure 3(a).

Tile evahlation of equation (24) is somewhat morn complicated. As an approximation, we

assume there is an energy E N such that ¢(x, E) Call be taken as zero for all E > E N. In this

case, equation (24) can be written as

H,. dE dE I+ Z H,. dE dE I (31)

' jE/=EiJE-=Ei j=i+lJEj Ei

where H,. = cr,.(E,U) 0(x,U). For example see figure 3(b).

Fquations (3(1) and (31) mw then be written for the case where El+ 1 = Ei/o as

* f Ei+lIs'i = i _(E,E i ) dE _i(x) dE+ J_E,*+I o',(E,Ei+ 1) dE _i+l(X) dE

(32)

and

EY N-1 /Ei+ 1= [' aF+I,.,i.t E i j=i+l J Ei

E,E ) dE ,j.) dE (33)

where E* = Ol(Ei+l-Ei) and Ei+l = 02(Ei+2-Ei+l)forsome 01 and 02 such that O < 01,02 < 1

by once again using the previously stated mean value theorem. The special partitioning of the

energy as illustrated in figure 2(c) enables us to obtain from equation (22) a system of ordinarydifferential equations of the form

[i01 .1.1[.01@1 a22 a23 a2N _1

d . . .

= a,33 +

--O--

-1 aNNJ ¢I',_,'-l L,_N- 1

(34)

where each equation is associated with an energy group. This is where the t.erm nmltigroupmethod originates. In equation (34), tile coefficient matrix has the elements

j_E 'U*a,:,;= ' + de -i

: / E/+I _, / El+2

ai,i+ 1 JaE_+l a_.(E,Ei+ 1 ) de + YEi+I

Ei+j+ I= ,Zna i, i+j J Ei+j

cr,.(E,EL1) de

(j=2,3,...)

Further asstmle that for some large value of N, Oi equals 0 for all i _> N. This assunlption

gives rise to the following system of ordinary differential equations:

subject to the initial conditions _'(0) = 0". Here _ is the colmnn vector of (I)i values,

co1(¢0, (I)1 ..... (I)N_ 1), the matrix A is an N by" N upper triangular matrix, and _ is the column

vector col(_0,_l ..... _N-I). Tlfis system can be solved by using back substitution. In a similar

manner, the integrals in equation (29) and (31) can be evaluated for other kinds of energy par-

titioning, and a system of equations having the same form of equation (34) obtained. How the

elements of the matrix A are calculated will depend upon the elastic scattering as determined

by" the type of energy partition. (See, for example, fig. 3(a).)

For our purposes the system of equations (eq. (34)) is used to discuss some of the problems

associated with the multigroup method. Of prime concern is how an energy grid is to be

constructed and how this energy grid controls the size of the matrix in equation (34). Consider

the construction of the energy partition

E0 E0 E0 "_E0,

where E0 = 0.1 MeV, for the selected elements of lithium, aluminum, and lead. Table 1

illustrates integer values of N necessary to achieve energies greater than 30 MeV. These values

of N repre_nt the size of the matrix associated with the number of energy groups. The value

E0 = 0.1 MeV, in terms of hmnan exposure, represents a lower bomM where lower energies are

not important. The value of 30 MeV represents an upper limit for the evaporation particles and

could be adjusted for other source terms.

TabLe1.EnergyPartitionSizeN

Element o N O.1/a x

Lithimn 0.563 10 31.53

Alum inum 0.862 39 32.75

Lead 0.981 298 30.38

Observe that for energy partitions where Ei+l < Ei/a the values of N are larger, and

if Ei+l > Ella the values of N are smaller. The cases where Ei+l > Ei/(_ give rise to

problems associated with the integration of the elastic scattering terms over the areas A 1 and

A2 of figure 2(d) when the order of integrat.ion is interchanged. In this figure, the area ,41 is

associated with the integral defilfing _i, and the area ,42 is a remaining area associated with

an h_l.egral which is some fraction of the integral defining q_i+l which is outside the range of

integration; therefore, some approxilnation nmst be made to define this fractional part. This type

of parl itioning produces errors, due to any approximations, bill. it has the advantage of greatly

reducing the size of the A' by N matrix A at. the cost of introducing errors into the systeln of

equatiolts. A more detailed analysis of the energy partition can be found in reference 22.

The case of neutron penetration into a. composite material gives rim to the case where there

is more than one value of I;/in equat, ion(16). In this special case, equation (23) becomes

[z,+, [E/,,,, (z,z,) de' dEI_,i =3 JEi .]E

(as)

an d equation (24) beco rues

jft E i+ 1 /£?,l,.,i = _ Ei or,. _(E,E') O(x,E') dE' dE

(36)

In order to avoid the errors introduced when an energy grid is selected such that Ei+l > Ei/o.

we select, ct = max(ol, o2,..., 0:3) and construct the energy partition where Ei+ 1 = Fi/o so

that Ei+l <_ Ei/o_ for all _3. Obtain a system of differential equations having the same upper

triangular form but with ela_stic scattering contributions for off-diagonal elements. Observe that

for some arbitrary energy grouping we have, for the element hydrogen, a case where the value

of a_ is zero and Ei/a,_ is therefore infinite. In this situation, we mttst integrate over many

energy groups. In this case, the area of integration is similar to that shown in figure 3(b).

For any composite material, depending upon the selected energy partitioning, some type of

approximations nmst be made when the order of integration is interchanged in equation (35).

Also the problem of selecting the mean values associated with each of these integratioILs exksts

and now addressed.

Mean Value Determination

A realistic test. case wassolved analytically and numerically (with and without the nmltigroup

approximation) for which the mean values were found empirically for several single element

materials (ref. 19). The values determined are

ET.* = E i + 01(Ei+ 1 - El)

Ei*+l = Ei+I + 02( Ei+2 - Ei+I)

9

where

and

where

Ol={71 + roll(E- Ell) -- 61

71 + m12(E-- Ell) -- 61

73 + ml3(E- E22) - fl

71 = 0.93

3'2 = 0.90

73 = 0.30

_4 = 0.27

72 + m21(E - Ell)

72 + rn22(E - Ell)

74 + m23(E - E22)

roll = 0.0030485

m12 = 0.2490258

m13 = --0.3937186

Ell = 3.037829

(E > Ell )

(E22 < E < E_I)

(E < E22)

(E 2> Ell )

(E22 < E < Ell)

m21 = 0.004355

m22 = 0.249026

rn23 = -0.255920

E22 = 0.5079704

and 61 is 0.0 for lead, 0.02 for aluminum, and 0.075 for lithium. These values of 0 for the mean

value theorems were determined by trial and error so that. the nmltigroup curves would have the

correct shape and agree with tile numerical solution. These selections for the mean values are

not unique.

Application to Evaporation Source in A1-H20 Shield-Target Configuration

Apply the previous development to an application of the nmltigroup method associated

with an aluminum-water shield-target configuration. In particular, consider the ca_ where

the source term g(E,x), in equation (16), represents evaporation neutrons produced per unit

mass per MeV and is specified as a numerical array of values corresponding to various shield-

target thicknesses and energies. The numerical array of values is produced hy the radiation code

HZETRN developed by Wilson et al. (ref. 4). This nmnerical array of source term values is

actually given in the form g(Ei,xj,yk) in units of particles/g-MeV, where Yk represents discrete

values for various target, thicknesses of water in g/cm 2, xj represents discrete values for various

shield thicknesses of aluminum, also in units of g/cm 2, and E i represents discrete energy values

hi units of MeV. These discrete source term values are used in the following way. Consider

first the solution of equation (1 6) by the multigroup method for an all-alumhmm shield with no

target, material, that is, target, thickness Yk = 0. The HZETRN program was run to simulate

the solar particle event, of February 23, 1956, and the source term g(Ei,xj,yk) associated with

an aluminuin-water shield-target configuration was generated for these conditions. Using this

source term, we solved equation (16) by the multigroup method.

For a single shield material with only one value of/3, equation (16) becomes

0 ] _(E,U) 0(x,U) dE'+ O(x,E)= [El,.,JE

+ _,.(E,U) _(xY) dE' + g(E,_) (37)

10

whereall integrationof equation(37)from Ei to Ei+l produces

EEi+ 1

i

= [Ei+l[ E/'_J E i J E

Ei +1 ,:)c,

J E i

Ei+la._£_dF + _(E) ¢(.,E) dEOx jE i

a_.(E E I) O(x,E r) dE r dE

a,.(E,E _) 0(x,U) dE I dE+ [ Ei+l

a Eig(E,x) dE (3S)

We define the quantities

_'i : [u_+__O(.,E) dE ]

dEi (3.q)

_i = [Ei+t .q(E,x) dE

J E i

and interchange tile order of int_grat, ion of the double integral l.erms in equation (38). If the

energy grid is chosen so that Ei+ 1 = Ei/o, a. Ineedl va.lue theorem is applied to obtain the r¢.'su]t

d_ i Ei+l _t E_77-,.+_"__= [J E i • E=E i

+

+

_,,(E,E') dE o(.,ff) d#.

/,,E_+.e[E_+, ..,(E,#) dE O(_',E') dE'• El+ 1 JI:'=_l Et

jEEi+l _' crr(E._, ) dEO(x,E4) dE ti = E i

N--I [Ej+I [Ei+l+ Z G,.(E,U) dE O(.c,E') dE' + _i

j=i+l dE) JE i

(4o)

over the energy group Ei < E r < Ei+l. The first, double integral in equation (40) represents

integration over the lower triangle illustrated ha figure 2(c). The second double integrM in

equation (40) represents integration over the upper triangle illustrated in figure 2(c). Define

gl(E') = [ cr,.(E,E') dEJE = E i

Ei+l92(E') G,.(E,E') dEJ E= o 1Et

(41)

_II d

cr,.( E,E _) dE:"l(E') =. =_:_

i [ El+ I,-2,.,,,(v_,,,)= ,_,.(E,ff,,,) dEJE i

(42)

11

thenemployanotherapplicationof ameanvaluetheoremfor integralsto writeequation(40) illthe foml

d_bi

d"-x- + "ffdPi = gl [El + Ol( Ei+l - El)] 4Pi + g2 [Ei+I + 02(Ei+2 - E'i+I)] (I)i+l

+ rl [El + Ol(Ei+l - El)] (_i+

N-1

2 ,j[Ej +o2(Ej+ - Ej)] ¢jj=i+l

(43)

Tiffs produces the matrix coefficients associated with tile energy group Ei to Ei+l so that,

ai,i = Yl + 7"1-- "ffl

ai,i+l = g2 + r2i,i+l

ai,i+j = r2i,i+j

(j = 2, 3 .... ) (44)

In this way', the diagonal and off-diagonal elements of the coefficient, matrix in equation (37) arecalculated. "

For a compound target material made up of more than one type of atom, we modify slightly

the .solution technique given ill reference 19. For a target, material comprised of component 1

and component 2, there are two values for a. A value a'l is detemlhmd for component 1 and a

value c_2 is determined for component 2 of the compound lnaterial. In this case, equation (37)takes on the form

eta,E)= )¢(x,d)dE'E

+ if

ff+ a,.2(E,E') 4,(x,E') dE'+g(E,x)

O'rl(E,E# ) ¢(X,E#) dE'

(45)

where ers 1 and as2 are scattering terms and rrrl and (Tr:_ are reaction terms associated with therespective components of the compound material. These terms are calculated in the HZETRN

code. We consider two cases. Case I requires that the E/c_2 line be above the E/ct 1 line. In

case II, ct2 equals 0 (the hydrogen case), and the limit of integration for the second integral goes

to infiuity. Each case is considered separately.

For case I, we assume that. o 1 > 0(2 > 0 and select, the energy spacing Ei+ 1 = Ei/a 1. We

then proceed as we did using the single component shield material. Integrate equation (45)

from Ei to Ei+ 1 and interchange the order of integration on the double integral terms. Define

_i = jE it'L;'+lg(E,x) dE and obtain the equation

dO id---_- + _'_; = HI1 + H12+ H21 +H22 +Ell + K12+ K21 + K22 + _i (46)

where H dl and H32 represent the elastic scattering caused by collisious with /3 type atoms and

Kdl and h'32 represent, the nonelastic scattering. Note that H31 and K31 are integrals over the

energy range (E i, Ei+I) for ;7 = 1, 2, and H32 and K32 represent integrals over higher energies.

12

The integralsHll, H12, Kll, and 1712 are the easiest to evaluate because of the exact spacing

of t.he energy part.it.ion. These integrals have tile forms

f E,'+I rE'Hll = o-_I(E,E' ) dE O(z,E') dE'E i E--E i

(47)

an d

Ei+2 fEi+lH12 = aEi+l E=_IE,_.,.I(E,E') dE 0(z,E')dE t

(48)

or,.l (E,E') dE O(z,E') dE' (49)I_'11 = JEi . Ei

N-I [Ej+ l [<+,1712 = Z cr,.t(E,E I) dE O(x,E') dE' (50)j=i+l JEj JEi

Here the first sub,rip, represents the material component. A second subscripl of 1 represents

integration over the lower triangle in figure 2(c). A second subscript, of 2 represents integration

over upper triangles, like figure 2(c), or higher rect.angles, like figures 37a) and (b). Defining the

1,er ms

L't

E _r_,_3(E,E') dE (13 = 1,2) (51)

h2(1)(Et) = [<+' or,. l(E,E I) dE (52)JE=_ I E t

kl(j)(E' ) = a,..j(E,E')dE (3 = 1.2) (53).E=Ei

jf Ei+ lk2(a)(E' ) = _,.,:3(E,E')dE ('3= 1.2) (54)E:E i

and tLsing the mean value t.heorenl for integrals we obtain from equat, ions (,17) through (50)

Hll = hl(1)[Ei + 01(Ei+I - Ei)]Oi

H12 : h2(1)[Ei+l + 02(Ei+2 -- Ei+l )]_i+1

Kll = kl(1)[Ei + Ol(Ei+l - Ei)]_i (55)

1712 :

N - 1

Zj=i+l

_,2(1)[Ej + O_(Es+_ - E5)]¢5

where 01,0"2 and 01,02 define intermediate energy values associated with the mean value theorem.

The integrals H21 and H22 are associated with integration lilnits (E,E/c_2) and energy

intervals dictated by the select.ion of ¢_1 for deternfilfing the energy spacings. The integral

H21 is as_ciat.ed with l.he triangular area. shown in figure 3(a) and takes l.he form

: cr_.,(£',E') dE O(a:,E') dE'H"21 .: Ei Ei .

(56)

13

The integral H22 is associated with the remaining shaded area shown in figure 3(a). Thisremaining area is made up of a number of rectangles, trapezoids, and triangles. We approximate

the integral over each of these rectangles, trapezoids, and triangles as a fraction r/j of the integral

over the whole rectangle with area Aij = (Ei+ 1 - Ei)(Ej+ 1 - Ej). Integral H22, therefore, takesthe form

N-1 /Ej+I/Ei+IH2,e = _ ,jj ,_.e(E,E') dE ¢,(.,U) dE' (57)j=i+ 1 J Ej J Ei

where

L1 (/%-+1 < _2)

- - E_)(Ej+I - E,-/_2)

qJ = 0.5[(Ei+l - a2Ei,l,) + (Ei+lAij - °_2Ei)]('E'i+l - El) (o_< Ej < Ej+ 1 < _)

Aij

Defining the term

Ei+lha(2)(E')= jE=E i

H21 and H22 can be written as

c%,2( E ,E' ) dE

(58)

(59)

H21 = hl(2)¢iN-I }H22 Z qjh3(2) _jj=i+l

(60)

Similar to /_'11 and A'12 , integra[s /(21 and A22 are given by'

K21 -- kl(2)_iN-1 }K22 Z k2(2)_J

j=i+l

(61)

The coefficients for our system of differential equations (eq. (341)) are then given by

ai, i = hl(l ) + hl(2) + kl(1) + kl(2) - g

ai.i+l = h2(1) + r/i+1h3(2)+ k2(1) + k2(2)

ai,i+ 2 = 71i+2h3(2) q- k2(1) q- k2(2)

ai,i+3 = r/i+3h3(2) q- a:2(l) q- ]_'2(2)

(62)

where evaluation at the appropriate mean energies is implied.

14

In case II, the second component is hydrogen which means thai. c_2 = 0; therefore, one of the

limits of integration becomes infinite. We once again let, a I determine the energy spacing and

integrat,e equations (47) through (50) over an energy interval (Ei, Ei+I) which is determined by

the E' = E'/O'l line. Using the definitions given by equations (39) we integrate equation (45)

over the interval (Ei, Ei+I) and then interchange the order of integration in the resulthlg double

integrals to obtaindO id'7 + _'q_i = HI + H_ + Iq_ + lx_ + _i (63)

W lie re

---- O,l(E,E/) dE O(x,E') dE': 71dE i E E_

[ E,+2 [ Ei+, (E,E') de ¢)(.r,r') dE' ((54)+ cr_ 1JEi+l JE=_tl Et

_/Ei+l iE_= dE O(x,E') at;"Ei i

N IEi+j+l IEi+l+ E cr_._(E,E') dE 0(x,PJ) dE' (65)j=l dEi+J JEt

f Ei+l iE S-¢K*_ = o-,.,.,_(E,E') dE ¢,(x,E') dE'Ei i

N El+j+ I

+ E IE cr,.,,_(E,E') dE O(z,E') dE' (66)j=l ' i+j

where for all N* greater than some integer N > 0, we know that O(_',E) will be taken as zero.

Define

E /

h4(E') =iE _si(E,E') dEi

Ei+lhs(E') = obl(E,E') dEJ (_l Et

E I

h6(E') = iE °%2(E'E') dEi

DT(j)(Et ) = i Ei+j+lJEi+J a_2(E,E') dE

E'

k:_(j)(E') = JE o',.,,_(E,E') dEi

k4(_)(E' ) = i Ei+l o-r,,.j(E,];J ) dEd E i

15

(Ei < E' < E/+I) (67)

(Ei+I < Et< Ei+2) (68)

(El < E' < Ei+I) (69)

(El+ j < E( < Ei+j+l) (70)3

(71)

(72)

then write the coefficients associated with the system of differential equations as

ai'i = h4 + h6 + t'3(1) + k3(2) - _" /

/

ai, i+l h5 + h7(1) + k4(1) + k4(2)

ai, i+2 h7(2) + t"4(1) + k4(2)(73)

where evaluation at. the appropriate mean energies is implied.

In this way, we generate a system of equations having the triangular form given by

equation (34). We again use the source temls g(E i,xj ,y/,.) obtained from the H ZETRN simulationof tile solar particle event of February 23, 1956, associated with an aluminmn-water shield-

target configuration. Note that now we must solve the multigroup equation (34) associated withequation (40) for the multiple atom target, material of water. We consider the cases of discrete

shield thickness x,2,xa, ... and apply the multigroup method to the solution of equation (16)applied to all target material y > 0. For each xFvalue considered, the initial conditions are

obtained from the previous solutions generated where y = 0. This represents the application of

the nmltigroup method to two different regions: region 1 of all shield material and region 2 of alltarget material. We then continue to apply the multigroup method to region 2 for each discrete

value of shield thickness, where the initial conditions on the start of the second region representexit conditions from the shield region 1. This provides for continuity of the solutions for the

fluence between the two regions.

Reaction Effects on Evaporated Neutron Fields

In the present calculations, the two-stream bidirectional version of tlLe multigroup method isalways used because of its improved physical description and improved accuracy, especially near

the boundaries of the incident radiation. Here, the assumption is made that half the evaporationsource neutroi_s move in tile forward direction and the other half move in the backward direction.

The multigroup equations are, therefore, solved twice, once for the forward half of the source

term and again for the backward half of the source. We evaluate the radiation fields for tile

solar particle event of February 23, 1956, in an Muminum slab 100 g/cm 2 deep with the results

shown in figure 4 using the computational code of Heinbockel, Clowdsley, and Wilson (ref. 19) inwhich the evaporation neutrons are transported under elastic scattering only. Also shown in the

figure are results obtained with the nonelastic process ms described by the present calculation.

A general decrease occurs in the 5 to 25 MeV neutron flux with a corresponding increase below2 MeV. As one would expect, the more reactive energetic neutrons are removed from the field

by the reactions with the appearance of lower energy neutrons as reaction products. Also shown

are results from the MCNPX Monte Carlo code. It is clear that the discrepancies reported byClowdsley (ref. 22) are not from neglect of nonelastic processes. Similar results are shown in

figure 5 for a water target along with Monte Carlo calculations using the LAHET code (ref. 6).The discrepancies observed in our earlier calculations are clearly due to factors other than effects

of reactive processes associated with the transport of evaporation neutrons.

High-Energy Backward Produced Neutrons

Although the two-body interactions of nucleons are limited to the forward scattering, themultiple scattering of nucleons in nuclei can produce a nucleon in the backward direction after

several scattering events. In addition the Fermi motion within the nucleus will enhance this

16

effect.. Tile angular distribution of nucleons from nuclear reactions was estflnated by Ranfl

(ref. 23) to be given by the approximate function

(o < 0 < _/2) [

J(Otherwise)

(74)

where )_ = (120 + 0.36AT)/E, with E the secondary particle energy in MeV, A T the a.tonlic

weight of tile struck nucleus, and N a normMiza.tion constant. The fraction of neutrons producedin the forward direction ks

/,

&'or = 2rr[ g(Ar,E,0) d(cos0) (7:i)J for

The corresponding backward-produced neutron fraction is

Fbac = 1-Ftbr (76)

The apl)roxinlat.e isotropic component of the interaction can be taken as

_o = 2El,a(. (77)

In earlier developlnent of the multigroup method, we assumed thai. the direct reaction products

would mainly be of high energy and in the forward direction and, therefore, adequately solved

by the HZETRN code. The assumed isotropic evaporation source was then treated with

the multigroup method by assuming half of the evaporation source was propagating in theforward direction and the second half in" the backward direction. In similar fashion, we replace

evaporation and direct, reaction spectra in the code as follows:

I_(E' E_) --&s°[f_(E'E')+fll(E'E')] ! (78)

fd( E,E') F-- HZETRN[f (/=J,E') + fd(E,E')] J

where FttZETR N = 1- Fi_o. These replacements (eqs. (78)) were made in the new version

of HZETRN/multigroup code which is now only a minor modification. The terms on the

right-hand side of equations (78) are shown in figure 6 and should be compared with figure 1.

The inlportauce of the reactive channels are accentuated because of the higher energim of the

backward propagating neutrons.

Results for Ranft Modified Source

We have reevaluated the neutron fiel&s in aluminum and water for the solar particle event

of February 23, 1956, with the angular dependence of Ranft and the separations into HZETRN

and isotropic components. The results are shown in figures 7 and 8 along with MCNPX (ref. 22)

and LAHET (ref. 6) derived Monte Carlo results. The addition of the high-euergy backward

component is essential in reaching agreeinent with the MCNPX and LAHET codes. It is clear

that the di_repa.ncies observed by' Shinn et al. (ref. 16) in the 50 to 200 MeV region are due

to the energetic neutrons produced in the backward direction as reasonably described by the

Ranft formula. The Ranft formula appears to overestimate the backward component for oxygen

t)ecau_" agreement is improved for the omnidirectional flux a.t larger depths although forward

and backward components may be somewhat incorrect. Quite satisfactory agreement is obtained

a.t. the largest, depths. Still many of the cross sections in the HZETRN code are crude and a

continued effort, to improve them is expected to further enhance the calculated results.

17

Improved Cross Sections

A programof improvedcrosssectionshas beenin progressfor severalyears. Greatestattentionhasbeengivento fragmenteventsfor heavyionswhichhavebeensignificantlyimproved(refs.24and25). Recentyearsof researchhaveresultedin improvedabsorptioncross sections

(ref. 26) and improved production cross sections for the present study using the LAHET code

with results in table 2. The effects of these new cross sections are shown in figure 9.

Table 2. Number of Nucleons Produced in Nuclear Collisions With Aluminum Atoms

Cascade Nucleons Evaporation Nucleons

Energy, MeV _ -- p p ---*p n ---*p p -- p

25

200

400

1000

2000

3000

0.13

0.73

0.91

1.481.97

2.32

rt--?l p---_

0.26 0.09

1.26 0.801.58 0.99

2.12 1.59

2.58 2.112.92 2.49

0.24

1.181.49

2.02

2.45

2.76

0.38

0.75

0.87

1.30

1.44

1.48

n -- T_ p -- n

1.13 0.431.22 0.90

1.29 1,04

1.69 1.55

1.83 1.701.86 1.72

0.961.11

1.13

1.44

1.581.62

Concluding remarks

The methods described herein greatly improve the HZETRN computer code's neutron

transport predictions. To summarize, a bidirectional nmltigroup solution of the straight-ahead

Boltzmann equation for elastic and nonelastic transport of tow-energy evaporation neutrons has

been implemented. The resulting computer code was added to the existing HZETRN computer

code which was developed at the Langley Research Center. With the new modified code, various

simulations were conducted to test its accuracy. The Monte Carlo codes LAHET and MCNPX

were used as benchmarks of accuracy. The modified code with and without the inclusion of

nonelastic scattering processes for the evaporation neutrons is compared with these benctmmrks.

The neutron fluences are calculated at depths in aluminmn of 1, 10, and 30 g/cln 2. These depths

were selected because they represent typical values of shielding associated with the constantly

changing space environment encountered by astronauts. The shield material of aluminum is

typical because of weight considerations in space. The neutron fluences are calculated at

depths of l, 10, and 30 g/cm 2 in water. Water is used to model human tissue. Including

nonelastic scattering processes in the calculation of the transport of low-energy evaporation

neutrons slightly improves the prediction of neutron fluence at low energies, but. the prediction

of neutron fluence at slightly higher energies, around l0 MeV, is decreased by this change.

The HZETRN code underestimates the fluence of neutrons in the range of 5 to 200 MeV. In

an effort to fix this problem, a formula by Raifft was used to estimate the number of isotropic

neutrons at each energy. Using the bidirectional multigroup method to propagate all the isotropic

neutrons greatly improved the neutron fluence predictions.

The addition of improved production cross sections to the code showed only modest improve-

ment to the predicted neutron fluence. In the future, more accurate absorption cross sections

will be added to the code, but this is expected to also have only a modest effect.

18

Referen ces

13. Wilson, Johl, W.:

14. WiL,_on, John W.;

pp. 231- 237.

1. WiLson, John W.; Cucinot, ta, F. A.: Shinn, J. L.; Simcllsen, L. C.; Dubey, R. R.; Jordan, W. R.; Jones, T. D.;

Chang, C. K.: and Ki,n, M. Y.: Shielding From Solar Part.icle Even( Exposures in l_'e l) Space. Rad. Meas.,

vol. 30, 1999, pp. 361 382.

2. Wilson, J. W.: Cucinotta, F. A.; and SimoF,seu, L. C.: Proton Target Fragmentation Effects in Space Exposures.

Adv. ,qpace Reds., To be Published.

3. Wilson, John W.; Nealy, John E.; Wood, James S.; Quails, Garry D.; At, well, William: Sldnn, Judy L.; and

Simoasen, Lisa ('.: Variations in Astronaut Radiation Exposure Due (.o Anisotropic Shiehl Distribution. Health

Phys., vol. 69, no. 1, 1995, pp. 34 45.

4. Wilson, John W.; Badavi, Francis F.; (:uchFot.ta, Francis A.; Shinn, Judy L.; Badhwar, (;autam D.; Sill)erberg,

R.; Tsao, C. H.; Towlrsend, Lawrence W._ and Tril)athi, Ram K.: HZETRN: Descrq_liol_ o/Free-Space Ion and

Nucleon Transporl and 5'hi¢_ldin9 Compule'r PmgTttm. NASA TP-3495, 1995.

5. Alsnfilier, R. (_., .]r._ Irving, D. ('.: Kmuey, W. E.: and Moran, H. S.: The Validity of ihe Straightahead

Approximation in Space Vehicle Shielding Studies. ,q_.co,d 5ymposi_lm o1_ Pmtt.clios_ Agaim_l RadiatimJ._ i'll

5,_tmct, Arthur ]_eet.z, .lr., ed.. NASA SP-71, 1965, pp. 177 186.

6. Prael, R. E.; and Lichte|tstein, Henry: [rser (;aide to L('5: Tb¢ L.4HET (:od_ Systtm. LA-UR-89-3011, Los

Alamos Nat. Lab., 1989.

7. Nealy, John E.; Quails. (;arty D.; and SimolLsen, Lisa C.: Inl_grated Shield Design Methodology: Ai)plieation to

a Satelfite Instrument. Shielding Strategies for HumaT_ :;ImC_ Es'ploTrllimJ, .l.W. Wilson, J. Miller, A. Konradi,

and F. A. Cuciuotta, eds.,NASACP-3360, 1997, pp. 383 396.

8. Wikson, John W.: (_uciuotta, F. A.; Shhln, J. L.; Shnonsen, L. ('.; and Badavi, F. F.: Ov_ rvie_v oJ HZETRN t_d

BRNTRN ,s'pac¢ Radialim_ 5'hidding Codes. SHE Paper No. 2811-08, 1996.

9. Wi[soll, .John _,V._ aud Khandelwal, (;. S.: Proton Dose ApproxhtJat, ion m Arbitrary (:onvex (_ometry. Natl.

T_cbTwl., vol. 23, no. 3, _pt. 1974, pp. 298 3(}5.

10. WiLson, John W.I _uFd Lamkin, Stanley L.: Perturbation Theory for (:harged-Particle Transport in ()Fie

Dimension. Natl. ,s'ct. /:' Eng., vol. 57, no. 4, Au K. 1975, pp. 292 299.

11. Lamkin, Stanley Lee: A Theory lbr High-Energy Nucleon '/rauspor! in One Dhnension. M.S. Thesis, Ohl

Ek)miuion Univ., Dec. 19L_I.

12. |,claw, John: Tsao, C. H.: and Silberberg, R.: Matrix Methods of(:osmic Ray Propagation. ('omposilio_ and

Oriqin of Cosmic Rttys, Maurice M. Shapiro, ed.. D. R¢,iel Publ. (',o., 1983, pp. 337 342.

Analysis of the Theory of High-Energy Io_ Transport. NASA TN D-8381, 1977.

and Badavi, F. F.: Methods of (;alact,i<" Heavy Ion Transport. Radar. R¢'s., vol. 108, 1985,

15. WiLson, John W.; Townsend, Lawrence W.; Nealy, John E.; Chu.n, Sang Y.; Hong, B. S.: Buck, Warren W.:

Latakia, S. L.; (;anapol, Barry D.; Khan, Ferdous; and Cucinolt, a, Franci,s A.: BRYNTRN: ,4 Ba ryon "D'a_sporl

Mod_l. NASA TP-2887, 1989.

16. Shmu, Judy L.; Wilson, .]ohn W.; Nealy, John E.; and Cucinotlra, Francis A.: Comt_rimm of Do.s_ Eslimatt._

Using the Bu ildup- F_tctor Melh od an d a Baryo _ T_a_sporl Code (BRYNTR N) Wilh Mo_d _ ('a rio Re sulls. NASA

TP-3021, 1990.

17. Lamkin, Stanley L.: Khandelwal, (;ovhFd S.; ShhH_, Judy L.; and Wilson, John W.: Space Proton Transport in

One Dime_tsion. Nurl. Sci. _:'; Eng., vol. 116, no. 4, t99,'1, pp. 291 299.

18. ShhH_, Judy L.; Wilson, John "_V.: Lone, M. A.; Wong, P. Y.; al_d (',osten, Robert. ('.: Preliminary Eslimahs

oJ" Nuclton Flus:¢s i_ a Water Target EsToscd to Solar-Flare Protons: BRYYTRN _ *_ur_"Moult Carlo ('od¢.

NASA T,M-4_5, 19_.

19. Heintx)ckel, John H.; Clowdsley, Mart.ha S.; and Wilson, John W.: An lmprov_d Neulro_ 7_a_,sporl Algorithm

for Space Radialw n. NASA/TP-2009-209865, 2000.

20. V_*i[son, .]o]ltl _V.; Towusend, Lawrence "_V.; Schinunerling, Walt.¢,r S.; Khaadelwal, (;oviHd S.: Khaa, Ferdous S.;

Ne'aly, John E._ Cucinotta, Francis A.; Simoasen, Lisa (!.: ShhFn, Judy L.; and Norbury, John W.: Transport

Methods an d In te _ulio_s for ,_'pac¢ Rad_alio ns. NASA R P-1257, 1991.

19

21. Haffner, James W.: Radiation and Shielding in Spact. Acadeltfic Press, 1967.

22. Clowdsley, Martha Sue: A Numerical Solution of tile Low Energy Neutron Boitzmaan Equation. Ph.D. Thesis,

Old Dollfinion Univ., May 1999.

23. Ranft, J.: Lecture 22: The FLUKA and KASPRO Hadronic Cascade (',odes. (;omputer Techniques in RadialioT_

Ttan.sport and Dosimelry, Walter 1C Nelson and Theodore M. Jeiflcins, eds., Plenum Press, 1980, pp. 339-371.

24. Wilson, J. W.; Tripathi, R. K.; Cucinotta. F. A.: Shhln, J. L.; Badavi, F. F.; Chtm, S. Y.; Norbury. J. W.; Zeitlhi,

('.. J.; Heilt)ronn, L.; and Miller, J.: NUCFRG2: An EvalualioTJ oJ" the Semiempirical Nuclear FTtrgme_,lalion

Dalabast. NASA TP-3533, 1995.

25. Cucinotta, F. A.: Cluster Abrasion ot' Large Fragment_ in Relativistic Heavy Ion Fragmentation. Bull. Am. Phys.

Sot., _vl. 39, no. 5, 1994, p. 1401.

26. Tripathi, R. K.; Wilson, .1. W. ; al|d (',ucillotta, F. A.: Acc urate Universa.[ Parameterization of Absorption Cross

SectiotLs: II--Neutron Absorption (',ross Sections. Nucl. hlstvum. U Methods Phys. Res. B, vol. 129, no. 1, 1997,

pp. 11 15.

2O

10-I _

fe .AI(E,E' )

Figure 1. Evaporation and direct, cascading neutron spectral effects for collision of 500 MeVneutrons in aluminum.

21

E' = EI_

E'

/Ei Et+ I

(a) General.

E'=E

El+ I[(_

El+ l"

E

E' = E/_x

E. , El+ I

(b) /;/+1 < _-_0 '

E'=E

E

E'

El+2

Ei+ I

E i/

E i Ei+ I

E'

= E/or / E'=E

-E

El+2

g'

El+2

Ei+ I

E i

E i El+ 1

(c) Ei+l =(_ '

(d) El+ 1 > E'ioL "

Figure 2. Various energy partitioning schemes.

= E/(x

El+2

=E

22

E'

E,_,

El+

Ei+ I_El

EI, I

Eilo_L)

E,÷ ,

El+ I"

Ei

E'=E

E

(a) E;+I < E__/.

E'

E,,,

El+ I}o_Ek

El+ t

Ei

E' = Elct

E'=E

E

(b) Nonelastic sca, tl, ering.

Figure 3. Multigroup energy partition.

23

i011

i

E

e-¢0

r7

10 9

10 7

10 5

10 3

"% ._

"\.

\.\ :i_::

,.;:i::

N'-_.

I 01 '_::

10-t - Elastic '_I- - - Elastic and inelastic '_

i2..10-3 , ,,,,,,, , ,,,,ml , _i,,,,,,

10 -2 10 -I 10 0 l01 10 2 10 3 10 4

Energy. MeV

(a) Depth of 1 g/cm 2.

I0 II

eq I

E

e-

Lt.

10 9 ":"":'" .....

10 7

10 5

10 3

10 ]

°

\.

\'\

10 -I -- Elastic i

Elastic and inelastic[ I

10-3 ........ , ,J,.,. . ,! ......

10 -2 10 -I 10 0 101 10 2 10 3 10 4

Energy. MeV

(b) Depth of 10 g/cm 2.

Figure 4. Energy spectra of neutron fluence in ahmfinum calculated by HZETRN program with

bidirectional multigroup method used t,o transport evaporation neutrons.

24

10 I1

10 9 * " . .

>_ 107

°'E-_ lo5e...

_- 103g

IOt

10 -I _ --

t0-310-2

°

"\'4 ..... .

Elastic- - - Elastic and inelastic

MCNPXI I illlll I I IIIlll I I IIIIll I I IIIlll

10-I 100 10 j 102

Energy, MeV

(c) Depth of 30 g/cm 2

eFigure 4. (.onclud d.

,,<

7

10 3

ii

I lil IIIll

104

25

10 8

10 6

t--i i

_ 104e-

_ 102

E

10 °

- - - Elastic and inelastic............ LAHET

Elastic

10-2 , ....... I , t,,,,,,l , ,,,,,,,

I0 -] I0 o I01 102

Energy+ MeV

\\ ,

\

+v':.+

'\,i

M

i i i 111111

103

(a) Depth of 1 g/cm 2.

4(:

\

I I",I IIIII

I0 +

10 8

>

it',-+

E

E

10 6

10 4

10 2

I0 0 ....

::<£Z .....

_d++--.. :

"<_<7."

Elastic and inelastic............ LAHET

Elastic

3

+,<:....

N'"

.k.

i i i iiiii

103

10-2 , k,,,,.t J +_,,t.,I . , ,.,,.,I j ,ii.,,+.

i0 -1 i0 o 101 10 2 10 4

Energy. MeV

(b) Depth of 10 g/cm e.

Figure 5. Energy spectra of neutron fluence in water calculated by HZETRN program with

bidirectional nmltigroup method used to transport evaporation neutrons.

26

lOs

10 6

._ 104e-

_ I0 ?

10°

Elastic and inelastic............. LAHET

Elastic

.... _ .... ,,'-L-l,, .......

10 0 101 10 2

Energy, MeV

";%,..

I I I IIIII

103

),

,\:%

(c) Depth of a0 g/cm 2.

Figure 5. Concluded.

I

I lil IIIII

lo4

27

10° -

lO-1

10-2

10-30

Fis o [fe.AI(E,E') +.fdAI(E,E')]

,AI(E,E ) +fdnn,Al(E,E')]

100 200 300 400 500

Energy, MeV

Figure 6. Forward moving and isot, ropic neutron spectral effects for collision of 500 MeV neutronsin alumhmm.

28

10 II

>z_

e.i I

E

¢-.

,5

e-

10 9 _ " * .--

i0 3 ........._:

,_.C"

iO5

h:

10 3

101

10 -] -- Elastic- - - Elastic and inelastic

10-2 10 -1 10 0 101 10 2

Energy. MeV

(a) Depth of 1 g/cm _.

\,\

i i i 11111

10 3

i I!,l IIIII

I (I4

10 I1

i

E

e-,

_d

LI.

10 9

10 7

10 5

L .-L 5" ._

10 3

I01

I0 -I Elastic

- - - Elastic and inelastic

o MCNPX

]0_3 I I Illlll I I IIIIIt I I _ IIIll I I I ]Jill J I [till

10 -2 10 -I 10 0 101 10 2 10 3

Energy. MeV

(b) Depth of 10 g/cm _.

'\

i

'j

l tllJ J LLii

10 4

Figure 7. Energy spectra of neutron fluence iu ahmfinum calculated by HZETRN program with

bidirect, ional multigroup method used to transport all isotropic neutrolL'_.

29

i0 li

10 9

107

E105

f-

g 103

lO I

lO-I

t

..... --L-_L-._.IL_:.-*--

Elastic- - - Elastic and inelastic

MCNPX

I I IIII11 I I IIIIII I I IIIIII I I IIIIII

10 -I 10 0 101 10 2

Energy, MeV

(c) Depth of 30 g/cm 2.


\

I I I IIIII

103

\,,t

t

\

, Jh ,,,,,I

10 4

30

ae-i

e-

_a

gr,

10 8

10 6

10 4

10 2

i0 °

Elastic and inelastic

............ LAHET-- Elastic

,, ,,,,--_,, .... ,,-71,, , ,

10 ° 10 ] 10 2

Energy. MeV

\

i i i iiiii

10 3

l I_1 IILI

io4

(a) Depth of 1 g/era 2.

>

,¢-q

¢..

g

e-

rr

10 8

10 6

10 4

10 2

10 °

7:e->,vz;_,__

- - - Elastic and inelastic

............ LAHET

Elastic

..... ,,-'51", .... ,,--'71,, ....... t

10 o l0 t 10 2

Energy, MeV

%:

i i J IIIII

1(13

N:

'k

i i Ii ] ILlJ

10 4

(b) Depth of 10 g/cm 2.

Figure 8. Energy spectra of neutron fluence in wa,ler calculated by' HZETIR,N progr;tm wilhbidirectional mu]t, igmup method used to transport all isotropic neut, rons.

31

10 8

10 6

i

104

10 2

e-

10 °

- - - Elastic and inelastic............ LAHET

Elastic I

10° 101 i0 _

Energy, MeV

\;

i i i Iltll

10 3

\,\



;i

, ,!,,,,,,I

10 4

32

I011:

E

r.-

_5

,'r"

109

I07

105

103

I0 _

10-I

10 -310 -2

-- Old cross sections- - - New cross sections

MCNPXI I IIitll I I IIIIIL I I III1[I I I III111

I(y I 10 ° 101 10 2

Energy, MeV

(a) Depth of 1 g/ctn 2.

\

\

k,,

\,

i

i

I I I Illll I I !1 IIIIll

1I)3 10 4

10 i t

7::

?.

e-

i()_ ..'.°,.....

107

105

103

I01

.....:£i.: ,

"<,.N

"v,

10 -i Old cross sections [- - - New cross sections I

10 -3 L i lllllta J L JtLHII t t ,l,m' J ' IILLU

10-2 10 -1 10 o l0 t 10 2

Energy, MeV

I I iII111

l0 4

(b) Depth of l0 g/cm 2.

Figure 9. Energy spectra of neutron fluence in aluminutn calculated with new cross sectiolas.

33

iO fl

10 9 .... _ '-*-. .........

10 7

it_

E_ lo5e-

_ io3_J

_ 10 i

10 -1

10 -310 -2

..>.. ....

Old cross sections

-- - - Ne.w cross sections

i i i iiii1

10-I i0 o l0 t 10 2

Energy, MeV



\

10 3

I Ill IIIIi

10 4

34

Form ApprovedREPORT DOCUMENTATION PAGE OMBNo.0704-0188

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

July 2000 Technical Public ttion

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

An Improved Elastic and Nonelastic Neutron Transport Algorithm for SpaceRadiation WU 101-21-23-03

6. AUTHOR(S)

Martha S. Clowdsley, John W. Wilson, John H. Heinbockel, R. K. Tripathi,Robert C. Singleterry, Jr., and Judy L. Shinn

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

NASA Langley Research CenterHampton, VA 23681-2199

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

National Aeronautics and Space AdministrationWashington, DC 20546-0001

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17971

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TP-2000-210299

11. SUPPLEMENTARY NOTES

Clowdsley: NRC-NASA Resident Research Associate, Langley Research Center, Hampton, VA; Wilson, Tripathi,Singleterry, and Shinn: Langley Research Center, Hampton, VA; Heinbockel: Old Dominion University, Norfolk,VA.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

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Subject Category 93 Distribution: StandardAvailability: NASA CASI (301 ) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

A neutron transport algorithm including both elastic and nonelastic particle interaction processes for use in spaceradiation protection for arbitrary shield material is developed. The algorithm is based upon a multiple energygrouping and analysis of the straight-ahead Boltzmann equation by using a mean value theorem for integrals. Thealgorithm is then coupled to the Langley HZETRN code through a bidirectional neutron evaporation source term.Evaluation of the neutron fluence generated by the solar panicle event of February 23, 1956, for an aluminum-water shield-target configuration is then compared with MCNPX and LAHET Monte Carlo calculations for thesame shield-target configuration. With the Monte Carlo calculation as a benchmark, the algorithm developed in thispaper showed a great improvement in results over the unmodified HZETRN solution. In addition, a high-energybidirectional neutron source based on a formula by Ranft showed even further improvement of the fluence resultsover previous results near the front of the water target where diffusion out the front surface is important. Effects ofimproved interaction cross sections are modest compared with the addition of the high-energy bidirectional sourceterm s.

14. SUBJECT TERMS

Multigroup; Secondary neutrons; Neutron transport; HZETRN; Radiation shielding

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