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NASA/TP-2000-209865
An Improved Neutron Transport Algorithm
for Space Radiation
John H. Heinbockel and Martha S. Clowdsley
Old Dominion University, Norfolk, Virginia
John W. Wilson
Langley Research Center, Hampton, Virginia
!
i
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March 2000
https://ntrs.nasa.gov/search.jsp?R=20000031890 2018-05-25T01:12:02+00:00Z
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NASA/TP-2000-209865
An Improved Neutron Transport Algorithm
for Space Radiation
John H. HeinbockeI and Martha S. Clowdsley
Old Dominion University, Norfolk, Virginia
John W. Wilson
Langley Research Center, Hampton, Virginia
National Aeronautics and
Space Administration
Langley Research CenterHampton, Virginia 23681-2199
March 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
Abstract
A low-energy neutron transport algorithm for use in space radi-ation protection is developed. The algorithm is based upon a multi-group analysis of the straight-ahead Boltzmann equalion by using amean value theorem for integral_. This analysis is accomplished bysolvzng a realistic but simplified neutron transport test problem. Thelest problem is analyzed by using numerical and analytical proce-dure.s to obtain an accurate solution within specified error bounds.Results from the test problem are then used for determining meanvalues associated with rescallering terms that are associated with amultigroup solution of the straight-ahead Boltzmann equation. The
algovithm is then coupled to the Langley HZETRN code through theevaporation source term. Evaluation of the neutron fluence gener-ated by the solar particle event of February 23, 1956, for a waler and
an alu rain urn-ware r sh ie Id-ta rget configu _ lion i.s th eu compa red wit hLAHET and MCNPX Monte Carlo code calculations for the same
shield-target configuration. Th.e algorithm developed showed a greatimprovement in results over the unmodified HZETRN solution. In
addition, a two-dwectional solution of the evaporation sourc_ sho wedeven further improvement of the fluence near the front of the watertarget where diffusion from the front surface is important.
Introduction
The purpose of this paper is to present an improved algorithm for the analysis ofthe transport "
of low-energy neutrons arising in space radiation protection studies. The design and operational
processes in space radiation shMding and protection require higtfly efficient, computational
procedures to adequately characterize time-dependent environments, time-dependent geometricfactors, and to address shield evaluation issues in a nmltidisciplinary integrated engineering
design environment. One example is the recent study of the biological response in e.xposures
to .space solar particle events (SPE's) in which the changing quality of the radiation fields atspecific tissue sites is followed over 50 hours of satellite data. to evaluate time-dependent fac|ors
in biological respon._ of the hematopoietic system (ref. 1). Similarly, the study of cellularrepair dependent effects on the neopla.stic cell transformation of a C3HIOT½ population in low
Earth orbit, where trapped radiations and galactic cosmic raysvary continuously in intel_sity and
spectral content about the orbital path (ref. 2), requires computationally efficient codes to matchtime-dependent boundary conditions around the orbital path. But. even in a steady environment
which is homogeneous and isotropic, the radiation fields within a spacecraft have large spatial
gradients and highly anisotropic factors so that. the mapping of the radiation fields wittfin theastronaut's tissues depends on the astronaut, timeline of location and orientation within the
spacecraft, hltefior where large differences in exposure patterns that depend on the activity of
the astronaut have been found (ref. 3). Obvious cases exist where rapid evaluation of exposurefields of specific tissues are required to describe the effects of variations in the time-dependent
exterior environment or changing geometric arrangement. A recent study of the time-dependent
response factors for 50 hours of exposure to the SPE of August ,I, 1972, required 18 CPU hours
on a VAX 4000/500 computer by using the nucleon-light ion section of the deterministic high
charge and energy transport code HZETRN. The related calclflation with a standard MonteCarlo code such as HETC or LAHET, which only handles neutrons, protons, pions, and alphas,
would have required approximately 2 years of computer time to complete the study. The designenvironment alto requires rapid evaluation of the radiation fields to adequately determine effects
of multiparameter design changes on system performance (reg. 4 and 5). These effects are the
drivingfactorsin thedevelopmentandu_ of determiltisticcodesandinparticulartheHZETRNcodesystemthat handlesall naturallyoccurringatomic ionsandneutrons.
Thebasicphilosophyfor the developmen!of tile deterministictIZET_N codebeganwith thestudy by Msmiller et al. (ref. 6) with an early version of IIETC, wherein they dexnonstratedthat the straight-ahead appro_mation for broad beam exposures was adequate for evaluation of
exposure quantities. Wilson and Khandelwal (ref. 7) examined the effects of beam divergenceon the estimation of exposure in arbitrary convex geometries and demonstrated thai the errors
in the straight.ahead a.pproxfimation are proportional to the square of the ratio of the beam
divergence to the radius of curvature, which is small in typical space applications. From ashielding perspective, the straight-ahead approximation overestimates the transmitted flux, and
the error is found to be small in space radiation exposure quantities. Our first implementationof a numerical procedure was performed by Wilson and Lamkin (ref. 8) as a numerical iterative
procedure of the charged components perturbation series expansion of the Boltzmann transport
equation and showed good agreement with Monte Carlo calculations for modest penetratio_s to
where neutrons play an important role. The neutron component was added by Lamkin (ref. 9);
this closed the gap betw_n 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
handling large data arrays, wtfich would be solved in the next generation of codes.
The transport of high-energy ions is well adapted to tile straight-ahead approximation. In
fact., a more common assmnption that secondary ion fragments are produced with tile same
velocity as the primary initial ion (ref. 10) is inferior to the straight-ahead approximation
contrary to intuition (ref. 11). The Boltzmann transport, equation for the particle fields _j(x, E)_s given in the straight-ahead and continuous slowing down approximations as
[_ C_T2b'j(E)-I-o'j(E)] q_)j(a:,E) = J50"jk(E'L_) 4)k(x'Er) dE_ (1)
where x is the depth of penetration, E is the particle kinetic energy, ,S'j(E) is the particle
stopping power, cj(E) is the macroscopic interaction cro_ section, and o'j_.(E,E') is themacroscopic cross_ section for particle k of energy E _ produced as a result of the interaction
with a particle j of energy E. At Langley Research Center for all the code development, it.
has been customary to invert the differential operator and implement it exactly as a marchingprocedure 0yr. 12), and the remaining issue has been in approximating the integral t_rm on the
right.hand side of equation (1). The implementation for the heavy fragments was facilitated bythe assumption that, the fragment velocity ks the _me as the primary ion which is inadequate
for the description of the coupled nucleonic and light ion components. A compatible nucleonictransport procedure was developed by Wilson et al. (ref. 13) and showed good agreement with
expomlre quantities evaluated by Monte Carlo transport procedures (ref. 14). The traILsport of
the nucleonic component, was developed by assuming that the nfidpohlt energy within the step
was the appropriate energy to evaluate the integral term. Thus, the residual range of the protonwill reduce by h/2 before the interaction and the secondary proton residual range will reduce
by h/2 before arriving at the next marching step. Neut.roi_ show no loss in residual range astheir stopping power is zero. Tiffs choice was shown to minimize the second-order corrections to
the marching procedure (ref. 15). Although reasonable agreement on exposure quantities fromMonte 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. 16). Analysis concluded that the problem was in the rescattering terms inwtfich the number of elastic scattered neutrons was underestimated numerically, which must be
addressed as suggesiJed by Slfilm et al. (ref. 16).
2
Theissueofevaluationoftheintegraltern,oftheBoltzmannequationfor tileelasticscatteringksthe next issueto be resolvedin thedevelopmentof the tIZETRN deterministiccode.Oncethe elasticscatteringeventsareadequatelyrepresentedandtheassociatedimprovementsin theneutronflux aremade,onestill needsto addresstheissueof tile adequacyofthenucleardatabasefor nucleonictransport,in the HZETRNcodesystem(ref. 13).
Formulation of Transport Equations
Define the differential operator B as
o sj(F)+ _j(E)] 0(.,E)cgE
o_(x,E) o0. oE ['%(E) 4(.,z)] + _;(E) O(.,E) (2)
and consider the following one-dilneltsional Bollzlnann equation from reference 17
k
where %j is the differential flux spectrum for the type j particles, Sj(E) is the stopping powerof the type j particles, and crj(E) is the total macroscopic cross section. The term ajt,(E,E'),a macroscopic differential energy cross section for redistribution of particle type and energy, iswritten as
,_j k(U,E') = y]_ Pd _v(E') & ,._(E,E')
_3
where fjk,.3(E,E _) is the spectral redistribution, o'fl is a micro_opic cross section, and p3 is thenumber density of fl type atoms per unit mass. The spectral terms are expressed as
el r dsj .j = + +
where fief.,9 represents the elastic redistribution in energy, f]'k,d represenl.s evaporation terms,
and fat,,, 3 represents direct knockout temas. The elastic tern1 is generally linfited to a. smallenergy range near that of the primary particle. The evaporation process donfinates over the low
energies (/?7 < 25 MeV) and the direct cascading effect, dominates over lhe high energy range
(E > 25MeV) as illustrated in figure 1.
Equation (3) is then written for j = n as
t_a[ l el e du{o,,]: _ _ pe_ (u)(I,,e+ +t3 _ ,._., f,,.k,3 f,,£d) 0k(x, E') dE_JE
(4)
wtfich is expanded to the form
r-x),/ r el e dB[O,,] : E Pd a_(E )(fm,,a + f.r,,d + f,,,,.,,3) O,,(x,E') dE'JE
3
'/E _'(rff(E )(f,&3 + f.,k,,d q- f.k,J) Or,(x,E') dE'
1_.¢,, 13
(s)
3
Definethe integraloperatorsI as
i(_) £__[_] =
i,I:)[,,]:
t,,3_a(U) _ dUf,,k.,J_(. ,E')
_Q p.a_v(E') I,;_.,aO(_,r') dE'
p..o(L') ,,L,_-,,,34,(x,E') _E'a
where k = n denotes coupling to neutron collisions and k = p denotes the neutron source from
proton collisions. When considering only neutrons and protons, equation (,5) can be written in
the linear operator form as
_") /;')[0,,]+ ;_"_[_.1+ I,_')[o.] (6)B[4,.]= I_')[4,,,]÷ i!:'.)[0,,]+/_l [_"]÷ .I
Note that l_')[&p] does not contribute to the neutron field; therefore, equation (6), with &,,.
replaced by _, is written as
Assume asolution to equa.tion (7) of the form _ = _b¢+0d, where Ot is the solution for evaporationsources and contributes over the low-energy range and 0d is the solution for t.he direct, knockout
sources and contributes mainly over the high-energy range as suggested by figure 1. Substitute
this assumed solution into equation (7) and find
-(")r__ 1B[0] = /314,_]+ B[qJd] = l.l topeI + I_i')[0d] + I[n)[_be] + I_n)[Od ]
+4'% 1+ I0.1+ +4"'I .l (*)(.) , (.) ,
The terms I_ [0el and Ia [0_] are near zero and are _gnored because evaporat.ion neutrons at.low energies do not produce additional evaporation neutrons, and the direct, cascade effects hax,e
very small cross sections over the low-energy range of .be. and hence does not contribute- anypl_oduction over the low- or high-energy range, Further asstmle t.hat 4_8 is calculated by the
IIZETILN program so that Od is a solution of tile equation
B[0d] = 'el tq)dJ--k
This assumption simplifies equation (8) to the form
B[¢,_] = I[[')[_,:1 + I/")[0,/1 + I_")[4,v] (10)
Define the elastic seatt.ering temas
O's, d PJ O.,d(_t) el !, = , /]_.,a(_,_ )
with units of cm2/g-MeV, and note that for neutrons the stopping power @(E) [s zero and
equation (10) reduces to the integro-differential transport, equation with source term
[0 ] /:_+ _(_) _.:(.,e) = _ _.,_(_,s') _(_,,_') aE' + _(_,.) (1_)
4
Equation(11) representsthe steady-statelow-energyneutronfluenceCe(x,E) at depth x andenergy E. The various terms in equation (11) are energy E with unit.s of M eV, depth ill medium
is x with units of g/era 2, Ce(x,E) (in particles/cm2-MeV) ks the evaporation neutron fluence,
and g(E, x) = I('_)r_ I+I(P)e tV'dJ e' [¢p] (in particles/g-MeV) is a volume source term to be eva.htal.ed by
the HZETR.N algorithm. Equation (11) is further reduced by considering the neutron energiesbefore and after a collision. The neutron energy E,, after an elastic collision with a nucleus of
mass number .4T3, initiMly at. rest, is, from reference 18,
+ 1)2 (12)
where E is the neutron energy' before the collision, AT.,. is t.he atomic weighl of the ith type of
atom being bombarded, and 0 is the angle ofscatl.er. Define the ratio
2
(13)
a.s a constant less than 1 and note t.hat when 0 = 0, E, = E, and when 0 = 7r, E, = Eag.
Therefore, change the limits of integration in equation (11) to (E, E/_:d) which represent thekinetically allowed energies for the scatt.ered neutron to result, in an energy E. Equation (1 1)then is written as
(14)
The quantity a in cm2/g is a macroscopic cross section given by
,a
where Pd is the number of atoms per gram and crd is a microscopic cross section in cm2/atom.Reference 19 provides approximate Maxwellian averages of cross-section values in barns which
are used herein for studies of solution tectmiques. These values are listed in table 1 Mong wilh
other parameters of interest for selected elements. Other units for equation (11) are obtained
from the previous units by ttsing the scale factor representing the density of the material in unitsof g/era 3 .
5
Table1. ParameterValuesfor SelectedElements
Elasticcrosssection,
Element AT3 ba.rns a
Lithium, Li 7 1.050
Carbon, C 12 4.739
Aluminuna, AI 27 1.348
Calcium, Ca 40 2.99
Iron, Fe 56 11.40
Lead, Pb 207 11.19,1
Density,
g/cm 3 a tJ
0.534 0.563
0.352 0.716
2.7 0.862
1.54 0.905
7.85 0.931
11.342 0.981
_'Maxwellian averages (ref. 19).
Mean Value Theorem
Throughout. the remaining discu_im_s, the following mean value theorem is used for integrals.
Mean Value Theorem: For O(x,E) and f(E) continuous over an interval a _< E _< b such
that (1) _(x,E) does not change sign over the interval (a, b), (2) O(x,E) is integrable over theinter_aal (a, b), and (3) f(E) is bounded over the interval (a, b), there exists at least one point csuch that
b _,bI"
/. I(E),(,,,E)dr.: l o(.,z) _<b)Ja g(/
In particle tralzsport, this mean value approach is not. commonly used. In reactor neutroncalculations, an assumed spectral dependence for (_(,,E) is used to approximate the integral over
energy groups. The present use of the mean value theorem is free of these assumptions; thus,more flexibility is allowed in the HZETRN code, and the result, is a fast. and efficient algorithm
for low neutron analysis.
Multigroup Method
Consider the case where there is only one value of/3 wtfich represents neutron penetration
into a single element material and let Ce be denoted by ¢. Equation (14) is integrated from Ei
to Ei+ 1 with respect to the energy E to obtain
f [Ei+ tE;+_06(.,E) aE + _.(E) 0(.,F) aE : r_+ _,.JE i Oa: ,lEt
(10)
wtie r e
I_= [E_+, "f_/_'__,_._(E,E' O(.,E') at:' dE (17)J Ei
an d
f Ei+ I_ : _,z_ g(E,_) aE (18)
As a test, case for developing solution techniques, we u_ the approximate source and scatteringterms taken from subroutine FBERT of the HZETRN code (ref. 5), g = g(E,x) = KEe -E/T
6
with K and T constants, and the elastic scattering term from subroutine ELSPEC of the
HZETILN code (ref. 5),
:1 - C(l-n)rEl
with r con_.ant, so thal equation (18) is easily integrated to obtain
(19)
The quantity
*,'(x) _ --[Ei+l 0('_', E) dE (20)-- dE i
is associated with the ith energy group, so t,hal 1 _i(x) represents an average fluence forEl+ 1 -- Ei
each energy group. Then equation (16) can be written in t.enns of q_i(x) as follows. In the first
term of equation (16), interchange the order of integration and differentiation to obtain
Ei+ I &)(x,E) d E _ dq_i(x)i O* dx
('21)
With the previously stated mean value theorem for. integrals, the second term in equation (16)
can be expressed as
u + dE = a"¢,.(x) (22)i
where a" = or[El + O(Ei+l - Ei)], for some value of 0 between 0 and 1.
For the term I i in equation (17), the order of integration is interchanged. Various partitioning
schemes are illustrated in figure 2. The integration of equation (17) depends upon the energypartition _lected. For example, figure 2(b) illustrates an energy partition where Ei+ 1 < Ei/a,
and hi this ease, equation (17)can be written as
Ii [Ei+ 1 fEi+ [ wLi+I/t_ f&+l= HdEdE _+ HdEdE r+ HdEdE r (23)JEt=El =Ki dEt=Ei+l dE=El JEt=Ella JE=aE t
where H = as(E,E') q_(x,E'). Figure 2(c) depicts the case where E;+t = E,:/a exactly for all i.
In this special cam, equation (17) reduces to
Ii H dE dE'+ H dEdE I (24)aU=Ei Ei ,/U=Ei+l dE=_U
The selection of an energy partition can lead to two or more distinct groups associated witheach interchange in the order of integration (for example, see fig. 3). The integrand H can be
integrated with respect to E and the results expressed in terms of the quantities
and
j[abF(b, a) = r c rE dE = e rb - e
a(E') =1 - e-(1-") rE'
T(1
and equation (24) call be written in the form
Ii -- i Ei+lJE'=Zi G(E') F(#,E D ¢(x,#) d/T'
I Ei+ 1/o:+ G(/7') dE'J Et =El+ 1
(25)
To illustrate tile basic idea behixld the multigmup method, tLse the same mean value theorem
for integrals and write equation (25) in tlle form
where/7i < E.[ < Ei/a and Ei+ 1 < /7_+1 < /7i+1/c'" The special partitioning of the energy asillustrated in figure 2(c) enables us to obtain from equation (16) a system of ordinary differentialequations as follows:
d _1 a22 a23
Ta. @_-2 -O-
k4,)_;_l J
--O--
aN-1,N-I aN-1,N
aN N
v-2 _N-2L4';V-lJ G-,
(26)
where ai,i = G(/7[) F(ld_,Ei) --# and ai, i+l = (7(/7"+i) F(/Ti+l,ol=Ti*+l). Fm'ther assume thatfor large values of N, q'i = 0 for all i > N. This assuniption gives rise to the following system ofordinary differential equations:
d-2= ,47+dx
subject, to the initial conditions g(0) = 0". IIere _" is the cohmm vector of q_i values,col ((I) 0, _1,"', (I)N-1), the matrix A is an N by N upper triangular matrix, and b-is the column
vector col (_0,_1,... ,_:v-l). In a similar manner, the integrals in equation (23) can be evaluated
for other kin4s of energy partitioning and a system of equations having the form of equation(26) obtained. However, for the_ other energy partitions, the structure of the N by N _uare
matrLx ,4 will change. It remains upper triangular but with more off-diagonal elements wliieh
depend upon the type of energy partition. (See, for example fig. 3.) For our purposes the system
of equations (eq. (26)) is used to discum mine of the problems associated with the multigroupmethod.
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 (26). Cottsider the construction of the energy partition
{Eo,eo Eo Eolc_ ' c_2 '" " a,N J
where /70 = 0.1 MeV, for the selected elements of lithium, aluminum, and lead. Table 2illustrat_ integer values of N necemary to achieve energies greater than 30 MeV. These ,"a.lues
of N repre_nt the size of the matrLx associated with the number of energy groups. The value/70 = 0.1 MeV, in temas of htmaan exposure, represents a lower bomad where lower energies are
not. important. The value of 30 MeV represents an upper limit for the evaporation particles.
8
Table2.EnergyPartition Size N
Element a N 0.1/o ,N
Lithium
AI u minu m
Lead
0.563
0.862
0.981
10
39
298
31.53
32.75
30.38
Observe that for energy partitions where Ei+l < Ei/o the values of N are larger, and if
Ei+l > Ei/a' the values of N are smaller. Tile cases where Ei+l > Ei/a give rise to problemsassociated with the integration over the areas A 1 and A 2 of figure 2(d) when tile order of
integration is interchanged. In this figure, the area A1 is associated with the integral definingOi, and the area A 2 is a remaining area amociated with an integral that is some fraction of the
integral defining q_i+l which is outside the range of integration. Therefore, some approximationmust be made to define this fractional part. Tlfis type of partitioning produces errors, due to any
approximations, but it. has the advantage of greatly reducing the size of the N by N matrkx A
at the cost of h_troducing errors into the system of equations. A more detailed analysis of the
energy partition can be found in reference 20.
The ca_ of neutron penetration into a composite material gives rise to the case where fl > 1
in equation (11). In this special case, equation (17)becomes
h= £ El'b• aEi E o',j(E,E ) ¢(x,E') dE' dE'.I.
Select. a = max (al, o_2.... , a j) and construct the energy partition where El+ 1 = E'i/o. Thenobtain a system of differential equations having the upper triangular form:
[.][11a1213O1 a22 a23 a 2N • 1
d : z : : "+ (27)
• -1 aNN • -1 k( -1
Observe that for some arbitrary energy grouping we have, for the element hydrogen, a ce._
where the value of oj is zero• In this situation we nmst integrate over many energy groups asillustrated in figure 3. Some type of approximations umst be made when the order of integrationks hlterchanged, depending upon the selected energy partitioning. Also the problem of .selecting
the mean values associated with each of these integrations extsts.
Mean Value Determination
Consider the case of neutron fluence in a single shield material with the energy partitioning as
illustrated hi figure 2@). This case is where successive energy values are given by Ei+l = El�o*for all values of the index i as it ranges fi'om 0 to N. Select a finite value for N large enough
that the assumption ON = 0 holds true. The system of equations in equation (26) is thena closed system and we can solve for the last term O N and then march backwards to solve
ON-l, ON-2 .....
The nonzero elements ai,j for matrix A in equation (26) consists of the diagonal elementsand the first diagonal above the main diagonal. This gives the values
aii =G(E*) F(E* Ei) - _"
ai,i+l =G(Ei*+I) F(Ei+I,OE*+I )
for i = 1,..., N, where E* and El+ 1 are selected mean values associated with the lower andupper triangles illustrated in figure 2(c). These mean values vary with energy and were selected
so that the multigroup ,solution agrees with the numerical solution of the test problem. The
values determined empirically were
_" he r e
and
whe re
E._ = E_ + O_(E_+_ - E_)
E*+I = Ei+ 1 -4-02(Ei+2 - Fi+I)
71 + roll(E- Ell)- 51
71 -4-ml2(E- Ell) - 51
")3 + ml3(E- E2"2) - 51
(E > Ell )
(E22 < E < E11)
(E < E22)
72 + m21(E-- Ell)
_'2 -4-m22(E- Ell)
(E > Ell)
(E22 < E' < Ell)
(E < E22)
3"rI = 0.93
72 = 0.90
73 = 0.30
74 = 0.27
r/ill = 0.0030485
mr2 = 0.2490258
ml3 = -0.3937186
El1 = 3.037829
rn21 = 0.004355
m22 = 0.249026
m23 = -0.255920
E22 = 0.5079704
and _1 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 muMgroup curves_ would have the
correct shape and agree with the numerical solution. These selections for the mean values are
not, unique.
Solution Method in Shield Materials
Consider the energy partition Ei+I = Ei/a and the resulting system of differential equations
(eq. (26)). The solution ofth_s system of equations is obtained by first, solving the last equation
of the system. This equation has the form
d_N_ldx - aNNr_N-1 + _N-l(X) (¢x_l(o) = o)
and ha.s the solution
[ /0 ]_N-I(X) = e aNNa" _N-l(O) + _N-l(S) c -aNNs ds
which implies
x0+Ax¢N_l(X0 + 2._x) = e ctN/vAx (I)N-1 (x0) + ¢ aNN(x0+Ax) _N-1 (s) e -aNNs d8,z x 0
10
Now consider each of the reinaining equations above the last equation in equation (26). A typical
equation from this stack has the form
d(b i- 1- aiie_i-1 + fi(x) ((I'i-1 (0) = O) (28)
dx
where fi(x) = _i(x)+ai, i+lOi(x) is known, since (I)i(x) is calculated before Oi-l(X). This typicalequation has the solution
• i_l(X)=eaiix [d2i_l(O)+_oXfi(_) e-aiisds]
which implies
_/_l(X0-[- _x) eaii-Mc @i-1 (xo) + eaii(xO+'hx) f x0+'-kx f() c--aiis ds= . _isa x 0
Observe that for the system of equa.tions in equation (27), the solution technique is essenl.ially
the same with the exception that the right-hand side of equation (28) is replaced by a summation
of the previously calculated terms, so that fi(x) = 4i(x) + _ aid q'j-l(x).
j=i+l
Numerical Solution
The solulions obtained from the system of equat.io_rs (eq. (26) or (27)) depend upon the
. selection of metal values associated with each energy interval. The selection of l.hew mean
values is determined by examining the numerical solution in certain special cases. We obtain a
numerical solution of equation (1 1) in the special case given by
g = .q(E, .) = KEe -E/T
where K (particles/cma-MeV) and T (Meg) are constants. We construct the solution over the
spatial domain x >__0 and energy range 0.1 _< E _< 80 Me\7. This domain is discretized by
const.ructing a set of grid points x i = i Ax and Ej = j AN for some grid spacing defined by
Ax and AE va.hles being used. For i,j integers, define uij = O(xi,Ej), then the transportdifferential-integrM equation (11) can be written in a discrete form as follows, with the starting
values u0, j = 0 and v0, j = 0 being used. For the first, step in Ax, approximate the flux by theaccumulation of the source over the first, intervM as
'Ul,j = AxKEj e-EJ/T (29)
followed by the numerical calculation of the rescattering term
[Za/', _(£')r e-"(E'-Za )vi,j = JEj 1 -- e -(1-°)rE, u(xi,E t) dE'
(30)
for i = 1. After this first, and each successive step, integrals of the type vi, j given by equation (30)
are evaluated with Simpson's one-third rule. Evaluate equation (30) for all energies j = 0, 1,...,and then ltse a two-step Mgorithm in a repetitive fashion to advance the solution. For _Tdues of a
near 1, the numerical solution of equation (11) requires that _E become small. The low-energy
spectrum then becomes difficult to cMculate without special procedures, as cited in reference 17.
In this case, a two-step modified Euler prediclor-corrector scheme is used (refs. 21 and 22), which
ks defined 1)3"
11
_cond step:
Third step:
fl,j -- Vl,j q- Ej e-Ej -- CrUl,j
Ul,j -I- Ax fl,ju2,j = ½ (ul,j_ 1 + Ulj+l ) + Ax fl0"
/(j = 0)
(j >o) J(31a)
f2j = v2j + Ej e-KJ - cru2d "_ (31b)u3j Ulj + 2Ax f2,j J
Tile second step is an adoption of the Fredrichs method from reference 21. The third step is a
central difference second-order step in Ax. Afl.er 100 applications of this two-step a.lgorithm, we
apply the following stat)ility correction as suggested in reference 22:
faj = va'Jl + Ej e-gJ -- eru3,j }uad = _" (uad + u2,j) + Ax f3j"
(32)
Note equations (32) are to be tmderstood in an iterative sense and not strictly algebraic sense.
Recursive S oluti on
In the special case g(E,x)= g(E), a, solution t.o equation equation (11) is assumed of theform
¢(x,E) = E On(E) f,,(x) =¢l(E) fl(x) + 02(E) f2(x)+ " "" (33)TI: 1
Substitute this series into equation (ll) and obtain a solution by requiring that ¢ an(] f satksfy
CITE) -- g(L-_
/Ed,+I(E) = f.(E,E') ¢.(E') dE'(34)
f_(x) + (_ fl(x) = 1
f_,(x)-l- cr f,(x)= fn-l(X)
for r_ = 1,2,3, .... where the differential equations are subject, to the initial condition that
f,,(0) = 0 for all ,,. _ere the terms for ¢,,(E) are defined recursively aim take a great deal ofcomputational time for large values of n. The differential equations have the solutions given bythe recursive relations
1 (1- e-qX) }
fl(x) =7 (35)
/0"f.(x) = f,,-1 (u) e-'(x-u) du
which are easily evaluated for as large a value of n as desired. We find numerically that If,(x)l
decrea_s with increasing _ for x < 1 and increases for x > 1 so that the series solution does
not converge in 1.hks case. For I*1_< 1, we calculated the solution given by equation (35) for
terms through n = 5 and n = 6 and compared them with the numerical solution. The meanvalues associated with the multigroup method were then adjusted so that the muMgroup method
agreed with the numerical solution and recursive solution for this special test. problem. We thenused these same mean va.lues which where associated with numerical source temls as provided
by the HZETRN code.
12
Comparison of Multigroup and Other Solutions
Thenumericalsolutionsandrecursivesolutionsof tile testproblemwerethencomparedwiththe multigroupsolutionfor neutronpenetrationin lithium, aluminum,andleadmediums.Tileresultsare illustrated in figures4, 5, and 6. Excellentagreementis obtainedin thesethreecases.In thesefigures,thesolidline representsthe numericalsolution.The circlesrepresentlherecursivesolutionandthe trianglesrepresent,the multigroupsolution.The variouscurves werecalculated for depths x of 0.1, 0.5, 1.0, 5.0, 10:0, 50.0 and 100.0 g/cm 2.
" The multigroup method h_s huge advantage in its very short computational time needed to
calculate the solution without loss of accuracy. The nmltigroup method takes less than 1 rain
of computational tilne, whereas the Monte Carlo methods require many hours of computa-tional time.
Application for A1-H20 Shield-Target Configuration
The previous development is now applied to an application of the multigroup method
associated with an aluminum-water shield-target configuration. In particular, consider the case
where the source term g(E,x) in equation (11) represents evaporation neutrons produced per unitmass per MeV and is specified as a numerical array of values corresponding to various shield-
target, thicknesses and energies. The numerical array of va.lues is produced by the radiationcode HZETRN &veloped by Wilson et al. (ref. 23). The ntmlerical array of values are actually
given in the fore1 g(Ei,xj,yt. ) in milts of particles/g-MeV, where y/,. represents di_rete values
for various target thicknesses of water in g/cm 2, xj represents discrete values for various shield
ttficknesses of aluminum, also in units of g/cIn 2, and Ei represents ¢fiscrete energy values in
units of MeV. These discrete source term values are used in the following way. Consider flint
the solution of equation (11) by the multigroup method for an all-aluminum shield with no
target material; i.e., target, thickness Yr. = 0. The HZETRN program was run to sinmlate the
solar particle event of February 23, 1956, and the source term .g(Ei,xj,yt.) associated with analuminum-water shield was generated for these conditions. With this source term, equation (11)
was solved by the nmltigroup method.
For a single shield material, _ = 1, equation (11) becomes
[° ] :F"+ ¢(x,E) .E o-.,.,(E,r') ¢(.,E') dE' + .u(E, (36)
where an integration of equation (36)from E i to El+ 1 produces
Ei + I Ei+ 1
O.e j Ei
[Ki+l [ E/al= tr, j (E,E') ¢(x,E') dE' dE+ [ Ei+lJ Ei d E d E i
.q(E,.) dE (37)
D,_ define the quantities
:: [ <+' g(z,. )O(*'z) aF}JE i
(38)
13
and interchangethe order of integrationof the doubleintegral termsin equation(37). Thenapply the meanvaluetheoremto obtaintile restllt
(E,E') JE dE'J E i Ei
fEi+2fEi+ 1+ a_I(E,E')dE %(x,E') dE' + biJEt+ t JE={_I Er
(39)
over the energy group E i < E t < Ei+ 1. For the energy spacing El+ 1 = Ei/o , the first, double
hategral in equation (39) represents integration over the lower triangle illustrated in figure 2(c).
The second double integral in equation (39) represents integration over the upper triangle
illustrated in figure 2(c). Define
=fie °bl(E'E')dEgl ( El ) :Ei
g2(E') =[ Ei+l (E,E')dEjE:o I Er O'sl
(40)
and then employ another application of a mean value theorem for integrals to _n'ite equation (39)in the form
d_id7 + _q)i = gl[Ei + 01(Ei+I - Ei)]_i + 92[Ei+1 + 02(Ei+2 - Ei+l)]q)i+l + bi (41)
This produces the coefficients associated with the energy group Ei to El+l, which are given by
aii = .ql (42)ai,i+l = g2
In tiffs way, the diagonal and off-diagonal elements of the coefficient nmtrkx in equation (26) are
cal cu lat ed.
For a compound target material, comprised of material 1 and material 2, there are two "_'alues
of a. A value o I is selected for material 1 and a value cr2 is selected for material 2 of the
compound material. In this case, equation (36) takes on the form
[0 ] j:/cH e_(x,E ) dEr+ ,,( F) : ,% (E,E') ' '
j:+ c_2(E,E') O(x,E')dE'+ g(E,x) (43)
where _ and rr,._ are scattering terms associated with the respective materials. These termsare calculated in the tIZETRN code. Two cases are considered. The first case requires that
the E/_2 line be above the E/nl line. (See fig. 2(d).) The second ease is where _2 = 0 (the
hydrogen case) and the limits of integration for the second integral goes to infinity. Each ca._
is considered _pa.rately.
For the first, case, assume that _1 > ct2 > 0 and select the exact energy spacing dictated
by the E/c_ 2 line. Then proceed as for the single shield material. Integrate equation (43)
14
from Ei to Ei+l and interchange the order of integration on the double integral terms. Define
bi rEi+l g(E,x) dE and obtain the equation_-- JEi
d(I)_...j.i+ _'_i = I11 + I12 + I21 +/22 + bi (44)dx
where now the 121 and I22 integrMs have, because of the exact spacings, the forms
Ei+ I j_t121 = ob2(E,E' ) dE ¢(x,E') dE'Ei =gi
(45)
JEt+ I J E=_2 EI
Defining the t,erms
t_i(E,E') dE (i = 1,2)
hl(i)(/_" ) = =Ei
f Ei+1 c%i(E, lff)dE (i 1,2)1_2(.i) (E/) jE=a.2E t
and ttsing the mean vMue theorem for integrals gives from equations (45)
/21 = h,l(2)[Ei + 01(Ei+I - Fi)]@i
an (t
/22 = h2(2)[Ei+l + 02(Ei+2 -- Fi+I)]_i+I
where 01 and 02 define intermediaJe energy values associated with the mean value theorem.
The integrals [ H and I12 are associated with integration limits (E, E/al) and energy intervals
dictated by the selection of a2 for determining the exact, energy spacings. These integrals are
associated with the trapezoidal area. 1 (A1) and triangular area 2 (A2) illustrated in figure 2(d).
These areas are a fraction of the triangle areas associated with the line E I = E/_2. The._.
fractions are given by
k = ½(E,.+_.- -E")2- ½(Ei+_-](E,.---_I -7:i_---- .,,- E,./(u)(E_+I --1E_+l) ](46)fI_ = (Ei+ff,_l - &'+0(E,+l -- o'lE;+l)
(E'i+ 1 -- Ei)(E'i+ 2 - Ei+I)
and we write
Ill =flhl(1)@i }I12 = f2h2(1)(I'i+l
The coefficient_ for the system of differential equations in equation (27) are then gwen
all =hl(2) + flhl(1) - _'/
/a12 =h2(2) + f2h2(.1)
15
(47)
by
(48)
For the second case, of hydrogen, a2 equals 0; therefore one of the linfits of integration be-
comes infinite. Let a, 1 determine tile energy spacing in this ease. Again integrate equations (45)
over the energy interval (El,El+l), which is determined by t]le E I -- E/Ctl line. With the def-
initions given by equations (38), integrate equation (43) over the interval (El,El+l) and then
interchange the order of integration in the resulting double integrals to obtain
dOid--7+ "_'_ = s_ + I._ + _
where
I.;+,j[It = _r_ =Ei _.l(E,#) de ¢(.,U) dU+ J E;+_ J E=<,_E' _._(E,#) de ¢(_,#) dff
and
i.,+,l;' ki?+,+,i.,+,: _.._(E,#) dE ¢(x,E') de'+ _,._(E,#) de ¢(.,_)Ur.'J Ei i j =1 Ei +j a Ei
and for all N* greater than some integer N > 0, it is known that ¢(x,E) will be zero. Define
E I
Da(E_) =iE[ °'sl(E'E') dE
_ i Ei+l oL,.l(E,E') dEh4 (E#) --dale'
E #
#,s(U) :j; %(E,#) dei
- / E;+/+l cr,2(E,E' ) deho(J) -- a El+ j
(E i < E' < Ei+l)
(Ei+l < E' < Ei+2)
(E,: < L" < E,+l)
(Ei+j < L_ < Ei +j+ 1)
and then write the coefficients associated with the system of differential equations as
aid= h3 + h5- "_
ai,i+ 1= h 4 + h6(1)
ai,i+2 = h6(2)
ai,i+3 = h6(3 )
ai,i+ n = ho(n)
hi this way a system of equations is generated that has the triangular form given by the
system of equations in equation (27).
Again use the source term g(Ei,xj,yk) obtained from the HZETRN simulation of thesolar particle event of February 23, 1956, associated with an aluminum-water shield-target
configuration. Note that now the multigroup system of equations (eq. (27)) associated with
equation (39) must be solved for the nmltiple atom target material of water. Comider thecases of discrete shield thickness x2, x3 .... and apply the multigroup method to the solution of
16
equation(11) appliedto all targetmaterialy > 0. For each value of xi considered, the initial
conditions are obtained from the previous solutions generated where y = 0. This represents the
application of the nmltigroup lnethod to two different regioi_s: region 1 of all shield materialand region 2 of all target material. 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 tlle second
region represents exit conditions from the shield region 1. This provides for continuity of the
sohitions for the fluence between the two regions.
Results and Discussion
The present formalism was used to evaluate the neutron fluence for various Muminmn shield
and water target combinations. Figure 7 ilhtst.rates the low-energy neutron fluence due to the
scattering of evaporation neutrolts in an aluminum shield for various thicknesses with yl, = 0
(i.e., no target material). Figure 8 illustrates the total neutron fluence for various aluminumshield thicknesses. This fluence consists of the tlZETRN-generated neutron fluence phls the
multigroup-generated low-energy neutron fluence. Figures 9, 10, and 11 are graphs of the neutronfluence in depths of 1, 10, and 100 g/cm 2 of Muminum generated from the HZETRN code both
with and without the addition of the multigroup evaporation neutrons.
Typical results for no shield before the water target, are illuslrated in figures 12, 13, and 14
where a comparimn of the multigroup method wMl the previous IIZETRN results for thicknesses
of 1, 10, and 30 g/cm 2 can be made. Note that in the calculations of the multigroup method, the.
source terms g(E,x), the scattering term o'_(E,E_), and cro._ section or(E) of equation (11) are
all given as numerical output from the IIZETRN code for the solar particle event of February 23,1956. AI_ note that these calculations were compared with the LAttET MonW Carlo results
from reference 24. Figures 12, 13, and 14 illustrate this comparison for neutron fluences versusenergy at. water depths of 1, 10, and 30 g/era 2, respectively. Figure 15 is a graph of neutronfluence versus depth in a shMd-target configuration of 100 g/era 2 of Muminnm followed by
100 g/cm 2 of water. Observe the incream in the low-energy neutron fluence at the alunainum-
water boundary. This increase is caused by high-energy neutrons colliding with hydrogen a.toms,
which results in large energy lomes. In these types of collisions, the neutrons of modest energies
give up one half of their energy on the average; thus, the lower energy neutron fluence is increased.
In figures 12, 13, and 14, note the distinct improvement of the fluence by using the nmltigroup
evaporation neutrons over that of the previous HZETRN resull.s. These improved results arestill a little lower than the results predicted by the Monte Carlo simulation. These figures
show that the multigroup method is more accurate at. the higher target, depths compared with
results at the lower depths. This is due to the straight-ahead approximation assumptionsused in the one-dimeI_sional Boltzmann equation, where all secondaries produced by nuclear
collisiol_s are assumed to move in the same direction as the primary nucleon which caused the
collision. This assumption is true for secondaries which are high-energy particles. This straight-
ahead approximation is not true for low-energy neutrons produced by evaporation because these
neutrons are generally isotropically distributed. The_ neutrons make up the source tenns in the
multigroup method. The straight-ahead assumption causes errors at the smaller target depthsbecause it=fails to accounl for all the low-energy neutrons trartsported back from larger depths of
the material. In an attempt to improve the performance of the multigroup method for simulatinglow-energy neutrons, the assumption was made thai. only one half the source terms moved inthe forward direction while the other half moved in the backward direction. The solution of the
multigroup system of equations (eq. (27)) was then modified. Using one half the source terms
g(Ei,xj,yk), system of equations (eq. (27)) was marched first through the shield material andthen through the target material. By using the end boundary condition generated, the equationswere then marched backwards through the target, and then the shield material. The fluences
from the forward and backward marching were then added to obtain a total fluence. Tlfis process
17
is referred to in the figures as the two-dimensional multigroup method. Figures 16, 17, and 18
illustrate the results of the two-directional multigmup method applied to the case of no shield
and a target of water only for nominal dept.trs for an exposure to the solar particle event of
February 23, 1956. Figure 19 ilhrstrates the fluence in a depth of 10 g/cm 2 of water when t,he
two-dimensional method was applied to a 100 g/cm 2 a.luminun_ shield followed by a 100 g/cm 2
target of water when exposed to the solar particle event of February 23, 1956. Observe that the
two-directional nmltigroup method greatly improves the low-energy fluence predictions at the
smaller depths.
Research is contilming to clo_, the remaining gap between transport, code predictions and
Monte Carlo results. Possible errors from various sources are being investigated. The nuclear
cross sections u_d are believed to be one source of error because only elastic cross sections
were used in the muMgroup simulation. The elastic cross ,sections are much larger than the
nonelastic cross sections at low energies. Nonelastic cascading does occur and il is believed
that the muMgroup method would be improved by incorporating both types of cross sections.
Other sources of errors reside in the IIZETRN program itself. The nuclear cross sections used
by ttZETRN are interpolated from a large database that was developed experimentally many
years ago; this database needs to be updated. The ttZETRN code is a one-dimensional tra.i;sport
code using the straight-ahead approximation. The improvement of the nmltigmup method in
going from the straight-ahead approximation to the two-directional nmMgroup appm_mat.ion
suggests that similar type changes be incorporated into the HZETRN code in order to reflect
the nonisotropic character oft.he events.
Con cluding Remarks
These preliminary studies have shown that the multigroup method developed for the study of
low-energy neutron transport has made significant improvements ill and is compatil_le with the
current ttZETRN code developed at Langley Research Center. It. has proven to be a fast. and
efficient, algorithm for the inclusion of low-energy neutrons into the HZETRN code. The addition
of nonelastic processes in the low-energy neutron transport is expected to further improve the
result,.
References
1. Wilson, John W.; Cueinotta, F. A.; Shinn, .l.L.; Silnonsen, L. C.; Dubey, R. R.; Jordan, W. R.; Jones, T. D.;Chang, C. K.; and Kim, M. Y.: Shielding From Solar Particle Event Exposures in Deep Space. Rad. Meas.,vol. 30, 1999, pp. 361-382.
2. Wilson, J. W.; Cuchmtta, F. A.; and Simonsen, L. C.: Proton Target Fragmentation Effects in Space Exposures.
Adv. Spac_ Res., To be published.
3. Wilson,-J[,hn i;C; Nealy, John E.; WoodTJam-es S.; Quails, Garry D.; At.;ve[l, William; Stfinn, audy L.; andSimonsen, Lisa C.: Variations in Astronaul Radiation Exposure Due to Anisolropic Shiehl Distribution. Health
Phys., ml. 69, no. 1, 1995, pp. 34 45.
4. Nealy, John E.; Quails, Garry D.; and Simonsen, Lisa C.: Inlegrat.ed Shield Design Metl,odology: Application t.o
a Sal.ellite Instrument. Shieldin 9 .q_lrategies for Huma_ Spac_ _EaTdoralion, J. W. Wilson, J. Miller, A. Konradi,and F. A. Cucinotta, eds., NASA CP-3360, 1997, pp. 383-396.
5. Wilson, John W.; Cucivotta, F. A.; Shillli, J. L.; Simonsel|, L. C.; and Badavi, F. F.: Overview of HZETRN andBRNTRN 5"pac_ Radiation Y,hielding Codes. SPIE Paper No. 2811-08, 1996.
6. Alsnfiller, R. G., Jr.; Irving, D. C.; Kinney, W. E.; and Moran, I-I. S.: The Validity of the Straightahead
Approximation in Space Vehk:le SI,idding Studies. Second Symposium on Pwteclion Against Radiations in.qpacc, Arthur Reetz, Jr., ed., NASA SP-71, 1965, pp. 177-186.
7. WiLson, John W.; and Khandelwal, G. S.: Proton Dose Approxhnation hi Arbitrary Convex Geometry. Nucl.Technol., vol. 23, no. 3, Sept. 1974, pp. 298--305.
8. WiLson, John _V.; aad Lamkin, Stanley L.: Perturbation Theory for Charged-Particle Transport in One
Dimonsion. Nucl. Sci. &"Eng., vol. 57, no. 4, Aug. 197;5, pp. 292-299.
18
9. Lantkin, Stanley Lee: A Theory for High-Energy Nucleon Transport in One Dhnension. M.S. Thesis, Old
Dominion Univ., Dec. 1974.
I0. Letaw, John; Tsao, C. II.; and Silberl)erg, R.: Matrix Methods of Cosmic Ray Propagation. CompositioTi and
(.higin of Cosmic Rays, Maurice M. Shapiro, ed., D. Reie[ Pub]. Co., 1983, pp. 337-342.
11. Wilson, John W.: Analysis of th_ Theory of High-E1_ergy Io_ Tvansporl. NASA TN D-8381, 1977.
12. Wilson, John W.; and Badavi, F. F.: Methods of Galact.ic Heavy Ion Transport. Radial. Res., vol. 108, 1986,
pp. 231-237.
13. WiLson, John W.; Townsend, Lawrence W.; Nealy, John E.; Chun, Sang Y.; Hong, B. S.; Buck, t_Varren W.;
Lamkin, S. L.; Ganalx)l , Barry D.; Khan, Ferdous; and Cucinotta, Francis A.: BRYNTRN: A Baryon 7_'ansporl
Mod_l. NASA TP-2887, 1989..
14. Shinn, Judy L.; Wilson, John W.; Nealy, John E.; and CucinotLa, Francis A.: Comparisol_ of Dos( Estimates
Using Ih( Buildup-Faclor M_thod and a Bar'you Transport ('ode (BR}NTR._ IVitb Moul( Carlo R¢sults. NASA
TP-3021, 1990.
15. Lannkin, Stanley L.; Khandelwal, Goviud S.; Shinn: Judy L.; and Wilson, John W.: Space Proton Tra,,sporl iu
One Dimension. Nucl. 5"ci. _" Eng., vol. 116, no. 4, 1994, pl ). 291 299.
16. Shinn, Judy L.; Wilson, John W.; Lone, M. A.; Wong, P. Y.; and Cost.en, Robert C.: Preliminary Eslimal¢s
of Nucl_ou Fluxes in a Waler Tt_rg_l E_Toscd to Solar-Flar_ Pwlons: BRYNTRN V_rsus Moult Carlo Uod_.
NASA TM-4565, 199_1.
17. Wilson, John W.; Townsond, Lawrence W.; Schi|nmerling, Walter S.; Khandelwal, Govind S.; Khan, Ferdous S.;
Nealy, John E.; Cucinotta, Francis A.; Simon_n, Lisa C.; Shinn, Judy L.; attd Norbury, Jol, n W.: It'ansporl
Methods and It_le_uclions for Space R_tdiatiot_s. NASA RP-1257, 1991.
18. Haffner, James W.: Radiation and Shi¢ldin 9 in Space. Acadenfic Press, 1967.
19. Brookhaven Nat. Lab.: Guidebook for the ENDF/B-V Nuclear Dalt_ Files. EPRI NP-2510, Electric Power Res.
hist., 1982.
20. Clowdsley, Mart.ha Sue: A Numerical Solution of the Low Energy Neutron Boltzmann Equation. Ph.D. Thesis,
Old Donfinion Universily, May 1999.
21. Hall, Charles A.; and Porsclfing, Thomas A: Numerical Analysis of ParliaI Diff_renlial EqualioT_s. Prentice Hall,
1990.
22. Dahlquisl, Germund; and BjSrck, Ake: ._v)tmerical M¢thMs. Prentice Hail, 197-4.
23. Wilson, John W.; Badavi, Francis F.; Cucinotta, Francis A.; Slfinn, Judy L.; Badhwar, Gautam D.; Silberberg, R.;
Tsao, C. 1I.; Townsend, Lawrence W.; and Trii)athi, Ram K.: HZETRN: Dtscriplion of F_'t_-,q'pacc lo'll and
Nucleon 2q'ansporl alld Shielding Compul(r P_og'mm. NASA TP-3495, 1995.
24. Prael, R. E.; and Lichtenstehl, Henry: User Guid_ to LCS: The LAHET Cod_ _qystem. LA-UR-89-3014, Los
Alamos Nat. Lab., 1989.
19
10-1
10-2
E
I0-3
fd I(E,E, )
10 -40 100 200 300 400 500
Energy, MeV
Figure 1. Evaporation and cascading neutron spectral effects for collision of 500 MeV neutrons
in aluminum.
20
E'
/I
E,l
E' = EIo_
El+ I
(a) General.
E'= E
E
E _
(b) Ei+l < Lt_t '
E'=E
E'
El+2
El+ 1
E.t
V
E,1
(c) Ei+ 1 = F4
/ E'=E
E'
E+2
E+I
E' = E/o_
-EEi+l E+2 Ei E+ I E.+2
(d) E_+l > E,.
Figure 2. Various energy partitioning schemes.
=E
21
E'=E
Bm
/VEi
//E
E i Ei+ 1
Figure 3. Mult, igroup energy partit,ion where Ei+I < E/¢Jt "
10 2Depth, g/cm 2
_. A I00.0
A A
10.0
5.0I
A
A
-- Numerical method0 Recursive method
A Multigroup method
o A0
I i t t I Illl I I I I I t Itl , I I I I ] i ill10 ° 101 10 2
Energy, MeV
Figure 4. Numerical and multigroup .solutions for lithium.
9.2
101
Depth, g/cm 2
A A A A 100,0
10.0
-- Numerical methodO Recursive method
A Multigroup method
l0 -6 I I i I I__t III I I I I I I III
10-i 10° 10i 102
Energy, MeV
Figure 5. NumericM solution arid multigroup solution for <_|unlilnim.
10 2 A AAA
Depth, g/cm 2I00.0
50.0101 10.0
10-5
-- Numerical methodO Recursive method
A Multigroup method
10 -6 I i i i i ilil i i i i i ill] I I _U
10 -I 100 101 102
Energy, MeV
Figure 6. Numerical solution and multigroup sohttion for lead.
23
107
6
105
¢_ 10 4
eaJ
103
10 9 __
108 _ Thickness,
_ glcm2
102 __ .... \
% ...............ioo......'l<O,.....7o:'"";'o-'........-2 i0-! 10 4
Energy, McV
Figure 7. Low-energy neutron fluences due to scattering of evaporation neutrons in alunfinum
shield exposed to solar particle event of February 23, 1956.
10 9
108
10 7>
e.,' 10 6E
105
_5 104
103
g
g
0. I
10 2
101 J t_Jtm _ J_Nu10 -2 10 -I 10 o
IThickness,
g/cm 2
N
i i iiiii1 i 1 iltll
101 10 2
Energy, MeV
i i iiii11_ I t1111t
10 3 10 4
Figure 8. Total neut.ron fluences in aluminum shield exposed to solar particle event, of
February 23, 1956.
24
>
g.
_5¢)
©
G
-- HZETRN fluence with multigroup neutrons
- - - HZETRN fluence without evaporation neutrons..... HZETRN fluence with evaporation neutrons
IO9
10 8
10 7
10 6 .............. _ .........
10 5 _.
\10 4
Io3 F
LI0 2
101 _ _H.lO -2 10-I
i i iiiiii i i 1iiiii 1 I iiiiii
I0 ° I0 t 102
1 1 ii1111\ i i llllll
10 3 04
Energy, MeV
Figure 9. Energy spectra of neutron fluence at, depth of 1 g/cm 2 in aluminum exposed to solar
particle event of February 23, 1956.
>
g
10 9
10 8
10 7
10 6
10 5
104
10 3
10 2
l01 ' .,,...10 -2 10 -I
-- HZETRN fluence with multigroup neutrons
- - - HZETRN fluence without evaporation neutron,,
..... HZETRN fluence with evaporation neutrons
1I I IIIitl I I I11111 I I II1111/
10 o 101 10 2
Energy, MeV
i\\
,\I IUUll I IIIIi
10 3 104
Figure 10. Energy spectra of neutron fluence a.t depth of 10 g/cm 2 in almnhmm exposed to solarparticle event, of February 23, 1956.
9.5
6
109
10 8
10 7
10 6
10 5
10 4
10 3
-- HZETRN fluence with multigroup neutrons
- - - HZETRN fluence without evaporation neutrons
..... HZETRN fluence with evaporation neutrons
: -.....
.--4\\
\
102 _
'°',o-_.....;"o,......;'oO......;'o,................ _.....,o,
\
10 2 103
Energy. MeV
Figure 11. Energy spectra of neutron fluence at depth of 100 g/cm 2 in aluminum exposed to
..solar particle event, of February 23, 1956.
>
O
108
106
104
10 2
I 0°
"'"""'""_.,,,,_ ....
1 I
- HZETRN fluence with multigroup neutrons
- - - HZETRN fluence without evaporation neutrons
..... HZETRN fluence with evaporation neutrons
'a._.....] ..................Monte Carlo fluence ref. 25)
• 1.........!.......
_ ;"',"'!
10_2 , I I tllll J | t lllll
10 -t 10 o i01 102
Energy, MeV
....
_1 i iiii
103 104
Figure 12. Energy spectra of neutron fluence at depth of 1 g/cm 2 in water exposed to solar
particle event of February 23, 1956.
26
108 J '-
>,D
g.t-
2
eoe=
'DCa
10 6
104
102
10°
10-- 2 l I _k3_ I III I I I IIIII
10 -l 10o 101 102
Energy, MeV
-- HZETRN fluence with multigroup neutronsHZETRN fluence without evaporation neutrons
HZETRN fluence with evaporation neutrons
Monte Carlo fluence _ref. 25)
i i i iiii1 i ] i i1111 i
103 104
Figure 13. Energy spectra, of neutron fluence at depth of 10 g/cm "2 in wal.er exposed to solar
pa.rticle event of February 23, 19,56.
>
{D
108
106
104
10 2
I 0°
J/
10_ 2 , , ,,,H,10-1 100 l01 102 103 10 a
-- HZETRN fluence with multigroup neutrons- - - HZETRN fluence without evaporation neutrons
HZETRN fluence with evaporation neutronsMonte Carlo fluence ref. 25)
i i i 11111 i I I iiiii I i i iiiii iiiii
Energy, MeV
Figure 14. Energy spectra of neutron fluence at. depth of 30 g/cm 2 in water exposed t.o solar
particle event of February 23, 1956.
27
i0 Io
109>tO
i
E
e-,
_ 108
e-
d
_ |0 7
106 _ , , , 1 , . . , I J L_ _ I , J _ _ t
0 50 100 150 200
Depth, g/cm 2
Figure 15. Neutron fluence a-s function of depth ha 100 g/cm 2 aluminum shield followed by
100 g/cm 2 water target.
108
106
>09
_'E 104
e,,
_ 10 2©
100
10--2 I i Itlttt I I IIII11 t I t/Itll
10-1 10 o 101 102
Energy, MeV
I !
HZETRN fluence with two-directional multigroup neutronsHZETRN fluence with multigroup neutrons
HZETRN fluence without evaporation neutrons
HZETRN fluence with evaporation neutronsMonte Carlo fluence (ref. 25)
<
I 1 l[ttll_t iiiii
10 3 10 4
Figure 16. Energy spectra of neutron fluence at depth of 1 g/cm 2 in water exposed to solar
particle event of February 23, 1956, and calculated with two-directional multigroup method.
28
108 t t
>
.gI.
e-,
_Dr.,)
¢J
E
106
104
10 2
10 °
t
10- 2 I I Illlll I I ttl]l_ t t Itlltl
10 -I 10o 101 102
-- -- - HZETRN fluence with two-directional multigroup neutrons
-- HZETRN fluence with multigroup neutrons
- - - HZETRN fluence without evaporation neutrons
..... HZETRN fluence with evaporation neutrons
..................Monte Carlo fluence (ref. 25)
i i J i iiii i i_lllll
103 104
Energy, MeV
Figure 17. Energy spectra of neutron fluence at. depth of 10 g/cm 2 in water exposed to solar
particle event of Febrt]ary 23, 1956, and calculated with two-directional multigroup n]ethod.
• 108
106
"_ 104
.gt=
g
= 102
,D
E
I0 °
10-2 , , _[,H,10 -] 10 o
i [ i i1111 I i i iiiii
10 ] 102
HZETRN fluence with two-directional multigroup neutrons
HZETRN fluence with multigroup neutrons
HZETRN fluence without evaporation neutrons
..... HZETRN fluence with evaporation neutrons
Monte Carlo fluence (ref. 25)
\
103 104
Energy, MeV
Figure 18. Energy spectra of neutron fluence at depth of 30 g/cm 2 in water exposed to solar
particle event of February 23, 1956, and ca.lculated with two-directional nmltigroup method.
29
lO 11
10 9
10 7
6105
2
103
_J(3
101
10 -I
1"0-3
- l .... _- I - '-2 HZE-1_RNfluende with two-directional multigroup neutrons| l I --- HZETRN nuence with multigroup neutrons
- | | { - -- - HZETRN fluence without evaporation neutrons
..... HZETRN fluence with evaporation neutrons
____ --_._] MCNpX fluence
I
I
I
1
I _LI lIE . I I I II1111 _ I IIIIIII
10-2 10 -I 10 o 101 10 2 10 3 10 4
Energy, MeV
Figure 19. Energy spectra of neutron fluence at, depth of 10 g/cm 2 in target of 100 g/cm 2alumimma shield with target of 100 g/era 2 of water behind it, when exposed to solar particle
event of February 23, 1956, and calculated with two-direcl.ional muMgroup met.hod.
30
Form ApprovedREPORT DOCUMENTATION PAGE OMB No.0704-0188
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March 2000 Technical Publication
4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
An Improved Neutron Transport Algorithm for Space Radiation
6. AUTHOR(S)
John H. Heinbockel, Martha S. Ciowdsley, and John W. Wilson
17. 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
WU I01-15-01-51
8. PERFORMING ORGANIZATIONREPORT NUMBER
L-17876
10, SPONSORING/MON_ORINGAGENCY REPORT NUMBER
NASA/TP-2000-209865
11. SUPPLEMENTARY NOTES
Heinbockel and Clowdsley: Old Dominion University, Norfolk, VA; Wilson: Langley Research Center, Hampton,VA.
12a. DISTRIBUTIDN/AVAILABILITY STATEMENT
Unclassified-UnlimitedSubject Category 93 Distribution: StandardAvailability: NASA CASI (301) 621-0390
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
A low-energy neutron transport algorithm for use in space radiation protection is developed. The algorithm is basedupon a muhigroup analysis of the straight-ahead Bohzmann equation by using a mean value theorem for integrals.This analysis is accomplished by solving a realistic but simplified neutron transport test problem. The test problemis analyzed by using numerical and analytical procedures to obtain an accurate solution within specified errorbounds. Results from the test problem are then used for determining mean values associated with rescatteringterms that are associated with a multigroup solution of the straight-ahead Boltzmann equation. The algorithm is
then coupled to the Langley HZETRN code through the evaporation source term. Evaluation of the neutron flu-ence generated by the solar particle event of February 23, 1956, for a water and an aluminum-water shield-targetconfiguration is then compared with LAHET and MCNPX Monte Carlo code calculations for the same shield-tar-get configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRNsolution. In addition, a two-directional solution of the evaporation source showed even further improvement of thefluence near the front of the water target where diffusion from the front surface is important.
14. SUBJECT TERMS
Multigroup; Low-energy neutrons; Neutron transport; Evaporation neutrons;HZETRN; Radiation shielding
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