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A comparison of the multigroup and collocation methods for solving the low-energy neutron Boltzmann equation M.S. Clowdsley, J.H. Heinbockel, H. Kaneko, J.W. Wilson, R.C. Singleterry, and J.L. Shinn
Abstract: A low-energy neutron transport algorithm for use in space-radiation protection is developed. The algorithm is based upon a multiple energy group analysis of the straight ahead Boltzmann equation utilizing a mean value theorem for integrals. The algorithm developed is then verified by using a collocation method solution on the same straight ahead Boltzmann equation. This algorithm was then coupled to the existing NASA Langley HZETRN (high charge and energy transport) code through the evaporation source term. Evaluation of the neutron fluence generated by the February 23, 1956 solar particle event for an aluminum–water shield–target configuration is then compared with the LAHET Monte Carlo calculation for the same shield–target configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRN solution. A bidirectional modification of the evaporation source produced further improvement of the fluence. PACS Nos.; 87.50N, 25.40D, 28.20G Résumé : Nous développons un algorithme de transport de neutrons de basse énergie pour le calcul de protections contre les radiations dans l’espace. L’algorithme est basé sur une analyse à énergies multiples de l’équation de Boltzmann en direction avant utilisant un théorème de valeur moyenne pour les intégrales. L’algorithme est alors vérifié en utilisant une solution par méthode de classement sur la même équation de Boltzmann. L’algorithme a ensuite été couplé, via les termes d’évaporation de la source, à un programme déjà existant de la NASA, le HZETRN (High Charge and Energy Transport) développé à Langley. L’évaluation de la fluence de neutrons générés par l’orage solaires du 23 février 1956 pour une cible d’aluminium avec un écran d’eau est alors comparé avec le résultat du calcul Monte-Carlo LAHET pour la même configuration de cible. Le nouvel algorithme montre une amélioration importante des résultats lorsque comparés à ceux obtenus par HZETRN sans modification. Une modification bidimensionnelle de la source d’évaporation a amélioré encore plus le calcul de fluence. [Traduit par la rédaction]
Received July 9, 1999. Accepted February 8, 2000. M.S. Clowdsley, J.H. Heinbockel, and H. Kaneko. Department of Mathematics, Old Dominion University, Norfolk, VA 23529-0077, U.S.A. Telephone: (757) 863-3887; FAX: (757) 683-3885; e-mail:
[email protected];
[email protected];
[email protected];
[email protected];
[email protected] J.W. Wilson, R.C. Singleterry, and J.L. Shinn. NASA Langley Research Center, Hampton, VA 23681-0001, Mail Stop 188B, U.S.A. Telephone: (757) 864-1414, FAX: (757) 864-8094; e-mail:
[email protected];
[email protected];
[email protected] Can. J. Phys. 78: 45–56 (2000)
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1. Introduction The purpose of this paper is to present an improved algorithm for the analysis and computation of the transport of low-energy secondary neutrons arising in space-radiation protection studies using a multigroup method. A collocation method of solution to the same problem is presented. This provides an alternative solution method as well as a verification of the multigroup solution technique. The results obtained are then used to improve output from the current HZETRN (high charge and energy transport) code in use at NASA Langley Research Center. The two main sources of radiation hazards to manned space missions are energetic solar particle events (SPEs) and galactic cosmic rays (GCRs). Methods for accurately describing the interactions and transport of GCR particles and SPE particles through shielding materials have been developed by NASA Langley Research Center and implemented in the HZETRN computer code, which handles all naturally occurring atomic ions and neutrons [1]. An analysis of a space craft’s radiation field and environment depends upon the craft’s geometry along with the thickness and material composition of its shielding. The design and operational processes in space-radiation shielding and protection require highly efficient computational procedures to adequately characterize time-dependent radiation fields that vary continuously in intensity and spectral content along an orbital path or on a space mission [2]. Also, time-dependent geometric factors resulting from crew movement and use of consumables must be considered in these environments to address shield evaluation issues in a multidisciplinary integrated engineering design environment. Simulations of radiation fields using standard Monte Carlo codes such as LAHET (Los Alamos high-energy transport code), MCNP (Monte Carlo N-particle transport code), and MCNPX (LAHET and MCNP code merger), are restricted to neutrons, protons, pions, and alpha particles and require an excessive amount of computer time. In comparison, the HZETRN code is deterministic, computationally fast and can simulate radiation fields associated with all naturally occurring atomic ions and neutrons. It is computationally efficient and rapid enough to handle many types of simulations. The research presented in this paper addresses the problem of comparing neutron fluences in the low-energy range, from 0.01 to several MeV, produced by the HZETRN code and Monte Carlo codes. In the past, the HZETRN neutron fluences were substantially below the Monte Carlo results in this low-energy range. Analysis suggested that the discrepancy was in the rescattering terms in which the number of elastic scattered neutrons was underestimated. We present a multiple energy group analysis for the representation of low-energy evaporation neutrons that leads to a great improvement in the HZETRN code simulation results when benchmarked with the Monte Carlo results. We include in our presentation elastic processes as well as neutron production at backward angles to account for this improvement of results from the HZETRN code.
2. Formulation of transport equations The HZETRN code calculates solutions to the one-dimensional Boltzmann transport equation [3] XZ ∞ σj k (E, E 0 )φk (x, E 0 ) dE 0 (1) B[φj ] = k
0
where B[ ] is the differential operator ∂ ∂ − Sj (E) + σj (E) φ(x, E) B[φ] = ∂x ∂E ∂ ∂φ(x, E) − Sj (E)φ(x, E) + σj (E) φ(x, E) = ∂x ∂E
(2)
In (1), the term φj is the differential flux spectrum for the type j particles, Sj (E) is the stopping power of type j particles and σj (E) is the total macroscopic cross section. The term σj k (E, E 0 ) is the ©2000 NRC Canada
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Fig. 1. Comparison of evaporation and cascading neutron effects for collision of 500 MeV neutrons in aluminum.
macroscopic differential-energy cross section for redistribution of particle type and energy given by σj k (E, E 0 ) =
X
ρm σm (E 0 )fj k,m (E, E 0 )
(3)
m
where fj k,m (E, E 0 ) is the spectral redistribution, σm is a microscopic cross section and ρm is the number density of m type atoms per unit mass of material. The spectral terms are expressed as fj k,m = fjelk,m + fjek,m + fjdk,m
(4)
where fjelk,m represents the elastic redistribution in energy, fjek,m represents the evaporation terms, and fjdk,m represents the 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 the high-energy range (E > 25 MeV) as illustrated in the Fig. 1, which was obtained from the HZETRN code [1]. The neutron component of (1) is then rewritten for j = n as B[φn ] =
XZ k
∞X
E
m
el e d ρm σm (E 0 )(fnk,m + fnk,m + fnk,m )φk (x, E 0 ) dE 0
(5)
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which we expand to the form Z ∞X el e d ρm σm (E 0 )(fnn,m + fnn,m + fnn,m )φn (x, E 0 ) dE 0 B[φn ] = E
+
XZ
m ∞X
k6=n E
el e d ρm σm (E 0 )(fnk,m + fnk,m + fnk,m )φk (x, E 0 ) dE 0
m
We define the integral operator Z ∞X a ρm σm (E 0 )fnk,m φ(x, E 0 ) dE 0 Ia(k) [φ] = E
(6)
(7)
m
where k = n denotes coupling to neutron collisions, k = p denotes the neutron source from proton collisions, and a can be one of the labels “el” for elastic redistribution, “e” for evaporations processes, or “d” for direct knockout terms. Considering only neutrons and protons we can write (6) in the linear operator form (n)
(p)
(n)
(p)
(p)
B[φn ] = Iel [φn ] + Ie(n) [φn ] + Id [φn ] + Iel [φp ] + Ie [φp ] + Id [φp ]
(8)
(p)
This equation is further simplified by setting Iel [φp ] = 0 since it does not contribute to the neutron field. For simplicity of notation we drop the subscript of the field by replacing φn by φ and we write 8) as (n)
(p)
(n)
(p)
B[φ] = Iel [φ] + Ie(n) [φ] + Id [φ] + Ie [φp ] + Id [φp ]
(9)
We assume a solution to (9) of the form φ = φe + φd where φe is the solution for evaporation neutron sources and contributes over the low-energy range and φd is the solution for the direct knockout neutron sources and contributes mainly over the high energy range as suggested by the Fig. 1. We substitute this assumed solution into (9) and find (n)
(n)
(n)
(n)
B[φe ] + B[φd ] = Iel [φe ] + Iel [φd ] + Ie(n) [φe ] + Ie(n) [φd ] (p)
(10)
(p)
+ Id [φe ] + Id [φd ] + I(e) [φp ] + Id [φp ] (n)
(n)
We observe that for elastic scattering, the terms Ie [φe ] and Id [φe ] are near zero and can be ignored since evaporation neutrons at low energies do not produce additional evaporation neutrons and direct cascade effects have very small cross sections over the low-energy range of φe and do not contribute any production over the low- or high-energy range. We assume that the fluence φd is calculated by the HZETRN program because for the direct cascade neutrons the straight ahead approximation is valid. Consequently, we treat φd as a solution of the equation (n)
(n)
(p)
B[φd ] = Iel [φd ] + Id [φd ] + Id [φp ]
(11)
This simplifies (10) to the form (n)
(p)
B[φe ] = Iel [φe ] + Ie(n) [φd ] + Ie [φp ]
(12)
el (E, E 0 ) and note that for neutrons We define the elastic-scattering terms in (10) σs,m = ρm σm (E 0 )fnn,m the stopping power Sj (E) is zero and so (12) reduces to the integro-differential transport equation with the source term as XZ ∞ ∂ + σ (E) φe (x, E) = σs,m (E, E 0 )φe (x, E 0 ) dE 0 + g(E, x) (13) ∂x e m
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Equation (13) represents the steady-state evaporation neutron fluence φe (x, E) at depth x and energy (p) (n) E. The volume source term g(E, x) = Ie [φd ] + Ie [φp ] together with the cross sections σ (E) and σs,m (E, E 0 ) are determined from the HZETRN algorithm. Equation (13) 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 ATm , initially at rest, is from ref. 4 " E=E
0
A2Tm + 2ATm cos θ + 1
# (14)
(ATm + 1)2
where E 0 is the neutron energy before the collision, ATm is the atomic weight of the mth type of atom being bombarded and θ is the angle of scatter. Note that for forward scattering θ = 0, E = E 0 and for backward scattering θ = π , E = E 0 αm where αm is the ratio αm =
ATm − 1 AT m + 1
2 (15)
which is a constant less than one. Therefore, we can change the limits of integration in (13) to [E, E/αm ], which represent the kinetically allowed energies for the elastic scattered neutron to result in an energy E. Equation (13) then is written
X Z E/αm ∂ + σ (E) φe (x, E) = σs,m (E, E 0 )φe (x, E 0 ) dE 0 + g(E, x) ∂x e m
(16)
where σ (E) =
X m
ρm σmel (E)
(17)
where ρm is the number of atoms per gram, and σmel (E) is a microscopic elastic cross section. For simplicity, we only consider in this paper, the case where there is one value of m, which represents neutron penetration into a single-element material, and we let φe be denoted by φ. Similarly, we denote σs,m and αm by σs and α, respectively. We now consider solutions to the equation
Z E/α ∂ σs (E, E 0 )φ(x, E 0 ) dE 0 + g(E, x) + σ (E) φ(x, E) = ∂x E
(18)
by the multigroup and the collocation methods.
3. Multigroup method Throughout the remaining discussion we employ the following mean-value theorem for integrals. 3.1. Mean-value theorem For φ(x, E) and f (E) to be continuous over an interval a ≤ E ≤ b such that (i) φ(x, E) does not change sign over the interval [a, b], (ii) φ(x, E) is integrable over the interval [a, b], and (iii) f (E) is bounded over the interval [a, b], ©2000 NRC Canada
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then there exists at least one point such that Z b Z b f (E)φ(x, E) dE = f () φ(x, E) dE, a
a
a≤≤b
(19)
In particle transport this mean-value approach is not commonly used. In reactor-neutron calculations an assumed spectral dependence for φ(x, E) is used to approximate the integral over energy groups. The present use of the mean-value theorem is free of these assumptions allowing more flexibility in the HZETRN code and results in a fast and efficient algorithm for low neutron energy analysis. To solve (18) we introduce an energy partition {E0 , E1 , . . . , Ei , Ei+1 , . . . , EN } and for each i we integrate (18) from Ei to Ei+1 with respect to the energy E to obtain Z Ei+1 Z Ei+1 ∂φ(x, E) dE + σ (E)φ(x, E) dE = Is,i + ξi ∂x Ei Ei where
Z
Is,i = and ξi (x) =
Ei+1
Z
Ei
E/α
E
Z
Ei+1
Ei
σs (E, E 0 )φ(x, E 0 ) dE 0 dE
(20)
(21)
g(E, x) dE
(22)
We define the quantity Z Ei+1 φ(x, E) dE 8i (x) =
(23)
Ei
associated with the ith energy group (Ei , Ei+1 ), so that Ei+11−Ei 8i (x) then represents an average fluence for the ith energy group. We can write (20) as a system of ordinary differential equations in terms of 8i (x), i = 0, . . . , N as follows. In the first term in (20), we interchange the order of integration and differentiation to obtain Z Ei+1 ∂φ(x, E) d8i (x) dE = (24) ∂x dx Ei Using the previously stated mean-value theorem for integrals, the second term in (20) can be expressed as Z Ei+1 σ φ(x, E) dE = σ 8i (x) (25) Ei
where σ = σ (Ei + θ(Ei+1 − Ei )), is a mean value associated with some value of θ between 0 and 1. For the term Is,i in (21), we interchange the order of integration. The limits of integration then depend upon energy partitioning. The author of ref. 5 investigated various energy partitions and concluded that the partitioning defined by Ei+1 = Ei /α produced the simplest form for the resulting equation which then has the form, [5,6] Z Ei+1 Z E 0 Z Ei+1 /α Z Ei+1 σs (E, E 0 )φ(x, E 0 ) dE dE 0 + σs (E, E 0 )φ(x, E 0 ) dE dE 0 Is,i = E 0 =Ei
E=Ei
E 0 =Ei+1
E=αE 0
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Fig. 2. Comparison of multigroup and collocation methods for neutron fluence in an aluminum shield exposed to the February 23, 1956 solar energetic particle event. (Continuous lines represent the multigroup method.)
and can be simplified using the above mean-value theorem to the form "Z ∗ "Z # # Is,i =
Ei
Ei
σs (E, Ei∗ ) dE
8i (x) +
Ei+1
∗ αEi+1
∗ σs (E, Ei+1 ) dE
8i+1 (x)
(26)
∗ = θ (E where Ei∗ = θ1 (Ei+1 − Ei ) and Ei+1 2 i+2 − Ei+1 ) for 0 < θ1 , θ2 < 1 giving mean values over the intervals. These mean values are not unique and were selected to satisfy a test problem exactly [5]. For some large value of N we assume the term 8i (x) = 0 for all i ≥ N . Then for i = 0, . . . , N − 1 there results from (20) the system of ordinary differential equations 80 a11 a12 80 ξ0 81 81 ξ1 a22 a23 −0− d .. .. .. . . .. .. (27) . = . + . dx 8N−2 −0− aN −1,N−1 aN −1,N 8N −2 ξN −2 8N−1 aN N 8N −1 ξN −1
subject to the initial conditions 8i (0) = 0. We observe that the coefficient matrix is banded with nonzero elements on the main and first upper diagonal. These elements are given by Z E∗ Z Ei+1 i ∗ σs (E, Ei∗ ) dE − σ¯ , ai,i+1 = σs (E, Ei+1 ) dE ai,i = Ei
∗ αEi+1
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Can. J. Phys. Vol. 78, 2000 Fig. 3. Comparison of multigroup and collocation methods for neutron fluence in an aluminum–water, shield–target configuration from exposure to the February 23, 1956 solar energetic particle event. Water depth 1.0 gm/cm2 , aluminum depths vary from 0 to 100 gm/cm2 . (Continuous lines represent the multigroup method.)
for i going from 1 to N. In the system of equations (27), we integrate the last equation and use back substitution to solve the remaining equations above it. Here the trapezoidal rule was used for the integrations. This treatment appears to be approximately the same as the standard multigroup method as annotated in ref. 7. However, because of the nature of the available data and the method being used by HZETRN, a fundamentally different treatment of the cross section data is needed. The standard method assumes that functional or evaluated cross-section data exist and can be evaluated outside of the solution of the transport equation and appear in the transport equation as group constants. These group constants are evaluated in ref. 7 (eqs. 5.32 and 5.33) as R Ei+1 σ (x, E)φ(x, E) dE Ei (28) σi (x) ≡ R Ei+1 φ(x, E) dE Ei and 0
σs (x; E, E ) ≡
R E 0 i+1 E0 i
RE φ(x, E 0 ) Eii+1 σs (x; E, E 0 ) dE dE 0 R Ei+1 φ(x, E) dE Ei
(29)
The standard method determines these constants by assuming a flux spectrum and integrating with evaluated (or calculated) cross-section data. These constants are put in libraries that depend on the ©2000 NRC Canada
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Fig. 4. Comparison of multigroup and collocation methods for neutron fluence in an aluminum–water, shield–target configuration from exposure to the February 23, 1956 solar energetic particle event. Water depth 10.0 gm/cm2 , aluminum depths vary from 0 to 100 gm/cm2 . (Continuous lines represent the multigroup method.)
number of energy groups, flux spectrum, and energy spectrum intervals and are used for a multitude of different calculations. The problem with this method is that a particular library is only valid for a particular calculation as long as the final flux spectrum is not too different from the flux spectrum used in the library evaluation. This is fine for most reactor-type calculations; however, for space radiation, this assumption is inappropriate. Also, HZETRN uses a continuous energy-based method and determines its cross sections when a calculation is performed. Fundamental material constants are stored, read, and used when needed. Therefore, the reported method was developed to fit the requirements needed by HZETRN.
4. Collocation method Consider (18) over the interval E0 ≤ E ≤ EN . Partition this interval into n energy subintervals defined by Ei , i = 0, 1, 2, . . . , n with some convenient spacing for the subintervals. We approximate the solution to (18) by the series φ(x, E) =
n X
φj (x, Ej )Bj (E)
(30)
j =0
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Can. J. Phys. Vol. 78, 2000 Fig. 5. Energy spectra of neutron fluence at 10 gm/cm2 depth in water exposed to the February 23, 1956 solar energetic particle event.
with functions φj (x, Ej ) to be determined and the Bj (E) are selected as linear splines with support [Ej −1 , Ej +1 ] defined by E−Ej −1 , j −1 Bj (E) = EEjj−E +1 −E ,
on [Ej −1 , Ej ] on [Ej , Ej +1 ]
Ej +1 −Ej
(31)
such that det[Bj (Ei )] 6 = 0. This condition is required for the collocation method to work. Substituting 0≤i≤n 0≤j ≤n
(30) into (18) gives n X dφj (x, Ej ) j =0
dx
Bj (E) + =
n X
σ (E)φj (x, Ej )Bj (E)
j =0 n Z E/α X j =0 E
σs (E, E 0 )φj (x, Ej )Bj (E 0 ) dE 0 + g(E, x)
(32)
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In (32), we make E = Ei so that for each i we obtain n X
n
Bj (Ei )
j =0
dφj (x, Ej ) X = −σ (Ei )Bj (Ei ) dx j =0 Z Ei /α σs (Ei , E 0 )Bj (E 0 ) dE 0 φj (x, Ej ) + g(Ei , x) +
(33)
Ei
This produces the system of ordinary differential equations n
X dφi aij φj + gi , = dx
i = 0, 1, . . . , n
where gi = g(Ei , x) and Z aij = −σ (Ei )Bj (Ei ) +
Ei /α
(34)
j =0
Ei
σs (Ei , E 0 )Bj (E 0 ) dE 0
(35)
where use of the mean-value theorem is not needed. The system of equations (34) is subject to the boundary conditions φj (0, Ej ) = 0 for each value of j . Again, the cross section σ , scattering cross sections σs , and source terms gi = g(Ei , x) were obtain from the HZETRN code and we used a fourth-order Runge–Kutta method to solve this system of equations. With additional work the above multigroup and collocation methods can be modified to solve (16) involving more than one material [6,8]. This leads to upper triangular coefficient matrices in the system of differential equations (27) and (34) [6,8]. For comparison, we used the same energy grid on both methods (n = 36) with Ei+1 = Ei /αk for the simulation of neutron radiation on an aluminum–water, shield–target configuration. Here αk is associated with the compound element with the largest atomic weight. The multigroup and collocation methods take about the same time to execute on the computer. However, in using the collocation method care should be taken in the placement of the support functions.
5. Concluding remarks Figures 2–4 illustrate some of the comparisons obtained from the implementation of the two solution methods discussed above. Each set of calculations was done in less than 1 min on a 200 MHZ personal computer using 36 energy groups for both the multigroup and collocation methods. Figure 2 illustrates neutron fluence as a function of energy for an aluminum shield only, while Figs. 3 and 4 illustrate neutron fluence for water depths of 1.0 and 10.0 gm/cm2 , respectively, for various aluminum shield thickness. The collocation method results are illustrated by the symbols while the multigroup method is illustrated by the continuous-line curves. All three figures show a good comparison between the two methods. Figure 5, from ref. 8, shows a great improvement in the output of the fluence from the HZETRN code when the low-energy neutron component was adjusted using the multigroup method for calculating the low-energy neutron fluence from the Boltzmann equation. The collocation method was developed as a check on these improvements. A further improvement resulted from a bidirectional implementation of the boundary conditions and source terms. Using one-half the volume source term and starting fluence of zero at the left edge, the solution was marched to the right boundary. Using the right-side boundary condition generated, the solution was then marched back to the left edge using the other one-half of the volume source term. This bidirectional implementation is an attempt to incorporate neutron directional dependence into the one-dimensional model and represents a diffusion approximation. This diffusion approximation will not adequately treat highly anistropic scattering that requires the development of a more directionally dependent neutron transport code. The improved results from this bidirectional implementation is also illustrated in Fig. 5. ©2000 NRC Canada
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Acknowledgments This research was supported under NASA research grant NCC-1-320.
References 1. J.W. Wilson, F.F. Badavi, F.A. Cucinotta, J.L. Shinn, G.D. Badhwar, R. Silberberg, C.H. Tsao, L.W. Townsend, and R.K. Tripathi. NASA Tech. Paper, 3495. (1995). 2. J.W. Wilson, F.A. Cucinotta, J.L. Shinn, L.C. Simonsen, R.R. Dubey, W.R. Jordan, T.D. Jones, C.K. Chang, and M.Y. Kim. Rad. Measur. 30, 361 (1999). 3. J.W. Wilson, L.W. Townsend, W. Schimmerling, G.S. Khandelwal, F.Kahn, J.E. Nealy, F.A. Cucinotta, L.C. Simonsen, J.L. Shinn, and J.W. Norbury. NASA Publ. 1257. (1991). 4. J.W. Haffner. Radiation and shielding in space. Academic Press, New York and London. 1967. 5. M.S. Clowdsley. Ph.D. thesis, Old Dominion University, Virginia. 1999. 6. M.S. Clowdsley, J.W. Wilson, J.H. Heinbockel, R.K. Tripathi, R.C. Singleterry, and J.L. Shinn. NASA Tech. Paper, In press. 7. G.I. Bell and S. Glasstone. Nuclear reactor theory. Robert E. Krieger Publishing Co. Malabar, Flor. 1985. 8. J.H.Heinbockel, M.S. Clowdsley, and J.W. Wilson. NASA Tech. Paper, 2000-209865, March 2000. In press.
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