Abstract
Methods for approximately accounting for the terms neglected in a finite (L’th-order) Legendre expansion of the scattering source in the transport equation are called transport corrections. This paper derives adjoint-based sensitivities of a neutron or gamma-ray transport response for problems that use diagonal, Bell-Hansen-Sandmeier (BHS), or n’th-Cesàro-mean-of-order-2 (Cesàro) transport corrections in the discrete-ordinates method. For diagonal and BHS transport corrections, there is a sensitivity to the L + 1ʹth scattering cross-section moment, and the sensitivity to nuclide and material densities requires this contribution. For the Cesàro transport correction, the sensitivities to the scattering cross section for the l’th moment are multiplied by a simple function of l and the scattering expansion order L. Numerical results for a keff problem and a fixed-source problem verify the derivation and implementation of the sensitivity equations into the SENSMG multigroup sensitivity code. The Cesàro transport correction yields inaccurate responses for both problems.
I. INTRODUCTION
Truncating the spherical harmonics expansion of the scattering source in the Boltzmann transport equation leads to truncation error. Methods for approximately accounting for the neglected terms are called transport corrections.Citation1 This paper deals with three widely used transport corrections in the discrete-ordinates method: diagonal,Citation2 Bell-Hansen-SandmeierCitation3 (BHS), and n’th-Cesàro-mean-of-order-2 (Cesàro).Citation4 These transport corrections modify the scattering cross sections and (except for Cesàro) the total cross sections used in the transport calculation. Another transport correction for several applications, including discrete ordinates, was derived by StaceyCitation5 from a variational principle for the quantity of interest. Transport corrections are also used in nuclear reactor physics.Citation6–8
Transport corrections were developed in a time when including even one scattering cross-section moment beyond isotropic (i.e., a linearly anisotropic expansion) was challenging, both in terms of computation time and memory. As fast and large as modern computers are, modern multigroup discrete-ordinates transport codes still have transport correction options. PARTISN (CitationRef. 9) has diagonal, BHS, Cesàro, and the default of none. These options, including the coding, were inherited from PARTISN’s predecessor, ONEDANT (CitationRef. 10). Denovo (CitationRef. 11) has diagonal, Cesàro, and none, with diagonal being the default. ARDRA (CitationRef. 12) has diagonal and none, with none being the default. Attila (CitationRef. 13) has diagonal, BHS, and none, with diagonal being the default. (ARDRA has historically offered two additional transport corrections that have not been published.Citation14 Attila also offers an “optimized diagonal” transport correction.Citation15,Citation16) Of course, users who supply their own nuclear data to the deterministic transport code can apply a transport correction in the nuclear data preparation process. The nuclear data formatting utility codeCitation17 TRANSX offers the diagonal and BHS transport corrections as well as the consistent-P approximationCitation3 (and two other approximations described only briefly in CitationRef. 1 and the various versions of the TRANSX and NJOY code manualsCitation17,Citation18). ARDRA relies on data processing and formatting codes for its transport corrections.Citation14
Transport corrections are well known, but to this author’s knowledge, the adjoint-based sensitivity of a computed response to the transport-corrected scattering and total cross sections and nuclide densities has never been presented. Calculating adjoint-based sensitivities is enormously efficient compared to using direct perturbations. This paper discusses the issues associated with computing sensitivities with adjoint methods when transport corrections are used. The three transport correction options available in PARTISN have been implemented in the SENSMG multigroup neutron sensitivity code.Citation19,Citation20 SENSMG has a similar role as the old SWANLAKE (CitationRef. 21), FORSS (CitationRef. 22), and SENSIT (CitationRef. 23) codes. It has capabilities similar to those of SUSD3D (CitationRefs. 24 and Citation25), which also uses PARTISN. SUSD3D relies on TRANSX for its nuclear data, and its transport corrections are applied in TRANSX rather than the transport code.
This paper is organized as follows. The diagonal, BHS, and Cesàro transport corrections are presented in Sec. II. Transport-corrected adjoint sensitivities are derived in Sec. III. Section IV presents numerical results for a keff test problem, and Sec. V presents numerical results for an inhomogeneous (fixed-source) test problem. Section VI is a summary and conclusions. The Appendix provides a corrigendum for CitationRef. 19.
II. TRANSPORT CORRECTIONS
The forward Boltzmann transport equation for neutral particles is
where
ψ = = | = | the angular flux of particles at position r, direction , and energy group g |
q = = | = | the source rate density at position r, direction , and energy group g |
λ = | = | multiplication factor; λ = 1 if q ≠ 0 and is the eigenvalue if q = 0 |
F = | = | the fission production operator, which will not be discussed in this paper |
A = | = | the loss operator, defined by |
In EquationEq. (2(2) (2) ),
= | = | the total macroscopic cross section at position r and energy group g |
= | = | the l’th Legendre moment of the macroscopic scattering cross section from energy group g′ to g at position r |
= | = | the l’th Legendre moment of the angular flux in energy group g′ at position r |
G = | = | the number of energy groups |
L = | = | the scattering expansion order. |
EquationEquation (1)(1) (1) has appropriate boundary conditions such as vacuum (appropriate forward boundary conditions are any that, with the adjoint boundary conditions mentioned in Sec. III, cause the bilinear concomitant to vanishCitation26). The macroscopic cross section for reaction x in group g at position r is
where
= | = | the microscopic cross section for nuclide i |
= | = | the atom density of nuclide i at position r |
I = | = | the number of nuclides in the material at position r. |
The diagonal transport correctionCitation2 corrects the macroscopic total and self-scattering cross sections using
and
where the tilde represents the transport-corrected cross section that is used in EquationEq. (2)(2) (2) instead of the cross section without the tilde. (In this section, the translation from the original sources’ notation to that of this paper was facilitated by chapter 3 of the PARTISN manual.Citation9) Strictly speaking, Pendlebury and UnderhillCitation2 defined the diagonal transport correction only for L = 0. Bell et al.Citation3 provided the natural extension to an arbitrary scattering expansion order. Pendlebury and Underhill wrote that the diagonal transport correction “can be expected to be fairly reliable as long as the major contribution to elastic scattering retains the neutrons in the same energy group.”Citation2
The BHS transport correctionCitation3 corrects the macroscopic total and self-scattering cross sections using
and
Following Lathrop,Citation27 Bell et al.Citation3 called their method the “extended transport correction,” an appellation that MacFarlaneCitation1 retained in addition to the more common one. Comparing the diagonal transport correction with the BHS (or extended) transport correction, Bell et al. wrote, “… [F]or a coarse group structure, this approximation [i.e., the diagonal transport correction] approaches the extended transport approximation, while for G → ∞, the ‘transport correction term’ … vanishes altogether.”Citation3 Attila (CitationRef. 13) requires L ≥ 1 for use of the BHS correction.
The Cesàro transport correctionCitation4 corrects the macroscopic scattering cross sections using
The Cesàro transport correction has no effect for l = 0 (isotropic scattering), and the modified scattering cross sections do not affect the total cross section . The n in the phrase “n’th Cesàro mean of order 2” is in our notation L, the order of the last term in the scattering expansion. While the diagonal and BHS transport corrections seek to make the SN form of the transport equation as close as possible to the PN form,Citation1 the Cesàro transport correction seeks to approximate the infinite sum
with the partial sumCitation28,Citation29
In EquationEqs. (9)(9) (9) and (Equation10(10) (10) ), μ0 is the cosine of the scattering angle and Pl is the Legendre polynomial of order l. Albert and Nelson recognized that the Cesàro transport correction would only be accurate for relatively large degrees of anisotropy, whereas the usual partial sum,
“for relatively low degrees of anisotropy … tends to be much better than the [Cesàro transport correction],”Citation4 but they sought to exploit the fact that is nonnegative. Denovo requires L ≥ 2 for use of the Cesàro correction.Citation11 The PARTISN manualCitation9 advises this, but the code does not require it.
PARTISN applies the selected transport correction to the macroscopic material cross sections. The output cross-section file macrxs contains uncorrected cross sections. However, the cross sections output to the standard output file using xsectp > 0 in block 5 are transport corrected.
III. TRANSPORT-CORRECTED SENSITIVITIES
The relative sensitivity of a response R to input parameter x is defined to be
where subscript 0 indicates a specific state of the system. The sensitivity is evaluated at a particular system state. Henceforth, subscript 0 will be assumed.
Briefly, the adjoint-based relative sensitivity of a response R to the total cross section of nuclide i in energy group g is obtained by multiplying EquationEq. (2)(2) (2) by the adjoint angular flux and integrating over volume and angle.Citation30 The adjoint flux satisfies
where
= = | = | the adjoint source rate density at position r, direction , and energy group g |
λ = | = | the same multiplication factor as in EquationEq. (1(1) (1) ) |
= | = | the adjoint fission production operator, which will not be discussed in this paper |
= | = | the adjoint loss operator, defined by |
where is the l’th Legendre moment of the adjoint angular flux in energy group g′ at position r. EquationEquation (13)(13) (13) has appropriate boundary conditions, such as vacuum (appropriate adjoint boundary conditions are any that, with the forward boundary conditions mentioned in Sec. II, cause the bilinear concomitant to vanishCitation26). The sensitivity is
The right side of EquationEq. (15)(15) (15) may have a denominator, depending on the response and whether q = 0 in EquationEq. (1)(1) (1) . (For example, for an eigenvalue problem in which R = keff is the response, is EquationEq. (15)(15) (15) divided by .) Likewise, the relative sensitivity of R to the scattering cross section of nuclide i for expansion order l from energy group g to g′ is
where is the Kronecker delta function (see Appendix). The second term on the right side appears because the isotropic scattering cross sections are addends in the total cross section .
The relative sensitivity in group g of R to the density of nuclide i is
where the relative sensitivity to total scattering within and out of group g is
(the effect of isotropic scattering on the total cross section is accounted for in ). is the contribution from the fission terms and is not discussed further in this paper; see CitationRef. 20. The relative sensitivity in group g of R to the mass density ρ of the material is
where I is the number of nuclides in the material, as before.
When the BHS transport correction is used, EquationEqs. (6)(6) (6) and Equation(7)(7) (7) are used in EquationEq. (2)(2) (2) to yield
The same derivation that leads to EquationEqs. (15)(15) (15) and (Equation16(16) (16) ) yields the relative sensitivity of R to the scattering cross section of nuclide i for expansion order L + 1 from energy group g to g′:
The first term on the right side of EquationEq. (21)(21) (21) appears because of EquationEq. (7(7) (7) ); the second term appears because of EquationEq. (6)(6) (6) . Comparing EquationEqs. (4)(4) (4) and (Equation5(5) (5) ) with EquationEqs. (6)(6) (6) and (Equation7(7) (7) ), when diagonal transport correction is used, the relative sensitivity of R to the scattering cross section is , or just the diagonal elements of the scattering sensitivity matrix. Note that the index on the adjoint flux in the first term on the right side of EquationEq. (21)(21) (21) is g, not the usual g′ for scattering sensitivities—e.g., see the first term in EquationEq. (16)(16) (16) —because only the within-group scattering cross section is transport corrected [EquationEq. (7)(7) (7) ].
When diagonal transport correction is used, the relative sensitivity in group g of R to the density of nuclide i is
and when the BHS transport correction is used, the relative sensitivity in group g of R to the density of nuclide i is
When the Cesàro transport correction is used, EquationEq. (8)(8) (8) is used in EquationEq. (2)(2) (2) to yield
The same derivation that leads to EquationEqs. (15)(15) (15) and Equation(16)(16) (16) yields the modified relative sensitivity of R to the scattering cross section of nuclide i for expansion order l from energy group g to g′:
Likewise, the modified relative sensitivity to total scattering within and out of group g is
Thus, when the Cesàro transport correction is used, the relative sensitivity in group g of R to the density of nuclide i is
For diagonal and BHS transport corrections, the macroscopic total and self-scatter cross sections are modified [EquationEqs. (4)(4) (4) through (Equation7(7) (7) )] for the transport, but the sensitivity of R to the microscopic total and self-scatter cross sections is unaffected by the transport correction, except insofar as the forward and adjoint fluxes are affected. In other words, EquationEqs. (15)(15) (15) and (Equation16(16) (16) ) are still used for and . The calculation of these sensitivities is unaffected by the transport correction.
For the diagonal and BHS transport corrections, EquationEq. (21)(21) (21) is used for . For the diagonal transport correction, the sensitivity matrix is diagonal, .
For the diagonal and BHS transport corrections, a new term is added to the nuclide and material density sensitivities, the last term on the right side of EquationEqs. (22)(22) (22) and (Equation23(23) (23) ). An analogous term is added to the interface- and boundary-location derivatives.Citation19,Citation20
It is sometimes advantageous to approximate the second term on the right side of EquationEq. (21)(21) (21) using adjoint and forward flux moments instead of angular fluxesCitation25; i.e.,
Unfortunately, when this approximation is used, is zero because the two terms on the right side of EquationEq. (21)(21) (21) cancel.
IV. TEST PROBLEM FOR keff
The test configuration for keff was Flattop-Pu, otherwise known as PU-MET-FAST-006 (CitationRef. 31), a 6.06-kg ball of δ-phase plutonium reflected by 19.6 cm of natural uranium (NU) (the evaluationCitation31 and its antecedent reportCitation32 refer to this material as normal uranium). The geometry and materials are given in . The density and composition were converted from the standard International Criticality Safety Benchmark Evaluation Project specification in atom density using modern values of physical constants. The problem was run with various scattering expansion orders and each of the transport corrections available in PARTISN (including none).
An angular quadrature of S256 and fine mesh spacing of 0.005 cm were used. Thirty-group ENDF/B-VIII.0 cross sections having upscattering in the bottom five groups were used. The maximum scattering expansion order available in this library is 4 (indexed from 0). PARTISN version 8.32.54 was used.
IV. A. keff
The keff’s are compared in and . The keff’s for L = 4 (R4) are identical for no transport correction, BHS, and diagonal transport correction because that is the highest expansion order in the cross-section library; there is no to use in EquationEqs. (4)(4) (4) through (Equation7(7) (7) ). PARTISN writes the message “isct too large for transport correction” and solves the problem with no correction. SENSMG writes the message “no transport correction for nuclide [nuclide specified].” The keff’s for L = 0 are identical for no transport correction and Cesàro transport correction because in EquationEq. (8)(8) (8) .
Assuming that R4 with no transport correction (or diagonal or BHS) is the most accurate value, each of the keff’s is compared to R4 in . For L ≥ 1, diagonal and BHS results are essentially identical. Diagonal- and BHS-corrected P1 calculations are slightly more accurate than a P3 calculation with no transport correction.
The conventional wisdom that a P1 scattering expansion is more accurate than P2 is not true for this problem using these nuclear data, even without transport correction.
The Cesàro transport correction is wildly inaccurate for this problem, even for L ≥ 2.
IV.B. keff Sensitivities
In order to verify the derivations in Sec. III as well as the implementation in code, central-difference (CD) estimates of sensitivities were compared with adjoint-based calculations.
How important is it to account for the transport correction when using adjoint-based sensitivities? In other words, is it really necessary to use EquationEqs. (22)(22) (22) , (Equation23(23) (23) ), or (Citation27) (depending on the transport correction) instead of EquationEq. (17(17) (17) )? The adjoint-based (SENSMG) relative sensitivity of each of the material densities, computed without accounting for the effect of the transport corrections on the sensitivities [i.e., using EquationEq. (17(17) (17) )], was compared with a CD estimate for various scattering expansion orders and each of the transport corrections (including none). The CD estimate of the relative sensitivity of response R when input parameter x is perturbed by is
where p is the relative perturbation . For the CDs, each material mass density was perturbed ±0.1% (separately). The adjoint-based material density sensitivity is the sum of the nuclide density sensitivities over nuclide and energy group [EquationEq. (19)(19) (19) ]. Results are shown in . These results are shown for information only; SENSMG has no user option to disable the correct accounting for any transport correction. Again, the point of this comparison is to assess the importance of accounting for the transport correction when computing adjoint-based sensitivities.
shows that adjoint sensitivities can be in error by several percent for low scattering orders if transport correction is not accounted for. Also, results for L = 4 for no transport correction, BHS, and diagonal transport correction are all identical, as in , and for the same reason.
The proper adjoint-based (SENSMG) relative sensitivity of each of the material densities, which accounts for the effect of the transport corrections on the sensitivities, was also compared with the CD estimates. Results are shown in . verifies EquationEqs. (22)(22) (22) , (Equation23(23) (23) ), and (Equation27(27) (27) ) for the diagonal, BHS, and Cesàro transport corrections, respectively.
EquationEquation (21)(21) (21) for the relative sensitivity of R to the L + 1’th scattering cross section for the BHS transport correction and the quantity for the diagonal transport correction were verified using CDs. For each scattering expansion order L, all 238U L + 1 elastic scattering cross sections were perturbed by ±10%. The computed sensitivity is for BHS and for diagonal, with given by EquationEq. (21)(21) (21) . Results are shown in . The adjoint sensitivities are in excellent agreement with the CDs. (Elastic, and later, inelastic scattering cross sections for use in SENSMG and in the CD were obtained from an output file written by the cross-section processing codeCitation33 NJOY’s GROUPR module.)
Kodeli and SlavičCitation25 used this problem and computed the total relative sensitivity of keff to 238U elastic and inelastic scattering. They used ENDF/B-VII.0 nuclear data in a 33-group structure, so their results are not directly comparable with ours.
The total relative sensitivity of R to scattering reaction x of nuclide i when the diagonal or BHS transport correction is used is
where for 0 ≤ l ≤ L is from EquationEq. (16)(16) (16) and is from EquationEq. (21)(21) (21) . The total relative sensitivity of R to scattering reaction x of nuclide i when the Cesàro transport correction is used is
where is from EquationEq. (25(25) (25) ) [but of EquationEq. (25)(25) (25) is equal to of EquationEq. (16)(16) (16) ].
The adjoint formulas for the sensitivity of keff to 238U elastic scattering are compared with CDs in . All 238U elastic scattering cross sections were perturbed ±10%. The adjoint sensitivities are in excellent agreement with the CDs. It is noteworthy that for L ≥ 1, sensitivities are essentially identical when the diagonal and BHS corrections are used. Kodeli and SlavičCitation25 reported a sensitivity of 0.1431%/% for L = 3 and 0.1399%/% for L = 5.
The adjoint formulas for the sensitivity of keff to 238U inelastic scattering are compared with CDs in . All 238U inelastic scattering cross sections were perturbed ±10%. The adjoint sensitivities are in excellent agreement with the CDs. Sensitivities when the diagonal and BHS corrections are used are similar but differ noticeably. Kodeli and SlavičCitation25 reported a sensitivity of 0.06776%/% for L = 3 and 0.06451%/% for L = 5.
It should be noted that there is a longstanding bug in the publicly available version of PARTISN, version 8.29.32, that affects adjoint calculations that use the BHS transport correction. The BHS correction is made after the scattering matrix is transposed; it should be made before. This bug was inherited from ONEDANT (CitationRef. 10). It has been fixed as of PARTISN version 8.32.52.Footnotea (A slightly newer version was used for this paper.)
V. TEST PROBLEM FOR INHOMOGENEOUS SOURCE
The test configuration for an inhomogeneous (fixed-source) transport problem was a 4.5-kg sphere of α-phase, weapons-grade plutonium (the BeRP ballCitation34) reflected by 38.1 mm of high-density polyethylene (HDPE). The geometry and materials (simplified from the compositions given in CitationRef. 34—in particular, all impurities in the plutonium were turned into gallium) are given in . The response of interest was the total leakage of neutrons from the outer surface. The problem was run with various scattering expansion orders and each of the transport corrections (including none) available in PARTISN.
An angular quadrature of S256 and fine mesh spacing of 0.005 cm were used. The 79-group Kynea3 cross sectionsCitation35 having upscattering in the bottom 35 groups were used. The maximum scattering expansion order available in this library is 7 (indexed from 0). The Citation1H cross sections include thermal scattering. PARTISN version 8.32.54 was used.
SOURCES4C (CitationRef. 36) was used to compute the neutron source rate density from spontaneous fission.
V.A. Neutron Leakage
The leakages are compared in and . Scattering order 0 (isotropic scattering only) is not shown because that causes the system to be supercritical for all transport corrections. Scattering orders 1 and 2 also cause the system to be supercritical with the Cesàro transport correction. The leakages for L = 7 (R7) are identical for no transport correction, BHS, and diagonal transport correction because that is the highest expansion order in the cross-section library. The leakages for L = 0 are identical for no transport correction and the Cesàro transport correction because in EquationEq. (8)(8) (8) .
Assuming that R7 with no transport correction (or diagonal or BHS) is the most accurate value, each of the leakages is compared to R7 in . Whether the diagonal or BHS transport correction is more accurate depends on the scattering order. Interestingly, for L ≥ 4, using no transport correction is more accurate than using the diagonal or BHS transport correction.
The conventional wisdom that P3 scattering is more accurate than P4 is not true for this problem using these nuclear data.
For L > 2, the Cesàro transport correction is wildly inaccurate for this problem.
V.B. Neutron Leakage Sensitivities
Again, in order to verify the derivations in Sec. III as well as the implementation in code, CD estimates of sensitivities were compared with adjoint-based calculations. The adjoint-based (SENSMG) relative sensitivity of each of the material densities, computed without accounting for the effect of the transport corrections on the sensitivities, was compared with a CD estimate for various scattering expansion orders and each of the transport corrections (including none). For the CDs, each material mass density was perturbed ±0.1% (separately). The adjoint-based material density sensitivity is the sum of the nuclide density sensitivities over nuclide and energy group [EquationEq. (19)(19) (19) ]. Results are shown in . These results are shown for information only; SENSMG has no user option to disable the correct accounting for any transport correction. Again, the point of this comparison is to assess the importance of accounting for the transport correction when computing adjoint-based sensitivities.
shows that adjoint sensitivities can be in error by several percent for low scattering orders if transport correction is not accounted for. Also, results for L = 7 for no transport correction, BHS, and diagonal transport correction are all identical, as in , and for the same reason.
The proper adjoint-based (SENSMG) relative sensitivity of each of the material densities, which accounts for the effect of the transport corrections on the sensitivities, was also compared with the CD estimates. Results are shown in . verifies EquationEqs. (22)(22) (22) , (Equation23(23) (23) ), and (Equation27(27) (27) ) for the diagonal, BHS, and Cesàro transport corrections, respectively.
As stated in Sec. IV.B, there is a longstanding bug in the publicly available version of PARTISN, version 8.29.32, that affects adjoint calculations that use the BHS transport correction. It has been fixed as of PARTISN version 8.32.52. (A slightly newer version was used for this paper.)
VI. SUMMARY AND CONCLUSIONS
Sensitivities for neutron or gamma-ray transport responses have been derived for problems that use the diagonal, BHS, or Cesàro transport corrections in the discrete-ordinates method. For the BHS transport correction, the response is sensitive to the L + 1’th scattering cross sections of all nuclides. For the diagonal transport correction, the response is sensitive to the within-group scattering cross sections . For the BHS and diagonal transport corrections, the sensitivities to the total cross sections and to the scattering cross sections for l ≤ L have their usual formulas [EquationEqs. (15)(15) (15) and (Equation16(16) (16) )]. Also for the BHS and diagonal transport corrections, the sensitivities to material and nuclide densities must account for the L + 1’th term separately. For the Cesàro transport correction, the sensitivities to the total cross sections and the isotropic scattering cross sections have their usual formulas [EquationEqs. (15)(15) (15) and (Equation16(16) (16) )], and the sensitivities to the scattering cross sections for l ≥ 1 are the usual formulas [EquationEq. (16(16) (16) )] multiplied by a simple function of l and L. Also for the Cesàro transport correction, the sensitivities to nuclide and material densities have their usual formulas [EquationEqs. (17)(17) (17) and (Equation19(19) (19) )].
It should be emphasized that in EquationEq. (21)(21) (21) for the relative sensitivity for the diagonal and BHS transport corrections, the index on the adjoint flux in the first term on the right side is g, the same as the forward flux, not g′ as usual for the scattering term.
These formulas have been implemented in the multigroup sensitivity code SENSMG, which uses PARTISN for the neutron transport. Numerical test results confirm the derivations and implementation.
While the intent of this paper is not necessarily to compare the performance of the transport corrections themselves, it was noteworthy that the Cesàro transport correction yielded extremely inaccurate results for keff and for the neutron leakage in a fixed-source problem. Caution is advised when using the Cesàro transport correction in PARTISN, Denovo, or any other transport code that offers it.
Acknowledgments
The author would like to thank D. K. Parsons (LANL) for providing the GROUPR file with the elastic and inelastic scattering cross sections for 238U. The author would like to thank the following people for discussions about the transport corrections used in their codes: M. A. Jessee (Oak Ridge National Laboratory), B. R. Beck and P. N. Brown (Lawrence Livermore National Laboratory), G. A. Failla (Silver Fir Software, Inc.), J. M. McGhee [Varian Medical Systems (retired)], and D. K. Parsons (LANL). This work was supported by the National Nuclear Security Administration.
Notes
a M. I. Ortega and R. J. Zerr at Los Alamos National Laboratory (LANL) fixed this bug.
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APPENDIX
CORRIGENDUM FOR REF. 19
In Table I of CitationRef. 19, the entries for isotropic scattering had typos, and the entries for anisotropic scattering had typesetting errors. The correct entries are given in . The notation of CitationRef. 19 is retained.