Transport Corrections and Sensitivities in the Discrete-Ordinates Method

Figures & data

TABLE I Flattop-Pu Model

Region	Material	Outer Radius (cm)	Density (g/cm^3)	Composition (Weight Fraction)
1	Plutonium	4.5332	15.52990779	^239Pu, 9.380012881E−01 ^240Pu, 4.799879422E−02 ^241Pu, 2.999857897E−03 ^69Ga, 6.536531152E−03 ^71Ga, 4.463528618E−03
2	NU	24.142	18.99986673	^234U, 5.407778609E−05 ^235U, 7.109665361E−03 ^238U, 9.928362569E−01

TABLE II Flattop-Pu k_eff

Transport Correction	Scattering Order (L)	R	(R – R4)/R4
No	0	1.149628	14.998%
	1	0.974799	−2.491%
	2	1.005704	0.601%
	3	0.997479	−0.222%
	4	0.999698	0.000%
Diagonal	0	1.010732	1.104%
	1	0.997655	−0.204%
	2	0.999657	−0.004%
	3	0.999066	−0.063%
	4	0.999698	0.000%
BHS	0	1.009837	1.014%
	1	0.997628	−0.207%
	2	0.999660	−0.004%
	3	0.999070	−0.063%
	4	0.999698	0.000%
Cesàro	0	1.149628	14.998%
	1	1.097266	9.760%
	2	1.072832	7.316%
	3	1.058075	5.839%
	4	1.048259	4.858%

Fig. 1. k_eff versus scattering expansion order for Flattop-Pu for four transport corrections (including none). The LANL standard 30-group structure was used. The PARTISN manual⁹ advises against using the Cesàro transport correction for L < 2.

TABLE III Relative Sensitivity (%/%) of Flattop-Pu k_eff to Material Density When SENSMG Does Not Account for Transport Correction

Transport Correction	L	Plutonium			NU
		SENSMG	CD	Difference^a	SENSMG	CD	Difference^a
No	0	0.65012	0.65012	0.000%	0.22134	0.22133	0.002%
	1	0.68385	0.68385	0.000%	0.23843	0.23842	0.002%
	2	0.67558	0.67555	0.004%	0.21800	0.21801	−0.003%
	3	0.67784	0.67783	0.000%	0.22813	0.22813	0.000%
	4	0.67736	0.67734	0.004%	0.22404	0.22404	0.000%
Diagonal	0	0.71552	0.67481	6.033%	0.33116	0.22083	49.960%
	1	0.67593	0.67789	−0.289%	0.20210	0.22661	−10.818%
	2	0.67733	0.67733	0.001%	0.23095	0.22444	2.901%
	3	0.67747	0.67748	−0.001%	0.22376	0.22544	−0.745%
	4	0.67736	0.67734	0.004%	0.22404	0.22404	0.000%
BHS	0	0.71598	0.67481	6.100%	0.33200	0.22083	50.343%
	1	0.67594	0.67790	−0.289%	0.20215	0.22664	−10.805%
	2	0.67733	0.67735	−0.002%	0.23095	0.22444	2.900%
	3	0.67747	0.67748	−0.001%	0.22374	0.22544	−0.750%
	4	0.67736	0.67734	0.004%	0.22404	0.22404	0.000%
Cesàro	0	0.65012	0.65012	0.000%	0.22134	0.22133	0.002%
	1	0.63646	0.66096	−3.707%	0.15210	0.22766	−33.188%
	2	0.64714	0.66539	−2.743%	0.18388	0.22809	−19.382%
	3	0.65298	0.66791	−2.235%	0.18839	0.22825	−17.462%
	4	0.65694	0.66959	−1.889%	0.19526	0.22805	−14.375%

^aRelative to CD.

TABLE IV Relative Sensitivity (%/%) of Flattop-Pu k_eff to Material Density When SENSMG Accounts for Transport Correction

Transport Correction	L	Plutonium			NU
		SENSMG	CD	Difference^a	SENSMG	CD	Difference^a
Diagonal	0	0.67480	0.67481	−0.001%	0.22084	0.22083	0.006%
	1	0.67789	0.67789	−0.001%	0.22662	0.22661	0.003%
	2	0.67734	0.67733	0.002%	0.22444	0.22444	0.000%
	3	0.67748	0.67748	−0.001%	0.22544	0.22544	0.003%
	4	0.67736	0.67734	0.004%	0.22404	0.22404	0.000%
BHS	0	0.67484	0.67481	0.004%	0.22083	0.22083	0.000%
	1	0.67790	0.67790	0.000%	0.22664	0.22664	0.002%
	2	0.67734	0.67735	−0.001%	0.22444	0.22444	0.000%
	3	0.67748	0.67748	0.000%	0.22544	0.22544	0.001%
	4	0.67736	0.67734	0.004%	0.22404	0.22404	0.000%
Cesàro	0	0.65012	0.65012	0.000%	0.22134	0.22133	0.002%
	1	0.66093	0.66096	−0.005%	0.22765	0.22766	−0.003%
	2	0.66534	0.66539	−0.007%	0.22813	0.22809	0.017%
	3	0.66791	0.66791	0.000%	0.22823	0.22825	−0.006%
	4	0.66958	0.66959	−0.001%	0.22808	0.22805	0.014%

^aRelative to CD.

TABLE V Relative Sensitivity (%/%) of Flattop-Pu k_eff to ²³⁸U Elastic Scattering Cross Section L + 1’th Moment*

Transport Correction	L	SENSMG	CD	Difference^a
Diagonal	0	−0.10663	−0.10663	−0.003%
	1	0.024510	0.024516	−0.025%
	2	−0.0064855	−0.0064869	−0.021%
	3	0.0016742	0.0016749	−0.042%
BHS	0	−0.10662	−0.10662	−0.003%
	1	0.024515	0.024521	−0.025%
	2	−0.0064894	−0.0064910	−0.024%
	3	0.0016791	0.0016798	−0.043%

*SENSMG accounts for transport correction. Sum of self-scattering and out-scattering.

^aRelative to CD.

TABLE VI Relative Sensitivity (%/%) of Flattop-Pu k_eff to ²³⁸U Total Elastic Scattering*

Transport Correction	L	SENSMG	CD	Difference^a
No	0	0.17814	0.17819	−0.030%
	1	0.12226	0.12227	−0.002%
	2	0.13164	0.13168	−0.031%
	3	0.13047	0.13049	−0.018%
	4	0.13024	0.13027	−0.023%
Diagonal	0	0.13159	0.13163	−0.026%
	1	0.13002	0.13005	−0.020%
	2	0.13037	0.13040	−0.023%
	3	0.13041	0.13043	−0.022%
	4	0.13024	0.13027	−0.023%
BHS	0	0.13191	0.13195	−0.027%
	1	0.13007	0.13010	−0.020%
	2	0.13036	0.13039	−0.023%
	3	0.13041	0.13043	−0.022%
	4	0.13024	0.13027	−0.023%
Cesàro	0	0.17814	0.17819	−0.030%
	1	0.16704	0.16707	−0.019%
	2	0.15949	0.15952	−0.019%
	3	0.15448	0.15451	−0.019%
	4	0.15090	0.15093	−0.019%

*SENSMG accounts for transport correction. Sum of self-scattering and out-scattering.

^aRelative to CD.

TABLE VII Relative Sensitivity (%/%) of Flattop-Pu k_eff to ²³⁸U Total Inelastic Scattering*

Transport Correction	L	SENSMG	CD	Difference^a
No	0	0.032464	0.032382	0.253%
	1	0.083017	0.083034	−0.020%
	2	0.061841	0.061816	0.040%
	3	0.069775	0.069774	0.002%
	4	0.067132	0.067119	0.019%
Diagonal	0	0.064187	0.064174	0.020%
	1	0.069120	0.069113	0.009%
	2	0.067268	0.067257	0.016%
	3	0.067973	0.067965	0.012%
	4	0.067132	0.067119	0.019%
BHS	0	0.063824	0.063810	0.022%
	1	0.069093	0.069087	0.009%
	2	0.067273	0.067262	0.016%
	3	0.067968	0.067960	0.012%
	4	0.067132	0.067119	0.019%
Cesàro	0	0.032464	0.032382	0.253%
	1	0.044115	0.044049	0.150%
	2	0.049555	0.049500	0.111%
	3	0.053055	0.053008	0.089%
	4	0.055423	0.055382	0.075%

*SENSMG accounts for transport correction. Sum of self-scattering and out-scattering.

^aRelative to CD.

TABLE VIII Simplified BeRP Ball Model

Region	Material	Outer Radius (cm)	Density (g/cm^3)	Composition (Weight Fraction)
1	Plutonium	3.794	19.6	^239Pu, 9.38039E−01 ^240Pu, 5.94113E−02 Ga, 2.54981E−03
2	HDPE	7.604	0.95	^1H, 1.43701E−01 C, 8.56299E−01

TABLE IX BeRP Ball Leakage

Transport Correction	Scattering Order (L)^a	R (10^6 neutrons/s)	(R – R7)/R7
No	1	1.380973	−17.703%
	2	1.719439	2.467%
	3	1.675059	−0.178%
	4	1.677911	−0.008%
	5	1.678125	0.005%
	6	1.678039	0.000%
	7	1.678046	0.000%
Diagonal	1	1.475439	−12.074%
	2	1.692591	0.867%
	3	1.680865	0.168%
	4	1.676516	−0.091%
	5	1.678480	0.026%
	6	1.677936	−0.007%
	7	1.678046	0.000%
BHS	1	1.610892	−4.002%
	2	1.715937	2.258%
	3	1.671834	−0.370%
	4	1.677877	−0.010%
	5	1.678270	0.013%
	6	1.678037	−0.001%
	7	1.678046	0.000%
Cesàro	3	12.439907	641%
	4	5.957554	255%
	5	4.350959	159%
	6	3.622205	116%
	7	3.205996	91%

^aL = 0 for all transport corrections and L = 1 and 2 for Cesàro did not converge.

Fig. 2. Neutron leakage versus scattering expansion order for the simplified BeRP ball for four transport corrections (including none). The Kynea3 79-group structure³⁵ was used. The PARTISN manual⁹ advises against using the Cesàro transport correction for L < 2.

TABLE X Relative Sensitivity (%/%) of BeRP Ball Leakage to Material Density When SENSMG Does Not Account for Transport Correction

Transport Correction	L^a	Plutonium			HDPE
		SENSMG	CD	Difference^b	SENSMG	CD	Difference^b
No	1	5.67597	5.67502	0.017%	1.02083	1.02091	−0.008%
	2	7.10436	7.10325	0.016%	1.35770	1.35803	−0.024%
	3	6.91909	6.91841	0.010%	1.32045	1.32043	0.001%
	4	6.93067	6.92945	0.018%	1.32370	1.32313	0.043%
	5	6.93157	6.93033	0.018%	1.32351	1.32276	0.057%
	6	6.93124	6.93002	0.018%	1.32346	1.32273	0.055%
	7	6.93127	6.92991	0.020%	1.32348	1.32291	0.044%
Diagonal	1	6.07378	6.08883	−0.247%	1.05099	1.10892	−5.225%
	2	6.99099	6.98967	0.019%	1.35225	1.33532	1.268%
	3	6.94359	6.94256	0.015%	1.32011	1.32360	−0.264%
	4	6.92485	6.92361	0.018%	1.32404	1.32316	0.066%
	5	6.93302	6.93202	0.014%	1.32344	1.32385	−0.031%
	6	6.93085	6.92929	0.023%	1.32348	1.32255	0.070%
	7	6.93127	6.92991	0.020%	1.32348	1.32291	0.044%
BHS	1	6.62953	6.64660	−0.257%	1.09946	1.26461	−13.059%
	2	7.08943	7.08799	0.020%	1.35580	1.35387	0.143%
	3	6.90550	6.90472	0.011%	1.32084	1.31912	0.130%
	4	6.93053	6.92920	0.019%	1.32369	1.32358	0.009%
	5	6.93216	6.93199	0.003%	1.32348	1.32383	−0.027%
	6	6.93124	6.92977	0.021%	1.32346	1.32339	0.005%
	7	6.93127	6.92991	0.020%	1.32348	1.32291	0.044%
Cesàro	3	51.4798	53.2008	−3.235%	10.9120	15.7975	−30.926%
	4	24.6345	25.2504	−2.439%	5.12752	7.07026	−27.478%
	5	17.9835	18.3638	−2.071%	3.69359	4.91125	−24.793%
	6	14.9671	15.2400	−1.791%	3.04391	3.93149	−22.576%
	7	13.2449	13.4573	−1.578%	2.67354	3.36780	−20.615%

^aL = 0 for all transport corrections (including none) and L = 1 and 2 for Cesàro did not converge.

^bRelative to CD.

TABLE XI Relative Sensitivity (%/%) of BeRP Ball Leakage to Material Density When SENSMG Accounts for Transport Correction

Transport Correction	L^a	Plutonium			HDPE
		SENSMG	CD	Difference^b	SENSMG	CD	Difference^b
Diagonal	1	6.08866	6.08883	−0.003%	1.10799	1.10892	−0.084%
	2	6.98995	6.98967	0.004%	1.33652	1.33532	0.090%
	3	6.94259	6.94256	0.000%	1.32361	1.32360	0.000%
	4	6.92385	6.92361	0.003%	1.32321	1.32316	0.004%
	5	6.93202	6.93202	0.000%	1.32365	1.32385	−0.015%
	6	6.92985	6.92929	0.008%	1.32342	1.32255	0.066%
	7	6.93027	6.92991	0.005%	1.32348	1.32291	0.044%
BHS	1	6.64668	6.64660	0.001%	1.26496	1.26461	0.028%
	2	7.08840	7.08799	0.006%	1.35380	1.35387	−0.005%
	3	6.90450	6.90472	−0.003%	1.31893	1.31912	−0.015%
	4	6.92953	6.92920	0.005%	1.32367	1.32358	0.007%
	5	6.93116	6.93199	−0.012%	1.32357	1.32383	−0.020%
	6	6.93024	6.92977	0.007%	1.32346	1.32339	0.005%
	7	6.93027	6.92991	0.005%	1.32348	1.32291	0.044%
Cesàro	3	53.0060	53.2008	−0.366%	15.7895	15.7975	−0.051%
	4	25.2359	25.2504	−0.057%	7.06522	7.07026	−0.071%
	5	18.3565	18.3638	−0.040%	4.90546	4.91125	−0.118%
	6	15.2367	15.2400	−0.021%	3.92665	3.93149	−0.123%
	7	13.4558	13.4573	−0.011%	3.36814	3.36780	0.010%

^aL = 0 for all transport corrections and L = 1 and 2 for Cesàro did not converge.

^bRelative to CD.

TABLE A.I Reaction-Rate Ratio Sensitivities to Scattering for λ-Mode Flux,* Indirect Effect

Sensitivity	Label	Equation
In-scattering, P0	in-scat-0	$\begin{array}{r}{\left(S_{R,{\sigma }_{s,0,i}}^{g{ }^{\mathrm{\prime }}\to g}\right)}_{indirect}={\int }_{V}dV\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}_0^{\ast g}(r)N_i{\sigma }_{s,0,i}^{g{ }^{\mathrm{\prime }}\to g}{\varphi }_0^{g{ }^{\mathrm{\prime }}}(r)\\ -{\int }_{V}dV{\int }_{4\pi }d\hat{\mathbf{\Omega }}\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}^{\ast g{ }^{\mathrm{\prime }}}(r,\hat{\mathbf{\Omega }})N_i{\sigma }_{s,0,i}^{g{ }^{\mathrm{\prime }}\to g}{\psi }^{g{ }^{\mathrm{\prime }}}(r,\hat{\mathbf{\Omega }})\end{array}$
Self-scattering, P0	self-scat-0	$\begin{array}{r}{\left(S_{R,{\sigma }_{s,0,i}}^{g\to g}\right)}_{indirect}={\int }_{V}dV{\mathrm{\Gamma }}_0^{\ast g}(r)N_i{\sigma }_{s,0,i}^{g\to g}{\varphi }_0^g(r)\\ -{\int }_{V}dV{\int }_{4\pi }d\hat{\mathbf{\Omega }}{\mathrm{\Gamma }}^{\ast g}(r,\hat{\mathbf{\Omega }})N_i{\sigma }_{s,0,i}^{g\to g}{\psi }^g(r,\hat{\mathbf{\Omega }})\end{array}$
Out-scattering, P0	out-scat-0	$\begin{array}{r}{\left(S_{R,{\sigma }_{s,0,i}}^{g\to g{ }^{\mathrm{\prime }}}\right)}_{indirect}={\int }_{V}dV\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}_0^{\ast g{ }^{\mathrm{\prime }}}(r)N_i{\sigma }_{s,0,i}^{g\to g{ }^{\mathrm{\prime }}}{\varphi }_0^g(r)\\ -{\int }_{V}dV{\int }_{4\pi }d\hat{\mathbf{\Omega }}\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}^{\ast g}(r,\hat{\mathbf{\Omega }})N_i{\sigma }_{s,0,i}^{g\to g{ }^{\mathrm{\prime }}}{\psi }^g(r,\hat{\mathbf{\Omega }})\end{array}$
In-scattering, Pl, 1 ≤ l ≤ L	in-scat-[l]	${\left(S_{R,{\sigma }_{s,l,i}}^{g{ }^{\mathrm{\prime }}\to g}\right)}_{indirect}={\int }_{V}dV\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}_l^{\ast g}(r)N_i(2l+1){\sigma }_{s,l,i}^{g{ }^{\mathrm{\prime }}\to g}{\varphi }_l^{g{ }^{\mathrm{\prime }}}(r)$
Self-scattering, Pl, 1 ≤ l ≤ L	self-scat-[l]	${\left(S_{R,{\sigma }_{s,l,i}}^{g\to g}\right)}_{indirect}={\int }_{V}dV{\mathrm{\Gamma }}_l^{\ast g}(r)N_i(2l+1){\sigma }_{s,l,i}^{g\to g}{\varphi }_l^g(r)$
Out-scattering, Pl, 1 ≤ l ≤ L	out-scat-[l]	${\left(S_{R,{\sigma }_{s,l,i}}^{g\to g{ }^{\mathrm{\prime }}}\right)}_{indirect}={\int }_{V}dV\sum _{\genfrac{}{}{0ex}{}{g{ }^{\mathrm{\prime }}=1}{g{ }^{\mathrm{\prime }}\ne g}}^G{\mathrm{\Gamma }}_l^{\ast g{ }^{\mathrm{\prime }}}(r)N_i(2l+1){\sigma }_{s,l,i}^{g\to g{ }^{\mathrm{\prime }}}{\varphi }_l^g(r)$

* These equations are also used for α-mode and fixed-source flux, but $\lambda =1/k_{eff}$ is set to 1 and a divisor is used where needed.

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