Estimation of uncertainty in lead spallation particle multiplicity and its propagation to a neutron energy spectrum

Figures & data

Figure 1. Comparison of DDX for $p$(800 MeV) + ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb reaction between the PHITS calculations and experimental data

Figure 2. Histograms of relative difference from the PHITS calculation of DDXs for 20, 34, 48, 62.9, 63, 78, 113, 256, 597, and 800 MeV, as well as 1.0, 1.2, 1.5, 1.6, and 3.0 GeV $p$ + ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}/208}$Pb reactions, where the values in each bin are normalized such that the integral value is unity. In each panel, the datapoint with the error bar indicates $q_{\mathrm{M}}$ and uncertainty width ${ }_{-v_x}^{+u_x}$ of the distribution

Table 1. Summary of DDX experiments for ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}/208}$Pb$(p,xn)$ reactions and evaluation results for each incident proton energy

$E_p$ (MeV)	Target	Reference no.	Detector angle (°)	Sample size	$d_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{D}\mathrm{D}\mathrm{X}}$ dist.	$x_{pn,\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}(n/p)$
20	^natPb	[36]	0	13	$-{0.684}_{-0.279}^{+0.414}$	${0.263}_{-0.263}^{+0.344}$
34	^natPb	[36]	0	17	$-{0.761}_{-0.689}^{+0.142}$	${0.528}_{-1.520}^{+0.314}$
48	^natPb	[36]	0	35	$-{0.498}_{-0.471}^{+0.057}$	${1.40}_{-1.31}^{+0.16}$
62.9	^208Pb	[43]	24, 35, 55, 80, 120	272	${0.170}_{-0.506}^{+0.656}$	${3.92}_{-1.70}^{+2.20}$
63	^natPb	[36]	0	19	$-{0.320}_{-0.261}^{+0.079}$	${2.28}_{-0.88}^{+0.26}$
78	^natPb	[36]	0	29	$-{0.431}_{-0.191}^{+0.300}$	${2.14}_{-0.72}^{+1.13}$
113	^natPb	[44]	7.5, 30, 60, 150	292	${0.471}_{-0.276}^{+0.321}$	${6.72}_{-1.26}^{+1.47}$
256	^natPb	[45]	7.5, 30, 60, 150	339	${0.211}_{-0.231}^{+0.172}$	${8.25}_{-1.57}^{+1.17}$
597	^natPb	[46]	30, 60, 120, 150	395	${0.071}_{-0.152}^{+0.147}$	${11.1}_{-1.6}^{+1.5}$
800	^natPb	[37–42]	0, 10, 15, 25, 30, 40, 55, 60, 85,
			90, 120, 130, 145, 150, 160	1110	${0.027}_{-0.250}^{+0.266}$	${12.2}_{-3.0}^{+3.2}$
1000	^natPb	[42]	15, 30, 60, 90, 120, 150	231	${0.002}_{-0.196}^{+0.184}$	${13.1}_{-2.4}^{+2.6}$
1200	^natPb	[40]	0, 10, 25, 40, 55, 70, 85, 100,
			115, 130, 145, 160	563	$-{0.050}_{-0.332}^{+0.285}$	${13.3}_{-4.7}^{+4.0}$
1500	^natPb	[39]	30, 60, 90, 120, 150	124	${0.275}_{-0.347}^{+0.634}$	${19.2}_{-5.2}^{+9.5}$
1600	^natPb	[40,42]	0, 10, 15, 25, 30, 40, 55, 60,
			85, 90, 100, 115, 120, 130,
			145, 150, 160	806	$-{0.040}_{-0.468}^{+0.402}$	${14.7}_{-7.2}^{+6.2}$
3000	^natPb	[39]	15, 30, 60, 90, 120, 150	121	$-{0.264}_{-0.185}^{+0.420}$	${12.3}_{-3.1}^{+7.0}$

Figure 3. Proton-induced ^natPb spallation neutron multiplicity (top panel: linear scale, bottom panel: logarithmic scale). The dot-dashed lines are the calculations with PHITS. The solid line is the JENDL-4.0/HE evaluation; the dotted line is the TENDL-2017 evaluation; the datapoints with error bars indicate $x_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}$

Figure 4. Histogram of relative difference from the PHITS calculations for incident proton energies above 100 MeV. The dashed and solid curves indicate a fit to the histogram with Gaussian distribution and Student’s t-distribution, respectively (top panel: linear scale, bottom panel: logarithmic scale)

Figure 5. Histogram of relative difference from Equation.(12)(12) $x_{pn,\mathrm{f}\mathrm{i}\mathrm{t}}(E_p)=a(E_p-c{)}^b(n/p),$(12) for incident energies below 100 MeV (top panel: linear scale, bottom panel: logarithmic scale)

Figure 6. Same as Figure 3, but including the uncertainty estimated from $x_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}$, $x_{\mathrm{f}\mathrm{i}\mathrm{t}}$, and $x_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}$ (top panel: linear scale, bottom panel: logarithmic scale). The dotted lines and the dark and light bands are $q_{\mathrm{M}}$ and 50% and 90% UIs, respectively

Table 2. Summary of statistical analyses for samples of $d_{\mathrm{r}\mathrm{e}\mathrm{l}}^x$ ($E_p\ge 100$ MeV) and ${\delta }_{\mathrm{r}\mathrm{e}\mathrm{l}}^x$ ($E_p<100$ MeV)

Incident energy	Statistics	Probability density function
$E_p\ge 100$ MeV	$q_{\mathrm{L}}=-0.170$, $q_{\mathrm{M}}=0.049$, $q_{\mathrm{U}}=0.270$,	Student’s t ($\mu =0.049$, $\lambda =5.72$, $\nu =4.01$)
$E_p<100$ MeV	$q_{\mathrm{L}}=-0.060$, $q_{\mathrm{M}}=0.485$, $q_{\mathrm{U}}=1.436$,	N/A

Figure 7. Neutron-induced spallation proton multiplicity ($x_{nn}$) (left panels), proton-induced spallation proton multiplicity ($x_{pp}$) (middle panels), and neutron-induced spallation proton multiplicity ($x_{np}$) (right panels) inferred from the estimated $x_{pn}$ uncertainty and the PHITS nuclear reaction calculations (top panels: linear scale, bottom panels: logarithmic scale). The dark and light bands are 50% and 90% UIs, respectively

Table 3. Summary of DDX experiments for ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}/208}$Pb$(p,xp)$ reactions and evaluation results for each incident energy

$E_p$ (MeV)	Target	Reference no.	Detector angle (°)	Sample size	$d_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{D}\mathrm{D}\mathrm{X}}$ dist.	$x_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}(p/p)$
392	${ }^{208}$Pb	[52]	20, 25, 30, 40, 45, 50, 60, 75, 90, 105	287	$-{0.255}_{-0.284}^{+0.295}$	${1.76}_{-0.50}^{+0.52}$
558	${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb	[53]	20, 30, 40, 50, 60	498	${0.111}_{-0.184}^{+0.170}$	${3.12}_{-0.57}^{+0.53}$

Table 4. Estimates of $x_{pn}$, $x_{nn}$, $x_{pp}$, and $x_{np}$ for ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb at representative incident energies

	$x_{pn}$ $(n/p)$		$x_{nn}$ $(n/n)$		$x_{pp}$ $(p/p)$		$x_{np}$ $(p/n)$
Incident energy (MeV)	$q_{\mathrm{M}}$	50% UI	$q_{\mathrm{M}}$	50% UI	$q_{\mathrm{M}}$	50% UI	$q_{\mathrm{M}}$	50% UI
35	1.3	(0.8, 2.1)	–	–	–	–	–	–
40	1.5	(1.0, 2.5)	–	–	–	–	–	–
50	1.9	(1.2, 3.2)	–	–	–	–	–	–
80	3.1	(1.9, 5.0)	–	–	–	–	–	–
150	5.7	(4.5, 6.7)	7.1	(5.6, 8.6)	2.1	(1.7, 2.5)	0.6	(0.5, 0.7)
200	6.5	(5.0, 7.7)	7.7	(6.1, 9.3)	2.0	(1.6, 2.4)	0.8	(0.6, 0.9)
400	9.0	(6.9, 10.6)	10.3	(8.1, 12.4)	2.5	(2.0, 3.0)	1.2	(1.0, 1.5)
600	11.1	(8.6, 13.2)	12.7	(10.1, 15.4)	3.0	(2.4, 3.7)	1.8	(1.4, 2.2)
800	12.8	(9.9, 15.1)	14.4	(11.4, 17.4)	3.5	(2.8, 4.3)	2.2	(1.8, 2.7)
1000	14.0	(10.9, 16.6)	15.7	(12.4, 19.0)	3.9	(3.1, 4.7)	2.6	(2.1, 3.2)
1200	15.1	(11.6, 17.8)	16.6	(13.1, 20.1)	4.2	(3.3, 5.1)	2.9	(2.3, 3.6)
1500	16.2	(12.5, 19.1)	17.6	(14.0, 21.4)	4.6	(3.6, 5.6)	3.3	(2.6, 4.0)
2000	17.2	(13.3, 20.4)	18.6	(14.7, 22.5)	5.0	(4.0, 6.1)	3.7	(3.0, 4.5)
3000	18.0	(13.9, 21.3)	19.2	(15.2, 23.3)	5.4	(4.3, 6.5)	4.1	(3.3, 5.0)

Figure 8. Responses of $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples to ${\sigma }_{pn}$ and ${\sigma }_{pp}$ for the 800 MeV proton + ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb reaction (top panel: histogram of the $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples, right panel: histogram of the ${\sigma }_{pn}$ and ${\sigma }_{pp}$ output values, middle panel: relationship between $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ and ${\sigma }_{pn}$ [${\sigma }_{pp}$]). The datapoints with error bars are the calculation values with 1$\sigma$ statistical uncertainties; the solid line is the least-squares linear fit for a set of the points with error bars. The sample size is 300

Figure 9. A schematic of the experimental condition [55]

Table 5. Energy group structure for tally

	Energy range (MeV)
Group number $\mathrm{\ell }$	lower boundary	upper boundary
1	$100$	$500$
2	$10$	$100$
3	$1$	$10$
4	$0.1$	$1$

Figure 10. Convergence of ${\psi }_{\mathrm{\ell }}$ (left panels) and their 1$\sigma$-equivalent uncertainties (right panels) as a function of the sample size ($N$) for the detector angle of 30${ }^{\circ }$. For the left panels, the dots with error bars indicate the calculated ${\psi }_{\mathrm{\ell }}$ samples, the solid line is median of the ${\psi }_{\mathrm{\ell }}$ sample distributions, and the dashed line is a reference calculated without random sampling. For the right panels, the dashed line indicates $\stackrel{‾}{{\left(\delta {\psi }_{\mathrm{\ell }}\right)}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}}$, which is the 1$\sigma$ statistical uncertainty of ${\psi }_{\mathrm{\ell }}$ averaged over $N$ samples; the dot-dashed line is ${\left(\delta {\psi }_{\mathrm{\ell }}\right)}_{\mathrm{o}\mathrm{b}\mathrm{s}}$, which is the observed statistical uncertainty obtained from $N$ PHITS runs; the solid line is ${\left(\delta {\psi }_{\mathrm{\ell }}\right)}_x$, which is the uncertainty due to $x$

Figure 11. Estimated $\psi$ along with experimental data, which is drawn using the logarithmic scale. The dark and light bands indicate 90% and 50% UIs, respectively, propagated from the estimated $x$ uncertainty above 100 MeV

Figure 12. Same as Figure 11, but drawn with the linear scale

Figure 13. Relationships between $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples and ${\psi }_{\mathrm{\ell }}$ ($\mathrm{\ell }=1,..,4$) for the six detector angles in the sampling energy range from 100 to 500 MeV. For each panel, the datapoints with error bars indicate the calculation values with 1$\sigma$ statistical uncertainties; the solid line is the least-squares linear fit for a set of points with the error bars; the dashed line with the dark and light bands indicates $q_{\mathrm{M}}$ with observed 50% and 90% UIs of $\psi$, respectively. The sample size is 200

Figure 14. Response of $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples to ${\psi }_4$ (top panel: histogram of the $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples, right panel: histogram of the ${\psi }_4$ output values, middle panel: relationship between $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ and ${\psi }_4$). The datapoints with error bars are the calculation values with 1$\sigma$ statistical uncertainties; the solid line is the least-squares linear fit for a set of points with error bars. The sample size is 800

Table 6. Energy group structure for random sampling

	Energy range (MeV)
Group number $m$	lower boundary	upper boundary
1	$400$	$500$
2	$300$	$400$
3	$200$	$300$
4	$100$	$200$
5	$20$	$100$

Figure 15. Energy breakdown of the response of $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ random samples to ${\psi }_{\mathrm{\ell }}^{(m)}$ for the 30${ }^{\circ }$ detector angle

Figure 16. Relative sensitivity matrix of ${\psi }_{\mathrm{\ell }}^{(m)}$ to $x_{\mathrm{r}\mathrm{e}\mathrm{l}}$ for each detector angle

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