Using the ${\Delta }_3$ statistic to test for missed levels in mixed sequence neutron resonance data

The ${\Delta }_3(L)$ statistic is studied as a tool to detect missing levels in the neutron resonance data where two sequences are present. These systems are problematic because there is no level repulsion, and the resonances can be too close to resolve. ${\Delta }_3(L)$ is a measure of the fluctuations in the number of levels in an interval of length $L$ on the energy axis. The method used is tested on ensembles of mixed Gaussian orthogonal ensemble spectra, with a known fraction of levels $(x\%)$ randomly depleted, and can accurately return $x$. The accuracy of the method as a function of spectrum size is established. The method is used on neutron resonance data for 11 isotopes with either $s$-wave neutrons on odd-$A$ isotopes, or $p$-wave neutrons on even-$A$ isotopes. The method compares favorably with a maximum likelihood method applied to the level spacing distribution. Nuclear data ensembles were made from 20 isotopes in total, and their ${\Delta }_3(L)$ statistics are discussed in the context of random matrix theory.

Figure 1

${\Delta }_3(L,0.25,x)$ shown for $x=0,2,4,\dots 10$ (upper lines). The calculations were for 500 spectra, with $N=400$. The standard deviation, $\sigma (400,L;x)$, corresponding to each value of $x$ (lower lines), grows rapidly with $L$. This makes the statistic much less sensitive for large $L$ values.

Figure 2

${\Delta }_3(L)$ for the seven odd-$A$ isotopes of Table (thin lines). The dashed line is the average of these thin lines. The lower thick line is the GOE value for $\alpha =0.4$ with no depletion ($x=0\%$). The upper thick line is for spectra with Poissonian statistics. In cases where an isotope has two entries in Table the low-energy subset is used here.

Figure 3

${\Delta }_3(L)$ and its standard deviation vs $L$ for five randomly chosen GOE spectra (dashed lines), with 4% depletion, $\alpha =0.33$, and $N=200$. The upper solid line is the ensemble average. The lower solid line is the corresponding σ.

Figure 4

${\Delta }_3(L)$ for the four data sets of $p$-wave neutrons of Table  (thin lines). The dashed line is the average of these thin lines. The lower thick line is mixed GOE spectra with $\alpha =0.4$ and 10% depletion. The upper thick line is for spectra with Poissonian statistics. In cases where an isotope has two entries in Table the low-energy subset is used here.

Figure 5

${\Delta }_3(L)$ for the even-even NDE of the nine isotopes described in Ref. [7], the dashed line being the average value. Compare this with the pure GOE case (lower thick line). The upper thick line is the Poissonian case.

Figure 6

$N(E)$ vs $E$ for the raw ${}^{58}\mathrm{Cr}$, ${}^{58}\mathrm{Ni}$, and ${}^{60}\mathrm{Ni}$ data. Notice the decrease in slope after the first 100 levels in the ${}^{58}\mathrm{Cr}$ data, suggesting that more levels were missed after the first 100 levels.

Figure 7

$N(E)$ vs. $E$ for the raw ${}^{235}\mathrm{U}$ data. Notice the decrease in slope after the first 950 levels. There are two other kinks, at $E=1000$ eV and 2200 eV.

Figure 8

Histogram of the raw ${}^{235}\mathrm{U}$ data. This is the slope of Fig. 7. The bin size is 75 eV. The first 950 levels span the range $0<E<500$ eV. There is a drop in the level density at 550 eV and a sharp increase at 2000 eV, where the level density nearly doubles. This is almost certainly artificial, because the data are the union of data from different facilities.

Figure 9

${\Delta }_3(L)$ vs $L$ for the lowest 950 levels in the ${}^{235}\mathrm{U}$ (upper dots) and the lowest 112 levels of the ${}^{147}\mathrm{Sm}$ (lower dots). The dashed lines are the GOE values for a mixed ($\alpha =0.4$) spectra, depleted by $x=0\%$, 3%, 5%, 8%, and 10%, starting from the lowest curve. The ${\Delta }_3(L)$ analysis suggests that there are no levels missed in the ${}^{147}\mathrm{Sm}$ data, and 3% of the ${}^{235}\mathrm{U}$ data are missed.

Figure 10

$N(E)$ vs $E$ for the ${}^{103}\mathrm{Rh}$, ${}^{147}\mathrm{Sm}$, and ${}^{197}\mathrm{Au}$ data. Notice the change in slope after the 112th level in the ${}^{147}\mathrm{Sm}$ case.

Figure 11

$P(s)$ vs $s$ for the combined ${}^{50}\mathrm{Cr}$, ${}^{58}\mathrm{Ni}$, and ${}^{60}\mathrm{Ni}$ data. The histogram has a bin width of 0.3 and has been normalized so that its area is unity. Also shown are the GOE results for $x=0\%$ (dotted line) and $x=10\%$ (dashed line).

Figure 12

$P(s)$ vs $s$ for the ${}^{238}\mathrm{U}$ data. As in Fig. 11, the GOE results for $x=0\%$ (dotted line) and $x=10\%$ (dashed line) are shown.