Calculation of Gamow-Teller and Two-Neutrino Double-Beta Decay Properties for 130Te and 136Xe with a realistic nucleon-nucleon potential
L. Coraggio, L. De Angelis, T. Fukui, A. Gargano, and N. Itaco
TL;DR
This study calculates Gamow-Teller and double-beta decay properties for nuclei near 132Sn using a realistic shell model, deriving interactions without empirical quenching, and compares results with experimental data to validate the approach.
Contribution
It introduces a method to compute decay properties with a realistic interaction without empirical adjustments, enhancing predictive reliability for neutrinoless double-beta decay.
Findings
Results agree well with experimental data.
Method avoids empirical quenching factors.
Establishes a validated approach for future decay studies.
Abstract
We report on the calculation of Gamow-Teller and double-beta decay properties for nuclei around 132Sn within the framework of the realistic shell model. The effective shell-model Hamiltonian and Gamow-Teller transition operator are derived by way of many-body perturbation theory, without resorting to empirical effective quenching factor for the Gamow-Teller operator. The results are then compared with the available experimental data, in order to establish the reliability of our approach. This is a mandatory step, before we apply the same methodology, in forthcoming studies, to the calculation of the neutrinoless double-beta decay nuclear matrix element for nuclei that are currently considered among the best candidates for the detection of this process.
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Figure 13| Proton SP spacings | Neutron SP spacings | |||
|---|---|---|---|---|
| I | II | I | II | |
| 0.0 | 0.0 | 0.0 | 0.0 | |
| 0.3 | 0.4 | 0.6 | 0.7 | |
| 1.2 | 1.4 | 1.5 | 2.1 | |
| 1.1 | 1.3 | 1.2 | 1.9 | |
| 1.9 | 1.6 | 2.7 | 3.0 |
| 1.66 | 1.00 | |
| 1.70 | 1.07 | |
| 1.65 | 1.00 | |
| 1.71 | 1.00 | |
| 1.52 | 0.63 | |
| 1.50 | 0.64 | |
| 1.53 | 0.62 | |
| 1.63 | 0.97 | |
| 1.48 | 0.66 | |
| 1.51 | 0.69 | |
| 1.55 | 0.68 | |
| 1.52 | 0.63 | |
| 1.56 | 0.67 | |
| 1.50 | 0.68 |
| GT | quenching factor | |
|---|---|---|
| -1.239 | 0.50 | |
| -0.139 | ||
| 0.017 | ||
| 1.864 | 0.64 | |
| -1.747 | 0.56 | |
| 1.942 | 0.63 | |
| -1.023 | 0.66 | |
| -0.118 | ||
| 0.095 | ||
| 1.598 | 0.65 | |
| 2.597 | 0.69 |
| GT | quenching factor | |
|---|---|---|
| -1.239 | 0.50 | |
| -0.019 | ||
| 0.131 | ||
| 1.864 | 0.64 | |
| -1.891 | 0.61 | |
| 1.794 | 0.58 | |
| -1.023 | 0.66 | |
| -0.093 | ||
| 0.117 | ||
| 1.598 | 0.65 | |
| 2.597 | 0.69 |
| Nucleus | I | II | ||
|---|---|---|---|---|
| 130Te | ||||
| 430 | 420 | |||
| 220 | 200 | |||
| 130Xe | ||||
| 954 | 876 | |||
| 136Xe | ||||
| 300 | 300 | |||
| 9 | 11 | |||
| 1.58 | 2.42 | |||
| 136Ba | ||||
| 590 | 520 |
| Nucleus | Expt. | I | II | |
|---|---|---|---|---|
| 130Te | ||||
| GT strength | 0.842 | 0.873 | ||
| NME | 0.044 | 0.046 | ||
| 136Xe | ||||
| GT strength | 0.94 | 1.13 | ||
| NME | 0.0285 | 0.0287 |
| TBME | |||
|---|---|---|---|
| 0 | 1 | -0.426 | |
| 0 | 1 | -0.660 | |
| 0 | 1 | -0.333 | |
| 0 | 1 | -0.253 | |
| 0 | 1 | 1.751 | |
| 0 | 1 | -0.660 | |
| 0 | 1 | -0.394 | |
| 0 | 1 | -1.060 | |
| 0 | 1 | -0.344 | |
| 0 | 1 | 0.650 | |
| 0 | 1 | -0.333 | |
| 0 | 1 | -1.060 | |
| 0 | 1 | 0.067 | |
| 0 | 1 | -0.284 | |
| 0 | 1 | 0.771 | |
| 0 | 1 | -0.253 | |
| 0 | 1 | -0.344 | |
| 0 | 1 | -0.284 | |
| 0 | 1 | -0.514 | |
| 0 | 1 | 0.372 | |
| 0 | 1 | 1.751 | |
| 0 | 1 | 0.650 | |
| 0 | 1 | 0.771 | |
| 0 | 1 | 0.372 | |
| 0 | 1 | -0.500 | |
| 1 | 1 | 0.254 | |
| 1 | 1 | -0.030 | |
| 1 | 1 | -0.014 | |
| 1 | 1 | -0.030 | |
| 1 | 1 | 0.304 | |
| 1 | 1 | -0.022 | |
| 1 | 1 | -0.014 | |
| 1 | 1 | -0.022 | |
| 1 | 1 | 0.285 | |
| 2 | 1 | -0.014 | |
| 2 | 1 | 0.002 | |
| 2 | 1 | -0.149 | |
| 2 | 1 | -0.105 | |
| 2 | 1 | -0.152 | |
| 2 | 1 | -0.187 | |
| 2 | 1 | -0.120 | |
| 2 | 1 | 0.076 | |
| 2 | 1 | 0.367 | |
| 2 | 1 | 0.397 | |
| 2 | 1 | 0.163 | |
| 2 | 1 | 0.034 | |
| 2 | 1 | 0.080 | |
| 2 | 1 | 0.116 | |
| 2 | 1 | 0.055 | |
| 2 | 1 | -0.018 | |
| 2 | 1 | -0.238 | |
| 2 | 1 | 0.012 | |
| 2 | 1 | -0.114 | |
| 2 | 1 | -0.154 | |
| 2 | 1 | -0.192 | |
| 2 | 1 | -0.120 | |
| 2 | 1 | 0.172 | |
| 2 | 1 | 0.433 |
| 2 | 1 | -0.105 | |
| 2 | 1 | 0.034 | |
| 2 | 1 | -0.114 | |
| 2 | 1 | -0.026 | |
| 2 | 1 | -0.179 | |
| 2 | 1 | -0.246 | |
| 2 | 1 | -0.243 | |
| 2 | 1 | 0.200 | |
| 2 | 1 | 0.247 | |
| 2 | 1 | -0.152 | |
| 2 | 1 | 0.080 | |
| 2 | 1 | -0.154 | |
| 2 | 1 | 0.218 | |
| 2 | 1 | -0.082 | |
| 2 | 1 | -0.252 | |
| 2 | 1 | 0.239 | |
| 2 | 1 | 0.089 | |
| 2 | 1 | -0.187 | |
| 2 | 1 | 0.116 | |
| 2 | 1 | -0.192 | |
| 2 | 1 | -0.149 | |
| 2 | 1 | -0.208 | |
| 2 | 1 | 0.485 | |
| 2 | 1 | 0.191 | |
| 2 | 1 | -0.120 | |
| 2 | 1 | 0.055 | |
| 2 | 1 | -0.120 | |
| 2 | 1 | -0.243 | |
| 2 | 1 | -0.252 | |
| 2 | 1 | -0.208 | |
| 2 | 1 | 0.200 | |
| 2 | 1 | 0.073 | |
| 2 | 1 | 0.146 | |
| 2 | 1 | 0.076 | |
| 2 | 1 | -0.018 | |
| 2 | 1 | 0.172 | |
| 2 | 1 | 0.200 | |
| 2 | 1 | 0.239 | |
| 2 | 1 | 0.485 | |
| 2 | 1 | 0.099 | |
| 2 | 1 | -0.245 | |
| 2 | 1 | 0.367 | |
| 2 | 1 | -0.238 | |
| 2 | 1 | 0.433 | |
| 2 | 1 | 0.247 | |
| 2 | 1 | 0.089 | |
| 2 | 1 | 0.191 | |
| 2 | 1 | 0.146 | |
| 2 | 1 | -0.245 | |
| 2 | 1 | -0.315 | |
| 3 | 1 | 0.261 | |
| 3 | 1 | 0.058 | |
| 3 | 1 | -0.033 | |
| 3 | 1 | -0.003 | |
| 3 | 1 | -0.018 | |
| 3 | 1 | 0.281 | |
| 3 | 1 | -0.004 | |
| 3 | 1 | -0.023 | |
| 3 | 1 | -0.022 | |
| 3 | 1 | 0.335 | |
| 3 | 1 | -0.015 | |
| 3 | 1 | 0.024 | |
| 3 | 1 | -0.003 | |
| 3 | 1 | -0.023 | |
| 3 | 1 | -0.015 | |
| 3 | 1 | 0.248 | |
| 3 | 1 | -0.042 | |
| 3 | 1 | -0.018 | |
| 3 | 1 | -0.022 |
| 3 | 1 | 0.024 | |
| 3 | 1 | 0.250 | |
| 4 | 1 | 0.219 | |
| 4 | 1 | 0.049 | |
| 4 | 1 | -0.068 | |
| 4 | 1 | 0.031 | |
| 4 | 1 | -0.058 | |
| 4 | 1 | -0.146 | |
| 4 | 1 | 0.214 | |
| 4 | 1 | 0.303 | |
| 4 | 1 | 0.161 | |
| 4 | 1 | -0.139 | |
| 4 | 1 | 0.046 | |
| 4 | 1 | 0.128 | |
| 4 | 1 | -0.182 | |
| 4 | 1 | 0.244 | |
| 4 | 1 | 0.102 | |
| 4 | 1 | -0.037 | |
| 4 | 1 | -0.099 | |
| 4 | 1 | 0.167 | |
| 4 | 1 | 0.156 | |
| 4 | 1 | 0.076 | |
| 4 | 1 | 0.150 | |
| 4 | 1 | -0.191 | |
| 4 | 1 | -0.058 | |
| 4 | 1 | 0.046 | |
| 4 | 1 | -0.037 | |
| 4 | 1 | 0.076 | |
| 4 | 1 | 0.169 | |
| 4 | 1 | -0.337 | |
| 4 | 1 | 0.115 | |
| 4 | 1 | -0.146 | |
| 4 | 1 | 0.128 | |
| 4 | 1 | -0.099 | |
| 4 | 1 | 0.150 | |
| 4 | 1 | -0.292 | |
| 4 | 1 | 0.216 | |
| 4 | 1 | 0.214 | |
| 4 | 1 | -0.182 | |
| 4 | 1 | 0.167 | |
| 4 | 1 | -0.191 | |
| 4 | 1 | 0.115 | |
| 4 | 1 | 0.216 | |
| 4 | 1 | 0.031 | |
| 5 | 1 | 0.252 | |
| 5 | 1 | 0.021 | |
| 5 | 1 | 0.386 | |
| 6 | 1 | 0.343 | |
| 6 | 1 | 0.051 | |
| 6 | 1 | 0.137 | |
| 6 | 1 | -0.012 | |
| 6 | 1 | -0.246 | |
| 6 | 1 | 0.137 | |
| 6 | 1 | -0.246 | |
| 6 | 1 | 0.141 | |
| 8 | 1 | 0.192 | |
| 10 | 1 | 0.261 | |
| 2 | 1 | 0.024 | |
| 3 | 1 | 0.135 | |
| 3 | 1 | 0.082 | |
| 3 | 1 | 0.082 | |
| 3 | 1 | -0.225 | |
| 4 | 1 | 0.231 | |
| 4 | 1 | 0.063 | |
| 4 | 1 | -0.003 | |
| 4 | 1 | 0.063 | |
| 4 | 1 | 0.210 | |
| 4 | 1 | 0.000 | |
| 4 | 1 | -0.003 |
| 4 | 1 | 0.000 | |
| 4 | 1 | 0.271 | |
| 5 | 1 | 0.187 | |
| 5 | 1 | 0.088 | |
| 5 | 1 | -0.052 | |
| 5 | 1 | 0.046 | |
| 5 | 1 | 0.088 | |
| 5 | 1 | 0.111 | |
| 5 | 1 | 0.169 | |
| 5 | 1 | -0.198 | |
| 5 | 1 | -0.052 | |
| 5 | 1 | 0.169 | |
| 5 | 1 | 0.246 | |
| 5 | 1 | 0.163 | |
| 5 | 1 | 0.046 | |
| 5 | 1 | -0.198 | |
| 5 | 1 | 0.163 | |
| 5 | 1 | 0.045 | |
| 6 | 1 | 0.235 | |
| 6 | 1 | 0.038 | |
| 6 | 1 | -0.017 | |
| 6 | 1 | 0.012 | |
| 6 | 1 | 0.038 | |
| 6 | 1 | 0.308 | |
| 6 | 1 | 0.030 | |
| 6 | 1 | -0.012 | |
| 6 | 1 | -0.017 | |
| 6 | 1 | 0.030 | |
| 6 | 1 | 0.249 | |
| 6 | 1 | -0.010 | |
| 6 | 1 | 0.012 | |
| 6 | 1 | -0.012 | |
| 6 | 1 | -0.010 | |
| 6 | 1 | 0.343 | |
| 7 | 1 | 0.049 | |
| 7 | 1 | 0.134 | |
| 7 | 1 | -0.187 | |
| 7 | 1 | 0.134 | |
| 7 | 1 | 0.218 | |
| 7 | 1 | 0.236 | |
| 7 | 1 | -0.187 | |
| 7 | 1 | 0.236 | |
| 7 | 1 | -0.088 | |
| 8 | 1 | 0.257 | |
| 8 | 1 | 0.008 | |
| 8 | 1 | 0.008 | |
| 8 | 1 | 0.369 | |
| 9 | 1 | -0.602 | |
| 0 | -1 | -0.739 | |
| 0 | -1 | -0.680 | |
| 0 | -1 | -0.417 | |
| 0 | -1 | -0.291 | |
| 0 | -1 | 1.616 | |
| 0 | -1 | -0.680 | |
| 0 | -1 | -0.724 | |
| 0 | -1 | -1.058 | |
| 0 | -1 | -0.385 | |
| 0 | -1 | 0.717 | |
| 0 | -1 | -0.417 | |
| 0 | -1 | -1.058 | |
| 0 | -1 | -0.317 | |
| 0 | -1 | -0.333 | |
| 0 | -1 | 0.713 | |
| 0 | -1 | -0.291 | |
| 0 | -1 | -0.385 | |
| 0 | -1 | -0.333 | |
| 0 | -1 | -0.735 | |
| 0 | -1 | 0.399 | |
| 0 | -1 | 1.616 |
| 0 | -1 | 0.717 | |
| 0 | -1 | 0.713 | |
| 0 | -1 | 0.399 | |
| 0 | -1 | -0.915 | |
| 1 | -1 | 0.024 | |
| 1 | -1 | -0.033 | |
| 1 | -1 | -0.017 | |
| 1 | -1 | -0.033 | |
| 1 | -1 | 0.051 | |
| 1 | -1 | -0.019 | |
| 1 | -1 | -0.017 | |
| 1 | -1 | -0.019 | |
| 1 | -1 | 0.055 | |
| 2 | -1 | -0.280 | |
| 2 | -1 | 0.016 | |
| 2 | -1 | -0.178 | |
| 2 | -1 | -0.116 | |
| 2 | -1 | -0.158 | |
| 2 | -1 | -0.200 | |
| 2 | -1 | -0.134 | |
| 2 | -1 | 0.093 | |
| 2 | -1 | 0.339 | |
| 2 | -1 | 0.130 | |
| 2 | -1 | 0.161 | |
| 2 | -1 | 0.039 | |
| 2 | -1 | 0.078 | |
| 2 | -1 | 0.113 | |
| 2 | -1 | 0.051 | |
| 2 | -1 | -0.024 | |
| 2 | -1 | -0.221 | |
| 2 | -1 | -0.281 | |
| 2 | -1 | -0.104 | |
| 2 | -1 | -0.163 | |
| 2 | -1 | -0.192 | |
| 2 | -1 | -0.134 | |
| 2 | -1 | 0.186 | |
| 2 | -1 | 0.428 | |
| 2 | -1 | -0.116 | |
| 2 | -1 | 0.039 | |
| 2 | -1 | -0.104 | |
| 2 | -1 | -0.291 | |
| 2 | -1 | -0.157 | |
| 2 | -1 | -0.261 | |
| 2 | -1 | -0.241 | |
| 2 | -1 | 0.218 | |
| 2 | -1 | 0.270 | |
| 2 | -1 | -0.158 | |
| 2 | -1 | 0.078 | |
| 2 | -1 | -0.163 | |
| 2 | -1 | -0.042 | |
| 2 | -1 | -0.080 | |
| 2 | -1 | -0.240 | |
| 2 | -1 | 0.238 | |
| 2 | -1 | 0.074 | |
| 2 | -1 | -0.200 | |
| 2 | -1 | 0.113 | |
| 2 | -1 | -0.192 | |
| 2 | -1 | -0.417 | |
| 2 | -1 | -0.195 | |
| 2 | -1 | 0.460 | |
| 2 | -1 | 0.210 | |
| 2 | -1 | -0.134 | |
| 2 | -1 | 0.051 | |
| 2 | -1 | -0.134 | |
| 2 | -1 | -0.241 | |
| 2 | -1 | -0.240 | |
| 2 | -1 | -0.195 | |
| 2 | -1 | -0.047 | |
| 2 | -1 | 0.106 |
| 2 | -1 | 0.134 | |
| 2 | -1 | 0.093 | |
| 2 | -1 | -0.024 | |
| 2 | -1 | 0.186 | |
| 2 | -1 | 0.218 | |
| 2 | -1 | 0.238 | |
| 2 | -1 | 0.460 | |
| 2 | -1 | -0.159 | |
| 2 | -1 | -0.241 | |
| 2 | -1 | 0.339 | |
| 2 | -1 | -0.221 | |
| 2 | -1 | 0.428 | |
| 2 | -1 | 0.270 | |
| 2 | -1 | 0.074 | |
| 2 | -1 | 0.210 | |
| 2 | -1 | 0.134 | |
| 2 | -1 | -0.241 | |
| 2 | -1 | -0.541 | |
| 3 | -1 | 0.029 | |
| 3 | -1 | 0.069 | |
| 3 | -1 | -0.030 | |
| 3 | -1 | -0.006 | |
| 3 | -1 | -0.022 | |
| 3 | -1 | 0.047 | |
| 3 | -1 | -0.001 | |
| 3 | -1 | -0.027 | |
| 3 | -1 | -0.027 | |
| 3 | -1 | 0.109 | |
| 3 | -1 | -0.008 | |
| 3 | -1 | 0.027 | |
| 3 | -1 | -0.006 | |
| 3 | -1 | -0.027 | |
| 3 | -1 | -0.008 | |
| 3 | -1 | 0.013 | |
| 3 | -1 | -0.048 | |
| 3 | -1 | -0.022 | |
| 3 | -1 | -0.027 | |
| 3 | -1 | 0.027 | |
| 3 | -1 | -0.006 | |
| 4 | -1 | -0.027 | |
| 4 | -1 | 0.060 | |
| 4 | -1 | -0.081 | |
| 4 | -1 | 0.045 | |
| 4 | -1 | -0.066 | |
| 4 | -1 | -0.160 | |
| 4 | -1 | 0.203 | |
| 4 | -1 | 0.060 | |
| 4 | -1 | 0.157 | |
| 4 | -1 | -0.145 | |
| 4 | -1 | 0.046 | |
| 4 | -1 | 0.127 | |
| 4 | -1 | -0.164 | |
| 4 | -1 | -0.005 | |
| 4 | -1 | 0.124 | |
| 4 | -1 | -0.033 | |
| 4 | -1 | -0.101 | |
| 4 | -1 | 0.161 | |
| 4 | -1 | -0.094 | |
| 4 | -1 | 0.069 | |
| 4 | -1 | 0.149 | |
| 4 | -1 | -0.185 | |
| 4 | -1 | -0.066 | |
| 4 | -1 | 0.046 | |
| 4 | -1 | -0.033 | |
| 4 | -1 | 0.069 | |
| 4 | -1 | -0.077 | |
| 4 | -1 | -0.338 | |
| 4 | -1 | 0.129 | |
| 4 | -1 | -0.160 |
| 4 | -1 | 0.127 | |
| 4 | -1 | -0.101 | |
| 4 | -1 | 0.149 | |
| 4 | -1 | -0.506 | |
| 4 | -1 | 0.208 | |
| 4 | -1 | 0.203 | |
| 4 | -1 | -0.164 | |
| 4 | -1 | 0.161 | |
| 4 | -1 | -0.185 | |
| 4 | -1 | 0.129 | |
| 4 | -1 | 0.208 | |
| 4 | -1 | -0.213 | |
| 5 | -1 | 0.026 | |
| 5 | -1 | 0.025 | |
| 5 | -1 | 0.161 | |
| 6 | -1 | 0.105 | |
| 6 | -1 | 0.066 | |
| 6 | -1 | 0.135 | |
| 6 | -1 | -0.262 | |
| 6 | -1 | -0.232 | |
| 6 | -1 | 0.135 | |
| 6 | -1 | -0.232 | |
| 6 | -1 | -0.100 | |
| 8 | -1 | -0.039 | |
| 10 | -1 | 0.034 | |
| 2 | -1 | -0.293 | |
| 3 | -1 | -0.168 | |
| 3 | -1 | 0.092 | |
| 3 | -1 | 0.092 | |
| 3 | -1 | -0.524 | |
| 4 | -1 | -0.017 | |
| 4 | -1 | 0.063 | |
| 4 | -1 | -0.004 | |
| 4 | -1 | 0.063 | |
| 4 | -1 | -0.032 | |
| 4 | -1 | -0.034 | |
| 4 | -1 | -0.004 | |
| 4 | -1 | -0.034 | |
| 4 | -1 | -0.008 | |
| 5 | -1 | -0.079 | |
| 5 | -1 | 0.075 | |
| 5 | -1 | -0.068 | |
| 5 | -1 | 0.047 | |
| 5 | -1 | 0.075 | |
| 5 | -1 | -0.152 | |
| 5 | -1 | 0.154 | |
| 5 | -1 | -0.225 | |
| 5 | -1 | -0.068 | |
| 5 | -1 | 0.154 | |
| 5 | -1 | -0.019 | |
| 5 | -1 | 0.170 | |
| 5 | -1 | 0.047 | |
| 5 | -1 | -0.225 | |
| 5 | -1 | 0.170 | |
| 5 | -1 | -0.203 | |
| 6 | -1 | 0.003 | |
| 6 | -1 | 0.045 | |
| 6 | -1 | -0.008 | |
| 6 | -1 | 0.027 | |
| 6 | -1 | 0.045 | |
| 6 | -1 | 0.082 | |
| 6 | -1 | 0.029 | |
| 6 | -1 | -0.023 | |
| 6 | -1 | -0.008 | |
| 6 | -1 | 0.029 | |
| 6 | -1 | 0.043 | |
| 6 | -1 | -0.002 | |
| 6 | -1 | 0.027 | |
| 6 | -1 | -0.023 |
| 6 | -1 | -0.002 | |
| 6 | -1 | 0.101 | |
| 7 | -1 | -0.178 | |
| 7 | -1 | 0.117 | |
| 7 | -1 | -0.176 | |
| 7 | -1 | 0.117 | |
| 7 | -1 | -0.038 | |
| 7 | -1 | 0.244 | |
| 7 | -1 | -0.176 | |
| 7 | -1 | 0.244 | |
| 7 | -1 | -0.316 | |
| 8 | -1 | 0.033 | |
| 8 | -1 | 0.035 | |
| 8 | -1 | 0.035 | |
| 8 | -1 | 0.152 | |
| 9 | -1 | -0.760 | |
| 0 | 0 | -0.817 | |
| 0 | 0 | -0.632 | |
| 0 | 0 | -0.403 | |
| 0 | 0 | -0.269 | |
| 0 | 0 | 1.612 | |
| 0 | 0 | -0.632 | |
| 0 | 0 | -0.688 | |
| 0 | 0 | -1.067 | |
| 0 | 0 | -0.340 | |
| 0 | 0 | 0.754 | |
| 0 | 0 | -0.403 | |
| 0 | 0 | -1.067 | |
| 0 | 0 | -0.279 | |
| 0 | 0 | -0.296 | |
| 0 | 0 | 0.721 | |
| 0 | 0 | -0.269 | |
| 0 | 0 | -0.340 | |
| 0 | 0 | -0.296 | |
| 0 | 0 | -0.795 | |
| 0 | 0 | 0.412 | |
| 0 | 0 | 1.612 | |
| 0 | 0 | 0.754 | |
| 0 | 0 | 0.721 | |
| 0 | 0 | 0.412 | |
| 0 | 0 | -0.977 | |
| 1 | 0 | -0.378 | |
| 1 | 0 | -0.163 | |
| 1 | 0 | 0.144 | |
| 1 | 0 | 0.134 | |
| 1 | 0 | -0.043 | |
| 1 | 0 | 0.044 | |
| 1 | 0 | -0.138 | |
| 1 | 0 | -0.069 | |
| 1 | 0 | 0.065 | |
| 1 | 0 | 0.004 | |
| 1 | 0 | -0.799 | |
| 1 | 0 | -0.699 | |
| 1 | 0 | 0.746 | |
| 1 | 0 | -0.013 | |
| 1 | 0 | 0.279 | |
| 1 | 0 | -0.278 | |
| 1 | 0 | -0.147 | |
| 1 | 0 | -0.248 | |
| 1 | 0 | 0.232 | |
| 1 | 0 | -0.080 | |
| 1 | 0 | -0.196 | |
| 1 | 0 | 0.144 | |
| 1 | 0 | 0.746 | |
| 1 | 0 | -0.686 | |
| 1 | 0 | 0.023 | |
| 1 | 0 | -0.275 | |
| 1 | 0 | 0.266 | |
| 1 | 0 | 0.137 |
| 1 | 0 | 0.222 | |
| 1 | 0 | -0.230 | |
| 1 | 0 | 0.079 | |
| 1 | 0 | 0.168 | |
| 1 | 0 | 0.134 | |
| 1 | 0 | -0.013 | |
| 1 | 0 | -0.313 | |
| 1 | 0 | 0.741 | |
| 1 | 0 | -0.722 | |
| 1 | 0 | 0.634 | |
| 1 | 0 | 0.046 | |
| 1 | 0 | -0.044 | |
| 1 | 0 | -0.256 | |
| 1 | 0 | 0.407 | |
| 1 | 0 | -0.043 | |
| 1 | 0 | 0.279 | |
| 1 | 0 | -0.912 | |
| 1 | 0 | 0.945 | |
| 1 | 0 | -0.033 | |
| 1 | 0 | 0.314 | |
| 1 | 0 | -0.325 | |
| 1 | 0 | 0.478 | |
| 1 | 0 | -0.284 | |
| 1 | 0 | 0.044 | |
| 1 | 0 | -0.278 | |
| 1 | 0 | 0.266 | |
| 1 | 0 | -0.722 | |
| 1 | 0 | 0.945 | |
| 1 | 0 | -0.854 | |
| 1 | 0 | 0.045 | |
| 1 | 0 | -0.307 | |
| 1 | 0 | 0.298 | |
| 1 | 0 | -0.465 | |
| 1 | 0 | 0.290 | |
| 1 | 0 | -0.138 | |
| 1 | 0 | -0.147 | |
| 1 | 0 | 0.137 | |
| 1 | 0 | 0.634 | |
| 1 | 0 | -0.033 | |
| 1 | 0 | -0.172 | |
| 1 | 0 | -0.271 | |
| 1 | 0 | 0.266 | |
| 1 | 0 | -0.003 | |
| 1 | 0 | -0.268 | |
| 1 | 0 | -0.069 | |
| 1 | 0 | -0.248 | |
| 1 | 0 | 0.222 | |
| 1 | 0 | 0.046 | |
| 1 | 0 | 0.314 | |
| 1 | 0 | -0.443 | |
| 1 | 0 | 0.504 | |
| 1 | 0 | -0.125 | |
| 1 | 0 | -0.090 | |
| 1 | 0 | 0.065 | |
| 1 | 0 | 0.232 | |
| 1 | 0 | -0.230 | |
| 1 | 0 | -0.044 | |
| 1 | 0 | -0.325 | |
| 1 | 0 | 0.298 | |
| 1 | 0 | 0.266 | |
| 1 | 0 | 0.504 | |
| 1 | 0 | -0.450 | |
| 1 | 0 | 0.115 | |
| 1 | 0 | 0.081 | |
| 1 | 0 | 0.004 | |
| 1 | 0 | -0.080 | |
| 1 | 0 | 0.079 | |
| 1 | 0 | -0.256 | |
| 1 | 0 | 0.478 |
| 1 | 0 | -0.465 | |
| 1 | 0 | -0.003 | |
| 1 | 0 | -0.125 | |
| 1 | 0 | -1.209 | |
| 1 | 0 | 0.222 | |
| 1 | 0 | -0.799 | |
| 1 | 0 | -0.196 | |
| 1 | 0 | 0.168 | |
| 1 | 0 | 0.407 | |
| 1 | 0 | -0.284 | |
| 1 | 0 | 0.290 | |
| 1 | 0 | -0.268 | |
| 1 | 0 | -0.090 | |
| 1 | 0 | 0.081 | |
| 1 | 0 | 0.222 | |
| 1 | 0 | -0.574 | |
| 2 | 0 | -0.279 | |
| 2 | 0 | 0.011 | |
| 2 | 0 | -0.137 | |
| 2 | 0 | -0.014 | |
| 2 | 0 | -0.107 | |
| 2 | 0 | -0.107 | |
| 2 | 0 | -0.134 | |
| 2 | 0 | -0.136 | |
| 2 | 0 | 0.106 | |
| 2 | 0 | -0.131 | |
| 2 | 0 | 0.060 | |
| 2 | 0 | -0.128 | |
| 2 | 0 | -0.060 | |
| 2 | 0 | 0.335 | |
| 2 | 0 | -0.390 | |
| 2 | 0 | 0.330 | |
| 2 | 0 | -0.533 | |
| 2 | 0 | 0.022 | |
| 2 | 0 | -0.049 | |
| 2 | 0 | -0.016 | |
| 2 | 0 | -0.150 | |
| 2 | 0 | -0.122 | |
| 2 | 0 | 0.032 | |
| 2 | 0 | -0.078 | |
| 2 | 0 | 0.115 | |
| 2 | 0 | -0.066 | |
| 2 | 0 | -0.177 | |
| 2 | 0 | -0.244 | |
| 2 | 0 | 0.131 | |
| 2 | 0 | -0.042 | |
| 2 | 0 | 0.058 | |
| 2 | 0 | 0.024 | |
| 2 | 0 | -0.038 | |
| 2 | 0 | 0.214 | |
| 2 | 0 | -0.093 | |
| 2 | 0 | 0.196 | |
| 2 | 0 | -0.172 | |
| 2 | 0 | 0.032 | |
| 2 | 0 | 0.316 | |
| 2 | 0 | -0.014 | |
| 2 | 0 | -0.533 | |
| 2 | 0 | 0.131 | |
| 2 | 0 | -0.386 | |
| 2 | 0 | -0.011 | |
| 2 | 0 | -0.121 | |
| 2 | 0 | -0.116 | |
| 2 | 0 | -0.302 | |
| 2 | 0 | -0.029 | |
| 2 | 0 | -0.035 | |
| 2 | 0 | -0.060 | |
| 2 | 0 | 0.005 | |
| 2 | 0 | -0.067 | |
| 2 | 0 | 0.169 |
| 2 | 0 | -0.107 | |
| 2 | 0 | 0.022 | |
| 2 | 0 | -0.042 | |
| 2 | 0 | -0.290 | |
| 2 | 0 | -0.119 | |
| 2 | 0 | -0.180 | |
| 2 | 0 | -0.062 | |
| 2 | 0 | 0.111 | |
| 2 | 0 | -0.242 | |
| 2 | 0 | 0.137 | |
| 2 | 0 | -0.173 | |
| 2 | 0 | -0.138 | |
| 2 | 0 | 0.293 | |
| 2 | 0 | -0.107 | |
| 2 | 0 | -0.049 | |
| 2 | 0 | 0.058 | |
| 2 | 0 | -0.602 | |
| 2 | 0 | -0.226 | |
| 2 | 0 | -0.229 | |
| 2 | 0 | -0.560 | |
| 2 | 0 | -0.176 | |
| 2 | 0 | -0.108 | |
| 2 | 0 | 0.151 | |
| 2 | 0 | -0.351 | |
| 2 | 0 | 0.051 | |
| 2 | 0 | -0.134 | |
| 2 | 0 | -0.016 | |
| 2 | 0 | 0.024 | |
| 2 | 0 | -0.301 | |
| 2 | 0 | -0.194 | |
| 2 | 0 | -0.153 | |
| 2 | 0 | -0.146 | |
| 2 | 0 | -0.173 | |
| 2 | 0 | -0.110 | |
| 2 | 0 | -0.645 | |
| 2 | 0 | 0.157 | |
| 2 | 0 | -0.136 | |
| 2 | 0 | -0.150 | |
| 2 | 0 | -0.038 | |
| 2 | 0 | -0.302 | |
| 2 | 0 | -0.062 | |
| 2 | 0 | -0.229 | |
| 2 | 0 | -0.194 | |
| 2 | 0 | -0.237 | |
| 2 | 0 | -0.054 | |
| 2 | 0 | -0.092 | |
| 2 | 0 | -0.028 | |
| 2 | 0 | 0.020 | |
| 2 | 0 | -0.191 | |
| 2 | 0 | 0.295 | |
| 2 | 0 | 0.106 | |
| 2 | 0 | -0.122 | |
| 2 | 0 | 0.214 | |
| 2 | 0 | -0.029 | |
| 2 | 0 | 0.111 | |
| 2 | 0 | -0.560 | |
| 2 | 0 | -0.153 | |
| 2 | 0 | -0.591 | |
| 2 | 0 | 0.184 | |
| 2 | 0 | -0.351 | |
| 2 | 0 | 0.225 | |
| 2 | 0 | -0.108 | |
| 2 | 0 | -0.052 | |
| 2 | 0 | -0.131 | |
| 2 | 0 | 0.032 | |
| 2 | 0 | -0.093 | |
| 2 | 0 | -0.035 | |
| 2 | 0 | -0.242 | |
| 2 | 0 | -0.176 |
| 2 | 0 | -0.146 | |
| 2 | 0 | -0.047 | |
| 2 | 0 | 0.065 | |
| 2 | 0 | -0.132 | |
| 2 | 0 | -0.068 | |
| 2 | 0 | 0.134 | |
| 2 | 0 | 0.060 | |
| 2 | 0 | -0.078 | |
| 2 | 0 | 0.196 | |
| 2 | 0 | -0.060 | |
| 2 | 0 | 0.137 | |
| 2 | 0 | -0.108 | |
| 2 | 0 | -0.173 | |
| 2 | 0 | -0.336 | |
| 2 | 0 | 0.665 | |
| 2 | 0 | -0.179 | |
| 2 | 0 | -0.187 | |
| 2 | 0 | -0.128 | |
| 2 | 0 | 0.115 | |
| 2 | 0 | -0.172 | |
| 2 | 0 | 0.005 | |
| 2 | 0 | -0.173 | |
| 2 | 0 | 0.151 | |
| 2 | 0 | -0.110 | |
| 2 | 0 | 0.020 | |
| 2 | 0 | 0.225 | |
| 2 | 0 | -0.132 | |
| 2 | 0 | 0.665 | |
| 2 | 0 | -0.291 | |
| 2 | 0 | 0.165 | |
| 2 | 0 | 0.152 | |
| 2 | 0 | -0.060 | |
| 2 | 0 | -0.066 | |
| 2 | 0 | 0.032 | |
| 2 | 0 | -0.067 | |
| 2 | 0 | -0.138 | |
| 2 | 0 | -0.351 | |
| 2 | 0 | -0.645 | |
| 2 | 0 | -0.191 | |
| 2 | 0 | -0.108 | |
| 2 | 0 | -0.068 | |
| 2 | 0 | -0.179 | |
| 2 | 0 | -0.331 | |
| 2 | 0 | 0.181 | |
| 2 | 0 | 0.335 | |
| 2 | 0 | -0.177 | |
| 2 | 0 | 0.316 | |
| 2 | 0 | 0.169 | |
| 2 | 0 | 0.293 | |
| 2 | 0 | 0.051 | |
| 2 | 0 | 0.157 | |
| 2 | 0 | 0.295 | |
| 2 | 0 | -0.052 | |
| 2 | 0 | 0.134 | |
| 2 | 0 | -0.187 | |
| 2 | 0 | 0.152 | |
| 2 | 0 | 0.181 | |
| 2 | 0 | -0.559 | |
| 3 | 0 | -0.091 | |
| 3 | 0 | -0.111 | |
| 3 | 0 | -0.147 | |
| 3 | 0 | -0.091 | |
| 3 | 0 | 0.101 | |
| 3 | 0 | 0.009 | |
| 3 | 0 | 0.061 | |
| 3 | 0 | -0.007 | |
| 3 | 0 | -0.135 | |
| 3 | 0 | -0.059 | |
| 3 | 0 | -0.148 |
| 3 | 0 | 0.077 | |
| 3 | 0 | -0.005 | |
| 3 | 0 | -0.344 | |
| 3 | 0 | -0.184 | |
| 3 | 0 | -0.136 | |
| 3 | 0 | -0.212 | |
| 3 | 0 | 0.236 | |
| 3 | 0 | -0.093 | |
| 3 | 0 | 0.112 | |
| 3 | 0 | -0.129 | |
| 3 | 0 | -0.202 | |
| 3 | 0 | -0.105 | |
| 3 | 0 | -0.109 | |
| 3 | 0 | 0.179 | |
| 3 | 0 | -0.095 | |
| 3 | 0 | 0.019 | |
| 3 | 0 | -0.093 | |
| 3 | 0 | -0.242 | |
| 3 | 0 | 0.222 | |
| 3 | 0 | 0.054 | |
| 3 | 0 | 0.022 | |
| 3 | 0 | 0.002 | |
| 3 | 0 | -0.164 | |
| 3 | 0 | -0.057 | |
| 3 | 0 | -0.154 | |
| 3 | 0 | 0.231 | |
| 3 | 0 | 0.041 | |
| 3 | 0 | -0.125 | |
| 3 | 0 | -0.277 | |
| 3 | 0 | 0.196 | |
| 3 | 0 | 0.072 | |
| 3 | 0 | 0.034 | |
| 3 | 0 | -0.059 | |
| 3 | 0 | -0.230 | |
| 3 | 0 | -0.027 | |
| 3 | 0 | -0.182 | |
| 3 | 0 | 0.391 | |
| 3 | 0 | -0.088 | |
| 3 | 0 | -0.044 | |
| 3 | 0 | 0.101 | |
| 3 | 0 | 0.236 | |
| 3 | 0 | 0.222 | |
| 3 | 0 | 0.196 | |
| 3 | 0 | -0.188 | |
| 3 | 0 | 0.090 | |
| 3 | 0 | -0.098 | |
| 3 | 0 | 0.097 | |
| 3 | 0 | 0.136 | |
| 3 | 0 | 0.103 | |
| 3 | 0 | 0.098 | |
| 3 | 0 | -0.205 | |
| 3 | 0 | 0.121 | |
| 3 | 0 | -0.024 | |
| 3 | 0 | 0.009 | |
| 3 | 0 | -0.093 | |
| 3 | 0 | 0.054 | |
| 3 | 0 | 0.072 | |
| 3 | 0 | -0.299 | |
| 3 | 0 | 0.390 | |
| 3 | 0 | -0.374 | |
| 3 | 0 | 0.066 | |
| 3 | 0 | -0.380 | |
| 3 | 0 | 0.125 | |
| 3 | 0 | -0.076 | |
| 3 | 0 | -0.355 | |
| 3 | 0 | 0.209 | |
| 3 | 0 | 0.061 | |
| 3 | 0 | 0.112 | |
| 3 | 0 | 0.022 |
| 3 | 0 | 0.034 | |
| 3 | 0 | -0.176 | |
| 3 | 0 | 0.165 | |
| 3 | 0 | 0.045 | |
| 3 | 0 | 0.180 | |
| 3 | 0 | 0.353 | |
| 3 | 0 | -0.034 | |
| 3 | 0 | 0.207 | |
| 3 | 0 | -0.081 | |
| 3 | 0 | -0.007 | |
| 3 | 0 | -0.129 | |
| 3 | 0 | 0.002 | |
| 3 | 0 | -0.059 | |
| 3 | 0 | -0.567 | |
| 3 | 0 | 0.041 | |
| 3 | 0 | -0.235 | |
| 3 | 0 | 0.045 | |
| 3 | 0 | 0.068 | |
| 3 | 0 | -0.555 | |
| 3 | 0 | 0.189 | |
| 3 | 0 | -0.135 | |
| 3 | 0 | -0.202 | |
| 3 | 0 | -0.164 | |
| 3 | 0 | -0.230 | |
| 3 | 0 | 0.136 | |
| 3 | 0 | 0.066 | |
| 3 | 0 | 0.045 | |
| 3 | 0 | 0.041 | |
| 3 | 0 | -0.094 | |
| 3 | 0 | -0.018 | |
| 3 | 0 | -0.143 | |
| 3 | 0 | 0.225 | |
| 3 | 0 | 0.005 | |
| 3 | 0 | -0.119 | |
| 3 | 0 | -0.059 | |
| 3 | 0 | -0.105 | |
| 3 | 0 | -0.057 | |
| 3 | 0 | -0.027 | |
| 3 | 0 | 0.103 | |
| 3 | 0 | -0.380 | |
| 3 | 0 | 0.180 | |
| 3 | 0 | -0.235 | |
| 3 | 0 | -0.156 | |
| 3 | 0 | -0.341 | |
| 3 | 0 | 0.030 | |
| 3 | 0 | -0.149 | |
| 3 | 0 | 0.081 | |
| 3 | 0 | -0.148 | |
| 3 | 0 | -0.109 | |
| 3 | 0 | -0.154 | |
| 3 | 0 | -0.182 | |
| 3 | 0 | 0.098 | |
| 3 | 0 | 0.125 | |
| 3 | 0 | 0.353 | |
| 3 | 0 | 0.045 | |
| 3 | 0 | -0.695 | |
| 3 | 0 | 0.201 | |
| 3 | 0 | 0.067 | |
| 3 | 0 | -0.100 | |
| 3 | 0 | 0.077 | |
| 3 | 0 | 0.179 | |
| 3 | 0 | 0.231 | |
| 3 | 0 | 0.391 | |
| 3 | 0 | -0.205 | |
| 3 | 0 | -0.076 | |
| 3 | 0 | -0.034 | |
| 3 | 0 | 0.068 | |
| 3 | 0 | 0.225 | |
| 3 | 0 | 0.030 |
| 3 | 0 | 0.201 | |
| 3 | 0 | -0.271 | |
| 3 | 0 | 0.052 | |
| 3 | 0 | 0.034 | |
| 3 | 0 | -0.005 | |
| 3 | 0 | -0.095 | |
| 3 | 0 | 0.041 | |
| 3 | 0 | -0.088 | |
| 3 | 0 | 0.121 | |
| 3 | 0 | -0.355 | |
| 3 | 0 | 0.207 | |
| 3 | 0 | -0.555 | |
| 3 | 0 | 0.005 | |
| 3 | 0 | -0.149 | |
| 3 | 0 | 0.067 | |
| 3 | 0 | -0.549 | |
| 3 | 0 | 0.185 | |
| 3 | 0 | -0.344 | |
| 3 | 0 | 0.019 | |
| 3 | 0 | -0.125 | |
| 3 | 0 | -0.044 | |
| 3 | 0 | -0.024 | |
| 3 | 0 | 0.209 | |
| 3 | 0 | -0.081 | |
| 3 | 0 | 0.189 | |
| 3 | 0 | -0.119 | |
| 3 | 0 | 0.081 | |
| 3 | 0 | -0.100 | |
| 3 | 0 | 0.034 | |
| 3 | 0 | 0.185 | |
| 3 | 0 | -0.146 | |
| 4 | 0 | -0.018 | |
| 4 | 0 | 0.041 | |
| 4 | 0 | -0.059 | |
| 4 | 0 | 0.033 | |
| 4 | 0 | -0.040 | |
| 4 | 0 | -0.062 | |
| 4 | 0 | -0.108 | |
| 4 | 0 | -0.057 | |
| 4 | 0 | 0.105 | |
| 4 | 0 | -0.031 | |
| 4 | 0 | 0.199 | |
| 4 | 0 | -0.230 | |
| 4 | 0 | 0.106 | |
| 4 | 0 | -0.278 | |
| 4 | 0 | -0.302 | |
| 4 | 0 | 0.026 | |
| 4 | 0 | 0.016 | |
| 4 | 0 | 0.058 | |
| 4 | 0 | -0.088 | |
| 4 | 0 | -0.118 | |
| 4 | 0 | -0.129 | |
| 4 | 0 | -0.008 | |
| 4 | 0 | 0.076 | |
| 4 | 0 | -0.063 | |
| 4 | 0 | -0.008 | |
| 4 | 0 | 0.039 | |
| 4 | 0 | 0.011 | |
| 4 | 0 | 0.120 | |
| 4 | 0 | -0.050 | |
| 4 | 0 | 0.119 | |
| 4 | 0 | -0.130 | |
| 4 | 0 | -0.127 | |
| 4 | 0 | 0.038 | |
| 4 | 0 | -0.081 | |
| 4 | 0 | 0.054 | |
| 4 | 0 | -0.214 | |
| 4 | 0 | -0.046 | |
| 4 | 0 | -0.149 |
| 4 | 0 | -0.040 | |
| 4 | 0 | -0.302 | |
| 4 | 0 | -0.063 | |
| 4 | 0 | -0.127 | |
| 4 | 0 | -0.229 | |
| 4 | 0 | -0.020 | |
| 4 | 0 | -0.084 | |
| 4 | 0 | -0.095 | |
| 4 | 0 | 0.029 | |
| 4 | 0 | -0.259 | |
| 4 | 0 | 0.123 | |
| 4 | 0 | -0.062 | |
| 4 | 0 | 0.026 | |
| 4 | 0 | -0.008 | |
| 4 | 0 | 0.038 | |
| 4 | 0 | -0.075 | |
| 4 | 0 | -0.247 | |
| 4 | 0 | -0.019 | |
| 4 | 0 | 0.237 | |
| 4 | 0 | -0.043 | |
| 4 | 0 | 0.140 | |
| 4 | 0 | -0.108 | |
| 4 | 0 | 0.016 | |
| 4 | 0 | 0.039 | |
| 4 | 0 | -0.081 | |
| 4 | 0 | -0.827 | |
| 4 | 0 | -0.123 | |
| 4 | 0 | -0.291 | |
| 4 | 0 | -0.219 | |
| 4 | 0 | 0.158 | |
| 4 | 0 | -0.057 | |
| 4 | 0 | 0.058 | |
| 4 | 0 | 0.011 | |
| 4 | 0 | 0.054 | |
| 4 | 0 | -0.095 | |
| 4 | 0 | -0.019 | |
| 4 | 0 | -0.123 | |
| 4 | 0 | -0.016 | |
| 4 | 0 | -0.038 | |
| 4 | 0 | -0.070 | |
| 4 | 0 | 0.110 | |
| 4 | 0 | 0.105 | |
| 4 | 0 | -0.088 | |
| 4 | 0 | 0.120 | |
| 4 | 0 | -0.214 | |
| 4 | 0 | 0.029 | |
| 4 | 0 | 0.237 | |
| 4 | 0 | -0.291 | |
| 4 | 0 | -0.815 | |
| 4 | 0 | -0.073 | |
| 4 | 0 | -0.158 | |
| 4 | 0 | -0.031 | |
| 4 | 0 | -0.118 | |
| 4 | 0 | -0.050 | |
| 4 | 0 | -0.046 | |
| 4 | 0 | -0.259 | |
| 4 | 0 | -0.043 | |
| 4 | 0 | -0.219 | |
| 4 | 0 | -0.070 | |
| 4 | 0 | -0.073 | |
| 4 | 0 | -0.126 | |
| 4 | 0 | 0.136 | |
| 4 | 0 | 0.199 | |
| 4 | 0 | -0.129 | |
| 4 | 0 | 0.119 | |
| 4 | 0 | -0.149 | |
| 4 | 0 | 0.123 | |
| 4 | 0 | 0.140 | |
| 4 | 0 | 0.158 |
| 4 | 0 | 0.110 | |
| 4 | 0 | -0.158 | |
| 4 | 0 | 0.136 | |
| 4 | 0 | -0.217 | |
| 5 | 0 | -0.264 | |
| 5 | 0 | -0.134 | |
| 5 | 0 | -0.259 | |
| 5 | 0 | 0.124 | |
| 5 | 0 | -0.104 | |
| 5 | 0 | -0.235 | |
| 5 | 0 | -0.186 | |
| 5 | 0 | -0.022 | |
| 5 | 0 | -0.209 | |
| 5 | 0 | 0.072 | |
| 5 | 0 | -0.114 | |
| 5 | 0 | -0.232 | |
| 5 | 0 | 0.041 | |
| 5 | 0 | -0.426 | |
| 5 | 0 | 0.259 | |
| 5 | 0 | 0.027 | |
| 5 | 0 | -0.599 | |
| 5 | 0 | -0.074 | |
| 5 | 0 | 0.124 | |
| 5 | 0 | 0.072 | |
| 5 | 0 | 0.259 | |
| 5 | 0 | -0.034 | |
| 5 | 0 | 0.106 | |
| 5 | 0 | 0.204 | |
| 5 | 0 | -0.042 | |
| 5 | 0 | -0.104 | |
| 5 | 0 | -0.114 | |
| 5 | 0 | 0.027 | |
| 5 | 0 | -1.024 | |
| 5 | 0 | 0.038 | |
| 5 | 0 | 0.240 | |
| 5 | 0 | -0.235 | |
| 5 | 0 | -0.232 | |
| 5 | 0 | -0.599 | |
| 5 | 0 | 0.204 | |
| 5 | 0 | 0.038 | |
| 5 | 0 | -0.404 | |
| 5 | 0 | -0.067 | |
| 5 | 0 | -0.186 | |
| 5 | 0 | 0.041 | |
| 5 | 0 | -0.074 | |
| 5 | 0 | -0.042 | |
| 5 | 0 | 0.240 | |
| 5 | 0 | -0.067 | |
| 5 | 0 | -0.105 | |
| 6 | 0 | 0.118 | |
| 6 | 0 | 0.046 | |
| 6 | 0 | -0.043 | |
| 6 | 0 | 0.133 | |
| 6 | 0 | -0.406 | |
| 6 | 0 | -0.150 | |
| 6 | 0 | -0.183 | |
| 6 | 0 | -0.043 | |
| 6 | 0 | -0.150 | |
| 6 | 0 | -0.397 | |
| 6 | 0 | 0.174 | |
| 6 | 0 | 0.133 | |
| 6 | 0 | -0.183 | |
| 6 | 0 | 0.174 | |
| 6 | 0 | -0.106 | |
| 7 | 0 | -1.119 | |
| 7 | 0 | -0.096 | |
| 7 | 0 | -0.096 | |
| 7 | 0 | -0.172 | |
| 8 | 0 | -0.047 |
| 9 | 0 | -0.350 | |
| 10 | 0 | 0.028 | |
| 11 | 0 | -1.148 | |
| 2 | 0 | -1.420 | |
| 2 | 0 | 1.066 | |
| 2 | 0 | 1.066 | |
| 2 | 0 | -1.349 | |
| 3 | 0 | -0.746 | |
| 3 | 0 | 0.306 | |
| 3 | 0 | -0.541 | |
| 3 | 0 | -0.206 | |
| 3 | 0 | 0.306 | |
| 3 | 0 | -0.434 | |
| 3 | 0 | 0.183 | |
| 3 | 0 | -0.095 | |
| 3 | 0 | -0.541 | |
| 3 | 0 | 0.183 | |
| 3 | 0 | -0.685 | |
| 3 | 0 | -0.282 | |
| 3 | 0 | -0.206 | |
| 3 | 0 | -0.095 | |
| 3 | 0 | -0.407 | |
| 4 | 0 | -0.370 | |
| 4 | 0 | -0.094 | |
| 4 | 0 | -0.226 | |
| 4 | 0 | 0.359 | |
| 4 | 0 | -0.157 | |
| 4 | 0 | 0.236 | |
| 4 | 0 | -0.094 | |
| 4 | 0 | -0.129 | |
| 4 | 0 | -0.363 | |
| 4 | 0 | 0.176 | |
| 4 | 0 | -0.098 | |
| 4 | 0 | 0.307 | |
| 4 | 0 | -0.226 | |
| 4 | 0 | -0.363 | |
| 4 | 0 | -0.546 | |
| 4 | 0 | 0.224 | |
| 4 | 0 | -0.284 | |
| 4 | 0 | 0.525 | |
| 4 | 0 | 0.359 | |
| 4 | 0 | 0.176 | |
| 4 | 0 | 0.224 | |
| 4 | 0 | -0.355 | |
| 4 | 0 | 0.088 | |
| 4 | 0 | -0.230 | |
| 4 | 0 | -0.157 | |
| 4 | 0 | -0.098 | |
| 4 | 0 | -0.284 | |
| 4 | 0 | -0.108 | |
| 4 | 0 | 0.360 | |
| 4 | 0 | 0.236 | |
| 4 | 0 | 0.307 | |
| 4 | 0 | 0.525 | |
| 4 | 0 | -0.536 | |
| 5 | 0 | -0.474 | |
| 5 | 0 | 0.103 | |
| 5 | 0 | -0.240 | |
| 5 | 0 | 0.147 | |
| 5 | 0 | -0.373 | |
| 5 | 0 | -0.035 | |
| 5 | 0 | -0.161 | |
| 5 | 0 | -0.104 | |
| 5 | 0 | 0.103 | |
| 5 | 0 | -0.086 | |
| 5 | 0 | 0.107 | |
| 5 | 0 | -0.166 | |
| 5 | 0 | 0.021 | |
| 5 | 0 | -0.071 |
| 5 | 0 | -0.047 | |
| 5 | 0 | -0.055 | |
| 5 | 0 | -0.240 | |
| 5 | 0 | 0.107 | |
| 5 | 0 | -0.212 | |
| 5 | 0 | 0.222 | |
| 5 | 0 | -0.161 | |
| 5 | 0 | 0.043 | |
| 5 | 0 | -0.183 | |
| 5 | 0 | -0.045 | |
| 5 | 0 | 0.147 | |
| 5 | 0 | -0.166 | |
| 5 | 0 | 0.222 | |
| 5 | 0 | -0.154 | |
| 5 | 0 | 0.092 | |
| 5 | 0 | -0.055 | |
| 5 | 0 | 0.034 | |
| 5 | 0 | -0.043 | |
| 5 | 0 | -0.373 | |
| 5 | 0 | 0.021 | |
| 5 | 0 | -0.161 | |
| 5 | 0 | 0.092 | |
| 5 | 0 | -0.432 | |
| 5 | 0 | -0.095 | |
| 5 | 0 | -0.226 | |
| 5 | 0 | -0.138 | |
| 5 | 0 | -0.035 | |
| 5 | 0 | -0.071 | |
| 5 | 0 | 0.043 | |
| 5 | 0 | -0.055 | |
| 5 | 0 | -0.070 | |
| 5 | 0 | -0.107 | |
| 5 | 0 | -0.165 | |
| 5 | 0 | -0.161 | |
| 5 | 0 | -0.047 | |
| 5 | 0 | -0.183 | |
| 5 | 0 | 0.034 | |
| 5 | 0 | -0.192 | |
| 5 | 0 | -0.211 | |
| 5 | 0 | -0.104 | |
| 5 | 0 | -0.055 | |
| 5 | 0 | -0.045 | |
| 5 | 0 | -0.043 | |
| 5 | 0 | -0.144 | |
| 6 | 0 | -0.160 | |
| 6 | 0 | -0.099 | |
| 6 | 0 | -0.090 | |
| 6 | 0 | -0.067 | |
| 6 | 0 | 0.173 | |
| 6 | 0 | -0.152 | |
| 6 | 0 | 0.095 | |
| 6 | 0 | -0.109 | |
| 6 | 0 | -0.099 | |
| 6 | 0 | -0.170 | |
| 6 | 0 | -0.204 | |
| 6 | 0 | -0.287 | |
| 6 | 0 | 0.169 | |
| 6 | 0 | -0.251 | |
| 6 | 0 | 0.225 | |
| 6 | 0 | -0.262 | |
| 6 | 0 | -0.090 | |
| 6 | 0 | -0.204 | |
| 6 | 0 | -0.090 | |
| 6 | 0 | -0.195 | |
| 6 | 0 | 0.087 | |
| 6 | 0 | -0.207 | |
| 6 | 0 | 0.133 | |
| 6 | 0 | -0.196 | |
| 6 | 0 | -0.067 |
| 6 | 0 | -0.287 | |
| 6 | 0 | -0.195 | |
| 6 | 0 | -0.276 | |
| 6 | 0 | 0.118 | |
| 6 | 0 | -0.255 | |
| 6 | 0 | 0.212 | |
| 6 | 0 | -0.370 | |
| 6 | 0 | 0.173 | |
| 6 | 0 | 0.169 | |
| 6 | 0 | 0.087 | |
| 6 | 0 | 0.118 | |
| 6 | 0 | -0.158 | |
| 6 | 0 | 0.097 | |
| 6 | 0 | -0.093 | |
| 6 | 0 | 0.072 | |
| 6 | 0 | -0.152 | |
| 6 | 0 | -0.251 | |
| 6 | 0 | -0.207 | |
| 6 | 0 | -0.255 | |
| 6 | 0 | -0.150 | |
| 6 | 0 | 0.203 | |
| 6 | 0 | -0.285 | |
| 6 | 0 | 0.095 | |
| 6 | 0 | 0.225 | |
| 6 | 0 | 0.133 | |
| 6 | 0 | 0.212 | |
| 6 | 0 | -0.082 | |
| 6 | 0 | 0.196 | |
| 6 | 0 | -0.109 | |
| 6 | 0 | -0.262 | |
| 6 | 0 | -0.196 | |
| 6 | 0 | -0.370 | |
| 6 | 0 | -0.263 | |
| 7 | 0 | -0.499 | |
| 7 | 0 | 0.076 | |
| 7 | 0 | -0.343 | |
| 7 | 0 | -0.299 | |
| 7 | 0 | 0.036 | |
| 7 | 0 | -0.165 | |
| 7 | 0 | 0.076 | |
| 7 | 0 | -0.039 | |
| 7 | 0 | 0.129 | |
| 7 | 0 | -0.039 | |
| 7 | 0 | -0.006 | |
| 7 | 0 | -0.107 | |
| 7 | 0 | -0.343 | |
| 7 | 0 | 0.129 | |
| 7 | 0 | -0.420 | |
| 7 | 0 | -0.161 | |
| 7 | 0 | 0.113 | |
| 7 | 0 | -0.092 | |
| 7 | 0 | -0.299 | |
| 7 | 0 | -0.039 | |
| 7 | 0 | -0.161 | |
| 7 | 0 | -0.456 | |
| 7 | 0 | -0.073 | |
| 7 | 0 | -0.325 | |
| 7 | 0 | 0.036 | |
| 7 | 0 | -0.006 | |
| 7 | 0 | 0.113 | |
| 7 | 0 | -0.027 | |
| 7 | 0 | -0.132 | |
| 7 | 0 | -0.165 | |
| 7 | 0 | -0.107 | |
| 7 | 0 | -0.092 | |
| 7 | 0 | -0.397 | |
| 8 | 0 | -0.018 | |
| 8 | 0 | -0.125 | |
| 8 | 0 | 0.069 |
| 8 | 0 | -0.180 | |
| 8 | 0 | -0.125 | |
| 8 | 0 | -0.546 | |
| 8 | 0 | 0.198 | |
| 8 | 0 | -0.704 | |
| 8 | 0 | 0.069 | |
| 8 | 0 | 0.198 | |
| 8 | 0 | -0.022 | |
| 8 | 0 | 0.125 | |
| 8 | 0 | -0.180 | |
| 8 | 0 | -0.704 | |
| 8 | 0 | -0.516 | |
| 9 | 0 | -1.057 | |
| 9 | 0 | -0.255 | |
| 9 | 0 | -0.255 | |
| 9 | 0 | -0.989 |
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Calculation of Gamow-Teller and Two-Neutrino Double-
Decay Properties for 130Te and 136Xe with a realistic nucleon-nucleon potential
L. Coraggio
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
L. De Angelis
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
T. Fukui
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
A. Gargano
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
N. Itaco
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, viale Abramo Lincoln 5 - I-81100 Caserta, Italy
Abstract
We report on the calculation of Gamow-Teller and double- decay properties for nuclei around 132Sn within the framework of the realistic shell model. The effective shell-model Hamiltonian and Gamow-Teller transition operator are derived by way of many-body perturbation theory, without resorting to empirical effective quenching factor for the Gamow-Teller operator. The results are then compared with the available experimental data, in order to establish the reliability of our approach. This is a mandatory step, before we apply the same methodology, in forthcoming studies, to the calculation of the neutrinoless double- decay nuclear matrix element for nuclei that are currently considered among the best candidates for the detection of this process.
pacs:
21.60.Cs, 21.30.Fe, 27.60.+j, 23.40-s
I Introduction
The detection of neutrinoless double- decay () is nowadays one of the main targets in many laboratories all around the world, triggered by the search of “new physics” beyond the Standard Model. The observation of such a process would be the evidence of a lepton number violation and shed more light on the nature and properties of the neutrino (see Refs. Avignone et al. (2008); Vergados et al. (2012) and references therein).
It is well known that the expression for the half life of the decay can be written in the following form:
[TABLE]
where is the so-called phase-space factor (or kinematic factor), is the effective neutrino mass that takes into account the neutrino parameters associated with the mechanisms of light- and heavy-neutrino exchange, and is the nuclear matrix element (NME) directly related to the wave functions of the parent and grand-daughter nuclei.
From the expression (1), it is clear that a reliable estimate of the NME is a keypoint to understand which are the most favorable nuclides to be considered for the search of the decay, and how to link the experimental results to a measurement of . It is therefore incumbent upon the theoretical nuclear structure community to make an effort to provide calculations of the NME as much reliable as possible.
Currently, the nuclear structure models which are largely employed in this research field are the Interacting Boson Model (IBM) Barea and Iachello (2009); Barea et al. (2012, 2013), the Quasiparticle Random-Phase Approximation (QRPA) Šimkovic et al. (2008, 2009); Fang et al. (2011); Faessler et al. (2012), Energy Density Functional methods Rodríguez and Martínez-Pinedo (2010), and the Shell Model (SM) Caurier et al. (2008); Menéndez et al. (2009a, b); Horoi (2013); Horoi and Brown (2013); Neacsu and Horoi (2015); Brown et al. (2015); Frekers et al. (2017). All of them have different advantages and drawbacks, that make one model more suitable than another for a certain class of nuclei, but nowadays the results obtained employing these approaches agree within a factor (see Ref. Barea et al. (2015) and references therein).
A common feature in all the many-body models applied to systems with mass number ranging from to 150 is that the parameters upon which they depend need to be determined fitting some spectroscopic properties of the nuclei under investigation. In particular, since the Hilbert space considered in these approximated models is a truncated one, it is necessary to introduce quenching factors of the axial and vector coupling constants and that appear in the NME expression. Besides of the excluded degrees of freedom in the many-body calculation, the quenching operation has to take into account the subnucleonic structure of the nucleons too. The free value of , that is obtained by the measurement of from the neutron decay Nakamura et al. (2010), is 1.269, and its quenching factor is usually fixed fitting the observed Gamow-Teller (GT) and two-neutrino double- decay () properties, that are experimentally available.
We remark that the structure of the two operators, corresponding to the and decays, is quite different, and the quenching operation may be effective to calculate the GT strengths and NME, but not consistent with the renormalization of the -decay operator.
As a matter of fact, there are two main open questions about this problem. The first one is related to the fact that in the decay essentially the states of the intermediate odd-odd nucleus are involved in the process, while all multipoles come into play in the decay. So there is no precise prescription if should be quenched only for the multipole, the quenching factor being fitted on -decay properties, or all the multipole channels should be equally quenched Barea et al. (2013).
Besides this, there is another question to be addressed. In the decay the term associated with the vector current of the electroweak lagrangian and its coupling constant plays a negligible role, but this might not be the case for the decay. So, it may be necessary also to renormalize this factor in order to take into account the many-body effects and the neglected subnucleonic degrees of freedom. Actually, there is no experimental evidence for an underlying mechanism for the renormalization of , namely if the same quenching factor used for , fixed by fitting decay data, should be used to quench too.
Our framework to tackle these problems is the realistic shell model, where all the parameters appearing in the SM Hamiltonian and in the transition operators are derived from a realistic free nucleon-nucleon () potential by way of the many-body theory Kuo and Osnes (1990); Suzuki and Okamoto (1995). In this way the bare matrix elements of the potential and of any transition operator are renormalized with respect to the truncation of the full Hilbert space into the reduced SM model space, to take into account the neglected degrees of freedom without resorting to any empirical parameter. In other words, in our approach we do not employ effective charges to calculate electromagnetic transition strengths, and we do not quench empirically the axial and vector current coupling constants.
It is a mandatory step, however, to check this approach to calculate properties related to the GT and decays of nuclei involved in possible , and compare the results with the available data. This is the content of present work, where we present the outcome of SM calculations for nuclei around 132Sn, focussing our attention on the GT strengths and -decay of 130Te and 136Xe.
These two nuclei are currently considered as candidates for the observation of neutrinoless double-beta decay by some large experimental collaborations. The decay of 130Te is targeted by the CUORE collaboration at the INFN Laboratori Nazionali del Gran Sasso in Italy Alfonso et al. (2015), while the decay of 136Xe is investigated both by the EXO-200 collaboration at the Waste Isolation Pilot Plant in Carlsbad, New Mexico, Auger et al. (2012), and by the KamLAND-Zen collaboration in the Kamioka mine in Japan Gando et al. (2016).
Our starting point is the high-precision potential CD-Bonn Machleidt (2001), whose repulsive high-momentum components are smoothed out using the approach Bogner et al. (2002). Then, from this realistic potential we have derived, within a model space spanned by the five proton and neutron orbitals outside the doubly-closed 100Sn, the effective shell-model Hamiltonian , effective electromagnetic and GT transition operators. The derivation of the effective Hamiltonian and operators has been performed by way of the time-dependent perturbation theory Kuo et al. (1971); Coraggio et al. (2009a), including diagrams up to the third-order in .
The following section is devoted to the presentation of some details about the derivation of our shell-model Hamiltonian and of the effective transition and decay operators. In Section III, we report the results of our calculations for the spectroscopic properties of 130Te, 130,136Xe, and 136Ba, electromagnetic and GT transition strengths for 130Te and 136Xe, and their NMEs for the decay. Theoretical results are compared with available experimental data. In the last section we sketch out a summary of the present work and an outlook of our future program. In the Supplemental Material sup , the calculated two-body matrix elements (TBME) of our SM Hamiltonian can be found.
II Outline of calculations
We start our calculations by considering the high-precision CD-Bonn potential Machleidt (2001). Because of the non-perturbative behavior induced by the repulsive high-momentum components of CD-Bonn potential, we have renormalized the latter by way of the so-called approach Bogner et al. (2001, 2002). This procedure provides a smooth potential that can be employed directly in the many-body perturbation theory, and that preserves exactly the onshell properties of the original potential up to a cutoff momentum . We have chosen its value, as in many of our recent papers Coraggio et al. (2009b, c); Coraggio et al. (2014); Coraggio et al. (2015a); Coraggio et al. (2016), to be equal to fm*-1*, because we have found that the larger the cutoff the smaller the role of the missing three-nucleon force (3NF) Coraggio et al. (2015b). The Coulomb potential is explicitly taken into account in the proton-proton channel.
The next step is to derive an effective Hamiltonian for SM calculations employing a model space spanned by the five proton and neutron orbitals outside the doubly-closed 100Sn core. To this end, an auxiliary one-body potential is introduced in order to break up the Hamiltonian for a system of nucleons as the sum of a one-body term , which describes the independent motion of the nucleons, and a residual interaction :
[TABLE]
Once has been introduced, the reduced model space is defined in terms of a finite subset of ’s eigenvectors. In our calculation we choose as auxiliary potential the harmonic oscillator (HO) potential.
Since the diagonalization of the many-body Hamiltonian (2) in an infinite Hilbert space is obviously infeasible, our eigenvalue problem is then reduced to the solution of that one for an effective Hamiltonian in a truncated model space.
In this paper, we derive by way of the Kuo-Lee-Ratcliff (KLR) folded-diagram expansion Kuo et al. (1971); Kuo and Osnes (1990) in terms of the vertex function box, that is defined as
[TABLE]
The box may be expanded perturbatively in terms of irreducible valence-linked one- and two-body Goldstone diagrams through third order in Hjorth-Jensen et al. (1995). We have reviewed the calculation of our SM effective Hamiltonian in Ref. Coraggio et al. (2012), where details of the diagrammatic expansion of the box and its perturbative properties are also reported.
In terms of the box, the effective SM Hamiltonian can be written in an operator form as
[TABLE]
where the integral sign represents a generalized folding operation, and is obtained from by removing terms at the first order in Kuo et al. (1971); Kuo and Osnes (1990). The folded-diagram series is then summed up to all orders using the Lee-Suzuki iteration method Suzuki and Lee (1980).
From we obtain both single-particle (SP) energies and TBME for our SM calculations. As already mentioned in the Introduction, in the Supplemental Material sup our calculated TBME are reported, and in Table LABEL:spetab our calculated SP energies. There, the latter (labelled as I) are compared with a set of empirical SP energies (labelled as II) that are needed to fit the observed SP states in 133Sb and 131Sn ens ; xun .
As regards the effective transition and decay operators, namely the effective charges of the electric quadrupole operators and the matrix elements of the effective GT operator, we have derived them consistently with SM , within an approach that is strictly based on the one presented by Suzuki and Okamoto in Ref. Suzuki and Okamoto (1995).
In that paper, it has been demonstrated that a non-Hermitian effective operator can be written in the following form:
[TABLE]
where is the box defined by the expression (3), and
[TABLE]
being the eigenvalue of the degenerate model-space of the unperturbed Hamiltonian , that, as mentioned before, we have chosen to be the HO one.
The operators are defined as follows:
[TABLE]
where , have the following expressions:
[TABLE]
with
[TABLE]
being the bare operator.
In our calculations for the one-body operators we arrest the series to the leading term , and the latter is expanded perturbatively including diagrams up to the third order in the perturbation theory, consistently with the perturbative expansion of the box.
In Fig. 1 we have reported all the single-body diagrams up to the second order, the bare operator being represented with an asterisk. The first-order -insertion is represented by a circle with a cross inside, which arises in the perturbative expansion owing to the presence of the term in the interaction Hamiltonian (see for example Ref. Coraggio et al. (2012) for details).
Using this approach we have calculated proton and neutron effective state-dependent charges, which are reported in Table 2. It should be pointed out that our results are close to the usual empirical values ().
In Tables 3 and 4, the matrix elements of the proton-neutron GT+ and neutron-proton GT- effective operators, respectively, are reported.
The breaking of the proton-neutron symmetry is due to the fact that we include in the perturbative calculation of and also the effect of the Coulomb potential between the interacting protons. In the last column the quenching factors that should be employed in order to obtain the corresponding matrix element are also reported. The quenching factor is not reported for those matrix elements that are forbidden for the bare GT operator.
III Results
This section is devoted to the presentation of the results of our SM calculations.
We compare the calculated low-energy spectra of 130Te, 130Xe, 136Xe, and 136Ba, and their electromagnetic transition strengths with the available experimental data, that are reported in Table 5. It should be mentioned that in Ref. Vietze et al. (2015) shell-model calculations for 130,136Xe isotopes have been performed using the empirical shell-model Hamiltonian GCN5082 Menéndez et al. (2009b).
We show also the results of the GT- strength distributions of 130Te and 136Xe, which are defined as follows:
[TABLE]
where indices refer to the parent and daughter nuclei, respectively, and the sum is over all interacting nucleons.
In the following subsections, we also report the results of the calculated NME of the decays Xeg.s. and Bag.s., via the following expression:
[TABLE]
where is the excitation energy of the intermediate state, , and being the value of the decay and the mass difference between the daughter and parent nuclei, respectively. In the expression of Eq. (15) the sum over index runs over all possible intermediate states of the daughter nucleus. The NMEs have been calculated using the ANTOINE shell-model code, using the Lanczos strength function-method as in Ref. Caurier et al. (2005). The theoretical values are then compared with the experimental counterparts, that are directly related to the observed half life
[TABLE]
In connection with the decay, we show also the comparison between our calculated proton/neutron occupancies/vacancies and the recent data.
All the calculations have been performed employing both theoretical and empirical SP energies, reported in Table LABEL:spetab, in order to provide an indicator of the sensitivity of our SM results on the choice of the SP energies.
III.1 130Te strengths and
decay
In Figs. 2 and 3, we show the experimental ens ; xun and calculated spectra of 130Te and 130Xe up to an excitation energy of 2 MeV. As can be seen, these results are scarcely sensitive to the choice of the SP energies, those of 130Te being in a very good agreement with the experimental data, while the reproduction of the observed 130Xe low-lying states is less satisfactory.
From inspection of Table 5, it can be seen that our calculated electric-quadrupole transition rates compare well with the observed values for both nuclei, testifying the reliability of our SM wavefunctions and of the effective electric-quadrupole transition-operator. Its matrix elements are reported in Table 2. It is worth noting that the calculated s do not show a relevant dependence on the choice of the SP energies, their values being very close each other.
In Fig. 4, our calculated running sums of the Gamow-Teller strengths () as a function of the excitation energy for 130Te are shown. The comparison of the calculated GT strength distributions with the observed ones is a very relevant point when trying to assess the reliability of a many-body approach to the description of the decay.
The single decay GT strengths, defined by Eq. (14), can be accessed experimentally through intermediate energy charge-exchange reactions. As a matter of fact, the GT strength can be extracted, following the standard approach in the distorted-wave Born approximation (DWBA), from the GT component of the cross section by way of the relation Goodman et al. (1980); Taddeucci et al. (1987)
[TABLE]
where is the distortion factor, and is the volume integral of the effective interaction.
In the following, we compare our results with the GT- distributions obtained in recent high-resolution studies on 130Te Puppe et al. (2012).
In Fig. 4, the data are reported with a red line, while the results obtained with SP energies (I) and (II) define the blue and black areas, respectively, for the bare and effective operators. It can be seen that the renormalized GT operator is able to reproduce quite well the behavior of the experimental running GT strength.
As a matter of fact, if we shift the calculated distributions in order to reproduce the position of the first state in 130I, the theoretical total strengths up to 3 MeV excitation energy are equal to 0.842 and 0.873, for the calculations with SP energies I and II respectively, which should be compared with the experimental value . The crucial role of the many-body renormalization is evident when considering the results obtained using the bare GT operator. In this case the total strength is equal to 2.554 and 2.408 with SP energies from set (I) and (II), respectively.
As regards the decay of 130Te, we have calculated NME, as defined by expression (15), and the results, compared with value obtained from the experimental half life of the Xe decay Barabash (2010), are reported in Fig. 5.
The theoretical results are reported as a function of the maximum excitation energy of the intermediate states included in the sum of expression (15). As can be seen, the calculated values saturate when including at least intermediate states up to 8 MeV excitation energy.
As in the case of the theoretical GT strength distributions, the NMEs calculated with the effective GT operator are in a good agreement with the experimental datum MeV*-1* Barabash (2010), our results being 0.044 MeV*-1* and 0.046 MeV*-1* with SP energies (I) and (II), respectively. Actually, the NMEs calculated with the bare GT operator are 0.131 MeV*-1*(I) and 0.137 MeV*-1* (II), which are far away from the experimental one.
It is worth mentioning now the results obtained by two recent SM calculations Caurier et al. (2012); Neacsu and Horoi (2015), where the shell model Hamiltonians are based on realistic potentials but empirically modified in order to reproduce some spectroscopic properties of nuclei around 132Sn. In Ref. Caurier et al. (2012) the calculated NME is 0.043 MeV*-1*, with a quenching factor of 0.57 of , while in Ref. Neacsu and Horoi (2015) a value of 0.0328 MeV*-1*, using a quenching factor of 0.74, is reported.
Another important indicator of the quality of the calculated NME, both for and decay, may be provided by the comparison of the theoretical occupancies of valence nucleons in the ground states of the parent and grand-daughter nuclei with the observed ones. Recently, those quantities have been determined by measuring the cross sections of one-proton stripping and one-neutron pick-up reactions, for proton occupancies and neutron vacancies, respectively Entwisle et al. (2016); Kay et al. (2013). These data are reported in Figs. 6 and 7 and compared with our calculations.
The calculations with SP energies (I) and (II) give very close results, which are in nice agreement with experiment, bearing in mind the experimental uncertainties that are up to for the change in occupancy of proton orbital Entwisle et al. (2016).
III.2 136Xe strengths and
decay
This subsection is organized like the previous one, so we start from the inspection of Figs. 8 and 9, where the experimental ens ; xun and calculated spectra of 136Xe and 136Ba up to an excitation energy of 2 MeV are reported. The calculated spectra are again in a good agreement with experiment, and results are rather insensitive to the SP energies I and II.
From inspection of Table 5, it can be seen that our calculated s are very close to the observed value, while for 136Xe the theoretical s and s are less satisfactory when compared with available data ens ; xun , calculations with SP energies (I) and (II) underestimating the observed and overestimating the experimental .
The calculated for 136Xe, as a function of the excitation energy, can be found in Fig. 10, where they are compared with the observed GT- distributions extracted from high-resolution reactions on 136Xe Frekers et al. (2013).
As in the case of 130Te, we observe that the renormalized GT operator reproduces satisfactorily the observed running GT strength, the theoretical total strengths up to 4.5 MeV excitation energy being equal to 0.94 (I) and 1.13 (II) and to be compared with an experimental value of .
We have calculated the NME related to the decay of 136Xe into 136Ba, whose values are 0.091 MeV*-1* (I) and 0.094 MeV*-1* (II) with bare GT operator, and 0.0285 MeV*-1* (I) and 0.0287 MeV*-1* (II) with the effective operator . The experimental value, obtained from the experimental half life of the Ba decay Albert et al. (2014), is MeV*-1*, that compares well with the theoretical values derived employing the effective operator (Fig. 11).
For the sake of completeness, we mention that in Ref. Caurier et al. (2012) the calculated value of the NME is 0.025 MeV*-1* with a quenching factor equal to 0.45, and in Ref. Neacsu and Horoi (2015) they obtain 0.0256 MeV*-1*, having quenched by a factor 0.74. The latter has been employed also in Ref. Horoi and Brown (2013) as a quenching factor to calculate the matrix element of decay of 136Xe, resulting in a calculated NME of 0.062 MeV*-1*. In this paper, the authors have used a different shell-model Hamiltonian, derived by way of the KLR folded-diagram expansion from the realistic N3LO potential Entem and Machleidt (2002), which however seems to describe the nuclear structure of nuclei around equally as well as in Ref. Neacsu and Horoi (2015). This evidences the tight relationship between the shell-model Hamiltonian and the choice of the quenching factor.
Finally, in Figs. 12 and 13 the theoretical occupancies of valence nucleons in the ground states of the parent and grand-daughter nuclei are shown and compared with those obtained in Ref. Kay et al. (2013); Entwisle et al. (2016) from the experimental cross sections of the and reactions. The poor reproduction of the experimental neutron vacancies, as can be seen in Fig. 13, is due to the fact that in our model space the neutron component of 136Xe is frozen, having its 32 valence neutrons totally filled the 50-82 shell.
IV Summary and Outlook
In the present work, we have presented the results of a realistic SM calculation of GT decay properties for 130Te and 136Xe. Our aim has been to test an approach to the calculation of the NME of the decay of these nuclei, where the SM Hamiltonian and the related transition operators are derived, starting from a realistic potential, from the many-body theory. This means that the need to resort to empirical parameters is drastically reduced, so enhancing the predictiveness of the nuclear structure calculations.
The first step toward this goal has been to test the reliability of our theoretical framework, that cannot be done for the . In particular we have calculated the GT strengths and the NMEs of the , and compared the results with the available experimental ones. This is reported in Section III, and the overall agreement with the data is quite good. As summarized in Table 6, the quality of our results is similar to, and even better than, that obtained with recent calculations available in literature, which employ SM parameters that have been empirically fitted to reproduce some selected observables.
Our results are encouraging for our next steps towards an (almost) parameter-free calculation of the NME of 130Te and 136Xe, making us confident of a positive outcome of a fully microscopical approach to this problem.
Finally, it should be pointed out that in our calculations the renormalization of the bare operators by way of the many-body perturbation theory takes into account the degrees of freedom that are not explicitly included within the reduced SM model space.
There are also two other main effects that should be included in the renormalization of the GT operators, namely the blocking effect and the role of the subnucleonic degrees of freedom.
The blocking effect is responsible for taking into account the correlations among the active nucleons in systems with many interacting valence particles, within the derivation of the effective operators. We are currently investigating the role played by these correlations in the calculation of GT and properties.
Another contribution to the renormalization of the GT operators is associated to the quark structure of nucleons. Since a realistic potential is our starting point, we do not consider in such a picture the role played by the nucleon resonances () - that are also responsible for three-nucleon forces - whose contribution should lead to renormalized values of and .
Nowadays, the derivation of nuclear potentials by way of the chiral perturbation theory Epelbaum et al. (2005); Machleidt and Entem (2011) allows a consistent treatment of this approach to the renormalization of the axial- and vector-current constants, that has been already explored in Ref. Menéndez et al. (2011). We are investigating this subject, which will be the topic of a forthcoming paper.
Acknowledgments
The authors gratefully acknowledge useful comments and suggestions from Francesco Iachello and Frederic Nowacki.
Appendix
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1Avignone et al. (2008) F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod. Phys. 80 , 481 (2008).
- 2Vergados et al. (2012) J. D. Vergados, H. Ejiri, and F. Šimkovic, Rep. Prog. Phys. 75 , 106301 (2012).
- 3Barea and Iachello (2009) J. Barea and F. Iachello, Phys. Rev. C 79 , 044301 (2009).
- 4Barea et al. (2012) J. Barea, J. Kotila, and F. Iachello, Phys. Rev. Lett. 109 , 042501 (2012).
- 5Barea et al. (2013) J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C 87 , 014315 (2013).
- 6Šimkovic et al. (2008) F. Šimkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C 77 , 045503 (2008).
- 7Šimkovic et al. (2009) F. Šimkovic, A. Faessler, H. Müther, V. Rodin, and M. Stauf, Phys. Rev. C 79 , 055501 (2009).
- 8Fang et al. (2011) D.-L. Fang, A. Faessler, V. Rodin, and F. Šimkovic, Phys. Rev. C 83 , 034320 (2011).