Shape Phase Transitions in Even–Even 176–198Pt: Higher-Order Interactions in the Interacting Boson Model †
Abstract
:1. Introduction
2. The CQ Formalism and Its Extension
3. The Modified Soft Rotor Model
4. Shape Phase Transition in Pt
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Nucleus | (MeV) | N | ||||||
---|---|---|---|---|---|---|---|---|
Pt | −0.008 | 0.6216 | 0.4795 | 0 | 0 | 0.1287 | −1.1218 | 10 |
Pt | 0 | 0.8363 | 0.4594 | 0 | 0 | 0.1841 | −1.0001 | 11 |
Pt | 0.0002 | 0.9624 | 0.4483 | −0.0449 | 0 | 0.0499 | −0.8625 | 12 |
Pt | 0.0006 | 0.9996 | 0.4462 | 0 | −0.1648 | 0 | −0.6456 | 13 |
Pt | 0.0007 | 0.9481 | 0.4530 | 0.1747 | −0.0775 | 0.2886 | −0.6085 | 12 |
Pt | 0.001 | 0.8079 | 0.4690 | 0 | −0.2471 | 0 | −0.4868 | 11 |
Pt | 0.0011 | 1.0696 | 0.1061 | −0.2197 | −0.5142 | 0 | −0.0159 | 10 |
Pt | 0.00201 | 1.0708 | 0.1225 | −0.3707 | −0.2638 | 0.1009 | 0.0265 | 9 |
Pt | 0.0009 | 1.0514 | 0.1389 | −0.2587 | −0.4614 | 0.2395 | 0.0635 | 8 |
Pt | 0.0014 | 1.0113 | 0.1553 | −0.2083 | −0.3281 | 0.2547 | 0.0741 | 7 |
Pt | 0.001 | 0.9507 | 0.1717 | −0.2007 | −0.3380 | 0.2777 | 0.0846 | 6 |
Pt | 0.00148 | 0.8694 | 0.1882 | −0.2085 | −0.3810 | 0.2553 | 0.0953 | 5 |
Nucleus | c (MeV) | N | ||
---|---|---|---|---|
Pt | 0.9814 | 0.5240 | −1.2382 | 10 |
Pt | 1.0296 | 0.4761 | −1.1959 | 11 |
Pt | 1.0402 | 0.4457 | −0.9631 | 12 |
Pt | 1.0133 | 0.4329 | −0.7461 | 13 |
Pt | 0.9490 | 0.4376 | −0.7302 | 12 |
Pt | 0.8471 | 0.4598 | −0.6456 | 11 |
Pt | 1.3784 | 0.3609 | −0.1058 | 10 |
Pt | 1.7264 | 0.2848 | −0.0212 | 9 |
Pt | 1.9379 | 0.2182 | 0.0159 | 8 |
Pt | 2.0130 | 0.1612 | 0.0688 | 7 |
Pt | 1.9515 | 0.1138 | 0.1111 | 6 |
Pt | 1.7535 | 0.0759 | 0.1217 | 5 |
87(8) | 87.011 | 87.008 | 76 | 130.180 | 130.184 | ||
163(15) | 148.646 | 159.366 | 22.2 | 40.214 | 37.509 | ||
174(16) | 191.204 | 197.981 | 11.2 | 16.237 | 15.681 | ||
192(25) | 217.843 | 215.919 | 4.7 | 6.512 | 6.570 | ||
195(17) | 195.068 | 195.009 | 37.5 | 65.430 | 65.450 | ||
186(14) | 231.816 | 232.173 | 10.9 | 13.993 | 13.971 | ||
206(23) | 248.684 | 249.205 | – | 4.789 | 4.779 | ||
154(15) | 154.051 | 154.027 | 374 | 1091.599 | 1091.769 | ||
(4) | 247.620 | 251.145 | 22.9 | 49.808 | 49.109 | ||
≥50 | 290.043 | 296.687 | ≤35 | 9.662 | 9.446 | ||
114(8) | 114.002 | 114.004 | 479 | 1361.953 | 1361.929 | ||
192(12) | 176.723 | 184.507 | 32.5 | 61.294 | 58.708 | ||
292(20) | 203.217 | 219.257 | 5.28 | 12.124 | 11.237 | ||
127(5) | 127.009 | 127.014 | 360 | 936.807 | 936.770 | ||
> | 2.960 | 1.385 | ≤ | 10.800 | 23.081 | ||
> | 47.809 | 74.740 | ≤ | 52.991 | 33.897 | ||
210(8) | 202.506 | 207.042 | 25.3 | 44.583 | 43.606 | ||
226(12) | 236.014 | 246.825 | 6.1 | 9.331 | 8.923 | ||
271(18) | 251.128 | 267.136 | 2.15 | 3.623 | 3.406 | ||
94(5) | 94.010 | 94.426 | 240 | 550.146 | 547.722 | ||
89(4) | 89.167 | 89.014 | 66 | 112.028 | 112.220 | ||
150(+4-5) | 117.39 | 129.247 | 5.1 | 10.400 | 9.446 | ||
158(15) | 132.752 | 148.733 | 1.53 | 2.820 | 2.517 | ||
118(17) | 137.830 | 155.836 | 0.97 | 1.262 | 1.116 | ||
56(3) | 56.010 | 56.018 | 62.3 | 103.070 | 103.055 | ||
57.2(12) | 57.199 | 57.198 | 43.7 | 70.653 | 70.654 | ||
0.55(4) | 1.911 | 0.004 | 26.5 | 78.849 | 37,670.042 | ||
109(7) | 69.760 | 77.975 | 26.5 | 81.611 | 73.013 | ||
89(5) | 77.345 | 78.108 | 4.2 | 7.450 | 7.377 | ||
0.68(7) | 1.525 | 0.007 | 21.3 | 105.526 | 22,989.550 | ||
102(10) | 46.093 | 60.795 | 21.3 | 101.257 | 76.770 | ||
70(30) | 84.382 | 84.901 | 1.8 | 2.316 | 2.302 | ||
38(10) | 26.257 | 24.342 | 21.3 | 10,589.454 | 11,422.533 |
49.2(8) | 41.7 | 68.261 | 68.274 | ||||
0.29(4) | 1.008 | 0.137 | 35 | 135.900 | 999.908 | ||
89(11) | 62.116 | 63.507 | 35 | 93.474 | 91.427 | ||
85(5) | 65.238 | 66.106 | 3.7 | 7.437 | 7.339 | ||
0.36(7) | 1.279 | 0.0003 | 3.8 | 16.793 | 71,595.730 | ||
21(4) | 36.687 | 36.299 | 3.8 | 4.219 | 4.229 | ||
14 | 24.721 | 31.944 | 3.8 | 40.428 | 31.287 | ||
0.63(14) | 3.269 | 0.396 | 6.1 | 5.344 | 44.117 | ||
8.4(19) | 87.311 | 67.085 | 6.1 | 1.308 | 1.703 | ||
<75 | 21.289 | 19.870 | – | 35,551.848 | 38,090.754 | ||
100 | 42.538 | 50.148 | – | 121.345 | 102.931 | ||
< | 0.766 | 0.190 | – | 225.150 | 907.709 | ||
67(21) | 69.313 | 69.965 | 1.6 | 2.347 | 2.325 | ||
50(14) | 64.454 | 66.068 | 1.1 | 1.284 | 1.252 | ||
53(10) | 40.362 | 42.628 | 0.61 | 1.222 | 1.157 | ||
40.60(20) | 34.15 | 54.244 | 54.248 | ||||
(4) | 1.373 | 0.309 | 33.8 | 59.081 | 262.519 | ||
5(5) | 0.003 | 11.079 | – | 7,121,171.886 | 1928.289 | ||
0.0025(24) | 0.064 | 0.033 | – | 42.145 | 81.735 | ||
18(10) | 69.587 | 51.888 | 4.2 | 10.143 | 13.602 | ||
2.8(15) | 3.072 | 0.545 | 4.2 | 14.201 | 80.047 | ||
< | 1.046 | 0.380 | 1.6 | 9.571 | 26.345 | ||
< | 0.009 | 1.218 | 1.6 | 7541.910 | 55.728 | ||
54(+11-12) | 45.337 | 50.195 | 33.8 | 31.224 | 28.203 | ||
0.26(23) | 0.277 | 0.086 | – | 329.352 | 1060.819 | ||
60.0(9) | 53.551 | 53.566 | 3.55 | 6.127 | 6.125 | ||
29(+6-29) | 28.837 | 28.323 | 2.6 | 5.389 | 5.487 | ||
0.56(+12-17) | 0.691 | 0.002 | 2.6 | 25.103 | 8673.028 | ||
0.13(12) | 0.600 | 0.370 | – | 782.267 | 1268.541 | ||
17(6) | 18.093 | 23.829 | 2.6 | 55.878 | 42.427 | ||
73(+4-73) | 54.709 | 54.661 | 0.98 | 2.000 | 2.001 | ||
0.48(14) | 0.029 | 0.0003 | 0.77 | 234.695 | 22,687.186 | ||
49(13) | 0.036 | 0.008 | 0.77 | 1885.478 | 8484.649 | ||
16(5) | 0.007 | 0.020 | 0.77 | 69,886.211 | 24,460.174 | ||
78(+10-78) | 47.562 | 48.542 | 0.42 | 1.304 | 1.278 | ||
31.81(22) | 22.25 | 34.950 | 34.945 | ||||
0.038(12) | 1.035 | 0.148 | 27 | 42.912 | 300.093 | ||
0.05(3) | 0.066 | 0.009 | 9.7 | 54.756 | 401.545 | ||
0.6(4) | 0.003 | 0.0001 | 9.7 | 8209.613 | 246,288.402 | ||
2.2(15) | 0.469 | 0.060 | 9.7 | 806.114 | 6301.127 | ||
37(7) | 34.484 | 37.277 | 27 | 54.085 | 50.033 | ||
26(7) | 2.225 | 0.302 | – | 166.593 | 1227.382 | ||
38(4) | 40.519 | 40.567 | 3.3 | 4.750 | 4.745 | ||
>57 | 38.899 | 39.092 | 1.550 | 1.543 |
Experiment | This Work | CQ | ||
---|---|---|---|---|
+0.55(21) | +0.478 | +0.051 | ||
+0.409 | +0.348 | +0.267 | ||
−0.303 | −0.322 | −0.249 | ||
+0.752 | +0.624 | +0.268 | ||
−0.06(11) | +0.111 | −0.106 | ||
+0.195 | +0.794 | +0.263 | ||
−0.06, 0.28 | +0.908 | +0.262 | ||
+0.62 | +0.503 | +0.399 | ||
−0.39 | −0.474 | −0.372 | ||
+1.03 | +0.662 | +0.397 | ||
−0.18 | +0.762 | +0.380 | ||
+0.42(12), 0.54(12) | +0.431 | +0.305 |
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Li, D.; Wang, T.; Pan, F. Shape Phase Transitions in Even–Even 176–198Pt: Higher-Order Interactions in the Interacting Boson Model. Symmetry 2022, 14, 2610. https://doi.org/10.3390/sym14122610
Li D, Wang T, Pan F. Shape Phase Transitions in Even–Even 176–198Pt: Higher-Order Interactions in the Interacting Boson Model. Symmetry. 2022; 14(12):2610. https://doi.org/10.3390/sym14122610
Chicago/Turabian StyleLi, Dongkang, Tao Wang, and Feng Pan. 2022. "Shape Phase Transitions in Even–Even 176–198Pt: Higher-Order Interactions in the Interacting Boson Model" Symmetry 14, no. 12: 2610. https://doi.org/10.3390/sym14122610
APA StyleLi, D., Wang, T., & Pan, F. (2022). Shape Phase Transitions in Even–Even 176–198Pt: Higher-Order Interactions in the Interacting Boson Model. Symmetry, 14(12), 2610. https://doi.org/10.3390/sym14122610