Prerpints.org logo
Preprint
Article

Muonic Hyperfine Structure and the Bohr-Weisskopf Effect

Altmetrics

Downloads

139

Views

25

Comments

0

This version is not peer-reviewed

Submitted:

10 August 2023

Posted:

11 August 2023

You are already at the latest version

Alerts
Abstract
A review is given of the experimental values of the magnetic hyperfine structure and the Bohr-Weisskopf effect in muonic atoms. The need for more measurements and systematic calculations is discussed in the light of distinguishing between different models of the Bohr-Weisskopf effect in nuclei.
Keywords: 
Subject: Physical Sciences  -   Atomic and Molecular Physics

1. Introduction

Muons have since their discovery in 1936 been extensively studied and have been used in the study of nuclear properties[1] and different applications, for example, muon spin spectroscopy [2] and muon tomography [3]. Since the muon is in many aspects similar to the electron, with the same charge but ≈ 207 times heavier, this means that when a muon comes close to an atom it might be captured and form a hydrogen-like muonic atom. The lifetime of the muon ( 2.2 μ s ) is long enough for being able to cascade down to the lowest orbit and can be considered a stable particle in the atomic timescale. This will give rise to large effects on the energy levels due to the extension of the nucleus, i.e. the distribution of charge and magnetisation, especially for heavier nuclei, as well as relativistic effects. As a muonic atom can be considered to be hydrogen-like it is possible to do high-accuracy calculations of the level structure which in combination with experimental results can give information on different nuclear parameters like nuclear charge radii [4,5], quadrupole moments [6] and magnetic dipole interaction constants [7]. In this paper, we focus on the use of muons to probe the nucleus and more specifically the distribution of magnetisation. The finite-nucleus effect of magnetisation or the Bohr-Weisskopf (BW) effect [8,9] is essential in the determination of accurate values of the nuclear magnetic dipole moment if it cannot be determined directly but rather through the hyperfine structure (hfs) as is the case in short-lived isotopes [10]. Knowledge of the distribution of magnetisation is also important in studies of QED effects as well as studies of atomic parity violation [11], where there is a need for high-quality modelling of the atomic wave functions within the nucleus. This means that the nuclear models used must take both the charge and magnetisation distribution into account. Since the magnetic hyperfine interaction splitting in the 2 p 1 / 2 − 1 s 1 / 2 transitions in heavy muonic atoms are in the same order of magnitude as the energy resolution in the detectors used, the splitting has to be determined from the line broadening. This line shape analysis is complicated and requires the use of simultaneously recorded calibration lines. The magnetic hyperfine splitting in transitions between higher-lying states is usually too small to be detected, why the 2 p 1 / 2 − 1 s 1 / 2 transitions are the only option. Due to the experimental difficulties, there does not exist many measurements of the magnetic hyperfine structure and the majority were done during a rather short period during the 1960s and 1970s. The interest in measuring the nuclear charge radii continued and indicate that measurements of the magnetic hyperfine structure are possible, with the muX project [12] as an example. There have been published at least two reviews [4,13] with the last in 1984, why it is justified with an updated review.
The published experimental A constants are presented in Table 1. It should be noted that the A constants in previous reviews [4,13] were defined in different ways. In this paper we follow the normal definition of the A constant, also used in [13]:
A = Δ E F
with Δ E being the magnetic hfs splitting and F = I + 1 2 , where I being the nuclear spin. The definition used in [4] is:
A = Δ E 2 I + 1 I
Something that might cause some confusion.
Even if the number of measurements is quite small, they might still give valuable information on the quality of the different nuclear models used to calculate the BW-effect. However, we will argue that there is a need for more measurements, but also systematic calculations with different nuclear models to expand calculations to atomic systems[14].

2. Hyperfine Structure in Muonic Atoms

The hyperfine structure in muonic atoms is due to the electric quadruple (E2) and magnetic dipole (M1) interactions. The electric quadrupole moment depends on the effective shape of the nucleus while the magnetic dipole interaction depends on the magnetic dipole moment of the nucleus. Since the magnetic dipole moment of the muon is ≈ 207 times smaller than the magnetic moment of the electron, the magnetic dipole interaction is normally two orders of magnitude smaller than the quadrupole interaction. This implies that the magnetic part may only be studied in cases where the total angular momenta are 1/2 for either I or J, making the transitions 2 p 1 / 2 , 3 / 2 → 1 s 1 / 2 and the 1 s 1 / 2 state best suited for studies. Considering that the transition energies are in the MeV region and the A constants are in the keV region, the experimental difficulties are quite challenging. This can in some respect explain why the technique fell out of fashion, especially with the possibility to do measurements on hydrogen-like highly charged ions. The difficulty is also evident from the errors in Table 1. Based on the available data We have done an assessment and present recommended values for the experimental A constants in Table 2

3. Hyperfine Anomaly in Muonic Atoms

For the low-lying states, the point-like nucleus is not a good approximation and the finite size must be taken into account in calculations of the hyperfine structure. This means the relativistic Dirac equation must be solved with a reasonable nuclear potential. The effect of an extended charge distribution is normally referred to as the Breit-Rosenthal (BR) effect [30] and is calculated with different models. The most common charge distribution used is the two-parameter Fermi distribution given as
ρ ( r ) = ρ 0 1 + e x p ( 4 l n 3 r − c t ) − 1
where t is the skin thickness and c is the half-density radius of the nuclei. This distribution is normally not a good approximation for deformed nuclei where a multipole expansion should be used. The distribution of magnetisation or BW-effect is normally extracted from experimental data and calculations of the A constant for a distributed charge (including the BR-effect) and an assumed point-like dipole ( A 0 , p . d . ). The experimental A constant is thus described as
A e x p = A 0 ( 1 + ϵ B R ) ( 1 + ϵ B W ) = A 0 , p . d . ( 1 + ϵ B W )
where the BW-effect should be independent of the charge distribution used for the BR-effect. However, when extracting the BW-effect from experimental data, the values will depend on the calculations why it is important to have both a reasonable charge distribution and high-quality calculations. To obtain the experimental BW-effect ( ϵ B W , e x p ) one divides the experimental A constant with the calculated point dipole A constant ( A 0 , p . d . ).
ϵ B W , e x p = A e x p A 0 , p . d . − 1
As this value depends on the calculation it is important that this is done with high precision and that the magnetic dipole moment, charge distribution and parameters used are presented in detail. In Table 3 the experimental BW-effect is presented with different calculations of the point dipole A constant and the values of the magnetic dipole moment used. As the calculated point magnetic hyperfine A constant ( A 0 , p . d . ) differs in the original papers one obtains slightly different values of the BW-effect.

4. Calculations of the BW-Effect

There exist several different models for calculating the BW-effect, see [13] for an overview. One would expect that muonic atoms are more sensitive to the distribution of magnetisation and relatively easy to do calculations on, why muonic atoms are expected to be better to use in comparing different models. In the case of muonic atoms, only a few different models have been used in cases where there exist experimental values. It is therefore not possible to check how good the different models are at reproducing the BW-effect. The paper of Johnson and Sorensen [22] used different versions of the configuration mixing model while Fujita and Arima [31] applied their microscopic model. Recently, Sanamyan et al. [11] applied both a uniform magnetisation distribution and single-particle models to the 133 C s , 203 T l , 205 T l and 209 B i muonic atoms to find the BW contribution to the A constant. As the calculations of Sanamyan[11] are the most recent, these isotopes are presented separately in Table 4 from the isotopes with only older calculations that are presented in Table 3.
It should be noted that the BW-effect has been obtained with the calculated point dipole A 0 constant given in the original paper. As these values differ both should be considered in the comparison. As the experimental errors are quite large most calculated values with the models are within the error bars. As can be seen in Table 3 there are some exceptions where the models fail to reproduce the BW-effect. In E u the nucleus is deformed which might explain the deviation. In the case of I r , it is not entirely clear why. Still, one should observe that it seems to be connected to the low value of the nuclear spin indicating that the distribution of magnetisation might be significantly different than for high spin nuclei. When considering the recent calculations with older calculations and experiments in Table 4 the trend of difficulties for low spin continues. In the case of Tl (I=1/2), the single-particle and homogeneous magnetisation model is equally good. Since the homogeneous magnetisation (ball) model fails for C s and I r it is clear that this model is not appropriate for calculations of the BW-effect, this has also been observed in atoms [32]. However, it would be interesting to see how well a shell magnetisation can reproduce the BW-effect in muonic atoms and maybe replace the homogeneous magnetisation model as a simplistic model to use. The single-particle model seems to reproduce the BW-effect reasonably well, with some possible issues for low-spin heavy nuclei, therefore, it would be interesting to perform systematic calculations for all measured nuclei to assess the model in detail. It is also possible to conclude that the calculated point dipole A 0 constant differs in the published papers, why it might also be important to perform systematic calculations with up-to-date programs. It is also clear that the simple model of a uniform magnetisation distribution is too crude to be used for muonic atoms, possible with a few exceptions. As the single-particle model seem to be the best to describe the BW-effect systematic calculations in muonic atoms will also be valuable.

5. Discussion

This review of measurements and theoretical calculations of the BW-effect in muonic atoms has unearthed the need for renewed systematic studies of the magnetic hyperfine interaction in the 1 s 1 / 2 states in muonic atoms. As the use of muonic atoms opens the possibility to obtain values of the BW-effect directly, new or improved measurements will make it possible to compare different nuclear models for the distribution of magnetisation. The muX project at PSI [12] is such a project which might be able to do measurements in the lead and trans-lead regions. In addition to the need for systematic measurements, there exists a need for high-precision calculations of the point dipole A 0 constant as illustrated by the difference in the published values presented in Table 3 and Table 4, this will also give more certain values of the BW-effect ( ϵ ). To find which of the nuclear models of magnetisation is best, the experimental values of the BW-effect should be compared with the values calculated using different models. This will possibly give a better comparison as one would expect that the muonic BW-effect is more sensitive to the distribution of magnetisation than electronic atoms. As of today, systematic calculations in muonic atoms only exist for the configuration mixing model [22] and the microscopic model [31] with a few exceptions [11]. It is important to do systematic calculations using other models to find which models are valid and for which isotopes. This is especially important in the case of high Z and low spin isotopes, where the models seem inadequate.

Funding

This research received no external funding

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Knecht, A.; Skawran, A.; Vogiatzi, S.M. Study of nuclear properties with muonic atoms. The European Physical Journal Plus 2020, 135, 1–18. [Google Scholar] [CrossRef]
  2. Blundell, S.J.; De Renzi, R.; Lancaster, T.; Pratt, F.L. Muon Spectroscopy: An Introduction; Oxford University Press, 2021.
  3. Alvarez, L.W.; Anderson, J.A.; Bedwei, F.E.; Burkhard, J.; Fakhry, A.M.; Girgis, A.H.; Goneid, A.; Hassan, F.; Iverson, D.; Lynch, G.; et al. Search for hidden chambers in the pyramids. Science 1970, 167 3919, 832–9. [Google Scholar] [CrossRef]
  4. Engfer, R.; Schneuwly, H.; Vuilleumier, J.; Walter, H.; Zehnder, A. Charge-distribution parameters, isotope shifts, isomer shifts, and magnetic hyperfine constants from muonic atoms. Atomic data and nuclear data tables 1974, 14, 509–597. [Google Scholar] [CrossRef]
  5. Piller, C.; Gugler, C.; Jacot-Guillarmod, R.; Schaller, L.; Schellenberg, L.; Schneuwly, H.; Fricke, G.; Hennemann, T.; Herberz, J. Nuclear charge radii of the tin isotopes from muonic atoms. Physical Review C 1990, 42, 182. [Google Scholar] [CrossRef] [PubMed]
  6. Tanaka, Y.; Laubacher, D.; Steffen, R.; Shera, E.; Wohlfahrt, H.; Hoehn, M. Ground-state quadrupole moments of 155, 157Gd determined with muonic X-rays. Physics Letters B 1982, 108, 8–10. [Google Scholar] [CrossRef]
  7. Rüetschi, A.; Schellenberg, L.; Phan, T.; Piller, G.; Schaller, L.; Schneuwly, H. Magnetic hyperfine structure in muonic 209Bi. Nuclear Physics A 1984, 422, 461–479. [Google Scholar] [CrossRef]
  8. Bohr, A. Nuclear magnetic moments and atomic hyperfine structure. Phys. Rev. 1951, 81, 331. [Google Scholar] [CrossRef]
  9. Bohr, V.W.A. The influence of nuclear structure on the hyperfine structure of heavy elements. Phys. Rev. 1950, 77, 94. [Google Scholar] [CrossRef]
  10. Persson, J.R. Table of hyperfine anomaly in atomic systems. Atomic Data and Nuclear Data Tables 2013, 99, 62–68. [Google Scholar] [CrossRef]
  11. Sanamyan, G.; Roberts, B.; Ginges, J. Empirical determination of the Bohr-Weisskopf effect in cesium and improved tests of precision atomic theory in searches for new physics. Physical Review Letters 2023, 130, 053001. [Google Scholar] [CrossRef]
  12. Wauters, F.; Knecht, A. The muX project. SciPost Physics Proceedings 2021, p. 022.
  13. Büttgenbach, S. Magnetic hyperfine anomalies. Hyperfine Interactions 1984 20:1 1984, 20, 1–64. [Google Scholar] [CrossRef]
  14. Jönsson, P.; Godefroid, M.; Gaigalas, G.; Ekman, J.; Grumer, J.; Li, W.; Li, J.; Brage, T.; Grant, I.P.; Bieroń, J.; et al. An introduction to relativistic theory as implemented in GRASP. Atoms 2022, 11, 7. [Google Scholar] [CrossRef]
  15. Cheng, S.; Chen, M.; Kast, J.; Lee, W.; Macagno, E.; Rushton, A.; Wu, C. Magnetic dipole hyperfine structure in muonic atoms. Physics Letters B 1971, 34, 615–617. [Google Scholar] [CrossRef]
  16. Povel, H. M1 and E2 hyperfine effects in muonic 9341Nb. Nuclear Physics A 1973, 217, 573–588. [Google Scholar] [CrossRef]
  17. Lee, W.; Bernow, S.; Chen, M.; Cheng, S.; Hitlin, D.; Kast, J.; Macagno, E.; Rushton, A.; Wu, C.; Budick, B. Finite Distributions of Nuclear M 1 and E 2 Moments in Muonic In 115, Cs 133, and Pr 141. Physical Review Letters 1969, 23, 648. [Google Scholar] [CrossRef]
  18. Link, R.; Backe, N.; Engfer, R.; Fluri, L.; Michaelsen, R.; Petitjean, C.; Schneuwly, H.; Walter, H. Magnetic hyperfine splitting of the 2s12- 2p12 transition in muonic 115In. Physics Letters B 1971, 37, 50–52. [Google Scholar] [CrossRef]
  19. Link, R. Magnetic hyperfine interaction in muonic indium, osmium and mercury. Zeitschrift für Physik 1974, 269, 163–171. [Google Scholar] [CrossRef]
  20. Lee, W.; Bernow, S.; Chen, M.; Cheng, S.; Hitlin, D.; Kast, J.; Macagno, E.; Rushton, A.; Wu, C. The resonance process and the intensity anomaly in muonic 127I. Nuclear Physics A 1971, 167, 652–666. [Google Scholar] [CrossRef]
  21. Thompson, A.C. The Electromagnetic Sizes and Shapes of Barium-136,137,138, LANTHANUM-139, CERIUM-140,142, Praseodymium-141 and NEODYMIUM-142 Nuclei from Muonic X-Rays. Ph. D. Thesis 1969. [Google Scholar]
  22. Johnson, J.; Thompson, A.; Adler, D.; Sorensen, R.; Sutton, R. Magnetic hyperfine structure of the muonic X-rays of 141Pr. Physics Letters B 1969, 29, 420–422. [Google Scholar] [CrossRef]
  23. Carrigan Jr, R.; Gupta, P.; Sutton, R.; Suzuki, M.; Thompson, A.; Coté, R.; Prestwich, W.; Gaigalas, A.; Raboy, S. Muonic X Rays from Separated Isotopes of Europium. Physical Review Letters 1968, 20, 874. [Google Scholar] [CrossRef]
  24. Link, R.; Schellenberg, L.; Backe, H.; Engfer, R.; Kankeleit, E.; Michaelsen, R.; Schneuwly, H.; Schröder, W.; Vuilleumier, J.; Walter, H.; et al. Magnetic hyperfine interaction of the 2+ rotational states in muonic 190,192 Os. Physics Letters B 1972, 42, 57–59. [Google Scholar] [CrossRef]
  25. Backe, H.; Engfer, R.; Kankeleit, E.; Link, R.; Michaelsen, R.; Petitjean, C.; Schellenberg, L.; Schneuwly, H.; Schröder, W.; Vuilleumier, J.; et al. Nuclear excitation and isomer shifts in muonic atoms:(i). experiment and evaluation. Nuclear Physics A 1974, 234, 469–503. [Google Scholar] [CrossRef]
  26. Backe, H.; Engfer, R.; Jahnke, U.; Kankeleit, E.; Pearce, R.; Petitjean, C.; Schellenberg, L.; Schneuwly, H.; Schröder, W.; Walter, H.; et al. Study of X-rays and nuclear γ-rays in muonic thallium. Nuclear Physics A 1972, 189, 472–512. [Google Scholar] [CrossRef]
  27. Powers, R.J. Hyperfine Effects in Muonic Bi and Pb. Physical Review 1968, 169, 1. [Google Scholar] [CrossRef]
  28. Carrigan, R.; Gupta, P.; Sutton, R.; Suzuki, M.; Thompson, A.; Cote, R.; WV, P.; Gaigalas, A.; Raboy, S. HYPERFINE EFFECTS IN MUONIC X RAYS OF BISMUTH. In Proceedings of the BULLETIN OF THE AMERICAN PHYSICAL SOCIETY. AMER INST PHYSICS CIRCULATION FULFILLMENT DIV, 500 SUNNYSIDE BLVD, WOODBURY …, 1968, Vol. 13, p. 65.
  29. Lee, W.; Chen, M.; Cheng, S.; Macagno, E.; Rushton, A.; Wu, C. Nuclear excitation in muonic 209Bi. Nuclear Physics A 1972, 181, 14–24. [Google Scholar] [CrossRef]
  30. Rosenthal, J.E.; Breit, G. The Isotope Shift in Hyperfine Structure. Physical Review 1932, 41, 459. [Google Scholar] [CrossRef]
  31. Fujita, T.; Arima, A. Magnetic hyperfine structure of muonic and electronic atoms. Nuclear Physics A 1975, 254, 513–541. [Google Scholar] [CrossRef]
  32. Roberts, B.; Ginges, J. Hyperfine anomaly in heavy atoms and its role in precision atomic searches for new physics. Physical Review A 2021, 104, 022823. [Google Scholar] [CrossRef]
Table 1. Experimental values of the A constant in the muonic 1 s 1 / 2 states.
Table 1. Experimental values of the A constant in the muonic 1 s 1 / 2 states.
Isotope Nuclear spin I A constant (keV) reference
93 N b 9/2 0.716(49) [15]
0.693(22) [16]
115 I n 9/2 0.733(32) [17]
0.716(14) [18,19]
127 I 5/2 0.712(72) [20]
133 C s 7/2 0.634(103) [17]
139 L a 7/2 0.691(52) [21]
0.771(46) [15]
141 P r 5/2 1.22(13) [17]
1.216(48) [22]
151 E u 5/2 0.64(22) [23]
190 O s 2 0.266(32) [24]
191 I r 3/2 0.44(8) [19]
191 m I r 5/2 0.584(64) [19]
192 O s 2 0.320(32) [19]
193 I r 3/2 0.307(54) [19]
193 m I r 5/2 0.416(48) [19]
199 H g 1/2 0.47(12) [19]
200 H g 2 0.262(42) [19]
203 T l 1/2 2.340(80) [25]
205 T l 1/2 2.300(32) [26]
2.44(12) [15]
209 B i 9/2 1.11(23) [27]
0.933(90) [28]
0.96(7) [29]
0.888(30) [7]
209 m B i 13/2 0.414(59) [7]
209 m B i 9/2 0.420(77) [7]
209 m B i 15/2 0.300(70) [7]
Table 2. Recommended values of the A constant in the muonic 1 s 1 / 2 states.
Table 2. Recommended values of the A constant in the muonic 1 s 1 / 2 states.
Isotope Nuclear spin I A constant (keV)
93 N b 9/2 0.697(20)
115 I n 9/2 0.719(13)
127 I 5/2 0.712(72)
133 C s 7/2 0.634(103)
139 L a 7/2 0.736(40)
141 P r 5/2 1.216(45)
151 E u 5/2 0.64(22)
190 O s 2 0.266(32)
191 I r 3/2 0.44(8)
191 m I r 5/2 0.584(64)
192 O s 2 0.320(32)
193 I r 3/2 0.307(54)
193 m I r 5/2 0.416(48)
199 H g 1/2 0.47(12)
200 H g 2 0.262(42)
203 T l 1/2 2.340(80)
205 T l 1/2 2.309(35)
209 B i 9/2 0.905(27)
209 m B i 13/2 0.414(59)
209 m B i 9/2 0.420(77)
209 m B i 15/2 0.300(70)
Table 3. Experimental and calculated values of the Bohr-Weisskopf effect ϵ in the muonic 1 s 1 / 2 states. CM:configuration mixing model; PM:Pairing model; PQM:Pairing plus quadrupole; MT: Microscopic Model I and II.
Table 3. Experimental and calculated values of the Bohr-Weisskopf effect ϵ in the muonic 1 s 1 / 2 states. CM:configuration mixing model; PM:Pairing model; PQM:Pairing plus quadrupole; MT: Microscopic Model I and II.
Isotope I μ N A 0 ϵ e x p CM PM PQM MT I MT II ref.
93 N b 9/2 6.1671 1.031 -0.324(30) -0.362 -0.362 -0.332     [22]
1.013 -0.312(30)       -0.301 -0.345 [31]
115 I n 9/2 5.5351 1.156 -0.378(18) -0.392 -0.392 -0.362     [22]
5.5348 1.147 -0.373(18)       -0.335 -0.384 [31]
127 I 5/2 2.8091 1.120 -0.364(101) -0.329 -0.336 -0.207     [22]
1.145 -0.378(101)       -0.333 -0.368 [31]
133 C s 7/2 2.5789 0.766 -0.172(162) -0.097 -0.119 -0.097     [22]
0.780 -0.187(162)       -0.131 -0.163 [31]
139 L a 7/2 2.7781 0.857 -0.141(54) -0.140 -0.133 -0.133     [22]
0.872 -0.156(54)       -0.154 -0.189 [31]
141 P r 5/2 4.28 1.928 -0.369(37) -0.390 -0.390 -0.386     [22]
4.16 1.971 -0.383(37)       -0.366 -0.403 [31]
151 E u 5/2 3.463 1.632 -0.608(334) -0.397 -0.392 -0.299     [22]
1.707 -0.625(334)       -0.375 -0.413 [31]
190 O s 2 0.68 0.474 -0.439(120)       -0.370 -0.426 [31]
191 I r 3/2 0.1454 0.140 2.143(182)       2.910 3.058 [31]
191 m I r 5/2 0.5 0.337 0.733(82)       -0.246 -0.280 [31]
192 O s 2 0.78 0.567 -0.436(100)       -0.371 -0.426 [31]
193 I r 3/2 0.468 0.152 1.020(176)       2.630 2.765 [31]
193 m I r 5/2 0.6 0.422 -0.014(115)       -0.286 -0.320 [31]
199 H g 1/2 0.49787 1.503 -0.687(255)       -0.811 -0.838 [31]
200 H g 2 0.9 0.321 -0.184(160)       -0.302 -0.338 [31]
203 T l 1/2 1.61169 4.760 -0.508(34) -0.355 -0.350 -0.347   [22]
1.6115 4.846 -0.517(34)       -0.419 -0.451 [31]
205 T l 1/2 1.62754 4.760 -0.515(15) -0.350 -0.345 -0.345     [22]
1.6274 4.868 -0.526(15)       -0.425 -0.459 [31]
209 B i 9/2 4.0794 1.351 -0.330(30) -0.322         [22]
4.08 1.385 -0.347(30)       -0.266 -0.314 [31]
Table 4. Experimental and calculated values of the Bohr-Weisskopf effect ϵ in the muonic 1 s 1 / 2 states. CM:configuration mixing model; PM:Pairing model; PQM:Pairing plus quadrupole; MT: Microscopic Model I and II; Ball:Homogeneous; SP:single-particle; SP-WS: Single-particle with Wood-Saxon.
Table 4. Experimental and calculated values of the Bohr-Weisskopf effect ϵ in the muonic 1 s 1 / 2 states. CM:configuration mixing model; PM:Pairing model; PQM:Pairing plus quadrupole; MT: Microscopic Model I and II; Ball:Homogeneous; SP:single-particle; SP-WS: Single-particle with Wood-Saxon.
Isotope A 0 ϵ e x p CM PM PQM MT I MT II Ball SP SP-WS ref.
133 C s 0.766 -0.172(162) -0.097 -0.119 -0.097     [22]
0.780 -0.187(162)   -0.131 -0.163 [31]
0.762 -0.172(162) -0.413 -0.155 -0.155 [11]
203 T l 4.760 -0.508(34) -0.355 -0.350 -0.347   [22]
4.846 -0.517(34)       -0.419 -0.451 [31]
4.712 -0.503(34) -0.527 -0.527 -0.439 [11]
205 T l 4.760 -0.515(15) -0.350 -0.345 -0.345     [22]
4.868 -0.526(15)       -0.425 -0.459 [31]
4.744 -0.513(15) -0.527 -0.527 -0.439 [11]
209 B i 1.351 -0.330(30) -0.322         [22]
1.385 -0.347(30)       -0.266 -0.314 [31]
1.339 -0.324(30) -0.533 -0.310 -0.347 [11]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

facebook logotwitter logolinkedin logo
MDPI Initiatives
Important Links
Subscribe

Disclaimer

Terms of Use

Privacy Policy

© 2024 MDPI (Basel, Switzerland) unless otherwise stated