pith. sign in

arxiv: 1906.08465 · v1 · pith:MQTK7P4Snew · submitted 2019-06-20 · ⚛️ physics.atom-ph

Reevaluation of the nuclear electric quadrupole moment for 87Sr by hyperfine structures and relativistic atomic theory

Benquan Lu , Tingxian Zhang , Hong Chang , Jiguang Li , Yong Wu , Jianguo Wang This is my paper

Pith reviewed 2026-05-25 19:12 UTC · model grok-4.3

classification ⚛️ physics.atom-ph
keywords nuclear quadrupole moment87Srhyperfine structureelectric field gradientDirac-Hartree-Fockatomic spectraelectron correlation
0
0 comments X

The pith

The nuclear electric quadrupole moment of 87Sr is 328(4) millibarns from hyperfine structures and atomic calculations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a revised value for the nuclear electric quadrupole moment of strontium-87 by computing the electric field gradient at the nucleus in the 5s5p 3P1,2 states of the neutral atom. Calculations in the multi-configuration Dirac-Hartree-Fock framework account for electron correlations to limit uncertainty in the gradient to roughly one percent. These theoretical gradients are combined with measured hyperfine constants to extract Q(87Sr) = 328(4) mb. The result differs from the current recommended value but matches earlier theoretical work, leading the authors to propose it as the new reference. A sympathetic reader would care because nuclear quadrupole moments enter interpretations of atomic spectra, nuclear structure tests, and precision measurements involving strontium.

Core claim

Using the multi-configuration Dirac-Hartree-Fock method with systematic inclusion of electron correlations, the electric field gradients at the strontium nucleus are obtained for the 5s5p 3P1,2 states; these gradients together with experimental hyperfine structure data yield Q(87Sr) = 328(4) mb, a value recommended as the reference for 87Sr over the existing tabulated figure.

What carries the argument

Multi-configuration Dirac-Hartree-Fock computation of the electric field gradient produced by electrons at the nucleus, with electron correlations included to control uncertainty near the one-percent level.

If this is right

  • Analyses of strontium hyperfine data should adopt the new quadrupole moment instead of the prior recommended value.
  • The multi-configuration Dirac-Hartree-Fock approach achieves roughly one-percent accuracy for electric field gradients in these states when correlations are treated systematically.
  • Nuclear data tables for 87Sr require updating to reflect the revised moment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same computational strategy could be applied to extract quadrupole moments for other strontium isotopes or neighboring elements where data remain inconsistent.
  • Precision experiments that rely on the quadrupole moment, such as those in optical lattice clocks, would shift by several percent if the new value is adopted.
  • A direct laboratory measurement of the quadrupole moment through an independent technique such as Coulomb excitation would provide a decisive cross-check.

Load-bearing premise

The electric field gradients at the nucleus can be computed with uncertainties held to about one percent once electron correlations are systematically included in the multi-configuration Dirac-Hartree-Fock treatment of the 5s5p 3P1,2 states.

What would settle it

An independent experimental measurement of Q(87Sr) or a separate high-accuracy calculation of the electric field gradient that yields a value outside the stated 328(4) mb range would falsify the recommended reference value.

Figures

Figures reproduced from arXiv: 1906.08465 by Benquan Lu, Hong Chang, Jianguo Wang, Jiguang Li, Tingxian Zhang, Yong Wu.

Figure 1
Figure 1. Figure 1: FIG. 1. Contributions from occupied orbitals to the calcula [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Contributions from occupied orbitals to the calcula [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The values of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The absolute differences, [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

The values of nuclear electric quadrupole moment are different by about 7% for 87Sr nucleus between the recommended value [N. J. Stone, At. Data Nucl. Data Tables 111-112, 1 (2016); P. Pyykko, Mol. Phys. 116, 1328 (2018)] and earlier results [e.g. A. M. Matensson-Pendrill, J. Phys. B: At. Mol. Opt. Phys. 35, 917 (2002); K. Z. Yu et al., Phys. Rev. A 70, 012506 (2004)]. In this work, we reported a new value, Q(87Sr) = 328(4) mb, making use of our calculated electric field gradients produced by electrons at nucleus in combination with experimental values for hyperfine structures of the 5s5p 3P1,2 states of the neutral Sr atom. In the framework of the multi-configuration Dirac-Hartree-Fock theory, the electron correlations were taken into account systematically so as to control the uncertainties of the electric field gradient at about 1% level. The present result is different from the recommended value, but in excellent agreement with those by Matensson-Pendrill and Yu et al.. We would recommend the present Q value as a reference for 87Sr.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reports a reevaluation of the nuclear electric quadrupole moment Q(87Sr) = 328(4) mb. This value is obtained by combining experimental hyperfine constants A and B for the 5s5p 3P1,2 states of neutral Sr with electric field gradients (EFGs) computed via multi-configuration Dirac-Hartree-Fock (MCDHF) theory; the authors state that systematic inclusion of electron correlations controls the EFG uncertainty at the ~1% level, yielding a result 7% different from the Stone/Pyykkö recommendation but in agreement with earlier calculations by Matensson-Pendrill and Yu et al.

Significance. If the claimed 1% EFG uncertainty control is substantiated, the result would supply a more precise reference value for Q(87Sr) and help resolve the existing discrepancy with the recommended value, with direct utility for nuclear structure studies and atomic parity-violation experiments.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'electron correlations were taken into account systematically so as to control the uncertainties of the electric field gradient at about 1% level' is load-bearing for the quoted 4 mb uncertainty on Q(87Sr), yet no convergence tables, basis-set progression data, or explicit error budget for the EFG in the 5s5p 3P1,2 states are supplied to support this precision.
  2. [Computational Methods] Computational section: without quantitative demonstration (e.g., EFG values as successive correlation layers are added, or comparison of active-space sizes) that residual core-valence and higher-order relativistic contributions remain below 1%, the uncertainty assignment and the recommendation to adopt 328(4) mb as reference cannot be verified.
minor comments (1)
  1. The manuscript would benefit from an explicit table listing the final EFG values for both 3P1 and 3P2 states together with the experimental hyperfine B constants used to extract Q.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address the major points below and will revise the manuscript to strengthen the presentation of our uncertainty analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'electron correlations were taken into account systematically so as to control the uncertainties of the electric field gradient at about 1% level' is load-bearing for the quoted 4 mb uncertainty on Q(87Sr), yet no convergence tables, basis-set progression data, or explicit error budget for the EFG in the 5s5p 3P1,2 states are supplied to support this precision.

    Authors: We agree that the abstract claim requires explicit supporting data. In the revised manuscript we will insert convergence tables for the electric field gradients of the 5s5p 3P1,2 states, showing values obtained as successive correlation layers and active-space sizes are increased, together with a concise error budget that quantifies the residual contributions at the ~1% level. revision: yes

  2. Referee: [Computational Methods] Computational section: without quantitative demonstration (e.g., EFG values as successive correlation layers are added, or comparison of active-space sizes) that residual core-valence and higher-order relativistic contributions remain below 1%, the uncertainty assignment and the recommendation to adopt 328(4) mb as reference cannot be verified.

    Authors: We accept that the computational section must contain quantitative evidence. The revision will add tables that display the EFG progression with added correlation layers and enlarged active spaces, explicitly demonstrating that residual core-valence and higher-order relativistic contributions lie below 1%. These additions will allow independent verification of the quoted uncertainty and of the recommended Q(87Sr) value. revision: yes

Circularity Check

0 steps flagged

No circularity: Q derived from independent EFG computation and experimental hyperfine data

full rationale

The central result Q(87Sr) = 328(4) mb is obtained by dividing experimental hyperfine constants B for the 5s5p 3P1,2 states by an electric field gradient computed via multi-configuration Dirac-Hartree-Fock theory with systematic electron correlation inclusion. This is a standard division of measured quantity by independently calculated atomic property; the EFG calculation does not reference or reduce to the target Q value, nor does any equation in the paper make the result tautological. No self-citations are load-bearing for the derivation, and the 1% EFG uncertainty claim is an assertion about computational convergence rather than a definitional loop. The derivation is self-contained against external benchmarks (experiment + ab initio atomic theory).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the MCDHF method with systematic correlation inclusion yields electric field gradients accurate to 1%. No free parameters or invented entities are introduced in the abstract. Standard mathematical axioms of relativistic quantum mechanics are presupposed but not listed as novel.

axioms (1)
  • domain assumption Multi-configuration Dirac-Hartree-Fock theory with systematic inclusion of electron correlations controls uncertainties in the electric field gradient at the 1% level for the 5s5p 3P states of Sr.
    Directly invoked to justify the reported uncertainty on Q and the reliability of the new value.

pith-pipeline@v0.9.0 · 5805 in / 1420 out tokens · 24509 ms · 2026-05-25T19:12:30.821262+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Neyens, Rep

    G. Neyens, Rep. Prog. Phys. 66, 633 (2003)

    work page 2003

  2. [2]

    Heyde and J

    K. Heyde and J. L. Wood, Rev. Mod. Phys. 83, 1467 (2011)

    work page 2011

  3. [3]

    Campbell, I

    P. Campbell, I. D. Moore, and M. R. Pearson, Prog. Part. Nu cl. Phys. 86, 127 (2016)

    work page 2016

  4. [4]

    Lievens, R

    P. Lievens, R. E. Silverans, L. Vermeeren, W. Borchers, W . Neu, R. Neugart, K. Wendt, F. Buchinger, and E. Arnold, Phys. Lett. B 256, 141 (1991)

    work page 1991

  5. [5]

    J. Park, A. B. Garnsworthy, R. Kr¨ ucken, C. Andreoiu, G. C . Ball, P. C. Bender, A. Chester, A. Close, P. Finlay, P. E. Garrett, J. Glister, G. Hackman, B. Hadinia, K. G. Leach, E. T. Rand, S. Sjue, K. Starosta, C. E. Svensson, and E. Tardiff, Phys . Rev. C 93, 014315 (2016)

    work page 2016

  6. [6]

    Cl´ ement et al

    E. Cl´ ement et al. , Phys. Rev. Lett. 116, 022701 (2016)

    work page 2016

  7. [7]

    Buchinger, E

    F. Buchinger, E. B. Ramsay, E. Arnold, W. Neu, R. Neugart, K. Wendt, R. E. Silverans, P. Lievens, L. Vermeeren, D. Berdichevsky, R. Fleming, D. W. L. Sprung, and G. Ulm, Phys. Rev. C 41, 2883 (1990)

    work page 1990

  8. [8]

    zu Putlitz, Z

    G. zu Putlitz, Z. Phys. 175, 543 (1963)

    work page 1963

  9. [9]

    S. M. Heider and G. O. Brink, Phys. Rev. A 16, 1371 (1977)

    work page 1977

  10. [10]

    A. M. M ˚ artensson-Pendrill, J. Phys. B: At. Mol. Opt. Ph ys. 35, 917 (2002)

    work page 2002

  11. [11]

    K. Z. Yu, L. J. Wu, B. C. Gou, and T. Y. Shi, Phys. Rev. A 70, 012506 (2004)

    work page 2004

  12. [12]

    B. K. Sahoo, Phys. Rev. A 73, 062501 (2006)

    work page 2006

  13. [13]

    I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules (Springer, New York, 2007) 15 p. 424

    work page 2007

  14. [14]

    C. F. Fischer, M. Godefroid, T. Brage, P. J¨ onsson, and G . Gaigalas, J. Phys. B: At. Mol. Opt. Phys. 49, 182004 (2016)

    work page 2016

  15. [15]

    Beloy, A

    K. Beloy, A. Derevianko, and W. R. Johnson, Phys. Rev. A 77, 012512 (2008)

    work page 2008

  16. [16]

    A. K. Singh, D. Angom, and V. Natarajan, Phys. Rev. A 87, 012512 (2013)

    work page 2013

  17. [17]

    Biero´ n, P

    J. Biero´ n, P. Pyykk¨ o, D. Sundholm, V. Kell¨ o, and A. J. Sadlej, Phys. Rev. A 64, 052507 (2001)

    work page 2001

  18. [18]

    J. G. Li, M. Godefroid, and J. G. Wang, J. Phys. B: At. Mol. Opt. Phys. 49, 115002 (2016)

    work page 2016

  19. [19]

    B. O. Roos, P. R. Taylor, and P. E. M. Siegbahn, Chem. Phys . Lett. 48, 157 (1980)

    work page 1980

  20. [20]

    Olsen, B

    J. Olsen, B. O. Roos, P. Jorgensen, and J. A. Jensen, J. Ch em. Phys. 89, 2185 (1988)

    work page 1988

  21. [21]

    C. F. Fischer, T. Brage, and P. J¨ onsson, Computational Atomic Structure An MCHF Approach (Institute of Physics Publishing, London, 1997) p. 90

    work page 1997

  22. [22]

    J. G. Li, P. J¨ onsson, M. Godefroid, C. Z. Dong, and G. Gai galas, Phys. Rev. A 86, 052523 (2012)

    work page 2012

  23. [23]

    J¨ onsson, X

    P. J¨ onsson, X. He, C. F. Fischer, and I. Grant, Comput. P hys. Commun. 177, 597 (2007)

    work page 2007

  24. [24]

    Verdebout, P

    S. Verdebout, P. Jnsson, G. Gaigalas, M. Godefroid, and C. F. Fischer, J. Phys. B: At. Mol. Opt. Phys. 43, 074017 (2010)

    work page 2010

  25. [25]

    Verdebout, P

    S. Verdebout, P. Rynkun, P. Jnsson, G. Gaigalas, C. F. Fi scher, and M. Godefroid, J. Phys. B: At. Mol. Opt. Phys. 46, 085003 (2013)

    work page 2013

  26. [26]

    Froese Fischer, S

    C. Froese Fischer, S. Verdebout, M. Godefroid, P. Rynku n, P. J¨ onsson, and G. Gaigalas, Phys. Rev. A 88, 062506 (2013)

    work page 2013

  27. [27]

    J¨ onsson, G

    P. J¨ onsson, G. Gaigalas, J. Biero´ n, C. F. Fischer, and I. Grant, Comput. Phys. Commun. 184, 2197 (2013)

    work page 2013

  28. [28]

    Biero´ n, C

    J. Biero´ n, C. F. Fischer, P. J¨ onsson, and P. Pyykk¨ o, J. Phys. B: At. Mol. Opt. Phys. 41, 115002 (2008)

    work page 2008

  29. [29]

    T. X. Zhang, L. Y. Xie, J. G. Li, and Z. H. Lu, Phys. Rev. A 96, 012514 (2017)

    work page 2017

  30. [30]

    Santra, K

    R. Santra, K. V. Christ, and C. H. Greene, Phys. Rev. A 69, 042510 (2004)

    work page 2004

  31. [31]

    S. G. Porsev and A. Derevianko, Phys. Rev. A 69, 042506 (2004)

    work page 2004

  32. [32]

    M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. Blatt, T. Zano n-Willette, S. M. Foreman, and J. Ye, Phy. Rev. A 76, 022510 (2007)

    work page 2007

  33. [33]

    H. J. Kluge and H. Sauter, Z. Phys. 270, 295 (1974). 16

    work page 1974

  34. [34]

    B. A. Bushaw and W. N¨ ortersh¨ auser, Spectrochim. Acta, Part B: At. Spectrosc. 55, 1679 (2000)

    work page 2000

  35. [35]

    Biero´ n, P

    J. Biero´ n, P. Pyykk¨ o, and P. J¨ onsson, Phys. Rev. A71, 012502 (2005)

    work page 2005

  36. [36]

    Biero´ n and P

    J. Biero´ n and P. Pyykk¨ o, Phys. Rev. Lett. 87, 133003 (2001)

    work page 2001

  37. [37]

    N. J. Stone, At. Data Nucl. Data Tables 111-112, 1 (2016)

    work page 2016

  38. [38]

    Pyykk¨ o, Mol

    P. Pyykk¨ o, Mol. Phys. 106, 1965 (2008)

    work page 1965

  39. [39]

    Pyykk¨ o, Mol

    P. Pyykk¨ o, Mol. Phys. 116, 1328 (2018)

    work page 2018

  40. [40]

    Grundevik, M

    P. Grundevik, M. Gustavsson, I. Lindgren, G. Olsson, T. Olsson, and A. Ros´ en, Z. Phys. A 311, 143 (1983)

    work page 1983

  41. [41]

    B. A. Bushaw, H. J. Kluge, J. Lantzsch, R. Schwalbach, J. Stenner, H. Stevens, K. Wendt, and K. Zimmer, Z. Phys. D 28, 275 (1993)

    work page 1993

  42. [42]

    Stellmer and F

    S. Stellmer and F. Schreck, Phys. Rev. A 90, 022512 (2014). 17

    work page 2014