Pith. sign in

REVIEW 3 major objections 6 minor 62 references

Ab initio Calculations of Electric Dipole Polarizabilities in the Li, Na and K Atoms

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Relativistic coupled-cluster theory supplies ab initio scalar and tensor polarizabilities for six low-lying states each of Li, Na, and K.

desk verdict A useful systematic RCCSD polarizability table for three alkalis, but the quoted error bars don't cover the D-state discrepancies the paper itself shows, so the 'accurate' claim is overstated. read the letter →

arxiv 2502.10312 v2 pith:37PRLWQL submitted 2025-02-14 physics.atom-ph

A. Chakraborty , B. K. Sahoo This is my paper · ORCID
classification physics.atom-ph
keywords electricdipolepolarizabilityrelativisticcoupled-clustertheorylinearresponsealkaliatomslithiumsodiumpotassiumscalarandtensorpolarizabilities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that a purely ab initio relativistic coupled-cluster calculation, using no experimental energy input, can determine the static electric dipole polarizabilities of lithium, sodium, and potassium for the ground state and five low-lying excited states. The claim matters because these quantities control Stark shifts and blackbody-radiation shifts in atomic clocks and cold-atom experiments, and most earlier excited-state values relied on semi-empirical sum-over-states input with core and continuum contributions added separately. Across all four methods used, the authors find that pair-correlation effects, not core polarization, dominate the correlation corrections, and they quote recommended singles-and-doubles coupled-cluster values with uncertainties estimated from the leading triple-excitation contributions.

What carries the argument

The central object is the linear-response relativistic coupled-cluster singles-and-doubles (RCCSD) formula, which expresses the polarizability as $$\$\alpha$ = 2\,\frac{\langle\Phi_v|\{1+$S_v^{{\dagger}}$\}\,\bar{\tilde D}\,\{$T^{{(1)}}$(1+S_v)+$S_v^{{(1)}}$\}|\Phi_v\rangle}{\langle\Phi_v|\{$S_v^{{\dagger}}$+1\}\,\bar N\,\{1+S_v\}|\Phi_v\rangle},$$ with $\bar{\tilde D}=e^{T^{\dagger}}\tilde D e^{T}$ the dressed dipole operator. Here $\tilde D$ encodes the scalar and tensor angular factors from the reduced-matrix-element expression, so one calculation yields both $\alpha^S_d$ and $\alpha^T_d$. The machinery solves the first-order perturbation equations for the core ($T^{(1)}$) and valence ($S_v^{(1)}$) cluster amplitudes against the Dirac-Coulomb Hamiltonian, so that core, valence, and continuum intermediate states enter on an equal footing without a sum-over-states truncation. Triple excitations are not included in the wave operators; a selected set of leading triple diagrams is added perturbatively, and the difference from the RCCSD result is used as the uncertainty estimate.

What would settle it

Take the same RCCSD calculation, add full iterative triple excitations in a systematically enlarged Gaussian basis, and check whether the scalar polarizabilities move by more than the quoted uncertainties; a shift larger than the error bars would falsify the uncertainty prescription. A direct Stark-shift measurement of the Li 3D states, where this work gives about $-15003(16)$ a.u. for $3D_{3/2}$ versus $-14925(8)$ a.u. in an earlier high-precision calculation, would also decide whether the one-percent discrepancy is real.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the recommended values from the relativistic coupled-cluster singles-and-doubles (RCCSD) method in Table V are accurate ab initio scalar and tensor static dipole polarizabilities for the ground and five low-lying excited states of Li, Na, and K. The comparison across methods shows a consistent pattern: Dirac-Hartree-Fock results sit close to random-phase-approximation results, while third-order many-body perturbation theory results sit close to RCCSD results, which the paper reads as evidence that pair correlations, not core polarization, carry the important many-body physics for these atoms. The reported uncertainties come from comparing RCCSD with a perturbative treatment of the leading triple excitations, and the final values agree with available experiments for the ground and low-lying P states. For the Li 3D states, the paper's values differ from earlier sum-over-states results by about one percent, which it attributes to the more complete treatment of core and continuum intermediate states in the ab initio approach.

Load-bearing premise

The quoted error bars assume that the neglected electron-correlation terms, especially terms that move three electrons at once, plus basis-set incompleteness and small relativistic or QED effects, are all smaller than the estimated triple-excitation shift; if any of those neglected errors is comparable in size, the reported RCCSD values overstate their accuracy.

Editorial extensions

If this is right

  • The reported values give clock and cold-atom experiments direct ab initio input for Stark-shift and blackbody-radiation systematic corrections in Li, Na, and K.
  • For the Li $3D_{3/2,5/2}$ states, the paper predicts polarizabilities about one percent larger in magnitude than earlier sum-over-states values, so a future measurement can discriminate between the two treatments.
  • For the K $4P$ and $3D$ states, the calculation narrows the uncertainty far below existing experiments, giving specific targets for new measurements.
  • The method-level pattern (DHF close to RPA, MBPT(3) close to RCCSD) implies that any accurate alkali polarizability calculation must include pair correlations beyond the random-phase approximation; core polarization alone is not enough.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same linear-response RCC machinery should become more necessary for the heavier alkalis Rb, Cs, and Fr, where the paper's own core and core-valence contributions grow with atomic number and will be harder to treat by simplified methods.
  • Editorial inference: the paper's finding that core-valence contributions are negligible for light alkalis suggests that model potentials tuned to reproduce these polarizabilities can safely omit core-valence coupling for Li and Na, but not for K; this is a testable prediction for effective-potential builders.
  • Editorial inference: the uncertainty protocol, which uses only selected triple-excitation diagrams, could be stress-tested by computing the same quantities with a different basis set and with the Breit interaction; agreement would confirm the quoted error bars, while disagreement would show the error budget is too narrow.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript reports ab initio calculations of the static electric dipole scalar and tensor polarizabilities of six low-lying states of Li, Na, and K, using Dirac-Hartree-Fock (DHF), third-order many-body perturbation theory (MBPT(3)), random-phase approximation (RPA), and relativistic coupled-cluster singles and doubles (RCCSD). The RCCSD values are recommended and assigned uncertainties estimated from selected perturbative triple-excitation diagrams, and the final numbers are compared with earlier calculations and experiments. The paper also gives a term-by-term decomposition of core, core-valence, and valence contributions for each method.

Significance. If the quoted uncertainties were reliable, the recommended RCCSD values would be a useful systematic dataset for these three alkali-metal atoms: the calculations are genuinely ab initio, no parameters are fitted to polarizability data, and the comparison across DHF, MBPT(3), RPA, and RCCSD gives a clear picture of how correlation effects enter. The extensive tabulations of core, core-valence, and valence contributions, and of individual coupled-cluster terms, are a useful asset. However, the central accuracy claim is not supported for several D states because the quoted uncertainties are inconsistent with the paper's own comparisons to high-precision references. The uncertainty budget is incomplete (no basis-set convergence study and no estimate of several neglected contributions), so the reported 'accurate ab initio values' go beyond what the evidence establishes.

major comments (3)
  1. [§IV, Table V (Li 3D states)] The recommended Li 3D values are not consistent with the uncertainties quoted. For 3D3/2 the recommended value -15003(16) differs from Ref. [30] (-14925(8)) by 78 a.u., which is about 4.4σ when the two quoted errors are combined; for 3D5/2 the difference is 82 a.u., about 4.7σ. The text dismisses this as a limitation of the sum-over-states approach, but no calculation is shown that demonstrates that sum-over-states truncations actually produce a shift of this size. The discrepancy could equally be due to basis-set incompleteness or omitted triple excitations in the present RCCSD calculation. Because the error bars in Table V are the basis of the 'accurate values' claim, this issue is load-bearing and needs to be addressed quantitatively.
  2. [§IV, Table V (Na 3D states)] The statement that the Na 3D3/2,5/2 scalar polarizabilities 'are in agreement with Ref. [48]' is contradicted by the numbers in the same table. The scalar differences are 14.6 a.u. (about 3.6σ) for 3D3/2 and 16.9 a.u. (about 5.6σ) for 3D5/2, using the quoted uncertainties. For the tensor polarizability of 3D5/2 the difference is 10.3 a.u. against a quoted error of 1.3 a.u., about 7.9σ. If the quoted uncertainties are standard deviations, these are not agreements. The comparison claims and the uncertainty estimates cannot both stand as written; either the uncertainties must be enlarged to cover the spread of benchmark results, or the text must be revised to report the discrepancies honestly.
  3. [§IV (uncertainty estimation) and Table V] The uncertainty budget is incomplete in a way that directly affects the central claim. The text says the uncertainties are 'derived from the leading order triple excitations' and shows only selected diagrams in Fig. 2, but there is no basis-set convergence study in the manuscript. Since polarizabilities of diffuse excited states are sensitive to the Gaussian basis, tests with systematically varied numbers of GTOs and/or GTO parameters are needed. In addition, contributions from omitted triple-excitation diagrams, higher excitations, and Breit/QED effects are not estimated or added to the quoted errors. Without such a convergence study, the reported error bars cannot be taken as reflecting the actual accuracy of the RCCSD method.
minor comments (6)
  1. [§III, Eq. (12) and surrounding text] 'Block equation' should be 'Bloch equation'; the same misspelling appears in the sentences introducing Eqs. (12).
  2. [Reference [1]] The publisher location is misspelled: 'Signapore' should be 'Singapore'.
  3. [Table III] Several entries in the K rows contain malformed spacing, such as '0 .02' and '0 .03'; please fix the formatting.
  4. [Abstract and §IV (K core-valence contribution)] The abstract states that core-valence contributions are negligibly small in all methods and atoms, but §IV says that for the K ground state the core-valence contribution 'cannot be disregarded for accurate calculations.' These statements should be reconciled.
  5. [Table IV] In the Li 'Others' row, the tensor polarizability entry for 2P3/2 appears as '-3.54 3.54', which looks like a formatting or alignment error; please verify the entry.
  6. [Fig. 1 caption] The caption reads 'Ratios of scalar and tensor polarizability values from different many-body methods and their DHF values'; this should be '... to their DHF values'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the RCCSD linear-response polarizabilities are computed from first-principles equations with no fitted inputs; self-citations are comparisons, not load-bearing.

full rationale

The derivation chain is self-contained. The paper defines polarizabilities through the standard second-order expression (Eq. 5) but does not evaluate it from fitted matrix elements; instead it solves the first-order response equation (Eq. 8) for the Dirac-Coulomb Hamiltonian and evaluates Eq. 7 with RCCSD wave operators, whose amplitudes are obtained from the coupled-cluster amplitude equations (Eqs. 22-25). No experimental or previous-theory polarizability value is used as an input, and no parameter is adjusted to reproduce the quoted Table V numbers. The uncertainty estimate is generated by adding selected leading triple-excitation diagrams perturbatively and comparing with RCCSD; this is a physics-based error estimate, not a fit to the final results. Self-citations (Refs. [14], [17], [18], [32], [39], [40], [42]-[44]) appear as methodological precedents and as comparison values in Table V, but the central RCCSD calculation would stand without them. No uniqueness theorem or prior work by the same authors is invoked to forbid alternatives or to force the final values. The paper's internal consistency issues, such as the Li 3D disagreement with Ref. [30] and the Na 3D comparisons with Ref. [48], are accuracy and uncertainty problems rather than circularity: those comparisons are external benchmarks, and the paper does not redefine its results in terms of them.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

No invented entities and no data fitting. The calculation imports the standard Dirac-Coulomb Hamiltonian, many-body perturbation theory, RPA, and RCCSD from prior literature; the only hand-set inputs are basis parameters.

free parameters (2)
  • GTO basis parameters eta0 and beta per symmetry = eta0: s=0.0009, p=0.0008, d=0.001, f=0.004, g=0.005, h=0.005, i=0.005; beta: s,p,d=2.15, f=2.25, g,h,i=2.35 (Table I)
    Chosen by hand to define the even-tempered Gaussian basis; no convergence study is reported, so final values depend on this choice.
  • Number of GTO functions per symmetry = 40, 39, 38, 37, 36, 35, 34 for s, p, d, f, g, h, i
    Finite basis truncation chosen for the calculation; no test of convergence with respect to the number of functions is shown.
assumptions (3)
  • domain assumption The Dirac-Coulomb Hamiltonian with a point nucleus and no Breit or QED terms describes these systems at the target accuracy.
    Used as HDC in Sec. III; no estimate of Breit, QED, or finite-nuclear-size corrections is provided.
  • domain assumption The RCCSD truncation, with triples only as a perturbative uncertainty check, captures the correlation effects needed for accurate polarizabilities.
    All final values use Eq. (26) at singles-and-doubles level; triple contributions are not fully included in the wave functions.
  • standard math The linear-response (Dalgarno-Lewis) formulation in Eq. (8) gives the complete second-order energy shift for a weak static field.
    Sec. II; standard second-order perturbation theory for a uniform static electric field.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ab initio Calculations of Electric Dipole Polarizabilities in the Li, Na and K Atoms." pith.science (2026). https://pith.science/paper/37PRLWQL

@misc{pith2026250210312,
  author       = {Pith},
  title        = {Pith review of: Ab initio Calculations of Electric Dipole Polarizabilities in the Li, Na and K Atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37PRLWQL}},
  note         = {Machine review of arXiv:2502.10312}
}
read the original abstract

We carry out first-principle calculations of scalar and tensor components of the static electric dipole polarizabilities of six low-lying states of lithium (Li), sodium (Na) and potassium (K) alkali atoms in the linear response approach. Results are compared from the Dirac-Hartree-Fock (DHF) method, third-order many-body perturbation theory (MBPT(3) method), random phase approximation (RPA) and singles and doubles approximated relativistic coupled-cluster theory (RCCSD method). We find the DHF and RPA results are close to each other, while the MBPT(3) and RCCSD results are close to each other. This suggests that pair-correlation effects play significant roles over core-polarization effects to determine these quantities accurately in the above alkali atoms. We also compare contributions arising through the core, core-valence and valence correlations through all the methods in Li, Na and K, which show that the core-valence contributions are negligibly small in all the methods and there is no particular trends of the core and valence correlation contributions with the size of the atom. Uncertainties to the RCCSD results are estimated to quote the final values, and they are compared with the previous calculations and experimental results.

Figures

Figures reproduced from arXiv: 2502.10312 by the authors.

Figure 1
Figure 1. FIG. 1. Ratios of scalar and tensor polarizability values from different many-body methods and their DHF values. These plots [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Few important Goldstone diagrams that contribute to the unperturbed [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 58 canonical work pages

  1. [30]

    Miffre, M

    A. Miffre, M. Jacquey, M. B¨ uchner, G. Trenec, and J. Vigue, Phys. Rev. A 73, 011603(R) (2006)

    work page 2006

  2. [48]

    Mitroy, Phys

    Li-Yan Tang, Zong-Chao Yan, Ting-Yun Shi, and J. Mitroy, Phys. Rev. A 81, 042521 (2010)

    work page 2010

  3. [1]

    K. D. Bonin and V. V. Kresin, Electric Dipole Polariz- abilities of Atoms, Molecules and Clusters, World Scien- tific, Signapore (1997)

    work page 1997

  4. [2]

    Haken and H

    H. Haken and H. C. Wolf, Atoms in an Electric Field. In: The Physics of Atoms and Quanta, Springer (Berlin, Heidelberg), pp. 251-279 (1996)

    work page 1996

  5. [3]

    N. L. Manakov and V. D. Ovsiannikov and L. P. Rapoport, Phys. Rep. 141, 320 (1986)

    work page 1986

  6. [4]

    A. D. Buckingham, Adv. Chem. Phys. 12, 107 (1967)

    work page 1967

  7. [5]

    Maroulis, George, J

    G. Maroulis, George, J. Comput. Methods Sci. Eng. 4, 235 (2004)

    work page 2004

  8. [6]

    V G Pal’chikov, Yu S Domnin and A V Novoselov, J. Opt. B: Quantum Semiclass. Opt. 5, S131 (2003)

    work page 2003

Show all 62 references
  1. [7]

    B. K. Sahoo, Pramana 83, 255 (2014)

    2014

  2. [8]

    M. S. Safronova, M. G. Kozlov, and C. W. clark, IEEE Trans. on Ult. Ferro. and Freq. cntrl. , 59, 439 (2012)

    2012

  3. [9]

    Burrow, J

    P. Burrow, J. Michejda, J. Comer, J. Phys. B At. Mol. Phys. 9, 3225 (1976)

    1976

  4. [10]

    Jain, Phys

    A. Jain, Phys. Rev. A 41, 2437 (1990)

    1990

  5. [11]

    Tenfen, M

    W. Tenfen, M. V. Barp, and F. Arretche, Phys. Rev. A 99, 022703 (2019)

    2019

  6. [12]

    K. T. Tang, Phys. Rev. 177, 108 (1969)

    1969

  7. [13]

    Derevianko, W

    A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, Phys. Rev. Lett. 82, 3589 (1999)

    1999

  8. [14]

    [30] applied a sum- over-states method using a mix of experimental and the- oretical energy values

    used a hybrid approach of combining numerical and analytical basis functions, while Ref. [30] applied a sum- over-states method using a mix of experimental and the- oretical energy values. In Ref. [48], a semi-empirical DHF + CP potential approach was adopted to estimate polar...

    2023

  9. [15]

    L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans, B. P. Das, and D. Mukherjee, Phys. Rev. A 78, 012515 (2008)

    2008

  10. [16]

    Arora, and B

    B. Arora, and B. K. Sahoo, Phys. Rev. A 89, 022511 (2014)

    2014

  11. [17]

    Monin and G

    J. Monin and G. -A. Boutry, Phys. Rev. B9, 1309 (1974)

    1974

  12. [18]

    B. K. Sahoo, Chem. Phys. Lett. 448, 144 (2007)

    2007

  13. [19]

    B. K. Sahoo and B. P. Das, and D. Mukherjee, Phys. Rev. A 79, 052511 (2009)

    2009

  14. [20]

    Oymak and S

    H. Oymak and S. Erkoc, Int. J. Mod. Phys. B 26, 1230003 (2012)

    2012

  15. [21]

    Schwerdtfeger and J

    P. Schwerdtfeger and J. K. Nagle, Mol. Phys. 117, 1200 (2019)

    2019

  16. [22]

    K. Ravi, S. Lee, A. Sharma, G. Werth, and S. Rangwala, Nat. Comm. 3, 1126 (2012)

    2012

  17. [23]

    F. H. J. Hall and S. Willitsch, Phys. Rev. Lett. 109, 233202 (2012)

    2012

  18. [24]

    Catani, P

    J. Catani, P. Maioli, L. De Sarlo, F. Minardi, and M. Inguscio, Phys. Rev. A 73, 033415 (2006)

    2006

  19. [25]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010)

    2010

  20. [26]

    Joachim, J

    C. Joachim, J. K. Gimzewski, and A. Aviram, Nature (London) 408, 541 (2000)

    2000

  21. [27]

    R. W. Molof, H.L. Schwartz, T.M. Miller, B. Bederson, Phys. Rev. A 10, 1131 (1974)

    1974

  22. [28]

    C. R. Ekstrom, J. Schmiedmayer, M.S. Chapman, T.D. Hammond, and D.E. Pritchard, Phys. Rev. A 51, 3883 (1995)

    1995

  23. [29]

    W. D. Hall and J. C. Zorn, Phys. Rev. A 10, 1141 (1974)

    1974

  24. [31]

    M. S. Safronova, U. I. Safronova, and Charles W. Clark, Phys. Rev. A 86, 042505 (2012)

    2012

  25. [32]

    M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999)

    1999

  26. [33]

    D. K. Nandy, Y. Singh, B. P. Shah, and B. K. Sahoo, Phys. Rev. A 86, 052517 (2012)

    2012

  27. [34]

    Marrus and J

    R. Marrus and J. Yellin, Phys. Rev. 177, 127 (1969)

    1969

  28. [35]

    Dalgarno and J

    A. Dalgarno and J. T. Lewis, Proc. R. Soc. London 233, 70 (1955)

    1955

  29. [36]

    B. K. Sahoo, S. Blundell, A. V. Oleynichenko, R. F. Gar- cia Ruiz, L. V. Skripnikov and B. Ohayon, J. Phys. B58, 042001 (Topical review) (2025)

    2025

  30. [37]

    Katyal, A

    V. Katyal, A. Chakraborty, B. K. Sahoo, B. Ohayon, C.-Y. Seng, M. Gorchtein and J. Behr, arXiv:2412.05932 (Unpublished)

    arXiv

  31. [38]

    M. G. Kozlov and S. G. Porsev, Eur. Phys. J. D 5, 59 (1999)

    1999

  32. [39]

    J. E. Stalnaker, D. Budker, S. J. Freedman, J. S. Guz- man, S. M. Rochester, and V. V. Yashchuk, Phys. Rev. A 73, 043416 (2006)

    2006

  33. [40]

    Arora, D

    B. Arora, D. K. Nandy, and B. K. Sahoo, Phys. Rev. A 85, 012506 (2012)

    2012

  34. [41]

    J. Kaur, D. K. Nandy, B. Arora, and B. K. Sahoo, Phys. Rev. A 91, 012705 (2015)

    2015

  35. [42]

    Lindgren and J

    I. Lindgren and J. Morrison, in Atomic Many-Body Theory, edited by G. Ecker, P. Lambropoulos, and H. Walther (SpringerVerlag, Berlin, 1985)

    1985

  36. [43]

    Singh, B

    Y. Singh, B. K. Sahoo, and B. P. Das, Phys. Rev. A 89, 030502(R) (2014)

    2014

  37. [44]

    Chakraborty and B

    A. Chakraborty and B. K. Sahoo, J. Phys. Chem. A 127, 7518 (2023)

    2023

  38. [45]

    B. K. Sahoo, Y. Singh, Phys. Rev. A 95, 062514 (2017)

    2017

  39. [46]

    Mukherjee, Pramana 12, 203 (1979)

    D. Mukherjee, Pramana 12, 203 (1979)

    1979

  40. [47]

    A. K. Mohanty and E. Clementi, Chem. Phys. Lett. 157, 348 (1989)

    1989

  41. [49]

    Tang Yong-Bo, Li Cheng-Bin and Qiao Hao-Xue, Chin. Phys. B 23, 063101 (2014)

    2014

  42. [50]

    S. I. Themelis and C. A. Nicolaides, Phys. Rev. A 51, 2801 (1995)

    1995

  43. [51]

    Windholz, M

    L. Windholz, M. Musso, G. Zerza, and H. Jager, Phys. Rev. A 46, 5812 (1992)

    1992

  44. [52]

    L. R. Hunter, D. Krause, D. J. Berkeland, and M. G. Boshier, Phys. Rev. A 44, 6140 (1991)

    1991

  45. [53]

    W. R. Johnson, U. I. Safronova, A. Derevianko, and M. S. Safronova, Phys. Rev. A 77, 022510 (2008)

    2008

  46. [54]

    Rerat, M

    M. Rerat, M. Merawa, and B. Honvault-Bussery, J. Chem. Phys. 109, 7246 (1998)

    1998

  47. [55]

    Windholz and C

    L. Windholz and C. Neureiter, Phys Lett. 109A, 155 (1985)

    1985

  48. [56]

    Windholz and M

    L. Windholz and M. Musso, Phys Rev. A39, 2472 (1989)

    1989

  49. [57]

    W. F. Holmgren, M. C. Revelle, V. P. A. Lonij, and A. D. Cronin, Phys. Rev. A 81, 053607 (2010)

    2010

  50. [58]

    U. I. Safronova and M. S. Safronova, Phys. Rev. A 78, 052504 (2008)

    2008

  51. [59]

    M. S. Safronova, U. I. Safronova and C. W. Clark, Phys. Rev. A 87, 052504 (2013)

    2013

  52. [60]

    Arora, M

    B. Arora, M. S. Safronova, and C. W. Clark, Phys. Rev. A 76, 052509 (2007)

    2007

  53. [61]

    Schmieder, A

    R. Schmieder, A. Lurio, and W. Happer, Phys. Rev. A 3, 1209 (1971)

    1971

  54. [62]

    Krenn, W

    C. Krenn, W. Scherf, O. Khait, M. Musso, and L. Wind- holz, Z. Phys. D 41, 229 (1997)

    1997

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.