Isotope shift for total electron binding energy of atoms

V. A. Dzuba; V. V. Flambaum

REVIEW 4 major objections 4 minor 24 references

Isotope shift for total electron binding energy of atoms

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper establishes a systematic tabulation of isotope shifts of the total electron binding energy for neutral atoms and singly charged ions up to Z=120, with field-shift coefficients accurate to about 1% for heavy atoms, and shows…

desk verdict Useful first tabulation of total-binding-energy isotope shift coefficients up to Z=120, but the percent-level accuracy claim for the field shift factors needs quantitative support. read the letter →

arxiv 2507.21410 v1 pith:UWNFRBAE submitted 2025-07-29 physics.atom-ph

V. A. Dzuba , V. V. Flambaum This is my paper · ORCID

classification physics.atom-ph

keywords isotopeshifttotalelectronbindingenergyfieldmasssuperheavyelementsrelativisticHartree-Focknuclearradius

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 provides the first systematic tabulation of isotope shifts of the total electron binding energy for all neutral atoms and singly charged ions up to Z=120, computed with relativistic Hartree-Fock including the Breit interaction. The central result is that the field-shift part of the isotope shift is accurately captured by two tabulated coefficients F and G in the formula $F\delta\langle r^2\rangle + G\delta\langle r^2\rangle^2$, together with a simple power-law interpolation $F=bZ^k$ that reproduces the calculated values within about 1%. This matters because isotope mass differences enter nuclear mass measurements, searches for new forces via isotope shift spectroscopy, and predictions of superheavy element stability, where subtracting the electron binding energy at percent accuracy is needed. The paper also shows that the mass shift dominates below $Z\approx 38$ and the field shift above, and that the difference between neutral atoms and singly charged ions stays at the few percent level, so the tables extend to higher charge states.

What carries the argument

The central object is the parametrization of the isotope shift as a field shift plus mass shift, with field-shift coefficients F and G extracted by computing total binding energies at three nuclear radii and fitting a parabola. The power-law interpolation $F(Z)=bZ^k$, with $k = \ln(F_1/F_2)/\ln(Z_1/Z_2)$, is the device that extends the closed-shell results to open-shell atoms with about 1% accuracy. The normal mass shift uses the virial theorem $E_k=-E_{total}$ from earlier total-energy calculations, and the specific mass shift $\langle \sum_{i<j} \mathbf{p}_i\cdot \mathbf{p}_j\rangle$ is evaluated in the relativistic Hartree-Fock ground state, relying on the fact that inner shells dominate so correlation corrections are small.

What would settle it

A direct test would be to recalculate the isotope shift of the total electron binding energy for a heavy closed-shell atom such as Hg or No using a high-precision relativistic many-body method that includes electron correlation and QED corrections, and compare the resulting F and G to the tabulated values; a deviation clearly exceeding the claimed percent-level accuracy would invalidate the assumption that inner-shell dominance makes such corrections negligible.

Watch

Extended reading notes

Core claim

The paper establishes that the isotope shift $\Delta E_{IS}$ of the total electron binding energy can be parameterized as $\Delta E_{IS} = F\delta\langle r^2\rangle + G\delta\langle r^2\rangle^2 + (m_e/M_1 - m_e/M_2)(K_{NMS} + K_{SMS})$, with tabulated F and G for closed-shell systems from Ne to Og and benchmark open-shell cases, and that the dependence of F on Z follows $F=bZ^k$ to within about 1% between neighboring closed shells. The effective exponent k grows from about 5 near $Z\approx 50$ to 12.6 at $Z=118\text{--}120$, reflecting the growing contribution of deep s and p_{1/2} shells. The calculations are done with relativistic Hartree-Fock including the Breit interaction, varying the nuclear radius to extract the field-shift coefficients; the normal mass shift is obtained from the virial theorem, while the specific mass shift is evaluated as a Hartree-Fock expectation value. The result makes the electron-binding contribution to isotope mass differences available for essentially the whole periodic table.

Load-bearing premise

The load-bearing premise is that correlation and QED corrections to the isotope shift are negligible because deep inner-shell electrons dominate the total-energy shift; the paper asserts this but provides no quantitative estimate, so if these corrections are not small the tabulated coefficients carry an unquantified systematic error.

Editorial extensions

If this is right

For any isotope pair, the electron-binding contribution to the atomic mass difference can now be estimated to about 1% accuracy in heavy atoms using the tabulated F and G and the power-law interpolation, with no additional atomic-structure calculation.
Nuclear mass evaluations for elements beyond $Z\approx 38$ will need to treat the field-shift part of the electron-binding correction as a leading term, since the mass-shift contribution becomes small there.
The tabulated coefficients extend directly to singly charged ions and, with few-percent adjustments, to higher charge states, because the isotope shift is dominated by inner shells.
The alternative coefficient a parametrizes the field shift in terms of $A^{1/3}$, providing a way to estimate isotope shifts for superheavy isotopes when only mass numbers are known, not radii.
The crossover near $Z\approx 38$ identifies where the dominant uncertainty in isotope mass differences shifts from atomic electron motion to nuclear charge radii.

Reading between the lines

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

Because the paper does not quantify correlation or QED corrections, a natural next step is to recompute F and G for a few heavy atoms with a many-body method that includes those corrections; the comparison would either confirm the percent-level estimate or bound the omitted terms.
The steep rise of the exponent k from about 5 to 12.6 could serve as a compact diagnostic for how strongly the field shift concentrates into the innermost s and p_{1/2} orbitals, and might be used to extrapolate field-shift coefficients for elements beyond Z=120.
The near-independence of the isotope shift on ionization state suggests that the same tables could anchor total-energy isotope shifts for trace elements in astrophysical plasmas, where ionization levels are not singly charged, though explicit high-charge calculations would be needed to verify this extension.

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.

Desk Editor's Note

read the letter

Short version: this is a useful data paper, the first systematic tabulation of isotope shift coefficients for the total electron binding energy up to Z=120. If you work on nuclear mass accounting, superheavy elements, or isotope shift applications, the tables of F, G, and a are worth having. The interpolation formula F = bZ^k with a Z-dependent exponent is simple and seems to work at the 1% level for reproducing their RHF numbers. The crossover between mass and field shift near Z=38 is cleanly presented.

The paper computes at the relativistic Hartree-Fock level with the Breit interaction, which is fine for a first pass. The quantity is genuinely new; they cite no prior calculation of it. Keeping the quadratic G term for large radius changes is sensible, and the A^(1/3) parametrization for cases where radius changes are unknown is a nice addition.

The soft spots are about what is not shown. The 1% interpolation accuracy is asserted rather than demonstrated: there is no residual table for the open-shell benchmarks, and those open-shell cases are exactly what the formula is supposed to predict. That is easy to fix. More important, the claim that the tabulated F and G are accurate at the percent level for heavy atoms rests on the assertion that deep shells dominate and correlation/QED corrections are small. They do not quantify either. For Z~100 the 1s QED corrections are hundreds of eV; the finite-nuclear-size derivative of those corrections can feed directly into F, and the reader gets no estimate of whether that is below 1% of F (which is a few eV/fm^2 for heavy atoms). Same for correlation derivatives. This is a load-bearing gap in the error budget, not a style issue. It does not kill the RHF-level tables, but it means the percent-level physical accuracy claim is unsubstantiated. A concrete estimate, even a rough one, would change the paper's value.

There is also a small inconsistency in the mass shift treatment. They say SMS is 10-20% of NMS, then neglect it in Table II. For light atoms, where mass shift dominates, 10-20% of the dominant term is not negligible. The heavy-atom conclusions are unaffected, but the wording should be tightened.

Bottom line: worth a serious referee. The dataset is new and useful; the requested changes are quantitative error estimates, a residual table for the interpolation, and a cleaned-up mass shift discussion. I would engage with it.

Referee Report

4 major / 4 minor

Summary. This paper computes the isotope shift of the total electron binding energy for neutral atoms and selected ions up to Z=120 using relativistic Hartree-Fock including the Breit interaction. Field-shift coefficients F and G are extracted from total-energy calculations at three nuclear radii and tabulated for closed-shell systems, together with a coefficient a for an A^(1/3) parametrization. The normal mass shift is estimated from the total kinetic energy via the virial theorem, the specific mass shift is evaluated at the RHF level but described as unimportant, and a power-law interpolation F(Z)=bZ^k is proposed with a claimed accuracy of about 1%. The paper concludes that the field shift dominates above Z≈38 and that differences between neutral atoms and singly charged ions are at the few-percent level.

Significance. The paper fills a genuine gap: no comprehensive tabulation of isotope shifts of total electron binding energies currently exists, and the results would be useful for nuclear mass evaluations, drip-line studies, and isotope-shift searches for new physics. The method is transparent, the three-point fit retains a quadratic term for superheavy elements, and the bZ^k interpolation is simple and potentially useful. However, the central percent-level accuracy claim is not supported by the evidence actually presented: no residual analysis for the interpolation is shown, no numerical estimate is given for omitted correlation and QED contributions to the field shift, and at least one tabulated configuration appears not to be the neutral ground state. The approach is plausible, but the validation depth is insufficient for the stated accuracy.

major comments (4)

[Section III (Conclusion)] The percent-level uncertainty claim for F and G rests entirely on the sentence "Isotope shift in the total electron energy is dominated by the contribution of deep shells, therefore correlation corrections are expected to be small." This is load-bearing because F and G are extracted from RHF total energies that omit correlation and QED. For heavy atoms the 1s self-energy and vacuum polarization are hundreds of eV, and their finite-nuclear-size dependence enters the field shift alongside the RHF contribution; no estimate is given of whether the derivatives of these corrections are small compared with F, which is a few eV/fm^2 near Z~100. Please provide a quantitative estimate, for example by evaluating the finite-size derivative of the QED corrections from Ref. [19] for representative nuclei such as Pb, No, and Og, and state explicitly whether the tabulated F and G include any QED or correlation contribution. Without this, the claimed percent-level accuracy is unsupported.
[Section III and Eq. (5)] The claim that bZ^k reproduces calculated field shifts "to within about 1%" is not evidenced: no residual table, plot, or numerical comparison is shown, despite the text saying that open-shell benchmark cases were used to test the formula. Please include (F_fit - F_calc)/F_calc for all tabulated points and for the open-shell benchmark cases, and define precisely which neighboring points are used for each interpolation interval. This is needed because the 1% figure is the paper's main quantitative deliverable and is not currently verifiable from the manuscript.
[Section II, Eqs. (1)-(2) and Table II] The treatment of the specific mass shift is internally inconsistent for light elements. The text states that the SMS is 10-20% of the NMS in Xe and that it is not important because the mass shift is only relevant for light atoms, but Table II shows that for Z<38 the mass shift dominates the field shift. For a light atom such as Ne or Mg, neglecting a 10-20% SMS correction means the total isotope shift is wrong by 10-20%, which is not a modest error if the quantity of interest is the total binding-energy difference. Either tabulate KSMS values or explicitly restrict the paper's final formula to the field-shift part and remove the mass-shift term from Eq. (1) for light atoms.
[Table III, Z=78 (Pt)] The listed ground-state configuration for Pt, [Xe]4f14 5d10, is not the neutral Pt ground state, which is [Xe]4f14 5d9 6s1. The listed configuration has 78 electrons but an empty 6s shell, and it is an excited configuration, not the ground state. If the calculation used this d10 configuration, the resulting F and G do not correspond to neutral Pt, and the few-percent effect of removing an outer s electron makes this matter for the table and for the interpolation around Z=78. If the configuration is a typographical error, please correct it and verify the reported values; otherwise, clarify why this configuration was used.

minor comments (4)

[Abstract and Table IV] The abstract claims calculations for "singly charged ions up to element Z=120," but Table IV lists only ten ions (In+, Cs+, La+, Lu+, Au+, Tl+, Fr+, Lr+, Nh+, E119+), not the full set of singly charged ions. Please either extend the table or revise the abstract and introduction to say "selected singly charged ions."
[Table I caption] The notation such as 4.86[-6] is not defined; please add a note in the Table I caption that [n] denotes multiplication by 10^n.
[Eq. (1)] Equation (1) is split across two lines without a single equation number or a clear alignment; please reformat so that the two lines are visually one equation or are numbered separately.
[Section II, mass shift discussion] The paper states that the SMS is evaluated but does not report any SMS values or a table of KSMS; since the mass shift is included in Eq. (1), please either provide the SMS values used or state clearly that Eq. (1) is given for completeness and that the numerical results do not include the SMS.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: F and G are extracted from independent RHF total-energy calculations, and the power-law interpolation is an empirical fit tested on open-shell benchmark cases, not a derivation from the target result.

full rationale

The paper's central quantities F and G are obtained by solving the RHF equations with the Breit interaction at three nuclear radii and fitting the resulting total-energy changes to Eq. (1); this is an independent computation, not an input assumed in the conclusion. The interpolation F(Z)=bZ^k is fitted to the tabulated closed-shell values, and its claimed 1% accuracy is checked against open-shell atoms computed with fractional occupations, so the agreement is not forced by construction. The normal mass shift uses total binding energies from the authors' previous Ref. [19] via the virial theorem, but that prior work is an external input for total energies, not the isotope-shift result being derived; in any case the mass shift is secondary for heavy atoms where the field shift dominates. The statement that correlation and QED corrections to the total-energy isotope shift are small because deep shells dominate is an unquantified physical assumption and a genuine correctness risk, but it is not circular: the tabulated coefficients come from RHF calculations, and the assumption does not define F or G in terms of the claimed result. No equation or parameter in the paper is equivalent by construction to the output, and no load-bearing argument reduces to a self-citation chain. Score 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the RHF model and two approximations: negligible correlation/QED corrections and the empirical interpolation. No new entities are introduced.

free parameters (3)

Effective exponent k in F=bZ^k = ~5 at Z~50 to 12.6 at Z=118-120
Determined from neighboring computed F values via k = ln(F1/F2)/ln(Z1/Z2); used for interpolation.
Normalization b in F=bZ^k = Derived from F1 and k
Set by requiring the power law to pass through a computed F value; slope parameter in the empirical interpolation.
Coefficient a in A^(1/3) parametrization = e.g., 44.6770 a.u. for Og
Fitted to computed energy changes using formula (4) for superheavy elements.

assumptions (6)

domain assumption Relativistic Hartree-Fock including Breit interaction is an adequate approximation for the total binding energy isotope shift.
Used for all calculations; no higher-order correlation or QED corrections are included.
domain assumption Correlation and QED corrections to the isotope shift are negligible because deep shells dominate.
Stated in Sections II and III; supports the percent-level accuracy claim.
standard math The normal mass shift is given by the virial theorem, Ek = -Etotal, using total binding energies from Ref [19].
Used to obtain kinetic energy without separate calculation.
domain assumption The specific mass shift can be computed as an RHF expectation value with negligible correlation corrections.
Used to estimate SMS; correlation sensitivity noted in transition frequencies is ignored.
domain assumption A Fermi nuclear charge distribution with RN = 1.1 A^(1/3) gives delta<r^2> for the mass-shift comparison in Table II.
Used to convert between A and delta<r^2>; different nuclear models may change field shift values.
domain assumption The field shift is adequately represented by F delta<r^2> + G delta<r^2>^2.
Used to extract F and G from three radii; the quadratic term is retained for large radius changes.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Isotope shift for total electron binding energy of atoms." pith.science (2026). https://pith.science/paper/UWNFRBAE

@misc{pith2026250721410,
  author       = {Pith},
  title        = {Pith review of: Isotope shift for total electron binding energy of atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWNFRBAE}},
  note         = {Machine review of arXiv:2507.21410}
}

abstract

We compute the isotope shifts of the \emph{total} electron binding energy of neutral atoms and singly charged ions up to element $Z=120$, using relativistic Hartree-Fock method including the Breit interaction. Field shift coefficients are extracted by varying the nuclear charge radius; a small quadratic term is retained to cover large radius changes relevant to superheavy nuclei. We tabulate isotope shift coefficients for closed shell systems from Ne to Og and benchmark selected open shell cases, used to test the interpolation formula. A simple power law interpolation $bZ^k$ reproduces calculated field shifts to within about 1\% across the table, with the effective exponent $k$ growing from roughly 5 near $Z \sim 50$ to about 12 at $Z \sim 118$. Due to the domination of inner shells, differences between neutrals and singly charged ions does not exceed few percent, becoming noticeable mainly when an outer $s$ electron is removed. Therefore, these results may also be used for higher charge ions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[19]

V. A. Dzuba, V. V. Flambaum, and A. V. Afanasjev, Calculation of the correlation, relativistic, and QED cor- rections to the total electron binding energy in atoms and their nuclear charge dependence, Phys. Rev. A 110, 052810 (2024)

work page 2024
[1]

Mittig, A

W. Mittig, A. L´ epine-Szily and N. A. Orr, Mass Mea- surements far from stability, Annu. Rev. Nucl. Sci., 47, 27 (2003)

work page 2003
[2]

Dilling, K

J. Dilling, K. Blaum, M. Brodeur and S. Eliseev, Penning-trap mass measurements in atomic and nuclear physics, Annu. Rev. Nucl. Sci., 68, 45 (2018)

work page 2018
[3]

Erler, N

J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac and M. Stoitsov, The limits of the nuclear landscape, Nature 486, 509 (2012)

work page 2012
[4]

A. V. Afanasjev, S. E. Agbemava, D. Ray and P. Ring, Nuclear landscape in covariant density functional theory, Phys. Lett. B 726, 680 (2013)

work page 2013
[5]

S. E. Agbemava and A. V. Afanasjev and D. Ray and P. Ring, Global performance of covariant energy density functionals: Ground state observables of even-even nuclei and the estimate of theoretical uncertainties, Phys. Rev. C 89, 054320 (2014)

work page 2014
[6]

B. Osei, A. V. Afanasjev, A. Taninah, A. Dalbah, U. C. Perera, V. A. Dzuba, and V. V. Flambaum, Further steps towards next generation of covariant energy den- sity functionals, arXiv:2507.17082, Phys. Rev. C, to be published

work page Pith review arXiv
[7]

Chamel and P

N. Chamel and P. Haensel, Physics of Neutron Star Crusts, Living Rev. Relativity 11, 10 (2008)

work page 2008

Show all 24 references

[8]

F. -K. Thielemann and M. Eichler and I. V. Panov and B. Wehmeyer, Neutron Star Mergers and Nucleosynthe- sis of Heavy Elements, Ann. Rev. Nucl. Part. Sci.67, 253 (2017)

2017
[9]

B. A. Brown, A. Derevianko, V. V. Flambaum, Calcu- lation of the neutron skin and its effect in atomic parity violation, Phys. Rev. C 79, 035501 (2009)

2009
[10]

Antypas, A

D. Antypas, A. Fabricant, J. E. Stalnaker, K. Tsigutkin, V. V. Flambaum and D. Budker, Isotopic variation of parity violation in atomic ytterbium, Nature Physics 15, 120 (2018)

2018
[11]

Y. V. Stadnik and V. V. Flambaum, Axion-induced effects in atoms, molecules and nuclei: parity non- conservation, anapole moments, electric dipole moments, and spin-gravity and spin-axion momentum couplings, Phys. Rev. D 89, 043522 (2014)

2014
[12]

B. M. Roberts, Y. V. Stadnik, V. A. Dzuba, V. V. Flam- baum, N. Leefer and D. Budker, Limiting P-odd inter- actions of cosmic fields with electrons, protons and neu- trons, Phys. Rev. Lett. 113, 081601 (2014)

2014
[13]

Berengut, Dmitry Budker, Cedric Delaunay, Victor V

Julian C. Berengut, Dmitry Budker, Cedric Delaunay, Victor V. Flambaum, Claudia Frugiuele, Elina Fuchs, Christophe Grojean, Roni Harnik, Roee Ozeri, Gi- lad Perez, and Yotam Soreq, Probing new light force- mediators by isotope shift spectroscopy. Phys. Rev. Lett. 120, 103202 (2018)

2018
[14]

Arnould and S

M. Arnould and S. Goriely and K. Takahashi, The r- process of stellar nucleosynthesis: Astrophysics and nu- clear physics achievements and mysteries, Phys. Reports 450, 97 (2007)

2007
[15]

L. D. Landau, E. M. Lifshitz. Quantum Mechanics (Perg- amon Press, Oxford, 1965)

1965
[16]

Aoyagi, M

M. Aoyagi, M. H. Chen, B. Crasemann, and H. Mark, Neutral-atom electron binding energies from relaxed- orbital relativistic Hartree-Fock-Slater Calculations, 2 ≤ Z ≤ 106, Atom. Data Nuc. Data Tables, 18, 243 (1976)

1976
[17]

Rodrigues, P

G.C. Rodrigues, P. Indelicato, J.P. Santos, P. Patt´ e, F. Parente, Systematic calculation of total atomic energies of ground state configurations, Atom. Data Nuc. Data Tables, 86, 117 (2004)

2004
[18]

Lunney, J

D. Lunney, J. M. Pearson, and C. Thibault, Recent trend in the determination of nuclear masses Rev. Mod. Phys. 75, 1021 (2003)

2003
[20]

I. I. Sobelman, Atomic Spectra And Radiative Transi- tions, (Springer-Verlag, Berlin, 1979)

1979
[21]

V. V. Flambaum, A. J. Geddes, and A. V. Viatkina, Iso- tope shift, nonlinearity of King plots, and the search for new particles. Phys. Rev. A 97, 032510 (2018)

2018
[22]

M. S. Safronova and W. R. Johnson, Third-order isotope- shift constants for alkali-metal atoms and ions Phys. Rev. A 64, 052501 (2001)

2001
[23]

J. C. Berengut, V. A. Dzuba, and V. V. Flambaum, Isotope-shift calculations for atoms with one valence elec- tron, Phys. Rev. A 68, 052502 (2003)

2003
[24]

V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Isotope shift and search for metastable superheavy elements in astrophysical data, Phys. Rev. A 95, 062515 (2017)

2017

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [19]

V. A. Dzuba, V. V. Flambaum, and A. V. Afanasjev, Calculation of the correlation, relativistic, and QED cor- rections to the total electron binding energy in atoms and their nuclear charge dependence, Phys. Rev. A 110, 052810 (2024)

work page 2024

[2] [1]

Mittig, A

W. Mittig, A. L´ epine-Szily and N. A. Orr, Mass Mea- surements far from stability, Annu. Rev. Nucl. Sci., 47, 27 (2003)

work page 2003

[3] [2]

Dilling, K

J. Dilling, K. Blaum, M. Brodeur and S. Eliseev, Penning-trap mass measurements in atomic and nuclear physics, Annu. Rev. Nucl. Sci., 68, 45 (2018)

work page 2018

[4] [3]

Erler, N

J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac and M. Stoitsov, The limits of the nuclear landscape, Nature 486, 509 (2012)

work page 2012

[5] [4]

A. V. Afanasjev, S. E. Agbemava, D. Ray and P. Ring, Nuclear landscape in covariant density functional theory, Phys. Lett. B 726, 680 (2013)

work page 2013

[6] [5]

S. E. Agbemava and A. V. Afanasjev and D. Ray and P. Ring, Global performance of covariant energy density functionals: Ground state observables of even-even nuclei and the estimate of theoretical uncertainties, Phys. Rev. C 89, 054320 (2014)

work page 2014

[7] [6]

B. Osei, A. V. Afanasjev, A. Taninah, A. Dalbah, U. C. Perera, V. A. Dzuba, and V. V. Flambaum, Further steps towards next generation of covariant energy den- sity functionals, arXiv:2507.17082, Phys. Rev. C, to be published

work page Pith review arXiv

[8] [7]

Chamel and P

N. Chamel and P. Haensel, Physics of Neutron Star Crusts, Living Rev. Relativity 11, 10 (2008)

work page 2008