Analysis of Bohr formula of momentum of inertia for even-even atomic nuclei

Abdurahim Okhunov; Mohd Kh. M. Abu El-Sheikh

REVIEW 3 major objections 4 minor 16 references

Analysis of Bohr formula of momentum of inertia for even-even atomic nuclei

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Including second and third terms in the R-expansion raises the hydrodynamical moment of inertia of even-even deformed nuclei from about 0.2 to nearly 0.7 of the experimental value.

desk verdict A genuine but flawed extension of Bohr's moment-of-inertia formula: the claimed improvement rests on a fitted scaling factor and a mixed-order truncation, so the paper is a candidate for heavy revision rather than acceptance. read the letter →

arxiv 1908.01944 v1 pith:CD6TMJNN submitted 2019-08-06 nucl-th

Mohd Kh. M. Abu El-Sheikh , Abdurahim Okhunov This is my paper · ORCID

classification nucl-th

keywords momentofinertiahydrodynamicalmodelirrotationalflowdeformedeven-evennucleiBohrformulaR-expansionrotationalbandsnucleardeformation

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

The paper argues that the well-known shortfall of the hydrodynamical, irrotational-flow moment of inertia, where Bohr's formula gives values only about a fifth of measured moments, is largely an artifact of keeping only the first term in the expansion of the nuclear surface radius inside the kinetic-energy integral. Keeping the second and third terms, which carry the deformation parameters $\beta$ and $\gamma$, adds positive corrections to each principal moment of inertia. The resulting totals reach about 0.7 of the experimental values for even-even deformed nuclei, and after multiplication by $0.5/\beta$ they form a lower envelope to the experimental band when plotted against $\beta$. This matters because it locates the discrepancy inside the model's own expansion rather than in an entirely new mechanism.

What carries the argument

The engine of the calculation is the Taylor expansion of $R^5(\theta,\phi)$ about the spherical radius $R_0$, where $R(\theta,\phi)=R_0[1+\sum_{\lambda\mu}\alpha_{\lambda\mu}Y_{\lambda\mu}]$ is the deformed nuclear surface. Inserting the first three terms into the radial integral $\int_0^R r^4\,dr$ turns the kinetic energy into sums of spherical-harmonic products; orthogonality and angular-momentum recoupling identities select the rotational pieces proportional to the angular velocities, giving the Bohr term together with two additive corrections. The key feature is that the higher terms carry the deformation parameters, so the corrections are not fitted constants but predicted functions of $\beta$ and $\gamma$.

What would settle it

Compute the fourth and fifth terms of the same expansion and check whether their contributions to the total moment of inertia are smaller than the third; if they are not, the 0.7 ratio is an artifact of the truncation point. A direct experimental check: for a nucleus at $\beta\approx0.33$, where the corrections are largest, the predicted total plus the $(0.5/\beta)$ scaling should hug the lower edge of the measured band; a clear violation for several nuclei would rule out the proposed lower-limit behavior.

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Extended reading notes

Core claim

The central claim is that the first three terms in the $R$-expansion of the kinetic energy of an irrotational, incompressible nuclear droplet produce a moment of inertia $\mathcal{I}_{kk} = \mathcal{I}^{(0)}_{kk} + \mathcal{I}^{(1)}_{kk} + \mathcal{I}^{(2)}_{kk}$ whose components, for both axially symmetric and triaxial even-even nuclei, are substantially larger than the one-term Bohr value. The first correction comes from the quadratic term in the deformation and the second correction from the cubic term. In the authors' tables the third-order contribution exceeds the second-order one and in some cases even the first-order Bohr contribution. For the axial case the total reaches nearly 0.7 of the experimental moment, against 0.2 for Bohr's formula, and the quantity $(0.5/\beta)\mathcal{I}_{\rm tot}$ tracks the lower edge of the experimental band as a function of the deformation $\beta$.

Load-bearing premise

The load-bearing premise is that the Taylor expansion of $R^5(\theta,\phi)$ can be cut after the third term in the deformation parameters; the authors themselves note in their summary that the third term contributes more than the second or even the first, so the series convergence is not established.

Editorial extensions

If this is right

For axially symmetric even-even nuclei, including the second and third terms raises the hydrodynamical moment of inertia to roughly 0.7 of the experimental ground-band value, so the old factor-of-five discrepancy is reduced to about 30 percent within the same model.
The third-order term is the dominant correction, so any calculation that stops at second order will underestimate the improvement.
After the $(0.5/\beta)$ rescaling, the theoretical curve is a lower limit of the experimental moment-of-inertia band, suggesting that the missing remainder is approximately proportional to the deformation.
For triaxial nuclei, each principal-axis component computed with the three-term expansion is closer to the empirical values than the corresponding Bohr component, so the correction scheme is not restricted to axial symmetry.

Reading between the lines

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

Beyond the paper, the fact that the third term already dominates the second suggests the $R$-expansion may be asymptotic rather than convergent; computing the fourth and fifth terms would tell whether the 0.7 ratio is stable or an artifact of where the series is cut.
Beyond the paper, the observed lower-envelope behavior implies an empirical bound $I_{\rm exp} \gtrsim (0.5/\beta)\,I_{\rm tot}$, which could be tested against a wider set of nuclei and used to quantify how much rigid-body or pairing enhancement remains to be explained.
Beyond the paper, because the corrections are explicit functions of deformation, the same expansion could be used to predict moments of inertia along a rotational band rather than only for the bandhead, connecting the correction to centrifugal stretching.

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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

The one thing to know: this is a real, checkable extension of the classic Bohr hydrodynamical moment of inertia. The authors compute the second and third terms in the R-expansion and show that, on paper, they raise the predicted moment of inertia from about 20% to about 70% of the experimental band. That is a legitimate exercise, and the paper is honest enough to report that the third term dominates the second, which is itself a red flag that the expansion is not converged.

What is actually new: Acker and Marchal kept only the first two terms; this paper pushes to the third term and isolates each correction. The derivation is self-contained and the tables are organized so you can see the separate contributions. That is useful, and the authors deserve credit for not hiding the convergence problem.

Now the soft spots, in proportion. The most damaging is the factor (0.5/beta) in the last column and in Figure 1. It appears from nowhere, is not derived from the model, and is applied to the total theoretical value to make the curve sit at the lower edge of the experimental band. That is parameter fitting dressed up as comparison, and it undercuts the central claim that the higher-order terms improve agreement. The stress-test note is also right: the boundary condition is linearized at R≈R0 in Eq. (8), so if you are keeping terms of order α^2 and α^3 in the R^5 expansion, you must re-expand the boundary condition to the same order. The paper never does that. The corrections are therefore a mixture of orders, not a controlled higher-order calculation. The third-term-larger-than-second observation reinforces the point: the truncation is not demonstrably convergent.

A practical issue: the text is badly garbled. Many equations are missing or misrendered, which makes it impossible to fully verify the algebra. The authors should provide a clean manuscript and ideally a Mathematica/Maple notebook.

Who is this for? Nuclear structure people who care about the hydrodynamic model's historical shortcomings. It is not a major contribution to modern nuclear theory, where microscopic models already do much better, but it is a useful case study in why naive series expansions in deformation parameters need care.

Recommendation: I would send this to peer review rather than desk reject, because the derivation is serious and the consistency issue is worth a referee's time. But I would expect major revision: derive or remove the (0.5/beta) factor, fix the boundary-condition expansion to consistent order, and benchmark against known microscopic calculations. Without those changes, the paper does not support its own conclusions.

Referee Report

3 major / 4 minor

Summary. The paper reexamines Bohr's hydrodynamical formula for the moment of inertia of even-even deformed nuclei. The authors expand the integrand R^5 in Eq. (11) in a Taylor series and, beyond Bohr's leading term, retain the second and third terms. They derive the corresponding kinetic-energy corrections in Sections C and D, express the total moment of inertia as a sum of three contributions, and compare the result with experimental moments for axially symmetric nuclei (Table 1) and triaxial nuclei (Table 3). The paper claims that including the first three terms raises the theoretical moment of inertia to about 0.7 of the experimental values, and that after multiplying by the factor (0.5/beta) the theoretical curve forms a lower limit of the experimental band in Fig. 1.

Significance. If the calculation were internally consistent and free of fitted parameters, the result would constitute a notable improvement over Bohr's leading-order estimate, which typically gives only about 0.2 of the experimental moment of inertia. The paper is also creditable for attempting a self-contained derivation of the higher-order terms and for using published experimental data. However, the central claim is undermined by three issues: an unexplained multiplicative factor applied only to the final result, an apparently non-convergent expansion in which the third-order contribution exceeds the second, and a mixed-order treatment of the boundary condition. As presented, the comparison to experiment is not a parameter-free prediction, and the principal numerical conclusion is not established.

major comments (3)

[Section F, Table 1] The factor (0.5/beta) is introduced without derivation and applied only to the total theoretical moment of inertia. No justification is given for this factor within the hydrodynamical model. Since the text states that this multiplication makes the theoretical curve 'a lower limit' of the experimental band, the final comparison in Fig. 1 is effectively a fit rather than a predictive test. The paper should either derive this factor from the model or remove it and discuss the comparison without it.
[Section F, Table 1] The reported third-term contribution (Correc. 2) is systematically larger than the second-term contribution and in many cases larger than the first-term (Bohr) contribution. For example, at beta=0.336, Correc. 2 is 0.206 while the Bohr value is 0.092. This shows that the Taylor expansion in Eq. (11) is not controlled, and truncating at the third term is unjustified. The paper needs to demonstrate that higher-order terms are small or provide a resummation, otherwise the computed total depends on where the series is cut.
[Eqs. (6)-(8), Sections C and D] The velocity potential coefficients are determined by linearizing the boundary condition, replacing R by R0 in Eq. (8). The second and third corrections in Sections C and D are derived from terms in the R^5 expansion that are of relative order alpha and alpha^2. Yet the same deformation that changes the upper integration limit in Eq. (11) also changes the boundary condition at relative order alpha. Omitting these boundary-condition terms while retaining the R-dependence in the upper limit mixes orders, so the computed corrections are not a consistent higher-order result. A consistent calculation to the claimed order must re-expand the boundary condition as well.

minor comments (4)

[Throughout] The manuscript is heavily garbled: many equations are missing or displayed as blank spaces, variables are not defined, and equation numbers are inconsistent (e.g., references to eq. (C.7) and eq. (D.1)). This makes the derivation impossible to verify.
[Abstract and Section E] The phrase 'too much better' in the abstract should read 'much better', and the sentence in Section E claiming that the results 'are much better than the experimental ones' is confusing—the intended meaning is presumably that the theoretical values are closer to the experimental ones.
[Table 3] The caption and column headers of Table 3 are unclear: the columns labeled 'A', 'Before', 'After', and 'Exp.' are not adequately defined, and the row entries are not self-explanatory.
[References] Reference 11 is incomplete: it lists 'Bohr, A. J. S., MOMENTS OF INERTI A OF ROTATING NUCLE I. 1955, 1, 0.' but omits the journal name, volume, and page numbers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the higher-order R-expansion corrections are derived algebraically from the same deformation inputs, not from the target moments of inertia.

full rationale

The paper's derivation chain starts from the nuclear surface parametrization (Eq. 1), the irrotational-flow velocity potential (Eq. 5), and the radial boundary condition (Eq. 6). The kinetic-energy integral is expanded by Taylor-expanding R^5 (Eq. 11), and the first, second, and third terms are evaluated separately to give Bohr's contribution and two corrections (Eqs. 25, 29, 37). The deformation parameters beta and alpha are inputs extracted from B(E2) data; the moments of inertia are outputs of the derived formulas. No step defines the expansion in terms of the moment of inertia being predicted, and no parameter is fitted to the experimental moments of inertia before comparison. The only ad hoc element is the multiplicative factor (0.5/beta) applied to the final curve in Fig. 1, but the paper's central claim about reaching nearly 0.7 of experiment uses the un-scaled total theoretical column, and the factor is presented as a visual scaling rather than as a model-derived prediction. The observation that the third-term contribution exceeds the second (Section F) is a convergence or consistency concern about truncating the Taylor series, not evidence that the derivation reduces to its inputs. No load-bearing self-citation chain appears. Therefore, while the expansion may be physically questionable, the paper's core calculation is self-contained and not circular by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model uses no new entities. The main free knob is the ad hoc 0.5 factor in the (0.5/beta) multiplier. The axioms are the standard liquid-drop assumptions plus an unverified convergence assumption about the R-expansion.

free parameters (1)

Scale factor 0.5 in multiplier (0.5/beta) = 0.5
Applied to the total theoretical moment of inertia in Table 1 and Fig 1 to place the curve at the lower edge of the experimental band; no derivation is provided; the choice of 0.5 is empirical and not part of the model.

assumptions (4)

domain assumption Nuclear matter is an incompressible, irrotational fluid (curl-free velocity field).
Central to the hydrodynamical model used in Section B; quoted as 'The hydrodynamical model assumes an irrotational flow for the nuclear matter and the incompressibility'.
standard math The nuclear surface radius can be expanded as R(theta,phi) = R0 [1 + sum_{lambda mu} alpha_{lambda mu} Y*_{lambda mu}], and the kinetic-energy integral uses a Taylor expansion of R^5 about R0.
Eq. (1) and Eq. (11); the Taylor expansion is standard for small deformations, but the paper does not establish convergence at the beta values used.
domain assumption The velocity at the nuclear surface is purely radial (boundary condition Eq. 6).
Used to fix the coefficients in the velocity potential (Eqs. 7-8); a standard but non-trivial modeling choice.
ad hoc to paper The third term in the R-expansion is a legitimate small correction despite its computed magnitude exceeding the second term.
Section F reports 'The unusual conclusion is that the contribution comes from the third term is larger than that comes from the second term or even that of the first terms', which contradicts the expectation of a converging expansion; the paper proceeds as if three-term truncation is valid without proving convergence.

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Cite this review

Pith. "Pith review of Analysis of Bohr formula of momentum of inertia for even-even atomic nuclei." pith.science (2026). https://pith.science/paper/CD6TMJNN

@misc{pith2026190801944,
  author       = {Pith},
  title        = {Pith review of: Analysis of Bohr formula of momentum of inertia for even-even atomic nuclei},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CD6TMJNN}},
  note         = {Machine review of arXiv:1908.01944}
}

abstract

The moment of inertia of even-even deformed nuclei which are derived on the basis of hydrodynamical model yield values that are too small compared with the experimental [Davidson 1965] ones. We expect that these contradictions come from the consideration only the first term in $R$- expansion in spite of not containing parameters indicating deformity of the nucleus, and neglecting all other terms which include the deformation parameters $\alpha\lambda\mu$. In this work, the first three terms in $R$- expansion are taken into account. The results are more realistic and too much better the previous ones.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Bohr, A.; Mottelson, B. R. J. P. R., Rotational states in even -even nuclei. 1953, 90 (4), 717

work page 1953
[2]

Griffin, J.; Rich, M. J. P. R., Moments of Inertia of Even -Even Rare Earth Nuclei. 1960, 118 (3), 850

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[3]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bohr, A.; Mottelson, B. R. J. P. R., Rotational states in even -even nuclei. 1953, 90 (4), 717

work page 1953

[2] [2]

Griffin, J.; Rich, M. J. P. R., Moments of Inertia of Even -Even Rare Earth Nuclei. 1960, 118 (3), 850

work page 1960

[3] [3]

Gupta, R. K. J. P. R. C., Irrotational or Rotational Flow Motion in Deformed Nuclei. 1972, 6 (2), 426

work page 1972

[4] [4]

Rayleigh, J. W. S., On progressive waves. CF Hodgson and Son: 1877

work page

[5] [5]

Villars, F. J. N. P., A note on rotational energy levels in nuclei. 1957, 3 (2), 240-254

work page 1957

[6] [6]

M.; Greiner, W., Nuclear theory

Eisenberg, J. M.; Greiner, W., Nuclear theory. 1972

work page 1972

[7] [7]

M.; Greiner, W., Nuclear theory

Eisenberg, J. M.; Greiner, W., Nuclear theory. 1987

work page 1987

[8] [8]

A., Nuclear models

Greiner, W.; Maruhn, J. A., Nuclear models. Springer: 1996

work page 1996