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arxiv: 1907.09071 · v2 · pith:G65AO7NOnew · submitted 2019-07-22 · ⚛️ nucl-th

The n-th order Moment of the Nuclear Charge Density and Contribution from the Neutrons

Haruki Kurasawa , Toshio Suzuki This is my paper

Pith reviewed 2026-05-24 18:17 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords nuclear charge densitypoint neutron densitymean square radiusrelativistic effectselectron scatteringnuclear moments
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The pith

The relativistic n-th order moments of the nuclear charge density for n ≥ 4 include contributions from the point neutron density.

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

The paper derives a relativistic expression for the n-th order moment of the nuclear charge density. For the mean square radius, it also provides a consistent non-relativistic version up to order 1/M² using the Foldy-Wouthuysen transformation. It shows that moments of order 4 and higher depend on the point neutron density, unlike lower moments. This dependence means the fourth-order moment can give information on the mean square radius of the point neutron density. Such information is relevant for analyzing electron scattering experiments on neutron-rich nuclei.

Core claim

The relativistic expression for the n-th order moment of the nuclear charge density depends on the point neutron density when n is at least 4. The fourth-order moment in particular yields useful information on the mean square radius of the point neutron density.

What carries the argument

The relativistic expression for the n-th order moment of the nuclear charge density, derived using the Foldy-Wouthuysen transformation to connect to the non-relativistic limit up to 1/M².

Load-bearing premise

The derivations assume that the Foldy-Wouthuysen transformation provides a consistent non-relativistic limit up to 1/M² and that the relativistic framework for nuclear charge density applies without additional higher-order corrections that would alter the neutron dependence.

What would settle it

A precise measurement of the fourth-order moment of the nuclear charge density in a neutron-rich nucleus via electron scattering that deviates from predictions assuming no neutron contribution would falsify the dependence claim.

Figures

Figures reproduced from arXiv: 1907.09071 by Haruki Kurasawa, Toshio Suzuki.

Figure 1
Figure 1. Figure 1: The contribution of the n-th order moment of the charge density to the form factor squared for elastic electron scattering off 48Ca and 208Pb in PWBA[38]. For details, see the text. Fig.1 shows the contribution of the n-th order moment of the charge density to the form factor squared for elastic electron scattering off 48Ca and 208Pb [38], which is defined as Sk(q) = [PITH_FULL_IMAGE:figures/full_fig_p011… view at source ↗
read the original abstract

The relativistic expression for the $n$-th order moment of the nuclear charge density is presented. For the mean square radius(msr) of the nuclear charge density, the non-relativistic expression, which is equivalent to the relativistic one, is also derived consistently up to $1/M^2$ with use of the Foldy-Wouthuysen transformation. The difference between the relativistic and non-relativistic expressions for the msr of the point proton density is also discussed. The $n(\ge 4)$-th order moment of the nuclear charge density depends on the point neutron density. The 4-th order moment yields a useful information on the msr of the point neutron density, and is expected to play an important role in electron scattering off neutron-rich nuclei.

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 / 2 minor

Summary. The paper presents the relativistic expression for the n-th order moment of the nuclear charge density. For the mean square radius (msr), it derives the equivalent non-relativistic expression consistently up to 1/M² via the Foldy-Wouthuysen transformation. It concludes that moments with n ≥ 4 depend on the point neutron density, and that the 4th-order moment in particular yields useful information on the msr of the point neutron density, with relevance to electron scattering on neutron-rich nuclei.

Significance. If the central claim holds, the work supplies a relativistic framework linking higher moments of the charge density to neutron distributions, which could aid interpretation of scattering data from exotic nuclei. The explicit, consistent derivation of the msr equivalence up to 1/M² is a clear strength, as is the direct identification of the neutron term in the relativistic n ≥ 4 expressions.

major comments (2)
  1. [Derivation of the n-th order moment and Foldy-Wouthuysen transformation] The claim that the 4th-order moment depends on the point neutron density and provides its msr rests on the relativistic charge-density operator. Equivalence between relativistic and non-relativistic expressions is demonstrated only up to O(1/M²) for the msr via the Foldy-Wouthuysen transformation. Because the 4th moment weights the radial integrand by an extra r², any O(1/M³) or higher pieces of the transformed operator that carry neutron form-factor or spin-orbit structure could shift or cancel the claimed neutron contribution at the same nominal order. This truncation must be justified or extended for the n=4 case.
  2. [Discussion of relativistic vs. non-relativistic msr for point proton density] The paper discusses the difference between relativistic and non-relativistic expressions for the msr of the point proton density, but does not show how (or whether) this difference extends to the neutron-dependent term in the 4th moment. An explicit propagation of the proton-density difference to n=4 would be required to confirm that the neutron contribution remains cleanly isolated.
minor comments (2)
  1. The abstract would be clearer if it stated the precise coefficient or functional form of the neutron-density term that appears in the 4th-order moment.
  2. All equations should be numbered and cross-referenced consistently throughout the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Derivation of the n-th order moment and Foldy-Wouthuysen transformation] The claim that the 4th-order moment depends on the point neutron density and provides its msr rests on the relativistic charge-density operator. Equivalence between relativistic and non-relativistic expressions is demonstrated only up to O(1/M²) for the msr via the Foldy-Wouthuysen transformation. Because the 4th moment weights the radial integrand by an extra r², any O(1/M³) or higher pieces of the transformed operator that carry neutron form-factor or spin-orbit structure could shift or cancel the claimed neutron contribution at the same nominal order. This truncation must be justified or extended for the n=4 case.

    Authors: The relativistic n-th moment is obtained directly from the expectation value of the relativistic charge-density operator; the Foldy-Wouthuysen transformation is invoked only to establish the non-relativistic equivalence for the n=2 (msr) case up to O(1/M²). The neutron contribution for n≥4 enters at the same relativistic order through the Darwin-Foldy and spin-orbit terms in the charge operator and is not an artifact of the 1/M² truncation. Because the additional r² weighting in the n=4 integrand multiplies the same operator, any O(1/M³) corrections remain higher order and do not cancel the leading neutron term at the precision relevant for nuclear scales. We will add a short paragraph in the revised manuscript explicitly stating this order counting for the n=4 case. revision: yes

  2. Referee: [Discussion of relativistic vs. non-relativistic msr for point proton density] The paper discusses the difference between relativistic and non-relativistic expressions for the msr of the point proton density, but does not show how (or whether) this difference extends to the neutron-dependent term in the 4th moment. An explicit propagation of the proton-density difference to n=4 would be required to confirm that the neutron contribution remains cleanly isolated.

    Authors: The relativistic-nonrelativistic difference for the point-proton msr originates from the same Foldy-Wouthuysen corrections that generate the neutron term in the charge-density operator. Because the neutron contribution to the n=4 moment is produced by precisely those corrections (with the point-neutron density replacing the proton density), the same difference propagates directly; the neutron term therefore remains isolated at the order considered. An explicit algebraic substitution for n=4 is straightforward from the operator structure already given in the manuscript and does not alter the conclusion. We will nevertheless insert one additional equation in the revised text showing the n=4 neutron term with the relativistic correction made explicit. revision: yes

Circularity Check

0 steps flagged

Derivation from relativistic charge-density operator is self-contained

full rationale

The paper starts from the standard relativistic expression for the nuclear charge density operator, expands its n-th moment, and applies the Foldy-Wouthuysen transformation to obtain the non-relativistic form up to O(1/M²). The neutron-density dependence for n≥4 follows algebraically from the operator structure (neutron form-factor and spin-orbit terms) without any parameter fitting, self-referential definition, or load-bearing self-citation. No step reduces the claimed result to an input by construction; the derivation remains independent of the target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from relativistic nuclear theory without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption Nuclear charge density can be treated using relativistic quantum mechanics with the Foldy-Wouthuysen transformation providing the non-relativistic limit up to 1/M².
    Invoked to derive equivalent non-relativistic expressions for the mean square radius.

pith-pipeline@v0.9.0 · 5658 in / 1173 out tokens · 25039 ms · 2026-05-24T18:17:33.749812+00:00 · methodology

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