Patterns of spin and pseudo-spin symmetries in nuclear relativistic mean-field approaches

E. Khan; J.P. Ebran; L. Heitz

REVIEW 2 major objections 6 minor 40 references

Patterns of spin and pseudo-spin symmetries in nuclear relativistic mean-field approaches

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper derives closed-form first-order perturbative formulas for spin and pseudo-spin energy gaps in relativistic mean-field nuclear models, and shows how each gap scales with mass number.

desk verdict Eq. (12) has a sign error in the ΔEg term that inverts the paper's main A-scaling interpretation; the total first-order expression is fine and the paper is salvageable. read the letter →

arxiv 2501.19037 v1 pith:MQD66D7C submitted 2025-01-31 nucl-th

L. Heitz , J.P. Ebran , E. Khan This is my paper · ORCID

classification nucl-th PACS 21.10.Pc21.60.Jz

keywords spinsymmetrypseudo-spinrelativisticmean-fieldperturbationtheoryspin-orbitsplittingnuclearshellstructuremassdependenceharmonicoscillator

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 proposes a common perturbative framework for two related symmetries in atomic nuclei: spin symmetry and pseudo-spin symmetry. Working at first order around a spin-symmetric relativistic harmonic oscillator reference state, it obtains explicit closed expressions for the energy gaps between spin doublets and pseudo-spin doublets. For spin doublets, the gap depends only on the lower component of the Dirac wavefunction and on the potential difference $\Delta = S - V$, and it reproduces the well-known $A^{-2/3}$ mass scaling as the product of two separate $A^{-1/3}$ effects. For pseudo-spin doublets, both upper and lower components and both potentials $\Sigma = S + V$ and $\Delta$ contribute: the upper-component term grows roughly linearly with $A$ while the lower-component term stays nearly constant. The result is a unified, parameter-light explanation of how these symmetries are broken across the nuclear chart.

What carries the argument

The central object is the spin-symmetric relativistic harmonic oscillator (SS-RHO) reference state: a Dirac Hamiltonian with harmonic-oscillator central potential $\Sigma_{HO} = c_0 + c_2 r^2$ and constant $\Delta_{HO} = d_0$, whose eigenfunctions $g$ (upper) and $f$ (lower) are known analytically. The perturbation operator $W = \mathrm{diag}(\Sigma - \Sigma_{HO}, d_0 - \Delta)$ is applied with the three reference parameters fixed by minimizing a perturbation parameter; in practice only the oscillator frequency is optimized. The derivation then expresses the energy gap at first order as integrals over the integrated density differences $F_i(r) = \int_0^r dr'\, r'^2 f_i(r')^2$ and $G_i(r) = \int_0^r dr'\, r'^2 g_i(r')^2$, weighted by the surface derivatives $\Sigma'$ and $\Delta'$. These integrated densities and their overlaps with the potential derivatives carry the whole argument about $A$-dependence.

What would settle it

Compute the first-order closed-form gaps using the actual self-consistent $\Sigma$ and $\Delta$ potentials without Woods-Saxon fitting for several isotonic chains and several pseudo-spin pairs; if the upper-component term no longer grows roughly linearly with $A$ or the lower-component term no longer stays roughly constant, the claimed $A$-dependence pattern is an artifact of the fits.

Watch

Extended reading notes

Core claim

At first order in perturbation theory around the spin-symmetric relativistic harmonic oscillator, the spin-orbit splitting of a doublet collapses to $\Delta E_{SS} \approx (F_\downarrow(R) - F_\uparrow(R)) \Delta_0$, where $F_i(r)$ is the integrated lower-component density and $\Delta_0$ is the depth of $\Delta = S - V$. The pseudo-spin splitting takes the analogous but two-term form $\Delta E_{PSS} = \int dr\, (G_a - G_b) \Sigma' + \int dr\, (F_a - F_b) \Delta'$, where $G_i$ is the integrated upper-component density. The paper argues that these closed forms capture the dominant mechanisms: the spin gap inherits its $A^{-2/3}$ scaling from the relativistic weakening of the lower component (norm $\sim \omega \sim A^{-1/3}$) and from the decreasing overlap between the surface-peaked $\Delta'$ and the integrated density difference (another $A^{-1/3}$); the pseudo-spin gap is governed by the competition between a $\Sigma'$-driven upper-component term that grows roughly linearly with $A$ and a $\Delta'$-driven lower-component term that remains almost constant. This gives a common footing for spin and pseudo-spin symmetry breaking and explains the decomposition $\Delta E_{PSS} = \Delta E_{HO} - \Delta E_{SO}$, where the central-potential term and the spin-orbit term have opposite signs.

Load-bearing premise

The conclusions about how pseudo-spin splitting changes with mass number depend on the assumption that Woods-Saxon fits to the self-consistent potentials faithfully reproduce the surface slopes of $\Sigma$ and $\Delta$, and that the behavior seen in one neutron orbital pair along the $N=50$ chain is generic.

Editorial extensions

If this is right

If the first-order closed forms are correct, the spin-orbit splitting of any doublet can be computed directly from the integrated lower-component density and the surface slope of $\Delta$, without solving the full Dirac equation.
The $A^{-2/3}$ scaling of spin-orbit splittings is explained as the product of two kinematic effects: the relativistic weakening of the lower component and the geometric mismatch between $F_\downarrow - F_\uparrow$ and $\Delta'$, making the scaling a consequence of relativistic dynamics rather than a phenomenological input.
For pseudo-spin doublets, the near-constancy of the lower-component term and the roughly linear growth of the upper-component term imply that the total pseudo-spin splitting can change sign along a chain, which would show up as accidental (re)appearance of pseudo-spin degeneracy.
Because the perturbation is controlled ($W_m \ll 1$) for both spin and pseudo-spin partners, the same reference state supports both calculations, so spin and pseudo-spin phenomena can be studied with one consistent expansion.

Reading between the lines

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

Since the paper's A-dependence analysis rests on a single orbital pair in the N=50 chain, an immediate test is whether the linear-in-A growth of $\Delta E_g$ and the constancy of $\Delta E_f$ survive for other pseudo-spin pairs (e.g., in N=82 or N=126 isotones) and for proton doublets, where the Coulomb potential modifies $\Sigma$ and $\Delta$.
The framework suggests a practical way to predict shell evolution far from stability: if the surface diffusivity of $\Sigma$ changes (e.g., in neutron-rich nuclei with thick neutron skins), the linear term $\Delta E_g$ will grow, pushing pseudo-spin partners apart; the paper's mechanism would then predict where magic gaps weaken.
The same perturbative machinery could be carried to second order systematically (the paper shows second order shifts values but preserves trends) to build an analytic map of where pseudo-spin symmetry is accidentally restored across the nuclear chart.

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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

Honest take: this paper has a useful framework but one of its central formulas, Eq. (12), is printed with the wrong sign in the ΔEg term. The total first-order expression, Eq. (4), is correct, so the N=50 benchmark in Fig. 2 is not compromised. The sign error sits specifically in the decomposition that drives the paper's interpretation of why ΔEg grows with A.

What is genuinely new here is the closed-form first-order decomposition of pseudo-spin splitting into a g/Σ part and an f/Δ part, and the identification of two distinct A-behaviors. The spin-symmetry part is well executed: the recovery of the A−2/3 scaling as a product of a norm decrease and an overlap decrease is clean and gives a mechanistic picture that goes beyond quoting the usual non-relativistic derivation. The optimization of the HO reference against the perturbation parameter follows Ref. [24] and is done properly.

The soft spot is the sign. Starting from Eq. (4), with A = Ga−Gb, the g contribution is ∫ r²(ga²−gb²)Σ dr = ∫ A' Σ dr = −∫ A Σ' dr, after dropping the ΣHO piece that vanishes by degeneracy. So the first term in Eq. (12) should be −∫(Ga−Gb)Σ' dr, not +. The f term is correctly +∫(Fa−Fb)Δ' dr because the boundary term cancels. The consequence is material: the text around Fig. 6 argues that as A increases, Σ' increasingly probes the positive lobe of Ga−Gb and therefore ΔEg increases. With the minus sign, that same overlap would make ΔEg decrease. The story inverts. Fig. 5, if computed with the printed plus sign, would not actually represent the separate g and f contributions.

Two other things are softer. The scaling conclusions for ΔEg and ΔEf rest on Woods-Saxon fits to the self-consistent potentials along a single isotonic chain, with no fit uncertainties or sensitivity test reported. That makes the A-scaling claims suggestive rather than established. Also, no code or data are provided, which limits reproducibility.

The paper is aimed at nuclear structure theorists working on shell evolution, bubble nuclei, and relativistic mean-field. It deserves a serious referee: the framework is relevant, the error is local and fixable, and the SS part alone is worth publishing. I would send it to review with a request that the referee verify the signs and ask for a robustness check on the fits.

Referee Report

2 major / 6 minor

Summary. The paper develops a first- and second-order perturbative treatment of spin-symmetry (SS) and pseudo-spin-symmetry (PSS) breaking in relativistic mean-field models, using the spin-symmetric relativistic harmonic oscillator as the reference. It derives closed-form, first-order expressions for the SS and PSS energy splittings, analyzes their mass-number dependence along the N=50 isotonic chain, and concludes that the SS splitting scales as A^(-2/3) through two separate A^(-1/3) factors, while the PSS splitting is governed by a g/Σ contribution that grows with A and an f/Δ contribution that is approximately constant. The results are benchmarked against RHB calculations and experimental data.

Significance. If the sign issue in Eq. (12) is resolved, the unified perturbative framework is a valuable qualitative tool: it is not fitted to the energy gaps, it cleanly separates the g/Σ and f/Δ contributions to PSS breaking, and it provides a transparent covariant account of the known A^(-2/3) spin-orbit trend. The N=50 benchmark and the explicit checks of the perturbation parameter are strengths. The generality of the PSS scaling claims, however, currently depends on a single orbital pair and on Woods-Saxon fits whose quality is not reported, so the quantitative conclusions should be treated with caution until those points are addressed.

major comments (2)

[IV.B, Eq. (12)] The upper-component term in Eq. (12) has the wrong sign. From Eq. (4), the g-dependent contribution is I_g = ∫ r²(g_a²−g_b²)(Σ−Σ_HO) dr. The Σ_HO part vanishes for degenerate PSS partners, as stated in Section IV.B. Defining A(r)=∫_0^r r′²(g_a²−g_b²) dr′ (so A′=r²(g_a²−g_b²)), integration by parts gives I_g = ∫ A′Σ dr = [AΣ]_0^∞ − ∫ A Σ′ dr = −∫(G_a−G_b)Σ′ dr, because Σ(∞)=0 and A(0)=0. The printed Eq. (12) has +∫(G_a−G_b)Σ′ dr. This is not a cosmetic sign: with the shapes displayed in Fig. 6, where G_a−G_b is negative in the core and positive near the surface while Σ′ is surface-peaked, the corrected formula produces an overlap trend with A that is opposite to the mechanism described in Section IV.C.1. Please correct the sign and re-evaluate the ΔE_g curve in Fig. 5, the interpretation in Section IV.C.1, and the decomposition in Eq. (15).
[IV.C] The A-scaling conclusions for ΔE_g and ΔE_f rely on fitting the self-consistent RHB potentials Σ and Δ to Woods-Saxon forms, but the paper reports no fit parameters, residuals, or uncertainties, and no sensitivity test of the surface derivatives Σ′ and Δ′. Because the closed-form overlaps in Eq. (12) weight precisely the surface region, a small misfit in the Woods-Saxon form could change the sign or magnitude of the integrated contributions. The generalization is also drawn from a single PSS doublet, ν(2d5/2−1g7/2), along the N=50 chain. Please provide fit-quality information (e.g., parameter errors or χ²), a direct comparison of the fitted derivatives with the RHB derivatives, and at least one additional isotonic/isotopic chain or orbital pair before claiming generic scaling.

minor comments (6)

[Abstract] The abstract contains a typo: "theses spin" should read "these spin".
[Introduction and Fig. 1] The definition of Δ is inconsistent: the text defines Δ ≡ V − S, while the caption of Fig. 1 states Δ = S − V. Please unify the convention, since the sign of Δ′ enters Eqs. (8) and (12).
[Eq. (9)] Please clarify that Δ0 denotes the integrated jump of Δ across the surface, not necessarily Δ(r=0), to make the sign in Eq. (9) unambiguous.
[Fig. 2] The in-figure title labels the plotted gap as ν(1g7/2 − 1g9/2), whereas the text defines the benchmark PSS splitting as ν(2d5/2 − 1g7/2); please correct the label and verify that the plotted quantity is the intended PSS gap.
[II.C] The condition W_m < 1 is not quantified for the PSS partners used in the figures; please report the numerical values of W_m for the orbitals shown in Figs. 5–7.
[III.B.1] The numerical rescaling used to isolate effect ii) and to infer its A^(-1/3) scaling is not described; please specify the procedure and the fit used to extract the exponent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the perturbative gaps are derived from input potentials and benchmarked externally, not fitted.

full rationale

The paper's derivation chain is internally consistent and self-contained in the sense required by the circularity test. The first-order energy splitting in Eq. (4) is computed from the perturbation operator W in Eq. (3), with the SS-RHO reference parameters fixed by minimizing the perturbation parameter Wm in Eq. (6), not by matching the target energy gaps. The spin and pseudo-spin closed forms, Eqs. (8)-(9) and (12), are obtained by integration by parts from Eq. (4), so they are algebraic rearrangements of the same input potentials and reference wavefunctions rather than additional fitted relations. The comparisons shown in Figs. 2, 4, and 5 are benchmarks against independent RHB calculations and experimental data. The Woods-Saxon fits in Section IV.C are fits to the RHB potentials used as inputs; they do not target the energy splittings, and the resulting Delta_Eg / Delta_Ef decomposition is an analysis of the integrands, not a fit to the gaps. The only self-citations, Refs. [19], [23], and [28], provide background relations and a specific covariant energy functional; they are not load-bearing for the perturbative derivation, which is carried out explicitly in the present paper. No fitted parameter is renamed as a prediction, and no equation is defined in terms of the quantity it purports to predict. Therefore no circular step is identified.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the validity of first-order perturbation theory around an SS-RHO reference, on the adequacy of RHB potentials as input, and on the faithfulness of Woods-Saxon fits. No new physical entities are introduced.

free parameters (5)

c0 = -73 MeV
Constant offset of the harmonic reference central potential; chosen by minimizing the perturbation parameter Wm, not to data. Converges to the same value for all states.
d0 = -588 MeV
Constant value of the reference spin-orbit potential Delta_HO; chosen by minimizing Wm.
c2 (or omega) = not given explicitly
Harmonic oscillator strength optimized for each calculation to minimize Wm; assumed to scale as A^-1/3.
Delta0 (Woods-Saxon depth for SS) = not given
Depth of a Woods-Saxon fit to the RHB Delta potential, used in Eq. (9) to compute spin-symmetry gaps; fitted to the potential, not to the gap.
Woods-Saxon parameters for Sigma and Delta (PSS) = not given
Approximate the RHB potentials to isolate DeltaEg and DeltaEf contributions; fit fidelity is not quantified.

assumptions (6)

domain assumption The perturbation series in W converges; Wm < 1 for the states considered
Section II C states Wm ~ 10^-2 for SS partners and checks Wm < 1, but no proof of convergence is given for all chains.
domain assumption Spherical symmetry and self-consistent RMF potentials at the RHB level with DD-MEV parametrization adequately describe the nucleus
Section II A/B uses spherical RHB potentials; this is a standard domain assumption in nuclear structure.
standard math The non-relativistic identification of the upper component g with the Schrodinger wavefunction and V_ls ~ (1/r) dDelta/dr
Section II A, Eq. (2); used to interpret spin-symmetry scaling.
ad hoc to paper Delta'(r) is sharply peaked at r ~ R and can be approximated by a delta distribution
Section III B 1, below Eq. (8); this approximation is load-bearing for Eq. (9) and the authors note it is not valid for bubble nuclei.
domain assumption The harmonic oscillator frequency omega scales as A^-1/3
Used in Eq. (11) to infer that the norm of the lower component f decreases as A^-1/3; standard nuclear matter scaling.
ad hoc to paper The radial structures of F_down - F_up and Ga - Gb are generic for the orbitals considered
Section IV C relies on the sign change of Ga - Gb and the shape of Fa - Fb; only demonstrated for the N=50 chain.

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Pith. "Pith review of Patterns of spin and pseudo-spin symmetries in nuclear relativistic mean-field approaches." pith.science (2026). https://pith.science/paper/MQD66D7C

@misc{pith2026250119037,
  author       = {Pith},
  title        = {Pith review of: Patterns of spin and pseudo-spin symmetries in nuclear relativistic mean-field approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQD66D7C}},
  note         = {Machine review of arXiv:2501.19037}
}

read the original abstract

The behavior of spin doublets is known to play a major role in nuclear structure and shell effects. Pseudo-spin doublets are also known to impact the single-particle spectrum. The covariant framework, having these two effects encoded in its approach, is an excellent tool to understand the main mechanism driving theses spin and pseudo-spin symmetries and their breaking. A perturbative expansion of the degeneracy raising related to spin and pseudo-spin effects is proposed, up to second order. It allows to understand the main behavior of spin and pseudo-spin energy doublets, such as their A dependence, as well as their common footing and differences. In the case of the spin symmetry, only the lower component of the Dirac bi-spinor is involved, whereas in the case of the pseudo-spin one, both the upper and lower components are involved. Their interplay with the covariant potentials is also analyzed.

Figures

Figures reproduced from arXiv: 2501.19037 by the authors.

**Figure 2.** Neutron 1 ˜f PSS gap in N = 50 isotonic chain. Grey squares correspond the RHB calculation output, red up triangles to first order in perturbation (Eq. 4) with RHB Σ and ∆ potentials, and orange down triangles to second order in perturbation (Eq. 5). A. General scaling of SS breaking In non-relativistic approaches the scaling of SS breaking with mass number A can be derived by approximating the spin-orbit potential… view at source ↗

**Figure 3.** Evolution along N = 50 of F↓ − F↑ (Eq. (9)). Fi = R r 0 dr′ r ′2 f(r′ ) 2 is the integral of the lower component density. ∆′ is the spatial derivative of ∆. quantum number j (see section II.A), Eq. (4) reduces to : ∆E (1) SS ≡ ∆E (1) = Z dr r2 (f2 ↓ − f 2 ↑ ) (d0 − ∆(r)) (7) which yields, after integration by parts : ∆E (1) = Z dr (F↓ − F↑) d∆ dr , (8) where Fi(r) ≡ R r 0 dr′ r ′2 fi(r′ ) 2 is the integrated lowerc… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Perturbation at first order for spin symmetry [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Contribution to the PSS gap of the upper [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: Respective evolution of normalised ∆′ (dashed curves) and Fa − Fb the integrated lower component densities (plain curves) along N = 50 isotonic chain, for 1 ˜f partners. cording to equation (11). The second effect is geometrical and tends to increase the value of ∆Ef :…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 37 canonical work pages

[24]

Delafosse, D

C. Delafosse, D. Verney, P. Marevi´ c, A. Gottardo, C. Michelagnoli, A. Lemasson, A. Goasduff, J. Ljung- vall, E. Cl´ ement, A. Korichi, G. De Angelis, C. Andreoiu, M. Babo, A. Boso, F. Didierjean, J. Dudouet, S. Fran- choo, A. Gadea, G. Georgiev, F. Ibrahim, B. Jacquot, T. Konstantinopoulos, S. M. Lenzi, G. Maquart, I. Matea, D. Mengoni, D. R. Napoli, T....

work page 2018
[1]

Let us now derive the energy gap at first order

General scaling It should be first reminded that at zeroth order, ∆E(0) = 0, as the HO reference state is spin-symmetric. Let us now derive the energy gap at first order. Using the independence of the upper component g from the 4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 r/R 0.0 0.2 0.4 0.6 0.8 1.0 A ↗ ∆ ′ F↓ − F↑ N = 50, ν(1g7/2 − 1g9/2) A ↗ ∆ ′ F↓ − ...

work page
[2]

The origin of this decrease can also be qualitatively explained

We now prove i) : FN (R) ≤ Z ∞ 0 dr r2f 2 N(r) ∝ ω eM N + 3 2 ∝ ω ∼ A−1/3, (11) with fM = M + (c0 − d0)/2 and ω the harmonic oscillator frequency that generically scales as A−1/3. The origin of this decrease can also be qualitatively explained. As A increases,the binding potential extends over a larger region, confining nucleons in a broader region L. Us-...

work page
[3]

4 shows the experimental trend of SS split- ting [30], the output of RHB calculations as well as those obtained with Eq

Typical trends and deviations Fig. 4 shows the experimental trend of SS split- ting [30], the output of RHB calculations as well as those obtained with Eq. (8). The trend of SS split- ting has been probed experimentally [30] and leads to ∆E ∼ 24.5(ℓ + 1/2)A−2/3. This bulk law has been tested for n = 1 for several ℓ. Here, Eq. (9) has been applied, with a ...

work page
[4]

It is possible to understand the typical scaling with A by analyzing the typical behaviours of the two integrands of ∆ Eg (see Eq

∆Eg We first analyze the role of the term related to the up- per components. It is possible to understand the typical scaling with A by analyzing the typical behaviours of the two integrands of ∆ Eg (see Eq. (12)), displayed on Fig. 6 for increasing value of A. In the case of SS , F↓ −F↑ was always positive and featured a clear peak. In the present case, ...

work page
[5]

∆Ef We now focus on the second term of the r.h.s. of Eq. (12). In the case of ∆ Ef , the lower components wave- functions fa and fb have similar structures, leading to a smoother behavior: in the limit case of exact PSS, fa = −fb [25, 32]. Both integrands of ∆ Ef are depicted on Fig. 7. There are two competing effects leading to an approxi- mate constant ...

work page
[6]

General mechanisms to PSS breaking In light of the present analysis, we offer a systematic framework to better understand PSS splitting, grounded in a perturbative approach. It should be noted that a previous study [33] made a significant progress in ex- ploring the mechanisms of PSS breaking, though it re- lied on a non-perturbative method where energy s...

work page
[7]

Baranger, A definition of the single-nucleon potential, Nuclear Physics A 149, 225 (1970)

M. Baranger, A definition of the single-nucleon potential, Nuclear Physics A 149, 225 (1970)

work page 1970

Show all 40 references

[8]

Duguet and G

T. Duguet and G. Hagen, Ab initio approach to effec- tive single-particle energies in doubly closed shell nuclei, Physical Review C 85, 034330 (2012)

2012
[9]

Duguet, H

T. Duguet, H. Hergert, J. D. Holt, and V. Som` a, Nonob- servable nature of the nuclear shell structure: Meaning, illustrations, and consequences, Physical Review C 92, 034313 (2015)

2015
[10]

Som` a and T

V. Som` a and T. Duguet, On the calculation and use of effective single-particle energies: The example of the neu- tron 1d3/2–1d5/2 splitting along n = 20 isotones, Philo- sophical Transactions of the Royal Society A: Mathemat- ical, Physical and Engineering Sciences 382, 2023...

2024
[11]

M. G. Mayer, On closed shells in nuclei. ii, Phys. Rev. 75, 1969 (1949)

1949
[12]

M. B. R. Bohr A, Nuclear Structure, Volume 1 , w. a. benjamin ed. (1969)

1969
[13]

Hecht and A

K. Hecht and A. Adler, Generalized seniority for favored J ̸= 0 pairs in mixed configurations, Nuclear Physics A 137, 129 (1969)

1969
[14]

Arima, M

A. Arima, M. Harvey, and K. Shimizu, Pseudo LS cou- pling and pseudo SU3 coupling schemes, Physics Letters B 30, 517 (1969)

1969
[15]

J. N. Ginocchio, Pseudospin as a relativistic symmetry, Phys. Rev. Lett. 78, 436 (1997)

1997
[16]

Dudek, W

J. Dudek, W. Nazarewicz, Z. Szymanski, and G. A. Le- ander, Abundance and systematics of nuclear superde- formed states; relation to the pseudospin and pseudo- SU(3) symmetries, Physical Review Letters 59, 1405 (1987)

1987
[17]

Bahri, J

C. Bahri, J. Draayer, and S. Moszkowski, Pseudospin symmetry in nuclear physics, Physical Review Letters68, 2133 (1992)

1992
[18]

Nazarewicz, M

W. Nazarewicz, M. A. Riley, and J. D. Garrett, Equi- librium deformations and excitation energies of single- quasiproton band heads of rare-earth nuclei, Nuclear Physics A 512, 61 (1990)

1990
[19]

Nazarewicz, P

W. Nazarewicz, P. J. Twin, P. Fallon, and J. D. Garrett, Natural-parity states in superdeformed bands and pseudo SU(3) symmetry at extreme conditions, Physical Review Letters 64, 1654 (1990)

1990
[20]

J. Y. Zeng, J. Meng, C. S. Wu, E. G. Zhao, Z. Xing, and X. Q. Chen, Spin determination and quantized alignment in the superdeformed bands in Dy 152 , Tb 151 , and Gd 150, Physical Review C 44, R1745 (1991)

1991
[21]

von Neumann-Cosel and J

P. von Neumann-Cosel and J. N. Ginocchio, L -forbidden M 1 transitions and pseudospin symmetry, Physical Re- view C 62, 014308 (2000)

2000
[22]

Troltenier, W

D. Troltenier, W. Nazarewicz, Z. Szyma´ nski, and J. Draayer, On the validity of the pseudo-spin concept for axially symmetric deformed nuclei, Nuclear Physics A 567, 591 (1994)

1994
[23]

J. N. Ginocchio, Implications of pseudospin symmetry on relativistic magnetic properties and Gamow-Teller tran- sitions in nuclei, Physical Review C 59, 2487 (1999)

1999
[25]

Ebran, E

J.-P. Ebran, E. Khan, A. Mutschler, and D. Vretenar, Spin–orbit coupling rule in bound fermion systems, Jour- nal of Physics G: Nuclear and Particle Physics43, 085101 (2016)

2016
[26]

Heitz, J.-P

L. Heitz, J.-P. Ebran, E. Khan, and D. Verney, A uni- fied mechanism for the origin and evolution of nuclear magicity, arXiv 10.48550/arXiv.2411.15562 (2024)

2024 doi arXiv
[27]

Liang, J

H. Liang, J. Meng, and S.-G. Zhou, Hidden pseudospin and spin symmetries and their origins in atomic nuclei, Physics Reports 570, 1 (2015)

2015
[28]

J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, and A. Arima, Pseudospin symmetry in relativistic mean field theory, Physical Review C 58, R628 (1998)

1998
[29]

Ebran, A

J.-P. Ebran, A. Mutschler, E. Khan, and D. Vretenar, Spin-orbit interaction in relativistic nuclear structure models, Physical Review C 94, 024304 (2016)

2016
[30]

Liang, P

H. Liang, P. Zhao, Y. Zhang, J. Meng, and N. V. Giai, Perturbative interpretation of relativistic symmetries in nuclei, Physical Review C 83, 041301 (2011)

2011
[31]

J. N. Ginocchio, Relativistic harmonic oscillator with spin symmetry, Physical Review C 69, 034318 (2004)

2004
[32]

P. G. Reinhard, The relativistic mean-field description of nuclei and nuclear dynamics, Reports on Progress in Physics 52, 439 (1989)

1989
[33]

Niksic, N

T. Niksic, N. Paar, D. Vretenar, and P. Ring, DIRHB – a relativistic self-consistent mean-field framework for atomic nuclei, Computer Physics Communications 185, 1808 (2014), arxiv:1403.4039

2014 arXiv
[34]

Mercier, J.-P

F. Mercier, J.-P. Ebran, and E. Khan, Covariant energy density functionals with and without tensor couplings at the Hartree-Bogoliubov level, Physical Review C 107, 034309 (2023)

2023
[35]

Bohr and B

A. Bohr and B. Mottelson, Nuclear Structure (World Sci- entific Publishing Company, 1998)

1998
[36]

Mairle, Scaling of measured nuclear spin-orbit split- tings, Physics Letters B 304, 39 (1993)

G. Mairle, Scaling of measured nuclear spin-orbit split- tings, Physics Letters B 304, 39 (1993)

1993
[37]

Otsuka, A

T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, and Y. Utsuno, Evolution of shell structure in exotic nuclei, Rev. Mod. Phys. 92, 015002 (2020)

2020
[38]

J. N. Ginocchio, Pseudospin symmetry and relativistic mean field eigenfunctions, Physical Review C 66, 064312 (2002)

2002
[39]

Alberto, M

P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, and M. Chiapparini, Pseudospin symmetry as a relativistic 9 dynamical symmetry in the nucleus, Physical Review C 65, 034307 (2002)

2002
[40]

Dobaczewski, I

J. Dobaczewski, I. Hamamoto, W. Nazarewicz, and J. A. Sheikh, Nuclear shell structure at particle drip lines, Phys. Rev. Lett. 72, 981 (1994)

1994

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [24]

Delafosse, D

C. Delafosse, D. Verney, P. Marevi´ c, A. Gottardo, C. Michelagnoli, A. Lemasson, A. Goasduff, J. Ljung- vall, E. Cl´ ement, A. Korichi, G. De Angelis, C. Andreoiu, M. Babo, A. Boso, F. Didierjean, J. Dudouet, S. Fran- choo, A. Gadea, G. Georgiev, F. Ibrahim, B. Jacquot, T. Konstantinopoulos, S. M. Lenzi, G. Maquart, I. Matea, D. Mengoni, D. R. Napoli, T....

work page 2018

[2] [1]

Let us now derive the energy gap at first order

General scaling It should be first reminded that at zeroth order, ∆E(0) = 0, as the HO reference state is spin-symmetric. Let us now derive the energy gap at first order. Using the independence of the upper component g from the 4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 r/R 0.0 0.2 0.4 0.6 0.8 1.0 A ↗ ∆ ′ F↓ − F↑ N = 50, ν(1g7/2 − 1g9/2) A ↗ ∆ ′ F↓ − ...

work page

[3] [2]

The origin of this decrease can also be qualitatively explained

We now prove i) : FN (R) ≤ Z ∞ 0 dr r2f 2 N(r) ∝ ω eM N + 3 2 ∝ ω ∼ A−1/3, (11) with fM = M + (c0 − d0)/2 and ω the harmonic oscillator frequency that generically scales as A−1/3. The origin of this decrease can also be qualitatively explained. As A increases,the binding potential extends over a larger region, confining nucleons in a broader region L. Us-...

work page

[4] [3]

4 shows the experimental trend of SS split- ting [30], the output of RHB calculations as well as those obtained with Eq

Typical trends and deviations Fig. 4 shows the experimental trend of SS split- ting [30], the output of RHB calculations as well as those obtained with Eq. (8). The trend of SS split- ting has been probed experimentally [30] and leads to ∆E ∼ 24.5(ℓ + 1/2)A−2/3. This bulk law has been tested for n = 1 for several ℓ. Here, Eq. (9) has been applied, with a ...

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[5] [4]

It is possible to understand the typical scaling with A by analyzing the typical behaviours of the two integrands of ∆ Eg (see Eq

∆Eg We first analyze the role of the term related to the up- per components. It is possible to understand the typical scaling with A by analyzing the typical behaviours of the two integrands of ∆ Eg (see Eq. (12)), displayed on Fig. 6 for increasing value of A. In the case of SS , F↓ −F↑ was always positive and featured a clear peak. In the present case, ...

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[6] [5]

∆Ef We now focus on the second term of the r.h.s. of Eq. (12). In the case of ∆ Ef , the lower components wave- functions fa and fb have similar structures, leading to a smoother behavior: in the limit case of exact PSS, fa = −fb [25, 32]. Both integrands of ∆ Ef are depicted on Fig. 7. There are two competing effects leading to an approxi- mate constant ...

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[7] [6]

General mechanisms to PSS breaking In light of the present analysis, we offer a systematic framework to better understand PSS splitting, grounded in a perturbative approach. It should be noted that a previous study [33] made a significant progress in ex- ploring the mechanisms of PSS breaking, though it re- lied on a non-perturbative method where energy s...

work page

[8] [7]

Baranger, A definition of the single-nucleon potential, Nuclear Physics A 149, 225 (1970)

M. Baranger, A definition of the single-nucleon potential, Nuclear Physics A 149, 225 (1970)

work page 1970