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K. Neergård

Publications and source records attributed to K. Neergård.

18 recordsLinked to original sources
arXiv2026

Relation to a property of the angular momentum zero space of states of four fermions in an angular momentum $j = 9/2$ shell unexpectedly found to be stationary for any rotationally invariant two-body interaction

The existence of states with angular momenta $I = 4$ and~6 of four fermions in an angular momentum $j = 9/2$ shell that are stationary for any rotationally invariant two-body interaction despite the presence of other states with the same angular momentum, the Escuderos-Zamick states, is shown to be equivalent to the invariance to any such interaction of the span of states generated from $I = 0$ states by one-body operators. This invariance is verified by exact calculation independently of previous verifications of the equivalent statement. It explains the occurrence of the Escuderos-Zamick states for just $I = 4$ and 6. The action of an arbitrary interaction on the invariant space and its orthogonal complement is analyzed, leading to a relation of the Escuderos-Zamick energy levels to levels with $I = 10$ and 12. Parts of the observed spectra of $^{72}$Ni, $^{74}$Ni, $^{94}$Ru, $^{96}$Pd, $^{212}$Rn, and $^{214}$Ra are discussed in the light of this relation.

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arXiv2023

"Onishi" formulas

The term "Onishi" formula refers to a family of related formulas for the overlap amplitude of two Bogolyubov quasi-fermion vacua. As a common feature, these formulas display a square root, which gives rise to an apparent sign ambiguity. For some members of the family, this sign ambiguity is real and unavoidable while in some other cases it is not. The relation between the different Onishi formulas is discussed.

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arXiv2023

Fock space representations of Bogolyubov transformations as spin representations

The representation on a Fock space of the group of Bogolyubov transformations is recognized as the spin representation of an orthogonal group. Derivations based on this observation of some known formulas for the overlap amplitude of two Bogolyubov quasi-fermion vacuum states that are in some cases more complete than those in the literature are shown. It is pointed out that the name of an "Onishi formula" is assigned in the literature to two different expressions which are related but have different scopes. One of them has what has been described as a sign problem, the other one, due to Onishi and Yoshida, has a more limited scope and no sign problem. I give a short proof of the latter, whose derivation is missing in the paper by Onishi and Yoshida, and a new, combinatoric, proof of an equivalent formula recently derived by Robledo.

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arXiv2023

Fock space dualities

Several cases of Fock space duality occurring in the theory of many-body systems in general and nuclei in particular are discussed. All of them are special cases of a general duality theorem proved in mathematics by Howe in the 1970s. Dualities on a fermion Fock space between orthogonal Lie algebras and related groups, including an o-Pin duality recently discovered by the author, present a nice, symmetric pattern.

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arXiv2020

Fock space dualities

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal-orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half a century ago in the analogous case of symplectic-symplectic duality, one can base a proof of the orthogonal-orthogonal duality theorem and a precise characterization of the relation between the equivalence classes of the dually related irreducible representations on a calculation of characters by combining it with an analysis of the representation of a reflection. Young diagrams for the description of equivalence classes of irreducible representations of orthogonal Lie algebras are introduced. The properties of a reflection of the number non-conserving part in the dual relationship between orthogonal Lie algebras corroborate a picture of an almost perfect symmetry between the partners.

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arXiv2019

Proof by characters of the orthogonal-orthogonal duality and relations of Casimir invariants

The theorem of orthogonal-orthogonal duality of Rowe, Repka, and Carvalho is proven by a method based on characters that is very different from theirs and akin to Helmers's half a century earlier proof of the analogous sympletic-symplectic duality. I demonstrate how three duality theorems listed by Rowe, Repka, and Carvalho allow very brief derivations of linear relations between the Casimir invariants of the connected representations based on the geometry of their Young diagrams, and discuss for which physical systems other than such already considered in the literature an analysis in terms of the orthogonal-orthogonal duality might be useful.

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arXiv2019

Role of pair-vibrational correlations in forming the odd-even mass difference

In the random-phase-approximation-amended (RPA-amended) Nilsson-Strutinskij method of calculating nuclear binding energies, the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers $N$ and $Z$ of neutrons and protons and even mass number $A = N + Z$. Here it is applied to odd-$A$ nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with $Z = N - 1$ and $25 \le A \le 99$. (ii) The odd-$A$ isotopes of In, Sn, and Sb with $46 \le N \le 92$. (iii) The odd-$A$ isotopes of Sr, Y, Zr, Nb, and Mo with $60 \le N \le 64$. The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper $sd$ shell this correction accounts for almost the entire odd-even mass difference for odd $Z$ and about half of it for odd $N$. The size and sign of the RPA contribution varies, which is explained qualitatively in terms of a closed expression for a smooth RPA counter term.

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arXiv2018

Closed expression for the pair vibrational correlation energy of a uniform distribution of single-nucleon levels

A closed expression is derived for the pair vibrational correlation energy generated in the random phase approximation by the isovector pairing force in the case when Kramers and charge degenerate single-nucleon levels are uniformly distributed in an interval. The expression is used to analyze the spectral density of pair vibrational frequencies relative to that of two-quasinucleon energies. Applications to the analysis of the symmetry energy of the isovector pairing model and to a Strutinskij renormalization of this model are discussed.

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arXiv2017

Symmetry energies for $A = 24$ and $48$ and the USD and KB3 shell model Hamiltonians

Calculations in the sd and pf shells reported some time ago by Satuła\etal\ [Phys.~Lett.~B~407, 103 (1997)] are redone for an extended analysis of the results. As in the original work, we do calculations for one mass number in each shell and consider in each case the sequence of lowest energies for isospins 0, 2, and 4, briefly the symmetry spectrum. Following further the original work we study how this spectrum changes when parts of the two-nucleon interaction are turned off. The variation of its width is explored in detail. A differential combination $ε_\text{W}$ of the three energies was taken in the original work as a measure of the so-called Wigner term in semi-empirical mass formulas, and it was found to decrease drastically when the two-nucleon interaction in the channel of zero isospin is turned off. Our analysis shows that the width of the symmetry spectrum experiences an equally drastic decrease, which can be explained qualitatively in terms of schematic approximations. We therefore suggest that the decrease of $ε_\text{W}$ be seen mainly as a side effect of a narrowing of the symmetry spectrum rather than an independent manifestation of the two-nucleon interaction in the channel of zero isospin.

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arXiv2017

Pairing Theory of the Wigner Cusp

Conclusions: (1) Calculations with an RPA correction added to the BCS pairing correction conventionally employed in Nilsson-Strutinskij calculations account well for the variation with A of the pattern of masses near N=Z. (2) The RPA correction is insignificant for reproducing the doubly even masses and hence for the shape of the Wigner cusp. (3) It is important, however, that the macroscopic (liquid drop) symmetry energy be proportional to T(T+1). (4) This form of the macroscopic symmetry energy is understood microscopically, in terms of the RPA, to result from the nuclear superfluidity. (5) The variation of the shape of the Wigner cusp is dominated by shell effects. (6) The RPA correction significally reduces the T=0 binding in doubly odd nuclei, thus reducing the required pair coupling constant.

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arXiv2015

Conservation laws in the $1f_{7/2}$ shell model of $^{48}$Cr

Conservation laws in the $1f_{7/2}$ shell model of $^{48}$Cr found in numeric studies by Escuderos, Zamick and Bayman [A. Escuderos, L. Zamick, and B. F. Bayman, arXiv:0506050 (2005)] and me [K. Neergård, Phys. Rev. C \textbf{90}, 014318 (2014)] are explained by symmetry under particle-hole conjugation and the structure of the irreps of the symplectic group Sp(4). A generalization is discussed.

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arXiv2014

Symplectic group structure of the 48Cr, 88Ru, and 92Pd ground states

The ground states of 48Cr, 88Ru, and 92Pd are studied in the 1f7/2 or 1g9/2 shell model with effective interactions from the literature. They are found to be composed, quite independently of the shell and the interaction, roughly of 75% of (s,t)=(0,0) and 25% of (s,t)=(4,0), where s is the seniority and t the reduced isospin. Other irreps of the symplectic group Sp(2j+1), where j is the single-nucleon angular momentum, make only very small contributions. The state chi obtained by antisymmetriziation and normalization of the ground state in the stretch scheme of Danos and Gillet [M. Danos and V. Gillet, Phys. Rev. 161, 1034 (1967)] has a very different structure where the Sp(2j+1) irreps other than (s,t)=(0,0) and (4,0) contribute 20% and 41% for j=7/2 and 9/2, respectively. The contributions of chi and the s=0 state to the calculated ground states state are about equal for 48Cr. For 88Ru and 92Pd the s=0 state is unambiguously a better approximation to the calculated states than chi. A state chi' obtained by antisymmetrization and normalization of the product of two stretch-scheme ground states of the system with two valence nucleons or nucleon holes of each type has much larger overlaps with the calculated ground states than chi but a deviating Sp(2j+1) decomposition.

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arXiv2014

Nuclear masses near N = Z from Nilsson-Strutinsky calculations with pairing corrections beyond BCS from an isospin-conserving pairing force

A model with nucleons in a charge-independent potential well interacting by an isovector pairing force is considered. For a 24-dimensional valence space, the Hartree-Bogolyubov (HB) plus random phase approximation (RPA) to the lowest eigenvalue of the Hamiltonian is shown to be accurate except near values of the pairing force coupling constant G where the HB solution shifts from a zero to a non-zero pair gap. In the limit G -> infinity the HB + RPA is asymptotically exact. The inaccuracy of the HB + RPA in the critical regions of G can be remedied by interpolation. The resulting algoritm is used to calculate pairing corrections in the framework of a Nilsson-Strutinsky calculation of nuclear masses near N = Z for A = 24-100, where N and Z are the numbers of neutrons and protons, and A = N + Z. The dimension of the valence space is 2A in these calculations. Adjusting five liquid drop parameters and a power law expression for the constant G as a function of A allows us to reproduce the measured binding energies of 112 doubly even nuclei in this range with a root mean square deviation of 0.95 MeV. Several combinations of the masses for different N, Z, and isospin T are considered and the calculations found to be in good agreement with the data. It is demonstrated by examples how fluctuations as a function of A of the constant X in an expansion of the symmetry energy of the form T(T+X)/(2 theta) can be understood from the shell structure.

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arXiv2013

Cooperation of anti-aligning and aligning shell-model forces for N=Z

For two neutrons and two protons or two neutron holes and two proton holes in a single j-shell, the state |phi> with isospin and seniority zero and the lowest angular momentum zero state |chi> produced by an attractive interaction of quasinucleon pairs with angular momentum 2j have a large overlap for all relevant j and large contents of quasinucleon pairs with angular momenta 2j and 0, respectively. In the 1f7/2 and 1g9/2 shells, the large negative matrix elements of the effective interaction in these two channels relative to most of the rest therefore_cooperate_ to produce a ground state which is essentially a linear combination of |phi> and |chi> with comparable coefficients. Interaction matrix elements in other channels influence significantly the_ratio_ of these coefficients. The state |phi> makes up about 80 % of the calculated ground states. The overlaps of the latter with |chi> are_less_ in the 1g9/2 shell than in the 1f7/2 shell.

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arXiv2009

Pairing theory of the symmetry energy

As a model which displays a picture of the symmetry energy as an energy of rotation in isospace of a Cooper pair condensate, a Hamiltonian with a pairing force and an interaction of isospins is analyzed in the Hartree-Bogolyubov (HB) plus Random Phase (RPA) approximation. The HB energy minus Lagrangian multiplier terms is shown to be locally minimized by a product of neutron and proton Bardeen-Cooper-Schrieffer states. Nambu-Goldstone RPA solutions appear due to global gauge invariance and isobaric invariance. In an idealized case of infinitely many equidistant single-nucleon levels, the symmetry energy is composed of contributions from the single-nucleon and isospin interaction energies and the RPA correlation energy. The contribution of the latter is dominated by a neutron-proton Nambu-Goldstone solution, which makes the total symmetry energy nearly proportional to T(T+1). Observations reported from Skyrme force calculations are discussed in the light of these results. Calculations with deformed Woods-Saxon single-nucleon levels give results similar to those of the idealized case, whereas a somewhat different behavior is found with spherical Woods-Saxon levels. The calculations with Woods-Saxon single-nucleon levels reproduce surprisingly well the empirical symmetry energy.

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arXiv2000

Moments of inertia for multi-quasiparticle configurations

Tilted-axis cranking calculations have been performed for multi-quasiparticle states in well deformed A$\approx$180 nuclei. In the limit of zero pairing, not only are the calculated moments of inertia substantially smaller than for rigid rotation, but also they are close to the experimental values. The moments of inertia are found to be insensitive to dynamic pair correlations.

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