The many-nucleon unified model and the dynamics of nuclear rotations

TL;DR

This paper demonstrates that the symplectic model uniquely extends the Bohr-Mottelson unified model to many nucleons, providing insights into nuclear rotation dynamics and revealing that rotational energies are mixtures of potential and kinetic energies.

Contribution

It establishes the symplectic model as the unique many-nucleon extension of the unified model, enabling detailed analysis of nuclear rotation dynamics.

Findings

Rotational energies are mixtures of potential and kinetic energies.

The symplectic model spans the Hilbert space of fully anti-symmetric nuclear states.

Provides a framework for examining nuclear rotation dynamics.

Abstract

It is determined that a many-nucleon version of the Bohr-Mottelson unified model that contains the essential observables of that model and has irreducible representations that span the Hilbert space of fully anti-symmetric states of nuclei, is given uniquely by the symplectic model. This model is shown to provide a framework for an examination of the dynamics of nuclear rotations. A first discovery is that rotational energies are mixtures of potential and kinetic energies even in an adiabatic limit.

Equations

$${\hat{Q}}_{ij}=\sum _{n=1}^A{\hat{x}}_{ni}{\hat{x}}_{nj},{\hat{P}}_{ij}=\sum _{n=1}^A({\hat{x}}_{ni}{\hat{p}}_{nj}+{\hat{p}}_{ni}{\hat{x}}_{nj}),$$

$$\displaystyle\hbar\hat{L}_{ij}=\sum_{n=1}^{A}\big{(}\hat{x}_{ni}\hat{p}_{nj}-\hat{x}_{nj}\hat{p}_{ni}\big{)},\quad\hat{K}_{ij}=\sum_{n=1}^{A}\hat{p}_{ni}\hat{p}_{nj}.$$

$$\hat{\mathcal{H}}(\omega )=\frac{1}{2}{\sum }_{n=1}^A{\sum }_{i=1}^3\hslash {\omega }_i(c_{ni}^{†}{c}_{ni}+c_{ni}{c}_{ni}^{†}),$$

$$\begin{array}{r}{\hat{\mathcal{A}}}_{ij}={\sum }_{n=1}^A{c}_{ni}^{†}{c}_{nj}^{†},{\hat{\mathcal{B}}}_{ij}={\sum }_{n=1}^A{c}_{ni}{c}_{nj},\\ {\hat{\mathcal{C}}}_{ij}={\sum }_{n=1}^A(c_{ni}^{†}{c}_{nj}+\frac{1}{2}{\delta }_{i,j}),\end{array}$$

$${\hat{\mathcal{B}}}_{ij}|\sigma ,\omega ⟩=0,$$

$${\hat{\mathcal{C}}}_{ij}|\sigma ,\omega ⟩=0,$$

$${\hat{\mathcal{C}}}_{ii}|\sigma ,\omega ⟩={\sigma }_i|\sigma ⟩,$$

$${\sigma }_1{\omega }_1={\sigma }_2{\omega }_2={\sigma }_3{\omega }_3,$$

Full text

PACS numbers: 21.60.Ev, 21.60.Fw, 21.60.Cs Keywords: Unified model, dynamics of rotations, algebraic mean-field theory, shape coexistence.

Nuclei have shell structures, similar to atoms, and rotational states with properties between those of molecules and superfluids. Such properties and the numerous bands of rotational states observed in nuclear spectra Morinaga (1956); Brown and Green (1966); Heyde and Wood (2011); J. L. Wood and K. Heyde (2016) (Guest Editors) motivate attempts to understand the dynamics of nuclear rotations and to extend shell-model methods for their description to heavy nuclei, for which the huge dimensions of realistic spherical shell-model spaces currently pose formidable problems.

A fundamental objective is to determine if the rotational energies of nuclei are purely kinetic or mixtures of kinetic and potential energies. It is generally assumed that, like translations, they are kinetic but perturbed by inertial Coriolis and centrifugal forces. In the Bohr-Mottelson unified model Bohr and Mottelson (1953, 1975), in its standard adiabatic limit, the states of a rotational band are assigned a common intrinsic state and, hence, common potential energies. A rotating nucleus has also been modelled G.Rosensteel and Graber (2002); Graber and Rosensteel (2003) as a rotating Riemann ellipsoid Chandrasekhar (1969) with rotational energies given by the kinetic energies of linear combinations of rigid-body and irrotational flows. In contrast, two model calculations Park et al. (1984); Bahri and Rowe (2000) showed the kinetic and potential energy components of rotational energies to be of comparable magnitude. However, because they used schematic interactions in truncated spaces, their results can be questioned. Recall that, as a result of being restricted to single spherical harmonic-oscillator shells, SU(3) model states have zero kinetic-energies and an effective interaction is required to obtain rotational spectra.whereas the rotations of a rigid rotor are 100% kinetic.

In an extension of the unified model, Nilsson replaced its intrinsic state with an axially symmetric independent-particle state Nilsson (1955). Such an intrinsic state is viewed in HF (Hartree-Fock) theory as one of a set of states generated by rotating it through all angles. An orthonormal basis of rotational states is then obtained by angular-momentum projection methods Löwdin (1964); Kamlah (1968); Lee and Cusson (1972); Hara and Sun (1995); Bender and Heenen (2008); Sun (2016) and rotational states emerge with mixed kinetic and potential energies. However, a many-nucleon version of the unified model based on HF theory is not ideal, most significantly because its dynamical group of one-body unitary transformations has a single irrep on the whole Hilbert space of a nucleus, whereas the unified model has many representations. Ideally, what is needed is a dynamical group that decomposes a nuclear Hilbert space into a sum of subspaces with the property that all matrix elements of essential rotor-model observables between the states of its different subspaces are zero. Essential observables include the nuclear monopole and quadrupole moments, the angular-momentum operators, the nuclear kinetic energy, and SU(2)S spin and SU(2)T isospin observables.

A remarkable fact is that these essential observables are elements of the already known Lie algebra Rosensteel and Rowe (1977, 1980); Rowe (1985). of the direct product group ${\rm Sp}(3,\mathbb{R})\times{\rm SU(2)}_{S}\times{\rm SU(2)}_{T}$, where, for an $A$-nucleon nucleus, the Sp$(3,\mathbb{R})$ Lie algebra is spanned by the operators, with $1\leq i,j\leq 3$,

[TABLE]

As required, a model with these observables has an infinite number of irreps each of which contains a complete set of states of a unified model irrep with the property that there are no isoscalar E2 transitions or any non-zero matrix elements of any model observable between states of its different irreps. Physical states are inevitably mixtures of states of different model irreps. But, a many-nucleon unified model, defined in this way, has the property that the probabilities, given by the squared matrix elements of an element of its Lie algebra between states of its mixed irreps are the averages of the contributions from the mixed irreps; they cannot be enhanced above these averages by any coherent mixing of the irreps.

An Sp$(3,\mathbb{R})$ irrep of an $A$-nucleon nucleus is spanned by the positive (or negative) parity eigenstates of a three-dimensional harmonic-oscillator Hamiltonian

[TABLE]

where $c^{\dagger}_{ni}$ and $c_{ni}$ are harmonic oscillator raising and lowering operators for which $[c_{ni},c^{\dagger}_{mj}]=\delta_{m,n}\delta_{i,j}$. Then, with corresponding Sp$(3.\mathbb{R})$ operators defined by

[TABLE]

for $1\leq i,j\leq 3$, it follows that a lowest-weight state for an Sp$(3.\mathbb{R})$ irrep is defined by the equations

[TABLE]

It is now observed that the unified model is obtained as a mean-field approximation to an Sp$(3,\mathbb{R})$ irrep in which the values of $\{\omega_{i}\}$ are chosen such that the energy $\langle\sigma,\omega|\hat{H}|\sigma,\omega\rangle$, is minimised for a rotationally invariant many-nucleon Hamiltonian $\hat{H}$. The subset of $\hat{\mathcal{C}}_{ij}$ operators that are angular-momentum operators and do not annihilate the lowest-weight state then act on it to generate a set of rotated states as in standard HF theory and the SU(3) model. The $\hat{\mathcal{A}}_{ij}$ raising operators likewise act to generate purely vibrational excitations. Thus, by choosing a minimum energy lowest-weight, the first order coupling between the rotational states and their vibrations, including that of the Coriolis and centrifugal forces, is eliminated as assumed in the unified model. But, of course, it is restored in the full symplectic model.

Recall that in mean-field theory, the minimum-energy lowest-weight state satisfies a self-consistency relationship in which the density of the lowest-energy state has essentially the same shape as that of the mean field of the single-particle Hamiltonian of which it is an eigenstate. This relationship was expressed by Bohr, Mottelson Bohr and Mottelson (1955, 1975) in a harmonic-oscillator approximation to the mean field by the equiation

[TABLE]

in an interpretation of the Inglis cranking model Inglis (1954). The remarkable result is that, with its restriction to an Sp$(3,\mathbb{R})$ irrep, the mean field is identically that of a harmonic oscillator and equation (8) becomes precise.

The strength of the interaction component of $\hat{H}$ should also be such that the volume of the shape-consistent state is that expected for near-incompressible nuclear matter. Thus, the minimum-energy lowest-weight state is essentially the same for any acceptable choice of the nuclear Hamiltonian.

To illustrate the possibilities, the precise kinetic energy contributions to the rotational energies of a symplectic model have been calculated for an axially symmetric ${\rm Sp}(3,\mathbb{R})\times{\rm SU(2)}_{S}\times{\rm SU(2)}_{T}$ irrep of spin $S=0$ and minimum isospin. This irrep was previously used Bahri and Rowe (2000) to fit the lower-energy rotational states of 166Er and, in particular, the E2 transitions between them were calculated without the use of an effective charge. The chosen irrep is undoubtedly not the most appropriate Jarrio et al. (1991). However, the results obtained are characteristic of those of an axially symmetric irrep that fits the observed quadrupole moments and E2 transition rates. The results shown in figure 1

were derived by use of explicit algebraic expressions for the angular-momentum states projected from an axially symmetric harmonic-oscillator ground state Rowe et al. (2000). Because the many-nucleon kinetic-energy operator is an element of the Sp$(3,\mathbb{R})$ Lie algebra, it was then possible to calculate the kinetic energies of each of the projected angular-momentum states algebraically. A remarkable result is that the kinetic energies derived are proportional to $L(L+1)$ to a high level of accuracy.in spite of the fact that the kinetic energy component accounted for only $\sim 35\%$ of the observed rotational energies. This is consistent with the results obtained by Bahri Bahri and Rowe (2000) but is now obtained without use of a truncated space or a schematic interaction.

The above results, call for further and more detailed investigation which are now much simplified by the use of a mean-field basis for an Sp$(3,\mathbb{R})$ irrep. Desirable calculations would be to compare the results of full many-nucleon calculations in spaces of single symplectic-model irreps with those of corresponding mean field approximations with the same Hamiltonian. Initial calculations might be for the axially symmetric rotational states based on the Hoyle state of 12C, the first excited state of 16O or the ground state of 20Ne with realistic interactions. Extensions to the more general triaxial irreps and irreps with spin-orbit interactions will be more challenging. Hopefully, with the major advances, in which it has become possible to identify the states of an Sp$(3,\mathbb{R})$ irrep in SA-NCSM (symmetry-adapted no-core shell model) calculations Dytrych et al. (2018), such calculations, and others for triaxial irreps will soon be feasible. A very satisfying many-nucleon theory of deformed nuclei will then have been achieved.