Electric Multipole Operators Research Articles

Implementation of the quasiparticle finite amplitude method within the relativistic self-consistent mean-field framework (II): The program DIRQFAM v2.0.0

We describe the new version 2.0.0 of the code DIRQFAM that calculates the multipole response of even-even axially symmetric deformed nuclei using the framework of relativistic self-consistent mean-field models. The response is calculated by implementing the finite amplitude method for relativistic quasiparticle random phase approximation. In the new version, we have implemented the following features: (i) meson-exchange interactions, (ii) low-rank method for calculating induced densities and currents, (iii) generalized minimal residual method (GMRES), (iv) evaluation of the nucleon localization function, (v) extraction of the QRPA transition matrix elements and eigenfrequencies, (vi) evaluation of the H11 part of the induced Hamiltonian, (vii) multipole excitation operators up to J≤5 are now included. Program summaryProgram title:DIRQFAM v2.0.0CPC Library link to program files:https://doi.org/10.17632/6gv8nxg4ns.2Licensing provisions: GPLv3Programming language: Fortran 90.External routines/libraries: BLAS/LAPACK, version 3.1. or higher.Journal reference of previous version: A. Bjelčić, T. Nikšić, Comp. Phys. Comm. 253, 107184 (2020).Does the new version supersede the previous version?: YesNature of problem: Multipole response of deformed even-even open-shell nuclei can be calculated using the quasiparticle finite amplitude method (QFAM), based on the relativistic self-consistent mean-field models. The particle-hole channel is described by a zero-range relativistic effective interaction or interaction based on meson exchange, while the particle-particle channel of the effective inter-nucleon interaction is described by a separable finite-range pairing force. The method can be applied to perform systematic studies of collective modes even in heavy deformed nuclei.Solution method: The current implementation computes the multipole response by solving the QFAM equations in a self-consistent iteration scheme. At each iteration the QFAM solutions are updated using the GMRES method. The QFAM amplitudes are expanded in the simplex-y harmonic oscillator basis.Summary of revisions:1.meson-exchange interaction are included in the code;2.calculation of the nucleon localization function;3.extraction of the QRPA transition matrix elements and eigenfrequencies;4.evaluation of the H11 part of the induced Hamiltonian;5.multipoles excitation operators up to J≤5 are included;6.low-rank method for calculating induced densities and currents;7.generalized minimal residual method;8.dynamic memory allocation;9.reduced memory requirements;Additional comments including restrictions and unusual features: Open-shell even-even nuclei with parity conserved axially symmetric ground states are considered. An electric multipole operator with J≤5 is used to calculate the response function.

Implementation of the quasiparticle finite amplitude method within the relativistic self-consistent mean-field framework: The program DIRQFAM

The DIRQFAM code calculates the multipole response of even–even axially symmetric deformed nuclei using the framework of relativistic self-consistent mean-field models. The response is calculated by implementing the finite amplitude method for relativistic quasiparticle random phase approximation. Program summaryProgram Title:DIRQFAMProgram Files doi:http://dx.doi.org/10.17632/6gv8nxg4ns.1Licensing provisions: GPLv3Programming language: Fortran 90/95, easily downgradable to FORTRAN 77.External routines/libraries: BLAS/LAPACK, version 3.6.0. or higher.Nature of problem: Multipole response of deformed even–even open-shell nuclei can be calculated using the quasiparticle finite amplitude method (QFAM), based on the relativistic self-consistent mean-field models. The particle–hole channel is described by a zero-range relativistic effective interaction, while the particle–particle channel of the effective inter-nucleon interaction is described by a separable finite-range pairing force. The method can be applied to perform systematic studies of collective modes even in heavy deformed nuclei.Solution method: The current implementation computes the multipole response by solving the QFAM equations in a self-consistent iteration scheme. At each iteration the QFAM solutions are updated using the modified Broyden’s method. The QFAM amplitudes are expanded in the simplex-y harmonic oscillator basis.Restrictions: Open-shell even–even nuclei with axially symmetric ground states are considered. An electric multipole operator with J≤3 is used to calculate the response function.Unusual features: The code should be recompiled before running it with different values of one of the following input parameters: number of oscillator shells in the expansion of nucleon spinors (n0f), number of Gauss–Hermite (NGH) or Gauss–Laguerre nodes (NGL), the multipolarity values that define the external perturbation operator (J_multipole, K_multipole).

Many-body electric multipole operators in extended systems

The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define $n$-th order multipole operators whose expectation values can be used to calculate the $n$-th multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n-1$-st multipole currents, and encode the adiabatic evolution of the system in the presence of an $n-1$-st gradient of the electric field. Finally, we test our operators on a set of tightbinding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, Lμ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), \(L_n^{l+(1/2)}(\alpha r^2)r^l Y_{lm} ({\hat {\bf r}}) e^{-\alpha r^2}\). The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator

Numerical and symbolic calculation of the multipole matrix elements in the axial and triaxial harmonic oscillator basis

General expressions for matrix elements of the electric multipole operators in axial and triaxial harmonic oscillator basis are given. In each case we present algorithms which are either well suited for efficient numerical evaluation or for algebraic manipulation, using symbols representing the quantum numbers and the oscillator lengths. A Mathematica program using the symbolic algorithm is also presented. In the axial case, a function which provides the selection rules is implemented, thus being able to generate a full table with all the allowed transitions for a given multipolarity.

Transverse electric form factors for electron scattering and violation of current conservation in nuclear models.

Comparison is made between calculations of transverse electric form factors using the standard expression for the electric multipole operator and those obtained by invoking current conservation in the long wavelength limit and for arbitrary momentum transfer. In all cases, only the free-nucleon one-body current and charge operators are explicitly used. Results are presented for select E2 transitions in $^{12}\mathrm{C}$, $^{20}\mathrm{Ne}$, $^{24}\mathrm{Mg}$, and $^{28}\mathrm{Si}$. It is found that the form factors yielded by the various operators differ significantly when the conventional 0\ensuremath{\Elzxh}\ensuremath{\omega} shell model wave functions are used, confirming that these do violate conservation of the usual free-nucleon current. For these cases, the data are best reproduced using the operator with current conservation invoked in the long-wavelength limit. However, the variation between results is much smaller when multishell models of nuclear structure are used, all three forms of the operator yielding good agreement with existing data, suggesting that the wave functions obtained from these still practicable models are significantly closer to being current conserving.

Isoscalar quadratic energy weighted sum rules and the quadrupole moment of the giant quadrupole resonance

Isoscalar sum rules homogeneous quadratic in energy weighting are derived for the electric multipole operators. Except for scaling factors the sum rule values for the pure quadrupole and monopole transitions are the same as those for the corresponding linear energy weighted sum rules. Through these sum rules the electric quadrupole moment of giant quadrupole resonance is found to be −2.7A 1 3 e fm 2 .

Degenerate states in optical rotation and time-reversal invariance.

The consequence of time-reversal symmetry in a general second-order optical process between degenerate initial and final states is elucidated. The principle is applied to the pseudoscalar natural-optical-rotation operator (${R}_{L}$\ensuremath{\cdot}${M}_{L}$), which involves the interference of an electric-multipole (${R}_{L}$) transition and a magnetic-multipole (${M}_{L}$) transition. In addition to deriving point-group selection rules, time-reversal effects on matrix elements between degenerate states of a chiral versus an achiral molecule are compared and contrasted. Other non--time-reversal-degenerate states in chiral versus achiral molecules (${D}_{3}$ versus ${C}_{3v}$ and ${D}_{4}$ versus ${D}_{2d}$) are also compared.The analogy to the Jahn-Teller effect, which also involves (albeit first-order vibronic) matrix elements over degenerate states, is mentioned, especially with regard to the formal ``splitting'' of the degenerate levels by the pseudoscalar optical-rotation operator. A comprehensive search among the point groups for degenerate-state self-products that will give rise to the pseudoscalar function is conducted. Detailed results for the coupling of (degenerate) irreducible representations for important point groups (and their subgroups) are worked out for ${T}_{d}$ (T), O, ${D}_{6d}$ (${D}_{6}$, ${D}_{3}$, ${C}_{6v}$, ${C}_{3v}$, ${C}_{6}$, ${C}_{3}$, ${D}_{2d}$), and ${D}_{4d}$. The couplings to give dipolar (L=1), quadrupolar (L=2), and octopolar (L=3) pseudoscalars are shown explicitly.Attention is called to the differences in the transformation of a magnetic-multipole operator and an electric-multipole operator, even though they both belong to the same irreducible representation. The resulting differences in coupling coefficients for degenerate states are given to help determine nonvanishing optical-rotation matrix elements. A summary and a discussion of the conservation of parity and time symmetry in optical rotation are given.

Current conservation and the transverse electric multipole field.

Different prescriptions for imposing the constraints of current conservation on the Fourier transform of the electromagnetic current operator generate alternative forms for the electric multipole fields. These forms lead to quite different predictions for electromagnetic form factors in nuclear calculations that violate current conservation either by adopting an approximate current operator or by diagonalizing the nuclear Hamiltonian in a truncated Hilbert space. For typical shell model calculations of electron scattering form factors, it appears that no existing prescription for imposing the constraints of current conservation on electric multipole operators is entirely satisfactory. Numerical results are given for electric transitions in $^{12}\mathrm{C}$.

Current conservation and electric multipoles

New forms of the electric multipole operators are derived. We have uniquely and optimally isolated those components of the nuclear current density which are constrained by current conservation, and replaced them by multipole fields based on the charge density. This generalization of the work of Siegert is shown by means of a sum rule to have a very different structure from conventional forms which were designed to be effective in the long wavelength limit. The new form puts the electric and magnetic multipoles on the same footing: The current ebters only as the magnetization density, μ(x) = x × J(x) 2 .

Selection rules for giant resonances

We obtain selection rules for the matrix elements connecting perfect giant electric multipole states to other states in even nuclei through the use of a sum rule for the electric multipole operators.

Magnetic scattering of neutrons by a relativistic atom

We examined the magnetic scattering of neutrons by an atom described by a relativistic Hamiltonian. It is shown that the magnetic-scattering amplitude can be expressed in terms of relativistic magnetic and electric multipole operators. An effective magnetic-scattering amplitude is then defined so that a relativistic result is obtained by taking matrix elements of this effective operator between nonrelativistic states of the atom. It is shown that the effective magnetic-scattering amplitude can be expressed in terms of relativistic radial integrals and the generalized Racah tensors ${W}^{(0,k)k}(nl,{n}^{\ensuremath{'}}{l}^{\ensuremath{'}})$ and ${W}^{(1,{k}^{\ensuremath{'}})k}(nl,{n}^{\ensuremath{'}}{l}^{\ensuremath{'}})({k}^{\ensuremath{'}}=k,k\ifmmode\pm\else\textpm\fi{}1)$. The relativistic radial integrals can be evaluated using one-electron radial wave functions, obtained by Hartree-Fock-Dirac calculations. The matrix elements of the Racah tensors can be obtained by standard techniques and their evaluation is considerably simplified by selection rules based on the symmetry properties of the atomic states. The formalism has been applied to the magnetic scattering of neutrons by an atom in the ${l}^{N}$ electronic configuration. In this case, explicit expressions for the relativistic radial integrals have been derived, and the nonrelativistic limit of the effective magnetic-scattering amplitude has been examined.

Kinematic effective charges in photonuclear reactions

It is shown that the Gartenhaus-Schwartz-transformation of the electric multipole operator Ω[L], i.e. expressing it in lab frame coordinates, not only leads to the well known concept of kinematic effective charges but in addition gives 2-, 3-,...,N-particle operator contributions whereN=min(A, L). In particular forE2 transitions the electric quadrupole operator contains one- and two-particle operators which contribute to the single particle transition rate and which will lead to two particle transitions.

Consequences of Guage Invariance for Radiative Transitions

Consideration is given to any system of particles whose behavior under the influence of an external electromagnetic field can be described by a guage invariant Schroedinger equation. Detailed restrictions on the form of the hamiltonian which are imposed by the condition of guage invariance are derived. These provide a simple means to the solution of many problems of the interaction of a system with the electromagnetic field. In particular the following consequences are established: (1) In multipole expansions for single photon processes the electric multipole operators have the usual form but the form of the magnetic multipole operators may depend in a detailed way on the interactions between particles and electromagnetic field. (2) The $f$-sum rule can be expressed in closed form in terms of the interactions. (3) A generalization of the $f$-sum rule to all electric multipole orders is given. (4) The cross section for scattering of a low energy photon can be expressed in terms of the electrostatic polarizability quite independently of the interactions. Applications of these methods to problems in nuclear physics are given in an accompanying paper.