An approximate projection technique for the calculation of electromagnetic properties of deformed nuclei

A three-dimensional angular momentum projection is carried out for cranking model wave functions. The projected matrix elements of electromagnetic operators are evaluated using a method originally developed by Kamlah for the case of projected energy, which is valid for large deformations and weakly triaxial nuclei. The calculated spectroscopic quadrupole moments deviate substantially from the predictions of a rigid rotor model with axial symmetry. For E2 transitions the deviations are small. Projected values of the magnetic moments are almost identical with those of a semiclassical calculation. Cranking model wave functions are decomposed into its components having good angular momentum.     

Construction of microscopic boson states and their relevance for IBM-2

We investigate four methods for the construction of collective shell model states which may be mapped onto boson states of the IBM-2. These methods use, as building blocks for the wave functions, particle-particle pair operators, particle-hole operators, pair operators with seniority projection and energy-weighted quadrupole operators. It is demonstrated that one obtains stronger collectivity with the energy-weighted quadrupole operator than with the other methods.On the basis of a comparison of calculated and empirical IBM-2 interaction parameters we can rule out the seniority projection method. This implies that particle-particle and particle-hole approaches difler.The ratios between quadrupole matrix elements of the microscopic boson states appear to be similar to the IBM predictions. For states corresponding to those with two d-bosons coupled to J = 0 there is a smaller quadrupole matrix element when subshells with small angular momenta dominate near the Fermi level. Especially for this type of states the collective quadrupole space will be larger than represented in the IBM, however, which may compensate the smaller proton-neutron quadrupole coupling.The calculated bare quadrupole interaction between like bosons is found to be weak.     

The spin density matrix for slowly reorienting molecules in zero field

The time evolution of the spin density matrix of a spin embedded in a slowly moving molecule is investigated using Gordon's extended diffusion model for molecular reorientation and allowing the spin to follow the molecular motion. The results for this model are identical with those previously obtained using a jump diffusional model and do not give the adiabatic limit in slow motion. This model is valid provided the molecular angular velocity correlation time is short compared to the spin precession period. The perturbed angular correlation of rays or the spin resonance line shape can be calculated for any value of the rotational diffusion constant by diagonalizing a finite matrix. The necessary matrix elements are given for the 247 keV state of ¹¹¹Cd (I = 5/2).     

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.     

A schematic model for monopole and quadrupole pairing in nuclei

A schematic Hamiltonian with a pairing interaction plus a quadrupole-quadrupole interaction between nucleons is presented. It is shown that all the states of the fermion system can be classified according to the number of nucleons u not coupled to coherent monopole or quadrupole pairs. The states with u = 0 are shown to have a one-to-one correspondence to the states of the interacting boson model. The spectra for these states are derived analytically for various limits of the pairing strength and the quadrupole strength. Analytical forms for the matrix elements of operators are derived for these limits. The operators in fermion space are mapped onto boson operators. The matrix elements of operators in the fermion space are shown to be equal to matrix elements of the boson operators multiplied by analytical Pauli factors which are state dependent. The two-nucleon transfer strength is calculated in two limits and is compared to experimental values.     

Elastic scattering of relativistic electrons by nuclei in the helicity representation

The partial-wave analysis of the scattering problem electron-nucleus is performed using helicities instead of the angular momentum of the electron. This proceeding leads to remarkable advantages in the treatment of the scattering process, especially in the numerical solution of the system of coupled radial differential equations: a) The matrix elements over the potential operators are independent of the total angular momentum. b) Instead of 6-j symbols etc., only very simple algebraic expressions are to be evaluated. c) Due to special properties of the system computing time is reduced for higher values of the nuclear spin by an amount proportional to about the order of the dimension of the system.     

Dirac Particle in an Aharonov-Bohm Potential: The Structure of the First Order S-matrix

The structure of the interaction Hamiltonian in the first order S-matrix element of a Dirac particle in an Aharonov-Bohm (AB) field is analyzed and shown to have interesting interesting algebraic properties. It is demonstrated that as a consequence of these properties, this interaction Hamiltonian splits both the incident and outgoing waves in the the first order components (eigenstates of the third component of the spin). The matrix element can then be viewed as the sum of two transitions taking place in these two channels of the spin. At the level of partial waves, each partial wave of the conserved total angular momentum is split into two partial waves of the orbital angular momentum in a manner consistent with the conservation of the total angular momentum quantum number.     

Calculation of the fine structure of the level in Rydberg state of lithium

The level shift and level formula of lithium atom in Rydberg states are achieved by means of the calculation of polarization of the atomic core (including the contribution of dipole moment, quadrupole moment and octupole moment);meanwhile, the effect of relativity theory, the orbital angular momentum L and the spin angular momentum S coupling (LS coupling), and high order correction of the effective potential are considered. The some fine structures (N=5～12,L=4～9,J=L±1/2) and the corresponding level intervals in Rydberg states can be calculated by the above-mentioned level formula and compared with correlated experimental data.     

Determining the energy barrier for decay out of superdeformed bands

An asymptotically exact quantum mechanical calculation of the matrix elements for tunneling through an asymmetric barrier is combined with the two-state statistical model for decay out of superdeformed bands to determine the energy barrier (as a function of spin) separating the superdeformed and normal-deformed wells for several nuclei in the 190 and 150 mass regions. The spin-dependence of the barrier leading to sudden decay out is shown to be consistent with the decrease of a centrifugal barrier with decreasing angular momentum. Values of the barrier frequency in the two mass regions are predicted.     

Nucleon internal structure: a new set of quark, gluon momentum, angular momentum operators and parton distribution functions

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

Can gluon spin contribute to that of nucleon?

The transformation of angular momentum to its Belinfante form requires the smooth behaviour of classical fields at infinity which for the case of quantum operators transforms to the smooth behaviour of matrix elements at small momentum transfers. For the case of quarks this provides the kinematical counterpart of U _A (1) problem while for gluons there is a contradiction between kinematics and dynamics governed by Kogut-Susskind pole. This may result in the violation of Equivalence Principle for nucleons or in the stringent constraints to the strange quark polarization in nucleons, while the most likely outcome would be the impossibility to separate gluon angular momentum to spin and orbital parts in the meaningful way.     

Spin Squeezing of One-Axis Twisting Model

We investigate spin squeezing of the one-axis twisting model. By using short-time approximation solutions of the angular momentum operators, we analytically and numerically calculate the spin squeezing parameter. It is shown that smaller linear interaction can produce a stronger spin squeezing and maintain a longer time interval. It is also shown that the stronger spin squeezing can be achieved by increasing the number of particles.     

Geometrical derivation of collective operators and paths in TDHF

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.     

Antipairing and antistretching in 22Ne

The ground state rotational band of ²²Ne has been investigated with the angular momentum and particle number projected Hartree-Bogoliubov theory. Variation before projections was performed in the framework of the constrained Hartree-Bogoliubov theory with the quadrupole moment and the degree of pairing as constraints. It is shown that there is a clear decrease of the pairing correlations (antipairing) and a decrease of the quadrupole deformation (antistretching) with increasing angular momentum. The antipairing effect appears to be essential to reproduce the experimentally known deviation of the spectrum from the J(J + 1) rule. The antistretching indicated by the B(E2) values is much stronger than that indicated by the calculated static quadrupole moments. This amounts to a breakdown of the rotational picture which is possibly connected with the low band cut-off values in the sd shell. The antipairing effect decreases the antistretching only slightly. The interaction between the pairing and quadrupole degrees of freedom is found to be too weak to change the earlier conclusions concerning antipairing and antistretching in the whole sd shell. No inert core was assumed in the calculations. The effective G-matrix elements of the Hamada-Johnston potential were used as an interaction.     

Irreducible spin precession theory applied to some topics in atomic and nuclear radio-frequency spectroscopy

Several problems from radio-frequency spectroscopy of atoms and nuclei are treated with irreducible spin precession theory. In the first part, effective field techniques are used to derive analytically single and multiple quantum double resonance lineshapes for atoms with a hyperfine structure in a high magnetic field. In the second part (as an extension to previous work), nuclear resonance signals are calculated for oriented nuclei subject to an electric hexadecapole interaction. Lineshapes of acoustically driven hexadecapole transitions are derived in closed form and compared to experiment. Further, multiple quantum NMR transitions within a hexadecapole shifted nuclear Zeeman structure are calculated, and some distinct features of hexadecapole effects on NMR lineshapes are pointed out. This last case is of current interest due to recent progress in NMR-line narrowing techniques. — In the Appendix, we give lineshape equations for single and double quantum NMR transitions on oriented (I=1)-nuclei subject to an electric quadrupole interaction; these equations are also being used in the atomic rf-spectroscopy calculations. The equations are exact to all orders of the interaction with the external fields.     

Polarized fluorescence of jet-cooled molecular iodine

The degree of polarized fluorescence of molecular iodine ¹²⁷I₂ cooled in a supersonic jet under rotationally selective excitation in the electronic transition B ³Π_0u ⁺-X ¹Σ _g ⁺ has been measured and calculated. It was found that the interaction of the angular momentum of the molecule with the nuclear spins of iodine atoms leads to a considerable depolarizing effect. This effect is most pronounced for small rotational quantum numbers of the angular momentum that are comparable with the total nuclear spin of the iodine molecule, which is equal to 5.     

Calculation of pulsed NMR signal in I=3/2 quadrupolar spin system

The rf pulse response of I=3/2 spin system experiencing first order quadrupolar splitting is studied using density matrix approach. A general expression is derived in terms of spin populations, quadrupole splitting and duration and amplitude of the rf pulse for calculating the NMR signal arising due to the centre line and satellite resonances for the situation where the impressed rf pulse excites the resonances selectively as well as non-selectively. The necessary 4×4 transformation matrix obtained analytically by diagonalyzing the Hamiltonian are used to get the expression for the centre line response. The satellite signals are obtained in the same way but by using the numerical values of the roots of the related quartics. The widths of the corresponding π/2-pulses are calculated for different initial spin populations. The variations of this pulse-width and the corresponding signal amplitude as a function of satellite splitting are studied.     

Gravitational radiation from n-body systems

An expression for the quadrupole moment of any two-body system with structure is derived from a “paralel axes” theorem. Within the weak-field limit of the theory of general relativity, expressions for the gravitational radiation flux of energy and angular momentum from two particles or two spherically symmetric bodies in arbitrary plane motion arising from any type of forces are consequently obtained in terms of time derivatives of the relative coordinates of the system. An estimate of the gravitational flux from any plane motion follows. In particular, the flux from systems with Keplerian and straight-line motion are deduced as special cases. For the general problem of a two-body system with intrinsic quadrupole moment (due to deviation from spherical symmetry), it is found that in addition to the flux from the orbital and the spin motion there is another source of flux—the interaction flux. This is shown explicitly in two special cases—the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid, and that of two spinning rigid rods in plane circular motion with parallel spin and orbital angular momentum. The interaction flux is regarded as the result of interaction of the bodies with gravitational waves. An outline of the method for the calculation of gravitational radiation flux from an n-body system is given. For a three-body system—an astrophysically interesting situation—this is worked out in detail. It is seen that the presence of an unsuspected third body can, by virtue of the interaction power term, increase the generation of gravitational waves significantly.     

The fine structure splitting of the level of lithium in Rydberg states

The Hamiltonian of the four-body problem for a lithium atom is expanded in series. The level shift and level formula of a lithium atom in Rydberg states are achieved by means of the calculation of polarization of the atomic core (including the contribution of dipole, quadrupole and octupole components). We also consider the effect of relativity theory, the orbital angular momentum L and the spin angular momentum S coupling scheme (LS coupling) and high-order correction of the effective potential to the level shift. The fine structure splitting (N=5-12, L=4-9, J=L±1/2) and level intervals in Rydberg states have been calculated by the above-mentioned formula and compared with recent experimental data.