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Journal of Physics B Atomic and Molecular Physics

Quantum-mechanical theory of transfer and relaxation of atomic multipole moments

Gerard Nienhuis

Published Feb 1, 1976• 37 citations
Publisher

Abstract

The author points out that the Waldmann-Snider equation leads directly to quantum-mechanical expressions for the rate constants for transfer and relaxation of atomic excited state multipole moments, both in the case of resonant interaction and of foreign-gas perturbation. The rate constants are expressed in terms of quantum-mechanical partial cross sections. The effect of the hyperfine interaction is briefly discussed.

DOI
10.1088/0022-3700/9/2/007

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The author points out that the Waldmann-Snider equation leads directly to quantum-mechanical expressions for the rate constants for transfer and relaxation of atomic excited state multipole moments, both in the case of resonant interaction and of foreign-gas perturbation. The rate constants are expressed in terms of quantum-mechanical partial cross sections. The effect of the hyperfine interaction is briefly discussed.
A theoretical treatment of the relaxation and transfer of atomic multipole moments due to interatomic collisions is presented, applicable to systems of emitters and perturbers with any angular momenta. Expressions are obtained for the rate constants, within the impact approximation, in terms of the elements of the scattering matrix evaluated as a function of impact parameter for an emitter-perturber collision. Processes in which the total electronic excitation energy is not conserved are excluded and collisions are assumed to occur isotropically. Explicit evaluation of rate constants has been carried out using the first-order dipole-dipole (resonance) interaction, assuming the classical straight line path approximation. Four special cases have been treated by numerical integration of the Schrodinger equation in the interaction representation. These are: J g (ground state angular momentum)=0, J e (excited state)=1; j g = 1 / 2 , J e = 1 / 2 ; J g = 1 / 2 , J e = 3 / 2 ; J g =1, J e =1. The ( 1 / 2 , 1 / 2 ) rate constants are shown to be simply related to those for the (0, 1) case. Within the context of the various approximations, numerical accuracy is expected to be better than 1%.
The determination of cross sections for the relaxation of atomic multipole moments from the intensities of the components of a resolved Zeeman fluorescence spectrum may be carried out using the density-matrix formalism. However, the spherical-tensor basis is not well adapted to state-selected measurements and can allow a significant propagation of error. We can reduce the error by determining the Q(j,m\ensuremath{\rightarrow}j',m') cross sections directly from the data taken at the lowest pressures. A comparison is given between recent experimental and theoretical relaxation cross sections for collisions of 5 2P^{2}P2P potassium atoms with ground-state He, Ne, and Ar atoms.
We derive algebraic expressions for the atomic multipole moments mk,q(F){m}_{k,q}(F)mk,q​(F) describing the steady-state polarization of an atomic ensemble with angular momentum FFF in a static magnetic field of arbitrary direction and general multipole relaxation rates ${\ensuremath{\Gamma}}_{q}^{(k)}$. The longitudinal moments mk,0{m}_{k,0}mk,0​ are given in terms of truncated continued fractions, while the transverse moments ${m}_{k,q\ensuremath{\ne}0}$, representing the ensemble coherence, are given by products of truncated continued fractions. The special case of isotropic relaxation leads to particularly simple and elegant expressions. Our results are relevant for all domains of physics that consider the evolution of a spin system interacting with vector-type perturbations.
This paper is devoted to the development of a rigorous theory of angular-momentum relaxation. It is shown that by incorporating quantum-dynamical semigroups in the quantum Markovian master equation, the theory becomes mathematically sound. The expansion of the initial relaxation process in terms of primary relaxation processes of rank l offers a clearer view of the relaxation process structure. A close look at experimental procedure (the system Hilbert space structure, symmetry of the environment, and multipolarities of the system-environment interaction) gives all the information needed to simplify the structure of the relaxation equation and renders it a powerful tool for considering specific physical effects. The quantum Markovian master equation in the state multipole representation is obtained and the physical meaning of the parameters involved is found. The general approach is used for the study of a Zeeman multiplet. The relaxation rates as functions of the spectral densities of correlations (describing the environment properties) are given in an explicit form. Cross relaxation and cross correlation are taken properly into account there. The important case of spin 1 is discussed in more detail. Some useful expressions and relations among the relaxation rates are given in the Appendixes.
Theoretical study is made of the transfer between orientation and alignment of excited-state atoms by anisotropic collisions with ground-state atoms. The anisotropy considered is due to the anisotropic velocity distribution of atoms excited by a single-mode laser beam. The transfer rate and relaxation rates for each multipole moment are calculated for the excited state with J=1J=1J=1 as a function of detuning of the laser frequency from the center of an absorption line and mass of perturbing ground-state atoms. Experiments to observe the transfer from alignment to orientation and from orientation to alignment in a weak magnetic field are considered, and corresponding signals are analyzed in terms of the transfer and relaxation rates.
A method of determining the relaxation constants of the multipole moments of gas atoms with non-zero nuclear spin has been proposed by using the polarisation properties and damping law of the non-stationary Raman scattering (NRS) of light. Correct account was taken of the depolarising atomic collisions and of the effect of the hyperfine structure (HFS) of the atomic levels. The NRS polarisation properties found have been assumed to be a basis method for the identification of optically forbidden transitions in both the electronic and total angular momenta. A possible method of determining the HFS constants through analysis of the NRS quantum beats has been established.