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Physical Review A

Classical and quantal atomic form factors for nlm→n′l′m transitions

M. Raymond Flannery, D. Vrinceanu

Published Jan 4, 2002• 5 citations
Publisher

Abstract

An analytical expression for the classical form factor or impulsive probability ${P}_{\mathrm{if}}(\mathbf{q})$ for $\mathrm{nl}\stackrel{\ensuremath{\rightarrow}}{m}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}m$ transitions is derived directly from the ``phase-space distribution'' method [Phys. Rev. A 60, 1053 (1999)] and is compared with quantal results. Exact universal scaling laws are derived for the classical probability for any $\stackrel{\ensuremath{\rightarrow}}{i}f$ transition. As n is increased, convergence of the quantal to classical results is obtained and it becomes even more rapid upon averaging in succession over the m and then the $\mathit{l}$ substates. The classical results reveal the basic reason for the underlying structure in the variation of ${P}_{\mathrm{if}}$ with momentum transfer $\mathbf{q}.$ Classical form factors can operate as an effective averaged version of the exact quantal counterpart.

DOI
10.1103/physreva.65.022703

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An analytical expression for the classical form factor or impulsive probability Pif(q){P}_{\mathrm{if}}(\mathbf{q})Pif​(q) for nlm\ensuremath→n\ensuremath′l\ensuremath′m\mathrm{nl}\stackrel{\ensuremath{\rightarrow}}{m}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}mnlm\ensuremath→n\ensuremath′l\ensuremath′m transitions is derived directly from the ``phase-space distribution'' method [Phys. Rev. A 60, 1053 (1999)] and is compared with quantal results. Exact universal scaling laws are derived for the classical probability for any i\ensuremath→f\stackrel{\ensuremath{\rightarrow}}{i}fi\ensuremath→​f transition. As n is increased, convergence of the quantal to classical results is obtained and it becomes even more rapid upon averaging in succession over the m and then the l\mathit{l}l substates. The classical results reveal the basic reason for the underlying structure in the variation of Pif{P}_{\mathrm{if}}Pif​ with momentum transfer q.\mathbf{q}.q. Classical form factors can operate as an effective averaged version of the exact quantal counterpart.
The classical form factor is deduced from exact correspondence with a phase-space representation of the quantal form factor. Analytical expressions are provided for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}},$ $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$, and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions in hydrogenic systems and for $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ in the one-dimensional harmonic oscillator. An efficient procedure for calculation of quantal form factors as analytical functions of momentum transfer, for arbitrary quantum numbers, is presented. The classical approach has the ability to explain quite succinctly interesting trends and various important aspects which remain hidden within the quantal treatment of form factors. The classical-quantal comparison ranges from being qualitatively good for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ transitions to close agreement for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$ and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions. Excellent agreement is obtained for the integrated form factor for all transitions.
Exact classical and quantal solutions are presented for the full array of intrashell transitions $n\mathcal{l}\ensuremath{\rightarrow}n{\mathcal{l}}^{\ensuremath{'}},$ between any angular momentum states, induced by slow distant collisions with a charged particle. The collisions are adiabatic with respect to the orbital frequency of the atomic electron and the transitions are induced by the weak ion-atom dipole field generated by the ion moving along a classical path. The rich symmetry of the problem allows a unified approach and is the source of the excellent agreement, beyond the usual Ehrenfest's correspondence principle, between the classical and quantal treatments. A classical transition probability is defined. Probabilities for transition between any angular momentum states within a high Rydberg energy level are derived in exact analytic forms and are analyzed for a large number of numerical examples. The transition probabilities obtained from the three methods---quantal and classical formulations and Monte Carlo classical simulations---are directly compared to provide excellent agreement.
Étude asymptotique des facteurs de formes des chaînes de spin quantiques Les systèmes intégrables quantiques restaient longtemps un domaine où des méthodes mathématiques modernes permettaient d’accéder aux résultats intéressants pour l‘étude de systèmes physiques. Le calcul exacte, numérique et asymptotique de fonction de corrélation reste un de sujets les plus importants de la théorie de modèles intégrables quantiques. Dans ce cadre l’approche basée sur le calcul des facteurs de forme s’est révélée la plus efficace. Dans ce thèse, une méthode alternative fondée sur l'ansatz de Bethe algébrique est développée pour calculer des facteurs de formes dans la limite thermodynamique. Elle est appliqué et décrit dans le contexte de chaîne de spin isotrope XXX, qui est un des cas plus intéressant des modèles critiques où la zone de fermi est non-compacte.
The very successful convergent close-coupling (CCC) method [I. Bray and D. V. Fursa, Phys. Rev. A 54,54,54, 2991 (1996)] for calculating cross sections for the ionization of atoms by electron impact is examined for the case of hydrogen ionization. The theory is recast in an operator theoretic form, allowing explicit formulas to be given for the solutions of the truncated CCC equations. The limit to which these solutions tend as ever more terms are included in the truncation is shown to be the exact transition operator for the system. This analytical result is not consistent with the numerical CCC results, which yield amplitudes that violate the symmetrization principle of quantum mechanics and which depend for success on treating the electrons as classically, rather than quantally, indistinguishable particles. This new result is also inconsistent with the ``step function hypothesis'' [I. Bray, Phys. Rev. Lett. 78,78,78, 4721 (1997)] that was introduced to restore the consistency of the CCC calculations with the symmetrization postulate. In addition, the CCC method does not include long-range effects of the Coulomb interaction that are expected to be important in certain kinematic configurations at low energies. Consequently, it is not justified yet to interpret the numerical successes of the CCC method as confirmation of a fundamental dynamical understanding of electron-induced atomic ionization.
An exact expression for the probability P(n)l'l of Stark mixing transitions between arbitrary angular momentum states l and l' within the same energy shell n, as a result of a collision with a slow charged projectile at large impact parameters is presented. The formula obtained is compact and easy to use for numerical evaluations even for very large quantum numbers (n~100). A classical approximation is directly obtained and compared with the exact quantal result in the limit of large n. Two distinct sets of quantal oscillations are predicted.