Classical and quantal atomic form factors fornlm→n′l′mtransitions

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.