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Physical Review A

Classical and quantal atomic form factors for arbitrary transitions

D. Vrinceanu, M. Raymond Flannery

Published Aug 1, 1999• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.60.1053

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The classical form factor is deduced from exact correspondence with a phase-space representation of the quantal form factor. Analytical expressions are provided for nl\ensuremath→n\ensuremath′l\ensuremath′,n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}},nl\ensuremath→​n\ensuremath′l\ensuremath′, nl\ensuremath→n\ensuremath′n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}nl\ensuremath→​n\ensuremath′, and n\ensuremath→n\ensuremath′\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}n\ensuremath→n\ensuremath′ transitions in hydrogenic systems and for n\ensuremath→n\ensuremath′\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}n\ensuremath→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 nl\ensuremath→n\ensuremath′l\ensuremath′n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}nl\ensuremath→​n\ensuremath′l\ensuremath′ transitions to close agreement for nl\ensuremath→n\ensuremath′n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}nl\ensuremath→​n\ensuremath′ and n\ensuremath→n\ensuremath′\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}n\ensuremath→n\ensuremath′ transitions. Excellent agreement is obtained for the integrated form factor for all transitions.
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We begin with some examples of one-dimensional vibration in a nonparabolic potential chosen to permit complete analytical solution of Schrödinger's time-independent equation. These examples are of isochronous vibrators which classically have a frequency independent of amplitude, and which might be expected therefore to have energy levels equally spaced at a separation of ħω 0 . This expectation turns out to be very nearly right and inspires a certain confidence in the semi-classical procedure developed by (among others) Bohr, Wilson and Sommerfeld. We therefore apply this procedure to some non-isochronous systems and find once more rather good agreement with the results of exact quantum mechanics. Periodic systems in fact can often be treated semi-classically with adequate accuracy, and significant economy of effort in comparison with strict quantum-mechanical analysis. This approach pays handsomely when we turn in the next chapter to the quantization of electron cyclotron orbits which, as already discussed in chapter 8, are closely related to harmonic oscillators. Conduction electrons in semi-conductors, and still more in metals, have their behaviour modified by the lattice through which they move, and a complete quantal treatment has never been achieved. It is clear, however, from approximate calculations, often of great complexity, that the semiclassical method describes most of the interesting physical processes correctly and very simply. In chapter 15 we shall describe in outline some of the effects which can be treated quite well enough for most purposes without even writing down Schrödinger's equation.
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Atoms in high-(n,l)(n,\mathcal{l})(n,l) states formed in cold Rydberg plasmas decay to the ground state in a succession of radiative transitions populating intermediate excited states. A classical treatment presents radiative cascade in a physically transparent way and reveals the ``trajectory'' in nln\mathcal{l}nl space obeyed by the cascade, scaling rules, and other aspects hidden within the quantal approach. Quantal-classical correspondence in radiative decay is directly demonstrated. Classical transition probabilities are also presented and are in excellent agreement with quantal transition probabilities, even for moderate quantum numbers.
A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for 1S1S1S to 2S2S2S transitions in hydrogen.