Theoretical Atomic Physics by Harald Friedrich

Author:Harald Friedrich
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

a finite scattering length does not exist even for l = 0. The leading near-threshold behaviour of the s-wave phase shift is [LK63],

(4.110)

so, towards threshold, the s-wave partial-wave scattering amplitude diverges logarithmically, (see (4.34) in Sect. 4.1.3),

(4.111)

and the differential scattering cross section also diverges logarithmically towards threshold.

The usefulness of the expressions (4.110), (4.111) is limited to a very small range in k, however, because the next term in the expansion is only of marginally higher order. For a repulsive inverse-cube potential, i.e. with a “+” sign on the right-hand side of (4.109), an extended formula was derived by Del Giudice and Galzenati [DG65] in 1965 and rederived by Gao [Gao99] in 1999. According to (12 ′ ) in [DG65],

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