Critical angles for axial channeling

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Abstract
The critical angle for axial proton channeling has been measured as a function of temperature for a number of different crystals. The results are consistently below the values predicted from simple analytical estimates. The magnitude of this discrepancy does not seem to depend on the specific lattice structure. In point of fact, all the results are consistent with a relationship ψ1/2 = α(p/a). ψ1, where ψ1 is Lindhard's characteristic angle, and α(p/a)is a function of the thermal vibrational amplitude p, measured in units of the Thomas-Fermi screening distance a. A formula for α(p/a), derived by Barrett from computer simulations, on the average predicts the magnitude of α quite well, but the observed functional dependence on p/a is considerably stronger than predicted.