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14.6: Ramsauer-Townsend effect

Last updated

Oct 11, 2020
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- 14.5: Partial Waves in the Classical Limit - Hard Spheres
- 15: The Quantum Mechanics Nobody Rreally Understands

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This is the name given to the fact that electrons with energy about 1 eV can pass almost freely through Xe, Kr, and Ar:- there is a sharp minimum in electron scattering cross-section for these noble gases.

Due to polarization of these atoms by the incoming electron the potential appears to increase as $K$ increases (more localized electrons are better able to polarise the atom). Thus $\delta_0(k \rightarrow 0) = n\pi$ , in accordance with Levinson’s theorem, and $\delta_0$ initially increases as $k$ increases, before eventually decreasing. Thus at a certain value of $k$ , the phase shift is again $\delta_0(k) = n\pi$ , and the total scattering cross section $\sigma_T$ has an abrupt minimum. Although there are subsequent $s$ -wave minima at e.g. $\delta_0(k) = (n − 1)\pi$ , these occur at sufficiently large values of $k$ that $s$ -wave scattering is no longer dominant.

By contrast, neon and helium have lower polarisability, due to fewer bound electrons. Thus the phase shift $\delta_0$ decreases monotonically with $k$ from $n\pi$ at $k = 0$ at there is no low-energy minimum.

Higher $l$ phase shifts may increase with $k$ because higher $k$ implies smaller impact parameter (classically, more chance of hitting the atom). The cross section increases more slowly due to the additional $K^{−2}$ dependence. The maximum in the Ar cross section at about 13 eV is mainly due to the $d$ -wave $\delta_2 = \pi /2$ .

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