The potential which an electron sees as it approaches an atom of a monatomic gas can be qualitatively represented by a square well. Slow particles are considered.
V(r) = −V0; r ≤ R
= 0; r > R
corresponding to an attractive potential. Scattering of slow particles for which kR << 1, is determined by the equation
(∇2 + k2 − 2μV/ℏ2) ψ2 = 0 (inside the well) ....(1)
with k2 = 2μE/ℏ2, and the wave number k = p/ℏ
Outside the well the equation is (∇2 + k2)ψ1 = 0 ...(2)
Further writing
k21 = k2 + k20
where k20 = 2μV/ℏ2 and V = −V0
The solutions are found to be ψ2 = A sink1r ....(3)
ψ1 = B sin(kr + δ0) ....(4)
ψ1(r) is the asymptotic solution at large distances with the boundary condition
ψ1(0) = 0 Matching the solutions (3) and (4) at r = R both in amplitude and first derivative,
A sin k1R = Bsin(k R + δ0) ....(5)
Ak1 cos k1R = Bk cos(k R + δ0) ....(6)
Dividing one equation by the other, and setting k1cot k1R = 1/D , and with simple algebraic manipulations we get
tanδ0 = (kD − tankR)(1 + kD tankR)−1 ...(7)
The phase shift δ0 determined from (7) is a multivalued function but we are only interested in the principle value lying within the interval −π/2 ≤ δ0 ≤ π/2 .
For small values of the energy of the relative motion
is satisfied, the phase shift and the scattering cross-section both vanish. This phenomenon is known as the Ramsauer-Townsend effect. The field of the inert gas atoms decreases appreciably faster with distance than the field of any other atom, so that to a first approximation, we can replace this field by a rectangular spherical well with sharply defined range and use Equation (10) to evaluate the cross-section for slow electrons.
Physically, the Ramasuer – Townsend effect is explained as the diffraction of the electron around the rare-gas atom, in which the wave function inside the atom is distorted in such a way that it fits on smoothly to an undistorted function outside.
Here the partial wave wave with l = 0 has exactly a half cycle more of oscillation inside the atomic potential then the wave in the force-free field, and the wavelength of the electron is large enough in comparision with R so that higher l phase-shifts are negligible.
