Boson and Fermion Effects in Helium-Helium Scattering

The scattering of two colliding beams of helium atoms provides a definitive illustration of the influence of quantum statistics on identical particles [1]. Helium has two stable isotopic forms, and , which are nearly identical in chemical and electromagnetic properties. The latter isotope has a natural occurrence of only 0.0002% but can be concentrated for use in experiments.
In the graphic, atoms are colored blue while atoms are green. Two beams collide at the center and are scattered. Detectors can be rotated by the angle , . The paths of the incident and scattered particles are lit up by light beams; particles scattered at angles other than are not visible. The counter shows the average number of scattered atoms per second per unit solid angle.
The low-energy (around 100 meV) elastic scattering of helium atoms can be treated by the Rutherford scattering formula. The scattering amplitude in the center-of-mass system is given by [2, 3]:
,
where .
( reduced mass of , electron charge, , helium nuclear charge and wavenumber, such that kinetic energy .)
The differential scattering cross-section (also written ) is determined from the amplitude by .
For scattering, the identities of the atoms can, in principle, be distinguished. The scattering cross section for all atoms is then equal to the sum
.
For scattering, since the scattered particles are identical bosons, the scattering amplitude is given by the sum and the cross section is then
.
scattering, by contrast, involves identical fermions. The scattering amplitude is then given by the difference and
.
In simplified arbitrary units, the three cross sections are given by
,
,
.
Interpret these as the number of helium atoms hitting the detectors per second per unit solid angle , as recorded by the counter.
The most dramatic effect occurs for scattering at . The cross section has twice the magnitude of that for , but the scattering vanishes completely at .

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References
[1] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 3. (Jun 15, 2022) www.feynmanlectures.caltech.edu/III_03.html#Ch3-S4.
[2] L. I. Schiff, Quantum Mechanics, 3rd ed., New York: McGraw-Hill, 1968 pp. 138ff.
[3] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, 3rd ed., Oxford: Clarendon Press, 1965 pp. 53–57.
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