Partial waves for $\rm ^{87}Rb$

We are considering collisions between gases of $\rm ^{87}Rb$ atoms in the $\vert F = 2, m_F = 2 \rangle$ hyperfine substate. Effectively, two colliding atoms are each identical composite bosons and the extended version of the Pauli principle then requires the total wave function to be symmetric. As a result, only terms of even

contribute to the partial wave expansion Eq. (3) and the sum acquires a degeneracy factor of 2. At low collision energies only the

term of Eq. (3) remains nonzero -- all elastic scattering has an isotropic (s-wave) nature. For increasing collision energies differential scattering via higher order partial waves sets in. The differential cross section in the special case of bosonic scattering restricted to s- and d-wave states is given by

$\displaystyle \frac{d\sigma} {d\Omega}$	$\textstyle =$	$\displaystyle k^{-2}\vert \overbrace{(e^{2i\eta_0}-1)}^{\rm s}+\overbrace{5(e^{2i\eta_2}-1)(3\cos^2\theta-1)/2}^{\rm d}\vert^2$
	$\textstyle =$	$\displaystyle k^{-2}[4\sin^2 \eta_0 +25\sin^2\eta_2(3\cos^2\theta-1)^2 +\underb... ...sin\eta_2\cos(\eta_0-\eta_2)(3\cos^2\theta-1)}_{\rm s+d interference term}]$	(7)

which is Eq. (1)

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nk 2004-11-02