13.7: Low energy Scattering - Partial Waves

The Born Approximation is a perturbation method based on the Fermi Golden Rule and is therefore valid when the incoming particle energy is large compared to the potential. An alternative approach is needed at low energy. For a central potential, scattering geometry plane wave in, radial wave out, implies a wavefunction:

$$|\Psi \rangle = \text{IncidentWave} + \text{ScatteredWave} = = e^{ikz} + f(\theta )e^{ikr}/r \nonumber$$

The incident flux is $I = ve^{iKz} e^{−iKz} = v = \hbar k/m$. The scattered flux must be a normalisable plane wave (hence $e^{−iKr}/r$), with a $\theta$ dependence arising from the scattering. By symmetry, there is no $\phi$ dependence. Thus the scattered flux per unit area will be: $vf^* (\theta )f(\theta )/r^2$. The cross section $d\sigma /d\Omega = S(\theta )/I = f^* (\theta )f(\theta )$, and all we need do is solve the Schrödinger equation and calculate $f(\theta )$.

For a spherically symmetric potential, the angular parts of the wavefunction are simply spherical harmonics, so scattering is described by the radial equation:

$$\frac{d^2u_l(r)}{dr^2} − \frac{l(l + 1)}{r^2} u_l r) + \frac{2\mu}{\hbar^2} [E − V (r)]u_l(r) = 0 \nonumber$$

where $u_l(r) = rR_l(r)$, the same substitution as in the atomic hydrogen problem. Assuming a short range potential, $V (r \rightarrow \infty) = 0$, $R_l(Kr \rightarrow \infty)$ describes a free particle, with some phase $l \pi / 2−\delta_l$.

$$R_l(Kr) = \sin(Kr − l\pi /2 + \delta_l)/Kr \nonumber$$

Thus the effect of the scattering at long range can be described by a set of phase shifts $\delta_l$. To solve further, we expand a plane wave into angular momentum components using a complete set of spherical harmonics and Bessel Functions:

$$\text{exp}(iKr \cos \theta ) = \sum^{\infty}_{l = 0} i^l j_l (Kr)(2l + 1)P_l(\cos \theta ) \nonumber$$

so that we can write:

$$\Psi ({\bf r}) = e^{iKz} + f(\theta ) \frac{e^{iKr}}{r} = \sum^{\infty}_{l = 0} i^l j_l (Kr)(2l + 1)P_l(\cos \theta ) + f(\theta ) \frac{e^{iKr}}{r} = \sum^{\infty}_{l = 0} b_l R_l (Kr)P_l (\cos \theta ) \nonumber$$

where $b_l$ are expansion coefficients for the expression of $\Psi$ in the partial wave basis, which can be determined from the boundary $r \rightarrow \infty$, giving:

$$f(\theta ) = K^{−1} \sum^{\infty}_{l = 0} (2l + 1)e^{i\delta_l} \sin \delta_lP_l( \cos \theta ) \nonumber$$

From this we can calculate $d\sigma /d\Omega = |f(\theta )|^2$ and $\sigma = 2\pi R |f(\theta )|^2 d\theta$. Differential cross sections $d\sigma /d\Omega$ are complicated, involving many cross terms. However, when integrated over all \theta these cross terms vanish due to orthogonality of the Legendre polynomials $\langle P_l |P_{l'} \rangle = 0 (l \neq l')$, and

$$\sigma = \frac{4\pi}{ K^2} \sum^{\infty}_{l = 0} (2l + 1) \sin^2 \delta_l \nonumber$$

Hence scattering cross sections are completely determined by $|K|$ and the phase shifts $\delta_l$. This is most useful in the low energy limit (S-wave scattering) where any particle with $l > 0$ must be so far from the target (impact parameter $b = l/\hbar k )$ that it will miss.

Note the term (2l+1). This can be related to the classical ‘impact parameter’ mentioned above. The angular momentum of a particle of velocity $v$ is $mvb = \sqrt{l(l + 1)}\hbar$. Thus a classical (large $l$) particle with angular momentum $l\hbar$ would pass between a ring of radius $b = l\hbar /mv$ and one of radius $b = (l + 1)\hbar /mv$. The area between these rings is $(2l + 1)\pi (\hbar /mv)^2$ so for a uniform beam the probability of a particle having angular momentum $l$ is proportional to (2l+1).