Optical Theorem

The differential scattering cross-section $d\sigma/d{\mit\Omega}$ is simply the modulus squared of the scattering amplitude $f(\theta)$ . The total cross-section is given by

$$\sigma_{\rm total} = \oint d{\mit\Omega}\,\vert f(\theta)\vert^{\,2}$$

$$= \frac{1}{k^2} \oint d\varphi \int_{-1}^{1} d\mu \sum_l \sum_{l'... ...elta_l-\delta_{l'})]\, \sin\delta_l \,\sin\delta_{l'}\, P_l(\mu)\, P_{l'}(\mu),$$ (980)

where $\mu = \cos\theta$ . It follows that

$$\sigma_{\rm total} = \frac{4\pi}{k^2} \sum_{l=0,\infty} (2\,l+1)\,\sin^2\delta_l,$$ (981)

where use has been made of Equation (967). A comparison of this result with Equation (979) yields

$$\sigma_{\rm total} = \frac{4\pi}{k}\, {\rm Im}\left[f(0)\right],$$ (982)

since $P_l(1) = 1$ . This result is known as the optical theorem. It is a reflection of the fact that the very existence of scattering requires scattering in the forward ($\theta=0$ ) direction in order to interfere with the incident wave, and thereby reduce the probability current in this direction.

It is usual to write

$$\sigma_{\rm total} = \sum_{l=0,\infty} \sigma_l,$$ (983)

where

$$\sigma_l = \frac{4\pi}{k^2}\, (2\,l+1)\, \sin^2\delta_l$$ (984)

is the $l$ th partial cross-section: i.e., the contribution to the total cross-section from the $l$ th partial wave. Note that the maximum value for the $l$ th partial cross-section occurs when the phase-shift $\delta_l$ takes the value $\pi/2$ .