Scattering amplitude

When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have[2]

${\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ,\mathbf {n} '')\,d\Omega ''}$

Optical theorem follows from here by setting ${\displaystyle \mathbf {n} =\mathbf {n} '.}$

In the centrally symmetric field, the unitary condition becomes

${\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}$

where ${\displaystyle \gamma }$ and ${\displaystyle \gamma '}$ are the angles between ${\displaystyle \mathbf {n} }$ and ${\displaystyle \mathbf {n} '}$ and some direction ${\displaystyle \mathbf {n} ''}$. This condition puts a constraint on the allowed form for ${\displaystyle f(\theta )}$, i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if ${\displaystyle |f(\theta )|}$ in ${\displaystyle f=|f|e^{2i\alpha }}$ is known (say, from the measurement of the cross section), then ${\displaystyle \alpha (\theta )}$ can be determined such that ${\displaystyle f(\theta )}$ is uniquely determined within the alternative ${\displaystyle f(\theta )\rightarrow -f^{*}(\theta )}$.[2]

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[3]

${\displaystyle f=\sum _{\ell =0}^{\infty }(2\ell +1)f_{\ell }P_{\ell }(\cos \theta )}$,

where f_ℓ is the partial scattering amplitude and P_ℓ are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element S_ℓ (${\displaystyle =e^{2i\delta _{\ell }}}$) and the scattering phase shift δ_ℓ as

${\displaystyle f_{\ell }={\frac {S_{\ell }-1}{2ik}}={\frac {e^{2i\delta _{\ell }}-1}{2ik}}={\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}={\frac {1}{k\cot \delta _{\ell }-ik}}\;.}$

Then the total cross section[4]

${\displaystyle \sigma =\int |f(\theta )|^{2}d\Omega }$,

can be expanded as[2]

${\displaystyle \sigma =\sum _{l=0}^{\infty }\sigma _{l},\quad {\text{where}}\quad \sigma _{l}=4\pi (2l+1)|f_{l}|^{2}={\frac {4\pi }{k^{2}}}(2l+l)\sin ^{2}\delta _{l}}$

is the partial cross section. The total cross section is also equal to ${\displaystyle \sigma =(4\pi /k)\,\mathrm {Im} f(0)}$ due to optical theorem.

For ${\displaystyle \theta \neq 0}$, we can write[2]

${\displaystyle f={\frac {1}{2ik}}\sum _{\ell =0}^{\infty }(2\ell +1)e^{2i\delta _{l}}P_{\ell }(\cos \theta ).}$

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r₀.

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

A quantum mechanical approach is given by the S matrix formalism.

The scattering amplitude can be determined by the scattering length in the low-energy regime.

• Veneziano amplitude
• Plane wave expansion