Scattering amplitude

In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction

\psi(\mathbf{r}) = e^{ikz} + f(\theta)\frac{e^{ikr}}{r} \;, $$ where $$\mathbf{r}\equiv(x,y,z)$$ is the position vector; $$r\equiv|\mathbf{r}|$$; $$e^{ikz}$$ is the incoming plane wave with the wavenumber $k$   along the $z$ axis; $$e^{ikr}/r$$ is the outgoing spherical wave; $θ$ is the scattering angle (angle between the incident and scattered direction); and $$f(\theta)$$ is the scattering amplitude. The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,

d\sigma = |f(\theta)|^2 \;d\Omega. $$

The asymptotic form of the wave function in arbitrary external field takes the form
 * $$\psi = e^{ikr\mathbf n\cdot\mathbf n'} + f(\mathbf n,\mathbf n') \frac{e^{ikr}}{r}$$

where $$\mathbf n$$ is the direction of incidient particles and $$\mathbf n'$$ is the direction of scattered particles.

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

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

In the centrally symmetric field, the unitary condition becomes
 * $$\mathrm{Im} f(\theta)=\frac{k}{4\pi}\int f(\gamma)f(\gamma')\,d\Omega''$$

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

Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
 * $$f=\sum_{\ell=0}^\infty (2\ell+1) f_\ell P_\ell(\cos \theta)$$,

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 $S_{ℓ}$ ($$=e^{2i\delta_\ell}$$) and the scattering phase shift $δ_{ℓ}$ as
 * $$f_\ell = \frac{S_\ell-1}{2ik} = \frac{e^{2i\delta_\ell}-1}{2ik} = \frac{e^{i\delta_\ell} \sin\delta_\ell}{k} = \frac{1}{k\cot\delta_\ell-ik} \;.$$

Then the total cross section
 * $$\sigma = \int |f(\theta)|^2d\Omega $$,

can be expanded as
 * $$\sigma = \sum_{l=0}^\infty \sigma_l, \quad \text{where} \quad \sigma_l = 4\pi(2l+1)|f_l|^2=\frac{4\pi}{k^2}(2l+1)\sin^2\delta_l$$

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

For $$\theta\neq 0$$, we can write
 * $$f=\frac{1}{2ik}\sum_{\ell=0}^\infty (2\ell+1) e^{2i\delta_l} P_\ell(\cos \theta).$$

X-rays
The scattering length for X-rays is the Thomson scattering length or classical electron radius, $r$0.

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

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

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