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.^[1] At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction^[2]

\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^[2]

\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^[2]

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 $f(\theta )\rightarrow -f^{*}(\theta )$ .^[2]

Partial wave expansion

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

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^[4]

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

,

can be expanded as^[2]

\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^[2]

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.

References

[1]

[2]

[3]

[4]

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