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]

${\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}${\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}

where ${\displaystyle \mathbf {r} \equiv (x,y,z)}${\displaystyle \mathbf {r} \equiv (x,y,z)} is the position vector; ${\displaystyle r\equiv |\mathbf {r} |}${\displaystyle r\equiv |\mathbf {r} |}; ${\displaystyle e^{ikz}}${\displaystyle e^{ikz}} is the incoming plane wave with the wavenumber k along the z axis; ${\displaystyle e^{ikr}/r}${\displaystyle e^{ikr}/r} is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and ${\displaystyle f(\theta )}${\displaystyle 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,

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

The asymptotic form of the wave function in arbitrary external field takes the form^[2]

${\displaystyle \psi =e^{ikr\mathbf {n} \cdot \mathbf {n} '}+f(\mathbf {n} ,\mathbf {n} '){\frac {e^{ikr}}{r}}}${\displaystyle \psi =e^{ikr\mathbf {n} \cdot \mathbf {n} '}+f(\mathbf {n} ,\mathbf {n} '){\frac {e^{ikr}}{r}}}

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

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]

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

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

In the centrally symmetric field, the unitary condition becomes

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

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

${\displaystyle f=\sum _{\ell =0}^{\infty }(2\ell +1)f_{\ell }P_{\ell }(\cos \theta )}${\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 (${\displaystyle =e^{2i\delta _{\ell }}}${\displaystyle =e^{2i\delta _{\ell }}}) and the scattering phase shift δ_l as

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

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

can be expanded as^[2]

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

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

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

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

• Levinson's theorem
• Veneziano amplitude
• Plane wave expansion

• Neutron
• X-rays
• Electron
• Scattering
• Diffraction
• Quantum mechanics

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