Atomic scattering factors

The atomic scattering factors are measures of the scattering power of individual atoms. Each element has a different atomic scattering factor, which represents how strongly x-rays interact with those atoms.

The scattering factor has two components: f₁ and f₂, which describe the dispersive and absorptive components. In other words, f₂ describes how strongly the material absorbs the radiation, while f₁ describes the non-absorptive interaction (which leads to refraction).

Elemental dependence

Because x-ray interactions occur with an atom's electron cloud, the scattering factors increase with number of electrons, and thus with atomic number (Z). However, the relationship between f and Z is not monotonic, owing to resonant (absorption) edges.

Energy dependence

The atomic scattering factors vary with x-ray wavelength. In particular, a given element will have resonant edges at certain energies, where the absorption increases markedly. The dispersive component f₁ will also vary rapidly in the vicinity of an absorption edge (c.f. Kramers-Kronig relations). In general, absorption decreases with increasing energy (i.e. high-energy x-rays can penetrate more efficiently through materials).

Examples

silicon

gold

Elemental/Energy dependence

Related forms

There are a variety of quantities related to the material's x-ray interaction strength:

• Dispersive atomic scattering factor $f_{1}$, the intrinsic interaction of the material.
• Critical angle $\theta _{c}$, the angle below which the beam undergoes total external reflection.
• Critical wave-vector $q_{c}$, the momentum transfer (in reciprocal-space) corresponding to the critical angle.
• Real refractive index $\delta$, the refractive component of the refractive index.
• Real Scattering Length Density $\mathrm {Re} (\mathrm {SLD} )$, the primary (non-absorptive) component of the scattering contrast.
• Electron Density $\rho _{e}$, the number of electrons per unit volume.

$f_{1}$	$f_{1}={\frac {\pi M_{a}}{\rho N_{a}r_{e}\lambda ^{2}}}\theta _{c}^{2}$	$f_{1}={\frac {M_{a}}{16\pi \rho N_{a}r_{e}}}q_{c}^{2}$	$f_{1}={\frac {2\pi M_{a}}{\rho N_{a}r_{e}\lambda ^{2}}}\delta$	$f_{1}={\frac {M_{a}}{\rho N_{a}r_{e}}}\mathrm {Re} (\mathrm {SLD} )$	$f_{1}={\frac {M_{a}}{\rho N_{a}}}\rho _{e}$
$\theta _{c}={\sqrt {{\frac {\rho N_{a}r_{e}\lambda ^{2}}{\pi M_{a}}}f_{1}}}$	$\theta _{c}$	$\theta _{c}\approx {\frac {\lambda }{4\pi }}q_{c}$	$\theta _{c}={\sqrt {2\delta }}$	$\theta _{c}\approx {\sqrt {\frac {\lambda ^{2}\mathrm {SLD} }{\pi }}}$	$\theta _{c}={\sqrt {{\frac {r_{e}\lambda ^{2}}{\pi }}\rho _{e}}}$
$q_{c}={\sqrt {{\frac {16\pi \rho N_{a}r_{e}}{M_{a}}}f_{1}}}$	$q_{c}={\frac {4\pi }{\lambda }}\sin \theta _{c}$	$q_{c}$	$q_{c}\approx {\frac {4\pi }{\lambda }}{\sqrt {2\delta }}$	$q_{c}={\sqrt {16\pi \mathrm {SLD} }}$	$q_{c}=4{\sqrt {\pi r_{e}\rho _{e}}}$
$\delta ={\frac {\rho N_{a}r_{e}\lambda ^{2}}{2\pi M_{a}}}f_{1}$	$\delta ={\frac {\theta _{c}^{2}}{2}}$	$\delta \approx {\frac {\lambda ^{2}}{32\pi ^{2}}}q_{c}^{2}$	$\delta$	$\delta \approx {\frac {\lambda ^{2}}{2\pi }}\mathrm {Re} (\mathrm {SLD} )$	$\delta ={\frac {\lambda ^{2}r_{e}}{2\pi }}\rho _{e}$
$\mathrm {Re} (\mathrm {SLD} )={\frac {\rho N_{a}r_{e}}{M_{a}}}f_{1}$	$\mathrm {Re} (\mathrm {SLD} )\approx {\frac {\pi }{\lambda ^{2}}}\theta _{c}^{2}$	$\mathrm {Re} (\mathrm {SLD} )={\frac {q_{c}^{2}}{16\pi }}$	$\mathrm {Re} (\mathrm {SLD} )\approx {\frac {2\pi }{\lambda ^{2}}}\delta$	$\mathrm {Re} (\mathrm {SLD} )$	$\mathrm {Re} (\mathrm {SLD} )=r_{e}\rho _{e}$
$\rho _{e}={\frac {\rho N_{a}}{M_{a}}}f_{1}$	$\rho _{e}={\frac {\pi }{r_{e}\lambda ^{2}}}\theta _{c}^{2}$	$\rho _{e}={\frac {q_{c}^{2}}{16\pi r_{e}}}$	$\rho _{e}={\frac {2\pi }{r_{e}\lambda ^{2}}}\delta$	$\rho _{e}={\frac {\mathrm {Re} (\mathrm {SLD} )}{r_{e}}}$	$\rho _{e}$

See also absorption length for a comparison of the quantities related to f₂.

See Also

• Atomic Form Factor
• Periodic table of atomic scattering factors: Useful tool for looking up the values for any element.
• APS Python code for calculating f
• Online Dictionary of Crystallography: Atomic scattering factor