View source for 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: ''f''<sub>1</sub> and ''f''<sub>2</sub>, which describe the dispersive and absorptive components. In other words, ''f''<sub>2</sub> describes how strongly the material absorbs the radiation, while ''f''<sub>1</sub> describes the non-absorptive interaction (which leads to [[Refractive_index|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.

[[Image:Elements-f1.png|400px]][[Image:Elements-f2.png|400px]]

==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 ''f''<sub>1</sub> will also vary rapidly in the vicinity of an absorption edge (c.f. [http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations Kramers-Kronig relations]). In general, absorption decreases with increasing energy (i.e. high-energy x-rays can penetrate more efficiently through materials).

===Examples===

====[[Material:Silicon|silicon]]====

[[Image:Silicon-atomic scatt factor.png|400px|[[Atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>).]]

====[[Material:Gold|gold]]====

[[Image:Gold-atomic scatt factor.png|400px|[[Atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>).]]

==Elemental/Energy dependence==
[[Image:Elements2D-f1.png|400px]][[Image:Elements2D-f2.png|400px]]

==Related forms==
There are a variety of quantities related to the material's x-ray interaction strength:
* '''Dispersive atomic scattering factor''' <math>f_1</math>, the intrinsic interaction of the material.
* '''[[Critical angle]]''' <math>\theta_c</math>, the angle below which the beam undergoes [[Refractive_index#Total_external_reflection|total external reflection]].
* '''[[Critical_angle#Critical_q|Critical wave-vector]]''' <math>q_c</math>, the [[momentum transfer]] (in [[reciprocal-space]]) corresponding to the critical angle.
* '''Real [[refractive index]]''' <math>\delta</math>, the refractive component of the refractive index.
* '''Real [[Scattering Length Density]]''' <math>\mathrm{Re}(\mathrm{SLD})</math>, the primary (non-absorptive) component of the scattering contrast.

{| class="wikitable"
|-
| <math>f_1</math>
| <math>f_1 = \frac{ \pi M_a}{ \rho N_a r_e \lambda^2 } \theta_c^2</math>
| 
| <math>f_1 = \frac{2 \pi M_a}{ \rho N_a r_e \lambda^2 } \delta</math>
| <math>f_1 \approx \frac{M_a}{\rho N_a r_e} \mathrm{Re}(\mathrm{SLD}) </math>
|-
| <math>\theta_c = \sqrt{ \frac{ \rho N_a r_e \lambda^2 }{ \pi M_a} f_1 }</math>
| <math>\theta_c</math>
| <math>\theta_c \approx \frac{\lambda}{4 \pi} q_c </math>
| <math>\theta_c = \sqrt{ 2 \delta }</math>
| <math>\theta_c\approx \sqrt{\frac{\lambda^2 \mathrm{SLD} }{\pi}}</math>
|-
| 
| <math>q_c = \frac{4 \pi}{\lambda}\sin\theta_c</math>
| <math>q_c</math>
| 
| <math>q_c = \sqrt{ 16 \pi \mathrm{SLD} }</math>
|-
| <math>\delta = \frac{ \rho N_a r_e \lambda^2 }{2 \pi M_a} f_1</math>
| <math>\delta = \frac{\theta_c^2}{2}</math>
| 
| <math>\delta</math>
| 
|-
| <math>\mathrm{Re}(\mathrm{SLD}) \approx \frac{\rho N_a r_e}{M_a} f_1</math>
| <math>\mathrm{Re}(\mathrm{SLD}) \approx \frac{\pi}{\lambda^2}\theta_c^2</math>
| <math>\mathrm{Re}(\mathrm{SLD}) = \frac{q_c^2}{16\pi}</math>
| 
| <math>\mathrm{Re}(\mathrm{SLD})</math>
|}


See also '''[[Absorption_length#Related_forms|absorption length]]''' for a comparison of the quantities related to ''f''<sub>2</sub>.

==See Also==
* [http://henke.lbl.gov/optical_constants/asf.html Periodic table of atomic scattering factors]: Useful tool for looking up the values for any element.
* [https://subversion.xor.aps.anl.gov/trac/pyFprime/ APS Python code for calculating ''f'']
* [http://reference.iucr.org/dictionary/Atomic_scattering_factor Online Dictionary of Crystallography: Atomic scattering factor]