Atomic scattering factors

From GISAXS
Jump to: navigation, search

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.

Elements-f1.pngElements-f2.png

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

Atomic scattering factors (f1 and f2).

gold

Atomic scattering factors (f1 and f2).

Elemental/Energy dependence

Elements2D-f1.pngElements2D-f2.png

Related forms

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

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