Structure factor: Difference between revisions
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To calculate <math>S(k)</math> in computer simulations one typically uses: | To calculate <math>S(k)</math> in computer simulations one typically uses: | ||
:<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \exp(-i(r_i-r_j)) </math> | :<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} <\exp(-i(r_i-r_j))> </math> | ||
:<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left< \exp(-i(r_i-r_j)) \right></math> | :<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left< \exp(-i(r_i-r_j)) \right></math> | ||
Revision as of 17:27, 15 September 2011
The structure factor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k)} , for a monatomic system is defined by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~dr}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the scattering wave-vector modulus
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k= |\mathbf{k}|= \frac{4 \pi }{\lambda \sin \left( \frac{\theta}{2}\right)}}
The structure factor is basically a Fourier transform of the pair distribution function ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(|\mathbf{k}|)= 1 + \rho \int \exp (i\mathbf{k}\cdot \mathbf{r}) \mathrm{g}(r) ~\mathrm{d}\mathbf{r}}
At zero wavenumber, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{k}|=0} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T}
from which one can calculate the isothermal compressibility.
To calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k)} in computer simulations one typically uses:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S(k)={\frac {1}{N}}\sum _{i,j=1}^{N}<\exp(-i(r_{i}-r_{j}))>}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left< \exp(-i(r_i-r_j)) \right>}