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Structure factor

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The structure factor, S(k), for a monatomic system is defined by:

S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~dr

where k is the scattering wave-vector modulus

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 {\rm g}(r),

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. |\mathbf{k}|=0,

S(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T

from which one can calculate the isothermal compressibility.

To calculate S(k) in molecular simulations one typically uses:

S(k) = \frac{1}{N} \sum^{N}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle ,

where N is the number of particles and \mathbf{r}_n and \mathbf{r}_m are the coordinates of particles n and m respectively.

The dynamic, time dependent structure factor is defined as follows:

S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1}  \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle ,

The ratio between the dynamic and the static structure factor, S(k,t)/S(k,0), is known as the collective (or coherent) intermediate scattering function.


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