Dynamic structure factor
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In condensed matter physics, the dynamic structure factor (or dynamical structure factor) is a mathematical function that contains information about inter-particle correlations and their time evolution. It is a generalization of the structure factor that considers correlations in both space and time. Experimentally, it can be accessed most directly by inelastic neutron scattering or X-ray Raman scattering.
The dynamic structure factor is most often denoted S ( k β , Ο ) {\displaystyle S({\vec {k}},\omega )} , where k β {\displaystyle {\vec {k}}} (sometimes q β {\displaystyle {\vec {q}}} ) is a wave vector (or wave number for isotropic materials), and Ο {\displaystyle \omega } a frequency (sometimes stated as energy, β Ο {\displaystyle \hbar \omega } ). It is defined as:cite-ref-theoryliquids-1-0[1]
S ( k β , Ο ) β‘ 1 2 Ο β« β β β F ( k β , t ) exp β‘ ( i Ο t ) d t {\displaystyle S({\vec {k}},\omega )\equiv {\frac {1}{2\pi }}\int _{-\infty }^{\infty }F({\vec {k}},t)\exp(i\omega t)\,dt}
Here F ( k β , t ) {\displaystyle F({\vec {k}},t)} , is called the intermediate scattering function and can be measured by neutron spin echo spectroscopy. The intermediate scattering function is the spatial Fourier transform of the van Hove function G ( r β , t ) {\displaystyle G({\vec {r}},t)} :cite-ref-2[2]cite-ref-vineyard1958-3-0[3]
F ( k β , t ) β‘ β« G ( r β , t ) exp β‘ ( β i k β β
r β ) d r β {\displaystyle F({\vec {k}},t)\equiv \int G({\vec {r}},t)\exp(-i{\vec {k}}\cdot {\vec {r}})\,d{\vec {r}}}
Thus we see that the dynamical structure factor is the spatial and temporal Fourier transform of van Hove's time-dependent pair correlation function. It can be shown (see below), that the intermediate scattering function is the correlation function of the Fourier components of the density Ο {\displaystyle \rho } :
F ( k β , t ) = 1 N β¨ Ο k β ( t ) Ο β k β ( 0 ) β© {\displaystyle F({\vec {k}},t)={\frac {1}{N}}\langle \rho _{\vec {k}}(t)\rho _{-{\vec {k}}}(0)\rangle }
The dynamic structure is exactly what is probed in coherent inelastic neutron scattering. The differential cross section is :
d 2 Ο d Ξ© d Ο = a 2 ( E f E i ) 1 / 2 S ( k β , Ο ) {\displaystyle {\frac {d^{2}\sigma }{d\Omega d\omega }}=a^{2}\left({\frac {E_{f}}{E_{i}}}\right)^{1/2}S({\vec {k}},\omega )}
where a {\displaystyle a} is the scattering length.
Contents
β’ References
β’ Further reading
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The van Hove function
The van Hove function for a spatially uniform system containing N {\displaystyle N} point particles is defined as:cite-ref-theoryliquids-1-1[1]
G ( r β , t ) = β¨ 1 N β« β i = 1 N β j = 1 N Ξ΄ [ r β β² + r β β r β j ( t ) ] Ξ΄ [ r β β² β r β i ( 0 ) ] d r β β² β© {\displaystyle G({\vec {r}},t)=\left\langle {\frac {1}{N}}\int \sum _{i=1}^{N}\sum _{j=1}^{N}\delta [{\vec {r}}'+{\vec {r}}-{\vec {r}}_{j}(t)]\delta [{\vec {r}}'-{\vec {r}}_{i}(0)]d{\vec {r}}'\right\rangle }
It can be rewritten as:
G ( r β , t ) = β¨ 1 N β« Ο ( r β β² + r β , t ) Ο ( r β β² , 0 ) d r β β² β© {\displaystyle G({\vec {r}},t)=\left\langle {\frac {1}{N}}\int \rho ({\vec {r}}'+{\vec {r}},t)\rho ({\vec {r}}',0)d{\vec {r}}'\right\rangle }
References
cite-note-theoryliquids-11. β citerefhansenmcdonald1986Hansen, J. P.; McDonald, I. R. (1986). Theory of Simple Liquids. Academic Press.
cite-note-22. β citerefvan-hove1954van Hove, L. (1954). "Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles". Physical Review. 95 (1): 249. Bibcode:1954PhRv...95..249V. doi:10.1103/PhysRev.95.249.
Further reading
β’ ashcroft-merminAshcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (Appendix N). Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.
β’ Lovesey, Stephen W. (1986). Theory of Neutron Scattering from Condensed Matter - Volume I: Nuclear Scattering. Oxford University Press. ISBN 9780198520283.