Dynamic structure factor

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 → , ω ) {\displaystyle S({\vec {k}},\omega )} , where k → {\displaystyle {\vec {k}}} (sometimes q → {\displaystyle {\vec {q}}} ) is a wave vector (or wave number for isotropic materials), and ω {\displaystyle \omega } a frequency (sometimes stated as energy, ℏ ω {\displaystyle \hbar \omega } ). It is defined as:^[1]

S ( k → , ω ) ≡ 1 2 π ∫ − ∞ ∞ F ( k → , t ) exp ⁡ ( i ω 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 → , 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 → , t ) {\displaystyle G({\vec {r}},t)} :^[2][3]

F ( k → , t ) ≡ ∫ G ( r → , t ) exp ⁡ ( − i k → ⋅ r → ) d r → {\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 ρ {\displaystyle \rho } :

F ( k → , t ) = 1 N ⟨ ρ k → ( t ) ρ − k → ( 0 ) ⟩ {\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 σ d Ω d ω = a 2 ( E f E i ) 1 / 2 S ( k → , ω ) {\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.

The van Hove function for a spatially uniform system containing N {\displaystyle N} point particles is defined as:^[1]

G ( r → , t ) = ⟨ 1 N ∫ ∑ i = 1 N ∑ j = 1 N δ [ r → ′ + r → − r → j ( t ) ] δ [ r → ′ − r → i ( 0 ) ] d r → ′ ⟩ {\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 → , t ) = ⟨ 1 N ∫ ρ ( r → ′ + r → , t ) ρ ( r → ′ , 0 ) d r → ′ ⟩ {\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 }

• Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (Appendix N). Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.
• Lovesey, Stephen W. (1986). Theory of Neutron Scattering from Condensed Matter - Volume I: Nuclear Scattering. Oxford University Press. ISBN 9780198520283.

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