Dynamic structure factor

In <a href="/facts/Condensed_matter_physics/EBVI5nmL">condensed matter physics</a>, 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 <a href="/facts/Structure_factor/ulu9jjuc">structure factor</a> that considers correlations in both space and time. Experimentally, it can be accessed most directly by <a href="/facts/Inelastic_neutron_scattering/wMze4rvs">inelastic neutron scattering</a> or <a href="/facts/X-ray_Raman_scattering/Fb5tT3yj">X-ray Raman scattering</a>.
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:

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 <a href="/facts/Neutron_spin_echo/0XgbCvJR">neutron spin echo</a> spectroscopy. The intermediate scattering function is the spatial <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the van Hove function 
 
 
 
 G
 (
 
 
 
 r
 →
 
 
 
 ,
 t
 )
 
 
 {\displaystyle G({\vec {r}},t)}
 
:

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 <a href="/facts/Leon_van_Hove/TM6P3hau">van Hove</a>'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 <a href="/facts/Differential_cross_section/lo014h2D">differential cross section</a> 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 <a href="/facts/Scattering_length/6GlNnu6D">scattering length</a>.

Dynamic structure factor open-in-new

Dynamic structure factor