Ph.D. thesis (pdf) - dirac

64 Experimental techniques and observables
Coherent - linearized hydrodynamics
At small wave vectors and long times the liquid can be described as a continuum,
this limit is called the hydrodynamic limit. When dealing with fluctuations it is
sufficient to consider the linearized form of the equations.
The collective dynamics splits up in two uncorrelated parts, one is due to entropy
fluctuations, which are modes with no particular frequency giving rise to a central
peak in S coh (Q, ω). The other part corresponds to the sound waves in the liquid and
is due to pressure fluctuations. Each sound wave has a well defined frequency and
wave vector, and they therefore give rise to Brillouin peaks at non-zero ω-values in
S coh (Q, ω). The sound modes are damped due to the viscosity of the liquid and this
leads to a broadening of the corresponding Brillouin peaks.
Coherent - compressibility
The static structure factor S(Q) is a measure of the amount of density fluctuation
seen on the wavelength 2π/Q. In the limit of Q → 0 this becomes the macroscopic
density fluctuations, (〈ρ 2 〉 − 〈ρ〉 2 )/ρ. The macroscopic density fluctuations
are related to the macroscopic isothermal compressibility, κ T , via the fluctuation
dissipation theorem. This yields
lim S(Q) = ρκ Tk B T. (4.3.37)
Q→0
The ratio between the integral of the Brillouin peaks and the total integral S(Q) is
in the zero Q limit given by the so called Landau Placzek ratio:
I brill(Q)
lim
Q→0 S(Q)
= κ s
= v2 T
κ T vs
2 , (4.3.38)
where κ s is the adiabatic compressibility and v T and v s are respectively the isotherm
and adiabatic sound speeds.
Combining equation 4.3.38 with equation 4.3.37 it follows that the intensity in the
Brillouin peaks is proportional to the adiabatic compressibility, κ s , while the intensity
in the central peak is proportional to the difference between adiabatic and
isothermal compressibility; κ T −κ s . This difference is small in solids while it is large
in liquids particularly close to critical points.