By N. H. March, M. P. Tosi (auth.)
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Additional resources for Atomic Dynamics in Liquids
Since we want to relate S(k) and ;)>(k), we must somehow reduce (Pk+nPkPn) to S(k) and, if we pick out from the sum the term n = -k, this is"evidently related to S(k). This is, in fact, the random-phase approximation which will be discussed more fully in chapter 7. The qualitative argument is that unless k and n bear this simple relation, destructive interference between the various oscillatory components will tend to annul the other terms. Actually, such an argument turns out to be a long wavelength approximation.
R', t') at times t' later than t, x(r, t) must vanish when t < 0. 43). In turn, this mathematical statement implies the existence of a dispersion relation of the form x(k, w)=~ Joo dw' 7r - 00 ~"(k, w'). 47) namely, x"(k, w) serves as the spectral density for x(k, w). The above relation implies that x'(k, w) and x"(k, w) are related by Kramers-Kronig relations. 46). 20), we have the desired expressions for the moments. 20). 52) is given by ikt 2 k2t2 2! (x1(0)) -2! 55) Actually, from the definition of F(k, t) as the distinct term i 1- 1 vanishes to order t 2 , since the velocities oftwo atoms taken at the same instant of time are uncorrelated.
24) the usual (static) structure factor. 3 Relation to neutron scattering The process of inelastic scattering of a beam of incident particles (typically, neutrons) from a liquid serves as a microscopic probe of the dynamics of this many-particle system. More precisely, as first shown by van Hove (1954a), a measurement of coherent inelastic scattering yields S(k, w) (see, for example, Lomer and Low, 1965; Marshall and Lovesey, 1973). 5 below) in statistical mechanics (Callen and Welton, 1951), according to which a measurement of the dissipation of energy by an external probe weakly coupled to a many-particle system is directly related to correlations between equilibrium fluctuations in the system.
Atomic Dynamics in Liquids by N. H. March, M. P. Tosi (auth.)