Diffusive Motions in Water and Cold Neutron Scattering

Abstract

Using a model of liquid water in which a molecule, in its equilibrium position, performs an oscillatory motion for a mean time TO, and then diffuses by continuous motion for a mean time n, and repeats this sort of motion, the differential scattering cross section for cold neutrons has been calculated. It is found that the shape of the "quasi-elastic" scattering is, in general, not Lorentzian. The formula for the broadening of the quasi-elastic peak assumes a simple form in two limiting cases: In case (i) ri^>ro, it reduces to the formula derived on the simple diffusion theory; and in case (ii) TI<£> 0 , the broadening is the same as in case (i) if K 2 DTQ<K1, and it approaches the asymptotic value 2h/ro, if K 2 DTQ2>1, where %K is the momentum transferred to the system * Based on work performed under the auspices of the U. S. Atomic Energy Commission.t On leave of absence from the Atomic Energy Establishment, Trombay, Bombay.Step 2N, F 2N (r,t)=(-1) 2N Xp(T2N+i)g(%2N+i,T2N+l)q'(T2N)h(£w,T2N)' ' ' p'(ri)g(%i,Ti) exp[f (K'T -«/)].Expression (6) follows immediately by making the following change of variables for the time integrals; t~~faN == T2N+l, t 2 N-t 2 N-l =T2N, '" h~~ h-T 2 , h~ Tl\ and a similar change of variables for the space integrals. Hence, I dt j dr exp[;(K-r-co/)]F 2 iv(r,0= I dr I d% exp[i(K-?~cor)]^(r)^(?,r) X { -fdr'JdZ exp[i(K. r~co/)]^(rOg(r,rO [ XJ-J dr // Jde ,/ exp[f(K-r-«r // )](Z , (r ,, )A« ,, ,r ,, )| =AC»D»; (7)

Density variations in liquid tellurium: Roles of rings, chains, and cavities

“…͑4͒͒ and is given for four temperatures in Table II. Measurements of the self-diffusion coefficient in Te have been made using neutron scattering, 44 where the broadening of the elastic peak caused by diffusion of the atoms in a liquid can be related to D, 45 and by observing the motion of radioactive tracer atoms ͑such as Te 127 ͒ in a capillary geometry. 46 The experimental data are characterized by substantial error bars and difficulties in extrapolating values measured at higher temperatures to the melting point.…”

Diffusion à longue distance

“…11 souligne que la vitesse d'une molécule telle que I'on peut la déduire de l'équation (9) (6DIt) "' (12) ne peut être supérieure à la vitesse dans le gaz parfait :…”

“…Les autres modèles qui ont été proposés dans la littérature peuvent tous être retrouvés à partir de l'équation (20). Ainsi, Egelstaff semble avoir été le premier à montrer que si l'on définissait une distribution de la forme p(r) = r exp(-r/ro), on aboutissait à la version simplifiée du modèle de Singwi et Sjolander [12], en négligeant le temps de saut devant le temps de résidence [13]. Pour comparer les différents modèles, on utilisera plutôt pour le modèle SS une distribution normalisée représentée sur la Figure 4 en ligne pointillée La distance de saut carrée moyenne correspondant à cette distribution a pour valeur Le troisième modèle, qui a été publié par Hall et Ross (HR), est basé sur une distribution de la forme [7] Cette distribution, qui est normalisée, est tracée sur la Figure 4 en ligne continue.…”

Quasielastic small-angle neutron scattering from heavy water solutions of cyclodextrins

“…18 The solutewater structure factor can also be calculated; a way to do this is shown in Appendix C. Note an approximate equality in Eq. 19 In a two state model, both translational and rotational dynamics could be described by two sets of parameters, for the bulk and for the hydration water, respectively. Note that although the rotational dynamics of the hydration water is different from that of the bulk water, in our Q region the effect of using somewhat different rotational diffusion coefficients is small (the terms for l > 0 are negligible).…”