PHYSICAL REVIEW B VOLUME 46, NUMBER 18 1 NOVEMBER 1992-II Scattering delay and renormalization of the wave-diffusion constant Gabriel Cwilich' and Yaotian Fu Department of Physics, Box 1105, Washington University, Saint Louis, Missouri 63130 (Received 6 July 1992) Recent work by van Albada, van Tiggelen, Lagendijk, and Tip reveals the existence of a significant correction to the wave-di6'usion constant. We give a simple physical picture for this, along with a derivation for the case of scalar wave and weak disorder. The result of van Albada et al. is largely confirmed and its inevitability established using the notion of Wigner scattering time. In a random medium, a classical wave propagates on a short length scale but diffuses on a length scale much greater than the mean free path. ' The latter regime is characterized by the effective diffusion constant D. If the disorder is weak, D can be calculated using standard tech- niques and is given by the familiar expression vplr/3, where vo is the phase velocity and IT is the transport mean free path involving the weighted angular average of the differential cross section of a single scatterer 1T= [n f cr(8)[1 — cos(8)]] where n is the density of scatterers. In a recent paper van Albada et al. point out that this naive expression for D is, in general, not valid because a sizable correction has so far been overlooked. Using the model of a randomly placed set of identical Mie scatterers of radius R with a volume-filling fraction f, they find the correction at fre- quency E to be d+n d n 1+—, ' g (2n+1) + Do x „ i dX dX where Do is the "bare" diffusion constant volT/3, x =RE/uo is the size parameter, and a„and P„are the van de Hulst phase shifts. A derivation of the above re- sult was sketched, and good numerical agreement with experiments was found. In this paper we present a qualitative physical picture based on the notion of the scattering delay proposed by Wigner. This picture clearly shows why the difFusion- constant correction of the kind proposed by van Albada et al. should be expected on general ground; in addition, we present a detailed derivation of the diffusion-constant correction and find qualitative agreement with the results of van Albada et al. For simplicity, we restrict ourselves to the discussion of scalar ~aves; though we expect our results to be qualitatively useful to the case of vector waves, the precise quantitative result remains to be worked out. We also assume the disorder to be weak. Central to our discussion is the important observation by van Albada et al. that a clear distinction must be made between steady-state measurements and dynamical measurements. Typically, the mean free path can be measured under steady-state conditions whereas the diffusion constant must be measured dynamically. The physical picture underlying the diffusion of classical waves in a random medium is the same as that of a classi- cal particle undergoing Brownian motion; both are exam- ples of random walk. If, at t =0, a wave packet is intro- duced at the point r=0, then to reach a point ~r~ =L ))1 requires N-(L /1 ) steps and consequently an amount of time rN, where r=l/vo is the time it takes to freely propagate a distance I. The expression of the diffusion constant follows irnrnediately. At first sight, the above discussion is completely gen- eral and watertight. Note, however, that the operational definition of the diffusion constant requires that we keep track of the time evolution of the wave. As a result, we cannot use a monochromatic wave and must use a wave packet which involves at least two different frequencies. The use of wave packets brings out the new physics: As pointed out by Wigner, the scattering of a wave packet takes time. Very crudely speaking, we can think of the N scattering events in the following way. Between any two consecutive scatterings the wave propagates a distance l which takes time ~. Each scattering requires an addition- al Wigner "scattering time" ~~. Thus the ¹teprandom walk takes (r+ r ii )N to complete. The apparent diffusion constant is then given by D =Dos/(r+ ra, ). Wigner has given a simple derivation of ~~. Consider an incoming scalar wave packet which consists of two plane waves with slightly different frequencies and wave vectors @; = exp(ik r icot)[1— + exp(ihk r ihcot)], — where hco and hk are both small. The outgoing wave is a superposition of the scattered waves and is, in standard notation, e /k'f ONE 2i 5(( co) g (21+1)P,( cos8)(e ' — 1) 2l P' ( O 2t 5I (co+ b, co) X 1+ 1+5k/k '~l' ' ] e where the phase shifts 5I are, in general, functions of co. The motion of the center of the outgoing wave packet can be determined by looking at ~%~, and the 1th partial wave moves according to (bk)-r — (hto)(t d5&Idea)— =const. The center of the wave packet is seen to move 12 015 1992 The American Physical Society