VOLUME76, NUMBER10 P H Y S I C A L R E V I E W L E T T E R S 4 MARCH1996 Crevecoeur, de Schepper, and Montfrooij Reply: The
distinction made in the Comment [1] to our Letter [2] between the frequency regions "v ø kBT and "v ¿ kBT has indeed been proven to be useful in the past to
understand the neutron spectra Ssq, vd in purely classi-cal fluids s"v ø kBTd and superfluid He s"v ¿ kBTd.
However, as a consequence, this has led to two quite dif-ferent physical descriptions of Ssq, vd: the two-Lorentzian DHO (phonon) model for superfluid He and the three-Lorentzian model for classical fluids (i.e., the hydrody-namic Rayleigh-Brillouin triplet extended to larger q). In our Letter we consider He at 4 K (intermediate) purposely to determine the physical connection between both regimes using a unified theoretical description.
To do so, we use the exact Mori-Zwanzig projection operator formalism since it is valid for all q and v, for both classical as well as quantum fluids. In particular, it takes the asymmetry of Ssq, vd, which develops when one enters the quantum regime, exactly into account. In this formalism, dynamic correlation functions like Ssq, vd [or xnnsq, vd] are expressed in terms of (v-independent)
coupling parameters between a number of , relevant microscopic variables (microscopic density, longitudinal momentum, temperature, and so on), with , $ 2, since the continuity equation implies that a fluctuation in the number density is always (and exclusively) coupled to a momentum fluctuation which can then be coupled to other microscopic variables. These coupling parameters are given by the, 3 , matrix Hsqd. The effective number , of contributing parameters can be determined in principle from theory and experiment, by including increasingly more microscopic variables until an accurate description of Ssq, vd has been obtained. This formalism is derived in Ref. [13] of Ref. [2] for all q and v, and not only for the low frequency domain "v ø kBT as Griffin stated in
his Comment. From the good fits to our experimental data we have shown that, 2 for 4He at 4.0 K in the region covered by our neutron experiment, implying that the only two relevant microscopic variables are the density and longitudinal momentum. Therefore the observed fluctuations in density must necessarily be propagating (phonons) or diffusive (overdamped phonons). Thus, not only have we shown that the spectra can be well described by a damped harmonic oscillator form, we have also shown this form to be theoretically sound outside its hitherto assumed region of validity. By no means have we simply taken an expression for the low energy region s"v ø kBTd and assumed its validity throughout the
entire spectrum as suggested in the Comment. We note here that there is no theoretical derivation yet for the fact that, 2 for He at 4 K.
An illuminating discussion on the physical nature of overdamped phonons has been given by Kirkpatrick [3]. He considers for classical hard spheres a 2 3 2
FIG. 1. Half-widths of He at 4 K as functions of q: Gssqd (crosses; propagating phonons), Gs,1sqd (open circles;
overdamped phonons), Gs,2sqd (closed circles; overdamped
phonons), vHsqd (open squares).
model matrix Hsqd directly derived from the Liouville equation. For wave numbers q near qp where Ssqd has its first maximum, overdamped phonons are a common phenomenon for height densities. Their behavior near qp is determined by the “cage effect” (i.e., the particles are locked up in mutual cages of size 2pyqp) which causes the so-called de Gennes narrowing of Ssq, vd near qp. Thus, the overdamping of phonons we observe in He at 4 K is most likely a remnant of de Gennes narrowing in classical fluids and not an artifact of the fits; more so since the half-width at half-height vHsqd of Spsq, vd
xsq, vd in He at 4 K also shows a pronounced de Gennes minimum near qp 20 nm21 (cf. Fig. 1). We note that vHsqd , Gs,2sqd due to the negative amplitude of the
broad Lorentzian, as usual for overdamped harmonic oscillators.
R. M. Crevecoeur and I. M. de Schepper Interfaculty Reactor Institute
Delft University of Technology 2629 JB Delft, The Netherlands W. Montfrooij
ISIS Pulsed Source
DRAL Rutherford Appleton Laboratory Didcot, Oxon OX11 0QX, United Kingdom Received 28 November 1995
PACS numbers: 67.20. + k, 05.30. – d, 61.12.Ex, 61.25.Bi [1] A. Griffin, preceding Comment, Phys. Rev. Lett. 76, 1759
(1996).
[2] R. M. Crevecoeur, R. Verberg, I. M. de Schepper, L. A. de Graaf, and W. Montfrooij, Phys. Rev. Lett. 74, 5052 (1995).
[3] T. R. Kirkpatrick, Phys. Rev. A 32, 3130 (1985).
