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Submitted on 1 Jan 1981
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INTERPRETATION OF EXPERIMENTS IN THE
SOLIDON MODEL FOR ROTONS
M. Héritier, G. Montambaux
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque 66, suppzlment au no 12, Tome 4 2 , dlcembre 1981 page C6-534
INTERPRETATION OF EXPERIMENTS I N THE SOLIDON MODEL FOR ROTONS M. Hsritier and G. Montambaux
Laboratoire de Physique des SoZides, Universitg de Paris-Sud,91405 Orsay, France.
Abstract
-
A roton in superfluid Helium can be described as a phonon self-trapped in a region possessing the translation symme- tries of the metastable solid. This picture can account for va- rious experimental data, such as the roton wavevector in bulk Helium I1 and Helium films, the temperature de~endence of the ro- ton sap, the roton-roton collision time.It has been shown recently that, under certain conditions, a
longitudinal phonon in a quantum liquid can be self-trapped in a region of the liquid possessing the translation symmetries of the metastable solid (l). The first condition is a favourable balance between, on the one hand, the phonon energy lowering at a wavevector belonging to the reciprocal lattice of the solid, and, on the other hand, the phonon localization energy within the "solid-like" volume and the formation thermodynamic potential of this volume. (In principle, volume and sur- face terms are to be included. Another condition for the existence of a well-defined self-trapped excitation is a large enough lifetime. Helium 11, which is a "nearly solid" quantum liquid (2) (first condi- tion) and superfluid (second condition) is a good candidate. This pic- ture, given as an example of collective excitation self-trapping, led to a new description of the roton, somewhat analogous to the FeymIan's vortex ring ( 3 ) , but with a large "solid-like" core in which the atoms undergo a translation motion. At present time, this model (the "solidon model") seems to give the best account for various experimental data.
A very simple variational procedure gives the roton gap by minimi- zation of the solidon free energy with respect with the number of
atoms involved in the core radius N. Three terms are to be included :
the phonon localization energy a 2 / 2 ~ m (the "recoil" energy), where m is the atomic mass, and q the wavevector, the solidification free en- thalpy, proportional to N, and the surface tension N ~ / ~ .
Near enough to the solidification curve, the interface energy term predominates. A good estimate of the roton gap at low temperature can be obtained, simply from the knowledge of the solid-liquid surface tension. The corresponding values of N range from 25 to 40 depending on
the pressure.
The gap is now a thermodynamic quantity which can vary with tem- perature, even in the case of independent rotons. (In the Feynman model, the temperature dependence comes only from the roton-roton interactions and seems to be weaker than observed experimentally). In fact, the temperature dependence due to the variation of the solidification free enthalpy is weak. A larger effect is obtained when one takes into account the entropy flow associated with the atom dynamics. This entro- py flow is necessary to preserve entropy conservation because the solid and liquid phases have different entropy, uarticularly at temperatures above 1 K. Because the momentum of this flow depends on temperature, so does the roton gap. Then a good agreement with the observed varia- tion of the roton gap is obtained.
An important prediction of the model is the roton wavevector : it should correspond to the shortest reciprocal lattice wavevector of the metastable solid with the lowest free enthalpy. Both surface and vo- lume enthalpy terms are to be included in determining this lattice. This property is an essential and simple feature of the model. In bulk Helium 11, at low pressure, where the volume term ~redominates, the metastable solid with the lowest volume free enthalpy has a b.c.c.
lattice. This is no longer true near the solidification curve at low temperatures (the lowest volume free enthalpy lattice is h.c.~.), but in these conditions the interface energy term predominates and the lowest solid-liquid surface tension is obtained for a b.c.c. lattice. Therefore, one expects a solidon core with a b.c.c. structure at any temperature and at any pressure. Indeed, the observed roton wavevector agrees well with this prediction.
Thin Helium films exhibit a roton-like excitation at a wavevector
q = 2 A-' ( 4 ) , much larger than expected in the Feynman model ( 5 ) . In our model, two interpretations are possible. In the first, the excita- tion is a solidon bound state at the solid-liquid interface, due to image forces, stable because of the finite size of the solidon core. Then, q should correspond to a.b.c.c. lattice distorted by the Van der Waals attraction to the substrate. In the second interpretation, the excitation is a truly two-dimensional roton. Then, q should correspond to the metastable 2 D triangular lattice. Both interpretations give about the same values for q, in agreement with experiment.
C6-53 6 JOURNAL DE PHYSIQUE
mum at a wavevector q shows the existence of a finite size core, with a radius of order 2a/q, in agreement with the previous estimate. This departure from the dipolar behaviour, noted by Castaing and Libchaber (6) in the roton-roton interaction potential, should exist also in the velocity field of a single roton. Roton-roton scattering can ~rovide a test of the model. According to various experiments (see references in (6)), the roton-roton collision frequency can be written T ~ = ~BnR. - ~
nR is the thermal roton density and B
-
1.7 X 10-'~cm~/s (7). The con-tribution of the dipolar backflows to the collision frequency is small (7). Assuming a large core size, the interaction is written, in the hydrodynamic approximation as a core-core interaction. Using the Lipp- man-Schwinger method, the solidon core radius R is determined from B. We find R = 7.5 at 25 bars, i.e. about 35-40 atoms in the core. Castaing and Libchaber (6) calculation also fits the data. In fact, their theory is consistent with our model since they arrive to a roton- roton interaction non dipolar at distances shorter than 27r/q.
These ordered "grains" may have been observed in neutron experi- ments (8). A new structure was observed in the structure factor below the h transition, consisting of a single peak at about 2
X-'
momentum transfer. The intensity of this peak increases as approaches TX Or when the pressure is increased. It is tempting to ascribe these effects by a larger number of excited rotons (with their cores modulated at q 2 i - l ) , when T increases or when the roton gap decreases with
applied pressure.
1) M. HGritier, G. Montambaux and P. Lederer, J. Phys. (Paris),
40,
L. 493 (1979).
2) B. Castaing and P. NoziSres, J. Phys. Paris,
40,
257 (1979). 3) R.P. Feynman and M. Cohen, Phys. Rev.102,
1189 (1956).4) W. Thomlinson, J.A. Tarvin and L. Passell, Phys. Rev. Letters,
44,
266 (1980).5) W. Gijtze and M. Lucke, J. of Low Temp. Phys.,
25,
671 (1976).6) B. Castaing and A. Libchaber, J. of Low Temp. ~hys.,
