On the dispersion relation of crystallization-melting waves on the quantum crystal surface

SD 1

Physica 108B (1981) 1195-1196 North-Holland Publishing Company

ON THE DISPERSION RELATION OF CRYSTALLIZATION-~ELTING ON THE QUANTUM CR~/STAL SURFACE

WAVES

A. M. Kosevich Physico-Technical Institute of Low Temperatures, UkrSSR Academy of Sciences, Kharkov, USSR and Yu. A. Kosevich Moscow State University,

Moscow, USSR

We present the correction to the dispersion relation of surface waves on the quantum crystal-liquid phase boundary predicted by Andreev and Parshin and observable on solid helium surface. On the boundary between quantum crystal and liquid (its melt) there may be weakly attenuating waves caused by periodical melting-crystallization processes. Such waves were predicted and their dispersion relation calculated in ref° I. They are indeed observed on solid helium surface (see ref. 25. At T = O, the phase boundary movement does not disrupt phase equilibrium and may be interpreted as a purely mechanical process. The most important feature of it is the movement of the growth steps whose height equals the interatomic separation a. If crystallization proceeds as a plane wave, propagating along the X axis, then steps of the type represented in the figure will be infinitely large along the Y axis. If the wavelength ~ is macroscopic (akin1, k = 2 ~ r / ~ ) , then in the principal approximation the structure of an individual step may be neglected, as was done in ref. I, where the dispersion relation for surface capillary waves was derived. Here we shall show that since the steps are discrete, the kinetic energy of the moving liquid should be included in the treatment whence comes the principal correction to the said dispersion relation, as obtained in ref. 1. If within time St the step displaces by 8 X = V St, then along the step, a local change in the liquid volume takes place, ( Q 2 - ~ ) a V / g z (per unit step length), ~ w ~ e ~ Q4 a n d ~ 2 are the crystal and melt densities respectively. Let us replace the step by a semicylindrical surface of radius r o s a . Then the said change in the volume will result in the following flow velocity through this surface:

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l

/r o

~

.

Figure :

The growth step on the crystal surface

=

2-g~

r.9~

(I)

The "extra volume" source, eq. (I), will give rise in incompressible liquid to a stationary field of velocities v~ = v~ (r /r). The liquid kinetic ener6y generated by single step motion is (per unit length along the Y axis):

R

Based on eq. (2), one can introduce the effective mass of the unit length of an indivual step:

m.,: containing racteristic

g the logarithm~c f a c t o r

cha-

for the linear singularity field mass. Parameter R determines the distance from which, on one hand, the step is perceived as straight-linear and, on the other hand, the step 1195

1196

velocity field preserves its individual character. In case of long plane wave vibrations of the crystal surface, the step movement is similar to linear charge movement accompanying plasma oscillations. Steps of the opposite sign, when within a surface element which is small against the wave length, move in opposite directions, and their velocity fields in the bulk add up. As a result, in liquid at distances much larger than the average step separation io (io = I/~a ), though smaller than the wavelength, there arises a practically uniform field of velocities. The discreteness of steps is at such distances quite unimportant. Therefore the kinetic energy of the liquid may well be represented as the sum of kinetic energies of individual steps only in a subsurface layer of thickness close to lo and each step should be associated with an effective mass, eq. (3), assuming in ( R / r ) . ~ l n ( l o / r ) . The kinetic energies of deeper-lying liquid layers may be calculated on the assumption of continuous step distribution, and therefore it was correctly included in the surface oscillation dispersion law in ref. I. The additional contribution to the discrete step kinetic energy per unit length of the interphase boundary (along the X axis) is

We see that the movement of the interphase boundary unit area along the Z axis is characterized by the effective mass M = m ~ / 2 a ~ o = m~lo/2a 2, which is directly proportional to the average separation of steps (lo = I / ~ ) . The amount lo is convenient to represent as 1o = rof(T) , where f(T) specifies the temperature dependence of the equilibrium step density. The value of l@ varies from crystal to crystal. Since the crystallization-melting wave causes acceleration of the interphase boundary, then there arises an additional inertial force, which, when allowed for, changes one of the boundary conditions in ref. I to another:

z where p~ and PZ are pressures in crystal and liquid respectively, and ~ the surface tension coefficient.

Let us use eqs. (5) and (3) to derive the dispersion relation. By rather simple calculation we arrive at the following surface wave dispersion law (with(A) being the oscillation frequency):

,

-

~

(6)

In the derivation of eq. (6) we assumed that l@k ((I, and the correction is indeed small. However there may be some crystal faces for which l@>> a, and the correction becomes then important. Still, more important is the fact that the equilibrium step density 7" depends on the crystal state and therefore on temperature too: l@ = r@f(T). Thus, the dispersion relation (6) should be de-

p e n d e n t on t e m p e r a t u r e .

REFERENCES ~]

Andreev, A. F. and Parshin, A. Ya. 0 ravnovesnoy forms i kolebaniyakh poverkhnosti kvantovykh krlstallov, ZhETF 75 (1978) 1511-1516

~]

Keshishev, K. 0., Parshin, A. Ya. and Babkin, A. V., Eksperimentalnoe obnaruzhenie kristallizatsionnykh voln v He , Pis'ma v ZhETF 30 (1979) 63-67