Possible long-range step interaction in 4He due to step oscillation

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Possible long-range step interaction in 4He due to step oscillation

M. Uwaha

To cite this version:

M. Uwaha. Possible long-range step interaction in 4He due to step oscillation. Journal de Physique,

1990, 51 (24), pp.2743-2746. �10.1051/jphys:0199000510240274300�. �jpa-00212568�

Short Communication

Possible long-range step interaction in 4He due to step oscillation

M. Uwaha

Institute for Materials Research, ^Tohoku University, ^2-1-1 Katahira, Aoba-ku, ^Sendai 980, Japan (Received ²⁸ September ^1990, accepted ^{8 October} 1990)

Abstract.

Hydrodynamic interaction of steps is considered. Origin ^{of the} interaction is inter- ference of superfluid flow associated with the step oscillation. We calculate contribution of the step oscillation to the surface free energy ^at high temperatures, ^{and find that the} interaction is repulsive

and inversely proportional ^{to the} step ^distance.

Classification

Physics ^Abstracts

67.80G

82.65D

61.50C

In a previous paper [1] it has been shown that steps ^{at the} superfluid-solid interface of helium- 4 can interact each other via hydrodynamic superfluid ^{flow. At zero} temperature, ^it gives ^{a d-2} repulsion, ^{which has the same} power-law dependence ^{on the} step distance d as that of the elastic [2]

and the statistical interactions [3]. ^The purpose ^{of this} paper ^{is to} calculate its finite temperature

contribution to the surface free energy.

The model has been described in detail in reference [1], ^{and we} give only ^a brief sketch here. A

step ^is regarded ^{as an} oscillating string ^{which moves} freely ^{on a flat} singular ^face. Displacement

of a step, ^{that is} growth ^or melting, introduces superfluid ^{flow from a} melting part ^{to a} growing part. ^This liquid ^flow represents ^{a kinetic} energy, and line tension of a step represents ^a potential

energy ^{of the} step. ^{The motion of the} system ^{can be described} by ^the Lagrangian [4] :

where vs(r) ^{is the} velocity ^{of a} step ^element dl(r), and E is defined by

with the solid and liquid densities, ps and pl. Oscillation spectrum ^{of a} step lying along y-axis ^is given by

where Mlk is the effective mass of a single step ^{for the wave} number k,

^{k :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510240274300

2744

I{o( z) is the zero-th order modified Bessel function, and ~ ^{is a short} distance cutoff or a step ^width.

If there are two parallel steps ^{with a} distance d along y-axis, the interference of the liquid ^flow

associated with the step oscillation modifies the oscillation spectrum. ^7ivo eigen ^{modes of the} step

oscillation appear:

where ~2k ^{is a} interference mass term given by

In the interference mass term, ^{the short} range ^{cutoff is} replaced by the distance of the steps. ^The

difference of Wf: ^and w, is appreciable only ^at wavelength larger ^{than the} step distance, k 1/d.

The free energy ^{of the two} parallel steps ^can be calculated with (4) ^as

where Fl(T) ^{is the free} energy ^{of a} single step:

The interaction free energy is the first term of (7): ^{AF = F2 - 2Fi.} The dominant contribution in the integral ^{of AF comes from the} range k m 1/d. ^At high temperatures, ^{that is} 1ic.,;7r / d ~ ^T,

the interaction free energy ^becomes

where the integral

is weekly dependent ^{on z. The} integral ^{can be evaluated} numerically, and the result is shown in

figure ^1.

The interaction free energy (10) ^irs positive ^and proportional ^{to d~ ~} except ^{the weak} d-depen-

dence of 7i. ^{Thus we obtain a} d~ ~ repulsive interaction, ^which decays ^more slowly ^{than the elas-}

tic [2] and the statistical interactions [3]. ^{Note that} the interaction depends only ^on temperature ^and

Fig. ^1.

^The integrals h ^and 12 given by (11) ^and (19).

the step ^distance d, ^{and is} independent of any material constants. This universal feature is due to the fact that the origin of the interaction is the modification of the liquid ^{flow, which is} solely

determined by ^the geometrical configuration. ^{A similar} universality ^{has been found for a surface}

attraction due to acoustic and capillary ^waves [5].

A vicinal surface consists of an array ^of parallel steps. ^{If there are not} any direct interaction between steps, ^the oscillation spectrum of this surface is [1]

where Mk ^{is the effective mass of the} surface (1~ :.

with

Thus the interaction free energy ^{of this} system ^is

where N is the number density ^of steps: ^~V

1/d. ^{As before, at} high temperatures ^this equation

takes a simple ^form

2746

where the integral

is a slowly varying ^{function of z, and} its behavior is shown in figure ^1.

Basic features of (18) ^{is the same as that of two} steps, (10): ^the interaction free energy per step

is proportional ^{to d-1.}

The interference of the step oscillation brings ^{about a d-1} hydrodynamic step repulsion ^at

"high" temperatures, ⁱⁿ contrast to d-2 ^at zero temperature [1]. Actually, ^the "high" tempera-

ture is not so high. ^{If we} take d - 10a and assume ~ ~ ^{a, we obtain} w~~ d ~ 108 s-1 with an

experimental ^value [6] ^{of ,Q ~ 0.02} erg/cm, ^{and the} corresponding temperature ^{is T -} 10-3 K.

This hydrodynamic interaction is the first example ^{of d-1} type( 1 ), ^{which gives a quadratic term in}

the free energy expansion [7], ^{and most} long-ranged ^so far known in this system. It is still unset- tled that there is a quadratic ^{term in this system. There are two} observations of the equilibrium crystal shape ^{of 4He : the first one} [8] supports ^the existence of a quadratic ^{term, but the most}

recent one [9] is unfavorable to this term [10]. Measurement of the surface stiffness a has been done recently by observing ^the crystallization spectrum ^on vicinal faces [11, 12]. ^The experiment

indicates the existence of a long-range repulsive interaction. Unfortunately, ^our hydrodynamic

interaction is not strong enough ^to explain ^the experimental value of a. Its contribution to the surface stiffness is expected ^{to be}

which is rather small compared ^{with the observed} surface stiffness & - 10-1 erg/cm2. Whether the

present hydrodynamic interaction contributes to the surface stiffness can be judged by measuring

temperature dependence ^{of a.}

References

[1] ^{UWAHA M., J. Low} Temp. Phys. ⁷⁷ (1989) ^165.

[2] MARCHENKO VI. and PARSHIN A.Ya., ^Zh. Eksp. Teor. Fiz. 79 (1980) ²⁵⁷ (Sov. Phys.-JETP ⁵² (1980) 120).

[3] GRUBER E.E. and MULLINS W.W, ^J. Phys. Chem. Solids 28 (1967) ^875.

[4] NOZIÈRES P. and UWAHA M., ^J. Phys. ^{France 48} (1987) ^389.

[5] ^{CHERNOV A.A.} and MIKHEEV L.V, DokL Akad. Nauk SSSR 297 (1987) ³⁴⁹ (Sov. Phys. ^{Dokl. 32} ( 1987) 906).

[6] ^{GALLET F., BALIBAR S. and ROLLEY E., J.} Phys. ^{France 48} (1987) ^369.

[7] ^ANDREEV A.F., ^Zh. Eksp. Teor. Fiz. 80 (1981) ²⁰⁴² (Sov. Phys. -JETP ⁵³ (1982) 1063) .

[8] ^BABKIN A.V, KOPELIOVITCH D.B. and PARSHIN A.Ya., ^Zh. Eksp. ^{Teor. Fiz. 89} (1985) ²²⁸⁸ (Sov.

Phys. -JETP ⁶² (1985) 1322).

[9] ^CARMI Y., LIPSON S.G. and POLTURAK E., Phys. ^{Rev. B 36} (1987) ^1894.

[10] AVRON J.E. and ZIA R.K.P, Phys. ^{Rev. B 37} (1988) ^6611.

[11] ^{ANDREEVA O.A. and KESHISHEV} K.O., Pr,’s’ima Zh. Eksp. Teor. Fiz. 46 (1987) ¹⁶⁰ (JETP ^{Lett. 46} (1987) 200).

[12] ^ANDREEVA O.A., KESHISHEV K.O. and OSIP’YAN S.Yu., Pis’ima Zh. Eksp. ^{Teor. Fiz. 49} (1989) ⁶⁶¹ (JETP ^{Lett. 49} (1989) 759).

(1) ^{The weak} (logarithmic) d-dependence ⁱⁿ (10) ^and (18) ^is probably ^an artifact of the approximation

which takes into account only ^small amplitude oscillations.

Cet article a dtd imprimd ^{avec le Macro} Package "Editions de Physique Avril 1990".