Possible longrange step interaction in 4He due to step oscillation M. Uwaha To cite this version: M. Uwaha. Possible longrange step interaction in 4He due to step oscillation. Journal de Physique, 1990, 51 (24), pp.27432746. <10.1051/jphys:0199000510240274300>. <jpa00212568> HAL Id: jpa00212568 https://hal.archivesouvertes.fr/jpa00212568 Submitted on 1 Jan 1990 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
82.65D Hydrodynamic J. Phys. France 51 (1990) 27~.3274~ 15 DÉCEMBRE z 2743 Classification Physics Abstracts 67.80G 61.50C Short Communication Possible longrange step interaction in 4He due to step oscillation M. Uwaha Institute for Materials Research, Tohoku University, 211 Katahira, Aobaku, Sendai 980, Japan (Received 28 September 1990, accepted 8 October 1990) Abstract. interaction of steps is considered. Origin of the interaction is interference 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. In a previous paper [1] it has been shown that steps at the superfluidsolid interface of helium 4 can interact each other via hydrodynamic superfluid flow. At zero temperature, it gives a d2 repulsion, which has the same powerlaw 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 yaxis 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 zeroth 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 yaxis, 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 ddependence of 7i. Thus we obtain a d~ ~ repulsive interaction, which decays more slowly than the elastic [2] and the statistical interactions [3]. Note that the interaction depends only on temperature and
The 2745 Fig. 1. integrals h and 12 given by (11) and (19). material constants. This universal feature is due to the step distance d, and is independent of any 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 d1. The interference of the step oscillation brings about a d1 hydrodynamic step repulsion at "high" temperatures, in contrast to d2 at zero temperature [1]. Actually, the "high" temperature is not so high. If we take d 10a and assume ~ ~ a, we obtain w~~ d ~ 108 s1 with an experimental value [6] of,q ~ 0.02 erg/cm, and the corresponding temperature is T 103 K. This hydrodynamic interaction is the first example of d1 type( 1 ), which gives a quadratic term in the free energy expansion [7], and most longranged so far known in this system. It is still unsettled 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 longrange 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 & 101 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) ddependence 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".