On quantum effects on the surface of solid hydrogen
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AND LIQUIDS
On Quantum Effects on the Surface of Solid Hydrogen V. I. Marchenko Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, 119334 Russia Received April 12, 2013
Abstract—The lowfrequency spectrum of hypothetical superfluidity on the free surface of a quantum crystal of hydrogen is determined. In the quantumrough state of the surface, crystallization waves with a quadratic spectrum should propagate. In the atomically smooth state, the spectrum is linear. Crystallization waves propagating along elementary steps are also considered. DOI: 10.1134/S1063776113120054
Rapid variation in the shape of hydrogen crystals at a temperature of about 1.8 K was observed in [1]. Such a behavior was not confirmed in [2, 3]. However, the hydrogen surface at lower temperatures has not been investigated as yet, and the possibility of quantum effects in this case remains unclear, like for the inter face between solid and liquid helium prior to the dis covery of crystallization waves [4, 5]. Surface point defects in a quantum crystal are delo calized. Upon an increase in tunnel effects, zeropoint defects and their superfluidity appear [6]. Surface superfluidity in hydrogen was considered in [7] in con nection with observation [1]. Upon their further growth, quantum fluctuations lead to a quantum rough state of the surface [4]. Here, we pay attention to the considerable difference in the oscillation spectra of the surface superfluid liquid in the atomically smooth and quantumrough states of the free surface of crys tals. The equations for the dynamics of a superlfluid liq uid have the form (see [8], (139, 7)) ˜ s = 0, v· s + ∇μ (1) ˜ s = μs/m, μs is the chemical potential of the where μ superfluid liquid, m is the mass of a hydrogen mole cule, and vs is the velocity of the superfluid compo nent. The variation of μs is defined as ∂μ s δμ s = δn (2) s, ∂n s where ns is the density of the superfluid component. The continuity equation for a superfluid 2D liquid on an atomically smooth surface has the form n· s + divj = 0, (3) where j is the surface flux of particles. Disregarding the diffusion of the normal component, we have j = ns vs . (4) The chemical potential μs of a 2D superfluid liquid on an atomically smooth surface can be in equilibrium
with the bulk value only in the case of exchange of par ticles on linear surface defects (steps). In accordance with Eqs. (1)–(4), in the absence of steps, oscillations of the surface superfluid liquid on an atomically smooth face are characterized by the standard line spectrum. In the quantumrough state, chemical potential μs does not differ from the bulk value μ of the chemical potential. In this case, the part of μs depending on the variation of surface profile ζ(x, y) is defined by the Herring capillary correction [10] (5) δμ = – α Δζ, n where α is the surface energy (we disregard anisotropy to simplify the form of expressions), n is the number density of particles in the crystal, and Δ is the 2D Laplace operator. According to Mullins [
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