Phonons, rotons and ripplons at interfaces

We use dispersive hydrodynamics to describe thermal excitations of superfluid helium 4. Dispersion relation of the bulk quasiparticles, phonons and rotons, acts as an input parameter of the theory. Wiener and Hopf method is used to solve nonlocal equations of the fluid in half-space. The dispersion...

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Дата:	2012
Автори:	Tanatarov, I.V., Adamenko, I.N., Nemchenko, K.E., Wyatt, A.F.G.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Назва видання:	Вопросы атомной науки и техники
Теми:	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
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Цитувати:	Phonons, rotons and ripplons at interfaces / I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2012. — № 1. — С. 260-264. — Бібліогр.: 9 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1071532025-02-09T21:38:09Z Phonons, rotons and ripplons at interfaces Фононы, ротоны и риплоны на границах раздела Фонони, ротони і ріплони на границях розподілу Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases We use dispersive hydrodynamics to describe thermal excitations of superfluid helium 4. Dispersion relation of the bulk quasiparticles, phonons and rotons, acts as an input parameter of the theory. Wiener and Hopf method is used to solve nonlocal equations of the fluid in half-space. The dispersion relation helium surface excitations, ripplons, is derived and analyzed; numerical solution reveals its new unusual branch. The same method applies to the description of bulk quasiparticles’ interaction with the interface with a solid. All quasiparticles creation probabilities are derived and weak interaction of rotons with negative dispersion with interfaces is explained. Для описания тепловых возбуждений сверхтекучего гелия 4 используется нелокальная гидродинамика, в которой дисперсионное соотношение является входным параметром. Для решения нелокальных уравнений жидкости в полупространстве используется метод Винера-Хопфа. Вычислено дисперсионное соотношение поверхностных мод гелия, риплонов; численное решение выявляет существование новой ветви. Это же решение описывает взаимодействие фононов и ротонов с границей раздела с твердым телом. Найдены все вероятности рождения квазичастиц и объяснена слабость взаимодействия ротонов с отрицательной дисперсией с границей раздела. Для опису теплових збуджень надплинного гелію 4 використовується нелокальна гідродинаміка, в якій дисперсійне співвідношення є вхідним параметром. Для розв'язку нелокальних рівнянь рідини в півпросторі використовується метод Вінера-Хопфа. Обчислено дисперсійне співвідношення поверхневих мод гелію, ріплонів; чисельний розв'язок виявляє їх нову гілку. Те ж рішення описує взаємодію фононів і ротонів із границею розподілу із твердих тілом. Знайдені всі вірогідності народження квазічастинок та пояснена слабкість взаємодії ротонів з від'ємною дисперсією із границею розподілу. 2012 Article Phonons, rotons and ripplons at interfaces / I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2012. — № 1. — С. 260-264. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 47.37.+q, 62.60.+v, 67.25.+k, 67.80.bf. https://nasplib.isofts.kiev.ua/handle/123456789/107153 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
spellingShingle	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. Phonons, rotons and ripplons at interfaces Вопросы атомной науки и техники
description	We use dispersive hydrodynamics to describe thermal excitations of superfluid helium 4. Dispersion relation of the bulk quasiparticles, phonons and rotons, acts as an input parameter of the theory. Wiener and Hopf method is used to solve nonlocal equations of the fluid in half-space. The dispersion relation helium surface excitations, ripplons, is derived and analyzed; numerical solution reveals its new unusual branch. The same method applies to the description of bulk quasiparticles’ interaction with the interface with a solid. All quasiparticles creation probabilities are derived and weak interaction of rotons with negative dispersion with interfaces is explained.
format	Article
author	Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G.
author_facet	Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G.
author_sort	Tanatarov, I.V.
title	Phonons, rotons and ripplons at interfaces
title_short	Phonons, rotons and ripplons at interfaces
title_full	Phonons, rotons and ripplons at interfaces
title_fullStr	Phonons, rotons and ripplons at interfaces
title_full_unstemmed	Phonons, rotons and ripplons at interfaces
title_sort	phonons, rotons and ripplons at interfaces
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2012
topic_facet	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
url	https://nasplib.isofts.kiev.ua/handle/123456789/107153
citation_txt	Phonons, rotons and ripplons at interfaces / I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2012. — № 1. — С. 260-264. — Бібліогр.: 9 назв. — англ.
series	Вопросы атомной науки и техники
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fulltext	PHONONS, ROTONS AND RIPPLONS AT INTERFACES I.V. Tanatarov 1∗, I.N. Adamenko 2, K.E. Nemchenko 2 and A.F.G. Wyatt 3 1National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine 2V.N. Karazin Kharkov National University, 61077, Kharkov, Ukraine 3School of Physics, University of Exeter, Exeter EX4 4QL, UK. (Received October 24, 2011) We use dispersive hydrodynamics to describe thermal excitations of superfluid helium 4. Dispersion relation of the bulk quasiparticles, phonons and rotons, acts as an input parameter of the theory. Wiener and Hopf method is used to solve nonlocal equations of the fluid in half-space. The dispersion relation helium surface excitations, ripplons, is derived and analyzed; numerical solution reveals its new unusual branch. The same method applies to the description of bulk quasiparticles’ interaction with the interface with a solid. All quasiparticles creation probabilities are derived and weak interaction of rotons with negative dispersion with interfaces is explained. PACS: 47.37.+q, 62.60.+v, 67.25.+k, 67.80.bf. 1. INTRODUCTION The bulk and surface thermal excitations of super- fluid helium are well-defined and long-lived both for small wave vectors and for large, comparable with inverse average atomic separation. The distinctive dispersion relation of the bulk excitations, shown in Fig. 1, is measured in experiments on neutron scatter- ing [1]. There is an almost linear phonon part, and a parabola-like roton part. Rotons with negative group velocity, to the left from the minimum, are called R− rotons, and with positive group velocity R+ rotons. The bulk excitations were studied in a large series of experiments on pulses of quasiparticles in helium at very low temperatures (see [3, 4]). R− rotons, however, eluded direct detection, supposedly due to anomalously weak interaction with interfaces, as solid heaters and bolometers were usually used for creation and detection of pulses. This was further affirmed by their eventual detection in [3] by means of a spe- cial source and quantum evaporation, and the effect needed theoretical description. Ripplons are quantised capillary waves on the sur- face of He II. Experiments on neutron scattering in thin films [5] showed, that they are well-defined quasi- particles even at frequencies of the order of the roton gap. It was important to derive theoretically their dispersion relation for large wave vectors (see [6]). Both problems were solved using a model in which the quantum fluid is described as a continuous medium at all length scales, down to the interatomic distances [7, 8]. This approach naturally allows to consider a quantum fluid with essentially arbitrary nonlinear dispersion relation, and the two problems turned out to be two sides of one. In this paper we give a short review of the model and method used, and the main results of this approach. 2. DISPERSIVE HYDRODYNAMICS In a quantum fluid atoms are delocalized and values of variables of continuous medium velocity v, density ρ and pressure P can be introduced in every mathe- matical point in space. Then it can be described [7] by the linearized equations of ideal liquid with equi- librium density ρq v̇ = −ρ−1 q ∇P, ρ̇ = −ρq∇v, (1) complemented with a nonlocal equation of state ρ(r) = ∫ V d3r′ h(r, r′)P (r′). (2) The kernel h can only be the function of \|r− r′\| for a homogeneous and isotropic fluid; area of integration V is the volume filled with the fluid. In terms of one variable, for example pressure P , this system leads to a nonlocal wave equation of the form ΔP = ∫ d3r′ h (\|r − r′\|)P̈ (r′). (3) If the fluid fills the infinite space, the integrand is a convolution, and Fourier transform gives us the relation between the kernel’s Fourier image h(k) and the dispersion relation of the medium Ω(k): h(k) = k2 Ω2(k) . (4) ∗Corresponding author E-mail address: igor.tanatarov@gmail.com 260 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 260-264. 3. FLUID IN HALF-SPACE When the fluid fills a half-space z > 0, equation for P takes the form ΔP = ∫ z1>0 d3r′ h (\|r − r′\|)P̈ (r′), (5) so there is no convolution any more. Equations of this type can be solved by the Wiener and Hopf method, in which the equation is reduced to a Hilbert-Riemann boundary problem (see [9]), in our case P−(kz; ω,kτ )J(kz ; ω,kτ ) + P+(kz ; ω,kτ ) = 0 (6) for kz ∈ (−∞,∞), where J(kz ; ω,kτ ) = Ω2(k) − ω2 Ω2(k) . (7) Here kz and kτ are normal and tangential to the in- terface components of wave vector k = kzez + kτ ; P+(kz) and P−(kz) are two functions to be found, analytic in the upper half-plane C+ and lower half- plane C− of the complex variable kz correspondingly, so that the signs in the superscripts denote their do- mains of analyticity. The relation (6) is between their limiting values on the real line of kz, which separates those domains C±. P (r) at z > 0 is obtained as the inverse Fourier transform of P−(k). The general solution for P±, that takes into ac- count singular behavior of J at k = 0 and asymptotic boundary conditions for P− at z → ∞, utilizes fac- torization of an auxiliary density I(kz ; ω,kτ ) = J(kz; ω,kτ ) k2 k2 + A2 , (8) which is bounded and separated from zero on k ∈ � (here A is some arbitrary positive constant), i.e. its expression in the form I(kz) = I−(kz) I+(kz) , (9) where I−(kz) is a function analytic and with no zeros in C−, and I+ likewise in C+: P−(kz; ω,kτ ) = const (k − iA)I−(kz; ω,kτ ) . (10) Solution to the factorization problem is in the gen- eral case given by I− = exp { 1 2πi ∞∫ −∞ dk′ z k′ z − kz ln I(k′ z) } . (11) It can be essentially simplified, however, if we as- sume the function Ω2 is a polynomial of k2 (and thus a polynomial of k2 z). In this case the density is fac- torized in an obvious way in terms of all the roots ki z of polynomial equation Ω2(kz ,kτ ) = ω2 : (12) we obtain I− (I+) by collecting all the factors of I that have poles and zeros in C+ (C−). The solution is then brought to P−(kz ; ω,kτ ) = C(ω,kτ ) · P̃ (kz ; ω,kτ ), where P̃ (kz ; ω,kτ ) = ∏ i�=1 [ kz − ki z(0,kτ ) ] ∏ i [ kz − ki z(ω,kτ ) ] , (13) and the products are taken over all the roots ki z in C+. The product in the numerator misses the one factor with ki z which turns to zero when kτ = 0, and is eliminated from I before factorization (8). This root, which corresponds to the long-wavelength phonon branch, we denote by subscript 1. In the limit ω → 0 all the factors in the numera- tor and denominator of P̃ are subtracted except for the extra one (kz − k1 z) in the denominator, and we have P−(kz) ≈ const kz − k1 z , where k2 1 z ≈ ω2 s2 − k2 τ , (14) s is sound velocity in the limit of small ω. Then so- lution in the coordinate space with a given ω is P (r, t) ∼ eik1 zr−iωt, k1 z = k1 zez + kτ . (15) In the general case the inverse Fourier transform gives monochromatic in ω solutions in the coordinate space of the form P (r, t; ω) = C ∑ ki z∈C+ rie ikir−iωt, (16) where ki = ki zez + kτ , (17) and ri is the residue of P̃ (13) in ki z. Using the equations of continuous medium (1), we obtain for velocity v(r, t; ω) = C ρqω ∑ ki z∈C+ kirie ikir−iωt. (18) Each real root of ki z gives a traveling wave in the sum, and each complex root gives a damped wave. 4. RIPPLONS If we want to consider the surface mode of the super- fluid, we should assume there are no real roots ki z of the equation (12). This happens when k2 τ is greater than all the real roots of Ω2(k) = ω2 with respect to k2. Then all ki z are divided into complex conjugate pairs, and the solution (13) is unique. It should be accompanied by the boundary condi- tions that take account surface tension σ on the free surface. Writing that pressure at the surface is equal to the Laplace pressure σ/Rcurv, where Rcurv is the curvature radius, for small deviations of the surface from equilibrium we get P ∣∣∣ z=0 = σkτ2 · vz iω ∣∣∣ z=0 . (19) 261 On the other side, the solution in the fluid (16,18) gives vz\|z=0 = 1 ρqω ∑ ki zri∑ ri · P \|z=0. (20) As P−(kz) is by construction analytic in C−, the poles ki z constitute all of its poles in the finite part of the complex plane kz. The sums in the numerator and denominator are equal to residues of P̃ and kzP̃ at infinity correspondingly, and those can be calculated directly by expanding P̃ at kz → ∞: vz \|z=0 = 1 ρqω { k1 z + ∑ i>1 [ ki z −ki z(ω = 0) ]} ·P \|z=0. (21) Comparing (19) and (21), we obtain the equation of dispersion relation for the surface modes ω(kτ ): ω2 = σk2 τ i ρ0 { k1 z(ω, kτ )+ ∑ i>1 [ ki z(ω, kτ )−ki z(0, kτ ) ]} . (22) In the limit ω → 0 the sum tends to zero and equation turns into ω2 = σ ρ0 k3 τ √ k2 τ − ω2/s2. (23) This is also the form it takes when Ω(k) is linear. Ne- glecting compressibility, in the limit s → ∞ we get the classic relation ω2 ∼ k3 τ . Expanding ki z in powers of ω and kτ , and taking into account the asymptotic ω2 ∼ k3 τ , we can obtain further corrections: ω2(kτ ) = σ ρq · k3 τ { 1− σ 2ρqs2 kτ + σ2 8ρ2 qs 4 k2 τ+ (24) − i σ ρq ∑ i>1 ∂ki z(ω, 0) ∂(ω2) ∣∣∣ ω=0 · ( k2 τ − σ ρqs2 k3 τ ) +O ( k4 τ )} . The two summands after unity in the braces take into account compressibility, and those after them repre- sent contribution of non-phonon roots. They take into account non-linearity of Ω(k) and at small k give only small corrections. The consideration above applies to arbitrary dispersion relation Ω(k) which has linear long- wavelength phonon part. We are now particularly interested in superfluid helium’s non-monotonic dis- persion, and in what happens to ripplon’s dispersion relation close to the frequency of the roton gap Δrot. In this region dispersion of the two roton roots can be well approximated by a parabola Ω(k) ≈ Δrot + h̄ 2μ (k − krot)2. (25) Then expansion of the two corresponding roots, which we will denote as k2 z and k3 z, in terms of the small parameter (Δrot − ω), contains summands ∼ √ Δrot − ω, which determine the asymptotic be- havior of ω(kτ ) in the vicinity of the roton gap: kc − kτ = d · √ Δrot − ω. (26) Fig. 1. Bulk and surface excitations of He II. Dots are experimental data on neutron scattering [1, 5]; solid lines are numerical approximation to the bulk spectrum and the derived dispersion for the ripplons. Inset shows the new ripplon branch, with δT ≈1.6 · 10−3K, δk≈0.7 · 10−3Å−1. Thus the ripplon dispersion curve (Fig. 1) ends at a point (kc, Δrot) with zero derivative, where kc < krot is the root of ω(kc) = Δrot, which should be eval- uated numerically (see below). At higher frequencies energy-momentum conservation laws allow ripplons’ decay into rotons, and they are unstable. The con- stant d can be expressed through the parameters of the spectrum at Δrot: d = ∣∣∣ b a + c ∣∣∣; a = 2σ ρ0 Δ2 rot k3 c ; b = 2krot √ 2μ/h̄ k2 rot−k2 c ; c =ikc { k−1 1 z (Δrot, kc)+ + ∑ i≥4 k−1 i z (Δrot, kc) − ∑ i≥2 k−1 i z (0, kc) } . Numerical solution gives us the dispersion curve ω(kτ ) for all frequencies, and gives kc =1.27 Å−1. It also reveals an unexpected new branch of ripplons at very high frequencies, close to 2Δrot. This branch is glued to the bulk dispersion curve from below, and is stable with regard to decay into bulk excitations. 5. PHONONS AND ROTONS AT INTERFACE WITH A SOLID Now let us consider the problem of quasiparticles of the superfluid interacting with the interface with an- other medium. In this case the asymptote of the so- lution at z → ∞ should correspond to isolated trav- eling quasiparticles, and at least one root ki z should be real. The easiest way to make sense of the general solution (13) in this case is the usual roots shifting from the real line into the upper or lower complex half-planes. In our case, however, there is additional condition imposed: the number of real roots shifted down must be equal to the number of roots shifted up, so that the index of density I remains equal to zero. This was an essential ingredient for validity of the general solution1. 1The index of I in our case is essentially the difference between the number of its zeros in C+ and C−. For mathematical details we must refer the reader to [9]. 262 The real roots ki z go in pairs, and for superfluid helium their maximum number is N = 6. We will consider this most general case, as the others N = 2, 4 (N = 0 was discussed in the previous section) can be deduced in the same way, and the final results can be directly obtained from the general formulas. A solution is obtained from (13) by picking arbi- trary three out of the set of six real roots, which are then shifted in the upper half-plane and after inverse Fourier transform give traveling waves in the coordi- nate space. The overall number of linear-independent solutions is thus C3 6 = 20. Physically relevant for the problem of waves/quasiparticles’ interaction with an interface are four of the set, which contain no more than one particle traveling towards the interface, i.e. in the negative direction of axis z, at z → ∞. The basic, and most interesting, solution, is the one that contains all three quasiparticles traveling away from the interface. We will denote it as Pout. We denote by k1,2,3 the three positive roots of equation Ω2(k) = ω2 in the order of their absolute values, such that 1 stands for phonons, 2 for R− ro- tons, and 3 for R+ rotons. R− rotons have negative group velocity, and thus propagate in the opposite direction of their wave-vectors. Therefore the Pout solution will contain traveling waves corresponding to k1,2,3, with the signs defined as 0 < k1 z < (−k2 z) < k3 z. (27) We denote the remaining, initially complex roots of Eq. (12) in C+, by subscripts i > 3. Solid’s phonon incident. The Pout solution is the one that is realized in the fluid when a quasiparticle is incident on the interface from the solid. Its amplitude is derived by using boundary conditions discussed be- low, which gives us the total energy transmission co- efficient D and the partial ones Di, which describe the portions of the energy flow of the incident wave trans- ferred into each of the created ones in the fluid. But it is important, that the relative amplitudes and energy flows of the asymptotically distinct three waves are determined by the structure of solution Pout alone, with no regard to parameters of the interface. Calcu- lating the amplitudes of each wave as corresponding residues ri (16), (18) and group velocities, we derive the relative energy transmission coefficients: Di D = ki z ki z+kj z+kk z · ki z+kj z ki z−kj z ki z+kk z ki z−kk z , (28) where the subscripts {i, j, k} form a permutation of {1, 2, 3}. These expressions do not depend on ki z with i > 3. It can be verified directly that ∑ Di = D. Due to (27), D2 D1,2 (Fig. 2), so R− rotons are weakly created by a solid’s phonon. When we consider the full problem of a solid’s phonon incident on the interface with superfluid he- lium, we use boundary conditions P ∣∣ z=−0 = P ∣∣ z=+0 ; vz ∣∣ z=−0 = vz ∣∣ z=+0 . (29) They imply that ω and kτ are the same for all har- monic summands on both sides of the interface. For traveling waves this fixes all angles of transmission and reflection, measured from the normal, via a gen- eralization of Snell’s law sin θi si(ω) = sin θj sj(ω) ∀i, j = 0, 1, . . . (30) where si(ω) = ω/ki(ω) are phase velocities; subscript 0 denotes solid’s phonons. In the scalar model the solid’s phonon corresponds to a longitudinal wave; its dispersion is almost lin- ear in the frequency scale of helium dispersion curve. Then the solution in the solid is just a sum of incident and reflected waves, and in the fluid it is Pout. Then using the b.c. (29), we express the amplitudes of all the waves through the amplitude of the incident one, and calculating energy flows in each of them, derive full energy transmission coefficient D(ω, kτ ) = 4 Re kskq \|ks + kq\|2 , where (31) kq =k1 z+ ∑ i=2 [ ki z−ki z(ω=0) ] ; ks = ρq ρ0 k0 z, (32) and ρ0 is equilibrium density of the solid. The partial transmission coefficients are then given by (28). Due to smallness of parameters ρq/ρ0 and si/s0 for helium and common solids, the transmission coef- ficients D and Di for real interfaces with helium are obtained in the effective limit ρq → 0. In this case the transmission angles θi are small due to (30) and Di(ω, θi) ≈ Di(ω, 0) cos θi. (33) Fig. 2. Relative transmission coefficients Di(ω)/D(ω) at normal incidence. They do not depend on the parameters of the interface, and are only determined by dispersion curve Ω(k) A helium quasiparticle is incident. In this case the asymptote of the solution in z > 0 must contain one traveling wave traveling towards the interface. If the incident quasiparticle carries subscript i, then the solution can be presented as a linear combination of Pout and a solution, in which ki z in the denominator is formally replaced by (−ki z) (numerator stays the same): P (i) in = Pout ∣∣∣ ki z �→−ki z . (34) 263 Using the same scheme as used above, we de- rive the transmission and reflection coefficients. The transmission coefficients D′ i, when expressed through conserved quantities ω and kτ , obey the principle of detailed balance D′ i(ω, kτ ) = Di(ω, kτ ). There are nine reflection coefficients Rij , where the first subscript denotes the type of incident quasi- particle, and the second – of the reflected one: Rii = ∣∣∣∣ (ki z+kj z)(ki z+kk z) (ki z−kj z)(ki z−kk z) ∣∣∣∣ 2 · ∣∣∣∣1 − 2ki z ks + kq ∣∣∣∣ 2 ; (35) Rij = \|4ki zkj z\| (ki z−kj z)2 ∣∣∣∣(ki z+kk z)(kj z+kk z) (ki z−kk z)(kj z−kk z) ∣∣∣∣ × ∣∣∣∣1 − ki z+kj z ks + kq ∣∣∣∣ 2 for i �= j. (36) The formulas for Rij in the form (35,36) hold for the case when some of the roots ki z are complex. For example, if k1 z and k2 z are complex, then R11 is given by (35), where the first factor is reduced to unity. It is then trivial to show that R11 +D1 = 1. It can be also directly verified in each case, that energy is always conserved ∑ j Rij + Di = 1. 6. CONCLUSIONS We used the description of superfluid helium as a continuous medium at interatomic scales, obeying equations of nonlocal hydrodynamics. Their solu- tions apply both to description of surface excita- tions, ripplons, and interaction of the bulk excita- tions, phonons and rotons, with interfaces. Ripplons’ dispersion relation is derived, and conforms well with experimental data. It is shown to end in a point with zero derivative at the frequency of the roton gap; new ripplon branch is found at very high frequen- cies. Creation probabilities of all quasiparticles are derived when any quasiparticle at the interface be- tween superfluid helium and a solid. It is shown that R− rotons weakly interact with interfaces, which also explains experiments. References 1. R.J. Donnelly, J.A. Donnelly, and R.N. Hills // J. Low Temp. Phys., 1981, v. 44, p. 471-489. 2. A.F.G. Wyatt, N.A. Lockberie, and R.A. Sher- lock // Phys.Rev.Lett., 1974, v. 33, p. 1425-1428. 3. M.A.H. Tucker and A.F.G. Wyatt // Science, 1999, v. 283, p. 1150-1152. 4. D.H.S. Smith and A.F.G. Wyatt // Phys.Rev.B, 2010, v. 81, 134519. 5. H.J. Lauter et al. // Phys.Rev.Lett., 1992, v. 68, p. 2484. 6. F. Dalfovo et al. // Phys.Rev.B, 1995, v. 52, p. 1193. 7. I.N. Adamenko, K.E. Nemchenko, I.V. Tanatarov // Phys. Rev. B, 2003, v. 67, 104513. 8. I.V. Tanatarov, I.N. Adamenko, K.E. Nem- chenko, and A.F.G. Wyatt, // J. Low Temp. Phys., 2010, v. 159 (5/6), p. 549-575; Low Temp. Phys., 2010, v. 36 (7), p. 582. 9. F.D. Gahov. Boundary Value Problems. Dover Publications, Inc., New York, 1990. �� !��"�� # �� $�� % � �� "� � �� &�� '�� & � ��(�� &� �� &�� )* � �� + �� )�� )�� )� ��)� � ��)��)� �� + ��)� �� ,�� )� � � �)�� )�� ) �� + �� ) ��!��"�� -�� )� � ��)��)� �� )� �)�� )�# �� ,�� + .� �� )�� /� � �)�� + ��+��) "� � )� ) �� )� )� �� &�* ��)�� )� �� )�� 0 �� ) ��) �)��)� �� ) �� )�� )� � ��+��). �� )� � �)�,+� �* ��)+* )� �� &�* ��)�� 12�

Phonons, rotons and ripplons at interfaces

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