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Quantum Evaporation from the Free Surface of Superfluid 4He

The scattering of atoms and rotons at the free surface of superfluid 4He is studied in the framework of linearised time dependent mean field theory. The phenomenological Orsay-Trento density functional is used to solve numerically the equations of motion for the elementary excitations in presence of a free surface and to calculate the flux of rotons and atoms in the reflection, condensation, and evaporation processes. The probability associated with each process is evaluated as a function of energy, for incident angles such that only rotons and atoms are involved in the scattering. The evaporation probability for R+ rotons is predicted to increase quite rapidly from zero, near the roton minimum, to 1 as the energy increases. Conversely the evaporation from R- rotons remains smaller than 0.25 for all energies. Close to the energy of the roton minimum the mode-change process is the dominant one. The consistency of the results with general properties of the scattering matrix, such as unitarity and time reversal, is explicitly discussed. The condensation of atoms into bulk excitations is also investigated.

The condensation of atoms into bulk excitations is also investigated. The condensation probability is almost 1 at high energy in agreement with experiments, but it lowers significantly when the energy approaches the roton minimum in the phonon forbidden region. have been carried out in the last decades [1][2][3][4][5][6][7][8][9][10] to explore the phenomenon of quantum evaporation and condensation (see also Ref. [11] for a recent review). From the theoretical viewpoint [12][13][14][15][16][17][18][19] these processes are very appealing, being clean examples of scattering of excitations in a highly correlated many-body quantum system. A good theory of quantum evaporation and condensation is however a difficult task and a clear understanding of the fundamental mechanism underlying these phenomena has still to come, even though relevant steps in this direction have been accomplished in the last years.
In this paper we present a time dependent mean field theory, based on the density functional formalism, which allows one to calculate the scattering matrix elements associated with the scattering of rotons, phonons and atoms at the free surface. Preliminary results of this theory have been recently published in Ref. [19]. Here we discuss the theory in more detail and we present results in a wider range of energy and wave vectors.
A discussion about quantum evaporation and condensation requires first a detailed knowledge of the spectrum of elementary excitations of superfluid 4 He. A schematic picture is given in Fig. 1. The liquid-vacuum interface is supposed to be in the xy-plane. The system is translational invariant in those directions, so that the parallel wave vector q x is a conserved quantity (we choose q y = 0 without any loss of generality). The minimum energy needed to produce free atom states outside the liquid at zero temperature is the chemical potential |µ| = 7.15 K. The threshold for evaporation for given values of q x is then |µ| +h 2 q 2 x /2m (dashed line). Phonons, rotons and atoms can propagate at different angles with respect to the surface, that is with different q z for a given q x . The dispersion law for phonons and rotons propagating in the bulk liquid and in the direction parallel to the surface (q z = 0) is plotted as a function of q x . Excitations having q z = 0 fill different regions of the spectrum in Fig. 1 with a continuum of states. For instance each point in region V corresponds to a roton R + (positive group velocity) propagating with q = q 2 x + q 2 z such thathω(q) is the R + roton dispersion. Since q x ≤ q, roton states with q z = 0 fill the region at the left In Refs. [17,18] the effect of the surface on the propagation of the elementary excitations has been studied within a local density approximation. The dispersion law of the excitations was assumed to be the same as for a uniform liquid of density equal to the local density.
Maris [17] used a simple interpolation between the phonon-roton dispersion law in bulk liquid and the dispersion law of the free atoms. He studied the behavior of classical trajectories for quasiparticles along the z-direction assuming the conservation of energy and parallel momentum. From that analysis one finds some interesting constraints on the structure of the classical trajectories crossing the free surface. For instance only phonons and rotons above the maxon energy (about 14 K) are found to evaporate atoms. Vice-versa, the theory predicts no evaporation from rotons with energy smaller than the maxon energy, because of the occurrence of a barrier at the interface. The experimental evidence [11] for quantum evaporation induced by rotons even below the maxon energy is consequently an important indication of the crucial role played by quantum effects. Quantum effects were partially included in the calculations of Ref. [18] by means of perturbation theory based on WKB states; the density profile in that case was approximated by a suitable analytic function and the dispersion law for the excited states, in local density approximation, was calculated using Beliaev's theory with an effective interaction.
For an accurate treatment of quantum effects one has to go beyond the local density approximation and the semiclassical treatment. To accomplish the task we proceed in two distinct steps: • We first evaluate the ground state and the excited states of the inhomogeneous system within a self-consistent theory sufficiently accurate to reproduce the main features of the spectrum in Fig. 1, including the structure and the excitations of the free surface.
• We identify the current carried by the elementary excitations and use the numerical solutions of the equations of motion to calculate the asymptotic flux of excitations associated with a given scattering process. While this identification is trivial in the description of free atoms in vacuum, it becomes less obvious in the liquid where manybody effects, present in this highly correlated system, must be properly included.
Taking the ratios of incoming and outgoing fluxes one finally gets the evaporation and condensation probabilities, which are the final goal of the theory.
For the first step we use a time dependent mean field theory based on the least action principle applied to an energy functional drH[Ψ * , Ψ]. For the form of the functional we follow the recent proposal of Ref. [20]. We consider linear variation of Ψ, in the form (Ψ 0 + δΨ), and expand δΨ in plane waves along the parallel direction. We restrict ourselves to variations of the energy linear in δΨ. For this reason the theory describes only one-to-one processes (evaporation of one atom by one roton, creation of one roton by condensing one atom, etc.) and not processes involving more than two excitations (for example multi-phonon or multi-ripplon production in the atom condensation process). It is worth mentioning that some experiments by Wyatt and co-workers [5,6,11] support the idea that the evaporation of atoms from bulk excitations is essentially a one-to-one process, so that the use of a linear theory seems justified in this case. Conversely measurements of atom condensation [3,9] point out an important role played by non linear effects associated with the creation of ripplons. This asymmetry between evaporation and condensation is still basically unexplained. Quantitative predictions for the scattering rates in linear theory are essential in order to better understand the origin of this and other discrepancies.
As concerns the second step we derive the definition of flux (or current density) associated with the propagation of the elementary excitations, in terms of the linearised solutions of the equations of motion. We show that such a current density obeys the proper equation of continuity. We will devote detailed discussions about the meaning of the flux of excitations and about the properties of the scattering matrix elements, which are calculated starting from the knowledge of the fluxes involved in the scattering process.
The theory can be applied to all the relevant regions of the spectrum in Fig. 1. In the present work we present results for processes involving rotons and atoms in the region III. We chose this region for two basic reasons: first, phonons are not allowed there and thus the analysis is simpler; second, the arguments based on classical trajectories predict no evaporation in this region and this means that quantum effects should play a crucial role in the evaporation process. The analysis of the remaining parts of the spectrum will be the object of a future work.
The paper is organised as follows: in Sec. II we present the theoretical formalism of the density functional and the equations of motion. In Sec. III we introduce the concept of current of excitations. In Sec. IV we discuss general properties of the scattering matrix. In Sec. V we present the results for the probabilities of evaporation and condensation. Finally we will give a brief summary of the main results and discuss some future perspectives.

II. TIME DEPENDENT DENSITY FUNCTIONAL THEORY
The formalism used in the present work was already introduced in Ref. [20]. In that paper detailed discussions were devoted to the motivations of the theory and to the choice of the density functional. Results for static and dynamic properties of superfluid 4 He in different geometries were presented. In Refs. [21,22] the dynamics of the free surface and of droplets, respectively, were studied in detail within the same scheme. As a consequence we refer to the above-mentioned papers for details and we recall here simply the main points.
The starting assumption is that the energy can be written as a functional of the form where the complex function Ψ is written as The real function Φ is related to the diagonal one-body density by ρ = Φ 2 . The phase S fixes the velocity of the fluid through the relation v = (1/m)∇S, where m is the mass of the 4 He atoms. In the calculation of the ground state only states with zero velocity must be considered, so that the energy is a functional only of the particle density ρ(r). A natural representation is given by where the second term on the r.h.s. is a quantum pressure, corresponding to the kinetic energy of a Bose gas of non uniform density. The quantity E c [ρ] is a correlation energy; it incorporates the effects of dynamic correlations induced by the interaction. Ground state configurations are obtained by minimising the energy of the system with respect to the density. This leads to the Hartree-type equation where is an effective Hamiltonian. The quantity U[ρ, r] ≡ δE c /δρ(r) acts as a mean field, while the chemical potential µ is introduced in order to ensure the proper normalisation of the density to a fixed number of particles. The dynamics of the system can be studied by using the least action principle: By making variations with respect to Ψ or Ψ * one derives the equations of motion for the excited states, in the form of Schrödinger-like equations. The equation for Ψ is where H = δE/δΨ * is an effective Hamiltonian. We linearise the equation by writing where Ψ 0 (r) corresponds to the ground state. The change of wave function δΨ can be written in the form where the functions f (r) and g(r) are fixed, together with the frequency ω, by the solution of the equations of motion (7). The Hamiltonian H then takes the form and the equation of motion becomes The term δH is linear in δΨ and accounts for changes in the Hamiltonian induced by the collective motion of the system. Since H depends explicitly on the wave function Ψ, Equation (7) has to be solved using a self-consistent procedure.
The above scheme corresponds to a time dependent density functional (TDDF) theory.
A discussion about equivalent formulations of the same approach (hydrodynamic-like equations, Green's functions), as well as about the connection with other many body approaches, is given in Refs. [20][21][22]. Equations of the same type are obtained using Correlated Basis Functions in the context of Hypernetted Chain approximation [23]. Here we stress that the theory corresponds to a quantum mechanical treatment of the dynamics and consequently accounts for the interference and tunneling phenomena which are expected to play a crucial role in the evaporation process. Of course, due to linearisation, they do not include inelastic processes associated with multi-phonons or multi-ripplons. As already said in the Introduction, these effects lie beyond the present theory.
To proceed further one has to specify both the explicit form of the energy functional (1) and the geometry of the system under investigation. The energy functional is the one introduced in Ref [20]. It consists of two parts, a functional of the particle density only and a functional of both the particle density and the current density. It corresponds to a generalisation of a previous finite-range functional introduced by Dupont-Roc et al. [24]. Its explicit form is given in Appendix A. The functional is phenomenological, i.e., it contains parameters which are fixed to reproduce known properties of the bulk liquid. In particular the equation of state and the static response function in bulk liquid are reproduced by construction. The new functional, with respect to the one of Ref. [24], contains a currentcurrent effective interaction, which accounts for backflow-like correlations and allows one to reproduce the experimental phonon-roton spectrum in the uniform liquid. The latter is a crucial requirement for a theory whose goal is the prediction of evaporation and condensation probabilities, since these quantities depend dramatically on the form of the excitation spectrum.
As concerns the geometry, we assume translational invariance in the x-y plane. The density profile of the liquid in the ground state is a function of the orthogonal co-ordinate z.
The profile of the free surface obtained with the present density functional theory is about 6Å thick and compares well with the one coming from ab initio calculations as well as with the experimental data of Ref. [25] (see Ref. [22] for a discussion). The excited states are described by fluctuations of the particle density and velocity associated with the solutions (9) of the equations of motion. In the planar geometry the same solutions can be written in the form [26] δΨ( With a proper choice for the boundary conditions the functions f (z) and g(z) can be chosen real. Once the equations of motion are written explicitly for the unknowns f (z) and g(z), they assume the typical form of the equations of the random phase approximation (RPA) for bosons. In particular they account for both particle-hole [f (z)] and hole-particle [g(z)] transitions which are coupled by the equations of motion (6). This coupling is of crucial importance in order to treat the correlation effects associated with the propagation of elementary excitations in an interacting system. The equations of motion have also a structure formally identical to the one of the Bogoliubov equations for the dilute Bose gas [27] and to the one of the Beliaev equations for Bose superfluids [18,28]. With respect to those theories the present approach makes use of a finite ranged and momentum dependent effective interaction. The same theory gives a reliable description of surface modes (ripplons) both at small and high momenta [21]. In the vacuum the equations of motion coincide with the Schrödinger equation for the free atom wave function f (z), while g(z) vanishes.
The equations for f (z) and g(z) are solved numerically in a finite box. We work in a slab geometry, that is, with the liquid confined within two parallel surfaces, the slab being accuracy by fitting f (q z ). In the case of Fig. 3b the best fit is indistinguishable from f (q z ).
All these numerical results are used as input in the calculation of the flux of excitations in the scattering processes at the surface, as explained in the following section.

III. FLUX OF ELEMENTARY EXCITATIONS
In the standard theory of scattering the probability associated with a given output channel is written as the ratio between the outgoing flux in that channel and the incoming flux.
In the case of quantum evaporation the free surface acts as scattering region and one has to Far away from the surface and for a given energyhω the function δΨ is the sum of plane waves of the form where the index j = a, +, − refers to the type of excitations (atoms, R + and R − respectively) which contributes to δΨ. Each type of excitation obeys its own dispersion law ω(q) and contributes to δΨ with amplitudes f j and g j .
Well outside the liquid δΨ describes free atoms. The hole-particle part g a vanishes, while the atom density is given by ρ a = f 2 a . The density of current is given by where v is the group velocity which, for the free atoms, coincides withhq/m. These are the usual definitions of density and current density for free particles.
Inside the liquid the same definitions can be generalised to describe the elementary modes of the correlated system. The RPA (or, equivalently, Bogoliubov) formalism provides a natural generalisation for the density of elementary excitations which, in term of the components f and g of the time dependent solution (9), is given by This expression is consistent with the well known ortho-normalisation property of the RPA solutions: The total density of elementary excitations, ρ ex = j ρ ex j , satisfies the equation of continuity: The quantity j ex = j j ex j is identified with the current density of excitations, for which one expects the result where This result is also connected with the unitarity of the scattering matrix, as we will discuss in the next section.
Equations (15,17,18) emphasise a remarkable feature of the TDDF solutions. In a dilute system they reduce to the usual expressions holding for free particles, while in the interacting liquid they include the correlations associated with the propagation of the collective modes.
For instance in the long wavelength phonon regime (q → 0) the group velocity coincides with the sound velocity and g ≃ f .
is also a solution (the same holds for g(z)).
the normal reflection probability is and the mode-change reflection is The explicit formulae for this scattering process, including the geometry of the slab and the box, are given in Appendix C. By combining the stationary solutions with different coefficients one can select the process in which the incident excitation is a R − roton. In this case the ratios of the outgoing fluxes and the incoming R − flux provide the probabilities P −a (evaporation), P −− (normal reflection) and P −+ (mode-change reflection). Similarly, by selecting the process with an incident atom one gets the probabilities P aa (reflection), P a+ (condensation in R + ) and P a− (condensation in R − ). In region III of the spectrum in Fig. 1 these are the only one-to-one processes available. Generalisation to the other regions of the spectrum is straightforward.
The method we have used in Ref. [19] to derive the evaporation and reflection probabilities is simpler but less general. It corresponds to the case when one of the state in the linear combination has vanishing amplitude outside the liquid, due to destructive interference. The existence of such resonant states simplifies the calculation of the probabilities P ij .
However, the more general method used in the present work is not restricted to special states and provides results on a wider range of energy. Furthermore it can be easily extended to describe more scattering channels.

IV. THE SCATTERING MATRIX
The scattering matrix S characterises a scattering process through the definition Ψ out = SΨ in . The unitarity and the time reversal symmetry (t → −t) of S imply and respectively. Combining these two conditions one finds that the scattering matrix elements must satisfy the general property The scattering matrix elements are related to the probabilities P ij by Equation (25) then implies that the evaporation, reflection and condensation rates, introduced in the previous section, reduce to six quantities to be calculated: P +a , P −a , P ++ , P −− , P +− and P aa . Furthermore the unitarity condition takes also the form S * S = 1, or |S +a | 2 + |S −a | 2 + |S aa | 2 = 1 .
These conditions express the conservation of flux in the three processes where the incoming excitation is a R + roton, a R − roton and an atom, respectively. They reflect the fact that the present approach does not include inelastic channel. The unitarity of the S matrix gives three additional constraints: All these conditions for the scattering matrix elements can be used to obtain useful checks for the results of the numerical calculation.

V. RESULTS
The main results of the numerical analysis are shown in Fig. 5, where we report the probabilities P ij = |S ij | 2 as a function of energy. The data are obtained with several values of the parallel wave vector q x in order to span the phonon forbidden region (region III of the spectrum in Fig. 1). The data at lowest energies in Fig. 5 correspond to q x = 0.6Å −1 , the ones at highest energy to q x = 0.85Å −1 . From the knowledge of the orthogonal wave vector q z one extracts also the incidence angle. For R + and R − rotons it is approximately 15 • ÷ 20 • and 20 • ÷ 30 • , respectively, while for atoms is 50 • ÷ 90 • . The range of incidence angles for rotons is rather narrow, so that a discussion about the angular dependence of the evaporation and reflection probabilities is not significant in region III of the spectrum.
Further work is planned to explore the angular dependence including region II. The error bars arise from fluctuations of the numerical results. They originate mainly from the fact that the states in the linear combination (19) may be not enough linearly independent.
As explained at the end of Appendix C, this implies an amplification of small numerical uncertainties in the values of amplitudes, wave vectors and other parameters entering the expressions for P ij .
Looking at Fig. 5 one notices that the unitarity conditions (27)(28)(29) are satisfied within the error bars. Actually each point with error bar corresponds to several combinations of states at the same energy. The unitarity condition can be checked for each combination separately.
It turns out that it is satisfied within about 5% when the states in the combination are sufficiently linearly independent. An example of numerical values is given in Table I  The evaporation probabilities for R + and R − rotons behave quite differently. The one for R + rotons grows from 0 to 1 by increasing the energy. A similar behavior has been recently observed in experiments at normal incidence in the same range of energy [30].
Conversely the R − rotons are less effective in evaporating atoms. The ratio between the two probabilities has to be 1 at the energy ∆ of the roton minimum where the two excitations become identical. As soon as the energy grows the ratio P +a /P −a increases and becomes large. Results for P +a /P −a are shown in Fig. 6.
At low energies, approaching ∆, the mode-change is dominant in the reflection processes for rotons. This is also consistent with symmetry arguments (see discussions in Ref. [21]).
At higher energies the mode-change becomes less probable. An incident R − rotons has low probability of evaporating atoms, as discussed above, and it reflects as R + at low energy and as R − at high energy. The behavior of an incident R + roton at high energy is completely different: the normal reflection as R + is found to be practically zero everywhere. R + rotons prefer to evaporate atoms or to reflect as R − , the sum of the two probabilities being almost 1. This fact can be explained qualitatively by considering the change of momentum in a normal mode reflection. This should be twicehq z , where q z in this part of the spectrum is of the order of 2Å −1 . But it is difficult to imagine a mechanism for absorbing a momentum of the order of 4Å −1 at the surface, the latter being quite smooth (the Fourier transform of the density profile has vanishing components above 2Å −1 ). On the contrary, the momentum transfer for reflections including R − rotons is at least a factor two smaller and it is expected to be more probable.
The fact that P ++ is almost zero everywhere has an interesting consequence. If we assume S ++ = 0 and use the unitarity conditions (27)(28)(29)(30)(31)(32), we can express all the probabilities through a single parameter, for instance P +a . Simple algebra gives (see Appendix D): Note that the maximum value of P −a , from the last equation, is obtained for P +a = 1/2 and thus the following inequality holds: All the above relations, which follow from the assumption S ++ = 0, are well satisfied by the numerical results in Fig. 5.
Finally, the results for the reflection probability for incident atoms are more intriguing.
One knows from the experiments that the reflection probability is less than 1% apart from very small values of q z where the long range Van der Waals interaction dominates. Our results for the reflection probability are consistent with the experimental results only at high energy, where P aa drops to zero. At lower energy the roton channels become less and less active in the condensation process and the atom is reflected with large probability. This behavior is mainly the consequence of the fact that in the range of angles considered in the present work phonons cannot be excited. As soon as phonons are allowed (region I and II in Fig. 1) they are expected to provide the natural channel for the condensation of the incoming atoms. It has been also suggested that, as soon as the atom has enough energy to feel the external part of the density profile, it dissipates its energy by producing many low energy ripplons. This mechanism is not accounted for in our one-to-one theory.

VI. CONCLUSIONS AND PERSPECTIVES
In this paper we have investigated the evaporation and condensation processes taking place at the free surface of superfluid helium. The main achievements of the present work concern both the formal development of scattering theory, in the framework of linearised time dependent density functional formalism, and the numerical analysis, which provides first systematic results for the evaporation, condensation and reflection rates. The most relevant steps and achievements of our approach are here summarised: • iii) The numerical results, restricted in the present work to the region where only condensation of atoms into rotons is possible (region III of Fig. 1), reveal that rotons R + are systematically more effective than rotons R − in the evaporation mechanism. An interesting feature emerging from our analysis, is that the P ++ scattering probability is extremely small at all energies. This behavior, with the help of the formal properties of the scattering matrix, allows one to express all the evaporation and reflection rates in terms of a single parameter [see Eqs. (33-36)].
• iv) A major discrepancy between our predictions and the experimental data concerns the atomic reflectivity which is predicted to be significantly large when the energy approaches the roton minimum. This behavior is easily understood since our theory accounts only for one-to-one processes where condensation into ripplons is excluded by the conservation laws. Due to the absence of phonon excitations, which are excluded in region III, no active excitations are allowed in the condensation process when the energy approaches the roton minimum with the consequent full reflection of the incident atom.
Natural developments of the present work will include: • The study of the scattering process in different ranges of energy and incident angles, where phonons can participate in the scattering process.
• The study of multi-ripplon excitations whose effects can be taken into account through the use of an optical-type potential in the time dependent equations.
Work in these directions is in progress. We use the phenomenological density functional of Ref. [20]. It has the form: where ρ and v are the density and velocity of the atoms, respectively. The first term is the kinetic energy of the non interacting bosons, the correlation energy E (c) is given by finally, the backflow energy E (bf ) is The first term in the kinetic energy, which depends on gradient of the density, is a quantum pressure; it corresponds to the zero temperature kinetic energy of non-interacting bosons of mass m. The two-body interaction V l in the correlation energy E (c) is the Lennard-Jones interatomic potential, with the standard parameters α = 2.556Å and ε = 10.22 K, screened at short distance (V ≡ 0 for r < h, with h = 2.1903Å). The two terms with the parameters c 2 = −2.411857 × 10 4 KÅ 6 and c 3 = 1.858496 × 10 6 KÅ 9 account phenomenologically for short range correlations between atoms. The weighted densityρ is the average of ρ(r) over a sphere of radius h. Those terms are very similar to the functional of Ref. [24]. The last term in E (c) , depending on the gradient of the density in different points, has been added in order to improve the description of the static response function in the roton region.
where V is a generic two-body interaction. The generalisation to the more complex Orsay-Trento functional is conceptually straightforward but rather tedious in practice.
Taking variations with respect Ψ * in Eq. (6) one gets Here we are interested in the solutions well inside the liquid, where the ground state density ρ 0 is a constant and the above equation yields As in Eq. (9) we take a linear expansion of Ψ to describe the excited state. Let us include the time dependence in the definitions of f and g in the form f (r, t) = f e i(q·r−ωt) ; g(r, t) = ge i(q·r−ωt) , so that δΨ(r, t) = f (r, t) + g * (r, t) .
Inserting this expression in Eq. (B2) and isolating the terms oscillating with positive and negative frequencies, one gets two coupled equations for f and g: where we have used the short notation (1) ≡ (r 1 , t) and (2) This equation has the form of an equation of continuity if the quantity is interpreted as the density of elementary excitations and if the right hand side is identified with −ih∇ · j ex . Now we prove that the quantity j ex has indeed the form j ex = vρ ex , where v is the group velocity of the excitation. To this purpose we rewrite properly the various terms in Eq. (B8). We begin with the one coming from the kinetic energy. All terms with the laplacian can be written in the form −ih∇ · [j ex ] kin , with We remember that f (1) and g(1) are plane waves, as in Eq. (B4). Thus the expression for [j ex ] kin , in terms of the real amplitudes f and g, becomes To manage the terms containing the interaction is less trivial. Let us use the Fourier decomposition f (r, t) = dq 1 f (q 1 , t)e iq 1 ·r (B12) and take the limit f (q 1 ) = f δ(q 1 − q) at the end. We can rewrite the first two terms with the interaction in Eq. (B8) as where V (q) is the Fourier transform of the interaction. By doing the same with the terms containing g, one gets Now, by considering f (q) and g(q) as delta functions, one gets and summing [j ex ] pot to [j ex ] kin one finally obtains the result for the total current. At this point it is convenient to rewrite Eq. (B16) in terms of the dispersion law ω(q) coming from the solutions of the equations of motion (B6-B7) which, for plane waves of the form (B4) and after some algebra, yield with Inserting these results in Eq. (B16) one finally finds where v coincides with the usual definition of the group velocity: and g = 0. Close to z = L, inside the liquid, f (z) is the sum of two oscillating functions of the form in the antisymmetric case (a cosinus in the symmetric case) and the same for g. Here, for simplicity, the symbol q, for wave vectors, denotes the z-component only. The where the solution f (α) corresponds to the parameters L α and b α . The atomic part, well outside the slab, becomes the R + roton part, close to z = L, is and the R − roton part is The quantities in brackets represent the amplitudes of incoming and outgoing excitations and one can choose the coefficients a 2 and a 3 in order to make two of them vanish. As said at the beginning, we show here the case of an incident roton R + . So, we must have zero amplitude for incoming atoms and R − rotons (see Fig. 4). Keeping in mind that R − rotons travel with negative group velocity, one obtains the following algebraic equations for a 2 and a 3 : Since the ratio between the amplitude of the particle-hole f and the hole-particle g components depends only on the energy and not on the particular state in the combination, the above equations ensure that the incoming flux of atoms and R − rotons vanish. One has to recall the definition (18) of the current density and use the values of a 2 and a 3 coming from Eqs. (C7-C8) in order to calculate the flux of the incoming rotons and the ones of the outgoing excitations. The scattering rates come out to be: . One can also derive similar expressions using two antisymmetric states and one symmetric state. The other scattering rates can be evaluated by selecting different processes, i.e., by solving different equations for the coefficients a 2 and  (4) and 135Å (1 and 2). The quantity Σ is the sum of the probabilities for each process.