A convective model of a roton

A convective model describing the nature and structure of the roton is proposed. According to the model, the roton is a cylindrical convective cell with free horizontal boundaries. On the basis of the model, the characteristic geometric dimensions of the roton are estimated, and the spatial distribution of the velocity of the helium atoms and the perturbed temperature inside are described. It is assumed that the spatial distribution of rotons has a horizontally multilayer periodic structure, from which follows the quantization of the energy spectrum of rotons. The noted quantization allows us to adequately describe the energy spectrum of rotons. The convective model is quantitatively confirmed by experimental data on the measurement of the density of the normal component of helium II, the scattering of neutrons and light by helium II. The use of a convective model for describing the scattering of light by Helium II made it possible to estimate the dipole moment of the roton, as well as the number of helium atoms participating in the formation of the roton.

He ) in which neutrons are scattered.
The postulated dependence of energy on momentum in the short-wave part of the spectrum has been confirmed, according to the authors of a number of publications (see, for example, [2,3,4]), in experiments using neutron diffraction analysis when the probe beam of monochromatic neutrons with a specific wavelength is scattered on Helium II. In these experiments the values of the constants of Landau energy spectrum have been identified which are summarized in Table 1: where - 24 6.6464836122 10 He m   g the mass of the Helium atom [5], k -the Boltzmann constant.
According to [4], the width of the energy gap k  is not constant, but decreases with increasing temperature by the law 7 8.68 0.0084 k T     K. As shown in Table 1, the of the energy spectrum's constants values are different. This discrepancy points to the need to give the physical sense to the postulated short-wave part of the energy spectrum.

MODELS OF A ROTON'S STRUCTURE
A number of experimental data about the roton's thermophysical parameters is not yet completed by the establishment of its physical model.
Throughout the Helium II investigations by various methods do not stop attempts trying to describe the roton's structure and properties.
For example, R. Feynman proposed a roton's model as a vortex ring, which consists of six atoms 4 He arranged along the ring line with the gaps between them by the order of the Helium atom diameter. Each atom in the ring is rotated in synchronism, being located in the initial position or taking the neighboring, e.g., the left interval. The typical size of a vortex ring is about average atomic distance in a liquid Helium II [6].
Another model of roton's structure [7] is based on the assumption that at temperatures   N of atoms in such roton cluster can be determined by the free energy minimum, which is a condition for the stability of a roton. According to [7], the roton cluster must have nearly spherical shape, and the number of Helium atoms -13 c N  . The average radius of the mentioned spherical cluster is estimated at 5.22 Å. However, in the described roton's models (vortex and spherical) the physical nature describing the forces that hold the Helium atoms in the vortex or in the sphere is missing.

FREE ENERGY, THE NUMBER OF PHONONS AND PHONON DENSITY'S PART OF THE NORMAL COMPONENT OF HELIUM II
The Bose free energy -for the gas per unit volume is given by [1,8,9]  ph kT kT p F T n cp dp n cp do dp np dp npcdp kT dp The first term in square brackets at the upper and lower limits is zero. The integration of (2) gives the expression: 1  3  3  3  3  3  0 0  0   3  5   3   4  1  1  3 2  3 2   4  3 Phonon gas density is determined from the expression for flow in Helium II. The flow is determined by the momentum in the reference frame moving with the superfluid component: where vv ns w -relative velocity of the normal and superfluid components [9]. The same momentum, by definition equals to: For the small difference between velocities w the Planck distribution function in (7) can be expanded in a series pw and by retaining only the second term of the expansion (because the first term of the expansion due to asymmetric integrand in symmetrical limits of integration equals zero) one can obtain: x x y y z z x y z n cp pdp n cp p e p e p e dp dp dp For the calculation of the second term of the expansion (7) let's proceed as follows. Multiply (7) scalar left and right w . As a result, we have: Product in the right side pw of (9) in spherical system of momentum is as follows: Then, after reducing a common factor 2 w , the expression (9) takes the form: Thus, the existing ideas about the phonon component of the normal component of liquid Helium II are adequately described by the Planck distribution function for particles with energy   p  .

FREE ENERGY, THE ENERGY AND THE NUMBER OF ROTON DENSITY'S PART OF THE NORMAL COMPONENT OF HELIUM II
The calculation results shown in [1] for the roton density, in which the distribution function of the Planck energy of the roton is represented in the form: where  -energy roton gap, The number 4 in the denominator in (12) is necessary to ensure the transition to the classical recording of the dependence of energy on momentum in 0 0 p  . The possibility of the existence of a counter-motion of the atoms in Helium II will be discussed below. Let Expression (13) was first obtained by Landau [1].
Number of rotons per unit volume is calculated as in (13), and is given by:  (15) Expanding the exponential in a series of small w , we obtain a simpler expression for the roton part of the normal density: Thus, the number of rotons r N , and with it nr  , depend on the temperature exponentially. The above calculations of free energy, energy and the number of quasi-particles per unit volume of the normal component of Helium II match the original calculations for the equilibrium rotons distribution carried by Landau (12) [1].
A little later Landau changed to the non-equilibrium energy spectrum (11), adding at the same time the question about the reason for the non-equilibrium origin.
Thus it can be concluded that the above-noted difference in the experimental determination of the roton energy gap value, the momentum and its effective mass (see . Table 1), and the lack of description of the physical nature of the forces holding Helium atoms in a roton indicates the need of search of the models describing the roton's physical nature and structure.
Therefore, in the present article on the basis of the currently existing sufficiently comprehensive experimental data, a physical model of the roton's origin, defined its dimensions and internal structure, and provides estimates of its thermodynamic parameters.

ROTONS EQUILIBRIUM ENERGY SPECTRUM
Equilibrium rotons energy spectrum in the form (12) is possible, if it is formed by the volumetric vortex of Helium atoms moving in a cylindrical volume with a diameter r DD  and height z hL  . The vortex motion of the atoms is arranged so that close to the cylinder axis they moving, for example, along the axis direction (upward), and at the external border of the cylinder -in the opposite direction. Fig. 1 a) shows schematically the vortex motion of Helium II atoms inside a cylindrical cell.
The existence of a cylindrical convection cells with free boundary conditions in a viscous incompressible fluid first studied experimentally and theoretically described in [10]. It is shown that these cells have such geometrical dimensions that the ratio of diameter r D to cell's height z L is constant and equal 3.44 rz DL .
There are two liquid flows mutually opposite in the vertical section of such cell: at the cell's upper boundary flow velocity is directed out of the center and at the bottom -to the center or vice versa.
It should be noted that the roton's vortex nature was pointed out by Feynman [6 ,11]. He wrote about the roton as a vortex ring similar to the ring of smoke. Such vortex momentum is not associated with its translational movement as a whole, but with the movement around the cylindrical surface curved in a closed ring.
Thus roton can be regarded as a cylindrical convection cell. The use of roton energy spectrum of the form (12) is justified, and allows us to consider the Helium II as a whole, as a resting fluid. The spatial distribution of flows in roton refers to its internal structure and only characterizes its intrinsic properties.

ESTIMATION OF THE THERMODYNAMIC PARAMETERS OF THE ROTONS FORMATION IN HELIUM II
Let's assume the rotons in Helium II rotons as cylindrical convection cells. Then, to confirm this assumption we should assess the values of the Helium II thermodynamic parameters. These parameters are the coefficients of thermal expansion  , thermal conductivity  and the kinematic viscosity  [12]. They specify the value of the Rayleigh number, which, ultimately, will determine the conditions of the origin and existence of a stable cell -the roton.
Let's define dynamic and kinematic viscosity on the basis of data in the scientific literature. In the review [14] it is noted that the dynamic viscosity coefficient of Helium II is certainly less than 10 -11 Pa·s. Kapitsa's experiments determined the upper limit of the viscosity of the superfluid component [15]. Assuming laminar flow it was obtained that the value of dynamic viscosity coefficient is ≈ 10 -11 Pa·s.
In [16] the value of the dynamic viscosity coefficient is estimated to be about 10 -11 Pa·s.
In [17] the dynamic viscosity is determined by 14 Therefore, in further calculations we assume that the coefficient of the dynamic viscosity is 10 -12 Pa·s.
Thus, the above values of the thermodynamic parameters of Helium II correspond to those required to form a stable convection cell.

THE NATURE AND THE INTERNAL STRUCTURE OF A ROTON
Until now the nature and physical parameters of the rotons are not defined. Therefore, we shall present below the description of the above characteristics of this quasiparticle.
But before describing the nature and the internal structure of the roton, it is useful to trace the process of liquid Helium transition in the superfluid state.
To lower the temperature of liquid Helium below  -the point the permanent pumping of Helium vapor from the cryostat must be performed. Achieving a certain equilibrium temperature located below the  -point is provided by continuously maintaining a certain under pressure in a Helium cryostat [3].
In the process of establishing the thermodynamic equilibrium with falling pressure in the volume of Helium II the origin of the horizontally-layered structure of rotons spatial distribution becomes possible Lets describe in more detail the formation of the rotons spatial distribution.

Horizontal multi-layered structure of the rotons spatial distribution
Horizontal multilayered structure of the rotons spatial distribution apparently arises in connection with the formation of the monoatomic superfluid layer when passing through the  -point on the upper boundary of Helium. Under these conditions, apparently, the normal Helium II component located below forms a horizontal transition layer which thickness is sufficiently small and is comparable with the interatomic distance value in Helium II ( 0 3,579 z L Å). In this transition layer the temperature of the normal Helium II components will increase in the vertical direction from several K degrees at the lower boundary layer (at small deviations of the temperature from  -point) up to zero at the upper boundary layer.
Thus, based on the above-described conditions, we'll came across the problem of Rayleigh-Benard convection in the layer of the viscous, incompressible fluid heated from below [10,12,13].
In the above conditions with free boundaries in the transition layer normal Helium II components would occur cylindrical convective cells (read rotons) with a diameter-to-height respect ratio of the order of 3 is shown in Fig. 1. In such cell the Helium atoms move along the toroidal surface (Figure 1 a.)) with perturbed velocity   wherein , rz -the horizontal and longitudinal coordinates respectively, The toroidal surface inside the cylinder (Fig. 1, a)) corresponds to a certain Stokes line value (Fig. 1 b)) which is defined through the horizontal projection of the perturbed velocity of the substance in convective cell by relation From (19) it follows that the horizontal speed has its maximum  By increasing the number of cylindrical cells their packaging will occur such that each cell will abut with the six same cells. At the final stage of packaging cylindrical cells fill the entire volume of the transition layer, so that the boundaries between cells will have the form of Benard cells, i.e., hexagons.
The above is shown in Fig. 2. Here numeral 1 is a plan view of the upper boundary of the cylindrical cell in a regular hexagon inscribed. Within each hexagon the from its center arrows indicate schematically the direction of motion of the Helium atoms at the upper boundary of the cell. At the lower boundary of the cell Helium atoms move in the opposite direction. At points ,, abc of counter speed of the Helium atoms has maximum, Gh L -the period of the hexagonal lattice. Helium II volume filling by the underlying layer of the rotons would occur as follows. After forming the first layer of convective cells -rotons, below, at the depth of the interatomic distances in Helium II the monomolecular superfluid layer will begin to form again. This movement into deep of the superfluid Helium II can be explained as follows. By lowering the temperature of the Helium II upper boundary the normal flow (warm) of the Helium II components will be directed to the boundary. But according to the second sound theory, normal and superfluid components moving towards each other provide no net flow of the substance [1]. So long cooling of the Helium II upper boundary will contribute to the same duration of the superfluid component penetration in the depth, because time periods of the Helium II components oscillations are same.
Thus at the bottom of the first layer of the convective cells resulting in the above-described moving mechanism of the superfluid component motion the monomolecular layer of the Helium II superfluid components will begin to form. In this case superfluid and normal Helium II components are characterized with the commensurate geometric dimensions.
After the formation of a monomolecular layer of the superfluid component the Helium II normal component located below again forms a horizontal transition layer with a thickness of the interatomic distances in Helium II order. In this new transition layer the second layer of the rotons begins to form. After completion of the rotons second layer formation the superfluid component starts again to accumulate below the second layer and the process of forming the third and subsequent layers of convective cells -rotons is repeated.
Such process of the layered roton gas formation will occur until the thermodynamic equilibrium is achieved and the certain amount of the rotons corresponding given temperature (vapor pressure) is accumulated.   Fig.2 hexagonal structure of the upper boundary of the convective cells pattern repeats itself in depth with the period z L . The crystal structure of the rotons arrangement described above leads not to the roton-neutron scattering as was previously thought, but the neutron-Helium atoms, which form the rotons, scattering.
In Fig. 2. the lines represent the points of the cell's upper boundary, where the neutrons n are scattered by Helium atoms 4 He oncoming neutrons. At these points the Helium atoms have the maximum velocity He V  [10]. At points disposed symmetrically about the center of a cylindrical cell, the maximum velocity of the Helium atoms is equal to He V and is directed along the neutron velocity.
The distance of the noted points from the center of the cell is equal to -the first zero of the Bessel function of the first kind of orders zero and one respectively [19].
Thus, from the description given above it follows that the cold monoenergetic neutrons are scattered by the periodic lattice  (21) can be set on the basis of experimental data on the scattering of neutrons on cold rotons. Let's use the data about the rotons energy spectrum dependence on the wavenumber, which are summarized into a single curve in [20]. Based on this kind of curve we can conclude that the rotons energy spectrum is quantized by the wavenumber and energy. Fig. 3 shows the energy spectrum of the elementary excitations in Helium II as a function of wavenumber for temperatures below  -point. In this figure, data points from different sources are marked by 1, 2, 3, 4 and 5 [20].
Processing of these data allows the following conclusions regarding the energy spectrum of the quantization parameters rotons (21): -quantizing the spectrum is observed over the wavenumbers . It should be noted that the above values of the roton's mass and width of the roton's energy gap correspond to ones shown in Table 1.
Comparison of the experimental data and proposed model indicate the rotons energy spectrum quantization due to the periodic arrangement of rotons in the horizontal and vertical directions in the Helium II volume.
In the proposed model the rotons energy spectrum is represented in a classical form (20). Summing the energies of two rotons derives the energy which is postulated by Landau to justify the appearance of the roton minimum:

THE DENSITY OF THE HELIUM II NORMAL COMPONENT WITH THE STREAMLESS ROTONS ENERGY SPECTRUM
Initially, using the classical representation of the dependence of the rotons energy on a pulse in the form (20) Landau calculated the following thermodynamic quantities of the normal component of helium II: free energy, entropy, heat capacity (per unit mass), and density [8]. Then, in the same work re-published in Physics-Uspekhi, but already with the Appendix, Landau recounted the thermodynamic quantities obtained earlier for the energy of rotons, postulated in the form [1]: Therefore, there is a natural question about the validity of the application of the roton energy dependence on the pulse in the classical form (20) or in the form (22).
The ratio of the density of the normal component of Helium II to the total density of liquid Helium n  (by convention, we call this relation by the term "ro-n-to-ro" ) has the form [1]:  Fig. 4).
The answer to this question will be a comparative assessment of the coincidence of the theoretical "ro-n-to-ro" dependence on the temperature of the two kinds of roton energy (20) and (22) with experimental points. In this case it is necessary to take into account that the analytical expressions for the temperature dependence of "ro-n-to-ro" are inapplicable for the temperatures near the point  and in a neighborhood of zero [9].
To answer this question, let us use the experimental data on the temperature dependence of "ro-n-to-ro" [20]. In Fig. The 4 markers " × " give the recommended data for experimental measurements of the "ro-n-to-ro" values depending on temperature in the temperature range from 0.15 К to 1.95 К.
The curves I and II are constructed from the expression (23) for the number of rotons in a unit volume in the form (20) and (22) (24), (25). The optimal parameters were chosen according to the minimum standard deviation of the theoretical dependence on the experimental points. Fig. 4. Dependence of "ro-n-to-ro" on temperature T in the temperature range from 0.15 К to 1.95 К. The marks " × " indicate experimentally measured points. The curve I corresponds to the (1) r N , the Curve II - (2) r N .
The smallest standard deviation of the curve I from the experimental points is 1.772·10 -3 , while for the curve II it is 2.48 times larger -4.395·10 -3 .
Thus, in the range of temperatures considered, the "ro-n-to-ro" of Helium II is determined with the greatest degree of accuracy by the streamless rotons energy spectrum in the form (20).

THE ELASTIC SCATTERING OF NEUTRONS BY MOVING ATOMS OF HELIUM
Unlike the generally accepted concept of the scattering of slow neutrons by Helium II, as inelastic scattering by rotons, we present a different picture of this process. We shall assume that in convective cells the neutrons are elastically scattered by Helium atoms moving towards them. Scattering of neutrons on incidental Helium atoms is not considered as they are slowed down and not detected by the time-of-flight neutron spectrometer [3].
To confirm the validity of this assumption, let us compare the experimental data obtained in the Stockholm experiment on a time-of-flight neutron spectrometer with the results of the theoretical analysis, which are given below.
Let's calculate the neutron scattering parameters for a moving Helium atom.
Let's determine the angle of elastic scattering of a neutron  on an atom moving 4 He in a cylindrical convective cell. Let the helium atom prior to collision have a maximum horizontal velocity , / He He n n opposite points of the upper boundary of the cell (see Fig. 1, 2). The neutron mass is equal to 24 1,674 928 727 6·10 n m   g [21], which is 0.252 times less than the mass of the Helium atom. We assume that the neutron velocity before the collision is n V .
As a result of an elastic collision, the neutron will acquire velocity 1 n V , and the Helium atom -1, He V  . In the case of an elastic collision, and in the absence of external forces from the laws of conservation of momentum and kinetic energy it is not difficult to obtain the expression for the relative velocity of the scattered neutron In equation (26), the upper sign corresponds to the scattering of a neutron by the Helium atom moving in the opposite direction, and in the the same direction for the lower sign. nn VV  . Therefore, the value of the time shift of the Bragg truncation of the primary beam [3] for a given scattering angle will be minimal, since it is inversely proportional to the velocity of the scattered particle.
This conclusion is based on the following estimates. We assume that the initial velocity n V is the same as the neutron velocity after elastic scattering by Vanadium, since it dissipates the neutrons isotropically and without changing the energy [3]. In the Stockholm experiment this velocity is estimated by the value of 973.71 n V  m/s [3]. We set the initial velocity of the neutron in the form where L is the neutron transit distance in the spectrometer, 0 t is the transit time. The neutron velocity after its scattering by Helium II equals to  (29) From the expression (29) and the experimental data given in Table 2 [3], it is possible to determine the maximum horizontal counter-flow velocity    From the experimental results presented in Table 2 it follows that in the cell the velocity of the scattering Helium atom depends on the viewing angle  . At small observation angles, the velocity is small. With the approach of the observation angle to a certain value, in our case about 83 °, the velocity reaches its maximum value, and then decreases again.
The noted above change in the Helium atom's velocity as a function of the viewing angle can be explained on the basis of the dependence of the horizontal velocity of Helium II on the upper boundary of the convective cell on r : where r is the distance from the center of the cell to its outer boundary. Such an explanation is possible if we find the dependence of the radius r on the observation angle  .
To find this dependence, we write down the expression for Archimedes' spiral: where  -the polar radius,  -polar angle,    (27), (28). Therefore, further all calculations on the angular distribution of the scattered neutron will be carried out for a fast neutron emitted from a point , where the angle's magnitude 0  is determined from the experimental data.
The substitution of (31) in (19)  Substituting the experimental data of Table 2 in expression (33), and by optimizing the theoretical calculations and experimental data by the method of least mean-square deviation, we determine the values of the constants: 0.2603 A  , 0 0.9755   . The standard deviation is quite small, and is of the order of magnitude 3 3,919 10   . The solid line in Fig. 5 shows an optimized curve of the dependence of the relative horizontal velocity of the Helium atom , He n VV  on the viewing angle  . The inset shows the deviation of the theoretical curve from the experimental points. Calculations show that the deviation does not exceed 7.5%. Thus, in this section it is shown that neutron scattering on a roton can be represented as an elastic collision of a neutron with a Helium atom moving in a cylindrical convective cell. This is indicated by the quantitative agreement of the experimental data obtained earlier by other authors on the scattering of neutrons by Helium II with theoretical calculations of neutron scattering by a helium atom moving in a convective cell.

SCATTERING OF LIGHT FROM HELIUM II
Along with the neutron diffraction analysis described above, it is possible to study the physical properties of rotons using scattering of light. The data on the measurement of the spectrum, intensity and polarization of light by Raman scattering of argon laser light (514.5 nm wavelength, 1 Watt) by superfluid helium at temperatures in the interval 1.16 К and 2.14 К are given in [22].
In experiments, the incident linearly polarized laser beam and scattered light are located in a horizontal plane. The scattered light was collected at an angle of 90 ° to the incident radiation within a solid angle of about 0.08 sr. To ensure the maximum level of the detected signal, the vector of the electric field of the incident laser radiation was oriented in the horizontal direction [23]. Analysis of the intensity of the scattered radiation spectrum clearly showed the presence of an asymmetric sharp peak shifted by an energy of 18.5 ± 0.5 К relative to the energy of the incident radiation.
The following work on the Raman scattering of laser radiation by superfluid helium [24] used the same experimental setup as in the original paper [22]. However, instead of the diffraction monochromator, a Fabry-Perot spectrometer was used, which free spectral range was 48.6 K, which roughly corresponds to the triple shift of two-roton Raman scattered light. The light source was an argon ion laser (wavelength 488.0 nm). It was shown in this paper that the shift of the energy of the scattered radiation relative to the energy of the incident radiation is 17.022 ± 0.027 K at a temperature of 1.2 К.
The measurements carried out in [1 -3] have shown that the investigation of light scattering is more accurate, in comparison with the neutron diffraction tool, for describing elementary excitations in superfluid Helium. The very high resolution of this method makes it possible to accurately measure such characteristics of excitations in liquid helium, as energy and lifetimes of the roton. Let's describe the above results of experiments on Raman scattering of light on the basis of the convective model of the structure of the roton proposed in this paper.

The dipole moment of a roton
Let's define the number of the Helium atoms which are involved in the convective motion in a roton. It follows from the calculations that there are 7 Helium atoms in the volume of one roton: and represents two differently directed dipoles with a cylindrically symmetric distribution of the dipole moment in space.
To estimate the magnitude of the total dipole moment from (33) we determine the amplitude of the velocity A : 179.71 A  m/s. Then the maximum value of the total dipole moment for bound Helium atoms is reached on the roton axis, and has the order: Hence we can assume that in the convective motion in the roton three pairs of bound Helium atoms (37) are involved which are located in three planes shifted along the azimuth by 120 °.
Proceeding from the foregoing, let us estimate the maximum total dipole moment of the roton in the vertical direction 4 max 1 3 0.678 10 Dd In the horizontal direction, without taking into account the seventh Helium atom, the roton's dipole moment equals to zero.
Thus, the estimates given above show that the roton's dipole moment consists of two dipoles oriented in the vertical direction and directed in opposite directions (see (37)). The magnitude of the dipole moment on the roton axis is of the order of magnitude of the experimentally measured [27].