On the suprathermal distribution in an anisotropic phonon system in He II

The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to th...

Повний опис

Збережено в:
Бібліографічні деталі
Опубліковано в: :Физика низких температур
Дата:2003
Автори: Adamenko, I.N., Nemchenko, K.E., G.Wyatt, A.F.
Формат: Стаття
Мова:Англійська
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2003
Теми:
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/128779
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
_version_ 1860017318904463360
author Adamenko, I.N.
Nemchenko, K.E.
G.Wyatt, A.F.
author_facet Adamenko, I.N.
Nemchenko, K.E.
G.Wyatt, A.F.
citation_txt On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ.
collection DSpace DC
container_title Физика низких температур
description The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to the Bose-Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the dependence of S on the relevant parameters.
first_indexed 2025-12-07T16:45:53Z
format Article
fulltext Fizika Nizkikh Temperatur, 2003, v. 29, No. 1, p. 16–21 On the suprathermal distribution in an anisotropic phonon system in He II I. N. Adamenko1, K. E. Nemchenko1, and A. F. G. Wyatt2 1 V. N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine 2 School of Physics, University of Exeter, Exeter EX4 4QL, UK E-mail: adamenko@pem.kharkov.ua Received July 25, 2002, revised September 24, 2002 The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equa- tion enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribu- tion to the Bose—Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the de- pendence of S on the relevant parameters. PACS: 67.40.Db, 67.40.Fd 1. Introduction For a phonon system in superfluid 4He (He II) the rates of kinetic processes are determined by the un- usual form of the phonon energy—momentum depend- ence. At zero pressure the phonon dispersion curve in He II bends upwards [1–3], and this causes sponta- neous decay of phonons with energies � �� �c 10 K [4,5]. For these phonons, the energy and momentum conservation laws allow processes in which the pho- non numbers in the initial and final states do not equal one another. The fastest process among these is the three-phonon process (3pp), in which one phonon de- cays to two phonons or two phonons interact to create one phonon. The rate of these three-phonon processes was obtained in [6,7], in the two extreme limits; and the general case was calculated in [8]. At higher energies the phonon dispersion curve bends downwards, and at � �� c phonon spectrum be- comes nondecaying. Here the most rapid process is the four-phonon process (4pp), in which there are two phonons in the initial and final states. The rate of three-phonon processes �3pp is obtained using Landau’s Hamiltonian in first-order perturba- tion theory, and the rate of four-phonon �4pp pro- cesses is determined by second-order perturbation the- ory [9–11]. This is quantitatively evaluated and confirmed by experiment [12]. The difference between the orders of perturbation theory results in the strong inequality � �3 4pp pp�� . Thus the phonons of super- fluid 4He form two subsystems: one of low energy (l phonons) with � �� c, which very quickly attains equi- librium; and the other of high-energy phonons (h phonons), which goes to equilibrium relatively slowly. The angles between the phonons which take part in 3pp is small due to the smallness of the deviation of the energy—momentum dispersion from linearity, � � cp. Thus in isotropic phonon systems, when all di- rections in momentum space are uniformly occupied, equilibrium is not attained isotropically in the short term because interactions involve only phonons within a limited solid angle. Thus, all thermodynamic para- meters of the l-subsystem become functions of angle [11]. In the long term, four-phonon processes and dif- fusion in angular space bring about complete equilib- rium. This quasidiffusion is determined by fast three-phonon interactions involving small angles [13–15]. The situation is quite different in highly ani- sotropic phonon systems. Here the momenta of all phonons are in a narrow cone with solid angle � p , which has a value close to the typical angle for three-phonon processes. Such strongly anisotropic phonon systems have been created in liquid 4He [16–20]. This pure and isotropic superfluid can have such a low temperature that one can neglect the exis- tence of thermal excitations. Low-energy phonons are © I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, 2003 injected by a heater and then propagate along the di- rection normal to the surface of the heater. In momen- tum space all the phonons lie in a narrow cone of solid angle � p � 0125. sr. In such experiments [16–20] the unusual phenome- non of the creation of high-energy phonons was observed. These h phonons are created by a pulse of l phonons, which has a temperature an order of magni- tude less than the energy of the high-energy phonons that it creates. The theory of this unique phenomenon was proposed in [21,22]. A further development of the theory [23–28], shows that in strongly anisotropic phonon systems there exists an asymmetry between processes of creation and decay for the h phonons. Such an asymmetry causes the distribution function of h phonons in the anisotropic phonon system to be S times greater than that in the Bose—Einstein distribu- tion. Moreover, it is possible to have S �� 1. Using the notation of [23], we will call such an unusual distri- bution of h phonons a suprathermal distribution, and the parameter S, the suprathermal ratio. In this paper we derive an exact equation that al- lows us to calculate the suprathermal ratio and to determine its dependences on momentum, anisotropy parameter, and temperature. We analyze the contribu- tion of all possible processes that lead to the formation of suprathermal distributions in anisotropic phonon systems. From this equation we derive an estimate of the average value of the suprathermal ratio S: the ra- tio of the number density of h phonons in the anisotropic and isotropic phonon systems. 2. The inequalities for determining of the stationary distribution function of phonons in an anisotropic phonon system The attainment of equilibrium in the subsystem of h phonons can be described by the kinetic equation for the distribution functions, which can be written as dn dt N Nb d 1 1 1� �( ) ( )p p , (1) where Nb( )p1 � � � � �W n n n n b ( , | , ) ( )( )p p p p1 2 3 4 3 4 1 21 1 � � � � � � � �( )1 2 3 4 � � � ( )p p p p1 2 3 4 3 2 3 3 3 4d p d p d p (2) is the number of phonons with momentum p1 created per unit time as the result of four-phonon (4pp) inter- actions; N W n n n nd d ( ) ( , | , ) ( )( )p p p p p1 1 2 3 4 1 2 3 41 1� � � � � � � � � � � �( )1 2 3 4 � � � ( )p p p p1 2 3 4 3 2 3 3 3 4d p d p d p (3) is the number of phonons that decay per unit time; W W( , | , ) ( , | , )p p p p p p p p1 2 3 4 3 4 1 2� defines the tran- sition probability density for 4pp processes, which have been calculated [25]; �b and � d each represent a set of three solid angles, one for each momentum over which the function is integrated, i.e., �bi and � di (i = 2,3,4). These maximum angles are deter- mined by the anisotropy of the phonon system and the angles in the 4pp interactions. In the isotropic case � �bi di� . In relations (1)–(3) and below we con- sider p p k /cc B c1 � � � (i.e., phonon «1» is the h1 phonon), while the other three phonons can be l phonons or h phonons. The stationary distribution function is determined by the equality N Nb d� . (4) For isotropic phonon systems � �bi di� , and equality (4) gives: n n n n n n n n1 2 3 4 3 4 1 21 1 1 1( )( ) ( )( )� � � � � . (5) The solution of this equation is the Bose—Einstein distribution � �� �n /Ti i ( ) exp0 1 1� � � � . (6) In an anisotropic phonon system, when the initial phonons are inside a narrow cone with solid angle � p � 4 , the result differs from (5) and (6). In this case the limits of integration in (2) and (3) with re- spect to angular variables are defined by the inequali- ties � �i p� . (7) Here i = 3,4 for creation processes, and i = 2 for de- cay processes. The angular variables for the final phonons have no such restrictions. Such asymmetry for the initial and final states re- sults in the inequality � �b d� . In this case in anisotropic phonon systems the equality (4) cannot be satisfied by solution (5), and the Bose—Einstein di- stribution is not a solution of equation (4). In highly anisotropic phonon systems, when � p is less than the typical solid angle for four-phonon pro- cesses, the stationary distribution of h phonons will be essentially different from the Bose—Einstein distribu- tion (6) in magnitude and in momentum dependence. The integrals (2) and (3) can be written as a sum of five terms with definite ranges of integration. These On the suprathermal distribution in an anisotropic phonon system in He II Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 17 terms correspond to the different processes, which are possible for the interactions of h phonons between themselves and with l phonons: 1) h l l l1 2 3 4� � � ; 2) h l h l1 2 3 4� � � ; 3) h l h h1 2 3 4� � � ; 4) h h h l1 2 3 4� � � ; (8) 5) h h h h1 2 3 4� � � The arrow to the right indicates the decay of an h1 phonon and to the left creation. We define the rates �b d n , ( ) of creation (b) and decay (d) processes with dis- tribution function n for h phonons by the equalities: N nb b n � � �� 1 0( ) ( ); N nd d n � � �� 1 ( ); ( , , , , )� � 1 2 3 4 5 . (9) As Nb is the sum over all Nb� , we rewrite relation (4) as follows: n nb n d n 1 0 1 5 1 1 5 ( ) ( ) ( )� � � � � �� � � �� . (10) We take into account that 3pp processes instanta- neously establish equilibrium (on the time scale of h-phonon creation and propagation) in the subsystem of l phonons, which occupy the solid angle �3pp that is typical for 3pp. The phonon pulses in experiments [16–20] have � p close to �3pp [13–15]. Therefore in this case we can consider that the l phonons in the pulse have a Bose—Einstein distribution: n nl l( ) ( )p � 0 , at p pl c� . (11) For the stationary distribution of h phonons, the dis- tribution function can be written in the form: n S nh h h( ) ( ) ( )p p� 0 , at p ph c� . (12) Starting from equalities (10)–(12) we have � � � � � � b n d nS( ) ( )( ) � � � �� 1 5 1 1 5 p . (13) This equation is an integral equation with respect to the unknown function S( )p1 . For decay processes when h1 combines with an l or h phonon, the rate is independent or a linear functional of S respectively. For creation processes if initially there are zero, one, or two h phonons then the rate is independent, a li- near functional, or a quadratic functional, respec- tively. Therefore the desired function S( )p1 is absent in the rates �b n 1 ( ), �d n 1 ( ), �d n 2 ( ), �d n 3 ( ) if one takes into ac- count that 1 1� �nh . The rates �b n 2 ( ), �b n 4 ( ), �d n 4 ( ), �d n 5 ( ) include linear functionals, and �b n 3 ( ), �b n 5 ( ) include quadratic functionals. Using these facts, we present the rates from (9) as � �b n b1 1 0( ) ( )� ; � �d n d1 1 0( ) ( )� ; � �d n d2 2 0( ) ( )� ; � �d n d3 3 0( ) ( )� ; (14) � �b n b bS2 2 1 2 0( ) ( )( )� p ; � �b n b bS4 4 1 4 0( ) ( )( )� p ; � �d n d dS4 4 1 4 0( ) ( )( )� p ; � �d n d dS5 5 1 5 0( ) ( )( )� p ; and � �b n b bS3 3 2 1 3 0( ) ( )( )� p ; � �b n b bS5 5 2 1 5 0( ) ( )( )� p ; (15) where � � � �b d b d n S, ( ) , ( ) ( )0 0 1� � (16) are the rates calculated with distribution function (6). The above equations define Sb d, � which are functionals of the function S h( )p . Using (14)–(16) the relation (13) can be written as � � � �SS S SS Sd d d b d d b b5 5 0 5 2 5 0 4 4 0 3 2 3 0� � � �( ) ( ) ( ) ( )� � � � � � � �� � � � �S S S Sd b b d b b� � � �3 0 4 4 0 2 0 2 2 2 0( ) ( ) ( ) ( ) � �� �b dS1 0 1 0( ) ( ). (17) For isotropic phonon systems, when� �bi di� , re- lations (2), (3) give � � � � � �b d ( ) ( ) ( )0 0� � isotr . In this case Eq. (17) has the solution S h( )p � 1, and, according to (12), we get the Bose—Einstein distribution (6) for all phonons in an isotropic system. 3. Asymmetry of processes of h-phonon creation and decay, resulting in a suprathermal distribution on an anisotropic phonon system In anisotropic phonon systems, when � �bi di� the creation rate � �b ( )0 can be significantly different from the decay rate � �d ( )0 . In Refs. [25,26] the rates of all five processes of creation and decay are calculated for phonons with momentum p1 directed along the axis of symmetry of the pulse, chosen as the Z axis, so �1 0� . These rates we denote �b d, , where the super- script (0) is understood. The main role in (17) is played by a type-1 process that describes the exchange of phonons between the l and h systems. For a pulse typically used in the experi- ments [16–20] the values are: anisotropy parameter � p � 0125. sr and temperature T = 1 K. Then the min- imum value of the ratio � � �b d1 1 30� at p pc1 � ; it grows quickly with increasing p1 and becomes equal to infinity at p p1 0� (see Fig. 1). The limiting mo- mentum p0 is determined by the solid angle � p and 18 Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt the conservation laws of energy and momentum, which govern the interaction of l phonons with such h1 phonons. The corresponding analytical expressions and detailed discussion of the rates and their de- pendences on momentum shown in Fig. 1, are given in Refs. 25, 26. An infinite lifetime coupled with a finite creation rate of h phonons with p p1 0� means that in aniso- tropic phonon systems, type-1 processes cannot effect a dynamic equilibrium between the h and l subsys- tems. However such an equilibrium can be provided by type-4 processes, because � �d b4 4� in the momen- tum region where �d1 0� (see Fig. 1). Using the nu- merical values for rates calculated for the anisotropic phonon system (see Fig. 1), equation (17) can be sat- isfied for S �� 1. As a result, in anisotropic phonon systems, the stationary distribution function of such h phonons is many times greater than the Bose—Ein- stein one (6) and has a different energy dependence which is determined by the momentum-dependent rates shown in Fig. 1. In the left-hand side of (17) the rates that have the same power of S and describe mutually compensating processes are separated in curved braces. These com- pensating processes in (17) are of two fundamentally different types. The second and the third braces de- scribe processes which exchange phonons between the l and h systems. At the same time, type-4 decay pro- cesses are partly compensated by type-3 creation pro- cesses (but not by type-4 creation processes), and type-3 decay processes partly compensate type-4 cre- ation process (but not type-3 creation process). The first and fourth curved braces describe processes that conserve the number of h phonons. Here decay is com- pensated by creation in processes of the same type. The presence in (17) of processes that conserve and do not conserve the number of h phonons makes it use- ful to consider the expression obtained from (17) by averaging the anisotropic phonon system over p1. We define the average as A An d p n d p p p � � � 1 0 3 1 1 0 3 1 ( ) ( ) � � . (18) Then one should take into account the following equalities, which are obtained by reindexing the vari- ables of integration: N d p N d pd b p d 2 1 3 1 2 1 3 1 2 3 ( ) ( ) ( ) p p � � � �� , (19) N d p N d pd b p d 5 1 3 1 5 1 3 1 5 3 ( ) ( ) ( ) p p � � � �� , (20) N d p N d pd b p d 3 1 3 1 4 1 3 1 3 3 ( ) ( ) ( ) p p � � � �� , (21) N d p N d pd b p d 4 1 3 1 3 1 3 1 4 3 ( ) ( ) ( ) p p � � � �� , (22) where� d� ( )3 is the solid angle of the created p3 phonon in decay processes of the type �. From the conservation laws of energy and momen- tum it follows that at the typical momenta of the phonons taking part in the decay processes, the cre- ated p3 phonon remains inside the initial phonon pulse. This fact allows us to consider approximately that � �d p� ( )3 � . In this case the averaging of (17) ta- king into account (19)–(22) gives: SS S Sd d b b b d4 4 0 4 4 0 1 0 1 02 2� � � �( ) ( ) ( ) ( )� � � . (23) On the suprathermal distribution in an anisotropic phonon system in He II Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 19 Fig. 1. The momentum dependences of the creation rates �b and decay �d rates, at T = 1 K and �p � 0125. sr, for all the processes which exchange phonons between the l- and h-phonon systems. This result can be rewritten in the form S S Sd b b d 2 4 0 4 0 1 0 1 02 2~ ~( ) ( ) ( ) ( )� � � �� � � , (24) where � �b b1 0 1 0( ) ( )� , and the other average values of the rates ~ , ( )�b d1 4 0 are determined by the obvious equali- ties of the respective terms in Eqs. (23) and (24). The solution of Eq. (24) is S b d b d b d � � � � 4 2 8 2 1 0 1 0 4 0 2 4 0 1 0 1 � � � � � � ( ) ( ) ( ) ( ) ( ) (( ~ ~ ) ~ ~ 0 4 0) ( )~� �b . (25) Using this solution we can estimate the value of S if we substitute the rates in (25) (shown in Fig. 1) with their average values calculated over the range 10 201K K� �cp /kB , at �1 0� . These rates we de- note �b d, . In this case 2 81 4 1 4� � � �d b b d� �� (26) and from (25) we have S b d � �2 301 0 4 0 � � ( ) ( ) . (27) This relation has a simple physical meaning. The value of S is defined as the square root of the ratio of the rate of growth of the number of h phonons by type-1 processes to the rate of decrease of the number of h phonons by type-4 processes, which are partly compensated by type-3 processes. According to (17) and results obtained for �b d2 2, and �b d5 5, [25,26], it follows that S depends strongly on the angle �1 between the direction of the phonon p1 1 1 1( , , )p � � and the Z axis of symmetry of the anisotropic phonon system. Therefore, according to [25,26], over a relatively wide region of momentum p1 the creation rate of h phonons with �1 0� , for the se- cond and the fifth processes, is greater than decay rate. Since the total number of h phonons is separately conserved in processes 2 and 5, the h phonons will concentrate in momentum space near the Z axis. With increasing �1, the number of h phonons decreases, so that there will be relatively few h1 phonons at large � �1 � p in the pulse. This result agrees with the re- sults of experiments [17,19], where the h-phonon cone is found to be narrower than the l-phonon cone. The suprathermal ratio S is also a function of the temperature of the l phonons, because according to [25,26] the rates of creation and decay of all five pro- cesses become smaller, at different rates, with decreas- ing temperature. Thus, at � p � 0125. sr and p pc1 � , according to [26], with decreasing temperature from 1 K to 0.7 K the rates �b1 and �d1 becomes smaller by � 5 and � 6 times, respectively, the rates �b2 and �d2 by � 9 and � 6 times, the rates �b3 and �d3 by � 70 and � 65 times, the rates �b4 and �d4 by � 80 and � 95 times, and the rates �b5 and �d5 by � 85 and � 100 times, respectively. In general, the suprather- mal ratio increases as the temperature decreases. Although in this paper we are concerned with the value of S averaged over momentum, we do expect that S is strongly peaked just above pc, where � �d d4 3� is small and �b3 has a maximum [26]. How- ever, the situation is complicated, as S is also expected to vary with angle within the beam. 4. Conclusion In this paper we have shown that the asymmetry between the processes of decay and creation of high-energy phonons in long enough phonon pulses created in experiments [16–20] in superfluid helium results in a suprathermal distribution. Then the quasi- equilibrium distribution function of the h phonons dif- fers from the Bose—Einstein distribution by a factor S(p). We have obtained an equation (17) whose solution determines the value of suprathermal ratio S and its dependence on momentum p1, anisotropy parameter � p , and temperature T. Expressions that describe mu- tually compensating processes are separated in (17) by curved braces. These compensated processes have two different principal types: the first type describes pro- cesses that exchange phonons between the l and h sys- tems, and the second type conserves the number of h phonons. That is why we consider the expressions (23), obtained from (17) by averaging with respect to all p1 of the anisotropic phonon system. Starting from relation (23) and the available re- sults for the rates of creation and decay of phonons with momentum p1 1 1 10( , , )p � �� directed along the symmetry axis Z (see Fig.1), an estimate is made of the average value S of the suprathermal ratio. The full evaluation of the suprathermal ratio S and its de- pendence on the parameters p1, � p , and T will only be possible after calculation of all the rates in (17) at arbitrary angles �1. We plan to carry out this calcula- tion. At present we have only the values of all the rates �b d, for the case �1 0� . This estimation of and the analysis of Eqs. (17) and (23) indicates that the distribution function of h phonons can exceed the Bose–Einstein energy distribution by two orders of magnitude in anisotropic phonon systems. We find that S depends strongly on the parameters p1, � p , and T. Besides the creation S of a more complete the- ory of the suprathermal distribution, we plan to carry 20 Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt out experiments to observe this very unusual pheno- menon occuring in phonon pulses in He II. Acknowledgements We express our gratitude to EPSRC of the UK (grant GR/N18796 and GR/N20225), and to GFFI of Ukraine (grant N02.07/000372) for support for this work. 1. H. J. Marris and W. E. Massey, Phys. Rev. Lett. 25, 220 (1970). 2. J. Jackle and K. W. Kerr, Phys. Rev. Lett. 27, 654 (1971). 3. W. G. Stirling, 75th Jubilee Conference on 4He, J. G. M. Armitage (ed.), World Scientific, Singapure (1983), p. 109. 4. R. C. Dynes and V. Narayanamurti, Phys. Rev. Lett. 33, 1195 (1974). 5. A. F .G. Wyatt, N. A. Lockerbie, and R.A. Sherlock, Phys. Rev. Lett. 33, 1425 (1974). 6. S. Havlin and M. Luban, Phys. Lett. A42, 133 (1972). 7. H. J. Maris, Phys. Rev. A8, 1980 (1972). 8. M. A. H. Tucker, A. F. G. Wyatt, I. N. Adamenko, K. E. Nemchenko, and A. V. Zhukov, Fiz. Nizk. Temp. 25, 657 (1999); [Low Temp. Phys. 25, 488 (1999)]. 9. L. D. Landau, Zh. Eksp. Teor. Fiz. 11, 592 (1941). 10. S. G. Eckstein, Y. Eckstein, J. B. Ketterson, and J. H. Vignos, Phys. Acoustics 2, Academic Press, New York (1970), p. 244. 11. I. M. Khalatnikov, Theory of Superfluidity, Nauka, Moscow (1971) (in Russian). 12. M. A. H. Tucker and A. F. G. Wyatt, J. Phys.: Condens. Matter. 4, 7745 (1992). 13. I. N. Adamenko and M. I. Kaganov, Zh. Eksp. Teor. Fiz. 53, 886 (1967). 14. V. L. Gurevich and B. D. Likhtman, Zh. Eksp. Teor. Fiz. 69, 1230 (1975). 15. V. L. Gurevich and B. D. Likhtman, Zh. Eksp. Teor. Fiz. 70, 1907 (1976). 16. M. A. H. Tucker and A. F. G. Wyatt, Physica B194, 549 (1994). 17. M. A. H. Tucker and A. F. G. Wyatt, Physica B194, 551 (1994). 18. M. A. H. Tucker and A. F. G. Wyatt, J. Phys.: Condens. Matter. 6, 2813 (1994). 19. M. A. H. Tucker and A. F. G. Wyatt, J. Phys.: Condens. Matter. 6, 2825 (1994). 20. M. A. H. Tucker and A. F. G. Wyatt, J. Low Temp. Phys. 113, 621 (1998). 21. I. N. Adamenko, K. E. Nemchenko, A. V. Zhukov, M. A. H. Tucker, and A. F. G. Wyatt, Phys. Rev. Lett. 82, 1482 (1999). 22. A. F. G. Wyatt, M. A. H. Tucker, I. N. Adamenko, K. E. Nemchenko, and A. V. Zhukov, Phys. Rev. B6, 9402 (2000). 23. A. F. G. Wyatt, M. A. H. Tucker, I. N. Adamenko, K. E. Nemchenko, and A. V. Zhukov, Phys. Rev. B62, 3029 (2000). 24. A. F. G. Wyatt, M. A. H. Tucker, I. N. Adamenko, K. E. Nemchenko, and A. V. Zhukov, Physica B280, 36 (2000). 25. I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, J. Low Temp. Phys. 125, 1 (2001). 26. I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, J. Low Temp. Phys. 126, 1471 (2002). 27. I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, J. Low Temp. Phys. 126, 609 (2002). 28. I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, Fiz. Nizk. Temp. 28, 123 (2002); [Low Temp. Phys. 28, 85 (2002)]. On the suprathermal distribution in an anisotropic phonon system in He II Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 21
id nasplib_isofts_kiev_ua-123456789-128779
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 0132-6414
language English
last_indexed 2025-12-07T16:45:53Z
publishDate 2003
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
record_format dspace
spelling Adamenko, I.N.
Nemchenko, K.E.
G.Wyatt, A.F.
2018-01-13T19:49:20Z
2018-01-13T19:49:20Z
2003
On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ.
0132-6414
PACS: 67.40.Db, 67.40.Fd
https://nasplib.isofts.kiev.ua/handle/123456789/128779
The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to the Bose-Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the dependence of S on the relevant parameters.
We express our gratitude to EPSRC of the UK (grant GR/N18796 and GR/N20225), and to GFFI of Ukraine (grant N02.07/000372) for support for this work.
en
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Физика низких температур
Квантовые жидкости и квантовые кpисталлы
On the suprathermal distribution in an anisotropic phonon system in He II
Article
published earlier
spellingShingle On the suprathermal distribution in an anisotropic phonon system in He II
Adamenko, I.N.
Nemchenko, K.E.
G.Wyatt, A.F.
Квантовые жидкости и квантовые кpисталлы
title On the suprathermal distribution in an anisotropic phonon system in He II
title_full On the suprathermal distribution in an anisotropic phonon system in He II
title_fullStr On the suprathermal distribution in an anisotropic phonon system in He II
title_full_unstemmed On the suprathermal distribution in an anisotropic phonon system in He II
title_short On the suprathermal distribution in an anisotropic phonon system in He II
title_sort on the suprathermal distribution in an anisotropic phonon system in he ii
topic Квантовые жидкости и квантовые кpисталлы
topic_facet Квантовые жидкости и квантовые кpисталлы
url https://nasplib.isofts.kiev.ua/handle/123456789/128779
work_keys_str_mv AT adamenkoin onthesuprathermaldistributioninananisotropicphononsysteminheii
AT nemchenkoke onthesuprathermaldistributioninananisotropicphononsysteminheii
AT gwyattaf onthesuprathermaldistributioninananisotropicphononsysteminheii