Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes

An exact expression for the diffusion time which depends on the interaction rates for particles of not only different, but also of the same species, has been derived from the system of kinetic equations. The result is valid for particles with arbitrary statistics and energy-momentum relations. The...

Full description

Saved in:

Bibliographic Details
Date:	1999
Main Authors:	Adamenko, I.N., Nemchenko, K.E., Zhukov, A.V., George, T.F., Pandey, L.N., Um, Chung-In
Format:	Article
Language:	English
Published:	Інститут фізики конденсованих систем НАН України 1999
Series:	Condensed Matter Physics
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/119920
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes / I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, T.F. George, L.N. Pandey, Chung-In Um // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 63-74. — Бібліогр.: 23 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-119920
record_format	dspace
spelling	nasplib_isofts_kiev_ua-123456789-1199202025-02-23T18:53:42Z Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes Дифузія у двокомпонентних системах квазічастинок рідких та твердих сумішей ізотопів гелію Adamenko, I.N. Nemchenko, K.E. Zhukov, A.V. George, T.F. Pandey, L.N. Um, Chung-In An exact expression for the diffusion time which depends on the interaction rates for particles of not only different, but also of the same species, has been derived from the system of kinetic equations. The result is valid for particles with arbitrary statistics and energy-momentum relations. The derived general relations are valid for investigating diffusion in liquid and solid ³He-⁴He mixtures. The contribution of interaction between quasiparticles of the same type to the diffusion coefﬁcient and effective thermal conductivity of superﬂuid solutions is analyzed. The calculated values are compared with experimental data. The calculated diffusion coefﬁcient of ³He-⁴He solid solutions differs from the previous theoretical results. A comparison of the obtained diffusion coefﬁcient with experimental data makes it possible to determine the numerical value of the energy band width for impurity quasiparticles. Виходячи з системи кiнетичних рiвнянь для компонент сумiшi отримано точний вираз для дифузiйного часу, що залежить вiд взаємодiї частинок не тiльки рiзних, а й одного типу. Отриманий результ справедливий для частинок з довiльною дисперсiєю і статистикою. Одержаний точний вираз використовується для дослiдження дифузiї у рiдких та твердих квантових розчинах ³He-⁴He. Проаналiзовано внесок взаємодiї мiж квазiчастинками однакових типiв у коефiцiєнт дифузiї та теплопровiдностi надплинних розчинiв. Обчисленi результати порiвнюються з iснуючими експериментальними даними. Отриманий коефiцiєнт дифузiї для твердих квантових розчинiв ³He-⁴He iстотно вiдрiзняється від попереднiх результатiв теорiї. Порiвняння цього результату з експериментальними даними дозволило дiстати числове значення для енергетичної зони квазiчастинок домiшки. 1999 Article Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes / I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, T.F. George, L.N. Pandey, Chung-In Um // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 63-74. — Бібліогр.: 23 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.63 PACS: 51.20.+d, 67.80.Mg https://nasplib.isofts.kiev.ua/handle/123456789/119920 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	An exact expression for the diffusion time which depends on the interaction rates for particles of not only different, but also of the same species, has been derived from the system of kinetic equations. The result is valid for particles with arbitrary statistics and energy-momentum relations. The derived general relations are valid for investigating diffusion in liquid and solid ³He-⁴He mixtures. The contribution of interaction between quasiparticles of the same type to the diffusion coefﬁcient and effective thermal conductivity of superﬂuid solutions is analyzed. The calculated values are compared with experimental data. The calculated diffusion coefﬁcient of ³He-⁴He solid solutions differs from the previous theoretical results. A comparison of the obtained diffusion coefﬁcient with experimental data makes it possible to determine the numerical value of the energy band width for impurity quasiparticles.
format	Article
author	Adamenko, I.N. Nemchenko, K.E. Zhukov, A.V. George, T.F. Pandey, L.N. Um, Chung-In
spellingShingle	Adamenko, I.N. Nemchenko, K.E. Zhukov, A.V. George, T.F. Pandey, L.N. Um, Chung-In Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes Condensed Matter Physics
author_facet	Adamenko, I.N. Nemchenko, K.E. Zhukov, A.V. George, T.F. Pandey, L.N. Um, Chung-In
author_sort	Adamenko, I.N.
title	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
title_short	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
title_full	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
title_fullStr	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
title_full_unstemmed	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
title_sort	diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1999
url	https://nasplib.isofts.kiev.ua/handle/123456789/119920
citation_txt	Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes / I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, T.F. George, L.N. Pandey, Chung-In Um // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 63-74. — Бібліогр.: 23 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT adamenkoin diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT nemchenkoke diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT zhukovav diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT georgetf diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT pandeyln diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT umchungin diffusionintwocomponentquasiparticlesystemsofliquidandsolidmixturesofheliumisotopes AT adamenkoin difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû AT nemchenkoke difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû AT zhukovav difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû AT georgetf difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû AT pandeyln difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû AT umchungin difuzíâudvokomponentnihsistemahkvazíčastinokrídkihtatverdihsumíšejízotopívgelíû
first_indexed	2025-11-24T12:23:57Z
last_indexed	2025-11-24T12:23:57Z
_version_	1849674471523221504
fulltext	Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 63–74 Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes I.N.Adamenko 1 , K.E.Nemchenko 1 , A.V.Zhukov 1 , Thomas F. George 2 , Lakshmi N. Pandey 3 , Chung-In Um 4 1 Department of Physics and Technology, Kharkiv State University, 4 Svobody Sq., Kharkiv, 310077, Ukraine 2 Office of Chancellor, Departments of Chemistry, Physics and Astronomy, University of Wisconsin-Stevens Point, Wisconsin 54481-3897, U.S.A. 3 Departments of Chemistry and Physics, Washington State University, Pullman, Washington 99164-4630, U.S.A. 4 Department of Physics, College of Science, Korea University, Seoul 136-701, Korea Received December 5, 1997 An exact expression for the diffusion time which depends on the interac- tion rates for particles of not only different, but also of the same species, has been derived from the system of kinetic equations. The result is valid for particles with arbitrary statistics and energy-momentum relations. The derived general relations are valid for investigating diffusion in liquid and solid 3He-4He mixtures. The contribution of interaction between quasipar- ticles of the same type to the diffusion coefficient and effective thermal conductivity of superfluid solutions is analyzed. The calculated values are compared with experimental data. The calculated diffusion coefficient of 3He-4He solid solutions differs from the previous theoretical results. A com- parison of the obtained diffusion coefficient with experimental data makes it possible to determine the numerical value of the energy band width for impurity quasiparticles. Key words: diffusion, thermal conductivity, helium, quantum liquid, quantum solid, superfluids, 3He-4He mixtures, phonons, rotons, impuritons. PACS: 51.20.+d, 67.80.Mg 1. Introduction Investigation of diffusion processes in condensed media, in which a quasiparticle description is valid, is one of the most important problems of modern classical c© I.N.Adamenko, K.E.Nemchenko, A.V.Zhukov, T.F.George, L.N.Pandey, Chung-In Um 63 I.N.Adamenko et al. and quantum kinetics. The present paper deals with the diffusion of impurity excitations in weakly-concentrated solutions of 3He in superfluid and solid 4He. The general result obtained for a two-component gaseous system with arbitrary statistics and the dispersion law makes it possible to describe the quasiparticle systems in some limiting cases of interest. For every classical and quantum two-component gaseous system known to us, the diffusion coefficients can be written in the form D = u2DτD, (1) where uD is a typical velocity, whose analytical expression is determined by the dispersion law and statistics of the particles, and τD is a typical diffusion time. The solution of a set of two linearized kinetic equations for a mixture of α-type and β-type gases with any dispersion law and chemical potential takes the form (see [1]) τD = −〈ϕD \| (Ŝ + Î)−1 \| ϕD〉, (2) where Ŝ and Î are the matrices in the 2D space of component momenta. The matrix Ŝ = ( Jαα 0 0 Jββ ) (3) includes the operators of collisions between particles of the same type, and the matrix Î = ( Jαβ Jαβ Jβα Jβα ) (4) consists of the operators of collisions between particles of different types. The col- lision operators Jkl(k, l = α, β) are linearized integrals of the collision of k−type particles with l−type particles and in a usual way can be expressed by the tran- sition probability density function. 2. Diffusion time The time τD can be expressed as a scalar product determined in the following way: 〈ψ \| χ〉 = ∑ k=α,β 1〈ψk \| χk〉1 = − ∑ k=α,β ∫ ψ∗ kχkf ′ 0kdΓk. (5) Here and below, the subscript 1 on a bra- or a ket-vector denotes a 1D vector, and f ′ 0k is a derivative of the local equilibrium distribution function with respect to energy. The vector \|ϕD〉 that determines τD is normalized with respect to the scalar product (5) as \| ϕD〉 = (ρραρβ) − 1 2 ∣ ∣ ∣ ∣ ρβpαz −ραpβz 〉 , (6) 64 Diffusion in 3He-4He mixtures where pkz is the z-th component of the momentum (k = α, β), and ρk = 1〈pkz\|pkz〉1 (7) is the normal density of the k-th component, and ρ = ρα + ρβ is the total density of the mixture. To obtain an exact expression for the matrix elements (2), one should introduce a complete set of orthonormal two-component vectors \|ϕn〉 (n = 1, 2, ..). The first vector of this set corresponds to the total momentum of the two-component system of quasiparticles, \| ϕ1〉 = ρ−1 ∣ ∣ ∣ ∣ pαz pβz 〉 , (8) and the second one should be taken as \| ϕ2〉 =\| ϕD〉. (9) The remaining vectors can be constructed by using the standard procedure (see, e.g., [2]) and the definition of the scalar product (5). By constructing a complete set of vectors in this way, the expression for τD τD = − { [Ŝ + Î] −1 } 22 (10) can be rewritten as τD = − { I22 − ∞ ∑ n,n′=3 I2n[(Ĩ + S̃) −1 ]nn′In′2 }−1 . (11) Here the square matrices Ĩ and S̃ include the matrix elements (Ĩ)nn′ = Inn′, (S̃)nn′ = Snn′, (12) where Inn′ = 〈ϕn\|Î\|ϕ ′ n〉, Snn′ = 〈ϕn\|Ŝ\|ϕ ′ n〉. (13) The matrices Ĩ and S̃ are infinite and nondiagonal. Therefore, the exact solution (11) does not allow us to obtain an explicit analytical expression for τD. However, the solution (11) makes it possible to consider various limiting cases, to find min- imal and maximal values of τD, to obtain correct interpolation formulae and to carry out computer calculations for various physical systems. So, in the case of fast equilibrium established between particles of the same type, i.e. when the inequality Snn′≫Inn′ (14) is valid, one can obtain from equation (11) τD = τDmin = − 1 I22 = (τ (0) αβ −1 + τ (0) βα −1 ) −1 , (15) 65 I.N.Adamenko et al. where τ (0) kl −1 = −〈Jkl〉k; k, l = α, β; k 6=l. (16) Here and below, the normalized average of the arbitrary operator L̂ with respect to the quasiparticle momentum is denoted as 〈L̂〉k = ρ−1 k 1〈pkz\|L̂\|pkz〉1. (17) According to the momentum conservation law and definition (16), τ (0) αβ −1 = ρβ ρα τ (0) βα −1 . (18) Using the fact that the operators Ĩ and S̃ are Hermitian and defined as negative, it can be shown that τD>τDmin . (19) In the opposite limiting case of the slow establishing of equilibrium between iden- tical particles (Sββ′ ≪ Iββ′), τD reaches its maximum value τDmax = 〈ϕ2\|Î −1\|ϕ2〉. (20) When the density of the α−component is relatively low (ρα ≪ ρβ) and the relaxation time in the α−component is great (Jαα→0), from (11) we obtain τD = τ (∞) αβ , (21) where τ (∞) αβ = −〈J−1 αβ 〉α. (22) It should be noted that (16) is the average of the rate, but (22) is the average of the time, and one can prove that for arbitrary momentum dependence of Jαβ, the time (22) is greater than the time defined by (16). For a definite physical system, the rate τ (0) αβ −1 can be easily calculated, but to calculate τ (∞) αβ −1 (22) one must find the inverse operator to the integral operator Jαβ, which can be done only by using some approximations. The exact expression (11) and the limiting formulae (15), (20) and (22) make it possible to propose the correct relaxation-time approximation for a two-component system: Ŝ =     −t−1 αα + t−1 αα\|pαz〉1 τ (0) αα ρα 1〈pαz\|t −1 αα 0 0 −t−1 ββ + t−1 ββ \|pβz〉1 τ (0) ββ ρβ 1〈pβz\|t −1 ββ     , Î =    −t−1 αβ t−1 αβ \|pαz〉1ρ −1 α τ (0) αβ 1〈pβz\|t −1 βα t−1 βα\|pβz〉1ρ −1 β τ (0) βα 1〈pαz\|t −1 αβ −t−1 βα    . (23) 66 Diffusion in 3He-4He mixtures This approximation satisfies the conservation of the z-th component of the total momentum of the system: Î \|ϕ1〉 = 0 . (24) Here t−1 kl (pk) = νkl(pk) 〈νkl(pk)〉k τ (0) kl −1 (k 6=l), (25) and νkl is a transport scattering rate that is determined in a usual way from collision integrals. The model (23) can be used for the calculation of τD to give τD = τDmin + ρβ ρ (ταβ − τ (0) αβ ) + ρα ρ (τβα − τ (0) βα ), (26) where τkl = 〈Rk〉+ 〈Rkt −1 kk 〉 2 k〈Rkt −1 kk t −1 kl 〉 −1 k , (27) Rk = (t−1 kk + t−1 kl ) −1 . (28) The time τkl in (26) depends on the relaxation rates t−1 kk of the k-particles. Note, the expression (26) obtained in the limiting cases gives not only the formulae (16), (20) and (22) of this paper, but also the results of other theoretical investigations [3] and, in particular, the well-known Callaway formula [4]. 3. Thermal conductivity of quasiparticle systems of dilute so- lutions of 3 He in superfluid 4 He Because of the nature of thermal excitations, diffusion in such mixtures defines thermal conductivity of the matter that includes these thermal excitations [5]. Consider a superfluid mixture of helium isotopes. The kinetic properties of the mixtures are determined by a set of quasiparticles: phonons and rotons (thermal excitations of He II) and impuritons (3He quasiparticles). It is useful to separate three temperature regions which differ one from another by the types of physical processes which govern the thermal conductivity of the mixture. At low temperatures (T < 0.6 K), when the roton contribution can be ne- glected, the thermal conductivity of the mixtures is determined by diffusion in the phonon-impuriton system: κeff = Diph ( Sph ρph + ρi ρph )2 1 n3 + κ3 . (29) Here Diph = 4 9 1〈εi\|1〉 2 1 1〈1\|1〉1 ρph ρph + ρi 1 ρi τ (iph) D (30) is a coefficient of diffusion of impuritons in a phonon gas, Sph is the entropy of the phonon gas, n3 is the number density of impuritons, κ3 is a partial coefficient of thermal conductivity, and εi = p2i /2mi is the kinetic energy of impuritons. The 67 I.N.Adamenko et al. time τ (iph) D is defined by (26) where the subscripts α, β refer to ph, i, respectively. For nondegenerate mixtures one has 2 3 1〈εi\|1〉1 1〈1\|1〉1 = T. (31) Figure 1 shows the calculated (from equation (29)) and measured in ([6]) values of the effective coefficient of thermal conductivity for a mixture with the concen- tration x = 1.39 · 10−4. The contribution of κ3 is negligible. At this concentration τ (iph) D turns out to be equal to τ (iph) Dmin, which corresponds to fast relaxation in the phonon system of this mixture. 0.5 1.0 1.5 2.0 10 4 10 5 10 6 κ , er g/ (c m s K ) T , K Figure 1. Temperature dependence of the effective thermal conductivity of a mixture with the concentration x = 1.39 · 10−4, showing the contributions to the effective thermal conductivity from the diffusion in the impuriton-phonon (curve 1), roton-phonon (curve 2) and impuriton-roton (curve 3) systems. Curve 4 corresponds to the effective thermal conductivity calculated by taking into account contributions from all the quasiparticles. The experimental data obtained in [6] are represented by �. Figure 2 gives the observed ([6] and [7]) and calculated values of the effective coefficient of the thermal conductivity of a mixture with different concentrations. The figure shows that in the temperature region considered, with the increase in concentration the coefficient κeff with τ (iph) D = τ (iph) Dmin (dashed lines) differs from the values obtained from formula (26), which takes into account the finite values of the phonon-phonon relaxation times. The values calculated by formula (26) 68 Diffusion in 3He-4He mixtures are presented by solid curves and are in better agreement with the experimental data. The results for the considered temperature region correspond to the calcu- lations of [3, 8, 9]. In the region of intermediate temperature (0.7 K < T < 1 K), when the concentration of impuritons is small, the effective thermal conductivity is determined mainly by diffusion in the gas of thermal excitations. 0.5 1.0 1.5 2.0 10 3 10 4 10 5 10 6 κ , er g/ (c m s K ) T , K Figure 2. Temperature dependence of the effective thermal conductivity of the mixtures with different concentrations: x = 1.39 · 10−4 (curve 1), x = 1.32 · 10−3 (curve 2) and x = 1.36 ·10−2 (curve 3). The results of the calculations taking into account the contributions from all the types of quasiparticles are depicted by solid and dashed curves. The latter correspond to calculations with the assumption of instantaneous relaxation in the mixture components. The experimental results obtained in [6] and [7] are represented by � and ♦. Diffusion processes in a phonon-roton gas were first considered in [10]. Later, in [1], these processes were shown to be the reason for a thermal transfer caused by a difference of dispersion laws of phonons and rotons. The calculation from relation (26) is analogous to that made in [1] and gives κeff≈κ (rph) D = T−1 ρphρr ρph + ρr {SphT ρph − SrT ρr }2 τ (rph) D . (32) Here Sr and ρr are the entropy and normal density of rotons, and τ (rph) D is given by equation (26) in which the subscripts α, β should be substituted with r, ph, respectively. Expression (32) gives the result of [1] when the times tphph and trr are equal to zero. Curve 3 in figure 1 presents the calculations from equation (32) and 69 I.N.Adamenko et al. shows the existence of a wide enough temperature range, such that κD should be taken into account. At high enough temperatures (T > 1 K) the kinetic properties of superfluid mixtures are governed by rotons and impuritons. According to [11], in this tem- perature region, the coefficient of effective thermal conductivity can be written as κeff = Dir ( Sr ρr + ρi ρr )2 n−1 3 + κ3 + κr , (33) where κr is thermal conductivity of rotons. Using the relations (1) and (26), the diffusion coefficient of impuritons in a roton gas is given as Dir = ρr ρr + ρi T mi τ (ir) D . (34) Here τ (ir) D is given by relation (26), where the subscripts α, β should be substituted by r, i, respectively. The relation (34) gives the result of [1] if the times tii and trr are equal to zero. The rate of the roton-impuriton interaction can be written in the form [12] t−1 ir (pi) = A2nr ( ∫ ∞ 0 exp { − µv2r 2T } dvr ) −1 × ∫ ∞ 0 exp { − µv2r 2T } dvr 1 2 ∫ 1 −1 sin2θ (µ 2 (vr − vi) 2 + p2i 2mi sin2θ ) 1 2 d(cos θ) (35) where A is a scattering amplitude. The limiting relations for the relaxation rates refer to the absence of equilibrium in the impuritons (tii≫tir) and fast relaxation in the roton gas (trr≪tri). Under these conditions the general expression (26) gives τ (ir) D = τDmax (ir) = 〈tir〉i . (36) The results calculated from equation (34) for the effective thermal conductivity for a mixture with x = 1 · 10−4 are presented in figure 1 (curve 1). Here the contributions of the third and fourth terms in the right-hand side of equation (33) can be neglected. The account of the finite values of tii and trr makes the calculated values greater to the order of 10%, improving the agreement between theory and experiment. In figure 2, the dashed curves present calculations with τD = τDmin, and the solid curves correspond to τD calculated from equation (26). 4. Diffusion in solid 3 He- 4 He mixtures The methods used in the previous section allow us to calculate the contribution of the phonon-impuriton interaction to spin diffusion of impurities in solid 3He-4He mixtures as D (s) iph = 2 3 1〈εi\|1〉1 1〈1\|1〉1 1 mi τ (0) phi . (37) 70 Diffusion in 3He-4He mixtures 0.5 1.0 1.5 2.0 10 -7 10 -6 D s , c m /s 1/T , 1/K Figure 3. Dependence of the spin diffusion coefficient of solid 3He-4He mixtures on the reciprocal temperature for the concentration x = 6 · 10−5. � stands for the experimental data of [20–22], and the curve corresponds to the calculations by equations (39–42). Calculating the scalar products in (37), one should take integrals in the limit of the impurity energy band ∆ ≪ T , thus giving D (s) iph = 2∆ 5mi τ (0) iph . (38) This result differs from the diffusion coefficient in the phonon-impuriton system of a liquid 3He-4He mixture, especially by its temperature dependence. To examine this, we rewrite equation (38) by using the definition (18): D (s) iph = 4 25 ∆2 T ni ρph τ (0) phi . (39) The time τ (0) phi has a typical Rayleigh scattering temperature dependence T−4, and ρph is proportional to T 4, so that Ds iph ∼ T−9. Such a dependence was first obtained in [13] from phenomenological arguments. According to equation (38), this depen- dence is completely determined by τ (0) iph, and from equation (39) it follows that eight powers of temperature deal with phonons (normal density and scattering rate), and one power deals with the normal density of impuritons. The expressions (38) and (39) differ from the results of [13, 14-18]. Relation (39) includes param- eters which can be directly determined from independent experiments on thermal 71 I.N.Adamenko et al. conductivity in solid 3He-4He mixtures [19]. The numerical values of ∆ can be obtained from the experimental data [20–22] for a mixture with x = 6×10−5. The expression for spin diffusion can be written in the form Ds = (D (s) iph −1 +D−1 ii ) −1 , (40) where Dii is independent of the temperature contribution of the impuriton- impu- riton interaction to spin diffusion, which according to [23] is Dii = 2.67 · 10−7 cm2/s. (41) Figure 3 presents the experimental [18] and calculated (by formula (41)) values of the diffusion coefficients. An agreement between the calculated and observed data is achieved with ∆ = 3.5 · 10−4 K (42) The obtained value (42) refines the results of [17, 18] which give the order of magnitude only. References 1. Adamenko I.N., Nemchenko K.E. Diffusion of quasiparticles of 3He-4He mixtures and classical particles. // Fiz. Nizk. Temp., 1995, vol. 21, p. 498–508 (in Russian). 2. Fertziger J.H., Kaper H.G. Mathematical Theory of Transport Processes in Gases. Amsterdam, North-Holland, 1972. 3. Adamenko I.N., Rudavskii E.Ya., Tsyganok V.I., Chagovets V.K. New relaxation pro- cess in the phonon-impuriton system. // Pis’ma Zh. Eksp. Teor. Fiz., 1984, vol. 33, p. 404–407 (in Russian). 4. Callaway J. Model for lattice thermal conductivity at low temperatures. // Phys. Rev., 1959, vol. 113, p. 1046–1051. 5. Landau L.D., Pomeranchuk I.Ya. About the motion of outside particles in helium II. // Dokl. Ak. Nauk SSSR, 1948, vol. 59, p. 669–671 (in Russian). 6. Ptukha T.P. Thermal conductivity and diffusion of dilute 3He-4He solutions in the temperature range of from λ-point to 0.6 K. // Zh. Eksp. Teor. Fiz., 1961, vol. 40, p. 1583–1593 (in Russian). 7. Abel W.B., Wheatley J.C. Experimental thermal conductivity of two dilute solutions of 3He in superfluid 4He. // Phys. Rev. Lett., 1968, vol. 21, p. 1231–1234. 8. Adamenko I.N., Tsyganok V.I. Change of phonon kinetics due to dispersion law and impurities. // Zh. Eksp. Teor. Phys., 1984, vol. 87, p. 865–877 (in Russian). 9. Adamenko I.N., Tsyganok V.I. Collective modes and dissipative coefficients in solu- tions of 3He-4He quantum liquids. // Zh. Eksp. Teor. Phys., 1985, vol. 88, p. 1641–1655 (in Russian). 10. Um C.I., Jun C., Shin H., George T.F. Quasi-diffusion between phonon and roton gases in two- and three-dimentional liquid helium. // J. Low Temp. Phys., 1990, vol. 78, p. 51–61. 11. Khalatnikov I.M. The Theory of Superfluidity. Nauka, Moskow, 1971 (in Russian). 72 Diffusion in 3He-4He mixtures 12. Adamenko I.N., Zhukov A.V., Nemchenko K.E. Contribution of the interaction of identical particles to the diffusion of classical and quantum mixtures. // Fiz. Nizk. Temp., 1996, vol. 22, p. 1470–1473 (in Russian). 13. Andreev A.F., Lifshitz I.M. Quantum theory of defects in crystals. // Zh. Eksp. Teor. Fiz., 1969, vol. 56, p. 2057–2058 (in Russian). 14. Pushkarov I. To the theory of the motion of impurities in solid 4He. // Pis’ma Zh. Eksp. Teor. Fiz., 1974, vol. 19, p. 751–752 (in Russian). 15. Kagan Yu., Maksimov L.A. Theory of particle transport in extrimely narrow bands. // Zh. Eksp. Teor. Fiz., 1973, vol. 65, p. 622–639 (in Russian). 16. Kagan Yu., Klinger M.I. Theory of quantum diffusion of atoms in crystals. // J. Phys. C 1974, vol. 7, p. 2791–2808. 17. Slusarev V.A., Strzhemechnyi M.A., Burachovich I.A. Quantum diffusion of 3He in solid 4He. I. The role of impurity interaction anisotropy. // Fiz. Nizk. Temp., 1977, vol. 3, p. 1229–1240 (in Russian). 18. Slusarev V.A., Strzhemechnyi M.A., Burachovich I.A. Quantum diffusion of 3He in solid 4He. II. Tunneling-jumping approximation. // Fiz. Nizk. Temp., 1978, vol. 4, p. 698–705 (in Russian). 19. Golub A.A., Svatko S.V. Density dependence of the thermal conductivity of HCP 3He in 4He solutions. // Fiz. Nizk. Temp., 1981, vol. 7, p. 970–976 (in Russian). 20. Grigor’ev V.N., Esel’son B.N., Mikheev V.A. New peculiarities of quantum diffusion of 3He in solid helium. // Pis’ma Zh. Eksp. Teor. Fiz., 1973, vol. 18, p. 289–291 (in Russian). 21. Grigor’ev V.N., Esel’son B.N., Mikheev V.A., et al. 3He impurity excitations in solid 4He. // J. Low Temp. Phys., 1973, vol. 13, p. 65–86. 22. Grigor’ev V.N, Esel’son B.N., Mikheev V.A. Diffusion of 3He in BCC- and HCP-phases of solid solutions of helium isotopes. // Zh. Eksp. Teor. Fiz., 1974, vol. 66, p. 321–329 (in Russian). 23. Esel’son B.N., Grigor’ev V.N., Ivonsov V.G, et al. Solutions of Quantum Liquids 3He- 4He. Nauka, Moscow, 1973 (in Russian). 73 I.N.Adamenko et al. Дифузія у двокомпонентних системах квазічастинок рідких та твердих сумішей ізотопів гелію І.Н.Адаменко 1 , К.Е.Немченко 1 , А.В.Жуков 1 , Томас Ф. Джордж 2 , Лакшмі Н. Пенді 3 , Чанґ-Ін Ум 4 1 Харківський державний університет, 310077 Харків, пл. Свободи, 4 2 Університет Вісконсіна – Стівенс Пойнт, факультети хімії, фізики і астрономії, Вісконсін 54481–3897, США 3 Університет штату Вашінгтон, факультети хімії і фізики, Пуллмен, Вашінгтон 99164–4630, США 4 Університет Кореї, факультет фізики, коледж природничих наук, Сеул 136–701, Корея Отримано 5 грудня 1997 р. Виходячи з системи кiнетичних рiвнянь для компонент сумiшi отри- мано точний вираз для дифузiйного часу, що залежить вiд взаємодiї частинок не тiльки рiзних, а й одного типу. Отриманий результ спра- ведливий для частинок з довiльною дисперсiєю і статистикою. Одер- жаний точний вираз використовується для дослiдження дифузiї у рiд- ких та твердих квантових розчинах 3He-4He. Проаналiзовано вне- сок взаємодiї мiж квазiчастинками однакових типiв у коефiцiєнт ди- фузiї та теплопровiдностi надплинних розчинiв. Обчисленi результа- ти порiвнюються з iснуючими експериментальними даними. Отри- маний коефiцiєнт дифузiї для твердих квантових розчинiв 3He-4He iстотно вiдрiзняється від попереднiх результатiв теорiї. Порiвняння цього результату з експериментальними даними дозволило дiстати числове значення для енергетичної зони квазiчастинок домiшки. Ключові слова: дифузія, теплопровідність, гелій, квантова рідина, квантове тверде тіло, надплинність, суміш 3He-4He, фонони, ротони, домішки. PACS: 51.20.+d, 67.80.Mg 74

Diffusion in two-component quasiparticle systems of liquid and solid mixtures of helium isotopes

Institution

Similar Items