THE POSSIBILITY OF SECOND SOUND WAVES REGISTRATION IN ISOTOPIC ENRICHED DIAMOND SINGLE CRYSTAL
V. D. Khodusov1 and D. M. Litvinenko2
1Kharkiv V.N. Karazin National University, High-Technology Institute, 31 Kurchatov Ave, 61108, Kharkov, Ukraine; e-mail: [email protected]; 2Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; e-mail: [email protected]
Using the experimental data of diamond thermal conductivity we obtain the data of normal and resistive proc- esses of phonon interaction in Callaway approach. As the result we obtain the concentration values of 13C in dia- mond single crystal, under which the isotropic scattering is weak, and the temperature range of second sound waves existence in diamond. PACS: 63.10.+a, 63.20.1s, 63.20.Kr
1. INTRODUCTION 2. EQUATIONS OF GAS DYNAMICS Modern technologies give us opportunity to obtain OF QUASI-PARTICLES WITH TAKING isotopic pure perfect single crystals. It’s been experi- INTO ACCOUNT EXTERNAL FIELDS mentally established that the thermal conductivity coef- In the kinetic theory the state of gas of quasi- ficient of such crystals has maximum in low tempera- particles is characterized by distribution function of ture range [1]. Especially it becomes apparent in dia- quasi-particles NN≡ j (pr,,t), which satisfies ki- mond crystals. This fact is caused by a special role netic equation which has a form of Boltzmann equation: played by normal processes (where the conservation law of quasi-impulse is present) in interaction between ∂∂∂ε ∂ +−g NN=() , (1) phonons. In perfect single crystals the propagation of ∂∂t rr∂∂p ст weakly damping second sound waves, similar to HeII ones, is possible. The most probable at that are the nor- where gg≡=( j) ∂ε ( j) ∂p is the group velocity of mal processes of phonon interaction comparatively to quasi-particles; εε≡ ()j pr,,t is the Hamiltonian of resistive ones (where the conservation law of quasi- () impulse fails). The major contribution to second sound quasi-particle, which is equal to its local energy; N ()ст waves damping makes isotopic phonon scattering. For is the collision integral of quasi-particles, which takes the first time second sound waves were registered in into account the processes of collision, merge, fission NaF [2] single crystal and solid 4 He [3], where the and radiation of quasi-particles. isotopic damping is absent at the temperature range of If we designate τN as a character time of phonon in- 10…20 K. teraction concerned with N-processes (where phonon Using the data of normal and resistive processes of energy and quasi-impulse conservation laws are accom- phonon interaction (umklapp processes, phonon scat- plished), and τR as a character time of resistive proc- tering on impurities, isotopes and on bounds of sam- esses of phonon interaction (where these laws fail), the ple), obtained while the description of experimental condition of predomination of normal processes is writ- data of diamond thermal conductivity coefficient in ten as 13 Callaway model, the concentration values of C are ττNR<< . (2) obtained, under which the isotopic scattering is weak, If at some moment of time the system of quasi- the temperature range of second sound waves exis- particles moves out of its equilibrium state, then in time tence in diamond is established. Starting from 13 C τN quasi-local balance is established, which is character- isotope concentration equal to 0.1 % and less the sec- ized by distributive function N ()j , which turns to zero ond sound waves registration is possible in the tem- 0 the collision integral because of N-processes , and it is perature range near maximum of thermal conductivity written as : of diamond at T=104 K. This temperature is one order −1 ()j higher than the temperature of second sound waves ()j ε − ()pu N =−exp 1 , (3) registration in NaF and solid 4 He . 0 Т ()1+θ 0
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where u is drift velocity in gas of quasi-particles, 222 22 ΩΩ()−WkII −20iWII kΓII = . (9) Θ= (-TT0)/T0 is relative temperature.
In the sate of gas of quasi-particles, which is close to From the dispersion equation (9) follows that WII is local statistic balance, the solution of equation (1) in the the phase velocity of SW and quantity ГII is the damp- gas dynamical approximation we search as: ing coefficient of SW. Expressions for WII и ГII have a simple structure for SSW phonon gas dynamics in re- ()jj()j () ()j ()j duced isotropic crystal model [4]: NN=+0 δ N, δ NN<< 0 , (4) () 1 WT= S2 Cρ 2 ; ()j II () where N0 is the local equilibrium distributive func- tion, which depends on gas dynamical quantities, and r 14 1 2 (j) Γ=II +ηζ+ + κk , (10) δN depends on its gradients. 22ρρ3 2C If we input thermodynamical potential F as: 0 2 2 44 3Vt 2π kTB 3 ()j where CS==3(ρβ = 2+) is FT=− ln 1+N, (5) 515 33 0 ∑ ()0 (2 + β )TVt = k, j obtained in reduced isotropic crystal model in low tem- perature range. By low temperature range we mean the then knowing it as a function of T, u, A , which satis- j range of temperature where Θ>D T >1 ; fies thermodynamical identity: Θ=D2π =VktBa is the Debye temperature, Vl, Vt are dF00=−S dT −Pdu, (6) the velocities of transverse and longitudinal phonons, β = VV, a is the lattice parameter. Diamond single we obtain impulse density P , heat capacity C and en- tl crystal has the highest Debye temperature tropy S densities, components of density tensor of (Θ≈1845K ) among other crystals, that’s why the quasi-particles ρ : D ij temperature in order of 100 K and lower can be consid- ∂F ∂F P =− 0 ; S =− 0 ; ered as low for diamond. 0 If to introduce diffusive lengths of free path and re- ∂u TA, ∂T u,A lated with them diffusive times by following expres- ∂S ∂∂P2F sions: CT= 0 ; ρ ==i −0 . (7) il η ζ κ ρ ∂T u,A ∂∂uuli⋅∂ul τ = ; τ = ; τ = ; τ = , (11) TA, η 2 ζ 2 κ 2 R ρWII ρWII CWII r Applying the standard procedure [4] from kinetic Г can be written in the following way: equation (1) it is possible to obtain a system of gas dy- II namical equations, that describe an isotropic gas of 114 Γ= + ττ+ Ω2+τΩ2. (12) quasi-particles in linear approximation on drift velocity II ηζ κ 23τ R u in reduced isotropic crystal model: The condition of weak damping SSW existence η Pu+∇ST θη=−r +∆u+ζ+ ⋅∇divu; ( Γ<<Ω) leads to next condition imposed on fre- 3 ΙΙ quency Ω , known as SSW existence “window” [4]: CSθ+ divu =κ∆θ, (8) min{ννηκ, ζ ,ν} >> Ω >>νR , (13) where P=ρ ⋅u, ρ is the density of quasi-particles number, r is the external friction coefficient, caused by where the collision frequencies νi are related to diffu- processes of phonon interaction without quasi-impulse sive times τi by the expression ν=ii1 τ. conservation (umklapp processes, phonon scattering on The calculation of kinetic coefficients of phonon gas impurities, isotopes and sample boundaries), η, ζ are dynamics in the model of reduced isotropic crystal rela- the first and second viscosity coefficients, κ is the tively to elasticity coefficient in [4] shows that basic thermal conductivity coefficient. role plays phonon viscosity. The number, similar to Cη Prantl number in gas theory Pr = , for phonon gas 3. PROBLEM SETTING AND ITS SOLUTION ρκ IN REDUCED ISOTROPIC CRYSTAL for most of crystals is equal to 102 with a great degree AND CALLAWAY MODELS of accuracy. The secondary waves (SW) propagation in Bose gas Further on we’ll take into account that in reduced of quasi-particles in solids was studied in work [4] bas- isotropic crystal model energy and quasi-impulse con- ing on gas dynamics equations of quasi-particles, espe- servation laws allow tt+↔l+g and l+↔t l+g cially SSW in phonon gas in dielectric crystals. umklapp processes. The solution of equations (9) for all variables we’ll Using the Callaway [5] model for description of ex- search as exp it(Ω−kr ) with frequency Ω and wave perimental data of thermal conductivity of diamond in low temperature range [6], we can calculate frequencies vector k . From the compatibility condition of system of phonon collisions, caused by normal and umklapp of equations we obtain the dispersion equation for sec- processes and also by scattering on isotopes and sample ondary waves (SW) [4]: boundaries:
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kAB 4 esses. In high temperature range (higher than νζN = T 5! (5) ; T≈ 80 K) umklapp processes predominate, in tem- 2πρ= a6 perature range lower than T≈ 80 K normal processes 2 ka43 ∆M ν= cTB 4 8!ζ(8) ; (14) and processes of phonon scattering on impurities and iso i 43 boundaries predominate. 4π = vs M C kB2 − v νζ= B Te3 T 6! (6) ; νζ= s 4! (4) , U 2 b (2π =) vs D N where c = i is the isotope concentration, ∆M is i N the difference in isotope and the basic atom masses, M is the mass of basic atom, ζ(n) is the Riemann zeta- function, ρ is the diamond crystal density, AK=⋅7, 2 10−−11 3 , B =⋅1, 5 10 −12 cm/K, CK= 670 , D = 0,33 cm are phenomenological constants obtained from experiment [6]. −4 −4 Solid line – ci =⋅510 , dash-dot line – ci = 10 , −5 −6 4. RESULTS ANALYSIS AND CONCLUSION dash line – ci = 10 , dot line – ci = 10 For making numerical estimations and obtaining the Thus the carried out analysis gives us reason to af- condition of weakly damping second sound waves exis- firm that SSW in diamond crystal can be registered at tence in diamond single crystal we use the following temperatures in the range of TK≈ 100 , if 13 C isotope data: −4 2 6 concentration is lower than 10 . ρ=3, 512 g/cm , β = 0,68 , Vl =1.81⋅10 cm/s, V =1.235 ⋅106 cm/s, W =0.76 ⋅106 cm/s, t II REFERENCES where V , V and W were calculated in reduced iso- l t II 1. A.P. Jernov, A.V. Inushkin. Kinetic coefficients in tropic crystal model [4]. Using the Callaway model we isotopically disordered crystals //Phisics – Uspekhi. examine inequality (13) to obtain the “window” of SSW 2002, v. 172, N5, p. 573-599. existence or at least its low limit and the isotope 2. H.E. Jackson, C.T. Walker. Thermal conductivity, 13 concentrations (the main isotope is C , ∆M =1) of second sound and phonon-phonon interaction in such “window”. Substituting expressions (14) into NaF // Phys. Rev. B. 1971, v. 3, N 4, p. 1428-1439. inequality (13) we obtain SSW existence “window” in 3. С.С. Ackerman, B. Bertman, H.A. Fairbank, diamond single crystal. It’s convenient to introduce P.A. Guyer. Second sound in solid helium //Phys. function f(T) = νNU(ν +νiso+νb), which value range Rev. Lett. 1996, v. 16, p. 789-801. f(T) >> 1 will provide the SSW existence “window”. 4. A.I. Akhiezer, V.F. Aleksin, V.D. Khodusov. Gas The following figure gives temperature dependence of dynamics of quasiparticles //Low. Temp. Phys. 1994, function f(T) for different concentrations of 13 C iso- v. 20, p. 939-970. 5. R. Berman. Thermal conduction in solids. M.: tope: “Mir”, 1979, 286 р. (in Russian). In the figure we can see that diamond has the SSW 6. Lanhua Wei, P.K. Kuo, R.L. Thomas, T.R. Antony, existence “window” in the region of T≈ 80 K, start- W.F. Banholzer. Thermal conductivity of isotopi- −4 ing from ci ≈ 10 and less. Such form of curves is cally modified single crystal diamond provided by competition of normal and resistive proc //Phys.Rev.Lett. 1993, v. 70, p. 3764-3767.
ВОЗМОЖНОСТЬ НАБЛЮДЕНИЯ ВОЛН ВТОРОГО ЗВУКА В ИЗОТОПИЧЕСКИ ОБОГАЩЁННОМ МОНОКРИСТАЛЛЕ АЛМАЗА В. Д. Ходусов, Д.М. Литвиненко Используя экспериментальные данные по теплопроводности алмаза, в модели Калавэя получена информация о нор- мальных и резистивных процессах фононного рассеяния. Как следствие получены концентрации примеси изотопа 13C в монокристалле алмаза, при которых изотопическое рассеяние фононов становится незначительным, а также промежу- ток температур, в котором существуют волны второго звука.
МОЖЛИВІСТЬ СПОСТЕРЕЖЕННЯ ХВИЛЬ ДРУГОГО ЗВУКУ В ІЗОТОПІЧНО ЗБАГАЧЕНОМУ МОНОКРИСТАЛІ АЛМАЗУ В. Д. Ходусов, Д.М. Литвиненко Використовуючи експериментальні дані теплопровідності алмазу, в моделі Калавея здобуто інформацію про норма- льні та резистивні процеси фононного розсіяння. Як наслідок, здобуто величини концентрацій ізотопічних домішок 13C в монокристалі алмазу, за яких ізотопічне розсіяння фононів стає незначним, а також проміжок температур, в якому існують хвилі другого звуку.
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