First sound velocity in liquid ⁴He

First sound velocity in liquid ⁴He

Based on the many-boson system structure factor, which takes into account three- and four-particle direct correlations, there was found the first sound velocity temperature behaviour in liquid ⁴He in the post-RPA approximation. The expression received for the sound velocity matches with the well-kno...

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1. Verfasser: Hryhorchak, O.I.
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Zitieren:First sound velocity in liquid ⁴He / O.I. Hryhorchak // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43001: 1–7. — Бібліогр.: 33 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-1552642025-02-09T17:34:35Z First sound velocity in liquid ⁴He Швидкiсть першого звуку в рiдкому ⁴He Hryhorchak, O.I. Based on the many-boson system structure factor, which takes into account three- and four-particle direct correlations, there was found the first sound velocity temperature behaviour in liquid ⁴He in the post-RPA approximation. The expression received for the sound velocity matches with the well-known results in both low and high temperature limits. The results of this paper can be used to analyze the contributions of three- and four-particle correlations into thermodynamic and structural features of liquid ⁴He . На основi виразу для структурного фактора багатобозонної системи з урахуванням прямих три- i чотиричастинкових кореляцiй знайдено температурну поведiнку швидкостi першого звуку в рiдкому ⁴He в пост-RPA наближеннi. У границi як низьких, так i високих температур отриманий вираз переходить у вже вiдомий. Результати можуть бути застосованi для аналiзу внескiв три- та чотиричастинкових кореляцiй у термодинамiчнi та структурнi функцiї рiдкого ⁴He. I am greatful to my supervisor Prof. I.O. Vakarchuk for valuable remarks and suggestions, as well as to my colleague Dr. V.S. Pastukhov for longlasting debates and discussions concerning the topic of this paper. 2015 Article First sound velocity in liquid ⁴He / O.I. Hryhorchak // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43001: 1–7. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 05.30.Jp, 67.25.-k, 43.35.+d, 62.60.+v,67.25.dt DOI:10.5488/CMP.18.43001 arXiv:1512.07742 https://nasplib.isofts.kiev.ua/handle/123456789/155264 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 Based on the many-boson system structure factor, which takes into account three- and four-particle direct correlations, there was found the first sound velocity temperature behaviour in liquid ⁴He in the post-RPA approximation. The expression received for the sound velocity matches with the well-known results in both low and high temperature limits. The results of this paper can be used to analyze the contributions of three- and four-particle correlations into thermodynamic and structural features of liquid ⁴He .
format Article
author Hryhorchak, O.I.
spellingShingle Hryhorchak, O.I.
First sound velocity in liquid ⁴He
Condensed Matter Physics
author_facet Hryhorchak, O.I.
author_sort Hryhorchak, O.I.
title First sound velocity in liquid ⁴He
title_short First sound velocity in liquid ⁴He
title_full First sound velocity in liquid ⁴He
title_fullStr First sound velocity in liquid ⁴He
title_full_unstemmed First sound velocity in liquid ⁴He
title_sort first sound velocity in liquid ⁴he
publisher Інститут фізики конденсованих систем НАН України
publishDate 2015
url https://nasplib.isofts.kiev.ua/handle/123456789/155264
citation_txt First sound velocity in liquid ⁴He / O.I. Hryhorchak // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43001: 1–7. — Бібліогр.: 33 назв. — англ.
series Condensed Matter Physics
work_keys_str_mv AT hryhorchakoi firstsoundvelocityinliquid4he
AT hryhorchakoi švidkistʹperšogozvukuvridkomu4he
first_indexed 2025-11-28T18:46:56Z
last_indexed 2025-11-28T18:46:56Z
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fulltext Condensed Matter Physics, 2015, Vol. 18, No 4, 43001: 1–7 DOI: 10.5488/CMP.18.43001 http://www.icmp.lviv.ua/journal First sound velocity in liquid 4 He O.I. Hryhorchak Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov St., 79005 Lviv, Ukraine Received April 1, 2015, in final form June 25, 2015 Based on the many-boson system structure factor, which takes into account three- and four-particle direct correlations, there was found the first sound velocity temperature behaviour in liquid 4He in the post-RPA approximation. The expression received for the sound velocity matches with the well-known results in both low and high temperature limits. The results of this paper can be used to analyze the contributions of three- and four-particle correlations into thermodynamic and structural features of liquid 4He. Key words: boson systems, liquid 4 He, first sound velocity PACS: 05.30.Jp, 67.25.-k, 43.35.+d, 62.60.+v,67.25.dt 1. Introduction The study of the sound velocity in Fermi and Bose systems has been attracting attention of researchers for a long time and still remains relevant today [1–9]. There are a lot of various many-particle systems, in which these investigations take place nowadays, for example: mixtures of 3He and 4He [10–12], 4He in aerogel [1, 2], solid 3He [3], plasma [13], trapped Bose gas [9], etc. Experimental study of the sound velocity temperature behavior in many-boson system began almost 80 years ago. Barton [14] and a year later Findlay with colleagues [15, 16] using ultrasonic method made measurements in liquid 4He under saturated vapor pressure (experiments were also carried out at higher pressures). The range of the research began from temperature T = 1.75K. The precision ofmeasurements was estimated less than 0.5 percent by authors. The curve of the then obtained temperature behavior perfectly conforms with the current results [17]. Hronevold [18] interpreted these experimental data. He proved that the ultrasonic waves in helium II are adiabatic despite their high frequency and the large thermal conductivity of helium. He also made attempts to explain a much smaller gap of the sound velocity at the λ-point transition compared to the value which stems from the Ehrenfest equation. Another issue that caused interest of researchers was the sound attenuation in the 4He liquid. Pellet and Skvayyer [19] sought it experimentally and showed that the experimental data are in a good agree- ment (at least for the temperature region from 3.2 K to 4.2 K) with theoretical results obtained from the classical formulae in assumption that the reasons of sound attenuation are only viscosity and heat losses. Later on, the attenuation and the sound velocity in 4He were calculated using the Landau-Khalatnikov kinetic equations and the phonon Boltzmann equation [20]. In paper [21], the authors compared the the- oretical results of the critical behavior of the sound propagation in liquid 4He (as well as in other liquids) near the gas-liquid critical point which was derived within the field-theoretic renormalization group for- malism with the experimental data. Another phenomenon that is studied nowadays is the first sound reflection in liquid 4He [22]. Theoretical study of the sound velocity was also carried out by the collective variables representative method, but only in the limit of low temperatures [23–27]. Particularly in papers [24, 26], correlation was found between the first sound velocity at absolute zero temperature and the Fourier coefficient of pair interparticle interaction energy in the “one sum over the wave vector” approximation. This result was obtained in two different ways. The first one went through finding longwave asymptotics of structural © O.I. Hryhorchak, 2015 43001-1 http://dx.doi.org/10.5488/CMP.18.43001 http://www.icmp.lviv.ua/journal O.I. Hryhorchak quantities, which take into account direct three- and four-particle correlations, and the second one was based on the well-known thermodynamic relation: mc2/N = ∂2E0/∂N 2, where c is the first sound ve- locity, N is the number of particles, m is the particle mass, E0 is ground state energy taken in the paircorrelations approximation. The above mentioned relations in higher approximation (“two sums over the wave vector”) were found in paper [28]. The objective of this paper is to find the temperature behavior of the first sound velocity in liquid 4He. We proceed from the exact relation that links the sound velocity with the longwave limit of the structure factor [29], as well as from the results which we obtained earlier for the pair structure factor in the “one sum over the wave vector” approximation that takes into account three- and four-particles correlations [30]. It should be noted that the calculation of the sound velocity through the structure factor in pair correlations approximation only leads to a constant value in pre-critical region and gives a light growth in the above-critical region (this behavior does not conform with the experimental data). The results of the sound velocity calculation in the post-RPA approximation show a fairly good agreement with the experimental data. The resulting expression for the first sound velocity in the limit of low temperatures matches with the already known one [24, 26]. 2. Structure factor and first sound velocity in many-boson system It is known [29] that there is a relation between the sound velocity value in the wide temperature region and the longwave limit of the structure factor: lim q→0 S(q) = T mc2(T ) . (2.1) Sound velocity temperature behaviour obtained from the expression for the structure factor in pair cor- relations approximation yields incorrect results in the above-critical region, as it was already mentioned. Therefore, we should use the post-RPA approximation which takes into account three- and four-particle correlations [28]: Sq = S0(q) 1+ (λq +Πq )S0(q) , (2.2) where S0(q) is the two-particle structure factor of the ideal bose gas, Πq =− 1 2N 1 S2 0(q) ∑ k,0 λk S(4) 0 (q,−q,k,−k) 1+λk S0(k) + 1 2N 1 S2 0(q) ∑ k,0 λkλ|q+k| [ S(3) 0 (q,k,−q−k) ]2 [1+λk S0(k)][1+λ|q+k|S0(|q+k|)] +4C2(q)+ 12 N ∑ k,0 C3(q,k,−q−k)S(3) 0 (q,k,−q−k) S0(q)[1+λk S0(k)][1+λ|q+k|S0(|q+k|)] + 8 N ∑ k,0 C4(q,k)S0(k) 1+λk S0(k) + 72 N ∑ k,0 C 2 3 (q,k,−q−k)S0(k)S0(|q+k|) [1+λk S0(k)][1+λ|q+k|S0(|q+k|)] , (2.3) λq =αq tanh(βEq )− tanh(βεq ), αq = √ 1+ 2N V νq /ħ2q2 2m , (2.4) νq = ∫ e−iqrΦ(r )dr is the Fourier coefficient of the pair interparticle interaction energy; β = 1/T , T is temperature. Generally, we are not interested in the explicit form of the pair interparticle interaction energy be- cause finally we express its Fourier coefficient from the experimentally measured structure factor. At the same time, the existence and finiteness of the Fourier coefficient of the pair interparticle interaction energy in the 4He liquid (it is an important issue in our theory) follow from the fact of the existence of the investigated system, for which we have experimentally measured the scattering length. 43001-2 First sound velocity in liquid 4He The explicit look of the expressions for the quantities C2(q1), C3(q1,q2,q3), C4(q1,q2) is described in paper [30]. In the longwave limit, we obtain such expressions for these quantities as well as for λq : C 0 2 (T ) = lim q→0 C2(q) = 1 32N ∑ k,0 (α2 k −1)2 α4 k [ β2E 2 k sinh2 ( βEk ) +βEk coth ( βEk )−2 ] , (2.5) C 0 3 (k,T ) = lim q→0 C3(q,k,−q−k) = 1 24 α2 k +1 αk tanh ( β 2 Ek ) − 1 12 tanh ( β 2 εk ) + 1 48 βεk (1−α2 k ) cosh2 ( β 2 Ek ) , (2.6) C 0 4 (k,T ) = lim q→0 C4(q,k) = 1 32 β2E 2 k (α2 k −1)2 4α3 k tanh ( β 2 Ek ) cosh2 ( β 2 Ek ) + βEk cosh2 ( β 2 Ek ) (α2 k −1)2 +2(α4 k −1) 4α3 k − (α2 k −1)2 +2(α4 k −1) 2α3 k tanh ( β 2 Ek ) − 4 αk tanh ( β 2 Ek ) +4tanh ( β 2 εk ) , (2.7) lim q→0 λq =βρν0 , (2.8) where εk = ħ2k2/2m, Ek = εkαk , ρ is the equilibrium density of liquid 4He. Using expressions for pair, three- and four-particle structure factors of ideal bose-gas (see Appendix) we can also find the values for 1/S0(q), S(3) 0 (q,k,−q−k)/S0(q) and S(4) 0 (q,k)/S2 0(q) in longwave limit: S0 2(T ) ≡ lim q→0 1 S0(k) =  0, (T É Tc), 1 1+F1(T ) , (T > Tc), (2.9) S0 3(k,T ) ≡ lim q→0 S(3) 0 (q,k,−q−k) S0(q) =  2nk +1, (T < Tc), 1+2 S0(k)−1+F2(k,T ) 1+F1(T ) , (T > Tc), (2.10) S0 4(k,T ) ≡ lim q→0 S(4) 0 (q,k) S2 0(q) = { 0, (T É Tc), 2[S0(k)−1+2F2(k,T )+F3(k,T )], (T > Tc), (2.11) where Tc is the critical temperature, nk = [z−1 0 exp(βεk )−1]−1 are occupation numbers, z0 = exp(βµ) is activity, µ is chemical potential. The explicit look for F1(T ), F2(k,T ), F3(k,T ) functions is as follows: F1(T ) = 1 N ∑ p,0 n2 p = 1 2π2ρ ∞∫ 0 p2dp[ z−1 0 exp(βp2)−1 ]2 , (2.12) F2(k,T ) = 1 N ∑ p,0 n2 p n|p+k| = 1 4π2ρ ∞∫ 0 p2dp[ z−1 0 exp(βp2)−1 ]2 × ( 1 2βpk ln { z−1 0 exp [ β(p +k)2 ]−1 z−1 0 exp [ β(p +k)2 ]−1 } −2 ) , (2.13) 43001-3 O.I. Hryhorchak F3(k,T ) = 1 N ∑ p,0 n2 p n2 |p+k| = 1 8π2ρβk ∞∫ 0 p2dp[ z−1 0 exp(βp)−1 ]2 ( − ln { z−1 0 exp [ β(p +k)2 ]−1 z−1 0 exp [ β(p +k)2 ]−1 } +4βpk − 1 z−1 0 exp [ β(p +k)2 ]−1 + 1 z−1 0 exp [ β(p −k)2 ]−1 ) . (2.14) As a result we obtain: c2(T ) = ρν0 + T m { S0 2(T )+ 1 2N ∑ k,0 λk S0 4(k,T ) 1+λk S0(k) − 1 2N ∑ k,0 [ λk S0 3(k,T ) ]2 [1+λk S0(k)]2 +4C 0 2 (T )+ 12 N ∑ k,0 C 0 3 (k,T )S0 3(k,T ) [1+λk S0(k)]2 + 8 N ∑ k,0 C 0 4 (k,T )S0(k) 1+λk S0(k) + 72 N ∑ k,0 [ C 0 3 (k,T )S0(k) ]2 [1+λk S0(k)]2 } . (2.15) 3. Low and high temperature limit In the low temperature limit lim T→0 TC 0 2 (T ) = 1 32N ∑ k,0 εk ( α2 k −1 )2 α3 k , lim T→0 T C 0 3 (k,T ) = 0, lim T→0 TC 0 4 (k,T ) = 0, lim T→0 T S0 2(k,T ) = 0, lim T→0 T S0 3(k,T ) = 0, lim T→0 T S0 4(k,T ) = 0. (3.1) That is why c2 ≡ lim T→0 c2(T ) = ρν0 m − 1 8mN ∑ k,0 εk ( α2 k −1 )2 α3 k . (3.2) The same result we obtain if we take the second derivative of the energy (in the pair correlation approx- imation) on the number of particles: c2 = N m ∂2E ∂N 2 . (3.3) Taking the square root from the equation (3.2) and using the smallness of the second term compared with the first one, we have: c ≡ lim T→0 c(T ) = √ ρν0 m − 1 16N p mρν0 ∑ k,0 εk ( α2 k −1 )2 α3 k . (3.4) The other way to the same result goes through formula [24, 26]: c =− ħ m lim q→0 qã2(q), (3.5) where the value ã2(q) has such a look: ã2(q) =−1 2 (αq −1)+ 1 N ∑ k,0 [ k2 2q2αq a4(q,−q,k,−k) + (k,q+k) q2αq a3(q,k,−q−k) ] . (3.6) In the high temperature limit, the contributions of three- and four-particle correlations are equal to zero [28]. Taking into account that lim T→0 S0(q) = 1, we obtain a well-known classical expression for the first sound velocity in a high temperature region [29]: c(T ) = √ T m + ρν0 m . (3.7) 43001-4 First sound velocity in liquid 4He 4. Numeric calculations We use the expression (2.15) for the first sound velocity numeric calculation. The transition from the summation to integration is carried out quite simply: 1 N ∑ k,0 = 1 2π2ρ ∞∫ 0 k2dk. (4.1) We can find the unknown quantity ν0 using the value of the first sound velocity at zero temperature,which we obtain by extrapolating the experimental data. Therefore, from relation (3.2) we have: ν0 = m ρ [ c2 + 1 8mN ∑ k,0 εk ( α2 k −1 )2 α3 k ] . (4.2) We use the value for ν0 (4.2) only in the expressions that reproduce the approximation of pair corre-lations in order not to exceed accuracy. We use ν0 = mc2/ρ in expressions under the sum.We do numeric calculation using the effective mass of helium atom in liquid instead of real mass, in order to eliminate infra-red divergence [31]. The use of the effective mass is a phenomenological one, but it is a necessary step, which allows us to get rid of the essential consequences of approximate calculations (breaking of the series of the perturbation theory): the infra-red divergences and the incorrect Bose-Einstein condensation temperature. We get the temperature behaviour of the isothermal sound velosity (cT ) based on the experimentaldata for the adiabatic sound velocity (cσ) [17], specific heat at a constant volume (Cv ) and specific heat ata constant pressure (Cp ) [32] using the well-known relationship: cT = cσ √ Cv /Cp . 160 180 200 220 240 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 c (T ) [m /s ] T [K] Figure 1. Temperature dependence of the first sound velocity in liquid 4He. Dashed curve— pair corre- lations approximation, solid curve— post-RPA approximation, points— indirect experimental data (cT ). 5. Conclusions In this paper we have got a temperature dependence of the first sound velocity in liquid 4He. The received expression for the first sound velocity matches with the well-known results in both low and high temperature limits. As it can be seen in figure 1, the matching of the theoretical results with the experimental data is quite good but not sufficient. In order to improve it, we should take into account the next approximation for the sound velocity. It means that we need to use the expression for a structure factor in “two sums over the wave vector” approximation in our calculations. 43001-5 O.I. Hryhorchak Acknowledgements I am greatful to my supervisor Prof. I.O. Vakarchuk for valuable remarks and suggestions, as well as to my colleague Dr. V.S. Pastukhov for longlasting debates and discussions concerning the topic of this paper. A. Appendix Two-particle structure factor of the ideal bose gas [33]: S0(q) = 1+2 n0 N nq + 1 N ∑ p,0 np n|p+q| . 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Vakarchuk I.O., Hryhorchak O.I., J. Phys. Stud., 19, No. 1/2, 1005 (in Ukrainian); Preprint arXiv:1506.03987, 2015. 31. Vakarchuk I.O., Hryhorchak O.I., Pastukhov V.S., Prytula R.O., Preprint arXiv:1506.03317, 2015. 32. Arp V.D., McCarty R.D., Friend D.G., Natl. Inst. Stand. Technol. Tech. Note 1334 (revised), 1998. 33. Vakarchuk I.O., Prytula R.O., J. Phys. Stud., 2008, 12, 4001 (in Ukrainian). Швидкiсть першого звуку в рiдкому 4 He О.I. Григорчак Кафедра теоретичної фiзики, Львiвський нацiональний унiверситет iменi Iвана Франка, вул. Драгоманова, 12, 79005 Львiв, Україна На основi виразу для структурного фактора багатобозонної системи з урахуванням прямих три- i чоти- ричастинкових кореляцiй знайдено температурну поведiнку швидкостi першого звуку в рiдкому 4He в пост-RPA наближеннi. У границi як низьких, так i високих температур отриманий вираз переходить у вже вiдомий. Результати можуть бути застосованi для аналiзу внескiв три- та чотиричастинкових кореляцiй у термодинамiчнi та структурнi функцiї рiдкого 4He. Ключовi слова: Бозе системи, рiдкий 4 He,швидкiсть першого звуку 43001-7 http://dx.doi.org/10.1103/PhysRevE.57.705 http://dx.doi.org/10.1007/s10909-007-9534-3 http://dx.doi.org/10.1007/BF01019246 http://dx.doi.org/10.1007/BF01079297 http://dx.doi.org/10.1007/BF01019263 http://dx.doi.org/10.1007/BF01029225 http://arxiv.org/abs/1506.03707 http://arxiv.org/abs/1506.03987 http://arxiv.org/abs/1506.03317 Introduction Structure factor and first sound velocity in many-boson system Low and high temperature limit Numeric calculations Conclusions Appendix

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