Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K

A self-consistent approach is applied for calculations within the two-time temperature Green function formalism in the random phase approximation for superfluid ⁴He. The effective mass of the ⁴He atom is computed asm m * =1.58 . The excitation spectrum is found to be in a satisfactory agreement with...

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Дата:	2003
Автор:	Rovenchak, A.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2003
Назва видання:	Физика низких температур
Теми:	Квантовые жидкости и квантовые кpисталлы
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/128787
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K / A.A. Rovenchak // Физика низких температур. — 2003. — Т. 29, № 2. — С. 145-148. — Бібліогр.: 12. назв. — англ.

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spelling	irk-123456789-1287872018-01-14T03:04:02Z Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K Rovenchak, A.A. Квантовые жидкости и квантовые кpисталлы A self-consistent approach is applied for calculations within the two-time temperature Green function formalism in the random phase approximation for superfluid ⁴He. The effective mass of the ⁴He atom is computed asm m * =1.58 . The excitation spectrum is found to be in a satisfactory agreement with experiment. The sound velocity is calculated as 230 m/s. The Bose-condensation temperature with the effective mass taken into consideration is estimated as 1.99 K. 2003 Article Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K / A.A. Rovenchak // Физика низких температур. — 2003. — Т. 29, № 2. — С. 145-148. — Бібліогр.: 12. назв. — англ. 0132-6414 PACS: 67.40.-w, 67.40.Db http://dspace.nbuv.gov.ua/handle/123456789/128787 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы
spellingShingle	Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Rovenchak, A.A. Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K Физика низких температур
description	A self-consistent approach is applied for calculations within the two-time temperature Green function formalism in the random phase approximation for superfluid ⁴He. The effective mass of the ⁴He atom is computed asm m * =1.58 . The excitation spectrum is found to be in a satisfactory agreement with experiment. The sound velocity is calculated as 230 m/s. The Bose-condensation temperature with the effective mass taken into consideration is estimated as 1.99 K.
format	Article
author	Rovenchak, A.A.
author_facet	Rovenchak, A.A.
author_sort	Rovenchak, A.A.
title	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K
title_short	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K
title_full	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K
title_fullStr	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K
title_full_unstemmed	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K
title_sort	effective mass of atom and the excitation spectrum in liquid helium-4 at t = 0 k
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2003
topic_facet	Квантовые жидкости и квантовые кpисталлы
url	http://dspace.nbuv.gov.ua/handle/123456789/128787
citation_txt	Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K / A.A. Rovenchak // Физика низких температур. — 2003. — Т. 29, № 2. — С. 145-148. — Бібліогр.: 12. назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT rovenchakaa effectivemassofatomandtheexcitationspectruminliquidhelium4att0k
first_indexed	2025-07-09T09:53:45Z
last_indexed	2025-07-09T09:53:45Z
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fulltext	Fizika Nizkikh Temperatur, 2003, v. 29, No. 2, p. 145–148 Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K A.A. Rovenchak Department for Theoretical Physics, Ivan Franko National University of Lviv 12 Drahomanov Str., Lviv 79005, Ukraine E-mail: andrij@ktf.franko.lviv.ua Received April 22, 2002, revised August 20, 2002 A self-consistent approach is applied for calculations within the two-time temperature Green function formalism in the random phase approximation for superfluid 4He. The effective mass of the 4He atom is computed as m m* .� 158 . The excitation spectrum is found to be in a satisfactory agreement with experiment. The sound velocity is calculated as 230 m s/ . The Bose-condensation temperature with the effective mass taken into consideration is estimated as 1.99 K. PACS: 67.40.—w, 67.40.Db 1. Introduction The idea of the effective mass of the helium atom in superfluid phase was suggested by Feynman in 1953 [1]. He stated that one should insert the effective mass (slightly larger than the mass of a «pure» atom m) in the expressions for the density matrix. Isihara and Samulski [2] used the effective mass m m* .� 171 in order to obtain a good agreement of the sound branch of the excitation spectrum with the ex- perimental data on the sound velocity. While in both these cases the effective mass was in- troduced phenomenologically, it appeared to be possi- ble to obtain the value of this quantity on the basis of experimental data for the structure factor of liquid he- lium-4 [3], and a result of 1.70 m was calculated there by Vakarchuk. The main idea of the present paper is to give a way of obtaining the 4He atom effective mass by means of a self-consistent equation. The expressions are written within the collective variables formalism as described by Bogoliubov and Zubarev [4]. Two-time tempera- ture Green’s functions [5] are utilized for the calcula- tion of the thermodynamic averages. We also intend to show the possibility of an essen- tially simple approach to the problem of the many-boson system with strong interaction, such as liquid helium-4. The method applied does not require much computational effort. This advantage allows for the development of further approximations. In addition, if one accepts the assumption that the phenomena in liquid 4He are at least partly due to the Bose condensation being «spoiled» by the interatomic interaction, it turns out to be possible to estimate the lambda transition temperature as the critical tempera- ture of the ideal Bose gas. We show that such an ap- proach leads to very good agreement with experiment: if the effective mass is about 50 % larger than the pure one, the Bose condensation temperature decreases to a value of � 2 K. The Green function technique also provides a possi- bility of obtaining the excitation spectrum of the sys- tem. As a result of the mass renormalization, the exci- tation spectrum is found to be in better agreement with experiment than the Bogoliubov or Feynman spectrum, while the expressions are the same (the lat- ter two spectra suffer from the well-known problem of the so-called «roton» minimum overestimation if one considers the pure mass). A self-consistent approach was recently applied for the calculation of the 4He excitation spectrum by Pashitskii et al. [6]. The results of that paper are in excellent agreement with experiment. The authors used the «semitransparent spheres» potential for the calculation, with some adjustable parameters. A new interpretation of the roton region of the he- lium excitation spectrum was given by Kruglov and Collett [7]. They considered the so-called roton clus- ter as a bound state of 13 atoms and solved the nonli- near Schrödinger equation for this system surrounded by the condensate. This approach reflects the experi- mental data for the roton branch of the excitation curve very well. © A.A. Rovenchak, 2003 In our work, we utilize the potential of Ref. 8 as in- put information for the computations. This potential was obtained on the basis of the quantum-mechanical equations with the static structure factor as the only experimental datum. Since this quantity is quite easily measured directly in the scattering experiments, we consider this a good approach. Unfortunately, a direct calculation of the potential for the many-body prob- lem cannot be carried out at the present time. The paper is organized as follows. The calculation procedure is given in Sec. 2. The Hamiltonian is writ- ten and the equations of motion for the Green func- tions are solved in the random phase approximation (RPA), providing a self-consistent equation for the ef- fective mass extraction. The numerical results are ad- duced and discussed in Sec. 3. 2. Calculation procedure The Hamiltonian of the Bose system in the collec- tive variables representation reads [4]: � [ ] [ ]H � � � �� k k N Vk k k k k k k 2 1 0 � � � � ��1 2 2 0 0 0N m � q k q k k q k qkq � � , (1) where the operator � k k� � � �/ . Here �k is the ener- gy spectrum of a free particle, �k k / m� �2 2 2 , N is the total number of particles in the system, and V is the system volume, and �k is the Fourier transform of the interatomic potential in the thermodynamic limit, N/V �� const. The item with one summation over the wave vector k in Eq. (1) corresponds to the random phase approximation, and the second one is the correction. Let us assume that our system is described by exactly RPA Hamiltonian � ()H , i.e., � () * [ ] [ ] ,H � � � �� k k k k k k k k N V k2 1 0 (2) where �k * � �2 2 2k / m and m is the effective mass of the 4He atom. It is the only quantity suitable for the «effective» role, since we wish to preserve the inter- atomic potential as the initial information. Such a definition of the effective mass means that we partly transfer the interaction and the higher-order correction onto the kinetic term in the energy. This approach correlates with that of Feynman [1]. As was also shown by Vakarchuk [9], the mass renormali- zation obtained in a similar fashion leads to the ex- pression obtained for the effective mass of 3He impu- rity in 4He but with the «pure» mass of the 3He atom replaced by that of 4He atom. We define m* by demanding that the effective Hamiltonian (2) leads to the same ground-state en- ergy as the initial Hamiltonian (1), � () �H H� : � � � �k () ()[ ]k k k k k � � � �� 0 � � � � �� N V2 1� k k k[ ]() � �� k[ ]k k k k k 0 � � � � �� N V k2 1� [ ]k k � � � � ��1 2 2 0 0 0N m � q k q k k q k qkq � � , (3) where the superscript () near the angle brackets is introduced for convenience. One can find the operator product average by uti- lizing two-time temperature Green functions defined as follows [5]: A t B t i t t A t B t( ) ( ) ( ) [ ( ), ( )]� � � � �� , (4) with the operators given in the Heisenberg represen- tation, � is the Heaviside step function. AB GBA� �� ( )A �� i d G i G iBA BA � � � � � � � � � � � � � ( ) ( ) e 1 0 , (5) where GBA stands for B A and the operator �A is introduced for convenience. We set the time argu- ments in the operators A(t), B t( )� to coincide: t t� � � 0. This will provide the static properties of the system under consideration. In the above expression, � is the inverse temperature, � � 1/T. Now we proceed to the equations of motion for the Green functions G ( )k k k� � ,G �( )k � � � �k k , etc. It is easy to obtain the following set of equations in the RPA: ( ) ( ( )�� k kG Gk) k2 , ( ) (�� k G k) �� kG (k) + 1 2 , ( ) ( ( )�� k kG Gk) k2 1 2 , ( ) (�� k G k) �� kG (k) . (6) 146 Fizika Nizkikh Temperatur, 2003, v. 29, No. 2 A.A. Rovenchak Here� is equilibrium density. Next, let us consider the triple product average ABC . One can obtain it utilizing either the Green function G C ABC AB; � or G BC ABC A; � . We suggest the first possibility to fulfill ABC G C ABC AB� �� ;A . (7) In other words, we neglect the functions of the type G BC ABC A; � for the sake of simplicity (when applying this to Eq. (6) it means that only the RPA term of the Hamiltonian (1) is taken into conside- ration when constructing the equations of motion). Having performed a similar procedure with the func- tion G� � � �; ( , )k k , k k kk1 2 3 1 2 3 � , we obtain in the RPA the following set of equations: ( ) ( , , );�� k G 2 2 1 2 1k k k k �� k G g 2 2 1 2 1; ( , , ) ( )k k k k , ( ) ( , , );�� k G 2 2 1 2 1k k k k � � � � ��2 2 2 1 2 1� �� k G g; ( , , ) ( )k k k k , (8) where the quadruple Green functions were decoupled in such a way as to provide the inhomogeneous set of equations: AB CD BD A C CA B D� � � � �AD B C CB A D . (9) The inhomogeneous terms in Eq. (8) read: g m D Gk � �( ) [ ( )� � � �2 1 2 1 22 2 1 k k k k � ��k k kk +k1 2 11 2 S G\| \| ( ) + k k + k k + k2 1 2 1 21 ( ) ( )�� D Gk � � �� k k k kk k2 1 2 11 2 ( ) ( )]\| \|D G � , g m D Gk� � � � � �( ) [ ( ) ( ) �2 1 1 2 1 21 k k + k k + k � �� k k + k kk k1 1 2 11 2 ( ) ( )]\| \|D G . (10) The notations for the averages of pair products are listed below: � � � � � �k k S Tk k k k1 2 coth , � � � � � � �� k k D T Sk k k k k 1 2 1 2 1 1 2 1coth ( ), � � � � � � � �� k k D T Sk k k k k k k k 1 4 2 2 2 coth . (11) The quantity �k is defined as follows: � � �k k k / � � �� 1 2 1 2 . (12) Now, if we turn back to correlation (3), the mean- ing of the asterisk as a superscript becomes clear: one should substitute m with m* in the left hand side of this equation. In the ground state (T = 0 K), hyperbolic cotan- gents in Eq. (11) equal to unity. Therefore, a self-con- sistent equation for the extraction of m* becomes as follows: ! " ! "� � � � � � � �1 4 1 1 4 1 0 2 0 2 N N k k k k k � � � � * * k � � � �� # � ��1 22 2 2 00 0 N m q p � qk k k q kq � � � # � � � � � � � � � � � $ % & & �kp � � � � � � � � � � � � � k k p k k q q q p q q p p2 � � � � � � � � � � � �qp � � � � � � � � � � � � q q p k k q q k q q p p2 � � � � � � � � � � # 1 1 � � � � � � � �k k q q q q p p # � � � � � �� ' ( ) ) kq kp qp � � � � � � � � k q k q k k q q2 2 , (13) where p k q� � . We also consider the specific energy instead of the total one by introducing the factor of 1/N. One should notice that for the non-interacting system, when �k � 0, the above equation is satisfied trivially (� �k k� �* 1 in this case). In particular, the triple product average k q k q� � �� in (3) equals to zero providing a well-known expression for the ideal Bose-gas (IBG) energy: E TIBG k k k Tk � � $ %& ' () � � � �� 2 2 1 10 0k k coth /e . 3. Numerical results and discussion We use the previously obtained results [8] for the Fourier transform �k of the interatomic potential. The value of the equilibrium density is � � �0 02185 3. Å . The mass of the helium-4 atom is m = 4.0026 a.m.u. We pass from the summation over the wave vector to integration in the usual way: � � �k kV d /( )2 3� . Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K Fizika Nizkikh Temperatur, 2003, v. 29, No. 2 147 The value of the upper cutoff for the integration over the wave vector is 16.0 Å �1. The solution of Eq. (13) at the above-listed condi- tions is m* = 1.58 m. (14) One can also obtain the excitations spectrum using Green’s functions. The solutions of set (6) are propor- tional to 1 2 2 2 2/ k k( – )� � � � providing the spectrum Ek k k� * � � , which is a very well known result [4]. If one inserts the effective mass into the definitions of �k and � k, the obtained curve fits the experimental one in a quite satisfactory manner (see Fig. 1). The phonon branch is reflected quite well, provi- ding a sound velocity of approximately 230 m/s ver- sus the experimental one 238 m/s at T = 0.8 K [11] or 240 m/s at T = 0.1 K [12]. The so-called «roton» mi- nimum also has a value close to the experimental one. In addition, the value obtained for the effective mass shifts the temperature of the Bose condensation from Tc = 3.14 K for the pure mass to Tc = 1.99 K, versus the experimental temperature of the lambda transition T+ = 2.17 K. We consider the results dis- cussed above to be quite good for such a rough approx- imation as random phases. Acknowledgements The author is grateful to Prof. I. Vakarchuk for valuable discussions on the problem considered in this work. 1. R.P. Feynman, Phys. Rev. 91, 1291 (1953). 2. A. Isihara and T. Samulski, Phys. Rev. B16, 1969 (1977)}. 3. I. O. Vakarchuk, Visn. Lviv. Univer. Ser. Fiz. 26, 29 (1993) (in Ukrainian). 4. N.N. Bogoliubov and D.N. Zubarev, Zhurn. Eksp. Teor. Fiz. 28, 129 (1955); [Sov Phys.-JETP 1, 83 (1955)]. 5. D.N. Zubarev, Neravnovesnaia Statisticheskaia Ter- modinamika, Nauka, Moscow (1971) (in Russian); D.N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bureau, New York (1974). 6. E.A. Pashitskii, S.I. Vilchinskyy, and S.V. Mashke- vich, Fiz. Nizk. Temp. 28, 115 (2002) [Low Temp. Phys. 28, 79 (2002)]. 7. V.I. Kruglov and M.J. Collett, Phys. Rev. Lett. 87, 185302 (2001). 8. I.O. Vakarchuk, V.V. Babin, and A.A. Rovenchak, J. Phys. Stud. (Lviv) 4, 16 (2000). 9. I.O. Vakarchuk, J. Phys. Stud. (Lviv) 1, 25 (1996); 1, 156 (1997) (in Ukrainian). 10. R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49, 177 (1971). 11. V.D. Arp, R.D. McCarty, and D.G. Friend, Natl. Inst. Stand. Technol. (U.S.) Tech. Note 1334 (1998) (revised). 12. R.D. McCarty, Natl. Bur. Stand. (U.S.) Tech. Note 1029 (1980). 148 Fizika Nizkikh Temperatur, 2003, v. 29, No. 2 A.A. Rovenchak Fig. 1. Excitations spectrum of liquid helium-4. Filled circles — experimental data [10]; Solid line — calculated energy.

Effective mass of atom and the excitation spectrum in liquid helium-4 at T = 0 K

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