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Transmission probabilities of high-frequency phonons through a solid-He-II interface
Volume 45A,
number
2
10 September 1973
PHYSICS LETTERS
TRANSMISSION PROBABILITIES OF HIGH-FREQUENCY PHONONS THROUGH A SOLID-He-II INTERFACE H. HAUG* Philips Research Laboratories, Eindhoven,
The Netherlands
Received 23 July 1973 The transmission of monochromatic, high-frequency phonons from a solid into He-II by single-quantum processes is calculated. The roton minimum and especially pronounced the phonon maximum of the He-II spectrum show up as peaks in the transmission coefficient.
With the advent of monochromatic, high-frequency phonon generators and detectors [l] the interest revived in studying the transmission mechanism of phonons through a solid-liquid interface. Former investigations with long-wavelength thermal phonons, i.e. by means of the Kapitza resistance, did not yield a complete understanding of this mechanism [2]. It seems, however, that at least for long-wavelength phonons corresponding to temperatures of a few mK, the phonon transmission can be described by Khalatnikov’s acoustic impedance theory [3], provided that one accounts for a damaged surface layer in the solid by introducing a phonon absorption coefficient [4]. Experimental results with high-frequency phonons are up to now relatively scarce. Most measurements have. been done in reflection [5]. Astonishingly, Kinder [5] could not find any structure in the reflection coefficient between sapphire and He-II at frequencies which correspond to the roton minimum A, and the phonon maximum A, of the He-II spectrum. Because the reflection coefficient is relatively large, direct measurements in transmission would clearly be more sensitive. Only recently, Swanenburg and Wolter [6] reported the first direct measurement in transmission for longitudinal and transversal high-frequency phonons. However, pulsed heaters were used in their experiments which emitted a broad band of hot phonons, so that not information about the frequency dependence of the transmission coefficient could be obtained. In order to extend the theory of phonon transmission to higher frequencies, we use the coupling Hamil* Address from Sept. 1, 1973: Institut f. Theor. Physik, Universitat Frankfurt, Frankfurt/Main,
170
F.R. Germany.
4
r
3
& =,I 2
li 0
0
--/
I
5
10
415 fiwplk,
1/
20&
:
Iy.1
Fig. 1. The normalized impedance of He-II as a function of the frequency wp.
tonian formalism of Bowley, Sheard and Toombs [7], in which the excitations in the solid and in He-II are coupled by Hint = pi J dS, nziUi, where Ui is the displacement in the solid and IIZi is the stress tensor of He-II. The integration extends over the interface (see inset in fig. 1). In ref. [7], Ui and n,i have been expanded into running modes which is, strictly speaking, inconsistent with the geometry and introduced numerical errors. In order to include the geometrical restrictions of both subsystems correctly, we calculate the displacement u in Hint by assuming that both surfaces of the solid which are parallel to the x-y plane oscillate freely, i.e. the density p of He is in zeroth approximation taken to be zero. The other surfaces of the solid are for convenience taken to be rigidly fared and, to
Volume 45A, number 2
10 September 1973
PHYSICS LETTERS
simplify the formulae, only longitudinal phonons are considered. The quantum hydrodynamic theory wh' IC (QHT)[81, h provides a quantitative description of He-II, will be used to evaluate the stress tensor Il in dint. A direct calculation within the framework of the QHT is hindered by the fact that this theory is at present only formulated for systems with periodic boundaries. Therefore, the stress tensor will be calculated in three successive steps. Firstly, II is determined by assuming that He-II can be described as a quantized elastic medium (QEM), bounded by rigid walls. These boundaries are a good zeroth approximation because of the large acoustic mismatch between He-II and the solid. Secondly, II is evaluated within the inhomogeneous Bogolubov model (BM). The appropriate boundary conditions for the field operator are obtained by comparing the long-wavelength limit of the BM with the result of the QEM. Finally, it is known that the results of the BM and of the QHT are identical [8], as long as only single- quantum processes are considered and provided that the condensate density in the BM is replaced by the total density and that the Fourier transform of a realistic two-particle potential is substituted in the BM. Therefore, we can obtain Il in the QHT from the results of the BM by means of these substitutions. The resulting coupling Hamiltonian is Hi”* =
p,ps; a, as; k,q,Ek/&Uq and ck,a4 are the densities, volumes, wavevectors, frequencies and phonon annibilation operators of He-II and the solid, respectively. /cl, ql are the wavevector components parallel to the interface. The unrenormalized excitation spectrum of He-II is& = (T; t 2Tk pvk)1/2, where Tk = R2k2/2m. The structure factor hk is given by & = Tk/Ek. Hint contains the factor of two which was missing in the result of ref. [7]. In the next higher order the QHT gives a coupling Hamiltonian, which describes twoquantum processes in He, i.e. two-phonon or roton generation or the inelastic scattering of excitations at the interface. These higher order processes play an essential role for the total energy transmission and will be treated in a following investigation. However, with narrow band generation and detection one should be able to observe the single-quantum processes separately at low temperatures. The detection in He-II should take
place at a distance from the interface which is smaller than the mean free path of these excitations. The thermal heat current which follows from the long-wavelength limit of (1) in lowest order perturbation theory agrees with Khalatnikov’s result [3] (specialized to longitudinal phonons only) in the limit of weak coupling, i.e. Z,/Z, 4 1, where the acoustic impedances of He and the solid are given by Z, = cp/cos8 and Z, = c,ps/cos~,, respectively. f3 and ~9sare the angles which k and q enclose with the z-axis. Assuming that a current Q, of monochromatic, high-frequency phonons with a given wavevector p hits the interface from the solid, we obtain again in lowest order perturbation theory (golden rule) an energy current Q through the interface, which is given by Q = 4ZJZs Q,. The impedance Zp of He-II for singlequantum processes is obtained as zP=
&
P
’ c xk(EkIxk-TpJ)2 a! t
3
(2)
where k, are the solutions of hw = Ek. The impedance Z reduces in the long-wavelengt g limit to Z, = pc/cos& I# e result (2) is plotted in fig. 1 for normal incidence (pti = 0, case, = 1) using the parameters given in ref. [8]. The roton minimum A0 and still stronger pronounced the broad phonon maximum A1 show up as singularities in the impedance Zp. Note that the simple lowest order perturbation theory breaks down right at the singularities, causing the unphysical result Q/Q, > 1. The use of a renormalized energy spectrum [8] would shift A0 and A1 to the experimentally observed values of 8.65 K and 11.9 K. Another slngularity at 2Ao, the endpoint of the phonon-roton branch, is to be expected. For an isotropic solid, transverse phonons will only contribute for oblique incidence. Their contribution will modify the transmission coefficient, but will not change the above derived features, which are inherent to the properties of He-II.
References [l] W. Eisenmenger and A.H. Dayem, Phys. Rev. Letters 18 (1967) 125; H. Kinder, Phys. Rev. Letters 28 (1972) 1564.
171
Volume 45A, number 2
PHYSICS LETTERS
[2] G.L. Pollak, Rev. Mod. Phys. 41 (1969) 48. [3] I.M. Khalatnikov, An introduction to the theory of superfluidity, (N.Y., Benjamin 1965). 141 H. Haug and K. Weiss, Phys. Lett. 40 A (1972) 19; R.E. Peterson and A.C. Anderson, Phys. Lett. 40A (1972) 317. 15) Choreng-Jee Guo and H.J. Maris, Phys. Rev. Letters 29 (1972) 855; H.J. Trumpp, K. Lassmann and W. Eisenmenger, Phys. Letters41A (1972) 43i;
172
10 September 1973
H. Kinder, talk given at the spring conference of the German Physical Society, Miinster, 1973. ]6] T. Swanenburg and J. Wolter, preprint. [ 71 R.M. Bowley, F.W. Sheard and G.A. Toombs, Proc. Int. Conf. on Phonon scattering in solids (Paris, 1972) p. 380. [8] S. Sunakawa, S. Yamasaki and T. Kebukawa, Prog. Theor. Phys. 41 (1969) 919.
number
2
10 September 1973
PHYSICS LETTERS
TRANSMISSION PROBABILITIES OF HIGH-FREQUENCY PHONONS THROUGH A SOLID-He-II INTERFACE H. HAUG* Philips Research Laboratories, Eindhoven,
The Netherlands
Received 23 July 1973 The transmission of monochromatic, high-frequency phonons from a solid into He-II by single-quantum processes is calculated. The roton minimum and especially pronounced the phonon maximum of the He-II spectrum show up as peaks in the transmission coefficient.
With the advent of monochromatic, high-frequency phonon generators and detectors [l] the interest revived in studying the transmission mechanism of phonons through a solid-liquid interface. Former investigations with long-wavelength thermal phonons, i.e. by means of the Kapitza resistance, did not yield a complete understanding of this mechanism [2]. It seems, however, that at least for long-wavelength phonons corresponding to temperatures of a few mK, the phonon transmission can be described by Khalatnikov’s acoustic impedance theory [3], provided that one accounts for a damaged surface layer in the solid by introducing a phonon absorption coefficient [4]. Experimental results with high-frequency phonons are up to now relatively scarce. Most measurements have. been done in reflection [5]. Astonishingly, Kinder [5] could not find any structure in the reflection coefficient between sapphire and He-II at frequencies which correspond to the roton minimum A, and the phonon maximum A, of the He-II spectrum. Because the reflection coefficient is relatively large, direct measurements in transmission would clearly be more sensitive. Only recently, Swanenburg and Wolter [6] reported the first direct measurement in transmission for longitudinal and transversal high-frequency phonons. However, pulsed heaters were used in their experiments which emitted a broad band of hot phonons, so that not information about the frequency dependence of the transmission coefficient could be obtained. In order to extend the theory of phonon transmission to higher frequencies, we use the coupling Hamil* Address from Sept. 1, 1973: Institut f. Theor. Physik, Universitat Frankfurt, Frankfurt/Main,
170
F.R. Germany.
4
r
3
& =,I 2
li 0
0
--/
I
5
10
415 fiwplk,
1/
20&
:
Iy.1
Fig. 1. The normalized impedance of He-II as a function of the frequency wp.
tonian formalism of Bowley, Sheard and Toombs [7], in which the excitations in the solid and in He-II are coupled by Hint = pi J dS, nziUi, where Ui is the displacement in the solid and IIZi is the stress tensor of He-II. The integration extends over the interface (see inset in fig. 1). In ref. [7], Ui and n,i have been expanded into running modes which is, strictly speaking, inconsistent with the geometry and introduced numerical errors. In order to include the geometrical restrictions of both subsystems correctly, we calculate the displacement u in Hint by assuming that both surfaces of the solid which are parallel to the x-y plane oscillate freely, i.e. the density p of He is in zeroth approximation taken to be zero. The other surfaces of the solid are for convenience taken to be rigidly fared and, to
Volume 45A, number 2
10 September 1973
PHYSICS LETTERS
simplify the formulae, only longitudinal phonons are considered. The quantum hydrodynamic theory wh' IC (QHT)[81, h provides a quantitative description of He-II, will be used to evaluate the stress tensor Il in dint. A direct calculation within the framework of the QHT is hindered by the fact that this theory is at present only formulated for systems with periodic boundaries. Therefore, the stress tensor will be calculated in three successive steps. Firstly, II is determined by assuming that He-II can be described as a quantized elastic medium (QEM), bounded by rigid walls. These boundaries are a good zeroth approximation because of the large acoustic mismatch between He-II and the solid. Secondly, II is evaluated within the inhomogeneous Bogolubov model (BM). The appropriate boundary conditions for the field operator are obtained by comparing the long-wavelength limit of the BM with the result of the QEM. Finally, it is known that the results of the BM and of the QHT are identical [8], as long as only single- quantum processes are considered and provided that the condensate density in the BM is replaced by the total density and that the Fourier transform of a realistic two-particle potential is substituted in the BM. Therefore, we can obtain Il in the QHT from the results of the BM by means of these substitutions. The resulting coupling Hamiltonian is Hi”* =
p,ps; a, as; k,q,Ek/&Uq and ck,a4 are the densities, volumes, wavevectors, frequencies and phonon annibilation operators of He-II and the solid, respectively. /cl, ql are the wavevector components parallel to the interface. The unrenormalized excitation spectrum of He-II is& = (T; t 2Tk pvk)1/2, where Tk = R2k2/2m. The structure factor hk is given by & = Tk/Ek. Hint contains the factor of two which was missing in the result of ref. [7]. In the next higher order the QHT gives a coupling Hamiltonian, which describes twoquantum processes in He, i.e. two-phonon or roton generation or the inelastic scattering of excitations at the interface. These higher order processes play an essential role for the total energy transmission and will be treated in a following investigation. However, with narrow band generation and detection one should be able to observe the single-quantum processes separately at low temperatures. The detection in He-II should take
place at a distance from the interface which is smaller than the mean free path of these excitations. The thermal heat current which follows from the long-wavelength limit of (1) in lowest order perturbation theory agrees with Khalatnikov’s result [3] (specialized to longitudinal phonons only) in the limit of weak coupling, i.e. Z,/Z, 4 1, where the acoustic impedances of He and the solid are given by Z, = cp/cos8 and Z, = c,ps/cos~,, respectively. f3 and ~9sare the angles which k and q enclose with the z-axis. Assuming that a current Q, of monochromatic, high-frequency phonons with a given wavevector p hits the interface from the solid, we obtain again in lowest order perturbation theory (golden rule) an energy current Q through the interface, which is given by Q = 4ZJZs Q,. The impedance Zp of He-II for singlequantum processes is obtained as zP=
&
P
’ c xk(EkIxk-TpJ)2 a! t
3
(2)
where k, are the solutions of hw = Ek. The impedance Z reduces in the long-wavelengt g limit to Z, = pc/cos& I# e result (2) is plotted in fig. 1 for normal incidence (pti = 0, case, = 1) using the parameters given in ref. [8]. The roton minimum A0 and still stronger pronounced the broad phonon maximum A1 show up as singularities in the impedance Zp. Note that the simple lowest order perturbation theory breaks down right at the singularities, causing the unphysical result Q/Q, > 1. The use of a renormalized energy spectrum [8] would shift A0 and A1 to the experimentally observed values of 8.65 K and 11.9 K. Another slngularity at 2Ao, the endpoint of the phonon-roton branch, is to be expected. For an isotropic solid, transverse phonons will only contribute for oblique incidence. Their contribution will modify the transmission coefficient, but will not change the above derived features, which are inherent to the properties of He-II.
References [l] W. Eisenmenger and A.H. Dayem, Phys. Rev. Letters 18 (1967) 125; H. Kinder, Phys. Rev. Letters 28 (1972) 1564.
171
Volume 45A, number 2
PHYSICS LETTERS
[2] G.L. Pollak, Rev. Mod. Phys. 41 (1969) 48. [3] I.M. Khalatnikov, An introduction to the theory of superfluidity, (N.Y., Benjamin 1965). 141 H. Haug and K. Weiss, Phys. Lett. 40 A (1972) 19; R.E. Peterson and A.C. Anderson, Phys. Lett. 40A (1972) 317. 15) Choreng-Jee Guo and H.J. Maris, Phys. Rev. Letters 29 (1972) 855; H.J. Trumpp, K. Lassmann and W. Eisenmenger, Phys. Letters41A (1972) 43i;
172
10 September 1973
H. Kinder, talk given at the spring conference of the German Physical Society, Miinster, 1973. ]6] T. Swanenburg and J. Wolter, preprint. [ 71 R.M. Bowley, F.W. Sheard and G.A. Toombs, Proc. Int. Conf. on Phonon scattering in solids (Paris, 1972) p. 380. [8] S. Sunakawa, S. Yamasaki and T. Kebukawa, Prog. Theor. Phys. 41 (1969) 919.