- Email: [email protected]
Spectrum of spontaneous light scattering from a superfluid liquid surface
Volume
44A, number
4
PHYSICS
SPECTRUM
LETTERS
OF SPONTANEOUS
FROM A SUPERFLUID
18 June 1973
LIGHT SCATTERING
LIQUID
SURFACE
A.I. BOZHKOV P.N. Lebedev Physical Institute of the Academy of Sciences of the USSR, Moscow, USSR
Received An expression for the spectral surface is suggested.
intensity
1 May 1973
of light scattered
from the thermal
fluctuations
of free superfluid
liquid
In [l] the total intensity of light scattered in a given direction from the thermal fluctuations of the boundary surface of a superfluid helium was measured. On the other hand, there are some works on the spectral intensity measurements of such a scattering from the surface of an ordinary liquid [2]. It seems interesting to carry out similar measurements for liquid helium. In what follows an analytical expression for the spectral intensity of spontaneously scattered light from the surface of liquid helium is obtained. For this purpose two-fluid hydrodynamics of an incompressible superfluid liquid [3] $ and the fluctuation-dissipation theorem (FDT) [5] are usedS$. The equations of motion and the boundary conditions with due reference to “external” pressure jump II(r, t)$$$ have the form:
pn(avn/af)
= Avn-
Vpn , div Vi, = 0,
Aq = 0
v, = al+7plaz= auat , a vwy/az+av,/a(x,
y) = 0 , for
2rl(a v,laz)
- p, + ps@4at)
z=O.
- QAJ = W, 4
Here the plane (x, y) corresponds to a quiet interface, axis z is directed away from the liquid, {(r, t) is the deviation of the boundary surface from the plane one; the other designations are the same as in [3]. The Fourier components r(9, a) = (27r)-3 J dr dtt(r, t) exp {- i(9r - Rt)} and ?(9, R) are related by 7\9, R) = x(9, Sz) ‘$9, R), x(9, R) = (9~~1~) [~~r2+(ps/Pn)+(l-i~r)2-(~//4~l - 2iWp,lp)l -l, a0 =(w /PI”*, 7= GJw~/P)-~,Wd/) Thus, the spectral density of liquid helium surface fluctuations, according to FDT, is determined by [7(9, s2)12 = 2kTa-I(2n)-3 Im{X(9,52)} and expressed as follows 2kTqT2
I!?((494r2
Im { [( 1- iRr)2 - (p/p,)d
1 - 2iWpn/p)]
> 0.
*}
(1)
= (W3ps21(pJpn)
+ s2zr2 + (1 - iW2
-Wp,)
d 1 - 2iWpn/Ph2
where, in the case of necessity one should substitute kT for average energy of quantum oscillator (AR/2) coth(AS2/2kr). If ps = 0 eq. (1) coincides with the similar formula obtained in [7] Let a plane electromagnetic wave of light with intensity I, be incident on the surface of liquid helium and let it be polarized perpenducular to the plane of incidence E = Re {E. exp [i(k,r- k,z-w,t)]}. The intensity of light scattered into free space within solid angle sin0 dqd0 and in a frequency range dR can be written as dl =
$ Similar observations are easily extended to a more general [e.g. 41 description of helium II surface motion. $$ Eq. (1) (the main result of this work) can be obtained by another method using the phenomenological theory fluctuations and the well-known postulate by Onsager [ 6 1. $$t Such a choice of ‘Lexternal” force, energy-conjugated with ~(r, t), follows from general physical considerations result which is in good agreement with experiments [ 71.
of equilibrium and gives a
251
Volume
44A. number
4
PHYSICS
O~C- ‘I(k, +q, w,+SI) cos0 sin0 dqd0 dR, k,+q,
= w,,c
LETTERS
18 June
1973
’ sin0 cosp. k,,. +q,,. = CJ,,C ’ sin6 sir19 where
Eq. (2) is valid only for such values of q which provide scattering in the form of homogeneous
plane waves. 1.~.
kc”’ z should be real.
It is interesting that eq. ( 1) at .Clc)~> 1 and ps - p,, within the frequency ranges ii1 t 12,, 5 T ’ results III IWO ordinary Lorentzian lines in the spectrum of scattered light Bnd in this case the superfluid component is set aside at all:
In other cases, especially at R,r spectrum of scattered light.
5 p/p,,, the superfluid component
The author wishes to express his gratitude
must produce considerable
changes in the
to Professor F.V. Bunkin for useful discussions of the problem
References 1I ] [2] [3] [4] [5]
1,. Wagner. Phys. Lett. 42A (1972) 265. J. Meunier. D. Cruchon and M.A. Bouchiat, C’.R. Acad. SCI. Paris 268 (I 969) 92.422. L.D. Landau, Zh. Fksp. i. Tear. Fiz. 14 (1944) 112 (in RuGan) [J, Phys. USSR 8 t 19c14) J. Seiden, C.R. Acad. Sci. Paris 275 (1972) B-713. M.L. Levin and S.M. Rytov. Theory of thermal equilibrium fluctuations in electrodynamics Moscow, 1967), p. 277. [6] R.D. Mountain, Rev. Mod. Phys. 38 (1966) 205. [ 7 ] M.A. Bouchiat and J. Meunier. J. Physique 32 ( 197 I ) 56 1.
I]. (in Russian.
“Nauka”
Press.
44A, number
4
PHYSICS
SPECTRUM
LETTERS
OF SPONTANEOUS
FROM A SUPERFLUID
18 June 1973
LIGHT SCATTERING
LIQUID
SURFACE
A.I. BOZHKOV P.N. Lebedev Physical Institute of the Academy of Sciences of the USSR, Moscow, USSR
Received An expression for the spectral surface is suggested.
intensity
1 May 1973
of light scattered
from the thermal
fluctuations
of free superfluid
liquid
In [l] the total intensity of light scattered in a given direction from the thermal fluctuations of the boundary surface of a superfluid helium was measured. On the other hand, there are some works on the spectral intensity measurements of such a scattering from the surface of an ordinary liquid [2]. It seems interesting to carry out similar measurements for liquid helium. In what follows an analytical expression for the spectral intensity of spontaneously scattered light from the surface of liquid helium is obtained. For this purpose two-fluid hydrodynamics of an incompressible superfluid liquid [3] $ and the fluctuation-dissipation theorem (FDT) [5] are usedS$. The equations of motion and the boundary conditions with due reference to “external” pressure jump II(r, t)$$$ have the form:
pn(avn/af)
= Avn-
Vpn , div Vi, = 0,
Aq = 0
v, = al+7plaz= auat , a vwy/az+av,/a(x,
y) = 0 , for
2rl(a v,laz)
- p, + ps@4at)
z=O.
- QAJ = W, 4
Here the plane (x, y) corresponds to a quiet interface, axis z is directed away from the liquid, {(r, t) is the deviation of the boundary surface from the plane one; the other designations are the same as in [3]. The Fourier components r(9, a) = (27r)-3 J dr dtt(r, t) exp {- i(9r - Rt)} and ?(9, R) are related by 7\9, R) = x(9, Sz) ‘$9, R), x(9, R) = (9~~1~) [~~r2+(ps/Pn)+(l-i~r)2-(~//4~l - 2iWp,lp)l -l, a0 =(w /PI”*, 7= GJw~/P)-~,Wd/) Thus, the spectral density of liquid helium surface fluctuations, according to FDT, is determined by [7(9, s2)12 = 2kTa-I(2n)-3 Im{X(9,52)} and expressed as follows 2kTqT2
I!?((494r2
Im { [( 1- iRr)2 - (p/p,)d
1 - 2iWpn/p)]
> 0.
*}
(1)
= (W3ps21(pJpn)
+ s2zr2 + (1 - iW2
-Wp,)
d 1 - 2iWpn/Ph2
where, in the case of necessity one should substitute kT for average energy of quantum oscillator (AR/2) coth(AS2/2kr). If ps = 0 eq. (1) coincides with the similar formula obtained in [7] Let a plane electromagnetic wave of light with intensity I, be incident on the surface of liquid helium and let it be polarized perpenducular to the plane of incidence E = Re {E. exp [i(k,r- k,z-w,t)]}. The intensity of light scattered into free space within solid angle sin0 dqd0 and in a frequency range dR can be written as dl =
$ Similar observations are easily extended to a more general [e.g. 41 description of helium II surface motion. $$ Eq. (1) (the main result of this work) can be obtained by another method using the phenomenological theory fluctuations and the well-known postulate by Onsager [ 6 1. $$t Such a choice of ‘Lexternal” force, energy-conjugated with ~(r, t), follows from general physical considerations result which is in good agreement with experiments [ 71.
of equilibrium and gives a
251
Volume
44A. number
4
PHYSICS
O~C- ‘I(k, +q, w,+SI) cos0 sin0 dqd0 dR, k,+q,
= w,,c
LETTERS
18 June
1973
’ sin0 cosp. k,,. +q,,. = CJ,,C ’ sin6 sir19 where
Eq. (2) is valid only for such values of q which provide scattering in the form of homogeneous
plane waves. 1.~.
kc”’ z should be real.
It is interesting that eq. ( 1) at .Clc)~> 1 and ps - p,, within the frequency ranges ii1 t 12,, 5 T ’ results III IWO ordinary Lorentzian lines in the spectrum of scattered light Bnd in this case the superfluid component is set aside at all:
In other cases, especially at R,r spectrum of scattered light.
5 p/p,,, the superfluid component
The author wishes to express his gratitude
must produce considerable
changes in the
to Professor F.V. Bunkin for useful discussions of the problem
References 1I ] [2] [3] [4] [5]
1,. Wagner. Phys. Lett. 42A (1972) 265. J. Meunier. D. Cruchon and M.A. Bouchiat, C’.R. Acad. SCI. Paris 268 (I 969) 92.422. L.D. Landau, Zh. Fksp. i. Tear. Fiz. 14 (1944) 112 (in RuGan) [J, Phys. USSR 8 t 19c14) J. Seiden, C.R. Acad. Sci. Paris 275 (1972) B-713. M.L. Levin and S.M. Rytov. Theory of thermal equilibrium fluctuations in electrodynamics Moscow, 1967), p. 277. [6] R.D. Mountain, Rev. Mod. Phys. 38 (1966) 205. [ 7 ] M.A. Bouchiat and J. Meunier. J. Physique 32 ( 197 I ) 56 1.
I]. (in Russian.
“Nauka”
Press.