On the effect of superfluid flows on the interaction of microwaves with He II
The paper proposes a possible mechanism of interaction of microwaves with superfluid helium that results in an experimentally observed narrow peak of microwave absorption on the frequencies by the order of the roton frequency. The obtained microwave photon absorption coefficient depends on the local...
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| Цитувати: | On the effect of superfluid flows on the interaction of microwaves with He II / V.D. Khodusov, A.S. Naumovets // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43601:1-4. — Бібліогр.: 21 назв. — англ. |
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Khodusov, V.D. Naumovets, A.S. 2017-06-11T14:52:12Z 2017-06-11T14:52:12Z 2012 On the effect of superfluid flows on the interaction of microwaves with He II / V.D. Khodusov, A.S. Naumovets // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43601:1-4. — Бібліогр.: 21 назв. — англ. PACS: 67.25.du, 67.30.eh, 67.25.dt DOI:10.5488/CMP.15.43601 arXiv:1212.6346 https://nasplib.isofts.kiev.ua/handle/123456789/120302 The paper proposes a possible mechanism of interaction of microwaves with superfluid helium that results in an experimentally observed narrow peak of microwave absorption on the frequencies by the order of the roton frequency. The obtained microwave photon absorption coefficient depends on the local equilibrium distribution function which is established due to fast roton-roton and roton-phonon interactions. With the availability of superfluid flows, the local equilibrium distribution function depends on their velocity. The critical velocity of the flows, at which the absorption of microwaves is replaced by their radiation, is found. В роботi запропоновано механiзм взаємодiї надвисокочастотних хвиль iз надплинним гелiєм, який пояснює експериментально спостережуваний вузький пiк поглинання надвисокочастотних хвиль з частотою порядку ротонної частоти. Отриманий коефiцiєнт поглинання надвисокочастотних фотонiв залежить вiд локально-рiвноважної функцiї розподiлу, яка встановлюється завдяки швидких ротон-ротонних та ротон-фононних взаємодiй. При наявностi надплинних потокiв, локально-рiвноважна функцiя розподiлу залежить вiд їхньої швидкостi. Отримано критичну швидкiсть потокiв, за якої поглинання надвисокочастотних хвиль змiнюється на їхнє випромiнювання. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the effect of superfluid flows on the interaction of microwaves with He II Вплив надплинних потокiв на взаємодiю надвисокочастотних хвиль з Не II Article published earlier |
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On the effect of superfluid flows on the interaction of microwaves with He II |
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On the effect of superfluid flows on the interaction of microwaves with He II Khodusov, V.D. Naumovets, A.S. |
| title_short |
On the effect of superfluid flows on the interaction of microwaves with He II |
| title_full |
On the effect of superfluid flows on the interaction of microwaves with He II |
| title_fullStr |
On the effect of superfluid flows on the interaction of microwaves with He II |
| title_full_unstemmed |
On the effect of superfluid flows on the interaction of microwaves with He II |
| title_sort |
on the effect of superfluid flows on the interaction of microwaves with he ii |
| author |
Khodusov, V.D. Naumovets, A.S. |
| author_facet |
Khodusov, V.D. Naumovets, A.S. |
| publishDate |
2012 |
| language |
English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Вплив надплинних потокiв на взаємодiю надвисокочастотних хвиль з Не II |
| description |
The paper proposes a possible mechanism of interaction of microwaves with superfluid helium that results in an experimentally observed narrow peak of microwave absorption on the frequencies by the order of the roton frequency. The obtained microwave photon absorption coefficient depends on the local equilibrium distribution function which is established due to fast roton-roton and roton-phonon interactions. With the availability of superfluid flows, the local equilibrium distribution function depends on their velocity. The critical velocity of the flows, at which the absorption of microwaves is replaced by their radiation, is found.
В роботi запропоновано механiзм взаємодiї надвисокочастотних хвиль iз надплинним гелiєм, який пояснює експериментально спостережуваний вузький пiк поглинання надвисокочастотних хвиль з частотою порядку ротонної частоти. Отриманий коефiцiєнт поглинання надвисокочастотних фотонiв залежить вiд локально-рiвноважної функцiї розподiлу, яка встановлюється завдяки швидких ротон-ротонних та ротон-фононних взаємодiй. При наявностi надплинних потокiв, локально-рiвноважна функцiя розподiлу залежить вiд їхньої швидкостi. Отримано критичну швидкiсть потокiв, за якої поглинання надвисокочастотних хвиль змiнюється на їхнє випромiнювання.
|
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120302 |
| citation_txt |
On the effect of superfluid flows on the interaction of microwaves with He II / V.D. Khodusov, A.S. Naumovets // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43601:1-4. — Бібліогр.: 21 назв. — англ. |
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2025-11-25T23:52:48Z |
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Condensed Matter Physics, 2012, Vol. 15, No 4, 43601: 1–4
DOI: 10.5488/CMP.15.43601
http://www.icmp.lviv.ua/journal
On the effect of superfluid flows on the interaction
of microwaves with He II
V.D. Khodusov, A.S. Naumovets
Karazin Kharkov National University, 4 Svobody Sq., 61077 Kharkov, Ukraine
Received July 3, 2012, in final form August 22, 2012
The paper proposes a possible mechanism of interaction of microwaves with superfluid helium that results in
an experimentally observed narrow peak of microwave absorption on the frequencies by the order of the roton
frequency. The obtained microwave photon absorption coefficient depends on the local equilibrium distribution
function which is established due to fast roton-roton and roton-phonon interactions. With the availability of
superfluid flows, the local equilibrium distribution function depends on their velocity. The critical velocity of the
flows, at which the absorption of microwaves is replaced by their radiation, is found.
Key words: superfluid, rotons, microwaves
PACS: 67.25.du, 67.30.eh, 67.25.dt
1. Introduction
The experiments aimed at the study of the absorption of microwaves in the frequency range 40÷
200 GHz in superfluid helium have produced a number of unexpected results. Thus, a resonant absorp-
tion of microwaves has been revealed in the frequencies relevant to roton energy [1–4]; in this case, a
narrow absorption line near the minimum energy of rotons has been observed against the background
of a wide pedestal and, what is more, the results of the measurements of temperature dependence of this
narrow line correlated with the change in the minimum roton energy. Besides, the effect of the velocity
of the relative motion of normal Vn and superfluid Vs components on the character of the resonant in-
teraction of microwaves has been found. The microwave absorption coefficient decreased at an increase
of the velocity of the relative motion. When there was an excess of some critical velocity, the absorption
of waves was replaced by their radiation.Similar effects take place in plasma physics when one describes
linear and nonlinear Landau attenuation of plasma waves on particles as well as the excitation of these
waves by particle streams moving at the velocity higher than some critical value [5].
These experimental results are indicative of an intensified manifestation of electric properties in he-
lium at temperatures below Tλ. Moreover, they point out a special role, which the quasi-particles (i.e.,
phonons, rotons) should play in explaining these experiments. In [6–12], some attempts were made to
theoretically explain these features. In the above works, a possible mechanism of microwave interaction
with superfluid helium is proposed. It consists in taking into account the effect of its electric properties on
quasi-particles and their kinetics. In the temperature range, in which the experiments were carried out
(1.4÷2.3 K), rotons had a dominant role. They determined both thermodynamic and kinetic properties
of He II. Fast roton-roton and roton-phonon interactions provide the establishment of a hydrodynamic
regime in gas of quasi-particles. The explanation of these effects can be made if we take into account the
summands in the roton energy that are linear in the electric field [13], i.e., if one suggests that a roton has
a dynamic dipole moment.
© V.D. Khodusov, A.S. Naumovets, 2012 43601-1
http://dx.doi.org/10.5488/CMP.15.43601
http://www.icmp.lviv.ua/journal
V.D. Khodusov, A.S. Naumovets
2. Raman scattering of electromagnetic waves from quasiparticles in
He II
The interaction of electromagnetic waves with superfluid helium had been previously studied, both
theoretically and experimentally, while describing the Raman light scattering from phonons, second
sound and rotons [14–19]. In references [15, 16], the two-roton scattering of light was experimentally
observed. Since the roton momentum considerably exceeds (by some orders) the photon momentum
(visible light and microwaves) [17], in order to monitor the electromagnetic wave Raman scattering from
rotons, it is necessary to have two rotons, which follows from the momentum conservation. Energy and
momentum conservation permits a birth of two rotons by photon and photon scattering from rotons.
For the first process, the energy and momentum conservation laws are:
ħω1 =ħω2 +ε3 +ε4 , ħk1 =ħk2 +p3 +p4 , (2.1)
where ω1,2 and k1,2 are the frequencies and wave vectors of the incident and the reflected electromag-
netic wave accordingly, ε3,4 and p3,4 are the energies and the momenta of rotons, respectively. The scat-
tering in this case occurs with the excitation of two rotons with the opposite momenta p3 ≈−p4.
Taking into account that the roton energy is ε=∆+(p −p0)2/2µ , from (2.1) one can obtain for Stokes
(red) satellite:
ħ(ω1 −ω2) Ê 2∆+ (p3 −p0)2
µ
, (2.2)
where p0 is the value of a momentum, at which the energy or roton has a minimum equal to ∆, µ is the
effective mass of roton. The density of the number of roton energy states per a unit of volume is equal to:
ρ(ε) =
√
2µ
4π4ħ3
[
p0 +
√
2µ(ε−∆)
]2
p
ε−∆
. (2.3)
From equation (2.3) it follows that the basic role in these processes is played by rotons having minimal
energy ε ≈ ∆ . Then, as follows from equation (2.2), two-roton light absorption takes place, as observed
in works [15] and [16]. Theoretical explanation of these experiments as well as the detection of matrix
elements of photon and roton interactions are given in references [17] and [18]. For the second process,
describing photon scattering from rotons, conservation laws of energy and momentum are:
ħω1 +ε3 =ħω2 +ε4 , ħk1 +p3 =ħk2 +p4 . (2.4)
In contrast to the first case, here we have p3 ≈ p4.
As follows from the compatibility condition of the system (2.4), the next condition is imposed on
frequencies of the incident and the reflected electromagnetic waves, for Stokes satellite:
0 É (ω1 −ω2) É
2
(
p3 −p0
)
|k1 −k2|+ħ|k1 −k2|2
2µ
. (2.5)
This condition, as opposed to (2.2), determines an upper limit for (ω1 −ω2) and shows that frequencies
ω1 and ω2 are closer to each other than in the previous case. Indeed, under the conditions corresponding
to those for the experiments in [15], estimations give the following result for p3 ≈ p0, k1 ≈ −k2, k1 ≈
105 cm−1: (ω1 −ω2) ≈ 2·106 s−1. In the first case, under the same conditions, from equation (2.2) we obtain
(ω1 −ω2) ≈ 2.27 ·1012 s−1, if we take the value for the roton minimal energy at temperature T = 1.4 K
∆ = 8.65 K. Stokes line intensity of Raman light scattering on rotons due to the second process, as it
follows from the research [18], is e−∆/T times less than that one due the first process. Moreover, it is in
other frequency ranges (MHz). We hope that with the use of modern equipment it will be possible to
detect this line in spite of small intensity. A method of resonant combinational light scattering represents
a special interest in studying Raman scattering [20]. Thus, the instances are possible when light frequency
coincides with the own frequencies of elementary excitations. In this case, there is an imposition of two
effects: a forced resonant excitation of quasi-particles by electromagnetic wave and photon scattering
from quasi-particles. If an electromagnetic wave frequency tends to roton energy, the processes of the
birth of two rotons are forbidden, as it follows from energy conservation.
43601-2
On the effect of superfluid flows on the interaction of microwaves with He II
3. The narrow resonant line and the effect of the flows
Weassume that a narrow resonant line, which is observed on the background of the pedestal (Rayleigh
wings), is caused by the photons scattering from rotons. The quasi-local distribution function of rotons in
this case is established in a time 1/γr ∼ 10−11 s as:
nr 0 =
[
exp
(
ε+pW
T
)
−1
]−1
, (3.1)
where W = Vn −Vs is the relative velocity.
The change in a unit of time of a number of microwave photons ∆N1 with energies ħω1 due to the
induced processes of scattering of photons from rotons can be written in the form:
∂∆N1
∂t
= ∆N1
∫
|Φ (1,3;2,4)|2 N2 (n04 −n03)δ(ħ (ω1 −ω2)+ε3 −ε4) (3.2)
× δ
(
p4 −p3 −ħ (k1 −k2)
) d3p4d3p3
(2πħ)6
d3k2
(2π)3
.
|Φ (1,3;2,4)| is a matrix element of interaction of rotons and microwaves. It depends from an interaction
constant, which determines the interaction between the dipole moment of rotons and the electric com-
ponent of microwaves. This constant is small, because the electromagnetic field relaxation occurs during
the period of an order of seconds after turning off the field [21]. Taking into account that the momentum
of photons is much smaller than that of the rotons and p3 ≈ p4, we can write n04 as:
n04 = n03 +
n03
T
[−ħ (ω1 −ω2)+ħ (k1 −k2)W] . (3.3)
If we try to find a solution to this equation in the form ∆N ∼ e−γt , we obtain
γ = − 1
T
∫
|Φ (1,3;2,4)|2 n03N2 [(ω1 −ω2)− (k1 −k2)W ] (3.4)
× 2µ
|k1 −k2|
√
(p3 −p0)2 +2µħ(ω1 −ω2)
2πp2
3dp3
(2πħ)3
d3k2
(2π)3
.
By comparing this attenuation factor with the one experimentally measured, it is possible to obtain the
matrix element order estimates of interaction of microwave photons and rotons. The absorption coeffi-
cient of microwaves depends upon ω1. Assuming that the frequencies ω1, ω2 fall into the same range in
the vicinity of the resonance, it is possible to determine the boundary conditions from (2.5):
∆
ħ
−
2
∣
∣p3 −p0
∣
∣ |k1 −k2|+ħ|k1 −k2|2
4µ
Éω1 É
∆
ħ
+
2
∣
∣p3 −p0
∣
∣ |k1 −k2|+ħ|k1 −k2|2
4µ
.
From this relation it follows, that both the resonant frequency itself and the limiting values of the
permitted frequencies of microwaves change as a function of temperature basically in the same way as
the energy gap in a roton spectrum, which corresponds to the experimental results.
Besides, if k1 ≈ −k2 and
∣
∣p3 −p0
∣
∣ ≈
√
2µT for the thermal rotons, the width of a resonant curve at
zero intensity is defined by the expression:
∆ω≈ 2
(
√
2µT k1 +ħk2
1
)/
µ. (3.5)
If T = 1.4 K, k1 ≈ 37.7 cm−1, then ∆ω ≈ 4.7 ·105 Hz. It coincides by the order of the magnitude with
the one which has been observed in the experiment.
From the expression (3.4), for γ it is obvious that there exists such a relative critical velocity Wcr at
which γ= 0. At the velocities greater than Wcr, absorption of microwaves is replaced by their radiation.
After the calculation of the remaining integral, we obtain the critical velocity: Wcr = 8
3
ħk1
µ . Substituting
the values of k1 and µ, we obtain the following value of Wcr ∼ 10−2 cm/s.
43601-3
V.D. Khodusov, A.S. Naumovets
4. Conclusions
A possiblemechanism of interaction betweenmicrowaves and superfluidHe is proposed in the paper.
The obtained microwave phonon absorption coefficient depends on the local equilibrium distribution
function, which is established due to fast roton-roton and roton-phonon interactions. It is shown that the
resonant line changes with the temperature similarly to the minimum roton energy. The value of the
resonant line width is obtained, which is in agreement with the experimentally observed one. The local
equilibrium distribution function and, correspondingly, the microwave absorption coefficient depend on
the velocity of superfluid flows. There is found a critical velocity of flows at which the absorption of
microwaves is replaced by their radiation.
References
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Вплив надплинних потокiв на взаємодiю
надвисокочастотних хвиль з Не II
В.Д. Ходусов, А.С. Наумовець
Харкiвський нацiональний унiверситет iм. В.Н. Каразiна, пл. Свободи, 4, 61077 Харкiв, Україна
В роботi запропоновано механiзм взаємодiї надвисокочастотних хвиль iз надплинним гелiєм, який по-
яснює експериментально спостережуваний вузький пiк поглинання надвисокочастотних хвиль з часто-
тою порядку ротонної частоти. Отриманий коефiцiєнт поглинання надвисокочастотних фотонiв залежить
вiд локально-рiвноважної функцiї розподiлу, яка встановлюється завдяки швидких ротон-ротонних та
ротон-фононних взаємодiй. При наявностi надплинних потокiв, локально-рiвноважна функцiя розподiлу
залежить вiд їхньої швидкостi. Отримано критичну швидкiсть потокiв, за якої поглинання надвисокоча-
стотних хвиль змiнюється на їхнє випромiнювання.
Ключовi слова: надплиннiсть, ротони, надвисокочастотнi хвилi
43601-4
http://dx.doi.org/10.1103/PhysRevB.76.140503
http://dx.doi.org/10.1063/1.2911649
http://dx.doi.org/10.1063/1.2957000
http://dx.doi.org/10.1063/1.3266909
http://dx.doi.org/10.1063/1.1820356
http://dx.doi.org/10.1063/1.2126942
http://dx.doi.org/10.1063/1.2126950
http://dx.doi.org/10.1063/1.2911650
http://dx.doi.org/10.1103/PhysRevLett.17.294
http://dx.doi.org/10.1103/PhysRevLett.22.987
http://dx.doi.org/10.1103/PhysRevLett.25.1547
http://dx.doi.org/10.1103/PhysRev.181.338
http://dx.doi.org/10.1103/PhysRev.187.279
Introduction
Raman scattering of electromagnetic waves from quasiparticles in He II
The narrow resonant line and the effect of the flows
Conclusions
|