Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface
We solve the one-dimensional problem of quasiparticles’ transfer through the interface between a solid and superfluid helium. Superfluid helium is treated as a continuous medium with correlations. When a solid’s phonon is incident on the interface, phonons, R⁻ and R⁺rotons are created in helium, and...
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| Zitieren: | Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface / I.N. Adamenko, K.E Nemchenko, I.V. Tanatarov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 404-409. — Бібліогр.: 8 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1110602025-02-09T15:20:45Z Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface Народження ротонов надплинного гелію фононами твердого тіла, що падають нормально на границю Рождение ротонов сверхтекучего гелия фононами твердого тела, падающими нормально на границу Adamenko, I.N. Nemchenko, K.E Tanatarov, I.V. Physics of quantum liquids We solve the one-dimensional problem of quasiparticles’ transfer through the interface between a solid and superfluid helium. Superfluid helium is treated as a continuous medium with correlations. When a solid’s phonon is incident on the interface, phonons, R⁻ and R⁺rotons are created in helium, and their creation probabilities are obtained. When a quasiparticle of superfluid helium is incident, it can be reflected as any one of the three quasiparticles, and the corresponding probabilities are derived. The R⁻ rotons creation and detection probability are both shown to be small, and this explains why they could not be detected experimentally for a long time. Ми розв’язали одновимірну задачу про проходження квазічастинок через границю між твердим тілом та надплинним гелієм. Надплинний гелій описується як суцільне середовище із кореляціями. Отримано ймовірності того, що фонон твердого тіла при падінні на границю народжує фонон, R⁻ або R⁺ ротони надплинного гелію. Також обчислено ймовірності, з якими відбивається одна з трьох можливих квазічастинок при падінні на границю заданої квазічастинки надплинного гелію. Показано, що ймовірності народження й реєстрації R⁻ ротону малі, і, таким чином, дано пояснення тому, що протягом довгого часу вони не були експериментально зареєстровані. Мы решаем одномерную задачу о прохождении квазичастиц через границу между твердым телом и сверхтекучим гелием. Сверхтекучий гелий описывается как сплошная среда с корреляциями. Получены вероятности того, что фонон твердого тела при падении на границу рождает фонон, R⁻ или R⁺ ротон сверхтекучего гелия. Также вычислены вероятности, с которыми отражается одна из трех возможных квазичастиц при падении на границу заданной квазичастицы сверхтекучего гелия. Показано, что вероятности рождения и регистрации R⁻ ротона малы, и, таким образом, дано объяснение тому, что в течение долгого времени они не были экспериментально зарегистрированы. We would like to express our gratitude to A.F.G. Wyatt for many helpful discussions, and to EPSRC of the UK (grant EP/C 523199/1) for support for this work. 2007 Article Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface / I.N. Adamenko, K.E Nemchenko, I.V. Tanatarov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 404-409. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 67.40.Bz, 67.40.Hf, 67.40.Pm, 67.57.Np. https://nasplib.isofts.kiev.ua/handle/123456789/111060 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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| language |
English |
| topic |
Physics of quantum liquids Physics of quantum liquids |
| spellingShingle |
Physics of quantum liquids Physics of quantum liquids Adamenko, I.N. Nemchenko, K.E Tanatarov, I.V. Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface Вопросы атомной науки и техники |
| description |
We solve the one-dimensional problem of quasiparticles’ transfer through the interface between a solid and superfluid helium. Superfluid helium is treated as a continuous medium with correlations. When a solid’s phonon is incident on the interface, phonons, R⁻ and R⁺rotons are created in helium, and their creation probabilities are obtained. When a quasiparticle of superfluid helium is incident, it can be reflected as any one of the three quasiparticles, and the corresponding probabilities are derived. The R⁻ rotons creation and detection probability are both shown to be small, and this explains why they could not be detected experimentally for a long time. |
| format |
Article |
| author |
Adamenko, I.N. Nemchenko, K.E Tanatarov, I.V. |
| author_facet |
Adamenko, I.N. Nemchenko, K.E Tanatarov, I.V. |
| author_sort |
Adamenko, I.N. |
| title |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| title_short |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| title_full |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| title_fullStr |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| title_full_unstemmed |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| title_sort |
creation of superfluid helium rotons by a solid’s phonons incident normal to the interface |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Physics of quantum liquids |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111060 |
| citation_txt |
Creation of superfluid helium rotons by a solid’s phonons incident normal to the interface / I.N. Adamenko, K.E Nemchenko, I.V. Tanatarov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 404-409. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| first_indexed |
2025-11-27T08:30:47Z |
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2025-11-27T08:30:47Z |
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1849931597674971136 |
| fulltext |
CREATION OF SUPERFLUID HELIUM ROTONS BY A SOLID’S
PHONONS INCIDENT NORMAL TO THE INTERFACE
I.N. Adamenko1, K.E. Nemchenko1, and I.V. Tanatarov2
1Karazin Kharkov National University, 4, Svobody Sq., 61077, Kharkov, Ukraine;
2National Science Center "Kharkov Institute of Physics and Technology";
1, Academicheskaya Str., 61108, Kharkov, Ukraine
We solve the one-dimensional problem of quasiparticles’ transfer through the interface between a solid and su-
perfluid helium. Superfluid helium is treated as a continuous medium with correlations. When a solid’s phonon is
incident on the interface, phonons, R– and R+rotons are created in helium, and their creation probabilities are ob-
tained. When a quasiparticle of superfluid helium is incident, it can be reflected as any one of the three quasiparti-
cles, and the corresponding probabilities are derived. The R– rotons creation and detection probability are both
shown to be small, and this explains why they could not be detected experimentally for a long time.
PACS: 67.40.Bz, 67.40.Hf, 67.40.Pm, 67.57.Np.
1. INTRODUCTION
The dispersion curve of superfluid helium was
first sketched by L. D. Landau, and later it was meas-
ured in various experiments, particularly on neutron
scattering in helium. The curve has a specific form: it is
almost linear at small wave vectors , then reaches the
maximum called the “maxnon maximum”, goes down to
the “roton minimum”, then goes up again, and finally
becomes unstable. The quasiparticles that populate the
mostly linear part of the curve are called phonons, R
( )kΩ
k
–
rotons are on the downward section left from the mini-
mum, and R+rotons are to the right from the minimum.
The R–rotons are quasiparticles with negative dispersion
0<Ω dkd , i.e. they propagate in the direction opposite
to the one of the momentum they carry. However, for a
long time they could not be detected in direct experi-
ments, such as the experiments on creating beams of
quasiparticles in superfluid helium by a solid heater [1].
The first time they were registered was in 1999, when
the experimental group of A.F.G. Wyatt used a special
cunningly constructed source to create them and detec-
tion was achieved by means of quantum evaporation
[2]. This event made relevant some questions regarding
R–rotons, in particular the explanation of the failures to
detect them earlier became necessary.
In order to describe rotons and phonons in a unified
way, a model of quantum fluid was proposed [3], in
which it is treated as a continuous medium with correla-
tions. This theory is based on the fact that the thermal de
Broglie wavelength of a particle of a quantum fluid ex-
ceeds the average interatomic separation. Then the vari-
ables of the continuous medium can be assigned values
in each mathematical point of space in the probabilistic
sense, but the relations between them become nonlocal.
This nonlocality allows one to describe a continuous
medium with an arbitrary dispersion relation. The qua-
siparticles are described then as wave packets propagat-
ing in the medium.
In a series of papers [4-6] the theory built in [3] was
applied to solve the problem of waves’ transmission
through the interface between a solid and a quantum
fluid, for the case when the dispersion relation of the
latter is nonlinear. The considered dispersion relation
was the one of BEC in the approximation of point-like
interaction [7], which is essentially nonlinear, though
monotonic. The problem was solved in full, and the
reflection and transmission coefficients were derived as
functions of the incidence angle and frequency.
In the present work we consider the same problem
with the dispersion relation of the form that approxi-
mates well the specific dispersion curve of superfluid
helium with its phonons and rotons. We solve the prob-
lem of any quasiparticle incident on the interface from
either side in the one-dimensional case. This includes
creation and reflection of rotons on the interace.
Section 1 contains derivation of the solutions of the
equations describing the quantum fluid with the taken
dispersion relation in the half-space. The problem of a
solid’s phonon incident on the interface is solved in
section 2. The probabilities of creation of either the re-
flected phonon or the phonon or R–roton or R+roton in
superfluid helium are obtained as the corresponding
reflection and transmission coefficients for wave pack-
ets. The second part of this problem, when one of the
quasiparticles of superfluid helium is incident on the
interface, is solved in section 3. The probabilities of
each quasiparticle creation are obtained. It is shown that
the total reflection probability of an R–roton is close to
unity, while its creation probability on the interface is
very small compared to the other quasiparticles’ of su-
perfluid helium. This makes the detection by a solid
detector of R–rotons created by a solid heater almost
impossible, and explains why they were not detected
until 1999.
The results are also important for classical acoustics,
as an example of solution of the problem of creating
multiple waves lying on the same non-monotonic dis-
persion curve.
2. QUANTUM FLUID WITH ROTON-LIKE
DISPERSION RELATION
In the model of quantum fluid built in [3] it is de-
scribed by the linearized equations of ideal liquid with
nonlocal relation between pressure and density. In the
one-dimensional case, when the fluid fills the half-line
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 404-409. 404
0>x , the problem in terms of pressure can be
brought to the form
P
( ) ( ) ( )
( ) ( .,,,0
,,,
2
2
0
2
2
∞∞−∈∞∈
∂
′∂′−′=
∂
∂
∫
∞
tx
t
txPxxhxd
x
txP
)
(1)
The Fourier transform of the kernel is related
with the dispersion relation of the quantum fluid [3]
( )xh
( ) ( )
2
2
k
kkh Ω
= . (2) 2/3∈λ
In this work we consider the dispersion relation of
the form
( )
++=Ω 2
2
2
2
222 21
gg k
k
k
kksk λ . (3)
Here is the sound velocity at zero frequency;
is the quantity that defines the scale of wave vectors on
which becomes essentially nonlinear; parameter
determines the form of the dispersion curve. The
values give negative for a range of wave
vectors and therefore are not physically relevant;
s
(k
<
gk
)Ω
−λ
λ
1 ( )k2Ω
2/3−>λ provide monotonic function Ω that
does not differ in principle from the relation already
considered in [6]. In the interval
( )k2
( )2/3−
rotω
,1−∈λ the
dispersion curve is non-monotonic, with the “maxnon”
maximum and the “roton” minimum , and the
value of gives good approximation of the
dispersion relation of superfluid helium (see Fig.1).
max
0−
ω
= 98,λ
Fig. 1. The curves Ω for (dashed),
(thick solid) and
( )k 1−=λ
98.0−=λ 2/3−=
)
λ
rot
(dash-dotted).
The roton minimum and maxnon maximum
are shown for
ω
98.0
maxω
−=λ
In this paper we restrict our consideration to the last
case, that is most interesting, and to frequencies
. In this interval all the six roots
of dispersion equation
( max,ωωω rot∈ 3,2,1k±
( ) 22 ω=Ω ik , (4)
are real and can be put in the order 0 .
Then corresponds to a superfluid helium’s phonon,
to R
2
3
2
2
2
1 kkk <<<
1k
2k –roton and to R3k +roton.
The kernel that corresponds to the dispersion
relation (3) is obtained from (2) and (3):
( )xh
( ) ( ) ( )xikhxikhxh −−++ −+= expexp . (5)
Here
22
2
22 −+
± −
=
kk
k
s
ik
h g∓ (6)
and are the poles of , that in the considered case ±k ( )kh
( ),−1− can be written as
( ) 2/11 λλ +±−=± ikk g . (7)
The problem (1) was solved by Wiener-Hopf
method in [5] for Ω of polynomial form, and the
solutions were shown to be sums of waves that corre-
spond to all the roots of dispersion equation (4) in the
complex plane. Therefore we can search for solutions
of (1) with the kernel from (5) in the form
( 22 k
( )xh
)
)
k
( ) (∑ −=
i
ii tixiktxP ωα exp. . (8)
Substituting (7) and (5) into (1), we obtain that all
the of (8) really have to be the roots of equation (4),
and the amplitudes of the waves satisfy the two
equalities
ik
iα
∑∑ =
+
=
− −+ i i
i
i i
i
kkkk
0,0 αα . (9)
The number of waves in the solution (8) can be up to
six. However, when we solve a definite problem of
waves’ transmission through the interface, the given
asymptotes of the solution at infinity force some of the
amplitudes to be equal to zero. Then the boundary con-
ditions stated on the interface, together with equations
(9), provide enough equations for the problem to be
solved in full.
3. CREATION OF PHONONS
AND ROTONS OF SUPERFLUID HELIUM
BY A SOLID’S PHONONS
Let there be an interface between a solid and
the quantum fluid dispersion relation (3) and let a pho-
non of the solid be incident on the interface. The solid
with density and sound velocity occupies the
half-space , and is described as an ordinary con-
tinuous medium. The quantum fluid with density
and dispersion relation (3) fills the region , and is
treated as a continuous medium with correlations (1).
Quasiparticles of energy are wave packets propa-
gating in corresponding media with the carrier fre-
quency . It can be shown (see for example [5]), that
the wave packet’s interaction with the interface can be
described in the first approximation as that of a plane
wave with the carrier frequency. All the transmission
and reflection coefficients are obtained then as those
quantities for plane waves.
0=x
ω
solρ
0<
sols
x
0ρ
0>x
ω
In the problem formulated for plane waves, the solu-
tion in the solid is known to consist of the incident and
405
reflected waves. The solution in the quantum fluid can
consist only of waves that are constitutuents of wave
packets traveling away from the interface, i.e. with
0>Ω dkd
21 ,, kkk ±±±
, so there is only one of each pair
. We define the signs of roots of (4)
for them to be the wave vectors of this solution. As R
3 3,2,1k
0
–
rotons have negative dispersion, the R–roton wave of the
out-solution has negative wave vector , and the
roots are put in order as
2 <k
3210 kkk <−<< . (10)
With the help of relations (9) we can put down this
“out-solution” in the form with a single “amplitude”
( ) ( ) ( )( )
( ) ( )∑ ∏=
≠
−+ −
−
+−
=
3
1
exp0~,~
i
i
ij
ji
ii
outout tixik
kk
kkkkPtxP ω .(11)
The hydrodynamic velocity is found from (11) and
the linearized equations of ideal liquid
( ) ( ) ( )( )
( ) ( )∑ ∏=
≠
−+ −
−
+−
=
3
10
exp0~
,~
i
i
ij
ji
ii
i
out
out tixik
kk
kkkkkPtxV ω
ωρ
.(12)
The three plane waves in (11) and (12) correspond to
the phonon, R–roton and R+roton wave packets that
travel away from the solid, R–roton having momentum
directed towards the interface. 2k
The two boundary conditions, that demand continu-
ity of pressure and velocity on the interface, sew to-
gether the solutions in the quantum fluid (11), (12) and
in the solid. Then the amplitudes of the reflected and
transmitted waves are expressed in terms of the ampli-
tude of the incident wave. The reflection coefficient,
defined as the ratio of pressures in the reflected and in-
cident waves, is obtained
∆−+
∆+−
=→ ifZ
ifZr ~
~
0
0
χ
χ . (13)
Here is the impedance of the interface at zero
frequency, is dimensionless frequency, ∆ is a con-
stant, is a construction of wave vectors , that
can be shown to be the function of only and :
0Z
χ
f
~
3,2,1
λ
k
χ
( )
( )
( ) ( ) (
.3,2
,
;,
~
;12;;
213132321
2
3
0
0
=
−+−+−=
=
+=∆==
n
kkkkkkkkkf
fk
ff
sks
sZ
nnn
n
g
gsolsol
χλ
λωχ
ρ
ρ
)
(14)
When the solid’s phonon is incident on the interface,
it is either reflected with certain probability, or one of
the three possible quasiparticles is created in the super-
fluid helium. The probability of the phonon’s reflection
is equal to the reflected fraction of the incident wave’s
energy density 2
→r in the problem formulated in terms
of plane waves. The energy transmission coefficient is
the fraction of energy density that is transferred through
the interface 21 →
→ −= rD , so from (13) we have
( ) 22
0
0
~
~
4
∆++
=→
fZ
fZD
χ
χ . (15)
It is the creation probability of either a phonon or an
R+roton or an R–roton. The creation probability of each
of them is the corresponding partial transmission coeffi-
cient – the fraction of the energy of the incident wave
that is carried from the boundary by each of the three
waves of superfluid helium.
The energy fractions are proportional to the energy
density fluxes in each wave packet (not energy densi-
ties, as the packets differ in length due to difference in
group velocities) when they all are far enough from the
interface to be spatially separated. It was shown in [8]
for the quantities averaged that over quick oscillations,
the energy density flux in a wave packet is equal to
, where is the energy density and u is the
group velocity of the wave packet. The energy density
in a wave packet can be brought to the form
iQ
iii uQ ε= iε i
22
0 ii Vρε = [5], where V is the amplitude of hydro-
dynamic velocity. The relative amplitudes of velocities
in the three wave packets, as well as in plane waves, are
given by (12), and the group velocities are obtained
from the dispersion relation (3), so the energy fluxes in
the wave packets can be shown to be proportional to the
coefficients
i
( )( )( ,22222
kjkijiii kkkkkkk −−−=ξ ) (16)
where ( ) ( ) ( ) ( ).2,1,3,1,3,2,3,2,1,, =kji
Then the partial transmission coefficients for
are equal to
→
iD
3,2,1=i
→→
++
= DD i
i
321 ξξξ
ξ
. (17)
The structure of coefficients (16), together with
inequalities (10), leads to the fact that for
all frequencies. Also we can see that near to the roton
minimum
3,12 ξξ <
( ) rotkk ωω −+ ~32
3ξ
is a small parameter,
so and also tend to zero as 2ξ rotω−ω . In the
same way and tend to zero as 2ξ 1ξ ω−max
98.0−
ω
=λ
in the
neighborhood of maxnon maximum. So one can say
that these two asymptotes of at the ends of the
interval of frequencies, in which R
( )ωξ2
( )χ
–rotons exist, pull
the R–roton creation possibility curve to zero. The
qualitatively described behavior of the partial trans-
mission coefficients is well reflected by numerical
evaluation of the curves for different parame-
ters and . Fig. 2 shows the partial and total
transmission coefficients for and
, which is common impedance for a bound-
ary between superfluid helium and a solid.
→
iD
0
01.
Z
0=
λ
0Z
The smallness of compared even to
, which is small by itself for the considered
impedances, means that R
( )χ→
2D
( )χ→D
–rotons are barely created
406
in the experiments on creation of beams of quasipar-
ticles of superfluid helium by a solid heater.
Fig. 2. The total transmission coefficient for a
phonon incident on the interface from the solid (thick
solid curve) and partial transmission coefficients ,
equal to creation probabilities of the phonon
(dashed), R
→D
−=λ
→
iD
→
1D
→
3D
98.0
–roton (dash-dotted) and R→
2D +roton
(dotted), as functions of frequency. Here ,
01.00 =Z
4. REFLECTION OF PHONONS
AND ROTONS FROM THE INTERFACE
Now let us consider the problem when one of the
quasiparticles of superfluid helium is incident on the
interface with a solid. The solution in the solid is just
one transferred wave. About the solution in the quantum
fluid we know that there is only one wave that corre-
sponds to the wave packet traveling towards the inter-
face. Together with the wave packets traveling away
from the interface there are four waves in the solution.
Taking into account the two relations between the am-
plitudes (9), there are two free amplitudes. We take the
already considered out-solution (11) as the first of the
two linear-independent solutions, that constitute the full
solution in the quantum fluid with the given asymptotes
at infinity.
The second one is built this way: if, for example, a
phonon (wave 1) is incident, we constitute the solution
of waves with wave vectors , and , with the
amplitudes related through Eqs. (9). This way the linear
combination of the in- and out-solutions gives the full
solution in the quantum fluid when the incident qua-
siparticle is phonon, and consists of waves that give a
phonon wave packet traveling towards the interface and
three wave packets traveling away from it. The three
sorts of in-solutions, corresponding to the type of the
incident wave, can be written in the form
( )1k− 2k 3k
( ) ( ) 3,2,1,,, ==
−→
ntxPtxP
nn kkout
n
in . (18)
The two boundary conditions provide two relations
between the amplitudes of in- and out-solutions and the
transmitted wave in the solid. Then the relative ampli-
tudes of all the constituent waves are calculated with the
use of (11) and (18). The energy fluxes are obtained in
the same way as in the previous section.
When an i -th wave packet of superfluid helium is
incident on the interface, all the three waves are re-
flected. We denote the fraction of the energy of the in-
cident wave packet that is carried by the -th reflected
wave as for i . It is shown that
j
ijR 3,2,1, =j
( ) ( )( )
( )302
302
fZf
fZ i
+
+ −
χ
χkg=
(i
nf−
3,
ii kk
k
−→
]3
jiij R=
12R k
n k ,[ 1
ji ≠
4= χ
2
2
∆+
+
R
=
→ =
( ) ~3
)
1
13
33
O
−=
=
=
=
+2 k
1
32
22
=
=
=
= O
( )
2
2
222
2
2
222
fkk
fkf
R
gg
i
g
i
ii ∆+
∆+ −− . (19)
Here ) are modified combinations of wave vec-
tors , that are equal to from (14) in which one
of the three wave vectors enters with the opposite sign:
2,1k nf
( )i
nf− ,k2f= .
For and the expressions for them are
obtained from the expression for by cyclic permuta-
tion of subscripts in , and :
R
12R
3u3
( )
( ) .
2
2
22
302
2
303
2
3
2
62
12 fkfZfk
kkZ
s
u
k
k
kR
gg
gg
g +
∆+
χ
χ
(20)
In terms of quasiparticles the quantity gives the
probability the quasiparticle of type is reflected as
type , i.e. the probability that when a quasiparticle of
type is incident on the interface, the quasiparticle of
type is reflected, so can be called conversion
coefficients. The probability that the quasiparticle i is
reflected is , the probability it is transmitted
is its energy transmission coefficient .
When all types of quasiparticles coexist in the quantum
fluid in equilibrium, the fraction of the total energy flux
incident on the interface, that is transmitted into the
solid, can be shown to be equal to , and
thermodynamic equilibrium between the two media at
equal temperatures demands that . This
equality can be checked in a straightforward way by
simple, though a little cumbersome, calculations.
ij
iD←
∑3
i
D
i
←D
D
j
i
j ijR
∑=
j iji RR
i= 1
←
iD
←
R−
In the neighborhood of the roton minimum
the small parameters are
rotωω ≈
rotωω −kk +~ 23,2u ,
and the asymptotic behavior of conversion coefficients
is obtained by direct substitution into (i
nf± and then (19)
and (20):
( )
( )
( )
( ).
,
,
,1
23
,12
,22
11
rot
rot
rot
OR
OR
OR
R
ωω
ωω
ωω
−
−
−
(21)
In the neighborhood of the maxnon maximum
( ) ωω −max12,1 ~~ ku and we obtain in the same
way
( )
( )
( )
( ).
,
,
,1
max12
max,31
max,11
33
ωω
ωω
ωω
−−
−
−
OR
OR
OR
R
. (22)
407
The asymptotes (21) and (22) give us full qualitative
description of behavior on frequency. In particular,
the last equality of (21) means that when an R
ijR
0Z
–roton
with frequency close to the roton minimum is incident,
it is almost always reflected as R+roton, and vice versa.
The last equality of (22) means that when an R–roton
with frequency near to the maxnon maximum is inci-
dent, it is almost always reflected as phonon, and vice
versa. The asymptotic behavior of the conversion coef-
ficients is illustrated by Fig. 3, in which the curves
are shown for the same parameters as in Fig. 2,
and .
ijR
98
( )ωijR
0−=λ . 01.0=
Fig. 3. or and The
curves are denoted by the corresponding pairs of sub-
scripts. and are cloze but not equal to unity at
because of the small impedance, and the pic-
ture’s scale just does not let us see the finite difference
( )ωijR f
R
98.0−=λ 01.00 =Z .
11R 33
max,rotω
The total probability that R–roton is reflected is
, and so it tends to unity at both
ends of the interval . These two asymp-
totes pull to zero the curve of R
2322212 RRRR ++=
ω
←
2D
→
2D
( )ω←
iD
←
3D
( max,ωωrot∈
2R−
)
–roton detection prob-
ability , which is equal to 1 , in the same way
as in the previous section. Moreover, the depend-
ences are qualitatively the same as of ,
so the transmission coefficient for R
( )ω→
iD
←
2D–rotons is
much less than the ones for phonos and R←
1D +rotons
(see Fig. 2). This means that R–rotons are very
poorly detected by solid detectors, in comparison to
phonons and R+rotons. Put together with their small
creation probability, it makes detection of R–rotons
in the experiments on creating beams of quasiparti-
cles of superfluid helium by a solid heater almost
impossible. This is the reason they could not be di-
rectly detected until 1999, when the experimental
group of A.F.G. Wyatt used a special source and R–
rotons were finally registered by means of quantum
evaporation [2].
5. CONCLUSIONS
In this paper we considered the one-dimensional
problem of quasiparticles’ transfer through the interface
between a solid and superfluid helium. This problem
can be also formulated in terms of wave packets or
plane waves. The dispersion relation of superfluid he-
lium is non-monotonic, so there are multiple roots of
dispersion equation . The quasiparticles
corresponding to these roots
( ) 22 ω=Ω k
321 kkk <<
i
0 <
←
2D
, in as-
cending order of wave vectors, are phonons, R–rotons
and R+rotons. Creation probalilities of quasiparticles of
each type by a solid’s phonon are obtained (17).
The probabilities that a quasiparticle of type is
reflected when a quasiparticle of type is incident on
the interface are derived (18), (19). The R
→
iD
ijR j
–rotons crea-
tion and detection probability are both shown
to be small, and this explains why they could not be
detected until the experiments [2].
→
2D
ACKNOWLEDGEMENTS
We would like to express our gratitude to
A.F.G. Wyatt for many helpful discussions, and to
EPSRC of the UK (grant EP/C 523199/1) for support
for this work.
REFERENCES
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Liquid 4He: a tunable high-pass phonon filter //Phys.
Rev. Lett. 1974, v. 33, p. 1425.
2. M.A.H. Tucker and A.F.G. Wyatt. Direct evidence
for R– rotons having antiparallel momentum and ve-
locity //Science. 1999, v. 283, p. 1150.
3. I.N. Adamenko, K.E. Nemchenko and I.V. Tana-
tarov. Application of the theory of continuous media
to the description of thermal excitations in superfluid
helium //Phys.Rev. B. 2003, v. 67, 104513.
4. I.N. Adamenko, K.E. Nemchenko and I.V. Tanatarov.
Energy transfer through the interface into a quantum
fluid with nonlinear dispersion relation //J. of Low
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5. I.N. Adamenko, K.E. Nemchenko and I.V. Tana-
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lous dispersion through the interface of two
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6. I.N. Adamenko, K.E. Nemchenko, and I.V. Tana-
tarov. Transmission of phonons with anomalous dis-
persion through the interface of two continuous me-
dia //Published online J. of Low Temp. Phys. 2006,
v. 144, N 1/3, p. 13-34.
7. N.N. Bogoliubov. To the theory of superfluidity
//Izv. AN USSR, ser. fiz. 1947, v. 11, # 1, p. 77-90.
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408
РОЖДЕНИЕ РОТОНОВ СВЕРХТЕКУЧЕГО ГЕЛИЯ ФОНОНАМИ ТВЕРДОГО ТЕЛА,
ПАДАЮЩИМИ НОРМАЛЬНО НА ГРАНИЦУ
И.Н. Адаменко, К.Э. Немченко, И.В. Танатаров
Мы решаем одномерную задачу о прохождении квазичастиц через границу между твердым телом и
сверхтекучим гелием. Сверхтекучий гелий описывается как сплошная среда с корреляциями. Получены ве-
роятности того, что фонон твердого тела при падении на границу рождает фонон, R– или R+ротон сверхте-
кучего гелия. Также вычислены вероятности, с которыми отражается одна из трех возможных квазичастиц
при падении на границу заданной квазичастицы сверхтекучего гелия. Показано, что вероятности рождения и
регистрации R–ротона малы, и, таким образом, дано объяснение тому, что в течение долгого времени они не
были экспериментально зарегистрированы.
НАРОДЖЕННЯ РОТОНОВ НАДПЛИННОГО ГЕЛІЮ ФОНОНАМИ ТВЕРДОГО ТІЛА,
ЩО ПАДАЮТЬ НОРМАЛЬНО НА ГРАНИЦЮ
І.Н. Адаменко, К.Е. Немченко, І. В. Танатаров
Ми розв’язали одновимірну задачу про проходження квазічастинок через границю між твердим тілом та
надплинним гелієм. Надплинний гелій описується як суцільне середовище із кореляціями. Отримано ймові-
рності того, що фонон твердого тіла при падінні на границю народжує фонон, R– або R+ ротони надплинного
гелію. Також обчислено ймовірності, з якими відбивається одна з трьох можливих квазічастинок при падінні
на границю заданої квазічастинки надплинного гелію. Показано, що ймовірності народження й реєстрації R–
ротону малі, і, таким чином, дано пояснення тому, що протягом довгого часу вони не були експерименталь-
но зареєстровані.
409
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