The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He
We discuss a number of unusual phenomena, which have been discovered recently in anisotropic quasiparticle systems in superfluid ⁴He. These include the creation of high-energy phonons by a pulse of low-energy phonons, the suprathermal distribution of high-energy phonons in long phonon pulses, the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1209332017-06-14T03:06:12Z The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" We discuss a number of unusual phenomena, which have been discovered recently in anisotropic quasiparticle systems in superfluid ⁴He. These include the creation of high-energy phonons by a pulse of low-energy phonons, the suprathermal distribution of high-energy phonons in long phonon pulses, the mesa shape of the angular distribution of low-energy phonons, the creation of a «hot line» when two-phonon pulses cross. The thermodynamic properties of anisotropic quasiparticle systems of He II are derived for all degrees of anisotropy. 2007 Article The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He / I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1008–1015. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 67.40.Db, 67.40.Fd http://dspace.nbuv.gov.ua/handle/123456789/120933 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
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International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| spellingShingle |
International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He Физика низких температур |
| description |
We discuss a number of unusual phenomena, which have been discovered recently in anisotropic
quasiparticle systems in superfluid ⁴He. These include the creation of high-energy phonons by a pulse of
low-energy phonons, the suprathermal distribution of high-energy phonons in long phonon pulses, the mesa
shape of the angular distribution of low-energy phonons, the creation of a «hot line» when two-phonon
pulses cross. The thermodynamic properties of anisotropic quasiparticle systems of He II are derived for all
degrees of anisotropy. |
| format |
Article |
| author |
Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. |
| author_facet |
Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. |
| author_sort |
Adamenko, I.N. |
| title |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He |
| title_short |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He |
| title_full |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He |
| title_fullStr |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He |
| title_full_unstemmed |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He |
| title_sort |
unusual properties of anisotropic systems of quasiparticles in superfluid ⁴he |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120933 |
| citation_txt |
The unusual properties of anisotropic systems of quasiparticles in superfluid ⁴He / I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1008–1015. — Бібліогр.: 18 назв. — англ. |
| series |
Физика низких температур |
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| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1008–1015
The unusual properties of anisotropic systems of
quasiparticles in superfluid 4He
I.N. Adamenko, K.E. Nemchenko, and V.A. Slipko
V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: vaslipko@mail.ru
A.F.G. Wyatt
School of Physics, University of Exeter, Exeter EX4 4QL, UK
E-mail: a.f.g.wyatt@ex.ac.uk
Received October 1, 2006
We discuss a number of unusual phenomena, which have been discovered recently in anisotropic
quasiparticle systems in superfluid 4He. These include the creation of high-energy phonons by a pulse of
low-energy phonons, the suprathermal distribution of high-energy phonons in long phonon pulses, the mesa
shape of the angular distribution of low-energy phonons, the creation of a «hot line» when two-phonon
pulses cross. The thermodynamic properties of anisotropic quasiparticle systems of He II are derived for all
degrees of anisotropy.
PACS: 67.40.Db Quantum statistical theory; ground state, elementary excitations;
67.40.Fd Dynamics of relaxation phenomena.
Keywords: superfluid 4He, quasiparticle anisotropic systems, phonon, roton.
1. Introduction
In superfluid helium, it is possible to create quasi-
particle systems which are strongly anisotropic (see, for
example, Refs. 1–3). A phonon pulse is a strongly aniso-
tropic quasiparticle system and can be created by a short
current pulse of duration t p ~ 10 7� –10 5� s in a heater
which is immersed in the superfluid helium. If the bulk
temperature is ~ 50 mK, then the influence of any ambient
excitations, on the pulse, can be neglected, and the quasi-
particle pulse moves in a «quasiparticle vacuum» from
the heater to the detector. The quasiparticle pulse has a net
momentum along the direction normal to the heater which
defines the anisotropy axis. In momentum space, the dis-
tribution of occupied states is strongly anisotropic and
this contrasts with isotropic quasiparticle systems which
have no net momentum and have no special direction in
momentum space.
We shall discuss a number of properties of anisotropic
quasiparticle systems. In quasiequilibrium they can be
characterized by a temperature T and a drift velocity u. In
isotropic systems u � 0, and in weakly anisotropic quasi-
particle systems the drift velocity u is much smaller than
the maximum drift velocity u max. These have been stu-
died for many years, but the properties of strongly aniso-
tropic quasiparticle systems, when the drift velocity u is
near the maximum drift velocity u max, have not been
considered until now.
2. Quasiparticle anisotropic systems of superfluid
helium: experiment and theory
Anisotropic pulses of quasiparticles in superfluid he-
lium are a unique physical system which show interesting
and unusual behavior. One of the most amazing phenom-
ena is the creation of high-energy phonons (h-phonons)
with energy � h ~ 10 K from a pulse of low-energy pho-
nons � � ~ 1 K (�-phonons).
When a single short current pulse is applied to the
heater, two-phonon pulses are detected by the bolometer
(see Fig. 1). The faster pulse is due to �-phonons, and the
slower one is due to h-phonons. Several experiments, in-
cluding quantum evaporation of 4He atoms from the free
surface of liquid helium [4], unambiguously prove that
h-phonons are produced in the bulk of liquid helium and
not in the heater. The heater injects mainly �-phonons
with short high-power pulses, but with long low-power
pulses, it is possible to create rotons. The theory that a
pulse of cold phonons can produce hot phonons has been
presented in Refs. 5, 6.
© I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt, 2007
The h-phonons are created from the �-phonons as the
system tries to reach equilibrium. The �-phonons in the
pulse interact by four-phonon processes (4pp) and create
pairs of low- and high-energy phonons. When the energy
of the high-energy phonon is greater than 10 K, it is very
stable because it cannot spontaneously decay, unlike
phonons with � � 10 K. The created h-phonons are lost
from the back of the �-phonon pulse due to the difference
in group velocities between the �- and h-phonon subsys-
tems (238 and � 189 m/s, respectively), and create the
second pulse. Whereas h-phonons interact weakly,
�-phonon system attains quasiequilibrium very quickly by
fast three-phonon processes (3pp). This causes the
�-phonons to move as a whole with velocity which is close
to a sound velocity c � 238 m/s. These phonons create the
first pulse.
The theory accounts for the high efficiency of the
h-phonon creation process, when up to ~ 50 per cent of the
initial �-phonon energy along the anisotropy axis can be
transformed to h-phonon energy. Using the exact drift
distribution function, the energy density lost by the �-
phonons, due to h-phonon creation, relative to the initial
�-phonon density �( )t , is shown as a function of time in
Fig. 2.
Measurements of time-resolved angular distribution of
�-phonons at several powers and pulse durations were
made to investigate their temporal and spatial develop-
ment [7]. Figure 3 shows angular distributions of
�-phonon signal, integrated over time, for several input
powers for t p � 50 ns. We can clearly see the constant
central region with decreasing outer regions, in the shape
of a mesa. The theory for the evolution of �-phonon pulses
has been developed for the case of cool pulses [8], when
h-phonon creation is neglected. In this case the evolution
along and transverse to the anisotropy axis, are investi-
gated assuming instantaneous relaxation of the �-phonon
system. This is a good approximation, because the charac-
teristic time of three-phonon processes is less then any
relevant time in the problem by two orders of magnitude.
It was shown that in spite of the dispersion of �-phonon
velocities, the �-phonon pulse moves as a whole with only
a slow change in shape. This is in contrast to the case of
noninteracting phonon pulse, which shows considerable
dispersion.
The theory for the transverse evolution of long pulse
gives the angular distribution for �-phonons due to lateral
spreading. As it has a round shape, we suggest that mesa
shape arises from h-phonon creation, the hotter central re-
gion of �-phonon pulse burns more quickly, whereas the
The unusual properties of anisotropic systems of quasiparticles in superfluid
4
He
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1009
0
0.1
0.2
0.3
0.4
0 20 40 60 80 100 120
t, �s
l-phonons h-phonons
S
ig
n
al
p
o
w
er
,
n
W
Fig. 1. The measured phonon signal in liquid 4He showing the
�- and h-phonons produced by a single heater pulse: pulse du-
ration 10 7� s, heater power 10 mW/mm2, propagation length
15.7 mm, THe = 60 mK, at the saturated vapour pressure.
5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
�
t, s�
1
2
3
4
Fig. 2. The energy density lost by the �-phonons, due to
h-phonon creation, relative to the initial �-phonon energy den-
sity �( )t , as a function of time, calculated with � 002. for dif-
ferent values of T equal to 0.041 (1), 0.036 (2), 0.030 (3), and
0.025 (4) K.
In
te
g
ra
te
d
l-
p
h
o
n
o
n
si
g
n
al
,
ar
b
.
u
n
it
s 1600
1400
1200
1000
800
600
400
200
0
–20 –15 –10 –5 0 5 10 15 20
Angle, deg
25 mW
12.5
6.3
3.1
Fig. 3. The low-energy phonon signal, integrated over time,
versus heater angle for a range of heater powers is shown. The
pulse length is 50 ns, propagation time is 16.7 mm, and the
lines are guides for the eye.
cooler outer wings of �-phonon distribution are little
changed. This is due to the very strong dependence of the
h-phonon creation rate on �-phonon energy density. This
interpretation is supported by the fact that the sum of �-
and h-phonon angular distributions does not have a mesa
shape but is rounded. It should be noted that the whole
problem of the spatial evolution of the finite �-phonon
pulse, with h-phonon creation taken consistently into
account, is not yet theoretically solved.
Experiments have been performed in which two short
phonon pulses (phonon sheets) are forced to collide [9].
When the angle between the normals to the two sheets is
less then 13° , there is a strong interaction along the line
of intersection of the two sheets. The amplitude of the
�-phonon signal is greater than the sum of the signals from
pulses which move independently to the bolometer (see
Fig. 4). The interaction between the two sheets gets
weaker as the angle between the two normals is larger
then 13°. Also it has been found that the interaction is
negligible when the pressure is increased to 19 bar.
The theory of this phenomenon has been developed
[10]. If the time for the phonon pulses to cross the volume
of overlap of the two sheet is greater than the time for
phonon scattering
3pp , then phonons from the two pulses
have enough time to interact with each other. The angle
between the momenta of the two incoming phonons must
be small for three-phonon processes to occur. This inter-
action creates a new anisotropic system, the hot line,
which propagates along the total symmetry axis. The sig-
nals from the hot line are considerably greater than the
sum of the signals from pulses which move independently
to the bolometer (see Fig. 4) because energy is redirected,
from the shrinking phonon sheets, onto the bolometer in
the form of the hot line.
If the time for the phonon pulses to cross the volume of
overlap of the two sheets is less than the time for phonon
scattering
3pp , then the phonons from the two pulses
have no time to interact. This occurs when the angle be-
tween two sheets is large or when the 3pp scattering rate
is reduced by applying pressure. Then the hot line does
not form because the pulses pass through each other with-
out interaction.
In addition to the «dynamical phenomena» considered
above, there are very interesting «static properties», in
particular, thermodynamic properties of anisotropic
quasiparticle systems, which we shall discuss in the
following sections.
3. Quasiequilibrium distribution functions of
anisotropic quasiparticle systems
For isotropic quasiparticle systems there is no special
direction as all directions are equivalent. In this case the
distribution function depends on the modulus of momen-
tum p, and the momentum density of quasiparticles is
zero.
Anisotropic quasiparticle systems have a special direc-
tion in momentum space, and the distribution function n
depends on the vector of momentum p, and the momen-
tum density P,
P p p� �
�p
n
d p
( )
( )
3
32
�
,
(1)
is not zero (here � p is the region of integration in momen-
tum space).
The equilibrium distribution function of anisotropic
quasiparticle systems must make the collision integral
equal to zero. Also the momentum density should be non-
zero. These conditions are satisfied by the Bose–Einstein
distribution which is a function of the energy � �� ( )p of
quasiparticles, the temperature T and the drift velocity u:
n
p
k TB
( ) exp
( )
.p
p u
�
� ��
�
��
�
�
�� �
�
�
�
�
�
�
�
�
1
1
(2)
The distribution function n must be positive. As a re-
sult, the drift velocity u can change up to the minimum
value of the phase velocity of the quasiparticles, which
form the system:
0 � � �
�
�
��
�
�
��u u u
p
p
max max
min
( )
,, where
�
(3)
which is determined by the dependence of the quasi-
particle energy � on its momentum p.
1010 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
40 50 60 70 80
sij
s + si j
S
ig
n
al
,
1
0
V
4
�
7
6
5
4
3
2
1
0
–1
t, s�
Fig. 4. The bolometer signals for the two separate pulses
(dash-dotted lines) and for the double pulse (solid line). The
sharp peak at 41 �s is due to the �-phonons; the signal after �
44 �s is due to h-phonons. The sum of the signals from the two
separate pulses is shown as the dotted line. The input pulses
are 100 ns and 6.3 mW.
The energy–momentum relation for superfluid helium
is well known (see, for example, [11]). It contains a pho-
non part � ph ( )p , which has a nearly linear dependence
and the roton part � r p( ), which can be described by a
parabola:
� � �
�ph and ,( ) ( ( )), ( )
( )
p cp p p
p p
r� � � �
�
1
2
0
2
� (4)
where the parameters for the roton spectrum are
� / .k B � 8 71K, p0 1 91/ .� � �
�1, � � 0 161 4. m , and m4
is the 4He atomic mass, at zero pressure.
The function�( )p , which describes the deviation from
linearity for phonon energy–momentum relation, is small
(| ( )|� p �� 1). Nevertheless it completely determines the
type and strength of phonon interactions. According to
[12], at low pressure, the function �( )p is positive for
phonons with momentum 0 � �p pc . This is the momen-
tum range of the �-phonons where the fastest scattering is
3pp. For small p the function�( ) ~p p 2. At cp k B� � 7 K
the function�( )p reaches its maximum value � 0 04. . After
that the function �( )p decreases and becomes zero at
p pc� . For p pc� , �( )p is negative, so 3pp is prohibited
by the conservation laws. In this case the fastest scatter-
ing is 4pp. This is the momentum condition for h-phon-
ons. At zero pressure cp kc B� �10 K. We shall use for
numerical calculations a convenient analytical approxi-
mation of the function �( )p introduced in Ref. 13.
It follows from Eqs. (3) and (4) for the pure phonon
system, that the minimum phase velocity u ph is close to
the sound velocity c. For the phonon–roton system, the
minimum phase velocity u pr � � / 0 is determined by the
roton minimum.
Besides condition (3), the drift velocity u must have a
value which makes the system thermodynamically stable.
The general thermodynamic inequality [14] that can be
applied to superfluid helium, determines the line of ther-
modynamic stability u u T� st ( ). We have derived (see
Refs. 13, 15) the stability curves for anisotropic phonon
and phonon–roton systems. The results are presented on
Fig. 5. At T � 0 the respective maximum drift velocity
coincides with the Landau critical velocity for the corres-
ponding quasiparticle system. The critical velocity de-
creases monotonically with increasing temperature, but
remains close to the u max for temperatures up to T ~ 1 K.
So, strongly anisotropic phonon (Fig. 5,a) and phonon–
roton systems (Fig. 5,b) are thermodynamically stable up
to high temperatures.
The quasiequilibrium state is realized in the pulse for
phonons whose time of relaxation
�ph ( ) is smaller than
the current pulse duration t p . Our calculations show [6]
that, in long sufficiently pulse (t p � �10 7 s), there are
phonons with energy up to 11 K in the quasiequilibrium
state. So the quasiequilibrium phonon system can be char-
acterized by the Bose–Einstein distribution (2) up to some
phonon momentum p f � 11 K.
Let us introduce !� �1 cos , where the angle ! is bet-
ween the phonon momentum p and the anisotropy axis z,
directed along the vector of drift velocity u. Also, we in-
troduce the parameter of anisotropy � � �1 u c. Then the
quasiequilibrium phonon distribution function nph can be
rewritten in a form useful for analyzis:
" #n p
cp
k T
p
B
ph ( , ) exp ( ) ( ) . � � � � �
�
�
��
�
�
�� �
�
�
�
�
�
�
�
1 1
1
(5)
For the isotropic phonon system the parameter of ani-
sotropy is equal to unity, and the curly bracket, in the ex-
ponent of the expression (5), is close to unity. For the
strongly anisotropic phonon systems, the anisotropy pa-
rameter is much less than unity. �( )p is also small and
the curly bracket, in the exponent is small, when is
small. As a result, when the angle ! is small we have a
The unusual properties of anisotropic systems of quasiparticles in superfluid
4
He
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1011
0 0.4 0.8 1.2 1.6 2.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
stable
unstable
a
T, K
T, K
u
/u
p
h
u
/u
r
0 0.4 0.8 1.2 1.6 2.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
unstable
stable
b
Fig. 5. The maximum thermodynamically stable drift velocity
divided by umax for anisotropic phonon ( p f �11 K) (a) and for
anisotropic phonon–roton (b) as a function of temperature.
large number of phonons. This situation in momentum
space, is shown schematically on Fig. 6,a for typical val-
ues in experiments � 0 04. and T � 0 05. K; p f �11 K.
In directions close to the anisotropy axis, there are a
large number of high-energy phonons with energy higher
than10 K, because for these phonons�( )p is negative and
cancels in the distribution function (5). As a result, the
curly bracket, in the exponent, is smallest for h-phonons.
The large number of high-energy phonons in a long-
phonon pulse we call a suprathermal distribution (see
Refs. 16, 17).
In the direction perpendicular to the z axis there are the
same number of phonons as in the isotropic case at
T � 0 05. K. This is because, for the perpendicular direc-
tion, is equal to unity and the curly bracket, in the distri-
bution function (5), is also close to unity. At this tempera-
ture, the number of phonons is very small. So we can have
the unusual situation in a strongly anisotropic phonon
system; the temperature of the pulse can be lower than the
temperature of superfluid helium into which it is injected,
but the pulse has a much larger number of phonons than
the helium.
We can define a cone in momentum space, which is cut
from the isotropic Bose distribution for phonons (see
Fig. 6,b):
n pc
c
cp k TB p
( , )
( )
/
!
$ ! !
�
�
�e 1
. (6)
If this cone has a cone angle ! c of 12° and the temperature
T p of the isotropic distribution is1 K, then the energy den-
sity and momentum density, is equal to the anisotropic
system defined by the typical values of � 0 04. and
T � 0 05. K, which were mentioned above. The distribution
function nc , which includes the step function $, is the
Bose-cone approximation. Note that in this cone approxi-
mation, the suprathermal distribution is absent.
4. Energy and angular distributions of anisotropic
quasiparticle systems
Figure 7,a shows the momentum dependence for the
anisotropic phonon system with T � 0 05. K and � 0 04. of
the energy distribution function E p( ), which is deter-
mined for all quasiequilibrium quasiparticle systems by
the following expression:
E p n p p d
k Tp
u
p pu
k T
B
B
( ) ( , ) ln exp
( )
� � � � �
��
�
��
�
��
� �
0
2
2 1
�
��
�
�
�
�
�
�
%
&
'
('
�
� � �
��
�
��
�
�
��
�
�
�
�
�
�
)
*ln exp
( )
1
� p pu
k TB
'
+'
. (7)
The energy distribution for the Bose function, when the
drift velocity is equal to zero, has one maximum, as is
1012 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
a bpx px
pz pz
Fig. 6. The exact distribution function (a) is shown schemati-
cally for typical values in experiments � 004. , T � 005. K and
p f �11 K along with the conus distribution function (b) with
cone angle !c � 12° and the temperature Tp �1 K.
1 2 3 4 5 6 7 8 9 10 110
0.1
0.2
0.3
0.4
cp/kB, K
cp/kB, K
a
1
d
E
c
d
p
,
2
4
–
3
1
0
m
1
d
E
c
d
p
,
2
4
–
3
1
0
m
0 2 4 30 32 34 36 38 40
0.05
0.10
0.15
0.20
0.25
0.30
b
Fig. 7. The momentum dependence of the quasiparticle energy
distribution for the anisotropic phonon system with T � 0 05. K,
� 0 04. (a) and for the anisotropic phonon–roton system with
T � 0 3. K, u ur/ .� 0 63 (b), which corresponds the same energy
and momentum density as for Fig. 7,a.
well known. For a strongly anisotropic phonon system,
with typical experimental values and T , we see there are
two almost separate subsystems: a subsystem of low-en-
ergy phonons, which forms the first maximum and a sub-
system of high-energy phonons, which forms the second
maximum. The second maximum is caused by the nega-
tive value of dispersion function �( )p for h-phonons,
which cancels in Eq. (5), and is the suprathermal distri-
bution.
In Fig. 7,b we see the momentum dependence of the
energy distribution function E p( ) for the anisotropic
phonon–roton system, which has the same energy and
momentum density as the phonon system in Fig. 7,a, but
now T � 0.3 K and u u r� � 0 63. . Even though the temper-
ature is low, the roton energy density is very high com-
pared to that in the isotropic phonon–roton system with
u � 0 and T � 0 3. K. Anisotropic phonon–roton systems
are realized when we have superfluid flow in narrow
channels where the normal component is at rest. Also they
should be formed in very long pulses in superfluid he-
lium.
It is natural to define the normalized angular distribu-
tion function, which characterizes the anisotropy of a
quasiparticle subsystem, as follows:
W
n p p dp
d n p p dp
p p
p
p
p
p1 2
1
2
1
2
2
0
2
2
� �
�
� �
( )
( )
( )
�
�
,
,
. (8)
Figure 8,a shows angular distribution function
W p p1 2� ( ) for the �- and h-phonons, that form the first and
second maxima in Fig. 7,a. Note, that the h-phonons are
concentrated near the anisotropy axis, with a narrower an-
gular distribution than �-phonons. This behavior was ob-
served in [18].
In Fig. 8,b we see that phonons form a weakly aniso-
tropic background, while the roton subsystem is strongly
anisotropic. It is interesting to note that most of the en-
ergy is in the phonon subsystem, whereas the momentum
is mainly in the roton subsystem.
5. Thermodynamic functions of anisotropic
quasiparticle systems
The the rmodynamic func t ions o f the gas o f
quasiparticles can be found from the expression for the
free energy density F:
F k T
p
k T
d p
B
B
� � �
� ��
�
��
�
�
��
�
�
�
�
�
�� ln exp
( )
( )
1
2
3
3
�
p u
�
, (9)
which we obtain for all levels of anisotropy,
F
k T
u
p dp L
p pu
k T
L
p pu
k
B
B
� �
��
�
��
�
�
�� �
�
�
( ) ( ) ( )
2
2 3 2 2
4
� �
� BT
�
�
��
�
�
��
%
&
(
)
*
+
,
(10)
where
L x
nn
xn
, ,
( ) �
�
-
�. 1
1
e (11)
is a polylogarithmic function of ,th order.
At low values of x, which applies, for example, to a
strongly anisotropic roton gas, it is useful to rewrite the
series (11) as a series in powers of x:
The unusual properties of anisotropic systems of quasiparticles in superfluid
4
He
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1013
5 10 15 200
20
60
100
140
180
W0-10
N
o
rm
al
iz
ed
an
g
u
la
r
d
is
tr
ib
u
ti
o
n
!, deg
!, deg
W10-11 a
0 40 80 120 160
5
10
15
Wr
N
o
rm
al
iz
ed
an
g
u
la
r
d
is
tr
ib
u
ti
o
n
10Wph
b
25
Fig. 8. a — Angular distribution function W p p1 2� ( )! for the �-
and h-phonons, that form the first and second maxima in
Fig. 7,a. b — Angular distribution function W p p1 2� ( )! for
phonon and roton subsystems, that form the first and second
maxima in Fig. 7,b.
L x x
n
n
n
n
n
,
,
,
,
,
( ) ( )
( ) ( )
!( )
si
� � �
�
� � �
�
�
�
�
-
.
�
�
1
1 1
2
1
1
0
n ( ) ( ) ,
, ,
2
1n n x n�
�
��
�
��
� �
(12)
where � is Euler gamma-function; is Riemann zeta-func-
tion.
The entropy density S, heat capacity density C, the mo-
mentum density P, and the density of the normal compo-
nent /n can be obtained by differentiating the free energy
density F:
S
F
T
C T
F
T
F P
u
n� �
0
0
� �
0
0
� �
0
0
�, , ,
2
2
P
u
/ . (13)
For differentiation, one should note that dL x dx, ( ) � �
� � �L x, 1( ).
Our calculation shows that the temperature depend-
ence of the thermodynamic functions for a strongly aniso-
tropic phonon system, is essentially different to the iso-
tropic case. The values of the thermodynamic functions,
of strongly anisotropic phonon systems, are much larger
than for the isotropic system at the same temperature, be-
cause they occupy different volumes in momentum space.
So, for example, at the same low temperature, the energy
density of a strongly anisotropic phonon system with ty-
pical value is more than three orders larger than the iso-
tropic one. This fact is also illustrated by Fig. 9, where the
temperature dependence of entropy density S u uph st( )�
of the extremely anisotropic phonon system divided by
entropy density of isotropic phonon system (u � 0), is
shown.
The general expression (10) can be simplified for the
roton system:
F
p k T
u
L
u
k T
Lr
B
B
� �
�
�
�
�
�
�
�
� �
�
� � �
�
0
5 2
3 2 3 5 2 5 2
2
( )
( )
( )*
�
� �* ( )
,
��
�
�
�
�
�
�
�
%
&
'
('
)
*
'
+'
u
k TB
(14)
where we denote � �* ( ) /u p u u� � �0
2 2� .
It should be noted, that the third term in �* ( )u cannot
be omitted, because in highly anisotropic systems (at
u u� max) it can have a similar value to the difference of
the first and second terms.
If u u� max, then �* ( )u � 0, and the function L5 2 0� ( ) is
equal to ( )5 2� . In this case, when the temperature de-
creases, the free energy of the strongly anisotropic roton
subsystem, decreases as T 5 2� , according to Eq. (14). At
the same time the free energy of the weakly anisotropic
phonon system, varies as T 4 [11].
This temperature dependence gives an unusual prop-
erty: in strongly anisotropic phonon–roton system of
superfluid helium, all the thermodynamic properties of
helium, at all temperatures, are determined by rotons. The
dominance of the rotons over the phonons can also be
seen in the Fig. 10, where the temperature dependence of
the roton entropy density S r divided by the phonon en-
tropy density S ph are presented on the line of thermody-
namic stability, and also for the isotropic systems where
u � 0. This behavior is in strong contrast to the isotropic
quasiparticle system, where rotons «freeze out» faster
than phonons, and so for temperatures less then 0.5 K the
roton contribution can be neglected.
Conclusion
We see that the strongly anisotropic phonon and
phonon–roton systems are thermodynamically stable over
a wide temperature range and can be realized in experi-
1014 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
0 0.5 1.0 1.5 2.0
10
0
10
1
10
2
10
3
10
4
p = 10 Kf
S
(u
(T
))
/S
(u
=
0
)
p
h
st
p
h
T, K
Fig. 9. The temperature dependence of entropy density
S u uph st( )� of the extremely anisotropic phonon system divi-
ded by entropy density of the isotropic phonon system (u � 0).
0 0.5 1.0 1.5 2.0
10
–1
10
0
10
1
10
2
10
3
10
4
S
/S r
p
h
T, K
u = 0
u = u (T)st
Fig. 10. The temperature dependence of the roton entropy den-
sity Sr divided by the phonon entropy density Sph on the line
of thermodynamic stability and for the isotropic, u � 0, system.
ments. A number of unusual phenomena have been dis-
covered recently such as the phenomenon of the creation
of high-energy phonons by a pulse of low-energy phon-
ons; the suprathermal distribution of high-energy pho-
nons in long phonon pulses; the mesa shape angular dis-
tribution of low-energy phonons; the phenomenon of the
«hot line» formation, when two phonon pulses cross.
The theoretical results on anisotropic quasiparticle
systems considered in this paper are:
1. The energy distribution of a strongly anisotropic
phonon system has two maxima. The second maximum, is
formed by h-phonons, and it demonstrates a suprathermal
distribution, which is realized in long pulses.
2. The h-phonons are concentrated near the anisotropy
axis; they have smaller angular distribution than �-phon-
ons, which was observed in Ref. 18.
3. The values of the thermodynamic functions, of strong-
ly anisotropic phonon systems, are much larger than for
isotropic systems at the same temperature.
4. In the strongly anisotropic phonon–roton system, all
thermodynamic properties of helium, at all temperatures,
are determined by rotons.
5. The thermodynamic functions of strongly aniso-
tropic phonon and phonon–roton systems have unusual
temperature dependencies, which are essentially different
from the isotropic system.
Although a number of astonishing facts have already
been discovered, we think that the physics of anisotropic
quasiparticle systems will show more surprising phenom-
ena in the future.
We express our gratitude to EPSRC, UK (grant EP/C
523199/1) for supporting this work.
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