Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
The evolution of a pulse of noninteracting quasiparticles, caused by their different velocities and angular distribution of momenta, is studied theoretically. Equations are found that describe the shape of the pulse surface at any time. The time of the beginning, end and duration of the density o...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width / I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt // Физика низких температур. — 2004. — Т. 30, № 6. — С. 579-589. — Бібліогр.: 20 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1197302017-06-09T03:04:28Z Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. Квантовые жидкости и квантовые кpисталлы The evolution of a pulse of noninteracting quasiparticles, caused by their different velocities and angular distribution of momenta, is studied theoretically. Equations are found that describe the shape of the pulse surface at any time. The time of the beginning, end and duration of the density of the quasiparticle energy flux is determined at a general spatial point. The quasiparticle energy density is considered at all times and positions, and it is shown that the region of high energy density, in the middle of the pulse, is equal to the initial energy density under certain conditions. These theoretical results are discussed in relation to experimental data on the evolution of a pulse of noninteracting phonons in superfluid helium. 2004 Article Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width / I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt // Физика низких температур. — 2004. — Т. 30, № 6. — С. 579-589. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 67.40.Fd http://dspace.nbuv.gov.ua/handle/123456789/119730 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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English |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
| spellingShingle |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Adamenko, I.N. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width Физика низких температур |
| description |
The evolution of a pulse of noninteracting quasiparticles, caused by their different velocities
and angular distribution of momenta, is studied theoretically. Equations are found that describe
the shape of the pulse surface at any time. The time of the beginning, end and duration of the density
of the quasiparticle energy flux is determined at a general spatial point. The quasiparticle energy
density is considered at all times and positions, and it is shown that the region of high energy
density, in the middle of the pulse, is equal to the initial energy density under certain conditions.
These theoretical results are discussed in relation to experimental data on the evolution of a pulse
of noninteracting phonons in superfluid helium. |
| 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 |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| title_short |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| title_full |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| title_fullStr |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| title_full_unstemmed |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| title_sort |
evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119730 |
| citation_txt |
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width / I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt // Физика низких температур. — 2004. — Т. 30, № 6. — С. 579-589. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT adamenkoin evolutionofapulseofnoninteractingquasiparticleswithdispersionandinitialangularwidth AT nemchenkoke evolutionofapulseofnoninteractingquasiparticleswithdispersionandinitialangularwidth AT slipkova evolutionofapulseofnoninteractingquasiparticleswithdispersionandinitialangularwidth AT wyattafg evolutionofapulseofnoninteractingquasiparticleswithdispersionandinitialangularwidth |
| first_indexed |
2025-07-08T16:29:57Z |
| last_indexed |
2025-07-08T16:29:57Z |
| _version_ |
1837096971573133312 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6, p. 579–589
Evolution of a pulse of noninteracting quasiparticles with
dispersion and initial angular width
I.N. Adamenko1,2, K.E. Nemchenko1, V.A. Slipko1, and A.F.G. Wyatt3
1Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
E-mail: nemchenko@pht.univer.kharkov.ua
2Scientific and Technological Center of Electrophysics, NAN of Ukraine
28 Chernyshevskyi Str., P.O. Box 8812, Kharkov 61002, Ukraine
3School of Physics, University of Exeter, Exeter EX4 4QL, UK
Received December 18, 2003
The evolution of a pulse of noninteracting quasiparticles, caused by their different velocities
and angular distribution of momenta, is studied theoretically. Equations are found that describe
the shape of the pulse surface at any time. The time of the beginning, end and duration of the den-
sity of the quasiparticle energy flux is determined at a general spatial point. The quasiparticle en-
ergy density is considered at all times and positions, and it is shown that the region of high energy
density, in the middle of the pulse, is equal to the initial energy density under certain conditions.
These theoretical results are discussed in relation to experimental data on the evolution of a pulse
of noninteracting phonons in superfluid helium.
PACS: 67.40.Fd
1. Introduction
Quasiparticle pulses are studied theoretically and
experimentally in various fields of modern physics.
The variety of phenomena that appear are due to a
number of factors, the most important of which is the
role played by following:
1. The dependence of the quasiparticle energy on
momentum (the dispersion or energy—momentum rela-
tion). Classical particles can be considered as quasi-
particles with a quadratic energy—momentum relation.
2. The angular width of the pulse, which is deter-
mined by the angular range of the quasiparticles’
momenta.
3. Interactions between quasiparticles.
4. The interaction of quasiparticles with the me-
dium in which the pulse moves.
The above-mentioned factors determine, for exam-
ple, phenomena observed in photon pulses propagat-
ing in an optically transparent medium with variable
refraction index, e.g., optical waveguides, or in a
medium where the refractive index depends on photon
energy. Another interesting property of anisotropic
crystals is phonon focusing, which is caused by pho-
nons having different velocities in different directions.
Also a large variety of phenomena are observed in
pulses of common particles.
Superfluid helium (He II) is a unique medium for
considering quasiparticle propagation. This is due to a
number of factors.
1. In superfluid helium there exist quasiparticles
with different energy—momentum relations, phonons
with an almost linear relation, R � -rotons with a rela-
tion which is close to quadratic but at a finite energy
and momentum, and R �-rotons, whose velocity and
momentum have opposite directions. The results of ex-
periments on phonon pulses can be found in [1,2],
R � -rotons in [3,4], and R �-rotons in [5].
2. By changing the temperature of superfluid he-
lium, one can control the interactions of quasiparticles
with the ambient medium. So, at relatively high tem-
peratures, a pulse of quasiparticles moving in super-
fluid helium interacts with the gas of thermal excita-
tions of He II. At relatively low temperatures, when
the contribution of the thermal excitation can be ne-
glected, the pulse propagates in a «superfluid vacu-
um», and its evolution is just determined by the pro-
perties of the quasiparticles in the pulse.
© I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt, 2004
3. Interactions between quasiparticles in superfluid
helium can be altered by pressure. The use of pressure
allows us to reach two limiting cases: the creation of a
pulse of strongly interacting phonons and a pulse of
almost noninteracting phonons.
At the saturated vapor pressure, low-energy phon-
ons (l-phonons) with energy � � 10K have anomalous
dispersion [6]. In this case, fast three-phonon pro-
cesses are allowed and cause an almost instantaneous
equilibrium in the system of l-phonons that form the
pulse. This fast relaxation causes a unique phenome-
non; a pulse of low energy-phonons creates high-en-
ergy phonons (h-phonons). Their energy � � 10K, is an
order of magnitude greater than the temperature of
the l-phonons. This phenomena was discovered and de-
scribed in [7,8], and theory of this process was pre-
sented in [9–11]. The results of an experimental study
of the evolution of strongly interacting phonons was
presented in [12,13] and the theory for these pulses
was given in [14].
At a pressure of 19 atm, the dispersion of phonons
becomes normal [15]. In this case fast three-phonon
processes are prohibited by energy and momentum
conservation laws, and four-phonon processes are
weak. Thus, it is possible to create a pulse of almost
noninteracting phonons, that move in a «superfluid
vacuum», formed by He II. Results of experiments on
the evolution of such pulses formed by noninteracting
phonons are presented in [16,17].
The main aim of this paper is the theoretical study
of the evolution in time and space of a pulse of
noninteracting quasiparticles propagating in vacuum.
This is the first step to solving the inverse problem
which would enable us to derive the initial character-
istics of a phonon pulse, created by a heater, from de-
tected signals at different points in space. The results
of this theory are compared to the experimental data
of [16,17]. It complements the theory of the evolution
of a pulse of strongly interacting phonons (see [14])
and can stimulate new experiments. Moreover, the
methods presented in this paper can be used to de-
scribe the evolution of pulses formed by weakly inter-
acting particles and quasiparticles in other fields of
physics.
2. Description of a pulse of noninteracting
quasiparticles
A pulse of quasiparticles can be described by the
quasiparticle distribution function n t( , , )p r that deter-
mines the number of quasiparticles in a quantum state,
which includes the point of phase space p r, .
An important property of a pulse is its energy den-
sity
E t n
p dpd
p
p
p
( , )
( )
r � � �
0
1 2
32
�
�
�
, (1)
where � �� ( )p is the energy of the quasiparticle, p
is the solid angle in which the momenta of the
quasiparticles lie, and the limits p0 and p1 determine
the momentum interval for the quasiparticles under
consideration. However, the signal amplitude mea-
sured by a bolometer is determined by the energy flux
density I through the area da, which is equal to the
energy passing per unit time through a unit surface
area da situated at point r:
I t n
p dpd
p
p
p
( , )
( )
,r v Na� � �
0
1 2
32
�
�
gr
�
(2)
where v pgr �
� is the group velocity of the quasi-
particle, and Na is the unit vector of the area da.
The distribution function n, included in (1), (2)
can be found from the solution of the kinetic equation
�
�
n
t
n
Jv
rgr , (3)
where J is the collision integral, which determines
the change in the number of quasiparticles per unit
time in the quantum state under consideration, caused
by their interaction.
The kinetic equation (3) should be supplemented
by the initial condition
n t n( , , ) ( ,p r p r)� �0 0 , (4)
which determines the pulse in its initial state at t � 0.
In experiments this is the time of the end of the
heater pulse. Equation (3) together with the initial
condition (4) is the complete formulation of the
mathematical problem for deriving the function
n t( , , )p r . In the general case such a problem has no
analytical solution, since the collision integral J in
(3) is a nonlinear integral operator, including distri-
bution functions with different momenta. So various
model solutions are used to treat definite physical
problems. For example, in [14], a model solution for
a phonon system with fast relaxation allowed a com-
plete description of the evolution of a phonon pulse in
superfluid helium and gave results which compared
with experimental data [12,13].
Here we consider the opposite limiting case, when
the interactions between quasiparticles are so weak
that their contribution to the evolution of the pulse
can be neglected. In this case, the collision integral J,
in the kinetic equation (3), is equal to zero, and the
solution of the problem can be easily found and looks
like
580 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
n t n t( , , ) ( , )p r p r v� �0 gr . (5)
Using this solution we can rewrite (1) and (2) as fol-
lows:
E t n t
p dpd
p
p
p
( , ) ( )
( )
r pr v� �� �
0
1
0
2
32
�
�
gr
�
, (6)
I t n t
p dpd
p
p
p
( , ) ( )
( )
r v N p, r va� �� �
0
1
0
2
32
�
�
gr gr
�
. (7)
According to Eqs. (6) and (7) the temporal deforma-
tion of the pulse from its initial state is determined by
two factors:
1. The finite angular range of the momenta in the
pulse, which is defined by the solid angle
p p� 2�� , (8)
where
� �p p� �1 cos (9)
is the angular-width parameter. If � p
0, quasi-
particles in the pulse have different directions, and
this results in a deformation of the initial pulse.
2. The dispersion, when quasiparticles with differ-
ent momentum have different group velocities, results
in a temporal widening of initial pulse.
If dispersion is neglected, the group velocity can be
written in the form v pgr � c p, and it follows from
Eqs. (6) and (7) when n p0( , )r that
E t E ct
p
d
p
p
( , ) ( )r r
p
� �� 0
, (10)
I t c E ct
p p
d
p
p
( , ) ( )r r
p p
Na� �� 0
, (11)
where
E E t p
p dp
p
p
p
0 0
2
3
0
2
0
1
( ) ( , ) ( , )
( )
r r n r� � � � � �
�
(12)
is the energy density in the initial pulse.
Equations (10) and (11) have a simple physical
meaning which allows us to describe, the temporal
and spatial evolution of the phonon pulse without do-
ing the integration. In order to find the energy density
and the energy flux density at the point r and at time t
according to Eqs. (10) and (11), one should consider a
sphere of radius ct and center at r. The energy density
at the point r is equal to the average value of the ini-
tial density on the spherical surface of a segment of
this sphere with solid angle p .
By analogy, one can interpret the integrands in the
general expressions (6) and (7) as the partial contri-
bution from quasiparticles with momentum p to the
energy E t( , )r and to the flux I t( , )r .
3. The shape of the quasiparticle pulse at
different times
Let us consider a pulse, moving along the z axis,
with the initial distribution function
n r n p g
p
p
z
p0 0( , ) ~ ( , ( ) ( cos )p r) r� �� � , (13)
where
g L r L z z( ) ( ) ( ) ( )| |r � � ��� � � (14)
where � is the Heaviside step function, which is equal
to unity or zero when the argument is positive or ne-
gative, respectively. n r0( , )p is nonzero inside the cy-
linder (14) for all values of momentum, determined
by the limits of integration in Eqs. (1) and (2). This
distribution function describes the pulse considered
below. It consists of quasiparticles whose momenta lie
inside a cone with solid angle p , moving along an
axis parallel to the z axis of the cylinder (14).
Let us suppose that the moduli of the group veloci-
ties of the quasiparticles in this pulse, vary between
the maximum value
v c vmax � � 1 (15)
to the minimum value
v c vmin � � 2. (16)
Let us obtain the equation that describes the shape of
this pulse at different times. At time t, the length of
the quasiparticle’s path changes from its maximum
value
L v tmax max� (17)
to its minimum value
L v tmin min� . (18)
At time t, the cross section of the pulse which in-
cludes the z axis (see Fig. 1) is bounded by the front,
back, and side surfaces. These are formed by quasi-
paricles which were in the corresponding surfaces of
the initial pulse.
According to (6), the front and the side surfaces of
the pulse at time t are formed by the points z and r,
which are the centers of segments of spherical surfaces
with radius Lmax and solid angle p with axes di-
rected oppositely to the z axis. The spherical surface of
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 581
a segment should not intersect the front surface of ini-
tial pulse but just have a common point with the ini-
tial front surface. A plane section through the symme-
try axis is shown in Fig. 1. In other words, the points
that form the front and side surfaces are at the maxi-
mum distance Lmax from the initial pulse.
As a result, when r varies between the limits
0 � � �r L (19)
the front surface is the vertical plane with coordinate
z L LF � �| | max . (20)
As r increases in the range:
L r r tF� � � ( ), (21)
where
r t L LF p( ) sinmax� �� � , (22)
the front surface becomes an arc, which is described
by the equation
( ) ( )| | maxz L r L L� � � ��
2 2 2 . (23)
This equation determines the r and z coordinates of
the front surface of the pulse.
At r rF� the side surface is a cylinder of radius rF ,
with front surface at z L L p� �| | max cos� and back
surface at z L p� max cos� .
At points r rF� the signal is equal to zero. Here it
should be noted that the second term in Eq. (22) de-
scribes the widening of the pulse caused by its finite
angular width.
According to (6), the back surface of the pulse at
time t is defined by the points z and r, which are the
centers of segments of spherical surfaces with radius
Lmin and solid angle p with axes directed oppositely
to the z axis. The spherical surface of this segment
should intersect the back surface of the initial pulse and
the segment should include the maximum volume of
the initial pulse. In this case, the points that form the
back surface are at the minimum distance from the back
surface of the initial pulse, for a given Lmin; see Fig. 1.
As a result, at
t t
z
v p
� ��
�
�min cos
, (24)
where
z L p� �� � tan (25)
(see at Fig. 1 pulse at t t� 1), the back surface is the
vertical plane with the coordinates
z LB p� min cos� , (26)
and the coordinates r are confined by the following
inequality:
0 � �r r tB( ), (27)
where
r t L LB p( ) sinmin� �� � . (28)
At the time t t� � (see pulse at t t� 2 in Fig. 1) the
back surface is the vertical plane with coordinates zB ,
if the coordinates r are confined by the following in-
equality
r t r r ta B( ) ( )� � , (29)
where
r t L La p( ) sinmin� � �� . (30)
For smaller values of r, in the range
0 � �r r ta ( ), (31)
the back surface becomes an arc and is described by
the equation
z r L L2 2 2� � ��( ) min . (32)
This surface connects the points r ra� , z zB� with
the points
r z L L� � � �0 2 2; min . (33)
The points of the back vertical surface with coordi-
nates r rB� , z zB� are connected with the end of the
582 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
0 1 2 3 4 5 6 7 8 9
4
2
0
2
4
z, mm
r,
m
m
Fig. 1. The shape and form of the structure of the phonon
pulse with t sp � 4� and �p � 18�, and L� � 1mm at times
t s t1 83� �. � � and t s t2 19� �� � in the plane of the cross
section which includes the axis of the initial pulse. Solid
lines represent phonons which move with maximum velo-
city vmax � 363 m s, and dotted lines correspond to the
minimum velocity vmin � 265 m s. At time t1, the rectangu-
lar region, in the center of the pulse, has the initial value
of the energy density for phonons with a given velocity.
The intersection of all these regions gives the region with
the initial value of the energy density. At time t2 the region
in the center of the pulse has a relatively high energy den-
sity, but the dispersion has already led to its decrease.
side face, which has the following coordinates:
r r L LM p� � �� max sin� , z z LM p� � max cos� by a
cone-like surface formed by the dispersion.
The length of the pulse L t rz ( , ), at different t at
various r, is given by the relation
L t r z t r z t rz ( , ) ( , ) ( , )� �front back . (34)
At t t� � when r is changing between the limits
(19), using the relations (20), (26), and (34) we get
L t t r L L v v t ct v tz p p( , ) ( ) .| |� � � � � � � ��� � �0 1 2 2
(35)
The second term on the right-hand side of this equa-
tion describes the widening of the pulse caused by
dispersion, the third term is caused by the finite an-
gular width of the pulse, and the fourth term is due
to the product of dispersion and finite angular width.
At t t� � and r � 0 from relations (20), (33) and
(34) we obtain
L t t r L v v tz ( , ) ( )| |� � � � � �� 0 1 2
� �
�
�
�
�
�
�
�
�
�L
L
L
min
min
1
2
2
. (36)
At L L� ��2 2 1min in the third term of the right-hand
side of Eq. (36) one can extract the contribution due
to the finite angular width
L t t r L v v tz ( , ) ( )| |� � � � � �� 0 1 2
� � �
�
�
��
�
�
��
� �L
ct
L
ct
c
v
2 2
2 2
1
min
. (37)
Comparing the results of (35) with (36) and (37) it
follows that at any time, the widening of the pulse
caused by dispersion, grows. The influence of a finite
angular width has the opposite effect (near the axis of
the pulse). At t t� � the length of the pulse grows
linearly, and at t t� � the pulse width, caused by the
finite angular width, begins to decrease as 1 t.
4. The energy density of a phonon pulse
The energy density of a phonon pulse at different
points in space and time is determined by the initial
phonon distribution function (4). Let us consider the
case when the initial distribution (5) is given in the
following form:
n n p g
p
p
z
p0 0( , ) ( ) ( cosp r r� �
�
�
��
�
�
��)� � . (38)
In order to simplify this problem we will not take dis-
persion into account. Then the energy density at dif-
ferent points in space and time can be calculated by
Eq. (6) taking into account (38), can be rewritten as
follows:
E t E g ct
p
d
p
p
( , )r r
p
� �
�
�
��
�
�
��� 0
, (39)
where
E n p
p dp
p
p
p
0 0
2
3
0
1
2
� � � �
( )
( )
�
(40)
is the initial energy density.
According to Eq. (39), in order to find the energy
density at the point r at time t one should build a cir-
cular cone with apex at the point r and radius
L ctc � , (41)
solid angle p and with a spherical end cap of radius
Lc. The axis is directed oppositely to the z axis. Be-
low, we refer to this circular cone as simply the cone.
The energy density at the center of the pulse is de-
termined by that part of the spherical surface which
lies inside the initial pulse (38). If it lies completely
inside the initial pulse, then, according to Eq. (39),
the energy density at the point of the apex of this seg-
ment is equal to the initial value of the energy density
(40). If the spherical surface includes no points of the
initial pulse, then the energy density at the apex is zero.
This graphical method of explanation helps us to
consider the distribution of the energy. At times t t� �
(see pulse at t t� 1 on Fig. 1) the energy density has a
maximum value inside the circle with coordinates
z L LR c p� �| | cos� , and 0 � �r r za R( ), (42)
where
r z L
z L
za ( )
| |
� �
��
�
�
�
�
�
�
�� 1
�
(43)
for the case
L Lc p| | � � . (44)
The spherical surfaces of the cones whose apexes are
situated on the circle (42) are found to lie completely
inside the initial pulse, and so the energy density on
the circle (42) is equal to the initial value.
It should be noted that according to Eq. (35),
when the inequality (44) is valid, the length of the
initial pulse is greater than its longitudinal broaden-
ing caused by the finite angular width. In this case,
according to inequality (44), there exists a region in-
side the pulse, in which the energy density is maximal
and equal to the value of the energy density in the ini-
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 583
tial pulse (40) (see cylindrical region at t t� 1 on
Fig. 1). This region is closed by two circles at each
end, one of which is given by Eq. (42) and another is
given by the following expressions:
z LL c� , and 0 � �r r za R( ). (45)
The side surface is described by the equation
r r za R� ( ).
The length of this region is
z z L LR L c p� � �| | � , (46)
and the length of the pulse at this time is equal to
L L Lz c p� �| | � . (47)
As a result, the region with the maximum value of
energy density, E0, is separated from the front and
back surfaces by the distance Lc p� , which is equal to
the widening of the pulse. During the motion of the
pulse along the z axis, the size of this region with E0
shows a monotonic decrease and disappears at the
point z L z� �| | � , which is the point of intersection of
the lines r za ( ) with the z axis.
Here we should note that according to [14] a region
of the pulse with density E0 exists in the opposite li-
miting case, when the pulse is formed by strongly in-
teracting phonons. However, the physical reasons for
this region and its decrease are quite different in the
two cases. Moreover, a phonon pulse of strongly inter-
acting phonons, with the initial distribution function
(38), moves along the axis z with much less transverse
broadening than for noninteracting particles, in spite
of the angular width and dispersion (see [14]).
At t t� � , the integral (39) always has a region
which gives no contribution to the energy density,
i.e., where the initial pulse is absent (see pulse at
t t� 2 in Fig. 1). That is why at times t t� � , the en-
ergy density, at all points in the pulse, is less than the
initial energy density (40).
However, in this case the pulse has a region where
the energy density is greater than at other points in
the pulse. This region has coordinates
z L L LR c
�
�� � �| |
2 2 and r � 0, (48)
since the spherical surface of the cone, with apex at
the point (48), intersects the edges of the front sur-
face of the initial pulse at the point with coordinates
z L� | | and r L� � . Clearly this region contributes
most to the integral (39).
The situation corresponding to inequality (44) in
the case considered case takes place at
L L
L
L
c
c
| | � � �
�
�
�
�
�
�
�
�
�1 1
2
2
, (49)
when the length of initial pulse is greater than the
widening caused by finite angular width, described
by the third term in right-hand side of Eq. (36).
When the inequality (49) is satisfied, the pulse at
t t� � has a relatively high energy density, separated
by the surfaces described by Eq. (48) and the point
z L rL c
� � �and 0, (50)
which is the apex of the cone whose spherical surface
intersects the side face of the initial pulse and touches
its back face.
5. The beginning, end, and duration of the
quasiparticle pulse at different spatial points
Consider the pulse at t � 0, described at the begin-
ning of Sec. 3. Taking into account the symmetry of
the initial pulse, it is sufficient to consider the solu-
tion of the problem in the plane of the cross section
which includes the z axis, and coincides with the axis
of the cylinder (14), which describes the surface
boundaries of the initial pulse.
In experiments, the signals are detected with detec-
tors which are oriented so that all the quasiparticles in
the pulse have
v Nagr
0. (51)
So, below we will consider the condition (51) to be
valid.
The time of the start of the signal t z rbeg ( , ) detected
at the point with coordinates z and r can be found
from the geometric interpretation of formula (6) in
the way we now describe. One should again construct
a cone with apex at the point z, r and axis directed op-
positely to axis z, with solid angle p . The surface of
the cone’s spherical end cap should have just one com-
mon point with the front or side surfaces of the initial
pulse. Then the radius of this segment is calculated
and this value gives us the minimum value R z rmin( , ),
determined by position of point z, r. The time of the
start of the signal is given by the following formula:
t z r
R z r
vbeg ( , )
( , )min
max
� . (52)
Using this interpretation, the dependence on coordi-
nates z at any r is given by
t z r L
z L
vbeg ( , )
| |
max
0 � � �
�
� . (53)
The signals at time (53) is formed by phonons moving
parallel to z axis with velocity vmax from the front
surface of the initial pulse. On Fig. 1 at t t� 1 and
t t� 2 these phonons create the front surface.
584 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
Figure 2 shows the dependence on time of the sig-
nals at points with different values of r with a fixed
value of z. In Fig. 2 the time (53) at z z� 1 and z z� 2
is shown by the vertical lines which are determined by
the interval (19).
On further increasing r, when
L r r zF� � � ( ) , (54)
where
r z L z L L
z z L
zF p( ) ( )| |
| |
� � � �
� �
� �tan�
�
�
.
(55)
The time of the start of the signal is given by
t z L r r zFbeg ( , ( ))� � � �
� � � ��
1 2 2
v
r L z L
max
| |( ) ( ) . (56)
The signal at time (56) is formed by quasiparticles
that move at a definite angle 0 � �� �p with respect to
the z axis, with velocity vmax from the point z L� | | and
r L� � of the initial pulse. Figure 1 shows that at t t� 1
and t t� 2 these quasiparticles create the arc on the
front surface of the pulse (23). In Fig. 2 at z z� 1 and
z z� 2, the time (56) is shown by the curve (56).
And, finally, when r lies between the limits
r z r r zF B( ) ( )� � , (57)
where
r z L z L
z z
zB p( ) � � �
�
� �tan� �
�
, (58)
and the time of the signal start is given by
t z r z r r z
r L
vF B
p
beg ( , ( ) ( ))
sinmax
� � �
� �
�
. (59)
The signals at time (59) are formed by quasi-
particles, which move at angle �p with respect to z
axis with the velocity vmax from the side surface of
the initial pulse. In Fig. 1 at t t� 1 and t t� 2 these
quasiparticles form the cylindrical side surface
r r tF� ( ) (see Eq. (22) and the second sentence after
Eq. (23)). Figure 2 at z z� 1 and z z� 2 shows the time
(59) by the segment of line inside the interval (59).
At r r zB� ( ) there is no quasiparticle signal because
�p is the maximum angle of quasiparticle motion in
the initial pulse.
Using the geometric interpretation of expression
(6), let us find the time t z rend( , ) of the end of the sig-
nal at a definite point z, r. In order to do this, it is suf-
ficient to create the same cone as before but now this
segment must intersect the back face of the initial
pulse and include as many points of the initial pulse as
possible. Then we find the radius of this segment
R z rmax( , ) and calculate the desired time:
t z r
R z r
vend( , )
( , )max
min
� . (60)
Using this approach, for points with z z� � , when
the radius changes inside the interval
0 � �r r zB( ), (61)
we find the maximum value for the time of the signal
end, as will be shown below,
t
z
v p
end
max( )
min( )
�
�1 �
. (62)
The signal at time (62) is created by quasiparticles
which move at the angle �p with respect to the z axis,
with velocity vmin, from the back surface of the ini-
tial pulse. In Fig. 1, at t t� 1, these phonons form the
back surface of the pulse. Figure 2, at z z� 1, shows
the time (62) by the segment of the vertical line in-
side the interval (61).
In Fig. 2 at z z� 1 and z z� 2 an imaginary horizon-
tal line connects the end of the line describing the be-
ginning of the signal (59) with the end of the vertical
line, which describes the time of the end of the signal.
These correspond to v v� max and v v� min. In Fig. 1
at t t� 1 and t t� 2 the corresponding construction con-
nects the ends of the back vertical surfaces.
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 585
0 5 10 15 20 25 30
4
2
0
2
4
t,�s
r,
m
m
Fig. 2. The beginning, end, and duration of the same
phonon signal as in Fig. 1 at points with coordinates z z1 � �
and z z2 � � at different values of the radius r. Solid lines
represent phonons, moving with maximum velocities, and
dotted lines, with minimum velocities. The cone-like region
at z z� �1 3 mm represents the maximum possible ampli-
tude of the signal. At point z z� �2 7mm there is no region
of constant signal.
At z z� � , when the radius changes inside the inter-
val
r z r r zA B( ) ( )� � , (63)
where
r z L
z z
zA( ) �
�
�
�
�
, (64)
the time of the signal end is given by formula (62)
and this signal is formed by the same quasiparticles as
those at the end of the pulse at z z� 1. In Fig. 1 at
t t� 2 these quasiparticles form the back surface of the
pulse. In Fig. 2 at z z� 2, the time (62) is shown by
the segment of the right vertical line, which is limited
by inequalities (63).
At z z� � , when the radius changes inside the interval
0 � �r r zA( ), (65)
the time of the end of the signal is given by
t z z r r z
v
z r LAend( , ( )) ( )
min
� � � � � � �� 0
1 2 2 .
(66)
At this time, signals are created by quasiparticles that
move at a definite angle 0 � �� �p with respect to the
z axis with velocity vmin, from the point z � 0, r L� �
of the initial pulse. In Fig. 1 at t t� 2 these quasi-
particles form the arc of the back surface, and in
Fig. 2 at z z� 2, the time (66) is described by the
right arc of a hyperbola.
The duration of the pulse at the point z, r is deter-
mined by the equality
�t z r t z r t z r( , ) ( , ) ( , )� �end beg . (67)
According to (53), (56), (59), and (60) at z z� � the
duration of the pulse has a maximum value, when r is
inside the interval 0 � � �r L and equal to
�t t z
v v
v v
z
vp
p
p
max
max min min( )
� �
�
�
�
1 2
1
�
�
, (68)
where t L vp � | | max is the initial length of the pulse.
In the right-hand side of Eq. (68) the second term de-
scribes the increase in pulse length due to dispersion,
and the third term is caused by the finite angular
width.
Using relations (56), (62), and (67) at z z� �
when L r rF� � � the pulse duration is given by
� �t t
v
z LF � � � �max
max
| |(
1
� � � � �( ) ( ) )| |z L r L2 2 . (69)
At r L� � the increase in the pulse length, according
to (69), reaches its maximum value at �tmax. In-
creasing r causes the signal length to decrease so, that
at r rF� we have
�t r r
t z v v
v vF F
p
p p
( )
max min
� �
�
�
�
�
1 1
1 2
� �
. (70)
According to Eqs. (59), (62), and (67) at z z� �
when r r rF B� � , the pulse length is given by
�t
z
v
r L
vB
p p
� �
� �
min maxcos sin� �
. (71)
From this equation it follows that the length of the
pulse decreases from the maximum value (70) to the
minimum value at r rB� and is given by
�t r r
z v v
v vB B
p
( )
max min
� �
�
�
1
1 2
�
, (72)
where the length of the signal does not depend on tp
and is determined by the dispersion together with an
additional contribution from the finite value of the
angular width. At r rB� there is no signal. Expres-
sions (68)–(72) give the length of the pulse at z z� �
for all r.
At z z� � , from Eqs. (53), (66), and (67), for a
value of r inside the interval
0 � � �r L rAmin( , ), (73)
it follows that
�t t z
v v
v v
z
v
r L
zp�
�� �
�
� �
��
�
�
�
�
� �
�
�
�
��
�
�
1 2
2
1 1
max min min
�
��
.
(74)
At r LA � � expression (74) has a maximum value (68),
when r is changing inside the interval r r LA � � � .
At z z� � and r LA � � , when L r rF� � � , the
length of the pulse can be found from expression (69),
and for greater r (r r rF B� � ) by expression (71).
Using the equations (56), (62), and (66) at
z z� 2 �, for r inside the interval L r rA� � � for the
pulse length we get
�t t z
v v
v v
r L
zv
r L
z LA p� �
�
�
�
�
�
�
� �1 2
2 2
2 2max min min | |
( ) ( )
( )
.
maxv
(75)
On the right-hand side of this equation, the third and
the fourth terms describe the contribution of the finite
angular width, which decreases with increasing z.
586 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
At z z� 2 � and L r r rA F� � � � the length of the
signal is given by Eq. (69), and at r r rF B� � , by the
relation (71). At r rB� there is no signal.
The result (68)–(75) describes the length of the
pulse at any value of r and z. From an examination of
these results, it follows that the length of the pulse al-
ways grows linearly due to dispersion. As to the con-
tribution of the finite angular width, at first (when
z z� � and 0 � � �r L ) the signal length grows lin-
early with z. At z z� � such a dependence takes place
only at z z z� �� � 2 and r r LA � � � . In all other
cases, the contribution of the finite angular width to
the increase of the pulse length is either constant (or
grows when � p � 0 — see (70)), or decreases with in-
creasing z. For large z the contribution of the finite
angular width to the increase of pulse length can be
neglected (see (74),(75)).
However, it should be emphasized that according
to Eq. (58) it is the finite angular width that always
leads to the linear increase of the transverse width of
the pulse with increasing z. As energy is conserved,
this results in the signal decreasing monotonically
along the z axis, during the motion of the signal.
6. Temporal dependence of the signal amplitude
in space
Let us consider the case when the initial distribu-
tion is given by relation (38) and omit dispersion,
which is not so important for the signal amplitude. In
this case relation (7) can be written as
I ,t c E g ct
p p
d
p
p
p
(r r
p p
Na) � �
�
�
��
�
�
��� 0
. (76)
The temporal dependence of the signal amplitude at
the point z r, can be obtained from the geometrical in-
terpretation of the formula (76) which we used
above. To find the amplitude at the point z r, we cre-
ate a cone with its apex at this point, with solid angle
p and radius R, and with its axis directed oppositely
to the z axis. The signal amplitude at the apex is deter-
mined by the averaged value of (76) using the initial
energy density (40). The time t R z r( , , ) of the signal,
detected at the considered point z r, , is determined by
t R z r R c( , , ) � . (77)
If all of the spherical-end surface of the cone lies inside
the initial pulse, then, according to (76) at time (77)
the signal amplitude is equal to the initial value which
would be detected by the bolometer at t � 0, with the
given Na , situated inside the initial pulse. If the
curved surface of the cone has no points within the ini-
tial pulse, the signal at time (77) is equal to zero.
Using this approach we can find, without integra-
tion, the point z r, at which the amplitude will have
maximum value and find the time interval when the
maximum value will be observed.
During the initial stage of the evolution, there ex-
ists, for a definite time, a region (denoted by the sub-
script «spot») where the amplitude is equal to its ini-
tial value. It follows from (76) at time t that this
region is limited by two planes
z ct z z L ctL R p� � � � �| | cos� (78)
and the cylindrical surface which describes the
right-hand side of the inequality
0 � � � ��r r z L cta R p( ) sin� . (79)
From inequalities (78) and (79), we see that the time
duration that this region exists is determined as follows:
t t
L
c
L
cp p
� �
�
�
�
�
�
�
�
�
�
�
spot min
sin
,
( cos )
| |
� �1
. (80)
At t t� spot the region lies within the pulse. In the
case of a sufficiently long pulses, for which
L L
p p
| |
cos sin1 �
� �
� �
, (81)
the maximum values of z where one can observe the
maximum amplitude according to inequalities (78)
and (80) is equal to
z z t L L L zR pmax | | | |( ) cos� � � � ��spot � � . (82)
In the case of sufficiently short pulses
L L
p p
| |
cos sin1 �
� �
� � (83)
from the inequalities (78) and (80), we get
z z t
L
R
p
max
| |
( )
cos
� �
�spot 1 �
.
The evolution of the maximum-region for the case
(81) is presented in the Figs. 1, 2.
In the case (81), at fixed value z z z� �1 max in the
plane r t, (see Fig. 2) the high energy density region,
with the initial value of the signal amplitude, is deter-
mined by the inequalities (see inequalities (78))
z L
c
t
z
cp
1 1�
� �
| |
cos�
(84)
and limited by the conical side surface (see inequa-
lity (79))
Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 587
r L ct p� �� sin� . (85)
When z z� max the high energy density region is
absent and there is no region where amplitude of the
signal is constant (see Fig. 2).
7. Discussion
The results obtained above allow us to discuss the
experimental data presented in [16]. In these experi-
ments the amplitude of phonon signal was measured
by a bolometer, situated at the distance zB � 8 mm.
The helium had such a low temperature (THe K� 01. )
that the thermal excitations can be neglected. A
heater injects a pulse of phonons into this pure and
isotropic superfluid helium («the superfluid vacu-
um»). The duration of the pulse was 5�s and it moved
in the direction normal to the surface of the heater.
The helium was under 24 atm pressure at which
phonon interactions are determined by the relatively
slow four-phonon processes. This fact justified the au-
thors using a model of noninteracting phonons, mo-
ving ballistically. We start from this model, as it al-
lows us to use the results obtained in this paper.
Here we present the numerical values of all para-
meters that describe experiments [16]. The sound ve-
locity in helium at pressure of 24 atm is equal to
c � 363 m s. The heater was made from a gold film and
had dimensions L� � 0 5. mm. The velocity of longitu-
dinal sound in the gold is equal to cl � 3240 m s and
transverse sound, ct � 1200 m s.
Using the acoustical theory of transmission [18,19],
we find the maximum value �0 for a phonon radiated
by a perfectly plane heater into helium:
�0 0 307� �arcsin rad.c ct . (86)
Below, we use this numerical value of the parameter
� �p � 0 even though the evaporated gold film is
rough and the conditions for a critical cone are not
met. However, the phonon emission from a gold film
has been measured and is found to be strongly peaked
in the direction of the surface normal [20]. Then for
typical distance z� , according to (25) we have
z� � 158. mm, and for the length of the initial pulse
L t cp| | � in this case we have L| | .� 182 mm. The size of
bolometer was 1 1� mm.
The amplitude of the phonon signal was measured
at distance zB , which is five times more than z� . In
this case we will use expression (74) for calculating
the pulse length.
To calculate the velocitiesv1 andv2, included to (74),
we use the energy—momentum relation for phonons
� �� �cp p( )1 2 , (87)
where, according to [16], at a pressure of 24 atm, the
dispersion parameter is equal to � � � � �8 96 10 4 2. K .
According to (87) one can consider that v1 0� see
(15), (16), and v2 can be found from the relation
v c v2 � �( )gr , (88)
where the line denotes averaging with respect to mo-
mentum with an equilibrium function for the initial
phonons in the pulse. Integration of (88) with the
Bose—Einstein distribution function gives
v
c
T2
0
20 028� . , (89)
where T0 is the temperature that describes the distri-
bution of the initial phonon pulse.
Comparing the numerical values of the second and
the third terms in right-hand side of Eq. (74), which
describe the broadening of the pulse caused respec-
tively by dispersion and the finite angular width of
the initial pulse, we conclude that the contribution of
the latter is negligibly small. The numerical value of
the contribution of the dispersion is equal to
� �t
z
c
v
v
T
T
B
disp s� �
�
2 0
2
0
2
0 62
1 0 028min
.
.
. (90)
According to the result presented in Sec. 6, when
zB exceeds z� by a factor of five, even in the region of
maximum phonon energy density, the energy density
is much less than its initial value. Under the condi-
tions of the experiments considered, the intensity was
0.055 of its initial value.
Figure 3 shows the temporal dependence of signal
amplitude for a definite value of temperature T0 10� . K
in the initial Bose—Einstein distribution function.
The plots in the Fig. 3 are obtained from the formula
(7) with the numerical values of the parameters given
above. Figure 3 shows that the half width of the pulse
broadening due to dispersion is close to (90), and the
contribution of the angular width is negligible, in ac-
cordance with the result obtained from Eq. (74).
Estimations of the temperatures of the heater in the
experiments [16] was � 1K for heater powers � 0 3. mW.
However, the theoretical length of a pulse at T = 1 K ex-
ceed the observed ones for heater powers � 0 3. mW. The
theoretical values of the increase in pulse length, at dif-
ferent initial temperatures (see [16]) agree with those
calculated here, for initial temperatures in the interval
0.2–0.8 K. The same results were found in [16] using a
computer calculation. These calculations were made us-
ing the expression in [16], which describes the ballistic
motion of phonons moving strictly parallel to the z axis,
with group velocities given by the dispersion law (87).
588 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
This model phonon pulse changes its form just because
of dispersion.
The conclusions from both analyses is that the tem-
perature of the initial pulse of phonons is much lower
than the temperature of the heater which creates
them.
According to the experimental conditions in [16]
angular spreading does not contribute to the increase
of the length of the pulse. However, the finite value of
angle �p determines the transverse spreading of the
pulse. Under the experimental conditions [16], this
spreading results in a considerable decrease of the am-
plitude of the signal along the axis of the pulse. Thus,
the amplitude of the pulse at the detector is a factor
0.055 smaller than the amplitude of the initial pulse.
Undoubtedly, a series of experiments at high pres-
sure similar to experiments at the saturated vapor
pressure [13] are of great interest. This would give the
temporal and spatial dependence of the phonon energy
density for different pulse lengths. The comparison of
results of such experiments at high pressure and satu-
rated vapor pressure would make it possible to com-
pare the observed values with those calculated with
the model of strongly interacting phonons (see [14])
and the results obtained here for a system of phonons
without interactions. This comparison will give us im-
portant information about anisotropic systems of
phonons in superfluid helium.
We express our gratitude to EPSRC of the UK
(grant GR/S24855 and GR/N20225), and to GFFI
of Ukraine (grant N02.07/000372) for support for
this work.
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Evolution of a pulse of noninteracting quasiparticles with dispersion and initial angular width
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 589
16 18 20 22 24 26 28
0
0,2
0,4
0,6
0,8
1,0
R
el
at
iv
e
en
er
gy
de
ns
ity
flu
x
t,�s
10�
10�
Fig. 3. The temporal dependence of the energy density
flux at z � 8 mm, t sp � 5 � , �p 176. �, and L� � 05. mm,
relative to that of the initial pulse. The initial temperature
of the pulse is T0 10� . K. The dotted line represents the
signal when there is only dispersion, the dashed line repre-
sents the signal without dispersion but with a finite angu-
lar width in the initial pulse, and the solid line represents
the signal with both dispersion and finite angular width.
Note that the dashed and solid lines have had their y values
multiplied by 10.
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