A new ripplon branch in He II
We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using the dispersive hydrodynamics approach and find a new ripplon branch. We obtain analytical equation for the dispersion relation and analytic expressions for the limiting cases. The probabilities of decay of uns...
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nasplib_isofts_kiev_ua-123456789-1173612025-02-10T01:53:05Z A new ripplon branch in He II Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. Квантовые жидкости и квантовые кpисталлы We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using the dispersive hydrodynamics approach and find a new ripplon branch. We obtain analytical equation for the dispersion relation and analytic expressions for the limiting cases. The probabilities of decay of unstable ripplons above the roton gap into rotons are derived. A numerical solution for the ripplon dispersion curve is obtained. The new ripplon branch is found at energies just below the instability point of the bulk spectrum, and is investigated; its stability is discussed. 2010 Article A new ripplon branch in He II / I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt // Физика низких температур. — 2010. — Т. 36, № 7. — С. 731–739. — Бібліогр.: 25 назв. — англ. 0132-6414 PACS: 67.25.dg, 47.37.+q https://nasplib.isofts.kiev.ua/handle/123456789/117361 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. A new ripplon branch in He II Физика низких температур |
| description |
We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using the dispersive hydrodynamics approach and find a new ripplon branch. We obtain analytical equation for the dispersion relation and analytic expressions for the limiting cases. The probabilities of decay of unstable ripplons above the roton gap into rotons are derived. A numerical solution for the ripplon dispersion curve is obtained. The new ripplon branch is found at energies just below the instability point of the bulk spectrum, and is investigated; its stability is discussed. |
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Article |
| author |
Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. |
| author_facet |
Tanatarov, I.V. Adamenko, I.N. Nemchenko, K.E. Wyatt, A.F.G. |
| author_sort |
Tanatarov, I.V. |
| title |
A new ripplon branch in He II |
| title_short |
A new ripplon branch in He II |
| title_full |
A new ripplon branch in He II |
| title_fullStr |
A new ripplon branch in He II |
| title_full_unstemmed |
A new ripplon branch in He II |
| title_sort |
new ripplon branch in he ii |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
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https://nasplib.isofts.kiev.ua/handle/123456789/117361 |
| citation_txt |
A new ripplon branch in He II / I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, A.F.G. Wyatt // Физика низких температур. — 2010. — Т. 36, № 7. — С. 731–739. — Бібліогр.: 25 назв. — англ. |
| series |
Физика низких температур |
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| first_indexed |
2025-12-02T14:14:56Z |
| last_indexed |
2025-12-02T14:14:56Z |
| _version_ |
1850406230217981952 |
| fulltext |
© I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, and A.F.G. Wyatt, 2010
Fig. 1. The dots are experimental data for the dispersion relation
of bulk excitations ( )kΩ [22], and the thin line shows its analytic
approximation that is used, with S = 18. The thick line shows the
results of numerical solution of Eq. (21) for the ripplons' disper-
sion ( )kτω . The dashed lines are the approximations of ( )kΩ
used in Ref. 20 and the resulting curve ( )kτω obtained there.
Dotted line shows the 3/2k∼ law. Large dots with error bars are
experimental data for the ripplon dispersion [4]. Two black dots
in the high-energy part of the bulk spectrum designate the end
points of the very high energy ripplon solution, and the insert
graph shows its behavior schematically, where γ = 0.7⋅10–3 Å 1−
and δ = 1.6⋅10–3 K.
�(k)
�
�
�(k )�
2.5 2.6 2.7
17.3
17.0
16.7
0 0.5 1.0 1.27 1.5 2.0 2.5 3.0
17.5
15.0
12.5
10.0
7.5
5.0
2.5
�max
�rot
ph
kc
krot
R
–
R+
kmax
2�rot
k, k , �
–1
��
/k
, K
B
Fizika Nizkikh Temperatur, 2010, v. 36, No. 7, p. 731–739
A new ripplon branch in He II
I.V. Tanatarov1,2, I.N. Adamenko2, K.E. Nemchenko2, and A.F.G. Wyatt3
1National Science Center «Kharkov Institute of Physics and Technology»
1 Academicheskaya St., Kharkov 61108, Ukraine
2V. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
E-mail: igor.tanatarov@gmail.com
3School of Physics, University of Exeter, Exeter EX4 4QL, UK
Received February 25, 2010
We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using the dispersive hy-
drodynamics approach and find a new ripplon branch. We obtain analytical equation for the dispersion relation
and analytic expressions for the limiting cases. The probabilities of decay of unstable ripplons above the roton
gap into rotons are derived. A numerical solution for the ripplon dispersion curve is obtained. The new ripplon
branch is found at energies just below the instability point of the bulk spectrum, and is investigated; its stability
is discussed.
PACS: 67.25.dg Transport, hydrodynamics, and superflow;
47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids.
Keywords: superfluid, helium, ripplon, roton, dispersion.
1. Introduction
Superfluid helium is a unique medium, in which there
are well defined long-living excitations with very short
wavelengths, of the order of average interatomic distance,
— rotons. The continuous dispersion curve of the bulk
excitations contains both the long-wavelength phonon part,
which is almost linear, and the essentially nonlinear max-
on–roton part (see Fig. 1).
Ripplons are quantized capillary waves on the free sur-
face of superfluid 4He. At low frequencies their dispersion
law gives a good way of measuring the surface tension of
liquid helium [1,2]. The temperature dependence of the
surface tension is due to ripplons [3]. Ripplons can be de-
tected by neutron scattering, in a similar way to the bulk
modes, and have been shown to exist up to wavenumber of
1.5 Å–1, where the ripplon energy is close to that of the
roton minimum, 8.6 K [4,5]. Ripplons are the dominant
scatterer of surface state electrons on liquid helium (see
[6,7] and references therein). Ripplons play a significant
role in the condensation [8], evaporation and reflection of
atoms from liquid 4He [9,10]. It has been suggested that
ripplons are the most favourable excitations for the simula-
tion of general-relativistic effects related to horizons of
white holes [11].
It was shown [4,5], that the surface modes of He II, rip-
plons, exhibit the same properties as the bulk modes, being
well-defined quasiparticles even in the high-energy region,
close to the roton gap. Their dispersion relation in this re-
gion is determined by the bulk excitations, in particular
I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, and A.F.G. Wyatt
732 Fizika Nizkikh Temperatur, 2010, v. 36, No. 7
rotons. It was studied in theory in work [12], and in Ref. 13
the ripplon dispersion curve below the roton gap was ob-
tained in the framework of density functional approach,
and it was in good agreement with the experimental data.
In this paper we present a theoretical model of ripplons
which is good enough to account for the measured disper-
sion curve ( )kω but is simple enough to expose the under-
lying physics. It explains why ripplons only exist in certain
parts of the ( , )kω plane, and why the dispersion curve ap-
proaches the line rot=ω Δ at the top of a parabola, where
rotΔ is the energy of the roton minimum. Moreover the
model predicts a new roton branch at energies rot2 .Δ
The theoretical model is a nonlocal hydrodynamical
theory [12], developed by us in Ref. 14. The actual physi-
cal characteristics of the liquid are introduced into the
model by using the measured dispersion curve for bulk
excitations. The justification of application of nonlocal
hydrodynamics to the description of a quantum fluid at
interatomic distances is given in Ref. 14, and in more de-
tail in Ref. 15. This model was used in Refs. 15–17 to de-
scribe interaction of He II phonons and rotons with solid
interfaces. It is based on the fact, that in a quantum fluid
the atoms are delocalized, as their thermal de Broglie wa-
velength is greater than the atom spacing. The general idea
of description of a quantum fluid in terms of hydrodynamic
variables at interatomic scales is being widely used. One of
the first to exploit it was Atkins in 1959 [18], when he in-
troduced bubbles and snowballs of microscopic size in
order to describe the mobility of electrons and ions in He II
by means of methods of theory of continuous medium. The
idea is utilized now in the variations of density functional
theory (see, for example, Ref. 13), and is implicitly assumed
in other fields of research related to superfluidity. In particu-
lar, the variables of continuous media are used lately for
description of vortices in helium with dimensions of the
order of the interatomic distances (see, for example, [19]).
The quantum fluid, which is considered continuous at
any length scales, can be described in terms of the va-
riables of continuous media, i.e., density, pressure and ve-
locity, which satisfy the mass and momentum conservation
laws. The equations of ideal liquid, which follow from
them, do not form a closed system, and are supplemented
by the equation of state, which specifies the functional
relation between pressure and density. As shown in Refs.
14, 15, the equation of state for small deviations of the
system from equilibrium in the general case is nonlocal,
with some difference kernel ( )h r .
The closed linear system of equations is brought to a
nonlocal integro-differential wave equation with regard to
pressure. The dispersion relation of the fluid ( )kΩ then is
determined by the Fourier transform of the kernel ( )h r
and can contain arbitrary degrees of k. In this paper we
start from the dispersion relation that approximates the
experimental data for the spectrum of superfluid helium.
The chosen ( )kΩ gives explicit expression for the kernel
( )h r , which determines the equation of state and the non-
local wave equation.
In order to derive the dispersion relation of ripplons, we
look for the solution of the nonlocal wave equation (1) in
half-space, equipped with usual boundary conditions (3). The
use of simple model of superfluid helium allows us to fully
solve the problem of ripplons’ spectrum. The ripplons’ dis-
persion equation is obtained in algebraic form, and its analyt-
ic solutions are derived in limiting cases. The analytic and
numerical results of the paper in the region below the roton
gap are consistent with the experiments, numerical computa-
tions and qualitative estimates of other authors [4,13].
A preliminary and partial account of this analysis is
given in Ref. 20, where a very approximate function for
the dispersion curve for bulk phonons and rotons was used.
This allowed to derive in [20] the equation for the ripplon's
dispersion relation in elementary functions. Its solution
below the roton gap yielded good agreement with the expe-
riment and previous works. However, the simplicity of the
used function ( )kΩ also limited the obtained results to
semi-quantitative and restricted the investigation of surface
excitations to the energies below the roton gap.
In this work we follow the approach used in Ref. 20,
but now we use a much more detailed function ( ),kΩ
which gives approximation of the experimentally measured
curve in the whole interval of wave vectors with good ac-
curacy. In fact, the proposed scheme works for functions
( )kΩ that approximate the experimental data with any
given precision. This allows us to obtain accurate results in
the whole energy interval up to twice the roton gap energy,
which is the Pitaevskii instability point of the bulk excita-
tions spectrum [21], and in particular to search for the sur-
face solutions above roton gap, with wave vectors greater
than the R+ roton’s. In the second section we formulate
the mathematical problem. It is brought to the parametric
equation for the ripplons’ dispersion relation in the next
section. The equation is investigated both at small energies
and high energies, close to the roton gap and up to the in-
stability point. It is just below this energy where we find
the new unusual ripplon branch. The numerical solution is
given and discussed in the fourth section of this work, the
new branch is investigated and its stability is discussed.
2. Equations and boundary conditions
Let us consider the half-space z > 0 filled by superfluid
helium. In accordance with the approach of [14], it obeys
the ordinary linearized equations of an ideal liquid, but the
relation between the deviations P and ρ of pressure and
density from the respective equilibrium values is nonlocal,
with some difference kernel ( )h r . The problem in terms of
pressure can be expressed as a nonlocal wave equation.
When solving the problem in half-space, the integration
domain is limited to this half-space [15] and the problem
can be brought to the form
A new ripplon branch in He II
Fizika Nizkikh Temperatur, 2010, v. 36, No. 7 733
3
1 1 1
>01
( , ) = (| |) ( , ),
, , ( , ), (0, ).
z
P t d r h P t
x y t z
Δ −
∈ −∞ ∞ ∈ ∞
∫r r r r
(1)
We assume that the interface is sharp enough to consider
that the kernel ( )h r is the same in the presence of the in-
terface as it is in the bulk medium. We discuss possible
consequences of taking into account the smoothness of the
density profile of the free surface at the end of Sec. 4. The
kernel is related to the dispersion relation of the bulk exci-
tations ( )kΩ through its Fourier transform (see [14])
2
2( ) = .
( )
kh k
kΩ
(2)
Thus the equation describes a continuous medium with
dispersion relation ( )kΩ , which fills the half-space z > 0.
The relation ( )kΩ can be essentially nonlinear and have
the distinctive form of the dispersion of superfluid helium.
The latter is almost linear at small wave vectors, reaches
the maxon maximum maxΔ at wave vector maxk , then
after the decreasing part reaches the roton minimum rotΔ
at rotk , and increases again until the instability point at
frequency rot2Δ [21] (see Fig. 1).
It is important, that the problem (1) is stated in the half-
space (0, )z∈ ∞ , and cannot be solved by an even continu-
ation of the solution to negative z with extension of inte-
gration limits to infinity, as was done in Ref. 12. Such con-
tinuation gives a mathematical problem, which is com-
pletely different from (1) and is not physically consistent,
in the general case violating energy conservation at the
interface (see more in [16]). Indeed, if Eq. (1) was true on
( , )z∈ −∞ ∞ , then the values of P at z < 0 would be deter-
mined by the values of P at z > 0, and the resulting solu-
tion would not be even.
The problem (1) in one dimension was solved in [16]
for arbitrary dispersion with the help of Wiener and Hopf
method, in the case where only one root is real, as it is for
energies less than rotΔ . In [17] the solution was genera-
lized to 3D, and in [15] it was generalized to the case when
the function 2 2( )kΩ is a third degree nonmonotonic poly-
nomial. This is the simplest analytic form of 2 2( )kΩ ,
which can qualitatively reproduce the distinctive disper-
sion of superfluid helium, including both the phonon and
the maxon–roton parts. In Ref. 20 the same solution was
used for a study of the dispersion relation of ripplons. The
use of such simple approximation of 2 2( )kΩ allowed to
derive the equation for the ripplon dispersion relation in
elementary functions, but also limited the obtained results
to semi-quantitative. In this work we investigate the dis-
persion relation of ripplons, basing on this equation, but for
the case when 2 2( )kΩ is a polynomial of high power,
which can approximate the experimentally measured curve
with any needed precision.
The Eq. (1) is supplemented by the boundary condition
on the free surface. The pressure at the surface with surface
tension σ is the Laplace pressure = /LP Rσ , where R is
the surface curvature radius. For small deviations of the
surface from equilibrium position this can be rewritten as
2 2
2 2= ,d dP
dx dy
⎛ ⎞ξ ξ
σ +⎜ ⎟⎜ ⎟
⎝ ⎠
(3)
where ξ is the z-coordinate of the points of the surface (see,
for example, [12] or [23]). For a solution exp( ),i i t− ωkr∼
with given frequency ω and the projection τk of wave
vector on the plane of the free surface ( , )x y , this expression
turns into 2=P kτ− σ ξ . The z-component of velocity of the
surface in this case is = ,zv i− ωξ and therefore we can write
the boundary condition in the form
=0 =02= .z z z
iv P
kτ
ω
σ
(4)
Substitution of the solution of Eq. (1) into this boundary
condition will give us a parametric equation for the disper-
sion relation of ripplons ( ).kτω
It should be noted, that in reality the density profile of
the free surface of superfluid helium is rather smooth, with
a transition layer of several monolayers thickness. We dis-
cuss possible consequences of taking this into account at
the end of Sec. 4.
3. Equation for the ripplons’ dispersion and its
analytic solution
Let us assume now that the function 2 2( )kΩ is a poly-
nomial of degree S in powers of 2k , such that the only
real zero of 2 2( )kΩ is 2 = 0k , where 2 2.kΩ ∼ It was
shown in Ref. 17, that in this case the Fourier image in
terms of r and t of the solution of Eq. (1) of the simplest
form is
out
1
( = 0, )( , )
( ; , ) =
( , ) ( , )
z i z
z
z z z i zkiz
k kC
P k
k k k k
ττ
τ
τ τ∈ +
− ωω ′ω
− ω − ω∏
kk
k
k kC
.
(5)
Here = z zkτ +k k e is the wave vector. The product is tak-
en over all the roots =z i zk k of equation
2 2 2 2 2( = )zk k kτΩ + = ω in the upper half-plane +C of the
complex variable zk . The real roots are assumed to be
shifted from the real line in accordance to some selection
rules (see [16]). Different selection rules regarding to
which roots are shifted up and which down lead to differ-
ent linear independent solutions. There is a condition that
the number of roots shifted up and down should be equal,
so the full number of roots i zk in +C is S. The root
1 ( , )zk τω k is the phonon root, i.e., the one in +C , which
continuously turns to zero at = 0ω and 0kτ = . The prime
I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, and A.F.G. Wyatt
734 Fizika Nizkikh Temperatur, 2010, v. 36, No. 7
superscript on the product designates that 1zk is omitted in
it. Function outC is the amplitude.
Taking the inverse Fourier transform, we obtain
( , ),P tr and then velocity of continuous medium ( , )tv r is
found from the relation 0= /P−∇ ρv . The solutions with
given ω and τk are
( , ; , ) = e ;i i tii
kiz
P t r − ω
τ
∈ +
ω ∑ k rr k
C
0
1( , ; , ) = e .i i ti
i i
kiz
t r − ω
τ
∈ +
ω
ρ ω ∑ k rv r k k
C
(6)
Here =i z i zkτ +k k e , the sums are taken over the same
roots i zk as in Eq. (5) but including 1zk ; ir is the residue
of the right hand part of (5) in .i zk
In works [15–17], where the prime concern was the prob-
lem of interaction of waves with the interface between he-
lium and a solid, the solutions which contained at least one
running wave were sought. So there were from one to three
real roots ,i zk and different rules of shifting them from the
real line, that determine the set of real i zk in (5) and their
signs, were used to obtain different linear independent solu-
tions, of which the general solution was composed.
In the current work we are interested in ripplons, and
therefore in the surface solutions that are damped away
from the interface. So we have to take that all the ,i zk for
all i, are not real. Then the selection rules are no longer
needed, and the surface solution is unique. It still has the
form (5), but now we investigate it in a different region on
the plane of parameters ( , ),kτω in which 2kτ is greater
than any of the real roots 2 ( )ik ω of equation
2 2 2( ) = .kΩ ω
We are interested in the dispersion relation that approx-
imates the one of superfluid helium (see Fig. 1). Then the
mentioned region on the plane ( , )kτω can be divided into
two parts. The first is below the roton minimum rot<ω Δ
and on the high k side of the phonon dispersion curve
1>k kτ . The second is above the roton minimum rot>ω Δ
and on the high k side of the R+ roton curve 3>k kτ .
Here it is supposed that the subscripts are assigned to the
real roots ( )ik ω of equation 2 2( ) =kΩ ω in the ascending
order of their absolute values: there are only phonons 1k
below rot ,Δ and at rot>ω Δ we have 1 2 3< <k k k , so 2k
corresponds to R− rotons and 3k to R+ rotons.
The values of P and zv on the surface = 0z are ob-
tained from (6) as the sums of the written residues in all
the finite singular points of the corresponding expressions,
because the latter are built so that all of their singularities
lie in .+C When calculating ,zv we take into account that
res ( ) = res ( ( ))i z k k z k k z zz i z z i z
k P k k P k→ → . Then the val-
ues =0zP and =0z zv are equal to minus the residues of
( )zP k and 0( ) / ( )z zk P k ρ ω in infinity correspondingly,
and those are obtained directly by expanding P from (5).
Then it can be shown that
1=0 =0
0 =2
1= [ ( = 0)] .
S
z z i z i zz z
i
v k k k P
⎧ ⎫⎪ ⎪+ − ω⎨ ⎬
ρ ω ⎪ ⎪⎩ ⎭
∑ (7)
Comparing (7) with the boundary condition (4), we finally
obtain the equation for the dispersion relation ( )kτω of
ripplons:
2 2
1
0 =2
1= ( , ) [ ( , ) ( = 0, )] .
S
z i z i z
i
k k k k k k k
iτ τ τ τ
⎧ ⎫σ ⎪ ⎪ω ω + ω − ω⎨ ⎬
ρ ⎪ ⎪⎩ ⎭
∑
(8)
This equation is written in terms of ( , ) =i zk kτω
2 2= ( ) ,ik kτ +ω − ∈C where 2 ( )ik ω in turn are the roots of
the polynomial equation 2 2 2( ) =kΩ ω with real coeffi-
cients. Then taking into account the condition that all the
i zk are nonreal, we can show, that the right hand part of the
Eq. (8) is in fact real. Indeed, let us consider the equation
2 2 2 2 2( = ) =zk k kτΩ + ω with regard to 2.zk This equation is
also a polynomial equation with real coefficients. So its
roots with regard to 2
zk are either negative (no positive roots
exist or there would be real i zk ) or break up into complex-
conjugate pairs. The negative roots give imaginary i zk and
complex-conjugate pairs 2 2,i z j zk k give pairs ,i z j zk k in ,+C
such that *= ,i z j zk k− and for such a pair =i z j zk k+
= (Im Im ) = 2 Im .i z j z i zi k k i k+ Therefore the expression
inside the braces in (8) is imaginary and the right hand part
is real.
In the limit of small frequencies 0ω→ the sum in (8)
tends to zero and the equation turns into
2
2 2 2
2
0
= ,k k
s
τ τ
σ ω
ω −
ρ
(9)
where s is sound velocity at zero frequency. This is also
the form that the equation takes in the case when the dis-
persion relation of the fluid is linear. Its exact solution is
2
2 3
lin 2 2
0 0 0
= 1 .
2 2
k k k
s s
τ τ τ
⎧ ⎫⎛ ⎞σ σ σ⎪ ⎪ω + −⎜ ⎟⎨ ⎬⎜ ⎟ρ ρ ρ⎪ ⎪⎝ ⎠⎩ ⎭
(10)
The primary term, which is left if we neglect compressibil-
ity of the fluid (i.e., in the limit s →∞ ), gives the relation,
well-known from hydrodynamics [23]
2 3
0
= .kτ
σ
ω
ρ
(11)
In order to obtain further expansion of ω in powers of kτ ,
we take into account that for small frequencies, accurately
A new ripplon branch in He II
Fizika Nizkikh Temperatur, 2010, v. 36, No. 7 735
within 4( ),O ω 2 2 2
1 ( ) = /k sω ω and 2 2 2( ) = (0)i i ik kω +β ω
for 1i ≠ . Then we search for 2ω in the form of series by
,kτ and on substituting the expansions of all quantities by
small kτ and ω into the Eq. (8), obtain
2
2 3 2
2 2 4
0 0 0
= 1
2 8
k k k
s s
τ τ τ
⎧σ σ σ⎪ω − + +⎨
ρ ρ ρ⎪⎩
2 3 4
2
0 0
( ) ( ) ,k k O k
i s
τ τ τ
⎫σ β σ ⎪+ − + ⎬
ρ ρ ⎪⎭
(12)
where 1
=2
= (0,0) / 2
S
i i z
i
k−β β∑ .
The first two summands in the braces after unity are the
first summands of the expansion of (10) and take into ac-
count compressibility. The next summands in (12), propor-
tional to β , express the influence of roots ik with > 1i , in
particular of the roton ones. The latter give only small correc-
tion at small ω , but with the increase of frequency they be-
come of the order of the phonon summands and, as it will be
shown below, they determine the asymptotic behavior of the
curve ( )kτω in the proximity of rot=ω Δ . The nonlinearity
of dependence 1( )k ω itself, which determines whether the
dispersion relation ( )kΩ is normal or anomalous at small
,kτ gives to (8) the correction of the order 4( ).O kτ
Of particular interest is the behavior of the dispersion
curve close to rot=ω Δ . Numerical solution of Eq. (8)
shows, that the curve reaches the level of rotΔ at some
wave vector rot= <ck k kτ (see Fig. 1). Let us introduce
small parameters rot=ω ω −Δ and = ck k kτ − . In the
region close to the minimum the dispersion of bulk excita-
tions is well approximated by a parabola
2
rot rot( ) ( )
2
k k kΩ ≈ Δ + −
μ
, (13)
where μ is the «roton mass». Then for the roton roots at
rotω ≈ Δ
2,3 rot 2 /k k≈ ± ω⋅ μ (14)
and
rot
2,3 rot 2
rot
1 2 ,z z
z
k
k k
k
⎛ ⎞μ⎜ ⎟≈ + ω
⎜ ⎟
⎝ ⎠
∓ (15)
where 2 2
rot rot= > 0zk k kτ− . Here the branch of square
root is used, which gives = | |ω ω for > 0ω and
= | |iω ω for < 0ω . In this way above rotΔ the signs
of 2,3zk are such that 2 30 < ( ) <z zk k− , and thus the wave
packets comprised of these waves propagate away from the
interface (in accordance with the selection rules used in
Ref. 15 and taking into account negative group velocity of
R− rotons). Below rotΔ the roots are defined by continui-
ty and both give exponentially damped waves.
The nonlinear dependence of 2,3 ( )zk ω near rotΔ leads
to the same square-root singularity in the Eq. (8):
( )2 3
1 ,z zk k b
i
+ ≈ ω (16)
where
rot 2 2
rot
2 /= 2 = | | .
c
b ik i b
k k
μ
− −
−
The expansions of other quantities in Eq. (8) in powers of
k and ω are trivial, and on substituting there (16), we
obtain the asymptotic
2
rot
3
0
1 1 1
1 rot rot
=4 =2
= , where = ; = 2 ;
= ( , ) ( , ) (0, ) .
c
S S
c z c i z c i z c
i i
bk d d a
a c k
c ik k k k k k k− − −
Δσ
− ω
+ ρ
⎧ ⎫⎪ ⎪Δ + Δ −⎨ ⎬
⎪ ⎪⎩ ⎭
∑ ∑
(17)
Here the branch of ω is the same as before; quantities a
and c are real.
Thus for rot= 0ω Δ − , we can rewrite the asymptote in
original variables in the form
rot= | |.ck k dτ− Δ −ω (18)
We see from (18) that the curve ( )kτω approaches the
level rot=ω Δ at the top of an inverted parabola, and ends
in the adhesion point, with zero derivative. There is no
dispersion curve below rotΔ with > .ck kτ In Ref. 20 the
qualitative behavior of the curve was derived, but the result
was presented in an ambiguous form 2kω∼ . Even earlier
the asymptote was obtained in theory in Ref. 13 as one of
the possible variants, in a quite different approach, from
general quantum-mechanic considerations. However, as
opposed to that work, we have explicitly derived the coef-
ficient d , which depends only on the bulk excitations
spectrum ( )kΩ and surface tension σ . The constant ,ck
which is present in the coefficient as a parameter, is in turn
obtained by numerical solution of algebraic Eq. (8) at
rot=ω Δ (see next section).
It should be noted also, that close to max=ω Δ the
roots 1,2 ( )zk ω have the same behavior as 2,3 ( )zk ω in the
considered case close to rotΔ . Therefore, if there was a
common point of the ripplon dispersion curve with the line
of the maxon level, the curve would also approach it with
zero derivative and end at the adhesion point. However, the
numerical solution (see below) shows, that such points do
not actually exist.
Equation (17) gives also the asymptotic behavior of the
curve above rot .Δ When > 0,ω k is imaginary, which
means that the surface solution dissolves on the distances of
I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, and A.F.G. Wyatt
736 Fizika Nizkikh Temperatur, 2010, v. 36, No. 7
the order of 1 1| Im | =| Im |k k− −
τ , decomposing into rotons.
This is indicated by the structure that the solution (6) takes
in this case. At > 0ω the two roton waves 2,3exp ( )zik z∼
are running waves in the z direction. The correct choice of
signs of 2,3zk ensured that these waves carry energy away
from the surface, and wave packets comprised of them prop-
agate away from the surface. A different choice of signs
would correspond to the processes of rotons reflection from
the free surface, extensively studied in Ref. 15. The proba-
bilities of either R− or R+ roton's creation can be obtained
by calculating the relative z-components of energy flows in
the two corresponding waves.
Taking into account that energy is carried with group
velocity, and average energy density in a wave with veloci-
ty amplitude iv and group velocity iu is 2
0 | | ,i iu ρ v we
get that the z-component of energy flow in this wave is
2
2
0
| |
= | | .
2
i i z
i i i
k k
Q u r
ρ ω
(19)
As mentioned above, the roots i zk that lie in +C are either
real (i.e., which are shifted up from the real line; 2,3zk are
the only ones for rot>ω Δ and 2< ( )),ck k kτ ≈ ω or im-
aginary (as 1zk ), or break up into pairs that are related as
=i z j zk k∗− . When taking this into account and calculating
(19), we obtain that
2,3 2,3~ | | .z zQ k (20)
So in the zero approximation by ω the probabilities of the
surface excitation with energy greater than rotΔ (or «high-
energy ripplon» for short) decaying into an R− or R+
roton are equal to 1/2. With the increase of frequency, de-
cay into R+ roton becomes more probable.
The derived probabilities should be applicable to a
physical situation, when the free surface of superfluid he-
lium is excited externally with characteristic frequency a
little above rotΔ . Then high-energy ripplons should be
created and consequently decay into rotons on distances of
the order of 1| Im |=| Im |k k −
τ from the source.
Above rotΔ and between the phonon and R− roton
curves we have 2zk and 3zk real, and between the R−
and R+ roton curves only 3zk is real. Evidently, in both
cases, those real roots introduce nonzero imaginary parts
into Eq. (8), and it has no solution in the plane of real ω
and kτ . The imaginary part of kτ should be of the order
of the real part, which means that the excitations are unsta-
ble and quickly decompose into rotons.
4. Numerical solution and new ripplon branch
For numerical solution it is convenient to rewrite the
Eq. (8). As mentioned above, the structure of the roots i zk
is such, that only their imaginary parts contribute to (8), and
for them Im > 0i zk . Then we rewrite the sums including
= 1i , and taking into account that 1 ( = 0) = ,zk ikτω we can
represent Eq. (8) in the form
2 2 2 2 2 2
0 =1 =1
= Im ( ) Im (0) .
S S
i i
i i
k k k k k kτ τ τ τ
⎧ ⎫σ ⎪ ⎪ω + ω − − −⎨ ⎬ρ ⎪ ⎪⎩ ⎭
∑ ∑
(21)
These sums taken over all i zk in +C are now the same as the
sums over all the S roots 2
ik of equation 2 2 2( ) = .kΩ ω
There is no need to sort the roots and define whether each
lies in C+ or not. The functions 2 ( )ik ω are obtained nu-
merically as all the roots of the corresponding polynomial
equation. The Eq. (21) is valid for the dispersion relation
of ripplons in the regions where they are stable, i.e., where
all i zk are not real. Its numerical solution in the range of
0 3kτ≤ ≤ Å–1 gives the curves shown on Fig. 1.
We see that, as predicted, the computed curve at small
wave vectors is close to the classical 3/2kτ law, but deviates
from it at larger kτ and approaches the level of rotΔ at the
top of an inverted parabola at = 1.27ck Å–1. As predicted
in previous work [20], the use of better approximation of
the bulk dispersion relation stretched the curve in this re-
gion towards the roton minimum rot ,k if compared with
the results of [20]. The reason for this is that the rough
approximation used in [20] underestimated the «roton
mass» by factor of order of 1.5, so that the curvature radius
of the dispersion curve at the roton minimum was overes-
timated. As the ripplon dispersion curve near rotΔ is de-
termined by the asymptotes of the roton roots there, the
rough approximation overestimated also the curvature ra-
dius of the ripplon dispersion curve at the adhesion point.
There is no ripplon solution in the region on the high k
side of the R+ roton dispersion curve (see Fig. 1), for
rot max( , ).ω∈ Δ Δ If we rewrite the Eq. (8) in the form
( , ) = 0,F kτω then in this area the main contribution to
( , )F kτω can be shown to be provided by the summand
2 ,zk∼ which thus prevents it from turning to zero. How-
ever, if we further increase ω while moving along the
curve 3( ),k ω as we get close to the region where ( ) /k kΩ
reaches its maximum, the curve turns down and aims to the
point of instability [21] in almost straight line, the left hand
part of Eq. (8) starts to change slowly. At the same time,
in this region the structure of the roots 2 ( )ik ω is qualita-
tively the same as in the neighbourhood of an inflection
point of the curve 2 ( ).kΩ This leads to rapid changes of
some of the roots with > 3i (as near the inflection point
two of the complex-conjugate roots tend to the same real
value with the usual square-root asymptotic), which give
significant contribution to ( , ).F kτω Thus here the solution
is determined by the right-hand part of Eq. (8) and the
functions i zk .
Therefore, when we search for the solutions on the R+
roton curve, two are found above maxΔ . The first point is
at 2.52(2) Å–1 and 16.65(5) K, almost exactly at the maxi-
mum of ( ) / ,k kΩ and the second is at 2.75(5) Å–1 and
17.3(1) K, close to the point of instability rot= 2ω Δ (esti-
mation of errors is made by comparison of the results giv-
en by different approximation polynomials, see Appendix).
A new ripplon branch in He II
Fizika Nizkikh Temperatur, 2010, v. 36, No. 7 737
Fig. 2. The set A (black region) consists of the points, which cor-
respond to excitations which can decay into two collinear rip-
plons with energy rot< Δ , from energy-momentum conservation
considerations. As the new ripplon branch lies to the right of A, it
is stable with regard to this process.
0
17.5
15.0
12.5
10.0
7.5
5.0
2.5
0.5 1.0 1.27 1.5 2.0 2.5 3.0
k, k , �
–1
�max
�rot
2�rot
��
/k
, K
B
The ripplon dispersion curve ( )kτω between them sticks
closely to the bulk dispersion from below, and their two
common points are adhesion points. This is partly the rea-
son for the deviation between the curves being extremely
small. At = 2.6kτ Å–1, midway between the two end
points, the deviation is 1.6⋅10–3 K or 0.7⋅10–3 Å–1, which
is too small to see on the scale of the main graph. So in the
inset to Fig. 1 we show this region expanded, and the sepa-
ration of the ripplon dispersion curve from the roton dis-
persion curve is greatly exaggerated.
Let us show that the common points of the ripplon dis-
persion curve ( )kτω and the R+ roton dispersion curve
3( )k ω should indeed be adhesion points. There is the
summand 3zk in ( , ),F kτω which turns to zero as
3( )k kτ − ω on the R+ roton branch. So the gradient of
( , )F kτω on the plane ( , )kτω tends to infinity on the
curve 3= ( )k kτ ω and is directed normal to the curve.
From the other side, the gradient of ( , )F kτω is directed
normal to the curve ( ),kτω which is its level curve = 0.F
Therefore in the common points of the two curves the an-
gle between them is equal to zero, and those points are
adhesion points.
We see now, that in case there are common points of
the curve ( )kτω with the boundaries of its possible exis-
tence (i.e., curves 1( )k ω and 3( )k ω ) or the lines
rot,max=ω Δ of extremums of the bulk dispersion ( ),kΩ
those points can only be adhesion points (with the excep-
tion of point = 0)kτ . Numerical solution shows that there
are three such points altogether — two on 3( )k ω and one
at rot= .ω Δ
At = 2.6kτ Å–1 the value of 3zk is close to its maxi-
mum 0.06 Å–1 on this branch of the ripplon curve, and the
penetration depth of the ripplon solution, also determined
by the 3zk∼ summand here, to its minimum
1
3| | 16zk −δ∼ ∼ Å, tending to infinity at the end points.
The deviation between the two curves is of the second or-
der by the small parameter 3zk , as 3 3| | ~ .zk k kτ −
The relatively large penetration depth of the solution
means that macroscopic films or the surface of bulk helium
are needed to observe these ripplons. They will not be seen
on films of a few monolayers. The high-energy ripplons
should exist and be observable on saturated films of He II
[24], which have a typical thicknesses of 300 Å. In the
same way the penetration depth of the ripplons close to the
roton gap (see Fig. 1) tends to infinity when ω tends to
rot .Δ So the ending point of the dispersion curve
rot( , )ckΔ can also be observed only in thick enough films
of He II. Thus in the two most interesting regions of their
dispersion curve, the ripplon solutions have large penetra-
tion depths, much larger than the characteristic distances,
of several monolayers thickness, of the changes of the den-
sity profile at the free surface.
The numerical calculations were carried out for differ-
ent approximation polynomials SP of function 2 2( ),kΩ
with S = 18 and S = 21. Also different polynomials of
power 21 were obtained by approximating the experimen-
tal data supplemented by different subsets of the points,
extracted from the spline (see Appendix). The high-energy
ripplons solution exists in all cases, and for different SP
its ending points differ insubstantially. The deviation be-
tween the curves 3( )k ω and ( )kτω remains very small,
much less than the deviation between the approximation
curves themselves.
Stability of the new ripplon branch with regard to decay
into two ripplons with energies less than rotΔ can be
checked graphically. In the process the energy and momen-
tum should be conserved, so if we denote the initial high-
energy ripplon by index 0 and the resulting two quasipar-
ticles by 1 and 2, we have
0 1 2 0 1 2= and = .ω ω +ω +k k k (22)
Then 0 1 2< .k k k+ If we define the set A of points
( , )kτω that obey the conditions 1 2=ω ω +ω and
1 2=k k kτ + for all 1 1( , )kω and 2 2( , )kω satisfying the
ripplon dispersion relation below rot ,Δ then the point
0 0( , )kω should be situated to the left of A, in order for the
process of its decay into two ripplons to be allowed. This
set is shown on Fig. 2, and we see that A is to the left of the
new ripplon branch, thus it is stable with regard to decay
into two ripplons below rot .Δ In a similar way it can be
shown that it is stable with regard to decay into three and
more ripplons below rot .Δ
It is also natural to check for the stability of the solution
with regard to changes in surface tension .σ While in this
work we assume, for simplicity, that σ is constant
( = 3.544 /N mσ at zero temperature [22]), it is known
(see [25]) that better agreement with experiment is
I.V. Tanatarov, I.N. Adamenko, K.E. Nemchenko, and A.F.G. Wyatt
738 Fizika Nizkikh Temperatur, 2010, v. 36, No. 7
achieved if its curvature dependence is taken into account.
For small ω this should only change the numerical results
insignificantly. However, there is no data yet for surface
tension at energies ≥ 15 K, which correspond to the end of
bulk spectrum, so the only thing we can do is to solve the
equation with different values of σ and see if the very
high energy ripplons still exist. It appears, that if we de-
crease σ , the end points of the ripplon dispersion curve
slide apart along 3( )k ω and its length increases. At the
value = 1.75 /N mσ , the lower adhesion point reaches
16 K. If we, on the contrary, increase ,σ then the end
points slide towards each other and the solution disappears
when σ reaches the value of 8.7 /N m (2.5 times value at
T = 0). It should be noted, however, that any realistic de-
pendence ( )kτσ can be inserted in Eq. (8) without compli-
cating the numerical solution.
It should be noted, that as the initial equation for pres-
sure (1) was first obtained in order to describe the superflu-
id–solid interface [15], the interface was assumed to be
sharp, so this relatively simple generalization of nonlocal
wave equation to half-space could be used, with the mod-
ified area of integration but the same kernel as in the infi-
nite space. The same assumption of interface sharpness
was maintained in this work for the calculation of ripplon’s
dispersion relation. In reality the density profile of the free
surface of superfluid helium is rather smooth, and there is a
transition layer of several monolayers thickness. However,
when we consider ripplons with large penetration depths
— either those close to rotΔ or the ripplons of the new
branch, close to 3( )k ω — or slow rotons close to the roton
minimum, the major part of the energy of the wave is
stored outside of the transition layer. Therefore the latter
can be neglected, and the surface can be considered sharp
(i.e., the kernel h(r) in Eq. (1) is the same as in the bulk
fluid). This is also justified by the fact that the obtained
results are in rather good agreement with both experiment
and the results of calculations based on density-functional
approach [13], in which the smooth transition from liquid
to vapour is reproduced.
5. Conclusions
In this work we have investigated the dispersion rela-
tion of the surface excitations of superfluid helium, rip-
plons, in the dispersive hydrodynamics approach, in which
the only input «parameter» is the dispersion relation of the
bulk medium. We use the approximation of the dispersion
relation by a polynomial and derive the equation for the
ripplon’s dispersion in terms of the roots of the corres-
ponding polynomial equation. In this work, as opposed to
[20], we use the polynomial of arbitrary power S, which
can approximate the experimentally measured curve with
any given accuracy in the full range of wave vectors, from
zero to the point of instability.
The ripplon dispersion curve ( )kτω is investigated ana-
lytically both at small frequencies 0ω→ and close to the
level of roton minimum rot ,Δ and the expansions are de-
rived. It is shown to deviate slightly from the classic 3/2kτ
law at small kτ , but the deviation increases with greater
kτ and it approaches the level of rotΔ at the top of an in-
verted parabola branch at = 1.27kτ Å–1.
The probabilities of the decay of unstable ripplons above
rotΔ into R− and R+ rotons are calculated and in zero ap-
proximation by rot( )ω −Δ are shown to be equal to 1/2.
The equation is solved numerically for S = 18 and S = 21.
Besides the ripplon dispersion curve below rot ,Δ a high-
energy solution is found above the maxon level max .Δ
It follows closely the bulk dispersion curve from 2.5 Å–1 to
2.7 Å–1 on distances 310−∼ K and its end points on the
bulk dispersion curve are shown to be adhesion points. This
solution is shown to be stable with regard to variation of
approximation polynomials and to changes of surface ten-
sion in wide range. It is also stable with regard to decay into
two or more ripplons with energies rot< .Δ
We hope that evidence for existence of the unusual rip-
plon branch above the roton gap, found in this work, sti-
mulates new experiments, that would confirm the predic-
tions of the theory.
We are grateful to EPSRC of the UK (grant EP/F
019157/1) for support of this work.
Appendix. Fitting the dispersion curve of He II
In order to solve numerically Eq. (8), we have to pre-
pare the approximating polynomial 2 2( ).kΩ We use the
following scheme for that. First, we construct least-square
roots approximation of experimental data [22] for
2 2/ ,kΩ as function of 2 ,k by a polynomial of high
enough power:
2
2 2 2( 1)
1 0 1 12
( ) = ( ) = .S
S S
k M k a a k a k
k
−
− −
Ω
+ + +…
(23)
This way 2 2kΩ ∼ at small k regardless of approxima-
tion, which is important. The polynomials of powers 17
and 20 provide curves close enough, so we made the calcu-
lations for 1SM − polynomials with S = 18 and S = 21. In
order to suppress large amplitude high-frequency oscilla-
tions at high 2 ,k which occur because of sparseness of
experimental points in the maxon–roton region (except for
around the roton minimum), we supplement the data in this
region by additional points taken from its spline (using
different subsets of the points from the spline we obtain
slightly different approximation polynomials M; the results
of the calculations for the ripplon’s dispersion for different
M can be shown to differ insubstantially).
The resulting polynomial gives good approximation of
the data in the given interval of ,k which we choose as
[0,3]k ∈ Å–1, but at higher k it has large amplitude oscil-
A new ripplon branch in He II
Fizika Nizkikh Temperatur, 2010, v. 36, No. 7 739
lations, is nonmonotonic and turns into zero. The poly-
nomial has to be further corrected in our case, to make it
monotonic at > 3k Å–1. Otherwise additional real roots of
equation 2 2( ) =kΩ ω for the considered ω would appear,
which would lead to existence of running waves in solution
(6) at < 3kτ Å–1, which should be a surface solution. The
additional zeros of 2 ( )kΩ for real k also should not exist
for the solution given in Ref. 16 to be valid. In order to
correct the polynomial, we increase its highest power coef-
ficient 1Sa − until the oscillations vanish and beyond the
approximation interval M becomes monotonic. The high
power S ensures that, while beyond 3 Å–1 function
2 ( )kΩ now rises as 2( )Sk , the changes to it at < 3k Å–1
are negligible. The resulting polynomial approximates ex-
perimental data on the chosen interval [0,3]k ∈ Å–1 and
obeys all the necessary additional conditions. If 1Sa − is
negative, the trick does not work, and we just have to add
the summand of even higher power with the coefficient
that eliminates the oscillations at > 3k Å–1, but is small
enough for the summand to be negligible at < 3k Å–1.
The scheme can be used with a polynomial of any power,
which can in principle approximate the experimental data
with any given precision.
One of the used sets of coefficients for S =21 is pre-
sented in the table 1.
Table 1. Coefficients for S = 21
The number of significant digits is large because of
high powers of argument involved in Eq. (23). The result-
ing ripplon dispersion curves, obtained when using differ-
ent approximations, are indistinguishable below rotΔ and
the same branch manifests itself above max .Δ This serves
as the indication of the stability of Eq. (8) for the ripplons’
dispersion with regard to changes in the approximation
polynomial.
The actual calculations were carried out with precision
greater than given in the table, but the presented coeffi-
cients are accurate enough to reproduce the dispersion of
bulk excitation up to 3 Å–1.
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a0 334.38 a7 1536.6172803 a14 2.0431806958⋅10–4
a1 441.78 a8 –504.57796831 a15 –1.25670939443⋅10–4
a2 –2680.24 a9 124.559633213 a16 1.24161912937⋅10–5
a3 5515.4755 a10 –23.052039054 a17 –6.988403321⋅10–7
a4 –6881.11663 a11 3.13661307342 a18 2.434762814⋅10–8
a5 5809.52885 a12 –0.296185514695 a19 –4.918385291⋅10–10
a6 –3490.147348 a13 0.0157637822729 a20 4.43529347⋅10–12
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