Charged snowball in nonpolar liquid
The problem of correct definition of the charge carrier effective mass in superfluid helium is revised. It is demonstrated that the effective mass M of a such quasiparticle can be introduced without use of the Atkins’s idea concerning the solidification of liquid He in the close vicinity of ion. T...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Charged snowball in nonpolar liquid/ I. Chikina, V. Shikin, A. Varlamov // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1016–1022. — Бібліогр.: 14 назв. — англ. |
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Chikina, I. Shikin, V. Varlamov, A. 2017-06-13T10:40:33Z 2017-06-13T10:40:33Z 2007 Charged snowball in nonpolar liquid/ I. Chikina, V. Shikin, A. Varlamov // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1016–1022. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 67.40.–w, 72.20.Jv https://nasplib.isofts.kiev.ua/handle/123456789/120934 The problem of correct definition of the charge carrier effective mass in superfluid helium is revised. It is demonstrated that the effective mass M of a such quasiparticle can be introduced without use of the Atkins’s idea concerning the solidification of liquid He in the close vicinity of ion. The two-liquid scenario of the «snowball» mass formation is investigated. The normal fluid contribution to the total snowball effective mass, the physical reasons of its singularity and the way of corresponding regularization procedure are discussed. Within of two-liquid model the existence of two different effective snowball radiuses: Rid for superfluid flow component and Rn for the normal one, Rn > Rid is demonstrated. Agreement of the theory with the available experimental data is found. This work was partly supported by RFBR grant No 06 02 17121 and the Program of the Presidium of Russian Academy of Sciences «Physics of Condensed Matter». en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Charged snowball in nonpolar liquid Article published earlier |
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Charged snowball in nonpolar liquid Chikina, I. Shikin, V. Varlamov, A. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
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Charged snowball in nonpolar liquid |
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Charged snowball in nonpolar liquid |
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charged snowball in nonpolar liquid |
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Chikina, I. Shikin, V. Varlamov, A. |
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Chikina, I. Shikin, V. Varlamov, A. |
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International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The problem of correct definition of the charge carrier effective mass in superfluid helium is revised. It is
demonstrated that the effective mass M of a such quasiparticle can be introduced without use of the Atkins’s
idea concerning the solidification of liquid He in the close vicinity of ion. The two-liquid scenario of the
«snowball» mass formation is investigated. The normal fluid contribution to the total snowball effective
mass, the physical reasons of its singularity and the way of corresponding regularization procedure are discussed.
Within of two-liquid model the existence of two different effective snowball radiuses: Rid for
superfluid flow component and Rn for the normal one, Rn > Rid is demonstrated. Agreement of the theory
with the available experimental data is found.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120934 |
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Charged snowball in nonpolar liquid/ I. Chikina, V. Shikin, A. Varlamov // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1016–1022. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT chikinai chargedsnowballinnonpolarliquid AT shikinv chargedsnowballinnonpolarliquid AT varlamova chargedsnowballinnonpolarliquid |
| first_indexed |
2025-11-26T06:25:31Z |
| last_indexed |
2025-11-26T06:25:31Z |
| _version_ |
1850612450487959552 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1016–1022
Charged snowball in nonpolar liquid
I. Chikina
DRECAM/SCM/LIONS CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
V. Shikin
Institute of Solid State Physics RAS, Chernogolovka, Moscow District, 142432, Russia
E-mail: shikin@issp.ac.ru
A. Varlamov
INFM-CNR, COHERENTIA, via del Politecnico, 1, I-00133 Rome, Italy
Received November 29, 2006
The problem of correct definition of the charge carrier effective mass in superfluid helium is revised. It is
demonstrated that the effective mass M of a such quasiparticle can be introduced without use of the Atkins’s
idea concerning the solidification of liquid He in the close vicinity of ion. The two-liquid scenario of the
«snowball» mass formation is investigated. The normal fluid contribution to the total snowball effective
mass, the physical reasons of its singularity and the way of corresponding regularization procedure are dis-
cussed. Within of two-liquid model the existence of two different effective snowball radiuses: Rid for
superfluid flow component and Rn for the normal one, R Rn � id is demonstrated. Agreement of the theory
with the available experimental data is found.
PACS: 67.40.–w Boson degeneracy and superfluidity of
4
He;
72.20.Jv Charge carriers: generation, recombination, lifetime, and trapping.
Keywords: superfluid helium, two-liquid model, ion–dipole interaction.
1. The ion–dipole interaction between the inserted
charged particle and the induced electric dipoles of sur-
rounded atoms is one of the interesting phenomena which
take place in a nonpolar liquid. It is interaction which is
generally responsible for the different salvations phe-
nomena [1], while in nonpolar cryogenics liquids (like
He, Ne, Ar, etc.) it leads to so-called «snowball effect».
The latter consists in the formation of a nonuniformity
��( )r in a density of a liquid around the inserted charged
particle.
The ion–dipole interaction U ri d� ( ) between the in-
serted charged particle and solvent atoms in the simplest
form can be written as
U r
e
r
i d� � �( )
�
2
2
4
, (1)
where � is the polarization of a solvent atom and r is the
distance to the charged particle. The presence of attrac-
tion potential (1) in equilibrium has to be compensated by
the growth of the solvent density ��( )r in direction of the
charged particle. The latter can be estimated, using the re-
quirement of the chemical potential constancy:
��
�
( ) ( ) ,r P r v
e
r
s s� � �
2
0
2
4
��
�
( ) ( ) ( )r
P
P r
s
P rs s�
�
�
�
�
� �
1
2
, (2)
where P rs( ) is the local pressure around the ion, vs is the
volume of individual solvent atom and s is the sound ve-
locity. The distance at which P rs( ) reaches the value of
pressure of a solvent solidification Ps
s
R
e
P v
s
s
s
s
4
2
2
�
�
(3)
corresponds to the radius of rigid sphere which is called
«snowball». In the case of positive ions being in liquid
helium, where the inter-atomic distance a � 3 � and the
pressure Ps
s � 25 atm, the snowball radius is estimated as
Rs
He � 7�.
© I. Chikina, V. Shikin, and A. Varlamov, 2007
Described above so-called Atkins’s snowball model [2]
is quite transparent and it was found useful for the various
qualitative predictions. In particular, it provides by the nat-
ural definition and estimation for the effective mass M of a
such quasi-particle as the sum of the extra mass caused by
the presence of the density perturbation ��( )r
M r r dr(sol) �
�
�4 2� ��
�
( ) (4)
and so-called hydrodynamic associated mass, related to
the appearance of the velocity distribution around the
sphere moving in liquid
M R Rs s0
32
3
( )
( , )
ass � ��� . (5)
Both of these contributions turn out to be of the same or-
der. Calculated in this way snowball effective mass turns
out to be M m� 50 4 , what roughly corresponds to the ex-
perimental data [3].
Careful analysis shows that both expressions (4) and
(5) require more precise definitions and further develop-
ment of the Atkins’s model. Indeed, one can easily see
that the value of M (sol) turns out to be critically sensitive
to the lower limit of the integral. Atkins used the value of
helium inter-atomic distance a � 3� [2] as a rough cut-off
parameter only. The microscopic analysis shows (see Eq.
(21) below) that the real lower limit turns out to be less
that this value.
Principal revision requires the definition of the associ-
ated mass M ( )ass for the two-fluid model. The matter of
fact that its strong temperature dependence, which has
been systematically observed experimentally [4–7], can-
not be explained within the «ideal» flow picture. It is why
below we propose the hydrodynamics scenario of the as-
sociated mass M ( )ass formation that takes into account
the viscous part of this problem. The straightforward ac-
counting for the nonzero viscosity results in a dramatic
growth (compared to the ideal case) of kinetic energy of
the moving sphere and, consequently, its effective associ-
ated mass. This fact is caused by setting in motion of a
spacious domain of viscous liquid around the moving par-
ticle. Nevertheless, the arising divergency can be cut off
by accounting for the nonlinear effects in stationary flow.
Even more interesting is the flexible behavior of positive
ion’s dynamic responde versus of excitation conditions. It
(responde) cannot be described neither by means of effec-
tive mass approximation, nor using some alternative, which
are not sensitive to excitation conditions. Therefore below
we propose several snowball motion scenario and try to
overlap the details of its dynamic responde, when the excita-
tion conditions are close one to another.
2. Let us start from the definition of the stationary ef-
fective associated mass within the snowball approxima-
tion and using the two-fluid model. In that case
M M Ms n
( ) ( ) ( ) .ass ass ass� � (6)
The first one can be defined as follows:
M V T
ds s
s
R
( ) ( )
( ) ,
ass
id
2
2 3
2 2
�
�
�
�
v r r (7)
where �s T( ) is the superfluid density, V is the velocity of
a «snow-cloud» center of mass forward motion, v rs( ) is
the superfluid component local velocity distributions ap-
pearing in liquid due to charge carrier motion. Let us un-
derline that some auxiliary parameter R id instead of the
snowball radius Rs
He (see Eq. (3)) here is used as the
lower limit cut-off. Some physical reasons for the intro-
duction of R id beyond the Atkins model are presented be-
low (see Eq. (21)).
The superfluid flow has potential character [8,9]
v r r
An
r
A = Vs R /��� � � �( ), ( ) , .
2
3 2id (8)
Correspondingly, the superfluid part of associated mass
M s
( )ass (7) with the velocity distribution determined by
the Eq. (8) is reduced to the expression (5) with the simple
substitutions R Rs � id and � �� s .
As to the normal fluid part of the associated mass
M n
( )ass , its definition turns out to be more cumbersome. It
would be natural to define it in the same way as the
superfluid one, just substituting in Eq. (7) the subscript s
by n. But this programme runs against the considerable
obstacles in spite of the fact that the problem of rigid
sphere motion in a viscous liquid was considered a long
ago (see, for instance, Refs. 8, 9). Indeed, for small
Reynolds numbers Re /� ��� �VRn 1 and distances
R r Rn n� � / Re the gradient term ( )v v�� in correspond-
ing Navier–Stokes equation can be ignored. The solution
of such linearized equation in zero approximation turns out
to be independent on the liquid viscosity �. In the frame re-
lated with the sphere center of mass it has the form:
v r V
R
r
R
r
r
n n( , ) cos ,� �� � �
�
�
�
�
�
3
2
3
3
(9)
v r V
R
r
R
r
n n
� � �( , ) sin� �
�
�
�
�
�
3
4 4
3
3
. (10)
Here � is the polar angle counted from the x axes, which
coincides with the ion velocity direction.
One can easily see, that corresponding contribution to
the kinetic energy diverges at the upper limit of integra-
tion. To regularize this divergence it is necessary to use at
Charged snowball in nonpolar liquid
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1017
large distances r � l Rn� � / Re (l V� � �� / is the charac-
teristic viscous length) more precise, so-called Ossen, solu-
tion of the Navier–Stokes equation which was obtained tak-
ing into account the gradient term in Ref. 9. This solution
shows that almost for all angles, besides the domain re-
stricted by the narrow paraboloid � � � �( ) / | |x V x� � be-
hind the snowball (so-called «laminar trace») at large dis-
tances the velocity field decays exponentially. In latter
the velocity decays by power law [9] and it gives
logarithmically large contribution with respect to other
domain of disturbed viscous liquid. Such specification
permits to cut off formal divergency of the kinetic energy
and to find with the logarithmic accuracy the value of the
normal component associated mass for the stationary
moving in viscous liquid charge carrier:
M M R
l
R
l
R
R
Vn n n
n n
n( )
( ) ( )
( , ) ln
st
ass ass�
�
�
�
�
� �
0
2�
�� �
ln .
�
�VRn
(11)
Here the associated mass M Rn n0
( )
( , )
ass � is defined by
Eq. (5) with radius Rn and density �n . Since in our as-
sumptions l R� �� * the value of M
n( )
( )
st
ass
turns out much
larger than the value of the associated mass in the ideal
liquid. Moreover, when velocity V such definition, in
spite of the performed above regularization procedure,
fails since Eq. (11) diverges.
3. The simplest alternative scenario for snowball dy-
namics is it periodic oscillation. Let us consider the situa-
tion, when a periodic electric field is applied to a charge
carrier placed in normal liquid. The dynamic Stokes force
appearing when the sphere oscillates in the viscous liquid
with finite frequency has the form [8,9]:
F R
R
Vn
n( )
( )
( )� ��
� �
�� �
�
�
� �6 1
� �
�
�
�3
2
1
2
9
2�
��
� � �
� �R
R
i Vn
nn
( )
( ), (12)
where � � � � �( ) ( )� 2 1 2/ n
/ is so-called dynamic penetra-
tion depth. It is natural to identify the coefficient in front
of the Fourier transform of acceleration i V� �( ) with the
effective dynamic associated mass, that gives:
M R
R
n n
n( )( )
( )
ass n� �
��
� � �
� �
�
�
�3
2
1
2
9
2 . (13)
One can see that for high frequencies (� �( ) �� Rn ) the dy-
namic associated mass coincides with that one of a sphere
moving stationary in an ideal liquid, while when
� �� 0 M n
( )( )ass diverges as ��1 2/ . As we already have
seen above this formal divergence is related to fall down
of the linear approximation in the Navier–Stokes equa-
tion assumed in derivation of Eq. (12). It is clear that the
definition (13) is valid for high enough frequencies, until
� �( ) � l�(i.e., � �
~ / )� � �� n nR 2 . When � becomes lower
than ~� the penetration depth � �( ) in Eq. (13) has to be
substituted by l� and up to the accuracy of ln l /R� the dy-
namic definition Eq. (13) matches with the static one (see
Eq. (11)).
The additional possibility to formulate the beginning
of nonlinear situation in M n ( )� behavior is reorganiza-
tion of requirement following from (11)
l /R� �1
to synonym
�� �1, (14)
where � is express in terms �andV is presented asV R�� .
To finish the discussion (12)–(14) it is reasonably to
add the following speculations. If instead of purely oscil-
lation regime, we have for, e.g., the cyclotron snowball
motion, in this case the overlap between (11), (13) has to
be correct, including the logarithm correction (in CR pic-
ture there are oscillations, but the velocity never goes to
zero; as a result the stationary v r( ) picture can be saved
permanently, just the laminar trace maybe tilted with
respect to its conventional position).
Thus, the following qualitative picture arises. The gen-
eral expression for the normal effective mass definition
could be presented in a form, similarly (7). But any time
the velocity distribution in this integral has to be written
for the real dynamic scenario, e,g., stationary motion,
CR-motion, simple oscillations, relaxation phenomena
etc. There is no general M Tn
(ass)( ) definition, which is in-
dicated in all experimental papers [4–7]. Even, there is no
M Tn
(ass)( , )� -presentation without indexation who is re-
sponsible for the motion excitation. And this many pic-
ture situation is not the fables of the theory. It corresponds
to the nature of V t( ) snowball motion in viscose liquid.
4. The next question comes, how sensitive are the in-
troduced above definitions of the associated mass
Eqs. (13), ( 7) to the real shape of the liquid density per-
turbation in the vicinity of the charge carrier? The exam-
ples are already known when the snowball Atkins’s model
and more realistic snow-cloud model lead to qualitatively
different predictions for the value of positive ion mobility
in liquid 4He (see Refs. 10–12 ). In order to clear up this
problem let us consider the hydrodynamic picture of a
snow-cloud motion. The latter we assume as the density
compression which decays with distance by power law:
� � �� �( ) ( )r r
C
r
� � � �
4
(15)
(constant C is expressed in terms of Eqs. ( 2)). The conti-
nuity equation is read as
� �( ) ( ) .r rdiv v + v �� � 0 (16)
1018 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I. Chikina, V. Shikin, and A. Varlamov
For an ideal (superfluid) liquid the velocity field is poten-
tial and it can be presented in the form Eq. (8). Substitut-
ing Eqs. (8) and (15) to Eq. (16) one finds
� �
��
�
�( )
( )
r
r4
0
5
C
r � ��
. (17)
The Eq. (17) has to be solved with the additional require-
ment
� � � �( ) cosr rV � . (18)
In Born approximation Eq. (17) is reduced to the Pois-
son equation. Supposing that � � � �( ) cos ( ),r rV r� 1 it can
be rewritten in the form
� �
�
� � � � �1 5 1
4
0 0
C
r
V
r
cos
, ( )
�
� ��
, (19)
which solution can be written down as
� �
�
�
�
��1 3 6
3( ) ,
( )
( )
x
d x
d
r V r
r.
r
C
r r
d
� � ��
(20)
The Eq. (20) shows, that far enough from a snow-cloud
center the potential of velocity distribution looks like a
dipole one, exactly in the same way as for the flow around
the rigid sphere of the radius R id . Using Eqs. (8) and (20)
one finds
R C
dr
r
id
3
2
0
2�
�
�
��
� ��( )
. (21)
It is easy to see that in spite of presence of r 2 in denominator
of the integrand this integral converges at the lower limit.
The above analysis demonstrates that the problem of
the associated mass definition in the snowball and the
snow-cloud models for an ideal liquid are qualitatively
identical. It is just enough to renormalize the effective ra-
dius R* (8) to the hydrodynamic dipole radius R id (20).
The same analysis has to be done basing on the Na-
vier–Stokes formalism for the motion of the snow-cloud
in the viscous liquid. The problem is to solve the continu-
ity equation, similar (16)
� �( ) ( )r rdiv v + v �� � 0 (22)
and a stationary Navier–Stokes equation
� v ��p r( ) , (23)
here � is the first liquid viscosity, p r( ) is the pressure dis-
tribution.
Acting (23) by operator div and using (22), we have fi-
nally
p r
r
r
( )
( )
( )
� �
��
�
�
�
v
. (24)
Using this definition (and the suitable Born simplifica-
tion: v � Vs) as right part of (23), we have the possibility
to describe v distribution. First, it is evident such a distri-
bution is not sensitive to �. Secondly, it is convenient to
build the velocity components in cartesian coordinates
v
V
r x
x
x�
�
��
�
( )
2
2
, (25)
v
V
r x y
v
V
r x z
y
x
z
x�
�
� �
�
�
� ��
�
�
�
( )
,
( )
2 2
. (26)
The Eqs. (25), (26) show the reasons for r �1 anomaly
in velocity (9), (10) distributions. Both (25) and (26)
definitions correspond to Poison equations. If, like in
Eq. (25),
d r
x
3
2
2
0�
�
�
!
�
,
in this case the solution for v rx ( ) looks like the potential
distribution around point-like charge (i.e., v r rx ( )" �1).
In two other equations (26)
d r
x y
d r
x z
3
2
3
2
0 0� �
�
� �
�
�
� �
�
� �
, .
Therefore the solution of these equations cannot have r �1
behavior. They could excite the distributions with the
asimptotes v r v r ry z( ) ( )" " �3.
Go back to Eq. (25). Its solution has the form
v xyz
V
dx dy
dz
R x y z
x y
x
x( ) ' '
'
( ' ' ' )
( '
�
�
�� �� ��
� � �4
0 0 0
2
� �
� ' ' )
( ' )
z
x� 2
,
(27)
R x x y y z z2 2 2 2� � � � � �( ' ) ( ' ) ( ' ) .
At the big distances r Rn��
v xyz
V R
r
x
x n( ) �
4�
, (28)
R dx dy
dz
x y z
x y z
x
n � ' '
'
( ' ' ' )
( ' ' ' )
( ' )
0 0 0
2
2
�� �� ��
� � �
�
��
�
. (29)
As in R id definition (21), the main contribution in Rn (29)
follows from the small distances r Rn' �� . For these r' the
divergency (29) looks much stronger, than the same one
in (21). Such a qualitative difference leads to conclusion
R Rn � id . (30)
Therefore the discussing model of snowball effective
mass is not only two-liquid one. It contains also two ra-
diuses: R id (21) and Rn (29) with the qualitative predic-
tion (30).
Charged snowball in nonpolar liquid
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1019
5. Let us pass to discussion of the experimental situa-
tion. First of all it is necessary to stress that the effective
mass temperature dependence for positive charge carriers
in liquid 4He has been observed in the interval from
20 mK up to 2 K (see [4–7] and Fig. 1). The original data
[4] for M T(ass)( ) in interval 1 K � �T 2 K have been ex-
tracted from the impedance behavior of 2D-ion pool. In
addition to measurements this paper contains a reason-
able scenario for M T(ass)( ) interpretation. The following
publications: [5–7]-deal with the 2D-plasma resonances,
which are sensitive to M. These measurements show
M T(ass)( ) dependence until � 20 mK. Below 20 mK the
temperature influence in M (ass)-structure is negligible.
Limiting value
M T( )� 0 � 30m4, R id < 6 � (31)
corresponds to modern level of total snowball effective
mass limitation, caused by helium compression and
superfluid flow association contribution in low-tempera-
ture regime. Estimation (31) for R id corresponds to re-
quirement, that the total mass M T( ) has a finite contribu-
tion from M (sol) (4).
From other side dc-mobility data in «Stokes area» from
Ref. 4, when radius Rn is important, lead to estimation
Rn � 7 � > R id
which is consistent with (30).
It is interesting to note that all authors [4–7] indicate
the temperature influence in snowball effective mass be-
havior. And nobody discuss M (ass)( )� dependence. But
all information above leads to the conclusion: both
M T(ass)( ) and M (ass)( )� are coupled. To demonstrate at
least qualitatively M (ass)( )� sensitivity, we can use the
experiments with 2D-plasma resonance �( , )l m excita-
tions [5]. Plasma excitations in an infinite 2D-ion system
have the dispersion law �( )q
�
�
�
2
22
( )
( )
| |q
e n
M
s� q . (33)
Here ns is the 2D ion density; �, q are the frequency and
the wave vector, respectively; M( )� is the ion effective
mass which generally depends on �. The knowledge of
the spectrum�( )q obviously allows one to extract the ion
effective mass.
In the experiments of Ref. 6 the 2D-ion pool had the
shape of a disk of radius R. In that case the continuous dis-
persion law �( )q (33) is replaced by a set of discrete
eigenfrequencies �( , )l m where the integers l and m label
the radial and azimuthal wave numbers
q q l m q l q m2 2 2 2� � �( , ) ( ) ( ) .
The eigenfrequencies observed in Ref. 6 correspond to
the following pairs: (0,1); (0,2); (0,3). Their measured
relative values are (information from Fig. 1 in Ref. 6)
�
�
�
�
2
2
2
2
0 3
0 1
5 849
0 3
0 2
1 878
( , )
( , )
. ,
( , )
( , )
.� � (34)
and depend on both the ratio of the wave numbers q l m( , )
and the ratio of the effective masses M l m M l m( ( , )) ( , )� � .
Thus, for example,
�
�
2
2
0 3
0 1
0 3
0 3
0 1
0 1
0 1 0
( , )
( , )
( , )
( , )
( , )
( , )
, ( , ) (� �
q
M
M
q
M M , ).3
Further, one can calculate the same frequencies (0,1);
(0,2); (0,3) for the cell geometry of Ref. 6 and find their
ratios assuming M � const. In that case, e.g.,
�
�
2
2
0 3
0 1
0 3
0 1
( , )
( , )
( , )
( , )
theor
�
q
q
yielding
�
�
�
�
2
2
2
2
0 3
0 1
5 115
0 3
0 2
1 76
( , )
( , )
. ,
( , )
( , )
.
theor theor
� � 8. (35)
If the mass M were frequency independent, the esti-
mates (34) and (35) would give identical figures. How-
ever, actually the ratios (34) exceed the corresponding
numbers in (35) suggesting M M( , ) ( , ),0 1 0 3� in full agree-
ment with (13).
Here several words should be said concerning the
method used in Ref. 4 to determine M T(ass)( , )� . In that
paper (relevant to the viscous scenario of the M T(ass)( , )�
description) the authors declare the possibility to measure
both dc and two ac components for ion mobility� �( ) with
the same 2D-ion density distribution. In this case within
the Drude approximation we have the reasons for the fol-
lowing definitions:
� �( ) ,0 � e /M (36)
Im ( , )� � �
� ��
� �
�
�
e
M 1 2 2
, (37)
Re ( , )� � �
�
� �
�
�
e
M
1
1 2 2
, (38)
where � is the corresponding relaxation time, � is the
ac-ion mobility.
The ratio
Im ( , )
Re ( , )
� � �
� � �
��� (39)
has to be linear function of�. If so, we have from (39) the
� definition. Finally, using � (39) and information (36) for
1020 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I. Chikina, V. Shikin, and A. Varlamov
�( )0 the way is open for M estimations (see (19) from
Ref. 4):
M T
e e
( )
( ) ( )
Im ( , )
Re ( , )
� �
��
��
��
� � �
� � �0 0
. (40)
Drude–Stokes modification of the ion-motion equa-
tion with an additional information on M n ( )� (13) leads
to evident reorganization of the definitions (36)–(39).
Nevertheless, the authors [4] believe the relationship (40)
is still valid when �� 0. The corresponding M T( ) data
are presented in Fig. 1.
On the other side, using (13), we have in the limit
�� 0
Im ( , )
Re ( , )
( )
( )
� � �
� � �
� �
�� �
��
�
"
M
R R/
n
6 1
. (41)
Within this scenario, the combination (40) cannot be
finite when�, and therefore, the definition of M n ( )� (22)
alone cannot be used for interpretation of experimental
data presented in Fig. 1. To resolve this problem, we re-
member the speculations above (see the discussion
around (14)). The idea is to normalize the limiting process
(40), using instead of (13), (41) some modification
M n x( )� , where �x follows from the condition (14). In
our model, such a requirement evidently corresponds to
the situation when both the real and imaginary parts of the
Stokes force (12) are of the same order. Equating them
one can evaluate the corresponding frequency as
�
�
�
x
n n
T
x T
T R
x /( )
( )
( )
� �
2
2
2
2
9 2 . (42)
If so the value of associated mass (13) is reduced to
M R
x
xn x n n
( )( )ass � � �� �
�
�
�3
1
1
2
9
3 . (43)
The fitting of the total effective mass Eq. (6) with the
definitions (7), (8) for M s
( )ass and (43) for M n
( )ass plus the
simplifications R R Mnid
(sol)
� , � 0, where Rn is adjust-
able parameter, demonstrates a good agreement with the
experimentally observed temperature dependence in the
wide range of temperatures (see Fig. 1). This fact qualita-
tively supports the idea to use two-liquid scenario and to
account for the viscous velocity distribution in the frame-
works of the Navier–Stokes flow picture.
It is necessary to mention that understanding of possi-
ble viscous origin of the temperature dependence
M Tn
( )( )ass was already contained in the early paper [4].
We support ideas from Ref. 4, demonstrate the basic rea-
sons for unusual features in M Tn ( , )� behavior, liberate
the theory from snowball Atkins [2] assumptions, intro-
duce two different radiuses: R id and Rn , repair the fitting
program [4] to explain the data Fig. 1 and discuss briefly
the modern (after [4]) experimental situation.
In conclusion, it should be emphasized that the ob-
served temperature (actually, frequency) dependence of
the snowball effective mass is interesting not only in it-
self. It also turns out sensitive indicator revealing qualita-
tive difference in the velocity distribution field around
the moving sphere in either viscous or ideal regime. This
difference has been known for a long time. However, it
did not attract much attention since in the applications
(calculation of the Stokes drag force) slow decrease (of
the 1/r type) of the velocity field does not result in any di-
vergence in its real part. The situation is quite different in
the calculation of the imaginary part, and the above out-
lined insight in this field is one of important conclusions
of this paper.
Certainly, the hydrodynamic treatment of the M(T,�)
dependence imposes some restrictions on explanation of
the available experimental data obtained at low tempe-
ratures (in the vicinity of several mK). However, the alter-
native kinetic language suitable for description of the ballis-
tic regime confronts in substantial difficulties when the
sphere effective mass is calculated even in the laminar limit.
Details of this formalism will be reported elsewhere.
This work was partly supported by RFBR grant No 06
02 17121 and the Program of the Presidium of Russian
Academy of Sciences «Physics of Condensed Matter».
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Charged snowball in nonpolar liquid
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1021
0
30
60
90
120
1.1 1.3 1.5 1.7 1.9 2.1
T, K
M
/m
as
s
4
Fig. 1. The temperature dependencies of M Ts
( )( )ass according
to Eqs. (7), (8) (triangles), M Tn
( )( )ass (see Eq. (43), black dots)
and their sum M Ts
( )( )ass � M Tn
( )( )ass (crosses) are presented se-
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1022 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
I. Chikina, V. Shikin, and A. Varlamov
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