Manifestation of the collective effects in the rotational motion of molecules in liquids

journal of MOLECULAR

LIQUIDS ELSEVIER

Journal of Molecular Liquids 93 (2001) 95-108 www.elsevier.com/locate/molliq

Manifestation of the Collective Effects in the Rotational Motion of Molecules in Liquids T.V.Lokotosh, N.P.Malomuzh D e p a r t m e n t of Theoretical Physics, Odessa State U n i v e r s i t y 2 Dvorjauskaya Str., 65026, Odessa, Ukraine The Lagrange approach to the theory of thermal hydrodynamic fluctuations is developed. The rotational motion of a Lagrange particle is studied in detail. The apparent expression for the angular velocity autocorrelation function (AVACF) at arbitrary times is derived. The obtained results are used to explain the specificity of the behaviour of the AVACF for a molecule. The collective contribution to the rotational self-diffusion coefficient as well as the manifestation of collective effects in the frequency dispersion of the dielectric permitlivity are discussed. The peculiarities of the AVACF for two-dimensional liquids are considered. © 2001 Elsevier Science B.V. All rights reserved.

1. Introduction The angular velocity autocorrelation function for a molecule and colloidal particle has repeatedly been a subject of computer experiments [1-5] and theoretical works [6-15]. In these works special attention was paid to the explanation of the fraction-power long-time asymptotics of the A V A C F . It was clearly shown that the specificity of the last is caused by the long-living transversal hydrodynamic modes, which are also responsible for the long-time behaviour of the velocity autocorrelation function (VACF) for a molecule [16-19]. The possibility of the non-trivial generalization of these results was proposed by I.Fisher in [20]. In that work it was supposed that the velocity of a molecule could be presented as a sum V(t) = V'(t) + lYe(t)

(I.i)

of the relative velocity I~' of the molecule about its nearest surroundings and the collective velocity ~YcOf the latter. It is clear that Vccorresponds to the slowest motions of the molecule. Besides, from the physical point of view it is possible to identify Vcwith the velocity Lagrangc particle,including the considered molecule. As a result, one can expect that

< ~(t)~(0) >Loo =< ~L (t)~L(0)>

VL(t) of the

(1.2)

To calculate the V A C F of the Lagrangc particle,the principles of fluctuation theory for the Lagrangc hydrodynamic variables were formulated in [20]. In such a way the behaviour of ( VL (t)VL(0)) was investigated for both t --+ oo and t ~ 0. On this basis the collective part D cof the self-diffusion 0167-7322/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII S0167-7322(01) 00214-8

96 coefficient D for a molecule had been defined. The analysis of the ratio D c / D demonstrates an essential role of the collective effects. This approach was used in [21] for detailed analysis of AVACF for a molecule. The collective part of the rotational diffusion coefficient was estimated. Unfortunately, the formal apparatus of the theory used in works [20, 21] was not good enough. There the central role is played by the special differential operator performing the transformation from the correlation functions of Euler variables to that of Lagrange variables. However, usage of this operator is quite satisfactory only at large t. In the limiting case t ~ 0 it leads to the singular contributions to autocorrelation functions. This difficulty was removed in the recent work [22]. In it the integral operator was proposed instead of the differential one. It leads to the finite values of Lagraage autocorrelation functions at t = 0 which are consistent with the demands of equilibrium statistical mechanics. In the present work this modernized approach is applied to study the rotational motion of molecules. In Section 2 the AVACF for a Lagrange particle is investigated in detail. In Section 3 the obtained results are applied to the rotational motion and rotational diffusion of a molecule. In particular, in Section 3 the frequency dispersion of the dielectric permittivity is considered. The peculiarities of the AVACF in the two-dimensional case are discussed in Section 4 and the Appendix. In the conclusions some new problems are formulated.

2. Lagrange autocorrelation function for the angular velocity To determine the angular velocity of a Lagrange particle we can use the nonlinear hydrodynamic equations in the Lagrange form [23] or try to establish the interconnection between the correlation functions of Lagrange and Euler variables, as it had been made in [20-22]. The second w a y seems to be preferable. In this case we should start from the equation connecting the angular velocity of a Lagrange particle with the corresponding Euler variable. The simplest equation of such a kind is:

~L)(t) = 1 [

~(E)(~+ ~;t)d~

Jvz

.

(2.3)

¢=~',(t)

where 1

-

~(E)(~., t) = ~rotV(~', t)

(2.4)

is the angular velocity of visible rotation in the Euler approach to hydrodynamics , tT(~',t) is the velocity field, VL is the volume of the Lagrange particle, 7/is its radius-vector. Nonlocul character of the equation (2.3) differs our definition of ~L(t) from that used in [23]. However, in spite of this it leads to results having restricted applicability. Indeed, let us consider the consequences of this step in more detail. According to [22] the Lagrange autocorrelation function of the angular velocity is equal to:

~blL)(t) ----< ~!L)(t)~L)(0) >----

(2.5)

=~fvL

d~'~fvL d~'2f de < ~(E)(~+ ~ _ ~-~,t)~(E)(6,0) > .wG(r's]r(t))

97 where the angular brackets denote the averaging over possible realizations of the hydrodynamic fields and

w~a~(~lr(t)) = (~r(t))2 z 1 2 ~xp • ~

(2.6)

is the Gaussiaa distribution for the mean square displacement I'(t) of the Lagrange particle. Because of (2.4) the Euler correlation functions < ~(E)(~., t),7(g)({~, 0) > and < V(~', t)~'(O,O) > satisfy the equation: 1

< ~(E)(~, t~(E)(6, 0) >---- --~ < Ag,(~, 09,(6, 0) >

(2.7)

where A is the Laplacian and ~ is the transversal component of the velocity field. Using the formula < ~(~', t)~,(6, 0) > =

p (47rut)3/2 .e=p

-

(2.8)

established in the theory of thermal hydrodynamic fluctuations [24, 25], we obtain the consecutive algorithm to calculate OL(t). The concrete calculations become more convenient in the Fourier representation. In this case

Introducing the Lagrange autocorrelation function 4~(t) of the velocity field, we can rewrite (2.9) in the form:

L = - ~ d A(L)tAtt~ ¢~(t) dAy(t) 'el' ~ ~ JJ'

A~(t) = ut + r(t) 6

(2.m)

At last substituting the evident expression of 0~(t) from [22] into (2.10) we get ~b(Ll(t) = -~:v/~kBT'x'/2[~4

2r~A~ Au2"exP(-~-~) (r~+2A~)]

(2.11)

The formula (2.11) leads to the long-time asymptotics: ¢(L)(t) =

3 ksT _1 32r 3/2 P0 A~/2+

(2A2)

exactly coinciding with that obtained in the works [7-12,1-4]. At the same time the asymptotics of (2.12) at t ~ 0 is wrong. In fact, from (2.11) it follows: 2

2mL~2

~r

V h~

To correct this result we must focus our attention on the definition of the angular velocity of the Lagrange particle given above. The expressions (2.3) and (2.4) are correct only for large-scaie

98 hydrodynamic flows, responsible for the long-time behaviour of gp(~L)(t). In the general case it is appropriate to use the integral definition: (2.14) It is clear that in the case of large-scale hydrodynamic fluctuations lT(Y+ fi; t) can be approximated by the first term in its expansion in powers of F-t: IT(F'4" F-l;t ) = IT(F,t)4" [~(E)(y,t), F-l]4"0 ( L )

(2.15)

where L is the characteristic spatial scale of the velocity field. Up to 0 (~Lt-) the formulas (2.3) and (2.4) lead to the same expression:

~iL(t)=

~(E)(F-,t)÷.=ri(O+ 0 (--~)

(2.16)

The integral definition of ~}L)(t) allows us to agree the short-time asymptotics of ¢(~L)(t) with the demands of statistical mechanics [26]. Indeed, constructing the Lagrange autocorrelation function of the rotational velocity with the help of (2.14) and using the equation: 1

< Vi(fi,t)Vk(F-2,0) >= ~t~k < ¢(lfi - ~'~l,t)IT(6,0) >

(2.17)

reflecting the spatial isotrop~, we obtain: C(,)

=

" rldrl)

-5

L L drl L,L dF-2fdr'W(G)(r'alr(t))(F-1F-2)× (2.18)

× < ~;(F-+ fi - ~'~,t)~(6,0) >

Since at t = 0 the Euler correlation function for the transversal components of the velocity field reduces to < 17t(~;0)iTt(0, 0) >= 2kB.T..b(r-') p0 and

w(a)(~lr(t))[,=o

= ~(r-3

(2.19)

(2.20)

from (2.18)it follows:

~b~L)(0)----<(t.~(L)(0)) 2 >_-- 3kBT IL

(2.21)

where IL = g~mL rL2 is the inertia moment of a spherical Lagrange particle. Statistical mechanics[26 leads to the same result. To calculate ¢(L)(t) at arbitrary t we substitute (2.6) and (2.8) into (2.18) and use the equality: r. l r_ 2 =

¢2.22)

99 Then 2p0

24

~a + 2-~a t ; )

~r L

"eZP (-a(~l - r'2)2)

(2.23)

where a = 4-~" The integral /~\3/2

t

"(°):/. "/."t:: ""~-°("-< .

.

determines the behaviour of the Lagrange velocity autocorrehtion function and according to [22] is equal to: /,(-} =

1 2 2_ ['(2v~r.)V. 9-~ (3-aa-2rO ÷exP(-4ah)3~a T-(2r. I)] 1

(2.24)

ir

The second integral I2(cO

=

/,,L#X~/,,L~ 2 t ; )

I'a\312

...(-.,-~2~')

has analogous structure: [1 /~'/'4 5

_~)

(

,) ')1

lr~

1~

+

(2.25)

2g~

Here 2

¢ ( x ) = ~ f ° e~p(-;)du is the error function. Substituting (2.24) and (2.25) into (2.22) we obtain the foHowing final result:

po

vzr~ g~L"L¢(2V~"~) l

,~

i"6d2 +

,~

7r

31

~

(2.26)

At t --* 0 and t --* oo the function ~ ) ( t ) has the following asymptotics:

[

l - 7; v

,~

(

I-

~

T r~L] + 0

( ('))l exp

t<
¢4L)(t) = 3 k B T ~

1 4 A v + 6,3A~ - -

'

t > > - ~It

100 As it is easy to see, the main asymptotic terms of

~b(~L)(t)at

small and large t coincide with

(2.21) and (2.12). To obtain the time dependence of ~(L)(t) in apparent form, it is necessary to substitute into (2.26) and (2.27) an expression for the mean-square displacement r(t). Its behaviour was discussed in [21,22]. Here we use only the asymptotic expression for F ( t ) at t --+ oo which is relatively simple and does not depend on details of the hydrodynamic model. In accordance with [22]

r(t)

=

6DLt -- 6Bt 1/2 + ....

for t > ~

1

( baT ~ 2, \poDL]

baT

B = 3p [~r(t, + DL)]3/2' (2.28)

where DL is the self-diffusion coetficient of a Lagrange particle. Combining (2.27) and (2.28), for long times we obtain: (L)

_ 3~'kaT

~b~ ( t ) 0

1

p (4~v(y.i_OL)t)$/2'

fL1 +

~(~+D'~-)t~-~

(2.~)

~-~

From (2.26) it follows that details of the hydrodynamic model essentially influence the behaviour of d)(~L)(t)~only at t ,,~ -~v• Therefore with satisfactory accuracy the Lagrange AVACF (LAVACF) for all t can be approximated by the expression:

~(L~lt~ 3kBT

1

At t ~ 0 and t -~ ~ it has correct asymptotics and for intermediate t it smoothly and monotonously decreases likely to the exact function (2.26). Using (2.30) and the additional assumption: F(t) = 6DLt at all times, we can calculate the spectral function of the LAVCF useful for many applications: 7(L)(~ ) = 1

foCCf~(L)(t)cos(~t)dt (2.31)

In such a way we obtain:

has the meaning of the rotational self-diffusion coefficient d L . Writing it in the standard form

kaT dL = 8~rrlV~l~----

{2.33)

we get that the effective volume ~ f l = 2- - -531 - ~ (8r)2/-----~ 1 +

VL

(2.34)

differs from the real one, V e H / V L ,,~ 0.3. This circumstances reflects the difference in the boundary conditions for the rotation o f a Lasranse particle and a Brownian one. Besides, it is worth to note the fraction-power contribution to the spectral function 7(L)(~) at small w:

It ]s caused by the leading asymptotic term of LAVACF ~b(~L)(t)and like the last has universal character, not depending on the size of the Lagraage particle.

3. Collective c o n t r i b u t i o n to t h e r o t a t i o n a l self-diffusion coeffic i e n t o f a molecule The obtained above results give us the basis to estimate the Ions-dine behaviour of the AVACF for a molecule and the collective part of the rotational self-diffusion coefficient for it. In fact the rotational velocity ~ ( t ) o f a molecule can be represented as a sum of the rotational velocity a3(L) ( t ) o f the Lagrange particle, including the considered molecule, and the relative rotational velocity of the latter:

(3.36)

~(t) = ~(L)(t) + ~'(t)

In other words, we take into account that the molecule turns about its nearest surroundings and together with it turns in the field of the hydrodynamic fluctuations [20, 21]. In accordance with Fisher's central idea we assume that the long-time asymptotics of the AVACF for a molecule is determined by the AVACF for a Lasrange panicle

~,,~(t)l,_~ = ,~"'(t)

(3.37)

at suitable choice of the Lagraage particle size. In work [22] the corresponding arguments are given and we take: rL = ~

3

VV/-Pr-~.~

(3,38)

where FM is the Maxwell relaxation time for shear tensions. To motivate this relation we take into account that in our consideration the diffusion mechanism of the time evolution for the transversal mode is applicable only up to tD "~ ivl " On the other hand,• the diffusion and elastic excitations in the molecular system are separated by the time FM. Identifying t o and TM we obtain the formula (3.38). Note that the numerical coefficient in it is borrmved from the asymptotic expansion of the Lagrange VACF at t --* 0. Substituting (3.38) into (3.37) we obtain: ¢$(t)lt__ ~

= 3rk~T P

1 {4r{p+D£

)t) 51~"

(3.39}

102 To estimate the collective contribution d, to the rotational diffusion coefficient d, we take into account that different terms in (3.36) are physically independent.In accordance with this the rotational self-diffusion coefficient can be represented in the form: (3.40)

d,=d~+dc

where the one-particle contribution dl is determined by the expression dl = g

< aT'(t)~7'(0) > dt

(3.41)

and dc is equal to (2.33) at rL given by the formula (3.38). In such a way we get:

,~k~T

d c - ~/(VrM)3/-----------~ + 0

(_~)

,

5z/5

)~ = 33/5614/~rrl/lo,

r/= pv.

(3.42)

Here we additionally < p_ ,,, 10-~, in which D is the molecular self, used the inequality: DL p diffusion coefficient. Note that the estimate (3.42) for the collective part of the rotational self-diffusion coefficient is applicable for liquids of anisotropic molecules with a maximum molecular size rmbeing smaller compared to r L . _For typical liquids of such a kind o ~ l O - 2 c m / s a n d r M ~, 10-12s [27], therefore r L I0 cm. The ratao d c of the value d r measured m experiments has the following order of magnitude: d,

~ r,

~3

(3.43)

dc is about(0.03:l:0.1)at the melting temperature. But, the fraction of the collective Numerically d-~-

contribution to d r increases noticeably if the temperature becomes higher. It becomes evident if we op

take into account the Maxwell assumption: r M - -~--, in which G is the shear elastic modulus. Since p changes slightly with temperature and G behaves like p , we expect that: d-~-~

m

(3.44)

where symbol m denotes the value of the parameters at the melting temperature. From (3.44) it follows that the ratio ~ is a monotonously increasing function of temperature. The collective motions in a liquid should also be manifested in the regularities of spectra studied in experiments. For example, let us consider the fine details of the dielectric relaxation. Usually, the relaxation equation for the averaged value of the molecular dipole moment/~(t) is taken in the simplest form [28]: -

1

/z = - - /r~ ~.

.

(3.45)

It is known [29], that the inverse value r d I of the relaxation time and the rotational self-diffusion coefficient dr are connected by the relation : 1 rd = 2d~.

-

-

(3.46)

103 Using now (3.40) and assuming the frequency dispersion of d~ in accordance with the formula

de(o;)= ;7(L)(w)]

" ~r

for the spectral density < fi(t)fi(O) >o, we can write:

< p(t)p(o) > , = < Z~ > 1

gt(~) 2 @ + g=(~))2 + g~(~)

(3.47)

where gl(w) and g2(w) are real and imaginary parts of the function

g(~) = 2(d, + 4(~))-

(3.4s)

The frequency dependent part of g(u~) arises as a result of the interaction of rotational degrees of freedom with long-living hydrodynamic modes. One more remark should be made about interrelation between rd and r~. In accordance with the theory of hydrodynamic fluctuations (see[31D for liquids with nonsphefical molecules the Maxwell relaxation time r u is equal to the relaxation time r~ of the anisotropy tensor: r~ = r,. In its turn, accordingly to [29] 1/r~ = 6do~. Therefore, r u = 3rd and the formula (3.47) cam be rewritten in the form: 1

< p(t)p(o) > , ~ = < ~ 2 ( o ) > -~" (,,., +

1/'rd + h,(3~rd)

-. ,~(~,'d)) ~ + (1/,-,, -; A,(~,-~)i"

(3.4o)

where A(w) = d~(~,,)- d~. Though tile magnitudes A t and A2 are relatively small they can be registered due to nontrivial fraction-power dependencies on the frequency. So, at 3~rd << 1 we have < ,~(t)p(O) >,~=< ~ ( 0 ) > ,-~,[1 - B(~,,.)~/2

_

(~,.d)~ + ...]

(3.50)

where ~ is an easily calculated coefficient. Analogous manifestation of the collective effects takes place in spectra of molecular fight scattering. Note that the applicability region of the modified spectral functions is limited by the frequency ~ ,,, 1/3rd N 1/v~t, which is the only characteristic parameter in the simplest hydrodynamic model. (L) To define more exactly the behaviour of the LAVACF f/~ (t) at t = r M it is necessary to apply to more fine hydrodynamic models. So the model with the shear viscosity relaxing accordingly to:

1 + hZrM

(3.51) leads to the following Euler correlation function for the transversal velocity [30]:

2vrMZ

(3.52)

104

where Z = ~/'~"S'~'~, c 2 = u/rM, H(z) is the Heaviside function, Jl(x) and J2(z) are the Bessel's functions of the first and the second orders. The ~-like terms in (3.52) describe the elastic reaction of a liquid at t < r~, the other terms generalize the purely diffusive contribution (2.8). At t > > r ~ the expressions (3.52) and (2.8) become identical. One can show that ~-like terms in (3.52) in the time interval 0 _< t < t~, where t~ ,,, r~, generate the nonmonotouic polynomial contributions to the Lagrange VACF and AVACF. In such a way the oscillations of AVACF for a molecule [1, 2] can be connected with the spread of high-frequency transversal waves in the system. However the full consideration of this possibility leaves the frames of the paper.

4. Rotational motion in a two-dimensional liquid Generally, the definition (2.18) of the Lagrange AVACF remains invariable:

£

£

JSL

•(F1. ~'2)1:¢°(: IF(t))

(4.53)

As earlier dF denotes the volume element in the two-dimensional space: di: = rdrdx, where Z is the polar angle, SL = Irr2L• The Euler VACF and the Gaussian distribution for the mean-square displacement F(t) are: <

k,T 4~rc,ut exp

~(F,t)~((~,0) > =

Wa(:lr(t)) =

r--~exp

-"~ (:)

(4.54)

r-¢

Here ~r is the surface mass density. After integration over F and transformation, analogous to (2.22), we obtain:

O,~(t) = kBT ( ~ d~2d~-2 [

dF1f dF2 [~ -

1

1 d

_

]

where a = 4-~L,A = vt + F/4. The result of accurate integration in (4.56) is very cumbersome and is not suitable for the further use. In this situation we will consider the leading asymptotic terms of ~(L)(t) at t --~ 0 and t ~ ~ and will construct the relevant approximate formula for o(~L)(t). It is not difficult to see, that:

--

k~-~T4-~

[

]

1 -- 31-:~q- 97.:.:~ -I6.~ - - "'"

A --~ 00,

,'-,

where [z = i1m z r 2L is the inertia moment of a Lagrange particle (mL = crSL). From (4.57) it follows, that ~(z)(t) can be approximated by the formula.:

ksT

1

105 The time dependence of the mean-square displacement F ( t ) i s discussed in the Appendix. Using these results we can estimate the rotational diffusion coefficient d r for a Lagrange particle in the twodimensional case and some other characteristics of the rotational motion. In particular for d r we obtain: (4.59)

1 knT d~ - 2 ~ / ~ avr2L

To describe the rotational motion of a molecule it is necessary to choose the suitable size of a Lagrange particle. The diffusion regime in the time evolution of correlation functions forms at t > (>>)tn ~ r~/v. Assuming the existence of the Maxwell relaxation time r ~ for the transversal modes in the two- dimensional liquid and attracting the reasons used earlier we suppose tD ~ r u . Taking the numerical coefficient from the expansion (4.57) at t --* 0, we get rL = 2 VX/'~-~M

(4.60)

Substituting this value of rL into (4.59) we can obtain the estimate for the collective contribution d, to the rotational diffusion coefficient of a molecule. These results can be used, for example, to describe the rotational motion of the director field in smectic liquid crystals.

A p p e n d i x . Mean-square displacement of a Lagrange particle in the two-dimensional case In accordance with [21, 22] the mean-square displacement the equation: d2F(t) dt 2

= 2~(t),

r(t)

of a Lagrange particle satisfies

(4.61)

where A(L)(t) is the Lagrange velocity autocorrelation function, determined bv the expression:

The latter has the asymptotics: ~-~-[lmL+ 0(exp(---~t ))1 ,

A(t) -+ 0

q$7"i(t) =

(4.63) .1-i~--~+gex(0]"

~(t)

Combining these small-time and long-time asymptotics we obtain the approximate version of ¢(L)(t): ¢~)(t) -~

2ksT

1

mL 1 + 8r--~. (vt + r4__~)

(4.64)

106 allowing us to carry out all needed calculations. From (4.61) and (4.64) it follows: a Z'(t) = Z-~, a = 4~akBT -------5~, mL

(4.65)

Wbere Z(t) = 1 + 87t a (Vt + F(4t)]. The equation (4.65) completed by the initial conditions: mL 1-(0) = 0,

F'(0) = 0 or

Z(0) = 1, Z'(O) = 8~ ow _ b mL leads to the following integral:

(4.66)

= 2act,

where

(4.67)

c = exp ~ a

" The meun-square displacement F ( t )

satis.lying (4.67) has the following

as~nptoties: kBT 2

r(t) = ~ mL

4 v tr

t---) 0

3 rE

.J

~

| r ~ lff~n~il-1

llnln_t

1 =!. l n c

[

4 lnt

V4(lnTr+j

[_

ln t

(4.68)

] -4vt'

t ----~ oo

where t = 2 4 ~ . Note that the leading divergent term in F ( t ) at t --~ oo, as it should be, coincides with that obtained earlier in works [19,20]. The existence of such an asymptoties is equivalent to the logarithmic divergence of the self-diffusion coefficient. However this conclusion seems to be premature(see also [32,33]). In fact, the main contribution into the long-time behaviour of the Euler correlation function < Vt (r,

t)Vt (6,0)

> : kBT - - 1 0

f d~e.-(v~:t+i~)

(2Zt) 2

is given by the integration over small I~¢[. The time dependence in (4.69) is taken from the linearized Navier-Stokes equation for the Fourier-transforms of the vclocit3" vector. However, at small k it is necessary to take into account the nonlinear contributions too. We plan to discuss this problem in a separate work.

CONCLUSIONS The main attention in this work is paid to the investigation of collective effects in the rotational motion of molecules in liquids. All estimates are obtained with the help of the simplest hydrodynamic model for incompressible liquids. Within such an approach we are able to reproduce satisfactorily the long-time behaviottr of the AVACF for a molecule as well as some details at small times. However to describe the oscillations of the AVACF at t,< t < h , where h,h < VM, we should refine the hydrodynamic models. For example we can take the model with relaxing shear viscosity. It is essential that the applicability region of the obtained results is restricted by the inequality: rm -< rL, where rm is the maximum molecular size. In the opposite case it is necessary to develop another approach (some details can be found in [ 12-1 5]). We discussed the manifestation of collective effects in the frequency dispersion of the dielectric permittivity only. However, analogous manifestations will take place in

107 the spectra of molecular fight scattering and incoherent scattering of slow neutrons in liquids with anisotropic molecules. We hope to consider these questions in the future. We are cordially thankful to Prof.L. Bulavin, Prof.M.Emst, Prof.N.P.Kovalenko, Prof.S.Magasu, Prof.J.Teixeira and Prof.A.Zatovsky for fruitful discussions of the obtained results.

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