- Email: [email protected]
Thermal fluctuations of the chain end
Volume 120, number 2
PHYSICS LETTERS A
2 February 1987
THERMAL FLUCTUATIONS OF THE CHAIN END E.V. K H O L O P O V The Institute of Inorganic Chemistry of the Siberian Division of the Academy of Sciences of the USSR, 630090 Novosibirsk, USSR Received 30 September 1986; revised manuscript received 24 November 1986;accepted for publication 26 November 1986
In the framework of the simplest one-dimensional elastic model an algorithm making use of recurrence properties is proposed for obtaining the temperature Green function for the chain-end atom. As a result, the quantum temperature fluctuations of the chain-end atom can be described without any expansion in eigenfrequencies.
1. Introduction
In the last few years systems containing onedimensional constituents as the structural elements [1,2] have been the subjects o f intense interest. On the other hand, the remarkable interest in onedimensional systems is also stimulated by the possibility of obtaining some rigorous theoretical results for them (see, e.g., refs. [3,4] and also ref. [5]). The situation we consider takes both o f the indicated features into account. In the present paper a straightforward algorithm for deriving the value of the thermal fluctuations of the chain-end atom is proposed. As the simplest example of its realization, the linear elastic chain where only longitudinal atomic motion is admissible is studied. As a result, the reconstruction o f the excitation spectrum by the edge effects is not required. As far as the application is concerned, the value in question may be essential for estimating the potential barriers determining the rotation o f some linear ion which plays the role o f a decorated link in a chain, that occurs in the T1HF2 c o m p o u n d [2,6 ]. This method is also suitable for investigating the corresponding localized edge states.
2. Model
Let a linear chain of atoms located in equidistantly distributed parabolic potential wells be infinite but have an origin. Let every pair o f nearest neighbour0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ing atoms be joined by an elastic linkage as well. The corresponding hamiltonian has the form
H = ~ [p2/2m+½~u2 +½g(ui+l-U,)2],
(1)
i=l
where Pi and ui are the longitudinal m o m e n t u m and displacement o f the ith atom, respectively, rn is the atomic mass, which is the same for all atoms of the chain, E and g are strength parameters, and the summation is over all the atoms beginning from the original one at i = 1. The temperature Green function is determined in the conventional manner as Gij(r) = - ((u~(r) uj )) ~,
(2)
where ui(r) = exp ( r H ) ui exp ( - zH),
(3)
the double angular brackets with the subscript r here and below denote the irreducible average of the arranged set of the factors contained, which are regarded as the functions of different z, so that the value of z increases upon moving along the set from the right to the l e f t , - f l <~z <~fl, T= l/fl is the temperature in energy units. According to definition (2), the value o f (( u 2 )) we are interested in is specified by the expression
<
(4)
Hence, the Green function GII(z) is especially important for our purposes, and we restrict ourselves to its calculation. 69
Volume 120, number 2
PHYSICS LETTERS A
3. Condition of recurrence
Before finding the value of G. ~(r), it is useful to carry out some preliminary calculations. We introduce the auxiliary hamiltonian by adding the term ½gul to expression (1) and representing the value obtained in the form H,~ = H[I + Hint +
Hb,
{5)
where
2 February 1987
for the chain corresponding to H b and, furthermore. the hamiltonian Hb determining G = ( r ) coincides with tt,,:, if the appropriate change in the atomic numbering is performed. Thus, the subscripts of the Green functions in expression (10) may be omitted so that the recurrence relationship specifying (7(r) follows from formula (10). Let us introduce the Fourier transform [ 7 ] (~,,= ~(~(r) e x p ( i e G r ) d r .
H ( ~ = p ~ 1 2 m + (½ ~ + g) u~,
t6)
Hi,t = - gul tt2,
(7)
lib = S [ p ;~/ 2 m + ~ e1u ~ + ½ g ( u , . ,
.... u,) ~]
+ ½gu~.
18 )
The auxiliary Green functions will be labelled by a tilde. The Green function G ~ ( r ) corresponding to hamiltonian (5) can be built by making use of the perturbation series with respect to H . , . The series of interest is the expansion in irreducible averages of the conventional form [ 7 ] : B
(7,,(r)--~(-l)/+'f l:o l!
drl ...dr/ 0
× ~ ul(r) ulHm,(rl)...Hint(Z/)~'>~.
~
(a+ a
)
g21f drl...dr21
(12t
where a ~ and a are respectively the creation and annihilation phonon operators describing the isolated local vibration of the first atom with the frequency (~ + 2 g ] ' (o = \ - - ~ i t - i '
I I3 }
corresponding to Hi, (here and below we assume h = 1 ). According to expressions (2), (12 ), and the relationship for (7°(r) similar to (11), the Fourier transform C7~,~,is of the form (14) Upon transforming to the Fourier transforms, the right-hand side of eq. (10) is reduced to a geometric progression. The appropriate sum is
&,', G " - I - g 2 G-o, & ,
fl
/=0
(2re(o)
(9)
Due to the harmonicity o f hamiltonian (5), in the given sum only the terms with even I have nonzero values and the averages o f products of the displacements that arise are split into all the possible products of the pair averages giving rise to
G l l ( r ) = ~.
where oJ,, = 27t Tn, n is an integer, and C~,j, is defined similarly. For calculating fT~,~, we use the following representation for u~: u~
12
(11)
tt
llS)
On solving eq. ( 15 ) for G,,, we have
0
1
G , , - 292 0,0 { 1 - [ 1 - 4g-(O,,)-'] "2 }.
xG°,(r-r~) G~(':,-~:~) G,°,(r~-r~)..xG22(r2/_, -r2/) ~°~(r2/).
(10)
Here ~oj (z) is the Green function averaged with the help of the hamiltonian H0. As far as the Green function ~22(r) is concerned, it coincides w i t h G l i ( r ) inasmuch as the second atom is now the original one 70
(16)
4. Application
Now we return to the initial hamiltonian ( 1 ), which can be represented in the form
Volume 120, number 2
PHYSICS LETTERS A
H=Ho+Him+Hb,
2 February 1987 ~b0
(17)
where
Ho=pZ~/2m+½(~+g) u~,
(18)
Hint and Hb are determined by formulae (7) and (8). The appropriate Green function G. specifying the motion of the first atom of the chain at hand is determined by a formula similar to (15 ):
Re~
6o G. = 1
o~ , - g 2GnG.
(19)
where the Green function G o is of the form (14) but with t5 replaced by ~
1/2
in agreement with expression (18). Making use of formula (16) and the corresponding one for G o, from relationship (19 ) we finally have
2/m G.= where
to2 + A 2 + 4 ( t o 2
+ A 2 ) (to2 +B2),
(21)
A2=~./m and B2= (~+4g)/m. Formula (21)
solves in principle the task of determining the thermal averages of interest. On making use of the transformation inverse to (11 ), one may write ((u~))=-T
~ n=
G.
Fig. I. The plane of the complex variable to of integration in expression (23). The circle maps the circular contour of integration of the infinite radius. The bold-faced points on the imaginary axis map the countable set of the poles of the integrand. The bold-faced segments on the real axis are the branch cuts which confine the integrand to the required analytic branch.
the residues of the integrand at all its poles plus the integrals along closed curves surrounding the cuts connecting the branch points of the denominator of the integrand. In this case, we must obviously choose the analytic branch of the denominator which is characterized by the zero value of its argument at co = 0 in agreement with expressions (21) and (22). On performing some simple calculations, expression (22) is then reduced to the form 1 IBI B 2
(22)
toE 1/2
--oo
in agreement with expression (4). For the further reduction of expression (22) it is instructive to consider the integral
(24) Putting T--0, one can in particular obtain the contribution of the zero-point vibrations, which is equal to
~ d t o [exp(flto) - 1 ]
((u 21)) I T=o =12-~g [K(q) X [09 2 - A 2 - 4 ( A
E(q)],
(25)
2 - c o 2) ( B 2 - 0 9 2 ) ] - i
where =0,
(23)
which is taken along the circular contour of the infinite radius, having the origin as its centre and transversed, for definiteness, in counter-clockwise fashion. The plane of the complex variable to with respect to which the integration in formula (23) is carried out is shown in fig. 1. According to the theorem of residues, the integral (23) is determined by the sum of
q=
,
(26)
K(q) and E(q) are the complete elliptic integrals of the first and second kind, respectively. It is worth comparing the derived value of the fluctuation given by expression (24) with that relevant to the case of an inner atom of the infinite chain. The regular phonon spectrum of the infinite chain 71
Volume 120, number 2
PHYSICS LETTERS A
i
f
2
I
//" @
05
"~0
t5
!
Fig. 2. The values IPl'of the mean square atomic fluctuations measured in units of I~ mA ] ' versus the temperature T - ;"/[. t I at R-=- IB/AI =2. Curve 1 corresponds to (( u~ ~). curve 2 is relevant to ((u~)). The dashed lines corresponding to W~ - 2rt T/( 1 + R) and W2= ~ TIR are the asymptotes.
2 February 1987
ments, which, t o t instance, h a p p e n s in the chains o f the T 1 H F : c o m p o u n d [6] or in the m o l e c u l a r chains in c h a n n e l s [9]. A s i m i l a r c a l c u l a t i o n for a t o m s next after the original one can be p e r f o r m e d as well. In this case, however, the i n t e r a c t i o n b e t w e e n all the p r e v i o u s a t o m s m u s t be taken into a c c o u n t side by side with t e r m s ( 5 ) where H,', n o w describes the m o t i o n o f the a t o m in question. I n a s m u c h as the c o m p l e t e solution o f the spectral task for those first a t o m s is required, such a p r o g r a m rapidly b e c o m e s rather c o m p l i c a t e d with g r o w i n g n u m b e r o f the a t o m c o n c e r n e d . T h e p r o p o s e d m e t h o d for calculating the t h e r m a l f l u c t u a t i o n s can be a c c o m p l i s h e d too w h e n all the three o r t h o g o n a l d i r e c t i o n s are a d m i s s i b l e for the a t o m i c m o t i o n . In the latter case, h o w e v e r , the a p p e a r i n g system o f coupled e q u a t i o n s similar to ( 15 ) can be solved only n u m e r i c a l l y . T h e a p p r o a c h develo p e d is also suitable for i n v e s t i g a t i n g the peculiarities o f the existence o f the localized states located at the c h a i n end, that will be the subject o f the further publication.
d e s c r i b e d by the h a m i l t o n i a n s i m i l a r to ( 1 ) takes the form
o)~ = ( ~ + 4 g s i n 2 ( ½ k d ) ) ,
m
''2
Acknowledgement
(27)
,
where d is the i n t e r a t o m i c d i s t a n c e along the chain i n v o l v e d and the w a v e v e c t o r k varies f r o m -- n / d to Tc/d. T h e m e a n s q u a r e a t o m i c f l u c t u a t i o n is then specified by the usual p h o n o n e x p r e s s i o n [8] which has the f o r m iBI
1
f
coth (½flo)) do)
14! T h e natural i n e q u a l i t y
<> > <>
129)
follows i m m e d i a t e l y f r o m c o m p a r i n g (24) with (2 8 ). T h e c o r r e s p o n d i n g t e m p e r a t u r e b e h a v i o u r is shown in fig. 2. N o t e that the h a r m o n i c external c o n s t r a i n t tbr the original a t o m can be c o n s i d e r e d in the s a m e m a n n e r as the case o f the free c h a i n e n d above. T h e latter can also be u s e d for calculating b o t h the energy b a r r i e r for the r o t a t i n g d e c o r a t e d linkage a n d the barriers specifying the d i f f u s i o n o f the d o m a i n walls p e r t u r b ing the o r d e r e d state o f the l o n g i t u d i n a l displace72
I a m grateful to Dr. V,R. B e l o s l u d o v a n d Dr. N.A. N e m o v for helpful discussions.
References
1] L.N. Bulaevskii, Usp. Fiz. Nauk 115 11975) 263 [So< Ptbs Usp. 18119761 131]. 2] E.V. Kholopov, A.M. Panich, N.K. Moroz and ~u.G. Knger, Zh. Eksp. Teor. Fiz. 84 (1983~ 1(191 [Soy. Phys. JETP 57 (19831 632], 3] E.H. Kerner, Phys. Rev. 95 11954) 087. [4] J.A. Krumhansl and J.R. Schrieffer. Phys. Rev. B I I ( 1975 3535. [5] E.V. Kholopov, Solid State (ommun. 53 ll985) 681. [6] E.V. Kholopov, Solid State (ommun. 47 (19831 187. [ 7 ] A.A. Abrikosov, L.P. GoFkov and I.E. Dzyaloshinski, Quantum field theory in statistical physics (Prentice-Hall, Englewood Cliffs, 19631. [8] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon. New York. 1968). [9] Yu.G. Kriger, S.G. Kozlova, G.N. Chekhov& Yu.A. Dyadin and S.P. Gabuda, in: Abstracts of the Symposium on lnhomogeneous electronic states (Novosibirsk, USSR, 19841 p. 192.
PHYSICS LETTERS A
2 February 1987
THERMAL FLUCTUATIONS OF THE CHAIN END E.V. K H O L O P O V The Institute of Inorganic Chemistry of the Siberian Division of the Academy of Sciences of the USSR, 630090 Novosibirsk, USSR Received 30 September 1986; revised manuscript received 24 November 1986;accepted for publication 26 November 1986
In the framework of the simplest one-dimensional elastic model an algorithm making use of recurrence properties is proposed for obtaining the temperature Green function for the chain-end atom. As a result, the quantum temperature fluctuations of the chain-end atom can be described without any expansion in eigenfrequencies.
1. Introduction
In the last few years systems containing onedimensional constituents as the structural elements [1,2] have been the subjects o f intense interest. On the other hand, the remarkable interest in onedimensional systems is also stimulated by the possibility of obtaining some rigorous theoretical results for them (see, e.g., refs. [3,4] and also ref. [5]). The situation we consider takes both o f the indicated features into account. In the present paper a straightforward algorithm for deriving the value of the thermal fluctuations of the chain-end atom is proposed. As the simplest example of its realization, the linear elastic chain where only longitudinal atomic motion is admissible is studied. As a result, the reconstruction o f the excitation spectrum by the edge effects is not required. As far as the application is concerned, the value in question may be essential for estimating the potential barriers determining the rotation o f some linear ion which plays the role o f a decorated link in a chain, that occurs in the T1HF2 c o m p o u n d [2,6 ]. This method is also suitable for investigating the corresponding localized edge states.
2. Model
Let a linear chain of atoms located in equidistantly distributed parabolic potential wells be infinite but have an origin. Let every pair o f nearest neighbour0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ing atoms be joined by an elastic linkage as well. The corresponding hamiltonian has the form
H = ~ [p2/2m+½~u2 +½g(ui+l-U,)2],
(1)
i=l
where Pi and ui are the longitudinal m o m e n t u m and displacement o f the ith atom, respectively, rn is the atomic mass, which is the same for all atoms of the chain, E and g are strength parameters, and the summation is over all the atoms beginning from the original one at i = 1. The temperature Green function is determined in the conventional manner as Gij(r) = - ((u~(r) uj )) ~,
(2)
where ui(r) = exp ( r H ) ui exp ( - zH),
(3)
the double angular brackets with the subscript r here and below denote the irreducible average of the arranged set of the factors contained, which are regarded as the functions of different z, so that the value of z increases upon moving along the set from the right to the l e f t , - f l <~z <~fl, T= l/fl is the temperature in energy units. According to definition (2), the value o f (( u 2 )) we are interested in is specified by the expression
<
(4)
Hence, the Green function GII(z) is especially important for our purposes, and we restrict ourselves to its calculation. 69
Volume 120, number 2
PHYSICS LETTERS A
3. Condition of recurrence
Before finding the value of G. ~(r), it is useful to carry out some preliminary calculations. We introduce the auxiliary hamiltonian by adding the term ½gul to expression (1) and representing the value obtained in the form H,~ = H[I + Hint +
Hb,
{5)
where
2 February 1987
for the chain corresponding to H b and, furthermore. the hamiltonian Hb determining G = ( r ) coincides with tt,,:, if the appropriate change in the atomic numbering is performed. Thus, the subscripts of the Green functions in expression (10) may be omitted so that the recurrence relationship specifying (7(r) follows from formula (10). Let us introduce the Fourier transform [ 7 ] (~,,= ~(~(r) e x p ( i e G r ) d r .
H ( ~ = p ~ 1 2 m + (½ ~ + g) u~,
t6)
Hi,t = - gul tt2,
(7)
lib = S [ p ;~/ 2 m + ~ e1u ~ + ½ g ( u , . ,
.... u,) ~]
+ ½gu~.
18 )
The auxiliary Green functions will be labelled by a tilde. The Green function G ~ ( r ) corresponding to hamiltonian (5) can be built by making use of the perturbation series with respect to H . , . The series of interest is the expansion in irreducible averages of the conventional form [ 7 ] : B
(7,,(r)--~(-l)/+'f l:o l!
drl ...dr/ 0
× ~ ul(r) ulHm,(rl)...Hint(Z/)~'>~.
~
(a+ a
)
g21f drl...dr21
(12t
where a ~ and a are respectively the creation and annihilation phonon operators describing the isolated local vibration of the first atom with the frequency (~ + 2 g ] ' (o = \ - - ~ i t - i '
I I3 }
corresponding to Hi, (here and below we assume h = 1 ). According to expressions (2), (12 ), and the relationship for (7°(r) similar to (11), the Fourier transform C7~,~,is of the form (14) Upon transforming to the Fourier transforms, the right-hand side of eq. (10) is reduced to a geometric progression. The appropriate sum is
&,', G " - I - g 2 G-o, & ,
fl
/=0
(2re(o)
(9)
Due to the harmonicity o f hamiltonian (5), in the given sum only the terms with even I have nonzero values and the averages o f products of the displacements that arise are split into all the possible products of the pair averages giving rise to
G l l ( r ) = ~.
where oJ,, = 27t Tn, n is an integer, and C~,j, is defined similarly. For calculating fT~,~, we use the following representation for u~: u~
12
(11)
tt
llS)
On solving eq. ( 15 ) for G,,, we have
0
1
G , , - 292 0,0 { 1 - [ 1 - 4g-(O,,)-'] "2 }.
xG°,(r-r~) G~(':,-~:~) G,°,(r~-r~)..xG22(r2/_, -r2/) ~°~(r2/).
(10)
Here ~oj (z) is the Green function averaged with the help of the hamiltonian H0. As far as the Green function ~22(r) is concerned, it coincides w i t h G l i ( r ) inasmuch as the second atom is now the original one 70
(16)
4. Application
Now we return to the initial hamiltonian ( 1 ), which can be represented in the form
Volume 120, number 2
PHYSICS LETTERS A
H=Ho+Him+Hb,
2 February 1987 ~b0
(17)
where
Ho=pZ~/2m+½(~+g) u~,
(18)
Hint and Hb are determined by formulae (7) and (8). The appropriate Green function G. specifying the motion of the first atom of the chain at hand is determined by a formula similar to (15 ):
Re~
6o G. = 1
o~ , - g 2GnG.
(19)
where the Green function G o is of the form (14) but with t5 replaced by ~
1/2
in agreement with expression (18). Making use of formula (16) and the corresponding one for G o, from relationship (19 ) we finally have
2/m G.= where
to2 + A 2 + 4 ( t o 2
+ A 2 ) (to2 +B2),
(21)
A2=~./m and B2= (~+4g)/m. Formula (21)
solves in principle the task of determining the thermal averages of interest. On making use of the transformation inverse to (11 ), one may write ((u~))=-T
~ n=
G.
Fig. I. The plane of the complex variable to of integration in expression (23). The circle maps the circular contour of integration of the infinite radius. The bold-faced points on the imaginary axis map the countable set of the poles of the integrand. The bold-faced segments on the real axis are the branch cuts which confine the integrand to the required analytic branch.
the residues of the integrand at all its poles plus the integrals along closed curves surrounding the cuts connecting the branch points of the denominator of the integrand. In this case, we must obviously choose the analytic branch of the denominator which is characterized by the zero value of its argument at co = 0 in agreement with expressions (21) and (22). On performing some simple calculations, expression (22) is then reduced to the form 1 IBI B 2
(22)
toE 1/2
--oo
in agreement with expression (4). For the further reduction of expression (22) it is instructive to consider the integral
(24) Putting T--0, one can in particular obtain the contribution of the zero-point vibrations, which is equal to
~ d t o [exp(flto) - 1 ]
((u 21)) I T=o =12-~g [K(q) X [09 2 - A 2 - 4 ( A
E(q)],
(25)
2 - c o 2) ( B 2 - 0 9 2 ) ] - i
where =0,
(23)
which is taken along the circular contour of the infinite radius, having the origin as its centre and transversed, for definiteness, in counter-clockwise fashion. The plane of the complex variable to with respect to which the integration in formula (23) is carried out is shown in fig. 1. According to the theorem of residues, the integral (23) is determined by the sum of
q=
,
(26)
K(q) and E(q) are the complete elliptic integrals of the first and second kind, respectively. It is worth comparing the derived value of the fluctuation given by expression (24) with that relevant to the case of an inner atom of the infinite chain. The regular phonon spectrum of the infinite chain 71
Volume 120, number 2
PHYSICS LETTERS A
i
f
2
I
//" @
05
"~0
t5
!
Fig. 2. The values IPl'of the mean square atomic fluctuations measured in units of I~ mA ] ' versus the temperature T - ;"/[. t I at R-=- IB/AI =2. Curve 1 corresponds to (( u~ ~). curve 2 is relevant to ((u~)). The dashed lines corresponding to W~ - 2rt T/( 1 + R) and W2= ~ TIR are the asymptotes.
2 February 1987
ments, which, t o t instance, h a p p e n s in the chains o f the T 1 H F : c o m p o u n d [6] or in the m o l e c u l a r chains in c h a n n e l s [9]. A s i m i l a r c a l c u l a t i o n for a t o m s next after the original one can be p e r f o r m e d as well. In this case, however, the i n t e r a c t i o n b e t w e e n all the p r e v i o u s a t o m s m u s t be taken into a c c o u n t side by side with t e r m s ( 5 ) where H,', n o w describes the m o t i o n o f the a t o m in question. I n a s m u c h as the c o m p l e t e solution o f the spectral task for those first a t o m s is required, such a p r o g r a m rapidly b e c o m e s rather c o m p l i c a t e d with g r o w i n g n u m b e r o f the a t o m c o n c e r n e d . T h e p r o p o s e d m e t h o d for calculating the t h e r m a l f l u c t u a t i o n s can be a c c o m p l i s h e d too w h e n all the three o r t h o g o n a l d i r e c t i o n s are a d m i s s i b l e for the a t o m i c m o t i o n . In the latter case, h o w e v e r , the a p p e a r i n g system o f coupled e q u a t i o n s similar to ( 15 ) can be solved only n u m e r i c a l l y . T h e a p p r o a c h develo p e d is also suitable for i n v e s t i g a t i n g the peculiarities o f the existence o f the localized states located at the c h a i n end, that will be the subject o f the further publication.
d e s c r i b e d by the h a m i l t o n i a n s i m i l a r to ( 1 ) takes the form
o)~ = ( ~ + 4 g s i n 2 ( ½ k d ) ) ,
m
''2
Acknowledgement
(27)
,
where d is the i n t e r a t o m i c d i s t a n c e along the chain i n v o l v e d and the w a v e v e c t o r k varies f r o m -- n / d to Tc/d. T h e m e a n s q u a r e a t o m i c f l u c t u a t i o n is then specified by the usual p h o n o n e x p r e s s i o n [8] which has the f o r m iBI
1
f
coth (½flo)) do)
14! T h e natural i n e q u a l i t y
<
129)
follows i m m e d i a t e l y f r o m c o m p a r i n g (24) with (2 8 ). T h e c o r r e s p o n d i n g t e m p e r a t u r e b e h a v i o u r is shown in fig. 2. N o t e that the h a r m o n i c external c o n s t r a i n t tbr the original a t o m can be c o n s i d e r e d in the s a m e m a n n e r as the case o f the free c h a i n e n d above. T h e latter can also be u s e d for calculating b o t h the energy b a r r i e r for the r o t a t i n g d e c o r a t e d linkage a n d the barriers specifying the d i f f u s i o n o f the d o m a i n walls p e r t u r b ing the o r d e r e d state o f the l o n g i t u d i n a l displace72
I a m grateful to Dr. V,R. B e l o s l u d o v a n d Dr. N.A. N e m o v for helpful discussions.
References
1] L.N. Bulaevskii, Usp. Fiz. Nauk 115 11975) 263 [So< Ptbs Usp. 18119761 131]. 2] E.V. Kholopov, A.M. Panich, N.K. Moroz and ~u.G. Knger, Zh. Eksp. Teor. Fiz. 84 (1983~ 1(191 [Soy. Phys. JETP 57 (19831 632], 3] E.H. Kerner, Phys. Rev. 95 11954) 087. [4] J.A. Krumhansl and J.R. Schrieffer. Phys. Rev. B I I ( 1975 3535. [5] E.V. Kholopov, Solid State (ommun. 53 ll985) 681. [6] E.V. Kholopov, Solid State (ommun. 47 (19831 187. [ 7 ] A.A. Abrikosov, L.P. GoFkov and I.E. Dzyaloshinski, Quantum field theory in statistical physics (Prentice-Hall, Englewood Cliffs, 19631. [8] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon. New York. 1968). [9] Yu.G. Kriger, S.G. Kozlova, G.N. Chekhov& Yu.A. Dyadin and S.P. Gabuda, in: Abstracts of the Symposium on lnhomogeneous electronic states (Novosibirsk, USSR, 19841 p. 192.