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Theory of disordered Coulomb system: High-frequency transverse conductivity in liquid metals
Solid State Communications, Printed in Great Britain.
Vo1.56,No.l,
pp.21-27,
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
THEORY OF DISORDERED COULOMB SYSTEkl:HIGH-FREQUEECY TRANSVERSE CONDUCTIVITY IN LIQUID kETALS V.B.Bobrov and S.A.Trigger Institute of High Temperatures, Academy of Sciences of the USSR Izhorskaya 13/19, Bnoscow127412, USSR ( Received 27 June 1985 by G.S.ZManov > A general expression for the long-wave frequency-dependent obtained previously transverse conductivity dtr (k+O,O) for disordered Coulomb system is applied to description of optical properties of liquid metals. Using the COPW basis one can separate contributions to d "(k+O,ti) from localized and delocalized electron states. The calculations performed for liquid Na explain frequency dependences of the effective mass m*(o) and the collision frequency V(l)),
dent conductivity was obtained4 for arbitrary frequencies, but with a statically screened electron-ion potential in the random-phase approximation (RPA) for electrons. Dynamical polarization effects in the adiabatic approximation for ions and using RPA for electrona were considered later596. The most consistent description of the frequency-dependent and correlational effects in the high-frequency conductivity of simple liquid metals has been given7 within the framework of the pseudopotential method. A model Hamiltonian has been used where localized electron states have not been taken into account explicitly. The present work is based upon a general approach using the COPW method (cf. the preceding communication) which is applied to the exact expression for the transverse conductivity of the Coulomb system. Thus localized electron states and the effects of the environment are involved in the consideration explicitly. Though these effects are negligible for alkali metals far from the critical point, they are dominating
1. Detailed experimental information on optical properties of liquid metals has been obtained in the past few years'. The results on alcali metals are especially important, as they are an object most simple and suitable for theoretical investigation. New theoretical schemes developed for more complicated (say, transition ) metals must be tested first on alkali metals. Frequency dependence6 of the real and imaginary parts of conductivity of liquid Na were measured2 comprehensively at a temperature of 120' C in the fre3.8 eV. A frequency range 0.6
22
THEORY OF DISORDERED COULOMB SYSTEM
for transition metals, as well as for simple metals at high pressures. 2. The linear response method has been applied' to derive general expressions for the long-wave limits of the longitudinal and transverse dielectric permittivities, E'(k+O,W), ftr(k+O,c)), for a disordered Coulomb system at arbitrary frequencies end the interaction intensity. Unlike the standard results of kinetic theory, it was shown that &l(k+O,O) # &tr(k+O,O), even though at certain frequencies these quantities are equal approximately. The longitudinal dielectric permittivity, describing the response to the longitudinal field induced by external sources,is
Vol. 56, No. I
where f(t) =idFf($,t); $(F,t) is the Heisenberg operator of the electric current density, <...> fi stands for averaging over the Gibbs ensemble with the exact Coulomb system Hamiltonian H. It follows from (1) that damping of the plasma oscillations in the conventional kinetic theory results from the use of perturbation theory in Y(O) and I((C9) nearO=Wp , where it is never valid. The static conductivity is determined from (1) and the relation
An expression for the transverse dieleot-
= L&&!!pq~- v$ + &+k-cO,G,) This expression predicts a non-damped plasmon mode,O=WpP (F4KZt&,,.,.,J%
ric permittivity has been obtained8 in terms ofY(O) and l((O) ,
(4)
related to the total average densities @a of electrons and ions in the system. For the Coulomb system with no localized states one has i.e. the usual plasma frequency, corresponding to delocalized charges Real functionsU(O) and J
It is shown that since y(Q) is finite at O+O the quantities &tr(k+O,W) and btr(keO,ti) have no finite static limits. One should bear in mind, however, that the O+O limit for Etr(k+O,O) is a formal operation, as the spatial dispersion must be taken into account at low frequencies. 3. Let us consider the transverse conductivity 6tr(k*0,0) for liquid metals, using the construction of the COFW basis developed previously. Emp-
(2)
Vol. 56, No. I
THEORY OF DISORDERED COULOMB SYSTEM
23
loying the Hamiltonian ijeff,proposed in the preceding work for description of liquid metals, the transverse conductivity is represented as follows
is the energy level of a localized state and In) = In,{ii,,,,j> is its wave function, <...>core means averaging over positions of nuclei.
where F =-ihOT is the momentum operator. As the subsystem of nuclei is treated within the framework of classical statistical mechanics and we neglect dynamics of nuclei, averaging in (6) is performed in two stages. First we average over the electron subsystem Aeff, then the with the Hamiltonian He resulting expression is averaged over positions of nuclei interacting by means of potential UEEfe . Using Wick's theorem, one can verify that
Note that the fluctuating field must be taken into account in (7) to make the conductivity finite. In simple liquid metals
6 loc,loc(~)~
O ’
(8)
because the localized electron states are completely filled in such metals, 80 f(&.,+l. The zeroth approximation in the electron-ion interaction pseudopotenI
6 loc,loc(~) =
f(&&(
-
0
I~lm>W7Vloc I S)>,,,
where 6 =+O, f(&) is the Fermi-Dirac energy distribution, &n= &,({$,,,~)
x '$exP(-q)(% 0s -00
1mockQE o+ &+!!Z
+ i&
(7)
tial for the delocalized states leads to
-'y)2(f(En)Gfree(k,q)
- (I - f$.,>)x
24
THEORY OF DISORDERED COULOMB SYSTEM
X
GfLree(k,q)) + it*
exp(-ic$(f+
-
Vol. 56, No. I
~~)(f($)o,ee(k,q) -
-00
-
f(&n)X;ree(k,q,)>core
(1 -
(9)
where
Gfree(Wd =
-i~dtexp(~~~t(ag6(tI)a~6(t2))>lleff Itzt _ (10) e I t2
-00
The function 6 mined analogously. one-centre approximation for localized
equality (12) ia reduced (in the onecentre approximation for the localized states) to
( 14)
W6 free,loc(t)) states is reasonable in (9). We shall compare the results with the experimental data on liquid alkali metals, where the external field frequency L3 does not exceed a few eV, so equalities (9) and (10) can be rewritten under the condition
where
In obtaining 6 free,free(O) 'Ipe
(II)
Re( 6 free,loc(") + 610c,free(0))530
Im( 6 free,loc(0)
+ 6loo,free(a))zs
~~@~~I’>12 e
x
,
(12)
> core where e, is the Permi level. In deriving (12) the one-electron approximation has been used for the delocalized states enabling one to get a simple numerical estimate which is given below. We have also assumed that f(&n)%l, f(Ei;)SO for
Er;>&F
and
f(&%)"-1
(15)
forc@&F.
6 free,free(O)
Setting
Plot = IXV
neglect a difference between the COPW states and the corresponding plane waves, as well as a difference between the plane wave sets lg> and}z) . Integrating 6 free,free(*) by parts twice, one gets
f(En)Z n6
ree(0)-Zfree(O)l*
Vol. 56, No. 1
THEORY OF DISORDERED COULOMB SYSTEM
25
where
(16)
Here wei is a model local peeudopotential describing interaction of delocalized electrona with ions, and =~~ree(r,t)9free(~,t) P free(%) is the density operator for delocalized electrons. Retaining only the eecond perturbative order in the paeudopotential wei, (15) and (16) can be written as
Thus, with account of (5)-(20), the transverse conductivity of liquid alkali metale,d~&(k+O,~), is (at low frequenciesO<<(&n- &,1/h )
Re6igf (k+o,o)-~LVfree(Ol
2 W,(2l)
Im6t,~f(k420,0)=~OP~ee(0)/0
Re6free,free(O) = Yfree(W)WLee 4K&
(17)
Im6free,free(@) =
(18)
,(22)
where
(IS)
Itfree@) = 2 3me@
s
'L (2K)3
k21wei(k)12Score(k)Re( pee('@)
-
free(k@)
17free(k*O) & free(kr0)) 3 , Zi is the Ufree'(4Ke2Pfree/me) ion charge, S,,,,(k) is the static structure factor for the nuclear subaystem,efree (k&J) =I-(4ne2/k2)nfree ie the dielectric permittivity, II free(ke@) is the exact polarization operator of the system of interacting electrons with the density pfree in the compensating positive medium.
where 0 m*(o)
Gm 2 pz,,=Ofreeme/m*(O
1 and
= me/(l+d+)(free(W))*
4. The modified Aehcroft pseudopotential -4fie2Zicoa(kQ/k2, kb2kF2kp
, (23) ,
26
THEORY OF DISORDERED COULOMB SYSTEM
has been used in the computation of the conductivity 6 iFf(k+O,&j). The modification of the Aahcroft pseudopotential at large wave vectors is due to the practically complete absence of screening of the Coulomb interaction at small dietancea. On the other hand, as shown by calculations, the region of large wave vectors contribute6 to 8 free(O) substantially. The structure factor Score(k) has been taken as that for the hard sphere potentialg. The packing parameter is taken from the limiting relation
;~oScore(k)
= -ijcoreTfi,
-
(z-1= (“ii:l$lB3 exp(-r&l, (27) no * has been used to repreeent the localized state in the numerical estimate. Here Ri ie radius of the metal ion14. It is 8BSumed that the level energy EO equals the eecond ionization poteitial of the metal atom. The result for liquid Ba at a temperature T=120°C is d Z 0.05. The calculated magnitudes of the real part of the transverse conductivity Re6bFf(k+0,0) and the effective mass m*(W) and their experimental values are shown in Figs.1 and 2. The account for electron-electron correlations produces an appreciable effect in Re6$f(k+0,0) and enables one to get a good agreement with the experiment. The dependence of the effective mass m*(O) on the external field frequency W observed experimentally can be described reasonably within the framework of the developed formalism. The results obtained for liquid Ha suggest an expectation that the present
(24)
KI
RPA
+ fW$(k@)n
It was shown12 that G(k,O)"NG(k,O) .
The main contribution to the coefficient d in (14) is that from the localized level with the leaat modulus of the energyJ&noI. A hydrogen-like wave function,
‘Q
where>,=-V"( bV/ bp)T is the isothermic compressibility. The influence of the exchangecorrelational effects in the dielectric permittivity of degenerate electron gas upon the conductivity Re6~~f(k-r0,ti) has been studied in the calculations. As was shown previously on the basis of kinetic equationl',ll, this influence is quite substantial in the longitudinal static conductivity $l(O+O,k/a+O). The exchange-correlation8leffect8 are described usually by means of 8 function G(k,O),
&free(%W)= I
Vol. 56, No. 1
(26)
An expression obtained recently13 for G(k,O) is used in the present work.
(k,U)
approach will be effective when applied to more complicated met.818for whiah the influence of medium upon localized states is considerable.
Vol. 56, No. I
27
ThEORY OF DISORDERED COULOMB SYSTEM
m’(w)/%
2
1
u
f
2
3
4
hw(ev) Fig.1. Real part of the tranaverae conductivity Re6tr(k+0 .0): o - experimental values; the calculated magnitudee (rA= 1.8 a.u.): I - G(k,O)=O (RPA), II - G(k,o) = k2/2(k2+k$ (the Hubbard approximation), III - an approximated G(k,O) from Ref.13.
Fig.2. Effectiv; mass m*(W): 0- experimental values ;- - the calculated magnitudes (rA= 1.8 a.u., an approximated G(k,O) from Ref.13).
REFERENCES 5246 (1981). INAGAKI, T. et al., Phys. Rev. w, INAGAKI, T. et al., Phys. Rev. a, 5610 (1976). 2. Pie. %,,,I771 (1961). Zhurn. Exp. Teor. SILIE, V.P., 3. phys. Cond. Matt. 2, 60 (1966). HELM.AU, J.S. and BALTENSPERGER, W., 4. PETCHIK, C.J., Phys. Rev. B2, 1789 (1970). 5. STURM, K., J. Phys.F: Metal Phys. 2, 199 (1973). 6. BOBROV, V.B. and TRIGGER, S.A., Zhurn. tixp.Teor. Fiz. 86, 514 (1984). 7. 3OBROV, V.B. and THIGGER, S.A., Inst. High Temp. Preprint ~0.1-163 (1985) 8. WERTHEIM, D., Phys. Rev. Lett. IO, 321 (1963). 9. KLYUCHNIKOV, M.I. and TRIGGER, S.A., Teor. kat. Fiz. 26, 256 (1976); 10. 2, 368 (1979). KLYUCHNIKOV, N.I. and TRIGGER, S.A., Doklady AN SSSR B, 565 (1978). 11. TOIGO, F. and WOODRUFF, T.O., Phys. Rev. 2, 4312 (1971). 12. UTSIkI, K. and ICHIk&ZU, S., Phys. Rev. 826, 603 (1982). 13. 14. KITTEL, C., "Introduction to solid State Physics",4th edition (J.Wiley, New York - London - Sydney - Toronto, 1976).
1.
Vo1.56,No.l,
pp.21-27,
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
THEORY OF DISORDERED COULOMB SYSTEkl:HIGH-FREQUEECY TRANSVERSE CONDUCTIVITY IN LIQUID kETALS V.B.Bobrov and S.A.Trigger Institute of High Temperatures, Academy of Sciences of the USSR Izhorskaya 13/19, Bnoscow127412, USSR ( Received 27 June 1985 by G.S.ZManov > A general expression for the long-wave frequency-dependent obtained previously transverse conductivity dtr (k+O,O) for disordered Coulomb system is applied to description of optical properties of liquid metals. Using the COPW basis one can separate contributions to d "(k+O,ti) from localized and delocalized electron states. The calculations performed for liquid Na explain frequency dependences of the effective mass m*(o) and the collision frequency V(l)),
dent conductivity was obtained4 for arbitrary frequencies, but with a statically screened electron-ion potential in the random-phase approximation (RPA) for electrons. Dynamical polarization effects in the adiabatic approximation for ions and using RPA for electrona were considered later596. The most consistent description of the frequency-dependent and correlational effects in the high-frequency conductivity of simple liquid metals has been given7 within the framework of the pseudopotential method. A model Hamiltonian has been used where localized electron states have not been taken into account explicitly. The present work is based upon a general approach using the COPW method (cf. the preceding communication) which is applied to the exact expression for the transverse conductivity of the Coulomb system. Thus localized electron states and the effects of the environment are involved in the consideration explicitly. Though these effects are negligible for alkali metals far from the critical point, they are dominating
1. Detailed experimental information on optical properties of liquid metals has been obtained in the past few years'. The results on alcali metals are especially important, as they are an object most simple and suitable for theoretical investigation. New theoretical schemes developed for more complicated (say, transition ) metals must be tested first on alkali metals. Frequency dependence6 of the real and imaginary parts of conductivity of liquid Na were measured2 comprehensively at a temperature of 120' C in the fre3.8 eV. A frequency range 0.6
22
THEORY OF DISORDERED COULOMB SYSTEM
for transition metals, as well as for simple metals at high pressures. 2. The linear response method has been applied' to derive general expressions for the long-wave limits of the longitudinal and transverse dielectric permittivities, E'(k+O,W), ftr(k+O,c)), for a disordered Coulomb system at arbitrary frequencies end the interaction intensity. Unlike the standard results of kinetic theory, it was shown that &l(k+O,O) # &tr(k+O,O), even though at certain frequencies these quantities are equal approximately. The longitudinal dielectric permittivity, describing the response to the longitudinal field induced by external sources,is
Vol. 56, No. I
where f(t) =idFf($,t); $(F,t) is the Heisenberg operator of the electric current density, <...> fi stands for averaging over the Gibbs ensemble with the exact Coulomb system Hamiltonian H. It follows from (1) that damping of the plasma oscillations in the conventional kinetic theory results from the use of perturbation theory in Y(O) and I((C9) nearO=Wp , where it is never valid. The static conductivity is determined from (1) and the relation
An expression for the transverse dieleot-
= L&&!!pq~- v$ + &+k-cO,G,) This expression predicts a non-damped plasmon mode,O=WpP (F4KZt&,,.,.,J%
ric permittivity has been obtained8 in terms ofY(O) and l((O) ,
(4)
related to the total average densities @a of electrons and ions in the system. For the Coulomb system with no localized states one has i.e. the usual plasma frequency, corresponding to delocalized charges Real functionsU(O) and J
It is shown that since y(Q) is finite at O+O the quantities &tr(k+O,W) and btr(keO,ti) have no finite static limits. One should bear in mind, however, that the O+O limit for Etr(k+O,O) is a formal operation, as the spatial dispersion must be taken into account at low frequencies. 3. Let us consider the transverse conductivity 6tr(k*0,0) for liquid metals, using the construction of the COFW basis developed previously. Emp-
(2)
Vol. 56, No. I
THEORY OF DISORDERED COULOMB SYSTEM
23
loying the Hamiltonian ijeff,proposed in the preceding work for description of liquid metals, the transverse conductivity is represented as follows
is the energy level of a localized state and In) = In,{ii,,,,j> is its wave function, <...>core means averaging over positions of nuclei.
where F =-ihOT is the momentum operator. As the subsystem of nuclei is treated within the framework of classical statistical mechanics and we neglect dynamics of nuclei, averaging in (6) is performed in two stages. First we average over the electron subsystem Aeff, then the with the Hamiltonian He resulting expression is averaged over positions of nuclei interacting by means of potential UEEfe . Using Wick's theorem, one can verify that
Note that the fluctuating field must be taken into account in (7) to make the conductivity finite. In simple liquid metals
6 loc,loc(~)~
O ’
(8)
because the localized electron states are completely filled in such metals, 80 f(&.,+l. The zeroth approximation in the electron-ion interaction pseudopotenI
6 loc,loc(~) =
f(&&(
-
0
I~lm>W7Vloc I S)>,,,
where 6 =+O, f(&) is the Fermi-Dirac energy distribution, &n= &,({$,,,~)
x '$exP(-q)(% 0s -00
1mockQE o+ &+!!Z
+ i&
(7)
tial for the delocalized states leads to
-'y)2(f(En)Gfree(k,q)
- (I - f$.,>)x
24
THEORY OF DISORDERED COULOMB SYSTEM
X
GfLree(k,q)) + it*
exp(-ic$(f+
-
Vol. 56, No. I
~~)(f($)o,ee(k,q) -
-00
-
f(&n)X;ree(k,q,)>core
(1 -
(9)
where
Gfree(Wd =
-i~dtexp(~~~t(ag6(tI)a~6(t2))>lleff Itzt _ (10) e I t2
-00
The function 6 mined analogously. one-centre approximation for localized
equality (12) ia reduced (in the onecentre approximation for the localized states) to
( 14)
W6 free,loc(t)) states is reasonable in (9). We shall compare the results with the experimental data on liquid alkali metals, where the external field frequency L3 does not exceed a few eV, so equalities (9) and (10) can be rewritten under the condition
where
In obtaining 6 free,free(O) 'Ipe
(II)
Re( 6 free,loc(") + 610c,free(0))530
Im( 6 free,loc(0)
+ 6loo,free(a))zs
~~@~~I’>12 e
x
,
(12)
> core where e, is the Permi level. In deriving (12) the one-electron approximation has been used for the delocalized states enabling one to get a simple numerical estimate which is given below. We have also assumed that f(&n)%l, f(Ei;)SO for
Er;>&F
and
f(&%)"-1
(15)
forc@&F.
6 free,free(O)
Setting
Plot = IXV
neglect a difference between the COPW states and the corresponding plane waves, as well as a difference between the plane wave sets lg> and}z) . Integrating 6 free,free(*) by parts twice, one gets
f(En)Z n6
ree(0)-Zfree(O)l*
Vol. 56, No. 1
THEORY OF DISORDERED COULOMB SYSTEM
25
where
(16)
Here wei is a model local peeudopotential describing interaction of delocalized electrona with ions, and =~~ree(r,t)9free(~,t) P free(%) is the density operator for delocalized electrons. Retaining only the eecond perturbative order in the paeudopotential wei, (15) and (16) can be written as
Thus, with account of (5)-(20), the transverse conductivity of liquid alkali metale,d~&(k+O,~), is (at low frequenciesO<<(&n- &,1/h )
Re6igf (k+o,o)-~LVfree(Ol
2 W,(2l)
Im6t,~f(k420,0)=~OP~ee(0)/0
Re6free,free(O) = Yfree(W)WLee 4K&
(17)
Im6free,free(@) =
(18)
,(22)
where
(IS)
Itfree@) = 2 3me@
s
'L (2K)3
k21wei(k)12Score(k)Re( pee('@)
-
free(k@)
17free(k*O) & free(kr0)) 3 , Zi is the Ufree'(4Ke2Pfree/me) ion charge, S,,,,(k) is the static structure factor for the nuclear subaystem,efree (k&J) =I-(4ne2/k2)nfree ie the dielectric permittivity, II free(ke@) is the exact polarization operator of the system of interacting electrons with the density pfree in the compensating positive medium.
where 0 m*(o)
Gm 2 pz,,=Ofreeme/m*(O
1 and
= me/(l+d+)(free(W))*
4. The modified Aehcroft pseudopotential -4fie2Zicoa(kQ/k2, kb2kF
, (23) ,
26
THEORY OF DISORDERED COULOMB SYSTEM
has been used in the computation of the conductivity 6 iFf(k+O,&j). The modification of the Aahcroft pseudopotential at large wave vectors is due to the practically complete absence of screening of the Coulomb interaction at small dietancea. On the other hand, as shown by calculations, the region of large wave vectors contribute6 to 8 free(O) substantially. The structure factor Score(k) has been taken as that for the hard sphere potentialg. The packing parameter is taken from the limiting relation
;~oScore(k)
= -ijcoreTfi,
-
(z-1= (“ii:l$lB3 exp(-r&l, (27) no * has been used to repreeent the localized state in the numerical estimate. Here Ri ie radius of the metal ion14. It is 8BSumed that the level energy EO equals the eecond ionization poteitial of the metal atom. The result for liquid Ba at a temperature T=120°C is d Z 0.05. The calculated magnitudes of the real part of the transverse conductivity Re6bFf(k+0,0) and the effective mass m*(W) and their experimental values are shown in Figs.1 and 2. The account for electron-electron correlations produces an appreciable effect in Re6$f(k+0,0) and enables one to get a good agreement with the experiment. The dependence of the effective mass m*(O) on the external field frequency W observed experimentally can be described reasonably within the framework of the developed formalism. The results obtained for liquid Ha suggest an expectation that the present
(24)
KI
RPA
+ fW$(k@)n
It was shown12 that G(k,O)"NG(k,O) .
The main contribution to the coefficient d in (14) is that from the localized level with the leaat modulus of the energyJ&noI. A hydrogen-like wave function,
‘Q
where>,=-V"( bV/ bp)T is the isothermic compressibility. The influence of the exchangecorrelational effects in the dielectric permittivity of degenerate electron gas upon the conductivity Re6~~f(k-r0,ti) has been studied in the calculations. As was shown previously on the basis of kinetic equationl',ll, this influence is quite substantial in the longitudinal static conductivity $l(O+O,k/a+O). The exchange-correlation8leffect8 are described usually by means of 8 function G(k,O),
&free(%W)= I
Vol. 56, No. 1
(26)
An expression obtained recently13 for G(k,O) is used in the present work.
(k,U)
approach will be effective when applied to more complicated met.818for whiah the influence of medium upon localized states is considerable.
Vol. 56, No. I
27
ThEORY OF DISORDERED COULOMB SYSTEM
m’(w)/%
2
1
u
f
2
3
4
hw(ev) Fig.1. Real part of the tranaverae conductivity Re6tr(k+0 .0): o - experimental values; the calculated magnitudee (rA= 1.8 a.u.): I - G(k,O)=O (RPA), II - G(k,o) = k2/2(k2+k$ (the Hubbard approximation), III - an approximated G(k,O) from Ref.13.
Fig.2. Effectiv; mass m*(W): 0- experimental values ;- - the calculated magnitudes (rA= 1.8 a.u., an approximated G(k,O) from Ref.13).
REFERENCES 5246 (1981). INAGAKI, T. et al., Phys. Rev. w, INAGAKI, T. et al., Phys. Rev. a, 5610 (1976). 2. Pie. %,,,I771 (1961). Zhurn. Exp. Teor. SILIE, V.P., 3. phys. Cond. Matt. 2, 60 (1966). HELM.AU, J.S. and BALTENSPERGER, W., 4. PETCHIK, C.J., Phys. Rev. B2, 1789 (1970). 5. STURM, K., J. Phys.F: Metal Phys. 2, 199 (1973). 6. BOBROV, V.B. and TRIGGER, S.A., Zhurn. tixp.Teor. Fiz. 86, 514 (1984). 7. 3OBROV, V.B. and THIGGER, S.A., Inst. High Temp. Preprint ~0.1-163 (1985) 8. WERTHEIM, D., Phys. Rev. Lett. IO, 321 (1963). 9. KLYUCHNIKOV, M.I. and TRIGGER, S.A., Teor. kat. Fiz. 26, 256 (1976); 10. 2, 368 (1979). KLYUCHNIKOV, N.I. and TRIGGER, S.A., Doklady AN SSSR B, 565 (1978). 11. TOIGO, F. and WOODRUFF, T.O., Phys. Rev. 2, 4312 (1971). 12. UTSIkI, K. and ICHIk&ZU, S., Phys. Rev. 826, 603 (1982). 13. 14. KITTEL, C., "Introduction to solid State Physics",4th edition (J.Wiley, New York - London - Sydney - Toronto, 1976).
1.