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EPR lineshape studies of low-temperature vanadium 3d1 polaron dynamics in the 70V2O5-30P2O5 binary glass
CHEMICAL
Volume 160, number 56
EPR LINESHAPE STUDIES OF LOW-TEMPERATURE VANADIUM POLARON DYNAMICS IN THE 70V,05-30P,O, BINARY GLASS P. RAGHUNATHAN
25 August 1989
PHYSICS LETTERS
3d’
’ and B.B. DA8 ’ ofTechnology, Kanpur 208016. India
Department of Chemistry, Indian Institute Received
15 June 1989
In a specimen of 70V,0s-3OP,Or glass, EPR lineshapes of the vanadium 3d’ polaron have been studied between 4 and 77 K. At the lowest temperature the unpaired electron is localized at a single“V site, and values ofg,,= 1.959, g, = 1.989,A, = 156.6x 10m4 cm-’ and Al =53.8x 10s4 cm-’ have been measured. A Markovian small-step rotational diffusion model consistent with the random structure of the glass network is proposed for the polaron dynamics at the higher temperatures up to 77 K. This motion has a small activation energy barrier of 114 peV.
1. Introduction Vanadium phosphate glasses are known to be ntype semiconductors, their semiconductivity arising from thermal hopping of the 3d’ electron from a V4+ to a Vs + site [ l-81. The unpaired electron induces a polarization of the lattice around it, whereby the strongly phonon-coupled charge carrier actually becomes a “small polaron” [ 9,101. To date, several cursory reports have also appeared regarding the EPR of paramagnetic vanadium in V205-PZ05 systems over a range of compositions [ 1l-l 71. Although the hopping motion of the polaron between the reduced and oxidized forms of the transition metal ion in the oxide network has been established to be thermally activated from all these studies, to our knowledge no detailed analysis of the EPR lineshapes aimed at probing the low-temperature limiting polaron dy namics at the localized glass-forming sites has been carried out. In this communication we report the fully resolved g and hyperfine anisotropies of the localized 3d’ polaron in V,O, (70 mol%)-P205 (30 mol%) glass, as well as an analysis of the rotational diffusion of the paramagnetic site which modulates the EPR line’ To whom correspondence should be addressed. * Present address: Department of Chemistry, Pondicherry versity. JIPMER Campus, Pondicherry 605 006, India.
shapes. We propose a Markovian model for the slowmotional polaron dynamics which is compatible with the random structure of the glass network.
2. Experimental The VzOs (70 molO/o)-P,O, (30 mol%) glass was prepared by melting a mixture of appropriate weights of reagent grade oxides in an alumina crucible and then quenching in air. The glassy structure was ascertained by its X-ray diffraction pattern as well as by a variety of other physical techniques [ 181. Over a range of compositions of the binary glasses prepared, the nominal batch composition 70VZ0530P20, ensured the “cleanest” formation of a singlecomponent amorphous system without any phase separation. Variable temperature X-band EPR spectra between 4 and 77 K were recorded on a Bruker spectrometer system using Heli-Tran (Air Products Co. ) cryogenics, while for higher temperatures a Varian E-109 EPR spectrometer was used. Both spectrometers were equipped with 100 kHz magnetic field modulation. The measured g-values were calibrated with respect to the resonance line of DPPH (g,,,,,=2.00354).
Uni-
0 009-26 14/89/$ 03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division )
B.V.
627
Volume
CHEMICAL
160, number 5,6
3.EPR spectrum of the glass at 4 K In fig. 1 we present the EPR spectrum of the glass recorded at 4 K. It exhibits well-resolved parallel and perpendicular features of the hyperfine interaction, showing that the 3d unpaired electron is localized at a single 5’V site (1=7/2). The spectrum is typically axially symmetric, and is described by the spin Hamiltonian
PHYSICS
ulation [ 19-2 11, we have deduced the spin Hamiltonian parameters. The “best-fit” between the experimental and simulated spectra (shown in fig. 1) led to
(1)
where the principal axes of the g- and A-interaction matrices are taken to lie along the z axis. Relating the experimental spectra to eq. ( 1) requires that the spin Hamiltonian be diagonalized to yield the ‘Yesonance condition” H( 8, g), which is an expression involving the magnetic field Ho at which resonance occurs as a function of the principal values of the interaction metrices, the resonance frequency w and the magnetic quantum numbers MS and A4,. Using the usual techniques of amorphous state lineshape sim-
g, = 1.989kO.001
Our EPR parameters, while reflecting the general trend of those reported for splat-cooled amorphous V205 [ 221 as well as amorphous compounds formed between VZOs and other oxides [ 12,151, identifies the paramagnetic site as the oxovanadium, VO*+, ion, which is in a severely tetragonally compressed octahedral (i.e. VO,-...O) environment. Assuming C4” point group symmetry appropriate for this coordination we have used MO theory elsewhere [ 181 to rationalize the bonding picture from our EPR results. For the purpose of the present study, the important point to note is that, at low temperatures at any rate, the vanadium coordination in our glass is closer to six than to five.
low-temperature
I
'0
3090
I 3360
I 3670
I 3960
4250
MAGNETIC FIELD,tj (GAUSS) Fig. I, (a) Experimental and (b) computer-simulated EPR spectrum of 70Vz05-30PL05 glass at 4 K.
628
X-band
,
=(53.8&0.9)x10-4cm-‘.
4. Temperature
2
,
g,, = 1.959+0.001
A,
+ A, C&L +w,?) ,
25 August 1989
LETTERS
dependence of the EPR lineshapes polaron dynamics
-
We present in fig. 2 the EPR spectra of the VZ05P,05 glass recorded at a number of temperatures between 4 and 77 K, which clearly reveal the progressive onset of line-narrowing due to a dynamic modulation process. An interpretation of the observed lineshapes may be sought by invoking Markovian small-step rotational diffusion whereby the small polaron (i.e. the 3d’ electron localized at a V-O poiyhedral unit) changes its orientation at random. This motion is thought to be brought about by an appropriate acoustic lattice mode, which is in principle an allowable excitation at the temperatures of our experiment. Let the orientation of the paramagnetic site bedefined by B,, 4, at zero time. After an appropriate phonon impulse, the site acquires a new oriefitation defined by &, &_ The Markovian model for this stepwise excitation is depicted in fig. 3. H,, is defined
Volume 160, number
CHEMICAL
5,6
25 August 1989
PHYSICS LETTERS
Ho
I
c c
Fig. 3. Markovian
-
IAGNETIC
FIELD
(Gauss)
Fig. 2. Experimental EPR spectra (solid lines) of 70VZ05-30PzOr glass at several temperatures: (a) 10 K: (c) 20 K; (e) 30 K; (g) 50 K and (i) 77 K. The simulated lineshapes (b, d, f, h and j), shown by dotted lines, correspond to various appropriate values of R,the rotational rate index (see text ).
along the z. axis of the (xo~Ozo) reference coordinate system, and an arbitrary initial position of the principal axis of the V-O polyhedron in this coordinate system is obtained by the random numbers xl, x2 generated by a Monte Carlo process case, =x, ,
(25, =x2.
(2)
Subsequent evolution of the slow-diffusional dynamics is then computed by generating a random number x and using the relationship [ 231 cos& =cos6, cosJ+sin@
sins cosx ,
$I = @,+ arcsin (sins sir&sir&
) ,
(3)
rotational
diffusion model for the VOs...O site.
where 6 is defined as the fixed length of a rotational “step” on the surface of a spherical grid. Since J2 is basically related to the diffusion constant for the motion, a convenient rotational rate index R = d2/ht Aw is defined in our computations, where At is the duration of localization of a polaron between rotational steps, and Aw is the overall frequency spread of the spectrum which is being diffusionally modulated. The new resonance condition will be given by
and consequently the resonance frequency eigenvalues of the A - ‘X operator, which are now modulated in time, appear as w ( t ) . In the adiabatic approximation, we treat all the mr= +7/2 to ml= - 7/2 hyperfine absorption regions independently. These absorptions may be expressed as +7/2
Z(w-co,,)=
C
--7/1
exp[ -i(w-wo)t]
In eq. (5 ), G(t) is the relaxation expressible as
G(f) dt function
.
(5)
which is
629
Volume 160, number 5,6
c(i)=cerp(i/
CHEMICAL
PHYSICS LETTERS
25 August 1989
[a(1)--O]df)), 0
where the outermost angular brackets denote an appropriate averaging over the Monte Carlo motional process. For a sequence of M independent Markovian jumps described by our model (Mz 1000)) the temporal fluctuations of the relaxation function, eq. ( 6), are averaged to I
G(t)=(lIM)
Cexp
.I(
is
[o(t)-o)]dt 0
>
(7)
and the resultant frequency domain spectrum is readily obtained by a Fourier transformation of G(t). In our computation, the eigenvalue problem corresponding to this small polaron dynamics is solved by setting up the Hamiltonian matrix, eq. ( 1)) and then carrying out the calculations corresponding to eqs. (2)-( 7) for various trial values of the rotational rate index, R, between 0.0001 (virtually rigid limit ) and IO (nearly free rotational diffusion). The function G(t) is sampled using a Tukey spectral window, so that spurious “aliasing” effects (which are often inherent artefacts of the Fourier transformation technique) are suppressed. We note here that more elaborate ensemble-averaging procedures are also available for calculating the EPR spectra of rotationally diffusing species in amorphous substances [24,25]. In fig. 2, lineshape simulations for various R values are compared with experimental spectra recorded at several temperatures between 4 and 77 K. Over the temperature range of these modulated lineshapes, a logarithmic plot of the spectral frequency width against T-‘, fig. 4, yields a straight line, the slope of which gives a barrier of 1.14 X IO- 4 eV. This value may be identifikd with the energy needed for the vanadium 3d’ sites to become rotationally “disordered” over short range.
5. Conclusions For the 7OV,O,-3OP,O, binary glass, our EPR measurement at 4 K shows not only that the odd electron is localized at a single 5’V site, but that the 630
,I 5
100
200 103/~(K-')
Fig. 4. Plot of the logarithmic EPR spectral width versus T70V,05-30PI05 between 4 and 77 K_
IO
’ for
vanadium coordination corresponds to a severely compressed V05...0 octahedral unit. Variable temperature EPR above 4 K probes the local dynamics, and the small activation energy of 114 peV measured in the 4-77 K range may be understood to be the energy needed by the localized electron to overcome the crystal-field energy associated with shortrange order in the glass. An alternative, and perhaps more appealing, rationalization of the above results would be that when pyramidally distorted octahedral V-O units pack together to optimize the short-range order of the glass network structure, some covalency contribution to the V-O-V bonds linking adjacent structural units would not only facilitate the packing but also lead to overlap of the 3d’ electron wavefunction over adjacent sites. The measured values of 114 peV (or 28 GHz) would then be a measure of how frequently the electron wavefunction overlap (or “tunneling”) is interrupted by lattice phonon modes as the polaron dynamics gradually go over into the adiabatic hopping regime. For the semiconducting glass we have studied, this is essentially a band model at low temperatures where rotational polaron dynamics leads to a disorder energy which is much less than the polaron bandwidth.
Volume 160, number 5,6
CHEMICAL
PHYSICS LETTERS
References
[ 1 ] G.W. Anderson and W.D. Compton, I. Chem. Phys. 52 (1970) 6166. [2] F.R. Landsberger and P.J. Bray, J. Chem. Phys. 53 (1970) 2757. [3] G.S. Linsley, A. Owen and F.M. Hayatee, J. Non-Cryst. Solids 4 (1970) 208. [4] M. Sayer and A. Mansingh, Phys. Rev. B 6 (1972) 4629. [ 51 G.N. Greaves, J. Non-Cryst. Solids 11 (1973) 427. [6] H. Harper and P.W. McMiBan, Phys. Chem. Glasses 15 (1974) 148. [7] L. Muraski, C.H. Chung and J.D. McKenzie, J. Non-Cryst. Solids 32 (1979) 208. [S] J.O. Isard, J. Non-Cryst. Solids 42 (1980) 371. [9] N.F. Mott, J. Non-Cryst. Solids 1 ( 1966) 1. [ IO] LG. Austin and N.F. Mott, Advan. Phys. 18 ( 1969) 41. [ 11 ] V.M. Nagiev, Soviet Phys. Solid State 7 (1966) 2204. [ 121 E.J. Friebele, L.K. Wilson and D. Kinsener, J. Am. Ceram. Sot. 55 (1972) 164. [ 13 ] G.F. Lynch, M. Sayer, S.L. Segel and G. Rosenblatt, J. Appl. Phys. 42 (1971) 2587.
25 August 1989
[ 141 J. Livage, P. Pineau, M.C. Leroy and M. Michaud, Phys. Stat. Sol. 39a (1977) 73. [ 151 G. Sperlich, Z. Physik 250 ( 1972) 335. [ 161 L.D. Bogomolova, T.F. Dolgolenko, V.A. Jachlin and V.N. Lazukin, J. Magn. Reson. 15 ( 1974) 283. [I?‘] E.E. Khawaja, M.S. Hussain, M.A. Khan and J.S. Hwang, J. Mater. Sci. 21 (1986) 2812. [ 181 B.B. Das, Ph.D. Dissertation, Indian Institute of Technology, Kanpur (1987). [ 191 P. Raghunathan and SK. Sur, J. Am. Chem. Sot. 106 (1984) 8014. [20] P. Raghunathan and SC. Sivasubramanian, Proc. Indian Acad. Sci. Chem. Sci. 96 ( 1986) 565. [2 I ] P. Raghunathan, in: Electron magnetic resonance ofthe solid state, ed. J.A. Weil (Can. Sot. Chem., Ottawa, 1987) ch. 13. [ 221 L. Rivoalen, A. Revcolevschi, J. Livage and R. Collongues, J. Non-Cryst. Solids 2 1 ( 1976 ) 17 1. [ 231 J.B. Pedersen, J. Chem. Phys. 57 (1972) 2680. [24] D. Kivelson and S. Lee, J. Chem. Phys. 76 (1982) 5746. [25]S.LeeandD.P.Ames, J.Chem. Phys.80 (1984) 1766.
631
Volume 160, number 56
EPR LINESHAPE STUDIES OF LOW-TEMPERATURE VANADIUM POLARON DYNAMICS IN THE 70V,05-30P,O, BINARY GLASS P. RAGHUNATHAN
25 August 1989
PHYSICS LETTERS
3d’
’ and B.B. DA8 ’ ofTechnology, Kanpur 208016. India
Department of Chemistry, Indian Institute Received
15 June 1989
In a specimen of 70V,0s-3OP,Or glass, EPR lineshapes of the vanadium 3d’ polaron have been studied between 4 and 77 K. At the lowest temperature the unpaired electron is localized at a single“V site, and values ofg,,= 1.959, g, = 1.989,A, = 156.6x 10m4 cm-’ and Al =53.8x 10s4 cm-’ have been measured. A Markovian small-step rotational diffusion model consistent with the random structure of the glass network is proposed for the polaron dynamics at the higher temperatures up to 77 K. This motion has a small activation energy barrier of 114 peV.
1. Introduction Vanadium phosphate glasses are known to be ntype semiconductors, their semiconductivity arising from thermal hopping of the 3d’ electron from a V4+ to a Vs + site [ l-81. The unpaired electron induces a polarization of the lattice around it, whereby the strongly phonon-coupled charge carrier actually becomes a “small polaron” [ 9,101. To date, several cursory reports have also appeared regarding the EPR of paramagnetic vanadium in V205-PZ05 systems over a range of compositions [ 1l-l 71. Although the hopping motion of the polaron between the reduced and oxidized forms of the transition metal ion in the oxide network has been established to be thermally activated from all these studies, to our knowledge no detailed analysis of the EPR lineshapes aimed at probing the low-temperature limiting polaron dy namics at the localized glass-forming sites has been carried out. In this communication we report the fully resolved g and hyperfine anisotropies of the localized 3d’ polaron in V,O, (70 mol%)-P205 (30 mol%) glass, as well as an analysis of the rotational diffusion of the paramagnetic site which modulates the EPR line’ To whom correspondence should be addressed. * Present address: Department of Chemistry, Pondicherry versity. JIPMER Campus, Pondicherry 605 006, India.
shapes. We propose a Markovian model for the slowmotional polaron dynamics which is compatible with the random structure of the glass network.
2. Experimental The VzOs (70 molO/o)-P,O, (30 mol%) glass was prepared by melting a mixture of appropriate weights of reagent grade oxides in an alumina crucible and then quenching in air. The glassy structure was ascertained by its X-ray diffraction pattern as well as by a variety of other physical techniques [ 181. Over a range of compositions of the binary glasses prepared, the nominal batch composition 70VZ0530P20, ensured the “cleanest” formation of a singlecomponent amorphous system without any phase separation. Variable temperature X-band EPR spectra between 4 and 77 K were recorded on a Bruker spectrometer system using Heli-Tran (Air Products Co. ) cryogenics, while for higher temperatures a Varian E-109 EPR spectrometer was used. Both spectrometers were equipped with 100 kHz magnetic field modulation. The measured g-values were calibrated with respect to the resonance line of DPPH (g,,,,,=2.00354).
Uni-
0 009-26 14/89/$ 03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division )
B.V.
627
Volume
CHEMICAL
160, number 5,6
3.EPR spectrum of the glass at 4 K In fig. 1 we present the EPR spectrum of the glass recorded at 4 K. It exhibits well-resolved parallel and perpendicular features of the hyperfine interaction, showing that the 3d unpaired electron is localized at a single 5’V site (1=7/2). The spectrum is typically axially symmetric, and is described by the spin Hamiltonian
PHYSICS
ulation [ 19-2 11, we have deduced the spin Hamiltonian parameters. The “best-fit” between the experimental and simulated spectra (shown in fig. 1) led to
(1)
where the principal axes of the g- and A-interaction matrices are taken to lie along the z axis. Relating the experimental spectra to eq. ( 1) requires that the spin Hamiltonian be diagonalized to yield the ‘Yesonance condition” H( 8, g), which is an expression involving the magnetic field Ho at which resonance occurs as a function of the principal values of the interaction metrices, the resonance frequency w and the magnetic quantum numbers MS and A4,. Using the usual techniques of amorphous state lineshape sim-
g, = 1.989kO.001
Our EPR parameters, while reflecting the general trend of those reported for splat-cooled amorphous V205 [ 221 as well as amorphous compounds formed between VZOs and other oxides [ 12,151, identifies the paramagnetic site as the oxovanadium, VO*+, ion, which is in a severely tetragonally compressed octahedral (i.e. VO,-...O) environment. Assuming C4” point group symmetry appropriate for this coordination we have used MO theory elsewhere [ 181 to rationalize the bonding picture from our EPR results. For the purpose of the present study, the important point to note is that, at low temperatures at any rate, the vanadium coordination in our glass is closer to six than to five.
low-temperature
I
'0
3090
I 3360
I 3670
I 3960
4250
MAGNETIC FIELD,tj (GAUSS) Fig. I, (a) Experimental and (b) computer-simulated EPR spectrum of 70Vz05-30PL05 glass at 4 K.
628
X-band
,
=(53.8&0.9)x10-4cm-‘.
4. Temperature
2
,
g,, = 1.959+0.001
A,
+ A, C&L +w,?) ,
25 August 1989
LETTERS
dependence of the EPR lineshapes polaron dynamics
-
We present in fig. 2 the EPR spectra of the VZ05P,05 glass recorded at a number of temperatures between 4 and 77 K, which clearly reveal the progressive onset of line-narrowing due to a dynamic modulation process. An interpretation of the observed lineshapes may be sought by invoking Markovian small-step rotational diffusion whereby the small polaron (i.e. the 3d’ electron localized at a V-O poiyhedral unit) changes its orientation at random. This motion is thought to be brought about by an appropriate acoustic lattice mode, which is in principle an allowable excitation at the temperatures of our experiment. Let the orientation of the paramagnetic site bedefined by B,, 4, at zero time. After an appropriate phonon impulse, the site acquires a new oriefitation defined by &, &_ The Markovian model for this stepwise excitation is depicted in fig. 3. H,, is defined
Volume 160, number
CHEMICAL
5,6
25 August 1989
PHYSICS LETTERS
Ho
I
c c
Fig. 3. Markovian
-
IAGNETIC
FIELD
(Gauss)
Fig. 2. Experimental EPR spectra (solid lines) of 70VZ05-30PzOr glass at several temperatures: (a) 10 K: (c) 20 K; (e) 30 K; (g) 50 K and (i) 77 K. The simulated lineshapes (b, d, f, h and j), shown by dotted lines, correspond to various appropriate values of R,the rotational rate index (see text ).
along the z. axis of the (xo~Ozo) reference coordinate system, and an arbitrary initial position of the principal axis of the V-O polyhedron in this coordinate system is obtained by the random numbers xl, x2 generated by a Monte Carlo process case, =x, ,
(25, =x2.
(2)
Subsequent evolution of the slow-diffusional dynamics is then computed by generating a random number x and using the relationship [ 231 cos& =cos6, cosJ+sin@
sins cosx ,
$I = @,+ arcsin (sins sir&sir&
) ,
(3)
rotational
diffusion model for the VOs...O site.
where 6 is defined as the fixed length of a rotational “step” on the surface of a spherical grid. Since J2 is basically related to the diffusion constant for the motion, a convenient rotational rate index R = d2/ht Aw is defined in our computations, where At is the duration of localization of a polaron between rotational steps, and Aw is the overall frequency spread of the spectrum which is being diffusionally modulated. The new resonance condition will be given by
and consequently the resonance frequency eigenvalues of the A - ‘X operator, which are now modulated in time, appear as w ( t ) . In the adiabatic approximation, we treat all the mr= +7/2 to ml= - 7/2 hyperfine absorption regions independently. These absorptions may be expressed as +7/2
Z(w-co,,)=
C
--7/1
exp[ -i(w-wo)t]
In eq. (5 ), G(t) is the relaxation expressible as
G(f) dt function
.
(5)
which is
629
Volume 160, number 5,6
c(i)=cerp(i/
CHEMICAL
PHYSICS LETTERS
25 August 1989
[a(1)--O]df)), 0
where the outermost angular brackets denote an appropriate averaging over the Monte Carlo motional process. For a sequence of M independent Markovian jumps described by our model (Mz 1000)) the temporal fluctuations of the relaxation function, eq. ( 6), are averaged to I
G(t)=(lIM)
Cexp
.I(
is
[o(t)-o)]dt 0
>
(7)
and the resultant frequency domain spectrum is readily obtained by a Fourier transformation of G(t). In our computation, the eigenvalue problem corresponding to this small polaron dynamics is solved by setting up the Hamiltonian matrix, eq. ( 1)) and then carrying out the calculations corresponding to eqs. (2)-( 7) for various trial values of the rotational rate index, R, between 0.0001 (virtually rigid limit ) and IO (nearly free rotational diffusion). The function G(t) is sampled using a Tukey spectral window, so that spurious “aliasing” effects (which are often inherent artefacts of the Fourier transformation technique) are suppressed. We note here that more elaborate ensemble-averaging procedures are also available for calculating the EPR spectra of rotationally diffusing species in amorphous substances [24,25]. In fig. 2, lineshape simulations for various R values are compared with experimental spectra recorded at several temperatures between 4 and 77 K. Over the temperature range of these modulated lineshapes, a logarithmic plot of the spectral frequency width against T-‘, fig. 4, yields a straight line, the slope of which gives a barrier of 1.14 X IO- 4 eV. This value may be identifikd with the energy needed for the vanadium 3d’ sites to become rotationally “disordered” over short range.
5. Conclusions For the 7OV,O,-3OP,O, binary glass, our EPR measurement at 4 K shows not only that the odd electron is localized at a single 5’V site, but that the 630
,I 5
100
200 103/~(K-')
Fig. 4. Plot of the logarithmic EPR spectral width versus T70V,05-30PI05 between 4 and 77 K_
IO
’ for
vanadium coordination corresponds to a severely compressed V05...0 octahedral unit. Variable temperature EPR above 4 K probes the local dynamics, and the small activation energy of 114 peV measured in the 4-77 K range may be understood to be the energy needed by the localized electron to overcome the crystal-field energy associated with shortrange order in the glass. An alternative, and perhaps more appealing, rationalization of the above results would be that when pyramidally distorted octahedral V-O units pack together to optimize the short-range order of the glass network structure, some covalency contribution to the V-O-V bonds linking adjacent structural units would not only facilitate the packing but also lead to overlap of the 3d’ electron wavefunction over adjacent sites. The measured values of 114 peV (or 28 GHz) would then be a measure of how frequently the electron wavefunction overlap (or “tunneling”) is interrupted by lattice phonon modes as the polaron dynamics gradually go over into the adiabatic hopping regime. For the semiconducting glass we have studied, this is essentially a band model at low temperatures where rotational polaron dynamics leads to a disorder energy which is much less than the polaron bandwidth.
Volume 160, number 5,6
CHEMICAL
PHYSICS LETTERS
References
[ 1 ] G.W. Anderson and W.D. Compton, I. Chem. Phys. 52 (1970) 6166. [2] F.R. Landsberger and P.J. Bray, J. Chem. Phys. 53 (1970) 2757. [3] G.S. Linsley, A. Owen and F.M. Hayatee, J. Non-Cryst. Solids 4 (1970) 208. [4] M. Sayer and A. Mansingh, Phys. Rev. B 6 (1972) 4629. [ 51 G.N. Greaves, J. Non-Cryst. Solids 11 (1973) 427. [6] H. Harper and P.W. McMiBan, Phys. Chem. Glasses 15 (1974) 148. [7] L. Muraski, C.H. Chung and J.D. McKenzie, J. Non-Cryst. Solids 32 (1979) 208. [S] J.O. Isard, J. Non-Cryst. Solids 42 (1980) 371. [9] N.F. Mott, J. Non-Cryst. Solids 1 ( 1966) 1. [ IO] LG. Austin and N.F. Mott, Advan. Phys. 18 ( 1969) 41. [ 11 ] V.M. Nagiev, Soviet Phys. Solid State 7 (1966) 2204. [ 121 E.J. Friebele, L.K. Wilson and D. Kinsener, J. Am. Ceram. Sot. 55 (1972) 164. [ 13 ] G.F. Lynch, M. Sayer, S.L. Segel and G. Rosenblatt, J. Appl. Phys. 42 (1971) 2587.
25 August 1989
[ 141 J. Livage, P. Pineau, M.C. Leroy and M. Michaud, Phys. Stat. Sol. 39a (1977) 73. [ 151 G. Sperlich, Z. Physik 250 ( 1972) 335. [ 161 L.D. Bogomolova, T.F. Dolgolenko, V.A. Jachlin and V.N. Lazukin, J. Magn. Reson. 15 ( 1974) 283. [I?‘] E.E. Khawaja, M.S. Hussain, M.A. Khan and J.S. Hwang, J. Mater. Sci. 21 (1986) 2812. [ 181 B.B. Das, Ph.D. Dissertation, Indian Institute of Technology, Kanpur (1987). [ 191 P. Raghunathan and SK. Sur, J. Am. Chem. Sot. 106 (1984) 8014. [20] P. Raghunathan and SC. Sivasubramanian, Proc. Indian Acad. Sci. Chem. Sci. 96 ( 1986) 565. [2 I ] P. Raghunathan, in: Electron magnetic resonance ofthe solid state, ed. J.A. Weil (Can. Sot. Chem., Ottawa, 1987) ch. 13. [ 221 L. Rivoalen, A. Revcolevschi, J. Livage and R. Collongues, J. Non-Cryst. Solids 2 1 ( 1976 ) 17 1. [ 231 J.B. Pedersen, J. Chem. Phys. 57 (1972) 2680. [24] D. Kivelson and S. Lee, J. Chem. Phys. 76 (1982) 5746. [25]S.LeeandD.P.Ames, J.Chem. Phys.80 (1984) 1766.
631