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Thermally stimulated currents in Rocks. II.
175
Tectonophysics, 224 (1993) 175-180
Elsevier Science Publishers B.V., Amsterdam
Thermally stimulated currents in rocks. II. E. Dologlou University of Athens, Department of Geology, Geophysics-Geothermy Division, Knossou Street 36, Ano Glyfada, Athens 4’6561, Greece
(Received March 20, 1991; revised version accepted January 1, 1992)
ABSTRACT Dologlou, E., 1993. Thermally stimulated currents in Rocks. II. In: P. Varotsos and 0. Kulhanek (Editors), Measurement and Theoretical Models of the Earth’s Electric Field Variations Related to Earthquakes. Tectonophysics, 224: 175-180. Thermally stimulated currents have been observed in eight rock samples, selected intentionally to represent the mean crustal stucture in Greece. For each sample we have determined the (reknrientation parameters rc and h,, from in the usual Arrhenius expression r = r,, exp(h,/kT). In this paper we indicate that, for all rock samples investigated, a linear connection exists between In rn and h,.
1. Introduction
Since 1982, eighteen telemetric stations in Greece have been continuously monitoring variations in the electrotelluric field. Transient changes in this field of no magnetic or atmospheric origin, so-called seismic electric signals @ES), have been observed several hours to few weeks before an earthquake (Varotsos and Alexopoulos, 1984a,b, 1986). A plausible physical model to explain the above observations is the following: rocks contain dipoles, due to aliovalent impurities (see section 2.1.) and piezoelectric inclusions (e.g., quartz) in a non-perfect directional distribution. Before an earthquake, the stress gradually increases in the focal area and an electrical field is generated. These stress variations may decrease the relaxation time of the dipoles (whenever the migration volume is negative, see below). When the stress reaches a certain value, the so-called critical stress, ucr (which is below the stress value for rock failure), a transient current is emitted (Varotsos et al., 1982). If the basic assumptions of the above model are correct, the rocks should also emit a transient current when heated under a constant temperature rate, since a gradual increase of temperature causes a decrease in the relaxation time of dipoles. 0040-1951/93/$06.00
A few years ago, Dologlou and Varotsos (1986) presented experimental results on the detection of thermally stimulated currents (I) in various rock samples, without the application of any external electric field. The aim of the present paper is to draw attention to the fact that, for all samples investigated, the parameters which govern the (relorientation of dipoles are interconnected. 2. Theory 2.1. Relaxation time of dipoles Consider an ionic crystal of the form A+Bwith an aliovalent impurity ion M2+. For reasons of charge compensation, a number of cation vacancies appear in the crystal and, attracted by the aliovalent impurities, form complex dipoles. When an external electric field is applied, the dipoles can change orientation within time, T, the socalled relaxation time. The variation in the relaxation time T with temperature, T, and pressure, P, is given by the formula (Varotsos and Alexopoulos, 1981): T( T, p) = 7. exp(h,/kT) where 7. = the pre-exponential ‘factor; h, = the enthalpy for the orientation process; and k =
0 1993 - Elsevier Science Publishers B.V. Ah rights reserved
17h
Boltzmann’s constant. The relaxation time, T, always decreases with increasing temperature because r0 and h, are positive, while its variation with increasing pressure depends on the algebraic sign of the quantity:
b
(index m = migration) where g, = the Gibbs’ energy for the re-orientation process; and U, = the migration volume Warotsos and Alexopoulos, 1980). 2.2. Themwcurrents
l-_.__
1
c
:
:
* *
,
-t
i
1 According to the procedure for the application of an external electric field, the dipoles can be either polarized or depolarized. The technique for the emission of a thermally stimulated polarization current (McKeever and Hughes, 1975) can be described in the following steps (Fig. 1): Consider a solid at a relatively low temperature, T,, in which the relaxation time, r(Ta), of the dipoles is high (e.g. a few hours). At this temperature, an external electric DC field is applied (Fig. la) for a time t QL:T(TJ and thus the
E t
b
Fig. 1. Schematic representation of the therma& stimulated polarization current procedure. See text for details.
:
j:
I
I
4
t
Fig. 2. Schematic representation of the thermally stimulated depolarization current procedure. See text for details.
orientation of the dipoles remains random. By increasing the temperature T (Fig. lb) at a constant rate b (e.g., a few degrees per minute), the relaxation time gradually decreases. At a certain temperature, the so called “stimulated temperature”, TM, the dipoles become suffkiently mobile and they tend to align with the constantly applied electric field, E thus giving rise to a transient current, j (Fig. lc), the thermally stimulated polarization current (TSCP). A thermally stimulated depolarization current can be observed by carrying out the following procedure: (1) At high temperature T (e.g., room temperature) an electric DC field, E, is applied (Fig. 2, stage a) for a time t > T(T). The dipoles (due to their small relaxation time) can easily align to the direction of the electric field and a current appears from this rotation. (2) The solid is now cooled down to an appreciably low temperature To (Fig. 2, stage b). The relaxation time dTo) becomes rather high and the dipoles are “frozen”. At this low temperature the external electric field is removed and the dipoles remain align to the field.
--OS
1.7
1.9
given. For each sample, as mentioned above, WC give two pairs (rO, h,), which correspond to the low and high temperature range, respectively. An inspection of Figure 5 indicates a linear correlation of In rd with h’,. A least-squares fit to a straight line gives a correlation coefficient of 0.993 and an intercept of approximately 6. whereas the corresponding slope is - 42 eV- ’ . Table 1 shows that, for various rock samples, the h, values vary by a factor of 5 (i.e., from 0.4 to 2.0 eV) and the r0 values vary by 27 orders of magnitude (i.e. from lo-’ to 10e3“ s>. In spite of the latter large variation, Figure 5 indicates a clear interconnection between the (reknientation parameters rb and h’,. This result is not unexpected if we consider the following aspects of the review by Varotsos and Alexopoulos (1986). The relaxation time, T, is given by Varotsos and Alexopoulos (1986) in the formula:
21
h:, 4ev
Fig. 5. The quantityIn T; versus h& for eight rock samples at low and high temperatures. The corresponding values come from Table 1.
been applied. Considering eqn. Cl), the intercept and the slope lead to the following values:
7 =
low temperature: r. = 9 x 10T7 s h, = 0.4 eV high temperature : r. =3x10-i’s
&exp( -s,/k)
exp( h,/kT)
where A = geometrical factor; Y= the attempt frequency for the (re)orientation process; s, = the migration entropy; h, = the migration enthalpy; and k = Boltzmann’s constant. A comparison of eqns. (2) and (3) indicates that the pre-exponential factor, ro, is given by:
h m =06eV *
The same procedure was applied to the other rock samples reported in Dologlou and Varotsos (1986). The resulting values are given in Table 1. 4. In&Seo~n of the t ters of the various rock samples
7. =
&exp( -Mk)
According to Varotsos and Alexopoulos (19861, for a given material (i) the quantity s,/h, for various reorientation processes (j) should be a
In Figure 5 a plot of In ~5 against h’, for various rock samples 6) included in Table 1 is TABLE 1 Relaxation parameters of the various rock samples Rock sample
r; (s) Granite Mica-schist Sipohne Crystalline limestone Marble Gneiss Limestone Serpentinite
High temperature
Low temperature WI&)x 10-33 (l$J x 10-2’ (2:i.J x 10-34 (0.2:;:) x (1 7_+;,3)x ;4’&) x wy2p,) x (9:;,,) x
10-23 10-M lo-= 10-2’ 10-7
h’, (eV)
rb (s)
hh (eV)
2.0 f 0.2 1.3 f 0.1 2.0 f 0.1
(2’&f x lo-‘4 (3+&)x 10-2’ (2$J x lo-l4
0.9 f 0.1 1.6_+0.1 0.8 f 0.1
1.4 f 1.8 f 1.6 f 1.5 f 0.4 f
(I?&> (1 +;.‘I w;.,) (l+&) (3+yI
1.2fO.l 1.2 f 0.1 0.9 f 0.1 1.2 f 0.1 0.6 f 0.1
0.1 0.1 0.1 0.2 0.1
x x x x x
10-20 lo-so 10-16 lo-” lo-lo
THERMALLY
STIMULATED
CURRENTS
IN ROCKS.
17Y
II
TABLE 2
constant, given by: /3’B’
S&
* m’p
B’ - TP’B’ - TaB’pT
ci=-
Q values for various rocks
F’
1p
1p = - l_ (5)
with F’ = pi + (l/B’)aB’/aT I p where pi = the thermal expansion coefficient and B’ = the isothermal bulk modulus of the matrix material. The quantity given by the right-hand side of eqn. (5) may vary only a little among the rock samples investigated. This suggestion may not be far from reality as this quantity is closely connected to the usual Gruneisen constant, y (see Varotsos and Alexopoulos, 1986). By assuming that the term hv does not vary appreciably among the rock samples studied, we conclude that a plot of In ~6 against h’, should be a straight line, the slope of which is equal to the right-hand side of eqn. (5) divided by k. By considering that: B’=
B;
(6)
1 + y@T
where B6 = the adiabatic bulk modulus; yPT K 1 for rock; and T m 270 K; a combination of eqns. (5) and (6) gives:
(aB;,aT + pi) - yp’
F’ = ; s
It can be easily shown that:
=
2vJ avpaq
-
:v,( abyaT)
vp2- ;v,”
(8)
where 4 = the seismic parameter; V, = the P wave seismic velocity; and V, = the S wave seismic velocity. Therefore, we have:
s; -= h’,
and: In
Tb + ln( Au) = k
Q (eV_‘)
Granite
Peridotite
Amphibolite
Quartzite
- 1.25
- 1.47
- 0.84
- 1.88
The following values for the parameters I$,, V,, avJaT, al/,/aT, y, p and the estimation of the quantity Q = l/k(l/4 Q/dT - y/?‘) for different materials, were kindly forwarded to the author by Prof. J. Zschau (pers. commun., 1988). For polycrystalline forsterite: I$ = 7.586 km/s; v, = 4.359 km/s; avr/aT = -4.1 x 1O-4 km/s/gK, al/,/U = -2.9 x 1O-4 km/s/gK; p = 24 x 10e6 K-‘; y = 0.92 and k = 8.616 X 1O-5 eV/ K, thus, we obtain Q = - 1.283 eV_‘. In a similar way, and considering that rP N 4 x lo-‘. K-’ the quantity, Q, was estimated for some rocks and the resulting values are listed in Table 2. The above Q values differ significantly from that indicated by the experimental plot of In 70 against h,. This difference might be attributable to the so-called compensation effect. According to Peacock-Lopez and Suhl (1982) it frequently happens that Arrhenius plots of different members of a family of reactions are straight lines with different slopes, which (when extrapolated if necessary) intersect at a common point. The temperature coordinate of this point is called the compensation temperature or the isokinetic temperature, T,. In the present experimental case, the temperature T, is calculated from: kT, = (41.8 eV-I))’
(11)
and hence T, N 281 K. Although a precise physical meaning of the compensation temperature is still lacking, we note that in our case, T, is comparable with the temperature TM at which the maximum TSC is observed by Dologlou and Varotsos (1986) for the various members of the series of Table 1. In summary, we can state that a close interconnection exists between the (rejorientation parameters ~5 and h’, of the dipoles detected by means of the thermally stimulated current techniques in a series of rock samples.
Acknowledgements
We are grateful to Prof. P. Varotsos for the suggestion of the subject and his critical comments.
Refkrences Bucci, C., Fieschi, R. and Guide, G., 1966. Ionic thermocurrents in dielectrics. Phys, Rev., 148: 816-823. Dologlou, E. and Varotsos, P., 1986. Thermally stimulated currents in rocks. J. Geophys., 59: 177-182. Mckeever, S. and Hughes, D., 1975. Thermally stimulated currents in dielectrics. J. Phys. D: Appl. Phys., 8: 15201.529. Peacock-Lopez, E. and Suhl, H., 1982. Compensation effect in thermally activated processes. Phys. Rev. B, 26: 3774. Varotsos, P. and Alexopoulos, K., 1980. On the question of
the calculation of migration volumes in ionic crystals Philos. Mag., 42: 13-18. Varotsos, P. and Alexopoulos, K., 1981. Migration parameters for the bound fluorine motion in alkaline earth fluoridc~ 11. J. Phys. Chem. Sol., 42: 409-410. Varotsos, P., Alexopoulos, K. and Nomicos, K.. lW2. (‘onrments on the pressure variation of the Gibbs energy for bound and unbound defects. Phys. Status Solidi fb). 1I I: 581-590. Varotsos, P. and Alexopoulos, K., 1984a. Physical properties of the variations of the electric field of the earth preceding earthquakes, I. Tectonophysics, 110: 73-98. Varotsos, P. and Alexopoulos, K., 1984b. Physical Properties of the variations of the electric field of the earth preceding earthquakes. II. Determination of epicenter and magnitude. Tectonophysics, 110: 99-125. Varotsos, P. and Alexopoulos, K., 1986. Simulated current emission in the earth: piezosimulated currents and related geophysical aspects. In: S. Amelinckx, R. Gevers and J. Nihoul (Editors), Thermodynamics of Point Defects and their Relation with Bulk Properties. North-Holland. Amsterdam, pp. 355-356, 361-362.
Tectonophysics, 224 (1993) 175-180
Elsevier Science Publishers B.V., Amsterdam
Thermally stimulated currents in rocks. II. E. Dologlou University of Athens, Department of Geology, Geophysics-Geothermy Division, Knossou Street 36, Ano Glyfada, Athens 4’6561, Greece
(Received March 20, 1991; revised version accepted January 1, 1992)
ABSTRACT Dologlou, E., 1993. Thermally stimulated currents in Rocks. II. In: P. Varotsos and 0. Kulhanek (Editors), Measurement and Theoretical Models of the Earth’s Electric Field Variations Related to Earthquakes. Tectonophysics, 224: 175-180. Thermally stimulated currents have been observed in eight rock samples, selected intentionally to represent the mean crustal stucture in Greece. For each sample we have determined the (reknrientation parameters rc and h,, from in the usual Arrhenius expression r = r,, exp(h,/kT). In this paper we indicate that, for all rock samples investigated, a linear connection exists between In rn and h,.
1. Introduction
Since 1982, eighteen telemetric stations in Greece have been continuously monitoring variations in the electrotelluric field. Transient changes in this field of no magnetic or atmospheric origin, so-called seismic electric signals @ES), have been observed several hours to few weeks before an earthquake (Varotsos and Alexopoulos, 1984a,b, 1986). A plausible physical model to explain the above observations is the following: rocks contain dipoles, due to aliovalent impurities (see section 2.1.) and piezoelectric inclusions (e.g., quartz) in a non-perfect directional distribution. Before an earthquake, the stress gradually increases in the focal area and an electrical field is generated. These stress variations may decrease the relaxation time of the dipoles (whenever the migration volume is negative, see below). When the stress reaches a certain value, the so-called critical stress, ucr (which is below the stress value for rock failure), a transient current is emitted (Varotsos et al., 1982). If the basic assumptions of the above model are correct, the rocks should also emit a transient current when heated under a constant temperature rate, since a gradual increase of temperature causes a decrease in the relaxation time of dipoles. 0040-1951/93/$06.00
A few years ago, Dologlou and Varotsos (1986) presented experimental results on the detection of thermally stimulated currents (I) in various rock samples, without the application of any external electric field. The aim of the present paper is to draw attention to the fact that, for all samples investigated, the parameters which govern the (relorientation of dipoles are interconnected. 2. Theory 2.1. Relaxation time of dipoles Consider an ionic crystal of the form A+Bwith an aliovalent impurity ion M2+. For reasons of charge compensation, a number of cation vacancies appear in the crystal and, attracted by the aliovalent impurities, form complex dipoles. When an external electric field is applied, the dipoles can change orientation within time, T, the socalled relaxation time. The variation in the relaxation time T with temperature, T, and pressure, P, is given by the formula (Varotsos and Alexopoulos, 1981): T( T, p) = 7. exp(h,/kT) where 7. = the pre-exponential ‘factor; h, = the enthalpy for the orientation process; and k =
0 1993 - Elsevier Science Publishers B.V. Ah rights reserved
17h
Boltzmann’s constant. The relaxation time, T, always decreases with increasing temperature because r0 and h, are positive, while its variation with increasing pressure depends on the algebraic sign of the quantity:
b
(index m = migration) where g, = the Gibbs’ energy for the re-orientation process; and U, = the migration volume Warotsos and Alexopoulos, 1980). 2.2. Themwcurrents
l-_.__
1
c
:
:
* *
,
-t
i
1 According to the procedure for the application of an external electric field, the dipoles can be either polarized or depolarized. The technique for the emission of a thermally stimulated polarization current (McKeever and Hughes, 1975) can be described in the following steps (Fig. 1): Consider a solid at a relatively low temperature, T,, in which the relaxation time, r(Ta), of the dipoles is high (e.g. a few hours). At this temperature, an external electric DC field is applied (Fig. la) for a time t QL:T(TJ and thus the
E t
b
Fig. 1. Schematic representation of the therma& stimulated polarization current procedure. See text for details.
:
j:
I
I
4
t
Fig. 2. Schematic representation of the thermally stimulated depolarization current procedure. See text for details.
orientation of the dipoles remains random. By increasing the temperature T (Fig. lb) at a constant rate b (e.g., a few degrees per minute), the relaxation time gradually decreases. At a certain temperature, the so called “stimulated temperature”, TM, the dipoles become suffkiently mobile and they tend to align with the constantly applied electric field, E thus giving rise to a transient current, j (Fig. lc), the thermally stimulated polarization current (TSCP). A thermally stimulated depolarization current can be observed by carrying out the following procedure: (1) At high temperature T (e.g., room temperature) an electric DC field, E, is applied (Fig. 2, stage a) for a time t > T(T). The dipoles (due to their small relaxation time) can easily align to the direction of the electric field and a current appears from this rotation. (2) The solid is now cooled down to an appreciably low temperature To (Fig. 2, stage b). The relaxation time dTo) becomes rather high and the dipoles are “frozen”. At this low temperature the external electric field is removed and the dipoles remain align to the field.
--OS
1.7
1.9
given. For each sample, as mentioned above, WC give two pairs (rO, h,), which correspond to the low and high temperature range, respectively. An inspection of Figure 5 indicates a linear correlation of In rd with h’,. A least-squares fit to a straight line gives a correlation coefficient of 0.993 and an intercept of approximately 6. whereas the corresponding slope is - 42 eV- ’ . Table 1 shows that, for various rock samples, the h, values vary by a factor of 5 (i.e., from 0.4 to 2.0 eV) and the r0 values vary by 27 orders of magnitude (i.e. from lo-’ to 10e3“ s>. In spite of the latter large variation, Figure 5 indicates a clear interconnection between the (reknientation parameters rb and h’,. This result is not unexpected if we consider the following aspects of the review by Varotsos and Alexopoulos (1986). The relaxation time, T, is given by Varotsos and Alexopoulos (1986) in the formula:
21
h:, 4ev
Fig. 5. The quantityIn T; versus h& for eight rock samples at low and high temperatures. The corresponding values come from Table 1.
been applied. Considering eqn. Cl), the intercept and the slope lead to the following values:
7 =
low temperature: r. = 9 x 10T7 s h, = 0.4 eV high temperature : r. =3x10-i’s
&exp( -s,/k)
exp( h,/kT)
where A = geometrical factor; Y= the attempt frequency for the (re)orientation process; s, = the migration entropy; h, = the migration enthalpy; and k = Boltzmann’s constant. A comparison of eqns. (2) and (3) indicates that the pre-exponential factor, ro, is given by:
h m =06eV *
The same procedure was applied to the other rock samples reported in Dologlou and Varotsos (1986). The resulting values are given in Table 1. 4. In&Seo~n of the t ters of the various rock samples
7. =
&exp( -Mk)
According to Varotsos and Alexopoulos (19861, for a given material (i) the quantity s,/h, for various reorientation processes (j) should be a
In Figure 5 a plot of In ~5 against h’, for various rock samples 6) included in Table 1 is TABLE 1 Relaxation parameters of the various rock samples Rock sample
r; (s) Granite Mica-schist Sipohne Crystalline limestone Marble Gneiss Limestone Serpentinite
High temperature
Low temperature WI&)x 10-33 (l$J x 10-2’ (2:i.J x 10-34 (0.2:;:) x (1 7_+;,3)x ;4’&) x wy2p,) x (9:;,,) x
10-23 10-M lo-= 10-2’ 10-7
h’, (eV)
rb (s)
hh (eV)
2.0 f 0.2 1.3 f 0.1 2.0 f 0.1
(2’&f x lo-‘4 (3+&)x 10-2’ (2$J x lo-l4
0.9 f 0.1 1.6_+0.1 0.8 f 0.1
1.4 f 1.8 f 1.6 f 1.5 f 0.4 f
(I?&> (1 +;.‘I w;.,) (l+&) (3+yI
1.2fO.l 1.2 f 0.1 0.9 f 0.1 1.2 f 0.1 0.6 f 0.1
0.1 0.1 0.1 0.2 0.1
x x x x x
10-20 lo-so 10-16 lo-” lo-lo
THERMALLY
STIMULATED
CURRENTS
IN ROCKS.
17Y
II
TABLE 2
constant, given by: /3’B’
S&
* m’p
B’ - TP’B’ - TaB’pT
ci=-
Q values for various rocks
F’
1p
1p = - l_ (5)
with F’ = pi + (l/B’)aB’/aT I p where pi = the thermal expansion coefficient and B’ = the isothermal bulk modulus of the matrix material. The quantity given by the right-hand side of eqn. (5) may vary only a little among the rock samples investigated. This suggestion may not be far from reality as this quantity is closely connected to the usual Gruneisen constant, y (see Varotsos and Alexopoulos, 1986). By assuming that the term hv does not vary appreciably among the rock samples studied, we conclude that a plot of In ~6 against h’, should be a straight line, the slope of which is equal to the right-hand side of eqn. (5) divided by k. By considering that: B’=
B;
(6)
1 + y@T
where B6 = the adiabatic bulk modulus; yPT K 1 for rock; and T m 270 K; a combination of eqns. (5) and (6) gives:
(aB;,aT + pi) - yp’
F’ = ; s
It can be easily shown that:
=
2vJ avpaq
-
:v,( abyaT)
vp2- ;v,”
(8)
where 4 = the seismic parameter; V, = the P wave seismic velocity; and V, = the S wave seismic velocity. Therefore, we have:
s; -= h’,
and: In
Tb + ln( Au) = k
Q (eV_‘)
Granite
Peridotite
Amphibolite
Quartzite
- 1.25
- 1.47
- 0.84
- 1.88
The following values for the parameters I$,, V,, avJaT, al/,/aT, y, p and the estimation of the quantity Q = l/k(l/4 Q/dT - y/?‘) for different materials, were kindly forwarded to the author by Prof. J. Zschau (pers. commun., 1988). For polycrystalline forsterite: I$ = 7.586 km/s; v, = 4.359 km/s; avr/aT = -4.1 x 1O-4 km/s/gK, al/,/U = -2.9 x 1O-4 km/s/gK; p = 24 x 10e6 K-‘; y = 0.92 and k = 8.616 X 1O-5 eV/ K, thus, we obtain Q = - 1.283 eV_‘. In a similar way, and considering that rP N 4 x lo-‘. K-’ the quantity, Q, was estimated for some rocks and the resulting values are listed in Table 2. The above Q values differ significantly from that indicated by the experimental plot of In 70 against h,. This difference might be attributable to the so-called compensation effect. According to Peacock-Lopez and Suhl (1982) it frequently happens that Arrhenius plots of different members of a family of reactions are straight lines with different slopes, which (when extrapolated if necessary) intersect at a common point. The temperature coordinate of this point is called the compensation temperature or the isokinetic temperature, T,. In the present experimental case, the temperature T, is calculated from: kT, = (41.8 eV-I))’
(11)
and hence T, N 281 K. Although a precise physical meaning of the compensation temperature is still lacking, we note that in our case, T, is comparable with the temperature TM at which the maximum TSC is observed by Dologlou and Varotsos (1986) for the various members of the series of Table 1. In summary, we can state that a close interconnection exists between the (rejorientation parameters ~5 and h’, of the dipoles detected by means of the thermally stimulated current techniques in a series of rock samples.
Acknowledgements
We are grateful to Prof. P. Varotsos for the suggestion of the subject and his critical comments.
Refkrences Bucci, C., Fieschi, R. and Guide, G., 1966. Ionic thermocurrents in dielectrics. Phys, Rev., 148: 816-823. Dologlou, E. and Varotsos, P., 1986. Thermally stimulated currents in rocks. J. Geophys., 59: 177-182. Mckeever, S. and Hughes, D., 1975. Thermally stimulated currents in dielectrics. J. Phys. D: Appl. Phys., 8: 15201.529. Peacock-Lopez, E. and Suhl, H., 1982. Compensation effect in thermally activated processes. Phys. Rev. B, 26: 3774. Varotsos, P. and Alexopoulos, K., 1980. On the question of
the calculation of migration volumes in ionic crystals Philos. Mag., 42: 13-18. Varotsos, P. and Alexopoulos, K., 1981. Migration parameters for the bound fluorine motion in alkaline earth fluoridc~ 11. J. Phys. Chem. Sol., 42: 409-410. Varotsos, P., Alexopoulos, K. and Nomicos, K.. lW2. (‘onrments on the pressure variation of the Gibbs energy for bound and unbound defects. Phys. Status Solidi fb). 1I I: 581-590. Varotsos, P. and Alexopoulos, K., 1984a. Physical properties of the variations of the electric field of the earth preceding earthquakes, I. Tectonophysics, 110: 73-98. Varotsos, P. and Alexopoulos, K., 1984b. Physical Properties of the variations of the electric field of the earth preceding earthquakes. II. Determination of epicenter and magnitude. Tectonophysics, 110: 99-125. Varotsos, P. and Alexopoulos, K., 1986. Simulated current emission in the earth: piezosimulated currents and related geophysical aspects. In: S. Amelinckx, R. Gevers and J. Nihoul (Editors), Thermodynamics of Point Defects and their Relation with Bulk Properties. North-Holland. Amsterdam, pp. 355-356, 361-362.