A Dislocation Model for Seismic Electric Signals


L. M. SLIFKIN

Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA

Introduction

Charged Dislocations
The Model


It is suggested that a possible mechanism for production of seismic electric signals involves the motion of charged dislocation loops upon rapid change in a shear component of the stress acting on blocks of rock in the earthquake preparation zone. A calculation, which uses likely values of the relevant parameters indicates that such a process could produce significant voltages at the surface of the Earth.

Introduction
Detection of telluric voltages at the Earth's surface has been reported by several groups, such as the VAN team in Greece. In order to relate these signals to impending earthquakes, it would be helpful to understand the mechanisms of both their production and also their transmission in the highly anisotropic and heterogeneous media found in the Earth's crust. A feasible mechanism for transmission of the signals over long distances has been proposed by Lazarus, and is further discussed in a companion paper in these proceedings; the present article is concerned only with the process of the generation of the voltage.

It has been experimentally demonstrated some years ago, by Hoenig, that severe loading of laboratory specimens of rocks can result in the appearance of very large voltages at the surface. Piezoelectric effects and stress induced orientation of impurity-vacancy electric dipoles do not appear capable of giving rise to large enough signals in the usual polycrystalline materials found in nature. Hoenig proposed that his observations were the result of a sequence of microfractures, and somewhat later it was shown by Gershenzon et al that the production and motion of electrically charged dislocations at the tip of the fracture crack would produce voltage differences. They were able to estimate the magnitudes of electromagnetic emission to be expected from this process during crustal fracturing, and concluded that this phenomenon could produce significantly large signals. 

A number of possible mechanisms, for production of seismic electric signals have recently been examined by Bernard. He finds that, given the right conditions, one could perhaps, understand these effects in terms of electrokinetic phenomena: he suggests that a change of stress may trigger fluid instabilities that would give rise to large streaming potentials. One would still require some "channeling" of the electric signal in order to explain the detection at distances as great as involved in the VAN experiments. The model proposed by Lazarus would seem to be a possible explanation of such preferential channeling. 

It has been suggested by the present author that, even in the absence of large amounts of plastic flow or of fracture, an abrupt change in the shear stress —either increase or decrease— should cause a voltage difference to appear between opposite surfaces of a specimen of most minerals. In such materials as silicates, oxides, and the like, lattice vacancies and aliovalent impurity ions carry effective charges. Moreover, single-stepped jogs on edge dislocations (essentially, the edges of partial lattice planes which do not extend completely through the crystal) are also electrically charged. Because these jogs must establish simultaneous equilibria with all of the different types of point defects within the crystal, it follows that, in general, the edge dislocations in ionic crystals carry a net electric charge. 

Now, although the dislocation may not be able to move through large distances in response to the applied shear stress, segments of it can indeed bow out between the points at which the dislocation is pinned, as has been discussed by Granato and Lücke. The pins may be impurity ions, clusters, or points of intersection with other dislocations. The dislocation loops between the pinning points respond to applied shear stresses much as if they were non-Hookean elastic bands. Upon a change in stress, the relaxation rate of the loops is primarily a function of the Peierls stress— the stress required to cause dislocation displacement in an otherwise perfect lattice. At high temperatures, the dislocation loops can be quite mobile. 

The purpose of this paper is to make an estimate of the magnitude of the electric field produced by the motion of these dislocation loops when the stress is rapidly increased or decreased, causing the loops to expand or contract, respectively. The numerical results are quite sensitive to the values chosen for the relevant parameters, some of which are uncertain. At best, then, this discussion is a feasibility test of the question: given a reasonable set of values for these parameters, is it possible for the proposed mechanism to account for signals comparable to those that have been reported?

Charged Dislocations
Since the proposed mechanism depends heavily on the net electrical charge of dislocations in non-metallic mineral crystals, it seems worthwhile to review briefly the origin and extent of the phenomenon. We are only concerned with edge dislocations, since screw dislocations are not charged. Moreover, the points of particular interest to us along the dislocation are the "jogs", where the edge of the extra half-plane that produces the dislocation makes an abrupt step from one slip plane to an adjacent, parallel one. In a crystal composed of ions, as in the minerals of interest to us, such a step leaves an exposed ion at the jog. Seitz showed that the jog thereby has an effective charge equal to half of that of the exposed ion. It was earlier pointed out by Frenkel that jogs on surface terraces and also, by implication, on dislocations must act as sources and sinks for each of the individual types of intrinsic point defect, such as cation and anion vacancies. The process of emitting or absorbing a vacancy causes the jog to change electrical sign and to move along the dislocation by one interatomic spacing. 

The jogs must therefore establish multiple equilibria —i.e., with each of the species of point defect. This results in the presence of more jogs of one sign than the other, so that the dislocations carry a net charge. The linear charge density on the dislocations is compensated by an approximately cylindrical space charge around the dislocation, consisting of an excess of point defects of the opposite sign. The electric potential difference that is thereby established between the dislocation and the distant interior just compensates for the differing formation free energies of the different species of intrinsic point defects. 

In materials with significant concentrations of impurities, as in the case of geophysically interesting minerals, the space charge consists largely of aliovalent dopant ions. These ions will, in general, be very much less mobile than dislocation segments bowed out between pinning points. Hence, after any abrupt change in stress, the bowed loops will quickly respond, leaving the space charge distribution unrelaxed; the center of the space charge no longer coincides with the line of the dislocation, and an electric dipole has been produced. This is the basis of the mechanism proposed here for the generation of seismic electric signals.

The Model
First, we estimate the dipole moment per unit length of dislocation that can be expected to result from a large and sudden change of shear stress. For the purposes of this rough calculation, it is convenient to ignore the various cosine factors that arise from the fact that the slip planes are not necessarily aligned with the major faces of the rock specimen or parallel to the axis of any folding to which the specimen had previously been subjected. 

The stress-induced dipole moment lies in the slip plane and is oriented perpendicular to the dislocation line. Its magnitude per unit length of dislocation is given by the product of the dislocation charge density and the change in mean displacement of the bowed segments, caused by the change in applied stress. For a number of alkali and silver halides, experimental values have been deduced for the charge density, as a function of purity, temperature, and dislocation dynamics; these studies have been reviewed by Whitworth. A typical value is of the order of 0.1 e/b, where e is the electronic charge and b the lattice spacing. If we assume comparable numbers for the rocks of interest to the present problem, and take b to be about 5 x 10-10 m, then the net charge density is of the order of 3 x 10-11 C/m. 

The second factor determining the dipole moment is the -mean displacement of the dislocation loops bowed out between pairs of pins. If the applied stress is a significant fraction of the flow or fracture stress, then the loops will bow out through a distance large compared to b, and we can approximate the mean displacement by one-fourth the spacing between pins (for example, if the loop were nearly semicircular, then its maximum displacement would be half the interpin distance). We now need some reasonable estimate of this interpin spacing, which must depend on the crystal structure, the presence and state of aggregation of impurities, the dislocation distribution, and —overall— the thermal and mechanical histories of the specimen. Analyses of various mechanical properties of a range of solids have suggested-that the interpin spacing is often of the order 10- 100 b. If we choose a value in the middle of this range, then our mean dislocation loop displacement under the maximum supportable stress is about (50/4) b » 6x10-9 m. 

If we now consider a sudden change in stress of magnitude comparable to the stress at failure, the dipole moment produced by the motion of the charged loops, per unit length of dislocation, is equal to the product of these two factors:

(3 x 10-11 C/m) x (6 x 10-9 m) » 2 x 10-19 C-m/m. The resulting electric polarization, the dipole moment per unit volume, is obtained by multiplying this quantity by the appropriate density of dislocations (i.e., in the simplest case, the total dislocation length per unit volume, or the number intersecting a surface of unit area). 

The dislocation density, of course, depends on the mechanical history of the specimen. While it is possible to grow crystals of such semiconductors as very pure silicon with virtually no dislocations, most annealed simple crystals have densities in the range 108- 1010/m2, and heavily deformed material contains 103 - 104 times this. These values, however, are not appropriate to our problem, because some dislocations have their extra half-planes coming from above, while others are from below (one says that they have opposite mechanical signs). Under an applied stress, the interpin segments of these two types bow out in opposite directions; hence, they make opposite contributions to the polarization. Only the net excess of one type over the other produces a net signal. If the distribution were quite random, this excess would be approximately equal to the square root of the total number. 

The distribution of dislocations in most rock strata is not, however, at all random, because of the folding they have undergone in earlier history. Plastic bending of crystalline material is accomplished through introduction of a set of edge dislocations, all of the same mechanical sign, to accommodate the differing lengths of the upper and lower surfaces. Consider, for example, the specimen sketched in Fig. 1, a cube of edge length 1 m. Suppose that it has been bent through an angle of 1° (» 0.02 rad). Then the upper edge must be longer than the lower one by 0.02 m. If we again take b = 5 x 10-10 m, then the bending must have resulted from the introduction of 0.02 / (5 x 10-10) = 4 x 107 new edge dislocations, all of the same mechanical sign, as shown in the diagram. In this case, the excess density of dislocations is then 4 x 107/m2. This is a quite modest value, and probably greatly underestimates the dislocation densities in naturally occurring rock. 

If, nevertheless, we use this result for our illustrative example, and recall our earlier estimate of the stress-induced dipole moment per unit length of dislocation (i.e., 2 x 10-19 C-m/m), we then obtain a volume polarization of (4 x 107/m2) x (2 x 10-19 C-m/m) = 8 x 10-12 C/m2 this is just the charge density that appears on the surfaces at the ends of the slab (ignoring such factors as the cosine of the angle between the normal of the end surfaces and the direction of displacement of the dislocation loops). 

We are now ready to calculate the electric field appearing at the Earth's surface. The numerical result, depends sensitively on the choice of such parameters as the size and depth of the block. It will also be greatly perturbed by heterogeneities and anisotropy in the dielectric properties of the surrounding medium. In particular, the decay of electric field with the cube of the distance from the dipole is only applicable for a homogeneous, isotropic medium; this is considered at greater length in the companion paper in these proceedings by D. Lazarus. With these limitations in mind, the following estimate can only be taken as an illustrative example, at best. 

For concreteness, consider a horizontal block 1000 m on each side, and 100 m thick. Suppose that it has been folded about a horizontal axis perpendicular to one of the end faces, as shown in Fig. 2. If we take the angle through which the block has been bent to be 1°, as in the calculation relating the Fig. 1, so that the density of excess dislocations is 4 x 107/m2, and if we use the previously estimated stress-induced dipole moment per unit length of dislocation (i.e., 2 x 10-19 C-m/m), then the resulting dipole moment is (2 x 10-19 C-m/m) x (4 x 107/m2) x (105 m2) x (103 m); the last two factors are the cross-sectional area and the length of the dipole, respectively. The stress change has thus produced an electric dipole moment of magnitude 8 x 10-4 C-m. 

To estimate the effect of this dipole moment on observations made at the Earth's surface, we must know the depth of the block and the dielectric structure of the intervening material. Let us first assume that the medium is homogeneous, isotropic, and has a dielectric coefficient of unity. Then, for a dipole of length L, the electric field at a distance d perpendicular to its axis is given by: 

E = (1/4p e0) p (d2 + L2/4)-3/2 

where p is the dipole moment, estimated above to be 8 x 10-4 C-m, and the depth d is taken, for sake of illustration, to be 10 km = 104 m. Ignoring the term L2/4 as compared to d2, and inserting e0 = 9 x 10-12 C2/Nm2, we obtain an electric field at the earth's surface of 7 x 10-6 V/m, or 7 mV/km. This is in the range of signals obtained for large earthquakes by the VAN group, but, of course, their electrodes are not located directly above the active regions. For large distances from the dipole (d > L), E falls off as l/d3, and one can understand the experimental observations only in terms of some sort of channeling by dielectric inhomogeneities. This problem is not unique to the present mechanism; virtually any reasonable model would have the same requirement An attractive solution to this puzzle is proposed by D. Lazarus elsewhere in these proceedings. 

Another question relates to the rather long durations of the signals recorded by the VAN group, as compared to the quite short RC time constant for relaxation of electric fields in typical wet minerals. It is possible that the recorded signals consist of unresolved superpositions of many rapid pulses, generated by the propagation of mechanical relaxations in a sequence of neighboring blocks. 

It thus appears that the charged dislocation loop mechanism is a likely candidate but certainly not the only possibility, for explaining stress-induced voltages in rocks. For example, if significant microplasticity occurs, then the long-range motion of the charged dislocations would greatly enhance the electric signal. In any case, ifs wave-guiding mechanism for long-range transmission, such as that suggested by Lazarus, proves to be an acceptable means of precluding the l/d3 decay, then the voltages estimated here are indeed consistent with the observations that have been made in Greece. It is clear, however, that the predicted signal is quite sensitive to the chemistry, morphology, and mechanical history of the relevant strata. It thus cannot be expected to be a general phenomenon, independent of the local geology.

 

Fig. 1. Introduction of edge dislocations by plastic bending. If the angle q = 0.02 radians, the upper edge of the block is longer than the lower by 0.02 m. The dislocations arc indicated by upside-down "T"s, with the base representing the slip plane and the stem the end of the extra half-plane. (Not to scale)
Fig. 2. Production of an electric dipole in a previously plastically bent specimen, upon application of a shear stress parallel to the slip plane of the dislocations.




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