A SINGLE DOMAIN GRAIN MODEL FOR THE LOW - FIELD CONSTRICTED HYSTERESIS LOOPS OF SOME BASALTIC ROCKS BY C. RADHAKRISHNAMURTY AND N. P.

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1 A SINGLE DOMAIN GRAIN MODEL FOR THE LOW - FIELD CONSTRICTED HYSTERESIS LOOPS OF SOME BASALTIC ROCKS BY C. RADHAKRISHNAMURTY AND N. P. SASTRY (Tata Institute of Fundamental Research, Bombay-5, India) Received January 31, 1970 (Communicated by Prof. D. Lal, F.A.sc.) ABSTRACT Basaltic rocks contain magnetic grains oriented randomly in a nonmagnetic matrix. Experimental studies of low-field hysteresis of such rocks have shown partially and totally constricted hysteresis loops for some specimens. A possible origin of these constricted loops on the basis of interacting single domain behaviour is suggested. 1. INTRODUCTION BASALTIC ROCKS contain a few per cent of magnetic mineral grains which are usually of magnetite or titan omagnetite. The range of Curie temperatures of these rocks is 120 to 580 C., the lesser values corresponding to large amounts of titanium in solid solution with magnetite. Neel' pointed out that such rocks may obey the Rayleigh laws in magnetic fields upto 10 Oe. The low-field hysteresis loops, generally termed Rayleigh loops, reported earlier 2,1 for basalts, are shown in Fig. 1. The magnetic state of all the rocks showing constricted loops like the one in Fig. 1 d, and that of some showing almost no hysteresis like that in Fig. 1 a, has been found to be totally disturbed when subjected to strong magnetic fields (> 50 Oe) or temperature changes of ± 20 C. or even mechanical shocks such as due to hammering. The change of state shows up as enhanced hysteresis and isothermal remanence as indicated in Fig. 2. The ii,itial state is recovered in times characteristic of the specimens ranging from a few minutes to several weeks. For the rocks showing wide loops (Fig. 1 c) the dimensionless paramethr BHe/A, where B and A are the quadratic and linear coefficients of Rayleigh relation and H e is the bulk coercive force, is of the order of

2 Loops of Some Basaltic Rocks 95 After disturbing the state of a rock showing constricted loop, the constriction in the loop disappears and in this state, BHc/A lies between 5 and 20. L a F 10) Z a0 W FIG. 3 REPRESENTATIVE POIIYTS "T'\ z N (.b ) W ir a 0 W K U- 0 Z W (C) 0 LOCAL FIELD h ^ FIG. 5 THREE POSSIBLE DENSITY, DISTRIBUTIONS OF REPRESENTATIVE FIG.4 BANDING OF REPRESENTATIVE POINTS DUE TO INTERACTION POINTS Earlier, Brissoneau observed similar constricted loops which showed time dependence for a-iron containing a very small amount of carbon in solid solution, at a temperature of 21.3 C. He explained the constriction as arising out of stabilization of Bloch walls in their new positions aided by a diffusion after effect. However, such an explanation does not appear to be valid in the case of rocks because of the following_ reasons (1) Neel' calculated that BH c/a should lie in the range 0.1 to 0.03 for rocks containing large ferromagnetic grains which can sustain domain walls. The observed value is 20 or more for rocks which show wide loops. Hence the behaviour of such rocks is unlikely to be due to the_ presence of a large number of multidomain grains dispersed in a non-magnetic matrix. (2) Rocks with different Curie temperatures ( C.) thus indicating a wide variety of magnetic constituents have been found to show a A5

3 96 C. RADHAKRISHNAMURTY AND N. P. SASTRY large overall low-field hysteresis whose measure is BH c/a. This is in contrast to the rather narrow range of impurity of carbon in a-iron where large BHc/A in conjunction with disaccommodation has been observed. (3) The Q ratio, which is the ratio of thermoremanent magnetic intensity to the intensity induced in a field of about 0.5 Oe, for basaltic rocks has been found to range from 1 to 100. Stacey' concluded basing on Parry's works that high Qn values (> 10) indicate unambiguously predominance of single domain grains in the rocks irrespective of the composition of the magnetic grains (whether magnetite or titan omagnetite). Most of the rocks that show wide loops occur among those that show very high (> 50) value for Qu denoting the presence of a large number. of single domain grains. Thus we are led to consider a model other than the domain wall disaccommodation and diffusion, which can account for the above-mentioned facts. The one model that suggests itself is a single domain grain structure which is examined here. 2. Low-FIELD HYSTERESIS OF AN ASSEMBLY OF SINGLE DOMAIN GRAINS Let us consider a single domain grain of uniaxial symmetry whose easy axis makes an angle 00, with respect to an externally applied magnetic field H. The acquired magnetization lies in the plane determined by H and the easy axis and makes an angle 0 with the latter. The free energy expression K sin 28 JH cos (8 0) where K and J are the anisotropy constant and saturation magnetization of the grain, leads to the coercive force relations he J(2 sin 20) when 0 < 0 <4 (1) and = ± (sin 280) when 4 <0 < 2. (2) The two expressions agree at 0 = 4/4. According to (1) and (2) the coercive force decreases as 0 0 increases and becomes zero at O o = 42.

4 Loops of Some Basaltic Rocks 97 In an assembly of randomly oriented single domain grains which are not interacting with each other, those grains, with their easy axes nearly parallel to the applied field H, will have coercive forces of the order of' 2K/J and their magnetic moments cannot be rotated by H when H <I 2K/J f. However, even in this field there will be grains with he < H and from (2) their easy axes are nearly perpendicular to H. Only these grains contribute to hysteresis in H. Defining a = h, the higher and lower coercive forces of a grain oriented at 90 are (a, a). Let us represent this grain by a point whose co-ordinates are, a, along X-axis and, a, along Y-axis (Fig. 3). Each grain now corresponds to a point on the diagonal line Oa and the entire assembly can be described by a density of such points, which varies along Oa. According to the definition, a > 0 and hence there are no points in the upper. half-plane Y > 0. This diagram is similar to that of Preisach. 7 In a small field H, only those grains whose easy axes depart by small angles S or less than 7T/2 contribute to hysteresis. Thus H = J sin G 8) J 28. If we now assume that all possible orientations of the easy axis are equally probable, then the number of grains with is proportional to S < 00 <2 6' 27TS sin ( r I'JH Z) K This number being proportional to H, we see that the density of points near the origin along the diagonal O a is a constant. Now, if a small alternating magnetic field is applied the assembly will show hysteresis. It is easy to see that the magnetization of such grains does not obey Rayleigh's second law because the magnetization retains its maximum value until the field reverses sign as pointed out by Neel.' However, this need not be strictly true if there are grains with small enough volumes

98 C. RADHAKRISHNAMURTY AND N. P. SASTRY because then the thermal energy can flip the direction of magnetization over the potential barrier created by the anisotropy energy.

5 98 C. RADHAKRISHNAMURTY AND N. P. SASTRY because then the thermal energy can flip the direction of magnetization over the potential barrier created by the anisotropy energy. In rocks the magnetic grains are dispersed and in some they may foim aggregates. in which case we have to consider different types of interactions. A single domain grain with t3. it/2, or equivalently, a small coercive force is subject to a constant local field component h arising from its magnetic environments. A large part of this h may be contributed by neighbours with large coercive forces which cannot therefore be subject to changes of magnetization by an external field. The effect of h on the grain is to shift the representative point from (a, a) to (a h, a h), that is to an off-diagonal position. If h. 0, the point is pushed towards Y direction and if h < 0, it will be towards + X (Fig. 3). Both signs are equally probable, thus the distribution of points is symmetrical about Oa. The magnitude of the local field h has a distribution due to the state of aggregation of the grains, which may then show a maximum at some average value and fall off on either way. In such a case it is obvious that the representative points near the origin (hence small coercive force) fall neatly into two bands like that shown in Fig. 4. The dotted line represents schematically the density of distribution of representative points near the origin, as a function of the displacement from the diagonal Oa. In Fig. 5 the different possibilities for the distribution of points are shown. Fig. 5 c corresponds to that of Fig. 4. At this point, it might be worth considering how, in spite of spherical symmetry with which the magnetic dipoles of grains surround the grain considered, a non-vanishing field at this grain can be obtained. Even in a purely superparamagnetic single domain grain system, Evdokimov 8 has. shown how a " molecular field" can be expected. The estimated value of this field ranges upto a maximum of 1,000 gauss,8 and the value depends mainly on the density of dispersion of the magnetic particles in a nonmagnetic matrix. Superparamagnetic clusters treated by Evdokimov are fast - relaxing systems. By extending his ideas to relatively slowly relaxing clusters of ferromagnetic particles in the rock, it is not at all difficult to obtain a field of stabilization whose modulus averages to a finite value instead of to zero. This average field is a result of the co-operative nature of the magnetic dipolar interactions among grains belonging to a cluster. It should be noted, however, that the field itself averages to zero, unless disturbed by external means such as a large field, which aligns all " local " fields in one - direction, Then it takes a certain time to relax to the disw

6 Loops of Some Basaltic Rocks 99 oriented state, which corresponds to equilibrium. As experimentally observed, this time shows wide dynamic range, and reflects the state of aggregation (or strength of coupling between clusters). Neel' estimated that the magnitude of the local fluctuating field, h, is of the order of 10 Oe, the field in which we studied the hysteresis for various basaltic rocks. It is obvious without further argument that the cases of Fig. 5 a, b, c correspond to unconstricted, partially constricted and totally constricted loops respectively. In Fig. 5 a, if h is large the density of points near the origin is constant and Rayleigh's second law may be obeyed as explained by Neel.' In fields of the order of 10 Oe, we are naturally concerned with the hysteresis of grains with small coercive forces. Whatever the origin of this small he of the grains, it is not surprising to expect a case similar to Fig. 5 c which physically corresponds to a state of aggregation where local interaction plays a predominant role. Such a situation can either naturally occur in fine-grained basalts or subsequently created by weathering in somewhat coarser-grained rocks where constricted hysteresis loops are often encountered. 3. DISCUSSION OF RESULTS If we have a rock with a distribution of Fig. 5 c, we can change it to assume that of Fig. 5 a by some external force like magnetic field which orients if D.C. or randomises if A.C., all the magnetization directions of the grains temporarily, both of which result in merging the bands because the stabilization field is lost in these cases. In this state the rock shows an unconstricted loop and relaxes back to its original stable state. If the state of aggregation in a rock produces a distribution like 5 a or b, no external influence can make it into a distribution corresponding to 5 c. This again corresponds to the fact that no rock shows a constricted loop when disturbed if it shows a normal loop to start with, nor a totally constricted loop if it was a partially constricted one. An important distinction between the interacting single domain and multidomain pictures is that, whereas in the former the recovery of the initial magnetic state is entirely reversible, in the latter the recovery if at all, may be to a state similar to but not identical with the initial state. Also, as the domains always display their respective saturation magnetization on

7 100 C. RADHAKRISHNAMURTY AND N. P. SASTRY which the interaction and thereby the distribution depends, their equilibrium state is practically independent of the magnetic state of the bulk rock such as saturated or demagnetized. This has been confirmed by experimental observations. 4. LINE-LOOP-LINE TRANSITION The type of transition namely a rock showing almost no hysteresis in the equilibrium state and a large hysteresis on disturbing it (Fig. 2 c and d) does not seem to have been observed in ferrites so far. Even this can be accounted for on the basis of the present model. Suppose the local mean molecular field, h, seen by the grains in the clusters is large compared to the applied field. The band gap 2 h can then be of the order of 100 Oe. To get a constricted loop for such a rock a field of 100 Oe or more has to be applied. But it is a fact that rocks contain other individual grains which show hysteresis in fields of this order and hence mask the hysteresis behaviour of the clusters. Such a situation manifests experimentally as a line-to-loop transition; a line initially when the field is small and a loop when the equilibrium is disturbed by using a large enough field. Thus the constricted loops of such rocks would automatically be prevented from being observed. 5. EFFECTS OF TEMPERATURE Let us consider the configuration where the band gap corresponds to the magnitude of the applied field whereby the constricted loop is seen. When the temperature of such a rock is increased, the molecular field, h, decreases$ and hence the bands get closer or even merge when the constriction in the loop disappears. As more representative points are getting closer to the diagonal Oa, the susceptibility as well as the amount of hysteresis shown by the rock- increase. If, on the other hand, the temperature of the rock is decreased, h increases and when the band gap becomes large compared to th,; applied field again the constriction in the loop disappears. However, the susceptibility decreases on account of the representative points moving away from Oa. The experimental observation 3 precisely correspond to such changes. On the basis of the present model the " mean molecular field " seems to be rapidly varying with temperature which implies that the room temperature itself is quite close to the "critical point " region where the molecular field would vanish, 8 Thus a small variation in temperature is able to bring

8 C. Radhakrishnamurty Proc. Ind. Acad. Sci., A, Vol. LXXII, Pl. V and N. P. Sastry FIG, 1. Types of low field hysteresis loops shown by basalts. Scale: X-axis 1 small division = 0.5 Oe; Y-axis 1 small division = 17 x 10-3 e.m.u.

9 C. Radhakrishnamurty Proc. Ind. Acad. Sci., A, Vol. LXXII, Pl. VI and N. P. Sastry FIG. 2. a and b constricted loop to loop transition shown by a basalt specimen. c and d line-to-loop transition shown by a specimen. Scale same as for Fig. 1

10 Loops of Some Basaltic Rocks 101 about dramatic changes in the shapes of the observed hysteresis loops of the rocks. In the model proposed by Neel 9 the effects of temperature on the interaction mechanism of the grains were not clearly brought out. 6. SOME ORDER OF MAGNITUDE ESTIMATES The experimental observation is that when constriction disappears due to a change of temperature (of the order of ) the remanence acquired is about 15 x 10-3 e.m.u. We can estimate (a) the range of volumes of single domain particles and (b) their concentration from these figures, if we assume that the observed behaviour is in fact due to single domain grains. (a) The relation v = kt/2hcj defines a blocking volume' above which the relaxation times are too long and below which too short to matter experimentally. If we assume that h, and J are the same for all the single domain grains in the specimen (that is, identical composition assumed), we get dv_dt 60 _1 N -- v T and for the corresponding diameters dd ldv 1 D 3 v - 15' that is 7% variation about the mean diameter. The actual size variations may be as high as even 100%, due to the spread in h e and J values corresponding to a range of composition for the magnetic constituents to be found in the same specimen of rock. For magnetite, Neel' estimated the blocking diameter to be about 640 A at room temperature. (b) Assuming the composition to be magnetite, we have J (300 K) Js (0 K) = 500 Oe, and the remanence 15 X 10-3 e.m.u. = Nv J, where N is the number of grains with an average volume v in the specimen, which alone contribute to hysteresis. Thus Nv 3 X The magnetic constituents form I % of the rock volume 8 c.c. Thus the ratio of the volume contributing to hysteresis to the total volume of the magnetic grains is 3x10-5/8X10-2 N 04%. Since J/K N 1/1000 Oe and the applied field H = 10 Oe, the number of grains oriented such that their coercive force is less than the applied

11 102 C. RADHAKRISHNAMURTY AND N. P. SASTRY field is nearly (J/K) H = 3%, which, is nearly a hundred times more than what is required if we assume all of the magnetic constituents to be single domain grains. It is enough to assume, therefore, that only about 1% of the total volume is in the form of single domain grains to explain the observed facts. (According to Neel's calculation, the multidomain grains would give too low a value of B H e/a and would therefore not contribute to low-field hysteresis normally.) ACKNOWLEDGEMENTS We record our thanks to Professor D. Lal for his keen interest in this work and to Professor L. Neel for the valuable comments he made on an earlier version of the manuscript. REFERENCES 1. Neel, L. 2. Likhite, S. D. and Radhakrishnamurty, C. 3. Radhakrishnamurty, C. and Sahasrabudhe, P. W. 4. Brissoneau, P. 5. Stacey, F. D. 6. Parry, L. G. 7. Preisach, F. 8. Evdokimov, V. B. 9. Neel, L. Add,. Phys., 1955, 4, 191. Curr. Sci., 1966, 35, 534. Ibid., 1967, 36, 251. J. Phys. Cheer. Solids, 1958, 7, 22. Earth Planet Sci. Letters, 1967, 2, 67. Phil. Mag., 1965, 11, 303. Z. Phys., 1935, 94, 277. Russ. J. Phys. Chem., 1963, 37, 1018, 1153 ; Ibid., 1964, 38, Comptes Rendus, 1970 (In press) Printed at The Bangalore Press, Bangalore-18, by V. J. F. Jesudason, L.P.T., Superintendent. Published by, B. S. Venkatachar, Editor, 'Proceedings of the Indian Academy of Sciences", Bangalore

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