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Field-dependence of AC susceptibility in titanomagnetites
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Earth and Planetary Science Letters 157 (1998) 129–139
Field-dependence of AC susceptibility in titanomagnetites Mike Jackson a,Ł , Bruce Moskowitz a , Joe Rosenbaum b , Catherine Kissel c a
Department of Geology and Geophysics and Institute for Rock Magnetism, University of Minnesota, Minneapolis, MN 55455, USA b US Geological Survey Denver, Denver, CO 80225, USA c Centre des Faibles Radioactivite ´ s, Laboratoire Mixte CEA=CNRS, Gif-sur-Yvette, France Received 17 July 1997; revised version received 5 February 1998; accepted 11 February 1998
Abstract AC susceptibility measurements as a function of field amplitude Hac and of frequency show a strong field dependence for a set of synthetic titanomagnetites (Fe3 x Tix O4 ) and for certain basalts from the SOH-1 Hawaiian drill hole and from Iceland. In-phase susceptibility is constant below fields of about 10–100 A=m, and then increases by as much as a factor of two as Hac is increased to 2000 A=m. Both the initial field-independent susceptibilities and the field-dependence of susceptibility are systematically related to composition: initial susceptibility is 3 SI for a single-crystal sphere of TM0 .x D 0/ and decreases with increasing titanium content; field-dependence is nearly zero for TM0 and increases systematically to a maximum near TM60 .x D 0:6/. This field dependence can in some cases be mistaken for frequency dependence, and lead to incorrect interpretations of magnetic grain size and composition when titanomagnetite is present. 1998 Elsevier Science B.V. All rights reserved. Keywords: titanomagnetite; magnetic susceptibility; magnetic minerals
1. Introduction Low-field susceptibility is among the most commonly measured magnetic properties of geologic materials, with applications in a wide range of research areas, including paleoclimate [1–3], petrofabric and deformation [4,5], and crustal composition and thermal structure [6–8]. Despite its familiarity, however, susceptibility is a rather recondite property, depending in nontrivial ways upon measurement parameters such as frequency [9,10] and field amplitude [11,12] as well as upon material properties such as grain size and composition. The dependence of susceptibility on these measurement parameters Ł Corresponding
author. Fax: C1 612 625 7502; E-mail: [email protected]
can, if not properly taken into account, lead to erroneous geological interpretations; here we focus on the field-dependence of AC susceptibility. Magnetization of multidomain (MD) materials in low fields (less than the coercivity) is often described by the empirical law first formulated by Rayleigh in 1887: M D k H C ÞH 2
(1)
where k is the initial susceptibility and α is the Rayleigh coefficient [13]. In applied fields much less than k=Þ the quadratic term is negligible, and dM=dH has a constant (field-independent) value. Above this threshold, dM=dH increases rapidly, and therefore so does apparent AC susceptibility. The measured susceptibility of strongly magnetic multidomain materials is determined by intrinsic sus-
0012-821X/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 1 2 - 8 2 1 X ( 9 8 ) 0 0 0 3 2 - 6
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ceptibility and by self-demagnetization according to 1=ko D 1=ki C N
(2)
where subscripts o and i refer to observed and intrinsic susceptibilities and N is the demagnetizing factor. In the limit ki ! 1, ko approaches a constant value of 1=N . This ‘self-demagnetization limit’ applies to most multidomain ferro- and ferrimagnetic materials. In the other limiting case, as ki ! 0, ko becomes very nearly equal to ki . For pure magnetite, measured AC susceptibility is to a good approximation field-independent up to at least 80 A=m .¼0 H ³ 100 µT), although Smith and Banerjee [14] observed decreasing DC susceptibility with increasing applied fields (to 80 A=m). Worm and others [12] have recently discovered that coarse-grained ferrimagnetic pyrrhotite has a strongly field-dependent AC susceptibility in applied fields larger than about 8 A=m, increasing by about 50% per decade of field amplitude. They conclude that the dependence is not well described by the Rayleigh law, but instead increases according to H 1=4 in fields larger than the threshold. Pyrrhotite also differs from magnetite in having a susceptibility significantly lower than the self-demagnetization limit, even for large multidomain particles. According to Worm et al. [12], the specific initial susceptibility of pyrrhotite (0.3 SI) is about a factor of ten lower than that of magnetite. In this paper we report new results for titanomagnetites, showing that they too exhibit a strong fielddependence of AC susceptibility. This has relevance in anomaly modeling, paleoclimate studies, magnetic anisotropy, and other areas of geological investigation using magnetic susceptibility measurements.
2. Materials and methods 2.1. Sample descriptions AC susceptibility measurements were made on three sets of specimens: synthetic titanomagnetites (Fe3 x Tix O4 ) with compositions ranging from TM0 to TM78 .0 x 0:78/; natural basaltic intrusive samples from the SOH-1 Hawaiian drill hole; and specimens from a basaltic dike from Iceland (the same samples were used in the study described in the
accompanying paper on low-temperature magnetic properties [15]). The synthetic samples comprise two sets: single-crystal spheres approximately 1–2 mm in diameter (TM0, TM05, TM28, TM41, TM55, and TM78) and irregularly-shaped polycrystals (TM40, TM60). Previous studies on the Hawaiian basalts included susceptibility measurements made on a Sapphire Instruments model SI2, using two different frequencies with respective applied AC fields of approximately 90 and 300 A=m. Significant differences in the resulting susceptibility values were originally interpreted as a strong frequency dependence [16], but are here shown to be produced instead by field dependence. Susceptibility measurements at elevated temperature indicate a single magnetic phase with a Curie temperature of 200–250ºC, corresponding to an approximate composition of TM50–TM60. Thermomagnetic behavior and inferred composition of the Icelandic samples vary with position across the dike. Samples from the margins have relatively high Curie temperatures (500–550ºC) corresponding to compositions near TM10; those from the dike interiors have significantly lower Tc , in the range 200º to 300ºC, corresponding to (unoxidized) compositions near TM50. 2.2. Complex susceptibility measurements For a more detailed study of the field- and frequency-dependent susceptibility of the basalts and synthetics, measurements were made on a LakeShore Model 7130 AC susceptometer. In a sinusoidallyvarying applied field H .t/, the magnetic response M.t/ is determined by factors including field-dependence M.H /, time- or frequency-dependence (i.e., viscosity), and electromagnetic induction (in sufficiently conductive material). If M.H / is a nonlinear function, M.t/ is not a pure sine function but also contains higher (primarily odd) harmonics. When M.H / is irreversible (i.e., with hysteresis), there is a phase lag between M.t/ and H .t/, and M.t/ can be resolved into in-phase and out-of-phase (a.k.a. quadrature or imaginary) components. Both in-phase .k 0 / and quadrature .k 00 / susceptibilities were measured at two frequencies (625 Hz and 6000 Hz), with AC field amplitudes between 0.1 A=m and 2000 A=m (at 6000 Hz the maximum
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attainable field was 100 A=m). The first harmonic was measured for all samples, and the third harmonic for selected samples. Low-field hysteresis was also measured for selected samples using a Princeton MicroMag VSM, starting from an AC demagnetized state and using peak fields increasing from 80 to 2000 A=m.
3. Results 3.1. AC susceptibility For the synthetic titanomagnetites, AC susceptibility varies systematically with composition (Fig. 1). In applied fields below about 10 A=m, susceptibilities of the single-crystal spheres are independent of H , and they diminish systematically with increasing Ti content (Fig. 1a). The TM0 sample has a susceptibility very close to the self-demagnetization limit (for a sphere, N D 1=3 and k D 3 SI). For TM05 and the more titaniferous compositions, the initial susceptibility .H < 10 A=m) is significantly lower than 1=N . The polycrystalline samples (Fig. 1b) are somewhat irregular in shape, and susceptibilities (along their long dimensions) are significantly larger than those of the single-crystal spheres. The field-independent susceptibilities give way to strongly field-dependent values at threshold fields between 10 and 100 A=m; the threshold field magnitude is not sharply defined and does not appear to vary systematically with composition. The increase in susceptibility at higher fields is strongly dependent on composition, with a maximum field dependence in the range 0:4 x 0:6. For these compositions susceptibility nearly doubles between 100 and 2000 A=m (Fig. 1). The single-crystal spheres show inflected k(H) curves, with a sharp increase followed by a flattening and a slow approach to the self demagnetization limit of 3 SI (Fig. 1a). None of the single-crystal spheres reaches a susceptibility exceeding 3.05 SI. The polycrystalline samples do not show such a flattening; the susceptibility increases in a qualitatively Rayleigh-like way, with AC susceptibilities reaching as high as 8 SI. There is no detectable frequency dependence of either susceptibility or its field-dependence for these
Fig. 1. In-phase susceptibility, k 0 (625 Hz), as a function of AC field amplitude H ac for synthetic titanomagnetites: (a) singlecrystal spheres; (b) irregular polycrystals.
samples. The quadrature susceptibilities are generally negligible below about 100 A=m, rising in higher fields to about 10% of the in-phase components. Third harmonics were smaller than first harmonics by at least 3 orders of magnitude and noisy enough that no clear pattern was visible as a function of Hac . The susceptibilities of powdered rock samples of Hawaiian and Icelandic intrusions show field dependence similar to that of the synthetic samples (Fig. 2). Susceptibility in AC fields below about 80 A=m is both field- and frequency-independent (dual frequency data are shown in Fig. 2a). For the low Curie-temperature samples (i.e., those with high Ti-content), susceptibility above the threshold field increases linearly with log.Hac/, approximately doubling between 80 and 2000 A=m for both the
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that remained after AF demagnetization appears to be progressively reset during the sequence of minor loops; this accounts for the slight lack of rotational symmetry in Fig. 3a. Direct comparison of the low-field hysteresis measurements and the AC susceptibility data can be made by transforming the hysteresis data to the frequency domain. Although low-field hysteresis can be considered as low-frequency (a few mHz) AC behavior, the applied field H .t/ is non-sinusoidal (more closely approximating a triangular wave, by virtue of the nearly uniform sweep rate). The measured M.H / values are therefore first mapped into the time domain through an assumed h.t/ function .h.t/ D H0 cos.!t// to produce a magnetization waveform (e.g., [13]) M.t/ D M.h.t//
Fig. 2. (a) In-phase (k 0 ) and quadrature .k 00 / susceptibilities as functions of Hac for a powdered basalt sample from Hawaiian drill hole SOH-1; solid symbols indicate measurement at 625 Hz, open at 6000 Hz; (b) in-phase susceptibilities as functions of Hac for four powdered Icelandic basalt samples; solid squares indicate samples from the dike interior; open circles represent chilled margin samples.
Hawaiian sample and the dike interior samples from Iceland. 3.2. Low-field hysteresis Low-field hysteresis measurements on the Hawaiian sample show irreversible behavior in fields well below 1000 A=m (Fig. 3a). A sequence of loops with peak fields increasing in 100 A=m increments show increasing openness or irreversibility and increasing average slope. The magnetizing loop segments (in which the field is increasing) tend to superimpose over one another, whereas the relaxing segments (decreasing fields) tend to rotate out progressively in a counterclockwise sense; this resembles the modeled Rayleigh behavior in Fig. 3b (with k D 3 SI and Þ D 2 ð 10 3 m=A). A small initial remanence
(3)
and then into the frequency domain by Fourier transformation, yielding in-phase and out-of-phase susceptibilities for the fundamental and harmonic frequencies. The in-phase AC susceptibilities derived in this way from the low-field hysteresis measurements are generally very similar to those measured directly with the AC susceptometer. Low-field hysteresis measurements are not feasible for peak fields lower than about 80 A=m, so the comparison is restricted to the field-dependent region between 80 and 2000 A=m. Both data sets show an increase in k 0 by roughly a factor of two over this interval, for a polycrystalline TM60 measured along the long dimension of the specimen (Fig. 4). The values derived from the low-field loops are systematically lower than those measured with the susceptometer, by roughly 10%. The cause of this difference is not immediately apparent; instrumental calibrations do not differ by more than a few percent.
4. Discussion 4.1. Initial susceptibility of titanomagnetites In the low-field regime, susceptibility decreases systematically with increasing titanium content, in broad agreement with previous experimental results [17,18] (Fig. 5). For the x D 0 endmember, suscepti-
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Fig. 3. (a) Low-field hysteresis loops for the Hawaiian basalt sample; (b) model loops for a material following the Rayleigh law with k D 3 SI and a quadratic coefficient of 2 ð 10 3 m=A.
bility is essentially identical to the self-demagnetization limit for a sphere, 3 SI. The moderate decrease in measured susceptibility with increasing x implies a much more rapid decrease in intrinsic susceptibility. Intrinsic susceptibilities for the single-crystal spheres can be calculated from the measured values and from the appropriate value of N (Table 1), for those samples with ko less than the self-demagnetization limit .x ½ 0:05/. Intrinsic initial susceptibility in large multidomain particles is controlled primarily by reversible domain wall displacements e.g., [19]. In the absence of an
applied field, domain walls occupy local energy minimum positions determined by the spatial distribution of crystal defects. An externally applied field lowers the magnetostatic energy of domains magnetized parallel to the field, and raises that of antiparallel domains; total energy is reduced when domain walls move so as to enlarge the favorably-oriented domains at the expense of the unfavorably-oriented ones. The equilibrium wall displacements, in the absence of self-demagnetization, are proportional to the domain (saturation) magnetization Ms . The induced magnetization is proportional to the displacement and to
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M. Jackson et al. / Earth and Planetary Science Letters 157 (1998) 129–139 Table 1 Initial and intrinsic susceptibilities, single-crystal spheres x
k0 , SI
ki , SI
0 0.05 0.28 0.41 0.55 0.78
3.02 2.86 2.60 2.24 1.92 0.05
61.27 19.50 8.78 5.36 0.05
Fig. 4. In-phase susceptibilities as functions of Hac for a polycrystalline TM60 specimen: circles represent values measured directly with an AC susceptometer; squares indicate values derived from low-field hysteresis measurements, as explained in the text.
Fig. 6. Comparison of calculated intrinsic susceptibility (solid circles) and squared saturation magnetization (open squares) for the spherical titanomagnetites, showing an approximate proportionality.
quence of the variation with x in the magnetocrystalline anisotropy constant K 1 and magnetostriction coefficients [20]. Wall motions are impeded by interactions with defects, and in general, ki .x; T / / Ms2 .x; T /=.K 1a .x; T /½b .x; T // Fig. 5. Dependence of in-phase AC susceptibility on composition for synthetic titanomagnetites: solid triangles: mm-sized spheres (this study); open circles: fine-grained (¾0.1 µm) ball-milled material [17]; open squares: ¾30 µm material [18].
Ms , and therefore intrinsic susceptibility is expected to vary in proportion with Ms2 .x; T / e.g., [19]. As shown in Fig. 6, the variation of room-temperature Ms with x appears, to first order, to account for the dependence of intrinsic initial susceptibility on x. Because the samples were not systematically oriented with respect to their crystallographic axes during these measurements, the variation in ki .x/=Ms2 .x/ that can be discerned in Fig. 6 may be due in part to orientation differences. However, some variation in this ratio is expected as a conse-
(4)
where the constants a and b depend on domain state and on pinning mechanism. Both ½111 and K 1 have (absolute value) maxima near x D 0:2 at room temperature [21,22], which should (and apparently does) depress the ratio ki =Ms2 for compositions near TM20. ½100 and ½s both increase continuously with increasing x, and the stress-sensitivity of ki therefore also becomes more significant for more titaniferous compositions. 4.2. Irreversibility in low fields Susceptibility of geological materials is commonly measured in alternating fields of a few hundred A=m, and the assumption is implicit that Rayleigh-like irreversible behavior is negligible in
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such fields. Although this assumption appears to be valid for pure magnetite, our results show that it is not so for titanomagnetites, particularly in the composition range 0:4 x 0:6. Both the fielddependent AC susceptibility and low-field hysteresis measurements show that irreversible magnetization changes begin in fields below 100 A=m. The micromagnetic processes responsible for the low-field hysteresis and field-dependent susceptibility of titanomagnetite must involve irreversible domain wall translations, with walls crossing energy barriers to new local energy minimum positions. The wall displacement-energy function E w .d/ depends on the details of the defect distribution and on pinning mechanisms, but a qualitative understanding of the observed field-dependence can be obtained through a simple model in which E w .d/ varies sinusoidally (similar to the TRM=pTRM models of Schmidt [23] and Dunlop and Xu [24] with repeated identical energy barriers). The other relevant energies are the magnetostatic self-energy E m and the magnetostatic energy associated with the external field E h e.g., [25]. Imagine a particle divided by planar domain walls, and a field applied parallel to the walls and to the magnetization in the domains. Domain walls are displaced independently of one another and they do not interact; we also assume rigid walls that do not bow. E h .d/ decreases linearly with displacement d, and the gradient is proportional to Ms . This energy gradient, as described previously, is equivalent to a force acting on the walls, causing them to move and expand the volume of the favorably-oriented domains. Opposing this force are: first, the gradient in E m .d/, which pushes the walls in the opposite direction (to reduce the net magnetic moment for the particle); and second, the gradients in E w .d/, which push the walls in the direction of the nearest local energy minimum. E m .d/ varies in proportion with N Ms2 d 2 e.g., [25], so the magnetostatic restoring force becomes rapidly larger with increasing displacement, and it is also much larger for low x, due to the strong dependence on Ms . When E w .d/ is small enough to be neglected, the balance of the other two energy terms results in susceptibility equal to 1=N , i.e., the self-demagnetization limit. With increasing x, both magnetostatic terms are reduced, due to their depen-
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dence on Ms , and the wall energy profile assumes a greater importance, resulting in both depression of susceptibility to less than 1=N and irreversible wall displacements in small fields. E w .d/ in general depends upon various material constants (as well as the spatial distribution and types of crystal defects), including Ms (e.g., pinning of walls to voids by reduction of magnetostatic energy of the void), magnetocrystalline anisotropy constants (e.g., diminishing wall area and wall energy by pinning to voids), and magnetostriction constants (through interaction of walls and microstresses). Thus, E w .d/ may vary considerably with composition, due to the variation in these material constants with x, but it is not possible to specify an exact dependence without detailed knowledge of the defect distribution. For the present qualitative model, we hold the amplitude and wavelength of E w .d/ constant for all x. The sum of the three energy terms as functions of x and d, shown in Fig. 7, provide a basis for qualitative explanation of the low-field behavior of the titanomagnetite spheres. With an applied field (Fig. 7a), the energy variation at small displacements is dominated by E m for low-Ti (high Ms ) compositions, and by E w for large x. In TM0, walls move easily across the E w maxima and minima, driven by the large gradient in E h , until at sufficiently large displacements the self-energy becomes large enough to stop the advance of the walls; self-demagnetization is thus the limit for TM0. In more titaniferous compositions with lower Ms , the slope of the field energy is less, and the fluctuations due to E w are more significant, stopping the walls in local energy minima far short of their equilibrium (absolute energy minimum) displacement; for these compositions susceptibility is significantly less than 1=N . When the applied field is removed, only the wall and self-energies remain (Fig. 7b), tending to push the walls back toward their initial locations. For TM0, the .Ms d/2 dependence of E m creates a steep parabolic energy well at d D 0, and the fluctuations due to E w are too small to prevent the wall from returning all the way. The reduction in Ms with increasing x results in shallower energy parabolas, and E w fluctuations are sufficient to impede the domain walls and prevent their return to their initial locations. Thus for those compositions, M.H D 0/ > 0, accounting for the low-field hysteresis and the large quadrature
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Fig. 7. A simple model of energy variation with wall displacement, including a parabolic magnetostatic self energy and a sinusoidal wall-defect interaction energy. (a) also includes a linear term due to an external applied field, with dots indicating metastable equilibrium (local energy minimum) wall displacements; (b) represents the zero-field case, with dots indicating wall displacements after removal of the field in (a).
susceptibilities in fields larger than the irreversibility threshold. The initial field-independent susceptibilities below this threshold represent reversible domain wall displacement within a single E w well. For the single-crystal spheres, the measured k 0 .H / curves were inflected, rising rapidly from the initial field-independent value and then flattening and slowly approaching 1=N (i.e., 3 SI). In the fields attainable, none reached a susceptibility significantly exceeding 1=N . With increasing fields, the energy gradients shown in Fig. 7a become proportionally steeper, pushing the domain walls more effectively over the local energy maxima towards the equilibrium displacement. All the synthetic spheres except TM78 have susceptibilities very close to 3 SI in alternating fields of 2000 A=m. For the nonspherical specimens (TM40 and TM60), measured along their long dimensions, N is significantly less than 1=3. The steepness of the self-energy parabola varies in proportion with N , so that for the elongate specimens, equilibrium wall displacements are much larger, and wall energy fluctuations are comparatively more significant than for the spheres. For these reasons, equilibrium displacements (self-demagnetization limits) are not attained for the nonspherical samples, and k 0 .H / increases continuously in the available fields. Similarly, the powdered basalts contain randomly-oriented particles of various shapes, and for elongate particles aligned with the applied field, k 0 increases continuously to the maximum available field.
Low-field hysteresis properties of basalts have previously been investigated in a limited number of studies [26,27]. Bloemendal et al. [26] interpreted Rayleigh-type behavior as a result of magnetic viscosity in ultrafine particles at the superparamagnetic=stable-single-domain (SPM– SSD) boundary, and related it to frequency-dependent and quadrature susceptibility. Although they did not explicitly relate low-field hysteresis and composition, the data in their paper appear to be consistent with compositional rather than grain-size control (i.e., where compositions are noted, those near TM60 have the most irreversibility). For the samples in the present study, superparamagnetism cannot play any role in the observed behavior of the large single-crystal synthetics, and it is also unlikely to play a significant role in the natural basalts, where frequency-dependence of susceptibility is very low (approximately 2% per decade of frequency). Multidomain-state viscosity may still play a role, but the lack of frequency dependence suggests that it is not a major one. 4.3. Comparison with pyrrhotite The field-dependence of AC susceptibility in these titanomagnetites is similar in many respects to that observed by Worm et al. [12] for coarse-grained pyrrhotite, with an initial range of field-independent susceptibility giving way to a strong field dependence at a threshold field between about 10 and 100
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A=m. There are also several significant differences between the pyrrhotite and titanomagnetite results. First, for pyrrhotite, the field dependence is a strong function of grain size, most clearly seen in millimeter-sized crystals and suppressed (or shifted to higher fields) for particles less than about 20 µm. In contrast, the millimeter-sized synthetic titanomagnetites and the powdered basalt behave in very similar ways, even though high-field hysteresis measurements on the basalts yield Mrs =Ms ratios ranging from about 0.1 (for the Icelandic samples) to 0.3 (for the Hawaiian samples), indicating a mean size in the fine PSD range (i.e., not larger than a few µm). Second, electrical eddy current effects are more significant for pyrrhotite due to its greater electrical conductivity. The out-of-phase response for the titanomagnetites is dominantly due to low-field hysteresis, whereas that for pyrrhotite is apparently dominated by the eddy current component at kHz frequencies. As noted for pyrrhotite [12], the AC magnetic response of the titanomagnetites qualitatively resembles Rayleigh behavior, but is not in quantitative accord with the Rayleigh law, except perhaps for a rather narrow range of AC fields. In-phase AC susceptibility for true Rayleigh behavior varies with applied field according to: k 0 D k0 C a Hac
(5)
Both the synthetic polycrystalline samples and the natural basalts have AC susceptibilities that increase more nearly in proportion to the logarithm of Hac (above the threshold of irreversibility), and the synthetic single-crystal spheres yield inflected susceptibility curves. In each case true Rayleigh behavior is apparently limited to AC fields below about 200 A=m; in higher applied fields, AC susceptibility increases much more slowly than predicted by the Rayleigh law. The H 1=4 power-law dependence found previously for pyrrhotite [12] also adequately describes the behavior of these polycrystalline and natural titanomagnetites.
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ological research that use magnetic properties. In environmental magnetism, frequency-dependence of susceptibility has become a standard tool, but it is sometimes measured using separate instruments or separate coils for each frequency, and these commonly also use different field amplitudes. Even in measurements on a single instrument such as the LakeShore, there is some incentive to use larger amplitudes to compensate for the reduced sensitivity at lower frequencies. In either of these situations, field dependence can be easily mistaken for frequency dependence, and incorrect interpretations of grainsize and compositional parameters may result. Similarly, the field-dependence observed in these titanomagnetites closely resembles that reported by Worm et al. [11,12] for pyrrhotite. Intermediate composition titanomagnetites have Curie temperatures similar to those of ferrimagnetic sulfides, and can in some circumstances be mistakenly identified as such. Although field-dependent susceptibility may have seemed a diagnostic characteristic of pyrrhotite, it is now clear that this property is also shared by intermediate-composition TMs. Finally, as pointed out by Moskowitz et al [15], the loss of remanence by pyrrhotite on warming through its low-temperature transition is closely mimicked by low-x titanomagnetites with isotropic points in the same temperature range. The distinction of titanomagnetite from pyrrhotite using solely magnetic measurements remains a difficult problem. Anisotropy of magnetic susceptibility is described mathematically by a tensor [k] relating the vectors M and H, which is valid only for the linear regime. Many popular commercial instruments operate at higher field amplitudes (e.g., Kappabridge models KLY-2 and KLY-3 use 300 A=m), significantly above the linear threshold for titanomagnetites as well as for pyrrhotites, and a proper mathematical treatment requires separate handling of the linear and quadratic terms [28].
4.4. Implications
5. Conclusions
As emphasized by Worm et al. [11,12], the fielddependence of susceptibility in the range of AC fields commonly used for measuring susceptibility has significant implications in all fields of ge-
In-phase initial AC susceptibility k 0 decreases systematically with increasing Ti-content in synthetic millimeter-sized homogeneous titanomagnetite spheres. For TM0 (pure magnetite) the suscep-
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tibility is exactly equal (within experimental error) to the self-demagnetization limit of 3 SI for a sphere. For all Ti-bearing specimens, k 0 < 1=N . Intrinsic susceptibilities ki0 .x/ vary in approximate proportion with Ms2 .x/. Irreversible M.H / behavior begins in AC fields larger than a few tens of A=m, and low-field hysteresis results in significant quadrature susceptibilities as well as a significant field-dependence of in-phase susceptibility. Variations in domain wall energy as a function of wall displacement become more significant with increasing x (and decreasing Ms ), in comparison to magnetostatic field and self-energies. This accounts qualitatively for both the composition-dependence of initial (field-independent) susceptibility and for the field dependence for amplitudes up to 2000 A=m. When titanomagnetites are present, estimates of frequency-dependence may be significantly in error if measurements are made with different field amplitudes. Anisotropy of susceptibility should be determined with amplitudes of no more than a few tens of A=m to assure that M.H / is linear and the standard tensorial mathematics are valid.
Acknowledgements This is IRM contribution #9703. The IRM is supported by grants from the National Science Foundation and the W.M. Keck Foundation. We are pleased to acknowledge the indispensable advice and contributions of Jim Marvin and Subir Banerjee, measurement help from Patrick Jackson, and constructive reviews by Horst Worm, Mark Hudson, Steve Harlan, John Dearing, and an anonymous reviewer. [RV]
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Res. Lett. 13 (1986) 1169–1172. [4] G.J. Borradaile, Magnetic susceptibility, petrofabrics, and strain – a review, Tectonophysics 156 (1988) 1–20. [5] R. Kligfield, W. Lowrie, I.W.D. Dalziel, Magnetic susceptibility as a strain indicator in the Sudbury Basin, Ontario, Tectonophysics 40 (1977) 287–308. [6] C.M. Schlinger, Magnetization of lower crust and interpretation of regional magnetic anomalies: example from Lofoten and Vestera˚len, Norway, J. Geophys. Res. B 90 (1985) 11484–11504. [7] P.B. Toft, S.E. Haggerty, Limiting depth of magnetization in cratonic lithosphere, Geophys. Res. Lett. 15 (1988) 530– 533. [8] P.J. Wasilewski, D.M. Fountain, The Ivrea Zone as a model for the distribution of magnetization in the continental crust, Geophys. Res. Lett. 9 (1982) 333–336. [9] J.A. Dearing, R.J.L. Dann, K. Hay, J.A. Lees, P.J. Loveland, B.A. Maher, K. O’Grady, Frequency dependent susceptibility measurements of environmental materials, Geophys. J. Int. 124 (1996) 228–240. [10] C.E. Mullins, M.S. Tite, Magnetic viscosity, quadrature susceptibility, and frequency dependence of susceptibility in single-domain assemblies of magnetite and maghemite, J. Geophys. Res. 78 (1973) 804–809. [11] H.-U. Worm, Multidomain susceptibility and anomalously strong low field dependence of induced magnetization in pyrrhotite, Phys. Earth Planet. Inter. 69 (1991) 112–118. [12] H.-U. Worm, D. Clark, M.J. Dekkers, Magnetic susceptibility of pyrrhotite: grain size, field and frequency dependence, Geophys. J. Int. 114 (1993) 127–137. [13] S. Chikazumi, S.H. Charap, Physics of Magnetism, Wiley, New York, 1964, 554 pp. [14] G.M. Smith, S.K. Banerjee, The dependence of weak field susceptibility on applied magnetic field, Phys. Earth Planet. Inter. 46 (1987) 71–76. [15] B.M. Moskowitz, M.J. Jackson, C. Kissel, Low-temperature magnetic behavior of titanomagnetites, Earth Planet. Sci. Lett. 157 (1998) 141–149. [16] J.G. Rosenbaum, R.L. Reynolds, T. Hildebrand, Insight into the structure of the east rift zone of Kilauea, Hawaii, from a rock magnetic study of drill core, in: VIIth International Symposium on the Observation of Continental Crust Through Drilling, 1994, pp. 142–145. [17] R. Day, M. Fuller, V.A. Schmidt, Hysteresis properties of titanomagnetites: grain-size and compositional dependence, Phys. Earth Planet. Inter. 13 (1977) 260–266. [18] B.W. Robins, Remanent Magnetization in the Spinel Iron Oxides, Ph.D., University of New South Wales, 1972. [19] F.D. Stacey, S.K. Banerjee, The Physical Principles of Rock Magnetism, Elsevier, Amsterdam, 1974, 195 pp. [20] E. Appel, H.C. Soffel, Domain state of Ti-rich titanomagnetites deduced from domain structure observations and susceptibility measurements, J. Geophys. 56 (1985) 121– 132. [21] Z. Kakol, J. Sabol, J.M. Honig, Magnetic anisotropy of titanomagnetites Fe3 x Tix O4 0 x 0:55, Phys. Rev. B 44 (1991) 2198–2204.
M. Jackson et al. / Earth and Planetary Science Letters 157 (1998) 129–139 [22] Y. Syono, Magnetocrystalline anisotropy and magnetostriction of Fe3 O4 –Fe2 TiO4 series, with special application to rock magnetism, Jpn. J. Geophys. 4 (1965) 71–143. [23] V.A. Schmidt, A multidomain model of thermoremanence, Earth Planet. Sci. Lett. 20 (1973) 440–446. [24] D.J. Dunlop, S. Xu, Theory of partial thermoremanent magnetization in multidomain grains: 1. Repeated identical barriers to wall motion (single microcoercivity), J. Geophys. Res. B 99 (1994) 9005–9023. [25] W. O’Reilly, Rock and Mineral Magnetism, Blackie, Glasgow, 1984, 220 pp. [26] J. Bloemendal, C.E. Barton, C. Radhakrishnamurty, Cor-
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relation between Rayleigh loops and frequency-dependent and quadrature susceptibility: application to magnetic granulometry of rocks, J. Geophys. Res. B 90 (1985) 8789– 8792. [27] C. Radhakrishnamurty, Magnetism and Basalts, Geological Society of India, Bangalore, 1993, 208 pp. [28] H. Markert, A. Lehmann, Three-dimensional Rayleigh hysteresis of oriented core samples from the German Continental Deep Drilling Program: susceptibility tensor, Rayleigh tensor, three-dimensional Rayleigh law, Geophys. J. Int. 127 (1) (1996) 201–214.
Earth and Planetary Science Letters 157 (1998) 129–139
Field-dependence of AC susceptibility in titanomagnetites Mike Jackson a,Ł , Bruce Moskowitz a , Joe Rosenbaum b , Catherine Kissel c a
Department of Geology and Geophysics and Institute for Rock Magnetism, University of Minnesota, Minneapolis, MN 55455, USA b US Geological Survey Denver, Denver, CO 80225, USA c Centre des Faibles Radioactivite ´ s, Laboratoire Mixte CEA=CNRS, Gif-sur-Yvette, France Received 17 July 1997; revised version received 5 February 1998; accepted 11 February 1998
Abstract AC susceptibility measurements as a function of field amplitude Hac and of frequency show a strong field dependence for a set of synthetic titanomagnetites (Fe3 x Tix O4 ) and for certain basalts from the SOH-1 Hawaiian drill hole and from Iceland. In-phase susceptibility is constant below fields of about 10–100 A=m, and then increases by as much as a factor of two as Hac is increased to 2000 A=m. Both the initial field-independent susceptibilities and the field-dependence of susceptibility are systematically related to composition: initial susceptibility is 3 SI for a single-crystal sphere of TM0 .x D 0/ and decreases with increasing titanium content; field-dependence is nearly zero for TM0 and increases systematically to a maximum near TM60 .x D 0:6/. This field dependence can in some cases be mistaken for frequency dependence, and lead to incorrect interpretations of magnetic grain size and composition when titanomagnetite is present. 1998 Elsevier Science B.V. All rights reserved. Keywords: titanomagnetite; magnetic susceptibility; magnetic minerals
1. Introduction Low-field susceptibility is among the most commonly measured magnetic properties of geologic materials, with applications in a wide range of research areas, including paleoclimate [1–3], petrofabric and deformation [4,5], and crustal composition and thermal structure [6–8]. Despite its familiarity, however, susceptibility is a rather recondite property, depending in nontrivial ways upon measurement parameters such as frequency [9,10] and field amplitude [11,12] as well as upon material properties such as grain size and composition. The dependence of susceptibility on these measurement parameters Ł Corresponding
author. Fax: C1 612 625 7502; E-mail: [email protected]
can, if not properly taken into account, lead to erroneous geological interpretations; here we focus on the field-dependence of AC susceptibility. Magnetization of multidomain (MD) materials in low fields (less than the coercivity) is often described by the empirical law first formulated by Rayleigh in 1887: M D k H C ÞH 2
(1)
where k is the initial susceptibility and α is the Rayleigh coefficient [13]. In applied fields much less than k=Þ the quadratic term is negligible, and dM=dH has a constant (field-independent) value. Above this threshold, dM=dH increases rapidly, and therefore so does apparent AC susceptibility. The measured susceptibility of strongly magnetic multidomain materials is determined by intrinsic sus-
0012-821X/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 1 2 - 8 2 1 X ( 9 8 ) 0 0 0 3 2 - 6
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ceptibility and by self-demagnetization according to 1=ko D 1=ki C N
(2)
where subscripts o and i refer to observed and intrinsic susceptibilities and N is the demagnetizing factor. In the limit ki ! 1, ko approaches a constant value of 1=N . This ‘self-demagnetization limit’ applies to most multidomain ferro- and ferrimagnetic materials. In the other limiting case, as ki ! 0, ko becomes very nearly equal to ki . For pure magnetite, measured AC susceptibility is to a good approximation field-independent up to at least 80 A=m .¼0 H ³ 100 µT), although Smith and Banerjee [14] observed decreasing DC susceptibility with increasing applied fields (to 80 A=m). Worm and others [12] have recently discovered that coarse-grained ferrimagnetic pyrrhotite has a strongly field-dependent AC susceptibility in applied fields larger than about 8 A=m, increasing by about 50% per decade of field amplitude. They conclude that the dependence is not well described by the Rayleigh law, but instead increases according to H 1=4 in fields larger than the threshold. Pyrrhotite also differs from magnetite in having a susceptibility significantly lower than the self-demagnetization limit, even for large multidomain particles. According to Worm et al. [12], the specific initial susceptibility of pyrrhotite (0.3 SI) is about a factor of ten lower than that of magnetite. In this paper we report new results for titanomagnetites, showing that they too exhibit a strong fielddependence of AC susceptibility. This has relevance in anomaly modeling, paleoclimate studies, magnetic anisotropy, and other areas of geological investigation using magnetic susceptibility measurements.
2. Materials and methods 2.1. Sample descriptions AC susceptibility measurements were made on three sets of specimens: synthetic titanomagnetites (Fe3 x Tix O4 ) with compositions ranging from TM0 to TM78 .0 x 0:78/; natural basaltic intrusive samples from the SOH-1 Hawaiian drill hole; and specimens from a basaltic dike from Iceland (the same samples were used in the study described in the
accompanying paper on low-temperature magnetic properties [15]). The synthetic samples comprise two sets: single-crystal spheres approximately 1–2 mm in diameter (TM0, TM05, TM28, TM41, TM55, and TM78) and irregularly-shaped polycrystals (TM40, TM60). Previous studies on the Hawaiian basalts included susceptibility measurements made on a Sapphire Instruments model SI2, using two different frequencies with respective applied AC fields of approximately 90 and 300 A=m. Significant differences in the resulting susceptibility values were originally interpreted as a strong frequency dependence [16], but are here shown to be produced instead by field dependence. Susceptibility measurements at elevated temperature indicate a single magnetic phase with a Curie temperature of 200–250ºC, corresponding to an approximate composition of TM50–TM60. Thermomagnetic behavior and inferred composition of the Icelandic samples vary with position across the dike. Samples from the margins have relatively high Curie temperatures (500–550ºC) corresponding to compositions near TM10; those from the dike interiors have significantly lower Tc , in the range 200º to 300ºC, corresponding to (unoxidized) compositions near TM50. 2.2. Complex susceptibility measurements For a more detailed study of the field- and frequency-dependent susceptibility of the basalts and synthetics, measurements were made on a LakeShore Model 7130 AC susceptometer. In a sinusoidallyvarying applied field H .t/, the magnetic response M.t/ is determined by factors including field-dependence M.H /, time- or frequency-dependence (i.e., viscosity), and electromagnetic induction (in sufficiently conductive material). If M.H / is a nonlinear function, M.t/ is not a pure sine function but also contains higher (primarily odd) harmonics. When M.H / is irreversible (i.e., with hysteresis), there is a phase lag between M.t/ and H .t/, and M.t/ can be resolved into in-phase and out-of-phase (a.k.a. quadrature or imaginary) components. Both in-phase .k 0 / and quadrature .k 00 / susceptibilities were measured at two frequencies (625 Hz and 6000 Hz), with AC field amplitudes between 0.1 A=m and 2000 A=m (at 6000 Hz the maximum
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attainable field was 100 A=m). The first harmonic was measured for all samples, and the third harmonic for selected samples. Low-field hysteresis was also measured for selected samples using a Princeton MicroMag VSM, starting from an AC demagnetized state and using peak fields increasing from 80 to 2000 A=m.
3. Results 3.1. AC susceptibility For the synthetic titanomagnetites, AC susceptibility varies systematically with composition (Fig. 1). In applied fields below about 10 A=m, susceptibilities of the single-crystal spheres are independent of H , and they diminish systematically with increasing Ti content (Fig. 1a). The TM0 sample has a susceptibility very close to the self-demagnetization limit (for a sphere, N D 1=3 and k D 3 SI). For TM05 and the more titaniferous compositions, the initial susceptibility .H < 10 A=m) is significantly lower than 1=N . The polycrystalline samples (Fig. 1b) are somewhat irregular in shape, and susceptibilities (along their long dimensions) are significantly larger than those of the single-crystal spheres. The field-independent susceptibilities give way to strongly field-dependent values at threshold fields between 10 and 100 A=m; the threshold field magnitude is not sharply defined and does not appear to vary systematically with composition. The increase in susceptibility at higher fields is strongly dependent on composition, with a maximum field dependence in the range 0:4 x 0:6. For these compositions susceptibility nearly doubles between 100 and 2000 A=m (Fig. 1). The single-crystal spheres show inflected k(H) curves, with a sharp increase followed by a flattening and a slow approach to the self demagnetization limit of 3 SI (Fig. 1a). None of the single-crystal spheres reaches a susceptibility exceeding 3.05 SI. The polycrystalline samples do not show such a flattening; the susceptibility increases in a qualitatively Rayleigh-like way, with AC susceptibilities reaching as high as 8 SI. There is no detectable frequency dependence of either susceptibility or its field-dependence for these
Fig. 1. In-phase susceptibility, k 0 (625 Hz), as a function of AC field amplitude H ac for synthetic titanomagnetites: (a) singlecrystal spheres; (b) irregular polycrystals.
samples. The quadrature susceptibilities are generally negligible below about 100 A=m, rising in higher fields to about 10% of the in-phase components. Third harmonics were smaller than first harmonics by at least 3 orders of magnitude and noisy enough that no clear pattern was visible as a function of Hac . The susceptibilities of powdered rock samples of Hawaiian and Icelandic intrusions show field dependence similar to that of the synthetic samples (Fig. 2). Susceptibility in AC fields below about 80 A=m is both field- and frequency-independent (dual frequency data are shown in Fig. 2a). For the low Curie-temperature samples (i.e., those with high Ti-content), susceptibility above the threshold field increases linearly with log.Hac/, approximately doubling between 80 and 2000 A=m for both the
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that remained after AF demagnetization appears to be progressively reset during the sequence of minor loops; this accounts for the slight lack of rotational symmetry in Fig. 3a. Direct comparison of the low-field hysteresis measurements and the AC susceptibility data can be made by transforming the hysteresis data to the frequency domain. Although low-field hysteresis can be considered as low-frequency (a few mHz) AC behavior, the applied field H .t/ is non-sinusoidal (more closely approximating a triangular wave, by virtue of the nearly uniform sweep rate). The measured M.H / values are therefore first mapped into the time domain through an assumed h.t/ function .h.t/ D H0 cos.!t// to produce a magnetization waveform (e.g., [13]) M.t/ D M.h.t//
Fig. 2. (a) In-phase (k 0 ) and quadrature .k 00 / susceptibilities as functions of Hac for a powdered basalt sample from Hawaiian drill hole SOH-1; solid symbols indicate measurement at 625 Hz, open at 6000 Hz; (b) in-phase susceptibilities as functions of Hac for four powdered Icelandic basalt samples; solid squares indicate samples from the dike interior; open circles represent chilled margin samples.
Hawaiian sample and the dike interior samples from Iceland. 3.2. Low-field hysteresis Low-field hysteresis measurements on the Hawaiian sample show irreversible behavior in fields well below 1000 A=m (Fig. 3a). A sequence of loops with peak fields increasing in 100 A=m increments show increasing openness or irreversibility and increasing average slope. The magnetizing loop segments (in which the field is increasing) tend to superimpose over one another, whereas the relaxing segments (decreasing fields) tend to rotate out progressively in a counterclockwise sense; this resembles the modeled Rayleigh behavior in Fig. 3b (with k D 3 SI and Þ D 2 ð 10 3 m=A). A small initial remanence
(3)
and then into the frequency domain by Fourier transformation, yielding in-phase and out-of-phase susceptibilities for the fundamental and harmonic frequencies. The in-phase AC susceptibilities derived in this way from the low-field hysteresis measurements are generally very similar to those measured directly with the AC susceptometer. Low-field hysteresis measurements are not feasible for peak fields lower than about 80 A=m, so the comparison is restricted to the field-dependent region between 80 and 2000 A=m. Both data sets show an increase in k 0 by roughly a factor of two over this interval, for a polycrystalline TM60 measured along the long dimension of the specimen (Fig. 4). The values derived from the low-field loops are systematically lower than those measured with the susceptometer, by roughly 10%. The cause of this difference is not immediately apparent; instrumental calibrations do not differ by more than a few percent.
4. Discussion 4.1. Initial susceptibility of titanomagnetites In the low-field regime, susceptibility decreases systematically with increasing titanium content, in broad agreement with previous experimental results [17,18] (Fig. 5). For the x D 0 endmember, suscepti-
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Fig. 3. (a) Low-field hysteresis loops for the Hawaiian basalt sample; (b) model loops for a material following the Rayleigh law with k D 3 SI and a quadratic coefficient of 2 ð 10 3 m=A.
bility is essentially identical to the self-demagnetization limit for a sphere, 3 SI. The moderate decrease in measured susceptibility with increasing x implies a much more rapid decrease in intrinsic susceptibility. Intrinsic susceptibilities for the single-crystal spheres can be calculated from the measured values and from the appropriate value of N (Table 1), for those samples with ko less than the self-demagnetization limit .x ½ 0:05/. Intrinsic initial susceptibility in large multidomain particles is controlled primarily by reversible domain wall displacements e.g., [19]. In the absence of an
applied field, domain walls occupy local energy minimum positions determined by the spatial distribution of crystal defects. An externally applied field lowers the magnetostatic energy of domains magnetized parallel to the field, and raises that of antiparallel domains; total energy is reduced when domain walls move so as to enlarge the favorably-oriented domains at the expense of the unfavorably-oriented ones. The equilibrium wall displacements, in the absence of self-demagnetization, are proportional to the domain (saturation) magnetization Ms . The induced magnetization is proportional to the displacement and to
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M. Jackson et al. / Earth and Planetary Science Letters 157 (1998) 129–139 Table 1 Initial and intrinsic susceptibilities, single-crystal spheres x
k0 , SI
ki , SI
0 0.05 0.28 0.41 0.55 0.78
3.02 2.86 2.60 2.24 1.92 0.05
61.27 19.50 8.78 5.36 0.05
Fig. 4. In-phase susceptibilities as functions of Hac for a polycrystalline TM60 specimen: circles represent values measured directly with an AC susceptometer; squares indicate values derived from low-field hysteresis measurements, as explained in the text.
Fig. 6. Comparison of calculated intrinsic susceptibility (solid circles) and squared saturation magnetization (open squares) for the spherical titanomagnetites, showing an approximate proportionality.
quence of the variation with x in the magnetocrystalline anisotropy constant K 1 and magnetostriction coefficients [20]. Wall motions are impeded by interactions with defects, and in general, ki .x; T / / Ms2 .x; T /=.K 1a .x; T /½b .x; T // Fig. 5. Dependence of in-phase AC susceptibility on composition for synthetic titanomagnetites: solid triangles: mm-sized spheres (this study); open circles: fine-grained (¾0.1 µm) ball-milled material [17]; open squares: ¾30 µm material [18].
Ms , and therefore intrinsic susceptibility is expected to vary in proportion with Ms2 .x; T / e.g., [19]. As shown in Fig. 6, the variation of room-temperature Ms with x appears, to first order, to account for the dependence of intrinsic initial susceptibility on x. Because the samples were not systematically oriented with respect to their crystallographic axes during these measurements, the variation in ki .x/=Ms2 .x/ that can be discerned in Fig. 6 may be due in part to orientation differences. However, some variation in this ratio is expected as a conse-
(4)
where the constants a and b depend on domain state and on pinning mechanism. Both ½111 and K 1 have (absolute value) maxima near x D 0:2 at room temperature [21,22], which should (and apparently does) depress the ratio ki =Ms2 for compositions near TM20. ½100 and ½s both increase continuously with increasing x, and the stress-sensitivity of ki therefore also becomes more significant for more titaniferous compositions. 4.2. Irreversibility in low fields Susceptibility of geological materials is commonly measured in alternating fields of a few hundred A=m, and the assumption is implicit that Rayleigh-like irreversible behavior is negligible in
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such fields. Although this assumption appears to be valid for pure magnetite, our results show that it is not so for titanomagnetites, particularly in the composition range 0:4 x 0:6. Both the fielddependent AC susceptibility and low-field hysteresis measurements show that irreversible magnetization changes begin in fields below 100 A=m. The micromagnetic processes responsible for the low-field hysteresis and field-dependent susceptibility of titanomagnetite must involve irreversible domain wall translations, with walls crossing energy barriers to new local energy minimum positions. The wall displacement-energy function E w .d/ depends on the details of the defect distribution and on pinning mechanisms, but a qualitative understanding of the observed field-dependence can be obtained through a simple model in which E w .d/ varies sinusoidally (similar to the TRM=pTRM models of Schmidt [23] and Dunlop and Xu [24] with repeated identical energy barriers). The other relevant energies are the magnetostatic self-energy E m and the magnetostatic energy associated with the external field E h e.g., [25]. Imagine a particle divided by planar domain walls, and a field applied parallel to the walls and to the magnetization in the domains. Domain walls are displaced independently of one another and they do not interact; we also assume rigid walls that do not bow. E h .d/ decreases linearly with displacement d, and the gradient is proportional to Ms . This energy gradient, as described previously, is equivalent to a force acting on the walls, causing them to move and expand the volume of the favorably-oriented domains. Opposing this force are: first, the gradient in E m .d/, which pushes the walls in the opposite direction (to reduce the net magnetic moment for the particle); and second, the gradients in E w .d/, which push the walls in the direction of the nearest local energy minimum. E m .d/ varies in proportion with N Ms2 d 2 e.g., [25], so the magnetostatic restoring force becomes rapidly larger with increasing displacement, and it is also much larger for low x, due to the strong dependence on Ms . When E w .d/ is small enough to be neglected, the balance of the other two energy terms results in susceptibility equal to 1=N , i.e., the self-demagnetization limit. With increasing x, both magnetostatic terms are reduced, due to their depen-
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dence on Ms , and the wall energy profile assumes a greater importance, resulting in both depression of susceptibility to less than 1=N and irreversible wall displacements in small fields. E w .d/ in general depends upon various material constants (as well as the spatial distribution and types of crystal defects), including Ms (e.g., pinning of walls to voids by reduction of magnetostatic energy of the void), magnetocrystalline anisotropy constants (e.g., diminishing wall area and wall energy by pinning to voids), and magnetostriction constants (through interaction of walls and microstresses). Thus, E w .d/ may vary considerably with composition, due to the variation in these material constants with x, but it is not possible to specify an exact dependence without detailed knowledge of the defect distribution. For the present qualitative model, we hold the amplitude and wavelength of E w .d/ constant for all x. The sum of the three energy terms as functions of x and d, shown in Fig. 7, provide a basis for qualitative explanation of the low-field behavior of the titanomagnetite spheres. With an applied field (Fig. 7a), the energy variation at small displacements is dominated by E m for low-Ti (high Ms ) compositions, and by E w for large x. In TM0, walls move easily across the E w maxima and minima, driven by the large gradient in E h , until at sufficiently large displacements the self-energy becomes large enough to stop the advance of the walls; self-demagnetization is thus the limit for TM0. In more titaniferous compositions with lower Ms , the slope of the field energy is less, and the fluctuations due to E w are more significant, stopping the walls in local energy minima far short of their equilibrium (absolute energy minimum) displacement; for these compositions susceptibility is significantly less than 1=N . When the applied field is removed, only the wall and self-energies remain (Fig. 7b), tending to push the walls back toward their initial locations. For TM0, the .Ms d/2 dependence of E m creates a steep parabolic energy well at d D 0, and the fluctuations due to E w are too small to prevent the wall from returning all the way. The reduction in Ms with increasing x results in shallower energy parabolas, and E w fluctuations are sufficient to impede the domain walls and prevent their return to their initial locations. Thus for those compositions, M.H D 0/ > 0, accounting for the low-field hysteresis and the large quadrature
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Fig. 7. A simple model of energy variation with wall displacement, including a parabolic magnetostatic self energy and a sinusoidal wall-defect interaction energy. (a) also includes a linear term due to an external applied field, with dots indicating metastable equilibrium (local energy minimum) wall displacements; (b) represents the zero-field case, with dots indicating wall displacements after removal of the field in (a).
susceptibilities in fields larger than the irreversibility threshold. The initial field-independent susceptibilities below this threshold represent reversible domain wall displacement within a single E w well. For the single-crystal spheres, the measured k 0 .H / curves were inflected, rising rapidly from the initial field-independent value and then flattening and slowly approaching 1=N (i.e., 3 SI). In the fields attainable, none reached a susceptibility significantly exceeding 1=N . With increasing fields, the energy gradients shown in Fig. 7a become proportionally steeper, pushing the domain walls more effectively over the local energy maxima towards the equilibrium displacement. All the synthetic spheres except TM78 have susceptibilities very close to 3 SI in alternating fields of 2000 A=m. For the nonspherical specimens (TM40 and TM60), measured along their long dimensions, N is significantly less than 1=3. The steepness of the self-energy parabola varies in proportion with N , so that for the elongate specimens, equilibrium wall displacements are much larger, and wall energy fluctuations are comparatively more significant than for the spheres. For these reasons, equilibrium displacements (self-demagnetization limits) are not attained for the nonspherical samples, and k 0 .H / increases continuously in the available fields. Similarly, the powdered basalts contain randomly-oriented particles of various shapes, and for elongate particles aligned with the applied field, k 0 increases continuously to the maximum available field.
Low-field hysteresis properties of basalts have previously been investigated in a limited number of studies [26,27]. Bloemendal et al. [26] interpreted Rayleigh-type behavior as a result of magnetic viscosity in ultrafine particles at the superparamagnetic=stable-single-domain (SPM– SSD) boundary, and related it to frequency-dependent and quadrature susceptibility. Although they did not explicitly relate low-field hysteresis and composition, the data in their paper appear to be consistent with compositional rather than grain-size control (i.e., where compositions are noted, those near TM60 have the most irreversibility). For the samples in the present study, superparamagnetism cannot play any role in the observed behavior of the large single-crystal synthetics, and it is also unlikely to play a significant role in the natural basalts, where frequency-dependence of susceptibility is very low (approximately 2% per decade of frequency). Multidomain-state viscosity may still play a role, but the lack of frequency dependence suggests that it is not a major one. 4.3. Comparison with pyrrhotite The field-dependence of AC susceptibility in these titanomagnetites is similar in many respects to that observed by Worm et al. [12] for coarse-grained pyrrhotite, with an initial range of field-independent susceptibility giving way to a strong field dependence at a threshold field between about 10 and 100
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A=m. There are also several significant differences between the pyrrhotite and titanomagnetite results. First, for pyrrhotite, the field dependence is a strong function of grain size, most clearly seen in millimeter-sized crystals and suppressed (or shifted to higher fields) for particles less than about 20 µm. In contrast, the millimeter-sized synthetic titanomagnetites and the powdered basalt behave in very similar ways, even though high-field hysteresis measurements on the basalts yield Mrs =Ms ratios ranging from about 0.1 (for the Icelandic samples) to 0.3 (for the Hawaiian samples), indicating a mean size in the fine PSD range (i.e., not larger than a few µm). Second, electrical eddy current effects are more significant for pyrrhotite due to its greater electrical conductivity. The out-of-phase response for the titanomagnetites is dominantly due to low-field hysteresis, whereas that for pyrrhotite is apparently dominated by the eddy current component at kHz frequencies. As noted for pyrrhotite [12], the AC magnetic response of the titanomagnetites qualitatively resembles Rayleigh behavior, but is not in quantitative accord with the Rayleigh law, except perhaps for a rather narrow range of AC fields. In-phase AC susceptibility for true Rayleigh behavior varies with applied field according to: k 0 D k0 C a Hac
(5)
Both the synthetic polycrystalline samples and the natural basalts have AC susceptibilities that increase more nearly in proportion to the logarithm of Hac (above the threshold of irreversibility), and the synthetic single-crystal spheres yield inflected susceptibility curves. In each case true Rayleigh behavior is apparently limited to AC fields below about 200 A=m; in higher applied fields, AC susceptibility increases much more slowly than predicted by the Rayleigh law. The H 1=4 power-law dependence found previously for pyrrhotite [12] also adequately describes the behavior of these polycrystalline and natural titanomagnetites.
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ological research that use magnetic properties. In environmental magnetism, frequency-dependence of susceptibility has become a standard tool, but it is sometimes measured using separate instruments or separate coils for each frequency, and these commonly also use different field amplitudes. Even in measurements on a single instrument such as the LakeShore, there is some incentive to use larger amplitudes to compensate for the reduced sensitivity at lower frequencies. In either of these situations, field dependence can be easily mistaken for frequency dependence, and incorrect interpretations of grainsize and compositional parameters may result. Similarly, the field-dependence observed in these titanomagnetites closely resembles that reported by Worm et al. [11,12] for pyrrhotite. Intermediate composition titanomagnetites have Curie temperatures similar to those of ferrimagnetic sulfides, and can in some circumstances be mistakenly identified as such. Although field-dependent susceptibility may have seemed a diagnostic characteristic of pyrrhotite, it is now clear that this property is also shared by intermediate-composition TMs. Finally, as pointed out by Moskowitz et al [15], the loss of remanence by pyrrhotite on warming through its low-temperature transition is closely mimicked by low-x titanomagnetites with isotropic points in the same temperature range. The distinction of titanomagnetite from pyrrhotite using solely magnetic measurements remains a difficult problem. Anisotropy of magnetic susceptibility is described mathematically by a tensor [k] relating the vectors M and H, which is valid only for the linear regime. Many popular commercial instruments operate at higher field amplitudes (e.g., Kappabridge models KLY-2 and KLY-3 use 300 A=m), significantly above the linear threshold for titanomagnetites as well as for pyrrhotites, and a proper mathematical treatment requires separate handling of the linear and quadratic terms [28].
4.4. Implications
5. Conclusions
As emphasized by Worm et al. [11,12], the fielddependence of susceptibility in the range of AC fields commonly used for measuring susceptibility has significant implications in all fields of ge-
In-phase initial AC susceptibility k 0 decreases systematically with increasing Ti-content in synthetic millimeter-sized homogeneous titanomagnetite spheres. For TM0 (pure magnetite) the suscep-
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tibility is exactly equal (within experimental error) to the self-demagnetization limit of 3 SI for a sphere. For all Ti-bearing specimens, k 0 < 1=N . Intrinsic susceptibilities ki0 .x/ vary in approximate proportion with Ms2 .x/. Irreversible M.H / behavior begins in AC fields larger than a few tens of A=m, and low-field hysteresis results in significant quadrature susceptibilities as well as a significant field-dependence of in-phase susceptibility. Variations in domain wall energy as a function of wall displacement become more significant with increasing x (and decreasing Ms ), in comparison to magnetostatic field and self-energies. This accounts qualitatively for both the composition-dependence of initial (field-independent) susceptibility and for the field dependence for amplitudes up to 2000 A=m. When titanomagnetites are present, estimates of frequency-dependence may be significantly in error if measurements are made with different field amplitudes. Anisotropy of susceptibility should be determined with amplitudes of no more than a few tens of A=m to assure that M.H / is linear and the standard tensorial mathematics are valid.
Acknowledgements This is IRM contribution #9703. The IRM is supported by grants from the National Science Foundation and the W.M. Keck Foundation. We are pleased to acknowledge the indispensable advice and contributions of Jim Marvin and Subir Banerjee, measurement help from Patrick Jackson, and constructive reviews by Horst Worm, Mark Hudson, Steve Harlan, John Dearing, and an anonymous reviewer. [RV]
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