Multidomain and single-domain relations between susceptibility and coercive force

Physics of the Earth and Planetary Interiors, 49 (1987) 181—191 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

181

Multidomain and single-domain relations between susceptibility and coercive force David J. Dunlop, Ozden Ozdemir and Randolph J. Enkin

*

Geophysics Laboratory, Department of Physics, University of Toronto, Toronto, M5S JA 7 (Canada) (Received November 21, 1986; revision accepted January 26, 1987)

Dunlop, D.J., Ozdemir, O. and Enkin, R.J., 1987. Multidomain and single-domain relations between susceptibility and coercive force. Phys. Earth Planet. Inter., 49: 181—191. Theoretically, multidomain (MD) initial susceptibility (X~)jmtand coercive force H~are related by l/(X~)jmt= k’H~/j~ + N, in which j, is spontaneous magnetization J, normalized to its room-temperature value J,~, N is a demagnetizing factor, and . k is a sample-dependent constant of the order of 45 emu cm~ (4.5 X 1O’~A rn~) for magnetite. The corresponding single-domain (SD) relation is l/(X~)imt= k ~ with k = 0.319J,~(for magnetite 150 emu cm _3). The slopes and intercepts of linear plots of 1/(Xo ) ~ and H~/j,data measured at different temperatures should thus reveal domain structure and give a value of N in the MD case. We have tested the equations using data for SD and small MD magnetites, maghemites and titanomaghemites. In most cases the relations were linear, but many of the slopes and intercepts were at variance with theory. Some of the discrepancy between theory and experiment may result from mixtures of SD and MD material and non-uniaxial amsotropies, but most is due to thermal fluctuations. We develop a generalized method of thermal fluctuation analysis (TFA) to deal with such effects in the thermal excitation range below the unbiocking temperature.

1. Infroduction The ferromagnetic domain structure responsi-

grains, x~is the initial value of observed susceptibility Xo’ but for MD grains, it is the initial intrinsic susceptibility x1~ which is not directly

ble for stable remanent magnetization in rocks is still hotly debated. Initial observed susceptibility, (Xo)~t, and coercive force, H~,are two easily measured properties that reflect domain structure. Single-domain (SD) grains generally have low (Xo)mit and high H~,while multidomain (MD) grains have high (X~)~t and low H~. Theory (Stoner and Wohlfarth, 1948; Stacey, 1963) predicts

measurable. Most tests of (1) (e.g. Kittel, 1949; Dunlop, 1974; Ozdemir and O’Reilly, 1981a) have compared room-temperature data from grains of varying sizes or compositions. Hodych (1986) has proposed using instead data determined for a single sample at different temperatures. The present paper tests his procedure, using high-temperature susceptibility and coercive force measurements for SD and small MD magnetites and

ximtHc kj~ (1) where k is a constant dependent on domain structure and j~ is spontaneous magnetization J~nor-

titanomagbemites.

=

malized to its room temperature value J0. For SD *

Now at Institut de Physique du Globe, Université de Paris 7, Place Jussieu, 75252 Paris, France.

0031-9201/87/$03.50

© 1987 Elsevier Science Publishers B.V.

2. Theory ~

.

.

Singie-domain susceptibility and coercive force Assuming uniaxial shape anisotropy and coherent rotation of SD magnetization, Stoner and

182

Wohlfarth

(1948) find

H~= 0.479~NJ~and

(xo)~~,12/3~N, ~N

susceptibilities, for example at point B on the

= being the difference between demagnetizing factors in the easy and hard directions. Combining equations and making ternperature dependences explicit

descending major ioop where H0 = H~= —He. Susceptibility is not often measured under these conditions, but (Xo ) DL can be approximated using standard hysteresis loop parameters. If the seg-

(Xo),mt(T) = O.319J~(T)/H~(T) (2) It is also useful to write the inverse equation

ment CB between J~(the and —H~ is nearly linear,saturation we haveremanence) (Xo)DL ~ Thus

1/(X0)jmt(T)

Hc/Jrs

=

(0.319J~0)’H~(T)/j~(T)

(3)

Equations 2 and 3 are valid for umaxial anisotropies of crystalline and magnetoelastic as well as shape origin. Note that k = 0.319J~0for uniaxial anisotropy; k will have a different value for cubic or other multiaxial anisotropies.

1/(xI)DL

=

+

N

(6)

where (x1 ) DL is the intrinsic susceptibility of the descending ioop at B’, given approximately by the slope of C’B’. Similarly, if the segment BA of the descending major ioop is nearly linear, then (Xo)DL JD/(Hcr Hc) and (Her Hc)/JD = 1/(x~)DL+ N (7) x1 now being the slope 0J/8H~of the corresponding intrinsic loop segment B’A’. —

2.2. Multidomain susceptibility

—

The intrinsic susceptibility of MD grains, X~= (0J/8H~)~, is the response of the magnetization J to the internal field H~.It is deduced from the observed susceptibility Xo = (aJ/aH0)~, the response to an externally applied field H0, as follows. Since the average internal field in an MD grain can be approximated as H, = H0 NJ, where N is the demagnetizing factor —

1

—

0H1

~~aJaJ

8H0

N=

-~--

—N

(4)

Xo

Equation 4 expresses in differential form Néel’s (1955) procedure for ‘shearing’ the external-field curve to obtain the corresponding internal-field curve. It can be applied to any part of the magnetization curve. Consider, for example, the measured curve of a dispersion of magnetite crystals with mean size 0.22 zm (Fig. 1, this issue, enlarged from fig. 3 of Dunlop, 1984). To give a picture of what the internal-field curve looks like, a demagnetizing factor N = 2 (Ns1 = 0.16) has been assumed and the measured curve ‘sheared’ accordingly. In the initial weak-field region of either curve, eq. 4 predicts 1/(Xo)imt = 1/(Xi)imt + N (5) (Xo)imt and (Xj)jmt being the slopes of tangents OE, OE’ to the measured and intrinsic initial magnetization curves, respectively, Equation 4 applies equally well to strong-field

2.2. Multidomain demagnetizing factor N A simple rearrangement of eqs. 5, 6 and 7 gives the following equations for demagnetizing factor N, derived originally by Smith and Merill (1982) and Dunlop (1984)

N= 1/(x~)~~— ~ N=Hc/J~S—1/(xI)DL

N = (He.

—

Hc)/JD

—

1/(xI)DL

(8) (9) (10)

In each equation, Xo and x~must be determined for matching sections of the measured and intnnsic hysteresis loops. Hodych (1986) attempts to test eq. 9 using initial x1 values, which are smaller than the proper descending-loop susceptibilities (segment C’B’ in Fig. 1) and lead to low N values. His conclusion that eqs. 9 and 10 are invalid is unjustified. Hodych accepts eq. 8, but dismisses eqs. 9 and 10 as being tautologies. In reality, all three are equally valid restatements of eq. 4 for different sections of the magnetization curve. In actual calculations of N, x1 is not measurable. One needs an independent estimate of X~ Since it is (X~)~~ that is normally estimated (see below), (8) is the most useful of the three equations. To solve (9) and (10), (~)DL is often taken

183 ‘I

/

M~ç~~e//t~

(J~0. 22,,am)

/dr/

-Q’-°

/ I,

C’

/

‘V

V 4,5 -

C

,-‘

O~dIOOP Hc,

OF’ 8,8’

0

—c~7ku/ci~d ,ntr,nsic k~ (ib- Nr2) A’

A’

Fig. 1. Measured hysteresis loop (heavy lines) for sample 1 and hypothetical corresponding internal-field loop for an assumed demagnetizing factor N = 2 (lighter lines).

to be infinite (e.g. Stacey and Banerjee, 1974). As Smith and Merrill (1982) and Hodych (1986) point out, this approximation yields an upper bound on N. Dunlop (1984) made more reasonable estimates

on the initial guess made for N. Hodych uses the upper bound H~/J~,which is a considerable overestimate of N. It leads to much too high a value for J1~(see Fig. 1), so that (Xj)DL may in

of (x~) DL’ but his procedure is circular, in that a value of N calculated using eq. 8 is needed. Thus his determinations of N using (9) and (10) are not independent estimates, as originally claimed. The numerical difference between them results from the non-linearity of segment CBA. Hodych (1986) proposes J~/H~as an estimate of (XI)DL at B’. Since segment D’B’ of the descending intrinsic curve is always strongly curved compared to C’B’ (Fig. 1), (xI)DL would be considerably underestimated if the actual value of J~ were used. However, J,.~is unknown a priori, and must be estimated by Néel’s construction: the intersection of a line OF of slope 1/N with the measured major loop CD. Since OF and CD are nearly parallel, the value obtained for J, depends

fact be overestimated rather than underestimated. It follows that Hodych’s suggested lower bound on N, (H~/J~) (H~/J~), may actually be > N rather than
2.3. Multidomain coercive force To solve eq. 8, Dunlop (1984) relied on eq. 1 to estimate ~ using a value for k of 45 emu cm ~ (4.5 X iO~A m’), appropriate to micron-

184

size magnetite (Stacey and Banerjee, 1974, eq. 4.33). Hodych (1986) has greatly improved on this procedure by combining eqs. 1 and 5 to give (ii’~ 1 //(~Xoi~ (T~= k1H (T~~ (T’1 + N 1n1t~ / //Js~ / ‘ / Equation 11 has four advantages over all previous approaches. (1) It yields a value for N without assuming a value for k. (2) It requires data from a single sample only. (3) It contains a built-in test of eq. 1. (4) Since it has the same form as eq. 3, but a different slope and intercept, it should distinguish MD from SD carriers, Hodych tested eq. 11 using low-temperature data for seven mafic rocks and one magnetite dispersion. In the remainder of this paper, we report high-temperature measurements, displayed in the form of eqs. 2, 3 and 11 for 13 dispersions of magnetite, maghemite and oxidized titanomagnetite grains. C’~

3. Expenmental results 3.1. Samples and techniques

The compositions, estimated grain sizes, and basic magnetic properties (Curie temperature 1~ and room-temperature values of J~and J~/J~) of the experimental samples are listed in Tables I and II. All are dispersions (— 1% by volume) of synthetic magnetites and aluminum-substituted

titanomaghemites, except for SD1 and SD2, which contain concentrates from natural rocks. Judging by particle size and values of jrs/1~, samples 4, SD1, SD2contain and SD3 and all titanomaghemite samples mainly SDthegrains. Magnetite samples 1, 2 and 3 contain mainly small MD (so-called pseudo-SD) grains. For a detailed description of the preparation and properties of these samples, see Dunlop (1973, 1986) and Ozdemir and O’Reilly (1981b, 1982). In the case of the magnetite and maghemite samples, hysteresis was measured at temperatures up to 575°Cin fields ~ 2400 Oe (240 mT) using a ballistic magnetometer, non-inductive furnace, and solenoid (West and Dunlop, 1971). (xo)jmt was determined from the initial magnetization curve in the Rayleigh region (H0 < 100 Oe or 10 mT). A detailed presentation of the data and methods is given in Dunlop (1986, 1987). In the case of the titanomaghemites, (xo ) ~ was measured continuously up to the Curie point or the inversion ternperature with a recording susceptibility bridge (Stephenson and de Sa, 1970), and saturation hysteresis was measured using a vibrating-sample magnetometer. A brief description of the hightemperature H~data appears in Ozdemir and O’Reilly (1982). The susceptibility data have not been published previously. 3.2. Magnetite and maghemite results Experimental values of l/(Xo ) ~ and H~/j~ are plotted in Fig. 2 for the magnetite and mag-

TABLE I Basic properties of the magnetite and maghemite samples, and inverse slopes and intercepts of l/(X~)jmt— H~/j~ graphs No.

Magnetic
.1,, J,

k (emu cm

0.112 0.202 0.242 0.289 0.296 0.291 0.363

34.0 66.0 75.0 65.5

3)

Intercept 0.66 1.07 1.14 0.68

185 136

0 ~

15

7’

10

,/

,‘

‘4,

—

~

0.5 0

I

0

I

40

I

I

80

I

I

120

160

200

240

Fig. 2. Experimental l/(X~),,,~t versus H~/j, plots using high-temperature data for samples 1—4

and SD1, SD2.

Hc /j~ (Oe)

hemite samples. The temperature of each determination is given as a parameter along the curves. Samples 1—4 obey the proposed MD eq. 11, but samples SD1 and SD2 follow non-linear trends totally at variance with SD eq. 3. The data for SD3 (not shown), which contains elongated SD grains, plot near a single point for all T < 560°C. Since H~/j~ is temperature independent for pure shape anisotropy, this result is as expected. Inverse slopes and intercepts of the sample 1—4 graphs are listed in 3Table I. The valuesare of (3.4—7.5 X inferred i04 A m1), k, 34—75to emu cm (1986, table II) values for similar Hodych’s magnetite-bearing rocks. The intercepts, however, range from 0.66 to 1.14. These values are too small to be demagnetizing factors; possible values of N range from 1.6 (N 51 = 0.13) for two-domain grains

to 4sr/3 (N51 1983).

=

1/3) for large MD grains (Dunlop,

3.3. Titanomaghemite results The high coercive forces and low initial susceptibilities of the titanomaghemite samples, together with J~/J~ values between 0.52 and 0.59 (Table II), indicate SD behaviour with mixed umaxial and cubic anisotropies. One would expect (X~)jmt and H~to follow eq. 3. When the data are plotted in the than manner of zero, Fig. are 2, however, the intercepts, rather being large. In this situation, for reasons we explain below, it is more appropriate to plot (X~)jmtversus following eq. 2. Figure 3 presents data for the TM6O and ATM6O/10 samples, plotted in this form. The ATM6O/20 samples behave similarly

186 TABLE II Basic properties of the titanomaghemite samples, and slopes and intercepts of (X~)imt— ~ m for all samples Magnetic material

~r,

J,~ 3) (emu cm

(°C)

H~ (Oe)

.1,

JS/HC

Low-temperature slope intercept

graphs. Particle sizes are 0.02—0.05 High-temperature slope intercept

TM6O a 320 103 0.589 2015 0.179 0.004 0.024 z = 0.29 ATM6O/10 271 85 0.516 1930 0.074 0.008 0.048 z = 0.32 ATM6O/10 304 75 0.559 1970 0.153 0.005 0.077 z = 0.41 ATM6O/20 242 59 0.545 1560 0.088 0.007 0.028 z = 0.35 ATM6O/20 261 46 0.550 1710 0.114 0.004 0.068 z = 0.47 ATM6O/20 275 41 0.537 1390 0.154 0.004 0.072 z = 0.59 a TM6O, ATM6O/10 and ATM6O/20 mean titanomagnetite containing 60 mol% Ti and 0, 10 mol% and 20 mol%

0.029 0.012 0.014 0.011 0.007 0.007 Al, respectively.

The oxidation parameter z indicates the degree of low-temperature oxidation.

(see Table II). Each plot has two linear segments. The high-temperature segment has a large intercept and a small slope, much less than expected from eq. 2. Figure 4 depicts the temperature variation of ~ for the same samples. At lower tempera-

0.04

-

i::: ¼

0.01

tures, J~,/i~ decreases gradually because of the declining importance of crystalline anisotropy, but at higher temperatures there is a sharp decrease due to remanance unbiocking. The temperature at which the first superparamagnetic (SP) particles appear, as indicated by the breakpoint in the

-

0 0

/

)Z21M~::Eo:4,

4t 6!

- --

/

S-W

o ATM6O~’/O,z = 032

meory I

0.04

I

0.08

I

0.12

s/s ~

I

0.16

I

0.20

I

0.24

I

0.28

I

0.32

(emu/cm’Oe)

Fig. 3. Experimental (X~),,,,t versus J,/H~ plots using above-room-temperature data for three of the titanomaghemite samples. Dashed lines are projections of the lower-temperature data trends. The dash-dot line has the theoretical Stoner—Wohlfarth slope 0.319.

187

4. Discussion

0.6

:

~76~c

,js~co\

0.3

21cn \

~

4.1.Fseudo~Dsampks(1-4) 1/(x~)~~~ against H~/j,is in principle a good way of determining k and N. The data usually cover a good span of either parameter (Fig. 2), so that linearity is easily judged, slopes can be obtained precisely, and not

\\

much extrapolation is needed to frnd intercepts. The high-temperature k values for samples 1—4 (Table I) are in the same range as Hodych’s (1986)

\

\ \ \ \ \\

\

0.2

o.

~C ‘~,

00

I

00

~

3()()

Tempera/ure

~

r(=C)

Fig. 4. Experimental high-temperature J,.,/J, data for the same titanomaghemite samples as in Fig. 3. The breakpoints represent the onset of remanence unblocking.

J~,/J, T data, matches quite well the breakpoint between low-temperature and high-temperature segments in Fig. 3. We therefore ascribe the anomalously small slopes of the high-temperature segments in Fig. 3 to thermal unblocking of partide moments. The lower-temperature segments in Fig. 3 generally have small intercepts and slopes closer to the Stoner—Wohlfarth value of 0.319 (eq. 2). In view of the fine particle size (0.02—0.05 ~tm) and low Curie temperatures (240~3200C), these data are probably also affected to some extent by thermal excitations (see Discussion), although we are below the unblocking temperatures of most partides. A hypothetical third segment (shown as a dashdot line in Fig. 3), joining the room-temperature data to the origin, has almost exactly the Stoner—Wohlfarth slope. The data for each sample thus seem to follow a 3-segment curve, eq. 2 defining the lowest-temperature segment. The intercept of a l/(Xo)~t versus Hc/fs plot —

is a high-temperature limit. At these temperatures, the finer SD grains are much affected by thermal agitation and some may be unblocked. For this reason, such a plot is usually unsuitable for cornparison with Stoner—Wohlfarth theory.

low-temperature determinations for magnetitebearing rocks. However, our intercepts, like Hodych’s, are anomalously low for MD grains. Hodych appealed to two-domain structures or interactions among lamellar or rod-like subgrains to explain the implied low values of N, although rocks with low coercive forces, presumably containing large homogeneous grains with many domains, also gave unreasonably low N values. We have the advantage of knowing the domain structures in our particles. All samples contain varying proportions of SD and small MD crystals. For example, sample 1 contains magnetite cubes averaging 0.22 ,sm in size and having mainly two-domain structures (Dunlop, 1977, 1986, 1987). Dunlop (1984) concluded, by applying eq. 8 to room-temperature hysteresis and initial susceptibility data for this sample, that N is 1.8, in good agreement with the theoretical N of 1.6—2.0 (Dunlop, 1983). The high-temperature plot for this sample (Fig. 2), however, has an intercept of 0.66, in total disagreement with the isothermal determination of N and with theory. The fact that samples 1—4 contain a mixture of SD and MD grains suggests that their k values and intercepts may be weighted averages of SD and MD values. This hypothesis explains the k values quite well. Most are intermediate between the MD value for micron-size magnetites, k 45 emu cm3 (Stacey and Banerjee, 1974), and the value for SD magnetite, k 0.319J~ 0 150 emu cm ~ (eq. 3). Furthermore they tend to decrease with increasing mean particle size, i.e. with decreasing SD/MD ratio. The intercepts are also intermediate between the expected values of zero for SD grains (eq. 3) and N = 1.6—2.0 for 2D

188

grains. However, sample 1 does not fit the expected trend of increasing intercept with increasing grain size. Moreover the isothermal determination of N = 1.8 for this sample implies that it contains almost no SD material. 4.2. SD samples (SDJ-3 and titanomaghemites) The SD samples did not obey eqs. 2 or 3. The ‘/(X~)imt versus H~/j~ plots for SD1 and SD2 were convex downward (Fig. 2), while the (Xø)jmt versus J~/H~ graphs for all titanomaghemite samples were convex upward, consisting of two approximately linear segments. The most plausible explanation of these results is thermal excitation. Many of the data were measured at temperatures approaching, or in, the main unblocking range. This is particularly so for the titanomaghemites, whose Curie temperatures are low. Initial susceptibility, being a weak-field property, is not significantly enhanced by thermal agitation until the unblocking temperature is reached and particles become other hand, measuresSP. SDCoercive reversal force, underonthethecombined influence of temperature and a moderately strong field, and can be significantly reduced at temperatures well below the weak-field unblocking ternperature TUB (Dunlop, 1976; Dunlop and Bina, 1977). As a result, for T < TUB, there is a decrease in H~ that is not matched by a corresponding increase in Xo~ This decrease is small near room temperature unless the sample contains a large fraction of nearly SP grains, but substantial at high temperatures. The net effect is to move hightemperature points to the left on a 1/( Xo) versus H~/j~ plot, producing convex down curvature as seen in Fig. 2. On a (Xo ) ~ versus .J~/H~ graph Fig. 3, anomalously points will be the right, like producing lowmoved slopesto and non-zero intercepts, The effect of thermal fluctuations is most pronounced within the unblocking range (T> TUB), corresponding for the titanomaghemites to the highest-temperature segments of Figs. 3 and 4. As particles unblock, their susceptibilities increase to large SP values, while their coercive forces vanish, Points on a (Xo)~tversus J~/HC graph will shift

up, but will shift even more strongly to the right, resulting in an çxtreme reduction in slope compared with the Stoner—Wohlfarth value (cf. Table II). An exact analysis within the unblocking range requires a detailed knowledge of the unbiocking temperature spectrum, which we lack for these samples. However, an exact analysis within the thermal excitation range below the lowest TUB can be carried out with the aid of thermal fluctuation analysis (TFA). For this purpose, we now develop a generalized method of TFA. 4.3. Generalized thermal fluctuation analysis (TFA)

Thermal fluctuation analysis (Dunlop, 1976) is based on the well-known equation for SD grains (e.g. Bean and Livingston, 1959) H~( T) = HK (7~) Hq (T) —

[2kBTHK ln( f0t )/ v.j~ 11/2 2 (12) HK y(HKT/J~)V In eq. 12, microscopic coercive force HK = 2K/J for uniaxial anisotropy K, Hq is the ‘fluctuation field’ (Née!, 1955), k 8 is Boltzmann’s constant, 1,t is time, V is the frequency factor f~ iO~° s ~ particle volume, and y = (2kg ln(f 1”2. For pure shape anisotropy, HK (T) ~ 0t)/V) ./~( T), so that (12) can be linearized and solved graphically for V. For mixed anisotropies, which occur in all our SD samples except SD3, (12) is unworkable as a method of finding V since HK(T) is unknown. To avoid this problem, we define a = HK (Xo ) ~ = k/J~ 0, which according to eq. 1 is temperature independent. Then eq. 12 can be recast as 2[T/xo]~2H~1= 1 (13) a[J,/H~~o] ya or aa b,ya1”2 = 1 =

HK

—

—

—

—

in which Xo means the initial susceptibility (Xo ) mit’ The a, b. are experimentally measured values at a series of temperatures 7
—

189

solves numerically the equation i ~ [a1a b~’ya~’2 i][a~ b.y/2&”2]

0

(14)

after substituting the explicit solution for y = [a 2]/~b~ 1b1a”~ bja_1~’

(15)

—

—

=

—

~

—

Best-fit values of ‘y and a were determined for the six titanomaghemite samples using H~,(Xo ) ~ and J, data measured below high-temperature breakpoints in .J~S/J~ T graphs like those of Fig. 4. The average values with their standard deviations were a = 0.311 ±0.071 (close to a = 0.319, the Stoner—Wohlfarth value) and y = 4.2 ±0.7 Oe K 1/2 In Fig. 5, we illustrate these fits by plotting the parameter aa 1”2b 1(T) ya 1(T) as a function of temperature. Experimental values are relatively temperature insensitive, and are close to the ideal value 1 (eq. 13) for T < TUB. They fall rapidly to zero in the unblocking range. The best-fit average value for y yields an estimate (d) = 0.070 ±0.011 tim for mean grain size. The mean size determined from electron micrographs is only 0.03—0.04 ,sm (Ozdemir and 0’ Reilly, 1981b). However, the theoretical critical SP size for TM6O is 0.07—0.08 ~sm (Butler and Banerjee, 1975), so that many of the observed particles should behave superparamagnetically, —

—

.2

0.8

L~ 0.6 IC

~ 0160 z’029

I

o A016O//0~,.O.3

I

0.4o 0.2

•jATM6,,~59 ~ 40160/10. z.0.4 ATM6O/20.,-0.47

0

50

~

00 150 TempeourejYC)

\\I ~ 200

250

Fig. 5. Results of thermal fluctuation analysis (TFA) for the titanomaghemite samples. The parameter aa 1~2b 1 — ya 1, calculated using best-fit values of a and -y from TFA together with experimental a1, b. determined from values of (X~)imt, H, and J, at various T, is compared to its ideal value of 1.

unless they act collectively because of strong interactions. The good fit obtained by using the Stoner— Wohlfarth value 0.319 for a implies that uniaxial amsotropy is dominant over the temperature range examined. This result is a little surprising in view of the fact that cubic magnetocrystalline anisotropy is a major source of the high coercive forces and J~/J~ values > 0.5 measured at and just above room temperature. For magnetization confined to a plane, a is about 0.15 for trigonal anisotropy and about 0.09 for triaxial anisotropy (Dunlop, 1971, table 2.3), but the cubic anisotropy problem has never been solved in full (Johnson and Brown, 1961). It may be that a is not greatly reduced, since the planar constraint accounts for much of the reduction in a in the trigonal and triaxial cases. 5. Conclusions We reach the following conclusions. (1) The exact eqs. 8—10 (Smith and Merrill, 1982; Dunlop, 1984) remain the best basis for an isothermal determination of demagnetizing factor N. The upper and lower bounds proposed by Hodych (1986) do not bracket N very closely, and there may be cases where the suggested lower bound is > N rather than
agree 1/(Xa)jmt’k~Hc/fs+N withsmall Hodych’s range of values forvalues magnetite-bearing (3) Equation rocks. 11 yields anomalously for N well for our MD magnetites, as low also found by Hodych for some of his rocks. A quasi-isothermal determination, combining k determined from variable-temperature data using (11) with roomtemperature values of (Xo)mit, H~and j~in eqs. 1 and 8, gives a more reasonable value of N for the largest of our MD magnetites.

190

(4) Using the SD analog to (11) 1H~/j ~ = k 5

Butler, R.F. and Baneijee, S.K., 1975. Theoretical single-domain grain-size range in magnetite and titanomagnetite. J.

(eq. 3) with variable-temperature data is not recommended. None of our SD magnetites, maghemites, or titanomaghemites obeyed (3), mainly because thermal agitation causes the high-temperature points to deviate from Stoner—Wohlfarth theory, which ignores thermal fluctuations. Hodych’s (1986) use of below-room-temperature data therefore has some advantages over high-temperature fits, since MD grains may also be affected to some extent by thermal excitation of domain walls. (5) A better approach for SD grains is to fit to the direct Stoner—Wohlfarth equation (Xo = (k/i;0) i;/H~= ai;/H~

Dunlop, D.J., 1971. Magnetic properties of fine-particle hematite. Ann. Géophys., 27: 269—293. Dunlop, D.J., 1973. Superparamagnetic and single-domain threshold sizes in magnetite. J. Geophys. Res., 78: 1780—1793.

)~

(eq. 2), which appears to be tangent to our titanomaghemite data at sufficiently low temperatures.

(6) Good fits, yielding a = k/i;0 0.319 (the Stoner—Wohlfarth value) but a mean particle size about twice that observed in electron micrographs, resulted from using a new method of thermal fluctuation analysis (eqs. 13—15) with the titanomaghemite data measured within the thermal excitation range but below the main thermal unblocking range. (7) Non-uniaxial amsotropies will yield different values for a = k/i;0 in the SD case and ~ lead to deviations from eq. 2. However, although our titanomaghemites clearly have mixed uniaxial and cubic magnetocrystalline anisotropies near room temperature, the purely uniaxial value a = k/i;0 = 0.319 gave acceptable fits.

Acknowledgments

RJE is grateful to the National Research Council of Canada for a Postgraduate Fellowship. This research was supported by NSERC Operating Grant A7709 to DJD.

References Bean, C.P. and Livingston, J.D., 1959. Superparamagnetism. J. AppI. Phys., 30: 120S—129S.

Geophys. Res., 80: 4049—4058.

Dunlop, D.J., 1974. Thermal enhancement of magnetic ceptibility, J. Geophys., 40: 439—451. Dunlop, D.J., 1976. Thermal fluctuation analysis: a new technique in rock magnetism. J. Geophys. Res., 81: 3511—3517. Dunlop, D.J., 1977. The hunting of the ‘psark’. J. Geomagn. Geoelectr., 29: 293—318. Dunlop, D.J., 1983. On the demagnetizing energy and demagnetizing factor a multidomain ferromagnetic cube. Geophys. Res. Lett.,of10: 79—82. Dunlop, DJ., 1984. A method of determining demagnetizing factor from multidomain hysteresis. J. Geophys. Res., 89: 553—558. Dunlop, DJ., 1986. Hysteresis properties of magnetite and their dependence on particle size: a test of pseudo-single domain remanence models, J. Geophys. Res., 91: 9569—9584. Dunlop, DJ., 1987. Temperature dependence of hysteresis in 0.04—0.22 ~sm magnetites and implications for domain structure, Phys. Earth Planet. Inter., 46: 100—119. Dunlop, DJ. and Bina, M-M., 1977. The coercive force spectrum of magnetite at the highblocking temperatures: evidence for thermal activation below temperature. Geophys. J. R. Astron. Soc., 51: 121—147. Hodych, J.P., 1986. Determination of self-demagnetizing factor N for multidomain magnetite grains in rock. Phys. Earth Planet. Inter., 41: 283—291. Johnson, C.E. and Brown, W.F., 1961. Theoretical magnetization curves for particles with cubic anisotropy. J. Appi. Phys., 32: 2435-244S. Kittei, C., 1949. Physical theory of ferromagnetic domains. Rev. Mod. Phys., 21: 541—583. Ned, L., 1955. Some theoretical aspects of rock magnetism. Adv. Phys., 4: 191—243. Ozdemir, O. and O’Reilly, W., 1981a. High-temperature hysteresis and other magnetic properties of synthetic monodomain titanomagnetites. Phys. Earth Planet. Inter., 25: 406-418. Ozdemir, O. and O’Reilly, W., 1981b. Laboratory synthesis of aluminum-substituted titanomaghemites and their characteristic properties, J. Geophys., 49: 93—100. Ozdemir, O. and O’Reilly, W., 1982. Magnetic hysteresis properties of synthetic monodomain titanomaghemites. Earth Planet. Sci. Lett., 57: 437—447. Smith, G. and Merrill, R.T., 1982. The determination of the internal field in magnetic grains. J. Geophys. Res., 87: 9419—9423. Stacey, F.D., 1963. The physical theory of rock magnetism. Adv. Phys., 12: 45—133.

191 Stacey, F.D. and Baneijee, S.K., 1974. The Physical Principles of Rock Magnetism. Elsevier, Amsterdam, 195 pp. Stephenson. A. and de Sa, A., 1970. A simple method for the measurement of the temperature variation of initial magnetic susceptibility between 77 and 1000 K. J. Phys. Earth Sci. Instr., 3: 59—61.

Stoner, E.C. and Wohlfarth, E.P., 1948. A mechanism of magnetic hysteresis in heterogeneous alloys. Philos. Trans. R. Soc., London, Ser. A, 240: 599—642. West, G.F. and Dunlop, D.J., 1971. An improved ballistic magnetometer for rock magnetic experiments. J. Phys. Earth Sci. Instr., 4: 37—40.