Gubbins D., Herrero-Bervera E. Encyclopedia of Geomagnetism and Paleomagnetism

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glacial perturbation (e.g., Jessop, 1971). The uncertainties involved in

estimating the paleogeothermal gradient are significant and can lead to

relatively large uncertainties in the depth of burial, even if T

amb

is well

constrained (see Schwarz et al., 1985; Hyodo et al., 1993).

Kenneth L. Buchan

Bibliography

Adam, E., 1990. Temperature ambiante de la roche hôte deduite de

l’histoire thermique d’un dyke: apport du paleomagnetisme.

B.Eng. thesis, École Polytechnique, Montreal: Université de

Montréal, 62 pp.

Figure M124 Orthogonal component plots showing thermal demagnetization results for selected samples from the Indin dike profile of

Figure M122 (after Schwarz et al., 1985). Peak temperatures are indicated in degree Celsius. Closed and open symbols indicate

projections onto the horizontal and N-S vertical planes, respectively.

606 MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES

Buchan, K.L., and Schwarz, E.J., 1981. Uplift estimated from rema-

nent magnetization: Munro area of Superior Province since 2150

Ma. Canadian Journal of Earth Sciences, 18: 1164–1173.

Buchan, K.L., and Schwarz, E.J., 1987. Determination of the maxi-

mum temperature profile across dyke contacts using remanent

magnetization and its application. In Halls, H.C., and Fahrig, W.F.,

(eds.), Mafic Dyke Swarms. Geological Association of Canada

Special Paper 34, pp. 221–227.

Buchan, K.L., Schwarz, E.J., Symons, D.T.A., and Stupavsky, M.,

1980. Remanent magnetization in the contact zone between

Columbia Plateau flows and feeder dykes: evidence for ground-

water layer at time of intrusion. Journal of Geophysical Research,

85: 1888–1898.

Carslaw, H.S., and Jaeger, J.C., 1959. Conduction of Heat in Solids,

2nd edn. New York: Oxford University Press, 510 pp.

Delaney, P.T., 1982. Rapid intrusion of magma into wet rock: ground-

water flow due to pore pressure increases. Journal of Geophysical

Research, 87: 7739–7756.

Delaney, P.T., 1987. Heat transfer during emplacement and cooling of

mafic dykes. In Halls, H.C., and Fahrig, W.F. (eds.), Mafic Dyke

Swarms. Geological Association of Canada Special Paper 34, pp.

31–46.

Delaney, P.T., 1988. Fortran 77 programs for conductive cooling of

dikes with temperature-dependent thermal properties and heat of

crystallization. Computers and Geosciences, 14: 181–212.

Delaney, P.T., and Pollard, D.D., 1982. Solidification of basaltic

magma during flow in a dike. American Journal of Science, 282:

856–885.

Dodson, M.H., 1973. Closure temperature in cooling geochronological

and petrological systems. Contributions to Mineralogy and Petrol-

ogy, 40: 259–274.

Dunlop, D.J., 1979. On the use of Zijderveld vector diagrams in

multicomponent paleomagnetic studies. Physics of the Earth and

Planetary Interiors, 20:12–24.

Dunlop, D.J., and Özdemir, Ö., 1997. Rock magnetism: fundamen-

tals and frontiers. Cambridge: Cambridge University Press,

573 pp.

Halls, H.C., 1986. Paleomagnetism, structure and longitudinal correla-

tion of Middle Precambrian dykes from northwest Ontario and

Minnesota. Canadian Journal of Earth Sciences, 23: 142–157.

Hyodo, H., York, D., and Dunlop, D., 1993. Tectonothermal history in

the Mattawa area, Ontario, Canada, deduced from paleomagnetism

and

Ar/

Ar dating of a Grenville dike. Journal of Geophysical

Research, 98: 18001–18010.

Jaeger, J.C., 1964. Thermal effects of intrusions. Reviews of Geophy-

sics, 2(3): 711–716.

Jessop, A.M., 1971. The distribution of glacial perturbation of heat

flow. Canadian Journal of Earth Sciences, 8: 162–166.

Jessop, A.M., and Lewis, T., 1978. Heat flow and heat generation in

the Superior Province of the Canadian Shield. Tectonophysics,

50:55–77.

McClelland Brown, E., 1981. Paleomagnetic estimates of temperatures

reached in contact metamorphism. Geology, 9:112– 116.

McClelland Brown, E., 1982. Discrimination of TRM and CRM by

blocking-temperature spectrum analysis. Physics of the Earth and

Planetary Interiors, 30: 405–411.

Nagata, T., 1961. Rock Magnetism. Tokyo, Japan: Maruzen Company

Ltd., 350 pp.

Oveisy, M.M., 1998. Rapakivi granite and basic dykes in the Fennos-

candian Shield: a palaeomagnetic analysis, PhD thesis, Luleå,

Sweden: Luleå University of Technology.

Pullaiah, G., Irving, E., Buchan, K.L., and Dunlop, D.J., 1975. Magne-

tization changes caused by burial and uplift. Earth and Planetary

Science Letters, 28: 133–143.

Schwarz, E.J., 1976. Vertical motion of the Precambrian Shield from

magnetic overprinting. Bulletin of the Canadian Association of

Physicists, 32:3.

Schwarz, E.J., 1977. Depth of burial from remanent magnetization: the

Sudbury Irruptive at the time of diabase intrusion (1250 Ma).

Canadian Journal of Earth Sciences, 14:82–88.

Schwarz, E.J., and Buchan, K.L., 1982. Uplift deduced from remanent

magnetization: Sudbury area since 1250 Ma ago. Earth and Plane-

tary Science Letters, 58:65–74.

Schwarz, E.J., and Buchan, K.L., 1989. Identifying types of remanent

magnetization in igneous contact zones. Physics of the Earth and

Planetary Interiors, 68: 155–162.

Schwarz, E.J., Buchan, K.L., and Cazavant, A., 1985. Post-Aphebian

uplift deduced from remanent magnetization, Yellowknife area

of Slave Province. Canadian Journal of Earth Sciences

, 22:

1793–1802.

Symons, D.T.A., Hutcheson, H.I., and Stupavsky, M., 1980. Positive

test of the paleomagnetic method for estimating burial depth

using a dike contact. Canadian Journal of Earth Sciences, 17:

690–697.

Touloukian, Y.S., Judd, W.R., and Roy, R.F., (eds.), 1981. Physical

Properties of Rocks and Minerals. New York: McGraw-Hill,

548 pp.

Cross-reference

Baked Contact Test

MAGNETIZATION, REMANENT, FOLD TEST

Establishing the age of the remanence acquisition with respect to the

origin of the rock unit or age of structural deformation events is critical

in the interpretation of paleomagnetic data. It is also important to

establish that the magnetic minerals carry a stable magnetization over

geological timescales. These two related questions remain difficult

problems, which often affect the application of paleomagnetism to tec-

tonic, stratigraphic, and paleogeographic problems (Cox and Doell,

1960; Irving, 1964). It is therefore not surprising that early in the

development of the paleomagnetic method, several field and labora-

tory tests were developed and applied to a range of geological con-

texts. Among them, an elegant simple answer was put forward to

constrain the age of remanence acquisition by using field information

by Graham (1949). He realized that in folded rocks a simple test based

on rotation of the magnetization directions about the local strike could

determine if the remanence was acquired before the deformation event

—if magnetization directions were dispersed in present-day coordi-

nates and clustered upon tilt rotation (Figure M125a). Magnetization

directions that cluster in present-day coordinates and disperse upon tilt

rotation indicate that remanence acquisition occurred after the defor-

mation event (Figure M125b). At the time Graham (1949) developed

this test, the multivectorial nature of natural remanent magneti-

zation (NRM) had not been properly recognized, and the demagnetiza-

tion techniques for isolating remanent components had not been

developed. The fold test nevertheless worked well as applied to the

deformed sedimentary strata of the Rose Hill Formation from Maryland,

mainly because the NRMs are univectorial and the strata are tilted

to large angles without major structural complexities (French and

Van der Voo, 1979). The Graham fold test can be applied to study

single deformed structures, including a fold within an undeformed

sequence, or over wide deformed areas with sampling at sites with

varying bedding tilts.

The impact of paleomagnetic data in the study of regional tectonic

processes, particularly in the establishment of the continental drift

theory, and the increasing use of demagnetization and statistical

techniques in investigating multi-vectorial remanences made the use

of field tests fundamental tools for paleomagnetists (Irving, 1964;

McElhinny, 1973). Field tests to investigate the magnetization stability

and the timing and mode of remanent magnetization acquisition include

MAGNETIZATION, REMANENT, FOLD TEST 607

the (a) fold test, (b) conglomerate test, (c) baked contact test, (d) rever-

sal test, and (e) consistency test. Looking at the many applications

developed over the years can easily assess the importance of the field

tests in paleomagnetic research.

Development of the fold test in paleomagnetism has involved incor-

poration of statistical criteria to evaluate the significance of the test,

advances in separating multicomponent magnetizations, advances in

paleomagnetic instrumentation providing more sensitive instrumenta-

tion and methods to measure weak magnetizations, and application

to a wide range of deformed rock units as well as local and regional

tectonic structures.

Statistical criteria

Application of statistical criteria in evaluating the test significance was

first attempted by McElhinny (1964), by using the concentration para-

meter k of the Fisher (1953) statistics to distinguish between pre- and

postfolding magnetization data sets. The criterion is an application of

the statistical test for comparing the precisions using the Fisherian con-

centration parameter of two separate groups of sample directions with

the precision of unfolded directions (k

) compared with the precision

for in situ directions (k

). The ratio k

is referred to F-ratio tables

at given confidence limits and with equal degrees of freedom

(2N  1), where N is the number of sample directions studied.

The application of statistical criteria permitted the use of the fold

test in a wide range of deformed structures, and McElhinny (1964)

fold test is still used in paleomagnetic studies. The test is applied in

spite of the problem noticed by McFadden and Jones (1981) that the

use of Fisher statistics for directional data sets, which are not Fisher

distributed, is not valid. This occurs either way if magnetizations are

pre- or postfolding. For instance, if directions are Fisher distributed

in in situ coordinates, then they are not tilt-corrected coordinates.

McFadden and Jones (1981) proposed a test based on comparison of

Fisher distributed data sets derived from opposite limbs of a fold in

in situ and tilt-corrected coordinates. The new test again implies no

internal distortion of magnetization and more intensive field sampling,

but it can be applied under less stringent conditions than the McElhinny

(1964) test.

Two further statistical significance tests based on comparison of dis-

tributions of magnetization directions, which require less demanding

sampling strategies than the test by McFadden and Jones (1981), have

been proposed by McFadden (1990) and Bazhenov and Shipunov

(1991). The test of Bazhenov and Shipunov (1991) considers different

structural attitudes by making division of bedding plane data into

groups, which gives greater flexibility in the sampling strategy. Never-

theless, the tests have been little used in paleomagnetic studies mainly

because their statistical significance is relatively weak (e.g., Weil and

Van der Voo, 2002).

Evaluation of fold test data sets has also been examined in terms of

a parameter estimation problem, by applying simple bootstrap or

Monte Carlo approaches. In these tests, large numbers of sample data

are generated to determine confidence limits and significance of the

maximum clustered directional distribution (Fisher and Hall, 1990;

Tauxe et al., 1991; Watson and Enkin, 1993; Tauxe and Watson,

1994). Tauxe and Watson (1994) developed a test using eigen analysis

of directional data sets that estimates the degree to which directions

from different structural attitude sites are parallel. In this way, direc-

tional data sets do not need to conform to a Fisher distribution, and

confidence limits are derived from bootstrap and parametric bootstrap

techniques according to data set size.

McFadden (1998) discussed the underlying assumption of para-

metric estimation fold tests that the magnetization directions show

maximum clustering with the strata in the position magnetization

was acquired. This is not the case when folding does not conform to

Figure M125 Paleomagnetic fold test.

608 MAGNETIZATION, REMANENT, FOLD TEST

simple cylindrical horizontal axis geometry and additional factors are

involved like plunging folds, unaccounted overprints, etc.

Jaime Urrutia-Fucugauchi

Bibliography

Bazhenov, M.L., and Shipunov, S.V., 1991. Fold test in paleomagnet-

ism: new approaches and reappraisal of data. Earth and Planetary

Science Letters, 104:16–24.

Chan, L.S., 1988. Apparent tectonic rotations, declination anomaly

equations, and declination anomaly charts. Journal of Geophysical

Research, 93: 12151–12158.

Cogné, J.P., and Perroud, H., 1985. Strain removal applied to paleo-

magnetic directions in an orogenic belt; the Permian red slates of

the Alpes Maritimes, France. Earth and Planetary Science Letters,

72: 125–140.

Cox, A., and Doell, R.R., 1960. Review of paleomagnetism. Bulletin

of the Geological Society of America, 71: 647–768.

Facer, R.A., 1983. Folding, Graham’s fold tests in paleomagnetic

investigations. Geophysical Journal of the Royal Astronomical

Society, 72: 165–171.

Fisher, N.I., and Hall, P., 1990. New statistical methods for directional

data—I. Bootstrap comparison of mean directions and the fold

test in paleomagnetism. Geophysical Journal International, 101:

305–313.

Fisher, R.A., 1953. Dispersion on a sphere. Proceedings of the Royal

Society of London Series A, 217: 295–305.

French, A.N., and Van der Voo, R., 1979. The magnetization of the

Rose Hill Formation at the classic site of Graham’s fold test. Jour-

nal of Geophysical Research, 48: 7688–7696.

Graham, J.W., 1949. The stability and significance of magnetism in

sedimentary rocks. Journal of Geophysical Research, 54: 131–167.

Irving, E., 1964. Paleomagnetism and Its Application to Geological

and Geophysical Problems. New York: Wiley.

Kodama, K.P., 1988. Remanence rotation due to rock strain during

folding and the stepwise application of the fold test. Journal of

Geophysical Research, 93: 3357–3371.

MacDonald, W.D., 1980. Net rotation, apparent tectonic rotation, and

the structural tilt correction in paleomagnetic studies. Journal of

Geophysical Research

, 85: 3659–3669.

McElhinny, M.W., 1964. Statistical significance of the fold test in

palaeomagnetism. Geophysical Journal of the Royal Astronomical

Society, 8: 338–340.

McElhinny, M.W., 1973. Paleomagnetism and Plate Tectonics. Cam-

bridge Earth Science Series, Cambridge University Press, 358 pp.

McFadden, P.L., 1990. A new fold test for paleomagnetic studies.

Geophysical Journal International, 103: 163–169.

McFadden, P.L., 1998. The fold test as an analytical tool. Geophysical

Journal International, 135: 329–338.

McFadden, P.L., and Jones, D.L., 1981. The fold test in paleomagnet-

ism. Geophysical Journal of the Royal Astronomical Society, 67:

53–58.

Stamatakos, J., and Kodama, K.P., 1991a. Flexural flow folding and

the paleomagnetic fold test: an example of strain reorientation of

remanence in the Mauch Chunk Formation. Tectonics, 10: 807–819.

Stamatakos, J., and Kodama, K.P., 1991b. The effects of grain-scale

deformation on the Bloomsburg Formation pole. Journal of Geo-

physical Research, 96: 17919–17933.

Stewart, S., 1995. Paleomagnetic analysis of plunging fold structures:

errors and a simple fold test. Earth and Planetary Science Letters,

130:57–67.

Tauxe, L., and Watson, G.S., 1994. The fold test: an eigen analysis

approach. Earth and Planetary Science Letters, 122: 331–341.

Tauxe, L., Kylstra, N., and Constable, C., 1991. Bootstrap statistics

for paleomagnetic data. Journal of Geophysical Research, 96:

11723–11740.

Van der Plujim, B.A., 1987. Grain-scale deformation and fold test-

evaluation of syn-folding remagnetization. Geophysical Research

Letters, 14: 155–157.

Watson, G.S., and Enkin, R.J., 1993. The fold test in paleomagnetism

as a parameter estimation problem. Geophysical Research Letters,

20: 2135–2137.

Weil, A.B., and Van der Voo, R., 2002. The evolution of the paleomag-

netic fold test as applied to complex geologic situations, illustrated

by a case study from northern Spain. Physics and Chemistry of the

Earth, 27: 1223–1235.

MAGNETIZATION, THERMOREMANENT

Introduction

Thermoremanent magnetization (TRM) is acquired when magnetic

minerals cool in a weak magnetic field H from above their Curie tem-

peratures. TRM is the most important remanent magnetization used in

paleomagnetism. It is almost always close to parallel to the field which

produced it, and its intensity is proportional to the strength of the field

for weak fields like the Earth’s. The TRM of rocks is therefore a vast

storehouse of recorded information about past movements of the

Earth’s lithospheric plates and the history of the geomagnetic field.

The primary natural remanent magnetization of an igneous rock or a

high-grade metamorphic rock is a TRM. Newly erupted seafloor lavas

at mid-ocean ridges acquire an intense TRM on cooling below the

Curie temperature T

. Rapid cooling also results in fine grain size,

which makes the TRM highly stable, so that oceanic basalts are excel-

lent recorders of the paleomagnetic field. However, the TRM is largely

replaced within at most a million years of formation by chemical

remanent magnetization (CRM) of reduced intensity.

The magnetic signal recorded in archeological materials such as pot-

tery, bricks, and the walls and floors of the ovens in which they were

fired at high temperature is also a pure TRM residing in single-domain

(SD) and pseudosingle-domain (PSD) grains. These materials should

be ideal for determinations of the paleointensity of the Earth’s field

at the time of firing because they are strongly magnetized and the

minerals have been stabilized physically and chemically well above

their Curie temperatures before their initial cooling.

TRM results from thermally excited changes in magnetization.

In the case of SD grains, it is a frozen high-temperature equilibrium

distribution between two microstates in which the spins of all

atoms are either parallel or antiparallel to an applied field H. At tem-

peratures below T

, there are large perturbations of a crystal’s spin

structure, leading to transitions between structures of different types.

With cooling, transitions become more difficult, as evidenced by a

rapidly increasing relaxation time t. Eventually, decreasing thermal

energy and increasing energy barriers E

between states prevent

further transitions and TRM is frozen in very abruptly at a blocking

temperature T

The TRM of multidomain (MD) grains results from the blocking of

domain walls at positions determined by the externally applied field H,

the internal demagnetizing field H

of the grain, and the pinning effect

of lattice defects.

Theories of TRM and reviews of experimental data have been given

by Néel (1949, 1955), Everitt (1961, 1962), Stacey (1958), Schmidt

(1973), Day (1977), and Dunlop and Özdemir (1997).

Relaxation time and single-domain TRM

Néel’s (1949, 1955) theory of relaxation time deals with noninter-

acting uniaxial SD grains with saturation magnetization M

and

volume V. M

results from the parallel exchange coupling of atomic

spins and is a strong function of temperature T near T

, but is weakly

dependent on T at low temperatures. Néel’s SD theory makes use of

MAGNETIZATION, THERMOREMANENT 609

Stoner and Wohlfarth’s (1948) results for coherent rotation in spheroi-

dal SD grains aligned with a weak field H. The magnetic moment

has a choice of orientations controlled by the easy axes of aniso-

tropy. Uniaxial shape anisotropy is dominant in minerals such as mag-

netite with high M

. Shape anisotropy produces two energy minima,

corresponding to spins in one or the other direction along the longest

axis of the grain. These two minima define the two SD microstates.

At 0 K, spins would be exchange coupled exactly parallel; but at

ordinary temperatures, thermal excitations perturb the spin lattice,

resulting in a steady decrease in M

(T ). At high temperatures, rever-

sals of grain moments are also excited as M

(T ) drops rapidly toward

zero at T

At ordinary temperatures, spontaneous reversals of an SD grain are

unlikely because the energy barrier between microstates due to shape

anisotropy is much larger than the available thermal energy (25kT

for experimental times of a few minutes, where k is Boltzmann’s

constant). Close to T

, where E

is small, the energy barrier becomes

comparable to 25kT and SD moments can reverse with the help of

thermal excitations. This thermally excited condition is called super-

paramagnetism. The transition from a superparamagnetic state to

a stable magnetic state is quite sharp and defines the blocking

temperature T

At any time, SD particles are either in state 1, in which moments are

aligned with H, or state 2, in which moments are antiparallel to H. The

transition probability set by the energy barrier DE

between states

leads to a relaxation time for exponential magnetization decay

1=t ¼1=t

expðE

=kTÞ¼1=t

exp½ðm

=2kTÞ

ð1H=H

 (Eq: 1)

where t

¼ 10

9

–10

10

s is the atomic reorganization time for transi-

tions between microstates and microcoercivity H

¼ (N

 N

for

shape anisotropy. N

and N

are the demagnetizing factors when M

directed parallel or perpendicular, respectively, to the long axis. The

thermally activated transitions between states 1 and 2 cause the aver-

age moment of a grain ensemble to relax to a thermal equilibrium

value

MðtÞ¼M ð0Þexpðt=tÞþMð1Þ½1  expðt=tÞ: (Eq. 2)

The thermal equilibrium magnetization above and at the blocking tem-

perature T

¼ M ð1Þ ¼ M

tanhðm

H=kT Þ: (Eq. 3)

The dependence of t on temperature by Eq. (1) is very significant.

As T decreases, the energy barriers increase and t grows exponentially,

changing from a few seconds to millions of years over a narrow

temperature interval. This rapid change leads to the concept of the

blocking temperature at which t ¼ t (experimental time).

¼½m

=2k lnðt=t

Þð1  H=H

; (Eq. 4)

where M

and H

are the values of M

and H

at the blocking tem-

perature.

Equation (1) also predicts a strong dependence of t on grain size.

SD grains only slightly above superparamagnetic size have relaxation

times greater than the age of the earth, but relaxation times of only a

few minutes at T

of a few hundred degrees Celsius.

Thus, on cooling below T

in an applied field, all of the orientations

of the SD moments remain fixed as their M

increases and TRM is

frozen in:

TRM

¼ M

ðTÞtanh m

H=kT

ðÞ; (Eq. 5)

where M

is the saturation value of TRM. Single-domain M

TRM

should be relatively intense because tanh(a) saturates rapidly as

a ¼ m

H/kT

increases and TRM then approaches the SD satura-

tion remanence M

The high thermal stability of TRM is also accounted for, because

TRM

can only be unfrozen and reset to zero by reheating to the

unblocking temperature T

in H ¼ 0. According to Eq. (4),ifH 

, then T

 T

. TRM can only be demagnetized by reheating to

its original high blocking temperature.

Experimental single-domain TRM

According to Eq. (5), the intensity of weak-field TRM is

proportional to the applied field strength H, since tanh(a) / a

for small a. The proportionality between M

TRM

and H in weak

fields (Figure M126) has been verified by measuring TRM

acquisition curves in the weak-field region for synthetic oxidized SD

titanomagnetites Fe

2.3

0.1

0.6

over a range of oxidation para-

meters 0.15 < z < 0.41 (Özdemir and O’Reilly, 1982). Figure M126

also shows a decrease in the intensity of TRM with increasing degree

of maghemitization. This fall in intensity is about the same as that

reported to take place in submarine basalts with increasing distance

from mid-ocean ridges.

Although Eq. (5) gives a reasonable match to observed absolute

TRM intensity in the weak-field region, the Néel SD theory is less suc-

cessful in explaining TRM acquisition for higher fields, H 01Oe

or 0.1 mT. The TRM intensity predicted by Eq. (5) reaches saturation

at quite low fields. Experimentally, SD TRM usually saturates in

much larger fields, on the order of 100 Oe or 10 mT (Figure M127).

This discrepancy has been attributed to either angular dispersion of

particle axes (Stacey and Banerjee, 1974) or particle interactions.

Dunlop and West (1969) modified Eq. (5) by introducing an interac-

tion field distribution that was estimated from experimental Preisach

diagrams.

Figure M126 The acquisition of TRM in weak fields for single

domain titanomaghemites with composition Fe

2.5

0.1

0.4

(ATM/60). TRMs are linear with H and decrease with increasing

oxidation parameter z (after O

zdemir and O’Reilly, 1982).

610 MAGNETIZATION, THERMOREMANENT

TRM intensity varies strongly with grain size d. Eq. (5) predicts an

increase in M

TRM

in proportion to increasing grain volume V. Figure

M128 shows the grain-size dependence of TRM intensity for SD

hematites. M

TRM

increases in almost exact proportion to grain size d,

not d

. A similar dependence is observed for magnetite grains smaller

than the critical SD size d

 0.1 mm, the threshold between SD and non-

SD states (Figure M129). Above d

, M

TRM

decreases with increasing

grain size, as d

–1

below 1 mmandasd

–0.55

above 1 mm. Thus, the Néel

SD theory does not explain experimental TRM data for magnetite over

most of its natural grain-size range. However, Néel’s SD theory is more

relevant to TRM in SD hematites because hematite has a small M

(2

kA m

–1

)andalarged

One of the important properties of SD TRM is the equality of the

blocking temperature T

during field cooling and the unblocking

temperature T

during zero-field reheating. Experimentally, as in

Figure M126, and theoretically according to Eq. (5), the intensity of a

weak-field TRM, and the intensities of all the partial TRMs with differ-

ent T

values that compose it, are proportional to the applied field

strength. The equality of T

and T

for weak fields and the proportion-

ality of TRM and pTRM intensities to field H form the basis for deter-

mining paleofield intensity by the Thellier and Thellier (1959) method.

Field dependence of blocking temperature has been examined

by Sugiura (1980) and Clauter and Schmidt (1981). T

(H)as

predicted by Eq. (4) has been verified directly by measuring pTRMs

covering the entire blocking-temperature spectrum for SD magnetites.

Sugiura (1980) obtained a close match of theoretical and experimental

spectra by assuming a single value of V and subdividing the

spectrum into 11 fractions (Figure M130). The blocking-tempera-

ture spectrum shifts to lower temperature as H increases. Unblocking

temperatures during zero-field heating are therefore always higher than

blocking temperatures observed during in-field cooling (Dunlop and

West, 1969).

Equation (4) predicts that blocking and unblocking temperatures

vary with grain volume V. Experimentally determined blocking tem-

peratures (Dunlop, 1973) agree well with the theoretical blocking

temperatures calculated with a 3D micromagnetic model (Winklhofer

et al., 1997). Small SD grains have lower blocking and unblocking

temperatures than large SD grains. The predicted quadratic depen-

dence of T

on the applied field is confirmed experimentally.

The dependence of TRM intensity on the rate of cooling is of prac-

tical interest to paleomagnetists studying slowly cooled orogens.

Figure M127 Experimental TRM acquisition curves for a number of synthetic magnetites with mean grain sizes from 0.04 to 0.22 mm

compared to the prediction of Ne

el’s (1949) theory of noninteracting SD grains. The theory cannot explain the rapid initial rise of the

experimental curves combined with their gradual approach to saturation (after Dunlop and West, 1969).

Figure M128 Grain-size dependence of TRM intensity for the

single-domain hematites. The dashed line has slope 1.04 showing

that M

TRM

is very nearly proportional to d . Open circles:

experimental data points; crosses: theoretical TRM values from

el (1949) SD theory (after O

zdemir and Dunlop, 2002).

MAGNETIZATION, THERMOREMANENT 611

Dodson and McClelland-Brown (1980) and Halgedahl et al. (1980)

predicted that the intensity of TRM in SD grains should increase for

longer cooling times, possibly by as much as 40% between laboratory

and geological settings. Experiments by Fox and Aitken (1980) for

baked clay samples containing SD magnetites showed that there was

about a 7% decrease in the intensity of TRM when the cooling time

changed from 2.5 h to 3 min.

Partial TRM and paleofield intensity determination

According to Eq. (4), T

depends on grain volume V and microscopic

coercivity H

. Any real rock has a distribution f (V, H

) of both V and

, and as a result will have a spectrum of blocking temperatures.

Therefore, the total TRM of a rock will consist of a spectrum of partial

TRMs, each carried by grains with similar (V, H

). When a rock cools

from T

to T

in a field H, SD ensembles with T

 T

pass from

the unblocked to the blocked condition, acquiring a partial TRM,

PTRM

, T

, H).

Thellier (1938) showed that SD partial TRMs follow experimental

laws of additivity, reciprocity, and independence. Independence: Par-

tial TRMs acquired in different temperature intervals are mutually

independent in direction and intensity. Each partial TRM disappears

over its own blocking-temperature interval. Additivity: Partial TRMs

produced by the same H have intensities that are additive. This is

expected theoretically because the blocking-temperature spectrum can

be decomposed into nonoverlapping fractions, each associated with

one of the partial TRMs. The total TRM is the sum of partial TRMs cov-

ering the entire blocking-temperature interval from T

to room tempera-

ture because each partial TRM contains a unique part of the T

spectrum

of the total TRM. Reciprocity: The partial TRM acquired between T

and T

during cooling in H is thermally demagnetized over the interval

, T

) when heated in zero field. In other words, the blocking and

unblocking temperatures are identical for weak fields.

Since partial TRM acquired at T

during cooling is erased at T

dur-

ing heating, one can replace the natural remanent magnetization

(NRM) M

NRM

of a rock in a stepwise fashion with a laboratory

TRM. This is done in practice by a series of double heatings, the first

in zero field and the second in a known laboratory field H

LAB

. If the

NRM of a rock was acquired as a TRM in an ancient geomagnetic

field H

, M

NRM

should be proportional to H

. The intensity of the

ancient geomagnetic field is then given by

¼ M

NRM

LAB

TRM

;

Figure M129 Weak-field TRM as a function of magnetite grain size d. M

TRM

decreases approximately as d

–1

between 0.1 and

1 mm and less strongly as d

–0.55

above 1 mm (after Dunlop and Argyle, 1997).

Figure M130 Continuous cumulative TRM spectra measured

during cooling of single domain magnetite (points and solid

curves) compared with theoretical stepwise spectra for H ¼ 2,

100, and 400 Oe (after Sugiura, 1980).

612 MAGNETIZATION, THERMOREMANENT

each pair of heating steps giving an independent estimate of H

. This

is the Thellier and Thellier (1959) method of paleointensity determina-

tion as modified by Coe (1967).

Multidomain TRM

In large grains containing many domain walls, TRM acquisition results

from several processes: domain wall pinning, domain nucleation or

denucleation, and nucleation failure. A multidomain grain contains

crystal imperfections such as inclusions, voids, and line defects such

as dislocations, all of which create surrounding stress fields. These

defects tend to pin domain walls. Voids and inclusions reduce the

volume of a wall and create magnetic poles, affecting both the wall

energy and demagnetizing energy. Dislocations pin walls by the mag-

netoelastic interaction between their stress fields and the spins in the

wall, which are rotated out of easy axes (Özdemir and Dunlop, 1997).

Pinned domain walls can be thermally activated just as SD moments

can. Wall displacements in response to a weak field applied at high

temperatures, like reversals of SD grains, can be frozen in by cooling

to room temperature, resulting in TRM. The Néel (1955), Everitt

(1962), and Schmidt (1973) multidomain TRM theories consider the

wall-defect interaction and the field dependence of the blocking tem-

perature. These theories follow Néel (1949) SD theory fairly closely

except that V becomes the volume of one Barkhausen jump of a

domain wall and H

becomes the critical field for such a Barkhausen

jump.

McClelland and Sugiura (1987) observed that during cooling below

the blocking range, after H had been zeroed, multidomain partial

TRM did not remain constant but decreased in real terms, i.e., M(T)/

(T). This observation compromises the concept of the blocking

temperature, because evidently, magnetization can partially relax dur-

ing cooling below T

. Later Shcherbakov et al. (1993) developed a

kinetic model that invokes nucleation or denucleation of domain walls

as a way of explaining changes in M(T ) below T

The only direct observations of domain nucleation during zero-field

warming or cooling are by Heider et al. (1988) and Ambatiello et al.

(1999) for magnetite and by Metcalf and Fuller (1987) and Halgedahl

and Fuller (1983) for titanomagnetite. In each case, any changes in the

number or widths of domains occurred either close to T

or in a range

of T where anisotropy changes rapidly, just above room temperature in

magnetite. In intermediate T ranges, where most observations of partial

TRM relaxation have been made, there is no evidence for nucleation.

A related phenomenon is nucleation failure. Titanomagnetite grains

field-cooled from T

sometimes nucleate fewer than the equilibrium

number of domain walls, and occasionally none at all (Halgedahl

and Fuller, 1980). The latter structure is of particular interest because

single-domain TRM is so intense that a small fraction of such grains

in metastable SD states could account for multidomain TRM. In parti-

cular, the size dependence of TRM in grains larger than SD size is then

explicable.

There are fundamental differences between SD and most MD TRM

models. The internal demagnetizing field H

¼ –NM favors a config-

uration of domain walls with minimum net moment. Loosely pinned

domain walls may undergo a series of Barkhausen jumps during heat-

ing and cooling. Therefore, there is no single blocking temperature for

a given domain wall or for a grain containing many walls since a jump

by one wall alters the internal field at the location of every other wall.

This is an alternative way of explaining TRM relaxation below T

el and Schmidt two-domain TRM theories

Néel’s (1955) 2D theory is based on the temperature dependence of

the wall-defect interaction and the wall displacement driven by the

internal demagnetizing field H

¼N M where M is the local magne-

tization vector. A domain wall, which has been displaced by H, will

reequilibrate its position at each new temperature during cooling under

the influences of H

and of local potential wells created by interaction

between the domain walls and lattice defects (Dunlop and Özdemir,

1997). The pinning strength is measured by the microcoercivity H

and determined by the barrier height between adjacent local energy

minimums (LEMs). Figure M131 shows the total energy in a two-

domain grain with many identical wall energy (E

) barriers, each with

the same coercivity H

for positive and negative jumps. The central

position of the wall is favored by the parabolic energy well due to the

demagnetizing energy E

. As the applied H increases, the parabolas

are tilted to the right by the field energy E

and the wall jumps from

one LEM to another in a series of Barkhausen jumps. Repeated jumps

of the wall generate the ascending part of the hysteresis loop, which

has a slope 1/N. The barriers to wall motion are not symmetric in

increasing and decreasing H.AsH decreases, the wall is pinned in a dif-

ferent set of LEMs and generates a descending loop with the same slope

1/N. When H becomes zero, the grain is left with a displaced wall and a

net moment, which is an isothermal remanent magnetization.

As the temperature decreases, the demagnetizing field and the bar-

riers between wells grow. At the blocking temperature T

, the potential

barriers E

begin to grow more rapidly than the demagnetizing field

pushing the wall back toward the demagnetized state and the wall

is trapped. TRM is frozen in. The basic result of Néel’s 2D theory is

that

TRM

¼ 2H

1=2

=N: (Eq. 6)

Equation (6) predicts absolute TRM intensity at room temperature.

The TRM is blocked by the growth of energy barriers to wall motion

Figure M131 Total energy, E

þ E

, as a function of wall

displacement x for various fields (after Schmidt, 1973).

Downward arrows show successive local energy minima in

which the wall is trapped as H increases from 0 to 50 Oe. Upward

arrows indicate minima where the wall is pinned as H decreases

from 50 to 0 Oe. The wall occupies different minima in increasing

and decreasing fields of the same strength, giving rise to magnetic

hysteresis and remanence.

MAGNETIZATION, THERMOREMANENT 613

and thermal fluctuations are ignored. This is referred to as field-

blocked TRM.

Néel (1955) pointed out that field blocking of TRM occurs very

close to T

if H is small. At such high temperatures, thermal fluctua-

tions can unpin a domain wall below its blocking temperature T

. Néel

introduced a thermal fluctuation field H

such that a wall cannot be

blocked until

ðT

Þ¼H

; (Eq. 7)

where H

) is the thermal fluctuation blocking temperature. TRM

intensity in the presence of thermal fluctuations is given by

TRM

¼ H

1=2

H=N : (Eq. 8)

Thermally blocked TRM should be proportional to H as observed

experimentally for small fields.

Schmidt’s (1973) MD model is based on the temperature varia-

tion of total energy in a two-domain grain with a single wall (Figure

M131). The model has the same physical basis as Néel’s 2D theory,

but the coercivity appears only as a derived quantity. The total

energy has contributions due to the interactions between the domain

wall and the applied field, the demagnetizing field, and lattice

defects. The temperature dependence of wall-defect interaction is con-

tained in a term m

, where m is the reduced domain magnetization

m(T) ¼ M/M

(T). The index p takes a value between 2 and 10

depending on the relative importance of magnetostriction or magne-

tocrystalline anisotropy in the wall pinning. The basic results of the

Schmidt model are that

ðTÞ/mðTÞ

p1

; (Eq. 9)

TRM

/ H

11=ðp1Þ

; (Eq. 10)

TRM

/ H

1=ðp1Þ

: (Eq. 11)

These are identical to Néel’s predictions (Eq. (6)) when p ¼ 3.

Stacey multidomain TRM theory

Stacey (1958) considered the blocking temperature T

to be indepen-

dent of the applied field and neglected the interaction between the

domain wall and lattice defects. At high temperatures, T

< T < T

the walls are highly mobile and take up positions in which the internal

field H

¼ H  NM is zero. Above T

, the magnetization is given by

M ¼ H/N. During cooling below T

, the walls’ displacements are fro-

zen, so that M changes only by the reversible increase in M

(T ). At

room temperature, the TRM is given by

TRM

¼ HM

=½NM

ð1 þ N w

Þ; (Eq. 12)

where w

is the initial volume susceptibility. Equation (12) seemingly

predicts the linear dependence of TRM on the applied field observed

experimentally for weak fields. In reality, as Néel (1955) showed,

blocking temperature T

is itself dependent on H. TRM as given by

Eq. (12) actually varies approximately as H

1/2

, a dependence observed

experimentally only when H > 1 mT.

Fabian phenomenological TRM model

Fabian (2000) has proposed a phenomenological approach in which a

sample is described in terms of a distribution function w(T

, T

). This

formulation establishes no direct link between (T

, T

) and physical

properties like grain size and coercivity, but recognizes that for

MD grains T

6¼ T

in general, i.e., Thellier reciprocity does not

hold. With this simple approach, it is possible to explain many

observed properties of multidomain partial TRMs and to justify the

kinematic equation for partial TRM relaxation during cooling or heat-

ing (Shcherbakov et al., 1993). Although not a theory of TRM acqui-

sition in the fundamental sense, the Fabian model is useful in

describing and connecting a body of experimental data and making

predictions about the TRM properties of a particular sample based

on its measured w(T

, T

Experimental multidomain TRM

There are relatively few testable predictions of multidomain TRM the-

ories. The prediction that TRM intensity should increase with increas-

ing H

(Eqs. (6), (8), (11)) was verified in a general way in Néel’s

original paper and the test has not been improved upon since. The pre-

dicted dependence on applied field H has received more attention.

Tucker and O’Reilly (1980) showed that in the low-field region (H

 1kAm

–1

, where 80 A m

–1

 1Oe¼ 0.1 mT), the TRM of large

titanomagnetite crystals, as predicted by Eq. (8), was more or less pro-

portional to H (Figure M132). For intermediate field strengths, the

field dependence was as a power of H < 1(Eqs. (6) or (10)), and

saturation occurred around 10 kA m

–1

. Similar results for magnetite

and magnetite-bearing rocks were reported by Dunlop and Wadding-

ton (1975) and for pyrrhotite by Menyeh and O’Reilly (1998).

Figure M132 TRM acquisition curves for millimeter-size single

crystals of titanomagnetite with composition Fe

2.4

0.6

(TM60),

plotted bilogarithmically (after Tucker and O’Reilly, 1980). The

experimental data have been fitted by a series of linear segments

corresponding to power law dependences of TRM on applied

field. TRM intensity is proportional to H for fields below about

2 mT, then increases approximately as H

1/2

until it saturates at

the value M

¼ H/N, in agreement with Ne

el’s (1955) theory

of TRM due to domain wall displacement limited by

self-demagnetization.

614 MAGNETIZATION, THERMOREMANENT

An interesting sidelight on multidomain TRM behavior is the fact

that for H of the order of the Earth’s magnetic field, weakly magnetic

hematite (M

 2 kA/m) in a multidomain state has a much stronger

TRM than MD magnetite (M

¼ 480 kA m

–1

) or SD hematite (Figure

M133). The reason is that the weak M

of hematite results in a negli-

gible demagnetizing field H

, so that even a small H is sufficient to

drive walls to their limiting positions and saturate the TRM (Dunlop

and Kletetschka, 2001). MD magnetite does not approach saturation

TRM except in fields 3 orders of magnitude larger because of its cor-

respondingly greater H

The model of Fabian (2000) was a response to recent experimental

work on multidomain partial TRMs. The thermal demagnetization

data of Figure M134 are one example. Partial TRMs produced by

applying H over a narrow blocking interval T

¼ 370–350



C are

demagnetized over increasingly broad intervals of unblocking tem-

perature T

as the grain size increases. While partial TRMs of

1 and 6 mm magnetites demagnetized mainly over the original T

interval, partial TRMs of 110 and 135 mm grains demagnetized

almost entirely outside the 370



C–350



C interval. This almost total

violation of reciprocity led Fabian (2000) to treat T

and T

independent variables.

Özden Özdemir

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Figure M134 Stepwise thermal demagnetization of partial TRMs

produced by a small field applied during cooling from 370 to

350



C in magnetites ranging from small PSD to MD size. About

50%–90% of the remanence unblocks at temperatures below or

above the PTRM blocking temperature range. Tailing of thermal

demagnetization curves is not confined to large MD grains but

is significant even for fairly small PSD grains (after Dunlop and

zdemir, 2001).

Figure M133 Comparison of experimental (points and solid

curves) and theoretical (dashed curves) TRM values for SD and

MD hematites and magnetites (Dunlop and Kletetschka, 2001).

The contrast in TRM intensities for two minerals is due to the

internal demagnetizing field H

¼NM. At saturation,

 200 mT for magnetite, but for hematite H

 1 mT,

making it much easier for a small field like the Earth’s to push

the magnetic domain walls to their limiting positions.

MAGNETIZATION, THERMOREMANENT 615