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

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Here, N

FID

is the number of intensity-inclination-declination triplets;

is the number of inclination-declination pairs, etc. Maximizing L

is accomplished numerically (Press et al., 1992), an exercise yielding

a particular paleomagnetic vectorial mean and variance. Insofar as

the Gaussian distribution is an appropriate description of the paleovec-

tor field, then in the limit of large number of data, the maximum-like-

lihood method yields unbiased estimates of the vectorial mean

intensity and the vectorial mean direction (Love and Constable,

2003), which is not usually the case with the traditional method of

making separate numerical averages of intensity data and unit direc-

tional vectors (Creer, 1983).

Applic ation example

In Figure S47 we show the results (after Love and Constable, 2003) of

a maximum-likelihood analysis using the 3D bi-Gaussian distribution

and fitting paleomagnetic data covering the past 5 Ma from both

Hawaii and Réunion. Comparison of data from these two sites is of

interest since they are on almost opposite latitudes, and therefore the

asymmetry seen in the data, most prominently in inclination, is indica-

tive of mean-field ingredients other than a simple geocentric axial

dipole. The Réunion data are fitted much better than the Hawaiian data

by the bi-Gaussian distribution with scalar variance, and we can say,

therefore, that the Réunion data are relatively “ isotropic” in their vec-

torial variance, while the Hawaiian data display an “anisotropy ” in

vectorial variance. Better fits to the Hawaiian data would require the

introduction of covariance into the underlying Gaussian distribution

functions; nonetheless this comparative analysis is enlightening. Soft-

ware for fitting the 3D Gaussian distributions to paleomagnetic data

can be obtained at http://geomag.usgs.gov.

Jeffrey J. Love

Bibliography

Bingham, C., 1964. Distributions on the sphere and on the projective

plane. Ph.D. thesis, Yale University, New Haven.

Bingham, C., 1983. A series expansion for the angular Gaussian distri-

bution. In Watson, G.S. (ed.) Statistics on Spheres. New York:

John Wiley and Sons, pp. 226–231.

Creer, K.M., 1983. Computer synthesis of geomagnetic palaeosecular

variations. Nature, 304: 695–699.

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

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

Khokhlov, A., Hulot, G., and Carlut, J., 2001. Towards a self-

consistent approach to paleomagnetic field modelling. Geophysical

Journal International, 145: 157–171.

Love, J.J., and Constable, C.G., 2003. Gaussian statistics for palaeo-

magnetic vectors. Geophysical Journal International , 152:

515–565.

McFadden, P.L., and McElhinny, M.W., 1982. Variations in the

geomagnetic dipole 2: Statistical analysis of VDMs for the past 5

million years. Journal of Geomagnetism and Geoelectricity, 34:

163–189.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P.,

1992. Numerical Recipes. Cambridge, UK: Cambridge University

Press.

Stuart, A., Ord, K., and Arnold, S., 1999. Kendall’s Advanced Theory

of Statistics, Volume 2A, Classical Inference and the Linear Model.

London: Arnold.

Cross-references

Bingham Statistics

Dipole Moment Variation

Fisher Statistics

Geocentric Axial Dipole Hypothesis

Magnetization, Remanent

Nondipole Field

Paleomagnetic Secular Variation

Paleomagnetism

STORMS AND SUBSTORMS, MAGNETIC

Introduction

Magnetic storms were first defined in the mid-19th century when large

variations in the horizontal intensity of the magnetic field were mea-

sured at a variety of locations on the surface of the Earth. Although

much work was subsequently initiated on this topic, it has been the

space age that has led to a more detailed understanding of the phenom-

ena involved in magnetic storms. It is now established that magnetic

storms are an element of the interaction between processes that occur

on the Sun, the coupling between the solar wind and the Earth’s mag-

netosphere, and the subsequent energization of particles within the

Earth’s magnetosphere (Tsurutani et al., 1997). Furthermore, it is clear

that storms occur as a result of abnormal conditions at the Sun and in

the solar wind. As a result, the effects during magnetic storms on the

space environment surrounding the Earth can be serious in terms of

human activities in space and on the ground.

The term “magnetospheric substorm” is used to describe a range of

associated phenomena that occur in the magnetosphere. The word sub-

storm was initially used in the early part of the 1960s to portray rapid,

repeatable variations of the polar magnetic field during magnetic

storms. In order to characterize the overall phenomenology of auroral

disturbances, the term was modified to the auroral substorm (Akasofu,

1964), before becoming more widely incorporated as the magneto-

spheric substorm in the 1970s. Substorms can be considered to be part

of the normal solar wind magnetosphere interaction.

Magnetic storms

Magnetic storms are most easily observed in equatorial or low-latitude

magnetograms as a depression in the magnetic field. In order to pro-

duce a simple means of identifying a magnetic storm, the Dst index

was derived, which is based on the change in the horizontal compo-

nent of the magnetic field measured at a number of low-latitude sta-

tions, which are separated in longitude. A schematic representation

of the Dst index for an individual storm is given in Figure S48, which

illustrates the three specific intervals or phases of a storm, each of

which has different timescales.

The initial phase, often but not always, follows a rapid enhancement in

the Dst index, over a timescale of a few minutes, which is referred to as a

storm sudden commencement (SSC) and is caused by a rapid increase

in the solar wind pressure incident on the Earth’s magnetosphere.

Figure S48 A schematic representation of the Dst index during a

typical magnetic storm.

926 STORMS AND SUBSTORMS, MAGNETIC

The initial phase lasts up to a few hours and is the period during which

the Dst remains at a high level following the SSC. The second phase is

the main phase of the storm, which is the time frame over which the

Dst index reduces to its minimum value and represents the period of

the main energy transfer and storage into the Earth’s magnetosphere.

The main phase of the storm typically lasts for 5–10 h. The final phase

of the storm is the recovery phase, during which the Dst returns to its

prestorm values and this phase lasts for about 10–15 h. Thus, the time-

scale for the whole storm is typically 24–30 h. Minimum values of Dst

which can be reached are typically several hundred nanoteslas.

The Dst index, however, is just a means to identify the storm. The

cause of these changes in the magnetic field is in the magnetosphere

and is referred to as the ring current. This current flows in a westward

direction in the equatorial plane of the magnetosphere and during

storms ultimately surrounds the whole Earth at a distance between

about 2 and 7R

from the center of the Earth (where R

is the radius

of the Earth and is approximately 6380 km). The ring current is carried

by ions in the energy range 20–200 keV and the main phase of the

storm is the time during which the ring current increases in intensity

while during the recovery phase the ring current decays.

The initiators of magnetic storms are solar phenomena such as cor-

onal mass ejections, solar flares, and coronal holes. These phenomena

result in unusual conditions in the solar wind, for example, high values

of the solar wind density and velocity (and hence pressure) and the

interplanetary magnetic field (IMF). The orientation of the IMF is cen-

tral to the energy transfer from the solar wind to the magnetosphere

such that when the IMF has a southward component, the energy trans-

fer is most efficient and occurs through the process of magnetic recon-

nection between the IMF and the geomagnetic field.

Magnetospheric substorms

A magnetospheric substorm comprises three separate phases: the

growth, expansion, and recovery phases (e.g., Baker et al., 1996).

The growth phase is the interval during which energy transfer from

the solar wind is the main process and this results in storage of that

energy in the tail of the magnetosphere together with a small amount

of dissipation in the ionosphere. The expansion phase is the interval

during which the energy that has been stored in the tail is explosively

released, with some energy appearing in the ionosphere, some in the

ring current, and some being lost to the solar wind. Finally, the recov-

ery phase is an interval when the magnetosphere-ionosphere system

relaxes back to a quiescent state. The overall timescale of a magneto-

spheric substorm is typically 2– 4h.

The growth phase of the magnetospheric substorm begins with the

onset of a period of southward IMF at the dayside magnetopause,

which leads to magnetic reconnection between the IMF and the geo-

magnetic field, thereby creating open magnetic flux, which is con-

nected at one end to the Earth and the other to the solar wind

plasma. This newly opened magnetic flux is transported by the motion

of the solar wind into the magnetotail where it is added to the lobes. In

this way, magnetic energy, B

/2m

per volume, is stored in the tail

as the lobe magnetic field intensity increases. A consequence of the

addition of this magnetic flux to the tail lobes is the reorientation of

the magnetic field in the near-Earth tail, for example at geostationary

orbit, where the field becomes less dipolar-like and more like the

stretched mid- and far-tail field. This reorientation results from the for-

mation and subsequent intensification of a thin current sheet in the

inner magnetosphere termed “crosstail current.”

The expansion phase of the magnetospheric substorm is normally

accepted as starting with the brightening of the equatorward most

auroral arc in the nightside ionosphere. This brightening begins in a

localized region near midnight and propagates along the length of

the arc both westward and eastward at phase velocities of up to

10 km s

1

. This is then followed by a poleward expansion of the aur-

oral luminosity at speeds of typically a few hundred m s

1

. This is

known as the auroral breakup because of the rapid deformation of

the original quiet arc. Accompanying the auroral breakup is a host of

phenomena of which the most important with regard to expansion

phase onset identification are the Pi2 pulsation and the westward elec-

trojet, both observed in the magnetic field. Pi2 pulsations are MHD

waves with periods between 40 and 200 s, which are impulsive and

last for a few cycles. The westward electrojet is the current that flows

in the ionosphere primarily in the enhanced conductivity region asso-

ciated with the auroral activity. This electrojet is sensed on the ground

by magnetometers and has characteristic spatial variations in the hori-

zontal and vertical intensities of the magnetic field. In the near-Earth

tail, the enhanced crosstail current decreases, resulting in a change in

the orientation of the field back to more dipolar like. The cause of this

change in the crosstail current remains uncertain and is the source of

much controversy. The energy stored in the tail is released in a bursty

fashion, with the trigger for the release as yet unknown, and there may

be more than one burst of energy release, resulting in secondary sub-

storm intensifications after the expansion phase onset.

The recovery phase is the phase during which the system returns to

a quiescent state. Overall, this results in the magnetic field becoming

less disturbed and the auroral activity declining. Despite this, however,

there are still dynamic variations in the field during the recovery

phase, which illustrate that the process is by no means simple.

Outstanding questions

Regarding magnetic storms, the outstanding questions concern the

growth and decay of the ring current. The mechanisms by which par-

ticles can be energized to increase the ring current intensity remain

unclear. Furthermore, once the ring current is formed, the mechanisms,

which lead to the loss of particles from the ring current and the conse-

quent decay of the ring current, are still debated. It is likely that waves

over a range of frequencies play central roles in both these processes.

Debate also continues over the solar cause of magnetic storms in terms

of which phenomenon is the most important.

The outstanding questions regarding magnetospheric substorms cen-

ter on the expansion phase onset. Several models exist for this and can

be summarized thus. Reconnection at a near-Earth neutral line in the

tail (approximately 20R

downstream from Earth) leads to a disruption

of the enhanced tail current formed during the growth phase. This dis-

ruption propagates earthward, resulting in signatures in the ionosphere

and at geosynchronous orbit (Baker et al., 1996). Alternatively, the

current disruption could be caused nearer to the Earth (approximately

6–10R

) as a result of some instability, which is responsible for the

signatures in the ionosphere and geosynchronous orbit, and then pro-

pagates tailward, causing the reconnection at a near-Earth neutral line

(Lui, 1996). In addition, there is a question over what triggers the

expansion phase onset (Lyons, 1996). It is unclear if a threshold has

to be reached in the state of the tail or whether external triggers are

sufficient to drive the process. Finally, the linkage between storms

and substorms is more complex than previously thought and requires

further attention (Kamide et al., 1998). These and many other ques-

tions concerning magnetic storms and substorms will be answered

over the next decade of space research.

Mark Lester

Bibliography

Akasofu, S.I., 1964. The development of the auroral substorm. Plane-

tary and Space Science, 12: 273.

Baker, D.N., Pulkkinen, T.I., Angelopoulos, V., Baumjohann, W., and

McPherron, R.L., 1996. Neutral line model of substorms: past

results and present view. Journal of Geophysical Research, 101:

12975.

Kamide, Y., et al., 1998. Current understanding of magnetic storms:

storm-substorm relationships. Journal of Geophysical Research,

103: 17705.

STORMS AND SUBSTORMS, MAGNETIC 927

Lui, A.T.Y., 1996. Current disruption in the earth’s magnetosphere:

observations and models. Journal of Geophysical Research, 101:

13067.

Lyons, L.R., 1996. Substorms: fundamental observational features,

distinction from other disturbances, and external triggering. Jour-

nal of Geophysical Research, 101: 13011.

Tsurutani, B.T., Gonzalez, W.D, Kamide, Y., and Arballo, J.K., 1997.

Magnetic Storms. Washington, DC: American Geophysical Union.

Cross-references

Archeomagnetism

Magnetization, Natural Remanent (NRM)

Paleomagnetism

Rock Magnetism

SUPERCHRONS, CHANGES IN REVERSAL

FREQUENCY*

The geomagnetic reversal timescale appears to contain intervals of

four distinct durations. They are called cryptochrons, subchrons,

chrons, and superchrons, with durations of the order of thousands of

years, tens of thousands, hundreds of thousands, and tens of millions.

Cryptochrons are observed as excursions and are interpreted as aborted

reversals; empirically they are said to differ from secular variation if

the geomagnetic pole moves more than 45



from the geographic pole.

Theoretically an excursion differs from a pair of reversals if the inter-

mediate state fails to completely reverse, that is change from B to B

(Gubbins, 1999). There is probably no theoretical distinction between

subchrons and chrons. Superchrons are distinct periods of one polarity

lasting longer than any natural timescale for the geodynamo. Since this

implies a cause related to some longer term evolution of the Earth,

such as mantle convection or inner core growth, any theory aimed at

explaining superchron behavior must also address long-term changes

in the frequency of reversals, unless it invokes some sudden cata-

strophic change in the state of the Earth’s core.

Theories for long-term changes in reversal rate can be listed under

five categories:

1. Natural nonlinear behavior of the geodynamo

2. Growth and changes of the inner core

3. Changes in lower mantle electrical conductivity

4. Changes in total heat flow from the core

5. Changes in the geographical pattern of heat flow from the core

All theories are in a very rudimentary state as it has only recently

become possible to test speculations against dynamo theory.

There are two clear periods in the past when the field remained the

same polarity for a considerable length of time: from about 119 to

84 million years ago (Ma) the field was normal (the Cretaceous Nor-

mal Superchron—CNS) while from about 312 to 262 Ma the field

was reversed (the Permo-Carboniferous Superchron—PCRS). The

CNS is established beyond doubt because it is recorded on the ocean

floors; any but the shortest of reversed intervals would have been dis-

covered by now. The PCRS is older than the oceans and therefore nor-

mal intervals may remain to be discovered.

312 Myr still represents a small fraction of the age of the Earth and

of the geomagnetic field. A number of authors have suggested the pos-

sible existence of another superchron during the Ordovician (see, e.g.,

Gallet et al. (1992)). Using a global paleomagnetic database, Johnson

et al. (1995) derived a model of long-term reversal behavior for the

last 570 Myr. In addition to the two established superchrons (the

CNS and PCRS), they found a long period of very low reversal rate

in the Ordovician (502–470 Ma), which they suggested may be a pre-

viously unidentified superchron.

Johnson et al. (1995) also found a short period of anomalously low

reversal rate around the Jurassic-Triassic boundary (212–185 Ma).

However, Kent and Olsen (1999) in their analysis of cores from the

Newark rift basin in North America found no evidence for an interval

of low reversal rate around this time. They suspect Johnson et al.’s

results may be an artifact of sampling bias. Algeo (1996) carried

out a study of geomagnetic bias patterns through the Phanerozoic.

In addition to the CNS and PCRS, he found four other first order

polarity features—an interval of reversed polarity bias in the middle

Cambrian-middle Ordovician and three intervals of normal polarity

bias in the late Ordovician-late Silurian, in the early Jurassic, and in

the middle Jurassic.

Gallet and Pavlov (1996), in an analysis of data from the Moyera

river section of northwestern Siberia, found very few reversals, the

field being dominantly reversed from the upper Cambrian to the mid-

dle Ordovician, and normal from the upper Ordovician to the lower

Silurian. However only the Avenig (140 samples) has an acceptable

time resolution (100 Myr???) and they suggested the possibility of

a single reversed interval lasting at least 15 Myr during this time. This

is in agreement with the result from a composite sequence for the same

period by Trench et al. (1991), but not with those of Piper (1978). In a

later paper, Pavlov and Gallet (1998), using data from the Kulumbe

river section in the same area suggested a 25–30 Ma-long period dur-

ing the lower and middle Ordovician, dominated by reversed polarity

with very few reversals. Algeo (1996) also found strong polarity bias

during the first half of the Ordovician. Pavlov and Gallet (2001) later

returned to the Kulumbe river section, sampling older rocks, and found

a sequence of 28 magnetic polarity intervals, eight of which were

defined by only one sample. Since the section had a duration of

5 Myr, they proposed that the magnetic reversal frequency was high

(4–6 reversals per Myr) during part of the middle Cambrian, which

is amongst the highest rates known within the Phanerozoic. This rever-

sal frequency may also have existed during the lower Cambrian. They

further suggested a drastic decrease in the reversal frequency between

the lower-middle Cambrian and the end of the Tremodoc (Ordovician)

when a superchron may have occurred.

The reversal record suggests a gradual increase in reversal fre-

quency since the CNS, perhaps a result of slow changes on the time-

scale of mantle convection. McFadden and Merrill (1984) found an

approximately linear trend in the mean rate of reversals, increasing

from zero at 84 Ma at the end of the CNS to over four per Myr in

recent times. Visual inspection of the reversal sequence suggests that,

superimposed on this linear trend, there are oscillations with a period

of 

30 Myr There has been much controversy over the reality of a

30 Ma periodicity (or 15 Myr proposed by some authors) but on the

whole the claims of periodicities are not convincing (see Jacobs

(1994) for a review of the debate). Merrill and McFadden (1990) later

showed the pattern of reversals to have a linear decrease in reversal

frequency from about 165 to 119 Ma, the commencement of the

CNS. In a further paper, McFadden and Merrill (1997) showed that

the reversal rate prior to the CNS is larger than the rate after the CNS.

Gallet and Courtillot (1995) adopted a different approach, plotting

the successive lengths of polarity intervals as a function of their order

of occurrence in the sequence. The longer duration polarity intervals

become shorter, fewer and further part to indicate a change in behavior

at 15–30 Ma. Flat “white noise” characterizes the behavior from this

time to the present, while a quasiperiodic structure in magnetic interval

lengths is observed between 85 and 25 Ma. Such behavior is similar

to that proposed by Dubois and Pambrun (1990)—low-dimensional

deterministic chaos from 85–30 Ma and a random process from

25 Ma to the present.

Gallet and Hulot (1997) offered an alternative interpretation to that

of McFadden and Merrill (1997) for the evolution of the geomagnetic

*This article was left by Jack Jacobs in manuscript form and com-

pleted by the editor D. Gubbins, who takes responsibility for all errors

and omissions in the final article.

928 SUPERCHRONS, CHANGES IN REVERSAL FREQUENCY

field over the last 160 Myr. They plotted the duration of individual

polarity intervals as a function of their order of occurrence (as Gallet

and Courtillot (1995) had done) and suggested that the geomagnetic

polarity timescale could be split into three segments with different char-

acteristics. For the time period 160–130 Ma, they proposed that the

duration of magnetic polarity intervals was constant, i.e., during this

period, the reversal process, unlike that of McFadden and Merrill was

stationary. They suggested that at 130 Ma, a change occurred leading

to a discontinuous decrease in reversal frequency up to the CNS. The

CNS is followed by a period of gradually increasing reversal frequency

until 25 Ma, when the reversal process again became steady.

McFadden and Merrill (2000) derived a new statistic to distinguish

between the two extreme alternatives of themselves (1997) and of

Gallet and Hulot (1997). They showed that the data between 160 and

130 Ma require a linear trend towards longer polarity intervals, this trend

continuing to the beginning of the CNS, i.e., a discontinuity at 130 Ma

(as proposed by Gallet and Hulot) is neither required nor supported by

the data. Moreover the trend towards shorter polarity intervals from 85

Ma continues to at least 12 Ma, there being no need for a change in the

reversal process at 25 Ma as demanded by Gallet and Hulot (1997).

Constable (2000) obtained a new estimate for the reversal rate as a

function of time and was able to obtain confidence bounds on the

temporal probability density function for geomagnetic reversals. For

the period 0–158 Ma, she found that the derivative of the rate functions

of reversals must have changed sign at least once. The timing of this

sign change is constrained to be between 152 and 22 Ma at the 95%

confidence level. Unfortunately she was unable to accept or reject

either of the models of McFadden and Merrill and Gallet and Hulot.

Reversal frequency estimates for earlier times are hampered by the

lack of precise dating. There are very few records showing a sequence

of several reversals in rocks older than 1 Ga. Opdyke and Channel

(1996) compiled a histogram of the lengths of polarity intervals from

330 Ma to the present. Apart from the CNS and PCRS, the majority

of polarity intervals lie in the time range from 0.1 to 1 Myr, and there

are very few with durations longer than 2 Myr.

A related question is the relationship (if any) between the strength of

the Earth’s magnetic field and the frequency of reversals— in particu-

lar with superchrons. Fuller and Weeks (1992) have given a summary

of the two opposing views—either the field is strong, or it is weak dur-

ing a superchron. Pick and Tauxe (1993) found extremely low average

intensity for the CNS, and low values of the geomagnetic field during

the PCRS were found by Prévot and Perrin (1992) from a list of

paleointensity data determined from magmatic rocks up to 3.5 Ga,

and by Harcombe-Smee et al. (1996) from lavas in S.W.

The latest contribution to this controversial question has been given

by Tarduno et al. (2001), who found high geomagnetic intensities dur-

ing the mid-Cretaceous. They used single crystals of plagioclase feld-

spar from eight independent lava flows that erupted at different times

within 0.1–1 Ma from basalts of the Raymahal Traps of India

(113 –116 Ma). The average value from all the flows and the 56 crys-

tals with reproducible data are taken to represent the true dipole field

during the CNS. The averaging process removes possible sources of

error in the magnitude of the virtual dipole moment from the highly

variable (in space and time) nondipole components of the field, which

vary on a timescale shorter than 10 kyr and hence cancel out unless

data are averaged over 0.1–1 Myr. Their value of the mean intensity

of the field is significantly higher than other values for the past

180 Myr, and the times of the mean paleointensity for the mean Cen-

ozoic and early Cretaceous-late Jurassic when reversals were frequent.

The question is still open.

Turning to theory, we must first ask whether the geodynamo could

move spontaneously between reversing and nonreversing states,

whether in fact no special external changes are required at all to pro-

duce superchrons. The longest natural mescale is the magnetic diffu-

sion time (see Geodynamo, dimensional analysis and timescales),

25 kyr. If this Ordovician low reversal rate is substantiated, the

200 Ma spacing between the CNS, PCRS, and the Ordovician

superchron would be a fundamental time constant for long-term geo-

magnetic field behavior, simulation for anything like this time, but stu-

dies of simpler systems suggest it is highly unlikely (Jones and

Wallace, 1992). Barring some dramatic new findings, we can be fairly

confident that some change is needed external to the liquid core.

The inner core is known to stabilize numerical dynamo simulations

(Hollerbach and Jones, 1993). It is growing slowly as the Earth cools

(see Geodynamo, energy sources), and was therefore smaller in the

past. This may have made superchrons less likely in the distant past

(Gubbins, 1999). Some dynamo calculations have more stable solu-

tions without an inner core (Morrison and Fearn, 2000) because

the two hemispheres are better connected, but this may be a special

feature of this weakly driven model. For the inner core to account

for the changes since the Ordovician superchron it would have to

change size very significantly in 100 Myr, and repeatedly, with

implied warming of the entire core. There is as yet no evidence for

this, so the theory must remain a long shot.

This leaves changes in the lower mantle, which controls the boundary

conditions for the geodynamo. The electrical conductivity of the lower

mantle could have an effect if it were large enough, for example in sub-

stantial regions of entrained iron. This idea appears in theories of short-

term core-mantle coupling (see Core-mantle coupling) and of reversal

transition fields (Herrero-Bervera and Runcorn, 1997), but to affect the

geodynamo seriously the highly conductive layer would have to be of

10–100 km thick, and this would affect the short term secular variation

in a way that would surely have been observed by now.

This leaves thermal effects, which have received most attention to

date. One idea is that a change in the style of mantle convection draws

more heat from the core, as might follow for example from a major

reorganization of the plate boundaries at the Earth’s surface (it is inter-

esting to note that Venus had a resurfacing event at 500 Ga and no

longer has plate tectonics or magnetic field, but it may have had both

in the past). Naïvely one might expect the geomagnetic field to be

stronger and also to reverse in unstable fashion if the geodynamo is

being driven hard, and weak and stable if the driving is weak. How-

ever, Pal and Roberts (1988) and Larson and Olson (1991) argued per-

suasively that the field is strong during a superchron, arguing that a

thin D

layer at the bottom of the mantle would increase heat flow

from the outer core (OC) to the mantle and so create more vigorous

convection in the OC. The associated heat loss would lead to a strong,

stable, nonreversing field (a superchron). Loper and McCartney

(1986), McFadden and Merrill (1986) and Courtillot and Besse

(1987), on the other hand, had argued earlier against a strong magnetic

field during a superchron, which they claimed represents a low energy

state with infrequent instabilities corresponding to inactive periods in

the D

later when it is thick and heat transfer across the CMB rela-

tively small. This view receives some support from dynamo studies

(Jones and Wallace, 1992), which now include some sophisticated

3D numerical simulations (Kutzner and Christensen, 2002), although

their reversing solutions do not resemble the geomagnetic field very

much. Further support for low field strength during a superchron has

been given by Hide (2000), who carried out a mathematical investiga-

tion of nonlinear processes in self-exciting dynamos. In his models the

maximum strength of the magnetic field could be systematically higher

when reversals are frequently than when they are infrequent. He sug-

gested that a superchron is associated with a long quiescent period

when convection in the lower mantle is comparatively feeble.

The second idea invokes changes not in the average heat flux from

the core but the lateral variations in heat flux into the lower mantle

(see Core-mantle coupling, thermal). These lateral variations almost

certainly affect convection in the core, and there have been a number

of geodynamo simulations that incorporate inhomogeneous thermal

boundary conditions. A change in this pattern, which at present has

high heat flow around the Pacific and low below Africa and the central

Pacific, may change the style of convection from one that allows

reversals to one that does not. There is some evidence for this in recent

simulations (Glatzmaier et al., 1999). The theory has the appeal of a

SUPERCHRONS, CHANGES IN REVERSAL FREQUENCY 929

natural process of evolution of the Earth with the right timescale. This

study also indicates correlations between the frequency of reversals,

the direction, and the strength of the magnetic field. The most stable

geodynamo (i.e., nonreversing, low SV) is the most efficient with

the highest dipole moment. The near future will surely bring some

exciting new insights into this problem.

Jack A. Jacobs

Bibliography

Algeo, T.J., 1996. Geomagnetic polarity bias patterns through the Pha-

nerozoic. Journal of Geophysical Research-Solid Earth, 101:

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Cross-references

Core-Mantle Boundary, Heat Flow Across

Core-Mantle Coupling, Electromagnetic

Core-Mantle Coupling, Thermal

as a Boundary Layer

Dipole Moment Variation

Geodynamo

930 SUPERCHRONS, CHANGES IN REVERSAL FREQUENCY

Geodynamo, Dimensional Analysis and Timescales

Geodynamo, Energy Sources

Geomagnetic Polarity Reversals, Observations

Geomagnetic Polarity Timescales

Reversals, Theory

SUSCEPTIBILITY

What is magnetic susceptibility? This is a question I use to ask stu-

dents, and I have got many different answers. One even said that it

is the number on the display of instrument used. It is true that one

can find several definitions of this physical parameter, more or less

correct. For example, Encyclopedia Britannica gives quite nice descrip-

tion, saying that magnetic susceptibility is “quantitative measure of the

extent to which a material may be magnetized in relation to a given

applied magnetic field.” Let us improve this definition, saying that mag-

netic susceptibility is a parameter characterizing the nature and intensity

of response of material to external magnetic field.

Since magnetic properties of materials are mostly studied by apply-

ing a magnetic field (H) and measuring the magnetization (M) induced

in these materials, magnetic susceptibility is defined by equation relat-

ing M and H:

M ¼ w

which yields

¼ M=H;

where the subscript v denotes “volume.” Thus, w

is a material para-

meter—transfer function, linking induced magnetization and applied

magnetic field.

Magnetization induced by the applied magnetic field is due to the

effect of the field upon moving charged particles. Magnetism is

characteristic of all materials containing charged particles. Since con-

tribution of nucleons is very small and can be neglected, magnetism

can be considered as practically entirely of electronic origin. Thus,

magnetic susceptibility can be used to classify materials in terms of

their response to external magnetic field as diamagnetics, paramag-

netics, and a group of ferrimagnetics (for more details, reader is

referred to, e.g., Dekker (1957) or Dunlop and Özdemir (1997)).

However, the above definition of magnetic susceptibility is only

valid for linear behavior of induced magnetization in dependence of

the applied magnetic field. In other words, it can only be applied to

diamagnetic or paramagnetic materials, or ferrimagnetics in the range

of magnetic saturation. In all other cases, and these are more common

in natural rocks and minerals, more general definition applies



H;M

or w

¼ lim

H!0

M

H



H;M

Mathematically speaking, it represents an angle between the tangent

to the magnetization curve M(H) at given H and the H axis, and in

physics literature a term reversible susceptibility is sometimes used,

or differential for H ¼ M ¼ 0. From this it is clear, that magnetic

susceptibility depends not only on the material and the applied field,

but also on the way it is measured.

Diamagnetic susceptibility rises from the distortion of electronic

orbit due to applied magnetic field, forcing the electrons to additional

precession motion. It is very small, negative, and independent of tem-

perature and magnetic field

¼

;

where z is the number of electrons in atom, n is the number of atoms in

unit volume,

is the median of square of distances of electrons from

the nucleus. The magnitude of diamagnetic susceptibility is very small,

in the range of 10

–6

cgs Most organic compounds are diamagnetic

only. Among minerals, diamagnetism is expressed in quartz, plagio-

clase, calcite, and apatite.

In paramagnetic substances, the constituent atoms have unpaired

electrons, resulting in a net magnetic moment. At room temperature,

the individual atomic moments are randomly oriented. If an external

magnetic field is applied, they will tend to become aligned in a direc-

tion parallel to the external field. The resulting induced magnetization

is called paramagnetic, and the corresponding susceptibility is defined

by Curie law (first empirically derived by P. Curie in 1895):

;

where C ¼

3nm

and M

¼ nM ; k being the Boltzman constant:

The variation of the inverse susceptibility with temperature is linear,

with slope 1/C, which is related to the number of unpaired electrons

in the substance (n). The magnitude of paramagnetic susceptibility is in

the range of 10

–3

–10

–6

cgs and is temperature dependent (Figure S49).

Figure S49 Temperature behavior of paramagnetic susceptibility.

SUSCEPTIBILITY 931

In nature , paramagnetic miner als may include clay minerals (chlorite,

smectite, and glauconite), iron and manganese carbonates (siderite,

rhodochrosite), ferromagn esia n silicates (olivine, amphiboles, pyrox-

enes, etc.), as well as a variety of ferric-oxyhydroxide mineraloids.

In paramagnetic substances, individual atomic magnetic moments

do not interact with each other. However, in many materials, interac-

tions (of quantum-mechanic nature) between atomic moments result

in spontaneous parallel (or antiparallel) alignment of atomic magnetic

moments in the absence of external magnetic field, typical for a group

of ferromagnetic, ferrimagnetic, and antiferromagnetic materials (all

called herein as ferrimagnetics). In these materials, Curie’s law is

not obeyed and magnetization curve M(H) is not linear, but shows

hysteresis , instead. In this case, magnetic susceptibility cannot be

expressed on the basis of classical theory, as in the case of dia- or para-

magnetism. However, above the transition Curie (Néel) temperature,

material behaves paramagnetically, and the inverse value of magnetic

susceptibility follows a linear line defined by Curie-Weiss law

(Figure S50):

T  T

ðÞ

;

where T

is Weiss constant, often called paramagnetic Curie tempera-

ture. Note that mathematically T

can be < 0 K, which is physically

impossible, but typical for antiferromagnetics. Typical ferrimagnetics

are, for instance, magnetite and maghemite, while hematite is antifer-

romagnetic. There are no true ferromagnetic minerals in nature.

Magnetic response of ferrimagnetics becomes stronger as the num-

ber of unpaired electrons increases. Therefore, it is customary to

measure magnetic susceptibility of a substance as a function of its

weight, yielding mass-specific susceptibility. In order to distinguish

the volume and mass-specific susceptibility, the former one is com-

monly denoted with k (unlike most physics literature), while w is used

for the latter one:

w ¼

;

where k replaced w

and r is density.

Mass-specific susceptibility is related to molar susceptibility w

;

where W

is molar weight of the substance and Z is the number of

moles. After some more calculations and simplifications, one can

arrive at an expression for the effective magnetic moment of an ion:

eff

¼ 2:828



T  T

ðÞ



1=2

Regarding the fact that magnetization curve of ferrimagnetics is not

linear, experimental determination of magnetic susceptibility depends

also on the intensity of magnetic field (or point of the M (H) curve,

at which it is measured), and on the way how it is measured.

Therefore, in physics literature one can find terms initial, reversible,

and differential magnetic susceptibility (Figure S51).

In terms of the applied magnetic field, frequency of the AC field has

significant effect in case of presence of ultrafine super-paramagnetic

Figure S50 Temperature dependence of inverse susceptibility for paramagnetics (left), ferrimagnetics (middle) and antiferromagnetics

(right).

Figure S51 Reversible and differential susceptibility as a function

of the magnetic field at which the slope of magnetization curve is

determined. w

stands for initial susceptibility.

Figure S52 Schematic dependence of frequency-dependent

magnetic susceptibility on grain size (after Dearing, 1994).

932 SUSCEPTIBILITY

particles. At higher frequencies, energy of thermal excitations over-

comes the alignment effect of the applied field and, consequently,

magnetic moments of such particles are not aligned parallel to the

applied field and do not contribute to magnetic susceptibility. There-

fore, magnetic susceptibility measured at higher frequencies is always

equal or lower than that measured at lower frequencies. Difference,

expressed as percentage, is called frequency-dependent magnetic sus-

ceptibility (Figure S52),

 w

While w

depends strongly on grain size (e.g., Eyre, 1997; Dearing

et al., 1996), and is used as indicator of significance of SP particles,

magnetic susceptibility itself depends mostly on mineralogy, and for

a given mineral on its concentration.

Eduard Petrovsky

Bibliography

Dearing, J.A., 1994. Environmental Magnetic Susceptibility––Using

the MS2 Bartington System. Chi Publishing, Kenilworth, UK.

Dearing, J.A., Dann, R.J.L., Hay, K., Lees, J.A., Loveland, P.J.,

and O’Grady, K., 1996. Frequency-dependent susceptibility

measurements of environmental materials. Geophysical Journal

International, 124: 228–240.

Dekker, A.J., 1957. Solid State Physics. Englewood Cliffs, NJ:

Prentice-Hall.

Dunlop, D., and Özdemir, Ö., 1997: Rock Magnetism––Fundamentals

and Frontiers. Cambridge: Cambridge University Press.

Eyre, J.K., 1997. Frequency dependence of magnetic susceptibility

for populations of single-domain grains. Geophysical Journal

International, 129: 209–211.

SUSCEPTIBILITY, MEASUREMENTS OF SOLIDS

Introduction

Magnetism is a property characteristic of all materials that contain

electrically charged particles. A moving electrical charge (i.e., electric

current) gives rise to a magnetic field in a material. The contribution of

nucleons to the magnetic field in substances is small and requires spe-

cial instrumentation to observe. For the purpose of this laboratory the

effect from nucleons is neglected, thus magnetism may be considered

to be entirely of electronic origin. In an atom, the magnetic field is due

to the coupled orbital and spin magnetic moments associated with the

motion of electrons. The orbital magnetic moment is due to the motion

of electrons around the nucleus whereas the spin magnetic moment is

due to the precession of the electrons about their own axes. The resul-

tant of the orbital and spin magnetic moments of the constituent atoms

of a material gives rise to the observed magnetic properties.

Knowledge about the magnetic properties of materials is an impor-

tant aspect in the study and characterization of substances. Unlike

many others, commonly employed analytical methods, magnetic prop-

erty characterization is a nondestructive technique.

Theory

Often magnetic properties of materials are studied by applying a mag-

netic field and measuring the induced magnetization in these materials.

The magnetic induction, B, that a substance experiences when placed

in an applied external magnetic field, H, is given by the expression

B ¼ H þ 4pM ; (Eq. 1)

where M is the magnetic moment of the compound per unit volume or

the Magnetization. The volume susceptibility of the compound, w

,is

defined as the ratio of the induced magnetic moment per unit volume

to the applied field:

¼ M =H : (Eq. 2)

The two most important responses observed are characterized as dia-

magnetic and paramagnetic moment (Figure S53). As shown in this

figure, the applied magnetic field remains unaffected in a vacuum

but is attracted by a paramagnetic, and repelled by an ideal diamag-

netic media.

In diamagnetic materials, all the electrons in the atoms are paired

(e.g., H

2(g)

, NaCl

(s)

) and the resultant magnetic moment is zero due

to the absence of unpaired electrons. The external magnetic field

induces a current whose associated magnetic field is directed opposite

to the applied field. The induced magnetic moment associated with

this field is called a diamagnetic moment and is negative relative

to the applied magnetic field and independent of temperature. The

magnitude of the diamagnetic susceptibility is usually small, in the

range of 10

6

cgs units. All inert gases and most organic compounds

are examples of diamagnetic materials.

In paramagnetic materials, the constituent atoms have unpaired elec-

trons resulting in a net magnetic moment. Generally at room tempera-

ture, the individual magnetic moments are randomly oriented as shown

in Figure S54 above. If an external magnetic field is applied to these

randomly oriented intrinsic magnetic moments, they will tend to

become aligned in a direction parallel to the external magnetic field.

The magnetic moment induced in such materials by the application

of an external magnetic field is called a paramagnetic moment. The

magnitude of the susceptibility of paramagnetic substances is in the

range of 10

3

–10

6

cgs units and is positive. Typical paramagnetic

compounds include gaseous compounds like molecular oxygen (O

)

and nitric oxide (NO), vapors of alkali metals and certain salts of tran-

sition and rare-earth metals.

The alignment of the magnetic moments of a paramagnetic sub-

stance under an applied magnetic field is opposed by the thermal

motion of the magnetic ions which tends to randomize the moments.

Hence the observed paramagnetic susceptibility of a substance

increases with decreasing temperature since the effect of thermal

Figure S53 Behavior of applied magnetic field in different media:

(a) vacuum, (b) paramagnet, and (c) ideal diamagnet.

SUSCEPTIBILITY, MEASUREMENTS OF SOLIDS 933

motion is minimized at lower temperatures (Figure S55). According to

Curie’s Law,

(Eq. 3)

the susceptibility of a paramagnetic substance is inversely proportional

to temperature where C is the Curie constant. The variation of

the inverse magnetic susceptibility ðw

1

Þ with temperature is linear.

The slope of this line is equal to 1/C. The value of C is related to

the number of unpaired electrons present in the compound by the fol-

lowing expression

ð8CÞ

1=2

¼ g½SðS þ 1Þ

1=2

¼½nðn þ 2Þ

1=2

; (Eq. 4)

where Lande’s factor g  2:0 for an electron or ion with no orbital

contribution to the magnetic moment and S is the resultant spin quan-

tum number (it is the total spin angular momentum of all the unpaired

electrons in the system), and n is the number of unpaired electrons in

the system (i.e., S ¼ n=2). Equation (4) is generally valid for com-

pounds of most of the first row transition metals, where the orbital

contribution to the magnetic moment is completely quenched and only

the spin contribution, S need be considered. This will be discussed in

more detail subsequently.

Figure S55 shows the plot of w versus T and 1=w versus T (T in abso-

lute temperature, K) typical for a paramagnetic substance in which

orbital contribution to the susceptibility is negligible.

Paramagnetic materials, besides having unpaired electrons, contain

paired electrons in the inner (closed) shells of the constituent atoms.

The presence of these paired electrons makes diamagnetism an inher-

ent property of all materials. Thus the magnetic moment measured is

in fact the sum of both paramagnetic (positive quantity) and the asso-

ciated diamagnetic (negative quantity) moment. Since the presence of

an intrinsic magnetic moment in a substance results in a large para-

magnetic moment, the diamagnetic effects are often neglected in the

calculations. However, if the paramagnetic and diamagnetic moments

of the substance under investigation are of comparable magnitude,

and where accurate measurements are desired, corrections due to

the diamagnetic contributions are made to the measured magnetic

moments.

So far we have considered paramagnetic substances in which there

are no significant interactions between the magnetic moments of its

atoms. However, magnetic moments of individual atoms in many sub-

stances do interact with each other in several different ways. In these

materials, Curie’s law is not obeyed; often, the magnetic behavior of

these substances is best described by a Curie-Weiss law which takes

into account the interactions among the individual magnetic moments

ðT  yÞ

; (Eq. 5)

where y is the Weiss constant (or paramagnetic Curie temperature).

Interactions of magnetic moments in condensed systems (i.e., solids)

will in most cases lead to different types of magnetic ordering, charac-

teristic of the substance. This is shown in Figure S56 where the net

spin on each atom is represented by an arrow aligned “with” or

“against” the applied external magnetic field. The type of magnetic

interactions present in a particular substance is primarily determined

by the nature of the constituent ions and chemical bonds. The mag-

netic interactions between the individual magnetic moments result in

a net stabilization energy. The two most common types of magnetic

ordering or interactions are called ferromagnetism and antiferro-

magnetism. In ferromagnetic substances, the magnetic moments of

adjacent atoms are aligned parallel to one another below a certain cri-

tical temperature (known as the Curie temperature, T

) as shown in

Figure S56a. This type of magnetic ordering is characterized by a

spontaneous magnetization of the substance below T

even in the

absence of an external magnetic field. Iron, nickel, and cobalt and

some rare-earth metals, as well as some compounds and alloys of these

elements are typical examples that show ferromagnetic ordering.

In antiferromagnetic substances, the individual magnetic moments

are aligned antiparallel to one another below a certain critical tempera-

ture (known as Néel temperature, T

). This interaction gives no net

magnetization when the neighboring atoms have magnetic moments

of identical magnitude as shown in Figure S56b. Most compounds

of the transition elements such as MnO, MnF

, FeCl

, and elements

such as chromium, a-manganese and some rare-earth metals such

as cerium, praseodymium, neodymium, samarium, and europium dis-

play antiferromagnetic ordering. Compounds that exhibit ferromag-

netic or antiferromagnetic type of magnetic ordering often obey the

Figure S54 Random orientation of magnetic moments in

paramagnetic material.

Figure S55 (a) Magnetic susceptibility and (b) inverse magnetic

susceptibility dependence on temperature of a paramagnetic

material.

Figure S56 Types of magnetic ordering: (a) ferromagnetic,

(b) antiferromagnetic, and (c) ferrimagnetic.

934 SUSCEPTIBILITY, MEASUREMENTS OF SOLIDS

Curie-Weiss law (5) at temperatures well above the transition

temperatures (ordering temperature, T

or T

Ferrimagnetism is another prevalent type of ordering of the mag-

netic moments. In a ferrimagnetic substance, although the individual

magnetic moments of adjacent atomic particles are aligned antiparallel,

they are not of equal magnitude (i.e., when the type of interacting

neighboring atoms are different), e.g., Fe

ðFeO  Fe

Þ.

As expected, the magnetic response becomes stronger, as the num-

ber of unpaired electrons present increases. Hence it is customary to

measure experimentally the magnetic susceptibility of a substance as

a function of its weight, w

¼ w

=r; (Eq. 6)

where w

, the mass susceptibility of the compound, is the induced

magnetic moment per unit mass per unit applied magnetic field, and

r is the density of the material.

The mass susceptibility of the compound is related to the molar

susceptibility, w

¼ w

 MW=Z; (Eq. 7)

where MW is the molecular weight of the compound and Z is the num-

ber of moles of magnetic ions per formula weight of the compound.

The molar magnetic susceptibility of a compound having isolated

magnetic ions with no magnetic interactions is given by (assuming

total quenching of orbital moments)

SðS þ 1Þ

: (Eq. 8a)

In the above expression, N

is Avogadro’s number, k

is the

Boltzmann constant, and b is a constant unit called the Bohr magneton

(BM). Note that (8a) is similar to that shown by Curie’s law which was

derived from experimental observations, giving the Curie constant, C

C ¼

SðS þ 1Þ

: (Eq. 8b)

As discussed above, in condensed systems, e.g., liquids and solids,

generally there are some magnetic interactions between the magnetic

ions such that a modified Curie’s law known as the Curie-Weiss law

is applied

SðS þ 1Þ

ðT  yÞ

: (Eq. 9)

In substances with interacting magnetic moments and where the orbital

contribution to the magnetic moments is significant, the molar suscept-

ibility is given by

JðJ þ 1Þ

ðT  yÞ

; (Eq. 10)

here the resultant total angular momentum, J, is given by

J ðJ þ 1Þ¼LðL þ 1ÞþSðS þ 1Þ; (Eq. 11)

and the value of g factor is given by Landé’s equation:

g ¼ 1 þ

SðS þ 1ÞLðL þ 1ÞþJ ðJ þ 1Þ

2J ðJ þ1Þ

: (Eq. 12)

The quantum number L is the resultant orbital angular momentum of

all unpaired electrons present in the atom (i.e., L is zero for a com-

pletely filled orbital). However for most compounds of the first row

transition metal series, the orbital contribution to the magnetic moment

is negligible, such that J ¼ S and (10) is identical to (9). In these and

other compounds of the transition metal series the metal ions are typi-

cally surrounded by strong ligands that quench the orbital contribution

to the magnetic moment.

Equation (9) can be further simplified by defining m

eff

SðS þ 1Þ and combining all of the constants to give:



eff

ðT  yÞ

 0:125

eff

ðT  yÞ

: (Eq. 13)

This equation allows us to calculate the effective magnetic moment of

an ion, m

eff

, which can be rewritten as

eff

¼ 2:828½w

ðT  yÞ

1=2

; (Eq. 14)

where w

, the molar susceptibility, is obtained from (7) and T is the

temperature in Kelvin.

Instrumentation

In this experiment, the method to be used in the measurement of the

magnetic susceptibility of substances is based on the principle of sta-

tionary sample but moving magnet. Here, the balance (i.e., Johnson

Matthey Magnetic Susceptibility Balance, MSB), measures the force

that the sample exerts on a suspended permanent magnet, unlike the

Gouy balance which measures the equal and opposite force which a

magnet exerts on the sample.

In the experiment that follows, the mass susceptibility of a com-

pound, w

, to be used in (7) is calculated from the quantities obtained

using the MSB by the following formula

bal

ðR  R

ðm  10

; (Eq. 15)

where C

bal

is the calibration constant of the balance, L is the length in

centimeters (cm) of a well-packed sample in the tube, m is sample

mass in grams (g), R is the reading obtained for the tube plus sample,

is the empty tube reading (normally a negative value, i.e., diamag-

netic).

The MSB must be calibrated with the standard compound,

HgCo(SCN)

which has a w

¼ 16:44  10

6

cgs at 20



C. This first

measurement is required for the calculation of the calibration constant

of the balance, C

bal

, in (15).

Experimental procedure

1. Turn the Range Knob to 1 scale and let the MSB warm up for.

2. Adjust the MSB to read 000 using the Zero knob when no tube is

inside the tube guide.

Many of the following steps may not be required if we supply

preprepared samples.

3. Place the empty tube into the tube guide of MSB, and record the

reading, R

(normally a negative value, i.e., diamagnetic).

4. Remove the empty tube from the MSB and weigh it carefully on a

balance to the nearest milligram; record the mass of the empty

tube in grams, m

5. Prepare your sample for measurement by grinding (if not already

in a powdered form). Sample must be ground to a fine powder

in a mortar and pestle.

6. Introduce your sample into the preweighed empty tube and tap

(the sample tubes are expensive, made of brittle quartz materials;

so tap them gently to avoid breakage) on a wooden bench several

times to achieve densely packed (i.e., minimum empty space

SUSCEPTIBILITY, MEASUREMENTS OF SOLIDS 935