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Geomagnetism
Part 1:
Basic Principles and
Material Properties
Magnetism
Like a lot of phenomena in Physics, understanding magnetism requires
an understanding of quantum theory, but perhaps more than most.
We’ll need to get into this a bit, but there are some useful ideas we can
discuss without going too deeply.
Scientists first investigating magnetism
noticed a lot of similarities between magnetic
fields and electrical fields, and so presumed
they were due to the same physical
mechanism.
In fact, Gauss proposed that Coulomb’s law
for the forces between electrical charges could
be modified for magnetic force, except that the
property eo, the electrical permittivity, is
Carl Friedrich Gauss
replaced by something called the magnetic
permeability - mo.
Thus, the electric force:
F
1 Q1Q2
4eo r 2
becomes the magnetic force
m PP
F  o 1 22
4 r
Carl Marks?
Where P1 and P2 are called magnetic poles. The main difference is
that the magnetic field always looks like there are two poles of
opposite sign in some proximity to each other, but as far as we know
the concept of a magnetic pole is a pure fiction.
Nevertheless, it turns out to be a useful one. Like the E field, we can
define the B field as the field produced by a single (fictional) pole:
F mo P
B 
P 4r 2
and define a magnetic potential W as
the work done to bring an second pole
in from infinity:
r
mo P
mo P
dr

2
4

r
4r

r
W    Bdr   

Since poles always occur in pairs (+ and -) it is useful to compute
the potential field of a dipole; it is simply the sum of the potentials
of individual poles. In the following we will consider an analogy
to Electrical Dipoles, where we use charges q of opposite sign.
Dipole Moment
It will be very useful to define a quantity called the Dipole Moment,
defined as the magnitude of the poles times the distance between them
and the defined direction is toward the positive pole.
For magnetic dipoles the dipole moment m = pd, where p is the pole
strength. At points far from the dipole (r >> d), we have
mo dP cos mo m cos 
W

2
4r
4r 2
where  is the angle between d and r
Electric Dipole. For Magnetic,
substitute p for q and m for P
Potential V for an Electric Dipole. For the Magnetic
equivalent, substitute W for V, p for q, m for k, and m for p
Torque on a Magnetic Dipole
Suppose a magnetic field B is oriented at an angle  with respect to
the dipole.
The Magnetic force is Bp on the positive pole and –Bp on the
negative pole.
Thus there are equal and opposite forces a distance dsin/2 from the
center. The total torque is then
t = 2Bpdsin/2 = Bpdsin = m x B
We will come back to the magnetic torque concept in a
different context in a bit.
If Poles are Fictitious, What’s Really Going On?
Well, a very realistic way to think about magnetism is that it is always
produced by moving charges (currents).
These can be macroscopic currents in wires, or microscopic currents
associated with electrons in atomic orbits, or even the individual
spinning of electrons or an atomic nucleus.
Then, the magnetic field B is defined in terms of force on moving
charge in the Lorentz force law. Both the electric field and magnetic
field can be defined from this law:
The electric force is straightforward, being in the
direction of the electric field if the charge q is
positive, but the direction of the magnetic part of
the force is given by the right hand rule.
Hendrick Antoon
Lorentz
The SI unit for magnetic field is the Tesla,
which can be seen from the magnetic part of
the Lorentz force law Fmagnetic = qvB to be
composed of (Newton x second)/(Coulomb x
meter).
A smaller magnetic field unit is the Gauss (1
Tesla = 10,000 Gauss), and an even smaller
one often used in Geophysics is the gamma
(g), which is 10-5 Gauss or 1 nanoTesla.
Nikola Tesla
Magnetic Dipole Moment Revisited
Let’s use this way of thinking to revisit the idea of a magnetic dipole
moment.
Consider a charge q moving through a magnetic field B with velocity v.
Lorentz says:
F  q (v  B )
If we consider the current as being the total charge of N charges that
passes through a volume of area A and length dl in a unit time, then the
total force dF on this element is
dF  NAdlq (v  B)  NAvq(dl  B)  I (dl  B)
In the case of a wire of length L perpendicular to the magnetic
field:
F  ILB
The torque exerted by the magnetic force on a wire (see figure below)
is given by
t= BILWsin
The magnetic moment of a loop is m = IA where A is the area of the
loop, or m = NIA for N loops.
Since A = LW, the torque is then written as
t= mBsin
The direction of the magnetic moment is perpendicular to the current
loop in the right-hand-rule direction.
Considering torque as a vector quantity, this can be written as the
vector product
t= m X B
Since this torque acts perpendicular to the magnetic moment, then it
can cause the magnetic moment to precess around the magnetic field
at a characteristic frequency called the Larmor frequency.
If you exerted the necessary torque to overcome the magnetic torque
and rotate the loop from angle zero to 180 degrees, you would do an
amount of rotational work given by the integral


0
0
W    td    mBsin d  mBcos  |0  2mB

As seen in the geometry of a current loop, this torque tends to line
up the magnetic moment with the magnetic field B, so this
represents its lowest energy configuration. The potential energy
associated with the magnetic moment is
U    m  B
so that the difference in energy between aligned and anti-aligned is

DU = 2mB
These relationships for a finite current loop extend to the magnetic
dipoles of electron orbits and to the intrinsic magnetic moment
associated with electron spin. Also important are nuclear magnetic
moments.
Magnetic Domains
It will be essential for Geomagnetism to understand the magnetic
properties of solids, and in particular how secondary fields are induced
and how permanent magnets are made. We begin with the idea of
Magnetic Domains.
The “long range order” which creates magnetic domains in
ferromagnetic materials arises from a quantum mechanical interaction
at the atomic level. This interaction is remarkable in that it locks the
magnetic moments of neighboring atoms into a rigid parallel order
over a large number of atoms in spite of the thermal agitation which
tends to randomize any atomic-level order.
Sizes of domains range from a 0.1 mm to a few mm. When an external
magnetic field is applied, the domains already aligned in the direction
of this field grow at the expense of their neighbors.
If all the spins were aligned in a piece of iron, the field would be about
2.1 Tesla. A magnetic field of about 1 T can be produced in annealed
iron with an external field of about 0.0002 T, a multiplication of the
external field by a factor of 5000! For a given ferromagnetic material
the long range order abruptly disappears at a certain temperature which
is called the Curie temperature for the material. The Curie temperature
of iron is about 1043 K.
The microscopic ordering of electron spins characteristic of
ferromagnetic materials leads to the formation of regions of magnetic
alignment called domains.
The main implication of the domains is that there is already a high
degree of magnetization in ferromagnetic materials within individual
domains, but that in the absence of external magnetic fields those
domains are randomly oriented.
A modest applied magnetic field can cause a larger degree of
alignment of the magnetic moments with the external field, giving a
large multiplication of the applied field.
These illustrations of domains are conceptual only and not meant to
give an accurate scale of the size or shape of domains.
The microscopic evidence about magnetization indicates that the net
magnetization of ferromagnetic materials in response to an external
magnetic field may actually occur more by the growth of the
domains parallel to the applied field at the expense of other domains
rather than the reorientation of the domains themselves as implied in
the sketch below.
Some of the more direct evidence we have about domains comes
from imaging of domains in single crystals of ferromagnetic
materials.
They suggest that the effect of external magnetic fields is to cause the
domain boundaries to shift in favor of those domains which are
parallel to the applied field. It is not clear how this applies to bulk
magnetic materials which are polycrystalline.
Keep in mind the fact that the internal magnetic fields which come
from the long range ordering of the electron spins are much stronger,
sometimes hundreds of times stronger, than the external magnetic
fields required to produce these changes in domain alignment.
The effective multiplication of the external field which can be
achieved by the alignment of the domains is often expressed in terms
of the relative permeability.
Domains may be made visible with the use of magnetic colloidal
suspensions which concentrate along the domain boundaries. The
domain boundaries can be imaged by polarized light, and also with
the use of electron diffraction.
Observation of domain boundary movement under the influence of
applied magnetic fields has aided in the development of theoretical
treatments. It has been demonstrated that the formation of domains
minimizes the magnetic contribution to the free energy.
Magnetic Field Strength H
The magnetic fields generated by currents and calculated from Ampere's
Law or the Biot-Savart Law are characterized by the magnetic field B
measured in Tesla.
Illustration
of the BiotSavart Law
But when the generated fields pass through magnetic materials which
themselves contribute internal magnetic fields, ambiguities can arise
about what part of the field comes from the external currents and what
comes from the material itself.
It has been common practice to define another magnetic field quantity,
usually called the "magnetic field strength" designated by H. It can be
defined by the relationship
H = B0/m0 = B/m0 - M
and has the value of unambiguously designating the driving magnetic
influence from external currents in a material, independent of the
material's magnetic response.
The quantity M is called the magnetization of the material.
The relationship for B can be written in the equivalent form
B = m0(H + M)
H and M will have the same units, amperes/meter. To further
distinguish B from H, B is sometimes called the magnetic flux density
or the magnetic induction.
You can think of H as the ambient field, M as the field induced in a
material by the presence of H (for example by realigning the magnetic
domains) and B as the sum of the two.
Another commonly used form for the relationship between B and H
is
B = mmH
Where
mm = Kmm0
m0 being the magnetic permeability of space and Km the relative
permeability of the material.
If the material does not respond to the external magnetic field by
producing any magnetization, then Km = 1.
Another commonly used magnetic quantity is the magnetic
susceptibility which specifies how much the relative permeability
differs from one.
Magnetic susceptibility km = Km – 1
For paramagnetic and diamagnetic materials the relative permeability
is very close to 1 and the magnetic susceptibility very close to zero.
For ferromagnetic materials, these quantities may be very large.
Note that
B = m0(H + M) = mmH = Kmm0H
So
M = (Km – 1)H = kmH
Relative Permeability
The magnetic constant m0 = 4 x 10-7 T m/A is called the permeability
of space. The permeabilities of most materials are very close to m0
since most materials will be classified as either paramagnetic or
diamagnetic.
But in ferromagnetic materials the permeability may be very large and
it is convenient to characterize the materials by a relative permeability.
When ferromagnetic materials are used in applications like an ironcore solenoid, the relative permeability gives you an idea of the kind
of multiplication of the applied magnetic field that can be achieved
by having the ferromagnetic core present.
So for an ordinary iron core you might expect a magnification of
about 200 compared to the magnetic field produced by the solenoid
current with just an air core.
This statement has exceptions and limits, since you do reach a
saturation magnetization of the iron core quickly, as illustrated in
the discussion of hysteresis.
Hysteresis
When a ferromagnetic material is magnetized in one direction, it will
not relax back to zero magnetization when the imposed magnetizing
field is removed. It must be driven back to zero by a field in the
opposite direction.
If an alternating magnetic field is applied to the material, its
magnetization will trace out a loop called a hysteresis loop.
The lack of retraceability of the magnetization curve is the property
called hysteresis and it is related to the existence of magnetic
domains in the material. Once the magnetic domains are reoriented, it
takes some energy to turn them back again.
This property of ferrromagnetic materials is useful as a magnetic
"memory". Some compositions of ferromagnetic materials will retain
an imposed magnetization indefinitely and are useful as "permanent
magnets".
The magnetic memory aspects of iron and chromium oxides make
them useful in audio tape recording and for the magnetic storage of
data on computer disks.
It is customary to plot the magnetization M of the sample as a function
of the magnetic field strength H, since H is a measure of the externally
applied field which drives the magnetization .
Variations in Hysteresis Curves
There is considerable variation in the hysteresis of different magnetic
materials.
Hysteresis in Magnetic Recording
Because of hysteresis, an input signal at the level indicated by the dashed line could give a
magnetization anywhere between C and D, depending upon the immediate previous history of
the tape (i.e., the signal which preceded it).
This clearly unacceptable situation is remedied by the bias signal which cycles the oxide grains
around their hysteresis loops so quickly that the magnetization averages to zero when no signal
is applied. The result of the bias signal is like a magnetic eddy which settles down to zero if
there is no signal superimposed upon it. If there is a signal, it offsets the bias signal so that it
leaves a remnant magnetization proportional to the signal offset.
Coercivity and Remanence in Permanent Magnets
A good permanent magnet should produce a high magnetic field with
a low mass, and should be stable against the influences which would
demagnetize it. The desirable properties of such magnets are typically
stated in terms of the remanence and coercivity of the magnet
materials.
To Review:
When a ferromagnetic material is magnetized in one direction, it will
not relax back to zero magnetization when the imposed magnetizing
field is removed. The amount of magnetization it retains at zero
driving field is called its remanence.
It must be driven back to zero by a field in the opposite direction; the
amount of reverse driving field required to demagnetize it is called its
coercivity.
If an alternating magnetic field is applied to the material, its
magnetization will trace out a loop called a hysteresis loop. The lack
of retraceability of the magnetization curve is the property called
hysteresis and it is related to the existence of magnetic domains in the
material. Once the magnetic domains are reoriented, it takes some
energy to turn them back again. This property of ferrromagnetic
materials is useful as a magnetic "memory". Some compositions of
ferromagnetic materials will retain an imposed magnetization
indefinitely and are useful as "permanent magnets".
The table below contains some data about materials used as permanent magnets. Both the
coercivity and remanence are quoted in Tesla, the basic unit for magnetic field B.
Besides coercivity and remanence, a quality factor for permanent magnets is the quantity
(BB0/m0)max. A high value for this quantity implies that the required magnetic flux can be
obtained with a smaller volume of the material, making the device lighter and more compact.
Coercivity
(T)
Remanence
(T)
(BB0/m0)max
(kJ/m3)
BaFe12O19
0.36
0.36
25
Alnico IV
0.07
0.6
10.3
Alnico V
0.07
1.35
55
Alcomax I
0.05
1.2
27.8
MnBi
0.37
0.48
44
Ce(CuCo)5
0.45
0.7
92
SmCo5
1.0
0.83
160
Sm2Co17
0.6
1.15
215
Nd2Fe14B
1.2
1.2
260
Material
The alloys from which permanent magnets are made are often very
difficult to handle metallurgically. They are mechanically hard and
brittle.
They may be cast and then ground into shape, or even ground to a
powder and formed. From powders, they may be mixed with resin
binders and then compressed and heat treated.
Maximum anisotropy of the material is desirable, so to that end the
materials are often heat treated in the presence of a strong magnetic
field.
The materials with high remanence and high coercivity from which
permanent magnets are made are sometimes said to be "magnetically
hard" to contrast them with the "magnetically soft" materials from
which transformer cores and coils for electronics are made.
Rare Earth Magnets
The permanent magnets which have produced the largest magnetic
flux with the smallest mass are the rare earth magnets based on
samarium and neodynium. Their high magnetic fields and light
weight make them useful for demonstrating magnetic levitation
over superconducting materials.
The samarium-cobalt combinations have been around longer, and the
SmCo5 magnets are produced for applications where their strength and
small size offset the disadvantage of their high cost. The more recent
neodynium materials like Nd2Fe14B produce comparable performance,
and the raw alloy materials cost about 1/10 as much. They have begun to
find application in microphones and other applications which exploit the
high field and light weight. The production is still quite costly since the
raw allow must be ground to powder, pressed into the desired shape and
then sintered to make a durable solid.
Material
Coercivity
(T)
Remanence
(T)
(BB0/m0)max
(kJ/m3)
SmCo5
1.0
0.83
160
Sm2Co17
0.6
1.15
215
Nd2Fe14B
1.2
1.2
260
Ferromagnetism
Iron, nickel, cobalt and some of the rare earths (gadolinium,
dysprosium) exhibit a unique magnetic behavior which is called
ferromagnetism because iron (ferric) is the most common and most
dramatic example.
Samarium and neodynium in alloys with cobalt have been used to
fabricate very strong rare-earth magnets.
Ferromagnetic materials exhibit a long-range ordering phenomenon at
the atomic level which causes the unpaired electron spins to line up
parallel with each other in a region called a domain. Within the
domain, the magnetic field is intense, but in a bulk sample the material
will usually be unmagnetized because the many domains will
themselves be randomly oriented with respect to one another.
Ferromagnetism manifests itself in the fact that a small externally
imposed magnetic field, say from a solenoid, can cause the magnetic
domains to line up with each other and the material is said to be
magnetized. The driving magnetic field will then be increased by a
large factor which is usually expressed as a relative permeability for
the material. There are many practical applications of ferromagnetic
materials, such as the electromagnet.
Ferromagnets will tend to stay magnetized to some extent after being
subjected to an external magnetic field. This tendency to "remember
their magnetic history" is called hysteresis.
The fraction of the saturation magnetization which is retained when
the driving field is removed is called the remanence of the material,
and is an important factor in permanent magnets.
All ferromagnets have a maximum temperature where the
ferromagnetic property disappears as a result of thermal agitation.
This temperature is called the Curie temperature.
Ferromagntic materials will respond mechanically to an impressed
magnetic field, changing length slightly in the direction of the
applied field. This property, called magnetostriction, leads to the
familiar hum of transformers as they respond mechanically to 60 Hz
AC voltages.
Magnetic Properties of Ferromagnetic Materials
Material
Treatment
Initial
Relative
Permeability
Iron, 99.8%
pure
Annealed
150
5000
1.0
13,000
Iron, 99.95%
pure
Annealed in hydrogen
10,000
200,000
0.05
13,000
78 Permalloy
Annealed, quenched
8,000
100,000
.05
7,000
Annealed in hydrogen, controlled
cooling
100,000
1,000,000
0.002
7,000
Cobalt, 99%
pure
Annealed
70
250
10
5,000
Nickel, 99%
pure
Annealed
110
600
0.7
4,000
Steel, 0.9% C
Quenched
50
100
70
10,300
Steel, 30% Co
Quenched
...
...
240
9,500
Alnico 5
Cooled in magnetic field
4
...
575
12,500
Silmanal
Baked
...
...
6,000
550
Iron, fine
powder
Pressed
...
...
470
6,000
Superpermalloy
Maximum
Relative
Permeability
Coerciv
e
Force
Remanent
Flux
Density
Diamagnetism
The orbital motion of electrons creates tiny atomic current loops,
which produce magnetic fields. When an external field is applied, it
produces a torque which induces a precision of the magnetic moment
at the Larmor frequency.
The direction of the precession is in the opposite direction of the
electron orbit, and produces a field that opposes the applied field
(Lenz’s Law).
The result is a material with a negative susceptibility.
All materials are inherently diamagnetic, but if the atoms have some
net magnetic moment as in paramagnetic materials, or if there is longrange ordering of atomic magnetic moments as in ferromagnetic
materials, these stronger effects are always dominant.
Diamagnetism is the residual magnetic behavior when materials are
neither paramagnetic nor ferromagnetic.
Any conductor will show a strong diamagnetic effect in the presence of
changing magnetic fields because circulating currents will be generated
in the conductor to oppose the magnetic field changes. A
superconductor will be a perfect diamagnet since there is no resistance
to the forming of the current loops.
Paramagnetism
Some materials exhibit a magnetization which is proportional to the
applied magnetic field in which the material is placed. These materials
are said to be paramagnetic and follow Curie's law:
All atoms have inherent sources of magnetism because electron spin
contributes a magnetic moment and electron orbits act as current
loops which produce a magnetic field.
In most materials the magnetic moments of the electrons cancel, but
in materials which are classified as paramagnetic, the cancelation is
incomplete.
Magnetic Susceptibilities of Paramagnetic and
Diamagnetic Materials at 20°C
Material
km=Km-1
(x 10-5)
Paramagnetic
Iron aluminum alum
66
Uranium
40
Platinum
26
Aluminum
2.2
Sodium
0.72
Oxygen gas
0.19
Diamagnetic
Bismuth
-16.6
Mercury
-2.9
Silver
-2.6
Carbon (diamond)
-2.1
Lead
-1.8
Sodium chloride
-1.4
Copper
-1.0
For ordinary solids and liquids at room temperature, the relative
permeability Km is typically in the range 1.00001 to 1.003.
We recognize this weak magnetic character of common materials by
the saying "they are not magnetic", which recognizes their great
contrast to the magnetic response of ferromagnetic materials.
More precisely, they are either paramagnetic or diamagnetic, but that
represents a very small magnetic response compared to
ferromagnets.
Magnetostriction
Why does a transformer hum?
You may have noticed the humming sound associated with a
transformer or a fluorescent light ballast. For U.S. circuits, that hum
will be at 120 Hz since the iron material associated with the
transformer core responds mechanically to the magnetic field which is
impressed upon it.
The effect is called magnetostriction, and it is one of the magnetic
properties which accompanies ferromagnetism. For 60 Hz applied
magnetic fields in AC electrical devices such as transformers, the
maximum length change happens twice per cycle, producing the
familiar and sometimes annoying 120 Hz hum.
In formal treatments, a magnetostrictive coefficient L is defined as the
fractional change in length as the magnetization increases from zero to
its saturation value. The coefficient L may be positive or negative, and
is usually on the order of 10-5.
There is an elastic strain energy associated with the deformation,
leading to some dissipation of energy in transformer cores. If the
magnetostriction acts to contract a specimen, then this will act against
any tensile stress on the material and leads to a larger value for the
Young's modulus for the material.
Two examples of measurements of this phenomenon are included in
the table below.
Material
Crystal axis
Saturation
magnetostriction
L (x 10-5)
Fractional change of
Young's modulus
DE/E
Fe
100
+(1.1-2.0)
...
Fe
111
-(1.3-2.0)
...
Fe
Polycrystal
-.8
0.002-0.003
Ni
100
-(5.0-5.2)
...
Ni
111
-2.7
...
Ni
Polycrystal
-(2.5-4.7)
0.07
Co
Polycrystal
-(5.0-6.0)
...
It is also observed that applied mechanical strain produces some
magnetic anisotropy.
If an iron crystal is placed under tensile stress, then the direction of the
stress becomes the preferred magnetic direction and the domains will
tend to line up in that direction.
Ordinarily the direction of magnetization in iron is easily changed by
rotating the applied magnetic field, but if there is tensile stress in the
iron sample, there is some resistance to that rotation of direction.
Bulk solid samples may have internal strains which influence the
domain boundary movement.
Magnetostriction can be used to create vibrators, where usually some
lever action is used in conjunction with the magnetic deformation to
increase the resultant amplitude of vibration.
Magnetostriction is also used to produce ultrasonic vibrations either
as a sound source or as ultrasonic waves in liquids which can act as a
cleaning mechanism in ultrasonic cleaning devices.
Remanent magnetization in rocks
Primary secondary -
acquired at time of formation (like cooling igneous)
acquired later (weathering, lightening).
Several types
TRM – Thermal remanent magnetism
Melting point of igneous rx ~ 1000 degrees
Curie temp for minerals ~ 600 degrees
So grains form but still magnetic chaos.
Eventually cools below the “blocking temp” and domains will
tend to align with ambient field.
Very stable magnetization – very important for lavas; basalts.
DRM – Depositional or detrital remanent mag.
Grains settling and rotating.
CRM – Chemical remanent mag
Chemical modification, such as precipitation of hematite, or
other grain growth.
IRM – isothermal
Exposure to strong ambient field (from lab instruments, for
example).
VRM – viscous
a slow coersion by an ambient field.
When describing the magnetic properties of rocks in the field, we
need to have some idea of which part of the field if induced and
which is remanent.
We quantify this by the Konigsberger ratio, which is the ratio
between the remanent magnetization to the induced magnetization.