Induced and remanent magnetism

BACKGROUND: For a review of basic quantum mechanics and statistical mechanics, read relevant chapters from an introductory Chemistry text book.

Scientists in the late 19th century thought that it might be possible to exploit the magnetic record retained in accidental records to study the geomagnetic field in the past. Work in the mid 20th century provided the theoretical and experimental basis for presuming that such materials might retain a record of past geomagnetic fields. There are several books and articles that describe the subject in detail (see e.g., the supplemental readings). We present here a brief overview of theories on how rocks get and stay magnetized. We will begin with magnetism at the atomic level caused by electronic orbits and spins giving rise to induced magnetizations. Then we will see how electronic spins working in concert give rise to permanently magnetized substances (like magnetic minerals) making remanent magnetization possible.

We learned in Chapter 1 that magnetic fields are generated by electric currents. Given that there are no wires leading into or out of permanent magnets, you may well ask, “Where are the currents?” At the atomic level, the electric currents come from the motions of the electrons. From here quantum mechanics quickly gets esoteric, but some rudimentary understanding is helpful. In this chapter we will cover the bare minimum necessary to grasp the essentials of rock magnetism.

In Chapter 1 we took the classical (pre-quantum mechanics) approach and suggested that the orbit of an electron about the nucleus could be considered a tiny electric current with a correspondingly tiny magnetic moment. But quantum physics tells us that this “planetary” view of the atom cannot be true. An electron zipping around a nucleus would generate radio waves, losing energy and eventually would crash into the nucleus.

Apparently, this does not happen, so the classical approach is fatally flawed and we must turn to quantum mechanics.

In quantum mechanics, electronic motion is stabilized by the fact that electrons can only have certain energy states; they are quantized. The energy of a given electron can be described in terms of solutions, Ψ, to something called Schr�dinger’s wave equation. The function Ψ(r,θ,ϕ) gives the probability of finding an electron at a given position. [Remember from Chapter 2 that r,θ,ϕ are the three spherical coordinates.] It depend on three special quantum numbers (n,l,m):

| (3.1) |

The number n is the so-called “principal” quantum number. The R_{n}^{l}(r) are
functions specific to the element in question and the energy state of the electron
n. It is evaluated at an effective radius r in atomic units. The Y _{l}^{m} are a fully
normalized complex representation of the spherical harmonics introduced in
Section 2.2. For each level n, the number l ranges from 0 to n-1 and m from l
backwards to -l.

The lowest energy of the quantum wave equations is found by setting n equal to unity and both l and m to zero. Under these conditions, the solution to the wave equation is given by:

| (3.2) |

where Z is the atomic number and ρ is 2Zr∕n. Note that at this energy level, there is no dependence of Y on ϕ or θ. Substituting these two equations into Equation 3.1 gives the probability density Ψ for an electron as a function of radius of r. This is sketched as the line in Figure 3.1. Another representation of the same idea is shown in the inset, whereby the density of dots at a given radius reflects the probability distribution shown by the solid curve. The highest dot density is found at a radius of about one atomic unit, tapering off the farther away from the center of the atom. Because there is no dependence on θ or ϕ the probability distribution is a spherical shell. All the l,m = 0 shells are spherical and are often referred to as the 1s, 2s, 3s shells, where the numbers are the energy levels n. A surface with equal probability is a sphere and example of one such shell is shown in Figure 3.2a.

For l = 1, m will have values of -1, 0 and 1 and the Y _{l}^{m}(ϕ,θ)s are given
by:

As might be expected, the shells for l = 2 are even more complicated that for l = 1. These shells are called “d” shells and two examples are shown in Figure 3.2c and d.

Returning to the tiny circuit idea, somehow the motion of the electrons in their
shells acts like an electronic circuit and creates a magnetic moment. In quantum
mechanics, the angular momentum vector of the electron L is quantized, for
example as integer multiples of ℏ, the “reduced” Planck’s constant (or h _
2π where
h = 6.63 x 10^{-34} Js). The magnetic moment arising from the orbital angular
momentum is given by:

| (3.3) |

This is known as the Bohr magneton.

So far we have not mentioned one last quantum number, s. This is the “spin” of
the electron and has a value of �1
2. The spin itself produces a magnetic moment
which is given by 2sm_{b}, hence is numerically identical to that produced by the
orbit.

Atoms have the same number of electrons as protons in order to preserve charge balance. Hydrogen has but one lonely electron which in its lowest energy state sits in the 1s electronic shell. Helium has a happy pair, so where does the second electron go? To fill in their electronic shells, atoms follow three rules:

- No two electrons may have the same set of quantum numbers. This is
Pauli’s exclusion principle. Because spin (s) can be �1
2, two electrons
fit in one orbital. When a single electron occupies a given orbital, it is
called “unpaired” and has a magnetic moment of 1 m
_{b}. - Orbitals are filled in order of increasing energy. The energy state of a given orbital is dependent on the context (whether the atom is bound to other atoms or not), but in general they will be filled according to the scheme shown in Figure 3.3.
- Electrons are added so that the spins remain as parallel as possible (Hund’s rule). Notice in Figure 3.3 that when filling the third energy level (n = 3), all five d shells are filled with one kind of spin (say, all up, or +1 2), before the electrons begin to pair up. Also, because the energies of the shells change somewhat according to the context they are in, the 4s shell will actually give up an electron to a d shell, before the d shells begin to pair up. Hund’s rule gives the atoms with some d shell electrons (the so-called “transition elements”, e.g., Cr, Mn, Fe, Co and Ni) the possibility of large magnetic moments.

Each unpaired spin has a moment of one Bohr magneton m_{b}. The elements with
the most unpaired spins are the transition elements which are responsible for
most of the paramagnetic behavior observed in rocks. For example, in Figure 3.3
we see that Mn has a structure of: (1s^{2}2s^{2}2p^{6}3s^{2}3p^{6})3d^{5}4s^{2}, hence has
five unpaired spins and a net moment of 5 m_{b}. Fe has a structure of
(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6})3d^{6}4s^{2} with a net moment of 4 m_{b}, In minerals, the
transition elements are in a variety of oxidation states. Fe commonly
occurs as Fe^{2+} and Fe^{3+}. When losing electrons to form ions, transition
metals lose the 4s electrons first, so we have for example, Fe^{3+} with a
structure of (1s^{2}2s^{2}2p^{6}3s^{2}3p^{6})3d^{5}, or 5 m_{b}. Similarly Fe^{2+} has 4 m_{b}
and Ti^{4+} has no unpaired spins. Iron is the main magnetic species in
geological materials, but Mn^{2+} (5 m_{b}) and Cr^{3+} (3 m_{b}) occur in trace
amounts.

We have learned that there are two sources of magnetic moments in electronic motions: the orbits and the (unpaired) spins. These moments respond to external magnetic fields giving rise to an induced magnetization, a phenomenon alluded to briefly in Chapter 1. We will consider first the contribution of the electronic orbits.

The angular momentum of electrons is quantized in magnitude but also has
direction (see L in Figure 3.4). The angular momentum vector has an associated
magnetic moment vector m_{b}. A magnetic field H exerts a torque on the moment,
which nudges it (and the momentum vector associated with it) to the side (ΔL).
L therefore will precess around the magnetic field direction, much like a spinning
top precesses around the direction of gravity. The precession of L is called
Larmor precession.

The changed momentum vector from Larmor precession in turn results in a
changed magnetic moment vector Δm. The sense of the change in net moment is
always to oppose the applied field. Therefore, the response of the magnetic
moments of electronic orbitals creates an induced magnetization M_{I}
that is observable outside the substance; it is related to the applied field
by:

We learned in Chapter 1 that the proportionality between induced magnetization
and the applied field is known as the magnetic susceptibility. The ratio M_{I}∕H for
the response of the electronic orbitals is termed the diamagnetic susceptibility
χ_{d}; it is negative, essentially temperature independent and quite small. This
diamagnetic response is a property of all matter, but for substances whose atoms
possess atomic magnetic moments, diamagnetism is swamped by effects
of magnetic fields on the atomic magnetic moments. In the absence of
unpaired electronic spins, diamagnetic susceptibility dominates the magnetic
response. Common diamagnetic substances include quartz (SiO_{2}), calcite
(CaCO_{3}) and water (H_{2}O). The mass normalized susceptibility of quartz
is -0.62 x 10^{-9} m^{3}kg^{-1} to give you an idea of the magnitudes of these
things.

In many geological materials, the orbital contributions cancel out because they
are randomly oriented with respect to one another and the magnetization arises
from the electronic spins. We mentioned that unpaired electronic spins behave as
magnetic dipoles with a moment of one Bohr magneton. In the absence of an
applied field, or in the absence of the ordering influence of neighboring spins
which are known as exchange interactions, the electronic spins are essentially
randomly oriented. An applied field acts to align the spins which creates a net
magnetization equal to χ_{p}H where χ_{p} is the paramagnetic susceptibility. For any
geologically relevant conditions, the induced magnetization is linearly dependent
on the applied field. In paramagnetic solids, atomic magnetic moments react
independently to applied magnetic fields and to thermal energy. At any
temperature above absolute zero, thermal energy vibrates the crystal
lattice, causing atomic magnetic moments to oscillate rapidly in random in
orientations. In the absence of an applied magnetic field, atomic moments are
equally distributed in all directions with a resultant magnetization of
zero.

A useful first order model for paramagnetism was worked out by P. Langevin in 1905. (Of course in messy reality things are a bit more complicated, but Langevin theory will work well enough for us at this stage.) Langevin theory is based on a few simple premises:

- Each unpaired spin contributes a dipole moment.
- In the absence of an applied field, the moments are essentially randomly oriented, i.e., all directions are equally likely to occur.
- Application of a magnetic field exerts an aligning torque on the atomic
magnetic moments. The magnetic energy E
_{m}(see also Section 1.4 in Chapter 1) of a magnetic moment m at an angle θ with an external magnetic field B = μ_{o}H is given by:(3.4) Magnetic energy is at a minimum when the magnetic moment is lined up with the magnetic field.

- There is competition between the magnetic energy E
_{m}and the thermal energy kT where k is Boltzmann’s constant (1.38 x 10^{-23}m^{2}kg s^{-2}K^{-1}) and T is temperature in kelvin).

Consider an atomic magnetic moment, (m = 2m_{b} = 1.85×10^{-23} Am^{2}), in a
magnetic field of 10^{-2} T, (for reference, the largest geomagnetic field at the
surface is about 65 μT – see Chapter 2). The aligning energy is therefore mB =
1.85 × 10^{-25} J). However, thermal energy at 300K (traditionally chosen as a
temperature close to room temperature providing easy arithmetic) is
Boltzmann’s constant times the temperature, or about 4 x 10^{-21} J. So thermal
energy is several orders of magnitude larger than the aligning energy and the net
magnetization is small even in this rather large (compared to the Earth’s field)
magnetizing field.

Using the principles of statistical mechanics, we find that the probability density
of a particular magnetic moment having a magnetic energy of E_{m} is given
by:

| (3.5) |

From this we see that the degree of alignment depends exponentially on the ratio
of magnetic energy to thermal energy. The degree of alignment with
the magnetic field controls the net magnetization M. When spins are
completely aligned, the substance has a saturation magnetization M_{s}. The
probability density function leads directly to the following relation (derived in
Appendix A.2.1):

| (3.6) |

where a = mB∕kT. The function enclosed in square brackets is known as the Langevin function ().

Equation 3.6 is plotted in Figure 3.5a and predicts several intuitive results: 1)
M = 0 when B = 0 and 2) M∕M_{s} = 1 when the applied magnetic field is infinite.
Furthermore, M is some 90% of M_{s} when mB is some 10-20 times kT.
When kT >> mB,(a) is approximately linear with a slope of ~ 1∕3. At
room temperature and fields up to many tesla, (a) is approximately
mB∕3kT. If the moments are unpaired spins (m = m_{b}), then the maximum
magnetization possible (M_{s}) is given by the number of moments N, their
magnitude (m_{b}) normalized by the volume of the material v or M_{s} = Nm_{b}∕v,
and

Please note that we have neglected all deviations from isotropy including quantum mechanical effects as well as crystal shape, lattice defects, and state of stress. These complicate things a little, but to first order the treatment followed here provides a good approximation. We can rewrite the above equation as:

| (3.7) |

To first order, paramagnetic susceptibility χ_{p} is positive, larger than
diamagnetism and inversely proportional to temperature. This inverse T
dependence (see Figure 3.5b) is known as Curie’s law of paramagnetism. The
paramagnetic susceptibility of, for example, biotite is 790 x 10^{-9} m^{3} kg^{-1}, or
about three orders of magnitude larger than quartz (and of the opposite
sign!).

We have considered the simplest case here in which χ can be treated
as a scalar and is referred to as the bulk magnetic susceptibility χ_{b}. In
detail, magnetic susceptibility can be quite complicated. The relationship
between induced magnetization and applied field can be affected by crystal
shape, lattice structure, dislocation density, state of stress, etc., which
give rise to possible anisotropy of the susceptibility. Furthermore, there
are only a finite number of electronic moments within a given volume.
When these are fully aligned, the magnetization reaches saturation. Thus,
magnetic susceptibility is both anisotropic and non-linear with applied
field.

Some substances give rise to a magnetic field in the absence of an applied field. This magnetization is called remanent or spontaneous magnetization, also loosely known as ferromagnetism (sensu lato). Magnetic remanence is caused by strong interactions between neighboring spins that occur in certain crystals.

The so-called exchange energy is minimized when the spins are aligned parallel or anti-parallel depending on the details of the crystal structure. Exchange energy is a consequence of the Pauli exclusion principle (no two electrons can have the same set of quantum numbers). In the transition elements, the 3d orbital is particularly susceptible to exchange interactions because of its shape and the prevalence of unpaired spins, so remanence is characteristic of certain crystals containing transition elements with unfilled 3d orbitals.

In oxides, oxygen can form a bridge between neighboring cations which are
otherwise too far apart for direct overlap of the 3d orbitals in a phenomenon
known as superexchange. In Figure 3.6 the 2p electrons of the oxygen are shared
with the neighboring 3d shells of the iron ions. Pauli’s exclusion principle means
that the shared electrons must be antiparallel to each of the electrons in the 3d
shells. The result is that the two cations are coupled. In the case shown in
Figure 3.6 there is an Fe^{2+} ion coupled antiparallel to an Fe^{3+} ion. For
two ions with the same charge, the coupling will be parallel. Exchange
energies are huge, equivalent to the energy associated with the same
moment in a field of the order of 1000 T. [The largest field available
in the Scripps paleomagnetic laboratory is about 2.5 T, and that only
fleetingly.]

As temperature increases, crystals expand and exchange becomes weaker. Above
a temperature characteristic of each crystal type (known as the Curie
temperature T_{c}), cooperative spin behavior disappears entirely and the material
becomes paramagnetic.

While the phenomenon of ferromagnetism results from complicated interactions
of neighboring spins, it is useful to think of the ferromagnetic moment as
resulting from a quasi-paramagnetic response to a huge internal field. This
imaginary field is termed the Weiss molecular field H_{w}. In Weiss theory, H_{w} is
proportional to the magnetization of the substance, i.e.,

where β is the constant of proportionality. The total magnetic field that the substance experiences is:

where H is the external field. By analogy to paramagnetism, we can substitute
a = μ_{o}m_{b}(H_{tot})∕kT) for H in Langevin function:

| (3.8) |

For temperatures above the Curie temperature T_{c} (i.e. T - T_{c} > 0) there is by
definition no internal field, hence βM is zero. Substituting Nm_{b}∕v for M_{s}, and
using the low-field approximation for (a), Equation 3.8 can be rearranged to
get:

| (3.9) |

Equation 3.9 is known as the Curie-Weiss law and governs ferromagnetic susceptibility above the Curie temperature (dashed line in Figure 3.7).

Below the Curie temperature H_{w} >> H; we can neglect the external field H and
get:

Substituting again for M_{s} and rearranging, we get:

| (3.10) |

where T_{c} is the Curie temperature and is given by:

We have treated ferromagnetism from a classical point of view and this is strictly
incorrect because ferromagnetism results primarily from quantum mechanical
phenomena. The primary difference between the classical derivation and the
quantum mechanical one lies in the fact that in quantum mechanics, only certain
angles of the magnetic moments are allowed, as opposed to all directions in
Langevin theory. In the end, the predictions of magnetization as a function of
temperature are different in detail. The end product of the quantum
mechanical treatment (see Dunlop and �zdemir, 1997) is that the variation of
saturation magnetization as a function of temperature can be reasonably well
approximated (near the Curie Temperature, T_{c}) by a normalized power law
variation:

| (3.11) |

where γ is 0.5 from simple molecular field theory and T_{o} is absolute zero (in
kelvin). Dunlop and �zdemir (1997) cite a value of around 0.43 for γ, but
the data sets cited by Dunlop and �zdemir (1997; e.g., Figure 3.5 on
page 52) are actually best-fit with values for γ of about 0.36 – 0.39 (see
Figure 3.8). These curves have been normalized by their inferred curie
Temperatures which are around 565^{∘}C (data of B. Moskowitz, cited in Banerjee,
1991).

As we have seen, below the Curie temperature, certain crystals have a permanent (remanent) magnetization resulting from the alignment of unpaired electronic spins over a large area within the crystal. Spins may be either parallel or anti-parallel; the sense of spin alignment is controlled entirely by crystal structure. The energy term associated with this phenomenon is the exchange energy. There are three categories of spin alignment: ferromagnetism (sensu stricto), ferrimagnetism and antiferromagnetism (see Figure 3.9).

In ferromagnetism (sensu stricto, Figure 3.9a), the exchange energy is minimized when all the spins are parallel, as occurs in pure iron. When spins are perfectly antiparallel (antiferromagnetism, Figure 3.9b), there is no net magnetic moment, as occurs in ilmenite. Occasionally, the antiferromagnetic spins are not perfectly aligned in an antiparallel orientation, but are canted by a few degrees. This spin-canting (Figure 3.9c) gives rise to a weak net moment, as occurs in hematite, a common magnetic mineral (see Chapter 6). Also, antiferromagnetic materials can have a net moment if spins are not perfectly compensated owing to defects in the crystal structure, as occurs in fine-grained hematite. The uncompensated spins result in a so-called defect moment (Figure 3.9d). We note in passing that the temperature at which spins become disordered in antiferromagnetic substances is termed the N�el temperature. In ferrimagnetism, spins are also aligned antiparallel, but the magnitudes of the moments in each direction are unequal, resulting in a net moment (Figure 3.9e).

In figures like Figure 3.9, electronic spins are depicted as being simply aligned with some minimum energy direction (aligned with the field, or along some easy axis). Yet we already know about the paramagnetic effect of misalignment through random thermal fluctuations. We learned that an external magnetic field generates a torque on the electronic spins, and in isolation, a magnetic moment will respond to the torque in a manner similar in some respects to the way a spinning top responds to gravity: the magnetic moment will precess about the applied field direction, spiraling in and come to a rest parallel to it (Figure 3.10a). Because of the strong exchange coupling in ferromagnetic phases, spins tend to be aligned parallel (or antiparallel) to one another and the spiralling is done in a coordinated fashion, with neighboring spins as parallel as possible to one another (Figure 3.10b). This phenomenon is known as a spin wave.

SUPPLEMENTAL READINGS: O’Reilly (1984), Chapter 3.1; Dunlop and �zdemir (1997), Chapter 2.1 to 2.7.

Problem 1

a) Given one Bohr magneton (m_{b}) in the Earth’s field (40 μT), write a program
using Python that calcuates magnetostatic interaction energy (-m_{b}B cosθ) for
angles 0→ 180^{∘}. Make a plot of this with the matplotlib module in
Python.

b) Calculate the thermal energy at room temperature (300K). How does this compare with the interaction energy?

Problem 2

Fayalite (Fe_{2}SiO_{4}) is a paramagnetic solid with magnetic susceptibility χ =
4.4 x 10^{-4} (cgs units) at 0^{∘}C (= 273K). A single crystal of fayalite has
a volume of 2 cm^{3}. This crystal is placed in a magnetic field, H = 10
oe at 0^{∘}C. What is the resulting induced magnetic moment m of this
crystal?

a) Do this problem first in cgs units. Then convert your answer to SI using the conversion factors in Table 1.1 in Chapter 1.

b) Do the problem again by first converting all the parameters into SI units. Check your answer by converting the SI answer that you get back to cgs. You should get the same answer (but you would be surprised how many people do this wrong).

Problem 3

If fayalite is placed in a magnetic field H= 100 oe at a temperature of 500^{∘}C (=
773K), what is the resulting magnetization, M?

Problem 4

MnS is a paramagnetic solid. At 300K there are 4 x 10^{28} molecules of MnS per
m^{3}. Look up the number of unpaired spins for the cationic magnetic moment of
Mn^{2+} in the text and find the paramagnetic susceptibility, χ, of MnS at
300K?

Problem 5

a) Read into a Pandas DataFrame the datafile Chapter_3/BMoskinBan91.txt provided. Make a plot of magnetization versus temperature. What is the Curie temperature of the material?

b) Using this Equation 3.11 from the chapter, find the value for γ between 0.35
and 0.43 at intervals of 0.01 that fits the best. Plot the data as in Figure 3.8 in
the chapter, i.e. M_{s}(T)∕Ms(T_{o}) against T∕T_{c}.