The magnetic moment of a material is the incomplete cancellation of the atomic magnetic moments in that material. Electron spin and orbital motion both have magnetic moments associated with them (Figure 1), but in most atoms the electronic moments are oriented usually randomly so that overall in the material they cancel each other out (Figure 2), this is called diamagnetism.

Figure 1: Orbital magnetic moment.

Figure 2: Magnetic moments in a diamagnetic sample.

If the cancellation of the moments is incomplete then the atom has a net magnetic moment. There are many subclasses of magnetic ordering such as para-, superpara-, ferro-, antiferro- or ferrimagnetisim which can be displayed in a material and which usually depends, upon the strength and type of magnetic interactions and external parameters such as temperature and crystal structure atomic content and the magnetic environment which a material is placed in.

The magnetic moments of atoms, molecules or formula units are often quoted in terms of the Bohr magneton, which is equal to the magnetic moment due to electron spin

The magnetisim of a material, the extent that which a material is magnetic, is not a static quantity, but varies compared to the environment that a material is placed in. It is similar to the temperature of a material. For example if a material is placed in an oven it will heat up to a temperature similar to that of the ovens. However the speed of heating of that material, and also that of cooling are determined by the atomic structure of the material. The magnetization of a material is similar. When a material is placed in a magnetic field it maybe become magnetized to an extent and retain that magnetization after it is removed from the field. The extent of magnetization, and type of magnetization and the length of time that a material remains magnetized, depends again on the atomic makeup of the material.

Measuring a materials magnetisim can be done on a micro or macro scale. Magnetisim is measured over two parameters direction and strength. Thus magnetization has a vector quantity. The simplest form of a magnetometer is a compass. It measures the direction of a magnetic field. However more sophisticated instruments have been developed which give a greater insight into a materials magnetisim.

So what exactly are you reading when you observe the output from a magnetometer?

The magnetisim of a sample is called the magnetic moment of that sample and will be called that from now on. The single value of magnetic moment for the sample, is a combination of the magnetic moments on the atoms within the sample (Figure 3), it is also the type and level of magnetic ordering and the physical dimensions of the sample itself.

Figure 3: Schematic representations of the net magnetic moment in a diamagnetic sample.

The "intensity of magnetization", M, is a measure of the magnetization of a body. It is defined as the magnetic moment per unit volume or

M = m/V

with units of Am (emucm³ in cgs notation).

A material contains many atoms and their arrangement affects the magnetization of that material. In Figure 4 (a) a magnetic moment m is contained in unit volume. This has a magnetization of m Am. Figure 4 (b) shows two such units, with the moments aligned parallel. The vector sum of moments is 2m in this case, but as the both the moment and volume are doubled M remains the same. In Figure 4 (c) the moments are aligned antiparallel. The vector sum of moments is now 0 and hence the magnetization is 0 Am.

Figure 4: Effect of moment alignment on magnetization: (a) Single magnetic moment, (b) two identical moments aligned parallel and (c) antiparallel to each other. Adapted from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

Scenarios (b) and (c) are a simple representation of ferro- and antiferromagnetic ordering. Hence we would expect a large magnetization in a ferromagnetic material such as pure iron and a small magnetization in an antiferromagnet such as γ-Fe₂O₃

When a material is passed through a magnetic field it is affected in two ways:

Magnetic susceptibility

The concept of magnetic moment is the starting point when discussing the behavior of magnetic materials within a field. If you place a bar magnet in a field it will experience a torque or moment tending to align its axis in the direction of the field. A compass needle behaves in the same way. This torque increases with the strength of the poles and their distance apart. So the value of magnetic moment tells you, in effect, 'how big a magnet' you have.

Figure 5: Schematic representation of the torque or moment that a magnet experiences when it is placed in a magnetic field. The magnetic will try to align with the magnetic field.

If you place a material in a weak magnetic field, the magnetic field may not overcome the binding energies that keep the material in a non magnetic state. This is because it is energetically more favorable for the material to stay exactly the same. However, if the strength of the magnetic moment is increased, the torque acting on the smaller moments in the material, it may become energetically more preferable for the material to become magnetic. The reasons that the material becomes magnetic depends on factors such as crystal structure the temperature of the material and the strength of the field that it is in. However a simple explanation of this is that as the magnetic moment strength increases it becomes more favorable for the small fields to align themselves along the path of the magnetic field, instead of being opposed to the system. For this to occur the material must rearrange its magnetic makeup at the atomic level to lower the energy of the system and restore a balance.

It is important to remember that when we consider the magnetic susceptibility and take into account how a material changes on the atomic level when it is placed in a magnetic field with a certain moment. The moment that we are measuring with our magnetometer is the total moment of that sample.

where χ = susceptibility, M = variation of magnetization, and H = applied field.

Magnetic permeability

Magnetic permeability is the ability of a material to conduct an electric field. In the same way that materials conduct or resist electricity, materials also conduct or resist a magnetic flux or the flow of magnetic lines of force (Figure 6).

Figure 6: Magnetic ordering in a ferromagnetic material.

Ferromagnetic materials are usually highly permeable to magnetic fields. Just as electrical conductivity is defined as the ratio of the current density to the electric field strength, so the magnetic permeability, μ, of a particular material is defined as the ratio of flux density to magnetic field strength. However unlike in electrical conductivity magnetic permeability is nonlinear.

μ = B/H

Permeability, where μ is written without a subscript, is known as absolute permeability. Instead a variant is used called relative permeability.

μ = μ_o x μ_r

Absolute permeability is a variation upon 'straight' or absolute permeability, μ, but is more useful as it makes clearer how the presence of a particular material affects the relationship between flux density and field strength. The term 'relative' arises because this permeability is defined in relation to the permeability of a vacuum, μ₀.

μ_r = μ/μ_o

For example, if you use a material for which μ_r = 3 then you know that the flux density will be three times as great as it would be if we just applied the same field strength to a vacuum.

Initial permeability

Initial permeability describes the relative permeability of a material at low values of B (below 0.1 T). The maximum value for μ in a material is frequently a factor of between 2 and 5 or more above its initial value.

Low flux has the advantage that every ferrite can be measured at that density without risk of saturation. This consistency means that comparison between different ferrites is easy. Also, if you measure the inductance with a normal component bridge then you are doing so with respect to the initial permeability.

Permeability of a vacuum in the SI

The permeability of a vacuum has a finite value - about 1.257 × 10^-6 H m^-1 - and is denoted by the symbol μ₀. Note that this value is constant with field strength and temperature. Contrast this with the situation in ferromagnetic materials where μ is strongly dependent upon both. Also, for practical purposes, most non-ferromagnetic substances (such as wood, plastic, glass, bone, copper aluminum, air and water) have permeability almost equal to μ₀; that is, their relative permeability is 1.0.

The permeability, μ, the variation of magnetic induction,

with applied field,

μ = B/H

A single measurement of a sample's magnetization is relatively easy to obtain, especially with modern technology. Often it is simply a case of loading the sample into the magnetometer in the correct manner and performing a single measurement. This value is, however, the sum total of the sample, any substrate or backing and the sample mount. A sample substrate can produce a substantial contribution to the sample total.

For substrates that are diamagnetic, under zero applied field, this means it has no effect on the measurement of magnetization. Under applied fields its contribution is linear and temperature independent. The diamagnetic contribution can be calculated from knowledge of the volume and properties of the substrate and subtracted as a constant linear term to produce the signal from the sample alone. The diamagnetic background can also be seen clearly at high fields where the sample has reached saturation: the sample saturates but the linear background from the substrate continues to increase with field. The gradient of this background can be recorded and subtracted from the readings if the substrate properties are not known accurately.

When a material exhibits hysteresis, it means that the material responds to a force and has a history of that force contained within it. Consider if you press on something until it depresses. When you release that pressure, if the material remains depressed and doesn’t spring back then it is said to exhibit some type of hysteresis. It remembers a history of what happened to it, and may exhibit that history in some way. Consider a piece of iron that is brought into a magnetic field, it retains some magnetization, even after the external magnetic field is removed. Once magnetized, the iron will stay magnetized indefinitely. To demagnetize the iron, it is necessary to apply a magnetic field in the opposite direction. This is the basis of memory in a hard disk drive.

The response of a material to an applied field and its magnetic hysteresis is an essential tool of magnetometry. Paramagnetic and diamagnetic materials can easily be recognized, soft and hard ferromagnetic materials give different types of hysteresis curves and from these curves values such as saturation magnetization, remnant magnetization and coercivity are readily observed. More detailed curves can give indications of the type of magnetic interactions within the sample.

The intensity of magnetization depends upon both the magnetic moments in the sample and the way that they are oriented with respect to each other, known as the magnetic ordering.

Diamagnetic materials, which have no atomic magnetic moments, have no magnetization in zero field. When a field is applied a small, negative moment is induced on the diamagnetic atoms proportional to the applied field strength. As the field is reduced the induced moment is reduced.

Figure 7: Typical effect on the magnetization, M, of an applied magnetic field, H, on (a) a paramagnetic system and (b) a diamagnetic system. Adapted from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

In a paramagnet the atoms have a net magnetic moment but are oriented randomly throughout the sample due to thermal agitation, giving zero magnetization. As a field is applied the moments tend towards alignment along the field, giving a net magnetization which increases with applied field as the moments become more ordered. As the field is reduced the moments become disordered again by their thermal agitation. The figure shows the linear response M v H where μH << kT.

The hysteresis curves for a ferromagnetic material are more complex than those for diamagnets or paramagnets. Below diagram shows the main features of such a curve for a simple ferromagnet.

Figure 8: Schematic of a magnetization hysteresis loop in a ferromagnetic material showing the saturation magnetization, M_s, coercive field, H_c, and remnant magnetization, M_r. Virgin curves are shown dashed for nucleation (1) and pinning (2) type magnets. Adapted from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

In the virgin material (point 0) there is no magnetization. The process of magnetization, leading from point 0 to saturation at M = M_s, is outlined below. Although the material is ordered ferromagnetically it consists of a number of ordered domains arranged randomly giving no net magnetization. This is shown in below (a) with two domains whose individual saturation moments, M_s, lie antiparallel to each other.

Figure 9: The process of magnetization in a demagnetized ferromagnet. Adaped from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

As the magnetic field, H, is applied, (b), those domains which are more energetically favorable increase in size at the expense of those whose moment lies more antiparallel to H. There is now a net magnetization; M. Eventually a field is reached where all of the material is a single domain with a moment aligned parallel, or close to parallel, with H. The magnetization is now M = M_sCosΘ where Θ is the angle between M_s along the easy magnetic axis and H. Finally M_s is rotated parallel to H and the ferromagnet is saturated with a magnetization M = M_s.

The process of domain wall motion affects the shape of the virgin curve. There are two qualitatively different modes of behavior known as nucleation and pinning, shown in Figure 8 as curves 1 and 2, respectively.

In a nucleation-type magnet saturation is reached quickly at a field much lower than the coercive field. This shows that the domain walls are easily moved and are not pinned significantly. Once the domain structure has been removed the formation of reversed domains becomes difficult, giving high coercivity. In a pinning-type magnet fields close to the coercive field are necessary to reach saturation magnetization. Here the domain walls are substantially pinned and this mechanism also gives high coercivity.

As the applied field is reduced to 0 after the sample has reached saturation the sample can still possess a remnant magnetization, M_r. The magnitude of this remnant magnetization is a product of the saturation magnetization, the number and orientation of easy axes and the type of anisotropy symmetry. If the axis of anisotropy or magnetic easy axis is perfectly aligned with the field then M_r = M_s, and if perpendicular M_r= 0.

At saturation the angular distribution of domain magnetizations is closely aligned to H. As the field is removed they turn to the nearest easy magnetic axis. In a cubic crystal with a positive anisotropy constant, K1, the easy directions are <100>. At remnance the domain magnetizations will lie along one of the three <100> directions. The maximum deviation from H occurs when H is along the <111> axis, giving a cone of distribution of 55^o around the axis. Averaging the saturation magnetization over this angle gives a remnant magnetization of 0.832 Ms.

The coercive field, H_c, is the field at which the remnant magnetization is reduced to zero. This can vary from a few Am for soft magnets to 10⁷Am for hard magnets. It is the point of magnetization reversal in the sample, where the barrier between the two states of magnetization is reduced to zero by the applied field allowing the system to make a Barkhausen jump to a lower energy. It is a general indicator of the energy gradients in the sample which oppose large changes of magnetization.

The reversal of magnetization can come about as a rotation of the magnetization in a large volume or through the movement of domain walls under the pressure of the applied field. In general materials with few or no domains have a high coercivity whilst those with many domains have a low coercivity. However, domain wall pinning by physical defects such as vacancies, dislocations and grain boundaries can increase the coercivity.

Figure 10: Shape of hysteresis loop as a function of Θ _H, the angle between anisotropy axis and applied field H, for: (a) Θ_H, = 0°, (b) 45° and (c) 90°. Adaped from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

The loop illustrated in Figure 10 is indicative of a simple bi-stable system. There are two energy minima: one with magnetization in the positive direction, and another in the negative direction. The depth of these minima is influenced by the material and its geometry and is a further parameter in the strength of the coercive field. Another is the angle, Θ_H, between the anisotropy axis and the applied field. The above fig shows how the shape of the hysteresis loop and the magnitude of H_c varies with Θ_H. This effect shows the importance of how samples with strong anisotropy are mounted in a magnetometer when comparing loops.

A hysteresis curve gives information about a magnetic system by varying the applied field but important information can also be gleaned by varying the temperature. As well as indicating transition temperatures, all of the main groups of magnetic ordering have characteristic temperature/magnetization curves. These are summarized in Figure 11 and Figure 12. At all temperatures a diamagnet displays only any magnetization induced by the applied field and a small, negative susceptibility.

The curve shown for a paramagnet (Figure 11) is for one obeying the Curie law,

and so intercepts the axis at T = 0. This is a subset of the Curie-Weiss law,

where θ is a specific temperature for a particular substance (equal to 0 for paramagnets).

Figure 11: Variation of reciprocal susceptibility with temperature for: (a) antiferromagnetic, (b) paramagnetic and (c) diamagnetic ordering. Adaped from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

Figure 12: Variation of saturation magnetization below, and reciprocal susceptibility above T_c for: (a) ferromagnetic and (b) ferrimagnetic ordering. Adaped from J. Bland Thesis M. Phys (Hons)., 'A Mossbauer spectroscopy and magnetometry study of magnetic multilayers and oxides.' Oliver Lodge Labs, Dept. Physics, University of Liverpool

Above T_N and T_C both antiferromagnets and ferromagnets behave as paramagnets with 1/χ linearly proportional to temperature. They can be distinguished by their intercept on the temperature axis, T = Θ. Ferromagnetics have a large, positive Θ, indicative of their strong interactions. For paramagnetics Θ = 0 and antiferromagnetics have a negative Θ.

The net magnetic moment per atom can be calculated from the gradient of the straight line graph of 1/χ versus temperature for a paramagnetic ion, rearranging Curie's law to give

where A is the atomic mass, k is Boltzmann's constant, N is the number of atoms per unit volume and x is the gradient.

Ferromagnets below T_C display spontaneous magnetization. Their susceptibility above T_C in the paramagnetic region is given by the Curie-Weiss law

where g is the gyromagnetic constant. In the ferromagnetic phase with T greater than T_C the magnetization M (T) can be simplified to a power law, for example the magnetization as a function of temperature can be given by

where the term β is typically in the region of 0.33 for magnetic ordering in three dimensions.

The susceptibility of an antiferromagnet increases to a maximum at T_N as temperature is reduced, then decreases again below T_N. In the presence of crystal anisotropy in the system this change in susceptibility depends on the orientation of the spin axes: χ (parallel)decreases with temperature whilst χ (perpendicular) is constant. These can be expressed as

where C is the Curie constant and Θ is the total change in angle of the two sublattice magnetizations away from the spin axis, and

where n_g is the number of magnetic atoms per gramme, B’ is the derivative of the Brillouin function with respect to its argument a’, evaluated at a^’₀, μ_H is the magnetic moment per atom and γ is the molecular field coefficient.