Magnetisation M - is the vector quantity describing response of matter to applied magnetic field. Magnetisation is the sum of all magnetic dipole moments of individual atoms, molecules or subatomic particles, per unit volume of a given matter.¹⁾²⁾³⁾ In the SI system, the unit of magnetisation is ampere per metre or A/m, the same as the unit of magnetic field strength H.⁴⁾⁵⁾⁶⁾

The physical quantity of magnetisation M should not be confused with the process of magnetisation, which is also commonly referred to a “magnetisation”. The difference is usually clear from the context in a given publication.⁷⁾

Magnetisation M is the sum of magnetic dipole moments m_i per unit of volume V

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Magnetisation $M$ (A/m) represents an averaged quantity (“smoothed” over distances spanning across many individual magnetic moments⁸⁾) and is equivalent to magnetic polarisation $J$ (T) - the two represent the same physical quantity, with numerical values and units scaled by the permeability of vacuum $μ_0$.⁹⁾

Magnetic field is a vector field in space, and is a type of energy whose full quantification requires the knowledge of the vector fields of both magnetic field strength $H$ and flux density $B$, or other values correlated with them, such as magnetisation $M$ (or polarisation $J$). In vacuum, $M$ = 0, and at each point the $H$ and $B$ vectors are oriented along the same direction and are directly proportional through permeability of free space, but in other media they can be misaligned (especially in highly anisotropic materials).

From engineering viewpoint, magnetic field strength $H$ can be thought of as excitation, and the magnetic flux density $B$ as the response of the magnetised medium. The quantity $B$ encompasses all of the magnetic response of the medium (including $M$ or $J$, as well as any non-magnetic contribution arising from the applied $H$).¹⁰⁾

The four vector quantities (H, B, M, J) are interlinked such that:^11)12)13)14)

Expressing the equation with magnetisation M is known as the Sommerfeld convention, and with polarisation J as the Kennelly convention. These are mathematically equivalent, and can be used depending on the easy of application in a given magnetic problem.¹⁵⁾

There are many other names which are used in the literature, but refer to the same physical quantity:

• magnetisation M ¹⁶⁾
• magnetic polarisation M ¹⁷⁾
• intensity of magnetization M ¹⁸⁾
• magnetization field M ¹⁹⁾
• induced magnetic field M ²⁰⁾
• magnetization density M ²¹⁾
• field M ²²⁾
• M-field ²³⁾
• intrinsic magnetisation M ²⁴⁾
• and probably several more.

Magnetic dipole moment is defined as the product of the current (measured in amperes) flowing in the loop and the area of the loop (measured in square metres), so the resulting unit is (A·m²). Subatomic particles such as electrons exhibit intrinsic moments due to spin and orbital movement.

Magnetic moment $\vec m$: $I$ - current, $A$ - area, $\vec a$ - unit vector (normal to $A$)

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The vector sum of many such dipoles over some volume is therefore (A·m² / m³) ≡ (A/m), so the unit for M is the same as for magnetic field strength H.

In some publications M is expressed in tesla or gauss²⁵⁾, which are the same units as for magnetic flux density B or magnetic polarisation J. This is simply because the values are scaled by the permeability of vacuum μ₀ which itself has the units of H/m.²⁶⁾

Therefore, the resulting unit of M·μ₀ is (A/m)·(H/m) ≡ (T).

The scaling of M (or H) by μ₀ is not always explicitly stated or explained in the publications, but if the units are (T) then the scaling must have been applied, for the SI units to match.

Similar logic applies to the CGS units, if M is compared to B.

Depending on the application, magnetisation can also be expressed per unit weight (mass magnetisation), mole (molar magnetisation), per unit cell of crystal, Bohr magnetons, or even in atomic mass (atomic magnetisation).²⁷⁾

Magnetisation M is linked to the definition of volume magnetic susceptibility χ_vol, such that:²⁸⁾²⁹⁾

The value of relative permeability μ_r can be used, such that μ_r = χ_vol + 1.

Illustration of magnetic field strength H, flux density B, magnetisation M, and polarisation J in vacuum

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In the assumed ideal vacuum there is no matter of any kind (the illustration shows an “empty box” with volume V).

Therefore, there are no particles, atoms or molecules which could contribute some magnetic moments (intrinsic or induced), so their sum is zero over any volume, regardless the level of magnetic excitation,³⁰⁾ hence it is always M_vacuum = 0 A/m.

Magnetic polarisation J is M scaled by permeability of vacuum μ₀, therefore also always J_vacuum = 0 T.

Magnetic susceptibility of vacuum is precisely zero, and relative permeability is precisely unity.

In vacuum, magnetic field strength H and magnetic flux density B are equivalent, proportional to each other by the magnetic constant μ₀.

Thus, B and H vectors are parallel, and in case of periodic excitation (e.g. such as in alternating or rotating process of magnetisation) there is no lead/lag between the vectors, owing to the fact that vacuum is a lossless medium.

Illustration of magnetic field strength H, flux density B, magnetisation M, and polarisation J in a diamagnet

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In diamagnetic materials, the intrinsic magnetic moments of electrons in each atom cancel out, for example because of the even number of electrons which are paired such that on each orbital there are two electrons, with opposing spins.³¹⁾

Therefore, without any magnetic field applied from the outside, the material remains non-magnetised, because of sum of magnetic moments over some unit volume is still zero.

However, from a classical physics viewpoint, electrons orbiting in atoms can be treated as loops of superconducting current.³²⁾. Application of any external magnetic field changes trajectory of each orbit, equivalent to inducing a current in it. The direction is such that the magnetic field associated with it opposes the external field (due to Lenz's law).

The effect per atom equates to a small magnetic moment, and their sum adds up vectorially to a non-zero magnetisation M over the given volume, such that M always opposes the direction of of the applied magnetic field B. Therefore, susceptibility of diamagnets is negative and relative permeability is less than unity.³³⁾³⁴⁾ The effects are linear and do not depend on temperature in a significant way.

The effect is quite weak, but it happens for all atoms, also in other types of materials such as paramagnetic or even ferromagnetic, it is simply masked by the stronger effects of para- and ferromagnets (described below).

Any change of the applied magnetic field changes the induced magnetic moment. And therefore removal of the field un-induces precisely the amount of current which was initially induced, so that in zero applied field the value of M, J and therefore also B returns to zero as well (M = 0).

For most engineering applications diamagnets are treated simply as “non-magnetic” (similar to vacuum). For example, copper has the relative permeability of 0.999994 (compared to vacuum's 1.000000).

Illustration of magnetic field strength H, flux density B, magnetisation M, and polarisation J in a paramagnet

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In paramagnetic materials, there are atoms with at least one unpaired electron with the contribution of its intrinsic magnetic moment (spin and orbital).

Because of thermal agitation, the individual magnetic moments point in random directions, so that without any applied external field there is no net magnetisation (M = 0).³⁵⁾

When some magnetic field is applied, the individual magnetic moments begin to align to it, producing some net vector sum of M, in the same direction as the applied field.

For paramagnets susceptibility is positive, but also weak, yet stronger than the co-existing diamagnetic effect, so that the paramagnetic contribution dominates.

At a constant temperature, the effects are linear with respect to the applied magnetic excitation. However, at lower temperature it is easier for the moments to align with the applied field and the susceptibility increases, and can even diverge for the material to become ferromagnetic or antiferromagnetic.³⁶⁾

After removal of magnetic excitation, all the individual moments return to the normal thermally-agitated state and there is no retained magnetisation (M = 0).

For most engineering applications paramagnets are treated simply as “non-magnetic” (similar to vacuum). For example, aluminium has the relative permeability of 1.000022 (compared to vacuum's 1.000000).

Magnetisation process involves changes in the configuration of magnetic domains

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If the interactions between the individual moments become strong enough, they can align spontaneously so that very long-range ordering occurs between many individual atoms, forming macroscopic structures called magnetic domains, in some materials extending even over tens of millimetres. Such phenomenon is called ferromagnetism.³⁷⁾

Magnetic domains (with arrows showing their orientation)

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In ferromagnetic materials, all the moments in a given domain are fully aligned, so that they are locally magnetically saturated (therefore locally M = M_sat). The domains are separated by domain walls, and the neighbouring domains can point in other directions (such as 180° or 90°), so that the net magnetisation of the whole material can be zero, even though each domain is saturated.

The domain walls can move and if some external magnetic field is applied the domains with directions similar to the field will grow at the expense of other - this is the process of magnetisation (which involves all the fundamental quantities: M, H and B). Large movements of magnetic domains are lossy (e.g. due to domain wall pinning) and thus irreversible, making the process history-dependent. The effects are non-linear, and under alternating excitation result with a hysteresis loop, when plotting the instantaneous values of M versus H (also B or J vs. H). The area of such loop is proportional the the total magnetic losses occurring in the material.³⁸⁾

Illustration of magnetic field strength H, flux density B, magnetisation M, and polarisation J in a ferromagnet; all three vector quantities H, M and B can be at different angles due to energy of anisotropy

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The interactions between the individual magnetic moments are affected by the distance between the atoms, and these are dictated for example by the crystal structure of the material. Most crystals have some privileged directions, along which the spacing between the atoms is smaller and the alignment of moments is easier because of the minimisation of energy. This effect is called anisotropy and can “anchor” the vector of magnetisation M at a direction different from the applied field. If this happens then all the vectors of H, M (or J) and B can point in different directions even under DC excitation, as illustrated schematically. Additional energy is required to overcome the anisotropy energy, and higher excitation needs to be applied to change the angular position of the M vector.

The spontaneous alignment of moments acts against the thermal agitation, and increased temperature weakens the magnetic ordering. At sufficiently high temperature (Curie temperature) the magnetic alignment can no longer be sustained, and the material becomes paramagnetic. This is the case for all ferromagnetic and other ordered structures (described below).

Depending on the definition, the value of susceptibility can diverge to infinity (because of the way it is defined mathematically), and therefore especially in engineering a more useful figure of merit is the permeability.³⁹⁾ Materials which are easy to magnetise (and demagnetise) are called soft magnetic materials and their relative permeability can reach very high values, even exceeding 1 000 000, as compared to non-magnetic materials which is 1.

However, the internal structure of permanent magnets is designed to have the energy storage capability maximised, by increasing the value of coercivity. The magnets with the highest energy capability have the permeability also very close to unity, e.g. 1.05, even though they are ferromagnetic.⁴⁰⁾

Demagnetising field $H_d$ in a ferromagnetic body magnetised with uniform magnetisation $M$⁴¹⁾

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In a magnet, even if the actual magnetisation is uniform, in an open magnetic circuit the distribution of magnetic flux through the magnet and the surrounding medium leads still to non-uniform flux density B. Some demagnetising field H_d arises and points in the direction opposite to M. Therefore, the value of B is lower that if the demagnetising effect was not present.⁴²⁾

The value of the unitless demagnetising coefficient N_d (contributing to the demagnetising field) is linked to the shape and dimensions of the magnetised body. For closed magnetic circuit N_d = 0 (no demagnetising field).

But for infinitely large thin sheet magnetised perpendicularly to its surface N_d = 1, and the resulting B is the same as if the material had permeability equal to vacuum (regardless of the actual value). Such shape anisotropy will favour for M to be aligned within the plane of such thin sheet.⁴³⁾

In ferrimagnetic materials, there are two interleaved crystal lattices, each aligned in magnetically opposing directions. The resultant magnetisation is therefore lower, but still the material behaves macroscopically as a ferromagnet, including having magnetic domains, exhibiting hysteresis loop and high permeability.

At sufficiently high temperature (Néel temperature) ferrimagnets also become paramagnetic.

There are several types of magnetic ordering, which also result with strong interactions and ferromagnetic-like macroscopic behaviour. They differ mostly by the way the alignment of the individual moments follows some rules slightly different from classical ferromagnetism. They are known as types of magnetisms, and for example in the helical ferromagnetism the alignment of spins follows a helical order.⁴⁴⁾ Practical applications of such materials is limited.⁴⁵⁾

Under sufficiently high temperature all these materials transition to paramagnets.

Antiferromagnetic materials have the internal structure similar to ferrimagnets, but the two interleaved lattices have almost the same contributions so they almost perfectly cancel one another, such that the material is macroscopically indistinguishable from a paramagnet (“non-magnetic”).⁴⁶⁾

However, antiferromagnets are magnetically highly ordered structures, linked to the phenomenon of ferromagnetism. This is evident for example from Bethe-Slater curve, in which the type of interaction is plotted versus interatomic spacing. Closely packed structure can be ferromagnetic, with increasing distances could become antiferromagnetic, only for even larger distances to default to paramagnetic.

Some antiferromagnets can even form magnetic domains.⁴⁷⁾ Above Néel temperature antiferromagnets become paramagnetic.

Antiferromagnets are used in special types of magnetic sensors such as spin valves.⁴⁸⁾

Illustration of magnetic field strength H, flux density B, magnetisation M, and polarisation J in a superconductor

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Superconductors do not exhibit magnetic ordering, and are therefore “non-magnetic”. Above the critical temperature, in a normal resistive state, superconductors can be paramagnetic or diamagnetic, with susceptibility close to zero and permeability close to unity.

However, when the temperature is lowered and the material enters the superconducting state, the magnetic field is expelled from the inside of the material (Meissner effect).

Additionally, any change of external magnetic field induces macroscopic superconducting currents $I_{sup}$ flowing on the surface. These currents are equivalent to magnetic moment and hence to magnetisation associated with it. However, it is the macroscopic superconducting currents which sustain the response to the magnetic field, rather then the individual magnetic moments associated with each atom (see also the description of “bound” and “free” currents in the next section).

The magnetisation in superconductors always opposes the applied external magnetic field, such that it cancels it, with the effect of zero magnetic field penetrating to the inside of the superconductor. This is equivalent to the susceptibility being χ = -1 and relative permeability μ_r = 0, so that a superconductor can be referred to as an “ideal diamagnet”.⁴⁹⁾⁵⁰⁾

See also the main article: Magnetic saturation.

The magnetic response of all magnetic materials (ferromagnetic and similar) increases with the applied magnetic field, but eventually reaches magnetic saturation at sufficiently high amplitude.

Beyond a certain level of excitation the value of magnetisation saturation M_sat is reached when all the internal magnetic moments are aligned (all domain walls disappear and there remains a single domain). Further increase of excitation does not cause for the M to increase any more, but the flux density B keeps increasing with the excitation H, as dictated by the equation:

Therefore, there is no limiting value of “saturation flux density” B_sat or “saturation induction” even though some professional literature refers to such concepts.⁵¹⁾⁵²⁾ Nevertheless, in some cases it useful to use the concept of technical saturation defining some other point such as state of magnetic domains or value of permeability.⁵³⁾⁵⁴⁾

(The term “saturation flux density” in the CGS nomenclature is only correct with the implicit assumption that the name refers to the “intrinsic flux density”, which is equivalent to the magnetic polarisation J and hence also to the magnetisation M.)

Magnetic saturation of single-crystal iron-silicon alloy is the same regardless of direction of excitation (the image shows the equivalent graph, expressed with magnetic polarisation J_sat)⁵⁵⁾

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Magnetic saturation means that the material reaches saturation magnetisation M_sat (which can be expressed equivalently by saturation polarisation J_sat)⁵⁶⁾

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Saturation magnetisation is mostly dictated by the chemical composition and physical state of a given material.⁵⁷⁾⁵⁸⁾ For anisotropic materials it does not depend on the direction of the applied field, but it might require larger amplitude to reach saturation.⁵⁹⁾

Theoretically, it should be possible to saturate non-magnetic materials (such as paramagnetic and antiferromagnetic) by aligning all their magnetic moments. However, in practice this is extremely difficult to achieve, as it would require fields at the order of 80 MA/m or 1000 T.⁶⁰⁾ Such fields can be generated only by explosive means.

Normalised spontaneous magnetisation M/M_sat as a function of temperature for Fe, Co, Ni and Gd;⁶¹⁾⁶²⁾ dashed lines show room temperature relative to the Curie temperature T/T_C, J_e denotes the exchange integral coefficient

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The value of saturation magnetisation reduces with increasing temperature, and above the critical point of Curie temperature T_C the ferromagnets (and other magnetic materials) become paramagnetic. The value of T_C depends on the chemical composition of a given material.

At room temperature only three chemical elements are ferromagnetic: iron, cobalt and nickel, whereas for gadolinium T_C = 20 °C, so it is paramagnetic.⁶³⁾ There are many alloys of Fe, Co and Ni comprising additional “non-magnetic” additions which remain ferromagnetic above room temperature. However, Heusler alloys can be ferromagnetic at room temperature even though they are made from non-magnetic elements, due to the complex interaction of the electron spins.⁶⁴⁾

The value of spontaneous magnetisation can be calculated from the Curie-Weiss law, by using the Brillouin equation which is a function of the exchange integral J_e. From a practical viewpoint, the value of spontaneous magnetisation at a given temperature represents the magnetisation saturation limit at that temperature.

The classical physics approach ($J_e = ∞$) fails to accurately reflect the shape of the function. Ferromagnetism is linked to the quantum effects acting between the spin and orbital moments of the electrons. For Fe, Co and Ni the curves are close to 1/2 or 1 and $J_e = 1/2$ means that the exchange would be caused completely just by the spin moments.⁶⁵⁾

Rare earth metals have unpaired electrons on different shells and therefore the exchange integral has a significantly different value.⁶⁶⁾

Iron is inexpensive and has large saturation magnetisation, which is exploited in electrical steels used for large power transformers and generators in power plants. Although Co has lower saturation than Fe, the alloy 50Co-50Fe (known as Permendur) has the highest known saturation of 1.96 MA/m or 2.46 T.⁶⁸⁾

Magnetisation of a body can be reduced to an analysis of microscopic bound currents, which cancel internally

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The analysis of magnetisation M in matter can be carried out from the viewpoint of bound and free currents.⁷¹⁾⁷²⁾

For such analysis, a given body which is assumed to be magnetised uniformly with M, can be first divided into thin slices, and then each slice into many small volumes, as illustrated. Because the magnetisation is uniform everywhere, on the edge of each small fragment there is the same current flowing around each individual loop (there is no current on the top or bottom surface⁷³⁾).

The edges of the neighbouring elements have the same length, with the same current, but pointing in the opposite directions. Therefore, everywhere inside the whole body the effects of these currents cancel (blue), and only the outside edges (red) remain not cancelled. Therefore, the magnetisation is mathematically equivalent to a surface current flowing on the outside surface of the given body (magnetised uniformly).

Because M is magnetisation per unit volume⁷⁴⁾ and each infinitesimal fragment has an area of da and thickness dz, and observing that I = M·dz its magnetic dipole moment m is:

The surface current density for each element can be defined as some current per unit width $J_{bound} = I/dz$, which will have the same units as magnetisation. Hence, for amplitudes of values $J_{bound} = M$, and geometrically the vector of current density circulates around the vector of magnetisation hence:

Therefore, a uniformly magnetised body can be represented by an equivalent current flowing on its surface, even though in reality the electrons are bound to atoms and do not physically circulate on the surface of the whole body, but the magnetic moment arises due to electrons spins and orbital movements.⁷⁵⁾⁷⁶⁾ This is why the name bound current is used, to distinguish it from free current flowing due to macroscopic movement of electric charges.

For non-uniform magnetisation the same analysis applies, but because currents of each infinitesimal elements would not cancel completely, then the situation is equivalent to some volume current (rather than just an equivalent current flowing on the surface of the whole body). The same equation $\vec{J_{bound}} = curl ~ \vec{M}$ still holds, but it has to be evaluated locally.⁷⁷⁾

Magnetic flux density B encompasses the effects of all currents “bound” and “free” (e.g. macroscopic eddy currents), whereas magnetisation M can be expressed in terms of only “bound” currents. Therefore, magnetic field strength H can be linked to the “free” currents. The following equations are used for the definition of H from a theoretical viewpoint:⁷⁸⁾

Uniformly magnetised body can be represented as an equivalent current sheet⁷⁹⁾

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The equivalency between J_surf and M can be used for certain calculations. For example, a cylindrical permanent magnet can be theoretically replaced by a cylindrical solenoid of the same dimensions and appropriate current sheet. This approach is sometimes taken when including magnets in finite-element simulations.⁸⁰⁾

Magnetisation can be measured in several ways, of which just a few examples are briefly described below.

Paramagnets are attracted into a magnetic field and diamagnets are expelled from it. Therefore, by using a very sensitive mechanical balance it is possible to measure the amount of force and deduce the susceptibility and thus magnetisation of the sample, because only the M component contributes to the force under static conditions.

Two important methods are the Curie method (sometimes called the Faraday method) and the Gouy method, which differ in the way the electromagnet poles are designed and how the force is measured.⁸¹⁾

The distribution of magnetic field around a magnetic dipole is independent of the shape or size of the body with the magnetic moment, when measured at a large distance (as compared to the dimensions of the analysed magnetic dipole).⁸²⁾ Therefore, a sufficiently small sample can be treated as a “point source dipole”.⁸³⁾

The value of magnetisation M is an average of magnetic moment m per unit volume V:

Therefore, assuming uniform magnetisation within the whole sample, scaling M by the volume or mass gives direct relation to the magnetic moment:

Simplified drawing of a vibrating-sample magnetometer VSM

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This proportionality can be employed for a measurement of M in small samples, by means of a vibrating-sample magnetometer⁸⁴⁾ (VSM) or vibrating-coil magnetometer. Both methods allow measuring the same quantity, but VSM is more robust and hence it is more widely used.⁸⁵⁾.

For example, in the VSM, the sample is attached to a rod, which is made to vibrate in a sinusoidal manner. DC magnetic field can be applied from an electromagnet. Because of the Faraday's law of induction, the DC field does not induce any voltage in the sensing coils. Each data point is measured with the static field from the electromagnet, so the measurement is not made during a change of the electromagnet field.

The magnetisation M in the sample generates a stray field. When the sample is vibrating the variations in the stray field induce voltage in the sensing coils.⁸⁶⁾ This voltage is directly proportional to the magnetic moment of the sample:⁸⁷⁾

However, the “point source” approximation can be assumed only for samples which are small in comparison to the sensing coils. For larger samples the measurement error increases even though the induced voltage also increases due to larger volume. In any case, the VSM needs to be calibrated by a sample of a known magnetic moment, for example made as a palladium cylinder or a nickel sphere.⁸⁸⁾

Smaller samples produce less signal. The measurement is difficult because the signal is low even for larger samples, and therefore sophisticated signal processing is required to improve the signal-to-noise ratio of the apparatus. The sensing coils can be wound in a gradiometer configuration and the signals are filtered through lock-in amplifiers. The limit of resolution is dictated by the noise, and for high-precision scientific systems can be at the level of 10^-6 emu or 10^-9 A·m²,⁸⁹⁾ which for a sample 1 cm³ is equivalent to a resolution of 0.001 A/m. Smaller samples give lower resolution.

Commercial VSM devices can measure the magnetic moment in the SI units of A·m² per mass or volume of the sample, but also in the unit of emu/g, which is equivalent to A·m²/kg. Therefore, to calculate magnetisation the volume and mass or density of the sample is required.⁹⁰⁾

The method is very sensitive and therefore measurements on very small samples can be affected by contamination of the sample holder. For example, even colouring agents in plastics can contain small amount of magnetic particles, or ordinary dust can contribute to magnetic contamination.⁹¹⁾

Periodic excitation of a sample will induce a voltage in the coil wrapped around it. The voltage is proportional to the flux density B which comprises a vector sum of magnetisation M (equivalent to polarisation J) and magnetic field strength H.

Measurement of polarisation J: a) without compensation the induced voltage is proportional to the sum of J and H, b) with air flux compensation the induced voltage is proportional only to J (or M)

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However, an additional winding can be installed and exposed to the same excitation field, but without the sample. This can be achieved for example by using a mutual inductor whose parameters are adjusted to the main magnetising and sensing windings. With correct settings, in such additional coil the induced voltage will not be affected by any value of J, but it will comprise the component proportional to H.

Such auxiliary winding can be then connected in opposition to the winding on the sample, and therefore the H component will be compensated out, leaving only the voltage proportional to J (and hence to M).

This method is used in the Epstein frames and single sheet testers for characterisation of soft magnetic materials such as electrical steels.⁹²⁾

• Magnetic flux density B
• Magnetic field strength H
• Magnetic polarisation J