In the study of magnetism there are two systems of units currently in use: the mks (metres-kilograms-seconds) system, which has been adopted as the S.I. units and the cgs (centimetres-grams-seconds) system, which is also known as the Gaussian system. The cgs system is used by many magnets experts due to the numerical equivalence of the magnetic induction (B) and the applied field (H).

When a field is applied to a material it responds by producing a magnetic field, the magnetisation (M). This magnetisation is a measure of the magnetic moment, m, per unit volume of material, but can also be expressed per unit mass, the specific magnetisation (s).

The Total Magnetic field (B) or Magnetic Induction, is the total flux of magnetic field lines through a unit cross sectional area of the material

Another way to deal with the magnetic fields which arise from magnetization of materials is to introduce a quantity called magnetic field strength, H . It can be defined by the relationship

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 relationship for B above can be written in the equivalent form

Considering both lines of force from the applied field and from the magnetisation of the material. B, H and M are related by equation 1a in S.I. units and by equation 1b in cgs units.

In equation 1a, the constant m_o is the permeability of free space (4π x 10^-7Hm^-1) (which is the ratio of B/H measured in a vacuum). In cgs units the permeability of free space is unity and so does not appear in equation 1b. The units of B, H and M for both S.I. and cgs systems are given in table 1. Note that in the cgs system 4π M is usually quoted as it has units of Gauss and is numerically equivalent to B and H.

Another equation to consider at this stage is that concerning the magnetic susceptibility (c),, equation 2, this is the same for S.I. and cgs units. The magnetic susceptibility is a parameter that demonstrates the type of magnetic material and the strength of that type of magnetic effect.

Sometimes the mass susceptibility (c_m), is quoted and this has the units of m³kg^-1 and can be calculated by dividing the susceptibility of the material by the density.

Another parameter that demonstrates the type of magnetic material and the strength of that type of magnetic effect is the permeability (m) of a material, this is defined in equation 3 (the same for S.I. and cgs units).

In the S.I. system of units, the permeability is related to the susceptibility, as shown in equation 4 and can also be broken down into ľ_o and the relative permeability (ľ_r ), as shown in equation 5.

If the material does not respond to the external magnetic field by producing any magnetization, then ľ_r = 1.

Finally, an important parameter (in S.I. units) to know is the magnetic polarisation (J), also referred to as the intensity of magnetisation (I). This value is effectively the magnetisation of a sample expressed in Tesla, and can be calculated as shown in equation 6.

Quantity	Gaussian (cgs units)	S.I. Units	Conversion factor (cgs to S.I.)
Magnetic Induction (B)	G	T	10^-4
Applied Field (H)	Oe	Am^-1	10³ / 4p
Magnetisation (M)	emu cm^-3	Am^-1	10³
Magnetisation (4π M)	G	-	-
Magnetic Polarisation (J)	-	T	-
Specific Magnetisation (s)	emu g^-1	JT^-1kg^-1	1
Permeability (ľ)	Dimensionless	H m^-1	4 p . 10^-7
Relative Permeability (ľ_r)	-	Dimensionless	-
Susceptibility (χ)	emu cm^-3 Oe^-1	Dimensionless	4 p
Maximum Energy Product (BH_max)	M G Oe	k J m^-3	10² / 4 p

Table 1: The relationship between some magnetic parameters in cgs and S.I. units.
(Where: G = Gauss, Oe = Oersted, T = Tesla)

The intrinsic properties of a magnetic material are those properties that are characteristic of the material and are unaffected by the microstructure (e.g. grain size, crystal orientation of grains). These properties include the Curie temperature, the saturation magnetisation and the magnetocrystalline anisotropy.

Saturation Magnetisation
The saturation magnetisation (M_S) is a measure of the maximum amount of field that can be generated by a material. It will depend on the strength of the dipole moments on the atoms that make up the material and how densely they are packed together. The atomic dipole moment will be affected by the nature of the atom and the overall electronic structure within the compound. The packing density of the atomic moments will be determined by the crystal structure (i.e. the spacing of the moments) and the presence of any non-magnetic elements within the structure.

For ferromagnetic materials, at finite temperatures, M_S will also depend on how well these moments are aligned, as thermal vibration of the atoms causes misalignment of the moments and a reduction in M_S. For ferrimagnetic materials not all of the moments align parallel, even at zero Kelvin and hence M_S will depend on the relative alignment of the moments as well as the temperature.

The saturation magnetisation is also referred to as the spontaneous magnetisation, although this term is usually used to describe the magnetisation within a single magnetic domain. Table 2 gives some examples of the saturation polarisation and Curie temperature of materials commonly used in magnetic applications.

Material	Magnetic Structure	J_S at 298K (T)	T_C (şC)
Fe	Ferro	2.15	770
Co	Ferro	1.76	1131
Ni	Ferro	0.60	358
Nd₂Fe₁₄B	Ferro	1.59	312
SmCo₅	Ferro	1.14	720
Sm₂Co₁₇	Ferro	1.25	820
BaO.6Fe₂O₃	Ferri	0.48	450
SrO.6Fe₂O₃	Ferri	0.48	450
Fe 3wt% Si	Ferro	2.00	740
Fe 4wt% Si	Ferro	1.97	690
Fe 35wt% Co	Ferro	2.45	970
Fe 78wt% Ni	Ferro	0.70	580
Fe 50wt% Ni	Ferro	1.55	500
MnO.Fe₂O₃	Ferri	0.51	300

Table 2: The saturation polarisation (J_S) and Curie temperature (T_C) of a range of magnetic materials.

Magnetic Anisotropy
In a crystalline magnetic material the magnetic properties will vary depending on the crystallographic direction in which the magnetic dipoles are aligned. Figure 4 demonstrates this effect for a single crystal of cobalt. The hexagonal crystal structure of Co can be magnetised easily in the [0001] direction (i.e. along the c-axis), but has hard directions of magnetisation in the <1010> type directions, which lie in the basal plane (90° from the easy direction).

A measure of the magnetocrystalline anisotropy in the easy direction of magnetisation is the anisotropy field, Ha (illustrated in figure 1), which is the field required to rotate all the moments by 90° as one unit in a saturated single crystal. The anisotropy is caused by a coupling of the electron orbitals to the lattice, and in the easy direction of magnetisation this coupling is such that these orbitals are in the lowest energy state.

The easy direction of magnetisation for a permanent magnet, based on ferrite or the rare earth alloys, must be uniaxial, however, it is also possible to have materials with multiple easy axes or where the easy direction can lie anywhere on a certain plane or on the surface of a cone. The fact that a permanent magnet has uniaxial anisotropy means that it is difficult to demagnetise as it is resistant to rotation of the direction of magnetisation.

ľ_r = χ + 1	Equ.4

ľ = ľ_o ľ_r	Equ.5


B = H + 4 π M	Equ.1b