Table of Contents
Molar magnetic susceptibility
| Stan Zurek, Molar magnetic susceptibility, Encyclopedia Magnetica, https://e-magnetica.pl/doku.php/molar_magnetic_susceptibility, {accessed: 2025-04-20} |
Molar magnetic susceptibility, denoted typically by χ (chi) or κ (kappa) - a type of magnetic susceptibility calculated on the basis of molar magnetisation M.1)
Susceptibility expresses the ratio of magnetisation M in a given material to the magnetic field strength H, but depending on the mathematical and physical definition there can be many types of magnetic susceptibility.2)
S. Zurek, E-Magnetica.pl, CC-BY-4.0
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Volume susceptibility
 See also the main article: Magnetic susceptibility. |
Before defining molar susceptibility, it is beneficial to first define the volume magnetisation M and the volume susceptibility χ. If M is defined as the vector sum of magnetic dipole moments over unit volume4) then it can be referred to as “volume magnetisation”:
| (volume) Magnetisation M | |
|---|---|
| $$M = \frac{\sum m_i}{V}$$ | (A·m2)/(m3) ≡ (A/m) |
| where: $m_i$ - individual magnetic moments (A·m2), $V$ - unit volume (m3) | |
Therefore, if such volume magnetisation is taken for calculation of magnetic susceptibility then the volume magnetic susceptibility is obtained:5)
| Volume magnetic susceptibility χ | |
|---|---|
| $$χ_\text{vol} = \frac{M}{H}$$ | (A/m)/(A/m) ≡ (unitless) |
| where: $M$ - magnetisation (A/m), $H$ - magnetic field strength (A/m) | |
Mass susceptibility
If the volume susceptibility described above is divided by density of material, then the values are expressed per unit mass, thus obtaining mass magnetic susceptibility: 6)
| Mass magnetic susceptibility χ | |
|---|---|
| $$ χ_\text{mass} = \frac{M}{H·ρ} = \frac{M_ρ}{H} = \frac{χ_\text{vol}}{ρ} $$ | (m3/kg) |
| where: $M$ - magnetisation (A/m), $H$ - magnetic field strength (A/m), $χ_\text{vol}$ - volume magnetic susceptibility (unitless), $ρ$ - mass density of the material (kg/m3), $M_ρ$ - mass magnetisation (A·m2/kg) | |
Alternatively, the mass magnetisation $M_ρ = M / ρ$ (which is the volume magnetisation divided by the density of the material) value can be used to achieve the same calculation.
Molar susceptibility
If the mass susceptibility described above is multiplied by the molar mass mmol, then the values are expressed as volume per mol, thus obtaining molar magnetic susceptibility: 7)
| Mass magnetic susceptibility χ | |
|---|---|
| $$ χ_\text{mol} = \frac{M · m_\text{mol}}{H·ρ} = \frac{χ_\text{vol} ·m_\text{mol}}{ρ} $$ | (m3/mol) |
| where: $M$ - magnetisation (A/m), $H$ - magnetic field strength (A/m), $χ_\text{vol}$ - volume magnetic susceptibility (unitless), $ρ$ - mass density of the material (kg/m3), $m_\text{mol}$ - molar mass (kg/mol) | |
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Confusion of units
This article focuses on SI units. However, In the CGS system of units the dimension for volume susceptibility is the same as in SI (also unitless), but the numerical value differs by a factor $4π$, so care must be taken when comparing and analysing the values.
This difference in the numerical factor is important, because the same physical units can be valid in both systems (e.g. unitless for volume susceptibility, or cm3/mole for molar susceptibility). The multiplicity of units and approaches causes errors even in reputable data sources.8)
A converter for the values between some SI and CGS units is provided below.
Since the question of the units of susceptibilities is often confusing, let us emphasize the point here. For the susceptibility χ, the definition is M=χH, where M is the magnetization (magnetic moment per unit of volume) and H is the magnetic field strength. This χ is dimensionless, but is expressed as emu/cm3. The dimension of emu is therefore cm3. The molar susceptibility χ, is obtained by multiplying with the molar volume, υ (in cm3/mol). So, the molar susceptibility leads to M = HχN/υ, or Mυ= χNH, where Mυ is now the magnetic moment per mol. The dimension of molar susceptibility is thus emu/mol or cm3/mol.
Further confusion is introduced by the fact that the symbol “$emu$” (from: electromagnetic unit)) widely used in some publications does not denote a proper physical unit, but rather it is an indication that the CGS units are used.10)11) The “emu” symbol could be even used instead of an actual unit.12)
This type of confusing notation is used both for the magnetisation M and for susceptibility, as listed in the table below.13)14) This leads to the possibility that the two quantities, magnetisation and susceptibility, could be expressed with the same “unit”.
| Multiplicity of CGS units used in some publications15)16) | |
|---|---|
| Magnetisation | Susceptibility |
| G, Oe, emu/g, μB/atom, μB/impurity, G·cm3/g, emu/cm3, emu | emu/g, emu/cm3, emu/mole, emu/(g·kOe), emu·At/(g·V), emu/(Oe·mole) |
If the proper SI units are used no confusion can arise, especially when re-calculating or converting the numerical values. It is unfortunate that some physicists continue to use the confusing CGS notation, and even some modern professional measurement equipment reports the values in “emu units”.17)
| See also: Confusion between B and H. |
Calculator of magnetic susceptibility
| See also: Magnetic units. |
Magnetic susceptibility can be calculated from magnetisation M and magnetic field strength H, with the SI units:
| Quantity | Equation | SI units | CGS units | Conversion factor F such that SI = F·CGS |
|---|---|---|---|---|
| Volume magnetic susceptibility $M$ - magnetisation (A/m) $H$ - magnetic field strength (A/m) | $$ χ_\text{vol} = \frac{M}{H} $$ | (unitless) ≡ (A/m)/(A/m) | (unitless) | $4π$ |
| Mass magnetic susceptibility $ρ$ - mass density (kg/m3) | $$ χ_\text{mass} = \frac{χ_\text{vol}}{ρ} $$ | (m3/kg) | (cm3/g) | $4π×10^{-3}$ |
| Molar magnetic susceptibility $ρ$ - mass density (kg/m3) $m_\text{mol}$ - molar mass (kg/mol) | $$ χ_\text{mol} = \frac{χ_\text{vol}·m_\text{mol}}{ρ} $$ | (m3/mol) | (cm3/mol) | $4π×10^{-6}$ |