# Encyclopedia Magnetica™

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volume_magnetic_susceptibility

# Volume magnetic susceptibility

 Stan Zurek, Volume magnetic susceptibility, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/volume_magnetic_susceptibility

Volume magnetic susceptibility, denoted typically by χ (chi) or κ (kappa) - a type of magnetic susceptibility calculated on the basis of volume 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)

Magnetic susceptibility χ expresses the proportionality between magnetisation M and magnetic field strength H: ferro - ferromagnets (and ferrimagnets), para - paramagnets, dia - diamagnets, vac - vacuum, sup - superconductors3)
Magnetisation M is the sum of magnetic dipole moments mi per unit of volume V

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)
 → → → Helpful page? Support us! → → → PayPal ← ← ← Help us with just $0.10 per month? Come on… ← ← ← ## 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.6) A converter for the values between some SI and CGS units is provided below. Richard L. Carlin, Magnetochemistry:7) 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.8)9) The “emu” symbol could be even used instead of an actual unit.10) This type of confusing notation is used both for the magnetisation M and for susceptibility, as listed in the table below.11)12) 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 publications13)14) 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”.15)  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}\$

Convert from:
(use 2.2e-5 for 2.2×10-5)

Additional data required for conversion of mass and molar values:
Mass density:
Molar mass:

Convert to: