Encyclopedia Magnetica™

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====== Volume magnetic susceptibility ======

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| //[[user/Stan Zurek]], Volume magnetic susceptibility, Encyclopedia Magnetica//, \\ @PAGEL@  |

**Volume magnetic susceptibility**, denoted typically by **χ** (//chi//) or **κ** (//kappa//) - a type of **[[magnetic susceptibility]]** calculated on the basis of [[volume magnetisation]] //M//.[(Cullity>[[https://books.google.co.uk/books?isbn=9780471477419|B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials, 2nd edition, Wiley, IEEE Press, 2009, ISBN 9780471477419]])] 

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]].[(Cullity)]


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Magnetic susceptibility χ expresses the proportionality between [[magnetisation]] //M// and [[magnetic field strength]] //H//: //ferro// - [[ferromagnetism|ferromagnets]] (and [[ferrimagnetism|ferrimagnets]]), //para// - [[paramagnetism|paramagnets]], //dia// - [[diamagnetism|diamagnets]], //vac// - [[vacuum]], //sup// - [[superconductivity|superconductors]][(Cullity)]
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[[Magnetisation]] //M// is the sum of [[magnetic dipole moment|magnetic dipole moments]] //m<sub>i</sub>// per unit of volume //V//
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If //M// is defined as the [[vector]] sum of [[magnetic dipole moment|magnetic dipole moments]] over unit [[volume]][(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])] then it can be referred to as "volume magnetisation":

^  [[Volume magnetisation|(volume) Magnetisation]] //M//  ^^
|  $$M = \frac{\sum m_i}{V}$$ |  (A·m<sup>2</sup>)/(m<sup>3</sup>) ≡ (A/m)  |
| where: $m_i$ - individual [[magnetic dipole moment|magnetic moments]] (A·m<sup>2</sup>), $V$ - unit [[volume]] (m<sup>3</sup>)   ||

Therefore, if such volume magnetisation is taken for calculation of magnetic susceptibility then the **volume magnetic susceptibility** is obtained:[(Purcell)]

^  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)   ||


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| {{/wiki/logo.png?20&nolink}} //See also the main article: **[[Magnetic susceptibility]]**.// |


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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 cm<sup>3</sup>/mole for molar susceptibility). The multiplicity of units and approaches causes errors even in reputable data sources.[(ChemEurope>[[https://www.chemeurope.com/en/encyclopedia/Magnetic_susceptibility.html|Magnetic susceptibility, Encyclopedia, ChemEurope.com]], {accessed 2021-09-05})]

A converter for the values between some SI and CGS units is provided below.

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Richard L. Carlin, //Magnetochemistry//:[(Carlin>[[https://onlinelibrary.wiley.com/doi/10.1002/bbpc.198800101|R.L. Carlin: Magnetochemistry, Springer-Verlag, Berlin, 1986, ISBN ‎ 9783642707353]])]
> //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/cm<sup>3</sup>. The dimension of emu is
therefore cm<sup>3</sup>. The molar susceptibility χ, is obtained by multiplying with the molar volume, υ (in cm<sup>3</sup>/mol). So, the molar susceptibility leads to M = Hχ<sub>N</sub>/υ, or Mυ= χ<sub>N</sub>H, where Mυ is now the magnetic moment per mol. The dimension of molar susceptibility is thus emu/mol or cm<sup>3</sup>/mol.//
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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.[(Bennett>[[https://nvlpubs.nist.gov/nistpubs/jres/83/jresv83n1p9_A1b.pdf|L.H. Bennett, C.H. Page, L.J. Swartzendruber, Comments on Units in Magnetism, Journal of Research of the National Bureau of Standards Vol. 83, No.1 , Jan-Feb 1978]], {accessed 2021-08-23})][(IEEE_units>[[https://www.ieeemagnetics.org/index.php?option=com_content&view=article&id=118&Itemid=107|Magnetic Units, IEEE Magnetics, www.ieeemagnetics.org]], {accessed 2021-08-25})] The "emu" symbol could be even used //instead// of an actual unit.[(Reed>[[https://books.google.co.uk/books?id=4t9RAAAAMAAJ|R.P. Reed, A. F. Clark (ed.), Materials at Low Temperatures, American Society for Metals, 1983, ISBN 9780871701466]])]

This type of confusing notation is used both for the magnetisation //M// and for susceptibility, as listed in the table below.[(Bennett)][(Reed>)] This leads to the possibility that the two quantities, magnetisation and susceptibility, could be expressed with the same "unit".

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^   Multiplicity of CGS units used in some publications[(Bennett)][(Reed)]  ^^
|  Magnetisation  |  Susceptibility  |
|  G, Oe, emu/g, μ<sub>B</sub>/atom, μ<sub>B</sub>/impurity, G·cm<sup>3</sup>/g, emu/cm<sup>3</sup>, emu  |  emu/g, emu/cm<sup>3</sup>, 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//".[(Quantum>[[http://www.mrl.ucsb.edu/sites/default/files/mrl_docs/instruments/VSM%20manual%20B0.pdf|Vibrating Sample Magnetometer (VSM) Option User’s Manual, Part Number 1096-100, B0, Quantum Design, February 2011]], {accessed 2021-08-11})]

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| {{/wiki/logo.png?20&nolink}} //See also: [[Confusion between B and H]].// |

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===== See also =====
  * [[Types of magnetic susceptibility]]
  * [[Magnetic susceptibility]] //μ<sub>r</sub>//
  * [[Magnetic permeability]] //μ<sub>r</sub>//
  * [[Magnetisation]] //M//
  * [[Magnetic field strength]] //H//

===== References =====
~~REFNOTES~~

{{tag> Magnetic_susceptibility Magnetic_permeability Volume_susceptibility Mass_susceptibility Molar_susceptibility Magnetisation Magnetic_moment Counter}}