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In [[../Electromagnetism/|Electromagnetism]], the ''volume magnetic susceptibility'' , represented by the symbol <math> \chi_{v} </math> is defined by the following equation

<math> \vec{M} = \chi_{v} \vec{H},</math>
where in SI units <math>\vec{M}</math> is the ''magnetization''  of the material (defined as the magnetic dipole moment per unit [[../Volume/|volume]], measured in amperes per meter), and H is the ''strength of the [[../NeutrinoRestMass/|magnetic field]]'' <math>\vec{H}</math>, also measured in amperes per meter.

On the other hand, the magnetic induction <math>\vec{B}</math> is related to <math>\vec{H}</math> by the equation

<math>\vec{B} \ = \ \mu_0(\vec{H} + \vec{M}) \ = \ \mu_0(1+\chi_{v}) \vec{H} \ = \ \mu \vec{H},</math>
where <math>\mu_0</math> is the magnetic constant, and <math> \ (1+\chi_{v}) </math> is the relative permeability of the material.

Note that the magnetic susceptibility <math>\chi_v</math> and the magnetic permeability
<math>\mu</math> of a material are related as follows:

<math> \mu = \mu_0(1+\chi_v) \, .</math>

There are two other measures of susceptibility, the ''mass magnetic susceptibility'' , <math>\chi_g</math> or <math>\chi_m</math>, and the
''molar magnetic susceptibility'' , <math>\chi_{mol}:</math>

<math> \chi_{=mass= }= \chi_v/\rho ,</math>
<math> \chi_{mol} \, = \, M\chi_m = M \chi_v / \rho, </math>

where <math>\rho</math> is the density and M is the molar [[../CosmologicalConstant/|mass]].

====Susceptibility Sign convention====
If <math>\chi</math> is positive, then <math>(1+\chi_v)> 1</math> (or, in cgs units,
<math>(1+4 \pi \chi_v) > 1)</math> and the material can be paramagnetic, ferromagnetic, ferrimagnetic, or anti-ferromagnetic; then, the magnetic field inside the material is strengthened by the presence of the material, that is, the magnetization value is greater than the external H-value.

On the other hand there are certain materials--called ''diamagnetic'' -- for which <math>\chi</math> negative, and thus <math>(1+Ï‡v) < 1</math> (in SI units).

===Magnetic Susceptibility Tensor, ''χ''===

The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a [[../Vectors/|scalar]], but it is instead representable by a [[../Tensor/|tensor]] '''<math>\chi</math>''' . Then, the crystal magnetization <math>\vec{M}</math> is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field <math>\vec{H}</math>. Note that even non-crystalline materials may have a residual anisotropy, and thus require a similar treatment.

In all such magnetically anisotropic materials, the volume magnetic susceptibility tensor is then defined as follows:

<math> M_i=\chi_{ij}H_j , </math>

where <math>i</math> and <math>j</math> refer to the directions (such as, for example, x, y, z in Cartesian coordinates) of, respectively, the applied magnetic field and the magnetization of the material. This rank 2 tensor (of dimension (3,3)) relates the component of the magnetization in the <math>i</math>-th direction, <math>M_i</math> to the component <math> H_j</math> of the external magnetic field applied along the <math>j</math>-th direction.

==All Sources==
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<ref name='OMM80'/>
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<ref name='EHB73'/>
<ref name='KCBHDH2k3'/>
<ref name='KB62'/>
<ref name='H2k3'/>

==References==
<references>
====Magnetic Properties of Materials====
</ref><ref name='AMM68'>
G. P. Arrighini, M. Maestro, and R. Moccia (1968). Magnetic Properties of Polyatomic Molecules: Magnetic Susceptibility of <math>H_2O, NH_3, CH_4, H_2O_2</math>.
''J. Chem. Phys.''  49: 882-889. doi:10.1063/1.1670155.

</ref><ref name='OMM80'>
S. Otake, M. Momiuchi and N. Matsuno (1980). Temperature Dependence of the Magnetic Susceptibility of Bismuth. J. Phys. Soc. Jap. 49 (5): 1824-1828. doi:10.1143/JPSJ.49.1824.

</ref><ref name='HOM94'>
J. Heremans, C. H. Olk and D. T. Morelli (1994). Magnetic Susceptibility of Carbon Structures. ''Phys. Rev. B''  49 (21): 15122-15125. doi:10.1103/PhysRevB.49.15122.

</ref><ref name='OMM80'>
R. E. Glick (1961). On the Diamagnetic Susceptibility of Gases.
''J. Phys. Chem.''  65 (9): 1552-1555. doi:10.1021/j100905a020.

</ref><ref name='DF73'>
R. Dupree and C. J. Ford (1973). Magnetic susceptibility of the noble metals around their melting points. ''Phys. Rev. B''  8 (4): 1780â€“1782. doi:10.1103/PhysRevB.8.1780.

====Magnetic Moments and Nuclear Magnetic Resonance Spectrometry====
</ref><ref name='ZF57'>
J. R. Zimmerman, and M. R. Foster (1957). Standardization of NMR high resolution spectra. ''J. Phys. Chem.''  61: 282-289. <math>doi:10.1021/j150549a006</math>.

</ref><ref name='EHB73'>
Robert Engel, Donald Halpern, and Susan Bienenfeld (1973). Determination of magnetic moments in solution by nuclear magnetic resonance spectrometry. Anal. Chem. 45: 367-369. doi:10.1021/ac60324a054.

</ref><ref name='KCBHDH2k3'>
P. W. Kuchel, B. E. Chapman, W. A. Bubb, P. E. Hansen, C. J. Durrant, and M. P. Hertzberg (2003). Magnetic susceptibility: solutions, emulsions, and cells. ''Concepts Magn. Reson.''  A 18: 56-71. <math>doi:10.1002/cmr.a.10066</math>.

</ref><ref name='KB62'>
K. Frei and H. J. Bernstein (1962). Method for determining magnetic susceptibilities by NMR. J. Chem. Phys. 37: 1891-1892. <math>doi:10.1063/1.1733393</math>.

</ref><ref name='H2k3'>
R. E. Hoffman (2003). Variations on the chemical shift of TMS.
''J. Magn. Reson.''  163: 325-331. <math>doi:10.1016/S1090-7807(03)00142-3</math>.

</ref></references>

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[[Category:Electromagnetism]]