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magnetic susceptibility (Topic)
Definition 0.1   In Electromagnetism, the volume magnetic susceptibility, represented by the symbol $\chi_{v} $ is defined by the following equation

$\displaystyle \vec{M} = \chi_{v} \vec{H},$
where in SI units $\vec{M}$ is the magnetization of the material (defined as the magnetic dipole moment per unit volume, measured in amperes per meter), and H is the strength of the magnetic field $\vec{H}$, also measured in amperes per meter.

On the other hand, the magnetic induction $\vec{B}$ is related to $\vec{H}$ by the equation

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

Note that the magnetic susceptibility $\chi_v$ and the magnetic permeability $\mu$ of a material are related as follows:

$\displaystyle \mu = \mu_0(1+\chi_v) \, .$
Remark 0.1   There are two other measures of susceptibility, the mass magnetic susceptibility, $\chi_g$ or $\chi_m$, and the molar magnetic susceptibility, $\chi_{mol}:$

$\displaystyle \chi_{\text{mass}}= \chi_v/\rho ,$

$\displaystyle \chi_{mol} \, = \, M\chi_m = M \chi_v / \rho, $

where $\rho$ is the density and M is the molar mass.

Susceptibility Sign convention

If $\chi$ is positive, then $(1+\chi_v)> 1$ (or, in cgs units, $(1+4 \pi \chi_v) > 1)$ 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 $\chi$ negative, and thus $(1+\uml I‡v) < 1$ (in SI units).

Magnetic Susceptibility Tensor, $\chi$

The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a scalar, but it is instead representable by a tensor $\chi$. Then, the crystal magnetization $\vec{M}$ is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field $\vec{H}$. 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:

$\displaystyle M_i=\chi_{ij}H_j , $

where $i$ and $j$ 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 $i$-th direction, $M_i$ to the component $H_j$ of the external magnetic field applied along the $j$-th direction.

Bibliography

Magnetic Properties of Materials

1
G. P. Arrighini, M. Maestro, and R. Moccia (1968). Magnetic Properties of Polyatomic Molecules: Magnetic Susceptibility of $H_2O, NH_3, CH_4, H_2O_2$. J. Chem. Phys. 49: 882-889. doi:10.1063/1.1670155.
2
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.
3
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.
2
R. E. Glick (1961). On the Diamagnetic Susceptibility of Gases. J. Phys. Chem. 65 (9): 1552-1555. doi:10.1021/j100905a020.
4
R. Dupree and C. J. Ford (1973). Magnetic susceptibility of the noble metals around their melting points. Phys. Rev. B 8 (4): 1780–1782. doi:10.1103/PhysRevB.8.1780.

Magnetic Moments and Nuclear Magnetic Resonance Spectrometry

5
J. R. Zimmerman, and M. R. Foster (1957). Standardization of NMR high resolution spectra. J. Phys. Chem. 61: 282-289. $doi:10.1021/j150549a006$.
6
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.
7
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. $doi:10.1002/cmr.a.10066$.
8
K. Frei and H. J. Bernstein (1962). Method for determining magnetic susceptibilities by NMR. J. Chem. Phys. 37: 1891-1892. $doi:10.1063/1.1733393$.
9
R. E. Hoffman (2003). Variations on the chemical shift of TMS. J. Magn. Reson. 163: 325-331. $doi:10.1016/S1090-7807(03)00142-3$.



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Also defines:  diamagnetic susceptibility, magnetic field strength, volume magnetic susceptibility, mass magnetic susceptibility, \chi_g, \chi_m, molar magnetic susceptibility, $\chi_(mol), magnetic susceptibility tensor, \chi, magnetization, magnetic induction vector
Keywords:  magnetic moments in solution by nuclear magnetic resonance spectrometry, NMR

Cross-references: tensor, scalar, mass, magnetic field, volume, Electromagnetism
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This is version 11 of magnetic susceptibility, born on 2009-02-17, modified 2009-02-19.
Object id is 535, canonical name is MagneticSusceptibility.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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