Magnetic susceptibility

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In electrical engineering, the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. The dimensionless volume magnetic susceptibility, represented by the symbol [\ \chi_] (also represented in the literature by [\kappa] or [\Kappa]), is defined by the relationship

[\mathbf = \chi_ \mathbf]

where

M is the magnetization of the material (the magnetic dipole moment per unit volume), measured in amperes per meter, and

H is the applied field, also measured in amperes per meter.

The magnetic induction B is related to H by the relationship

[\mathbf \ = \ \mu_0(\mathbf + \mathbf) \ = \ \mu_0(1+\chi_) \mathbf \ = \ \mu \mathbf ]

where μ₀ is the permeability of free space (see table of physical constants), and [ \ (1+\chi) ] is the relative permeability of the material. Note that this definition is according to SI conventions. However, many tables of magnetic susceptibility give cgs values that rely on a different definition of the permeability of free space. The cgs value of susceptibility is multiplied by 4[\pi] to give the SI susceptibility value. For example, the cgs volume magnetic susceptibility of water at 20°C is [-7.19x10^] which is [-9.04x10^] using the SI convention.

There are two other measures of susceptibility, the mass magnetic susceptibility in [cm^3g^] ([\chi] or [\chi_g]) and the molar magnetic susceptibility ([\chi_m]) in [cm^3mol^] that are defined as follows where [\rho] is the density in [gcm^] and M is molar mass in [gmol^].

[\chi_g=\chi_v/\rho]

[\chi_m=M\chi_g=M\chi_v/\rho]

If χ is positive, then (1+χ) > 1 and the material is called paramagnetic. In this case, the magnetic field is strengthened by the presence of the material. Alternatively, if χ is negative, then (1+χ) < 1, and the material is diamagnetic. As a result, the magnetic field is weakened in the presence of the material.

Volume magnetic susceptibility is measured by the force change felt upon the application of a magnetic field (L. N. Mulay in Techniques of Chemistry, eds. A. Weissberger and B. W. Rossiter, Wiley-Interscience, New York, 4, 431 (1972)). Early measurements were made using the Gouy method where a sample is hung between the poles of an electromagnet. The change in weight when the electromagnet is turned on is proportional to the susceptibility. Today, high-end measurement systems use a supercondictive magnet. An alternative is to measure the force change on a strong compact magnet upon insertion of the sample. This system, widely used today, is called the [Evan's balance]. For liquid samples, the susceptibility can be measured from the dependence of the NMR frequency of the sample on its shape or orientation (J.R. Zimmerman, M.R. Foster, J. Phys. Chem. 61 (1957) 282-289; R. Engel, D. Halpern, S. Bienenfeld, Anal. Chem., 45 (1973) 367-369; P.W. Kuchel, B.E. Chapman, W.A. Bubb, P.E. Hansen, C.J. Durrant, M.P. Hertzberg, Conc. Magn. Reson. A 18 (2003) 56-71;K. Frei, H.J. Bernstein, J. Chem. Phys. 37 (1962) 1891-1892; R.E. Hoffman, J. Magn. Reson. 163 (2003) 325-331).

The magnetic susceptibility of a ferromagnetic substance is not a scalar. Response is dependent upon the state of sample and can occur in directions other than that of the applied field. To accommodate this, a more general definition using a tensor derived from derivatives of components of M with respect to components of H

[\chi_ = \frac]

called the differential susceptibility describes ferromagnetic materials, where i and j refer to the directions (e.g., x, y and z in Cartesian coordinates) of the applied field and magnetization, respectively. The tensor is thus rank 2, dimension (3,3) describing the response of the magnetization in the j-th direction from an incremental change in the i-th direction of the applied field.

When the coercivity of the material parallel to an applied field is the smaller of the two, the differential susceptibility is a function of the applied field and self interactions, such as the magnetic anisotropy. When the material is not saturated, the effect will be nonlinear and dependent upon the domain wall configuration of the material.

When the magnetic susceptibility is studied as a function of frequency, the permeability is a complex quantity and resonances can be seen. In particular, when an ac-field is applied perpendicular to the detection direction (called the "transverse susceptibility" regardless of the frequency), the effect has a peak at the ferromagnetic resonance frequency of the material with a given static applied field. Currently, this effect is called the microwave permeability or network ferromagnetic resonance in the literature. These results are sensitive to the domain wall configuration of the material and eddy currents.

In terms of ferromagnetic resonance, the effect of an ac-field applied along the direction of the magnetization is called parallel pumping.

The magnetic susceptibility and the magnetic permeability (μ) are related by the following formula:

[\mu = \mu_0(1+\chi) \,]

where [ \ (1+\chi) ] is the relative permeability of the material.

Magnetic susceptibility

See also