- Magnetic susceptibility
In electromagnetism, the magnetic susceptibility χm (latin: susceptibilis “receptiveness”) is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field. A related term is magnetizability, the proportion between magnetic moment and magnetic flux density.
Definition of volume susceptibility
- See also Relative permeability.
The volume magnetic susceptibility, represented by the symbol χv (often simply χ, sometimes χm — magnetic, to distinguish from the electric susceptibility), is defined by the relationship
where, in SI units,
- M is the magnetization of the material (the magnetic dipole moment per unit volume), measured in amperes per meter, and
- H is the magnetic field strength, also measured in amperes per meter.
The magnetic induction B is related to H by the relationship
where μ0 is the magnetic constant (see table of physical constants), and (1 + χv) is the relative permeability of the material. Thus the volume magnetic susceptibility χv and the magnetic permeability μ are related by the following formula:
This allows an alternative description of all magnetization phenomena in terms of the quantities I and B, as opposed to the commonly used M and H.
Conversion between SI and CGS units
Note that these definitions are according to SI conventions. However, many tables of magnetic susceptibility give CGS values (more specifically emu-cgs, short for electromagnetic units, or Gaussian-cgs; both are the same in this context) that rely on a different definition of the permeability of free space:
Mass susceptibility and molar susceptibility
There are two other measures of susceptibility, the mass magnetic susceptibility (χmass or χg, sometimes χm), measured in m3·kg−1 in SI or in cm3·g−1 in CGS and the molar magnetic susceptibility (χmol) measured in m3·mol−1 (SI) or cm3·mol−1 (CGS) that are defined below, where ρ is the density in kg·m−3 (SI) or g·cm−3 (CGS) and M is molar mass in kg·mol−1 (SI) or g·mol−1 (CGS).
- χmass = χv / ρ
- χmol = Mχmass = Mχv / ρ
Sign of susceptibility: diamagnetics and other types of magnetism
If χ is positive, the material can be paramagnetic . In this case, the magnetic field in the material is strengthened by the induced magnetization. Alternatively, if χ is negative, the material is diamagnetic. As a result, the magnetic field in the material is weakened by the induced magnetization. Generally, non-magnetic materials are said para- or diamagnetic because they do not possess permanent magnetization without external magnetic field. Ferromagnetic, ferrimagnetic, or antiferromagnetic materials, which have positive susceptibility, possess permanent magnetization even without external magnetic field.
Experimental methods to determine susceptibility
Volume magnetic susceptibility is measured by the force change felt upon the application of a magnetic field gradient. Early measurements were made using the Gouy balance 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 superconductive 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 Evans balance. For liquid samples, the susceptibility can be measured from the dependence of the NMR frequency of the sample on its shape or orientation.
The magnetic susceptibility of most crystals is not a scalar. Magnetic response M is dependent upon the orientation of the sample and can occur in directions other than that of the applied field H. In these cases, volume susceptibility is defined as a tensor
- Mi = χijHj
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 component of magnetization in the i-th direction from the external field applied in the j-th direction.
In ferromagnetic crystals, the relationship between M and H is not linear. To accommodate this, a more general definition of differential susceptibility is used
where is a tensor derived from partial derivatives of components of M with respect to components of H. 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.
Susceptibility in the frequency domain
When the magnetic susceptibility is measured in response to an AC magnetic field (i.e. a magnetic field that varies sinusoidally), this is called AC susceptibility. AC susceptibility (and the closely related "AC permeability") are complex quantities, and various phenomena (such as resonances) can be seen in AC susceptibility that cannot in constant-field (DC) susceptibility. 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.
For a tutorial with more information on AC susceptibility measurements, see here (external link).
Magnetic susceptibility of some materials Material Temperature Pressure χmol (molar susc.) χmass (mass susc.) χv (volume susc.) M (molar mass) ρ (density) Units (°C) (atm) SI
vacuum Any 0 0 0 0 0 0 0 – 0 water  20 1 −1.631×10−10 −1.298×10−5 −9.051×10−9 −7.203×10−7 −9.035×10−6 −7.190×10−7 18.015 0.9982 bismuth  20 1 −3.55×10−9 −2.82×10−4 −1.70×10−8 −1.35×10−6 −1.66×10−4 −1.32×10−5 208.98 9.78 Diamond  R.T. 1 −7.4×10−11 −5.9×10−6 −6.2×10−9 −4.9×10−7 −2.2×10−5 −1.7×10−6 12.01 3.513 Graphite  (to c-axis) R.T. 1 −7.5×10−11 −6.0×10−6 −6.3×10−9 −5.0×10−7 −1.4×10−5 −1.1×10−6 12.01 2.267 Graphite  χ | | R.T. 1 −3.2×10−9 −2.6×10−4 −2.7×10−7 −2.2×10−5 −6.1×10−4 −4.9×10−5 12.01 2.267 Graphite  χ | | -173 1 −4.4×10−9 −3.5×10−4 −3.6×10−7 −2.9×10−5 −8.3×10−4 −6.6×10−5 12.01 2.267 He  20 1 −2.38×10−11 −1.89×10−6 −5.93×10−9 −4.72×10−7 −9.85×10−10 −7.84×10−11 4.0026 0.000166 Xe  20 1 −5.71×10−10 −4.54×10−5 −4.35×10−9 −3.46×10−7 −2.37×10−8 −1.89×10−9 131.29 0.00546 O2  20 0.209 4.3×10−8 3.42×10−3 1.34×10−6 1.07×10−4 3.73×10−7 2.97×10−8 31.99 0.000278 N2  20 0.781 −1.56×10−10 −1.24×10−5 −5.56×10−9 −4.43×10−7 −5.06×10−9 −4.03×10−10 28.01 0.000910 Al 1 2.2×10−10 1.7×10−5 7.9×10−9 6.3×10−7 2.2×10−5 1.75×10−6 26.98 2.70 Ag  961 1 −2.31×10−5 −1.84×10−6 107.87
Sources of confusion in published data
There are tables of magnetic susceptibility values published on-line that seem to have been uploaded from a substandard source, which itself has probably borrowed heavily from the CRC Handbook of Chemistry and Physics. Some of the data (e.g. for Al, Bi, and diamond) are apparently in cgs Molar Susceptibility units, whereas that for water is in Mass Susceptibility units (see discussion above). The susceptibility table in the CRC Handbook is known to suffer from similar errors, and even to contain sign errors. Effort should be made to trace the data in such tables to the original sources, and to double-check the proper usage of units.
- Curie constant
- Electric susceptibility
- Magnetic constant
- Magnetic flux density
- Maxwell's equations
- Permeability (electromagnetism)
- Susceptibility weighted imaging
References and notes
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- ^ a b c d R. E. Glick (1961). "On the Diamagnetic Susceptibility of Gases". J. Phys. Chem. 65 (9): 1552–1555. doi:10.1021/j100905a020.
- ^ R. Dupree and C. J. Ford (1973). "Magnetic susceptibility of the noble metals around their melting points". Phys. Rev. B 8 (4): 1780–1782. Bibcode 1973PhRvB...8.1780D. doi:10.1103/PhysRevB.8.1780.
- ^ "Magnetic Properties Susceptibilities Chart from". READE. 2006-01-11. http://www.reade.com/Particle_Briefings/magnetic_susceptibilities.html. Retrieved 2011-11-08.
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