In electromagnetism, the magnetic susceptibility (Latin: susceptibilis, "receptive"; denoted χ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization M (magnetic moment per unit volume) to the applied magnetizing field intensity H. This allows a simple classification, into two categories, of most materials' responses to an applied magnetic field: an alignment with the magnetic field, χ > 0, called paramagnetism, or an alignment against the field, χ < 0, called diamagnetism.
Magnetic susceptibility indicates whether a material is attracted into or repelled out of a magnetic field. Paramagnetic materials align with the applied field and are attracted to regions of greater magnetic field. Diamagnetic materials are anti-aligned and are pushed away, toward regions of lower magnetic fields. On top of the applied field, the magnetization of the material adds its own magnetic field, causing the field lines to concentrate in paramagnetism, or be excluded in diamagnetism. Quantitative measures of the magnetic susceptibility also provide insights into the structure of materials, providing insight into bonding and energy levels. Furthermore, it is widely used in geology for paleomagnetic studies and structural geology.
The magnetizability of materials comes from the atomic-level magnetic properties of the particles of which they are made. Usually, this is dominated by the magnetic moments of electrons. Electrons are present in all materials, but without any external magnetic field, the magnetic moments of the electrons are usually either paired up or random so that the overall magnetism is zero (the exception to this usual case is ferromagnetism). The fundamental reasons why the magnetic moments of the electrons line up or do not are very complex and cannot be explained by classical physics (see Bohr–van Leeuwen theorem). However, a useful simplification is to measure the magnetic susceptibility of a material and apply the macroscopic form of Maxwell's equations. This allows classical physics to make useful predictions while avoiding the underlying quantum mechanical details.
Magnetic susceptibility 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. A closely related parameter is the permeability, which expresses the total magnetization of material and volume.
The volume magnetic susceptibility, represented by the symbol χv (often simply χ, sometimes χm – magnetic, to distinguish from the electric susceptibility), is defined in the International System of Units — in other systems there may be additional constants — by the following relationship:
- 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.
χv is therefore a dimensionless quantity.
where μ0 is the vacuum permeability (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.
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 (SI) and the molar magnetic susceptibility (χmol) measured in m3/mol that are defined below, where ρ is the density in kg/m3 and M is molar mass in kg/mol:
In CGS units
Note that the definitions above 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). These units rely on a different definition of the permeability of free space:
For example, the cgs volume magnetic susceptibility of water at 20 °C is 7.19×10−7, which is 9.04×10−6 using the SI convention.
In physics it is common to see cgs mass susceptibility given in cm3/g or emu/g·Oe−1, so to convert to SI volume susceptibility we use the conversion 
where ρcgs is the density given in g/cm3, or
The molar susceptibility is measured cm3/mol or emu/mol·Oe−1 in cgs and is converted by considering the molar mass.
Paramagnetism and diamagnetism
If χ is positive, a 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. In this case, the magnetic field in the material is weakened by the induced magnetization. Generally, nonmagnetic materials are said to be para- or diamagnetic because they do not possess permanent magnetization without external magnetic field. Ferromagnetic, ferrimagnetic, or antiferromagnetic materials possess permanent magnetization even without external magnetic field and do not have a well defined zero-field susceptibility.
Volume magnetic susceptibility is measured by the force change felt upon a substance when a magnetic field gradient is applied. Early measurements are 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. Another method using NMR techniques measures the magnetic field distortion around a sample immersed in water inside an MR scanner. This method is highly accurate for diamagnetic materials with susceptibilities similar to water.
The magnetic susceptibility of most crystals is not a scalar quantity. 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
where i and j refer to the directions (e.g., x and y in Cartesian coordinates) of the applied field and magnetization, respectively. The tensor is thus rank 2 (second order), dimension (3,3) describing the component of magnetization in the ith direction from the external field applied in the jth 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
ij 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.
Several experimental techniques allow for the measurement of the electronic properties of a material. An important effect in metals under strong magnetic fields, is the oscillation of the differential susceptibility as function of 1/. This behaviour is known as the de Haas–van Alphen effect and relates the period of the susceptibility with the Fermi surface of the material.
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 number quantities, and various phenomena, such as resonance, 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.
|Material||Temp.||Pressure||Molar susc., χmol||Mass susc., χmass||Volume susc., χv||Molar mass, M||Density,|
|(103 kg/m3 |
|Paraffin oil, 220–260 cSt||22||1||−1.01×10−8||−8.0×10−7||−8.8×10−6||−7.0×10−7||0.878|
|Fused silica glass||22||1||−5.12×10−9||−4.07×10−7||−1.128×10−5||−8.98×10−7||2.20|
|Graphite χ∥ (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|
Sources of confusion in published data
The CRC Handbook of Chemistry and Physics has one of the few published magnetic susceptibility tables. Some of the data (e.g., for aluminium, bismuth, and diamond) is listed as cgs, which has caused confusion to some readers. "cgs" is an abbreviation of centimeters–grams–seconds; it represents the form of the units, but cgs does not specify units. Correct units of magnetic susceptibility in cgs is cm3/mol or cm3/g. Molar susceptibility and mass susceptibility are both listed in the CRC. Some table have listed magnetic susceptibility of diamagnets as positives. It is important to check the header of the table for the correct units and sign of magnetic susceptibility readings.
Application in the geosciences
Magnetism is a useful parameter to describe and analyze rocks. Additionally, the anisotropy of magnetic susceptibility (AMS) within a sample determines parameters as directions of paleocurrents, maturity of paleosol, flow direction of magma injection, tectonic strain, etc. It is a non-destructive tool, which quantifies the average alignment and orientation of magnetic particles within a sample.
- Roger Grinter, The Quantum in Chemistry: An Experimentalist's View, John Wiley & Sons, 2005, ISBN 0470017627 page 364
- Tauxe, Lisa (2019). Essentials of Paleomagnetism: Fifth Web Edition. UC Press.
- "magnetizability, ξ". IUPAC Compendium of Chemical Terminology—The Gold Book (2nd ed.). International Union of Pure and Applied Chemistry. 1997. Archived from the original on 2016-03-04. Retrieved 2011-10-13.
- O'Handley, Robert C. (2000). Modern Magnetic Materials. Hoboken, NJ: Wiley. ISBN 9780471155669.
- Richard A. Clarke. "Magnetic properties of materials". Info.ee.surrey.ac.uk. Retrieved 2011-11-08.
- Bennett, L. H.; Page, C. H. & Swartzendruber, L. J. (1978). "Comments on units in magnetism". Journal of Research of the National Bureau of Standards. NIST, USA. 83 (1): 9–12. doi:10.6028/jres.083.002.
- "IEEE Magnetic unit conversions".
- L. N. Mulay (1972). A. Weissberger; B. W. Rossiter (eds.). Techniques of Chemistry. 4. Wiley-Interscience: New York. p. 431.
- "Magnetic Susceptibility Balances". Sherwood-scientific.com. Retrieved 2011-11-08.
- J. R. Zimmerman, and M. R. Foster (1957). "Standardization of NMR high resolution spectra". J. Phys. Chem. 61 (3): 282–289. doi:10.1021/j150549a006.
- Robert Engel; Donald Halpern & Susan Bienenfeld (1973). "Determination of magnetic moments in solution by nuclear magnetic resonance spectrometry". Anal. Chem. 45 (2): 367–369. doi:10.1021/ac60324a054.
- P. W. Kuchel; B. E. Chapman; W. A. Bubb; P. E. Hansen; C. J. Durrant & M. P. Hertzberg (2003). "Magnetic susceptibility: solutions, emulsions, and cells". Concepts Magn. Reson. A 18: 56–71. arXiv:q-bio/0601030. doi:10.1002/cmr.a.10066.
- K. Frei & H. J. Bernstein (1962). "Method for determining magnetic susceptibilities by NMR". J. Chem. Phys. 37 (8): 1891–1892. Bibcode:1962JChPh..37.1891F. doi:10.1063/1.1733393.
- R. E. Hoffman (2003). "Variations on the chemical shift of TMS". J. Magn. Reson. 163 (2): 325–331. Bibcode:2003JMagR.163..325H. doi:10.1016/S1090-7807(03)00142-3. PMID 12914848.
- Wapler, M. C.; Leupold, J.; Dragonu, I.; von Elverfeldt, D.; Zaitsev, M.; Wallrabe, U. (2014). "Magnetic properties of materials for MR engineering, micro-MR and beyond". JMR. 242: 233–242. arXiv:1403.4760. Bibcode:2014JMagR.242..233W. doi:10.1016/j.jmr.2014.02.005. PMID 24705364.
- R. E. Glick (1961). "On the Diamagnetic Susceptibility of Gases". J. Phys. Chem. 65 (9): 1552–1555. doi:10.1021/j100905a020.
- John F. Schenck (1993). "The role of magnetic susceptibility in magnetic resonance imaging: MRI magnetic compatibility of the first and second kinds". Medical Physics. 23 (6): 815–850. Bibcode:1996MedPh..23..815S. doi:10.1118/1.597854. PMID 8798169.
- G. P. Arrighini; M. Maestro & R. Moccia (1968). "Magnetic Properties of Polyatomic Molecules: Magnetic Susceptibility of H2O, NH3, CH4, H2O2". J. Chem. Phys. 49 (2): 882–889. Bibcode:1968JChPh..49..882A. doi:10.1063/1.1670155.
- J. Heremans, C. H. Olk and D. T. Morelli (1994). "Magnetic Susceptibility of Carbon Structures". Phys. Rev. B. 49 (21): 15122–15125. Bibcode:1994PhRvB..4915122H. doi:10.1103/PhysRevB.49.15122.
- N. Ganguli & K.S. Krishnan (1941). "The Magnetic and Other Properties of the Free Electrons in Graphite". Proceedings of the Royal Society. 177 (969): 168–182. Bibcode:1941RSPSA.177..168G. doi:10.1098/rspa.1941.0002.
- Nave, Carl L. "Magnetic Properties of Solids". HyperPhysics. Retrieved 2008-11-09.
- R. Dupree & 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.
- S. Otake, M. Momiuchi & N. Matsuno (1980). "Temperature Dependence of the Magnetic Susceptibility of Bismuth". J. Phys. Soc. Jpn. 49 (5): 1824–1828. Bibcode:1980JPSJ...49.1824O. doi:10.1143/JPSJ.49.1824. The tensor needs to be averaged over all orientations: χ = 1/χ∥ + 2/χ⊥.
- Borradaile, Graham John (December 1988). "Magnetic susceptibility, petrofabrics and strain". Tectonophysics. 156 (1–2): 1–20. doi:10.1016/0040-1951(88)90279-X.