Magnetic susceptibility
In electromagnetism, the magnetic susceptibility (latin: susceptibilis “receptive”) 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.^{1} A closely related parameter is the permeability, which expresses the total magnetization of material and volume.
Contents
 1 Definition of volume susceptibility
 2 Conversion between SI and CGS units
 3 Mass susceptibility and molar susceptibility
 4 Sign of susceptibility: diamagnetics and other types of magnetism
 5 Experimental methods to determine susceptibility
 6 Tensor susceptibility
 7 Differential susceptibility
 8 Susceptibility in the frequency domain
 9 Examples
 10 Sources of confusion in published data
 11 See also
 12 References and notes
Definition of volume susceptibility

 See also Relative permeability.
The volume magnetic susceptibility, represented by the symbol (often simply , sometimes – 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, it is same as residual magnet.
where
 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 μ_{0} is the magnetic constant (see table of physical constants), and is the relative permeability of the material. Thus the volume magnetic susceptibility and the magnetic permeability are related by the following formula:
 .
Sometimes^{2} an auxiliary quantity, called intensity of magnetization (also referred to as magnetic polarisation J) and measured in teslas, is defined as
 .
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 emucgs, short for electromagnetic units, or Gaussiancgs; both are the same in this context) that rely on a different definition of the permeability of free space:^{3}
The dimensionless CGS value of volume susceptibility is multiplied by 4π to give the dimensionless SI volume susceptibility value:^{3}
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.
Mass susceptibility and molar susceptibility
There are two other measures of susceptibility, the mass magnetic susceptibility (χ_{mass} or χ_{g}, sometimes χ_{m}), measured in m^{3}·kg^{−1} in SI or in cm^{3}·g^{−1} in CGS and the molar magnetic susceptibility (χ_{mol}) measured in m^{3}·mol^{−1} (SI) or cm^{3}·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).
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, 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, 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.^{4} 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, highend 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.^{5} For liquid samples, the susceptibility can be measured from the dependence of the NMR frequency of the sample on its shape or orientation.^{6}^{7}^{8}^{9}^{10}
Tensor susceptibility
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.
Differential susceptibility
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 constantfield (DC) susceptibility. In particular, when an acfield 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 acfield 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).
Examples
Material  Temperature  Pressure  (molar susc.)  (mass susc.)  (volume susc.)  M (molar mass)  (density)  

Units  (°C)  (atm)  SI (m^{3}·mol^{−1}) 
CGS (cm^{3}·mol^{−1}) 
SI (m^{3}·kg^{−1}) 
CGS (cm^{3}·g^{−1}) 
SI 
CGS (emu) 
(10^{−3} kg/mol) or (g/mol) 
(10^{3} kg/m^{3}) or (g/cm^{3}) 
water ^{11}  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 ^{12}  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 ^{13}  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 ^{14} (to caxis)  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 ^{14}  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 ^{14}  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 ^{15}  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 ^{15}  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 
O_{2} ^{15}  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 
N_{2} ^{15}  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 ^{16}  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 online that seem to have been uploaded from a substandard source,^{17} 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 doublecheck the proper usage of units.
See also
 Curie constant
 Electric susceptibility
 Iron
 Magnetic constant
 Magnetic flux density
 Magnetism
 Magnetochemistry
 Magnetometer
 Maxwell's equations
 Paleomagnetism
 Permeability (electromagnetism)
 Quantitative susceptibility mapping
 Susceptibility weighted imaging
References and notes
 ^ "magnetizability, ξ". IUPAC Compendium of Chemical Terminology—The Gold Book (2nd ed.). International Union of Pure and Applied Chemistry. 1997.
 ^ Richard A. Clarke. "Magnetic properties of materials". Info.ee.surrey.ac.uk. Retrieved 20111108.
 ^ ^{a} ^{b} Bennett, L. H.; Page, C. H.; and Swartzendruber, L. J. (1978). "Comments on units in magnetism". Journal of Research of the National Bureau of Standar (NIST, USA) 83 (1): 9–12.
 ^ L. N. Mulay (1972). A. Weissberger and B. W. Rossiter, ed. Techniques of Chemistry 4. WileyInterscience: New York. p. 431.
 ^ "Magnetic Susceptibility Balances". Sherwoodscientific.com. Retrieved 20111108.
 ^ {{ SAM_ @dreamlyf10 cite journal  author=J. R. Zimmerman, and M. R. Foster  title=Standardization of NMR high resolution spectra  journal=J. Phys. Chem.  volume=61  year=1957  pages=282–289  doi=10.1021/j150549a006  issue=3}}
 ^ Robert Engel, Donald Halpern, and 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, and M. P. Hertzberg (2003). "Magnetic susceptibility: solutions, emulsions, and cells". Concepts Magn. Reson. A 18: 56–71. doi:10.1002/cmr.a.10066.
 ^ K. Frei and 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/S10907807(03)001423. PMID 12914848.
 ^ G. P. Arrighini, M. Maestro, and R. Moccia (1968). "Magnetic Properties of Polyatomic Molecules: Magnetic Susceptibility of H_{2}O, NH_{3}, CH_{4}, H_{2}O_{2}". J. Chem. Phys. 49 (2): 882–889. Bibcode:1968JChPh..49..882A. doi:10.1063/1.1670155.
 ^ S. Otake, M. Momiuchi and N. Matsuno (1980). "Temperature Dependence of the Magnetic Susceptibility of Bismuth". J. Phys. Soc. Jap. 49 (5): 1824–1828. Bibcode:1980JPSJ...49.1824O. doi:10.1143/JPSJ.49.1824. The tensor needs to be averaged over all orientations: .
 ^ 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.
 ^ ^{a} ^{b} ^{c} N. Ganguli and 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.
 ^ ^{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. 20060111. Retrieved 20111108.
HPTS  Area Progetti  EduSoft  JavaEdu  N.Saperi  Ass.Scuola..  TS BCTV  TS VideoRes  TSODP  TRTWE  