In electromagnetism, the magnetic susceptibility (Latin: susceptibilis, "receptive"; denoted χ) is one measure of the magnetic properties of a material. The susceptibility indicates whether a material is attracted into or repelled out of a magnetic field, which in turn has implications for practical applications. Quantitative measures of the magnetic susceptibility also provide insights into the structure of materials, providing insight into bonding and energy levels. If the magnetic susceptibility is greater than zero, the substance is said to be "paramagnetic"; the magnetization of the substance is higher than that of empty space. If the magnetic susceptibility is less than zero, the substance is "diamagnetic"; it tends to exclude a magnetic field from its interior.  Mathematically it is the ratio of magnetization M (magnetic moment per unit volume) to the applied magnetizing field intensity H.Contents1 Definition of volume susceptibility 2 Mass susceptibility and molar susceptibility 3 Sign of susceptibility: diamagnetics and other types of magnetism 4 Experimental methods to determine susceptibility 5 Tensor Tensor susceptibility 6 Differential susceptibility 7 Susceptibility in the frequency domain 8 Examples 9 Sources of confusion in published data 10 See also 11 References and notes 12 External linksDefinition of volume susceptibility See also: Permeability (electromagnetism) Permeability (electromagnetism) § Relative permeability and magnetic susceptibility 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 χ v displaystyle chi _ v (often simply χ displaystyle chi , sometimes χ m displaystyle chi _ m  – 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 = χ v H . displaystyle mathbf M =chi _ v mathbf H . HereM 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. χ v displaystyle chi _ v is therefore a dimensionless quantity. Using SI units, the magnetic induction B is related to H by the relationship B   =   μ 0 ( H + M )   =   μ 0 ( 1 + χ v ) H   =   μ H displaystyle mathbf B = mu _ 0 left(mathbf H +mathbf M right) = mu _ 0 left(1+chi _ v right)mathbf H = mu mathbf H where μ0 is the vacuum permeability (see table of physical constants), and ( 1 + χ v ) displaystyle (1+chi _ v ) is the relative permeability of the material. Thus the volume magnetic susceptibility χ v displaystyle chi _ v and the magnetic permeability μ displaystyle mu are related by the following formula: μ = μ 0 ( 1 + χ v ) displaystyle mu =mu _ 0 left(1+chi _ v right), .Sometimes an auxiliary quantity called intensity of magnetization (also referred to as magnetic polarisation J) and measured in teslas, is defined as I = μ 0 M displaystyle mathbf I =mu _ 0 mathbf M , .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. 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). These units rely on a different definition of the permeability of free space: B cgs   =   H cgs + 4 π M cgs   =   ( 1 + 4 π χ v cgs ) H cgs displaystyle mathbf B ^ text cgs = mathbf H ^ text cgs +4pi mathbf M ^ text cgs = left(1+4pi chi _ v ^ text cgs right)mathbf H ^ text cgs The dimensionless CGS value of volume susceptibility is multiplied by 4π to give the dimensionless SI volume susceptibility value: χ v SI = 4 π χ v cgs displaystyle chi _ v ^ text SI =4pi chi _ v ^ text cgs 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 (in older literature) to see CGS mass susceptibility given in emu/g, so to convert to SI volume susceptibility we use the conversion  χ v SI = 4 π ρ cgs χ m cgs displaystyle chi _ v ^ text SI =4pi ,rho ^ text cgs ,chi _ m ^ text cgs where ρ cgs displaystyle rho ^ text cgs is the density given in g/cm3, or χ v SI = ( 4 π × 10 − 3 ) ρ SI χ m cgs displaystyle chi _ v ^ text SI =left(4pi times 10^ -3 right),rho ^ text SI ,chi _ m ^ text cgs where ρ SI displaystyle rho ^ text SI is the density given in kg/m3. 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). χ mass = χ v ρ χ mol = M χ mass = M χ v ρ displaystyle begin aligned chi _ text mass &= frac chi _ v rho \chi _ text mol &=Mchi _ text mass = frac Mchi _ v rho end aligned Sign of susceptibility: diamagnetics and other types of magnetism 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, non-magnetic materials are said to be para- or diamagnetic because they do not possess permanent magnetization without external magnetic field. Ferromagnetic, ferrimagnetic, or antiferromagnetic materials have positive susceptibility and possess permanent magnetization even without external magnetic field. Experimental methods to determine 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 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 MRI/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. Tensor 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 M i = H j χ i j displaystyle M_ i =H_ j chi _ ij 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 i-th direction from the external field applied in the j-th 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 χ i j d = ∂ M i ∂ H j displaystyle chi _ ij ^ d = frac partial M_ i partial H_ j where χ i j d displaystyle chi _ ij ^ d 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 number 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). Examples Magnetic susceptibility of some materialsMaterial Temp. Pressure Molar susc., χ mol displaystyle chi _ text mol Mass susc., χ mass displaystyle chi _ text mass Volume susc., χ v displaystyle chi _ v Molar mass, M Density, ρ displaystyle rho (°C) (atm) SI (m3·mol−1) CGS (cm3·mol−1) SI (m3·kg−1) CGS (cm3·g−1) SI CGS (emu) (10−3 kg/mol = g/mol) (103 kg/m3 = g/cm3)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 0.000166Xe 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 0.00546O2 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 0.000278N2 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 0.000910Air (NTP) 20 13.6×10−7 2.9×10−8 28.97 0.00129Water 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 0.9982Paraffin oil, 220–260 cSt 22 1−10.1×10−9 −8.0×10−7 −8.8×10−6 −7.0×10−70.878PMMA 22 1−7.61×10−9 −6.06×10−7 −9.06×10−6 −7.21×10−71.190PVC 22 1−7.80×10−9 −6.21×10−7 −10.71×10−6 −8.52×10−71.372Fused silica glass 22 1−5.12×10−9 −4.07×10−7 −11.28×10−6 −8.98×10−72.20Diamond 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 3.513Graphite χ ‖ displaystyle chi _ Vert (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 2.267Graphite χ ‖ displaystyle chi _ Vert 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 2.267Graphite χ ‖ displaystyle chi _ Vert −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 2.267Al1 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 2.70Ag 961 1−2.31×10−5 −1.84×10−6 107.87Bismuth 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 9.78Copper 20 1−9.63×10−6 −7.66×10−7 63.546 8.92Nickel 20 1600 48 58.69 8.9Iron 20 1200,000 15,900 55.847 7.874Sources of confusion in published data The CRC Handbook of Chemistry and Physics CRC Handbook of Chemistry and Physics has one of the only published magnetic susceptibility tables. Some of the data (e.g., for Al, Bi, and diamond) is listed as CGS. CGS 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. See alsoCurie constant Electric susceptibility Iron Magnetic constant Magnetic flux density Magnetism Magnetochemistry Magnetometer Maxwell's equations Paleomagnetism Permeability (electromagnetism) Quantitative susceptibility mapping Susceptibility weighted imagingReferences and notes^ Roger Grinter, The Quantum in Chemistry: An Experimentalist's View, John Wiley & Sons, 2005, ISBN 0470017627 page 364 ^ "magnetizability, ξ". 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Bibcode:1980JPSJ...49.1824O. doi:10.1143/JPSJ.49.1824.  The tensor needs to be averaged over all orientations: χ = ( 1 / 3 ) χ + ( 2 / 3 ) χ ⊥ displaystyle chi =(1/3)chi _ +(2/3)chi _ perp .External linksLinear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN 978-3

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