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. [1] Mathematically it is the ratio of magnetization M (magnetic moment per unit volume) to the applied magnetizing field intensity H. Contents 1 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
Definition of volume susceptibility[edit]
See also:
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:[3] M = χ v H . displaystyle mathbf M =chi _ v mathbf H . Here 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. 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[4] 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:[5] 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:[5] 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 [6] 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[edit] 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[edit]
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[edit]
Volume magnetic susceptibility is measured by the force change felt
upon a substance when a magnetic field gradient is applied.[7] Early
measurements are made using the
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[edit] 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[edit] 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[edit]
Material 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[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 O2[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 N2[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 Air (NTP)[16] 20 1 3.6×10−7 2.9×10−8 28.97 0.00129 Water[17] 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 Paraffin oil, 220–260 cSt[14] 22 1 −10.1×10−9 −8.0×10−7 −8.8×10−6 −7.0×10−7 0.878 PMMA[14] 22 1 −7.61×10−9 −6.06×10−7 −9.06×10−6 −7.21×10−7 1.190 PVC[14] 22 1 −7.80×10−9 −6.21×10−7 −10.71×10−6 −8.52×10−7 1.372 Fused silica glass[14] 22 1 −5.12×10−9 −4.07×10−7 −11.28×10−6 −8.98×10−7 2.20 Diamond[18] 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[19] 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 2.267 Graphite[19] 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 2.267 Graphite[19] 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 2.267 Al[20] 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[21] 961 1 −2.31×10−5 −1.84×10−6 107.87 Bismuth[22] 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 Copper[16] 20 1 −9.63×10−6 −7.66×10−7 63.546 8.92 Nickel[16] 20 1 600 48 58.69 8.9 Iron[16] 20 1 200,000 15,900 55.847 7.874 Sources of confusion in published data[edit]
The
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χ = ( 1 / 3 ) χ
+ ( 2 / 3 ) χ ⊥ displaystyle chi =(1/3)chi _ +(2/3)chi _ perp . External links[edit] Linear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN |