In
electromagnetism the
magnetic susceptibility (
latin:
susceptibilis “receptiveness”) is the degree of
magnetization of a material in response to an applied
magnetic field.
Definition of volume susceptibility
 See also Relative permeability.
The
volume magnetic susceptibility, represented by the symbol
$chi\_\{v\}$ (often simply
$chi$, sometimes
$chi\_m$ — magnetic, to distinguish from the
electric susceptibility), is defined by the relationship
 $$
mathbf{M} = chi_{v} mathbf{H}
where, in SI units,
 M is the magnetization of the material (the magnetic dipole moment per unit volume), measured in amperes per meter, and
 H is the magnetic field strength, also measured in amperes per meter.
The magnetic induction B is related to H by the relationship
 $$
mathbf{B} = mu_0(mathbf{H} + mathbf{M}) = mu_0(1+chi_{v}) mathbf{H} = mu mathbf{H}
where μ_{0} is the magnetic constant (see table of physical constants), and
$(1+chi\_\{v\})$ is the relative permeability of the material.
The magnetic susceptibility χ_{v} and the magnetic permeability μ are related by the following formula:
 $mu\; =\; mu\_0(1+chi\_v)\; ,$ .
Sometimes an auxiliary quantity, called
intensity of magnetization and measured in
tesla, is defined as
 $mathbf\{I\}\; =\; mu\_0\; mathbf\{M\}\; ,$ .
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.
Conversion between SI and cgs units
Note that these definitions are according to
SI conventions. However, many tables of magnetic susceptibility give
cgs values (often denoted by "
emu" or "
e.m.u.", short for
electromagnetic unit) that rely on a different definition of the permeability of free space:
 $$
mathbf{B}^{text{cgs}} = mathbf{H}^{text{cgs}} + 4pimathbf{M}^{text{cgs}} = (1+4pichi_{v}^{text{cgs}}) mathbf{H}^{text{cgs}}
The dimensionless cgs value of volume susceptibility is multiplied by 4π to give the dimensionless SI volume susceptibility value:
 $chi\_v^\{text\{SI\}\}=4pichi\_v^\{text\{cgs\}\}$
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.
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).
 $chi\_\{text\{mass\}\}=chi\_v/rho$
 $chi\_\{text\{mol\}\}=Mchi\_\{text\{mass\}\}=Mchi\_v/rho$
Sign of susceptibility: diamagnetics and other types of magnetism
If χ is positive, then (1+χ
_{v}) > 1 (or, in
cgs units, (1+4πχ
_{v}) > 1) and the material can be
paramagnetic,
ferromagnetic,
ferrimagnetic, or
antiferromagnetic. In this case, the magnetic field is strengthened by the presence of the material. Alternatively, if χ is negative, then (1+χ
_{v}) < 1 (or, in
cgs units, (1+4πχ
_{v}) < 1), and the material is
diamagnetic. As a result, the magnetic field is weakened in the presence of the material.
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, 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
Evan's balance. For liquid samples, the susceptibility can be measured from the dependence of the
NMR frequency of the sample on its shape or orientation.
Tensor susceptibility
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 $M\_i=chi\_\{ij\}H\_j$
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 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
 $chi^\{d\}\_\{ij\}\; =\; frac\{part\; M\_i\}\{part\; H\_j\}$
where $chi^\{d\}\_\{ij\}$ 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.
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 closelyrelated "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
Magnetic susceptibility of some materials
Material
 Temperature
 Pressure
 $chi\_\{text\{mol\}\}$ (molar susc.)
 $chi\_\{text\{mass\}\}$ (mass susc.)
 $chi\_\{v\}$ (volume susc.)
 M (molar mass)
 $rho$ (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})

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
 −6.9×10^{−11}
 −5.5×10^{−6}
 −5.8×10^{−9}
 −4.6×10^{−7}
 −2.0×10^{−5}
 −1.6×10^{−6}
 12.01
 3.513 
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 
O_{2}
 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}
 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 online 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 doublecheck the proper usage of units. Use them at your own risk!
See also
References and notes