Magnetic susceptibility

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In physics and applied disciplines such as electrical engineering, the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. 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 applied field, 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 μ₀ is the permeability of free space (see table of physical constants), and $\ (1+\chi_{v})$ is the relative permeability of the material. The magnetic susceptibility and the magnetic permeability (μ) are related by the following formula:

$\mu = \mu_0(1+\chi_v) \,$ .

Note that these definitions are according to SI conventions. However, many tables of magnetic susceptibility give cgs values that rely on a different definition of the permeability of free space. The cgs value of susceptibility is multiplied by 4π to give the SI susceptibility value. 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.

There are two other measures of susceptibility, the mass magnetic susceptibility (χ_mass or χ_g, sometimes χ_m), measured in m³·kg^-1 in SI or in cm³·g^-1 in cgs and the molar magnetic susceptibility (χ_mol) measired 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 m a s s = χ v / ρ

χ m o l = M χ m a s s = M χ v / ρ

If χ is positive, then (1+χ) > 1 and the material is called paramagnetic. In this case, the magnetic field is strengthened by the presence of the material. Alternatively, if χ is negative, then (1+χ) < 1, and the material is diamagnetic. As a result, the magnetic field is weakened in the presence of the material.

Volume magnetic susceptibility is measured by the force change felt upon the application of a magnetic field ^[1]. 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, 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 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^[2]^[3]^[4].

The magnetic susceptibility of a ferromagnetic substance is not a scalar. Response is dependent upon the state of sample and can occur in directions other than that of the applied field. To accommodate this, a more general definition using a tensor derived from derivatives of components of M with respect to components of H

$\chi_{ij} = \frac{\part M_j}{\part H_i}$

called the differential susceptibility describes ferromagnetic materials, 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 response of the magnetization in the j-th direction from an incremental change in the i-th direction of the applied field.

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.

When the magnetic susceptibility is studied as a function of frequency, the permeability is a complex quantity and resonances can be seen. 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.

[edit] Examples

Magnetic susceptibility of some materials
Material	$χ m o l$ (SI)	$χ m o l$ (cgs)	Tc
vacuum	0	0
water ^[5]	-1.6*10^-10	-1.3*10^-5
Bi ^[6]	-3.5*10^-9	-2.79*10^-4
Diamond ^[7]	-7.4*10^-11	-5.8*10^-6
He ^[8]	-2.38*10^-11	-1.89*10^-6
Xe ^ibid.	-5.7*10^-10	-4.54*10^-5
O₂ ^ibid.	4.3*10^-8	3.42*10^-3
Al	2.2*10^-10	1.7*10^-5
Ag ^[9]	-2.38*10^-10	-1.89*10^-5
Fe		?200	774°C
Co		?70	1131°C
Ni		?110	354°C

The following table seems to have been uploaded from a substandard source ^[10], which itself has probably borrowed heavily from the CRC Handbook of Chemistry and Physics. Some of the data below (e.g. for Al, Bi, and diamond) is 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 below to the original sources, and to double-check the proper usage of units. Use at your own risk!

Substance	Formula	Mass Susceptibility (10^-6 c.g.s. units)
Aluminum	Al	+16.5 (Molar?)
Aluminum oxide	Al₂O₃	-37.0
Antimony	Sb	-99.0
Antimony oxide	Sb₂O₃	-69.4
Barium	Ba	+20.6
Barium oxide	BaO	-29.1
Beryllium	Be	-9.0
Beryllium oxide	BeO	-11.9
Bismuth	Bi	-280.1 (Molar?)
Bismuth oxide	BiO	-110.0
Boric acid	H₃BO₃	-34.1
Boron	B	-6.7
Boron oxide	B₂O₃	-39.0
Cadmium	Cd	-19.8
Cadmium oxide	CdO	-30.0
Cadmium sulfide	CdS	-50.0
Calcium	Ca	+40.0
Calcium carbonate	CaCO₃	-38.2
Calcium oxide	CaO	-15.0
Carbon, diamond	C	-5.9 (Molar?)
Carbon, graphite	C	-6.0
Cerium (alpha)	Ce	+5,160.0
Cerium oxide	CeO₂	+26.0
Cesium	Cs	+29.0
Cesium oxide	CsO₂	+1,534.0
Chromium	Cr	+180.0
Chromium oxide	Cr₂O₃	+1,960.0
Cobalt	Co	ferro
Cobalt oxide	CoO	+4,900.0
Copper	Cu	-5.46
Copper oxide	CuO	+259.6
Dysprosium	Dy	+103,500.0
Dysprosium oxide	Dy₂O₃	+89,600.0
Erbium	Er	+44,300.0
Erbium oxide	Er₂O₃	+73,920.0
Europium	Eu	+34,000.0
Europium oxide	Eu₂O₃	+10,100.0
Gadolinium	Gd	+755,000.0
Gadolinium oxide	Gd₂O₃	+53,200.0
Gallium	Ga	-21.6
Gallium oxide	Ga₂O	-34.0
Germanium	Ge	-76.84
Germanium oxide	GeO	-28.8
Germanium sulfide	GeS	-40.9
Gold	Au	-28.0
Hafnium	Hf	~75.0- 104.0
Hafnium oxide	HfO₂	-23.0
Holmium oxide	Ho₂O₃	+88,100.0
Indium	In	-64.0
Indium oxide	In₂O	-47.0
Indium sulfide	InS	-28.0
Iridium	Ir	+32.1
Iridium oxide	IrO₂	+224.0
Iron	Fe	ferro
Iron oxide	FeO	+7,200.0
Iron oxide (red)	Fe₂O₃	+3,586.0
Iron sulfide	FeS	+1,074.0
Lanthanum	La	+118.0
Lanthanum oxide	La₂O₃	-78.0
Lanthanum sulfide	La₂S₃	-100.0
Lead	Pb	-23.0(Molar)
Lead oxide	PbO	-42.0
Lead sulfide	PbS	-84.0
Lithium	Li	+14.2
Magnesium	Mg	>0.0
Magnesium oxide	MgO	-10.2
Manganese (alpha)	Mn	+529.0
Manganese (beta)	Mn	+483.0
Mercury (liquid)	Hg	-33.4
Mercury (solid)	Hg	-24.1
Molybdenum	Mo	-96.5
Molybdenum oxide	MoO₂	+41.0
Molybdenum sulfide	MoS₃	-63.0
Neodymium	Nd	+5,628.0
Neodymium oxide	Nd₂O₃	+10,200.0
Neodymium sulfide	Nd₂S₃	+5,550.0
Nickel	Ni	ferro
Nickel oxide	NiO	+6,60.0
Nickel sulfide	NiS	+190.0
Niobium	Nb	+195.0
Niobium oxide	Nb₂O₅	-10.0
Nitric acid	HNO₃	-19.9
Nitrogen oxide (solid)	NO	+19.8
Osmium	Os	+9.9
Palladium	Pd	+567.4
Phosphorus, black	P	-26.6
Phosphorus, red	P	-20.8
Platinum	Pt	+201.9
Platinum oxide	Pt₂O₃	-37.70
Potassium	K	+20.8
Potassium oxide	KO₂	+3,230.0
Potassium sulfide	K₂S	-60.0
Praseodymium	Pr	+5,010.0
Praseodymium oxide	PrO₂	+1,930.0
Praseodymium sulfide	Pr₂S₃	+10,770.0
Rhenium	Re	+67.6
Rhenium oxide	ReO₂	+44.0
Rhodium	Rh	+111.0- +123.0
Rhodium oxide	Rh₂O₃	+104.0
Rubidium	Rb	+17.0
Rubidium oxide	RbO₂	+1,527.0
Rubidium sulfide	Rb₂S₂	-90.0
Ruthenium	Ru	+43.2
Ruthenium oxide	RuO₂	+162.0
Samarium	Sm	+1,860.0- +2,230.0
Samarium oxide	Sm₂O₃	+1,988.0
Samarium sulfide	Sm₂S₃	+3,300.0
Scandium	Sc	+315.0
Selenium	Se	-25.0
Selenium oxide	SeO₂	-27.2
Silicon	Si	-3.9
Silicon carbide	SiC	-12.8
Silicon oxide	SiO₂	-29.6
Silver	Ag	-19.5 (Molar)
Silver oxide	AgO	-19.6
Sodium	Na	+16.0
Sodium oxide	Na₂O	-19.8
Sodium sulfide	Na₂S	-39.0
Strontium	Sr	+92.0
Strontium oxide	SrO	-35.0
Sulfur (alpha)	S	-15.5
Sulfur (beta)	S	-14.9
Tantalum	Ta	+124.0- +154.0
Tantalum oxide	Ta₂O₅	-32.0
Technetium	Tc	+250.0- +290.0
Technetium oxide	Tc₂O₇	-40.0
Tellurium	Te	-106.0
Terbium	Tb	+146,000.0
Terbium oxide	Tb₂O₃	+78,340.0
Thallium (alpha)	Tl	-50.9
Thallium (beta)	Tl	-32.3
Thallium oxide	Tl₂O₃	+76.0
Thallium sulfide	Tl₂S	-88.8
Thorium	Th	+132.0
Thorium oxide	ThO₂	-16.0
Thulium	Tm	+25,500.0
Thulium oxide	Tm₂O₃	+51,444.0
Tin (gray)	Sn	-37.0
Tin oxide	SnO	-19.0
Tin oxide	SnO₂	-41.0
Titanium	Ti	+150.0- 153.0
Titanium carbide	TiC	+8.0
Titanium oxide	TiO₂	+5.9
Tungsten	W	+59.0
Tungsten carbide	WC	+10.0
Tungsten oxide	WO₂	+57.0
Vanadium	V	+255.0
Vanadium oxide	V₂O₃	+1,976.0
Water	H₂O	-7.2×10^-7 emu
Ytterbium	Yb	+249.0
Yttrium	Y	+2.43
Yttrium oxide	Y₂O₃	+44.4
Zinc	Zn	-11.4
Zinc oxide	ZnO	-46.0
Zinc sulfide	ZnS	-25.0
Zirconium	Zr	+119.0- 122.0
Zirconium carbide	ZrC	-26.0
Zirconium oxide	ZrO₂	-13.8