# Magnetic susceptibility

The magnetic susceptibility (v. Latin susceptibilitas " ability to take over") is a unitless physical quantity that indicates the magnetizability of matter in an external magnetic field . In the simplest case, it is a constant of proportionality characteristic of the respective substance , namely the ratio of the magnetization to the magnetic field strength . In other cases it can depend on different sizes, e.g. B. from the location, from the frequencies of the magnetic field and from the previous magnetization. Their possible values ​​range from −1 to almost infinite, with negative values ​​indicating magnetization against the external magnetic field ( diamagnetism ). ${\ displaystyle \ chi}$

The magnetic susceptibility is closely related to the magnetic permeability .

The comparable relationship between polarization and electrical field is described by the (di) electrical susceptibility .

## definition

The most common form, the magnetic volume susceptibility (often also ; is the Greek letter Chi ), describes in the simplest case a constant of proportionality between the magnetization and the magnetic field strength : ${\ displaystyle \ chi}$${\ displaystyle \ chi _ {V}}$${\ displaystyle \ chi}$ ${\ displaystyle {\ vec {M}}}$ ${\ displaystyle {\ vec {H}}}$

${\ displaystyle {\ vec {M}} = \ chi _ {V} {\ vec {H}}}$

This definition is only correct if the magnetic field strength and magnetization have a simple linear relationship.

More generally, the magnetic susceptibility can be defined as a derivative:

${\ displaystyle \ chi _ {ij} = {\ frac {\ partial M_ {i}} {\ partial H_ {j}}},}$

thus as a change in magnetization when the magnetic field strength changes. The indices denote the components of the spatial orientation ( in Cartesian coordinates) of the corresponding fields. In this form, the susceptibility is a tensile quantity and takes into account that magnetization and magnetic field can point in different directions ( magnetic anisotropy ). ${\ displaystyle i, j}$${\ displaystyle x, y, z}$

## Relationship to related quantities

### Molar and mass susceptibility

Two other measures are used for magnetic susceptibility:

• the magnetic mass susceptibility (also or (!)) in m 3 · kg -1 indicates the susceptibility per density${\ displaystyle \ chi _ {\ text {mass}}}$${\ displaystyle \ chi _ {g}}$${\ displaystyle \ chi _ {m}}$ ${\ displaystyle \ rho:}$
${\ displaystyle \ chi _ {\ text {mass}} = {\ frac {\ chi _ {V}} {\ rho}} = \ chi _ {V} \ cdot {\ frac {V} {m}}}$
with mass and volume${\ displaystyle m}$ ${\ displaystyle V.}$
• the molar magnetic susceptibility in m 3 · mol -1 distinguished by the use of the molecular weight or the molar volume :${\ displaystyle \ chi _ {\ text {mol}}}$ ${\ displaystyle M}$ ${\ displaystyle V _ {\ mathrm {m}}}$
${\ displaystyle \ chi _ {\ text {mol}} = M \ cdot \ chi _ {\ text {mass}} = M \ cdot {\ frac {\ chi _ {V}} {\ rho}} = \ chi _ {V} \ cdot {\ frac {m} {n}} \ cdot {\ frac {V} {m}} = \ chi _ {V} \ cdot {\ frac {V} {n}} = \ chi _ {V} \ cdot V _ {\ mathrm {m}}}$
with the amount of substance ${\ displaystyle n.}$

### Magnetic permeability

The constant magnetic susceptibility is simply related to the relative magnetic permeability :

${\ displaystyle \ chi _ {V} = \ mu _ {r} -1}$

This follows from the dependence of the magnetic flux density  on the magnetization and the magnetic field strength : ${\ displaystyle B}$ ${\ displaystyle M}$ ${\ displaystyle H}$

${\ displaystyle B = \ mu _ {0} \ cdot (H + M) = \ mu _ {0} \ cdot (1+ \ chi _ {V}) \ cdot H = \ mu _ {0} \ cdot \ mu _ {r} \ cdot H}$

with the magnetic field constant ${\ displaystyle \ mu _ {0}.}$

### Conversion between SI and CGS units

All of the above definitions refer to the International System of Units  (SI) prescribed in Germany . Since the permeability constant of the vacuum is defined differently in the Gaussian CGS system, the conversion factor is 4π:

${\ displaystyle \ chi _ {V} ^ {\ text {CGS}} = {\ frac {1} {4 \ pi}} \ cdot \ chi _ {V} ^ {\ text {SI}}}$

Since the (volume) susceptibility is also without units in the CGS system, the system of units used must be observed, especially when using older table values. For example, the susceptibility of water at 20 ° C in the CGS system is, which corresponds to a value in the SI. ${\ displaystyle -7 {,} 19 \ cdot 10 ^ {- 7}}$${\ displaystyle -9 {,} 04 \ cdot 10 ^ {- 6}}$

## Classification of magnetic materials

### Constant magnetic susceptibility / no magnetic order

All substances react to a certain degree to magnetic fields. In the simplest case of constant magnetic susceptibility, a distinction is made between two effects that occur in every physical state . Since they are usually very weak, many of these substances are also shown as "non-magnetic".

#### Paramagnetism ${\ displaystyle \ chi> 0}$

Paramagnetic materials have permanent magnetic dipoles that are distributed over all spatial directions without an external magnetic field due to the thermal movement, so that the mean magnetization is zero. In the external magnetic field, the atomic magnetic moments align parallel to the external field and thus strengthen the magnetic field inside the substance. The magnetization is therefore positive and so is the susceptibility. In the inhomogeneous magnetic field, a paramagnetic body is drawn into the area of ​​great field strength. The temperature dependence of the susceptibility is determined by Curies law . Paramagnetism can also have other causes, for example conduction electrons from metals make a temperature-independent contribution ( Pauli paramagnetism ). Examples of paramagnetic substances: aluminum , sodium , α- manganese , oxygen O 2 .

#### Diamagnetism ${\ displaystyle \ chi <0}$

Diamagnetic substances tend to push the magnetic field out of their interior. They do not have a permanent magnetic dipole moment . In the magnetic field, however, dipoles are induced that are opposite to the external field, so that the resulting field inside the material is smaller than outside. Since the magnetization is set against the direction of an external magnetic field, the susceptibility is negative. In the inhomogeneous magnetic field, a diamagnetic body is pushed out of the area of ​​great field strength. Diamagnetic contributions are generally independent of temperature and result from the principle of Lenz's rule . They are therefore present in all materials, even if they are usually not dominant. Examples of diamagnetic substances: hydrogen H 2 , noble gases , nitrogen N 2 , copper , lead , water .

Superconductors represent a special case . They behave as ideal diamagnets in a constant magnetic field . This effect is called the Meißner-Ochsenfeld effect and is an important part of superconductivity. ${\ displaystyle \ chi = -1}$

### Variable magnetic susceptibility / with magnetic order

Solids with a magnetic order respond very strongly to magnetic fields. Their magnetic susceptibility shows a complicated behavior. Above a threshold temperature it behaves paramagnetically, below it depends on other factors:

#### Ferromagnetism

Ferromagnets align their magnetic moments parallel to the external magnetic field, but do so in a strongly amplifying manner. It is often possible to completely magnetize a ferromagnet so that the susceptibility shows a saturation effect. The saturation also depends on the previous magnetization; they are said to have a memory. The behavior is described by a hysteresis loop . Examples of ferromagnets are α- iron , cobalt , nickel .

#### Ferrimagnetism

As with ferromagnets, the susceptibility of ferrimagnets depends on the previous magnetization. The reason for their magnetic behavior is an anti-parallel alignment of differently sized magnetic moments in a crystal. The crystal lattice of a ferrimagnetic substance can be described by two sublattices placed one inside the other. Without an external magnetic field, the magnetic moments of the sublattices are exactly antiparallel; but they have a different amount, so that without an applied field there is a spontaneous magnetization. The magnetization curve is similar to that of ferromagnets, but with a significantly lower saturation magnetization. An example of a ferrimagnetic material is magnetite (Fe 3 O 4 ).

#### Antiferromagnetism

Antiferromagnets are magnetically anisotropic ; That is, their susceptibility depends on the orientation of the solid in the magnetic field. If the external magnetic field is in the same plane as the elementary magnetic moments , the relationship between susceptibility and temperature is approximately linear. If the magnetic field is perpendicular to that plane, the susceptibility is approximately independent of temperature. The crystal lattice of an antiferromagnetic substance can be described by two sublattices placed one inside the other. Without an external magnetic field, the magnetic moments of the sublattices are exactly antiparallel; but they have the same amount, so that without an applied field the magnetization disappears. The temperature dependence is described by the Néel temperature . Examples of antiferromagnets: Metals with built-in paramagnetic ions such as MnO or MnF 2 .

#### use

Ferric and ferromagnetic materials can be used as permanent magnets if they have a large residual magnetization after the external magnetic field has been switched off . Soft magnetic materials , on the other hand, can be (re) magnetized very easily and are therefore used, for example, for generators and transformers .

## Calculation using Gouy's balance

For the Gouy balance, see magnetochemistry .

The changes in two forces can be measured with a Gouy's balance:

${\ displaystyle \ Delta F_ {g} = \ Delta m \ cdot g}$
• By introducing a para- or diamagnetic substance into a magnetic field  , the field lines are drawn together or spread. This changes the force (before air: after material:) :${\ displaystyle H}$${\ displaystyle \ chi _ {v1} \ approx 0,}$${\ displaystyle \ chi _ {v2} \ neq 0}$
{\ displaystyle {\ begin {aligned} \ Delta F_ {z} & = - {\ frac {1} {2}} \ cdot (\ chi _ {v2} - \ chi _ {v1}) \ cdot \ mu \ cdot H ^ {2} \ cdot A \\ & \ approx - {\ frac {1} {2}} \ cdot \ chi _ {v2} \ cdot \ mu \ cdot H ^ {2} \ cdot A \ end { aligned}}}
with the area of  the substance to be examined that is penetrated by the magnetic field.${\ displaystyle A}$

The volume susceptibility can be determined from the balance on the balance: ${\ displaystyle \ Delta F_ {g} = \ Delta F_ {z}}$

${\ displaystyle \ chi _ {v} = - 2 \ cdot {\ frac {\ Delta m \ cdot g} {\ mu \ cdot H ^ {2} \ cdot A}}}$

From the relationship

${\ displaystyle B = \ mu _ {0} \ cdot (1+ \ chi _ {v}) \ cdot H = \ mu \ cdot H}$

for the magnetic field, the magnetizing field for the vacuum ( ) can be determined. For a neodymium magnet with a magnetic flux density  T , for example, a magnetic field strength A / m  Oe results  directly on the surface of a pole . ${\ displaystyle H = B / \ mu _ {0}}$${\ displaystyle \ chi _ {v} = 0}$${\ displaystyle B = 0 {,} 29}$ ${\ displaystyle H = 230.781}$ ${\ displaystyle \ approx 2.899}$

The magnetizing field is the same as the magnetic field depending on position and distance from the current -carrying conductors or magnets, and may by circular integral calculus be accurately determined.

## Magnetic susceptibility of some materials

material ${\ displaystyle T}$ ${\ displaystyle \ chi _ {\ text {mol}}}$ ${\ displaystyle \ chi _ {\ text {mass}}}$ ${\ displaystyle \ chi _ {V}}$
° C (SI)
10 -9 m 3 · mol -1
(cgs)
10 −6 cm 3 mol −1
(SI)
10 -9 m 3 · kg -1
(CGS)
10 -6 cm 3 · g -1
(SI)
10 −6
(cgs)
10 −6
vacuum evil 0 0 0 0 0 0
water 20th −0.163 −13 −9.05 −0.72 −9.035 −0.719
Bi 20th −3.55 −282 −17.0 −1.35 −166 −13.2
diamond RT −0.069 −5.5 −5.8 −0.46 −20 −1.6
Hey −0.0238 −1.89 −5.93 −0.472
Xe −0.57 −45.4 −4.35 −0.346
O 2 43 3420 2690 214
Al 0.22 17th 7.9 0.63
Ag −0.238 −18.9 −2.20 −0.175

The tensor must be averaged over all spatial directions: .${\ displaystyle \ chi = (1/3) \ chi _ {||} + (2/3) \ chi _ {\ perp}}$