居里点

居里-韦斯定律是居里定律的修正版本，是基于平均场论近似的简单模型，在材料温度T远高于其对应居里温度T_C（即T ≫ T_C）时较为适用，但却因原子间的局部波动作用无法在居里点附近对磁化率χ进行描述^[23]。而T < T_C的情况下，在居里定律和居里-韦斯定律均不成立。

Curie's law for a paramagnetic material:^[24]

${\displaystyle \chi ={\frac {M}{H}}={\frac {M\mu _{0}}{B}}={\frac {C}{T}}}$

${\displaystyle C={\frac {\mu _{0}\mu _{\mathrm {B} }^{2}}{3k_{\mathrm {B} }}}N_{A}g^{2}J(J+1)}$ ^[25]

${\displaystyle N_{A}}$	Avagadro常数
µ0	the permeability of free space. Note: in CGS units is taken to equal one.[26]
g	the Landé g-factor
J(J + 1)	the eigenvalue for eigenstate J2 for the stationary states within the incomplete atoms shells (electrons unpaired)
µB	the Bohr Magneton
kB	Boltzmann's constant
total magnetism	is N number of magnetic moments per unit volume

The Curie–Weiss law is then derived from Curie's law to be:

${\displaystyle \chi ={\frac {C}{T-T_{\mathrm {C} }}}}$

where:

${\displaystyle T_{\mathrm {C} }={\frac {C\lambda }{\mu _{0}}}}$

λ 是Weiss分子场常数。^[25][27]

For full derivation see 居里-韦斯定律.

As the Curie–Weiss law is an approximation, a more accurate model is needed when the temperature, T, approaches the material's Curie temperature, T_C.

Magnetic susceptibility occurs above the Curie temperature.

An accurate model of critical behaviour for magnetic susceptibility with critical exponent γ:

${\displaystyle \chi \sim {\frac {1}{(T-T_{\mathrm {C} })^{\gamma }}}}$

The critical exponent differs between materials and for the mean-field model is taken as γ = 1.^[28]

As temperature is inversely proportional to magnetic susceptibility, when T approaches T_C the denominator tends to zero and the magnetic susceptibility approaches infinity allowing magnetism to occur. This is a spontaneous magnetism which is a property of ferromagnetic and ferrimagnetic materials.^[29][30]

Magnetism depends on temperature and spontaneous magnetism occurs below the Curie temperature. An accurate model of critical behaviour for spontaneous magnetism with critical exponent β:

${\displaystyle M\sim (T-T_{\mathrm {C} })^{\beta }}$

The critical exponent differs between materials and for the mean-field model as taken as β = 1/2 where T ≪ T_C.^[28]

The spontaneous magnetism approaches zero as the temperature increases towards the materials Curie temperature.

The spontaneous magnetism, occurring in ferromagnetic, ferrimagnetic and antiferromagnetic materials, approaches zero as the temperature increases towards the material's Curie temperature. Spontaneous magnetism is at its maximum as the temperature approaches 0 K.^[31] That is, the magnetic moments are completely aligned and at their strongest magnitude of magnetism due to no thermal disturbance.

In paramagnetic materials temperature is sufficient to overcome the ordered alignments. As the temperature approaches 0 K, the 熵 decreases to zero, that is, the disorder decreases and becomes ordered. This occurs without the presence of an applied magnetic field and obeys the 热力学第三定律.^[15]

Both Curie's law and the Curie–Weiss law fail as the temperature approaches 0 K. This is because they depend on the magnetic susceptibility which only applies when the state is disordered.^[32]

硫酸钆 continues to satisfy Curie's law at 1 K. Between 0 and 1 K the law fails to hold and a sudden change in the intrinsic structure occurs at the Curie temperature.^[33]

The Ising model is mathematically based and can analyse the critical points of phase transitions in ferromagnetic order due to spins of electrons having magnitudes of ±1/2. The spins interact with their neighbouring dipole electrons in the structure and here the Ising model can predict their behaviour with each other.^[34][35]

This model is important for solving and understanding the concepts of phase transitions and hence solving the Curie temperature. As a result, many different dependencies that affect the Curie temperature can be analysed.

For example, the surface and bulk properties depend on the alignment and magnitude of spins and the Ising model can determine the effects of magnetism in this system.

Materials structures consist of intrinsic magnetic moments which are separated into domains called Weiss domains.^[36] This can result in ferromagnetic materials having no spontaneous magnetism as domains could potentially balance each other out.^[36] The position of particles can therefore have different orientations around the surface than the main part (bulk) of the material. This property directly affects the Curie temperature as there can be a bulk Curie temperature T_B and a different surface Curie temperature T_S for a material.^[37]

This allows for the surface Curie temperature to be ferromagnetic above the bulk Curie temperature when the main state is disordered, i.e. Ordered and disordered states occur simultaneously.^[34]

The surface and bulk properties can be predicted by the Ising model and electron capture spectroscopy can be used to detect the electron spins and hence the magnetic moments on the surface of the material. An average total magnetism is taken from the bulk and surface temperatures to calculate the Curie temperature from the material, noting the bulk contributes more.^[34][38]

The angular momentum of an electron is either +ħ/2 or −ħ/2 due to it having a spin of 1/2, which gives a specific size of magnetic moment to the electron; the Bohr magneton.^[39] Electrons orbiting around the nucleus in a current loop create a magnetic field which depends on the Bohr Magneton and magnetic quantum number.^[39] Therefore, the magnetic moments are related between angular and orbital momentum and affect each other. Angular momentum contributes twice as much to magnetic moments than orbital.^[40]

For terbium which is a rare-earth metal and has a high orbital angular momentum the magnetic moment is strong enough to affect the order above its bulk temperatures. It is said to have a high anisotropy on the surface, that is it is highly directed in one orientation. It remains ferromagnetic on its surface above its Curie temperature while its bulk becomes ferrimagnetic and then at higher temperatures its surface remains ferrimagnetic above its bulk Néel Temperature before becoming completely disordered and paramagnetic with increasing temperature. The anisotropy in the bulk is different from its surface anisotropy just above these phase changes as the magnetic moments will be ordered differently or ordered in paramagnetic materials.^[37]

Composite materials, that is, materials composed from other materials with different properties, can change the Curie temperature. For example, a composite which has silver in it can create spaces for oxygen molecules in bonding which decreases the Curie temperature^[41] as the crystal lattice will not be as compact.

The alignment of magnetic moments in the composite material affects the Curie temperature. If the materials moments are parallel with each other the Curie temperature will increase and if perpendicular the Curie temperature will decrease^[41] as either more or less thermal energy will be needed to destroy the alignments.

Preparing composite materials through different temperatures can result in different final compositions which will have different Curie temperatures.^[42] Doping a material can also affect its Curie temperature.^[42]

The density of nanocomposite materials changes the Curie temperature. Nanocomposites are compact structures on a nano-scale. The structure is built up of high and low bulk Curie temperatures, however will only have one mean-field Curie temperature. A higher density of lower bulk temperatures results in a lower mean-field Curie temperature and a higher density of higher bulk temperature significantly increases the mean-field Curie temperature. In more than one dimension the Curie temperature begins to increase as the magnetic moments will need more thermal energy to overcome the ordered structure.^[38]

The size of particles in a material's crystal lattice changes the Curie temperature. Due to the small size of particles (nanoparticles) the fluctuations of electron spins become more prominent, this results in the Curie temperature drastically decreasing when the size of particles decrease as the fluctuations cause disorder. The size of a particle also affects the anisotropy causing alignment to become less stable and thus lead to disorder in magnetic moments.^[34][43]

The extreme of this is superparamagnetism which only occurs in small ferromagnetic particles and is where fluctuations are very influential causing magnetic moments to change direction randomly and thus create disorder.

The Curie temperature of nanoparticles are also affected by the crystal lattice structure, body-centred cubic (bcc), face-centred cubic (fcc) and a hexagonal structure (hcp) all have different Curie temperatures due to magnetic moments reacting to their neighbouring electron spins. fcc and hcp have tighter structures and as a results have higher Curie temperatures than bcc as the magnetic moments have stronger effects when closer together.^[34] This is known as the coordination number which is the number of nearest neighbouring particles in a structure. This indicates a lower coordination number at the surface of a material than the bulk which leads to the surface becoming less significant when the temperature is approaching the Curie temperature. In smaller systems the coordination number for the surface is more significant and the magnetic moments have a stronger affect on the system.^[34]

Although fluctuations in particles can be minuscule, they are heavily dependent on the structure of crystal lattices as they react with their nearest neighbouring particles. Fluctuations are also affected by the exchange interaction^[43] as parallel facing magnetic moments are favoured and therefore have less disturbance and disorder, therefore a tighter structure influences a stronger magnetism and therefore a higher Curie temperature.

Pressure changes a material's Curie temperature. Increasing pressure on the crystal lattice decreases the volume of the system. Pressure directly affects the kinetic energy in particles as movement increases causing the vibrations to disrupt the order of magnetic moments. This is similar to temperature as it also increases the kinetic energy of particles and destroys the order of magnetic moments and magnetism.^[44]

Pressure also affects the density of states (DOS).^[44] Here the DOS decreases causing the number of electrons available to the system to decrease. This leads to the number of magnetic moments decreasing as they depend on electron spins. It would be expected because of this that the Curie temperature would decrease however it increases. This is the result of the exchange interaction. The exchange interaction favours the aligned parallel magnetic moments due to electrons being unable to occupy the same space in time^[15] and as this is increased due to the volume decreasing the Curie temperature increases with pressure. The Curie temperature is made up of a combination of dependencies on kinetic energy and the DOS.^[44]

The concentration of particles also affects the Curie temperature when pressure is being applied and can result in a decrease in Curie temperature when the concentration is above a certain percent.^[44]

Orbital ordering changes the Curie temperature of a material. Orbital ordering can be controlled through applied strains.^[45] This is a function that determines the wave of a single electron or paired electrons inside the material. Having control over the probability of where the electron will be allows the Curie temperature to be altered. For example, the delocalised electrons can be moved onto the same plane by applied strains within the crystal lattice.^[45]

The Curie temperature is seen to increase greatly due to electrons being packed together in the same plane, they are forced to align due to the exchange interaction and thus increases the strength of the magnetic moments which prevents thermal disorder at lower temperatures.

In analogy to ferromagnetic and paramagnetic materials, the term Curie temperature (T_C) is also applied to the temperature at which a ferroelectric material transitions to being paraelectric. Hence, T_C is the temperature where ferroelectric materials lose their spontaneous polarisation as a first or second order phase change occurs. In case of a second order transition the Curie Weiss temperature T₀ which defines the maximum of the dielectric constant is equal to the Curie temperature. However, the Curie temperature can be 10 K higher than T₀ in case of a first order transition.^[46]

Figure 4. (Below T₀) Ferroelectric polarisation P in an applied electric field E

Figure 5. (Above T₀) Dielectric polarisation P in an applied electric field E

Materials are only ferroelectric below their corresponding transition temperature T₀.^[48] Ferroelectric materials are all pyroelectric and therefore have a spontaneous electric polarisation as the structures are unsymmetrical.

Ferroelectric materials' polarization is subject to hysteresis (Figure 4); that is they are dependent on their past state as well as their current state. As an electric field is applied the dipoles are forced to align and polarisation is created, when the electric field is removed polarisation remains. The hysteresis loop depends on temperature and as a result as the temperature is increased and reaches T₀ the two curves become one curve as shown in the dielectric polarisation (Figure 5).^[49]

A modified version of the Curie–Weiss law applies to the dielectric constant, also known as the relative permittivity:^[46][50]

${\displaystyle \epsilon =\epsilon _{0}+{\frac {C}{T-T_{\mathrm {0} }}}.}$

A heat-induced ferromagnetic-paramagnetic transition is used in magneto-optical storage media, for erasing and writing of new data. Famous examples include the Sony Minidisc format, as well as the now-obsolete CD-MO format. Curie point electro-magnets have been proposed and tested for actuation mechanisms in passive safety systems of fast breeder reactors, where control rods are dropped into the reactor core if the actuation mechanism heats up beyond the material's curie point.^[51] Other uses include temperature control in soldering irons,^[52] and stabilizing the magnetic field of tachometer generators against temperature variation.^[53]