Curie Weiss Law & Curie Constant

MS Chaudhry

10 months ago

In a paramagnetic material the magnetization of the material is (approximately) directly proportional to an applied magnetic field. However, if the material is heated, this proportionality is reduced; for a fixed value of the field, the magnetization is (approximately) inversely proportional to temperature. This fact is encapsulated by Curie’s law:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75eee02c221efbde2b67821aedc97ba69c9c07e" alt="{\mathbf {M}}=C\cdot {\frac {{\mathbf {B}}}{T}}," width="5367" height="2300.3">

where

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e499ae5946af9c09777ada933051b3669d3372c2" alt="\mathbf {M} " width="1092.5" height="936.9">

is the resulting magnetisation

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" alt="\mathbf {B} " width="818.5" height="936.9">

is the magnetic field, measured in teslas

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" alt="T" width="704.5" height="936.9">

is absolute temperature, measured in kelvins

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" alt="C" width="760.5" height="936.9">

is a material-specific Curie constant.

This relation was discovered experimentally (by fitting the results to a correctly guessed model) by Pierre Curie. It only holds for high temperatures, or weak magnetic fields. As the derivations below show, the magnetization saturates in the opposite limit of low temperatures, or strong fields.

The Curie–Weiss law describes the magnetic susceptibility $χ$ of a ferromagnet in the paramagnetic region above the Curie point:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8f8b8e29ee77cb725a2cab3284c362fa654763" alt="\chi ={\frac {C}{T-T_{{c}}}}" width="5239" height="2443.8">

where $C$ is a material-specific Curie constant, $T$ is absolute temperature, measured in kelvins, and $T c$ is the Curie temperature, measured in kelvin. The law predicts a singularity in the susceptibility at $T = T c$ . Below this temperature the ferromagnet has a spontaneous magnetization.

The magnetic moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field. A loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments.

The magnetization or magnetic polarization of a magnetic material is the vector field that expresses the density of permanent or induced magnetic moments. The magnetic moments can originate from microscopic electric currents caused by the motion of electrons in individual atoms, or the spin of the electrons or the nuclei.

Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic moment that may be present even in the absence of the external magnetic field; for example, in sufficiently cold iron. We call the latter spontaneous magnetization. Other materials that share this property with iron, like Nickel and magnetite, are called ferromagnets. The threshold temperature below which a material is ferromagnetic is called the Curie temperature and varies between materials.

Limitation of Curie-Weiss Law

In many materials the Curie–Weiss law fails to describe the susceptibility in the immediate vicinity of the Curie point, since it is based on a mean-field approximation. Instead, there is acritical behavior of the form

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ad82e7f202cefdfcbb2768b239cbf8c5ba3eb9" alt="\chi \sim {\frac {1}{(T-T_{{c}})^{\gamma }}}" width="6502.3" height="2587.3">

with the critical exponent $γ$ . However, at temperatures $T ≫ T c$ the expression of the Curie–Weiss law still holds, but with $T c$ replaced by a temperature $Θ$ that is somewhat higher than the actual Curie temperature. Some authors call $Θ$ the Weiss constant to distinguish it from the temperature of the actual Curie point.

Curie Constant

The Curie constant is a material-dependent property that relates a material’s magnetic susceptibility to its temperature.

The Curie constant, when expressed in SI units, is given by

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700a196537a4b45fb15f988778ba62f40d979991" alt="C={\frac {\mu _{0}\mu _{B}^{2}}{3k_{B}}}Ng^{2}J(J+1)" width="10345.8" height="2802.6">

where

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" alt="N" width="888.5" height="936.9">$ is the number of magnetic atoms (or molecules) per unit volume,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" alt="g" width="480.5" height="865.1">$ is the Landé g-factor,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3f8b1c93e8a761c8619ab15e3a5fb1fd1e2120" alt="\mu _{B}" width="1240.5" height="936.9">$ (9.27400915e-24 J/T or A·m²) is the Bohr magneton,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" alt="J" width="633.5" height="936.9">$ is the angular momentumquantum number and

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade" alt="k_{B}" width="1158.5" height="1080.4">$ is Boltzmann’s constant. For a two-level system with magnetic moment $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" alt="\mu " width="603.5" height="936.9">$ , the formula reduces to

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef3fd67f904b5a53fd35c31c10c0c8d1f8c6572" alt="C={\frac {1}{k_{B}}}N\mu _{0}\mu ^{2}" width="6616.4" height="2443.8">

The constant is used in Curie’s Law, which states that for a fixed value of a magnetic field, the magnetization of a material is (approximately) inversely proportional to temperature.

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d811f3b981878f635681a71ea67e7f944f8c0acd" alt="{\mathbf {M}}={\frac {C}{T}}{\mathbf {B}}" width="4365.6" height="2300.3">

This equation was first derived by Pierre Curie.

Because of the relationship between magnetic susceptibility $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" alt="\chi " width="626.5" height="865.1">$ , magnetization $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae54b24dc1d087a28995b8e02bd85df6c4d1446" alt="\scriptstyle \mathbf {M} " width="772.5" height="721.6">$ and applied magnetic field $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27550f7f8078b57b43c73abe2eba545d93e526d" alt="\scriptstyle {\mathbf {H}}" width="636.7" height="721.6">$ :

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137e9030da22cd1db5602d71c455c154469b160d" alt="\chi ={\frac {{\mathbf {M}}}{{\mathbf {H}}}}" width="3413.1" height="2300.3">

this shows that for a paramagnetic system of non-interacting magnetic moments, magnetization $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" alt="T" width="704.5" height="936.9">$ .

Curie Weis Law Video Explaination – Curie’s Law and Curie – Weiss Law

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