 # Curie Weiss Law

Curie-Weiss Law Explanation

Curie-Weiss Law refers to one of the most important laws in the field of electromagnetism. It states that the magnetic susceptibility of a material above a specific temperature (also known as the Curie Temperature), becomes ferromagnetic. With this feature, the object's magnetic moment helps in understanding the torque of a magnet in response to an external magnetic field. For substances above the Curie temperature, the moments can be oriented at random, causing the net magnetic polarization to be zero. The formula can be expressed as:

χ = $\frac{C}{T-T_{C}}$  … eqn. 1

Here C represents the Curie Constant, T depicts absolute temperature, and T$_{C}$ is the Curie Temperature.

The above graph represents that at the Curie Temperature, the paramagnetic properties still exist as the magnetization is zero (because of the absence of a magnetic field). The internal field increases the susceptibility of the element, and the plot of 1produces a straight line in a zero magnetic field; however, it can turn to zero as the temperature approaches Curie Temperature.

Understanding Ferromagnetism and Weiss Law

Ferromagnetism is known to be the phenomenon of spontaneous magnetization, where magnetization appears in a substance when there's a complete absence of applied magnetic field. Some of the most popular ferromagnets are known to be Fe, Co, Ni, few alloys that show ferromagnetism properties. It occurs when there's an alignment of the molecular moments in a suitable direction.

For ferromagnetism to appear, there's a threshold temperature (also known as the ferromagnetic transition temperature), which can go as high as 1000K for elements like Fe, Co, Gd, etc. It occurs as there is the presence of atomic magnetic dipoles in parallel directions within the complete absence of an external field. For example, in Iron, the induced magnetic moment depends on the spinning of the electrons in the nuclei's outer shell. According to Pauli's exclusion principle, no two electrons present in the exact location can have similar spins directed in the same direction. It creates an absolute repulsion between the two electrons. For electrons having counter-direction spins can exhibit attractive interaction with magnetization. Therefore, such an attractive effect found in oppositely spinning electrons can make the iron atoms align with each other. This can be expressed in the following equation:

In this formula, the influence of exchange forces yield and effective molecular field H$_{int}$, that depends on the size of magnetization M;

H$_{int}$ = λM … eqn. 2

Where, λ is the Weiss Constant.

The yielding magnetization (represented by M) can also be represented as a sum and product of the magnetic susceptibility, χ$_{p}$

χ$_{p}$(H + λM) = M   ...eqn. 3

The above equation serves as the base for the Curie-Weiss Law equation.

Limitations of the Curie-Weiss Law

χ = ($\frac{1}{T-T_{c}}$)$^{γ}$    ...eqn. 4

To answer the question of what happens to a ferromagnetic substance heated above Curie temperature, the Curie Weiss Law fails to provide an explanation for the susceptibility of certain elements. It is because, when the temperature (Θ) gets to a place where it is at a really higher value than the Curie Temperature and replaces T$_{c}$, the entire susceptibility becomes infinite.

Relationship of the Curie Law with the Curie-Weiss Law

According to the Curie Law, the magnetization of any paramagnetic element is directly proportional to the applied magnetic field. Often represented as:

M = C x ($\frac{B}{T}$)

here M = Magnetization, B = Magnetic Field, T = absolute temperature, C = Curie Constant.

The Curie Constant is represented as:

C = $\frac{μ_{0}μ_{B}^{2}}{3k_{B}}$ ng²J(J + 1)

here, k$_{B}$ represents the Boltzmann's constant (1.380649 x 10⁻²³), n represents the magnetic atoms per unit volume, g is Landé factor,  μ$_{B}$ is Bohr magneton, and J = angular momentum quantum number.

The fluctuations that occur in the Curie temperature is because of the deviations in the magnetic moments of an element as it reaches the phase transition temperature. Therefore, in a more accurate way, the Curie law can be represented in the modified Curie Weiss Law equation:

χ = $\frac{M}{H}$ = $\frac{Mμ_{0}}{B}$ = $\frac{C}{T}$, where μ$_{0}$ is the permeability of free space.

Therefore, taking from eqn 2, the new equation would be,

χ = $\frac{Mμ_{0}}{B+λM}$ = $\frac{C}{T}$

Since

χ = $\frac{C}{T-\frac{Cλ}{μ_{0}}}$ and χ = $\frac{C}{T-T_{c}}$

Therefore,

T$_{c}$ = $\frac{Cλ}{μ_{0}}$  …. eqn. 5

## Here are the Curie Temperatures for a Few Ferromagnetic Substances

 Substance Name Curie Temperature Iron (Fe) 1,043K Gadolinium (Gd) 293K Nickel (Ni) 631K

The following graph shows the saturation in magnetization observed in Nickel at a high magnetic field. With an increase in temperature, the saturation magnetization decreases till it reaches zero at Curie temperature. Here, Nickel becomes paramagnetic.

Differentiating the equation,

χ = $\frac{M}{H}$

in terms of temperature represents the maximum susceptibility of any substance at Curie temperature.

It proves that the magnetic moment can be effortlessly increased for any transition material with the application of a magnetic field in its transition. The graph above represents the susceptibility of Nickel reaching infinity, as the Curie temperature gets closer to Curie temperature.

Q1. What is the Weiss Theory of Ferromagnetism?

The Weiss theory of ferromagnetism represents that for ferromagnetic materials, the neighboring atoms can cause several mutual exchanges in several closed regions known as domains. It can also be further divided into many points:

1. The domains that are aligned according to the direction of the magnetic field can get more prominent than those aligned in the opposite direction.

2. The domains can rotate and place themselves in the direction of the applied external magnetic field.

Q2. What is the Difference Between the Curie and the Curie Weiss Temperature?

The Curie Temperature (Tc) is the temperature at which the susceptibility of the material gets blown up:

χ = C / T-Tc and T ~ Tc

The Curie - Weiss temperature, on the other hand, holds for T >> T0, and is close to T0 ~ Tc for transitions in the first order.