# Magnetic susceptibility (Ising model)

• garyman
In summary, the conversation discusses the confusion about proving the equality between two equations involving temperature, magnetic susceptibility, and magnetization. The speaker expresses their method of solving the problem and asks for clarification on the variable "M_s". They also suggest using the partial derivative symbol to simplify the calculation.

#### garyman

Hi,

I'm slightly confused about how to prove that:

$$\chi$$=$$\vartheta$$<M>/$$\vartheta$$H

is equal to...

$$\chi$$=(<M2>-<M>2 )/ T

I've expressed <M> as $$\sum$$Msexp(-E/kBT) /
$$\sum$$exp(-E/kBT)

and know that E=-J$$\sum$$SiSj-H$$\sum$$Si

But seem to get lost in the differentiation. I am going aobut this the correct way?

garyman said:
Hi,

I'm slightly confused about how to prove that:

$$\chi$$=$$\vartheta$$<M>/$$\vartheta$$H

is equal to...

$$\chi$$=(<M2>-<M>2 )/ T

I've expressed <M> as $$\sum$$Msexp(-E/kBT) /
$$\sum$$exp(-E/kBT)

and know that E=-J$$\sum$$SiSj-H$$\sum$$Si

But seem to get lost in the differentiation. I am going aobut this the correct way?

what is "$M_s$"?

Also, you can use "\partial", i.e., "$\partial$" for partial derivatives.

Firstly $$M=\sum_i S_i \neq M_s$$? I can't really envision what $M_s$ would be at all. Secondly I personally find it easiest to solve this problem by first noticing that $$\langle M \rangle= k_b T \frac{\partial \log Z}{\partial h}$$ then compute $$\frac{\partial \langle M \rangle}{\partial h}$$.

## What is magnetic susceptibility?

Magnetic susceptibility is a measure of how easily a material can be magnetized in response to an external magnetic field. It is defined as the ratio of the magnetization (magnetic moment per unit volume) to the applied magnetic field strength.

## What is the Ising model?

The Ising model is a mathematical model used to study the behavior of interacting particles or spins in a lattice system. It was originally developed to understand the properties of ferromagnetic materials, but has since been applied to a wide range of physical systems.

## What does the Ising model predict about magnetic susceptibility?

The Ising model predicts that as temperature decreases, the magnetic susceptibility of a material will increase until it reaches a critical temperature, known as the Curie temperature. Below this temperature, the material will exhibit spontaneous magnetization.

## How does the Ising model account for different types of magnetic ordering?

The Ising model takes into account the interactions between neighboring spins in a lattice, allowing for the simulation of various types of magnetic ordering such as ferromagnetism, antiferromagnetism, and paramagnetism. The strength of these interactions is determined by the coupling constant in the model.

## What are the limitations of the Ising model?

The Ising model is a simplified representation of real physical systems and does not take into account many factors that can influence magnetic behavior, such as thermal fluctuations, defects, and external fields. It also assumes a regular lattice structure, which may not accurately represent the complexity of real materials.