transition metal based magnetic materials

• Name: 贺一珺
• Student Number: 2014302290002
• Class: 物理学弘毅班

Holand）

# Abstract

In statistical physics, we are interested in multi-particle and
many-state systems particularly where the interactions between particles
play an essntial role, which can exhibit a phenomenon known as a phase
transition. This phenomenon can be observed universally in nature such
as the appearance of ferromagnetism in materials such as iron. The
transition like this require the concept of temperature, which is quite
significant in condensed matter physics.
In this article I will give a brief introduction to Ising model and try
to use it to investigate several properties of solid like Magnetization,
Energy per spin, Specific heat per spin and susceptibility of particles.
To do all of these works, I will use Mean Field Theory which is widely
used in the study of statistical mechanics to symplify the calculation
and use the Monte Carco Method which is useful when solving some
numerical problems to simulate the process of thermalequibulum in real
world. What’s more, phase transitions will also be investigated in this
article so that one could have a deeper understanding about the
equilibrium state under various tempreture and appreciate the tremendous
beauty of the real world.
All of my codes are uploaded
here

# Background

## Ising Model

Magnetism is an inherently quantum phenomena which cannot be exhibited
classically. To decribe the behavior of a magnetic material, we should
introduce the electron’s spin and the associated magnetic moment in
quantum mechanics. The Ising model is a mathematical model of
ferromagnetism in statistical mechanics. In Ising Model, the spins are
arranged in a graph, usually a lattice, allowing each spin to interact
with its neighbors. Ferromagnetism arises when a collection of such
spins conspire so that all of their magnetic moments point in the same
direction, yielding a total moment that is macroscopic in size. Each
spin is able to point at two directions and thus can be denoted by two
numbers. After we introduce the concept of spin, the energy of the
system is given by these neighbouring interactions and the interaction
between the spins and an external magnetic field: Energy

For simplicity, we assume that: .
for neighberhood spins.
Less energy is prefered by a system, thus it can be derived that for a
ferromagnetic material all spin are aligned.
Assuming that our spin system is in equilibrium with a heat bath at
temperature T, so that we can use the conclusion of statistical
mechanics directly. The probability of finding the system in any
particular state is proportional to the Boltzmann factor Boltzmann Distribution

And then we get the measured magnetization of the system: Magnetization

where  Ising quench(From wikipedia)

## Mean Field Theory

Mean field theory is a useful approach for calculating the properties of
a spin system. The magnetization is related to the average spin
alignment. For an infinitely large system, the spins will all have the
same average alignment.Hence all spins must have the same average
properties. The total magnetization at temprature T for a system of N
spins will then be Total Magnetization

Thus we can derive M if we can calculate <si>. An exact
computation of <si> would require the probabilities of all
possible microstates. We cannot do this completely constricted by our
ability to calculate. The only thing we can do is to consider an
approximate alternative known as mean field theory.
Using the result of statistical mechanics, the thermal average of si can
be calculated as <Si>

This is the exact result for the behavior of a single spin in a magnetic
field. Now consider an approximate method: The interaction of a spin
with its neighbering spins is equivalent to an effective magnetic field
acting on si.So the thing we need to do first is to calculate the
effective field Heff.
The energy function can be rewritten in this form: Total Energy

which shows that the term involving J has the form of a magnetic field
with Then we have the following relationship and ## The Monte Carlo Method

The mean field theory we mensioned before is not always valid. A
critical example is the values of the critical exponent. A more powerful
approach is the Monte Carlo Method. Rayleigh Taylor instability(From Wikipedia)

To simulate how a spin system interacts with its environment, we will
consider the particular case of a collection of Ising spins. The Monte
Carlo method uses a stochastic approach to simulate the exchange of
energy between the spin system and the heat bath. A spin is chosen and
the energy required to make it flip is calculated. If Eflip is negative,
the spin is flipped and the system moves into a different microstate. If
Eflip is positive, a decision must be made. The core of the Monte Carlo
method is to use the computer to generate some randon numbers which
satisfy the Boltzmann distribution. The approach to realize this is to
compare the numbergenerated with the function derived from the Boltzmann
factos which we mensioned before.

2017年12月4日

# Realization of progrem

I use more than 10 progrem to finish this article and I will describe
some of them so that readers could have a deeper understanding about the
theory I mensioned before.
All of my codes are uploaded
here

• Getting the solution of the equation of <s>

1. A direct method to do this is to plot both the right side and
the left side of the equation in one picture. The solution is
the x axis of their cross point. I take J=4 here for two
dimensional cases. Temperature and volue of z can be modefied
directly before we runthe progrem as the initial condition.
2. Also, we can use the Newton-Raphson method to get the solusion which is useful when soving many equation. But this method requires a good property of converience. And sometimes this method is pretty relay on the choice of initial conditions.
• Ising Model using the Monte Carlo Method

1. First I creat the function to generate the matrex and random
number automatedly.
2. Each site in the matrex will have some contribution to the total
energy. I define the function to calculate this by adding up the
energy generated by the interaction of spin and field of the
four sites near the point.
3. Define the function to calculate the total energy by simply
adding up the energy of all points in the matrix.
4. Calculate the Eflip. Note that it can be expressed as the
difference of the energy of two spin in the same site.
5. In this step I will try to use the Metropolis algorithm. This is
the most crutial and difficult part in this progrem.

``````- Firstly I tried to simply calculate the flip energy in each site of the matrix and generate the random number to do the Monte Carlo process in order to renew the states of the whole matrix, but I found that the value generated by the progrem is not random as I expected. Here I show my code: ##
``````
• Then I tried to improve my progrem to solve this problem, what I
did is to consider the concept of equabulium. I choose the site
randomly until the progrem run about 100000 times to garentee
that each site is equally considered. Then I use ‘if’ paragraph
to decide weather the system is in equalibrium or not. This time
the problem is solved.
• To see the details, please see the progrem I uploaded on Github.

# Results

All of my codes are uploaded
here

Ekkes Brück教授，1991年获阿姆斯特丹大学物理学博士学位（Ph.
D）。1992年开始在阿姆斯特丹大学实验物理系工作，历任研究员、副教授。2008年开始任代尔夫特理工大学应用科学学院教授，并任应用科学学院材料和能源基础系主任，同时也是荷兰皇家科学及人文学会会员。2014年1月开始与我校联合培养博士研究生，主要从事新能源材料及器件研发制备及表征等方面的研究，是磁制冷技术方面国际知名专家，全球磁制冷行业的领跑者之一，Thermag、Intermag等国际会议的组委，著名国际学术期刊Journal
of Magnetism and Magnetic
Materials等的合作编辑，H-index指数为36。截至目前为止，已在权威期刊发表论文400余篇，被引用次数超过7200次（ISI）。与BASF公司及FOM合作，他的团队(FAME)是全球最好的新能源材料研究团队之一。

## The Ising Model, Statistical Mechanics and Mean Field Theory

### The first method to work out the equation Using the cross point to work out the equation

• Here is the two side of equation under various tempreture. It can be
derived from the picture that Tc=4 is the critical temperature which
denote the temperature when phase transition occurs.