Narits Is Ing Model Demo

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The Ising Model of a Ferromagnet

Narit Pidokrajt1

Fysikum, AlbaNova

Stockholm University

SE-10691 Stockholm

Kingdom of Sweden

February 27, 2008

Abstract

This note summarizes what is written about a two-dimensional Ising model of

a ferromagnet in Section 8.2 of Schroders book. The main purpose of this class is

the Ising model simulation. This note is downloadable from the Teaching section of

the URL www.physto.se/~narit .

1E-mail: [email protected]

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1 Quick facts about magnets

In an ideal paramagnet, each microscopic magnetic dipole is affected only by external

magnetic ( B) fields, not by one another. In an applied magnetic field these dipoles

start to align parallel to the field such that the magnetisation of the material is

proportional to the applied field.

Figure 1: Schematic diagram showing the magnetic dipole moments randomly aligned in

a paramagnetic sample.

The magnetic moments in a ferromagnet have the tendency to become aligned par-

allel to each other under the influence of a magnetic field. However, unlike the

moments in a paramagnet, these moments will then remain parallel when a mag-

netic field is not applied. An example is Fe.


a ferromagnetic sample.

Adjacent magnetic moments from the magnetic ions tend to align anti-parallel to

each other without an applied field. In the simplest case, adjacent magnetic mo-

ments are equal in magnitude and opposite therefore there is no overall magnetisa-

tion. Examples are Cr, NiO, FeO.

In the class we will focus only on the ferromagnet. Recap: at low T the mag-

netization (M) is nonzero, at high T thermal fluctuations reduce M. There is a

critical tempeature known as Curie temperature at which M becomes zero (with

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a antiferromagnetic sample.

no external B field),

M(TC) = 0 (1)

Above TC a ferromagnet becomes a paramagnet. TC for Fe is 1043 K.

Below TC You will not notice M because the iron divides itself into microscopic

domains (so small but still contains billions of dipoles!). If you now heat the iron

in the presence of B and then remove the field after the iron has cooled to room

temperature then you obtain a permanent magnet.

2 The Ising model

Named after the physicist Ernst Ising who did it in 1920s. It is a mathematical

model in statistical mechanics. It has since been used to model diverse phenomena

in which bits of information, interacting in pairs, produce collective effects. Serious

applications include complicated models of ferromagnets, fluids, alloyds, interfaces,

nuclei, subnuclear particles.

For us we will use it to model the behavior of a ferromagnet, just a single domain

within it.

In this class we will account for the tendency of neighboring dipoles to align parallel

to each other while neglecting interactions between dipoles.

Notation: For spin-up () we have si = 1 and spin-down () we have si = 1.

The energy due to interaction: if parallel and + if anti-parallel.

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Figure 4: Born in Cologne, Germany on May 10, 1900, Died in Peoria, IL, USA on May

11, 1998. Obtained his PhD at age 24 but dismissed from job when Hitler came to power.

Became school teacher, immigrated to Luxembourg but soon fled to USA when Germany

invaded Luxembourg. Known as an excellent teacher. He published just a few physics

papers. Between 1966-2000, at least 16,000 articles with contents related to the Ising

model published.

We write the energy as

U =

neighboring pairs i,j

sisj (2)

which is negative if parallel and positive if anti-parallel.

To predict the thermal behavior we need the partition function,

Z =

All possible sets of dipole alignements

eU (3)

For N dipoles the number of terms in the sum is 2N.

In 1D it is possible to solve by hand as shown by Ingemar in the lecture.

In 2D you can find exact solution but it is complicated. Solved in 1940s by Lars

Onsager, a Norwegian who won The Nobel Prize in Chemistry 1968. Can you

imagine working with a 10 10 lattice which has 2100 1030 possible states?!

NO exact solution ever found in 3D we need approximations!

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Figure 5: A 2D Ising model on a 8 8 lattice.

Figure 6: Lars Onsager from Christiania, Oslo, Norway. He did his PhD under Peter

Debye. He was the first to solve the 2D Ising model exactly and later won the Nobel

Prize Winner in Chemistry in 1968. He died in Florida, USA at age 72.

3 Monte Carlo Simulation

A Monte Carlo method is a technique that involves using random numbers and

probability to solve problems. It is a random sampling (as many states as possible)

process basically. So, you need a computer!

The term Monte Carlo Method was coined by S. Ulam and N. Metropolis in reference

to games of chance, a popular attraction in Monte Carlo, Monaco back in the early

1950s.

For our purpose we just do as many sampling of states as possible, compute the

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Figure 7: A game of chances!!! Not quite what we are going to do in this class.

Boltzmann factors for these states, then use this random sample to compute the

average energy, M and other thermodynamic quantities.

If we just do ramdom sampling purely it may not be enough even for a 10 10

lattice because at low T when the system wants to magnetize, the important states

(with nearly all dipoles in the same direction) makes up such a small fraction of the

total which means that we are likely to miss them.

We use Metropolis algorithm instead. This means that we use the Boltzmann

factors as a guide during random generation. Specifically

1. Start with any state.

2. Choose a dipole at random + consider a possibility of flipping it.

3. Computer U (the energy difference) that comes from the flipping.

4. If U < 0 the systems energy decreases or remains unchanged, then flip thisdipole and generate the next state.

5. If U > 0 the energy increases, then decide at random when to flip it with the

probability of flipping being eU/kT.

6. If the dipole does not get flipped, then the new system state will be the same

as before.

7. Go back to 2.

The Metropolis algorithm generates a subset of system states where low-energy

states occur more often than high-energy states.

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Lets see how the algorithm works:

1. Consider 2 states, 1 and 2.

2. Let U1 and U2 be the energies of these states. Let number the states so that

U1 U2. If the system is initially in state 2, then the probability of transition

to state 1 is 1N

, symbolically

P(2 1) =1

N(4)

3. We have

P(1 2) =

1

Ne

(U2U1)kT

(5)

4. The ratio of these 2 transition probabilities is just the ratio of two Boltzmann

factorsP(1 2)

P(2 1)=

eU2/kT

eU1/KT(6)

5. This applies only after the algorithm has been running infinitely long. But we

have just a little time for doing this, so most states are never generated.

6. Our main concerns are that the randomly generated staes give an accurate

picture of the systems energy and magnetization.7. It is noticeable that in the low T regime, the algorithm will rapidly push the

system into a metastable state where almost all the dipoles are parallel to

their neighbors. This resembles the real world in that a large system never

has time to explore all possible microstates, and relaxation time2 to reach

thermodynamic equilibrium can be very long.

4 The Program

You can download the program written by L. Wulff at

http://www.physto.se/~linus/files/Ising.exe

Unfortunately it is only available for Windows platform.

2Relaxation time is a general concept in physics for the characteristic time in which a system changes

to an equilibrium condition from a non-equilibrium condition.

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Figure 8: A program for a Monte Carlo simulation of a 2D Ising model.

It is pretty much the game of chances when initial state is generated, just like in

Monte Carlo!

It employs Metropolis algorithm.

In this program a white square represents an up-spin, and the black one is a down-

spin.

It always starts with random states.

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Figure 9: The Ising programs graphical output for 2020 lattice. There is physics behind

these figures!

These are just snapshots which is less fun. So we run the program at various temper-

atures.

At T=10 the final state is almost random.

As T decreases the dipoles become larger and larger clusters of positive and negative

M.

At T= 2.5 the clusters are as big as the lattice.

At T = 2 a single cluster has taken over the whole lattice and that is when the

system is magntized. Some dipoles would flip but last very briefly.

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At T = 1.5 run we have the opposite M.

At T =1 we would expect that it is completely magnetized. However it is more orless in metastable (quasi stable) with 2 domains.

Conclusion: This system has a critical temperature somewhere between 2.0 and

2.5 (in units of /k)

5 Some exercises

5.1 Estimate of the largest cluster

We run the program with 20 20 lattice at different T for at least 100 iterations. Well,

we are going to run far more than 100. Here is the estimation of the largest cluster

T Size of the largest cluster

10 < 10

5 10+

4 15+

3 20+

2.5 40+

2.4 60+

2.27 (TC!) 80-100

So at the critcal temperature, the cluster size is about the size of the whole lattice. As

long as T is high enough, the behavior of the system is pretty much independent of the

lattice size. For a 2D Ising model his behavior can be extracted from Onsagers exact

solution.

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5.2 Average magnetization

T Magnetization

2 90% (20/400 5 %)

1.5 99% (5/400 1 %)

1 99% (6/400 1 %)

So what do you see? At low T, it is completely magnetized.

5.3 Average energy of the system

We calculate average energy < E > (per dipole) over all interations.

We do 20 20 run as before

T < E >

4 -0.88

3.5 -0.653 -0.81

2.5 -1.1

2 -1.7

1.5 -1.8

1 -1.9

Lets plot the < E > and the heat capacity.

Figure 10: Average energy and heat capacity of the system over iterations.

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5.4 Total magnetization of the system

T M

1 1

1.5 0.95

2 0.9

2.27 0

2.5 0

3 0

Figure 11: Total magnetization the system over iterations.

6 Last words

Hope you enjoyed the random numbers.

Good luck with your exam!

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