PHY 341/641 Thermodynamics and Statistical Mechanics MWF:...

PHY 341/641 Thermodynamics and Statistical Mechanics

MWF: Online at 12 PM & FTF at 2 PM

Discussion for Lecture 35:Treatment of particle interactions in statistical mechanics;

electron spin effects

Reading: Sections 8.2

1. Effects of spin interactions – Ising model

2. Mean field solution

3. Exact solution for special cases

Record!!!

4/26/2021 PHY 341/641 Spring 2021 -- Lecture 35 1


Important dates: Final exams available < May 6; due May 14Outstanding work due May 14

Instead of a HW set, please send me your thoughts on topics, examples, etc. that you would like to see in our review starting on Wed. 4/28/2021.


Comment on homework (generally and also #26)It is important to check your units when you are doing numerical evaluations.

( )1/

NkT B TPV V N

= +

Specifically for #26, you need to evaluate B(T) which modifies the ideal gas equation of state..

What are the units of B?


Your questions –From Mike -- I'm assuming the Ising model is only an accurate approximation for ferromagnets. In this case, how do we deal with antiferromagnets?

From Kristen -- 1. How is it that the system discussed can become more ordered as the temperature gets higher, doesn't this violate entropy?2. Could you go over how we get the average expected spin values (equation 8.49) from the partition function, just in this case especially?


Some results from Friday focusing on multi-electron atoms with net composite spin S (half integer or integer values) in the presence of a magnetic field B and nearest neighbor spin interactions (J) modeled according to the Heisenberg model:

( )

( )

=

assuming that B is in the z-direction and alsoassuming that the z component of spin dominatesthe dot product --

ij neighbors

ij nei

a i i ji

a izg

iz jzi hbors

J

B J S SS

µ

µ

− − ⋅

≈ − −

⋅ ∑

∑

∑

∑

S S SBH

Hi j


( )

(neighbors of i)

Review from Friday, continued --2

= 2

(canonical average)

Me

Approximation using mean field approach --

a iz iz jzi

iz a jz

z

ij neighbors

ji

jz j

B J S S

B J S

S S

S

S

µ

µ

≈ − −

− +

→

∑ ∑

∑∑

H

( ) (neighbors of i)

e

a

= 2 wh

re

H

i

d

n fiel am ltonian --

a iz neigh neigh jzi a

MFj

SJS BB Bµµ

≡− + ∑∑H

Note that in some equations presented Friday, J should have been written 2J. Apologies.


( )( )( )

12

12

Evaluating the canonical partition function for N atoms in mean field approximation:

sinh ( , ) where

sinh

N

aneigh

a

tottot

tot

S BZ T N B

BB B

βµβµ

+= ≡ +

( )( )( ) ( )( )( )

( ) ( )( ) ( )( )

1 12 2

1 1 1 12 2 2 2

Helmholtz free energy:

( , ) = ln sinh ln sinh

Average magnetic moment along the magnetic field direction:

coth coth

tot tot

tot tot

a a

a a aN

k

F

F T N N T

B

S B B

N S S B B

βµ βµ

µ µ βµ βµ

− + −

= − = + + − ∂∂

( ) ( )( ) ( )( )

( ) ( )( )

1 1 1 12 2 2 2

1 1 1 12 2 2 2

Note that in this approximation the average spin is the same at all sites:

coth coth

For neighbors, this can be written

coth cot2 h

tot tot

neigh tot

jz a aa

aa

S S S B BN

n

B BnJ S S

µβµ βµ

µ

βµ βµµ

= = + + −

⇒ = + + − ( )( )a totB


Comment on n neighbors --

Atoms in a line -- … …… n=2

Atoms in a plane -- … …… n=4

n=6


( ) ( ) ( )( ) ( )( )( )1 1 1 12 2 2 2

Solving the self-consistent mean field equations --

coth coth

Transcendental equation that can be solved numerically at each and .

2neigh n aeigh neigha

a

B S S B B B B

B T

nJ βµ βµµ

= + + + − +

Note, that for some materials it is possible to find solutions for zero external magnetic field.

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

1 1 1 12 2 2 2

1 1 1 12 2 2 2

coth coth

Let Equation to solve:

2 coth co h

2

t

g

a aneigh n ia

eigh ne gh

nei ha

nJ

nJ x x

B S S B B

x B

x S S

βµ βµµ

βµ

β

⇒ = + + −

=

= + + −

In this case, we found that solutions can be found for where the Curie temperature is:

2 ( 1)3

c

c

T TS SkT nJ +

≤

=

For T ≤ Tc the material is in a ferromagnetic phase.


( ) ( )( ) ( )( )

( ) ( )

1 1 1 12 2 2 2

1 12 2

The Ising model is a special case of this treatmentwith 1/ 2 --

2

2 coth coth

becomes or tanh

Notation

Self-consistent equation: for

tanh

neigh iza

iz iz

nJ

nJ x x

nJ x nJ

Sx B S

x S S

x S S

βµ β

β

β β

=

= =

= + + −

= =

12

Here Schroeder 2

izS sJ


Critical temperature for this case

( )

o

Using Schroeder's notation tanhFor small values of the argument of the tanh, the condition becomes

which defines the critical temperature

1

c

c

s s

n

s s

n

n

β

β

β

=

=

=

a

r

n

Alternate form of self-consistency con

a

t h

t nh for Summary

0

dition:

-- fo

r

c

c

cc

c

nTk

Ts s

s TT

T

TT T

sT

=

≤ = >

=


tanh c sTsT

=

, tanh cTsT

s

s

T=0.1Tc

T=0.5Tc

T=Tc


s

T/Tc

Self-consistent solutions to the mean field approximation to the Ising model in the absence of an external magnetic field.


How accurate is the mean field approximation?

Answer – The mean field approximation is generally qualitatively correct except in certain cases

The Ising model can be solved exactly in some cases and used to access the accuracy of the mean field approximation.

The following slides use material from previous course notes based on the textbook Statistical and Thermal Physics by Gould and Tobochnik


Some exact treatments using the Ising model 1

, ( ) 1

2

1 2 1 2

M

)

odel Hamiltonian with magnetic field

1 respresenting up and down spin

F (or =2,

N N

i

e

ii j ij nn i

s

H

s s H s s

N

B

s H s s

µ

=

= − − ⇒ ±=

= −

≡

− +

∑ ∑H

H

-ε-2H -ε+2H εε

( 2 ) ( 2 )

Canonical partition function( ) 2H HZ T e e eβ β β+ − −= + +


Exact solution for 2 particle Ising model -- continued

1 2 1 2 ) (s s H s s− −= +H

( )

( 2 ) ( 2 )

( 2 ) ( 2 )1 2

Canonical partition function( ) 2

Calculation of the average spin for this system1 1 2 22 2 ( )

H H

H Hs

Z T e e e

s s e eZ T

β β β

β β

+ − −

+ −

=

+

+ +

= = −


Exact solution for 2 particle Ising model -- continuedPlots of s

kT/ε H/ε

H/ε=1

kT/ε=1


Now consider an infinite system where we treat N particles with periodic boundary conditions

……… ……………

i= 0 1 2 N-1 N N+1

The index i ranges between 1 and N; N+1 maps to 1 0 maps to N


One dimensional Ising model

11 1

One-dimensional system with periodic boundary conditions:N N

i i ii i

s s H sε += =

= − −∑ ∑H

Partition function for 1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1)

( ){ }

1 2 3

1 11

1 2 2 3 1 1, ,

( , ) exp2

( , ) ( , ) ( , ) ( , )N

N

i i i is i

N N N Ns s s s

HZ T N s s s s

f s s f s s f s s f s s

ββε + +=

− +

= + +

≡

∑ ∑

∑

Clever trick for performing sum over si – write f factors as 2x2 matrices and perform matrix multiplication


( ){ }

( ) ( )

( ) ( )

1 2 3

1 11

1 2 2 3 1 1, ,

( , ) exp2

( , ) ( , ) ( , ) ( , )

where:(1,1) (1, 1)

( , ')( 1,1) ( 1, 1)

N

N

i i i is i

N N N Ns s s s

H

H

HZ T N s s s s

f s s f s s f s s f s s

f ff s s

f f

e e

e e

βε β βε

βε βε β

ββε + +=

− +

+ −

− −

= + +

≡

− = − − −

≡ ≡

∑ ∑

∑

T


( ) ( )

( ) ( )

1 2 3

1 2 2 3 3 4 4 5 11 2 3

1 2 2 3 1 1, ,

, ,

( , ) ( , ) ( , ) ( , ) ( , )

where:

N

N NN

N N N Ns s s s

s s s s s s s s s ss s s s

H

H

Z T N f s s f s s f s s f s s

T T T T T

e e

e e

βε β βε

βε βε β

+

− +

+ −

− −

=

=

≡

∑

∑

T

1-dimensional Ising system of N spins with periodic boundary conditions (sN+1=s1) (continued)

( )Because all of the matrices are identical:

( , ) Tr NZ T N =

T

T



)(Tr)(Tr)(Tr 3. 2.

.

000

0000

type theof

ation transformaby eddiagonaliz becan matrix symmetricAny 1.:algebralinear from tricksSome

1

111

2

1

1

ΛΛΛTUTTTUTTTTTUTUTUUTUUTTTT

ΛTUU

T

==

=

≡=

−

−−−

−

nλ

λλ

Nn

NNNN λλλλ 321)(Tr ++=⇒ T



( ) ( )

( ) ( )

( ) ( ){ }( ) ( ){ }( )

1

2

1/22 41

1/22 42

1 2

In this case:

00

cosh sinh

cosh sinh

( , ) Tr

H

J H

N N N

e e

e e

e H H e

e H H e

Z T N

βε β βε

β ε βε β

βε βε

βε βε

λλ

λ β β

λ β β

λ λ

+ −

− −

−

−

≡

=

= + +

= − +

= = +

T

Λ

T



( )

( ) ( )

21 2 1

1

21

1

21

1

1/2 4

( , ) Tr 1

( , ) ln ( , ) ln ln 1

ln because 1

ln cosh sinh

NN N N N

N

Z T N

F T N kT Z T N NkT kT

NkT

N kT H H e βε

λλ λ λλ

λλλ

λλλ

ε β β −

= = + = +

= − = − − +

≈ − <

= − − + +

T

( )( )

2

1/22 4

sinh( , )

sinh

N HFM T NH H e βε

β

β −

∂= − =

∂ +


( )( ) 1/22 4

sinh( , )

sinh

N HM T N

H e βε

β

β −= +

βH

Μ/Ν βε=0

βε=1

βε=2


( )

11 1

1 1

1

1

Exact model for Ising model

Mean approximate Hamiltonian:

N N

i i ii i

N N

MF i i ii i

N

i ii

N

eff ii

s s H s

s s H s

s H s

H s

ε

ε

ε

+= =

= =

=

=

= − −

= − −

= − +

≡ −

∑ ∑

∑ ∑

∑

∑

H

H

Mean field approximation for 1-dimensional Ising model


( ) ( )

( )

1 1

1

ln ln ln 2cosh( )

Consistency condition:1 tanheff i

i

Neff

eff i

H si i i

s

F kT Z NkT Z NkT H

H s H

s s e s HZ

β

β

ε

β ε−

= − = − = −

= +

= = + ∑

Mean field partition function and Free energy:


( )

Mean field solution:

tanhi is s Hβ ε = + ( )

( ) 1/22 4

Exact solution:sinh

sinhi

HMsN H e βε

β

β −≡ =

+

β=1ε=1

H

is

One dimensional Ising model with periodic boundary conditions:

28


Comment, while the solution for H>0 is qualitatively similar for the exact and mean field solutions, the solution for H=0 is qualitatively different.

Exact solution in one dimension has no average spin for H=0 while the mean field solution has a finite average spin solution.

PHY 341/641 Thermodynamics and Statistical Mechanics MWF:...

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