2/20/2014PHY 770 Spring 2014 -- Lecture 121 PHY 770 -- Statistical Mechanics 12:00-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course

PHY 770 Spring 2014 -- Lecture 12 12/20/2014

PHY 770 -- Statistical Mechanics12:00-1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 12 -- Chapter 5Equilibrium Statistical Mechanics

Canonical ensemble

Magnetic effects Ising model

http://www.wfu.edu/~natalie/s14phy770



Summary of results for the canonical ensemble

ˆexp

ˆˆ where Tr exp

ˆˆˆln Z exp exp ( )

T ˆ ˆln

ˆˆTr

r

B

B

BB

B

Hk H

Z TZ T k

H AA k T T H A

k

S

U

T

T

T

k

H


Canonical ensemble – example including magnetic effectsConsider a system of N particles in a box of volume V , treated in the semiclassical limit. Since we are in the semiclassical limit, we can use the relationship

1

11 1

2

1 1( ) 1(mag)

ˆˆwhere Tr exp Tr exp

ˆ ˆ ˆ2

here, denotes the magnetic field strength

denotes the magnetic moment factor

1=

!

N

N

B

trans

Z Z TTN

HZ T H

k

H sB H HmB

T

p

1 denotes the intrinsic spin2s


Canonical ensemble – example including magnetic effects -- continued

1 1( ) 1(mag) 1( ) 1( )

1( ) 3

/2 /21( )

3

, ,,

2

ˆ ˆTr exp

/ 2

1 2/ 2

!

cosh

cosh

ln 3 1tanh / 2

22

trans trans mag

transT

B Bmag

N

NN

T

NB

V N BV N

Z T H H Z T Z T

VZ T

Z T e e B

VZ T B

N

Z TT N BU Nk

UC

B

T

,

22

, ,

3 sech

2 2 2

lnMagnetization: tanh

2 2

B B

B N

T V

B

N

NB B

T Z T B

k Nk

k NM

B



22

, ,

3 sech

2 2 2V N B B BC NkB

NB

k

B=1

B=5 B=10



, ,

ln tanh

2 2

B N

T V N

T Zk BNM

B

T


Statistical mechanics of the Ising modelSpin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field

2412

Spin alignment contribution (convenient redefinition of " ")

where 1, spin alignment energy

9.28 10 / (for an electr

ˆ

on)

i ii

B

B

s

s B s B

g J T

H

1

1 2 3

1

1

1 1 1 1

11

Partition function for for non-interactingˆ term al spins:

one

N

ii

N

B s

Ns s s s

N

NB

s

B

s

N

Z e

e Z

H


Calculation of Z1

)cosh(21

1

1

1 BeeeZ BB

s

Bs

Thermodynamic functions:

1 1

1

2 2

ln ln ln 2cosh( )

lntanh

sech

N

B

A kT Z NkT Z NkT B

ZU N N B B

UC kN B B

T


Statistical mechanics of the Ising model -- continuedSpin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field

, ( )

Nearest-neighbor interaction m:

ˆ

terN

iint ji j nn

ijs sH ò

3 71 2 654

e12 e23 e34 e45 e56e67

, ( ) 1

ˆ ˆ

Ising mod :

ˆ

el

Ising int B

N N

i ji j nn i

ij iB sH sH H s

ò


, ( )

ˆN

i ji j n

in

nt s sH ò

2

2

4cosh

lntanh

Z e e e e

ZU

Ising model – effects of interaction term alone for N=2; eij=e

ˆ intH ò ò ò ò


2 22 ( , )

2 cosh 2 2

B BZ B e e e e

e B e

Ising model -- full model for N=2; eij=e2 2

, ( ) 1Ising

ˆi j

i j ii

nn

H Bs s s

ò

ˆ intH ò ò ò ò


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

1 2 3

1 11 1

1 2 2 3 1 1, ,

Tr exp2

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

where:

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

( 1,1) ( 1, 1)

N

N N

N i i i ii i

N N N Ns

B

s

B

s s

Z 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

B

e

P


1 2 3

1 2 2 3 3 4 4 5 1

1 2 3

1 2 2 3 1 1, ,

, ,

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

where:

N

N N

N

N N N N Ns s s s

s s s s s s s s s ss

B

B

s s s

Z f s s f s s f s s f s s

P P P P P

e e

e e

P

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

Tr NNZ P



)(Tr)(Tr)(Tr 3.

2.

.

000

00

00

type theof

ation transformaby eddiagonaliz becan matrix symmetricAny 1.

:algebralinear from tricksSome

1

111

2

1

1

ΛΛΛTUTTTUTTTT

TUTUTUUTUUTTTT

ΛTUU

T

n

1 2Tr N N N P



1

2

1/22 41

1/22 42

1 2

In this case:

0

0

cosh sinh

cosh sinh

Tr N N NN

B

B

e e

e e

B B

B B

e e

e e

Z

P

Λ

P



1/22 41

1/22 42

cosh sinh

cosh sinh

e e

e

B

e

B

B B

l1

l2

l 1,l 2

bmB

=1be



21 2 1

1

21

1

1

1/22 4

Tr 1

( , , ) ln ln ln 1

ln

ln cosh sinh

si( , , )

N

N N N NN

N

N

Z

A T B kT Z NkT kT

NkT

N kT e

NAM T B

B

B B

P

1/22 4

nh

sinh e

B

B


1/22 4

sinh( , , )

sinh

NM T J H

e

B

B

bmB

/M N be=0

be=1

=2be


I

11 1

1 1

1

s

1

ing

Exact Hamiltonian:

Mean field approximation:

ˆ

ˆ

Ising

N N

i i ii i

N N

i i ii i

N

i ii

N

eff ii

MF

s s s

s s s

H B

H B

Bs s

H s

Mean field approximation for 1-dimensional Ising model


1 1

1

ln ln ln 2cosh( )

Consistency condition:

1tanheff i

i

N

eff

eff i

H si i i

s

A kT Z

B

NkT Z NkT H

H s

s s e sZ

B

Mean field partition function and Free energy:


Consistency condition:

tanhi i Bs s

s

tanh i Bs


Mean field solution:

tanhi is s B

1/22 4

Exact solution:

sinh

sinhi

B

B

Ms

N e

b=1e=1

mB

is

One dimensional Ising model with periodic boundary conditions:


, ( ) 1 , ( ) 1

Ising m del:

ˆ

oN N N N

i j i ji j n

Ising ij i in i i j nn i

s s s s sB B sH

ò ò

Extension of mean field analysis to more complicated geometries

1

1 1

Ising model in mean field approximation:

ˆ2

number of nearest ne ighbors

ˆ2

MFIsing i

MFIs

N N

i ji i

N N

jing i eff ii i

s s s

s

B

B s

H

H s H

ò

ò


Extension of mean field analysis to more complicated geometries -- continued

1 1

1

ln ln ln 2cosh( )

2Consistency condition:

1tanh

2eff i

i

N

eff

eff i

H si i i

s

A kT Z NkT Z NkT H

H s

s s e s

B

BZ

Mean field partition function and Free energy:



1

Consistency condition for =0:

1tanh

2eff i

i

H si i i

s

B

s s e sZ

<s>

tanh2 is


0 0


Define: tanh2

B

s s


0

20

2

2cosh( / 2)

for 0

for

ln( ) 1

2

N

N

N

N

s

s

Z

Z

N

s

sU


0 0


Define: tanh2

B

s s


2 00

2 2 2 20

20

Heat capacity:

2co

ln( )

( / 2)sh

NN

N N

N

UC N

N

Z ss

T

k s

sC


0 0For: tanh2

Note that there is no solution for 1:2

s s


1:2

12

Define critical temperature

2 cTk



220

20

Heat capacity in terms of critical temperature:

2for

cosh

0 f

/

r

/

o

/

cc

N c c

c

Nks TT T

C T

T

s T TT

T T

2cT k

Documents

2/20/2014PHY 770 Spring 2014 -- Lecture 121 PHY 770 -- Statistical Mechanics 12:00-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course