3.7. Two Theorems: the “Equipartition” & the “Virial”

Let

; 1, ,6 , ; 1, ,3 ,j i ix j N q p i N q p

6 3 3N N Nd d x d q d p 1 Hi i

j j

H Hx d e x

x Z x

HZ d e

max

min

1 j

j

xH H ii jj x

j

xd e x d x e

Z x

1 H

ij

ed x

Z x

iji jj

dd d x

d x

1 H

j i jjd d x eZ

k kx extreme xH

1 Hi j d e

Z

i i jj

Hx kT

x

i i jj

Hx kT

x

i i ii

Hq q p kT

q

i i ii

Hp p q kT

p

3i i ii ii

Hq q p NkT

q

3i i ii ii

Hp p q NkT

p

Equipartition Theorem

Quadratic Hamiltonian :

3

1

3N

i i ii ii

Hq q p NkT

q

3

1

3N

i i ii ii

Hp p q NkT

p

2 2

1 1

QPnn

i i j ji j

H A P B Q

generalized coord. & momenta

2 j jj

HA P

P

2 j j

j

HB Q

Q

1 1

2QP

nn

i ji ji j

H HP Q H

P Q

1

2H f kT Equipartition Theorem

f = # of quadratic terms in H.

Fails if DoF frozen due to quantum effects

1

2 P QH n n kT

Virial Theorem

i ii

r fVVirial = 3N kT Virial theoremj jj

q p

Ideal gas: f comes from collision at walls ( surface S ) :

SP d r SVP S P f n S

Gaussian theorem : P dV rV 3 P V PV N kT

Equipartition theorem : 1

32

U K N kT 2KV

d-D gas with 2-body interaction potential u(r) :

i ji j i j

ud PV r

r

V d N kT 11 i j

i j i j

P ur

N kT d N kT r

Virial equation of stateProb.3.14

3.8. A System of Harmonic Oscillators

See § 7.3-4 for applications to photons & phonons.

2 2 21 1,

2 2i i i iH q p p m qm

System of N identical oscillators :

1, ,i N

2 2 21

1 1 1exp

2 2Q dq dp p m q

h m

2

1 2 2m

h m

1

1Q

kT

1

N

NQ Q Oscillators are distinguishable :N

kT

N

N

kTZ Q

ln lnA kT Z N kTkT

,

lnT V

AkT

N kT

,

0T N

AP

V

U A T S N kT

,

lnN V

AS N k N k

T kT

ln 1

kTN k

,

V

N V

UC N k

T

H U PV N kT ,

P

N P

HC N k

T

Equipartition :1

22

U N kT N kT

N

N

kTZ Q

1

2

i E

ig E d e Z

i

'

'

1 1

2

Ei

N Ni

eg E d

i

0

1

0

Res 011 !

0 0

E N

NN

e EE

N

E

contour closes on the left

contour closes on the right

1

ln ln1 !

N

N

ES k g E k

N

ln ln

N

N

Ek N N N

ln 1E

S N kN

,

1

N V

S N k

T E E

ln 1kT

S N k

as before

Quantum Oscillators

1

2n n

0,1,2,n

10

1exp

2n

Q n

1

2 1

1e

e

11

2sinh2

1

2

11

N

NN

eZ Q

e

12sinh

2

N

lnA kT Z1

ln 2sinh2

N kT 1

ln 12

N N kT e

,T V

A A

N N

,

0T N

AP

V

1

2 1

NU A T S N

e

,

ln 11

N V

A eS N k e N

T T e

2

2

, 1V

N V

N k eUC

T e

H U PV U

,

P

N P

HC

T

Equipartition :1

22

U N kT N kT

1ln 1

2A N N kT e 1

ln 2sinh2

N kT

1ln 1

2kT e

ln 11

N k ee

1 1 1ln 2sinh coth

2 2 2N k N

T

1 1coth

2 2N

221 1

csch2 2

N k

fails

/

/

1 1

2 11

1

k T

k T

Schrodingere

Plancke

kTClassical

quantum classicalC C

g ( E )

1

2

11

N

NN

eZ Q

e

1

2

0

1 !

1 ! !

N R

R

N Re e

N R

0

1 1exp

2R

N RZ N R

R

0

Ed E g E e

0

1 1

2R

N Rg E E N R

R

Microcanonical Version

Consider a set of N oscillators, each with eigenenergies1

2n n

0,1,2,n

Find the number of distinct ways to distribute an energy E among them.

Each oscillator must have at least the zero-point energy disposable energy is

1

2E E N R R Positive integers

= # of distinct ways to put R indistinguishable quanta (objects)

into N distinguishable oscillators (boxes).

= # of distinct ways to insert N1 partitions into a line of R object.

1 !

1 ! !

N R

N R

1 !

1 ! !

N R

N R

N = 3, R = 5

# of distinct ways to put R indistinguishable quanta

(objects) into N distinguishable oscillators (boxes).

Number of Ways to Put R Quanta into N States

S

1 !

1 ! !

N R

N R

lnS

k ln ln lnN R N R N R N N N R R R

ln ln lnN R N R N N R R

1

N

S

T E

1

N

S

R

1

2E R N

ln 1 ln 1

kN R R

1ln

k N R

T R

12ln12

E Nk

E N

/

1212

k TE N

eE N

/

/

1 1

2 1

k T

k T

E e

N e

1

2

EN R N

1

2

ER N

/

1 1

2 1k Te

same as before

Classical Limit

Classical limit :E

N

12

R N

N

R N

1 !

1 ! !

N R

N R

1 2 1

1 !

N R N R R

N

1

1 !

NR

N

ln ln lnS k k N R N N N

!

NR

N

ln 1R

k NN

ln 1E

S k NN

E R

1

N

S

T E

k N

E

E N kT 12

2N kT

equipartition

3.9. The Statistics of Paramagnetism

System : N localized, non-interacting, magnetic dipoles in external field H.

1

N

ii

E E

1

N

ii

μ H1

cosN

ii

H

cos1

HQ e

1

N

NZ Q Q Dipoles distinguishable

coszM N ˆHH z

cos

cos

cos H

H

eN

e

T

A

H

ln

T

ZkT

H

1ln

T

QN kT

H

Classical Case (Langevin)

Dipoles free to rotate.

2 1 cos

1 0 1cos HQ d d e

2 H He eH

4sinh H

H

1

4ln ln sinhA N kT Q N kT H

H

zz

M

N 1ln

T

QkT

H

cosh 1

sinh

HkT

H H

1cothz H

H

H

LkT

1cothL x x

x Langevin function

z L x

zM NL x

V VMagnetization =

Hx

kT

1

cothL x xx

Strong H, or Low T : 1x 211 xL x O e

x

zM N

V Vz

Weak H, or High T : 1x 3

5

3 45

x xL x O x

3z

H

kT

2

3zM N

HV V kT

00

lim zT H

H

M

V H

Isothermal susceptibility :

2

3

N

V kT

C

T Curie’s law

C = Curie’s const

CuSO4 K2SO46H2O 

Quantum Case

μ J

= gyromagetic ratio2

eg

mc

= Lande’s g factor

2 1J J J J = half integers, or integers

1 13

2 2 1

S S L Lg

J J

g = 2 for e ( L= 0, S = ½ )

2 2 2 2J 2 2 1Bg J J

2B

e

mc

= Bohr magneton

z m Bg m , 1, , 1,m J J J J

ˆHH z z BH g m H

Bgμ J

Bg m H

1B

Jm g H

m J

Q e

/J

m x J

m J

e

Bx g J H

2 1 /

/

1

1

J x Jx

x J

ee

e

2/

0

Jx m x J

m

e e

2 1 / 2 2 1 / 2

2 1 / 2 2 1 / 2

J x J J x Jx

J x J J x J

e ee

e e

2 1 / 2 2 1 / 2

/ 2 / 2

J x J J x J

x J x J

e e

e e

1sinh 1

2

sinh2

xJ

xJ

1lnsinh 1 lnsinh

2 2

xA N kT x

J J

1lnsinh 1 lnsinh

2 2

xA N kT x

J J

z

T

AM

H

1 1 11 coth 1 coth

2 2 2 2B

xN g J x

J J J J

B

xkT g J

H

zz B J

Mg J B x

N

1 1 11 coth 1 coth

2 2 2 2J

xB x x

J J J J

= Brillouin function

Bx g J H

Limiting Cases

1 1 11 coth 1 coth

2 2 2 2J

xB x x

J J J J

2 2

1

1 1 1 1 11 1 0

3 2 2 3J

x

B xx x x

J J J

cothy y

y y

e ey

e e

21 2

10

3

ye y

yy

y

2

2

1

1

y

y

e

e

z B Jg J B x 21

1 03

BB

B BB

Hg J g J

kT

H HJ J g g J

kT kT

Curie’s const =

2 211

3J B

NC g J J

V k 21

3

N

V k

Bx g J H

2 2 2 1Bg J J

Dependence on J

J ( with g 0 so that is finite ) :

Bx g J H

2 2 2 1Bg J J

x , 1JB x L x J ~ classical case

J = 1/2 ( “most” quantum case ) :

g = 2

1/2 2coth 2 cothB x x x

1 1 11 coth 1 coth

2 2 2 2J

xB x x

J J J J

2coth 1coth

coth

xx

x

tanh x

z B Jg J B x tanhB x0

BB

B BB

H

kT

H H

kT kT

2

1/2BN

CV k

2

3

N

V k

1/2J JC

 KCr(SO4)2

J = 3/2, g = 2

 FeNH4(SO4)2 · 12H2O,J = 5/2, g = 2

Gd2(SO4)3 · 8H2OJ = 7/2, g = 2

3.10. Thermodynamics of Magnetic Systems: Negative T

J = ½ , g = 2 m Bg m H 1

2m

N

NZ Q e e 2coshN

ln lnA kT Z N kT e e ln 2coshN kTkT

,H N

AS

T

, ,A T H N

d A SdT M d H d N M is extensive; H, intensive.

ln 2cosh tanhN k NkT T kT

,T N

AM

H

tanhBNkT

B H

1m

m

Q e e e

tanhU A T S NkT

M H , ,U S H N

22

2

,

sechH N

UC N

T kT kT

2N

Ordered Disordered

(Saturation) (Random)

22

2sechC N

kT kT

2

22 / /

4k T k T

NkT e e

2 /

22 / 1

k T

k T

eN

kT e

2 energy gap

Peak near / kT ~ 1

( Schottky anomaly )

T < 0

0E

E

Z e E Z finite T 0 if E is unbounded.

0B H

T < 0 possible if E is bounded.

tanhU NkT

e.g., Usually T > 0 implies U < 0.

But T < 0 is also allowable if U > 0.

ln 2cosh tanhS

N k kT kT kT

11tanh

k U

T N ln

2

k N U

N U

1 1 1

tanh ln2 1

xx

x

1

ln2

N U

kT N U

2

1cosh

1 tanhx

x

2

1cosh

1kT U

N

N

N U N U

2

ln ln2

N U N U

N N UN U N U

2

ln ln2

S N U N U

N k N N UN U N U

1 1ln 2 1 ln 1 ln

2 2

U UN N U N U

N N

ln ln2 2 2 2

N U N U N U N U

N N N N

Experimental Realization

Let t1 = relaxtion time of spin-spin interaction.

t2 = relaxtion time of spin-lattice interaction.

System is 1st saturated by a strong H, which is then reversed.

Lattice sub-system has unbounded E spectrum so its T > 0 always.

For t2 < t , spin & lattice are in equilibrium T > 0 & U < 0 for both.

For t1 < t < t2 , spin subsystem is in equilibrium but U > 0, so T < 0.

Consider the case t1 << t2 , e.g., LiF with t1 = 105 s, t2 = 5 min.

T 300K

T 350KNMR

T >> max

maxkT 1n n

1N

NZ Q Q n

N

n

e 2 21

12

N

n nn

Let g = # of possible orientations (w.r.t. H ) of each spin

___

1

g

n nn n

g

___

2 211

2

N

Z g

___2 21

ln ln ln 12

Z N g

2___ ___2 2 2 21 1 1

ln2 2 2

N g

2 31 1ln 1

2 3x x x x

___2 2 21

ln2

N g

___2 2 21

ln ln2

Z N g

___________

21 1, ln ln

2

NA N Z g N N

2

N N

A AS k

T

___________

222

1ln

2

Nk g N

___2 2 21

ln2

S N k g

___________

221ln

2N g

___________

2U A T S N N

2,N N

U UC N k

T

___________

22 0N k

0

0max lnS S N k g

Energy flows from small to large negative T is hotter than T = +