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 xx Z x

HZ d e

max

min

1 j

j

xH H ii jj x

j

xd e x d x eZ x

1 H

ij

ed xZ x

iji jj

dd d xd x

1 H

j i jjd d x eZ

k kx extreme xH

1 Hi j d e

Z

i i j

j

Hx kTx

i i jj

Hx kTx

i i ii

Hq q p kTq

i i ii

Hp p q kTp

3i i ii ii

Hq q p NkTq

3i i ii ii

Hp p q NkTp

Equipartition Theorem

Quadratic Hamiltonian :

3

1

3N

i i ii ii

Hq q p NkTq

3

1

3N

i i ii ii

Hp p q NkTp

2 2

1 1

QP nn

i i j ji j

H A P B Q

generalized coord. & momenta

2 j jj

H A PP

2 j j

j

H B QQ

1 1

2QP nn

i ji ji j

H HP Q HP Q

12

H f kT Equipartition Theoremf = # of quadratic terms in H.

Fails if DoF frozen due to quantum effects

12 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 rr

V d N kT 11 i ji j i j

P urN kT d N kT r

Virial equation of stateProb.3.14

3.8. A System of Harmonic OscillatorsSee § 7.3-4 for applications to photons & phonons.

2 2 21 1,2 2i i i iH q p p m q

m

System of N identical oscillators :

1, ,i N

2 2 21

1 1 1exp2 2

Q dq dp p m qh m

2

1 2 2mh m

11Q

kT

1

N

NQ Q Oscillators are distinguishable :N

kT

N

NkTZ Q

ln lnA kT Z N kT

kT

,

lnT V

A kTN kT

,

0T N

APV

U A T S N kT

,

lnN V

AS N k N kT kT

ln 1kTN k

,V

N V

UC N kT

H U PV N kT ,

PN P

HC N kT

Equipartition :122

U N kT N kT

N

NkTZ Q

1

2i E

ig E d e Z

i

'

'

1 12

Ei

N Ni

eg E di

0

1

0

Res 01 1 !

0 0

E N

NN

e E EN

E

contour closes on the left

contour closes on the right

1

ln ln1 !

N

NES k g E k

N

ln ln

N

NEk N N N

ln 1ES N kN

,

1

N V

S N kT E E

ln 1kTS N k

as before

Quantum Oscillators

12n n

0,1,2,n

10

1exp2n

Q n

12 1

1e

e

112sinh2

12

11

NN

NeZ Q

e

12sinh2

N

lnA kT Z1ln 2sinh2

N kT 1 ln 1

2N N kT e

,T V

A AN N

,

0T N

APV

12 1

NU A T S Ne

,

ln 11N V

A eS N k e NT T e

2

2, 1

VN V

N k eUCT e

H U PV U

,P

N P

HCT

Equipartition :122

U N kT N kT

1 ln 12

A N N kT e 1ln 2sinh2

N kT

1 ln 12

kT e

ln 11

N k ee

1 1 1ln 2sinh coth2 2 2

N k NT

1 1coth2 2

N

221 1csch

2 2N k

fails

/

/

1 12 1

11

kT

kT

Schrodingere

Plancke

kT Classical

quantum classicalC C

Mathematica

g ( E )

12

11

NN

NeZ Q

e

12

0

1 !1 ! !

N R

R

N Re e

N R

0

1 1exp2R

N RZ N R

R

0

Ed E g E e

0

1 12R

N Rg E E N R

R

Microcanonical Version

Consider a set of N oscillators, each with eigenenergies12n 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

12

E 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 RN R

1 !1 ! !

N RN 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

Mathematica

S

1 !1 ! !

N RN R

lnSk

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

ST E

1

N

SR

12

E R N

ln 1 ln 1k N R R

1 lnk N RT R

12ln 12

E Nk

E N

/

1212

kTE N

eE N

/

/

1 12 1

kT

k T

E eN e

12

EN R N

12

ER N

/

1 12 1kTe

same as before

Classical Limit

Classical limit :EN

12

R N

N

R N

1 !1 ! !

N RN R

1 2 1

1 !N R N R R

N

1

1 !

NRN

ln ln lnS k k N R N N N

!

NRN

ln 1Rk NN

ln 1ES k NN

E R

1

N

ST E

k NE

E N kT 122

N 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

zT

GMH

ln

T

ZkTH

1ln

T

QN kT

H

, lnG T H k T Z

( E = 0 set at H = 0 )

(Zrot cancels out )

Classical Case (Langevin)

Dipoles free to rotate.

2 1 cos

1 0 1cos HQ d d e

2 H He eH

4 sinh HH

14ln ln sinhG N kT Q N kT H

H

zz

MN

1ln

T

QkT

H

cosh 1sinh

HkT

H H

1cothz HH

HLkT

1cothL x xx

Langevin function

Mathematica

( Q , G even in H )

( c.f. Prob 2.2 )

z L x

zM N L xV V

Magnetization =

HxkT

1cothL x xx

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

zM NV V

z

Weak H, or High T : 1x 3

5

3 45x xL x O x

3zHkT

2

3zM N H

V V kT

00

lim zT H

H

MV H

Isothermal susceptibility :( paramagnetic )

2

3NV kT

C

T Curie’s law

C = Curie’s const

CuSO4 K2SO46H2O 

Quantum Case

μ J

= gyromagetic ratio2

egmc

= Lande’s g factor

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

1 132 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

2Bemc

= (signed) Bohr magneton

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

ˆHH z z BH g m H

L S Bg g g μ L S 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 /

/

11

J x Jx

x J

eee

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 eee e

2 1 / 2 2 1 / 2

/ 2 / 2

J x J J x J

x J x J

e ee e

1sinh 12

sinh2

xJ

xJ

1ln sinh 1 lnsinh2 2

xG N kT xJ J

( Q , G even in e & H )

1lnsinh 1 lnsinh2 2

xG N kT xJ J

zT

GMH

1 1 11 coth 1 coth2 2 2 2B

xN g J xJ J J J

BxkT g J

H

zz B J

M g J B xN

1 1 11 coth 1 coth2 2 2 2J

xB x xJ J J J

= Brillouin function

Bx g J H

Mathematica

( M is even in e & // H )

Limiting Cases

1 1 11 coth 1 coth2 2 2 2J

xB x xJ J J J

2 2

1

1 1 1 1 11 1 03 2 2 3

J

x

B xx x x

J J J

cothy y

y y

e eye e

21 21 0

3

ye yy y

y

2

2

11

y

y

ee

z B Jg J B x

2

21 1 03 3

BB

B BB

Hg J g JkT

H H HJ J g g JkT kT kT

Curie’s const =

2

3JNCV 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 coth2 2 2 2J

xB x xJ J J J

2coth 1 cothcoth

x xx

tanh x

z B Jg J B x tanhB x0

BB

B BB

HkT

H HkT kT

2

1/2BNC

V k

2

3NV k

1/2J JC

2 23 B

 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 12

m

N

NZ Q e e 2cosh N

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

,H N

GST

, ,G T H N

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

ln 2cosh tanhN k NkT T kT

,T N

GMH

tanhBNkT

B H

1m

m

Q e e e

tanhU G T S NkT

M H , ,U S H N

22

2,

sechH N

UC NT kT kT

U here is the “enthalpy”.

Note: everything except M is even in H.

2N

Ordered Disordered(Saturation) (Random)

Mathematica

22

2 sechC NkT kT

2

22 / /

4kT kT

NkT e e

2 /

22 / 1

kT

kT

eNkT e

2 energy gap

Peak near / kT ~ 1

( Schottky anomaly )

Absolute T

Two equivalent ways to define the absolute temperature scale :

1.Ideal gas equation.

2.Efficiency of a Carnot cycle.

P V n R T 0 0T P

1 C

H

TeT

0 1T e Violation of the Kelvin & Clausius versions of the 2nd law.

Definition of the temperature of a system : UTS

U is any thermodynamic potential with S as an independent variable.

0T S as U

Dynamically unstable.

Impossible if Er is unbounded above.

T < 0

0E

E

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

BH

T < 0 possible if E is bounded.

tanhU NkTe.g.,

Usually T > 0 implies U < 0.

But T < 0 is also allowable if U > 0.U = 0 set at H = 0

( U is even in H )

ln 2cosh tanhSN k kT kT kT

2

1cosh1 tanh

xx

2

1cosh

1kT U

N

N

N U N U

12ln tanhN U US

N NN U N U

tanhU NkT

1tanh UkT N

1 1 1tanh ln

2 1xxx

1 ln2

N UN U

Also

2ln ln2

S N U N UN k N N UN U N U

BH

2ln ln

2S N U N U

N k N N UN U N U

1 1ln 2 1 ln 1 ln2 2

U UN N U N UN N

ln ln2 2 2 2

N U N U N U N UN N N N

Mathematica

1 ln2

N UkT N U

Heat Flow

: small to large

Flow of U (as Q) : High to low.

T : 0 0+

tanhU NkT

BH

Mathematica

Experimental Realization

Let t1 = relaxation time of spin-spin interaction.

t2 = relaxation time of spin-lattice interaction.

System is 1st saturated by a strong H ( US = HM < 0 ). H is then reversed.

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

For t1 < t < t2 , spin subsystem in equilibrium; M unchanged US = HM > 0 TS < 0.

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

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

T 300K

T 350KNMR

T ( + ) K

T < 0 requires E bounded above:

Usually, K makes E unbounded T < 0 unusual

T > 0 requires E bounded below :

Uncertainty principle makes E bounded below T > 0 normally

T >> max

maxkT 1n n

1N

NZ Q Q n

N

n

e 2 211

2

N

n nn

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

___

1

g

n nn n

g

___

2 2112

N

Z g

___2 21ln ln ln 1

2Z N g

2___ ___2 2 2 21 1 1ln

2 2 2N g

2 31 1ln 12 3

x x x x

___2 2 21ln

2N g

___2 2 21ln ln

2Z N g

___________

21 1, ln ln2

NA N Z g N N

2

N N

A AS kT

___________

222

1ln2

Nk g N

___2 2 21ln

2S N k g

___________

221ln2

N g

___________

2U A T S N N

2,N N

U UC N kT

___________

22 0N k

0

0max lnS S N k g

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

U is larger for smaller