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3.7.Two Theorems: the “Equipartition” & the “Virial”. Let. . . Equipartition Theorem. generalized coord. & momenta. Quadratic Hamiltonian :. . . . Fails if DoF frozen due to quantum effects. Equipartition Theorem f = # of quadratic terms in H. Virial Theorem. Virial =. - PowerPoint PPT Presentation
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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 = +