1. 3.7. Two Theorems: the “Equipartition” & the “Virial”
Let 

2. 

3. Equipartition Theorem
Quadratic Hamiltonian :
generalized coord. & momenta   
Fails if DoF frozen due to quantum effects
Equipartition Theorem
f = # of quadratic terms in H.

4. Virial Theorem Virial = Virial theorem
Ideal gas: f comes from collision at walls ( surface S ) : 
Gaussian theorem : 
Equipartition theorem : 
d-D gas with 2-body interaction potential u(r) : 
Prob.3.14
Virial equation of state

5. 3.8. A System of Harmonic Oscillators
See § for applications to photons & phonons.
System of N identical oscillators :
Oscillators are distinguishable :

6. 
Equipartition :

7. contour closes on the left
contour closes on the right  
as before

8. Quantum Oscillators

9. Equipartition :
fails

10.   
Mathematica

11. g ( E ) 

12. Microcanonical Version
Consider a set of N oscillators, each with eigenenergies
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
R  Positive integers
 = # of distinct ways to put R indistinguishable quanta (objects)
into N distinguishable oscillators (boxes).
= # of distinct ways to insert N1 partitions into a line of R object. 

13. Number of Ways to Put R Quanta into N States
N = 3, R = 5
Number of Ways to Put R Quanta into N States
# of distinct ways to put R indistinguishable quanta (objects) into N distinguishable oscillators (boxes).
Mathematica

14. S     
same as before

15. Classical Limit
Classical limit :    
equipartition

16. 3.9. The Statistics of Paramagnetism
System : N localized, non-interacting, magnetic dipoles in external field H.
( E = 0 set at H = 0 )
(Zrot cancels out )
Dipoles distinguishable   

17. Classical Case (Langevin)
Dipoles free to rotate.
( c.f. Prob 2.2 )
( Q , G even in H )
Langevin function
Mathematica

18. CuSO4 K2SO46H2O  
Magnetization =
Strong H, or Low T :
Weak H, or High T :
Isothermal susceptibility :
( paramagnetic )
Curie’s law
C = Curie’s const

19. Quantum Case J = half integers, or integers = gyromagetic ratio
= Lande’s g factor
g = for e
( L= 0, S = ½ )
= (signed) Bohr magneton 

20. ( Q , G even in e & H )

21. ( M is even in e & // H )
= Brillouin function
Mathematica

22. Limiting Cases 
 Curie’s const =

23. Dependence on J J   ( with g  0 so that  is finite ) :  x  ,
~ classical case
J = 1/2 ( “most” quantum case ) :
g = 2  

24. Gd2(SO4)3 · 8H2O
J = 7/2, g = 2
 FeNH4(SO4)2 · 12H2O,
J = 5/2, g = 2
 KCr(SO4)2
J = 3/2, g = 2

25. 3.10. Thermodynamics of Magnetic Systems: Negative T
J = ½ , g = 2   
M is extensive; H, intensive.
Note: everything except M is even in H.
U here is the “enthalpy”.

26. (Saturation) (Random)
Ordered Disordered
(Saturation) (Random)
Mathematica

27. Peak near  / kT ~ 1 ( Schottky anomaly )

28. Absolute T
Two equivalent ways to define the absolute temperature scale :
Ideal gas equation.
Efficiency of a Carnot cycle.
Dynamically unstable.
Violation of the Kelvin & Clausius versions of the 2nd law.
U is any thermodynamic potential with S as an independent variable.
Definition of the temperature of a system :
Impossible if Er is unbounded above.

29. T < 0 Z finite  T  0 if E is unbounded.
 T < 0 possible if E is bounded.
e.g.,
( U is even in H )
Usually T > 0 implies U < 0.
But T < 0 is also allowable if U > 0.
U = 0 set at H = 0

30.   
Also

31. Mathematica

32. Heat Flow Flow of U (as Q) : High to low. T : 0      0+
 :    small to large
Mathematica

33. Experimental Realization
Let t1 = relaxation time of spin-spin interaction.
t2 = relaxation time of spin-lattice interaction.
Consider the case t1 << t2 , e.g., LiF with t1 = 105 s, t2 = 5 min.
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.
T  300K
T  350K
NMR
T  (   + ) K

34. 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

35. T >> max  
Let g = # of possible orientations (w.r.t. H ) of each spin  

36. 
U is larger for smaller 
Energy flows from small  to large 
 negative T is hotter than T = +