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

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 = 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) :  Virial equation of state Prob.3.14

3.8. A System of Harmonic Oscillators See § 7.3-4 for applications to photons & phonons. System of N identical oscillators : Oscillators are distinguishable :

 Equipartition :

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

Quantum Oscillators

Equipartition : fails

  

g ( E ) 

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. 

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

S      same as before

Classical Limit Classical limit :     equipartition

3.9. The Statistics of Paramagnetism System : N localized, non-interacting, magnetic dipoles in external field H. Dipoles distinguishable  

Classical Case (Langevin) Dipoles free to rotate. Langevin function

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

Quantum Case J = half integers, or integers = gyromagetic ratio = Lande’s g factor g = 2 for e ( L= 0, S = ½ ) = Bohr magneton 

= Brillouin function

Limiting Cases   Curie’s const =

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

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

3.10. Thermodynamics of Magnetic Systems: Negative T J = ½ , g = 2    M is extensive; H, intensive.

OrderedDisordered (Saturation)(Random)

Peak near  / kT ~ 1 ( Schottky anomaly )

T < 0 Z finite T  0 if E is unbounded.  T < 0 possible if E is bounded. Usually T > 0 implies U < 0. But T < 0 is also allowable if U > 0. e.g.,    

Experimental Realization Let t1= relaxtion time of spin-spin interaction. t2= relaxtion 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, 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. T 300K T 350K NMR

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

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

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