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



Equipartition Theorem Quadratic Hamiltonian : generalized coord. & momenta    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) :  Prob.3.14 Virial equation of state

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

   Mathematica

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. 

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

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. ( E = 0 set at H = 0 ) (Zrot cancels out ) Dipoles distinguishable   

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

CuSO4 K2SO46H2O   Magnetization = Strong H, or Low T : Weak H, or High T : Isothermal susceptibility : ( paramagnetic ) 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 ) = (signed) Bohr magneton 

( Q , G even in e & H )

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

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. Note: everything except M is even in H. U here is the “enthalpy”.

(Saturation) (Random) Ordered Disordered (Saturation) (Random) Mathematica

Peak near  / kT ~ 1 ( Schottky anomaly )

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.

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

   Also

Mathematica

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

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

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

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