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Thermodynamics and Statistical Mechanics Camille Bélanger-ChampagneLuis Anchordoqui LehmanMcGill College University City University of New York ThermodynamicsCharged Particle and CorrelationsStatistical Mechanics in Minimum Bias Events at ATLAS Statistical Mechanics IV • ● ThermodynamicsPhysics motivation of magnetism ● Minbias event and track selection November 2014 • Spin interaction th February 26 2012,• ● SecondAzimuthal order correlation phase resultstransition WNPPC 2012 ● Forward-Backward correlation results Thursday, November 13, 14 1 HYPERBOLIC FUNCTIONS Hyperbolic functions areOverview defined by x x x x e e− e + e− (65) sinh (x) − cosh(x) ⌘ 2 ⌘ 2 sinh(x) cosh(x) 1 tanh (x) coth (x) = ⌘ cosh(x) ⌘ sinh(x) tanh(x) (66) derivatives sinh (x)0 = cosh(x) cosh (x)0 =sinh(x) (67) expansions x, x 1 x, x 1 sinh (x) = tanh (x) = (68) ⇠ ex/2,x⌧ 1 ⇠ 1,x⌧ 1 ⇢ ◆ ⇢ ◆ C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 2 ELECTRON’S MAGNETIC MOMENT Overview Angular momentum of point particle in motion ☛ L = r p = mr v ⇥ ⇥ magnetic moment ☛ ~⌧ = µ~ B~ ⇥ vector relating the aligning torque on the object from externally applied B ~ -field to the field vector itself 1 Magnetic moment of point particle in motion ☛ µ~ = qr v 2 ⇥ If electron is visualized as classical charged particle rotating about an axis e µ~ = L −2me Classical result is off by proportionality factor for spin magnetic moment S µ~ = gµB − ~ 24 Bohr’s magneton ☛ µB = e~/(2me)=9.27 10− J/T ⇥ g 2 ' C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 3 SPINS IN MAGNETIC FIELD Magnetism of solids is dueOverview to intrinsic angular momentum of electron ☛ spin Value of electronic angular momentum ☛ s =1 / 2 eigenvalue of square of electron’s angular momentum (in units of ~ ) sˆ2 = s(s + 1) = 3/4 Eigenvalues of sˆ z projection ☛ sˆz = mm= 1/2 (69) ± This results in magnetic moment of electron ☛ µˆ = gµBˆs (70) C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 4 ZEEMAN HAMILTONIAN Hamiltonian of electron spinOverview in a magnetic field B has form Hˆ = gµ ˆs B (71) − B · minimum of energy corresponds to spin pointing in B direction choose z axis in direction of B ☛ Hˆ = gµ sˆ B − B z using (69) energy eigenvalues of electronic spin in magnetic field become " = gµ mB (72) m − B for electron m = 1/2 ± For atoms spins of all electrons usually combine in collective spin S that can be greater than 1/2 (71) becomes ☛ Hˆ = gµ Sˆ B − B · Eigenvalues of projections of S ˆ on direction of B m = S, S +1,...,S 1,S, − − − altogether 2 S +1 different energy levels of spin are given by (72) C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 5 PARTITION FUNCTION AND FREE ENERGY Physical quantities of spinOverview are determined by partition function S S gµ B "m my B ZS = e− = e ☛ y βgµB B = (73) ⌘ kBT m= S m= S X− X− (73) can be reduced to a finite geometrical progression n (1 x) xk =(1 x)(x0 + x1 + x2 + + xn) − − ··· kX=0 = x0 + x1 + x2 + x3 + + xn ··· x1 x2 x3 xn xn+1 − − − −···− − =1 xn+1 − 2S (2S+1)y (S+1/2)y (S+1/2)y Sy y k yS e 1 e e− sinh[(S +1/2)y] ZS = e− (e ) = e− y − = y/2 − y/2 = (74) e 1 e e− sinh (y/2) kX=0 − − For S =1 / 2 ☛ using sinh (2 x )=2sinh( x ) cosh ( x ) (74) becomes Z1/2 = 2 cosh (y/2) (75) C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 6 FREE ENERGY & AVERAGE SPIN POLARIZATION Partition function beingOverview found ☛ easily obtain thermodynamic quantities e.g. ☛ free energy ↴ sinh[(S +1/2)y] F = Nk T ln Z = Nk T ln − B S − B sinh (y/2) S 1 " Average spin polarization ☛ S = m = me− m h zi h i Z m= S X− 1 @Z @ ln Z using (73) ☛ S = = h zi Z @y @y using (67) and (74) ☛ Sz = bS(y) (76) h i ☟ 1 1 1 1 b (y)= S + coth S + y coth y S 2 2 − 2 2 ⇣ ⌘ h⇣ ⌘ i h i C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 7 BRILLOUIN FUNCTION bS(y)=SBS(Sy)Overview ☟ 1 1 1 x B (x)= 1+ coth 1+ x coth S 2S 2S − 2S 2S ✓ ◆ ✓ ◆ h i 1 y For S =1 / 2 from (74) ☛ Sz = B1/2(y)= tanh (77) h i ☟ 2 2 B1/2(x) = tanh x b ( )= S and BS( )= 1 S ±1 ± ±1 ± At zero magnetic field y =0 ☛ spin polarization vanishes This is immediately seen in (77) but not in (76) y to clarify behavior of b S ( y ) at small ☛ use coth x ⇠= 1 /x + x/ 3 ↴ 1 1 1 y 1 1 1 y bS(y) = S + + S + + ⇠ 2 (S + 1 )y 2 3 − 2 1 y 2 3 ⇣ ⌘h 2 ⇣ ⌘ i h 2 i y 1 2 1 2 S(S + 1) = S + = y (78) 3 2 − 2 3 C.Luis B.-Champagne Anchordoqui h⇣ ⌘ ⇣ ⌘ i 2 Thursday, November 13, 14 8 21 BRILLOUIN FUNCTION B S ( x ) FOR VARIOUS S Overview BS(x) S = 1/2 1 5/2 S = ∞ x FIG. 5: Brillouin function BS (x) for different S. C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 9 With the use of Eq. (157) and the derivatives sinh(x)! = cosh(x), cosh(x)! =sinh(x) (162) one obtains S = b (y), (163) ! z" S where 1 1 1 1 b (y)= S + coth S + y coth y . (164) S 2 2 − 2 2 ! " #! " $ # $ Usually the function bS(y) is represented in the form bS(y)=SBS(Sy), where BS(x) is the Brillouin function 1 1 1 1 B (x)= 1+ coth 1+ x coth x . (165) S 2S 2S − 2S 2S ! " #! " $ # $ For S =1/2 from Eq. (158) one obtains 1 y S = b (y)= tanh (166) ! z" 1/2 2 2 and B (x) = tanh x. One can see that b ( )= S and B ( )= 1. At zero magnetic field, y =0, the spin 1/2 S ±∞ ± S ±∞ ± polarization should vanish. It is immediately seen in Eq. (166) but not in Eq. (163). To clarify the behavior of bS(y) at small y, one can use coth x ∼= 1/x + x/3 that yields 1 1 1 y 1 1 1 y bS(y) = S + + S + + ∼ 2 S + 1 y 2 3 − 2 1 y 2 3 ! " % 2 ! " ( # 2 $ y 1 & 2 '1 2 S(S + 1) = S + = y. (167) 3 2 − 2 3 %! " ! " ( In physical units, this means S(S + 1) gµBB Sz = ,gµBB kBT. (168) ! " ∼ 3 kBT & MAGNETIZATION Overview S(S + 1) gµBB In physical units ☛ Sz = gµBB kBT h i ⇠ 3 kBT ⌧ Average magnetic moment of spin can be obtained from (70) µ = gµ S (79) h zi B h zi Magnetization of sample M is defined as magnetic moment per unit volume If there is 1 magnetic atom per unit cell and all of them are uniformly magnetized µz gµB M = h i = Sz v0 v0 h i ☟ unit-cell volume C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 10 INTERNAL ENERGY Overview Internal energy U of a system of N spins @ ln Z U = N − @ Calculation can be however avoided since from (72) simply follows U = N " = Ngµ B m = Ngµ B S = Ngµ Bb (y)= Nk Tyb (y) h mi − B h i − B h zi − B s − B S (80) at low magnetic fields (or high temperatures) 2 S(S + 1) (gµBB) U = N ⇠ − 3 kBT C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 11 MAGNETIC SUSCEPTIBILITY @ µz Magnetic susceptibility per spin ☛ χ = h i Overview @B 2 (gµB) From (76) and (79) ☛ χ = bS0 (y) (81) kBT 2 2 dbS(y) S +1/2 1/2 b0 (y) = + S ⌘ dy − sinh [(S +1/2)y] sinh [y/2] ⇣ ⌘ 1 ⇣ ⌘ For S =1 / 2 from (76) ☛ b0 (y)= 1/2 4 cosh2 (y/2) From (77) follows b S0 (0) = S ( S + 1) / 3 thus one obtains high-temperature limit 2 S(S + 1) (gµB) χ = kBT gµBB 3 kBT In opposite limit y 1 function b 0 ( y ) and thus susceptibility becomes small S Physical reason for this is that at low temperatures ☛ k T gµ B B ⌧ B spins are already strongly aligned by magnetic field ☛ S = S h zi ⇠ so that S becomes hardly sensitive to small changes of B h zi C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 12 HEAT CAPACITY Heat capacity C can beOverview obtained from (80) as @U @y 2 C = = Ngµ Bb0 (y) = Nk b0 (y)y (82) @T − B S @T B S C has a maximum at intermediate temperatures ☛ y 1 ⇠ At high temperatures C decreases as 1 /T 2 2 S(S + 1) gµBB C = NkB ⇠ 3 kBT ⇣ ⌘ At low temperatures C becomes exponentially small except for case S = 1 (81) and (82) imply a simple relation between χ and C Nχ T = C B2 that does not depend on spin value S C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 13 TEMPERATURE DEPENDENCE OF HEAT CAPACITY Overview 23 C /(NkB ) S = ∞ 5/2 1 S = 1/2 kBT /(SgµB B) FIG. 6: Temperature dependence of the heat capacity of spins in a magnetic fields with different values of S. C.Luis B.-ChampagneXIV. Anchordoqui PHASE TRANSITIONS AND THE MEAN-FIELD APPROXIMATION 2 Thursday, November 13, 14 14 As explained in thermodynamics, with changing thermodynamic parameters such as temperature, pressure, etc., the system can undergo phase transitions.
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