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 gµB 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 = ∞

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 ☛ = h i Overview @B 2 (gµB) From (76) and (79) ☛ = 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) = 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 and C N 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

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. In the first-order phase transitions, the chemical potentials µ of the two competing phases become equal at the phase transition line, while on each side of this line they are unequal and only one of the phases is thermodynamically stable. The first-order trqansitions are thus abrupt. To the contrast, second- order transitions are gradual: The so-called order parameter continuously grows from zero as the phase-transition line is crossed. Thermodynamic quantities are singular at second-order transitions. Phase transitions are complicated phenomena that arise due to interaction between particles in many-particle systems in the thermodynamic limit N . It can be shown that thermodynamic quantities of finite systems are non-singular and, in particular, there are→∞ no second-order phase transitions. Up to now in this course we studied systems without interaction. Including the latter requires an extension of the formalism of statistical mechanics that is done in Sec. XV. In such an extended formalism, one has to calculate partition functions over the energy levels of the whole system that, in general, depend on the interaction. In some cases these energy levels are known exactly but they are parametrized by many quantum numbers over which summation has to be carried out. In most of the cases, however, the energy levels are not known analytically and their calculation is a huge quantum-mechanical problem. In both cases calculation of the partition function is a very serious task. There are some models for which thermodynamic quantities had been calculated exactly, including models that possess second-order phase transitions. For other models, approximate numerical methods had been developed that allow calculating phase transition temperatures and critical indices. It was shown that there are no phase transitions in one-dimensional systems with short-range interactions. It is most convenient to illustrate the physics of phase transitions considering magnetic systems, or the systems of spins introduced in Sec. XIII. The simplest forms of interaction between different spins are Ising interaction J Sˆ Sˆ − ij iz jz including only z components of the spins and Heisenberg interaction Jij Sî Sˆj . The exchange interaction Jij follows from quantum mechanics of atoms and is of electrostatic origin. In most− cases· practically there is only interaction J between spins of the neighboring atoms in a crystal lattice, because Jij decreases exponentially with distance. For the Ising model, all energy levels of the whole system are known exactly while for the Heisenberg model they are not. In the case J>0 neighboring spins have the lowest interaction energy when they are collinear, and the state with all spins pointing in the same direction is the exact ground state of the system. For the Ising model, all spins in the ground state are parallel or antiparallel with the z axis, that is, it is double-degenerate. For the Heisenberg model the spins can point in any direction in the ground state, so that the ground state has a continuous degeneracy. In both cases, Ising and Heisenberg, the ground state for J>0isferromagnetic. The nature of the interaction between spins considered above suggests that at T 0 the system should fall into its → ground state, so that thermodynamic averages of all spins approach their maximal values, Sˆ S. With increasing i → !" #! ! ! ! ! PHASE TRANSITIONS

We have seen that with Overviewchanging thermodynamic parameters such as T,P, system can undergo phase transitions···

1st-order phase transitions chemical potentials µ of two competing phases become equal at phase transition line while on each side of this line they are unequal and only one of phases is thermodynamically stable 1st-order transitions are thus abrupt

To contrast ☛ 2nd-order transitions are gradual: order parameter continuously grows from zero as phase-transition line is crossed

Thermodynamic quantities are singular at second-order transitions

Phase transitions are complicated phenomena that arise due to interaction between particles in many-particle systems in thermodynamic limit N !1

C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 15 SPIN INTERACTION Simplest forms of interactionOverview between different spins are: Heisenberg interaction ☛ J Sˆ Sˆ ij i · j Ising interaction ☛ J Sˆ Sˆ ij iz jz including only z components of spins

Exchange interaction ☛ Jij follows from quantum mechanical properties of crystal lattice

In most cases there is only interaction J between spins of neighboring atoms because J ij decreases exponentially with distance

For Ising model ☛ all spins in ground state are parallel or antiparallel with z axis that is ☛ it is double-degenerate

For Heisenberg model ☛ spins can point in any direction in ground state so that ground state has a continuous degeneracy

C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 16 AVERAGE SPIN VALUE AS ORDER PARAMETER Nature of interaction between spins suggests that at T 0 Overviewsystem should fall into its! ground state so that thermodynamic averages of all spins approach their maximal values Sˆ S |h ii|! With increasing temperature ☛ excited levels of system become populated and average spin value decreases Sˆ

If there is no external magnetic field acting on spins average spin value should be exactly zero because there as many spins pointing in one direction as there are spins pointing in other direction This is why high-temperature state is called symmetric state

~~If now temperature is lowered there should be a phase transition temperature TC below which order parameter (average spin value) becomes nonzero~~

~~Below T C symmetry of state is spontaneously broken ☛ ordered state~~

C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 17 MEAN-FIELD APPROXIMATION Idea is to reduce originalOverview many-spin problem to effective self-consistent one-spin problem by only considering spin i and replacing other spins in interaction by their average thermodynamic values J Sˆ Sˆ J Sˆ Sˆ ij i · j ) ij i ·h ji Effective one-spin Hamiltonian 1 Hˆ = Sˆ (gµ B + Jz Sˆ )+ Jz Sˆ 2 · B h i 2 h i z ☛ number of nearest neighbors for spin in lattice z =6for simple cubic lattice For Heisenberg model ☛ if weak B -field is applied ordering will occur in direction of B Choosing z axis in this direction ˆ ˆ ˆ 1 ˆ 2 H = (gµBB + Jz Sz ) Sz + Jz Sz (83) h i 2 h i Same mean-field H ˆ is valid for Ising model if B field is applied along z axis

~~C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 18 FREE ENERGY PER SPIN~~

~~Within MFA Ising and HeisenbergOverview models are equivalent (but not in general!)~~

~~Energy levels corresponding to (83) are~~

~~1 " = (gµ B + Jz Sˆ ) m + Jz Sˆ 2 m = S, S +1,...,S 1,S m B h zi 2 h zi~~

~~Straightforward calculation of partition function (73) leads to free energy per spin~~

~~1 sinh[(S +1/2)y] F = k T ln Z = Jz Sˆ 2 k T ln B 2 h zi B sinh (y/2) gµ B + Jz Sˆ y B h zi ⌘ kBT~~

To find actual value of order parameter S ˆ at any T and B h zi minimize F with respect S ˆ h zi C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 19 CURIE-WEISS EQUATION Overview@F 0= = Jz Sˆz JzbS(y) @ Sˆ h i h zi Rearranging terms one arrives at transcendental Curie-Weiss equation

gµBB + Jz Sˆz Sˆz = bS h i (84) h i kBT ⇣ ⌘ For B =0 ☛ (84) has only solution Sˆ =0 at high temperatures h zi As b S ( y ) has maximal slope at y =0 it is sufficient that this slope (with respect to S ˆ ) becomes smaller than 1 h zi to exclude any solution other than Sˆ =0 h zi Using (78) ☛ only solution S ˆ =0 is realized for T T C h zi S (S + 1) Jz TC = ☛ Curie temperature within mean-field approximation 3 kB C.Luis B.-Champagne Anchordoqui 2 Thursday, November 13, 14 20 GRAPHIC SOLUTION OF CURIE-WEISS EQUATION T b (y) Sˆz Below C slope of S Overview with respect to h i exceeds 1 25 so that there are three solutions for Sˆ h zi • T = 0.5TC T = T lhs and rhs of (84) C

~~T = 2TC~~

~~• ˆ Sz~~

~~S = 1/2~~

•

One solution Sˆ z =0 FIG.and 7:twoGraphic symmetric solution of solutions the Curie-Weiss S ˆ equation. =0 h i h zi6 C.Luis B.-Champagne Anchordoqui 2 To find the actual value of the order parameter Sˆz at any temperature T and magnetic field B, one has to minimize Thursday, November 13, 14 21 ! " F with respect to Sˆz , as explained in thermodynamics. One obtains the equation ! " ∂F 0= = Jz Sˆz JzbS(y), (186) ∂ Sˆ − z ! " ! " where bS(y)isdefinedbyEq.(164). Rearranging terms one arrives at the transcedental Curie-Weiss equation

~~ˆ gµBB + Jz Sz Sˆz = bS (187)  kBT ! " ! "   that deﬁnes Sˆz . In fact, this equation could be obtained in a shorter way by using the modiﬁed argument, Eq. (185), in Eq.! (163").~~

For B =0, Eq. (187) has the only solution Sˆz = 0 at high temperatures. As bS(y) has the maximal slope at ! " y =0, it is suﬃcient that this slope (with respect to Sˆz ) becomes smaller than 1 to exclude any solution other than Sˆ =0. Using Eq. (167), one obtains that the only! solution" Sˆ = 0 is realized for T T , where z z ≥ C ! " ! " S(S + 1) Jz TC = (188) 3 kB is the Curie temperature within the mean-ﬁeld approximation. Below TC the slope of bS(y)withrespectto Sˆz exceeds 1, so that there are three solutions for Sˆ : One solution Sˆ = 0 and two symmetric solutions Sˆ =0(see! " z z z # ! " ! " ! " Fig. 7). The latter correspond to the lower free energy than the solution Sˆz = 0 thus they are thermodynamically stable (see Fig. 8).. These solutions describe the ordered state below TC . ! " 3 Slightly below TC the value of Sˆz is still small and can be found by expanding bS(y)uptoy . In particular, ! " 1 y 1 1 b (y)= tanh = y y3. (189) 1/2 2 2 ∼ 4 − 48 FREE ENERGY WITHIN MEAN FIELD APPROXIMATION ˆ solutions S z =0 lowerOverview free energy than solution Sˆ =0 26 h i6 h i thus they are thermodynamically stable

~~These solutions describe ordered state below TC~~

~~2F/(Jz) T = 2TC~~

~~T = TC~~

~~ˆ Sz T = 0.8TC~~

~~S = 1/2 T = 0.5TC~~

~~(arbitrary vertical shift) FIG. 8: Free energy of a ferromagnet vs Sˆz within the MFA. (Arbitrary vertical shift.) C. B.-Champagne 2 Luis Anchordoqui ! " Thursday, November 13, 14 22~~

~~Using TC =(1/4)Jz/kB and thus Jz =4kBTC for S =1/2, one can rewrite Eq. (187)withB = 0 in the form~~

~~T 4 T 3 3 Sˆ = C Sˆ C Sˆ . (190) z T z − 3 T z ! " ! " # $ ! "~~

One of the solutions is Sˆz = 0 while the other two solutions are ! " ˆ Sz T 1 = 3 1 ,S= , (191) ! S " ± − T 2 % # C $ where the factor T/TC was replaced by 1 near TC . Although obtained near TC , in this form the result is only by the factor √3off at T =0. The singularity of the order parameter near TC is square root, so that for the magnetization critical index one has β =1/2. Results of the numerical solution of the Curie-Weiss equation with B = 0 for different S are shown in Fig. 9. Note that the magnetization is related to the spin average by Eq. (170). Let us now consider the magnetic susceptibility per spin χ defined by Eq. (173) above TC . Linearizing Eq. (187) with the use of Eq. (167), one obtains

ˆ gµBB + Jz Sz S(S + 1) S(S + 1) gµBB TC Sˆz = = + Sˆz . (192) 3 kBT ! " 3 kBT T ! " ! " Here from one obtains ˆ ∂ µ ∂ Sz S(S + 1) (gµ )2 χ = # z$ = gµ = B . (193) ∂B B !∂B " 3 k (T T ) B − C In contrast to non-interacting spins, Eq. (177), the susceptibility diverges at T = TC rather than at T =0. The inverse 1 susceptivility χ− is a straight line crossing the T axis at TC . In the theory of phase transitions the critcal index for γ the susceptibility γ is deﬁned as χ ∝ (T TC )− . One can see that γ = 1 within the MFA. More precise methods than the MFA yield− smaller values of β and larger values of γ that depend on the details of the model such as the number of interacting spin components (1 for the Ising and 3 for the Heisenberg) and the dimension of the lattice. On the other hand, critical indices are insensitive to such factors as lattice structure and the value of the spin S. On the other hand, the value of TC does not possess such universality and depends on all parameters of the problem. Accurate values of TC are by up to 25% lower than their mean-ﬁeld values in three dimensions.