Chemistry 593: Statistical Mechanics. David Ronis McGill University

Transcription

1 Albert Einstein Josiah Willard Gibbs Ludwig Boltzmann Chemistry 593: Statistical Mechanics David Ronis McGill University John Gamble Kirkwood Lars Onsager Robert W. Zwanzig Irwin Oppenheim Ryogo Kubo Winter, 28

2 Chemistry 593: Statistical Mechanics David Ronis 28 McGill University All rights reserved. In accordance with Canadian Copyright Law, reproduction of this material, in whole or in part, without the prior written consent the author is strictly prohibitied. Last modified on 8 April 28 Winter, 28

3 Chemistry Problem Set 4 Table of Contents. General Information Stirling s Formula Ideal Bose & Fermi Gases Introduction and General Considerations Results for Particles in a Cubic Box The Case For Boltzmann Statistics Results Specific to Fermions or Bosons The Semi-Classical Limit Dense Gases: Virial Coefficients Correlation Functions and the Pressure Normal Mode Analysis Quantum Mechanical Treatment Normal Modes in Classical Mechanics Force Constant Calculations Normal Modes in Crystals Normal Modes in Crystals: An Example The q Limit of the Structure Factor Gaussian Coil Elastic Scattering Some Properties of the Master Equation Critical Phenomena Liouville s Equation Nonequilibrium Systems Langevin Equations Projection Operators Inelastic Light Scattering Problem Sets Problem Set Problem Set Problem Set Problem Set

4 General Information -3- Chemistry 593. General Information Professor David Ronis Otto Maass, Room 426 Lectures: MWR :35-2:25 P.M. in BURN B24 Course Website: Case sensitive username and password needed for total acccess. Username: chem593 Password: Boltzmann Chemistry 593: Statistical Mechanics Course Description Intermediate topics in statistical mechanics, including: modern and classical theories of real gases and liquids, critical phenomena and the renormalization group, time-dependent phenomena, linear response and fluctuations, inelastic scattering, Monte Carlo and molecular dynamics methods... TEXT D. A. McQuarrie, Statistical Mechanics.... SUPPLEMENTARY TEXTS. L.K. Nash, Elements of Statistical Thermodynamics (Introductory) 2. T. L. Hill, An Introduction to Statistical Thermodynamics (Dover Publications, introductory, cheap) D. A. McQuarrie, Statistical Mechanics. 3. T. L.Hill, Statistical Mechanics. (Dover, advanced, slightly old fashioned) 4. K. Huang, Statistical Mechanics. (Advanced). 5. L. E. Reichl, AModern Course in Statistical Physics. (Intermediate)...2. General References Old Classics J. W. Gibbs, Elementary Principles in Statistical Mechanics R. C. Tolman, The Principles of Statistical Mechanics H. S. Green, The Molecular Theory of Liquids J. Frenkel, Kinetic Theory of Liquids Advanced References Winter, 28

5 Chemistry General Information A. Munster, Statistical Thermodynamics Volumes I and II (encyclopedic) Landau and Lifshitz, Statistical Physics R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics Hirschfelder, Curtiss and Bird, Molecular Theory of Gases and Liquids The Liquid State Hansen and MacDonald, Theory of Simple Liquids Frisch and Lebowitz, Classical Liquids Critical Phenomena S. K. Ma, Modern Theory of Critical Phenomena H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena Time Dependent Phenomena P. Resibois and M. DeLeener, Classical Kinetic Theory of Fluids D. H. Zubarev, Nonequilibrium Statistical Mechanics S. Chapman and T. Cowling, The Mathematical Theory of Nonuniform Gases D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions.2. Grading Scheme The grade in this course will be computed as follows: Note: Grading Scheme: CHEM 593 Homework % Midterm 4% Term Paper 5% McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offenses under the code of student conduct and disciplinary procedures (see for more information). In accord with McGill University s Charter of Students Rights, students in this course have the right to submit in English or in French any written work that is to be graded. In the event of extraordinary circumstances beyond the University s control, the content and/or evaluation scheme in this course is subject to change. Winter, 28

6 General Information -5- Chemistry Course Outline Lecture Outline: Chemistry 593, Winter, 28 Topic Lecture Quantum corrections & the classical limit Lecture 2 " Lecture 3 Introduction to dense gases Lecture 4 The 2nd virial coefficient Lecture 5 Introduction to reduced distribution functions Lecture 6 Reduced distribution functions and thermodynamics Lecture 7 " Lecture 8 " Lecture 9 Elastic scattering and the structure factor Lecture Theories of reduced distribution functions I. YBG and Kirkwood equations and the superposition approximation Lecture " Lecture 2 " Lecture 3 Theories of reduced distribution functions II: The direct correlation function, Ornstein-Zernike equation, and HNC, PY and MSA closures. Lecture 4 " Lecture 5 " Lecture 6 Monte Carlo methods Lecture 7 " Lecture 8 Perturbation theory of liquids Lecture 9 WCA vs BH (Canada/USA Round I). Lecture 2 Introduction to critical phenomena: Landau-Ginzberg and Weiss Mean field theories, critical exponents and universality. Lecture 2 " Lecture 22 " Lecture 23 Scaling and renormalization group: Survey of experimental results, the Rushbrooke inequality, Widom s scaling hypothesis, Kadanoff s renormalization group, fixed points and scaling, real space renormalization method. Lecture 24 " Lecture 25 " Lecture 26 " Lecture 27 Time dependent phenomena: Brownian motion and Langiven equations, classical and quantum Liouville Theorem, Langevin equations, linear response theory: susceptibilities and response functions, fluctuation dissipation theorem, projection operators and separation of time scales, hydrodynamic equations and Green-Kubo formulas. Lecture 28 " Lecture 29 " Lecture 3 " Lecture 3 " Lecture 32 " Lecture 33 Molecular dynamics methods Lecture 34 " Lecture 35 Inelastic light scattering: fluctuations and scattering, detection methods, optical filters, optical mixing: heterodyne and homodyne methods, the Ralyeigh-Brillouin spectrum. Lecture 36 " Lecture 37 " Lecture 38 " Lecture 39 " Winter, 28

7 Chemistry Stirling s Formula 2. Stirling s Formula There is a simple way to obtain an asymptotic expansion for the factorial, n!, for large arguments. First define a new function, called the gamma-function, as Γ(x + ) x! ds sx e s = x x+ ds exp{x[ln(s) s]}, (2.) where the last equality follows by making the change of variables s xs. The function Γ(x) is called the gamma function. It is easy to show, either by integrating by parts or by noting that Γ(x + ) = xγ(x), that x! = n! when x = n, aninteger. The integrand appearing in Eq. (2.) is shown in the following figure. Fig. 2.. The integrand in Eq. (2.). When x is large, the integrand is large and sharply peaked around the maximum. In this case, it is profitable to Taylor expand the integrand around the maximum of the argument to the exponential, i.e., about s = ineq. (2.). Hence, we write x[ln(s) s] = x x Σ ( )n (s ) n n n=2. (2.2) If we only keep the n = 2term in the sum in Eq. (2.2), it follows that the integrand in Eq. (2.) decays like agaussian centered at s =, with width x /2. Remembering that x is large, we can now extend the lower limit of integration in Eq. (2.) to with exponentially small error. Finally, wemake the change of variable s + /x /2 and rewrite Eq. (2.) as x! = x x+/2 e x d exp 2 Σ ( )n n 2 nx (n 2)/ 2. (2.3) n=3 Winter, 28

8 Stirling s Formula -7- Chemistry 593 The terms in the sum are at least as small as x /2,and hence, to leading order can be dropped. The resulting integral is easily done, and hence, x! (2π ) /2 x x+/2 e x, (2.4a) which implies that ln(x!) x ln(x) x + ln (2π x)/2. (2.4b) Clearly, for x O( 23 ), the last logarithm is negligible, and the simple form of Stirling s formula ln(x!) x[ln(x) ], is obtained. A comparison of the approximate and exact results is shown in Fig Fig Comparing Stirling s approximations to ln(x!). The inset shows smaller x. Finally, some of the numerical results used to make Fig. 2.2 are shown in the following table: x Table 2.: Testing Stirling s Approximation ln(x!) x[ln(x) ] x[ln(x) ] + ln[(2π x) /2 ] (exact) Value % Value % With a little extra effort you can work out the next term in the expansion, by Taylor expanding the exponential of the sum in Eq. (2.3) and keeping the next to leading order terms which Winter, 28

9 Chemistry Stirling s Formula give nonvanishing integrals. If you were to carry out this procedure to arbitrarily large order, would you expect the resulting series to converge? Why? Winter, 28

10 Ideal Bose & Fermi Gases -9- Chemistry Ideal Bose & Fermi Gases 3.. Introduction and General Considerations When we consider quantum statistical mechanics of identical particles, we abandon the conventional, classical, notion of states (i.e., as lists of quantum numbers assigned to the particles) and instead focus on "occupation numbers", n i,the number of particles in -particle state i (e.g., a single hydrogenic electron or a single particle in a box). Indistinguishably implies that the quantum probability density, ψ (, 2, 3, 4,...) 2,besymmetric under exchange of any particle labels, and, in turn, this means that ψ (, 2,...) =±ψ (2,,...). Quantum field theory and special relativity show that the + sign is used for integer spin particles, generically called Bosons, while the sign is used for half-odd integer spins, referred to as Fermions. Tw o see how this works, consider the following 2-particle example: each particle can be in one of two states, α, i >or β, i >, i =, 2, with energies ε α and ε β,respectively. Ifweignore the consequences of indistinguishability we can construct 4 possible states for this system as shown in the following table: Table 3.: Possible States and Energies for a Two Level System with Two Particles State Wavefunction Energy n α n β Particle α, > α,2 > 2ε α 2 Bosons 2 2 /2 ( α, > β,2 >+ β, > α,2 >) ε α + ε β Bosons 3 2 /2 ( α, > β,2 > β, > α,2 >) ε α + ε β Fermions 4 β, > β,2 > 2ε β 2 Bosons 2 e ε α /k B T + e ε β /k B T, for distinguishable particles (4 states) Q(N = 2, T, V ) = e (ε α +ε β )/k B T, for Fermions ( state) e 2ε α /k T B + e (ε α +ε β )/k T B + e 2ε β /k T B, for Bosons (3 states). For systems with more than two particles we simply symmetrize or anti-symmetrize the wavefunction (i.e, add all possible permutations of the particle labels together) for Bosons or use the so-called Slater determinant for Fermions. This shows that n i =,, 2,... for Bosons, while For Bosons this means Ψ(, 2,..., N) = Σ P[ψ a ()... ψ z (N)], Permutations where, P permutes the particle labels and where ψ a () is an orbital for one particle in -particle state a (e.g., for atoms, an atomic orbital). For Fermions we construct the so-called Slater determinant, i.e., (3.) ψ a () ψ b () Ψ(, 2,..., N) =... ψ z () ψ a (2) ψ b (2)... ψ z (2) ψ a (N) ψ b (N).... ψ z (N)

11 Chemistry Ideal Bose & Fermi Gases n i =, for Fermions, as required by the Pauli Exclusion Principle. In either case, the wavefunction is determined, up to an overall phase factor, byspecifying the n i s. Since E = Σi n iε i,itfollows that the canonical partition function is Q(N, V, T ) = or Σ or Σ... exp Σ ε i n i /k B T i, (3.2) n α = n β = iσ n i = N where the upper limits of the sums depend on whether Fermions or Bosons are under consideration. The sums aren t easy to do when constrained to have Σi n i = N. Since N fluctuates in the grand canonical ensemble, we expect that the constraint on N to disappear. Moreover, we ve shown that the choice of ensemble does not matter when calculating thermodynamic quantities. Hence, we switch to the grand canonical ensemble, with partition function Ξ= Σ e β µn N = or Σ or n = n 2 = iσ n i = N Σ... e β iσ ε in i = or Σ or Σ... e β n = n 2 = iσ (ε i µ)n i = Π or i Σ e β (µ ε i)n i n i = = Π ± i ± eβ (µ ε i), (3.3) where we ve gone back to our earlier notation of naming states by a roman index, β and µ have their usual meanings, and where the last equality follows by explicitly carrying out the sums for Fermions (upper signs) or Bosons (lower signs). Once we have the grand canonical partition function, thermodynamic functions are obtained in the usual manner, e.g., β pv = ln(ξ) =± Σ ln i ± eβ (µ ε i), (3.4) and N = ln(ξ) = β µ T,V where the average occupation numbers are Σ n i = i Σ i e β (ε i µ) ±, (3.5) E = ln(ξ) = β Σ n i ε i, (3.6) β µ,v i Note that neither expression is normalized. The Slater determinant is odd under exchange of any two rows and vanishes if two or more rows are the same, thereby having the required symmetry and Pauli exclusion principle. Winter, 28

12 Ideal Bose & Fermi Gases -- Chemistry 593 n i = e β (ε i µ) ±, (3.7) and are known as the Fermi-Dirac (+ sign) or Bose-Einstein (- sign) distributions. Finally consider the heat capacity. From Eqs. (3.6) and (3.7) it follows that C V = k B 2 Σ n j 2 e β (ε j µ) n i 2 e β (ε i µ) [β (ε i ε j )] 2 i, j Σ n j 2 e β (ε j µ) j, (3.8) where the details of calculation are given inappendix A. Note that C V is positive, asexpected. The quantity n i 2 exp[β (ε i µ)] plays a key role, and is shown in Figs. 3. and 3.2, for Fermions and Bosons, respectively. Anapproximate expression for Fermions for 4 n i 2 e x can be obtained by noting that the Taylor expansion of ln(4 n i 2 e x ) x 2 /4 + O(x 4 ), where x β (ε i µ); hence, 4 n i 2 e x e x2 /4 Fig. 3.. Mean occupation numbers, n i,and 4 n i 2 e β (ε µ) for Fermions. The spin degeneracy isnot included. Notice how the orbitals change from filled to empty over a range of β (ε µ) ± 5 or, for T = 3K, over a range (ε µ) ± 2. 5k B T =±. 65 ev. The plot of n i 2 β (ε µ) e shows that the main contributions to the heat capacity come from a narrow band of -particle states near the Fermi energy. Finally, the dotted brown line is the approximation to 4 n i 2 e β (ε µ) discussed in the text. Winter, 28

13 Chemistry Ideal Bose & Fermi Gases Fig Mean occupation numbers, n i,and n i 2 e β (ε µ) for Bosons. The spin degeneracy isnot included. Equation (3.8) leads to an approximate expression for C V. Appendix A shows a more rigorous approach Results for Particles in a Cubic Box In order to proceed we need a concrete model for the -particle states and energies. The simplest is probably the particle with mass m in a cubical box of side length L. This has energies: ε nx,n y, n z = h2 8mL 2 (n2 x + n 2 y + n 2 z), where n i =, 2, 3,.... (3.9) Note that the ground state energy, ε,, = 3h 2 /(8mL 2 ) as L. When the sum over single particle states (i.e., over positive n i, i = x, y, z,) is approximated as an integral and the result transformed to polar coordinates, we find that all the examples at the end of the previous section can be written as (2S + ) π 2 dn n2 f (λ, βε n ), (3.) where the factor of (2S + ) accounts for the spin degeneracy ofaparticle with spin S, the factor of π /2 = 4π /8 is the area of the unit sphere in the octant where n i >, i = x, y, z, and λ e β µ is the proper activity. Next, change variables by letting n = 8m/h 2 Lε /2 and rewrite Eq. (3.) as dε g(ε ) f (λ, βε), (3.) 2 See J.E. Mayer and M.G Mayer, Statistical Mechanics, (John Wiley &Sons, Inc., 94), Sec. 6g. Winter, 28

14 Ideal Bose & Fermi Gases -3- Chemistry 593 where g(ε ) 2π (2S + ) 3/2 2m V h 2 ε /2 (3.2) is known as the density of states and can be interpreted as the number of states per unit energy with energies between ε and ε + dε. With this, the general results given atthe end of the preceding section become β pv =± dε g(ε )ln ± λe βε, (3.3a) and N = dε g(ε ) λe βε, (3.3b) ± λe βε E = dε g(ε )ε λe βε. (3.3c) ± λe βε Note that Eqs. (3.3b) and (3.3c) can be obtained by taking the usual derivatives of β pv,i.e., ln(ξ), cf. Eq. (3.3a). Next we expand the integrands into Taylor series in λ, specifically, ± ln ± λe βε = + Σ ( + λ) j j j = e jβε (3.4a) and λe βε ± λe =± ln( ± λe βε ) = + βε β µ Σ ( + λ) j e jβε, β,v,n j = (3.4b) and change variables yet again by letting ε (k B T / j)x.this gives: β pv = + 2 π /2 V Λ (2S + ) 3 Σ ( + λ) j Γ(3 / 2), (3.5a) j 5/2 j = and N = + 2 π /2 E = + 2 π /2 V Λ (2S + ) 3 Σ ( + λ) j Γ(3 / 2), (3.5b) j 3/2 j = V Λ k BT (2S + ) Σ ( + λ) j Γ(5 / 3), (3.5c) 3 j 5/2 j = Winter, 28

15 Chemistry Ideal Bose & Fermi Gases where Λ h/ 2π mk B T is the thermal de Broglie wavelength, and have expressed the remaining integrals as Γ-functions, 3 specifically, Γ( 3) = π /2,for β pv and N, and Γ( 5) = 3 π /2 for E When these are used in Eqs. (3.5a) (3.5c) and the results rearranged, we find that ρλ 3 = (2S + )G 3/2 (λ) (3.6a) and where β pλ 3 = 2β E Λ3 3V = (2S + )G 5/2 (λ), (3.6b) G z (λ) Γ(z) dx x z λe x ± λe = λφ( x + λ, z,) = + Σ j z ( + λ) j, j= (3.6c) generalizes Hill s F z (α ), 4 the function Φ(a, b, c) isdiscussed in Gradshetyn and Ryzhik, 5 and ρ = N /V is the number density. The first equality in Eq. (3.6b) holds for the individual states for the particle in a box, i.e., p n = 2ε n /3V. These sums converge absolutely for λ < (µ<) and can be summed numerically, although the effort to compute the integrals numerically is comparable and must be used for Fermions for λ >. Some results are shown in Fig When λ = (i.e., µ = ), the sums in Eq. (3.6c) reduce to the Riemann zeta functions as shown in the following table: Table 3.2: Connections to the Riemann- Zeta Functions G z (): Formulas and Numerical Values z Fermions: ( 2 z )ζ (z) Bosons: ζ (z) 3/ / One of the ways of defining the Γ function is via its integral representation Γ(z) dx x z e x, which includes all of the remaining integrals. For more properties of these functions, see, e.g., the Handbook of Mathematical Functions, tenth edition, M. Abramowitz and I.S. Stegun, eds., ch T. L.Hill, An Introduction to Statistical Thermodynamics (Dover Publications), p I.S. Gradshetyn and I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press. New York, 98), Sec Winter, 28

16 Ideal Bose & Fermi Gases -5- Chemistry 593 Fig Proper activities, λ, and equations of state for ideal Boltzmann, Fermi-Dirac, and Bose-Einstein gasses. The spin degeneracy factor has not been included. Note that the figure was actually obtained by choosing a range of λ s,calculating ρλ 3,and transposing the plot. The first thing to notice is that the curves are universal, in that all ideal systems properties, once corrections for the spin degeneracy are made, will fall on the same curves. Also notice the positive (Fermions) and negative deviations (Bosons) from ideal gas behavior. For Fermions, these positive deviations arise because the Pauli exclusion principle which disallows two noninteracting particles to occupy the same state or space, and thus has an effect similar to steric repulsion). Bosons don t have to obey the exclusion principle and can have any number of particles in the same state, thereby acting like anattraction The Case For Boltzmann Statistics It turns out that there is a simple fix that allows us to treat the particles as if they were distinguishable. Consider aseparable Hamiltonian which has -particle energy levels, ε i = i, for i =, 2, 3,.... Let s examine the states for a system comprised of three identical particles, where HΨ(a, b, c) = 9Ψ(a, b, c) Winter, 28

17 Chemistry Ideal Bose & Fermi Gases Table 3.3: States where HΨ(a, b, c) = 9Ψ(a, b, c) -Particle States Nonzero n i s Degeneracy Ω FD Ω Distinguishable Ω BE (7,, ) n 7 = and n = 2 3 (6, 2, ) n 6 =, n 2 =, and n = 3! (5, 3, ) n 5 =, n 3 = and n = 3! (5, 2, 2) n 5 = and n 2 = 2 3 (4, 4, ) n 4 = 2and n = 3 (4, 3, 2) n 4 =, n 3 =, and n 2 = 3! (3, 3, 3) n 3 = 3 3! The degeneracy for Fermions, Ω FD,vanishes whenever the Pauli exclusion principle is violated, i.e., whenever n i >,asexpected. Since the wavefunction for Bosons can always be symmetrized, any choice of the n i s gives asingle wavefunction, and Ω BE =. Finally, the degeneracies for the distinguishable cases have < Ω Distinguishable N!, the equality holding when all the -particle states are different. This discussion can be summarized by noting that Ω FD Ω Distinguishable N! Ω BE, where equality holds when all the -particle states are different. If the number of available states is much larger than the number of particles, then most of the states included in the partition function obey the Pauli exclusion principle, and the distinguishable calculation over-counts these by the same factor of N!. Hence, a corrected partition function becomes Q Boltzmann = Q Distinguishable N!. This approach is known as Boltzmann statistics. Note that each component of a mixture contributes a N! factor. Returning to the particle in a box example, we now have and thus Q Boltzmann = (V Λ 3 ) N N! β A = log(q) = N log(v Λ 3 ) + N[log(N) ] = N[log(ρΛ 3 ) ], where we have used Stirling s formula and ρ N/V is the number density. 6 The Gibbs paradox is resolved! The ideal gas model makes it fairly straightforward to quantify the condition necessary for Boltzmann statistics to be used, namely, that the number of accessible states, N States,greatly exceeds the number of particles, N. As we have just shown, when this holds, the resolution of 6 Hill writes this as a single logarithm as N log(ρλ 3 /e). Winter, 28

18 Ideal Bose & Fermi Gases -7- Chemistry 593 Gibbs paradox is to simply divide the partition function, calculated as if the particles were distinguishable, by N!. The argument is simple and is essentially that presented in Hill. 7 To begin, note that the particle in a box quantum numbers, n i,i=x,y,z,which when plotted, form a three dimensional cubic lattice in the positive octant, each lattice point corresponding to a state. In terms of the lattice cells, each has unit volume and each corresponds to a single state (to be sure, there are eight lattice points per cell, but each is shared with 8 neighbors), Thus, if we ignore edge effects, the number of states with energies less than ε max is just the volume of of an eighth of a sphere with radius, n, less than the n corresponding to the ε max. Byusing Eq. (3.9), the cutoff radius is n = (8mε max /h 2 ) /2 L,and the corresponding number of states is simply N States 8 4π n 3 3 = 4π 3 2mε max h 2 3/2 V. By taking ε max = 3k B T /2 and requiring that N States >> N, wesee that ρλ 3 << 6 π /2 (3.7) in order to use Boltzmann statistics. Perhaps more physically, the number of particles per volume corresponding a size comparable to the thermal de Broglie wavelength is small, and hence, the number of particles within a thermal de Broglie wavelength, the scale at which quantum effects (e.g., wave-particle duality) become important, is small. Note that raising temperature and/or mass, and/or lowering the density favor the use of Boltzmann statistics. Fig The temperature dependence of the thermal de Broglie wavelength for various gases. Note that with the exception of the electron, Λ de Broglie is comparable to or smaller than the size of an atom at room temperature. 7 T.L. Hill, op. cit., Sec. 4.. Winter, 28

19 Chemistry Ideal Bose & Fermi Gases Some specific results are given inthe following table: 8 Table 3.4: Is Boltzmann Statistics Valid? Element (State) T(K) (π /6) /2 ρλ 3 He (l) 4.6 He (g) 4. He (g) He (g) Ne (l) Ne (g) Ne (g) 3. 6 Ar (l) Ar (g) Kr (l) Kr (g) electrons in metals (Na) Thus, we see that except for ultra-low temperatures and the lightest elements (both probably aren t of much interest to chemists), Boltzmann Statistics should be an excellent approximation. There is one major exception, namely electrons around room temperature, something that has major ramifications for bonding, etc Results Specific to Fermions or Bosons Fermions As we saw in the preceding section, cf. Fig. 3., the Fermi-Dirac distribution is basically flat for ε <µ F and falls off rapidly for ε >µ F,where µ F is the chemical potential, known as the Fermi energy to honor Enrico Fermi; hence, a reasonable approximation is to cut off the integrations at ε = µ F and let n i 2S + for ε <µ F. For example, by using Eqs. (3.3b) and (3.3c) it follows that N µ F dε g(ε ) = 4π (2S + ) 3 2mµ F h 2 3/2 V (3.8) or µ F h2 8m 6ρ π (2S + ) 2/3. (3.9) Similarly, 8 From, D.A. McQuarrie, Statistical Mechanics, (Harper and Row Pub., Inc., New York, NY, 973), Table 4-, p. 72. Winter, 28

20 Ideal Bose & Fermi Gases -9- Chemistry 593 E µ F dε g(ε )ε = 4π (2S + ) 5 3/2 2m V µ 5/2 h 2 F = 3 5 N µ F. (3.2) As the temperature is reduced, the approximations we ve just made become more accurate; hence, the simple results just obtained become more valid and represent the ground state configuration of the Fermi (ideal) gas. Finally, the approximations leading these results don t lend themselves well to quantities like like the heat capacity. Aswesaw above, C V,arises from a small band of energy levels where ε µ F ;these make C V nonzero and its calculation more complicated. By using Eq. (3.8) and the approximate expression for n i e β (ε i µ) it follows that C V k BT k B 8 dx g(k BTx + µ)e x2 /4 dy g(k BTy + µ)e y2 /4 (x y) 2 dx g(k BTx + µ)e x2 /4 k BTg(µ) 4 dx /4 x 2 = k e x2 B Tg(µ)π /2 = 2(2S + ) 3/2 2π mk B T V h 2 where the last expression was obtained by using Eq. (3.2). By using Eq. (3.8) to eliminate the volume in favor of N it follows that C V π /2 3k B T = 3τ N k B 2µ F 2π, /2 where τ π k B T /µ F is a reduced temperature. The more rigorous calculation given inthe Appendix, cf. Eq. (A3), gives πτ/2, i.e., a 46% error. Inany event, both calculations show that that C V T as T, which is consistent with the Third Law ofthermodynamics. Also note that k B T /µ F << ;e.g., if µ F = ev, and T = 3K then C V /k B Bosons The Bose gas thermodynamic properties can be obtained from the results given in Eqs. (3.3a c) or (3.4 7). In particular, from Eq. (3.3b), i.e., µ k B T /2, N = dε g(ε ) e β (ε µ), (3.2) noting that the particle in a box model has ε as L, itfollows that µ <or λ <so that the integrand in Eq. (3.2) be positive and integrable. However, from the data shown in Fig. 3.3, it follows that λ = for ρλ For 4 He, using the experimental density of liquid helium (. 45g/cm 3 )this occurs at T = 3. 4K. Experiment shows that there is a transition at 2.8K. (What contributes to the discrepancy?) In order to resolve this problem, note that the exact number of particles in the ground state (ε = ) in the grand canonical ensemble is Winter, 28

21 Chemistry Ideal Bose & Fermi Gases (2S + ) λ ( λ). (3.22) On the other hand, the factor of g(ε ) ε /2 in Eq. (3.2) shows that the ground state doesn t contribute at all. This makes sense if the states form a continuum; the probability of finding any exact state (not a narrow band of them) vanishes, and we must introduce probability densities. On the other hand, the ground state contribution to the partition function is the most divergent! The way out of this seeming paradox is to postulate what is known as the two fluid model. We write N 2S + = λ λ + 2π 2m 3/2V h 2 dε ε /2 e βε (3.23a) = λ λ + V ζ (3/ 2), (3.23b) Λ3 where the first term is the number of particles in the ground state, the so called Bose condensate, while the second accounts for those in excited states. Finally, ifweassume that µ, and expand λ in the denominator of the first term in Eq. (3.23b) we find that β µ Λ 3 V ρλ3 ζ (3 / 2), (3.24) where we have taken S = with ρλ 3 ζ (3 / 2). Thus, the number of particles in the ground state is λ N λ = N ζ (3/ 2) ρλ 3 = N T 3/2 T, (3.25) where T is the transition temperature (i.e., where ρλ 3 = ζ (3 / 2) = ). Notice that when T = T no particles are in the ground state, but this rises as temperature is lowered until % of them are in the ground state at absolute zero. Having macroscopic occupation of a single state drastically changes the physical properties of the system. Have a look at the demonstrations by Alfred Leitner for an experimental survey of some of the more striking changes that arise in superfluids or at for superconductors Photons and Black Body Radiation Photons can be wave-like or particle-like. The former means that each photon has a frequency, ω, a wave-vector, k, and amplitudes for the components of the photon s electric, E, and magnetic, H, fields. The amplitudes are arbitrary, except that the radiation must be transverse, i.e., k E =, while the wave-vector and frequency are related by the well known dispersion relation Winter, 28

22 Ideal Bose & Fermi Gases -2- Chemistry 593 ω = kc, (3.26) where c is the speed of light in vacuum. The condition of transversality means that radiation is made up of two independent polarizations, e.g., left and right circular polarization. From the selection rule for absorption and emission, we know that the photon behaves as a spin- particle, modulo transversality, since a molecule s angular momentum changes by l =±h on absorption or emission in all linear spectroscopies. Hence, photons are Bosons. Finally, the energy of the photon is just ε = h ω. Note that the number of photons of a given frequency isn t conserved (unlike the atoms in achemical reaction) and, in particular, processes like M photons at frequency ω N photons at frequency ω (3.27) are possible. At equilibrium, the free energy is a minimum, and the usual argument requires that (N M)µ Photon =, or µ Photon =. With this introduction, consider the radiation in a closed cubical container of side length L. The radiation in the box comes to equilibrium with the walls and forms standing waves in different directions and having different frequencies. If we assume that the electric field vanishes at the walls and consider the field as sin-like (just like the particle in a box) it follows that k = π /L(n x, n y, n z ) T,where n i =, 2, 3,... Byrepeating the steps used above, itfollows that n(ω )the density of radiation between frequencies ω and ω + dω is n(ω ) = V ω 2 π 2 c 3 e β h ω, (3.28) where remember that the photon energy states are doubly degenerate owing to the two polarization possibilities. The main differences between this and our earlier result are caused by the fact that ε = h ω = h kc. Bymultiplying this result by the photon energy, h ω,gives E(ω ) = V h ω 3 π 2 c 3 e β h ω, (3.29) where E(ω )dω is the energy found in the frequency interval ω to ω + dω and is known as Planck s formula. Limiting behaviors are easy to find; specifically, V k BTω 2 for h π E(ω ) 2 c 3 ω << k B T V h π 2 c 3 ω 3 e β h ω when h ω >> k B T. (3.3) The low frequency (or high temperature) behavior is known as the Raleigh-Jeans radiation law, and was known before the discovery of quantum mechanics. One of the problems with the classical theory, according to the Raleigh-Jeans formula, is that the total radiation energy in the box diverges; this is known as the ultraviolet catastrophe, and posed a very difficult problem for classical mechanics and electrodynamics. Winter, 28

23 Chemistry Ideal Bose & Fermi Gases By integrating the full result, we find that E total V = h π 2 c 3 dω ω 3 e β h ω = (k BT ) 4 (ch ) 3 π 2 dx x 3 e x = 6ζ (4)(k BT ) 4 (ch ) 3 π 2, (3.3) where the last two results were obtained by letting ω = k B T /h x. The T 4 dependence was deduced classically using thermodynamics by Stephan and Wein. Wein actually proved something more general called Wein s Law; namely, E(ω )/V = ω 3 f (ω /T ), where f is not known. Note that Wein s Law is obeyed by the Planck formula Appendix A: C V in an Ideal Bose or Fermi Gas The mathematical details leading to the constant volume heat capacity, C V,cf. Eq. (3.8), are now presented. First, from thermodynamics, recall that C V E = E + E β µ T N,V T β µ,v β µ β,v T N,V = E E T β µ,v β µ β,v N T β µ,v, (A) N β µ β,v where the second equality follows when we consider E to be a function of T or β, β µ, and volume, V,while the last equality is obtained when we use the cyclic rule (implicit function differentiation). Next we use Eqs. (3.5) (3.7) to evaluate the derivatives that appear in Eq. (A); i.e., E = T β µ,v k B T 2 Σ n i 2 e β (ε i µ) ε i 2, i E = β µ Σ n i 2 e β (ε i µ) ε i, β,v i N = T β µ,v k B T 2 Σ n i 2 e β (ε i µ) ε i, i N = β µ β,v Σ n i 2 e β (ε i µ), i which when used in Eq. (A) gives C V = k B T 2 Σ n i 2 e β (ε i µ) ε i 2 ε i i Σ n j 2 e β (ε j µ) ε j j Σ n j 2 e β (ε j µ) j = i, j Σ n i 2 e β (ε i µ) n j 2 e β (ε j µ) ε i (ε i ε j ) k B T 2 Σ n j 2 e β (ε j µ) j. Finally, we take half the double sum in the numerator of Eq. (A2), exchange the dummy sum indices, i j and add it to the unmodified half, resulting in Eq. (3.8). Note that there are (A2) Winter, 28

24 Ideal Bose & Fermi Gases -23- Chemistry 593 alternate ways of writing Eq. (A2). For example, n i 2 e β (ε i µ) = sech( β (ε i µ) / 2) 4 csch( β (ε i µ) / 2) for Fermions for Bosons. (A3) An alternate route to C V,istoeliminate µ in favor of N in the energy. From Eq. (3.8) we see that ρ = D dε ε /2 e β (ε µ) +, where ρ N /V and D 2π (2S + )(2m/h 2 ) 3/2. Byintegrating by parts, we can rewrite Eq. (A4) as ρ = 2D 3 ε 3/2 e β (ε µ) + ) + 2D 3 β dε ε 3/2 β (ε µ) e [(e β (ε µ) + ], 2 (A4) (A5a) = 2Dµ3/2 3 dx β µ + k BTx µ 3/2 e x (e x + ) 2, (A5b) where the boundary term in Eq. (A5a) vanishes and where we have changed variables to x β (ε µ) to get Eq. (A5b). By using the Taylor expansion ( + x) α = + Σ α (α )...(α n + ) n! n= x n, (A6) we see that where ρ 2Dµ3/2 3 + Σ 3 n= n k B T µ n Z n, (A7) Z n n! dx x n e x (e x + ) 2 = for n odd 2( 2 n )ζ (n), for n even, where ζ (n) isthe Riemann zeta function. 9 Note that the odd n terms vanish because e x /(e x + ) 2 is even in x. In obtaining Eq. (A7) we ve extended the lower integration limit to. Onone hand, this only introduces exponentially small errors, since β µ dx xn e x... is exponentially small when β µ>>;nonetheless, while this makes sense for each term in the Taylor expansion, the sum 9 See I.S. Gradshetyn and I.M. Ryzhik, op. cit., Eq. (3.4.3), p (A8) Winter, 28

25 Chemistry Ideal Bose & Fermi Gases must (and does) diverge since ( + k B Tx/µ) 3/2 is imaginary when x < β µ. Our approach generates what is known as an asymptotic expansion. If you plot the individual terms, they will decrease until some minimum value is obtained, and increase thereafter causing the series to diverge. The trick is to sum no further. Another example of this is the Stirling formula for n!. let The goal of this calculation is to obtain a low temperature expansion for µ; accordingly, we µ µ ( + c τ + c 2 τ ), (A9) where µ h 2 (6ρ/π (2S + )) 2/3 /8m is the chemical potential at absolute zero (cf. Eq. (A9)) and where τ π k B T /µ is a reduced temperature. When Eq. (A9) is substituted into Eq. (A7),the result expanded into a series in τ,and the coefficients of τ n, n >,set to zero, it follows that µ = µ 2 τ 2 8 τ τ τ τ (A) By starting with the general expression for the energy, i.e., E = DV dε ε 3/2 e β (ε µ) +, and repeating the steps that led from Eq. (A4) to (A7), it follows that E = 2DV µ5/2 5 + Σ 5 n= 2 Finally, byusing Eq. (A) to eliminate µ it follows that n k B T µ n Z n. (A) which implies that E = 3 5 N µ τ 2 6 τ τ τ τ +..., (A2) C V = πτ N k B 2 3 τ τ τ τ (A3) Note that C V T as T, which is consistent with the 3rd Law ofthermodynamics. The first correction is O(T 3 ), just like the Debye-T 3 law for vibrations. That being the case, how can we distinguish between vibrations and electronic contributions? The algebra becomes horrendous if you want to go much beyond the first correction; nonetheless, it s easy for a symbolic algebra program, e.g., Mathematica or Maxima ( to do so. Maxima was used here. Winter, 28

26 Ideal Bose & Fermi Gases -25- Chemistry 593 Fig The low temperature, constant volume, heat capacity for a system of ideal Fermions in a box as a function of the reduced temperature, τ, introduced in the text.. The different curves correspond to the number of terms kept in Eq. (A3). As expected, all the curves reduce to the linear one as τ. This doesn t happen at higher temperatures, and moreover, it is obvious that something is breaking down since C V >. Winter, 28

27 Chemistry The Semi-Classical Limit 4. The Semi-Classical Limit 4.. The Semi-Classical Limit: Quantum Corrections Here we will work out what the leading order contribution to the canonical partition function in the so-called Boltzmann Statistics approximation to account for quantum exchange symmetries. For concreteness, and in order to not make a cumbersome notation even worse, we ll consider an interacting set of structureless particles. Our starting point is a general, albeit formal, expression for the canonical partition function, i.e., Q = Tr e β H, (4.) where H is the Hamiltonian and β /k B T. The assumption that the particles are structureless allows us to consider a Hamiltonian of the form H h 2 2m N N + V (r N ). (4.2) The trace is invariant under unitary transformations and so we can use any representation in evaluating it; here we will use the momentum eigenfunctions as a basis. That is where p N are the momentum eigenvalues. It follows that Q = ψ p N (r N ) h 3N eipn r N /h, (4.3) h 3N N! dx N e ipn r N /h e β H e ipn r N /h, (4.4) where dx N dp...dp N dr...dr N and the factor of N! approximately accounts for the exchange symmetries in what is known as Boltzmann Statistics. The calculation simplifies slightly if we consider the β Laplace transform of Q; i.e., or, byusing Eq. (4.4), Q(z) d β e β z Q(β ), (4.5) where h 3N N! Q(z) = dx N e ipn r N /h A B eipn r N /h, (4.6) * Our approach follows that of J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, NY, 976) Sec. 6.. Winter, 28

28 The Semi-Classical Limit -27- Chemistry 593 A z + H classical (4.7a) and B 2m h 2 N N + p N p N. (4.7b) The classical Hamiltonian H classical p N p N /2m + V (r N )and B is the difference between the classical and quantum mechanical kinetic energies. Clearly B and must be small if a classical description is justifiable. The operator /(A B) isknown as the resolvent operator and is easily shown to obey the following equality A B = A + A B A B (4.8a) = A + A B A + A B A B A +... (4.8b) where the second equality is obtained by iterating the first and is nothing more than an operator generalization of the geometric series. When Eq. (4.8b) is used in Eq. (4.6) we see that the first term results in the Laplace transform of the classical partition function, and thus, h 3N N![ Q(z) Q classical (z)] = dx N e ipn r N /h By using Eqs. (4.7a) and (4.7b) we see that A B A + A B A B A +... eipn r N /h. (4.9) A B A eipn r N /h = eipn rn /h A 2m h 2 N N A + 2ih p N N A (4.) Note that we got both st and 2nd (in h )order terms. The terms O(h )cancel! When this result is used in the first term in Eq. (4.9) we obtain 2 h 2 2m dx N N, (4.) A where we have integrated by parts to move one of the N s to the left. The purely imaginary terms linear in p N vanish when integrated since the integrand is odd in the momenta. (Remember that the partition function is real and so we expect this to happen more generally). Unfortunately, we re not quite done, since the B 2 terms in Eq. (4.9) also contribute at O(h 2 ). By noting that B is Hermitian, it follows that dx N e ipn r N /h A B A B A eipn r N /h = dx N A B 2 r N A eipn Winter, 28

29 Chemistry The Semi-Classical Limit By combining these terms we find that = h 2 m dx N 2 A pn N A h 3N N![ Q(z) Q classical (z)] = h 2 2m dx N N A O(h 3 ). (4.2) 2 ma pn N + O(h A 2 4 ). (4.3) We can now inv ert the Laplace transforms by making use of the convolution theorem, which states that the Laplace transform of a convolution integral, f * g τ f (t τ )g(τ ) is f * g = f g. Thus, returning to Eq. (4.3), we see that h 3N N![Q Q classical ] = h 2 2 m β 2m dx N β d β e β H (β β )β F N F N 2 d β 2 pn F N (β β )(β β 2 ) + O(h 4 ), where F N N H is force acting on the particles. The various β integrals are easy to do and we find that h 3N N![Q Q classical ] = h 2 β 3 2m dx N e β H F N F N β 2m (pn F N ) 2. (4.4) Since momentum and coordinates are separable classically, and moreover the momentum distribution is just the Maxwell-Boltzmann distribution, we can carry out all the momentum integrations and find that and finally, where Q Q classical 2π mk B T 3N/2 h 2 β 3 h 2 24mN! drn e βv F N F N, (4.5) Q Q classical h 2 β 3 24m FN F N, (4.6) Winter, 28

30 The Semi-Classical Limit -29- Chemistry 593 Q classical = 2π mk B T 3N/2 Z N h 2 N! (4.7) with which is known as the configurational partition function. Z N drn e βv (4.8) This result is strange. If the particles are identical we can write F N F N = Σ F j F j = N F F, (4.9) j where F is the total force exerted on particle. The correction is huge, unless F F O(N )which seems unlikely. Fortunately, the explicit factor of N is just what is needed to make the extensive thermodynamic functions extensive. Tosee this, remember that the Helmholtz Free Energy, A, isrelated to the partition function as A = k B T ln(q), which from Eq. (4.6), can be written as A = A classical k B T ln N h 2 β 3 24m F F + O(h 4 ) (4.2) The expansion we just carried out above assumes that h is small parameter and has O(h 4 )corrections that were dropped. The logarithm also has similar corrections; specifically, by noting that the Taylor expansion of ln( x) is ln( x) = x x xn n..., which when applied to Eq. (4.2) shows that the x 2 term is O(h 4 ). Given that we ve already dropped O(h 4 )terms in arriving at Eq. (4.2), consistency requires that we not keep them in the logarithm. Thus, β (A A classical ) N h 2 β 3 24m F F + O(h 4 ) Λ2 β 2 48π F F + O(h 4 ), (4.2) where the last expression arises when we note that h 2 =Λ 2 mk B T /2π,where Λ is the thermal de Broglie wavelength. Of course, what remains to be shown is that the O(h 4 )terms that arise both from the expansion of the trace and the logarithm have enough cancellations to give a result that is O(N). This is the case, as we will see later in a more general context. We can make several observations about our result. First, note that the leading order free energy correction is positive, no matter what the interaction potential. Second, Eq. (4.2) can be rewritten as β (A A classical ) N Λ2 β 48π drn Z N F e βv (rn ) = Λ2 β 48π drn e βv (rn ) Z N F, (4.22) Winter, 28

31 Chemistry The Semi-Classical Limit where the last equality is obtained by integrating by parts. To go further, we ll assume that the interactions are pairwise additive; i.e., F i = Σ F i, j (r i, j ), (4.23) j i where r i, j r i r j and F i, j is the force particle j exerts on i. With this, given that the particles are identical, we can rewrite Eq. (4.22) as β (A A classical ) N (N ) Λ2 β 48π dr F,2 dr 2 r,2 dr... e βv (rn ) 3 dr N, (4.24) Z N where the factor of (N ) is just the number of ways of choosing the particle interacting with particle. By introducing the so-called generic reduced pair correlation function, ρ (2) (r, r 2 )and pair correlation function, g (2) (r, r 2 ), i.e., ρ (2) (r, r 2 ) ρ 2 g (2) (r 2 ) N(N ) dr 3... dr N e βv (rn ) where ρ N/V is the number density, wecan rewrite Eq. (4.24) as Z N, (4.25) β (A A classical ) N Λ2 ρ 2 48π N dr F,2 dr 2 r,2 g(2) (r,2 ) = Λ2 βρ 48π dr,2 F,2 r,2 g(2) (r,2 ), (4.26) where the last equality is obtained by going to a coordinate system centered on r 2 for the r integration and assuming that the system is translationally invariant. The subsequent integration in r 2 gives afactor of volume. Finally, bywriting F,2 (r,2 ) =,2 u,2 (r,2 ), where u,2 (r,2 )isthe pair potential and only depends on the distance between particles and 2, Eq.(26) can be rewritten as β (A A classical ) N Λ2 βρ 48π dr,2 [ 2,2u,2 (r,2 )]g (2) (r,2 ) (4.27a) Λ2 βρ 2 dr r2 2 u,2 (r) r r u,2 (r) r g (2) (r),(4.27b) where 2,2 is the Laplacian for r,2. The 2nd equality is obtained by changing the integration and Laplacian to polar coordinates and letting r,2 r. The energies that matter are those O(k B T ); hence, a crude estimate of size of our correction is Λ 2 ρσ = ρλ 3 σ /Λ,where σ is the molecular interaction length scale. The use of Boltzmann Statistics already requires that ρλ 3 <<,and thus, unless σ /Λ >>,the semi-classical Winter, 28

32 The Semi-Classical Limit -3- Chemistry 593 approximation should be valid. The proceeding estimates are crude. Hansen and Weis * performed monte carlo simulations for a model of neon near the triple point and showed that the corrections are about % of the Boltzmann statistic s classical result. * J.P. Hansen and J.J Weis, Phys. Rev. 88, 34 (969). Winter, 28

33 Chemistry Dense Gases: Virial Coefficients 5. Dense Gases: Virial Coefficients One way to correct the ideal gas equation of states is to use the virial expansion, namely, β p = ρ + B 2 (T )ρ 2 + B 3 (T )ρ , (5.) where ρ N/V is the number density and where B n (T )isknown as the n th virial coefficient. Our goal is to obtain a molecular expression for the virial coefficients (at least for B 2 (T )). To proceed, we need to identify a useful small parameter that can be used to expand the equation of state. To do this, note that in Boltzmann statistics and ignoring all interactions between molecules, the canonical partition function for a system of N molecules, Q N (V, T ) = [ q (T )V ] N /N!, where V q (T )isthe molecular partition function. Hence, the grand partition function becomes Ξ= Σ eβ µn [V q (T )] N N! N= = exp λv q (T ), (5.2) where λ e β µ,isthe activity. Bytaking the appropriate derivatives oflog(ξ), it follows that ρ N V = V log(ξ) = λ β µ β,v V log(ξ) = λ q λ (T ), (5.3) β,v where the last equality is obtained when Eq. (5.2) is used. The ideal gas equation of state arises since β p = V log(ξ) = λ q (T ) = ρ. Thus, we see that λ ρ as ρ, where ideal gas behavior is expected. Since, in general, Ξ= + Σ λ N Q N (V, T ), (5.4) N= we can view the grand partition function as a power series in the small parameter λ, atleast for low densities. Unfortunately, λ is not a particularly convenient quantity to control experimentally (unlike the density). To deal with this, assume that the activity can be written as a power series in density; i.e., From Eqs. (5.3) and (5.4) it follows that λ = ρc + ρ 2 c 2 + ρ 3 c (5.5) ρv = Σ N λ N Q N (T, V ) N= + Σ, (5.6) λ N Q N (T, V ) N= or equivalently, Winter, 28

34 Dense Gases: Virial Coefficients -33- Chemistry 593 ρv + Σ λ N Q N (T, V ) = Σ N λ N Q N (T, V ). (5.7) N= Equation (5.5) is used to rewrite Eq. (5.7) as ρv + Σ ρ N+ (c + ρc 2 + ρ 2 c ) N VQ N (T, V ) = Σ N ρ N (c + ρc 2 + ρ 2 c ) N Q N (T, V ), N= N= N= and similar to what is done in the Frobenius method for ODE s, we equate the coefficients of each power of ρ; i.e., (5.8) V = c Q for O(ρ), (5.9a) c VQ = c 2 Q + 2c 2 Q 2 for O(ρ 2 ), (5.9b) c 2 VQ + c 2 VQ 2 = c 3 Q + 4c c 2 Q 2 + 3c 3 Q 3 for O(ρ 3 ), (5.9c) etc.. These equations are easily solved, giving c = V Q, (5.a) c 2 = (2Q 2 Q 2 )V 2 Q 3, (5.b) c 3 = (3Q Q 3 8Q Q 2 Q 2 Q 4 )V 3 Q 5, (5.c) etc.. By substituting Eqs. (5.a-c) into Eq. (5.5) we see that λ = ρv Q (2Q 2 Q 2 )ρv Q 2 (3Q Q 3 8Q Q 2 Q 2 Q 4 )ρ 2 V 2 Q , (5.) which gives β µ = log(λ) = log ρv Q 2Q 2 Q 2 V ρ + Q 2 Q4 6Q 2 Q 2 6Q 3 Q + 2Q 2 2 V 2 ρ 2 2Q , (5.2) expressing the non-ideal corrections to the Gibbs free energy (recall that µ = G in a one component system) in terms of canonical partition functions for systems of N = 2, 3,... particles; in addition, we ve made no assumptions about using classical or quantum mechanics. Similarly, Winter, 28

35 Chemistry Dense Gases: Virial Coefficients remembering that β pv = log Ξ, we see that β p = ρ (2Q 2 Q 2 )V ρ 2 2Q 2 (6Q Q 3 2Q Q 2 Q 2 Q 4 )V 2 ρ 3 3Q (5.3) Note that in obtaining Eqs. (5.2) and (5.3) we have had to re-expand the logs in order to be consistent with the omitted terms. * Our results can be simplified if we assume that the internal electronic, vibrational, and rotational degrees of freedom are separable, and that the translational ones are classical. Thus Q N q N Z N N!, where, as above, q is the molecular partition function for all intra-molecular degrees of freedom, including translation, divided by volume, and Z N dr... dr N e βu(n) (r,...,r N ), (5.4) where U (N) (r,...,r N )isthe interaction potential for a system of N molecules and Z = V. Z N is known as the configurational partition function. Note that the separability assumption is reasonable for electronic (in the absence of chemical reactions) and for vibrational degrees of freedom, but is problematic for rotations if the molecule is aspherical. Fortunately, the rotational degrees of freedom can be treated classically for most of the cases of interest, and can be incorporated into the definition of Z N,leaving only the rotational kinetic energy contributions in q (this is analogous to what we do with the translational contributions in q or with the rotational partition function when T >> Θ rot ). where With this change of notation, it follows that β µ = log ρ q Σ β n ρ n, (5.5) n= β Z 2 V 2 V, (5.6a) * Specifically, weuse the Taylor expansion log( + x) = x 2 x2 + 3 x3 +...= Σ ( )n+ n n= x n, where x λ/(ρv /Q ), cf. Eq. (). Thus, e.g., if we only keep terms to O(ρ) in λ we should only keep terms through the same order in the series for the logarithm; i.e., β µ = log(ρv /Q ) (2Q 2 Q 2 )V ρ/q 2 + O(ρ 2 ). Winter, 28

36 Dense Gases: Virial Coefficients -35- Chemistry 593 β 2 Z 3V + 3Z 2 V 2 V 4 3Z 2 2 2V 2, (5.6b) etc.. The β n s are referred to as n th order irreducible cluster integrals. By expressing the density expansion of the pressure in terms of the configurational partition functions and the irreducible cluster integrals, we see that Hence, B 2 (T ) = β /2, B 3 (T ) = 2β 2 /3. β p = ρ β 2 ρ2 2β 2 3 ρ (5.7) A general connection between the irreducible cluster integrals and the virial coefficients can easily be obtained from the Gibbs-Duhem relation, SdT Vdp + Ndµ =, specifically, β p = ρ β µ = ρ T ρ Σ nβ n ρ n, (8) T where the second equality is obtained when (5.5) is used. By integrating, remembering that p = when ρ =, we see that β p = ρ d ρ p = ρ ρ Σ T n= n= n n + β n ρ n+, (9) and thus, B n+ = n n + β n. (5.2) It turns out that even though our definitions of β n explicitly contain the volume V, the volume dependence disappears. For example, consider the second virial coefficient, B 2 (T ). According to Eqs. (5.6a) and (5.2), B 2 (T ) = Z 2 V 2 2V = 2V dr dr 2 e β u 2(r 2 ), (5.2) where u 2 (r 2 )isthe interaction potential of a pair of molecules separated by a distance r 2. For neutral molecules, the interaction rapidly decays to zero once r 2 is much larger than a few tens of angstroms, as will the integrand in Eq. (5.2); hence, it is useful to change one of the integration variables, say r to a basis centered on r 2.This allows Eq. (5.2) to be rewritten as B 2 (T ) = 2V dr 2 dr 2 e β u 2(r 2 ) = 2 dr 2 e β u 2(r 2 ). (5.22a) Similar manipulations can be used in Eq. (5.6b) to show that B 3 only depends on temperature. More generally, there are some sophisticated mathematical tools that prove this in general, but they are well beyond the level ofthis class. * * See, e.g., J.P. Hansen and I.R. MacDonald, Theory of Simple Liquids, (Academic Press, London, 976), ch. 4. Winter, 28

37 Chemistry Dense Gases: Virial Coefficients For spherically symmetric interactions we can switch to polar coordinates, giving B 2 (T ) = 2π dr 2 r 2 2 e β u 2(r 2 ). (5.22b) If the pair interactions are predominantly repulsive, i.e., u 2 (r 2 )>,the quantity in the parenthesis will be positive, aswill B 2 (T )and the deviation from ideal gas behavior will be positive. If the interactions are predominantly attractive, the reverse will happen. Consider a more realistic potential where steric repulsions dominate at short distances and attractions dominate a larger distances, e.g., as for the so called square well potential for r < σ u 2 (r) = ε for σ < r < γσ otherwise, where ε and γ. The potential is shown in Fig. 5.. Fig. 5.. The square well (solid) and more realistic Lennard-Jones (L- J) 6-2 (dashed) potentials, respectively. The L-J potential, u(r) = 4ε [(σ /r) 2 (σ /r) 6 ], is more realistic, in that it includes the effects of London dispersion forces at larger distances. The integral in Eq. (5.22b) is easily done and gives B 2 (T ) = 2πσ3 3 (γ 3 )(e βε ). (5.23) From this it is easy to show that the Boyle temperature, where B 2 (T B ) =, is T B = ε k B log( γ 3 ). The van der Waals model gives B 2 (T ) = b β a, where a and b are the van der Waals a and b coefficients. Both models show that the effects of steric repulsion and attraction are additive. Winter, 28

38 Dense Gases: Virial Coefficients -37- Chemistry 593 Other than that, the temperature dependence of the attractive terms differ slightly, although they approach each other when βε where e βε βε. The integration in Eq. (5.22b) for the more realistic Lennard-Jones 6-2 potential is more complicated and is given inthe Appendix. Fig The reduced second virial coefficient for the Lennard-Jones potential. B * 2 B 2 /B 2, hard sphere,where B 2, hard sphere 2πσ 3 /3, isthe second virial coefficient for the hard-sphere potential, and where the reduced temperature is T * k B T /ε. Asexpected B * 2 >(<) for high (low) reduced temperatures, where repulsions (attractions) dominate. The reduced Boyle temperature is approximately Note that B * 2 (T * ) /4 as T * and is not constant (why?). The integral in our expression for B 2 (T ), cf. Eq. (5.22b), imposes some important limitations on the interaction potential, u 2 (r); specifically, for large separations u 2 (r), and thus r 2 { exp[ β u 2 (r)]} r 2 β u 2 (r) showing that u 2 (r) must decay faster than /r 3 in order that the integral converges. Charge-charge, charge-dipole, and dipole-dipole interactions decay like r, r 2,and r 3,respectively! Charge-dipole and dipole-dipole interactions depend on the orientation of the dipole moments (i.e., angular degrees of freedom), and when these are included properly the integral converges. Ionic interactions are more problematic; in short, the system must be electrically neutral, and even then, the basic assumption about the nature of the virial expansion (a power series in density) breaks down, giving rise to ρ /2 terms. 5.. Appendix: B 2 (T ) for the Lennard-Jones potential This section is based on the extensive discussion of the viral expansion by Hirshfelder, Curtiss, and Bird * First, express B 2 in a dimensionless form by writing * J.O. Hirshfelder, C.F. Curtiss, and R.P. Bird, Molecular Theory of Gases and Liquids, 2nd ed. (John Wiley and Sons Inc, N.Y., 964), ch. 3. Winter, 28

39 Chemistry Dense Gases: Virial Coefficients B 2 (T ) = 2πσ3 3 B * 2(T * ), (5.24) where we have let r σ r, T * k B T /ε is the reduced temperature and where the reduced 2nd virial coefficient is B * 2(T * ) 3 dr r2 exp 4 T * (r 2 r 6 ), (5.25a) = 24 T * dr r2 (2r 2 r 6 )exp 4 T * (r 2 r 6 ), (5.25b) = 24 T * dr r2 (2r 2 r 6 )exp 4 T * r 2 Σ n! n= 4r 6 n,(5.25c) T * where the second equality is obtained by integration by parts and where the last expression is obtained by expanding exp(4r 6 /T * )inataylor series (i.e., we treat the attractions in the Boltzmann factor as a perturbation). Next we change variables by letting thereby implying that z 4 T * r 2 or r = T * /2 4 z, (5.26) dr = T * 4 /2 z 3 / 2 2 dz. (5.27) When these are used in Eq. (5.25c), the latter becomes B * 2(T * ) = Σ 2n+/2 (T * ) (2n+3) / 4 n! dz z(2n 3) / 4 ( z T * )e z (5.28a) n= = Σ 2n+/2 (T * ) (2n+3) / 4 T n= n! * Γ 2n Γ 2n + 4 (5.28b) Σ = n= 2 n 3/2 Γ 2n 4 n! T * (2n+) / 4, (5.28c) where Winter, 28

40 Dense Gases: Virial Coefficients -39- Chemistry 593 Γ(z) = dt t z e t, (5.29) is known as the Γ-function. The last equality was obtained by noting that Γ(x + ) = xγ(x) and redefining the sum index. Winter, 28

41 Chemistry Correlation Functions and the Pressure 6. Correlation Functions and the Pressure In order to calculate the pressure for liquids it is tempting to imagine that we could simply resum the virial expansion in some way; unfortunately, this doesn t work and an alternate approach is required. We ll show two methods here. 6.. The Virial Form of The Pressure In the canonical ensemble, the pressure can be determined from the partition as β p = ln(q) = ln(z N ), (6.) V N,β V N,β where we are using classical statistical mechanics, with Q = Z N /(N!Λ 3N ), Z N is the configurational partition function, and Λ=h/(2π mk B T ) /2 is the thermal de Broglie wavelength. The volume only appears in the integration limits, and when differentiated, pins the position of one of the particles to one of the faces of the cubical box. Although the result is correct, it is not what we want, namely, anexpression where the walls of the container do not play an explicit role. An alternative expression can be found by first changing the integration variables to explicitly scale the particle coordinates by the size of the box, thereby removing the volume from the limits of integration, i.e., by letting r N V /3 r N,*.With this, we see that When this is used in Eq. (6.) we find that Z N = V N dx*... dz* N e βu(rn,* V /3). (6.2) β p = ρ + βv N Z N 3 dx*... dz* N F N r N e βu(rn ) (6.3a) = ρ + β 3V FN r N = ρ + β N 3V F r (6.3b) = ρ + βρ2 3V dr dr 2 F,2 r g (2) (r, r 2 ), (6.3c) where ρ = N/V,and where the last equality assumes a pairwise additive potential. Newton s action-reaction law demands that F,2 = F 2.,thereby allowing us to switch the integration variables in Eq. (6.3c) in half the integral, which becomes β p = ρ + βρ2 6 dr,2 F,2 r,2 g (2) (r,2 ). (6.4) This result is known as the virial form of the pressure (not to be confused with the virial expansion). Winter, 28

42 Correlation Functions and the Pressure -4- Chemistry The Compressibility Form of the Pressure There is an alternate route to the pressure based on the reduced generic distribution functions in the grand canonical ensemble. First, note that ρ (m) GC (r,...,r m ) Σ P(N)ρ (m) C (r,...,r m ), (6.5) N where P(N) = e β µn Q(N, V, T )/Ξ is the probability of having a system with N particles in the grand canonical ensemble, and ρ (m) C is the generic reduced distribution function in the canonical ensemble with N particles. In general, dr... dr m ρ (m) C (r,...,r m ) = N! (N m)! (6.6a) and dr... dr m ρ (m) GC (r,...,r m ) =< N! (N m)! >GC. (6.6b) When m = 2, remembering that ρ (2) GC (r, r 2 ) ρ 2 g (2) GC (r, r 2 ), we rewrite Eq. (6.6b) as ρ 2 dr dr 2 g (2) GC (r, r 2 ) = < N(N ) > GC = < N 2 > GC < N > GC. (6.7) By adding and subtracting to the integrand and rearranging the result, Eq. (6.7) becomes < N > +ρ 2 dr dr 2 g(2) (r, r 2 ) = < N 2 > < N > 2, (6.8) where the GC labels have been dropped; henceforth, all averages and correlation functions are grand canonical. The right hand side of the equation is the variance of the number distribution in the grand canonical ensemble; i.e., <(N < N >) 2 > = < N 2 > < N > 2 = ρk B Tκ < N >, where κ V ( V / P) N,T is the isothermal compressibility. Thus, Eq. (6.8) becomes + ρ dr,2 By noting that κ = ρ ( ρ/ P) T,wecan rewrite Eq. (6.9) as g(2) (r,2 ) = ρk BTκ, (6.9) + ρ dr,2 g(2) (r,2 ) = ρ, β P β (6.a) or equivalently, Winter, 28

43 Chemistry Correlation Functions and the Pressure β P = ρ T + ρ dr,2 g(2) (r,2 ). (6.b) This is known as the compressibility form of the equation of state (it is easily integrated over density to get the pressure). Note that we have not made any assumptions about the form of the interaction potential, e.g., pairwise additivity. As we shall see later, the integral involving the pair correlation function can be measured in a variety of scattering experiments, which when used in Eq. (6.b), yield the compressibility (and hence the pressure). In the end, the two approaches considered here should yield equivalent results, assuming, of course, that the pair correlation function is correct. This is never the case owing to inaccuracies in the theories used to calculate the pair correlation function. Nonetheless, approximate theories of g (2) (r) can be judged on how many virial coefficients of either form are correct. Alternately, several "generalized" approaches add an adjustable parameter that is used to force equivalency of the two expressions. It is still not clear how or why this leads to a more accurate pressure. Winter, 28

44 Normal Mode Analysis -43- Chemistry Normal Mode Analysis 7.. Quantum Mechanical Treatment Our starting point is the Schrodinger wave equation: Σ h 2 2 2m i N i= r 2 i + U( r,..., r N ) Ψ( r,..., r N ) = E Ψ( r,..., r N ), (7..) where N is the number of atoms in the molecule, m i is the mass of the i th atom, and U( r,..., r N )isthe effective potential for the nuclear motion, e.g., as is obtained in the Born- Oppenheimer approximation. If the amplitude of the vibrational motion is small, then the vibrational part of the Hamiltonian associated with Eq. (7..) can be written as: H vib Σ N h 2 2 2m i i= 2 i + U + 2 N Σ K i, j : i j, (7..2) i, j= R i is the equilibrium posi- where U is the minimum value of the potential energy, i r i R i, tion of the i th atom, and K i, j 2 U (7..3) r i r j r k = R k is the matrix of (harmonic) force constants. Henceforth, we will shift the zero of energy so as to make U =. Note that in obtaining Eq. (7..2), we have neglected anharmonic (i.e., cubic and higher order) corrections to the vibrational motion. The next and most confusing step is to change to matrix notation. We introduce a column vector containing the displacements as: [ x, y, z,..., x N, y N, z N ]T, (7..4) where "T "denotes a matrix transpose. Similarly, we can encode the force constants or masses into 3N 3N matrices, and thereby rewrite Eq. (7..2) as H vib = h 2 2 M + 2 K, (7..5) where all matrix quantities are emboldened, the superscript " " denotes the Hermitian conjugate (matrix transpose for real matrices), and where the mass matrix, M, is a diagonal matrix with the masses of the given atoms each appearing three times on the diagonal. The Hamiltonian given by Eq. (7..5) is the generalization of the usual harmonic oscillator Hamiltonian to include more particles and to allow for "springs" between arbitrary particles. Winter, 28

45 Chemistry Normal Mode Analysis Unless K is diagonal (and it usually isn t) Eq. (7..5) would seem to suggest that the vibrational problem for a polyatomic is non-separable (why?); nonetheless, as we now show, it can be separated. This is first shown using using the quantum mechanical framework we ve just set up. Later, we will discuss the separation using classical mechanics. This is valid since, for harmonic oscillators, you get the same result for the vibration frequencies. We now make the transformation i m /2 i i,which allows Eq. (7..5) to be reexpressed in the new coordinates as H vib = h K, (7..6) where K M /2 KM /2. (7..7) Since the matrix K is Hermitian (or symmetric for real matrices), it is possible to find a unitary (also referred to as an orthogonal matrix for real matrices) matrix which diagonalizes it; i.e., you can find a matrix P which satisfies where λ is diagonal (with real eigenvalues) and where P KP = λ or K = PλP, (7..8) PP = (7..9) where is the identity matrix. Note that P is the unitary matrix who s columns are the normalized eigenvectors. (See a good quantum mechanics book or any linear algebra text for proofs of these results). We now make the transformation in Eq. (7..5). This gives P (7..) H vib = h 2 2 P P + 2 λ, (7..) Finally, Eq. (7..8) allows us to cancel the factors of P in this last equation and write H vib = 3N Σ 2 h 2 i= 2 2 i + 2 λ i 2 i. (7..2) Thus the two transformations described above have separated the Hamiltonian into 3N uncoupled harmonic oscillator Hamiltonians, and are usually referred to as a normal mode transformation. Remember that in general the normal mode coordinates are not the original displacements from the equilibrium positions, but correspond to collective vibrations of the molecule. Winter, 28

46 Normal Mode Analysis -45- Chemistry Normal Modes in Classical Mechanics The starting point for the normal mode analysis in classical mechanics are Newton s for a system of coupled harmonic oscillators. Still, using the notation that led to Eq. (7..2), we can write the classical equations of motion as: M (t) = K (t), (7.2.) where components of the left-hand-side of the equation are the rates of change of momentum of the nuclei, while the right-hand-side contains the harmonic forces. Since we expect harmonic motion, we ll look for a solution of the form: (t) = cos(ω t + φ)y, (7.2.2) where φ is an arbitrary phase shift. If Eq. (7.2.2) is used in Eq. (7.2.) it follows that: (Mω 2 K)Y =, (7.2.3) where remember that Y is a column vector and M, and K are matrices. We want to solve this homogeneous system of linear equations for Y. In general, the only way to get a nonzero solution is to make the matrix [Mω 2 K] singular; i.e., we must set det(mω 2 K) = det(ω 2 M /2 KM /2 )det(m) =. (7.2.4) Since, det(m) = (M M 2...M N ) 3 wesee that the second equality is simply the characteristic equation associated with the eigenvalue problem: Ku = λu, (7.2.5) with λ ω 2 and with K given byeq. (7.2.7) above. The remaining steps are equivalent to what was done quantum mechanically Force Constant Calculations Here is an example of a force constant matrix calculation. We will consider a diatomic molecule, where the two atoms interact with a potential of the form: U(r, r 2 ) 2 2 K r r 2 R ; (7.3.) i.e., a simple Hookian spring. It is easy to take the various derivatives indicated in the preceding sections; here, however, we will explicitly expand the potential in terms of the atomic displacements, i. Bywriting, r i = R i + i (where R i is the equilibrium position of the i th nucleus), Eq., (7.3.) can be rewritten as: U(r, r 2 ) = /2 2 2 K R R 2 ( 2 ) R, (7.3.2) Winter, 28

47 Chemistry Normal Mode Analysis where R 2 R R 2. Clearly, the equilibrium will have R 2 = R. Moreover, we expect that the vibrational amplitude will be small, and thus, the terms in the s in the square root in Eq. (7.3.2) will be small compared with the first term. The Taylor expansion of the square root implies that we can write A + B A + B +..., (7.3.3) 2 A U(r, r 2 ) = 2 K R 2 + 2R 2 ( 2 ) R 2 2 R, (7.3.4) which can be rewritten as U(r, r 2 ) = 2 K[ ˆR 2 ( 2 )] 2, (7.3.5) where the ˆ denotes a unit vector and where all terms smaller than quadratic in the nuclear displacements have been dropped. If the square is expanded, notice the appearance of cross terms in the displacements of and 2. It is actually quite simple to finish the normal mode calculation in this case. To doso, define the equilibrium bond to point in along the x axis. Equation (7.3.5) shows that only x- components of the displacements cost energy, and hence, there will no force in thy y or z directions (thereby resulting in 4 zero eigen-frequencies). For the x components, Newton s equations become: m m 2 x = K K x 2 K K x 2 x, (7.3.6) where m i is the mass of the i th nucleus. This in turn leads to the following characteristic equation for the remaining frequencies: = det m m 2 ω 2 K K K K = (m ω 2 K)(m 2 ω 2 K) K 2 and = ω 2 (µω 2 K), where µ m m 2 /(m + m 2 )isthe reduced mass. Thus we pick up another zero frequency and a nonzero one with ω = K/µ, which is the usual result. (Note that we don t count ± roots twice--why?). Winter, 28

48 Normal Mode Analysis -47- Chemistry Normal Modes in Crystals The main result of the preceding sections is that the characteristic vibrational frequencies are obtained by solving for eigenvalues, cf., Eq. (7.2.5). For small to mid-size molecules this can be done numerically, onthe other hand, this is not practical for crystalline solids where the matrices are huge (3N 3N with N O(N A ))! Crystalline materials differ from small molecules in one important aspect; they are periodic structures made up of identical unit cells; specifically, each unit cell is placed at R = n a + n 2 a 2 + n 3 a 3, (7.4.) where n i is an integer and a i is a primitive lattice vector. The atoms are positioned within each unit cell at positions labeled by an index, α. Thus, the position of any atom in entire crystal is determined by specifying R and α.hence, Eq. (7.2.5) can be rewritten as λu R,α = Σ R α Σ K R,α ;R,α u R,α. (7.4.2) The periodicity of the lattice implies that only the distance between unit cells can matter; i.e., K R,α ;R,α = K R R ;α,α,and this turns Eq. (7.4.2) into a discrete convolution that can be simplified by introducing a discrete Fourier transform, i.e., we let ũ k,α When applied to both sides of Eq. (7.2.5), this gives * where λũ k,α K k;α,α Σ e ik R u R,α. (7.4.3) R Σ K k;α,α ũ k,α, (7.4.4) α Σ e ik R K R;α,α. (7.4.5) R The resulting eigenvalue problem has rank 3N cell,where N cell is the number of atoms in a unit cell, and is easily solved numerically. We haven t specified the values for the k s. Itturns out that a very convenient choice is to use k s expressed in terms of the so-called reciprocal lattice vectors, G,i.e., G m b + m 2 b 2 + m 3 b 3, (7.4.6) where the m i s are integers, and where the reciprocal lattice basis vectors are defined as * Strictly speaking, in obtaining Eq. (7.4.5) we have assumed some special properties of the lattice. We don t use a finite lattice, rather one that obeys periodic boundary conditions (see below). Here we re using the definition of C. Kittel, Introduction to Solid State Physics (Wiley, 966), p. 53. There are others, see, e.g., M. Born and K. Huang, Dyamical Theory of Crystal Lattices (Oxford, 968), p. 69. Winter, 28

49 Chemistry Normal Mode Analysis a 2 a 3 b 2π a ( a 2 a 3 ), a b 2 3 a 2π a 2 ( a 3 a ), and b a 3 a 2 2π a 3 ( a a 2 ). (7.4.7) Note the following: a) The denominators appearing in the definitions of the b i are equal; i.e., a ( a 2 a 3 ) = a 2 ( a 3 a ) = a 3 ( a a 2 ) = ν cell, (7.4.8) where ν cell is the volume of the unit cell. The volume of the reciprocal lattice unit cell is (2π ) 3 /ν cell. b) By construction, where δ i, j is a Kronecker-δ. a i b j = 2πδ i, j, (7.4.9) c) The real and reciprocal lattices need not be the same, even ignoring how the basis vectors are normalized; e.g., they are for the SCC, but the reciprocal lattice for the BCC lattice is a FCC lattice. d) For any lattice vector, R,and reciprocal lattice vector, G, e i R G = e 2π i(m n +m 2 n 2 +n 3 m 3 ) =, (7.4.) cf. Eqs. (7.4.), (7.4.6), and (7.4.9). Hence, adding any reciprocal a lattice vector to the k in Eq. (7.4.4) changes nothing, and so, we restrict k to what is known as the First Brillouin Zone; specifically, k = k b + k 2 b 2 + k 3 b 3, (7.4.) where k i m i /N i with m i =,, 2,..., N i. Note that N N 2 N 3 = N, the total number of cells in the crystal. The N i s are the number of cells in the a i direction. With this choice *,consider N k Σ e i k Rũ k,α = N k Σ Σ e i k ( R R) 3 u = R,α Σ u Π R,α R R i= N i N i Σ e 2π im i(n i n i )/N i.(7.4.2) m i = The sums in parenthesis are geometric series and give N i N i Σ e 2π im i(n i n i )/N i m i = = e 2π i(n i n i ) N i e 2π i(n i n i )/N i = δ ni,n i, (7.4.3) which when used in Eq. (7.4.2) shows that * The more common choice for the range of the m i s is m i = N i /2,...,,..., N i /2, which, cf. Eq. (7.4.), means that /2 k i /2. Winter, 28

50 Normal Mode Analysis -49- Chemistry 593 N k Σ e i k Rũ k,α = u R,α. (7.4.4) The sums over k are really sums over the m i s, and thus, k i hardly changes as we go from m i m i + for large N i,cf. Eq. (7.4.). This allows the sums over m i to be replaced by integrals; i.e., Σ... k 3 Π N i i= dm i...= V (2π ) d k..., (7.4.5) 3 st Brillouin zone where V = Nν cell is the volume of the system. The main goal of this discussion is to obtain the vibrational density of states. By using Eq. (7.4.5), it is easy to show that g(ω ) = V (2π ) d k 3 δ (ω ω (k)), (7.4.6) st Brillouin zone where δ (x) is the Dirac δ -function. Once the frequencies are known, it is relatively easy to numerically bin them by frequency asafunction of k Normal Modes in Crystals: An Example Consider a crystal with one atom of mass m per unit cell and nearest neighbor interactions of the type considered in Sec Newton s equations of motion for this atom becomes m,, = K[ê x( x,, + x,, 2 x,,) + ê y ( y,, + y,, 2 y,, ) +ê z ( x,, + x,, 2 x,,)], (7.5.) where ê i is a unit vector in the i direction. Note that this model implies that vibrations in the x, y, z, directions are separable. By using this in Eq. (7.4.5) we see that sin 2 (k π ) K k = 4ω 2 sin 2 (k 2 π ) sin 2 (k 3 π ), (7.5.2) where ω (K/m) /2 and where the identity, cos(x) = 2sin 2 (x/2) was used. Since K k is already diagonal, we see that ω i ( k) = 2ω sin(k i π ), i =, 2, 3, which is also the result for a one dimensional chain (as was expected, given the separability of the vibrations in the x, y, and z directions). The normal mode is just ê i,which shows that our model potential is too simple. Consider the first eigen-vector. Itcorresponds to an arbitrary displacement in the x direction, with an eigenvalue that is independent of k y and k z. When k x = wedon t get a single zero frequency, as expected, but one for any ofthe N y N z values of (k y, k z ). Thus, with 3N 2/3 zero Winter, 28

51 Chemistry Normal Mode Analysis frequencies, not 6, the crystal is unstable! The problem is easily fixed by modifying the interaction potential, i.e., we replace Eq. (7.3.5) by U K 2 ( 2 ) 2, (7.5.3) which is invariant under rigid translations and rotations, and allows us to rewrite Eq. (7.5.) as m,, = K(,, +,, 2,, +,, +,, 2,, +,, +,, 2,, ). (7.5.4) With this, it follows that K k = 4ω 2 [sin 2 (k π ) + sin 2 (k 2 π ) + sin 2 (k 3 π )], (7.5.5) where is a 3 3identity matrix and where we have expressed k in terms of the reciprocal lattice basis vectors, cf. Eq (7.5.). The triply degenerate vibrational frequencies are simply ω i ( k) = 2ω [sin 2 (k π ) + sin 2 (k 2 π ) + sin 2 (k 3 π )] /2. (7.5.6) If all the k i, Eq. (7.5.6) becomes ω ( k) 2ω kπ,where k (k 2 + k k 2 3) /2 which is the expected linear dispersion law for long wevelength, acoustic vibrations. Some constant frequency surfaces in the First Brillouin Zone are shown in Figs Winter, 28

52 Normal Mode Analysis -5- Chemistry 593 Fig A constant normal mode frequency surface for ω /ω =. The reciprocal lattice basis was used which need not be orthogonal (although it is for the SCC lattice). Also note that we ve switched to the other definition of the first Brillouin zone, with /2 k i /2. The surface is roughly spherical, as expected for long wavelength acoustic phonons. Fig As in Fig but with ω /ω = 2. Notice the C 4 axises through the centers of any ofthe unit cell faces. Fig ω /ω = 3. As in Fig but with Normal mode frequencies for this model were computed numerically for a uniform sample of k i s in the First Brillouin Zone and binned in order to get the vibrational density of of states as shown in Fig Winter, 28

53 Chemistry Normal Mode Analysis Fig Vibrational density of states for the isotropic model defined by Eq. (7.5.3). Any single atom per unit cell lattice will give the same result. The data was obtained by numerically binning the frequencies in the First Brillouin Zone. Note that the maximum frequency for this model is 2 3ω. By noting that where f (x) has zeros at x = x i, i =,..., and that δ ( f (x)) = Σ δ (x x i), (7.5.7) i df (x i ) dx i we can rewrite Eq. (7.4.6) as g(ω ) 3N δ (x) = ds 2π eixs, (7.5.8) = 2ω dk dk 2 dk 3 δ (ω 2 ω 2 (k)) = ω π = ω π ds dk dk 2 dk 3 e is(ω 2 ω 2 (k)) ds eisω 2 Φ 3 (s), where the extra factor of 3 is due to the triple degeneracy ofeach mode and where Winter, 28

54 Normal Mode Analysis -53- Chemistry 593 Φ(s) dk e 4iω 2 s sin2 (π k) When we use Eq. (7.5.7) for the frequencies, it follows that the wavevector integrations separate and where g(ω ) = 2Vω Φ(s) dk 2 2 2π e is4ω sin 2 (π k ) = e 2ω is 2π 2 ds 2π eisω 2 Φ 3 (s), (7.5.9) π dz e2ω 2 is cos(z) 2π dz e 2ω 2 isz 2 2 ( z 2 ) = e 2ω /2 2π = e 2ω 2 is is J (2ω 2 s), (7.5.) where J (x) isabessel function of the first kind. * When this is used in Eq. (7.5.), we see that g(ω ) = 2Vω (2π ) 4 ds eis(ω 2 6ω 2 ) J 3 (2ω 2 s) {7.5.) * The integral leading to the last equality can be found in I.S. Gradsheyn and I.M. Ryzhik, Table of Integrals, Series, and Products, A.Jeffrey editor, (Academic Press, 98), Eq. ( ) on p. 32. Winter, 28

55 Chemistry The q Limit of the Structure Factor 8. The q Limit of the Structure Factor For auniform fluid, we have shown that the structure factor, NS q = N + ρ dr 2 e iq r 2 [g(r 2 ) ], (8.) where, strictly speaking, g(r) is the pair correlation function for the canonical ensemble. In order to show how the q limit comes about, note that the integral in Eq. (8.) depends mainly on distances r< σ,where σ characterizes the range of the pair correlation function; in most cases, this length is microscopic. Moreover, on this finite length scale, the correlations should be equivalent to those in an open system (i.e., in a grand canonical ensemble) since the rest of the liquid can be viewed as playing the role of a particle reservoir. In a grand canonical ensemble, the generic pair distribution function, ρ (2) (r, r 2 )isdefined as the average over the number of particles, N, of the canonical ones; i.e., ρ (2) (r, r 2 ) Σ eβ µn Q(N, V, T ) Ξ(µ, V, T ) N=2 N(N ) dr 3...dr N e βu Z c (N, V, T ). (8.2) From this definition, it is easy to see that the grand canonical partition function satisfies: In order to proceed, we rewrite Eq. (8.) as: dr dr 2 ρ (2) (r, r 2 ) = < N(N ) >. (8.3) < N > S q = < N > + dr dr 2 e iq r 2 ρ (2) (r, r 2 ) < N > 2 V, (8.4) where it should be remembered that the density of the grand canonical ensemble equals that of the canonical one. If we now let q and use Eq. (8.3), we find that lim < N > S q = < N > + < N(N ) > < N > 2 q + = <(N < N >) 2 > = < N > k B T ργ P, where the isothermal compressibility is defined by γ P V V. P N,T Winter, 28

56 The q Limit of the Structure Factor -55- Chemistry 593 What happens to the scattering intensity as the system approaches a critical point? What does this imply for g(r)? Winter, 28

57 Chemistry Gaussian Coil Elastic Scattering 9. Gaussian Coil Elastic Scattering Here is a more exact treatment of elastic scattering from coils with Gaussian segment distributions, that is, where the probability that a pair of monomers, i and j, are separated by a distance R is given by where from the central-limit theorem, P i, j (R) = exp[ R2 /2 < R 2 i, j >] [2π < R 2 i, j >]/2, (9.) < R 2 i, j >= i j <l 2 >/3 (9.2) where < l 2 >isthe RMS average bond length (and becomes the bond length in the freelyjointed chain). The structure factor * for the chain is just S(q) N 2 Σ e iq r i, j, (9.3) i, j where N is the number of monomers in the polymer. When the distribution given byeq. (9.) is used, Eq. (9.3) becomes S(q) = N 2 Σ e q2 i j <l 2 >/6 i, j (9.4a) = N + 2N 2 N Σ i i=2 j= Σ e q2 (i j)<l 2 >/6, (9.4b) where the sum in Eq. (9.4a) has been split into three parts in order to obtain Eq. (9.4b), i.e., the term with i = j, i < j, and i > j, where the last two sums are equal. The sum over j in Eq. (9.4b) is simply a geometric series and when summed gives S(q) = N + 2N 2 Σ N e q2 i<l2 >/6 e q2 <l2 >/6 e q2 <l 2 >/6 i=2 (9.5a) = N + 2N 2 e q2 (N+)<l 2 >/6 e 2q2 <l 2 >/6 (e q2 <l 2 >/6 ) 2 (N )e q2<l2>/6 e q2 <l 2 >/6 (9.5b) = N + e q2 <l 2 >/6 e q2 <l 2 >/6 + 2N 2 e q2<l2>/6 (e q2n<l2>/6 ) (e q2 <l 2 >/6 ) 2, (9.5c) * Actually, since inter-chain effects will not be included, this is really the form-factor. Winter, 28

58 Gaussian Coil Elastic Scattering -57- Chemistry 593 where (9.5b) is obtained by summing the series in Eq. (9.5a). In light scattering experiments, q 2 < l 2 ><<. Onthe other hand, R 2 N < l 2 >can be large (i.e., comparable to the wavelength of light) and hence, qr is not necessarily small. To see how Eq. (9.5b) simplifies for light scattering, let y q 2 R 2 /6; this allows Eq. (9.5b) to be rewritten as + e y/n S(q) = N e + 2N 2 e y/n (e y ). (9.6) y/n (e y/n ) 2 Since N is large, the exponentials can be expanded in Taylor series. When the leading order terms are kept, we find that S(q) = 2 y 2 (e y + y ) + O(N ). (9.7) Thus, a universal scattering function is obtained for large chains. Note that when y, S(q), as it must from Eq. (9.3). Finally, S(q) isshown below. The figure also shows the result for rigid rod-like molecules (which can be treated in a manner analogous to that presented above). Winter, 28

59 Chemistry Some Properties of the Master Equation. Some Properties of the Master Equation In the continuum limit the master equation can be written as * P n (t) t = Σ[W n n P n (t) W n n P n (t)], (.) n where P n (t) isthe probability of finding the system in state n (here assumed discrete) at time t, W i j is the transition probability per unit time for switching from state i to state j, and where the two terms in the sum are so-called birth (growth, gain) and death (decay, loss) terms, respectively. Atpresent, W n n is non-negative but otherwise arbitrary. Inorder that Eq. (.) make sense, it must conserve probability; i.e., Σ P n (t) = n or Σ P n(t) n t =. (.2) By using Eq. (.) in Eq. (.2) it follows that Σ [W n n P n (t) W n n P n (t)] n,n must vanish, as is easily shown by exchanging the dummy summation indices in one of the terms. Next we will show that the master equation has an equilibrium solution, denoted as P eq n. Clearly, cf. Eq. (.), a necessary condition is that which is satisfied if we impose detailed balance, i.e., Σ n W n np eq n W n n P eq n =, (.3) W n n = Peq n W n n P eq n = e E n n/k B T. (.4) The last equality is obtained for a canonical distribution, where E n n E n E n is the energy difference between states n and n and is the basis of the Metropolis Monte Carlo method. Returning to the general case, we multiply both sides of Eq. (.3) by arbitrary numbers ψ n and sum over n; this gives = Σ n,n W n np eq n ψ n W n n P eq n ψ n = Σ W n n P eq n (ψ n ψ n ), (.5) n,n * The discussion in this section is based on that of N.G. van Kampen, Stochastic Processes in Physics and Chemistry, (North-Holland Pub. Co., Amsterdam, 984). N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller, and E. Teller, J.Chem. Phys. 2, 87 (.953). Winter, 28

60 Some Properties of the Master Equation -59- Chemistry 593 where the last equality is obtained by exchanging the dummy sum variables in the second term. Next, consider the following quantity H(t) Σ P eq n f n P n (t) P eq n, (.6) where f (x) is, as yet, an arbitrary function. By using Eq. (.) in (.6) we see that dh(t) dt = Σ f ( P n (t)/p eq n ) [W n n P n (t) W n n P n (t)] n,n (.7a) = Σ W n n P eq n [x n f (x n ) x n f (x n )], (.7b) n,n where x n P n (t)/p eq n. Finally, let ψ n f (x n ) x n f (x n )and add the result to Eq. (.7b). This gives dh(t) dt = Σ W n n P eq n (x n x n ) f (x n ) + f (x n ) f (x n ). (.8) n,n Given that W n n and P n are positive the sign of each term in Eq. (.8) is determined by the sign of the terms in the ( ) s. These terms are negative, and hence, H(t) isadecreasing function, if we take f (x) to be convex on [, ). For example, let f (x) x log(x). In this case, ( ) = (x n x n )[log(x n ) + ] + x n log(x n ) x n log(x n ) (.9a) = x n log x n x n + x n x n, (.9b) which vanishes for x n = x n and is otherwise negative. Thus, H(t) will decrease until x n = x n for all n and n, i.e., at equilibrium when P n = P eq n.note that we have implicitly assumed that there is a unique equilibrium state. Finally, note the similarity between H(t) and the entropy in equilibrium statistical mechanics; we can make two observations:. We v e shown that there is a second law; 2. H(t) isn t unique (what other properties of the entropy would further constrain our choice of f (x)?) This is a consequence of the Klein inequality, namely, x log x x log y x + y, for x, y [, ) and vanishes only for x = y. Prove the Klein inequality. Winter, 28

61 Chemistry Critical Phenomena. Critical Phenomena.. Introduction In these lectures some of the general features of the phenomena of phase transitions in matter will be examined. We will first review some of the experimental phenomena. We then turn to a discussion of simple thermodynamic and so-called mean-field theoretical approaches to the problem of phase transitions in general and critical phenomena in particular, showing what they get right and what they get wrong. Finally, wewill examine modern aspects of the problem, the scaling hypothesis and introduce the ideas behind a renormalization group calculation. Fig... Phase Diagram of Water. Fig..2. Liquid-Vapor P-V phase diagram isotherms near the critical point. 2 Consider the two well known phase diagrams shown in Figs.. and.2. Along any of the coexistence lines, thermodynamics requires that the chemical potentials in the coexisting phases be equal, and this in turn gives the well known Clapeyron equation: dp = H dt coexistence T V, (..) where H and V are molar enthalpy and volume changes, respectively, and T is the temperature. Many of the qualitative features of a phase diagram can be understood simply by using the Clapeyron equation, and knowing the relative magnitudes and signs of the enthalpy and volume changes. Nonetheless, there are points on the phase diagram where the Clapeyron equation cannot be applied naively, namely at the critical point where V vanishes. G. W. Castellan, Physical Chemistry, 3rd ed., (Benjamin Pub. Co., 983), p R.J. Silbey and R.A. Alberty, Physical Chemistry, 3rd ed., (John Wiley &Sons, Inc. 2) p. 6. Winter, 28

62 Critical Phenomena -6- Chemistry 593 The existence of critical points was controversial when it was first considered in the 9th century because it means that you can continuously transform a material from one phase (e.g., a liquid) into another (e.g., a gas). We now have many experimental examples of systems that have critical points in their phase diagrams; some of these are shown in Table.. In each case, the nature of the transition is clearly quite different (from the point of view of the qualitative symmetries of the phases involved). TABLE.. Examples of critical points and their order parameters 3 Critical Point Order Parameter Example T c ( o K) Liquid-gas Density H 2 O Ferromagnetic Magnetization Fe 44. Anti-ferromagnetic Sub-lattice FeF magnetization Super-fluid 4 He-amplitude 4 He.8-2. Super- Electron pair Pb 7.9 conductivity amplitude Binary fluid Concentration CCl 4 -C 7 F mixture of one fluid Binary alloy Density of one Cu Zn 739 kind on a sub-lattice Ferroelectric Polarization Triglycine sulfate The cases in Table.I are examples of so-called 2nd order phase transitions, according to the naming scheme introduced by P. Eherenfest. More generally, an nth order phase transition is one where, in addition to the free energies, (n ) derivatives of the free energies are continuous at the transition. Since the first derivatives of the free energy give entropy and volume, all of the freezing and sublimation, and most of the liquid-vapor line would be classified as first-order transition lines; only at the critical point does it become second order. Also note that not all phase transitions can be second order; in some cases, symmetry demands that the transition be first order. At a second order phase transition, we continuously go from one phase to another. What differentiates being in a liquid or gas phase? Clearly, both have the same symmetries, so what quantitative measurement would tell us which phase we are in? We will call this quantity (or 3 S.K. Ma, Modern Theory of Critical Phenomena, (W.A. Benjamin, Inc., 976), p. 6. Winter, 28

63 Chemistry Critical Phenomena quantities) an order parameter, and adopt the convention that it is zero in the one phase region of the phase diagram. In some cases there is a symmetry difference between the phases and this makes the identification of the order parameters simpler, in others, there is no obvious unique choice, although we will show later that for many questions, the choice doesn t matter. For example, at the liquid-gas critical point the density (molar volume) difference between the liquid and vapor phases vanishes, cf. Fig..2, and the density difference between the two phases is often used as the order parameter. Second order transitions are also observed in ferromagnetic or ferroelectric materials, where the magnetization (degree or spin alignment) or polarization (degree of dipole moment alignment) continuously vanishes as the critical point is approached, and we will use these, respectively, as the order parameters. Other examples are given intable.. At a second order critical point, many quantities vanish (e.g., the order parameter) while others can diverge (e.g., the isothermal compressibility, V ( V / P) T,N cf. Fig..2). In order to quantify this behavior, we introduce the idea of a critical exponent. For example, consider a ferromagnetic system. As we just mentioned, the magnetization vanishes at the critical point (here, this means at the critical temperature and in the absence of any externally applied magnetic field, H), thus near the critical point we might expect that the magnetization, m might vanish like or at the critical temperature, in the presence of a magnetic field, m T c T β, when H =, (..2) m H /δ. (..3) The exponents β and δ are examples of critical exponents and are sometimes referred to as the order parameter and equation of state exponents, respectively; we expect both of these to be positive. Other thermodynamic quantities have their own exponents; for example, the constant magnetic field heat capacity (or C P in the liquid-gas system) can be written as C H T c T α, (..4) while the magnetic susceptibility, χ,(analogous to the compressibility) becomes χ T c T γ. (..5) Non-thermodynamic quantities can also exhibit critical behavior similar to Eqs. (..2) (..5). Perhaps the most important of these is the scattering intensity measured in light or neutron scattering experiments. As you learned in statistical mechanics (or will see again later in this course), the elastic scattering intensity at scattering wave-vector q is proportional to the static structure factor NS(q) < N(q) 2 >, (..6) where N(q) is the spatial Fourier transform of the density (or magnetization density), and <... > denotes an average in the grand canonical ensemble. In general, the susceptibility or compressibility and the q limit of the structure factor 4 are proportional, and thus, we expect the Winter, 28

64 Critical Phenomena -63- Chemistry 593 scattered intensity to diverge with exponent γ as the critical point, cf. Eq. (..5). This is indeed observed in the phenomena called critical opalescence. At T = T c and non-zero wavevectors, we write, S(q). (..7) q2 η Finally, we introduce one last exponent, one that characterizes the range of molecular correlations in our systems. The correlation-length is called ξ,and we expect that where we shall see later that the correlation-length exponent, ν >. ξ T c T ν, (..8) Some experimental values for these exponents for ferromagnets are given in Table.2. The primes on the exponents denote measurements approaching the critical point from the twophase region (in principle, different values could be observed). What is interesting, is that even though the materials are comprised of different atoms, have different symmetries and transition temperatures, the same critical exponents are observed, to within the experimental uncertainty. TABLE.2. Exponents at ferromagnetic critical points 5 Material Symmetry T ( o K) α, α β γ, γ δ η Fe Isotropic 44. α = α = ±. ±.2 ±.5 ±.7 Ni Isotropic α = α = ±.3 ±.3 ±.2 ±. EuO Isotropic α = α =. 9 ±. YFeO 3 Uniaxial γ =. 33 ±.5 ±.4 γ =. 7 ±. Gd Anisotropic γ = ±. Of course, this behavior might not be unexpected. After all, these are all ferromagnetic transitions; a phase transition where "all" that happens is that the spins align. What is more interesting are the examples shown in Table.3. Clearly, the phase transitions are very different physically; nonetheless, universal values for the critical exponents seem to emerge. 4 See, e.g., 5 S.K. Ma, op. cit., p.2. Winter, 28

65 Chemistry Critical Phenomena TABLE.3. Exponents for various critical points 6 Critical Points Material Symmetry T c ( o K) α, α β γ, γ δ η Antiferro- CoCl 2 6H 2 O Uniaxial 2.29 α..23 magnetic α. 9 ±. 2 FeF 2 Uniaxial α = α =. 2 ±. 44 RbMnF 3 Isotropic 83.5 α = α = γ = ±. 7 ±. 8 ±. 34 ±. Liquid-gas CO 2 n = 34.6 α / γ = γ = ±. 7 ±. 2 Xe α = α = γ = γ = ±. 2 ±. 3 ±. 2 ±. 4 3 He 3.35 α γ = γ =. 5 α. 2 ±. ±. 3 4 He α = γ = γ =. 7 α =. 59 ±. 28 ±. 5 Super-fluid 4 He α = α < Binary CCl 4 C 7 F 4 n = γ =. 2 4 Mixture ±. 2 Binary Co Zn n = γ =. 25 alloy ±. 5 ±. 2 Ferro- Triglycine n = γ = γ =. electric sulfate ±. 5 Our goals in these lectures are as follows:. To come up with some simple theory that results in phase transitions in general, and second order phase transitions in particular. 2. To show how universal critical exponents result. 3. To beable to predict the correct values for the critical exponents. It turns out the. and 2. are relatively easily accomplished; 3. is not and Kenneth G. Wilson, won the 982 physics Nobel Prize for showing how tocalculate the critical exponents..2. Thermodynamic Approach.2.. General Considerations Other than the already mentioned Clapeyron equation, cf. Eq. (..), and its generalizations to higher order phase transitions (not discussed), thermodynamics has relatively little to say about the critical exponents. One class of inequalities can be obtained by using thermodynamic stability requirements (e.g., that arise by requirements that the free energy be a minimum at 6 S.K. Ma, op. cit., pp Winter, 28

66 Critical Phenomena -65- Chemistry 593 equilibrium). As an example of how this works, recall the well known relationship between the heat capacities C P and C V,namely, C P = C V + TVγ 2 T γ P, (.2.) where γ T V ( V / T ) P,N is the thermal expansion coefficient and γ P V ( V / P) T,N is the isothermal compressibility. Since thermodynamic stability requires that C V and γ P be positive, it follows that C P TVγ 2 T γ P. (.2.2) By using the different exponent expressions, Eqs. (..2) (..4) and (..5), this last inequality implies that, as T T c, where τ T T c /T c.the inequality will hold at T c only if τ α positive constant τ 2(β )+γ, (.2.3) α + 2β + γ 2. (.2.4) This is known as the Rushbrook inequality. If you check some of the experimental data given in Tables S2 and.3, you will see that in most of the cases, α + 2β + γ 2, and to within the experimental error, the inequality becomes an equality. This is no accident!.2.2. Landau-Ginzburg Free Energy We now try to come up with the simplest model for a free energy or equation of state that captures some of the physical phenomena introduced above. For example, we could analyze the well known van der Waals equation near the critical point. It turns out however, that a model proposed by Landau and Ginzburg is even simpler and in a very general manner shows many of the features of systems near their critical points. Specifically, they modeled free energy difference between the ordered and disordered phases as G HΨ+ A 2 Ψ2 + B 3 Ψ3 + C 4 Ψ4 +..., (.2.5) where A, B, C, etc., depend on the material and on temperature, and where H plays the role of an external field (e.g., magnetic or electric or pressure). In some cases, symmetry can be used to eliminate some of the terms in G; for example, in systems with inversion or reflection symmetry (magnets), in the absence of an external field either Ψ or Ψ must give the same free energy. This means that the free energy must be an even function of Ψ in the absence of an external field, and from Eq. (.2.5) we see that this implies that B =. Examples of the Landau free energy for ferromagnets are shown in Figs..3 and.4. Winter, 28

67 Chemistry Critical Phenomena Fig..3. The Landau-Ginzberg free energy (cf. Eq. (.2.5)) for ferromagnets (B =. ) at zero external magnetic field, and C =.. Fig..4. The Landau-Ginzberg free energy, cf. Eq. (.2.5), for ferromagnets (B =. ) at non-zero external magnetic field (H =. ), and C =.. The minima of the free energy correspond to the stable and metastable thermodynamic equilibrium states. In general, we see that an external field induces order (i.e., the free energy has a minimum with Ψ when H ) and that multiple minima occur for A <. When the external field is zero, there are a pair of degenerate minima when A <. This is like the behavior seen at the critical point, where we go from a one- to two-phase region of the phase diagram, cf. Fig..2. To make this more quantitative, weassume that A T T c,as T T c, (.2.7) with a positive proportionality constant, while the other parameters are assumed to be roughly constant in temperature near T c. In order to extract the critical exponents, the equilibrium must be analyzed more carefully. The equilibrium state minimizes the free energy, and hence, Eq. (.2.5) gives: H = AΨ+BΨ 2 + CΨ 3. (.2.8) For ferromagnets with no external field, B =, and Eq. (.2.8) is easily solved, giving and Ψ= (.2.9a) Ψ=± A C. (.2.9b) Clearly, the latter makes physical sense only if A <, i.e., according to the preceding discussion, when T < T c. Indeed, for A <the it is easy to see that the nonzero roots correspond to the minima shown in Fig..3, while Ψ=isjust the maximum separating them, and is thus not the Winter, 28

68 Critical Phenomena -67- Chemistry 593 equilibrium state. With the assumed temperature dependence of A, cf. Eq. (.2.7), we can easily obtain the the critical exponents. For example, from Eqs. (.2.7) and (.2.9b), it follows that Ψ (T c T ) /2,and thus, β = /2. Inthe absence of a magnetic field, the free energy difference in the equilibrium state is easily shown to be Since, when A >(T > T c ) G = A2 (.2.) 4C, otherwise. C H = T S = T 2 G, (.2.) T H,N T 2 H,N it follows that the critical contribution to the heat capacity is independent of temperature, and hence, α = α =. An equation for the susceptibility can be obtained by differentiating both sides of Eq. (.2.8) with respect to magnetic field and solving for χ ( Ψ/ H) T,H=.This gives: χ = A + 2BΨ+3CΨ = A, for T > T c 2 2 A, for T < T c, (.2.2) which shows, cf. Eq. (..5), that γ = γ =, and also shows that the amplitude of the divergence of the susceptibility is different above and below T c. Finally, bycomparing Eq. (.2.8) at T = T c (A = ) with Eq. (..3) we see that δ = 3. These results are summarized in Table.4. Note that the Rushbrook inequality is satisfied as an equality, cf. Eq. (.2.4). Table.4. Mean-Field Critical Exponents Quantity Exponent Value Heat Capacity α Order Parameter β /2 Susceptibility γ Eq. of State at T c δ 3 Correlation length ν /2 Correlation function η The table also shows the results for the exponents η and ν, which strictly speaking, don t arise from our simple analysis. They can be obtained from a slightly more complicated version of the free energy we ve just discussed, one that allows for thermal fluctuations and spatially Winter, 28

69 Chemistry Critical Phenomena nonuniform states. This is beyond the scope of present discussion and will not be pursued further here. Where do we stand? The good news is that this simple analysis predicts universal values for the critical exponents. We ve found values for them independent of the material parameters. Unfortunately, while they are in the right ball-park compared to what is seen experimentally, they are all quantitatively incorrect. In addition, the Landau-Ginzburg model is completely phenomenological and sheds no light on the physical or microscopic origin of the phase transition..3. Weiss Mean-Field Theory The first microscopic approach to phase transitions was given by Weiss for ferromagnets. As is well known, a spin in an external magnetic field has a Zeeman energy given by E = γ h H S, (.3.) where S is the spin operator, γ is called the gyromagnetic ratio, and H is the magnetic field at the spin (which we use to define the z axis of our system). First consider a system of non-interacting spins in an external field. This is a simple problem in statistical thermodynamics. If the total spin is S, the molecular partition function, q, is given by q = S Σ e α S z = S z = S sinh[α (S + /2)] sinh(α /2), (.3.2) where α γ h H/(k B T ), k B is Boltzmann s constant, and where the second equality is obtained by realizing that the sum is just a geometric series. With the partition function in hand it is straightforward, albeit messy, to work out various thermodynamic quantities. For example, the average spin per atom, < s >, is easily shown to be given by where < s >= ln q α = B S(α ), (.3.3) B S (α ) (S + /2)coth[α (S + /2)] coth(α /2)/2 (.3.4) is known as the Brillouin function. The average energy per spin is just γ h H < s >, while the Helmholtz free energy per spin is k B T ln q, asusual. The spin contribution to the heat capacity is obtained by taking the temperature derivative of the energy and becomes: C H = α 2 Nk B 4sinh 2 (α /2) (S + /2) 2 sinh 2 [(S + /2)α ]. (.3.5) Other thermodynamic quantities are obtained in a similar manner. The magnetization and spin contributions to the heat capacity are shown in Figs..5 and.6. Winter, 28

70 Critical Phenomena -69- Chemistry 593 Fig..5. Spin polarization of an ideal spin in an external magnetic field. Fig..6. Spin contribution to the constant magnetic field heat capacity, C H. While this simple model gets many aspects of a spin system correct (e.g., the saturation values of the magnetization and the high temperature behavior of the magnetic susceptibilities), it clearly doesn t describe any phase transition. The magnetization vanishes when the field is turned off and the susceptibility is finite at any finite temperature. Of course, the model didn t include interactions between the spins, so no ordered phase should arise. Weiss included magnetic interactions between the spins by realizing that the magnetic field was made up of two parts: the external magnetic field and a local field that is the net magnetic field associated with the spins on the atoms surrounding the spin in question. In a disordered system (i.e., one with T > T c and no applied field) the neighboring spins are more or less randomly oriented and the resulting net field vanishes, on the other hand, in a spin aligned system the neighboring spins are ordered and the net field won t cancel out. To be more specific, Weiss assumed that H = H ext + λ < s >, (.3.6) where H ext is the externally applied field and λ is a parameter that mainly depends on the crystal lattice. In ferromagnets the field of the neighboring atoms tends to further polarize the spin, and thus, λ > (it is negative in anti-ferromagnetic materials). Note that the mean field that goes into the partition function depends on the average order parameter, which must be determined self-consistently. When Weiss s expression for the magnetic field is used in Eqs. (.3.3) and (.3.4) a transcendental equation is obtained, i.e., < s >= B S (γ h (H ext + λ < s >) / (k B T )). (.3.7) In general, while it is easy to show that there are at most three real solutions and a critical point, Eq. (.3.7) must be solved graphically or numerically. Nonetheless, it can be analyzed analytically close to the critical point since there < s >and H ext are small as is α. Wecan use this by noting the Taylor series expansion, Winter, 28

71 Chemistry Critical Phenomena which when used in Eq. (.3.7) gives coth(x) = x + x 3 x3 +..., (.3.8) 45 < s >= α 3 [(S + /2)2 ( / 2) 2 ] α 3 45 [(S + /2)4 ( / 2) 4 ]+.... (.3.9) If the higher order terms are omitted, Eq. (.3.9) is easily solved. For example, when H ext =, we see that in addition to the root < s >=, we have: < s >= ± 45T 2 (T c T ) [(S + /2) 4 (/ 2) 4 ](γ h λ/k B ), (.3.) 3 where the critical temperature (known as the Curie temperature in ferromagnets) is T c γ h λs(s + ) 3k B. (.3.) When T < T c the state with the nonzero value of < s >has the lower free energy. Thus we ve been able to show that the Weiss theory has a critical point and have come up with a microscopic expression for the critical temperature. By repeating the analysis of the preceding section, one can easily obtain expressions for the other common thermodynamic functions. The Weiss mean field theory is the simplest theory of ferro-magnetism, and over the years many refinements to the approach have been proposed that better estimate the critical temperature. Unfortunately, they all fail in one key prediction, namely, the critical exponents are exactly the same as those obtained in preceding section, e.g., compare Eqs. (.2.9b) and (.3.). This shouldn t be too surprising, given the similarity between Eqs. (.2.8) and (.3.9), and thus, while we ve been able to answer some of our questions, the matter of the critical exponents still remains..4. The Scaling Hypothesis When introducing the critical exponents, cf. Table.4, we mentioned the exponent ν associated with the correlation length, that is ξ T T c ν. What exactly does a diverging correlation length mean? Basically, it is the length over which the order parameter is strongly correlated; for example, in a ferromagnet above its Curie temperature, if we find a part of the sample where the spins are aligned and pointing up, then it is very likely that all the neighboring spins out to a distance ξ will have the same alignment. In the disordered phase, far from the critical point the correlation-length is microscopic, typically a few molecular diameters in size. At these scales, all of the molecular details are important. What happens as we approach the critical point and the correlation length grows? Winter, 28

72 Critical Phenomena -7- Chemistry 593 Fig..7. A snapshot of the spin configuration in a computer simulation of the 2D Ising model of a ferromagnet slightly above its critical temperature. Dark and light regions correspond to spin down and spin up, respectively. Figure.7 shows the spin configuration obtained from a Monte Carlo simulation of the Ising spin system (S = /2 with nearest-neighbor interactions) close to its critical point. We see large interconnected domains of spin up and spin down, each containing roughly 3 4 spins. If this is the case more generally, what determines the free energy and other thermodynamic quantities? Clearly, two very different contributions will arise. One is associated with the shortrange interactions between the aligned spins within any given domain, while the other involves the interactions between the ever larger (as T T c ) aligned domains. The former should become roughly independent of temperature once the correlation length is much larger than the molecular lengths and should not contain any ofthe singularities characteristic of the critical point. The latter, then, is responsible for the critical phenomena and describes the interactions between large aligned domains. As such, it shouldn t depend strongly on the microscopic details of the interactions, and universal behavior should be observed. The next question is how do these observations help us determine the structure of the quantities measured in thermodynamic or scattering experiments? First consider the scattering intensity or structure factor, S(q), introduced in Eq. (..6). The scattering wave-vector, q, Winter, 28