Crystals Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Lecture 1, Statistical Thermodynamics, MC26P15, 5.1.216 If you find a mistake, kindly report it to the author :-)
Preliminaries Ideal crystal: Particles fixed at their lattice positions distinguishable Small vibrations around energetic minimum 3N 6 3N vibrational DOF Normal mode analysis: 3N oscillators 3N eigenfrequencies Normal coordinates: ξ i = at minimum P. Košovan Lecture 1: Crystals. 1/21
Harmonic approximation Taylor expansion around the minimum U(ξ 1, ξ 2,..., ξ N ) = U(,,..., ) + + 1 2 N i=1 j=1 N ( 2 U First derivative vanishes at minimum: ξ i ξ j U(ξ 1, ξ 2,..., ξ N ) = U(,,..., ) + 1 2 N ( ) U j=1 ) ξ j ξ j ξ i ξ j + N i=1 j=1 N k ij ξ i ξ j U(,,..., ) = U(, ρ) is a function of lattice spacing, i. e., of density: V/N = ρ U(ξ 1, ξ 2,..., ξ N ) is a quadratic function of displacements ξ i, force constants k ij are functions of density. P. Košovan Lecture 1: Crystals. 2/21
The partition function 3N 6 3N eigenfrequencies: http://www.natur.cuni.cz/chemie/fyzchem ν j = 1 2π ( kj µ j ) 1/2, j = 1, 2,..., 3N 6 Normal vibrational modes are independent (orthogonal), they contribute independently to Q(N, V, T ): 3N 6 Q(N, V, T ) = e βu(,ρ) j=1 3N 6 ( e q vib,j = e βu(,ρ) βhν j ) /2 1 e βhν j N = O(1 2 ) assume continuous distribution of eigenfrequencies and introduce the frequency distribution g(ν): g(ν)dν = 3N 6 3N j=1 P. Košovan Lecture 1: Crystals. 3/21
Thermodynamic functions in terms of g(ν) Partition function in terms of frequency distribution ln Q(N, V, T ) = U(, ρ) k B T + Internal energy: ( ) ln Q E = E + k B T 2 = U(, ρ) + T N,V Heat capacity: C v = k B (βhν) 2 e βhν (1 e βhν ) 2 g(ν)dν ( ln ( 1 e βhν) + hν ) g(ν)dν 2k B T ( ) hνe βhν hν + g(ν)dν 1 e βhν 2 Up to this point, the derivations were exact (within O(ξ 3 )) Challenge: How to determine g(ν) P. Košovan Lecture 1: Crystals. 4/21
C v of crystals: classical theory versus experiment The law of Dulong and Petit: C v /Nk B = R http://www.natur.cuni.cz/chemie/fyzchem Follows from equipartition: k B /2 per DOF Experiment: C v /Nk B T 3 as T Figure from McQuarrie: Statistical Mechanics, Univresity Science books (2) P. Košovan Lecture 1: Crystals. 5/21
The Einstein theory of C v of crystals (197) Einstein s approximation in terms of g(ν): http://www.natur.cuni.cz/chemie/fyzchem g(ν) = 3Nδ(ν ν E ) The original argument was formulated along the lines of Planck s argument for black body radiation: vibrational energy has to be quantized Heat capacity in Einstein s approximation: ( ) 2 hνe e βhν E C v = 3Nk B ( ) k B T 1 e βhν 2 E Introducing Einstein temperature Θ E = hν E /k B T : ( ) 2 ΘE e Θ E/T C v = 3Nk B ( T ) 1 e Θ E /T 2 P. Košovan Lecture 1: Crystals. 6/21
The Einstein theory of C v of crystals (197) Einstein s approximation in terms of g(ν): http://www.natur.cuni.cz/chemie/fyzchem g(ν) = 3Nδ(ν ν E ) The original argument was formulated along the lines of Planck s argument for black body radiation: vibrational energy has to be quantized Heat capacity in Einstein s approximation: ( ) 2 hνe e βhν E C v = 3Nk B ( ) k B T 1 e βhν 2 E Introducing Einstein temperature Θ E = hν E /k B T : ( ) 2 ΘE e Θ ( ) E/T 2 C v = 3Nk B ( T ) T ΘE 1 e Θ E /T 2 = 3Nk B e Θ E/T T P. Košovan Lecture 1: Crystals. 6/21
Einstein theory versus experiment Reproduces the general trend of drop in C v as T http://www.natur.cuni.cz/chemie/fyzchem Fails to reproduce T 3 law deviations at low T Theorem of corresponding states: C v is a universal function of Θ E /T Figure from McQuarrie: Statistical Mechanics, Univresity Science books (2) P. Košovan Lecture 1: Crystals. 7/21
The Debye theory of C v of crystals (1912) From spectroscopy: ν j [ : 1 13 ] Hz http://www.natur.cuni.cz/chemie/fyzchem Low frequencies (long wavelengths) dominate at low T Quantization standing waves compatible with size of the crystal Wave traveling in a direction k with frequency ω = 2πν and speed v = ω/k = vλ: u(r, t) = Ae i(k r ωt) Standing wave superposition of two waves in opposite directions u(r, t) = 2Ae i(k r) cos(ωt) Imaginary part must vanish at fixed points (edges of the crystal) For a cube of length L: k x L = n x π, k y L = n y π, k z L = n z π k = π L n where n x, n y, n z are positive integers P. Košovan Lecture 1: Crystals. 8/21
The Debye theory of C v of crystals (1912) Frequency depends only upon the magnitude of k: ω = νk ( ) π 2 k 2 = (n 2 x + n 2 y + n 2 L z) Analogy with particle in a box: number of wave vectors k < k: Φ(k) = π ( ) Lk 3 = L3 k 3 6 π 6π 2 = V k3 6π 2 The frequency distribution ω(k)dk = dφ(k) = V k2 dk 2π 2 dk Use ν = v/λ = vk/2π to obtain the Debye approximation to g(ν) g(ν)dν = 4πV ν2 v 3 dν P. Košovan Lecture 1: Crystals. 9/21
The Debye theory of C v of crystals (1912) Frequency depends only upon the magnitude of k: ω = νk ( ) π 2 k 2 = (n 2 x + n 2 y + n 2 L z) Analogy with particle in a box: number of wave vectors k < k: Φ(k) = π ( ) Lk 3 = L3 k 3 6 π 6π 2 = V k3 6π 2 The frequency distribution ω(k)dk = dφ(k) = V k2 dk 2π 2 dk Use ν = v/λ = vk/2π to obtain the Debye approximation to g(ν) ( 4πV ν2 2 g(ν)dν = v 3 dν = vt 3 + 1 ) vl 3 4πV ν 2 dν Accounting for degeneracy of transverse and longitudinal direction P. Košovan Lecture 1: Crystals. 9/21
The Debye theory of C v of crystals (1912) Introduce an average velocity to arrive at 3 v 3 = 2 vt 3 + 1 4πV ν2 vl 3 g(ν)dν = v 3 dν Valid at low frequencies atomic structure of the crystal unimportant Debye theory uses this expression for all frequencies Introduce a maximum frequency ν D such that ν D This results in g(ν)dν = 3N ν D = v ( 3N 4πV g(ν)dν = { 9N ν 3 D ν 2 dν for ν ν D for ν > ν D ) 1/3 P. Košovan Lecture 1: Crystals. 1/21
Thermodynamic functions from the Debye theory The heat capacity where C v = 9Nk B ( T Θ D ) 3 Θ D = hν D k B, Θ D /T x = hν k B T x 4 e x (e x 1) 2 dx The integral depends only on the upper limit The Debye function ( ) ( ) Θ T T 3 D /T x 4 e x ( T D = 3 Θ D Θ D (e x 1) 2 dx C v = 3Nk B D Can be evaluated numerically Θ D ) P. Košovan Lecture 1: Crystals. 11/21
The limits High T limit (x ): Θ D /T x 4 e x (e x 1) 2 dx C v 3Nk B Θ D /T Low T limit: (x ): ( ) ( T T D = 3 Θ D Θ D ( ) C v 12π4 T 3 5 Nk B Θ D x 4 (1 + x + ) 1 + x + 1) 2 dx = Θ D /T ) 3 x 4 e x ( T (e x 1) 2 dx 3 x 2 dx = 1 3 Θ D ) 3 4π 4 5 ( ) 3 ΘD T P. Košovan Lecture 1: Crystals. 12/21
Debye theory and the law of corresponding states Also in Debye theory, C v is a universal function of Θ D /T http://www.natur.cuni.cz/chemie/fyzchem The Debye temperature Θ D is the only material constant characterizing the crystal Θ D from experiments depends on temperature Figure from McQuarrie: Statistical Mechanics, Univresity Science books (2) P. Košovan Lecture 1: Crystals. 13/21
Beyond Debye The Debye theory is correct for long wavelengths (low frequencies) where atomic structure of the lattice can be neglected. Wavelengths have to be multiples of lattice spacing a importatnt especially for short wavelengths (high frequency) Dispersion curve: relation between frequency and wave vector ω(k): For a 1-dimensional crystal: ( ) sin ω ka max 2 discrete lattice ω(k) = ( ) ω ka max 2 Debye continuum Both match for k ω(k) 2 1.5 1.5 continuum lattice -2π/a -π/a π/a 2π/a P. Košovan Lecture 1: Crystals. k 14/21
Energy in terms of dispersion curve Periodic boundary conditions k = 2πj/Na, j = ±1, ±2,..., ± N 2 http://www.natur.cuni.cz/chemie/fyzchem Unique solutions only for k [ π/a, π/a] periodic wave Internal energy: E = j ω j e β ω j 1 Na π π/a ω(k) e β ω(k) 1 dk substitution: dk = dk dω dω = d ( 2 dω a sin 1( ω )) dω = ω max E = 2π N π/a ω ( e β ω 1 )( ω 2 max ω 2) 1/2 dω, g(ν) = 2N π 2 a(ωmax 2 ω 2 dω ) 1/2 1 (ν 2 max ν 2 ) 1/2 dω P. Košovan Lecture 1: Crystals. 15/21
Spectroscopically determined g(ν) http://www.natur.cuni.cz/chemie/fyzchem Figure from McQuarrie: Statistical Mechanics, Univresity Science books (2) P. Košovan Lecture 1: Crystals. 16/21
Phonons Rearrange the vibrational energy of a crystal E({n j }) = = 3N j=1 3N j=1 ( hν j n j + 1 ) = 2 hν j n j + E 3N j=1 hν j n j + Can be interpreted as the energy of a system of independent particles occupying states 1, 2,..., 3N where hν j are the energies and n j the occupation numbers These particles are called phonons 3N j=1 hν j 2 Full set {n j } uniquely determines the state of the system No restriction on n j, j [, 3N] Follow Bose-Einstein statistics within the harmonic approximation they behave like non-interacting bosons P. Košovan Lecture 1: Crystals. 17/21
Phonons Total number of phonons is not conserved: http://www.natur.cuni.cz/chemie/fyzchem Various sets of {n j } can yield the same E({n j }) but different n = j n j This implies µ = λ = 1 Mean occupation number from Bose-Einstein statistics: n j = λe βε j 1 λe βε = 1 j λ 1 e βε j 1 = 1 e βε j 1 Internal energy: E = 3N j=1 hν j n j + E = 3N j=1 hν j e βν j 1 + E Introduce g(ν) to obtain same expression as earlier g(ν) E = E + e βν j 1 dν P. Košovan Lecture 1: Crystals. 18/21
Phonons http://www.natur.cuni.cz/chemie/fyzchem Phonons are an important concept in contemporary solid state theories Analogy: Photons: quanta of electromagnetic vibrations Phonons: quanta of lattice vibrations (sound waves) If you want to learn more: Read chapters 1 and 11 of McQuarrie s textbook Read a monograph on solid state physics (e. g. Kittel) P. Košovan Lecture 1: Crystals. 19/21
Point defects in crystals http://www.natur.cuni.cz/chemie/fyzchem Real crystals contain defects Lattice vacancies (Schottky defects) Interstitial atoms (Frenkel defects) Dislocations Defects increase entropy Equilibrium at the minimum of free energy Schottky Frenkel P. Košovan Lecture 1: Crystals. 2/21
Point defects in crystals Schottky defects Free energy with n defects: A(n) = E T S = nε v k B T ln N! n!(n n)! Number of defects at equilibrium: ( ) A(n) = n Ne βεv n T Take ε v = 1 ev: n/n 1 17 at 3 K n/n 1 5 at 1 K P. Košovan Lecture 1: Crystals. 21/21
Point defects in crystals Schottky defects Free energy with n defects: A(n) = E T S = nε v k B T ln N! n!(n n)! Number of defects at equilibrium: ( ) A(n) = n Ne βεv n T Take ε v = 1 ev: n/n 1 17 at 3 K n/n 1 5 at 1 K Frenkel defects Free energy with n defects, N interstitial sites: N! A(n) = nε I k B T ln n!(n n)! N! k B T ln n!(n n)! Number of defects at equilibrium: n (NN ) 1/2 e βε I P. Košovan Lecture 1: Crystals. 21/21