Crystals. Peter Košovan. Dept. of Physical and Macromolecular Chemistry

Size: px

Start display at page:

Download "Crystals. Peter Košovan. Dept. of Physical and Macromolecular Chemistry"

Lizbeth Daniels
6 years ago
Views:

1 Crystals Peter Košovan Dept. of Physical and Macromolecular Chemistry Lecture 1, Statistical Thermodynamics, MC26P15, If you find a mistake, kindly report it to the author :-)

2 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

3 Harmonic approximation Taylor expansion around the minimum U(ξ 1, ξ 2,..., ξ N ) = U(,,..., ) 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(,,..., ) 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

4 The partition function 3N 6 3N eigenfrequencies: 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

5 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

6 C v of crystals: classical theory versus experiment The law of Dulong and Petit: C v /Nk B = R 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

7 The Einstein theory of C v of crystals (197) Einstein s approximation in terms of g(ν): 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

8 The Einstein theory of C v of crystals (197) Einstein s approximation in terms of g(ν): 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

9 Einstein theory versus experiment Reproduces the general trend of drop in C v as T 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

10 The Debye theory of C v of crystals (1912) From spectroscopy: ν j [ : 1 13 ] Hz 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

11 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

12 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 ) vl 3 4πV ν 2 dν Accounting for degeneracy of transverse and longitudinal direction P. Košovan Lecture 1: Crystals. 9/21

13 The Debye theory of C v of crystals (1912) Introduce an average velocity to arrive at 3 v 3 = 2 vt πV 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

14 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

15 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

16 Debye theory and the law of corresponding states Also in Debye theory, C v is a universal function of Θ D /T 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

17 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) continuum lattice -2π/a -π/a π/a 2π/a P. Košovan Lecture 1: Crystals. k 14/21

18 Energy in terms of dispersion curve Periodic boundary conditions k = 2πj/Na, j = ±1, ±2,..., ± N 2 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

19 Spectroscopically determined g(ν) Figure from McQuarrie: Statistical Mechanics, Univresity Science books (2) P. Košovan Lecture 1: Crystals. 16/21

20 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

21 Phonons Total number of phonons is not conserved: 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

22 Phonons 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

23 Point defects in crystals 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

24 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

25 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

1+e θvib/t +e 2θvib/T +

1+e θvib/t +e 2θvib/T + 7Mar218 Chemistry 21b Spectroscopy & Statistical Thermodynamics Lecture # 26 Vibrational Partition Functions of Diatomic Polyatomic Molecules Our starting point is again the approximation that we can treat