Vibration Excitations of the Lattice Static vs dynamic
Vibration Excitations of the Lattice
Static vs. dynamic picture Static lattice • Give good explanation for diffraction patterns • Fail to explain : • Thermal expansion • Melting • Heat conduction • Specific heat • Acoustic conduction • Optical absorption Ionic crystal Metallic crystal
Static vs. dynamic picture Static lattice • Give good explanation for diffraction patterns • Fail to explain : • Thermal expansion • Melting • Heat conduction • Specific heat • Acoustic conduction • Optical absorption Ionic crystal In metals we can attribute some of this properties to the electrons Metallic crystal
• For T>0, atoms are in relative motion Let’s try to describe this motion! the harmonic approximation Equation of motion: harmonic oscillator Exercise: Calculate the vibration frequency around the equilibrium point in the Lennard-Jones potential
Lattice vibrations Defining the matrix elements r’ R’ R Equation of motion: D r D’
Motion in 1 D Lattice of identical atoms s-2 s-1 s consider only nearest neighbor interaction s+1 s+2
• Equation of motion is: C is force constant per atom • Look for a wave solution:
• Traveling wave solution: a is spacing between planes • Substitute in and divide by u 0 exp(is. Ka):
Wave-vectors with in the first Brillouin zone
• Phase difference between successive planes: unique values only for -p/a < K < p/a Recognize these limits? • Why? – Because it is a discrete system For continuum limit, a→ 0 and Kmax → ∞
• Group velocity (what is group velocity? ): For K = p/a, vg = 0 standing wave Alternately, No position dependence
Long Wavelength Limit • Substitute in to
Long wavelength limit Velocity of sound is independent of frequency for low notes
Two Atom Basis a vs-1 us-1 Dx us vs us+1 vs+1
• Look for a wave solution with same w, K for both u, v (but different amplitudes) • Substitute into equation of motion
• Solving: • Either Trivial or
Limiting Cases: • Ka << 1 2 roots: • K = p/a 2 roots: cos(Ka) 1 - K 2 a 2/2
Character of Solutions Put solution back into equation • Ka << 1 acoustical branch
• Whole cell moves together like a sound wave hence, acoustical branch • Now let’s consider second solution in Ka<<1 case
• Ka<<1 solution for w:
• Atoms vibrate opposite each other with a fixed center of mass • For a cell with oppositely charged atoms, this is the type of motion that could be excited by a varying electrical field (light) Hence, optical branch
Acoustic vs Optical modes Longitudinal wave Transverse wave
Multiple Atom Basis in 3 D lattice • For a given direction and polarization, there are multiple branches in the dispersion relation, w(k) • For p atoms in primitive cell, there are 3 p phonon branches p longitudinal, 2 p transverse 3 acoustical, 3 p-3 optical
Multiple Atom Basis in 3 D lattice
Specific heat of the lattice due to vibrations To explain the low T behavior we need quantum description of vibrations – phonons
Longer range interactions • Allow non-nearest neighbor interactions (though still linear) • Force law becomes: • Equation of motion becomes:
• Substitute in: • Divide by: deviation from simple theory
• How do we evaluate influence of longrange interactions? – Evaluate Cn Take empirical data for w 2(K) • Then:
Measuring w(K) • How do we measure w(K)? Scatter neutrons off the crystal – Neutrons see the atomic nucleus, which in crystal, also constitutes a periodic potential – Neutron scattering in static lattice therefore follows k+G = k´ – In dynamic lattice neutron can interact with phonons – either creating or destroying
– Phonon modifies “conservation” equation of scattering calculation. k + G = k´ becomes k + G = k´ + K for phonon creation, and k + G + K = k´ for phonon absorption – Often convenient to think in terms of crystal momentum of phonon (p = ħK) – Crystal momentum is different from linear momentum
neutron scattering • Conservation equation of crystal momentum k + G = k´ + K for phonon creation, and k + G + K = k´ for phonon absorption • However, phonons also have energy (ħw as we will see), so scattering involving neutrons is inelastic – Conservation of energy:
• Determine energy gain or loss of scattered neutron as a function of scattering direction, k-k´, and we get: w(K)
- Slides: 34