Lattice Dynamics related to movement of atoms about
Lattice Dynamics related to movement of atoms about their equilibrium positions determined by electronic structure Physical properties of solids Sound velocity Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors) Hardness of perfect single crystals (without defects) Reminder to the physics of oscillations and waves: Harmonic oscillator in classical mechanics: Equation of motion: Hookes law m x Fspring Example: spring pendulum m x D x 0 or where x D ~ ~ x 0
m x( t ) Re( ~ x( t )) ~ x( t ) A ei t Solution with ~ x( t ) A cos(t ) where D m Dx X=A sin t 1 Epot D x 2 2 X x D m y( t ) A cos (t kx ) ~
~ y( t ) A ei( t kx ) Traveling plane waves: or Y 0 in particular y( t ) A cos (t kx ) 0 X X=0: y( t ) A cos t t=0: y( x ) A cos kx Particular state of oscillation Y=const travels according ~ ~ y( t ) A e solves wave equation i( t kx ) 1 2 y 2 y
2 v t 2 x 2 d t kx d const. 0 dt dt 2 x v v k 2 / Transverse wave Longitudinal wave Standing wave ~ ~ y1 A ei(kx t ) ~ ~ y 2 A ei(kx t ) ~
~ y s ~ y1 ~ y 2 A ei(kx t ) ei(kx t ) ~ ~ A eikx eit e it 2 A eikx cos t ys Re( y s ) 2 A cos kx cos t Large wavelength k 2 0 >10-8m 10-10m 8 Crystal can be viewed as a continuous medium: good for 10 m Speed of longitudinal wave: v (ignoring anisotropy of the crystal)
Bs where Bs: bulk modulus with 1 2 Bs determines elastic deformation energy density U Bs2 (click for details in thermodynamic context) dilation compressibility V V E.g.: Steel v Bs Bs=160 109N/m2 =7860kg/m3 v 160 109 N / m2 7860 kg / m3
1 Bs 4512 m s < interatomic spacing continuum approach fails In addition: vibrational modes quantized phonons Vibrational Modes of a Monatomic Lattice Linear chain: Remember: two coupled harmonic oscillators Symmetric mode Anti-symmetric mode Superposition of normal modes generalization Infinite linear chain How to derive the equation of motion in the harmonic approximation n-2 D un-2
n-1 a un-1 n n+1 un un+1 n+2 un+2 Fnr Dun un1 Fnl Dun un 1 un-2 un-1 un fixed un+1 un+2 ?
Total force driving atom n back to equilibrium Fn Dun un 1 Dun un1 n n Dun1 un 1 2un n Fn mu D n un1 un 1 2un u m equation of motion Solution of continuous wave equation u A ei(kx t ) i(kna t ) approach for linear chain un A e n 2 A ei(kna t ) , u 2 D ika e e ika 2 m
un1 A ei(kna t )eika 2 2 D 1 cos ka m ? , Let us try! un 1 A ei(kna t )e ika D 2 sin(ka / 2) m D 2 sin(ka / 2) m 2 D m Note: here pictures of transversal waves although calculation for the longitudinal case k Continuum limit of acoustic waves:
sin ka / 2 ka / 2 ... k 2 0 D ak m D v a k m k k h k 2 , here h=1 a 2 i (( k h
) na t ) a un A ei(k na (k )t ) A ei(k na t ) A e A ei(k na t )ei2 h n A ei(k na t ) (k ) (k ) k k h 2 a 1-dim. reciprocal lattice vector Gh Region un (k , (k)) un (k, (k )) k a a ei2 h n 1 is called first Brillouin zone
Brillouin zones We saw: all required information contained in a particular volume in reciprocal space a first Brillouin zone 1d: r n n a e x Gh r n 2m 2 a In general: first Brillouin zone 2 Gh h e a x where m=hn integer 1st Brillouin zone Wigner-Seitz cell of the reciprocal lattice Vibrational Spectrum for structures with 2 or more atoms/primitive basis Linear diatomic chain: 2n-2 D 2n-1 a
2n 2n+1 2n+2 2a u2n-2 u2n u2n-1 u2n+1 2n Equation of motion for atoms on even positions: u u2n+2 D u2n1 u2n 1 2u2n m 2n1 Equation of motion for atoms on even positions: u i( 2kna t ) Solution with: u2n A e and D u2n2 u2n 2u2n1 M
u2n1 B ei(( 2n1)ka t ) D A2 B(eika e ika ) 2 A m D B2 A(eika e ika ) 2B M D D A 2 2 2 B cos ka m m A 2 D B cos ka m D 2 2 m D D B 2 2 2 A cos ka M M Click on the picture to start the animation M->m D the movie
1 1 note wrong axis2 in 2D m m M 2 D 2 2 D 2 4 D cos 2 ka 2 m M Mm D2 D 2 D 2 D2 4 4 2 2 4 cos2 ka Mm M m Mm 2
D M D D D2 4 2 2 2 4 1 cos ka 0 Mm m M 2 2 sin2 ka k : 2a 2 1 1 4 sin2 ka 1 1 2 D D Mm m M
m M 1 1 1 1 D D m M m M 2 2 D m , 2 D M 2 Atomic Displacement B m k 0 A M Optic Mode
B k 0 1 A Atomic Displacement Click for animations Acoustic Mode Dispersion curves of 3D crystals 3D crystal: clear separation into longitudinal and transverse mode only possible in particular symmetry directions sound waves of elastic theory Every crystal has 3 acoustic branches 1 longitudinal 2 transverse acoustic Every additional atom of the primitive basis further 3 optical branches again 2 transvers 1 longitudinal p atoms/primitive unit cell ( primitive basis of p atoms): 3 acoustic branches + 3(p-1) optical branches = 3p branches 1LA +2TA
(p-1)LO +2(p-1)TO z y Intuitive picture: 1atom 3 translational degrees of freedom x 3+3=6 degrees of freedom=3 translations+2rotations +1vibraton # atoms in primitive basis # of primitive unit cells Solid: p N atoms 3p N vibrations no translations, no rotations Part of the phonon dispersion relation of diamond 2 fcc sublattices vibrate against one another However, identical atoms no dipole moment Longitudinal Optical Transversal Optical degenerated Longitudinal Acoustic
(0,0,0) 1 1 1 ( , , ) 4 4 4 diamond lattice: fcc lattice with basis Transversal Acoustic degenerated P=2 2x3=6 branches expected Phonon spectroscopy Inelastic interaction of light and particle waves with phonons Constrains: conservation law of energy momentum Condition for in elastic scattering in elastic sattering k k 0 q Ghkl phonon wave vector sc att ere d
incoming wave 0 (q) 0 for photon scattering ic e t t a l cal o r p i Rec vector wa ve 2k 2 2k 0 2 (q) 0 2Mn 2Mn quasimomentum k k 0 q Ghkl for neutrons k
0 (q) k 0 q Triple axis neutron spectrometer @ ILL in Grenoble, France Very expensive and involved experiments Table top alternatives ? Yes, infra-red absorption and inelastic light scattering (Raman and Brillouin) However only q 0 accessible see homework #8 Lonely scientist in the reactor hall
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