1、Solid state physics(SAITO)1Solid state physicsTohoku UniversityRiichiro Saito Solid state physics(SAITO)2Types of solids4Crystal Translational symmetry4Amorphous Disordered structures4Quasicrystal Ordered structure No translation symmetryparacrystal（Al,Pd,Mn alloy）crystal（oxide super conductor)Amorp
2、hous material（example:ceramics）Multicrystal,surface,and particleSolid state physics(SAITO)3Solid state physics?4Properties of existing materials Understand experimental results Give direction to the experiment 4Predict by calculation Non-existing material Create interesting theories Study difficult
3、conditions High pressure,high temperature,rapid coolingSolid state physics(SAITO)61.Review of vibration-Harmonic oscillations-Particles placed periodically-“wave number”and“density of state”Solid state physics(SAITO)7Review of vibration4Oscillation of one particle Equation of motionKxxm2 Initial con
4、dition0)0(xInherent oscillationmKtAx2,sinxSolid state physics(SAITO)8Solution of one particle vibration4Simple harmonic motion(inherent oscillation)Forces of two springs operating in phaseKxxm2 Normal modemKtAx2,sin0)0(xtxxSolid state physics(SAITO)9Two particle vibration4Applying symmetry()+()and()
5、-()（What symmetry？）and()is invariant under translating Simultaneous differential equation)(2111xxKKxxm 211xxX)(1222xxKKxxm 212xxX1x1x2x2x()()()Solid state physics(SAITO)10Solution of two particle oscillationAdd each side of()substract each side of()Two degree of freedomTwo normal modesSimultaneous d
6、ifferential equations)(2111xxKKxxm)(1222xxKKxxm)()(2121xxKxxm )(3)(2121xxKxxm 11KXXm 223KXXm mK1mK321x2x()Solid state physics(SAITO)11Normal modesCenter of spring doesnt stretch or shrinkCenter of spring shrinks twice as long as others“acoustic”“optical”11KXXm 1x223KXXm mK1mK311x2x2xSolid state phys
7、ics(SAITO)12N-1 particles vibrationDefine walls as fixed particles-equivalent equations(Here we assume to take only real part of )Use)()(11 xxKxxKxm)1,1(n00nxx2x1x0 xnx1nx)(tkaiAex,2 xxxexika1xSolid state physics(SAITO)13Solution(1)dispersion relation：dispersion relation)(Re)(tkaiAex2x1x0 xnx1nx)()(
8、11 xxKxxKxmxeKxeKxmikaika)1()1(2)cos1(22kaKm2sin42kaK)(k2sin2)(kamKk Solid state physics(SAITO)14Solution(2)wave vectorBoundary conditionapply atdegrees of freedom?(Corresponding to translational motion)2x1x0 xnx1nx)(Re)(tkaiAex00nxx0t,0Re0AxibA n0p)1,2,1,0(,npnpka0)sin()Re(nkabibexinkan)1(,2,0nnka2
9、sin2)(kamKk mKk2)(kaexceptSolid state physics(SAITO)15Check!(1)In the case of(2)In the case of 11n2sin2)(kamKk)1,2,1,0(,npnpka2x1x0 xnx1nx21n2kamKk2)(32,3kamKmKk3,)(1x2x1xSolid state physics(SAITO)16Wave number and wavelength4 discrete,periodicIn the case ofOne state for everyInvariant for)1,2,1(,np
10、npka0t)(Re)(tkaiAex)sin(kab21n32,3kaaak32,62nakakk21x1x2x2xa6a3kSolid state physics(SAITO)17Density of states(One state for everystates betweenstates between)Flat dispersionDivergence（）ddNnak2sin2)(kamKk kakamKk2)(,nadkdNdkkkdNdknadNdNdddkdkdNddN2cos1kamKnmKnddNSolid state physics(SAITO)18Actual lat
11、tice vibration4Longitudinal wave oscillation in the z(direction)4Transverse wave 2 in x and y direction43N eigen-values（：atom number）yzxyxzlongitudinaltransversalIn the case of molecules:eigenvalues translation degrees+rotation degrees=6Solid state physics(SAITO)19Summary of N-1 atoms4Solution of 4W
12、ave number k has equal distance4Wave number k:periodicity4Dispersion relation 4Density of statesSimilar to electronic states of solids)(Re)(tkaiAexakk2nak2sin2)(kamKk ddNSolid state physics(SAITO)20Problems for“review of oscillation”42N-1 particles(mass:m,M)are placed alternately 4Two springs(spring
13、 constant:K,K )are placed alternately4Transverse wave is represented by other spring constant4More than three kinds of particles are placed periodically4Springs are placed in two or three dimension4Springs are placed except square shape at the previous problemDerive the dispersion relation and density of states in following cases.Use the same conditions as in this class.12
