1、Solid state physics (SAITO)1Solid state physicsTohoku UniversityRiichiro Saito Solid state physics (SAITO)1SoSolid state physics (SAITO)2Types of solidsCrystalTranslational symmetryAmorphousDisordered structuresQuasicrystalOrdered structureNo translation symmetryparacrystal（Al,Pd,Mn alloy）crystal（ox

2、ide super conductor)Amorphous material（example: ceramics）Multicrystal, surface, and particleSolid state physics (SAITO)2TySolid state physics (SAITO)3Solid state physics?Properties of existing materialsUnderstand experimental resultsGive direction to the experiment Predict by calculationNon-existing

3、 materialCreate interesting theoriesStudy difficult conditionsHigh pressure, high temperature, rapid coolingSolid state physics (SAITO)3So(日本東北大學固體物理學課件)Types-of-solids(日本東北大學固體物理學課件)Types-of-solidsSolid state physics (SAITO)61. Review of vibrationHarmonic oscillationsParticles placed periodically“w

4、ave number” and “density of state”Solid state physics (SAITO)61.Solid state physics (SAITO)7Review of vibrationOscillation of one particleEquation of motionInitial conditionInherent oscillationSolid state physics (SAITO)7ReSolid state physics (SAITO)8Solution of one particle vibrationSimple harmonic

5、 motion (inherent oscillation)Forces of two springs operating in phaseNormal modeSolid state physics (SAITO)8SoSolid state physics (SAITO)9Two particle vibrationApplying symmetry()+() and ()-()（What symmetry？）and() is invariant under translating Simultaneous differential equation()()()Solid state ph

6、ysics (SAITO)9TwSolid state physics (SAITO)10Solution of two particle oscillationAdd each side of ()substract each side of ()Two degree of freedomTwo normal modesSimultaneous differential equations()Solid state physics (SAITO)10SSolid state physics (SAITO)11Normal modesCenter of spring doesnt stretc

7、h or shrinkCenter of spring shrinks twice as long as others“acoustic”“optical”Solid state physics (SAITO)11NSolid state physics (SAITO)12N-1 particles vibrationDefine walls as fixed particles- equivalent equations(Here we assume to take only real part of )UseSolid state physics (SAITO)12NSolid state

8、 physics (SAITO)13Solution(1) dispersion relation：dispersion relationSolid state physics (SAITO)13SSolid state physics (SAITO)14Solution(2) wave vectorBoundary conditionapply atdegrees of freedom?(Corresponding to translational motion)exceptSolid state physics (SAITO)14SSolid state physics (SAITO)15

9、Check !(1) In the case of (2) In the case of Solid state physics (SAITO)15CSolid state physics (SAITO)16Wave number and wavelength discrete, periodicIn the case ofOne state for everyInvariant forSolid state physics (SAITO)16WSolid state physics (SAITO)17Density of states(One state for everystates be

10、tweenstates between)Flat dispersionDivergence（）Solid state physics (SAITO)17DSolid state physics (SAITO)18Actual lattice vibrationLongitudinal wave oscillation in the z (direction)Transverse wave 2 in x and y direction3N eigen-values（：atom number）yzxyxzlongitudinaltransversalIn the case of molecules

11、 : eigenvalues translation degrees + rotation degrees = 6Solid state physics (SAITO)18ASolid state physics (SAITO)19Summary of N-1 atomsSolution of Wave number k has equal distanceWave number k : periodicityDispersion relation Density of statesSimilar to electronic states of solidsSolid state physic

12、s (SAITO)19SSolid state physics (SAITO)20Problems for “review of oscillation”2N-1 particles (mass : m, M) are placed alternately Two springs (spring constant : K , K ) are placed alternatelyTransverse wave is represented by other spring constantMore than three kinds of particles are placed periodicallySprings are placed in two or three dimensionSprings are plac