Thermal fluctuation at finite temperature u--atomic displacement, -potential energy Home Page Newton equation Title Page ∑ Isa,I's'Bul's'B (5.1.3) Contents Force constant-PIsa, 's'B=a2/aulsaauvs'Blo, periodicity → PIsa, ''B=osa,Is,B 4 with Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 6 of 41 Go Back Full Screen Close Quit Thermal fluctuation at finite temperature u—atomic displacement, Φ—potential energy Newton equation Ms ∂ 2 ∂t2 ulsα = − X l 0s 0β Φlsα,l0s 0βul 0s 0β (5.1.3) Force constant—Φlsα,l0s 0β ≡ ∂ 2Φ/∂ulsα∂ul 0s 0β|0, periodicity Φlsα,l0s 0β = Φ0sα,¯ls0β , (5.1.4) with ¯l = l 0 − l
Electromagnetic waves, Maxwell equations 10B V×E= V·B=0 at V×H1D4xr +-,V.D=4丌p Four coefficients related to materials: 6, A,x,Xm Home Page Title Page D=EE=E+4TP=(1+4TXE Contents B=H=H+4丌M=(1+4xxm)H In general, 6, u, x and Xm, tensors; Strong fields, nonlinear polarization → Go Back Full Screen
Home Page Title Page Contents JJ II J I Page 7 of 41 Go Back Full Screen Close Quit Electromagnetic waves, Maxwell equations ∇ × E = − 1 c ∂B ∂t , ∇ · B = 0 ∇ × H = 1 c ∂D ∂t + 4π c j, ∇ · D = 4πρ Four coefficients related to materials: , µ, χ, χm D = E = E + 4πP = (1 + 4πχ)E B = µH = H + 4πM = (1 + 4πχm)H In general, , µ, χ and χm, tensors; Strong fields, nonlinear polarization
Electromagnetic waves, Maxwell equations 10B e=cOt, vB=0 VXH、1D4丌 Four coefficients related to material." V·D=4丌p E, p,X, Xm Home Page Titie D=E=E+4P=(1+4丌x)E Contents B=H=H+4M=(1+4丌xm)H In general, 6, u,x and xm, tensors; Strong fields, nonlinear polarization Electric displacement vector D, wave equation → 1 0-D +V× V (5.1.5 T Go Back u(r) and e(r), second-order tensors scalars Full Screen 14(r+)=1(),∈(r+U)=∈(7) Close hree other field quantities, E, B, H, equivalent
Home Page Title Page Contents JJ II J I Page 7 of 41 Go Back Full Screen Close Quit Electromagnetic waves, Maxwell equations ∇ × E = − 1 c ∂B ∂t , ∇ · B = 0 ∇ × H = 1 c ∂D ∂t + 4π c j, ∇ · D = 4πρ Four coefficients related to materials: , µ, χ, χm D = E = E + 4πP = (1 + 4πχ)E B = µH = H + 4πM = (1 + 4πχm)H In general, , µ, χ and χm, tensors; Strong fields, nonlinear polarization Electric displacement vector D, wave equation 1 c 2 ∂ 2D ∂t2 + ∇ × 1 µ(r) ∇ × D (r) = 0 (5.1.5) µ(r) and (r), second-order tensors → scalars µ(r + l) = µ(r), (r + l) = (r) (5.1.6) Three other field quantities, E, B, H, equivalent
5.1.2. Bloch Waves Some common traits for wave propagation in periodic structures Tuning condition Lattice separation a and Characteristic wavelengths A Electrons: A=(h2/2mE)/, ranges from lattice spacing to bulk size Home Page Lattice vibrations Title Page acoustic branches and optical branches Contents Wavelengths ranges from lattice spacing to bulk size Electromagnetic radiation r-ray(<0.4 A), X-ray(0.4 50 A), ultraviolet ray(50 x 4000 A), vis-4l ible light(4000 x 7000 A), infrared ray(0.76 n 600 um), to microwave and radio wave(>0. 1 mm) Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 8 of 41 Go Back Full Screen Close Quit 5.1.2. Bloch Waves Some common traits for wave propagation in periodic structures Tuning condition: Lattice separation a and Characteristic wavelengths λ Electrons: λ = (~ 2 /2mE) 1/2 , ranges from lattice spacing to bulk size Lattice vibrations: acoustic branches and optical branches Wavelengths ranges from lattice spacing to bulk size Electromagnetic radiation: γ-ray (< 0.4 ˚A), X-ray (0.4 ∼ 50 ˚A), ultraviolet ray (50 ∼ 4000 ˚A), visible light (4000 ∼ 7000 ˚A), infrared ray (0.76 ∼ 600 µm), to microwave and radio wave (> 0.1 mm)
Periodic structures, Bloch waves, an electron stationary equation 2m (r)vr)=Ev(r) ak(r)=uk(r)fi(r) (5.1.8) Home Page Titie k(r+l)=uk(r) (5.1.9) To determine f(r), consider k(r)l2,note k(r)2=|vk(r+D)2 Ifk(r+lI=If r) Go Back Bloch function Full Screen (5.1.10) loch theorem ak(r+l)=k(r)el (5.1.11)
Home Page Title Page Contents JJ II J I Page 9 of 41 Go Back Full Screen Close Quit Periodic structures, Bloch waves, an electron stationary equation − ~ 2 2m ∇ 2 + V (r) ψ(r) = Eψ(r) (5.1.7) ψk(r) = uk(r)fk(r) (5.1.8) uk(r + l) = uk(r) (5.1.9) To determine fk(r), consider |ψk(r)| 2 , note |ψk(r)| 2 = |ψk(r + l)| 2 get |fk(r + l)| 2 = |fk(r)| 2 Bloch function ψk(r) = uk(r)eik·r (5.1.10) Bloch theorem ψk(r + l) = ψk(r)eik·l (5.1.11)