ëoã…éqä‘ëäå›çÏóp.pdf

1 2 1.1 2 1.2 4 1.3 10 1.4 13 2 14 2.1 16 2.1.a 1 16 2.1.b 18 2.1.c 2 20 2.1.d 24 2.1.e 24 2.1.f 25 2.2 Scheffler 25 2.3 Person Ryberg - 1 27 2.4 Person Ryberg - 2 Coherent Potential Approximation 31 2.5 33 2.6 own image 33 2.7 35 3 SFG 35 3.1 IRAS 35 3.2 36 3.3 40 3.4 40 3.5 Own Image 40 3.6 40 4 40 A. 42 B. image charge image dipole 44 C. image 47 D. image charge 49 E. 53 F. 54-1

1 1 1 1.1. m x(t) -κx β dx dt F(t) m d 2 x dt 2 = κx β dx dt + F (t) (1.1a) κ m = ω 2 0 β = 2γ F(t) = f (t) m m d 2 x dt 2 + 2γ dx dt + ω 2 0x = f (t) (1.1b) q Ecos(ωt) f (t) = qe m cosωt = qe m exp(iωt) + exp( iωt) 2 (1.1b) x(t) x(t) = qe 2m = qe m exp(iωt) ω 2 o ω 2 + 2iγω + exp( iωt) ω 2 o ω 2 2iγω 1 (ω 2 ω 0 2 ) 2 + (2γω ) 2 (ω 2 ω 0 2 ) cosωt (2γω) sinωt [ ] + q - q x q x qx - q + q - 2

p(t) = qx (t) = q 2 E 2m = q2 E m exp(iωt) ω 2 o ω 2 + 2iγω + exp( iωt) ω o 2 ω 2 2iγω (1.2a) 1 (ω 2 ω 0 2 ) 2 + (2γω ) 2 (ω 2 ω 0 2 ) cosωt (2γω) sinωt [ ] (1.2b) x(t) q m N N exp(+iωt) v.s. exp(-iωt) (1.2a) [ ] 2 cos(ωt) sin(ωt) exp(+iωt) exp(-iωt) cos(ωt) x(t) (1.2) (1.2b) (1.2b) Acosωt - Bsinωt = (A 2 +B 2 ) 1/2 cos(ωt + δ) tanδ = B/A δ = tan -1 [2γω/(ω 2 - ω 0 2 )] (1.2b) δ (1.2a) 2 +ω -ω +ω -ω * exp(+iωt) exp(-iωt) exp(+iωt) exp(-iωt) -iωt i j +iωt exp(-iωt) * (1.2a) exp(iωt) ω 2 0 ω 2 + 2iγω = [cos(ωt) + isin(ωt)][(ω 2 ω 2 ) 2iγω] 0 (ω 2 0 ω 2 ) 2 + 4γ 2 ω 2 [(ω 2 ω 0 2 ) cos(ωt) 2γωsin (ωt)] (ω 0 2 ω 2 ) 2 + 4γ 2 ω 2 -ω (1.2b) a = a + ia a* = a - ia (1/2)(a + a*) = a Re(a) = a - 3

1.2. (1.2) ω χ ij (1 ) (ω) = P i (1) (ω) / E j (ω) = N e2 h (r i ) ng (r j ) gn [ (r j) ng (r i ) gn ] ω + ω ng + iγ ng ω ω ng + iγ ρ ( 0), g ng gn [Y. R. Shen: "The Principles of Nonlinear Optics" (Wiley, 1984), Sec. 2.2] (1.3) Shen χ ij (1) (ω) (ω) χ ij (1) (ω) exp(-iωt) hω ng g n (E n - E g ) (1.3) exp(-iωt) (1.2a) [ ] 2 (r i ) ng (r j ) gn (1.3) [ (r ) (r ) j ng i gn ] 2 ω + ω ng + iγ ng ω ω ng + iγ ng q 2 r q 1 (ω + ω ng ) + iγ ng 1 (ω ω ng ) + iγ ng = 2ω ng (ω + iγ ng ) 2 (ω ng ) 2 2ω ng = ω 2 ω 2 2 ng + Γ ng ( ) + 2iωΓ ng (1.4) (1.2a) exp(iωt) ω 2 o ω 2 + 2iγω + exp( iωt) ω 2 o ω 2 2 exp(-iωt) 2iγω (1.2a) 2 (1.4) ω ω 2 0 = Γ 2 2 ng + ω ng 1.5 (1.4) (1.2a) 1 (1.4) (1.3) - 4

dρ dt [ ] 1 ( Γρ + ργ ) 2 (1.6) = 1 ih H, ρ (1.6) 1 (1.3) ( ) FID y v = i( ρ ng ρ gn ) (1.7) v + 2Γ ng v + ω 2 2 ( ng + Γ ng )v = 0 (1.8) e Γ ng t cos(ω ng t) (1.9) (ω ng ) 2 7.3 ~ 7.5 Γ ng = Γ gn γ γ = 2Γ ng Bloembergen "Nonlinear Optics" (Benjamin Press) 2.1 ~ 2.2 C. P. Slichter Principles of Magnetic Resonance (Springer-Verlag, 2 nd Ed. 1978) 5.4 ~ 5.8 Liouville Y. R. Shen (2.1 ) ρ n,n ' t relax = Γ n,n ' ρ n,n' - 5

n" Γ n,n" ρ n",n Γρ (1/2)( Γρ + Γρ) (1.1b) e γt cos( ω 2 0 γ 2 t) (1.10) (1.9) (1.10) γ Γ ng (1.11) ω 2 0 Γ 2 2 ng + ω ng (1.12) ( ) (1.4) ω 2 = ω 2 2 2 ( ng + Γ ng ) 2Γ ng (1.13) 1 ω 2 o ω 2 2iγω (1.14) ω ω 2 = ω 0 2 2γ 2 (1.15) (1.15) (1.11) (1.12) ω ω 0 ω + ω 0 2ω 0 1 (ω ω 0 ) iγ - 6

ω ω = ω 0 ω 2 2 - ω 0 ω 2 (ω 2 ng + Γ 2 ng ) ω 2 γ 2 ω 2 ng 2 2 γ Γ ng ω 0 ω ng + Γ ng 2 (1.1b) dx dt 2γ Γ ng ( ) dρ dt [ ] 1 ( Γρ + ργ ) 2 = 1 ih H, ρ ρ ng 1 ρ ng Γ ng Tr[ρx] e γt cos( ω 2 0 γ 2 t) γ Γ ng γ Γ ng 2ω ng α(ω) = α e + α v ω 2 ω 2 ng Γ 2 (1.16) ng + 2iωΓ ng (1.4) (1.3) ω e v e v ω ng 2 + Γ ng 2 ω 0, Γ ng γ (1.16) - 7

2ω α( ω) ng = α e + α V ω 2 ω 2 0 + 2iωγ = α 2ω ng e + α V ω 2 0 ω 2 2iωγ α V 2ω ng = α e + ω 2 0 1 ω ω 0 [ ( ) 2 2i( ω ω 0 )( γ ω 0 )] = α e + 2 α V 2ω ng ω 0 1 ( ω ω 0 ) 2 2i ω ω 0 = α e + 2 α V 2ω ng ω 0 1 ( ω ω 0 ) 2 2i ω ω 0 α = α e + V 1 ( ω ω 0 ) 2 i γ ( ω ω 0 ) [ ( )( γ ω 0 )] [ ( )( γ ω 0 )] (1.17) α v 2ω ng /ω 0 2 α v 2γ /ω 0 γ 2 (1.3) (1.16) (1.3) 2 ω g n 1 2 ω - ω ng Γ (1.2a) (1.3) 2 1 2 (1.2b) (1.4) 2 (1.2a) (1.3) 1 2 1 2 2 (1.2a) (1.3) 10 14 Hz ~ 10 15 Hz 100 cm -1-8

= 100 30 10 9 Hz = 3 10 12 Hz 2 (1.3) Maxwell-Schroedinger (ω - ω 0 ) (ω + ω 0 ) (ω + ω 0 ) (ω - ω 0 ) (ω - ω 0 ) (ω - ω 0 ) 1 - (ω /ω 0) 0 ω ω 0 (ω /ω 0) 1 1 - (ω /ω 0) 2 = (1 + ω /ω 0)(1 - ω /ω 0) 2(1 - ω /ω 0) (1.17) α( ω) α e + α V / 2 1 (ω /ω 0 ) i γ / 2 = α e + ω 0 α V / 2 ω 0 ω ω 0 iγ /2 α V 2ω ng ω 0 2 α V 2γ ω 0 γ ω ng ω 0 α α( ω) α e + V ( ω 0 ω) iγ (1.18) (1.3) 2 (1.2a) (1.2a) - 9

N. Bloembergen "Nonlinear Optics" (Benjamin, 3rd Ed.:1977) 1 e m x ω 0 +Vx 2 ω 1 ω 2 (1.1b) d 2 x dt 2 + 2Γ dx dt + ω 2 0x + Vx 2 = 2e m Re [ E 1 exp(ik 1 z iω 1 t) + E 2 exp(ik 2 z iω 2 t) ] (1.19) Y. R. Shen, "The Principles of Nonlinear Optics" (John Wiley & Sons, 1984) 1.3 2 1 2 ) 2 2 2 2 Maxwell-Schrödinger 2 2 2 1.3. ( ) - 10

2 E(t) = E 0 cos(ωt) x(t) = x 0 cos(ωt) ωt -π/2 +π/2 -π/2 0 x(t) 0 +π/2 x(t) = x 0 cos(ωt + π/2) = x 0 sin(ωt) ωt -π/2 +π/2 x(t) x(t) = x c cos(ωt) + x s sin(ωt) x s sin(ωt) x(t) = Re[A 0 exp(-iωt)] A 0 A 0 = x c + ix s x(t) = Re{(x c + ix s )[cos(ωt) isin(ωt)]} = x c cos(ωt) + x s sin(ωt) A 0 x s E q Eqx qx (Bloembergen's textbook 1.3 3.1 ) E P W W = 1 T T 0 P(t) [E(t) ]dt t (1.20) - 11

ω exp(+iωt) exp(-iωt) (1.3) (1.16) exp(-iωt) E(ω; t) = E 0 (ω)[exp(-iωt) + exp(+iωt)], P(ω; t) = P 0 (ω)exp(-iωt) + P 0 (ω)* exp(+iωt) = [P 0 (ω) + P 0 (ω)*]cosωt + i[p 0 (ω) - P 0 (ω)*]sinωt (1.21) E(ω; t) E 0 (ω) P(ω; t) P(ω; t) P 0 (ω)* P 0 (ω) P 0 (ω) P(ω; t) / t = -iω[p 0 (ω)exp(-iωt) - P 0 (ω)* exp(+iωt)] W = -iωe 0 (ω)(1/t) 0 T dt{p 0 (ω)[exp(-2iωt) 1} - P 0 (ω)* [exp(+2iωt) 1)] W = -iωe 0 (ω)[p 0 (ω) - P 0 (ω)*] = ωe 0 (ω)im[p 0 (ω)] = ωim[α(ω)] E 0 (ω) 2 = ωim[α(ω)]i 0 (ω) (1.22) (1.22) 90 E 90 P α(ω) 2 mv 2 /2 kx 2 /2 2 2-12

ω 2 2 2 Im[P 0 (ω)] P 0 (ω) 2 1.4. 4 6 (1.22) SFG IR VIS UV UV IR VIS SFG IR + VIS UV Bloembergen 1 2 2 SFG 1 SFG I SFG (ω SF ) β SFG (ω SF ) 2 (1.23) SFG VIS IR SFG SFG χ (2) SFG E S E L 2 E S + E L = ISFG + I L + ( E S E L + c.c. ) - 13

E L E L SFG E L E S SFG SFG SFG SFG SFG E L E L E S E L E L E L E L SFG SFG α e SFG α e (ω SFG ) SFG 2 1 R. A. Hammaker et al., Spectrochim. Acta 21, 1295(1965); Crossley and King, Surf. Sci. 68, 528(1977); Scheffler, Surf. Sci. 81, 562 (1979) Persson Ryberg Phys. Rev. B, 24, 6954 (1981) A B ~ D Persson & Ryberg E F p d f 3 5 7 1 CO H 2 O H 2 CO 2 CH 4-14

1 1 J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 2nd Ed., 1975) 4.4-15

- ( ) 2.1 2.1.a. 1 1 m κ β F(t) m d 2 x dt 2 = κx β dx + F (t) (2.1.1) dt κ m = ω 2 0, β F(t) = 2γ, = f (t) q Ecos(ωt) m m f (t) = qe m cosωt = qe m exp(iωt) + exp( iωt) 2 2.1.2 d 2 x dt 2 + 2γ dx dt + ω 2 0x = f (t) 2.1.3 (1.2a) (1.2b) - 16

p(t) = qx (t) = q 2 E 2m = q2 E m exp(iωt) ω 2 o ω 2 + 2iγω + exp( iωt) ω o 2 ω 2 2iγω 2.1.4 1 (ω 2 ω 0 2 ) 2 + (2γω ) 2 (ω 2 ω 0 2 ) cosωt (2γω) sinωt [ ] 2.1.5 p(t) exp(-iωt) 1 α e p(t) = α (ω )E exp( iωt) 2.1.6 ˆ 2 α α( ω) α e + V ω 0 α ˆ ω 2 0 ω 2 iωω 0 γ ˆ = α e + V 1 ω ω ω 0 + iγ ˆ ω 0 2.1.7 (2.1.7) 2γ γ ˆ ω 0 q 2 2m 2 α ˆ V ω 0 α(ω) α e + α e + α ˆ V ω 0 2(ω 0 ω) iω 0 γ ˆ = α e + α V ( ω 0 ω) iγ ω ~ ω 0 α ˆ V ω 0 2 ( ω 0 ω) iω 0 γ ˆ 2 2.1.8 2.1.9 α V ˆ α V ω 0 2 = q 2 (4mω 0 ), 2.1.10 (2.1.9) (1.17) (2.1.9) γ (2.1.3) γ α e (1) (2) (3) (3a) π * (3b) π/2 * (3c) * (2.1.9) (3) (2.1.9) ω (3) P & R - 17

F 0 cos(ωt) ωt = 0, 2π, ωt = π/2, 3π/2, x(t) = x 0 sin ωt = x 0 cos(ωt - π/2) π/2 x(t) = -x 0 cos ωt = x 0 cos(ωt - π) π x(t) = x 0 cos(ωt - δ) δ (3a) (3b) (3c) π π/2 (a) π (b) π/2 π/2 2.1.b. 2 1 p 1 = q 1 x 1 x 1 m d 2 x 1 dx dt 2 = κ 1 x 1 β 1 1 + F dt 1 (t) + C 12 x 2 2.1.11 (2.1.1) p 2 = q 2 x 2 1 q 1 A 2 E r r p = 2 d 3 C 12 x 2 = q 1 E = q 1 q 2 x 2 d 3 C 12 = q 1q 2 d 3 C 12 = C 21 = -C C * x 2 x 1 x 1 * (1) (2) - 18

m κ β 2 m d 2 x 1 dt 2 = κx 1 β dx 1 dt + F (t) Cx 2 2.1.12 m d 2 x 2 dt 2 = κx 2 β dx 2 + F (t) Cx dt 1 2.1.13 (2.1.12) (2.1.13) x 1 x 2 3 x 1 x 2 x + = x 1 + x 2 x = x 1 x 2 (2.1.12) (2.1.13) x + x - m d 2 x + dt 2 = ( κ + C) x + β dx + dt + 2F(t) 2.1.14 m d 2 x dt 2 = ( κ C) x β dx dt 2.1.15 (2.1.14 ) (2.1.15) (1) CO 2 CO (2) ( κ + C) m ( κ C) m (3) ω 0 = κ m 2 ω 0 C m ω 0 C κ ( ) C d 2 (4) (2.1.14) x + 2 (2.1.15) x - (5) x + ω 0 (ω 0 / 2)C κ 1 (4) (5) x + 2-19

in-phase x - 2 out-of-phase 2 (1) 2 (2) ω 0,± 2 = m 1 + m 2 2m 1 m 2 κ ± 1 2 m 1 m 2 m 1 m 2 κ 2 + 2 C m 1 m 2 2 2.1.16 (3) 2 (4) (5) (6) 2 2.1.c 2 A i i E i E i 2 B j p j i -U ij p j p i = α A [E i - U ij p j ] 2.1.17a α A A U ij = 1/ r i - r j 3 = 1/d 3 ( B) 1 ( A) p j = α B [E j - U ji p i ] 2.1.17b j E j j B U ij = U ji u 2 p j p i = α A [E i - uα B [E j - up i ]] - 20

= α A E i - α A α B ue j + α A α B u 2 p i 2.1.18 p i = (α A E i - α A α B ue j )(1 - α A α B u 2 ) -1 2.1.19 i j E i p i = (α A - α A α B u)e i (1 - α A α B u 2 ) -1 = α A E i (1 - α B u)(1 - α A α B u 2 ) -1 2.1.20 2 α A = α B p i = α A E i (1 - α A u)(1 - α A 2 u 2 ) -1 = α A E i (1 + α A u) -1 2.1.21 α A α B 2 2 (2.1.20) P = p 1 + p 2 = [α A (1 - α B u) + α B (1 - a A u)](1 - α A α B u 2 ) -1 E i = [(α A + α B ) - 2α A α B u](1 - α A α B u 2 ) -1 E i 2.1.22 2 2 (2.1.21) P = p 1 + p 2 = [α A (1 - α A u) + α A (1 - α A u)](1 - α A 2 u 2 ) -1 E i = 2α A (1 - α A u)](1 - α A 2 u 2 ) -1 E i = 2α A (1 + α A u) -1 E i 2.1.23 (2.1.23) 1 α A α 0 = α A 1 + α A u 2.1.24 ( SCF Self-consistent Field SCF CPA Coherent Potenshal Approximation ) 2.1.a 2.1.b 2.1.a ω - 21

p(t) = qx (t) = q 2 E 2m exp( iωt) ω 2 A ω 2 2iγω 2.1.25 α A = q 2 1 α 2m ω 2 A ω 2 = V 2iγω 1 ω ω A ( ) 2 i ( ) γ ω ω A 2.1.26 α q 2 V = 2 2.127 2mω A 2.1.b E = q 2 x 2 d 3 = C 12 x 2 q 1 = C x 2 q 1 E = C x 2 q 1 = C q 2 x 2 ( ) ( q 2 q 1 ) = C p 2 ( q 1 q 2 ) = up 2 i 1 j 2 q 1 = q 2 = q u = C /q 2 α q 2 V = 2 = C u 2 2mω A 2mω A = C u 2κ 2.1.28 (2.1.24) (2.1.29) P & R Ω 2 = ω A 2 (1+ α V u) 2.1.29) u P & R P & R (2.1.28) (2.1.29) Ω 2 = ω A 2 (1+ α V u) = ω A 2 (1+ C 2κ ) 2.1.30 2.1.b 1 2 2 SCF 2 (2.1.17a) p i = α A [E i - U ij p j ] p = α A [E i - up] 2.1.17c - 22

p p + α A up = (1 + α A u)p =α A E i α 0 = α A 1 + α A u (2.1.24) (2.1.24) p (2.1.17c) (1) (CPA) (2) 2 p A = α A [E i - U Aj p j ] = α A [E i - U Aj α j (E j U ja p A )] = α A {E i - U Aj α j [E j - U ja α A (E i - U Aj )] = α A {E i [1 + U Aj U ja α j α A + + (U Aj U ja α j α A ) n ]+ (U Aj U ja α j α A ) n +1 α n A p A - U Aj α j {E j [1 + U Aj U ja α j α A + + (U Aj U ja α j α A ) n ] - (U Aj U ja α j α A ) n+1 α n j p j } p j (U Aj U ja α j α A ) n+1 n α j U Aj U ja α j α A < 1 U Aj = U ja p A = [α A /(1 - U Aj 2 α j α A )](E i - U Aj α j E j ) 2.131a) p j = [α j /(1 - U Aj 2 α j α A )](E j - U Aj α A E i ) 2.1.31b) U Aj 2 α j α A 1 (2.1.24) - 23

2.1.d H = r=1,f [ 1 2 mx 2 r + 1 2 kx 2 r + ( qx r )E] 2.1.32 x r H dd = 1 2 ( qx r )( qx s ) r,s 1 r r r s 3 2.1.33 x r x s (2.1.33) x r x s r s 3 2 1/8 d d 2 1/4 (2.31) d 3 21.e N N j sin jnπ /N ( ) (j = 1,, N) j = 1 j = 2 n = 1 ~ N/2 n = N/2 ~ N j = 3-24

2/3 1/3 1/3 1, 0, 1/3, 0.1/5, 0, 1/7, sin -1 (ω - ω ) 2.1.f CPA(coherent potential approximation) 2 2.2 Scheffler 2.1 Scheffler M. Scheffler Surf. Sci., 81, 562(1975) R i ω p(r i, ω, t) = p st + α(ω)e local (R i, ω, t) (2.2.1) p st 1 α(ω) 2 E local (R i, ω, t) i 1 exp[+iωt] E(r, ω, t) = E 0 exp[i(ωt - k r)], k = 2π/λ 1 E local E - 25

1 E local E i own E image i other E dipole i other E image i i E local i = E i + E i own image + E i other dipole + E i other image (2.2.2) Persson & Ryberg A ( ) (2.2.1) p(r i, ω, t) = p st /[1 + α(ω)(s(θ) - 1/4d 3 )] + α(ω)e j (R i, ω, t)/[1 + α(ω)(s(θ) - 1/4d 3 )] (2.2.3) S(θ) = k i 1/ R k - R i 3 + 1/( R k - R i 2 +4d 2 ) 3/2-12d 2 /( R k - R i 2 +4d 2 ) 5/2 (2.2.4) (2.2.3) 1 2 p(ω) = E 0 j(ω)α(ω)/[1 + α(ω)(s(θ) - 1/4d 3 ) (2.2.5) IRAS far field R R = <[Re(r p E 0 + E dipole + E images )] 2 > - <[Re(r p E 0 )] 2 > = 2<Re(r p E 0 (ω, t))(e dipole (ω, t) + E images (ω, t))> + <Re(ω dipole (ω, t) + E images (ω, t)) 2 > (2.2.6) < > r p E 0 p(r i, ω, t) r -{d 2 /dt 2 [p(r i, ω, t - r - R i /c)]}/(c 2 r - R i ) + [{d 2 /dt 2 [p(r i, ω, t - r - R i /c)]}(r - R i )](r - R i )/(c 2 r - R i 3 ) (2.2.6) 1 R ~ nω 2 [Re(p(ω))Re(r p ) - Im(p(ω))Im(r p )] (grazing angle) Re(r p ) ~ 0-26

R ~ -nω 2 Im(p(ω))Im(r p ) Im(p(ω)) (2.2.7) (2.2.5) (2.2.7) Scheffler α(ω) = α e + α v /[ω - ω ng + iγ] (2.2.8) α(ω)/[1 + α(ω)(s(θ) - 1/4d 3 )] = [1/( S(θ) - 1/4d 3 )]{1 - (S(θ) - 1/4d 3 )/[ α(ω) + 1/( S(θ)- 1/4d 3 )] (2.2.9) [1/(1 + α e (S(θ) - 1/4d 3 )] {α e + [α v /(1 + α e (S(θ) - 1/4d 3 )]/[ ω - ω ng + α v (S(θ) - 1/4d 3 )/(1 + α e (S(θ) - 1/4d 3 )) + iγ]} (2.2.10) (2.2.8) α(ω) ω = ω n 2 α e ω n (2.2.9) ω ω n α v (S(θ) 1/4d 3 )/(1+α e (S(θ) 1/4d 3 )) IRAS S(θ) SFG SFG 2 2.3 Persson & Ryberg - 1 Persson Ryberg B. N. J. Persson and R. Ryberg, Phys. Rev. B 24, 6954(1981) 100 % 2 2.3. 2.5 2.6 (1) 1 1-27

(2) SCF 2.6 1 (own image) real dipole image dipole P & R U(q) (a) (b) own image P & R SCF SCF P & R i p i j p j (2.17a) p i p j p i = α A [E i U ij p j ] (2.3.1) j i α A A, B U ij j p j p j * i p i p i,z p i,r A -p i,z / r i - r j 3 +2p i,r / r i - r j 3 (2.3.2) r i r j i j 2.3.e image dipole p j * r i - r j P & R U ij - 28

p j p j * r i -U ij p j P & R q- p i = q p q exp(iq r i - iωt), E i = q E q exp(iq r i - iωt) (2.3.3) exp(-iωt) q q (2.32) (2.3.3) exp(-iωt) p i = q p q exp[ iq x i ], E i = E q q exp[ iq i x ] (2.3.4) (2.3.1) p q e iq x i = α A E q e iq x i U ij p q e iq x i q q j q = α A E q U ij p q exp[ i(q x i q x j )] q j e iq x i U ij (2.3.3) U q r [ ] = U ij exp iq (x i x j ) (2.3.5) ( ) = U ij exp i(q x i q x j ) j j [ ] exp(iq x i ) p q = α A [ E q U ( q ) p q ], (2.3.6) 2 2-29

(2.3.6) (1) 1 q q (2) 2 q q q q SCF p q α A p q = 1+ α A U q ( ) E q α 0 q,ω ( )E q 2.3.7 (2.3.5) U(q) q U ij i (2.3.5) U(q) q q = 0 U(0) 2.1.a ˆ 2 α α( ω) α e + V ω 0 α ˆ ω 2 0 ω 2 iωω 0 γ ˆ = α e + V 1 ω ω ω 0 + iγ ˆ ω 0 (1) ω A ω A 1+ α VU ( 0) 1+α e U (0) ω α A 1+ 1 V U ( 0) 2 1+α ω A 1+ 1 2 α V U 0 e U (0) [ ( )] 2.3.8-30

α V U ( 0) <<1 α e U ( 0) <<1 2.1. (2) (3) α v U/(1 + α e U) 2 U U TPD Clausius-Mossoti 2.4 Persson & Ryberg - 2 Coherent Potential Approximation Persson & Ryberg (1) (2) q = 0 1 P & R 2.1.c 2 2 (1) 3 (2) 4 1950 coherent potential approximation (CPA ) CPA (1) i α i α i (2) (stochastic) P & R CPA (1) i r i (2) i A B i A B (a) i - 31

i (b) c A :c B (c A + c B = 1) (c) (d) iteration P & R CPA CPA p A p B c A p A + c B p B r i p P & R p A p B A, B r i (1) 2 2 1 (2) 12 CO 13 CO Pd(110) 12 CO 30 % 2 ( M. W. Urban, "Vibrational Spectroscopy of Molecules and Macromolecules on Surfaces", John Wiley & Sons, 1993, Sec. 2.4 ) q Q P & R c 2 2 CO/Cu(100) U(q) = U 0 [1 + A(q/q 0 ) + B(q/q 0 ) 2 ] CO Q = 1 α {1 ξ 0.2 [ln(ξ 1.2 ξ + 1 ) + 1 ξ 1) ln((ζ )]} (2.4.1) (ζ + 1) ξ = 1 αu 0, ζ = 0.2 ξ 1.2 (2.4.2) c 2 2 CO/Cu(100) A Q = 1 α {1 ξ + 1+ A + B [ln(ξ ) A 1 (ζ 1 B ξ + 1 2B ξ ln( 2B )(ζ + A 2B ) (ζ + 1+ A 2B )(ζ A )]} (2.4.3) 2B ) ξ = 1, ζ = 1 αu 0 B ( A 2 1 ξ) (2.4.4) 4B - 32

(2.4.1) (2.4.2) c 2 2 CO/Cu(100) U 0 = 0.3 Å -3 A = -2.4 B = 1.2 (2.4.3) (2.4.4) q a = 3.6 Å [πq 2 0 = (2π/a) 2 ] d = 0.8 Å Ni(111) U(q) 2 SFG P & R (1) Cu(100) 2π/a q 0 (2) q = (1, 0) q = (1, 1) U(q) U 0 A B (3) U/(1 + au) q x z 2.5 2 detailed balance 2.4 A B A 2 P & R intensity borrowing intensity suppression 2.6 own image own image P & R explicit 2.1 2-33

p A kp A r i Xkp A p A p A = α A {E i [1 + Xkα A + + (Xkα A ) n ]+ (Xkα A ) n+1 p A } (2.6.1) Xkα A < 1 p A = [α A /(1 - Xkα A )]E i (2.6.2) A k = 1, X = 1/4d 3 k = -1, X = -1/8d 3 p A, z = [α A /(1 - α A /4d 3 )]E i, p A, r = [α A /(1 - α A /8d 3 )]E i k = -(ε 1 - ε 2 )/(ε 1 + ε 2 ) X = +1/4d 3 k = +(ε 1 - ε 2 )/(ε 1 + ε 2 ) X = -1/8d 3 α A = α e + α v /[ω - ω A + iγ] (2.6.2) p A (ω) = (1/(1 - kxα e )){ α e + [α v /(1 - kxα e )]/[ ω - ω A - kxα e /(1 - kxα e ) + iγ]}e i (2.6.3) (1) 1 kxα e /(1 - kxα e ) (2) RAS kxα e /(1 - kxα e ) 2 (2.3.1) U ij j p j p j * i A (A.1a) p i p i,z p i,r -p i,z / r i - r j 3 +2p i,r / r i - r j 3 r i r j i j image dipole p j * r i - r j P & R p j p j * r i p j E j - 34

U ij p j p j * r i -U ij p j 1/10 ( ) 2.7 1 2 3 3 SFG 3.1 IRAS IRAS SFG 2 SFG SFG SFG 3 (1) (2) IRAS SFG SFG (3) SFG (a) - 35

p i = α i (E i - U ij p j ) 1 SFG SFG (b) SFG 3.2 ω 1 ω 2 E ext ext 1 E 2 ω 1 ω 2 1, 2 SFG ω 3 = ω 1 + ω 2 3 i p i,1 p i, 2 i A α A,1 p i,1 = α 1 E i,1 U ij,1 p j,1 (3.1a) j p i,2 = α 2 E i,2 U ij,2 p j,2 (3.1b) j i modify E ext ext i,1, E i,2 E i,1 = E ext i,1 U ij,1 p j,1 j (3.2a) E i,2 = E ext i,2 U ik,2 p k,2 k (3.2b) SFG E i,3 SFG E i,3 = U il,3 p l,3 l (3.3) i SFG p i,3 ω 1 + ω 2 2 SFG i p i,3 = β 3 E i,1 E i,2 + α 3 E i,3 (3.4) (3.2a) (3.2b) (3.3) p i,3 = β 3 E ext ext i,1 U ij,1 p j,1 E i,2 j U ik,2 p k,2 k + α 3 U il,3 p l,3 (3.5) l 2 n = 1,2,3 p i,n = m = 1,2 E ext i,m = q p q,n e i( qr i ωt) E ext q',me i( q 'r i ωt) q ' - 36

(3.5) p q,3 e i( qr i ω 3 t) = β 3 E ext q',1e i( q'r i ω 1 t) U ij p q,1 e i(qr j ω 1 t) q q' j q E ext q',2e i( q'r i ω 2 t) U ik p q,2 e i( qr k ω 2 t) + α 3 U il p q,3 e i( qr l ω 3 t ) q' k q l q (3.6) U q = U ij e i( qr i qr j ) j U (q) n j q p q,1 e i( qr j ωt) = U ij e i( qr i qr j ) e +i(qr i qr j ) p q,1 e i( qr j ωt) U ij j q q j = ( U ij e i( qr i qr j ) ) p q,1 e i( qr i ωt ) = U q p q,1 e i( qr i ωt ) q j q (3.7) q 1 q 1 p q,3 e i( qr i ω 3 t) = β 3 E ext q,1e i( qr i ω 1 t) U q p q,1 e i( qr i ω 1 t) q q q E ext q',2e i( q'r i ω 2 t) U q' p q ',2 e i( q 'r i ω 2 t) + α 3 U q p q,3 e i( qr i ω 3 t) q' q' q (3.8) ω 1 + ω 2 = ω 3 ext E q,1e i(qr i ) E ext q ',2e i(q 'r i ) E ext q,1e i(qr i ) U q' p q ',2 e i( q 'r i ) p q,3 e i( qr i ) q q' q q' = β 3 q U q p q,1 e i( qr i ) E ext q',2e i( q'r i ) + U q p q,1 e i( qr i ) U q' p q ',2 e i( q'r i ) q q ' q q' + α 3 U q p q,3 e i( qr i ) q (3.9) e i( qr i ) E ext q 0,1e i( q 0r ) i E ext q', 2e i (q'r i ) E ext q 0,1e i (q 0r i ) U q' p q',2 e i(q'r i ) q' q' p q0,3 = β 3 U q0 p q 0,1e i (q 0r i ) E ext q', 2e i (q'r i ) + U q 0 p q0,1e i (q 0r i ) U q' p q',2 e i(q'r i ) q' q' +α 3 U q 0 p q0, 3e i( q 0r i ) ( ) (3.10) exp[-iq 0 r i ] q δ exp (argument) - 37

q e i( q 0 r i ) e i(q 'r i ) 1 p 0,3 = β 3 [ E ext 0,1 E ext 0,2 E ext 0,1U 0 p 0,2 U 0 p 0,1 E ext 0,2 + U 0 p 0,1 U 0 p 0,2 ] + α 3 ( U 0 p 0,3 ) (3.11) E p 0 p 3 +α 3 U 0 p 3 ( ) = [ 1+ α 3 U 0 ] p 3 = β 3 [ E ext 1 E ext 2 E ext 1U 0 p 2 U 0 p 1 E ext 2 +U 0 p 1 U 0 p 2 ] 3.12 (2.1.23) (2.1.24) p n = α 0,n E n ext, α 0,n = α n 1+ α n U 0 (n = 1, 2) 3.13 SFG [ 1+ α 3 U 0 ] p 3 ext ext ext ext = β 3 E ext 1 E ext 2 E ext α 1U 2 E 2 0 E ext α 2U 1 E 1 α 1 + α 2 U 0 + U 1 E 1 α 0 1+ α 1 U 0 U 2 E 2 0 1 + α 1 U 0 0 1 + α 2 U 0 = β 3 1 α U 1 0 α U 2 0 + α U 1 0 α 2 U 0 E ext 1 E ext 2 1 + α 1 U 0 1 + α 2 U 0 1 + α 1 U 0 1+ α 2 U 0 = β 3 1 + α 1 U 0 = β 3 ( )( 1 + α 2 U 0 ) α 2 U 0 ( 1+ α 1 U 0 ) α 1 U 0 ( 1 + α 2 U 0 ) + α 1 U 0 α 2 U 0 ( 1 + α 1 U 0 )( 1+ α 2 U 0 ) ( 1 + α 1 U 0 + α 2 U 0 + α 1 U 0 α 2 U 0 ) ( α 2 U 0 + α 1 U 0 α 2 U 0 ) ( α 1 U 0 + α 1 U 0 α 2 U 0 ) + α 1 U 0 α 2 U 0 ( 1 + α 1 U 0 )( 1 + α 2 U 0 ) 1 = β 3 1 + α 1 U 0 E ext 1 E ext 2 E ext 1 E ext 2 ( )( 1 + α 2 U 0 ) E ext 1 E ext 2 (3.14) 1 p 3 = β 3 1+ α 1 U 0 ( )( 1+ α 2 U 0 )( 1+ α 3 U 0 ) E ext 1 E ext 2 3.15 E ext n /(1+ α n U 0 ) SFG [E ext 1 / (1+ α 1 U 0 )][E ext 2 / (1+ α 2 U 0 )]β 3 SFG 1/(1+α 3 U 0 ) Y. R. Shen (2.52) - 38

SFG ω 1 E 1 ω 2 E 2 SFG ω 3 E 3 SFG 1) α 1 = α e (ω 1 ) 2) SFG α 3 = α e (ω 3 ) ω 3 α 1 α 3 3 α 3 3) IR 2.1 α 2 α e ( ω 2 ) + α V ( ) iγ ω A ω 2 (3.16) β 3 = β background + β A (ω A ω 2 ) iγ (3.17) SFG β p 3 = β background + A (ω A ω 2 ) iγ ( 1 +α e (ω 1 )U 0 ) 1 +α e ω 2 1 α V ( ) + ( ) iγ U 0 ( 1 + α (ω )U 3 0) ω A ω 2 = β A β background + (ω A ω 2 ) iγ 1 +α e ( ω 2 ) + α V ( ) iγ U 0 ω A ω 2 E ext 1 E ext 2 ( 1+ α e (ω 1 )U 0 )1 ( +α (ω 3 )U 0 ) 3.18 IR SFG 1 β background 2 p 3 2 SFG SFG β e (3) p 3 2 P & R - 39

derivation 3.3 (5.7) 2.4 SFG 2c 3.4 3.3 2.5 SFG 3.5 own image 2.6 SFG 2.6 2.6 negative entropy system 3.6 4. A α A SFG β A α A (ω) = α e α v ω ω v + iγ v = α e + β v α v ω v ω iγ v β A (ω ) = β e = β ω ω v + iγ e + v ω v ω iγ v β v 2 1 2 1 α A0 (ω) = (1 + k)α A α = (1 + k) α A e + Av ω 0 ω iγ V - 40

β β A0 (ω) = A (ω) [1- k SF Xα A,SF (ω)][1- k vis Xα A,vis (ω vis )][1- k IR Xα A,IR (ω IR )] = 1 (1- k SF Xα e,sf )(1- k vis Xα e,vis )(1 - k IR Xα e,ir ) β e + [β α β / (1- k Xα )] v v e IR e,ir α ω ω v v + iγ 1- k IR Xα v e,ir k = -(ε 1 - ε 2 )/( ε 1 + e 2 ) k = +(ε 1 - ε 2 )/( ε 1 + ε 2 ) ε 2 k = +1 k = -1 2 SFG f ( ) φ kp p // = p(1 - k)cosφ p = p (1 + k)sinφ α A0 (ω) = α A (ω ) 1 +α A (ω) 1 = 1 +α e U α + α v /(1+α e U) e ω ω v α vu 1+α e U + iγ v β β A0 (ω) = A (ω) [1 + α A,SF U][1 + α A,vis U ][1 + α A,IR (ω IR )U] 1 = [ [1 + α e,sf U ][1 + α e,vis U][1 + α e,ir U ] ] β + [β α β /(1+ α U)] v v e e,ir e α ω ω v v 1+ α e,ir U + iγ v - 41

U 2 2 α A (ω) A α B (ω) B α(ω) α(ω) c α(ω) = A α A (ω ) 1+ [α A (ω) α (ω)]q + c B α B (ω ) 1+ ([α B (ω) α (ω )]Q Q = U 1 +α(ω)u Q α (ω) α 0 (ω) = 1+α (ω )U = c Aα A (ω ) 1+α A (ω )U + c Bα B (ω) 1+α B (ω)u SFG SFG suppression IRAS U A (1) r' p(r') r E(r; p(r )) E(r; p(r' )) = p(r') 3(r r')[ p(r') (r r')] r r' 3 + r r' 5 (A.1a) (2) r p(r) r' E(r ; p(r)) - 42

E(r'; p(r)) = p(r) 3(r r')[ p(r) (r r')] r r' 3 + r r' 5 (A.1b) (3) r p(r) r = 0 E(0;p(r)) = p(r) 3r[ p(r) r] r 3 + r 5 (A.1c) (electrostatics) d 2 +q -q 2 qd r r' d qd B (x, y.z) z (0, 0, d) p (own image) p* (0, 0, d) ρ φ [ (rcosφ, rsinφ, d) ] (other dipole) p' (other image) p'* p p* p = p ρ + p z, p* = p* ρ + p* z x xz the dipole (0, 0, d) p ρ : (p ρ, 0, 0), p z ; (0, 0, p z ) own image (0, 0, -d) p* ρ : (p* ρ, 0, 0), p* z ; (0, 0, p* z ) other dipole (rcosφ, rsinφ, d) p' ρ : (p ρ, 0, 0), p' z ; (0, 0, p z ) other image (rcosφ, rsinφ, -d) p'* ρ : (p * ρ, 0, 0), p'* z ; (0, 0, p * z ) (A.2) p p* p* ρ = p ρ, p* z = -p z insulator surface: 0 < p* z < p z, -p ρ < p* ρ < 0 (when p ρ, p z > 0) own image p* other dipole p' other image p'* (0, 0, d) E own image = p* z /4d 3 - p* ρ /8d 3, [on ideal metal; E own im = p z /4d 3 + p ρ /8d 3 ) (A.3) E other dipole = -(p' z + p' ρ )/ρ 3 + 3p' ρ e r cosφ/ρ 3 (A.4) (e r = (cosφ, sinφ, 0), unit vector from (0, 0, d) to p') - 43

E other image = -(p'* z + p'* ρ )/(ρ 2 + 4d 2 ) 3/2 + (-6dp'* z + 3ρp'* ρ cosφ)e' r /(ρ 2 + 4d 2 ) 3 (e' r = (1/(ρ 2 + 4d 2 ) 1/2 )(ρcosφ, ρsinφ, 2d), unit vector from (0, 0, d) to p'*) [on ideal metal; E other image = -(p' z - p' ρ )/(ρ 2 + 4d 2 ) 3/2 + (-6dp' z - 3ρp' ρ cosφ)e' r /(ρ 2 + 4d 2 ) 3 ] (A.5) other dipoles z C 3 z (A.4) other dipoles cosφ sinφ p 1 p 2 2 V = p 1 p 2 r 3 ( )( p 2 r) + 3 p 1 r r 5 p 1 p 2 θ 1 p 1 r θ 2 p 2 r p 1 r p 2 φ θ 1 r θ 2 r V = p 1 p 2 r 3 ( 2 cosθ 1 cosθ 2 sinθ 1 sinθ 2 cos φ) B image charge image dipole (J. D. Jackson, Classical Electrodynamics, John Wiley and Sons Inc., NY, 2nd Ed., 1975, Chaps. 1~4 ) q image charge (1) (2) (2a) (2b) (Dirichlet (Neumann) - 44

image charge ε ε 1 1 q d 2 ε 2 q d q' 1 ε 1 q q' q' q (B.1a) 2 q (B.1b) q" ε 2 q" Sect. 4.4 of Jackson's textbook q' = [(ε 1 - ε 2 )/(ε 1 + ε 2 )]q q" = [2ε 2 /(ε 1 + ε 2 )]q (B.1a) (B.1b) cgs esu MKSA ε 1 ε 2 ε 1 q q q" Z d (ε 1 ) (ε 2 ) q' ε 2 1 (1) (B.1b) (2) ε 2 ε 1 q' = -q image charge Jackson 149-45

ε q R Φ E D cgs esu Φ = q/(εr) E = Φ D = ee = E + 4πP MKSA Φ = q/(4πεr) E = Φ D = εe = ε 0 E + P 1 2 1 E 1 2 E 2 ρ z cgs esu MKSA ε 1 E 1z = -ε 2 E 2z (D 1z = -D 2z = D z ) E 1ρ = E 2ρ (= E ρ ) (B.2a) (B.2b) P = [(ε -1)/4π]E cgs esu P = [ε - ε 0 ]E MKSA P 0 cgs esu P 1z = [(ε 1-1)/4π]D z /ε 1, P 2z = -[(ε 2-1)/4π]D z /ε 2 (B.3a) P 1ρ = [(ε 1-1)/4π]E ρ /ε 1, P 2ρ = [(ε 2-1)/4π]E ρ /ε 2 (B.3b) MKSA 1 ε 0 4π 1 D = 4πρ D = E + 4πP P 1 E = 4πρ E = 4π[ρ - P] ρ - P (B.3a) (B.3b) polarization surface charge density σ pol + + + + n 21 + + + + + + + + + + σ pol = -(P 1 - P 2 ) n 21 (B.4) P 2 n 21 2 1 P - - - - - - - - - - 1 q σ pol 1 q σ pol image charge q σ pol 2 d q - 46

-d -q +q -q = (-d, -q) ( ) cgs esu σ pol = -(q/2π)[(ε 2 ε 1 )/{ε 1 (ε 2 + ε 1 )}][d/(ρ 2 + d 2 ) 3/2 ] (B.5a) MKSA σ pol = -(q/2π)[(ε 2 ε 1 )/{(ε 1 /ε 0 ) (ε 2 + ε 1 )}][d/(ρ 2 + d 2 ) 3/2 ] (B.5b) σ pol q q' ε 1 2 q q" ε 2 C image 2 (1) (2) p 1 p 1 E D p 2 p 1 + p 2 1-47

( ) p r r local 1 = α A 2 p r 1 = p r 1,// + r p 1, p r 2,// = kp r 1,// r p 2, = +kp r 1, E 1i k = ε 1 ε 2 ε 1 +ε 2 = ε 2 ε 1 ε 2 +ε 1 (C.1) r p 1 d A.1 p r 2 p r 1 r r p E 1 2 = 2 r + 3 r ( p r 2 r ) = ( r p 2,// + r p 2, ) + 3(2d r n )(( p r 2,// + r p 2, ) (2dn r )) 3 r 5 (2d) 3 (2d) 5 = (r p 2,// + p r 2, ) + 3(2d r r n )( p 2, 2d) = ( r p 2,// + p r 2, ) + 3( r r n )( p 2, ) (2d) 3 (2d) 5 (2d) 3 (2d) 3 = (r p 2,// + p r 2, ) + 3( r n ) r (C.2) ( p 2, ) = 1 [ p r (2d) 3 (2d) 3 8d 3 2,// + 2p r 2, ] = k 8d 3 r p 1,// + 2 r [ p 1, ] p r r 1 p 1 = α A ( E r ext + E r 1 2 ) 2 r p 1,// + p r 1, = α A ( E r ext + k r [ 8d 3 p 1,// + 2p r 1, ]) (C.3) 1 kα A 8d 3 r p 1,// + 1 kα A 4d 3 r r p 1, = α A E ext // + r ext ( E ) (C.4) α A α A α A,// = 1 k, α 8d α A, = 3 A 1 k 4d α 3 A (C.5) - 48

1 r P = r p 1 + r p 2 = (1 k) r p 1,// + (1 + k) r p 1, = (1 k)α A 1 k 8d α 3 A r E // ext + (1+ k)α A 1 k 4d α 3 A r E ext (C.6) ε 2 i k = ε 1 ε 2 ε 1 + ε 2 +1 r P = r p 1 + v p 2 = 2 v p 1, = 2α A 1 1 4d α 3 A r E ext (C.7) s 0 2α A 2 2α A 1 + α A (4d 3 ) 1 D. 3 2 image charge - 49

Born & Wolf 3 h ε* 1 2 1 SFG image charges 2 1 1 2 2-1 1 d q ε 1 d q ε* h ε 2 1 q 1 σ pol (1) 1 q' q" 2 q" 2 σ pol (2) q 1 * 2 q 1 ** 3 q 1 * 1 σ pol '(1) q 2 ' 1 q 2 " 4 q 2 ' 2 σ pol '(2) q 3 * 2 q 3 ** z 1 z = 0 α 12 = (ε 1 - ε*)/(ε 1 + ε*), α 21 = (ε* - ε 1 )/(ε 1 + ε*), α 23 = (ε* - ε 2 )/(ε 2 + ε*) (D.1a) - 50

β 12 = 2ε*/(ε 1 + ε*), β 21 = 2ε 1 /(ε 1 + ε*), β 23 = 2ε 2 /(ε 2 + ε*) (D.1b) z 1 ε 1 q(d) = q q 1 '(-d) = α 12 q q 2 "(-d - 2h) = β 21 q 1 * = β 12 β 21 α 23 q q 3 "(-d - 4h) = β 21 q 2 * = α 12 α 21 β 12 β 21 α 23 q q 4 "(-d - 6h) = β 21 q 3 * = (α 12 α 21 ) 2 β 12 β 21 α 23 q q n "(-d - 2(n - 1)h) = β 21 q (n-1) * = (α 12 α 21 ) n-2 β 12 β 21 α 23 q (n 2) (D.2a) h 0 q*(-d) = (ε 1 - ε 2 )/(ε 1 + ε )q (D.2b) 2 ε 2 q 1 **(d) = β 23 q 1 " = β 12 β 23 q q 2 **(d + 2h) = β 23 q 2 ' = α 21 α 23 β 12 β 23 q q 3 **(d + 4h) = β 23 q 3 ' = (α 21 α 23 ) 2 β 12 β 23 q q n **(d + 2(n - 1)h) = β 23 q n * = (α 21 α 23 ) n-1 β 12 β 23 q (n 1) (D.3a) h 0 q*(d) = 2ε 2 /(ε 1 + ε 2 )q (D.3b) ε* q 1 "(d) = β 12 q q 1 *(-d - 2h) = α 23 q 1 " = β 12 α 23 q q 2 '(d + 2h) = α 21 q 1 * = α 21 β 12 α 23 q q 2 *(-d - 4h) = α 23 q 2 ' = α 21 β 12 α 2 23 q q 3 '(d + 4h) = α 21 q 2 * = α 2 21 β 12 α 2 23 q q 3 *(-d - 6h) = α 23 q 2 ' = α 2 21 β 12 α 3 23 q q n '(d + 2(n - 1)h) = α 21 q n-1 * = (α 21 α 23 ) n-1 β 12 q (n 2) - 51

q n *(-d - 2nh) = α 23 q 2 ' = (α 21 α 23 ) n-1 β 12 α 23 q (n 1) (D.4a) h 0 q*(+d) = 2(ε* + ε 2 )/(ε 1 + ε )q q*(-d) = 2(ε* - ε 2 )/(ε 1 + ε )q (D.4b) 3-2 d q ε 1 d ε* q h ε 2 q 1 2 1 q 1 "(-d), q 3 "(-2h - d), q 5 "(-4h - d),, q 2n+1 " = (α 21 α 23 ) n β 21 q,, (n 0) q 2 "(-2h + d), q 4 "(-4h + d), q 6 "(-6h + d),, q 2n " = (α 21 α 23 ) n-1 α 23 β 21 q,, (n 1) (D.5a) h 0 q*(+d) = 2[ε 1 (ε*-ε 2 )]/[ε*(ε 1 + ε )]q q*(-d) = 2[ε 1 (ε*+ε 2 )]/[ε*(ε 1 + ε )]q (D.5b) 2 q 1 **(-d), q 3 **(2h - d), q 5 **(4h - d),, q 2n+1 **= (α 21 α 23 ) n β 23 q,, (n 0) q 2 **(+d), q 4 **(2h + d), q 6 **(4h + d),, q 2n **= (α 21 α 23 ) n-1 α 21 β 21 q,, (n 1) (D.6a) - 52

h 0 q*(+d) = 2[ε 2 (ε*- ε 1 )]/[ε*(ε 1 + ε )]q q*(-d) = 2[ε 2 (ε*+ε 2 )]/[ε*(ε 1 + ε )]q (D.6b) q(-d) = q q 1 '(+d), q 3 '(3h + d), q 5 '(4h + d),, q 2n+1 ' = (α 21 α 23 ) n α 21 q,, (n 0) q 2 '(2h - d), q 4 '(4h - d), q 6 '(6h - d),, q 2n ' = (α 21 α 23 ) n q,, (n 1) q 1 *(-2h + d), q 3 *(-4h + d), q 5 *(-6h + d),, q 2n+1 *= (α 21 α 23 ) n α 23 q,, (n 0) q 2 *((-2h - d), q 4 *(-4h - d), q 6 *(-6h - d),, q 2n *= (α 21 α 23 ) n q,, (n 1) (D.7a) h 0 q*(+d) = 2(ε* 2 - ε 1 ε 2 )/[ε*(ε 1 + ε )]q q*(-d) = 2(ε* - ε 1 )(ε* - ε 2 )/[ε*(ε 1 + ε )]q (D.7b) E δ(ax) = δ (x) a δ(x 2 a 2 ) = [δ (x a) + δ (x + a)] 2 a 1 α δ(x) = lim α 0 π α 2 + x 2 δ( x) = δ(x) xδ (x) = 0 (def): δ + (x) = δ * (x) = 1 2iπ lim 1 α 0 x iα, δ + (x) + δ (x) = δ(x) δ + (x) δ (x) = 1 iπ lim x α 0 α 2 + x 2 then δ(x)dx = 1 F(x)δ (x a)dx = F(a) - 53

1 + r + r 2 + + r n = 1 r n+1 1 r F SCF(self-consistent field) MFA(mean field approximation) molecular field approximation CPA(coherent field approximation) P.Soven (1967 ) CPA CPA, A x B 1-x (x A ) A B 1 x H 1 H A ( B ) H A ( B ) A A x B B (1 - x) 0 H CPA Clausius-Mosotti s relation - 54

F F = E + P/(3ε o ) (E. P ) ε α ε 1 ε + 2 = 1 Nα 3ε 0 N. ε ( ) (M/ρ) n 2 1 n 2 + 2 M ρ = 1 N 3ε A α R 0 0 n M ρ N A R 0 local field F E 0 ( ) F E 0 3 E 1 E 2 E 3 F = E 0 + E 1 + E 2 + E 3 E 1 E 1i = -N i P i /ε 0 (i = x, y, z N i P ε 0 ) E 2 E 2 = P/(3ε 0 ) E 3 E 3 = 0 E E 0 F E = E 0 + E 1 F = E + E 2 + E 3 (E 3 = 0) F = E + P/(3ε 0 ) F = E +γ P γ E E 1 F - 55