9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (

Size: px

Start display at page:

Download "9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint ("

としみひのと
4 years ago
Views:

1 9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3) ˆd x ( ) 1

2 ( ) ψ (r, t) = m C m (t) e iω mt ψ m (r). (4) (2) LHS = [ ] ω m C m e iωmt ψ m (r) + i Ċme iωmt ψ m (r), (5) m RHS = m [ ] ω m C m e iωmt ψ m (r) + C m e iωmt Ĥ int ψ m (r). (6) ψ n (r) ( n m = δ nm ) i C n (t) e iωnt = C m e iω mt drψn(r)ĥintψ m (r). (7) m C n (t) = i C m e i(ω n ω m )t m = ie cos ωt m C m e i(ω n ω m )t drψ n(r)ĥintψ m (r), drψ n(r) ˆdψ m (r), = ie cos ωt m C m e iω nmt d nm, (8) ω nm ω n ω m, (9) d nm drψn (r) ˆdψ m (r) (1) ψ 1 (r) ψ 2 (r) d 11 =d 22 = C 2 (t) = id 21E cos ωt e iω21t C 1 (t), (11) C 1 (t) = id 12E cos ωt e iω12t C 2 (t), (12) cos ωt = (e iωt + e iωt )/2 e ±i(ω21+ω)t ( ) C 2 (t) = id 21E 2 ei(ω 21 ω)t C 1 (t), (13)

3 C 1 (t) = id 12E 2 ei(ω 12+ω)t C 2 (t), (14) 1 ψ(r, t) = C 1 (t)e iω 1t ψ 1 (r) + C 2 (t)e iω 2t ψ 2 (r), (19) E = E = C 2 (t) = id 21E 2 ei(ω 21 ω)t C 1 (t) Γ 1 2 C 2 (t). (2) C 2 (t) 2 e Γ 1t, (21) τ 1 = Γ 1 1 P P (t) = drψ (r, t) ˆdψ(r, t) = C 1C 2 e iω 12t d 12 + C 2C 1 e iω 21t d 21. (22) C 1C 2 C 1C 2 (14), (2) d dt (C 1C 2 ) = C 1 C 2 + C 1 C 2, (23) d dt (C 1C 2 ) = id 21E 2 ei(ω 21 ω)t ( C 1 2 C 2 2 ) Γ 1 2 C 1C 2. (24) ( ) 1 ω = ω 21 C 2 (t) = id 21E 2 C 1 (t), (15) C 1 (t) = id 12E 2 C 2 (t). (16) ( ) 2 d21 E C 2 (t) = C 2 (t). (17) 2 ( ) C 2 (t) 2 = sin 2 d21 E 2 t. (18)

4 (τ 1 ) τ 2 = Γ 1 2 d dt (C 1C 2 ) = id 21E 2 ei(ω 21 ω)t ( C 1 2 C 2 2 ) Γ 1 2 C 1C 2 Γ 2 C 1C 2, = id 21E 2 ei(ω 21 ω)t ( C 1 2 C 2 2 ) Γ 2 C 1C 2. (25) Γ/2 = Γ 2 + Γ 1 /2 Γ 2 >> Γ 1 Γ/2 Γ 2 ( ) C 1 2 C 2 2 = 1 E = d dt (C 1C 2 ) = id 21E 2 ei(ω 21 ω)t Γ 2 C 1C 2. (26) C 1C 2 e Γt/2, (27) P ω 21 (Γ/2) 1 τ 2 C 1C 2 = d 21E 2 e i(ω 21 ω)t ω 21 ω iγ/2, (28) P P (t) = d 21 2 E 2 [ e iωt ] ω 21 ω iγ/2 + c.c., (29) c.c. E (t) = E cos ωt = E 2 χ P = ε χe P (t) = ε E 2 (29) (31) χ (ω) = d 21 2 ε ( e iωt + e iωt), (3) [ χ(ω)e iωt + χ( ω)e iωt]. (31) ε (ω) = ε (1 + χ (ω)) ε ε 1 ω 21 ω iγ/2. (32) ε (ω) = ε + d 21 2 ω 21 ω (ω 21 ω) 2 + (Γ/2) 2, (33)

5 ε ε κ ω ω 1: ε ε n(ω) κ(ω) R(ω) Γ =.2ω 21, d 21 2 = ω 21, ε = 1.5 ε (ω) = d 21 2 Γ/2 (ω 21 ω) 2 + (Γ/2) 2, (34) ϵ ϵ ε ω = ω 21 Γ ε ω = ω 21 Γ/2 ω = ω 21 + Γ/2 ( 1 ) 2.2 Γ << ω 21 lim Γ Γ/2 (ω ω 21 ) 2 + (Γ/2) 2 = πδ(ω ω 21), (35) δ(ω) ε (ω) = π d 21 2 δ(ω ω 21 ). (36) f 21 f 21 = 2m e 2 d 21 2 ω 21, (37)

6 m f m1 = 1, (38) m 2 1s 2p f 2p1s =.42 ε (ω) = πe2 2m f 21 ω 21 δ(ω ω 21 ). (46) 2 2 V N ε (ω) = πe2 2m V N i=1 f 21i ω 21i δ(ω ω 21i ). (47) 2 n [ˆx, Ĥ] g = (ϵ n ϵ g ) n ˆx g = ω ng e n ˆd g, (39) f ng = 2m g e ˆd n n ˆd g ω 2 ng = 2m g 2 e ˆd n n [ˆx, Ĥ] g = 2m g ˆx[ˆx, Ĥ] g, 2 n n n (4) [ˆx, Ĥ] = 2 (ˆx d2 2m dx d2 i ˆx) = ˆp 2 dx2 x. (41) m n f ng = 2i g ˆxˆp x g. (42) g [Ĥ, ˆx] n = ω ng e g ˆd n, (43) (42) (44) n f ng = 2i g ˆp xˆx g. (44) f ng = i g [ˆx, ˆp x] g = 1. (45) n

7 (f 21i = f 21 ) ω 2 pl = N e 2 /ε m V 3 ε (ω) = πε ω 2 pl f 21 2N N i=1 δ(ω ω 21i ) ω 21i. (52) 2.3 (µ=µ ) n n = c/v = ε/ε, (53) ñ = n + iκ = ε/ε, (54) ε (ω) ε (ω) n(ω) κ(ω) ε /ε = n 2 κ 2, (55) ε /ε = 2nκ, (56) 3 ω pl χ(ω) = d 21 2 ϵ i=1 2ω 21 ω 2 21 ω 2 (Γ/2) 2 iωγ. (48) f 21 ω pl V N χ(ω) = ω2 N pl f 21 N ω21 2 ω 2 (Γ/2) 2 iωγ. (49) ω f 21 f n1 = 1 χ(ω) = ω2 pl ω 2. (5) ϵ(ω) = ϵ (1 ω2 pl ). (51) ω2 ω=ω pl ϵ= 2.3 n= ω < ω pl n

8 1 [ n (ω) = ε 2ε (ω) + ] ε 2 (ω) + ε 2 (ω), (57) κ (ω) = ε (ω) 2n (ω) ε. (58) z E(z, t) = E exp[i(k z z ωt)], (59) k = nω/c ñ E(z, t) = E exp[i(ñω c z ωt)] = E exp[i( nω c ωκz z ωt) ], (6) c κ n κ I = 1 2 ε c E 2 I(z) = 1 2 ε ce 2 exp[ 2ωκ z]. (61) c α(ω) I(z) = I exp[ αz] α (ω) = 2ωκ c = ω ncε ε (ω), (62) R(ω) R = ñ 1 2 ñ + 1, (63) ( ) ε 2 (ω) + ε 2 (ω) 2ε ε (ω) + ε 2 (ω) + ε 2 (ω) + ε R (ω) = ε 2 (ω) + ε 2 (ω) + 2ε ε (ω) +, (64) ε 2 (ω) + ε 2 (ω) + ε n(ω) κ(ω) R(ω) 1 ω = ω 21 1 ε << ε n(ω) ε (ω)/ε, (65)

9 n b α(ω) ω n b cε ε (ω), (66) ω = ω 21 Γ << ω 21 (52) (69) α(ω) = πω2 pl f 21 2n b c N N f 21 = 2n bc πωpl 2 i=1 ωδ(ω ω 21i ) ω 21i. (67) α(ω)dω. (68) f 21 = 2n bc π( ω pl ) 2 α( ω)d ω. (69) ω α

10 3. (Ti:Al 2 O 3 ) (DCM) ϕ1 2 mm 1 mm 3.1 (69) I (ω) L I t (ω) I t (ω) = I (ω) exp ( α (ω) L), (7) α (ω) = 1 L ln I (ω) I t (ω), (71) T (ω) I I t T (ω) = I t(ω) I (ω), (72) 2: I t (ω) I (ω) PMT PMT MEA- SURE PMT mv ACQUIRE

11 128 S/N 4 COURSOR 1 R( ) G( ) B( ) (nm) 2 1. ( 2 mm) DCM( 1 mm) (cm 1 ) (ev) 2. (N /V ) cm 3 DCM cm 3 DCM ω pl ev ω 2 pl = N e 2 /ε m V 3. DCM n b = DCM f 21 f 21 = 2n bc α( ω)d ω. (73) π( ω pl ) 2 3: DCM 4-Dicyanmethylene-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran (C 19 H 17 N 3 ), 33.36

12 3.2 R(ω) (χ(ω) = χ (ω)+iχ (ω)) (Kramers-Kronig) χ (ω) = 1 π P χ (ω) = 1 π P χ (ν) dν, (74) ν ω χ (ν) dν, (75) ν ω P P (ω) E(ω) χ (ω) = χ( ω) χ (ω) = 2ω π P χ (ν) ν 2 ω2dν. (76) ( ) r(ω) = n + iκ 1 n + iκ + 1 = R(ω) exp[iθ(ω)]. (77) ln r(ω) = ln R(ω) + iθ(ω). (78) θ θ(ω) = 2ω π P ln R(ν) dν. (79) ν 2 ω2 (79) θ(ω) (77) n = 1 R 1 + R 2 R cos θ, (8) 2 R sin θ κ = 1 + R 2 R cos θ, (81) R(ω) θ(ω) ω = I R (ω) I R (ω)

13 4: R(ω) R (ω) = I R (ω) /I R (ω) 1 R(ω) (ev) 3.3 (luminescence; ) LED (EL) (PL) (19) DCM ( 1 ns ) 5:

14 I P L (ω) 55 nm (LPF) PD Nd:YAG 164 nm 532 nm 532 nm 164 nm (532 nm) PMT ( ) 6-7 BNC 5 1 M 4 BNC BNC M (1 M )

15 3 1. PD ch1 ( ) TRIG MENU > > ch1 PD ch1 5 PD 5 6: 2. T BNC ch1 3 m BNC ch2 ch1 5 ch2 5 ch1 ch2 BNC (m/s) 7: BNC 3. PMT ch1 5 PD ch2 TRIG MENU > > ch2 PD ch2 5 PMT -2 V -4 V ch1-1 V

16 PMT ch1 p-p (ev) 8: 4. ch1 5 PMT PMT -4 V -8 V ch1 τ 1 9: 5. DCM DCM ch1 5 PMT -2 V -4 V DCM (ev) 6. ch1 5 DCM

17 Ti Ti 1s 2 2p 6 3d 2 4s 2 3 Ti 3+ Al 3+ 3 (4s 3d ) O 2 Ti 3+ 3d 1 Ti 3+ 6 O 2 Ti 3+ x y z ±a Ze 6 v c v c = 6 i=1 Ze 2 4πε R i r. (82) R i i v c ( ) r < a v c = 1 6Ze Ze 2 4π 5 4πε a 4πε 2a 5 r4 9 {Y (Y Y 4)} 4 +. (83) v c V c 3d 3Ze 2 /2πϵ a 3d n=3 l=2 m=2,1,,-1,-2 m m Ĥ 3d ϵ 3d V c Schrödinger Ĥ m = ϵ 3d m. (84) (Ĥ + V c )ψ = ϵψ, (85) ψ m ψ = 2 m= 2 a m m, (86)

18 Schrödinger m 2 (ϵ 3d ϵ)a m + a m m V c m =, (87) m= 2 a m a m V c Y 4 Y 4 4 Y 4 4 m=m m=m ±4 2 V c 2 2 V c 2 = 1 7Ze 2 4π 4πε 2a 5 9 r 4 π 2π r 2 drr 3d(r)r 4 R 3d (r) sin θdθθ 22(θ)Θ 4 (θ)θ 22 (θ) dϕφ 2(ϕ)Φ (ϕ)φ 2 (ϕ) = 1 Ze 2 4πε 6a 5 r4 Dq. (88) r 4 = r 2 drr 3d(r)r 4 R 3d (r), (89) 3d r 4 2 V c 2 = 2 V c 2 = Dq, 1 V c 1 = 1 V c 1 = 4Dq, V c = 6Dq, 2 V c 2 = 2 V c 2 = 5Dq. ϵ 3d + Dq ϵ 5Dq ϵ 3d 4Dq ϵ ϵ 3d + 6Dq ϵ =, ϵ 3d 4Dq ϵ 5Dq ϵ 3d + Dq ϵ (9) 3 1 ϵ 3d + Dq ϵ 5Dq 5Dq ϵ 3d + Dq ϵ =, (91)

19 dγ ε ε 3Ze 2 2πε a 6Dq -4Dq dε 1: 6 3d ϵ (1) = ϵ 3d + 6Dq, ϵ (2) = ϵ 3d 4Dq. (92) ϵ (1) ϵ (2) ϵ 3d = ϵ 3d + 3Ze 2 /2πε a ϵ (1) ϵ (2), ( )/ 2, (93) 1, 1, ( 2 2 )/ 2. (94) 1 1 ( 2 2 )/ 2 i ϵ (1) 15 φ 3z2 r 2 = 3z 2 r 2 R 16π r 2 3d (r), (95) 15 φ x2 y 2 = x 2 y 2 R 16π r 2 3d (r). (96) ϵ (2) 15 yz φ yz = 4π r R 3d(r), (97) 2 15 zx φ zx = 4π r R 3d(r), (98) 2 15 xy φ xy = 4π r R 3d(r). (99) 2 φ yz φ zx φ xy d ε φ 3z2 r 2 φ x 2 y 2 d γ (92) 1Dq d γ x y z d ε x y z 2 2 O 2 d γ d ε

20 d 1Dq Ti 3+ O 2 (Z=2) a 2 Å d r 4 1 Å 1Dq = 5Ze2 r ev (1) 12πε a5 ( 2.4 ev) Ti 3+ 1 d d ε d γ (d d) Energy 2E µ 2T 2 Ti-O 11: Ti 3+ O 2 d ε (d 1 εd γ) 2 T 2 d γ (d εd 1 γ) 2 E Ti-O ( ) 11 Ti-O Ti-O d γ O 2 Ti-O ( ) Ti-O Franck Condon

21 4.2 1s 2p f 2p1s = 2m e 2 2p d 1s 2 ω 2p1s. (11) z d = er cos θ 3 2p 2pz f 2p1s = 2m e 2 f 2p1s = 2m e 2 2pz er cos θ 1s 2 ω 2p1s. (12) r 2 sin θϕ 2pzer cos θϕ 1s drdθdϕ ϕ 2pz = ϕ 1s = ω 2p1s = 1 32πa 3 1 πa 3 e r 2a 2 ω 2p1s. (13) e r a, (14) r a cos θ, (15) a = 4πϵ 2 m e 2, (16) m e 4 3 (4πϵ ) (17) f 2p1s = , (18) 39 Y 1 = 3 cos θ, (19) 4π z 4π d = er 3 Y 1, (11)

22 ϕ 1 = R n (r)y l m(θ, ϕ) ϕ 2 = R n (r)ym l (θ, ϕ) d 21 4π = n l m er 3 Y 1 nlm 4π = e r 2 drrn 3 (r)rr n(r) π 2π sin θdθdϕy l m Y 1 Y l m δ l,±1 δ m,, (111) l = l l m = m m l = ±1 m = x y m = ±1 l = ±1 m =, ±1 ( ) d 21 1s 2p DCM d Ti 3+ O 2 Ti 3+ 3d Ti 3+ 4s O 2 2p d ϵ Ti 3+ 4s O 2 2p d γ d ϵ

23 4.3 ( Γ 1 ) ( Γ R ) ( Γ NR ) (τ 1 ) τ 1 1 = Γ 1 = Γ R + Γ NR, (112) (Γ 1 ) Γ Γ 12: 2, {} s 1, {1 s } {1 s } s 1 ψ I = s C 1s (t) 1, {1 s } + C 2 (t) 2, {}, (113) C 2 () = 1 1 2, 1, 1 13:

24 Ĥ I = s g s [ˆσ + â s e i(ω s ω 21 )t + ˆσ â + s e i(ω s ω 21 )t ], (114) ˆσ ± ˆσ + 1 = 2, ˆσ + 2 =, (115) ˆσ 1 =, ˆσ 2 = 1, (116) â ± s s â + s {} = {1 s }, (117) â s {1 s } = δ s,s {}, (118) g s = d 21 ξ/ ξ = ω s /2ϵ V Schrödinger i d dt ψ I = ĤI ψ I, (119) 2, {} 1, {1 s } C 1s (t) C 2 (t) Ċ 2 (t) = i s g s e i(ω s ω 21 )t C 1s (t), (12) Ċ 1s (t) = ig s e i(ω s ω 21 )t C 2 (t), (121) (121) t C 1s (t) = ig s dt e i(ω s ω 21 )t C 2 (t ), (122) (12) Ċ 2 (t) = s t gs 2 dt e i(ω s ω 21 )(t t ) C 2 (t ), (123) s ω s D(ω)dω. (124)

25 D(ω) V D(ω) = V ω 2 /π 2 c 3 t Ċ 2 (t) = dω dt g 2 (ω)d(ω)e i(ω ω 21)(t t ) C 2 (t ), t dω dt g 2 (ω)d(ω)e i(ω ω 21)(t t ) C 2 (t), = dω dt g 2 (ω)d(ω)e i(ω ω 21)t C 2 (t), = lim dω dt g 2 (ω)d(ω)e i(ω ω 21)t +γt C 2 (t), γ + = dωg 2 1 (ω)d(ω)[πδ(ω ω 21 ) ip ]C 2 (t), ω ω 21 = πg 2 (ω 21 )D(ω 21 )C 2 (t) + ip dω g2 (ω)d(ω) C 2 (t), ω ω 21 = π 21ξ 2 D(ω 2d2 21 )C 2 (t) + ip dω g2 (ω)d(ω) C 2 (t). (125) ω ω 21 P 2 ω t=t C 2 (t ) C 2 (t) 5 C 2 (t) 2 = exp[ 2π 2 d2 21ξ 2 D(ω 21 )t]. (126) 2π 2 d2 21ξ 2 D(ω 21 ) = ω3 d 2 21 πϵ c3. (127) θ cos θ d 2 21ξ 2 cos 2 θ θ 5 Lamb Stark Lamb

26 cos 2 θ = π sin θ cos 2 θdθ π sin θdθ = 1 3, (128) Γ R = 2π d 2 21ξ D(ω 21) = ω3 21d πϵ c3, (129) f 21 Γ R = e2 ( ω 21 ) 2 f 21 6πϵ c 3 m 2. (13) Γ 1 R =τ R 2p 1s f 2p1s =.42 2p 3 τ R =1.6 ns 1-1 ns ( ) η Γ R η = = τ 1, (131) Γ R + Γ NR τ R f 21 τ R τ 1 η=8 %

27 2. 3. f 21 1s 2p DCM 4. DCM ω 1.5 ev 1.9 ev f 21 τ R τ 1 η I Loudon

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c