1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e ) e β(e µ) (9.3) *1 f f + f
2 9 9.1: (a) T = (b) T > µ T 1 = µ /k B T 2 = 2T 1 T 3 = 3T 1 µ 1 T = T = E > µ (9.1). (9.2) E > f(e ) = E = f() > E = E µ (9.2) µ E = µ = (9.4) T = (9.1) (9.4) f(e ) E = 9.1 δ (T > ) µ 1(b) E < E E = µ E > E >
9.1. 3 9.2: µ = µ = 9.2 µ = T = 9.1.2 E E s E s e β(e µ) 1 de x = E /k B T α = µ/k B T F (s, α) = 1 Γ(s) x s 1 e x+α dx (9.5) 1 F (s, α) (Bose-Einstein integral) µ α α *2 F (s, α) (9.5) *2 s > s 1 α 1 (John E. Robinson, Phys. Rev. 83, 678 (1951)) α
4 9 e x+α F (s, α) = 1 Γ(s) = 1 Γ(s) x s 1 n=1 e (x+α) 1 e nx = t *3 (x+α) dx x s 1 e n(x+α) dx (9.6) F (s, α) = 1 ( ) s 1 t s 1 e t nα dt = 1 ( ) s 1 Γ(s) e nα Γ(s) n=1 n Γ(s) n n=1 = n s e nα (9.7) n=1 (9.7) F (s, α) α α F (s, α) α = F (s, ) α = (9.7) F (s, ) = n s = ζ(s) (9.8) n=1 ζ(z) 9.3 s α 9.2 dk (8.16) dk 1 g = 1 *4 k dk gv 8π 3 dk = V dk (9.9) 8π3 dk 8π 3 (9.1) 1 α > 1 (9.7) X F (s, α) = Γ(1 s)α s 1 ( 1) n + ζ(s n)α n n! n= gcc (tgamma() ) n = 5 14 ζ(s) = 1 1 2 1 s X n= 1 2 n+1 nx ( 1) k n (k + 1) s k k= *3 Z Γ(z) = t z 1 e t dt *4 4 He 1 2
9.2. 5 9.3: s α i n i p(n i ) (7.111) p(n i ) = e β(ei µ)ni [1 e β(ei µ) ] (9.11) (7.97) p(n i ) ln p(n i ) p(n i ) ln p(n i ) n i= =[1 e β(ei µ) ] ln(1 e β(ei µ) ) e β(ei µ)ni [1 e β(ei µ) ] n i= = ln(1 e β(ei µ) ) [1 e β(ei µ) ]( β) β e β(ei µ) n i= e β(ei µ)ni = ln(1 + f) β(e i µ) 1 e = ln(1 + f) f ln( 1 β(ei µ) f + 1) β(e i µ)n i e β(ei µ)ni =f ln f (1 + f) ln(1 + f) (9.12) n i= S = k B {f(e i ) ln f(e i ) [1 + f(e i )] ln[1 + f(e i )]} (9.13) i S = k BgV 8π 3 {f(e (k)) ln f(e (k)) [1 + f(e (k))] ln[1 + f(e (k))]}dk (9.14) (7.87) Ξ = i (1 e β(ei µ) ) 1 (9.15)
6 9 S = J/ T = ( / T )k B T ln Ξ (9.15) S = k B ln[1 e β(ei µ) ] 1 T i i (E i µ)e β(ei µ) 1 e β(ei µ) = k B {ln(1 + f) + 1 T (E i µ)f} = k B {f ln f (1 + f) ln(1 + f)} (9.16) i (9.16) 1 P P = J V = k BT V ln Ξ = k BT ln(1 + f) (9.17) V S = P V T + V (E µn) (9.18) T (9.17) P P = k B T V i ln[1 e β(ei µ) ] = k BT g 8π 3 i i ln[1 e β(e (k) µ) ]dk (9.19) 9.3 9.3.1 E E = mv2 = p2 2 2m = h2 k 2 (9.2) 2m Ω(E ) 8.3 E E + de Ω(E )de k k = 2mE / h 2 m/2 h 2 E de Ω(E )de = gv 8π 3 dk = gv ( ) ( ) 3/2 2mE m 8π 3 4π gv 2m h 2 2 h 2 de = E 4π 2 h 2 E 1/2 de (9.21) E E +de g g = 1 1/2 9.3.2 N µ (9.1) N = Ω(E )de (9.22) = gv ( ) 3/2 2m E 1/2 4π 2 h 2 de (9.23) E β(e µ) 1 ( ) 3/2 2πmkB T 2 x 1/2 = gv π h 2 e x+α dx (9.24) 1
9.3. 7 N (9.24) µ (9.5) ( 2πmkB T N = gv h 2 ) 3/2 F ( 3 2, α) (9.25) 5.7 λ T = ( h 2 /2πmk B T ) 1/2 N = V λ 3 F ( 3 2, α) (9.26) T λ T α = µ/k B T N λ T F (s, α) µ (9.25) µ F (s, α) α N 1 (9.26) N 1 = V λ 3 F ( 3 2, ) (9.27) T N = N 1 F ( 3 2, α) F ( 3 2, ) (9.28) (9.27) F ( 1 2, ) N 1 λ T (T ) T 3/2 T N 1 (9.28) T N 1 F ( 3 2, α) (9.5) (9.7) T α (9.27) T N 1 T T = N 1 N 1 = N T c (9.26) N 1 = N F ( 3 2, ) = ζ( 3 2 ) N = T c gv λ 3 T (T c) ζ( 3 2 ) = gv h2 T c = 2πmk B ( 2πmkB T c h 2 ) 3/2 ζ( 3 2 ) (9.29) [ ] 2/3 n gζ( 3 2 ) (9.3) T c T > T c µ (9.23) N µ N T c T = T c µ = µ µ T T c µ = µ T T c (9.23) N T T c (9.22) N T T c (9.22) N T T c
8 9 9.4: T c (µ/k B T c ) (T/T c ) (9.22) (9.23) E = E = 1 ( g ) Ω() = (9.22) N (9.22) 1 1 T = 2 E > 1 T T c T > T c E = ( 9.1) E > E = T < T c (9.23) E = N N N = N + N (9.31) N = V λ 3 T ζ( 3 2 ) (T T c) (9.32) T > T c µ µ (9.25) (9.28) N V ( T T c ) 3/2 ζ( 3 2 ) = F ( 3 2, α) (9.33) 9.4 µ/k B T T/T c (9.31) F ( 3 2, ) =
9.3. 9 ζ( 3 2 ) T = T c α = µ/k B T = T = T c µ = (9.23) N T T c (9.33) T T c 9.4 T T c µ 9.1 E > 9.4 T < T c (9.33) µ µ = 9.3.3 E = = gv 4π 2 E f(e )Ω(E )de (9.34) ( ) 3/2 2m h 2 E 3/2 de (9.35) E β(e µ) 1 = 3gV 2λ 3 k B T F ( 5 2, α) (9.36) T α T < T c µ = E T T T c µ 9.4 T = T c E T < T c (9.37) (9.29) T < T c α = E = 3gV 2λ 3 k B T F ( 5 3gV 2, ) = T 2λ 3 k B T ζ( 5 2 ) (9.37) T E = 3 2 Nk BT c ( T T c ) 5/2 F ( 5 2, α) ζ( 3 2 ) (T T c ) (9.38) E = 3 2 Nk BT c ( T T c ) 5/2 ζ( 5 2 ) ζ( 3 2 ) (T < T c ) (9.39) T < T c E T 5/2 T T c E 9.5 T = E = (9.38) (9.39) C V = de/dt T T c [ ( ) 3/2 15 T F ( 5 2 C V = k B N, α) 4 ζ( 3 2 ) 3 ( ) 5/2 T 2 T F ( 1 2, α) ] dα c ζ( 3 2 ) (9.4) dt T c df (s, α)/dα = F (s 1, α). dα/dt (9.25) T T T c N dn/dt = T c dα dt = 3F ( 3 2, α) 2T F ( 1 2, α) (9.41)
1 9 9.5: T c (E/Nk B T c ) (T/T c ) 9.6: k B N = R (C V ) (T/T c ) dα/dt (9.4) (9.33) C V = k B N [ 15 F ( 5 2, α) 4 F ( 3 2, α) 9 4 F ( 3 2, α) ] F ( 1 2, α) T < T c α = (9.4) 2 C V = 15k BN 4 ( T T c (T T c ) (9.42) ) 3/2 ζ( 5 2 ) ( ) 3/2 T ζ( 3 2 ) 1.927R (T < T c ) (9.43) T c T < T c C V T 3/2 T T c 9.6 R = k B N C V 6 (3k B N/3)
9.3. 11 9.3.4 (9.14) (9.21) S S = k B [f ln f (1 + f) ln(1 + f)] (9.44) i = k B gv 4π 2 ( ) 3/2 2m h 2 E 1/2 [f ln f (1 + f) ln(1 + f)]de (9.45) ln(1 + f) = ln (9.44) β(e µ) e = β(e µ) + ln f (9.46) e β(e µ) 1 f ln f (1 + f) ln(1 + f) = f ln f (1 + f)[β(e µ) + ln f] (9.45) S = k B gv 4π 2 = fβ(e µ) [β(e µ) + ln f] = fβ(e µ) + ln[1 e β(e µ) ] (9.47) ( ) 3/2 2m [ ( h 2 E 1/2 fβ(e µ) ln 1 e β(e µ))] de (9.48) (E µn)/t P V/T J P = V J = k BT V ln Ξ = k BT V ln i = k B T ln[1 e β(e µ) ] = k B T V V = gk BT 4π 2 i ( 2m h 2 1 1 e β(e µ) ln[1 e β(e µ) ] gv 8π 3 dk ) 3/2 E 1/2 ln[1 e β(e µ) ]de (9.49) (9.48) P V/T S = E µn + P V (9.5) T T 9.2 (9.49) *5 P = 2 gk B T 3 4π 2 (9.37) ( ) 3/2 2m h 2 E 1/2 β e β(e µ) 1 de = gk BT λ 3 T F ( 5 2, α) (9.51) P = 2E (9.52) 3V (9.5) S = 1 T ( ) 5 3 E µn (9.53) *5 1 h E 3/2 ln(1 e β(e µ) ) i E ln(1 e β(e µ) ) e β(e µ)
12 9 9.7: (S/k B N T/T c ) E (9.38) (9.39) µ (9.33) 9.7 k B N T = S = 9.4 T T c 1 g J J = k B T ln Ξ = gk B T ln 1 1 e β(e µ) (9.54) J µ 9.8 µ J µ = J µ = µ = V
9.4. 13 9.8: k B T J (J/gk B T µ/k B T ) E /k B T = 1 9.9: J (J/gk B T µ/k B T ) T/T c =.5 k B T.
14 9 P J = k B T ln Ξ = k B T i (1 e β(e µ) ) (9.55) = 2 gv 3 4π 2 = 2 3 ( ) 3/2 2m h 2 E 3/2 de (9.56) e β(e µ) 1 gv λ 3 k B T F ( 5 T 2, µ k B T ) (9.57) 9.9 µ = J T < T c N N N N µ J µ = T < T c µ = 9.5 9.5.1 T c T < T c N N N N = N N E = 1 1 T c (Bose-Einstein condensation condensate) N (9.31) (9.32) (9.29) N = N N = N V ζ( 3 ( ) ] 3/2 T [1 λ T 2 ) = N T c (9.58) 9.1 1 2 (macroscopic quantum effect) ( 4 He)
9.5. 15 9.1: N /N. ( 87 Rb, 23 Na *6 ) 1% *7 2 9.5.2 T c, (transition temperature) (9-3) V N g [ ] 2/3 T c = h2 N 2πmk B ζ( 3 2 )gv (9.59) T c 3.1 K 2.17 K T c 1 K 1 K T c 13 K (9.59) λ 3 T (T c) = ζ( 3 2 )gv (9.6) N *6 17 nk 2 µk E. A. W. C. E. 21 *7 J. Klaers et al., Bose-Einstein condensation of photons in an optical microcavity., Nature 468, 545 (21).
16 9 λ T (T c ) T c g = 1 ζ( 3 2 ) 2.612 1 1 5.7 *8 9.6 (9.1) µ 1 1 f(e ) e β(e µ) (9.61) 9.1 µ µ µ λ 3 T V (9.62) N (9.6) ( ) T [gζ( 3 2 )]1/3 1 (9.63) T c *8 P. W. Anderson, More is different., Science 177, 393 (1972).
9.6. 17 9.11: α. T c 9.11 (9.33) α T/T c T c 1 T c = 3.1 K 3 K (9.62) α N r T cr (9.29) (9.24) λ 3 T (T cr) = gv N r ζ( 3 2 ) (9.64) ( ) 3/2 ( N T = N r T cr ) 3/2 F ( 3 2, α) ζ( 3 2 ) (9.65). 9.14 α N/N r T cr 1 1 α 6 1 1 6 1 6 α = ( )
18 9 9.12: α.n r T cr 9.7 9.7.1 19 3Nk B N 19 T 3 a 9.11(a) x 2a x 9.11(b) 2a 2a 2a 9.11(c) 1 5 1 2π/L 2π/2a
9.7. 19 9.13: (a) k = 2π/3a, (b) k = 2π/2a, (c)k = 2π/(6/7)a k = 2π/6a. 2(2π/2a)/(2π/L) = L/a = N x a N x x y z 3 N x N y N z = N N L 3 2 1 3 3 (phonon) 1 1 3 z 1-1 1 2 3N N 5.6 Ω ν Ω ν (ν)dν = 4πgV c 3 ( 1 ν 2 dν = 4πV + 2 c 3 t c 3 l ) ν 2 dν (9.66) g V = L 3 c l c t 3N 1 (Debye model) ( 1 Ω ν (ν)dν = 4πV + 2 ) c 3 ν 2 dν = 3N (9.67) t c 3 l
2 9 νd Ω ν (ν)dν = 9N 1 ν 2 dν (9.68) hν hν (phonon) µ = E E = νd C V = E/ T C V = νd hνf(hν)ω ν (ν)dν = 9N ν 3 D hνf(hν)ω ν (ν)dν = 9k BN ν 3 D = 9Nk B ( T Θ D ) 3 ΘD/T ν 3 D νd νd hν 3 e βhν dν (9.69) 1 h 2 ν 4 e βhν 1 (e βhν 1) 2 dν (9.7) (k B T ) 2 x 4 e x (e x dx (9.71) 1) 2 βhν = x hν D = k B Θ D T 1 T 1 T 1 e x 1 e x 1 x C V T Θ 3 D νd x 2 dx = 3k B N (9.72) 3 6 (9.71) F (4, ) = ζ(4) = π 4 /9 C V 9Nk B ( T Θ D ) 3 4x 3 e x 1 = 9Nk B ( T Θ D ) 3 Γ(4)ζ(4) = 12π4 5 k BN T 3 ( T Θ D (9.73) ) 3 (9.74) ν 3N E = 3Nhν(n + 1 2 ) (9.75) n 1 g = 1 (Einstein model)
9.7. 21 9.14: (9.67) 2 E = 3N hν e βhν 1 (9.76) ( ) 2 hν e βhν C V = 3Nk B k B T (e βhν 1) 2 = 3Nk B ( Θ T ) 2 e Θ/T (e Θ/T 1) 2 (9.77) Θ = hν/k B Θ /T 1 e Θ/T 1 e Θ/T 1 Θ /T (9.77) C V 3Nk B (9.78) 6 1 3 (9.77) C V 9.12 9.7.2 5 (E (λ)/λ)ω(λ) E (ν)ω(ν) Ω(ν) (5.77) Ω(ν) = 4πgV c 3 ν 2 (9.79) 1 g = 2 ν E (ν) = nhν n µ = U ν = E (ν)ω(ν) = nhν 8πV c 3 ν2 = 8πV c 3 hν 3 e βhν 1 (9.8) 5
22 9 9 9.1 (9.33) 9.4 9.2 (9.5) F (s, α) df dα = F (s 1, α) (9.81) 9.3.145 g cm 3 4. k B = 1.38 1 23 J/K 6.2 1 23 /mol h = 1.5 1 34 J s T c 9.4 1 9.5 ν hν hν/k B T 1