2 2.1? [ ] L 1 ε(p) = 1 ( p 2 2m x + p 2 y + pz) 2 = h2 ( k 2 2m x + ky 2 + kz) 2 n x, n y, n z (2.1) (2.2) p = hk = h 2π L (n x, n y, n z ) (2.3) n k p 1 i (ε i ε i+1 )1 1 g = 2S + 1 2 1/2 g = 2 ( p F k F ) (= p 2 F/2m) v F = p F /m 1 ε 1 N(ε) N(ε) = 2 4π 3 p3 F V 1 (2π h) 3 = V 3π 2 ( ) pf 3 V = h 3π 2 k3 F. (2.4) 1 (spinless fermion) 2 13
2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) 2 1 27 (1 8 ) 2 1 11 erg, (2.6) /k B 1 11 /1 16 1 5 K ε < 1 1 1 11 23..917g/cm 3 n = 6.2 1 23.917 23. 2.4 122 cm 3 (2.7) 14
k F = ( 3π 2 n ) 1/3 = (3 3.14 2 2.4 1 22 ) 1/3 = (3 3.14 2 24) 1/3 1 7 = 8.92 1 7 cm 1 (2.8) p F = hk F = 1.5 1 27 8.92 1 7 = 9.37 1 2 g cm/sec (2.9) v F = p F m e = 9.37 1 2 9.11 1 28 = 1.3 18 cm/sec (2.1) 1km/sec = h2 2m k2 F = (1.5 1 27 ) 2 2 9.11 1 (8.92 28 17 ) 2 = 1.52 8.92 2 1 12 2 9.11 = 4.81 1 12 erg (2.11) 3.1eV(= 4.81 1 12 /1.6 1 12 ) 3 T F = k B = [ ] E = 2V (2π h) 3 = V 5π 2 ( pf h pf : E = N p2 2m = N 2m 4.81 1 12 1.38 1 16 = 3.49 14 K (2.12) pf p 2 V 2m 4πp2 dp = 2π 2 h 3 p 4 dp = 1 V p 5 F m 1π 2 h 3 m ) 3 p 2 F 2m = 3 5 N. (2.13) pf pf p 2 p 2 dp p 2 dp = N p 5 F/5 2m p 3 F/3 = N 2m 3 5 p2 F. (2.14) E = 3 ( 5 N h2 3π 2 N ) 2/3 (2.15) 2m V : P = E V = [ 3 V 5 N h2 2m ( 3π 2 N ) 2/3 ] V = 2 h 2 5 2m ( ) 3π 2 2/3 ( ) N 5/3 V = 2 E 3 V = 2 N 5 V = 2 5 n. (2.16) 15
P = 2 E 3 V (2.17) 3 κ 1 κ P = V V = V V (3π2 ) 2/3 5 (2.18) h 2 ( ) N 5/3 (2.19) m V = (3π2 ) 2/3 h 2 ( ) N 5/3 3 m V (2.2) = 2 h 2 ( 3π 2 n ) 2/3 2 n = 3 2m 3 n (2.21) ( ) 1 κ = 2 3 2.4 122 4.81 1 12 = 7.7 1 1 erg/cm 3. (2.22) C 11 C 12 4 1 κ = C 11 + 2C 12 3 [ ] = 1 3 (7.3 11 + 2 6.2 1 1 ) = 6.6 1 1 dyn/cm 2 (2.23) (white dwarf) M R U = 3GM 2 5R GM 2 R (2.24) 3 4 (1968) 4.2 16
(a) (b) 2.2: (a) B (b) : (http://www.nationalgeographic.co.jp/science/photos/photo science.php?gallery VignVCMId=ba8d9f48ddfe411VgnVCM1ee2a8cRCRD&no=1) (http://ja.wikipedia.org/wiki/%e4%b8%ad%e6%8%a7%e5%ad%9%e6%98%9f) K 2K = i r i ṗ i = i r i i U (2.25) ( -1) 2K = U (2.26) m N K = E = 3 ( 5 N h2 3π 2 N ) 2/3 2m V (2.27) h2 M 5/3 mm 5/3 N R 2 (2.28) (2.24) (2.28) M 1/3 R h 2 Gmm 5/3 N (1.5 1 27 ) 2 6.67 1 8 9.11 1 28 (1.67 1 24 ) 5/3 1 2 g 1/3 cm (2.29) 17
ρ M/R 3 M 2 2 (2.28) (2.24) ( ) E(R) = h2 M 5/3 mm 5/3 N R GM 2 2 R (2.3) 2 R 1 R (2.29) (2.28) ( ) E = 3 ( M 4 Ncp F M V ) 1/3 M 4/3 R (2.31) 1.44 5 p + e n + ν e (2.28) m m N m n (2.29) M 1/3 R 1 17 g 1/3 cm (2.32) (2.29) (2.31) 2 1 33 g 1 4 km 1km 5 (Chandrasekhar) 18
1.4 1.2 1..8.6.4.2 f..5 1. 1.5 2. Μ 2.3: k B T/µ = 1/1, k B T/µ = 1/1 k B T/µ = 1 2.2 Ε [ ] f(ε) N = i f(ε i ), E = i ε i f(ε i ) (2.33) 1 i dε 1 S V i (2π h) S = S 3 d 3 p (2.34) g = 2S + 1 ε = p2 2m, i dε dp = p m = = = 2mε m = 2ε m (2.35) gv 4πp 2 dp (2π h) 3 gv 2π 2 h 3 2mε dp dε dε gv m3/2 dε 2 1/2 3 ε1/2 π2 h dεd(ε) (2.36) 19
1.4 1.2 1..8.6.4.2 D f..5 1. 1.5 2. Μ 2.4: 3 D(ε) D(ε)f(ε) k B T/µ = 1/1 µ D(ε) = 1 (density of states) 1 N = E = gv m3/2 2 1/2 π 2 h 3 ε1/2 (2.37) D(ε)f(ε)dε, (2.38) εd(ε)f(ε)dε (2.39) 1 1 ε(p) D( ) = 2V 4πp2 dp (2π h) 3 = V dε p=pf π 2 h 3 mp F = 3 V p 3 F 2m 2 3π 2 h 3 = 3N (2.4) 2 N(ε) ε 2/3 1 A n A A n n A = e βen A n n e βen 1 i N i {N i } n {N i } p 2 F Ε N = i N i, E n = i N i ε i 2
f i N i f i = f(ε i ) = N i A 1 a a i a 1 i A = i a i N i = i a i N i = i a i f i A L 1 i p = hk = 2π h L n = 2π h L (n x, n y, n z ) (n x, n y, n z ) n 1 n p n 2π h/l i (??) S S = S [ ] d 3 n S S = S ( ) L 3 2π h ε 1 1 N(ε) = ε D(ε )dε dn(ε) dε d 3 p (2.37) dn(ε) N = f(ε)dε dε = N(ε)f(ε) + N(ε) ( ddε ) f(ε) dε ( ddε ) f(ε) = N(ε) = D(ε) (2.41) dε (2.42) N() = N(ε)f(ε) (as ε ) f (ε) O(k B T ) (k B T/ ) 2 N = N(µ) + π2 6 D (µ)(k B T ) 2 + (2.43) = N(µ ) + D(µ )(µ µ ) + π2 6 D (µ )(k B T ) 2 + (2.44) 21
f 5 4 3 2 1..5 1. 1.5 2. Ε Μ 2.5: f (ε) k B T/µ = 1/1, k B T/µ = 1/1 k B T/µ = 1 N(µ ) = N µ µ = π2 6 µ = π2 6 (2.36) D (µ ) D(µ ) (k BT ) 2 + (2.45) D ( ) D( ) (k BT ) 2 + (2.46) D (ε) D(ε) = d ln D dε = 1 2ε (2.47) µ = π2 (k B T ) 2 + 12 ε F ( ) 2 ( ) 4 = 1 π2 kb T kb T + O (2.48) 12 2 ( ) 22
Sommerfeld ( F (ε) df(ε) ) dε = F (µ) + 2 C 2n (k B T ) 2n d2n F (ε) dε n=1 dε 2n (2.49) ε=µ f(ε) = 1/[exp{(ε µ)/k B T } + 1], F (ε) C 2n 1 x 2n 1 dx (2.5) Γ(2n) e x + 1 = ( 1 2 1 2n) ζ(2n) (2.51) Γ(m) x m 1 e x dx, (2.52) n! = Γ(n + 1) Riemann ζ(m) r=1 (2.48) 1 r m (2.53) F (ε)f (ε)dε = F (µ) + π2 6 (k BT ) 2 F (µ) + 7π4 36 (k BT ) 4 F (µ) + (2.54) F (ε) = ϕ(ε) ϕ(ε)f(ε)dε = µ ϕ(ε)dε + π2 6 (k BT ) 2 ϕ (µ) + 7π4 36 (k BT ) 4 ϕ (µ) + (2.55) [ ] (2.54) ϕ(ε) = εd(ε) µ = µ µ E = = µ + µ εd(ε)f(ε)dε εd(ε)dε + π2 6 (k BT ) 2 (D(µ) + µd (µ)) + = E + µ D(µ ) µ + π2 6 (k BT ) 2 (D(µ ) + µ D (µ )) + = E + π2 6 (k BT ) 2 D(µ ) +. (2.56) 23
(2.39) E = E + π2 6 ( ) 3N 2 (k B T ) 2 + = E + π2 4 N (k BT ) 2 + ε F = 3 ( ) 2 ( ) 4 5 N 1 + 5π2 kb T kb T + O. (2.57) 12 C = de dt = π2 2 Nk k B T B (2.58) k B T / k B T / k B T k B T (2.47) G = Nµ = N 1 π2 12 ( ) 2 ( ) 4 kb T kb T + O. (2.59) P V = (2/3)E G = E + P V T S = 5 3 E T S (2.6) : S = (5/3)E G T = π2 2 Nk k B T B + (2.61) (2.57) [ ] F = E T S = 3 5 N 1 5π2 12 ( ) 2 ( ) 4 kb T kb T + O (2.62) 2 24
2.6: T = D(ε)f(ε) ( ) ( ) (Lande g ) (Bohr ) ( )=1/2 2 µ B H = µ B H εp+ = p2 2m µ BH (2.63) εp = p2 2m + µ BH (2.64) M + M H = ε = M = M + M = µ B (N + N ) D(ε + µ B H) D(ε µ B H) = µ B f(ε)dε µ B f(ε)dε 2 2 D(ε) = µ B (f(ε µ B H) f(ε + µ B H)) dε 2 ( ε ) f(ε) µ 2 BH = µ 2 BH ( D(ε) dε D(µ) + π2 6 D (µ )(k B T ) 2 + ). (2.65) D(ε) 25
χ = M H = µ 2 BD( ) (2.66) H Pauli (paramagnetism) ( ) M = µ 2 BH D(µ ) + D (µ )(µ µ ) + π2 6 D (µ )(k B T ) 2 + [ ( = µ 2 BH D(µ ) + D (µ ) π2 D ) ] (µ ) 6 D(µ ) (k BT ) 2 + π2 6 D (µ )(k B T ) 2 + [ ( = µ 2 BH D(µ ) + π2 D (µ ) (D (µ )) 2 ) ] (k B T ) 2 + 6 D(µ ) = µ 2 BH D(µ ) + D(µ ) π2 d 2 ln D(ε) 6 dε 2 (k B T ) 2 + ε=µ ( ) 2 ( ) 4 = µ 2 BHD(µ ) 1 π2 kb T kb T + O. (2.67) 12 D(ε) ε 1/2 χ = χ 1 π2 12 ( ) 2 ( ) 4 kb T kb T + O. (2.68) 26