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2 固体物理の基礎サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.

3 ( FAX )

4 i Maple

5 ii Maple Fortran Maple Maple C BASIC David G. Pettifor

6 iii ( ) ( ) Schrödinger Schrödinger

7 iv 6 Carnot Carnot Gibbs Boltzmann Gibbs / AB

8 v Maple 127 A A A A

9 vi ev (= J) Å (=0.1 nm= cm) Ry ( ) a.u. ( ) 1Ry= ev 1a.u.= Å Boltzmann k B = J/K = ev/k Planck h = J s= ev s Avogadro N A = /mol R = J/mol K 1cal= J 1 ev/ = cal/mol K k B T = ev

10 1 1 Ti 1.1 Si ( ) Al ( ) Ti ( ) Ti PC Ti Ti Ti H-II 8

11 LE-7 Ti-5Al-2.5Sn 1 ( ) 1.1(a) Ti ( 1.1(b)) Ti hcp (α ) bcc (β ) ω (4.1.1 )

12 1.2 ( ) hcp bcc ω ( ) 2 Schrödinger ( ) Schrödinger Schrödinger Schrödinger Newton ( ) 20 ( ) Newton

13 4 1 E 1 m Schrödinger( ) 2 d 2 ψ(x)+v (x)ψ(x) =Eψ(x) (1.1) 2m dx2 V (x) x Plank ( ) h = h/2π ψ(x) 2 Schrödinger (a) V (x) (b) ψ(x) (c) 2 ψ(x) (a) { 0 (0 x L) V (x) = ( ) (1.2) (1.1) 2 d 2 ψ(x) =Eψ(x) (1.3) 2m dx2 A B ψ(x) =A sin kx + B cos kx (1.4)

14 1.2 ( ) 5 2 k d 2 dx 2 ψ(x) = Ak2 sin kx Bk 2 cos kx (1.5) 2 2m k2 = E (1.6) ψ(0) = 0 B =0 ψ(l) =0 kl = nπ (1.7) E n E n = 2 n 2 π 2 2m L 2 = h2 8mL 2 n2 (1.8) 1.3 (b) (c) ψ(x) 2 ψ(x) 2 Schrödinger (1.1) (a) A B (ψ A ψ B ) (b) (E A E B ) 2 ( (a) ) E + E (c) 2 2

15 (a) (b) (c) 2 2 ( ) Ti Ti Schrödinger V ψ 1.6 (a)

16 1.2 ( ) 7 Ti bcc hcp ω (b) hcp hcp ω P = de/dv 1.1(b) Ti hcp bcc hcp ( ) ω Ti bcc 0K Ti bcc hcp ω hcp 1.0

17 48 5 EAM N O 2 (cohesive energy) 5.1 ( )

18 ( 5-1 ) Si (dangling bond) (unsaturated bond) (grain) 5.2

19 50 5 (grain boundary) (orientation) ( ) (vacancy) (solute atom) (dislocation) (stress) (strain) 5.3 (a) 5.3

20 (stress-strain curve) (elastic) (plastic) (yield stress) (yield strength) 5.3(b) 5.4 (edge) (screw) 1 (fracture) (brittle) (ductile) (line tension) 5.4 (a) (b) (c) 1 ( ) ( 1999) ( 1999)

21 EAM EAM 2 1 E v E coh 5.5 E v =(3/8) (1/4)E coh 5.5 E v E coh

22 ( ) z E(z) E v = E final E initial = ze(z 1) ze(z) (5.1) Lennard-Jones Morse 2 ψ 0 (5.1) E(z) = E coh = zψ 0 (5.2) E v = z(z 1)ψ 0 z(z)ψ 0 = zψ 0 = E coh (5.3) 2 EAM EAM (5.2) E(z) =zψ 0 z h 0 (5.4) (5.3) (5.1) E v = z { (z 1)ψ 0 z 1 h 0 } z { zψ 0 z h 0 } (5.5) exp L = dln(h) p q dr ψ(r) =A exp( pr) (5.6) h(r) =B exp( qr) E coh r 0 h 0 = p zψ0 q ψ 0 = E (5.7) coh (1 p/q)z 0 E v = 2 p/q E coh 1 p/q 2 (5.8)

23 54 5 ( 5-3 ) p/q =3 5 E v =(3/8) (1/4)E coh (back bond strengthening) ( 5.7 (a) ) EAM z z 0 2 z ( 5.7 (b) ) E v E coh 5.7 z (unsaturation) C C N 2 2 G. Allan and M. Lannoo, J. Phys. Chem. Solids, 37(1976), 699.

25 A B 2 2 ( ) 11.1 (solution) (solid solution) G (9.11) N i x i G = i µ i x i (11.1) G

26 A B G 0 A G0 B A B 1mol A B x A x B 2 G 0 = x A G 0 A + x B G 0 B (11.2) 2 (segregation limit) A B 7.4 N A B N A N B W = N! N A!N B! (11.3) Stirling ( ) (ln N! N ln N N) ln W =(Nln N N) (N A ln N A N A ) (N B ln N B N B ) =(N A + N B )lnn N A ln N A N B ln N B (11.4) N = N A + N B x A = N A /N x B = N B /N Boltzmann S = k B ln W = R(x A ln x A + x B ln x B ) (11.5) N 1mol Avogadro R = k B N G G = G ideal AB G 0 = T S = RT (x A ln x A + x B ln x B ) (11.6) (ideal solution) G ideal AB 11.1(a) G 0 T S H (PV) (E) H = PV +E E

27 A A B B A B i j N ij e ij E AB = N AA e AA + N BB e BB + N AB e AB (11.7) z (coordination number) A B zx B A Nx A N AB Nzx A x B A A B B N AA =(1/2)Nzx 2 A N BB =(1/2)Nzx 2 B 1/2 E AB = 1 2 Nzx A 2 e AA Nzx B 2 e BB + Nzx A x B e AB = 1 2 Nzx A(1 x B )e AA Nzx B(1 x A )e BB + Nzx A x B e AB = 1 2 Nzx Ae AA + 1 ( 2 Nzx Be BB + Nzx A x B e AB e ) AA + e BB 2 (11.8) 2 ( H = Nz e AB e ) AA + e BB x A x B = Ωx A x B (11.9) 2 Ω (interaction parameter) (10.5) G m = x A G 0 A + x B G 0 B + Ωx A x B + RT (x A ln x A + x B ln x B ) (11.10)

28 (regular solution) Ω < 0 A B A A B B A B Ω > 0 A B Ω =0 Ω 11.1 (b) (excess free energy) G EX m = G m G ideal m = Ωx A x B (11.11) Ω Ω = Ω 0 + Ω 1 x B (sub-regular solution) 1mol G AB = x A µ A + x B µ B (11.12) 11.1(a) x =0.8 A B µ A µ B x =0 x =1 A µ A = G 0 A + RT ln x A (11.13) 1 ( 1997).

29 µ A = G 0 A + RT ln a A = G 0 A + RT ln γ A x A (11.14) a A (activity) γ A (activity coefficient) 11.2 ( ) (b) (b) (a) Ω < 0 (b) Ω > 0 (b) 1 G EX m = RT (x A ln γ A + x B ln γ B ) (11.15) A

30 T fa H fa = T fa S fa (11.16) T fa µ 0L A µ 0S A = H fa T S fa (11.17) 2 µ 0S A (T )=0 B µ 0L A (T )= H fa T S fa (11.18) µ 0L B (T )= H fb T S fb (11.19) A B T fa = 1500 K T fb = 1000 K Richards (9.2.3 ) H fa =12.55 kj/mol H fb =8.34 kj/mol 2 Ω L = Ω S = 0 kj/mol AB 2 (11.6) G S m 1mol Gibbs µ 0S A G S = x S Aµ 0S A + x S Bµ 0S B + G S m (11.20) G L = x L Aµ 0L A + x L Bµ 0L B + G L m (11.21) (T )=µ0s B (T )= (a) T = 1500 K A (b) T = 1300 K (d)

31 (a) (b) (c) (d) (b) (e) µ S A = µ L A µ S B = µ L B (11.22) 2 p q a p q 2 p q (c) B T = 1000 K

32 175 1 (first-order transition) (binomial coefficient) 80 2 (binomial distribution) 80 2 (second-order transition) (second moment approximation) (pair potential) 39 3 (triple point) 100 c/a (c over a ratio) 34 Boltzmann ( ) vi, 64 Boltzmann 85 Carnot ( ) 65 Cauchy (Cauchy restriction or relation 43 Clausius-Clapeyron ( ) 100 Compton ( ) 14 de Broglie ( ) 15 Dulong-Petit ( ) 9, 94 EAM 52 Einstein( ) 9, 83 Fermi ( ) (Fermi level) 36 Gauss ( ) 80 Gibbs-Duhem ( ) 98 Gibbs ( ) 70 Gibbs (canonical ensemble) 88 Gibbs (phase rule) 99 Γ 75 Hamiltonian 25 Hamiltonian ( ) 25 Helmholts ( ) 69 Lagrange ( ) (Lagrange multiplier) 90 Lennard-Jones ( ) 39 Markov ( ) (Markov process) 113 Markov (Markov chain) 113 Maxwell ( ) 71 Morse ( ) 39 Otto ( ) 74 Pauli ( ) (Pauli s exclusion principle) 20 Planck ( ) vi, 13 Richards ( ) 103, 121 Schrödinger( ) 4, Stirling ( ) 82 TTT (Time-Temperature- Transformation diagram) 106

33 176 (simulated annealing) 77 (ionicity) 30, 32 (heteronuclear) 2 30 (phase space) 75 4 (anisotropy factor) 43 (liquidus) 123 (droplet model) 104 (energy level) 16, 21 p 21 s 21 (energy barrier) (ergodic hypothesis) 85 (ductile) 51 (enthalpy) 69 (entropy) (stress-strain curve) (open system) 59, (chemical potential) 96, 97, 119 (reversible) 62 (nucleation frequency) 105 (compound) (overlap integral) 25 (excess free energy) 119, 120 (activation energy) 105 (activity) 120 (activity coefficient) 120 ( valence band) 35 (undercooled or supercooled liquid) 102 (rock salt or NaCl) (spherical harmonics) 19 (cohesive energy) 36, 48 (eutectic temperature) (eutectic composition) 124 (tight binding approximation) 44 (covalency) 31, 32 (curvature) 28 (homogeneous nucleation) 104 (metal) (rectangular) 36, 45 (driving force) 103 (system) 59 (empirical temperature) (bond energy) 28, 30, 37 (binding energy curve) 28 5, 27 (bond stiffness) 28 (crystal structure) 33 (orientation) 50 (atomic orbital) 19

34 177 LCAO (linear combinations of atomic orbitals) 25 (vacancy) 50 (embedded atom method: EAM) 43 (lattice) (photoelectric effect) 14 (yield stress) 51 (yield strength) 51 (black body radiation) 13 (solidus) 123 (Eigenfunction) (Eigenvalue) 16 (solute atom) 50 (solid solution) 116 (isolated system) (maximum solubility limit) (intensive variable) (exponential distribution) 80, (metastable equilibrium) 103 (sub-regular solution) (condition of detailed balance) 114 (crystallization) 103 (state function) 61 2, 116 (state variable) 61 (state quantity) 60 (sum over states) (extensive(additive) variable) 61 (normal distribution) 80 (canonical partition function) 90 (brittle) 51 (regular solution) 119 (precipitation) (zinc blende) 34 (transition probability) 113 (shear constant) 43 (line tension) 51 (latent heat) (interaction parameter) 118 2, 99 (phase transition) 101 (phase equilibria) 96 (phase transformation) 101 (plastic) 51 (diamond) 34 (the first principles calculation, ab initio calculation) 3, 38 (body centered cubic; bcc) 33 (bulk modulus) 42 (unit cell) 33 (dangling bond) 49

35 178 (elastic constant) 41 (elastic) 51 (adiabatic) 63, 65, 66 (central limit theorem) (harmonic oscillator) 18, 83 (grain) 49 (lever rule) 123 (dislocation) 50, 51 (transition temperature) 101 (electronegativity) 31 (density of states: DOS) ( conduction band) 35 (isothermal) 65, 66 (homonuclear) 2 27 (radial function) 19 (principle of equal apriori probability) 80, 87 (closed system) 59 (tunnel effect) 17 (wet) (thermodynamical absolute temperature) 62, , (fracture) 51 (edge) 51 (back bond strengthening) 54 (wave function) 16 (duality) , 27 (semiconductor) 36 35, 50 (strain) , 106 (irreversible) 62 (inhomogeneous nucleation) 106 matter wave 3, (unsaturated bond) 49, 54 (molecular orbital) 24 (average) 81 2 (mean square width) 81 (equilibrium distance) 28 (equilibrium state) 62 (translation vectors) 33 (stress) 50 (segregation limit) (potential energy cuve) (bond integral) 27 (configurational space) 76 (coordination number) 40, 44, ( : embryo) 104 (face centered cubic; fcc) 33

36 ( ) 8 77 (dielectrics) 36 (fluctuation) 80, 93 (solution) 116 (screw) (ideal gas temperature) 62, (ideal solution) 117 (grain boundary) 50 (critical point) 100 (critical radius) 105 (hexagonal close packed; hcp) 33

37 2006 JCLS Printed in Japan ISBN

iBookBob:Users:bob:Documents:CurrentData:ﬂMŠÍ…e…L…X…g:Statistics.dvi

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