磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論
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- れれ いくのや
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1 May 14, 2009
2 Outline / 262
3 Today s Lecture: Mode-mode Coupling Theory 100 / 262
4 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear Mode-Mode Coupling of Fluctuations Curie-Weiss Law of Magnetic Susceptibility Moriya-Kabata Theory Magnetic Excitations Effect of non-linearity Summary 101 / 262
5 Curie-Weiss Law of Magnetic Susceptibility N 0 3 (gµb)2 S(S + 1) = k BTχ(T) χ(t) N0(gµB)2 S(S + 1) 3k BT p eff = N0p2 eff 3k BT : = p (S + 1)/S 1 p s (p s = gµ BS: ) CW ( p eff /p s 1 ) 102 / 262
6 Fluctuation-Dissipation Theorem ( ) H = H 0 M qh z q (t) Mq(t) z = dt χ zz (q,ω)e iω(t t ) H q (t ) χ zz (q,ω) = i [M z q (0),Mz q (t)] eiωt dt ( ) 103 / 262 {M z q (0),Mz q (t)} eiωt dt = coth(βω/2)imχ zz (q,ω)
7 General Form of Equal-Time Correlation Functions (T > T c ) S i S i = 3 N0 2 S q S q = 3 ( ) dω βω N0 2 2π coth Imχ(q, ω) 2 q 3k BT N 2 0 q dω π q Imχ(q, ω) ω = 3k BT N 2 0 χ(q, 0) q : coth(βω/2) 2/βω Kramers-Kronig : Reχ(q, 0) = 1 π Reχ(q, ω) = 1 π Z dω Imχ(q, ω ) ω dω Imχ(q, ω ) ω ω, Imχ(q, ω) = 1 Z dω Reχ(q, ω ) π ω ω 104 / 262
8 Origin of Curie-Weiss Law Heisenberg Model Curie-Weiss J k B T c 1/χ(q,0) J : χ(q,0) χ(t) S(S + 1) = S i S i = 3k BT N0 2 χ(q,0) 3k BT χ(t) N 0 q 105 / 262
9 Moriya-Kawabata Theory Curie-Weiss 1970 : Curie-Weiss SCR Stoner-Wohlfarth (SW) (4 ) : SW + ( ) 106 / 262
10 Moriya-Kawabata Theory Curie-Weiss 1970 : Curie-Weiss SCR (SW ) 2 4 ( ) ( ) 106 / 262
11 Band Splitting and Magnetic Ordering : =? order T c magnetic moment T T m T c = T m? SW T ( ) Stoner ( ) T c?? 107 / 262
12 Effect of Collective Excitations ( ) C = γt + bt 3 + γ ( ) (Landau ) 108 / 262
13 Spin Waves in Itinerant Magnets MnSi : T = 5 K 109 / 262
14 Effect of Fluctuations H(x,p) = p2 2m mω2 x 2 H(x,p) exp( ω/k B T) k B T log(k B T/ ω), (Classical) F(T) = ω 2 + k BT log(1 e ω/kbt ), (Quantum) x 2 = 1 [ ] 1 H = mω2 mω e ω/kbt / 262
15 Free Energy with Spatial Fluctuations Stoner-Wohlfarth (q = 0) Mq (q 0) Ψ[{M q },M,T] = F SW (M,T) + q 0 1 2χ 0 (q) M q M q M q Curie-Weiss γ 111 / 262
16 Non-Linear Effect of Phonons ( ) F(T) = [ ] 1 2 ω qs + k B T log(1 e ω qs/k BT ), ω qs = v qs q qs ( ) ω(v ) : 1 2 mω2 (V )x / 262
17 Free Energy of SCR Theory SW Ψ[{M q },M,T] = F SW (M,T) + Φ({M q }) Φ({M q }) = 1 2χ q 0 (q) M q M q b {q i }M q1 M q2 M q3 M q4 + b: 1/χ(q) 113 / 262
18 A Simple Example of the Effect of Nonlinearity F(x) F(x) = x^2 F(x) = x^ *x^4 F(x) = c + 2*x^ x : F(x) = a 0 x 2 + b 0 x 4 a eff x 2 a { a 0, x 2 0 a eff = a 0 + a, x 2 a x 2 = a / 262
19 Thermodynamics of SCR Model exp[ F(M,T)/k B T] = {M q} exp[ Ψ({M q })/k B T] = e F SW (M,T)/k B T exp[ Φ({M q })/k B T] {M q} H = F(M,T) M = 1 χ(t) M / 262
20 Harmonic Approximation ( ) Φ({M q }) Φ 0 ({M q }) = q (Ω q M q 2 + Ω q M q 2 ) exp[ βφ({m q })] = e βφ0({mq}) exp( β[φ Φ 0 ]) {M q} {M q} = e 1 βf0 e βφ({mq}) exp(β[φ Φ 0 ]) Z {M q} = e βf0 exp[ β(φ Φ 0 )], Z = e βf0 = {M q} e βφ0({mq}) 116 / 262
21 Upper Bound of Free Energy F(Ω,Ω,M 0,T) = F SW (M,T) + F 0 + F exp[ F/k B T] = exp[ (Φ Φ 0 )/k B T] F Φ Φ 0 ( Ω q, Ω q, M) : F = F SW + F 0 + Φ Φ 0 e X = e x = 1 x x2 + = exp[ x ( x2 x 2 ) + ], x X 1 2 ( x2 x 2 ) / 262
22 A Simple Example of Non-linear Model φ(x) = 1 2 ax bx4, e βf = dxe βφ(x) (φ(x) ) F = 1 2 ax bx4 0, ax 0 + bx 3 0 = 0 x = x0 + δx ( ) φ(x) 1 2 ax bx aδx2, F = φ(x 0 ) + F 0 (a + 3bx0 2 ) F 0 (a) = 1 2 k BT log(2πk B T/a), e βf 0 = dxe βax2 /2 = 2π/βa 118 / 262
23 A Simple Example (2) (x = x 0 + δx) φ(x) φ(x0 ) + φ 0(x), φ 0 (x) = 1 2 a δx 2 e β[f φ(x 0 )] = e βf 0 e βf 0 dxe βφ0(x) e β[φ(x) φ(x 0 ) φ 0(x)] = e βf 0 e β[φ(x) φ(x 0 ) φ 0(x)] = e β(f 0+ F) = e βf0 dxe βφ0(x), e βf0 = ( dxe βa δx 2 /2 2πkB T = a ) 1/2 F = φ(x 0 ) + F 0(a ) + F, F = φ(x) φ(x 0 ) φ 0(x) 119 / 262
24 A Simple Example (3) F φ(x) δx φ(x) = a 2 (x 0 + δx)2 + b 4 (x 0 + δx)4 = φ(x 0 ) + a 2 δx2 + b 4 (6x 0 2 δx 2 + δx 4 ) + x 0 [a + b(x δx 2 )]δx δx F x0, a F = a 2 δx2 + b 4 (6x 0 2 δx 2 + δx 4 ) a 2 δx2 δx 2 = k ( ) 2 BT a, kb T δx4 = 3 a 120 / 262
25 A Simple Example (4) F (φ(x 0 ), F 0 (a ), F ) F = 1 2 ax bx ( ) 2 k 2πkB T BT log a ( kb T a ) + 14 b [ 6x 0 2 ( kb T a a ) ( kb T + 3 a ) 2 ] a ( a = a + 3b x0 2 + k ) BT k B T a, a = δx 2 x0 [ ( x0 a + 3b x0 2 + k )] BT a = / k BT
26 Summary Curie-Weiss Harmonic( ) 122 / 262
27 Part II SCR Spin Fluctuation Theory SCR Spin Fluctuation Theory New Origin of Curie-Weiss Law Experimental Verification of SCR Theory Inconvenience in the SCR Theory Summary 123 / 262
28 Approximate Free Energy F Ω q, Ω q F0 2 e βf 0 = X Z e βφ 0({M q}) = Π q dm qe βφ 0({M q}) = Π q {M q} F 0 = k B T X q "! 1 2 log πk B T Ω + log q!# πk B T Ω q 4 πk BT Ω q! 1/2! 3 πk B T 5 Ω q Φ 0 Φ 0 = ( ) Ω z q M q 2 + Ω q Mq 2 = 3 2 k BT q q 1 = 3 2 N 0k B T M q 2 = k BT 2Ω z, Mq 2 = k BT q Ω q 124 / 262
29 Approximate Free Energy (2) Φ Φ = q 1 2χ 0 (q) M q M q b M q1 M q2 M q3 M q4 + {q i } M q1 M q2 M q3 M q4 = M0 4 + [ M2 0 2 Mq M q + 4 Mq z Mz q ] {q i } + q,q [ M q M q M q M q + 2 α M q 2 = k BT 2Ω z, Mq 2 = k BT q Ω q q M α q M α q M α q Mα q ] 125 / 262
30 Minimum condition Ω 0 Ω 1 q = + 1 2χ q 2 bm bt 1 q Ω + 2 q Ω q bt q 1 Ω q Ω q = Ω 0 + (Ω q Ω 0 ) = Ω Aq2 (T > T c ): Ω q = Ω q Ω q = 1 2χ q bt q 1 Ω q Ω q : 126 / 262
31 Origin of Curie-Weiss Law in SCR Theory 1 2χ(0) = 1 2χ 0 (0) b p M p M p 2 2 : λ(t) : χ(q) = χ(0) 1 + q 2 /κ 2, κ = 1/λ SCR (Self-Consistent Renormalization) : χ(0) 2 χ(0) 127 / 262
32 Time-Dependence of Order Parameter = W k B T (W : ) : W = kb Θ : W J kb T c : kb T W 128 / 262
33 Self-Consistent Equation T = T c χ 1 (0) = 0: 0 = 1 2χ 0 (0) b p M p M p (T c ) M p M p (T) = p p M p M p (T c ) b χ 1 (0) M p M p 0 dωn(ω)imχ(q, ω) ωγ q Imχ(q,ω) = χ(0) κ 2 + q 2 ω 2 + Γ 2, Γ q = Γ 0 q(κ 2 + q 2 ), (κ = 1/λ) q 129 / 262 κ 2
34 Numerical Examples 130 / 262
35 Application of the SCR Theory NMR T / 262
36 Central Dogma of SCR Theory SCR 4 ( ) (2 4 ) ( ) ( ( ) ( ) : MnSi, Ni 3 Al, ZrZn 2, Sc 3 In 132 / 262
37 Comparison: SCR vs SW Theories SCR Stoner-Wohlfarth SCR Theory SW Theory (Boson) (Fermion) (T T c) (T 2 Tc 2 ) T/T c 1 M 2 Ms 2 T 2 M 2 = Ms 2 (1 T 2 /Tc 2 ) T/T c 1 M 2 (Tc 4/3 T 4/3 ) H = am + bm 3 (T > T c) 133 / 262
38 Experimental Verification of SCR Theory Curie-Weiss 2 K. R. A. Ziebeck (1982) Y. Ishikawa (1985) 134 / 262
39 Comparison with Experiments 3 1. : T 0, T A ωγ q Imχ(q,ω) = χ(q,0) ω 2 + Γ 2, Γ q = Γ 0 q(q 2 + κ 2 ) q : T 0 = Γ 0 q 3 B/2π, T A = N 0 [χ 1 (q B,0) χ 1 (0,0)]/2 T 0, T A Θ D 2. : b 135 / 262
40 Determination of Parameters (σ 0, T c, T 0 T A ) T 0, T A : T 0 : NMR T 1 b: Arrott ( ) σ = 5T 0C 4/3 3T A ( ) 4/3 Tc T / 262
41 Neutron Scattering v f 1 S(q, ω) Imχ(q, ω) 1 e ω/kbt { [1 + n(ω)]imχ(q, ω), ω 0 = n( ω )]Imχ(q, ω ), ω < 0 1 n(ω) = e ω/kbt 1 ω = ε(v i ) ε(v f ), ε(v) = Mv 2 /2 v i Sample q S(q,ω) ω = 0 ω < 0: (gain) ω > 0: (loss) 137 / 262
42 Neutron Scattering MnSi by Y Ishikawa (1985) 138 / 262 Figure: S(q, ω) (MnSi: 33 K, 100 K, 270 K )
43 Damping of Spin Fluctuations : (Ishikawa et al) Γ q = 1/τ q : Γ q q(κ 2 + q 2 ) 139 / 262
44 Temperature Dependence of Thermal Component 0 : S(q, ω)dω 140 / 262
45 Temperature Dependence of Squared Local Moments S loc S(S + 1): σs 2 (T = 0) S 2 therm = 3σs 2/5 (T = T c) S 2 therm 1/χ(T) (T > T c ) T/T c : 2 T = 0 S 141 / 262
46 Inconvenience involved in SCR Theory 1980 SCR 1 M(T) (T = T c ) Arrott T 2 ( )? ( ) T c 142 / 262
47 Summary SCR Curie-Weiss SCR Stoner-Wohlfarth 143 / 262
遍歴電子磁性とスピン揺らぎ理論 - 京都大学大学院理学研究科 集中講義
email: takahash@sci.u-hyogo.ac.jp August 3, 2009 Title of Lecture: SCR Spin Fluctuation Theory 2 / 179 Part I Introduction Introduction Stoner-Wohlfarth Theory Stoner-Wohlfarth Theory Hatree Fock Approximation
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磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論
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ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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8- My + Cy + Ky = f () t 8. C f () t ( t) = Ψq( t) () t = Ψq () t () t = Ψq () t = ( q q ) ; = [ ] y y y q Ψ φ φ φ = ( ϕ, ϕ, ϕ,3 ) 8. ψ Ψ MΨq + Ψ CΨq + Ψ KΨq = Ψ f ( t) Ψ MΨ = I; Ψ CΨ = C; Ψ KΨ = Λ; q
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最新耐震構造解析 ( 第 3 版 ) 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, 第 3 版 1 刷発行時のものです.
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9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
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