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1 No.7, No.8, No.9 Spring semester, 2012

2 Introduction (Critical Behavior) SCR ( b > 0) Arrott 2 Total Amplitude Conservation (TAC) Global Consistency (GC) TAC 2 / 25

3 Experimental Support of TAC Condition (<S i 2 >) 1/ I = q ωc ω c dω S(q, ω) π = 2I therm + I zp T/T N (MnSi) by Ziebeck et al (1982) 3 / 25

4 Global Consistency H ( H/M) M H/M, H/ M H/M H/ M H M = ( M H ) = H M M M + M ( ) H M M (SCR) = 4 / 25

5 Comparison between Two Approaches SCR ( ) TAC-GC SCR 5 / 25

6 Definition of Thermal and Zero-point Fluctuations ( ) S 2 i = 3 dω N0 2 coth(βω/2)imχ(q, ω) 0 π q coth(βω/2) = eβω + 1 e βω 1 = e βω 1 = 1 + 2n(ω) 6 / 25 S 2 i = S 2 Z + S 2 T S 2 i Z = 3 dω N0 2 Imχ(q, ω) 0 π S 2 i T = 3 N 2 0 q q 0 dω π 2 e βω Imχ(q, ω) 1

7 Spectral Shape in the Low-Energy Region (Double Lorentzian Form) ωγ q Imχ(q, ω) = χ(q, 0) ω 2 + Γ 2 q 1 χ(q, 0) = χ(0, 0) 1 + q 2, κ = 1/λ (λ ) /κ2 : Γq = Γ 0 q(κ 2 + q 2 ) κ 2 7 / 25

8 Frequency Dependence of Neutron Intensity ω/γ 1 S(q, ω) Imχ(q, ω) 1 e ħω/kbt { [1 + n(ω)]imχ(q, ω), ω 0 = n( ω )]Imχ(q, ω ), ω < 0 n(ω) = 1 e ħω/k BT 1 ω = 0 ω 8 / 25

9 Thermal Amplitude of MnSi : 0 S(q, ω)dω 9 / 25

10 Temperature Dependence of Total Amplitude (MnSi) by Ziebeck et al (1982) (<S i 2 >) 1/ T/T N I = q ωc ω c dω S(q, ω) π = 2I therm + I zero pt 10 / 25

11 Numerical Study of Spin Amplitudes for MnSi SCR ( ) Ziebeck et al S 2 L q Ec E c dωs(q, ω)?? 11 / 25

12 Change of Spectral Shape ω /γ κ 2 ( ) q 12 / 25

13 Behaviors of Thermal and Zero-point Amplitudes Parametrization of Excitation Spectrum κ 2 (T )χ(0, 0): : x = q/q B, t = T /T 0 N 0 χ(q B, 0) = N 0(κ 2 + qb 2 ) χ(0, 0)κ 2 = N 0(y + 1) χ(0, 0)y 13 / 25 Γ qb = Γ 0 q B (κ 2 + q 2 B) = Γ 0 q 3 B(y + 1) T A = N 0 2χ(0, 0)y T 0 = Γ 0q 3 B 2π y = κ2 q 2 B N 0 = 2χ(0, 0)T A

14 Spectral Form in Reduced Units ωγ q Imχ(q, ω) = χ(q, 0) ω 2 + Γ 2 q = N 0 1 νγ(x) 2T A y + x 2 ν 2 + γ 2 (x) γ(x) = x(y + x 2 ) 14 / 25 S 2 Z = 9T 0 T A = 9T 0 2T A S 2 T = 18T 0 A(y, t) = T A x 3 dx ζc x 3 ζ dx dζ 0 ζ 2 + γ 2 (x) [ ] x 3 dx ln(ζc 2 + γ 2 (x)) 2 ln γ(x) x 3 dx 0 [ ln u 1 2u ψ(u) ξ 1 dξ e 2πξ 1 ξ 2 + u = 9T 0 A(y, t) 2 T A ], u = x(y + x 2 )/t

15 Amplitude of Thermal Amplitude S 2 T = 9T 0 [A(0, t) π ] y + T A 2 1/t A(0, t) = t 4/3 duu 1/3 [ln u 1/2u ψ(u)] A(y, t) = t y(1 + y) + 0 ex. y y = 0 15 / 25

16 Properties of Digamma Function ( ) ψ(x) = 1 2x γ π 2 cot πx ζ(2n + 1)x 2n n=1 ( ) = 1 x γ + π2 6 x ζ(3)x 2 + π4 90 x 3 ζ(5)x 4 + ln x 1/2x ψ(x) 1 12x x x x 8 1/x + 1/2x 2 ψ (x) 1 6x x x x 8 16 / 25

17 Amplitude of Zero-point Amplitude y y S2 Z (y) = 3 N N 2 0 S 2 Z (y) = S 2 Z (0) 9T 0 T A cy + { y dω ω [χ(q)γ(q, ω)] q 0 π ω 2 + Γ 2 (q, ω) } 2ωΓ(q, ω) Γ(q, ω) [χ(q)γ(q, ω)] [ω 2 + Γ 2 (q, ω)] 2 y q χ(q) Γ q y 0 dω π 2ωΓ 2 q (ω 2 + Γ 2 q) 2 = 3 N 2 0 χ(q)γ(q, ω) y, ω/[ω 2 + Γ 2 ] 2 1/ω 3 q χ(q) Γ q y y=0 17 / 25

18 New Explanation of Curie-Weiss Law Total Amplitude Conservation (TAC) S 2 loc tot = δs2 loc T (y, y z, T ) + δs 2 loc Z (y, y z) + σ2 4 y = κ 2 /q2 B, y z = κ 2 /q2 B = y + σ y σ δs 2 loc T (y, y z, T ) = 3T 0 T A [2A(y, t) + A(y z, t)] δs 2 loc Z (y, y z) = δs 2 loc Z (0, 0) 3T 0 T A c(2y + y z ) + σ, y, y/ σ = (GC) 18 / 25

19 Another New Origin of Curie-Weiss Law SCR 9T 0 T A cy = S 2 loc T (y, T ) + S2 loc Z (0) S2 loc tot = S 2 loc T (y, T c) S 2 loc T (0, T c), (y χ 1 ) SCR 1 2χ(T ) = χ 0 3 b M p M p (T ) p = 5 3 b [ p M p M p (T ) p M p M p (T c ) ] 19 / 25

20 Difference between Two Origins 1/χ(T ) = g [ S 2 loc T (y, T ) S2 loc T (0, T c) ], y 1/χ(T ) SCR : g b: : H = am + bm 3 + : g 1/C: S 2 loc Z (y) = S2 loc Z (0) Cy, C = 9T 0 T A c C 20 / 25

21 New Origin of Magnetic Isotherm Magnetic Isotherm in the Ground State σ 2 4 3c T 0 T A (2y + y z ) = S 2 loc T (0, T c) y = κ 2 /q2 B, y = y 1 (σ 2 σ 2 0 ) y z = κ 2 /q2 B = y + σ y σ y z = y + σ y σ = y 1(3σ 2 σ 2 0), 2y + y z = y 1 (5σ 2 3σ 2 0) σ 2 21 / 25 15T 0 T A cy 1 = 1 4, σ2 0 = S 2 loc T (0, T c)

22 Expansion Coefficient of Free Energy h = T A σy = T A y 1 σ(σ 2 σ 2 0) H = T Ay 1 2N0 3 ( M µ M 2 )M = am + bm 3, b = B b T 0, T A (g = 2) M = N 0 µ B σ, H = h/(gµ B ) 1 2N 3 0 µ4 B T 2 A 60cT 0 M/H = N 0 gµ 2 B σ/h = (gµ B) 2 χ, N 0 χ = 2h σ = 2T Ay 22 / 25

23 Critical Magnetic Isotherm A(y, t c ) A(0, t c ) πt c 4 y, (tc = T c /T 0 ) σ 2 4 3c T 0 T A (2y + y z ) + S 2 loc T (y, y z, T c ) = S 2 loc T (0, 0, T c) σ 2 = 3πt c T A (2 y + y z ) + O(y, y z ) (y = yc σ 2β ) 23 / 25 β = 2, y c = { } 20cz y 2 { 10 πt c (2 + T A = 5) 3πT c (2 + 5) } 2

24 Summary 1 F (M 2, H/M, H/ M) = 0 4 : H M 5, (, δ = 5) SCR ( ) 24 / 25

25 Summary & Conclusions 2 (1) Global Consistency, (2) Total Amplitude Conervation (SCR ) 4 b Arrott SCR SCR ( ) 25 / 25

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