H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e iht = gµ B α (t)b e e iωt 44

Φ ν (t) = [ t ] i dt V H (t ) + Φ ν () t = t gµ B β (t) = gµb Ψ(t) β Ψ(t) = gµ B Φ(t)e ih t β e iht Φ(t) [ t ] [ t ] = gµ B Φ() + i dt V H (t ) + (t) β i dt V H (t ) + Φ() t igµ B dt [V H (t ), (t)] β t = i(gµ B ) B e dt e iωt [ (t α ), (t)] β = (gµ B ) χ(, ω)b e e iωt χ βα (, ω) = i t dt e iω(t t) [ β (t), α (t )] χ βα (, ω) = i t dt e iω(t t) { β (t), α (t )} 3.. Random Phase Approximation (RPA) H = kσ ε kσ c kσ c kσ + U i n i n i + gµ B H e e iωt ε kσ = h k m + σ χ(, ω iδ) χ (, ω iδ) χ(, ω iδ) = χ (, ω iδ) Iχ (, ω iδ) I I = U/N 3..3 χ (, ω iδ) χ χ(, ω iδ) = (, ω) + iχ (, ω) Iχ (, ω) iiχ (, ω) = χ (, ω) + iχ (, ω) Iχ (, ω) iω/γ(, ω) 45

Γ(, ω) [ ( )] ω Γ(, ω) = Iχ (, ω)[ Iχ (, ω)] = ωχ (, ) Iχ (, ω) χ I + I χ (, ω) (, ) χ (, ) [ ( )] = ωχ (, ) Iχ (, ω) χ (, ) + I χ (, ω) χ, χ (, ω) = χ (, ω) (, ) Iχ (, ω) Imχ(, ω) = χ (, ω) Iχ (, ω) ωγ(, ω) ω + Γ (, ω) ω Γ(, ω) ω Γ = Γ(, ) Imχ(, ω) = χ (, ) Γ ω Iχ (, ) Iχ (, ) ω + Γ = χ(, ) ωγ (, ) ω + Γ Reχ(, ) = dω π Imχ(, ω) ω (3.) /Iχ (, ) Iχ (3.) ωγ Imχ(, ω) = Reχ(, ) ω + Γ (3.) Q Reχ(Q +, ) = Reχ(Q, ) + A ij i j + N Γ = { Γ (κ + ) Γ (κ + ) i,j κ χ (Q) A ij Γ (3.3) 3. i αβ j = 3 dω N coth(βω/)imχ(, ω) (3.4) π (3.4) 46

3.. (3.4) ω coth(βω/) n(ω) coth(βω/) = eβω + e βω = + e βω = + n(ω) (3.4) (3.4) i = + Z T i Z = 3 N i T = 6 N dω Imχ(, ω) π dω π e βω Imχ(, ω) D(T, ω) πn (3.5) Imχ(, ω) (3.6) n(ω) κ CR (3.5) Imχ(, ω) (3.) Lorentz Q Q, ω = ωγ Imχ(Q +, ω) = χ(q +, ) ω + Γ 47

Γ - (F) (AF) N /χ(q +, ) = N /χ(q, ) + A + (3.7) ( ) N = A Aχ(Q, ) + + = A(κ + + ) { Γ (κ + ) (F ) Γ = (3.8) Γ (κ + ) (AF ) κ = N Aχ(Q, ) κ κ κ κ B B 4πB 3 N Ω = (π) 3 3 = Ω (π) π B (π) B (3D ) (D ) (D ) Ω B x = / B χ(q +, ) Γ (3.8) χ(q +, ) = N T A y + x Γ = πt x α (y + x ) = πt u(x) = πt v(x) u = x α (y + x )/t, v = x α (y + x ) T T A y t T = Γ α+ B /π, T A = A B, χ(q, )κ / B = N /T A y = κ / B, t = T/T T, T A Γ χ(q +, ) T T A J 48

α d N = d x d dx Imχ(, ω) = = π T ξx α πt T A ξ + u π T ζx α πt T A ζ + v ξ = ω/πt, ζ = ω/πt 3.3 y y (3.) = 6T d x d +α ξ dx dξ T T A e πξ ξ + u = 3dT A(y, t) T A [ A(y, t) = x d +α dx ln u ] u ψ(u), (u = x(y + x )/t) (3.9) ξ digamma ψ(u) (A.) (Appendix A. ) (3.9) y t 3.3. y x (3.9) x x = [ x d +α ln u ] u ψ(u) xd +α u x d = t y + x y = x d 3 d > y = d y = d t > y 49

d T A T /(3T d) 3 C(ν, t c ) πt y/ (t/4) ln(t /3 /y) πt/(y / ) 4: y C(ν, t c ) Appendix Appendix y 4 3.3. d > y = (3.9) y = u = x +α /t x u x d +α dx = (tu) ν du + α u = + α tν u ν du, ν = (d + α)/( + α) (3.) t T = 3T d T A A(, t) /t A(, t) = t ν duu ν [ln u /u ψ(u)], ν = (d + α)/( + α) u t A(, t) = C(ν) = + α tν C(ν), ν = (d + α)/( + α) (3.) πζ(α)γ(α) (π) α sin(απ/) (3.) C(ν) (A.3) (Appendix A. ) d = 3 (F) (AF) t T = { 3T T A C(4/3)t 4/3 9T T A C(3/)t 3/ (F ) (AF ) (3.3) y t A(y, t) = 3 C(4/3)t4/3 πt y + (F ) C(3/)t3/ πt (3.4) y + (AF ) 5

3.3.3 (3.9) digamma A(y, t) = dx xd +α u t = t dx xd α (y + x ) 4 y( + y), 3 t ( y tan ), 3 4 y + y 3.4 y CR (3.) Z = 3T T A d = 3T T A d v = x α (y + x ) ζc x d +α dx ζ dζ ζ + v x d +α dx [ ln(ζ c + v ) ln v ] Lorentz Lorentz (ω) ω ω y = x y Z(y) Z (y) = Z () 3T d T A Z(y) (3.5) Z(y) y- dxx d +α ln [ x α (y + x ) ] = α α = (d + α) + dxx d +α ln x + dxx d +α ln(y + x ) dxx d +α ln(y + x ) x y β = d + α 5

y y 4 ln( + y) /8 + y/4 4 ln( + /y) y/ β = 3 dxx β ln(y + x ) = 3 ln( + y) /9 + y/3 3 y3/ tan (/y / ) y β = [y ln( + /y) + ln( + y) ] y [ln(/y) + ] β = ln( + y) + y / tan πy / β = y Z(y) d + α y + d > α Z(y) = y[ln(/y) + ] + d = α πy / d = α y y y Z (y) = 3 { dω ω N [χ()γ(, ω)] π y ω + Γ (, ω) } ωγ(, ω) Γ(, ω) [χ()γ(, ω)] [ω + Γ (, ω)] y 3 N χ() Γ dω ωγ y π [ω + Γ ] = 3 N χ() Γ y χ()γ(, ω) y y, ω y y ω /ω 3 ω y t=t c 3.5 T c 3 T c /T J kt c J T T A J/k T c J/k kt c i = Z 3 N i T 6 N dω Imχ(, ω) π dω π kt ω Imχ(, ω) 5

i T O() i T i Z i Z O() T c /T T c /T T c /T 5: T = T c kt/ω T > T c T c /T 5 CR Hartee-Fock 3.6 (, ω) (, ω) Imχ(, ω) e βω () = ωc ω c dω(, ω) () () = = ωc ωc dω[ + n(ω)]imχ(, ω) dω[ + n(ω)]imχ(, ω) = ω c dωn( ω)imχ(, ω) ωc dω coth(βω/)imχ(, ω) 53

() () kt Mni Imχ(, ω) dω = Reχ(, ) ω Mni Ishikawa et al. (985) Γ (κ + ) Γ 5 mevå 3 (7 K 6.8 mevå 3 ) B =.3575Å T T = Γ 3 B /π = 9.9 mev (3 K) κ κ = κ (T/T c ), (κ =.35Å ) (gµ B ) χ = N p eff 3g (T T c ), T c = 3 K, p eff =.9 T A (g = ) T A = A B = N B χκ Ni 3 Al = 3g T c B p eff κ =.8 3 [K] Ni 3 Al Bernhoeft et al. (983, 985) Γ c, γ c =.5 5 Å, γ = 3.3 µ evå χ () = χ + c +, Γ = γχ () = γ(χ + c + ) Γ = γc Γ = 4.5 mevå 3 (5.74 3 KÅ 3 ) T B =.578 Å T = 3.59 3 K c κ κ = (χ obs c) χ obs T A χ χ = χ obs /(gµ B ) A A = N χ obs κ /(gµ B ) = c N (gµ B ) = cg d N A µ B w A [cc] d [g] d/w A N = dn A /w A (N A ) d = 7.465, w A = 67.74, N A µ B =.375 [erg K/G ] A A =.375.5 5 7.465 67.74 T A = A B = 3.9 4 [K] =.4 4 Å 3.6. ω ω = (gain) 54

(, ω)dω = n(ω)imχ(, ω)dω Mni Ishikawa NMR T 55