4 6 C345 ) ) 3) 4) Louvlle Schrödnger Hesenberg Redfeld NMR e-mal 37( 7-37) e-mal nobuyuk@scl.kyoto-u.ac.jp
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- かつかげ あきくぼ
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2 4 6 C345 ) ) 3) 4) Louvlle Schrödnger Hesenberg Redfeld NMR e-mal 37( 7-37) e-mal nobuyuk@scl.kyoto-u.ac.jp
3 . NMR [,]
4 . 3. Gbbs
5 ..3 g/cm 3.7 g/cm 3...
6 Gbbs [3].Gbbs Clausus-Clapeyron [] 999 [] 997 [3]
7 . (soluton) (mxture) (sold soluton) (solvent) (solute) (solublty) (saturated soluton) kg (molarty) (molalty) kg kg
8 5 5 (chemcal potental). dq P V dv U du=dq-pdv (..) dq S dq=ds (..) (.) (.) du=ds-pdv (..3) (..3) (..3) S V U (..3) S V S (..3) d (..3) V A=U-S (..4) (Helmholtz free energy) A (..3) da=-sd-pdv (..5)
9 V (..5) (..3) P G=U+PV-S=A+PV (..6) (Gbbs free energy) G dg=-sd+vdp (..7) P (..7) N N µ N dn (..7) dg=-sd+vdp+ µdn (..8) (..8) P N (..4) (..6) (..8) G A U µ = = = (..9) N N N, P, V µ S, V
10 Gbbs Helmholtz (component) N N µ µ (.8) dg=-sd+vdp+µ dn +µ dn (..) dn µ A A µ (µ -µ A )dn µ A > µ A µ A < µ A A µ A = µ A (reactant) (product)
11 .3 x R µ µ = Rlnx +µ (.3.) µ x= x (.3.) x Rlnx µ x µ µ µ µ µ ( ) µ ( ) µ ( )µ ( ) µ ( )µ ( ) A A (.3.) A knetcs Rlnx A µ A
12 .4 N V (canoncal ensemble) (Hamltonan)H (partton functon)q P X H Q = dp dxexp (.4.) fn N! h k k (oltzmann constant) h (Planck constant) /N! N h f x yz 3 f=3 h Q A A = -k lnq (.4.) (.4.) (.4.) U (.4.) P (.4.) X Q = N! Λ N U dx exp k (.4.3)
13 /Λ (thermal de rogle wavelength) Λ Λ U (.4.3) (.4.3) (.4.) (.4.3) n N U u n W n Λλ Q(n,N)n y n U + W + u n n (, N ) = dxdy n N Λ k n N n exp!! λ Q n (.4.4) n N n nn+ µ Q(n+,N)Q(n,N) (.4.) µ = k = k Q n + log log nλ (, N ) Q( n, N ) dxdy n+ dxdy U + Wn+ + u exp k U + Wn + un exp k n n+ (.4.5) V ρ s (.4.5)
14 ( ) = = k u W U d d k u W U d d k k k u W U d d V k u W U d d k V n k n n n n n n s n n n n n n exp exp log log exp exp log log y X y X y X y X λ ρ λ µ (.4.6) (.4.6) log n+ (.4.6) n+ (.4.6) ( ) + = k U d k u U d k k s exp exp log log X X λ ρ µ (.4.7) (.4.7) X u (.4.67) L A = k U d k U A d A exp exp X X (.4.8) (.4.7) = k u k k s exp ln ln λ ρ µ (.4.9) (.4.6)
15 (.4.6) Λ (.4.9) µ = k ln ρ sλ (.4.) (.4.) (.4.6) ρ g s (.4.)ρ s ρ g s ρ l s (.4.9)ρ s ρ l s ρ ρ l s g s u = exp k (.4.) (.4.) L λ (.4.) u ρ l s u ρ g s (.4.9) ρ s ρ s x (.3.) (.4.9) (.3.) (.4.9)
16 . (.3.)µ..3 A n A n.4 A n A n A.5 A+C.6 C S.7 p m p m ( ) h πmk M 5.8 I A I I C θφψ rot rot = I A + I sn θ sn θ {( p p cosθ ) cosψ p snθ snψ } φ {( p p cosθ ) snψ + p snθ cosψ } + p φ ψ ψ θ θ I C ψ 3 h dp θ dp φ dp ψ rot exp = k π 8π I Ak h 8π I k h 8π I h C k snθ θφψ f(θ,φ,ψ) rot dpθ dpφ dpψ dθdφdψ exp f 3 h k πi Ak πi k πi Ck = h h h ( θ, φ, ψ ) snθdθdφdψ f ( θ, φ, ψ ) dxdydz snθdθdφdψ Jacoban
17 θφψ snθdθdφdψ dθdφdψ dxdydz snθdθdφdψ (.4.3).9.4..oltzmann (E) (V) (N) S(E,V,N)k log(e,v,n ) S(E,V,N) f = f(e,v,n )..3 f oltzmann
18 x x x ρˆ x ρˆ ( x) δ ( x - ) = x ( ) (3.) x ( ) x ρˆ ( x) ρˆ ( x) dx ρˆ x d ( ) ρ x ρ ( x) ˆ ρ( x) = (3.) x ρ x ( )
19 ( ) ρ ρ x ρ( x) ρ ( ) ( x,y) ρ ( ) ( x, y) = δ ( x x ) δ ( y x j ) j j ρ( x) ρ( x) ρ x ρ x,y 3 ( ) ( ) ( ) ( ) ρ x ρ x r ρ x ρg(r) ( ) r g(r) 3. g(r) (3.)g(r) r 3. ( )
20 x 6 6 (H O) H O O-OO-HH-H 3 3 α γ g αγ (r) ( r) ρ( x) ρg αγ () r = dxδ rαγ ( x) (3.3) 4πr (3.3) r αγ (x) x α γ (3.3) 6 ρ( x) ρ( x) g αγ (r) (3.3) n m nm v(x) ρ e ε ( ) ( ε ) = dxδ ( v( x) ε ) ρ( x) e ρ (3.4) g αγ (r) ρ ( ρ e ε ) (3.4) ( x) g αγ (r)( ρ e ε )
21 [,] 3. 3.? v(x) v(x)( ρ x) 3.4 v(x) ( ) v αγ ( r) v x = αγ v αγ (r)g αγ (r) 3.5 ρ e ε ( ) 3.
22 v(x) v(x) µ µ µ = k ln exp v( x ) k (3.5) (3.5) k L v(x) x (couplng parameter) λλ uλ ( x) ( x) = ( x) = v( x) u (3.6) u (3.5) ( x) ρ ( x ) uλ µ = d λ dx ; uλ λ (3.7) ( ) ρ x; u λ uλ x (3.) (3.7) λ (3.6) u x µ µ ( ) (3.7)(Krkwood )chargng u x λ chargng v(x) λ ( ) ( )
23 v(x) v(x) ρ x;v v(x)w(x) v(x)-w(x)= v(x)w(x), k ln exp ( ) ( w( x ) v( x )) k < dx( w( x) v( x) ) ρ( x v) ; v (3.8) L v(x) v v w k ln exp ( v( x ) w( x )) k < dx( v( x) w( x) ) ρ( x w) ; w v(x)w(x) ( x; v ) ρ( x; w) (3.9) ρ = (3.) (3.8) (3.9) v(x)w(x) ρ( x;v) ( ρ x; w) (3.) v(x) ρ x;v ( ) v(x) ρ( x) v(x) v(x)+δv(x) ρ(x) ρ(x)+δρ(x) δv(x) δρ ( x) dyχ ( x y;v) δv( y) =, (3.) χ(x,y;v) χ ( ( x, y; v) δ ( x y) ρ( x; v) + ρ ) ( x, y; v) ρ( x; v) ρ( y; v) = (3.) χ(x,y;v) ( x; v) = dyχ ( x, y;) ( ρ( x; v) ρ( x;) ) c (3.3) c(x;v) (3.3)Ornsten-Zernke ρ( x;) (v=)
24 ( x) v ρ( x; ) exp v = ρ x k ( ;) (3.4) (3.4) ω x;v ( ) ( x; v) ( x;) ρ ω ( x; v) = k ln v( x) (3.5) ρ (3.7) (3.6) = k ( x; v) ω( x; ρ ) ρ( x; u ) ω λ dx ( ρ( x; v) ρ( x;) ) + ρ( x; v) dλ (3.6) k k λ λ µ ω ( x; ρ λ ) ( ω x; u λ ) (3.6) v ρ ( ) ω x; (3.6)λ ρ λ (3.4) (3.5) c(x;v)( ω x;v) v v (3.6) ρ c(x;v)( ω x;v) v ω ( x; ρ ) ( ρ( x; v) ρ( x;) ) Percus-Yevck PY hypernetted-chan HNC ω ( x; ρ ) ( ρ( x; v) ρ( x;) ) [3]( ω x; ρ ) (3.6) PY HNC
25 [4] [5] PY HNC [4,5] 3.6 (3.5) 3.7 (3.5) µ = k ln exp v x ( ) k L (3.5) 3.8 x y k ln ρ x ρ y ( ( ) ( )) 3.9 (3.7) 3. v(x) ( ) v αγ ( r) v x = chargng αγ ( ) u x? λ 3. chargng u x? 3. (3.8) () F()=F()= F () (x)< F(x) F(x)> <x< () x x <λ< λ f () (x)> f(x) λf λ ( ) ( x ) + ( λ) f ( x ) > f ( λx + ( λ) x ) () () n (v) X f () (x)> f(x) ( ) ( ) f X > f X L
26 (v) (3.8) 3.3 (3.) (3.8) (3.9) (3.) 3.6 (3.3)Ornsten-Zernke Ornsten-Zernke (3.3) 3.7 (3.4) 3.8 (3.4) 3.9 (3.4) 3. ω( x;v) (3.5)x 3. (3.6) 3.c(x;v)( ω x;v) v 3.3PY ( ω( ) k ) ( ρ( x; v) ρ( x;) ) exp x;v PY c ( x; v) v = exp k ( x) ρ( x; v) ρ( x;) ρ = ρ ( x; v) ( x;) 3.4HNC x;v ω( ) ( ρ( x; v) ρ( x;) ) c ( x; v) ρ = ρ ( x; v) ( x;) ω + k ( x; v) ( x; v) ω exp k HNC 3.5 PY HNC 3.6 PY HNC [6] [4] 3.8
27 (3.6) [4] [5] 3.3 (3.6) 3.3 PY HNC 3 [] W. A. Steele, J. Chem. Phys., 39, 397 (963). [] L. lum and A. J. orruella, J. Chem. Phys., 56, 33 (97). [3] J. K. Percus, Phys. Rev. Lett., 8, 46 (96). [4] D. Chandler, J. D. McCoy, and S. J. Snger, J. Chem. Phys., 85, 597 (986). [5] N. Matubayas and M. Nakahara, J. Chem. Phys., 3, 67 (). [6] J. P. Hansen and I. R. McDonald, heory of Smple Lquds, nd Ed. Academc Press (986). Lennard-Jones 3. g(r) β µ r/σ ρσ 3 =.85, k /ε=.35 ρσ 3 k /ε=.35 3.Lennard-Jones
28 4 4. f N fn (q, q fn )fn (p, p fn ) fn fn fn (q, q fn ) q(p, p fn ) p (q,p) (Hamltonan)H(q,p) (Hamlton) dq H = dt p (4.) dp H = dt q (q,p) A(q,p) da A H A H = = LA (4.) dt q p p q L (Louvlle) (4.) (4.) ( t) exp( Lt) A( ) A = (4.3) L (q(),p()) (q(t),p(t)) ( q( t), p( t) ) ( q(), p() ) det = (4.4) ρ(q,p;t) ρ(q,p;t) (q,p) 6N A(q,p) t
29 <A(t)> A ( t) = dqdpa( q, p ) ( q, p; ) t t ρ (4.5) (q t, p t ) (q,p) t (q t, p t ) (q,p) t (4.5) (q,p) t, p t ) (q,p) (4.4) () = dqtdpt A( qt, pt ) ρ( q, p;) = dqdpa( q, p) ρ( q, p ;) A t (q t, p t ) (q -t, p -t (q,p) -t t () t = dqdpa( q p) ( q, p t) A (4.8) t (4.6) A, ρ ; (4.7) ( q p; t) = ρ( q, p ;) ρ (4.8), t t ( q, p; t) ρ( q, p; t) H ρ( q, p; t) ρ t ρ = q + p ( q, p; t) = exp( Lt) ρ( q, p;) p H = Lρ q ( q, p; t) (4.9) (4.) (4.3) (4.9) t (4.) (4.3) (q,p) (4.9) (q,p) (4.9) t (q,p) (4.5) (4.7) (4.9) t (4.9) (4.5) <A(t)> A(q t, p t ) ρ(q,p;) <A(t)> (4.7) A(q,p) <A(t)> ρ(q,p;t) 4.
30 Α(q t,p t ) Α(q,p ) Α(q,p) ρ(q,p;) ρ(q,p;) ρ(q,p;t) q, p Hesenberg pcture q, p Schrödnger pcture H(q,p) p H = + m ( q, p) V ( q) (4.) 4. (4.) 4.3 (4.4) 4.4 L L ρ ( q, p) X Y dqdpx dqdpx ( LY ) ρ ( q, p) = dqdp( LX ) Yρ ( q, p) { exp( Lt) Y } ρ ( q, p) = dqdp{ exp( Lt) X } Yρ ( q, p) ω exp ( ω t) dqdpx { exp( Lt) X } ρ ( q, p) 4.5 (4.8) (4.9) 4.6
31 4. (4.9) F(t) A H H =AF(t) (4.) L L L U(t) du dt U () t = L () t U () t t () t = exp( L t) ds exp( L ( t s) ) L U() s (4.) L t U () t = exp( Lt) U(t) L L t exp t ( Lt) = exp( L t) ds exp( L ( t s) ) L exp( L s) (4.) t> ρ (q,p) (4.9) (4.) ( ) L ( q, p) ( q p; t) = ρ ( q, p) ds exp L ( t s) t ρ, ρ ρ(q,p;t) (4.3) (4.3) (q,p) t ( ) L ( q, p) () t = ds dqdp L ( t s) exp ρ (4.4) (4.4) <(t)> L
32 X Y X Y, = (4.5) q p p q { X Y } (Posson) (4.4) Φ A () t = { () t, A} (t) (4.6) Φ A (t) (4.) t () t = dsf() s Φ ( t s) A (4.7) (4.6) () t A& & () t A Φ A () t = = (4.8) k k n A A n n ( t ) A ( ) n t A L n 3 U() U = ( ) dt A( t) F t ( t) (4.9) F(t) F(t) = F cos(ωt+α) (4.) U ( ) ωf ωf = dtφ AA AA exp () t sn( ωt) = Im dtφ () t ( ωt) (4.)
33 Φ A (t) A ( ω ) = dtφ ( t) exp( ωt) χ (4.) A χ A (ω) (4.) 4.7 (4.) 4.8 L t (4.)U(t) U ( t) = + ( ) dt L( t ) U ( t ) = + t t ( ) dt L( t ) + ( ) dt dt L( t ) L( t ) + L t t U t = () t exp dsl( s) L(t) 4.9 (4.3)L (4.) 4. XYZ { X,{ Y, Z } + { Y,{ Z, X } + { Z, { X, Y } = 4. XYZ { Y, Z} = dqdpy{ Z, X } dqdpz{ X Y } = dqdpx, 4. (4.7) 4.3 (4.6) (4.8) 4.4 (4.7) F(s)s F
34 A ( ) = F dtφ () t 4.5 (4.9) 4.6 (4.) 4.7 (4.) F(t) F(t) (4.)? 4.8 Φ A (t)t χ A (ω) Kramers-Krong χ A ( ω ) ( κ ) Im χ A = P dκ + Im χ A ( ω ) P π κ ω 4.9 χ AA (ω)ωim χ AA (ω) > 4.3 Redfeld (q s,p s ) (q b,p b ) ρ s (q s,p s ;t) ρ(q s,p s,q b,p b ;t) s ρ s s b b s s b b ( q, p ; t) dq dp ρ( q, p, q, p ; t) = (4.3) (q s,p s ) (q b,p b ) L s (q s,p s ) L b (q b,p b ) V s s b b s s b b ( q p, q, p ) A ( q, p ) F ( q p ) V, =, (4.4)
35 L ρ s (q s,p s )ρ b (q b,p b ) s s b b b b ( q, p ) = L ρ ( q, p ) = s s ρ (4.5) ρ s (q s,p s )ρ b (q b,p b ) F (q b,p b ) ρ b (q b,p b ) L v (4.) ρ s s b b ( q, p, q, p ; t) s b s s b b v s s s b b b = ( L + L ) ρ( q, p, q, p ; t) L ρ ( q, p ) ρ ( q, p ) t + t v s s b b ( ) L ρ( q, p, q, p ; y) v s b ( ) L dy exp ( L + L )( t y) (4.6) V(q s,p s,q b,p b ) t b {, } F ρ (4.7) ρ b (q b,p b ) s s b b s s s b b b ( q, p, q, p ; t) ρ ( q, p ; t) ρ ( q, p ) ρ = (4.8) ( ) (4.7) F t = L ρ b (q b,p b ) (4.6) (q b,p b ) ρ s (q s,p s ;t) s ρ ( s s q, p ; t ) s s = ( s s L ρ q, p ; t ) t + t, j dy F s s s { ( ){ A, ρ ( q, p ; y) } s () t F ( y) A,exp L ( t y) j j (4.9) (4.9) (4.9) ρ s (q s,p s ;t) () () F F t t j
36 s ρ s s ( q, p ; t) s s s s = L ρ ( q, p ; t) t +, j dy F s s s s () F ( y) { A,exp( L y) { A, ρ ( q, p ; t) } j j (4.3) (4.3)Redfeld (q s,p s ) t ( ) s s s s s s s s s () t dq dp ( q, p ) ( q, p ; t) = ρ (4.3) (4.3) d () t dt s = s L s +, j dy F s () F ( y) { A,exp( L y){ A, } (4.3) j ρ s (q s,p s ;t) j s d dt d = dt +, j dy F s () F ( y) { A,exp( L y){ A, } (4.33) j j (4.33) L s exp( L s y) A (q s,p s ) (4.33) exp L s y (4.33) ( ) () () A (q s,p s ) F F t j 4. (4.6) 4. (4.7) (4.8) 4. (4.9) 4.3 (4.3) 4.4 (4.3)
37 4.4 Ψ > p ρ ρ = p Ψ Ψ Redfeld 4.3 [,] 4.5 H ρ ρ = [ H, ρ] t h { } 4.6 (4.5) X, Y h h { X, Y } [ X, Y ] = ( XY YX ) (4.8) (4.8) Φ A k () t = dλ () t exp( λh ) A& exp( λh ) [] C. P [] J. McConnell, he heory of Nuclear Magnetc Relaxaton n Lquds, Cambrdge Unversty Press (987), pages 3-46.
38 5 NMR NMR (nuclear magnetc resonance) NMR F-NMR Redfeld 5. M=(M x, M y, M z ) µ µ =γm (5.) γ H E E = µh = γmh (5.) {M x, M y } = M z {M y, M z } = M x {M z, M x } = M y (5.3) (5.) (5.3) dm = { M, E} = γm H dt (5.4) H z H= (,, H) (5.4) M M M x y z () t = cos( γht) M x( ) + sn( γht) M y ( ) ( t) = sn( γht) M x( ) + cos( γht) M y ( ) () t = M ( ) z (5.5) (5.5) M z M x M y x-y γh γh 5. x-y
39 γh NMR M z γ M y C 6 O 5. H (M x, M y, M z ) (M x, M y, M z ) M M M x y z M x () t = [ cos( γh t) M x( ) + sn( γh t) M y ( ) ] () t = [ sn( γh t) M x () + cos( γh t) M y () ] () t = M ( ) z (5.6) H (H H )/H [] (5.4) M NMR M H γm = χh (5.7) χ χ z H= (,, H) M z M z (χ/γ)h M x M y
40 (5.4) dm dt dm dt dm dt z x y = γ = γ = γ ( M H ) ( M H ) ( M H ) z x y M z M M x M y z (5.8) loch loch 5. (5.4) 5. γ MHz MHz 3 C 7 O H 4 N 5.5 loch dm/dt = M 5. F-NMR F F 4. NMR F z H= (,, H) H x H (t) E ( t) = M H ( t) E = M x H x (5.9)
41 4. M x Φ(t) Φ t () = t ( ) = t exp sn γh t M z exp sn( γh t) χ H γ (5.) χ H (t) δ(t) (4.7) Φ(t) γh Φ(t) δ(t) Φ(t) F 5.6 (5.) 5.7 (5.9) M y M z 5.8 H (t)δ(t) H (t) H (t) δ(t) (5.) 5.9 F-NMR Φ(t) t t α (4.) α Redfeld H= (,, H) (5.) V V = γ (M x h x + M y h y + M z h z ) (5.) (h x, h y, h z ) 4.3 (h x, h y, h z ) H
42 (4.33) dm dt dm dt dm dt z x y = γ = γ = γ ( x x y y ) ( M H ) γ M ds cos( γhs) h ( ) h ( s) + h () h () s z ( y y z z ) ( M H ) γ M ds h ( ) h ( s) + cos( γhs) h ( ) h ( s) x ( x x z z ) ( M H ) γ M ds h ( ) h ( s) + cos( γhs) h ( ) h ( s) y z x y (5.) x y loch M z loch Redfeld h h x x () hx () t = hy () hy () t = hz () hz () t C() t () h () t = h () h () t = h () h () t = y (5.) y z z x (5.3) = γ dt cos = γ dt ( γht) C( t) ( + cos( γht) ) C( t) (5.4) C(t) τ t C() t = C exp (5.5) τ NMR = γ Cτ + ( γhτ ) = γ Cτ + + ( γhτ ) (5.6) C τ τ /γh τ
43 τ /γh τps γh GHz (5.6) = = γ Cτ (5.7) γhτ τ C τ 5. (5.) 5. (5.6) 5.C(t) /γh (5.5) (5.7) = τ C(t)? 5.4 []
44 [] [3]NMR 5 [] C. P []. Yamazak, H. Sato, and F. Hrata, J. Chem. Phys., 5, (). [3] (a) N. Matubayas, C. Waka, and M. Nakahara, J. Chem. Phys. 7, (997). NMR (b) N. Matubayas, C. Waka, and M. Nakahara, J. Chem. Phys., 8-8 (999). (c) N. Matubayas, N. Nakao, and M. Nakahara, J. Chem. Phys. 4, (). NMR
5 36 5................................................... 36 5................................................... 36 5.3..............................
9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
24.15章.微分方程式
m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt
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