Lecture 6: Ensemble of non-interacting spins Classical interacting spin magnetic moments θ Classical interacting spin magnetic moments θ Classical interacting spin magnetic moments θ Classical interacting spin magnetic moments θ Classical interacting spin magnetic moments θ          B2,x B2,y B2,z          = µ0 4πr5            3r2 x − r2 3rxry 3rxrz 3rxry 3r2 y − r2 3ryrz 3rxrz 3ryrz 3r2 z − r2            ·         µ2,x µ2,y µ2,z         Classical interacting spin magnetic moments θ E = −µ1 · B2 = − µ1,x µ1,y µ1,z ·        B2,x B2,y B2,z        = − µ0 4πr5 µ1,x µ1,y µ1,z ·         3r2 x − r2 3rxry 3rxrz 3rxry 3r2 y − r2 3ryrz 3rxrz 3ryrz 3r2 z − r2         ·        µ2,x µ2,y µ2,z        Classical interacting spin magnetic moments θ E = − µ1,x µ1,y µ1,z · D ·        µ2,x µ2,y µ2,z        = − µ0 4πr5 µ1,x µ1,y µ1,z ·         3r2 x − r2 3rxry 3rxrz 3rxry 3r2 y − r2 3ryrz 3rxrz 3ryrz 3r2 z − r2         ·        µ2,x µ2,y µ2,z        QM interacting spin magnetic moments θ ˆHD = − ˆµ1,x ˆµ1,y ˆµ1,z · D ·        ˆµ2,x ˆµ2,y ˆµ2,z        = − µ0γ1γ2 4πr5 ˆI1,x ˆI1,y ˆI1,z ·         3r2 x − r2 3rxry 3rxrz 3rxry 3r2 y − r2 3ryrz 3rxrz 3ryrz 3r2 z − r2         ·         ˆI2,x ˆI2,y ˆI2,z         QM interacting spin magnetic moments θ ˆHD = − µ0γ1γ2 4πr5 ((3r2 x−r2)ˆI1,xˆI2,x+3rxryˆI1,xˆI2,y+3rxrzˆI1,xˆI2,z +3ryrxˆI1,yˆI2,x + (3r2 y − r2)ˆI1,yˆI2,y + 3ryrzˆI1,yˆI2,z +3rzrxˆI1,zˆI2,x + 3rzryˆI1,zˆI2,y(3r2 z − r2)ˆI1,zˆI2,z 1 particle: Ψ(x, y, z, cα) 1 particle: Ψ(x, y, z, cα) 1 particle: Ψ = φ(x, y, z) · ψ(cα) 1 particle: Ψ = φ(x, y, z) ·   cα cβ   2 particles = 1 pair: Ψ(x1, y1, z1, x2, y2, z2, cα,1, cα,2) 2 particles = 1 pair: Ψ(x1, y1, z1, x2, y2, z2, cα,1, cα,2) 2 particles = 1 pair: Ψ = φ(x1, y1, z1, x2, y2, z2) · ψ(cα,1, cα,2) 2 particles = 1 pair: Ψ = φ(x1 . . .) · ψ1(cα,1) · ψ2(cα,2) 2 particles = 1 pair: Ψ = φ(x1 . . .) ·   cα,1 cβ,1   · ψ2(cα,2) 2 particles = 1 pair: Ψ = φ(x1 . . .) ·   cα,1 · ψ2(cα,2) cβ,1 · ψ2(cα,2)   2 particles = 1 pair: Ψ = φ(x1 . . .) ·         cα,1   cα,2 cβ,2   cβ,1   cα,2 cβ,2           2 particles = 1 pair: Ψ = φ(x1 . . .) ·        cα,1cα,2 cα,1cβ,2 cβ,1cα,2 cβ,1cβ,2        2 particles = 1 pair: Ψ = φ(x1 . . .) ·        cαα cαβ cβα cββ        Direct product of wave functions (vectors)     cα,1 cβ,1     ⊗     cα,2 cβ,2     =              cα,1     cα,2 cβ,2     cβ,1     cα,2 cβ,2                  =             cα,1cα,2 cα,1cβ,2 cβ,1cα,2 cβ,1cβ,2             ≡            cαα cαβ cβα cββ            Direct product of matrices ˆA ⊗ ˆB =    A11 A12 A21 A22    ⊗    B11 B12 B21 B22    =             A11    B11 B12 B21 B22    A12    B11 B12 B21 B22    A21    B11 B12 B21 B22    A22    B11 B12 B21 B22                =            A11B11 A11B12 A12B11 A12B12 A11B21 A11B22 A12B21 A12B22 A21B11 A21B12 A22B11 A22B12 A21B21 A21B22 A22B21 A22B22            2 particles = 1 pair: ψ1ψ2 =        cαα cαβ cβα cββ        1 000 000 000 000 000 000 000 000 000 pairs : ˆρ =                 cααc∗ αα cααc∗ αβ cααc∗ βα cααc∗ ββ cαβc∗ αα cαβc∗ αβ cαβc∗ βα cαβc∗ ββ cβαc∗ αα cβαc∗ αβ cβαc∗ βα cβαc∗ ββ cββc∗ αα cββc∗ αβ cββc∗ βα cββc∗ ββ                 4 × 4 matrix ⇒ 16 4 × 4 basis matrices Products of spin operators in Hamiltonian θ ˆHD = − µ0γ1γ2 4πr5 ((3r2 x−r2)ˆI1,xˆI2,x+3rxryˆI1,xˆI2,y+3rxrzˆI1,xˆI2,z +3ryrxˆI1,yˆI2,x + (3r2 y − r2)ˆI1,yˆI2,y + 3ryrzˆI1,yˆI2,z +3rzrxˆI1,zˆI2,x + 3rzryˆI1,zˆI2,y(3r2 z − r2)ˆI1,zˆI2,z Product operators as basis 2 · It(1) ⊗ It(2) = It(12) (1) 2 · Ix(1) ⊗ It(2) = I1x(12) (2) 2 · Iy(1) ⊗ It(2) = I1y(12) (3) 2 · Iz(1) ⊗ It(2) = I1z(12) (4) 2 · It(1) ⊗ Ix(2) = I2x(12) (5) 2 · It(1) ⊗ Iy(2) = I2y(12) (6) 2 · It(1) ⊗ Iz(2) = I2z(12) (7) 2 · Ix(1) ⊗ Ix(2) = 2I1xI2x(12) (8) 2 · Ix(1) ⊗ Iy(2) = 2I1xI2y(12) (9) 2 · Ix(1) ⊗ Iz(2) = 2I1xI2z(12) (10) 2 · Iy(1) ⊗ Ix(2) = 2I1yI2x(12) (11) 2 · Iy(1) ⊗ Iy(2) = 2I1yI2y(12) (12) 2 · Iy(1) ⊗ Iz(2) = 2I1yI2z(12) (13) 2 · Iz(1) ⊗ Ix(2) = 2I1zI2x(12) (14) 2 · Iz(1) ⊗ Iy(2) = 2I1zI2y(12) (15) 2 · Iz(1) ⊗ Iz(2) = 2I1zI2z(12), (16) Product operators as basis: examples 2 · It(1) ⊗ It(2) = 2 · 1 2    1 0 0 1    · 1 2    1 0 0 1    = 1 2            1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1            ≡ It 2 · Ix(1) ⊗ It(2) = 2 · 1 2    0 1 1 0    · 1 2    1 0 0 1    = 1 2            0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0            ≡ I1x 2 · It(1) ⊗ Ix(2) = 2 · 1 2    1 0 0 1    · 1 2    0 1 1 0    = 1 2            0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0            ≡ I2x Density matrix: interpretation ˆρ =              cααc∗ αα cααc∗ αβ cααc∗ βα cααc∗ ββ cαβc∗ αα cαβc∗ αβ cαβc∗ βα cαβc∗ ββ cβαc∗ αα cβαc∗ αβ cβαc∗ βα cβαc∗ ββ cββc∗ αα cββc∗ αβ cββc∗ βα cββc∗ ββ              ˆρ = CtIt +C1xI1x +C1yI1y +C1zI1z +C2xI2x +C2yI2y +. . . Matrices It, I1x, I1y, I1z, I2x, I2y . . . : features (polarizations) of the mixed state Coefficients Ct, C1x, C1y, C1z, C2x, C2y . . . : how much individual features contribute to the actual state Density matrix: populations 4 populations (real numbers), 3 independent (cααc∗ αα + cαβc∗ αβ + cβαc∗ βα + cββc∗ ββ = 1): ˆρ =              cααc∗ αα cααc∗ αβ cααc∗ βα cααc∗ ββ cαβc∗ αα cαβc∗ αβ cαβc∗ βα cαβc∗ ββ cβαc∗ αα cβαc∗ αβ cβαc∗ βα cβαc∗ ββ cββc∗ αα cββc∗ αβ cββc∗ βα cββc∗ ββ              or It = 1 2        +1 0 0 0 0 +1 0 0 0 0 +1 0 0 0 0 +1        I1z = 1 2        +1 0 0 0 0 +1 0 0 0 0 −1 0 0 0 0 −1        I2z = 1 2        +1 0 0 0 0 −1 0 0 0 0 +1 0 0 0 0 −1        2I1zI2z = 1 2        +1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 +1        No polarization It = 1 2            +1 0 0 0 0 +1 0 0 0 0 +1 0 0 0 0 +1            Longitudinal polarization of µ1, regardless of µ2 I1z = 1 2            +1 0 0 0 0 +1 0 0 0 0 −1 0 0 0 0 −1            Longitudinal polarization of µ1, regardless of µ2 I1z = 1 2            +1 0 0 0 0 +1 0 0 0 0 −1 0 0 0 0 −1            Longitudinal polarization of µ2, regardless of µ1 I2z = 1 2            +1 0 0 0 0 −1 0 0 0 0 +1 0 0 0 0 −1            Coupled longitudinal polarizations of µ1 and µ2 2I1zI2z = 1 2            +1 0 0 0 0 −1 0 0 0 0 +1 0 0 0 0 −1            Density matrix: coherences 12 coherences (complex numbers: 12 real, 12 imaginary), 6 independent (cααc∗ αβ = cαβc∗ αα etc.): ˆρ =              cααc∗ αα cααc∗ αβ cααc∗ βα cααc∗ ββ cαβc∗ αα cαβc∗ αβ cαβc∗ βα cαβc∗ ββ cβαc∗ αα cβαc∗ αβ cβαc∗ βα cβαc∗ ββ cββc∗ αα cββc∗ αβ cββc∗ βα cββc∗ ββ              or purely real/imaginary 1 2      0 0 +1 0 0 0 0 +1 +1 0 0 0 0 +1 0 0      1 2      0 0 +1 0 0 0 0 −1 +1 0 0 0 0 −1 0 0      1 2      0 0 −i 0 0 0 0 −i +i 0 0 0 0 +i 0 0      1 2      0 0 −i 0 0 0 0 +i +i 0 0 0 0 −i 0 0      1 2      0 +1 0 0 +1 0 0 0 0 0 0 +1 0 0 +1 0      1 2      0 +1 0 0 +1 0 0 0 0 0 0 −1 0 0 −1 0      1 2      0 −i 0 0 +i 0 0 0 0 0 0 −i 0 0 +i 0      1 2      0 −i 0 0 +i 0 0 0 0 0 0 +i 0 0 −i 0      1 2      0 0 0 +1 0 0 +1 0 0 +1 0 0 +1 0 0 0      1 2      0 0 0 −1 0 0 +1 0 0 +1 0 0 −1 0 0 0      1 2      0 0 0 −i 0 0 +i 0 0 −i 0 0 +i 0 0 0      1 2      0 0 0 −i 0 0 −i 0 0 +i 0 0 +i 0 0 0      Transverse polarization of µ1 (direction x), regardless of µ2 I1x = 1 2            0 0 +1 0 0 0 0 +1 +1 0 0 0 0 +1 0 0            Transverse polarization of µ1 (direction x) coupled with longitudinal polarization of µ2 2I1xI2z = 1 2            0 0 +1 0 0 0 0 −1 +1 0 0 0 0 −1 0 0            Transverse polarization of µ1 (direction y), regardless of µ2 I1y = i 2            0 0 −1 0 0 0 0 −1 +1 0 0 0 0 +1 0 0            Transverse polarization of µ1 (direction y) coupled with longitudinal polarization of µ2 2I1yI2z = i 2            0 0 −1 0 0 0 0 +1 +1 0 0 0 0 −1 0 0            Transverse polarization of µ2 (direction x), regardless of µ1 I2x = 1 2            0 +1 0 0 +1 0 0 0 0 0 0 +1 0 0 +1 0            Transverse polarization of µ2 (direction x) coupled with longitudinal polarization of µ1 2I1zI2x = 1 2            0 +1 0 0 +1 0 0 0 0 0 0 −1 0 0 −1 0            Transverse polarization of µ2 (direction y), regardless of µ1 I2y = i 2            0 −1 0 0 +1 0 0 0 0 0 0 −1 0 0 +1 0            Transverse polarization of µ2 (direction y) coupled with longitudinal polarization of µ1 2I1zI2y = i 2            0 −1 0 0 +1 0 0 0 0 0 0 +1 0 0 −1 0            Coupled transverse polarization of µ1 and µ2 in direction x 2I1xI2x = 1 2            0 0 0 +1 0 0 +1 0 0 +1 0 0 +1 0 0 0            Coupled transverse polarization of µ1 and µ2 in direction y 2I1yI2y = 1 2            0 0 0 −1 0 0 +1 0 0 +1 0 0 −1 0 0 0            Transverse x polarization of µ1 coupled with transverse y polarization of µ2 2I1xI2y = i 2            0 0 0 −1 0 0 +1 0 0 −1 0 0 +1 0 0 0            Transverse y polarization of µ1 coupled with transverse x polarization of µ2 2I1yI2x = i 2            0 0 0 −1 0 0 −1 0 0 +1 0 0 +1 0 0 0            WHAT REMAINS? WHAT CHANGES? Hamiltonian In the absence of radio waves: ˆH = ˆH0 + ˆHD ˆH = −γ1B0ˆI1z − γ2B0ˆI2z + ˆHδ,1 + ˆHδ,2 ˆH0 + ˆHD In the presence of radio waves (B1 B2): ˆH ≈ ˆH1 Hamiltonian of chemical shift = + + ˆHδ = ˆHδ,i + ˆHδ,a + ˆHδ,r Hamiltonian of chemical shift anisotropy ˆHδ,a = − ˆµx ˆµy ˆµz · δa ·        0 0 B0        = −γδa ˆIx ˆIy ˆIz ·         3a2 x − 1 3axay 3axaz 3axay 3a2 y − 1 3ayaz 3axaz 3ayaz 3a2 z − 1         ·        0 0 B0        Hamiltonian of dipolar coupling θ ˆHD = − ˆµ1,x ˆµ1,y ˆµ1,z · D ·        ˆµ2,x ˆµ2,y ˆµ2,z        = − µ0γ1γ2 4πr5 ˆI1,x ˆI1,y ˆI1,z ·         3r2 x − r2 3rxry 3rxrz 3rxry 3r2 y − r2 3ryrz 3rxrz 3ryrz 3r2 z − r2         ·         ˆI2,x ˆI2,y ˆI2,z         Chemical shift Hamiltonian Isotropic component (independent of orientation): ˆHδ,i = −γB0δi(ˆIz) Anisotropic (axially symmetric) component (depends on ϑ, ϕ): ˆHδ,a = −γB0δa(3 sin ϑ cos ϑ cos ϕˆIx+3 sin ϑ cos ϑ sin ϕˆIy+(3 cos2 ϑ−1)ˆIz) Rhombic (asymmetric) component (depends on ϑ, ϕ, χ): ˆHδ,r = −γB0δr( (− cos 2χ sin ϑ cos ϑ cos ϕ + sin 2χ sin ϑ cos ϑ sin ϕ)ˆIx + (− cos 2χ sin ϑ cos ϑ sin ϕ − sin 2χ sin ϑ cos ϑ cos ϕ)ˆIy + ((cos 2χ sin2 ϑ)ˆIz) Secular approximation and averaging Recall chemical shift Hamiltonian: Isotropic component: ˆHδ,i = −γB0δi(ˆIz) Anisotropic (axially symmetric) component: ˆHδ,a = −γB0δa(3 sin ϑ cos ϑ cos ϕˆIx+3 sin ϑ cos ϑ sin ϕˆIy+(3 cos2 ϑ−1)ˆIz) Rhombic (asymmetric) component: ˆHδ,r = −γB0δr( (− cos 2χ sin ϑ cos ϑ cos ϕ + sin 2χ sin ϑ cos ϑ sin ϕ)ˆIx + (− cos 2χ sin ϑ cos ϑ sin ϕ − sin 2χ sin ϑ cos ϑ cos ϕ)ˆIy + ((cos 2χ sin2 ϑ)ˆIz) Hamiltonian in presence of dipolar coupling In the absence of radio waves in isotropic liquid (on time scales ns, not relaxation!): ˆH = −γ1(1 + δi,1)B0 − γ2(1 + δi,2)B0 = ˆH0 ˆH = ˆH0 In the presence of radio waves (B1 B2): ˆH ≈ ˆH1 Liouville - von Neumann equation dˆρ dt = i(ˆρH − H ˆρ) = i[ˆρ, H ] = −i[H , ˆρ] If ˆρ = cIj, H = ωIl, and [Ij, Ik] = ±iIl, ˆρ evolves as ˆρ = cIj −→ cIj cos(ωt) ± cIk sin(ωt) THE SAME FORM ⇒ new commutators: [Inx, Iny] = iInz, [Iny, Inz] = iInx, [Inz, Inx] = iIny [Inj, 2InkIn l] = 2[Inj, Ink]In l [2InjIn l, 2InkIn m] = [Inj, Ink]δlm + [In l, In m]δjk Ij, Ik, Il are product operators ⇒ rotation in 3D subspace of 16D operator space Discussed later in the course (not needed now because dipolar coupling averages to zero in isotropic liquids) Operator of measured quantity M+ = M1+ + M2+ = M1x + iM1y + M2x + iM2y ˆM+ = N γ1(ˆI1x + iˆI1y) + γ2(ˆI2x + iˆI2y) ˆM+ = N γ1ˆI1+γ2ˆI2+ MINOR INTUITIVE MODIFICATION Thermal equilibrium as the initial state ˆρeq =            1 4 + γ1B0 8kBT + γ2B0 8kBT 0 0 0 0 1 4 + γ1B0 8kBT − γ2B0 8kBT 0 0 0 0 1 4 − γ1B0 8kBT + γ2B0 8kBT 0 0 0 0 1 4 − γ1B0 8kBT − γ2B0 8kBT            = 1 4            1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1            + γ1B0 8kBT            +1 0 0 0 0 +1 0 0 0 0 −1 0 0 0 0 −1            + γ2B0 8kBT            +1 0 0 0 0 −1 0 0 0 0 +1 0 0 0 0 −1            = 1 2 It + κ1I1,z + κ2I2,z , SAME APPROACH AS FOR ISOLATED NUCEI, just applied to both magnetic moments No effect of dipolar coupling (exact in isotropic liquids) Relaxation due to dipolar coupling Bloch-Wangsness-Redfield theory applicable dipolar coupling: different Hamiltonian, large effect dipolar b = − µ0γ1γ2 4πr3 d∆ M1z dt = − b2 8 (6J(ω0,1)+2J(ω0,1 − ω0,2) + 12J(ω0,1 + ω0,2))∆ M1z + b2 8 (2J(ω0,1 − ω0,2) − 12J(ω0,1 + ω0,2))∆ M2z = −Ra1∆ M1z − Rx∆ M2z d∆ M2z dt = − b2 8 (6J(ω0,2)+2J(ω0,1 − ω0,2) + 12J(ω0,1 + ω0,2))∆ M2z + b2 8 (2J(ω0,1 − ω0,2) − 12J(ω0,1 + ω0,2))∆ M1z = −Ra2∆ M2z − Rx∆ M1z d M1+ dt = − b2 8 (4J(0) + 3J(ω0,1)+6J(ω0,2) +J(ω0,1 − ω0,2) + 6J(ω0,1 + ω0,2)) M1+ = −  R0,1 + 1 2 Ra1   M1+ = −R2,1 M1+ Differences of dipole-dipole relaxation Hamiltonian of dipolar coupling vs. chemical shift: 2I1xI2z, 2I1yI2z, 2I1zI2x, 2I1zI2y like I1x, I1y, I2x, I2y 2I1xI2x, 2I1yI2y, 2I1xI2y, 2I1yI2x, 2I1zI2z new b2 = µ0γ1γ2 4πr3 2 (γ1B0δa,1)2 if 2 is attached proton +2J(ω0,1 − ω0,2) + 12J(ω0,1 + ω0,2) in Ra1 and Ra2 The whole Rx ∝ 2J(ω0,1 − ω0,2) − 12J(ω0,1 + ω0,2) cross-relaxation (mutual dependence) Nuclear Overhauser Effect (NOE) +6J(ω0,2) in R0,1