Lecture #8 Nuclear Spin Hamiltonian* • Topics – Introduction – External Interactions – Internal Interactions • Handouts and Reading assignments – van de Ven: Chapters 2.1-2.2 – Biographies: Ramsey, Zeeman – Levitt, Chapters 7 (optional) * many figures in this lecture from Levitt. 1 The Spin Density Operator • Spin density operator: σˆ (t) – Completely describes the state of a spin system. Isnt the trace a matrix • The expectation of any observable A ˆ : rather than operator € operation? ˆ ˆ ˆ pure state: A = Tr{σ ψ A} inner product in Liouville space € statistical mixture of states: Aˆ = Tr{σˆ Aˆ } € • Example: x, y, and z magnetization: ˆ ˆ My = γ!Tr{σ I y} ˆ ˆ ˆ Mx = γ! I x = γ!Tr{€σ I x} similarly ˆ ˆ Mz = γ!Tr{σ I z} 2 € € € € Liouville-von Neuman Equation • Time evolution of σ ˆ ( t ) : ∂ ˆ σˆ = −i[Hˆ ,σˆ ] = −iHˆ σ ˆ ∂t • Hamiltonian Hˆ - operator corresponding to energy of the system € • If H ˆ time independent:€ Hˆ (t) = Hˆ ˆ € I z σˆ ˆ I y −iHˆ t iHˆ t ˆ σˆ (t) = e σˆ (0)e = e−iHt σˆ (0) σˆ rotates around in operator space € Superoperator€ notation textbook notation ˆ € I x € € € • Key: find the Hamiltonian! € 3 The Spin Hamiltonian Revisited • In general, H ˆ is the sum of different terms representing different physical interactions. ˆ ˆ ˆ ˆ H = H1 + H 2 + H 3 +! Examples: 1) interaction of spin with B0 € 2) interaction with dipole field of other nuclei 3) spin-spin coupling • Life€ is easier if: € ˆ – H i are time independent. Terms that depend on spatial orientation may average to zero with rapid molecular tumbling (as if often seen in vivo). ˆ ˆ € – H i, H j terms commute, ˆ then rotations e − i H tσ ˆ can be computed in any order. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ −i( H 1 + H 2 )t −iH 1t −iH 2t −iH 2t −iH 1t [H i,H j ] = 0 e = e e = e e € 4 € € € Electromagnetic Interactions • Think of an atomic nucleus as a lumpy magnet with non- uniform positive electric charge • Nuclear spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy Nuclear magnetic moment interacts with magnetic fields Nuclear electric charge interacts with electric fields ˆ ˆ elec ˆ mag H = H + H 5 € Electromagnetic Interactions quadrapole • Electric Interactions monopole dipole Nuclear electric charge distributions can be expressed as a sum of ! ! ! ! multipole components. (0) (1) (2) C (r ) = C (r ) + C (r ) + C (r ) +" Based on symmetry arguments relating the shape of a nucleus and its spin: C(n)=0 for n>2I, and, within experimental error, odd terms disappear (not obvious) Hˆ elec = 0 (for spin I =1/2) Hence, for spin-1/2 nuclei there are no electrical € energy terms that depend on orientation or internal nuclear structure, i.e. Spin-1/2 nuclei behaves exactly like point charges! • Magnetic Interactions (weve seen this before) € Nuclear magnetic moment is primarily just the dipole moment. magnetic moment ! ! ! ! Hˆ mag ˆ B Iˆ B = −µ ⋅ = −γ" ⋅ local magnetic field 6 € Spin Hamiltonian: Overview Relative magnitudes Is tissue a solid or liquid? 7 External Magnetic Fields • Static Field B0 ! ! B = B e Hˆ static = −γB Iˆ = −ω Iˆ 0 z 0 z 0 z unit vector in z direction € • With RF €excitation (assuming B1 along the x axis) ! ! ! B = B cosωt e − sinωt e RF 1( x y ) Hˆ ext Hˆ static Hˆ RF ˆ ˆ ˆ = + = −γB0I z −γB1(I x cosωt − I y sinωt ) € In the rotating frame, near resonance (ω ≈ ω 0 ) € ˆ ext ˆ € H ≈ −ω1I x 8 € € Internal Interactions • In general, the form of the internal nuclear spin Hamiltonian is quite complicated. • Well make two simplifying approximations. Secular approx. Molecular motion ˆ ˆ 0 ˆ 0 H int H int H int 1. Secular approximation: large B0 field dominate some of the internal spin interactions. € 2. Motional averaging: € with rapid molecular tumbling,€ some interaction terms fluctutate with time and can be replaced by their motionally averaged values. Terms with zero time-average are dropped. • Discarded internal spin Hamiltonian terms are responsible for spin relaxation. 9 Motional Averaging • Molecular motion • Rotation Molecular orientation depends on time, hence secular Hamiltonian terms can be ˆ 0 written as H int ( Θ ( t ) ). These terms can be replaced by their time averages: τ ˆ 0 1 ˆ 0 ergodicity ˆ 0 ˆ 0 H int = τ ∫ H int (Θ(t))dt H int = H int p(Θ)dΘ 0 ∫ € p(Q)=probability density for molecule having orientation Q Isotropic materials: Hˆ isotropic 1 Hˆ 0 d int = N ∫ int (Θ) Θ € € normalization 10 € Motional Averaging • Diffusion On the timescale of an NMR experiment, molecules in a liquid largely diffuse within a small spherical volume a few tens of microns in diameter (known as a diffusion sphere). Intermolecular interactions Long-range interactions dont average to within the diffusion sphere zero, but are very small. average out to zero. 11 B0-Electron Interactions When a material is placed in a magnetic field it is magnetized to some degree and modifies the field… Electrons in an atom circulate about B0, Shielding: generating a magnetic moment opposing the applied magnetic field. • Global effects: magnetic susceptibility s B0 = (1− χ)B0 field inside sample bulk magnetic susceptibility applied field Hereafter well use B0 to refer to the internal field (to be revisited when we talk€ about field inhomogeneities and shimming). • Local effect: Chemical Shift shielding constant (Dont confuse with the Different atoms experience spin density operator!) different electron cloud densities. B = B0(1−σ) 12 € The Zeeman Hamiltonian ! • The interaction energy! between! the magnetic field, B , and the magnetic moment, µ = γ I , is given by the Zeeman Hamiltonian. Whos this ! ! ! !ˆ Zeeman guy? Classical: E = −γB ⋅ I QM: Hˆ = −γB ⋅ I zeeman€ • The formal€ correction for chemical shielding is: !ˆ ! € Hˆ I (1 )B € 3 3 shielding tensor zeeman = −γ −σ where σ = × • In vivo, rapid molecular tumbling averages out the non-isotropic components. € σ = σ iso = Tr(σ /3) (anisotropic€ components do contribute to spin relaxation) ! ˆ ˆ • Hence for B = [0,0,B0 ]: H Zeeman = −γ(1−σ)B0I z = −ωIˆ € z 13 € € € Chemical Shift • Energy diagram: water fat H2O ...(CH2)nCH3 e- ... E e- oxygen is electrophilic B = 0 B = B0 Frequency (Hz) ν = γB0(1−σ)/2π water 220 Hz @ 1.5 T € € fat First complete theory of chemical shift€ was developed by Norman Ramsey in 1950. Usually plotted as 4.7 1.3 0.9 ppm relative frequency shielding 6 δ =10 (ν −ν ref )/ν ref frequency 14 € Magnetic Dipoles • Nuclei with spin ≠ 0 act like tiny magnetic dipoles. # µ0 &# µ & Bµx = % (% (( 3sinθ cosθ) $ 4π '$ r3 ' permeability of free space Bµy = 0 # µ &# µ & B = % 0 (% ( 3cos2 θ −1 € µz $ 4π '$ r3 '( ) falls off as r3 Dipole at origin € Magnetic Field in y=0 plane € Lines of Force Bµz Bµx 15 Dipolar Coupling • Dipole fields from nearby spins interact (i.e. are coupled). • Rapid fall off with distance causes this to be primarily a intramolecular effect. Spins remain aligned with B0 Water with molecule tumbling in a magnetic field Interaction is time variant! 16 The Dipole Hamiltonian • Mathematically speaking, the general expression is: µ γ γ & "ˆ "ˆ 3 "ˆ " "ˆ " ) ! ˆ 0 I S where vector from H dipole = − 3 !( I ⋅ S − 2 (I ⋅ r )(S ⋅ r )+ r 2πr ' r * spin I to spin S • Secular approximation: $ !ˆ !ˆ ' µ γ γ 2 Hˆ = d& 3Iˆ Sˆ − I ⋅ S ) where d = − €0 I S " 3cos Θ −1 € dipole % z z ( 3 ( IS ) 4πr dipole coupling angle between B0 constant and vector from spins I and S € ˆ - with isotropic tumbling (e.g. liquid): H dipole = 0 What is line splitting? - without tumbling (e.g. solid): line splitting Note: big effects on relaxation for both cases.€ 17 J-Coupling • The most obvious interactions between neighboring nuclei is their mutual dipole coupling. • However this anisotropic interaction averages out to zero for freely tumbling molecules. • Consider the following NMR spectrum: Line splitting ?! Ethanol CH3 OH CH2 Chemical Shift (ppm) 18 J-Coupling: Mechanism • At very small distances (comparable to the nuclear radius), the dipolar interaction between an electron and proton is replaced by an isotropic interaction called Fermi contract interaction. Energy Diagram A simple model of J-coupling less stable I e e S I spin senses more stable polarization of S spin I e e S !ˆ !ˆ independent of molecular orientation • Interaction energy ∝−γ γ I ⋅ S e n Scalar coupling • What’s in a name? J-coupling Spin-spin coupling € Through-bond (vs through- space) interaction Indirect interaction 19 J-Coupling: Energy Diagram populations Zeeman Splitting J-Coupling Single quantum coherences ?? Energy Were now Alternative energy considering pairs of diagram spins • The J-Coupling Hamiltonian J-coupling constant (Hz) !ˆ !ˆ !ˆ !ˆ Hˆ I JS = 2πJI ⋅ S = 2πJ(Iˆ Sˆ + Iˆ Sˆ + Iˆ Sˆ ) J = x x y y z z product operators 20 € € € Summary: Nuclear Spin Hamiltonian • Free Precession ignore for now: effect small or time average=0 vanishes for spin=1/2 ˆ ˆ ˆ ˆ ˆ H = H Zeeman + H dipole + H J + H quadrupole WI and WS are resonant frequencies in rotating frame: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Ω = γ(1−σ )B −ω = ω −ω H = −ΩI I z + −ΩS S z + 2πJ(I xS x + I yS y + I zS z ) I I 0 rf I ΩS = γ(1−σ S )B0 −ω rf = ω S −ω € • With RF excitation (assuming B1 along the€ x axis) € € ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ I ˆ S ˆ H = −ΩI I z − ΩS S z + 2πJ(I xS x + I yS y + I zS z ) −ω1 I x −ω1 S x %ω I >> Ω and ω S >> Ω ˆ I ˆ S ˆ 1 I 1 S (secular approx.) H ≈ −ω1 I x −ω1 S x for & I S 'ω 1 , ω1 >> 2πJ € Above equations hold for both homonuclear and heteronuclear cases.