Lecture #3 Nuclear Spin Hamiltonian • Topics – Liouville-von Neuman equation – Time-averaged versus instantaneous spin Hamiltonian – Chemical shift and J, dipolar, and quadrupolar coupling • Reading assignments – van de Ven: Chapters 2.1-2.2 – Levitt, Chapters 7 (optional)

1 Coherences • Nuclear spins can be thought of as weak magnetic dipoles, each with a well defined direction of polarization.

• Summing the polarization across many spins can result in net phase coherences.

ˆ S z ˆ ˆ I 2I zS z S

€ Net tendency for S spins to be €+z No net tendency for I or S spins to be ±z No net tendency for I spins in any direction If I or S is ±z, increased probability paired spin is ±z

2 The Spin Density Operator • Spin density operator, σ ˆ ( t ), is the mathematical quantity that describes a statistical mixture of spins and the associated phase coherences that can occur, as encountered in a typical NMR or MRI experiment. € • σ ˆ ( t ) is usually expressed as a linear combination of basis 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (the 16 two-spin product operators) operators, e.g. { 2 E, Ix, Sx, Iy, Sy,…, 2IzSz } • Coherences (signals) observable with an Rf coil:

€ M = γ Tr σˆIˆ = γ Iˆ ˆ ˆ ˆ x ! { x } ! x M y = γ!Tr{σ Iy } = γ! Iy • Some coherences not directly observable with an Rf coil:

ˆ ˆ ˆ ˆ ˆ M z = γ! Iz Cxz = γ! 2IxSz Czz = γ! 2IzSz “antiphase x” “longitudinal two-spin order”

• The goal is to describe how these quantities evolve in time. 3 Liouville-von Neumann Equation • Time evolution of σ ˆ ( t ) : Iˆ z σˆ (0) ˆ t ˆ σ ( ) I y ˆ ∂ ˆ ˆ ˆ H is the Hamiltonian operator σ = −iHσ and corresponds to the energy € € ∂t of the system (E / ! ). € ˆ I x • H ˆ can often be expressed as sum of a large static component plus ˆ ˆ ˆ € a small time-varying perturbation: H = H 0 + H 1 ( t ) , leading to… ˆ Relaxation superoperator (a function of H 1 ) ∂ ˆ ˆ ˆ ˆ ˆ ˆ σ = −iH0σ − Γ(σ −σ B ) (Compare with Bloch’s equations) ∂t Rotations Relaxation • The key, yet again, is finding the Hamiltonian!

4 Spin-Lattice Disconnect

• Complete QM description of a molecule involves lots of terms in the Hamiltonian (nuclear spin, molecular motion, electron-nucleus interactions, etc). ˆ ˆ ˆ ˆ H = H l + H s + H i lattice spin interaction term

• Previously, we assumed a negligible interaction between the nuclear spins and the lattice: ˆ ˆ ˆ ∂ ˆ H ≈ H l + H s and just solved σˆ = −iH σˆ ∂t s • Now we need to take a closer look at the interaction term, which includes effects such as spin operator coefficients that depend on molecular orientation, etc.

5 The Nuclear Spin Hamiltonian • 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) interactions with dipole fields of other nuclei 3) nuclear-electron couplings • In general,€ we can think of an atomic€ nucleus as a lumpy magnet with a (possibly non-uniform) positive electric charge • The spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy The nuclear magnetic moment interacts with magnetic fields Hˆ = Hˆ elec + Hˆ mag The nuclear electric charge interacts with electric fields 6 € Electromagnetic Interactions • Magnetic interactions magnetic moment ! ! ! ! ˆ mag ˆ H = −µˆ ⋅ B = −γ"I ⋅ B local magnetic field

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 ) +" Symmetry properties: C(n)=0 for n>2I and odd interaction terms disappear Hence, for spin-½ nuclei there are no electrical ˆ elec H = 0 (for spin I€= 1/2) energy terms that depend on orientation or internal nuclear structure, and they behaves exactly like ˆ elec H ≠ 0 (for spin I >1/2) point charges! Nuclei with spin > ½ have electrical quadrupolar moments. 7 € Motional Averaging

• Molecular motion

• Previously, we used averaging to simplify the Hamiltonian Molecular orientation depends on time and Hamiltonian terms can be written ˆ 0 as H int ( Θ ( t ) ). These terms were 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(Θ)=probability € Secular ˆ isotropic 1 ˆ 0 Hamiltonian Isotropic materials: H H d density for molecule int = N ∫ int (Θ) Θ normalization having orientation Θ What does € • We no longer want to make this€ approximation. “secular” mean? Instead, the time variations will be analyzed as perturbations. € 8 Time-averaged Spin Hamiltonian Relative magnitudes Relative

9 Instantaneous Spin Hamiltonian Relative magnitudes Relative

10 Simplifications • In general, the nuclear spin Hamiltonian is quite complicated.

• We’ll regularly make use of two simplifications. 1. For terms in the Hamiltonian that are periodic, we use a change of basis. ˆ ˆ ˆ Hˆ ! = e−iωtIz Hˆ = e−iωtIz HˆeiωtIz rotating frame laboratory frame

Hˆ t Iˆ Iˆ cos t Iˆ sin t Hˆ Iˆ Iˆ ( ) = −ω0 z −ω1 ( x ω − y ω ) eff = −(ω0 −ω) z −ω1 x

2. The secular approximation

11 The Secular Approximation

• The large B0 field dominates some of the internal spin interactions. • Consider the general case of Hˆ = Aˆ + Bˆ “large” “small” ˆ • Choosing the eigenkets, A i = a i i , as an orthonormal basis set,

then A = diag(a1, a2..., an ). • Example: let the eigenvalues of A have the following pattern:

In general, B is ! * * * * * * * $ Energy # & non-zero * * * * * * * everywhere. # & a5, a6, a7 # * * * * * * * & # & a nearly degenerate. B = * * * * * * * 1 # & # * * * * * * * & a a 2, 3 degenerate # * * * * * * * & # & a4 * * * * * * * " % 12 The Secular Approximation • The secular approximation of B is: Eigenvalues of A ! * 0 0 0 0 0 0 $ # & Energy # 0 * * 0 0 0 0 & # 0 * * 0 0 0 0 & a5, a6, a7 B0 = # 0 0 0 * 0 0 0 & # & a1 # 0 0 0 0 * * * & # & 0 0 0 0 * * * a2, a3 # 0 0 0 0 * * * & " % a4 ` 0 • Mathematically B = ∑bnn n n + ∑bmn m n n m≠n where the` indicates summation only over terms which connect degenerate or nearly degenerate states of A.

• This is equivalent to omitting terms for which bmn << am − an . • The secular approximation for the Hamiltonian is: Hˆ ≈ Aˆ + Bˆ 0. 13 Secular Approximation Example

• Consider a spin in a large field B0 to which we add an x gradient Gx. ! • We actually can’t create the field G x x z alone (see Laplace’s Eqn) ! ! ! ! ! • We really have: B = B0z +Gx xz + ΔBx (x, y, z) x + ΔBy (x, y, z) y In MRI, these are often called “Maxwell” or “concomitant gradient” terms. ! ! ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Hence: H = −γB⋅ I = −γB0Iz −γGx xIz −γΔBx Ix −γΔByIy = A + B # & " % γG x γ ΔB +iΔB γB0 0 % x ( x y ) ( ˆ A 1 $ ' Bˆ B = 1 where A = 2 and 2 % ( $ 0 −γB0 ' γ ΔB −iΔB −γG x # & $% ( x y ) x '(

• However, B 0 > > G x x , Δ B x , Δ B y è a1 − a2 = γB0 >> γ ΔBx ±iΔBy

" γG x 0 % hence B0 = 1 $ x ' 2 0 G x #$ −γ x &' ! ! ˆ ˆ ˆ ˆ 0 ˆ • The secular approximation is: H = −γB⋅ I ≈ A + B = −γ (B0 +γGx x)Iz i.e. we can safely ignore the “Maxwell” terms as is routinely done in MRI (with just a few exceptions, particularly at low field) 14 B0-Electron Interactions When a material is placed in a magnetic field it is magnetized to some degree and this 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 we’ll use “B0” to refer to the internal field. € • Local effect: Chemical Shift shielding constant (Don’t confuse with the spin density operator!) Different atoms experience B B (1 ) different electron cloud densities. = 0 −σ

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€ The Zeeman Hamiltonian !

• The interaction energy! between! the magnetic field, B , and the magnetic moment, µ = γ I , is given by the Zeeman Hamiltonian. ! ! ! !ˆ Classical: E = −γB ⋅ I QM: Hˆ = −γB ⋅ I zeeman€ • The formal€ correction for chemical shielding is: !ˆ ! Hˆ I (1 )B σ = 3× 3 shielding tensor € zeeman = −γ −σ € where • In vivo, rapid molecular tumbling averages out the non-isotropic

components. σ = σ iso = Tr(σ /3) € ! € • Hence for B = [0, 0, B0 ]: E ˆ ˆ €H Zeeman = −γ(1−σ )B0Iz

16 B = 0 B = B0

€ € A little foreshadowing… Chemical Shielding Tensor • Electron shielding is in general anisotropic, i.e. the degree of shielding depends on the molecular orientation. • The shielding tensor can be written as the sum of three terms: ! $ σ xx σ xy σ xz ! 1 0 0 $ # & # & See Kowalewski, pp # & (1) (2) 105-6 for details. σ = σ yx σ yy σ yz = σ iso # 0 1 0 &+σ +σ # & # 0 0 1 & symmetric and # σ zx σ zy σ zz & " % traceless " % antisymmetric 1 = 3 (σ xx +σ yy +σ zz ) • Both σ(1) and σ(2) are time-varying due to molecular tumbling. • σ(2) gives rise to a relaxation mechanism called chemical shift anisotropy (CSA). (to be discussed later in the course) • σ(1) causes only 2nd order effects and is typically ignored. 17 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”. A simple model of J-coupling Energy Diagram less stable I e e S I spin “senses” polarization of more stable S spin I e e S !ˆ !ˆ independent of molecular orientation • Interaction energy ∝ −γ eγ n I ⋅ S Scalar coupling • What’s in a name? J-coupling Spin-spin coupling € Through-bond (vs through- Indirect space) interaction interaction 18 J-Coupling: Energy Diagram Zeeman Splitting J-Coupling Energy

We’re now considering pairs of spins

• The J-Coupling Hamiltonian J-coupling constant (Hz) !ˆ !ˆ !ˆ !ˆ ˆ 2 JI S 2 J(Iˆ Sˆ Iˆ Sˆ Iˆ Sˆ ) H J = I JS = π ⋅ = π x x + y y + z z

“product operators” 19

€ € € A little foreshadowing… J-Coupling and Relaxation

• Because J is unchanged with molecular tumbling, J-coupling typically does not contribute to relaxation. ! ! ˆ ˆ ˆ J Θ t = J H J = 2π JI ⋅ S where ( ( ))

• However, there are a few cases where J can become “effectively” time-varying.

- Case 1: the S spin is engaged in chemical exchange

- Case 2: the T1 of the S spin itself is << 1/J.

• These cases are called scalar relaxation of the first and second kind respectively, and both are important for the study of MRI

contrast agents. 20 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

€

21 Lines of Force Bµz Bµx 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! 22 The Nuclear Dipolar Coupling Hamiltonian • Mathematically speaking, the general expression is: µ γ γ # "ˆ !ˆ 3 !ˆ ! !ˆ ! & ! ˆ 0 I S where r vector from Hdipole = − 3 !%I ⋅ S − 2 (I ⋅ r)(S ⋅ r)( 4πr $ 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 1 dipole = z z − ⋅ = − 3 "( ΘIS − ) % ( 4πr dipole coupling angle between B0 constant and vector from spins I and S ˆ € - With isotropic tumbling, the time average of Hdipole = 0

ˆ - However, the temporal variations of H d ip o le ( t ) are typically the dominant source of T1 and T2 relaxation in vivo. 23 Quadrupolar Interactions • Nuclei with spin I > ½ have a electrical quadrupolar moment due to their non-uniform charge distribution.

• This electrical quadrupole moment interacts with local electric field gradients - Static E-field gradients results in shifts of the resonance frequencies of the observed peaks. - Dynamic (time-varying) E-field gradients result in relaxation. • Quadrupolar coupling Hamiltonian (secular approximation): Coupling constant ! ! Looks like an ˆ 3eQ ˆ2 ˆ ˆ HQ = V0 3Iz − I ⋅ I interaction of a spin 4I (2I −1)! ( ) with itself. What’s the spin of Gd3+ with its 7 unpaired electrons? Electric field gradient – dependent on molecular orientation 24 A little foreshadowing… Nucleus-unpaired electron couplings • Both nuclear-electron J and dipolar coupling occur. • Important for understanding MR contrast agents.

Magnevist 7 unpaired electrons, I = 7/2, non-zero quadrupolar moment

J coupling Dipolar coupling

25 Summary: Nuclear Spin Hamiltonian

!ˆ ! !ˆ ! !ˆ !ˆ !ˆ !ˆ 2 !ˆ !ˆ Hˆ = −γ I (1−σ )B −γ S(1−σ )B + 2π J I ⋅ S + d 3Iˆ Sˆ − I ⋅ S +ηQ 3Iˆ − I ⋅ I I I S S ( ) ( z z ) ( z ) Zeeman terms J-coupling Dipolar Quadrupolar coupling coupling (+S spin term) • Major relaxation mechanisms important for MRI (+ contrast agents) - d(t) gives rise to dipolar relaxation - σ(t) gives rise to chemical shift anisotropy (CSA) - η(t) gives rise to quadrupolar relaxation - “J(t)” gives rise to scalar relaxation of the 1st and 2nd kind - Plus, we also need to figure out how include chemical exchange effects ˆ I ˆ S ˆ - At the end, we’ll add Rf excitation when computing T1ρ: H Rf = −ω1 Ix −ω1 Sx

26 Next lecture: Basics of Relaxation

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