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NMR Literature review (Basic Concepts (Basic Design (High field! (larger…

- - - - Larmor frequency: \( \omega_0 = \dfrac{2\mu B_0}{\hbar} \)
        Radiofrequency range
      - Pauli matrices. Orthogonal. Complete set. Raising, lowering operators
      - Energy separation between two states: \( \Delta E = \gamma B_0 \)
        \( \gamma \) - gyromagnetic ratio
      - Nuclear magnetic moment
        \( \vec{\mu} = \gamma\vec{L} \); QM: \( \hat{\mu}_p = \gamma\hat{L}_p = \gamma \hbar \hat{I}_p \)
    - - Spin Precession with Larmor frequency
      - Netto Magnetization through spin allignment
      - RF puls (flip angle) - Magnetization nutation and decay (T1,T2). Phase coherence
      - Rotating Frame: easier Math
  - - - \( B = B_0 (1-\sigma) \), where \(\sigma\) is a shielding tensor
      - liquids - trace averaging, solids - no (anisotropy of dipol-dipol interactions)
      - \( \Delta \sigma = \Delta \sigma^{loc} + \Delta \sigma^m + \Delta \sigma^e + \Delta \sigma^{vW} + \Delta \sigma^{med} \)
    - - J coupling
        
        \( H = H_{\delta} + H_J = - \sum 2\pi \nu_i I_z(i) + H_J \)
        Iz is z comp. of spin angular momentum
        
        Weak AX / strong AB coupling
        \(\nu_a-\nu_x ?? J_{ax}\)
        *usually deal only with weak
        
        Off diagonal zero quantum operators
        \( \Delta H = \frac{1}{2}J_{AX}[I_A^+I_X^+ + I_A^-I_X^- ] \)
        *only for strong coupling
        
        Spin-spin coupling. J(ij) - scalar coupling constant
        \( H_J = 2\pi J_{ij}I_i I_j \)
        *indep. of magn. field, through joint el. cloud (bond)
      - Dipolar coupling
        
        through space
        \( B_D^j = \gamma_i (3\cos^2\Theta_{ij})/r_{ij}^3 \)
        
        \( H_{ij}=d (I_iI_j-3I_{zi}I_{zj}) \)
        *only high field spectra, no affect on relaxation
        
        Solids: \( \Delta \mu = 3\mu(3\cos^2\Theta_{ij}-1)r_{ij}^3(\mu_0/4\pi) \)
        *in liquids averages out (if anisotrop.). HR solid state NMR
      - Quadrupolar coupling
        
        High degeneration level D = 2I+1
        
        Nuclear charge distr. not spherically symm.
        
        Quadrupole moment changes en. levels with el. field gradient
        
        Strongest. Dominate relaxation
        
        Averages out by isotrop. motion (D2)
        
        Low el. gradient (symm. env) -> sharp line
      - El. - nucleus
        
        EPR
        
        Paramagnetic compounds!
        Relaxation is dominated by el. - nucl. interaction
        
        not all el. paired
        
        el. - very high spin magn. moment
  - - - Bloch (first order) longitud. relax.:
        \( \dfrac{dM}{dt} = R_1(M_0-M_z) \)
        \( M_z(t) = M_0(1-\exp[-t/T_1]) \)
        *exchange of E with env. (spin - lattice), DD interaction
      - Transvers. relax.:
        \( \dfrac{dM_{xy}}{dt} = - R_2 M_{xy} \)
        \( M_{xy}(t) = M_{xy}(0)exp[-t/T_2] \)
        *spin-spin relaxation, faster then T1
    - - Mol. motion -> loc. field fluct. -> frequency
      - Major cause - DD interaction (inter/intramolecular)
        Other: chemical shift anisotropy (CSA), scalar coupling (SC), spin rotation (SR)
      - Paramagn. material enhances relax. (high magn. moment)
      - Nuclei with s>1/2 relax mainly through quadrupole inter.
    - - Fluct. field - H perturbation
        \( \dfrac{d\rho}{dt} = -\frac{i}{\hbar} [H_0+H_1(t),\rho] \)
      - Correlation time
        \( \tau_c = 4\pi \nu r^3/3kT \)
      - Spectral density:
        
        Fast tumbling \(\omega\tau_c >>1\) -> both T1&2 very low
        Small molecules, extreme narrowing.
        \( R_1=R_2 = \tau_cd^2 \)
        
        \( J(\omega) = \dfrac{2\tau_c}{1+\omega^2\tau_c^2} \)
        *distribution of frequencies of mol. motion
        
        \(\int J(\omega)d\omega = \pi/2\)
        
        Intermediate tumbling \(\omega\tau_c \approx1\) -> most efficient relaxation for T1&2. Large contrib. from single, double and zero-quantum transitions. Medium size, viscous solution.
        
        Slow tumbling \(\omega\tau_c << 1\) -> both T1 high, T2 may vary. Small molecules, extreme narrowing
        \( R_1 = 1/T_1 \neq R_2 \)
        
        Cross section:
        \( \sigma_{ij} = (1/8) d^2 (\mu_0/4\pi) {12J(\omega_i+\omega_j)-2J(\omega_i-\omega_j)}\)
      - Dipolar relaxation
        \( R_{DD} = \gamma_i^2\gamma_j^2 \hbar^2\tau_c\sum r_{ij}^{-6}\)
    - - \( {\frac {dM_{x}(t)}{dt}}=\gamma ({\mathbf {M}}(t)\times {\mathbf {B}}(t))_{x}-{\frac {M_{x}(t)}{T_{2}}} \)
      - \(\frac{dM_y(t)}{dt} =\gamma ({\mathbf {M}}(t)\times {\mathbf {B}}(t))_{y}-{\frac {M_{y}(t)}{T_{2}}} \)
      - \(\frac{dM_z(t)}{dt} = \gamma ({\mathbf {M}}(t)\times {\mathbf {B}}(t))_{z}-{\frac {M_{z}(t)-M_{0}}{T_{1}}}\)
  - - - Slow exchange. Distinct A and B signals observed with intensities equal to their relative populations. Eq. constant and Free energy difference can be measured.
        \(k_{ex} << 1/\delta v\)
      - Fast exchange. NMR does not resolve conformations. Time-average.
        \(k_{ex} > 1/\delta v\)
      - Coalescence. Broad signal. Line shape - calculate exchange rate and free energy.
  - - - J coupling - loss in STR. Crucial for rare spins as C13
      - Radiation of B2 to the spin resonance removes all couplings to this spin. Example - remove all H couplings (broad band decoupling, off-resonance). Alternative - RF train (all H peaks)
      - Problem: retention of low BW at high field -> increased radiation power -> sample heating. Uniform irradiation (resid. split compared to line-width). Low power dissipation, T controll. Insens. to RF pulse width and phase shift. Side band signals.
    - - Through space.
      - Enhancement of signal intensity when a spin in close spat proximity is saturated -> polarization of nuclear spin distribution
      - System tends to return to thermal equilibrium through single, zero and double quantum transitions
      - Cross-relaxation rate \( \sigma_{AX}=W_2-W_0\)
        Auto-relaxation rate \( \rho=2W_1+W_2+W_0\)
        \( NOE = \dfrac{\sigma_{AX}}{\rho}\)
      - All transitions have their probabilities (see relaxation rates). If put together - NOE depends on correlation time. If relaxation is dominated by intramol. DD interactions and molecule is in the fast motion regime one can measure distances.
        \( \mu_{AX}/\mu_{AY} = (r_{AX}/r_{AY})^6 \)
      - Limited to 5A. Need reference distances.
  - - - Chemical shift
        \( \sum \Omega_k \hat{I}_{kz} \)
      - J coupling
        Strong: \( \sum 2\pi J_{kl} \hat{I}_k \hat{I}_l \)
        Weak: \( \sum 2\pi J_{kl} \hat{I}_{kz} \hat{I}_{lz} \)
      - Gradient
        \( \gamma_I G_z z \hat{I}_z \)
    - - Definition
        \( \hat{\sigma}_{\psi}(t) = \vert \psi_s(t) \rangle \langle \psi_s(t) \vert \)
      - Time dep.
        \( \frac{d}{dt}\hat{\sigma} = -i [\hat{H}_s,\hat{\sigma}] \)
      - Time indep H
        \( \hat{\sigma}(t) = e^{-i \hat{\hat{H_s}}t}\sigma(0) \)
      - Observable
        \( \langle \hat{A} \rangle = Tr[\hat{\sigma}\hat{A}] \)
      - Master equation
        \( e^{i \Theta \hat{A}}\hat{B} = \hat{B}\cos(\Theta) + i[\hat{A} \hat{B}] \sin(\Theta) \)
- - - - No need in single crystalls and heavy atom derivatives.
      - Difficult to look at big molecules (<50 kDa)
      - Need C13 and N15 labelled molecules.
      - Need high concentration. Aggregation problem.
      - Ability to study dynamics (bio function)
    - - High res. solid state NMR
      - Multi-dimensional NMR
      - In vivo (cell/tissue) studies
    - - Does not pass in 600ul 5mm NMR tube
      - No ionising radiation. Non invasive.
      - NMR signal of water protons. Spatial map.
  - - - \( M_z = M_z(0)[1-\exp(-TR/T_1)] \)
      - Wait \( \tau \sim 5T_1 \) for equilibration of z magnetization. If fast TR used: steady state reached
      - Smaller flip angle allows shorter longitudinal relaxation via reduction of transverse signal. Optimal Ernst angle:
        \( \cos(\theta) = \exp(-TR/T_1) \)
    - - Presaturation. Saturation with low RF power. Reduces SNR because of the magnetization transfer from the solvent.
      - Jump and Return. Two pi/2(y) pulses separated by time delay. Solvent resonances stay in X axis, then deleted to Z axis. Others spread and detected. Pi phase difference on other sides of solvent peak
    - - Spin-echo. \( (\pi/2)_y-\tau-\pi_x-\tau-FID \)
      - Pi flip over X axis inverses the dephasing and it recovers during next relax time. However other factors causing decay of transverse magnetization still act.
      - SE modulates J coupling (phase modulation) and refocuses chemical shifts. In case of heteronuclear spliting, if SE acts only on 1 spin, there is no phase modulation, frequency labeling of other spin.
  - - - Scheme
        
        Evolution (under \(H_{\delta}+H_J\)) Time incremented in steps of \(\Delta t_1\) - second dimention (number of steps)
        
        Mixing. Coherence transfer. Spin interaction.
        
        Detection of separate FID(\(t_2\)) -> 2D dataset \(S(t_1,t_2)\)
        -> 2DFT: \(S(\Omega_1,\Omega_2)\). Homonuclear diagonal: 1D NMR trace. Off diagonal - important info about spin interactions
        
        Preparation(RD + pi/2 pulse + decoupler pulse?)
      - Analysis
        
        Right cut-off
        
        Homonuclear - more steps (30 min)
        Heteronuclear - less (2 min)
      - Simplest experiment.
        
        COrrelated SpectroscopY (COSY). 2 pi/2 pulses. Freq - CS, Cross peaks -J couplings.
        
        Diag. - dispersive. Cross - absorptive.
        
        Adv. Shows whole network of coupled spins
        
        Problem: axial peaks (z-magn. -> mixes into xy plane). Solution: phase cycling.
        
        Problem: t1 noise. Instrumental instabilities.
        
        Problem: diagonal peak tails, masking of cross peaks near diag. Solution: symmetrization.
      - Advantages
        
        Better chem. shifts
        
        Correlation parameters in one experiment.
        
        See coupled spin systems
        
        MQ transitions
        
        Heteronuclear correlations
        
        Solvent filtering
        
        spin decoupling
    - - Coherences
        
        Single-quantum \(\Delta m = \pm 1\) (allowed), zero-quantum, double-quantum etc. (described through magnetic quantum number change)
        
        \( 2I_{kx}I_{lz} \) and \( 2I_{kz}I_{lx} \) give rise to single-quantum absorptive anti-phase doublet coherence
        
        \( 2I_{ky}I_{lz} \) and \( 2I_{kz}I_{ly} \) give rise to dispersive single-quantum anti-phase doublet coherence
        
        Density matrix \(\rho = aI_x + bI_y + cI_z\)
        Expectation value of an operator \( \langle A \rangle = Tr[\rho A] \)
        
        Single transverse component: observable
        Multiple transverse components: multiple quantum coherences - not observed
        One transverse and some z comp: antiphase components
      - Evolution
        
        Under time independent Hamiltonian
        \( H = H_{\delta}+H_J=(\Omega_kI_{kz}+\Omega_lI_{lz}) + (2\pi J_{kl}I_{kz}I_{lz}) \)
        
        Composite pi pulse: \((\pi/2)_x-(\pi)_y-(\pi/2)_x\). Corrects inperfections of pi pulse in COSY fashion. Series of pi pulses on water proton - water suppression (no pop. difference on water proton)
        
        MQ evolution: same rules apply
      - Density matrix
        
        Size: \( 2^n\times2^n \), where n is number of spins
        
        Multidementional base vectors can be constucted as Product operators via cross product of less dimentional base vectors. One dimensional base set - Pauli matrices and unity matrix. 4 dim.: \( I_{kx} = 2I_{kx}\bigotimes \frac{1}{2}E \)
        
        Multidimensional matrix describing spin state of the system
        
        Number of base vectors: \( 4^n \)
    - - Homonuclear
        
        COSY
        
        AX system: \( \rho_1 = I_{kz}+I_{lz} \)
        
        Preparation: \( \rho_1 \xrightarrow{(\pi/2)_y} I_{kx}+I_{lx} \)
        
        Evolution
        
        \( I_{kx} \xrightarrow{H_{\beta}} I_{kx}\cos(\Omega_k t_1)+I_{ky}\sin(\Omega_k t_1) \)
        
        \( I_{kx} \xrightarrow{H_J} I_{kx}\cos(\pi J_{kl} t_1)+2I_{ky}I_{lz}\sin(\pi J_{kl} t_1) \)
        
        \( H_{\beta}I_{kx} = EI_{kx} \rightarrow I_{kx}(t) = I_{kx}\exp[-iEt]\)
        
        Phase Cycling
        
        PAPS
        
        CYCLOP
        
        Axial
        
        Relayed COSY - seen relayed peak AX, where AM and M'X coupled and M/M' are not resolved
        
        TOCSY. All networks of coupled spins (in phase diagonal and cross peaks)
        
        MQ filtered COSY
        
        Exclusive COSY
        
        ROESY, if NOE not observed
      - Heteronuclear
        
        INEPT
        
        basic
        
        reversed
        
        refocoused
        
        DEPT
        
        Broadband Decoupling
        
        HSQC, directly bonded HX
        
        basic
        
        enhanced
        
        HMQC
      - MQ . Forbidden fruit
        
        DQ
      - MD
        
        NOESY
        
        Double resonance
        
        TOCSY
      - Macromolecules
        
        TROSY
      - Field gradients
        
        PFG
        
        WATERGATE