NMR: Relaxation Measurements. How to measure relaxation rates ?. T 1 : Longitudinal or spin-lattice relaxation . M z is restored , the system goes back to equilibrium .
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Howtomeasurerelaxationrates? T1: Longitudinal orspin-latticerelaxation. Mzisrestored, thesystemgoes back toequilibrium. T2: Transverse orspin-spinrelaxation. Transverse magnetizationMx,yvanishes, the observable signaldisappears. Formeasurementspulsedmethodsshouldbeused
T2-Measurement In principlewecouldcalculate T2accordingto Dn1/2 = 1/pT2 fromthewidthoftheLorentzianlineshapeofthesignals in ourspectrum. But... • Thisvalueisstronglydepends on theinhomogeneityofour B0-field. • Weareratherinterested in the 'pure' spin-spinrelaxationcomponent (which, in contrasttotheB0-field, is a molecularproperty)!
D NMR spectrum w Homogeneous and inhomogeneous linewidth • However, transverse relaxation can also proceed due to statistical (and static) inhomogeneities in the precession frequency ω0. T • Resulting rate of Free Induction Decay is denoted as • The first contribution is the same for all molecules and thus defines the homogeneous linewidth. • The last contribution defines the inhomogeneous linewidth. • It is quite common that • There are methods of getting rid of the inhomogeneous linewidth!
echo p p/2 t 0 2t t t=0 j=0 j=Dwt j=p–Dwt+Dw(t–t) t=2t j=p Y´ Y´ Y´ Y´ X´ X´ X´ X´ Spin echo • Large inhomogeneous linewidth means very fast dephasing of the spin • However, dephased magnetization can be focused back by pulses • Let us consider pulse sequence π/2x- τ- π x • Explanation: let us divide system into isochromates having the same frequency ω0. Their offsets are Δω=ω0–ω. At certain time they all have different phases • But at t=2τall have the same phase – there is an ‘echo’!
Y´ Y´ Y´ Y´ Y´ Y´ Y´ X´ X´ X´ X´ X´ X´ X´ X´ Carr-Purcell method • Spin echo not only allows one to get rid of the inhomogeneous broadening but also to measure T2. To do this, however, the pulse sequence should be modified because the repetition rate of the echo experiment is <1/T1 (quite low) • Luckily, the whole echo decay can be measured while applying one pulse sequence (C-P). Let us apply the sequence π/2x- τ- πx – 2τ- πx – 2τ- … • At t=2τwe will have the first echo (negative phase). Then spins start dephasing again, the next πx-pulse again focuses them (positive phase!). Thus, there are echoes at times 2τ, 4τ, 6τ, 8τ,… amplitudes decay with T2. • Drawback: if the pulses are not set precisely, mistakes are accumulated with time. It is better to use CPMG sequence: π/2x- τ- πy- 2τ- πy- 2τ- … Y´
T2-Measurement 90o 180o T2 t t Spin-echo sequence I(t)=I(0)exp(-2t/T2)
T2-Measurement 90o 180o T2 t t Before acquisition Spin-echo sequence After 90o Before 180o After 180o Before 90o
T2-Measurement The spin-echo experiment: • Compensatesforthecomponentof T2thatoriginsfromfieldinhomogeneity • The relaxationcanbemeasuredselectively • Importantdynamicpropertiesofthemoleculecanbeextractedthatway
T2-Measurement The experiment is repeated a number of times with increasing delays t. T2isobtainedfrom a plotofI(t) againstt: I (0) I (t) I (t)=I (0)exp(-2t/T2) t
1 t –1 Inversion-recovery technique • Determination of T1 is often quite important as well • Standard method is inversion-recovery • First we turn the spin(s) by pulse (usually π/2 or π) and then look how system goes back to equilibrium (recovers Z-magnetization). If the pulse is a π-pulse magnetization will be inverted (maximal variation of magnetization) and then recovered • Equation for Mz is as follows: • The kinetic trace (t-dependence) gives T1-time • To detect magnetization at time t in NMR one more π/2-pulse is applied, sequence is then πx- t (variable) - πx/2 - measurement • For broad lines spin echo is used for detection, the pulse sequence is then πx- t (variable) - πx/2 - τ- πx- τ- measurement • Both sequences should be repeated many times at different delays t
T1-Measurement t = 0 180o 90o t t = ln(2)T1 Inversion recorvery Mz(t)=Mo[1-2exp(-t/T1)] t >> T1
Inversion-Recovery t Mz(t)=Mo[1-2exp(-t/T1)]
Mz +1 t 0 -1 t = ln(2)T1 t >> T1 Inversion-Recovery Mz(t)=Mo[1-2exp(-t/T1)]
Fast T1-Measurement 180o 90o t t = ln(2)T1 zero observable signal Inversion recorvery For a quick estimationof T1: directlysearchforthe time t, whichresults in zerointensity (tzero) andcalculate T1fromthis: T1 = tzero/ln(2)
What you see of it Magnet (probe, sample) Console (transmitter, receiver, interface) Computer (pulse- programming, data processing) Probe
What you (usually) don’t see of it 1 Bore tube 2 Filling port (N2) 3 Filling port (He) 4 Outer housing 5 Vacuum chambers/ radiation shields 6 Nitrogen reservoir 7 Vacuum valve 8 Helium reservoir 9 Magnet coil Shimming coils (not shown here) are also very important: One should resolve tiny splittings!!! Homogeneity of the order of 10–9 is necessary for NMR Inside a Magnet
Tesla and MegaHertz • The strength of a magnetic field is meassured in Tesla (for strong fields) or Gauss (for weaker fields). 1 Tesla corresponds to 10000 Gauss. The earth magnetic field is about 0.5 Gauss. • The strength of an NMR magnet is usually given in terms of its 1H resonance frequency in MHz:
Why go for stronger fields? Another reason is resolution: It is always better to work with AX-systems and only zz-parts of the scalar couplings spectra are much simpler and better resolved
Signal-To-Noise Ratio S/N S/N or the signal-to-noise ratio is a measure for the sensitivity of the NMR experiment: S/N ~ n g5/2B03/2(NS)1/2 Number of spins Number of scans Relative sensitivityandresolutionofourspectrometer
NMR probe • Locates the sample at homogeneous field; • RF curcuit and coil for irradiating the sample and detecting its subsequent response; • Additional functions (sample rotations, T stabilization, field gradients)
Transmitter, receiver, amplifiers • Transmitter section: produces RF irradiation; consists of RF-synthesizer, pulse gates and RF amplifier • S(t)=Acos(ωt+φ(t)) φ(t) can be rapuidly switched • Receiver section: preamplifier, quadrature reciever (comparison of the signals with a reference wave to get rid of fast oscillations) • Mx(t)=M0cos(ω0t) M0cos(Ω0t) where Ω0=ω0–ωref going from 300 MHz to 1 MHz • The procedure does not distinguish positive and negative Ω0 receiver supplies two signals: • SA(t)=M0cos(Ω0t) and SB(t)=M0sin(Ω0t) full information is retained but phasing is necessary
Hardware (Summary) Magnet (Dewar, coil, shims) Probe Transmitter, receiver, amplifiers Acquisitioncomputer (ADC) Sensitivity
NMR: NOE NOE=Nuclear Overhauser Effect Overhauser effect (EPR and NMR meet): NMR enhancement after pumping EPR transitions; works on dipolar relaxation of electron and nucleus NOE also works using dipolar relaxation of two nuclei Applications are quite different: not mainly enhancing NMR signals but rather measuring distances between spins
NMR: NOE RF RF RF B A
Nuclear Overhauser Effect B A regular spectrum RF NOE, small molecule RF NOE, large molecule RF
Nuclear Overhauser Effect describes the change in intensity of a signal due to the NOE
Energy Level Diagram Withpopulationdifferences forthe A and B transitions in theundisturbedsystem: bb W1A W1B W2 A0 = B0 = D ab ba W0 W0and W2involve simultan- eoustransitionsofspins A and B. Spins relax together in thismechanism. When spin A is off-equilibrium spin B will feel it. Difference of W0 and W2 is important W1B W1A aa
A W2 > W0 small molecules A W0 > W2 large molecules A A Nuclear Overhauser Effect A = 1.5 D W2 A W1A W1B W0 W1B A W1A A = A0 = D B = 0 A0 = B0 = D A = 0.5 D
In practicewe find the NOE ranging from +0.5 forsmallupto -1.0 for large molecules Signof NOE depends on whether W0 (minus) or W2 (plus) isdominating Nuclear Overhauser Effect 0.5 0.0 -0.5 -1.0 0.01 0.1 10 1.0 100 w0tc slow tumbling fast tumbling
Distancesfrom NOEs tc ~ r6 tc = rotational correlation time (size of molecule) r = distance between the two corresponding atoms
tc r6ref . = ref tcref r6 Distancesfrom NOEs ref r = rref 6 tc tcref
Information about short 1H-1H-distances in molecules (< 5Å) Translated into distance-constraints applied in Molecular Simulations Main source of structural information in NMR Applicationfor NOEs
Information about short 1H-1H-distances in molecules (< 5Å) Translated into distance-constraints applied in Molecular Simulations Main source of structural information in NMR It will be explained how it works Applicationfor NOEs
NMR: 2D-NMR Why is 1D (just NMR spectrum) not enough?
1-Dimensional NMR 1D FT-NMR (simplest case) preparation - detection FT S(t) S(w)
A 1D-Spectrum of a Protein For large proteins it is really hard to assign NMR signals and to obtain quantitative information from the spectra! Too many peaks Spectrum is a mess!
2-dimensional NMR FT1, FT2 2D FT-NMR S(t1,t2) S(w1,w2) t1 tm t2 Preparation - evolution - mixing - detection t2 – direct domain; t1 – indirect domain
Contour plot of the same Signal Compare: Topographical map (lines of equal height)
Now let us see how it works How to get to this second dimension???
The Second Time Domain FT (t2) The size of the signal depends on the evolution in t1: the signal is said to be 'modulated' with w1 t2=0 t2 t1 For simplicity we look at a single frequency w which is the same in t1 and in t2 (no mixing)!
The Second Time Domain FT (t2) w1 w2 t2=0 t2 t1 FT (t1)
hn The SCOTCH Experiment Spin COherence Transfer in (photo) CHemical reactions Reaction A B with a protonatwA in A whichresonatesatwB in B. t1 t2 l i g h t The corresponding pulse sequence
The SCOTCH Experiment The proton's magnetization is in t1 modulated with the frequency wA. After the light pulse, the same proton evolves with wB. t1 t2 l i g h t Subsequent FT of the both time domains results in a 2D spectrum with a peak at wA in F1 and wB in F2:
The SCOTCH Experiment The proton's magnetization is in t1 modulated with the frequency wA. After the light pulse, the same proton evolves with wB. t1 t2 l i g h t If A would not completely be converted to B by the light pulse, we would be able to observe a diagonal peak of A as well: