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Pulse/Fourier Transform NMRChemical Exchange
Summary
Biophysical Chemistry: NMR SpectroscopySpin Dynamics & Chemical Exchange
Lieven Buts
Vrije Universiteit Brussel
25th November 2011
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Quantum Description of a Spin-1/2
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Larmor Precession (1)
The interaction between an individual spin and a uniformexternal magnetic field leads to precession of the spin aroundthe direction of the external field:
The angle θ between the direction of the field and the directionof the spin remains constant throughout this motion. Defrequency of the precession is the Larmor frequencyν = γ(1−σ)B0
2π or ω = γ(1− σ)B0.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Larmor Precession (2)
The interaction between the spin and the external field is farstronger than all other interactions between the nucleus andother particles in its environment. Therefore, as a firstapproximation, the nucleus behaves like an isolated gyroscopewhich rotates independently, with no regard for its surroundingsor the motions of the molecule which it is part of.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Populations (1)
The ratio between the populations of the two energy levels (nαand nβ) is determined by the energy difference ∆E and thetemperature T:
nβnα
= e−∆EkT
from which we find that
nα − nβnα + nβ
=∆E2kT
The Boltzmann constant (k = kB = 1.38066× 10−23 JK ) functions
a conversion factor from temperature to thermal energy.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Populations (2)
At a temperature of T = 300K the average thermal energy iskT = 4.14× 10−21J .The energy difference between the two stationary states of aspin I = 1/2 is very small, even for 1H (which has the largestgyromagnetic ratio of all practically available nuclei) in a strongexternal field:
γ = 26.73× 107T−1s−1; B0 = 9.4T; ∆E = ~γB0 = 2.65× 10−25J
This implies that the difference between the two populations isvery small:
nα − nβnα + nβ
=∆E2kT
= 3.2× 10−5
In other words, about one low-energy spin out of every 105 hasno counterpart in the high-energy orientation.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Populations at Thermal Equilibrium
Each individual spin contributes a certain fraction of "αcharacter" (proportional to |cα|2) and a complementary fraction"β character" (proportional to |cβ|2 = 1− |cα|2) to the ensemble(the collection of all spins).
The populations nα and nβ ofthe two energy levels are theavarage values |cα|2 and|cβ|2 over all spins in theensemble.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Thermal Coupling
Very infrequently, the nucleus does interact with a surroundingparticle, which can lead to a change of its orientation withrespect to the external field, as expressed by the angle θ. Theenergy that drives these interactions comes from the thermalenergy of the atoms, that is associated with their randommotions. The minuscule energy changes of the nuclear spinsare associated with equally minuscule temperature changes ofthe system. Because of the energy difference between the αand β states there is a small preference for random flips thatmove the spin state towards the lower energy level. As a result,a thermal equilibrium between the α and β populations isslowly established. This equilibrium is described by theBoltzmann distribution.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Bulk Magnetisation at Equilibrium (1)
The individual dipole moments of all spins can be addedtogether to find the total or bulk magnetisation of thesample.men.In the x and y directions, the spins are oriented completelyrandomly:
which results in a net magnetisation of zero in these directions.Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Bulk Magnetisation at Equilibrium (2)
In the z direction there is a small preference for the low energystate, as reflected by the slightly larger population nα:
nα − nβnα + nβ
=∆neq
N=
∆E2kT
Because of this, a small net magnetisation remains in thedirection of the positiove z axis. The magnitude of thisremainder is proportional to the population difference ∆neq:
M0 =12γ~∆neq
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Larmor Precession at Equilibrium
At equilibrium, the x and y components of the spin dipolesremain randomly distributed throughout the precessionalmotion, and theirsum remains zero. The distribution of the αand β components is also unaffected by the precessionalmotion, and therefore the z component of the totalmagnetisation also remains constant.The bulk magnetisation vector therefore remains constant asthe individual spins precess around the z axis.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Design of a Modern NMR Spectrometer (1)
Design of a Modern NMR Spectrometer (2)
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Bulk Magnetisation at Equilibrium (3)
At thermal equilibrium the spins are almost equally distributedin all directions, with a small preference for the low-energystate:
(For the purpose of the illustration, the population differencehas been greatly exaggerated.)
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Design of a Modern NMR Spectrometer (3)
Effect of an RF Pulse
A well-tuned RF pulse coherently rotates all spins about the xaxis. The net effect is that the bulk magnetisation as a wholeundergoes the same rotation:
Design of a Modern NMR Spectrometer (4)
Return to Equilibrium (Relaxation)
When the excitation by the RF pulse ends, the system returnsto its equilibirium state. The oscillating variation of the netmagnetisation in the (x, y) plane is the source of the obervablesignal:
The Magnetic Field of an RF Pulse
Physics tells us that only the magnetic component of the RFradition coming from the excitation coil affects the spins.Because of the position of the coil with respect to the samplethis magnetic component ~B1 rotates in the x,y plane, with afrequency ωRF and a phase φRF determined by the operator:
B1,x = B1 cos(ωRFt + φRF)
B1,y = B1 sin(ωRFt + φRF)
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
The Rotating Frame
In order to simplify the description of the precession of spinsaround a field that is itself rotating, we introduce a new frame ofreference that rotates around the z axis at the frequence of theRF wave (ωRF):
~e′x = cos(Φ(t))~ex + sin(Φ(t))~ey
~e′y = cos(Φ(t))~ey − sin(Φ(t))~ex
~e′z = ~ez
Φ(t) = ωRFt + φRF
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Implications
In the rotating frame the magnetic field of the RF pulse appearsto be fixed on the x axis.The precession frequency ω0 of the spins has to be replaced bythe offset frequency Ω0:
Ω0 = ω0 − ωRF
The pulse frequency ωRF is generally chosen to lie in the middleof the natural frequency range of the spins in the sample.Therefore, offset frequencies can be both positive and negative.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
The Effect of Resonance
The offset frequency in the rotating frame corresponds toLarmor precession around a reduced magnetic field ∆B:
∆B =Ωγ
=ω0 − ωRF
γ
When ωRF = ω0, ∆B = 0 and the effective magnetic field Beff iscompletety determined by B1 along the x axis.This is a geometric representation of the resonance principle.
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Coherent Excitation (1)
At resonance the spins therefore precess around an effectivefield along the x axis. The angle of rotation βp around the x axisis determined by the intensity of the pulse (B1) and by itsduration (tp):
βp ∼ γB1tp
Since all individual spins are coherently rotated by this sameangle, the bulk magnetisation also rotates by this angle.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Coherent Excitation (2)
A 90-degree pulse converts the equilibrium populationdifference (on the z axis) completely into a coherent orientationin the x, y plane.
The magnitude of the measurable transverse signal is thereforedetermined by the original population difference.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Pulse Length Calibration
By executing a series of test pulses of increasing duration, onecan determine which duration tp corresponds to a flip angle βp
of 180 degrees. Once this value is known, the required durationfor any desired flip angle can be easily calculated.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Larmor Precession after Excitation
When the RF pulse ends, the spins resume their precessionaround the external field. Since they are all rotating at the sameLarmor frequency, the direction of their preferred orientation,and therefore the bulk magnetisation vector, also rotate at theLarmor frequency.This rotational change of the bulk magnetisation in the x,y planeis equivalent to a variable magnetic field and induces anobservable current in the detector coil.
-10
-5
0
5
10
0 2 4 6 8 10 12 14
Mx
t
-10
-5
0
5
10
0 2 4 6 8 10 12 14
My
t
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Mixed Ensembles
If there are different ensembles of spins with distinct Larmorfrequencies mixed together in the sample, all spins are excitedsimultaneously by the RF pulse. Subsequently each subgroupprecesses at its own Larmor frequency, and the total observedsignal is the sum of the contributions of all subgroups atdifferent frequencies:
-10
-5
0
5
10
0 2 4 6 8 10 12 14
Mx
t +-10
-5
0
5
10
0 2 4 6 8 10 12 14
Mx
t =-15
-10
-5
0
5
10
15
0 2 4 6 8 10 12 14
Mx
t
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Fourier Analysis
Pulse/Fourier Transform NMRChemical Exchange
Summary
Thermal EquilibriumEffect of RF PulsesThe Fourier Transform
Relaxation
Due to a number of relaxation mechanisms, the bulkmagnetisation ultimately returns to its equilibrium value, andthe oscillating signal gradually fades away.
-10
-5
0
5
10
0 2 4 6 8 10 12 14
Mx
t
-10
-5
0
5
10
0 2 4 6 8 10 12 14
My
t
Mx = M0 sin(Ω0t) exp(− tT2
)
My = −M0 cos(Ω0t) exp(− tT2
)
in which T2 is a characteristic time constant.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
The Lorentzian CurveThe Fourier transform of such an oscillating and exponentiallyfading signal is called a Lorentzian curve and can be desribedanalytically as
S(Ω) = Aλ
λ2 + (Ω− Ω0)2
where A is the amplitude of the signal is, and λ = 1T2
.
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Isomerisation of a Partial Double Bond
The bond between the two nitrogen atoms in the nitroso groupof N,N’-dimethylformamide has a partial double bond character.The cis and trans forms both occur and have identical energies,but there is a significant energy barrier for the transition of oneconformer to the other.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Slow Exchange
When the exchange between the two states is very slow (or,more accurately, very rare) each individual molecule is eiher instate A or in state B during the whole course of the NMRmeasurement, without switching. The sample can then beconsidered as a mixture of two distinct, unchanging molecularspecies, and the spectrum will simply consist of twoindependent signals at the respective frequencies νA and νB
corresponding to the A and B states.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Transition from Slow to Fast Exchange
−20
0
20
x
k (s−1)νA+νB
2
νA
νB
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Intermediate Exchange: k = 100 Hz
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Slow-Intermediate Exchange
In the slow intermediate exchange regime some molecules willundergo a small number of conformational changes during thecourse of the experiment.As long as the condition k < | δν2 |, where δν = νA − νB, issatisfied, the two signal remain centered around νA and νB.However, the lines gradually broaden by an amount∆ν = k
π = 1πτ , until they finally coalesce into one very wide,
very weak signal.τ = 1
k is the average lifetime of each state.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Intermediate Exchange: k = 1000 Hz
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Fast-Intermediate Exchange
In the fast-intermediate exchange regime, where k > |δν/2|, themerged signal starts to get sharper again, and is centeredaround the average position of the two frequencies:
νpeak = νaverage =νA + νB
2
The line broadening contribution ∆ν in this regime ∆ν = π(δν)2
2k ,and therefore decreases as k increases.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Intermediate Exchange: k = 20000 Hz
Phase Differences during Slow Exchange
0
20
40
60
80
100
120
140
160
180
200
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Phase Differences during Intermediate Exchange
0
100
200
300
400
500
600
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Phase Differences during Fast Exchange
0
100
200
300
400
500
600
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Asymmetric Exchange
In the case of asymmetric exchange there is an energydifference between the A and B states, and the rate constantsin both directions (kA and kB) are no longer equal. Theequilibrium mixture will contain more of the lower-energyconformer.
If pA and pB = 1− pA are the fractional populations of the twoforms, at equilibrium the relation pAkA = pBkB holds.
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Transition from Slow to Fast Exchange (1)
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Transition from Slow to Fast Exchange (2)
In slow exchange, two "normal" peaks at frequencies νA and νB
are observed, with relative intensities that are proportional tothe populations of states A and B in the mixture.In the intermediate regime, a broadening of the two lines (with∆νA = kA
π and ∆νB = kBπ ) is initially observed, followed by a
merging into a single broad line, which subsequently startsbecoming sharper again (with a term ∆ν = 4πpApB(δν)2
kA+kB). The
combination line is no longer exactly at the average freqeuncy,but is shifted towards the frequency of the more abundntconformation:
νpeak = pAνA + pBνB
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Outline
1 Pulse/Fourier Transform NMRThermal EquilibriumEffect of RF PulsesThe Fourier Transform
2 Chemical ExchangeSymmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
3 Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Energy Profile
The effect of temperature on the reaction rate is expressed bythe Arrhenius equation:
k(T) = A exp(− Eact
NAkBT)
where NA is Avogadro’s number, kB is the Boltzmann constant,and Eact is the activation energy of the process.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Measuring Rate Constants
For N,N’-dimethylformamidethe rate constant could bedetermined experimentallyover a wide range oftemperatures. Fitting theArrhenius equation resultedin values ofEact = 90.1kJmol−1 andA = 1.16× 1014s−1.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Rotation of Tyrosine Side Chains
A tyrosine side chain can inpriciple rotate freely aroundthe single bond between Cαand Cβ. In the tightly packedhydrophobic core of a proteinthis motion can however belimited, in which case thesignals of symmetricallypositioned hydrogen atomscan be distinguished.
Pulse/Fourier Transform NMRChemical Exchange
Summary
Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications
Effect of Exchange on Scalar Coupling
Very pure ethanol
Ethanol with a catalytic amount of acid
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Summary (1)
A realistic NMR sample contains vast numbers ofindividual spins, each in its own quantum superpositionstate and precessing at the Larmor frequency around thedirection of the external field.The total magnetisation of all spins can be represented bya bulk magnetisaion vector, which obeys a few relativelysimple rules.At thermal equilibrium the bulk magnetisatiom pointstowards the positive z axis, and has a magnitudedetermined by the population difference between the twoenergy levels of the spins.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Summary (2)
An RF pulse with a frequency close to the resonancefrequency of the spins can rotate the bulk magnetisationover any desired angle around the x or y axis. The mostcommonly used flip angles are 90 and 180 degrees.Once displaced from equilibrium, the bulk magnetisationitself precesses in x,y plane at the Larmor frequency. Thisoscillation of the transverse magnetisation gives rise to anobservable signal in the detector coil.The initial amplitude of the signal is proportional to theequilibrium magnetisation, and thus to the populationdifference between the two energy states.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Summary (3)
A number of relaxation mechanism cause the bulkmagnetisation to slowly return to its equilibrium value alongthen z axis. As a result, the observed signl becomesprogressively weaker, ans becomes a free induction decay(FID).The Fourier transform of an FID signal is a Lorentziancurve around the resonance frequency, with a line widthdetermined by the rate of relaxation.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Summary (4)
When a nucleus can change between two states withdistinct Larmor frequencies (due to chemical reactions orconformational changes), the appearance of the spectrumis determined by the rate of the exchange process.In the very slow exchange regime, two separate signals atthe two distinct Larmor frequencies are observed.In the very fast exchange regime, a single signal at theaverage frequency is observed.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Pulse/Fourier Transform NMRChemical Exchange
Summary
Summary (5)
Going from very slow to very fast transitions the signalpasses through a transition region. At first, the twoseparate signals, each at its own frequency, become widerand wider, until they flow together and start getting sharperagain around their average frequency.In the intermediate regime, an analysis of the spectra canprovide the rate constant of the transition process. Outsideof this regime, the only conclusion that can be drawn iswhether the process is occuring too fast or too slow foranalysis by NMR methods.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy