Biophysical Chemistry: NMR Spectroscopymore.vub.ac.be/LievenButs/slides/NMR_en_4.pdfLieven Buts Biophysical Chemistry: NMR Spectroscopy The Effect of Resonance The offset frequency

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


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



Summary

Thermal EquilibriumEffect of RF PulsesThe Fourier Transform

Outline



3 Summary


Quantum Description of a Spin-1/2


Summary


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.



Summary


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.



Summary


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.



Summary


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:


=∆E2kT

= 3.2× 10−5

In other words, about one low-energy spin out of every 105 hasno counterpart in the high-energy orientation.



Summary


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.



Summary


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.



Summary


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


Summary



In the z direction there is a small preference for the low energystate, as reflected by the slightly larger population nα:


=∆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



Summary


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.



Summary


Outline



3 Summary


Design of a Modern NMR Spectrometer (1)



Summary



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.)



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:


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)


Summary


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



Summary


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.


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.


Summary


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.



Summary


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.



Summary


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.



Summary


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



Summary


Outline



3 Summary



Summary


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


Fourier Analysis


Summary


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.


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

.


Summary

Symmetric Exchange Between Two SitesAsymmetric Two-Site ExchangeApplications

Outline



3 Summary



Summary


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.



Summary


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.



Summary


Transition from Slow to Fast Exchange

−20

0

20

x

k (s−1)νA+νB

2

νA

νB


Intermediate Exchange: k = 100 Hz


Summary


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.




Summary


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.



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


Summary


Outline



3 Summary


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.


Summary


Transition from Slow to Fast Exchange (1)



Summary


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



Summary


Outline



3 Summary



Summary


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.



Summary


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.


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.


Summary


Effect of Exchange on Scalar Coupling

Very pure ethanol

Ethanol with a catalytic amount of acid



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.



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.



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.



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.



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.


Biophysical Chemistry: NMR Spectroscopymore.vub.ac.be/LievenButs/slides/NMR_en_4.pdfLieven Buts Biophysical Chemistry: NMR Spectroscopy The Effect of Resonance The offset frequency

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