NMR InstrumentationNMR Instrumentation• An NMR machine is basically a big and expensive FM radio.

• Magnet - Normally superconducting. Some electromagnetsand permanent magnets (EM-360, EM-390) still around.

• Frequency generator - Creates the alternating current(at ωωωωo) that induces B1. Continuous wave or pulsed.

• Detector - Subtracts the base frequency (a constantfrequency very close to ωωωωo) to the output frequency. It islower frequency and much easier to deal with.

• Recorder - XY plotter, oscilloscope, computer, etc., etc.

N S

Bo

B1

Detector

FrequencyGenerator

Recorder

Magnet

Continuous Wave excitationContinuous Wave excitation

• It’s pretty de mode, and is only useful to obtain 1D spectra.

• The idea behind it is the same as in UV. We scan thefrequencies continuously (or sweep the magnetic field, which has the same effect - ωωωω = γγγγ B), and record successively howthe different components of Mo generate Mxy at differentfrequencies (or magnetic fields).

• We get a time domain effect in the frequency spectrum (thefamous ringing) because we cannot sweep slow enough.

time

ωωωωo or Bo

ωωωωo or Bo

Fourier Transform Fourier Transform -- Pulsed excitationPulsed excitation

• The way every NMR instrument works today.

• The idea behind it is pretty simple. We have two ways oftuning a piano. One involves going key by key on thekeyboard and recording each sound (or frequency). Theother, kind of brutal for the piano, is to hit it with a sledgehammer and record all sounds at once.

• We then need something that has all frequencies at once.A short pulse of radiofrequency has these characteristics.

• To explain it, we use another black box mathematical tool, theFourier transformation: It is a transformation of informationin the time domain to the frequency domain (and vice versa).

• If our data in the time domain is periodical, it basically givesus its frequency components. Extremely useful in NMR,where all the signals are periodical.

S(ωωωω) = ∫ s(t) e-iωωωωt dt

s(t) = 1/2 ππππ ∫ S(ωωωω) eiωωωωt dt-∞

∞-∞

∞

Fourier Transform of simple wavesFourier Transform of simple waves

• We can explain (or see) some properties of the FT withsimple mathematical functions:

• For cos( ωωωω * t )

• For sin( ωωωω * t )

• The cosines are said to give absorptive lines, while sines give dispersive lines. We’ll refer to these particularly whenspeaking about the phase of a signal or spectrum. Also important to remember this in order to understand detection.

−ωω

−ω ω

FT

FT

Back to pulsesBack to pulses

• Now that we ‘master’ the FT, we can see how pulses work.A radiofrequency pulse is a combination of a wave (cosine)of frequency ωωωωo and a step function:

• This is the time domain shape of the pulse. To see thefrequencies it really carry, we have to analyze it with FT:

• The result is a signal centered at ωωωωo which covers a widerange of frequencies in both directions. Depending on thepulse width we have wider (shorter tp) or narrower (longer tp)ranges. Remember that f ∝∝∝∝ 1 / t.

* =tp

FT

ωωωωo

Pulse widths and tip anglesPulse widths and tip angles

• The pulse width is not only associated with the frequencyrange (or sweep width), but it also indicates for how long theexcitation field B1 is on. Therefore, it is the time for whichwe will have a torque acting on the bulk magnetization Mo:

• As the pulse width for a certain flip angle will depend on the instrument (B1), we will therefore refer to them in terms of therotation we want to obtain of the magnetization. Thus, wewill have ππππ / 4, ππππ / 2, and ππππ pulses.

z

x

Mxyy

z

x

y

Mo

B1

θθθθttp

θθθθt = γγγγ * tp * B1

Some useful pulsesSome useful pulses

• The most commonly used pulse is the ππππ / 2, because it putsas much magnetization as possible in the <xy> plane (moresignal can be detected by the instrument):

• Also important is the ππππ pulse, which has the effect of invertingthe populations of the spin system...

• With control of the spectrometer we can basically obtain anypulse width we want and flip angle we want.

z

x

Mxyy

z

x

y

Mo ππππ / 2

z

x

-Moy

z

x

y

Mo ππππ

Free Induction Decay (FID)Free Induction Decay (FID)

• Now, we forgot about the sample a bit. We are interested inanalyzing the signal that appears in the receiver coil afterputting the bulk magnetization in the <xy> plane (ππππ / 2 pulse).

• We said earlier that the sample will go back to equilibrium (z)precessing. In the rotating frame, the frequency of thisprecession is ωωωω - ωωωωo. The relaxation of Mo in the <xy> planeis exponential (more next class). Therefore, the receiver coildetects a decaying cosinusoidal signal (single spin type):

ωωωω = ωωωωo

Mxy

ωωωω - ωωωωo > 0

time

Mxytime

FID (continued)FID (continued)

• In a real sample we have hundreds of spin systems, which allhave frequencies different to that of B1 (or carrier frequency).Since we used a pulse and effectively excited all frequenciesin our sample at once, we will se a combination of all of themin the receiver coil, called the Free Induction Decay (or FID):

• The FT of this signal gives us the NMR spectrum:

0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00t1 sec

Data acquisitionData acquisition

• That was kind of fast. There are certain things that we haveto take into account before and after we take an FID (or thespectrum, the FID is not that useful after all).

• Some concern to the detection system. Since a computerwill be acquiring the data, we can only take certain numberof samples from the signal (sampling rate). How many willdepend on the frequencies that we have in the FID.

• The Nyquist Theorem says that we have to sample at leasttwice as fast than the fastest (higher frequency) signal

• If we sample twice as fast as the frequency, where thedots are we, are in the clear.

• On the other hand, if we go too slow and sample at halfthe speed at we end up with a digitized signal in thecomputer at 1 / 2 of the real frequency. These peakswill fold over with the wrong phase in our spectrum.This is called aliasing.

SR = 1 / (2 * SW)

Quadrature Quadrature detectiondetection

• Usually the frequency of B1 (carrier) was somewhere were itwas higher than all other frequencies. This was done to avoidhaving frequencies faster (or slower) than the carrier, so thecomputer always new the sign of the frequencies in the FID.

• There are two problems. One, noise, which is always there, isnot sampled properly and its aliased into our spectrum. Also,in order to excite lines far from the carrier, we need very goodpulses, which is never the case.

• Considering this he best place to put the carrier is the centerof the frequency spectrum:

• There are several things we have to consider before doing it.

carrier

QuadratureQuadrature detection (continued)detection (continued)

• How can we tell which frequency is going faster or slowerrelative to the carrier? The trick is to put 2 receiver coils at 90degrees (with a phase shift of 90 degrees) of each other:

• The phase of the faster signals is opposite to that of theslower signals, and the computer is then able to sort this out.

ωωωω (B1)

F

F

S

S

PH = 0

PH =

90

PH = 0

PH = 90

F

F

S

S

SummarySummary

• Continuous wave excitation is like UV. Pulsed NMR can geta whole collection of signals in one shot.

• A short pulse of a single radiofrequency affects in practice arange of frequencies around the carrier frequency. Differentpulse widths have different effects on the bulk magnetization.

• The Fourier Transformation allows us to go back and forthfrom the time to the frequency domains, and is useful to findthe frequency components of periodic functions.

• The FID is the time dependent signal produced by the Mxymagnetization. We have to sample the FID sufficiently fastto avoid peak folding (Nyquist).

• Quadrature detection works by using two phase-shifted coilsto detect the relative speeds of signals.

Next ClassNext Class

• Digital filtering (apodization) and window functions.Zero filling.

• Relaxation phenomena - Bloch equations.

• Saturation, decoupling and population transfer.

Data processing Data processing -- Window functionsWindow functions

• Now we have the signal in the computer, properly sampled.There are some things we can do now a lot easier, and one ofthem is filtering. The real information in the FID is in the firstsection. As Mxy decays, we have more and more noise:

• The noise is generally high frequency, and that is why NMRspectra have this jagged baseline. What if we could filter allthe signals that were higher than a certain frequency?

• We use digital filtering. Intuitively, it means multiplying theFID by a function that makes the noise at the end smaller:

0 0.10 0.20 0.30 0.40 0.50t1 sec

Good stuff Mostly noise

1

Window functions (continued)Window functions (continued)

• In this case, it is called exponential multiplication, and hasthe form:

F(t) = 1 * e - ( LB * t ) or F(t) = 1 * e - ( t / τ τ τ τ )

• Why is that this removes high frequency noise? Actually, weare convoluting the frequency domain data with the FT of adecaying exponential. The FT of this function is a Lorentzianshaped peak with a width at half-height proportional to therate of decay, or line broadening (LB), in Hz.

• Convolution makes the contributionof everything with a WAHH thinnerthan LB smaller in the spectrum(the scale here is bogus…).

• If we use an LB with the opposite sign, the exponential growsinstead of decaying, letting signals with narrower widths topass, improving resolution but lowering signal to noise ratio.

LB

0 0.10 0.20 0.30 0.40 0.50t1 sec

0 0.10 0.20 0.30 0.40 0.50t1 sec0 0.10 0.20 0.30 0.40 0.50

t1 sec

Sensitivity and resolution enhancementSensitivity and resolution enhancement

• For the following raw FID, we can apply either a positive ornegative LB factor and see the effect after FT:

FT FT

LB = -1.0 HzLB = 5.0 Hz

Other useful window functionsOther useful window functions• Gaussian/Lorentzian: Improves resolution and does not

screw up sensitivity as bad as resolution enhancement alone.

• Hanning: Another resolution/sensitivity enhancement combo.

• Cosine - shifted cosine: Employed mostly for 2D spectra.

• The right window function depends on the experiment, and,as usual, there is a lot of fooling around involved...

F(t) = e - ( t * LB + σσσσ2 t2 / 2 )

F(t) = 0.5 + 0.5 * cos( ππππ t / tmax )

F(t) = cos( ππππ t / tmax )

Data size and ZeroData size and Zero--fillingfilling• Another important consideration is the size (in bytes) of our

data. Remember that it was related with the spectral width(sampling rate). It is also related to the time we will samplethe FID. Longer sampling times means more data.

• In the good old days, memory, and thus the size of the data,was awfully scarce. Most machines would only allow 16K(16384) points to be taken, which meant that if we wantedgood resolution, we could only sample for short periods.

• Even if we have plenty memory, more acquisition time limitsthe number of repetitions we can do in a certain period.

• We now define the digital resolution as the number of Hzper point in the FID for a given spectral width:

DR = SW / SI

• So, for a SW of 5 KHz and an FID of 16K, we have a digitalresolution of 0.305 Hz/point.

• One obvious problem from this is that if we have a large SWand a small SI, our resolution may not be able to pick someof the line separations in our spectra.

SW - spectral width (Hz)SI - data size (points)

ZeroZero--filling (continued)filling (continued)• Is there any way we can increase our digital resolution (I.e.,

the number of points) without having to acquire for longertimes? The trick is called zero-filling.

• What we do is increase the number of data points prior to theFT by adding zeroes at the end of the FID. We usually adda power of 2 number of zeroes.

• In this way, we increase the points per Hz ratio, and can inmany cases improve the spectrum. However, it does not helpif we have really crappy data from the start (we don’t getgood resolution if we did not sample enough…).

0 0.20 0.40 0.60 0.80 1.00 1.2 1.4 1.6 1.8 2.0 2.2t1 sec

8K data 8K zero-fill

8K FID 16K FID

Relaxation phenomenaRelaxation phenomena

• So far we haven’t said anything about the phenomena thatbrings the magnetization back to equilibrium. Relaxation iswhat takes care of this. There are two types of relaxation,and both are time-dependent exponential decay processes:

Longitudinal or Spin-Lattice relaxation (T1):

• It works for the components of magnetizationaligned with the z axis (Mz).

- Loss of energy in the system to thesurroundings (lattice) as heat.

- Dipolar coupling to other spins,interaction with paramagnetic particles, etc...

Transverse or Spin-Spin relaxation (T2):

• It acts on the components of magnetizationlying on the <xy> plane (Mxy).

- Spin-spin interactions dephase Mxy- Also by imperfections in the magnet

homogeneity (fanning out).- Cannot be bigger than T1.

• In order to understand relaxation from a phenomenological point of view, we have to introduce the Bloch equations,which describe the evolution of the spin system with timeunder the effects of magnetic fields as well as relaxation.

Mz

z

x

y

Mxy

z

x

y

(ωωωωeff = ωωωωo - ωωωω)

Bloch equationsBloch equations• We know that the magnetic field interacts with magnetization

(or the angular momentum) generating a torque that tips it.We usually deal with B1 in the <xy> plane and Mo in the zaxis. However, the Bloch equations are for any case, anddescribe variations of M with time:

dMx(t) / dt = γγγγ [ My(t) * Bz - Mz(t) * By ] - Mx(t) / T2

dMy(t) / dt = γ γ γ γ [ Mz(t) * Bx - Mx(t) * Bz ] - My(t) / T2

dMz(t) / dt = γ γ γ γ [ Mx(t) * By - My(t) * Bx ] - ( Mz(t) - Mo ) / T1

• The γγγγ appears because it’s L (average angular momentum)which generates the torque. Without trying to understandvery well were they come from, we can se that the variationof M in one axis depends on the other two.

• We’ll analyze the solution for the differential equations foran ideal case in which we have magnetization on the <xy>plane exclusively (after a π / 2 pulse and a certain ω…):

Mz(t) = Mo * cos( ωωωωefft ) * e - t / T2

My(t) = Mo * sin( ωωωωefft ) * e - t / T2

Mz(t) = Mo * ( 1 - e - t / T1 )

Bloch equationsBloch equations

• Graphically we get:

• From these equations we get that the best value for LB is 1 / T2...

Mx(t) = Mo * cos( ωωωωefft ) * e - t / T2

My(t) = Mo * sin( ωωωωefft ) * e - t / T2

Mz(t) = Mo * ( 1 - e - t / T1 )

Nuclear Nuclear Overhauser Overhauser Effect (NOE)Effect (NOE)

• The NOE is one of the ways in which the system (a certainspin) can release energy. Therefore, it is profoundly relatedto relaxation processes. In particular, the NOE is related toexchange of energy between two spins that are not scalarlycoupled (JIS = 0), but have dipolar coupling.

• The NOE is evidenced by enhancement of certain signals inthe spectrum when the equilibrium (or populations) of othernearby are altered. We use a two spin system energydiagram to explain it:

• W represents a transition probability, or the rate at whichcertain transition can take place. For the system in equilibriumwe can have W1I and W1S transitions, which represents singlequantum transitions.

• W0IS and W2IS are zero and double quantum transitions, areforbidden and have a much lower probability.

αα (∗∗∗∗)

(∗∗) αβ

ββ ()

W1S

W1S

W1I

W1I

βα (∗∗)W2IS

W0IS

Nuclear Nuclear Overhauser Overhauser Effect (continued)Effect (continued)• The W1I and W1S transitions, are related to spin-lattice or

longitudinal relaxation.

• Here we see that relaxation due to dipolar coupling takesplace when the spins give away energy by processes thatoccur at frequencies close to ωωωω = γγγγ * Bo, which include themovement (translation, rotation) and collision of spins.

• We now saturate the S transition, which means that wemake both its energy levels equal. The populations of the Stransitions are now the same:

• The W1S transitions are not possible (we have the samepopulations in these levels), and the W1I is not happening (wehave not affected the equilibrium for this spin). The W0IS andW2IS become the only way S can relax.

• These relaxation pathways for S also involve transitions of I,so thus the enhancement of this signal. W2IS will give positiveenhancement of I, and W0IS will give negative enhancements.

αα (∗∗∗)

(∗∗∗) αβ

ββ (∗)

W1S

W1S

W1I

W1I

βα (∗)W2IS

W0IS

Nuclear Nuclear Overhauser Overhauser Effect (even more…)Effect (even more…)

• We cannot detect W2IS or W0IS, but they affect the way thespin system relaxes. One has a rate close to twice ωωωω, whilethe other one is almost zero. So one will be related to veryslow motions, and the other one to fast tumbling...

• If we now put all this in a big equation (the Solomon equation)we get something that will help us see several things.

. We have:

• First, if the molecule tumbles rapidly (all small organic gunk)we have that under saturation of the I transitions W2IS willdominate, so the maximum enhancement for S is γγγγI / γγγγS. If weare looking at the 13C signal while decoupling (saturating)1H, we get an enhancement of ~ 4.

• If the molecule tumbles slowly, as a protein, W0IS dominates,and we have a maximum NOE of - γγγγI / γγγγS. Since here we areinterested in 1H - 1H NOE, the theoretical enhancement willbe ~ -1 (we wish…).

W2IS - W0IS

2 * W1S + W2IS + W0IS

ηηηη = γγγγI / γγγγS *