Transcript No Slide Title

Summary of last class...
• Excitation is accomplished by oscillating magnetic fields (in
practice, currents in coils…)
• We saw how magnetization (and what happens to it) can be
described in either laboratory or rotating frames. Due to its
simplicity we always use the rotating frame, which means we
basically spin together with the whole system.
• The origins of chemical shift (local differences in magnetic
field) and of spin coupling (doubling of the energy levels)
were briefly discussed.
• We saw how magnetization evolves with time under the effect
of chemical shift differences and spin coupling.
For chemical
shifts...
For coupling
constants...
f = (w - wo) * t
f=p*t*J
• We’ll see today how we actually make it all work, or, in other
words, how an NMR instrument works...
NMR Instrumentation
• An NMR machine is basically a big and expensive FM radio.
Bo
N
S
Magnet
B1
Recorder
Frequency
Generator
Detector
• Magnet - Normally superconducting. Some electromagnets
and permanent magnets (EM-360, EM-390) still around.
• Frequency generator - Creates the alternating current
(at wo) that induces B1. Continuous wave or pulsed.
• Detector - Subtracts the base frequency (a constant
frequency very close to wo) to the output frequency. It is
lower frequency and much easier to deal with.
• Recorder - XY plotter, oscilloscope, computer, etc., etc.
Continuous 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 the
frequencies continuously (or sweep the magnetic field, which
has the same effect - w = g B), and record successively how
the different components of Mo generate Mxy at different
frequencies (or magnetic fields).
wo or Bo
wo or Bo
time
• We get a time domain effect in the frequency spectrum (the
famous ringing) because we cannot sweep slow enough.
Fourier Transform - Pulsed excitation
• The way every NMR instrument works today.
• The idea behind it is pretty simple. We have two ways of
tuning a piano. One involves going key by key on the
keyboard and recording each sound (or frequency). The
other, kind of brutal for the piano, is to hit it with a sledge
hammer 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, the
Fourier transformation: It is a transformation of information
in the time domain to the frequency domain (and vice versa).

∫-s(t) e dt

s(t) = 1/2 p ∫ S(w) e dt
-
S(w) =
-iwt
iwt
• If our data in the time domain is periodical, it basically gives
us its frequency components. Extremely useful in NMR,
where all the signals are periodical.
Fourier Transform of simple waves
• We can explain (or see) some properties of the FT with
simple mathematical functions:
• For cos( w * t )
FT
-w
w
• For sin( w * t )
FT
-w
w
• The cosines are said to give absorptive lines, while sines
give dispersive lines. We’ll refer to these particularly when
speaking about the phase of a signal or spectrum. Also
important to remember this in order to understand detection.
Back 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 wo and a step function:
*
=
tp
• This is the time domain shape of the pulse. To see the
frequencies it really carry, we have to analyze it with FT:
FT
wo
• The result is a signal centered at wo which covers a wide
range of frequencies in both directions. Depending on the
pulse width we have wider (shorter tp) or narrower (longer tp)
ranges. Remember that f  1 / t.
Pulse widths and tip angles
• The pulse width is not only associated with the frequency
range (or sweep width), but it also indicates for how long the
excitation field B1 is on. Therefore, it is the time for which
we will have a torque acting on the bulk magnetization Mo:
z
Mo
z
x
qt
tp
x
B1
Mxy
y
y
qt = g * tp * B1
• As the pulse width for a certain flip angle will depend on the
instrument (B1), we will therefore refer to them in terms of the
rotation we want to obtain of the magnetization. Thus, we
will have p / 4, p / 2, and p pulses.
Some useful pulses
• The most commonly used pulse is the p / 2, because it puts
as much magnetization as possible in the <xy> plane (more
signal can be detected by the instrument):
z
Mo
z
x
p/2
x
Mxy
y
y
• Also important is the p pulse, which has the effect of inverting
the populations of the spin system...
z
Mo
y
z
x
p
x
y
-Mo
• With control of the spectrometer we can basically obtain any
pulse width we want and flip angle we want.
Free Induction Decay (FID)
• Now, we forgot about the sample a bit. We are interested in
analyzing the signal that appears in the receiver coil after
putting the bulk magnetization in the <xy> plane (p / 2 pulse).
• We said earlier that the sample will go back to equilibrium (z)
precessing. In the rotating frame, the frequency of this
precession is w - wo. The relaxation of Mo in the <xy> plane
is exponential (more next class). Therefore, the receiver coil
detects a decaying cosinusoidal signal (single spin type):
Mxy
w = wo
w - wo > 0
Mxy
time
time
FID (continued)
• In a real sample we have hundreds of spin systems, which all
have frequencies different to that of B1 (or carrier frequency).
Since we used a pulse and effectively excited all frequencies
in our sample at once, we will se a combination of all of them
in the receiver coil, called the Free Induction Decay (or FID):
• The FT of this signal gives us the NMR spectrum:
Data acquisition
• That was kind of fast. There are certain things that we have
to take into account before and after we take an FID (or the
spectrum, the FID is not that useful after all).
• Some concern to the detection system. Since a computer
will be acquiring the data, we can only take certain number
of samples from the signal (sampling rate). How many will
depend on the frequencies that we have in the FID.
• The Nyquist Theorem says that we have to sample at least
twice as fast than the fastest (higher frequency) signal
SR = 1 / (2 * SW)
• If we sample twice as fast as the frequency, where the
dots are we, are in the clear.
• On the other hand, if we go too slow and sample at half
the speed at we end up with a digitized signal in the
computer at 1 / 2 of the real frequency. These peaks
will fold over with the wrong phase in our spectrum.
This is called aliasing.
Quadrature detection
• Usually the frequency of B1 (carrier) was somewhere were it
was higher than all other frequencies. This was done to avoid
having frequencies faster (or slower) than the carrier, so the
computer always new the sign of the frequencies in the FID.
• There are two problems. One, noise, which is always there, is
not sampled properly and its aliased into our spectrum. Also,
in order to excite lines far from the carrier, we need very good
pulses, which is never the case.
• Considering this he best place to put the carrier is the center
of the frequency spectrum:
carrier
• There are several things we have to consider before doing it.
Quadrature detection (continued)
• How can we tell which frequency is going faster or slower
relative to the carrier? The trick is to put 2 receiver coils at 90
degrees (with a phase shift of 90 degrees) of each other:
PH = 0
B
F
PH = 90
B
w (B1)
F
PH = 0
F
S
F
S
PH = 90
• The phase of the faster signals is opposite to that of the
slower signals, and the computer is then able to sort this out.
Summary
• Continuous wave excitation is like UV. Pulsed NMR can get
a whole collection of signals in one shot.
• A short pulse of a single radiofrequency affects in practice a
range of frequencies around the carrier frequency. Different
pulse widths have different effects on the bulk magnetization.
• The Fourier Transformation allows us to go back and forth
from the time to the frequency domains, and is useful to find
the frequency components of periodic functions.
• The FID is the time dependent signal produced by the Mxy
magnetization. We have to sample the FID sufficiently fast
to avoid peak folding (Nyquist).
• Quadrature detection works by using two phase-shifted coils
to detect the relative speeds of signals.
Next Class
• Digital filtering (apodization) and window functions.
Zero filling.
• Relaxation phenomena - Bloch equations.
• Saturation, decoupling and population transfer.