Summary of last class...

1

• 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...
f (w - wo) t
For coupling constants...
f p t J
2

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

Bo
N
S
Magnet
B1
Recorder
Frequency Generator
Detector
3

• Continuous Wave excitation
• Its 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
4

• 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(w) ? s(t) e-iwt dt s(t) 1/2 p ? S(w) eiwt dt
-?
?
-?
5

• Fourier Transform of simple waves
• We can explain (or see) some properties of the
  FT with
• simple mathematical functions
• For cos( w t )
• For sin( w t )

FT
-w
w
FT
-w
w
6

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

tp
FT
wo
7

• 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
z
qt
Mo
tp
x
x
B1
Mxy
y
y
qt g tp B1
8

• Some useful pulses
• The most commonly used pulse is the p / 2,
  because it puts
• as much magnetization as possible in the ltxygt
  plane (more
• signal can be detected by the instrument)
• Also important is the p pulse, which has the
  effect of inverting
• the populations of the spin system...

z
z
Mo
p / 2
x
x
Mxy
y
y
z
z
Mo
p
x
x
-Mo
y
y
9

• 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 ltxygt
  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 ltxygt plane
• is exponential (more next class). Therefore,
  the receiver coil
• detects a decaying cosinusoidal signal (single
  spin type)

Mxy
w wo
time
Mxy
w - wo gt 0
time
10

• 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)

11

• 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)
12

• 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

carrier
13

• 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
B
w (B1)
F
PH 90
PH 0
F
S
PH 90
F
S
14

• 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.