Lecture 2. Notes Chem 40900 Dr Matthew Crump Feb 26th 2003-02-24
In lecture 1. we recapped some basic principles of NMR and the phenomena of chemical
shift and spin-spin coupling that make this technique so useful to the chemist or protein
spectroscopist.
In order to determine protein structure by NMR, a pulsed Fourier transform
spectrometer is used. Older CW (constant wave) instruments are easier to understand.
They measure a radio frequency absorption spectrum as the frequency is scanned with a
constant field strength from the magnet. These spectra are slow to record and are usually
repeated a few times before averaging to gain a good signal to noise ratio. They do
reveal chemical shift and coupling information.
Fourier Transform NMR. Following notes taken from ‘Spin Dynamics’ by Malcolm H.
Levitt
Before the field is switched on…..
Consider some proton nuclei in a sample of water. In the absence of a magnetic field, the
spin polarisations are isotropic, that is, uniformly distributed and pointing in all possible
directions in space. The total magnetic moment of the sample is very close to zero since
the total number of spins point towards a given direction as against it.
Build up of longitudinal magnetization when sample is put into the magnet.
If a magnetic field is suddenly switched on, all protons will begin to precess at their
Larmor frequency around the field. This precessional motion is essentially invisible. It
does nothing to change the total magnetic moment of the sample. The isotropic
distribution of spin polarisations makes no contribution to the magnetism of the material.
However, the proton spins are not alone. The water molecules which carry the protons
undergo vigorous motion. The orientation of each molecule in space changes constantly
but how does this effect the nuclear spins? Remarkably, not much at all, and to a good
approximation, each continues to precess about the field (on some cone) and is oblivious
to the molecules rotation. This is like a ship’s gyroscope in a rough sea – it keeps calm
although all around it is moving.
If this was truly the case however, we would never get any change and no
longitudinal magnetisation building up after we put the sample in the magnet. On a closer
look one finds that the violent molecular surroundings do slightly influence the nuclear
magnets. Each molecule is full of magnetic particles: the electrons and nuclei are all
sources of magnetic fields. These fields are small, and fluctuate rapidly because of the
thermal motion of the environment. At any given moment, the spin precesses around the
large external field and the small microscopic field, which varies in time and can have
any direction. Therefore the field that spin feels has a very slightly varied magnitude and
varied direction. In a 12 T magnet these fluctuations may sum up to only around a 10-4
degree variation of field direction but over time, these important small fluctuations allow
the isotropy of the nuclear spin polarisation to be broken and without it a macroscopic
nuclear magnetisms would not develop and ultimately, nuclear magnetism would not be
observable.
The fluctuating field means that over a long time the magnetic moment of each
spin wanders around and moves between different precession cones, perhaps eventually
sampling all different angles with the field (This may be ringing warning bells as I said
we could only be aligned at a certain angle with or against the field… this is an oversimplification of what quantum mechanics actually tells us. Although there are two states
predicted by quantum theory, we can have Eigenstates that are a superposition of these
two states – i.e. transverse magnetisation is an example of a mixture of  and  states).
The wandering motion is not completely isotropic and it is more probable that the spin is
driven towards and orientation with low magnetic energy than toward an orientation with
high magnetic energy. The thermal wandering is slightly biased to spin orientations with
magnetic moments parallel to the field.
Mz (t) = Mzeq [1- exp{- (t-ton) / T1}]
T1 is the longitudinal relaxation time constant. Longitudinal indicates that the
magnetisation builds up in the same direction as the applied magnetic field.
The term relaxation is widely used in NMR. In the absence of a field thermal equilibrium
is established where all nuclear spins are equally likely. Turn the field on and the system
‘relaxes’ to a new equilibrium, in which the spin polarisations are distributed
anisotropically.
If the field is suddenly turned off then the magnetization relaxes back to zero again.
The potential energy E of the precessing charged particle shown below is given by
E  z Bo  Bo cos 
where  is the angle between the magnetic field vector and the spin axis of the particle, 
is the magnetic moment of the particle and z is the component of  in the direction of
the magnetic field. Thus when radio-frequency energy is absorbed by the nucleus, its

angle of precession must change. Hence we imagine for a nucleus having a spin of 1/2
that absorption involves a flipping of the magnetic moment so that it now opposes the
field. This process is pictured below.
In order for the magnetic dipole to flip there must be a magnetic force at right angles to
the fixed field that moves in a circular path in phase with the precessing dipole. The
magnetic moment of circularly polarised radiation of a suitable frequency has these
necessary properties; that is, the magnetic vector of such radiation has a circular
component. If the frequencies are on resonance, flipping can occur.
But a coil in an NMR machine provides a plane polarised radiation as follows.
A plane polarised wave can be decomposed into two counter-rotating vectors oscillating
with frequency o and -o.
Now to simplify this picture we imagine that the laboratory rotates about the Z axis at the
Larmor frequency o. This is called the rotating frame. In this frame we fix one of the
counter rotating vectors (say along the x-axis) and let the other vector rotate at –2o.
Only the vector that lies along the x-axis (i.e. the one that oscillates in the same sense as
the sample magnetisation) interacts with the sample magnetisation and is absorbed.
Fourier Transform NMR.
In 1970, Richard Ernst found that we could obtain the same information by subjecting the
sample to a short high power pulse of radiofrequency. The build up and decay of this
‘hard’ pulse shape leads to the sample being irradiated over a wide range of frequencies,
exciting all the receptive nuclei at the same time. When the pulse is finished, the
radiation emitted is recorded as a function of time. This is called the Free Induction
Decay or FID. It is a complex interferogram of many different decay curves, generated
by the relaxation of excited nuclei with different chemical environments, all superposed.