**A Short Introduction to NMR**

The following is an excerpt from
a lecture by Dr. Jeffrey H. Simpson discussing the origins of NMR signals.
Some nuclei have spin. The most important
nuclide from a spectroscopic standpoint is Nuclear spin is a natural precipitation of the Heisenberg uncertainty principle. The principle prohibits the simultaneous specification of its position and momentum (hence energy) beyond certain minimum limits (i.e., the nucleus doesn't just sit there). As a result, when the nucleus spins, the non-symmetric charge distribution circulates, thereby creating a magnetic moment m that is either parallel (or anti parallel) to the angular momentum P of the nucleus. P = [I(I+1)] where I is the quantum number of nuclear spin
(I=1/2 for Quantum mechanics also dictates that the nuclear spin be quantized into discreet values or states. This results from the small size of the nucleus. The angular momentum of the nucleus P can be related to the nuclear magnetic moment m via a proportionality constant g. m = g P In most cases m and P are parallel, but in some cases they are antiparallel (if g is negative). A projection operator can be used to poll the value of m along a reference axis (typically the z axis). For a spin 1/2 nucleus, the projection operator will yield: m for the values of the quantum number m Nuclei with a large g are easier to observe, since P always comes in units of (h/2p). Some nuclei have I=0, meaning that they are
not NMR-active ( Normally, the energies of the various allowed
spin states are degenerate (the same). However, application of a
magnetic field (B DE = g(h/2p)B The Heisenberg uncertainty principle prevents
the nuclear magnetic moment m from being perfectly
aligned with the applied magnetic field (only a component can be either
parallel or antiparallel to B Because m has to
have a component in the xy plane, there is a torque exerted on it by B t = m
x B This torque causes the nuclear magnetic moment m to precess just as a spinning top precesses in the earth’s gravitational field. This precession frequency is the Larmor frequency and is the NMR frequency we refer to all the time. If we consider an ensemble of spins, we will have a random distribution of all the m vectors along two cones (a and b). Photons tuned to the energy gap between the a and b spin states will induce transitions between the two states. The frequency n of the photons is also the Larmor frequency. E = hn The energy of a photon tuned the a-b
energy gap DE is very small (on the order of
0.0002 kJ/mol.) h is Planck’s constant and is equal to 6.63 x 10 Because there is an energy difference between
the a and b spin
states, a Boltzmann distribution will be set up between the two states.
The number of spins in the higher energy (b)
state is denoted N N T is the temperature in degrees Kelvin; k
is the Boltzmann constant and is 1.38 x 10 N Rearranging and plugging in numbers for the
proton in a 500 MHz instrument (B
_{a}
– N_{b})/N_{total} = DN
/ N_{total }=
That means that only one spin out of every
~25,000 will be in the lower energy (a) state
compared to the b state when the Larmor frequency
is 500 MHz (B If we sum up all the a
and b spin vectors (do vector addition on all
of the m’s), then we find that all of the xy
components cancel out and most of the z-components of the a
cone are cancelled out by the z-components of the b
cone. What we are left with is the net magnetization vector or the
macroscopic magnetization M M Perturbation of M Three points about the applied rf are important to note: - The frequency must be well-tuned to the
Larmor frequency or no appreciable tipping will result,
- The phase of the rf is also important –
coherent tipping can only result if the torque applied by B
_{1}on M_{0}is well-controlled (i.e., if it has a specific phase), and
- the amplitude of the rf applied to the
sample will determine how much each pulse of the sinusoidal rf will
tip M
_{0}(a larger amplitude will require fewer beats, or a shorter overall rf pulse).
For a continuous wave (CW) instrument, the
field is usually varied and the frequency of the applied rf is kept constant.
As B On a Fourier transform (FT) NMR instrument,
M |