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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 1H, followed by 13C, 31P, 19F, 2H, 15N, 29Si.  All of these nuclei are spin 1/2 nuclei except deuterium (2H) which is spin 1.  For a complete listing of NMR active nuclei, see our NMR Frequency Table.

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)]1/2 (h/2p)

where I is the quantum number of nuclear spin (I=1/2 for 1H)

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:

mz = +1/2m, -1/2m

for the values of the quantum number mz of +1/2 and –1/2, respectively. When the z-component of m is along the +z axis, we say it is in the a (or parallel or spin-up) state, and when it is along the –z axis, we say it is in the b (or antiparallel or spin-down) state.

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 (12C, 16O).

Normally, the energies of the various allowed spin states are degenerate (the same).  However, application of a magnetic field (B0) splits the energies of the spin states.  This is the Zeeman effect. The energy difference of the two spin states is

DE = g(h/2p)B0

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 B0.

Because m has to have a component in the xy plane, there is a torque exerted on it by B0:

t = m x B0

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 10-34 J s.

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 Nb, and likewise the number of spins in the a state is Na.

Nb/Na = exp(-DE/kT)

T is the temperature in degrees Kelvin; k is the Boltzmann constant and is 1.38 x 10–23 J/K.  Since DE is small relative to kT, we can convert the exponential expression into the first two terms of a power series and get:

Nb/Na = 1 - DE/kT

Rearranging and plugging in numbers for the proton in a 500 MHz instrument (B0 = 11.7 Tesla), we have DE=g(h/2p)B0 so 

(Na – Nb)/Ntotal = DN / Ntotal =
{(2.67 x 108 rad T-1 s-1)(6.63 x 10-34 J s / 2p) (11.7 T)}
2{(1.38 x 10-23 J/K)(300 K)}
~ 1 / 25000

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 (B0 = 11.7T) and the temperature is 300 K.

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 M0. At equilibrium M0 points exactly along the z axis (parallel to the applied field).

M0 can be treated classically.  It can point in any direction, it precesses at the Larmor frequency when it is tipped so that it has a component in the xy plane, and its magnitude can change.

Perturbation of M0 is done with electromagnetic radiation in the radio frequency (rf) range.  The magnetic field component of the applied rf (B1) pushes on M0 to make it tip a little away from its equilibrium position.  If the frequency of the B1 field is timed with the precession frequency of M0, then coherent tipping can take place.  If the frequency of the applied rf is not well-tuned to the Larmor frequency, then no net tipping will occur.

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 B1 on M0 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 M0 (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 B0 changes, different spins come into resonance and absorb rf.  When they reemit the energy they have absorbed, a receiver detects this emission and that is how we detect the NMR signal on a CW instrument.

On a Fourier transform (FT) NMR instrument, M0 is tipped into the xy plane where it then freely precesses and induces a signal in the receiver coil as it decays back to equilibrium. This is where the term free induction decay (FID) comes from.