Theory of NMR Spectroscopy

Nuclear Spin and Quantization of Energy:

The phenomenon of magnetic resonance results from the interaction of the magnetic moment of an atomic nucleus (µ) with an external magnetic field. The cause of this magnetic moment is the quantum mechanical angular momentum (spin angular momentum) of all nuclei that are no nuclei (even number of protons and neutrons). In order to understand this, imagine the nucleus as a small charged particle which is spinning around its own axis thus representing an electric current. Due to this current the atomic nucleus behaves as a small electromagnet.

Of course, this picture is a classical model whis has nothing to do with reality. The quantum mechanical property 'spin' does not mean that the nucleus is spinning around its own axis (if it did its radial speed would be greater than the speed of light). It is therefore an unfortunate choice of words if we denominate the spin as the angular momentum of a nucleus, because spin is a pure quantum mechanical property which could as easily be called 'happiness' or 'peppermint flavor'.


The spin is quantized according to

J = h/(2π) × I(I+1)) ½

With J being the spin angular momentum, I the spin quantum number (which can have values of I=0,1/2,1,3/2,...,6. By convention it is simply called 'spin'.) and h the plank’s constant. The angular momentum and the magnetic moment are directly proportional:

µ = γ× J = γ × h/(2 π) ×(I(I+1)) ½

The constant gamma is characteristic for each isotope and is called the gyromagnetic ratio. The sensitivity of a nucleus in NMR depends on gamma (high gamma, high sensitivity).

In an external magnetic field the magnetic moment orients according to:

Jz = - m × h/(2 π)
=> µz = m × γ × h/(2 π)

The magnetic quantum number m can be an integer number between -I and +I. Thus, the external field leads to a splitting of the energy levels. For spin 1/2 nuclei (e.g. protons, see table) two energy levels exist according to a parallel or antiparallel orientation of the magnetic moment with respect to the magnetic field:

The energy of these levels is given by the classical formula for a magnetic dipole in a homogenous magnetic field of the strength B0:

E = - µz ×B0 = - m × γ × h/(2 π) × B0

The magnetic moment of each nucleus precesses around B0. The frequency of this precession is the larmor frequency (w0) which is equivalent to the resonance frequency of the nucleus and the energy difference between the two levels.

γ× h/(2 π) × B0 =Δ
                        = h × nu
                       = h/(2 π) × w0
            => w0 = γ  × B0

The larmor frequency depends on the gyromagnetic ratio and the strength of the magnetic field (see picture), i.e. it is different for each isotope. At a magnetic field of 18.7 T the larmor frequency of protons is 800 MHz.

Continuous wave spectroscopy:

Transitions between different energy levels occur if the frequency of radiation is equivalent to the energy difference between the two levels: In the old days of NMR, experiments were carried out by varying the frequency of radiation at constant magnetic field ('frequency sweep') and measuring the absorption of radiation by the different nuclei. Equivalently, the magnetic field strength could be varied at constant radiation frequency ('field sweep'). Until the nineteen-seventies all NMR spectrometers worked with this continuous wave technique.

Pulsed Fourier Transform NMR Spectroscopy:

A far better resolution and sensitivity in NMR was achieved by the introduction of pulsed Fourier transform techniques (FT-NMR). In FT-NMR the resonances are not measured one after another but all nucleis are excited at the same time by a radio frequency pulse: Normally a radio emitter works at fixed frequency nu0. However, if the radiation is emitted as a very short pulse (some µs in NMR) the pulse frequency becomes 'uncertain'. A short radio frequency pulse contains many frequencies in a broad band around nu0 and thus excites the resonances of all spins in a sample at the same time.

The excited spins emit the absorbed radiation after the pulse. The emitted signal is a superposition of all excited frequencies. Its evolution in time is recorded. The intensities of the several frequencies, which give the observed signal in their superposition, are calculated by a mathematical operation, the Fourier transformation, which translates the time data into the frequency domain. The resulting NMR spectrum looks like an ordinary cw spectrum but its resolution is several orders of magnitudes better.

The FT method can be compared to the tuning of a bell. In principle, you could measure each of the tones which make up the sound of a bell in a 'cw experiment': Excite the bell with all frequencies from the deepest tones to the edge of ultrasound and measure the reaction of the bell with a microphone. But this method is extremely complicated and every bell founder knows a much faster way: Take a little hammer (or perhaps a bigger one) and - BOIIINGGGG.....

The sound of the bell contains each tone at the same time an every person can analyze it directly with his or her ears (which are a cleverly 'constructed' instrument for FT). The advantages of this 'pulse FT method' over the 'cw method' are clearly obvious.

Populations and equilibrium magnetization:

A NMR sample contains many identical molecules (usually in a concentration range of mM for proteins). The spins of these molecules align indepently of each other parallel or antiparallel to the external field. The ratio of parallel spins to the antiparallel ones is given by the

Boltzmann distribution:

Np / Nap = exp (Δ E/(kT)) = exp (γ ×h/(2 π) × B0 / (kT))

Both energy levels are nearly equally populated, because the energy difference is in the order of magnitude of thermic movements (kT). At T=300 K and a magnetic field of 18.7 T (800 MHz) the excess in the lower energy level is only 6.4 of 10000 particles for protons. This is the main reason for the inherently low sensitivity of NMR when compared to optical spectroscopic methods.
The magnetic moments of the individual spins sum up to a macroscopic magnetization M0 which can estimated according to Curie's law:

M0 = N × γ 2 × (h/(2 π))2 ×B0 ×I(I+1) / (3kT)

=> M0 = N × γ 2 × (h/(2 π))2 ×B0 / (4kT), with I=1/2

It is the evoution of this macroscopic magnetization which is recorded in the spectrometer. The classical theory of NMR also deals with this quantity. In thermal equilibrium only magnetization along the axis of the magnetic field exists (by definition z), because the x and y components sum up to zero.

The Bloch equations:
For the mathematical description of NMR spectroscopy a rotating coordinate frame is used, the rotation frequency of which equals the larmor frequency of the nuclei. All nuclei rotating with the larmor frequency are fixed in this coordinate frame.

This concept should be very familiar to us, because we all live in a rotating coordinate frame - the earth. To an observer in a spaceship a person 'standing' on the equator is moving at a speed of about 1700 km/h. A ball which is thrown 'vertically' up in the air comes down again in a straight vertical line. However, our observer in space sees this ball moving on a complex parabola.

Mathematically, the time dependency of the macroscopic magnetization M is described by the Bloch equation:
dM/dt = γ × [M × Beff]
Beff = (B0 + (w0/ γ)) + B1 = B1
     --------------
=0

The time dependency of the magnetization vector M results from the interaction of the magnetization with the effective external magnetic field Beff. In the rotating frame the contribution of B0 to Beff is cancelled out (for nuclei with the larmor frequency w0), i. e. Beff equals zero as long as only the static external field B0 is applied.                                                       
                                                                                                                                                                           
Transversal magnetization  can now be created by applying an additional magnetic field B1  which is perpendicular to B0. This B1 field is the radiofrequency pulse mentioned above. If the radiation frequency is equal to the larmor frequency of the nuclei the field causes a rotation of the equilibrium magnetization Mz  around the x axis (cross product, Beff=B1). You can completely transform the z magnetization to y magnetization if the duration of the pulse is sufficiently long. In this case the pulse is called an 'excitation pulse' or '90° pulse' (obviously because the rotation angle of the magnetization is 90°).

 The state of x (or y) magnetization can be explained in the single spin model: The two energy levels explained above are equally populated (Mz = 0). Additionally, the magnetization dipoles of the spins are not statistically distributed around the z axis. A small part of them precesses 'focussedly' in phase around the z axis. They sum up to the macroscopic x magnetization. Therefore, states with transversal magnetization are also called 'phase coherence'.

Relaxation:
The Bloch equation is not complete because it predicts an infinite precession of transversal magnetization. In reality, transversal magnetization is a non equilibrium state and the system returns to thermal equilibrium within short time. Therefore, Bloch introduced two empirical relaxation times (T1, T2) in his equation. He assumed that these relaxation processes are of first order:

dMz/dt = γ × [M × Beff]z + ((M0 - Mz)/T1)

dMx, y/dt = γ × [M × Beff]x, y - (Mx,y/T2)

T1 and T2 are called the longitudinal and transversal relaxation times, respectively. The transversal components of magnetization (Mx, My) approach zero whereas the longitudinal component Mz approaches M0 with time. Relaxation is caused by several time dependent interactions between different spins (T2) and between spins and the surrounding lattice (T1). Therefore, T1 is also called spin-lattice relaxation time and T2 spin-spin relaxation time.