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 ist 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 nuleus, 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*/(2pi) * sqrt(*I*(*I*+1))

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 planck's
constant. The angular momentum and the magnetic moment are
directly proportional:

*µ* = *gamma* * *J* = *gamma* * h/(2pi) * sqrt(*I*(*I*+1))

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:

*J _{z}* = -

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

*E* = - *µ*_{z} * *B _{0}*
= -

The magnetic moment of each nucleus precesses around *B _{0}*. The
frequency of this precession is the larmor frequency (

*gamma* * *h*/(2pi) * *B _{0}* =

=>

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.

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.

Andrew Derome suggested in his book
[5]
the following analogy for FT-NMR which
is really beautiful:

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 extremly 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 analyse it directly with
his or her ears (wich are a cleverly 'constructed' instrument
for FT). The advantages of this 'pulse FT method' over the 'cw
method' are clearly obvious.

*N _{p}* /

Both energy levels are nearly eqally 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 enery 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 *M _{0}* which can estimated according to
Curie's law:

*M _{0}* = N *

=>

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

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* = *gamma** [*M* × *B _{eff}*]

*B _{eff}* = (

--------------The time dependecy of the magnetization vector

=0

Transversal magnetization (dark blue)
can now be created by applying an additional magnetic field *B _{1}* (red)
which is perpendicular to

The state of x (or y) magnetization can be explained in the single spin
model: The two energy levels explained above are equally
populated (M_{z} = 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'.

*dM _{z}*/

*dM _{x,y}*/

T1 and T2 are called the longitudinal and transversal relaxation
times, respectively. The transversal components of magnetization
(*M _{x}*,

[Introduction] [Index] [PPS2 Projects] [1D NMR Spectroscopy]

Determination of Protein Structure with NMR Spectroscopy

last updated 281196