Page 302 - Electrical Properties of Materials

284 Magnetic materials

Since both the shape of the resonant curve and the exact value of the res-
onant frequency depend on the environment in which a nucleus finds itself,
nuclear magnetic resonance can be used as a tool to investigate the properties
of crystals. An important application in a different direction is the precision
measurement of magnetic fields. The proton resonance of water is generally
6
used for this purpose. The accuracy that can be achieved is about 1 part in 10 .
The most important application of Nuclear Magnetic Resonance is of course
in imaging, which was not even mentioned in the first edition of this book.
Maybe it should have been mentioned because the principle was already known
but its first, rather crude demonstration was still two years away. We shall atone
for our past omission by discussing Magnetic Resonance Imaging (MRI) in
more detail in Appendix VII devoted to Medical Imaging. The importance
of MRI was acknowledged by the award of the Nobel Prize in 2003 to Paul
Lauterbur and Peter Mansfield, who made it practical.

11.8.5 Cyclotron resonance
We have already discussed the phenomenon of cyclotron resonance from a
classical point of view, and we shall now consider it quantum mechanically.
For resonant absorption one needs at least two energy levels or, even better,
many energy levels equally spaced from each other. What are the energy levels
of an electron in a solid? Remember that in our earlier model we neglected
the interaction between electrons and simply assumed that the solid may be
regarded as an infinte potential well. The possible energy levels were then given
by eqn (6.2),

2 2 2 2 2 2 2 2
E = (k + k + k )= (n + n + n ),
y
z
z
x
x
y
2m 8 m(2a) 2
where n x , n y , n z are integers.
When a magnetic field is applied in the z-direction, then the above equation
∗ The effect of the magnetic field may modifies to ∗
be taken into account by replacing
2
2
p by (p – eA) in the Hamiltonian of 1 2
2
Schrödinger’s equation (where A is the E = λ + ω c + k , (11.50)
z
vector potential). 2 2m
where λ is an integer and ω c is the cyclotron frequency. For constant k z ,the
difference between the energy levels (called Landau levels)is ω c . Hence, we
may look upon cyclotron resonance as a process in which electrons are excited
by the incident electromagnetic wave from one energy level to the next.
11.9 The quantum Hall effect
Strictly speaking this does not belong to magnetic resonance (although Landau
levels are involved) and may be a little out of place in an engineering textbook.
The argument for including it is that there might be some relationship to high
temperature superconductivity (see Section 14.9) which is of great practical
significance, and it is also true that the effect would have never been discovered
had engineers not invented field-effect transistors, whose operation depended
on a two-dimensional electron gas (see Section 9.14).