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Scattering of X-rays by a Single Electron |
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Scattering of X-rays by a Single Electron
It is useful to consider the scattering of X-rays by an electron to
be a fundamental scattering process (later we will also consider the scattering
of neutrons). The unit of X-ray energy is termed an X-ray photon and can
be imagined as a finite travelling wave of electromagnetic displacement.
Many of the diagrams we will use will be highly schematic (and should not
always be taken too literally); the first diagram falls into this class,
illustrating the basic features of the electric displacement of an X-ray
photon:
The terms used are important: amplitude (the maximum displacement), wavelength
denoted by l (the distance between successive
maxima), direction of propagation of the X-ray photon (horizontal from
left to right as it appears on your screen), and polarisation (the direction
of electric displacement) which appears vertical on your screen.
An X-ray can be scattered by an electron. This can be perceived by the
change in direction of the X-ray after it has "passed" the electron (so
to speak), as illustrated in the following schematic diagram:
Again this diagram is somewhat naive (and perhaps questionable on true
quantum mechanical grounds) but it gets the essential ideas over. It is
not so important to understand exactly what goes on as the X-ray interferes
with and is scattered by the electron, but the net
end result is important: both the direction and polarisation of the X-ray
photon have been altered by an angle 2q.
There is nothing to worry about by the use of 2q
as the angle of scattering (rather than, say,
q);
this is simply a matter of convention at this stage. We might also point
out that the X-ray photon also suffers a phase shift of 180 degrees (p
radians), but this is not important either at this stage.
This whole process is often summarised by an important vector diagram
and accompanying vector equation, though you do not really need to understand
much about vector algebra to follow these: since a vector is something
which has both magnitude and direction we can represent two items, the
absolute momentum of the X-ray photon and its direction of propagation,
by a vector. One other fact you should accept is that the momentum of a
wave is proportional to the inverse of its wavelength (a result discovered
earlier in the 20th century by Planck); so we make the length of the vector
equal to 1/l and the direction of the vector
the same as the propagation direction of the X-ray. Then we represent the
scattering event by a vector diagram, which follows, in which so
represents the direction and momentum of the incident X-ray photon (i.e.
before it is scattered) and s represents the direction and momentum
of the X-ray photon after being scattered. An important aspect of diffraction
is that we only consider elastic scattering; by that we mean that there
is no loss of X-ray photon energy through scattering so that the energy
(and absolute momentum) of the X-ray photon remains unchanged. This is
why the length of
s in the diagram (1/l)
has been made the same as the length of so.
Finally the vector
S, which bridges s and so,
represents the net change as a result of the scattering process and is
an important item, often referred to as the scattering vector S;
its length is equal to
2 sinq / l
(you might like to try proving this from the properties of the isosceles
triangle in the diagram).
The vector equation that summarises the above figure is:
s = so + S
which basically re-iterates that the scattered wave (s) is the result
of the incident wave (so) being subjected
to the scattering process (+ S).
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