
Scattering of Xrays by a 1Dimensional Chain of Atoms
or Molecules 
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Scattering of Xrays by a 1Dimensional Chain of Atoms or
Molecules
Moving just slightly closer to reality, we next consider the consequences of
Xray scattering by a chain of atoms or molecules; the situation is idealised in
the following diagram (the experiment is again difficult to imagine ever being
performed, though perhaps not so impossible as the single electron experiment):
The key aspect of this case is that there is a repeating unit (be it an atom or
a molecule) and that the unit repeats regularly with a separation, a. It
turns out when we mathematically sum up the scattering contributions from all
these repeating atoms/molecules one has to take into account that only one of
these atoms/molecules can be placed at the space origin, the others being
displaced from the origin by a, 2a, 3a, 4a and so
on. This results in a scattering formula as follows:
F(S) = 
Σ n 
f_{n}e^{2π i
S.rn} 
= 
f  e^{2π i
S.r1} + f e^{2π i
S.(r1 + a )} +
f e^{2π i S.(r1 +
2a)} + ....

We can sum such a series and square the result to obtain an expression for the
intensity, I, observed on the photographic film (the expression is
squared because the intensity is always proportional to the square of
an amplitude):
I 
= f ^{2}
{sin N π a.S /
sin π a.S}^{2} 
The formula has two terms: f ^{2} which is the scattering
from the repeating atom/molecule and the part in brackets which represents the
interference (be it destructive or constructive) between the total number, N,
of atoms/molecules. We should first consider how this "interference function"
changes as we increase the number of atoms/molecules  this is shown below:
The characteristics of this interference function are very interesting. For
small N the pattern is somewhat "lumpy" but as N increases to very high values
the repeating peaks become sharper and the background becomes lower and
flatter; indeed this function begins to acquire some of the features we would
recognise in a diffraction pattern; in fact we only have to multiply this
function by f ^{2} to obtain the intensity (as in the first
equation) which is in effect a formula for the diffraction pattern, as
illustrated below:
Even though this is a somewhat primitive case, it holds some object lessons for
later:
 if the number of atom/molecule repeats (N) is small then the diffraction
pattern peaks are broad;
 the diffraction pattern becomes sharper as the number of atom/molecule
repeats (N) increases;
 the final diffraction pattern is a convolution of two effects: the
scattering factor of the repeating (atomic or molecular) unit and the
interference resulting from scattering by all the repeating units.
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