If we look carefully at this
equation, it
is easy to realize that:
- No fractional
cooperative dispersion orders can occur, since m
is an integer number, and
- If the incident
angle
μ
is maintained constant, the last equation will also be valid for any
scattered beam that maintains the same angle of scattering
(ν) shown
in the figure, and therefore all scattered beams
around the row of atoms (with the same ν)
will fulfill that equation and
will form a conical surface coaxial with the row of atoms (see figure
below)...
Scattering
from a row of atoms separated at regular distances
Cooperative
scattering (diffraction) from two rows of atoms regularly spaced
In the
two-dimensional case, ie if
we consider two rows of atoms, such as those shown with the letters a
and b
(figure above), each one will scatter the X-rays in the same
manner as described above, that is in the form of cones coaxial with
those atoms rows. But if both dispersions have to be cooperative
(diffraction) they will have to perform simultaneously two equations
equivalent to the one shown above, ie:
a (cos ν1 - cos μ1) =
m λ
b (cos ν2 - cos μ2) = n λ
The fact that these two
equations are
fulfilled simultaneously is equivalent as to consider graphically as
valid only the common points of both cones, ie their intersections,
which are two straight lines (shown as arrows). In other words, cooperative
scattering
(diffraction) from two non-parallel rows of atoms (and in
general from a
plane of atoms) is reduced to discrete diffraction lines only.
Generalizing
to any periodically ordered three-dimensional distribution of atoms, we
must
consider that three equations of the type shown above must be
fulfilled simultaneously:
a (cos
ν1 - cos μ1)
=
m λ
b (cos
ν2 - cos μ2)
= n λ
c (cos
ν3 - cos μ3)
= p λ
where m,
n, p
represent three integer numbers. These are the so called Laue equations.
The Max
von Laue
(1879-1960) equations
relate the incoming waves to the
outgoing waves in the
process of diffraction by a crystal lattice. They can be derived in a
somewhat less formal mode than made by the author,
but intuitively and in a simple way...