The
resolution of a crystal structure, through any of the methods
described
above,
requires previous knowledge regarding:
- Its approximate chemical
composition
- The unit
cell dimensions, to be
deduced from the diffraction pattern (the reciprocal unit
cell),
- The crystal
symmetry, to be deduced
from the diffraction symmetry (object of this subchapter), and
- The diffracted
intensities
It is
not the aim of these pages to deal with the composition of
crystals, and information on points 2, 3 and 4 has already
been
given elsewhere. However, from a practical point of view, the reader
will realize that
although there is a chapter dedicated to the internal symmetry of crystals, nothing
has been said about how
to determine it in
a specific crystal. On the other hand, it also probably seems
obvious that the visual inspection of
a crystal does not
answer many questions about the internal symmetry operations that describe how molecules
(or in general, atoms) pack together to build crystals.
The reader can also imagine that the only reliable information we have
about
symmetry is the one provided by the diffraction pattern, that
is,
the
symmetry of the reciprocal lattice. Therefore, it should now be clear
that the problem is to derive the crystal symmetry from the reciprocal
lattice symmetry, in the hope that the
crystal symmetry will be shown somehow in
the diffraction pattern.
FRIEDEL'S LAW
However, there can be some confusion regarding one of the possible
symmetry operators in crystals: the
centre of symmetry.
The existence of the centre of symmetry in the crystal cannot be
inferred from the existence of the same operation in the diffraction
pattern, because the reciprocal space is (always?)
centro-symmetric
as was deduced by Georges
Friedel (1865-1933), which gave us Friedel's
Law.
Left:
A graphic representation of Friedel's Law
in colours that represent the intensities associated with the
reciprocal points around the origin. Equal colours mean identical
intensities.
Right: Friedel's
Law
shown in terms of Bragg's model: reflections produced by
opposite
surfaces of a "mirror" have equal intensities.
Friedel's
Law
states that the
intensities associated with two reciprocal points, given by the indices
(h,
k, l) and (-h, -k, -l),
are almost equal. That is:
I
(h, k, l) ≈ I (-h, -k, -l)
This is
equivalent to saying that
the reciprocal lattice is always centro-symmetric, and thus the
"apparent" symmetry of the crystals will be one of the 11 Laue groups.
A pair of
reflections such as (h, k, l)
and (-h,
-k, -l)
are
known as a Friedel
Pair. In the
presence of anomalous
scattering,
these two reflections will
show a small difference in intensity, which is very useful to determine
the absolute
configuration
of the molecules and in solving the phase problem through the MAD methodology.
As a consequence of Friedel's Law we can state that the
crystal can contain lower symmetry than the one displayed by the
diffraction pattern. In other words, there will be some
type of uncertainty when trying to determine the crystal
symmetry from the
diffraction experiment.
Fortunately, however, some
crystal symmetry operations show
their "footprints" in the reciprocal space. For example, some symmetry
elements, or in certain
lattice types, the arrangement and spacing of lattice planes produces
diffractions from certain classes of planes in the structure which are
always exactly 180° out of phase producing a phenomenon called
systematic
extinction. In these cases, certain types of reflections from valid lattice planes (recognizable by simple rules
in their hkl
indices) will produce
no visible diffraction spots.
Systematic
absences (or systematic extintions) in hkl
reflections arise when symmetry elements containing translational
components are present, such as in the following cases:
- lattice centering
(translational operations derived from the lattice
type),
- screw axes (symmetry axes
that imply rotation and an additional translation), and
- glide
planes (mirror planes that imply reflection
and an additional translation).
All possible systematic absences in the diffraction patterns, produced
by this type of symmetry elements,
can be found in the table shown
through this link.
The examples below demonstrate how symmetry operations of this
type produce zero intensity associated with structure factors
whose indices are related by simple rules.
CENTERED LATTICES
Consider a crystal lattice (shown in 2-dimensions), such as the one
shown in the
figure below (left), with axes a
and b.
This lattice is called primitive because it contains one lattice node
inside the unit cell (actually in terms of 4 quarters of a point).
If for some reason
we have interpreted
this lattice in terms of another unit cell, with axes A
and B (figure
on the right), the lattice will become what we call a centered lattice
(non primitive) because it contains more than one lattice point inside
the unit cell (in this case 2 points: 1/4 in each corner + 1
in
the center).
The transformation of axes inherent to this change of unit cell is
given at the upper right part of the figure. The new A axis is a vector obtained by adding two times
the old a
axis and the old b
axis (2a
+ b),
and the new B axis is identical to b.
This transformation (from one cell to the other) can be represented by
the matrix shown above
(obtained with the coefficients of the terms of the second member of
the equations).
It can be demonstrated that this cell-to-cell transformation matrix can
also be applied to the hkl
indices of the original lattice to obtain the new HKL
indices that, according to the new unit cell, interpret the lattice.
If we do this in this example (as is shown in the lower right
corner of the figure) we will obtain the equations which relate old and
new Miller indices. After adding both equations we will
discover that the
new Miller indices (HK)
are such that its sum (H+K)
is always an even number. In other words, if we interpret a diffraction
pattern
in terms of a reciprocal cell and we only see intensity at those points
given by
H+K=2n, we can be sure that the crystal lattice is
centered
(in this particular case a C-centered lattice).
Other systematic absences, which also apply to all hkl
reciprocal points, are indicated in the table above.
SCREW AXES
Two-fold
screw axes, such as the one shown in the figure on the left (a
screw axis parallel to
c),
also leave their footprints in the diffraction pattern...
The reason why such a symmetry operator cancels certain intensities
is very simple to deduce if we look at the structure factors that would
result from the cooperative scattering of these two atoms:
F
= ƒ
cos 2π
( hx + ky + lz ) +
ƒ cos 2π ( - hx - ky + l
[(1/2) + z] )
However, taking into
account the
well-known relation:
cos a + cos b = 2
[cos (a+b)/2 ] [cos (a-b)/2]
the formula above can be
rewritten as:
F
= 2 ƒ
[cos π (2 l
z + l/2)]
[cos π (2hx
+ 2ky -
l/2)]
This
expression vanishes (F=0) for those hkl reflections
with h=0, k=0 and l=2n+1. Therefore, diffraction patterns
showing systematic absences of this type, or in other words, showing
intensity only for reflections of type 00l
with l=2n, indicate
the existence of a screw axis parallel to the c
axis.
Generalizing
for other two-fold screw axes, and depending of their direction, the
corresponding rules for systematic absences are:
Two-fold
screw axis
Existing
parallel to:
reflections
a
h00 h=2n
b
0k0
h=2n
c
00l
h=2n
For systematic absences produced by other screw axis types
see
the table appearing through this link. See also the
International Tables for X-ray
Crystallography.
GLIDE PLANES
Glide
planes, which are
mirror planes
which contain an additional translation (see the figure on the left),
are also responsible for some systematic extinctions in the reciprocal
lattice as seen in the table below:
Glide
plane
Existing
parallel
to: Translation:
reflections:
a
b/2
0kl
k = 2n
a
c/2
0kl
l = 2n
etc...
For systematic absences produced by other glide plane types
see
the table appearing through this link. See also the
International Tables for X-ray
Crystallography.
Summarizing: through the observation of systematic extinction
rules, such as those shown above, one can confirm the presence of
different lattice types (centered lattices) or symmetry
elements such as screw axes and glide planes, which provide a
very valuable tool for determining the space group (the
symmetry) of the crystal.
CENTRE
OF SYMMETRY
As
discussed in the beginning, there is only uncertainty regarding
symmetry elements, and it is the centre of symmetry, as shown by
Friedel's Law. However, there are situations where the presence of this
symmetry element is fixed by the combination of other symmetry elements
whose existence is evident through systematic extinctions.
This is the case of a very frequent space group, P2_{1}/c,
in which
screw axes parallel to the b
axis coexist with glide planes perpendicular to b,
as shown below:
The
symmetry elements shown in the left figure (which relate the positions
of the
black circles), such as the screw axes parallel to b,
and the glide planes (dotted lines) perpendicular to b,
are responsible for the systematic absences shown below:
0k0 k =
2n ---> 2_{1}
h0l l = 2n
---> c
The big circles in the left figure, which for instance represent atoms,
are repeated by
these operators (screw axes and glide planes). But looking
carefully at all the big circles, one can conclude
that they
are
also related through centres of symmetry (inversion centres) located on
the
cell corners (and at half the cell axes), as shown on the right figure
as small circles.
These inversion centres are thus generated by combining the screw axes
and the glide planes, and therefore
one can also conclude that P2_{1}/c is a centro-symmetric space group.
However, in other cases, making conclusions about the presence or
absence of
the centre of symmetry requires additional information from:
- some physical
measurements
(piezoelectric and/or pyroelectric effects)
- statistical tests on the intensity
distribution, or
- even the structure
solution
By the way, readers will have also probably noticed that although the
cell repeats atoms after applying complete translations of the cell
axes, the symmetry elements repeat themselves at a half of the
translations of the unit cell axes.
But let's go
back...