Unit cell determination and reJinement
43
7
while in the high symmetry cases it should be set at
4
(tetragonallcubic) to 6
(hexagonalltrigonal)
dm,.
When indexing superlattices, in which many
possible reflections are missing, higher limits on the maximum unit cell
dimensions may be required.
This is the simplest but also the slowest indexing method. Obviously,
each crystal system should be tested separately, as the number of free
variables has a critical influence on the computation time. For example, a
total of
4x
lo6 unit cells must be checked assuming a tetragonal or hexagonal
crystal system with unit cell dimensions in the range between 2 and 22
A
using 0.01
A
increment. In a triclinic crystal system, with unit cell edges
between 2 and 12
A
and angles between 90 and 120•‹, a total of 2.7~10'~
combinations should be tested using 0.01 and 0.1" increments,
respectively. Assuming that 1,000,000 unit cells can be tested in 1 second,'
an unrestricted and exhaustive search in the tetragonal or hexagonal case will
take
-4
seconds, but one will have to wait nearly 860 years to test all
possible combinations and see the answer in a triclinic crystal system.
Modem high-speed computers can handle the problem in high symmetry
cases, especially taking into account that other restrictions are applicable.
For example, the maximum expected unit cell volume can be evaluated
from
the density of diffi-action peaks observed in a certain range of Bragg angles.
Furthermore, the following additional restrictions can be imposed: in the
monoclinic crystal system
a
<
c and in the orthorhombic and triclinic crystal
systems a
I
b
I
c, because in these cases the solution is invariant to a
permutation of unit cell edges, except for the need to convert to a standard
setting after the indexing was judged successful.
The most effective is the reciprocal space approach, in which several low
Bragg angle peaks are chosen as a basis set, and then an exhaustive
permutation-based assignment of various combinations of hkl triplets to each
peak fkom the basis set is carried out. Index permutation algorithms are more
complex in realization than direct space algorithms but the former are many
orders of magnitude faster than the latter.' This occurs because the indices of
This assumption is unrealistic using even the most powerful single processor
PC
available
in late 2002. A more rational estimate is between -lo2 and -lo3 unit cells per second for a
well optimized computer code.
*
Consider a triclinic crystal system, where a minimum of six independent Bragg reflections
are required to determine the unit cell. Assuming that the maximum value of each of the
three indices is
1
and recalling that two of them should vary from
-1
to
1
(see
Table
5.7),
a
total number of possible combinations for one Bragg reflection is 3x3~2
-
I
=
17
[the set
(000) cannot be observed and is excluded from the consideration]. In an exhaustive search
without imposing any limitations, a total of
z
2.4~10' combinations among all six
reflections result. This represents about
8
orders
(!)
of magnitude reduction in the
computation time when compared to the mentioned above unrestricted exhaustive search
in direct space. The same example also highlights the critical role of the lowest Bragg