Chemistry Reference
In-Depth Information
“reciprocal space”, given by H
h
a*+
k
b*+
l
c*. The basis vectors a*, b* and c*
are called the reciprocal lattice vectors, and depend on the crystal structure. The
three-dimensional space defining the crystal structure is called “direct space”.
A given diffraction maximum H is completely defined by the structure factor
F
(H), which has amplitude |
F
(H)| and phase
¼
a
(H). In XRD, the structure factor
F
(H) is related to the electron density
r
(r) within the unit cell by the following
equation:
ð
F
ð
H
Þ¼
j
F
ð
H
Þ
j
exp
ð
i
að
H
ÞÞ ¼
rð
r
Þ
exp
½
2
p
i
H
r
dr
;
(1)
where r is the vector r
x
a +
y
b +
z
c in direct space (a, b and c are the lattice
vectors defining the periodicity of the crystal structure) and the integration is over
all vectors r in the unit cell. It follows from (
1
) that
¼
X
H
F
rð
r
Þ¼
ð
1
=
V
Þ
j
ð
H
Þ
j
exp
½
i
að
H
Þ
2
p
i
H
r
;
(2)
where
V
is the volume of the unit cell and the summation is over all vectors H with
integer coefficients
h
,
k
and
l
. If both the amplitude |
F
(H)| and phase
a
(H) of the
structure factor could be measured directly from the experimental XRD pattern,
then
(r) (i.e. the “crystal structure”) could be determined directly from (
2
)by
summing over the measured diffraction maxima H. However, while the values of
|
F
(H)| can be obtained experimentally from the measured diffraction intensities
I
(H), the values of the phases
r
(H) cannot be determined directly from the experi-
mental diffraction pattern, which constitutes the so-called “phase problem” in
crystallography. To determine a crystal structure from experimental XRD data by
using (
2
), it is necessary to use techniques (e.g. direct methods or the Patterson
method) that provide estimated values of the phases
a
a
(H). Using the estimated
phases
(H) together with the experimentally determined |
F
(H)| values in (
2
)
allows the electron density
a
(r) and hence the crystal structure to be elucidated
(at least approximately). More details of the techniques for overcoming the “phase
problem” are given elsewhere [
1
,
21
].
Importantly, the reverse procedure of calculating the diffraction pattern for any
given structure [using (
1
)] is an “automatic” calculation. Thus, the diffraction
pattern (|
F
(H)| data) can be calculated automatically for
any
crystal structure
using the positions of the atoms in the crystal structure in (
1
), employing a form
of (
1
) in which the electron density
r
(r) is approximated by a function that depends
on the positions of the atoms in the unit cell. This type of calculation is the basis of
the
direct-space strategy
for structure solution. In the direct-space strategy, a large
number of trial crystal structures are generated by computational procedures; the
XRD pattern for each trial structure is then calculated automatically using (
1
), and
these calculated XRD patterns are then compared with the experimental XRD
pattern in order to assess the degree of “correctness” of each trial structure. More
details of the direct-space strategy for structure solution are given in Sect.
2.6.2
.
r
