Voor meer informatie of aanvraag van een exemplaar. Summary
Single crystal X-ray diffraction is a powerful tool for determination
of the 3-dimensional structures of anorganic and organic compounds. In this
context, the physics of X-ray diffraction reduce to the Fourier transform
of the electron density function in the crystal unit cell, which yields a
function of the reciprocal vector. Because of the periodicity of the electron
density in the single crystal, the reciprocal vector is restricted to a reciprocal
lattice, the so called reflections. The corresponding values of the Fourier
transform are the so called structure factors. The measured diffraction intensities
are determined by the squared absolute values of these structure factors,
and by the thermal motion of the atoms which is corrected for by a least
squares normalisation procedure. For reconstruction of the electron density
by an inverse Fourier series, also the phases of the structure factors, which
are lost in the experiment, are needed.
The phases of the structure factors are dependent on the choice of origin
in the single crystal unit cell. However, linear combinations of phases of
structure factors of which the reciprocal vectors sum up to the zero vector,
the so called multiplets, are independent on the choice of origin. The triplet
phase sum for example consists of three such phases, the quartet phase sum
of four such phases, and so on. Estimations of these multiplets as functions
of some set of observed absolute values of structure factors have already
been derived. The practical use of these estimations for obtaining a phase
set that leads to the actual electron density in the unit cell, is the so
called Direct Method procedure. Many such programs that reveal the 3-dimensional
structure of thousands of compounds, have been developed in the last decades.
A mathematical aspect of Direct Methods is the derivation of joint probability
distributions of structure factors. The unknown positions of the atoms in
the unit cell are the primitive random variables, and the derivation results
in joint probability distributions of the multiplets and the absolute values
of corresponding sets of structure factors. As these absolute values are
known by measurement, conditional probability distributions of the multiplets
given the absolute values are derived. Finally, from these, estimations of
multiplets as functions of observed absolute values are derived. With these
estimations and a number of consistency criteria, Direct Method procedures
can in a number of cases determine a phase set that leads to the actual positions
of atoms in the unit cell. Also inequalities exist that restrict the possible
values of multiplets. In chapter 2, an outline of these derivations and their
mutual implications is provided.
The width of the conditional probability distributions of multiplets increases
with the number of atoms in the unit cell. As a consequence, Direct Method
procedures break down for single crystals with much more than a hundred atoms
in the unit cell. This is why much effort is spent on improving the mathematical
basis of the derivation of joint probability distributions of structure factors.
A mathematical theory that deals with large numbers of random variables,
is the Maximum Entropy theory. This theory has succesfully been applied to
many fields of science, especially in statistical gas theory. In chapter
3, the Maximum Entropy theory is introduced in analogy with statistical gas
theory, using Lagrange multipliers.
In chapter 4, a full theory for derivation of joint probability distributions
of structure factors with Maximum Entropy is provided. A well known tool
from statistical gas theory, the irreducible cluster integral, deals with
interatomic correlations of the positions of the atoms in the unit cell.
Well known exponential probability distributions of the triplet and the quartet
phase sums are derived, and from the irreducible cluster integrals important
observations are made concerning the normalisation procedure.
In chapter 5, the normalisation procedure that corrects for thermal motion
of the atoms, is studied in more detail. Usually, a least squares straight
line in the Wilson curve is used to determine the width of the overall thermal
motion. A new fit function is presented that accounts for other effects,
Bike librations along the bonds and interatomic distances and interatomic
angles. More realistic overall temperature factors are found for some small
structures. However, these values did not improve the performance of a direct
methods program like Simpel88.
In chapter 6, the effect of triples of atoms in the actual structure on the
joint probability distributions is investigated. Also, an iterative method
for finding these triangles is outlined. A first double Patterson function
is generated from the single Patterson function. After selecting peaks, triplets
Ire calculated using these peaks. The double Patterson function is recalculated,
and the process is continued iteratively. More and more triangles in the
actual structure may be found, thus finding more and more reliable triplet
values. The enantiomorph is fixed by the choice of the first peak out of
the two largest enantiomorphous peaks. This procedure requires a fast peak
search algorithm in 6-dimensional space, which can handle up to order N3
peaks. Also, all triplets may be used for recalculation of the
double Patterson function. The development of such a program is a subject
of further research.
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