Voor meer informatie of aanvraag van een exemplaar. Summary
Although the currently available direct-method computer programs solve most
crystal structures in a routine way, in some cases standard application fails
and human intervention is necessary. The phase problem involved in solving
the crystal structure consists of calculating a large number of non-measurable
variables, the phases, from an even much more numerous number of equations,
the invariant n-phase relationships, each consisting of a sum of n of the
unknown phases. So once the phase sum of a phase relation and all but one
of its phases are known, the remaining can be calculated. However, in general
none of the invariant phase-sum values are known exactly. Instead, under
general statistical assumptions, they can be estimated with a certain probability.
The direct-method phase-determination procedure consists then of expressing
all unknowns via the most reliable estimated phase sums in some starting
values. This thesis deals with two problems involved in this process. Firstly
with the problem of the reliability of the phase-sum estimates. The more
reliable the phase-sum estimates are, the more likely the phasing process
will be successful. The second problem is the order in which the unknown
phases must be calculated, via the estimated phase sums, and how many starting
values should be used. Because of the nature of the estimates of the phase
sums, statistically each calculated phase will be in error. In order to avoid
a propagation and cumulation of phase errors, a correct calculation order
is of utmost importance. The reliability of phase-sum estimates are based
on joint probability distributions (jpd's) of structure factors. Hitherto
all jpd's were derived by hand, following certain well-established mathematical
rules. The resulting distributions can be expressed in either a series-expansionform,
containing usually only a limited number of terms due to the length of the
derivations, or in an exponential form, which is a low-order approximation
as well.
In chapter two a new derivation procedure is described, valid for the triplet
phase sum in space group P1 for equal-atom structures. The resulting distribution,
a series expansion containing a selection of the higher-order terms, is not
derived by hand. Instead, the derivation algorithm has been programmed, resulting
in a computer-aided derivation. Test results show that with the new distribution
the absolute value of the triplet phase sum is estimated with less systematical
error.
In chapter three this technique is generalized to jpd's of any number of
structure factors in P1 for equal-atom structures and it is applied to the
distribution of the quartet phase sum. Test results show that with this computer-derived
distribution the quartet phase sums are estimated with considerably less
systematical error.
The technique to obtain computer-aided derivation of jpd's is further developed
in chapter four : a general theoretical and practical procedure is presented
to derive jpd's of structure factors in any space group. The series-expansion
distributions are correct up to any preset order. Optionally, the program
transforms the series expansion into an exponential expression. With a few
exceptions, in low-order approximation these exponential expressions turn
out to be identical to expressions known from literature.
In chapter five extensively test results are discussed obtained with the
computer-derived distributions for, amongst others, the triplet, quartet
and quintet phase sums. The phase determination technique is discussed in
chapter six. A new symbolic phase determination procedure is introduced,
based on the dynamic-programming technique, which calculates under the given
basic probabilistic assumptions the statistically most reliable phasing order
and starting values. In contrast to existing methods the starting set of
phases is not determined a priori and kept fixed during the phasing process
but build-up gradually. Test results show that this more flexible and systematic
technique is an enormous improvement over the conventionally employed symbolic
addition phasing procedure.
Voor meer informatie of aanvraag van een exemplaar.
Overzicht proefschriften kristallografie Amsterdam.
Hoofdpagina van Kristallografie.