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SYMPOSIUM FRANCO – FINLANDAIS
RANSKALAIS – SUOMALAINEN SYMPOSIO
STRUCTURE DE LA MATIERE
REPARTITIONS ELECTRONIQUES DANS LES CRISTAUX
STRUCTURE OF
MATTER
ELECTRON
DENSITIES IN
AINEEN RAKENNE
KITEIDEN ELEKTRONIJAKAUMAT
PARIS 4 – 5 – 6 MARS 1993 PARIISI 4 – 6 MAALISKUUTA 1993
THE OTHER PHILOSOPHY OF CHARGE DENSITY
Kaarle KURKI‑SUONIO
Department of Physics, P.O. Box 9
SF‑00014 University of Helsinki, Finland
Voir
traduction française SYMPOSIUM
Résumé
L'article discute les idées de base d'une approche
optionnelle pour la densité de charge, qui a été le thème principal de la
contribution de l'auteur dans la coopération scientifique Franco‑Finlandaise.
Elle est caractérisée comme une approche empirique* ou Fourier en
comparaison avec les études conventionnelles de la densité de charge décrites
comme l'approche théorique ou d'ajustement. Le pour et le contre des deux
approches sont discutés en principe et à la lumière des exemples en soulignant
les possibilités complémentaires offertes par l'approche empirique.
* Au
sens de l'adjectif: qui s'appuie exclusivement sur l'expérience et
l'observation.
Abstract
The article discusses the basic ideas of an alternative
approach to charge density, which has been the main theme of the author's
contribution to the French‑Finnish scientific co‑operation. It is characterized
as the empirical or Fourier approach in comparison with the conventional charge
density studies described as the theoretical or fitting approach. The pros and
cons of both approaches are discussed in principle and in light of examples
emphasizing the complementary possibilities offered by the empirical approach.
The
two approaches
The
interaction of theory and experiment is the driving force of all science. Their relation is the core of the
scientific method and the basic problem of the working philosophy of the
scientist. His ways of doing research reflects his attitude on this problem.
The two counterparts are obvious in the macroscopic
division of science into theoretical and experimental research, but it is
important to realize that both are present in every element of science. Their
relation can be traced back to the basic interaction
of observation and human mind. Therefore they are inseparably interwoven;
every concept or process is at the same time both empirical and theoretical.
There are neither any purely experimental experiments nor purely theoretical
theories. Each experiment, observation, even sensation, is based on a
structuring principle or background theory. All theories, models and concepts,
even mental pictures, arise from their empirical meanings.
In spite of this inseparability of theory and experiment
there are two directions of inference, named here the empirical approach and the theoretical
approach. The question is not about doing experimental or theoretical work,
nor about the amount of theory or experimental data involved. The distinction
comes from the direction of the logical processes dominating the thinking or
method applied, whether the mental arrow is pointing from theory towards
observation or the other way. The direction can be recognized in the treatment
of any single problem, small or large.
The
empirical approach is
based on perception of Gestalts,
recognition of relevant structural features of the observations or
measurements. The empirical Gestalts are conceptualized
as characteristic properties of entities or phenomena of Nature and quantified into quantities, quantitative
measures of the qualities. The empirical approach starts from the simple and
specific and proceeds to the structural and general. It proceeds from
representation of phenomena towards interpretation and understanding.
The
theoretical approach
starts from theory, from the interpretation of the phenomenon in terms of a
theoretical model. It proceeds from the general towards the specific. The model
yields predictions which can be
tested through experimental investigation of the phenomenon. To meet the
experimental test it may be necessary to allow flexibility of the model e.g.
through parametrization. The test
then leads to fitting of parameters.
A good fit is understood to confirm the original theoretical interpretation and
to yield "experimental values" for the parameters. Further, the great
physical theories give the understanding of phenomena in principle but exact
predictions can be calculated for trivial cases only. For any realistic systems
simplifying approximations are needed and the experiment gets the extra task to
check the validity of the approximation.
In the development of science empirical approach is the
primary process. It is building the Giant of theoretical understanding from
conceptual representations of observations through successive steps of
generalization. Deeper understanding is reached through more general concepts
representing wider structural features of the observations. The great
achievements of science are unifying ideas combining different classes of
phenomena into one structural whole. Theoretical approach is the process of the
Giant. Still, it is a secondary process made possible by this understanding.
Existence
of atoms
The game called charge density was initiated about 80
years ago. Both approaches were present right from the beginning.
Laue's approach was theoretical. On the basis of the well
established theoretical model of a crystal as a 3‑dimensional periodic array of
atoms and the recently confirmed idea of X‑rays as short‑wavelength
electromagnetic radiation he predicted the phenomenon of X‑ray diffraction from
crystals. The prediction was immediately verified.
The Braggs,
father and son, started the empirical approach. They attacked the problem, how
to determine the structure on the basis of the measured positions and intensities
of the diffraction peaks.
Both deserved their Nobel. Laue standing on the shoulders of the Giant saw the promised
land with its immense potentialities. The Braggs
found the beginning of a path leading to that land and prepared the first tools
to cut their way through its unexplored jungles.
The theoretical
approach found the basic connection between diffraction and structure,
1. the representation of crystal charge density as a
Fourier series
2. the relation
between the lattice dimensions and the scattering vectors Sj of the diffraction peaks
3. the relation between structure factors Fj of the crystal and the
peak intensities .
These relations made possible experimental determination
of the lattice and the charge density. Since then the empirical approach to charge density would aim at finding and
characterization of significant features of the experimental charge density or
of its deviations from the theoretical model and at their quantification in
terms of quantities which are functionals of the charge density and can, thus,
be calculated from the experimental structure factors.
The first great finding was that the constituent atoms
were visible as peaks in the charge density. For the first time in the history
of science, the atoms appeared as observable entities of nature. The ancient
hypothesis had recieved its first direct verification.
Atomic
definition - partitioning
The
existence of atoms as identifiable
structural entities within the crystals justified the idea of treating matter
as a system of interacting atoms. This divides the problem into two largely
independent phases: 1. the structure
and 2. the charge density. The
interactions of atoms would (1) determine the structure, i.e. the lattice, mutual positions and motions of the atoms and (2)
modify the atoms. It became obvious to represent both the crystal charge
density and the structure factors as sums of atomic contributions
where the first sum runs through all atoms of the crystal
and the second through the atoms of the unit cell.
The theoretical
approach applies the eqs. (2) from right to left. It proceeds from
theoretical evaluation of atomic charge densities and atomic factors to
prediction of the composite charge density and the structure factors, while the
empirical approach proceeds from left
to right, from the experimental structure factors and crystal charge density
towards evaluation and characterization of the atomic contributions and their
comparison with theory.
Quantum mechanics is the present starting point of the
theoretical approach. The work of the Hartrees
- again father and son - gave the theoretical means for treating the atomic
contributions in eqs. (2). In principle, quantum mechanics offers the basis for
predicting both the structure and the charge density on the basis of the known
composition of the crystal - at least we believe so. In practice, it gives
solutions for trivially simple systems only and has led us to an endless
succession of tedious small steps towards the treatment of less simple systems
and less crude approximations. Even today, we are still not very good in
predicting structures or - knowing the structure - the charge density.
The theoretical approach becomes, thus, reduced to a fitting approach. The background theory
is restricted to the basic structural idea of Laue
and that behind the theoretical atomic contributions, plus - particularly for
the charge density - various general principles guiding the parametrization.
And the fit of the theoretical structure factors with the experimental ones
acts as the main criterion for the validity of results.
Atomic superposition model built of theoretical free
atoms in harmonic motion with free parameters determined by the symmetry turned
out to be sufficient for structure determination. This has become routine,
except for extreme cases. At present the number of new structures reported is
of the order of
The empirical
approach has to work with the crystal charge density corresponding to the
experimental structure factors and represented by the Fourier series (1). It
becomes, thus, reduced to a Fourier
approach aiming at characterization of modifications of the atoms in terms
of empirical features to represent in proper way "what has happened to the
atoms when forming the crystal".
Here we meet the problem of partitioning. It is clear that no unique division of the composite
charge distribution into individual atomic contributions is possible. Any
result, qualitative or quantitative, concerning a crystal atom includes an
uncertainty, which is not an experimental inaccuracy but a result in its own
value, since it reflects the basic conceptual inaccuracy of the atom itself.
Therefore, any empirical statement concerning atomic properties must consist of
two parts: 1. the statement itself plus 2. uncertainty of the statement due to
the conceptual inaccuracy. Both parts include their own experimental
inaccuracies.
The view of the theoretical approach on this problem is
different. A superposition model in itself defines the atoms. Its parameters
define the limits within which the atoms are allowed to be modified. It seems
controversial that on one hand the theoretical approach thus neglects the
partitioning problem completely, but on the other hand it has led to crude
overestimations of the problem.
Atomic
size - locality
The first quantification problem in atomic charge
densities was determination of the ionic state. This problem has been discussed
repeatedly since the first propositions by Debye
and Scherrer in 1918 [7] and by Compton in 1926 [5], cf. [21, 15].
Debye and Scherrer
[7] were the first to apply reciprocal‑space
partitioning. In the case of some simple structures it seemed possible to
derive "experimental atomic factors" from the structure factors by an
experimental approach which required only straight‑forward interpolation and
simple algebra and to obtain "experimental atomic charges" by
subsequent extrapolation to sinθ/λ = 0. The partitioning problem is evident in the apparent
arbitrariness of the interpolation and extrapolation.
The conventional theoretical
approach inverts this procedure. It starts from theoretical atomic factors
representing the atoms in different ionic states and compares the resulting
structure factors with the experimental ones. The problem of partitioning is
hidden in the assumed theoretical atomic shapes and becomes, thus, easily
neglected. At worst the reciprocal‑space partitioning - independent of the
direction of approach - has led to the completely negative conclusion, that the
ionic state cannot be determined at all by X‑ray diffraction, as first argued
by Bijvoet and Lonsdale [2].
The alkali halides offer a good example. The structure
factors derived from neutral atoms and those corresponding to atoms in
different ionic states differ so little that it is far beyond the most
optimistic experimental accuracy. This is discouraging. If X‑ray diffraction
cannot tell the obvious, that sodium chloride is ionic, is it worth anything?
Compton [5] was the first to propose direct‑space partitioning, i.
e. simple counting of electrons from the atomic charge density peaks of the
experimental charge density. This procedure corresponds to the idea of the empirical Fourier approach. It yields a
definite outcome within certain conceptual inaccuracy. In this way also the
ionicity of the alkali halides becomes obvious, cf. [14, 15, 21, 27].
The apparent contradiction between the two methods calls
for conceptual clarity as was emphasized already by Cochran [3, 4, 5]. The idea of atoms as constituents requires
a certain degree of locality. The
definition of an atom should involve the charge density peak at the atomic
position as its main part and it should not engage distant parts of the charge
distribution. Several proposals have been made to define more exactly the
nature of this requirement to reduce the conceptual inaccuracy of the results,
either in terms of strict spatial partitioning or through principles governing
the overlapping of neighbouring atoms. In this context just the general
principle is important.
The nature of the direct‑space partitioning problem is
clearly reflected by fig. 1. While the degree of conceptual uncertainty of the
atom is different in different cases, the nature of bonding as well as the
ionic state can be discussed in terms of such curves.
Figure 1. Radial charge density 4πr2 ρ0(r) of C in Diamond and Si in Silicon [20], Mg in
metallic Magnesium [23], Be in BeO, O in BeO after subtraction of Be++
[26], Li in LiH and H in LiH after subtraction of Li+ [29].
The requirement of locality limits correspondingly the
nature of the reciprocal‑space partitioning, but it is difficult to define it
in terms of the interpolation‑extrapolation procedure. However, it should be
obvious that use of the theoretical free atoms in different ionic states
involves an invalid principle of extrapolation since any bonding effects as
well as the requirement of locality affect the atomic factors most strongly in
the region of small sinθ/λ The
impossibility to determine the state of ionization through such an approach is
a consequence of violation of locality. Any conclusions on the ionic state
obtained in this way are questionable. In a study of bonding the free‑atom
superposition model assumes what should be determined experimentally and makes
a wrong assumption.
Atomic
shape ‑ multipoles
Bonding is expected to affect in the first place the
shape of the atomic charge distributions. The changes as compared to the free
atoms are however not arbitrary. There are both physical and mathematical
grounds to believe that they can be represented reasonably well by functions
with low‑order harmonic angular behaviour. This leads to the idea of
representing the crystal charge density in terms of site‑symmetric harmonic
expansions
around each atomic center, first proposed by Atoji in 1958 [1] . The idea was
adopted by Dawson in 1967 [6] as the
guiding principle of the theoretical approach, which has then been expanded
into multipole analysis of charge densities through fitting. The first
formulations of an empirical approach in terms of multipole expansions were
presented in the same year [12, 18]. Development of these principles has been a
central theme in the French‑Finnish co‑operation on charge density [25 - 29] .
The principles of symmetrization were discussed in detail by the present author
[16]. The most complete tables of the resulting site‑symmetric harmonics have
been presented in the context of the extention to the treatment of rigid
molecular motions [10].
Representation of crystal charge density by the series
(3) requires very many terms already at the distance of nearest neighbours.
However, for representation of the central atom itself low‑order terms are
sufficient. This is a qualitative requirement supported by quantum mechanical
considerations and by the empirical fact that crystal charge density is closely
represented by the free atom superposition model.
The argument is enhanced by the requirement of locality.
Because of a mathematical reciprocity principle higher order terms of local
objects become unobservable unless they are immensely strong, cf. fig. 2.
Figure 2. Radial charge densities of different orders and
the correponding scattering factors [13].
Representation of charge density by eq. (3) differs essentially
from the conventional map representations in that each multipole shows a full
three‑dimensional feature of the atomic charge distribution. Some imagination
is needed to form a clear mental picture of their shapes. Otherwize this is
ideal, because just few one‑dimensional radial densities are required, each
coupled to a definite well known angular behaviour, to form a full picture of
the shape of the atom, cf. fig. 3.
Figure 3. Difference density at the Oxygen position in
LiOH in the multipole and the Fourier representations [22].
The multipole representation
by eq. (3) guides interpretation of the charge density features in several ways.
Different multipoles refer to
different parameters and to different physical properties of the atom. Any need
to correct the atomic positions or motions in the reference model is clearly
seen, each parameter corresponding to its own multipolar component [17] as is
demonstrated by comparison of figs. 4 and 3a.
Figure 4. The effect on the radial densities at the
Oxygen position in LiOH a) of the change Dz = 0,01 Å of the Oxygen position b) of introducing
prolateness < uz2 > ‑ < u2 > = 0,002 Å2 of the Oxygen thermal
motion [22].
If some feature observed in the map is seen to be a part of
a consistent three‑dimensional behaviour of a neighbouring atom it is much more
likely to be real and can be immediately assigned to that atom. It may happen
that such a detail between two atoms arises from the low‑order multipolar
behaviour by just one of the atoms. One can then conclude that this detail also
is due to the electrons belonging to that atom. The opposite may occur. A
feature may be shared by two atoms in the sence that it is composed by
contributions due to low order multipoles of both. This certainly hints to
interpretation of the feature as a bonding effect between the atoms, cf. fig. 5.
Figure 5. Difference density at O in BeO and the
corresponding low‑order multipole expansion [26].
see DVD or Réalité expérimentale de BeO -
CERIMES - Vidéo - Canal-U the text
is in english
It is also clear that interpretation will depend on the
radial nature of the multipoles, whether they represent features at small or
large sinθ/λ in reciprocal space. Any possible additional information
like thermal dependence and complementary results from other types of
experiments will help.
Quantification
- integral quantities
In the theoretical
approach the problem of quantification does not occur. The
"experimental information" is expressed in terms of parameters which
have predefined physical meanings given by the theory or model. Parameters
attached to the atoms represent properties of the atoms. Fitting is understood
to yield their quantitative values. Here some care is required since, as
indicated by the example of the ionic state, this is not always justified. The
nature of the model parameter or its value may contradict the principles to be
taken into account in the partitioning.
In the empirical
approach perception of significant features is followed by the problem of
quantification, representation of the features in terms of quantities, which
are functionals of the charge density, reflect properly the nature of the
features and yield, thus, empirical measures for their "strengths".
This process must be guided by the requirements of physicality and reliability.
The quantities should offer the opportunity to proceed towards a physically
reasonable interpretation, and it must be possible to derive them reliably from
the data. Considerations of the size and shape of the atoms serve the first
purpose and indicate that the multipole expansion eq. (3) is a proper tool to
guide the perception.
The second requirement can be discussed in terms of the
general class of linear functionals of
the charge density [14, 15]. Such quantities have similar representations in
the real space and in the reciprocal space defined by its real and reciprocal
distribution function gX(r) and qX(S)
Charge density ρ(r) at any point r, multipolar radial
charge densities ρj(r) at any distance r from the center of the expansion (3), electron count ZV of any volume V, multipolar electron counts Zj(R) = Bj
∫0R ρnj(r)r2dr , within any distance R from the center, where Bj is the angular
normalization coefficient, more generally, any moment Zpj(R) = Bj
∫0R ρj(r)r2+pdr , scattering factor fV(S) of any volume V partitioned from the charge distribution
and radial scattering factors fj(S;R)
corresponding to the multipolar radial charge densities up to some radius R, are different relevant examples of
linear functionals with evident real distributions.
The two distributions fulfill the normal reciprocity
theorem: a narrow real distribution corresponds to a broad real distribution
and vice versa, and the degree of
singularity of the one determines the asymptotic behaviour of the other, as
demonstrated by fig. 6.
Figure 6. Reciprocal distributions of 1. the charge
density at a point, 2. the average charge density on a spherical surface of
radius 0,5 Å and 3. the electron count in the same sphere [14].
In case of a crystal the reciprocal representation eq.
(4) becomes a series
in terms of crystal structure
factors.
In view of the finite number of observed structure
factors Fj it is obvious
how the reliability of the empirical values obtained for different quantities
depends on their distribution functions. The dependence of conclusions on the
unknown structure factors at large sinθ/λ becomes minimized
when based on integral quantities with wide and smooth real distributions and,
hence, narrow reciprocal distributions. The listed examples are, thus, roughly
in the order of increasing reliability.
Charge density at a point has the most singular real
distribution, a d‑function at that point. Correspondingly, its values
depend critically on the residual term. This gives rise to the old paradox. On
statistical grounds the experimental inaccuracy of charge density at any point
increases roughly like N½ with the number N
of the observed structure factors. Thus, in the empirical approach, "the
more we know, the less we know".
The radial charge densities have d‑function distributions of the form of a
spherical shell. Thus, their reliability increases with the distance from the
center and they are proper tools for discussion of bonding effects.
Electron counts, moments and scattering factors are
integrals over finite volumes. The singularity of their real distributions is
not worse than a discontinuity at the boarder. They are therefore the most
reliable quantities on the list. Their reliability increases with the volume.
However, the reciprocal distributions of the scattering factors fV(S) and fj(S;R) are peaked at S
and S, respectively, and their
reliability therefore drops steeply at the experimental cut‑off in sinθ/λ [13, 15].
Further, it should be noted that the "angular
smoothness" of the real distributions of all multipolar quantities is
reduced and, hence, the reciprocal distributions are broadened, with the
increasing multipole order j.
It has been argued that multipolar information represents
such a degree of detail that it is not realistic as compared to conclusions
made on the basis of density maps. This turns the basic argumentation upside
down. Any charge density values, hills and valleys visible in the maps, are
details, local features, which represent the uttermost unreliability as basis
of conclusions, while the multipolar radial densities are angular integrals of
the charge density over the full 4p solid angle. They do
not represent details but consistent integral or large‑scale features.
In this respect the radial scattering factors fj(S;R) are still better since
they are integrals over spherical volumes. They are therefore the most
sensitive measures for the presence and for the significance of different
multipole components of the atomic peaks.
For the same reason comparison of the significances and
strengths of different multipoles is most conveniently done in terms of
contributing electron counts. It follows from the integral nature of such
quantities that they can be experimentally significant even when their presence
is difficult to realize in the charge density, cf. fig. 7.
Figure 7. The radial scattering factors and the
corresponding radial densities of some third order multipole components of
equal observability [13].
In the conventional theoretical approach these questions
look different. Due to the definition of atoms in terms of analytic basis
functions, the charge distribution corresponding to the fitted theoretical
model is bound to such smoothness conditions that there are no special problems
in considering the density values at different points. The above arguments
become, however, valid when discussing the residual difference maps.
Accuracy
and interpretation
The questions of accuracy and interpretation of results
look completely different in the two approaches.
In the fitting
approach the two problems are coupled indistinguishably together. The model
is a set of predefined physical meanings expressed in terms of parameters
coupled together. Fitting means always fitting of a model as a whole, not of
individual parameters. The goodness of fit measures in the first place the
validity of the model, i.e. the validity of the whole set of physical
significances involved.
A good model leads to a good fit and to results, which
are far more accurate and present much finer details of the charge distribution
than can be discussed in the empirical approach. The experimental information
becomes expressed in terms of the model parameters. The experimental errors or
inaccuracies are transformed into error limits of the parameters. However, in
principle, the value obtained for any single parameter is not an independent
experimental result concerning some definite physical property of the charge
density. Both the value and its error are conditional, they have a meaning only
as a part of the model, i. e. on the condition that the whole set of the
predefined physical significances is valid. Correlations of the parameters
indicate the extent to which the significances overlap within the model.
In careful studies the dependence of results on the model
must be discussed. This is understood to give some idea of their genuine
experimental reliability. The problem is still present. The physical
significances of parameters are defined and coupled together by the model and
they vary with the variations of the model.
In the empirical
approach each quantity is defined and determined separately as a measure of
some systematic feature of the experimental charge distribution. Its value and
conceptual inaccuracy together with the experimental inaccuracies of both are
estimated independently of other quantities considered and can, thus, be
understood to be genuine empirical results. Since no model is involved no fits
can be presented to support them or to reduce the error limits, which are
always very large as compared to what is normal in the fitting approach.
Interpretation of the results or conclusions of their
physical meanings is a matter of separate discussion, although expectations on
possible interpretations may conduct the choice of quantities to be calculated.
Notes
on the residual term
The finite number of observed structure factors causes
the residual‑term problem, present always when empirical values of any
quantities (1), (3), (5) are evaluated.
In the fitting
approach the problem is apparently avoided, since the fit can be made on
the basis of any subset of structure factors. The internal coherence of the
model replaces the lacking information on unmeasurable or neglected
reflections. The lack of information becomes visible only in the dependence of
the accuracies and correlations on the subset applied. The problem is, however,
present in the discussion of the residual information possibly contained by the
difference density.
In the empirical
approach the problem requires estimation of the residual term. Use of
integral quantities, as discussed above, reduces the uncertainties involved but
it does not eliminate the problem. The experimental value of any quantity
derived from the data must include the residual term contribution, because any
charge density feature depends on all structure factors, except for specific
extinction rules which apply on certain multipoles at certain symmetries.
This is quite evident from the problem of ionic state.
The residual term is seen to contribute significantly to the empirical values
of any integrated charges [14, 21]. Differences of atomic parameters as
determined from difference charge densities are not sufficient. It is not clear
what they possibly mean, because one cannot add them to values corresponding to
the theoretical model atoms just because they correspond to an essentially
different definition of the atom. One should apply the same atomic definition
on the theoretical composite charge density to derive a model parameter on
equal basis.
The only possibility to evaluate the residual terms is
the use of a theoretical model. Therefore, a theoretical reference model is always
necessary as the starting point. It represents the minimum amount of
theoretical basis needed in the empirical approach. For this purpose an asymptotical model is required, i.e. a model which is reliable
asymptotically at large sinθ/λ [11]. The free atom
superposition model with harmonic thermal motion is often sufficient, when
fitted to the data at large sinθ/λ. Even the correct ionic state of the atoms is not
important. This is based on the assumption that the free atom core is not
disturbed by the crystal environment and that it is well represented by the
theory.
To obtain the experimental value Xobs for the quantity X one has to calculate the value Xref for the reference charge density and to add the
value DX
corresponding to the deviations of observed structure factors from those of the
reference model
To avoid infinite series it is practicable to use an
analytical reference model where the theoretical atomic factors are replaced e.g. by Gaussian representations, as
suggested already by Hosemann and
Bagchi [8, 9]. Again it is
essential that the Gaussians are fitted to the theoretical atomic factors
asymptotically at large sinθ/λ [19, 24] .
It should be obvious that the same definition of the
quantity X must be used for both
terms of eq. (6). Still this is an old trap as demonstrated by the problem of
ionic state. There is a temptation to use the parameter of the model atom, in
this case the atomic factor values at sinθ/λ = 0, instead of the integrated value. This seems to be
an easy way to avoid extra calculation but it leads to a misconception
comparable to the "nightingale liver pate" of the well known old
story, largely responsible for the frustrating negative conclusions referred to
earlier.
This procedure does not solve the problem caused by
unmeasured reflections at small sinθ/λ. In the fitting approach they need no special treatment but in the empirical
approach a separate discussion is necessary. If they are not too many, the
locality and the low‑order multipolar shape of atoms are sufficient to produce
an internal coherence which makes it possible to observe the significant
multipoles and to correct the structure factors of the reference model to yield
some estimates for the unobserved ones. In principle this leads to an iterative
procedure [20]. The same argument applies to the lacking information on the
phases in noncentric structures, and it is possible to take it into account
through an analogous iterative procedure [26]. See The
phase problem
References
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[4] Cochran,
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[5] Compton,
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264, 379, 395.
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[14] Kurki‑Suonio,
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MULTIPOLE – ANALYSIS GROUP
VISUALISATION 3D
mag3D Figure1 Figure2 Figure3 PRESENTATION SYMPOSIUM FRANCO FINLANDAIS SYMPOSIUM Problème des phases The phase problem DVD Semi-conducteur organométallique organosilicié TiO2-rutile SiO2-stishovite Cu2O-cuprite K2PtCl6 Be-metal bases and 3D views Bases et vues 3D Si-et-Ge CaF2-fluorite NbC-2023