mag3D Figure1 Figure2 Figure3 PRESENTATION SYMPOSIUM FRANCO FINLANDAIS SYMPOSIUM Problème des phases The phase problem DVD Semiconducteur 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
the
Direct Multipole Analysis
CONCEPTION AND 3D-VISUALIZATION
The
fitting procedures in assigning an analytic representation from different
models introduce several parameters and different sets of such parameters can
fit the experimental data in the same satisfying way. Moreover, they modify
the experimental data in course of analysis, replacing them by their
mathematical representation.
Direct
Multipole Analysis considers only experimental features without a priori statement on the
experimental behaviour, as described in “The other philosophy” SYMPOSIUM
FRANCO FINLANDAIS . It is in direct connexion with
experimental accuracy so that all the quantities found in course of analysis
have a physical meaning easily interpretable in terms of crystallographic
properties.
The
theoretical free atom superposition model accounts only for the total charge
density of the crystal and it is not obvious that the free atom parameters are
the same as those of the crystal atoms.
Since
the models are elaborated from free atoms, they are physically unable to
describe atoms enclosed in a lattice, especially to describe the outer shells
of the atoms.
The
comparison between the models and the crystal cannot give any real estimate for
the ionic state (Kurki-Suonio & Salmo (1971) (1))
and is just equivalent to an extrapolation method.
Studies
in reciprocal and direct spaces
The
atomic deformations affect mainly the outer electronic shells. The main part of
the information about deformations will be contained in the lower order
reflections. Thus, few measured reflections are necessary in order to collect
all the deformation effects.
A
real advantage of the Direct Multipole Analysis is the possibility to treat the
analysis in reciprocal space as well as in the real space.
The
direct space analysis gives estimation about ionicity and bonding nature. Also,
it reveals the average picture of the atom in the lattice. The reciprocal space
analysis shows a more specific behaviour, say, the electronic charge transfer
directions.
By
the Fourier invariance theorem of Spherical Harmonics, a Fourier-Bessel
transform connects the radial charge densities to the radial atomic factors
(Kurki-Suonio (1967) (2)).
Residual
term
To
determine the physical experimental quantities, such as the electron counts
Z(R) or the total charge density ρ(r), the entire series needs be computed. We must find a
reasonable continuation of the experimental data and be careful not to
introduce any extra wrong features in the results. It is the residual term
problem.
The
experimental series used truncate in a point of sinθ/λ value so near that the
residual terms of the experimental series are quite considerable. It is then
necessary to know these residual terms as accurately as possible. To have an
accurate residual term, it is necessary (Kurki-Suonio (1962) (3)):
-
to find a reasonable continuation of
the structure factors measured,
-
to determine the limits in which we
can modify this continuation without loss of reasonability.
A
reasonable continuation means that the atomic center is known and that the calculated atomic
factor gives a reasonable continuation of the true atomic factor beyond the cut
off e.g. in the core electron region.
It
is generally admitted that the quantum mechanical calculations give a good
picture of the inner electronic shells. If the atomic model has the same movement
and correct atomic positions as the inner part of the real atom, its atomic factor can be
considered correct with large sinθ/λ, and we can write:
f
= fth + Δf
with
Δf leading to zero when sinθ/λ increases. Only Δf makes the real atomic factors
to differ from the theoretical ones and this contribution is only responsible
of the differences between the true residual term and the correct model one.
But, to be consistent with the criterion of locality, the correct model
residual term must
be split in atomic contributions. This requirement is satisfied by using two
important characteristics of the Fourier series:
-
their convergence is local: the
convergence in a given point (or region) depends only on the characteristics of
this point (or region),
-
the higher rate of convergence, the
more regular function is, and the Fourier transform of a regular function
varying slowly shows a rapid decrease when sinθ/λ increases. Then, the only
hypothesis is that the true residual term can be split in atomic contributions.
A
characteristic property of the residual term is that its accuracy is seen
differently as the analysis is performed in real or reciprocal spaces.
In real space analysis,
the error due to the inaccuracy in the residual term value is spread over all
the space and a correct analysis depends strongly on the exact estimation of
this residual term. The Gaussian method with correct Gaussians:
without
constant terms and without B (I)<0 to
respect the physical nature of f (b) to have f (b) =
0 at infinite, improves largely this situation in avoiding the cut off
error.
A
Gaussian representation is best suited because of the Fourier invariance of a
Gaussian.
So,
the residual term is accurate enough to give a reasonable continuation of the
experimental data beyond the cut off. The cut off errors are eliminated by the
use of correct Gaussians. The Gaussian representations must not involve
positive exponents and no constant terms like those given e.g. in International
Tables, otherwise this leads to a physical nonsense.
Accuracy
of the residual term calculation is an order of magnitude better than the
statistical accuracy.
In reciprocal space
analysis, the inaccuracy of the residual term affects locally
the results near the cut off and does not disturb the results before this point.
So, the residual term estimation is not so important in reciprocal space as in
the real space analysis. This is a helpful point to understand the accuracy of
the results.
Expansion
in Spherical Harmonics
A
quantitative description in all directions simultaneously is obtained if the
atom is expanded in terms of different order harmonics components (Kurki-Suonio
& Meisalo (1967) (4)). For each atom of the
crystal, there is a complete orthogonal set of lattice harmonics that give a
good description of structures such as electronic distributions or atomic
factors of separate atoms.
When
non negligible very high order components are present, it means that the data
are not accurate enough to contain any information about non sphericities. In
fact, the atomic high angular momentum states are weakly occupied.
So,
for the analysis, two criteria have to be applied:
-
the atomic locality,
-
the low order harmonics behaviour of
non sphericities.
The
results obtained by the Direct Multipole Analysis take into account:
-
the correct reference,
-
the radius R used to represent the
atom.
The
Spherical Harmonics expansion presents important advantages (Kurki-Suonio
(1971) (5)) as compared to the other methods:
-
the information has a more concise form showing immediately the behaviour of
the radial atomic factors, in all directions with a rather small number of
terms in the development,
-
the information has a more concrete form, it is better analyzed, and it
separates the spherical effects from the non spherical ones due to the
crystalline field effect on the atoms or due to the bonding nature.
A
main advantage of this separation is to give some information about systematic
errors. The main part of systematic errors affects only the spherical term
because they depend on the diffraction angle but not on its direction,
-
spheres large enough can be used without a large overlapping effect of the
neighbouring atoms because of an average behaviour in all the directions.
Determination
of the spherical harmonic components
The
different components Δfn(b) are computed until n=10. In fact, the 10th
order is not required because it corresponds obviously to non visible
deformations. Nevertheless, this calculation is necessary to see if the
consistency criterion on experimental data is fulfilled or not.
The
non spherical components characterize the atomic properties inside the lattice,
e.g. they show the crystalline field effect on the atom.
Then,
the spherical harmonics significance is largely influenced by the radius R
determination (see later on: Radius of best separation). However, because of
the non observability criterion of the higher order components, a significance
limit can be pointed out. The magnitude of higher order components is
understood to give a measure of the inconsistency of non spherical information.
Physically, this magnitude has not any reality.
Integration
method by the Direct Multipole Analysis
Integration
methods require a division of the charge density ρ(r) into atomic contributions.
A
sphere centered on the atom is the most natural way to separate the atom from
its neighbouring (Kurki-Suonio & Salmo (1971) (1))
just because it considers the concept of locality isotropically. But, in this case,
some charge density can spread out inside the regions between the spheres that
are the regions of outer electrons. In fact, this electron density loss is
mainly due to the atomic deformations and represents, generally, very small
quantity as compared to the total atomic electron density except some
particular cases as Be-metal.
If
one considers the volume bound by the minimum of ρ(r) in all directions simultaneously, this leads to
complicated volumes not well adapted for integration purpose and the guarantee
of its efficacy on the electron count in the atomic region becomes disputable.
Generally,
the theoretical atomic factors used in the difference series are those
calculated in the case of a free atom (or free ion) given in the literature.
Local
axes and anisotropic atomic motions in Å2
The
experimental structure factors are analysed by the Direct Multipole Analysis as
described by Kurki-Suonio (1968) (6) with the physical criteria of Kurki-Suonio (1977) (7) TiO2-rutile
SiO2-stishovite Cu2O-cuprite K2PtCl6 Be-metal.
In the real crystal, the site-symmetries result from the positions of
the surrounding atoms.
With the iterative local refinement of parameters, we “purify” the
values of δF = Fobs ─ Fcalc from the parameter errors
left by the R-factor fitting so that the resulting δF are pure information on
real charge density difference. R-factor is an unspecified overall criterion,
which is not able to distinguish errors of parameters from electronic
deformation effects.
The natural criterion for “correct” positions and temperature factors
would be that the derivatives for the difference series are zero at the origin
when the centers of the theoretical and experimental atoms coincide.
Zeroth order concerns the average isotropic temperature factor Bave,
a correct average motion would make δρ0(r)
= zero at r = 0.
In case of non-atomic
positions, zeroth order shows the significance of the possible accumulation of
charge at those positions.
The position parameters affect the first order derivative radial
difference densities.
If the positions of the reference atoms are correct, the first
derivatives of all first order components are “flat” at small r. (10+) (11+)
and (11─) tell the possible deviations from the directions of the local z, x and y axes.
Correct parameters of the anisotropic motion i-e Bpr, Bna
and correct orientation of the axes of the thermal ellipsoid would make the
second derivatives of all second order components “flat” at small r.
The behaviour of the second order radial difference densities close to
origin tells about the possible necessity to refine iteratively the anisotropic
parameters.
(20+) refers to the prolateness i-e the deviation of uz2
from the average u2, and (22+) refers to the non-axiality or the
difference of the amplitudes in the x and y directions.
No multipole analysis of the anisotropic motion is possible without knowing
how changes of the parameters, one at a time, affect the relevant component.
All multipolar components of an atom must be calculated in the same
local coordinate system, because the angular functions depend on the choice of
axes see TiO2-rutile (fig.1) and MgF2
(Vidal et al. (1981) (8)). An instructive example of the choice of local axes is
given in the paper (Ahtee, Kurki-Suonio et al. (1980) (9)).
Radius
of best separation
The
radius of best separation gives the distance within which the electronic charge
can be treated as concentrated around the atomic center.
The
distribution inside the sphere of radius of best separation acts as a
distribution centered on the considered atom and defines the “effective inner
part of the atom or ion” in an X-Ray diffraction sense.
Determination of the radius of best
separation RM for each atom
The
spherical average electronic density ρ0(r)
or better 4πr2ρ0(r)
is computed in order to see the accumulation of charge around the atomic
positions. ρ0(r) is the first or spherical term of the Spherical
Harmonics expansion of ρ(r).
By plotting 4πr2ρ0(r)
in terms of r, the separability of the ion from its neighbours is defined by
the radius of best separation RM, minimum of the curve 4πr2ρ0(r).
The
electron count in a sphere of radius R is represented by the area under the curve 4πr2ρ0(R).
The
value of the radius of best separation RM indicates the appropriate
limit around which the electron count must be computed to estimate the ionic
state. It determines also the radius R of the sphere necessary in the Spherical
Harmonics calculations.
A
radius R slightly larger than RM (Kurki-Suonio & Ruuskanen
(1971) (10)) is chosen. This little “too large”
R radius is so that the very small artificial overlap created by this choice
must give an electronic contribution negligible as compared to the electronic
contribution of the charge density spread out in the “empty space”. This choice
is an important point of the Direct Multipole Analysis because a radius R too
small makes no deformations to appear while a too large radius R can evidence
non real deformations of the atom considered due to neighbouring influence.
Phase
problem
The
theoretical crystal can be compared with the real crystal only if the exact
crystalline structure is known, particularly the atomic positions and the
phases of the structure amplitudes. The well known phase problem concerns the
non centrosymmetric structures such as BeO (Vidal et al. (1987) (11)) belonging to a polar group that makes imaginary
the atomic factor. Then, it is necessary to fix accurately the phases before
any analysis, or to combine some iterative phase determination to the analysis The phase
problem.
Another kind of phase
problem is encountered in centrosymmetric structures as in
MgF2 (Vidal et al. (1981) (8)) where
the distance Mg-F is not a structure invariant. In this case, it is not an
imaginary atomic factor but an uncertainty due to this structure parameter that
provokes a local
phase problem in determination of scattering factor of a fluorine
atom.
Applications
of the Direct Multipole Analysis
As
already mentioned, the Direct Multipole Analysis is used to different and
various applications.
In
particular, the determinations of:
-
thermal parameters in adequation with
the experimental data,
-
the phase problem in non
centrosymmetric structures (BeO, GaAs,
InP, ZnTe).
In
addition, specific physical problems can be solved such as:
-
isotopic effects and breakdown of
Born-Oppenheimer approximation in LiH/LiD
-
charge density and magnetic phases in
MnO, CoO and NiO
-
several crystalline properties as
bonding nature, ionic state, atomic deformations, … provided by atomic charge
densities.
1)
Isotopic effects
The
problem was to see if the change of H in D can affect the charge distribution
in the unit cell.
Our
study on LiH/LiD (Vidal et al. (1991) (12))
shows that the charge density is largely dependent on the nature of the atom.
Differences
between the charge densities of LiH and LiD can only occur if the
electron-phonon coupling is strong enough. Thus, if observed, they would yield
a phenomenological measure of the violation of the Born-Oppenheimer
approximation.
The
significant deviation of the charge distribution of the anion H─ in
LiH from that D─ in LiD indicates breakdown of the Born-Oppenheimer
approximation due to coupling of the vibrations and the electronic states,
which is much stronger in LiH.
This
is the first case where such a breakdown can be seen by X-ray diffraction. It
could be the only opportunity to observe this breakdown in a crystal.
The
2D visualization of charge density difference has shown clearly the isotopic
effect and its consequence.
2)
Magnetic states Figure1 Figure2 Figure3
The
interest to study different magnetic states of a magnetic compound is to see
the influence of the magnetic state on the charge distribution.
We
decided to make an attempt to find evidence of the magnetic state by the Direct
Multipole Analysis and some indications to the bonding nature and the
electronic mechanisms of the magnetic ordering.
Our
study on MnO, CoO (para- and antiferromagnetic states) and NiO
antiferromagnetic state (Vidal et al.
(2002), (2004), (2005) (13)) has revealed
the rearrangement of the charge distribution according to the magnetic state
considered.
Particularly,
MnO (para- and antiferromagnetic states) displays a striking difference in
charge density according to the magnetic state and also the role played by the
oxygen ion in the establishment of the antiferromagnetic state.
3)
Three dimensional visualization
The
2D-visualization on maps based on the Fourier representation were the most
common way to show the different planes of the charge density difference.
Anyway,
such section-representations are not easily transformed in a 3D mental
visualization and can occult some essential features in the charge density of
the crystal.
Indeed,
ρ(r) represents the three dimensional
charge distribution and must be calculated in space to account for the spatial
behaviour of a crystal.
So,
we have elaborated the 3D visualization of the charge density difference.
The
3D visualization of the Direct Multipole Analysis leads to follow directly the
behaviour of an atom (or ion) enclosed in the crystal in all directions
simultaneously to attain the effect of the crystalline field.
It
is easier to have a realistic picture of the environment of each atom (or ion)
on the basis of the Direct Multipole
Analysis. A real and great advantage of the 3D-visualization is to show the
electronic equivalue surfaces with their respective volumes.
The
corresponding 3D illustrations made on the unit cell with the help of the
Fourier representation give the possibility to understand the structure of the
crystal.
The
comparison of the two 3D representations (Direct Mutipole Analysis and Fourier)
gives the genesis of the different bondings and the electronic mechanisms, so
the bondings are evidenced and their natures lead to a correct interpretation
of the characteristics of the crystal.
With
these 3D representations, it is also possible to investigate the empty space
and to detect the important points out of the atomic positions. In case of
structure like Be-metal,
these representations are very helpful to visualize the different equivalue
surfaces in the empty space and to understand the charge distribution net
around the Be atom.
Moreover,
3D-visualization is necessary to detect the dense packing of overlapping ions.
The
3D-visualization obtained by the Direct Mutipole Analysis produces an efficient
help to the knowledge of the mechanisms of the charge distributions atom by
atom. It allows to restore quickly the pertinent information and to lower the
abstraction degree on the results.
Then,
the physical phenomena of equivalue surfaces and of the localization by one
electronic equivalue surface are fully perceived and evidenced.
For
a same equivalue surface, there is no correspondance between the Multipole and
Fourier representations. The first representation corresponds to one atom
contribution, the second concerns the contribution of the totality of the
atoms.
An
animation of 3D-dimensional views DVD in streaming completes the 3D-information.
REFERENCES
(1)
K. Kurki-Suonio & Salmo
(1971) Ann. Acad. Scient. Fenn. A VI, 369
(2)
K. Kurki-Suonio (1967) Ann. Acad.
Scient. Fenn. A VII, 263
(3)
K. Kurki-Suonio (1962) Ann. Acad.
Scient. Fenn. A I, 93
(4)
K. Kurki-Suonio & Meisalo
(1967) Ann. Acad. Scient. Fenn. A VI, 241
(5)
K. Kurki-Suonio (1971) Italian
Crystallographic Association Meeting,
(6)
K. Kurki-Suonio (1968) (1968) Acta Cryst. , A24, 379-390
(7)
K. Kurki-Suonio (1977) Isr. J. chem. 16, 115-129; 132-136
(8) J-P. VIDAL, G. VIDAL-VALAT & K. KURKI-SUONIO (1981)
Acta Cryst., A37, 826-837
(9) M. AHTEE, K. KURKI-SUONIO,
A. VAHVASELKA, A.W. HEWATT, J. HARADA and S.
HIROTSU (1980) Acta Cryst., B36, 1023-1028
(10) K. Kurki-Suonio & A.
Ruuskanen (1971) Ann. Acad. Scient. Fenn. A VI, 358
(11)
G. Vidal‑Valat, J.‑P. Vidal, K. Kurki‑Suonio
and R. Kurki‑Suonio, (1987) Acta Cryst., A43, 540
(12) G. Vidal-Valat, J-P. Vidal, K. Kurki-Suonio, R.
Kurki-Suonio (1992) Acta Cryst. A48, 46-60
(13) J.‑P.
Vidal, G. Vidal‑Valat, K. Kurki‑Suonio
and R. Kurki‑Suonio, (2002),
Indications of Magnetic State in the Charge Distributions in MnO, CoO
and NiO.
I:
Kristallografiya, Vol. 47, N°3, 2002, pp. 391-405 (in Russian).
Crystallographic Reports, Vol. 47, N°3, 2002, pp. 347-361(in English).
J.‑P. Vidal G. Vidal‑Valat, and K. Kurki‑Suonio,
(2004)
Indicators of Magnetic State in the Charge Distributions in MnO, CoO and
NiO.
II:
Kristallografiya vol. 49, N° 3,
pp357-369, 2004 (in Russian).
Crystallographic Reports vol. 49, N° 3,
pp424-434, 2004 (in English).
J.‑P. Vidal, G. Vidal‑Valat and K. Kurki‑Suonio,
(2005)
Indications of Magnetic State in the Charge Distributions in MnO, CoO
and NiO.
III: Antiferromagnetism of
NiO
Kristallografiya vol. 50, N°1
pp23, 2005 (in Russian)
Crystallographic Reports vol. 50, N° 1, pp20, 2005 (in English)
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