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THE PHASE PROBLEM IN NON-CENTROSYMMETRIC STRUCTURES

1. The problem

In general, the crystal structure factors are complex quantities, F=A+iB=|F|eⁱ^φ where the phases φ, depend on the choice of the origin (figure1). The structure factors of a centrosymmetric crystal, with respect to the centre of symmetry, are all real, φ = 0 or π and F = A = ±|F|, while those of a non-centrosymmetric crystal are genuinely complex, independent of the origin.

For determination of crystal charge density, experimental structure factors F_obs are needed. However, from measurement of diffraction intensities, only absolute values |F_obs| can be derived. Conventional refinement of atomic positions and temperature factors yields theoretical structure factors F_theor close enough to the 'true' values to serve as a starting point for discussion of the phases. For centrosymmetric structures this is sufficient to fix the signs of F_obs, except for "weak reflections". Normally this is a minor problem. For non-centrosymmetric crystals the problem is acute. This is known as the phase problem.

Even if the theoretical phases are close to the experimental ones, φ_obs~ φ_theor adoption of φ_theor for φ_obs may be severely misleading. The point is, that experimentally founded corrections to the model are based on the differences ΔF=F_obs − F_theor=|ΔF|eⁱ^φ and depend critically on the phases F of these differences. As is easily gathered from figure 2, even small differences φ_obs − φ_theor can cause large deviations of F from the value φ_theor.

2. Phase refinement

A better approximation for the phases φ_obs can be obtained through an iterative procedure coupled to the "direct multipole analysis" SYMPOSIUM FRANCO-FINLANDAIS of crystal charge density. Figure 3 represents this procedure schematically.

The theoretical phases are taken as the starting point, and the low-order multipolar corrections Δf_nmp(b) to the atomic scattering factors are calculated in the 0^th approximation using ΔF= (|F_obs| − |F_theor|)eⁱ^φ_theor . The theoretical model is then modified by adding the multipolar corrections obtained to the theoretical atomic factors. This yields first-order modified theoretical structure factors F_mod1. In the next step, the phases of the modified structure factors are adopted and the multipolar corrections are calculated using ΔF=|F_obs|eⁱ^φ_mod1 − F_theor. Now, these corrections are again applied to the theoretical atomic factors to yield second-order modified structure factures F_mod2. The procedure can be repeated iteratively using ΔF=|F_obs|eⁱ^φ_modn − F_theor with n = 1, 2, 3,… until convergence is reached. The final values of the modified phases can then be adopted as the final approximate φ_obs~ φ_modn.

3. Example BeO

EXPERIMENTAL REALITY OF BERYLLIUM OXIDE

See « Multipole Analysis of X-Ray Diffraction Data on BeO » Vidal-Valat,G., Vidal, J.P., Kurki-Suonio, K. & Kurki-Suonio, R. (1987). Acta Cryst. A43, 540-550.

see DVD or Réalité expérimentale de BeO - CERIMES - Vidéo - Canal-U the text is in english

The study of Beryllium Oxide, Multipole Analysis of X-ray diffraction data on BeO, leads to a new stage of methodology. The crystal of BeO has high-symmetry but no centre of symmetry. Its structure factors are genuinely complex and the possibilities of treating the phase problem due to ionic deformation effects must be studied. In fact, the phase problem consists of 2 parts :

- the determination of the phases due to atomic positions

- and the determination of the phases specific to ionic deformation effects.

The phases due to the atomic positions are straightforward. The real problem that is not simple to overcome is the phase due to ionic deformation effects. The procedures developed are specific to Direct Multipole Analysis, as this analysis determines the scattering power of each atom involved in the electronic interaction of matter. Direct Multipole Analysis showed that the problem connected with the phases due to ionic deformation effects could be handled if the data are good enough- after that, there is no doubt that these procedures are applicable within any symmetry. In this work, the very diffuse nature of the O^2–ion posed another problem, that could be solved by developing techniques for subtracting the contribution of the compact cation from the data. The Be²⁺ cations were seen to be compact "ionic" structural entities possessing no observable dipole deformation. Subtraction of their contribution leaves a picture of layers of covalent Oxygen anions with the cations acting as electrostatic stabilising agents.

The geometry of the surroundings of an O^2–ion with the local coordinate axes

In this schematic representation, the OX, OY, OZ orthogonal axes used to partition space are indicated, and a few numerical values. Only a plane with oxygen atoms O₁ is seen.

The central atom O₁ is taken as the origin O.

The positions of the beryllium ions are noted. Planes of beryllium cations are parallel to the planes of oxygen anions. The Be₁ are on the OZ axis.

Effect of ionic deformation on the phases

Three-dimensional behaviour of the charge density of O^2–

Here we see the (0.01, 0.02, 0.03, 0.04) e/Å³ isosurfaces. By definition, the electron represents here an electric quantity of 1.6 10¹⁹ coulombs.

The YOZ half-space with negative x is seen. We place ourselves in front of this half-space, and lie on the positive x axis.

In order to see better the different isovalues we exclude the OX, OY, OZ axes.

On the right hand side the views show the experimental electronic distribution of O^2–ions. This distribution possesses genuine phases, due to the ionic deformations, never obtained by conventional density studies.

On the left hand side the views show the theoretical electronic distribution. Because of the real ionic deformations, the theoretical distribution cannot reveal the phases. In the view to the left the electron distribution is clearly symmetric on both sides of the OX, OY plane.

At present we examine the YOZ half-space with positive x and settle on the negative x half-axis.

Here we see the XOZ half-space with positive y. We face this half-space and are placed along the negative y half-axis.

Here we show the +0.02 e/Å³ isovalue. As before, the right hand view displays the true spatial distribution. The external structure is visible. It provides a significant gain in information and in a form ready for immediate and accurate interpretation of the results.

As we have already noticed, in the view to the left the electron distribution is clearly symmetric on both sides of the OX, OY plane.

Conclusion

On the left views a symmetrization is seen distinctly, due to the absence of knowledge of the phases due to the genuine ionic deformations.

4. Examples : GaAs, InP, ZnTe Semiconducteur

The geometry of the surroundings of an O2– ion with the local coordinate axes

Three-dimensional behaviour of the charge density of O2–

The geometry of the surroundings of an O^2–ion with the local coordinate axes

Three-dimensional behaviour of the charge density of O^2–