Institut Jean Barriol
Plateforme de mesures de diffraction X - Service Commun de RMN - Logiciel MoPro
École thématique "Analyse Structurale par Diffraction des Rayons X et Applications à l'Ingénierie Cristalline" (2-6 juillet 2012)
Université de Lorraine
CNRS
ANR;

Proposal for a PhD subject in mathematical crystallography

Co-operation ("cotutelle") between the CRM2 laboratory, Université Henri Poincare Nancy 1 (France) and the Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen (The Netherlands)
Supervisors: Prof. Massimo Nespolo (Nancy) and Dr. Bernd Souvignier (Nijmegen)

Title: Group-theoretical investigation of the structural basis for the formation of twinned crystals.

Description of the research work

Twinned crystals (twins, hereafter) are crystalline edifices composed of two or more individual crystals (the "individuals") of the same phase whose relative orientations in space are not equivalent under the symmetry operations of the crystals. The operation mapping the orientation of a crystal to that of another crystal of the twin is a crystallographic operation that fixes a sublattice common to the individuals. This mapping may be exact or approximate; in the latter case, a certain mismatch between the orientation produced by the mapping and the actual orientation of the individuals is observed.

Twins can form by different mechanisms. When a phase transition between polymorphs of different symmetry occurs, a homogeneous crystal develops a domain structure where the orientations of different domains are related by the operations that are lost at the transition (transformation twins). When a mechanical action is performed on a homogeneous crystal, a slip along preferential directions may occur, which gives rise to a twin (mechanical twins). More complex is the case of growth twins, where a twin forms as a response to a mistake in the normal growth of the individual, or by oriented attachment of crystals having already developed a precise morphology.

The occurrence of twins gives rise to serious problems in the structural study of materials and biomaterials. On the one hand, potential technological applications are hindered (e.g. a piezoelectric effect may be annihilated by the presence of twins); on the other hand, the presence of twinning reduces the amount of details that can be obtained from a structural study of large molecules, for which the resolution that can be achieved is already limited by the size of the unit cell. While the factors governing transformation and mechanical twins are clear, the formation of a growth twin is far from being well-understood, and this poses severe limitations to the synthesis of crystals of technological and biological interest. The elucidation of the factors governing the formation of twins is therefore of paramount importance and an absolute prerequisite for developing crystal engineering protocols capable of reducing, if not completely suppressing, the formation of twins.

The reticular theory of twinning, developed from the mid-XIX century onwards, provides - with few exceptions - the necessary conditions for the formation of twins in terms of the degree of overlap of the lattices of the crystals forming the twin. This theory is however unable to provide the sufficient conditions, i.e. to discriminate between candidates for twinning which indeed form and others which instead are never or seldom observed. The sufficient conditions for the formation of twins can be provided only by a general structural theory of twinning, whose development is the subject of this PhD programme.

For the formation of twins, a structural continuity through the interface is required. To study the conditions for this continuity in a general framework, without reducing the investigation to a case-by-case study, two approaches can be adopted.

  1. The interface, possibly extending to a few atomic layers in each crystal, must have a structural symmetry compatible with the symmetry of the crystals in their respective orientations, and must be invariant under the operation mapping the orientation of the individuals. The structure of the interface can be analysed in terms of the intersection of the space groups of the crystals, which is a sectional subperiodic group, augmented by the twin operation. The application of group-subgroup relations for sectional subgroups is expected to provide results of general validity.
  2. In a homogeneous crystal, the whole crystal structure - making abstraction of defects - is continuous across any interface that one may draw through the crystal. In a twin, the interface separating two individuals is a region of structural discontinuity. Nevertheless, part of the structure, i.e. part of the atoms forming the structure of the crystal (called a substructure) must be continuous or almost continuous across this interface, otherwise the contact between the individuals would be unstable and the twin would not form. When a substructure is continuous across an interface, its eigensymmetry is higher than that of the crystal and in particular it is invariant under the operation mapping the orientation of one crystal on that of the other. Each atom in a crystal structure gives rise to a crystallographic orbit, which may either have the same symmetry as the whole structure (characteristic orbit) or a higher symmetry (non-characteristic orbit). The study of non-characteristic crystallographic orbits in the context of twins is expected to provide the key for understanding the structural continuity necessary for the formation of twins.

Profile of the candidate

The candidate must have a strong mathematical background, in particular in algebra and group theory. Fundamental knowledge of crystallographic groups is also required.

Language

The partner institutions agreed that the PhD thesis will be written in English.

Financial support

The PhD program will be supported either by a CNRS grant (period January 2011-December 2013) or by a grant of the French Ministry of Higher Education (October 2011-September 2014). The 3-year research period will be considered as professional experience (CDD: "contrat a durée determinée").

Selected English bibliography

Contact

Prof. Massimo Nespolo
CRM2 UMR-CNRS 7036
Nancy-Université
BP 70239
54506 Vandoeuvre-les-Nancy Cedex, France

phone: +33-383-684885
fax: +33-383-406492