Symmetrical polyhedra (simple crystal forms) as orbits of noncrystallographic point symmetry groups

Simple crystal forms are analyzed as the orbits of noncrystallographic point symmetry groups on a set of smooth or structured (“hatched”) planes of crystal space. Polyhedra with symmetrically equivalent faces, obtained using noncrystallographic point symmetry groups, are considered. All possible versions of simple forms for all noncrystallographic groups are listed in a unified table.     

Combination of colored patterns and cubes for crystallographic point group visualization

A concept based on the application of a cube as a general polyhedron is presented for the visualization of point group symmetry. The cube is used to represent both the highest and lowest crystal symmetries, with differences observable as patterns characteristic to each point group. Patterns are generated using direction‐specific color keys, which enable the recognition of point group‐specific distribution of vectors in an external reference frame. For the visualization of the incoherent hexagonal crystal classes, two twinned cubes are applied in order to generate symmetry operators that would otherwise be missing. The resulting hexagonal dipyramid is described in the frame of a cube, reducing the number of used symmetry operators from 72 to 60. The complete set of 32 polyhedra are suitable, for example, as a visual aid for understanding the crystal symmetry and/or sub‐ and supergroup relationships.     

On a number of some categories of the limiting groups of multidimensional plane crystallographic point groups of symmetry

Many categories of multidimensional crystallographic plane point groups of symmetry are established with the aid of two-and three-dimensional point groups of symmetry and the groups of simple and multiple antisymmetry generated by these point groups and the groups of rosette, tablet, hypertablet, crystallographic, and hypercrystallographic P symmetries. The numerical characteristics of all the categories of such groups are given and, for each of these groups, the numbers of the limiting groups of multidimensional symmetry (whose subgroups they are) are determined for each category based on the limiting symmetry groups of the classical point groups and the generalizing groups of l-tuple antisymmetry and the indicated particular cases of the P symmetry.     

晶体表面结构和负离子配位多面体生长基元

本文讨论了晶体表面结构和负离子配位多面体结晶方位的关系.指出了晶体表面结构,显示了负离子配位多面体在晶体生长过程中的结晶轨迹.因此,运用负离子配位多面体生长基元理论模型,晶体的生长机制可以通过对晶体表面结构的分析得以解释.     

Structure information from morphological data of organic crystals with the help of Patterson methods

The morphological points of a space group are special or general point configurations, respectively combinations of point configurations. There are mainly point configurations to be found, which in three-dimensional space or at least in one projection, posses a higher translational symmetry than the crystallographic ones. The morphological lattice of organic structures with one molecule in the asymmetric unit e. g. corresponds to the lattice of the centers of gravity of molecules. Starting from the faces appearing on crystal, a possibility for deriving structure information with the help of Patterson methods will be demonstrated.     

Three‐dimensional thermoelastic stresses in off‐axis oriented single crystals with hexagonal symmetry

A three‐dimensional (3D) thermoelastic stress analysis is carried out on a single crystal with axisymmetric geometry but with a hexagonal crystallographic symmetry. The crystallographic orientation is off‐axis with respect to the cylindrical coordinate system. By applying a Fourier series expansion with respect to the rotational angle ϕ of the cylindrical coordinates, the 3D boundary value problem is reduced to a sequence of 2D ones on the meridian plane, which are solved by the finite‐element method. In our example, the off‐axis orientation is towards a direction of high symmetry, and therefore only four of the six stress tensor components are non‐zero. In the end, the stress tensor is projected onto the slip system of the crystal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some geometric characteristics of four-dimensional crystallographic point symmetry groups and their subgroups

Using three-dimensional point and tablet antisymmetry groups and also rosette and zero-dimensional groups of generalized antisymmetry that model all possible subgroups of four-dimensional crystallographic point groups, we determined the number of various symmetry groups forming the four-dimensional crystallographic classes for each category of such subgroups. These results allowed us to establish (without the use of the complete catalogue of the groups themselves) that 168 four-dimensional crystallographic classes preserve not only the point but also some other geometrical objects. The remaining 103 classes preserve only one invariant point of the four-dimensional Euclidean space.     

Structural conditionality of the piezoelectric properties of langasite family crystals

The atomic displacements upon isomorphic substitutions in crystals of the langasite family have been analyzed. The thermal parameters are determined and the probability density function of atoms is analyzed. Local potential energy minima are found which can be occupied by atoms under external effects. The contributions of cations in all four independent crystallographic positions and anions in all three such positions to the piezoelectric properties are established. One specific structural feature is the constant (at isomorphic substitutions) or possible (under external effects) but always opposite displacements of two cations along symmetry axis 2. Large cations in eight-vertex polyhedra make the main contribution to the piezoelectric properties. The cations in the tetrahedra on symmetry axis 2 weaken these properties. The cations in the octahedra in the origin of coordinates and in the tetrahedra on symmetry axes 3 only slightly affect the piezoelectricity.     

The uniqueness of determination of the space group symmetry by means of the vibrational spectroscopy methods

Vibrational spectroscopy methods are used to solve the most fundamental crystallographic problem—determination of crystal symmetry. It is shown that if X-ray diffraction methods do not make it possible to unambiguously choose the symmetry group, it can be successfully done on the basis of joint analysis of the vibrational spectroscopy and X-ray diffraction data. Irreducible representations of vibrations for all systems are systematized, and the cases are indicated where the ambiguity in assignment of crystal structures to the space groups with unified diffraction extinction rules can be avoided.     

Structural relationships between single atom type crystal structures and intermetallic compound structures,as illustrated by Laves‐ and Hume‐Rothery phases

In comparison to the single atom type crystal structures, which are relatively simple and are concentrated in four types of structure (Cu, W, Mg, diamond), the crystal structures of intermetallic phases are manifold and often very complicated. At first sight there is no obvious connection between these types of crystal structures, however, a transition mechanism must exist for the solid‐state transition from the simple single atom type crystal structures to the intermetallic phase structures. Such a mechanism would show the inner connection between the structures. In this work the topic is dealt with from a crystallographic point of view. The transition between the structures can be described by the three steps (1) substitution of atoms, (2) systematically ejection of particular atoms from the emerging intermediate structure and (3) spatial relaxation especially of the atoms around the newly formed interstices. This general structural phase transition path will be discussed for two of the intermetallic classes, the Laves phases and the Hume‐Rothery phases. A suitable way of showing this is by making a comparison between the coordination polyhedra (atomic environments) of the single atom type crystal structures and those of the derived intermetallic crystal structures.     

Reciprocal crystal and P.B.C.-vector method. Crystal growth of a-sulphur

The morphological lattice of α-sulphur (F ddd) has coordinates 1/8, 1/8, 0. The reciprocal crystal is a polyhedron with 15 faces and point symmetry 112. The arrangements of the polyhedra parallel to the crystal faces confirm that (100), (101), (010), (011), (111), (113), (001) are F-faces and (110), (112) S-faces according to Hartman and Perdok. Under consideration of the extinction rules of the morphological lattice (101) and (011) have to be regarded as S-faces. The size of the polyhedra faces is proportional to the interaction energy given by Hartman. It will be shown that the face area of the net planes, divided by the number of molecules per growth layer, represent a good approximation to the attachment energy calculated according to Hartman. The average deviation for α-sulphur is 3%, for olivine and SnJ₄ 8.5% and for anthracene 10.5%. The Wulff-plots correspond fully to the experimental observations.     

石榴子石晶体碱腐蚀像及其与酸腐蚀像的对比

本文采用NaOH-KOH混合熔融物和KOH溶液对石榴子石晶体各种不同的结晶学方向的晶面(切面)进行腐蚀实验,建立了石榴子石碱腐蚀像的立体模型,并与酸(HF溶液)腐蚀像模型进行对比.研究发现碱腐蚀像与酸腐蚀像一样能很好的反映晶体的对称特点,并发现在{ 100}、{110}晶面(切面)上碱腐蚀像与酸腐蚀像相同,而在{120}、{221}、{111}、{211}晶面(切面)上,碱腐蚀像与酸腐蚀像不同.该研究可以用来对石榴子石族矿物进行结晶学定向,同时具有揭示矿物所处地质环境酸碱性的指示意义.     

Using the <Emphasis Type="Italic">P</Emphasis>-symmetry space groups of crystals to investigate 6D symmetry groups

All P symmetry space groups of crystals in the geometric classification that are allowed by the general theory of P symmetry have been investigated. The number of different symmetry groups of a 6D Euclidean space that retain the invariant 3D plane has been revealed using the different (disregarding enantiomorphism) symmetry space groups and crystallographic P symmetries generated by these groups.     

X-ray diffraction from ideal mosaic crystals in external fields of certain types, part III: Changes in space symmetry and symmetry of diffraction pattern

The change in the space symmetry and the symmetry of the corresponding diffraction pattern occurring under the effect of uniform electric and magnetic fields applied to a crystal have been analyzed. The tables of point symmetry groups for various directions of applied magnetic fields are constructed. It is shown that in the absence of an external field, the diffraction patterns are described by nine possible point groups; in an electric field, they are described by eleven point groups; and in a magnetic field, by only five point groups. The theoretically calculated diffraction parameters are compared with the corresponding experimental values.     

Curie symmetry principle and growth of single crystals

Application of the symmetry principle, proposed by Pierre Curie, to the growth of single crystals is considered. The basic concepts of this principle and the rules for alignment of the crystallographic symmetry of a seed crystal with the external symmetry of the thermal field formed by the crucible with a melt are shown. The modified Stepanov method, developed at the General Physics Institute, Russian Academy of Sciences, on the basis of the Czochralski and Stepanov methods, is described, and its advantages are demonstrated. Examples of application of this technique to the growth of complex oxide crystals are given.     

晶体形态一些基本概念的实际意义分析

本文对晶体学中关于晶体形态一些古老的基本概念(单形的正形与负形、左形与右形、一般形与特殊形)进行了深入思考,并从实际晶体形态上发育这些单形的角度,分析了它们的本质区别与内在含义:单形的正形与负形是属于同一几何单形、同一对称型(点群)的两个结晶单形,它们可以相聚;单形的左形与右形也是属于同一几何单形、同一对称型(点群)的两个结晶单形,但它们却具有不同的内部结构,所以它们不能够相聚;单形的一般形与特殊形只有结晶单形意义,无几何单形意义.这些认识阐明了过去我们多年来并没有足够重视的一些理论与实际问题,这有助于晶体形态对称理论的发展,也有助于晶体学教学思路的明确.     

Accumulated distribution of material gain at dislocation crystal growth

A model for slowing down the tangential growth rate of an elementary step at dislocation crystal growth is proposed based on the exponential law of impurity particle distribution over adsorption energy. It is established that the statistical distribution of material gain on structurally equivalent faces obeys the Erlang law. The Erlang distribution is proposed to be used to calculate the occurrence rates of morphological combinatorial types of polyhedra, presenting real simple crystallographic forms.     

Complete derivation of space groups of magnetic symmetry of crystals by numerical construction

The solution is given to the problem regarding the derivation of the so-called space groups of magnetic symmetry with crystallographic angles of rotation of the spin vector, which was formulated as far back as the early 1970s. The crystallographic P-symmetries that should be used in the generalization with the three-dimensional space Fedorov groups G ₃ are revealed. The space groups G ₃ are generalized with each of these P-symmetries.     

Infinite symmetry applied to crystals and crystal aggregates

Infinite symmetry is referred to for describing in a more rational manner the morphology of certain crystals and crystal, particularly mineral aggregates. Their non-crystallographic forms depend on the peculiarities of their structure, enhanced growth along preferred directions and on the symmetry and composition of the growth environment. Apart from the seven Curie forms others, fitting the corresponding point groups are suitably added as variants of the growth geometry, applicable also to the property of matter.