Crystal morphology
The crystal morphology is a term used in crystallography and mineralogy , and describes the shape of the crystal is that determined from geometric surfaces, edges and corners. Two abutting crystal surfaces form a crystal edge and at least three edges form a crystal corner. Depending on the crystal system and crystal class , the edges enclose certain angles that are characteristic of the crystal class in question.
In order to mathematically describe the position of crystal surfaces and edges in space, the mineralogist uses various indices. With the Miller indices (hkl) the position of the surfaces is described in relation to the axis system of the crystal, with the direction indices [uvw] the direction of the edges.
Crystal face
The crystal faces form the outer boundary of the crystal body and lie parallel to the lattice or lattice planes of the crystal structure inherent in the crystal , which in turn depends on its chemical composition. Crystallographic surfaces that can be converted into one another by means of symmetry operations are called form (see below: Section crystal form ). The combination of the developed forms is called traditional costume . This term should not be confused with habitus , which describes the geometric expansion of a crystal. The habit can e.g. B. stalky, fibrous, platy or isometric.
Crystal faces are formed preferentially on the network planes that have the closest packing (largest number of atoms ) and at the same time have as few free, open valences as possible (chemical neutrality). With ideal crystals, the surfaces have a clear geometry (triangle, square, hexagon) and together form regular solids (see platonic solids , catalan solids ). Crystals that have been able to fully develop their own shape without interference are also referred to as idiomorphic crystals. Which shape this is, determines the respective crystal class to which the crystal belongs. An idiomorphic crystal is therefore not to be equated with a holohedral crystal.
However, natural crystals rarely form in the ideal form, due to disruptions in the supply of substances and the anisotropic behavior when the crystal grows during crystallization or due to mutual hindrance when a large number of crystals are formed at the same time, for example when cooling down quickly. Crystal faces can therefore be distorted or even completely absent. On the one hand, however, they are always convex to one another, which means that they “bend away” from the crystal center and, on the other hand, the angles between the surfaces always remain constant despite all deformations. Crystals that were only partially able to develop their own shape due to the aforementioned disturbances are referred to as hypidiomorphic crystals.
If a crystal is disturbed so much during growth that it could not develop its own shape, one speaks of xenomorphic crystals.
Crystal shape
In crystallography, the term form (more rarely crystal form or surface form ; English form , crystal form , face form , French forme ) refers to the entirety of all symmetry- equivalent crystal surfaces . A crystal shape is denoted by the symbol {hkl}, i.e. the Miller indices (hkl) of one of the surfaces in curly brackets. Note that the crystallographic meaning of the term crystal form differs significantly from the colloquial meaning: the “shape” of a crystal in the colloquial sense is more likely to be described by the terms costume and habitus. A single crystal has exactly one costume and habitus, but usually several (crystallographic) forms.
Crystal shapes can form closed bodies ( polyhedra ) such as cubes or octahedra. There are also open forms, such as the pinacoid , but also prisms and pyramids . Such open forms must necessarily be combined with other forms on a crystal. Note that the base surfaces of prisms or pyramids - unlike in school geometry - do not belong to the shape; they are not symmetry equivalent to the actual prism or pyramid surfaces. A tetragonal (square) prism has four, not six faces in crystallography.
There are a total of 17 open (18 if the dihedron is divided into doma and sphenoid) and 30 closed varieties of crystallographic shapes.
A distinction is also made between general forms and special forms , as well as borderline forms . The general form {hkl} is derived from a "surface of general position" (hkl), that is, the surface is not parallel or perpendicular to a mirror plane or perpendicular to an axis of rotation; the indices h, k, l are generally not zero and are different in pairs. Otherwise, they are special shapes. Boundary forms occupy an intermediate position: They have the same number of areas and symmetry as the general form. For example, the general shape of crystal class 3 is the trigonal pyramid {hkil}; If the index l becomes smaller and smaller, the surfaces become steeper, the trigonal prism {hki0} is the limit shape. A special shape in this crystal class would be the base surface (the pedion) (0001).
The following tables give an overview of all forms of the 32 crystal classes. Next to each other are the general form {hkl}, then the special and borderline forms. The name of the shape with synonyms and the number of surfaces are given.
Crystal class | {hkl} | |
---|---|---|
triklin-pedial |
Pedion (1) |
|
triclinic-pinacoidal |
Pinacoid (2) |
Crystal class | {hkl} | {h0l} | {010} | |
---|---|---|---|---|
monoclinic-sphenoid |
Sphenoid (2) |
(Clino-) pinacoid (2) | Pedion (1) | |
monoclinic |
Doma (2) |
Pedion (1) | (Ortho) pinacoid (2) | |
monoclinic prismatic | monoclinic prism (4) | (Clino-) pinacoid (2) | (Ortho) pinacoid (2) |
Crystal class | {hkl} | {hk0} | {h0l} and {0hl} | {100} and {010} | {001} | |
---|---|---|---|---|---|---|
rhombic-pyramidal |
rhombic pyramid (4) |
rhombic prism (4) |
Doma (2) | Pinacoid (2) | Pedion (1) | |
rhombic-disphenoidal |
rhombic disphenoid (4) |
rhombic prism (4) |
rhombic prism (4) |
Pinacoid (2) | ||
rhombic-dipyramidal |
rhombic dipyramid (8) |
rhombic prism (4) |
rhombic prism (4) |
Pinacoid (2) |
Crystal class | {hkl} | {h0l} | {hhl} | {hk0} | {100} and {110} | {001} | |
---|---|---|---|---|---|---|---|
tetragonal-pyramidal |
tetragonal pyramid (4) |
tetragonal prism (4) |
Pedion (1) | ||||
tetragonal-disphenoidal |
tetragonal disphenoid (4) |
tetragonal prism (4) |
Pinacoid (2) | ||||
tetragonal-dipyramidal |
tetragonal dipyramid (8) |
tetragonal prism (4) |
Pinacoid (2) | ||||
ditetragonal-pyramidal |
ditetragonal pyramid (8) |
tetragonal pyramid (4) |
ditetragonal prism (8) |
tetragonal prism (4) |
Pedion (1) | ||
tetragonal-scalenohedral |
tetragonal scalenohedron (8) |
tetragonal dipyramid (8) |
tetragonal disphenoid (4) |
ditetragonal prism (8) |
tetragonal prism (4) |
Pinacoid (2) | |
tetragonal-trapezoidal |
tetragonal trapezoid (8) |
tetragonal dipyramid (8) |
ditetragonal prism (8) |
tetragonal prism (4) |
Pinacoid (2) | ||
ditetragonal-dipyramidal |
ditetragonal dipyramid (16) |
tetragonal dipyramid (8) |
ditetragonal prism (8) |
tetragonal prism (4) |
Pinacoid (2) |
Crystal class | {hkil} | {h0 h l} | {hh 2h l} | {hki0} | {10 1 0} | {11 2 0} | {0001} | |
---|---|---|---|---|---|---|---|---|
trigonal-pyramidal |
trigonal pyramid (3) |
trigonal prism (3) |
Pedion (1) | |||||
rhombohedral |
Rhombohedron (6) |
hexagonal prism (6) |
Pinacoid (2) | |||||
ditrigonal-pyramidal |
ditrigonal pyramid (6) |
trigonal pyramid (3) |
hexagonal pyramid (6) |
ditrigonal prism (6) |
trigonal prism (3) |
hexagonal prism (6) |
Pedion (1) | |
trigonal-trapezoidal |
trigonal trapezoid (6) |
Rhombohedron (6) |
trigonal dipyramid (6) |
ditrigonal prism (6) |
hexagonal prism (6) |
trigonal prism (3) |
Pinacoid (2) | |
ditrigonal-scalenohedral |
ditrigonal scalenohedron (12) |
Rhombohedron (6) |
hexagonal diyramid (12) |
dihexagonal prism (12) |
hexagonal prism (6) |
Pinacoid (2) |
Crystal class | {hkil} | {h0 h l} | {hh 2h l} | {hki0} | {10 1 0} | {11 2 0} | {0001} | |
---|---|---|---|---|---|---|---|---|
hexagonal-pyramidal |
hexagonal pyramid (6) |
hexagonal prism (6) |
Pedion (1) | |||||
trigonal-dipyramidal |
trigonal dipyramid (6) |
trigonal prism (3) |
Pinacoid (2) | |||||
hexagonal-dipyramidal |
hexagonal dipyramid (12) |
hexagonal prism (6) |
Pinacoid (2) | |||||
dihexagonal-pyramidal |
dihexagonal pyramid (12) |
hexagonal pyramid (6) |
dihexagonal prism (12) |
hexagonal prism (6) |
Pedion (1) | |||
ditrigonal-dipyramidal |
ditrigonal dipyramid (12) |
trigonal dipyramid (6) |
hexagonal diyramid (12) |
ditrigonal prism (6) |
trigonal prism (3) |
hexagonal prism (6) |
Pinacoid (2) | |
hexagonal-trapezoidal |
hexagonal trapezoid (12) |
hexagonal dipyramid (12) |
dihexagonal prism (12) |
hexagonal prism (6) |
Pinacoid (2) | |||
dihexagonal-dipyramidal |
dihexagonal dipyramid (24) |
hexagonal dipyramid (12) |
dihexagonal prism (12) |
hexagonal prism (6) |
Pinacoid (2) |
Crystal class | {hkl} | {hhl} (h> l) | {hll} (h> l) | {hk0} | {111} | {110} | {100} | |
---|---|---|---|---|---|---|---|---|
tetrahedral-pentagon-dodecahedral |
tetrahedral pentagon dodecahedron (pentagon step-tetrahedron) |
Deltoid dodecahedron (tetragon step-tetrahedron) (12) |
Triakis tetrahedron (Tristetraeder, trine-Tritetraeder) (12) |
Pentagon Dodecahedron (Pyrite Hedron) (12) |
Tetrahedron (4) |
Rhombic dodecahedron (Granatoeder) (12) |
Cube (hexahedron) (6) |
|
disdodecahedral |
Disdodecahedron (dyakis dodecahedron, diploedron, diploid) (24) |
Triakisoctahedron ( Trisoctahedron , Trigon-Trioctahedron) (24) |
Deltoidicositetrahedron (icositetrahedron, tetragon-trioctahedron, trapezoahedron, leucitoeder) (24) |
Pentagon Dodecahedron (Pyrite Hedron) (12) |
Octahedron (8) |
Rhombododecahedron (Granatoeder) (12) |
Cube (hexahedron) (6) |
|
pentagonicositetrahedral |
Pentagonikositetrahedron (Gyroeder, Gyroid) (24) |
Triakis octahedron (...) (24) |
Deltoidicositetrahedron (...) (24) |
Tetrakis Hexahedron (Tetrahexahedron) (24) |
Octahedron (8) |
Rhombododecahedron (Granatoeder) (12) |
Cube (hexahedron) (6) |
|
hexakistrahedral |
Hexakistrahedron (Hexatetrahedron) (24) |
Deltoid dodecahedron (tetragon step-tetrahedron) (12) |
Triakis tetrahedron (...) (12) |
Tetrakis Hexahedron (Tetrahexahedron) (24) |
Tetrahedron (4) |
Rhombododecahedron (Granatoeder) (12) |
Cube (hexahedron) (6) |
|
hexakisoctahedral |
Disdyakis dodecahedron (Hexaoktaeder) (48) |
Triakis octahedron (...) (24) |
Deltoidicositetrahedron (...) (24) |
Tetrakis Hexahedron (Tetrahexahedron) (24) |
Octahedron (8) |
Rhombododecahedron (Granatoeder) (12) |
Cube (hexahedron) (6) |
Morphological laws
The law of constancy of angles
The law of constancy of angles says:
All single crystals belonging to the same type of crystal always form the same angle between analogous surfaces - assuming the same pressure, the same temperature and chemical composition.
The Dane Niels Stensen (Latin Nicolaus Steno) noticed around 1669 when studying quartz that the surfaces of the crystals - regardless of their size and shape - always form the same angles. He suspected that this was a property of all mineral crystals. After further preparatory work by Torbern Olof Bergman , this assumption was finally confirmed by Jean-Baptiste Romé de L'Isle . Romé de L'Isle and his assistant Arnould Carangeot systematically measured crystals with the goniometer developed by Carangeot . In 1783, Romé de L'Isle published a detailed description of 500 types of crystals based on these measurements. He was able to empirically confirm that the law of angular constancy - as suggested by Steno - applies to every type of crystal. Romé de L'Isle's systematic measurements and the resulting inductive proof of the law of angular constancy are the first example of a scientific (methodological- empirical ) approach in crystallography. In this respect, Romé de L'Isle can be considered the founder of scientific crystallography.
Given the way crystals grow, this law is only logical. The chemical composition and the type of bonding of the basic building blocks of a mineral determine the formation of the crystal system with the corresponding atoms and molecules at the intersection points of the space lattice. Further atoms are always built in parallel to the individual levels of the space lattice. Convection currents within the mineral solution lead to an irregular distribution of the constituent atoms and thus to the preference or disadvantage of individual surfaces. Nevertheless, the angles between the planes of the space lattice are necessarily retained by the predetermined crystal system.
The law of rationality
The law of rationality (also the principle of rationality, law of rational relationships or law of rational indices) states that all crystal faces and all edges can be represented by rational indices. The indices are always small whole numbers. This applies both to the Weiss indices m: n: p and to their reciprocal values, the later introduced Miller indices (hkl). The law of rationality in this formulation was introduced in 1809 by Christian Samuel Weiss .
In the approach to this law is already in Dekreszenzgesetz ( "loi de décrescence") by René Just Haüy (1801). It says: In the successive stacking of the smallest structural units, each subsequent layer or a step of m layers parallel to an edge or surface diagonal recedes by a fixed number of n = 1, 2, 3, 6 rows of subtractive molecules. Haüy himself noticed that it is impossible to create the regular dodecahedron from cube-shaped building units. He calculated an irrational ratio for the regular dodecahedron according to the golden ratio .
Bravais law
With Auguste Bravais , attempts began to find laws with which the crystal morphology from the internal crystal structure can be predicted (and vice versa). Bravais predicted around 1848 that the "relative importance" of a crystal face is proportional to its population density, that is, that shapes are more likely to occur on the crystal, the more lattice points per unit area are on the corresponding lattice plane. This means that the morphological importance of an area is inversely proportional to the distance between the lattice levels.
Donnay-Harker's rule
In the 20th century, the Bravais law (also Bravais principle, French "loi de Bravais", English "Bravais rule") was taken up again by Georges Friedel and by Joseph DH Donnay and David Harker . While Bravais only considered centering (the Bravais lattice ) in his considerations , Donnay and Harker also included other symmetry elements (glide planes and screw axes) that lead to changed population densities of lattice planes. In this way, they were able to assign an area ranking to each room group, which they called the morphological aspect .
Goldschmidt's rule of complications
Victor Mordechai Goldschmidt established the Law of Complications (Complication Rule) in 1897 , which states that two crystal surfaces, e.g. B. (100) and (010), by repeatedly adding or subtracting their Miller indices, all other areas of this zone can be derived. Through extensive statistical studies Goldschmidt was able to show that areas in general occur less frequently the more “complicated” this derivation is, i.e. the larger its indices become.
Periodic Bond Chain
The periodic bond chain theory or Hartman-Perdok theory derives the crystal morphology from the intermolecular bonds between the crystal building blocks. This theory was introduced by Hartman and Perdok from 1955 .
literature
- Paul Ramdohr , Hugo Strunz : Textbook of Mineralogy (16th edition) , Ferdinand Enke Verlag (1978), ISBN 3-432-82986-8
- Hans-Joachim Bautsch , Will Kleber , Joachim Bohm : Introduction to crystallography . Oldenbourg Wissenschaftsverlag, 1998 ( limited preview in Google book search).
Individual evidence
- ↑ a b IUCr Online Dictionary of Crystallography: Form .
- ↑ Paul Niggli : On the topology, metrics and symmetry of simple crystal forms. Switzerland. Mineral. u. Petrogr. Mitt. 43 (1963) pp 49-58.
- ^ Nicolaus Steno : De solido intra solidum naturaliter contento dissertationis prodromus. (Forerunner of a dissertation on solid bodies naturally enclosed within other solid bodies.) Florence 1669.
- ^ Jean-Baptiste Romé de L'Isle : Cristallographie, ou Déscription des formes propres à tous les corps du règne minéral. 1783.
- ↑ CS Weiss : De indagando formarum crystallinarum charactere geometrico principali dissertatio. Lipsiae [Leipzig] 1809.
- ^ René-Just Haüy : Traité de mineralogie etc. Tome 1–5. Paris 1801. / Dt .: Textbook of Mineralogy etc. Paris and Leipzig 1804-10, Volume 1, p. 34ff.
- ^ Georges Friedel : Études sur la loi de Bravais. Bull. Soc. Franc. Miner. 30 (1907), pp. 326-455.
- ↑ JDH Donnay, David Harker: A new law of crystal morphology Extending the law of Bravais. Amer. Miner. 22 (1937), p. 446.
- ↑ Victor Mordechai Goldschmidt : About the development of crystal forms. Z. crystal. 28 (1897), pp. 1-35, 414-451.