Crystal morphology

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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

Combination of cube {100} with octahedron {111}, cubic habit
same costume: {100} and {111}, but octahedral habit

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

Example of the combination of an open shape (hexagonal prism) with a closed (rhombohedron)

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.

Triclinic crystal system
Crystal class {hkl}
triklin-pedial Monoèdre.svg
Pedion (1)
triclinic-pinacoidal Pinacoid with symmetry center.png
Pinacoid (2)
Monoclinic crystal system
Crystal class {hkl} {h0l} {010}
monoclinic-sphenoid Diedre.png
Sphenoid (2)
(Clino-) pinacoid (2) Pedion (1)
monoclinic Diedre.png
Doma (2)
Pedion (1) (Ortho) pinacoid (2)
monoclinic prismatic monoclinic prism (4) (Clino-) pinacoid (2) (Ortho) pinacoid (2)
Orthorhombic crystal system
Crystal class {hkl} {hk0} {h0l} and {0hl} {100} and {010} {001}
rhombic-pyramidal Pyramid rhombique.png
rhombic pyramid (4)
Prisme rhombique.png
rhombic prism (4)
Doma (2) Pinacoid (2) Pedion (1)
rhombic-disphenoidal Disphenoide rhombique.png
rhombic disphenoid (4)
Prisme rhombique.png
rhombic prism (4)
Macrodiagonal Doma.png
rhombic prism (4)
Pinacoid (2)
rhombic-dipyramidal Bipyramid rhombique.png
rhombic dipyramid (8)
Prisme rhombique.png
rhombic prism (4)
Macrodiagonal Doma.png
rhombic prism (4)
Pinacoid (2)
Tetragonal crystal system
Crystal class {hkl} {h0l} {hhl} {hk0} {100} and {110} {001}
tetragonal-pyramidal Pyramid tetragonale.png
tetragonal pyramid (4)
Prisme tetragonal.png
tetragonal prism (4)
Pedion (1)
tetragonal-disphenoidal Disphenoide tetragonal.png
tetragonal disphenoid (4)
Prisme tetragonal.png
tetragonal prism (4)
Pinacoid (2)
tetragonal-dipyramidal Square pyramid.png
tetragonal dipyramid (8)
Prisme tetragonal.png
tetragonal prism (4)
Pinacoid (2)
ditetragonal-pyramidal Pyramid ditetragonale.png
ditetragonal pyramid (8)
Pyramid tetragonale.png
tetragonal pyramid (4)
Prisme ditetragonal.png
ditetragonal prism (8)
Prisme tetragonal.png
tetragonal prism (4)
Pedion (1)
tetragonal-scalenohedral Scalenoedre tetragonal.png
tetragonal scalenohedron (8)
Square pyramid.png
tetragonal dipyramid (8)
Disphenoide tetragonal.png
tetragonal disphenoid (4)
Prisme ditetragonal.png
ditetragonal prism (8)
Prisme tetragonal.png
tetragonal prism (4)
Pinacoid (2)
tetragonal-trapezoidal Trapézoèdre tétragonal.svg
tetragonal trapezoid (8)
Square pyramid.png
tetragonal dipyramid (8)
Prisme ditetragonal.png
ditetragonal prism (8)
Prisme tetragonal.png
tetragonal prism (4)
Pinacoid (2)
ditetragonal-dipyramidal Eight-sided pyramid.png
ditetragonal dipyramid (16)
Square pyramid.png
tetragonal dipyramid (8)
Prisme ditetragonal.png
ditetragonal prism (8)
Prisme tetragonal.png
tetragonal prism (4)
Pinacoid (2)
Trigonal crystal system
Crystal class {hkil} {h0 h l} {hh 2h l} {hki0} {10 1 0} {11 2 0} {0001}
trigonal-pyramidal Pyramid trigonale.png
trigonal pyramid (3)
Prisme trigonal.png
trigonal prism (3)
Pedion (1)
rhombohedral Rhomboedre.png
Rhombohedron (6)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
ditrigonal-pyramidal Pyramid ditrigonale.png
ditrigonal pyramid (6)
Pyramid trigonale.png
trigonal pyramid (3)
Pyramid hexagonale.png
hexagonal pyramid (6)
Prisme ditrigonal.png
ditrigonal prism (6)
Prisme trigonal.png
trigonal prism (3)
Prisme hexagonal.png
hexagonal prism (6)
Pedion (1)
trigonal-trapezoidal TrigonalTrapezohedron.svg
trigonal trapezoid (6)
Rhomboedre.png
Rhombohedron (6)
Bipyramid trigonale.png
trigonal dipyramid (6)
Prisme ditrigonal.png
ditrigonal prism (6)
Prisme hexagonal.png
hexagonal prism (6)
Prisme trigonal.png
trigonal prism (3)
Pinacoid (2)
ditrigonal-scalenohedral Skalenoeder.png
ditrigonal scalenohedron (12)
Rhomboedre.png
Rhombohedron (6)
Hexagonal pyramid.png
hexagonal diyramid (12)
Prisme dihexagonal.png
dihexagonal prism (12)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
Hexagonal crystal system
Crystal class {hkil} {h0 h l} {hh 2h l} {hki0} {10 1 0} {11 2 0} {0001}
hexagonal-pyramidal Pyramid hexagonale.png
hexagonal pyramid (6)
Prisme hexagonal.png
hexagonal prism (6)
Pedion (1)
trigonal-dipyramidal Bipyramid trigonale.png
trigonal dipyramid (6)
Prisme trigonal.png
trigonal prism (3)
Pinacoid (2)
hexagonal-dipyramidal Hexagonal pyramid.png
hexagonal dipyramid (12)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
dihexagonal-pyramidal Pyramid dihexagonale.png
dihexagonal pyramid (12)
Pyramid hexagonale.png
hexagonal pyramid (6)
Prisme dihexagonal.png
dihexagonal prism (12)
Prisme hexagonal.png
hexagonal prism (6)
Pedion (1)
ditrigonal-dipyramidal Bipyramid ditrigonale.png
ditrigonal dipyramid (12)
Bipyramid trigonale.png
trigonal dipyramid (6)
Hexagonal pyramid.png
hexagonal diyramid (12)
Prisme ditrigonal.png
ditrigonal prism (6)
Prisme trigonal.png
trigonal prism (3)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
hexagonal-trapezoidal Trapezoedre hexagonal.png
hexagonal trapezoid (12)
Hexagonal pyramid.png
hexagonal dipyramid (12)
Prisme dihexagonal.png
dihexagonal prism (12)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
dihexagonal-dipyramidal Twelve-sided pyramid.png
dihexagonal dipyramid (24)
Hexagonal pyramid.png
hexagonal dipyramid (12)
Prisme dihexagonal.png
dihexagonal prism (12)
Prisme hexagonal.png
hexagonal prism (6)
Pinacoid (2)
Cubic crystal system
Crystal class {hkl} {hhl} (h> l) {hll} (h> l) {hk0} {111} {110} {100}
tetrahedral-pentagon-dodecahedral Tetartoide.png
tetrahedral pentagon dodecahedron (pentagon step-tetrahedron)
Tetragonotritetraedre.png
Deltoid dodecahedron (tetragon step-tetrahedron) (12)
Trigonotritetraedre.png
Triakis tetrahedron (Tristetraeder, trine-Tritetraeder) (12)
Pentagon dodecahedron.svg
Pentagon Dodecahedron (Pyrite Hedron) (12)
Tetrahedron.svg
Tetrahedron (4)
Rhombendodecahedron.png
Rhombic dodecahedron (Granatoeder) (12)
Hexahedron-MKL4.svg
Cube (hexahedron) (6)
disdodecahedral Diploedre.svg
Disdodecahedron (dyakis dodecahedron, diploedron, diploid) (24)
Pyramids octahedron.png
Triakisoctahedron ( Trisoctahedron , Trigon-Trioctahedron) (24)
Trapezoeder.png
Deltoidicositetrahedron (icositetrahedron, tetragon-trioctahedron, trapezoahedron, leucitoeder) (24)
Pentagon dodecahedron.svg
Pentagon Dodecahedron (Pyrite Hedron) (12)
Octahedron.svg
Octahedron (8)
Rhombendodecahedron.png
Rhombododecahedron (Granatoeder) (12)
Hexahedron-MKL4.svg
Cube (hexahedron) (6)
pentagonicositetrahedral Gyroide.png
Pentagonikositetrahedron (Gyroeder, Gyroid) (24)
Pyramids octahedron.png
Triakis octahedron (...) (24)
Trapezoeder.png
Deltoidicositetrahedron (...) (24)
Pyramid cube.png
Tetrakis Hexahedron (Tetrahexahedron) (24)
Octahedron.svg
Octahedron (8)
Rhombendodecahedron.png
Rhombododecahedron (Granatoeder) (12)
Hexahedron-MKL4.svg
Cube (hexahedron) (6)
hexakistrahedral Hexatetraedre.png
Hexakistrahedron (Hexatetrahedron) (24)
Tetragonotritetraedre.png
Deltoid dodecahedron (tetragon step-tetrahedron) (12)
Trigonotritetraedre.png
Triakis tetrahedron (...) (12)
Pyramid cube.png
Tetrakis Hexahedron (Tetrahexahedron) (24)
Tetrahedron.svg
Tetrahedron (4)
Rhombendodecahedron.png
Rhombododecahedron (Granatoeder) (12)
Hexahedron-MKL4.svg
Cube (hexahedron) (6)
hexakisoctahedral Forty-eight flats.png
Disdyakis dodecahedron (Hexaoktaeder) (48)
Pyramids octahedron.png
Triakis octahedron (...) (24)
Trapezoeder.png
Deltoidicositetrahedron (...) (24)
Pyramid cube.png
Tetrakis Hexahedron (Tetrahexahedron) (24)
Octahedron.svg
Octahedron (8)
Rhombendodecahedron.png
Rhombododecahedron (Granatoeder) (12)
Hexahedron-MKL4.svg
Cube (hexahedron) (6)

Morphological laws

The law of constancy of angles

Angular constancy with ideal and distorted crystal growth

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

Construction of an (irregular) dodecahedron from cube-shaped units. Illustration from Haüy's Traité de Minéralogie , 1801

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

Individual evidence

  1. a b IUCr Online Dictionary of Crystallography: Form .
  2. Paul Niggli : On the topology, metrics and symmetry of simple crystal forms. Switzerland. Mineral. u. Petrogr. Mitt. 43 (1963) pp 49-58.
  3. ^ 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.
  4. ^ Jean-Baptiste Romé de L'Isle : Cristallographie, ou Déscription des formes propres à tous les corps du règne minéral. 1783.
  5. CS Weiss : De indagando formarum crystallinarum charactere geometrico principali dissertatio. Lipsiae [Leipzig] 1809.
  6. ^ 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.
  7. ^ Georges Friedel : Études sur la loi de Bravais. Bull. Soc. Franc. Miner. 30 (1907), pp. 326-455.
  8. JDH Donnay, David Harker: A new law of crystal morphology Extending the law of Bravais. Amer. Miner. 22 (1937), p. 446.
  9. Victor Mordechai Goldschmidt : About the development of crystal forms. Z. crystal. 28 (1897), pp. 1-35, 414-451.

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