Crystals - Basic Terms

 

last modified: Friday, 21-May-2010 23:28:56 CEST

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To describe the geometrical characteristics of a crystal, 5 terms are usually distinguished: crystal system, crystal class, habit, form, and twinning.
These notions are in turn all based on the concepts of the unit cell and the point lattice.

 

Unit Cell and Point Lattice

NaCl_unit_cell.jpg

Fig. 1

In a crystalline substance molecules and atoms are arranged in a regular manner, forming a body with specific geometrical characteristics (for example, table salt, sodium chloride, forms little cubes). The atoms in a crystal form a virtual three-dimensional grid. The smallest building block of this grid that reflects the geometry and symmetry properties of the crystal is the unit cell. Table salt has a unit cell of 13 sodium and 14 chlorine atoms that form a perfect cube (3 by 3 by 3, as shown in Fig. 1, with chlorine symbolized by green balls, and sodium by blue-gray ones[1]). The relation of sodium to chlorine in table salt is 1:1, so obviously a unit cell does not necessarily reflect the chemical formula. The reason is that neighboring unit cells share atoms that lie on their edges and corners. And for the same reason it is wrong to think of unit cells as bricks forming a crystal[2], the unit cell is just an abstract geometrical concept.


grid_unit_cell.png

Fig. 2

The edges of a unit cell correspond to axes in a three-dimensional grid [3]. This grid is called a point lattice, again an abstract concept. The relation between the unit cell, the lattice axes, and the crystal lattice is depicted in Fig. 2, with a unit cell highlighted. The crystal or space lattice is the coordinate system that is used to define crystal forms (see below). The crystal faces are named according to their relative position to that coordinate system.


 

Crystal System

The angles between the axes of the point lattice and the relative length of the edges of the unit cell are used to define various crystal systems. Every mineral (except those that are amorphous) belongs to one of the 7 crystal systems (or 6, as sometimes the rhombohedral/trigonal and the hexagonal system are grouped together). The following table gives a short overview.

Angles Axes Crystal System
α = β = γ = 90° a = b = c Cubic or Isometric
α = β = γ = 90° a = b , c ≠ a , c ≠ b Tetragonal
α = β = γ = 90° a ≠ b ≠ c Orthorhombic
α = β = γ ≠ 90° a = b = c Trigonal or Rhombohedral
α = β = 90° , γ = 120° a = b , c ≠ a , c ≠ b Hexagonal
α ≠ β ≠ γ , β = 90° a ≠ b ≠ c Monoclinic
α ≠ β ≠ γ ≠ 90° a ≠ b ≠ c Triclinic

crs_terms_fig3.png
Fig. 3
Quartz belongs to the trigonal crystal system [4].
A trigonal unit cell looks like an oblique cube - the lengths of all axes a, b, and c are equal, and the angles in the corresponding corners are equal but not rectangular (Fig. 3).


crs_terms_fig4".png"
Fig. 4
But although quartz belongs to the trigonal system, its unit cell is hexagonal.
Thus when it comes to describing quartz crystals, the terminology used is one for describing hexagonal crystals (Fig. 4):
Four axes are distinguished, a1, a2, a3, and c. All a axes are of equal length and lie in one plane, with 120° angles between them. When referring to all of them or one of them and it doesn't matter which one, people use the symbol a0 or simply a. The c axis runs perpendicular to a0. Its length is different from the a axes, for quartz it's 1.100 times the length of a0.


crs_terms_fig5.png
Fig. 5
Figure 5 depicts the relation between the hexagonal symmetry and the hexagonal unit cell. A hexagonal unit cell is not a hexagonal prism, it is a rhomb.


crs_terms_fig6.png
Fig. 6
A trigonal unit cell can also be defined in hexagonal coordinates - the hexagonal axes are easily projected into the unit cell (Fig. 6). This also works with a perfect cube, and if you look at a cube from the correct angle, you will see a hexagon as its outline.


 

Crystal or Symmetry Class

The term "crystal class" refers to the rotation and inversion symmetry properties of a mineral's crystal lattice. Every mineral (except those that are amorphous) belongs to one of the 32 symmetry classes.
Quartz is member of the class 32, trigonal-trapezohedral - that reads "three - two", not thirty two. "32" is the so called Hermann-Maugin symbol, indicating that quartz has a 3-fold rotational symmetry on one and a two-fold rotational symmetry on another axis.

The notion of mirror symmetry is related to the notion of handedness or enantiomorphy. A unit cell can lack mirror symmetry: in that case, if you mirror the atomic coordinates around an axis, the positions do not match each other. Such a unit cell - and consequently the entire crystal - is either left- or right-handed, similar to our left and right hands.


 

Crystal Form

The term "crystal form" is a bit misleading, because it does not say anything about the actual shape of the crystal as a whole. A crystal form is a virtual geometrical body that is enclosed by lattice planes with identical symmetry properties.

crystal_forms_1.png
Fig. 7
You can think of lattice planes as straight cuts through a grid. Figure 7 shows a two-dimensional view of a sodium chloride lattice[5], with the thin lines being the borders of the unit cells. The orange and yellow lines are lattice planes which enclose a diamond-shaped form. Of course, there are many ways to cut through a grid, and to be a valid crystal form, the planes need to run through lattice points - the corners of unit cells. In addition, only lattice planes with identical symmetry properties are to be used in one form. In a cubic grid, with a cubic unit cell, the most basic form is a cube.


crystal_forms_2.png
Fig. 8
Figure 8 shows a few possible forms in a two-dimensional grid. The rhomb with a dashed outline is not a valid form, because it is not enclosed by lattice planes with identical symmetry properties with respect to the underlying lattice (one plane shows mirror symmetry while the other does not; in addition, one runs through 3 atoms while the other runs through 5).

All faces of a crystal form in a three-dimensional grid have the same shape, they are, for example, all squares or triangles. In certain crystal classes some forms are not "complete" bodies, but open at one or two ends, and are called open forms (e.g. all prisms), as opposed to the complete ones, which are called closed forms, a cube being an example [6].

The actual shape of a crystal can be viewed as a combination of intersecting forms. The shape thus depends both on presence and size of the different crystal forms. Crystals that look very different (e.g. one being elongated and the other short-prismatic) can still show the same crystal forms.

Because of this relation, the term "crystal form" also sometimes refers to the types of  crystal faces that are visible on a crystal. Crystals with different types of crystals present on them are said to have a different form[7].


NaCl_111.jpg
Fig. 9
Which of the many possible forms manifest as faces on a crystal depends on inherent structural details and on environmental conditions during growth. For example, while you could think of a cut through a sodium chloride lattice like the one shown in Fig. 9, which corresponds to the diamond-shaped form in Fig. 8, you will never see such a form on a rock salt crystal in nature. While the crystal grows in a watery solution, such a plane made of negatively charged chlorine ions will strongly attract positively charged sodium ions and will quickly grow into a regular corner of a cube. This crystallographic form is thermodynamically not stable in rock salt. The faces of a cube are electrically balanced and tend to grow and dissolve slower in a watery solution (they are more stable), and accordingly they are the preferred crystal form in rock salt.

A form that is not stable in rock salt may very well be stable in other minerals of the cubic crystal system that have a different internal structure, like fluorite or garnet. Some minerals prefer to wear the same clothes every day, like rock salt, others show an astounding variety of forms and shapes that vary with the growth conditions, like pyrite, fluorite or, with about 200 different forms and more than 1000 known combinations, calcite.

There is a mathematical terminology for describing crystallographic forms, the so called Miller indices for crystal systems with three axes and the extension for crystal systems with four axes, the Miller-Bravais indices. The vector annotation used in this terminology (for example, {1 0 1 1} for the positive rhombohedral form r in quartz) exactly describes the relation of forms to the underlying crystal lattice. An introductory explanation of this awkward nomenclature is in preparation, but meanwhile I have to direct you to other websites and to textbooks of mineralogy to get into this rather complex and abstract topic:

  • English Wikipedia Article on Miller Indices
  • German Wikipedia Article on Miller Indices
  • Crystallography: Miller Indices on www.gly.uga.edu
  • Miller Indices (hkl) on www.chem.qmul.ac.uk
  • Miller Indices on www.uwgb.edu
  • Miller Indices on www.keele.ac.uk
  • 3.2. Richtungen und Ebenen im Gitter on geometrie.diefenbach.at
  • Millersche Indizes on www.mineralienatlas.de


    •  

    Twinning

    twin_types.jpg
    Fig. 10
    A crystal is said to be twinned if it is composed of 2 or more crystal subindividuals whose crystal lattices are oriented differently but nevertheless are intergrown in a law-like, specific manner. The subindividuals are not just put together at an arbitrary angle; they share atoms along their border. For this to be possible the border always corresponds to a specific crystal form: it cuts the crystal lattice of the subindividuals at the same respective angles, and the subindividuals touch each other as if two separate crystals touched each other with identical crystal faces. The virtual plane that corresponds to these crystal faces is called the composition plane.

    With respect to the crystal lattice, two kinds of geometrical operations can be performed to get a twinned crystal:

    So each twinning law is defined by the kind of geometrical operation and the specific crystallographic plane or axis. The twinning axes and planes do not have to coincide with the main crystallographic axes.
    Note that even if the twin has a twinning axis and not a twinning plane, it still has a composition plane.

    One further distinguishes penetration twins and contact twins:

    Contrary to common belief the terms contact twin and penetration twin do not define if subindividuals are macroscopically visible or not.

    In theory, there are almost as many different twinning laws as crystal forms, but in nature only very few can actually be observed in a mineral species. Quartz, for instance, only three types of twins are frequently found.

    Parallelly intergrown crystals might look like twins to most people, but in most cases they are not twins.


     

    Crystal Habit

    "Crystal habit" refers to the overall shape of the crystal, leaving aside its inner structure.
    The crystal habit can be described in obvious terms like "barrel-like", "needle-like", "elongated", or "short-prismatic".
    However, the notion of crystal habit also includes less obvious terms as "trigonal habit" or terms which are specific to individual minerals, like "Dauphiné habit". The habit of a crystal is dependent on the relative sizes of crystal faces. For example, the dominance of certain forms on a quartz crystal might cause a cube-like look that is accordingly called "pseudo-cubic habit".

    Note:
    One can also use the term "habit" to describe the general appearance of mineral aggregates, as opposed to a single crystal. Typical terms in that context are "druzy", "compact", "dendritic", "fibrous", or "granular".  

    Further Information, Literature, Links

    Any textbook on mineralogy will discuss this topic extensively.

    Again, I would recommend the very nice explanation that can be found at this website: Introduction to Crystallography and Mineral Crystal Systems.


    Footnotes

    1 There is no distance between the atoms, but to visualize their geometric relationship, I′ve left some space between them. If one could actually see the atoms, they would perhaps look more like a cloudy sphere with no clear border or surface, the clouds consisting of quickly moving electrons.

    2 Although when René Just Haüy came up with the idea of the unit cell in 1784, he did indeed think of unit cells as fundamental bricks that form a crystal.

    3 Usually 3 axes, but the hexagonal unit cell is an exception, it has 4 axes forming a three-dimensional grid.

    4 Some authors put quartz into the hexagonal crystal system. There is nothing wrong with that, it′s just that they choose to distinguish 6 crystal systems and regard the trigonal system as a special case of the hexagonal system. The unit cell of quartz is not trigonal, but hexagonal.

    5 To demonstrate it in a quartz lattice is very confusing because the structure is very complex and for the same reason beyond my technical possibilities.

    6 A form does not have to be a three-dimensional body, it can be a simple two-dimensional plane.

    7 In German there is an appropriate term for the fact that two crystals show equivalent crystal faces: they are said to have the same "Tracht", which is an old-fashioned expression for clothing. There is probably also a corresponding expression in English that I don′t know.

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