Crystallography: The Tetragonal System
Last Updated: 16th Nov 2018By Donald Peck & Alfred Ostrander
The crosssection of minerals normal to the caxis in the Tetragonal System is usually square, although the corners may be modified. The crosssectional shape, together with the 4fold axis of rotational symmetry make minerals reasonably easy to identify with the system. There are of course some problems. One of them occurs when an Orthorhombic crystal has an aaxis and a baxis that are very nearly equal in length. Another involves a slightly misshapen Isometric crystal that appears to be longer in one dimension than the other two. Both problems require critical observation in order to arrive at a correct solution, but the most common problem probably involves chalcopyrite. The crystal appears to be Isometric Tetrahedral . . . in reality one of the three axes is slightly longer, and the crystal is Tetragonal Scalenohedral.
There are some 300 minerals that crystallize in the Tetragonal System (2018). Approximately 175, more than half, are classified in the Ditetragonal Dipyramidal Class. About 40 minerals crystallize in the Tetragonal Scalenohedral Class; and fewer than to 25 in each of the other classes.
You may find it helpful to review the following articles before attempting this one:
Determining Symmetry of Crystals: An Introduction
Miller Indices
HermannMauguin Symmetry Symbols
Determining Symmetry of Crystals: An Introduction
Miller Indices
HermannMauguin Symmetry Symbols
The Unique Symmetry Element of the Tetragonal Crystal System
The single 4fold axis of rotational symmetry is unique to the tetragonal crystal system.
The only other crystal system with a 4fold axis is the Isometric System, which often has three of them mutually perpendicular to each other. The Isometric system always has four 3fold axes of rotational symmetry. Finding only one 4fold axis and no 3fold axes assures that one is observing a Tetragonal crystal.
Crystallographic Axes
The symmetry of the Tetragonal System is applied to a set of 3 axes with two of the axes equal in length and the third axis either longer or shorter than the other two. The three axes are mutually perpendicular to each other. The two axes, equal in length are labelled a_{1} and a_{2}. The third axis is labelled c and is usually viewed as the vertical axis.
The top end of the caxis is designated as positive, c+, and the bottom end as negative,c. One of the aaxes is set horizontally from front to back with a_{1}+ as the front end and a_{1} as the rear. The a2 axis is horizontal from a_{2}+ on the right to a_{2} on the left. Note the conventions on the Axial Diagram.
General and Special Forms
There are two types of forms, general forms and special forms.
Any form which is not a general form is a special form. Most often, the general form is the form for which the crystal class is named. It usually appears only in that crystallographic class or in classes of higher symmetry. A general form has the maximum number of faces of any form in its crystal class. Special forms may appear in any crystal class of the system.
The general form intercepts all axes, each at a different distance. Since the two aaxes are the same length in the Tetragonal System, a given face of the general form must intercept them at different fractions of the unit length (i.e. the Miller Index for each of the first two intercepts must have different values). The c axis, having a different unit length, may have an index of any value other than zero (0). The form is not always expressed on the crystal.
Forms in the Tetragonal System: Click any image to enlarge it.  

1st, 2nd, & 3rd Order Forms in the Tetragonal Crystal System
Prism faces of the 1st order form, {hh0}/{110}, intercept both aaxes (see diagram at right, below). Each face of the 2nd order prism, {h00}/{100}, is perpendicular to and intercepts one aaxis and is parallel to the other. Faces of the 3rd order form, {hk0}/{120} intercept both aaxes, but are neither perpendicular nor parallel to either one. In all cases, the prism faces are parallel to and surround the caxis. Pyramid faces of the same order are located above, or below, the corresponding faces of prisms.
Pyramids & Dipyramids:
^{. } Tetragonal Pyramids: 1st, 2nd and 3rd order: 4 faces. Positive, the faces converge on and meet at the upper end of the caxis; and negative, the faces converge and meet on the lower end of the caxis. A 1st order pyramidal face, {hhl}/{111}, intercepts both aaxes at equal distances and the caxis. A 2nd order face, {h0l}/{101} intercepts one aaxis, the caxis, and is parallel to the remaining aaxis. A 3rd order pyramidal face, {hkl}/{121}, intercepts all three axes, each at a different distance. Faces, ideally, are isosceles triangles. 3rd order faces are small and rarely seen,
^{. } Ditetragonal Pyramids: 4 faces. There are two of them. Four faces of the positive form converge and meet on the positive end of the caxis. Those of the negative form converge on the negative end of the caxis.
^{. } Tetragonal dipyramids: 8 faces. 1st, 2nd, and 3rd order. The dipyramids have the same general symbols as the pyramids.
^{. } Ditetragonal dipyamids: 16 faces. Eight faces on the upper half and eight on the lower half. Ideally, isosceles triangles. The symbol is {hkl}/{211}.
Prisms & Diprisms:
^{. } Tetragonal Prisms: 4 faces; parallel to and enclosing the caxis. An open form (requires other forms to enclose space); 1st order prism {hh0}/{110} intercepts both aaxes equally. The 2nd order prism {h00}/{100) intercepts and is perpendicular to one aaxis and is parallel to the other. The 3rd order prism {hk0}/{120} intercepts both aaxes and is neither parallel to nor perpendicular to either of them (See figure).
^{. }Ditetragonal Prism: 8 Faces, surrounding and parallel to the caxis; each face of a pair intercepts the two aaxes at unequal distances. The symbol for the form is {hk0}; {120}, {130}, {210}, {310}, etc. .
Trapezohedrons:
^{. }Tetragonal Trapezohedron: 8 faces, closed form (the form can enclose space); Left and Right forms result in enantiomorphism (left and right handed crystals).. The trapezohedron has eight 4 sided faces, each with edges not parallel to the others.
Disphenoid & Scalenohedron
^{. }Tetragonal Disphenoid: 4 faces {hhh}/{111}. Resembles a tetrahedron, but has one longer axis. There is a positive and a negative form. When positive and negative forms appear on the same crystal, they generally are different in size.
^{. }Tetragonal Scalenohedron: 8 Faces, each is a scalene triangle. A closed form, but not found itself as a crystal. It is found modifying chalcopyrite crystals. The symbol is {hkl}/{322}. There is a positive and a negative form.
Pinacoid & Pedion: ^{. }Pinacoid: 2 faces. {001}, perpendicular to the caxis, one at the positive end and one at the negative end.
^{. }Pedion: 1 face, basal, intercepts the caxis, usually at its lower end. The symbol for the form is {001}
General Morphology:
Of the 300 or so tetragonal minerals, somewhat more than half are prismatic/pyramidal (33%), equant (16%), or tabular (28%). In these cases, when viewing along the caxis one usually sees an essentially square crosssection. There are relatively few that show an acicular habit, although roughly 10% form radiating groups.
Crystal Classes:
Ditetragonal Dipyramidal Class:
HermannMauguin Symbol:
4/m 2/m 2/m . . . . 4fold principal axis of rotation; perpendicular to two 2fold axes that are each perpendicular to mirror planes; and two 2/fold axes in the alternate positions (bisecting the angle between the aaxes) that are each perpendicular to mirror planes.
Symmetry Elements:
1A_{4}, 4A_{2}, 5P, C . . . . 1 4fold axis (the caxis); 4 2fold axes (the aaxes and axes in the same plane but bisecting the angles between the aaxes); 5 mirror planes (1 horizontal; 2 vertical and coincident with the aaxes and caxis, and 2 coincident with the alternate/diagonal axes and caxis; a center of symmetry.
General Form:
Ditetragonal dipyramid: 16 faces {hkl}/212 Most common on zircons.
Special Forms:
Tetragonal prisms: 1st and 2nd order
Ditetragonal prism
Tetragonal dipyramids: 1st and 2nd order
Pinacoid: basal
Look For:
Essentially square crosssection. Most often prismatic and dipyramidal, but may be tabular.
Problems:
There may be no lower pyramid due to the crystal forming on matrix (look for a possible reentrant face at the base).
Model: Cassiterite
The Miller Indices, when shown on this model, are very close together. They are best observed by orienting the caxis horizontally, left to right, and spinning the crystal vertically.
..Symmetry:
....1 4fold axis of rotational symmetry
....4 2fold axes of rotational symmetry (all perpendicular to the 4fold axis)
....4 vertical mirror planes (each contains the caxis, and 1 axis or alternate axis.)
....1 horizontal mirror plane (in the plane of the aaxes)
....Center of symmetry
..Forms:
....1 1st order prism {110}
....1 2nd order prism {100}
....2 ditetragonal prisms {210} {320}
....1 1st order dipyramid {111}
....1 2nd order dipyramid {101}
Representative Minerals: Click to see Mineral Page
Anatase, Apophyllite. Cassiterite, Metaautunite, Rutile, Uraninite, Zircon
Tetragonal Trapezohedral Class:
HermannMauguin Symbol:
422 . . . principal axis is a 4fold axis of rotational symmetry, two horizontal axes of 2fold symmetry (the aaxes); and two horizontal axes of 2fold symmetry in the alternate positions (bisecting the angles between the aaxes).
Symmetry Elements:1A_{4},4A_{2} . . . 1 4fold axis of symmetry, 4 2fold axes of symmetry, no mirror planes, no center of symmetry
General Form:
Tetragonal Trapezohedron: 8 faces.
Special Forms:
Tetragonal Prisms: 1st and 2nd order
Ditetragonal Prism
Tetragonal Dipyramids: 1st and 2nd order
Pinacoid: basal
Tetragonal Trapezohedron: Right or Left; Very seldom seen
Problems:
Look for the 4fold axis of rotation and the essentially square crosssection. But, without the 3rd order pyramid it is virtually impossible to nail down this crystal class.
Model: Mellite
The general form, tetragonal trapezohedron, is not shown on this model and is very rarely seen on mineral crystals. Thus this crystal class usually cannot be distinguisched from the Tetragonal Dipyramidal Class.
..Symmetry:
....1 4fold axis of rotation
....4 2fold axes of rotation
Because the trapezohedral form is not present, this model falsely appears to have a center of symmetry and 5 mirror planes. The model is typical of the mineral species, and deceptively appears to belong to the Tetragonal Dipyramidal Class.
..Forms:
....Prism: 1st order, {110}
....Pyramid: 2nd order, {101}
....Pinacoid: {001)
Representative Minerals: Click to see Mineral Page
Ekanite, Lemanskiite, Mellite, Wardite
Tetragonal Pyramidal Class:
This class is hemimorphic (forms on the upper termination differ from those on the lower termination) and mineralogically is rather unimportant. It is comprised of only about fifteen minerals, all microscopic and fairly rare.
HermannMauguin Symbol:
4 . . . . 4fold axis of rotational symmetry as the principal axis
Symmetry Elements:1A_{4} . . . .1 4fold axis (caxis)
General Form:
The Tetragonal Pyramid: 3rd order {hkl}
Tetragonal Scalenohedral Class:
The Tetragonal Scalenohedral Class holds (2018) about 40 minerals, the most important of which is chalcopyrite. The axial ratio of chalcopyrite is very close to that of two stacked cubes. This results in a crystal that closely resembles an isometric tetrahedron, but is very slightly longer in the cdimension.
HermannMauguin Symbol:
_4 2 m . . . .4fold axis of rotatory inversion as the principal axis; two 2fold axes coincident with the aaxes; 2 mirror planes in the alternate positions (vertical between the aaxes)
Symmetry Elements:
3A_{2} 2P . . . .3 2fold axes (the 4 axis appears to be a 2fold axis); 2 vertical mirror planes bisecting the angles between the aaxes.
General Form:
The Tetragonal Scalenohedron: Very rarely, it may be observed as modifying the chalcopyrite crystal.
Special Forms:
Tetragonal disphenoids: positive and negative
Tetagonal dipyramid: 2nd order
Pinacoid: basal
Look For:
With chalcopyrite, look for the generally wedge shaped disphenoid, or the apparently octahedral combination of the positive and negative disphenoids. (see models below)
Model: Chalcopyrite
..Symmetry:
....3 2fold axes (a, b, & c), the caxis is also a 4fold rotatory inversion axes (rotate 90^{o} to the right and invert counter clockwise, 4 times)
....2 mirror planes, coincident with the caxis and diagonal to the aaxis and baxis
..Forms:
....Tetragonal disphenoid: positive, 4 faces, {111}
....Tetragonal disphenoid: negative, 4 faces, {112}
.. Forms
....Tetragonal disphenoid: positive, 4 faces, {112}
....Tetragonal disphenoid: negative, 4 faces, {112}
....Tetragonal dipyramid: 2nd order, 8 faces, {201}
....Pinacoid: 2 faces, basal, {001}
Representative Minerals: Click to see Mineral Page
Åkermanite, Chalcopyrite, Hardystonite, Stannite
Tetragonal Dipyramidal:
There are about 60 minerals that crystallize in the Tetragonal Dipyramidal Class.
HermannMauguin Symbol:
4/m . . . . A 4fold axis of rotational symmetry, as the principal axis, perpendicular to a mirror plane.
Symmetry Elements:
1A_{4} 1P C . . . . 1 4fold axis of symmetry (caxis), 1 mirror plane (horizontal), center of symmetry.
General Form:
Tetragonal Dipyramid, 3rd Order: the 3rd order form is often known simply by its symbol,{hkl}/{121}. This form is rarely seen, making the class usually impossible to differentiate from the ditetragonal dipyramidal class.
Special Forms:
Tetragonal dipyramids: 1st and 2nd Order
Tetragonal prisms: 1st and 2nd Order
Pinacoid: basal
Look For:
The square crosssection and dipyramids are often easily recognized.
Problems:
The 3rd order dipyramid is usually missing.
Model: Wulfenite
....4fold axis of rotational symmetry
....1 horizontal mirror plane
....Center of symmetry
..Forms:
....Tetragonal pyramids: 2 1st order; 8 faces each, {112} and {114}
....Tetragonal pyramids: 2 2nd order; 8 faces each, {011} and {013}
....Pinacoid: 2 faces, basal
Model: Scheelite
....4fold axis of rotational symmetry
....1 horizontal mirror plane
.. Forms
....Tetragonal pyramid: 1st order, 8 faces, {112}
....Tetragonal pyramid: 2nd order; 8 faces each, {101}
....Tetragonal pyramids: 3rd order; 8 faces each, {123} and {211}
Representative Minerals: Click to see Mineral Page
Leucite, Powellite, Scapolite, Scheelite, Wulfenite
Tetragonal Pyramidal:
There are fewer than a half dozen minerals classified in this class.
HermannMauguin Symbol:
4 . . . . principal axis is a 4fold axis of rotational symmetry
Symmetry Elements:1A_{4} . . . . 1 4fold axis of rotational symmetry (caxis). There are no mirror planes and no center of symmetry. The class is hemimorphic.
General Form:
Tetragonal pyramid, 3rd order; 4 faces {hkl}
Special Forms:
Look For:
Crystals, when they occur, are often prismatic, blocky, and pyramidal. Sometimes acicular.
Problems:
Hemimorphism is only occasionally visible.
Representative Minerals: Click to see Mineral Page
Pinnoite, Piypite, Percleveite(Ce)
Tetragonal Disphenoidal:
There are fewer than 10 minerals in this class, all of them rare. The best known are Cahnite and Tugtupite.
HermannMauguin Symbol:
4 . . . . A 4fold rotatory inversion axis as the principal axis.
Symmetry Elements:
1A_{2} . . . . A 2fold axis of rotational symmetry (while it is a 4fold inversion axis, it is more recognizable as a 2fold axis of rotational symmetry).
General Form:
The Tetragonal disphenoid: 4 faces, 3rd order. There are four of them: Right positive, Left positive, Right negative, and Left negative (but you don't want to know that.)
Special Forms:
Pinacoid
Pedion
Representative Minerals: Click to see Mineral Page
Cahnite, Kesterite, Tugtupite
Links to the Crystallography of Each of the Systems.
Crystallography: The Triclinic System
Crystallography: The Monoclinic System
Crystallography: The Orthohombic System
Crystallography: The Trigonal System
Crystallography: The Hexagonal System
Crystallography: The Tetragonal System
Crystallography: The Isometric System
References:
Mason, Brian and Berry, L.G. (1968) Elements of Mineralogy. W. H. Freeman and Company, San Francisco.
Dana,Edward Salisbury; Foord, William E. (editor); A Textbook of Mineralogy. John Wiley & Sons, Inc., New York
Smith, Jennie R. (1991) Understanding Crystallography. The Rochester Mineralogical Symposium.
Sinkankas, John: Mineralogy: A First Course. A great book with which to start.
Peck, Donald B. (2007) Mineral Identification: A Practical Guide for the Amateur Mineralogist. Mineralogical Record, Tucson, Arizona.
A.E.H Tutton,Crystallography and Practical Crystal Measurement Volume 1 Form and Structure; 2018. An incredibly thorough text. Not for the beginner.
Klein, Cornelis & Hurlbut, Cornelius S., Jr.: Manual of Mineralogy after J. D. Dana;20th Edition
http://www.minsocam.org/ammin/AM20/AM20_838.pdf: Rogers, Austin F. (1935) A historical discussion of the names of crystal forms.
http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm: A good explanation and depiction of crytallographic forms.
https://en.wikipedia.org/wiki/Monoclinic_crystal_system , Short explanation of lattices, space groups, hemimorphic & enantiomorphic structure.
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Comments
Don
Another great article and one of my favourite minerals is happy (Cassiterite)
Thank you
Keith Compton
8th Nov 2018 10:36pm
Another great article and one of my favourite minerals is happy (Cassiterite)
Thank you
Keith Compton
8th Nov 2018 10:36pm
Thank you , Keith. Al and I have enjoyed working on this series. It has forced us to review what we thought we knew and I believe the process has served to clarify both of our understandings.
Donald B Peck
14th Nov 2018 2:47am
Donald B Peck
14th Nov 2018 2:47am
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