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Crystallography: The Tetragonal System

Last Updated: 16th Nov 2018

By Donald Peck & Alfred Ostrander

Apophyllite: Ditetragonal Dipyramidal Class
Chalcopyrite: Tetragonal Scalenohedral Class
Apophyllite: Ditetragonal Dipyramidal Class
Chalcopyrite: Tetragonal Scalenohedral Class
Apophyllite: Ditetragonal Dipyramidal Class
Chalcopyrite: Tetragonal Scalenohedral Class
Diaboleite: Tetragonal Dipyramidal Class
Wulfenite: Tetragonal Pyramidal Class
Lemanskiite: Tetragonal Trapezohedral Class
Diaboleite: Tetragonal Dipyramidal Class
Wulfenite: Tetragonal Pyramidal Class
Lemanskiite: Tetragonal Trapezohedral Class
Diaboleite: Tetragonal Dipyramidal Class
Wulfenite: Tetragonal Pyramidal Class
Lemanskiite: Tetragonal Trapezohedral Class


The cross-section of minerals normal to the c-axis in the Tetragonal System is usually square, although the corners may be modified. The cross-sectional shape, together with the 4-fold 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 a-axis and a b-axis 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
Hermann-Mauguin Symmetry Symbols

The Unique Symmetry Element of the Tetragonal Crystal System


The single 4-fold axis of rotational symmetry is unique to the tetragonal crystal system.

The only other crystal system with a 4-fold axis is the Isometric System, which often has three of them mutually perpendicular to each other. The Isometric system always has four 3-fold axes of rotational symmetry. Finding only one 4-fold axis and no 3-fold axes assures that one is observing a Tetragonal crystal.

Crystallographic Axes

Tetragonal 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 a1 and a2. The third axis is labelled c and is usually viewed as the vertical axis.

The top end of the c-axis is designated as positive, c+, and the bottom end as negative,c-. One of the a-axes is set horizontally from front to back with a1+ as the front end and a1- as the rear. The a2 axis is horizontal from a2+ on the right to a2- 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 a-axes 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.

Pyramids & Dipyramids
Tetragonal Dipyramid 1st Order
Tetragonal Pyramid 2nd Order Positive
Ditetragonal Dipyramid
Prisms & Diprisms
Tetragonal Prism 1st Order
Tetragonal Prism 2nd Order
Ditetragonal Prism
Trapezohedrons; Disphenoid; & Scalenohedron
Tetragonal Trapezohedron Left
Tetragonal Trapezohedron, Right
Tetragonal Disphenoid
Tetragonal Scalenohedron Positive
Pinacoid & Pedion
Tetragonal Pinacoid
Tetragonal Pedion, basal


1st, 2nd, & 3rd Order Forms in the Tetragonal Crystal System
Prism faces of the 1st order form, {hh0}/{110}, intercept both a-axes (see diagram at right, below). Each face of the 2nd order prism, {h00}/{100}, is perpendicular to and intercepts one a-axis and is parallel to the other. Faces of the 3rd order form, {hk0}/{120} intercept both a-axes, but are neither perpendicular nor parallel to either one. In all cases, the prism faces are parallel to and surround the c-axis. Pyramid faces of the same order are located above, or below, the corresponding faces of prisms.
Pyramids & Dipyramids:
Tetragonal Prisms: Order
. Tetragonal Pyramids: 1st, 2nd and 3rd order: 4 faces. Positive, the faces converge on and meet at the upper end of the c-axis; and negative, the faces converge and meet on the lower end of the c-axis. A 1st order pyramidal face, {hhl}/{111}, intercepts both a-axes at equal distances and the c-axis. A 2nd order face, {h0l}/{101} intercepts one a-axis, the c-axis, and is parallel to the remaining a-axis. 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 c-axis. Those of the negative form converge on the negative end of the c-axis.
. 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 c-axis. An open form (requires other forms to enclose space); 1st order prism {hh0}/{110} intercepts both a-axes equally. The 2nd order prism {h00}/{100) intercepts and is perpendicular to one a-axis and is parallel to the other. The 3rd order prism {hk0}/{120} intercepts both a-axes and is neither parallel to nor perpendicular to either of them (See figure).
. Ditetragonal Prism: 8 Faces, surrounding and parallel to the c-axis; each face of a pair intercepts the two a-axes 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 c-axis, one at the positive end and one at the negative end.
. Pedion: 1 face, basal, intercepts the c-axis, 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 c-axis one usually sees an essentially square cross-section. There are relatively few that show an acicular habit, although roughly 10% form radiating groups.

Crystal Classes:

Ditetragonal Dipyramidal Class:



Hermann-Mauguin Symbol:
4/m 2/m 2/m . . . . 4-fold principal axis of rotation; perpendicular to two 2-fold axes that are each perpendicular to mirror planes; and two 2/fold axes in the alternate positions (bisecting the angle between the a-axes) that are each perpendicular to mirror planes.

Symmetry Elements:
1A4, 4A2, 5P, C . . . . 1 4-fold axis (the c-axis); 4 2-fold axes (the a-axes and axes in the same plane but bisecting the angles between the a-axes); 5 mirror planes (1 horizontal; 2 vertical and coincident with the a-axes and c-axis, and 2 coincident with the alternate/diagonal axes and c-axis; 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 cross-section. 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 re-entrant 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 c-axis horizontally, left to right, and spinning the crystal vertically.
Cassiterite: Ditetragonal Dipyramidal Class

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{110}, {111}, modified
Bernhardi, 1809; Presl, 1837. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923 ('Zinnerz').
.
..Symmetry:
....1 4-fold axis of rotational symmetry
....4 2-fold axes of rotational symmetry (all perpendicular to the 4-fold axis)
....4 vertical mirror planes (each contains the c-axis, and 1 a-xis or alternate axis.)
....1 horizontal mirror plane (in the plane of the a-axes)
....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, Meta-autunite, Rutile, Uraninite, Zircon

Tetragonal Trapezohedral Class:


Hermann-Mauguin Symbol:
422 . . . principal axis is a 4-fold axis of rotational symmetry, two horizontal axes of 2-fold symmetry (the a-axes); and two horizontal axes of 2-fold symmetry in the alternate positions (bisecting the angles between the a-axes).

Symmetry Elements:1A4,4A2 . . . 1 4-fold axis of symmetry, 4 2-fold 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 4-fold axis of rotation and the essentially square cross-section. 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.
Mellite: Tetragonal Trapezohedra Class

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{101}, {110}, {001}
Locality: Artern, Thüringen, Germany
Haüy, 1801, 1823, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923 ('Mellit').
.
..Symmetry:
....1 4-fold axis of rotation
....4 2-fold 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.

Hermann-Mauguin Symbol:
4 . . . . 4-fold axis of rotational symmetry as the principal axis

Symmetry Elements:1A4 . . . .1 4-fold axis (c-axis)

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 c-dimension.

Hermann-Mauguin Symbol:
_4 2 m . . . .4-fold axis of rotatory inversion as the principal axis; two 2-fold axes coincident with the a-axes; 2 mirror planes in the alternate positions (vertical between the a-axes)

Symmetry Elements:
3A2 2P . . . .3 2-fold axes (the 4 axis appears to be a 2-fold axis); 2 vertical mirror planes bisecting the angles between the a-axes.

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
Chalcopyrite: Tetragonal Scalenohedral Class

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{112} modified by {11-2}
Naumann, 1828, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923 ('Kupferkies').

..Symmetry:
....3 2-fold axes (a, b, & c), the c-axis is also a 4-fold rotatory inversion axes (rotate 90o to the right and invert counter clockwise, 4 times)
....2 mirror planes, coincident with the c-axis and diagonal to the a-axis and b-axis
..Forms:
....Tetragonal disphenoid: positive, 4 faces, {111}
....Tetragonal disphenoid: negative, 4 faces, {112}

Chalcopyrite: Tetragonal Scalenohedral Class

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{11-2} + {112}, modified
Locality: Wurtzborough, Sullivan County, USA (Beck)
Dana, 1837; Beck, 1842. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923 ('Kupferkies').

.. 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.

Hermann-Mauguin Symbol:
4/m . . . . A 4-fold axis of rotational symmetry, as the principal axis, perpendicular to a mirror plane.

Symmetry Elements:
1A4 1P C . . . . 1 4-fold axis of symmetry (c-axis), 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 cross-section and dipyramids are often easily recognized.

Problems:
The 3rd order dipyramid is usually missing.

Model: Wulfenite
Wulfenite: Tetragonal Dipyramidal Class

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Tabular {001}
Locality: Red Cloud Mine, Yuma County, Arizona
Dana, 1892. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
..Symmetry:
....4-fold 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
Scheelite: Tetragonal Dipyramidal Class

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{101}, modified
Locality: Schlaggenwald
Mohs, 1824, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
..Symmetry:
....4-fold 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.

Hermann-Mauguin Symbol:
4 . . . . principal axis is a 4-fold axis of rotational symmetry

Symmetry Elements:1A4 . . . . 1 4-fold axis of rotational symmetry (c-axis). 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.

Hermann-Mauguin Symbol:
4 . . . . A 4-fold rotatory inversion axis as the principal axis.

Symmetry Elements:
1A2 . . . . A 2-fold axis of rotational symmetry (while it is a 4-fold inversion axis, it is more recognizable as a 2-fold 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
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

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