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Crystallography: The Trigonal System

Last Updated: 11th Oct 2018

By Donald Peck & Alfred Ostrander

According to the International Union for Crystallography
Calcite: Hexagonal Scalenoheral Class
Quartz: Trapezohedral Class
Calcite: Hexagonal Scalenoheral Class
Quartz: Trapezohedral Class
Calcite: Hexagonal Scalenoheral Class
Quartz: Trapezohedral Class
Elbaite: Ditrigonal Pyramidal Class
Willemite: Rhombohedral Class
Parisite-(Ce): Trigonal Pyramidal Class
Elbaite: Ditrigonal Pyramidal Class
Willemite: Rhombohedral Class
Parisite-(Ce): Trigonal Pyramidal Class
Elbaite: Ditrigonal Pyramidal Class
Willemite: Rhombohedral Class
Parisite-(Ce): Trigonal Pyramidal Class


Traditionally, the Trigonal Crystallographic System was considered to be a division of the Hexagonal System. However, in recent years, crystallographers have considered it to be a separate system. It is a growing movement, but still debated. The reasons for separation include the separate sets of unit cells for hexagonal and trigonal minerals and the unique 3-fold principal axis of rotation for trigonal minerals versus the 6-fold axis for hexagonal minerals.

As of 2018, there are a few more than 500 minerals that crystallize in the trigonal system. That is almost 10% of all minerals. But more importantly, it includes two of the most common minerals in the earth's crust, quartz and calcite. Two hundred eight of the species occur in the hexagonal scalenohedral class, 112 in the ditrigonal pyramidal class, and nearly 80 in the rhombohedral class. The trapezohedral and pyramidal classes have fewer than 50 each.

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

Unique Symmetry of the Trigonal System


The trigonal system is unique, with a 3-fold axis of rotational symmetry or a 3-fold axis of rotatory inversion as its principal axis. A 6-fold inversion axis in the hexagonal system is easily mistaken for the trigonal 3-fold axis. So too, an isometric 3-fold axis, when only the corner of a cube is visible, can be misinterpreted as trigonal when in fact it is part of isometric symmetry.

Crystallographic Axes

Trigonal System Crystallographic System

The triclinic system is usually thought of as a 4-axis system. Three of the axes; commonly labeled a1, a2, and a3; are equal in length and lie in a single horizontal plane that is perpendicular to the principal c axis. The angle between the positive ends of the three a axes, is 120o. The c axis may be either longer or shorter than an a axis.

Take careful notice of the manner in which the a axes are designated. Positive and negative ends of the axes alternate around the c axis. The angle between the positive end of the a1+ and the positive end of the a2+ is 120o, not 60o.

Miller Indices

Miller Indices for the trigonal and hexagonal system customarily have four digits indicating intercepts on the a1, a2, a3, and c axes, respectively. In the four axis system crystallographers increasingly are using a three digit index, dropping the third, a3, index. In any case, the a3 index is redundant. It is the negative sum of the a1 and a2 indices. Given four digit indices such as (1,2,1,0), the resulting 3 digit indices are (1,2,0) [i.e. -( 1+ (-2) ) = 1 ]. Both give us the same information but for beginning crystallographers the four digit system is easier with which to work.

General & 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 all three a axes are the same length, 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 three 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 Trigonal Crystal System (Clck an image to enlarge it)

Pyramids and Dipyramids
Trigonal Dipyramid: 1st Order
Trigonal Dipyramid 2nd Order
Ditrigonal Pyramid 1st Order
Ditrigonal Dipyramid, 1st Order
Hexagonal Pyramid: 2nd Order
Prisms & Diprisms
Trigonal Prism: 1st Order
Trigonal Prism: 2nd Order
Ditrigonal Prism
Hexagonal Prism 1st Order
Hexagonal Prism 2nd Order
Scalenohedrons Rhombohedrons & Trapezohedrons
Trigonal Scalenohedron Positive
Trigonal Scalenohedron Negative
Rhombohedron: Positive
Rhombohedron: Negative
Trigonal Trapezohedron: Left Positive
Pinacoids & Pedions
Hexagonal Pinacoid (001)
Hexagonal Pedion


Pyramids in the Trigonal System: Pyramids consist of 3, 6, or 12 faces, all converging on and meeting at a point on the c axis. Dipyramids have twice the number of faces, half meeting at a point on the positive c axis and half at a point on the negative c axis. Pyramids are an open form, dipyramids are a closed form; and a pyramid is equivalent to half a dipyramid. Pyramids are limited to hemimorphic classes.
Prisms in the Trigonal System: Prisms are open forms with vertical, parallel edges. Trigonal prisms have 3 faces while trigonal diprisms exhibit 6 faces. Hexagonal prisms show 6 faces and hexagonal diprisms show 12. Prisms of the 1st order are oriented with the a axes emerging at the edge of the vertical faces. Those of the 2nd order emerge from the center of the vertical faces. 2nd order prisms are the result of a rotation of the 1st order prism on the crystal axes. It is not possible to differentiate 1st from 2nd order unless other forms on the crystal indicate the position of the a axes.
Scalenohedrons: The scalenohedron in the trigonal system is comprised of 12 faces. Each face is a scalene triangle. It is a closed form with 6 triangles on the upper half and 6 on the lower half. The "belt line" is a zigzag line around the crystal. Multiple scalenohedrons, with different inclinations to the c axis, can exist on the same crystal. There are positive and negative scalenohedrons. With the a1+ axis pointing toward the left shoulder of the observer, the base of the pair upper of scalene triangles facing the observer point downward. For the negative form they point upward. If there is no indication of where the a1 axis is located, the two forms cannot be differentiated.
Rhombohedrons The rhombohedron is a six sided solid with each side being a rhomb (or parallelogram). The body diagonal through two opposite points is either longer or shorter than the other three. This longer of shorter diagonal is the c axis. There are 3 facees on the upper half and 3 on the lower half. Like the scalenohedron, the "belt line" is zigzag. And there are positive and negative forms. With the axes in their standard positions, the positive form has an upper rhomb facing the observer; for the negative form it is a lower rhomb. Otherwise the positive and negative forms are geometrically identical. For every scalenohedron there is a comparable rhombohedron, and they may be found on the same crystal. Cleavage directions in trigonal minerals, notably calcite, often are parallel to the rhombohedral form.
Trapezohedrons: The trapezohedron is a closed form with six trapezoidal faces. There are four forms: Right positive, Right negative, Left positive, and Left negative. It is found on trigonal enantiomorphic crystals, especially quartz.
Pinacoids and Pedions: A pinacoid is a set of two parallel faces. The only pinacoids on trigonal crystals are c pinacoids (001), perpendicular to the c axis. A pedion is a single face and on crystals of the trigonal system is always basal, perpendicular to the negative end of the c axis.

A Note on Prism and Pyramid Order in the Trigonal System
There are 1st, 2nd, and 3rd order hexagonal prisms and pyramids in the trigonal system.
..1st order prism: each face intercepts two of the a-axes at unit distances and is parallel to the third a-axis.
..2nd order prism: each faces intercept all three a-axes and is perpendicular to one of them.
..3rd order prism: faces intercept all three a-axes at different distances.
..The corresponding pyramid faces are directly above the prism faces (1.e. the 1st order hexagonal pyramid face (1011) would be directly above prism face {1010).)

General Morphology


Trigonal mineral crystals usually show a hexagonal or a triangular cross section. Quartz and calcite, the two most common minerals, most often show the hexagonal cross section, with quartz showing a prismatic habit while calcite is often scalenohedral or rhombic. Some minerals in the trigonal system have either an equant or a tabular habit. The equant habit is often rhombic and will show the 3-fold symmetry when observed along c-axis, which is either the shortest or longest body diagonal. Tabular crystals may be triangular, or when hexagonal, have alternate edges with the same modifying faces. Observing the 3-fold symmetry of the trigonal classes will differentiate the trigonal system from the 6-fold symmetry of the hexagonal system.

The Trigonal Crystal Classes



There are five crystallographic classes in the Trigonal System. The Hexagonal Scalenohedral class holds approximately 200 mineral species. The Trapezohedral Class has about 36 minerals, the Ditrigonal Pyramidal Class about 112, the Rhombohedral Class about 80, and the Pyramidal Class around 40.

Hexagonal Scalenohedral Class: The Holohedral Class


This is the holohedral class, that is, it has the highest symmetry of the trigonal classes.

Hermann-Mauguin Symbol
3 2/m The principal axis is a three-fold inversion axis perpendicular to 3 two-fold axes, each of which is perpendicular to a (vertical) mirror plane.

Symmetry Elements
1A3 3A2 3P C (1 3-fold principal axis perpendicular to 3 2-fold secondary axes, each perpendicular to a mirror plane. And, a center of symmetry)

The General Form
The general form is the scalenohedron. A crystal may show both positive and negative scalenohedrons. Further, there may be multiple examples having different inclinations to the axes.

Special Forms
Rhombohedrons: positive,6 Faces; negative; 6 Faces
Hexagonal prism: 1st order: 6 Faces; 2nd order: 6 Faces
Trigonal prism: 3 faces
Dihexagonal prism: 12 Faces
Hexagonal dipyramid: 12 Faces
Trigonal dipyramid: 6 Faces
Trigonal trapezohedron: 6 Faces
Basal pinacoid: 2 Faces
Pedion: 1 Face

To Look For
Look for a triangular geometry in the termination of the crystal and a trigonal or hexagonal cross section to the crystal. If the crystal is tabular and hexagonal in outline, inspect alternate edges of the hexagon for similar faces, showing the trigonal structure. The 3-fold principle axis is usually discernible. Likewise, the 3 vertical mirror planes are easily seen from either a termination or side view of the crystal. Scalenohedrons and/or rhombohedrons with different inclinations to the c axis may be found as a zig-zag belt around the crystal.

Problems
The lower termination often in matrix is not visible. Thus the three 2-fold axes can not be see.


Smithsonite: Trigonal; Scalenohedral Class

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{401}, modified
Locality: Rezbnya, Hungary
Mohs, 1824, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
.
Symmetry
3-fold rotational axis: 1; c-axis
2-fold rotational axes: 3; midway between a-axes
Center: midway along any line between similar points on opposite sides of the crystal.
Faces
Recall that the third digit of the Miller Indices in the trigonal system is the negative sum of the first two digits and is dropped in the 3 index format displayed on the model.
Rhombohedron: {401} or {4041}; 6 faces (To be a scalenohedron, the symmetry requires 12 faces)
Trigonal dipyramid: {101} or {1011} ; 6 faces
Basal Pinacoid: {001} or {0001} ; 2 faces

Representative Minerals: Brucite, Calcite, Hematite, Rhodochrosite, Siderite, Smithsonite

Trigonal Trapezohedral Class


The trapezohedral class is enantiomorphic. That is, the crystals are either left handed or right handed.

Hermann-Mauguin Symbol
32

Symmetry Elements1A3 3A2 (1 3-fold principal axis of symmetry perpendicular to 3 2-fold axes of symmetry. There are no mirror planes and no center of symmetry)

The General Form
The general form is the trigonal trapezohedron, most often seen on enantiomorphic quartz crystals.

Special Forms
Trigonal dipyramids: 6 Faces
Trigonal prisms: 3 Faces
Rhombohedrons: 6 Faces
Ditrigonal prisms: 6 Faces (rarely seen)
Hexagonal prisms: 6 Faces
Pinacoid: 2 Faces (rarely seen)

To Look For
Of the 32 minerals that crystallize in this class, only quartz and cinnabar are at all common. The termination of quartz crystals usually show the 3 faces of the positive rhombohedron and 3 of the negative rhombohedron. The positive rhombohedral faces (usually larger) alternate with the negative faces. There may be a small, usually rhombic face of the trigonal pyramid at the lower corner of the negative rhombohedron, and diagonally below it, a usually larger trapezoidal face of the trigonal trapezohedron. If the trapezohedral face (and the pyramidal face if present) are at the upper right corner of the hexagonal prism face that is immediately below it, the quartz crystal is right handed. If it is at the upper left hand corner, the crystal is left handed. (See the excellent Mindat page: Quartz, for more information.)

Problems
Some quartz crystals do not show the difference between the positive and negative rhombohedral forms, nor the trigonal pyramidal or the trapezohedral form. However, quartz is enantiomorphic (left or right handed) and often twinned.

Quartz: Trigonal; Trapezoidal Class

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Right-handed Dauphiné twin (mimetic twin on [001])
Weiß, 1816. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
.

Symmetry:
This is a twinned crystal and as such symmetry elements appear to be doubled.
There are two parallel c axes, that appear to be one 6-fold axis.
There appear to be 6 2-fold axes instead of 3.
Forms:
Prisms: twinned {0110} and {1010}; there may be an indistinct etch line down the middle or a mismatch in striations across the face (Neither is shown on the model).
Rhombohedrons: both positive and negative(twinned) are parallel and coincident. They appear as single faces on a Dauphiné twin. There may be an indistinct wavy etch line down the middle of the face (not shown).
Trapezohedron: Positive right {5161}; above and right of prism face.
Pyramids: Positive trigonal pyramid {{1121},(small parallelogram); above and to the right of the prism face, with the trapezohedron below it.
Notice that turning this crystal to any position shows the trapezohedron on (clipping) the top, right corner of the prism face below it. Thus it is a right-handed crystal.

Representative Minerals: Calciohilairite, Cinnabar, Quartz, Tincalconite

Ditrigonal Pyramidal Class

Hermann-Mauguin Symbol

3m
Symmetry Elements

1A3 3P (1 3-fold axis of symmetry, parallel to and coincident with 3 mirror planes. There is no horizontal mirror plane and no center of symmetry; thus the class is hemimorphic.)

The General Form
Ditrigonal pyramid.

Special Forms
The forms of this class match those of the scalenohedral class, except they have half as many faces.
Pyramids: Trigonal dipyramids- 2 top (Positive & Negative) and 2 bottom (Positive & Negative), Trigonal pyramids- 2 top (Pos & Neg) and 2 bottom (Pos & Neg), Hexagonal prism 2nd Order, Hexagonal pyramid 2nd Order.
Prisms: Ditrigonal prisms, Trigonal prisms- Pos. & Neg.
Pedion.

To Look For
Hemimorphism. The upper termination is usually distinctly trigonal and different from the lower termination. On tourmaline the trigonal pyramid(s) is most often distinctive and the triangular cross section diagnostic. Tourmaline often shows multiple prisms producing an almost striated appearance.
Problems
Most commonly, the lower termination cannot be observed.
Tourmaline: Ditrigonal Pyramidal Class

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{100}, {110}, {011}, {-101}, modified
Locality: Madagascar; Greenland
Haüy, 1801, and others ('Turmalin'). In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
.
Symmetry:
1 3-fold axis: The principal, or c, axis.
3 mirror planes: vertical, coincident with the c-axis,bisecting the angles between the a-axes.
Forms:
Rhombohedrons: Positive- {101} or {1011}, 3 Faces top; Negative- (101} or {1011}, 3 faces bottom
Trigonal Pyramid: {021} or {0221}, 3 faces top.
Prisms: Trigonal Prism- {010} or {0110}, 3 vertical faces; Ditrigonal Prism- {110} or {1120}, 6 vertical faces
Notice that the trigonal prism faces are directly beneath the trigonal pyramid faces.

Representative Minerals: Dravite, Elbaite, Pyrargyrite, Spangolite, Whitlockite

Rhombohedral Class

Hermann-Mauguin Symbol

3
Symmetry Elements

1A3 C (1 3-fold axis and a center of symmetry)

The General Form
The rhombohedron is the general form.

Special Forms
Hexagonal Prisms: 1st order, 2nd order
Pinacoid: basal

To Look For
Ddolomite, dioptase, ilmenite, loweite, phenacite and willemite are relatively common minerals in this class. The rhombohedral shape can be indicative, but is not definitive. Crystals are often blocky.

Problems
The rhombohedron, as with dolomite, is easily confused with that of calcite which is in the scalenohedral class. Dolomite often shows slightly curved faces.
Tourmaline: Rhombohedral Class

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{101}
Locality: Many Locations
Naumann, 1828, and others. In: V.M. Goldschmidt, Atlas der Krystallformen, 1913-1923.
.
Symmetry:
1 3-fold axis of symmetry: Note that it is coincident with the short body diagonal or the crystal (the c axis).
3 mirror planes: vertical (they bisect the 60o angles between the a axes.)
No horizontal mirror plane; therefore the mineral is hemimorphic.
Forms:
Rhombohedron: {101} or {1011}; 6 faces.

Representative Minerals: Ankerite, Dioptase, Dolomite, Ilmenite, Phenakite, Willemite

Trigonal Pyramidal Class

Hermann-Mauguin Symbol

3
Symmetry Elements

1 3-fold axis
The General Form
The general form is the Trigonal Pyramid: 1st Order (Positive & Negative), 2nd Order (Pos. & Neg.): 3 faces.

Special Forms
Trigonal Prisms: 1st Order (Positive & Negative), 2nd Order (Pos. & Neg.); 3 faces
Pedion:

To Look For
There are no common minerals in this class. The most common are jarosite, parisite, and rontgenite. Look for the trigonal pyramid on the upper termination and hemimorphism.

Representative Minerals: Jarosite, Parisite-(Ce)


Links to the Crystallography of each of the systems.


The the pages for the Hexagonal, Tetragonal and Triclinic, Systems are under construction.
    Determining Symmetry of Crystals: An Introduction
    Miller Indices
    Hermann-Mauguin Symmetry Symbols
    Crystallography: The Monoclinic System
    Crystallography: The Orthohombic System
    Crystallography: The Trigonal 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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