Exploring Crystallography: Crystal Classes and Structures

1. CRYSTALLOGRAPHY_2 IMPORTANT: We list and describe all the crystal classes/minerals

2. Triclinic System Characterized by only 1-fold or 1-fold rotoinversion axis Symmetry content - i Pinacoidal Class,  One centre of symmetry (pairs of faces are related to each other through the centre).  Such faces are called pinacoids, pinacoidal class.  Egs. microcline (K-feldspar), plagioclase, turquoise, and wollastonite.

3. Monoclinic System Characterized by having only  mirror plane(s) or a single 2-fold axis. A MAJORITY OF ROCK FORMING MINERALS ARE INCLUDED IN THIS CLASS Normal Class or Prismatic Class, 2/m; Symmetry content - 1A2, m , i One 2-fold axis perpendicular to a single mirror plane.  This class has pinacoid faces and prism faces.  A prism = 3 or more identical faces that are all parallel to the same line.  In the prismatic class, these prisms consist of 4 identical faces, 2 of which are shown in the diagram on the front of the crystal.  The other two are on the back side of the crystal.  Egs. micas (biotite and muscovite), azurite, chlorite, clinopyroxenes, epidote, gypsum, malachite, kaolinite, orthoclase, and talc.

4. Orthorhombic System Characterized by having only three 2- fold axes/3 m or a 2-fold axis and 2 mirror planes. Normal Class or Barytes type or Rhombic- dipyramidal Class, 2/m2/m2/m, Symmetry content - 3A2, 3m, i This class has 3 perpendicular 2-fold axes that are perpendicular to 3 mirror planes.  The dipyramid faces consist of 4 identical faces on top and 4 identical faces on the bottom that are related to each other by reflection across the horizontal mirror plane or by rotation about the horizontal 2-fold axes. Egs. andalusite, anthophyllite, aragonite, barite, cordierite, olivine, sillimanite, stibnite, sulphur, and topaz.

5. Tetragonal System Characterized by a single 4-fold or 4-fold rotoinversion axis. Normal Class, Zircon Type Ditetragonal-dipyramidal Class, 4/m2/m2/m, Symmetry content - 1A4, 4A2, 5m, i It has a single 4-fold axis that is perpendicular to four 2-fold axes.  All of the 2-fold axes are perpendicular to mirror planes.  Another mirror plane is perpendicular to the 4-fold axis.  The mirror planes are not shown in the diagram, but would cut through all of the vertical edges and through the centre of the pyramid faces.  The fifth mirror plane is the horizontal plane.  Note the ditetragonal-dipyramid consists of the 8 pyramid faces on the top and the 8 pyramid faces on the bottom.   Common minerals that occur with this symmetry are anatase, cassiterite, apophyllite, zircon, and vesuvianite.

6. Hexagonal System Characterized by having either a 6-fold or a 3-fold axis Common form: Dihexagonal-dipyramidal Class, 6/m2/m2/m, Symmetry content - 1A4, 6A2, 7m, i It has a single 6-fold axis that is perpendicular to six 2-fold axes.  All of the 2-fold axes are perpendicular to mirror planes.  Another mirror plane is perpendicular to the 6-fold axis. Totalling 7 mirror planes. Beryl

7. Isometric System Characterized by having either a 6-fold or a 3-fold axis Common form: cube 4/m bar 3 2/m, Symmetry content - 3A4, 3¯ A3, 6A2, 9m note (1¯ A3 = 1A3 + i) Most symmetrical of a 3-D system. It has a four 3-fold axes, three 4-fold axes and six 2-fold axes.  All of the 2-fold axes. 9 mirror planes and a centre. Halite Egs. Gold. Galena, diamond, copper, silver, lead

8. Crystal Morphology, Crystal Symmetry, Crystallographic Axes Crystal Morphology and Crystal Symmetry: Recall: symmetry observed in crystals as exhibited by their crystal faces is due to the ordered internal arrangement of atoms in a crystal structure, as mentioned previously.  This arrangement of atoms in crystals is called a lattice.  Crystals, are made up of 3-dimensional arrays of atoms.  Such 3-D arrays are called space lattices. Crystal faces develop along planes defined by the points in the lattice.  In other words, all crystal faces must intersect atoms or molecules that make up the points.   A face is more commonly developed in a crystal if it intersects a larger number of lattice points.  This is known as the Bravais Law.

9. The angle between crystal faces is controlled by the spacing between lattice points. Since all crystals of the same substance will have the same spacing between lattice points (they have the same crystal structure), the angles between equivalent faces of the same mineral, measured at constant temp., are constant.  This is known as the Law of constancy of interfacial angles. Crystallographic Axes: The crystallographic axes are imaginary lines that we can draw within the crystal lattice.  These will define a co-ordinate system within the crystal.  3-D space lattices will have 3 or in some cases 4 crystallographic axes that define directions within the crystal lattices.

10. Crystallographic axes Triclinic Where a ≠ b ≠ c; α ≠ β ≠ γ Monoclinic Where a ≠ b ≠ c; α = γ = 900 & β > 900

11. Crystallographic axes cont’d Orthorhombic Where a ≠ b ≠ c; α = β = γ = 900 Tetragonal Where a1 =a2 ≠ c; α = β = γ = 900

12. Crystallographic axes cont’d Hexagonal Where a1 = a2 = a3 ≠ c; α = β = 900 ; γ = 1200 Isometric Where a1 = a2 = a3 ; α = β = γ = 900

13. Unit Cells The "lengths" of the various crystallographic axes are defined on the basis of the unit cell.  When arrays of atoms or molecules are laid out in a space lattice we define a group of such atoms as the unit cell.  This unit cell contains all the necessary points on the lattice that can be translated to repeat itself in an infinite array.  In other words, the unit cell defines the basic building blocks of the crystal, and the entire crystal is made up of repeatedly translated unit cells. The relative lengths of the crystallographic axes, or unit cell edges, can be determined from measurements of the angles between crystal faces.  We will consider measurements of axial lengths, and develop a system to define directions and label crystal faces. In defining a unit cell The edges of the unit cell should coincide with the symmetry of the lattice. The edges of the unit cell should be related by the symmetry of the lattice. The smallest possible cell that contains all elements should be chosen. 

14. The 7 Crystal systems: Unit cells (WIKEPEDIA) The 14 Bravais Lattices: triclinic(none monoclinic(1 diad) tetragonal(1 tetrad) cubic(4 triads) orthorhombic(3 perpendicular diads)

15. Axial Ratios, Parameters, Miller Indices • RECALL: • The lengths of the crystallographic axes are controlled by the dimensions of the unit cell upon which the crystal is based. • The angles between the crystallographic axes are controlled by the shape of the unit cell. • The relative lengths of the crystallographic axes control the angular relationships between crystal faces.  This is true because crystal faces can only develop along lattice points.  • The relative lengths of the crystallographic axes are called axial ratios

16. Axial Ratios Axial ratios are defined as the relative lengths of the crystallographic axes.  They are normally taken as relative to the length of the b crystallographic axis.  Thus, an axial ratio is defined as follows: Axial Ratio =  a/b : b/b : c/b Where a is the actual length of the a crystallographic axis, b, is the actual length of the b crystallographic axis, and c is the actual length of the c crystallographic axis. The end of the axis facing an observer is designated as the positive end, and the away end is referred to as the negative end.

17. For Triclinic, Monoclinic, and Orthorhombic crystals, where the lengths of the three axes are different, this reduces to:a/b : 1 : c/b (this is usually shortened to a : 1 : c) For Tetragonal crystals where a=b, this reduces to: 1 : 1 : c/b (or 1 : c) For Isometric crystals where the length of the a= b= c this becomes 1 : 1 : 1 (or 1) For Hexagonal crystals where there are three equal length axes (a1, a2, and a3) perpendicular to the c axis this becomes: 1 : 1 : 1: c/a (usually shortened to 1 : c)  Modern crystallographers can use x-rays to determine the size of the unit cell, and thus can determine the absolute value of the crystallographic axes inAngstrom units. 1 Å= 0.0000000001 m (10 -10 m).

18. Example For quartz which is hexagonal, the following unit cell dimensions determined by x-ray crystallography: a1 = a2 = a3 = 4.913Å ; c = 5.405Å Thus the axial ratio for quartz is: 1 : 1 : 1 : 5.405/4.913 Or1: 1 : 1 : 1.1001 which simply says that the c axis is 1.1001 times longer than the a axes. For orthorhombic sulphur the unit cell dimensions as measured by x-rays are: a  = 10.47Å, b = 12.87Å, c = 24.39Å Thus, the axial ratio for orthorhombic sulphur is: 10.47/12.87 : 12.87/12.87 : 24.39/12.87 or 0.813 : 1 : 1.903

19. Intercepts of Crystal Faces (Weiss Parameters) • Crystal faces can be defined by their intercepts on the crystallographic axes. • For non-hexagonal crystals, there are three cases. • A crystal face intersects only one of the crystallographic axes. • As an example the top crystal face shown here • intersects the c axis but • does not intersect the a or b axes.  • If we assume that the face intercepts the c axis at a distance of 1 unit length, • then the intercepts, sometimes called Weiss Parameters, are • Infinity a,  infinity b, 1c

20. 2. A crystal face intersects two of the crystallographic axes.As an example, the darker crystal face shown here intersects the a and b axes, but not the c axis.  Assuming the face intercepts the a and c axes at 1 unit cell length on each, the parameters for this face are: 1 a, 1 b, infinity c

21. 3. A crystal face that intersects all 3 axes.In this example the darker face is assumed to intersect the a, b, and c crystallographic axes at one unit length on each.  Thus, the parameters in this example would be: 1a, 1b, 1c

22. Two very important points about intercepts of faces: The intercepts or parameters are relative values, and do not indicate any actual cutting lengths. Since they are relative, a face can be moved parallel to itself without changing its relative intercepts or parameters. Note the dimensions of the unit cell is unknown. Therefore one face is assign to have intercept 1 Thus, the convention is to assign the largest face that intersects all 3 crystallographic axes the parameters  - 1a, 1b, 1c.  This face is called the unit face. Faces may make intercepts on all of the –ve or all of the +ve ends of axes or on one –ve and two +ve ends or two –ve and one +ve ends etc.

23. Miller Indices • The Miller Index for a crystal face is found by • first determining the parameters • second inverting the parameters, and • third clearing the fractions. • For example, • if the face has the parameters 1 a, 1 b, infinity c • inverting the parameters would be 1/1, 1/1, 1/ infinity • this would become 1, 1, 0 • the Miller Index is written inside parentheses with no commas - thus (110)

24. The face [labelled (111)] that cuts all three axes at 1 unit length has the parameters 1a, 1b, 1c.  Inverting these, results in 1/1, 1/1, 1/1 to give the Miller Index (111).   The square face that cuts the positive a axis, has the parameters 1 a, infinity b, infinity c.  Inverting these becomes 1/1, 1/infinity, 1/infinity to give the Miller Index (100). The face on the back of the crystal that cuts the negative a axis has the parameters -1a, infinity b, infinity c.  So its Miller Index is ( ¯100).  This would be read "minus one, zero, zero".  The 6 faces seen on this crystal would have the Miller Indices (00minus1),(001), (010), and (0minus10)(100)(minus100).

25. Since the hexagonal system has three "a" axes perpendicular to the "c" axis, both the parameters of a face and the Miller Index notation must be modified.  The modified parameters and Miller Indices must reflect the presence of an additional axis.  This modified notation is referred to as Miller-Bravais Indices, with the general notation (hkil). Let's derive the Miller indices for the dark shaded face in the hexagonal crystal shown. This face intersects the positive a1 axis at 1 unit length, the negative a3 axis at 1 unit length, and does not intersect the a2 or c axes.  This face thus has the parameters:1 a1, infinity a2, -1 a3, infinity c Inverting and clearing fractions gives the Miller-Bravais Index: (10 minus10). An important rule to remember in applying this notation in the hexagonal system, is that whatever indices are determined for h, k, and i, h + k + i = 0

26. For a similar hexagonal crystal, having the shaded face cutting all three axes, the parameters are 1 a1, 1 a2, -1/2 a3,  infinity c.  Inverting these intercepts gives: 1/1, 1/1, -2/1, 1/infinity resulting in a Miller-Bravais Index of (1 1 minus2 0) Note "h + k + i = 0" rule applies here!

27. Crystal Forms A crystal form is a set of crystal faces that are related to each other by symmetry.  To designate a crystal form (which could imply many faces) we use the Miller Index, or Miller-Bravais Index notation enclosing the indices in curly braces, i.e.{hkl} or {hkil} Such notation is called a form symbol. There are 48 possible forms that can be developed as the result of the 32 combinations of symmetry.  We discuss some, but not all of these forms. (Thirty (30) close and eighteen (18) open forms).

28. Open Forms and Closed Forms Open form An open form is one or more crystal faces that do not completely enclose space. A crystal with open-form faces also requires some additional closed-form facets to complete a structure. Open-forms include: Pedion, Pinacoid, Dome, Sphenoid, Pyramid, Prism

29. A Pedion is a flat face that is not parallel, or geometrically linked to any other faces. A Pinacoid is composed of only two parallel faces, forming tabular crystals such as ruby. A Dome is found in monoclinic and orthorhombic minerals Two intersecting faces that are caused by mirroring (topaz) commonly forms domes. Sphenoid’s are found in monoclinic and orthorhombic minerals, and have two-fold rotational axes. A Pyramid's multiple facets converge on a single crystallographic axis, and pyramid forms are not possible on minerals from the isometric, monoclinic or triclinic systems.

30. Open Hexagonal & Triangular Prisms Prisms have a set of facets that run parallel to an axis of a crystal, yet never converge with it. Eg. Quartz forms two sets of three sided prisms. Prisms are not possible in isometric or triclinic minerals. A Hexagonal (trigonal) prism is comprised of two hexagonal bases connected by a set of six rectangular faces that run parallel to, and never converge with an axes in the crystal. A triangular (trigonal) prism is comprised of two triangular bases connected by a set of three rectangular faces that run parallel to, and never converge with an axes in the crystal. This form is similar to a light-refracting 60º prism.

31. closed form A closed form is a set of crystal faces that completely enclose space.  Thus, in crystal classes that contain closed forms, a crystal can be made up of a single form. There are two types of closed forms (closed isometric and non-isometric forms) A crystal may comprise more than one form, called a combination. 

32. There are several crystal forms in the cubic crystal system that are common in diamond,garnet, spinel and other "symmetrical“ gemstones. A hexahedron (cube) has eight points, six faces, and twelve edges that are perpendicular to each other, forming 90 degree angles. An octahedron has two four sided pyramids lying base to base, and is totally symmetrical with no top, or bottom and has eight faces. A tetrahedronhas four equilateral triangular faces. Adodecahedronhas 12 faces

33. There are four types of dodecahedrons listed in order of descending symmetry: • Symmetrical pentagonal (five edged polygons) dodecahedrons, • 2. Asymmetrical (tetartoid) pentagonal dodecahedrons, • 3. Delta (four edged polygons) dodecahedrons, and • 4. Rhombic dodecahedrons. • Note: A Hexoctahedron is a multi-faceted dodecahedron with • 48 triangular faces.

34. Closed Non-Isometric Forms • 1. Hexagonal (Trigonal) Closed Forms • Hexagonal Pyramid • Hexagonal Bipyramid (Apatite) • Dihexagonal bipyramid (Beryl) • Hexagonal Trapezohedron • Hexagonal Scalenohedron • Tetrahexahedron • 2. Tetragonal Closed Forms • Tetragonal Disphenoid • Tetragonal Scalenohedron • Tetragonal Trapezohedron • Tetragonal Trapezohedral Trisoctahedron • Tetragonal Ditetragonal Bipyramidal (Rutile)

35. . Rhombohedral Closed Forms • Rhombohedral Trapezohedral (Quartz) • Rhombohedral Hemimorphic (Tourmaline) • Rhombohedral Holohedra (Calcite) • Rhombohedral Dodecahedron (Garnet, Fluorite) • Rhombohedral Trisoctahedron • 4. Orthorhombic Closed Forms • Rhombic Prism • Rhombic Pyramid • Rhombic Dipyramid • Rhombic Hemimorphic • Rhombic Sphenoid • Rhombic Pyramid

36. 5. Monoclinic Closed Forms Prism Monoclinic Clinopinacoid 6. Triclinic Closed Forms Prism Triclinic Dipyramid

37. Understanding Miller Indices, Form Symbols, and Forms We define the crystallographic axes in relation to the elements of symmetry in each of the crystal systems. Triclinic - Since this class has such low symmetry there are no constraints on the axes, but the most pronounced face should be taken as parallel to the c axis. Monoclinic - The 2 fold axis is the b axis, or if only a mirror plane is present, the b axis is perpendicular to the mirror plane. Orthorhombic - The current convention is to take the longest axis as b, the intermediate axis is a, and the shortest axis is c. 

38. Tetragonal - The c axis is either the 4 fold rotation axis or the rotoinversion axis. Hexagonal - The c axis is the 6-fold or 3-foldaxis Isometric - The equal length a axes are either the 3  4-fold rotation axes, rotoinversion axes, or, in cases where no 4-fold axes are present, the 3 2-fold axes. ZONES- A zone is defined as a group of crystal faces that intersect in parallel edges.  Since the edges will all be parallel to a line, we can define the direction of the line using a notation similar to Miller Indices. 

39. Crystal Habit The faces that develop on a crystal depend on the space available for the crystals to grow.  The term used to describe general shape of a crystal is habit. Cubic - cube shapes Octahedral - shaped like octahedrons, as described above. Tabular -  rectangular shapes. Equant - a term used to describe minerals that have all of their boundaries of approximately equal length. Fibrous -  elongated clusters of fibres. Acicular -  long, slender crystals. Prismatic -  abundance of prism faces. Bladed -  like a wedge or knife blade

40. Twinning in Crystals During the growth of a crystal (not in all cases), or if the crystal is subjected to stress or temperature/pressure conditions different from those under which it originally formed, two or more intergrown crystals are formed in a symmetrical fashion.  These symmetrical intergrowths of crystals are called twinned crystals.  Twinning is important to recognize, because when it occurs, it is often one of the most diagnostic features enabling identification of the mineral.

41. Types of Twinning Contact Twins - have a planar composition surface separating 2 individual crystals. These are usually defined by a twin law that expresses a twin plane (i.e. an added mirror plane). An example shown here is a crystal of orthoclase twinned on the Baveno Law,with {021} as the twin plane. Penetration Twins - have an irregular composition surface separating 2 individual crystals.  These are defined by a twin centre or twin axis.  Shown here is a twinned crystal of orthoclase twinned on the Carlsbad Law with [001] as the twin axis. Contact twins can also occur as repeated or multiple twins.  If the compositions surfaces are parallel to one another, they are called polysynthetic twins.  Plagioclase commonly shows this type of twinning, called the Albite Twin Law, with {010} as the twin plane.  Such twinning is one of the most diagnostic features of plagioclase.

42. Next lecture will be on sterographic projection

43. Pinacoids A Pinacoid is an open 2-faced form made up of two parallel faces. Domes Domes are 2- faced open forms where the 2 faces are related to one another by a mirror plane.  In the crystal model shown here, the dark shaded faces belong to a dome.  The vertical faces along the side of the model are pinacoids (2 parallel faces). Prisms A prism is an open form consisting of three or more parallel faces.  Depending on the symmetry, several different kinds of prisms are possible. Rhombic prism:  A form with four faces, with all faces parallel to a line that is not a symmetry element.  In the drawing to the right, the 4 shaded faces belong to a rhombic prism.  The other faces in this model are pinacoids (the faces on the sides belong to a side pinacoid, and the faces on the top and bottom belong to a top/bottom pinacoid).

44. Tetragonal prism: 4 - faced open form with all faces parallel to a 4-fold rotation axis.  The 4 side faces in this model make up the tetragonal prism.  The top and bottom faces make up the a form called the top/bottom pinacoid. Hexagonal prism: 6 - faced form with all faces parallel to a 6-fold rotation axis. The 6 vertical faces in the drawing make up the hexagonal prism.  Again the faces on top and bottom are the top/bottom pinacoid form.  Pyramids: A pyramid is a 3, 4, 6, 8 or 12  faced open form where all faces in the form meet, or could meet if extended, at a point. Hexagonal pyramid: 6-faced form where all faces are related by a 6 axis. If viewed from above, the hexagonal pyramid would have a hexagonal shape. Dipyramidsare closed forms consisting of 6, 8, 12, 16, or 24 faces.  Dipyramids are pyramids that are reflected across a mirror plane. Dihexagonal dipyramid: 24-faced form with faces related by a 6-fold axis with a perpendicular mirror plane.

45. Hexahedron: A hexahedron is the same as a cube. 3-fold axes are perpendicular to the face of the cube, and 4-fold axes run through the corners of the cube. Note that the form symbol for a hexahedron is {100}, and it consists of the following 6 faces. (100), (010), (001), (minus1 00), (0minus1 0), and (00 minus1). Example: Galena, Halite Octahedron: An octahedron is an 8 faced form that results form three 4-fold axes with perpendicular mirror planes.  The octahedron has the form symbol {111}and consists of the following 8 faces: (111),  ( minus1minus1minus1), (1 minus11), (1minus1minus1 ), (minus1minus1 1), (minus11minus1), (11minus1 ), (minus111).  Note that four 3-fold axes are present that are perpendicular to the triangular faces of the octahedron (these 3-fold axes are not shown in the drawing). Example: Diamond. Dodecahedron: A dodecahedron is a closed 12-faced form.  Dodecahedrons can be formed by cutting off the edges of a cube.  The form symbol for a dodecahedron is {110}.  As an exercise, you figure out the Miller Indices for these 12 faces. Example: Garnet