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• Unit-4 A Crystal systems and crystal classes • The rotation, reflection, inversion, and rotoinversion symmetry operations may be combined in thirty-two different ways. • Thirty-two different crystal classes are therefore defined so that each crystal class corresponds to a unique set of symmetry operations. • Each of the crystal classes is named according to the typical crystal form which it displays (or) based on the mineral crystallizing in that class (or) based on the basic symmetry elements as per the HERMANN and MAUGUIN symbols. • For example, the isometric hexoctahedral class belongs to the isometric crystal system and demonstrates the hexoctahedral crystal form. • The tetragonal pyramidal, hexagonal pyramidal, trigonal pyramidal and rhombic pyramidal classes each display a variation of the crystal form which is called a pyramid. • Each crystal class is a member of one of the crystal system according to which characteristic symmetry operation it possesses. • Note that the 32 crystal classes are divided into 6 crystal systems. • The Triclinic System has only 1-fold or 1-fold rotoinversion axes. • The Monoclinic System has only mirror plane(s) or a single 2-fold axis. • The Orthorhombic System has only two fold axes or a 2-fold axis and 2 mirror planes. • The Tetragonal System has either a single 4-fold or 4-fold rotoinversion axis. • The Hexagonal System has no 4-fold axes, but has at least 1 6-fold or 3-fold axis. • The Isometric System has either 4 3-fold axes or 4 3fold rotoinversion axes. • The class which possesses the highest possible symmetry within each crystal system is termed the Holomorphic or Holohedral or Normal class of that system. Other classes with less symmetry are known as Hemihedral classes. • The characteristic symmetry element of each crystal system is listed in bold type. • When listing the symmetry of each crystal class an axis of rotational symmetry is represented by the capital letter A. • Whether this axis is a 2-fold, 3-fold, or 4-fold axis is indicated by a subscript following the letter A. • The number of such axes present is indicated by a numeral preceding the capital A. 1A2, 2A3, and 3A4 thus represent one 2-fold axis of rotation, two 3-fold axes, and three 4-fold axes respectively. • A center of symmetry is noted by the lowercase letter 'i' while a mirror plane is denoted by 'm'. • The numeral preceding the m indicates how many mirror planes are present. • Axes of rotary inversion are indicated by a subscript following the letter R. Isometric Crystal System Crystal Class Symmetry elements Isometric Crystal Form Tetaroidal (Ullmannite) 23 3A2, 4A3 2/m ˉ3 Diploidal i,3A2, 3m, 4 3 (Pyrite) Gyroidal 3A4, 4A3, 6A2 (Cuprite) Hextetrahedr i ,3 4 , 4A3, 6m al (Tetrahedrite) i,3A4, 4 3, 6A2, Hexoctahedra 9m l (Galena) 432 3m 4/m 2/m The Isometric System has either 4 3-fold axes or 4 3-fold rotoinversion axes. Tetragonal Crystal System Crystal Class 4 Symmetry elements 1A4 4 4/m Tetragonal 422 4mm 2m 4/m2/m2/m i, 1A4, 1m Crystal Form TetragonalPyramidal Tetragonaldisphenoidal Tetragonaldipyramidal 1A4, 4A2 Tetragonaltrapezohedral 1A4, 4m Ditetragonalpyramidal 1 2A2, 2m Tetragonalscalenohedral i, 1A4, 4A2, 5m Ditetragonaldipyramidal 4, Hexagonal Crystal System Crystal Class Symmetry elements 6 1A6 1 Hexagonal Division 6 6/m i, 1A6, 1m 622 1A6, 6A2 6mm 1A6, 6m m2 6/m2/m2/m 1 6 , 3A2, 3m i, 1A6, 6A2, 7m Crystal Form Hexagonal-pyramidal Trigonal-dipyramidal Hexagonaldipyramidal Hexagonaltrapezohedral Dihexagonalpyramidal Ditrigonaldipyramidal Dihexagonaldipyramidal Hexagonal Crystal System Crystal Class 3 Symmetry elements Crystal Form Trigonalpyramidal 1A3 1 32 Trigonal Division 3m 2/m Rhombohedral 3 1A3, 3A2 Trigonaltrapezohedral 1A3, 3m Ditrigonalpyramidal 1 Hexagonalscalenohedral 3, 3A2, 3m Orthorhombic Crystal System Crystal Class 222 Symmetry elements 3A2 Orthorhombic mm2 (2mm) 1A2, 2m 2/m2/m2/m i, 3A2, 3m Crystal Form Rhombicdisphenoidal Rhombicpyramidal Rhombicdipyramidal •The Orthorhombic System has only two fold axes or a 2-fold axis and 2 mirror planes. Crystal Class Symmetry elements Triclinic Crystal System 1 none Pedial i Pinacoidal Monoclinic Crystal System 2 1A2 Sphenoidal m 1m Domatic 2/m i, 1A2, 1m Prismatic Crystal Form The Triclinic System has only 1-fold or 1-fold rotoinversion axes. The Monoclinic System has only mirror plane(s) or a single 2-fold axis. • Isometric System • The Isometric System, also known as the 'Cubic' and 'Regular' System, has three axes. • All three axes are of equal length and are at right angles (90°) to each other. • Because of these properties, the three axes are all interchangeable, and are therefore all designated as 'a'. Isometric Crystal System Crystal Class Symmetry elements Isometric Crystal Form Tetaroidal (Ullmannite) NiSbS 23 3A2, 4A3 2/m ˉ3 Diploidal i,3A2, 3m, 4 3 (Pyrite) FeS2 Gyroidal 3A4, 4A3, 6A2 (Cuprite) Cu O Hextetrahedral i ,3 4 , 4A3, 6m (Tetrahedrite) 432 2 3m (Cu,Fe,Zn,Ag)12Sb4S 13 4/m 2/m i,3A4, 4 9m 3, 6A2, Hexoctahedral (Galena) The Isometric System has either 4 3-fold axes or 4 3-fold rotoinversion axes. I. Normal Class - Hexoctahedral class -Galena Type • This Class is called 'Normal as it is the most common crystal class that occurs in nature. • The Normal Class has thirteen (13) axes of rotational symmetry. These axes are grouped into three sets: • - 3 axes of four fold symmetry. • - 4 axes of three fold symmetry. • - 6 axes of two fold symmetry. • It has 9 mirror Planes: • Principle Symetry Planes (Crystallographic planes) - 3 planes of the Isometric System. • Diagonal Symetry Planes (Auxiliary) - 6 planes of symmetry. • There are seven crystallographic forms to the Normal Class: • 1. Hexahedron or Cube 2.Octahedron 3.Dodecahedron 4.Tetrahexahedron 5.Trisoctahedron 6.Trapezohedron 7.Hexoctahedron • 1. Cube (Hexahedron) - (example: Galena, Fluorite) • It has six identical faces. Each face is: • (a) Parallel to two of the Primary Axes of the cube, and thus to one of the Principle Symetry Planes, and perpendicular to the third Primary Axis; • (b) At right angles (90°) to its four conjunctive faces; • • • • Octahedron - (example: chromite) {111} It has eight identical faces. Each face : (a) Meets all three Primary Axes of a cube, and is not parallel to any of them • (b) Is an equilateral triangle (each apex is 60°); • Minerals commonly exhibiting the simple octahedral form are magnetite, chromite, spinel, pyrochlore, cuprite, gold, and diamond. Sometimes fluorite, pyrite, and galena take this form. • 3. Dodecahedron - (example: Garnet) (110) • Geometrically, there are several forms of dodecahedron are recognized. The regular (pentagonal) dodecahedron being the most important. • The dodecahedron has the following properties: • Bounded by 12 faces. Each face • (a) Meets two Primary Axes, and is parallel to the third Primary Axis; • (b) Is parallel to one of the six Diagonal Planes of Symetry. • (c) Is a rhomb (four equal sides’ planar angles of other than 90°) with planar angles of 70.5° and 109.5°. • 4. Tetrahexahedron - (example: Magnetite, Fluorite) • The general symbol of the Tetrahexahedron is (hk0) i.e., meets two primary axes and Parallel to one of the Primary Axes of a cube;(210) • It has 24 faces: Each face is: • (a) An isosceles triangle; An isosceles triangle is a triangle with (at least) two equal sides. • (b) Part of a set of four conjunctive faces, there being six sets of four faces; • (c) The common meeting point for all four faces of each set being at one of the Primary Axes of a cube; • A set of four faces can be thought of as replacing one face of a cube (the alternate term for a cube is hexahedron, and from this and these sets of four conjunctive faces, the name Tetrahexahedron is derived); • 5. Trisoctahedron Also called: Trigonal Trisoctahedron. {221} (example: Fluorite, Magnetite and Diamond) • General symbol of Trisoctahedron is (hh1) i.e., each face meets two of the axis at less than unity distance and meets the third axis at unity distance • It is bounded by 24 faces. Each face is: • (a) An isosceles triangle: • (b)Three faces, with their shorter, equal, sides in conjunction, create a set of faces. • Each set: (i) forms a pyramid with an equilateral triangle for a base; • (ii) this pyramid occupies one face of what would be an underlying octahedron; and from this fact the name trisoctahedron has been given. • 6. Trapezohedron(211) (Example Leucite) • General symbol of Trapezohedron is (h11) i.e., meets two primary axes at unit length and the other primary axis at different length. • (a) (211) common form alone and in combination. • (b) (311) commonly seen in combination. • It is bounded by 24 faces • (a)Each face is a quadrilateral, also known as a trapezium. (A quadrilateral (a shape with four sides) with two of its sides parallel) • (b)Each face may be viewed as bevelling one edge of a Dodecahedron, and replacing half of each of its two adjacent faces. • (c)The faces may be viewed as being in six sets of four: each set may be viewed as replacing a face of a cube. • (d)The faces may also be viewed as being in eight sets of three: each set of three may be viewed as replacing a face of an octahedron, or the corner of a cube. • 7. Hexoctahedron (321) (Example:Diamond) • General symbol of Hexoctahedron is (hkl) i.e., meets all the three primary axes at different lengths. • It is bounded by 48 similar faces • (a)Each face is a scalene triangle (a triangle having three unequal sides). • (a)The faces may be viewed as being in sets of six conjunctive faces: each set of eight conjunctive faces may be seen as replacing one face of an octahedron, hence the name Hexoctahedron; • (c)The faces may also be viewed as being in sets of eight conjunctive faces: each set of eight conjunctive faces may be seen as replacing one face of a cube. • Isometric Normal Class Combinations of crystal forms: • Cube and Octahedron : 14 faces • Cube and Dodecahedron: 18 faces • Octahedron and Dodecahedron: 20 faces • Cube, Octahedron and Dodecahedron: 26 faces • Cube and Tetrahexahedron: 30 faces • Octahedron and Tetrahexahedron: 32 faces • Dodecahedron and Tetrahexahedron: 36 faces • Cube and Hexoctahedron: 54 faces • Octahedron and Hexoctahedron: 56 faces • Dodecahedron and Hexoctahedron: 60 faces II. Diploidal class (Pyrite type) i, 3A2, 4A3, 3m. (1)PYRITOHEDRON (Pentagonal dodecahedron) (Example: Pyrite) {210} • There are 12 pentagonal faces, each of which intersects one crystallographic axis at unity, intersects a second axis at some multiple of unity, and is parallel to the third axis. • Note that there are no 4-fold axes in this class. • Pyrite is the only common mineral that displays this form. (2) DIPLOID (Didodecahedron){321} i, 3A2, 4A3, 3m. (Example: Pyrite) • There are 24 faces (fig.), each face corresponding to one-half of the faces of a hexoctahedron. This is a rare form. • The diploid looks like a pyritohedron where two faces are made from each pentagonal face of the pyritohedron. • The resulting faces are trapezia. Pyrite is the only common mineral that exhibits the diploid form. • III. Hextetrahedral class (Tetrahedrite ) 3A2, 4A3, 6m 1.TETRAHEDRON .(Example: tetrahedrite, diamond, helvite, and sphalerite) • The tetrahedron includes both a positive and negative form with the notation {111} and {1-11}, respectively. These are simple mirror images of one another. A tetrahedron is a 4-faced form, each face being an equilateral triangle. • Each face intersects all 3 crystallographic axes at the same distance. 2. Trigonal Tristetrahedron (211) (Example: ) Take a tetrahedron and raise 3 isoceles triangle-shaped faces on each of the 4 tetrahedral faces. So this form has 12 triangular faces. Just like the tetrahedron, there are both positive and negative forms. This is only a relatively common form on tetrahedrite, usually subordinant to the tetrahedron , but has also been reported on sphalerite and boracite. (3)Tetragonal Tristetrahedron or Deltoid Dodecahedron – (Example: tetrahedrite or sphalerite) {221} • This is a 12-faced form, derived by raising three sided faces on the each face of a tetrahedron (fig.). • The shape of the resultant faces are rhombic. There are both positive and negative forms, designated as {hll} and {h-ll}, respectively. • This form is sometimes seen as a subordinate one on tetrahedrite or sphalerite. 4. Hexakistetrahedon or Hextetrahedron{321} (Example: tetrahedrite, rarely on sphalerite. Also a possible on diamond. ) • Again, we take a tetrahedron and, in similar manner as the hexoctahedron, we raise 6 triangular faces having a common apex from the center of the equilateral triangular face of the tetrahedron. • Repeating this on the entire tetrahedron results in 24 faces (fig). There are both positive and negative forms, designated as {hkl} and {h-kl}, respectively. This form has been reported on tetrahedrite, but rarely on sphalerite. Also a possible solution form on diamond.