# 4Cry Unit-4 A Isometric

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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.
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