Isometric tetragonal system

• Introduction about crystal system
• Introduction of Isometric system
• Introduction of Isometric division
• Classes of Isometric division
• Form of each classes
2ISOMETRIC SYSTEM & TETRAGONAL SYSTEM
• Introduction of Tetragonal system
• Introduction of Tetragonal division
• Classes of Tetragonal division
• Form of each classes

 A Crystal is the regular polyhedral form bounded by smooth surfaces which is
assumed by a chemical compound under the action of its internal atomic forces when
passing under suitable conditions from the state of the liquid or gas to that of a solid.
 There are mainly six system in the crystallography which are as follow-
1. Isometric System
2. Tetragonal System
3. Hexagonal System
4. Orthorhombic System
5. Trigonal system
6. Monoclinic System
7. Triclinic System

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2-Fold 3-Fold 4-Fold 6-Fold
Isometric Tetartoidal 3 4 - - - - 23
Diploidal 3 4 - - 3 yes 2/m ͞3
Hextetrahedral 3 4 - - 6 - ͞4 3m
Gyroidal 6 4 3 - - - 432
Hexoctahedral 6 4 3 - 9 yes 4/m ͞3 2/m
Tetragonal Disphenoidal - 1 - - - 4͞
Pyramidal - - 1 - - - 4
Dipyramidal - - 1 - 1 yes 4/m
Scalenohedral 2 - 1 - 2 - ͞4 2m
Ditetragonal pyramidal - - 1 - 4 - 4mm
Trapezohedral 4 - 1 - - - 422
Ditetragonal-Dipyramidal 4 - 1 - 5 yes 4/m 2/m 2/m
Orthorhombic Pyramidal 1 - - - 2 - mm2
Disphenoidal 3 - - - - - 222
Dipyramidal 3 - - - 3 yes 2/m 2/m 2/m
Hexagonal Trigonal Dipyramidal - - 1 1 - 6͞
Pyramidal - - - 1 - - 6
Dipyramidal - - - 1 1 yes 6/m
Ditrigonal Dipyramidal 3 - 1 4 - 6m2
Dihexagonal Pyramidal - - - 1 6 - 6mm
Trapezohedral 6 - - 1 - - 622
Dihexagonal Dipyramidal 6 - - 1 7 yes 6/m 2/m 2/m
Trigonal Pyramidal - 1 - - - - 3
Rhombohedral - 1 - - - yes 3͞
Ditrigonal Pyramidal - 1 - - 3 - 3m
Trapezohedral 3 1 - - - - 32
Hexagonal Scalenohedral 3 1 - - 3 yes ͞3 2/m
Monoclinic Domatic - - - - 1 - m
Sphenoidal 1 - - - - - 2
Prismatic 1 - - - 1 yes 2/m
Triclinic Pedial - - - - - - 1
Pinacoidal - - - - - yes 1͞
AXES Planes Center
Hermann-
Maugin
Symbols
System Class Name
ISOMETRIC SYSTEM & TETRAGONAL SYSTEM

 Isometric system includes all the crystals which
are referred to three equal and mutually
perpendicular axes. As the axes are equal, they
are interchangeable and hence designated by a1,
a2 and a3. a1 is front to back, a2 is right to left
and a3 is top to bottom axes.
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a1 = a2 = a3
α = β = γ = 90o

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 All crystals of the isometric system possess four 3-fold axes of symmetry.
 Crystals of the isometric system may also demonstrate up to three separate 4-fold
axes of rotational symmetry.
 Furthermore crystals of the isometric system may possess six 2-fold axes of
symmetry .
 Minerals of this system may demonstrate up to nine different mirror planes.
 There are five crystal classes found in this system: Tetroidal, Diploidal,
Hextetrahedral, Gyroidal and Hexoctahedral.
 Examples of minerals which crystallize in the isometric system are halite,
magnetite, and garnet.
Isometric Tetartoidal 3 4 - - - - 23
Diploidal 3 4 - - 3 yes 2/m ͞3
Hextetrahedral 3 4 - - 6 - ͞4 3m
Gyroidal 6 4 3 - - - 432
Hexoctahedral 6 4 3 - 9 yes 4/m ͞3 2/m
Hermann
Maugin
Symbols
System Class Name AXES Planes CenterTetartohedral
Pyritohedral
Tetrahedral
Plagiohedral
Normal

 Symmetry:4/m  2/m
3 xl. Ax. 4, 4diagonal Ax. 3, 6 diag. Ax. 2.
3 xl. P, 6 diag. P.
 Centre Present
 Seven distinct forms occur in this class.
(1) Cube or Hexahedron (100).
(2) Dodecahedron or Rhombic dodecahedron (110).
(3) Octahedron(111).
(4) Tetrahexahedron (210) (310) (410) (hk0).
(5) Trapezohedron or Tetragonal trisoctahedron (211) (311) (322) (hll).
(6) Trisoctahedron or Trigonaltrisoctahedron (221) (331) (332) (hhl).
(7) Hexoctahedron (321) (421) (432) (hkl).
 Fluorite, Halite, Magnetite, Garnet, Analcite, Spinel, Galena etc.
are the minerals crystallizing in this class.

Symmetry:2/m 
 3 xl. Ax. 2, 4 diag. Ax. 3.
 3 xl. P.
 Centre Present
 Seven forms occur of which two are unique.
(1) Cube (100).
(2) Dodecahedron (110).
(4) Pyritohedron - pentagonal dodecahedron (210) (hk0) (Developed from the alternate
faces of Tetrahexahedron, so +ve and –ve).
(5) Trapezohedron (211) (hll).
(6) Trisoctahedron (221) (hhl).
(7) Diploid (321) (231) (hkl) (Developed from the alternate faces of hexoctahedron, so
+ve and –ve).
Pyrite is a mineral crystallizing in this class. 9ISOMETRIC SYSTEM & TETRAGONAL SYSTEM

 Symmetry:  3 m
 3 xl. Ax. 2, 4 diagonal Ax. 3.
 6 diagonal P.
 Forms:
(1) Cube (100).
(2) Dodecahedron (110).
(3) Tetrahedron (111) (1 ī 1) (hhh) (Developed from the alternate octants of
Octahedron, so +ve and –ve).
(4) Tetrahexahedron (210).
(5) Tristetrahedron or Trigonaltristetrahedron (211) (2 ī 1) (hll).
(6) Deltoiddodecahedron or Tetragonal tristetrahedron (221) (221) (hhl).
(7) Hexatetrahedron (321) (321) (hkl).
Tetrahedrite, Sphellerite and Boracite are the minerals crystallizing in this class.

(4) Plagiohedral or Gyroidal class: Cuprite type:
 Symmetry: 4 3 2
 3 xl. Ax. 4, 4 diagonal Ax. 3, 6 diag. Ax. 2.
 Seven distinct forms occur in this class.
(1) Cube or Hexahedron (100).
(2) Dodecahedron or Rhombic dodecahedron (110).
(4) Tetrahexahedron (210) (310) (410) (hk0).
(5) Trapezohedron or Tetragonal trisoctahedron (211) (311) (322) (hll).
(6) Trisoctahedron or Trigonaltrisoctahedron (221) (331) (332) (hhl).
(7) Pentagonal icositetrahedron (Icosi means 20).
Right handed (312) and Left handed (321).
This class is rare among the minerals. It is represented by cuprite and Sal
ammoniac. 11ISOMETRIC SYSTEM & TETRAGONAL SYSTEM

Symmetry: 2 3
3 xl. Ax. 2, 4 diag. Ax. 3.
Forms:
(1) Cube (100).
(2) Rhombdodecahedron (110).
(3) Tetrahedron (111).
(4) Pyritohedron (210) (hk0) (Developed from the alternate faces of
Tetrahexahedron, so +ve and –ve).
(5) Tristetrahedron (211) (hll).
(6) Deltoiddodecahedron (221) (hhl).
(7) Tetrahedral pentagonal dodecahedron (321) derived from hexoctahedron,
one quarter of the faces and hence four types,
Right handed +ve& -ve.
Left handed +ve&-ve.

ISOMETRIC SYSTEM & TETRAGONAL SYSTEM 13

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● Tetragonal system includes all crystals which are referred to three mutually
perpendicular axes. Two horizontal planes are equal and interchangeable. They are
called a1 and a2. The third is the vertical or C axis and it may be shorter or
longer than the horizontal axes. Length of a C axis is referred to the unit length of
axes.

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Minerals of the Tetragonal System all possess a single 4-fold symmetry axis.
They may possess up to four 2-fold axes of rotation, a center of inversion, and up to
five mirror planes.
There are seven crystal classes found in this system: Disphenoidal, Pyramidal,
Dipyramidal , Scalenohedral, Ditetragonal pyramidal, Trapezohedral and
Ditetragonal-Dipyramidal.
Some examples of minerals that crystallize in this system include Common minerals
that occur with this symmetry are anatase, cassiterite, apophyllite, zircon, and
vesuvianite.
Tetragonal Disphenoidal - 1 - - - 4͞
Pyramidal - - 1 - - - 4
Dipyramidal - - 1 - 1 yes 4/m
Scalenohedral 2 - 1 - 2 - ͞4 2m
Ditetragonal pyramidal - - 1 - 4 - 4mm
Trapezohedral 4 - 1 - - - 422
Ditetragonal-Dipyramidal 4 - 1 - 5 yes 4/m 2/m 2/m
Hermann
Maugin
Symbols
System Class Name AXES Planes Center
Tetartohedral
Pyramidal
hemimorphic
Tripyramidal
Sphenoidal
Hemimorphic
Trapezohedral
Normal

Symmetry: 4/m 2/m 2/m
1 verti. Ax. 4, 4 hori. Ax. 2.
1 hori. P, 4 verti. P.
C.
Forms: Various possible forms under the normal class of this system are as follows:
(1) Base or Basal pinacoid (001).
(2) 1st order prism (110).
(3) 2nd order prism (100).
(4) Ditetragonal prism (210) (310) (320) (hk0).
(5) 1st order bipyramid (111) (221) (223) (hhl).
(6) 2nd order bipyramid (101) (201) (203) (h0l).
(7) Ditetragonalbipyramid (321) (421) (122) (hkl).
Minerals crystallizing in normal class are Zircon, Rutile, Vesuvianite, Cassiterite,
Apophyllite, Octahedrite.

Symmetry: 4 mm
1 verti. Ax. 4,
4 verti. P.
This class differs from the normal class in having no horizontal axes or plane of
symmetry hence the forms are hemimorphic. It is not known to be represented among
minerals but it shown on the crystal of Iodosuccinimide.
Forms:
(1) Basal plane or pedion (001) & (00 ī).
(4) Ditetragonal prism (210) (310) (320) (hk0).
(5) 1st order pyramid upp. (111), Lr. (11 ī).
(6) 2nd order pyramid upp. (101), Lr. (10 ī).
(7) Ditetragonal pyramid upp. (321), Lr. (32 ī).

Symmetry: 4
1 verti. Ax. 4.
This class is similar to tripyramidal class having the same symmetry axis but
no horizontal plane of symmetry hence the forms are hemimorphic.
Forms:
(1) Basal plane or Pedion (001) or (00 ī).
(4) 3rd order prism Rh. (120), Lt. (210).
(5) 1st order pyramid upp. (111), Lr. (11 ī).
(6) 2nd order pyramid upp. (101), Lr. (10 ī).
(7) 3rd order pyramid upp. Rh. (122), upp. Lt. (212), Lr. Rh. (122), Lr. Lt. (212).
Wulfenite is the mineral crystallizing in this class.

Symmetry:  2 m
3 xl. Ax. 2
2 verti. Diag. P.
The typical forms of this are sphenoid and the tetragonal scalenohedron.
FORMS:
(1) Basal pinacoid (001).
(4) Ditetragonal prism (210).
(5) Tetragonal sphenoid +ve (111), -ve (1 ī 1).
(6) Bipyramid of 2nd order (101).
(7) Tetragonal scalenohedron +ve (212), –ve (2 ī 2).
Chalcopyrite is a mineral crystallizing in this class.

Symmetry: 4 2 2
1 verti. Ax. 4, 4 hori. Ax. 2.
Characteristic form Tetragonal Trapezohedron is derived from the ditetragonal
bipyramid of the normal class by the extension of the alternate faces. Therefore two
types + ve or right handed and –ve or left handed.
FORMS:
Tetragonal Trapezohedron
+ve or Rh. (212) and –ve or Lh. (122).
Nickel Sulphate and few other artificial salts belong to this class.

Symmetry: 
1 verti. Ax. 2.
Forms:
(1) Basal pinacoid (001).
(4) 3rd order prism Rh. (120), Lt. (210).
(5) 1st order sphenoid +ve (111), -ve (1 ī 1).
(6) 2nd order sphenoid Rh. or +ve, Lt. or -ve.
(7) 3rd order sphenoid Rh. +ve (122), Rh. –ve (ī 22), Lt. +ve (212), Lt. –ve (2 ī 2).

 A TEXTBOOK OF MINERALOGY
BY: EDWARD SALISBURY DANA

Isometric tetragonal system

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Isometric tetragonal system