(1) Symmetry features are all 4L3, and there are five symmetry types 3L4 or 3L2 and 3L4i*** perpendicular to each other. The most common crystals are distributed in 3L44L36L29PC, 3L24L33PC and 3L4i4L36P.
(2) The crystal orientation takes three mutually perpendicular L4 or L2 and L4i as crystal axes, and the crystal constants are characterized as a=b=c, α = β = γ = 90.
(3) There are 15 kinds of common simplex and equiaxed crystal system, and only 6 kinds of common simplex. Their directions and shapes are shown in Figure 4- 13.
Figure 4- 13 Common Simplex Orientation and Shape Number of Equiaxed Crystal System
The common simplex of the above six kinds of equiaxed crystal systems often polymerize with each other to form poly. Among the mineral crystals produced in nature, the most common aggregate is shown in Figure 4- 14.
Fig. 4- 14 common polycrystalline cubic a{ 100} and octahedral o {1165438} of galena; B- the aggregate of rhombic dodecahedron d{ 1 10} and cube a{ 100} of fluorite; The aggregate of rhombic dodecahedron d{ 1 10} and octahedron O {11} of c- magnetite; D— polyhedron of pentagonal dodecahedron e{2 10} and cube a{ 100} of pyrite; E- garnet clusters of rhombic dodecahedron d{ 1 10} and tetragonal octahedron n{2 1 1}; Tetrahedral forms of f- sphalerite P {11} and p 1
2. Tetragonal crystal system
(1) symmetry must have an L4 or L4i, while * * * has seven symmetry types, and most of the common crystals are distributed in L44L25PC and L4PC.
(2) The crystal orientation takes L4 or L4i as the Z axis, and takes 2L2 which is perpendicular to L4 and perpendicular to each other or the normal of the symmetry plane and the crystal edge direction as the X and Y axes. The characteristics of crystal constant are a=b≠c, α = β = γ = 90.
(3) There are 1 1 geometric simplex in the tetragonal crystal system. Only four are the most common, and their directions and shapes are shown in Figure 4- 15. In a tetragonal crystal, there can be two or more simplex with the same name but different orientations, such as tetragonal column {100}, tetragonal column {1 10} and tetragonal column {hk0}. The same is true for square bipyramid (Figure 4- 15).
Figure 4- 15 Common Simplex Orientation and Shape Number of Tetragonal System
The most common tetragonal polycrystal is shown in Figure 4- 16.
Fig. 4- 16 common polymorphic crystals in tetragonal system
3. Tripartite and hexagonal crystal systems
(1) Symmetric cubic system has L3, and hexagonal system has L6 or L6 i.. Common trigonal and hexagonal mineral crystals are mainly distributed in the following symmetrical types:
Tri-crystal L33L23PC, L33L2, L33P hexagonal system L66L27PC
(2) Crystal Orientation According to the characteristics of symmetry, four crystal axes should be selected for the crystals of trigonal system and hexagonal system. Take the unique high-order axis (L3, L6, L6i) as the Z axis, and choose the normal of 3L2 or 3P which is perpendicular to the Z axis and forms an angle of120 with the Z axis, or the directions of the three crystal edges are X, Y and U axes. The spatial positions of the four crystal axes are shown in Figure 4- 17. The characteristics of crystal constant are a=b≠c, α = β = 90, γ = 120.
Crystal face symbols have four indexes, which are arranged in the order of X, Y, U and Z axes, such as {hkil}. The algebraic sum of the first three indicators must be equal to zero, that is, h+k+i=0, which is now proved as follows:
In fig. 4- 18, the intercept of crystal plane MM' is P 1 on the x axis, P2 on the y axis and P3 on the u axis. Make an auxiliary line KM' parallel to the U axis and an equilateral triangle with each side equal to P2. because
Fig. 4- 17 Four Crystal Axes of Tripartite and Hexagonal Crystal Systems
Figure 4- 18 Zero Algebraic Sum of the First Three Exponents of Triangular and Hexagonal Crystal Face Symbols
Why do cubic and hexagonal crystals have to adopt four-axis orientation?
Because both cubic and hexagonal crystals are repeated with the rotation of 120 or 60, for such symmetrical features, if triaxial orientation is still adopted, the intersection relationship between the crystal plane and the crystal axis on the same simplex is bound to be inconsistent, which means that the sum of the exponential absolute values of the crystal plane symbols is not equal, so the simplex symbol cannot be determined. Take the comparison of three-axis orientation and four-axis orientation of hexagonal column simplex as an example (Figure 4- 19 and Figure 4-20).
Figure 4- 19 Crystal Face Symbol of Hexagonal Prism Simplex Triaxial Oriented Hexagonal Cylinder
Figure 4-20 Crystal plane symbols of six cylinders when the hexagonal cylinder simplex is oriented on four axes.
As shown in fig. 4- 19, when the hexagonal column is oriented in three axes, from the actual intersection relationship and crystal plane symbol, the intersection relationship between crystal planes ①, ③, ④ and ⑥ and crystal axis is: intersecting with one crystal axis and parallel with two crystal axes; However, the intersection relationship between crystal planes ② and ⑤ and crystal axis is that they intersect with two crystal axes and are parallel to one crystal axis. Obviously, the intersection relationship between crystal plane and crystal axis on the same simplex is inconsistent, which is inconsistent with the properties of the simplex mentioned above. To solve this contradiction, just add a horizontally placed auxiliary crystal axis (U axis) on the bisector of the positive included angle (120) between the X axis and the Y axis, and the contradiction will be solved (as shown in Figure 4-20). Obviously, the U axis exists as an auxiliary crystal axis attached to the X axis and the Y axis.
(3) There are 18 kinds of common simplex and hexagonal systems, but there are only 7 kinds of common simplex. Their directions and shapes are shown in Figure 4-2 1. Similar to tetragonal crystal system, simplex with the same shape but different orientation can also appear on the same mineral crystal, such as hexagonal column and hexagonal column (Figure 4-2 1).
The most common cubic and hexagonal polycrystals are shown in Figure 4-22.
4. Oblique crystal system
The symmetry of (1) has no higher order symmetry axis, and the number of L2 or P is more than one. * * * * There are three symmetry types, and most of the common crystals are distributed in 3L23PC symmetry type.
(2) The crystal orientations of 3L23PC and 3L2 perpendicular to each other are regarded as X, Y and Z axes. For L22P symmetric type, L2 is the z axis, and the normals of 2P are the x and y axes. The crystal constant is characterized by a≠b≠c, α = β = γ = 90.
Fig. 4-2 1 Common Simplex Orientation and Shape Number of Tripartite and Hexagonal Crystal Systems
Figure 4-22 Common Trilateral and Hexagonal Polycrystals
(3) There are seven common simplex and orthorhombic crystal systems, and only three are the most common. Their directions and shapes are shown in Figure 4-23.
The most common orthorhombic polycrystals are shown in Figure 4-24.
5. Monoclinic system
(1) symmetry has no higher-order symmetry axis, and the number of L2 or P is not more than one; There are three types of * * * symmetry, and most common crystals are distributed in L2PC symmetry.
(2) The crystal orientation takes the normal of L2 or P as the Y axis, and the crystal edge directions perpendicular to the Y axis are the X and Z axes. The characteristics of crystal constant are a≠b≠c, α = γ = 90, β > 90.
(3) There are four kinds of common simplex and shape-number monoclinic systems, the most common ones are parallel double-sided and rhombic columns. Their direction and shape numbers are shown in Figure 4-25.
Figure 4-23 Orientation and Shape Number of Common Simplex in Orthogonal System
Figure 4-24 Common Polymorphic Crystals in Orthogonal Crystal System
Figure 4-25 Common simplex orientations and their shape numbers in monoclinic systems
The most common monoclinic polycrystals are shown in Figure 4-26.
6. Triclinic system
(1) symmetry has no symmetry axis and symmetry plane, * * * has two symmetry types, and most common crystals are distributed in C symmetry type.
(2) Crystal Orientation Three almost vertical crystal edge directions are selected as X, Y and Z axes. The characteristics of crystal constant are: a ≠ b ≠ c; α≠β≠γ≠90 。
(3) The common simplex and triclinic crystal systems with shape numbers can only have two types: simplex and parallel duplex, of which parallel duplex is the most common. Due to different orientations, their shape numbers can be {00 1}, {0 10}, {100} and {1/0.
Figure 4-26 Common polycrystals of monoclinic system and triclinic system are shown in Figure 4-27.
Figure 4-27 Common Polycrystallines in Triclinic System