The document discusses various topics related to crystals, including:
- Crystals have a periodic arrangement of atoms and can have regular shapes categorized by symmetry. Pharmaceutical crystals often have irregular dendritic shapes.
- There are seven crystal systems categorized by symmetry elements like axes and planes. The main systems discussed are cubic, tetragonal, orthorhombic, monoclinic, triclinic, rhombohedral, and hexagonal.
- Unit cells are the smallest repeating units that make up the overall crystal structure. They can be primitive, containing one lattice point, or non-primitive with multiple points.
An Introduction to Crystallography, Elements of crystals crystal systems: Cubic (Isometric) System,Tetragonal System, Orthorhombic System, Hexagonal System; Trigonal System, Monoclinic System, Triclinic System
Crystal Material, Non-Crystalline Material, Crystal Structure, Space Lattice, Unit Cell, Crystal Systems, and Bravais Lattices, Simple Cubic Lattice, Body-Centered Cubic Structure, Face centered cubic structure, No of Atoms per Unit Cell, Atomic Radius, Atomic Packing Factor, Coordination Number, Crystal Defects, Point Defects, Line Defects, Planar Defects, Volume Defects.
An Introduction to Crystallography, Elements of crystals crystal systems: Cubic (Isometric) System,Tetragonal System, Orthorhombic System, Hexagonal System; Trigonal System, Monoclinic System, Triclinic System
Crystal Material, Non-Crystalline Material, Crystal Structure, Space Lattice, Unit Cell, Crystal Systems, and Bravais Lattices, Simple Cubic Lattice, Body-Centered Cubic Structure, Face centered cubic structure, No of Atoms per Unit Cell, Atomic Radius, Atomic Packing Factor, Coordination Number, Crystal Defects, Point Defects, Line Defects, Planar Defects, Volume Defects.
In this lecture , the focus is on geological aspects of minerals. We are going to discuss about the crystal symmetry of minerals,axis of rotation , axis of rotoinversion symmetry y, and mirror plane. The crystal classes may be sub-divided into one of 6 crystal systems namely Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric (cubic).
Crystallographic axis
The identification of specific symmetry operations enables one to orientate a crystal according g to an imaginary set of reference lines known as the crystallographic axis.With the exception of the hexagonal system, the axes are designated designated The ends of each a, b, and c. The ends of each axes are designated axes are designated + or -. This is important for the derivation of Miller Indices.
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2. A Crystal is a three-dimensional solid which consists of a periodic arrangement of
atoms.
A crystal is a solid form of substance (ice)
Some crystals are very regularly shaped and can be classified into one of several shape
categories such rhombic, cubic, hexagonal, tetragonal, orthorhombic, etc.
With pharmaceuticals, crystals normally have very irregular shapes due to dendritic
growth which is a spiky type appearance like a snowflake. It can be difficult to
characterise the size of such a crystal.
Crystals are grown to a particular size that is of optimum use to the manufacturer.
Typical sizes in pharmaceutical industry are of the order of 50µm.
What is Crystal
3. Classification of Solids
Crystalline materials
Non-crystalline
materials
Glass
Rubber and
Plastics
Poly-crystal
(It has an aggregate of many
small crystals that are
separated by well-defined
boundaries)
Single crystal
(The entire solid consists of only
one crystal)
Example
Non-metallic
crystals
Metallic crystals
Germanium
Silicon
crystalline carbon etc.,
Iron
Copper
Silver
Aluminium
Tungsten etc.,
4. Crystal SymmetryCrystal Symmetry
Crystals have inherent symmetry
The definite ordered arrangement of the faces and edges of a crystal is known as
`crystal symmetry’.
It is a powerful tool for the study of the internal structure of crystals.
Crystals possess different symmetries or symmetry elements.
The seven crystal systems are characterized by three symmetry elements. They are
Centre of symmetry
Planes of symmetry
Axes of symmetry.
5. CeCentrentre of Symmetryof Symmetry
It is a point such that any line drawn through it will meet the
surface of the crystal at equal distances on either side.
Since centre lies at equal distances from various symmetrical
positions it is also known as `centre of inversions’.
It is equivalent to reflection through a point.
A Crystal may possess a number of planes or axes of symmetry but
it can have only one centre of symmetry.
For an unit cell of cubic lattice, the point at the body centre
represents’ the `centre of symmetry’ and it is shown in the figure.
6. PlaPlanene of Symmetryof Symmetry
A crystal is said to have a plane of symmetry, when it is divided by an imaginary plane into two
halves, such that one is the mirror image of the other.
In the case of a cube, there are three planes of symmetry parallel to the faces of the cube and six
diagonal planes of symmetry .
7. Axis of SymmetryAxis of Symmetry
This is an axis passing through the crystal such that if the crystal is rotated around it through
some angle, the crystal remains invariant. The axis is called `n-fold, axis’.
If equivalent configuration occurs after rotation of 180º, 120º and 90º, the axes of rotation are
known as two-fold, three-fold and four-fold axes of symmetry respectively.
If equivalent configuration occurs after rotation of 180º, 120º and 90º, the axes of rotation are
known as two-fold, three-fold and four-fold axes of symmetry.
If a cube is rotated through 90º, about an axis normal to one of its faces at its mid point, it brings
the cube into self coincident position.
Hence during one complete rotation about this axis, i.e., through 360º, at four positions the cube
is coincident with its original position. Such an axis is called four-fold axes of symmetry or
tetrad axis.
8. Axis of SymmetryAxis of Symmetry
If n=1, the crystal has to be rotated through an angle = 360º, about an axis to achieve self
coincidence. Such an axis is called an `identity axis’. Each crystal possesses an infinite number
of such axes.
If n=2, the crystal has to be rotated through an angle = 180º about an axis to achieve self
coincidence. Such an axis is called a `diad axis’. Since there are 12 such edges in a cube, the
number of diad axes is six.
If n=3, the crystal has to be rotated through an angle = 120º about an axis to achieve self
coincidence. Such an axis is called is `triad axis’. In a cube, the axis passing through a solid
diagonal acts as a triad axis. Since there are 4 solid diagonals in a cube, the number of triad axis
is four.
9. Axis of SymmetryAxis of Symmetry
If n=4, for every 90º rotation, coincidence is achieved and the axis is termed `tetrad axis’. It is
discussed already that a cube has `three’ tetrad axes.
If n=6, the corresponding angle of rotation is 60º and the axis of rotation is called a hexad axis.
A cubic crystal does not possess any hexad axis.
Crystalline solids do not show 5-fold axis of symmetry or any other symmetry axis higher than
`six’, Identical repetition of an unit can take place only when we consider 1,2,3,4 and 6 fold
axes.
10. Symmetrical ElemeSymmetrical Elements of cubents of cube
(a) Centre of symmetry 1
(b) Planes of symmetry 9 (Straight planes - 3, Diagonal planes - 6)
(c) Diad axes 6
(d) Triad axes 4
(e) Tetrad axes 3
----
Total number of symmetry elements = 23
----
Thus the total number of symmetry elements of a cubic structure is 23.
11. Lattice is as an array of points in space in which the environment about each point is the same i.e., every
point has identical surroundings to that of every other point in the array.
Lattice or Space lattice
The crystal structure is obtained by adding a unit assembly of atoms to each lattice point. This unit assembly
is called basis.
A basis may be a single atom or assembly of atoms which is identical in composition, arrangement and orientation.
Space lattice + Basis Crystal structure
Example
• Aluminium and Barium
•Sodium chloride NaCl
• Potassium chloride KCl
Basis
12. The unit cell is defined as the smallest geometric figure which is repeated to drive the actual crystal structure
Unit cell
Primitive cell
A primitive cell is the simplest type of unit cell which contains only one lattice point per unit cell (contains
lattice points at its corner only)
Example:
SC – Simple cubic
Non-primitive cell
If there are more than one lattice point in an unit cell, it is called a non-primitive cell (contains more than one
lattice point per unit cell)
Example:
BCC and FCC
15. Cubic crystal systemCubic crystal system
In this crystal system, all the three axial lengths of
the unit cell are equal and they are perpendicular to
each other
a = b = c and α = β = γ = 90o
Example:
Iron, copper, Sodium Chloride (NaCl),
Calcium Fluoride (CaF2)
a
a
a
90o
90o
90o
Cubic crystal system
Y
X
Z
17. TetragoTetragonal crystal systemcrystal system
In this crystal system, two axial lengths of the unit
cell are equal and third axial length is either longer
or shorter. All the three axes are perpendicular to
each other.
a = b ≠ c and α = β = γ = 90o
Example:
Ordinary white tin, Indium.
Tetragonal crystal system
90o
90o
90o
b
a
c
Y
X
Z
19. Orthorhombic crystal systemOrthorhombic crystal system
In this system, three axial lengths of the unit cell are
not equal but they are perpendicular to each other
a ≠ b ≠ c and α = β = γ = 90o
Example:
Sulphur, Topaz
a
b
c
90o
90o
90o
Orthorhombic crystal system
Y
X
Z
21. MoMonoclinic crystal systemcrystal system
In this system, three axial lengths of the unit cell are
not equal. Two axes are perpendicular to each other
and third axis is obliquely inclined.
a ≠ b ≠ c and α = β = 90o
, γ ≠ 90o
Example:
Sodium sulphite (Na2SO3),
Ferrous sulphate (FeSO4).
a
b
c
α
꞊ 90
o
β 90꞊ o
γ ≠ 90o
Monoclinic crystal system
Y
X
Z
23. TriTriclinic crystal systemcrystal system
In this system, three axial lengths of the unit cell are
not equal and all the three axes are inclined
obliquely to each other.
a ≠ b ≠ c and α ≠ β ≠ γ ≠ 90o
Example:
Copper sulphate (CuSO4),
Potassium dichromate (K2Cr2O7).
a
b
c
α
β
γ
Triclinic crystal system
Y
X
Z
24. RhombohedralRhombohedral crystal systemcrystal system
In this system, three axial lengths of the unit cell are
equal. They are equally inclined to each other at an
angle other than 90o
.
a = b = c and α = β = γ ≠ 90o
Example:
Calcite.
a
a
a α
β γ
Rhombohedral crystal system
Y
X
Z
25. HexagoHexagonal crystal systemcrystal system
In this crystal system, two axial lengths of the unit cell
(horizontal) are equal and lying in one plane at angle 120o
with each other
The third axial length (vertical) is either longer or shorter
than other two and it is perpendicular to this plane.
a = b ≠ c and α = β = 90o
γ = 120o
Example:
Quartz, Tourmaline
b
c
a
90o
120o
90o
Hexagonal crystal system
120o
26. Crystal plaCrystal planeses
Miller introduced a set of three numbers to designate a
plane in a crystal. This set of three numbers is called
Miller indices of the concerned plane.
A
B
C
Y
X
Z
2
2
1