Crystallography

Matter
• Any substance which has mass and occupies space
• All physical objects are composed of matter.

Solids:
• Objects with definite size and shape are
known as solids.
• Incompressible, Rigid, Mechanically
strong, Atoms are closely packed.
Liquids & Gases:
• Atoms or molecules are not fixed and
cannot form any shape and size. They gain
the shape and size of the container.
• Atoms are loosely packed.

i) Crystalline Solids:
The solids in which atoms or molecules are arranged in a
regular and orderly manner in three dimension are called
Crystalline Solids.
Ex: i) Metallic: Gold, Silver, Aluminium
ii) Non-Metallic: Diamond, Silicon,
Quartz,Graphite etc.,
Solids are classified into two categories

ii) Amorphous Solids
The solids in which atoms or molecules are not arranged in
a regular and orderly manner in three dimension are called
Amorphous Solids
Ex: Glass, Plastic, rubber

Crystalline Solids Amorphous Solids
1.Atoms or molecules have regular periodic
arrangements
2.They are anisotropic in nature.
3. They exhibit directional properties.
4.They have sharp melting points.
5. Crystal breaks along regular crystal planes and
hence the crystal pieces have regular shape
Ex: Copper, Silver, Aluminium etc
Atoms or molecules are not arranged in a regular
periodic manner. They have random arrangement.
They are isotropic in nature.
They do not exhibit directional properties.
They do not possess sharp melting points
Amorphous solids breaks into irregular shape due to
lack of crystal plane.
Ex: Glass, Plastic, rubber, etc.
Differences between
Crystalline solid and
Amorphous solid

Crystal lattice (Space lattice)
Defined as 3-dimensional array of points are spatially arranged in
specific pattern.
Or
It is geometric arrangement of matters (atoms, ions, molecules)
Characters of lattice

Unit Cell
• The smallest possible portion or geometrical figure
of crystal lattice and which buildup by repetition in
three dimensions, is called unit cell.
(or)
• It is a fundamental elementary pattern.
• This unit cell is basic structural unit or building
blocks of the crystal structure

UNIT CELL TYPESUNIT CELL TYPES
Primitive lattice (P)
Centered lattice (I):
• Primitive lattice (P) :
In this lattice the unit cell
consists of eight corner
atoms.

Centered lattice (I):
• Body Centered Unit Cells
• Face Centered Unit Cells
• End Centered Unit Cells
Body Centered Unit Cells(B)

• Face Centered lattice (F):

• Base Centered lattice (C):
• In this lattice along with the corner atoms,
the base and opposite face will have centre
atoms

Crystal Element Face
Edge
Solid angle
These 3 crystal elements be an a mathematical relationship is called a Euler’s Formula
Euler’s formula is F + A=E+2

Euler’s Formula:
F+A=E+2
Where, F=Number of Face.
A=Number of solid angle.
E=Number of Edges.
Name
Faces Solid angles Edges F+A=E+2
Tetrahedron 4 4 6 8=8
Cube 6 8 12 14=14
Octahedron 8 6 12 14=14
Dodecahedro 12 20 30 32=32

The angle found between a pair of adjacent faces of a crystal..
Interfacial angle

Crystal Symmetry
It explains how similar atoms or group of
atoms (motif) repeated symmetrically in
space to produce ordered structure .
It can be studied with the help of operation
than can be operated on the crystals known as
Symmetry Operation

Symmetry operation
Is an operation that can be performed either physically or
imaginatively on crystal with reference to Plane, Axis, Point
within its mass.
Symmetry operation achieved by
•Rotating the crystal in particular axis
•Select the plane which shows mirror image
•Select the point which shows equidistance

This is a line about which the crystal may be rotated so as to show the same view
of the crystal more than once per rotation.

There are four axis of symmetry :-
• On the rotation about the axis, if the same faces or same view occurs 2 times, the axis termed
as Diad axis or two fold axis. D:2fold.avi
• On the rotation about the axis, , if the same faces or same view occurs 3 times, the axis termed
as Triad axis or three fold axis. D:3 fold.avi
• On the rotation about the axis, if the same faces or same view occurs 4 times, the axis termed as
Tetrad axis or four fold axis. D:4 fold.avi
• On the rotation about the axis, , if the same faces or same view occurs 6 times, the axis termed
as Hexad axis or six fold axis.

Symmetry CharactersSymmetry Characters
Centre of SymmetryCentre of Symmetry Plane of SymmetryPlane of Symmetry Axes of SymmetryAxes of Symmetry
PresentPresent 0909
II fold axesII fold axes III fold axesIII fold axes IV fold axesIV fold axes VI fold axesVI fold axes
0606 0404 0303 --

Centre of Symmetry:-
It is a imaginary point in the crystal that any line drawn
through it intersects the surface of the crystal at equal
distance on either side
Center of symmetry is the point from
which all similar faces are
equidistant.

Crystallographic Axes
A set of reference axes in a
crystal that are used to
describe the crystal systems

Crystal System or Lattice system:
Crystal system refers to Geometry of crystal and crystal structure and
which can be described by set of reference axes (crystallographic axes) .
Types Of Crystal System

ISOMETRIC OR CUBIC SYSTEM
Centre of SymmetryCentre of Symmetry
Plane of
Symmetry
Plane of
Symmetry
Axes of SymmetryAxes of Symmetry
PresentPresent 0909
0606 0404 0303 --

TETRAGONAL CRYSTAL SYSTEMS
Plane of
Symmetry
Plane of
Symmetry
PresentPresent 0505
0404 -- 0101 --

ORTHORHOMBIC CRYSTAL SYSTEMS
Plane of
Symmetry
Plane of
Symmetry
PresentPresent 0303
0303 -- -- --

MONOCLINIC CRYSTAL SYSTEMS
Centre of
Symmetry
Centre of
Symmetry
Plane of
Symmetry
Plane of
Symmetry
PresentPresent 0101
II fold
axes
II fold
axes
III fold
axes
III fold
axes
IV fold
axes
IV fold
axes
VI fold
axes
VI fold
axes
0101 -- -- --

TRICLINIC CRYSTAL SYSTEMS
Centre of
Symmetry
Centre of
Symmetry
Plane of
Symmetry
Plane of
Symmetry
PresentPresent --
II fold axesII fold axes
III fold
axes
III fold
axes
IV fold
axes
IV fold
axes
VI fold
axes
VI fold
axes
-- -- -- --

HEXAGONAL CRYSTAL SYSTEMS
Plane of
Symmetry
Plane of
Symmetry
PresentPresent 0707
0606 -- -- 0101
Symmetry Characters(trigonal)Symmetry Characters(trigonal)
Plane of
Symmetry
Plane of
Symmetry
PresentPresent 0303
0303 0101 -- ----

CRYSTALLOGRAPHIC PLANES
z
x
y
a b
c
Crystal plane may be defined as imaginary plane (or indeed any
parallel plane) in the crystal which intersects the main
crystallographic axes of the crystal

CRYSTALLOGRAPHIC NOTATION
Crystallographic notation is the method or symbolic representation
of relationship of any crystal face to crystallographic axes .
This crystallographic notation system can be explained by two
methods.
• Weiss parameter
• Miller indices

WEISS PARAMETER (Weiss notation/ Weiss indices)
These are the relative numbers of at which given crystal face cuts
the crystallographic axes.
Or
The distance from the origin at which crystal face intercept the
crystallographic axes.
z
x
y
a b
c

The most general expression for Weiss parameter is
na:mb:pc
Where n, m, p are the length cut off by the face on the a, b, & c axes
respectively

• Miller Indices are a symbolic representation for the orientation of plane in a
crystal & are defined as the reciprocals of the fractional intercepts
which the plane makes with the crystallographic axes.
• Most common Millerian symbol is h,k,l

To find the Miller indices of a plane, take the following steps:
1. Determine the intercepts of the plane along each of the three crystallographic
directions.
2. Take the reciprocals of the intercepts.
3. If fractions result, multiply each by the denominator of the smallest fraction.

61
CRYSTALLOGRAPHIC PLANES
example a b c
z
x
y
a b
c
4. Miller Indices (200)
1. Intercepts 1/2 ∞ ∞
2. Reciprocals 1/½ 1/∞ 1/∞
2 0 0
3. Reduction 2 0 0
z
x
y
a b
c
4. Miller Indices (110)
1. Intercepts 1 1 ∞
2. Reciprocals 1/1 1/1 1/∞
1 1 0
3. Reduction 1 1 0
example a b c

62
Axis X Y Z
Intercept
points 1 ∞ ∞
Reciprocals 1/1 1/ ∞
1/
∞
Smallest Ratio 1 0 0
Miller İndices (100)
Example-1
(1,0,0)

63
Axis X Y Z
Intercept points 1 1 ∞
Reciprocals 1/1 1/ 1 1/ ∞
Example-2
(1,0,0)
(0,1,0)

64
Axis X Y Z
Intercept points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1
(1,0,0)
(0,1,0)
(0,0,1)
Example-3

65
Axis X Y Z
Intercept points 1/2 1 ∞
Reciprocals 1/(½) 1/ 1 1/ ∞
(1/2, 0, 0)
(0,1,0)
Example-4

Crystal formA crystal form is a set of crystal faces that are related to each other by symmetry.
Types of crystal form
Crystal forms are broadly classified into two types
1. Closed Forms 2. Open Forms

There are 48 possible forms, those are mainly classified as
1.Common form in non-isomeric system (38)
2.Common form in isometric system(10)

1. Pedions
2. Pinacoids
3. Domes (dihedron)
4. Sphenoids
5. Prisms(7)
6. Pyramids(7)
7. Dipyramids or Bipyramids(7)
8. Trapezohedron(3)
9. Scalenohedrons(2)
10. Rhombohedrons
Common form in non-isomeric system
Trigonal prism: Ditrigonal prism: Rhombic prism: Tetragonal prism: Ditetragonal prism:
Hexagonal prism: Dihexagonal prism:
Trigonal pyramid: Ditrigonal pyramid: Rhombic pyramid: Tetragonal pyramid: Ditetragonal
pyramid: Hexagonal pyramid; Dihexagonal pyramid:
Trigonal dipyramid: Ditrigonal dipyramid: Rhombic dipyramid:
Tetragonal dipyramid: Ditetragonal dipyramid: Hexagonal dipyramid:
Dihexagonal dipyramid:
Trigonal Trapezohedron: Tetragonal Trapezohedron: Hexagonal Trapezohedron
Tetragonal Scalenohedron: Hexagonal Scalenohedron
Tetragonal Disphenoid

Common form in isometric system
1. Hexahedron or cube
2. Dodecahedron (rhombic dodecahedron)
3. Octahedron
4. Tetra hexahedron
5. Trapezohedron
6. Tetrahedron
7. Gyroid
8. Pyritohedron
9. Diploid
10. Tetartoid

1. Pyramids(7) Trigonal pyramid: Ditrigonal pyramid: Rhombic pyramid: Tetragonal pyramid:
Ditetragonal pyramid: Hexagonal pyramid; Dihexagonal pyramid:

Rhombic pyramid:
A pyramid is a open form composed 3, 4, 6, 8 or 12 non-parallel face, where all faces in the form
meet at common point and each face intersect all three crystallographic axis

Trigonal pyramid: Ditrigonal pyramid: Rhombic pyramid: Tetragonal pyramid:

Trigonal pyramid: 3-faced form where all faces are
related by a 3-fold rotation axis.
Ditrigonal pyramid: 6-faced form where all faces are
related by a 3-fold rotation axis.

Trigonal pyramid: Ditrigonal pyramid: Rhombic pyramid: Tetragonal pyramid:

Rhombic pyramid: 4-faced form where the faces are related by
mirror planes..
Tetragonal pyramid: 4-faced form where the faces are related by a 4
axis. In the drawing the small triangular faces that cut the corners
represent the tetragonal pyramid.

Ditetragonal pyramid: 8-faced form where all faces are related
by a 4 axis. In the drawing shown here, the upper 8 faces belong to
the ditetragonal pyramid form.
• Hexagonal pyramid: 6-faced form where all faces
are related by a 6 axis. If viewed from above, the
hexagonal pyramid would have a hexagonal shape.
• Dihexagonal pyramid: 12-faced form where all faces are
related by a 6-fold axis. This form results from mirror planes
that are parallel to the 6-fold axis.

pyramids or Bipyramids(7)
Trigonal dipyramid: Ditrigonal dipyramid: Rhombic dipyramid
Tetragonal dipyramid: Ditetragonal dipyramid: Hexagonal dipyrami
Dihexagonal dipyramid:
Rhombic dipyramid:
Dipyramids are closed forms consisting of 6, 8, 12, 16, or 24 faces.
Dipyramids are pyramids that are reflected across a mirror plane.

Prisms(7) Trigonal prism: Ditrigonal prism: Rhombic prism: Tetragonal prism: Ditetragonal
prism: Hexagonal prism: Dihexagonal prism:
Rhombic prism

Trigonal prism: 3 - faced form with all faces parallel to a 3 -fold
rotation axis
Trigonal prism: Ditrigonal prism: Rhombic prism: Tetragonal prism: Ditetragonal
Ditrigonal prism: 6 - faced form with all 6 faces parallel to a
3-fold rotation axis. i.e. it does not have 6-fold rotational
symmetry.

Rhombic prism: 4 - faced form with all faces parallel to a line
that is not a symmetry element. In the drawing to the right, the 4
shaded faces belong to a rhombic prism.

Tetragonal prism: 4 - faced open form with all faces parallel to a
4-fold rotation axis. The top and bottom faces make up the a form
called the top/bottom pinacoid.

• Ditetragonal prism: 8 - faced form with all faces parallel to
a 4-fold rotation axis. In the drawing, the 8 vertical faces
make up the ditetragonal prism.
• Hexagonal prism: 6 - faced form with all faces parallel to
a 6-fold rotation axis. Again the faces on top and bottom
are the top/bottom pinacoid form.

Dihexagonal prism: 12 - faced form with all faces parallel to a 6-
fold rotation axis. Note that a horizontal cross-section of this model
would have apparent 12-fold rotation symmetry. The dihexagonal
prism is the result of mirror planes parallel to the 6-fold rotation axis.
Trigonal prism: Ditrigonal prism: Rhombic prism: Tetragonal prism: Ditetragonal prism:
Hexagonal prism: Dihexagonal prism:

8. Scalenohedrons(2) Tetragonal Scalenohedron: Hexagonal Scalenohedron
Tetragonal Scalenohedron: Hexagonal Scalenohedron

8. Trapezohedron(3) Trigonal Trapezohedron: Tetragonal Trapezohedron: Hexagonal Trapezohedron
Trigonal Trapezohedron: Tetragonal Trapezohedron: Hexagonal Trapezohedron

Hexahedron or cube
A hexahedron is a general form with 6 squares at 900
angle to
each other. Each faces intersection crystallographic axis and
parallel to others. 4-fold axes are perpendicular to the face of
the cube.
Octahedron
An octahedron is an 8-faced form, faces are equilateral triangle
shape, and these are meeting at common point. That results
form a three 4-fold axes with perpendicular mirror planes.


Dodecahedron (rhombic dodecahedron)
A dodecahedron is a closed 12- rhomb shape faced form. Each
faces intersect two crystallographic axes are parallel to other one
axis.
Tetra hexahedron
The tetra hexahedron is a 24-isosceles triangular faced form with all
faces are parallel to one of the a axes, and intersect the other 2 axes
at different lengths.


Trapezohedron
An isometric trapezohedron is a 12-faced closed form with all faces
intersect two of the a axes at equal length and intersect the third
a axis at a different length.
Tetrahedron
The tetrahedron is a form composed 4equilateral triangular
faces, each of which intersects all the three crystallographic axes
at equal lengths.


Gyroid
A gyroid is a form in the crystal that has note no mirror planes and
This form has no center of symmetry.
Pyritohedron
The pyritohedron is a 12-faced form that occurs in the crystal and
each of the faces that make up the form have 5 sides.


Diploid
The diploid is the general form for the diploid.
Tetartoid
Tetartoids are general forms in the tetartoidal class which only
has 3-fold axes and 2-fold axes with no mirror planes.

Crystallography

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