Introduction to Crystallography

Md. Hussain Monsur
Professor
Dept. of Geology
University of Dhaka
CRYSTALLOGRAPHY

Varieties of Beautiful Crystals

Crystals are so natural! So beautiful!

Crystals are so natural!
So beautiful!

CRYSTALLOGRAPHY
Crystallography is the branch of science which deals with the crystal: their development
and growth, external form, internal arrangement and physical properties.
The word "crystallography" derives from the Greek words crystallon. Combination of two
Words: Cold and congeal, means congealed by cold or "cold drop, frozen drop“.
J. Kepler (1619) Astronomer
Robert Hooke (1665) Inventor microscope
– Concept of crystals as regular arrangement of spherical particles.
Christian Huygens (1690): Studied Calcite crstal- Regular internal arrangement.
Nicolaus Stensen (1669): Law of constancy of angles between crystal faces.

A crystalline solid: atomic resolution image
of strontium titanate. Brighter atoms
are strontium and darker ones are titanium
Crystallography is the experimental science of determining the arrangement of
atoms in the crystalline solids.
Crystal structure of sodium
chloride (table salt)

The Koh-I-Noor was
mounted on the
Peacock Throne, the
Mughal throne of
India. It is said that
Shah Jahan, the ruler
who commanded the
building of the throne
and that of the Taj
Mahal was imprisoned
by his son and he
could only ever see
the Taj Mahal again
through the reflection
of the diamond.
Later, Shah’s son, Aurangazeb brought the Koh-I-Noor to the Badshahi Mosque in
Lahore. It was robbed from there by Nadir Shah who took the diamond to Persia in
1739, but the diamond found its way back to Punjab in 1813 after the deposed ruler
of Afghanistan, Shuja Shah Durrani took it to India and made a deal to surrender
the diamond in exchange for help in winning back the Afghan throne.
The Brits came across the gem when they conquered Punjab in 1849, and Queen
Victoria got it in 1851. The stone was then at 186 carats as before this point, the
diamond was not cut

Legend says that the diamond is 5000 years old and was
referred to in Sanskrit writings as the Syamantaka jewel
Raja of Gwalior in the 13th
centuryBabur documented (1526)Wt.
before cutting 186 carat (37gm). After
cutting (by Albert) 108.93 carats.
Oval shape.
Queen Elizabeth (later Queen Mother) wearing the Koh-I-Noor set in her
crown on the balcony of Buckingham Palace, after the coronation of King
George VI, with daughter Princess Elizabeth, now Queen Elizabeth II.

(1s)2(2s)2(2p)2
Carbon-Carbon bond - Hybridization
sp3-Hybridization (Diamond)
Sp2= Hybridization (graphite)
Carbon atom

sp3-Hybridization (Diamond)
Sp2= Hybridization (graphite)

DIAMOND (Multi faceted ball)DIAMOND ATOMIC STRUCTURE
GRAPHITE ATOMIC STRUCTURE
There are no covalent bonds between the layers and so the layers
can easily slide over each other making graphite soft and slippery
and a good lubricant.DIAMOND ATOMIC STRUCTURE

VIDEO BONDING OF
DIAMOND AND GRAPHITE
Carbon-Carbon bond - Hybridization

DEFINITIONS
MINERAL:Mineral is a naturally occurring homogeneous solid having external form, regular arrangement of
internal structure and a chemical formula.
CRYSTAL: A crystal is a body that is formed by the solidification of a chemical element, a compound, or a
mixture and has a regularly repeating internal arrangement of its atoms and often external plane faces.
Congealed by cold. Old English cristal "clear ice, clear mineral, “from
Latin crystallus "crystal, ice," from Greek krystallos,from kryos "frost," from PIE root *kru(s)-
"hard, hardouter surface"
CRYSTALLOGRAPHY: Crystallography is the branch of science which deals with crystals, their growth and
development, external form and internal structures.

Crystals are found in three forms
1. Euhedral, 2. Subhedral, 3.
Anhedral
SOME DEFINITIONS
1. FACES – Crystals are bounded by smooth plane surfaces
(some varieties of diamond have curve faces),
these are called Faces. a) Like faces &
a) Unlike faces
2. EDGES – The intersecting line of two adjacent faces is
called Edge.
3. ZONE AND ZONE AXIS
4. INTERFACIAL ANGLE – The interfacial angle between two crystal faces
as the angle between lines that are perpendicular to the faces. Such a lines are
called the poles to the crystal face. The interfacial angle is the angle between
two normal to two intersecting faces. The interfacial angles between
corresponding faces of the same mineral will be the same. This is known as
the Law of constancy of interfacial angles,

Space Lattice
A space lattice is an array of points showing how particles (atoms, ions or molecules) are arranged at different sites in three
dimensional spaces. Crystals, of course, are made up of 3-dimensional arrays of atoms. Such 3-
dimensional arrays are called space lattices. The ordered internal arrangement of
atoms in a crystal structure is called a LATTICE.
Unit Cell
The unit cell may be defined as, “the smallest repeating unit in space lattice which, when repeated over again, results in a crystal of
the given substance. The unit cell may also be defined as the unit parallelepiped which is repeated throughout the crystal by
translation along any lattice row.
Space Lattice and Unit Cell

CRYSTALLOGRAPHIC AND COORDINATE AXES
Crystallographic Axes
The crystallographic axes are imaginary lines within the crystal lattice. These
define a coordinate system within the crystal. Depending on the symmetry of the
lattice, the directions may or may not be perpendicular to one another, and the
divisions along the coordinate axes may or may not be equal along the axes. The
INTERCEPTS are distances between the centre of the crystal (point of intersection
of the crystallographic axes) and the points of intersection of the face and axes.
PARAMETERS are the ratios of the intercepts. This is known as parameter system
of Weiss.
Symmetry axes are equal to Coordinate axes.
The angles between the axes are equal to

A lattice system is generally identified as a set of lattices with the same shape according
to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ).

Crystallographic axes
Coincide with coordinate axes
Crystallographic axes
do not coincide with
coordinate axes
CRYSTALLOGRAPHIC AND
COORDINATE AXES
Lower Systems : Monoclinic, Triclinic and Orthorhombic.
Intermediate Systems: Tetragonal and Hexagonal.
Higher System: Isometric (Cubic).

Crystal family Lengths Angles Common
examples
Isometric a=b=c α=β=γ=90°
Garnet, halite,
pyrite
Tetragonal a=b≠c α=β=γ=90° Rutile, zircon,
andalusite
Orthorhombic a≠b≠c α=β=γ=90°
Olivine,
aragonite,
orthopyroxenes
Hexagonal a=b≠c α=β=90°,
γ=120°
Quartz, calcite,
tourmaline
Monoclinic a≠b≠c α=γ=90°, β≠90°
Clinopyroxenes,
orthoclase,
gypsum
Triclinic a≠b≠c α≠β≠γ≠90° Anorthite, albite
kyanite

CRYSTAL SYSTEMS WITH UNIT CELLS

a b
c
a b
c
a
b
c
a
b
c
a
b
c
a b
c

Videoof crystal Lattice
and Unit Cell

Videoof crystal Lattice structures
The Simple Cubic Lattice

A face is more commonly
developed in a crystal if it
intersects a larger number of
lattice points. This is known as
the Bravais Law (1848).
Faces are more commonly
Develops and 1 &2.

Symbols
•P - Primitive: simple unit cell
•F - Face-centered: additional point in the center of each face
•I - Body-centered: additional point in the center of the cell
•C - Centered: additional point in the center of each end
•R - Rhombohedral: Hexagonal class only
Auguste Bravais 14 arrangement of space lattices
(born Aug. 23, 1811, Annonay, Fr.—died March 30, 1863, Le Chesnay)

Isometric Cells
The F cell is very important because it is the pattern for cubic closest packing.
There is no C cell because such a cell would not have cubic symmetry.

Tetragonal Cells
A C cell would simply be a P cell with a
smaller cross-section, as shown below. An
F cell would reduce to a network of I cells.

Hexagonal Cells
The R cell is unique to hexagonal crystals. The two interior points
divide the long diagonal of the cell in thirds. This is the only Bravais
lattice with more than one interior point. A rhombohedron can be
thought of as a cube distorted along one of its diagonals.

Orthorhombic Cells
The orthorhombic class is the only one with all
four types of Bravais lattice

Monoclinic and Triclinic Cells
Monoclinic F or I cells could also be represented as C cells. Any other
triclinic cell can also be represented as a P cell.
Monoclinic Monoclinic Triclinic

Bravi’s 14 Different Types Of Space Lattices

Symmetry is a special characteristic of crystal. Crystals are subdivided into 32 classes on the basis of symmetry elements. Two
figures (parts) or two bodies (parts) are said to be symmetrical when they coincide if they are matched. The transformation or
matching of two symmetrical bodies are called symmetry operation and the object by which two bodies or figures become
symmetrical is called Element of Symmetry. Hence, symmetry operation is spatial transformation (rotations, reflections and
inversions)
SYMMETRY
1. Plane of symmetry (P)
2. Axis of symmetry (A)
3. Rotation-reflection Axis ( )
4. Cetre of symmetry (C)
Symmetry Elements

Equilateral Triangle Isosceles Triangle
Scalene Triangle
(No symmetry)
Reflection Symmetry
(Plane of Symmetry-P)

Rotational Symmetry
A shape has Rotational Symmetry when it still looks the same after a rotation. Sometimes a figure turns into congruent
position when it rotates about an axis. The line about which a figure rotates and turns into congruent position is called Axis of
Symmetry. The angle of rotation by which a figure turns into congruent position is called Elementary angle of rotation. The
number of turning of congruent position through a complete rotation is called Fold of an axis.

Axis of Symmetry (A)
There may be:
Two Fold Symmetry Axis (A2)
Three Fold Symmetry Axis (A3)
Four Fold Symmetry Axis (A4)
Six Fold Symmetry Axis (A6)
Axis of Rotary Inversion (A4
2 & A6
3)
Five Fold Symmetry Axis and more than Six
Fold Symmetry axes can not axist in crystals

CRYSTALLOGRAPHIC AND COORDINATE AXES
Crystallographic Axes
The crystallographic axes are imaginary lines within the crystal lattice. These define a
coordinate system within the crystal. Depending on the symmetry of the lattice, the
directions may or may not be perpendicular to one another, and the divisions along
the coordinate axes may or may not be equal along the axes. The intercept, made by
the unit cell on three crystallographic axes are called PARAMETERS.
Symmetry axes are equal to Coordinate axes.
The angles between the axes are equal to

Step 2 : Specify the intercepts in fractional co-ordinates
Co-ordinates are converted to fractional co-ordinates by dividing by the respective cell-dimension - for example, a point
(x,y,z) in a unit cell of dimensions a x b x c has fractional co-ordinates of ( x/a , y/b , z/c ). In the case of a cubic unit cell
each co-ordinate will simply be divided by the cubic cell constant , a . This gives
Fractional Intercepts : a/a , ∞/a, ∞/a i.e. 1 , ∞ , ∞
Step 3 : Take the reciprocals of the fractional intercepts
This final manipulation generates the Miller Indices which (by convention) should then be specified without being separated
by any commas or other symbols. The Miller Indices are also enclosed within standard brackets (….) when one is specifying
a unique surface such as that being considered here.
The reciprocals of 1 and ∞ are 1 and 0 respectively, thus yielding
Miller Indices : (100)
So the surface/plane illustrated is the (100) plane of the cubic crystal.
Step 1 : Identify the intercepts on the x- , y- and z- axes.
In this case the intercept on the x-axis is at x = a ( at the point (a,0,0) ), but the surface is parallel to the y- and z-axes -
strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at infinity ( ∞ ) for the
special case where the plane is parallel to an axis. The intercepts on the x- , y- and z-axes are thus
Intercepts : a , ∞ , ∞
procedure Of
DETERMInation
OF MILLER INDICES

procedure OF DETERMINe OF MILLER INDICES
Solution:
1.Since the plane passes through the existing origin the new origin must be
selected at the corner of adjust unit cell.
2.As related to new origin the following intercepts (in terms of lattice parameters
a, b, and c) with x, y, z axes can be referred: (plane is // to x-axis), -1, 1/2
3.The reciprocal of these numbers are: 0, -1 and 2 and they are already integer!
4.Thus the Miller indices of the consider plane are: (0-12)

1. The (110) surface
Assignment
Intercepts : a , a , ∞
Fractional intercepts : 1 , 1 , ∞
Assignment
Intercepts : a , a , a
Fractional intercepts : 1 , 1 , 1
Assignment
Intercepts : ½ a , a , ∞
Fractional intercepts : ½ , 1 , ∞
Exercises

MILLER INDEX NOTATION
The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a
crystal form with the unit-cell axes a, b, c (see Figure 1) are inversely proportional to prime
integers, h, k, l. They are called the MILLAR INDICES (hkl) of the face. They are usually
small because the corresponding lattice planes are among the densest and have therefore a
high inter-planar spacing and low indices.
The Miller indices of the Planes ABC' , ABC,ABC" AA"BB" are (112) ,
(111), (221), (110),respectively. These planes have AB , or, as common
zone axis.
Exercises

procedure OF DETERMInation OF MILLER INDICES
(Lecture)

LAW OF RATIONAL INDICES
The intercepts, made by a unit cell on three crystallographic axes
are called parameters. The parameters are denoted by small letters,
a, b and c. Rational Indices are reciprocal of the parameters. Any crystal face in
space can be represented by three whole numbers, if the
intercept, made by a unit cell on three crystallographic axes
are taken as unit of measurement (Law of Rational Indices).
The law of rational indices was deduced by Haüy (1784, 1801)

DETERMINATION OF MILLER’S INDICES
Some exercises

DETERMINATION OF MILLER’S
INDICES
Some exercises

DETERMINATION OF
MILLER’S INDICES
Some exercises

MILLAR INDICES FOR HEXAGONAL AND TRIGONAL SYSTEMS

LAW OF RATIONAL INDICES: Any crystal face in space can be represented by three whole
numbers, If the crystallographic axes are taken as coordinate axes and if the intercepts make by a unit
cell on crystallographic axes are taken as the unit of measurement.
LAW OF CONSTANCY OF INTERFACIALANGLES: The angles between
corresponding faces of the same mineral will be the same. This is known as the Law of constancy of
interfacial angles,
BRAVAIS LAW: A face is more commonly developed in a crystal if it intersects a larger
number of lattice points. This is known as the Bravais Law.
SOME IMPORTANT LAWS

CRYSTALFORMS
A CRYSTAL FORM is a set of crystal like faces that are related
to each other by symmetry, i.e. a set of like faces in a crystal
make a specific form. There are debates in writing symbols of
face and form. To avoid confusions, we shall represent the
face and form symbols in the following manner:
For example, in the case of a Cube,
Face : 001 - indicate top face of a cube, without bracket.
Form: (001)6 - Symbol of top face of a cube, with bracket.
Number 6 means the forms is composed of six
like faces. General symbol (hkl)n ‘h’ is less than ‘l’
less than ‘k’, n is the number of faces in a form.
The number of faces in a form depends
on the symmetry of the crystal.

The 48 Special Crystal Forms
Any group of crystal faces related by the same symmetry is called a form. There are 47 or 48
crystal forms depending on the classification used.
There are two kinds of forms in crystals
1. Open form (17 or 18)
2. Closed forms (30)
Open forms are those groups of like faces all related by symmetry
that do not completely enclose a volume of space.
Closed forms are those groups of like faces all related by symmetry
that completely enclose a volume of space.
Crystals are bounded by SIMPLE FORMS (all are like Faces) and
COMBINATION (bounded by Like and Unlike Faces).

Pedion: A single face unrelated to any other by symmetry.
Pinacoid: A pair of parallel faces related by mirror plane or twofold symmetry axis.
Dihedron: A pair of intersecting faces related by mirror plane or twofold symmetry axis. Some
crystallographers distinguish between domes (pairs of intersecting faces related by mirror plane)
and sphenoids (pairs of intersecting faces related by twofold symmetry axis).
Pyramid: A set of faces related by symmetry and meeting at a common point. All are Open form
SIMPLE FORMS OF LOWER SYSTEMS
Triclinic, Monoclinic and Orthorhombic Systems

Simple forms of the intermediate System
(Tetragonal and Hexagonal Systems)
3-, 4- and 6-Fold Prisms
A collection of
faces all are
parallel to a
symmetry axis.
All are open.
Prismatic (Gypsum)

3-, 4- and 6-Fold Pyramids
PYRAMIDS
A Pyramid: A set of triangular
like faces intersecting at a point
on a symmetry axis. All are open.
The base of the pyramid would
be a pedion.

3-, 4- and 6-Fold Bipyramids
(Beryl)

Disphenoid: A solid with four congruent triangle faces, like a distorted tetrahedron. Midpoints of
edges are twofold symmetry axes. In the tetragonal disphenoid the faces are isoceles triangles and a
fourfold inversion axis joins the midpoints of the bases of the isoceles triangles.
Scalenohedron: A solid made up of scalene triangle faces (all sides unequal)
Trapezohedron: A solid made of trapezia (irregular quadrilaterals)
Rhombohedron: A solid with six congruent parallelogram faces. Can be considered a cube distorted
along one of its diagonal three-fold symmetry axes
Scalenohedra and Trapezohedra

Six square faces. Each
intersects one cryst. Axis
and parallel to other two.
Form symbols: (001)6
8 equilateral triangular
faces. Each face cuts
all cryst. axes at equal
distance. symbol: (111)8
12 rhombo shaped
faces. Each face cuts
two cryst. axes at equal
Distance and parallel to
third. symbol: (011)12
24 isosceles traiangular
faces. Each face cuts two
cryst. axes at unequal
lengths and parallel to
third. symbol (okl,012)24
24 trapezoid faces. Each face
cuts two cryst. axes at equal
lengths and third a smaller
length. symbol: (hhl, 112)24
24 isosceles traiangular faces.
Each face cuts two cryst. axes at
equal lengths and third at greater
length. symbol: (hll, 122)24
24 scalene traiangular faces.
Each face cuts all cryst. axes
At unequal lengths.
Symbol (hkl,012)48
SIMPLE FORM
NORMAL CLAS
CUBIC SYSTE
HigherSystem,CUBIC

Tetrahedron: Four equilateral
triangle faces (111)
Trapezohedral
Tristetrahedron:
12 kite-shaped faces (hll)
Trigonal Tristetrahedron:
12 isoceles triangle faces (hhl).
Like an tetrahedron with a low
triangular pyramid built
on each face.
Hextetrahedron: 24 triangular
faces (hkl) The general form.
Simple forms of higher System
(Cubic Systems)
Hextetrahedral Forms

SIMPLE FORMS OF HIGHER SYSTEM
(Isometric System)
Tetartoidal, Gyroidal and Diploidal Forms
Tetartoid: The general form for symmetry class 233. 12 congruent irregular pentagonal faces. The name
comes from a Greek root for one-fourth because only a quarter of the 48 faces for full isometric
symmetry are present.
Gyroid: The general form for symmetry class 432. 24 congruent irregular pentagonal faces.
Diploid: The general form for symmetry class 2/m3*. 24 congruent irregular quadrilateral faces. The
name comes from a Latin root for half, because half of the 48 faces for full isometric symmetry are
present.
Pyritohedron:Special form
(hk0) of symmetry class 2/m3*.
Faces are each perpendicular
to a mirror plane, reducing
the number of faces to 12
pentagonal faces. Although
this superficially looks like
the Platonic solid with 12
regular pentagon faces,
these faces are not regular.

Carl's Gold Figures
A schematic diagram relating the various form types
within the holohedral isometric crystal class

Isometric - This system comprises crystals with three axes, all
perpendicular to one another and all equal in length.
Basic
wooden model Halite (salt)

Basic wooden model Apophyllite
Tetragonal - This system comprises crystals with
three axes, all perpendicular to one another; two
are of equal length.

Orthorhombic - This system comprises crystals with three
mutually perpendicular axes, all of different lengths.
Golden Topaz
Basic
wooden model

Monoclinic - This system comprises crystals with three axes of
unequal lengths, two of which are oblique (that is, not
perpendicular) to one another, but both of which are
perpendicular to the third
GypsumBasic wooden model

Triclinic - This system comprises crystals with three axes, all
unequal in length and oblique to one another
Orthoclase
Basic
wooden model

Hexagonal - This system comprises crystals with four axes.
Three of these axes are in a single plane, symmetrically spaced,
and of equal length. The fourth axis is perpendicular to the
other three.
Sapphire
Basic wooden model

Introduction to Crystallography

Introduction to Crystallography

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