An Introduction to Crystallography

CONTENTSCrystallography:
 Why we study Crystallography?
 Definition
 External characteristics of crystals
• Elements of crystals
 Crystal elements
 Crystal symmetry
 Crystal systems
 Crystal classes
 Axial ratios-crystal parameters and Miller indices
 Methods of Crystal Drawing
 Crystal habit and forms
• General Outlines of the crystal systems
 Cubic (Isometric) System
 Tetragonal System
 Orthorhombic System
 Hexagonal System
 Trigonal System
 Monoclinic System
 Triclinic System

 Atomic structure
 Central region called the nucleus
 Consists of protons (+ charges) and neutrons
(- charges)
 Electrons
 Negatively charged particles that surround
the nucleus
 Located in discrete energy levels called
shells

 Chemical bonding
 Formation of a compound by combining
two or more elements
 Ionic bonding
 Atoms gain or lose outermost (valence)
electrons to form ions
 Ionic compounds consist of an orderly
arrangement of oppositely charged ions

 Covalent bonding
 Atoms share electrons to achieve
electrical neutrality
 Generally stronger than ionic bonds
 Both ionic and covalent bonds typically
occur in the same compound

Covalent bond Model
- diamond (Carbon)

 Polymorphs
 Minerals with the same composition but
different crystalline structures
 Examples include diamond and graphite
 Phase change = one polymorph changing
into another

 Crystal form
 External expression of a mineral’s
internal structure
 Often interrupted due to competition for
space and rapid loss of heat

Why we study Crystallography?
It is useful for the identification of minerals. The
later are chemical substances formed under natural
conditions and have crystal forms.

Study of crystals can provide new chemical
information. In laboratories and industry, we can
prepare pure chemical substances by
crystallization process.
It is very useful for solid state studies of materials.
Crystal heating therapy
Crystallography is of major importance to a wide
range of scientific disciplines including physics,
chemistry, molecular biology, materials science and
mineralogy.

DEFINITION
• CRYSTALLOGRAPHY is simply a fancy
word meaning "the study of crystals"
• The study of crystalline solids and the principles
that govern their growth, external shape, and
internal structure
• Crystallography is easily divided into 3 sections -
- geometrical, physical, and chemical.
• We will cover the most significant geometric
aspects of crystallography

Classification of substances
• Crystalline Substances
• Amorphous substances

Properties of Crystalline Substances
1- Solidity 2- Anisotropy X Isotropy
3- Self-faceting ability 4- Symmetry
space lattice skeleton
The crystalline substances are characterise by the following properties:

Amorphous substances
(in Greek amorphous means “formless”) do not
have overall regular internal structure; their
constituent particles are arranged randomly; hence,
they are isotropic, have no symmetry, and cannot be
bounded by faces. Particles are arranged in them in
the same way as in liquids, hence, they are sometimes
referred to as supercooled liquids. Examples of
amorphous substances are glass, plastics. Glue, resin,
and solidified colloids (gels).

Curve of cooling of amorphous
substances
0
20
40
60
050100
time, min
To
Curve of cooling of a crystalline
subsatnce
0
10
20
30
40
50
60
050100
time, min
To
ab
In distinction to crystalline
substances, amorphous ones
have no clearly defined
melting point. Comparing
curves of cooling (or heating)
of crystalline substances and
amorphous substances, one
can see that the former has
two sharp bend-points (a
and b), corresponding to the
beginning and end
crystallization respectively,
whereas the latter is smooth.

Definition of Crystal
• A CRYSTAL is a regular polyhedral form,
bounded by smooth faces, which is assumed by
a chemical compound, due to the action of its
interatomic forces, when passing, under suitable
conditions, from the state of a liquid or gas to
that of a solid.

• A polyhedral form simply means a solid bounded
by flat planes (we call these flat planes
CRYSTAL FACES).
• A chemical compound" tells us that all minerals
are chemicals, just formed by and found in
nature.
• The last half of the definition tells us that a
crystal normally forms during the change of
matter from liquid or gas to the solid state.

Classification of crystals according
to the degree of crystallization
• Euhedral crystals
• Subhedral crystals
• Anhedral crystals
Euhedral Crystal Subhedral Crystal Anhedral Crystal

External characteristices of crystals
• Crystal faces
• Edge
• Solid angle
• Interfacial angle
• Crystal form
• Crystal habit

• Crystal faces: The crystal is bounded by
flat plane surfaces. These surfaces
represent the internal arrangement of
atoms and usually parallel to net-planes
containing the greatest number of lattice-
points or ions.
• Faces are two kinds, like and unlike.

• Edge: formed by the intersection of any two
adjacent faces.The position in space of an edge
depends upon the position of the faces whose
intersection gives rise to it.
• Solid Angles: formed by intersection of three
or more faces.
A
F
E
Edges………….E
Solid Angles (apices)…..A
Crystal Faces….F
Can you conclude mathematical
relation between them?

•Interfacial angle
we define the interfacial angle between two crystal
faces as the angle between lines that are perpendicular
to the faces. Such lines are called the poles to the
crystal face. Note that this angle can be measured
easily with a device called a contact goniometer.

Nicholas Steno (1669) a Danish physician and natural
scientist, found that, the angles between similar
crystal faces remain constant regardless of the size or
the shape of the crystal when measured at the same
temperature, So whether the crystal grew under ideal
conditions or not, if you compare the angles between
corresponding faces on various crystals of the same
mineral, the angle remains the same Steno's law is
called the CONSTANCY OF INTERFACIAL
ANGLES and, like other laws of physics and
chemistry, we just can't get away from it.

• Crystal forms: are a number of corresponding
faces which have the same relation with the
crystallographic axes.
• A crystal made up entirely of like faces is termed
a simple form. A crystal which consists of two or
more simple forms is called combination.
• Closed form: simple form occurs in crystal as it
can enclose space.
• Open form: simple forms can only occur in
combination in crystal
•The term general form has specific meaning in crystallography. In
each crystal class, there is a form in which the faces intersect each
crytallographic axes at different lengths. This is the general form {hkl}
and is the name for each of the 32 classes (hexoctahedral class of the
isometric system, for example). All other forms are called special
forms.

• Crystal Habit: the general external shape
of a crystal. It is meant the common and
characteristic form or combination of forms
in which a mineral crystallizes.(Tabular
habit, Platy habit, Prismatic habit, Acicular
habit, Bladed habit)

Elements of Crystallization
Crystal Notation
• Crystallographic axis
• Axial angles

Crystallographic axis
• All crystals, with the exception of those
belonging to the hexagonal and trigonal
system, are referred to three
crystallographic axis.

Axial angles
• ∝ is the angle between b axis and c axis
• β is the angle between a axis and c axis
• is the angle between a axis and b axis

Crystal Systems
• We will use our crystallographic axes which we just
discussed to subdivide all known minerals into these
systems. The systems are:
(1) CUBIC (ISOMETRIC) - The three crystallographic axes
are all equal in length and intersect at right angles (90
degrees) to each other.
β
Ɣ
α
a1 a2
a3

(2) TETRAGONAL - Three axes, all at right angles, two of
which are equal in length (a and b) and one (c) which is
different in length (shorter or longer).
(3) ORTHORHOMBIC - Three axes, all at right angles, and
all three of different lengths.
β
Ɣ
α
c
a1 a2
β
Ɣ
α
c
a b
TETRAGONAL ORTHORHOMBIC

• (4) HEXAGONAL - Four axes!
Three of the axes fall in the same plane and
intersect at the axial cross at 120 degrees
between the positive ends. These 3 axes,
labeled a1, a2, and a3, are the same
length. The fourth axis, termed c, may be
longer or shorter than the a axes set.

• (5) MONOCLINIC - Three axes, all unequal in
length, two of which (a and c) intersect at an
oblique angle (not 90 degrees), the third axis (b)
is perpendicular to the other two axes.
• (6) TRICLINIC - The three axes are all unequal
in length and intersect at three different angles
(any angle but 90 degrees).
c
a b
β
Ɣ
α
c
a
b
β
Ɣ
α
MONOCLINIC TRICLINIC

ELEMENTS OF SYMMETRY
• PLANES OF SYMMETRY
• Rotation AXiS OF SYMMETRY
• CENTER OF SYMMETRY.

PLANE OF SYMMETRY
• Any two dimensional surface (we can call it flat)
that, when passed through the center of the
crystal, divides it into two symmetrical parts that
are MIRROR IMAGES is a PLANE OF
SYMMETRY.
• In other words, such a plane divides the crystal
so that one half is the mirror-image of the other.
Horizontal planeVertical planeDiagonal plane

AXIS OF SYMMETRY
• An imaginary line through the
center of the crystal around
which the crystal may be rotated
so that after a definite angular
revolution the crystal form
appears the same as before is
termed an axis of symmetry.
• Depending on the amount or
degrees of rotation necessary,
four types of axes of symmetry
are possible when you are
considering crystallography

four types of axis of symmetry
• When rotation repeats form every 60 degrees, then we
have sixfold or HEXAGONAL SYMMETRY. A filled
hexagon symbol is noted on the rotational axis.
have fourfold or TETRAGONAL SYMMETRY. A filled
square is noted on the rotational axis.
have threefold or TRIGONAL SYMMETRY. A filled
equilateral triangle is noted on the rotational axis.
have twofold or BINARY SYMMETRY. A filled oval is
noted on the rotational axis.

Types of axis of symmetry
• BINARY SYMMETRY
Two fold system (180º)

• TRIGONAL SYMMETRY
Three fold system(120º)

• TETRAGONAL SYMMETRY
Four fold system(90º)

Six fold system(60º)
HEXAGONAL SYMMETRY

Symmetry Axis of rotary inversion
• This composite symmetry element combines a rotation
about an axis with inversion through the center.
• There may be 1, 2, 3, 4, and 6-fold rotary inversion axes
present in natural crystal forms, depending upon the
crystal system we are discussing.
- - - -

CENTER OF SYMMETRY
• Most crystals have a center of
symmetry, even though they
may not possess either planes of
symmetry or axes of symmetry.
Triclinic crystals usually only
have a center of symmetry. If
you can pass an imaginary line
from the surface of a crystal face
through the center of the crystal
(the axial cross) and it intersects
a similar point on a face
equidistance from the center,
then the crystal has a center of
symmetry.

Complete Symmetrical Formula
• We can use symbol to write the
symmetrical formula as following:
1- Plane of symmetry: m
2- Axis of symmetry: 2, 3, 4, 6 and we can
write the number of the axis at up left as 3
4
3- Center of symmetry: n
For example: the complete symmetrical
formula of hexoctahedral class of
Isometric system: 3
4/m 4
3 6
2/m n

Intercepts, Parameters and Indices
• Absolute Intercepts:The distances from
the center of the crystal at which the face
cuts the crystallographic axes.
• Relative Intercepts: divided the absolute
intercepts by the intercept of the face with
b axis.
• Ex: if the absolute intercepts (a:b:c)are
1mm : 2mm : ½ mm, the relative intercepts
will be ½ : 2/2 : ¼ = o.5 : 1 : o.25

Parameters
• The parameters of the crystal face are the
intercepts of this face divided by the axes
lengths.

-Parameters
Unit Face
oc
oc:
ob
ob:
oa
oa=
1:1:1
abc
def
2
1
:
3
1
:
4
1
=
oc
of
:
ob
oe
:
oa
od
anm
2:
3
4:
1=
oc
om:
ob
on:
oa
oa
If the face parallel to the axis,
Its intercept = ∞
Its Parameter=∞

Indices
• The Miller indices of a face consist of a series of
whole numbers which have been derived from
the parameters by their inversion and if
necessary the subsequent clearing of fractions.
• If the parameters are 111 so the indices will be
111
• If the parameters are 11∞ and on inversion 1/1,
1/1, 1/ ∞ woud have (110) for indices.
• Faces which have respectively the parameters 1,
1, ½ would on inversion yield 1/1, 1/1, 2/1 thus
on clearing of fractions the resulting indices
would be respectively (112)

• It is sometimes convenient when the exact
intercepts are unkown to use a general
symbol (hkl) for the miller indices.

c
ba
O
YX
Z
A
B
C
3-D Miller Indices (an unusually complex example)
a b c
unknown face (XYZ)
reference face (ABC)
2
1 4
Miller index of
face XYZ using
ABC as the
reference face
3
invert 1
2
4 3
clear of fractions (1 3)4

Miller indices
• Always given with 3 numbers
– A, b, c axes
• Larger the Miller index #, closer to the
origin
• Plane parallel to an axis, intercept is 0

What are the Miller Indices of face Z?
b
a
w
(1 1 0)
(2 1 0)
z

The Miller Indices of face z using x as the reference
b
a
w
(1 1 0)
(2 1 0)
z
a b c
unknown face (z)
reference face (x)
1
1 1
Miller index of
face z using x (or
any face) as the
reference face
1
invert 1
1
1 1
clear of fractions 1 00
(1 0 0)

b
a
(1 1 0)
(2 1 0)
(1 0 0)
What do you do with similar faces
on opposite sides of crystal?

b
a
(1 1 0)
(2 1 0)
(1 0 0)
(0 1 0)
(2 1 0)(2 1 0)
(2 1 0)
(1 1 0)(1 1 0)
(1 1 0)
(0 1 0)
(1 0 0)

Methods of Crystal Drawing
• Clingraphic Projection
• Orthogonal Projection
• Spherical Projection
• Stereographic Projection

3-Spherical Projection
Imagine that we have a crystal
inside of a sphere. From each
crystal face we draw a line
perpendicular to the face
(poles to the face).
Note that the angle is measured in the vertical plane
containing the c axis and the pole to the face, and the
angle is measured in the horizontal plane, clockwise
from the b axis.
The pole to a hypothetical (010) face will coincide
with the b crystallographic axis, and will impinge on
the inside of the sphere at the equator.

4-Stereographic Projection
Stereographic projection is a method used to depict the
angular relationships between crystal faces.
This time, however we
will first look at a cross-
section of the sphere as
shown in the diagram. We
orient the crystal such that
the pole to the (001) face
(the c axis) is vertical and
points to the North pole of
the sphere.
N
EW
(010)
(001)
(011)
(0-10)
(0-11)
ρ
ρ/2
Imagine that we have a crystal inside of a sphere.

For the (011) face we
draw the pole to the
face to intersect the
outside the of the
sphere. Then, we draw
a line from the point
on the sphere directly
to the South Pole of
the sphere.
N
EW
(010)
(001)
(011)
(0-10)
(0-11)
ρ
ρ/2
Where this line intersects the equatorial plane is
where we plot the point. The stereographic projection
then appears on the equatorial plane.

In the right hand-diagram we see the stereographic projection
for faces of an isometric crystal. Note how the ρ angle is
measured as the distance from the center of the projection to
the position where the crystal face plots. The Φ angle is
measured around the circumference of the circle, in a
clockwise direction away from the b crystallographic axis or
the plotting position of the (010) crystal face
N
EW
(010)
(001)
(011)
(0-10)
(0-11)
ρ
ρ/2
EW
(010)
(001)
(0-10) (011)(0-11)
ρ

1- The Primitive Circle is the circle that cross cuts
the sphere and separates it into two equal parts
(North hemisphere and South hemisphere). It is
drawn as solid circle when represents a mirror
plane.
The following rules are applied:
2- All crystal faces are plotted as poles (lines
perpendicular to the crystal face. Thus, angles
between crystal faces are really angles between
poles to crystal faces.
3- The b crystallographic axis is taken as the
starting point. Such an axis will be perpendicular to
the (010) crystal face in any crystal system. The
[010] axis (note zone symbol) or (010) crystal face
will therefore plot at Φ = 0° and ρ = 90°.

4- Mirror planes are shown as solid lines and curves.
The horizontal plane is represented by a circle
match with the primitive circle.
5- Crystal faces that are on the top of the crystal ρ <
90°) will be plotted as "+" signs, and crystal faces on
the bottom of the crystal (ρ > 90°) will be plotted as
open circles “ " .
6- The poles faces that parallel to the c
crystallographic axis lie on the periphery of the
primitive circle and is plotted as "+" signs.
7- The poles faces that perpendicular to the c
crystallographic axis lie on the center of the
primitive circle.
8- The pole face parallels to one of the horizontal
axes will plotted on the plane that perpendiculars to
this axis.

9- The Unit Face (that met with the positive ends of
the three or four crystallographic axes will be
plotted in the lower right quarter of the primitive
circle.
a
b
++
- +
+ -
- -
As an example all of the faces, both upper and
lower, for a crystal in the class 4/m2/m in the forms
{100} (hexahedron, 6 faces) and {110}
(dodecahedron, 12 faces) are in the stereogram to
the right
+
(001)(00-1)
+
++
+
+
(100)
(-100)
(010)(0-10)
+
++
++
+
+
(-110)
(-1-10)
(110)(1-10)
(101)(10-1)
(011)(01-1)(0-11)(0-1-1)
(-101)(-10-1)

Crystallographic forms
1- Pedion
It is an open form made up of a single face

1- Pinacoid
It is an open form made up of two parallel faces
Front pinacoid
Side
pinacoid
Basal pinacoid

3- Dome
It is an open form made up of two
nonparallel faces symmetrical with
respect to a symmetry plane
4- Sphenoid
It is an open form made up of two
nonparallel faces symmetrical with
respect to a 2-fold or 4-fold
symmetry axis

5- Disphenoid
It is an closed form composed of a four-faced form in which two
faces of the upper sphenoid alternate with two of the lower
sphenoid.

Bipyramid-6
It is an closed form composed of 3, 4, 6, 8 or 12 nonparallel faces
that meet at a point
Orthorhombic bipyramed
Ditetragonal bipyramid
Tetragonal bipyramid
Dihexagonal bipyramidHexagonal bipyramid

7- Prism
It is an open form composed of 3, 4, 6, 8 or 12 faces, all of which are
parallel to same axis.
Orthorhombic prism
Tetragonal prism
Ditetragonal prism
Hexagonal prism Dihexagonal prism

8- Rhombohedron
It is an closed form composed of 6
rhombohedron faces,
9- Scalenohedron
It is an closed form composed of 12 faces,
each face is a scalene triangle. There are
three pairs of faces above and three pairs
below in alternating positions

Crystallographic systems
Isometric system
β
Ɣ
α
a1 a2
a3
a3a2a1
Ɣ = 90βα
Class

1-Axis of symmetry
3
Isometric system
4
6

2- Center of symmetry
Isometric system

4 vertical plane
3- Plane of symmetry
Isometric system
1 horizontal plane 4 diagonal plane

43
34______
m n
62______
m
Isometric system

a
b (E)(W)
Stereographic Projection of Symmetry elements of the Isometric System

+
+
+
++
(100)
(010)
(-100)
(0-10)
1- Cube (Hexahedron)
Cubic form [100]
Crystal form
Isometric system
Stereographic Projection

+
+
+
++
(100)
(010)
(-100)
(0-10)
+
++
+
(111)
2- Octahedron
++
+
+
+
Crystal form
Isometric system
Octahedron [111] Stereographic Projection

Crystal form
Isometric system
Rhombic dodecahedron [110]+
+ ++
(100)
(010)10)
++
(111)
2- Octahedron
+
++
+
+
+
+
+
(110)
3- Rhombic dodecahedron
ereographic projection of Cubic System
rms.

Isometric system
Tetrahexahedron [hk0]
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
(210)
4- Tetrahexahedron
+
+ +
++ +

Isometric system
Trapezoctahedron [hll] +
++
+
+
++
+
+
+
+
(210)
4- Tetrahexahedron
+
+
+
+ ++
+
++ +
+
+
(211)
6-Trapezohedron

Trisoctahedron [hhl]
Isometric system
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
(210)
4- Tetrahexahedron
+
++
+
+
+
+
+
+
+
+
+
(221)
5- Trisoctahedron
+
+ +
+ +
++
+
+
+ +
+

Hexaoctahedron [hkl]
Isometric system
+
++
+
+
(210)
4- Tetrahexahedron
++ ++ (221)
5- Trisoctahedron
+
+
+
+ ++
+
++ +
+
+
(211)
6-Trapezohedron
+
+
+++
+
+
+
++
++
+
+
+ ++
+
+
+ +
+++
(321)
7- Hexaoctahedron

systemTetragonal
β
Ɣ
α
ca2a1 /
c
a1 a2
Ɣ = 90βα
Ditetragonal –
Bipyramid [hkl]
Class

1
4
systemTetragonal
1-Axis of symmetry

systemTetragonal

systemTetragonal
4 vertical plane
1 horizontal plane

4______
m
42______
m
n
systemTetragonal

Stereographic Projection of Symmetry elements of the Tetragonal System
a
b (E)(W)

Basal - pinacoid [001]
systemTetragonalCrystal form
+
1- Basal Pinacoid
(001)
(00-1)

Tetragonal prism of
first order [110]
systemTetragonal
+
1- Basal Pinacoid
(001)
(00-1)
2- Tetragonal prism of 1st order
+
++
+ (110)
+
+
+

Tetragonal prism of
second order [100]
systemTetragonal
ographic projection of Tetragonal
em Forms.
+
1- Basal Pinacoid
(001)
(00-1)
2- Tetragonal prism of 1st order
+
++
+ (110)
3- Tetragonal Prism of 2nd Order
+
+
+
+
(100)

Ditetragonal prism [hk0]
systemTetragonal
4- Ditetragonal prism
+
+
+
++
+
+
+
(210)
a
b
5-
b
+
+
++
+
+

systemTetragonal
Tetragonal – Bipyramid
of first order [hhl]
+
+
+
++
+
+
+
(210)
a
b
5- Tetragonal bipyramid of 1st Order
a
b
+
++
+
+
+
++
+
(111)

systemTetragonal
Tetragonal – Bipyramid
of second order [h0l]
+
++
+
(210)
a
b
5- Tetragonal bipyramid of 1st Order
a
b
++
6- Tetragonal bipyramid of 2nd Order
a
b
+
+
+
+
7- Ditetragonal bipyramid
a
b
+
+
+
++
+
+
+
(111)
(101)
(211)

systemTetragonal
Ditetragonal –
Bipyramid [hkl]
+
++
+
(210)
a
b
5- T
7- Ditetragonal bipyramid
a
b
+
+
+
++
+
+
+
(211)

Orthorhombic system
β
Ɣ
α
cba / /
c
a b
Ɣ = 90βα
Orthorhombic Bipyramid [hkl]
Class

7- Orthorhom bic Bipyram id {hkl}
 Exit
hkl
It is a closed form
com poses of 8 triangular
faces. It is the general
form of the orthorhom bic
holosym m etrical class.
Each face m et with the
crystallographic axes at
different distances {111}
or {hkl}.

3
1-Axis of symmetry
Orthorhombic system
2 vertical plane 1 horizontal plane

32______
m n
Orthorhombic system

Stereographic Projection of Symmetry elements of the Orthorhombic
System.
a
b (E)(W)

Orthorhombic system
Crystal form
Side
pinacoid
[010]
Front pinacoid [100]
Basal Pinacoid [001]
1- Basal Pinacoid
a
b+
2- Front Pi
a
b
+
+
(100)
(001)
b++
(010)
Stereographic projection of the Orthorhombic
System Forms.
1- Basal Pinacoid
a
b+
2- Front Pinacoid
a
b
+
+
(100)
(001)
3- Side Pinacoid
a
b++
(010)

Orthorhombic prism [hk0]
Orthorhombic system
4- Orthorhombic prism
a
b
+
++
+ (110)
b
++

Orthorhombic system
Orthorhombic front dome [h0l]
a
b
+
++
+ (110)
5- Front dome (b-Dome)
a
b
+
+
(101)
bb
++

Orthorhombic side dome [0kl]
Orthorhombic system
a
b
++ (110)
5- Front dome (b-Dome)
a
b
+
+
(101)
6- Side dome (a-Dome)
a
b++
(011)
7- orthorhombic bipyramid
a
b
+
++
+

Orthorhombic Bipyramid [hkl]
Orthorhombic system
a
b
++ (110)
7- orthorhombic bipyramid
a
b
+
++
+

Compound form
5- O rthorhom bic front dom e (b-dom e) or M acro
dom e {10l}
6- O rthorhom bic side dom e (a-dom e) or Brachy
dom e {01l}
01l
01l10l
100
Pinacoid

Hexagonal system
/ca3a2a1
a1
a2
-a
3
c
Ɣ
β α
90βα
Ɣ
Class
Dihexagonal bipyramid [hkwl]

61
Hexagonal system
1-Axis of symmetry 2- Center of symmetry

Hexagonal system
6 vertical plane 1 horizontal plane

Apatite
Ca5(PO4)3(OH, Cl,F)
-hexagonal structure
-prismatic habit
-major component teeth

6______
m
62______
m
n
Hexagonal system

a1
a2 (E)(W)
Stereographic Projection of Symmetry elements of the Hexagonal System
-a3

Hexagonal prism of
first order [1010]
-1010
-
a1
-a3 a2
0001
Hexagonal system
Crystal form
Basal pinacoid [0001]
Stereographic projection of the Hexagonal
System Forms.
a1
a2
-a3
+
1- Hexagonal Pinacoid
(0001)
a1
a2
-a3
2- Hexagonal prism of first order
(10-10)
+
+
+
+
+
+
a1
a2
-a3
3- Hexagonal prism of second order
+
+
++
+
+ (11-20)

hhw0
-
a1
-a
3
a2
Hexagonal systemHexagonal prism of
second order [hhw0]
-
Stereographic projection of the Hexagonal
System Forms.
a1
-a3
1- Hexagonal Pinacoid
a1
-a3
2- Hexagonal prism of first order
(10-10)
+
++
a1
a2
-a3
3- Hexagonal prism of second order
+
+
++
+
+ (11-20)

hkw0
-
Hexagonal system-Dihexagonal prism [hkw0]
a1
a2
-a3
4- Hexagonal Bipyramid of first order
+
(10-11)
++
a1
a2
-a3
6- Dihexagonal prism
(21-30)
+
+
+
+
+
++
+
+
+
+
+

Hexagonal Bipyramid of
first order [h0hl]
-
h0hl
-
a1
-a3
a2
Hexagonal system
a1
a2
-a3
+
(10-11)
+
+
+
+
+
a1
a2
+
+
+
+
+
++
+
+
+
+
+

Hexagonal Bipyramid of
second order [hhwl]
-
hhwl
-
a1 -a3
a2
Hexagonal system
a1
a2
-a3
+
(10-11)
+
+
+
+
+
a1
a2
-a3
5- Hexagonal Bipyramid of second order
+
(11-21)
+
++
+
+
a2
+
+
++
+
+
+ a2
+
+
++
+
+

Dihexagonal bipyramid [hkwl]
-
hkwl
-
Hexagonal system
a1
a2
-a3
+
(10-11)
++
a1
a2
-a3
5- Hexagonal Bipyramid of second order
+
(11-21)
++
+
a1
a2
-a3
6- Dihexagonal prism
(21-30)
+
+
+
+
+
++
+
+
+
+
+ a1
a2
-a3
7- Dihexagonal bipyramid
(21-31)
+
+
+
+
+
++
+
+
+
+
+

Compound form
Hexagonal prism (m = 6)
Hexagonal bipyramid (m = 12)

Trigonal system
Ɣ
β α
a1
a2
-a3
c
/ ca3a2a1
90βα
120Ɣ
ditrigonal scalenohedron
Class

31
Trigonal system

Trigonal system
3 vertical plane

______
32
m n3
Trigonal system

a1
a2 (E)(W)
Stereographic Projection of Symmetry elements of the Triagonal System
-a3

Forms
Basal Pinacoid
First Order Prism
Second Order Prism
Dihexagonal prism
Second Order bipyramid
Trigonal rhombohedron
Ditrigonal scalenohedron

Positive trigonal
rhombohedron [h0hl]
-
h0hl
-
a1 -a3
a2
Trigonal system
Crystal form
a1 -a3
a2
Positive rhombohedron {10-11}
+
++
a1
a2
+
+
+
+
+
+

Negative trigonal
rhombohedron [0kkl]
-
0kkl
-
a1 -a
3
a2
Trigonal system
a1 -a3
a2
a1 -a3
a2
Positive rhombohedron {10-11} Negative rhombohedron {01-11}
+
++
+
+
+
a1 -a3
a2
a1
a2
+
+
+
+
+
+
+
+
+ +
+
+

Positive ditrigonal
scalenohedron [hkwl]
-
hkwl
-
a1 -a3
a2
Trigonal system
Stereographic projection of the Tr
a1 -a3
Positive rhombohedron {10-11}
+
a1 -a3
a2
Positive Scalenohedron {21-31}
+
+
+
+
+
+

Negative ditrigonal
scalenohedron [hkwl]
-
hkwl
-
a1 -a3
a2
Trigonal system
Stereographic projection of the Triagonal System Forms.
a1 -a3 a1 -a3
Positive rhombohedron {10-11} Negative rhombohedron {01-11}
+
+
a1 -a3
a2
a1 -a3
a2
Negative Scalenohedron {12-31}Positive Scalenohedron {21-31}
+
+
+
+
+
+
+
+
+ +
+
+

Monoclinic system
90Ɣα
cba / /
β 90/
c
a b
β
Ɣ
α
Class

1
Monoclinic system

Monoclinic system
1 vertical plane

2______
m n
Monoclinic system

Stereographic Projection of Symmetry elements of the Monoclinic System
a
b (E)(W)

Monoclinic front pinacoid [100]
Monoclinic side pinacoid [010]
Monoclinic basal pinacoid [001]
Monoclinic system
• pinacoid
Crystal form
1- Basal Pinacoida
+(001)
2- Side Pinacoida
++
+
(00-1)
Stereographic projection of the Monoclinic
System Forms.
1- Basal Pinacoida
+(001)
2- Side Pinacoida
++
3- Front pinacoid
a
+
+
(00-1)

Monoclinic prism [hk0]
Monoclinic system
m {hk0} or {110}
4- Monoclinic Prism
a
+
++
+
++
(-111)

Positive hemibipyramid [hkl]
Monoclinic system
Positive Hemibipyramid {hkl} or {111}
Negative Hemibipyramid {-hkl} or {-111}
hkl
7- Hemibipyramid
Front View Back View
{111} {-111}
• hemibipyramid
Negative hemibipyramid [hkl]
-
54- Monoclinic Prism
a
++
a
a
+
Positive
(101)
7-Hemibipyramid
a a
++
(111)
++
(-111)
4- M
a
++
7
a
++
(-111)
NegativePositive

5- Side Dome (a-dome)4- Monoclinic Prism
a
+
+
a
++
a a
+ (101)
+(-101)
++
(111)
+
(-111)
-
011
Monoclinic system
• Dome
- hemi-orthodome
Positive hemi-orthodome [h0l]
Negative hemi-orthodome [h0l]
- side dome [0kl]
-
101
011
side dome [0kl]Positive
hemidome [h0l]
5- Side Dome (a-dome)4- Monoclinic Prism
a a
6- Hemi-orthodome
a a
+
Positive
(101)
+
Negative
(-101)
7-Hemibipyramid
a a
++
(111)
++
(-111)
hemi-orthodome

Triclinic system
cba / /
c
a
b
β
Ɣ
α
/α β Ɣ 90//
Class
Pinacoid

Triclinic system
1-Axis of symmetry = -
2- Center of symmetry = n
3- Plane of symmetry = -

n
Triclinic system

Stereographic Projection of Symmetry elements of the Triclinic System

front pinacoid [100]
side pinacoid [010]
basal pinacoid [001]
Triclinic system
Crystal form
Stereographic projection of theTriclinic System
Forms.
1- Basal Pinacoida a
a
2- Side Pinacoid
3- Frontl Pinacoid
+
+
+
+
+
Stereographic projection of theTriclinic System
Forms.
1- Basal Pinacoida a
a
2- Side Pinacoid
3- Frontl Pinacoid
+
+
+
+
+

Right hemi-prism [hk0]
Left hemi-prism [hk0]
Triclinic system
-
a a a a
a a
a a
a a
+
+
+
+
+
+ +
+ +
5- Hemi-b-dome {h0l}: two forms
{101} and {-101}
4- Hemi-a- dome { 0kl} : two forms
{011} and {0-11}
6- Hemi-prism{hk0} and {h-k0}
Upper left quarter bipyramid Upper right quarter bipyramid
Lower left quarter bipyramid Lower right quarter bipyramid

Hemi-brachydome(0kl) Hemi-macrodome(h0l)
Triclinic system
a a a a
+ +
+
+
5- Hemi-b-dome {h0l}: two forms
{101} and {-101}
{011} and {0-11}

Upper right quarter bipyramid [hkl]
Upper left quarter bipyramid [hkl]
Lower right quarter bipyramid [hkl]
Lower left quarter bipyramid [hkl]
-
-
--
Triclinic system
a a
a a
a a
+ +
+ +
+ +
{011} and {0-11}
Upper left quarter bipyramid Upper right quarter bipyramid
Lower left quarter bipyramid Lower right quarter bipyramid

Crystal Morphology
• The angular relationships, size and shape of
faces on a crystal
• Bravais Law – crystal faces will most commonly
occur on lattice planes with the highest density
of atoms
Planes AB and AC will be the most
common crystal faces in this cubic
lattice array

Unit Cell Types
in Bravais Lattices
P – Primitive; nodes at
corners only
C – Side-centered; nodes
at corners and in
center of one set of
faces (usually C)
F – Face-centered; nodes at
corners and in center
of all faces
I – Body-centered; nodes at
corners and in center
of cell

An Introduction to Crystallography

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