Crystallography (Geo 221)
Unit One
Introduction to Crystallography
and Mineral Crystal Systems
Course Director
Dr. Bassam A. Abuamarah
Why Crystallography in Geosciences?
*Most of the Earth is made of solid rock. The basic units
of these units are made of minerals.
*Minerals are natural crystals, and so the geological
world is largely a crystalline world.
* The properties of rocks are ultimately determined by the properties of the
constituent minerals that appear via microscopic studies.
For example,
 Rocks’-forming minerals formed via geological processes of sedimentation,
igneous or metamorphism activities are controlled by a small-scale process of the
movement of the atom (diffusion), and have packed (arranged) as a crystal form
(dislocation movement), throughout new crystal growth (nucleation,
crystallization), and phase transformations.
Why Crystallography in Geosciences?
 An understanding of mineral structures and properties
allows us to answer questions, such as why quartz and
diamond are so hard, and why solid granite rock is
intended to become soft, sticky clay.
Therefore
 Minerals are natural resources, providing raw materials
for many industries. Therefore, understanding minerals
have geological as well as economic applications.
 Definition of “MINERAL”: A mineral is a naturally
occurring, inorganic solid with characteristic chemical
composition and a crystalline structure.
Definition of Crystallography
CRYSTALLOGRAPHY is the study of minerals’ crystals.
Crystallography is the science of determining the arrangement
of atoms within any solid structure.
Crystals are natural solids material (mineral) that form by a
regular ordered arrangement of atoms in all directions with a
definite internal & external solid form with a symmetrical
arrangement of plane faces
 In some solids (Crystal/Mineral) the arrangements of its
building blocks (atoms and molecules) can be random or very
different throughout the material.
In crystals, however, It is a collection of atoms called the Unit Cell.
The Unit cell is repeated in exactly the same arrangement over and over
throughout the entire material.
 Very slow cooling of a liquid allows atoms to arrange themselves
into an ordered pattern, which may extend of a long-range (millions
of atoms). This kind of solid is called crystalline.
Structural Properties of a Crystal
Thus
Crystalline structures of all minerals are made of different
elemental compounds and is defined as a framework of its body.
The shape of a crystal is based on the atomic structure of these
elemental building blocks.
Atoms in minerals arranged in an ordered geometric pattern called
a "motif" which determines its "crystal structure.“
A minerals’ crystal structure will determine their symmetry, optical
properties, cleavage planes, and overall geometric shape.
 A crystal’s growth pattern is referred to as its “Crystal Habit.“ i.e.
External shape of the crystal.
I- Crystal Structure (CS)
1.
Crystal Lattice :
is naturally obtained by attaching atoms or ions or molecules groups in an
orderly repeated structure along the principal direction to form a 3-dimensional
symmetric pattern inside crystalline solid as a point in minerals, not their
external appearance called Crystal Lattice.
 Characteristics of Crystal Lattice
• In a crystal lattice, each atom, molecule or ions (constituent
particle) is represented by a single point.
• These points are called lattice sites or lattice points.
• Lattice sites or points are together joined by a straight line in a crystal lattice.
• When we connect these straight lines we can get a three-dimensional view of
the structure. This 3D arrangement is called Crystal Lattice also known as
Crystal Lattice.
I- Crystal Structure (CS)
2. Unit cell
 Unit Cell is the smallest part (portion) of a
crystal lattice. It is the simplest repeating unit in
a crystal structure.
 The entire lattice is generated by the repetition
of the unit cell in different directionc
1. Parameters of a Unit Cell
 There are six parameters of a unit cell.
 These are the 3 edges which are a, b, c
and the angles between the edges which
are α, β, γ.
 The edges of a unit cell may be or may not
be perpendicular to each other
I- Crystal Structure (CS)
2. Types of Unit Cell
I.
Primitive Unit Cells
• When the constituent particles occupy only the corner positions,
it is known as Primitive Unit Cells.
II. Centered Unit Cells
• When the constituent particles occupy other positions in addition
to those at corners, it is known as Centered Unit Cell. There are 3
types of Centered Unit Cells:
 Body Centered: When the constituent particle at the centre
of the body, it is known as a body-centred Unit cell.
 Face Centered: When the constituent particle is present at
the center of each face, it is known as the Face Centered
Unit cell.
 End Centered (Base Unit): When the constituent particle is
present at the centre of two opposite faces, it is known as
an End Centered Unit cell.
 Crystal Crystalline solid materials Structures materials involve the
following:
I.
Crystalline
Is a solid material whose constituents are atoms, molecules
or ions that are arranged in an orderly repeating pattern.
The crystalline structure formation process is formed via a
crystallization/solidification of fluid process
For example:
Quartz mineral
Types of Crystals structures: there are 4 types of crystals, as below:
(1) ionic crystal,
(2) metallic crystal,
(3) covalent network crystal,
and (4) molecular crystal
Therefore,
Crystal structure divided on:
1. Perfect crystal: formed by a slowly cooling rate of material, it often
enables a more perfect crystal structure to form which has stronger
material properties.
2. An imperfect crystal is a crystal in which has a regular,
periodic structure that is interrupted by various defects.
1- Ionic crystals -are solid crystals with a high melting point, of an attracting
positive cations to negative anions held together by electrical
attraction and arranged together in a regular, geometric
structure is called a crystal lattice.
 Examples of such crystals are the alkali halides,
which include: potassium fluoride (KF), and NaCl
2- Metallic crystals
 are made of atoms in minerals lose electrons to
form cations by delocalizing electrons
surrounding the ions creating Metallic bonds
(electrostatic interactions between the ions and
the electron cloud) holding metallic solid
together.
 Atoms arranged in the closely packed sphere
which are responsible for conductivity.
 Due to the tightly-packed crystal lattice of the metallic structure.
 These crystals have a shiny of metallic lustre on their outer surface
appearances such as copper, gold, aluminium, and iron.
Crystal Structure
3- Covalent network crystals -is a crystal consisting of atoms at the lattice points of the
crystal, with each atom being covalently bonded to its nearest
neighbor atoms.
The covalently bonded network is three-dimensional and contains
a very large number of atoms. (Diamond and Graphite).
4- Molecular crystals -consist of molecules held together rather than atoms or ions by
relatively weak intermolecular forces ( Van Der Waals forces)
rather than by ionic or covalent bonds.
Note: A molecule is the smallest particle in a chemical element
or compound that has the chemical properties of that element or
compound.
II. Polycrystalline
 Is solids that are composed of many crystallites/ grains
of varying size and orientation.
 Most crystalline solids are composed of a collection of
many small crystals or grains of varying size and
orientation.
 These have random crystallographic orientations
formed when the metal starts with crystallization,
 In their phase change as launched in a small crystals
(grains) known as crystallites, that grow until they fuse,
forming a polycrystalline structure.
Crystal Structure
III. Amorphous solid (Non-crystalline solid): is material that does not
have a definite geometric or crystalline shape, atoms are not an
orderly repeating pattern, atoms and molecules are not organized in a
definite lattice pattern such as Glass.
Therefore,
 Crystalline solids have well-defined edges and
faces, diffract x-rays, and tend to have sharp
melting points.
 Amorphous solids have irregular or curved
surfaces, do not give well-resolved by x-ray
diffraction patterns and melt over a wide range of
temperatures.
II – Elements of Crystal Structures
II. Crystal elements are:
1. Faces : is the internal atomic plane of structure bounded by a flat surface and
has shape.
2.
Edge: is a line of intersection between 2 adjacent planes or faces .
3. Solid Angle: is the point 3 edges or more meet 3 or more adjacent faces..
4.
Crystal Form: is a group of faces which have 9 position with the reference of
the different or equal crystallographic axes indicating growing history
Crystal Structures
III. Types of Crystals Forms:
1- Simple form and combined forms:
a)Simple form of Crystals is made of all like crystal faces
such as Cubic all its faces are square.
b) Combination crystal’s form: When a crystal is made up of two or more
simple forms, such as when it consists of basal pinacoid and prism faces,
each of which in itself is a simple
2- Open Form:
• Pinacoids form is an open 2-faced form made up of two parallel
opposites faces their faces cannot enclose space all by themselves,
as they do not have an adequate number of faces to close, as a
result, they occur only as a combination with other forms, They
are open,
•
Pedions is one faced form that occurs in the (Triclinic System), and is not
related to any other face by symmetry, each form symbol refers to a
single face.
Crystal Structures
3 Closed forms:
is a set of crystal faces that completely enclose space. Thus, in
crystal classes that contain closed forms, a crystal can be
made up of a single form., which can enclose a volume of
space.
4 -General form:
A general form is a form in a particular crystal class that contains faces
that intersect all crystallographic axes at different lengths..
 Besides to the above classification, forms have also been classified as:
Crystal Structures
(a) Holohedral forms:
These forms having as many planes faces exhibit the highest
degree of symmetry possible in a system.
(b) Hemihedral forms:
These crystal forms have only half planes or faces required a full
maximum symmetry of the system to which it belongs, e.g., the
tetrahedron is a hemihedral form of an octahedron.
(c) Hemimorphic forms:
Crystal forms have no transverse plane of symmetry and no centre of
symmetry and are composed of forms belonging to only one end of the
axis of symmetry.. Hemimorphic forms lack a centre of symmetry.
(d) Tetartohedral forms:
Crystal have only a one fourth of the number of faces or planes of
the corresponding holohedral form. The form have neither plane nor
centre of symmetry.e, the number of planes required complete
symmetry
Crystal Structures
(e) Enantiomorphic forms:
is composed of faces placed on two crystals of the same mineral in such a
way that faces on one crystal become the mirror image of the form of faces
on the other crystal.
Common Forms in Crystallography:
(i) Pedion:
It is represented by one face only.
(ii) Pinacoid:
Prisms
It is an open form, consisting of two faces which Trigonal
3 faces
cuts one crystallographic axis and remain
parallel to the remaining axes.
(iii) Prism:
It is also an open form, consisting of three or
parallel faces. Depending on the symmetry,
several different kinds of prisms are possible, Tetragonal Prisms
4faces
which essentially parallel the vertical axis and
cuts one or more horizontal axes.
Ditrigonal Prisms
6 faces
Ditetragonal
Prisms 4faces
Crystal Structures
(iv) Pyramids:
A pyramid is a 3, 4, 6, 8 or 12 faced open form where all faces in the form meet or
could meet if extended, at a point.
Rhombic pyramid: 4-faced form
(v) Domes:
are 2- faced open forms that are related to one another by a
mirror plane.
 In the crystal model shown here, the dark shaded faces belong to a dome.
 The vertical faces along the side of the model are Pinacoids (2 parallel faces).
 The faces on the front and back of the model are not related to each other by
symmetry, and are thus two different pedions.
VI. Sphenoid: two nonparallel faces symmetrical to a 2- or 4-fold axis of
symmetry; Disphenoid: four-faced closed form in which the two faces of a
sphenoid alter
Shapes of Crystals
In rocks the shapes of crystalline
grains/crystals are often classified as
Euhedral, Anhedral and Subhedral
Euhedral: grains bounded by their
own perfect to near-perfect crystal
growth faces (C)
Anhedral: irregular; little or no
evidence for its own growth faces (no
faces in the crystal) (A)
Subhedral: partly bound by its own
growth faces, only some of well
moderately crystal’s faces (B)
Individual Crystal Systems and the Axial System
* The crystal system is a grouping of crystal
structures that are categorized according to the
axial system used to describe their atomic "lattice"
structure.
* A crystal's lattice is a three-dimensional network
of atoms that are arranged in a symmetrical
pattern
Each crystal system consists of a set of three crystallographic axes (a, b, and c) in
a particular geometrical arrangement.
The seven unique crystal systems, listed in order of decreasing symmetry, are:
1. Cubic System,
2. Hexagonal System,
3. Tetragonal System,
4. Rhombohedric (Trigonal) System,
5. Orthorhombic System,
6. Monoclinic System,
7. Triclinic System.
Cubic Crystal System
This is also known as the isometric crystal system
 The cubic (Isometric) crystal system is characterized by its total
symmetry
The Cubic system has three crystallographic axes that are all
perpendicular to each other, and equal in length.
 The three crystallographic axes in cubic systems a1, a2, a3 (or a, b,
c) are all equal in length and intersect at right angles (90 degrees)
to each other.
The Hexagonal Crystal System
The Hexagonal Crystal System has four crystallographic
axes consisting of three equal horizontal or equatorial
(a, b, and d) axes at 120°, and one vertical (c) axis that is
perpendicular to the other three.
The (c) axis can be shorter, or longer than the horizontal axes.
Tetragonal Crystal System
A tetragonal crystal is a simple cubic shape that is stretched along
its (c) axis to form a rectangular prism.
The tetragonal crystal will have a square base and top, but a
height that is taller.
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)
Note: If C was equal in length to a or b, then we would be in the
cubic system!
The Rhombohedral (Trigonal)Crystal System
A Rhombohedron (Trigonal) has a three-dimensional shape that is
similar to a cube, but it has been skewed or inclined to one side
making it oblique.
It is also known as the Trigonal crystal system.
A rhombohedral crystal has six faces, 12 edges, and 8 vertices.
If all of the non-obtuse internal angles of the faces are equal (flat
sample, below), it can be called a trigonal-trapezohedron.
The Orthorhombic (Rhombic) Crystal System
The orthorhombic Mineral (aka rhombic) crystal system have three
mutually perpendicular axes, all with different, or unequal lengths.
The orthorhombic crystal system is also known as the Rhombic
crystal system.
Three axes, all at right angles, and all three of different lengths.
Note: If any axis was of equal length to any other, then we would
be in the tetragonal system!
The Monoclinic Crystal System
Crystals form in the monoclinic system have three unequal axes.
The (a) and (c) crystallographic axes are inclined toward each
other at an oblique angle, and the (b) axis is perpendicular to (a).
The (b) crystallographic axis is called the "ortho" axis.
 Note: If (a) and (c) crossed at 90 degrees, then we would
be in the orthorhombic system!
The Triclinic Crystal System
Crystals form in the triclinic system have three unequal
crystallographic axes, all of which intersect at oblique angles.
Triclinic crystals have a 1-fold symmetry axis with nearly no
noticeable symmetry, and no mirrored or prismatic planes.
Note: If any two axes crossed at 90 degrees, then we would be
describing a monoclinic crystal!
The 7 Crystal Systems
Rhombohedron
similar to Triclinic
system
Crystal Symmetry
 Symmetry is a property of an object to stay unchanged (invariant)
after a certain type of movement.
There are three forms of symmetry for a crystal, These include :
1. Axes of symmetry,
2. Plane of symmetry,
3. Center of symmetry
 These symmetry forms may be or may not be combined in the
same crystal. Indeed, we will find that one crystal class or system
has only one of these forms.
Crystal Symmetry
Crystals, minerals, have an ordered internal arrangement of atoms.
These atoms of atoms are ordered and arranged in a symmetrical
crystal form on a three-dimensional shape referred to or called as a
lattice.
* Crystal faces form smooth planar boundaries that make up the
surface of the crystal in an environment where there are no
impediments (Disordered) to its growth.
* Thus, crystal faces reflect the atom’s order toms build crystal
internal arrangement, moreover, the faces reflect the symmetry of
the crystal lattice.
Crystal Systems (Solids)
O
M
C
H
R
T
0
0
3
2
3
2
T=
Tetrag
onal
O=
Orthor
ombic
Angle Define (α. β, γ
Axes Define
Orthorombic
C=
cubic
a#b#c
ab c
α = β = γ = 90 o
Monoclinic
a#b#c
α β γ
A &c
α = β = 90 o , γ # 120
Cubic
a=b=c
α = β = γ = 90 o
Hexagonal
a=b#c
α = β = 90 o , γ = 120o
Rohombohedron a = b = c
α = β = 90 o , γ # 90 o
Teragonal
Triclinic
a=b#c
a#b#c
α = β = γ = 90 o
α # β # γ # 90 o
=
90 o
Crystal Symmetry
To see this, imagine a small 2-dimensional crystal composed of atoms in an
ordered internal arrangement as shown below.
Although all of the atoms in this lattice are the same, one of them is dark so that
its position can be tracked.
If we rotate the simple crystals by 90o notice that the lattice and crystal look the
same as what we started with.
Rotate it another 90o and again it’s the same. Another 90o rotation again
results in an identical crystal, and another 90o rotation returns the crystal
to its original orientation.
Thus, in 1 360o rotation, the crystal has repeated itself, or looks
identical 4 times. Thus, We say this object (Crystal) has 4-fold
rotational symmetry.
Types of Crystal Symmetry
1- Axes of Symmetry/Rotational Symmetry
i.e. If an object (Crystal) can be rotated around an axis and repeats
itself every 90o of rotation then it is said to have an axis of 4-fold
rotational symmetry.
Thus,
The symmetrical rotation is performed along the axis.
Therefore
We can say, the symmetry of the crystal is referred to as symmetrical axis
rotation.
Axes of Symmetry/Rotational Symmetry Low
* Thus, when the rotation repeats itself (form) every 60 degrees around
point, then we have 6-fold or HEXAGONAL SYMMETRY. A filled
hexagon symbol is noted on the rotational axis, (360o/60o= thus
ordered 6).
* And when rotation repeats it self (form) every 90o degrees, then we
have fourfold or TETRAGONAL SYMMETRY. A filled square is noted on
the rotational axis, (360o/90o= ordered 4).
* And when rotation repeats its form every 120 degrees, then we
have threefold or TRIGONAL SYMMETRY. A filled equilateral triangle
is noted on the rotational axis (360o/120o=order 3).
* And when rotation repeats its form every 180 degrees, then we have
twofold or BINARY SYMMETRY. A filled oval is noted on the
rotational axis (360o/180o=2).
o
* And when rotation repeats its form every 360 degrees, with a
filled circle as notation. This one I rotation consider optional to
list as almost any object has this symmetry. If you really want to
know the truth, this means NO SYMMETRY!
Axes of Symmetry/Rotational Symmetry
The following types of rotational symmetry axes are possible in
crystals:
 1-Fold RAxisotation - An has object that requires rotation of a
full 360o in order to restore it to its original appearance no
rotational symmetry.
Since it repeats itself 1 time every 360o you said to have a 1-fold
axis of rotational symmetry.
Axes of Symmetry/Rotational Symmetry
II. 2-fold Rotation Axis - If an object congruent (similar) identical
after a rotation of 180o, that is twice in a 360o rotation, then it is
said to have a 2-fold rotation axis (360o/180o = 2).
For instance, the axes we are referring to are imaginary lines
that extend toward you perpendicular to the page or
blackboard.
A filled oval shape represents the point where the 2-fold
rotation axis intersects the page.
Axes of Symmetry/Rotational Symmetry
III. 3-Fold Rotation Axis- Objects that repeat
themselves upon rotation of 120o are said to
have a 3-fold axis of rotational symmetry
(360/120 =3), and they will repeat 3 times in
a 360o rotation.
IV. A filled triangle is used to symbolize the of
3-fold rotation axis.
V. 4-Fold Rotation Axis-If an object repeats
itself after 90o of rotation, it will repeat 4
times in a 360o rotation, as illustrated
previously.
VI. A filled square is used to symbolize the
4-fold axis of rotational symmetry.
Equilateral
rotatation of
symmetry
Square
rotation of
symmetry
Axes of Symmetry/Rotational Symmetry
V. 6-Fold Rotation Axis - If rotation of 60o around an axis
causes the object to repeat itself, then it has a 6-fold
axis of rotational symmetry (360o/60o=6).
filled hexagon is used as the symbol for a 6-fold
rotation axis.
A
2. Plane of Symmetry/ Mirror Symmetry
* Any two dimensional surface when passed through the center axis
of the crystal, divides it into two symmetrical parts that are
MIRROR or reflected IMAGES is a PLANE OF SYMMETRY Reflection
Symmetry)
i.e. the reflection of an object in the mirror reproduces
the other half of the its object, then the object is said
to have mirror symmetry.
 The plane of the mirror is an element
of symmetry referred to as a mirror (Reflected )plane
and is symbolized with the letter m
No reflection
symmetry
 For instance: the human body is an object that approximates
mirror symmetry,
Plane of Symmetry/ Mirror Symmetry
Plane of Symmetry for the crystals in cubic system
Plane of Symmetry/ Mirror Symmetry
* The rectangles here show two planes of a mirror symmetry.
The rectangle on the left has a
mirror plane that runs vertically on
the page and is perpendicular to
the page.
 The rectangle on the right has a mirror plane
that runs horizontally and is perpendicular to
the page so the parts match each other.
 The dashed parts of the rectangles above show
the part the rectangles that would be seen as
a reflection in a mirror
Plane of Symmetry/ Mirror Symmetry
* The rectangles shown above have two planes of mirror symmetry:
 Three dimensional and more complex objects could have
more. For example, the hexagon shown above, not only has
a 6-fold rotation axis, but has 6 mirror planes.
Annotation that a rectangle does not have mirror
symmetry along the diagonal lines (“cut the
rectangle along a diagonal such as that labeled
"m” ???“), as shown in the upper diagram,
reflected the lower half in the mirror, then we
would see what is shown by the dashed lines in
the lower diagram.
 Since this does not reproduce the original
rectangle, the line "m???" does not represent a
mirror plane.
3. Center of Symmetry
Most crystals have a centre of symmetry called (i), even
though they may not possess either planes of symmetry or
axes of symmetry. i
i
The imaginary line from the surface of a crystal face through the
centre of the crystal (the axial cross) and intersects a similar point
on a face equidistance from the centre, then the crystal has a
centre of symmetry.
In this operation lines are drawn from all points on the
object through a point in the center of the object, called a
Centre symmetry (symbolized with the letter "i").
i
i
Both lines of each have lengths that are
equidistant from the original points (i).
i.e. the ends of the lines are connected, the original
object is repeated overturned or inverted from its
original appearance.
i
Rotoinversion
Inversion centre Symmetry (i) is a
property of an object to stay
unchanged (invariant) after a certain
type of the same movement
The equilateral does not have
inversion symmetry.
i
 Groupings of rotation with a centre of symmetry perform the
symmetry operation of Rotoinversion.
 Objects (Crystal) that have Rotoinversion symmetry are called a
Rotoinversion axis.
Rotoinversion
 Rotoinversion is a combination of rotation and an
inversion i.e. rotation axis + inversion center
 2-fold Rotoinversion –
 The operation of 2-fold Rotoinversion involves
first rotating the object by 180o then inverting it
through an inversion centre.
 A 2-fold Rotoinversion axis is symbolized as 2
with a bar over the top 2̅, and would be
pronounced as "bar 2"
 3-fold Rotoinversion –
This involves rotating the object by 120o (360o/3 =
120o), and inverting through a Centre.
 A 3-fold Rotoinversion axis is denoted and (pronounced "bar 3“ 3̅).
Rotoinversion
* 4-fold Rotoinversion –
o
 This involves rotation of the object by 90 then
inverting through a centre.
 A four-fold Rotoinversion axis is symbolized as bar 4̅.
* 6-fold Rotoinversion A 6-fold Rotoinversion axis involves rotating
the object by 60o and inverting through a
center and is called bar (6̅ ).
Combinations (CrystalClasses)of Symmetry Operations
 Thus, by now, the three-dimensional objects, such as crystals, and
symmetrical elements may be present in several different
combinations.
 In fact, the crystals systems are involving 32 possible
combinations of symmetry elements.
 These 32 crystal different combinations define as 32 Crystal
Classes.
 Every crystal system must belong to one of these 32 crystal
classes.
Combinations of Symmetry Operations
* For instance, the Tetragonal System shows how the various
symmetry elements are combined in somewhat a completed crystal.
 Thus, the crystal on the right side has the following symmetry
elements:
 1 - 4-fold rotation axis (A4)
 4 - 2-fold rotation axes (4 A2),
 2 cutting the faces & 2 cutting the edges
√ 5 mirror planes (5m),
√ 2 cutting across the faces,
√ 2 cutting through the edges, and one cutting
horizontally through the center.
Note also that there is a center of symmetry (i).
* The symmetry content of this crystal is thus: i, 1A4, 4A2, 5m
Hermann-Mauguin (International) Symbols
(Notation)
 The crystal aside shows symmetrical content :
3A2 3m or as
2/m 2/m
2/m
a) A notation is Internationally accepted as crystal Notation.
b) Different symbols are used:
 Notation of Rotation is giving symbols 1, 2 , 3, 4, and 6
• Mirror is giving symbol m
• Rotoinversion 3 Bar (3̅), bar 4 (4̅ )
• Center of symmetry is given by (i) (i is equal to bar1).
c) Crystal’s Geometry means are the symmetrical elements
parallel ( elements goes side by side i.e. 2m) to each other or
are they perpendicular to each other 2/m ( 2 over m means
elements perpendicular).
Accordingly, there are 32 crystal combinations (called crystal
classes) to symmetry elements in nature:
Hermann-Mauguin (International) Symbols
Before going into the 32 crystal classes, we used to describe the
crystal classes by their symmetry content.
We'll start with a simple crystal then look at some more
complex examples.
The rectangular block shown here has :
• 3 2-fold rotation axes (3 A2),
• 3 mirror planes (3 m), and a centre of symmetry (i).
The rules for deriving the Hermann- Mauguin symbol are as
follow:
•So, we need to know first how to derive the Hermann-Mauguin
Symbols (called the International Crystal Notation Symbols):
Combinations of Symmetry Operations
To write a number representing each of the unique rotation axes present:
* A Unique rotation axis is one that exists by itself and is not produced by
another symmetry operation.
*In this case, in all three 2-fold axes are unique, because each axe is
perpendicular to a different shaped face, so we write a 2 (for 2-fold axes) for
each axis
2 2 2
Next, we write "m" for each unique mirror plane.
Again, a unique mirror plane is one that is not
produced by any other symmetry operation.
In this example, we can tell that each mirror is unique
because each one cuts a different looking face.
So, we write:
2m 2m 2m
Hermann-Mauguin (International) Symbols
 If any of the crystal axes are perpendicular to a mirror plane:
 we put a slash (/) between the axis and the symbol of
the mirror plane.
In this case, each of the 2-fold axes are
perpendicular to mirror planes,
Therefore,

we wrote Herman Mauguin’s symbol as follows:
2/m 2/m 2/m
Hermann-Mauguin (International) Symbols
Our Second example involves the model having one 2-fold axis
and 2 mirror planes. For the 2-fold axis, we write:
2
 Each of the mirror planes is unique. We can tell
that each one cuts a different looking face.
 So, we write 2 "m"s, one for each mirror plane:
As
2 mm
Note that the 2-fold axis is not perpendicular to a mirror plane,
so we need no slashes. Our final symbol is then: 2mm
Hermann-Mauguin (International) Symbols
 The cubic system is the most complex.
 Note:
 Cubic system has 3 of 4-fold rotation axes, each one is
perpendicular to a square-shaped face.
 4 of 3-fold Rotoinversion axes ( 4 A3) (some of which are
not shown in the diagram to reduce complexity), each
sticking out of the corners of the cube.
 6 of 2-fold rotation axes (^ A2) (again, not all are shown),
sticking out of the edges of the cube.
In addition, the crystal has 9
mirror planes and a centre of
symmetry.
Hermann-Mauguin (International) Symbols
 There is only 1 unique 4 fold axis, because each is
perpendicular to a similar looking face (the faces of the cube).
 There is only one unique 3-fold Rotoinversion axes, because
all of them stick out of the corners of the cube, and all are
related by the 4-fold symmetry.
 And, there is only 1 unique 2-fold axis because all of the
others stick out of the edges of the cube that are related by
the mirror planes the other set of 2-fold axes.
So, we write A4, 3̅, and A2 for each of the unique rotation
axes.
There are 3 mirror planes that are perpendicular to the 4 fold axes, and 6 mirror planes
that are perpendicular to the 2-fold axes. No mirror planes are perpendicular to
the 3-fold Rotoinversion axes. So, our final symbol becomes: 4/m 2/m
The 32 Crystal Classes
The 32 crystal classes represent combinations of symmetry
operations.
Each crystal class will have crystal faces that uniquely define
the symmetry of the class.
These faces or groups of faces are called crystal forms, They
are distributed among the 7 crystal systems as shown here
Classify Crystal in classes based on crystal symmetrical elements
Example
m
Thus.
Center of
inversion I
not to be
mentioned
One m
one 2 perpendiculars to m 2-fold axis
2
2
m
 Mirror plane is shown in grey colour which cut the
shape into two identical halves
 There is the two-fold axis of rotation
perpendicular to the mirror plane
There is also a Centre of inversion (i)
So, it is automatically generated by 2/m
Crystal is called prismatic
Speak 2 over m
The
crystallographic
notation can be
written in this
way 2/m
Thus the crystal class is 2/m
Crystal Class
Symmetry
content
No
i
1A2
m
Crystal Class
2/m
222
i, 1A2, 1m
2mm
1A2, 2m parallel
to each others
1
Bar 1
2 fold rotation
m
crystal has mirror
3A2
2/m 2/m 2/m
i, 3A2, 3m
4 fold rotation
Bar 4
1A4
1 Bar 4
4/m
422
Symmetry
content
I, 1A4, and m
1A4, and 4A2
4mm
1A4 ,4m
4bar 2m
1A4, 2 A2, 2m
4/m 2/m 2/m
i, 1A4, 1A4,
5m
3 fold
3 Bar
3 2
3m
1A3
1A3, 3 A2
1A3 and 3m
3bar 2/m
1Bar A3, 3A2, 3m
6
System
The 32 Crystal Classes
Isometric
Class Name
Tetartoidal
Diploidal
Hextetrahedral
Gyroidal
Hexoctahedral
Tetragonal
Disphenoidal
Pyramidal
Dipyramidal
Scalenohedral
Ditetragonal pyramidal
Trapezohedral
Ditetragonal-Dipyramidal
Orthorhombic
Pyramidal
Disphenoidal
Dipyramidal
Hexagonal
Trigonal Dipyramidal
Pyramidal
Dipyramidal
Ditrigonal Dipyramidal
Dihexagonal Pyramidal
Trapezohedral
Dihexagonal Dipyramidal
Trigonal
Pyramidal
Rhombohedral
Ditrigonal Pyramidal
Trapezohedral
Hexagonal Scalenohedral
Monoclinic
Domatic
Sphenoidal
Prismatic
Triclinic
Pedial
Pinacoidal
AXES
2-Fold
3
3
3
6
6
2
4
4
1
3
3
3
6
6
3
3
1
1
-
3-Fold
4
4
4
4
4
1
1
1
1
1
-
4-Fold
3
3
1
1
1
1
1
1
1
-
6-Fold
1
1
1
1
1
1
1
-
Planes
Center
HermannMaugin
Symbols
3
6
9
1
2
4
5
2
3
1
1
4
6
7
3
3
1
1
-
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
23
2/m ͞3
͞4 3m
432
4/m ͞3 2/m
4͞
4
4/m
͞4 2m
4mm
422
4/m 2/m 2/m
mm2
222
2/m 2/m 2/m
6͞
6
6/m
6m2
6mm
622
6/m 2/m 2/m
3
3͞
3m
32
͞3 2/m
m
2
2/m
1
1͞
The 32 Crystal Classes
Note that the 32 crystal classes are divided into 7 crystal systems
The Triclinic System has only 1-fold or 1-fold rotoinversion axes.
Since class has only a center of symmetry, pairs of faces related
to each other through the center. Two crystal faces are called
Pedial and pinacoids.
 Crystals of this system possess no mirror planes and may have one
center of inversion symmetry.
◦ There are two crystal class found in this system: Pedial and
Pinacoidal .
◦ For instance; minerals found in this system are microcline
(K-feldspar), plagioclase, turquoise, and wollastonite.
System Class Name
Triclinic
Pedial
Pinacoidal
AXES
2-Fold
-
3-Fold
-
4-Fold
-
Planes
6-Fold
-
-
Hermann
Center Maugin
Symbols
yes
1
1͞
The 32 Crystal Classes
Monoclinic crystals demonstrate a single 2-fold rotation axis and/or a single
mirror plane.
◦ The crystals may have a center of symmetry.
◦ There are three crystal class found in this system: Domatic
(class m), Sphenoidal (monoclinic class 2) and Prismatic .
◦ The minerals crystallize in this system are including Micas (biotite and
muscovite), azurite, chlorite, clinopyroxenes, epidote, gypsum,
malachite, kaolinite, orthoclase, and talc.
System
Monoclinic
Class Name
Domatic
Sphenoidal
Prismatic
AXES
2-Fold
1
1
3-Fold
-
4-Fold
-
6-Fold
-
Planes
Center
Hermann
Maugin
Symbols
1
1
yes
m
2
2/m
The 32 Crystal Classes
The Orthorhombic System has only two fold axes or a
2-fold axis and/or up to 3 mirror planes.
◦ There are three crystal classes found in this system:
Pyramidal, Disphenoidal and Dipyramidal.
Disphenoidal
The 32 Crystal Classes
Some minerals crystallize in this system include andalusite, anthophyllite,
aragonite, barite, cordierite, olivine, sillimanite, stibnite, sulfur, and topaz.
System
Orthorhombic
Class Name
Pyramidal
Disphenoidal
Dipyramidal
AXES
2-Fold
1
3
3
3-Fold
-
4-Fold
-
Disphenoidal
6-Fold
-
Planes
Center
Hermann
Maugin
Symbols
2
3
yes
mm2
222
2/m 2/m 2/m
The 32 Crystal Classes
 Minerals of the Tetragonal System all possess a single 4-fold
symmetry axis.
◦ They may possess up to four 2-fold axes of rotation, a center of inversion, and
up to five mirror planes.
◦ There are seven crystal classes found in this system: Disphenoidal, Pyramidal,
Dipyramidal , Scalenohedral, Ditetragonal pyramidal, Trapezohedral and
Ditetragonal-Dipyramidal.
◦ The minerals that crystallize in this system include Common minerals that occur
with this symmetry are anatase, cassiterite, apophyllite, zircon, and
vesuvianite.
Scalenohedral
Trapezohedral
The 32 Crystal Classes
System
Class Name
AXES
2-Fold
Tetragonal
Disphenoidal
Pyramidal
Dipyramidal
Scalenohedral
Ditetragonal pyramidal
Trapezohedral
Ditetragonal-Dipyramidal
2
4
4
3-Fold
-
4-Fold
1
1
1
1
1
1
1
6-Fold
-
Planes
Center
Hermann
Maugin
Symbols
1
2
4
5
yes
yes
4͞
4
4/m
͞4 2m
4mm
422
4/m 2/m 2/m
The 32 Crystal Classes
All crystals of the hexagonal system possess a single 6-fold axis of rotation
◦ Crystals of the hexagonal system may possess up to six 2fold axes of rotation.
◦ They may demonstrate a center of inversion symmetry and up
to seven mirror planes.
◦ There are seven crystal classes found in this system: Trigonal Dipyramidal,
Pyramidal, Dipyramidal, Ditrigonal Dipyramidal, Dihexagonal Pyramidal,
Trapezohedral and Dihexagonal Dipyramidal.
Minerals species which crystallize in the hexagonal division are apatite, beryl, and
high quartz.
The 32 Crystal Classes
System
Hexagonal
Class Name
2-Fold 3-Fold
Trigonal Dipyramidal
Pyramidal
Dipyramidal
Ditrigonal Dipyramidal
3
Dihexagonal Pyramidal
Trapezohedral
6
Dihexagonal Dipyramidal
6
-
AXES
4-Fold
-
6-Fold
1
1
1
1
1
1
1
Planes
Center
Hermann
Maugin
Symbols
1
1
4
6
7
yes
yes
6͞
6
6/m
6m2
6mm
622
6/m 2/m 2/m
The 32 Crystal Classes
All crystals of the trigonal system have a single 3-fold axis of rotation
Crystals of this division may possess up to three 2-fold axes of rotation
and may have a center of inversion and up to three mirror planes.
There are five crystal classes found in this system: Pyramidal,
Rhombohedral, Ditrigonal Pyramidal, Trapezohedral and Hexogonal
Scalenohedral. Dipyramidal.
The minerals which crystallize in the Trigonal system are
calcite, dolomite, low quartz, and tourmaline.
The 32 Crystal Classes
System
Trigonal
Class Name
2-Fold 3-Fold
Pyramidal
1
Rhombohedral
1
Ditrigonal Pyramidal
1
Trapezohedral
3
1
Hexagonal Scalenohedral 3
1
AXES
4-Fold
-
6-Fold
-
Planes
Center
Hermann
Maugin
Symbols
3
3
yes
yes
3
3͞
3m
32
3͞ 2/m
The 32 Crystal Classes
All crystals of the isometric system possess four 3-fold axes of symmetry.
Crystals of the isometric system may also demonstrate up to three separate 4-fold
axes of rotational symmetry.
Furthermore crystals of the isometric system may possess six 2-fold axes of
symmetry .
Minerals of this system may demonstrate up to nine different mirror planes.
There are five crystal classes found in this system: Tetroidal, Diploidal (isometric system),
Hextetrahedral, Gyroidal and Hexoctahedral.
Examples of minerals that crystallize in the isometric system are halite, magnetite,
and garnet.
The 32 Crystal Classes
System
Isometric
Class Name
Tetartoidal
Diploidal
Hextetrahedral
Gyroidal
Hexoctahedral
AXES
2-Fold 3-Fold
3
4
3
4
3
4
6
4
6
4
4-Fold
3
3
6-Fold
-
Planes
Center
Hermann
Maugin
Symbols
3
6
9
yes
yes
23
2/m ͞3
͞4 3m
432
4/m ͞3 2/m
Crystal Forms
A crystal form is a set of crystal faces that are related to each other by symmetry
or any group of crystal faces related by the same symmetry is called a form.
There are 48 possible forms that can be developed as the result of the 32
combinations of symmetry.
I.
Closed forms are those groups of faces all related by symmetry that
completely enclose a volume of space.
It is possible for a crystal to have faces entirely of one closed form.
II.
Open forms are those groups of faces all related by symmetry that does not
completely enclose a volume of space.
A crystal with open form faces requires additional faces as well.
There are 18 open forms and 30 closed forms.
Crystal Forms
Monohedron: The monohedral crystal form is also called a pedion.
• It consists of a single face that is geometrically unique for the crystal and is not
repeated by any set of symmetry operations.
• Members of the triclinic crystal system produce monohedral crystal forms.
• These are open crystal forms.
Parallelohedron: The parallelohedral crystal form is also called a pinacoid.
•It consists of two and only two geometrically equivalent faces which occupy
opposite sides of a crystal.
•The two faces are parallel and are related to one another only by a reflection or
an inversion.
•Members of the triclinic crystal system produce parallelohedral crystal forms.
•These are open crystal forms.
Crystal Forms
Dihedron: The dihedron consists of two and only two nonparallel geometrically
equivalent faces.
The two faces may be related by a reflection or by a rotation.
The dihedron is termed a dome if the two faces are related only by reflection
across a mirror plane.
If the two faces are related by a 2-fold rotation axis then the dihedron is termed a
sphenoid.
Members of the monoclinic crystal system produces dihedral crystal forms.
These are open crystal forms.
Disphenoid: Members of the orthorhombic and tetragonal crystal systems
produce rhombic and tetragonal disphenoids, which possess two sets of
nonparallel geometrically equivalent faces, each of which is related by a 2-fold
rotation.
The faces of the upper sphenoid alternate with the faces of the lower sphenoid in
such forms.
These are closed crystal forms.
Crystal Forms
Crystal Forms
Prism: A prism is composed of a set of 3, 4, 6, 8, or 12 geometrically equivalent
faces which are all parallel to the same axis.
•Each of these faces intersects with the two faces adjacent to it to produce a set of
parallel edges.
•Variants of the prism form include the rhombic prism, tetragonal prism, trigonal
prism, and hexagonal prism.
•Prisms are associated with the members of the monoclinic crystal system.
◦ These are open crystal forms.
Pyramid: A pyramid is composed of a set of 3, 4, 6, 8, or 12 faces which are not
parallel but instead intersect at a point.
•The orthorhombic, tetragonal and hexagonal crystal systems all produce
pyramids. Variants of the pyramid include rhombic pyramid, tetragonal pyramid,
trigonal pyramid, and hexagonal pyramid.
•These are open crystal forms.
Crystal Forms
PRISM
PYRAMID
Crystal Forms
Dipyramid: The dipyramidal crystal form is
composed of two pyramids placed baseto-base and related by reflection across a
mirror plane which runs parallel to and
adjacent to the pyramid bases.
The upper and lower pyramids may each
have 3, 4, 6, 8, or 12 faces; the dipyramidal
form therefore possesses a total of 6, 8,
12, 16, or 24 faces.
* The orthorhombic, tetragonal and hexagonal crystal systems all
produce dipyramids.
* They can be called rhombic dipyramid, trigonal dipyramid,
tetragonal dipyramid, and hexagonal dipyramid.
* They have closed crystal forms.
Crystal Forms
Trapezohedron: A trapezohedron is a crystal form possessing 6, 8, or 12 trapezoidal faces.
The
Tetragonal crystal system, the trigonal and hexagonal crystal system produce
trapezohedral crystal forms.
Trigonal trapezohedra possess three trapezoidal faces on the top and three on the
bottom for a total of six faces;
Tetragonal trapezohedra have four faces on top and four on the bottom
for a total of eight faces; and hexagonal trapezohedra have six faces on
top and six on the bottom, resulting in twelve faces total.
They are closed crystal forms.
Crystal Forms
Scalenohedron:
It consists of 8 or 12 faces, each of
which is a scalene triangle.
* The faces appear to be grouped into
symmetric pairs.
*
The tetragonal, trigonal and hexagonal crystal systems produce the scalenohedral
crystal form, of which examples may be further described as trigonal, tetragonal and
hexagonal scalenohedral
They are closed crystal forms
Crystal Forms
Rhombohedron: (i.e.rhombohedron, a crystal with three surfaces)s
The rhombohedral crystal form possesses six rhombus-shaped faces.
 The Rhombohedral crystal form is produced only by members of the trigonal
/rhombohedral crystal system.
 They have closed crystal forms.
Tetrahedron:
A tetrahedron is composed of four triangular faces.
The tetrahedron crystal form is produced by the members of the isometric crystal system.
They have closed crystal forms.
Crystal forms –opened
The Pedion
The Pinacoid
The Dome
The Sphenoid
The Prism
The Pyramid
An open form is one or more crystal faces that do not completely
enclose space.
•Example 1.
•Pedions are single-faced forms. Since there is only one face in the
form a pedion cannot completely enclose space. Thus, a crystal
that has only pedions, must have at least 3 different pedions to
completely enclose space.
•Example 2.
•A prism is a 3 or more faced form where in the crystal faces are all
parallel to the same line. If the faces are all parallel then they
cannot completely enclose space. Thus crystals that have prisms
must also have at least one additional form to completely enclose
space.
•Example 3.
•A dipyramid has at least 6 faces that meet in points at opposite ends of
the crystal. These faces can completely enclose space, so a dipyramid
is closed form. Although a crystal may be made up of a single
dipyramid form, it may also have other forms present.
Crystal Forms-CLOSED (Isometric)
The Cube
The Octahedron
The Pyritohedron
The Dodecahedron
The Tetartoid
The Tetrahedron
The Diploid
The Gyroid
The Tetartoid
The Trapezohedron
The Hexoctahedron
The Tetrahexahedron
The Tristetrahedron
The Trisoctahedron
The Hextetrahedron
A closed-form is a set of crystal faces that
completely enclose space.
Thus, in crystal classes that contain closed
forms, a crystal can be made up of a single
form.
Crystal Forms-CLOSED (Non-isometric)
The Scalenohedron
The Rhombohedron
The Trapezohedrons
The Dipyramids
The Disphenoid
No of Faces
Open Form
1
PEDION
PINAKOID
2
SPHENOID
DOME
TRIGONAL PRISM
3
TRIGONAL PYRAMID
RHOMBIC PRISM
RHOMBIC PYRAMID
4
TETRAGONAL PRISM
TETRAGONAL PYRAMID
HEXOGONAL PRISM
HEXAGONAL PYRAMID
6
DITRIGONAL PRISM
DITRIGONAL PYRAMID
DITETRAGONAL PRISM
DITETRAGONAL PYRAMID
8
DIHEXAGONAL PRISM
DIHEXAGONAL PYRAMID
12
16
24
48
Closed Form
RHOMBIC DISPHENOID
TETRAGONAL DISPHENOID
TETRAHEDRON
CUBE
RHOMBOHEDRON
TRIGONAL DIPYRAMID
TRIGONAL TRAPEZOHEDRON
OCTAHEDRON
RHOMBIC DIPYRAMID
TETRAGONAL DIPYRAMID
TETRAGONAL SCALENOHEDRON
TETRAGONAL TRAPEZOHEDRON
DITRIGONAL DIPYRAMID
HEXAGONAL DIPYRAMID
HEXAGONAL DIPYRAMID
HEXAGONAL SCALENOHEDRON
HEXAGONAL TRAPEZOHEDRON
DODECAHEDRON
PYRITOHEDRON
TRISTETRAHEDRON
DELTOHEDRON
TETARTOID
DITETRAGONAL DIPYRAMID
DIHEXAGONAL DIPYRAMID
TRAPEZOHEDRON
TRISOCTAHEDRON
TETRAHEXAHEDRON
DIPLOID
GYROID
HEXOCTAHEDRON
Crystal Morphology
The symmetry observed in crystals as exhibited by their crystal faces
is due to the ordered internal arrangement of atoms in a crystal
structure,
This arrangement of atoms in crystals is called a lattice.
Crystal faces develop along planes and are defined by the points
in the lattice.
In other words, all crystal faces must intersect atoms or
molecules that make up the points.
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.
 The Bravais law state that Crystals are
formed by the repetition in 3
dimensions configurations into which
atoms can be arranged in crystal as a
unit-cell structure defined
by lattice points in space
Crystal Morphology
The angle between crystal faces is controlled by the spacing between
lattice points.
 Since all crystals of the same substance will have the same spacing between
lattice points (they have the same crystal structure),
The angles between corresponding faces of the same mineral will
be the same.
This is known as the Law of constancy of interfacial angles.
Crystal Morphology
In two dimensions, there are five Bravais lattices, called oblique,
rectangular, centered rectangular (rhombic), hexagonal, and square.[
Crystal Morphology
Although there are only 7 crystal systems or shapes, there are 14 different
crystal lattices, called Bravais Lattices in 3 dimesion. They are :
3 different cubic types,
2 different tetragonal types,
4 different orthorhombic types,
2 different monoclinic types,
1 rhombohedral,
1 hexagonal,
1 triclinic
https://youtu.be/MnFFqAPca40