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Introduction to Crystallography
and Mineral Crystal Systems
Prof. Dr. Andrea Koschinsky
Geosciences and Astrophysics
Why Crystallography in Geosciences?
• Most of the Earth is made of solid rock. The basic units from
which rocks are made are 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, and many geological
processes represent the culmination - on a very grand scale - of
microscopic processes inside minerals.
• For example, large-scale processes, such as rock formation,
deformation, weathering and metamorphic activity, are controlled
by small-scale processes, such as movement of atoms (diffusion),
shearing of crystal lattices (dislocation movement), growth of new
crystals (nucleation, crystallization), and phase transformations.
Why Crystallography in Geosciences?
• An understanding of mineral structures and properties also allows
us to answer more immediate questions, such as why quartz and
diamond are so hard, and why solid granite rock is destined to
become soft, sticky clay.
• Minerals are natural resources, providing raw materials for many
industries. Therefore, understanding minerals has geological as
well as economic applications.
• Definition of the term “MINERAL”: a solid body, formed by natural
processes, that has a regular arrangement of atoms which sets
limits to its range of chemical composition and gives it a
characteristic crystal shape.
Definition of Crystallography
• CRYSTALLOGRAPHY is the study of crystals.
• CRYSTALLOGRAPHY is a division of the entire study of
• Geometrical, physical, and chemical CRYSTALLOGRAPHY
• 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 from the state of a
liquid or gas to that of a solid.
• Polyhedral form: solid bounded by flat planes (CRYSTAL FACES).
• 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.
• Example: The chemical composition of window glass is virtually
identical with that of quartz (a crystalline material): both are forms
of SiO2. Window glass is glassy because it is made by chilling
molten SiO2 very quickly; quartz crystals form when molten SiO2 is
cooled very slowly or by precipitation from solution.
Crystal Forms
• During the process of crystallization, crystals assume various
geometric shapes dependent on the ordering of their atomic
structure and the physical and chemical conditions under which
they grow.
• These forms may be subdivided, using geometry, into six systems.
6 large groups of crystal
The three crystallographic axes
a1, a2, a3 (or a, b, c) are all
equal in length and
intersect at right angles (90
degrees) to each other.
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!
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!
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.
The c axis also passes through the
intersection of the a axes set at right angle to
the plane formed by the a set.
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.
Note: If a and c crossed at 90 degrees,
then we would be in the orthorhombic
The three axes are all unequal in length
and intersect at three different angles
(any angle but 90 degrees).
Note: If any two axes crossed at 90
degrees, then we would be describing a
monoclinic crystal!
Mathematical system for describing any crystal face or group of similar
faces (forms) developed by William H. Miller (1801-1880).
Face of an octahedron using Miller's
An octahedron is an eight-sided crystal
form that is the simple repetition of an
equilateral triangle about our 3
crystallographic axes. The triangle is
oriented so that it crosses the a1 (or a), a2
(or b), and a3 (or c) axes all at the same
distance from the axial cross. This unit
distance is given as 1. So the Miller indices
is (111) for the face that intercepts the
positive end of each of the 3 axes.
Note: A bar over the number tells me that
the intercept was across the negative end
of the particular crystallographic axis.
Face of a cube using Miller's
A cube face that intercepts the a3
(vertical) axis on the + end will not
intercept the a1 and a2 axes. If the
face does not intercept an axis, then
we assign a mathematical value of
infinity to it. So we start with Infinity,
Infinity, 1 (a1, a2, a3). So the Miller
indices of the +a3 intercept face
equals (001).
Any two dimensional surface that, when passed through the center of the
crystal, divides it into two symmetrical parts that are MIRROR IMAGES
A cube has 9 planes of symmetry, 3 of one set and 6 of another.
In the left figure the planes of symmetry are parallel to the faces of the cube
form, in the right figure the planes of symmetry join the opposite cube edges.
Any 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:
When rotation repeats form every 60 degrees, then we have sixfold or
When rotation repeats form every 90 degrees, then we have fourfold or
When rotation repeats form every 120 degrees, then we have threefold or
When rotation repeats form every 180 degrees, then we have twofold or
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.
The crystal face arrangement symmetry of any given crystal is simply
an expression of the internal atomic structure. The relative size of a
given face is of no importance, only the angular relationship or position
to other given crystal faces.
HABIT is the correct term to indicate outward appearance. Habit, when
applied to natural crystals and minerals, includes such descriptive terms
as tabular, equidimensional, massive, reniform, drusy, and encrusting.
A FORM is a group of crystal faces, all having the same relationship to the
elements of symmetry of a given crystal system. These crystal faces display
the same physical and chemical properties because the ATOMIC
ARRANGEMENT (internal geometrical relationships) of the atoms composing
them is the same.
Note: Crystals, even of the same mineral, can have differing
CRYSTAL FORMS, depending upon their conditions of growth.
Example: Various Crystal Forms of
Peruvian Pyrite
Pyrite is a common mineral which often
exhibits several forms on a single crystal.
One form is usually dominant, presenting
the largest faces on the crystal. Peruvian
pyrite commonly has cubic, octahedral,
and dodecahedral forms on a single
crystal. Crystals with the same forms
present, but with different dominant forms
will each appear very different.
There are 32
forms in the
(noncubic) crystal
systems and
another 15 forms
in the isometric
(cubic) system.
Introduction to the atomic arrangement of crystal forms
Crystal Lattice Structures
Cubic and
Schematic diagram of an atom
of the element carbon.
The nucleus contains six protons and six
neutrons. Electrons orbiting the
nucleus are confined to specific
orbits called energy-level shells.
Three-dimensional representation
showing the first energy-level shells.
The first shell can contain two
electrons, the second eight.
B. Two-dimensional representation
of the carbon atom to show the
number of protons and neutrons in
the nucleus and the number of
electrons in the energy-level shells.
The first energy-level shell is full
because it contains two electrons.
The second shell contains four
electrons and so is half full.
Model for the ionic compound LiF
To form the compound lithium fluoride, an atom of the element lithium
combines with an atom of the element fluorine. The lithium atom transfers
its lone outer-shell electron to fill the fluorine atom's outer shell, creating an
Li+ cation and a F- anion in the process. The electrostatic force that keeps
the lithium and fluorine ions together in the compound lithium fluoride is an
ionic bond.
Mineral structure of PbS
The arrangement of ions in
the most common lead
mineral, galena (PbS). Lead
forms a cation with a charge
of 2+, and sulfur forms an
anion with a charge of 2-. To
maintain a charge balance
between the ions, there must
be an equal number of Pb and
S ions in the structure.
The packing arrangement of
ions is repeated continuously
through a crystal. The ions are
shown pulled apart along the
black lines to demonstrate
how they fit together.
The tetrahedron-shaped silicate anion SiO4(-4)
Anion with the four oxygens touching each other in natural position.
Silicon (dashed circle) occupies central space.
Exploded view showing the relatively large oxygen anions at the
four corners of the tetrahedron, equidistant from the relatively small
silicon cation.
Polymerization of complex silicate anions
Polymeriz ation of complex sili cate anions
Summary of the way silicate anions polymerize to form the common silicate
minerals. The most important polymerizations are those that produce chains,
sheets, and three-dimensional networks.
Snow Crystals
Snow crystals: Individual ice crystals, often with six-fold symmetrical
shapes. They grow directly from the condensing water vapor in the air, size
microscopic to at most a few mm in diameter
Snowflakes: Collections of snow crystals, loosely bound together into a puffball. Can grow to large sizes (up to 10 cm across)
Plate forms:
Simple sectored plate
Dendritic sectored plate
fern-like stellar dendrite
Columns forms:
Hollow column
(sheet-like crystal)
Needle crystal