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GEO1020
MINERALOGI-del
Dette er tileggslitteratur til Kap.3 i
læreboken – Understanding Earth
De deler som er berørt i forelesninger er
pensum
Introduction to Crystallography and
Mineralogy
Written by
Andrew Putnis, University of Münster
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SOME RELEVANT INTERNET SITES
Amethyst galleries. Classical classification of mineral groups with photos and mineral
properties
http://mineral.galleries.com/minerals/by_class.htm
Young R.B. and Mehl, M. Crystal lattice structures. Also links to other sites
http://cst-www.nrl.navy.mil/lattice/
A few nice sites for looking at two-dimensional plane patterns
http://mathmuse.sci.ibaraki.ac.jp/pattrn/PatternE.html
http://web.inter.nl.net/hcc/Hans.Kuiper/17system.htm
http://www.clarku.edu/~djoyce/wallpaper/
http://xahlee.org/Wallpaper_dir/c5_17WallpaperGroups.html
or search in google.com under “Wallpaper patterns”
Another more advanced site for crystallographic information
http://www-sphys.unil.ch/escher/
Heyes, S.J. University of Oxford. An introductory course on crystal structures
http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html
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MINERALS
A mineral is a crystalline solid formed by natural processes, with a regular atomic
arrangement which sets limits to its range of chemical composition and gives it
characteristic shape and properties. A mineral name thus gives information about both
the internal crystalline structure as well as its chemical composition range. Rocks are
made up of assemblages of minerals.
Mineralogy is the study of minerals and extends back to ancient civilizations. The
earliest preserved book on Mineralogy is by Theophrastus (ca. 300 BC) called On
Stones. Pliny the Elder (77 AD) wrote an encyclopaedic review of mineralogy as it
was known at the time. The beginning of modern mineralogy is associated with Georg
Bauer (Georgious Agricola) in 1556 who provided detailed descriptions of mineral
properties that are still the basis of identifying minerals in hand-specimen.
CRYSTALLOGRAPHY
The systematic description of the regular internal arrangement and the structure of
crystals is more recent. Kepler (1611) explained the structure of snow crystals in
terms of packing of spheres, and Steno (1669) observed the constancy of interfacial
angles (see below). Haüy (1801) presented geometrical crystallography in Traité de
Minéralogie which then developed during the 19th century to the detailed theory of
crystal systems, patterns and symmetry. The discovery of X-rays (Röntgen, 1895) and
the demonstration in 1912 by von Laue that they are diffracted in a regular pattern by
minerals began the modern study of crystallography. The first X-ray determination of
a crystal structure of a mineral was made by WL Bragg in 1913.
MINERALOGY AND CRYSTALLOGRAPHY IN THE 21ST CENTURY.
Although new minerals and their crystal stuctures are still being discovered, the focus
nowadays is on processes in the earth and how mineral compositions and crystal
structures are modified by these processes. The study of Mineralogy includes a very
broad range of topics and is the underlying discipline for:
Petrology – the study of the processes which produce specific mineral assemblages in
rocks.
Geochemistry – the study of the processes which control the distribution of elements
and their isotopes in minerals and rocks.
Ore Geology – the discovery and extraction of minerals as raw materials for industry,
which is related to the broader subject of Technical (or Applied) Mineralogy.
Other aspects of Mineralogy include Environmental Mineralogy (how minerals
interact with the man-made environment e.g. the adsorption of pesticides in the soil),
Biomineralogy (the way in which organic processes in animals and plants produce
minerals e.g. bone formation, sea-shells, magnetobacteria), Cosmic Mineralogy (the
study of minerals in meteorites, cosmic dust and other planets).
An understanding of Mineralogy depends on an understanding of the basic ideas of
Crystallography.
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MINERALS AS CRYSTALS
The external shape (morphology) of a well-formed crystal can illustrate the symmetry
which is a result of the regular internal atomic structure. The flat external surfaces of a
crystal are called faces and they are usually symmetrically arranged. However the
faces of a crystal may grow at different rates which can affect the morphology but it is
the angular relationship between the faces that defines the symmetry. An interfacial
angle is defined as the angle between the normals (perpendiculars) to two faces,
regardless of whether these faces meet at an edge.
Law of constancy of interfacial angles (Steno, 1669) : For all different crystals of the
same substance the angles between corresponding faces have a contant value,
whatever the shape of the crystal. By measuring the interfacial angles, it is possible to
determine the symmetry and characterise the substance. This was the basis of
geometrical crystallography.
Figure 1
SYMMETRY OF THE EXTERNAL FORM OF CRYSTALS
The symmetry of an ideal crystal can be described in terms of two types of symmetry
elements: mirror (reflection) planes and rotation axes.
Mirror planes divide the crystal into two equal halves which are mirror images of
each other, reflected in the symmetry plane.
Rotation axes are axes around which the the crystal might be rotated so that it presents
the same appearance to a fixed observer, before and after the rotation. They are
defined in terms of the rotation angle required to repeat the appearance of the crystal.
The rotational symmetry elements in crystals are:
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Figure 2
In a crystal there are various combinations of symmetry elements possible. It should
be stressed again that it is the angular relations of the faces which are operated on by
the symmetry elements – the sizes of the individual faces may vary and this may
make the symmetry less obvious. That is why we use ideal crystals to describe the
symmetry more clearly.
The octahedron has mirror planes, tetrad axes, triad axes and diad axes.
THE UNIT CELL
The law of the constancy of interfacial angles led early mineralogists to postulate that
crystals were composed of small brick-like building blocks stacked in three
dimensions. They could account for all the crystal shapes and interfacial angles by
using different shapes of ‘bricks’. Modern X-ray analysis of crystal structures shows
that these ‘bricks’ can be defined as unit cells which contain the basic element of the
structure which is repeated along three crystallographic axes to produce the
macroscopic crystal.
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The shape of the unit cell is defined by the crystallographic axes and the interaxial
angles, as shown in the figure below.
z

c


x
y
a
b
Figure 3
a,b,c are the cell edge lengths parallel to the crystallographic axes x,y, and z.
is the angle between y and z
 is the angle between x and z
 is the angle between x and y.
SEVEN CRYSTAL SYSTEMS
A study of the external symmetry of crystals leads to the conclusion that crystals
belong to one of seven groups known as the crystal systems. The minimum symmetry
of each system defines the choice of the unit cell.
The crystal systems are illustrated overleaf.
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z
c


Cubic
Minimum symmetry
a=b= c
 
4 triad axes along cube diagonals
y

a
b
x
z
Minimum symmetry
a=b°c
 
1 tetrad axis parallel to z
c


Tetragonal
y

a
b
x
z
Orthorhombic
a ° b °c
 
c


Minimum symmetry
3 diad axes parallel to x,y, and z
y

a
b
x
z
Monoclinic


y

c
Minimum symmetry
a ° b °c
1 diad axis parallel to y or
1 mirror plane perpendicular to y
 
a
z
x
b
Triclinic
Minimum symmetry
a ° b °c
° ° ° 
none



y
c
a
x
b
Hexagonal
Minimum symmetry
a=b°c
= =1 hexad axis parallel to z

z
c
y
120 o
x
a
a
Trigonal
Minimum symmetry
a=b°c
= =1 triad axis parallel to z

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MILLER INDICES
Crystal planes or faces are described by their Miller indices which are a set of three
integers corresponding to the reciprocals of the intercepts that the face makes on each
of the three axes, x, y and z.
z
c
y
a
x
b
Figure 6
The plane drawn above makes an intercept of +1 on the x axis, +1 on the y axis and
+2 on the z axis.
The reciprocals of these intercepts are therefore +1, +1 and +0.5 respectively.
The Miller indices are these reciprocals multiplies to integers i.e. (2 2 1).
A general face may be written with indices (h k l ) and the intercepts on the x, y and z
axes are then in the ratios a/h : b/k : c/l. When referred to the external faces of a
crystal Miller indices contain no multiples e.g. (442) is the same as (221) etc. A zero
index means that that the plane is parallel to the axis.
z
y
(100)
(010)
(001)
(011)
(111)
(121)
(112)
(122)
(111)
(021)
Figure 7
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FORMS and HABIT
The form {hkl} refers to the group of crystal faces which are related to the face (hkl)
by symmetry. For example, in the cubic system the form {111} refers to a shape with
8 equivalent faces i.e. an octahedron, whereas (111) just refers to one face.
The habit describes the overall shape of the crystal and depends on the relative
development of faces in the various forms present. A mineral can grow with different
habits depending on the growth conditions. In the example below crystals in the cubic
system may grow as cubes from pure solutions, but the presence of impurities in the
solution may promote the growth of corner faces, leading ultimately to the dominance
of the octahedral form.
Figure 8
THE STEREOGRAPHIC PROJECTION OF CRYSTAL FACES
Crystal planes can be conveniently illustrated in two dimensions using a stereographic
projection, the principle of which is shown below:
P
O
P'
Projection
point
Figure 9
The projection of any direction OP from the centre of the sphere can be represented
on the equatorial plane by the point P’. In the stereographic projection of a crystal
face the line perpendicular to the crystal face is represented as a point in the
stereographic projection, as shown in the examples below.
10
z
100
y
z
010
001
010
y
x
(a)
100
x
z
100
y
z
010
100
001
010
y
010
001
z
x
011
101
110
(b)
110
100
x
z
100
x
100
y
z
010
100
001
x
010
y
010
z
001
111
100
x
100
x
z
111
y
111
z
111
y
111
(d)
x
Figure 10
010
y
111
(c)
x
010
y
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In the same way, symmetry elements can also be represented on a sterographic
projection. In this case the rotation axes are represented as the projections of the axes,
while the mirror planes are represented as great circles or projections of planes rather
than directions. In the example below, the symmetry elements of a tetragonal prism
are shown on a stereographic projection. The great circles drawn in heavy line
indicate the mirror planes.
Figure 11
THE 32 POINT GROUPS
Earlier it was mentioned that the symmetry elements shown by the faces of a crystal
(i.e. mirror planes and rotation axes) can be combined in a specific number of ways.
There are 32 ways of combining these symmetry elements, within the 7 crystal
systems. The 32 point groups as they are called can be represented on stereographic
projections. These are shown in Figures 12a,b on the following two pages, and should
be referred to when examining the wooden models of crystal faces.
In later courses the relevance of this systematic language of crystallography will be
explored in more detail.
PERIODICITY AND SYMMETRY IN THE INTERNAL STRUCTURE
In the following pages we explore the concept of a crystal as a periodic structure.
Periodicity implies the existence of a lattice, which is an array of points in space, all
of which have the same identical environment. The lattice therefore defines the
translational symmetry of the structure, and the unit cell is a parallelogram with
lattice points at the corners. The shapes of the different unit cells leads to the
definition of the 7 crystal systems, just as discussed above.
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The concept of lattice and unit cell can be illustrated more easily in 2 dimensions than 3,
and the Figure 13 shows a pattern which can represent a crystal structure. Any point in the
pattern can be defined as a lattice point, and all other lattice points must have the identical
environment.
O
y
x
Figure 13
If point O is chosen as a lattice point, the array of identical points defines a hexagonal
unit cell as shown in bold. The motif consists of 2 atoms with coordinates (2/3, 1/3) and
(1/3, 2/3). The whole pattern can be generated by specifying the hexagonal lattice, and
applying this motif to each lattice point. Note that only 2 atoms are needed for the motif,
but that their coordinates depend on the choice of origin for the unit cell.
Lattice + Motif = Structure
Choice of unit cell.
The choice of unit cell is made such that it is the smallest possible, but has the maximum
symmetry of the structure. When a unit cell has only lattice points at the corners it is
termed primitive, while a unit cell which has more lattice points is non-primitive. This
idea is demonstrated in Figure 14 which shows an array of lattice points and two possible
unit cells. In this pattern the distribution of lattice points is such that the smallest
primitive unit cell (shown dashed : a ≠ b ,  ≠ 120o) has a lower symmetry, and a
rectangular unit cell (solid line) with higher symmetry is conventionally chosen to
describe the lattice. This rectangular cell, with a lattice point at the centre as well as at the
corners is non-primitive, and is obviously larger than the smallest repeat unit.
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
b
a
Figure 14
LATTICE PLANES
Lattice planes connect lattice points in a crystal. Each set of lattice planes can be assigned
Miller indices just like crystal faces. The distance between lattice planes is an important
property and therefore there is a difference between (1 0 0) and (2 0 0) planes.
y
x
(010)
(100)
(210)
(110)
(120)
(120)
y
d200
x
d100
Figure 15
TWO-DIMENSIONAL LATTICE SYMMETRIES
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To understand some of the ways in which the three dimensional structures of minerals are
described in crystallographic terms it is convenient to begin with two dimensional
patterns. There are only five different types of two dimensional patterns (two dimensional
crystal systems) shown in Figure 16.
Figure 16
The complete symmetry of a pattern is not just defined by the symmetry of the lattice, but
also by the symmetry of the motif (the pattern or arrangment of atoms associated with
each lattice point). When we consider two or three dimensional patterns or crystal
structures we also need to introduce another symmetry element in addition to the mirrors
and rotation axes shown in Figure 2. This is because we now need also to consider
translational symmetry and a new symmetry operator called a glide plane. Figure 17
shows the way in which symmetry elements operate in two dimensions.
The symmetry of the motif must also be consistent with the symmetry of the lattice.
Figure 18 shows the 10 different ways of combining the symmetry elements of a motif
around a lattice point. These are the 10 two dimensional point groups.
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Figure 17
Figure 18
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The final step in the development of the symmetry of 2 dimensional patterns is the
combination of the 10 types of motif, with the 5 two-dimensional lattices. All “wall-paper
patterns” can be assigned to one of the 17 two-dimensional space groups shown in Figure
19. There are many internet sites illustrating beautiful patterns and their symmetries e.g.
http://mathmuse.sci.ibaraki.ac.jp/pattrn/PatternE.html and
http://web.inter.nl.net/hcc/Hans.Kuiper/17system.htm
To study the crystallography of mineral structures it is obviously necessary to extend
these ideas to three dimensions. The concepts are the same, but the number of
combinations of symmetry elements is greater. We have already seen that there are 32
point groups when we considered the symmetry of the external shapes of crystals.
Because these were combinations of symmetery elements without involving any
translation (i.e. around a point), they are exactly equivalent to the 10 two-dimensional
point groups shown in Figure 18. When the 32 point groups are combined with the
different types of lattices in three dimensions (in two dimensions there are 5; in three
dimensions there are 14), the result is 230 three-dimensional space groups (compared
with 17 in two dimensions). These topics are beyond the scope of this course and will be
considered in more detail in later courses in Crystallography.