Transcript CRYSTALLOGRAPHY GL10B IMPORTANT: Collect lecture notes …

IMPORTANT:
We list and describe all the crystal
classes/minerals
Triclinic System
Characterized by only 1-fold or 1-fold rotoinversion axis
Symmetry content - i
Pinacoidal Class,
One centre of symmetry (pairs of
faces are related to each other
through the centre).
Such faces are called pinacoids,
pinacoidal class.
Egs. microcline (K-feldspar),
plagioclase, turquoise, and
wollastonite.
Monoclinic System
Characterized by having only mirror plane(s) or a single 2-fold axis.
A MAJORITY OF ROCK FORMING MINERALS ARE INCLUDED IN THIS CLASS
Normal Class or Prismatic Class, 2/m; Symmetry content - 1A2, m , i
One 2-fold axis perpendicular to a single
mirror plane.
This class has pinacoid faces and prism
faces.
A prism = 3 or more identical faces
that are all parallel to the same line.
In the prismatic class, these prisms consist
of 4 identical faces, 2 of which are shown in
the diagram on the front of the crystal. The
other two are on the back side of the crystal.
Egs. micas (biotite and muscovite), azurite, chlorite, clinopyroxenes,
epidote, gypsum, malachite, kaolinite, orthoclase, and talc.
Orthorhombic System
Characterized by having only three 2- fold axes/3 m
or a 2-fold axis and 2 mirror planes.
Normal Class or Barytes type or
Rhombic- dipyramidal Class, 2/m2/m2/m,
Symmetry content - 3A2, 3m, i
This class has 3 perpendicular 2-fold
axes that are perpendicular to 3 mirror
planes.
The dipyramid faces consist of 4 identical
faces on top and 4 identical faces on the
bottom that are related to each other by
reflection across the horizontal mirror
plane or by rotation about the horizontal
2-fold axes.
Egs. andalusite, anthophyllite, aragonite,
barite, cordierite, olivine, sillimanite,
stibnite, sulphur, and topaz.
Tetragonal System
Characterized by a single 4-fold or 4-fold rotoinversion axis.
Normal Class, Zircon Type
Ditetragonal-dipyramidal Class,
4/m2/m2/m,
Symmetry content - 1A4, 4A2, 5m, i
It has a single 4-fold axis that is
perpendicular to four 2-fold axes.
All of the 2-fold axes are perpendicular to
mirror planes. Another mirror plane is
perpendicular to the 4-fold axis.
The mirror planes are not shown in the
diagram, but would cut through all of the
vertical edges and through the centre of
the pyramid faces. The fifth mirror plane
is the horizontal plane.
Note the ditetragonal-dipyramid consists
of the 8 pyramid faces on the top and the
8 pyramid faces on the bottom.
Common minerals that occur
with this symmetry are anatase,
cassiterite, apophyllite, zircon,
and vesuvianite.
Hexagonal System
Characterized by having either a 6-fold or a 3-fold axis
Common form: Dihexagonal-dipyramidal
Class,
6/m2/m2/m,
Symmetry content - 1A4, 6A2, 7m, i
It has a single 6-fold axis that is
perpendicular to six 2-fold axes.
All of the 2-fold axes are
perpendicular to mirror
planes. Another mirror plane is
perpendicular to the 6-fold
axis. Totalling 7 mirror planes.
Beryl
Isometric System
Characterized by having either a 6-fold or a 3-fold axis
Common form: cube
4/m bar 3 2/m,
Symmetry content - 3A4, 3¯ A3, 6A2, 9m note
(1¯ A3 = 1A3 + i)
Most symmetrical of a 3-D system.
It has a four 3-fold axes, three 4-fold
axes and six 2-fold axes.
All of the 2-fold axes. 9 mirror planes
and a centre.
Egs. Gold. Galena, diamond, copper, silver, lead
Halite
Crystal Morphology, Crystal Symmetry, Crystallographic
Axes
Crystal Morphology and Crystal Symmetry:
Recall: symmetry observed in crystals as exhibited by their crystal
faces is due to the ordered internal arrangement of atoms in a crystal
structure, as mentioned previously. This arrangement of atoms in
crystals is called a lattice.
Crystals, are made up of 3-dimensional arrays of atoms.
Such 3-D arrays are called space lattices.
Crystal faces develop along planes 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 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 equivalent
faces of the same mineral, measured at
constant temp., are constant. This is
known as the Law of constancy of
interfacial angles.
Crystallographic Axes:
The crystallographic axes are imaginary
lines that we can draw within the crystal
lattice. These will define a co-ordinate
system within the crystal. 3-D space
lattices will have 3 or in some cases 4
crystallographic axes that define
directions within the crystal lattices.
Crystallographic axes
Triclinic
Where a ≠ b ≠ c; α ≠ β ≠ γ
Monoclinic
Where a ≠ b ≠ c;
α = γ = 900 & β > 900
Crystallographic axes cont’d
Orthorhombic
Where a ≠ b ≠ c;
α = β = γ = 900
Tetragonal
Where a1 =a2 ≠ c;
α = β = γ = 900
Crystallographic axes cont’d
Hexagonal
Where a1 = a2 = a3 ≠ c;
α = β = 900 ; γ = 1200
Isometric
Where a1 = a2 = a3 ;
α = β = γ = 900
Unit Cells
The "lengths" of the various crystallographic axes are defined on the
basis of the unit cell.
When arrays of atoms or molecules are laid out in a space lattice we
define a group of such atoms as the unit cell.
This unit cell contains all the necessary points on the lattice that can
be translated to repeat itself in an infinite array.
In other words, the unit cell defines the basic building blocks of the
crystal, and the entire crystal is made up of repeatedly translated unit
cells.
The relative lengths of the crystallographic axes, or unit cell edges,
can be determined from measurements of the angles between crystal
faces. We will consider measurements of axial lengths, and develop a
system to define directions and label crystal faces.
In defining a unit cell The edges of the unit cell should coincide with
the symmetry of the lattice. The edges of the unit cell should be related
by the symmetry of the lattice. The smallest possible cell that contains
all elements should be chosen.
The 7 Crystal systems: Unit cells The 14 Bravais Lattices:
(WIKEPEDIA)
hexagonal
(1 hexad)
triclinic
(none
rhombohedra
l
(1 triad)
monoclinic
(1 diad)
tetragonal
(1 tetrad)
cubic
(4 triads)
orthorhombic
(3 perpendicular diads)
Axial Ratios, Parameters, Miller Indices
RECALL:
•The lengths of the crystallographic axes are controlled by
the dimensions of the unit cell upon which the crystal is
based.
•The angles between the crystallographic axes are
controlled by the shape of the unit cell.
•The relative lengths of the crystallographic axes control
the angular relationships between crystal faces. This is
true because crystal faces can only develop along lattice
points.
The relative lengths of the crystallographic axes are called
axial ratios
Axial Ratios
Axial ratios are defined as the relative lengths of the
crystallographic axes.
They are normally taken as relative to the length of the b
crystallographic axis.
Thus, an axial ratio is defined as follows:
Axial Ratio = a/b : b/b : c/b
Where
a is the actual length of the a crystallographic axis,
b, is the actual length of the b crystallographic axis,
and
c is the actual length of the c crystallographic axis.
The end of the axis facing an observer is designated as the positive end,
and the away end is referred to as the negative end.
For Triclinic, Monoclinic, and Orthorhombic crystals,
where the lengths of the three axes are different, this
reduces to: a/b : 1 : c/b (this is usually shortened to a : 1 : c)
For Tetragonal crystals where a=b, this reduces to: 1 : 1 : c/b
(or 1 : c)
For Isometric crystals where the length of the a= b= c this
becomes 1 : 1 : 1 (or 1)
For Hexagonal crystals where there are three equal length
axes (a1, a2, and a3) perpendicular to the c axis this
becomes: 1 : 1 : 1: c/a (usually shortened to 1 : c)
Modern crystallographers can use x-rays to determine the
size of the unit cell, and thus can determine the absolute
value of the crystallographic axes in Angstrom units.
1 Å= 0.0000000001 m (10 -10 m).
Example
For quartz which is hexagonal, the following unit cell
dimensions determined by x-ray crystallography:
a1 = a2 = a3 = 4.913Å ;
c = 5.405Å
Thus the axial ratio for quartz is: 1 : 1 : 1 : 5.405/4.913
Or 1: 1 : 1 : 1.1001
which simply says that the c axis is 1.1001 times longer
than the a axes.
For orthorhombic sulphur the unit cell dimensions as
measured by x-rays are:
a = 10.47Å, b = 12.87Å, c = 24.39Å
Thus, the axial ratio for orthorhombic sulphur is:
10.47/12.87 : 12.87/12.87 : 24.39/12.87
or
0.813 : 1 : 1.903
Intercepts of Crystal Faces (Weiss Parameters)
Crystal faces can be defined by their intercepts on the crystallographic axes.
For non-hexagonal crystals, there are three cases.
1. A crystal face intersects only one of the crystallographic axes.
As an example the top crystal face shown here
intersects the c axis but
does not intersect the a or b axes.
If we assume that the face intercepts the c axis at a distance of 1 unit length,
then the intercepts, sometimes called Weiss Parameters, are
Infinity a, infinity b, 1c
2. A crystal face intersects two of the crystallographic
axes.
As an example, the darker crystal face shown here
intersects the a and b axes, but not the c axis.
Assuming the face intercepts the a and c axes at 1 unit
cell length on each,
the parameters for this face are: 1 a, 1 b, infinity c
3. A crystal face that intersects all 3 axes.
In this example the darker face is assumed to
intersect the a, b, and c crystallographic axes at
one unit length on each.
Thus, the parameters in this example would be:
1a, 1b, 1c
Two very important points about intercepts of faces:
The intercepts or parameters are relative values, and do not
indicate any actual cutting lengths. Since they are relative, a
face can be moved parallel to itself without changing its
relative intercepts or parameters.
Note the dimensions of the unit cell is unknown. Therefore
one face is assign to have intercept 1
Thus, the convention is to assign the largest face that
intersects all 3 crystallographic axes the parameters 1a, 1b, 1c. This face is called the unit face.
Faces may make intercepts on all of the –ve or all of the +ve
ends of axes or on one –ve and two +ve ends or two –ve
and one +ve ends etc.
Miller Indices
The Miller Index for a crystal face is found by
•first determining the parameters
•second inverting the parameters, and
•third clearing the fractions.
For example,
if the face has the parameters 1 a, 1 b, infinity c
inverting the parameters would be 1/1, 1/1, 1/ infinity
this would become 1, 1, 0
the Miller Index is written inside parentheses with no
commas - thus (110)
The face [labelled (111)] that
cuts all three axes at 1 unit
length has the parameters
1a, 1b, 1c. Inverting these,
results in 1/1, 1/1, 1/1 to give
the Miller Index (111).
The square face that cuts the positive a axis, has the
parameters 1 a, infinity b, infinity c. Inverting these
becomes 1/1, 1/infinity, 1/infinity to give the Miller Index
(100).
The face on the back of the crystal that cuts the negative a
axis has the parameters -1a, infinity b, infinity c.
So its Miller Index is ( ¯100).
This would be read "minus one, zero, zero". The 6 faces seen
on this crystal would have the Miller Indices (00minus1),(001),
(010), and (0minus10)(100)(minus100).
Since the hexagonal system has three "a" axes perpendicular to the
"c" axis, both the parameters of a face and the Miller Index notation
must be modified.
The modified parameters and Miller Indices
must reflect the presence of an additional axis.
This modified notation is referred to
as Miller-Bravais Indices,
with the general notation (hkil).
Let's derive the Miller indices for the dark
shaded face in the hexagonal crystal shown.
This face intersects the
positive a1 axis at 1 unit length,
the negative a3 axis at 1 unit length, and
does not intersect the a2 or c axes. This face thus has the
parameters:1 a1, infinity a2, -1 a3, infinity c
Inverting and clearing fractions gives
the Miller-Bravais Index: (10 minus10).
An important rule to remember in applying this notation in the hexagonal system, is that
whatever indices are determined for h, k, and i, h + k + i = 0
For a similar hexagonal
crystal, having the shaded
face cutting all three axes,
the parameters are 1 a1, 1 a2,
-1/2 a3, infinity c.
Inverting these intercepts
gives:
1/1, 1/1, -2/1, 1/infinity
resulting in a Miller-Bravais
Index of (1 1 minus2 0)
Note "h + k + i = 0" rule
applies here!
Crystal Forms
A crystal form is a set of crystal faces that are related to each other by
symmetry.
To designate a crystal form (which could imply many faces) we use the
Miller Index, or Miller-Bravais Index notation enclosing the indices in
curly braces, i.e. {hkl} or {hkil}
Such notation is called a form symbol.
There are 48 possible forms that can be developed as the result of the
32 combinations of symmetry. We discuss some, but not all of these
forms. (Thirty (30) close and eighteen (18) open forms).
Open Forms and Closed Forms
Open form
An open form is one or more crystal faces that do not completely
enclose space.
A crystal with open-form faces also requires some additional
closed-form facets to complete a structure. Open-forms include:
Pedion, Pinacoid, Dome, Sphenoid, Pyramid, Prism
A Pedion is a flat face that is not parallel, or geometrically linked
to any other faces.
A Pinacoid is composed of only two parallel faces, forming
tabular crystals such as ruby.
A Dome is found in monoclinic and orthorhombic minerals Two
intersecting faces that are caused by mirroring (topaz)
commonly forms domes.
Sphenoid’s are found in monoclinic and orthorhombic minerals,
and have two-fold rotational axes.
A Pyramid's multiple facets converge on a single
crystallographic axis, and pyramid forms are not possible on
minerals from the isometric, monoclinic or triclinic systems.
Open Hexagonal & Triangular Prisms
Prisms have a set of facets that run parallel to an axis of a crystal,
yet never converge with it. Eg. Quartz forms two sets of three sided
prisms. Prisms are not possible in isometric or triclinic minerals.
A Hexagonal (trigonal) prism is comprised of two hexagonal bases
connected by a set of six rectangular faces that run parallel to, and
never converge with an axes in the crystal.
A triangular (trigonal) prism is comprised of two triangular bases
connected by a set of three rectangular faces that run parallel to,
and never converge with an axes in the crystal. This form is
similar to a light-refracting 60º prism.
closed form
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.
There are two types of closed forms (closed isometric and non-isometric
forms)
A crystal may comprise more than one form, called a combination.
There are several crystal forms in the cubic crystal system
that are common in diamond,garnet, spinel and other "symmetrical“
gemstones.
A hexahedron (cube) has eight points, six faces, and twelve edges
that are perpendicular to each other, forming 90 degree angles.
An octahedron has two four sided pyramids lying base to base, and
is totally symmetrical with no top, or bottom and has eight faces.
A tetrahedronhas four equilateral triangular faces.
A dodecahedron has 12 faces
There are four types of dodecahedrons listed in order of
descending symmetry:
1. Symmetrical pentagonal (five edged polygons)
dodecahedrons,
2. 2. Asymmetrical (tetartoid) pentagonal dodecahedrons,
3. 3. Delta (four edged polygons) dodecahedrons, and
4. 4. Rhombic dodecahedrons.
Note: A Hexoctahedron is a multi-faceted dodecahedron with
48 triangular faces.
Closed Non-Isometric Forms
1. Hexagonal (Trigonal) Closed Forms
•Hexagonal Pyramid
•Hexagonal Bipyramid (Apatite)
•Dihexagonal bipyramid (Beryl)
•Hexagonal Trapezohedron
•Hexagonal Scalenohedron
•Tetrahexahedron
2. Tetragonal Closed Forms
•Tetragonal Disphenoid
•Tetragonal Scalenohedron
•Tetragonal Trapezohedron
•Tetragonal Trapezohedral Trisoctahedron
•Tetragonal Ditetragonal Bipyramidal (Rutile)
. Rhombohedral Closed Forms
•Rhombohedral Trapezohedral (Quartz)
•Rhombohedral Hemimorphic (Tourmaline)
•Rhombohedral Holohedra (Calcite)
•Rhombohedral Dodecahedron (Garnet, Fluorite)
•Rhombohedral Trisoctahedron
4. Orthorhombic Closed Forms
•Rhombic Prism
•Rhombic Pyramid
•Rhombic Dipyramid
•Rhombic Hemimorphic
•Rhombic Sphenoid
•Rhombic Pyramid
5. Monoclinic Closed Forms
Prism
Monoclinic Clinopinacoid
6. Triclinic Closed Forms
Prism
Triclinic Dipyramid
Understanding Miller Indices, Form Symbols, and Forms
We define the crystallographic axes in relation to the elements of
symmetry in each of the crystal systems.
Triclinic - Since this class has such low symmetry there are no
constraints on the axes, but the most pronounced face should be taken
as parallel to the c axis.
Monoclinic - The 2 fold axis is the b axis, or if only a mirror plane is
present, the b axis is perpendicular to the mirror plane.
Orthorhombic - The current convention is to take the longest axis as b,
the intermediate axis is a, and the shortest axis is c.
Tetragonal - The c axis is either the 4 fold rotation axis or the
rotoinversion axis.
Hexagonal - The c axis is the 6-fold or 3-fold axis
Isometric - The equal length a axes are either the 3 4-fold rotation axes,
rotoinversion axes, or, in cases where no 4-fold axes are present, the 3 2fold axes.
ZONES- A zone is defined as a group of crystal faces that intersect in
parallel edges. Since the edges will all be parallel to a line, we can define
the direction of the line using a notation similar to Miller Indices.
Crystal Habit
The faces that develop on a crystal depend on the space available for the
crystals to grow. The term used to describe general shape of a crystal is habit.
Cubic - cube shapes
Octahedral - shaped like octahedrons, as described above.
Tabular - rectangular shapes.
Equant - a term used to describe minerals that have all of their boundaries of
approximately equal length.
Fibrous - elongated clusters of fibres.
Acicular - long, slender crystals.
Prismatic - abundance of prism faces.
Bladed - like a wedge or knife blade
Twinning in Crystals
During the growth of a crystal (not in all cases), or if the crystal
is subjected to stress or temperature/pressure conditions
different from those under which it originally formed, two or
more intergrown crystals are formed in a symmetrical fashion.
These symmetrical intergrowths of crystals are called twinned
crystals.
Twinning is important to recognize, because when it occurs, it is
often one of the most diagnostic features enabling identification
of the mineral.
Types of Twinning
Contact Twins - have a planar composition surface
separating 2 individual crystals. These are usually defined by
a twin law that expresses a twin plane (i.e. an added mirror
plane). An example shown here is a crystal of orthoclase
twinned on the Baveno Law, with {021} as the twin plane.
Penetration Twins - have an irregular composition surface
separating 2 individual crystals. These are defined by a twin
centre or twin axis. Shown here is a twinned crystal of
orthoclase twinned on the Carlsbad Law with [001] as the
twin axis.
Contact twins can also occur as repeated or multiple twins.
If the compositions surfaces are parallel to one another, they
are called polysynthetic twins. Plagioclase commonly shows
this type of twinning, called the Albite Twin Law, with {010}
as the twin plane. Such twinning is one of the most
diagnostic features of plagioclase.
Next lecture will be on sterographic projection
Pinacoids
A Pinacoid is an open 2-faced form made up of two parallel faces.
Domes
Domes are 2- faced open forms where the 2 faces 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).
Prisms
A prism is an open form consisting of three or more parallel
faces. Depending on the symmetry, several different kinds of
prisms are possible.
Rhombic prism: A form with four faces,
with all faces parallel to a line that is not a symmetry
element. In the drawing to the right, the 4 shaded faces
belong to a rhombic prism. The other faces in this model
are pinacoids (the faces on the sides belong to a side
pinacoid, and the faces on the top and bottom belong to a
top/bottom pinacoid).
Tetragonal prism:
4 - faced open form with all faces parallel to a 4-fold
rotation axis.
The 4 side faces in this model make up the tetragonal
prism. The top and bottom faces make up the a form
called the top/bottom pinacoid.
Hexagonal prism:
6 - faced form with all faces parallel to a 6-fold rotation
axis. The 6 vertical faces in the drawing make up the
hexagonal prism. Again the faces on top and bottom are
the top/bottom pinacoid form.
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.
Hexagonal pyramid: 6-faced form where all faces are
related by a 6 axis. If viewed from above, the hexagonal
pyramid would have a hexagonal shape.
Dipyramids are closed forms consisting of 6, 8, 12, 16, or
24 faces. Dipyramids are pyramids that are reflected
across a mirror plane. Dihexagonal dipyramid: 24-faced
form with faces related by a 6-fold axis with a
Hexahedron:
A hexahedron is the same as a cube. 3-fold axes are
perpendicular to the face of the cube, and 4-fold axes run
through the corners of the cube. Note that the form symbol for a
hexahedron is {100}, and it consists of the following 6 faces.
(100), (010), (001), (minus1 00), (0minus1 0), and (00 minus1).
Example: Galena, Halite
Octahedron:
An octahedron is an 8 faced form that results form three 4-fold
axes with perpendicular mirror planes. The octahedron has the
form symbol {111}and consists of the following 8 faces: (111),
( minus1minus1minus1), (1 minus11), (1minus1minus1 ),
(minus1minus1 1), (minus11minus1), (11minus1 ), (minus111).
Note that four 3-fold axes are present that are perpendicular to
the triangular faces of the octahedron (these 3-fold axes are not
shown in the drawing). Example: Diamond.
Dodecahedron:
A dodecahedron is a closed 12-faced form. Dodecahedrons can
be formed by cutting off the edges of a cube. The form symbol
for a dodecahedron is {110}. As an exercise, you figure out the
Miller Indices for these 12 faces. Example: Garnet