48919799 gl10b crystallography 2new

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IMPORTANT:

We list and describe all the crystal

classes/minerals



Triclinic SystemCharacterized by only 1-fold or 1-fold rotoinversion axis

Pinacoidal Class,

Symmetry content - i

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, andwollastonite.



Monoclinic SystemCharacterized by having only mirror plane(s) or a single 2-fold axis.

AMAJORITY OF ROCK FORMING MINERALS ARE INCLUDED IN THIS CLASSNormal Class or Prismatic Class, 2/m; Symmetry content - 1A2, m , i

One 2-fold axis perpendicular to a singlemirror plane.This class has pinacoid faces and prismfaces.A prism = 3 or more identical facesthat are all parallel to the same line.In the prismatic class, these prisms consistof 4 identical faces, 2 of which are shown in

the diagram on the front of the crystal. Theother 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 SystemCharacterized 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-foldaxes that are perpendicular to 3 mirror planes.The dipyramid faces consist of 4 identicalfaces on top and 4 identical faces on thebottom that are related to each other by

reflection across the horizontal mirror plane or by rotation about the horizontal2-fold axes.Egs. andalusite, anthophyllite, aragonite,barite, cordierite, olivine, sillimanite,stibnite, sulphur, and topaz.



Tetragonal SystemCharacterized 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 isperpendicular to four 2-fold axes.

All of the 2-fold axes are perpendicular tomirror planes. Another mirror plane isperpendicular to the 4-fold axis.

The mirror planes are not shown in the

diagram, but would cut through all of thevertical edges and through the centre of the pyramid faces. The fifth mirror planeis the horizontal plane.Note the ditetragonal-dipyramid consistsof 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-dipyramidalClass,6/m2/m2/m,Symmetry content - 1A4, 6A2, 7m, i

Beryl

It has a single 6-fold axis that isperpendicular to six 2-fold axes.All of the 2-fold axes areperpendicular to mirror planes. Another mirror plane is

perpendicular to the 6-foldaxis. Totalling 7 mirror planes.



Characterized by having either a 6-fold or a 3-fold axis

Common form: cube4/m bar 3 2/m,Symmetry content - 3A4, 3¯ A3, 6A2, 9m note(1¯ A3 = 1A3 + i)

Halite

Most symmetrical of a 3-D system.

It has a four 3-fold axes, three 4-foldaxes and six 2-fold axes.All of the 2-fold axes. 9 mirror planes

and a centre.

Isometric System

Egs. Gold. Galena, diamond, copper, silver, lead



Crystal Morphology, Crystal Symmetry, CrystallographicAxes

Crystal Morphology and Crystal Symmetry:

Recall: symmetry observed in crystals as exhibited by their crystalfaces is due to the ordered internal arrangement of atoms in a crystalstructure, as mentioned previously. This arrangement of atoms incrystals 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 intersectsa larger number of lattice points.This is known as the Bravais Law .



The angle between crystal faces iscontrolled by the spacing between latticepoints.

Since all crystals of the same substancewill have the same spacing between latticepoints (they have the same crystalstructure), the angles between equivalentfaces of the same mineral, measured at

constant temp., are constant. This isknown as the Law of constancy of

interfacial angles.

Crystallographic Axes:The crystallographic axes are imaginarylines that we can draw within the crystallattice. These will define a co-ordinatesystem within the crystal. 3-D spacelattices will have 3 or in some cases 4crystallographic axes that definedirections within the crystal lattices.



Where a b c; Į ȕ Ȗ

Triclinic

Monoclinic

Where a b c;

Į = Ȗ = 900 & ȕ > 900

Crystallographic axes



Orthorhombic

Where a b c;

Į = ȕ = Ȗ = 900

Tetragonal

Where a1 =a2 c;

Į = ȕ = Ȗ = 900

Crystallographic axes cont¶d



Hexagonal

Where a1 = a2 = a3 c;Į = ȕ = 900 ; Ȗ = 1200

Isometric

Where a1 = a2 = a3 ;

Į = ȕ = Ȗ = 900

Crystallographic axes cont¶d



Unit Cells

The "lengths" of the various crystallographic axes are defined on thebasis of the unit cell.When arrays of atoms or molecules are laid out in a space lattice wedefine a group of such atoms as the unit cell.

This unit cell contains all the necessary points on the lattice that canbe translated to repeat itself in an infinite array.

In other words, the unit cell defines the basic building blocks of thecrystal, and the entire crystal is made up of repeatedly translated unitcells.The relative lengths of the crystallographic axes, or unit cell edges,can be determined from measurements of the angles between crystalfaces. We will consider measurements of axial lengths, and develop asystem to define directions and label crystal faces.In defining a unit cell The edges of the unit cell should coincide withthe symmetry of the lattice. The edges of the unit cell should be relatedby the symmetry of the lattice. The smallest possible cell that contains

all elements should be chosen.



monoclinic

(1 diad)

triclinic(none

The 7 Crystal systems: Unit cells

(WIKEPEDIA)The 14 Bravais Lattices:

orthorhombic

(3 perpendicular diads)

hexagonal(1 hexad)

rhombohedral

(1 triad)tetr agonal

(1 tetr ad)

cubic

(4 triads)



Axial Ratios, Parameters, Miller Indices

RECALL:

The lengths of the crystallographic axes are controlled bythe dimensions of the unit cell upon which the crystal isbased.

The angles between the crystallographic axes are

controlled by the shape of the unit cell.

The relative lengths of the crystallographic axes controlthe angular relationships between crystal faces. This istrue because crystal faces can only develop along latticepoints.

The relative lengths of the crystallographic axes are calledaxial ratios



Axial RatiosAxial ratios are defined as the relative lengths of thecrystallographic axes.They are normally taken as relative to the length of the bcrystallographic axis.

Thus, an axial ratio is defined as follows:

Axial Ratio = a/b : b/b : c/b

Wherea is the actual length of the a crystallographic axis,b, is the actual length of the b crystallographic axis,andc 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, thisreduces 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 thisbecomes 1 : 1 : 1 (or 1)

For Hexagonal crystals where there are three equal lengthaxes (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 thesize of the unit cell, and thus can determine the absolutevalue of the crystallographic axes in Angstrom units.

1 Å= 0.0000000001 m (10 -10 m).



ExampleFor quartz which is hexagonal, the following unit celldimensions determined by x-ray crystallography:

a1 = a2 = a3 = 4.913Å ;c = 5.405ÅThus the axial ratio for quartz is: 1 : 1 : 1 : 5.405/4.913Or 1: 1 : 1 : 1.1001which simply says that the c axis is 1.1001 times longer

than the a axes.

For orthorhombic sulphur the unit cell dimensions asmeasured by x-rays are:

a = 10.47Å, b = 12.87Å, c = 24.39ÅThus, the axial ratio for orthorhombic sulphur is:10.47/12.87 : 12.87/12.87 : 24.39/12.87or 0.813 : 1 : 1.903



Intercepts of Crystal Faces (Weiss Parameters)

Crystal f aces can be defined by their intercepts on the crystallogr aphic axes.

For non-hexagonal crystals, there are three cases.

1. A crystal f ace inter sects only one of the crystallogr aphic axes.

As an example the top crystal f ace shown here

inter sects th

e c axis butdoes not inter sect the a or b axes.

If we assume that the f ace intercepts the c axis at a distance of 1 unit length,

then the intercepts, sometimes called Weiss Par ameter s, are

Infinity a, infinity b, 1c



2. A crystal face intersects two of the crystallographic

axes.

As an example, the darker crystal face shown hereintersects 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 tointersect the a, b, and c crystallographic axes atone 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 notindicate any actual cutting lengths. Since they are relative, aface can be moved parallel to itself without changing itsrelative intercepts or parameters.Note the dimensions of the unit cell is unknown. Therefore

one face is assign to have intercept 1Thus, the convention is to assign the largest face thatintersects 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 +veends of axes or on one ±ve and two +ve ends or two ±veand one +ve ends etc.



Miller IndicesThe Miller Index for a crystal face is found byfirst determining the parameterssecond inverting the parameters, andthird 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/ infinitythis would become 1, 1, 0

the Miller Index is written inside parentheses with nocommas - thus (110)



The face [labelled (111)] thatcuts all three axes at 1 unitlength has the parameters

1a, 1b, 1c. Inverting these,results in 1/1, 1/1, 1/1 to givethe Miller Index (111).

The square face that cuts the positive a axis, has theparameters 1 a, infinity b, infinity c. Inverting thesebecomes 1/1, 1/infinity, 1/infinity ҏto 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, zer o, zer o". The 6 f aces 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 notationmust be modified.

The modified parameters and Miller Indicesmust reflect the presence of an additional axis.This modified notation is referred toasMiller-Bravais Indices,with the general notation (hkil).

Let's derive the Miller indices for the darkshaded face in the hexagonal crystal shown.This face intersects thepositive a1 axis at 1 unit length,

the negative a3 axis at 1 unit length, anddoes not intersect the a2 or c axes. This face thus has theparameters:1 a1, infinity ҏa2, -1 a3, infinity ҏcInverting and clearing fractions givestheMiller-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 hexagonalcrystal, having the shaded

face cutting all three axes,the parameters are 1 a1, 1 a2,-1/2 a3, infinity ҏc.Inverting these intercepts

gives:1/1, 1/1, -2/1, 1/infinityresulting in a Miller-BravaisIndex of (1 1 minus2 0)

N ote "h + k + i = 0" ruleapplies here!



Crystal Forms

A crystal form is a set of crystal faces that are related to each other bysymmetry.

To designate a crystal form (which could imply many faces) we use theMiller Index, or Miller-Bravais Index notation enclosing the indices incurly 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 the32 combinations of symmetry. We discuss some, but not all of theseforms. (Thirty (30) close and eighteen (18) open forms).



Open form

An open form is one or more crystal faces that do not completelyenclose space.

Open Forms and Closed Forms

A crystal with open-form faces also requires some additionalclosed-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 linkedto any other faces.

A Pinacoid is composed of only two parallel faces, formingtabular crystals such as ruby.

A Dome is found in monoclinic and orthorhombic minerals Twointersecting 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 singlecrystallographic axis, and pyramid forms are not possible onminerals from the isometric , monoclinic or triclinic systems.



Open Hexagonal & Triangular PrismsPrisms have a set of f acets that run par allel to an axis of a crystal,

yet never conver ge with it. Eg. Quartz f or ms two sets of three sided

prisms. Prisms are not possible in isometric or triclinic miner als.

A Hexagonal (trigonal ) prism is comprised of two hexagonal bases

connected by a set of six rectangular f aces that run par allel to, and

never conver ge with an axes in the crystal.

A triangular (trigonal ) prism is comprised of two triangular bases

connected by a set of three rectangular f aces that run par allel to,

and never conver ge with an axes in the crystal. This f or m is

similar to a light-refr acting 60º prism.



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 madeup of a single form.

There are two types of closed forms (closed isometric and non-isometricforms)

A crystal may comprise more than one form, called a combination.

closed form



There are several crystal forms in the cu bic crystal systemthat are common in diamond,garnet, spinel and other "symmetrical³gemstones.

A hexahedron (cube) has eight points, six faces, and twelve edgesthat are perpendicular to each other, forming 90 degree angles.An octahedron has two four sided pyramids lying base to base, andis totally symmetrical with no top, or bottom and has eight faces.

A tetrahedronhas four equilateral triangular faces.

A dodecahedronhas 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, and4. 4. Rhombic dodecahedrons.

Note: A Hexoctahedron is a multi-faceted dodecahedron with48 triangular faces.



Closed Non-Isometric Forms

1. Hexagonal (Trigonal) Closed FormsHexagonal Pyr amid

Hexagonal Bipyr amid ( Apatite)Dihexagonal bipyr amid (Beryl )

Hexagonal Tr apezohedr on

Hexagonal Scalenohedr on

Tetr ahexahedr on

2. Tetragonal Closed FormsTetr agonal Disphenoid

Tetr agonal Scalenohedr on

Tetr agonal Tr apezohedr on

Tetr agonal Tr apezohedr al Trisoctahedr on

Tetr agonal Ditetr agonal Bipyr amidal (Rutile)



5. Monoclinic Closed FormsPrismMonoclinic Clinopinacoid

6. Triclinic Closed FormsPrism

Triclinic Dipyr amid



Tetragonal - The c axis is either the 4 fold rotation axis or therotoinversion 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 2-fold axes.

ZONES- A zone is defined as a group of crystal faces that intersect inparallel 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 thecrystals to grow. The term used to describe general shape of a crystal is habit.

C u bic - cube shapesOctahedral - 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.

Fi brous - 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 crystalis subjected to stress or temperature/pressure conditionsdifferent 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 twinnedcrystals.

Twinning is important to recognize, because when it occurs, it isoften 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 bya twin law that expresses a twin plane (i.e. an added mirror plane). An example shown here is a crystal of orthoclasetwinned on the Baveno Law, with {021} as the twin plane.

Penetration Twins - have an irregular composition surfaceseparating 2 individual crystals. These are defined by a twincentre or twin axis. Shown here is a twinned crystal of orthoclase twinned on the Carlsbad Law with [001] as thetwin axis.

Contact twins can also occur as repeated or multiple twins.If the compositions surfaces are parallel to one another, theyare called polysynthetic twins. Plagioclase commonly showsthis 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 ster ogr aphic pr ojection



PinacoidsA Pinacoid is an open 2-faced form made up of two parallel faces.

DomesDomes are 2- faced open forms where the 2 faces are related toone another by a mirror plane. In the crystal model shown here,the dark shaded faces belong to a dome. The vertical facesalong the side of the model are pinacoids (2 parallel faces).

PrismsA prism is an open form consisting of three or more parallelfaces. 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 facesbelong to a rhombic prism. The other faces in this modelare pinacoids (the faces on the sides belong to a sidepinacoid, and the faces on the top and bottom belong to atop/bottom pinacoid).



Tetragonal prism:4 - faced open form with all faces parallel to a 4-foldrotation axis.The 4 side faces in this model make up the tetragonal

prism. The top and bottom faces make up the a formcalled the top/bottom pinacoid.Hexagonal prism:6 - faced form with all faces parallel to a 6-fold rotationaxis. The 6 vertical faces in the drawing make up thehexagonal 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 allfaces in the form meet, or could meet if extended, at apoint.

Hexagonal pyramid: 6-faced form where all faces arerelated by a 6 axis. If viewed from above, the hexagonalpyramid would have a hexagonal shape.Dipyramids are closed forms consisting of 6, 8, 12, 16, or 24 faces. Dipyramids are pyramids that are reflectedacross 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 areperpendicular to the face of the cube, and 4-fold axes runthrough 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-foldaxes with perpendicular mirror planes. The octahedron has theform 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 tothe triangular faces of the octahedron (these 3-fold axes are notshown in the drawing). Example: Diamond.

Dodecahedron:A dodecahedron is a closed 12-faced form. Dodecahedrons canbe formed by cutting off the edges of a cube. The form symbolfor a dodecahedron is {110}. As an exercise, you figure out the

Miller Indices for these 12 faces Example: Garnet

48919799 gl10b crystallography 2new

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