Fig. 2-23Fig. 2-23

It is very easy to visualize the location of a simple face given miller index, or to derive a miller index from simple faces

a

b

c

-b

-c

Hexagonal Miller indexHexagonal Miller index There need to be 4 intercepts (hkil)There need to be 4 intercepts (hkil)

h = ah = a11

k = ak = a22

i = ai = a33

l = cl = c Two a axes have to have opposite Two a axes have to have opposite

sign of other axis so thatsign of other axis so that h + k + i = 0h + k + i = 0

Possible to report the index two ways:Possible to report the index two ways: (hkil)(hkil) (hkl)(hkl)

Klein and Klein and Hurlbut Fig. 2-Hurlbut Fig. 2-

3333

(100)(1010)

(110) (111)

(1120) (1121)

Assigning Miller indicesAssigning Miller indices Prominent (and common) faces have Prominent (and common) faces have

small integers for Miller Indicessmall integers for Miller Indices Faces that cut only one axisFaces that cut only one axis

(100), (010), (001) etc(100), (010), (001) etc Faces that cut two axesFaces that cut two axes

(110), (101), (011) etc(110), (101), (011) etc Faces that cut three axesFaces that cut three axes

(111)(111) Called Called unit faceunit face

Zones, Forms, HabitsZones, Forms, Habits

Quantitative description of orientation in Quantitative description of orientation in minerals – use Miller indices:minerals – use Miller indices: ZoneZone - Lines, or linear directions within - Lines, or linear directions within

mineralsminerals Form - Form - Shapes of three dimensional objectsShapes of three dimensional objects

Qualitative description of mineral shapes: Qualitative description of mineral shapes: HabitHabit

Crystal HabitCrystal Habit Qualitative terminology to describe Qualitative terminology to describe

individual minerals and aggregates individual minerals and aggregates of mineralsof minerals Shape of individual mineralsShape of individual minerals Intergrowths of several mineral grainsIntergrowths of several mineral grains Shape of masses of grainsShape of masses of grains

ColloformColloform finely crystalline, concentric mineral layerfinely crystalline, concentric mineral layer

Globular – (spherulitic)Globular – (spherulitic) radiating, concentrically arranged acicular mineralsradiating, concentrically arranged acicular minerals

ReniformReniform kidney shapedkidney shaped

BotryoidalBotryoidal like a bunch of grapeslike a bunch of grapes

MammillaryMammillary similar, but larger than botryoidal, breast-like or portions of similar, but larger than botryoidal, breast-like or portions of

spheresspheres DrusyDrusy

Surface covered with layer of small crystalsSurface covered with layer of small crystals

Globular hematite

Drusy quartz

Fig. 2-Fig. 2-4747

Terminology useful for describing general shapes of minerals

(table-like)

(asbestos: amphiboles and pyroxenes)

(knife like – kyanite)

(Mica)

Bladed kyaniteBladed kyanite

AlAl22SiOSiO55

Fibrous tremoliteFibrous Fibrous tremolite: tremolite: amphiboleamphibole

Ca(Mg,Fe)Ca(Mg,Fe)55SiSi88OO22 22

(OH)(OH)22

ZonesZones

Collection of common facesCollection of common faces Parallel to some common lineParallel to some common line Line called the Line called the zone axiszone axis Identified by index [hkl]Identified by index [hkl] Zone axis parallels intersection of Zone axis parallels intersection of

edges of facesedges of faces

Fig 2-Fig 2-3030

a

c

b

Faces = (110), (110), (110), (110)

Zone axis intesects (001) lattice nodes = [001]

Note typo in first edition

Intersection of facesIntersection of faces= [001] Zone= [001] Zone

-a

-c

-b

Fig 2-27Fig 2-27

Other linear crystallographic Other linear crystallographic directionsdirections

Includes crystallographic Includes crystallographic axesaxes

Referenced to intersection Referenced to intersection of lattice nodesof lattice nodes

For example: location of For example: location of rotation axes or other linear rotation axes or other linear featuresfeatures

Lattice node

FormForm

Formal crystallographic Formal crystallographic nomenclature of the shape of nomenclature of the shape of mineralsminerals

DescriptionDescription Collection of crystal facesCollection of crystal faces Related to each other by symmetryRelated to each other by symmetry Identified by index: {hkl}Identified by index: {hkl} Values for h, k and l are determined by Values for h, k and l are determined by

one of the facesone of the faces

ExampleExample There are six faces in a cube (a kind of There are six faces in a cube (a kind of

form):form): (100), (010), (001), (100), (010), (001)(100), (010), (001), (100), (010), (001) All faces parallel two axes and are All faces parallel two axes and are

perpendicular to one axisperpendicular to one axis Form is written with bracketsForm is written with brackets

Uses miller index of one faceUses miller index of one face Generally positive faceGenerally positive face E.g., {001}E.g., {001} a

b

c(001)

Isometric form{001}

(010)(100)

Possible to determine the shape of a Possible to determine the shape of a form with:form with:1)1) Miller index of one face in formMiller index of one face in form

2)2) Point symmetry of the crystal classPoint symmetry of the crystal class The form is created by operating The form is created by operating

point symmetry on the initial facepoint symmetry on the initial face Number of faces in a form depends Number of faces in a form depends

on crystal classon crystal class

Fig. 2-29Fig. 2-29

{011} form in crystal class with point {011} form in crystal class with point symmetry 2/m 2/m 2/m (Orthorhombic)symmetry 2/m 2/m 2/m (Orthorhombic)

– – called a called a rhombic prismrhombic prismRhombus – an equilateral parallelogramRhombus – an equilateral parallelogramPrism – a crystal form whose faces are Prism – a crystal form whose faces are parallel to one axisparallel to one axis

Mirror parallel to (010) Mirror parallel to (001)

Face parallel to a axis

Triclinic system:Triclinic system: Point group (i.e. crystal class) = 1Point group (i.e. crystal class) = 1 Symmetry content = (1ASymmetry content = (1A11)) {111} has only 2 faces{111} has only 2 faces

Isometric system:Isometric system: Point group (crystal class) = 4/m 3 2/mPoint group (crystal class) = 4/m 3 2/m Symmetry content = 3ASymmetry content = 3A44, 4A, 4A33, 6A, 6A22, 9m, 9m {111} has 8 faces{111} has 8 faces Form is an octahedronForm is an octahedron

Isometric system Isometric system Point group (crystal class) = 4Point group (crystal class) = 4 Symmetry content = 1ASymmetry content = 1A44

{111} has 4 faces{111} has 4 faces Form is a tetrahedronForm is a tetrahedron C

b

a

(111)

Two types of forms:Two types of forms: Open formOpen form – does not enclose a volume – does not enclose a volume Closed formClosed form – encloses a volume – encloses a volume

Minerals must have more than one form Minerals must have more than one form if they have an open formif they have an open form

Minerals may have only one closed formMinerals may have only one closed form Mineral could have more than 1 form, Mineral could have more than 1 form,

closed or openclosed or open

Open FormOpen Form PrismPrism Requires additional formsRequires additional forms

Closed FormClosed Form CubeCube Does not require Does not require

additional forms, but additional forms, but may contain themmay contain them

Example of multiple formsExample of multiple forms

Cube {001}, octahedron {111}, and Cube {001}, octahedron {111}, and 3 prisms{110}, {101}, {011}3 prisms{110}, {101}, {011}

All forms have 4/m 3 2/m symmetryAll forms have 4/m 3 2/m symmetry

Two combined closed forms, plus 3 additional open forms

{001} = cube

{111} = octahedron

Prisms{110}{101}{011}

a

c

b

Isometric formsIsometric forms

15 possible forms15 possible forms 4 common ones4 common ones

CubeCube {001} – 4/m 3 2/m symmetry {001} – 4/m 3 2/m symmetry OctahedronOctahedron {111} – 4/m 3 2/m {111} – 4/m 3 2/m

symmetrysymmetry TetrahedronTetrahedron {111} – 4 symmetry {111} – 4 symmetry DodecahedronDodecahedron {110} {110}

OctahedronOctahedron

a

c

b

c

b

a

Both isometric forms:Both isometric forms:

TetrahedronTetrahedron

{111}{111}{111}{111}

Crystal class = 4/m 3 2/mCrystal class = 4/m 3 2/m Crystal class = 4Crystal class = 4

(111)

Non-isometric formNon-isometric form 10 types of forms10 types of forms PedionPedion (open) (open)

Single faceSingle face No symmetrically identical faceNo symmetrically identical face

PinacoidPinacoid (open) (open) Two parallel facesTwo parallel faces Related by mirror plane or inversionRelated by mirror plane or inversion

Dihedron Dihedron (open - 2 types)(open - 2 types) Two non-parallel faceTwo non-parallel face Related by mirror (Related by mirror (domedome) or 2-fold ) or 2-fold

rotation (rotation (sphenoidsphenoid))

Fig. 2-Fig. 2-3131

Note: dome switches handednessSphenoid retains handedness

PrismPrism (open) (open) 3, 4, 6, 8 or 12 faces3, 4, 6, 8 or 12 faces Intersect with mutually parallel edges Intersect with mutually parallel edges

forming a tubeforming a tube PyramidPyramid (open) (open)

3, 4, 6, 8, or 12 faces3, 4, 6, 8, or 12 faces Intersect at a pointIntersect at a point

DipyramidDipyramid (closed) (closed) 6, 8, 12, 16, or 24 faces6, 8, 12, 16, or 24 faces Two pyramids at each end of crystalTwo pyramids at each end of crystal

All of these forms are named on the All of these forms are named on the basis of the shape of the cross sectionbasis of the shape of the cross section Total of 21 different formsTotal of 21 different forms

Fig. 2-Fig. 2-3232

Pri

sms

Pyra

mid

s

Dip

yra

mid

s

Cross section

Open

Closed

Open

Dihexagonal

Hexagonal

Ditrigonal

Trigonal

Ditetragonal

TetragonalRhombic

Three types – seven modifiers – total of 21 formsThree types – seven modifiers – total of 21 forms

TrapezohedronsTrapezohedrons (closed) (closed) 6, 8, 12 faces6, 8, 12 faces each a trapezoid (plane shape with 4 each a trapezoid (plane shape with 4

unequal sides)unequal sides) Named according to number of facesNamed according to number of faces

ScalenohedronScalenohedron (closed) (closed) 8 or 12 faces8 or 12 faces Each a scalene triangle (no two angles Each a scalene triangle (no two angles

are equal)are equal)

RhombohedronsRhombohedrons (closed) (closed) 6 faces, each rhomb shaped (4 equal 6 faces, each rhomb shaped (4 equal

sides, no 90 angles)sides, no 90 angles) Looks like a stretched or shortened cubeLooks like a stretched or shortened cube

TetrahedronTetrahedron (closed) (closed) 4 triangular faces4 triangular faces

Fig. 2-Fig. 2-3333

Combining formsCombining forms

Restrictions on types of forms within a Restrictions on types of forms within a crystalcrystal All forms must be in the same crystal All forms must be in the same crystal

systemsystem All forms must have symmetry of one All forms must have symmetry of one

crystal class, for example:crystal class, for example: Tetragonal prism has a single 4-fold rotation, Tetragonal prism has a single 4-fold rotation,

only found in tetragonal crystal class with only found in tetragonal crystal class with single 4-fold rotation axissingle 4-fold rotation axis

Pedions never occur in mineral with center of Pedions never occur in mineral with center of symmetrysymmetry

Multi-faced forms are not composed Multi-faced forms are not composed of several simpler formsof several simpler forms A cube is not 6 pedions or 3 pinacoidsA cube is not 6 pedions or 3 pinacoids

Special relationships Special relationships between formsbetween forms

Enantiomorphous formsEnantiomorphous forms Positive and negative formsPositive and negative forms

Enantiomrophous FormEnantiomrophous Form

Enantiomorphic forms contain a Enantiomorphic forms contain a screw axisscrew axis Axis may rotate to the right or leftAxis may rotate to the right or left

The two forms generated are mirror The two forms generated are mirror images of each otherimages of each other

Fig. 2.20Fig. 2.20

3-fold screw axis

May be 2-fold, 4-fold, or 6-fold

• Atomic scale rotation• Enantiomorphous forms

result from either right or left spiral of screw axis

Amino acids:Amino acids: Almost always left Almost always left

handedhanded Through time convert to Through time convert to

right handedright handed Age-dating tool 0 < D/L < Age-dating tool 0 < D/L <

11

Enantiomorphous FormsEnantiomorphous Forms Must lack center of symmetry and Must lack center of symmetry and

mirrorsmirrors Forms are related to each other by a Forms are related to each other by a

mirrormirror right and left handed formsright and left handed forms Individual crystal either right or left Individual crystal either right or left

handed, but not bothhanded, but not both Quartz is common exampleQuartz is common example

Fig. 2-Fig. 2-3434

Crystal are mirror images of each other, but there are no mirror images in the crystals

Enantiomorphous FormsEnantiomorphous Forms

Positive and Negative formsPositive and Negative forms

Created by rotation of a formCreated by rotation of a form Rotation not present in the form itselfRotation not present in the form itself

Two forms related to each other by Two forms related to each other by mirror planesmirror planes Mirror planes missing within the form itselfMirror planes missing within the form itself

Two possible rotations:Two possible rotations: 60º on 3-fold rotation axis60º on 3-fold rotation axis 90º on 4- or 2-fold rotation axis90º on 4- or 2-fold rotation axis

Fig. 2-35Fig. 2-35

Positive and Positive and negative faces in negative faces in quartz crystalquartz crystalQuartz lacks Quartz lacks center of center of symmetrysymmetry

Quartz Crystal

Forms in the Six Crystal Forms in the Six Crystal SystemSystem

Forms control orientation of Forms control orientation of crystallographic axes of the 6 crystal crystallographic axes of the 6 crystal systemsystem

Systematic relationship between Systematic relationship between form, symmetry present, and form, symmetry present, and Hermann-Mauguin symbolsHermann-Mauguin symbols

Following slides show these Following slides show these relationships relationships

TriclinicTriclinic Common symmetry: 1-fold rotationCommon symmetry: 1-fold rotation

Table 2.2Table 2.2 c-axis parallels prominent zone axisc-axis parallels prominent zone axis b and a axes parallel crystal edgesb and a axes parallel crystal edges and and typically > 90º typically > 90º Single Hermann-Mauguin symbolSingle Hermann-Mauguin symbol

Common minerals: plagioclase and Common minerals: plagioclase and microclinemicrocline

Fig. 2-Fig. 2-3636

Pedions Pinacoid

1 1

Triclinic

a

b

c = zone axis

MonoclinicMonoclinic Common symmetry: 2-fold rotation and/or Common symmetry: 2-fold rotation and/or

single mirror planesingle mirror plane b axis commonly parallel the 2-fold rotation b axis commonly parallel the 2-fold rotation

and/or perpendicular to mirror planeand/or perpendicular to mirror plane c axis parallel to prominent zonec axis parallel to prominent zone a axis down and to front so a axis down and to front so > 90 > 90 Single H-M symbol (2, m, or 2/m)Single H-M symbol (2, m, or 2/m)

Common minerals: amphiboles, pyroxenes, Common minerals: amphiboles, pyroxenes, micasmicas

Fig. 2-37Fig. 2-37

Monoclinic

2-fold rotation axis

OrthorhombicOrthorhombic

Common symmetry: 3 2-fold Common symmetry: 3 2-fold rotations and/or 3 mirror planesrotations and/or 3 mirror planes

Crystal axes are parallel to 2-fold Crystal axes are parallel to 2-fold rotations or perpendicular to mirror rotations or perpendicular to mirror planes, or bothplanes, or both

Any axis could have any symmetryAny axis could have any symmetry Reported in H-M notation:Reported in H-M notation:

11stst = a axis, 2 = a axis, 2ndnd = b axis, 3 = b axis, 3rdrd = c axis = c axis E.g. mm2 – a E.g. mm2 – a mirror, b mirror, b mirror, c mirror, c

parallel 2-fold rotationparallel 2-fold rotation

Fig. 2-38Fig. 2-38

Orthorhombic

a

c

b

mm2 2/m2/m2/m

222 aa

bb

cc

TetragonalTetragonal Common symmetry: single 4-fold Common symmetry: single 4-fold

rotation, or 4-fold rotoinversionrotation, or 4-fold rotoinversion c axis always the single 4-fold rotation axisc axis always the single 4-fold rotation axis

a and b coincide with 2-fold rotation or a and b coincide with 2-fold rotation or mirror (if present)mirror (if present)

H-M symbol:H-M symbol: 11stst = c axis = c axis 22ndnd = b and a axes = b and a axes 33rdrd = symmetry on [110] and [110] axis at = symmetry on [110] and [110] axis at

45º to a and b axes45º to a and b axes

ExampleExample

42m42m C = 4-fold rotoinversionC = 4-fold rotoinversion a and b axes [100] and [010] are 2-fold a and b axes [100] and [010] are 2-fold

rotationrotation There are mirrors There are mirrors to [110] and [110]to [110] and [110]

Fig. 2-Fig. 2-3939

42m

Positive and negative tetragonal tetrahedron

a

c

b Note – tetragonal so a = b ≠ c, this is not an isometric form

HexagonalHexagonal Common symmetry: 1 3-fold axis Common symmetry: 1 3-fold axis

(trigonal division) or 1 6-fold axis (trigonal division) or 1 6-fold axis (hexagonal division)(hexagonal division) c axis parallel to 6-fold or 3-fold rotationc axis parallel to 6-fold or 3-fold rotation a axes parallel to 2-fold rotation or a axes parallel to 2-fold rotation or

perpendicular to mirrorperpendicular to mirror H-M symbols written with 1H-M symbols written with 1stst = c axis, 2 = c axis, 2ndnd

parallel a axes, 3parallel a axes, 3rdrd perpendicular to a perpendicular to a

Figure 2-41Figure 2-41

a1

-a3

a2

c

A prism and multiple dipyramids

6/m2/m2/m

IsometricIsometric Common symmetry 4 3-fold axesCommon symmetry 4 3-fold axes 3 equivalent symmetry axes coincide 3 equivalent symmetry axes coincide

with crystallographic axeswith crystallographic axes (e.g. for cube, it’s the 4 fold rotations)(e.g. for cube, it’s the 4 fold rotations)

Symmetry either 2-fold or 2-foldSymmetry either 2-fold or 2-fold H-M symbols;H-M symbols;

11stst crystallographic axes crystallographic axes 22ndnd diagonal axes [111] diagonal axes [111] 33rdrd center of one edge to center of another center of one edge to center of another

edge [110]edge [110]

Fig. 2-44Fig. 2-44

4/m32/m - Isometric

4/m

3

2/m

ab

c