An Introduction to Crystallography

An Introduction to Crystallography

CONTENTSCrystallography:

Why we study Crystallography? Definition External characteristics of crystals

• Elements of crystals Crystal elements Crystal symmetry Crystal systems Crystal classes Axial ratios-crystal parameters and Miller indices Methods of Crystal Drawing Crystal habit and forms

• General Outlines of the crystal systems Cubic (Isometric) System Tetragonal System Orthorhombic System Hexagonal System Trigonal System Monoclinic System Triclinic System

Atomic structureCentral region called the nucleus

Consists of protons (+ charges) and neutrons (- charges)

ElectronsNegatively charged particles that surround

the nucleusLocated in discrete energy levels called

shells

Composition of minerals

Structure of an atom

Figure 3.4

Figure 3.6

Chemical bondingFormation of a compound by combining

two or more elementsIonic bonding

Atoms gain or lose outermost (valence) electrons to form ions

Ionic compounds consist of an orderly arrangement of oppositely charged ions


Halite (NaCl) – An example of ionic bonding

Covalent bondingAtoms share electrons to achieve

electrical neutralityGenerally stronger than ionic bonds Both ionic and covalent bonds typically

occur in the same compound


Covalent bond Model- diamond (Carbon)

PolymorphsMinerals with the same composition but

different crystalline structuresExamples include diamond and graphite

Phase change = one polymorph changing into another

Structure of minerals

Diamond and graphite – polymorphs of carbon

Crystal formExternal expression of a mineral’s

internal structureOften interrupted due to competition for

space and rapid loss of heat

Physical properties of minerals

Why we study Crystallography?It is useful for the identification of minerals. The later are chemical substances formed under natural conditions and have crystal forms.

Study of crystals can provide new chemical information. In laboratories and industry, we can prepare pure chemical substances by crystallization process.

It is very useful for solid state studies of materials.

Crystal heating therapy.

Crystallography is of major importance to a wide range of scientific disciplines including physics, chemistry, molecular biology, materials science and mineralogy.

DEFINITION • CRYSTALLOGRAPHY is simply a fancy

word meaning "the study of crystals" • The study of crystalline solids and the principles

that govern their growth, external shape, and internal structure

• Crystallography is easily divided into 3 sections -- geometrical, physical, and chemical.

• We will cover the most significant geometric aspects of crystallography

Classification of substances

• Crystalline Substances• Amorphous substances

Properties of Crystalline Substances

1- Solidity 2- Anisotropy X Isotropy

3- Self-faceting ability 4 -Symmetry

space lattice skeleton

The crystalline substances are characterise by the following properties:

Amorphous substances

(in Greek amorphous means “formless”) do not have overall regular internal structure; their constituent particles are arranged randomly; hence, they are isotropic, have no symmetry, and cannot be bounded by faces. Particles are arranged in them in the same way as in liquids, hence, they are sometimes referred to as supercooled liquids. Examples of amorphous substances are glass, plastics. Glue, resin, and solidified colloids (gels).

Curve of cooling of amorphous substances

0

20

40

60

050100

time, min

To

Curve of cooling of a crystalline subsatnce

0

10

20

30

40

50

60

050100

time, min

Toab

In distinction to crystalline substances, amorphous ones have no clearly defined melting point. Comparing curves of cooling (or heating) of crystalline substances and amorphous substances, one can see that the former has two sharp bend-points (a and b), corresponding to the beginning and end crystallization respectively, whereas the latter is smooth.

Definition of Crystal• 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, under suitable conditions, from the state of a liquid or gas to that of a solid.

• A polyhedral form simply means a solid bounded by flat planes (we call these flat planes CRYSTAL FACES).

• A chemical compound" tells us that all minerals are chemicals, just formed by and found in nature.

• The last half of the definition tells us that a crystal normally forms during the change of matter from liquid or gas to the solid state.

Classification of crystals according to the degree of crystallization

• Euhedral crystals• Subhedral crystals• Anhedral crystals

Euhedral Crystal Subhedral Crystal Anhedral Crystal

External characteristices of crystals

• Crystal faces• Edge• Solid angle• Interfacial angle• Crystal form• Crystal habit

• Crystal faces: The crystal is bounded by flat plane surfaces. These surfaces represent the internal arrangement of atoms and usually parallel to net-planes containing the greatest number of lattice-points or ions.

• Faces are two kinds, like and unlike.

• Edge: formed by the intersection of any two adjacent faces.The position in space of an edge depends upon the position of the faces whose intersection gives rise to it.

• Solid Angles: formed by intersection of three or more faces.

A

F

E

Edges………….E

Solid Angles (apices)…..A

Crystal Faces….F

Can you conclude mathematical relation between them?

• Interfacial anglewe define the interfacial angle between two crystal faces as the angle between lines that are perpendicular to the faces. Such lines are called the poles to the crystal face. Note that this angle can be measured easily with a device called a contact goniometer.

Nicholas Steno (1669) a Danish physician and natural scientist, found that, the angles between similar crystal faces remain constant regardless of the size or the shape of the crystal when measured at the same temperature, So whether the crystal grew under ideal conditions or not, if you compare the angles between corresponding faces on various crystals of the same mineral, the angle remains the same. Steno's law is called the CONSTANCY OF INTERFACIAL ANGLES and, like other laws of physics and chemistry, we just can't get away from it.

• Crystal forms: are a number of corresponding faces which have the same relation with the crystallographic axes.

• A crystal made up entirely of like faces is termed a simple form. A crystal which consists of two or more simple forms is called combination.

• Closed form: simple form occurs in crystal as it can enclose space.

• Open form: simple forms can only occur in combination in crystal

• The term general form has specific meaning in crystallography. In each crystal class, there is a form in which the faces intersect each crytallographic axes at different lengths. This is the general form {hkl} and is the name for each of the 32 classes (hexoctahedral class of the isometric system, for example). All other forms are called special forms.

Closed form

Open form

• Crystal Habit: the general external shape of a crystal. It is meant the common and characteristic form or combination of forms in which a mineral crystallizes.(Tabular habit, Platy habit, Prismatic habit, Acicular habit, Bladed habit)

Elements of CrystallizationCrystal Notation

• Crystallographic axis• Axial angles

Crystallographic axis

• All crystals, with the exception of those belonging to the hexagonal and trigonal system, are referred to three crystallographic axis.

Axial angles

• ∝ is the angle between b axis and c axis• β is the angle between a axis and c axis• is the angle between a axis and b axis

Crystal Systems• We will use our crystallographic axes which we just

discussed to subdivide all known minerals into these systems. The systems are:

(1) CUBIC (ISOMETRIC) - The three crystallographic axes are all equal in length and intersect at right angles (90

degrees) to each other.

β

Ɣ

α

a1 a2

a3

(2) TETRAGONAL - 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).

(3) ORTHORHOMBIC - Three axes, all at right angles, and all three of different lengths.

β

Ɣ

α

c

a1 a2

β

Ɣ

α

c

a b

TETRAGONAL ORTHORHOMBIC

• (4) HEXAGONAL - 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.

• (5) MONOCLINIC - 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.

• (6) TRICLINIC - The three axes are all unequal in length and intersect at three different angles (any angle but 90 degrees).

c

a b

β

Ɣ

α

c

ab

β

Ɣ

α

MONOCLINIC TRICLINIC

ELEMENTS OF SYMMETRY

• PLANES OF SYMMETRY• Rotation AXiS OF SYMMETRY• CENTER OF SYMMETRY.

PLANE OF SYMMETRY• Any two dimensional surface (we can call it flat)

that, when passed through the center of the crystal, divides it into two symmetrical parts that are MIRROR IMAGES is a PLANE OF SYMMETRY.

• In other words, such a plane divides the crystal so that one half is the mirror-image of the other.

Horizontal plane Vertical planeDiagonal plane

AXIS OF SYMMETRY

• An imaginary 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

four types of axis of symmetry

• When rotation repeats form every 60 degrees, then we have sixfold or HEXAGONAL SYMMETRY. A filled hexagon symbol is noted on the rotational axis.

• When rotation repeats form every 90 degrees, then we have fourfold or TETRAGONAL SYMMETRY. A filled square is noted on the rotational axis.

• When rotation repeats form every 120 degrees, then we have threefold or TRIGONAL SYMMETRY. A filled equilateral triangle is noted on the rotational axis.

• When rotation repeats form every 180 degrees, then we have twofold or BINARY SYMMETRY. A filled oval is noted on the rotational axis.

Types of axis of symmetry

• BINARY SYMMETRY

Two fold system (180º)


• TRIGONAL SYMMETRY

Three fold system(120º)


• TETRAGONAL SYMMETRY

Four fold system(90º)


Six fold system(60º)

HEXAGONAL SYMMETRY

Symmetry Axis of rotary inversion• This composite symmetry element combines a rotation

about an axis with inversion through the center.• There may be 1, 2, 3, 4, and 6-fold rotary inversion axes

present in natural crystal forms, depending upon the crystal system we are discussing.

- - - -

CENTER OF SYMMETRY• 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.

Complete Symmetrical Formula• We can use symbol to write the

symmetrical formula as following:

1- Plane of symmetry: m

2- Axis of symmetry: 2, 3, 4, 6 and we can write the number of the axis at up left as 34

3- Center of symmetry: n

For example: the complete symmetrical formula of hexoctahedral class of Isometric system: 34/m 43 62/m n

Intercepts, Parameters and Indices

• Absolute Intercepts:The distances from the center of the crystal at which the face cuts the crystallographic axes.

• Relative Intercepts: divided the absolute intercepts by the intercept of the face with b axis.

• Ex: if the absolute intercepts (a:b:c)are 1mm : 2mm : ½ mm, the relative intercepts will be ½ : 2/2 : ¼ = o.5 : 1 : o.25

Parameters• The parameters of the crystal face are the

intercepts of this face divided by the axes lengths.

Parameters: اإلحداثيات - 2

Unit Face وجه الوحدة

oc

oc:

ob

ob:

oa

oa =

1:1:1 abc

def 2

1:

3

1:

4

1 =

oc

of:

ob

oe:

oa

od

anm

2: 3

4:

1 =oc

om:

ob

on:

oa

oa

If the face parallel to the axis,Its intercept = ∞Its Parameter=∞

Indices

• The Miller indices of a face consist of a series of whole numbers which have been derived from the parameters by their inversion and if necessary the subsequent clearing of fractions.

• If the parameters are 111 so the indices will be 111

• If the parameters are 11∞ and on inversion 1/1, 1/1, 1/ ∞ woud have (110) for indices.

• Faces which have respectively the parameters 1, 1, ½ would on inversion yield 1/1, 1/1, 2/1 thus on clearing of fractions the resulting indices would be respectively (112)

• It is sometimes convenient when the exact intercepts are unkown to use a general symbol (hkl) for the miller indices.

c

ba

O

YX

Z

A

B

C

3-D Miller Indices (an unusually complex example)

a b c

unknown face (XYZ)

reference face (ABC)21

24

Miller index of face XYZ using

ABC as the reference face

23

invert 12

42

32

clear of fractions (1 3)4

Miller indices

• Always given with 3 numbers – A, b, c axes

• Larger the Miller index #, closer to the origin

• Plane parallel to an axis, intercept is 0

What are the Miller Indices of face Z?

b

a

w(1 1 0)

(2 1 0)

z

The Miller Indices of face z using x as the reference

b

a

w(1 1 0)

(2 1 0)

z

a b c

unknown face (z)

reference face (x)11

¥1

Miller index offace z using x (or any face) as the reference face

¥1

invert 11

1¥

1¥

clear of fractions 1 00

(1 0 0)

b

a

(1 1 0)

(2 1 0)

(1 0 0)

What do you do with similar faceson opposite sides of crystal?

b

a

(1 1 0)

(2 1 0)

(1 0 0)

(0 1 0)

(2 1 0)(2 1 0)

(2 1 0)

(1 1 0)(1 1 0)

(1 1 0)

(0 1 0)

(1 0 0)

Methods of Crystal Drawing

• Clingraphic Projection• Orthogonal Projection• Spherical Projection• Stereographic Projection

Clingraphic Projection

Orthogonal Projection

3-Spherical Projection

Imagine that we have a crystal inside of a sphere. From each crystal face we draw a line perpendicular to the face (poles to the face).

Note that the angle is measured in the vertical plane containing the c axis and the pole to the face, and the angle is measured in the horizontal plane, clockwise from the b axis.

The pole to a hypothetical (010) face will coincide with the b crystallographic axis, and will impinge on the inside of the sphere at the equator.

4-Stereographic Projection

Stereographic projection is a method used to depict the angular relationships between crystal faces.

This time, however we will first look at a cross-section of the sphere as shown in the diagram. We orient the crystal such that the pole to the (001) face (the c axis) is vertical and points to the North pole of the sphere.

N

EW(010)

(001)

(011)

(0-10)

(0-11)

ρ

ρ/2

Imagine that we have a crystal inside of a sphere.

For the (011) face we draw the pole to the face to intersect the outside the of the sphere. Then, we draw a line from the point on the sphere directly to the South Pole of the sphere.

N

EW(010)

(001)

(011)

(0-10)

(0-11)

ρ

ρ/2

Where this line intersects the equatorial plane is where we plot the point. The stereographic projection then appears on the equatorial plane.

In the right hand-diagram we see the stereographic projection for faces of an isometric crystal. Note how the ρ angle is measured as the distance from the center of the projection to the position where the crystal face plots. The Φ angle is measured around the circumference of the circle, in a clockwise direction away from the b crystallographic axis or the plotting position of the (010) crystal face

N

EW(010)

(001)

(011)

(0-10)

(0-11)

ρ

ρ/2

EW(010)

(001)(0-10) (011)(0-11)

a

ρ

1- The Primitive Circle is the circle that cross cuts the sphere and separates it into two equal parts (North hemisphere and South hemisphere). It is drawn as solid circle when represents a mirror plane.

The following rules are applied:

2- All crystal faces are plotted as poles (lines perpendicular to the crystal face. Thus, angles between crystal faces are really angles between poles to crystal faces.

3- The b crystallographic axis is taken as the starting point. Such an axis will be perpendicular to the (010) crystal face in any crystal system. The [010] axis (note zone symbol) or (010) crystal face will therefore plot at Φ = 0° and ρ = 90°.

4- Mirror planes are shown as solid lines and curves. The horizontal plane is represented by a circle match with the primitive circle.5- Crystal faces that are on the top of the crystal ρ <90°) will be plotted as "+" signs, and crystal faces on the bottom of the crystal (ρ > 90°) will be plotted as open circles “ " .6- The poles faces that parallel to the c crystallographic axis lie on the periphery of the primitive circle and is plotted as "+" signs.7- The poles faces that perpendicular to the c crystallographic axis lie on the center of the primitive circle.8- The pole face parallels to one of the horizontal axes will plotted on the plane that perpendiculars to this axis.

9- The Unit Face (that met with the positive ends of the three or four crystallographic axes will be plotted in the lower right quarter of the primitive circle.

a

b

++

- +

+ -

- -

As an example all of the faces, both upper and lower, for a crystal in the class 4/m2/m in the forms {100} (hexahedron, 6 faces) and {110} (dodecahedron, 12 faces) are in the stereogram to the right

+ (001)(00-1)

+

++

+

+

(100)

(-100)

(010)(0-10)

+

++

++

+

+

(-110)(-1-10)

(110)(1-10)

(101)(10-1)

(011)(01-1)(0-11)(0-1-1)

(-101)(-10-1)

Crystallographic forms1- Pedion

It is an open form made up of a single face

Crystallographic forms1- Pinacoid

It is an open form made up of two parallel faces

Front pinacoid

Side pinacoid

Basal pinacoid

Crystallographic forms3- Dome

It is an open form made up of two nonparallel faces symmetrical with respect to a symmetry plane

4- Sphenoid

It is an open form made up of two nonparallel faces symmetrical with respect to a 2-fold or 4-fold symmetry axis

Crystallographic forms5- Disphenoid

It is an closed form composed of a four-faced form in which two faces of the upper sphenoid alternate with two of the lower sphenoid.

Crystallographic forms6 -Bipyramid

It is an closed form composed of 3, 4, 6, 8 or 12 nonparallel faces that meet at a pointOrthorhombic bipyramed

Ditetragonal bipyramid

Tetragonal bipyramid

Dihexagonal bipyramidHexagonal bipyramid

Crystallographic forms7- PrismIt is an open form composed of 3, 4, 6, 8 or 12 faces, all of which are parallel to same axis.

Orthorhombic prismTetragonal prism

Ditetragonal prism

Hexagonal prism Dihexagonal prism

Crystallographic forms8- Rhombohedron

It is an closed form composed of 6 rhombohedron faces,

9- Scalenohedron

It is an closed form composed of 12 faces, each face is a scalene triangle. There are three pairs of faces above and three pairs below in alternating positions

Crystallographic systemsIsometric system

β

Ɣ

α

a1 a2

a3a3 = a2 = a1

Ɣ = 90° = β = α

Class

1-Axis of symmetry

3

Isometric system

4

6

2- Center of symmetry

Isometric system

4 vertical plane

3- Plane of symmetry

Isometric system

1 horizontal plane 4 diagonal plane

4

3

3

4______

m n6

2______

m

Isometric system

Complete Symmetrical Formula

a

b (E)(W)

Stereographic Projection of Symmetry elements of the Isometric System

+

+

+

++

(100)

(010)

(-100)

(0-10)

1- Cube (Hexahedron)

+

++

+(111)

2- Octahedron

+

++

++

+

++

(110)

3- Rhombic dodecahedron

Stereographic projection of Cubic SystemForms.

Cubic form [100]

Crystal formIsometric system

Stereographic Projection

+

+

+

++

(100)

(010)

(-100)

(0-10)


+

++

+(111)

2- Octahedron

+

++

++

+

++

(110)




Octahedron [111] Stereographic Projection


Rhombic dodecahedron [110]+

+

+

++

(100)

(010)

(-100)

(0-10)


+

++

+(111)

2- Octahedron

+

++

++

+

++

(110)




Crystals of pyrite

Isometric system

Tetrahexahedron [hk0]

+

+

+

++

+

+

++

+

+

++

++

+

(210)

4- Tetrahexahedron

+

++

++

+

++

+++

+

(221)

5- Trisoctahedron

+

++

+ ++

+

++ +

+

+

(211)

6-Trapezohedron

+++++

+

++

++++

+++ ++

+

++ ++++

(321)

7- Hexaoctahedron


Isometric system

Trapezoctahedron [hll] +

+

+

++

+

+

++

+

+

++

++

+

(210)

4- Tetrahexahedron

+

++

++

+

++

+++

+

(221)

5- Trisoctahedron

+

++

+ ++

+

++ +

+

+

(211)

6-Trapezohedron

+++++

+

++

++++

+++ ++

+

++ ++++

(321)

7- Hexaoctahedron


Trisoctahedron [hhl]

Isometric system

+

+

+

++

+

+

++

+

+

++

++

+

(210)

4- Tetrahexahedron

+

++

++

+

++

+++

+

(221)

5- Trisoctahedron

+

++

+ ++

+

++ +

+

+

(211)

6-Trapezohedron

+++++

+

++

++++

+++ ++

+

++ ++++

(321)

7- Hexaoctahedron


Hexaoctahedron [hkl]

Isometric system+

+

+

++

+

+

++

+

+

++

++

+

(210)

4- Tetrahexahedron

+

++

++

+

++

+++

+

(221)

5- Trisoctahedron

+

++

+ ++

+

++ +

+

+

(211)

6-Trapezohedron

+++++

+

++

++++

+++ ++

+

++ ++++

(321)

7- Hexaoctahedron


Tetragonal system

β

Ɣ

α

c = a2 = a1/

c

a1 a2

Ɣ = 90° = β = α

Ditetragonal– Bipyramid [hkl]

Class

1

4

Tetragonal system1-Axis of symmetry

Tetragonal system2- Center of symmetry

Tetragonal system3- Plane of symmetry

4 vertical plane1 horizontal plane

4______

m4

2______

mn

Tetragonal system


Stereographic Projection of Symmetry elements of the Tetragonal System

a

b (E)(W)

Basal - pinacoid [001]

Tetragonal systemCrystal form

Stereographic projection of Tetragonal System Forms.

+

1- Basal Pinacoid

(001)

(00-1)

2- Tetragonal prism of 1st order

+

++

+(110)

3- Tetragonal Prism of 2nd Order

+

+

+

+

(100)


Tetragonal prism of first order [110]

Tetragonal system


+

1- Basal Pinacoid

(001)

(00-1)


+

++

+(110)


+

+

+

+

(100)


Tetragonal prism of second order [100]

Tetragonal system


+

1- Basal Pinacoid

(001)

(00-1)


+

++

+(110)


+

+

+

+

(100)


Ditetragonal prism [hk0]

Tetragonal system

4- Ditetragonal prism

+

+

+

++

+

+

+(210)

a

b

5- Tetragonal bipyramid of 1st Ordera

b

+

++

+

6- Tetragonal bipyramid of 2nd Order

a

b

+

+

+

+

7- Ditetragonal bipyramid

a

b

++

+

+++

++

(111)

(101)(211)


Tetragonal system

Tetragonal – Bipyramid of first order [hhl]


+

+

+

++

+

+

+(210)

a

b


b

+

++

+


a

b

+

+

+

+


a

b

++

+

+++

++

(111)

(101)(211)


Tetragonal system

Tetragonal – Bipyramid of second order [h0l]


+

+

+

++

+

+

+(210)

a

b


b

+

++

+


a

b

+

+

+

+


a

b

++

+

+++

++

(111)

(101)(211)


Tetragonal system

Ditetragonal– Bipyramid [hkl]


+

+

+

++

+

+

+(210)

a

b


b

+

++

+


a

b

+

+

+

+


a

b

++

+

+++

++

(111)

(101)(211)


Compound form

Orthorhombic system

β

Ɣ

α

c = b = a/ /

c

a b

Ɣ = 90° = β = α

Orthorhombic Bipyramid [hkl]

Class

7-Orthorhombic Bipyramid {hkl}Exit

hkl

It is a closed form composes of 8 triangularfaces. It is the general form of the orthorhombic holosymmetrical class. Each face met with the crystallographic axes at different distances {111} or {hkl}.

3

1-Axis of symmetry

Orthorhombic system

2- Center of symmetry


2 vertical plane 1 horizontal plane

3

2______

m n

Orthorhombic system


Stereographic Projection of Symmetry elements of the Orthorhombic System.

a

b (E)(W)

Orthorhombic systemCrystal form

Side pinacoid

]010 [

Front pinacoid [100]

Basal Pinacoid [001]

Stereographic projection of the Orthorhombic System Forms.

1- Basal Pinacoid

a

b+

2- Front Pinacoid

a

b

+

+

(100)

(001)

3- Side Pinacoida

b++(010)

Stereographic projection of the Orthorhombic System Forms.

1- Basal Pinacoid

a

b+

2- Front Pinacoid

a

b

+

+

(100)

(001)

3- Side Pinacoida

b++(010)


Orthorhombic prism [hk0]

Orthorhombic system

4- Orthorhombic prisma

b

+

++

+(110)

5- Front dome (b-Dome) a

b

+

+

(101)

6- Side dome (a-Dome)

a

b++(011)

7- orthorhombic bipyramid

a

b

+

++

+


Orthorhombic system

Orthorhombic front dome [h0l]


b

+

++

+(110)


b

+

+

(101)


a

b++(011)


a

b

+

++

+


Orthorhombic side dome [0kl]

Orthorhombic system


b

+

++

+(110)


b

+

+

(101)


a

b++(011)


a

b

+

++

+


Orthorhombic Bipyramid [hkl]

Orthorhombic system


b

+

++

+(110)


b

+

+

(101)


a

b++(011)


a

b

+

++

+


Compound form

5-Orthorhombicfrontdome (b-dome) or Macro dome {10l}

6-Orthorhombicsidedome (a-dome) or Brachydome {01l}

01l

01l10l

100

Pinacoid

Hexagonal system

/c = a3 = a2 = a1

a1

a2

-a3

c

Ɣ

β α

90° = β = α°120 = Ɣ

ClassDihexagonal bipyramid [hkwl]

61

Hexagonal system

1-Axis of symmetry 2- Center of symmetry


Hexagonal system

6 vertical plane 1 horizontal plane

Apatite

Ca5(PO4)3(OH, Cl,F)-hexagonal structure -prismatic habit-major component teeth

6______

m6

2______

mn


Hexagonal system

a1

a2 (E)(W)

Stereographic Projection of Symmetry elements of the Hexagonal System

-a3

Hexagonal prism of first order [1010]-

1010-

a1 - a3a2

0001

Hexagonal systemCrystal form

Basal pinacoid [0001]

Stereographic projection of the Hexagonal System Forms.

a1

a2

-a3

+

1- Hexagonal Pinacoid

(0001)

a1

a2

-a3

2- Hexagonal prism of first order(10-10)

+

+

+

+

+

+

a1

a2

-a3

3- Hexagonal prism of second order

+

+

++

+

+ (11-20)


hhw0-

a1 -a3

a2

Hexagonal systemHexagonal prism of

second order [hhw0]-

Stereographic projection of the Hexagonal System Forms.

a1

a2

-a3

+

1- Hexagonal Pinacoid

(0001)

a1

a2

-a3

2- Hexagonal prism of first order(10-10)

+

+

+

+

+

+

a1

a2

-a3

3- Hexagonal prism of second order

+

+

++

+

+ (11-20)


hkw

0-

Hexagonal system-Dihexagonal prism [hkw0]


a1

a2

-a3

4-Hexagonal Bipyramid of first order

+(10-11)

+

++

+

+

a1

a2

-a3

5-Hexagonal Bipyramid of second order

+(11-21)

+

++

+

+

a1

a2

-a3

6-Dihexagonalprism

(21-30)+

+

++

+++

+

+

++

+ a1

a2

-a37-Dihexagonalbipyramid

(21-31)+

+++

++++

+++

+

Hexagonal Bipyramid of first order [h0hl]-

h0hl-

a1 - a3

a2

Hexagonal system

a1

a2

-a3


+(10-11)

+

++

+

+

a1

a2

-a3


+(11-21)

+

++

+

+

a1

a2

-a3

6-Dihexagonalprism

(21-30)+

+

++

+++

+

+

++

+ a1

a2


(21-31)+

+++

++++

+++

+


Hexagonal Bipyramid of second order [hhwl]-

hhwl-a1 - a3

a2

Hexagonal system

a1

a2

-a3


+(10-11)

+

++

+

+

a1

a2

-a3


+(11-21)

+

++

+

+

a1

a2

-a3

6-Dihexagonalprism

(21-30)+

+

++

+++

+

+

++

+ a1

a2


(21-31)+

+++

++++

+++

+


Dihexagonal bipyramid [hkwl]-

hkw

l-

Hexagonal system


a1

a2

-a3


+(10-11)

+

++

+

+

a1

a2

-a3


+(11-21)

+

++

+

+

a1

a2

-a3

6-Dihexagonalprism

(21-30)+

+

++

+++

+

+

++

+ a1

a2


(21-31)+

+++

++++

+++

+

Compound form

Hexagonal prism (m = 6)

Hexagonal bipyramid (m = 12)

Trigonal system

Ɣ

β α

a1

a2

-a3

c

/c = a3 =a2 = a1

90° = β = α

120° = Ɣ

ditrigonal scalenohedron

Class

31

Trigonal system1-Axis of symmetry 2- Center of symmetry

Trigonal system3- Plane of symmetry

3 vertical plane

______3

2mn3

Trigonal system


a1

a2 (E)(W)

Stereographic Projection of Symmetry elements of the Triagonal System

-a3

FormsBasal Pinacoid

First Order Prism

Second Order Prism

Dihexagonal prism

Second Order bipyramid

Trigonal rhombohedron

Ditrigonal scalenohedron

Positive trigonal rhombohedron [h0hl]-

h0hl-

a1 - a3

a2

Trigonal systemCrystal form

Stereographic projection of the Triagonal System Forms.

a1 -a3

a2

a1 -a3

a2

Positive rhombohedron {10-11} Negative rhombohedron {01-11}

+

++

+

+

+

a1 -a3

a2

a1 -a3

a2

Negative Scalenohedron {12-31}Positive Scalenohedron {21-31}

+

++

+

++

+

+

+ +

+

+


Negative trigonal rhombohedron [0kkl]-

0kkl-

a1 -a3

a2

Trigonal system


a1 -a3

a2

a1 -a3

a2


+

++

+

+

+

a1 -a3

a2

a1 -a3

a2


+

++

+

++

+

+

+ +

+

+


Positive ditrigonal scalenohedron [hkwl]

-

hkwl-

a1 - a3

a2

Trigonal system


a1 -a3

a2

a1 -a3

a2


+

++

+

+

+

a1 -a3

a2

a1 -a3

a2


+

++

+

++

+

+

+ +

+

+


Negative ditrigonal scalenohedron [hkwl]-

hkwl-

a1 - a3

a2

Trigonal system


a1 -a3

a2

a1 -a3

a2


+

++

+

+

+

a1 -a3

a2

a1 -a3

a2


+

++

+

++

+

+

+ +

+

+


Monoclinic system

90° = Ɣ = α

c = b = a/ /

β = 90°/

c

a b

β

Ɣ

α

Class

1

Monoclinic system

1-Axis of symmetry 2- Center of symmetry

Monoclinic system


1 vertical plane

2______

m n

Monoclinic system


Stereographic Projection of Symmetry elements of the Monoclinic System

a

b (E)(W)

Monoclinic front pinacoid [100]

Monoclinic side pinacoid [010]

Monoclinic basal pinacoid [001]

Monoclinic system

Stereographic Projection• pinacoidCrystal form

Stereographic projection of the Monoclinic System Forms.

1- Basal Pinacoida

+(001)

2- Side Pinacoida

++

3- Front pinacoida+

+

(00-1)

Stereographic projection of the Monoclinic System Forms.

1- Basal Pinacoida

+(001)

2- Side Pinacoida

++

3- Front pinacoida+

+

(00-1)

Monoclinic prism [hk0]

Monoclinic systemStereographic Projection

4-Monoclinic prism {hk0} or {110}

hk0

5- Side Dome (a-dome)4- Monoclinic Prisma

+

++

+

a

++

6- Hemi-orthodome

a a

+

Positive

(101)

+

Negative

(-101)

7-Hemibipyramid

a a

++(111)

++(-111)

Positive hemibipyramid [hkl]

Monoclinic system

Positive Hemibipyramid {hkl} or {111}

NegativeHemibipyramid{-hkl} or {-111}

hkl

7-Hemibipyramid

Front View Back View

{111} {-111}

• hemibipyramid

Negative hemibipyramid [hkl]-


+

++

+

a

++

6- Hemi-orthodome

a a

+

Positive

(101)

+

Negative

(-101)

7-Hemibipyramid

a a

++(111)

++(-111)



+

++

+

a

++

6- Hemi-orthodome

a a

+

Positive

(101)

+

Negative

(-101)

7-Hemibipyramid

a a

++(111)

++(-111)

NegativePositive


+

++

+

a

++

6- Hemi-orthodome

a a

+

Positive

(101)

+

Negative

(-101)

7-Hemibipyramid

a a

++(111)

++(-111)

-011

Monoclinic system• Dome

- hemi-orthodome Positive hemi-orthodome [h0l] Negative hemi-orthodome [h0l]

- side dome [0kl]

-

101011

side dome [0kl]Positive hemidome [h0l]


+

++

+

a

++

6- Hemi-orthodome

a a

+

Positive

(101)

+

Negative

(-101)

7-Hemibipyramid

a a

++(111)

++(-111)

hemi-orthodome


Triclinic system

c = b = a/ /

c

a

b

β

Ɣ

α/α = β = Ɣ = 90°//

ClassPinacoid

Triclinic system

1-Axis of symmetry = -

2- Center of symmetry = n

3- Plane of symmetry = -

n

Triclinic system


Stereographic Projection of Symmetry elements of the Triclinic System

front pinacoid [100]

side pinacoid [010]

basal pinacoid [001]

Triclinic systemCrystal form

Stereographic projection of theTriclinic System Forms.

1- Basal Pinacoida a

a

2- Side Pinacoid

3- Frontl Pinacoid

+

+

++

+

Stereographic projection of theTriclinic System Forms.

1- Basal Pinacoida a

a

2- Side Pinacoid

3- Frontl Pinacoid

+

+

++

+


Right hemi-prism [hk0]

Left hemi-prism [hk0]

Triclinic system

-

a a a a

a a

a a

a a

+ +

+

+

+

+

+

++ +

+ +

5- Hemi-b-dome {h0l}: two forms{101} and {-101}

4- Hemi-a- dome { 0kl} : two forms{011} and {0-11}

6- Hemi-prism{hk0} and {h-k0}

Upper left quarterbipyramid Upper right quarterbipyramid

Lower left quarterbipyramid Lower right quarterbipyramid

Hemi-brachydome(0kl) Hemi-macrodome(h0l)

Triclinic system

a a a a

a a

a a

a a

+ +

+

+

+

+

+

++ +

+ +






Upper right quarter bipyramid [hkl]

Upper left quarter bipyramid [hkl]

Lower right quarter bipyramid [hkl]

Lower left quarter bipyramid [hkl]

-

-

- -

Triclinic systema a a a

a a

a a

a a

+ +

+

+

+

+

+

++ +

+ +






Crystal Morphology

• The angular relationships, size and shape of faces on a crystal

• Bravais Law – crystal faces will most commonly occur on lattice planes with the highest density of atoms

Planes AB and AC will be the most common crystal faces in this cubic lattice array

Unit Cell Types in Bravais Lattices

P – Primitive; nodes at corners only

C – Side-centered; nodes at corners and in center of one set of faces (usually C)

F – Face-centered; nodes at corners and in center of all faces

I – Body-centered; nodes at corners and in center of cell

Date post:	19-Jan-2015
Category:	Education
Upload:	hassan-harraz
View:	1,815 times
Download:	6 times

An Introduction to Crystallography

Education