part 2-1: crystal morphology - uni- · pdf fileprojection plane. easy measurement of angle in...

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Part 2-1: Crystal morphology

2.1.1 Morphology and Crystal Structure

2.1.2 Law of constancy of the angle

2.1.3 Stereographic Projection

2.1.4 Wulff net

2.1.5 Goniometer

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Crystal structure and Morphology

a) Every crystal face lies parallel to a set of lattice planes;

parallel crystal faces correspond to the same set of planes;(hkl)

b) Every crystal edge is parallel to a set of lattice lines.

Morphology: the set of faces and edges which enclosea crystal

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Crystal growth - Law of constancy of the angleCrystals of different shapes can result from the same nucleus. But:

In different sample of the same crystal, the angles between correspondingfaces will be equal. (Nicolaus Steno 1669)

Nucleation → growth of a nucleus to a crystalsingle crystal → morphology

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Principle:

Winkeltreue:Application:

Represent lattice direction and plane normal as point in theprojection plane.

easy measurement of angle in one plane

Stereographic Projection - 1

Crystal orientation, Morphology, Symmetry

PE

North/south pole

p on equatorial plane is thestereographic projection of thepole P on the sphere.

stereographic projectionof a crystal morphology

S

p

PN

Oequator

N

S

polepole

pole

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Stereographic projection - Example galena(PbS)

Great circles as zone

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Stereography - great circle, angle between faces

Poles of the faces

The angle between two poles means theangle between the normals n in a greatcircle.

Meridian planeequator/equatorial plane

1. great circle: with radius of the sphere;

2. Those faces whose poles lie on a single greatcircle belong to a single zone;

3. The zone axis lie perpendicular to the plane ofthe great circle.

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Reflecting Goniometer

Prinzip eines Reflexionsgoniometers:(einkreisig oder zweikreisig)

Lampe

Goniometertischmit Kristall

Fernrohr

The angles between crystal facesmay be measured with areflecting goniometer angle scale

Zone axis

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Wulff Net - 1

Principle:

Element:

The Wulff net is the projection of grid net of a globe with North-South-axis (N-S) in the projection plane.

great- und small- circle, equator, pole, azimuth, pole distance

Pole distance

Azimuthal angle

NW’ E’

N’

S’

N

S

ρ

ϕ

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Wulff Net - Angle between faces

Steps:

1. The angle between two faces is the angle between their normals.

2. The two normals define the plane of a great circle.

3. The arc of the great circle between the two normals is the measuredangular value.

Angles are measured only from great circles.

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Wulff Net - Example: Pinacoid

The faces of the tetragonal pyramid have ϕ-coordinates of 0°, 90°, 180° and 270°respectively;

All faces have the same ρ-value.

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Wulff Net - Application 1

Angle between two polesRotate two poles lie in a great circle:

1. cover the stereogram with Wulffnet with the same circle center

2. Rotate the stereogram using thecircle center.

Red poles → blue poles

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Wulff Net - Application 2

Zone defined by two faces

Case 1: draw zone pole corresponding to a zonecircle;

1. Rotate the zone circle onto a great circle;2. Zone pole is 90° from the zone circle along theequator.

1. Rotate the pole onto the equator;2. The zone circle is the meridian 90° away fromthe pole

Case 2: draw zone circle corresponding to a givenzone pole

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Wulff Net - Indexing of crystal 1

[110]

[111]

[001]

(001)

(010)

Known lattice parameters, and consider only faces with low Miller indices

Zone-circle

angle

[010] zone [100] zone

1. Six faces <100>2. From (001), find (101) by δ and δ’3. Find other symmetric faces of <110>4. Find (111) by interaction of[(100)/(011)] and [(101)/(010)]

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Summary - Stereographic projection

1. The angle between faces.

2. Index the crystal zone in stereogram.

3. Great circle

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Gnomonic Projection

Prinzip: • Mittelpunkt des Kristalls bzw. der Polkugel alsProjektionszentrum

• Projektion auf die Tangentialebene durch den Nordpol

PE

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Part 2-2: Principles of Symmetry

2.2.1 Rotation axes

2.2.2 The mirror plane

2.2.3 The inversion center

2.2.4 Compound Symmetry operations

2.2.5 Rotoinversion axes

2.2.6 Rotoreflection axes

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Symmetry - Translation

Allen Gittern gemeinsam ist die Translationssymmetrie.(Einwirkung von 3 nicht komplanaren Gitter-Translationen auf einen Punkt

⇒⇒⇒⇒ Raumgitter)⇓⇓⇓⇓

Andere Symmetrieeigenschaften treten nicht notwendigerweise in jedemGitter auf.

⇓⇓⇓⇓Die Translationssymmetrie schränkt die Zahl denkbarer

Symmetrieelemente drastisch ein.

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Symmetry - Basic conceptions

All repetition operations, lattice tranlations, rotations and reflections, are calledsymmetry operations.

When a symmetry operation has a “locus”, that is a point, a line or a plane thatis left unchanged by the operation, this locus is referred to as the symmetryelement.

Mirror plane Rotation axis Inversion center

Symmetry element:

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A

Rotation axes - 1

A

BC

D

CA

B D

BC

D

A CB D

180°

180°

The symmetry element corresponding to the symmetry operation of rotationis called a rotation axis.

The order of the axis is given by X where X=360°/ε and ε is the minimumrotation angle.

Objects are said to be equivalent to one another if they can be brought intocoincidence by a symmetry operation.

X=360°/180°=2represented as if normal to the planeand if parallel to it.

Normal to plane Parallel to plane

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Rotation axes - 2

III

III IV

Three-fold rotation axis: 3( )

120°

Fourth-fold rotation axis: 4 ( )

Six-fold rotation axis: 6 ( )

90°

60°

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No 5- or 7-fold rotation

Parallele Gittergeraden müssen gleiche Translationsperiode haben.

Five-fold rotation does not constitute a lattice plane.

Rotation axes order higher than 6 do not constitute a lattice plane.

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Rotation - Short summary

Rotation angle: 180°Symbol: 2Graphic Symbol:

Rotation angle : 120°Symbol: 3Graphic Symbol:

Rotation angle : 360°Symbol: 1 (nach Hermann-Mauguin)Graphic Symbol: -

Rotation angle : 90°Symbol: 4Graphic Symbol:Rotation angle : 60°Symbol: 6Graphic Symbol:

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Mirror Plane

The symmetry element of reflection is called mirror plane or plane of symmetrysymbol is: m

Any point or object on one side of a mirror plane is matched by the generationof an equivalent point or object on the other side at the same distance from theplane along a line normal to it.

m

A’ A

m

A’ A

b

a

b

a

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Inversion Center

The symmetry operation, inversion, relates pairs of points or objects which areequidistance from and on opposite sides of a central point.The symbol is: 1

Each space lattice has inversion operation;All lattices are centrosymmetric.

Inversion at 1/2, 1/2, 1/2.

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2

Compound symmetry operations

Compound symmetry operationTwo symmetry operations are performed in

sequence as a single event.This produces a new symmetry operation but

the individual operations of which it is composed are lost.

e.g. Rotation of 90° about an axisFollowed by an inversion through a point on the axis.

Combination of symmetry operationTwo or more symmetry operations are

combined which are themselves symmetry operations.

e.g.4-fold rotationinversion

1 3

4

4

1 3

4

2

m4

56 7

8

Parallel 2

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Compound symmetry - Rotoinversion axesThe compound symmetry operation of rotation and inversion is calledrotoinversion. Its symmetry elements are the rotoinversion axes,general symbol: X

1

2

1

2

1

3

42

5

6

1≡ inversion 2 ≡ m 3 ≡ 3 + 1 6 ≡ 3 ⊥ m

1 3

4

2

5

6

4

Note:Only rotoinversion axes of odd order imply the presence of an inversion center,1 and 3.

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Compound symmetry - Rotoreflection axes

Rotoreflection implies the compound operation of rotation and reflection in aplane normal to the axis.

S1≡ m; S2 ≡ 1; S3 ≡ 6; S4 ≡ 4; S6 ≡ 3.

Symmetry elements:proper rotation axes X(1, 2, 3, 4 and 6) androtoinversion or improper axes X(1 ≡ inversion center, 2 ≡ m, 3, 4

and 6)

The axes rotation axes X(1,2,3,4 and 6) and rotoinversion axes X(1≡ inversioncenter, 2 ≡ m, 3,4 and 6), including 1 and m, are called point-symmetryelements, since their operation always leave at least one point unmoved.

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Summary - Symmetry basis

Mirrorrotation inversion

rotoinversion=rotation + inversionrotoreflection=rotation + mirror