Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in...
Transcript of Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in...
![Page 1: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/1.jpg)
Functional Materials SaarlandUniversity
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
![Page 2: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/2.jpg)
Functional Materials SaarlandUniversity
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
![Page 3: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/3.jpg)
Functional Materials SaarlandUniversity
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
![Page 4: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/4.jpg)
Functional Materials SaarlandUniversity
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
![Page 5: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/5.jpg)
Functional Materials SaarlandUniversity
Stereographic projection - Example galena(PbS)
Great circles as zone
![Page 6: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/6.jpg)
Functional Materials SaarlandUniversity
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.
![Page 7: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/7.jpg)
Functional Materials SaarlandUniversity
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
![Page 8: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/8.jpg)
Functional Materials SaarlandUniversity
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
![Page 9: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/9.jpg)
Functional Materials SaarlandUniversity
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.
![Page 10: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/10.jpg)
Functional Materials SaarlandUniversity
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.
![Page 11: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/11.jpg)
Functional Materials SaarlandUniversity
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
![Page 12: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/12.jpg)
Functional Materials SaarlandUniversity
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
![Page 13: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/13.jpg)
Functional Materials SaarlandUniversity
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)]
![Page 14: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/14.jpg)
Functional Materials SaarlandUniversity
Summary - Stereographic projection
1. The angle between faces.
2. Index the crystal zone in stereogram.
3. Great circle
![Page 15: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/15.jpg)
Functional Materials SaarlandUniversity
Gnomonic Projection
Prinzip: • Mittelpunkt des Kristalls bzw. der Polkugel alsProjektionszentrum
• Projektion auf die Tangentialebene durch den Nordpol
PE
![Page 16: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/16.jpg)
Functional Materials SaarlandUniversity
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
![Page 17: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/17.jpg)
Functional Materials SaarlandUniversity
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.
![Page 18: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/18.jpg)
Functional Materials SaarlandUniversity
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:
![Page 19: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/19.jpg)
Functional Materials SaarlandUniversity
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
![Page 20: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/20.jpg)
Functional Materials SaarlandUniversity
Rotation axes - 2
III
III IV
Three-fold rotation axis: 3( )
120°
Fourth-fold rotation axis: 4 ( )
Six-fold rotation axis: 6 ( )
90°
60°
![Page 21: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/21.jpg)
Functional Materials SaarlandUniversity
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.
![Page 22: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/22.jpg)
Functional Materials SaarlandUniversity
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:
![Page 23: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/23.jpg)
Functional Materials SaarlandUniversity
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
![Page 24: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/24.jpg)
Functional Materials SaarlandUniversity
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.
![Page 25: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/25.jpg)
Functional Materials SaarlandUniversity
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
![Page 26: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/26.jpg)
Functional Materials SaarlandUniversity
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.
![Page 27: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/27.jpg)
Functional Materials SaarlandUniversity
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.
![Page 28: Part 2-1: Crystal morphology - uni- · PDF fileprojection plane. easy measurement of angle in one plane ... draw zone pole corresponding to a zone ... symbol is: m Any point or object](https://reader033.fdocuments.us/reader033/viewer/2022051600/5a9f23dc7f8b9a0d158c739b/html5/thumbnails/28.jpg)
Functional Materials SaarlandUniversity
Summary - Symmetry basis
Mirrorrotation inversion
rotoinversion=rotation + inversionrotoreflection=rotation + mirror