II Crystal Structure 2-1 Basic concept

Slides:



Advertisements
Similar presentations
INTRODUCTION TO CERAMIC MINERALS
Advertisements

Fundamental Concepts Crystalline: Repeating/periodic array of atoms; each atom bonds to nearest neighbor atoms. Crystalline structure: Results in a lattice.
t1 t2 6 t1 t2 7 8 t1 t2 9 t1 t2.
Why Study Solid State Physics?
Stereographic projection
Introduction to Mineralogy Dr. Tark Hamilton Chapter 6: Lecture Crystallography & External Symmetry of Minerals Camosun College GEOS 250 Lectures:
3-Dimensional Crystal Structure
Reciprocal lattice How to construct reciprocal lattice
Crystal Structure Lecture 4.
CHAPTER 2 : CRYSTAL DIFFRACTION AND PG Govt College for Girls
II. Crystal Structure Lattice, Basis, and the Unit Cell
VI. Reciprocal lattice 6-1. Definition of reciprocal lattice from a lattice with periodicities in real space Remind what we have learned in chapter.
William Hallowes Miller
Crystallographic Planes
When dealing with crsytalline materials, it is often necessary to specify a particular point within a unit cell, a particular direction or a particular.
PHYS 430/603 material Laszlo Takacs UMBC Department of Physics
Wigner-Seitz Cell The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to.
Lec. (4,5) Miller Indices Z X Y (100).
Miller indices and crystal directions
Introduction and point groups Stereographic projections Low symmetry systems Space groups Deformation and texture Interfaces, orientation relationships.
Typical Crystal Structures
Expression of d-dpacing in lattice parameters
© Oxford Instruments Analytical Limited 2001 MODULE 2 - Introduction to Basic Crystallography Bravais Lattices Crystal system Miller Indices Crystallographic.
CONDENSED MATTER PHYSICS PHYSICS PAPER A BSc. (III) (NM and CSc.) Harvinder Kaur Associate Professor in Physics PG.Govt College for Girls Sector -11, Chandigarh.
Order in crystals Symmetry, X-ray diffraction. 2-dimensional square lattice.
The indeterminate situation arises because the plane passes through the origin. After translation, we obtain intercepts. By inverting them, we get.
Miller Indices And X-ray diffraction
Introduction to Crystallography
Solid State Physics Yuanxu Wang School of Physics and Electronics Henan University 双语教学示范课程.
Solid State Physics (1) Phys3710
EEE539 Solid State Electronics 1. Crystal Structure Issues that are addressed in this chapter include:  Periodic array of atoms  Fundamental types of.
Crystal Structure A “unit cell” is a subdivision of the lattice that has all the geometric characteristics of the total crystal. The simplest choice of.
W.D. Callister, Materials science and engineering an introduction, 5 th Edition, Chapter 3 MM409: Advanced engineering materials Crystallography.
Crystallographic Points, Directions, and Planes. ISSUES TO ADDRESS... How to define points, directions, planes, as well as linear, planar, and volume densities.
Bravais Lattices in 2D In 2D there are five ways to order atoms in a lattice Primitive unit cell: contains only one atom (but 4 points?) Are the dotted.
Miller Indices & Steriographic Projection
Crystal Structures Crystal is constructed by the continuous repetition in space of an identical structural unit. Lattice: a periodic array of mathematical.
Crystallography ll.
PHY1039 Properties of Matter Crystallography, Lattice Planes, Miller Indices, and X-ray Diffraction (See on-line resource: )
Crystalline Solids :-In Crystalline Solids the atoms are arranged in some regular periodic geometrical pattern in three dimensions- long range order Eg.
CRYSTAL STRUCTURE.
Crystallography lv.
ESO 214: Nature and Properties of Materials
Stereographic projection
Crystallography Lecture notes Many other things.
Crystallographic Axes
Basic Crystallography for X-ray Diffraction Earle Ryba.
DESCRIBING CRYSTALS MATHEMATICALLY Three directional vectors Three distances: a, b, c Three angles:   ll points can be described using coordinate.
Crystal Structure and Crystallography of Materials
Crystal Structure NaCl Well defined surfaces
Fundamentals of crystal Structure
Properties of engineering materials
Crystallographic Points, Directions, and Planes.
THE SPACE LATTICE AND UNIT CELLS CRYSTAL SYSTEMS AND BRAVAIS LATTICES.
CHAPTER 3: STRUCTURE OF CRYSTALLINE SOLIDS
Groups: Fill in this Table for Cubic Structures
Crystalline state Symmetry in nature Symmetry in arts and industry
Concepts of Crystal Geometry
Crystallographic Points, Directions, and Planes.
Crystal and Amorphous Structure in Materials
CRYSTAL STRUCTURE ANALYSIS
3-Dimensional Crystal Structure.
MSE420/514: Session 1 Crystallography & Crystal Structure
Why Study Solid State Physics?
MILLER PLANES Atoms form periodically arranged planes Any set of planes is characterized by: (1) their orientation in the crystal (hkl) – Miller indices.
3-Dimensional Crystal Structure
MODULE 2 - Introduction to Basic Crystallography
William Hallowes Miller
Crystal Structure Acknowledgement: This slides are largely obtained from Dr.Neoh Siew Chin UniMAP on the subject Material Engineering.
Presentation transcript:

II Crystal Structure 2-1 Basic concept  Crystal structure = lattice structure + basis  Lattice points: positions (points) in the structure which are identical. X X X X X X X X X X X X X X X X lattice point Not a lattice point

 Lattice translation vector  Lattice plane  Unit cell  Primitive unit cell【1 lattice point/unit cell】 Examples : CsCl Fe (ferrite) b.c.c (not primitive) Al f.c.c (not primitive) Mg h.c.p simple hexagonal lattice Si diamond f.c.c (not primitive)

 Rational direction integer Cartesian coordinate Use lattice net to describe is much easier! Rational direction & Rational plane are chosen to describe the crystal structure!

2-2 Miller Indices in a crystal (direction, plane) The direction [u v w] is expressed as a vector The direction <u v w>are all the [u v w] types of direction, which are crystallographic equivalent.

2-2-2 plane The plane (h k l) is the Miller index of the plane in the figure below. {h k l} are the (h k l) types of planes which are crystllographic equivalent.

You can practice indexing directions and planes in the following website http://www.materials.ac.uk/elearning/matter/Crystallography/IndexingDirectionsAndPlanes/index.html

Crystallographic equivalent? Example: (hkl) Individual plane {hkl} Symmetry related set {100} z y x z {100} Different Symmetry related set y x

2-2-3 meaning of miller indices >Low index planes are widely spaced. x [120] (110)

>Low index directions correspond to short lattice translation vectors. y x [120] [110] >Low index directions and planes are important for slip, cross slip, and electron mobility.

2-3 Miller Indices and Miller - Bravais Indices (h k l) (h k i l) 2-3-1 in cubic system Direction [h k l] is perpendicular to (h k l) plane in the cubic system, but not true for other crystal systems. y y x [110] x [110] (110) (110) |x| = |y| |x|  |y|

using Miller - Bravais indexing system: (hkil) and [hkil] 2-3-2 In hexagonal system using Miller - Bravais indexing system: (hkil) and [hkil] y [010] Miller indces x [110] [100] Reason (i): Type [110] does not equal to [010] , but these directions are crystallographic equivalent. Reason (ii): z axis is [001], crystallographically distinct from [100] and [010]. (This is not a reason!)

the Miller-Bravais indexing system (specific for hexagonal system) If crystal planes in hexagonal systems are indexed using Miller indices, then crystallographically equivalent planes have indices which appear dissimilar. the Miller-Bravais indexing system (specific for hexagonal system) http://www.materials.ac.uk/elearning/matter/Crystallography/IndexingDirectionsAndPlanes/indexing-of-hexagonal-systems.html

2-3-3 Miller-Bravais indices (a) direction The direction [h k i l] is expressed as a vector [ 1 1 20] [ 1 2 1 0] [2 1 1 0] 𝑎 3 [2 1 1 0] Note : is the shortest translation vector on the basal plane.

𝑎 3 [2 1 1 0] 𝑎 3 [ 1 2 1 0] [100] [010] You can check 𝑢

planes (h k i l) ; h + k + i = 0 Plane (h k l)  (h k i l) (010)  (01 1 0) (100) plane plane (100)  (10 1 0)

(h k l) (h k i l) ? plane ( 2 10) ( 2 110)

Proof for general case: For plane (h k l), the intersection with the basal plane (001) is a line that is expressed as Check back to page 5 𝑥 1 ℎ + 𝑦 1 𝑘 =1 line equation : Where we set the lattice constant a = b = 1 in the hexagonal lattice for simplicity. line equation :

( 1 ℎ+𝑘 , 1 ℎ+𝑘 ) The line along the 𝑢 axis can be expressed as line equation : 𝑥=𝑦 or 𝑥−𝑦=0 Intersection points of these two lines ℎ𝑥+𝑘𝑦=1 and 𝑥−𝑦=0 ( 1 ℎ+𝑘 , 1 ℎ+𝑘 ) is at The vector from origin to the point can be expressed along the 𝑢 axis as 𝑥 𝑥 +𝑦 𝑦 = 1 ℎ+𝑘 𝑥 + 1 ℎ+𝑘 𝑦 = 1 ℎ+𝑘 𝑥 + 𝑦 = 1 − ℎ+𝑘 𝑢 = 1 𝑖 𝑢  i = - (h + k)

( 1 ℎ+𝑘 , 1 ℎ+𝑘 ) 1 ℎ+𝑘 ? 1 𝑘 1 ℎ+𝑘 1 ℎ 1 ℎ+𝑘 1 1/ℎ : 1 1/𝑘 : −1 1/(ℎ+𝑘) =ℎ:𝑘:−(ℎ+𝑘)

(c) Transformation from Miller [x y z] to Miller-Bravais index [h k i l] rule ℎ= 2𝑥−𝑦 3 𝑘= 2𝑦−𝑥 3 𝑖= −(𝑥+𝑦) 3 𝑧=𝑙

The same vector is expressed as [x y z] in Proof: The same vector is expressed as [x y z] in miller indices and as [h k I l] in Miller-Bravais indices! 𝑥 𝑦 𝑧 =𝑥 𝑥 +𝑦 𝑦 +𝑧 𝑧 ℎ 𝑘 𝑖 𝑙 =ℎ 𝑥 +𝑘 𝑦 +𝑖 𝑢 +𝑙 𝑧 =ℎ 𝑥 +𝑘 𝑦 +𝑖 − 𝑥 − 𝑦 +𝑙 𝑧 = ℎ−𝑖 𝑥 + 𝑘−𝑖 𝑦 +𝑙 𝑧 𝑥=ℎ − 𝑖 𝑦 = 𝑘−𝑖  𝑧 = 𝑙

Moreover, ℎ+𝑘=−𝑖 𝑥=ℎ−𝑖=ℎ+ℎ+𝑘=2ℎ+𝑘  𝑦=𝑘−𝑖=𝑘+ℎ+𝑘=ℎ+2𝑘 𝑧=𝑙 ℎ= 2𝑥−𝑦 3 𝑘= 2𝑦−𝑥 3 𝑖= −(𝑥+𝑦) 3   𝑧=𝑙

2-4 Stereographic projections [𝑢 𝑣 𝑤] 2-4-1 direction [ℎ 𝑘 𝑙] Horizontal plane [ 𝑢 𝑣 𝑤 ] [ℎ 𝑘 𝑙] [𝑢 𝑣 𝑤] [ 𝑢 𝑣 𝑤 ]

A line (direction)  a point. Representation of relationship of planes and directions in 3D on a 2D plane. Useful for the orientation problems. A line (direction)  a point.

(100)

2-4-2 plane Great circle: the plane passing through the center of the sphere.

A plane (Great Circle)  trace http://courses.eas.ualberta.ca/eas233/0809winter/EAS233Lab03notes.pdf A plane (Great Circle)  trace

Small circle: the plane not passing through the center of the sphere. B.D. Cullity

Example: [001] stereographic projection; cubic Zone axis B.D. Cullity

2-4-2 Stereographic projection of different Bravais systems Cubic

How about a standard (011) stereographic projection of a cubic crystal? 𝑥 𝑦 𝑧 Start with what you know! What does (011) look like?

𝑥 [01 1 ] [ 1 00] (011) [100] 𝑧 𝑦 [0 1 1] [11 1 ] [1 1 1] 109.47o (011) 70.53o

[011] [001] [0 1 1] 𝑥 [01 1 ] [ 1 00] (011) [100] 𝑧 𝑦 [001] [0 1 1] 45o [011]

[011] [111] [100] [100] [111] 𝑥 [01 1 ] [ 1 00] (011) 𝑧 𝑦 [0 1 1] 35.26o [ 1 00] (011) 𝑧 𝑦 [0 1 1] [011]

1 10 111

Trigonal Hexagonal 3a [ 2 111] [0001] c tan −1 3𝑎 𝑐 3a [ 2 110]

Orthorhombic Monoclinic

2-5 Two convections used in stereographic projection (1) plot directions as poles and planes as great circles (2) plot planes as poles and directions as

2-5-1 find angle between two directions (a) find a great circle going through them (b) measure angle by Wulff net

Meridians: great circle Parallels except the equator are small circles

Equal angle with respect to N or S pole

Measure the angle between two points: Bring these two points on the same great circle; counting the latitude angle.

(i) If two poles up (ii) If one pole up, one pole down

2-5-2 measuring the angle between planes This is equivalent to measuring angle between poles Pole and trace http://en.wikipedia.org/wiki/Pole_figure

Angle between the planes of two zone circles is the angle between the poles of the corresponding

use of stereographic projections (i) plot directions as poles ---- used to measure angle between directions ---- use to establish if direction lie in a particular plane (ii) plot planes as poles ---- used to measure angles between planes ---- used to find if planes lies in the same zone