SlideShare a Scribd company logo

UCSD NANO106 - 03 - Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations

NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.

1 of 35
Download to read offline
Lattice Directions and Planes, Reciprocal
Lattice and Coordinate Transformations
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego
Lattice planes and
directions
NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2
Readings
¡Chapter 5 of Structure of Materials
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 1
Lattice Directions
¡ Directions in a lattice is denoted by
¡ E.g., denotes the direction parallel to the a-axis in
any lattice.
¡ Negative numbers are denoted with a bar above the
number, e.g.,
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
uvw[ ]
100[ ]
112!" #$≡ a − b+ 2c
Lattice Planes
¡ A lattice plane of a given Bravais lattice is a plane (or
family of parallel planes) whose intersections with the
lattice are periodic (i.e., are described by 2d Bravais
lattices) and intersect the Bravais lattice; equivalently, a
lattice plane is any plane containing at least three
noncollinear Bravais lattice points.
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
Miller indices
¡ Lattice planes are represented by Miller indices, denoted
as , where h, k and l are integers. Note the use of the
round brackets instead of the square brackets used for
lattice directions.
¡ The procedure for determining the Miller indices of a plane
is best illustrated using an example.
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
hkl( )

Recommended

Brillouin zone and wigner seitz cell
Brillouin zone and wigner  seitz cellBrillouin zone and wigner  seitz cell
Brillouin zone and wigner seitz cellPeter Believe Jr
 
Bravais lattices
Bravais  latticesBravais  lattices
Bravais latticesPramoda Raj
 
Statistical mechanics
Statistical mechanics Statistical mechanics
Statistical mechanics Kumar
 
Nuclear Shell models
Nuclear Shell modelsNuclear Shell models
Nuclear Shell modelsNumanUsama
 
nuclear physics,unit 6
nuclear physics,unit 6nuclear physics,unit 6
nuclear physics,unit 6Kumar
 

More Related Content

What's hot

Crystalography
CrystalographyCrystalography
Crystalographymd5358dm
 
Band structure
Band structureBand structure
Band structurenirupam12
 
Solid state physics lec 1
Solid state physics lec 1Solid state physics lec 1
Solid state physics lec 1Dr. Abeer Kamal
 
Brillouin zones newton sow
Brillouin zones newton sowBrillouin zones newton sow
Brillouin zones newton sowSoundar Rajan
 
Chapter3 introduction to the quantum theory of solids
Chapter3 introduction to the quantum theory of solidsChapter3 introduction to the quantum theory of solids
Chapter3 introduction to the quantum theory of solidsK. M.
 
Pertemuan 7 vibrational properties-lattice
Pertemuan 7   vibrational properties-latticePertemuan 7   vibrational properties-lattice
Pertemuan 7 vibrational properties-latticejayamartha
 
Directions, planes and miller indices
Directions, planes and miller indicesDirections, planes and miller indices
Directions, planes and miller indicesSrilakshmi B
 
Band theory of solids
Band theory of solidsBand theory of solids
Band theory of solidsutpal sarkar
 
Solid state__physics (1)by D.Udayanga.
Solid  state__physics (1)by D.Udayanga.Solid  state__physics (1)by D.Udayanga.
Solid state__physics (1)by D.Udayanga.damitha udayanga
 
6563.nuclear models
6563.nuclear models6563.nuclear models
6563.nuclear modelsakshay garg
 
Crystal structure notes
Crystal structure notesCrystal structure notes
Crystal structure notesPraveen Vaidya
 
Direct and in direct band gap-Modern Physics
Direct and in direct band gap-Modern PhysicsDirect and in direct band gap-Modern Physics
Direct and in direct band gap-Modern PhysicsChandra Prakash Pandey
 
The heat capacity of a solid
The heat capacity of a solid The heat capacity of a solid
The heat capacity of a solid Kumar
 
Sputtering process and its types
Sputtering process and its typesSputtering process and its types
Sputtering process and its typesMuhammadWajid37
 

What's hot (20)

Crystalography
CrystalographyCrystalography
Crystalography
 
Band structure
Band structureBand structure
Band structure
 
Solid state physics lec 1
Solid state physics lec 1Solid state physics lec 1
Solid state physics lec 1
 
crystalstructure
crystalstructurecrystalstructure
crystalstructure
 
Brillouin zones newton sow
Brillouin zones newton sowBrillouin zones newton sow
Brillouin zones newton sow
 
non linear optics
non linear opticsnon linear optics
non linear optics
 
Sputtering process
Sputtering processSputtering process
Sputtering process
 
Chapter3 introduction to the quantum theory of solids
Chapter3 introduction to the quantum theory of solidsChapter3 introduction to the quantum theory of solids
Chapter3 introduction to the quantum theory of solids
 
Pertemuan 7 vibrational properties-lattice
Pertemuan 7   vibrational properties-latticePertemuan 7   vibrational properties-lattice
Pertemuan 7 vibrational properties-lattice
 
Directions, planes and miller indices
Directions, planes and miller indicesDirections, planes and miller indices
Directions, planes and miller indices
 
Band theory of solids
Band theory of solidsBand theory of solids
Band theory of solids
 
Solid state__physics (1)by D.Udayanga.
Solid  state__physics (1)by D.Udayanga.Solid  state__physics (1)by D.Udayanga.
Solid state__physics (1)by D.Udayanga.
 
6563.nuclear models
6563.nuclear models6563.nuclear models
6563.nuclear models
 
Crystal structure notes
Crystal structure notesCrystal structure notes
Crystal structure notes
 
Chapter 3a
Chapter 3aChapter 3a
Chapter 3a
 
Direct and in direct band gap-Modern Physics
Direct and in direct band gap-Modern PhysicsDirect and in direct band gap-Modern Physics
Direct and in direct band gap-Modern Physics
 
The heat capacity of a solid
The heat capacity of a solid The heat capacity of a solid
The heat capacity of a solid
 
Sputtering process and its types
Sputtering process and its typesSputtering process and its types
Sputtering process and its types
 
Crystal defects
Crystal defectsCrystal defects
Crystal defects
 
Specific Heat Capacity
Specific Heat CapacitySpecific Heat Capacity
Specific Heat Capacity
 

Viewers also liked

UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal StructuresUCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal StructuresUniversity of California, San Diego
 
UCSD NANO106 - 08 - Principal Directions and Representation Quadrics
UCSD NANO106 - 08 - Principal Directions and Representation QuadricsUCSD NANO106 - 08 - Principal Directions and Representation Quadrics
UCSD NANO106 - 08 - Principal Directions and Representation QuadricsUniversity of California, San Diego
 
Crystal structure analysis
Crystal structure analysisCrystal structure analysis
Crystal structure analysiszoelfalia
 
Basic crystallography
Basic crystallographyBasic crystallography
Basic crystallographyMukhlis Adam
 
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...University of California, San Diego
 

Viewers also liked (20)

UCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to CrystallographyUCSD NANO106 - 01 - Introduction to Crystallography
UCSD NANO106 - 01 - Introduction to Crystallography
 
UCSD NANO106 - 04 - Symmetry in Crystallography
UCSD NANO106 - 04 - Symmetry in CrystallographyUCSD NANO106 - 04 - Symmetry in Crystallography
UCSD NANO106 - 04 - Symmetry in Crystallography
 
UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations
UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice ComputationsUCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations
UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations
 
UCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups
UCSD NANO106 - 05 - Group Symmetry and the 32 Point GroupsUCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups
UCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups
 
UCSD NANO106 - 06 - Plane and Space Groups
UCSD NANO106 - 06 - Plane and Space GroupsUCSD NANO106 - 06 - Plane and Space Groups
UCSD NANO106 - 06 - Plane and Space Groups
 
UCSD NANO106 - 07 - Material properties and tensors
UCSD NANO106 - 07 - Material properties and tensorsUCSD NANO106 - 07 - Material properties and tensors
UCSD NANO106 - 07 - Material properties and tensors
 
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal StructuresUCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures
 
UCSD NANO106 - 10 - Bonding in Materials
UCSD NANO106 - 10 - Bonding in MaterialsUCSD NANO106 - 10 - Bonding in Materials
UCSD NANO106 - 10 - Bonding in Materials
 
UCSD NANO106 - 11 - X-rays and their interaction with matter
UCSD NANO106 - 11 - X-rays and their interaction with matterUCSD NANO106 - 11 - X-rays and their interaction with matter
UCSD NANO106 - 11 - X-rays and their interaction with matter
 
UCSD NANO106 - 09 - Piezoelectricity and Elasticity
UCSD NANO106 - 09 - Piezoelectricity and ElasticityUCSD NANO106 - 09 - Piezoelectricity and Elasticity
UCSD NANO106 - 09 - Piezoelectricity and Elasticity
 
UCSD NANO106 - 08 - Principal Directions and Representation Quadrics
UCSD NANO106 - 08 - Principal Directions and Representation QuadricsUCSD NANO106 - 08 - Principal Directions and Representation Quadrics
UCSD NANO106 - 08 - Principal Directions and Representation Quadrics
 
UCSD NANO106 - 12 - X-ray diffraction
UCSD NANO106 - 12 - X-ray diffractionUCSD NANO106 - 12 - X-ray diffraction
UCSD NANO106 - 12 - X-ray diffraction
 
Crystal structure analysis
Crystal structure analysisCrystal structure analysis
Crystal structure analysis
 
Basic crystallography
Basic crystallographyBasic crystallography
Basic crystallography
 
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...
Creating It from Bit - Designing Materials by Integrating Quantum Mechanics, ...
 
Crystal structure
Crystal structureCrystal structure
Crystal structure
 
Miller indecies
Miller indeciesMiller indecies
Miller indecies
 
Solid state (2)
Solid state (2)Solid state (2)
Solid state (2)
 
NANO266 - Lecture 10 - Temperature
NANO266 - Lecture 10 - TemperatureNANO266 - Lecture 10 - Temperature
NANO266 - Lecture 10 - Temperature
 
NANO266 - Lecture 9 - Tools of the Modeling Trade
NANO266 - Lecture 9 - Tools of the Modeling TradeNANO266 - Lecture 9 - Tools of the Modeling Trade
NANO266 - Lecture 9 - Tools of the Modeling Trade
 

Similar to UCSD NANO106 - 03 - Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations

Crystallographic planes and directions
Crystallographic planes and directionsCrystallographic planes and directions
Crystallographic planes and directionsNicola Ergo
 
Materials Characterization Technique Lecture Notes
Materials Characterization Technique Lecture NotesMaterials Characterization Technique Lecture Notes
Materials Characterization Technique Lecture NotesFellowBuddy.com
 
Crystallography and structure
Crystallography and structureCrystallography and structure
Crystallography and structureCleophas Rwema
 
Crystallography and X-ray diffraction (XRD) Likhith K
Crystallography and X-ray diffraction (XRD) Likhith KCrystallography and X-ray diffraction (XRD) Likhith K
Crystallography and X-ray diffraction (XRD) Likhith KLIKHITHK1
 
Xray diff pma
Xray diff pmaXray diff pma
Xray diff pmasadaf635
 
bravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfbravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfsmashtwins
 
bravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfbravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfsmashtwins
 
Congruence of triangles /GEOMETRY
Congruence of triangles /GEOMETRYCongruence of triangles /GEOMETRY
Congruence of triangles /GEOMETRYindianeducation
 
Progressive collapse of flexural systems
Progressive collapse of flexural systemsProgressive collapse of flexural systems
Progressive collapse of flexural systemsDCEE2017
 
Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Ali Farooq
 
Ch 27.2 crystalline materials & detects in crystalline materials
Ch 27.2 crystalline materials & detects in crystalline materialsCh 27.2 crystalline materials & detects in crystalline materials
Ch 27.2 crystalline materials & detects in crystalline materialsNandan Choudhary
 
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
 
Crystallographic points directions and planes.pdf
Crystallographic points directions and planes.pdfCrystallographic points directions and planes.pdf
Crystallographic points directions and planes.pdfDrJayantaKumarMahato1
 

Similar to UCSD NANO106 - 03 - Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations (20)

Crystallographic planes and directions
Crystallographic planes and directionsCrystallographic planes and directions
Crystallographic planes and directions
 
Materials Characterization Technique Lecture Notes
Materials Characterization Technique Lecture NotesMaterials Characterization Technique Lecture Notes
Materials Characterization Technique Lecture Notes
 
X-Ray Topic.ppt
X-Ray Topic.pptX-Ray Topic.ppt
X-Ray Topic.ppt
 
A_I_Structure.pdf
A_I_Structure.pdfA_I_Structure.pdf
A_I_Structure.pdf
 
miller indices Patwa[[g
miller indices Patwa[[gmiller indices Patwa[[g
miller indices Patwa[[g
 
Crystallography and structure
Crystallography and structureCrystallography and structure
Crystallography and structure
 
Crystal Structure
Crystal StructureCrystal Structure
Crystal Structure
 
Miller indices
Miller indicesMiller indices
Miller indices
 
Crystallography and X-ray diffraction (XRD) Likhith K
Crystallography and X-ray diffraction (XRD) Likhith KCrystallography and X-ray diffraction (XRD) Likhith K
Crystallography and X-ray diffraction (XRD) Likhith K
 
Xray diff pma
Xray diff pmaXray diff pma
Xray diff pma
 
Structure of Solid Materials
Structure of Solid MaterialsStructure of Solid Materials
Structure of Solid Materials
 
bravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfbravaislattices-171022062152.pdf
bravaislattices-171022062152.pdf
 
bravaislattices-171022062152.pdf
bravaislattices-171022062152.pdfbravaislattices-171022062152.pdf
bravaislattices-171022062152.pdf
 
Congruence of triangles /GEOMETRY
Congruence of triangles /GEOMETRYCongruence of triangles /GEOMETRY
Congruence of triangles /GEOMETRY
 
Progressive collapse of flexural systems
Progressive collapse of flexural systemsProgressive collapse of flexural systems
Progressive collapse of flexural systems
 
Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1Electromagnetic theory EMT lecture 1
Electromagnetic theory EMT lecture 1
 
Ch 27.2 crystalline materials & detects in crystalline materials
Ch 27.2 crystalline materials & detects in crystalline materialsCh 27.2 crystalline materials & detects in crystalline materials
Ch 27.2 crystalline materials & detects in crystalline materials
 
Bell 301 unit II
Bell 301 unit IIBell 301 unit II
Bell 301 unit II
 
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
 
Crystallographic points directions and planes.pdf
Crystallographic points directions and planes.pdfCrystallographic points directions and planes.pdf
Crystallographic points directions and planes.pdf
 

More from University of California, San Diego

NANO281 Lecture 01 - Introduction to Data Science in Materials Science
NANO281 Lecture 01 - Introduction to Data Science in Materials ScienceNANO281 Lecture 01 - Introduction to Data Science in Materials Science
NANO281 Lecture 01 - Introduction to Data Science in Materials ScienceUniversity of California, San Diego
 
NANO266 - Lecture 12 - High-throughput computational materials design
NANO266 - Lecture 12 - High-throughput computational materials designNANO266 - Lecture 12 - High-throughput computational materials design
NANO266 - Lecture 12 - High-throughput computational materials designUniversity of California, San Diego
 
The Materials Project Ecosystem - A Complete Software and Data Platform for M...
The Materials Project Ecosystem - A Complete Software and Data Platform for M...The Materials Project Ecosystem - A Complete Software and Data Platform for M...
The Materials Project Ecosystem - A Complete Software and Data Platform for M...University of California, San Diego
 
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical Modeling
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical ModelingNANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical Modeling
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical ModelingUniversity of California, San Diego
 

More from University of California, San Diego (15)

A*STAR Webinar on The AI Revolution in Materials Science
A*STAR Webinar on The AI Revolution in Materials ScienceA*STAR Webinar on The AI Revolution in Materials Science
A*STAR Webinar on The AI Revolution in Materials Science
 
NANO281 Lecture 01 - Introduction to Data Science in Materials Science
NANO281 Lecture 01 - Introduction to Data Science in Materials ScienceNANO281 Lecture 01 - Introduction to Data Science in Materials Science
NANO281 Lecture 01 - Introduction to Data Science in Materials Science
 
NANO266 - Lecture 14 - Transition state modeling
NANO266 - Lecture 14 - Transition state modelingNANO266 - Lecture 14 - Transition state modeling
NANO266 - Lecture 14 - Transition state modeling
 
NANO266 - Lecture 13 - Ab initio molecular dyanmics
NANO266 - Lecture 13 - Ab initio molecular dyanmicsNANO266 - Lecture 13 - Ab initio molecular dyanmics
NANO266 - Lecture 13 - Ab initio molecular dyanmics
 
NANO266 - Lecture 12 - High-throughput computational materials design
NANO266 - Lecture 12 - High-throughput computational materials designNANO266 - Lecture 12 - High-throughput computational materials design
NANO266 - Lecture 12 - High-throughput computational materials design
 
NANO266 - Lecture 11 - Surfaces and Interfaces
NANO266 - Lecture 11 - Surfaces and InterfacesNANO266 - Lecture 11 - Surfaces and Interfaces
NANO266 - Lecture 11 - Surfaces and Interfaces
 
NANO266 - Lecture 8 - Properties of Periodic Solids
NANO266 - Lecture 8 - Properties of Periodic SolidsNANO266 - Lecture 8 - Properties of Periodic Solids
NANO266 - Lecture 8 - Properties of Periodic Solids
 
NANO266 - Lecture 7 - QM Modeling of Periodic Structures
NANO266 - Lecture 7 - QM Modeling of Periodic StructuresNANO266 - Lecture 7 - QM Modeling of Periodic Structures
NANO266 - Lecture 7 - QM Modeling of Periodic Structures
 
The Materials Project Ecosystem - A Complete Software and Data Platform for M...
The Materials Project Ecosystem - A Complete Software and Data Platform for M...The Materials Project Ecosystem - A Complete Software and Data Platform for M...
The Materials Project Ecosystem - A Complete Software and Data Platform for M...
 
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical Modeling
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical ModelingNANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical Modeling
NANO266 - Lecture 6 - Molecule Properties from Quantum Mechanical Modeling
 
NANO266 - Lecture 5 - Exchange-Correlation Functionals
NANO266 - Lecture 5 - Exchange-Correlation FunctionalsNANO266 - Lecture 5 - Exchange-Correlation Functionals
NANO266 - Lecture 5 - Exchange-Correlation Functionals
 
NANO266 - Lecture 4 - Introduction to DFT
NANO266 - Lecture 4 - Introduction to DFTNANO266 - Lecture 4 - Introduction to DFT
NANO266 - Lecture 4 - Introduction to DFT
 
NANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
NANO266 - Lecture 3 - Beyond the Hartree-Fock ApproximationNANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
NANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
 
NANO266 - Lecture 2 - The Hartree-Fock Approach
NANO266 - Lecture 2 - The Hartree-Fock ApproachNANO266 - Lecture 2 - The Hartree-Fock Approach
NANO266 - Lecture 2 - The Hartree-Fock Approach
 
NANO266 - Lecture 1 - Introduction to Quantum Mechanics
NANO266 - Lecture 1 - Introduction to Quantum MechanicsNANO266 - Lecture 1 - Introduction to Quantum Mechanics
NANO266 - Lecture 1 - Introduction to Quantum Mechanics
 

Recently uploaded

Plant Genetic Resources, Germplasm, gene pool - Copy.pptx
Plant Genetic Resources, Germplasm, gene pool - Copy.pptxPlant Genetic Resources, Germplasm, gene pool - Copy.pptx
Plant Genetic Resources, Germplasm, gene pool - Copy.pptxAKSHAYMAGAR17
 
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...Rabiya Husain
 
skeletal system complete details with joints and its types
skeletal system complete details with joints and its typesskeletal system complete details with joints and its types
skeletal system complete details with joints and its typesMinaxi patil. CATALLYST
 
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdfAynouraHamidova
 
Data Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdfData Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdfChristalin Nelson
 
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptx
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptxBBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptx
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptxProf. Kanchan Kumari
 
BTKi in Treatment Of Chronic Lymphocytic Leukemia
BTKi in Treatment Of Chronic Lymphocytic LeukemiaBTKi in Treatment Of Chronic Lymphocytic Leukemia
BTKi in Treatment Of Chronic Lymphocytic LeukemiaFaheema Hasan
 
DISCOURSE: TEXT AS CONNECTED DISCOURSE
DISCOURSE:   TEXT AS CONNECTED DISCOURSEDISCOURSE:   TEXT AS CONNECTED DISCOURSE
DISCOURSE: TEXT AS CONNECTED DISCOURSEMYDA ANGELICA SUAN
 
Food Web SlideShare for Ecology Notes Quiz in Canvas
Food Web SlideShare for Ecology Notes Quiz in CanvasFood Web SlideShare for Ecology Notes Quiz in Canvas
Food Web SlideShare for Ecology Notes Quiz in CanvasAlexandraSwartzwelde
 
Overview of Databases and Data Modelling-2.pdf
Overview of Databases and Data Modelling-2.pdfOverview of Databases and Data Modelling-2.pdf
Overview of Databases and Data Modelling-2.pdfChristalin Nelson
 
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in GhanaDecision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in GhanaPrince Armah, PhD
 
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...MohonDas
 
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdf
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdfDr.M.Florence Dayana-Cloud Computing-Unit - 1.pdf
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdfDr.Florence Dayana
 
Group_Presentation_Gun_Island_Amitav_Ghosh.pptx
Group_Presentation_Gun_Island_Amitav_Ghosh.pptxGroup_Presentation_Gun_Island_Amitav_Ghosh.pptx
Group_Presentation_Gun_Island_Amitav_Ghosh.pptxPooja Bhuva
 
Overview of Databases and Data Modelling-1.pdf
Overview of Databases and Data Modelling-1.pdfOverview of Databases and Data Modelling-1.pdf
Overview of Databases and Data Modelling-1.pdfChristalin Nelson
 
2.20.24 The March on Washington for Jobs and Freedom.pptx
2.20.24 The March on Washington for Jobs and Freedom.pptx2.20.24 The March on Washington for Jobs and Freedom.pptx
2.20.24 The March on Washington for Jobs and Freedom.pptxMaryPotorti1
 
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdfAynouraHamidova
 
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdf
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdfDiploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdf
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdfSUMIT TIWARI
 
ICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten NotesICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten NotesGauri S
 

Recently uploaded (20)

Plant Genetic Resources, Germplasm, gene pool - Copy.pptx
Plant Genetic Resources, Germplasm, gene pool - Copy.pptxPlant Genetic Resources, Germplasm, gene pool - Copy.pptx
Plant Genetic Resources, Germplasm, gene pool - Copy.pptx
 
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...
Barrow Motor Ability Test - TEST, MEASUREMENT AND EVALUATION IN PHYSICAL EDUC...
 
skeletal system complete details with joints and its types
skeletal system complete details with joints and its typesskeletal system complete details with joints and its types
skeletal system complete details with joints and its types
 
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 10-2023-Aynura-Hamidova.pdf
 
Data Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdfData Modeling - Entity Relationship Diagrams-1.pdf
Data Modeling - Entity Relationship Diagrams-1.pdf
 
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptx
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptxBBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptx
BBA 603 FUNDAMENTAL OF E- COMMERCE UNIT 1.pptx
 
BTKi in Treatment Of Chronic Lymphocytic Leukemia
BTKi in Treatment Of Chronic Lymphocytic LeukemiaBTKi in Treatment Of Chronic Lymphocytic Leukemia
BTKi in Treatment Of Chronic Lymphocytic Leukemia
 
DISCOURSE: TEXT AS CONNECTED DISCOURSE
DISCOURSE:   TEXT AS CONNECTED DISCOURSEDISCOURSE:   TEXT AS CONNECTED DISCOURSE
DISCOURSE: TEXT AS CONNECTED DISCOURSE
 
Food Web SlideShare for Ecology Notes Quiz in Canvas
Food Web SlideShare for Ecology Notes Quiz in CanvasFood Web SlideShare for Ecology Notes Quiz in Canvas
Food Web SlideShare for Ecology Notes Quiz in Canvas
 
Overview of Databases and Data Modelling-2.pdf
Overview of Databases and Data Modelling-2.pdfOverview of Databases and Data Modelling-2.pdf
Overview of Databases and Data Modelling-2.pdf
 
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in GhanaDecision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
Decision on Curriculum Change Path: Towards Standards-Based Curriculum in Ghana
 
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
BEZA or Bangladesh Economic Zone Authority recruitment exam question solution...
 
ThinkTech 2024 Prelims- U25 BizTech Quiz by Pragya
ThinkTech 2024 Prelims- U25 BizTech Quiz by PragyaThinkTech 2024 Prelims- U25 BizTech Quiz by Pragya
ThinkTech 2024 Prelims- U25 BizTech Quiz by Pragya
 
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdf
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdfDr.M.Florence Dayana-Cloud Computing-Unit - 1.pdf
Dr.M.Florence Dayana-Cloud Computing-Unit - 1.pdf
 
Group_Presentation_Gun_Island_Amitav_Ghosh.pptx
Group_Presentation_Gun_Island_Amitav_Ghosh.pptxGroup_Presentation_Gun_Island_Amitav_Ghosh.pptx
Group_Presentation_Gun_Island_Amitav_Ghosh.pptx
 
Overview of Databases and Data Modelling-1.pdf
Overview of Databases and Data Modelling-1.pdfOverview of Databases and Data Modelling-1.pdf
Overview of Databases and Data Modelling-1.pdf
 
2.20.24 The March on Washington for Jobs and Freedom.pptx
2.20.24 The March on Washington for Jobs and Freedom.pptx2.20.24 The March on Washington for Jobs and Freedom.pptx
2.20.24 The March on Washington for Jobs and Freedom.pptx
 
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf
11 CI SINIF SINAQLARI - 2-2023-Aynura-Hamidova.pdf
 
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdf
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdfDiploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdf
Diploma 2nd yr PHARMACOLOGY chapter 5 part 1.pdf
 
ICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten NotesICSE English Language Class X Handwritten Notes
ICSE English Language Class X Handwritten Notes
 

UCSD NANO106 - 03 - Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations

  • 1. Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations Shyue Ping Ong Department of NanoEngineering University of California, San Diego
  • 2. Lattice planes and directions NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2
  • 3. Readings ¡Chapter 5 of Structure of Materials NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 1
  • 4. Lattice Directions ¡ Directions in a lattice is denoted by ¡ E.g., denotes the direction parallel to the a-axis in any lattice. ¡ Negative numbers are denoted with a bar above the number, e.g., NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 uvw[ ] 100[ ] 112!" #$≡ a − b+ 2c
  • 5. Lattice Planes ¡ A lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice are periodic (i.e., are described by 2d Bravais lattices) and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
  • 6. Miller indices ¡ Lattice planes are represented by Miller indices, denoted as , where h, k and l are integers. Note the use of the round brackets instead of the square brackets used for lattice directions. ¡ The procedure for determining the Miller indices of a plane is best illustrated using an example. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 hkl( )
  • 7. Procedure for determining the Miller indices ¡ Let’s say we have a plane in the lattice specified by a, b and c. 1. If the plane pass through the origin, displace the plane by an arbitrary amount so that it does not pass through the origin (not required for worked example). 2. Determine the intercepts of plane with three lattice vectors, in units of the lattice vector length. If the plane is parallel to one or more of the axes, this corresponding number is ∞. In the example, these are 1:2:3. 3. Invert all three numbers. If the plane is parallel to one or more of the axes, this corresponding number is 1/∞ = 0. For the example, we get 1:½:1/3. 4. Reduce the numbers to the nearest integers (known as the relative primes). We get (632). NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a b c 2b 3c
  • 8. Example ¡Determine the Miller indices of the following planes NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a c b ac b 0.5
  • 9. ¡ When the lattice has symmetry (i.e., non-triclinic), certain planes are equivalent to each other under symmetry. Such families of planes are represented with a curly brackets, i.e. {hkl}. ¡ Similarly, families of directions are denoted by <uvw>. Families of planes or directions NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 {110} family of planes in a cubic lattice <111> family of directions in a cubic lattice
  • 10. Permutations of Miller indices ¡ From the cubic example, we may observe that all planes in the {110} family is given by permutations of the indices and their negatives: ¡ For lower symmetry systems, families are still given by permutations, though not all permutations belong to the same family. ¡ Rhombohedral: ¡ Orthorhombic: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 110{ }= 110( ), 110( ), 110( ), 101( ), 101( ), 101( ), 011( ), 011( ), 011( ){ } 100{ }= 100( ), 100( ), 010( ), 010( ), 001( ), 001( ){ } 100{ }= 100( ), 100( ){ }
  • 11. Miller-Bravais Indices of Hexagonal Crystal System ¡ Hexagonal system is defined by four basis vectors, three of which are co-planar. ¡ Miller-Bravais indices are given by intercepts with all four basis vectors . i is a redundant index and is given by i = -(h+k) ¡ The four-index representation allows families of planes for hexagonal systems to be represented as permutations, e.g., NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 1122( ) hkil( ) 1120{ }= 1120( ), 1210( ), 2110( ), 1120( ), 1210( ), 2110( ){ }
  • 12. Miller-Bravais Indices for Directions in Hexagonal System ¡ Denoted as [uvtw]. ¡ By convention, t = -(u+v), similar to indices for planes. ¡ It can be shown that the relationship between a three-Miller index [u’v’w’] and the corresponding four-Miller index [uvtw] is given by (please review proof on your own accord): NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 u = 1 3 (2u'− v') v = 1 3 (2v'− u') t = −(u+ v) w = w' u' = 2u+ v v' = 2v+u w' = w u, v, t, w èu’, v’, w’ u’, v’, w’ èu, v, t, w
  • 13. Examples ¡Determine the Miller indices of the following planes. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 0.5
  • 14. The reciprocal lattice NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2
  • 15. Readings ¡Chapter 6 of Structure of Materials NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 1
  • 16. Definition of the Reciprocal Lattice ¡ For a lattice given by basis vectors a, b and c, the reciprocal lattice is given basis vectors a*, b* and c* where: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a* = b× c a.(b× c) b* = a × c a.(b× c) c* = a × b a.(b× c) Properties of the Reciprocal Lattice Basis Vectors 1. a*.a = 1 (similarly for b* and c*) 2. a* is perpendicular to both b and c, i.e., a*.b = a*.c = 0 (similarly for b* and c*) 3. Using an alternative notation where a1, a2 and a3 represent the three lattice vectors, ai*.aj=δij where δij is the Kronecker delta. Important Note: In solid-state physics, the reciprocal lattice vectors have a factor of 2π.
  • 17. Reciprocal Lattice and Lattice Planes ¡ The reciprocal lattice is a lattice, just like the real space lattice. ¡ While real space vectors are represented by (u,v,w), reciprocal vectors are customarily represented by (h, k, l) (we will see the reason for this notation in a moment) ¡ Consider all real space vectors that are perpendicular to the reciprocal space vector above: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 g = ha* + kb* + lc* r = xa + yb + zc r.g = xa + yb + zc( ). ha* + kb* + lc* ( ) = hx + ky +lz = 0 Equation of plane passing through origin
  • 18. Reciprocal Lattice and Lattice Planes, contd. ¡ What is the equation of lattice planes with Miller indices (hkl)? Remember that h, k and l are the reciprocals of the intercepts with the intercepts with the three axes: ¡ Key result: The reciprocal lattice vector g with components (h, k, l) is perpendicular to lattice planes with Miller indices (hkl) and is usually denoted as . The reciprocal lattice therefore lets us describe plane normals with simple integers. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 ghkl x 1/ h + y 1/ k + z 1/ l = N Value of N determines distance of plane to origin
  • 19. Relationship between reciprocal lattice vector length and interplanar spacing ¡ We have seen earlier that is perpendicular to (hkl). ¡ Consider an arbitrary point in the plane (hkl) given by vector t from the origin. The distance from the plane to the origin is given by: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 ghkl a b c 2b 3c t g632 dhkl = 1 ghkl Inter-layer spacing is given by reciprocal of the reciprocal lattice vector length Blackboard proof
  • 20. Reciprocal lattice is just like any other lattice ¡ You can similarly define a reciprocal metric tensor ¡ And distances between points in the reciprocal space is given by the same relations as the crystal lattice. ¡ It can also be shown that the reciprocal and crystal metric tensors are inverses of each other, i.e., NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 g* = a* .a* a* .b* a* .c* b* .a* b* .b* b* .c* c* .a* c* .b* c* .c* ! " # # ## $ % & & && g* = g−1
  • 21. Coordinate Transformations NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 3
  • 22. Coordinate transformations ¡ Sometimes, we want to choose a different unit cell or set of basis vectors for a lattice. For example, we may want to use the primitive basis vectors instead of the conventional one, or vice versa. ¡ How do we derive geometric quantities such as positions, lengths, etc. in the new basis? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3
  • 23. Alternative notation for lattice vectors ¡Thus far, we have denoted the three lattice vectors as ¡We will here introduce an alternative indicial notation which will make it easier to represent certain kinds of operations, especially summations and matrix multiplications. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 a,b,c a1,a2,a3
  • 24. Notation ¡ Instead of using the conventional to represent basis vectors, let us choose to represent them as . ¡ Let’s say the new basis vectors are . This is known as a change of reference frame. ¡ In general, the relationship between the new basis vectors and original lattice vectors can be represented by a linear transformation: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 a, b,c{ } a1, a2,a3{ } a1 !1, a2 !,a3 !{ } a1 !1 ' a!2 a!3 " # $ $ $ $ $ % & ' ' ' ' ' = c11 c12 c13 c21 c22 c23 c31 c32 c33 " # $ $ $$ % & ' ' '' a1 a2 a33 " # $ $ $$ % & ' ' '' or A' = CA Note that in this case, the lattice vectors are written as rows in the matrix A and A’.
  • 25. How do you determine the transformation matrix? ¡It can be shown that ¡Usually, it is far simpler to do an “inspection” to determine the matrix. Example, how do you express each ai’ in terms of the original vectors ai for the lattice shown here? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 cij = ai !aj * Blackboard proof
  • 26. Transformation Relations for Direct Positions / Vectors NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard derivation
  • 27. Transformation Relations for Metric Tensor NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard derivation
  • 28. Special case – the reciprocal lattice ¡What if our new coordinate system is the reciprocal lattice? ¡Recall that we proved that ¡We simply have NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 cij = ai !aj * cij = ai * aj * = gij * = gij −1 , i.e., C = g−1 p1 * p2 * p3 * ( )= p1 p2 p3( ) a1 ⋅a1 a1 ⋅a2 a1 ⋅a3 a2 ⋅a1 a2 ⋅a2 a2 ⋅a3 a3 ⋅a1 a3 ⋅a2 a3 ⋅a3 " # $ $ $$ % & ' ' ''
  • 29. Summary of Coordinate Transformations NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Transformation Relationship Position/vector to position/vector Position/vector to reciprocal position/vector Reciprocal position/vector to Reciprocal position/vector p1 ' p2 ' p3 ' ( )= p1 p2 p3( )C−1 p1 p2 p3( )= p1 ' p2 ' p3 ' ( )C p1 *' p2 *' p3 *' ! " # # # # $ % & & & & = C p1 * p2 * p3 * ! " # # # # $ % & & & & p1 * p2 * p3 * ! " # # # # $ % & & & & = C−1 p1 *' p2 *' p3 *' ! " # # # # $ % & & & & p1 * p2 * p3 * ( )= p1 p2 p3( )g p1 p2 p3( )= p1 * p2 * p3 * ( )g−1
  • 30. Summary of Coordinate Transformations NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Transformation Relationship Metric Tensor g' = CgCT g = C−1 g' CT ( ) −1
  • 31. Expected knowledge and tips ¡It is not expected that you memorize how to derive all the coordinate transformation relations. The derivation was shown to give you a deeper appreciation of how it all works. ¡You only need to know how to apply the relations in doing transformations. ¡Be very very careful in noting whether it is a row x matrix or a matrix x column multiplication. The two are not the same! NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3
  • 32. Worked example: Conventional to Primitive Transformation of FCC ¡ What is the relationship between the primitive basis vectors and the conventional basis vectors? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard
  • 33. Worked example: Conventional to Primitive Transformation of FCC ¡ How does the [101] direction in the cubic frame transform in the rhombohedral frame? ¡ How does the (110) plane in the cubic frame transform in the rhombohedral frame? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard
  • 34. Worked example: C-centered Orthorhombic to Primitive ¡ What is the relationship between the primitive basis vectors and the conventional basis vectors? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard
  • 35. Worked example: C-centered Orthorhombic to Primitive ¡ How does the [111] direction in the conventional frame transform in the primitive frame? ¡ How does the (110) plane in the conventional frame transform in the primitive frame? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 3 Blackboard