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Transcript of Lecture 2 Symmetry and Structure - MIT OpenCourseWare · PDF fileStructure and Symmetry. 22.14...
![Page 1: Lecture 2 Symmetry and Structure - MIT OpenCourseWare · PDF fileStructure and Symmetry. 22.14 –Intro to Nuclear Materials. February 5, 2015. ... Symmetry and Structure, Slide 33.](https://reader031.fdocuments.in/reader031/viewer/2022030403/5a797f127f8b9a6c158c71e1/html5/thumbnails/1.jpg)
Structure and Symmetry
22.14 – Intro to Nuclear MaterialsFebruary 5, 2015
Scanned images, unless cited, are from Allen & Thomas, “The Structure of Materials,” 1999.
22.14 - Intro to Nuclear Materials
![Page 2: Lecture 2 Symmetry and Structure - MIT OpenCourseWare · PDF fileStructure and Symmetry. 22.14 –Intro to Nuclear Materials. February 5, 2015. ... Symmetry and Structure, Slide 33.](https://reader031.fdocuments.in/reader031/viewer/2022030403/5a797f127f8b9a6c158c71e1/html5/thumbnails/2.jpg)
Crystallography – The Common Language of Materials Science
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 2
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Crystalline vs. Amorphous
The difference is long-range order, and symmetry
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 3
http://physics.anu.edu.au/eme/research/amorphous.php
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Symmetry Evident in Materials
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 4
Etch pits in single crystal aluminum
Source: J. H. Seob, J.-H. Ryuc, D. N. Lee. “Formation of Crystallographic Etch Pits during AC Etching of Aluminum.” J. Electrochem Soc., 150(9):B433-B438 (2003).
![Page 5: Lecture 2 Symmetry and Structure - MIT OpenCourseWare · PDF fileStructure and Symmetry. 22.14 –Intro to Nuclear Materials. February 5, 2015. ... Symmetry and Structure, Slide 33.](https://reader031.fdocuments.in/reader031/viewer/2022030403/5a797f127f8b9a6c158c71e1/html5/thumbnails/5.jpg)
Simplest Operation: Translation
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 5
t1
t2
Move a point by two basis vectors, t1 & t2
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Higher Symmetry
Place restrictions on t1 and t2, and the angle between them.
How many combinations can you think of?
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 6
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Choosing Unit Cells
Draw a cell that does the following:– Contains fewest number of atoms– Has angles closest to 90 degrees– Exhibits the most symmetry
Try with different plane groups in class
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 7
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Choosing Unit Cells Example
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 8
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Choosing Unit Cells
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 9
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Choosing Unit Cells
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 10
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Choosing Unit Cells
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 11
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Choosing Unit Cells
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 12
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Miller Indices
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 13
t1
t2
Directions written as [hk]Multiples of t1 and t2
![Page 14: Lecture 2 Symmetry and Structure - MIT OpenCourseWare · PDF fileStructure and Symmetry. 22.14 –Intro to Nuclear Materials. February 5, 2015. ... Symmetry and Structure, Slide 33.](https://reader031.fdocuments.in/reader031/viewer/2022030403/5a797f127f8b9a6c158c71e1/html5/thumbnails/14.jpg)
Miller Indices
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 14
t1
t2
Can you name these crystal directions?
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Symmetry Operators in 2D
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 15
Rotational
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Symmetry Operators in 2D
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 16
Mirror
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Symmetry Operators in 2D
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 17
Glide
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Symmetry Operators in 2D
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 18
Mirror
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Symmetry Operators in 2D
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 19
Glide
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Square Lattice Symmetry
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 20
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Moving to 3D
Four new symmetry operators– Inversion– Rotoinversion (rotation & inversion)– Rotoreflection (rotation & reflection)– Screw axes (rotation & translation)
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 21
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Inversion
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 22
𝒉𝒉′𝒌𝒌′𝒍𝒍′
=−𝟏𝟏 𝟎𝟎 𝟎𝟎𝟎𝟎 −𝟏𝟏 𝟎𝟎𝟎𝟎 𝟎𝟎 −𝟏𝟏
𝒉𝒉𝒌𝒌𝒍𝒍
=−𝒉𝒉−𝒌𝒌−𝒍𝒍
Transformation matrix
Old coordinates
New coordinates
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Rotoreflection & Rotoinversion
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 23
What are the transformation
matrices?© John Wiley & Sons. All rights reserved. This content is excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
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Screw Axes
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 24
Rotation followed by translation
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Screw Axes
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 25
Rotation followed by translation
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Generalized Rotation Matrix
Or more concisely:
Where (ux, uy, uz) is a unit vector
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 26
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Miller Indices in 3D
Directions – [hkl]Families of directions – <hkl>Planes – (hkl)Families of planes – {hkl}
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 27
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Explore Some Examples
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 28
Done in class, using Crystalmaker
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Miller Indices – Lattice Parameter
Here, a=b =c– Not always the case
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 29
c
a
b
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Miller Indices – Directions
Drawing directions inside unit cell:
– [121]– 𝟎𝟎𝟏𝟏�𝟏𝟏 (1 means
negative– [331]
• Divide so largest index = 1 to get intercepts
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 30
c
b
a (1, 1, 1/3)
(1, 2, 1)
(½, 1, ½)
Origin
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Miller Indices – Direction Examples
Draw the following directions:– [001]– 00�1– [250]– 1�11– [441]– [632]– [633]
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 31
c
b
a
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Miller Indices – Planes
Example:– (234)
• Take reciprocals of indices (½, 1/3, ¼)
• Multiply so largest index is one (1, 2/3, ½)
• These are the plane intercepts on lattice axes
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 32
Origin
c
b
a
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Miller Indices – Directions and Planes
Example:– (234)– [234]
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 33
Origin
c
b
a
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Miller Indices – Plane Examples
Draw the following planes:– (001)– (001)– (251)– (111)– (441)– (632)– (633)
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 34
c
b
a
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Families of Directions & Planes
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 35
Family of [111] directions
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Miller Indices – Directions and Planes
In a cubic lattice directions are normal to planes. Example:
– (234)– [234]
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 36
Origin
c
b
a
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Miller Indices – Angle Between Planes in a Cubic Lattice
Example:– (234)– (110)
• 97.55 degrees
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 37
a = b = c = 3Å
c
b
a
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Miller Indices – Angle Between Planes in a Non-Cubic Lattice
Multiply vectors by lattice constants
Example:– (234)– (110)
• 108.44 degrees
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 38
a = c = 3Å, b = 5Å
c
b
a
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Miller Indices – Directions Common to PlanesDirection [uvw] common to planes (h1k1l1) and h2k2l2):
Check the Weiss Zone Law:
Example:– (234) and (110)
• [4,4,5]
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 39
a = b = c = 3Å
c
b
a
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Bravais Lattices
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 40
1) Characterize these systems in terms of a, b, c, and angles
2) Why is body-centered monoclinic equivalent to base-centered monoclinic?
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Packing Fraction
This slide intentionally left blank...
done in class!
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 41
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Space Groups
Unique combinations of symmetry, denoted by certain symbols
Find them in:The Int’l Tables for Crystallography
http://it.iucr.org/Or for free at the University College of London:
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 42
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Example: Triclinic (P1)
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 43
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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Example: Triclinic (P�𝟏𝟏)
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 44
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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Example Space Groups
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 45
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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P63/mmc
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 46
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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Example Space Groups
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 47
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 48
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Explore Some Examples
22.14 - Intro to Nuclear Materials Symmetry and Structure, Slide 49
Done in class, using Crystalmaker
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22.14 Materials in Nuclear EngineeringSpring 2015
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