Crystallography Tute 1 Miller Indices - Dr Greg's ...drgregsmaterialsweb.com/s/Crystallography Tute...
Transcript of Crystallography Tute 1 Miller Indices - Dr Greg's ...drgregsmaterialsweb.com/s/Crystallography Tute...
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Crystallography – Tutorial 1 Miller Indices and Indices of Direction Question 1
Determine the Miller Indices of two of the diagonal planes and deduce what family of planes they belong to.
Question 2
What are the Miller Indices of the following 2 planes?
Question3
What are the Miller indices for this plane?
2
3
4
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Question 4
a) What are the indices of direction for the lines in the following figure? Show all your working.
Question 5
What are the Miller indices and Miller‐Bravais Indices for the planes marked A and B.
A
B
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Question 6
What are the 3 and 4 digit indices of direction for vectors A and C?
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Question 7
Look at the body centred cubic (BCC) model. Lattice points at (0,0,0) and (½,½,½)
Atoms Per Unit Cell
Number of Nearest Neighbour Atoms To Each Atom
Number of Symmetry A2
Axes & their Indices A3
of Direction A4
A6
Symmetry Planes (Miller Indices)
Centre of Symmetry
Atomic Packing Factor? (APF)
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Now look at the face centred cubic (FCC) model. Lattice points at (0,0,0), (½,½,0) and (0,½,½).
Atoms Per Unit Cell
Number of Nearest Neighbor Atoms To Each Atom
What is the close packed direction?
Draw any of these on your diagram
Number of Symmetry A2
Axes & their Indices A3
of Direction A4
A6
Symmetry Planes (Miller Indices)
Centre of Symmetry
Atomic Packing Factor? (APF)
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Now look at the hexagonal close packed (HCP) model. Draw the unit cell:
Atoms Per Unit Cell
Number of Nearest Neighbour Atoms To Each Atom
What is the close packed direction?
Draw any of these on your diagram
Number of Symmetry A2
Axes A3
A4
A6
Symmetry Planes
Centre of Symmetry
Atomic Packing Factor? (APF)
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Question 8 ‐ Close‐Packed Directions
For each of the unit cells shown draw the close packed directions and in the table below list the indices of direction of those close packed directions (here it is useful to use the “family of directions” notation of
Unit Cell Close packed directions
BCC
FCC
HCP
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Question 9 ‐ Close‐Packed Planes
Using the sectioned models, perform the following:
For each of the two unit cells (FCC and HCP) below, please show a close‐packed plane and index the planes. As there is no close‐packed plane for BCC, draw and index the densest plane. It is useful to use the concept of the “family of planes”, {hkl}
Fill in this table:
Unit Cell Closest packed planes Indices Close packed planes Indices
BCC
FCC
HCP
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Draw the actual planes from above using circles to represent the atoms, do not use a "ball and stick" representation for this question. Label each diagram. On these diagrams indicate the close packed directions.
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Calculate the planar density (in terms of area occupied by atoms as a function of total area of the plane) for each of these planes.
Plane Close packed/closest packed? Planar Density
BCC {110}
FCC {110} NO 0.555
FCC {111}
HCP {001}
Question 10 Slip Systems.
We have now worked out the close and closest packed planes in 3 crystal systems along with the close packed directions. These close and closest packed planes are the preferred planes for slip to occur, and on those planes there are specific directions along which dislocations move. Yep, you guessed it the close packed directions. Thus we can now talk about slip planes and slip directions, both of which make up slip systems.
Fill in the table below and draw the slip system on the figures provided:
Crystal Structure
Slip Plane
# of Slip Planes
Slip Direction
# of Slip Directions
# of Slip Systems
Examples
BCC ‐Fe, Mo, W
FCC Al, Cu, ‐Fe,
Ni
HCP Cd, Mg, Zn
BCC FCC
Useful equations for those rusty on their geometry
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Unit cell of edge length “a”
Side diagonal length = 22 aad
Cube diagonal length = 22 adD
Area of a triangle = bh21 bh
43
From these simple relationships, we can show (R = atomic radius)
For BCC:
34Ra Rb
324
For FCC
Ra 22 Rb 4
Planar Density Example: (110) plane in FCC
1. Draw the plane
Area of atoms = 2 atoms x R2
Area of plane in terms of R = a x 4R = 22R x 4R = 82R2
555.0314.11283.6
282
282
2
2
RRPD
i.e. 55.5% only is taken up by atoms on these planes.
a b
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Symmetry
Symmetry elements consist of:
(i) Axes of symmetry
An axis of symmetry is a line drawn through the centre of the unit cell so that rotation of less than one full turn will result in superposition of the cell on itself (ie it doesn't look as though you have done anything to the cell). The amount turned defines the axis as follows.
A 'two fold axis of symmetry', or A2 axis, means that the crystal has to be rotated by 180°, or a ½ turn, to present a like face (superimpose on itself).
A 'three fold axis of symmetry', or A3 axis, means that the crystal has to be rotated by 120°, or a 31 turn, to present a like face.
A 'four fold axis of symmetry', or A4 axis, means that the crystal has to be rotated by 90°, or a ¼ turn, to present a like face.
A 'six fold axis of symmetry', or A6 axis, means that the crystal has to be rotated by 60°, or a 61 turn, to present a like face.
(ii) Planes of symmetry
Planes of symmetry are planes which cut a crystal into mirror images. This means that any point on one side of the plane can be reflected through the plane onto a similar point on the other side of the plane.
(iii) Centre of Symmetry
A centre of symmetry is a point. For a unit cell to have a centre of symmetry, the crystal must be totally symmetric about the point at the centre of the unit cell. For example, in 2 dimensions the letter S has a centre of symmetry while the letter B does not.