Crystallographic Points, Directions, and Planes'

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Crystallographic Points, Directions, and Planes.
ISSUES TO ADDRESS...
How to define points, directions, planes, as
well as linear, planar, and volume densities

• Define basic terms and give examples of each
• Points (atomic positions)
• Vectors (defines a particular direction - plane
  normal)
• Miller Indices (defines a particular plane)
• relation to diffraction
• 3-index for cubic and 4-index notation for HCP

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Points, Directions, and Planes in Terms of Unit
Cell Vectors

• All periodic unit cells may be described via
  these vectors and angles, if and only if
• a, b, and c define axes of a 3D coordinate
  system.
• coordinate system is Right-Handed!
• But, we can define points, directions and planes
  with a triplet of numbers in units of a, b, and
  c unit cell vectors.
• For HCP we need a quad of numbers, as we shall
  see.

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POINT Coordinates
To define a point within a unit cell. Express
the coordinates uvw as fractions of unit cell
vectors a, b, and c (so that the axes x, y, and
z do not have to be orthogonal).
pt. coord.
x (a) y (b) z (c)
0 0 0
1 0 0
1 1 1
origin
1/2 0 1/2
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Crystallographic Directions

• Procedure
• Any line (or vector direction) is specified by 2
  points.
• The first point is, typically, at the origin
  (000).
• Determine length of vector projection in each of
  3 axes in units (or fractions) of a, b, and c.
• X (a), Y(b), Z(c)
• 1 1 0
• Multiply or divide by a common factor to reduce
  the lengths to the smallest integer values, u v
  w.
• Enclose in square brackets u v w 110
  direction.

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Self-Assessment Example 1 What is
crystallographic direction?
Magnitude along X Y Z
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Self-Assessment Example 2
(a) What is the lattice point given by point P?
(b) What is crystallographic direction for the
origin to P?
Example 3 What lattice direction does the
lattice point 264 correspond?
The lattice direction 132 from the origin.
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Symmetry Equivalent Directions
Families of crystallographic directions e.g. lt1
0 0gt
Angled brackets denote a family of
crystallographic directions.
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Families and Symmetry Cubic Symmetry
(010)
Rotate 90o about z-axis
(100)
(001)
Rotate 90o about y-axis
Symmetry operation can generate all the
directions within in a family.
Similarly for other equivalent directions
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Designating Lattice Planes
Why are planes in a lattice important?

• (A) Determining crystal structure
• Diffraction methods measure the distance
  between parallel lattice planes of atoms.
• This information is used to determine the
  lattice parameters in a crystal.
• Diffraction methods also measure the angles
  between lattice planes.
• (B) Plastic deformation
• Plastic deformation in metals occurs by the
  slip of atoms past each other in the crystal.
• This slip tends to occur preferentially
  along specific crystal-dependent planes.
• (C) Transport Properties
• In certain materials, atomic structure in
  some planes causes the transport of electrons
• and/or heat to be particularly rapid in
  that plane, and relatively slow not in the plane.
• Example Graphite heat conduction is more in
  sp2-bonded plane.
• Example YBa2Cu3O7 superconductors Cu-O planes
  conduct pairs of electrons (Cooper pairs)
  responsible for superconductivity, but
  perpendicular insulating.
• Some lattice planes contain
  only Cu and O

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How Do We Designate Lattice Planes?
Example 1

• Planes intersects axes at
• a axis at r 2
• b axis at s 4/3
• c axis at t 1/2

How do we symbolically designate planes in a
lattice?
Possibility 1 Enclose the values of r, s, and
t in parentheses (r s t)

• Advantages
• r, s, and t uniquely specify the plane in the
  lattice, relative to the origin.
• Parentheses designate planes, as opposed to
  directions given by ...
• Disadvantage
• What happens if the plane is parallel to ---
  i.e. does not intersect--- one of the axes?
• Then we would say that the plane intersects that
  axis at 8 !
• This designation is unwieldy and inconvenient.

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How Do We Designate Lattice Planes?

• Planes intersects axes at
• a axis at r 2
• b axis at s 4/3
• c axis at t 1/2

How do we symbolically designate planes in a
lattice?
Possibility 2 THE ACCEPTED ONE

• Take the reciprocal of r, s, and t.
• Here 1/r 1/2 , 1/s 3/4 , and 1/r 2
• Find the least common multiple that converts all
  reciprocals to integers.
• With LCM 4, h 4/r 2 , k 4/s 3 , and
  l 4/r 8
• Enclose the new triple (h,k,l) in parentheses
  (238)
• This notation is called the Miller Index.
• Note If a plane does not intercept an axes
  (I.e., it is at 8), then you get 0.
• Note All parallel planes at similar staggered
  distances have the same Miller index.

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Self-Assessment Example
What is the designation of this plane in Miller
Index notation?
What is the designation of the top face of the
unit cell in Miller Index notation?
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Families of Lattice Planes

• Given any plane in a lattice, there is a infinite
  set of parallel lattice planes (or family of
  planes) that are equally spaced from each other.
• One of the planes in any family always passes
  through the origin.

• The Miller indices (hkl) usually refer to the
  plane that is nearest to the origin without
  passing through it.
• You must always shift the origin or move the
  plane parallel, otherwise a Miller index integer
  is 1/0, i.e.,8!
• Sometimes (hkl) will be used to refer to any
  other plane in the family, or to the family taken
  together.
• Importantly, the Miller indices (hkl) is the same
  vector as the plane normal!

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Crystallographic Planes in FCC (100)
z
y
Look down this direction (perpendicular to the
plane)
x
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Crystallographic Planes in FCC (110)
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Crystallographic Planes in FCC (111)
z
Look down this direction (perpendicular to the
plane)
y
x
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Note similar to crystallographic directions,
planes that are parallel to each other, are
equivalent
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Comparing Different Crystallographic Planes
For (220) Miller Indexed planes you are getting
planes at 1/2, 1/2, 8. The (110) planes are not
necessarily (220) planes!
Miller Indices provide you easy measure of
distance between planes.
For cubic crystals
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Directions in HCP Crystals

• To emphasize that they are equal, a and b is
  changed to a1 and a2.
• The unit cell is outlined in blue.
• A fourth axis is introduced (a3) to show
  symmetry.
• Symmetry about c axis makes a3 equivalent to a1
  and a2.
• Vector addition gives a3 ( a1 a2).
• This 4-coordinate system is used a1 a2 ( a1
  a2) c

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Directions in HCP Crystals 4-index notation
What is 4-index notation for vector D?
Example

• Projecting the vector onto the basal plane, it
  lies between a1 and a2 (vector B is projection).
• Vector B (a1 a2), so the direction is 110
  in coordinates of a1 a2 c, where c-intercept
  is 0.
• In 4-index notation, because a3 ( a1 a2),
  the vector B is since it is 3x farther
  out.
• In 4-index notation c 0001, which must be
  added to get D (reduced to integers) D

Check w/ Eq. 3.7 or just use Eq. 3.7
Easiest to remember Find the coordinate axes
that straddle the vector of interest, and follow
along those axes (but divide the a1, a2, a3 part
of vector by 3 because you are now three times
farther out!).
Self-Assessment Test What is vector C?
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Directions in HCP Crystals 4-index notation
Example
Check w/ Eq. 3.7 a dot-product projection in
hex coords.
What is 4-index notation for vector D?

• Projection of the vector D in units of a1 a2
  c gives u1, v1, and w1. Already reduced
  integers.
• Using Eq. 3.7

• In 4-index notation
• Reduce to smallest integers

After some consideration, seems just using Eq.
3.7 most trustworthy.
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Miller Indices for HCP Planes
4-index notation is more important for planes in
HCP, in order to distinguish similar planes
rotated by 120o.
As soon as you see 1100, you will know that it
is HCP, and not 110 cubic!
t
Find Miller Indices for HCP

• Find the intercepts, r and s, of the plane with
  any two of the basal plane axes (a1, a2, or a3),
  as well as the intercept, t, with the c axes.
• Get reciprocals 1/r, 1/s, and 1/t.
• Convert reciprocals to smallest integers in same
  ratios.
• Get h, k, i , l via relation i - (hk), where h
  is associated with a1, k with a2, i with a3, and
  l with c.
• Enclose 4-indices in parenthesis (h k i l) .

r
s
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Miller Indices for HCP Planes
What is the Miller Index of the pink plane?
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Yes, Yes.we can get it without a3!

• But note that the 4-index notation is
  unique.Consider all 4 intercepts
• plane intercept a1, a2, a3 and c at 1, 1/2, 1,
  and 8, respectively.
• Reciprocals are 1, 2, 1, and 0.
• So, there is only 1 possible Miller Index is

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Basal Plane in HCP
Name this plane

• Parallel to a1, a2 and a3
• So, h k i 0

• Intersects at z 1

a2
a3
a1
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Another Plane in HCP
z
a2
1 in a1
a3
a1
h 1,
l 0
i -(1-1) 0,
k -1,
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(1 1 1) plane of FCC
(0 0 0 1) plane of HCP
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SUMMARY

• Crystal Structure can be defined by space
  lattice and basis atoms (lattice decorations or
  motifs).
• Only 14 Bravais Lattices are possible. We focus
  only on FCC, HCP, and BCC, i.e., the majority in
  the periodic table.
• We now can identify and determined atomic
  positions, atomic planes (Miller Indices),
  packing along directions (LD) and in planes (PD).
• We now know how to determine structure
  mathematically.
• So how to we do it experimentally?
  DIFFRACTION.