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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
http://folk.uio.no/ravi/CMP2013
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
Structure & Symmetry in Solids
1
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystals
Atoms that are bound together, do so in a way that
minimizes their energy.
This most often leads to a periodic arrangement of the
atoms in space.
If the arrangement is purely periodic we say that it is
crystalline.
2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
3
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Write the prefix that represents each number below:
1
2
3
4
5
mono
di
tri
tetra
penta
6
7
8
9
10
hexa
hepta
octa
nona
deca
4
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Symmetry Operations
Translational
Reflection at a plane
Rotation about an axis
– Inversion through a point
Glide (=reflection + translation)
Screw (=rotation + translation)
5
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Rotations
6
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Mirror Symmetry
7
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Inversion
8
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
2D - Point Groups
9
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
3D Crystal Lattice 10
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
3D Lattice
7 crystal systems
14 Bravais lattices
230 Space groups
11
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
32 point symmetries
– 2 triclinic
– 3 monoclinic
– 3 orthorhombic
– 7 tetragonal
– 5 cubic
– 5 trigonal
– 7 hexagonal
Point groups & Space Groups
12
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
13
Each of the unit cells of the 14 Bravais lattices has one or more types of symmetry properties, such as inversion, reflection or rotation,etc.
SYMMETRY
INVERSION REFLECTION ROTATION
Elements Of Symmetry
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
14
Lattice goes into itself through
Symmetry without translation
Operation Element
Inversion Point
Reflection Plane
Rotation Axis
Rotoinversion Axes
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
15
Inversion Center
A center of symmetry: A point at the center of the molecule.
(x,y,z) --> (-x,-y,-z)
Center of inversion can only be in a molecule. It is not necessary to have an atom in the center (benzene, ethane). Tetrahedral, triangles, pentagons don't have a center of inversion symmetry.
Mo(CO)6
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
16
Reflection Plane
A plane in a cell such that, when a mirror reflection in
this plane is performed, the cell remains invariant.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
17
Examples
Triclinic has no reflection plane.
Monoclinic has one plane midway between and parallel to the bases, and so forth.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
18
We can not find a lattice that goes into itself
under other rotations
• A single molecule can have any degree of rotational
symmetry, but an infinite periodic lattice – can not.
Rotation Symmetry
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
19
Rotation Axis
This is an axis such that, if the cell is rotated around it through some angles, the cell remains invariant.
The axis is called n-fold if the angle of rotation is 2π/n.
90°
120° 180°
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Axis of Rotation 20
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
21
Axis of Rotation
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
22
Can not be combined with translational periodicity!
5-fold symmetry
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
23
5-fold symmetry
Kepler wondered why snowflakes have 6 corners, never 5 or 7.By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.
Empty space not
allowed
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
24
90°
Examples
Triclinic has no axis of rotation.
Monoclinic has 2-fold axis (θ= 2π/2 =π) normal to the
base.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Points, Directions, and
Planes
It is necessary to specify a particular
point/location/atom/direction/plane in a unit cell
We need some labeling convention. Simplest way is to
use a 3-D system, where every location can be
expressed using three numbers or indices.
– a, b, c and α, β, γ
x
y
z
β α
γ
25
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
(for now, focus on metals)
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?
1
Why do we care about crystal structures,
directions, planes ?
Physical properties of materials depend on the geometry of crystals
26
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Points, Directions, and Planes
Crystallographic direction is a vector [uvw]
– Always passes thru origin 000
– Measured in terms of unit cell dimensions a, b, and c
– Smallest integer values
Planes with Miller Indices (hkl)
– If plane passes thru origin, translate
– Length of each planar intercept in terms of the lattice parameters a, b, and c.
– Reciprocals are taken
– If needed multiply by a common factor for integer representation
27
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller indices - A shorthand notation to describe certain
crystallographic directions and planes in a material.
Denoted by [ ], <>, ( ) brackets. A negative number is
represented by a bar over the number.
Points, Directions and Planes in the Unit
Cell
28
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
• Coordinates of selected points in the unit cell.
• The number refers to the distance from the origin in terms of
lattice parameters.
Point Coordinates 29
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
30
Point Coordinates – Atom position
Point coordinates for unit cell
center are
a/2, b/2, c/2 ½ ½ ½
Point coordinates for unit cell corner are 111
Translation: integer multiple of lattice constants identical position in another unit cell
z
x
y
a b
c
000
111
y
z
2c
b
b
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Directions
1. Vector repositioned (if necessary) to
pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvw]
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1
families of directions <uvw>
z
x
Algorithm
where overbar represents a negative index [ 111 ] =>
y
31
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
32
Crystal Directions
Fig. Shows
[111] direction
We choose one lattice point on the line as an origin, say the point O. Choice of origin is completely arbitrary, since every lattice point is identical.
Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as;
R = n1 a + n2 b + n3c
To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets [ ...] is used.[n1n2n3]
[n1n2n3] is the smallest integer of the same relative ratios.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
For some crystal structures, several nonparallel
directions with different indices are
crystallographically equivalent; this means that
atom spacing along each direction is the same.
33
Families of Directions <uvw>
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal Directions
]100[],010[],001[],001[],010[],100[ 100
34
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller Index
Step 1 : Identify the intercepts on the x- , y- and z- axes.
Step 2 : Specify the intercepts in fractional co-ordinates
Step 3 : Take the reciprocals of the fractional intercepts
(i) in some instances the Miller indices are best multiplied or divided
through by a common number in order to simplify them by, for
example, removing a common factor. This operation of multiplication
simply generates a parallel plane which is at a different distance from
the origin of the particular unit cell being considered.
e.g. (200) is transformed to (100) by dividing through by 2 .
(ii) if any of the intercepts are at negative values on the axes then the
negative sign will carry through into the Miller indices; in such cases
the negative sign is actually denoted by overstriking the relevant
number.
e.g. (00 -1) is instead denoted by
100
35
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller index is used to describe directions and planes in a
crystal.
Directions - written as [u v w] where u, v, w.
Integers u, v, w represent coordinates of the vector in
real space.
A family of directions which are equivalent due to
symmetry operations is written as <u v w>
Planes: Written as (h k l).
Integers h, k, and l represent the intercept of the plane
with x-, y-, and z- axes, respectively.
Equivalent planes represented by {h k l}.
Miller Index For Cubic Structures 36
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
x y z
[1] Draw a vector and take components 0 2a 2a
[2] Reduce to simplest integers 0 1 1
[3] Enclose the number in square brackets [0 1 1]
z
y
x
Miller Indices: Directions 37
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
z
y
x
x y z
[1] Draw a vector and take components 0 -a 2a
[2] Reduce to simplest integers 0 -1 2
[3] Enclose the number in square brackets 210
Negative Directions 38
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller Indices: Equivalent Directions
z
y
x
1
2
3
1: [100]
2: [010]
3: [001]
Equivalent directions due to crystal symmetry:
Notation <100> used to denote all directions equivalent to [100]
39
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Directions
40
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
41
210
X = 1 , Y = ½ , Z = 0
[1 ½ 0] [2 1 0]
X = ½ , Y = ½ , Z = 1
[½ ½ 1] [1 1 2]
Examples
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
42
Negative directions
When we write the direction
[n1n2n3] depend on the origin,
negative directions can be
written as
R = n1 a + n2 b + n3c
Direction must be
smallest integers.
Y direction
(origin) O
- Y direction
X direction
- X direction
Z direction
- Z direction
][ 321 nnn
][ 321 nnn
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Directions of the form <110> in cubic
systems
43
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
44
X = -1 , Y = -1 , Z = 0 [110]
Examples of crystal directions
X = 1 , Y = 0 , Z = 0 [1 0 0]
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
45
Examples
X =-1 , Y = 1 , Z = -1/6
[-1 1 -1/6] [6 6 1]
We can move vector to the origin.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Directions
_
[1 2 1]
_
[1 0 1]
46
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
(A) Determining crystal structure
Diffraction methods directly measure the distance between parallel planes of lattice
points. This information is used to determine the lattice parameters in a crystal
and measure the angles between lattice planes.
(B) Plastic deformation
Plastic (permanent) deformation in metals occurs by the slip of atoms past each
other in the crystal. This slip tends to occur preferentially along specific lattice
planes in the crystal. Which planes slip depends on the crystal structure of the
material.
(C) Transport Properties
In certain materials, the atomic structure in certain planes causes the transport of
electrons and/or heat to be particularly rapid in that plane, and relatively slow away
from the plane.
Example: Graphite
Conduction of heat is more rapid in the sp2 covalently bonded lattice planes than in
the direction perpendicular to those planes.
Example: YBa2Cu3O7 superconductors
Some lattice planes contain only Cu and O. These planes conduct pairs of
electrons (called Cooper pairs) that are responsible for superconductivity. These
superconductors are electrically insulating in directions perpendicular to the Cu-O
lattice planes.
Why are planes in a lattice important? 47
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Atomic planes in BCC
{001} {011} {111}
48
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Atomic planes in FCC
{001} {011} {111}
49
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
(GPa)
Anisotropy 50
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
51
Crystal Planes
Within a crystal lattice it is possible to identify sets of
equally spaced parallel planes. These are called lattice
planes.
In the figure density of lattice points on each plane of a set
is the same and all lattice points are contained on each set
of planes.
b
a
b
a
The set of
planes in
2D lattice.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
52
Miller Indices of plane
Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
To determine Miller indices of a plane, take the following steps;
1) Determine the intercepts of the plane along each of the three crystallographic directions
2) Take the reciprocals of the intercepts
3) If fractions result, multiply each by the denominator of the smallest fraction
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
The intercepts of a crystal plane with the axis defined by a set of
unit vectors are at 2a, -3b and 4c. Find the corresponding Miller
indices of this and all other crystal planes parallel to this plane.
The Miller indices are obtained in the following three steps:
1. Identify the intersections with the axis, namely 2, -3 and
4.
2. Calculate the inverse of each of those intercepts,
resulting in 1/2, -1/3 and 1/4.
3. Find the smallest integers proportional to the inverse of
the intercepts. Multiplying each fraction with the
product of each of the intercepts (24 = 2 x 3 x 4) does
result in integers, but not always the smallest integers.
4. These are obtained in this case by multiplying each
fraction by 12.
5. Resulting Miller indices is
6. Negative index indicated by a bar on top. 346
53
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller Indices of plane
Reciprocal numbers are: 2
1 ,
2
1 ,
3
1
Plane intercepts axes at cba 2 ,2 ,3
Indices of the plane (Miller): (2,3,3)
(100)
(200)
(110) (111)
(100)
Indices of the direction: [2,3,3] a
3
2
2
bc
[2,3,3]
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
55
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
56
Family of Planes
Planes that are crystallographically equivalent have the
same atomic packing.
Also, in cubic systems only, planes having the same indices,
regardless of order and sign, are equivalent.
Ex: {111}
= (111), (111), (111), (111), (111), (111), (111), (111)
(001) (010), (100), (010), (001), Ex: {100} = (100),
_ _ _ _ _ _ _ _ _ _ _ _
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
57
Axis X Y Z
Intercept
points 1 ∞ ∞
Reciprocals 1/1 1/ ∞ 1/ ∞ Smallest
Ratio 1 0 0
Miller İndices (100)
Example-1
(1,0,0)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Axis X Y Z
Intercept
points 1 1 ∞
Reciprocals 1/1 1/ 1 1/ ∞ Smallest
Ratio 1 1 0
Miller İndices (110)
Example-2
(1,0,0)
(0,1,0)
58
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
59
Axis X Y Z
Intercept
points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1 Smallest
Ratio 1 1 1
Miller İndices (111) (1,0,0)
(0,1,0)
(0,0,1)
Example-3
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
60
Axis X Y Z
Intercept
points 1/2 1 ∞
Reciprocals 1/(½) 1/ 1 1/ ∞ Smallest
Ratio 2 1 0
Miller İndices (210) (1/2, 0, 0)
(0,1,0)
Example-4
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Axis a b c
Intercept
points 1 ∞ ½
Reciprocals 1/1 1/ ∞ 1/(½)
Smallest
Ratio 1 0 2
Miller İndices (102)
Example-5
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Axis a b c
Intercept
points -1 ∞ ½
Reciprocals 1/-1 1/ ∞ 1/(½)
Smallest
Ratio -1 0 2
Miller İndices (102)
Example-6
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
63
Example-7
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
64
Indices of a Family or Form
Sometimes when the unit cell has rotational symmetry,
several nonparallel planes may be equivalent by virtue of
this symmetry, in which case it is convenient to lump all
these planes in the same Miller Indices, but with curly
brackets.
Thus indices {h,k,l} represent all the planes equivalent to
the plane (hkl) through rotational symmetry.
)111(),111(),111(),111(),111(),111(),111(),111(}111{
)001(),100(),010(),001(),010(),100(}100{
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
z
y
x
z=
y=
x=a
x y z
[1] Determine intercept of plane with each axis a ∞ ∞
[2] Invert the intercept values 1/a 1/∞ 1/∞
[3] Convert to the smallest integers 1 0 0
[4] Enclose the number in round brackets (1 0 0)
Miller Indices of Planes 65
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
z
y
x
x y z
[1] Determine intercept of plane with each axis 2a 2a 2a
[2] Invert the intercept values 1/2a 1/2a 1/2a
[3] Convert to the smallest integers 1 1 1
[4] Enclose the number in round brackets (1 1 1)
Miller Indices of Planes
66
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
z
y
x
Planes with Negative Indices
x y z
[1] Determine intercept of plane with each axis a -a a
[2] Invert the intercept values 1/a -1/a 1/a
[3] Convert to the smallest integers 1 -1 -1
[4] Enclose the number in round brackets 111
67
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
• Planes (100), (010), (001), (100), (010), (001) are
equivalent planes. Denoted by {1 0 0}.
• Atomic density and arrangement as well as electrical,
optical, physical properties are also equivalent.
z
y
x
(100)
plane
(010)
plane
(001) plane Equivalent Planes
68
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
69
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
(0 1 1)
70
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
The (110) surface
Assignment
Intercepts : a , a ,
Fractional intercepts : 1 , 1 ,
Miller Indices : (110)
The (100), (110) and (111) surfaces considered above are
the so-called low index surfaces of a cubic crystal system
(the "low" refers to the Miller indices being small numbers -
0 or 1 in this case).
71
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller Indices (hkl)
reciprocals
Crystallographic Planes 72
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Planes
_
(1 1 1)
73
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
In the cubic system the (hkl) plane and the vector
[hkl], defined in the normal fashion with respect
to the origin, are normal to one another but this
characteristic is unique to the cubic crystal
system and does not apply to crystal systems of
lower symmetry
74
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Symmetry-equivalent surfaces
The three highlighted surfaces are
related by the symmetry elements of
the cubic crystal - they are entirely
equivalent.
In fact there are a total of 6 faces related by the symmetry
elements and equivalent to the (100) surface - any surface
belonging to this set of symmetry related surfaces may be
denoted by the more general notation {100} where the Miller
indices of one of the surfaces is instead enclosed in curly-
brackets.
75
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
In the hcp crystal system there are four
principal axes; this leads to four Miller Indices
e.g. you may see articles referring to an hcp
(0001) surface. It is worth noting, however, that
the intercepts on the first three axes are
necessarily related and not completely
independent; consequently the values of the first
three Miller indices are also linked by a simple
mathematical relationship.
76
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
In case of a cubic structure, the Miller index of a plane, in
parentheses such as (100), are also the coordinates of the direction
of a plane normal. It stands for a vector perpendicular to the family
of planes, with a length of d-1, where d is the inter-plane spacing.
Due to the symmetries of cubic crystals, it is possible to change the
place and sign of the integers and have equivalent directions and
planes:
•Coordinates in angle brackets or chevrons such as <100>
denote a family of directions which are equivalent due to
symmetry operations. If it refers to a cubic system, this
example could mean [100], [010], [001] or the negative of any
of those directions.
•Coordinates in curly brackets or braces such as {100} denote
a family of plane normals which are equivalent due to
symmetry operations, much the way angle brackets denote a
family of directions.
77
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
With hexagonal and rhombohedral crystal systems, it is possible
to use the Bravais-Miller index which has 4 numbers (h k i l)
i = -h-k
where h, k and l are identical to the Miller index.
The (100) plane has a 3-fold symmetry, it remains unchanged by a
rotation of 1/3 (2π/3 rad, 30°). The [100], [010] and
the directions are similar. If S is the intercept of the plane
with the axis, then
i = 1/S
i is redundant and not necessary.
78
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
in the hcp crystal system there are four principal axes; this leads
to four Miller Indices e.g. you may see articles referring to an
hcp (0001) surface. It is worth noting, however, that the
intercepts on the first three axes are necessarily related and not
completely independent; consequently the values of the first
three Miller indices are also linked by a simple mathematical
relationship.
79
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Distance between two planes (d spacing) 80
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
In the cubic crystal system, a plane and the direction normal to
it have the same indices.
[101] direction is normal to the plane (101)
Distance (d) separating adjacent planes
(hkl) of a cubic crystal of lattice constant
(a) is:
222 lkh
ad
Angle () between
directions [h1 k1 l1]
and [h2 k2 l2] of a
cubic crystal is:
))(()cos(
2
2
2
2
2
2
2
1
2
1
2
1
212121
lkhlkh
llkkhh
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
d spacing
2
2
2
2
2
2
c
l
b
k
a
h
ndhkl
Example
The lattice constant for aluminum is 4.041
angstroms. What is d220?
Answer
Aluminum has an fcc structure, so a = b = 4.041 Å
angstroms 43.1
041.4
22
11
2
22
2
22
a
kh
dhkl
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Miller-Bravais Indices
• For hcp crystal structure
• Planes (h k i l), directions [h k i l]
• Sum of the first three indices h + k + i = 0
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
To determine the Miller-Bravais indices of
the crystallographic direction indicated on
the a1-a2-a3 plane of an h.c.p. unit cell (ie.
c = 0), the vector must be projected onto
each of the three axes to find the
corresponding component, such that:
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
ex: linear density of Al in [110]
direction
a = 0.405 nm
Linear Density
Linear Density of Atoms LD =
a
[110]
Unit length of direction vector
Number of atoms
# atoms
length
1 3.5 nm a 2
2 LD - = =
86
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Planes
We want to examine the atomic packing of
crystallographic planes
Iron foil can be used as a catalyst. The atomic
packing of the exposed planes is important.
a) Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these
planes.
87
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
88
Planar Density of (100) Iron
Solution: At T < 912ºC iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
R 3
3 4 a
2D repeat unit
= Planar Density =
a 2
1
atoms
2D repeat unit
= nm2
atoms 12.1
m2
atoms = 1.2 x 1019
1
2
R 3
3 4 area
2D repeat unit
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
89
Planar Density of (111) Iron
Solution (cont): (111) plane 1 atom in plane/ unit surface cell
3 3 3
2
2
R 3
16 R
3
4
2 a 3 ah 2 area
atoms in plane
atoms above plane
atoms below plane
a h 2
3
a 2
1 = =
nm2
atoms 7.0
m2
atoms 0.70 x 1019
3 2 R
3
16
Planar Density =
atoms
2D repeat unit
area
2D repeat unit
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
90
HCP Crystallographic Directions
Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are
related to the direction indices (i.e., u'v'w') as
follows.
' w w
t
v
u
) v u ( + -
) ' u ' v 2 ( 3
1 -
) ' v ' u 2 ( 3
1 -
] uvtw [ ] ' w ' v ' u [
-
a3
a1
a2
z
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
91
HCP Crystallographic Directions
1. Vector repositioned (if necessary) to
pass through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no
commas [uvtw]
[ 1120 ] ex: ½, ½, -1, 0 =>
dashed red lines indicate
projections onto a1 and a2 axes a1
a2
a3
-a3 2 a 2
2 a 1
- a3
a1
a2
z Algorithm
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Procedure for finding Miller indices for hcp lattice in four-index notation:
1. Find the intersections, r and s, of the plane with any two of the basal plane axes.
2. Find the intersection t of the plane with the c axis.
3. Evaluate the reciprocals 1/r, 1/s, and 1/t.
4. Convert the reciprocals to smallest set of integers that are in the same ratio.
5. Use the relation i = -(h+ k), where h is associated with a1, k is associated with a2,
and i is associated with a3.
6. Enclose all four indices in parentheses: (h k i l)
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Example: What is the designation, using four
Miller indices, of the lattice plane shaded pink
in the figure?
• The plane intercepts the a1, a3, and c axes at r =
1, u = 1, and t = ∞, respectively.
• The reciprocals are
•1/r = 1
•1/u = 1
•1/t = 0
• These are already in integer form.
• We can write down the Miller indices as (h k i
l) = (1 k 1 0), where the k index is undetermined
so far.
• Use i = (h + k). That is, k = 2.
• The designation of this plane is (h k i l) =
)1021(
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
HCP and FCC structures
The hexagonal-closed packed (HCP) and FCC structures both have the ideal packing fraction of 0.74 (Kepler figured this out hundreds of years ago)
The ideal ratio of c/a for hcp packing is (8/3)1/2 = 1.633
Crystal c/a
He 1.633
Be 1.581
Mg 1.623
Ti 1.586
Zn 1.861
Cd 1.886
Co 1.622
Y 1.570
Zr 1.594
Gd 1.592
Lu 1.586
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Determine the Miller indices of directions A, B, and C.
Miller Indices, Directions 95
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
SOLUTION
Direction A
1. Two points are 1, 0, 0, and 0, 0, 0
2. 1, 0, 0, -0, 0, 0 = 1, 0, 0
3. No fractions to clear or integers to reduce
4. [100]
Direction B
1. Two points are 1, 1, 1 and 0, 0, 0
2. 1, 1, 1, -0, 0, 0 = 1, 1, 1
3. No fractions to clear or integers to reduce
4. [111]
Direction C
1. Two points are 0, 0, 1 and 1/2, 1, 0
2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1
3. 2(-1/2, -1, 1) = -1, -2, 2
2]21[ .4
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
97
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
98
Crystallographic Directions
1. Vector repositioned (if necessary) to
pass through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no
commas [uvw]
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1
z
x
Algorithm
where overbar represents a
negative index
[ 111 ] =>
y
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
If the plane passes thru origin, either:
– Construct another plane, or
– Create a new origin
– Then, for each axis, decide whether plane intersects or
parallels the axis.
Algorithm for Miller indices 1. Read off intercepts of plane with axes in terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no commas.
99
Crystallographic Planes
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
100
Crystallographic Planes
• Crystallographic planes are specified by 3 Miller Indices (h k l). All parallel planes have same Miller indices.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
FCC Unit Cell with (110) plane
101
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
BCC Unit Cell with (110) plane
102
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Planes
Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
Algorithm 1. If plane passes thru origin, translate 2. Read off intercepts of plane with axes in terms of a, b, c 3. Take reciprocals of intercepts 4. Reduce to smallest integer values 5. Enclose in parentheses, no commas i.e., (hkl)
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Planes
z
x
y a b
c
4. Miller Indices (110)
example a b c z
x
y a b
c
4. Miller Indices (100)
1. Intercepts 1 1
2. Reciprocals 1/1 1/1 1/ 1 1 0
3. Reduction 1 1 0
1. Intercepts 1/2
2. Reciprocals 1/½ 1/ 1/ 2 0 0
3. Reduction 2 0 0
example a b c
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystallographic Planes
z
x
y a b
c
4. Miller Indices (634)
example 1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4
(001) (010),
Family of Planes {hkl}
(100), (010), (001), Ex: {100} = (100),
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal = Lattice + Basis
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Unit Cell (2D)
10
7
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Face-centered Cubic Lattice
In the Face Centered Cubic
(FCC) unit cell there is one basis
atom at each corner and one
basis atom in each face.
There are 4 atoms per unit cell:
(1/8)8 + (1/2)6 = 4.
The simplest primitive cell is
actually rhombohedral. Its
volume is ¼ that of the cubic, as
we might expect.
108
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Primitive Cell of FCC Lattice
1
2
3
2
2
2
ˆ ˆa (x y)
ˆ ˆa (y z)
ˆ ˆa (z x)
a
a
a
The three interior angles formed between
unit cell edges are called:
a alpha, between edges a2 & a3
b (beta, between edges a1 & a3)
g (gamma, between edges a1 & a2)
In the FCC rhombohedral standard
reduced cell, it can be shown that a = b =
g = 60. Note that a cube is a just a
special rhombohedron, with a = b = g =
90.
0 1 2 3 a a a
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal Planes 110
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal Planes: Miller Indices 111
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal Planes 112
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystal Planes
113
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
FCC crystal structure
BCC crystal structure
HCP crystal structure
all edges = all angles =
11
4
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Types of Models to Describe Crystal Structures
Hard Sphere Model (a):
- “big atoms”
- Atoms are like spheres
- Atoms touch neighbor
- Radius of hard sphere = atomic radius
- Atomic radius: distance between two
nuclei of two touching atoms
r = d/2 d
Reduced Sphere Model (b)
-Center of atoms represented as
small circles
Aggregate of many atoms (c)
3 Predominate Crystal Structures for Metals: FCC, BCC, and HCP…
11
5
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Hexagonal close-packed crystal structure
(HCP)
Traditional HCP crystal structure
Expanded HCP crystal structure
G, H, E, F, J, A, B, C, D (only)
All atoms
11
6
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Crystalline Lattice
Repeating pattern of particles
Lattice point
117
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Unit cell: the smallest repeating unit in the pattern
in 3-D.
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
Equivalency of crystallographic directions of a form in
cubic systems.
11
9
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
12
0
Crystallographic Directions
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvw]
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1
Families of directions <uvw>
z
x
Algorithm
where overbar represents a negative index [ 111 ] =>
y
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
121
Equivalent crystallographic Planes
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
122
Crystallographic Planes
Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
123
Crystallographic Planes
z
x
y a b
c
4. Miller Indices (110)
example a b c z
x
y a b
c
4. Miller Indices (200)
1. Intercepts 1 1
2. Reciprocals 1/1 1/1 1/ 1 1 0
3. Reduction 1 1 0
1. Intercepts 1/2
2. Reciprocals 1/½ 1/ 1/ 2 0 0
3. Reduction 2 0 0
example a b c
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
124
Crystallographic Planes
z
x
y a b
c
4. Miller Indices (634)
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4
(001) (010),
Family of Planes {hkl}
(100), (010), (001), Ex: {100} = (100),
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
125
Crystallographic Planes (HCP)
In hexagonal unit cells the same idea is used
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1 -1 1 2. Reciprocals 1 1/
1 0 -1 -1
1 1
3. Reduction 1 0 -1 1
a2
a3
a1
z
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Structure & Symmetry in Solids
126
Crystallographic Planes
We want to examine the atomic packing of
crystallographic planes
Iron foil can be used as a catalyst. The atomic packing of
the exposed planes is important.
a) Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these planes.