Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Crystal Structure
Friday, July 1, 2016
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Crystal Structure
The Lattice and the Basis
The crystal can be thought of as consisting of two separate parts:
1) the LATTICE
and
2) the BASIS.
A crystal is a regular, ordered arrangement of atoms over a large scale.
The atoms may be of a single type or the repetition of a complex arrangement
of many different types of atoms.
Friday, July 1, 2016
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
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Translational Vector in Crystal Structure
Bravais lattice, studied by Auguste Bravais (1950) is an infinite array of discrete points
generated by a set of discrete translation operations described by:
T = n1a1 + n2a2 +n3 a3
where ni are any integers and ai are known as the primitive vectors which lie in
different directions and span the lattice. This discrete set of vectors must be
closed under vector addition and subtraction. For any choice of position
vector R, the lattice looks exactly the same.
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Friday, July 1, 2016
Bravais Lattice
Non-Bravais Lattice
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Lattice vectors are the shortest distances to the nearest neighbouring points on
the lattice and are conventionally denoted by a , b and c .
The angles between these vectors are given the symbols , α, β and γ.
The bases Vector
R = n1a + n2b
Lattice vectors are a linear combination
Of the basis vectors
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Unit Cell Smallest area occupied by the cell is called unit cell.
In the same lattice one can chose any basis vector where the area will be the same.
Primitive and Non-perimitive Unit cell.
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Primitive and Non-perimitive Unit cell.
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Primitive and Non-perimitive Unit cell.
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Seven crystal systems make up fourteen Bravais lattice types in three dimensions.
P - Primitive: simple unit cell
F - Face-centred: additional point in the centre of each face
I - Body-centred: additional point in the centre of the cell
C - Centred: additional point in the centre of each end
R - Rhombohedral: Hexagonal class only
Symmetries of crystals in 3 dimensions (All Crystals can be classified by):
• 7 Crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, cubic, hexagonal, trigonal)
• 14 Bravais Lattices (primitive, face-centered or body-centered for each of the 7 systems)
• 32 Points groups (rotations, inversion, reflection)
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Miller Indices and Notation
• Miller Indices are used to identify planes of atoms within a crystal structure.
• Miller indices are written as three digits between brackets, (100) for example.
• Given the intercepts of the plane with the crystallographic axes. (directions of the lattice vectors),
joining the intercepts with lines defines a plane which cuts through the crystal.
The Miller indices are calculated by taking the reciprocal of the intercepts and multiplying them by their highest common
factor.
Negative numbers are represented by placing a bar over the top of the digit.
If the intercept is at infinity, then the plane is parallel to that axis and the Miller index is zero, since,
Miller Planes (hkl)
Hexagonal
(hkil)
Parallel planes {hkl}
Direction
[uvw]
Equivalent directions <uvw>
Direction of planes
T=ua+vb+wc
Friday, July 1, 2016
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Miller Indices and Notation
• Miller Indices are used to identify planes of atoms within a crystal structure.
• Miller indices are written as three digits between brackets, (100) for example.
• Given the intercepts of the plane with the crystallographic axes. (directions of the lattice vectors),
joining the intercepts with lines defines a plane which cuts through the crystal.
The Miller indices are calculated by taking the reciprocal of the intercepts and multiplying them by their highest common
factor.
Negative numbers are represented by placing a bar over the top of the digit.
If the intercept is at infinity, then the plane is parallel to that axis and the Miller index is zero, since,
Miller Planes
(hkl)
Hexagonal
(hkil)
Parallel planes {hkl}
Direction
[uvw]
Equivalent directions <uvw>
Direction of planes
T=ua+vb+wc
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Stepwise Finding the Miller Indices
Step 1 : Identify the intercepts on the x- , y- and z- axes.
In this case the intercept on the x-axis is at x = a ( at the point (a,0,0) ), but the
surface is parallel to the y- and z-axes - strictly therefore there is no intercept on
these two axes but we shall consider the intercept to be at infinity ( ∞ ) for the
special case where the plane is parallel to an axis. The intercepts on the x- , yand z-axes are thus
Intercepts : a , ∞ , ∞
Step 2 : Specify the intercepts in fractional co-ordinates
Co-ordinates are converted to fractional co-ordinates by dividing by the
respective cell-dimension - for example, a point (x,y,z) in a unit cell of
dimensions a x b x c has fractional co-ordinates of ( x/a , y/b , z/c ). In the case
of a cubic unit cell each co-ordinate will simply be divided by the cubic cell
constant , a . This gives
Fractional Intercepts : a/a , ∞/b, ∞/c i.e. 1 , ∞ , ∞
Step 3 : Take the reciprocals of the fractional intercepts
This final manipulation generates the Miller Indices which (by convention) should
then be specified without being separated by any commas or other symbols. The
Miller Indices are also enclosed within standard brackets (….) when one is
specifying a unique surface such as that being considered here.
The reciprocals of 1 and ∞ are 1 and 0 respectively, thus yielding
Miller Indices : (100)
So the surface/plane illustrated is the (100) plane of the cubic crystal.
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Examples:- Stepwise finding the Miller Indices:
Step 1 : intercepts on the x , y and z- axes.
Intercepts : a , b , ∞
Step 2 : fractional Intercepts
Fractional Intercepts : a/a , b/b, ∞/c i.e.
Step 3 : Reciprocals of fractional intercepts
Fractional Intercepts : a/a , b/b, ∞/c i.e.
Step 4 : Redaction to common number
Miller Indices : (110)
1,1,∞
1, 1 ,∞
Step 1 : intercepts on the x , y and z- axes.
Intercepts : 1/2a, ∞ , ∞
Step 2 : fractional Intercepts
Fractional Intercepts : a/2a , ∞ /b, ∞/c i.e.
Step 3 : Reciprocals of fractional intercepts
Fractional Intercepts : a/2a , ∞ /b, ∞/c i.e.
Step 4 : Redaction to common number
Miller Indices : (200)
1/2 , ∞ , ∞
1/2 , ∞ , ∞
Step 1 : intercepts on the x , y and z- axes.
Intercepts : 1/2a, b , 3/4c
Step 2 : fractional Intercepts
Fractional Intercepts : a/2a , b/b, 3c/4c i.e. 1/2 , 1 , 3/4
Step 3 : Reciprocals of fractional intercepts
Fractional Intercepts : 1/2 , 1 , 3/4 i.e. 2 , 1 , 4/3
Step 4 : Redaction to common number
Miller Indices : (634)
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA
Majmaah University
Department of Physics
College of Science, Al-Zulfi
KINGDOM OF SAUDI ARABIA