The Simple Cubic Lattice
For simple cubic lattices, there is one
lattice point
per unit cell. The lattice may be described as a
three dimensional array of lattice points by either beginning with a lattice point at the position
(0,0,0), and adding additional lattice points by translating indefinitely in all three principal
directions by the lattice parameter,
a
, or alternatively, by allotting 1/8 of a lattice point at each of
the eight corners of the unit cell and stacking up unit cells in all three directions to build up the
lattice. (In reality, there is one lattice point at each cube corner, with each contributing 1/8 of
itself to each unit cell.) The physical surroundings of each lattice point in the unit cell of any
crystal lattice are identical.
The number of atoms contained within the unit cell is the number of lattice points per unit cell
times the number of atoms per lattice point. (For example, table salt is a simple cubic structure
with two atoms per lattice site.) The number of nearest neighbors an atom has is called the
coordination number
.
Exercise I:
A. Draw a cube.
i.
How many faces does the cube have?
6
ii.
How many edges?
12
iii.
How many corners?
8
B.
Draw a simple cubic crystal lattice with atoms represented by spheres.
i.
What is the number of atoms per unit cell ?
1 ; the lattice point for SC is (0,0,0)
ii.
What is the number of nearest neighbor atoms ?
6
iii.
What is the close-packed direction (direction of touching) for a simple cubic crystal structure?
(
The touching directions for SC are along X, Y and Z axis
)
iv.
If spheres with unit volume were centered on each of the corners, what would be the net
volume of sphere actually contained
inside
the cube?
1/8 of the sphere is inside the cube if
its center is located at the corner and we have a total of
8 corners in a cube
so 1 unit
sphere volume is inside the cube
v.
If spheres with unit volume were centered on each of the edges, what would be the net
volume of sphere actually contained
inside
the cube?
1/4 of the sphere is inside the cube if
its center is located on the edge and we have a total of 12 edges per cube; so 3 units
sphere volume is inside the cube
vi.
Draw the unit cell, indicate the position (x, y, z coordinates) of the center for the largest
hole
(called an interstitial position) within the unit cell.
(1/2,1/2,1/2)
vii.
How many lattice atoms surround the interstitial position?
8
C. The edge length of the cubic unit cell is called the lattice parameter and is denoted as
a
,
which would, for a given material, have some magnitude expressed in meters (or
nanometers). Cubic unit cells have only
one lattice parameter
. If the unit cell were not
cubic, we would need additional lattice parameters, some of which might be angles. If a
simple cubic unit cell were drawn as a hard-sphere model (the hard sphere model considers
the atoms to be spherical and touching their nearest neighbors), what would be
i.
the total edge length (in terms of the lattice parameter
,
a
);
12
a