CRYSTALLOGRAPHY
OBJECTIVES
• To build models of cubic structures.
• To determine the structural characteristics of various cubic crystals and compare them with their theoretical values.
• To answer the review questions based on the fundamental concepts of crystal structures.
INTRODUCTION
Atoms in crystalline solids are positioned in an orderly and repeated pattern that is in contrast to the random and distorted atomic distribution found in non-crystalline or amorphous materials. Atoms may be represented as solid spheres, and, for crystalline solids, crystal structure is just the spatial arrangement of these spheres. The various crystal structures are specified in terms of unit-cells that are characterized by geometry and atom positions within.
Most common crystalline solids exist in at least one of the three relatively simple crystal cubic structures: simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC). Fig. 1 shows the models of these three cubic structures which can be fully described by four structural characteristics, namely (a) the coordination number (CN), (b) the atomic packing factor (APF), (c) the number of a a atoms per cell (APC), and (d) the ratio of lattice parameter to atomic radius ( ). The values of ( )
R
R can be readily determined by using the close-packed directions along which the atoms are in continuous contact. Table 1 shows these characteristics for cubic structures of SC, BCC and FCC.
Once the crystal structure of a metal is known, its theoretical density (ρ) can be calculated using the relationship nA
(1)
ρ=
Vc N A where n is the number of atoms associated with each unit cell, A is atomic weight, Vc is volume of the unit cell and NA is Avogadro’s number (6.023 x 1023 atoms/mol).
Fig. 1: The models for SC, BCC, and FCC.
1
Table 1: Characteristics of cubic structures.
SC
BCC
a
R
2
4
Atoms per cell (APC)
1
2
Coordination number (CN)
6
8
Atomic Packing factor (APF)
0.52
0.68
FCC
3
2 2
4
12
0.74
Structure
Example Problem
Calculate the atomic packing factor (APF) for the FCC crystal structure.
APF = total volume of spheres in a unit cell / total unit cell volume
V
= s
Vc
(2)
4
Since there are four atoms per FCC unit cell and the volume of each atom or sphere is ( )πR3,
3
4
16
Hence, Vs = (4)( )πR3 = ( )πR3
3
3
Vc = 16R3 2
Therefore
Vs
16
= ( )πR3 / 16R3 2
3
Vc
= 0.74
APF =
EXPERIMENTAL PROCEDURE
In this lab you will be provided with polystyrene balls and tooth-picks. Your task is to construct models of the various cubic structures by following the steps carefully which will guide you to first obtain the measured values and then compare them with that of the theoretical values for each crystal model.
1. With the information provided in Table 1 as a guide, use the polystyrene balls provided to build the three cubic models as shown in Fig. 1.
2. Use a vernier caliper provided, measure the values of R and a, and insert these values in Table 2. a 3. Now, using the information provided in Table 1, calculate the values of , APC, CN, and APF for
R
each cubic model. Where relevant, include the values of uncertainty (%) for all measurements.
4. Compare your measured values with the theoretical values listed in Table 1.
*Note: Bonus marks will be given for accuracy and careful measurements
References:
W.D. Callister, Jr. (2006). “Materials Science and Engineering, An Introduction,” 7th Edn.
D.R. Askeland (1998). “The Science and Engineering of Materials,” 3rd Edn.
2
Table 2: Measured and calculated structural parameters.
Structure
Atomic radius
(R) cm
Lattice parameter (a) cm
a
R
Coordination Atoms per number cell
(CN)
(APC)
Sphere volume
(Vs)
Cell volume
(Vc)
cm3
cm3
SC
Atomic packing factor
(APF)
Vs
Vc
BCC
FCC
REVIEW QUESTIONS
Based on the experimental observations and results obtained as well as the important
