Page 87 - Materials Science and Engineering An Introduction

3.4 Metallic Crystal Structures • 59

Concept Check 3.1
(a) What is the coordination number for the simple-cubic crystal structure?
(b) Calculate the atomic packing factor for simple cubic.
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]

EXAMPLE PROBLEM 3.3
Determination of HCP Unit Cell Volume
z
a
(a) Calculate the volume of an HCP unit cell in
terms of its a and c lattice parameters.
(b) Now provide an expression for this volume in
terms of the atomic radius, R, and the c lattice
parameter.

Solution c
(a) We use the adjacent reduced-sphere HCP unit a
cell to solve this problem. D 2
Now, the unit cell volume is just the prod-
uct of the base area times the cell height, c. a 3 C E
This base area is just three times the area of A
the parallelepiped ACDE shown below. (This a 1
ACDE parallelepiped is also labeled in the
above unit cell.) C D
The area of ACDE is just the length of CD times
the height BC. But CD is just a, and BC is equal to 30º a = 2R
a13 60º
BC = a cos(30 ) = A B E
2
Thus, the base area is just a = 2R
2
a13 3a 13
AREA = (3)(CD)(BC) = (3)(a)a b =
2 2
a = 2R
Again, the unit cell volume V C is just the product of the
AREA and c; thus,
V C = AREA(c)
2
3a 13
= a b(c)
2
2
3a c13
= (3.7a)
2
(b) For this portion of the problem, all we need do is realize that the lattice parameter a is
related to the atomic radius R as
a = 2R
Now making this substitution for a in Equation 3.7a gives
2
3(2R) c13
V C =
2
= 6R c13 (3.7b)
2