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156 • Chapter 5 / Imperfections in SolidsSolution
(a)In order to determine the ASTM grain size number (n) it is necessary to
employ Equation 5.19. Taking logarithms of both sides of this expression
leads to
logN=(n−1)log 2
And solving fornyieldsn=logN
log 2+ 1
From the problem statement,N=45, and, thereforen=log 45
log 2+ 1 = 6. 5
(b)At magnifications other than 100×, use of the following modified form of
Equation 5.19 is necessary:NM(
M
100
) 2
= 2 n−^1 (5.20)In this expressionNM=the number of grains per square inch at magnifi-
cationM. In addition, the inclusion of the (M/100)^2 term makes use of the
fact that, while magnification is a length parameter, area is expressed in
terms of units of length squared. As a consequence, the number of grains
per unit area increases with the square of the increase in magnification.
Solving Equation 5.20 forNM, realizing thatM=85 andn=6.5, leads
toNM= 2 n−^1(
100
M
) 2
= 2 (6.^5 −1)
(
100
85
) 2
= 62 .6 grains/in.^2SUMMARY
Point Defects in Metals
All solid materials contain large numbers of imperfections or deviations from crys-
talline perfection. The several types of imperfection are categorized on the basis of
their geometry and size. Point defects are those associated with one or two atomic
positions; in metals these include vacancies (or vacant lattice sites), self-interstitials
(host atoms that occupy interstitial sites), and impurity atoms.Point Defects in Ceramics
With regard to atomic point defects in ceramics, interstitials and vacancies for each
anion and cation type are possible. These imperfections often occur in pairs as Frenkel
and Schottky defects to ensure that crystal electroneutrality is maintained.Defects in Polymers
Although the point defect state concept in polymers is different from in metals
and ceramics, vacancies, interstitial atoms, and impurity atoms/ions and groups of