Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

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2.5 Bonding Forces and Energies • 25

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(a)

(b)

Interatomic separation r

Interatomic separation r

Repulsive force FR

Attractive force FA

Net force FN

Attraction

Repulsion

Force

F

Repulsive energy ER

Attractive energy EA

Net energy EN

+

0

0

Attraction

Repulsion

Potential energy

E

r 0

E 0

Figure 2.8 (a)The

dependence of repulsive,

attractive, and net forces on

interatomic separation for

two isolated atoms. (b)The

dependence of repulsive,

attractive, and net potential

energies on interatomic

separation for two isolated

atoms.

exists between the two atoms. The magnitude of the attractive force varies with the

distance, as represented schematically in Figure 2.8a. Ultimately, the outer electron

shells of the two atoms begin to overlap, and a strong repulsive forceFRcomes into

play. The net forceFNbetween the two atoms is just the sum of both attractive and

repulsive components; that is,

FN=FA+FR (2.2)

which is also a function of the interatomic separation, as also plotted in Figure 2.8a.

WhenFAandFRbalance, or become equal, there is no net force; that is,

FA+FR= 0 (2.3)

Then a state of equilibrium exists. The centers of the two atoms will remain sepa-

rated by the equilibrium spacingr 0 , as indicated in Figure 2.8a. For many atoms,r 0

is approximately 0.3 nm. Once in this position, the two atoms will counteract any at-

tempt to separate them by an attractive force, or to push them together by a repulsive

action.

Sometimes it is more convenient to work with the potential energies between

two atoms instead of forces. Mathematically, energy (E) and force (F) are related as

E=

∫

Fdr (2.4)

Force-potential

energy relationship

for two atoms