Thermal Expansion of Solids

Thermal Expansion of Solids

When the temperature of a solid is raised, the atoms making up its crystal lattice vibrate about their

mean (equilibrium) positions with a larger average amplitude. Because the potential-energy curve between

two neighbouring atoms is not perfectly symmetric (it rises more steeply on the "too close" side than on

the "too far" side), a larger vibration amplitude also pushes the average separation between

neighbouring atoms slightly farther apart. Summed over the enormous number of atoms in a real solid, this

microscopic effect shows up macroscopically as thermal expansion -- almost every solid expands when

heated and contracts when cooled.

Linear expansion

For a rod, wire, or any solid whose change in a single dimension (its length) is of interest, the

relevant quantity is the coefficient of linear expansion, $\alpha$. If a solid of original length

$L_0$ (at some reference temperature) is heated through a small temperature change $\Delta T$, its new

length $L$ is found, to a very good approximation for the modest temperature ranges met in this chapter,

to be

with SI unit $^\circ\text{C}^{-1}$ (equivalently $\text{K}^{-1}$, since a change of one Celsius degree

equals a change of one kelvin). $\alpha$ is a property of the particular material; different solids have

very different values (see the table below).

Superficial (areal) and cubical (volume) expansion

A flat sheet or plate of solid, heated uniformly, expands in area, described by the coefficient of superficial (areal) expansion, $\beta$, defined analogously by $A = A_0(1 + \beta\,\Delta T)$. A solid

block, heated uniformly, expands in volume, described by the coefficient of cubical (volume) expansion, $\gamma$, defined by $V = V_0(1 + \gamma\,\Delta T)$.

For an isotropic solid -- one whose physical properties, including its expansion, are the same in

every direction -- these three coefficients are not independent quantities to be measured separately;

they follow directly from one another, because area scales as the square of a linear dimension and

volume scales as its cube:

So once $\alpha$ is known for an isotropic solid, both $\beta$ and $\gamma$ follow immediately without

further measurement.

Everyday consequences and applications

Thermal expansion of solids has to be allowed for in almost every large engineered structure. Railway

tracks are historically laid with a small expansion gap between successive rail lengths, so that the

rails have room to lengthen on a hot day without buckling sideways (modern continuously-welded track uses

other techniques, such as pre-stressing, to the same end). Long bridges are commonly supported on rollers

or expansion joints at one end, so the whole span can lengthen and shorten freely with the seasons without

cracking its supports. Overhead power-transmission cables are strung with a deliberate sag, since a taut

cable strung tightly in cold weather would be pulled dangerously tight -- or could even snap -- as it

contracted further in even colder conditions, or would sag excessively and might snap under its own

increased length in hot weather if strung too tightly to begin with.

A particularly useful application exploits the difference in $\alpha$ between two different metals: a …