Thermal Expansion

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5.5 Thermal Expansion

All substances change their volume or dimensions with temperature.

Although these changes are relatively small for solids, they are nonetheless of great technical importance, particularly in situations where one wishes to

permanently join materials with di€ering expansion coecients. In order to arrive at a definition that is independent of the length l of the sample, one defines the linear expansion coecient,, by

ˆ1

l dl

dT : …5:23†

For isotropic substances and cubic crystals, is equal to one-third of the volume expansion coecient

Vˆ3ˆ1 V

dV

dT : …5:24†

Typical values for linear expansion coecients of solids are of the order of 10^±5K^±1. The expansion coecient can clearly only be measured if the sam- ple is kept in a stress-free state. Thermodynamically, this means that the derivative of the free energy with respect to the volume, i.e., the pressure p, must be equal to zero for all temperatures:

@F

@V

T

ˆpˆ0: …5:25†

This equation can be used to calculate the thermal expansion coecient:

Provided one can express the free energy as a function of the volume, then the condition of zero stress for every temperature yields a relation between volume and temperature and thus the thermal expansion. We will use this approach and begin by considering the free energy of a single oscillator.

The generalization to a lattice is then straightforward.

The free energy of a system can be expressed in terms of the partition functionZ

Fˆ TlnZ with ZˆX

i

e ^{Ei= T}: …5:26†

The index i runs over all the quantum mechanically distinct states of the particular system. For a harmonic oscillator we have

ZˆX

n

e ^{hx…n‡1=2†= T}ˆ e ^{hx= T†=2} 1 e ^{hx= T}

: …5:27†

The vibrational contribution to the free energy is therefore Fsˆ1

2hx‡ Tln…1 e ^{hx= T}†: …5:28†

The total free energy also includes the valueUof the potential energy in the equilibrium position

FˆU‡1

2hx‡ T ln…1 e ^{hx= T}†: …5:29†

k k

k k

k

k k

k k

For aharmonicoscillator it is easy to convince oneself that the frequencyx is una€ected by a displacement u from the equilibrium position. Corre- spondingly, one finds that application of the equilibrium condition (5.25) yields no thermal expansion.

We now proceed to the case of the anharmonic oscillator in that we allow the frequency to change with a displacement from the equilibrium posi- tion. We assume that the energy levels are still given byE_n ˆ …n‡¹₂hx†. This procedure is known as the quasi-harmonic approximation. For asingleoscil- lator it is easy to express the frequency change in terms of the third coecient of the potential expansion (4.1). The actual calculation need not be performed here (Problem 5.6). For the simple calculation of the derivative (5.25) we con- sider the free energy expanded about the equilibrium position. The position of the potential minimum will be denoted bya0. In the anharmonic case, the time-averaged position of the oscillator is no longer equal to a0, and will be denoteda.Then, with force constantf, we obtain for the expansion

UˆU0…a0† ‡1

2f…a a0†²; F_sˆF_s…a0† ‡@Fs

@a

aˆa0

…a a0†: …5:30†

The equilibrium condition (5.25), together with (5.29), then yields f…a a0† ‡1

x

@x

@ae…x;T† ˆ0: …5:31†

With this equation we already have the relation between the average displa- cement and the temperature. The displacement is proportional to the ther- mal energy e…x;T† of the oscillator. Thus, for the linear expansion coe- cient, we obtain

…T† ˆ 1 a₀

da dTˆ 1

a²₀f

@lnx

@lna

@

@Te…x;T†: …5:32†

To generalize this to solids we simply need to replace ˆa₀¹…da=dT† by vˆV ¹…dV=dT† and to sum over all phonon wave vectors q and all branches j. In place of a²0f one has Vj, where j=V…@p=@V† is the bulk modulus of compressibility

1 V

dV…T†

dT ˆVˆ 1 Vj

X

q;j

@lnx…q;j†

@lnV

@

@Te‰x…q;j†;TŠ: …5:33†

This is the thermal equation of state of a lattice. One can immediately recognize that in the low- and high-temperature limits, the expansion coe- cient shows the same behavior as the specific heat capacity, i.e., it is propor- tional to T³ at low temperatures, and is constant (within this approxima- tion) at high temperatures. For many lattice types, even the ``GruÈneisen number ''

cˆ @lnx…q;j†

@lnV …5:34†

shows only weak dependence on the frequency x(q,j). The GruÈneisen num- ber can then be assigned an average value and taken out of the sum in (5.33). The expansion coecient thereby becomes approximately propor- tional to the specific heat at all temperatures. Typical values of this average GruÈneisen parameterhciare around 2, and are relatively independent of the material. On account of the bulk modulus appearing in the denominator of (5.33), one can claim, as a rule of thumb, that soft materials with their small bulk moduli have a high thermal expansion coecient.

The proportionality between V and the specific heat does not hold, however, for all crystal classes. For structures with tetrahedral coordination, the expansion coecient changes sign at low temperatures. The expansion coecient of silicon shown in Fig. 5.4 serves as an example.

We have implicitly assumed in our derivation of the thermal equation of state that we are dealing with a cubic structure. Hexagonal structures have di€erent expansion coecients parallel and perpendicular to the c-axis.

These coecients can even have di€erent signs as is the case for tellurium:

with increasing temperature a tellurium crystal expands perpendicular to the c-axis, but shrinks ± albeit only slightly ± in the direction parallel to the c-axis. Crystals with triclinic, monoclinic and rhombic lattices have three di€erent expansion coecients.