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Master USTH

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Master USTH

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Thermal expansion and thermoelasticity

I-1/ The general picture of a lattice energy U(r) is given in Fig. 1. In the case we consider it as harmonic (red curve in Fig. 1) with the simplified equation given by :

U(r)=U0+a. r2

can you explain, without equation, simplify by symmetry principles, why there is no thermal expansion ?

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Figure 1 : representation of the lattice energy U(r) versus the interatomic distance r. Uo is the binding energy.

I-2/ The average force that undergoes a solid is <F> with <F>=<-dU(r)/dr>. (the averge holds on time and space)

If the solid is isolated, what is the average force ? What is then the average interatomic position <r> ?

Deduce the value of the linear thermal expansion coefficient (α=d<r>/dT) ? I-3/ Do the same calculation for an anharmonic potential :

U(r)=U0+a. r2 + b. r 3 Deduce <F>,<r> and α.

I-4/ A laser interacts with a solid and gives rise to a lattice thermal heating. The energy provided by the laser is ΔE and the lattice heat capacity is Cv. What is the lattice temperature increase ?

I-5/ What is the consequence of this lattice heating on a the thermal expansion of a harmonic and anharmonic lattice ?

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