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This question already has an answer here:

"Thermal expansion arises from the asymmetrical nature of potential energy curve for atoms in a solid. If oscillators were truly harmonic separation would not change regardless of the amplitude of vibration."

I don't understand how a temperature increase might not affect the volume of a solid substance if the oscillations were harmonic?

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marked as duplicate by sammy gerbil, David Z May 11 '17 at 22:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ You appear to be quoting. Please can you identify the source. $\endgroup$ – sammy gerbil May 11 '17 at 19:04
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    $\begingroup$ Possible duplicate of Why solids expand on heating $\endgroup$ – sammy gerbil May 11 '17 at 19:07
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    $\begingroup$ @DavidZ I would like to suggest that this question is not a duplicate as the OP is asking why one would get no expansion if the potential was symmetrical not why solids expand. That is the reason that I provided an answer. $\endgroup$ – Farcher May 12 '17 at 7:56
  • $\begingroup$ @Farcher hm, maybe. I'd like to see this question edited by the OP to clarify that if that is really the case. $\endgroup$ – David Z May 12 '17 at 17:17
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If the interatomic potential were a perfect parabola, it is true that there would be no thermal expansion. Why? Well, as atoms vibrated more and more, the mean atom position would not change. Sure, it would wobble around more, but it would wobble as much to one side as to the other. The change in average atomic position would be zero. In contrast, if the potential is skewed (as it is for an interatomic potential), the time averaged atomic position will change as the amplitude of vibration increase. On average, it will (usually) move to a greater interatomic distance with increasing temperature.

Of course, this process is neither linear nor necessarily monotonic across all temperatures, as demonstrated by many real materials.

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These are potential energy against atomic separation graphs.

enter image description here

When the atom is not vibration the equilibrium separation between the atoms is $a_{\rm o}$.

As the temperature rises the kinetic energy, $E_!, E_2, E_3, E_4, E_5$, of the atoms increases and their amplitude of vibration increases.
For a potential that is symmetrical, as on the right, the average separation of the atoms, shown by the green blobs, stays the same at $a_{\rm o}$ and the material does not expand.

However for an asymmetrical potential graph, as shown on the left, the average separation of the atoms increases and progressively gets bigger than $a_{\rm o}$ and the material expands.

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