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Specific heat of nano particles is less than bulk specific heat of them. Why?

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The specific heat of the bulk is the integral of the specific of all the applicable energy carriers. Let us consider electrons and phonons for simplicity

$$ C_{v-total} = C_{v-electrons} + C_{v-phonons} = (\frac{\partial U}{\partial T}_{v-electrons}) + (\frac{\partial U}{\partial T}_{v-phonons})$$.

Taking into account that energy distributions, we can calculate the specific heat as $$ C_{v-total} =\frac{d}{dT}\int E f_{FD}(E)D(E)dE + \frac{d}{dT}\int E f_{BE}(E)D(E)dE $$

Where the distributions are the fermi-dirac and the Bose-Einstein distribution. Obvioulsy, when you have a bulk material, like a solid, the density of states is modified to accommodate the overalap of the electronic states resulting in band formation. This does not happen in free particls (nano-particles e.g).

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