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While considering the contribtution of the electronic specific heat capacity, it is stated that, when the temperature increases from 0 K , only the electrons with energy range of the order of $k_bT$ from the fermi energy are excited. How was this range determined? (Ref: Introduction to Solid State Physics, 8th Edition by Charles Kittel)

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This is an approximate statement, which follows from understanding the shape of the Fermi distribution: $$ n_F(E) = \frac{1}{e^{\frac{E - E_F}{k_B T}} + 1} $$ This distribution has a step-like form: it is approximately 1 for $E < E_F$ and approximately zero for $E > E_F$, except for the region around $E_F$ where $|E-E_F|\sim k_B T$. In the limit of very low temperatures this region is nearly non-existent and the distribution becomes a step-function $$ n_F(E) = \theta (E_F - E). $$ Broadening of this region when the temperature increases is what one refers to as the electrons excited above the Fermi surface (i.e. some states below $E_F$ are not fully filled, and some states above $E_F$ now contain electrons. But the states away from this region are still either filled (occupancy $1$) or empty (occupancy $0$).

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