Understanding Electronic Specific Heat Dependence
The electronic specific heat ($C$) of a solid is given by the formula $C = \gamma T$, where $\gamma$ is the electronic specific heat coefficient and $T$ is the temperature.
Analysis of Statements
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Statement A: $\gamma \propto m_{eff}$
This statement is correct. The coefficient $\gamma$ is directly related to the density of states at the Fermi level, $N(E_F)$. The density of states, in turn, depends on the effective mass ($m_{eff}$) of the electrons. A higher effective mass generally leads to a higher density of states and thus a larger $\gamma$. Mathematically, for a free electron gas, $\gamma = \frac{\pi^2 k_B^2}{3} N(E_F)$ and $N(E_F) \propto (m_{eff})^{3/2}$, leading to $\gamma \propto (m_{eff})^{3/2}$. While the exact relation depends on the band structure, a proportionality $\gamma \propto m_{eff}$ (or a similar direct dependence) is generally expected.
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Statement B: $m_{eff}$ is greater than free electron mass for all solids
This statement is incorrect. The effective mass ($m_{eff}$) can be smaller, equal to, or greater than the free electron mass ($m_e$), depending on the material's electronic band structure.
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Statement C: Temperature dependence of $C$ depends on the dimensionality of the solid
This statement is incorrect. The linear temperature dependence, $C = \gamma T$, is characteristic of the electronic contribution to specific heat in metals at low temperatures, derived from Fermi-Dirac statistics. While dimensionality affects the value of $\gamma$ (by altering $N(E_F)$), the fundamental linear relationship with temperature holds regardless of dimensionality in this regime.
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Statement D: The linear temperature dependence of $C$ is observed at $T \ll$ Debye temperature
This statement is correct. At temperatures significantly below the Debye temperature ($T_D$), the lattice (phonon) contribution to specific heat follows a $T^3$ dependence. In this low-temperature regime ($T \ll T_D$), the electronic contribution, $C_e = \gamma T$, becomes dominant over the lattice contribution. The linear dependence is characteristic of this low-temperature electronic behavior.
Conclusion
Based on the analysis, statements A and D are correct.
Important Questions from Free Electron Theory Fermi Energy Velocity
- Copper has an electron number density of $8.3 \times 10^{28} \text{ m}^{-3}$. Its Fermi energy in eV (rounded off to one decimal place) is _____
($\hbar = 1.06 \times 10^{-34} \text{ J.s}$, mass of electron $\text{m}_e = 9.10 \times 10^{-31} \text{ kg}$, charge of electron $= 1.60 \times 10^{-19} \text{ C}$) - Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is $E_F$. The energy corresponding to the filling of $\frac{N_A}{2}$ electrons, where $N_A$ is the Avogadro number, is $2^n E_F$. The value of $n$ is
Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant $a$. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures.
The electron concentrations in A and B are $n_A$ and $n_B$, respectively. The value of $(\frac{n_B}{n_A})$ isIf $X$ is the dimensionality of a free electron gas, the energy ($E$) dependence of density of states is given by $E^{½X-Y}$, where $Y$ is ________.
- Potassium metal has electron concentration of $1.4 \times 10^{28}m^{-3}$ and the corresponding density of states at Fermi level is $6.2 \times 10^{46}$ Joule$^{-1} m^{-3}$. If the Pauli paramagnetic susceptibility of Potassium is $n \times 10^{-k}$ in standard scientific form, then the value of $k$ (an integer) is __________ (Magnetic moment of electron is $9.3 \times 10^{-24}$ Joule T$^{-1}$; permeability of free space is $4\pi \times 10^{-7}$ T m A$^{-1}$)
