Dispersion Relation Analysis
The particle's energy ($E$) and wavevector ($k$) are related by the dispersion relation $E = Ck$ in two dimensions, where $C$ is a constant.
Density of States Calculation
The density of states $D(E)$ is defined as the number of states per unit energy interval. It can be calculated from the total number of states $N(E)$ with energy less than or equal to $E$ using the formula $D(E) = \frac{dN}{dE}$.
In a 2D system, the number of available states $N$ is proportional to the area in k-space. The area is given by $\pi k^2$, where $k$ is the magnitude of the wavevector.
From the dispersion relation $E = Ck$, we can express $k$ in terms of $E$: $k = \frac{E}{C}$
Substituting this into the expression for the area in k-space:
Area $\propto (\frac{E}{C})^2 = \frac{E^2}{C^2}$
Since the number of states $N(E)$ is proportional to this area:
N(E) \propto \frac{E^2}{C^2}
Thus, $N(E)$ is proportional to $E^2$.
Finding the Exponent p
Now, we find the density of states $D(E)$ by differentiating $N(E)$ with respect to $E$:
$D(E) = \frac{dN}{dE} \propto \frac{d(E^2)}{dE}$
$D(E) \propto 2E$
This shows that $D(E)$ is proportional to $E^1$.
The problem states that the density of states $D(E)$ is proportional to $E^p$, i.e., $D(E) \propto E^p$. Comparing this with our result $D(E) \propto E^1$, we find:
p = 1
Conclusion
The value of $p$ is 1.
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$^{-1}$)
