If $X$ 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 ________.
Density of States Dependence on Energy
The density of states, denoted as $g(E)$, describes the number of available electron states per unit energy interval. For a free electron gas in $X$ dimensions, the density of states is known to depend on energy $E$ with a specific power law.
Deriving the Energy Dependence
In $X$ dimensions, the number of electron states within a wavevector magnitude $k$ is proportional to $k^X$. The energy $E$ of a free electron is related to its wavevector magnitude $k$ by the equation $E = \frac{\hbar^2 k^2}{2m}$, where $\hbar$ is the reduced Planck constant and $m$ is the electron mass.
From the energy relation, we can express $k$ in terms of $E$: $k^2 \propto E$ $k \propto E^{\frac{1}{2}}$
Substituting this into the expression for the number of states $N(E)$: $N(E) \propto k^X \propto (E^{\frac{1}{2}})^X = E^{\frac{X}{2}}$
The density of states $g(E)$ is the derivative of $N(E)$ with respect to energy $E$: $g(E) = \frac{dN(E)}{dE}$ $g(E) \propto \frac{d}{dE} \left( E^{\frac{X}{2}} \right)$ $g(E) \propto E^{\left(\frac{X}{2} - 1\right)}$
Determining the Value of Y
The problem states that the energy dependence of the density of states is given by $E^{½X-Y}$. We compare this with our derived dependence, $E^{\left(\frac{X}{2} - 1\right)}$:
$ E^{\frac{X}{2}-1} = E^{\frac{1}{2}X-Y} $
Equating the exponents:
$ \frac{X}{2} - 1 = \frac{1}{2}X - Y $
Simplifying the equation:
$ -1 = -Y $
Therefore,
$ Y = 1 $
The value of $Y$ 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 $(\frac{n_B}{n_A})$ 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$^{-1}$)
- Electronic specific heat of a solid at temperature $T$ is $C = \gamma T$, where $\gamma$ is a constant related to the thermal effective mass ($m_{eff}$) of the electrons. Then which of the following statements are correct?
