Question

evaluate the fermi function for an energy KT above the fermi energy. find the temperature at which there is a 1% probability that a state,

Answer 1

Complete Question

Evaluate the Fermi function for an energy kT above the Fermi energy. Find the temperature at which there is a 1% probability that a state, with an energy 0.5 eV above the Fermi energy, will be occupied by an electron.

Answer:

a

The Fermi function for the energy KT is  $F(E_o) = 0.2689$

b

The temperature is  $T_k = 1261 \ K$

Step-by-step explanation:

From the question we are told that

   The energy considered is  $E = 0.5 eV$

Generally the Fermi  function is mathematically represented as

       $F(E_o) = \frac{1}{e^{\frac{[E_o - E_F]}{KT} } + 1 }$

    Here K is the Boltzmann constant with value $k = 1.380649 *10^{-23} J/K$

            $E_F$  is the Fermi energy

            $E_o$  is the initial energy level which is mathematically represented as

     $E_o = E_F + KT$

So

     $F(E_o) = \frac{1}{e^{\frac{[[E_F + KT] - E_F]}{KT} } + 1}$

=>   $F(E_o) = \frac{1}{e^{\frac{KT}{KT} } + 1}$

=>   $F(E_o) = \frac{1}{e^{ 1 } + 1}$

=>   $F(E_o) = 0.2689$

Generally the probability that a state, with an energy 0.5 eV above the Fermi energy, will be occupied by an electron is mathematically represented by the  Fermi  function as

     $F(E_k) = \frac{1}{e^{\frac{[E_k - E_F]}{KT_k} } + 1 } = 0.01$

Here$E_k$ is that energy level that is  0.5 ev above the Fermi energy  $E_k = 0.5 eV + E_F$

=>   $F(E_k) = \frac{1}{e^{\frac{[[0.50 eV + E_F] - E_F]}{KT_k} } + 1 } = 0.01$

=>   $\frac{1}{e^{\frac{0.50 eV ]}{KT_k} } + 1 } = 0.01$

=>   $1 = 0.01 * e^{\frac{0.50 eV ]}{KT_k} } + 0.01$

=>   $0.99 = 0.01 * e^{\frac{0.50 eV ]}{KT_k} }$

=>   $e^{\frac{0.50 eV ]}{KT_k} } = 99$

Taking natural  log of both sides

=>   $\frac{0.50 eV }{KT_k} } =4.5951$

=>    $0.50 eV =4.5951 * K * T_k$

Note eV is electron volt and the equivalence in Joule is     $eV = 1.60 *10^{-19} \ J$

So

     $0.50 * 1.60 *10^{-19 } =4.5951 * 1.380649 *10^{-23} * T_k$

=>   $T_k = 1261 \ K$