The temperature at which an electron occupies an energy state 0.1 eV above the Fermi level with a probability of 0.01 is approximately ___ K.

223

253

273

303

We can use the **Fermi-Dirac distribution** to solve this problem. The probability that an electron occupies an energy state \( E \) at a given temperature \( T \) is given by:

\[
f(E) = \frac{1}{1 + e^{(E - E_F) / (k_B T)}}
\]

Where:
- \( f(E) \) is the probability that the energy state is occupied.
- \( E - E_F = 0.1 \, \text{eV} \) is the energy difference between the state and the Fermi level.
- \( k_B \) is the Boltzmann constant, \( k_B = 8.617 \times 10^{-5} \, \text{eV/K} \).
- \( T \) is the temperature in Kelvin.
- \( f(E) = 0.01 \).

We need to solve for \( T \) when \( f(E) = 0.01 \).

### Rearranging the Fermi-Dirac equation:

\[
0.01 = \frac{1}{1 + e^{(0.1 \, \text{eV}) / (k_B T)}}
\]

\[
1 + e^{(0.1 \, \text{eV}) / (k_B T)} = 100
\]

\[
e^{(0.1 \, \text{eV}) / (k_B T)} = 99
\]

Now, take the natural logarithm on both sides:

\[
\frac{0.1 \, \text{eV}}{k_B T} = \ln(99)
\]

\[
T = \frac{0.1 \, \text{eV}}{k_B \ln(99)}
\]

Now, we calculate \( T \).

\[
T = \frac{0.1 \, \text{eV}}{(8.617 \times 10^{-5} \, \text{eV/K}) \ln(99)}
\]

Let me compute that for you.The temperature at which an electron occupies an energy state 0.1 eV above the Fermi level with a probability of 0.01 is approximately **253 K**.

Let me know if you'd like more details or have any questions!

Here are 5 related questions you might explore:
1. What is the Fermi-Dirac distribution used for in solid-state physics?
2. How does the probability of electron occupation change with temperature?
3. How is the Fermi level defined in a metal at absolute zero?
4. What is the significance of the Boltzmann constant in statistical mechanics?
5. How does the energy band structure affect electron occupancy in semiconductors?

**Tip:** The Fermi-Dirac distribution is most relevant for systems where quantum effects are significant, especially at very low temperatures.

The temperature at which an electron occupies an energy state 0.1 eV above the Fermi level with a probability of 0.01 is approximately ___ K.
223
253
273
303

Find the temperature at which an electron occupies an energy state 0.1 eV above the Fermi level with a probability of 0.01. Explore key concepts in statistical mechanics, including the Fermi-Dirac distribution and Boltzmann constant, with a detailed calculation of approximately 253 K.

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