At the Fermi temperature TFT_F, EF=kTFE_F = kT_F (see Exercise 42.2242.22). When T=TFT = T_F, what is the probability that a state with energy E=2EFE = 2E_F is occupied?

Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 42: Molecules and Condensed Matter

Problem 24

Chapter 42, Problem 24

At the Fermi temperature $T_{F}$ , $E_{F} = k T_{F}$ (see Exercise $42.22$ ). When $T = T_{F}$ , what is the probability that a state with energy $E equals 2 E sub F$ is occupied?

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Understand the problem: The Fermi-Dirac distribution function gives the probability that a quantum state with energy E is occupied at a given temperature T. The formula is:

f (E) = \frac{1}{1 + e^{(E - E F_{) / k T}}}

, where E is the energy of the state, E_F is the Fermi energy, k is the Boltzmann constant, and T is the temperature.

Substitute the given conditions: At the Fermi temperature T = T_F, and the energy of the state is E = 2E_F. Substitute these values into the Fermi-Dirac distribution formula:

f (2 E F_{) = \frac{1}{1 + e^{(2 E F_{- E F_{) / k T)}}}}}

Simplify the exponent: Since E = 2E_F and T = T_F, the exponent becomes:

(2 E F_{- E F_{) / k T)}}

. Simplify this to:

(E F_{/ k T)}

. Since T = T_F, and by definition

E F_{= k T F}

, the exponent simplifies further to 1.

Substitute the simplified exponent back into the Fermi-Dirac formula: The probability becomes:

f (2 E F_{) = \frac{1}{1 + e^{1}}}

Interpret the result: The probability is now expressed in terms of a simple fraction involving the exponential function. You can leave it in this form or calculate the numerical value if needed, but the key takeaway is that the probability decreases significantly for states with energy much higher than the Fermi energy at the Fermi temperature.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Fermi Temperature (T_F)

The Fermi temperature is a characteristic temperature associated with a system of fermions, such as electrons in a metal. It is defined as T_F = E_F / k, where E_F is the Fermi energy and k is the Boltzmann constant. At this temperature, the thermal energy is comparable to the energy levels of the particles, influencing their occupancy in quantum states.

Fermi Energy (E_F)

Fermi energy is the highest energy level occupied by fermions at absolute zero temperature. It represents the energy of the most energetic electrons in a system and is crucial for understanding the distribution of particles in a quantum system. At temperatures near T_F, the occupancy of states can be analyzed using the Fermi-Dirac distribution.

Fermi-Dirac Distribution

The Fermi-Dirac distribution describes the statistical distribution of particles over energy states in systems of indistinguishable fermions. It accounts for the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. The probability of occupancy of a state with energy E at temperature T is given by f(E) = 1 / (e^(E - μ)/(kT) + 1), where μ is the chemical potential.

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