At the Fermi temperature T F T_F , E F = k T F E_F = kT_F (see Exercise 42.22 42.22 ). When T = T F T = T_F , what is the probability that a state with energy E = 2 E F E = 2E_F is occupied?

Hi, everyone. In this practice problem, we are being asked to calculate the occupancy probability for a state with energy of E equals to three E F. The ferm temperature for metal is approximately the temperature at which molecules in a classical ideal gas will have the same kinetic energy as the fastest moving electron in the metal. Consider the ferm energy of E F equals to K T F where T F is the Fermi temperature and calculate the occupancy probability for a state with energy of E equals to three E F. The options given for this occupancy probabilities are a 0.129, B 0.1 oh nine, C 0.134 N D 0.119. The Fermi direct distribution gives the probability that a state with energy E will be occupied. The Fermi direct probability equation is given by F E which is a function of the energy will be equal to one divided by E to the power of E minus E F divided by K T plus one. So E or capital E is the energy state E F capital E F is the Fermi energy K is the bolt one constant and P is the temperature. In the problem statement, it is given that the capital E will be equals to three E F. So we can substitute that value into our Fermi direct distribution equation to then get the final equation that we will be able to calculate. So then F E will be equals to one divided by E to the power of three E F minus E F divided by K T plus one plus one. And um simplifying this, we will get F E to then be equal to one divided by E to the power of two E F divided by K T plus one. Um In this case, the T is then going to be equal to T F because at um the energy of E equals to three E F, then T is going to be at the temperature. So then I will uh add this subscript of F in the T. So it will be T F. And in this case, we know as well that E F will be equals to K T F. So we can employ this um equation or expression here to then re simplify our fermi direct probability distribution. So then F E will be equals to one divided by E to the power of two E F divided by, we know that it is K P F and K P F will be equals to E F. So we have E to the power of two E F divided by E F plus one and we can cross out the E F. So then we will get F E to then be equal to one divided by E to the power of two plus one, which will give us the uh occupancy probability or F E which will then come out to a value of 0.119. So 0.119 is going to be the uh occupancy probability for a state with energy of E equals to three E F. And that will give us the answer to this practice problem which will be equal to option D in our answer choices. So option D will be the answer to this particular practice problem and that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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Problem 24

Textbook Question

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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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?

Key Concepts

Fermi Temperature (T_F)

Fermi Energy (E_F)

Fermi-Dirac Distribution

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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?