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The table gives the occupation probabilities \(f(E)\) as a function of the energy \(E\) for a solid conductor at a fixed temperature \(T\). $$ \begin{array}{l|cccccc} f(\boldsymbol{E}) & 0.064 & 0.173 & 0.390 & 0.661 & 0.856 & 0.950 \\ \hline \boldsymbol{E}(\mathrm{eV}) & 3.0 & 2.5 & 2.0 & 1.5 & 1.0 & 0.5 \end{array} $$ To determine the Fermi energy of the solid material, you are asked to analyze this information in terms of the Fermi-Dirac distribution. (a) Graph the values in the table as \(E\) versus \(\ln \\{[1 / f(E)]-1\\} .\) Find the slope and \(y\) -intercept of the best-fit straight line for the data points when they are plotted this way. (b) Use your results of part (a) to calculate the temperature \(T\) and the Fermi energy of the material.

Step by step solution

01

Plotting the Transformed Data

First, start with transforming the given data in the table. For each occupation probability \(f(E)\), compute \(\ln \{[1 / f(E)]-1\}\). These newly computed values are to be plotted against their corresponding energy values \(E\) in a graph.

02

Calculating Slope and Y-intercept

Fit a straight line to the plotted points using statistical analysis and calculate the slope and y-intercept of this line. The slope and y-intercept are generally obtained from the straight-line equation given as \(y = mx + c\), where \(m\) is the slope and \(c\) is the intercept.

03

Computing Temperature

The slope which is obtained from the graph can be used to calculate the temperature. If the slope is represented by \(m\), the temperature \(T\) can be computed using the relationship \(T = -\frac{1}{mk}\), where \(k\) is the Boltzmann constant which equals \(8.6 \times 10^{-5} eV/K\).

04

Calculation of Fermi energy

The computed y-intercept from the fitted line can be used to calculate the Fermi energy of the material. If the intercept is denoted by \(c\), where \(c = -\frac{E_F}{kT}\), then Fermi energy \(E_F\) can be given by \(E_F = -ckT\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fermi-Dirac distribution

The Fermi-Dirac distribution is used to describe the statistical distribution of particles over energy states in systems comprising fermions, like electrons in a solid. This distribution is crucial for understanding how electrons are settled in different energy levels at a given temperature. It is represented by the equation:

• \( f(E) = \frac{1}{e^{(E-E_F)/(kT)} + 1} \)

Here, \( E \) is the energy level of interest, \( E_F \) is the Fermi energy, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.
This equation tells us the probability \( f(E) \) that a state with energy \( E \) is occupied by an electron. At absolute zero temperature, all states below \( E_F \) are fully occupied. As temperature increases, electrons can occupy higher energy levels due to increased thermal energy.

Occupation probabilities

Occupation probabilities are the chances that specific energy levels are filled with electrons. These probabilities are governed by the Fermi-Dirac distribution. In a simplified context, they manifest as values between 0 and 1, where 0 represents an unoccupied state and 1 signifies a fully occupied state.
In applications, such as the one presented in the exercise, you might find a table listing probabilities \(f(E)\) at different energies \(E\). This helps in analyzing material properties like electrical conductivity and heat capacity.

• When \(f(E)\) is close to 1, the energy level is mostly filled.
• When \(f(E)\) is close to 0, the level is scarcely occupied.

Slope and intercept calculation

To unveil important properties such as temperature and Fermi energy, the occupation probabilities from the exercise can be plotted in a specific format against energy levels. This requires plotting \( E \) versus \( \ln \{[1 / f(E)]-1\} \).
The resulting graph provides a linear relationship described by \( y = mx + c \), where:

• \( m \) represents the slope of the line.
• \( c \) represents the y-intercept.

The slope \( m \) directly relates to the temperature, while the intercept \( c \) assists in calculating the Fermi energy \( E_F \). The simplicity of fitting a straight line to the transformed data makes this approach both intuitive and powerful in extracting these parameters from experimental or tabulated data.

Temperature determination

Once the slope \( m \) from your plotted data is obtained, we can deduce the temperature of the solid conductor. The relationship used for this calculation is:

• \( T = -\frac{1}{mk} \)

Here, \( k \) is the Boltzmann constant (\(8.6 \times 10^{-5} \text{eV/K}\)).
This formula bridges the gap between the abstract data you have graphed and tangible thermal properties of the material being studied. A negative slope signifies that, as the energy increases, the probability falls, indicating how temperature influences the distribution. The smaller the magnitude of the slope, the higher the temperature, reflecting how energy levels become more closely packed as temperature increases.