Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.
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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
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from IPython.display import Image
from IPython.html.widgets import interact, interactive, fixed
In quantum statistics, the Fermi-Dirac distribution is related to the probability that a particle will be in a quantum state with energy $\epsilon$. The equation for the distribution $F(\epsilon)$ is:
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Image('fermidist.png')
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In this equation:
In the cell below, typeset this equation using LaTeX:
Define a function fermidist(energy, mu, kT)
that computes the distribution function for a given value of energy
, chemical potential mu
and temperature kT
. Note here, kT
is a single variable with units of energy. Make sure your function works with an array and don't use any for
or while
loops in your code.
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def fermidist(energy, mu, kT):
"""Compute the Fermi distribution at energy, mu and kT."""
H = (1/((np.e)**((energy - mu)/kT)+1))
return H
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assert np.allclose(fermidist(0.5, 1.0, 10.0), 0.51249739648421033)
assert np.allclose(fermidist(np.linspace(0.0,1.0,10), 1.0, 10.0),
np.array([ 0.52497919, 0.5222076 , 0.51943465, 0.5166605 , 0.51388532,
0.51110928, 0.50833256, 0.50555533, 0.50277775, 0.5 ]))
Write a function plot_fermidist(mu, kT)
that plots the Fermi distribution $F(\epsilon)$ as a function of $\epsilon$ as a line plot for the parameters mu
and kT
.
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?np.arange
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def plot_fermidist(mu, kT):
energy = np.linspace(0.,10.0,50)
plt.plot(fermidist(energy, mu, kT), energy)
plt.title('The Fermi Distribution')
plt.grid(True)
plt.xlabel('F [(Unitless)]')
plt.ylabel('Energy [(eV)]')
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plot_fermidist(4.0, 1.0)
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assert True # leave this for grading the plot_fermidist function
Use interact
with plot_fermidist
to explore the distribution:
mu
use a floating point slider over the range $[0.0,5.0]$.kT
use a floating point slider over the range $[0.1,10.0]$.
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interact(plot_fermidist, mu = [0.0,5.0], kT=[0.1,10.0]);
Provide complete sentence answers to the following questions in the cell below:
Use LaTeX to typeset any mathematical symbols in your answer.
If the temperature $kT$ is low, the energy stays relatively the same (about 5 eV). If the temperature $kT$ is high, then the energy decreases the higher the F is (it is negatively linear). Changing chemical potential $\mu$ changes the energy level. Decreasing $\mu$ decreases the energy, and increasing $\mu$ increases the energy. The chemical potential $\mu$ also affects the number of particles so that a higher chemical potential $\mu$ increases the number of particles.
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