# Interact Exercise 6

## Imports

Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.



In [13]:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np




In [14]:

from IPython.display import Image
from IPython.html.widgets import interact, interactive, fixed



## Exploring the Fermi distribution

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:



In [15]:

Image('fermidist.png')




Out[15]:



In this equation:

• $\epsilon$ is the single particle energy.
• $\mu$ is the chemical potential, which is related to the total number of particles.
• $k$ is the Boltzmann constant.
• $T$ is the temperature in Kelvin.

In the cell below, typeset this equation using LaTeX:

The Fermi-Dirac equation is given by: $$\large F(\epsilon)=\frac{1}{e^{\frac{(\epsilon-\mu)}{kT}}+1}$$ Where:

$\epsilon$ is the single particle energy. $\mu$ is the chemical potential, which is related to the total number of particles. $k$ is the Boltzmann constant. $T$ is the temperature in Kelvin.

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.



In [16]:

def fermidist(energy, mu, kT):
"""Compute the Fermi distribution at energy, mu and kT."""
return 1/(np.exp((energy-mu)/kT)+1)




In [17]:

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.

• Use enegies over the range $[0,10.0]$ and a suitable number of points.
• Choose an appropriate x and y limit for your visualization.
• Label your x and y axis and the overall visualization.
• Customize your plot in 3 other ways to make it effective and beautiful.


In [43]:

def plot_fermidist(mu, kT):
fermi = fermidist(np.linspace(0,10.0, 100), mu, kT)
f = plt.figure(figsize=(9,6))
ax = plt.subplot(111)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.plot(np.linspace(0,10.0, 100), fermi)
plt.xlabel(r"Energy $\epsilon$")
plt.ylabel(r"Probability $F(\epsilon)$")
plt.ylim(0,1.0)
plt.title(r"Probability that a particle will have energy $\epsilon$")




In [44]:

plot_fermidist(4.0, 1.0)







In [45]:

assert True # leave this for grading the plot_fermidist function



Use interact with plot_fermidist to explore the distribution:

• For mu use a floating point slider over the range $[0.0,5.0]$.
• for kT use a floating point slider over the range $[0.1,10.0]$.


In [46]:

interact(plot_fermidist, mu=(0,5.0, .1), kT=(0.1,10.0, .1));






Provide complete sentence answers to the following questions in the cell below:

• What happens when the temperature $kT$ is low?
• What happens when the temperature $kT$ is high?
• What is the effect of changing the chemical potential $\mu$?
• The number of particles in the system are related to the area under this curve. How does the chemical potential affect the number of particles.

a) When kT is low, a particle has a higher probability of having a lower energy $\epsilon < \mu$ and a lower probability of having an energy $\epsilon > \mu$. b) When kT is high, a particle has a more uniform probability of having any energy $\epsilon$. Though, it does still tend to favor lower $\epsilon$. c) The chemical potential $\mu$ affects where the curve $F(\epsilon)$ hits its inflection point. d) A higher chemical potential $\mu$ gives us more area under this curve and thus more particles.



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