Question

Consider a beam of monoenergetic free electrons (you can tune their energy) heading towards a potential energy barrier whose height is 100 eV and width of 1 A. Plot the transmission and reflection probabilities of the electron as a function of electron KE (for KE going from 0 to well over 100 eV-go high enough until it stops being interesting). 3.

Answer 1

Answer : Given above, the height of the potential barrier V is 100 eV and width = 1 A = 10^-10m. The Transmission probabilities of the electron as a function of KE is given by,

Case1: When E>V

$T = \frac{1}{(1+\frac{V^{2}\sinh(k_{1}a)^{2}}{4E(V-E)})}$ where $k_{1}= \sqrt{\frac{2m(V-E)}{\hbar^{2}}}$

Case 2 = When E<V

$T = \frac{1}{(1+\frac{V^{2}\sin(k_{1}a)^{2}}{4E(E-V)})}$ where, $k_{1}= \sqrt{\frac{2m(E-V)}{\hbar^{2}}}$

Using this equation we plot the transmission probability and reflection probality as a function of KE is given by,

1.0 0.8 Transmission Probability 0.6 0.4 0.2- 300 400 500 100 200 KE

1.0 Reflection probability 0.8 0.6 0.4 0.2 0.0 100 200 300 400 500 KE
Note: When KE = V , the probality of electron being reflected is very high.

**** Please find the Python code used to generate this graph,

import numpy as np
import matplotlib.pyplot as plt
E = np.linspace(0.01,100,1000)
E1 = np.linspace(101,500,1000)
a = 1
V = 100
k1 = np.sqrt(26.254*(V-E))
k2 = np.sqrt(26.254*(E1-V))
x = (V**2*np.sin(k2)**2/(4*E1*(E1-V)))
y = (V**2*np.sinh(k1)**2/(4*E*(V-E)))
T = 1/(1+y)
T1 = T = 1/(1+x)
R = 1-T
R1 =1 -T1
plt.plot(E,T,'k',E,R)
plt.show()
plt.plot(E1,T,E1,R1)