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The Wien (E x B) Filter

Developed by E. Behringer

This set of exercises guides the students to compute and analyze the behavior of a charged particle in a spatial region with mutually perpendicular electric and magnetic fields. It requires the student to determine the Cartesian components of hte forces acting on the particle and to obtain the corresponding equations of motion. The solutions to these equations are obtained through numerical integation, and the capstone exercise is the simulation of the E×B (Wien) filter.

Subject Area Electricity & Magnetism
Levels First Year and Beyond the First Year
Available Implementation Python
Learning Objectives

Students who complete this set of exercises will be able to:

  • generate equations predicting the Cartesian components of force acting on the charged particle and generate the equations of motion for the particle (Exercise 1);
  • calculate particle trajectories by solving the equations of motion (Exercise 2);
  • produce two-dimensional and three-dimensional plots of the trajectories (Exercise 2); and
  • simulate the operation of an E×B (Wien) filter (Exercise 3) and determine the range of particle velocities transmitted by the filter, and how these are affected by the geometry of the filter (Exercise 4).
Time to Complete 120 min
Exercise 4


As an extension of Exercise 3, now assume that the particles entering the field region at the origin have a normal distribution of velocities directed purely along the z-axis. The center of the distribution is vz,pass and its width is 0.1vz,pass.

(a) Allow 40,000 particles from this distribution to enter the field region at the origin. What is the resulting histogram of the scaled velocities vz/vz,pass of the particles transmitted through a circular aperture of radius R=1.0 mm centered on the z-axis? How does it compare to the histogram of the initial velocities?

(b) Repeat part (a) for an aperture of radius R=0.5 mm.

It is worth noting that an actual source of ions will not only be characterized by a distribution of velocities, but also distribution of directions (no ion beam is strictly mono-directional, just like a laser beam is not strictly mono-directional). This is an additional fact that would have to be considered to accurately simulate the performance of a real Wien filter. Bug Fix)




# This file is used to numerically integrate

# the second order linear differential equations

# that describe the trajectory of a charged particle through

# an E x B velocity filter.


# Here, it is assumed that the axis of the filter

# is aligned with the z-axis, that the magnetic field

# is along the +x-direction, and that the electric field

# is along the -y-direction.


# The numerical integration is done using the built-in

# routine odeint.


# Many particles selected from a normal distribution of

# z-velocities are sent through the filter and histograms

# of the z-velocities of the incident and transmitted particles

# are produced.


# By:

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799 (Office)

# This email address is being protected from spambots. You need JavaScript enabled to view it.


# Last updated:


# 20160624 ERB


from pylab import figure,xlim,xlabel,ylim,ylabel,grid,title,hist,show,text

from numpy import sqrt,array,arange,random,absolute,zeros,linspace

from scipy.integrate import odeint


# Initialize parameter values


q = 1.60e-19 # particle charge [C]

m = 7.0*1.67e-27 # particle mass [kg]

KE_eV = 100.0 # particle kinetic energy [eV]

Ex = 0.0 # Ex = electric field in the +x direction [N/C]

Ey = -105.0 # Ey = electric field in the +y direction [N/C]

Ez = 0.0 # Ez = electric field in the +z direction [N/C]

Bx = 0.002 # Bx = magnetic field in the +x direction [T]

By = 0.0 # By = magnetic field in the +x direction [T]

Bz = 0.0 # Bz = magnetic field in the +x direction [T]

R_mm = 0.5 # R = radius of the exit aperture [mm]

L = 0.25 # L = length of the crossed field region [mm]

Ntraj = 40000 # number of trajectories

transmitted_v = zeros(Ntraj) # array to save velocities of transmitted particles

n_transmitted = 0 # counter for the number of transmitted particles

# Derived quantities

qoverm = q/m # charge to mass ratio [C/kg]

KE = KE_eV*1.602e-19 # particle kinetic energy [J]

vmag = sqrt(2.0*KE/m) # particle velocity magnitude [m/s]

R = 0.001*R_mm # aperture radius [m]

vzpass = -Ey/Bx # z-velocity for zero deflection [m/s]

# Set up the distribution of incident velocities

mean = vzpass # the mean of the velocity distribution is vzpass

sigma = 0.1*vzpass # the width of the velocity distribution is 0.1*vzpass

vz = mean + sigma*random.randn(Ntraj) # the set of initial velocity magnitudes

scaled_vz = vz/vzpass # the set of scaled initial velocity magnitudes

# Set up the bins for the histograms

scaled_vz_min = 0.6

scaled_vz_max = 1.4

Nbins = 64

scaled_vz_bins = linspace(scaled_vz_min,scaled_vz_max,Nbins+1)

vz_bins = vzpass*scaled_vz_bins


# Over what time interval do we integrate?


tmax = L/vzpass;


# Specify the time steps at which to report the numerical solution


t1 = 0.0 # initial time

t2 = tmax # final scaled time

N = 1000 # number of time steps

h = (t2-t1)/N # time step size

# The array of time values at which to store the solution

tpoints = arange(t1,t2,h)

# Specify initial conditions that don't change

x0 = 0.0 # initial x-coordinate of the charged particle [m]

dxdt0 = 0.0 # initial x-velocity of the charged particle [m/s]

y0 = 0.0 # initial y-coordinate of the charged particle [m]

dydt0 = 0.0 # initial y-velocity of the charged particle [m/s]

z0 = 0.0 # initial z-coordinate of the charged particle [m]


# Here are the derivatives of position and velocity

def derivs(r,t):

# derivatives of position components

xp = r[1]

yp = r[3]

zp = r[5]

dx = xp

dy = yp

dz = zp

# derivatives of velocity components

ddx = qoverm*(Ex + yp*Bz - zp*By)

ddy = qoverm*(Ey + zp*Bx - xp*Bz)

ddz = qoverm*(Ez + xp*By - yp*Bx)

return array([dx,ddx,dy,ddy,dz,ddz],float)

# Start the loop over the initial velocities

for i in range (0,Ntraj-1):

# Specify initial conditions

dzdt0 = vz[i] # initial z-velocity of the charged particle [m/s]

r0 = array([x0,dxdt0,y0,dydt0,z0,dzdt0],float)

# Calculate the numerical solution using odeint

r = odeint(derivs,r0,tpoints)

# Extract the 1D matrices of position values

position_x = r[:,0]

position_y = r[:,2]

position_z = r[:,4]

# Extract the 1D matrices of velocity values and final velocity

v_x = r[:,1]

v_y = r[:,3]

v_z = r[:,5]

vxf = v_x[N-1]

vyf = v_y[N-1]

vzf = v_z[N-1]

vf = sqrt(vxf*vxf + vyf*vyf + vzf*vzf)

# If the particle made it through the aperture, save the velocity

if absolute(position_x[N-1]) < R:

if absolute(position_y[N-1]) < sqrt(R*R - position_x[N-1]*position_x[N-1]):

transmitted_v[n_transmitted] = vf

n_transmitted = n_transmitted + 1

# Only save the non-zero values for the histogram

transmitted_v_f = transmitted_v[0:n_transmitted]

scaled_transmitted_v_f = transmitted_v_f/vzpass

# Let the user know how many particles were transmitted

print("The number of incident particles is %d"%Ntraj) #Frem: Added Brackets

print("The number of transmitted particles is %d"%n_transmitted) #Frem: Added Brackets

# start a new figure


# plot the histogram of scaled initial velocities

n, bins, patches = hist(scaled_vz, scaled_vz_bins, normed=0, facecolor='orange', alpha=0.75)

xlabel('\(v_z/v_{z,pass}\) [m/s]',size = 16)

ylabel('\(N\)',size = 16)

title('Histogram of initial \(v_z/v_{z,pass}\) values')





# start a new figure


# plot the histogram of scaled final velocities (transmitted particles)

n, bins, patches = hist(scaled_transmitted_v_f, scaled_vz_bins, normed=0, facecolor='purple', alpha=0.75)

xlabel('\(v_z/v_{z,pass}\)',size = 16)

ylabel('\(N\)',size = 16)

title('Histogram of \(v_z/v_{z,pass}\) values for transmitted particles')




text(0.65,2750,"R = %.2f mm"%R_mm,size=16)




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