The starter code below models the scattering of an alpha particle from a gold nucleus. The initial kinetic energy of the alpha particle is set by the variable K0 to the equivalent of 5.5 MeV but in Joules. Notice that throughout the code all quantities are expressed in SI units. The initial position of the alpha particle is 100 fm (1 fm = 1 femtometer = 10-15 m) to the left of the nucleus but with an impact parameter determined by the variable b.

The code is complete except for the following lines which determine the force on the alpha particle by the gold and then update both the momentum and position of the alpha particle and the gold nucleus.

A Compute the force on the Alpha particle due to the Au nucleus B Update the momentum of the Alpha particle C Update the position of the Alpha particle D Update the momentum of the Au nucleus

E Update the position of the Au nucleus

Notice that the Alpha particle is described by the object Alpha and its charge and mass are given by the properties Alpha.q and Alpha.m respectively. Notice also how the initial momentum of the Alpha particle is computed from the initial kinetic energy by solving

K = p2/ 2m

For momentum and setting up the momentum vector to point initially in the +x direction.

Likewise, the gold nucleus is described by an object Au and its charge and mass are given by Au.q and Au.m. We assume the Au nucleus is initially at rest.

A variable named r is used in the code to give the position of the Alpha particle relative to the gold nucleus. The magnitude of this vector is the separation between the two particles.

Finally, there is a constant named oofpez that is equivalent to One Over Four Pi Epsilon Zero, or

1 =9x109Nm2/C2 40

You will know you got the correct equations when you find that the energy and momentum are both conserved throughout the collision as shown in the graphs.

Finally, the code does some things to be able to find the distance of closest approach. It does this by storing this distance in a variable d and comparing this saved value to the current separation mag(r) and finding which is smaller using a function called min.

please fix the code.

GlowScript 3.1 VPython # Rutherford Scattering Exercise

# Define Constants q_e = 1.6e-19 # Charge on a proton [Coulombs] m_p = 1.7e-27 # Mass of proton [kg]

oofpez = 9e9 # oofpez = One Over Four Pi Epsilon Zero [N m**2 / C**2]

K0 = 5.5e6 * 1.6e-19 # Energy equivalent of 5.5MeV in J b = 5e-15 # Impact Parameter [m]

# Model the Au nucleus as a yellow sphere of radius 4 fm, located at the origin Au = sphere(pos=vector(0,0,0), radius=4e-15, color=color.yellow, make_trail=True) Au.opacity = 0.7 # Make Au a little opaque so its trail can be seen eaiser.

Au.m = (79+118) * m_p # Mass of Gold nucleus, 79 protons and 118 neutrons [kg] Au.q = 79 * q_e # Charge of Gold nucleus [Coulombs]

# Define momentum of Au. Since initially at rest, velocity = <0,0,0> Au.p = Au.m*vector(0,0,0)

# Model the Alpha particle as a magenta sphere of radius 1 fm Alpha = sphere(radius=1e-15, color=color.magenta, make_trail=True)

# Set the initial position of the Alpha particle to be a large distance to the left # of the Au nucleus and above the x axis by an amount equal to the impact parameter. Alpha.pos = vector(-100e-15,b,0)

Alpha.m = (2+2) * m_p # Mass of Alpha particle, 2 protons and 2 neutrons [kg] Alpha.q = 2 * q_e # Charge of Alpha particle [Coulombs]

# Define initial momentum of Alpha particle using its initial kinetic energy. Alpha.p = vector(sqrt(2*Alpha.m*K0),0,0) # Initially moving in +x direction. Alpha.p1 = Alpha.p # Save the initial momentum for later

# Compute the initial separation of the Alpha and Au and store it in variable r0 r = Alpha.pos - Au.pos r0 = mag(r) # Save the initial separation

# Set up the graph displays gdx = gdisplay(width=600, height=200, title='x Momentum vs Time', background=vector(0.8, 0.8, 0.8)) p_Au_x_graph = gcurve(color=color.yellow, label="Au") p_Alpha_x_graph = gcurve(color=color.magenta, label="Alpha")

p_total_x_graph= gcurve(color=color.cyan, label="total") # Other curves as needed

gdy = gdisplay(width=600, height=200, title='y Momentum vs Time', background=vector(0.8, 0.8, 0.8)) p_Au_y_graph = gcurve(color=color.yellow, label="Au") p_Alpha_y_graph = gcurve(color=color.magenta, label="Alpha")

p_total_y_graph= gcurve(color=color.cyan, label="total")

g_Energy = gdisplay(width=600, height=200, title='Energy vs Time', background=vector(0.8, 0.8, 0.8)) p_Au_KE = gcurve(color=color.yellow, label="Au KE") p_Alpha_KE = gcurve(color=color.magenta, label="Alpha KE") p_PE = gcurve(color=color.black, label="Potential Energy") p_Total_E= gcurve(color=color.cyan, label="Total Energy")

# Create a variable d to hold the minimum separation between the Au and Alpha # Set it to the initial separation. d = r0

# Set up the time counter and time step t = 0 dt = 5e-25

# Continue until separation distance (r) is greater than the initial separation (r0), # that is until the Alpha has finished interacting with the Au and is at least # as far away as it was in the beginning.

while mag(r) <= r0: rate(10000) # Compute position of the alpha particle relative to the gold nucleus r = Alpha.pos - Au.pos # A - Compute the force on the alpha particle due to the gold nucleus F = # B - Update the momentum of the Alpha particle Alpha.p = # C - Update the position of the Alpha particle Alpha.pos = # D - Update the momentum of the Gold nucleus Au.p= # E - Update the position of the Gold nucleus Au.pos = t = t + dt # Update the distance of closest approach d = min(d,mag(r)) # Compute the kinetic and potential energies K_Au = mag(Au.p)**2/(2*Au.m) K_Alpha = mag(Alpha.p)**2/(2*Alpha.m) U = oofpez*Au.q*Alpha.q/mag(r) # Update the energy graph p_Au_KE.plot(pos = (t, K_Au)) p_Alpha_KE.plot(pos = (t, K_Alpha)) p_PE.plot(pos = (t, U)) p_Total_E.plot(pos = (t, K_Au + K_Alpha + U))

# Update the x-component of momentum graph p_Au_x_graph.plot(pos = (t, Au.p.x)) p_Alpha_x_graph.plot(pos = (t, Alpha.p.x)) p_total_x_graph.plot(pos = (t , Alpha.p.x + Au.p.x)) # Update the y-component of momentum graph p_Au_y_graph.plot(pos = (t, Au.p.y)) p_Alpha_y_graph.plot(pos = (t, Alpha.p.y)) p_total_y_graph.plot(pos = (t, Alpha.p.y + Au.p.y))

# Use the dot product to find the scattering angle D = acos(dot(Alpha.p , Alpha.p1) / (mag(Alpha.p) * mag(Alpha.p1))) # radians

# Print summary of results print("Initial kinetic energy of alpha: ", K0/1.6e-13, " MeV") print("Impact Parameter: ", b/1e-15, " fm") print("Distance of closest approach: " , d/1e-15, " fm") print("Scattering angle: ", degrees(D), " degrees")