Question

A proton and a deuteron (which has the same charge as the proton but 2.0 times the mass) are incident on a barrier of thickness \(10.0 \mathrm{fm}\) and "height" \(10.0 \mathrm{MeV} .\) Each particle has a kinetic energy of $3.0 \mathrm{MeV} .$ (a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.

Short answer

Answer: To find the answer, compare the tunneling probabilities of the proton and deuteron as calculated using the steps outlined in the provided solution. The higher tunneling probability will indicate which particle has a greater chance of passing through the barrier. Then, divide the proton's tunneling probability by the deuteron's tunneling probability to find the ratio of their tunneling probabilities.

Step-by-step solution

Find the reduced mass of each particle

To find the reduced mass of each particle, take the mass of the proton \((m_p)\) as the base unit and multiply it by a factor to obtain the deuteron mass \((m_d)\). Since the deuteron has 2.0 times the mass of the proton, we have \(m_d = 2m_p\).

Determine the potential energy and available kinetic energy for each particle

Both particles are incident on a barrier of height \(10.0 \mathrm{MeV}\). Their available kinetic energy is given as \(3.0 \mathrm{MeV}\).

Calculate the effective barrier penetration energy for each particle

For each particle, subtract the kinetic energy from the potential energy to obtain the effective energy that the particles need to overcome to penetrate the barrier. For both the proton and deuteron, this would be \(10.0 \mathrm{MeV} - 3.0 \mathrm{MeV} = 7.0 \mathrm{MeV}\).

Use the tunneling probability formula to determine the individual probabilities

The tunneling probability formula is given as: \(p_i = e^{-2k_i a}\), where \(i=p,d\) indicates the particle, \(a\) is the barrier width, and \(k_i = \frac{1}{\hbar} \sqrt{8\pi^2 m_i E_i}\), with \(\hbar\) being the reduced Planck constant and \(E_i\) being the effective energy calculated in Step 3.

Calculate \(k_p\) and \(k_d\) for the proton and deuteron

Now, calculate \(k_p\) and \(k_d\) for the proton and deuteron by substituting the respective values of \(m_p\), \(m_d\), and \(E_i\) into the formula for \(k_i\).

Calculate the tunneling probabilities for the proton and deuteron

Substitute the values of \(k_p\) and \(k_d\) calculated above, along with the value of the barrier width \(a\), into the formula for tunneling probability \(p_i\), and evaluate it for both the proton and the deuteron.

Compare the tunneling probabilities to answer part (a)

Compare the probabilities obtained for the proton and deuteron to determine which particle has a higher probability of tunneling through the barrier.

Find the ratio of the tunneling probabilities for part (b)

Take the ratio of the tunneling probabilities for the proton and deuteron, \(\frac{p_p}{p_d}\), to find the answer to part (b) of the exercise.