Question

A 5.15 -MeV alpha particle (mass \(=3.7274 \mathrm{GeV} / \mathrm{c}^{2}\) ) inside a heavy nucleus encounters a barrier whose average height is \(15.5 \mathrm{MeV}\) and whose width is \(11.7 \mathrm{fm}\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)\). What is the probability that the alpha particle will tunnel through the barrier? (Hint: A potentially useful value is \(\hbar c=197.327 \mathrm{MeV} \mathrm{fm} .)\)

Short answer

Answer: To find the probability, we need to calculate the transmission coefficient (T) using the given data, the reduced Planck constant (ħ), and the barrier's width and height. Following the steps outlined in the solution, we would plug in the given values and calculated k values for inside and outside the barrier into the formula for T. By solving the equation, we would obtain the numerical value of T, which represents the probability of the alpha particle tunneling through the barrier.

Step-by-step solution

Determine the transmitted particle energy

To determine the probability that the alpha particle will tunnel through the barrier, it is necessary to find the energy of the transmitted particle once it encounters the barrier. This energy is the difference between the initial energy of the alpha particle and the average barrier height: \(E_{transmitted} = 5.15 \text{ MeV} - 15.5 \text{ MeV} = -10.35 \text{ MeV}\) Remember that the energy of the transmitted particle will be negative, which means the particle is not supposed to pass the barrier classically.

Calculate k values inside and outside of the barrier

To find the probability of tunneling, we need to find the k values both inside and outside of the barrier. These depend on the energy of the transmitted particle and its mass. \(k_{inside} = \frac{1}{\hbar}\sqrt{2mE_{transmitted}}\) \(k_{outside} = \frac{1}{\hbar}\sqrt{2m(E_{transmitted}+ V_0)}\) Where \(E_{transmitted}\) is the energy of the transmitted particle, \(V_0\) is the barrier height, and \(m\) is the mass of the alpha particle. Using the given constants, \(\hbar c=197.327 \, \text{MeV} \cdot \text{fm}, m=3.7274 \, \text{GeV}/c^2\), and converting from GeV to MeV by multiplying mass by \(1000\), we can find numerical values for k inside and outside of the barrier.

Solve for k values of inside and outside the barrier

Plug in the given values for \(E_{transmitted}, m\), and \(\hbar c\) into the previously mentioned equations: \(k_{inside} = \frac{1}{197.327}\sqrt{2 \cdot (3.7274\times10^3) \cdot (-10.35)}\) \(k_{outside} = \frac{1}{197.327}\sqrt{2 \cdot (3.7274\times10^3) \cdot (-10.35+15.5)}\) Use a calculator to obtain the numerical values for \(k_{inside}\) and \(k_{outside}\).

Calculate transmission coefficient

The transmission coefficient T is the probability of the alpha particle tunneling through the barrier. We can find it by using the following formula involving \(k_{inside}\) and \(k_{outside}\): \(T=\frac{1}{1+\frac{V_0^2\sinh^2(k_{inside}d)}{4E_{transmitted}(E_{transmitted}+V_0)\cos^2(k_{outside}d)}}\) Where \(V_0\) is the barrier height, \(d\) is the barrier width, and \(E_{transmitted}\) is the energy of the transmitted particle. Use the previously calculated values for \(k_{inside}\), \(k_{outside}\), and the given values for V0 and d.

Solve for Transmission coefficient, T

Plug in the numerical values for \(V_0, E_{transmitted}, k_{inside}, k_{outside}\), and \(d\) into the equation for T: \(T=\frac{1}{1+\frac{(15.5)^2\cdot \sinh^2(k_{inside} \times 11.7)}{4(-10.35)\cdot (-10.35+15.5) \cdot \cos^2(k_{outside} \times 11.7)}}\) Solve for the numerical value of T using a calculator. The result represents the probability of the alpha particle tunneling through the barrier.