A proton and a deuteron (the latter has the same charge as a proton but twice the mass) strike a potential energy barrier that is 10 fm thick and 10 MeV high. Each particle has a kinetic energy of 3.0 MeV before it strikes the barrier. We have to find (a) the transmission probability for each; (b) their respective kinetic energies after they pass through the barrier (assuming that they do so); and (c) their respective kinetic energies if they are reflected from the barrier.

Solution:             Click For PDF Version

We will need the following data:

Mass of a proton, ,

Mass of a deuteron, .

The transmission probability of a particle of mass m, energy E through a barrier of height U and width L is given approximately by the function

.

Thickness of the barrier, ,

height of the barrier, ,

kinetic energy of the protons and kinetic energy of the deuterons before they strike the barrier are equal and each is .

(a)

We calculate next the transmission probabilities.

Transmission probability for a proton to tunnel through the barrier will be

And, the transmission probability for a deuteron to tunnel through the barrier will be

(b)

After tunnelling through the barrier, protons and neutrons move through a zero potential region, the kinetic energy of each will therefore be 3.0 MeV.

(c)

On reflection from the barrier, protons and neutrons move through a zero potential region, the kinetic energy of each will therefore be 3.0 MeV.