We are currently doing a lab on the application of conservation of energy and mo

Question

We are currently doing a lab on the application of conservation of energy and momentum in radioactive decays. In part a we had to calculate the decay energy of Plutonium-238 which I got as 8.950*10^-13 J. Next, in part b, we calculated the r minimum of the alpha particle from the daughter particle (Uranium-234) which was 4.742*10^-14 m. The question is: comparing your minimum radius to the Plutonium nuclear radius of r0~8 fm is disturbing in calssical mechanics: what bizarre thing does energy conservation and classical mechanics tell you about the kinetic energy (and hence the speed) of the alpha for distances smaller than your answer in part b) (but outside the nuclear potential at r0?). This is the why that region is called classically forbidden, and we say that the alpha particle quantum tunneled through the region between r0 and r min= r1. I am confused as to how to explain this, we were told to consider the meaning of total energy so I know I have to use the equation E= K + U if r0 < r< r min but I'm not sure what to do next. I am actually not sure what r0, r min and r really mean. Thanks in advance!

Explanation / Answer

Let me first help you in understanding the terms. r0 is the nuclear radius of Plutonium which means that all the nucleons (protons and neutrons) of the Plutonium atom (before decay) existed within a sphere of this radius from the center of the atom.

rmin as you have calculated is the minimum distance between the center of the daughter nucleus and the alpha particle (after decay). It is also being called r1 .

In a very simplistic picture, during the decay, the four nucleons comprising the alpha particle are moving from a distance less than r0 from the center of the atom to a distance more than r1 .

Now, think of the Plutonium atom initially. In its own frame of reference, the net kinetic energy of its nucleons is 0. If this weren't the case, it wouldn't be stable at all.

Also, let's say the initial kinetic energy is U1 . After the decay, the four nucleons comprising the alpha particle have clearly moved much farther than they were before, because r1>r0 . This means that the potential energy after the decay U2 is higher than U1 .

By conservation of total energy, U1+K1=U2+K2 . But since K1=0, K2=U1-U2 which is negative!!

Classically, a negative kinetic energy would mean an imaginary speed which is clearly bizarre. To explain this, we need to take recourse to quantum tunneling whereby a quantum particle (in this case the alpha particle) jumps over a potential barrier.

Hope things are clearer now!!

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