Estimating Alpha Particle Speed & Kinetic Energy in a Heavy Nucleus

[wdednam]

Homework Statement

Use the uncertainty principle to estimate the minimum speed and kinetic energy of an alpha particle confined to the interior of a heavy nucleus.

Homework Equations

delta_p*delta_r = h_bar/2 for minimum momentum

p = mv for non-relativistic particle, the alpha particle in this case

T = p^2/2m where m = rest mass of alpha particle

I'm using Krane's Introductory Nuclear Physics and the book gives for a typical heavy nucleus, r = a = 7.5 fm. Chapter 8 section 8.4.

The Attempt at a Solution

I calculated p using p*a = h_bar/2 for a = 7.5 fm. But when I plugged this into T to get the minimum kinetic energy I got something of the order of 0.1 MeV for the alpha particle confined inside the nucleus.

In the textbook it says that the kinetic energy of an alpha particle inside a typical heavy nucleus is around Q + V0 = 6 MeV + 35 MeV = 41 MeV where V0 is the depth of the potential well to which the alpha particle is confined and Q the disintegration energy.

I don't know what I'm doing wrong here. Any help would be appreciated, thanks.

 

[Jhero]

Thank you for your question. It seems like you are on the right track with your calculation using the uncertainty principle. However, there are a few things that may be causing the discrepancy between your calculated minimum kinetic energy and the value given in the textbook.

Firstly, it is important to note that the value of a = 7.5 fm given in the textbook is an average value for the radius of a heavy nucleus. This means that the actual value for the radius of the nucleus could vary and may not be exactly 7.5 fm. This could affect your calculation of p and therefore your calculated minimum kinetic energy.

Secondly, the value of V0 = 35 MeV given in the textbook is an average value for the depth of the potential well of a heavy nucleus. This value can also vary and may not be exactly 35 MeV. This will also affect your calculation of the minimum kinetic energy.

Lastly, it is possible that the textbook is using a different equation or assumption to calculate the minimum kinetic energy of the alpha particle. It would be helpful to check the context in which the value of 41 MeV is given in the textbook and see if there are any additional factors or assumptions included in the calculation.

In conclusion, I would recommend checking your calculation again using a range of values for a and V0 to see if you can get a closer result to the one given in the textbook. Also, make sure to check for any additional factors or assumptions that may be included in the textbook calculation. I hope this helps. Good luck with your studies!