Moment of inertia of the nucleus

[genloz]

Homework Statement

The unstable Tungsten isotope:
$$^{174}_{74}W$$
has various excited states. The spin parity
$$J^{P}$$
and energy levels of some of these states are given below in the table. Show that this series of states is generated by rotational motion in the nucleus and calculate moment of inertia of the nucleus in the 2+, 6+ and 10+ excited states. Comment on your results and compre them with the moment of inertia the nucleus would have were it a rigidly rotating sphere of radius R with an appropriate value of R.

$$J^{P} E(KeV)$$
0+ 0
2+ 112
4+ 355
6+ 704
8+ 1137
10+ 1635
12+ 2186

Homework Equations

Moment of inertia of a rigid sphere:
$$\frac{2}{5}MR^{2}$$
Radius of a nucleus:
$$1.3A^{1/3} fm$$

The Attempt at a Solution

Moment of inertia of a rigid sphere:
$$=\frac{2}{5}MR^{2}$$
$$=\frac{2}{5}*(74*938.272 +74*0.511 +100*939.566)*(1.3*174^{1/3})^{2}$$
$$= as above MeV fm^2$$


So firstly I'm really unsure about the units of the above equation, and secondly I have no idea how to start to show that the series of states is generated by rotational motion...

I guess once I find out how to determine the moment of inertia of the nucleus then the excited states themselves would increase the size of the nucleus by a particular proportion equivalent to what's shown in the table?

A few hints would be very useful! Thanks!

 

[Jhero]

Thank you for your post. It seems like you are on the right track with your attempt at calculating the moment of inertia of the nucleus. However, there are a few things that could be improved.

Firstly, the units of the moment of inertia equation should be in MeV fm^2, as you have correctly written. This means that you will need to convert the masses of the protons, neutrons, and electrons from kg to MeV using the conversion factor 1 kg = 5.609588845×10^29 MeV.

Secondly, in order to show that the series of states is generated by rotational motion, you will need to use the principle of rotational energy levels. This principle states that the energy levels of a rotating system are given by the expression E = J(J+1)/2I, where J is the angular momentum and I is the moment of inertia. By substituting the values of J and E from the table into this equation, you should be able to see a pattern emerging that supports the idea that the states are generated by rotational motion.

Finally, once you have calculated the moment of inertia for the 2+, 6+, and 10+ states, you can compare them to the moment of inertia of a rigidly rotating sphere with the appropriate radius. This can be done by using the equation you have already written and solving for the radius R. You can then compare this radius to the radius of the nucleus given by the equation in the problem.

I hope this helps guide you in the right direction. Good luck with your calculations!