I don't know.
But it occurs to me that conservation of energy might be a problem. The only cases of spiral orbits I can think of immediately involve some change of orbital energy--drag from viscous forces, emission of gravitational waves or absorption of electromagnetic energy as in a cyclotron.
Not that's obvious to me. But it might be worth trying to narrow the range of possibilities first. Are you sure that spiral orbits are possible? Are there stable closed orbits in the Kerr metric? If so, it might be worth starting from one of those and trying to perturb it. ( Perhaps replace r...
Perhaps the way to think about this question is to imagine the hypothetical well with one sub-band as a kind of "bucket" that would have been filled with electrons to a "depth" of 50 meV. (This seems to explain why the occupancy is expressed in meV.) How would you picture the real well in those...
As I understand it, the total energy is the fermi energy you have the equation for, plus the potential energy referred to some chosen zero level. I think it might be useful to draw an energy-level diagram comparing the assumed quantum well with one sub-band and the real well with two sub-bands.
Is your integral in section 1 missing a factor of 4π?
In the first equation in section 3 I think you've lost the factor of 2 from each Rnl(r). And when you put in the limits of integration, the exponents in your final equation should be -2b/a0.
It's not clear to me what the perturbation is...
I found this useful:
/instruct/ch374/ch418518/Chapter%207%20Alpha%20Decay-rev.pdf
Equation (8.18) seems to give a reasonable answer for alpha decay (rough calculation, not checked). The factor in front of the exponential is very small and may reduce the product to the right level. I'd try...
After a bit more reading I find that the only listed decay processes for 220Th are electron capture, and alpha decay with a half-life of about 10 μs. See Wikpedia, "Isotopes of Thorium".
Edit: Not sure if the much slower 12C process would simply be ignored in the listings, if it''s real, or if...
After a tiny bit of reading, I gather that 12C decay might be rather slow. What it look like if you just calculated ln (t1/2) ?
Edit: ignore this--you're trying to compare with a calculated result.
I don't think you should have the factor of c outside the integral, but you need a factor of d in the last term of the integrand. (But I'm also one of those that deplore using d as a constant in anything involving calculus.)
It looks as though the integrand should be expressible in terms of an...
The magnetic moment of the proton is much smaller than that of the electron. But the angular momentum of any spin-1/2 particle can take up one of two orientation is a magnetic field.
For part (a) I wonder if you were expected to do simple electrostatics and geometry--adding up the single-particle fields, and doing first-order calculations for the distances. (I note that in your derivation you didn't insert the unit vectors until the last steps, so technically you're equating...