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In my nuclear physics lecture we learned about the "fermi gas modell" of a nucleus with which I have some problems. First the potential for the nucleons is shown in the picture below and it makes somewhat sense to me.

potential as shown in lecture

But now the lecture goes on and says, we use the model of the free nucleon gas. Why does that make sense? There is a potential so the nucleons aren't free at all. To me it looks much more similar to a particle in a box for example. So first question: What das the free nucleon gas have to do with this potential?

Putting that aside, the lecture goes on saying:

To calculate the fermi energy and the depth of the potential, we look at the number of possible states in a volume V with a momentum between $p$ and d$p$:

$$dn=V\frac{4\pi p^2\text{d}p}{h^3}$$

My second question is: Why is that true? Wher does this formular come from?

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  • $\begingroup$ This is related: ictwiki.iitk.ernet.in/wiki/index.php/The_Fermi_gas_model and says that in nuclear physics proton and neutrons are considered as moving quasi-freely within the nuclear volume. Apologies if you have read it already but it might explain where the gas idea comes from. $\endgroup$ – user146020 Mar 8 '17 at 18:48
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    $\begingroup$ Free here means "non-interacting". $\endgroup$ – ZeroTheHero Mar 8 '17 at 20:40
  • $\begingroup$ Ok, so free does not mean, that the potential is ignored! I was confused there. $\endgroup$ – Thomas Mar 9 '17 at 13:16
  • $\begingroup$ Any help with the second question? $\endgroup$ – Thomas Mar 9 '17 at 13:17
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  1. "As spin-1/2 particles they naturally obey Fermi-Dirac statistics. It is assumed that the nucleon, inside those constraints imposed by the Paul I principle, can move freely inside the nuclear volume" Povh Particles and nuclei. Basically the potential is flat and nucleons are free inside the nuclei

  2. By the uncertainty principle ( Delta x Delta p >= hbar/2) one state occupies at least Delta x Delta p / (hbar/2) in 2D phase space . Here are are considering 3 spatial dimensions (x y z) and hence 6D phase space . Hence n = Volume_space x Volume_momentum_space / (hbar/2)^3 The factors of 2 pi in the denominator come from some more rigorous treatment I think, but you get the idea I hope. For the momentum phase space we consider a spherical shell.

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