Fermi Gas Model Heisenberg Uncertainty Principle Particle in dx will have a minimum uncertainty in p x of dp x dx pxpx Next particle in dx will have.

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Presentation transcript:

Fermi Gas Model

Heisenberg Uncertainty Principle Particle in dx will have a minimum uncertainty in p x of dp x dx pxpx Next particle in dx will have a momentum p x Particles with p x in dp x have minimum x-separation dx

Heisenberg Uncertainty Principle Identical conditions apply for the y, p y, and z, p z -- Therefore, in a fully degenerate system of fermions, (i.e., all fermions in their lowest energy state), we have 1 particle in each 6-dimensionl volume -- Phase space volume Momentum volume Spatial volume = 

Heisenberg Uncertainty Principle In some dV ps the maximum number dN of unique quantum states (fermions) is pxpx pzpz pypy p Number of states in a shell in p-space between p and p + dp Only Heisenberg uncertainty principle; completely general

FGM for the nucleus Treat protons & neutrons separately Consider a simple model for nucleus--

FGM for the nucleus Total energy eigenvalue unique states degenerate eigenvalues

FGM for the nucleus unique statesquantized momentum states pxpx pzpz pypy p from Heisenberg uncertainty relation

FGM for the nucleus  All momentum states up to p F are filled (occupied) pxpx pzpz pypy p Assume extreme degeneracy  all low levels filled up to a maximum -- called the Fermi level (E F ) We want to estimate E F and p F for nuclei -- The number N of momentum states within the momentum-sphere up to p F is -- one p-state per dp 3 1/8 of sphere because n x, n y, n z > 0

FGM for the nucleus Fermi energy (most energetic nucleon(s) Fermi momentum (most energetic nucleon(s) protons N = Z neutrons N = (A-Z) 2 spin states

FGM for the nucleus ProtonsNeutrons Assume Z = N

FGM for the nucleus ProtonsNeutrons Assume Z = N

FGM potential

Test of FGM not FGM