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Fermi Gas Model Heisenberg Uncertainty Principle h dpx dx h dpx dx Particle in dx will have a minimum uncertainty in px of dpx px Next particle in dx will have a momentum px dx h px px dpx px dx Particles with px in dpx have minimum x-separation dx dx dpx h Heisenberg Uncertainty Principle Identical conditions apply for the y, py, and z, pz -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 -- dV ps dx dp x dy dp y dz dpz h 3 dV ps dV dp 3 Phase space = volume Spatial Momentum volume volume Heisenberg Uncertainty Principle In some dVps the maximum number dN of unique quantum states (fermions) is pz dN dV ps h dN p 3 py dV 4 p 2 dp 2 3 px 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-V (x) 0 ; 0 x L V (x) ; x L V (y) 0 ; 0 y L V (z) 0 ; 0 z L V (y) ; y L V (z) ; z L V V (x) V (y) V (z) x (x) y (y) z (z) x (x) n x 2 sin L L 2 2M 2 V E E E x E y Ez Ex 2 2 n 2 2ML x FGM for the nucleus E E x E y Ez 2 E degenerate ni 1,2,3, eigenvalues x (x) y (y) z (z) unique states 2 2 n n y 2 x 2ML x (x) Ex n z2 n x 2 sin L L Total energy eigenvalue 2 2 n 2 x 2ML n x n y n z x, y,z 2 p x Ex px nx 2M L FGM for the nucleus 2 px n x p n x n 2y nz2 p n x n y nz L L unique states quantized momentum states p x , p x , p x n x n y n z x, y,z h dp L L 3 from Heisenberg uncertainty relation 3 3 px pz n x ,n y ,nz L p dp 3 p x , p x , p x py FGM for the nucleus Assume extreme degeneracy all low levels filled up to a maximum -- called the Fermi level (EF) All momentum states up to pF are filled (occupied) We want to estimate EF and pF for nuclei -The number N of momentum states within the momentum-sphere up to pF is -1 4 pF3 N 8 3 dp 3 pz p one p-state per dp3 1/8 of sphere because nx, ny, nz > 0 px dp 3 p x , p x , p x py FGM for the nucleus 1 4 pF3 1 4 pF3 N pF 2ME F N 3 3 8 3 dp 8 3 2 L 2 spin states 3/2 V 2ME 1 2 4 F 3 L3 V N N p V 2 2 F 3 3 8 2 3 EF pF 2 3N 2 / 3 2M V 1/ 3 1/ 3 N 2 3 Fermi energy nucleon(s) (most energetic Fermi momentum V (most energetic nucleon(s) protons N=Z neutrons N = (A-Z) FGM for the nucleus Protons pF Neutrons 1/ 3 1/ 3 Z 2 3 V pF 1/ 3 1/ 3 A Z 2 3 V Assume Z = N pF V 1/ 3 1/ 3 A /2 2 3 4 R3 3 R Ro A1/ 3 V pF 4 V Ro3A 4.18 Ro3A 3 1/ 3 1/ 3 A /2 2 3 V 3 2 1/ 3 pF Ro 2 4.18 FGM for the nucleus Protons pF Neutrons 1/ 3 1/ 3 Z 2 3 V pF 1/ 3 1/ 3 A Z 2 3 V Assume Z = N 3 2 1/ 197Mev 3 300 pF MeV /c Roc 2 4.18 Ro pF 231 MeV /c (Ro 1.3F) EF 2 pF 28 MeV 2M FGM potential Test of FGM not FGM