Central density. Consider nuclear charge density. Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987) QMPT 540

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1 Central density Consider nuclear charge density Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987)

2 Central density (A/Z* charge density) about the same for nuclei heavier than 16 O, corresponding to 0.16 nucleons/fm 3 Important quantity! Corresponds to k F =1.33 fm 1 Volume term from semiempirical mass formula corresponds to about 16 MeV binding per particle Nuclear density distribution Two most important numbers in nuclear physics to explain!

3 Nuclear matter Key quantities Saturation density: 0.16 nucleons per fm 3 interparticle spacing r fm k F =1.33 fm 1 ν =4 Energy per particle at saturation: ~-16 MeV No Coulomb; N=Z; thermodynamic limit Relation between V NN (including possible V NNN ) and these quantities still debated Bethe contributed ~10 years of his scientific life to this problem No global consensus on precise mechanism of saturation role of pions role of three-body interaction role of relativity if any many phenomenological ways to represent saturation properties

4 Old & fundamental problem: nuclear matter Figure adapted from Marcello Baldo <-- ONLY SRC: PRL 90, (2003) NucPhys

5 IPM for fermions in finite systems IPM = independent particle model Only consider Pauli principle Localized fermions Examples Hamiltonian many-body problem: with and Ĥ 0 = ˆT + Û Ĥ 1 = ˆV Û Suitably chosen auxiliary one-body potential Many-body problem can be solved for!! Also works with fixed external potential Ĥ = ˆT + ˆV = Ĥ0 + Ĥ1 Ĥ 0 U ext U Ĥ = ˆT + Ûext + ˆV = Ĥ0 + Ĥ1

6 Role of Can be chosen to minimize effect of two-body interaction Ground state of total Hamiltonian may break a symmetry (condensed matter systems) U Can speed up convergence of perturbation expansion in Ĥ 1 Spherical symmetry: sp problem straightforward but may have to be done numerically Assume solved: e.g. 3D-harmonic oscillator in nuclear physics For nuclei H 0 λ =(T + U) λ = ε λ λ λ = n(l 1)jm 2 j For atoms (include Coulomb attraction to nucleus) λ = nlm l 1 2 m s

7 Nucleons in nuclei Atoms: shell closures at 2,10,18,36,54,86 Similar features observed in nuclei Notation: # of neutrons # of protons # of nucleons N Z A = N + Z Equivalent of ionization energy: separation energy for protons for neutrons S p (N, Z) =B(N, Z) B(N, Z 1) S n (N, Z) =B(N, Z) B(N 1,Z) binding energy M(N, Z) = E(N, Z) c 2 = Nm n + Zm p B(N, Z) c 2

8 Shell closure at N=126 Odd-even effect: plot only even Z 8 7 S n (MeV) 6 5 Solid: N-Z=41 Dashed: N-Z= N Also at other values N and Z

9 Illustration of odd-even effect from Bohr & Mottelson Vol.1 (BM1)

10 BM1 figure Neutrons

11 BM1 figure Protons

12 Systematics excitation energies in even-even nuclei Ground states 0 + First excited state almost always 2 + Excitation energy in MeV

13 Heavy nuclei Magic numbers for nuclei near stability: Z=2, 8, 20, 28, 50, 82 N=2, 8, 20, 28, 50, 82, 126

14 Nuclear shell structure Ground-state spins and parity of odd nuclei provide further evidence of magic numbers Character of magic numbers may change far from stability (hot) Odd-N even Z T z =1/2 T z =9/2 Odd-N even odd Z T z =0 A. Ozawa et al., Phys. Rev. Lett. 84, 5493 (2000) T z =5 N=20 may disappear and N=16 may appear

15 Empirical potential Analogy to atoms suggests finding a sp potential shells + IPM Difference(s) with atoms? Properties of empirical potential overall? size? shape? Consider nuclear charge density Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987)

16 Nuclear density distribution Central density (A/Z* charge density) about the same for nuclei heavier than 16 O, corresponding to 0.16 nucleons/fm 3 Important quantity Shape roughly represented by ρ ch (r) = ρ exp ( ) r c z c 1.07A 1 3 fm z 0.55fm Potential similar shape

17 BM1 Empirical potential U = Vf(r)+V ls ( l s 2 ) r r d dr f(r) Central part roughly follows shape of density Woods-Saxon form Depth radius + neutrons - protons diffuseness f(r) = V = [ 1 + exp [ 51 ± 33 R = r 0 A 1/3 with a =0.67 fm ( r R a ( N Z A )] 1 )] r 0 =1.27 fm MeV

18 Analytically solvable alternative Woods-Saxon (WS) generates finite number of bound states IPM: fill lowest levels nuclear shells magic numbers reasonably approximated by 3D harmonic oscillator #! "! U HO (r) = 1! 2 mω2 r 2 A=100 V 0 H 0 = p2 2m + U HO(r) +1-, !"!!#!!$!!%! Eigenstates in spherical basis!&!!'!! " # $ % & ' ( ) * "! +,+-./0 H HO nlm l m s = ( ω(2n + l ) V 0) nlml m s

19 Filling of oscillator shells # of quanta N n l Harmonic oscillator N =2n + l # of particles magic # parity