Lesson 5 The Shell Model

Transcription

1 Lesson 5 The Shell Model

2 Why models? Nuclear force not known! What do we know about the nuclear force? (chapter 5) It is an exchange force, mediated by the virtual exchange of gluons or mesons.

3 Electromagnetic force Virtual exchange of photons ΔE Δt h ΔE Δt h h c R = cδt = ΔE hc E γ

4 Nuclear Force Virtual exchange of particles of mass m h Δt t 2 mc h R mc R = 1.4 fm m 140MeV / c 2 Assuming nuclear force is short range saturation property of nuclear forces

5 Nuclear Force Strongly attractive Repulsive core Not spherically symmetric (deuteron quadrupole moment), has symmetric central component and asymmetric tensor component. Spin dependent (deuteron ground state is triplet, singlet state is unbound)

6 Nuclear potential

7 Other potentials of note

8 Other potentials of note

9 Table 5-1 Charge independence of nuclear forces The nuclear force between a neutron and a proton is the same as the force between two protons or two neutrons. A Nucleus Total Binding Energy (MeV) Coulomb Energy (MeV) Net nuclear binding energy (MeV) 3 3 H He C N Na Ne Ca Sc

10 Isospin Consider that the neutron and the proton are just two states of the nucleon. Consider further that these two states are labeled by a quantum number, T, called isospin. For the nucleon, T=1/2. There are two projections of T in isospace, T 3 =+1/2 (proton) and T 3 =-1/2 (neutron) For a nucleus containing Z protons and N neutrons, T 3 =(Z-N)/2.

11 Example Consider the A=14 isobars, 14 C, 14 N and 14 O. 14 C and 14 O have T 3 =±1. Thus they must be part of an isospin multiplet, T=1. In 14 N, T 3 =0, but there must be a state with T=1. This state is called the isobaric analog of the ground states of 14 C and 14 O.

12 Getting back to the shell model Evidence for nuclear shell structure: Peaks in binding energy /nucleon Changes in separation energies for certain numbers of neutron and protons Magic numbers (2,8,20,28,50,82,.) Zero quadrupole moments Low neutron absorption cross sections Abundance systematics

13 More Examples

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15 Spin-orbit coupling

16 More on spin-orbit force Consider the Woods-Saxon potential

17 Aufbau Prinzip 1. Fill in lowest energy first 2. Pair up particles as you add them ( Katz s rule ) 3. Spin and parity of the ground state is the spin and parity of the last nucleon for odd A nuclei. For e e nuclei, J,π =0+. For o o nuclei, use Brennan Bernstein rules.

18 Reality Check

19 Brennan-Bernstein Bernstein Rules 1. When the odd nucleons are both particles or holes in their respective subshells, Rule 1 states that when j 1 = l 1 ½ and j 2 = l 2 ½, then J = j 1 j Rule 2 states that when j 1 = l 1 ½ and j 2 = l 2 ½, then J = j 1 j 2 3. Rule 3 states that for configurations in which the odd nucleons are a combination of particles and holes, such as 36 Cl, J=j 1 + j 2 +1

20 Example Consider the o-o nuclei, 38 Cl, 26 Al and 56 Co. Predict the ground state spin and parity for these nuclei. (a) 38 Cl has 17 protons and 21 neutrons. The last proton is in a d 3/2 level while the last neutron is in an f 7/2 level. (Figure 6-3). j p = 2 1/2, j n = 3 + 1/2 J = 7/2 3/2 = 2 π = (b) 26 Al has 13 protons and 13 neutrons. The last proton and the last neutron are in d 5/2 hole states, i.e., j p = j n = 2 + 1/2. J = 5/2 + 5/2 = 5 π = + (c) 56 Co has 27 protons and 29 neutrons. The last proton is in a f 7/2 hole state and the last neutron is in a p 3/2 state (1 + 1/2) J = 7/2 + 3/2 1 = 4 π = +

21 Statistics on B-BB rule.

22 Successes of Shell Model Ground state spins and parities of nuclei Some information about excited states

23 Magnetic Moments; Schmidt Limits Focus on a single particle μ i = g l L i + g s S i g l = lμ 0, g s = μ 5845μ 0 for protons and g l = 0, g s = μ 0 for neutrons where μ 0 is the nuclear magneton: μ 0 = eħ/2m p c

24 Application to Nuclei; Schmidt limits For j = l + s, μ = lg l + ½ g s For j = l -s, μ = (j/j+1)[(l + 1) g l ½ g s ]

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26 Islands of Isomerism What is a nuclear isomer? What causes an isomer? Where do they occur?

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29 Mirror nuclei Definition: nuclear-pairs in which the numbers of protons and neutrons are interchanged, for example, 3 He and 3 H

30 Failures of the Shell Model Positions and origin of low lying 2+ states in nuclei Rotational and vibration levels in deformed nuclei, like the rare earths and the actinides.