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Shell Model with residual interactions – mostly 2-particle systems

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### Shell Model with residual interactions – mostly 2-particle systems

Simple forces, simple physical interpretation

Lecture 2

Independent Particle Model

- Some great successes (for nuclei that are “doubly magic plus or minus 1”).
- Clearly inapplicable for nuclei with more than one particle outside a doubly magic “core”. In fact, in such nuclei, it is not even defined. Thus, as is, it is applicable to only a couple % of nuclei.

IPM cannot predict even these levels schemes of nuclei with only 2 particles outside a doubly magic core

IPM too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure

- Residual interactions – examples of simple forms
- Pairing – coupling of two identical nucleons to angular momentum zero. No preferred direction in space, therefore drives nucleus towards spherical shapes
- p-n interactions – generate configuration mixing, unequal magnetic state occupations, therefore drive towards collective structures and deformation
- Monopole component of p-n interactions generates changes in single particle energies and shell structure

So, we will have a Hamiltonian H = H0 + Hresid.where H0 is that of the Ind. Part. ModelWe need to figure out what Hresid. does.Since we are dealing with more than one particle outside a doubly magic core we first need to consider what the total angular momenta are when the individual ang. Mon. of the particles are vector-coupled.

Coupling of two angular momenta

j1+ j2 All values from: j1 – j2 to j1+ j2 (j1 =j2)

Example: j1 = 3, j2 = 5: J = 2, 3, 4, 5, 6, 7, 8

BUT: For j1 = j2: J = 0, 2, 4, 6, … ( 2j – 1) (Why these?)

/

How can we know which total J values are obtained for the coupling of two identical nucleons in the same orbit with total angular momentum j? Several methods: easiest is the “m-scheme”.

How can we understand the energy patterns

that we have seen for two – particle spectra

with residual interactions? Easy – involves

a very beautiful application of the Pauli

Principle.

This is the most important slide: understand this and all the key ideas about residual interactions will be clear !!!!!

Extending the Shell Model to 3-particle sysetms

- Consider now an extension of, say, the Ca nuclei to 43Ca, with three particles in a j= 7/2 orbit outside a closed shell?
- How do the 3 - particle j values couple to give final total J values?
- If we use the m-scheme for 3 particles in a 7/2 orbit, the allowed J values are 15/2, 11/2, 9/2, 7/2, 5/2, 3/2.
- For the case of J = 7/2, two of the particles must have their angular momenta coupled to J = 0, giving a total J = 7/2 for all three particles.
- For the J = 15/2, 11/2, 9/2, 5/2, and 3/2, there are no pairs of particles coupled to J = 0.
- What is the energy ordering of these 6 states? Think of the 2-particle system. The J = 0 lies lowest. Hence, in the 3-particle system, J = 7/2 will lie lowest.

Think of the three particles as 2 + 1. How do the 2 behave?

We have now seen that they prefer to form a J = 0 state.

43Ca

Treat as 20 protons and 20 neutrons forming a doubly magic core with angular momentum J = 0. The lowest energy for the 3-particle configuration is therefore

J = 7/2.

Note that the key to this is the result for the 2-particle system !!

Multipole Decomposition of Residual Interactions

We have seen that the relative energies of 2-particle systems affected by a residual interaction depend SOLELY on the angles between the two angular momentum vectors, not on the radial properties of the interaction (which just give the scale).

We learn a lot by expanding the angular part of the residual interaction,

Hresidual = V(q,f)

in spherical harmonics or Legendre polynomials.

Two mechanisms for changes in magic numbers and shell gaps

- Changes in the single particle potential – occurs primarily far off stability where the binding of the last nucleons is very weak and their wave functions extend to large distances, thereby modifying the potential itself.
- Changes in single particle energies induced by the residual interactions, especially the monopole component.

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