The Collective Model and the Fermi Gas Model

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Presentation transcript:

The Collective Model and the Fermi Gas Model Lesson 6 The Collective Model and the Fermi Gas Model

Evidence for Nuclear Collective Behavior Existence of permanently deformed nuclei, giving rise to “collective excitations”, such as rotation and vibration Systematics of low lying 2+ states in many nuclei Large transition probabilities for 2+0+ transitions in deformed nuclei Existence of giant multipole excitations in nuclei

Deformed Nuclei Which nuclei? A=150-190, A>220 What shapes?

How do we describe these shapes?

Rotational Excitations Basic picture is that of a rigid rotor

Are nuclei rigid rotors? No, irrotational rigid

Backbending

Rotations in Odd A nuclei The formula on the previous slide dealt with rotational excitations in e-e nuclei. What about odd A nuclei? Here one has the complication of coupling the angular momentum of the rotational motion and the angular momentum of the odd nucleon.

Rotations in Odd A nuclei (cont.) For K  1/2 I=K, K+1, K+2, ... For K=1/2 I=1/2, 3/2, 5/2, ...

Nuclear Vibrations In analogy to molecules, if we have rotations, then we should also consider vibrational states How do we describe them? Most nuclei are spherical, so we base our description on vibrations of a sphere.

Shapes of nuclei shaded areas are deformed nuclei Types of vibrations of spherical nuclei

Formal description of vibrations Suppose =0 Suppose =1 Suppose =2

Nuclear Vibrations

Levels resulting from these motions

You can build rotational levels on these vibrational levels

Net Result

Nilsson model Build a shell model on a deformed h.o. potential instead of a spherical potential Superdeformed nuclei Hyperdeformed nuclei

Successes of Nilsson model Nuclide Z N Exp. Shell Nilsson 19F 9 10 1/2+ 5/2+ 21Ne 11 3/2+ 21Na 23Na 12 23Mg

Real World Nilsson Model

Fermi Gas Model Treat the nucleus as a gas of non-interacting fermions Suitable for predicting the properties of highly excited nuclei

Details Consider a box with dimensions Lx,Ly,Lz in which particle states are characterized by their quantum numbers, nx,ny,nz. Change our descriptive coordinates to px,py,pz, or their wavenumbers kx,ky,kz where ki=ni/Li=pi/ The highest occupied level Fermi level.

Making the box a nucleus

Thermodynamics of a Fermi gas Start with the Boltzmann equation S=kBln Applying it to nuclei

Relation between Temperature and Level Density Lang and LeCouteur Ericson

What is the utility of all this?