Lecture 5
Nuclear Models, cont.
5.1 The Fermi gas model:
In this model, nuclei are considered to be composed of two fermion gases, a
neutron gas and a proton gas. The particles do not interact, but they are confined in
a sphere which has the dimension of the nucleus. The interactions appear implicitly
through the assumption that the nucleons are confined in the sphere.
The liquid-drop model is based on the saturation of nuclear forces and one
relates the energy of the system to its geometric properties. The Fermi model is
based on the quantum statistics effects on the energy of confined fermions. The
Fermi model provides a means to calculate the constants av, as and aa in the Bethe–
Weizs¨acker formula, directly from the density ρ of the nuclear matter. Its semiquantitative success further justifies for this formula.
The Fermi model is based on the fact that a spin 1/2 particle confined to a
volume V can only occupy a discrete number of states. In the momentum interval ,
the number of states is:
5.1
At temperature T = 0, i.e. for the nucleus in its ground state, the lowest states
will be filled up to a maximum momentum, called the Fermi momentum pF. The
number of these states follow from integrating eq.(5.1) from 0 to pF:
5.2
Since each state can be filled with two nucleons of the same type,
5.3
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where pF,n and pF,p are the Fermi momenta of the neutrons, respectively, the
protons. Using our prior knowledge from electron scattering, i.e.
5.4
we obtain for Z = N = A/2 and equal radius for the two separate potential wells of
the protons and the neutrons a Fermi momentum of
5.5
— The Fermi energy is the energy of the highest occupied nucleon level:
5.6
5.7
2
5.8
In the approximation Z ∼ N ∼A/2, the kinetic energy is of the form:
5.9
The second term is the surface coefficient of equation 5.9 in good agreement with
5.2 The shell model:
5.2.1 Magic Numbers
The binding energies predicted by the Liquid Drop Model underestimate the
actual binding energies of “magic nuclei” for which either the number of neutrons
N = (A − Z) or the number of protons, Z is equal to one of the following “magic
numbers”
2, 8, 20, 28, 50, 82, 126.
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(5.10)
This is particularly the case for “doubly magic” nuclei in which both the
number of neutrons and the number of protons are equal to magic numbers.
For example for 56 28Ni (nickel) the Liquid Drop Model predicts a binding energy
of 477.7 MeV, whereas the measured value is 484.0 MeV. Likewise for 132 50 Sn
(tin) the Liquid Drop model predicts a binding energy of 1084 MeV, whereas the
measured value is 1110 MeV.
There are other special features of magic nuclei:
• The neutron (proton) separation energies (the energy required to remove the last
neutron (proton)) peak if N (Z) is equal to a magic number.
• There are more stable isotopes if Z is a magic number and more stable isotones
if N is a magic number.
• If N is magic number then the cross-section for neutron absorption is much
lower than for other nuclides.
• The energies of the excited states are much higher than the ground state if either
N or Z or both are magic numbers.
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• Elements with Z equal to a magic number have a larger natural abundance than
those of nearby elements.
5.2.2 Simple Shell Model
The basic assumption of the shell model is that the effects of internuclear
interactions can be represented by a single-particle potential. One might think that
with very high density and strong forces, the nucleons would be colliding all the
time and therefore cannot maintain a single-particle orbit. But, because of Pauli
exclusion the nucleons are restricted to only a limited number of allowed orbits. A
typical shell-model potential is
where typical values for the parameters are Vo ~ 57 Mev, R ~ 1.25A1/3 F, a ~ 0.65
F.
For such a potential it turns out that the lowest level is 1s (i.e. n = 1, l = 0)
which can contain up to 2 protons or neutrons. Then comes 1p which can contain
up to a further 6 protons (neutrons). This explains the first 2 magic numbers (2 and
8). Then there is the level 1d, but this is quite close in energy to 2s so that they
form the same shell. This allows a further 2+10 protons (neutrons) giving us the
next magic number of 20.
The next two levels are 1f and 2p which are also quite close together and
allow further 6+14 protons (neutrons). This would suggest that the next magic
number was 40 – but experimentally it is known to be 50.
We note that the harmonic potential, like the Coulomb potential, has the
peculiarity that the energies depend only on the principal quantum number n and
not on the angular momentum quantum number l.
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The magic numbers corresponding to all shells filled below the maximum n,
as shown on the fourth line of Table , would then be 2, 8, 20, 40, 70, 112 and 168
in disagreement with observation (5.10). It might be expected that one could find
another simple potential that would give the correct numbers.
In general one would find that energies would depend on two quantum
numbers: the angular momentum quantum number l and a second giving the
number of nodes of the radial wave function. An example of such a l-splitting is
shown in Fig. 2.10. Unfortunately, it turns out that there is no simple potential that
gives the correct magic numbers.
The solution to this problem, found in 1949 by M. G¨oppert Mayer, and by
D. Haxel J. Jensen and H. Suess, is to add a spin orbit interaction for each nucleon:
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Without the spin-orbit term, the energy does not depend on whether the
nucleon spin is aligned or anti-aligned with the orbital angular momentum. The
spin orbit term breaks the degeneracy so that the energy now depends on three
quantum numbers, the principal number n, the orbital angular momentum quantum
number l and the total angular momentum quantum number j = l±1/2. We note that
the expectation value of ˆl.ˆs is given by
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5.4 Excited States
As in the case of Atomic Physics, nuclei can be in excited states, which
decay via the emission of a photon (ɤ-ray) back to their ground state (either
directly ore indirectly).
Some of these excited states are states in which one of the neutrons or
protons in the outer shell is promoted to a higher energy level.
However, unlike Atomic Physics, it is also possible that sometimes it is
energetically cheaper to promote a nucleon from an inner closed shell, rather than a
nucleon form an outer shell into a high energy state. Moreover, the excited state in
which more than one nucleon is promoted above its ground state is much more
common in Nuclear Physics than in Atomic Physics.
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