Physics 249 Lecture 31, Nov 19th 2012
Reading: Chapter 11
HW 9: due Friday Nov 30th
1) Nuclear size
Nuclear size is a fundamental parameter for nuclear physics. The physics of the nuclear
wave function should be determined by the potentials of the EM force and the strong
force. The functional form and strength of those potentials will determine the primary
characteristics of the nuclear wave function and thus the structure of the nucleus.
Another important factor is the spin of the nucleons, 1/2 , making them fermions. This
effects how they fill energy levels. The clearest observable feature of the nuclei is the
radii. Since the structure of the nucleus and the form of the strong potential were
unknown this type nuclear radii where used to understand the form of the potential.
Nuclear size can be experimentally determined in various ways.
Scattering of alpha particles. Observe the change in scattering behavior when you
transition from electromagnetic repulsion to strong attractions.
Scattering of neutrons. Scatter by purely strong attractions.
Scattering of electrons. Observe diffraction effects.
These experiments showed a consistent picture of the nucleus as spherical object with
radial size dependent on A:
𝑅 = 𝑅𝑜 𝐴1/3 where 𝑅𝑜 = 1.2 − 1.4 ± 0.3𝑓𝑚
Note that since the strong scattering indicated the similar size as the diffraction effects
this indicated that the strong force acted with significant strength at ranges only slightly
larger than the nucleus.
Another interesting way to experimentally measure the size of the nucleus involves
mirror nuclei. If you consider isobar with the same number of nucleons but different
numbers of protons and neutrons they should experience different electromagnetic
repulsions. If the strong force is the same for neutrons and protons and the strong for
dominates the size of the nucleus should be determined by the strong force. Then any
different in energy level will be due to the electromagnetic repulsion.
For this to work exactly they have be “mirrors” such that the numbers of protonsneutrons equals the number of neutrons-protons when they are interchanged. For instance
15
15
8𝑂 and 7𝑁
Approximating the nucleus as a uniform ball of charge the electrostatic energy where the
size for two isobars is the same since it is driven by the strong potential the electrostatic
potential energy is:
𝑈=
3 𝑘𝑍 2 𝑒 2
5 𝑅
and
∆𝑈 =
3 𝑘𝑒 2 2
(𝑍 − (𝑍 − 1)2 )
5 𝑅
As an example this energy different can be compared to the energy of a beta decay that
converts oxygen to nitrogen.
Any of these studies lead to a dependence of the nuclear size on A like:
𝑅 = 𝑅𝑜 𝐴1/3 where 𝑅𝑜 = 1.2 − 1.4 ± 0.3𝑓𝑚
Where the functional dependence is what you would expect from a spherical volume
packed by spherical balls.
More exact analysis of the data can be done to determine more detailed information on
the nuclear charge distribution. For instance when calculating the angular distribution of
the scattering you can account for the information that the charge is not concentrated at a
point, as was done in the Rutherford scattering calculation.
2) Nuclear shape.
The shape of the nucleus can be determined by studying the atomic orbitals, which are
dependent on the electric potential of the nucleus. In the hydrogen atom solution we
assumed a point like charge distribution. This was a good enough approximation to
understand all the qualitative features of the electron orbitals and get exact numerical
results in many cases. We were able understand atoms with more protons and multiple
electrons as having the same qualitative orbital features with deviations that could be
reasoned intuitively based on simple arguments and calculated using perturbation theory.
Similar arguments can be employed to study the shape of the nucleus. The finite size of
the nucleus will change the potential inside the radius of the potential. Any deviation in
the shape of the nucleus will result in a non-spherical potential, which can be
decomposed into electric field moments. For instance an elongation along any axis or
pair of axis’s (squashed) will lead a quadruple term in the electric field. Deviations in the
atomic orbitals due to these effects will be greatest for those orbitals that have the largest
radial and even nuclear penetrations. From this analysis we find in general the nuclei are
spherical while some larger nuclei are elongated or squashed.
3) Nuclear energy level filling and nuclear stability.
The filling of nuclear energy levels can be understood in terms of the following
considerations.
a) The nucleons in the nuclease are essentially fermions confined in a very small 3D
potential well. Some details will be different than in the electron orbitals since the
potential is not 1/r.
i) Differences in the radial wave functions can cause more drastic changes in the ordering
of the sub shells.
ii) Spin orbit coupling is stronger because of the proximity and can cause larger splitting.
iii) The magnetic moments of the neutron and proton are not the same changing the
splitting. Note that the neutron has a magnetic moment despite being neutral!
b) The Fermi exclusion principle will cause the cause each energy level to be filled in
turn with two spin ½ particles. The fact that the neutron and protons are different
particles results in alternating between pairs of the neutrons and protons.
c) It will be more favorable to fill energy levels with neutrons first and then protons since
the protons add electrostatic repulsion energy to the potential.
d) With more protons the electrostatic energy of repulsion can be large enough to exceed
the nuclear energy level spacing and it can be more favorable energetically to fill
neutrons first and then protons.
e) All of this assumes some sort of interaction that can convert a neutron to a proton or
vice versa if the other state is more favorable. This exists in beta decay.
This leads to a picture of energy level filling where levels are filled first with pair of
neutrons then a pair of protons and for larger elements neutrons start to exceed protons by
increasingly large amounts. As a function of the energy difference between a higher
energy unstable configuration of nucleons and the lowest energy configuration a nucleon
in the unstable configuration will decay to produce the other type of nucleon and a more
stable configuration. A range or proton and neutron configurations for a given A will be
approximately or fully stable.
Note that the difference in proton and neutron energy levels that leads to first two neutron
levels filling and then two protons levels will also make it most likely to see nuclei with
an even number of protons, neutrons or both and almost never an odd number of protons
and neutrons. This can happen if the proton repulsion for filling the second proton is
greater than the gap to the next neutron energy level. This factor increases at high Z
since the potential energy goes as Z2.
Higher energy orbitals will have greater number of equal energy orbitals even after
splitting. These can lead to larger numbers of stable isotopes for a given Z or isotones for
a given N. Many stable configurations are possible at 20, 28, 50. These stable regions
should be longer for a fixed proton number (vertically in the figure) than fixed neutron
number since proton repulsion breaks the energy symmetry for different numbers of
protons.