Download Physics 249 Lecture 32, Nov 21st 2012 Reading: Chapter 11 HW 9

Physics 249 Lecture 32, Nov 21st 2012
Reading: Chapter 11
HW 9: due Friday Nov 30th
(HW 9 will be reposted by Friday)
1) Nuclear energy level filling and nuclear stability.
Given the following considerations:
a) The nucleons nucleon are fermions with spin and magnetic moments
b) The nucleons are confined to in a 3D quantum potential well
c) Proton repulsion is a significant factor in the potential energy.
d) The nucleons have spin and magnetic moment and magnetic potential energy due to
spin orbit coupling and spin-spin (spin alignment) can is a significant factor.
(We will further review these factors to understand them better and qualitatively
understand all aspects of the nuclear energy levels.)
We expect:
A set of quantum orbitals and energy levels like the electron orbitals separately for
neutrons and protons. Since the potential is spherically symmetric the details of the
angular wave functions should be the same but since the potential is not 1/r the details of
the radial wave functions may be different. However, the moments of inertia are small
and thus the angular “energy” large requiring a simultaneous solution like the hydrogen
atom leading to a similar general structure for the energy levels.
(we will revisit the issue of a spherically symmetric potential and wave functions to
understand why the potential is spherically symmetric)
In multi nucleon atoms the difference in proton and neutron energy levels due to proton
repulsion leads to first two neutron levels filling and then two protons levels. This factor
increases at high Z since the potential energy goes as Z2. Eventually it can be more
energetically favorable to fill a neutron energy level than a proton level. At high Z
significantly more neutron levels will be filled than proton levels
These factors 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. Odd-Odd
proton and neutron nuclei can occur if the proton repulsion for filling the second proton is
greater than the gap to the next neutron energy level.
Since the strong potential is “strong” radial penetration can make a large difference in the
energy of a given orbital. This will lead to more reversals of standard ordering.
(well examine this structure in more detail later).
Splitting of energy levels will be large because the magnetic potential is large due to the
small distances between nucleons. They will be large enough to cause changes in orbital
ordering.
(we will revisit this issue to understand nuclear magnetic moments and potential energy)
Higher energy orbitals will have greater number of equal energy orbitals even after
splitting. Since splitting is due to spin you can only split into two energy levels. These
can lead to larger numbers of stable isotopes for a given Z and isotones for a given N. If
the energy levels for the protons and neutrons similar you can have a stable configuration
with i+j=k protons+neutrons=total in the energy level. There are many stable
configurations are possible at 20, 28, 50. Looking at specific numbers of protons or
neutrons 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.
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.
There is generally a line of stability in a N vs Z plot that starts with N=Z initially be
diverges from that line with N>Z increasingly with higher N. The line of stability
indicates the lowest potential energy filling of the nucleon energy states for the proton
and neutron for a give A. Around this line there are stable isotopes in cases where due to
orbitals with larger numbers of nucleons are approximately stable. Unstable isotopes can
be created and will live for varying lifetimes and eventually undergo nuclear decay
interactions such as beta decay to become stable isotopes (technically I should say
isobars). The further the isotope is from the line of stability the faster it tends to decay.
Typically a region of lifetime around a millisecond is defined to indicate the more
commonly observed isotopes.