PHYSICS 430
NUCLEAR PHYSICS
Chapter 9
SYNOPSIS:
sections.
The following comments should help you to focus on the major physics in these
9-1.
For each of the decays,  , and EC, you should be able:
• to write the equations for the decay.
• to describe the physics which gives rise to the observed energy distributions of the emitted
particles.
• to write the conservation of energy equation and the Q value for each of these three decays in
terms of the nuclear masses of the nuclei.
• to derive the Q value in terms of the isotopic masses of the neutral atoms and articulate the
assumptions made in getting to the workable equations.
• to understand how the Q value is modified if the decay is from an excited state of the parent.
• to understand how the Q value is modified if the decay is to an excited state of the daughter.
• what is the endpoint energy? That is, what is the decay configuration of the particles in the final
state which will give rise to the maximum energy carried away by the  particle?
9-2.
This section uses quantum mechanics and a set of (reasonable) assumptions to attempt to (a) derive
an expression for the decay rate and the lifetime for nuclei which are unstable against  decay. We
begin with Fermi's Golden Rule and focus on the two factors, the matrix element and the density of
final states. Along the way, we make assumptions to simplify the calculations. You must be able to
describe the assumptions and why each is reasonable. We will invoke the concept of phase space to
express the density of final states. This is described briefly in your text; a better discussion is
available in Segre and we will do this is detail later in the semester. I will not expect that you can
derive all of this on your own (i.e., closed book) but you should understand it well enough to
discuss the physics. For example:
• Why can you write the electron and neutrino wave functions as free particle wave functions as
shown in equation (9.18)?
• Why is the approximation (9.19) valid? What corrections later do we make for this?
• Why is it reasonable to write the Hamiltonian operator for the decay as a delta function, i.e.,
why is it reasonable to assume that this is a point interaction?
• What is the weak interaction coupling constant, g?
• Why are the measured curves in Fig. 9.3 different from the calculated curves in Fig. 9.2? That
is, what is the presumed physics which causes this difference?
• What is the Fermi Function and why is it included in the calculation of the decay rate?
• What is the Shape Factor and why is it included in the calculation of the decay rate?
9-3.
We take the results from section 2, and compare these with experiment. The emphasis here is on
seeing how the theory agrees or disagrees with measurements, and how we can use measurements
to extract better understanding of the physics.
• What is a Kurie Plot, i.e., what is plotted, what does it show about the agreement between
theory and experiment, and how is it useful in measuring the endpoint energy and the neutrino
mass?
• What is f (the Fermi integral)? i.e., how does it enter in the theory and how does it vary with the
endpoint energy?
• What do we learn from the comparative half-life or ft value?
• How can we extract a measurement of the weak interaction coupling, g? That is, what
measurements can we use and what assumptions might we make to get an estimate of g?
•
9.4.
How would a massive (i.e., m0.) alter the energy or momentum spectrum for the emitted 
particle and where in the spectrum is the effect of a non-zero neutrino mass most evident, and
why?
In this section, we look at physics which can affect the  decay rate but which we have nor formally
included in our development of the Fermi theory up to this point. In other terms, we have made
simplifying assumptions whereby these physics effects were not considered. Specifically, we had
neglected any considerations of angular momentum associated with the leptons (the  and the ).
This surely needs to be considered since at a minimum, all leptons (which are part of the larger
group of particles called Fermions) have an intrinsic angular momentum whose quantum number s
= 1/2. It is said that these spins can be either parallel or anti-parallel. To appreciate what this
means requires recalling some quantum mechanics. If there is no quantization axis chosen in space
by some physical effect (an interaction) then the spin of the particle can point in any direction and
no effect is observed which depends on the direction of the spin. However, if an axis is chosen by
some interaction along which the angular momentum can be quantized, i.e., chosen as the z-axis,
then only certain allowed projections along this axis are allowed and therefore, only certain values
of the z component of the spin angular momentum.
So, consider the case of the two leptons emitted in  decay. If their spin angular momenta couple
(interact) then the rules from quantum mechanics by which angular momenta couple are
appropriate and tell us that the net angular momentum carried off by the lepton system SL can be
calculated as:
ssSL ss
 





Since the spin quantum number for each of the leptons is 1/2, this means that the resulting
combined spin of the lepton system is either 0 or 1. These two allowed states (and only two allowed
states for the combined spins) are called anti-parallel (SL = 0) or parallel (SL = 1) spin states for the
lepton system. Since  decay can give rise to only these two final spin state combinations for the
leptons, these two final states are indications of two different physics interactions. The first (antiparallel) is called the Fermi (F) interaction; the second (parallel) is called the Gamow-Teller (GT)
interaction, named after the physicists who were most instrumental in articulating the underlying
physics (Enrico Fermi, George Gamow, and Edward Teller).
If there are two different interactions, then they should be characterized by two different matrix
elements and two different coupling strengths, thereby replacing
g2Mfi2  g2 MF2 + g 2 MGT2
F
GT
in our equation for the rate. It is, therefore, not surprising that one goal of the study of  decay is to
attempt to gather data which would be sufficient to characterize the nature of the fundamental
physics interactions which give rise to the decay.
Conservation of angular momentum requires that
IP = ID + JL , where JL = SL  LL .
The subscript L refers to the lepton system (the  and the ), and LL is the net "orbital" angular
momentum of the lepton system. We will learn how to make a useful table to assist in this analysis
so as to classify the decay schemes by the relevant physics which drives the transition. You should
appreciate the arguments made in the middle of page 291 concerning the semi-classical estimate of
the angular momentum and the implications for the probability of finding the lepton system in
quantum states of orbital angular momentum. The outline of the more rigorous treatment of this is
given in section 5.
An additional tool which is especially useful in this process is the conservation of parity. If the
nucleus makes a transition from one "stationary" state (parent) to another stationary state
(daughter) the transition is from a state of definite parity to another of definite parity. Moreover, if
the decay conserves parity, then one can write,
 = 0.0 P= D  (-1)l L
where lL is the net "orbital" angular momentum quantum number of the lepton system.
You should be able to apply these physics principles to the study of nuclei which undergo decay.
For each  decay, you should be able:
• to determine whether the decay is caused by a Fermi (F) interaction or a Gamow-Teller (GT)
interaction, or both (mixed).
• to determine whether the decay is "allowed" (or "superallowed"), "first forbidden", "second
forbidden", etc., and what these terms mean in a quantitative way in terms of the angular
momentum of the final state particles and the implications for the decay rate (see next section).
9.5.
Krane attempts to show you on a more sound quantum mechanical footing how "forbiddenness"
arises.
• Note that the first term in the expansion of the lepton wave function corresponds to the
"allowed" transition, i.e., it is the one we used initially when we ignored any angular
momentum.
• Note that the second term in the expansion is proportional to the angular momentum and its
expectation value is obtained by carrying out the integral between the initial and final state
wave functions. If the concepts in this section are unclear, we should discuss them; please ask
questions.
• In Fig. 9.8, note the strong dependence of f, the Fermi integral, on the endpoint energy of the
beta particle. You can use these curves to estimate the Fermi integral if you know Z and
(Te)max. Since f is defined to contain F, the Fermi function, where F accounts for the Coulomb
•
effects, you see different curves for  and  in Fig. 9.8.
From section 3 you should know how to solve for ft1/2 from the results of the Fermi theory.
The point here is that you measure t1/2 and you calculate f, and note that decays with similar
log (ft1/2 ) values correspond in general to a given "forbiddenness". Note especially the ft1/2
values on page 285, Table 9.2. These are all of order 3000, and are all at the low end of the
log(ft1/2 ) plot in Fig. 9.9, page 294.
•
9.6.
The several paragraphs on page 295 are an especially nice summary of this section.
Neutrino physics - watch out! I spent approximately 20 years working in this field...I love it!
Reading this section, you should:
• Understand how one can write the inverse  decay process from the observed  decay by
conserving all the relevant quantities (energy, linear momentum, angular momentum, electric
charge, baryon number, and lepton number) by simply moving one particle in the  decay
equation to the other side of the equation and turning it into its antiparticle.
• Follow the order-of-magnitude estimate of the neutrino cross section and the associated mean
free path <x>.


• Understand the source of  which Reines and Cowan, et al, used for their  beam and be able
to describe the experimental measurement which they made, i.e., of what did the detector
•
•
consist and what was the experimental signal which they observerved to convince themselves

that they were seeing the inverse  decay process and thereby detecting the  .

Be able to describe the experiment done by Davis, et al, to demonstrate that the  was a
different particle from the .

Know what is the significant difference between the  and the  , i.e., know how helicity is

defined and know what the helicity of the  and the  are. (You should also know that the 
particles from  decay also have a helicity.
9.7.
Why does  occur? Krane illustrates two reasons why it may occur. What is tricky about the
measurement to convince yourself that you are detecting  decay? What is neutrinoless  decay?
Has it been observed? What conservation law (as we know it) does this decay violates?
9.8.
Beta-delayed nucleon emission is a process whereby highly unstable nuclei, being created by 
decay, proceed toward stability by emitting a nucleon. The  decay proceeds in the usual manner
followed by a nucleon emitted from the daughter nucleus essentially immediately, competing with
photon de-excitation of the highly unstable excited nucleus. Much can be learned by studying this
process in terms of nuclear structure. And, this process is very important in fission systems since it
can be a source of neutrons after the neutrons from fission have been stopped. Your reading of this
section is commended; we will not discuss it in detail in class, but you are welcome to visit with me
to discuss it further.
9.9.
The detection of parity non-conservation was one of the great discoveries in modern physics since it
demonstrates what is an apparent difference between the microscopic world and the macroscopic
world. To detect parity non-conservation requires a rather clever approach since it is evidently a
small effect or we would have observed parity non-conservation in our studies of the common
weak interaction processes in  decay. In fact, we do not see evidence of parity non-conservation
unless we look very carefully as your author describes on page 315, and it was only seen much after
parity non-conservation was discovered. You should be able to describe the experiment done by
Wu, et al, to detect parity non-conservation.
• What function did the magnetic field serve?
• Why was it necessary to have the parent nuclei in a sample at very low temperatures?
• What would you expect to detect if parity was not conserved?
Because parity non-conservation in the weak interaction is so small, our use of parity conservation
for the analysis of  decay processes is essentially "always" correct.
9.10. As with  decay, this section is interesting and illustrates how nuclear physicists use data collected
on the energy spectra from  decay to elucidate nuclear structure. I encourage you to read this and
if you wish, to stop by and we can discuss it. There is too much detail here for our class schedule.