Nuclear Decay Forest NucPhys I
Alpha Decay
The spontaneous emission of an alpha particle
is the result of a natural decay process which can be described as the tunneling of energy ( in the form of the alpha particle) through the coulomb barrier. In other words, if a collection of nucleons within a nucleus finds itself sufficiently close to the nuclear force potential well limit, then a coulomb repulsion force can begin to dominant and facilitate the tunneling of this collection of nucleons ( an alpha particle) through the confining potential well.
The decay process can be represented by the following reaction notation
Q-value
The "Q-value" represents the net mass energy released in a nuclear reaction.
In the above example the Q value is calculated :
- : assume nucleus is initially at rest
A positive Q value (Q>0) identifies a reaction as exothermic (exoergonic) which means that energy is given off and that the reaction is spontaneous
A negative Q value (Q<0) identifies the reaction as endothermic (endoergonic) which means that energy is required to for the reaction to take place.
Example
The positive Q value (Q>0) identifies the reaction as exothermic (exoergonic) which means that energy is given off and that the reaction is spontaneous
A negative Q value (Q<0) identifies the reaction as endothermic (endoergonic) which means that energy is required to for the reaction to take place.
Kinetic energy of alpha
Since the original nucleus was at rest, the final nuclei will have the same momentum in opposite directions in order to conserve momentum.
Example
- Notice
- The alpha particle caries away most of the kinetic energy.
The nuclear fragment (Y) does have a non-negligible amount of energy which can be sufficient to escape the material it is embedded in if it is on the order of a few microns from the materials surface. Heavy nuclei loose energy quickly when traveling through material.
Kinetic energy of alpha
Geiger-Nuttal Law
In 1911 Geiger and Nuttal noticed that the decay half life (
of nuclei that emmitt alpha particles was related to the disentegration energy .It works best for Nuclei with Even
and Even . The trend is still there for Even-Odd, Odd-Even, and Odd-odd nuclei but not as pronounced.cluster decays
The Gieger-Nuttal Law has been extended to describe the decay of Large A (even-even and odd A) nuclei into clusters in which Silicon or Carbon are one of the clusters.
http://prola.aps.org/pdf/PRC/v70/i3/e034304
Theory of alpha emission
Barrier problem
Decay half life
The disintegration constant for \alpha emission may be expressed as
where
f = number of times the alpha particle tries to escape the well by interacting with the barrier P = probability that the alpha particle escapes when it hits the barrier
the half life
is then proportional to .Example
Curium
Gamma Decay
Beta Decay
Types of decay
- negative beta decay
- positive beta decay
- electron capture
negative beta decay
- let
- where
- = ith elctron binding energy
then
positive beta decay
electron capture
Conservation rules
baryon number is conserved
where
- is the number of constituent quarks, and
- is the number of constituent antiquarks.
beta decay just changes p to n or n to p so the number of quarks dont change just the flavor (isospin).
Up and down quarks each have isospin
, and isospin z-componentsAll other quarks have I = 0. In general
Lepton number is conserved
so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0.
Angular momentum
- Consider first that the net angular momentum is zero (only consider spins)
The \beta and neutrino are spin 1/2 objects, therefore their spins may be either parallel or anti-parallel.
Fermi decay
A
decay in which the and neutrino spins are anti-parallel is known as Fermi decay.This means
- no change in the spin of the nucleus
- Examples
also
parity is conserved: .- = excited state of N
Gamow-Teller decay
A
decay in which the and neutrino spins are parallel is known as Gamow-Teller decay. Interactions which take place are known as axial-vector.In terms of total angular momenum \vec{I} the transition is
- Examples
also
parity is conserved: the final Li-6 state have a radial component to its wave function which has negative parity.Mixed Fermi and Gamow-Teller decay
It is possible that
decay can be a mixture of the two decay types. Some of the time the remaining nucleus is in an exited state other times the decay is directly to the ground state.
- Examples
or
The above reaction involves "mirror" nuclei, nuclei in which the number of protons and neutrons is just interchange.
One can measure the angular distributions of
particles to determine what the mixture is between the two decay types (Fermi and Gamow-Teller).The mixture can be expressed as a ratio of matrix elements (Fermi's golden rule relates transitions to Matrix elements)
The interesting observation is that "y" for mirror nuclei is on the order of the value of "y" for neutron decay while nonmirror nuclear decays tend to be an order of magnitude less.
- What does this mean?
The CVC (Conservation of Vector Current) hypothesis was born. The Fermi decay is the result of a vector current and is dominant in the decay of the neutron to a proton while the Gammow-Terller decay is an an axial-current transition. CVC is the assumption that the weak vector current responsible for the decay is conserved. Another observation is that the mirror nuclei transition ( mostly Fermi) illustrates how the nucleons inside the nucleus interact as free particles despite being surrounded by mesons mediating the nuclear force.
Forbidden decays
The Fermi decays
are often times refered to as the superallowed decays which Gamow Teller are simple "allowed" decays.Forbidden decays are those which are substantially more improbable, due to parity violation, and as a result have long decay times.
Decay rate
A calculation of the
emmission decay rate is quite different from a calculation of decay. In \alpha decay the nucleons of the original nucleus are used to form the fnial state particle (He-4). In decay the and neutrino particles are the result of a nucleon transformation into its isospin complement . Below is a list of the differences- the and neutrino did not exist before the decay
- The and neutrino are relativistic (nuclear decay energy usually no enough to make heavy \alpha nucleus relativistic)
- The light decay products can have continuous energy distributions. (before assuming the carried away most of the eergy was usually
a good approximation)
The Fermi's Golden rule(see Forest_FermiGoldenRule_Notes) says that the transition rate is given by a transition matrix element (or "Amplitude") weighted by the phase space and Plank's constant such that
decay rate calculation was developed by Fermi in 1934 and was based on Pauli's neutrino hypothesis.- (Phase Space)
The underlying assumption is that the transition is a weak purturbation of the system. This assumption appears to be true based on the very short time scale (
sec) it takes for the formation of quasi-stationary nuclear states compared with the time it takes for a decay ( half lives ranging from seconds to days)