4th Week: Decays, Tunneling and Cross Sections B

Nuclear Decays

Nuclear decay occurs when an atom changes the composition of its nucleus. There are many different kinds of reactions. Some occur by an atom ejecting particles from its nucleus or when the nucleons change type.

Alpha Decay

Alpha decay occurs when the parent nucleus ejects an alpha particle (a helium-4 nucleus) and a daughter nucleus is left. This looks like:

$\left(Z,A\right)->\left(Z-2,A-4\right)+{^{4}He}$

Beta Decay

This is a very important process that occurs in stellar burning in part because it reduces the number of electrons available in the plasma and also because the neutrinos can take out some of the energy generated in the process. The most common weak interaction beta decays are:

$\beta ^{-}$ decay or electron emission

$\left(Z,A\right)\rightarrow \left(Z+1,A\right)+e^{-}+{\bar {\nu _{e}}}$

$\beta ^{+}$ decay or positron emission

$\left(Z,A\right)\rightarrow \left(Z-1,A\right)+e^{+}+\nu _{e}$

electron capture

$(Z,A)+e^{-}\rightarrow (Z-1,A)+\nu _{e}$

neutrino (antineutrino) capture

$(Z,A)+\nu \rightarrow (Z+1,A)+e^{-}$

$(Z,A)+{\bar {\nu }}\rightarrow (Z+1,A)+e^{+}$

These reactions occur on long time scales but do not need high energy to occur.

Electron Capture

This is also a form of a weak decay. This decay occurs when an electron enters the nucleus of an atom. It then reacts with a proton to form a neutron and an electron neutrino. Looking at the bigger picture one will see:

$\left(Z,A\right)+e^{-}->\left(Z-1,A\right)+\nu _{e}$

Reaction Nomenclature

When speaking of reactions it is nice to use the same language. Here is a generic reaction:

$A\left(B,C\right)D$ where:

• A is the "target" nucleus
• B is the "incoming projectile"
• C is the "outgoing particle"
• D is the "residual" nucleus

For example: ${^{12}C}\left(p,\gamma \right){^{13}N}$ means that a proton is captured by a Carbon-12 nucleus forming a Nitrogen-13 nucleus and an ejected a gamma ray.

Breit Wigner Cross Section

The effect of a resonance state on the cross section of a particular nuclear reaction can be determined via the Breit-Wigner cross section:

$\sigma (j,k)={\pi ^{2} \over k_{j}^{2}}{(1+\delta _{ij}) \over (2I_{i}+1)(I_{j}+1)}\sum _{n}(2J_{n}+1){\Gamma _{j,n}\Gamma _{o,n} \over (E-E_{n})^{2}+(\Gamma _{n}/2)^{2}}$