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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(Z,A\right) -> \left(Z-2, A-4\right) + {^4He}}
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 processes in nuclear beta decays are:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^-} decay or electron emission
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(Z,A\right) \rightarrow \left(Z+1, A\right) + e^- + \bar{\nu_e}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^+} decay or positron emission
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(Z,A\right) \rightarrow \left(Z-1, A\right) + e^+ + \nu_e}
electron capture
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z,A)+e^- \rightarrow (Z-1, A) + \nu_e}
neutrino (antineutrino) capture
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z,A)+\nu \rightarrow (Z+1, A) + e^- }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z,A)+\bar{\nu} \rightarrow (Z+1, A) + e^+ }
Notice how these processes conserve the mass number but change Z and N. These reactions occur on long time scales but do not need high energy to occur. They also are very slow compared to electromagnetic reactions.
Electron Capture
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.
Reaction Nomenclature
When speaking of reactions it is nice to use the same language. Here is a generic reaction:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {^{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.
Tunneling
Tunneling is a quantum mechanical effect in which a particle has a non-zero probability of passing through (tunnel) a potential barrier larger than the particle's total energy. For a classical particle this tunneling probability will be zero.
Step function potential
As a simple example of this consider a one dimensional problem with a step function potential
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=V0} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq x \leq a}
and V=0 elsewhere. Assume that the particle has energy E and mass m. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<0} the solution of Schrödinger's equation is a linear combination of the incident wave and the reflected wave. For convenience assume that the coefficient of the incident wave is 1 and for the reflected wave is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_R} , then the (non-normalized) solution is:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_1(x)=e^{ikx}+A_R e^{-ikx}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_2(x)=Ae^{-qx}+Be^{qx}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<x<a}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_3(x)=A_T e^{ikx}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>a}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_R, A_T, A, B} are constants and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\sqrt{2mE/\hbar^2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=\sqrt{2m(V_0-E)/\hbar^2}} . It can be shown by matching boundary conditions, that the transmission probability is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=|A_T|^2=\frac{4}{(1+\frac{k^2}{q^2})\sinh^2(qa)+(1+\frac{q^2}{k^2})\cosh^2(qa)}}
Finite potential well plus general potential
Assume that we have a finite potential well with depth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -V_0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<x<a} and a general potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>a} . Using WKB approximation we can write the tunneling probability as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\approx \exp(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx)}
Coulomb barrier
In Nuclear Astrophysics we are more interested in the tunneling through the Coulomb potential in nuclei, the so-called Coulomb barrier. As a first approximation one can assume that nucleons have a two particle interaction composed by a finite potential well and a modified Coulomb potential that is only effective at a certain separation between nucleons.
Cross Section
The cross section is a quantitative measurement of the probability of a reaction to occur. Is defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma=\frac{\mbox{number of reactions per target per second}}{\mbox{flux of incoming projectiles}}}
The relation of the total cross section and the tunneling probability is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma= \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)T_l}
Effects of Resonances on Cross Section
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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(j,k)={\pi^2 \over k^{2}_{j}}{(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}}}