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}}

By assuming that first the parent nucleus is at rest then according to the conservation of energy

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 M_{P}c^{2}=M_{D}c^{2}+T_{D}+M_{\alpha} c^{2}+T_{\alpha }} .

Beta Decay

Nuclei with more neutrons can attain a more stable nucleus by releasing an electron. 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. For a one dimensional time independent problem, a general way to write the tunneling probability (T) is with the ratio of the incoming and outgoing probability currents

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=\frac{j_{\mathrm{out}}}{j_{\mathrm{in}}}}

where the probability current is defined 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 \mathbf{j}=\frac{\hbar}{2mi}(\psi^*\nabla\psi-\psi\nabla\psi^*)}

Step function potential

As a simple example 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 \;\;\;\; \mbox{for} \;\; 0\leq x \leq a}

and V=0 elsewhere. Assume that the particle has energy E (here 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<E<V_0} ) 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} \;\;\;\; \mbox{for} \;\; 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} \;\;\;\; \mbox{for} \;\; 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} \;\;\;\; \mbox{for} \;\; 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} 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} 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. 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 T=e^{-2\pi\eta}}

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 \eta=\sqrt{m/(2E)}Z_1Z_2e^2/\hbar} is the Sommerfeld parameter. Here 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_1} is the charge of the nucleus 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 Z_2} is the charge of a particle with energy E and mass m.

Cross Section

The cross section is a quantitative measurement of the probability of a reaction to occur. In order to determine the cross section we need the incoming flux defined by a plane wave, scattering in all directions, and the penetration probability into the nucleus. The cross section 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}

Sometimes scientists are more interested in how does the cross section change if one changes the angle at which the scattered particles are detected. For this we use the differential 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 \frac{\partial \sigma}{\partial \Omega}} , 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 \partial \Omega} is the solid angle differential.

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}}}