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 types of nuclear decays. Some occur by ejecting particles from the nucleus or when the nucleons change its flavor by capturing an external particle (like an electron) or by decaying into lighter particles. A very important equation is the one called nuclear exponential decay. It can be obtained by the following reasoning. Suppose that initially we have 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 N_0} nuclei of the same type, let say type 1, and that the type 1 nuclei can only decay to type 2 nuclei. Let N=N(t) be the number nuclei of type 1 at a time t. If we assume that each type 1 nucleus has the same probability of decaying to a type 2 nucleus then this probability should be proportional to the time. In a differential form we have
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{dN}{N}=\lambda dt}
the minus sign comes from the fact the probability is positive definite and we have dN<0 (as the time pases we have less particles of type 1). The solution of this equation with the condition 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 N(0)=N_0} 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 N(t)=N_0 e^{-\lambda t}}
The factor 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 \lambda}
is the reciprocal of the time life 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 \tau}
and it can be determined experimentally.
Alpha Decay
Alpha decay is a form on nuclear fission and 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) \rightarrow \left(Z-2, A-4\right) + {^4He^{2+}}}
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 }} .
The typical kinetic energy of an alpha particle is 5 MeV and because of its relatively large mass this corresponds to a speed of around 0.05c. Alpha particles are very likely to interact with other atoms and lose their energy, so their forward motion is effectively stopped within a few centimeters of air.
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: 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.
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 A 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.
Gamma Decay
By far the most common type of decay in nuclei. It occurs when the nucleus changes its energy state by releasing a photon that carries the energy excess. Notice that in this transition neither the number of protons nor the number on neutrons change. To understand this better one should recall the nuclear Shell Model. From this model we know that inside the nuclei there are different energy levels (see Figure 2 of 3rd week summary). The transition from a higher level to a lower one will give an energy excess in the nucleus that is take out by a photon.
Several electromagnetic transitions can happen and they are labeled in relation to the 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 multipole} expansion for the electromagnetic potential (this expansion is related to the spherical harmonics). For example a magnetic dipole transition is called M1 transition because is the first term of the magnetic (vector) potential expansion, an electric quadrupole transition is called E2, etc.
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. The relation between the differential cross section and the total cross section 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 \sigma = \int_0^{4\pi}\frac{\partial \sigma}{\partial \Omega}d\Omega}
Hauser-Feshbach cross section
A statistical approach to calculating the total cross section is given by the Hauser-Feshbach formula 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_{J,\pi} (2J+1)\frac{T_{j}(E,J,\pi)T_o(E,J,\pi)}{T_{tot}(E,J,\pi) }}
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}}}