Binding Energy and Semi-Empirical Mass Formula

Binding energy is the minimum energy required to separate all the nucleons (protons and neutrons) in an atomic nucleus. Conversely, it is the energy released when these nucleons come together to form a nucleus. This energy arises from the strong nuclear force that holds the nucleus together, overcoming the electrostatic repulsion between protons.

The binding energy (BE) of a nucleus can be calculated using the mass defect. The mass defect ($\Delta m$) is the difference between the sum of the masses of the individual nucleons and the actual mass of the nucleus.

$\Delta m = [Z \cdot m_p + N \cdot m_n] - M_{nucleus}$

Where:

• $Z$ is the number of protons (atomic number).
• $m_p$ is the mass of a proton.
• $N$ is the number of neutrons (neutron number).
• $m_n$ is the mass of a neutron.
• $M_{nucleus}$ is the actual mass of the nucleus.

According to Einstein's mass-energy equivalence principle ($E = mc^2$), this mass defect is converted into binding energy:

$BE = \Delta m \cdot c^2$

A higher binding energy per nucleon indicates a more stable nucleus. The binding energy per nucleon is typically plotted against the mass number (A = Z + N). This plot shows that nuclei with mass numbers around iron (A ≈ 56) have the highest binding energy per nucleon, making them the most stable. Lighter nuclei can achieve greater stability by fusing, and heavier nuclei can achieve greater stability by fission.

Semi-Empirical Mass Formula (Bethe-Weizsäcker Formula)

The semi-empirical mass formula provides an approximation for the binding energy of a nucleus based on its mass number (A) and atomic number (Z). It's called "semi-empirical" because it combines theoretical concepts with experimental data. The formula expresses the binding energy as a sum of several terms, each accounting for a different physical effect:

$BE(A, Z) = a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A \frac{(A - 2Z)^2}{A} \pm \delta$

Let's break down each term:

• Volume Term ($a_V A$): This term assumes that the binding energy is proportional to the volume of the nucleus, which is proportional to the number of nucleons (A). It represents the attractive nuclear force acting between nucleons.
• Surface Term ($-a_S A^{2/3}$): Nucleons on the surface of the nucleus interact with fewer neighbors than those in the interior, leading to a reduction in binding energy. The surface area is proportional to $A^{2/3}$.
• Coulomb Term ($-a_C \frac{Z(Z-1)}{A^{1/3}}$): This term accounts for the electrostatic repulsion between protons. It is proportional to the number of proton pairs and inversely proportional to the nuclear radius (which is proportional to $A^{1/3}$). The $Z(Z-1)$ factor comes from considering the interaction between each pair of protons.
• Asymmetry Term ($-a_A \frac{(A - 2Z)^2}{A}$): This term arises from the observation that stable nuclei tend to have a roughly equal number of protons and neutrons, especially for lighter elements. Nuclei with a significant excess of either protons or neutrons are less stable. The term $(A - 2Z)$ represents the neutron excess ($N - Z$).
• Pairing Term ($\pm \delta$): This term accounts for the effect of nucleon pairing. Nuclei with even numbers of protons and neutrons (even-even) are more stable than those with odd numbers.
  • $\delta = +a_P A^{-1/2}$ for even-even nuclei.
  • $\delta = 0$ for odd-even or even-odd nuclei.
  • $\delta = -a_P A^{-1/2}$ for odd-odd nuclei.
  The constants $a_V, a_S, a_C, a_A,$ and $a_P$ are determined empirically by fitting the formula to experimental binding energy data.

  The semi-empirical mass formula is crucial for understanding nuclear stability, predicting the masses of unknown nuclei, and explaining phenomena like nuclear fission and beta decay.

Nuclear Stability

Nuclear stability refers to the tendency of an atomic nucleus to remain in its current state without undergoing radioactive decay. Stability is primarily determined by the balance between the strong nuclear force (attractive) and the electromagnetic force (repulsive between protons), as well as the ratio of neutrons to protons.

The Band of Stability

When plotting the number of neutrons ($N$) versus the number of protons ($Z$) for known stable isotopes, we observe a region known as the "band of stability" or "valley of stability."

• For light nuclei ($Z \le 20$), stable isotopes generally lie close to the line $N = Z$.
• As $Z$ increases, the number of neutrons required for stability increases more rapidly. Stable nuclei tend to lie above the $N = Z$ line, meaning they have more neutrons than protons. This is because the increasing Coulomb repulsion among protons requires more neutrons (which contribute only to the attractive nuclear force) to maintain stability.
• Nuclei outside the band of stability are radioactive and will decay to reach a more stable configuration.

Factors Affecting Stability

Several factors influence nuclear stability:

• Neutron-to-Proton Ratio ($N/Z$): As mentioned, heavier nuclei require a higher $N/Z$ ratio for stability due to the increasing Coulomb repulsion.
• Even-Odd Number of Nucleons:
  • Even-even nuclei (even Z, even N) are generally the most stable.
  • Odd-odd nuclei are generally the least stable.
  • Odd-even and even-odd nuclei have intermediate stability.
  This is related to the pairing energy term in the semi-empirical mass formula. Pairs of protons and pairs of neutrons are energetically favorable.
• Magic Numbers: Nuclei with certain "magic numbers" of protons or neutrons (2, 8, 20, 28, 50, 82, 126) exhibit exceptionally high stability, similar to the stability of noble gases due to their filled electron shells. Nuclei with magic numbers for both protons and neutrons are called "doubly magic" and are particularly stable (e.g., Helium-4, Oxygen-16, Calcium-40, Lead-208).
• Binding Energy per Nucleon: As discussed earlier, nuclei with higher binding energy per nucleon are more stable. The peak stability occurs around iron and nickel isotopes.

Radioactive Decay and Stability

Unstable nuclei decay to achieve a more stable configuration. The type of decay depends on where the nucleus lies relative to the band of stability:

• Nuclei with too many neutrons (above the band) tend to undergo beta-minus ($\beta^-$) decay, where a neutron converts into a proton, emitting an electron and an antineutrino. This decreases the $N/Z$ ratio.
• Nuclei with too many protons (below the band) tend to undergo beta-plus ($\beta^+$) decay (positron emission) or electron capture, where a proton converts into a neutron. This increases the $N/Z$ ratio.
• Very heavy nuclei (beyond lead) may undergo alpha ($\alpha$) decay, emitting an alpha particle (Helium nucleus), which reduces both $Z$ and $N$. They may also undergo spontaneous fission.

Understanding nuclear stability is fundamental to nuclear physics, nuclear medicine, and nuclear energy.

Nuclear Forces

The nuclear force is the fundamental interaction responsible for holding the atomic nucleus together. It is one of the four fundamental forces of nature, alongside gravity, electromagnetism, and the weak nuclear force. The nuclear force is extremely powerful but acts only over very short distances, typically within the nucleus.

Properties of the Nuclear Force:

• Strong Attraction: It is the strongest of the fundamental forces at nuclear distances, strong enough to overcome the immense electrostatic repulsion between protons.
• Short Range: Its influence extends only about $10^{-15}$ meters (1 femtometer or 1 fm). Beyond this range, the force drops off very rapidly. This short range is why the binding energy per nucleon doesn't increase indefinitely with mass number.
• Charge Independence: The nuclear force between two protons, two neutrons, or a proton and a neutron is approximately the same, provided they are in the same spin state. This means the force doesn't depend on the electric charge of the nucleons.
• Spin Dependence: The force is dependent on the relative spin orientations of the nucleons. The attraction is stronger when the spins of the nucleons are parallel (aligned) compared to when they are antiparallel.
• Repulsive Core: At extremely short distances (less than about 0.5 fm), the nuclear force becomes strongly repulsive. This prevents the nucleus from collapsing under its own forces.
• Exchange Force: The nuclear force can be understood as an exchange force, mediated by the exchange of virtual particles called mesons (like pions). A nucleon can emit a meson, and another nucleon can absorb it, leading to an interaction.

Meson Theory of Nuclear Force (Yukawa's Theory)

Hideki Yukawa proposed in 1935 that the nuclear force arises from the exchange of mesons between nucleons.

• Protons and neutrons are considered to be two states of the same particle, the nucleon.
• Nucleons exchange virtual pions ($\pi$-mesons). Pions have masses between those of electrons and protons.
• A proton can emit a positive pion ($\pi^+$) and become a neutron: $p \rightarrow n + \pi^+$.
• A neutron can absorb a positive pion ($\pi^+$) and become a proton: $n + \pi^+ \rightarrow p$.
• Similarly, neutrons can interact via neutral ($\pi^0$) and negative ($\pi^-$) pions.

The range of the force is related to the mass of the exchanged particle by the uncertainty principle. For a particle of mass $m$ exchanged over a time $\Delta t$, the distance it can travel is approximately $r \approx c \Delta t$. From the uncertainty principle $\Delta E \Delta t \approx \hbar$, and since $\Delta E \approx mc^2$ for the virtual particle, we get $\Delta t \approx \hbar / (mc^2)$. Thus, the range is $r \approx \hbar / (mc)$. Yukawa predicted a meson mass corresponding to a range of about 1-2 fm, which matched the size of the nucleus. The discovery of pions later confirmed this theory.

Modern View: Quantum Chromodynamics (QCD)

While meson theory provides a good phenomenological description, the fundamental understanding of the nuclear force comes from Quantum Chromodynamics (QCD).

• Protons and neutrons are not fundamental particles; they are composite particles made of quarks.
• Quarks are held together by the strong force, mediated by particles called gluons. This force acts between particles carrying "color charge."
• The residual effect of this strong force between quarks within separate nucleons manifests as the nuclear force between nucleons. This residual force is mediated by the exchange of mesons.

The nuclear force is thus a residual effect of the more fundamental strong force acting between quarks and gluons.

Ground State of Deuteron

The deuteron is the nucleus of deuterium, an isotope of hydrogen. It consists of one proton and one neutron ($^2_1H$). Studying the deuteron provides fundamental insights into the nature of the nuclear force.

Key Properties and Observations:

• Binding Energy: The deuteron is weakly bound, with a binding energy of approximately 2.22 MeV. This is relatively low compared to heavier nuclei, indicating the force is not as strong or as saturated as in larger nuclei.
• Spin: The deuteron has a total nuclear spin of $I=1$. Since both the proton and neutron have spin $s=1/2$, their spins must be aligned (parallel) to achieve a total spin of 1 ($1/2 + 1/2 = 1$). If their spins were antiparallel, the total spin would be 0, and calculations show that such a state would not be bound. This demonstrates the spin-dependence of the nuclear force.
• Magnetic Dipole Moment: The deuteron's magnetic dipole moment is approximately 0.857 nuclear magnetons ($\mu_N$). This value is slightly less than the sum of the magnetic moments of a free proton (≈ 2.79 $\mu_N$) and a free neutron (≈ -1.91 $\mu_N$), which sum to about 0.88 $\mu_N$. The slight difference suggests that the deuteron is not purely a state with aligned proton and neutron spins but also contains a small admixture of other states.
• Electric Quadrupole Moment: The deuteron has a small, positive electric quadrupole moment ($Q \approx +0.0028$ fm$^2$). This indicates that the deuteron's charge distribution is not perfectly spherical. It is slightly prolate (elongated along the spin axis). This observation is crucial because it implies that the nuclear force is not purely central (acting along the line connecting the centers of the two nucleons) but has a non-central component, often referred to as a tensor force. A purely central force would result in a spherical charge distribution ($Q=0$).

The Nature of the Nuclear Force from Deuteron Properties:

• Spin Dependence: The fact that the deuteron is bound only when the proton and neutron spins are aligned ($I=1$) strongly indicates that the nuclear force is spin-dependent. The force is stronger when spins are parallel.
• Tensor Force: The small quadrupole moment implies the existence of a tensor component in the nuclear force. This component depends on the orientation of the nucleons' spins relative to the line connecting them. It contributes to the binding and causes the slight elongation.
• Short Range: The low binding energy suggests the force is short-ranged, as expected.
• Charge Independence: While not directly shown by the deuteron alone, comparisons with other nuclei support the charge independence of the nuclear force.

Ground State Wave Function

The ground state wave function of the deuteron can be approximated. If we ignore the tensor force, the ground state would be an S-state (l=0, orbital angular momentum is zero), with total spin S=1. The wave function would be spherically symmetric.

However, the presence of the quadrupole moment means the ground state is not a pure S-state. It is predominantly an S-state but has a small admixture (about 4-7%) of a D-state (l=2, orbital angular momentum is 2) with the same total spin S=1. The total angular momentum J is given by $J = L + S$. For the deuteron, J=1.

• Pure S-state: L=0, S=1 => J=1
• Pure D-state: L=2, S=1 => J=1, 2, 3

The ground state of the deuteron is therefore a mixture: $\psi_{deuteron} \approx \alpha |S, J=1\rangle + \beta |D, J=1\rangle$, where $\alpha^2 \approx 0.93-0.96$ and $\beta^2 \approx 0.04-0.07$. The D-state admixture is necessary to explain the non-zero quadrupole moment.

The study of the deuteron was pivotal in establishing the complex nature of the nuclear force, revealing its spin dependence and the presence of a tensor component.

Alpha Decay

Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle ($\alpha$). An alpha particle is identical to a Helium-4 nucleus, consisting of two protons and two neutrons ($^4_2He$). Alpha decay occurs primarily in heavy nuclei (typically with atomic number $Z > 82$) because they have an excess of protons and neutrons, and the repulsive Coulomb forces become significant.

The Process:

When a nucleus undergoes alpha decay, its atomic number decreases by 2, and its mass number decreases by 4. The general equation is:

$ ^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2 He (\alpha)$

Where:

• $X$ is the parent nucleus.
• $Y$ is the daughter nucleus.
• $A$ is the mass number.
• $Z$ is the atomic number.

For example, Uranium-238 decays into Thorium-234:

$ ^{238}_{92} U \rightarrow ^{234}_{90} Th + ^4_2 He$

Energy Release (Q-value):

Alpha decay is a spontaneous process that releases energy. This energy, known as the Q-value, is the difference in mass between the parent nucleus and the decay products, converted to energy via $E=mc^2$.

$Q = [M_X - (M_Y + M_\alpha)] c^2$

The released energy appears primarily as the kinetic energy of the alpha particle and, to a lesser extent, the recoil kinetic energy of the daughter nucleus. Due to momentum conservation, the alpha particle carries away most of the kinetic energy.

Mechanism: Quantum Tunneling

The emission of an alpha particle is explained by the concept of quantum tunneling. Inside the nucleus, the alpha particle is bound by the strong nuclear force. However, there is a potential barrier due to the Coulomb repulsion from the remaining protons in the nucleus.

• Inside the nucleus: The alpha particle is attracted by the strong nuclear force.
• Outside the nucleus: The alpha particle is repelled by the Coulomb force.
• Potential Barrier: There exists a potential energy barrier between these two regions. Classically, an alpha particle with kinetic energy less than the barrier height could not escape.
• Quantum Tunneling: According to quantum mechanics, the alpha particle has a non-zero probability of "tunneling" through this potential barrier, even if its kinetic energy is less than the barrier height. This probability determines the half-life of the decaying nucleus.

The probability of tunneling is highly sensitive to the height and width of the barrier. Higher barriers and wider barriers lead to lower tunneling probabilities and thus longer half-lives. This explains the wide range of half-lives observed for alpha emitters.

Geiger-Nuttall Law

The Geiger-Nuttall law relates the half-life ($T_{1/2}$) of an alpha emitter to the range ($R$) of the alpha particles in air, which is directly related to their energy. A more common form relates the logarithm of the half-life to the energy of the alpha particle.

$\log(T_{1/2}) \propto -\frac{1}{E_\alpha^{1/2}}$ or $\log(T_{1/2}) \propto -\frac{1}{E_\alpha^{3/2}}$

This empirical law shows that higher energy alpha particles (which correspond to shorter half-lives) are emitted by nuclei that decay more rapidly. This relationship is a direct consequence of the quantum tunneling mechanism.

Characteristics of Alpha Particles:

• They have a discrete energy spectrum, meaning alpha particles emitted from a specific decay have specific, well-defined energies.
• They have a short range in matter (a few centimeters in air).
• They have a high ionizing power due to their charge (+2e) and mass.
• They can be easily stopped by a sheet of paper or the outer layer of skin.

Beta Decay

Beta decay is a type of radioactive decay in which a beta particle ($\beta$) is emitted from an atomic nucleus. This process transforms one type of subatomic particle into another and is mediated by the weak nuclear force. There are three main types of beta decay: beta-minus ($\beta^-$), beta-plus ($\beta^+$), and electron capture (EC). Beta decay occurs in nuclei that have an imbalance in the neutron-to-proton ratio relative to the stable "band of stability."

1. Beta-Minus ($\beta^-$) Decay:

This occurs in nuclei with an excess of neutrons. A neutron within the nucleus transforms into a proton, emitting an electron (the beta-minus particle) and an electron antineutrino ($\bar{\nu}_e$).

$n \rightarrow p + e^- + \bar{\nu}_e$

The general nuclear reaction is:

$ ^A_Z X \rightarrow ^A_{Z+1} Y + e^- + \bar{\nu}_e$

In $\beta^-$ decay:

• The mass number ($A$) remains unchanged.
• The atomic number ($Z$) increases by 1.
• The number of neutrons decreases by 1.
• The nucleus moves towards the band of stability by converting a neutron into a proton.

Example: Carbon-14 decaying into Nitrogen-14:

$ ^{14}_6 C \rightarrow ^{14}_7 N + e^- + \bar{\nu}_e$

2. Beta-Plus ($\beta^+$) Decay (Positron Emission):

This occurs in nuclei with an excess of protons (too few neutrons). A proton within the nucleus transforms into a neutron, emitting a positron (the beta-plus particle, which is the antiparticle of the electron) and an electron neutrino ($\nu_e$).

$p \rightarrow n + e^+ + \nu_e$

The general nuclear reaction is:

$ ^A_Z X \rightarrow ^A_{Z-1} Y + e^+ + \nu_e$

In $\beta^+$ decay:

• The mass number ($A$) remains unchanged.
• The atomic number ($Z$) decreases by 1.
• The number of protons decreases by 1.
• The nucleus moves towards the band of stability by converting a proton into a neutron.

Example: Fluorine-18 decaying into Oxygen-18:

$ ^{18}_9 F \rightarrow ^{18}_8 O + e^+ + \nu_e$

3. Electron Capture (EC):

This also occurs in proton-rich nuclei. Instead of emitting a positron, the nucleus captures one of its own atomic electrons (usually from an inner shell, like the K-shell). This captured electron interacts with a proton in the nucleus, converting it into a neutron and emitting an electron neutrino ($\nu_e$).

$p + e^- \rightarrow n + \nu_e$

The general nuclear reaction is:

$ ^A_Z X + e^- \rightarrow ^A_{Z-1} Y + \nu_e$

In Electron Capture:

• The mass number ($A$) remains unchanged.
• The atomic number ($Z$) decreases by 1.
• The number of protons decreases by 1.
• It achieves the same nuclear transformation as $\beta^+$ decay.

Example: Potassium-40 can decay by EC to Argon-40:

$ ^{40}_{19} K + e^- \rightarrow ^{40}_{18} Ar + \nu_e$

After EC, the atom has a vacancy in one of its inner electron shells. This vacancy is filled by an electron from a higher energy shell, resulting in the emission of characteristic X-rays or Auger electrons.

Energy Spectrum of Beta Particles:

Unlike alpha particles, which are emitted with discrete energies, beta particles are emitted with a continuous energy spectrum, ranging from nearly zero up to a maximum value ($E_{max}$). This continuous spectrum was initially a puzzle.

The reason for the continuous spectrum is that the decay energy is shared between the beta particle and the undetected neutrino (or antineutrino). The total energy released in the decay is constant, but it can be distributed in various ways between the electron/positron and the neutrino. The maximum energy occurs when the neutrino carries away almost no energy.

$E_{max} = Q$-value of the decay.

Wolfgang Pauli proposed the existence of the neutrino in 1930 to account for the missing energy and momentum in beta decay, preserving the laws of conservation of energy, momentum, and angular momentum. Enrico Fermi later incorporated the neutrino into his theory of beta decay.

Fermi Theory and Selection Rules for Beta Decay

The Fermi theory of beta decay, developed by Enrico Fermi in 1933, provides a fundamental explanation for the process, including the continuous energy spectrum of beta particles and the role of the weak nuclear force. It treats beta decay as a transformation of a nucleon (neutron to proton or vice versa) mediated by the weak interaction, involving the creation of a lepton (electron or positron) and its corresponding neutrino.

Key Postulates of Fermi Theory:

• Weak Interaction: Beta decay is a manifestation of the weak nuclear force, which is much weaker than the electromagnetic or strong nuclear forces and has a very short range.
• Point Interaction: The interaction responsible for beta decay is a point-like interaction, meaning it occurs at a single spacetime point. This implies a very short range, consistent with the weak force.
• Conservation Laws: The theory strictly adheres to the conservation of energy, momentum, angular momentum, and charge.
• Lepton Number Conservation: The theory upholds the conservation of lepton number. Electrons and neutrinos have a lepton number of +1, while positrons and antineutrinos have -1.
• Nucleon Transformation: A neutron transforms into a proton (or vice versa) by emitting or absorbing a lepton-antilepton pair.
• Interaction Strength: The probability of decay is proportional to the square of the coupling constant of the weak interaction ($G_F$, the Fermi constant).

The Beta Decay Hamiltonian:

Fermi constructed a Hamiltonian operator that describes the interaction. For example, in $\beta^-$ decay ($n \rightarrow p + e^- + \bar{\nu}_e$), the interaction term involves operators that can change a neutron into a proton and create an electron and an antineutrino.

The transition probability per unit time is proportional to $|M_{fi}|^2 \rho(E)$, where:

• $|M_{fi}|^2$ is the square of the matrix element, which depends on the overlap of the initial and final nuclear wave functions and the nature of the interaction.
• $\rho(E)$ is the density of final states, which accounts for the available phase space for the emitted particles (electron/positron and neutrino). This term is responsible for the continuous energy spectrum.

The Fermi constant $G_F$ determines the overall strength of the weak interaction.

Selection Rules for Beta Decay:

Selection rules govern which transitions are allowed or forbidden in beta decay, based on the conservation of angular momentum and parity. The nuclear spin ($J$) and parity ($\pi$) of the initial and final nuclear states play a crucial role.

The total angular momentum of the emitted particles (beta particle and neutrino) must account for the difference in angular momentum between the initial and final nuclear states.

Fermi Transitions:

• In Fermi transitions, the nuclear spin does not change ($ \Delta J = 0 $).
• The parity also does not change ($ \Delta \pi = + $).
• These transitions involve the change of a neutron to a proton (or vice versa) without a change in the orbital angular momentum of the nucleon involved. The emitted lepton and antilepton are created in an S-state (l=0) relative to each other.
• Example: $^6He \rightarrow ^6Li$ ($J^\pi=0^+ \rightarrow 0^+$)

Gamow-Teller Transitions:

• In Gamow-Teller transitions, the nuclear spin can change by 0, +1, or -1 ($ \Delta J = 0, \pm 1 $), but the transition $J=0 \rightarrow J=0$ is forbidden.
• The parity does not change ($ \Delta \pi = + $).
• These transitions involve a change in the orbital angular momentum of the nucleon involved or a spin flip. The emitted lepton and antilepton can be in an S-state (l=0) or a P-state (l=1) relative to each other.
• Gamow-Teller transitions are generally stronger (higher probability, shorter half-lives) than Fermi transitions for a given energy release.
• Example: $^3H \rightarrow ^3He$ ($J^\pi=1/2^+ \rightarrow 1/2^+$)

The matrix element $|M_{fi}|^2$ is different for Fermi and Gamow-Teller transitions. The Fermi matrix element depends only on the initial and final nuclear states, while the Gamow-Teller matrix element depends on both the states and the spins.

Modern Understanding (Electroweak Theory):

The Fermi theory was a highly successful precursor to the modern Glashow-Weinberg-Salam electroweak theory. This theory unifies the electromagnetic and weak forces and describes beta decay in terms of the exchange of massive vector bosons, $W^+$ and $W^-$, between quarks and leptons. The $W$ bosons are responsible for mediating the weak interaction.

• $\beta^-$ decay: $n(udd) \rightarrow p(uud) + W^-$. The $W^-$ then decays into $e^- + \bar{\nu}_e$.
• $\beta^+$ decay: $p(uud) \rightarrow n(udd) + W^+$. The $W^+$ then decays into $e^+ + \nu_e$.
• Electron Capture: $p(uud) + e^- \rightarrow n(udd) + \nu_e$. This is mediated by the $W^+$ boson interacting with the captured electron.

The Fermi constant $G_F$ is related to the mass of the $W$ boson and the electroweak coupling constant. The selection rules remain valid within this more complete framework.