Radioactive decay

• a) Are there differences between the photons emitted as a result of atomic versus nuclear transitions?

• Solution:

• Actually, photons released during atomic and nuclear transitions differ significantly from one another.
• Photons from atomic transitions are usually lower in energy, ranging from visible light to X-rays, whereas those from nuclear transitions are usually much more energetic, ranging from gamma rays.
• The energy, origin, and penetrating ability of photons emitted from nuclear transitions and those from atomic transitions differ significantly.
• Atomic Transitions (Electron energy levels):
• These entail modifications to an atom’s electrons’ energy levels. Since there aren’t many energy differences between these levels, photons with lower energies—like visible light or X-rays—are released.

•  Nuclear Changes (Nuclear energy levels):
• These entail modifications to the energy levels of the protons and neutrons that make up the nucleus. Nuclear energy levels emit much higher energy photons, frequently in the form of gamma rays, because they are much more tightly bound than atomic electron energy levels.

• b) How does equilibrium within a star compare to stability within the nucleus of an atom?

• Solution:

• Although both involve balancing forces, stability in an atom’s nucleus and equilibrium within a star are different ideas.
• A star maintains its size and structure by striking a balance between the pressures from nuclear fusion and inward gravity.
• Nuclear stability, on the other hand, is the ability of a nucleus to withstand radioactive decay, which is dictated by the strong nuclear force and the proton-neutron balance.
• 
• Figure 1 The sun and stellar structure
• Although they occur at quite different scales and are controlled by distinct interactions, both entail the balance of opposing forces.
• Let’s contrast them with respect to the forces at play, the kind of equilibrium, and the consequences of a balance failure.
• Stellar equilibrium:
• Balancing Forces:
• The outward pressure produced by nuclear fusion in a star’s core and the inward pull of gravity are balanced to keep the star in equilibrium.
• Hydrostatic equilibrium:
• When the inward gravitational force and the outward pressure force are equal at every location within the star, this balance is referred to as hydrostatic equilibrium.
• Preserving Size and Shape:
• This balance keeps the star from uncontrollably expanding or collapsing due to its own gravity, enabling it to preserve its size and shape for extended periods of time.
• Nuclear stability:
• Balancing Forces:
• The strong nuclear force, which opposes the electromagnetic repulsion between protons, holds protons and neutrons together inside the nucleus.
• Nuclear Binding Energy:
• A nucleus’s stability is correlated with its binding energy, which is the energy released when protons and neutrons combine to form the nucleus.
• Nuclei that are stable versus unstable:
• While some nuclei are unstable and decay spontaneously, releasing particles and energy, others are stable and do not undergo radioactive decay.

• c) Would a nucleus be able to exist if only gravitational and electric forces were considered?

• Solution:

• No, if only gravity and electricity were taken into account, there could not be a stable nucleus.
• Despite being an attractive force, gravity is insufficient to keep the nucleus together in the face of the protons’ electrostatic repulsion.
• The strong nuclear force is necessary to counteract this repulsion and bind the nucleus.
• This is because, at nuclear scales, the attractive gravitational forces are greatly outweighed by the repulsive electric forces between protons.
• In the absence of the strong nuclear force, electrostatic repulsion would cause the nucleus to fly apart.
• Electrostatic repulsion:
• Since protons are positively charged, the electric force causes them to naturally repel one another.
• [math]F_{electric} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r^2}[/math]
• Strongly repulsive of positively charged particles that are closely packed together, such as protons.
• Dominant force in the absence of any opposition.
• Gravitational force:
• Although gravity does exist, it is far weaker than the nuclear-level electric force. The repulsion between protons cannot be overcome by it.
• [math]F_{gravity} = G \cdot \frac{m_1 m_2}{r^2}[/math]
• In the nucleus, it is about ten times weaker than electric force.
• A nucleus would be extremely unstable and impossible to exist if it were solely dependent on gravity and electrostatic forces because:
• – It would be torn apart by the electric repulsion between protons.
• – There is not enough gravitational force to offset this.
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  d) How did conservation lead to experimental evidence of the neutrino? (NOS)

• Solution:

• In order to explain a conservation law, namely the conservation of energy in beta decay, the neutrino was first proposed.
• It was discovered that electrons in beta decay did not always release the same quantity of energy, which appeared to be against the principle of conservation of energy.
• Wolfgang Pauli proposed that the missing energy was carried away by another, undetected particle that was also released in order to address this. Later, this fictitious particle was given the name neutrino.
• The first experimental evidence for the existence of this elusive particle came from the later detection of neutrinos through experiments such as the Cowan-Reines neutrino experiment, which saw antineutrinos interacting with protons in a large tank of water.
• The Energy Crisis and Beta Decay:
• A radioactive nucleus undergoes beta decay when it transforms into a different nucleus and releases an electron (or positron).
• Experimental observations revealed a continuous spectrum of electron energies, contrary to the initial assumption that the electron would always carry away a fixed amount of energy.
• For physicists who held that energy must be conserved in all physical processes, this disparity raised concerns because it implied that energy was not conserved.
• 
• Figure 2 Discovery of the neutrino
• Pauli’s Proposal:
• Wolfgang Pauli suggested the existence of a new, neutral particle with very little or no mass that was also released in beta decay to solve the energy crisis and remove the energy that was lacking.
• He dubbed this particle the “neutron” (later renamed the neutrino to distinguish it from Chadwick’s discovery of the neutron).
• The Neutrino Hypothesis:
• Pauli’s theory introduced a particle that was very hard to detect because of its lack of charge and incredibly weak interaction with matter, thereby saving the law of conservation of energy in beta decay.
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  e) Which areas of physics involve exponential change? (NOS)

• Solution:

• In many branches of physics, exponential change—both growth and decay—occurs frequently. In particular, population growth (in some cases), radioactive decay, and capacitor discharge are excellent examples of changes that are modelled using exponential functions.
• Many aspects of physics, both classical and contemporary, involve exponential change, in which quantities increase or decrease at rates proportionate to their current values.
• Scientists can create predictive models and comprehend underlying laws by identifying these patterns, which represent universal mathematical structures.
• Exponential laws also show how complex real-world phenomena can be explained by simplified models (based on ideal assumptions).
• ⇒ Radioactive decay:
• Nuclei decay randomly, but large populations fellow:
• [math]N(t) = N_0 e^{-\lambda t}[/math]
• [math]N(t)[/math] : number of undecayed nuclei
  • [math]\lambda[/math] : Decay constant
• Half – life is derived from exponential decay
• The decay law demonstrates how deterministic laws can characterise probabilistic processes by statistically modelling randomness.
• 
• Figure 3 Type of radioactive decay
• ⇒ Capacitor discharge (Electric circuits):
• In RC circuits:
• [math]Q(t) = Q_0 e^{-\frac{t}{RC}} \\
  V(t) = V_0 e^{-\frac{t}{RC}}[/math]
• – Voltage and charge drop exponentially over time.
• – The charging behaviour is similar. (With [math]1 – e^{-\frac{t}{RC}}[/math])
• Simple differential equations, a crucial mathematical tool in physics modelling, give rise to exponential decay.
• 
• Figure 4 Capacitor charging and discharging
• ⇒ Cosmology: Exponential expansion:
• The early universe grew exponentially quickly during cosmic inflation.
• Exponential growth is also used to model the current accelerated expansion (caused by dark energy).
• 
• Figure 5 Cosmic inflation occur
• ⇒ Tunneling in Quantum mechanics:
• In classically forbidden regions, a particle’s probability amplitude decays exponentially:
• [math]\psi(x) \propto e^{-kx}[/math]
• 
• Figure 6 Tunneling in classical and quantum