Modern Physics: X-Rays, Radioactivity, and Nuclear Energy

Introduction to X-Rays

• Definition: X-rays are high-energetic electromagnetic radiation characterized by short wavelengths.
• Electromagnetic Spectrum Position: They are located between ultraviolet rays and gamma rays on the electromagnetic spectrum.
• The Full Spectrum Order: The complete sequence of electromagnetic radiation is as follows: Radio waves, micro-waves, infra-red, visible, ultra-violet, X-rays, and gamma rays.

Production of X-Rays

• Mechanisms: There are three common mechanisms for the production of X-rays:
  • Atomic transitions between discrete energy levels.
  • Radioactive decay of specific atomic nuclei.
  • The acceleration of a charged particle, typically utilizing a Coolidge tube.
• Process in a Coolidge Tube:
  • X-rays are generated when fast-moving electrons are suddenly stopped by a solid target.
  • Cathode: Consists of a tungsten filament heated by a low-tension battery.
  • Thermionic Emission: Electrons are emitted by the filament due to heat.
  • Acceleration: These electrons are accelerated toward the tungsten target by a high potential difference established between the filament and the target.
  • Deceleration: The accelerated electrons strike the target material and are suddenly stopped.
  • Energy Conversion: The loss of kinetic energy from these electrons is emitted as X-ray photons.

Factors Influencing X-Ray Production

• Material of Target Metal: Materials with a high melting point and a high atomic number (Z) are preferred. These materials produce more energetic and intense X-rays.
• Intensity: This is controlled by the number of electrons striking the target per second.
• Filament Current: The number of emitted electrons is proportional to the temperature of the filament. This temperature is varied by adjusting the current in the filament circuit.
• Voltage: The frequency of the emitted X-rays is dependent on the voltage applied between the cathode and the anode.

Types of X-Rays

• Bremsstrahlung X-rays:
  • The term "Bremsstrahlung" is German for "braking radiation."
  • Produced when electrons hit the anode, decelerate, and release energy.
  • This is most effectively produced when small, charged particles interact with large atoms, as seen in a Coolidge tube.
• Characteristic X-rays:
  • Produced when electrons transition from one atomic orbit to another.
  • The resulting photon energies are characteristic of the specific type of atom involved.
  • Because of this specificity, they can be used to identify very small quantities of particular elements.
• Radioactivity X-rays:
  • These are generated when radionuclides undergo disintegration.
  • Photons with X-ray energy levels are released as part of the radioactive process.

Radioactivity: Discovery and Definition

• Discovery: Understanding of the nucleus began with the discovery of radioactivity in early 1896 by the French physicist Henri Becquerel.
  • Experiment: He observed that uranium salt crystals emitted invisible radiation that could darken a photographic plate, even when the plate was covered to exclude light.
  • Observation: The radiation required no external stimulation. This spontaneous emission was named radioactivity.
• Definition: Radioactivity is the spontaneous disintegration of unstable atomic nuclei into more energetically stable atomic nuclei.
• Characteristics: It involves an unstable nucleus losing particles or releasing energy during the transformation.

Properties of Radioactivity

• Spontaneity: The process occurs spontaneously.
• External Agents: It is unaffected by external variables such as:
  • High temperature.
  • High pressure.
  • Large electric fields.
• Emissions: It is accompanied by the emission of:
  • Electrons ($e^-$).
  • Positrons ($e^+$).
  • Alpha particles ($^{+4}He_{2}$).
  • Photons of energy and other particles.
• Context: It occurs exclusively in unstable nuclei.

Nuclear Instability and Decay

• Proton-Neutron Balance:
  • In light nuclei, the neutron and proton proportion is approximately $1.2:1$.
  • Atoms with a large atomic number ($Z$) often require more neutrons than protons to overcome the electrostatic repulsion between protons.
• Mechanisms of Instability: As $Z$ increases, a point is reached where additional neutrons are insufficient to overcome the electrostatic repulsion of protons, leading to instability.
• Ground State: Energetic nuclei are unstable and seek to return to their ground state.
• Chain Decays: Unstable nuclei may undergo series or chain decays to eventually reach a state of stability.
• Conservation Laws: During radioactive decay, the principles of conservation apply to:
  • Energy.
  • Momentum.
  • Charge.
  • Nucleon number.

Types of Radioactivity and Decay Modes

• Alpha (̑) Decay:
  • An alpha particle ($2$ protons + $2$ neutrons) is emitted.
  • An unstable parent nucleus ($X$) disintegrates into a lighter daughter nucleus ($Y$) and an alpha particle ($^{4}He_{2}$).
  • General Equation: {^{A}X_{Z}} \rightarrow {^{A-4}Y_{Z-2}} + ̑ ({^{4}He_{2}}).
  • The atomic number ($Z$) decreases by $2$, and the mass number ($A$) decreases by $4$.
• Beta (̒) Decay:
  • Involves a change in the atomic number ($Z$) while the nucleon number ($A$) remains unchanged.
  • Beta Minus (̒^-): Occurs in light nuclei with excess neutrons. A neutron is converted into a proton, emitting an electron (̒^-).
  • General Equation: {^{A}X_{Z}} \rightarrow {^{A}Y_{Z+1}} + ̒^- + ̄{̐}.
  • Neutron Conversion: {^{1}n_{0}} \rightarrow {^{1}p_{1}} + {^{0}e_{-1}} + ̄{̐}.
  • Example: {^{12}B_{5}} \rightarrow {^{12}C_{6}} + ̒^- + ̄{̐}.
  • Beta Plus (̒^+): Occurs in light nuclei with excess protons. A proton is converted into a neutron, emitting a positron (̒^+).
  • General Equation: {^{A}X_{Z}} \rightarrow {^{A}Y_{Z-1}} + ̒^+ + ̐.
  • Proton Conversion: {^{1}p_{1}} \rightarrow {^{1}n_{0}} + {^{0}e_{1}} + ̐.
  • Condition: Only possible when the atomic mass of the parent ($m_x$) is greater than the daughter ($m_y$) by at least two electronic masses (equivalent to $1.02\,MeV$).
  • Example: {^{12}N_{7}} \rightarrow {^{12}C_{6}} + ̒^+ + ̐.
• Electron Capture:
  • Occurs when there are excess protons but insufficient energy for ̒^+ decay.
  • An inner atomic electron is captured by the nucleus to convert a proton into a neutron.
  • General Equation: {^{A}X_{Z}} + e^- \rightarrow {^{A}Y_{Z-1}} + ̐.
  • Proton Conversion: {^{1}p_{1}} + {^{0}e_{-1}} \rightarrow {^{1}n_{0}} + ̐.
  • Example: ${^{7}Be_{4}} + {^{0}e_{-1}} \rightarrow {^{7}Li_{3}}$.
  • Orbital electron capture can result in the emission of characteristic X-rays of the daughter element.
• Gamma (̓) Decay:
  • Emission of gamma radiation when a nucleus in an excited state transitions to a lower energy state.
  • Energy Equation: $hf(\text{MeV}) = E_h - E_l$, where $E_h$ is the excited state energy and $E_l$ is the lower state energy.
  • Gamma rays carry no charge and no mass.
  • Atomic number and mass number do not change.
  • General Equation: {^{A\ast}X_{Z}} \rightarrow {^{A}X_{Z}} + ̓.

Binding Energy and Mass Defect

• Atomic Example: If a proton and electron form a ground-state hydrogen atom, a photon of $13.6\,eV$ is emitted. The rest energy of the combined system is less than the rest energy of its constituents by $13.6\,eV$.
• Binding Energy (BE) Definition:
  • The extra energy obtained when an atom is assembled from its components.
  • OR, the energy that must be supplied to disassemble the atom into its components.
• Mass Defect ($\Delta m$): The total mass of a nucleus ($M_{nucl}$) is less than the sum of the masses of its constituent nucleons ($Z$ protons and $N$ neutrons).
  • Equation: $\Delta m = Zm_p + Nm_n - M_{nucl}$.
  • Nuclear binding energy is considered the "missing mass" converted into energy to keep the nucleus intact.

Nuclear Fusion and Fission

• Fusion:
  • Combination of light nuclei to form a heavier nucleus.
  • Example: Deuterium ($^{2}H$) and Tritium ($^{3}H$) fuse to form Helium and a free neutron.
  • Equation: ${^{2}H} + {^{3}H} \rightarrow {^{4}He} + {^{1}n} + 17.6\,MeV$.
  • Mass is converted to energy via $E = \Delta mc^2$.
• Fission:
  • A heavy nucleus (e.g., Uranium-235) splits into two or more lighter nuclei.
  • Occurs when a heavy nucleus absorbs a neutron and becomes unstable.
  • Yield: Releases energy, 2-3 free neutrons, and gamma radiation.
  • Example: ${^{235}U} + {^{1}n} \rightarrow {^{141}Ba} + {^{92}Kr} + 3{^{1}n} + \text{Energy}$.
  • Approximately $200\,MeV$ of energy is released per fission event.

Quantitative Examples

• Example 1: Mass Defect of Helium ($^{4}He_{2}$)
  • Given: $m_p = 1.00728\,u$, $m_n = 1.00867\,u$, actual nuclear mass = $4.00151\,u$.
  • Calculation:
  • $\Delta m = (2 \times 1.00728) + (2 \times 1.00867) - 4.00151$
  • $\Delta m = 4.03190 - 4.00151$
  • $\Delta m = 0.03039\,u$
• Example 2: Binding Energy of Carbon-12 ($^{12}C_{6}$)
  • Given: $Z = 6$, $N = 6$, nuclear mass = $11.99671\,u$, $c^2 = 931.5\,MeV/u$.
  • Calculation:
  • $\Delta m = (6 \times 1.00728) + (6 \times 1.00867) - 11.99671$
  • $\Delta m = 0.09899\,u$
  • $BE = 0.09899 \times 931.5\,MeV/u$
  • $BE \approx 92.20\,MeV$