Atomic Structure, Radioactivity, and Nuclear Physics

Rutherford Gold Foil Experiment

• The gold foil experiment was conducted by scientists Hans Geiger and Ernest Marsden, and its results were explained and interpreted by Ernest Rutherford.
• Experimental Setup:
  • An alpha source is enclosed within a heavy lead shield featuring a narrow opening. This produces a collimated, narrow beam of alpha particles and prevents backscattering of alpha particles.
  • The beam of alpha particles is directed toward the center of a thin gold sheet placed in its pathway.
  • A sensitive curved screen surrounds the setup to detect where alpha particles strike.
  • When an alpha particle strikes the screen, a flash of light called a scintillation is produced.

• Observations and Explanations:
  • Observation 1: The vast majority of alpha particles passed straight forward through the gold foil with no deviation or deflection, and flashes did not disappear from the central screen position.
  • Explanation: Most of the atom consists of empty space, allowing alpha particles to pass through unimpeded without colliding with dense matter.
  • Observation 2: A tiny fraction of alpha particles were deflected at large angles greater than $90^\circ$ (backscattered).
  • Explanation: There is a heavy, extremely tiny nucleus situated at the center of the atom that contains almost all of the atom's mass concentrated in a very small volume compared to the overall size of the atom.
  • Observation 3: A small number of alpha particles were deviated at small angles less than $90^\circ$.
  • Explanation: The nucleus carries a concentrated positive electric charge. Since alpha particles are also positively charged, they experience electrostatic repulsion when passing close to a nucleus, causing them to deviate from their straight path.

Atomic Models and Composition of Atoms

• Old Model of the Atom (Plum Pudding Model):
  • Described the atom as a solid sphere throughout which positive charge and negative electrons were evenly distributed.
  • Proven incorrect by Rutherford's gold foil experiment.
• Nuclear Model of the Atom (Rutherford Model):
  • Established after the gold foil experiment.
  • Describes the atom as a mostly empty sphere containing a tiny, dense, heavy, and positively charged nucleus at its center, with negatively charged electrons orbiting outside the nucleus.
• Subatomic Particles (Nucleons and Electrons):
  • Protons: Positively charged particles located inside the nucleus.
  • Neutrons: Electrically neutral particles (no charge) located inside the nucleus.
  • Nucleons: Collective term for particles situated within the nucleus (protons and neutrons combined).
  • Electrons: Negatively charged particles located outside the nucleus, orbiting in specific energy levels or shells.
• Mass Comparisons:
  • The mass of a single proton or a single neutron is approximately $1840$ times the mass of an electron.
  • Electrons contribute negligibly to the overall mass of an atom.
• Atomic Charge and Ion Formation:
  • A normal, uncharged atom is electrically neutral because its number of positive protons equals its number of negative electrons.
  • Positive Ion: Formed when a neutral atom loses one or more electrons, resulting in more protons than electrons.
  • Negative Ion: Formed when a neutral atom gains one or more electrons, resulting in more electrons than protons.

Atomic Notation and Isotopes

• Symbolic Representation of Elements:
  • Expressed in standard chemical notation as $^{A}_{Z}\text{X}$.
  • Proton Number / Atomic Number ($Z$): Represents the total number of protons in the nucleus. In a neutral atom, it also equals the number of orbiting electrons.
  • Mass Number / Nucleon Number ($A$): Represents the total number of protons plus the total number of neutrons in the nucleus ($\text{Nucleons} = \text{Protons} + \text{Neutrons}$).
  • Neutron Calculation: The number of neutrons in a nucleus is determined by subtracting the atomic number from the mass number:

$\text{Number of Neutrons} = A - Z$

• Isotopes:
  • Definition: Atoms of the same chemical element that possess the same number of protons (same atomic number $Z$) but a different number of neutrons (different mass number $A$).
  • Stability: Isotopes can be either stable or unstable (radioactive).
  • Carbon-12 ($^{12}_{6}\text{C}$): Stable isotope with $6$ protons and $6$ neutrons.
  • Carbon-14 ($^{14}_{6}\text{C}$): Unstable, radioactive isotope with $6$ protons and $8$ neutrons.
  • Cause of Instability: A nucleus becomes unstable and radioactive due to an excess number of neutrons or because the nucleus is too heavy and large.
  • Decay Process: To reach stability, an unstable isotope spontaneously undergoes radioactive decay by emitting radiation.

Characteristics of Alpha, Beta, and Gamma Radiation

• Radioactive Decay Definition:
  • Process by which an unstable nucleus emits alpha ($\alpha$), beta ($\beta$), or gamma ($\gamma$) radiation to transform into a more stable nucleus, often forming an isotope of an entirely different element.
• Fundamental Properties of Radioactive Decay:
  • Spontaneous: Occurs naturally without needing external energy, catalysts, or stimulation, and cannot be accelerated or slowed by external physical factors (such as temperature or pressure).
  • Random: Impossible to predict which specific unstable nucleus will decay next or precisely when it will decay; the decay rate fluctuates continuously over time.

• Alpha Particle ($\alpha$):

  • Composition: Consists of $2$ protons and $2$ neutrons bound together; identical to a Helium-4 nucleus ($^{4}_{2}\text{He}$ or $^{4}_{2}\alpha$).
  • Charge: $+2$
  • Mass: Approximately $4$ times the mass of a proton.
  • Ionization Power: Very strong.
  • Reason: Alpha particles are large, heavy, move relatively slowly, and carry a double positive charge ($+2$), allowing them to easily strip electrons from neutral atoms during collisions.
  • Penetrating Power: Very weak.
  • Shielding/Screening: Completely stopped by a single sheet of paper or human skin.
  • Range in Air: Short, typically only a few centimeters (about $5\,\text{cm}$ to $6\,\text{cm}$).

• Beta Particle ($\beta$):

  • Composition: A high-speed electron emitted directly from the nucleus ($^{0}_{-1}\text{e}$ or $^{0}_{-1}\beta$).
  • Charge: $-1$
  • Mass: Approximately $\frac{1}{1800}$ the mass of a proton.
  • Ionization Power: Weak.
  • Penetrating Power: Moderate.
  • Shielding/Screening: Completely stopped by a few millimeters (about $5\,\text{mm}$) of aluminum sheet.
  • Range in Air: Moderate, typically about $100\,\text{cm}$ ($1\,\text{m}$) or slightly more.

• Gamma Radiation ($\gamma$):

  • Composition: High-energy, short-wavelength electromagnetic radiation.
  • Charge: $0$ (neutral).
  • Mass: $0$
  • Ionization Power: Very weak (nearly zero).
  • Penetrating Power: Very high.
  • Shielding/Screening: Attenuated but not easily stopped completely; requires thick lead, concrete (about $1\,\text{m}$ thick), or steel to absorb significantly.
  • Range in Air: Extremely large, extending well beyond $100\,\text{m}$.

Deflection of Radiation in Electric and Magnetic Fields

• Behavior in Electric Fields:
  • When passed between a positively charged plate and a negatively charged plate:
  • Alpha Particles: Slightly deflected toward the negative plate due to their positive charge ($+2$). The deflection angle is small because alpha particles are heavy.
  • Beta Particles: Strongly deflected toward the positive plate due to their negative charge ($-1$). The deflection angle is large because beta particles are light electrons.
  • Gamma Rays: Pass completely straight through without deflection because they carry zero electrical charge.

• Behavior in Magnetic Fields:
  • Radiation direction in a magnetic field follows Fleming's Left-Hand Rule (treating moving positive alpha particles as conventional current and moving negative beta particles as opposite current):
  • Alpha Particles: Slightly deflected in one direction.
  • Beta Particles: Strongly deflected in the opposite direction.
  • Gamma Rays: Experience no deflection in magnetic fields.

Nuclear Equations

• Balancing Rules for Nuclear Reactions:
  • The sum of atomic numbers ($Z$) on the reactant side must equal the sum of atomic numbers on the product side.
  • The sum of mass numbers ($A$) on the reactant side must equal the sum of mass numbers on the product side.
• Alpha Decay Example:
  • Radium-226 ($^{226}_{88}\text{Ra}$) emits an alpha particle ($^{4}_{2}\alpha$) to form Radon-222 ($^{222}_{86}\text{Rn}$):

$^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + {}^{4}_{2}\alpha$

• Atomic numbers check: $88 = 86 + 2$
• Mass numbers check: $226 = 222 + 4$
  • Beta Decay Example:
• Sodium-24 ($^{24}_{11}\text{Na}$) emits a beta particle ($^{0}_{-1}\beta$) to form Magnesium-24 ($^{24}_{12}\text{Mg}$):

$^{24}_{11}\text{Na} \rightarrow {}^{24}_{12}\text{Mg} + {}^{0}_{-1}\beta$

• Atomic numbers check: $11 = 12 + (-1)$
• Mass numbers check: $24 = 24 + 0$
• Internal Mechanism of Beta Decay: Inside a nucleus undergoing beta decay, a single neutron splits into a proton (which remains in the nucleus) and a high-speed electron (which is ejected as a beta particle):

$^{1}_{0}\text{n} \rightarrow {}^{1}_{1}\text{p} + {}^{0}_{-1}\text{e}$

Radiation Detectors and Background Radiation

• Radiation Detectors:
  • Because nuclear radiation is invisible, specialized detectors are needed to observe and measure radiation.
  • Photographic Plates: Darken upon exposure to radiation (used historically to track alpha particle beams).
  • Geiger-Müller Detector (G-M Detector): The primary instrument used in modern laboratories; detects ionizing radiation and outputs numerical count values.
• Background Radiation:
  • Definition: The natural radiation present in the surrounding environment when no artificial radioactive sources are present.
  • Natural Sources of Background Radiation:
  • Earth and radioactive rocks in the ground.
  • Radioactive gases present in air, specifically Radon gas.
  • Space and cosmic rays.
  • The Sun.
  • Artificial Sources of Background Radiation:
  • Nuclear power generation stations.
  • Nuclear weapons testing and physics experimental facilities.
• Background Radiation Correction Calculation:
  • To calculate the true radiation emitted by a sample, background radiation must be subtracted from total measured counts:

$\text{Corrected Reading} = \text{Source Reading} - \text{Background Reading}$

• Calculation Example:
  • Background count rate reading = $20\,\text{count/min}$
  • Measured source count rate reading = $630\,\text{count/min}$
  • Corrected radiation count rate = $630 - 20 = 610\,\text{count/min}$
• Units of Radiation Measurement: Counts per minute ($\text{count/min}$) or Counts per second ($\text{count/s}$).

Experimental Investigations of Radiation Properties

• 1. Comparing Source Intensity and Strength:
  • Position different radioactive sources at identical fixed distances from a G-M detector.
  • Record count rates; the source yielding the highest corrected count rate possesses the highest strength or intensity.
• 2. Effect of Distance on Counting Rate:
  • Measure and record background count rate.
  • Position detector close to source and record count rate.
  • Gradually increase the distance between source and detector, taking readings at measured increments.
  • Observation: As distance increases, count rate drops sharply until reaching the background reading level.
• 3. Measuring Ranges in Air:
  • Alpha range: Limited to a few centimeters ($5\,\text{cm}$ to $6\,\text{cm}$).
  • Beta range: Usually around $1\,\text{m}$ or slightly more.
  • Gamma range: Extremely large, continuing across long distances.
• 4. Absorption by Materials:
  • Fix the distance between a radioactive source and a G-M counter, then insert test sheets of varying materials and thicknesses.
  • Record count rates with and without absorbing sheets:
  • Paper / human skin completely absorbs alpha particles.
  • A few millimeters of aluminum ($\approx 5\,\text{mm}$) completely absorbs beta particles.
  • Thick lead or concrete blocks attenuate gamma rays, though complete absorption is difficult to achieve.

• 5. Identifying Radiation Types using Electric Fields:
  • Place positive and negative electric plates near the path of emitted radiation.
  • Move the G-M detector up and down to locate peak count readings:
  • Peak reading near negative plate identifies an alpha emitter ($+2$ charge attraction).
  • Peak reading near positive plate identifies a beta emitter ($-1$ charge attraction).
  • Peak reading straight ahead identifies a gamma emitter (neutral electromagnetic radiation).

Nuclear Fission and Nuclear Fusion

• Nuclear Fission:
  • Definition: The process of splitting a large, heavy nucleus into two smaller daughter nuclei, releasing immense amounts of energy.
  • Fission Mechanism:
  • A slow-moving neutron ($^{1}_{0}\text{n}$) strikes a Uranium-235 nucleus ($^{235}_{92}\text{U}$).
  • The extra neutron is absorbed, forming highly unstable Uranium-236 ($^{236}_{92}\text{U}$).
  • The nucleus splits into Crypton-92 ($^{92}_{36}\text{Kr}$), Barium-141 ($^{141}_{56}\text{Ba}$), and emits $3$ free neutrons alongside a release of kinetic energy.
  • Balanced Nuclear Fission Equation:

$^{235}_{92}\text{U} + {}^{1}_{0}\text{n} \rightarrow {}^{92}_{36}\text{Kr} + {}^{141}_{56}\text{Ba} + 3\,{}^{1}_{0}\text{n} + \text{Energy}$

• Conservation Verification:
  • Atomic numbers: $92 + 0 = 36 + 56 + 3(0) = 92$
  • Mass numbers: $235 + 1 = 92 + 141 + 3(1) = 236$
• Chain Reaction: The $3$ released extra neutrons collide with neighboring Uranium-235 nuclei, triggering subsequent fission events in a self-sustaining chain reaction.

• Nuclear Fusion:
  • Definition: The process in which two light nuclei join together under extreme conditions to form a larger, heavier nucleus, releasing significantly more energy than nuclear fission.
  • Occurrence: Nuclear fusion is the core power process by which stars, including the Sun, generate light and heat energy.
  • Solar Fusion Reaction Equation:

$^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \rightarrow {}^{4}_{2}\text{He} + {}^{1}_{0}\text{n} + \text{Energy}$

• Conservation Verification:
  • Atomic numbers: $1 + 1 = 2 + 0 = 2$
  • Mass numbers: $2 + 3 = 4 + 1 = 5$
• Temperature Requirements: Fusion requires extremely high temperatures and pressures to provide positively charged light nuclei with sufficient kinetic energy to overcome electrostatic repulsion forces between them.
  • Mass-Energy Conversion in Nuclear Reactions:
• In both nuclear fission and nuclear fusion, total particle mass prior to reaction is slightly greater than total particle mass remaining after reaction.
• The lost mass (mass defect) is converted directly into released energy according to mass-energy equivalence.

Summary of Fundamental Particles in Nuclear Physics

• Proton: $^{1}_{1}\text{P}$ or $^{1}_{1}\text{p}$
• Neutron: $^{1}_{0}\text{n}$
• Electron: $^{0}_{-1}\text{e}$
• Beta Particle: $^{0}_{-1}\beta$
• Alpha Particle: $^{4}_{2}\alpha$ or $^{4}_{2}\text{He}$
• Gamma Ray: $\gamma$