Introduction to Modern Physics

Early Atomic Models and Their History

J.J Thompson Atomic Model (1897):
- This model is famously known as the "Plum-Pudding Model."
- Proposed that an atom is a neutral particle composed of a positive charge with lumps of negative charge embedded within it.
- Key measurements and statistics associated with this model:
  - Atomic radius: Approximately $1.0 \times 10^{-10}\,m$ .
  - Field strength on the surface of the atom: Approximately $1.4 \times 10^{11}\,v/m$ .
- Significant contribution: Thompson discovered the electron and successfully measured its charge-to-mass ratio ( $e/m$ ).
Rutherford Atomic Model:
- Rutherford discovered that positive charge is not spread out but is concentrated within a central region called the nucleus.
- He posited that electrons provide the properties and proper understanding of an atom's behavior.
- Key measurements associated with the Rutherford model:
  - Nucleus radius: Approximately $6.9 \times 10^{-5}\,m$ .
  - Field strength on the surface of the atoms: Approximately $3.0 \times 10^{19}\,v/m$ .

Structure and Dynamics of the Nucleus

Basic Particles of the Nucleus:
- Proton: A positively charged stable particle.
- Neutron: A neutral particle. It is only stable when inside the nucleus.
Elemental Composition:
- All nuclei contain both protons and neutrons, with the notable exception of hydrogen, which contains only one proton.
- In most cases, the number of neutrons in a nucleus exceeds the number of protons.
Nuclear Force:
- This is the force responsible for holding protons and neutrons together within the nucleus.
- Distinguishing characteristic: This force acts only over extremely small distances within the nucleus.
Nuclear Definitions:
- Atomic Number ( $Z$ ): The specific number of protons present in the nucleus of an atom.
- Nucleon Number (Mass Number, $A$ ): The total number of both protons and neutrons in the nucleus of an atom.

Electron Energy and Emission

The Electron Volt ( $eV$ ):
- Definition: The energy gained by an electron when it is accelerated through a potential difference of exactly one volt.
- Conversion factor: $1\,eV \approx 1.6 \times 10^{-19}\,J$ .
Thermionic Emission:
- Discovered by Edison in 1880.
- Definition: The process by which free electrons are emitted from a hot metal surface.
- Variable rate: The rate of this emission varies depending on the specific type of metal used.
Cathode Ray Tube (CRT) and Electron Velocity:
- Properties of cathode rays:
  - They exhibit rectilinear propagation (travel in straight lines).
  - They cause fluorescence upon impact with certain materials.
  - They possess kinetic energy.
  - They are deflected by both electric and magnetic fields.
- Electron Velocity Formula: The velocity ( $v$ ) of electrons from an electron gun is given by:
- $v = \sqrt{\frac{2eV}{m}}$
Specific Charge:
- Definition: The charge-to-mass ratio of any charged particle ( $q/m$ ).
- Specific term for electrons: This ratio is specifically denoted as $e/m$ .
Binding Energy:
- Mathematical assessment: The total energy in the nucleus (the bound system) minus the combined energy of the separated neutrons or the mass number.

Fundamentals of Radioactivity

Definition of Radioactivity:
- Refers to the spontaneous emission of radiation from the nucleus of an atom.
- This is a natural property of certain isotopes known as radioactive isotopes or radioisotopes.
- Unstable radioisotopes undergo radioactive decay to reach a more stable configuration.
Types of Radioactive Emissions:
- Alpha Particles: Identified as helium nuclei.
- Beta Particles: Can be electrons or positrons.
- Gamma Rays: High-energy photons.
- The emission of these rays and particles is accompanied by a release of energy.
Half-Life ( $T_{1/2}$ ):
- The time required for half of a radioactive material sample to decay.
- Each radioisotope has a unique half-life, ranging from small fractions of a second to billions of years.
Applications of Radioactivity:
- Medicine: Diagnostic imaging and cancer treatment.
- Industry: Gauging the thickness and density of materials.
- Scientific Research: Tracing biological processes and radiometric dating (determining the age of objects).

Health Risks and Safety Protocols

Health Hazards:
- Exposure to high levels or prolonged radioactivity is hazardous.
- Ionizing radiation damages living tissues and DNA, increasing risks for cancer and genetic mutations.
- Acute symptoms of high radiation exposure include nausea, vomiting, hair loss, and potentially death.
Mechanisms of Cellular Damage:
- Alpha and Beta particles: Damage cells via direct impact or by releasing energy when stopped by matter.
- Gamma rays: Damage cells by interacting with electrons, knocking them out of orbits and damaging DNA.
Safety Measures:
- Handling and storage are strictly regulated.
- Exposure is controlled via shielding and proper containment.
- Adhering to safety protocols and using protective gear.
- Regular medical checkups for exposed individuals.
- Lifestyle mitigations: Healthy diet, exercise, and avoiding smoking help reduce collective damage.

Mathematical Laws of Radioactive Decay

The Statistical Law of Chance:
- The disintegration of atomic nuclei obeys the statistical law of chance.
- The number of atoms disintegrating per second is directly proportional to the number of atoms present at that instant.
Decay Equations:
- The rate of change of the nucleus is proportional to the nucleus:
- $\frac{\Delta N}{\Delta t} \propto N$
- $\frac{\Delta N}{\Delta t} = -\lambda N$
- Where:
  - $\Delta N$ = change in number of nuclei.
  - $\Delta t$ = time interval.
  - $\lambda$ = Decay Constant (proportionality constant).
  - Negative sign indicates that $N$ decreases over time.
Exponential Decay Formula:
- Derived using calculus:
- $N = N_0 e^{-\lambda t}$
- Where:
  - $N$ = nuclei present at time $t$ .
  - $N_0$ = nuclei present at initial time $t = 0$ .
  - $e$ = Euler’s constant ( $\approx 2.718$ ).
Derivation of Half-Life ( $T_{1/2}$ ):
- At half-life, $N = \frac{N_0}{2}$ .
- $\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$
- $\frac{1}{2} = e^{-\lambda T_{1/2}}$
- $\ln(2) = \lambda T_{1/2}$
- $T_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693}{\lambda}$
Mean Life ( $T$ ):
- The average time taken for a particle to exist in a particular form.
- Expressed as the reciprocal of the decay constant:
- $T = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693}$

Wave-Particle Duality and Quantum Physics

Failure of Classical Physics:
- In the 20th century, scientists noted Classical/Newtonian physics could not explain specific phenomena involving sub-atomic particles:
  - Spectral lines of atoms.
  - X-ray production.
  - Photoelectric effect.
  - Compton effect (associated with Arthur Compton).
The Quantum Model:
- Physical quantities exist in discrete or integral values, a concept known as "quantized."
Principle of Complementarity (Bohr, 1928):
- States that wave and particle descriptions are complementary.
- Both are necessary for a complete understanding of light, but they are never used simultaneously to describe a single part of an occurrence.
- Wave nature evidence: Reflection, refraction, interference, diffraction.
- Particle nature evidence: Photoelectric effect, Compton effect.
De Broglie Hypothesis (1924):
- Extended duality to matter; if light is dualistic, particles like electrons/protons should behave as waves.
- A free particle with rest mass ( $m$ ) moving at non-relativistic speed ( $v \ll c$ ) has a wavelength ( $\lambda$ ) related to its momentum ( $P$ ):
- $\lambda = \frac{h}{P} = \frac{h}{mv}$
- Where $h$ is Planck's constant.
Application to Bohr’s Model:
- De Broglie suggested electron motion is guided by a "standing pilot wave" fitting the orbit's circumference.
- Condition for standing wave: $2\pi r = n\lambda$ , where $n = 1, 2, 3, \dots$
- Since $\lambda = \frac{h}{mv}$ , then $2\pi r = \frac{nh}{mv}$ .
- This leads to the quantization of angular momentum ( $L$ ):
- $mvr = \frac{nh}{2\pi} = nL$

Characteristics of Isotopes

Isotopes Definition:
- Variants of an element with the same number of protons but different numbers of neutrons.
- Identical atomic numbers but different mass numbers ( $A$ ).
Example: Carbon Isotopes:
- Carbon-12 ( $^{12}C$ ): 6 protons, 6 neutrons.
- Carbon-13 ( $^{13}C$ ): 6 protons, 7 neutrons.
- Carbon-14 ( $^{14}C$ ): 6 protons, 8 neutrons.
Properties and Significance:
- Similar chemical properties due to identical electron configurations.
- Differing physical properties (mass) and nuclear stability.
- Stable isotopes exist alongside radioactive ones.
- Represented with the mass number as a superscript before the symbol (e.g., $^{14}C$ ).

Nuclear Fission and Fusion

Nuclear Fission:
- Process: Splitting a large nucleus into two smaller nuclei.
- Nature: A chain reaction where one event triggers others.
- Applications: Nuclear weapons and nuclear power plants.
- Requirements: Low temperature and pressure compared to fusion.
Nuclear Fusion:
- Process: Combining two smaller nuclei to form a larger nucleus.
- Nature: Not a chain reaction.
- Energy: Releases significantly more energy per unit mass than fission.
- Applications: Powers stars; studied as a future Earth energy source.
- Requirements: Extremely high temperatures and pressures to overcome repulsive forces.
Comparison Summary:
- Energy Release: Both fission and fusion release energy.
- Efficiency: Fusion reactors are more challenging and expensive but offer clean, abundant energy without greenhouse gas emissions.

Examples and Exercises

Photoconductivity in Silicon Oxide: If illuminated with photons of energy $1.14\,eV$ or greater, calculate the corresponding wavelength.
Orange Light Photon: A photon of orange light has a wavelength of $610\,nm$ . Calculate the frequency and energy in $eV$ .
Quantum Energy: Calculate the energy of a quantum of light with a frequency of $5 \times 10^{14}\,Hz$ .
Photoelectron Ejection: Ultraviolet light of frequency $1.3 \times 10^{15}\,Hz$ shines on metal; photoelectrons are ejected with maximum energy $1.8\,eV$ . Find the work function ( $eV$ ) and frequency.
Half-Life Calculation: Radioactive sample activity reduces to $1/16$ of its initial value in $1\,hr\,20\,mins$ . Calculate its half-life.
Homework Question: A radioactive sample activity drops to $1/32$ of its initial value. If the half-life is $10.8\,hrs$ , calculate the total time taken.