Comprehensive Notes on Particle Physics

Particle Physics

Atoms, Nuclei, and Radiation

Alpha-Particle Scattering Experiment

• Inference from Results:

  • Existence of the nucleus.

  • Small size of the nucleus.

• Simple Model for the Nuclear Atom:

  • Protons.

  • Neutrons.

  • Orbital electrons.

• Distinguishing Nucleon Number and Proton Number

• Isotopes:

  • Forms of the same element.

  • Different numbers of neutrons in their nuclei.

• Notation for Nuclides:

  • ^A_ZX

• Conservation Laws:

  • Nucleon number and charge are conserved in nuclear processes.

• Composition, Mass, and Charge of Radiations:

  • Alpha ($\alpha$) particles.

  • Beta ($\beta$) particles:

    • $\beta^-$ (electrons).

    • $\beta^+$ (positrons).

  • Gamma ($\gamma$) radiation.

• Antiparticles:

  • Same mass but opposite charge.

  • Positron is the antiparticle of an electron.

• Neutrinos and Beta Decay:

  • (Electron) antineutrinos are produced during $\beta^-$ decay.

  • (Electron) neutrinos are produced during $\beta^+$ decay.

• Energy Spectra of Radiations:

  • Alpha particles have discrete energies.

  • Beta particles have a continuous range of energies because (anti)neutrinos are emitted in beta decay.

• Radioactive Decay Equations:

  • Alpha decay: {}^{238}{92}U \rightarrow {}^{234}{90}Th + \alpha

• Unified Atomic Mass Unit (u):

Rutherford/Geiger-Marsden Alpha Particle Scattering Experiment

• Significance:

  • Small size of nucleus (V \downarrow).

  • Greater mass of nucleus (m \uparrow).

  • Presence of a dense nucleus inside an atom.

• Why Alpha Particles?

  1. Massive particle:

     • Not deflected due to orbiting electrons.

     • Able to come closer to the nucleus.

  2. Emitted from a source with constant energy.

• Why Not Beta Particles?

  1. Light particle:

     • Bounced back due to the repulsive force of orbiting electrons.

     • Cannot come closer to the nucleus.

  2. Emitted from a source with a range of kinetic energies.

• Why Not Gamma?

  • No charge and passes straight through the nucleus without any change of energy.

Apparatus Components and Their Significance

• Lead Block as a Collimator:

  • Absorbs all other randomly emitted alpha particles.

  • Allows a single path directed towards the gold foil.

• Evacuated Chamber:

  • Used so that alpha particles do not transfer their energy in collisions with air/gas particles inside the chamber.

  • Ensures alpha particles reach the gold foil with constant energy.

• Gold Foil:

  • Thin foil allows alpha particles to pass through and hit the fluorescent screen.

• Fluorescent Screen/Boundary of Chamber:

  • Provides the position where alpha particles hit the screen as a light spot is formed when a particle hits a fluorescent material.

• Traveling Microscope:

  • Moves along the boundary of the chamber to locate positions where particles hit the screen.

Observations and Reasons

• Observation:

  • Most alpha particles pass through the foil without any deflection.

  • Very few are deflected at large angles.

  • Approximately 1 out of 8000 are bounced back at an angle greater than 150°.

• Reason:

  • Alpha particles are deflected due to repulsive forces existing between the positively charged alpha particles and the gold nucleus.

Results and Significance (Rutherford Scattering Experiment)

• Significance:

  1. Smaller size of the nucleus, V \downarrow

  2. Greater mass of the nucleus, m \uparrow

  • There is a dense nucleus present inside an atom.

• Results:

  1. Bouncing back of alpha particles provides evidence that the mass of the gold nucleus is much greater than the mass of the alpha particle.

  2. Deflection of alpha particles shows the smaller size of the nucleus.

Calculations and Definitions

Density of Gold Nucleus

• Given: Gold nucleus {}^{197}_{79}Au

• Formula: \rho = \frac{m}{V}

• Diameter of the nucleus = 4.2 \times 10^{-14} m

• Calculation:

  • \rho = \frac{(197)(1.66 \times 10^{-27})}{\frac{4}{3}(3.14)(2.1 \times 10^{-14})^3} = 2.67 \times 10^{17} kg/m^3

Components of an Atom

• Nucleus: Positive charge, greater mass, smaller volume, high density.

• Proton: Positive charge.

• Neutron: Neutral charge (\theta = 0).

• Electrons: Negative charge and orbiting in defined orbits.

• Nucleus size: 10^{-14} m

• Atom size: 10^{-10} m

• Nucleus is around 10,000 times smaller than the entire atom.

Mass Number/Nucleon Number (A)

• Definition: Number of protons and number of neutrons in the nucleus of an atom.

• Symbol: A

• Formula: A = Z + N

  • Z - Number of protons

  • N - Number of neutrons

Charge Number (Z)

• Definition: Number of protons in the nucleus of an atom.

• Symbol: Z

Nuclide

• Definition: An element identified by its charge and mass number.

• Notation: {}^A_ZX

• Examples: {}^42He, {}^11H, {}^{12}6C, {}^{16}8O

Isotope

• Definition: Nuclei/atoms of the same element with an identical number of protons but a different number of neutrons.

• Note: Despite having different numbers of neutrons, isotopes of the same element have very similar physical properties.

• Notation: {}^A_ZX (Variable A, Constant Z)

Examples of Hydrogen Isotopes

• Hydrogen has three isotopes:

  1. Protium ({}^1_1H): 1 proton, 0 neutrons.

  2. Deuterium ({}^2_1H): 1 proton, 1 neutron.

  3. Tritium ({}^3_1H): 1 proton, 2 neutrons.

Types of Decay

• Four types of radiations/particles emitted in radioactive decay:

  1. Alpha decay ({}^4_2He or \alpha)

  2. Negative Beta decay / electron decay ({}^0{-1}e or {}^0{-1}\beta)

  3. Positive Beta decay / positron decay ({}^0{+1}e or {}^0{+1}\beta)

  4. Gamma decay (\gamma)

Properties of Alpha, Beta, and Gamma Particles

Deflection of Alpha, Beta, and Gamma Particles in a Uniform Magnetic Field

Nuclear Decay Reactions

• General form:

  • Parent Nucleus → Daughter Nucleus + Emitted Particle + Energy

Conservation Laws

1. Nucleon number (A), proton number (Z), mass, energy, and momentum are all conserved in a nuclear reaction/process.

2. Neutron number (N) may not be conserved in a nuclear process.

Alpha Decay

• Alpha particle is identical to a Helium nucleus ({}^4_2He)

• General reaction: {}^AZX \rightarrow {}^{A-4}{Z-2}Y + {}^4_2He

• Example:

  • {}^{226}{88}Ra \rightarrow {}^{222}{86}Rn + {}^4_2He

  • {}^{238}{92}U \rightarrow {}^{234}{90}Th + {}^{4}_{2}\alpha

• Note: In alpha-decay, the proton number of the nucleus decreases by two, and the nucleon number decreases by 4 and a new element is formed.

Beta Decay

Negative Beta Decay (Electron Decay)

• General reaction: {}^AZX \rightarrow {}^{A}{Z+1}Y + {}^{0}_{-1}e + \overline{\nu} + Energy

• Example: {}^{214}{82}Pb \rightarrow {}^{214}{83}Bi + {}^{0}_{-1}e + \overline{\nu} + Energy

• Note:

  1. In negative beta decay, the proton number of the nucleus increases by 1, and the nucleon number remains constant, and a new element is formed.

  2. Another particle, called an antineutrino (\overline{\nu}) with no electrical charge and negligible mass, is also emitted from the nucleus at the same time.

  3. Here a neutron changes into a proton, a -ve electron and anti-neutrino are released: {}^10n \rightarrow {}^11p + {}^{0}_{-1}e + \overline{\nu}

Positive Beta Decay (Positron Decay)

• General reaction: {}^AZX \rightarrow {}^{A}{Z-1}Y + {}^{0}_{+1}e + {\nu} + Energy

• Example: {}^{30}{15}P \rightarrow {}^{30}{14}Si + {}^{0}_{+1}e + {\nu} + Energy

• Note:

  1. In the positive beta decay, the proton number of the nucleus decreases by 1, and the nucleus number remains constant, and a new element is formed.

  2. Another particle called a neutrino (\nu) with no electrical charge and negligible mass is also emitted from the nucleus at the same time.

  3. Here a proton in the nucleus changes into a neutron, a +ve electron and neutrino are released: {}^11p \rightarrow {}^10n + {}^{0}_{+1}e + {\nu}

Reason for Beta Decay

• Beta decay (electron or positron) is due to weak interaction/nuclear forces.

• The range of nuclear forces is within the nucleus, i.e., of the order of 10^{-18} m, and is the strongest fundamental force that exists in nature.

Gamma Decay

• General reaction: {}^AZX^* \rightarrow {}^AZX + {\gamma}

• Example: {}^{234}{90}Th^* \rightarrow {}^{234}{90}Th + {\gamma}

Note:

1. Parent nuclide {}^{234}{90}Th^* is unstable/excited to form a stable {}^{234}{90}Th nuclide by emitting a Gamma particle / ray.

2. In gamma emission, no particles are emitted, and there is, therefore, no change to the proton number or nucleon number of the parent nuclide.

Summary of Decay Reactions

Unified Atomic Mass Unit (u)

• Definition: It is the mass of (\frac{1}{12}) the part of the mass of the C-12 isotope.

• Symbol: u

• Value: 1u = 1.66 \times 10^{-27} kg

Examples:

1. Mass of Helium nucleus in u:

   • m = 4u = 4(1.66 \times 10^{-27}) = 6.64 \times 10^{-27} kg

   • So, the mass of Alpha particle is also 6.64 \times 10^{-27} kg

2. Mass of uranium nucleus:

   • M = 235u = 235 (1.66 \times 10^{-27}) = 3.901 \times 10^{-25} kg

Energy Associated with Mass

• E = mc^2

• Using uranium nucleus: E = (3.901 \times 10^{-25}) (3.00 \times 10^8)^2 = 3.5109 \times 10^{-8} J

Conversion to MeV

• 1 eV = 1.60 \times 10^{-19} J

• 1 MeV = (10^6) (1.60 \times 10^{-19}) = 1.60 \times 10^{-13} J

• E = \frac{3.5109 \times 10^{-8}}{1.60 \times 10^{-13}} \approx 219431.25 MeV

Fundamental Particles

Composition of Matter

• Displays a hierarchy from molecules to quarks, showing the size of each constituent

• Molecule: 10^{-10} \text{ to } 10^{-9} m

• Atom: 10^{-10} m

• Nucleus: 10^{-14} m

• Protons/Neutrons: 10^{-15} m

• Quarks: < 10^{-18} m

Definition of a Particle

• Classical Physics: Anything whose rest mass is defined.

• Modern/Quantum Physics: Anything whose momentum is defined.

Types of Particles

• Leptons

• Hadrons

Leptons

Definition

Particles that are considered fundamental and are not made of other particles.

Properties

1. Not affected by strong forces (electric or nuclear forces).

2. All leptons have very small masses.

   • Examples: Electron, neutrino, positron, and anti-neutrino.

Hadrons

Definition

These are not fundamental particles and are made of other particles called Quarks.

Properties

1. Hadrons are affected by strong nuclear forces.

2. Each Hadron is made of quarks.

Types of Hadrons

1. Baryons: Particles consisting of three quarks (e.g., protons and neutrons).

2. Mesons: Particles consisting of one quark and one antiquark.

Quark Model

Quarks as Fundamental Particles

• Particles which are combined to form Hadrons are Quarks.

• There are six types of Quarks. A Quack can never exist as independent in nature

Types of Quarks

Note: Top quark is the heaviest with a mass of approximately 200 times the mass of a proton. Protons and Neutrons are not fundamental

Composition of a Proton

A proton is an example of Baryons and is made of two up quarks, one down quark and a strange quark.

• Proton : uud

  • = u + u + d = (+\frac{2}{3}e) + (+\frac{2}{3}e) + (-\frac{1}{3}e) = +e = +1.60 \times 10^{-19} C

Composition of a Neutron

A proton is also an example of Baryons and is made of two down quarks, one up quark and a strange quark.

• Neutron : udd

  • = u + d + d = (+\frac{2}{3}e) + (-\frac{1}{3}e) + (-\frac{1}{3}e) = 0

Change in Quark Model in Beta Decay

1. Positron decay (the Beta decay):

   • p \rightarrow n + \beta^+ + \nu + Energy

   • uud → udd

   • Here a proton changes to a newton. So, an up quack changes to a down quack.

2. Electron decay (-ve Beta decay)

   • n \rightarrow p + \beta^- + \overline{\nu} + E

   • udd → uud

   • Here a neutron changes to a proton. So, a down Quack changes to an up quark.

Composition of Helium Nucleus in Terms of Quark Model

• {}^4He = 2p + 2n = 2(uud) + 2(udd) = 2(2u+d) + 2(u+2d) = 4u + 2d + 2u + 4d = 6u + 6d

  • = 6(+\frac{2}{3}e) + 6(-\frac{1}{3}e) = +4e - 2e = +2e = +2(1.60 \times 10^{-19}) = 3.20 \times 10^{-19} C

Fundamental Particles and Standard Model

Standard Model

• The standard model of particle physics asserts that there are 12 fundamental particles, divided into quarks and leptons.

Quarks

• Six types: up, down, strange, charm, top, and bottom. Protons and neutrons are made of different combinations of quarks.

Leptons

• Six types: electron, muon, tau, electron-neutrino, muon-neutrino, and tau-neutrino. All leptons have very small masses.

Charges of Quarks and Leptons

Quarks

• Quarks occur in groups of two or three, never separately. The top quark is the heaviest.

• As well as the 12 fundamental particles, there are 12 equivalent antiparticles.

Fundamental Forces

• There are four fundamental forces that control the interactions between fundamental particles.

Hadrons

Baryons

• Particles consisting of three quarks (e.g., proton (uud), neutron (udd)).

Mesons

• Particles consisting of one quark and one antiquark.

Antibaryons

• Consist of three antiquarks (e.g.,\overline{u}\overline{u}\overline{d} for an antiproton).

Quarks and Beta Decay

• In \beta^--decay, a down quark changes into an up quark in one of the neutrons in a nucleus, making it a proton, and in doing so emits an electron (the \beta-particle) and an electron antineutrino.

• In \beta^+-decay, one of the protons in a nucleus changes into a neutron by one of the up quarks changing into a down quark, emitting a positron (the \beta$$-particle) and an electron neutrino in the process.

• The force responsible for beta decay is the weak nuclear force.