Probing deep into matter
Module 6: Field and particle physics6.2 Fundamental particle |
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| 6.2.1 |
Probing deep into matter a) Describe and explain: I) Use of particle accelerators to generate high-energy beams of particles for scattering II) Evidence from scattering for a small massive nucleus within the atom III) Evidence of discrete energy levels in atoms IV) A simple model of the atom as the quantum behavior of electrons in a confined space V) Simple picture of the quark structure of protons and neutrons VI) Application of conservation of mass/energy, charge and lepton number in balanced nuclear equations VII) Relativistic calculations for particles travelling at very high speed, for example in particle accelerators or cosmic rays. b) Make appropriate use of: I) The terms: energy level, scattering, nucleus, proton, neutron, nucleon, electron, positron, quark, gluon, neutrino, hadron, lepton, antiparticle, lepton number by sketching and interpreting: II) Paths of scattered particles III) Electron standing waves in simple models of an atom c) Make calculations and estimates involving: I) Motion of a charged particle in magnetic field using [math]F = qvB[/math] II) Kinetic and potential energy of a scattered charged particle III) [math]E_{rest} = mc^2[/math] and relativistic factor IV) [math]E_{total} = γE_{rest}[/math] |
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a) Describe and explain:
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I. Use of Particle Accelerators to Generate High-Energy Beams of Particles for Scattering
- ⇒ Purpose:
- Particle accelerators are used to investigate the fundamental structure of matter by accelerating charged particles (e.g., electrons, protons) to high velocities and colliding them with target atoms or particles.
- High-energy beams allow us to probe deeper into matter, as shorter wavelengths (associated with higher energy) resolve smaller structures according to the de Broglie wavelength formula:
- [math]λ = \frac{h}{p}[/math]
- Where h is Planck’s constant and p is the momentum of the particle.
- Figure 1 De-Broglie wavelength
- ⇒ Mechanism:
- Particles are accelerated using electric fields and kept on precise paths using magnetic fields in devices like linear accelerators (LINACs), cyclotrons, and synchrotrons.
- The accelerated particles collide with a target or other particles in a collider. Scattered particles are detected to study the structure of matter.
- ⇒ Example:
- The Large Hadron Collider (LHC) accelerates protons to nearly the speed of light and collides them to study subatomic particles such as quarks, gluons, and the Higgs boson.
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II. Evidence from Scattering for a Small Massive Nucleus within the Atom
- ⇒ Rutherford’s Gold Foil Experiment (1909):
- A beam of alpha particles (α\alphaα, helium nuclei) was directed at a thin gold foil.
- Observations:
- – Most particles passed straight through the foil, indicating that atoms are mostly empty space.
- – A small fraction of particles were deflected at large angles, and some bounced back, indicating the presence of a dense, positively charged nucleus.
- Conclusion:
- – The atom consists of a tiny, massive, positively charged nucleus surrounded by electrons
- – The nucleus contains most of the atom’s mass and is incredibly small compared to the overall size of the atom.
- Figure 2 Rutherford’s Gold foil experiment
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III. Evidence of Discrete Energy Levels in Atoms
- ⇒ Observations:
- The emission and absorption spectra of atoms consist of distinct lines, indicating that electrons in atoms can only occupy specific energy levels.
- Figure 3 The line emission spectrum
- ⇒ Theoretical Explanation:
- Electrons transition between discrete energy levels, absorbing or emitting photons with energy:
- [math]∆E = hf[/math]
- where ΔE is the energy difference between levels, h is Planck’s constant, and f is the frequency of the photon.
- ⇒ Experimental Evidence:
- The hydrogen atom spectrum (Balmer series) provided strong evidence for quantized energy levels, leading to the development of the Bohr model of the atom.
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IV. A Simple Model of the Atom as the Quantum Behavior of Electrons in a Confined Space
- ⇒ Quantum Behavior of Electrons:
- Electrons in an atom behave as waves, and their behavior is described by the Schrödinger equation.
- The wavefunction (ψ) provides a probability distribution for finding the electron in a certain region, called an orbital.
- Figure 4 Quantum mechanics
- ⇒ Quantum Model:
- Energy Quantization:
- – Electrons can only occupy discrete energy levels, determined by the quantization of their wavefunctions.
- – Transitions between levels involve the absorption or emission of a photon with energy:
- [math]∆E = hf[/math]
- Where h is Planck’s constant and f is the frequency of the photon.
- Orbitals:
- – Instead of fixed orbits (as in the Bohr model), electrons occupy regions of space called orbitals (e.g., s, p, d, f).
- Uncertainty Principle:
- – According to Heisenberg’s uncertainty principle, it is impossible to simultaneously know the exact position and momentum of an electron.
- Confinement:
- – Electrons are confined within the potential well created by the positively charged nucleus, resulting in quantized energy states.
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V. Simple Picture of the Quark Structure of Protons and Neutrons
- ⇒ Quarks:
- Protons and neutrons are composed of elementary particles called quarks.
- Quarks interact via the strong nuclear force, mediated by particles called gluons.
- ⇒ Proton Structure:
- [math]\text{Proton (p)} = uud[/math]
- Composed of two up quarks (u) and one down quark (d).
- Total charge:
- [math]\left( +\frac{2}{3} \right) + \left( +\frac{2}{3} \right) + \left( -\frac{1}{3} \right) = +1[/math]
- Figure 5 Proton structure
- ⇒ Neutron Structure:
- [math]\text{Neutron (n)} = udd[/math]
- Composed of one up quark (u) and two down quarks (d).
- Total charge:
- [math]\left( +\frac{2}{3} \right) + \left( -\frac{1}{3} \right) + \left( -\frac{1}{3} \right) = 0[/math]
- Figure 6 Proton and neutron
- ⇒ Quark Properties:
- – Up Quark (u): Charge [math]+\frac{2}{3} e[/math].
- – Down Quark (d): Charge [math]-\frac{1}{3} e[/math].
- – Held together by gluons via the strong interaction.
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VI. Application of Conservation Laws in Nuclear Reactions
- ⇒ Conservation Laws:
- Conservation of Mass-Energy:
- – Total energy (including rest mass energy,[math]E = mc^2[/math]) remains constant in nuclear reactions.
- – Example: In nuclear fusion or fission, mass is converted into energy or vice versa.
- Conservation of Charge:
- – The total electric charge remains unchanged before and after a reaction.
- – Example: In beta decay:
- [math]n \rightarrow p + e^- + \bar{\nu}_e[/math]
- The charge is conserved:[math]0→ + 1 + (-1) + 0[/math]
- Conservation of Lepton Number:
- – Leptons (e.g., electrons, neutrinos) must balance on both sides of a reaction.
- – Example: In the same beta decay:
- [math]\text{Lepton number}∶0→0 + 1 – 1[/math]
- ⇒ Applications:
- Predicting reaction outcomes in nuclear physics, such as fusion in the Sun or radioactive decay.
- Ensuring charge, energy, and particle counts are balanced in reactions.
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VII. Relativistic Calculations for High-Speed Particles
- ⇒ Relativistic Effects:
- Time Dilation:
- – A particle moving at speeds close to the speed of light ([math]v ≈ c[/math]) experiences time dilation:
- [math]∆t’ = γ∆t[/math]
- Where [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math] is the Lorentz factor.
- Relativistic Momentum:
- – As velocity increases, the momentum is given by:
- [math]p = γmv [/math]
- Where γ increases significantly as v approaches c.
- Relativistic Energy:
- – The total energy of a particle is:
- [math]E = γmc^2[/math]
- – At rest (v=0), this reduces to the famous [math]E = mc^2[/math].
b) Make appropriate use of:
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I. The Terms
- ⇒ Energy Level:
- Definition:
- – A quantized state of energy that an electron can occupy in an atom. Transitions between levels correspond to the emission or absorption of photons.
- Example:
- – In hydrogen, the energy levels are given by:
- [math]E_n = -\frac{13.6}{n^2} \text{ eV}[/math]
- Use: Determines atomic spectra and chemical bonding properties.
- Scattering:
- Definition:
- – The deflection of particles (e.g., alpha particles or electrons) as they interact with other particles or nuclei.
- Example:
- – Rutherford’s gold foil experiment demonstrated scattering of alpha particles, revealing the presence of a small, dense nucleus.
- Nucleus:
- – Definition: The dense, positively charged core of an atom, composed of protons and neutrons.
- – Use: Contains almost all the atom’s mass and is studied through scattering experiments.
- Proton:
- – Definition: A positively charged particle found in the nucleus.
- – Composition: Made of two up quarks and one down quark (uud)
- – Charge: +1e.
- Neutron:
- – Definition: A neutral particle in the nucleus
- – Composition: Made of one up quark and two down quarks (udd).
- – Charge: 0.
- Nucleon:
- -Definition: A collective term for protons and neutrons, the components of the nucleus.
- Electron:
- – Definition: A negatively charged particle orbiting the nucleus.
- – Use: Responsible for chemical bonding and electric current.
- Positron:
- Definition: The antiparticle of the electron, with the same mass but a positive charge (+e).
- Quark:
- – Definition: A fundamental particle and constituent of protons and neutrons. Comes in six flavors: up, down, charm, strange, top, bottom.
- – Charge: Fractional (e.g., [math]+ \frac{2}{3} e[/math] for up quark, [math]- \frac{1}{3} e[/math] for down quark).
- Gluon:
- – Definition: The mediator particle of the strong nuclear force, binding quarks together.
- Neutrino:
- – Definition: A neutral, nearly massless particle involved in weak interactions, such as beta decay.
- – Types: Electron neutrino, muon neutrino, tau neutrino.
- Hadron:
- – Definition: A particle composed of quarks bound by the strong force. Includes protons and neutrons
- – Types: Baryons (3 quarks, e.g., protons, neutrons) and mesons (quark-antiquark pairs).
- Lepton:
- – Definition: A fundamental particle that does not experience the strong force (e.g., electrons, neutrinos).
- – Examples: Electron (e−), positron (e+), muon, tau.
- Antiparticle:
- – Definition: A particle with the same mass but opposite charge and quantum numbers to its corresponding particle (e.g., positron is the antiparticle of the electron)
- Lepton Number:
- – Definition: A quantum number conserved in particle interactions, where leptons have +1, and antileptons have −1.
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II. Sketching and Interpreting: Paths of Scattered Particles
- ⇒ Rutherford Scattering:
- Sketch: A beam of alpha particles directed at a thin gold foil, with most passing straight through, some deflected at small angles, and a few rebounding at large angles.
- ⇒ Interpretation:
- Most alpha particles pass through, showing that atoms are mostly empty space
- Large deflections indicate the presence of a small, dense, positively charged nucleus.
- Figure 7 Rutherford Scattering
- ⇒ Electron Scattering:
- Sketch: A beam of high-energy electrons scattered by nuclei, showing diffraction patterns.
- ⇒ Interpretation:
- The diffraction pattern provides information about the size and structure of the nucleus.
- Figure 8 Electron scattering
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III. Sketching and Interpreting: Electron Standing Waves in Atoms
- ⇒ Electron Standing Waves:
- Model: In the quantum mechanical model, electrons are confined to orbitals, where their wavefunctions form standing waves.
- Sketch:
- – For n=1 (ground state): A single wave peak around the nucleus.
- – For n=2: Two peaks, indicating a higher energy state.
- Interpretation:
- – The standing wave nature explains why only discrete energy levels are possible.
- – The wavelength of the electron is inversely proportional to its momentum ([math]λ = h/p[/math]).
- Figure 9 Electron Standing waves
- ⇒ Bohr Model Connection:
- Sketch: Circular orbits with integer multiples of de Broglie wavelengths fitting into the circumference.
- Interpretation:
- – Quantization arises because only orbits with whole-number multiples of wavelengths are allowed.
- Figure 10 Bohr Model
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Applications and Insights
- ⇒ Understanding Scattering:
- Scattering experiments are foundational in nuclear physics, revealing the structure of atoms and particles.
- Used in modern particle accelerators to study subatomic particles.
- ⇒ Electron Behavior:
- Standing wave models explain atomic spectra and the stability of electrons in atoms.
- Basis for quantum mechanics and modern atomic theory.
- ⇒ Conservation Laws:
- Fundamental in interpreting the results of scattering and particle interactions.
c) Make calculations and estimates involving:
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I) Motion of a Charged Particle in a Magnetic Field
- ⇒ Force Acting on a Charged Particle
- The force experienced by a charged particle moving in a magnetic field is given by:
- [math]F = qvBsinθ[/math]
- – q: Charge of the particle (Coulombs, C).
- – v: Velocity of the particle (m/s).
- – B: Magnetic field strength (Tesla, T).
- – θ: Angle between the velocity vector and the magnetic field.
- ⇒ Circular Motion
- If [math]θ = 90^0[/math], the motion of the particle is circular since the force is perpendicular to the velocity:
- [math]F = qvB[/math]
- This force provides the centripetal force:
- [math]\frac{mv^2}{r} = qvB[/math]
- Solving for the radius of the circular path:
- [math]r = \frac{mv}{qB}[/math]
- – r: Radius of the circular motion (m).
- – m: Mass of the particle (kg).
- ⇒ Cyclotron Frequency
- The angular velocity ω of the particle is:
- [math]\omega = \frac{v}{r} \\
\omega = \frac{qB}{m}[/math] - The frequency of revolution (cyclotron frequency) is:
- [math]f = \frac{\omega}{2\pi} \\
f = \frac{qB}{2\pi m}[/math] - ⇒ Example Calculation
- A proton ([math]1.6 × 10^{-19} C[/math]) moving with [math]v = 2 × 10^6 m/s[/math] in a magnetic field of B=0.5T:
- [math]r = \frac{mv}{qB} \\
r = \frac{\left( 1.67 \times 10^{-27} \right) \times \left( 2 \times 10^6 \right)}{\left( 1.6 \times 10^{-19} \right) \times 0.5} \\
r \approx 0.042 \text{ m}[/math] -
II) Kinetic and Potential Energy of a Scattered Charged Particle
- ⇒ Kinetic Energy
- The kinetic energy of a charged particle is given by:
- [math]KE = \frac{1}{2} mv^2[/math]
- If the particle gains energy in an electric field, the kinetic energy change can be calculated using the work-energy principle:
- [math]ΔKE = qΔV[/math]
- – ΔV: Potential difference (Volts, V).
- ⇒ Potential Energy
- The potential energy of a charged particle in an electric field is:
- [math]PE = qV[/math]
- – V: Electric potential (Volts, V).
- ⇒ Example Calculation
- An electron ([math]q = -1.6 × 10^{-19} C[/math]) accelerated through: [math]V = 10^3 V[/math]
- [math]KE = qV \\
KE = \left( 1.6 \times 10^{-19} \right) \times \left( 10^3 \right) \\
KE = 1.6 \times 10^{-16} \text{ J}
[/math] - Converting to electron-volts:
- [math]KE = 1.6keV[/math]
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III) Rest Energy and Relativistic Energy
- ⇒ Rest Energy
- The rest energy of a particle is:
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[math]E_{rest} = mc^2[/math]
- – m: Rest mass of the particle (kg).
- – c: Speed of light ([math]3 × 10^8 m/s[/math]).
- ⇒ Relativistic Factor (γ)
- For a particle moving at a significant fraction of the speed of light:
- [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
- – v: Velocity of the particle (m/s).
- ⇒ Total Energy
- The total relativistic energy is:
- [math]E_{total} = γE_{rest}[/math]
- – The relativistic kinetic energy:
- [math]KE_{\text{rel}} = E_{\text{total}} – E_{\text{rest}} \\
KE_{\text{rel}} = (\gamma – 1) mc^2[/math]