Physics with Astrophysics - MPhys

Introduction to Special Relativity:

Inadequacy of Galilean Transformation; Postulates of Relativity; Lorentz transformation; Time dilation, length contraction and simultaneity; Special relativity paradoxes; Invariant intervals; Momentum and energy in special relativity; Equivalence of mass and energy.

Introduction to Astronomy:

Astronomical coordinate systems and conversions; Positions and motions of stars; Timekeeping systems; Introduction to the distance scale.

Introduction to Astrophysics and Cosmology:

Stellar luminosity and magnitudes; Magnitude systems; Colour of stars; Stellar spectral classification; Evolution of stars, Hertzsprung-Russell diagram; Cosmological principle; Redshift; Hubble constant; Space expansion.

Derivatives and Integrals: Derivatives of elementary functions, chain rule, product rule, Integrals of elementary functions, Evaluation by substitution, Integration by parts, Area under the graph of a function.

Vectors: Basic properties, linear dependence, scalar and vector products, triple products, vector identities.

Matrices: Matrix representation, systems of equations, products, inverses, determinants, solution of linear systems, eigenvalues and eigenvectors, transformations.

Elementary Functions: Binomial coefficients, expansions and series, Maclaurin series, Taylor series, Exponential functions, Hyperbolic functions, Inverse functions.

Functions of a single variable: Linear and quadratic functions, polynomials, rational functions, limits, infinite series, approximation of functions.

Complex numbers: Quadratic equations, Argand diagram, modulus, Argument, complex exponential, de Moivre's theorem, roots of polynomials.

Differential Equations: Solving differential equations, separable equations, linearity, homogeneity, first and second order equations, particular integrals. Boundary and initial values, auxiliary equations with complex roots, coefficients and terms, examples from physics.

Partial Derivatives: functions of two variables , partial derivatives, directional derivatives, functions many variables, higher derivatives, function of a function, implicit differentiation, differentiation of an integral w.r.t a parameter, Taylor expansions, stationary points.

Elementary multivariate Calculus: the chain rule, Multiple integrals, integrals over rectangles/irregular areas in the plane, change of order of integration.

Polar Coordinates: Cylindrical polar coordinates in two and three dimensions, integrals, spherical coordinates, solid angle.

Introduction to Vector Calculus : Gradients, Divergence, Gauss's theorem, Curl, Stokes' theorem.

Measurement and motion; Dimensional analysis, Motion in one dimension: velocity, acceleration, motion with constant acceleration, Motion in a plane with constant acceleration, projectile motion, uniform circular motion, and Newton's laws of motion.

Work, Energy and Momentum; Work, kinetic energy, power, potential energy, relation between force and potential energy, conservation of energy, application to gravitation and simple pendulum, momentum, conservation of linear momentum, elastic and inelastic collisions.

Rotational Motion; Rotational motion: angular velocity, angular acceleration, rotation with constant angular acceleration, rotational kinetic energy, moment of inertia, calculation of moment of inertia of a rod, disc or plate, torque, angular momentum, relation between torque and angular momentum, conservation of angular momentum.

Concept of field; 1/r2 fields; Gravitational Field; Kepler's Laws, Newton's law of gravitation, Gravitational potential, the gravitational field of a spherical shell by integration.

Oscillations and Mechanical Waves; Vibrations of an elastic spring, simple harmonic motion, energy in SHM, simple pendulum, physical pendulum, damped and driven oscillations, resonance, mechanical waves, periodic waves, their mathematical representation using wave vectors and wave functions, derivation of a wave equation, transverse and longitudinal waves, elastic waves on a string, principle of superposition, interference and formation of standing waves, normal modes and harmonics, sound waves with examples of interference to form beats, and the Doppler Effect. Phase velocity and group velocity.

Properties of Light and Optical Images; Wave nature of light. Reflection, refraction, Snell’s law, total internal reflection, refractive index and dispersion, polarisation. Huygens' principle, geometrical optics including reflection at plane and spherical surfaces, refraction at thin lenses, image formation, ray diagrams, calculation of linear and angular magnification, magnifying glass, telescopes and the microscope.

Electric Field; Discrete charge distributions, charge, conductors, insulators, Coulomb’s law, electric field, electric fields lines, action of electric field on charges, electric field due to a continuous charge distribution, electric potential, computing the electric field from the potential, calculation of potential for continuous charge distribution.

Magnetic Field; Force on a point charge in a magnetic field, motion of a point charge in a magnetic field, mass spectrometer and cyclotron.

Electric current and Direct current circuits, electric current, resistivity, resistance and Ohm’s Law, electromotive force, ideal voltage and current sources, energy and power in electric circuits, theory of metallic conduction, resistors in series and in parallel, Kirchhoff’s rules and their application to mesh analysis, electrical measuring instruments for potential difference and current, potential divider and Wheatstone’s bridge circuits, power transfer theorem, transient current analysis in RC, RL, LC and LRC circuits using differential equations.

Alternating Current Circuits; Phasor and complex number notation introduced for alternating current circuit analysis, reactance and complex impedance for Capacitance and Inductance, application to LRC series and parallel circuits. Series and parallel resonance, AC potential dividers and filter circuits, Thevenin's theorem, AC bridge circuits to measure inductance and capacitance, mutual inductance, the transformer and its simple applications.

Static Equilibrium, Elasticity and fluids; Elasticity: stress, strain, Hooke's law, Young's modulus, shear modulus, forces between atoms or molecules, intermolecular potential energy curve, equilibrium separation, Morse and 6-12 potentials, microscopic interpretation of elasticity, relation between Young's modulus and parameters of the interatomic potential energy curve, the nature of interatomic forces, the ionic bond, calculation of the energy to separate the ions in an ionic crystal, viscosity of fluids, Poiseuille's law, Stokes' law.

Thermodynamics; Thermal equilibrium, temperature scales, thermal expansion of solids, relation between thermal expansion and the interatomic potential energy curve, the transfer of thermal energy: conduction, convection, radiation, the ideal-gas law, Boltzmann's constant, Avogadro's number, the universal gas constant. The kinetic theory of gases, pressure of a gas, molecular interpretation of temperature, molecular speeds, mean free path, specific heat, molar specific heat. The equipartition theorem, degrees of freedom. Heat capacities of monatomic and diatomic gases and of solids. Internal energy of a thermodynamic system, the first law of thermodynamics, work and the PV diagram of a gas., work done in an isothermal expansion of an ideal gas. Molar heat capacities of gases at constant pressure and at constant volume and the relation between them. Adiabatic processes for an ideal gas. Heat engines and the Kelvin statement of the second law of thermodynamics, efficiency of a heat engine. Refrigerators and the Clausius statement of the second law of thermodynamics. Equivalence of the Kelvin and Clausius statements. The Carnot cycle, the Kelvin temperature scale.

Atoms; The nuclear atom, Rutherford scattering and the nucleus, Bohr model of the atom, energy level calculation and atom spectra, spectral series for H atom. Limitation of Bohr theory. Photoelectric Effect. Blackbody Radiation. Compton scattering. X-ray diffraction. De Broglie hypothesis. Electron diffraction. Introduction to wavefunctions, Heisenberg's Uncertainty Principle.

Standard Lectures:

How Physical Sciences are taught at Kent.

Library use. Bibliographic database searches.

Error analysis and data presentation. Types of errors; combining errors; Normal distribution; Poisson distribution; graphs – linear and logarithmic.

Probability and Statistics. Probability distributions, laws of probability, permutations and combinations, mean and variance.

Academic integrity and report writing skills.

Laboratory experiments:

A number of experiments in weekly sessions; some of the experiments require two consecutive weeks to complete.

Experiments introduce students to test equipment, data processing and interpretation and cover subjects found in the Physics degree program which include the following topics:

Mechanics, Astronomy/Astrophysics, statistical and probability analysis, numerical simulations, electric circuits and Thermodynamics.

Computing Skills:

Introduction to the concept of programming/scripting languages. Introduction to operating systems: including text editors, the directory system, basic utilities and the edit-compile-run cycle.

Introduction to the use of variables, constants, arrays and different data types; iteration and conditional branching.

Modular design: Use of programming subroutines and functions. Simple input/output, such as the use of format statements for reading and writing, File handling, including practical read/write of data files.

Producing graphical representation of data, including histograms. Interpolating data and fitting functions.

Programming to solve physical problems.

Introduction to typesetting formal scientific documents.

SYLLABUS

Most practicing physicists at some point will be required to perform experiments and take measurements. This module, through a series of experiments, seeks to allow students to become familiar with some more complex apparatus and give them the opportunity to learn the art of accurate recording and analysis of data. This data has to be put in the context of the theoretical background and an estimate of the accuracy made. Keeping of an accurate, intelligible laboratory notebook is most important. Each term 3 three week experiments are performed. The additional period is allocated to some further activities to develop experimental and communications skills.

Revision of classical descriptions of matter as particles, and electromagnetic radiation as waves.

Some key experiments in the history of quantum mechanics. The concept of wave-particle duality.

The wavefunction. Probability density. The Schrodinger equation. Stationary states.

Solutions of the Schrodinger equation for simple physical systems with constant potentials: Free particles. Particles in a box. Classically allowed and forbidden regions.

Reflection and transmission of particles incident onto a potential barrier. Probability flux. Tunnelling of particles.

The simple harmonic oscillator as a model for atomic vibrations.

Revision of classical descriptions of rotation. Rotation in three dimensions as a model for molecular rotation.

The Coulomb potential as a model for the hydrogen atom. The quantum numbers l, m and n. The wavefunctions of the hydrogen atom.

Physical observables represented by operators. Eigenfunctions and eigenvalues. Expectation values. Time independent perturbation theory.

Atomic Physics

Review of previous stages in the development of quantum theory with application to atomic physics; Atomic processes and the excitation of atoms; Electric dipole selection rules; atom in magnetic field; normal Zeeman effect; Stern Gerlach experiment; Spin hypothesis; Addition of orbital and spin angular moments; Lande factor; Anomalous Zeeman effect; Complex atoms; Periodic table; General Pauli principle and electron antisymmetry; Alkali atoms; ls and jj coupling; X-rays. Lamb-shift and hyperfinestructure (if time).

Nuclear Physics

Properties of nuclei: Rutherford scattering. Size, mass and binding energy, stability, spin and parity.

Nuclear Forces: properties of the deuteron, magnetic dipole moment, spin-dependent forces.

Nuclear Models: Semi-empirical mass formula M(A, Z), stability, binding energy B(A, Z)/A. Shell model, magic numbers, spin-orbit interaction, shell closure effects.

Alpha and Beta decay: Energetics and stability, the positron, neutrino and anti-neutrino.

Nuclear Reactions: Q-value. Fission and fusion reactions, chain reactions and nuclear reactors, nuclear weapons, solar energy and the helium cycle.

SYLLABUS

Electromagnetism

Vectors: Review of Grad, Div & Curl; and other operations

Electrostatics: Coulomb's Law, electric field and potential, Gauss's Law in integral and differential form; the electric dipole, forces and torques.

Isotropic dielectrics: Polarization; Gauss's Law in dielectrics; electric displacement and susceptibility; capacitors; energy of systems of charges; energy density of an electrostatic field; stresses; boundary conditions on field vectors.

Poisson and Laplace equations.

Electrostatic images: Point charge and plane; point and sphere, line charges.

Magnetic field: Field of current element or moving charge; Div B; magnetic dipole moment, forces and torques; Ampere's circuital law.

Magnetization: Susceptibility and permeability; boundary conditions on field vectors; fields of simple circuits.

Electromagnetic induction: Lenz’s law, inductance, magnetic energy and energy density;

Optics

Field equations: Maxwell's equations; the E.M. wave equation in free space.

Irradiance: E.M. waves in complex notation.

Polarisation: mathematical description of linear, circular and elliptical states; unpolarised and partially polarised light; production of polarised light; the Jones vector.

Interference: Classes of interferometers – wavefront splitting, amplitude splitting. Basic concepts including coherence.

Diffraction: Introduction to scalar diffraction theory: diffraction at a single slit, diffraction grating.

Aims: To provide a basic but rigorous grounding in observational, computational and theoretical aspects of astrophysics to build on the descriptive course in Part I, and to consider evidence for the existence of exoplanets in other Solar Systems.

SYLLABUS:

Observing the Universe

Telescopes and detectors, and their use to make observations across the electromagnetic spectrum. Basic Definitions: Magnitudes, solid angle, intensity, flux density, absolute magnitude, parsec, distance modulus, bolometric magnitude, spectroscopic parallax, Hertzsprung-Russel diagram, Stellar Photometry: Factors affecting signal from a star. Detectors: Examples, Responsive Quantum Efficiency, CCD cameras. Filters, UBV system, Colour Index as temperature diagnostic.

Extra Solar Planets

The evidence for extrasolar planets will be presented and reviewed. The implications for the development and evolution of Solar Systems will be discussed.

Astrophysics

Basic stellar properties, stellar spectra. Formation and Evolution of stars. Stellar structure: description of stellar structure and evolution models, including star and planet formation. Stellar motions: Space velocity, proper motion, radial velocity, Local Standard of Rest, parallax. Degenerate matter: concept of degenerate pressure, properties of white dwarfs, Chandrasekhar limit, neutron stars, pulsars, Synchrotron radiation, Schwarzschild radius, black holes, stellar remnants in binary systems.

The aim of the module in Medical Physics is to provide a primer into this important physics specialisation. The range of subjects covered is intended to give a balanced introduction to Medical Physics, with emphasis on the core principles of medical imaging, radiation therapy and radiation safety. A small number of lectures is also allocated to the growing field of optical techniques. The module involves several contributions from the Department of Medical Physics at the Kent and Canterbury Hospital.

SYLLABUS:

Radiation protection (radiology, generic); Radiation hazards and dosimetry, radiation protection science and standards, doses and risks in radiology; Radiology; (Fundamental radiological science, general radiology, fluoroscopy and special procedures); Mammography (Imaging techniques and applications to health screening); Computed Tomography (Principles, system design and physical assessment); Diagnostic ultrasound (Pulse echo principles, ultrasound imaging, Doppler techniques); Tissue optics (Absorption, scattering of light in the tissue); The eye (The eye as an optical instrument); Confocal Microscopy (Principles and resolutions); Optical Coherence Tomography (OCT) and applications; Nuclear Medicine (Radionuclide production, radiochemistry, imaging techniques, radiation detectors); In vitro techniques (Radiation counting techniques and applications); Positron Emission Tomography (Principles, imaging and clinical applications); Radiation therapies (Fundamentals of beam therapy, brachytherapy, and 131I thyroid therapy); Radiation Protection (unsealed sources); Dose from in-vivo radionuclides, contamination, safety considerations.

Most physically interesting problems are governed by ordinary, or partial differential equations. It is examples of such equations that provide the motivation for the material covered in this module, and there is a strong emphasis on physical applications throughout. The aim of the module is to provide a firm grounding in mathematical methods: both for solving differential equations and, through the study of special functions and asymptotic analysis, to determine the properties of solutions. The following topics will be covered: Ordinary differential equations: method of Frobenius, general linear second order differential equation. Special functions: Bessel, Legendre, Hermite, Laguerre and Chebyshev functions, orthogonal functions, gamma function, applications of special functions. Partial differential equations; linear second order partial differential equations; Laplace equation, diffusion equation, wave equation, Schrödinger’s equation; Method of separation of variables. Fourier series: application to the solution of partial differential equations. Fourier Transforms: Basic properties and Parseval’s theorem.

Aims: After taking the classes students should be more fluent and adept at solving and discussing general problems in Physics (and its related disciplines of mathematics and engineering)

There is no formal curriculum for this course which uses and demands only physical and mathematical concepts with which the students at this level are already familiar. Instruction is given in:

Special Relativity: Limits of Newtonian Mechanics, Inertial frames of reference, the Galilean and Lorentz transformations, time dilation and length contraction, invariant quantities under Lorentz transformation, energy momentum 4-vector

Maxwell's equations: operators of vector calculus, Gauss law of electrostatics and magnetostatics, Faraday's law and Ampere's law, physical meanings and integral and differential forms, dielectrics, the wave equation and solutions, Poynting vector, the Fresnel relations, transmission and reflection at dielectric boundaries.

Modern Optics: Resonant cavities and the laser, optical modes, Polarisation and Jones vector formulation.

1. Thermodynamics

Review of zeroth, first, second laws. Quasistatic processes. Functions of state. Extensive and intensive properties. Exact and inexact differentials. Concept of entropy. Heat capacities. Thermodynamic potentials: internal energy, enthalpy, Helmholtz and Gibbs functions. The Maxwell relations. Concept of chemical potential. Applications to simple systems. Joule free expansion. Joule-Kelvin effect. Equilibrium conditions. Phase equilibria, Clausius-Clapeyron equation. The third law of thermodynamics and its consequences – inaccessibility of the absolute zero.

2. Statistical Concepts and Statistical Basis of Thermodynamics

Basic statistical concepts. Microscopic and macroscopic descriptions of thermodynamic systems. Statistical basis of Thermodynamics. Boltzmann entropy formula. Temperature and pressure. Statistical properties of molecules in a gas. Basic concepts of probability and probability distributions. Counting the number of ways to place objects in boxes. Distinguishable and indistinguishable objects. Stirling approximation(s). Schottkly defect, Spin 1/2 systems. System of harmonic oscillators. Gibbsian Ensembles. Canonical Ensemble. Gibbs entropy formula. Boltzmann distribution. Partition function. Semi-classical approach. Partition function of a single particle. Partition function of N non-interacting particles. Helmholtz free energy. Pauli paramagnetism. Semi Classical Perfect Gas. Equation of state. Entropy of a monatomic gas, Sackur-Tetrode equation. Density of states. Maxwell velocity distribution. Equipartition of Energy. Heat capacities. Grand Canonical Ensemble.

3. Quantum Statistics

Classical and Quantum Counting of Microstates. Average occupation numbers: Fermi Dirac and Bose Einstein statistics. The Classical Limit. Black Body radiation and perfect photon gas. Planck’s law. Einstein theory of solids. Debye theory of solids.

To provide an introduction to solid state physics. To provide foundations for the further study of materials and condensed matter, and details of solid state electronic and opto-electronic devices.

Structure

Aims: To provide, in combination with PH507, a balanced and rigorous course in Astrophysics for B.Sc. Physics with Astrophysics students, while forming a basis of the more extensive M.Phys. modules.

SYLLABUS

Physics of Stars

Introduction to Matlab

• Image representation,

• Image formation,

• Grey-scale transformation,

• Enhancement and extraction of image content,

• Fourier transforms and the frequency domain,

• Image restoration, geometrical transformations,

• Morphology and morphological transformations,

• Feature extraction,

• Segmentation.

Minimisation problems and the Euler-Lagrange equation;

Lagrange formulation of classical mechanics;

Link between symmetries and conservation laws (Noether's theorem);

Hamilton formulation of classical mechanics;

Semi-classical mechanics and the link to quantum mechanics;

Continuum mechanics and fluid dynamics;

Dynamical systems and chaos

Aims:

Students will develop a number of skills related to the Investigation and planning of research. Students will learn how to search and retrieve information from a variety of locations (databases, websites, journals, proceedings etc). They will learn how to compile professionally-produced documents such as the report of their own investigation in a direction of their choice. In addition, students will subsequently provide an outline proposal for funding for future research activity.

Through two Colloquium Reports, students will learn to write high-impact articles with a critical analysis of research presented by others. They will exercise presentation skills and present critical reviews and referee's reports of the research of others.

SYLLABUS:

The Research Project (60%)

Identification of a research area and the issues to tackle

Investigation of an unresolved issue comparing experiments and models, comparing approaches, assumptions and statistical methods.

Production of a dissertation

Proposal for future novel work as a short Case for Support for a PhD or research outside university environment

Project Management: Scheduling research programmes, Gantt, PERT charts.

Project Management: Costing of research, full economic cost, direct and indirect costs.

Poster presentation of the research

Research Review and Evaluation (40%)

Evaluation of Research: Colloquium attendance/viewing.

Science Communication: Preparation of two colloquium reports as a science magazine article with impact

Referee report on the colloquiums: strengths, weaknesses of both the speaker and the research quality.

Details of the work to be done will be announced by the convenor during the first two weeks of the academic year.

SYLLABUS:

Interstellar Medium

The major properties of the Interstellar Medium (ISM) are described. The course will discuss the characteristics of the gaseous and dust components of the ISM, including their distributions throughout the Galaxy, physical and chemical properties, and their influence the star formation process. The excitation of this interstellar material will be examined for the various physical processes which occur in the ISM, including radiative, collisional and shock excitation. The way in which the interstellar material can collapse under the effects of self-gravity to form stars, and their subsequent interaction with the remaining material will be examined. Finally the end stages of stellar evolution will be studied to understand how planetary nebulae and supernova remnants interact with the surrounding ISM.

Extragalactic astrophysics

Review of FRW metric; source counts; cosmological distance ladder; standard candles/rods.

High-z galaxies: fundamental plane; Tully-Fisher; low surface brightness galaxies; luminosity functions and high-z evolution; the Cosmic Star Formation History

Galaxy clusters: the Butcher-Oemler effect; the morphology-density relation; the SZ effect

AGN and black holes: Beaming and superluminal motion; Unified schemes; Black hole demographics; high-z galaxy and quasar absorption and emission lines;

Aims:

SYLLABUS:

Space Astronomy

Why use space telescopes; other platforms for non-ground-based astronomical observatories (sounding rockets, balloons, satellites); mission case study; what wavelengths benefit by being in space; measurements astronomers make in space using uv, x-ray and infra-red, and examples of some recent scientific missions.

Exploration of the Solar System

Mission types from flybys to sample returns: scientific aims and instrumentation: design requirements for a spacecraft-exploration mission; how to study planetary atmospheres and surfaces: properties of and how to explore minor bodies (e.g. asteroids and comets): current and future missions: mission case study; how space agencies liaise with the scientific community; how to perform calculations related to the orbital transfer of spacecraft.

Solar System Formation and Evolution

The composition of the Sun and planets will be placed in the context of the current understanding of the evolution of the Solar System. Topics include: Solar system formation and evolution; structure of the solar system; physical and orbital evolution of asteroids.

Extra Solar Planets

The evidence for extra Solar planets will be presented and reviewed. The implications for the development and evolution of Solar Systems will be discussed.

Life in Space

Introduction to the issue of what life is, where it may exist in the Solar System and how to look for it.

Flight Operations: Control of spacecraft from the ground, including aspects of telecommunications theory.

Propulsion and attitude control: Physics of combustion in rockets, review of classical mechanics of rotation and its application to spacecraft attitude determination and control.

Impact Damage: The mechanisms by which space vehicles are damaged by high speed impact will be discussed along with protection strategies.

Human spaceflight: A review of human spaceflight programs (past and present). Life-support systems. An introduction to some major topics in space medicine; acceleration, pressurisation, radiation, etc.

International Space Station: Status of this project/mission will be covered.

• Approximation Methods, perturbation theory, variational methods.

• Classical/Quantum Mechanics, measurement and the correspondence principle.

• Uncertainty Principle and Spin precession .

• Key Experiments in Modern Quantum Mechanics (Aharonov-Bohm, neutron diffractyion in a gravitational field, EPR paradox).

• Experimental methods in Particle Physics (Accelerators, targets and colliders, particle interactions with matter, detectors, the LHC).

• Feynman Diagrams, particle exchange, leptons, hadrons and quarks.

• Symmetries and Conservation Laws.

• Hadron flavours, isospin, strangeness and the quark model.

• Weak Interactions, W and Z bosons.