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The Larmor Formula (Chapters 18-19) T. Johnson 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 1 Outline • • • Brief repetition of emission formula The emission from a single free particle - the Larmor formula Applications of the Larmor formula – – – – Harmonic oscillator Cyclotron radiation Thompson scattering Bremstrahlung Next lecture: • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation and Lorentz transformations – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • 2017-02-28 Abraham-Lorentz force Dispersive Media, Lecture 12 - Thomas Johnson 2 Repetition: Emission formula • • The energy emitted by a wave mode M (using antihermitian part of the propagator), when integrating over the δ-function in ω – the emission formula for UM ; the density of emission in k-space Emission per frequency and solid angle – Rewrite integral: 𝑑 "𝑘 = 𝑘 % 𝑑𝑘𝑑 %Ω = 𝑘 % '() (+) %Ω 𝑑𝜔𝑑 '+ / is the unit vector Here 𝐤 in the 𝐤-direction 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 3 Repetition: Emission from multipole moments • Multipole moments are related to the Fourier transform of the current: Emission formula (k-space power density) 2017-02-28 Emission formula (integrated over solid angles) Dispersive Media, Lecture 12 - Thomas Johnson 4 Current from a single particle • Let’s calculate the radiation from a single particle – at X(t) with charge q. – The density, n, and current, J, from the particle: – or in Fourier space 5 𝐉 𝜔, 𝑘 = 𝑞 3 𝑑𝑡 𝑒68+9 3 𝑑 " 𝑘 𝑒8𝐤 :𝐱 𝐗̇ 𝑡 𝛿 𝐱 − 𝐗 𝑡 5 = 65 = 𝑞 3 𝑑𝑡 𝑒68+9 1 + 𝑖 𝐤 : 𝐗(𝑡) + ⋯ 𝐗̇ 𝑡 = 65 5 = −𝑖𝜔𝑞𝐗 𝜔 + 3 𝑑𝑡 𝑒 68+9 𝑖 𝐤 : 𝐗(𝑡) 𝐗̇ 𝑡 + ⋯ Dipole: d=qX 2017-02-28 65 Dispersive Media, Lecture 12 - Thomas Johnson 5 Dipole current from single particle • Thus, the field from a single particle is approximately a dipole field • When is this approximation valid? – Assume oscillating motion: - The dipole approximation is based on the small term: Dipole approx. valid for non-relativistic motion 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 6 Emission from a single particle • Emission from single particle; use dipole formulas from last lecture: ∗ : 𝐗(𝜔) % 𝑞% 𝐞 F M 𝑉F 𝜔, Ω = 𝑛 𝜔 2𝜋𝑐 " 𝜀K F 1 − 𝐞∗F : 𝛋 % – for the special case of purely transverse waves 𝑞% M 𝛋×𝐗(𝜔) 𝑉F 𝜔, Ω = 𝑛 𝜔 F 2𝜋𝑐 " 𝜀K % – Note: this is emission per unit frequency and unit solid angle • Integrate over solid angle for transverse waves • Note: there’s no preferred direction, thus 2-tensor is proportional to Kroneker delta ~δjm; but kjkj=k2, thus 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 7 Emission from single particle • Thus, the energy per unit frequency emitted to transverse waves from single non-relativistic particle 4𝜋 𝑉F 𝜔 𝑞% = % " 𝑛F 𝜔% 𝐗(𝜔) 6𝜋 𝑐 𝜀K % • An alternative is in terms of the acceleration 𝐚 and the net force 𝐅 2017-02-28 4𝜋 𝑉F 𝜔 𝑞% = % " 𝑛F 𝐚(𝜔) 6𝜋 𝑐 𝜀K 4𝜋 𝑉F 𝜔 𝑞% 𝐅(𝜔) = % " 𝑛F 6𝜋 𝑐 𝜀K 𝑚 Dispersive Media, Lecture 12 - Thomas Johnson 8 % % Larmor formula for the emission from single particle • Total energy W radiated in vacuum (nM=1) 5 % 𝑊 = 4𝜋 3 𝑑𝜔 𝑉F 𝜔 K 5 𝑞 = % " 3 𝑑𝜔 𝐚(𝜔) 6𝜋 𝑐 𝜀K % K • Rewrite by noting that 𝑎(𝜔) is even and then use the power theorem – Thus, the energy radiated over all time is a time integral • The average radiated power, Pave , will be given by the average acceleration aave The Larmor formula 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 9 Larmor formula for the emission from single particle • Strictly, the Larmor formula gives the time averaged radiated power • In many cases the Larmor formula describes roughly “the power radiated during an event” – Larmor formula then gives the radiated power averaged over the event • Therefore, the conventional way to write the Larmor formula goes one step further and describe the instantaneous emission – radiation is only emitted when particles are accelerated! 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 10 Outline • • • Brief repetition of emission formula The emission from a single free particle - the Larmor formula Applications of the Larmor formula – – – – 2017-02-28 Harmonic oscillator Cyclotron radiation Thompson scattering Bremstrahlung Dispersive Media, Lecture 12 - Thomas Johnson 11 Applications: Harmonic oscillator • As a first example, consider the emission from a particle performing an harmonic oscillation – harmonic oscillations – Larmor formula: the emitted power associated with this acceleration – oscillation cos2(ω0t) should be averaged over a period 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 12 Applications: Harmonic oscillator – frequency spectum • Express the particle as a dipole d, use truncation for Fourier transform • The time-averaged power emitted from a dipole 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 13 Applications: cyclotron emission • An important emission process from magnetised particles is from the acceleration involved in cyclotron motion – consider a charged particle moving in a static magnetic field B=Bzez – where is the cyclotron frequency 𝑧 – where we have the Larmor radius 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 14 Applications: cyclotron emission • Cyclotron emission from a single particle – where is the velocity perpendicular to B. • Sum the emission over a Maxwellian distribution function, fM(v) – where 𝑛 is the particle density and 𝑇 is the temperature in Joules. – Magnetized plasma; power depends on the density and temperature: • Electron cyclotron emission is one of the most common ways to measure the temperature of a fusion plasma! 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 15 Applications: wave scattering • Consider a particle being accelerated by an external wave field • The Larmor formula then tell us the average emitted power – Note: that this is only valid in vacuum (restriction of Larmor formula) • Rewrite in term of the wave energy density W0 – in vacuum : 𝑑𝑊K 8𝜋 𝑞% 𝑃\ ≡ − = 𝑐𝑊K % 𝑑𝑡 3 4𝜋𝜀K 𝑚𝑐 – Interpretation: this is the fraction of the power density that is scattered by the particle, i.e. first absorbed and then re-emitted 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 16 Applications: wave scattering • The scattering process can be interpreted as a “collision” – – – – Consider a density of wave quanta representing the energy density W0 The wave quanta, or photons, move with velocity c (speed of light) Imagine a charged particle as a ball with a cross section σT The power of from photons bouncing off the charged particle, i.e. scattered, per unit time is given by – thus the effective cross section for wave scattering is hω Cross section area σT of the particle r0 Density of incoming photons 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson Photons hitting this area are scattered 17 Applications: Thomson scattering • Scattering of waves against electrons is called Thomson scattering – from this process the classical radius of the electron was defined as – Note: this is an effective radius for Thomson scattering and not a measure of the “real” size of the electron • Examples of Thomson scattering: – In fusion devices, Thomson scattering of a high-intensity laser beam is used for measuring the electron temperatures and densities. – The cosmic microwave background is thought to be linearly polarized as a result of Thomson scattering – The continous spectrum from the solar corona is the result of the Thomson scattering of solar radiation with free electrons 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 18 Thompson scattering system at the fusion experiment JET Thompson scattering systems at JET primarily measures temperatures 2017-02-28 Laser beam Laser source in a different room Dispersive Media, Lecture 12 - Thomas Johnson 19 Thompson scattering system at the fusion experiment JET Detectors Scattered light scattering 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 20 Applications: Bremsstrahlung • • Bremsstrahlung (~Braking radiation) come from the acceleration associated with electrostatic collisions between charged particles (called Coulomb collisions) Note that the electrostatic force is long range E~1/r2 – thus electrostatic collisions between charged particles is a smooth continuous processes – unlike collision between balls on a pool table • Consider an electron moving near an ion with charge Ze – since the ion is heavier than the electron, we assume Xion(t)=0 – the equation of motion for the electron and the emitted power are – this is the Bremsstrahlung radiation at one time of one single collision • to estimate the total power from a medium we need to integrate over both the entire collision and all ongoing collisions! 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 21 Bremsstrahlung: Coulomb collisions • Lets try and integrate the emission over all times – where we integrate in the distance to the ion r – Now we need rmin and b • So, let the ion be stationary at the origin • Let the electron start at (x,y,z)=(∞,b,0) with velocity v=(-v0,0,0) • The conservation of angular momentum and energy gives – This is the Kepler problem for the motion of the planets! – Next we need the minimum distance between ion and electron rmin 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 22 Bremsstrahlung: Coulomb collisions • Coulomb collisions are mainly due to “long range” interactions, – i.e. particles are far apart, and only slightly change their trajectories (there are exceptions in high density plasmas) – thus and – we are then ready to evaluate the time integrated emission • This is the emission from a single collision – The cumulative emission from all particles and with all possible b and v0 has no simple general solution (and is outside the scope of this course) – An approximate: – Bremsstrahlung can be used to derive information about both the charge, density and temperature of the media 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 23 X-ray tubes • Typical frequency of Bremsstrahlung is in X-ray regime • Bremsstrahlung is the main source of radiation in X-ray tubes – electrons are accelerated to high velocity When impacting on a metal surface they emit bremsstrahlung Line radiation Counts per second • X-ray tubes may also emit line radiation. Wavelength, (pm) 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 24 Applications for X-ray and bremsstrahlung • X-rays have been used in medicine since Wilhelm Röntgen’s discovery of the X-ray in 1895 – Radiographs produce images of e.g. bones – Radiotherapy is used to treat cancer • for skin cancer, use low energy X-ray, not to penetrate too deep • for breast or cancer, use higher energies for deeper penetration • Crystallography: used to identify the crystal/atomic patterns of a material – study diffraction of X-rays • • X-ray flourescence: scattered X-ray carry information about chemical composition. Industrial CT scanner e.g. airport and cargo scanners 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 25 Applications of Bremsstrahlung • Astrophysics: High temerature stellar objects T ~ 107-108 K radiate primarily in via bremsstrahlung – Note: surface of the sun 103 – 106 K • Fusion: – Measurements of Bremsstrahlung provide information on the prescence of impurities with high charge, temperature and density – Energy losses by Bremsstrahlung and cyclotron radiation: • • • • Temperature at the centre of fusion plasma: ~108K ; the walls are ~103K Main challenge for fusion is to confine heat in plasma core Bremsstrahlung and cyclotron radiation leave plasma at speed of light! In reactor, radiation losses will be of importance – limits the reactor design – If plasma gets too hot, then radiation losses cool down the plasma. – Inirtial fusion: lasers shines on a tube that emits bremstrahlung, which then heats the D-T pellet 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 26 Summary • When charged particles accelerated they emits radiation • This emission is described by the Larmor formula 1 %𝑎% 𝑃= 𝑞 6𝜋𝜀K 𝑐 " • Important applications: – Cyclotron emission – magnetised plasmas – Thompson scattering – photons bounce off electrons – Bremstrahlung – main source of X-ray radiation • All these are used extensively for studying e.g. fusion plasmas 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 27