The Larmor Formula (Chapters 18-19)

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 1 The Larmor Formula (Chapters 18-19) T. Johnson

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 Abraham-Lorentz force 2017-02-28 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 U M ; the density of emission in k-space Emission per frequency and solid angle Rewrite integral: d " k = k % dkd % Ω = k % '( )(+) '+ dωd% Ω Here k/ is the unit vector 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) Emission formula (integrated over solid angles) 2017-02-28 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 e 68+9 3 d " k e 8k:x X t δ x X t = 5 = q 3 dt 65 65 e 68+9 1 + i k : X(t) + X t = 5 = iωqx ω + 3 dt e 68+9 i k : X(t) X t + Dipole: d=qx 65 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 5

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 6 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 7 Emission from a single particle Emission from single particle; use dipole formulas from last lecture: V F ω, Ω = q % 2πc " ε K n F ω M for the special case of purely transverse waves q % e F : X(ω) % 1 e F : κ % V F ω, Ω = 2πc " n ε F ω M κ X(ω) % K 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 ~δ jm ; but k j k j =k 2, thus

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 8 Emission from single particle Thus, the energy per unit frequency emitted to transverse waves from single non-relativistic particle 4π V F ω = q % 6π % c " ε K n F ω % X(ω) % An alternative is in terms of the acceleration a and the net force F 4π V F ω = q % 6π % c " ε K n F a (ω) % 4π V F ω = q % F (ω) 6π % c " n ε F K m %

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 9 Larmor formula for the emission from single particle Total energy W radiated in vacuum (n M =1) W = 4π 3 dω K 5 V F ω = q % 6π % c " ε K 3 dω Rewrite by noting that a(ω) is even and then use the power theorem K 5 a (ω) % Thus, the energy radiated over all time is a time integral The average radiated power, P ave, will be given by the average acceleration a ave The Larmor formula

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 10 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!

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 cos 2 (ω 0 t) should be averaged over a period 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 12

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 13 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

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=B z e z where is the cyclotron frequency z where we have the Larmor radius 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 14

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 15 Applications: cyclotron emission Cyclotron emission from a single particle where is the velocity perpendicular to B. Sum the emission over a Maxwellian distribution function, f M (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!

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 W 0 in vacuum : P\ dw K dt = 8π 3 q % 4πε K mc % cw K 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

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 17 Applications: wave scattering The scattering process can be interpreted as a collision Consider a density of wave quanta representing the energy density W 0 The wave quanta, or photons, move with velocity c (speed of light) Imagine a charged particle as a ball with a cross section σ T 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 σ T of the particle r 0 Density of incoming photons Photons hitting this area are scattered

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 18 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

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

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 20 Thompson scattering system at the fusion experiment JET Detectors Scattered light scattering

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/r 2 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 X ion (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

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 22 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 r min and So, let the ion be stationary at the origin Let the electron start at (x,y,z)=(,b,0) with velocity v=(-v 0,0,0) The conservation of angular momentum and energy gives b This is the Kepler problem for the motion of the planets! Next we need the minimum distance between ion and electron r min

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 23 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 v 0 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

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 X-ray tubes may also emit line radiation. Counts per second 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 ~ 10 7-10 8 K radiate primarily in via bremsstrahlung Note: surface of the sun 10 3 10 6 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: ~10 8 K ; the walls are ~10 3 K 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 P = 1 6πε K c " q% a % 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