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 • 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 UM ; the density of emission in k-space • Emission per frequency and solid angle () – Rewrite integral: �� = ��Ω = � ��Ω

Here � is the unit vector in the �-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 Emission formula (k-space power density) (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 � �, � = � �� ̇ � � � − � � = = � �� 1 + � � �(�) + ⋯ �̇ � = = −�� + �� (�) �̇ � + ⋯ Dipole: d=qX 2017-02-28 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: ∗ � � �(�) � �, Ω = �� ∗ 2�� 1 − � �

– for the special case of purely transverse waves � � �, Ω = �� ×�(�) 2�� – 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 2 Kroneker delta ~δjm; but kjkj=k , 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

� 4� � � = � � �(�) 6� � �

• An alternative is in terms of the acceleration � and the net force �

� 4� � � = � � (�) 6� � �

� � (�) 4� � � = � 6� � � �

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 8 Larmor formula for the emission from single particle

• Total energy W radiated in vacuum (nM=1) � � = 4� �� = �� (�) 6� � � • Rewrite by noting that �(�) 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

2017-02-28 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

2 – oscillation cos (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

�

– 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 � is the particle density and � 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 :

�� 8� � � ≡ − = �� 3 4�� – 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

hω – thus the effective cross section Cross section area σT of the particle for wave scattering is

Density of Photons hitting incoming this area are photons scattered

2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 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 Laser source in a different room

Thompson scattering systems at JET primarily measures temperatures Laser beam Laser

2017-02-28 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 b – Now we need rmin and • 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 • X-ray tubes may also emit line radiation. Line Line radiation Counts per second per Counts

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 � = � � 6��

• 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