The 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 1

The Larmor Formula(Chapters 18-19)

T. Johnson

• 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

Outline

2017-02-28 2Dispersive Media, Lecture 12 - Thomas Johnson

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

• 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% '()(+)'+ 𝑑𝜔𝑑%Ω

Repetition: Emission formula


Here 𝐤/ is the unit vector in the 𝐤-direction

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)

𝐉 𝜔, 𝑘 = 𝑞 3 𝑑𝑡5

65

𝑒68+9 3𝑑"𝑘 𝑒8𝐤:𝐱�̇� 𝑡 𝛿 𝐱 − 𝐗 𝑡 =

= 𝑞 3 𝑑𝑡5

65

𝑒68+9 1 + 𝑖 𝐤 : 𝐗(𝑡) +⋯ �̇� 𝑡 =

= −𝑖𝜔𝑞𝐗 𝜔 + 3 𝑑𝑡5

65

𝑒68+9 𝑖 𝐤 : 𝐗(𝑡) �̇� 𝑡 +⋯

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

Dipole: d=qX

Dipole current from single particle

• Thus, the field from a single particle is approximately a dipole field

• When is this approximation valid?


Dipole approx. valid for non-relativistic motion

– Assume oscillating motion:

- The dipole approximation is based on the small term:

Emission from a single particle

• Emission from single particle; use dipole formulas from last lecture:

– for the special case of purely transverse waves

– 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

𝑉F 𝜔,Ω =𝑞%

2𝜋𝑐 "𝜀K𝑛F𝜔M

𝐞F∗ : 𝐗(𝜔) %

1 − 𝐞F∗ : 𝛋 %

𝑉F 𝜔,Ω =𝑞%

2𝜋𝑐 "𝜀K𝑛F𝜔M 𝛋×𝐗(𝜔) %

Emission from single particle• Thus, the energy per unit frequency emitted to transverse waves from

single non-relativistic particle


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

4𝜋 𝑉F 𝜔 =𝑞%

6𝜋%𝑐"𝜀K𝑛F 𝜔%𝐗(𝜔) %


6𝜋%𝑐"𝜀K𝑛F 𝐚(𝜔) %


6𝜋%𝑐"𝜀K𝑛F

𝐅(𝜔)𝑚

%

Larmor formula for the emission from single particle• Total energy W radiated in vacuum (nM=1)


The Larmor formula

• 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

𝑊 = 4𝜋3 𝑑𝜔5

K

𝑉F 𝜔 =𝑞%

6𝜋%𝑐"𝜀K3 𝑑𝜔5

K

𝐚(𝜔) %

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!

• 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

Outline


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

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=Bzez


– where we have the Larmor radius

– where is the cyclotron frequency

𝑧

– 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: 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.

– Interpretation: this is the fraction of the power density that is scattered by the particle, i.e. first absorbed and then re-emitted

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 :

𝑃\ ≡ −𝑑𝑊K

𝑑𝑡=8𝜋3

𝑞%

4𝜋𝜀K𝑚𝑐%𝑐𝑊K

– The wave quanta, or photons, move with velocity c (speed of light)– Imagine a charged particle as a ball with a cross section σT

Applications: wave scattering• The scattering process can be interpreted as a “collision”

– Consider a density of wave quanta representing the energy density W0


– thus the effective cross section for wave scattering is

hω

Density of incomingphotons

Photons hittingthis area arescattered

r0

Cross section area σTof the particle

– The power of from photons bouncing off the charged particle, i.e. scattered, per unit time is given by

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 polarizedas 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


Lase

r bea

m

Laser source in a different room

Thompson scattering systems at JET primarily measures temperatures

Thompson scattering system at the fusion experiment JET


scattering

Detectors Scattered light

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!

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

• 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)

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

– 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 velocityWhen impacting on a metal surface they emit bremsstrahlung

• X-ray tubes may also emit line radiation.


Line radiation

Wavelength, (pm)

Cou

nts

per s

econ

d

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

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 emitsbremstrahlung, which then heats the D-T pellet

Summary

• When charged particles accelerated they emits radiation

• This emission is described by the Larmor formula

𝑃 =1

6𝜋𝜀K𝑐"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


Documents

The Larmor Formula · • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force