Post on 03-Jul-2020
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
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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
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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:
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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:
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– 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?
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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
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• 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
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• An alternative is in terms of the acceleration 𝐚 and the net force 𝐅
4𝜋 𝑉F 𝜔 =𝑞%
6𝜋%𝑐"𝜀K𝑛F 𝜔%𝐗(𝜔) %
4𝜋 𝑉F 𝜔 =𝑞%
6𝜋%𝑐"𝜀K𝑛F 𝐚(𝜔) %
4𝜋 𝑉F 𝜔 =𝑞%
6𝜋%𝑐"𝜀K𝑛F
𝐅(𝜔)𝑚
%
Larmor formula for the emission from single particle• Total energy W radiated in vacuum (nM=1)
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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
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• 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
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Applications: Harmonic oscillator• As a first example, consider the emission from a particle performing
an harmonic oscillation– harmonic oscillations
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– 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
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• 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
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– 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.
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• 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
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• 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
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– 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
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– 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
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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
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scattering
Detectors Scattered light
Applications: Bremsstrahlung• Bremsstrahlung (~Braking radiation) come from the acceleration
associated with electrostatic collisions between charged particles (called Coulomb collisions)
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• 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
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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)
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– 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.
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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
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• 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
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• 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
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