Download The Larmor Formula

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
•
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Abraham-Lorentz force
Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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Repetition: Emission from multipole moments
• Multipole moments are related to the Fourier transform of the current:
Emission formula
(k-space power density)
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Emission formula
(integrated over solid angles)
Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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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
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%
%
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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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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
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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!
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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Thompson scattering system at the fusion experiment JET
Thompson scattering
systems at JET primarily
measures temperatures
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Laser beam
Laser source in
a different room
Dispersive Media, Lecture 12 - Thomas Johnson
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Thompson scattering system at the fusion experiment JET
Detectors
Scattered light
scattering
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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!
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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
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Dispersive Media, Lecture 12 - Thomas Johnson
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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)
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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
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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
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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
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