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The Larmor Formula (Chapters 18-19)

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The Larmor Formula (Chapters 18-19) 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 5 � �, � = � 3 �� �68+9 3 �"� �8�:� �̇ � � � − � � = 65 5 = � 3 �� �68+9 1 + � � : �(�) + ⋯ �̇ � = 65 5 = −���� � + 3 �� �68+9 � � : �(�) �̇ � + ⋯ 65 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: % ∗ % � M �F : �(�) �F �, Ω = " �F� ∗ % 2�� �K 1 − �F : � – for the special case of purely transverse waves % � M % �F �, Ω = " �F� �×�(�) 2�� �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 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� �F � = % " �F � �(�) 6� � �K • An alternative is in terms of the acceleration � and the net force � % � % 4� �F � = % " �F � (�) 6� � �K �% � (�) % 4� �F � = % " �F 6� � �K � 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) 5 5 % � % � = 4� 3 �� �F � = % " 3 �� � (�) 6� � �K K K • 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 • 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 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� �% \ K � ≡ − = % ��K �� 3 4��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 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 r0 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
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