Emission from free particles (Chapters 18-19)

T. Johnson

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 1 Outline • Repetition: the emission formula and the multipole expansion • The emission from a free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation – Lorentz transformation and relativistic invariants – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force

13-02-27 Dispersive Media, Lecture 8 - 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

3 WM = ∑ ∫ d kUM (k) M

2 RM (k) * UM (k) = eM (k) • Jext (ωM (k),k) ε0 – the emission formula

– thus UM is a density of emission in k-space • €Alternatively, the emission per unit frequency and solid angle (dΩ) – for non-spatially dispersive media d 2Ωdω W = ∫ 3 ∑VM (ω,Ω) (2π) M

* 2 ωnM eM • Jext VM (ω,Ω) = − 3 2 c * ˆ (2π ) ε0 1− eM • k 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 3

€ Repetition: The current expressed in multipole moments

• Multipoles moments are related to Fourier transform of the current: J(ω,k) = ∫ dteiωt ∫ d 3x[e−ik•x ]J(t,x)

⎡ 1 2 ⎤ = dteiωt d 3x 1− ik • x − (k • x) + ... J(t,x) ∫ ∫ ⎣⎢ 2 ⎦⎥ = −iωd(ω) + ik × m(ω) − ωq • k(ω)/2 + ... Emission formula Emission spectrum (k-space power density) (integrated over solid angles) € ⎧ ⎧ 4 d RM (k) * 2 d n(ω)ω 2 ⎪U M (k) = ωM eM • d ⎪V rad (ω) = 2 3 d(ω) c ⎪ ε0 ⎪ 6π ε0 3 4 ⎪ 2 ⎪ m RM (k) * m n(ω) ω 2 ⎨U M (k) = eM • (k × m) ⎨V rad (ω) = 2 5 m(ω) ⎪ ε0 ⎪ 6π ε0c 3 6 ⎪ 2 ⎪ q RM (k) * q n(ω) ω * ⎪U M (k) = ωM eM ,iqij k j ⎪V rad (ω) = 2 5 dij dij ⎩ 4ε0 ⎩ 720π ε0c

13-02-27 Electromagnetic Processes In Dispersive Media, 4 Lecture 7

€ € Repetition: Emission in the time-domain • The total radiated power from a dipole T 1 ⎛ 1 2 ⎞ 1 2 P lim dt d˙˙ (t) d˙˙ (t) ave = 3 ⎜ T → ∞ ∫ ⎟ = 3 6πε0c ⎝ 2T −T ⎠ 6πε0c

– interpreted as a time average power – ideally the averaging should beperformed over all times € • for event, or periodic motion, averaging can be done over finite times

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 5 Dipole current from single particle

• Current from a single particle: J(t,x) = qX˙ ( t)δ(x − X(t)) ∞ J(ω,k) = q ∫ dte−iωt ∫ d 3k eik•x X˙ ( t)δ(x − X(t)) −∞ ∞ € = −iωq ∫ dte−iωt [1+ ik • X(t) + ...]X(t)δ(x − X(t)) −∞ ⎧ ∞ ⎫ = −iωq⎨ X(ω) + ∫ dte−iωt [ik • X(t) + ...]X(t)δ(x − X(t))⎬ ⎩ −∞ ⎭

Dipole: d=qX • Dipole approximation exp[ikx]~1 ; but what is the error? € – Assume oscillating motion: X˙ ( t) = vcos(ωt) ⇒ X ~ v/ω • for non-oscillating motion: emission of quanta ω occures on time-scale t~1/ω

v n ω v v Dipole valid for k • X(t) ~ k ~ M ~ n ω € c ω M c non-relativistic motion

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 6

€ Relation between emission and acceleration

• Emission from single particle; dipole current: J(ω,k) ≈ −iωqX(ω) 2 2 e* • ω2X(ω) ˆ q nM M VM ω,k = 3 2 ( ) * ˆ ε0(2πc) 1− e • k M € • Note that −ω 2 X ( ω ) = a ( ω ) is the acceleration: 2 2 e* • a(ω) ˆ q nM M VM ω,k = 3 2 € ( ) * ε0(2πc) 1 e kˆ € − M • – Thus emission from free particle is a response to acceleration!

•€ Power radiated for isotropic transverse waves:

truncated outside [-T,T]

2 2 1 T ˆ q nM (ω) 1 T Prad (ω) = lim dΩ VM ω,k = 2 3 lim a (ω) T → ∞ ∫ ∑ ( ) T → ∞ 2T M 12π ε0c 2T

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 7

€ The Larmor fomula • The emission integrated over all frequencies is related to time integral of the emitted power

– in vacuume (nM=1) ∞ ∞ T 1 2 2π 2 2π 2 P ∝ lim aT (ω) dω = lim aT (t) dt = lim a(t) dt T → ∞ ∫ T → ∞ ∫ T → ∞ ∫ 2T −∞ 4T −∞ T −T

The Larmor formula 2 q 2 P = a(t) € 3 6πε0c

– the Larmor formula related the averaged radiated power with the average acceleration • The time average€ has the same interpretation as for dipole and is sometimes written 2 q 2 P(t) = 3 a(t) 6πε0c – but should always be interpreted as an average! 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 8

€ Applications: Harmonic oscillator • As a first example, consider the emission from a particle performing an harmonic oscillation – harmonic oscillations in one dimesion X

˙˙ 2 X(t) = x0 cos(ω0t) , a(t) = X (t) = −x0ω0 cos(ω0t) – Larmor formula: the emitted power associated with this acceleration

2 4 2 q ω0 x0 2 € P(t) = 3 cos (ω0t) 6πε0c

2 – oscillation cos (ω0t) above is not physical; average over a period

€ 2 4 2 q ω0 x0 P = 3 12πε0c

€ 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 9 Applications: Harmonic oscillator – frequency spectum • Express the particle as a dipole d, use truncation for Fourier transform

d(ω) = qF{X(t)} = πqx 0[δ(ω − ω0) + δ(ω + ω0 )] = 1 ⎡ ⎛ (ω − ω )T ⎞ ⎛ (ω + ω )T ⎞ ⎤ lim Tqx sinc 0 sinc 0 = T → ∞ 0 ⎢ ⎜ ⎟ + ⎜ ⎟ ⎥ 2 ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ • The time-averaged power emitted from a dipole

2 1 1 ωd(ω) € P( ) lim V ( , )d lim ω = T → ∞ ∫∫ ω Ω Ω = T → ∞ 2 3 = T T 6π ε0c 2 1 1 1 ⎡ ⎛ (ω − ω )T ⎞ ⎛ (ω + ω )T ⎞ ⎤ lim Tqx sinc 0 sinc 0 = T → ∞ 2 3 ω 0 ⎢ ⎜ ⎟ + ⎜ ⎟ ⎥ = T 6π ε0c 2 ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ q2ω 4 x 2 {see Ch. 4.5}... 0 0 = = 3 [δ(ω − ω0 ) + δ(ω + ω0)] 12πε0c

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 10

€ 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 q qB a(t) = v˙ (t) = v × B = −Ωe × v , Ω = m z m ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ v˙ x 0 Ω 0 vx vx (t) v⊥sin(Ωt) x(t) ρL cos(Ωt) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˙ ⎢v y ⎥ = ⎢ −Ω 0 0⎥ ⎢v y ⎥ ⇒ ⎢v y (t)⎥ = ⎢v ⊥cos(Ωt)⎥ ⇒ ⎢ y(t)⎥ = ⎢ ρL sin(Ωt)⎥ ⎢ ⎥ ⎢ ⎥ ⎣ v˙ z ⎦ ⎣⎢ 0 0 0⎦⎥ ⎣ vz ⎦ ⎣⎢v x (t)⎦⎥ ⎣⎢ v|| ⎦⎥ ⎣⎢ z(t)⎦⎥ ⎣⎢ v||t ⎦⎥

– where we have the Larmor radius ρL = v⊥/Ω – This describe circular motion around magnetic field lines of force called € the Larmor- or cyclotron-motion 4 2 2 q B v⊥ – The period averaged radiation: P = 3 2 € 12πε0c m – Magnetized plasma; power depends on the temperature: 2 P ∝ v⊥ ∝ T • Electron cyclotron emission is one of the most common ways to measure the temperature of a fusion plasma! 13-02-27 Dispersive€ Media, Lecture 8 - Thomas Johnson 11 € Applications: wave scattering • Consider a particle being accelerated by an external wave field

E(t,x) = E0 cos(ω0t − k0 • x) ⇒ a(t) = qE(t,x)/m • The Larmor formula then tell us the average emitted power

2 q4 E € 0 P = 2 3 12πε0m c – Note: that this is only valid in vacuum (restriction of Larmor formula)

• Rewrite in term of the wave energy density W0 2 2 2 € – in vacuum : W0 = ε0 E0 /4 + ε0 k × E0 /cω /4 = ε0 E0 /2

2 8π ⎛ q2 ⎞ P = ⎜ 2 ⎟ cW0 3 ⎝ 4πε0mc ⎠ € – Shown in the next page: This describe the fraction of the power density that is scattered by the particle, i.e. first absorbed and then re-emitted € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 12 Applications: wave scattering • The scattering process can be interpreted as a collision

– Consider a density of wave quanta representing the power density W0 – The wave quanta, photons, move with velocity c (speed of light)

– Imagine a charged particle as a ball with a cross section σT – Then the power scattered per unit time is given by

P = σΤcW0 ω Cross section area σT • The effective cross section for of the particle wave scattering on electrons € r 8π q2 0 σ = r2 , r = Τ 3 0 0 4πε mc 2 0 Density of Photons hitting incoming this area are photons scattered

€

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 13 Applications: Thomson scattering • Scattering of waves against electrons is called Thomson scattering – from this process the classical radius of the electron was defined as 2 e −15 rele = 2 ≈ 2.8179 ×10 [m] 4πε0mec – Note: this is an effective radius for Thomson scattering and not a measure of the “real” size of the electron – in general quantum mechanics is needed to understand the behavior of € electrons at such short length scales • 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 continous spectrum from the solar corona is the result of the Thomson scattering of solar radiation with free electrons – The cosmic microwave background is thought to be linearly polarized as a result of Thomson scattering

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 14 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

• Derivation: 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 3 Ze2X(t) 2Z 2 ⎛ e2 ⎞ 1 m X˙˙ ( t) = − ⇒ P(t) = ⎜ ⎟ e 3 3m2c 3 4 4 4πε0 X(t) e ⎝ πε0 ⎠ X(t) – 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! €

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 15 Bremsstrahlung: Coulomb collisions • Lets try and integrate the emission over all times 3 ∞ 2Z 2 ⎛ e2 ⎞ ∞ dr W = P(t)dt = 2 ⎜ ⎟ rad ∫ 3m2c 3 4πε ∫ r4 r˙ (t) −∞ e ⎝ 0 ⎠ rmin – where we integrate in the distance to the ion r b – Now we need rmin and r˙ (t) • 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 2 ˙ mer θ = mebv0 2 2 2 2b0 b Ze Ze 1 ⇒ r˙ = v0 1+ − , b0 = 2 ˙ 2 2 ˙ 2 2 r r2 4πε m v me (r + r θ ) − = mev0 0 e 0 4πε0r 2 – This is the Kepler problem for the motion of the planets!

– Next we need the minimum distance between ion and electron rmin 2 € 2b0 b 2 2 2 2 € r˙ = 0 ⇒ 1+ − 2 = 0 ⇒ rmin + 2b0rmin − b = 0 ⇒ rmin = b0 + b0 + b r=rmin rmin rmin 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 16

€ 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 r min ≈ b and b0 << b – we are then ready to evaluate the time integrated emission 3 3 2Z 2 ⎛ e2 ⎞ ∞ dr πZ 2 ⎛ e2 ⎞ W = 2 ≈ ...≈ rad 2 3 ⎜ ⎟ ∫ 2 2 3 3 ⎜ ⎟ € €3 mec ⎝ 4πε0 ⎠ b 2b b 3mec v0b ⎝ 4πε0 ⎠ r4v 1+ 0 − 0 r r2 • This is now 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: 2 PBr ∝ Zi nine Te

• Thus it can be used to derive information about both the charge, density and temperature of the media 13-02-27 € Dispersive Media, Lecture 8 - Thomas Johnson 17 Industrial applications of Bremsstrahlung • Typical frequency of Bremsstrahlung is in X-ray regime • X-ray tubes: electrons are accelerated to high velocity When impacting on a metal surface they emit bremsstrahlung

X-ray tube

• X-ray tubes are also used in – CAT scanners – airport luggage scanners – X-ray crystallography – industrial inspection.

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 18 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 – Bremsstrahlung and cyclotron radiation power losses: • Temperature at the centre of fusion plasma: ~108K ; the walls are ~103K • Main issue for fusion is to confine heat in plasma core • However, both Bremsstrahlung and cyclotron radiation escape easily • In reactor, radiation losses will be of importance – limits the reactor design – If plasma gets too hot, then radiation losses cool down the plasma.

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 19 Outline • Repetition: the emission formula and the multipole expansion • The emission from a free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation – Lorentz transformation and relativistic invariants – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 20 Quick recap of special relativity • Next we’ll derive a Larmor formula that is valid at relativistic velocities • But first we’ll recap the some basic theory of special relativity

• Special relativity is based on two postulates – Principle of relativity: The laws of physics are the same for all observers in uniform motion relative to one another – The speed of light in a vacuum (c = 299 792 458 m/s) is the same for all observers, regardless of their relative motion, or of the motion of the source of the light • These postulates have many surprising consequences, e.g. – Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion. – Time dilation: Moving clocks are measured to tick more slowly than a "stationary" clock:

2 2 dTmoving = dTstationary 1− v /c – Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer. – Mass–energy equivalence: E = mc2, energy and mass are equivalent and transmutable. €

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 21 Background: Tensor formalism for special relativity

• The mathematical description of the “principle of relativity” is done with the so called Lorentz transform, which describe transformations between non-accelerated coordinate systems

• The Lorentz transform can be represented with the Minkowski formulation of spacetime [the 4D space spanned by time and real space (t,x,y,z)].

• In Minkowski spacetime every tensor has two representations

– a vector F can be represented by co-variant components, Fµ – or by contra-variant components, Fµ – think of them as e.g. “row vector” & “column vector” components, the “bra” & “ket” of quantum mechanics, or as “dual” spaces

• The Minkowski spacetime is a vector space with the inner product µ µ 1 2 3 4 F,G ≡ F Gµ = FµG = F1G + F2G + F3G + F4G – i.e. here summation over repeated indexes is implicit – in this tensor formalism, index can be repeated only for multiplication of pairs of co- and contra-variant tensor components € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 22 Background: Tensor formalism for special relativity • In the Minkowski formulation of spacetime the co- and contra- variant component of the position vector for the point (t, x, y, z)

xµ = [ct,−x,−y,−z] T x µ = [ct,x,y,z]

2 µ 2 2 2 – thus x = xµ x = c t − r

– Note: An alternative and equivalent definition is xµ = [−ct,x,y,z] € – Strange? …we’ll soon see why Minkowski choose this formulation •€ Transformations between co- and contra-variant forms are performed with the metric tensor gmn or g € mn 0 ⎡1 0 0 0 ⎤ ⎡ x ⎤ ⎡1 0 0 0 ⎤ ⎡ x0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −1 0 0 x1 0 −1 0 0 x ν ⎢ ⎥ ⎢ ⎥ ν νµ ⎢ ⎥ ⎢ 1 ⎥ xµ = gµν x = 2 ⇔ x = g xµ = ⎢0 0 −1 0 ⎥ ⎢ x ⎥ ⎢0 0 −1 0 ⎥ ⎢ x2 ⎥ ⎢ ⎥ ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 0 0 −1⎦ ⎣ x ⎦ ⎣0 0 0 −1⎦ ⎣ x3 ⎦

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 23

€ Invariance of coordinate transformation • Consider a moving object (moving in z-direction) viewed by an observer µ standing at the origin of a coordinate system x & xµ – The observer measures the speed of the object by measuring the position at two time points: v = [z(T) − z(0)]/T  x ✈ – The length of the 4-vector of the plane from µ x˜ the first to the second measurement point is µ dx 2 dx µ dx c 2dt 2 dz2 c 2T 2 v 2T 2 c 2T 2 1 v 2 /c 2 € = µ = − = − = ( − ) € – Note: that T 1 − v / c is the retarded time experienced€ by the object! µ • Second coordinate system x˜ µ & x ˜ , where the “moving” object is in rest – measure the speed in the new coordinate system using the same time as the € 2 observer, i.e. µ 2 ˜ 2 € dx˜ = dx˜ dx˜ µ = c dt − 0 2 – equation for time-dilation: dt˜ = T 1− v 2 /c 2 ⇒ dx˜ = c 2T 2 (1− v 2 /c 2 ) € • Thus, the length of a vector dxµ is independent of the coordinate system, despite€ time dilation and length contraction! – We say that dx µ dx is an invariant under Lorentz transformations € µ

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€ Lorentz transformations • Principle of relativity reformulated: The laws of physics are invariants under Lorentz transformations

µ • The transformation operator is therefore a 2-tensor L ν µ ν' ν' ν' µ – a transformation equation from x to x reads: x = L µ x – e.g. to a coordinate system moving in the x-direction with velocity v=βc : ⎛ γ −γβ 0 0⎞ t'= γ(t − βx) Repeat calculation ⎜ ⎟ from previous slide: −γβ γ 0€ 0 € x'=€γ x − βt Lν' = ⎜ ⎟ ( ) dx'= 0 ⇒ dx = βdt µ ⎜ ⎟ 0 0 1 0 y'= y ⇒ dt'= γ dt − ββdt ⎜ ⎟ ( ) ⎝ 0 0 0 1⎠ z'= z ... = dt /γ µ' λ • Invariance of inner product: a bµ' = a bλ L has to satisfy: aµ'b = g aµ'bν' = g Lµ' aλ Lν' bη € µ' µ'ν' µ'ν€' ( λ )( η ) µ' ν' € gµ'ν'L η L λ = gηλ aλb = g aλbη λ ηλ € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 25

€ € Representation of physical quantities in special relativity • For formulating physical laws we need 4-vector generalisation of common physical quantities; careful considerations yields µ – position x = [ct,x] µ – velocity u = [γ c,γ v] 2 – momentum pµ = [ε /c,p] , ε2 = m2c 4 + p c 2 – wave vector k µ = [ω /c,k] – current density J µ = [ρc,J] – 4-vector potential Aµ = [φ /c,A] – Force F µ = [γ v⋅ F /c,γ F] −1/ 2 where γ = (1− v 2 /c 2 ) ν • All these quantities has co-variant representations, e.g. Aµ = gµν A • Any inner €product between two of these quantities are invariant under Lorentz transformations! €

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson€ 26 Relativistic Larmor formula • In special relativity the velocity and acceleration is defined in the frame of an arbitrary observer experiencing a time τ T ∂x µ (τ) v µ (τ) = [c,x˙ ,y˙ ,z˙ ] v µ (τ) = ∂τ vµ (τ) = [c,−x˙ ,−y˙ ,−z˙ ] 2 µ ∂ x (τ) µ T aµ (τ) = a (τ) = [0,x˙˙ , y˙˙ , ˙z˙ ] ∂τ2 aµ (τ) = [0,−x˙˙ , −y˙˙ , −˙z˙ ] – Since inner products are invariant under Lorentz transformations, µ 2 2 2 2 thus a ( τ ) a µ ( τ ) = − x˙ ˙ − y˙ ˙ − ˙ z˙ = − a is an invariant! €• Consider a single particle,€ then pick a system where it is at rest at time τ, but being accelerated by a force – In this coordinate system the particle is non-relativistic and the € (average) emitted power is given by the Larmor formula 2 q µ P(τ) = − 3 a (τ)aµ (τ) 6πε0c µ – since a a µ is an invariant, so is P(τ) , i.e. above we have a relativistic Larmor formula!! 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 27 € € Relativistic Larmor formula • Rewrite the acceleration in term of the 3-vector velocity v ⎡ 2 2 ⎤ µ 2 γ γ a = γ ⎢ v • v˙ , v˙ + 2 v(v • v˙ ) ⎥ ⎣ c c ⎦ ⎛ 2 ⎞ µ 2 2 v × v˙ a aµ = −γ ⎜ v˙ − 2 ⎟ ⎝ c ⎠ 2 ⎛ 2 ⎞ q 2 2 v × v˙ ⇒ P = 3 γ ⎜ v˙ − 2 ⎟ 6πε0c ⎝ c ⎠

• Acceleration in special relativity is somewhat artificial € – instead use the force – Larmor formula in terms of the 3-vector force F

2 2 ⎛ 2 ⎞ Note: there’s a typo in the q γ 2 v × F book; the cross product is P = 2 2 ⎜ F − 2 ⎟ 6πε0c m c ⎝ c ⎠ replaced by scalar product

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€ Relativistic Larmor formula • Let the object move perpendicular to the force

v × F = v F γ-2 2 ⎛ 2 ⎞ 2 ✈ 2 q 2 v q 2 P = γ 3 2 F ⎜1 − 2 ⎟ = 3 2 F F=mg 6πε0c m ⎝ c ⎠ 6πε0c m

– No relativistic correction!

€• Let the object move along the force – i.e. F is parallel to v ✈ v × F = 0

2 2 q 2 P = γ 3 2 F F=mg 6πε0c m – relativitstic correction by γ2 !

€ 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 29 Outline • Repetition: the emission formula and the multipole expansion • The emission from a free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation – Lorentz transformation and relativistic invariants – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 30 Chapter 19: Alternative treatments of emission processes • The traditional treatment of emission; study the emitted Poynting flux – start with the scalar and vector potentials from single particle • Lorentz gauge: the potentials follows d’Alemberts equation (Ch. 5) ⎡ ⎤ ⎡ ⎤ A(t,x)/µ0 3 δ(t − t'− x − x' /c) J(t',x') ⎢ ⎥ = ∫ dt'd x' ⎢ ⎥ ⎣ φ(t,x)ε0 ⎦ 4π x − x' ⎣ρ (t',x')⎦ • When the sources is from a single particle ⎡ ⎤ ⎡ ˙ ⎤ A(t,x)/µ0 3 δ(t − t'− x − x'/ c) qX( t')δ(x'−X(t')) € ⎢ ⎥ = ∫ dt'd x' ⎢ ⎥ ⎣ φ(t,x)ε0 ⎦ 4π x − x' ⎣ qδ(x'−X(t')) ⎦

– integrate over x’ ⎡ ⎤ ⎡ ˙ ⎤ A(t,x)/µ0 δ(t − t'− x − X(t')/ c) qX( t') € ⎢ ⎥ = ∫ dt' ⎢ ⎥ ⎣ φ(t,x)ε0 ⎦ 4π x − X(t') ⎣ q ⎦

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 31 € The Lienard-Wiechert potentials • Be careful with integration over t’ ! – Note that t’ appear non-linearly in the Dirac delta • Remember: −1 ∫ dxδ(g(x)) = ∑ (g'(xr )) xr :g(xr )=0 −1 ⎡ ∂ ⎤ • thus ∫ dt'δ(t − t'− x − X(t')/ c) = ⎢ (t − tr − x − X(tr )/ c)⎥ ⎣ ∂tr ⎦ 1 € ⎡ ˙ ⎤ − (x − X(tr )) • X( tr ) = ... = ⎢1 − 2 ⎥ , where t − tr − x − X(tr )/ c = 0 ⎣⎢ c x − X(tr ) ⎦⎥

– the time tr is known as the retarded time • Note: t − t − x − X ( t ) / c = 0 is a non-linear equation for t r r € r • The€ potentials can then be written as the Lienard-Wiechert potentials ˙ ⎡A (t,x)/µ0 ⎤ q ⎡ X( t')⎤ ⎢ ⎥ = ⎢ ⎥ € t,x ˙ ⎣ φ( )ε0 ⎦ 4π x − X(tr ) − (x − X(tr )) • X( tr )/c ⎣ 1 ⎦

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€ The Lienard-Wiechert potentials • The Lienard-Wiechert potentials are simplified when choosing

coordinate to locate the source to the origin X(tr ) = 0

⎡ ⎤ ⎡ ˙ ⎤ A(t,x)/µ0 q X( tr ) ⎢ ⎥ = ˙ ⎢ ⎥ ⎣ φ(t,x)ε0 ⎦ 4π( x − x • X( tr )/c€) ⎣ 1 ⎦

˙ • Note that the term x • X ( t r ) / c ~ v / c is a relativistic term € • The Lienard-Wiechert potentials are derived from Maxwells equations, thus they are automatically relativistically correct! €

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 33 E and B from the Lienard-Wiechert potentials • Calculate E and B field from the Lienard-Wiechert potentials ∂A E = −∇φ + , B = ∇ × A ∂t

• Note functional dependence t r = t r ( t , x ) thus

∂φ(tr,x) ∂ ∇φ(tr (t,x),x) = ∇tr + e j φ(tr,x) € ∂tr ∂x j € ∂A(tr,x) ∂ ∇ • A(tr (t,x),x) = • ∇tr + A j (tr,x) ∂tr ∂x j

∂ ∂Ai (tr,x) ∂tr Ai (tr (t,x),x) = ∂t ∂tr ∂t

– After lengthy calculations, using {r ≡| x | , n ≡ x / | x | , a ≡ a(tr ) , v ≡ v(tr )} n E q (x − rv/c) 1− v 2 /c 2 + x • a /c 2 − ar /c 2 (r − x • v/c) € × ( ) ( ) B = , E = 3 c 4πε0 (r − x • v/c) € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 34

€ Radiative and Inductive fields • The electric field has term ~1/r and other ~1/r2 ; use x=nr

2 2 2 2 q [(n − v/c)(1− v /c + rn • a /c ) − r(a /c )((1− n • v/c))] E = 2 3 = 4πε0 r (1− n • v/c) 2 2 q ⎪⎧ (n − v/c)n • a − (1− n • v/c)a (n − v/c)(1− v /c )⎪⎫ = ⎨ + ⎬ 3 c 2r r2 4πε0(1− n • v/c) ⎩⎪ ⎭⎪ Radiative Inductive • Note: only the radiative terms depend on the acceleration € • Imagine radiation as emitted photons; • Radiate N photons at t=0, |x|=0, then at time T>0 you have N photons on the sphere with radius R=cT , i.e. the number of photons is independent of R. • What energy (~number of photons) reach the sphere of radius R?

2 2 2 2 • Radiative W ~ ∫ E R d Ω ~ 1 / R R ~ 1 for all R - free moving photons 2 2 • Inductive W ~ ∫ E R 2 d Ω ~ 1 / R 2 R 2 ~ R − 2 virtual photons bound to not leave the particle unless absorbed by receiver particle 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 35 € € Simplified form for the electric field • Simplify the radiative electric field (for improved interpretation) q ⎧ (n − v/c)n • a − (1− n • v/c)a ⎫ Erad = 3 ⎨ 2 ⎬ 4πε0(1− n • v/c) ⎩ c r ⎭

• Simplify the numerator: e j • {(n − v/c)n • a − (1− n • v/c)a} =

€ = {nkn j − nkv j /c − (1− nmvm /c)δ jk}ak =

= [nkn j − δ jk ]ak − [nkv j − nmvmδ jk ]ak /c • Use the vector identity: a × b × c = ε ε a b c = δ δ − δ δ a b c = a b − a b c [ ( )]i ijk klm j l m ( il jm im jl ) j l m ( j i j j ) j € ⇒ n n − δ a = n × n × a & n v − n v δ a = n × v × a [ k j jk ] k [ ( )] j [ k j m m jk ] k [ ( )] j

q n × [(n − v/c) × a] q n × {n × [(n − v/c) × a]} Erad = 3 B = 4 c 2 rad 3 3 € πε0 (1− n • v/c) r 4πε0c (1− n • v/c) r

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 36

€ € The Poynting flux • Poynting flux radiated by a single charge particle, which at the

retarded time tr had velocity v and acceleration a

2 E × B q2 n × [(n − v/c) × a] FEM = = ... = 2 2 6 2 n µ0 16π ε0c (1− n • v/c) r

• The non-relativistic Poynting flux: € 2 q2 n × [n × a] FEM = 2 2 2 n + O(v /c) 16π ε0c r

€

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 37 Radiated power – non-relativistic limit • The Poynting flux is an energy flux • The flux through a closed surface = the power leaving the enclosed region • Encircle the particle with a large sphere with radius R • The power leaving this sphere is 2 q 2 P = ∫∫ dS • FEM = 2 2 ∫∫ dΩn × [n × a] + O(v /c) r=R 16π ε0c

⎧ 2 2 2 ⎫ dΩn × [n × a] = dΩ nkn j − δ jk ak nm n j − δ jm am = dΩ a − (nkak ) = ⎪ ∫∫ ∫∫ [( ) ][( ) ] ∫∫ [ ] ⎪ ⎨ 2π 1 ⎬ € 4 ⎪ = let : n a = acos(v) = a2 dΩsin2 (v) = x = sin(v) = dφ dxx 2 = πa2 ⎪ ⎪ { k k } ∫∫ { } ∫ ∫ ⎪ ⎩ 0 −1 3 ⎭ • The non-relativistic power leaving the sphere is given by

2 2 € q a(tr ) P(t,R) = 2 + O(v /c) 6πε0c

the radiated power at the retarded time tr as given by the Larmor formula!

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 38 € Sketch of emission in dispersive media • The Poynting flux approach to emission can also be used in dispersive media, however this procedure may become tedious • Start with the photon propagator for the wave equation 2 µ c λij A (t,x) = dωd 3k 0 J (ω,k)e−iωt +ik•x i ∫ ω2 Λ j 2 2 iωt −ik•X(t ) • Particle source Ji (ω,k) = qω X i (ω,k) = qω ∫ dte X i (t) λ A (t,x) = qµ c 2 dt' X (t') dωd 3k ij e−iω(t −t')+ik•(x −X(t')) € i 0 ∫ i ∫ Λ • The emitted€ power is now given by the energy flux, which now include both an electro-magnetic term and a particle term (Ch. 15)

€ EM P ωM (k)⎧ 2 * * ∂Kij ⎫ Pflux = ∫∫ dΩ(FM + FM ) = ∫∫ dΩ ⎨ 2ℜ[A k − AA • k] − Ai A j ⎬ r=R r=R µ0 ⎩ ∂k ⎭ • However, this formalism is often more tedious then treating emission in terms of the “work” as outlined earlier in this lecture. € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 39 Outline • Repetition: the emission formula and the multipole expansion • The emission from a free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation – Lorentz transformation and relativistic invariants – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 40 Self-forces • Energy conservation: A particle emitting radiation losses energy (from previous lecture – from quantum mechanics) E 2 − E1 = ω • Interpretation: the emitted wave performs work on the particle: W = F • v = qE • v = E • J € – The force from one particle onto itself is called the self-force • Remember your first lecture on electrostatics € – two charged particles “1” and “2” exert a force on each other

q1q2 (r2 − r1) F1 = −F2 = 3 4πε0 r2 − r1 • This says nothing about the force from a single particle on itself, i.e. the self-force, that is needed to describe emission € – …this problem is only properly resolved in quantum mechanics! • Here we’ll derive the Abraham-Lorentz force – classical treatment of radiation conserving energy and momentum

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 41 Radiation reaction • This power emitted during radiation, according to the Larmor formula q2 v˙ ( t) 2 P(t) = 3 6πε0c • Construct a radiation-reaction force that describe the lost energy

– the work by this force: P(t) = −v(t) • Freact (t) • Time integration€ t2 2 t2 2 ⎧ t2 ⎫ q q ⎪ t ⎪ ˙ 2 ˙ 2 ˙˙ ∫ P(t)dt = 3 ∫ v (t) dt = − 3 ⎨[ v(t) • v( t)]t − ∫ dtv(t) • v( t)⎬ 6πε c 6πε c ⎪ 1 ⎪ t1 € 0 t1 0 ⎩ t1 ⎭ • To remove the first term, consider e.g. periodic motion, or events such that the acceleration is finite during event € t2 q2 t2 P(t)dt = − dtv(t) • ˙v˙ (t) ∫ 6πε c 3 ∫ t1 0 t1

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 42 € Abraham-Lorentz equation of motion

• Let the integrated power be represented by a reaction force t2 q2 t2 t2 P(t)dt = − dtv(t) • ˙v˙ (t) = − dtv(t) •F ∫ 6πε c 3 ∫ ∫ react t1 0 t1 t1 – Thus there is a force that recover the time integrated power q2 Freact = 3 ˙v˙ (t) € 6πε0c – This does not imply that it represents the instantaneous force!

• The equation of motion under the influence of a self-force F0(t) q2 2Z 2 r € ele −24 2 m[v˙ ( t) − τ˙v˙ (t)] = F0 , τ = 3 = ≈10 × Z [s] 6πε0c 3 c – this is the Abraham-Lorentz equation of motion – Note: the force has a time scale τ , which is roughly the time it takes for light to travel across the classical radius of the electron, rele € • But there are serious problems with the Abraham-Lorentz equation!

13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 43 Properties of the Abraham-Lorentz equation of motion • First, there’s a run-away solution possible in absence of any force ⎧ 0 m[v˙ ( t) − τ˙v˙ (t)] = 0 ⇒ v˙ ( t) ~ ⎨ t / ⎩ e τ run-away! • Rewrite the Abraham-Lorentz equation to avoid the run-away solution ∞ −x mv˙ ( t) = ∫ F0 (t + τx)e dx 0 € – This equation has no run-away solutions • However it includes pre-acceleration – the force is evaluated in future times t+τx € – the particle responds to the force before the force is applied – this is NOT causal model!! 2Z 2 r – however, the time scale of pre-acceleration is tiny: τ = ele 3 c

€ 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 44