, YerPhI Synchrotron Radiation Reflection from Outer Wall of Vacuum Chamber M.A. Aginian, S.G. Arutunian, E.G.Lazareva, A.V. Margaryan Yerevan Physics Institute The presentation is devoted to the eightieth anniversary of the birth of Garry Nagorsky and memory of Andrey Amatuni
Synchrotron radiation is produced by relativistic electrons rotating in a magnetic field. The spectral synchrotron radiation energy emitted by one electron [H. Wiedemann, Particle Accelerator Physics, Third Ed., Springer-Verlag Berlin Heidelberg, 2007] [http://xdb.lbl.gov/section2/sec_2-1.html]: 2 2 2 2 dw 3r0 mc 2ω 2 2 2 2 γψ 2 = γ (1 + γψ ) 2 2 K2/3( x) + K 2 2 1/3( x) dω dω 4π ωc 1+ γψ where dω= dθdψ (ψ is angle between orbit plane and radiation detection direction, θ is observation angle in the 2 horizontal plane), ω angular frequency of photon, γ is electron energy divided by mc e ( m e is electron mass, c velocity 2 2 ω 2 2 of light), r0 = e / mc e is electron classical radius, ( 1 ) 3/2 3 x = + γψ, ωc = 3 γ c /2ρ0 critical frequency, defined 2ω c as the frequency that divides the emitted power into equal halves. K1/3, K 2/3 are modified Bessel functions of the second kind. For ultrarelativistic electrons γ 1 the distribution is very wide and involves harmonics up to circulating frequency ω0 = c / ρ0 ( ρ 0 is radius of charge trajectory). 3 γ order of charge
Spectral distribution of SR as a function of the value of critical energy ε = γ ρ of the storage ring ( ρ 0 c 3 3 c / (2 0) radius of storage ring) [Synchrotron Radiation Basics, Methods and Applications, S. Mobilio, F. Boscherini, C. Meneghini Ed., Springer, Springer-Verlag Berlin Heidelberg 2015]
LEP synchrotron radiation [L. Rivkin, Electron dynamics with Synchrotron Radiation, SRBasicsCAS16_LRivkin.pptx].
Wide Fourier expansions correspond to narrow spatial structures. The pulse width detected by a detector has a duration 3 t 1/ γω, where ω0 = eb / γme is orbiting frequency. Therefore, synchrotron radiation is dominated by 0 3 frequency components in the range ω= γω 0. The wave fronts of synchrotron radiation for β = 0.9 [Hofmann].
Forward propagation of synchrotron radiation for β = 0.8 and 0.9 [Hofmann].
Pulse of synchrotron radiation: δt ρ γ 3 = 4 0 /3c [Wiedemann, Particle Accelerator Physics, Springer, 2007].
circulating electron with constant velocity β c in following coordinate system r () t = ρ e ( Φ) 0 0 1 Φ= βc t ρ 0
The electric and magnetic fields of a circulating charge are 2 γ ( n β) [ n [( n β) d β / dt]] Et () = e + 2 3 3 ρ (1 nβ) cρ 0 0(1 nβ) Ht () = [ n E] all values in right side are taken at retardation time t coupled with t by retardation equation ct t r r t ( ) = 0( )
Our concept since 1979 [С.Г. Арутюнян, Одновременная картина поля вблизи движущейся по кругу ультрарелятивистской заряженной частицы, препринт ЕФИ- 387(45)-79, Ереван, 1979] was to investigate direct Lienard Wiechert fields by solving retardation equation in vicinity of charge but far from Coulomb area. r = ρ (1 + δ) e ( Φ ) + ρζe 0 1 0 3 χ = ( Φ+ βct ( t )/ ρ )/2 0 4 2 2 2 2 2 χ + 3 χ ( γ δ) 3χΦ+ 3( Φ δ ζ ) / 4 = 0 Synchrotron radiation is concentrated in area near the line r 2/3 γ ( Φ ) = ρ0(1 + (3 Φ) /2) e1( Φ) ϕ = ( Φ (2 δ) / 3) γ 3/2 3
χ = δ /2+ η ( ) ( ) 2 2 9 1 3 9 1 3 1/3 1/3 η = ϕ + + ϕ ϕ + ϕ electric and magnetic field in synchrotron radiation area (in orbit plane): 2 4 2 ρ 0 2 2 γ (1 η ) e 1( φ) E = 2 e 2 3 1/2 2( Φ) e (1 + η ) δ 2 4 2 ρ 0 2 2 γ (1 η ) e3 H = 2 3 e (1 + η )
4 2 2 2 γ (1 η ) (1 + η ) 2 3 The main factor as function of angular (longitudinal) shift
To illustrate the field of synchrotron radiation we use electric field lines. First complete formula for lines of field of arbitrary moving charge was obtained in [M.A. Aginian, S.G. Arutunian, Force lines of electric and magnetic fields of arbitrary moving charge, Preprint YerPhI 684(74)-83, Yerevan, 1983]. r0( τ )/ ρ0 = e1( βτ ) L( τ )/ ρ = e1( βσ ) + ( τ σ )( ne1( βσ ) + ne2( βσ ) + ne3) n n n 1 2 3 = 0 1 2 3 = 1 c sin( βγ ( τ σ ) + c ) 2 2 1 γ (1+ β 1 c cos( βγ ( τ σ ) + c )) 2 2 1 2 β + 1 c2 sin( βγ ( τ σ ) + c1) = 1 + β 1 c cos( βγ ( τ σ ) + c ) 2 2 1 c γ (1+ β 1 c cos( βγ ( τ σ ) + c )) 2 2 2 1
γ = 3, scale = 2000
γ = 3, scale = 2000
γ = 3, scale = 2000
γ = 7, scale = 10000
γ = 20, scale = 50000, dt = 5e 5
γ = 20, scale = 50000, dt = 5e 5
γ = 20, scale = 50000, dt = 5e 5
This gamma-region in the seventies of the twentieth century in Yerevan Physics Institute is called as hair of Nagorsky γ = 50, scale = 600000
γ = 50, scale = 600000
On the outer wall of vacuum chamber, the "spot" of synchrotron radiation is running with angular velocity ω 0 = βc / ρ 0. If distance from charge trajectory to outer wall of vacuum chamber is δρ 0. the linear velocity of the "spot" is βc(1 + δ). For typical values of synchrotron light sources βc(1 + δ) is greater than c at very small values of δ. For CANDLE project E =3 GeV (γ =5.87E+03) and radius of trajectory in dipole magnet ρ 0 =7.4 m (corresponds to magnetic field B =1.35 T) [CANDLE Design Report, July 2002] the distance δρ 0 at which velocity of "spot" exceed the speed of light is only 0.5 µm. So, it is evidently that the "spots" of synchrotron radiation running on the outer wall of vacuum chamber with velocities greater than the light velocity.
The sizes of the "spot" are 3 ( ρ / γ ) ( ρ δ / γ). 0 0 q = σ S e q e γ ρ ρ δ e ρ γ γ = 4 0 0 2 3 0 δ [J.D. Jackson, Classical Electrodynamics, Third Ed., Wiley, 1999 /Models for the Molecular Polarizability]. x ( ee / m) δt en r e 0 2 Em ene x = γ 2 δ x 2 r γ 0 δ nrργ e 2 6 0 0 For values of CANDLE project this value is about 3.2 10-7 (for n_e =8.5 10 28 m -3 copper value). It means that charge induced in synchrotron radiation spot is about 3.2 10-7 e.
Superluminal synchrotron radiation Bτ r ( τ) = ρe1( ) ρ Bσ Bσ Bσ L ( τ) = ρe1( +Φ ) + ( τ σ)( ne 1 1( +Φ ) + ne 2 2( +Φ )) ρ ρ ρ B = ρβ = ρ γ > 2 1 1 G = B = ρ γ 2 2 2 1/ 1 1/ (1 ) 1 G = γ ρ = γ + γ 2 2 ( 1) / ( 1)
n n 1 2 = sinh( BG( τ σ) + c1 ) G(1 ± B cosh( BG( τ σ) + c )) = B ± sinh( BG( τ σ) + c1 ) 1 ± B cosh( BG( τ σ) + c ) 1 1 Here Bc Mach surface nb =1 is velocity of superluminal charge
γ = 3.5
γ = 3.5
γ = 3.5
γ = 3.5
γ = 3.5
γ = 3.5
γ = 3.5
γ = 7
γ = 7
γ = 7
γ = 7
γ = 7
γ = 7
γ = 7
γ = 7
γ =14
γ =14
γ =14
γ =14
γ =14
γ =14
, YerPhI Conclusion The main aim of the presentation is to emphasize the fine spatial structure of synchrotron radiation. In presence of vacuum chamber, especially walls placed outer from the beam orbit a lot of faster than lights "spots" contribute to beam self-produced field. The structure of the fields is so fine that for real bunches in accelerators it is thinner than interparticle distances. It means that these effects should be involved at least into dynamic consideration. We hope that new generation of accelerator physicists will continue this very interesting topic.
Thank you for attention