Synchrotron Radiation Reflection from Outer Wall of …candle.am/wp-content/uploads/2017/02/11.05-S.-Arutunian.pdf · Ultrafast Beams and Applications, 04-07 July 2017, CANDLE, Armenia

Ultrafast Beams and Applications, 04-07 July 2017, CANDLE, Armenia

Synchrotron Radiation Reflection from Outer Wall of Vacuum Chamber, M. Aginian, S. Arutunian, E.Lazareva, A. Margaryan, 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 Reflection from Outer Wall of Vacuum Chamber, M. Aginian, S. Arutunian, E.Lazareva, A. Margaryan

2 2 2 22 2 2 2 2 20

2/3 1/32 2 2 2

3 (1 ) ( ) ( )4 1c

r mcd W K x K xd d

ω γ ψγ γ ψω π ω γ ψ

= + + Ω +

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]:

where d d dθ ψΩ = (ψ is angle between orbit plane and radiation detection direction, θ is observation angle in the

horizontal plane), ω angular frequency of photon, γ is electron energy divided by 2em c ( em is electron mass, c velocity

of light), 2 20 / er e m c= is electron classical radius, ( )3/22 21

2 c

x ω γ ψω

= + , 303 / 2c cω γ ρ= critical frequency, defined

as the frequency that divides the emitted power into equal halves. 1/3 2/3,K K are modified Bessel functions of the second

kind. For ultrarelativistic electrons 1γ the distribution is very wide and involves harmonics up to 3γ order of charge circulating frequency 0 0/cω ρ= ( 0ρ is radius of charge trajectory).



Spectral distribution of SR as a function of the value of critical energy 303 / (2 )c cε γ ρ= of the storage ring ( 0ρ

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

01/t γ ω∆ ≈ , where 0 / eeB mω γ= is orbiting frequency. Therefore, synchrotron radiation is dominated by

frequency components in the range 30ω γ ω= .

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: 304 / 3t cδ ρ γ= [Wiedemann, Particle Accelerator Physics, Springer, 2007].



circulating electron with constant velocity cβ in following coordinate system

0 10( ) ( )r t eρ= Φ

0

c tβρ

Φ =



0

2

2 3 30

( ) [ [( ) / ]]( )(1 ) (1 )

n n n d dtE t en c n

γ β β βρ β ρ β

− − × − × = + − −

( ) [ ]H t n E= ×

The electric and magnetic fields of a circulating charge are

all values in right side are taken at retardation time t′ coupled with t by retardation equation

0( ) ( )c t t r r t′ ′− = −



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.

0 1 0 3(1 ) ( )r e eρ δ ρ ζ= + Φ +

0( ( ) / ) / 2c t tχ β ρ′= Φ + −4 2 2 2 2 23 ( ) 3 3( ) / 4 0χ χ γ δ χ δ ζ−+ − − Φ + Φ − − =

Synchrotron radiation is concentrated in area near the line

2/310( ) (1 (3 ) / 2) ( )r eγ ρΦ = + Φ Φ

3/2 3( (2 ) / 3)ϕ δ γ= Φ −



2 4 210

22 3 1/2

2 2 (1 ) ( ) 2 ( )(1 )

eE eeρ γ η φ

η δ − = − Φ +

( ) ( )1/3 1/32 29 1 3 9 1 3η ϕ ϕ ϕ ϕ= + + − + −

2 4 230

2 3

2 2 (1 )(1 )

eHeρ γ η

η−

= −+

/ 2χ δ η= +

electric and magnetic field in synchrotron radiation area (in orbit plane):



4 2

2 3

2 2 (1 )(1 )γ ηη−

+The main factor as function of angular (longitudinal) shift



0 10( ) / ( )r eτ ρ βτ=

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].

1 1 2 30 1 2 3( ) / ( ) ( )( ( ) ( ) )L e n e n e n eτ ρ βσ τ σ βσ βσ= + − + +

23 2

2 1(1 1 cos( ( ) ))cn

c cγ β βγ τ σ=

+ − − +

22 1

1 22 1

1 sin( ( ) )

(1 1 cos( ( ) ))

c cn

c c

βγ τ σ

γ β βγ τ σ

− − += −

+ − − +22 1

2 22 1

1 sin( ( ) )

1 1 cos( ( ) )

c cn

c c

β βγ τ σ

β βγ τ σ

+ − − +=

+ − − +



3, 2000scaleγ = =



3, 2000scaleγ = =



3, 2000scaleγ = =



7, 10000scaleγ = =



20, 50000, 5 5scale dt eγ = = = −



20, 50000, 5 5scale dt eγ = = = −



20, 50000, 5 5scale dt eγ = = = −



50, 600000scaleγ = =

This gamma-region in the seventies of the twentieth century in Yerevan Physics Institute is called as “hair of Nagorsky”



50, 600000scaleγ = =



On the outer wall of vacuum chamber, the "spot" of synchrotron radiation is running with angular velocity 0 0/cω β ρ= . If distance from charge trajectory to outer wall of vacuum chamber is 0δρ . the linear velocity of the "spot" is (1 )cβ δ+ . For typical values of synchrotron light sources (1 )cβ 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

0ρ =7.4 m (corresponds to magnetic field B =1.35 T) [CANDLE Design Report, July 2002] the distance 0δρ 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 30 0( / ) ( / )ρ γ ρ δ γ× .

eq Sσ= −4

0 02 30

eq eρ ρ δγρ γ γδ

− × × = −

20em e

en rE en x γδ

−×∆ =

2

0x r γδ

−

∆ 2( / )x eE m tδ∆

[J.D. Jackson, Classical Electrodynamics, Third Ed., Wiley, 1999 /Models for the Molecular Polarizability].

2 60 0en r ρ γ −

For values of CANDLE project this value is about 3.2×10-7 (for n_e =8.5×1028 m-3 copper value). It means that charge induced in synchrotron radiation spot is about 3.2×10-7e.



Superluminal synchrotron radiation

1( ) ( )Br e ττ ρρ

∗=

21 1B ρβ ρ γ −= = − >

2 2 21/ 1 1/ (1 ) 1G B ρ γ −= − = − −

2 2( 1) / ( 1)ρ γ γ= + −

1 1 21 2( ) ( ) ( )( ( ) ( ))B B BL e n e n eσ σ στ ρ τ σρ ρ ρ

∗ ∗ ∗ ∗= +Φ + − +Φ + +Φ

G γ=



12

1

sinh( ( ) )1 cosh( ( ) )B BG cn

B BG cτ στ σ

± − +=

± − +

1nB =

11

1

sinh( ( ) )(1 cosh( ( ) ))

BG cnG B BG c

τ στ σ

− +=

± − +

Here is velocity of superluminal charge Bc

Mach surface



3.5γ =



3.5γ =



3.5γ =



3.5γ =



3.5γ =



3.5γ =



3.5γ =



7γ =



7γ =



7γ =



7γ =



7γ =



7γ =



7γ =



7γ =



14γ =



14γ =



14γ =



14γ =



14γ =



14γ =


Synchrotron Radiation Reflection from Outer Wall of Vacuum Chamber, M. Aginian, S. Arutunian, E.Lazareva, A. Margaryan, 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

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Synchrotron Radiation Reflection from Outer Wall of …candle.am/wp-content/uploads/2017/02/11.05-S.-Arutunian.pdf · Ultrafast Beams and Applications, 04-07 July 2017, CANDLE, Armenia