1

On the spectrum of relativistic electrons’ beam passing through periodical medium

Minsk, 2016

N. Shul’ga, E. Bulyak National Science Center “Kharkov Institute of Physics and Technology”

Kharkov, Ukraine e-mail: shulga@kipt.kharkov.ua

• coherent radiation in crystals

• relativistic electrons’ spectrum in undulator

• destruction of coherent effect in undulator radiation

2

Some directions of our works

2

• Coherent Bremsstrahlung and scattering in crystals at high energies

• Electromagnetic processes with “half-bare” electrons

• Landau-Pomeranchuk-Migdal effect in radiation

• Beam-beam radiation

• Eikonal and WKB approximation

• Dynamical chaos phenomenon

• Electromagnetic showers in crystals

• Beam deflection by bent crystals

• etc.

Coherent length (Ter-Mikaelian, 1953)

22 '

cl mεε

ω=

R

min

2 22|| 2

||q

q

qd d q dq Uq

σ∞

⊥⊥≈ ∫ ∫

12|| ||

2 'eff eff cr q l m

εεω

−≈ ≈ =

1eff

effr Rq⊥

⊥≈ ≈

,ε ε ω= + = + +′ ′p p k q

100Mevε = 100kevω = 410cl cm−≈

100Mevε = 1cmλ ≈ 310 !!!cl m≈

22

cl mεε

ω′=

( ) ( )( )22

24i t kr tdE e k dt t e

d d

∞−

−∞

= × ∫ ω

υω ο π

( ) ~ 1t kr tϕ ω∆ = ∆ − ∆ <

2

2 2 2 2

2 1~1 t

t γω γ θ γ θ∆

∆+ +

2 2 2

2 2 2

2 1~

2 1

t

t

tγ ω γ θ

γ ω γ θ

∆

∆

<<∆ << >>

θ

1 iA e ϕ− ∆≈ +

1A ≈

0t =

( ) ( ) ( )20

1t 1 v t2

v v vt⊥ ⊥ ≈ − +

⋅

Coherent length (Landau-Pomeranchuk, 1953)

Coherent length (Ter-Mikaelian 1953 – crystal, Landau-Pomeranchuk 1953 – amorphous media)

Landau was agreed that Ter-Mikaelian’s results were correct, but he said that it is needed to use another way for describing this effect

Coherence + Interference Coherent effect

Coherent Bremsstrahlung in Born Approximation Ferretti 1950, Ter-Mikaelian 1952, Überall 1960

2 22 2 2 22 2

|| ||||

2 1 2 12

g ug

g

d e g U ed g gm g

σ δ ε ω δ δωω ε εε

−⊥ ′

= + − − ′∆ ∑

( )2|| ||/ 2 , cos sinz y xq m g g g gδ ω εε ψ α α δ′≥ = = + + ≥

Frascati ε=1 GeV, θ =4,6 mrad

DESY ε=4,8 GeV, θ =3,4 mrad

Experiment (1962 - 1965) ~1 5 GeVε −

Frascati, DESY,Kharkov, Protvino, Tomsk, Yerevan, SLAC, …

Generalization of CB theory

The main idea: -For

-The relative contribution of higher Born approximation can be also increased (A.Akhiezer, P.Fomin, N.Shul’ga 1971)

coh atomd dσ σ>>

Second Born approximation in CB theory A.Akhiezer, P.Fomin, N.Shul’ga (1970)

2

2 ,1Bornc coh

cd d= ⋅

±

θηθ

σ σ ω ε

1η θ − c crytical channelling angle

Higher Born Approximation in the CB Theory A.Akhiezer, N.Shul’ga (1975)

min ,ψ

cohcoh

a

l RNa

2 2 2

1 1ψ

→

cohZe R ZeN

c ae

cZ

c

PARADOX

This condition did not fulfill practically for experiments (1960-1970) on verification of F – T – Ü theoretical results.

But the experiments were in good agreement with this theory !!!

Why ???

0Quickly destroys for ψ →

Eikonal, Semiclassical, Classical CB Theory

Semiclassical approximation

2 2

1cN Ze R Zec a cψ

=

!!!

( ) ( ) σ σ=WKBcld d r t

Classical Electrodynamics

2

1,cZeN

cεω

• Radiation is determined by the classical trajectory !!! • It is necessary to know the types of particles’ motion in crystal • Same methods for description of CB and LPM effects !!!

New area of research

The interaction of high-energy particles with matter in conditions of effectively strong interaction of the particle with atoms of media (semiclassical, classical approximations)

2

1cZeN

c>>

Problems generated by the theory of coherent radiation in crystals

14

Planar Channeling Lindhard (1965)

This image cannot currently be displayed.

x y

p

z

( ) ( )1p n

ny z

U x dydz u r rL L

= −∑∫

( )1px U x

E x∂

= −∂

( )2

2p pE x U x⊥

= +

ε

z

x x

U(x)

maxp Uε⊥ <

maxp Uε⊥ >

2

max

max

22

pp

p

EU

UE

θε

θ

⊥

= =

=

pz=const≈p

• Above-barrier motion (Akhiezer, Shul’ga 1978)

Coherent and Channeling Radiation A. Akhiezer, N. Shul’ga (1978)

d

0 z

x

∫ ∑ −=n

nzy

p rrudydzLL

xU )(1)(

xxUex p

∂

∂−=

)(ε

Radiation at planar channeling

2/32max ~2 εγω oscΩ=

Channeling radiation in periodically deformed crystal plane

V. Boldyshev (1982) L. Grigorian, A. Mkrtchyan

et al (2001)

A. Korol, A. Solov’yov, W. Greiner (2004)

V. Biryukov et al (2006)

Radiation in the field of periodically deformed crystal planes of atoms at canalling

λ<<<< ad cmd 810~ − da 210..10~ λ45 10..10~ −−a

,

,

,

1.L.Sh.Grigoryan, A.R.Mkrtchyan, A.H.Mkrtchyan et al, Nucl. Instr. and Meth. in Physics Research B 173, 132 (2001). 2.A.V.Korol, A.V.Solov`yov and W.Greiner, Int. J. Mod. Phys. E 13, 867 (2004).

19

Coherence of undulator radiation for high energy electrons

E. Bulyak, N. Shul’ga

arxiv:1506.03255v2[physics.acc-ph] 11 Sep. 2015

(submitted to Phys. Rev.)

20

Undulator Radiation

( ) ( ) ||,t T t ⊥+ >> v = v v v

+ + +

+ + + +- - - -

- - - -x

( )( ) ||

sin 2mt T

z t t

πΩ Ω

≈

x = x t =

v

( ) 2

222

4 22

0 2 0

2

2

0

sin

sin

fo 2

~

r

D

D

Ti t

D D

NdE e d

dE N

td

d d

t e ω γ

θ

θ

ωπ

ω

ωπ

ω

ω

γω π

ω ω ω γ

ο =

⊥=

≈

≈ = Ω

∫ v

21

The Problem

2

0

~dE Nd d θω ο =

When the dependence is destroyed?

( ) ( ) ( )00

0, , , 1f d f zε δ ε ε ε ε∞

= − =∫

0ε

( ) ( )00,f ε δ ε ε= − ( ),f z ε

22

The Kinetic Equation Method

( ) ( ) ( ) ( )0

, , , ,d f z d w f z f zdz

ε ω ε ω ε ω ε∞

= + − ∫

Diffusion approximation ω ε<<

( ) ( ) ( )00

1, exp , 12

ipf z dp ip z d w e ωε ε ε ω ε ωπ

∞ = − − −

∫ ∫

( ) ( )20

22

1, exp22

zf z

zz

ε ε ωε

ωπ ω

− − ≈ −

( ) ( )2 2

0 0

, , , .d w d wω ω ω ε ω ω ω ω ε ω∞ ∞

= =∫ ∫

0ε ε

23

24

25

Poisson Distribution

- the mean number of photons

2~ 2effω ω γ∗ = Ω

( ) ( ) ( ) ( )2

00

,1 1, exp 12

ipd Ef z dp ip z d e

d dzωε ω

ε ε ε ωπ ω ω

∞ = − − −

∫ ∫

( ) ( )01 exp 1

2ipEdp ip e ωε ε

π ω∗

∗

∆ = − − − = ∫

Eξω∗

∆=

( ) ( )00

0

exp2 2 !

n inpipip

n

dp dp eip e en

ωε ε ξω ξε ε ξ ξ

π π

∗∗

∞− −

=

= − − + ≈ ⋅∑∫ ∫

( ) ( ) ( )00

, nn

f z f nε ξ δ ε ω ε∞

∗

=

= + −∑ ( ) ( ) 1,!

nnf z e

nξξ ε ξ −=

26

Coherent effect in undulator radiation

( ) ( )( ) ( ) 22

322

24 22

0 00

t tLL L i t tmdE e dt dt t t e e

d d

ωωωγγ

ϑ

γω ο π

′ −−′−

⊥ ⊥

=

′ ′≈ ⋅∫ ∫ v v

22

3 12

L Lm

ω ωγ

<<

( ) 2

222

4 22

2 00

sin

sin

LD i t

D

NdE e dt t e

d dω γ

ϑ

ωπω

γω ο πωπ

ω

⊥=

≈

∫v

2

0

~ for DdE N

d d ϑ

ω ωω ο

=

≈

27

Destruction of coherent effect in UR

maxwhere ,Dζ ω ω κ γϑ= =

22

3 ~ 12

L Lm

ω ωγ

>

( ) ( ) ( )2 2

20

1, 1 2 2 1ewTπ κε ω ζ ζ ζ ζ

ω= − + Θ −

2 2 23 27 16 ~ 1

60e N

mTπ κ π γ

6For 1, 1cm, 10Tκ γ= = =

240N ≈

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