yu.p. kunashenko 1,2 1 tomsk polytechnic university 2 tomsk state pedagogical university coherent...

Yu.P. Kunashenko1,2

1Tomsk Polytechnic University2Tomsk State Pedagogical University

CoherentBremsstrahlung from

planar channeled positron

«RREPS-13» and ''Meghri-13''

Coherent bremsstrahlung and coherent electron-positron pair photoproduction

was predicted by

Ter-Mikaelian High Energy Electromagnetic processes in Condensed Media, Wiley Interscience, New-York, 1972.

Uberall H. High-energy interference effect of bremsstrahlung and pair production in crystals. // Physical review volume 103, number 4. 1956. p.1055-1067.

Now coherent effects are divided into two types: A and B Saenz A.W. and Uberall H., Theory of coherent bremsstrahlung. -

in: Coherent Radiation Sourses, eds. A.W. Saenz and H. Uberall. - Berlin: Springer-Verlag.- 1985.- p.5-32.]


Review of experimental and theoretical work on coherent type

A bremsstrahlung and coherent electron-positron pair photoproduction in crystals was given by Diambrini - Palazzi G . // Rev.Mod.Phys., 1968, v.10, p.611.

Combined effect arise when channeled particle emits coherent bremsstrahlung and manifest it self in changing of coherent peaks

For the first time combined effect in coherent type B bremsstrahlung was observed in Tomsk

Amosov K.Yu., Vnukov I.E., Naumtnko G.F., Potulisin A.P., Saruchev V.P. Pis’ma v ZhETP 1992 v. 55 p. 587

.


Discreet axis

Planar continuous potential

Positron moves through crystal in the regime of planar channeling.Longitudinal positron momentum is parallel to crystal axis.


Photon scattering by channeled positron

;2

,12

3

3

||32

k

pd

dddJ

EEMdffififi

n ni

nifn

ni

nifnfi EEEE

M

ReAeAeRe 21

1

12

We use virtual photon methods therefore we need cross-section of Photon scattering by channeled positron. It is convenient to consider scattering of photons by channeled positron in a coordinate system moving with the longitudinal velositi of channeled positron. In the moving reference frame one can use non relativistic approach to the scattering process.

Cross - section

matrix element (n is intermediate state)

J initial positron flux



rrrkprR 2 dich

mc

elkkl exp

2ˆ

2

22

rrrkprA 1 dich

mc

elkkl exp

2ˆ

1

22

matrix element of the photon absorption

matrix element of photon emission


krr ich

ph exp2 22

||||exp rkr fififi ix

02

1 2

xxeVp

m fixfi 2

204

xd

VxeV

xHxCx nnn

2exp

22

!24/1

2/1

nC

nn

md

Vm 022

,

2

1nn

Photon wave function

Channeled positron wave function

Channeled positron transverse wave function



.

2

2

11

2

21

1

2

1

2

22

22

ddd

EE

EEEEcmd

f

fifi

nni

nifn

ni

nifn

||

||||||||

1

k

kkkk

ReAeAeRe


0

21

1

i

fi

fi

EE

||

||||||||

k

kkkk

From the δ function it is follows conservation laws:

In the initial state i positron at rest


m

kkc

k

ck

c

yfzffi

yf

zf

2

sin0

cos

2,||

2,||

21

,||

2,||

1


If the incident photon moves along the OZ axis then its wave vector is defined as follows:

ckk /,,0,0 k

the wave vector of a scattered photon

ckkkk /,cos,sinsin,cossin k

2

42

2

111

22

12

2

12

2

sin42cos41

cos

ctgcm

mcmcmc

mc

fi


Connected between scattered and initial photon energies

fi 12

(“(“Compton Compton formulaformula”)”)

d

dfi



The cross section of the coherent bremsstrahlung from channeled positron

.exp)(,|)(|)( 11

1

R

r

r

ZerVrrVrV

N

ii

electrostatic potential of the crystal axis

dSgku

cB

du

cBL

NeZ

dn

ngn

2

1

22

2

22

exp2

exp

)(

The spectrum of virtual photons of crystal axis

1,

/ln

22

2

aRc

aL

22

2

,1)()1()( uRc

xxEiexxB x

Here d is the lattice constant, ω is the frequency of virtual photon

/mc is Compton wave length

2u is the mean-square displacement of a crystal atom from equilibrium position

R is the screening radius, N is the number of atoms in the axis, Z is atomic number.



If we neglect the terms c in the functions L and B

than we obtain the virtual photons spectrum from well – known

Ter- Mikaelian book.

dngn

/2

Ei(-x) is the exponential integral function,

is the 1D reciprocal lattice vector,

S is the structure factor of a crystal axis, γ is the relativistic factor.


ddr

rfi,


For transform the photons scattering cross-section into the laboratory coordinate system we use the “Compton “Compton formula”formula” and the Lorentz transformation for the

photon energy (the Doppler shift): cos1 r

is the energy of emitted photon

is the energy of virtual photon

r



MAX

MIN

dnddd

r

rfi

r

r

CR

fi

)(,

The limits of integration are determined by the condition 1cos1

1

fir

MIN

1

1fir

MAX

Results of calculationResults of calculation

fir

r

CR

fi

d

r

rTM

d

0 .0 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 70

5 . 1 0 2 8

1 . 1 0 2 7

1 .5 1 0 2 7

2 . 1 0 2 7

2 .5 1 0 2 7

, M e V

st

,cm

2

W (100) planeW (220) axisNs=104

2


0 .0 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7

5 . 1 0 2 8

1 . 1 0 2 7

1 .5 1 0 2 7

2 . 1 0 2 7

2 .5 1 0 2 7

3 . 1 0 2 7

3 .5 1 0 2 7

, M e V

,

cm2

fir

r

CR

fi

r

rTM

dNd

1

Results of calculationResults of calculation



Conclusion

It is possible the coherent bremsstrahlung from planar channeled positron

The contribution to cross-section of the coherent bremsstrahlung from channeled positron result in splitting of coherent peek and appearance of fine structure of photon spectrum

yu.p. kunashenko 1,2 1 tomsk polytechnic university 2 tomsk state pedagogical university coherent...

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