Parametric X-ray radiation from composite bunches

Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx

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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Parametric X-ray radiation from composite bunches A.Yu. Savchenko, A.A. Tishchenko ⇑, M.I. Ryazanov, M.N. Strikhanov National Research Nuclear University ‘‘MEPhI’’, Moscow, Russia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 November 2014 Received in revised form 19 April 2015 Accepted 19 April 2015 Available online xxxx

Theory of parametric X-ray radiation (PXR) is developed for the case of composite bunch consisting of two fractions of charged particles with different charges and distributions. We suggest PXR as an instrument for the composite bunch diagnostics, for example in case of ion beams in crystal, consisting of two fractures of different ions. Also, for atto-second electron bunches the characteristics of coherent PXR are discussed. Ó 2015 Elsevier B.V. All rights reserved.

Keywords: Parametric X-ray radiation Beam diagnostics Composite bunch Attosecond bunches

1. Introduction Parametric X-ray radiation (PXR) occurring when charged particles move in a crystal was predicted theoretically by Ter-Mikaelian in 1972 [1], and then observed experimentally at the Tomsk Synchrotron Sirius in 1985 [2]. Since then PXR has been studied very well, both theoretically and experimentally (see [3–5]). However, up to now PXR is not used in practice owing to rather weak brightness. Nevertheless, PXR is intensive enough to be detected experimentally, and can serve as a source of information about bunches of charged particles.

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k ¼ xc eðxÞn ¼ xc eðxÞ rr is the wave-vector of the radiation. Let us consider PXR from a composite bunch of ions passing a crystal, on the basis of the clear physically conception of polarization currents (see, for example, Chapter 4 in monograph [6]). For M sorts of charged particles of quantity N i with different charges Z i ; i ¼ 1; 2; . . . ; M one can obtain:

Er ðr; xÞ ¼

Ni M X p ie eikr X X vg Z i eiðkgÞðRþDri Þ eðxÞ r g–0 i¼1 p¼1 " " ( )## g þ v eðxÞ cx2  k k dð x  ðk  gÞ v Þ 2 2 ðk  gÞ  k

ð3Þ

2. Polarization currents and field of X-ray parametric radiation Using the kinematic theory of scattering of X-rays, it is not hard to obtain an expression for the electric field intensity at a distance r from the scattering of the crystal [1]:

Er ðr; xÞ ¼

ieZ eikr X v eiðkgÞR eðxÞ r g–0 g " " ( )## g þ v eðxÞ cx2  k k dð x  ðk  gÞ v Þ ; 2 2 ðk  gÞ  k

ð1Þ

2

The composite bunch is thought to be mixture of two bunches of charged particles with different distributions and different properties of the single particles they consist of, see Fig. 1. The field of radiation from two particles is

Er ðr; xÞ ¼ Z 1

X

X

g–0

g0 –0

vg eihðR1 þDr1 Þ Ag dg þ Z 2

0

vg0 eih ðR1 þDr2 Þ Ag0 dg0 ;

ð4Þ

where

where

k ¼ eðxÞ

3. Parametric X-ray radiation from composite bunches

" " (

x

2

c2

;

x2 eðxÞ ¼ 1  p2 ; x

xp  x;

⇑ Corresponding author.

ð2Þ

Ag ¼ k k

dg ¼

g þ v eðxÞ cx2 2

ðk  gÞ  k

)##

2

ie eikr dðx  ðk  gÞv Þ eðxÞ r

ð5Þ

ð6Þ

E-mail address: [email protected] (A.A. Tishchenko). http://dx.doi.org/10.1016/j.nimb.2015.04.043 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: A.Y. Savchenko et al., Parametric X-ray radiation from composite bunches, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.04.043

2

A.Yu. Savchenko et al. / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx

distinguishing two different fractures of the charged particles in a beam: first fracture of particles with charge Z 1 in proportion a, and second fracture of particles with charge Z 2 in proportion ð1  aÞ. We suppose that the distribution of particles in beam is described by the Gaussian function 2

f ri ðrÞ ¼

r2 1 2r pffiffiffiffiffiffiffi 3 e i ; ðri 2pÞ

ð11Þ

centers of the beam distributions coincide, and ri ; i ¼ 1; 2 defines the width of distribution for the first and the second fractures, correspondingly. The average value of the first term in sum in Eq. (9) can be written in form

N21 Z 21 pffiffiffiffiffiffiffi 6 ðr1 2pÞ

Z

3 d rp1

2

Z

2

Fig. 1. Bunch consists of two fractures with different charges and different distributions.

Then the distribution of radiated energy per solid angle dX and frequencies is defined by

ð8Þ

i¼1 p¼1

where Nj Ni X 2 X 2 X X f g ðkÞ ¼ Z i Z j cosððg  kÞðDrpi  Drlj ÞÞ

¼

p¼1

þ Z 22 N2 N2

cosððg 

kÞðDrp1

þ 2N1 N2 Z 1 Z 2



cosððg  kÞðDrp2  Drl2 ÞÞ

l¼1; l–p

N2 X N1 X p¼1

cosððg  kÞðDrp2  Drl1 ÞÞ;

ð9Þ

l¼1; l–p

here Drpi describes position of the p-th particle of the i-th sort, and Drlj similarly. The first term in Eq. (9) is responsible for the field of radiation produced by the particles of the first fracture of the bunch, the second term, correspondingly, describes the contribution of the second fracture, and the last term reflects their interference.

Ni  1 ’ Ni ;

Z

3 d rp2

Z

p 2 ðr Þ

 22 3 d rl1 e 2r2



e

2 ðrl Þ 1 2r2 1

cosððg  kÞðrp2  rl1 ÞÞ

! 2 ðg  kÞ ðr21 þ r22 Þ ¼ 2N1 N2 Z 1 Z 2 exp  2

ð12Þ

F incoh ¼ N½aZ 21 þ ð1  aÞZ 22 ;

ð13Þ

 2 ðgkÞ2 r2 ðgkÞ2 r2 1 2 F coh ¼ N2 aZ 1 e 2 þ ð1  aÞZ 2 e 2

ð14Þ

r2

!

1

ð15Þ

This can be achieved for very small the beam size or for a comparatively big period of periodical structure. The former is feasible for attosecond bunches [7,8], the latter needs using of artificial periodic structures with a desired period. It is interesting that the expressions obtained show that even incoherent radiation gives possibility to determine the proportion a describing fracture of the constituents of the whole bunch. Indeed, if F coh is small we can still determine characteristics of the bunch by the following expression with help of Eq. (13):

a¼

I=N  Z 22 Z 21  Z 22

ð16Þ

where the value

Let us put

N1 ¼ aN; N2 ¼ ð1  aÞN;

cosððg  kÞðDrp2  Drl1 ÞÞ

p¼1 l ¼ 1;

ðg  kÞ N exp 2

Drl1 ÞÞ

l¼1; l–p

p¼1

+

N2 X N1 X

2

N2 X N2 X

!

Coherent part of radiation becomes comparable to the incoherent part when

l¼1; l–p

N1 X N1 X



rl1 ÞÞ5

where

Ni 2 X X N Z 2i þ f g ðkÞ

Z 21 N1 N1

cosððg 

F Ng ðkÞ ¼ F incoh þ F coh

i¼1 j¼1 p¼1 l¼1

p¼1

3 kÞðrp1

Thus, Eq. (8) takes the form:

Nj Ni X 2 X 2 X X Z i Z j cosððg  kÞðDrpi  Drlj ÞÞ

j¼1; j–i

*

2N1 N2 Z 1 Z 2 ¼ pffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffi 3 ðr1 2pÞ ðr2 2pÞ

For a bunch – mixture consisting of N1 particles of 1st sort and N2 particles of the 2nd sort, it is not hard to obtain the expression similar to Eq. (7) with

i¼1

e

2 ðrl Þ 1 2r2 1

l–p

F g ðkÞ ¼ Z 21 þ Z 22 þ 2Z 1 Z 2 cosððg  kÞðDr2  Dr1 ÞÞ:

¼

r21



The third term responsible for the field produced by the charges of the different bunches after averaging takes the form:

ð7Þ

where

F Ng ðkÞ ¼

ðg  kÞ exp  2

2N1 N2 Z 1 Z 2

2

X d Wðn; xÞ ¼ cr2 jEr ðr; xÞj2 ¼ jvg Adj2 F g dxdX g–0

¼

N21 Z 21

p 2 ðr Þ

 12 3 d rl1 4e 2r1

ð10Þ

I¼



   dN dN dX bunch dX single

ð17Þ

can be measured from the peak of PXR.

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Fig. 2. Angular distribution of PXR from Eq. (20) for different

3

r and parameters N ¼ 1010 , Z 1 ¼ 10, Z 2 ¼ 20, hB ¼ 15 , c ¼ 70, jgj ¼ 109 , r2 ¼ 0:85 r1 .

We would like to stress, that incoherent part of form-factor F incoh does not coincide with Gincoh from articles [9–11]. In these papers incoherent form-factor is due to the edge effect, whereas here we deal with infinite material, i.e., there is no boundary influence, and incoherent part of form-factor arises only owing to existence of two fractions of ions in the beam.

4. Spectral-angular distribution of PXR from composite bunches As a result, the distribution of radiated spectral-angular distribution of for PXR from composite bunch we have 2

Fig. 3. Angular distribution of PXR from Eq. (20) for parameters, Z 1 ¼ 10, Z 2 ¼ 20, hB ¼ 15 , c ¼ 70, jgj ¼ 109 , r ¼ 107 ; cm.

2

d W single ðn; xÞ d Wðn; xÞ ¼ F Ng dxdX dxdX

Fig. 4. Angular distribution of PXR from Eq. (20) for N ¼ 1010 , Z 1 ¼ 10, Z 2 ¼ 20, hB ¼ 35 , c ¼ 70, jgj ¼ 109 ,

ð18Þ

r ¼ 107 ; cm.

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A.Yu. Savchenko et al. / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx

x  ðk  gÞv ¼ 0;

ð22Þ

see delta-function in Eq. (3). As x2 does not satisfy Eq. (2), we shall use only x1 :

xg1 ¼ xg 

gv 1 þ nv=c

ð23Þ

The form of angular dependence of radiation is shown in Figs. 2 and 3. It is interesting to notice that Figs. 2 and 4, which differ in the width of distribution, demonstrate different asymmetry. This is due to the fact that for rather big r coherent radiation is suppressed, see Eq. (15). We use common designation sigma instead of two different ones because for big quantities difference between r1 and r2 does not play role. 5. Discussion

Fig. 5. Bunch of charged particles moves in the target with angle hB between g and v , and radiates near the angle um .

g 2  2gkg ¼ 0;

where

 2 2  X d W single ðn; xÞ ½k½kfg þ v x=c2 g N 2 2 ¼ ce jvg j dðx  ðg  kÞv Þ  F g ðkÞ 2 2   dxdX ðg  kÞ  k g–0 ð19Þ Angular distribution of PXR from composite bunches can be easily obtained from Eq. (18):

dNðnÞ c2 aT X ¼ jv j2  dX 2p i¼1;2 g

F Ng ðkÞ

x2p 1  nv  x2 c

gi

nv 2c

  ½k½kfg þ v xgi =c2 g2  2   g 2  2gk ð20Þ

where a ¼ 1=137 – the fine-structure constant, T – the time of duration of the particle’s motion in the crystal.

xg1;2 ¼

Peaks of PXR are defined both by numerator and denominator of Eq. (20). The numerator is responsible for splitting of the peak, see Figs. 2–4, whereas the denominator is responsible for the peak position and height. The denominator in Eq. (20) is

gv 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðgvÞ2 þ 4ð1 þ nv Þx2p nv c 2c 2ð1 þ nv Þ c

xgi , i ¼ 1; 2 are roots of the equation

ð21Þ

ð24Þ

which can be written as

g2  2

ðgnÞðgvÞ ¼ 0: c þ nv

ð25Þ

The root of Eq. (25) is

um ðhB Þ ¼ arccosððv =cÞ cos 2hB Þ;

ð26Þ

see Fig. 5. It is interesting that, judging from denominator, the maximum of radiation comes to Bragg condition not exactly, but the more exactly the closer v ! c; see. Fig. 6. In the ultrarelativistic case Eq. (26) turns into well-known equality

um ¼ 2hB :

ð27Þ

We also would like to note that for a bunch of particles of one sort our results coincide with the conventional results of kinematical approach in PXR, which is in good agreement with experiment [12,13]. Along with that interference between two fractions of charged particles of a composite bunch can be used for getting

Fig. 6. Function um ðhB Þ for different Lorentz factors c.

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information about the structure and ion composition of the bunch. With help of Eq. (16) we can determine the ratio of the number of particles of one fraction to another. Acknowledgements We are grateful to Dr. A.V. Shchagin for very useful criticism. The work was supported by the Ministry of Science and Education of the Russian Federation, Grant No. 3.1110.2014/K. References [1] M.L. Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media, Wiley-Interscience, New-York, 1972. [2] S.A. Vorobiev, B.N. Kalinin, S. Pak, A.P. Potylitsyn, Detection of monochromatic X-ray radiation at interaction of ultrarelativistic electrons with diamond single crystal, JETP Lett. 41 (1985) 3. [3] A.V. Shchagin, X.K. Maruyama, Parametric X-rays, in: S.M. Shafroth, J.C. Austin (Eds.), Accelerator-Based Atomic Physics Techniques and Applications, AIP Press, New York, 1997, pp. 279–307.

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[4] V.G. Baryshevsky, I.D. Feranchuk, A.P. Ulyanenkov, Parametric X-Ray Radiation in Crystals, Springer, Heidelberg, 2005. [5] A.P. Potylitsyn, Electromagnetic Radiation of Electrons in Periodic Structures, Springer, Berlin, 2011. [6] A.P. Potylitsyn, M.I. Ryazanov, M.N. Strikhanov, A.A. Tishchenko, Diffraction Radiation from Relativistic Particles, Springer, Heidelberg, 2011. [7] M.J.H. Luttikhof, A.G. Khachatryan, F.A. van Goor, K.-J. Boller, Generating ultrarelativistic attosecond electron bunches with laser Wakefield accelerators, Phys. Rev. Lett. 105 (2010) 124801. [8] A. Sell, F.X. Kärtner, Attosecond electron bunches accelerated and compressed by radially polarized laser pulses and soft-X-ray pulses from optical undulators, J. Phys. B At. Mol. Opt. Phys. 47 (2014) 015601. [9] D.Yu. Sergeeva, A.A. Tishchenko, M.N. Strikhanov, UV and X-ray diffraction and transition radiation from charged particles bunches, Nucl. Instr. Meth. B 309 (2013) 189. [10] D.Yu. Sergeeva, M.N. Strikhanov, A.A. Tishchenko, UV and X-ray diffraction radiation for submicron noninvasive diagnostics, Proc. IPAC 2013 (MOPME062) (2013) 616–619. [11] D.Yu. Sergeeva, A.A. Tishchenko, X-ray Smith–Purcell radiation from a beam skimming a grating surface, Proc. FEL2014 TUPO13 (2015) 378–383 . [12] A.V. Shchagin, Current status of parametric X-ray radiation research, Radiat. Phys. Chem. 61 (2001) 283. [13] Y. Takabayashi, A.V. Shchagin, Observation of parametric X-ray radiation by an imaging plate, NIMB 278 (2012) 78.

Please cite this article in press as: A.Y. Savchenko et al., Parametric X-ray radiation from composite bunches, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.04.043