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Spectrum and polarization of coherent and incoherent radiation and the LPM effect in oriented single crystal
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NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 3828–3834 www.elsevier.com/locate/nimb
Spectrum and polarization of coherent and incoherent radiation and the LPM effect in oriented single crystal V.N. Baier *, V.M. Katkov Budker Institute of Nuclear Physics, 11, Lavrenteiv Avenue, Novosibirsk 630090, Russia Received 31 October 2007; received in revised form 24 January 2008 Available online 1 February 2008
Abstract The spectrum and the circular polarization of radiation from longitudinally polarized high-energy electrons in an oriented single crystal are considered using the method which permits inseparable consideration of both the coherent and the incoherent mechanisms of photon emission. The spectral and polarization properties of radiation are obtained and analyzed. It is found that in some part of the spectral distribution the influence of multiple scattering (the Landau–Pomeranchuk–Migdal (LPM) effect) attains the order of 7%. The same is true for the influence of multiple scattering on the polarization part of the radiation intensity. The degree of the circular polarization of the total intensity of radiation is found. It is shown that the influence of multiple scattering on the photon polarization is similar to the influence of the LPM effect on the total intensity of radiation: it appears only for relatively low energies of radiating electron and has the order of 1%, while at higher energies the crystal field action excludes the LPM effect. Ó 2008 Elsevier B.V. All rights reserved. PACS: 78.70.g Keywords: Crystal; Radiation; Coherent; Incoherent
1. Introduction The study of processes with participation of polarized electrons and photons permits to obtain the important physical information. Because of this reason the experiments with use of polarized particles are performed and are planning in many laboratories (CERN, Jefferson Nat Accl Fac, SLAC, BINP, etc). In this paper the polarization effects are considered in the frame of the general theory developed by authors [1], which includes both the coherent and the incoherent mechanisms of radiation from highenergy electrons in an oriented single crystal. The influence of multiple scattering on the radiation process including polarization effects is analyzed. The study of radiation in oriented crystals is continuing and new experiments are performed recently see [2,3]. *
Corresponding author. Tel.: +7 383 329 4785; fax: +7 383 330 7163. E-mail address: [email protected] (V.N. Baier).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.01.046
The general expression for the energy loss of the longitudinally polarized electron in oriented crystal was found in [4] (see Eq. (2.7)) Z 3 am2 d3 k dr dEn ¼ 2 0 F ðr; #0 Þ V 8p ee Z x 1 e e0 iA ð1 þ nÞ 0 þ ð1 nÞ e 1þn þ e 4 e e 2 c2 ðv1 v2 Þ dt1 dt2 ; Z xe t2 1 2 A¼ 0 þ ðn vðtÞÞ dt: ð1Þ 2e t1 c2 where dEn ¼ xdwn , dwn is the probability of radiation, see e.g. Eq. (4.2) in [5], x and e are the photon and electron energy, e0 ¼ e x; c ¼ e=m, a ¼ e2 ¼ 1=137, the vector k is the photon momentum, n ¼ k= j k j,n ¼ kf; k ¼ 1 is the helicity of emitted photon, f ¼ 1 is the helicity of the initial electron, F ðr; #0 Þ is the distribution function of
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
electron in the transverse phase space depending on the angle of incidence #0 of the electron on crystal, V is the volume of a crystal, v1 ¼ vðt1 Þ is the electron velocity (see [5], Sec. 16.2). The degree of the circular polarization of radiation is defined by Stoke’s parameter n2 : n2 ¼ KðfvÞ;
K¼
dEþ dE ; dEþ þ dE
ð2Þ
where the quantity ðfvÞ defines the longitudinal polarization of the initial electrons, dEþ and dE is the energy loss for n ¼ þ1 and n ¼ 1 correspondingly. It should be noted that a few different spin correlations are known in an external field. But after averaging over directions of the crystal field only the considered here the longitudinal polarization survives. In [4] the polarization effects in the coherent radiation, which dominates at high electron energies (e 1 GeV for main axes of heavy elements, e.g. tungsten crystal), was studied. At intermediate energies the incoherent radiation contributes essentially and the contributions of both mechanisms should be taken into account. Recently authors developed the method which permits an indivisible consideration of both the coherent and the incoherent mechanisms of photon emission in oriented crystals [1]. Basing on Eqs. (18) and (19) of [1] (see also Eqs. (7.89) and (7.90) in [5]) and using Eq. (1) one can obtain the general expression for the intensity of radiation from the longitudinally polarized electrons which includes the coherent and incoherent contributions and the Landau–Pomeranchuk–Migdal (LPM) effect
4v2 ðxÞ ; u2 m20
y¼
x ; e
u¼
y ; 1y
ð3Þ
where 2
r2 ¼ 1 þ ð1 yÞ ; r3 ¼ 2ð1 yÞ;
r21 ¼ 2y y
where x0 ¼
1 ; pdna a2s
g1 ¼
2u21 ; a2s
x¼
.2 : a2s
ð6Þ
Here . is the distance from the axis, u1 is the amplitude of thermal vibration, d is the mean distance between atoms forming the axis, as is the effective screening radius of the potential. The parameters in Eq. (5) were determined by means of fitting procedure. The local value of the parameter vðxÞ which determines the radiation probability in the field Eq. (5) is pffiffiffi dU ð.Þ e 2 x vðxÞ ¼ ; ¼ vs d. m3 ðx þ gÞðx þ g þ 1Þ V 0e e vs ¼ 3 : ð7Þ m as e s For an axial orientation of crystal the ratio of the atom density nð.Þ in the vicinity of an axis to the mean atom density na is (see [1]) nðxÞ x0 ¼ nðxÞ ¼ ex=g1 ; na g1
e0 ¼
ee ; nð0Þ
ee ¼
m : 16pZ 2 a2 k3c na L0 ð8Þ
ee ðna Þ e0 ¼ ex=g1 ; nðxÞgðx; e; xÞ gðx; e; xÞ 1 L0 ¼ lnðmaÞ þ f ðZaÞ; 2 1 v2 ðxÞ 6Dr v2 ðxÞ gðx; e; xÞ ¼ g0 þ lnð1 þ 2 Þ þ ; 6L0 u 12u2 þ v2 ðxÞ 2 1 1 u 111Z 1=3 h 12 ; a ¼ ; g0 ¼ 1 þ L0 18 a m 1 X n2 f ðnÞ ¼ ; 2 2 n¼1 nðn þ n Þ 1 5 hðzÞ ¼ ½1 þ ð1 þ zÞez EiðzÞ; Dr ¼ Dsc 2 9 ec ðxÞ ¼
u1 ðtÞ ¼ ði 1Þt þ br ð1 þ iÞðu2 ðtÞ tÞ; pffiffiffi pffiffiffi m0 t 2 2m 0 pffiffiffi tanh pffiffiffi ; u3 ðtÞ ¼ u2 ðtÞ ¼ ; m0 2 sinhð 2m0 tÞ
1 r2n ¼ ðr2 þ nr21 Þ; 2 1 r3n ¼ ðr3 þ nr31 Þ; 2 1y e 2 ; m0 ¼ y ec ðxÞ
The situation is considered where the electron angle of incidence #0 (the angle between electron momentum p and the axis (or plane)) is small #0 V 0 =m. The axis potential (see Eq. (9.13) in [5]) is taken in the form 1 1 U ðxÞ ¼ V 0 ln 1 þ ln 1 þ ; ð5Þ xþg x0 þ g
The functions and values in Eqs. (3) and (4) are
dI n ðe; yÞ ¼ dI 0 ðe; yÞ þ ndI 1 ðe; yÞ Z x0 am2 ydy dx Grn ðx; yÞ; ¼ 2p 1 y 0 x0 Z 1 p Grn ðx; yÞ ¼ F rn ðx; y; tÞdt r3n ; 4 0 u ðtÞ 2 F rn ðx; y; tÞ ¼ Im e 1 ½r2n m0 ð1 þ ibr Þu2 ðtÞ þ r3n u3 ðtÞ ; br ¼
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2
r31 ¼ 2yð1 yÞ;
ð4Þ The intensity for the unpolarized electrons dI 0 ðe; yÞ was obtained in [1], the polarization term dI 1 ðe; yÞ is found here.
ð9Þ
where the function gðx; e; xÞ determines the effective logarithm using the interpolation procedure: L ¼ L0 gðx; e; xÞ, the function g0 is defined in Eq. (8) in [1], Dsc ¼ 2:30 is the constant entering in the radiation spectrum at v=u 1, see Eq. (7.107) in [5], Ei(z) is the integral exponential function, f ðnÞ is the Coulomb correction. It follows from Eqs. (2) and (3) that the circular polarization of radiation is n2 ¼
dI 1 ðe; yÞ ðfvÞ; dI 0 ðe; yÞ
ð10Þ
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2. The spectral distribution of radiation The radiation of a particle moving in an external field, if particle scattering does not taken into account, is called the coherent radiation. The incoherent radiation appears because of scattering of particle on separate atoms. Its intensity is proportional to the density of atoms in a medium (/ m20 ). Nonlinear intensity dependence on the density appears in consequence of multiple scattering of the radiating electron on the radiation formation length (the LPM effect). The expression for dI n Eq. (3) includes both the coherent and incoherent contributions as well as the influence of the multiple scattering (the LPM effect) on the photon emission process. The intensity of the coherent radiation dI coh 0 ðe; yÞ is the first term (m20 ¼ 0) of the decomposition of Eq. (3) over m20 . This intensity is contained in Eq. (17.7) of [5]. The polarization term in the intensity of the coherent radiation dI coh 1 ðe; yÞ is the first term of the decomposition of dI 1 in coh Eq. (3) over m20 . The expression dI coh 0 þ ndI 1 coincides with the term containing R0 ðkÞ in Eq. (3.5) of [4]. The intensity of the incoherent radiation dI inc 0 ðe; yÞ is the second term (/ m20 ) of the mentioned decomposition of dIðe; yÞ [1]. The expression for dI inc 0 ðe; yÞ follows also from
Fig. 1. The spectral distribution of radiation in tungsten, axis h1 1 1i, temperature T=100 K (solid lines) and T=293 K (dashed lines), with taking into account all mechanisms of photon emission. The spectral inverse radiation length (in cm1 ) dI 0 ðe; yÞ=edy, see Eq. (3), is shown versus y ¼ x=e for different energies: the curve 1 is for e ¼ 0:3 GeV, the curve 2 is for e ¼ 1 GeV, the curve 3 is for e ¼ 3 GeV, the curve 4 is for e ¼ 1 GeV, the curve 5 is for e ¼ 5 GeV and the curve 6 is for e ¼ 10 GeV.
Eq. (21.21) in ([5]). The polarization term dI inc 1 ðe; yÞ is correspondingly the second term (/ m20 ) of decomposition of dI 1 ðe; yÞ: Z x0 am2 e dx inc dI 0;1 ðe; yÞ ¼ gðx; e; xÞex=g1 dJ inc ; ð11Þ 0;1 ðv; yÞ x0 60p e0 0 here v ¼ vðxÞ, the notations is given in Eqs. (7)–(9), dJ inc 0;1 ðvÞ can be written as 2 dJ inc 0 ðv; yÞ ¼ ½y ðf1 ðzÞ þ f2 ðzÞÞ þ 2ð1 yÞf2 ðzÞdy; 2 dJ inc 1 ðv; yÞ ¼ ½y ðf1 ðzÞ f2 ðzÞÞ þ 2yf2 ðzÞdy; 2=3 y ; z¼ vð1 yÞ
ð12Þ
the functions f1 ðzÞ and f2 ðzÞ are defined in the just mentioned equation in [5]: f1 ðzÞ ¼ z4 !ðzÞ 3z2 !0 ðzÞ z3 ;
ð13Þ
f2 ðzÞ ¼ ðz4 þ 3zÞ!ðzÞ 5z2 !0 ðzÞ z3 ; here !ðzÞ is the Hardy function: Z 1 s3 sin zs þ !ðzÞ ¼ ds: 3 0
ð14Þ
For intermediate energies, where both the coherent and the incoherent contributions to the total intensity of radiation are essential, the spectral distribution of intensity dI 0 ðe; yÞ is shown in Fig. 1. The calculation was done for the axis h1 1 1i of tungsten at temperature T¼100 K and T¼293 K (the parameters of crystal are given in Table 1), et is the energy for which the coherent intensity becomes equal to the Bethe–Maximon one. These spectra describe radiation in thin targets when one can neglect the energy loss of projectile. It is seen that even for a moderate electron energy the photon spectrum spreads into the hard region, e.g. for T¼100 K and e ¼ 3 GeV the radiation intensity for y ¼ 0:2ðx ¼ 600 MeV) is still half of maximal one. The hard end of all spectral curves tends to the same value corresponding to the Bethe–Maximon intensity. For the energy e ¼1 GeV the spectrum at T¼100 K is harder than at T¼293 K. The next terms of decomposition of the intensity dI 0 ðe; yÞ over m20 describe the influence of multiple scattering on the radiation process, the LPM effect. The different contributions to that part of the spectrum, where the coherent and the incoherent contributions are comparable, are shown in Fig. 2. It is seen that for e ¼ 3 GeV, dI coh ’ 0 dI inc at y ’ 0:54ðx ’ 1:6 GeVÞ while for lower photon 0 energy the coherent contribution dominates and for higher photon energy the incoherent contribution dominates.
Table 1 Parameters of radiation process of the tungsten crystal, axis h1 1 1i and the germanium crystal, axis h1 1 0i for two temperatures T Crystal
T (K)
V 0 (eV)
x0
g1
g
e0 (GeV)
et (GeV)
es (GeV)
h
W W Ge Ge
293 100 293 100
417 355 110 114.5
39.7 35.7 15.5 19.8
0.108 0.0401 0.125 0.064
0.115 0.0313 0.119 0.0633
7.43 3.06 148 59
0.76 0.35 1.29 0.85
34.8 43.1 210 179
0.348 0.612 0.235 0.459
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
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All the curves in Fig. 3 have nearly the same height of the maximum and the position of the maximum is defined roughly by the expression um ’ 3e=e0 ðu y=ð1 yÞÞ. Such scaling in terms of u is the consequence of the following representation of the spectral inverse radiation length (the intensity spectrum Eq. (3)) e
e
dL1 1 dI 0 ðe; yÞ rad ¼ r2 ðyÞR2 ¼ þ r3 ðyÞR3 ð16Þ e dy u u dy
Fig. 2. The different contributions to the photon spectrum for electron energy e ¼ 3 GeV, axis h1 1 1i, temperature T¼100 K. The curve 1 is dI 0 ðe; yÞ, the curve 2 is the coherent spectrum dI coh 0 ðe; yÞ, the curve 3 is the coh incoherent spectrum dI inc 0 ðe; yÞ, the curve 4 is the sum dI 0 ðe; yÞþ inc coh dI 0 ðe; yÞ and the curve 5 is the difference ðdI 0 ðe; yÞ þ dI inc 0 ðe; yÞ dI 0 ðe; yÞÞ 10.
Calculation shows that for e ¼ 0:3 GeV, dI coh ’ dI inc at 0 0 coh y ’ 0:1ðx ’ 30 MeVÞ. For e ¼ 1 GeV, dI 0 ’ dI inc at 0 y ’ 0:28ðx ’ 280 MeVÞ. The difference shown by curve 5 arises due to the LPM effect. We define the contribution of the LPM effect into spectral distribution, by analogy with [1], as Ds ¼
inc dI 0 dI coh 0 dI 0 dI 0
ð15Þ
The function Ds ðyÞ is shown in Fig. 3. The curve 1 for e ¼ 0:3 GeV reaches the maximum 6.64% at y¼0.18, the curve 2 for e ¼ 1 GeV reaches the maximum 6.87% at y¼0.44 and the curve 3 for e ¼ 3 GeV reaches the maximum 7.32% at y¼0.7.
In the high-energy limit e e0 the maximum of the LPM effect is situated at the very end of the spectrum. In this limit r2 ’ 1 Oðe0 =eÞ and r3 ’ Oðe0 =eÞ and the scaling (the dependence on the combination e=u only) of each of the two R2;3 terms gets over into scaling of the whole expression for the spectral inverse radiation length. In the maximum of the LPM effect the coherent contribution into spectral radiation intensity is relatively small: less than 10%. Therefore the right slope of curves in Fig. 3 where the influence of the field still smaller is described by formulas of the LPM effect in a medium with small corrections due to action of the crystal field. Far from the maximum at u e=e0 one has Rcoh 2;3 ¼ 0 and the terms / m60 in the decomposition of the functions R2;3 (which includes the crystal field corrections) have the form ! Rinc e2 g20 v2 2 R2 ¼ 2 2 1 þ 377 2 ; u 3e0 u Rinc 2 ðv ¼ 0Þ ! Rinc e2 g20 31 2704 v2 3 R3 þ ¼ 2 2 ; 15 u2 3e0 u 63 Rinc 2 ðv ¼ 0Þ Z 1 Rinc 1 dx 3 ðv ¼ 0Þ 2 v2 ðxÞe3x=g1 ð17Þ ¼ ; v ¼ inc 3 g1 R2 ðv ¼ 0Þ 0 Here the terms independent on field coincide with the corresponding terms in Eq. (3.6) in [6], the corrections depending on crystal field are calculated in this paper. At the left slope of the curves in Fig. 3 the coherent contribution dominates (see Fig. 1), the relative contribution of incoherent radiation diminishes and the LPM effect is only its small part. The degree of the circular polarization of radiation Eq. (10) can be presented in a form (see Eqs. (3) and (16)) dI 1 ðe; yÞ r21 R2 þ r31 R3 ¼ dI 0 ðe; yÞ r 2 R2 þ r 3 R3 2 yð2 yÞ yð1 yÞ ¼ þ 2ð1 yÞ þ y 2 2ð1 yÞ þ y 2 1 y2 s 2R3 s 1þ ; s¼ : 2 2ð1 yÞ þ y 2 R 2 þ R3
K¼
Fig. 3. The LPM effect for spectral distribution of radiation in tungsten, axis h1 1 1i, temperature T¼100 K. The function Ds ðyÞ Eq. (15) (percent) is shown versus y ¼ x=e. The curve 1 is for e ¼ 0:3 GeV, the curve 2 is for e ¼ 1 GeV and the curve 3 is for e ¼ 3 GeV.
ð18Þ
Here the first term on the right-hand side dominates, it has an universal form. The first factor in the second term is very psmall numerically (its maximal value is ffiffiffi ð3 2 2Þ=4 ¼ 0:0429). The energy dependence of this term is defined by the rest factors which are smaller than sðeÞ > 0. In the region of low electron energy, where the incoherent radiation dominates, the value s ’ 1 and
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weakly depends on energy. In the high-energy region where the coherent radiation dominates one has s 1. In the intermediate region the dependence of the value s on energy is weak numerically. So the dependence of the value K on energy is very weak always. The value of K is shown in Fig. 4. In accord with the performed analysis, the curves for energies e ¼ 0:3; 1; 3 GeV coincide with each other inside the thickness of line. For any mechanism of radiation n2 ’ yðfvÞ for y 1 (see Eq. (2.9)) in [4] and n2 ! 1 for y ! ðfvÞ as a consequence of the helicity transfer from an electron to photon. The next terms of decomposition of the intensity dI 1 ðe; yÞ over m20 describe the influence of multiple scattering on the polarization part of the spectral intensity of radiation. We define the contribution of this effect into the polarization part as Ds1 ¼
inc dI 1 dI coh 1 dI 1 dI 1
ð19Þ
The function Ds1 ðyÞ for e ¼ 0:3 GeV reaches the maximum 6.78% at y¼0.18, for e ¼ 1 GeV reaches the maximum 7.09% at y¼0.44 and for e ¼ 3 GeV reaches the maximum 7.43% at y¼0.7. It is seen that the maximum positions are situated at the same photon energy as in Ds ðyÞ (see Fig. 3) and their values are very close to these values in the unpolarized part. All this means that the multiple scattering is affecting similarly on the unpolarized spectrum described by dI 0 ðe; yÞ and the polarization term described by dI 1 ðe; yÞ. The influence of the multiple scattering on the photon polarization degree may be also characterized by Dsn ¼ ncis2 ¼
nTs2 ncis2 ¼ Ds1 Ds ; ncis2
inc dI coh 1 þ dI 1 ; coh dI 0 þ dI inc 0
nTs2 ¼
dI 1 ðe; yÞ ; dI 0 ðe; yÞ ð20Þ
Since value Ds1 is very close to Ds the value of Dsn is much smaller than both Ds and Ds1 . 3. Effect for the total intensity of radiation Now we turn to the analysis of the polarization effects for the total intensities of radiation Z y¼1 I n ðeÞ ¼ I 0 ðeÞ þ nI 1 ðeÞ; I 0 ðeÞ ¼ dI 0 ðeÞ; I 1 ðeÞ ¼
Z
y¼0 y¼1
dI 1 ðeÞ:
ð21Þ
y¼0
The integral degree of the circular polarization of the radiation intensity in a crystal is given by the ratio nT2 ¼ I 1 ðeÞ=I 0 ðeÞ. In [1] it was shown that the total intensity IðeÞ contains both the coherent and incoherent contributions as well as the influence of the multiple scattering (the LPM effect) on the process under consideration. The same is true for the polarization part I 1 ðeÞ. The intensity of coherent radia2 tion I F ðeÞ I coh 0 ðeÞ is the first term (m0 ¼ 0) of the decom2 position of IðeÞ over m0 . Its explicit representation is given by Eqs. (25) and (26) in [1]. The coherent polarization part 2 I coh 1 ðeÞ is the first term (m0 ¼ 0) of the decomposition of 2 I 1 ðeÞ over m0 . Both can be written in the form Z x0 dx coh J coh ; I 0;1 ðeÞ ¼ 0;1 ðvÞ x0 0 s Z am2 kþi1 v2 coh J 0;1 ðvÞ ¼ i 2p ki1 3 ds ; Cð1 sÞCð3s 1Þð2s 1Þa0;1 cos ps 11 1 < k < 1: ð22Þ a0 ¼ s2 s þ 2; a1 ¼ ð1 sÞ; 6 3 coh where J coh 0 ðvÞ is the radiation intensity and J 1 ðvÞ is the contribution of the circular polarization of radiation in external field (see Eqs. (4.50), (4.51) and (4.84) in [5]). The representation (22) is convenient both for the analytical and numerical calculation. The degree of circular polarization of the coherent radiation in a crystal we define by coh the ratio ncoh ¼ I coh 2 1 =I 0 . In [1] the new representation of the function J inc 0 ðvÞ was obtained, which is suitable for both analytical and numerical calculation. The same procedure can be applied to J inc 1 ðvÞ. As a result we get Z ip kþi1 v2s Cð1 þ 3sÞ ds inc R0;1 ðsÞ 2 ; J 0;1 ðvÞ ¼ s 2 ki1 3 CðsÞ sin ps 1 ð23Þ
where Fig. 4. The degree of the circular polarization of radiation in the tungsten, axis h1 1 1i, temperature T¼100 K. The value of K Eq. (18) is shown versus y ¼ x=e. The curves for e ¼ 0:3; 1; 3 GeV coincide.
R0 ðsÞ ¼ 15 þ 43s þ 31s2 þ 28s3 þ 12s4 ; 25 109 2 þ 7s s 22s3 : R1 ðsÞ ¼ 3 3
ð24Þ
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
The integral degree of circular polarization of the incoherent radiation in a crystal we define by the ratio inc inc ninc 2 ¼ I 1 =I 0 . The integral degree of circular polarization in the tungsten crystal (axis h1 1 1i, the temperatures T¼100 K) nT2 ¼ I 1 ðeÞ=IðeÞ Eq. (21) is shown in Fig. 5 (the curve T), coh as well as the coherent degree ncoh ¼ I coh Eq. (22) 1 =I 0 2 inc inc (the curve 1) and the incoherent degree n2 ¼ I inc 1 =I 0 Eq. (24) (the curve 2) as a function of incident electron energy e. In low energy region e 6 1 GeV the contribution of the incoherent mechanism dominates (let us remind that the intensities of the incoherent and coherent radiation become equal at e ’ 0:4 GeV). At higher energies the intensity inc I coh 0 ðeÞ dominates while the intensity I 0 ðeÞ decreases monotonically [1]. Correspondingly the curve ncoh tends to the 2 curve nT2 . At extremely high-energy e > 106 GeV ncoh tends 2 to the external field limit: ncoh 2 =11/16 (see Eq. (4.88) in [5]). The next terms of decomposition of the total intensity IðeÞ over m20 describe the LPM effect in the radiation process. The contribution of the LPM effect in the total intensity of radiation I Eq. (21) is defined in [1] as I LPM ¼ inc I 0 I coh 0 I 0 . The relative contribution (negative since the LPM effect suppresses the radiation process) D ¼ I LPM =I 0 in the maximum is of the order of percent (see Fig. 3 in [1]). Similarly we define the relative influence of multiple scattering on the photon integral circular polarization as Dn ¼ D1 ¼
nT2 nci2 ¼ D1 D; nci2
inc I coh 1 þ I1 1: I1
nci2 ¼
inc I coh 1 þ I1 ; inc I coh 0 þ I0
ð25Þ
The function of D1 ðeÞ (percent) is shown in Fig. 6, it attains the maximum D1 ’ 2:0% at e ’ 1:4 GeV, while Dn ðeÞ attains the maximum Dn ’ 1:4% at e ’ 1:8 GeV and
Fig. 5. The integral degree of the circular polarization in tungsten, axis h1 1 1i, temperature T=100 K. The functions are shown versus electron energy in GeV. The curve 1 is for the coherent radiation coh ðncoh ¼ I coh 1 ðeÞ=I 0 ðeÞÞ, the curve 2 is for the incoherent radiation 2 T inc inc ¼ I ðeÞ=I ðninc 1 0 ðeÞÞ, the curve T is n2 ¼ I 1 ðeÞ=I 0 ðeÞ (see Eq. (21)). 2
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Fig. 6. The influence of multiple scattering on the circular polarization of emitted radiation described by the function D1 ðeÞ Eq. (25) (percent) in tungsten, axis h1 1 1i, temperature T¼100 K.
DðeÞ ’ 0:9% at e ’ 0:3 GeV [1]. One can see that maxima of corresponding functions are slightly shifted with respect each other and the highest maximum has D1 . From the other side, the behavior of all functions D, D1 and Dn are quite similar: just as in the total intensity of radiation the suppression of integral polarization due to the multiple scattering is concentrated in the interval of moderate energies e < 10 GeV and the scale of effect is of the order of percent. 4. Conclusion In this paper the spectrum of radiation from an electron of intermediate energy (a few GeV for heavy elements) moving in an oriented crystal is calculated for the first time. The interplay of the coherent and the incoherent parts is essential for formation of the spectrum. Just in this situation the effects of multiple scattering of projectile appear. The same is true also for the depending on polarization part of the spectral intensity. At motion of an electron near a chain of atoms (an axis) in an oriented crystal the atom density on the trajectory is much higher than in an amorphous medium. As a result, the parameter, characterizing the influence of multiple scattering on the radiation process in a medium in absence of an external field (m20 e=e0 ), becomes of the order of unity at enough low energy (values of e0 for tungsten and germanium are given in Table 1). From the other side, due to the high density of atoms at the trajectory near an axis, the strong electric field of the axis acts on the electron. This action diminishes the radiation formation length and expands the characteristic angles of photon emission and hence weakens the influence of multiple scattering on the radiation process. So, one has to use the general expression for the radiation intensity which takes into account both the crystal effective field (the coherent mechanism) and
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the multiple scattering (the incoherent mechanism) for study of the characteristics of radiation. Such expression for the unpolarized case dI 0 ðe; yÞ was obtained in [1] and the polarization term dI 1 ðe; yÞ derived here (see Eq. (3)). The two first terms of decomposition of dI n ðe; yÞ over the parameter m20 define the coherent and incoherent radiation. It should be noted that in the incoherent contribution the influence of crystalline field is taken into account. Other terms of the decomposition represent influence of the crystalline field on the multiple scattering effect (on the LPM effect). Since in an amorphous medium the LPM effect for the whole spectrum can be observed (for heavy elements) only in TeV energy range (see e.g. [7]) the possibility to study the influence of multiple scattering on radiation process in GeV energy range is evidently of great interest. In the present paper the detailed analysis of the spectral and the polarization properties of radiation is performed. The influence of different mechanisms of photon emission on the general picture of event is elucidated. At high-energy e e0 the influence of the multiple scattering on the radiation intensity is suppressed strongly (the coherent contribution dominates) and only in the very end of the spectrum at u P e=e0 1ð1 x=e 6 e0 =e 1Þ the incoherent radiation becomes essential. In this part of the spec2 trum the nearly complete (with accuracy ðe0 =eÞ ) helicity transfer from an electron to the emitted photon occurs and the scaling defined by Eq. (16) takes place. For any energy e the maximum value of the LPM effect is around 7% and its position is situated at um 3e=e0 ðxm ¼ eum =1 þ um Þ. For x ¼ xm the incoherent contribution dominates, its contribution is of one order of magnitude higher than the coherent one. So at e e0 the maximum of the LPM effect is
situated in the very end of the spectrum where the mentioned scaling holds. It should be noted the very right end of spectrum is described by the Bethe–Maximon formulae with independent on the electron energy crystal corrections (compare e.g. with Eq. (8) in [1]). For illustration of the discussed effect we considered the low energy region e 6 e0 , where both the coherent and incoherent contributions are essential, while the mentioned scaling is only approximate one. This energy region is suitable for the experimental study. The polarization effects in radiation for the intermediate energy is analyzed for the first time. It is shown that the influence of multiple scattering on the polarization part of the intensity spectrum dI 1 ðe; yÞ is very close to the LPM effect in the unpolarized part dI 0 ðe; yÞ. Acknowledgement The authors are indebted to the Russian Foundation for Basic Research supported in part this research by Grant 06-02-16226. References [1] V.N. Baier, V.M. Katkov, Phys. Lett. A 353 (2006) 91. [2] K. Kirsebom, U. Mikkelsen, E. Uggerhoj, K. Elsener, S. Ballestrero, P. Sona, Z.Z. Vilakazi, Phys. Rev. Lett. 87 (2001) 054801. [3] A. Baurichter et al., Phys. Rev. Lett. 79 (1997) 3415. [4] V.N. Baier, V.M. Katkov, Nucl. Instr. and Meth. B 234 (2005) 106. [5] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, World Scientific Publishing Co., Singapore, 1998. [6] V.N. Baier, V.M. Katkov, Phys. Rev. D 62 (2000) 036008. [7] V.N. Baier, V.M. Katkov, Phys. Rep. 409 (2005) 261.
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 3828–3834 www.elsevier.com/locate/nimb
Spectrum and polarization of coherent and incoherent radiation and the LPM effect in oriented single crystal V.N. Baier *, V.M. Katkov Budker Institute of Nuclear Physics, 11, Lavrenteiv Avenue, Novosibirsk 630090, Russia Received 31 October 2007; received in revised form 24 January 2008 Available online 1 February 2008
Abstract The spectrum and the circular polarization of radiation from longitudinally polarized high-energy electrons in an oriented single crystal are considered using the method which permits inseparable consideration of both the coherent and the incoherent mechanisms of photon emission. The spectral and polarization properties of radiation are obtained and analyzed. It is found that in some part of the spectral distribution the influence of multiple scattering (the Landau–Pomeranchuk–Migdal (LPM) effect) attains the order of 7%. The same is true for the influence of multiple scattering on the polarization part of the radiation intensity. The degree of the circular polarization of the total intensity of radiation is found. It is shown that the influence of multiple scattering on the photon polarization is similar to the influence of the LPM effect on the total intensity of radiation: it appears only for relatively low energies of radiating electron and has the order of 1%, while at higher energies the crystal field action excludes the LPM effect. Ó 2008 Elsevier B.V. All rights reserved. PACS: 78.70.g Keywords: Crystal; Radiation; Coherent; Incoherent
1. Introduction The study of processes with participation of polarized electrons and photons permits to obtain the important physical information. Because of this reason the experiments with use of polarized particles are performed and are planning in many laboratories (CERN, Jefferson Nat Accl Fac, SLAC, BINP, etc). In this paper the polarization effects are considered in the frame of the general theory developed by authors [1], which includes both the coherent and the incoherent mechanisms of radiation from highenergy electrons in an oriented single crystal. The influence of multiple scattering on the radiation process including polarization effects is analyzed. The study of radiation in oriented crystals is continuing and new experiments are performed recently see [2,3]. *
Corresponding author. Tel.: +7 383 329 4785; fax: +7 383 330 7163. E-mail address: [email protected] (V.N. Baier).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.01.046
The general expression for the energy loss of the longitudinally polarized electron in oriented crystal was found in [4] (see Eq. (2.7)) Z 3 am2 d3 k dr dEn ¼ 2 0 F ðr; #0 Þ V 8p ee Z x 1 e e0 iA ð1 þ nÞ 0 þ ð1 nÞ e 1þn þ e 4 e e 2 c2 ðv1 v2 Þ dt1 dt2 ; Z xe t2 1 2 A¼ 0 þ ðn vðtÞÞ dt: ð1Þ 2e t1 c2 where dEn ¼ xdwn , dwn is the probability of radiation, see e.g. Eq. (4.2) in [5], x and e are the photon and electron energy, e0 ¼ e x; c ¼ e=m, a ¼ e2 ¼ 1=137, the vector k is the photon momentum, n ¼ k= j k j,n ¼ kf; k ¼ 1 is the helicity of emitted photon, f ¼ 1 is the helicity of the initial electron, F ðr; #0 Þ is the distribution function of
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
electron in the transverse phase space depending on the angle of incidence #0 of the electron on crystal, V is the volume of a crystal, v1 ¼ vðt1 Þ is the electron velocity (see [5], Sec. 16.2). The degree of the circular polarization of radiation is defined by Stoke’s parameter n2 : n2 ¼ KðfvÞ;
K¼
dEþ dE ; dEþ þ dE
ð2Þ
where the quantity ðfvÞ defines the longitudinal polarization of the initial electrons, dEþ and dE is the energy loss for n ¼ þ1 and n ¼ 1 correspondingly. It should be noted that a few different spin correlations are known in an external field. But after averaging over directions of the crystal field only the considered here the longitudinal polarization survives. In [4] the polarization effects in the coherent radiation, which dominates at high electron energies (e 1 GeV for main axes of heavy elements, e.g. tungsten crystal), was studied. At intermediate energies the incoherent radiation contributes essentially and the contributions of both mechanisms should be taken into account. Recently authors developed the method which permits an indivisible consideration of both the coherent and the incoherent mechanisms of photon emission in oriented crystals [1]. Basing on Eqs. (18) and (19) of [1] (see also Eqs. (7.89) and (7.90) in [5]) and using Eq. (1) one can obtain the general expression for the intensity of radiation from the longitudinally polarized electrons which includes the coherent and incoherent contributions and the Landau–Pomeranchuk–Migdal (LPM) effect
4v2 ðxÞ ; u2 m20
y¼
x ; e
u¼
y ; 1y
ð3Þ
where 2
r2 ¼ 1 þ ð1 yÞ ; r3 ¼ 2ð1 yÞ;
r21 ¼ 2y y
where x0 ¼
1 ; pdna a2s
g1 ¼
2u21 ; a2s
x¼
.2 : a2s
ð6Þ
Here . is the distance from the axis, u1 is the amplitude of thermal vibration, d is the mean distance between atoms forming the axis, as is the effective screening radius of the potential. The parameters in Eq. (5) were determined by means of fitting procedure. The local value of the parameter vðxÞ which determines the radiation probability in the field Eq. (5) is pffiffiffi dU ð.Þ e 2 x vðxÞ ¼ ; ¼ vs d. m3 ðx þ gÞðx þ g þ 1Þ V 0e e vs ¼ 3 : ð7Þ m as e s For an axial orientation of crystal the ratio of the atom density nð.Þ in the vicinity of an axis to the mean atom density na is (see [1]) nðxÞ x0 ¼ nðxÞ ¼ ex=g1 ; na g1
e0 ¼
ee ; nð0Þ
ee ¼
m : 16pZ 2 a2 k3c na L0 ð8Þ
ee ðna Þ e0 ¼ ex=g1 ; nðxÞgðx; e; xÞ gðx; e; xÞ 1 L0 ¼ lnðmaÞ þ f ðZaÞ; 2 1 v2 ðxÞ 6Dr v2 ðxÞ gðx; e; xÞ ¼ g0 þ lnð1 þ 2 Þ þ ; 6L0 u 12u2 þ v2 ðxÞ 2 1 1 u 111Z 1=3 h 12 ; a ¼ ; g0 ¼ 1 þ L0 18 a m 1 X n2 f ðnÞ ¼ ; 2 2 n¼1 nðn þ n Þ 1 5 hðzÞ ¼ ½1 þ ð1 þ zÞez EiðzÞ; Dr ¼ Dsc 2 9 ec ðxÞ ¼
u1 ðtÞ ¼ ði 1Þt þ br ð1 þ iÞðu2 ðtÞ tÞ; pffiffiffi pffiffiffi m0 t 2 2m 0 pffiffiffi tanh pffiffiffi ; u3 ðtÞ ¼ u2 ðtÞ ¼ ; m0 2 sinhð 2m0 tÞ
1 r2n ¼ ðr2 þ nr21 Þ; 2 1 r3n ¼ ðr3 þ nr31 Þ; 2 1y e 2 ; m0 ¼ y ec ðxÞ
The situation is considered where the electron angle of incidence #0 (the angle between electron momentum p and the axis (or plane)) is small #0 V 0 =m. The axis potential (see Eq. (9.13) in [5]) is taken in the form 1 1 U ðxÞ ¼ V 0 ln 1 þ ln 1 þ ; ð5Þ xþg x0 þ g
The functions and values in Eqs. (3) and (4) are
dI n ðe; yÞ ¼ dI 0 ðe; yÞ þ ndI 1 ðe; yÞ Z x0 am2 ydy dx Grn ðx; yÞ; ¼ 2p 1 y 0 x0 Z 1 p Grn ðx; yÞ ¼ F rn ðx; y; tÞdt r3n ; 4 0 u ðtÞ 2 F rn ðx; y; tÞ ¼ Im e 1 ½r2n m0 ð1 þ ibr Þu2 ðtÞ þ r3n u3 ðtÞ ; br ¼
3829
2
r31 ¼ 2yð1 yÞ;
ð4Þ The intensity for the unpolarized electrons dI 0 ðe; yÞ was obtained in [1], the polarization term dI 1 ðe; yÞ is found here.
ð9Þ
where the function gðx; e; xÞ determines the effective logarithm using the interpolation procedure: L ¼ L0 gðx; e; xÞ, the function g0 is defined in Eq. (8) in [1], Dsc ¼ 2:30 is the constant entering in the radiation spectrum at v=u 1, see Eq. (7.107) in [5], Ei(z) is the integral exponential function, f ðnÞ is the Coulomb correction. It follows from Eqs. (2) and (3) that the circular polarization of radiation is n2 ¼
dI 1 ðe; yÞ ðfvÞ; dI 0 ðe; yÞ
ð10Þ
3830
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2. The spectral distribution of radiation The radiation of a particle moving in an external field, if particle scattering does not taken into account, is called the coherent radiation. The incoherent radiation appears because of scattering of particle on separate atoms. Its intensity is proportional to the density of atoms in a medium (/ m20 ). Nonlinear intensity dependence on the density appears in consequence of multiple scattering of the radiating electron on the radiation formation length (the LPM effect). The expression for dI n Eq. (3) includes both the coherent and incoherent contributions as well as the influence of the multiple scattering (the LPM effect) on the photon emission process. The intensity of the coherent radiation dI coh 0 ðe; yÞ is the first term (m20 ¼ 0) of the decomposition of Eq. (3) over m20 . This intensity is contained in Eq. (17.7) of [5]. The polarization term in the intensity of the coherent radiation dI coh 1 ðe; yÞ is the first term of the decomposition of dI 1 in coh Eq. (3) over m20 . The expression dI coh 0 þ ndI 1 coincides with the term containing R0 ðkÞ in Eq. (3.5) of [4]. The intensity of the incoherent radiation dI inc 0 ðe; yÞ is the second term (/ m20 ) of the mentioned decomposition of dIðe; yÞ [1]. The expression for dI inc 0 ðe; yÞ follows also from
Fig. 1. The spectral distribution of radiation in tungsten, axis h1 1 1i, temperature T=100 K (solid lines) and T=293 K (dashed lines), with taking into account all mechanisms of photon emission. The spectral inverse radiation length (in cm1 ) dI 0 ðe; yÞ=edy, see Eq. (3), is shown versus y ¼ x=e for different energies: the curve 1 is for e ¼ 0:3 GeV, the curve 2 is for e ¼ 1 GeV, the curve 3 is for e ¼ 3 GeV, the curve 4 is for e ¼ 1 GeV, the curve 5 is for e ¼ 5 GeV and the curve 6 is for e ¼ 10 GeV.
Eq. (21.21) in ([5]). The polarization term dI inc 1 ðe; yÞ is correspondingly the second term (/ m20 ) of decomposition of dI 1 ðe; yÞ: Z x0 am2 e dx inc dI 0;1 ðe; yÞ ¼ gðx; e; xÞex=g1 dJ inc ; ð11Þ 0;1 ðv; yÞ x0 60p e0 0 here v ¼ vðxÞ, the notations is given in Eqs. (7)–(9), dJ inc 0;1 ðvÞ can be written as 2 dJ inc 0 ðv; yÞ ¼ ½y ðf1 ðzÞ þ f2 ðzÞÞ þ 2ð1 yÞf2 ðzÞdy; 2 dJ inc 1 ðv; yÞ ¼ ½y ðf1 ðzÞ f2 ðzÞÞ þ 2yf2 ðzÞdy; 2=3 y ; z¼ vð1 yÞ
ð12Þ
the functions f1 ðzÞ and f2 ðzÞ are defined in the just mentioned equation in [5]: f1 ðzÞ ¼ z4 !ðzÞ 3z2 !0 ðzÞ z3 ;
ð13Þ
f2 ðzÞ ¼ ðz4 þ 3zÞ!ðzÞ 5z2 !0 ðzÞ z3 ; here !ðzÞ is the Hardy function: Z 1 s3 sin zs þ !ðzÞ ¼ ds: 3 0
ð14Þ
For intermediate energies, where both the coherent and the incoherent contributions to the total intensity of radiation are essential, the spectral distribution of intensity dI 0 ðe; yÞ is shown in Fig. 1. The calculation was done for the axis h1 1 1i of tungsten at temperature T¼100 K and T¼293 K (the parameters of crystal are given in Table 1), et is the energy for which the coherent intensity becomes equal to the Bethe–Maximon one. These spectra describe radiation in thin targets when one can neglect the energy loss of projectile. It is seen that even for a moderate electron energy the photon spectrum spreads into the hard region, e.g. for T¼100 K and e ¼ 3 GeV the radiation intensity for y ¼ 0:2ðx ¼ 600 MeV) is still half of maximal one. The hard end of all spectral curves tends to the same value corresponding to the Bethe–Maximon intensity. For the energy e ¼1 GeV the spectrum at T¼100 K is harder than at T¼293 K. The next terms of decomposition of the intensity dI 0 ðe; yÞ over m20 describe the influence of multiple scattering on the radiation process, the LPM effect. The different contributions to that part of the spectrum, where the coherent and the incoherent contributions are comparable, are shown in Fig. 2. It is seen that for e ¼ 3 GeV, dI coh ’ 0 dI inc at y ’ 0:54ðx ’ 1:6 GeVÞ while for lower photon 0 energy the coherent contribution dominates and for higher photon energy the incoherent contribution dominates.
Table 1 Parameters of radiation process of the tungsten crystal, axis h1 1 1i and the germanium crystal, axis h1 1 0i for two temperatures T Crystal
T (K)
V 0 (eV)
x0
g1
g
e0 (GeV)
et (GeV)
es (GeV)
h
W W Ge Ge
293 100 293 100
417 355 110 114.5
39.7 35.7 15.5 19.8
0.108 0.0401 0.125 0.064
0.115 0.0313 0.119 0.0633
7.43 3.06 148 59
0.76 0.35 1.29 0.85
34.8 43.1 210 179
0.348 0.612 0.235 0.459
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
3831
All the curves in Fig. 3 have nearly the same height of the maximum and the position of the maximum is defined roughly by the expression um ’ 3e=e0 ðu y=ð1 yÞÞ. Such scaling in terms of u is the consequence of the following representation of the spectral inverse radiation length (the intensity spectrum Eq. (3)) e
e
dL1 1 dI 0 ðe; yÞ rad ¼ r2 ðyÞR2 ¼ þ r3 ðyÞR3 ð16Þ e dy u u dy
Fig. 2. The different contributions to the photon spectrum for electron energy e ¼ 3 GeV, axis h1 1 1i, temperature T¼100 K. The curve 1 is dI 0 ðe; yÞ, the curve 2 is the coherent spectrum dI coh 0 ðe; yÞ, the curve 3 is the coh incoherent spectrum dI inc 0 ðe; yÞ, the curve 4 is the sum dI 0 ðe; yÞþ inc coh dI 0 ðe; yÞ and the curve 5 is the difference ðdI 0 ðe; yÞ þ dI inc 0 ðe; yÞ dI 0 ðe; yÞÞ 10.
Calculation shows that for e ¼ 0:3 GeV, dI coh ’ dI inc at 0 0 coh y ’ 0:1ðx ’ 30 MeVÞ. For e ¼ 1 GeV, dI 0 ’ dI inc at 0 y ’ 0:28ðx ’ 280 MeVÞ. The difference shown by curve 5 arises due to the LPM effect. We define the contribution of the LPM effect into spectral distribution, by analogy with [1], as Ds ¼
inc dI 0 dI coh 0 dI 0 dI 0
ð15Þ
The function Ds ðyÞ is shown in Fig. 3. The curve 1 for e ¼ 0:3 GeV reaches the maximum 6.64% at y¼0.18, the curve 2 for e ¼ 1 GeV reaches the maximum 6.87% at y¼0.44 and the curve 3 for e ¼ 3 GeV reaches the maximum 7.32% at y¼0.7.
In the high-energy limit e e0 the maximum of the LPM effect is situated at the very end of the spectrum. In this limit r2 ’ 1 Oðe0 =eÞ and r3 ’ Oðe0 =eÞ and the scaling (the dependence on the combination e=u only) of each of the two R2;3 terms gets over into scaling of the whole expression for the spectral inverse radiation length. In the maximum of the LPM effect the coherent contribution into spectral radiation intensity is relatively small: less than 10%. Therefore the right slope of curves in Fig. 3 where the influence of the field still smaller is described by formulas of the LPM effect in a medium with small corrections due to action of the crystal field. Far from the maximum at u e=e0 one has Rcoh 2;3 ¼ 0 and the terms / m60 in the decomposition of the functions R2;3 (which includes the crystal field corrections) have the form ! Rinc e2 g20 v2 2 R2 ¼ 2 2 1 þ 377 2 ; u 3e0 u Rinc 2 ðv ¼ 0Þ ! Rinc e2 g20 31 2704 v2 3 R3 þ ¼ 2 2 ; 15 u2 3e0 u 63 Rinc 2 ðv ¼ 0Þ Z 1 Rinc 1 dx 3 ðv ¼ 0Þ 2 v2 ðxÞe3x=g1 ð17Þ ¼ ; v ¼ inc 3 g1 R2 ðv ¼ 0Þ 0 Here the terms independent on field coincide with the corresponding terms in Eq. (3.6) in [6], the corrections depending on crystal field are calculated in this paper. At the left slope of the curves in Fig. 3 the coherent contribution dominates (see Fig. 1), the relative contribution of incoherent radiation diminishes and the LPM effect is only its small part. The degree of the circular polarization of radiation Eq. (10) can be presented in a form (see Eqs. (3) and (16)) dI 1 ðe; yÞ r21 R2 þ r31 R3 ¼ dI 0 ðe; yÞ r 2 R2 þ r 3 R3 2 yð2 yÞ yð1 yÞ ¼ þ 2ð1 yÞ þ y 2 2ð1 yÞ þ y 2 1 y2 s 2R3 s 1þ ; s¼ : 2 2ð1 yÞ þ y 2 R 2 þ R3
K¼
Fig. 3. The LPM effect for spectral distribution of radiation in tungsten, axis h1 1 1i, temperature T¼100 K. The function Ds ðyÞ Eq. (15) (percent) is shown versus y ¼ x=e. The curve 1 is for e ¼ 0:3 GeV, the curve 2 is for e ¼ 1 GeV and the curve 3 is for e ¼ 3 GeV.
ð18Þ
Here the first term on the right-hand side dominates, it has an universal form. The first factor in the second term is very psmall numerically (its maximal value is ffiffiffi ð3 2 2Þ=4 ¼ 0:0429). The energy dependence of this term is defined by the rest factors which are smaller than sðeÞ > 0. In the region of low electron energy, where the incoherent radiation dominates, the value s ’ 1 and
3832
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
weakly depends on energy. In the high-energy region where the coherent radiation dominates one has s 1. In the intermediate region the dependence of the value s on energy is weak numerically. So the dependence of the value K on energy is very weak always. The value of K is shown in Fig. 4. In accord with the performed analysis, the curves for energies e ¼ 0:3; 1; 3 GeV coincide with each other inside the thickness of line. For any mechanism of radiation n2 ’ yðfvÞ for y 1 (see Eq. (2.9)) in [4] and n2 ! 1 for y ! ðfvÞ as a consequence of the helicity transfer from an electron to photon. The next terms of decomposition of the intensity dI 1 ðe; yÞ over m20 describe the influence of multiple scattering on the polarization part of the spectral intensity of radiation. We define the contribution of this effect into the polarization part as Ds1 ¼
inc dI 1 dI coh 1 dI 1 dI 1
ð19Þ
The function Ds1 ðyÞ for e ¼ 0:3 GeV reaches the maximum 6.78% at y¼0.18, for e ¼ 1 GeV reaches the maximum 7.09% at y¼0.44 and for e ¼ 3 GeV reaches the maximum 7.43% at y¼0.7. It is seen that the maximum positions are situated at the same photon energy as in Ds ðyÞ (see Fig. 3) and their values are very close to these values in the unpolarized part. All this means that the multiple scattering is affecting similarly on the unpolarized spectrum described by dI 0 ðe; yÞ and the polarization term described by dI 1 ðe; yÞ. The influence of the multiple scattering on the photon polarization degree may be also characterized by Dsn ¼ ncis2 ¼
nTs2 ncis2 ¼ Ds1 Ds ; ncis2
inc dI coh 1 þ dI 1 ; coh dI 0 þ dI inc 0
nTs2 ¼
dI 1 ðe; yÞ ; dI 0 ðe; yÞ ð20Þ
Since value Ds1 is very close to Ds the value of Dsn is much smaller than both Ds and Ds1 . 3. Effect for the total intensity of radiation Now we turn to the analysis of the polarization effects for the total intensities of radiation Z y¼1 I n ðeÞ ¼ I 0 ðeÞ þ nI 1 ðeÞ; I 0 ðeÞ ¼ dI 0 ðeÞ; I 1 ðeÞ ¼
Z
y¼0 y¼1
dI 1 ðeÞ:
ð21Þ
y¼0
The integral degree of the circular polarization of the radiation intensity in a crystal is given by the ratio nT2 ¼ I 1 ðeÞ=I 0 ðeÞ. In [1] it was shown that the total intensity IðeÞ contains both the coherent and incoherent contributions as well as the influence of the multiple scattering (the LPM effect) on the process under consideration. The same is true for the polarization part I 1 ðeÞ. The intensity of coherent radia2 tion I F ðeÞ I coh 0 ðeÞ is the first term (m0 ¼ 0) of the decom2 position of IðeÞ over m0 . Its explicit representation is given by Eqs. (25) and (26) in [1]. The coherent polarization part 2 I coh 1 ðeÞ is the first term (m0 ¼ 0) of the decomposition of 2 I 1 ðeÞ over m0 . Both can be written in the form Z x0 dx coh J coh ; I 0;1 ðeÞ ¼ 0;1 ðvÞ x0 0 s Z am2 kþi1 v2 coh J 0;1 ðvÞ ¼ i 2p ki1 3 ds ; Cð1 sÞCð3s 1Þð2s 1Þa0;1 cos ps 11 1 < k < 1: ð22Þ a0 ¼ s2 s þ 2; a1 ¼ ð1 sÞ; 6 3 coh where J coh 0 ðvÞ is the radiation intensity and J 1 ðvÞ is the contribution of the circular polarization of radiation in external field (see Eqs. (4.50), (4.51) and (4.84) in [5]). The representation (22) is convenient both for the analytical and numerical calculation. The degree of circular polarization of the coherent radiation in a crystal we define by coh the ratio ncoh ¼ I coh 2 1 =I 0 . In [1] the new representation of the function J inc 0 ðvÞ was obtained, which is suitable for both analytical and numerical calculation. The same procedure can be applied to J inc 1 ðvÞ. As a result we get Z ip kþi1 v2s Cð1 þ 3sÞ ds inc R0;1 ðsÞ 2 ; J 0;1 ðvÞ ¼ s 2 ki1 3 CðsÞ sin ps 1 ð23Þ
where Fig. 4. The degree of the circular polarization of radiation in the tungsten, axis h1 1 1i, temperature T¼100 K. The value of K Eq. (18) is shown versus y ¼ x=e. The curves for e ¼ 0:3; 1; 3 GeV coincide.
R0 ðsÞ ¼ 15 þ 43s þ 31s2 þ 28s3 þ 12s4 ; 25 109 2 þ 7s s 22s3 : R1 ðsÞ ¼ 3 3
ð24Þ
V.N. Baier, V.M. Katkov / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3828–3834
The integral degree of circular polarization of the incoherent radiation in a crystal we define by the ratio inc inc ninc 2 ¼ I 1 =I 0 . The integral degree of circular polarization in the tungsten crystal (axis h1 1 1i, the temperatures T¼100 K) nT2 ¼ I 1 ðeÞ=IðeÞ Eq. (21) is shown in Fig. 5 (the curve T), coh as well as the coherent degree ncoh ¼ I coh Eq. (22) 1 =I 0 2 inc inc (the curve 1) and the incoherent degree n2 ¼ I inc 1 =I 0 Eq. (24) (the curve 2) as a function of incident electron energy e. In low energy region e 6 1 GeV the contribution of the incoherent mechanism dominates (let us remind that the intensities of the incoherent and coherent radiation become equal at e ’ 0:4 GeV). At higher energies the intensity inc I coh 0 ðeÞ dominates while the intensity I 0 ðeÞ decreases monotonically [1]. Correspondingly the curve ncoh tends to the 2 curve nT2 . At extremely high-energy e > 106 GeV ncoh tends 2 to the external field limit: ncoh 2 =11/16 (see Eq. (4.88) in [5]). The next terms of decomposition of the total intensity IðeÞ over m20 describe the LPM effect in the radiation process. The contribution of the LPM effect in the total intensity of radiation I Eq. (21) is defined in [1] as I LPM ¼ inc I 0 I coh 0 I 0 . The relative contribution (negative since the LPM effect suppresses the radiation process) D ¼ I LPM =I 0 in the maximum is of the order of percent (see Fig. 3 in [1]). Similarly we define the relative influence of multiple scattering on the photon integral circular polarization as Dn ¼ D1 ¼
nT2 nci2 ¼ D1 D; nci2
inc I coh 1 þ I1 1: I1
nci2 ¼
inc I coh 1 þ I1 ; inc I coh 0 þ I0
ð25Þ
The function of D1 ðeÞ (percent) is shown in Fig. 6, it attains the maximum D1 ’ 2:0% at e ’ 1:4 GeV, while Dn ðeÞ attains the maximum Dn ’ 1:4% at e ’ 1:8 GeV and
Fig. 5. The integral degree of the circular polarization in tungsten, axis h1 1 1i, temperature T=100 K. The functions are shown versus electron energy in GeV. The curve 1 is for the coherent radiation coh ðncoh ¼ I coh 1 ðeÞ=I 0 ðeÞÞ, the curve 2 is for the incoherent radiation 2 T inc inc ¼ I ðeÞ=I ðninc 1 0 ðeÞÞ, the curve T is n2 ¼ I 1 ðeÞ=I 0 ðeÞ (see Eq. (21)). 2
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Fig. 6. The influence of multiple scattering on the circular polarization of emitted radiation described by the function D1 ðeÞ Eq. (25) (percent) in tungsten, axis h1 1 1i, temperature T¼100 K.
DðeÞ ’ 0:9% at e ’ 0:3 GeV [1]. One can see that maxima of corresponding functions are slightly shifted with respect each other and the highest maximum has D1 . From the other side, the behavior of all functions D, D1 and Dn are quite similar: just as in the total intensity of radiation the suppression of integral polarization due to the multiple scattering is concentrated in the interval of moderate energies e < 10 GeV and the scale of effect is of the order of percent. 4. Conclusion In this paper the spectrum of radiation from an electron of intermediate energy (a few GeV for heavy elements) moving in an oriented crystal is calculated for the first time. The interplay of the coherent and the incoherent parts is essential for formation of the spectrum. Just in this situation the effects of multiple scattering of projectile appear. The same is true also for the depending on polarization part of the spectral intensity. At motion of an electron near a chain of atoms (an axis) in an oriented crystal the atom density on the trajectory is much higher than in an amorphous medium. As a result, the parameter, characterizing the influence of multiple scattering on the radiation process in a medium in absence of an external field (m20 e=e0 ), becomes of the order of unity at enough low energy (values of e0 for tungsten and germanium are given in Table 1). From the other side, due to the high density of atoms at the trajectory near an axis, the strong electric field of the axis acts on the electron. This action diminishes the radiation formation length and expands the characteristic angles of photon emission and hence weakens the influence of multiple scattering on the radiation process. So, one has to use the general expression for the radiation intensity which takes into account both the crystal effective field (the coherent mechanism) and
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the multiple scattering (the incoherent mechanism) for study of the characteristics of radiation. Such expression for the unpolarized case dI 0 ðe; yÞ was obtained in [1] and the polarization term dI 1 ðe; yÞ derived here (see Eq. (3)). The two first terms of decomposition of dI n ðe; yÞ over the parameter m20 define the coherent and incoherent radiation. It should be noted that in the incoherent contribution the influence of crystalline field is taken into account. Other terms of the decomposition represent influence of the crystalline field on the multiple scattering effect (on the LPM effect). Since in an amorphous medium the LPM effect for the whole spectrum can be observed (for heavy elements) only in TeV energy range (see e.g. [7]) the possibility to study the influence of multiple scattering on radiation process in GeV energy range is evidently of great interest. In the present paper the detailed analysis of the spectral and the polarization properties of radiation is performed. The influence of different mechanisms of photon emission on the general picture of event is elucidated. At high-energy e e0 the influence of the multiple scattering on the radiation intensity is suppressed strongly (the coherent contribution dominates) and only in the very end of the spectrum at u P e=e0 1ð1 x=e 6 e0 =e 1Þ the incoherent radiation becomes essential. In this part of the spec2 trum the nearly complete (with accuracy ðe0 =eÞ ) helicity transfer from an electron to the emitted photon occurs and the scaling defined by Eq. (16) takes place. For any energy e the maximum value of the LPM effect is around 7% and its position is situated at um 3e=e0 ðxm ¼ eum =1 þ um Þ. For x ¼ xm the incoherent contribution dominates, its contribution is of one order of magnitude higher than the coherent one. So at e e0 the maximum of the LPM effect is
situated in the very end of the spectrum where the mentioned scaling holds. It should be noted the very right end of spectrum is described by the Bethe–Maximon formulae with independent on the electron energy crystal corrections (compare e.g. with Eq. (8) in [1]). For illustration of the discussed effect we considered the low energy region e 6 e0 , where both the coherent and incoherent contributions are essential, while the mentioned scaling is only approximate one. This energy region is suitable for the experimental study. The polarization effects in radiation for the intermediate energy is analyzed for the first time. It is shown that the influence of multiple scattering on the polarization part of the intensity spectrum dI 1 ðe; yÞ is very close to the LPM effect in the unpolarized part dI 0 ðe; yÞ. Acknowledgement The authors are indebted to the Russian Foundation for Basic Research supported in part this research by Grant 06-02-16226. References [1] V.N. Baier, V.M. Katkov, Phys. Lett. A 353 (2006) 91. [2] K. Kirsebom, U. Mikkelsen, E. Uggerhoj, K. Elsener, S. Ballestrero, P. Sona, Z.Z. Vilakazi, Phys. Rev. Lett. 87 (2001) 054801. [3] A. Baurichter et al., Phys. Rev. Lett. 79 (1997) 3415. [4] V.N. Baier, V.M. Katkov, Nucl. Instr. and Meth. B 234 (2005) 106. [5] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, World Scientific Publishing Co., Singapore, 1998. [6] V.N. Baier, V.M. Katkov, Phys. Rev. D 62 (2000) 036008. [7] V.N. Baier, V.M. Katkov, Phys. Rep. 409 (2005) 261.