30
Nuclear
Instruments
and Methods
m Physics Research
B17 (1986) 30--36
North-Holland.
COMPUTER SIMULATION OF POLARIZATION RADIATION FOR RELATIVISTIC ELECTRONS E.G. VYATKIN,
Yu.L. PIVOVAROV
Ntrclear Physics Institute,
634050 Tomsk,
CHARACTERISTICS
Amsterdam
OF CHANNELING
and S.A. VOROBIEV
USSR
Received 31 July 1985 and in revised form 31 January
1986
A computation method for the polarization characteristics of relativistic particle radiation in a crystal has been developed. This method is based on the computer simulation of particle trajectories. Formation of the total radiation spectrum and the Stokes parameters by averaging over the possible electron trajectories under planar channeling has been analyzed. A comparison with experimental data has been made.
1. Introduction The linear polarization of the planar channeling radiation for relativistic particles was predicted by many authors [l-3] and was observed for the first time in experiments at the Tomsk synchrotron [4,5]. The observed degree of polarization was rather high, which stimulated detailed theoretical studies of the polarization characteristics. The detailed analysis of all the properties of the particles trajectories and their radiation in a crystal is easier using computer simulation with the binary collision model [6-91. This method may be used to analyze the polarization properties of the radiation [lO,ll]. The separation of the “forward” radiation into components with different polarization (parallel and perpendicular to the atomic planes) under planar channeling of 30-100 MeV positrons have already been done in reference [9], but the degree of linear polarization (and other Stokes parameters) has not been computed. The degree of linear polarization of the “forward” planar channeling radiation for positrons has been calculated using computer simulation in [IO] and that for 900 MeV electrons in diamond in ref. [ll]. Below we shall give a complete approach to the solution of this problem, based on the simulation of all Stokes parameters, and its application to the analysis of the polarization properties of radiation under planar channeling of relativistic electrons of different energies in diamond and silicon crystals.
2. Binary eoltision model in computing of radiation characteristics The radiation characteristics of relativistic particles in a crystal are obtained with the help of the standard 0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
binary collision model [12], modified to the range of high energies. With the help of this model (using the atomic Moliere potential) the trajectory r(f) and the particle velocity B(t) = u( t)/c in the crystal are found. These values are substituted into the well-known formulae of classical electr~ynami~s for the radiation intensity [13]. After partial integration of these formulae the follow expressions are obtained: I, I, = -iw
“dt / ‘I
+I,)
Xfo
exp[iu(t-n.r(t)/c)]fi(t),
(2)
(3) Here w and n are the frequency and the direction of photon emission, t2 and t, are the moments of the first and the last particle collision with the crystal atoms. In eq. (1) I, describes the radiation inside the crystal, I, describes the boundary effect and there is also an interference term between these two types of radiation. The procedure described above is analogous to that used in [6,9,10), but the boundary effect due to I, has not previously been taken into account. The po~a~zation properties of the radiation are completely described with a pola~zation matrix 1131 de,, = fopgk (here I, is the total radiation intensity). The matrix pIk may be expressed in terms of the Stokes parameters 5, (i = 1, 2, 3):
E.G. Vyatkin et al. / Polarization
of channelingradlatron-
31
termined by a simple addition of the radiation intensities from every trajectory (the electron radiation interference from different trajectories is absent). With allowance for the absence of radiation interference from different trajectories the averaging of 5, (w) over the trajectories must be done according to the scheme: (&(“,
‘>> = c<,k(o,
n).wk’
(7)
h
1
1
Fig. 1. The system of coordinates with the position of an initial velocity /3(t,) and a unit vector S. The direction of photon emission n and polarization vectors ek are given for 0 = 0.
where Wk=(IE:12+ IErk12)/(~.kIElk)‘+~.kIE~12) is the relative contribution to the total radiation intensity of k-trajectory and .$,k(w. n) is the Stokes parameter for electron radiation from a k-trajectory.
3. Computer simulation results These Stokes parameters are the squares of the components of the field strengths E, (A = 1, 2) with the definite linear polarization X: E,E; 5,=
+ E;E, I
E:E, ;
iZz=
- E, E; J
The maximum degree of linear polarization [13] is while the degree of circular polarization is (<: + s:)“’ characterized by t2. The angle CYdefining the direction of the maximum linear polarization in the plane perpendicular to the direction of photon emission is determined as tg(2a) = .$,/t3. The E, components are defined by means of polarization vectors ex. Within the accuracy of the numerical constants these components are defined by the formulae: E,=e,.(I,
+12).
(6)
Here I,, I, are found by computer simulation using the formulae (2, 3). Let us use the set of ehr adopted in the theory of synchrotron (undulator) radiation [14], since one can observe a definite analogy between particle radiation in a spiral (flat) undulator and that under axial (planar) channeling. Using the general formulae [14] for e, and choosing the unit vector [14] S JJOX (fig. 1) we obtain n = (0, 0, l}, e, = (0, 1, 0}, e, = { -1, 0, 0) for the “forward” radiation (0 = 0, computer experiment was carried out for this case). So, as one can see from (2) and (3) the components E, and E, for the “forward” radiation are determined by the transverse components of the particle velocity p,,( t) and &( t ) in the crystal. An important advantage of the computer simulation is the posibility of studying the formation of the observed radiation characteristics by summing over all possible particles trajectories in a crystal. The spectral summed over polarization, are decharacteristics,
The particle motion under planar channeling is usually described in theoretical work [l-3] using a continuum potential for the crystal planes V(u). In these works ,R,( t) = const, and radiation in a given direction 19= 0 is fully linearly polarized ((<,(a, 19= 0)) = 1) along the OY axes. This does not agree with the experimental data [4,5], where the degree of the linear polarization does not exceed (t3),,,,, = (0.6-0.8). The computer experiment is effective in understanding the causes of this divergence. Fig. 2 shows one of the characteristic trajectories for a 900 MeV electron channeled into the (110) planar direction in diamond, and the corresponding spectrum and the Stokes radiation parameters form this trajectory. It follows from the analysis of the trajectory projection on the X02 and XOY planes (figs. 2a and b), that the real trajectory is not a flat curve as in the continuum models [1,2]. Moreover, the amplitude and the frequency w* of the electron oscillations in a channel are changed appreciably (fig. 2a) during particle penetration due to the close collisions with atoms and multiple scattering. These peculiarities of the real trajectory influence the spectrum and the polarization characteristics of the radiation. Indeed, the theory [2] predicts for the radiation at the angle 0 = 0 the existence of the odd harmonics only (without the boundary effect). The width of these harmonics equals approximately r, - n2y2Aw*/N, (n = 1. 3 . .), where N is the total number of oscillations. The continuum approximation for the trajectory with the same r(t,), &t,) and N as in fig. 2a and b leads to widths of the harmonics of r, = 0.5-0.6MeV. The real spectrum (fig. 2c) is more complex due to the w* variation. Indeed, the widths of the first three observed harmonics near 5, 15, and 25 MeV are much greater then r,. Moreover, since the final o* value is twice the initial one, new harmonics appear in the range of every initial one. (The frequencies of these harmonics are up
32
E.G. Vyotkin et al. / Polarizatron oj‘channeling radiation
%Ld (Mev)
hw (w4
Fig. 2. Projections of a typical trajectory for (110) planar channeling of 900 MeV electrons in diamond on the planes YOZ (a) and XOY (b) (x, and y,, are the initial coordinates), the radiation spectrum (c) and the three Stokes parameters (d-f). The intensity spectrum is normalized to the maximum, the crystal thickness is 10 pm and the temperature 7’ = 0 K. The angle of incidence relative to the (110) axis is q,, = 40 mrad.
to twice as great according U* change), fig. 2c. For the real trajectory both /3,, and /I, depend on time and this leads to a more complicated dependence of the Stokes parameters on photon frequency, fig. 2d-f. The [t(w) and
angular size of the detector A@, = 0, in comparison with A&, - y -’ in the experiment. Some difference are also due to the fact that crystal thickness and temperature were h 3 10 pm, T = 0 K in the computer simulation, while the experiment [15] used h = 350 pm, T = 293 K. The position of the m~mum of the Stokes parameters (&( w )) averaged over ah trajectories correlates well with that in the radiation spectrum (cf. fig. 3a and fig. 3b). The radiation in this spectral region is linearly polarized with a high ((5s) 2: 0.75) degree of polarization. The Stokes parameters ([a(w)) and (ct (a)} averaged over all trajectories are appro~mately zero, figs. 3c and d. This is a consequence of the fast oscillations of
33
E. G. Vyatkin et al. / Polarrzation of channeling radiatiorr
electrons radiation under (110) channeling in a diamond is not circularly polarized ((t2) = 0) and the direction of the maximum linear polarization coincides with the OY axes. In contrast to the continuum models [1.3], where (t3( w, 0 = 0)) = 1 for all frequencies, the value ((a( o, 0 = 0)) obtained in the computer experiment, depends significantly on w and has a characteristic maximum (see however fig. 5 below). Fig. 4a-c show the radiation spectra of electrons of lower energies (600, 350, 54 MeV) at (110) planar channeling in diamond (600, 54 MeV) and silicon (350 MeV) averaged over all trajectories. The corresponding experimental data are published in refs. [15-171. The results of the computer simulation are also shown for the channeling radiation at the angle 0 = 0 and T= 0 K (the angular dimension of the detector is Ad, = 0). In the computer experiment the particle angle of incidence was 0, = 0 relative to the (110) planes and the angle ‘p,, in the channeling plane relative to the (110) axis was ‘p. = 0.3 rad, with the angular spread A0, = AT,, = 10m5 rad. The experimental and simulated spectra are normalized to the maxima of the intensity. For 54 MeV electrons in (110) Si the spectrum obtained by simulation only roughly duplicates the envelope of the discrete peak structure observed experimentally (171. The peaks in the measured spectrum [17], stipulated by the quantum transitions between the discrete sub-barrier levels, cannot be singled out by our computer simulation model, because it is based on a description of particle motion with in a framework of classical mechanics. At electron energies higher than 100 MeV a classical description of the planar channeling is applicable which explains the good agreement between the simulation results and those of the experiments at E, = 600 MeV [15] and E, = 350 MeV [16]. In these cases the position of the spectral maxima obtained by computer simulation coincides well with the experimental data (figs. 4b and c). The agreement of the calculated spectrum with the experimental results is rather better at E, = 350 MeV, because in the experiment [15] AB, = lop4 rad and in the experiment [16] At9, = 4.4 x lop6 rad, which is closer to the A@, = 0 of the simulation. Let us discuss the polarization characteristics of the radiation spectra considered above in figs. 4a-c. Here, as for the case of 900 MeV electrons (figs. 3d and e), the parameters (.$,(a)) and (t2( w)) averaged over all trajectories are practically zero. For this reason they are not shown below. The fast oscillations in t,(o) and Cz(o) for the separate trajectories are the basic cause that leads to (.$i( w)) = 0 and (.$*( 0)) - 0 after averaging over trajectories. In the continuum models [l-3] the values ti( w, 0 = 0), [z( U, 19= 0) are equal to zero for every trajectory and every U, therefore ((i(w)) = ($z( GJ )) = 0. Thus, the continuum model and computer simulation lead to a similar sequence ((5,) = (5,) = 0), but for quite different reasons.
Figs 5a-c show the variation of the Stokes parameters (t3(w)) calculated for the same electron energy as in figs. 4aac. From a comparison between figs. 4aac and figs. 5a-c, we obtain that the position of the maxima in the total radiation spectra correlates well with that of (t,(w)). In the region of the intensity maxima, the radiation is linearly polarized to a high degree ((4,) = 0.8550.92). The magnitude of (.$,(w)) depends essentially on the photon energy w and has a characteristic maximum., figs 5a-c, similar to that for 900 MeV electrons, fig. 3b. This differs essentially from the prediction of the continuum model, which leads to the value (.$s(w, 0 = 0)) = 1 for all photon frequencies emitted in planar channeling. A peculiarity in the behaviour of (t3(w)) in its dependence on w is the
0
ro
20
3
- xl
I.0
M bl
Fig. 3. Intensity spectrum (a) and Stokes parameters (b-d) for radiation from 900 MeV electrons under (110) panar channeling in diamond, as averaged over 900 trajectories. In (a) the crosses give experimental
values [15].
E.G. Vyatkin et al. / Polarization
34
change the sign of (t3( 0)) outside the regions characteristics for channeling radiation. This means that the radiation within these regions is linearly polarized in the OX direction (i.e. parallel to the channeling plane in contrary to the planar channeling radiation polarized along the OY direction (i.e. perpendicular to the channeling plane). The change of the sign of (t,(w)) is possible due to a high contribution to the radiation intensity from the E, component, being defined by the velocity&(t) and polarized in the OX direction (i.e., in the channeling plane). This effect, [3(w) < 0, is appreciable for many individual trajectories, see especially
of channeling radiation
fig. 2c. The behaviour of (t3( 0)) in this frequency regions has not yet been studied experimentally. The increase of the maximum value of ((a(w)),,, (figs 5a-c) with electron energy ((<3)max = 0.85 for E, = 54 MeV and (t3)max = 0.92 for 0, = 600 MeV) may be connected with the greater part of the channeled electrons over the total thickness h = 10 pm. Evidently, for each initial electron energy E, there exists an optimal value of the crystal thickness, at which ( t3( w)) is the greatest at the spectral maximum of the radiation intensity. In the first polarization experiment [4] the mean value o)= 0.65 f 0.15 of the degree of linear polarization within the spectral range Aw = 4-20 MeV is obtained for (110) planar channeling of 900 MeV electrons in diamond. As one can show, in the computer
a
1.0-
C
a.+
_ g++~ - e,++ c+ s_+ ’ % u1 t-
z e-4
-%
-
:
-A
+++
l 8*
%a
0.
t l
d
0
. ++ + +
5
1
* ..*..
t.....
IA
10 20 hi,, YPle”:
”
1
K)
I
35
Fig. 4. Total radiation spectra for electrons of different energies normalized to their maximum values (crosses - experiment, solid circles - simulation results): a) E, = 54 MeV, (110) C (experiment [17]); b) E, = 350 MeV, (110) Si (experiment [16]); c) E, = 600 MeV, (110) C (experiment (151)
Fig. 5. Stokes parameter (tj( w)), averaged over all trajectories for the cases in fig. 4 (a-c), which characterizes a linear polarization along the OY axis (6, > 0) and the OX axis (5, < 0).
E.G. Vyatkin et al. / Polarization of channeling radiation experiment
this value is detemined
by the equation:
k
k
CiE:l*+cI@i’ . k
k
(8)
I
Here (u(w, n) is the relative contribution of the radiation intensity at the frequency w in the n direction to the total radiation intensity in the range Aw = w,-wz and for a solid angle AQ. The Elk and E,k values are defined by eq. (6) and (E3(a, n)) by eq. (7). The calculation provided for the conditions of the experiment [4] according to (8) leads to the value (.$,(4-20 MeV)) = 0.677. This is in good agreement with the experiment [4].
35
intensity spectra. The position of these maxima coincides for electron energies less than 1 GeV with the region of the giant dipole resonance of the photonuclear reactions. This fact leads to the possibility of a detailed study of the nuclear structure via photonuclear reactions with the linearly polarized photons. The use of the linearly polarized photons emitted under planar channeling of relativistic electrons leads to the possibility of improved background conditions for photonuclear reaction experiments, activation analysis and Compton scattering. Both the above-mentioned cases and a more detailed study of the polarizationcharacteristics of the channeling radiation (temperature and thickness dependences, orientational dependence) are of further experimental and theoretical interest. Such investigations are also of importance for the creation of a possible intense source of monochromatic photons with a high degree of linear polarization and with easily changed photon energy for use in expermental nuclear physics.
4. Conclusion We have not considered two interesting cases in the investigated range of the electron energies. The first one is concerned with large angles of incidence S,,, ‘pe, when the channeling radiation transfers into coherent bremsstrahlung. As is known 1181, coherent bremsstr~lung radiation is polarized linearly at the characteristic maxima, which are in a harder region of the spectrum (e.g., for 900 MeV electrons the maxima of the coherent bremsstrahlung spectra are in the region of a hundred MeV). We shall study the transition from planar channeling radiation to coherent bremsstrahlung by means of computer simulations in a separate paper. The second case is a quantum region for electrons of low energies, E,, < 100 MeV. We expect the quantum peaks (fig. 4a) observed in the experiment [17] to appear for the degree of linear polarization if a correct quantitative quantal calculation is carried out. The quanta1 calculations [3] of the polarization characteristics of planar channeling radiation with the continuum planar potential should hence be modified with this purpose in mind. As to the results of computer simulations presented above, one can notice the following interesting results: The change of the oscillation frequency of electrons in a crystal channel (see for example fig. 2) during transmission through a crystal leads to a broadening of the radiation spectrum. On the other hand, this change of the channeling frequency complicates strongly the various possibilities for stimulation of channeling radiation by means of lasers or other external electromagnetic fields. Our calculations have shown that the planar channeling radiation is linearly polarized to a high degree in the range of the characteristics maxima of the
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E.G. Vyatkin et al. / Polarization
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