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Radiation of high-energy positrons channeled in bent crystals
Nuclear Instruments and Methods in Physics Research B31 (1988) 551-557 North-Holland, Amsterdam
RADIATION
OF HIGH-ENERGY
A.M. TARATIN
POSITRONS
CHANNELED
551
IN BENT CRYSTALS
and S.A. VOROBIEV
Nuclear Physics Institute,
Tomsk 634050, P.O. Box 25, USSR
Received 7 August 1987 and in revised form 25 November 1987
Computer calculations of radiation spectra of relativistic positrons planar channeled in a bent crystal were performed. It is observed that in the bent crystal, together with channeling radiation, additional quasi-synchrotronradiation occurs. The form of the total radiation spectra depends on the bending radius of the crystal. The intensity of the channeling radiation decreases with the increase of the crystal bend, while the intensity of quasi-synchrotron radiation increases. For a strong crystal bend the radiation intensities of the two types becomes comparable.
1. Introduction For channeling of relativistic electrons and positrons in a crystal hard electromagnetic radiation with high spectral intensity and directivity occurs [l]. Radiation emitted by particles planar channeled in crystals is similar to undulator radiation in its characteristics, but the period of transverse particle oscillations is determined by the planar crystal potential and depends on particle energy. It has been found in a Dubna experiment [2], that the channeled fraction of a swift proton beam follows bent planar channels and is deflected with a bent crystal. The deflection of a relativistic electron beam with a bent crystal axis was demonstrated experimentally at the Tomsk synchrotron [3]. It is obvious, that the additional curving of channeled particle trajectories due to a crystal bending should result in changes of channeling radiation spectra. The radiation of the particles moving along a circular trajectory, e.g., synchrotron radiation in cyclic accelerators and storage rings, is well studied and used for both scientific and applied purposes [4]. The synchrotron radiation spectrum is quasicontinuous and occupies mainly the frequency range (O-5) wc, where wc = !o,y3 is the characteristic frequency, 9, = u/R is the circular rotating frequency of the particle with the velocity u along the circle of the radius R, y is the particle Lorentz factor. The emission of radiation by relativistic electrons moving along the arc of a circle has been discussed in detail [5]. For an angular arc of dimensions A+ -< y-’ the radiation spectrum has the maximum at low frequencies near o = 0. The case where a particle completes the integer number N revolutions as well as the limit of high N, when the classical spectrum of synchrotron radiation is formed, have both been considered. For planar channeling in a uniformly bent crystal the particle moves along the arc of a circle simulta0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
neously performing radial oscillations in the channel field. For the first time the problem of quasi-synchrotron radiation of channeled particles in a bent crystal was discussed by Kaplin et al. [6]. The radiation in a bent crystal is of quasi-undulator type in the case I,, >> X, where X is the spatial period of particle oscillations in the channel, I,, = Ry-’ is the formation length of the radiation during the particle motion along the circular trajectory. This condition has been formulated by Bashmakov [7]. Otherwise, if I,, < X the radiation is of quasi-synchrotron type. This paper presents a computer analysis of radiation spectra of relativistic positrons planar channeled in a bent crystal. It was performed for positrons with y = lo4 channeled in th (110) planar channels of a bent silicon crystal. In the case under consideration the effective radiation angle &,, = y -’ is comparable to the critical channeling angle 0, and in a straight (unbent) crystal, radiation is formed by the full particle trajectory. The crystal bending causes a decrease of the radiation formation length, which leads to essential changes in the radiation spectra. For higher particle energies, when 0,/e,,, Z+ 1, the radiation is formed on a part of the trajectory, which is much less than the period of particle oscillations in the channel, and noticeable changes in the positron radiation spectra, as the crystal is bent, are not expected. The choice of particle energy was also stipulated by the consideration of the validity of classical electrodynamics for calculating the radiation spectra, when the number of energy levels in the transverse motion in the channel potential is rather large, but the quantum recoil and interaction of the particle spin with the effective radiation field can still be neglected [l]. The radiation spectra were calculated with particle trajectories in the averaged potential of planar channels. The particle flux redistribution in transverse energy due to multiple scattering by crystal electrons and nuclei was not taken into account. It is obvious. that the
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
552
calculated total radiation spectra should describe satisfactorily the experimentally observed ones in the direction, which is close to the tangent to the bent atomic planes at a crystal entrance.
where xi ‘X0, x2=2x,
i =c 2. Positron radiation intensity in a bent crystal
=
The radiative energy loss rate of a relativistic particle channeled in a crystal is determined by the averaged potential of a planar channel U(x) [l]:
After integrating we obtain x:-2x,1
=Z,,+Z,(R)+Z,(R),
2e2
-j,2,3Y21vw2>
(1)
where x, = ( pv/2u,)(Z2/R) is the coordinate of the minimum of the effective potential shifting to the outer wall of a bent channel as the crystal is bent, x is calculated from the center of a channel in the radial direction. The crystal bending results also in decreasing the critical transverse energy of channeling E,,(R) = v,,,(Z, R) PI. In (1) the radiation intensity should be averaged over the crystal thickness and particle ensemble =
f%(E,,)
dE,,$jX’Z(x) x2
x(x,
dx &I,
R) ’
(2) where T is the period of particle oscillations in a channel, E,., is the initial transverse energy of the particle, P(E,.,) is the particle distribution over transverse energies at the crystal entrance, x is the transverse particle velocity, x, and x2 are the coordinates of the particle turning points in the effective potential. For the particles entering a bent crystal parallel to bent planes E,, = Uefr(xo, R), where xc is the coordinate of the point of entry. Thus, in (2) the averaging over the particle ensemble (E,,) is reduced to that over x0
Z(R) = j+
l/2
R) - u,rr(xv RI) 1
6
where Z(x) is the instantaneous intensity of radiation, e and m are the charge and rest mass of the electron, c is the light velocity. In a first approximation the planar channel potential for positrons can be described as harmonic, U(x) = U,(X/~)~, where I is the channel half-width, U, is the potential barrier separating the channels in crystal. In a bent crystal a channeled particle is also influenced by the centrifugal force F, = pV/R, where p is the particle momentum, and the particle motion in the transverse (radial) direction can be described by the effective potential
Z(R)
[&&J,
~[(x1-x)(x-X2)y2.
f&x) = -Z(x) =
- xa,
/‘dx,+/X’Z(x)$, x”, x2
+x;
1
(4)
where Z,, is the radiation intensity accompanying particle channeling in L straight (unbent) crystal, Z, and Z, are the terms that describe the changes in the radiation intensity of the channeled particles as the crystal is bent due to the decrease of particle oscillation amplitude in a channel and the displacement of the equilibrium trajectories to the outer channel wall in the region with different from zero intensity of the electric field of the channel. The last term determines the radiation intensity due to the particle motion along the bent channel Zs( R)
= 3 $p4y4.
where /?= u/c. This is a known formula for the synchrotron radiation intensity, which increases with decreasing radius as R-2. Fig. 1 shows the dependence of the radiation intensity of a positron with y = lo4 on the radius of the crystal bending during planar channeling along (110) planes of a silicon crystal. This dependence is calculated according to eq. (4). Curves 1 and 2 show the intensity of channeling radiation in the straight crystal Z,, and quasi-synchrotron radiation Z,, respectively. With increasing crystal bend the total radiation intensity of the channeled positrons, eq. (4), curve 3, decreases at the beginning, but then it increases rapidly due to the synchrotron radiation. In the approximation considered for a channel potential a minimum of the intensity is observed at R = 7R,, where R,=(pv/2u,)l is the critical radius of the crystal bend for channeling. The efficiency of the particle capture into channeling regime also decreases with the crystal bend P,(R) = 1 x,(R)/I. As a result the radiation intensity per incident positron (curve 4) decreases more rapidly with increasing crystal bend, but has a local maximum at the bend close to a critical one, R=
-R,.
7
3+fi
The quasi-channeling particles, moving in an average potential of atomic planes and crossing them, also contribute to the radiation. The radiation intensity of
553
A.M. Taraiin, S.A. Vorobiev / Channeling radiation from bent crystals
crystal is bent at 0,~ 0, a moderate decrease of the integrated radiation intensity will be observed. Finally, at a large orientation angle Z?,,> 0, and a small crystal bend the mean intensity of positron radiation may be rather higher in the case when u,(O) < 0 than in a contrasting case, where u,(O) is the initial radial velocity of a particle. Besides, in the given case the effect of a volume capture of quasi-channeled particles into the channeling regime may be essential.
3. Positron radiation spectra in a bent crystal
‘...,,,f ‘.-. ... .. ... . .....
I
R1,
R(m) aa
Fig. 1. Dependence of the radiation intensity of positrons with y =104 on the radius R of the crystal bending at planar channeling along (110) planes in a Si crystal (curve 3). Curve 4 shows the same, but the capture efficiency in channeling regime is taken into account. Curve 1 - radiation intensity in a straight crystal; 2 - intensity of quasi-synchrotron radiation in a bent crystal.
quasi-channeled particles decreases as the orientation angle to the plane 0 increases. In unbent crystal it decreases from 3Z,, at B = 0, to 2Z,, when the angles exceed 0, by a few times. While a particle is passing through the bent crystal, the angle B varies, it increases with distance from a turning point in radial motion. Therefore, the radiation intensity of the quasi-channeled particles is maximum near the turning points. At a small crystal bend a particle remains near the turning points for a long time and a mean radiation intensity may be close to the maximum Zch= 3Z,,. On the contrary at a strong crystal bend it equals 2Z,, practically everywhere in the crystal: As the crystal is bent, the total integrated radiation intensity of the channeled and the quasi-channeled fraction Z,(R)
=Z(R)P,(R)
+Zch(R)(l
The general expression for the spectral-angular distribution of the radiation energy of a particle during the total time of motion on the trajectory r(t) has the form
[9,W d=W dttw d0
Here A, is the vector proportional to a Fourier-component of the intensity of electrical field of a particle at the observation point, II is the unit vector in the direction of radiation, (Y= l/137. Taking into account the changes of the longitudinal velocity of a particle in the harmonic approximation for the channel potential in an uniformly bent crystal of
t
ii
-P,)
changes from
to
for beam incidence parallel to the atomic planes. When the beam enters a crystal at an angle 0,~ e, to the planes the channeled fraction becomes less than 1 but the particles move with high amplitudes in the channels and their intensity is close to maximum as is the radiation intensity of the quasi-channeled fraction. As the
Fig. 2. Schematic planar channeling
representation of a positron trajectory for in a bent crystal. Here n is the unit vector in the “forward” direction.
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
554
radius R the equation of the positron trajectory in the planar channel (fig. 2) can be written as follows r(t)
= {r,(t)
r,(t)
sin 4(t),
O},
s(t)Isinw,t,
r,(t)=R+x,, +P(t)=R=i
cos $,(t),
1,
$t--$$sin2,,,
(7)
where s(t) is the particle path length along the channel, p = p(1 - /?i/4p2) is th e mean longitudinal velocity of the particle (the longitudinal velocity changes are related to the transverse particle oscillations in the channel field), & = xooO/c, w0 = (c/1)(2 u,/E)‘/~ is the oscillation frequency of the particle in a channel, x0 is the oscillation amplitude. 3.1. Radiation spectra several trajectories
in the “forward”
Substituting s = w,t lying in orbital plane A =L 0
As a0 / -As
(Cos ~-&)~y-(~in~-~y)/%
s-oocos~
-w.
direction for
we obtain for the directions
(1 - cos +.&-sin r,(x)
G(S) sin$,(s)
sin Cp
C
particle with oscillation amplitude x0 = 0 in a bent channel, the expression simplifies [S]:
+./3,)’ cos $,(s) C
ds,
where AS = woAt, At = L/2& L is the crystal length. The projections of the particle velocity and acceleration Is,, fix, /?,, & are defined from (7). In the case of particle motion along the arc of a circle, i.e., for the
Let us consider the positron radiation spectra for channeling in a bent crystal for the “forward” direction, i.e., tangent direction to the mid point of a bent crystal (see fig. 2). First, consider two limiting cases: (i) the case of radiation from a positron moving along the arc of a circle with the oscillation amplitude x,, = 0, and (ii) that with the oscillations in the channel with the curvature R- ’ = 0 (a straight channel). Fig. 3a shows the radiation spectra in the “forward” direction for a positron with y = lo4 moving along the arc of a circle with radius R = 10.8 cm. This value is by one order of magnitude greater than the critical bending radius of (110) channels. With increasing arc length (curves l-4) the energy emitted in the soft frequency range near w = 0 increase at first and then decreases. In consequence, the maximum is formed at w = wC (A wC= 2.74 MeV) in the radiation spectrum. The spectrum of quasi-synchrotron radiation is formed completely when the arc length exceeds I,, by more than an order. In a straight crystal the radiation spectrum from the channeled positrons in the forward direction at angle ~9= 0 with the channel axis has only the odd harmonics at frequencies of
c+.(fl=O)=k% 1-P
a,
=k
1+
0.5pzy2 ’
where wU= 2y2wo - y312. The dependence of the mean
a.
Fig. 3. (a) circle with (4) - 24X. Si crystal
Spectral distribution of the radiation energy in the “forward” direction of a positron with y = lo4 moving along an arc of a the radius R = 10.8 cm and the length 15. Curves (l)-(4) are. given for different arc lengths L: (1) - A, (2) - 2A, (3) - 4X, (b) Intensity radiation spectrum in the forward direction of a positron with y =104 at channeling along (110) planes in a of L = 32h thickness, the particle oscillation amplitude in a channel being x o = 0.81. Here A = 6.4 pm is the period of particle oscillations in (1lO)Si channel.
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
555
a
Fig. 4. Spectral distribution of radiation energy in the “forward” direction of a positron with y = lo4 channeled along (110) planes in bent silicon crystal with radius R and length L =128h, the particle oscillation amplitude in the channel being x,, = 0.41. (a) R = 76.38 cm, w,/w, =lOO, &,/A 211; b) R = 7.64 cm, w,/w, =lO, I,,,/h 21.
longitudinal velocity on the oscillation amplitude of a particle in the channel x0 results also in the radiation frequency ( wk) dependence upon x,,. The width of the radiation harmonics depends on the crystal length Aw/wk = 1/Nk, where N is the number of particle oscillations in a crystal channel. Fig. 3b shows the spectrum of the radiation intensity of a positron with y = lo4 in the forward direction at channeling along the (110) planes in a Si crystal of 200 pm thickness (n = 32, Aw, = 38.75 MeV). In the examples considered the characteristic radiation frequencies differ by more than one order of magnitude: w,/o, > 10. For channeling in a bent crystal the positrons moving along the arc of a bent channel undergo radial oscillations in a channel field and the radiation occurs both at frequencies typical for the channeling radiation and those for the radiation of the particle moving along the arc of a circle. Fig. 4 shows the radiation spectra in the “forward” direction for a positron with y = lo4 channeling along the (110) planes in a bent Si crystal with oscillation amplitude x0 = 0.41, being calculated according to (Q-(8). For a small bending with R x=-hy (fig. 4a) the maxima observed in the spectra can be easily identified with the radiation harmonics in a straight (unbent) crystal (marked by arrows). However, in contrast to a straight crystal the spectrum also shows the even harmonics. Moreover, due to the splitting of these harmonics the subharmonics at frequencies less than those of a straight crystal in the “forward” direction ~~(0) are observed. Apparently, these subharmonics can be considered as the corresponding radiation harmonics from the trajectory parts, which are at some angle to the above direction. Since the central trajectory section does not contribute to the radiation at even harmonics in the considered direction,
the non-central trajectory parts mainly form the even harmonics. For a large crystal bending, when the coherence length and the wavelength of the particle oscillations in a channel are comparable, I,, = A (fig. 4b), it becomes difficult to identify the maxima observed in the spectrum with the several radiation harmonics in a straight crystal. There is an increase in the width of separate maxima in a spectrum, while the main one still coincides with the first harmonic of the channeling radiation. The frequencies of the quasi-synchrotron radiation resulting from the particle motion along the arc of a bent channel approaches those of the radiation resulting from particle oscillations in a channel (for the above case w,/oC = 10). Energy loss by radiation of both types are also becoming comparable, the spectrum shows the maximum at low frequencies which appear due to quasisynchrotron radiation. Fig. 5 shows only those spectra fraction near w = 0, which resulted from quasi-synchrotron radiation at positron channeling in a bent channel. For a small bending, with a trajectory curvature due to particle oscillations in channel being much greater than that of a channel, changes occur only in a tail of the quasi-synchrotron spectrum (compare with the spectrum in fig. 3a). With increased bending of the crystal the total spectrum is distorted. 3.2. Integral
positron radiation spectrum in a bent crystal
Let us calculate the integral spectra of positron radiation for planar channeling in a bent crystal into the solid angle AD = y-’ in the tangent direction to the crystal at its entrance face. When the crystal length is much greater than the coherence length, L s=- lcohr which
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
556
a
w/w, Fig. 5. The same as in fig. 4, but for a quasi-synchrotron
region of the positron region (0-5)~~.
is realized for all the cases under consideration with R < 1 m and y = 104, there is no azimuthal dependence of the spectrum within the range of A$J = y-l, and for the calculation of the integral spectrum it is necessary to integrate (6) with respect to the angle 8’ only, the latter being measured from the positron orbit plane
g(“o)= dfi = Y-‘~-‘-$&(x~) d6”, l,,=_2&‘xo’
(11)
radiation
spectrum
in a bent crystal
in the frequency
frequencies. In addition, one can observe a small maximum of the quasi-synchrotron radiation in the frequency range (0-5)~~ together with the radiation in the intermediate frequency region between o, and w,, which changes the low-frequency edge of the quasi-undulator part of the spectrum. Increase of the crystal bending results in the decrease of the radiation yield at frequencies that correspond to radiation harmonics in straight (unbent) crystal. The maximum positions of the first and the second harmonics shift, the spectral maxima
To calculate an experimentally observed spectrum the formula (11) need be averaged over a particle ensemble. For a monodirected beam entering a bent crystal in the tangent direction this procedure is reduced to the averaging over the coordinates of entry points (amplitudes of particle oscillations in a channel) dW -=dAw
1 l-x,
‘-Xmg(xo) /0
dx,.
(12)
In (11) and (12) the numerical integration was performed. Fig. 6 shows the radiation spectra of positrons with y = lo4 for channeling along the (110) planes in a bent Si crystal, the spectra are integrated over the solid angle A52 = y-*. The features typical of the “forward” spectra from the several trajectories are well pronounced in the integral spectra. For small crystal bending (fig. 6a, b) the spectral maxima correspond to radiation harmonics in a straight crystal. The odd harmonics positions are determined by the radiation frequency in the “forward” direction for the positron with the oscillation amplitude being maximum possible in a channel Wk= w,(B=O, x0= 1 -x,). The maxima of the even harmonics are shifted from 5, to the smaller
4h
w/WC
Fig. 6. Integral radiation spectrum over a solid angle A0 = y-’ for positrons with y =104 channeled along (110) planes in a bent silicon crystal. The beam enters into the crystal in tangent direction with a zero angular divergence. (a) R = 76.38 cm, w,/w,=lOO, I,,/hhll; (b) R =38.19 cm, w,/w,=50, I,,/X>5; (c) R=15.28 cm, wU/wC=20, I,,/X>2; (d) R = 7.64 cm, wU/wC = 10,1,/X >- 1. Here Gj, = wk(8 = 0, x0 =1-x,).
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
broaden and merge into a single wide one as the crystal is strongly bent (fig. 6~). The quasi-synchrotron radiation maximum is broadened and, in the long run, has a “comb” structure. We have discussed above only the radiation spectra of the channeled fraction of a beam. Radiation emitted by quasi-channeled particles occurs at higher frequencies than for channeled particles because the frequencies of transverse particle oscillations are higher (they are determined by orientation angles to the planes). As the crystal is bent, the integrated radiation spectrum of channeled and quasi-channeled particles is broadened due to a quasi-channeled fraction since the range of the particle orientation angles to the atomic planes becomes wider. The multiple scattering of the particles by the crystal electrons and nuclei decreases the channeled fraction and the radiation intensity in a soft region of frequencies will be less than for the spectrum calculated without multiple scattering.
557
particle due to motion along the arc of a bent channel; it occurs together with the radiation in an intermediate frequency range between w, and wc. (4) With the increase of the crystal bending the radiation at frequencies corresponding to the radiation harmonics in a straight (unbent) crystal decreases and the quasi-synchrotron part of the radiation spectrum of channeled positrons is broadening. As a result at a strong crystal bending, R = Xv, the energy loss by radiation of both types are becoming comparable. (5) The quasi-synchrotron radiation spectrum of a particle moving along the arc of a circle, i.e., along the bent channels in crystal, acquires a “comb” structure as the radial particle oscillations occur near the equilibrium orbit.
References [l] Coherent Radiation Sources, eds. A.W. Saenz and H.
4. Conclusion (1) The channeling radiation intensity of positrons decreases with increasing crystal bending due to decrease in both the amplitudes of particle oscillations in the channels and the particle capture efficiency into channeling regime. (2) For a small bending of the crystal, R x=- Xy, the positron radiation spectrum is quasi-undulatory and the spectral maxima correspond to radiation harmonics in a straight (unbent) crystal wk - kw,. But the width of the harmonics increases and the subharmonics at frequencies less than those of an unbent crystal are observed. (3) In a bent crystal, quasi-synchrotron radiation occurs at low frequencies (0-5)~~. For a channeled
ijberall (Springer, Berlin, 1985). [2] A.F. Elishev, N.A. Filatova, V.M. Golovatyuk et al., Phys. Lett. B88 (1979) 387. [3] Yu.N. Adishchev, P.S. Ananyinm A.N. Didenko et al., Phys. Lett. A77 (1980) 263. (41 A.A. Sokolov and I.M. Temov, The Relativistic Electron (Nauka, Moscow, 1983) in Russian. [5] V.G. Bagrov, I.M. Temov and N.I. Fedosov, Zh. Eksp. Teor. Fiz. 82 (1982) 1442. [6] V.V. Kaplin and S.A. Vorobiev, Phys. Lett. A67 (1978) 135. [7] Yu.A. Bashmakov, Radiat. Eff. 56 (1981) 55. [8] A.M. Taratin and S.A. Vorobiev, Phys. Status Solidi (b) 107 (1981) 521. [9] L.D. Landau and E.M. Lifshitz The Classical Theory of Fields, 3rd ed. (Pergamon, Oxford, 1971). [lo] J. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
RADIATION
OF HIGH-ENERGY
A.M. TARATIN
POSITRONS
CHANNELED
551
IN BENT CRYSTALS
and S.A. VOROBIEV
Nuclear Physics Institute,
Tomsk 634050, P.O. Box 25, USSR
Received 7 August 1987 and in revised form 25 November 1987
Computer calculations of radiation spectra of relativistic positrons planar channeled in a bent crystal were performed. It is observed that in the bent crystal, together with channeling radiation, additional quasi-synchrotronradiation occurs. The form of the total radiation spectra depends on the bending radius of the crystal. The intensity of the channeling radiation decreases with the increase of the crystal bend, while the intensity of quasi-synchrotron radiation increases. For a strong crystal bend the radiation intensities of the two types becomes comparable.
1. Introduction For channeling of relativistic electrons and positrons in a crystal hard electromagnetic radiation with high spectral intensity and directivity occurs [l]. Radiation emitted by particles planar channeled in crystals is similar to undulator radiation in its characteristics, but the period of transverse particle oscillations is determined by the planar crystal potential and depends on particle energy. It has been found in a Dubna experiment [2], that the channeled fraction of a swift proton beam follows bent planar channels and is deflected with a bent crystal. The deflection of a relativistic electron beam with a bent crystal axis was demonstrated experimentally at the Tomsk synchrotron [3]. It is obvious, that the additional curving of channeled particle trajectories due to a crystal bending should result in changes of channeling radiation spectra. The radiation of the particles moving along a circular trajectory, e.g., synchrotron radiation in cyclic accelerators and storage rings, is well studied and used for both scientific and applied purposes [4]. The synchrotron radiation spectrum is quasicontinuous and occupies mainly the frequency range (O-5) wc, where wc = !o,y3 is the characteristic frequency, 9, = u/R is the circular rotating frequency of the particle with the velocity u along the circle of the radius R, y is the particle Lorentz factor. The emission of radiation by relativistic electrons moving along the arc of a circle has been discussed in detail [5]. For an angular arc of dimensions A+ -< y-’ the radiation spectrum has the maximum at low frequencies near o = 0. The case where a particle completes the integer number N revolutions as well as the limit of high N, when the classical spectrum of synchrotron radiation is formed, have both been considered. For planar channeling in a uniformly bent crystal the particle moves along the arc of a circle simulta0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
neously performing radial oscillations in the channel field. For the first time the problem of quasi-synchrotron radiation of channeled particles in a bent crystal was discussed by Kaplin et al. [6]. The radiation in a bent crystal is of quasi-undulator type in the case I,, >> X, where X is the spatial period of particle oscillations in the channel, I,, = Ry-’ is the formation length of the radiation during the particle motion along the circular trajectory. This condition has been formulated by Bashmakov [7]. Otherwise, if I,, < X the radiation is of quasi-synchrotron type. This paper presents a computer analysis of radiation spectra of relativistic positrons planar channeled in a bent crystal. It was performed for positrons with y = lo4 channeled in th (110) planar channels of a bent silicon crystal. In the case under consideration the effective radiation angle &,, = y -’ is comparable to the critical channeling angle 0, and in a straight (unbent) crystal, radiation is formed by the full particle trajectory. The crystal bending causes a decrease of the radiation formation length, which leads to essential changes in the radiation spectra. For higher particle energies, when 0,/e,,, Z+ 1, the radiation is formed on a part of the trajectory, which is much less than the period of particle oscillations in the channel, and noticeable changes in the positron radiation spectra, as the crystal is bent, are not expected. The choice of particle energy was also stipulated by the consideration of the validity of classical electrodynamics for calculating the radiation spectra, when the number of energy levels in the transverse motion in the channel potential is rather large, but the quantum recoil and interaction of the particle spin with the effective radiation field can still be neglected [l]. The radiation spectra were calculated with particle trajectories in the averaged potential of planar channels. The particle flux redistribution in transverse energy due to multiple scattering by crystal electrons and nuclei was not taken into account. It is obvious. that the
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
552
calculated total radiation spectra should describe satisfactorily the experimentally observed ones in the direction, which is close to the tangent to the bent atomic planes at a crystal entrance.
where xi ‘X0, x2=2x,
i =c 2. Positron radiation intensity in a bent crystal
=
The radiative energy loss rate of a relativistic particle channeled in a crystal is determined by the averaged potential of a planar channel U(x) [l]:
After integrating we obtain x:-2x,1
=Z,,+Z,(R)+Z,(R),
2e2
-j,2,3Y21vw2>
(1)
where x, = ( pv/2u,)(Z2/R) is the coordinate of the minimum of the effective potential shifting to the outer wall of a bent channel as the crystal is bent, x is calculated from the center of a channel in the radial direction. The crystal bending results also in decreasing the critical transverse energy of channeling E,,(R) = v,,,(Z, R) PI. In (1) the radiation intensity should be averaged over the crystal thickness and particle ensemble =
f%(E,,)
dE,,$jX’Z(x) x2
x(x,
dx &I,
R) ’
(2) where T is the period of particle oscillations in a channel, E,., is the initial transverse energy of the particle, P(E,.,) is the particle distribution over transverse energies at the crystal entrance, x is the transverse particle velocity, x, and x2 are the coordinates of the particle turning points in the effective potential. For the particles entering a bent crystal parallel to bent planes E,, = Uefr(xo, R), where xc is the coordinate of the point of entry. Thus, in (2) the averaging over the particle ensemble (E,,) is reduced to that over x0
Z(R) = j+
l/2
R) - u,rr(xv RI) 1
6
where Z(x) is the instantaneous intensity of radiation, e and m are the charge and rest mass of the electron, c is the light velocity. In a first approximation the planar channel potential for positrons can be described as harmonic, U(x) = U,(X/~)~, where I is the channel half-width, U, is the potential barrier separating the channels in crystal. In a bent crystal a channeled particle is also influenced by the centrifugal force F, = pV/R, where p is the particle momentum, and the particle motion in the transverse (radial) direction can be described by the effective potential
Z(R)
[&&J,
~[(x1-x)(x-X2)y2.
f&x) = -Z(x) =
- xa,
/‘dx,+/X’Z(x)$, x”, x2
+x;
1
(4)
where Z,, is the radiation intensity accompanying particle channeling in L straight (unbent) crystal, Z, and Z, are the terms that describe the changes in the radiation intensity of the channeled particles as the crystal is bent due to the decrease of particle oscillation amplitude in a channel and the displacement of the equilibrium trajectories to the outer channel wall in the region with different from zero intensity of the electric field of the channel. The last term determines the radiation intensity due to the particle motion along the bent channel Zs( R)
= 3 $p4y4.
where /?= u/c. This is a known formula for the synchrotron radiation intensity, which increases with decreasing radius as R-2. Fig. 1 shows the dependence of the radiation intensity of a positron with y = lo4 on the radius of the crystal bending during planar channeling along (110) planes of a silicon crystal. This dependence is calculated according to eq. (4). Curves 1 and 2 show the intensity of channeling radiation in the straight crystal Z,, and quasi-synchrotron radiation Z,, respectively. With increasing crystal bend the total radiation intensity of the channeled positrons, eq. (4), curve 3, decreases at the beginning, but then it increases rapidly due to the synchrotron radiation. In the approximation considered for a channel potential a minimum of the intensity is observed at R = 7R,, where R,=(pv/2u,)l is the critical radius of the crystal bend for channeling. The efficiency of the particle capture into channeling regime also decreases with the crystal bend P,(R) = 1 x,(R)/I. As a result the radiation intensity per incident positron (curve 4) decreases more rapidly with increasing crystal bend, but has a local maximum at the bend close to a critical one, R=
-R,.
7
3+fi
The quasi-channeling particles, moving in an average potential of atomic planes and crossing them, also contribute to the radiation. The radiation intensity of
553
A.M. Taraiin, S.A. Vorobiev / Channeling radiation from bent crystals
crystal is bent at 0,~ 0, a moderate decrease of the integrated radiation intensity will be observed. Finally, at a large orientation angle Z?,,> 0, and a small crystal bend the mean intensity of positron radiation may be rather higher in the case when u,(O) < 0 than in a contrasting case, where u,(O) is the initial radial velocity of a particle. Besides, in the given case the effect of a volume capture of quasi-channeled particles into the channeling regime may be essential.
3. Positron radiation spectra in a bent crystal
‘...,,,f ‘.-. ... .. ... . .....
I
R1,
R(m) aa
Fig. 1. Dependence of the radiation intensity of positrons with y =104 on the radius R of the crystal bending at planar channeling along (110) planes in a Si crystal (curve 3). Curve 4 shows the same, but the capture efficiency in channeling regime is taken into account. Curve 1 - radiation intensity in a straight crystal; 2 - intensity of quasi-synchrotron radiation in a bent crystal.
quasi-channeled particles decreases as the orientation angle to the plane 0 increases. In unbent crystal it decreases from 3Z,, at B = 0, to 2Z,, when the angles exceed 0, by a few times. While a particle is passing through the bent crystal, the angle B varies, it increases with distance from a turning point in radial motion. Therefore, the radiation intensity of the quasi-channeled particles is maximum near the turning points. At a small crystal bend a particle remains near the turning points for a long time and a mean radiation intensity may be close to the maximum Zch= 3Z,,. On the contrary at a strong crystal bend it equals 2Z,, practically everywhere in the crystal: As the crystal is bent, the total integrated radiation intensity of the channeled and the quasi-channeled fraction Z,(R)
=Z(R)P,(R)
+Zch(R)(l
The general expression for the spectral-angular distribution of the radiation energy of a particle during the total time of motion on the trajectory r(t) has the form
[9,W d=W dttw d0
Here A, is the vector proportional to a Fourier-component of the intensity of electrical field of a particle at the observation point, II is the unit vector in the direction of radiation, (Y= l/137. Taking into account the changes of the longitudinal velocity of a particle in the harmonic approximation for the channel potential in an uniformly bent crystal of
t
ii
-P,)
changes from
to
for beam incidence parallel to the atomic planes. When the beam enters a crystal at an angle 0,~ e, to the planes the channeled fraction becomes less than 1 but the particles move with high amplitudes in the channels and their intensity is close to maximum as is the radiation intensity of the quasi-channeled fraction. As the
Fig. 2. Schematic planar channeling
representation of a positron trajectory for in a bent crystal. Here n is the unit vector in the “forward” direction.
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
554
radius R the equation of the positron trajectory in the planar channel (fig. 2) can be written as follows r(t)
= {r,(t)
r,(t)
sin 4(t),
O},
s(t)Isinw,t,
r,(t)=R+x,, +P(t)=R=i
cos $,(t),
1,
$t--$$sin2,,,
(7)
where s(t) is the particle path length along the channel, p = p(1 - /?i/4p2) is th e mean longitudinal velocity of the particle (the longitudinal velocity changes are related to the transverse particle oscillations in the channel field), & = xooO/c, w0 = (c/1)(2 u,/E)‘/~ is the oscillation frequency of the particle in a channel, x0 is the oscillation amplitude. 3.1. Radiation spectra several trajectories
in the “forward”
Substituting s = w,t lying in orbital plane A =L 0
As a0 / -As
(Cos ~-&)~y-(~in~-~y)/%
s-oocos~
-w.
direction for
we obtain for the directions
(1 - cos +.&-sin r,(x)
G(S) sin$,(s)
sin Cp
C
particle with oscillation amplitude x0 = 0 in a bent channel, the expression simplifies [S]:
+./3,)’ cos $,(s) C
ds,
where AS = woAt, At = L/2& L is the crystal length. The projections of the particle velocity and acceleration Is,, fix, /?,, & are defined from (7). In the case of particle motion along the arc of a circle, i.e., for the
Let us consider the positron radiation spectra for channeling in a bent crystal for the “forward” direction, i.e., tangent direction to the mid point of a bent crystal (see fig. 2). First, consider two limiting cases: (i) the case of radiation from a positron moving along the arc of a circle with the oscillation amplitude x,, = 0, and (ii) that with the oscillations in the channel with the curvature R- ’ = 0 (a straight channel). Fig. 3a shows the radiation spectra in the “forward” direction for a positron with y = lo4 moving along the arc of a circle with radius R = 10.8 cm. This value is by one order of magnitude greater than the critical bending radius of (110) channels. With increasing arc length (curves l-4) the energy emitted in the soft frequency range near w = 0 increase at first and then decreases. In consequence, the maximum is formed at w = wC (A wC= 2.74 MeV) in the radiation spectrum. The spectrum of quasi-synchrotron radiation is formed completely when the arc length exceeds I,, by more than an order. In a straight crystal the radiation spectrum from the channeled positrons in the forward direction at angle ~9= 0 with the channel axis has only the odd harmonics at frequencies of
c+.(fl=O)=k% 1-P
a,
=k
1+
0.5pzy2 ’
where wU= 2y2wo - y312. The dependence of the mean
a.
Fig. 3. (a) circle with (4) - 24X. Si crystal
Spectral distribution of the radiation energy in the “forward” direction of a positron with y = lo4 moving along an arc of a the radius R = 10.8 cm and the length 15. Curves (l)-(4) are. given for different arc lengths L: (1) - A, (2) - 2A, (3) - 4X, (b) Intensity radiation spectrum in the forward direction of a positron with y =104 at channeling along (110) planes in a of L = 32h thickness, the particle oscillation amplitude in a channel being x o = 0.81. Here A = 6.4 pm is the period of particle oscillations in (1lO)Si channel.
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
555
a
Fig. 4. Spectral distribution of radiation energy in the “forward” direction of a positron with y = lo4 channeled along (110) planes in bent silicon crystal with radius R and length L =128h, the particle oscillation amplitude in the channel being x,, = 0.41. (a) R = 76.38 cm, w,/w, =lOO, &,/A 211; b) R = 7.64 cm, w,/w, =lO, I,,,/h 21.
longitudinal velocity on the oscillation amplitude of a particle in the channel x0 results also in the radiation frequency ( wk) dependence upon x,,. The width of the radiation harmonics depends on the crystal length Aw/wk = 1/Nk, where N is the number of particle oscillations in a crystal channel. Fig. 3b shows the spectrum of the radiation intensity of a positron with y = lo4 in the forward direction at channeling along the (110) planes in a Si crystal of 200 pm thickness (n = 32, Aw, = 38.75 MeV). In the examples considered the characteristic radiation frequencies differ by more than one order of magnitude: w,/o, > 10. For channeling in a bent crystal the positrons moving along the arc of a bent channel undergo radial oscillations in a channel field and the radiation occurs both at frequencies typical for the channeling radiation and those for the radiation of the particle moving along the arc of a circle. Fig. 4 shows the radiation spectra in the “forward” direction for a positron with y = lo4 channeling along the (110) planes in a bent Si crystal with oscillation amplitude x0 = 0.41, being calculated according to (Q-(8). For a small bending with R x=-hy (fig. 4a) the maxima observed in the spectra can be easily identified with the radiation harmonics in a straight (unbent) crystal (marked by arrows). However, in contrast to a straight crystal the spectrum also shows the even harmonics. Moreover, due to the splitting of these harmonics the subharmonics at frequencies less than those of a straight crystal in the “forward” direction ~~(0) are observed. Apparently, these subharmonics can be considered as the corresponding radiation harmonics from the trajectory parts, which are at some angle to the above direction. Since the central trajectory section does not contribute to the radiation at even harmonics in the considered direction,
the non-central trajectory parts mainly form the even harmonics. For a large crystal bending, when the coherence length and the wavelength of the particle oscillations in a channel are comparable, I,, = A (fig. 4b), it becomes difficult to identify the maxima observed in the spectrum with the several radiation harmonics in a straight crystal. There is an increase in the width of separate maxima in a spectrum, while the main one still coincides with the first harmonic of the channeling radiation. The frequencies of the quasi-synchrotron radiation resulting from the particle motion along the arc of a bent channel approaches those of the radiation resulting from particle oscillations in a channel (for the above case w,/oC = 10). Energy loss by radiation of both types are also becoming comparable, the spectrum shows the maximum at low frequencies which appear due to quasisynchrotron radiation. Fig. 5 shows only those spectra fraction near w = 0, which resulted from quasi-synchrotron radiation at positron channeling in a bent channel. For a small bending, with a trajectory curvature due to particle oscillations in channel being much greater than that of a channel, changes occur only in a tail of the quasi-synchrotron spectrum (compare with the spectrum in fig. 3a). With increased bending of the crystal the total spectrum is distorted. 3.2. Integral
positron radiation spectrum in a bent crystal
Let us calculate the integral spectra of positron radiation for planar channeling in a bent crystal into the solid angle AD = y-’ in the tangent direction to the crystal at its entrance face. When the crystal length is much greater than the coherence length, L s=- lcohr which
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
556
a
w/w, Fig. 5. The same as in fig. 4, but for a quasi-synchrotron
region of the positron region (0-5)~~.
is realized for all the cases under consideration with R < 1 m and y = 104, there is no azimuthal dependence of the spectrum within the range of A$J = y-l, and for the calculation of the integral spectrum it is necessary to integrate (6) with respect to the angle 8’ only, the latter being measured from the positron orbit plane
g(“o)= dfi = Y-‘~-‘-$&(x~) d6”, l,,=_2&‘xo’
(11)
radiation
spectrum
in a bent crystal
in the frequency
frequencies. In addition, one can observe a small maximum of the quasi-synchrotron radiation in the frequency range (0-5)~~ together with the radiation in the intermediate frequency region between o, and w,, which changes the low-frequency edge of the quasi-undulator part of the spectrum. Increase of the crystal bending results in the decrease of the radiation yield at frequencies that correspond to radiation harmonics in straight (unbent) crystal. The maximum positions of the first and the second harmonics shift, the spectral maxima
To calculate an experimentally observed spectrum the formula (11) need be averaged over a particle ensemble. For a monodirected beam entering a bent crystal in the tangent direction this procedure is reduced to the averaging over the coordinates of entry points (amplitudes of particle oscillations in a channel) dW -=dAw
1 l-x,
‘-Xmg(xo) /0
dx,.
(12)
In (11) and (12) the numerical integration was performed. Fig. 6 shows the radiation spectra of positrons with y = lo4 for channeling along the (110) planes in a bent Si crystal, the spectra are integrated over the solid angle A52 = y-*. The features typical of the “forward” spectra from the several trajectories are well pronounced in the integral spectra. For small crystal bending (fig. 6a, b) the spectral maxima correspond to radiation harmonics in a straight crystal. The odd harmonics positions are determined by the radiation frequency in the “forward” direction for the positron with the oscillation amplitude being maximum possible in a channel Wk= w,(B=O, x0= 1 -x,). The maxima of the even harmonics are shifted from 5, to the smaller
4h
w/WC
Fig. 6. Integral radiation spectrum over a solid angle A0 = y-’ for positrons with y =104 channeled along (110) planes in a bent silicon crystal. The beam enters into the crystal in tangent direction with a zero angular divergence. (a) R = 76.38 cm, w,/w,=lOO, I,,/hhll; (b) R =38.19 cm, w,/w,=50, I,,/X>5; (c) R=15.28 cm, wU/wC=20, I,,/X>2; (d) R = 7.64 cm, wU/wC = 10,1,/X >- 1. Here Gj, = wk(8 = 0, x0 =1-x,).
A.M. Taratin, S.A. Vorobiev / Channeling radiation from bent crystals
broaden and merge into a single wide one as the crystal is strongly bent (fig. 6~). The quasi-synchrotron radiation maximum is broadened and, in the long run, has a “comb” structure. We have discussed above only the radiation spectra of the channeled fraction of a beam. Radiation emitted by quasi-channeled particles occurs at higher frequencies than for channeled particles because the frequencies of transverse particle oscillations are higher (they are determined by orientation angles to the planes). As the crystal is bent, the integrated radiation spectrum of channeled and quasi-channeled particles is broadened due to a quasi-channeled fraction since the range of the particle orientation angles to the atomic planes becomes wider. The multiple scattering of the particles by the crystal electrons and nuclei decreases the channeled fraction and the radiation intensity in a soft region of frequencies will be less than for the spectrum calculated without multiple scattering.
557
particle due to motion along the arc of a bent channel; it occurs together with the radiation in an intermediate frequency range between w, and wc. (4) With the increase of the crystal bending the radiation at frequencies corresponding to the radiation harmonics in a straight (unbent) crystal decreases and the quasi-synchrotron part of the radiation spectrum of channeled positrons is broadening. As a result at a strong crystal bending, R = Xv, the energy loss by radiation of both types are becoming comparable. (5) The quasi-synchrotron radiation spectrum of a particle moving along the arc of a circle, i.e., along the bent channels in crystal, acquires a “comb” structure as the radial particle oscillations occur near the equilibrium orbit.
References [l] Coherent Radiation Sources, eds. A.W. Saenz and H.
4. Conclusion (1) The channeling radiation intensity of positrons decreases with increasing crystal bending due to decrease in both the amplitudes of particle oscillations in the channels and the particle capture efficiency into channeling regime. (2) For a small bending of the crystal, R x=- Xy, the positron radiation spectrum is quasi-undulatory and the spectral maxima correspond to radiation harmonics in a straight (unbent) crystal wk - kw,. But the width of the harmonics increases and the subharmonics at frequencies less than those of an unbent crystal are observed. (3) In a bent crystal, quasi-synchrotron radiation occurs at low frequencies (0-5)~~. For a channeled
ijberall (Springer, Berlin, 1985). [2] A.F. Elishev, N.A. Filatova, V.M. Golovatyuk et al., Phys. Lett. B88 (1979) 387. [3] Yu.N. Adishchev, P.S. Ananyinm A.N. Didenko et al., Phys. Lett. A77 (1980) 263. (41 A.A. Sokolov and I.M. Temov, The Relativistic Electron (Nauka, Moscow, 1983) in Russian. [5] V.G. Bagrov, I.M. Temov and N.I. Fedosov, Zh. Eksp. Teor. Fiz. 82 (1982) 1442. [6] V.V. Kaplin and S.A. Vorobiev, Phys. Lett. A67 (1978) 135. [7] Yu.A. Bashmakov, Radiat. Eff. 56 (1981) 55. [8] A.M. Taratin and S.A. Vorobiev, Phys. Status Solidi (b) 107 (1981) 521. [9] L.D. Landau and E.M. Lifshitz The Classical Theory of Fields, 3rd ed. (Pergamon, Oxford, 1971). [lo] J. Jackson, Classical Electrodynamics (Wiley, New York, 1975).