Planar channeling experiments with electrons at the 855 MeV Mainz Microtron MAMI

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NIM B Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 266 (2008) 3835–3851 www.elsevier.com/locate/nimb

Planar channeling experiments with electrons at the 855 MeV Mainz Microtron MAMI H. Backe *, P. Kunz, W. Lauth, A. Rueda Institut fu¨r Kernphysik der Universita¨t Mainz, Fachbereich Physik, Mathematik und Informatik, D-55099 Mainz, Germany Received 10 April 2008 Available online 22 May 2008

Abstract Planar channeling has been studied for silicon single crystals at a beam energy of 855 MeV at the Mainz Microtron MAMI. Complex channeling patterns were observed from which the crystal orientation can unambiguously be determined. Photon spectra at (1 0 0), (1 1 0) and (1 1 1) planar channeling were recorded with a 1000  1000 NaI detector. The planar (1 1 0) channeling process has been studied as function of the crystal thickness in the range between 7.9 and 270 lm from which a dechanneling length of 18.0 lm and the thickness dependent rechanneling lengths were deduced, employing solutions of the Fokker–Planck equation. A signal derived from high energy bremsstrahlung exhibits a characteristic length of (32 ± 4) lm which is tentatively interpreted as the occupation length of the lowest quantum states in the planar potential. Prospects are discussed to exploit channeling of high energy electrons in periodically bent silicon single crystals for production of radiation in the hundreds keV to multi MeV range. Ó 2008 Elsevier B.V. All rights reserved. PACS: 61.85.+p Keyword: Channeling phenomena

1. Introduction The possibility to produce undulator-like radiation in the hundreds of keV up to the MeV region by means of positron channeling is well known and was discussed in a number of papers, see e.g. [1–3]. However, the demonstration and utilization of such devices hampers from the fact that high quality positron beams in the GeV range are not easily available, in contrast to electron beams. Recently it was suggested [4,5] that by means of planar channeling of ultrarelativistic electrons in a periodically bent single crystal the production of undulator-like radiation should also be possible. The basic idea of this work is that similar dechanneling lengths as for positrons can also be achieved with electrons if their beam energy is chosen a factor of

*

Corresponding author. E-mail address: [email protected] (H. Backe).

0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.05.012

about 20 larger as for positrons. These considerations prompted the authors of this paper to commence planar channeling experiments at silicon single crystals with the 855 MeV electron beam of the Mainz Microtron MAMI. Only little is known experimentally on the dechanneling length of GeV- and multi-GeV electrons in single crystals. Moreover, nothing is known experimentally on channeling of electrons in periodically bent single crystals. Consequently, one of the questions of an ongoing experiment at MAMI is whether for such crystals a peak structure in the photon spectrum can be expected also with 855 MeV electrons, or not. The paper is organized as follows. In Chapter 2 the experimental setup at MAMI and the channeling signal generation is described. A survey of planar channeling experiments, including a novel method for a precise crystal alignment, is described in Chapter 3. In Chapter 4 photon spectra are presented and discussed which were taken at planar channeling with a 1000  1000 NaI detector. Chapter

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5 presents experimental results of (1 1 0) planar channeling signals as function of the target thickness from which in Chapter 6 the dechanneling length is extracted, employing solutions of the Fokker–Planck equation. The paper closes with Chapter 7 in which a conclusion and an outlook are given. 2. Experimental 2.1. Experimental setup The experimental setup at the Mainz Microtron MAMI is shown in Fig. 1. The low emittance electron beam of MAMI is well suited to prepare a beam with small angular divergence. Experiments were performed at beam energies of 600 and 855 MeV. At a beam energy of 600 MeV the horizontal and vertical emittances are eh = 3.0 p nm rad and ev = 0.5 p nm rad, respectively. Typical beam spots in our experiments had standard deviations rh = 340 lm, horizontally, and rv = 180 lm [6], vertically, resulting in standard deviations of the beam divergences of r0h ¼ 8:8 lrad and r0v ¼ 2:8 lrad, respectively. These numbers would correspond at a tilt angle of the beam with respect to the (1 1 0) plane, assumed to be equal to the critical angle wc = (2 U0/pc)1/2 = 0.27 mrad, to a spread of the transverse energy 0 dEkin ? ¼ 2:355  pv  wc  rh 6 3:4 eVðFWHMÞ. Here are p the momentum of the electron and U0 = 22 eV the depth of the (1 1 0) potential in silicon. At an electron energy of 855 MeV the emittance eh = 10.0 p nm rad in horizontal direction is somewhat larger, and ev = 0.3 p nm rad in vertical direction somewhat lower. The beam divergences and transverse energies scale accordingly.

As targets self supporting 10 mm  10 mm pieces of silicon single crystals were used, cut with their surfaces perpendicular to the [1 0 0] direction. The thicknesses of 34.5 lm, 98.7 lm, 199.5 lm, and 270.4 lm were measured with a digital thickness meter (Pim-100, Mauser). Thinner targets were prepared by anisotropically etching on a 30 wafer (nominal thickness of 466 lm, cut with the (1 0 0) plane parallel to the surface), areas of 3 mm  3 mm in a 30% KOH alkaline solution at 80 °C [7]. The resulting thicknesses of 7.9 lm and 14.7 lm were measured by means of the energy loss of a particles from a 239Pu, 241 Am, 244Cm mixed source (Amersham International) with an estimated accuracy of 4%. The arithmetical mean deviation of R lm the surface profile y(x) defined as Ra ¼ ð1=lm Þ x¼0 jyðxÞjdx, with lm a sufficient long path on the surface of the silicon film, was measured with an optical laser profilometer (Ulrich Breitmeier Mikroskop) to be Ra 0.1 lm. The crystals were mounted on goniometers with which rotations around three axes could be accomplished, see Fig. 2. The crystal can be rotated by (i) the azimuthal angle u around the vertical z-axis of the laboratory reference frame, (ii) the polar angle h around the y0 -axis which originates from the y-axis of the laboratory frame after rotation by the angle u and (iii) the tilt angle a around the x0 -axis of the crystal frame. Channeling signals were derived from the photon spectra of a 1000  1000 NaI detector. A signal which is also sensitive to channeling was derived from the 44°-ionization chamber behind the bending magnet BM2. While the signal generation from the former is straight forward, the latter needs to be explained in more detail. ZnS Screen

Beam Dump

Top View 44°- Ionization Chamber

e-

BM2

Si Target

34°

Camera

Pb 100 mm ∅ 20mm

BM1 44°

NaI Detector 10" ∅× 10"

-

e

7.633 m

Side View BM2

0

5m

44°-Ionization Chamber

7.2°

ZnS Screen

Beam Dump

Fig. 1. Experimental setup, top view above and side view below. Downstream the Si target the beam is deflected horizontally by the 44° bending magnet BM1 and vertically by a 7.2° bending magnet BM2. Just in front of the beam dump the beam spot can be monitored with a ZnS luminescent screen which is viewed by a CCD camera. Photon spectra are detected with a NaI detector of 1000 £ and 1000 length. The detector is shielded by a 1 0 0 mm thick lead wall with a 20 mm £ opening for the photons. The 44°-ionization chamber is employed to detect channeling. It is sensitive to emission of photons with energies between 5.8 and 14.5 MeV at a beam energy of 855 MeV.

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y

θ

(1 10) vertical planes

ϕ

α e

y′

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x′

-

x

(001) horizontal planes

θ

z y

0

α

ϕ

500 mm

Fig. 2. (a) Top view on the target setup with definition of angles and coordinate systems. The crystals can be aligned by a three-axis goniometer. Without any rotations the crystal frame (primed coordinates) coincides with the laboratory reference frame (unprimed coordinates). The x-axis shows into direction of the electron beam, the y-axis lies in the horizontal plane as indicated, and the z-axis is chosen to form a right handed coordinate system. (b) View to the silicon single crystal in opposite direction to the electron beam. The ½1 1 0 direction of the crystal point in direction of the x-axis which is the electron beam direction. The tilt angle a, azimuthal angle u, and polar angle h are also indicated.

2.2. The channeling signal from the ionization chamber at 44° As shown in Fig. 3, the ionization chamber is sensitive to charged particles of electromagnetic showers which are produced by electrons leaving the nominal beam direction due to scattering or after emission of multi-MeV photons in the silicon target crystal. This fact is demonstrated in Fig. 3(b) in which the ionization chamber signal is shown as function of the current IBM1 of the bending magnet BM1. The steep increase of the signal at currents of (586 ± 4) A originates from an obstacle which the beam hits. Ray trace calculations point to an obstacle in front of bending magnet BM2 of 47.2 mm horizontal clearance. This number agrees well with the diameter of the exit flange of an aluminum chamber in front of the bending magnet. An increase of the voltage signal of the ionization chamber is observed if electrons hit this exit flange. The decrease of the signal at currents IBM1 > (586 + 4) A and IBM1 < (586  4) A probably originates from the shielding of the ∅ 48 mm BM2

∅ 60 mm

ionization chamber by the iron yoke of the bending magnet BM2 with a gap width of 60 mm. The signal has a half width of about 3 A, meaning that in an experiment with target at nominal beam current IBM1 = 586 A the ionization chamber is sensitive to electrons which either suffer in the target an energy loss of 4.1 MeV < DE < 10.2 MeV, or a horizontal scattering with angles of 12.0 mrad >hh > 4.8 mrad. In the vertical direction ray trace calculations yield scattering angles hv > 3.9 mrad. In order to understand the 44°-ionization chamber signal in a channeling experiment, simulation calculations have been performed at a beam energy of 855 MeV. At this energy the accepted energy interval scales by a factor 855/ 600 to 5.8 MeV < DE < 14.5 MeV while the accepted angular scattering intervals remain unchanged. The calculations were done on the basis of the scattering formalism presented in [8, Eq. (3.14), p. 17] with #2 replaced by 2 h20 " # ! 1 #2 f ð1Þ ð#Þ : ð1Þ f# ð#Þ ¼ 2 exp  2 þ B 2p  2h20 2h0

Ionization Chamber e+ - e e

35 30

Shower

U [a.u.]

25 20 8A

15 10

3.80 m

Target

BM1

586 A

5 0 570

γ

580

590

600

IB M1 [A]

Fig. 3. Signal generation by the ionization chamber with an active volume of 450 mm  240 mm  48 mm. The chamber is filled with air at standard pressure. (a) Showers are produced by electrons, leaving the nominal direction due to scattering or energy loss in the target, at the exit flange of an aluminum chamber in front of the bending magnet BM2. (b) Voltage signal of the ionization chamber as function of the current IBM1 of the bending magnet BM1 at an electron beam energy of 600 MeV without target.

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Here h0 is the standard deviations of the scattering angle as calculated by the well known formula of the Particle Data Group rffiffiffiffiffiffi 13:6 MeV x ½1 þ 0:038 lnðx=X 0 Þ; ð2Þ h0 ¼ pv X0 with X0 = 0.0936 m the radiation length of silicon and x the target thickness. The quantity B varies between 3.7 and 4.3 for targets of thicknesses between 50 and 200 lm, qffiffiffiffiffiffiffi respectively. For angles #  2h20 the second term in Eq. (1) has the asymptotic behavior f ð1Þ ð#Þ  8h40 =#4 . The bremsstrahlung spectrum has been calculated according to the Bethe–Heitler formula from [8, Eq. (10.2), p. 68]. The results show that for a randomly oriented silicon target of 50 lm thickness a fraction 7.7  104 of scattered electrons contribute to the signal in the due energy range, while a somewhat smaller fraction of 3.2  104 of electrons are detected after emission of bremsstrahlung photons. For a 200 lm thick target the numbers are 30.0  104 and 13.0  104, respectively. At electron impacts well below the critical angle wc both fractions increase dramatically due to very intense coherent emission of channeling radiation and an enhanced scattering of the electrons in the target. The latter results in electrons in the 1/#4 tailings of the scattering distribution f(1)(#) of Eq. (1). In conclusion, the ionization chamber signal is well suited to study channeling phenomena as long as the target thickness is small enough that the Gaussian bulk of the scattering distribution does not contribute to the signal generation. The latter restriction was fulfilled for all target thicknesses of relevance for the experiments to be described below.

3. Survey of planar channeling 3.1. Experimental results In any kind of channeling experiment it is of utmost importance to identify unambiguously the crystal plane in which the electron channels. That this is not at all a trivial task is exemplified by means of Fig. 4 in which rich structures are observed. A silicon single crystal, cut with its surface perpendicular to the [1 0 0] direction, was aligned with its (1 1 0) orientation flat parallel to the horizontal y-axis. The different panels (a)–(e) show scans around the vertical z-axis for various polar angles h. If the electron beam direction coincides with the [1 0 0] crystallographic axis, only a single line is observed, see panel (a), which originates from axial channeling at the [1 0 0] atomic string. If the polar angle h is increased, a fine structure develops which gets better and better resolved the larger h is chosen, see panels (b)–(e). The various lines are attributed to planes which all have in common the [1 0 0] crystallographic axis. To identify the planes, simulation calculations have been performed which are described in the following subsection. 3.2. Simulation calculations of the planar channeling signals The simulation procedure includes the calculation of (i) the angle w\hkl(u, h, a) a certain crystallographic plane, characterized by the Miller indices h, k and l, makes with the electron beam direction at an azimuthal angle u for fixed angles h and a, and (ii) a calculation of the relative intensities of the signals at channeling. (i) The crystallographic planes are described by their reciprocal lattice vectors ~ g0 ¼ ð2p=aÞðh^e0x þ k^e0y þ l^e0z Þ in f)

100 50 0

g)

Intensity

20 0

h)

20 0

i)

20 0

k)

20 0 −60 −40 −20 0

20

40

60

Fig. 4. Signals of the 44°-ionization chamber behind the bending magnet BM2, see Fig. 1, at u scans around the vertical z-axis. The crystal with a thickness of 43.4 lm was cut with the [1 0 0] axis perpendicular to the surface. Coarse adjustment angles of the crystal u0 = 0, h0 = 0, and a0 = p/4. The ð0  1 1Þ plane was aligned along the horizontal y-axis, i.e. a = 0. Panels (a)–(e) show measurements at a beam current of 4 nA with increasing polar angle h = 0 (a), h = 0.1° (b), h = 0.2° (c), h = 0.5° (d), and h = 1.1° (e). Panels (f)–(k) show corresponding calculations.

H. Backe et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3835–3851

potential depth. The potential was calculated according to [16, Eqs. (9.5)–(9.7), p. 235] taking into account lat˚ . Since tice vibrations with an amplitude u1 = 0.07661 A lattice vibrations for small plane distances lead to rather shallow potential depths, the intensities of the lines were assumed to be proportional to U0,hkl rather than to d 2p for which lattice vibrations are not taken into account, i.e. I / U0,hkl. For the calculation of the spectra the line shapes were approximated by Lorentzians with a width between 0.5 and 0.8 mrad. All lines with Miller indices in the bounds 2 6 h 6 2, 10 6 k 6 10 and 10 6 l 6 10 which fell into the relevant u-scan interval were taken into account. The results of these calculations are shown in the right panels of Fig. 4. The similarity with the measurements shown in the left panels is striking. Measurements and calculation for h = 1.1° are shown once more in Fig. 5 together with the assignment of the crystal planes belonging to the strongest lines. These measurements fully supported the crystal manufacturers information that on a [1 0 0] wafer the [1 1 0] direction is made evident by a flat segment, also called the orientation flat. A cross check has been made by a measurement at which the crystal was rotated back into the ground position in which the crystallographic axes coincide with the laboratory frame coordinates, i.e. for u0 = 0, h0 = 0 and a0 = 0. The results are shown in Fig. 6. Both adjustments of the crystal can clearly be distinguised by their characteristic line pattern.

0

−40

−20

0 [mrad]

20

40

60

Fig. 5. Detailed comparison between measurements, upper panel, and calculations, lower panel, together with assignments of the corresponding planes. Data are the same as shown in Fig. 4(e) and (k) for h = 1.1°. Coarse adjustment angles of the crystal u0 = 0, h0 = 0, and a0 = p/4.

0

80

60

40

20

0

20

40

(015)

(013)

(012)

(031)

(012)

(015)

10

(013)

15

(031)

20

5 (021)

(031)

(051)

(021) (031) (051)

(010)

Intensity

(011)

−60

(015)

5

(013)

15

(015) (013) (012)

(001)

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30

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(011)

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(011)

Intensity

the crystallographic primed frame. The latter is the frame in which the x0 , y0 , and z0 axes coincide with the base vectors of the unit cell of the silicon lattice, having ˚ . In the starting position the a lattice constant a = 5.43 A three primed axis coincide with the x-, y- and z-axis of the laboratory frame. In the first step a mathematical coarse adjustment of the crystal is done which is described by the large rotation angles u0, h0 and a0 while, in a second step, the fine adjustment by the goniometers follows, described by the small angles u, h and a. The original reciprocal lattice vector ~ g0 is represented in the laboratory frame by ~ g ¼ DðuÞDðhÞDðaÞDða0 Þ Dðh0 ÞDðu0 Þ  ~ g0 , with D the usual rotation matrices. The wanted angle between electron beam direction, which coincides with the x-direction, and crystal plane is now simply given by sin w?hkl ¼ ^ex  ~ g=j~ gj. (ii) The relative intensities of the lines were modelled as follows. For the intensity of channeling radiation I / K2 is assumed, like in an undulator. The undulator parameter K is for an oscillation in the channel with a hypothetical maximal amplitude dp/2 given by K = c(dp/2) 2p/kU with c the relativistic factor of the electron, dp the distance between the planes, and kU the oscillation period at planar channeling. With 2p/kU = X/c and X the oscillation frequency in the channel, one obtains 2 for K 2 ¼ d 2p ð hx=4c hcÞ . In the latter expression 2 hx = 2c   hX is the photon energy observed in forward direction which is defined by the experiment. It can be treated as a constant. From this simple consideration it can be concluded that for the intensity I / d 2p holds. However, it was found empirically that for deep potentials with single minima d 2p / U 0;hkl with U0,hkl the

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60

80

[mrad] Fig. 6. Comparison between measurements, upper panel, and calculations, lower panel for coarse adjustment angles of the crystal u0 = 0, h0 = 0, and a0 = 0 meaning with respect to Fig. 5 just a rotation of 45° around the x-axis. In the lower panel h = 0.95° was assumed. The slight asymmetry originates from the small tilt angle a = 0.15°, see Section 3.3. The strongest planes are assigned by their Miller indices.

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3.3. Alignment procedure of the crystals for channeling experiments

h¼

In the preceding sections scans were discussed in which the tilt angle a was zero. If this restriction is abandoned a very useful new structure appears as will be demonstrated by means of the scans shown in Fig. 7. Beside the narrow structure at u = 0 broad satellite structures are observed in addition which can be assigned to the horizontal ð0 1 1Þ plane. The broad structure appears only if the crystal is tilted by an angle jaj > 0 with respect to the horizontal axis and, at the same time, nodded by a polar angle jhj > 0. In contrast to the narrow signal, to which in addition to the vertical (0 1 1) plane many other planes contribute, the broad structure signal originates only from the ð0 1 1Þ plane. Both, the polar angles h1 = 1.082 mrad, h2 = 1.361 mrad, and the corresponding peak positions u1 = 36.2 mrad, u2 = 46.1 mrad can precisely be measured. From these numbers the tilt angle a, which otherwise can not easily be measured, can be determined by the relation

a Du: 1 þ a2

ð5Þ

From the width Du1 = 33.8 mrad of the broad signal, the projected angular width can be calculated by the relation Dw?0 1 1 ¼ aDu ¼ 1:00 mrad. Notice that this number is somewhat larger as twice the critical angle wc = 0.23 mrad. Obviously, at larger depths in the crystal rechanneled electrons contribute to the signal, in accord with the fact that in a 200 lm thick amorphous target the scattering angle amounts to h0 = 0.563 mrad (standard deviation). The angle between the electron direction, projected on the (0 k l) plane, and the crystallographic [1 0 0] direction wk0kl ’

½ðk þ lÞ þ ðk þ lÞa  h þ ½ðk þ lÞ þ ðk þ lÞa  u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðk 2 þ l2 Þ ð6Þ

describing the angle between the electron trajectory and the (0 k l) plane. The result is a = 1.70°. Once this tilt angle has been determined, the polar angle h can be chosen from a well defined difference Du between the narrow and broad structure employing the relation

must fulfill the requirements for planar channeling. On the one hand, this angle must be reasonable large in comparison with the critical angle for axial channeling wc,a, which is certainly fulfilled for wk0kl  wc;a . On the other hand, it must not coincide with the direction of a periodic arrangement of atoms. It should be mentioned that this angle varies in general in a u scan. Finally a remark on the interpretation of scans like that one shown in Fig. 7 seems to be appropriate. In terms of the physically relevant angles w\0kl and wk0kl the abscissa has many scales which differ for the various crystallographic planes (0 k l), see Eqs. (4) and (6). Easily interpretable are the vertical (0 1 1) and the horizontal ð0  1 1Þ planes for which w\0 1 1 = (u + ah) and w?0 1 1 ¼ ðh  auÞ holds, respectively. An example of transformed broad structures into the transverse energy scale is shown in Fig. 8. The arrows indicate the critical energy which is just

Fig. 7. Voltage signals of the 44°-ionization chamber for u scans around the vertical z-axis for a crystal with a thickness of 200 lm cut with the [1 0 0] axis perpendicular to the surface. Coarse adjustment angles of the crystal u0 = 0, h0 = 0, and a0 = p/4, and tilt angle a = 1.70° were chosen. Panel (a) corresponds to a polar angle h1 = 1.082 mrad, panel (b) to h2 = 1.361 mrad. The broad ð0 1 1Þ structures are located at u1 = 36.2 mrad, u2 = 46.1 mrad. The weak narrow lines in the wings belong to higher order (±1 k l) planes as simulation calculations revealed.

Fig. 8. 44°-ionization chamber signal as function of the transverse energy E\ = (pc/2)  (aw\)2 for ð0 1 1Þ (a) and (0 0 1) (b) planar channeling at pc = 600 MeV. Crystal cut with [1 0 0] direction perpendicular to the surface, thickness 43.4 lm. The arrows indicate the potential depths 21.2 eV, and 11.7 eV for the ð0 1 1Þ and (0 0 1) planes, respectively.

a¼

h1  h 2 : u1  u2

ð3Þ

This relation can be derived from the more general formula w?0kl ’

½ðk  lÞðh  auÞ  ðk þ lÞðu þ ahÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðk 2 þ l2 Þ

ð4Þ

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the potential depths. Channeling and quasichanneling regions can clearly be distinguished. 4. Photon emission spectra 4.1. Photon spectra at planar channeling The ionization chamber signal provides a simple tool to align a crystal and to find the wanted planes for channeling experiments. However, the signal generation, based on scattering as well as photon emission, is somehow too unspecific if deeper inside into the nature of the channeling process is desired. In this section an energy resolved photon spectroscopy is described which can be exploited to this end. It should be mentioned that photon spectra at channeling of high energy electrons were measured earlier, for an overview see the articles of Fujimoto and Komaki [9, p. 271], Bak [9, p. 281], and Berman [10]. For example, for channeling of 110 MeV electrons in diamond and silicon see [11], for channeling of 900 MeV electrons in diamond see [12], and for channeling of 5–55 GeV electrons in silicon see [13–15]. Photon spectra were taken with the 1000  1000 NaI detector. A 175 lm thick crystals was aligned as described in

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Section 3.3. A u scan with an energy window 0.4 MeV < hx < 9.0 MeV set on the photon spectrum is shown in Fig. 9(a) and a simulation calculation in Fig. 9(b). The striking similarity of measurement and calculation corroborates the conjecture that the chosen energy window is associated with channeling radiation. Experimental photon spectra are presented in Figs. 10 and 11(a)–(c) for various selected u adjustments of the crystal as indicated in Fig. 9(a). The planes were assigned by means of the simulation calculation shown in Fig. 9(b). Depicted are unprocessed photon spectra, meaning that a deconvolution from the detector response function has not been performed. Although the peak-to-total ratio for the 1000  1000 NaI detector is rather high, such a deconvolution would result in a reduction of the intensity at low energies, in particular in the minimum at about 0.2 MeV, where indeed the intensity of channeling radiation is expected to be very low. Apparently an interesting correlation of the spectral shape with the potentials shown in panels (d)–(f) exists. It is well known that radiation is produced either at proper channeling, i.e. if the electron moves inside the potential well, or at quasichanneling, i.e. if the electron moves in an unbound state with an energy close to the barrier. In the latter case the unbound electron performs also an undulating movement under the influence of the potential. Proper channeling is predominantly associated with the low energy peak which increases the deeper and the wider the potential gets, quasichanneling to the long high energy tails. Notice that the tailings above 5.8 MeV contribute to the signal of the 44°-ionization chamber, see Section 2.2. Theoretical calculations of photon spectra on the basis of a properly modified formalism as described, e.g. in [19] would be highly desirable. 4.2. Coherent bremsstrahlung spectra An interesting feature is observed in the correlation between photon energy and angular alignment of the crys-

30

15

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10 5 −100

−50

10

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(001)

20

20

(551) (331)

Intensity

25

0 −150

(111)

(b)

(111)

30

(110)

35

0

50

100

[mrad] Fig. 9. Integrated intensity of the NaI spectrum in the energy interval 0.4 MeV <  hx < 9.0 MeV for a u scan around the vertical z-axis (a). A crystal with its surface normal in [1 1 0] direction and a thickness of 175 lm was aligned to the geometry shown in Fig. 2(b), i.e. coarse azimuthal angle u0 = p/4, and additionally rotated by a coarse tilt angle a0 = p/2. Simulation calculations (b) with azimuthal angle h = 0.685°, and tilt angle a = 6.4°.

Fig. 10. Spectra taken with the 1000  1000 NaI detector in the geometry of Fig. 1. Crystal as described in Fig. 9. The upper curve corresponds to planar ð1 1 0Þ channeling taken in point (2), the lower curve to the background taken with a randomly oriented crystal in points (B) of Fig. 9. The associated u angles are indicated in Fig. 9. Data collection time 60 s at a beam current of about 0.3 pA.

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hx ’ (c24phc/d0kl)  w\0kl, which is an approximation for photon energies small in comparison with the beam energy. The quantity w\0kl is the angle the electron makes with the (0 k l) plane, see Eq. (4). For both, the vertical (0 1 1) plane ˚ as well as the (0 1 0) or the (0 0 1) with d011 = 1.9198 A pffiffiffi planes with d 0 1 0 ¼ d 0 0 1 ¼ d 0 1 1 = 2 one obtains for the photon energy hx/MeV = ±36.1  (u/mrad). For u = 0.59 mrad the energy hx ’ 21.3 MeV is in good agreement with the half intensity point in the high energy shoulder of Fig. 12, lower panel. The intensity above 25 MeV is probably the second order effect. The shape of the shoulder indicates a line broadening which probably originates from the small angle scattering of the electrons. The V-shape structure in the scatter plot can be utilized to estimate the fine adjustment angles a and h of the crystal. As an example, from Fig. 12 right panel, the slope dhx=du ’ c2 ð4phc=d 0 1 1 Þ  a of the horizontally oriented ð0 1 1Þ planes the angle a can be determined. From the angle difference Du, i.e. the difference for the signal from the vertical (0 1 1) and the horizontal ð0 1 1Þ planes, the polar (nodding) angle h = a  Du follows. For the example shown in Fig. 12, right panel, the numbers are d hx/ du = 1.480 MeV/mrad, Du = 108.7 mrad, a = 2.35°, and a h = 0.255° follows. 5. Investigation of the dechanneling process Fig. 11. Background subtracted NaI spectra for planar channeling at the (0 0 1) plane (a), (0 1 1) plane (b), and (1 1 1) plane (c). Crystal and associated u angles as described resp. indicated in Fig. 9, i.e. for the (0 0 1) plane at point (1), for the (0 1 1) plane at point (2), for the (1 1 1) plane at points (3), and for the background at points (B). Corresponding planar potentials in inset panels (d)–(f), calculated according to [16, Eqs. (9.5)– (9.7), p. 235] with the Doyle–Turner parameters [18] ak = {2.1293, 2.5333, 0.8349, 0.3216}, bk = {57.7748, 16.4756, 2.8796, 0.3860}, and a thermal ˚ . Distances between two planes are vibration parameter u1 = 0.07661 A ˚ , d011 = 1.9198 A ˚ , and d111 = 3.1350 A ˚ , potential depths d001 = 1.3575 A 11.66 eV, 21.19 eV, and 23.07 eV, respectively.

tallographic planes with respect to the electron beam. The scatter plots of Fig. 12 reveals that the low energy peaks, which are associated with planar channeling, split at increasing photon energy into two well distinguishable branches with enhanced photon intensity. This phenomenon is observed just in the transition region between channeling radiation and coherent bremsstrahlung emission. As discussed in the literature, see [20] and references cited therein, channeling radiation and coherent bremsstrahlung originate from the same phenomenon and can be interpreted as transitions between bound states and between free states, respectively. A corresponding photon spectrum, taken at a fixed angle u = 0.59 mrad, is shown in Fig. 12, lower panel. Such a spectrum is interpreted in [8, Fig. 15, p. 92] as the coherent radiation-point effect which occurs due to the interference of electromagnetic waves emitted by an electron at the interaction with neighboring crystallographic planes, see also [11]. The calculated photon energy is given in forward direction by the relation

As already mentioned, very little is known on the dechanneling length of electrons with energies in the GeV range in planar channeling. Since this information is extremely important for experiments with crystalline undulators, channeling experiments were performed at MAMI with flat silicon single crystals. Of particular interest is channeling at the (1 1 0) planes since these planes can be deformed in graded composition strained layers, see [10,17] and references cited therein. The easiest way to get information on the dechanneling process is to carry out measurements of the fraction fch(x) of electrons trapped in the planar potential as function of the target crystal thickness R x x. In principle, in such an experiment the integral 0 fch ðnÞdn is measured from which fch(x) is obtained from the first derivative. However, the result depends strongly on the signal chosen for the measurement, as will be demonstrated next. A signal that is sensitive to high energy photons can be obtained with the experimental setup shown in Fig. 13, upper panel. An ionization chamber, located in forward direction, has been combined with a 1 cm thick iron converter. The assembly is most sensitive to photons with energies hx J 15 MeV, as is shown in GEANT 4 simulation, depicted in the lower panel of Fig. 13. Crystals of various thicknesses, all cut with the [1 0 0] direction perpendicular to the surface, were aligned as described in Section 3.3. A typical u scan is shown in Fig. 14(b). The signal, defined by the background subtracted maximum of the narrow

H. Backe et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3835–3851

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Fig. 12. Coherent bremsstrahlung. Upper panels scatter plots, lower panel photon energy spectrum taken at u = 0.59 mrad from which a background spectrum at u = 61.3 mrad was subtracted. A crystal with a thickness of 14.7 lm, cut with the [1 0 0] direction perpendicular to the surface, was employed. The full lines indicate hx = ±36.1 (MeV/mrad) (u  uk) with u0 = 0,u1 = 5.3 mrad, u2 = 4.9 mrad, left panel, and hx = ±1.480 (MeV/mrad) (u + 108.7 mrad), right panel.

BM2

Si Target

44°- Ionization Chamber 0°-Ionization Chamber

BM1 44°

Pb

BM3

-

e

8.3 m

0.2 m

7.5 m

Fe Converter

Fig. 13. Experimental setup with 0°-ionization chamber. In comparison to Fig. 1 the NaI detector has been replaced by a 0°-ionization chamber assembly to detect the integrated high energy photon emission spectra from the target. The ionization chamber is combined with a 1 cm iron converter. The assembly is sensitive to photons with energies hx J 15 MeV, see lower panel. The lead shield has a thickness of 200 mm with a rectangular clearance of 30 mm in width and 20 mm in height. The magnet BM3 prevents electrons and positrons, travelling in direction of the 1 cm thick iron converter, from detection.

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Fig. 15. High energy signals for ð0 1 1Þ planar channeling of crystals with various thicknesses, cut with the surface perpendicular to the [1 0 0] direction. Full circles indicate measurements. The full line is a best fit with the function U(x) = C0[1  exp(x/Ld,high)], with Ld,high = (32 ± 4) lm.

Fig. 14. Voltage signals of the 44°-ionization chamber (a) and (c) and of the 0°-ionization chamber (b) and (d) for u scans around the vertical zaxis. Measurements were performed simultaneously, i.e. the broad structure in panels (a) and (c) and the corresponding narrow structure in panel (b) and (d) both belong to the same ð0 1 1Þ plane. The crystal for measurements (a) and (b) had a thickness of 14.7 lm and for measurements (c) and (d) 199.5 lm, both cut with the [1 0 0] axis perpendicular to the surface. Coarse adjustment angles of the crystal for all measurements u0 = 0, h0 = 0, and a0 = p/4. Azimuthal angle h = 0.071°, tilt angle a = 2.78° for measurements (a), (b), and h = 0.061°, a = 1.70° for measurements (c) and (d).

ð0 1 1Þ structure, is shown in Fig. 15. Obviously, the signal saturates at large crystal thicknesses. A best fit with a function U(x) = C0[1  exp(x/Ld,high)] yields a dechanneling length Ld,high = (32 ± 4) lm. Measurements with low energy channeling radiation, including also electron scattering, were performed simultaneously with the 44°-ionization chamber. A corresponding scan is shown in Fig. 14(a). The signal is defined by the background subtracted maximum of the broad ð0 1 1Þ structure. Data were taken for various crystal thickness. The results are shown in Fig. 16 together with results derived from the NaI detector signal with an energy window 0.4 MeV <  hx < 9.0 MeV on the photon spectrum which should cover transitions inside the potential well (bound–bound transitions) sensitive to channeling. However, a certain fraction of the intensity, which might be in the order of about 30%, originates from coherent

Fig. 16. Signals for (0 1 1) planar channeling of [1 0 0]-cut crystals as function of the thickness. Stars and circles indicate measurements with the 44°-ionization chamber and the NaI detector in an energy window 0.4 MeV <hx < 9 MeV, respectively, with an estimated error of 5%. The full curve shows a calculation based on the solution of the Fokker–Planck equation described in Section 6.2.

bremsstrahlung (free–free transitions) which signals quasichanneling. Comparison of Figs. 15 and 16 reveals a significant different thickness dependence of the high and low energy signals. While the high energy signal saturates, the low energy signal does not. Further on, comparing the angular scans shown in Fig. 14(a) and (b) shows that the line widths of the clustered planes around u = 0 broadens for the high energy signal in comparison to the low energy one, and that at the same time the signal representing the horizontal ð0 1 1Þ plane narrows, exhibiting in addition undershooting in the wings. To gain more insight into this rather puzzling situation, in particular to find a measure to extract the dechanneling length from the thickness dependence of the signals as shown in Figs. 16 and 15, the solutions of the Fokker–Planck equation [21] were reinvestigated, as described in the next Chapter.

H. Backe et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3835–3851

6. Solution of the Fokker–Planck equation for (1 1 0) planar channeling

and Dð2Þ e ðE ? Þ ¼

6.1. Basic background The Fokker–Planck equation  oF ðx; E? Þ o2  ¼ 2 Deð2Þ ðE? ÞF ðx; E? Þ ox oE?  o  ð1Þ De ðE? ÞF ðx; E? Þ ;  oE?

ð7Þ

is a diffusion equation with a drift term which describes the evolution of the probability density F(x, E\) in time, or as the traversed distance x in the target. Details of the basic underlying formalism for the Fokker–Planck equation at planar channeling are described in, e.g. [16,21,26] and references cited therein and will not be repeated here. The quantity F = DP/DE\ is the probability DP per transverse energy interval DE\. The drift coefficient is the mean transverse energy increase   DE? Deð1Þ ðE? Þ ¼ : ð8Þ Dx T The diffusion coefficient is given by * +   2 1 ðDE Þ DE? ? ð2Þ ðE?  U ðyÞÞ : De ðE? Þ ¼ ¼ 2 2 Dx Dx T

ð9Þ

3845

E2s 4 2 2pvX 0 T ðE? Þc

Z

y max

dp pffiffiffiffiffiffi 2pu1 0 2 2 expðg =2u1 ÞðE?  U ðgÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dg; 2ðE?  U ðgÞÞ=E

ð13Þ

with Es = 13.6 MeV, E = cmec2 the total electron energy, ˚ the standard deviation of the thermal vibrau1 = 0.076 A tion amplitude, and ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  1  E? =U 0 d p =2 for E? =U 0 < 1 y max ¼ d p =2 for E? =U 0 P 1: ð14Þ For the calculation of the drift and diffusion coefficients only the Gaussian part of the scattering distribution Eq. (1) was taken into account while the 1/#4 term has been neglected. The standard deviation of the Gaussian scattering distribution was taken from Eq. (2), neglecting the term in square brackets. The time parameter T, drift coefficient Dð1Þ e , and diffusion coefficient Deð2Þ are shown in Fig. 17(a)–(c) as function of the reduced transverse energy E\/U0. Notice, that for E\ = U0, for which the time parameter diverges, 10

(a)

T

Both quantities are mean values with respect to one oscillation period in the potential which has been approximated by  2 jyj U ðyÞ ¼ U 0  U 0 1  ; ð10Þ d p =2 pffiffiffi ˚ the distance between adjacent with d p ¼ 2a=4 ¼ 1:92 A (1 1 0) planes, and U0 = 22 eV the corresponding potential depth. For convenience the potential was adjusted to zero at y = 0 by addition of the constant U0 at the right hand side of Eq. (10). The potential is depicted in Fig. 20(a) below. A time parameter 8 pffiffiffiffiffiffiffiffiffiffi 1þ E? =U 0 rffiffiffiffiffiffiffiffiffi> > pffiffiffiffiffiffiffiffiffiffi for E? =U 0 < 1 ln dp E < 1 E? =U 0 ð11Þ T ðE? Þ ¼ pffiffiffiffiffiffiffiffiffiffi c 2U 0 > 1þ E? =U 0 > : ln pffiffiffiffiffiffiffiffiffiffi for E? =U 0 P 1; 1þ

E? =U 0

has been introduced which is the oscillation period for E\/U0 < 1, see [26, Chapter 5], and twice the transit period over two succeeding potential maxima for E\/U0 P 1. Drift and diffusion coefficients have been calculated in the Kitagava–Ohtsuki approximation [25] by means of the integrals Z y max E2s 4 dp ð1Þ pffiffiffiffiffiffi De ðE? Þ ¼ 2pvX 0 T ðE? Þc 0 2pu1 2 2 expðg =2u1 Þ ffi dg;  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð12Þ 2ðE?  U ðgÞÞ=E

5

0 101

(b)

100

10 1 102

(c)

101

100

(d) 0.10

0.05

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 17. (a) Time parameter T  c, (b) drift coefficient Dð1Þ e , (c) diffusion coefficient Deð2Þ , and (d) initial distribution at x = 0, calculated with standard deviation r0y ¼ 150 lrad in the angular divergence of the electron beam, all as function of the reduced transverse energy E\/U0.

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the drift and the diffusion coefficients have deep minima. This is a consequence of the fact that the probability density of the electron at the potential maxima is very high. Since the nuclear density in the middle between the crystal planes is low, also the scattering probability is small and the drift coefficient has here its minimum, in contrast to the condition E\ = 0 for which the electron moves just inside a crystal plane. Consequently, the transverse energy increase reaches its maximum. For E\  U0 the drift coefficient approaches the constant value for amorphous matter, since the potential influences only very little the trajectory of an electron, while the diffusion coefficient increases linearly as function of E\. For the initial conditions at x = 0, required for the numerical solution of the Fokker–Planck equation, a uniform distribution of the electron across the transverse ycoordinate, and a Gaussian scattering distribution tilted by an angle w0in , and with standard deviation r0 y for the angular divergence were assumed. For a function with the two random variables yin and win, which are connected 2 to E\ by the relation E? =U 0 ¼ ð2y in =d p Þ þ ðwin =wc  2 0 win =wc Þ , the formalism described in [27, Chapter 6] was applied. This approach leads to the probability density

Eq. (18) can be proven by multiplying Eqs. (12) and (13) by T(E\) and differentiating Eq. (13) to E\. 6.2. Numerical results Eq. (19) has been numerically solved within the bounds 0 6 x 6 150 lm and 0 6 E\ 6 1 0 0U0 for which the program package Mathematica 5.1 [28] was employed. For the initial scattering distribution of the electron beam a standard deviation of r0y ¼ 150 lrad with inclination angle w0in ¼ 0 was assumed. The corresponding initial probability density is shown in Fig. 17(d). By this rather broad scattering distribution some ambiguities in the solution of the Fokker–Planck equation in the first couple of lm caused by large redistribution currents could be avoided. The probability that an electron is caught in the potential well amounts to 51%. As boundary conditions vanishing first derivatives oFT(x, E\)/oE\ = 0 were demanded for E\ = 0 and E\ = 100 U0, meaning that the probability current, introduced by Eq. (22), see below, must vanish at the bounds. Results are shown in Fig. 18. It is interesting to notice that the spike in the reduced initial probability

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 # g2 þE? =U 0 1þ E0? =U 0 1 Z 1 Bexp  2ðr0y =wc Þ2 B 1 B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p F 0 ðE? Þ ¼ pffiffiffiffiffiffi 2 2pðr0y =wc ÞU 0 y min B g2 þ E? =U 0  1 @ 0

"  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 #1  g2 þE? =U 0 1þ E0? =U 0 1 exp  C 2ðr0y =wc Þ2 C Cdg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C 2 g þ E? =U 0  1 A

ð15Þ

with y min

( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  E? =U 0 for E? =U 0 < 1 ¼ 0 for E? =U 0 P 1;

ð16Þ

and E0? =U 0  1 ¼ ðw0in =wc Þ2 . Both, the oscillation period T(E\) and the initial probability density diverge for E\/U0 = 1 which somehow complicates the numerical solution of the equation. Therefore, the Fokker–Planck equation was numerically solved by introducing the reduced probability density F T ðE? ; zÞ ¼

F ðE? ; zÞ : T ðE? Þ

ð17Þ

With the aid of the relation o ½T ðE? Þ  Deð2Þ ðE? Þ ¼ T ðE? Þ  Deð1Þ ðE? Þ: oE?

ð18Þ

Eq. (7) can be transformed into oF T ðx; E? Þ o2 F T ðx; E? Þ oF T ðx; E? Þ ¼ Deð2Þ ðE? Þ þ Deð1Þ ðE? Þ : 2 ox oE? oE? ð19Þ

Fig. 18. Upper panel: numerical solution of the Fokker–Planck Eq. (19) for the reduced probability density F(x, E\)/T(E\). The lower panel shows the corresponding probability density F(x, E\).

H. Backe et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3835–3851

0

2.0

(a)

(b)

1.5 Diffusion

6.3. Extraction of the dechanneling length It is well known that the Fokker–Planck Eq. (7) can be brought into the form oF ðx; E? Þ oJ ðx; E? Þ þ ¼ 0; ox oE?

ð21Þ

which actually is a continuity equation if the quantity

0.6

Diffusion

Drift

U (y)

0.5 0.0 −1.0

−0.5

0.0

0.5

1.0

−0.04 −0.02 0.00

0.02 0.04

Fig. 20. Panel (a) shows the potential U(y). The arrows indicate diffusion and drift currents across the channeling threshold E\/U0 = 0. Panel (b) shows diffusion, drift, and total currents as function of the reduced transverse energy E\/U0.

?

with h20 ¼ ð13:6 MeVÞ2 =ððpvÞ2 X 0 Þ. This function, which actually is the Gaussian part of the scattering distribution Eq. (1) transformed into the distribution function for the transverse energy E\, fits the numerical result in the interval 80 lm < x < 150 lm very precisely. For a comparison with the measurement presented in Fig. 16, the channeled fraction fch(x)Rmust be integrated over the target thickness. x This integral 0 fch ðnÞdn, also shown in Fig. 19, provides a nearly perfect fit to the experimental data, shown in Fig. 16. This finding provides confidence to extract the dechanneling length on the basis of the solution of the Fokker– Planck equation. The procedure to accomplish this and the result will be presented in the next subsection.

Drift

1.0

Total

density at x = 0 and E\/U0 = 1 disappears in a few lm, and that the reduced probability density spreads out across the E\/U0 coordinate very fast with increasing x. The rather wide transverse energy interval extending up to E\ = 100 U0 had to be chosen to avoid that at the largest x = 150 lm a significant reduced probability density reaches the boundary. The accuracy of the solution was checked by means of the normalization which deviated by less that 0.5% from unity. The fraction fch(x) of electrons in the channel, i.e. electrons in the transverse energy interval 0 < E\ < U0, is shown in Fig. 19. For x P 150 lm the numerical solution was extrapolated by a function proportional to ! 1 E? fE? ðE? Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  ; ð20Þ pvh20 ppvh2 E

3847

J ðx; E? Þ ¼ 

o ½Dð2Þ ðE? ÞF ðx; E? Þ oE? e

þ Deð1Þ ðE? ÞF ðx; E? Þ ¼ J diff ðx; E? Þ þ J drift ðx; E? Þ:

ð22Þ

is interpreted as a probability current, see e.g. [29, Eq. (4.46), p. 72]. It consists of two parts. The first term Jdiff is a diffusion current and the second Jdrift a drift current. In Fig. 20 these currents are shown for x = 50 lm, as an example, as function of the transverse energy E\. Notice that both, drift and diffusion currents, are rather large and nearly compensate each other, in particular deep in the potential well. To proceed further, bound and continuum states of the electron are interpreted as two states of one and the same channeling phenomenon. Both states, the channeling 1 and the off-channeling state 0, see Fig. 21, are connected to each other by the transition rates J drift ðx; E? ¼ U 0 Þ ; fch ðxÞ J diff ðx; E? ¼ U 0 Þ ¼ ; 1  fch ðxÞ

k1!0 ðxÞ ¼

k0!1 ðxÞ ð23Þ

with k1?0(x) and k0?1(x) the dechanneling and rechanneling rates, respectively. In Fig. 22(a) these rates are shown as function of x. While the dechanneling rate is more or less a constant, the rechanneling rate is much smaller and de-

0.5 0.4 0.3 0.2 0.1 0.0

0

50

100

150

200

250

300

RU Fig. 19. Fraction of channeled electrons fch ðxÞ ¼ 0 0 F ðx; E? ÞdE? as of the depth x in the crystal. Also shown is the integral Rfunction x gðnÞfch ðnÞdn assuming a constant photon generation function 0 g(n) = 0.01/lm.

Fig. 21. Two level system with channeling state 1 and off-channeling state 0, both connected by dechanneling rate k1?0(x) and rechanneling rate k0?1(x). Indicated are also the occupation probabilities fch(x) and 1  fch(x).

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Fig. 22. (a) Transition rates as function of the depth x in the crystal for an electron beam energy of 855 MeV. The corresponding mean time pattern are shown for an initial occupation of the channeling state 1 in panel (b), and for an initial occupation for state 0 in panel (c). Panels (d)–(f) show the same for an electron beam energy of 1.5 GeV. Notice that in reality channeling and off-channeling intervals fluctuate stochastically. Initial distribution according to Eq. (15) with E0? ¼ 0 and r0y ¼ 150 lrad (a), and r0y ¼ 113:2 lrad (d) which correspond both to an initial population of bound states of 51%.

creases continuously once it has reached its maximum value at x  15 lm. The corresponding mean channeling time pattern are shown in Fig. 22(b) and (c). From the dechanneling and rechanneling rates, dechanneling and rechanneling lengths Ld;FP ðxÞ ¼

1 ; k1!0 ðxÞ

Lr;FP ðxÞ ¼

1 ; k0!1 ðxÞ

ð24Þ

can be defined which amount for x = 100 lm to a dechanneling length Ld,FP = 18.0 lm and a rechanneling length Lr,FP = 105.7 lm. 6.4. Discussion As shown in the last subsection, the probability density F(x, E\) contains via the probability current information on the dechanneling length. Unfortunately, the connection to a physical observable is rather intricate which impedes a straight forward interpretation of the experimental results in term of the probability density or the dechanneling length. Actually, the experimental observable Z x Z hxmax Z E?max S E0? ;r0y ðxÞ ¼ Eð hxÞ n¼0

hx¼0 

E? ¼0

dGð hx; E? Þ F E0? ;r0y ðx; E? ÞdE? d  hx dn; dn

ð25Þ

is connected to the quantity of interest F E0 ;r0y ðx; E? Þ by a ? threefold integral. The subscript E0? ; r0y characterizes the

initial condition according to Eq. (15). The function dGðhx; E? Þ=dn describes the generation of a photon with energy hx at perpendicular energy E\ per target thickness element dn, and EðhxÞ the detection efficiency for the photon. Some general statements can be made on the photon generation function. At channeling, i.e. for E? K ðpv=2Þw2c , the function dGðhx; E? Þ=dn has a low energy component which originates from channeling radiation. At the same time the bremsstrahlung production should be enhanced with respect to amorphous matter since the electron moves in or close to a crystallographic plane. At quasichanneling, i.e. for ðpv=2Þw2c K E? K ðpv=2Þð2wc Þ2 , still low energy channeling radiation is emitted. However, the bremsstrahlung production should be reduced in comparison to amorphous matter because the position probability of the electron at the crystallographic planes reduces under the action of the planar potential. At the same time, coherent bremsstrahlung production emerges. For E? ggðpv=2Þw2c only ordinary bremsstrahlung like in amorphous matter is created. This crude classification allows an explanation of the signals shown in Fig. 14(a) and (b). The broadened line width of the clustered planes around u = 0 for the high energy signal, panel (b), in comparison to the low energy signal, panel (a), implies that the signal of the higher order planes in the wings are dying out faster in case of low energy channeling radiation production (a) as in case of high energy coherent bremsstrahlung production (b). For an explanation that, at the same time, the signal represent-

H. Backe et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3835–3851

ing the horizontal ð0  1 1Þ plane narrows and develops undershootings in the wings, it is important to realize that the undershooting minima appear at angles w?0 1 1 ¼ aDu ¼ 0:29 mrad which are very close to the critical angle wc = 0.227 mrad. This is the domain of quasichanneling with reduced bremsstrahlung production. For a detailed theoretical discussion of this effect see [24], for earlier experiments [22], and [23, Fig. 9] for axial channeling. However, it should be emphasized that the narrow signal is not situated directly on the bremsstrahlung background but on a broader structure, see Fig. 14(d), which might be associated with coherent bremsstrahlung emission. Since at quasichanneling still low energy channeling radiation is emitted as well, the corresponding low energy signals in Fig. 14(a) and (c) are broadened with respect to the high energy signal. Some additional remarks are required to explain the significant different slopes of the high and low energy signals shown in Figs. 15 and 16 as function of the target thickness. Let us first discuss Fig. 15 for the high energy signal which shows a saturation characteristics. The only plausible explanation is based on the assumption that for small E\ the bremsstrahlung production probability is enhanced. However, according to the solution of the Fokker–Planck equation the probability density at E\ = 0 vanishes, and bremsstrahlung production is expected to be suppressed. But it is very likely that these solutions are incorrect for E\  0 since thermal vibrations are neglected in the calculation of the potential. If these are taken into account, the potential flattens at the bottom, compare inset of Fig. 11(e) with Fig. 20(a), and for such a realistic potential the initial population at E\ = 0 does not vanish anymore. Moreover, the classical treatment may fail as well, and close to the ground state well separated quantum states exist, even at a beam energy of 855 MeV. A fraction of 5–10% of the electrons may populate this quantum states as estimated for a parallel electron beam. In contrast, for electrons which have left the low lying quantum states, and in particular for rechanneled electrons with a broad distribution in E\, the reoccupation probability of the quantum state is rather small, explaining why the signal saturates at larger depths in the crystal. Such saturation characteristics were observed experimentally at an electron beam energy of 54 MeV [30], 5.2 MeV [31], 350 MeV [32] and 1200 MeV [33], and were also found in a theoretical analysis of low energy channeling radiation [34]. In the papers [30,31] the parameter Locc in the [1  exp(x/Locc)] expression is interpreted as occupation length of a low lying quantum state. In case this interpretation would apply for our experimental results as well, a rather large characteristic occupation length Locc = Ld,high = (32 ± 4) lm results. It should be mentioned that in [32] measurements at large crystal thicknesses are missing. Therefore, it is not clear whether the measurements follow a characteristics of Figs. 15 or 16. Now let us turn to the discussion of the thickness dependence of the low energy signal shown in Fig. 16. Since the signal results from channeling and quasichanneling radiation,

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it is not obvious that a dechanneling length in the pure sense, meaning that the electron leaves the potential well, can be extracted. However, by inspecting once more Fig. 18, upper panel, one sees that already at about x = 50 lm the solution of the reduced Fokker–Planck equation gets rather flat in the regime of channeling and quasichanneling, and signals resulting from channeling or quasichanneling are expected to have the same thickness dependence. Therefore, the rather good fit of the integrated channeling fraction to the experimental data can be interpreted in terms of the above quoted dechanneling length Ld,FP = 18.0 lm. This number originates from a model-dependent analysis by means of the Fokker–Planck equation which takes into account the effect of rechanneling. It should be mentioned that the dechanneling rate is not a constant for small penetration depths for which rearrangement currents can be very large, and which, in turn, affect also the transition rates. In fact, the mean dechanneling length of 19.0 lm at the entrance of the crystal is somewhat longer, but also shorter values were found for rather narrow initial distributions. To avoid such ambiguities, the steady state value for x = 1 0 0 lm is taken as the dechanneling length. To estimate an error, some model assumptions should be varied, like the shape of the potential U(y), which, as already mentioned, is a rather crude approximation of the real one, and the constant Es in Eqs. (12) and (13). Such an enterprize was beyond the scope of this paper, and no errors are quoted for this dechanneling length. However, the errors may be estimated to be in the order of 15%. Finally, this dechanneling length is compared with the expression given in [16, Eq. (10.1)] Ld;B ¼

a 2U 0 E X 0; 2p ðme c2 Þ2

ð26Þ

with a the fine-structure constant. The result Ld,B = 15.7 lm for a beam energy of 855 MeV and U0 = 22 eV is somewhat lower as the experimental value. 7. Conclusion and outlook Planar channeling has been studied for silicon single crystals at the Mainz Microtron MAMI. A low energy channeling signal originating from the 1/#4 tails of the scattering distribution in combination with channeling radiation emission has been investigated. For this purpose the shower production at an aperture in the beam line downstream the target was exploited. Complex channeling patterns were disentangled with the aid of simulation calculations. These characteristic channeling patterns, as taken in a u scan around the vertical z-axis, allow to unambiguously identify the alignment angles of the crystal. Planar channeling in silicon single crystals has been studied with the low energy signal as a function of the crystal thickness in the range between 7.9 to 270 lm. Additional information has been obtained from the photon spectra of a 1000  1000 NaI detector, and from a high energy signal derived from the showers produced by high

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energy photons emitted in forward direction. The measurements with the low energy signal are very well described by calculations on the basis of a numerical solution of the Fokker–Planck equation. The interpretation of channeling as a two level system for bound and free states of the electron, which are connected by downward and upward transition rates, led to the extraction of a dechanneling length of 18.0 lm for channeling at 855 MeV in (1 1 0) planes. A beam energy scaling of the dechanneling length ld = 21.1 lm  E/GeV can be deduced from this result. A minor fraction of electrons in the order of about 5–10% may occupy well separated low lying quantum states. Employing high energy bremsstrahlung as a signal, a characteristic occupation length of (32 ± 4) lm has been derived from the thickness dependence. It should be mentioned that in [30, Chapter VI] and [11, Chapter VI. A] an enhancement of bremsstrahlung production at channeling in comparison to a randomly oriented crystal was, to the surprise of the authors, not observed. Experiments are in progress at MAMI to study the radiation emission from a 4-period crystalline undulator with a ˚ period of kU = 50 lm and an estimated amplitude of 9 A [35]. The aim of these experiments is to check the suitability of such a crystal for radiation production with positrons. At a beam energy of 855 MeV a peak at  hxU = 137 keV is expected. At a first glance such an enterprize might look hopeless because of the rather small dechanneling length in the order of 20 lm. However, one might speculate that rechanneled electrons contribute to the signal as well. Because of the stochastic character of the underlying dechanneling and rechanneling processes, the autocorrelation function of this amplitude modulated discrete-state stochastic process might therefore develop long ringing tails resulting in a peak structure at the spectral power density. However, also line broadening effects must be taken into account. The most important one is the coherence loss by multiple scattering in the dechanneling phase. In this respect, an increase of the beam energy does not help. As shown in Fig. 22(d)–(f) for 1.5 GeV, the highest energy achievable at MAMI, the ratio of rechanneling to dechanneling lengths remains essentially unchanged. For x = 100 lm the numbers are Ld,FP = 31.7 lm and Lr,FP = 149.1 lm. However, because of the increased dechanneling length at the increased beam energy the signal width should narrow which improves also the signal-tonoise ratio. In addition, the photon line shifts to hxU = 420 keV, where it can easier be separated from the low energy background if the 1000  100 NaI is used as detector. Nearly ideal conditions are expected if the dechanneling length approaches the total undulator length [4,5]. For instance, at an electron beam energy of E = 8 GeV a dechanneling length of about 182 lm can be expected. The photon line shifts to hxU = 12 MeV. The channeling radiation peak is now expected at  hx = 57 MeV, if scaled from Fig. 11(b) by (E/0.855 MeV)3/2, and should remain well separated from the photon peak sought for.

Acknowledgements We gratefully acknowledge experimental support by K. Aulenbacher, M. El-Ghazaly, H. Euteneuer, A. Jankowiak, K.-H. Kaiser, G. Kube, F. Hagenbuck, and T. Weber, and fruitful discussions with Ulrik I. Uggerhøj, A.V. Korol, A.V. Solov’yov, and W. Greiner. We thank A. Picard, and A. Sossalla from the Fachhochschule Kaiserslautern, Standort Zweibru¨cken, for the preparation of the anisotropically etched silicon targets. View graphs in the Mathematica 5.1 environment were generated with the LevelScheme figure preparation system of Capiro [36] version 3.21 (October 23, 2005). This work has been supported by the Sixth Framework Programme of the European Commission [FP6-2003NEST-A, the PECU project, Contract No. 4916]. References [1] A.V. Korol, A.V. Solov’yov, W. Greiner, Int. J. Mod. Phys. E 8 (1999) 49. [2] A.V. Korol, A.V. Solov’yov, W. Greiner, Int. J. Mod. Phys. E 13 (2004) 867. [3] A.V. Korol, A.V. Solov’yov, W. Greiner, Proc. SPIE–Int. Soc. Opt. Eng. 5974 (2005) 597405. [4] M. Tabrizi, A.V. Korol, A.V. Solov’yov, W. Greiner, Phys. Rev. Lett. 98 (2007) 164801. [5] M. Tabrizi, A.V. Korol, A.V. Solov’yov, W. Greiner, J. Phys. G: Nucl. Part Phys. 34 (2007) 1581. [6] A. Rueda, Diplomarbeit, Institut fu¨r Kernphysik, Universita¨t Mainz, Germany, 2005, unpublished. [7] M.J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, second ed., CRC Press LLC, Boca Raton, 2002. [8] A.I. Akhieser, N.F. Shul’ga, V.I. Truten, Phys. Rev. 19 (1998) 1. [9] R.A. Carrigan, J.A. Ellison (Eds.), Relativistic channeling, NATO ASI series B: physics, Proceedings of a NATO Advanced Research Workshop on Relativistic Channeling, Maratea, Italy, March 31– April 4, 1986, Plenum Press, New York, 1987, ISBN 0-306-42689-7. [10] B.L. Berman, Radiat. Eff. Def. Solids 122–123 (1991) 277. [11] M. Gouanere, D. Sillou, M. Spighel, N. Cue, M.J. Gaillard, R.G. Kirsch, J.-C. Poizat, J. Remillieux, B.L. Berman, P. Catillon, L. Roussel, G.M. Temmer, Phys. Rev. B 38 (1988) 4352. [12] Yu.N. Adishev, A.N. Didenko, V.V. Kaplin, A.P. Potylitsin, S.A. Vorobiev, Phys. Lett. 83A (1981) 337. [13] M. Atkinson, J.F. Bak, P.J. Bussey, P. Christensen, J.A. Ellison, R.J. Ellison, K.R. Eriksen, D. Giddings, R.E. Hughes-Jones, B.B. Marsh, D. Mercer, F.E. Meyer, S.P. Møller, D. Newton, P. Pavlopoulos, P.H. Sharp, R. Stensgaard, M. Suffert, E. Uggerhøj, Phys. Lett. 1 1 0B (1982) 162. [14] J.U. Andersen, E. Bonderup, R.H. Pantell, Ann. Rev. Nucl. Part Sci. 33 (1983) 453. [15] E. Uggerhøj, Phys. Scripta 28 (1983) 331. [16] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, World Scientific, Singapore, New Jersey, London, Hongkong, 1998. [17] M.B.H. Breese, Nucl. Instr. and Meth. B 132 (1997) 540. [18] P.A. Doyle, P.S. Turner, Acta Crystallogr. A 24 (1968) 390. [19] S.B. Dabagov, V.V. Beloshitsky, M.A. Kumakhov, Nucl. Instr. and Meth. B 74 (1993) 368. ¨ berall, in: A.W. Sa´enz, H. U ¨ berall (Eds.), Topics in [20] A.W. Sa´ens, H. U current physics, Coherent Radiation Sources, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. [21] V.V. Beloshitsky, Ch.G. Trikalinos, Radiat. Eff. 56 (1981) 71.

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