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Proceedings of a NATO Advanced Research Workshop on Relativistic Channeling, held March 31-April 4, 1986, at Villa Del Mare, Acquafredda di Maratea, ltaly
Library of Congress Cataloging in Publication Data
NATO Advanced Research Workshop on Relativistic Channeling (1986: Acquafredda di Maratea, ltaly) Relativistic channeling.
(NATO ASI series. Series 8: Physics; Voi. 165) "Published in cooperation with NATO Scientific Affairs Division." "Proceedings of a NATO Advanced Research Workshop on Relativistic
Channeling, held March 31-April 4, 1986, Acquafredda di Maratea, ltaly"-T.p. verso.
lncludes index. 1. Channeling (Physics)-Congresses. 2. Particles (Nuclear physics)-Con­
gresses. 1. Carrigan, R. A. 11 . Ellison, J. III. North Atlantic Treaty Organization. Scientific Affairs Division. IV. Title. V. Series: NATO advanced science insti­ tutes series. Series B, Physics; v. 165. QC794.6.C6N345 1986 539.T54 87-7318 ISBN 978-1-4419-3207-5 ISBN 978-1-4757-6394-2 (eBook) DOI 10.1007/978-1-4757-6394-2
© 1987 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1987
Ali rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
PREFACE
Channeling, by its nature, involves a wide and disparate range of disciplines. Crystal preparation, material science, accelerator physics, sophisticated theoretical analysis and, of course, channeling itself all must work in concert in a research program. In spite of the gulfs separating some of these activities, researchers have drawn together over the last decade to carry out remarkable experiments in relativistic channeling and channeling radiation.
Several informal workshops on high-energy channeling have been held over ~he years at Aarhus and Fermilab. However, with the vigorous progress in the field in the last several years it became clear that a more formal, comprehensive workshop was needed along with a book that covered the whole spectrum of the new developments, probed the future, and also laid out some of the foundations of the subject. This volume is the outcome of that process.
The organization and preparation of both the volume and the workshop owe much to several outstanding scientific committees. The membership of these included J. Andersen (Aarhus), S. Baker (Fermilab), B. Berman (G. Washington), G. Bologna (Torino), E. Bonderup (Aarhus), S. Datz (Oak Ridge), J. Forster (Chalk River), F. Fujimoto (Tokyo), W. Gibson (Albany), I. Mitchell (Chalk River), Y. Ohtsuki (Waseda), R. Pantell (Stanford), S. Picraux (Sandia), J. Remillieux (Lyon), A. Saenz (NRL), V. Schegelsky (Gatchina), C. Sun (Albany), H. tiberall (Catholic U.), E. Uggerh¢j (CERN), and R. Wedell (Humboldt). Others from across the spectrum of scientific disciplines agreed to serve as session chairmen. These included some of the committee members, authors in the present volume, and also A. Ugguzoni (Bologna), J. Spence (Arizona), G. Temmer (Rutgers), S. Salman (An-Najah), M. Toulemonde (Lourds), J . Poizat (Lyon), A. Kanofsky (Lehigh), B. Marsh (Albany), M. Thompson (East Anglia), and J . Davis (McMaster). We would like to acknowledge the contributions of all of these people as well as J . Lindhard, Y. Quere and the authors of the chapters in this volume. A refreshing aspect was the presence of three graduate students, H. S. Dumas (New Mexico), L. Hau (Aarhus) and J. Kephart (Stanford), working in the field of channeling.
The North Atlantic Treaty Organization Scientific Affairs Division and the Danish Com­ mittee for Accelerator Physics sponsored the workshop. We would particularly like to ac­ knowledge the vision of Mario Di Lullo of the NATO Scientific Affairs Division in approving this workshop. Unfortunately, Dr . Di Lullo passed away before he could see the publication of this volume completed. Both the University of New Mexico and Ferrnilab also provided generous assistance in the arrangements for the workshop and this volume.
Crucial to the workshop was a unique and interesting location with the necessary ambiance for nearly a week of intense activity. This was provided by Villa Del Mare in Acquafredda di Maratea, Italy. We particularly wish to acknowledge the manager, Mr. A. Guzzardi and Susy Travisano, who was in charge of conference arrangements.
v
A project like this ultimately succeeds on the basis of dedicated staff work. Jackie Damrau of the Department of Mathematics at the University of New Mexico handled the preparation of this volume and much of the conference work. Pat Oleck of Fermilab also contributed in both areas. Finally, Nancy Carrigan and Colleen Ellison ably assisted in the workshop ar­ rangements at Maratea.
R. Carrigan Fermilab
CONTENTS
Introduction
CHANNELING
Channeling and Related Phenomena for Ge V Particles E. Uggerh¢j
Experimental Demonstration of Reversibility Through Ge V Channeling J. S. Forster
Energy Loss and Straggling of Random and Channeled High-Energy Particles in Thin Targets
S. P. M¢ller
R. Wedell
Theory of Particle Motion in Straight and Distorted Crystals J. A. Ellison
Axial Channeling in Bent Crystals H. E. Schi¢tt
Deflection of Particle Beams Using Planar Channeling W. M. Gibson
The Theory of Volume Capture by a Curved Crystal in the Channeling Regime
O. 1. Sumbaev
1
5
39
49
59
69
79
89
101
117
129
vii
Quantum Theory of Channeling Radiation J. U. Andersen
Quantum Theory of Fast Charged Particles in Crystals G. Kurizki and J. K. McIver
Dirac Equation for Electron Channeling H. Pilkuhn and A. H. S¢rensen
Coherent Bremsstrahlung and Free Bound Transitions A. W. Saenz, A. Nagl and H. Uberall
Coherence Lengths for Emission of Classical Channeling Radiation o. Pedersen, J. U. Andersen and E. Bonderup
Measurement of Channeling Radiation E . Laegsgaard
Channeling Radiation Experiments Between 10 and 100 Me V B. Berman et al.
Channeling Radiation Experiments Between 100 and 1000 Me V F. Fujimoto and K. Komaki
Channeling Radiation Experiments at Energies Above 1 Ge V J. F. Bak
The Study of Electron Channeling and Channeling Radiation Using High- Voltage Electron Microscopes
F. Fujimoto
J. C. Kimball and N. Cue
Experimental Study of Pair Creation and Radiation in Ge Crystals at Ultrarelativistic Energies (30-200 Ge V)
A. Belkacem et al.
On the Angular Dependence of Bremsstrahlung and Pair Production in Single Crystals at Ge V Energies
A. H. S¢rensen
APPLICATIONS TO PARTICLE PHYSICS
The Application of Channeling in Bent Crystals to Charged Particle Beams
R. A. Carrigan, Jr.
Applications of Channeling to Particles Physics Experiments C . Sun
Radiation Damage Effects in Channeling Applications S. Baker
Application of Semiconductor Detectors in High-Energy Physics A. Menzione
Progress in High-Rate, High-Accuracy Detectors G . Charpak
APPLICATIONS TO MATERIAL SCIENCE
Growing Large Highly Perfect Single Crystals and Its Limitations A. Seeger
The Study of Material Properties Using Channeling Radiation R . Pant ell et al.
Crystal Potentials from Channeling Radiation-A First Principle Calculation
A. P. Pathak and S. Satpathy
Muon and Pion Stopping Sites in Crystals from Decay-Particle Blocking B . D. Patterson
Pion Diffusion Studies Using Muon Channeling at High Temperatures A. P. Pathak et al.
Lattice Location of Nuclear Probes by Electron and Positron Channeling H. Hofsass et al.
339
369
379
391
399
419
423
435
455
459
479
483
ix
HEAVY IONS
Stimulated and Cooperative Radiation from Channeled Particles and Ions G. Kurizki
Channeled Particle Acceleration by Plasma Waves in Metals P . Chen and R . J. Noble
Index
x
493
505
517
523
INTRODUCTION
Channeling is the process where charged particles are steered by the rows or planes of atoms in a perfect crystal. This can occur when a particle beam is aligned with major crystal axes or planes or when particles are created in a crystal. Related processes include coher­ ent bremsstrahlung and coherent pair production from aligned high-energy photon beams. Channeling has contributed in important ways to our understanding of particle motion in solids at the fundamental level and, just as significantly, has led to numerous applications in physics and technology. Furthermore, it presents a new source of interesting and challenging mathematical problems in dynamical systems and stochastic processes. Much of the early work focused on MeV energies and heavy particles; however, during the last decade there has been an increasing realization that the interaction of relativistic particles with aligned single crystals is interesting.
Pioneering work by an Aarhus-CERN group laid the foundation for such investigations. At Fermilab Tsyganov conceived the idea of using channeling in crystals to deflect particle beams and went on to demonstrate the technique at Dubna. During the same period, Kumakhov pointed out that electrons and positrons moving in channeling trajectories should have a characteristic radiation. By now Ge V scale channeling and channeling radiation experiments have been carried out at Brookhaven, CERN, Dubna, Fermilab, GANIL, Gatchina, Kharkov, LAMPF, SIN, SLAC, Serpukhov, Tokyo, Tomsk, and Yerevan.
While the low-energy work has been extensively reviewed there is no comprehensive sum­ mary of modern relativistic channeling. This volume has been designed to fulfill that need. It is the outgrowth of a NATO Advanced Research Workshop held at Villa Del Mare in Maratea, Italy, to review the field and appraise the future possibilities for channeling activ­ ity. The workshop was cosponsored by the Danish Committee for Acclerator Physics. This workshop was convened because of the recent advances in channeling at relativistic energies and because traditionally international conferences have emphasized low-energy channeling. The workshop topics included basic channeling phenomena, channeling radiation and coherent bremsstrahlung, strong field effects, applications to particle physics and to material science, heavy ion channeling, and stimulated and cooperative phenomena.
The present volume is divided into seven sections. The first section is devoted to basic channeling phenomena. UggerhjZlj reviews developments in the study of channeling and chan­ neling radiation concentrating particularly on apparent anomalies that have been clarified by recent work. Forster discusses reversibility in space and time reversibility in equilibrium as they arise in the master equation formulation of dechanneling and relates these to recent GeV channeling experiments at Fermilab. MjZlller considers energy loss, a-ray emission, and K-shell excitation by highly relativistic charged particles. Channeled particles dechannel due to electron multiple scattering and thermally vibrating nuclei. The understanding of this pro­ cess is fundamental in any comparison between theory and experiment. Ohtsuki and Nitta discuss diffusion dechanneling and give a detailed survey of the Japanese work on calculat­ ing diffusion coefficients associated with various dechanneling mechanisms. Wedell reviews dechanneling for electrons and positrons in the context of channeling radiation. He uses both a Fokker-Planck equation in phase space and a kinetic equation in transverse energy to treat
the non-equilibrium and equilibrium cases, respectively. Ellison's chapter includes discussions of the Hamiltonian random crystal model for lattice vibrational effects and the relation be­ tween phase space evolution and transverse energy evolution. Later in the second section Andersen considers dechanneling in the general context of the quantum theory of channeling radiation. The last five chapters of this first section deal with basic channeling processes in bent crystals. Ellison discusses the derivation of the axial and planar continuum models for bent crystals from the relativistic perfect crystal model using the method of averaging and notes some similarities to Hamiltonian systems in accelerator physics. Schiptt presents a Monte Carlo study of axial channeling in bent crystals and the associated feeding into planes. Gibson summarizes the experimental work of the last several years on channeling in curved crystals and then goes on to consider the possibilities in this area for high Z crystals in the future. Sumbaev and Samsonov review the Soviet theoretical and experimental work, respectively, on the capture of energetic charged particles into channeling trajectories in bent crystals. One particularly interesting facet of the Leningrad work is the careful crystal characterization of the silicon that was used in the experiment.
The second section of the book is devoted to the radiation emitted due to the special motion of channeled particles and the related phenomenon of coherent bremsstrahlung. The Andersen chapter discusses the quantum theory of MeV channeling radiation and develops a systematic approximation procedure through which a comprehensive and accurate description of photon energies is built up. This is followed by a review by Kurizki and McIver of the circumstances under which the full three-dimensional lattice needs to be considered. Pilkuhn and S¢rensen formulate the channeling problem in terms of the Dirac equation which allows a treatment of the fine structure and Zeeman splitting of channeling radiation. The basic formulae for the emission of channeling radiation are included. Saenz, Nagl, and Uberall discuss coherent bremsstrahlung and channeling radiation as two aspects of the same phenomena. They also consider the intermediate process involving free-bound transitions. Pedersen, Andersen and Bonderup complete the theoretical part with a classical description of channeling radiation in the Ge V region including a detailed analysis of the concept of a coherence length . The experimental portion begins with Laegsgaard who emphasizes the problems of measurement ranging from beam and target preparation to data analysis and calibration of experimental equipment. The next three chapters discuss channeling radiation experiments from 10 MeV to above 1 GeV. Berman and his colleagues discuss results in Si, C, W, GaAs and various alkali­ halide crystals which test the quality of potentials used in calculations and allow for parameter studies, such as temperature and thickness. Fujimoto and Komaki review the energy range from 100 to 1000 Me V where there are only a few experiments. Bak considers channeling radiation in the low Ge V range where a classical picture is valid. This section ends with a discussion of a novel approach to studying channeling radiation with high-powered electron microscopes.
Recently, a new front has developed and this is the subject of the third section. By empha­ sizing the electric field of a string of atoms rather than the electrostatic potential, both theory and experiment have shown that an aligned crystal becomes a laboratory for investigating strong-field quantum electrodynamic (qED) processes of ultrarelativistic particles. Kimball and Cue consider the special role a crystal plays in strong field qED at GeV energies. Un­ der nearly perfect alignment their theoretical developments predict enhanced radiation from electrons and positrons and enhanced pair creation from aligned energetic photons. They also present the possibility of photon splitting and pion creation. The first two of these effects have now been studied in a comprehensive set of measurements presented here by Belkacem and his colleagues. Sprensen completes this section by discussing the angular dependence of the coherent and incoherent contributions to the bremsstrahlung and the yield of the inverse process of photoproduction of electron-positron pairs. Particular attention is given to the breakdown of the first order Born approximation with decreasing angles to crystal axes or planes.
The next section considers some applications of channeling and channeling radiation in particle physics. Carrigan reviews the possibilities for use of bent crystals in charged particle
2
beams and discusses several recent applications at Fermilab. This section also reviews some of the areas of particle physics where crystal channeling might find applications. As a prelude, Diambrinni-Palazzi discusses charm particle physics. Sun considers possible applications of channeling to actual high-energy experiments including such topics as lifetime measurements and the study of charm particles . The impact of radiation damage to crystals must be consid­ ered in any application of channeling in an intense particle beam. Baker presents information showing that crystals are quite robust in such situations. Menzione considers other appli­ cations of semiconductor detectors while Charpak reviews progress in perfecting high-rate, high-accuracy detectors for high-energy experiments.
Relativistic channeling and channeling radiation have many attractive features for work in the material sciences . An important limitation is the problem of growing large, highly perfect crystals for these studies and the particle physics applications. Seeger reviews the possibilities for such crystals along with the limitations on growing them. Pant ell and his colleagues then discuss some of the material science studies that have already been carried out using channeling radiation. Pathak and Satpathy calculate the crystal potential including exchange and correlation effects among the crystal electrons using density functional theory combined with the linear muffin-tin orbital method. This is used to calculate channeling radiation frequencies which are compared with experiments. Patterson reviews the use of muon and pion probes to study stopping sites in crystals with the blocking phenomenon. Pathak and his collaborators study temperature effects of pion diffusion using the channeling/ blocking effect for the decay muons. Lattice locations of nuclear probes have also been extensively studied using implanted electron and positron sources . Hofsaess and collaborators summarize those studies.
Another interesting approach to material science studies is through the use of heavy ion channeling. In the next section Cohen and collaborators review some of the possibilities that are now opening up with the remarkable GANIL heavy ion accelerator in France.
Perhaps the most intruging possibilities for relativistic channeling have been the very spec­ ulative ideas on stimulated and cooperative phenomena. These include such devices as lasers using channeling radiation, discussed here by Kurizki, and solid state accelerators reviewed by Chen and Noble. While these ideas are exciting they also present extremely severe challenges to the materials that would be employed. Exploring ways around these material limitations may uncover other equally interesting subjects.
The focus of this book is strongly on recent channeling work at relativistic energies. More traditional channeling has been convered in a number of books, review articles and confer­ ence proceedings. Lindhard1 (1965) laid the theoretical foundation for a statistical treatment of channeling. This paper has been expanded in a set of Aarhus lecture notes which are available.2 A review article by Gemmel3 and a book on channeling edited by Morgan4 give general discussions of particle channeling and applications and an extensive bibliography of the literature up to 1974. Three books have appeared more recently. In "Materials Analy­ sis By Ion Channeling: Submicron Crystallography," Feldman, Mayer and Picraux5 review channeling as a material science tool. This book has an extensive bibliography of low-energy channeling through 1982. Ohtsuki6 discusses several theoretical aspects of channeling includ­ ing a detailed analysis of dechanneling in his book "Charged Beam Interaction with Solids". "Coherent Radiation Sources," edited by Saenz and Uberall7 is a collection of articles which combine to give a detailed review of the theoretical and experimental status of channeling radia­ tion and coherent bremsstrahlung. Proceedings from recent International Conferences on Atomic Collisions in Solids (1979,1981,1983, 1985) contain many channeling articles and have been published by Nuclear Instruments and Methods.
REFERENCES
1. J. Lindhard, Dansk. Vid. Selsk., Mat. Fys. Medd., 34(14) (1965).
2. J. U. Andersen, private communication.
3
3. D. S. Gemmell, Rev. Mod. Phys. 46 (1974) 129.
4. D. V. Morgan, editor, "Channeling" (Wiley, New York, 1973).
5. L. C. Feldman, J. W. Mayer and S. T. Picraux, "Materials Analysis by Ion Channeling: Submicron Crystallography" (Academic Press, New York, 1982).
6. Y. H. Ohtsuki, "Charged Beam Interaction with Solids" (Taylor and Francis, New York, 1983).
7. A. W. Saenz and H. Uberall, editors, "Coherent Radiation Sources," Topics in Current Physics 38 (Springer-Verlag, New York, 1985).
4
E. Uggerhf/lj
1. INTRODUCTION
Already in the original channeling paper by Lindhard1 it was shown that the correlated scatterings of a projectile incident along a crystalline row of atoms can be studied by classical mechanics even when the individual scattering events are not amenable to a classical treatment. The condition for the classical model is that the projectile mass is large compared to the rest mass of the electron. Although this was very surprising it was even more surprising that many aspects of channeling for keY-MeV electrons/positrons could be understood from the same classical model. This was based on the fact that in the transverse motion it is the relativistic mass that enters. Later it was shown that axial effects are more classical than planar ones and that positrons are more classical than electrons.
From this it followed that wide-angle scattering of positrons with energies of some hundred ke V should follow the classical predicted channeling dip for protons of the same momentum.
These surprising considerations led to the first channeling investigations using relativistic electrons and positrons derived either from implanted radioactive f3 emitters or beams from Van de Graaf accelerators. The axial results were in good agreement with Lindhard's classical theory, but clear diffraction effects were observed-especially for the planar cases.
Secondary high-energy beams in the energy region of 1-10 GeV on the other hand contain a variety of particle/ antiparticle types with different rest masses. This gives rise to an enormous range of values for the Lorenz factor ')'(1 - 104). This fact means that such beams are ideal for measuring the onset of relativistic effects. As all the particles are in the same beam, the investigations can be performed as relative measurements-relative to slow heavy particles like the proton.
Based on these arguments a strong interest arose in the mid-seventies in using such beams for channeling investigations. Especially prorrllsing was the possibility of examining relativistic channeling phenomena using positive and negative projectiles of the same kind, i.e., 71'+ /71'- , in the same set-up. The results could be compared to electron/positron results obtained in the MeV region and the influence of diffraction could be evaluated. Both scattering phenomena and energy loss could be compared for positive and negative particles. This had not been possible for energy loss measurement in the Me V regime because very thin crystals are needed for Me V electrons and positrons due to strong dechanneling. As dechanneling is inversely proportional to the particle energy, experiments in the Ge V region permit the use of millimeter­ thick crystals that can be turned into solid state detectors ("live targets"). Although the introduction of "live targets" into high-energy beams created many technical problems in the beginning it turned out to be a significant step forward . The technique is used all over in particle physics today.
5
A special motivation for high-energy channeling was the possibility of using strongly in­ teracting projectiles which could give new effects in coherent nuclear reactions and lead to measurements of ultrashort lifetimes of rare particles.
Based on these perspectives, GeV channeling investigations were started at CERN,2 in the USA,3 and in the USSR.4 In these first experiments channeling was looked for by its influ­ ence on scattering phenomena and incoherent bremsstrahlung. Completely new experimental techniques had to be introduced because of the low-intensity, divergent beams and the small critical channeling angles (100 Ilrad or less). This put very strong demands on crystal perfec­ tion (no "mosaic spread"), equipment stability and angular resolutions. Pronounced effects were found even in the first experiments, wHich motivated new investigations and led other laboratories to become involved.
Since then a large variety of subjects have been looked into, including 1) close encounter processes; 2) multiple scattering; 3) energy loss; 4) straggling; 5) the density effect; 6) doughnut scattering; 7) dechanneling; 8) inner shell excitation; 9) S-ray emission; 10) bending of GeV beams; 11) lifetime of short-lived particles; 12) channeling radiation and 13) coherent and incoherent particle production.
In the Proceedings from the Symposium on Selected Topics in Physics, in honor of Jens Lindhard's 60th birthdayS many of the above-mentioned subjects are reviewed (here­ after referred to as Ph.s.I.). The next three sections are drawn from that article and cover the general features of channeling and the experimental technique.
In Ph.S.1. the reader will find unsolved questions in certain fields such as: a) the lack of a density effect for inner shell excitations; b) the fact experimental straggling curves are wider than the Landau distributions; c) no peak in wide angle scattering yield for GeV negative pions; d) a disagreement between CERN and Fermilab data on dechanneling; e) the measured energies of the first harmonics in channeling radiation from positrons are 5-10% lower than the calculated ones.
In the following the experimental technique will be discussed and these open questions will be introduced and answered based upon later experiments. Finally the outlook for GeV channeling will be discussed. Subjects not covered here are treated in detail in other contri­ butions in this volume.
2. EXPERIMENTAL TECHNIQUE
In the MeV region, channeling experiments are normally performed with a beam diver­ gence much smaller than the critical angle of channeling, and the crystal is tilted through axial or planar directions by means of a goniometer. Since channeling angles in the Me V region are between 0.10 and 10, the technical requirements are not too severe. However, in the Ge V re­ gion, channeling angles are 50-100 Ilrad; thus this technique would entail strong requirements on the goniometer if parallel high-energy beams are used. Most secondary high-energy beams have divergences of about ±1 mrad, which makes it impossible to tilt through channeling con­ ditions. It was therefore necessary to introduce a new technique. For this, the development of high-accuracy drift chambers was a long step forward as these made it possible to measure particle positions with an accuracy of about 0 .1 mm. With sets of position-sensitive drift chambers in front of and behind the crystal studied, it was possible to simultaneously inves­ tigate a relatively wide range of angles of incidence and emergence. This kind of operation has permitted experiments with highly divergent secondary beams. In fact, the large angular spread of the beam provides a broad angular map of the channeling effects and eliminates the need for an extremely accurate alignment of the crystal and the accuracy of the remotely con­ trolled small-angle goniometers. On the other hand, the experimental data consist of millions of particle tracks, which require a considerable amount of computer time for their analysis.
A schematic drawing of a typical high-eqergy channeling experiment is shown in Fig. 1. The beam could be a low-intensity (~ 105 cm- 2 s-l) secondary, nonseparated charged beam
6
I No!
BEAM DUMP.
SC4 SC2
1 i I I VTI Y 'DC2
DCl ANNULAR VT2 DC3
1- 10m "" .. 10m
"I Fig. 1. Schematic drawing of experimental setups used at CERN. The beam en­ ters from the left. DC designates drift chambers, SC scintillators, and VT vacuum tubes. Above is the experimental setup used for channeling-radiation experiments. Here BMI (bending magnet) removes upstream incoherent-radiation background by a IO-mrad bend. The exit beam is bent away by BM2 from the radiation de­ tector through Cerenkov and lead-glass counters into the beam dump. The lower diagram shows the setup for conventional channeling.
with momentum adjustable between, for example, I and 15/GeV Ie. For the positive polarity, the beam consists of protons, deuterons, kaons, pions, and positrons with an angular spread of ±I mrad.
Particle identification was performed by scintillators placed between SCI and SC4 together with threshold Cerenkov and lead-glass counters. Additional scintillation counters (SC2 and SC3) in anti-coincidence with SCI and SC4 were used to define the usable fraction of the beam in order not to exceed the maximum size of the crystal sample at the focal point of the goniometer. The trajectories for incoming and outgoing particles were measured by a set of five drift chambers, DCI through DC5. Beam lines VTI and VT2 were evacuated to eliminate multiple scattering.
The goniometer allowed adjustment of the crystal axis to the center of the average beam cone although a precise alignment was not necessary with this method . In many cases, the target crystal was an intrinsic solid state detector; hence, a cooling system was attached to the goniometer.
When the channeling radiation is measured, a somewhat different setup, shown above in Fig. I, is used. The exit beam is bent into a dump by BM2, and a NaI detector is used for measuring the I rays, as indicated in the figure. A small magnet, BMI, is used to remove background up-stream radiation. This radiation, which gives an incoherent background, is produced by projectiles hitting material before reaching BMI. The present setup still gives
7
an incoherent background from DC2, which amounts to 200 /Jm amorphous silicon and is subtracted in all data.
The data acquisition is carried out as described in Ref. 6. In short, for each accepted event, the output from drift chambers, crystal detector, X- or I-ray detectors, the time correlation between these detectors and the SCI and the Cerenkov counters, or the TOF measurements, were stored on magnetic tape. In most cases, the CDC 7600 computer at CERN was used for the full data analysis, while a small on-line computer ensured correct behavior of the entire setup during runs.
Few elements form crystals suited for high-energy channeling experiments because chan­ neling angles are so small that even a moderate mosaic spread will smear the channeling effects. Hence, so far only silicon and germanium crystals, which can be produced with practically no mosaic spread, have been used. These elements are also well suited for the production of solid state detectors.
Target preparation and alignment are described in detail in Ref. 7. Here it should only be mentioned that the fastest and simplest way to align the crystal axis to the center of the beam cone is to prealign the crystal off-line, using X-ray techniques, and then adjust the alignment on-line by means of either channeling radiation or energy loss.
Fig. 2 is an example showing a case where a (110) germanium crystal is aligned in a lO­ Ge V I c secondary beam from the proton synchroton at CERN. The figure shows the intensity distribution in incident-angle space for those particles that have been transmitted through a 300-/Jm-thick crystal, and which have a scattering angle of less than 0.1 mrad. In this setup, the incident angular resolution was ~ 15/Jrad, in which case pronounced effects are seen even for high-order planes, for which the channeling angle is about 30 murad . Naturally, such transmission experiments are extremely sensitive to the stability of the entire system and are used on-line to check the system, especially the drift chambers.
3. WIDE-ANGLE SCATTERING AND BLOCKING
It was shown in Ref. 9 that the normal channeling picture also applies for relativistic particles provided the rest mass Ml of the projectile in the nonrelativistic description be replaced by the relativistic mass IMI and the projectile energy E = !MIV2 by !pv, where p is the relativistic momentum. Thus the critical angle I/Jl for axial channeling becomes
I/Jl = (I)
(2)
where N is the atomic density of target atoms, d is the distance between atoms in a string, dp is the distance between atomic planes, Zle and Z2e are the nuclear charges of the incident particle and target nucleus, respectively, and C is a constant of about )3. In Table 1, typical values of I/Jl are given. To a good approximation, I/Jp = ~I/Jl'
The application of a classical orbital picture to channeling phenomena was discussed in detail by Lindhard.1 The number of quantum states per unit cell for the ' transverse motion is proportional to (/MI)1/2 and IMI for planar and axial motion, respectively. While this number may be of the order of unity for low-energy electrons and also for low-energy positrons experiencing planar channeling, it is large for axially channeled positrons, even in the 100-ke V region. In addition, the BohrlO condition,
It-L = 2 (Ml Zl/2 aO) 1/2 > I mo 2 d
(3)
8
1.0
>- "3-
0.4
o
-G.4
-G.8
-1.2 ~ __ -..L ___ ...J... ___ .L....--=:"'-.....JI..-__ -L ___ ....I -1.2 -0.8 -D.4 D 0.4 0 8 1.2
TRANSMISSION o/x (MRAD)
Fig. 2. Stereogram of (llO}-oriented Ge crystal. The number of projectiles in two-dimensional angle space of beam of incidence (IO-Ge V Ic protons), which have been scattered less than 0.1 mrad, are plotted, normalized to the beam intensity. The highest intensity is represented by the darkest area. The axes and planes are very pronounced as multiple scattering is strongly reduced for these directions. The steering effect from planes is seen to weaken close to the axis.8
Table 1. The critical angle ,pI in mrad. The tabulated values are for (:J = 1.
Si (1l0) Ge (1l0)
0.32 0.48
0.19 0.28
0.12 0.18
for the applicability of orbital pictures in the description of deflections by an axis is also reasonably well fulfilled for ~ 100-ke V positrons. (Here mo is the mass of the electron and ao is the Bohr radius.) The same inequality, /C.L > 1, also ensures that tunneling to classically forbidden areas close to nuclei is very improbable.
Through the rule ofreversibility, as discussed by Lindhard,I it was shown that the critical angles ,pI and ,pp hold both for channeling and blocking.
9
•
~3 I • I I 0.. I • , I
I • :i I ' .... I ~ 0 I • .... IAI 0.6 N • ::; I \ . i c I • ::E !. i a:: 2 I
!l # \\ I I I
-,' -2' O· 2" 4' -6' -4' -2" O· 2' 4'
" EMISSION ANGLE
Fig. 3. Angular distribution around a (110) direction of electrons and positrons emitted from 64Cu implanted into a Cu crystal. The energies given are average ki­ netic energies, for which the dashed curves are calculated based on the continuum approximation. II
The first channeling experiments with positive, relativistic projectiles were performed by embedding radioactive nuclei into single crystals and then studying the blocking patterns as no good beams were available at that time . Fig. 3 illustrates such a blocking pattern for positrons emitted by 64Cu implanted into copper crystals (taken from Ref. 11). Since 64Cu also emits electrons in the lOO-keV region, this experiment could in a simple way show the general difference in channeling for positive and negative particles. The results are compared to classical calculations based on Lindhard's continuum approximation for perfect crystals. It should be noted that the experimental half-widths AtP are in good agreement with the calcu­ lated ones, and it is particularly interesting that the AtP value for positrons is more than twice that for electrons. This is also in agreement with calculations although some damage from implantation was present in the crystal. In early discussions of channeling, this comparison created heated discussions on the relationship between diffraction and channeling phenomena, especially for electrons, where information on penetration phenomena was known from electron microscopy. Discrepancies in peak heights and minimum yields were expected to be due to defects created during implantation. Thermal annealing of the defects smeared out the block­ ing patterns because of diffusion. Hence MeV beams of electrons and positrons were set up and used for wide-angle scattering experiments on thin perfect crystals. Figure 4(a)I2 shows a comparison between yields from wide-angle scattering of positrons and protons incident along the (110) axis in a 1200-A-thick gold crystal. Because of the term !pv in tPI, I -Me V positrons
10
are compared with 670-keV protons. The striking agreement for minimum yields and widths shows that the possible quantal corrections to the classical picture are small. Fig. 4b shows the same experiment but for I-MeV electrons.14 As in Fig. 3, the FWHM is only ~,pl' but the peak height is nearly the same. This might appear surprising since small defects should be present. Diffraction phenomena, on the other hand, might smear out the effects since the condition for a classical description is not so well fulfilled for I-MeV electrons. However, in the GeV region, this condition is well fulfilled for positive as well as for negative particles. Fig. 5a shows a dip for 15-GeV /c protons incident on a (110) 4.2-mm-thick germanium crys­ tal. The experimental points are compared to the calculated dip based on a modified Lindhard potential, for which thermal vibrations are taken into account. The angular resolution of the detector system was l,pl and hence to some extent influences the dip, especially the minimum. On the other hand, the agreement is good. In Fig. 5b a wide-angle normalized scattering dip is shown for 15-GeV /c protons incident on a 4.2-mm-thick germanium crystal along the (111) planes. Since ,pp here is 60 Ilrad, the minimum yield of ~50% is influenced by angular res­ olution on the incident side, which was about !,pp. The FWHM is also in good agreement with Lindhard's critical angle for planes. Experiments with up to 250-GeV protons15 and 55-GeV /c electrons/positrons have been performed so that channeling has been studied over a range of 5-6 orders of magnitude in momentum, and no discrepancies between theoretical and experimental critical angles have been found.
For negative particles, no detailed wide-angle scattering results exist for the GeV region although the good experimental angular resolution shown in Fig. 2 gave hope of such investi­ gations. From the Me V -electron data, it was clear that very thin targets should be used due to the increased multiple scattering for negative, channeled particles. However, in the GeV region, cross sections for close-encounter processes are small; hence such channeling investi­ gations would be very time consuming in divergent secondary beams, especially when large scattering angles are required. Therefore the condition on scattering angles with respect to ensuring close encounters was lowered to a few ,pl. Fig. 6 shows an example of yield curves for
a
b
~ 18 ~ gs 16 z
(110) Au
10hW----"' ...... I'-
0.8 -4' -3' -2' -I' 0' l' 2' 3' 4' TILT ANGLE
Fig. 4. Comparison of Rutherford-scattering yields around the (110) direction in 1500-A-thick Au crystals for (a) 1 MeV positrons and 0.67-MeV protons, (b) 1- MeV electrons. In (a), the abscissa scale for protons has been scaled from 1 MeV to 0.67 MeV.13,14
11
a b
1 .01---------~--~-....,..,
0 0.8 ...J W ;;: o . l:!l 06 ::::; « ::t a: o Z 0.4
0.2
ISGeV/c p-4.2mm {II11 Ge
SCATT. ANGLE> 10 IjIp
°0~---1L---~2L---~3L-~
INCIDENT ANGLE (1jJ/ljloJ
Fig. 5. Normalized yields of wide-angle scattering as a function of angle of inci­ dence to the (110) axis (a) and (110) plane (b) for 15-GeV Ic protons incident on a 4.2-mm-thick Ge crystal. The solid curve in (a) is calculated on the basis of a modified standard potential to approximate the thermally averaged string poten­ tial. Dechanneling and angular resolution are also taken into account.7,s
15-GeV Ic protons and negative pions transmitted along the (110) axis through a 4.2-mm-thick germanium crystal The scattering angle was 1 mrad (5.5 1/II). The huge increase in scattering probability obtained for angles of incidence as large as 15 ,pI clearly shows that some spe­ cial effects are involved, and such experiments cannot be used to investigate close-encounter processes . On the other hand, these results led to the discovery of so-called "doughnut" scat­ tering, which for the Ge V region has turned out to be very pronounced. This subject will be briefly discussed in the following section.
4. DOUGHNUT SCATTERING
If a parallel beam of momentum P is incident on a single crystal under axial-channeling conditions, correlated scattering will occur. In the transverse plane, the momentum vectors Pol only are rotated so that the original direction of Pol will be changed, and after a certain number of string collisions, Pol may be found in any direction in the transverse plane . This equalization process was discussed by Lindhard,l who derived an approximate analytical for­ mula, from which the crystal thickness necessary for such an equalization could be obtained. For thin crystals, where multiple scattering is negligible, the transverse energy Eol will not change appreciably by passing through the crystal, and the particles will exit by angles to the crystal axis close to those of incident Bin' Hence, the net result is that the incident parallel beam will appear as a ring-shaped (doughnut) distribution in angle space with a radius equal to Bin . In Fig. 7 scatter plots are shown for 15-GeV Ic protons and negative pions transmit­ ted through a O.9-mm-thick silicon crystal in an angular region 2-2.5 ,pI from the axis. The transmitted intensity is plotted as a function of the angle relative to the (110) axis. The incident beams had an azimuthal spread of 30° and a radial spread of 0.1 mrad . Both dis­ tributions clearly show the effect of doughnut scattering. These effects are found for angles of incidence very large compared to ,pI; hence apparently ,pI is not the limiting angle for doughnut formation . On the other hand, ,pI was calculated as the most restrictive condition,
12
i.e., only close-encounter processes with impact parameters equal to zero. However, correlated scattering from many atoms in the axis is still present for angles of incidence larger than tPI, especially in the GeV region, where channeling angles are small. This means that the contin­ uum description is valid for high-energy channeling for angles of incidence large compared to tPI.
In general, this doughnut scattering will strongly increase the multiple scattering. But this type of scattering is different from normal multiple scattering because in doughnuts, the angle
150
. '""'- . f"-
a a e
2.0
Random level
2.0
Fig. 6. Yield curves for 15-GeV Ic prdtons (a) and 7!'- (b) scattered more than 1 mrad by traversing a 4.2-mm-thick Ge crystal. The plots are given as a function of angle of incidence to the (UO) axis.
13
between projectile and axis is conserved, whereas normal multiple scattering will change this angle, Le., the radial distribution of the doughnut is smeared out. Multiple-scattering distri­ butions for 15-GeV Ic protons and negative pions traversing a 4.2-mm-thick germanium crystal are plotted in Fig. 8.7 The plots marked "random" correspond to directions of incidence far from axes and planes. For a detailed discussion of doughnut scattering, see Ref. 7. Doughnut scattering has also been found to play a dominant role in the bending of Ge V particles by channeling (see below).
14
6
-6
p
. ',' " I"
o 3 -3 o TRANSMITTED BEAM RELATIVE TO (110) AXIS
IjJ/ljJl
b 3
Fig. 7. Three-dimensional scatter plots in exit-angle space of 15-GeV Ic protons (a) and 1("- (b) transmitted through a O.9-mm (110) Si crystal. The incident-beam direction was in the region 2-2.5 tPl from the axis. The center of the plots is in the direction of the (110) axis.
.. 0-0.25 IjI I • 3-151j1, o~
'6Ge1lc p-Ge
15Ge11c TC-Ge
\ .. """" ... 0 ..
b)
a b "" " .. %~~~Q74~~Q8~~1~2~~1.~6 0~~~Q4~~~OB~~1~2~~~~
MIN. SCAT. ANGLE (mrod)
Fig. 8. Integrated intensity distribution as a function of scattering angle for 15- GeV Ic protons (a) and 1("- (b) transmitted through a 4.2-mm (110) Ge crystal. In both cases results are shown from three different incident regions given on plots. For comparison, the Bohr-Williams (dashed) and Moliere (solid) theoretical curves are shown.7
5. THE LACK OF A DENSITY EFFECT FOR INNER SHELL EXCITATIONS
When relativistic particles penetrate solid state detectors, the deposited energy can be measured in a very simple way. This energy deposit, however, does not directly give the stopping power, because very energetic electrons (delta rays) from close collisions may escape through the back of the target. The most probable energy loss (M.P.E.L.) on the other hand only depends on distant collisions and is a w~ll-defined experimental quantity. As a function of the Lorentz factor "I the M.P.E.L. has its minimum for "I ~ 5. From thereon it increases as log "I due to an increase proportional to "I in the action radius of the particle field. When this radius is large compared to the distance between target atoms b ~ 100 in solids), the particle field will be screened by a polarization of the solid. The screening is due to target atoms lying between the projectile path and the target atom under consideration. This so-called density effect saturates at "I ~ 1000 and therefore the M.P.E.L. saturates, because it only depends on distant collisions. The saturation level is called the "Fermi plateau" (Ph.S.I} Figs. 3 and 4).
In 1973, Dangerfield16 pointed out that the strong polarization effects found for M.P.E.L. should also be found in cross sections for excitations of individual electrons-the two effects are inherently the same. In spite of several experimental attempts,17 such a saturation was not found and the whole matter became a mystery for more than 10 years. In 1983, it was realized that the 1-10 Ge V channeling set-up at CERN would be ideal for such measurements because of the large spread in "I values. The onset of the density effect could be measured relative to protons, for which no effects were expected.
Already in the first experiment the effect was found. 18 A theoretical model was constructed which explained the lack of density effect in all other experiments. This was done by taking into account the interplay between the density effect and transition radiation (for details of this see, for example, Ref. 19) emitted upon entrance of the projectile into the target. In short, the model is as follows: for a particle penetrating a target, the adjustment of the projectile field from its unscreened vacuum value to the asymptotic screened limit to be reached (deep) inside the target results in the emission of the well-known transition radiation (TR). The TR intensity dITR /dw, all emitted in the forward direction, is essentially equal to the difference between the virtual photon spectrum in vacuum, dIv /dIN, and in an infinite medium, dIM /dw. As the real TR photons will be absorbed, the total photon intensity at depth z from the target surface is approximately given by
dITOT dIM dITR ~ ~ dIN + ~ exp[-z/Aa(w)] , (4)
dIM dITR dIv dIN +~~a;;;' (5)
where Aa denotes the absorption length. This model explains the absence of density effect in all previous experiments where targets thin compared to Aa have been used. On the other hand, targets used in the CERN investigations have a thickness t, where some density effect should be observed since t > Aa, because Aa = 3.8 J.Lm in Cu and the target thickness was 25 J.Lm.
Experimental tests of the model were performed during 1983. Various target thicknesses have been used, X-ray yields have been recorded on both the incident and the exit side, and extra TR producing foils have been put up in front of the target. All these measurements seem to confirm the validity of our model. All the ElIDPirical data from K-shell excitations in Al and Cu are shown in Fig. 9 including in the first CERN results. The pre-CERN results have been recorded for electrons up to 2 GeV. All data as well as theoretical curves have been normalized to the cross sections calculated20 for 5-GeV /c proton impact in order to compare with CERN results. The latter are relative data as described above. Clearly all the electron yields follow, or even lie above the dashed curve which emerges from the neglect of target polarization, i.e., it corresponds to the vacuum result for the range of fields. The saturating solid curve appears when the medium polarization is taken into account. The dot-dashed curve is calculated from the new model and using the actual thickness of the CERN target. The agreement between
15
experimental results from CERN and the new model is fair and especially convincing for the heavier target material.
The results from K-shell excitations in Ge using 1-10 GeV Ic protons and pions are shown in Fig. 10. The aim of the experiment was to measure the increasing influence of distant col­ lisions for increasing I values using channeling to vary the impact parameters. At that time the density effect on K-shell excitation was not known. On the other hand, there is essentially no influence of medium polarization for I values lower than the critical value19 IC = wlwp, where w is the frequency of the considered x-ray and wp is the plasma frequency of the target. For K-shell excitations in Ge, IC = 220, so the channeling results shown in Fig. 10 are not influenced by the density effect because the maximum I value is only 86.
6. STRAGGLING FOR THIN TARGETS
Experimental straggling curves are shown in Ph.S.I.5 (Figs. 5 and 6) for 740-/lm Ge, 280-/lm Ge and 95-/lm Si targets. The results are compared to Landau distributions. The experimental curves are seen to become wider and wider for decreasing target thickness and for a 95-/lm-thick Si target the Landau distribution is too narrow by almost a factor of two. The curves calculated by Bichsel and Saxon21 using the convolution method are in much better agreement concerning widths, but there is some disagreement in absolute values for M.P.E.L.
16
a ...J W ;;: Z o g [j x W ...J ...J W I ~ 3 :<: a w N
~ 2 ::E a: o z
AI
Cu
* ~", .... /{ ~-.-.-
~ o GI'nz 1'101. • Middll'monn 1'1 01. • Komiyo 1'1 01 o Bok 1'1 01.
o1L------1oL------1o41~----1~O~3-----1~074----~105
y
Fig. 9. K-shelJ excitation yield as a function of I for projectiles of unit charge impinging on solid targets of aluminum and copper. The full-drawn curves are calculated by inclusion of the dielectric response, whereas the dashed curves corre­ spond to neglect of density effects. Note that the large error bars on the points of Genz et al. 11 are due mainly to uncertainties in the fluorescence yield and target thickness and hence the relative positions of these points are much better known than the error bars indicate. The dot-dashed curves correspond to the theoretical yields obtained on the basis of the simple model, for foil thicknesses of 10 and 25 /lm for the case of aluminum and copper, respectively.
2.00 -.-----.------,-~-
1.50
c , / - ------ - . OC , 0 4 0 3 4 0 2 4
2.00
l50
1.25
<1'10/,
Fig. 10. Measured and calculated channeling dips for K-shell excitation by (a) 2- GeVjc, (b) 5-GeVjc, and (c) 11.9-GeVjc protons; (d) 2-GeVjc, (e) 5-GeVjc, and (f) 11.9-GeVjc 1('+. The calculated dips are based on the Komarov (solid lines) and Amundsen and Aashamar20 (dashed lines) cross sections. All points shown as open circles have been obtained on the same crystal. The closed circles in (a,d) were measured on another Ge target for comparison .8
4
Since then a new calculation22 has been performed in which the target electrons are rep­ resented as harmonic oscillators (one type for each shell). The collision cross section can then be divided into a resonant part and a Coulomb cross section. The different shells are treated separately, and the final distribution emerges as a convolution of the resonant and the Coulomb energy loss distribution for the different shells. The main contribution to the additional broadening comes from resonant collisions with strongly bound inner shell electrons (i.e., K-shell electrons).
The most probable energy loss and the FWHM of the distribution calculated for 2-Ge V jc pions have been plotted in Fig. 11 as a function of target thickness. The dotted curves are the Landau results. Finally the measured and calculated energy loss distributions for 2-GeV jc positrons traversing 1040- 290-, 174-, 51- and 32-J.lm of Si are shown in Fig. 12. The agreement
17
between the experimental distributions and the calculations is very good. The hump appear­ ing on the high-energy side of the model calculations is to some extent due to the harmonic oscillator representation of the K-shell electrons. For details see the paper by S. P. Moller in this volume. Here is also shown the effect of increasing 't value which should result in a narrowing of the distribution, because of the increasing number of collisions.
7. WIDE-ANGLE SCATTERING FOR CHANNELED GeV PARTICLES-POSITIVE AND NEGATIVE
Axial and planar wide-angle scattering dips for GeV protons are shown in Fig. 5. The widths of the dips are in agreement with the Lindhard critical angles.
18
300
..." ~
x(~m)
Fig. 11. The full drawn curves are the most probable energy loss (M.P.E.L.) !!.p/x and the widths of the energy-loss distributions (FWHM/4E) calculated from the Burenkov model. The curves are given as a function of target thickness. The dashed curves are obtained from the Landau model.
10 10-.3
0.24
0.2
" o 290m"" Si
25 50 7? 100 125 150 175 200 , , , ,
0174mm Si
00320mm Si
20 24 6 , keY
Fig. 12. Energy-loss distributions for 2-GeV Ic positrons traversing thin Si detec­ tors of varying thickness. The full drawn curves are calculated from the Burenkov model whereas the dashed curves are the Landau curves.
For Ge V negative particles no variation in wide-angle scattering yields have been found for the planar and axial directions. The normalized scattering yields for 15-GeV Ic (CERN) and 35-GeV Ic (Fermilab) protons and 11'- are illustrated in Fig. 13. The yields are given as a function of incident angle to the (110) axis in Ge crystals of 4 mm and 2 cm thickness, respectively. The required scattering angle in the Fermilab data is only ~ 3 ,pi, so the scat­ tering is still influenced by doughnut scattering that persist out to (15-20) ,pl. In doughnut scattering there is no change in the transverse energy ET, so it is not a close-encounter process although the scattering angle is large for a high-energy particle . Clearly there are pronounced dips for positive particles but no peaks for negative particles as was found for MeV electrons,
19
a
15
UJ Cl
b
10
Cl 08 ...J w >- 0 06 w ':c! -' « ~ 01. a:: 0
ISGeV/c p-42mm(110)Ge z
0 0 I
INCIDENT ANGLE I4JNII
seal. ang. > 3 mrad
.2
.I
TO CRYSTAL AXIS (MICRORAOIANS)
Fig. 13. Wide-angle scattering yields for 15-GeY Ic 11'- (a), protons (b) and 35- GeY Ie positive and negative pions (c) as a function of incident angle to the (110) axis in Ge crystals.
Bearing in mind, however, that there is a close connection between large energy loss events in a "live" target and nuclear reactions, we can look for integrated channeling effects in an­ other very simple way, namely by plotting "l~rge energy loss events" as a function of incident particle angle to a crystal axis or plane.
The energy loss curves for 6-GeY Ie 11'- transmitted through a thin "live" Si crystal along a random and an axial direction are shown in Fig. 14. The bottom curve illustrates the ratio between the channeled and random curves. Clearly channeled 11'- experience an increasing number of large energy loss events going from the M.P.E.L. at ~ 165 keY up to ~ 250 keY. From that point on the rati~ stays constant. The increase of large energy loss events is due
20
to the focusing of channeled 11"- to high electron density regions but also due to i) knock-out protons from the Si nuclei and to ii) charged particle production. In the following we define a "large energy loss" event as one where the energy deposited in the crystal is 3.5 times the M.P.E.L. This corresponds to the plateau of the bottom curve.
Large energy loss events are plotted in Fig. 15 as a function of incident angle to the (110) axis in Ge. The data are for 35-GeV Ic 11"+ and 11"- (Fermilab)lS and 15-GeV Ic 1I"-/protons (CERN)23 incident on 2-cm and 4-mm thick crystals, respectively. While protons give the expected dips for incident angles smaller than tPl, negative pions are rather seen to develop some sort of enhancement in the same region, but no clear peak is seen. In both experiments
10 3
.-----------------------------------------~
ENERGYLOSS OF PARTICLES INCIDENT ALONG R STRING .<110>-axis
50
o ~----~------~------_I------_I------_I------~~ so 100 150 200 250 300 ENERGYlOSS IN KEV
SCATTERING PRRTICLES DIVIDED BY RANDOM PRRTICl ES 6 GEV/C PI M1NUS,
Fig. 14. Energy-loss distributions for 6-GeV Ic 11"- traversing a 0.55-mm Si "live" target along a random (top) and an axial (middle) direction. The bottom curve is just the middle curve divided by the top curve.
21
2"10
b 15G~V Ie
15 > ~ I·'···~·· ! . ~ .. '·'-.~I-:--:--.--I- ~ 10 !
o 1V1 05 10 15 2.0 INCIDENT ANGLE (mrod)
Fig. 15. Large energy loss (2: 2.5 times M.P.E.L.) for 35-Ge V Ic positive and negative pions traversing a 2.O-cm (UO) Ge crystal and 15-GeV Ic protonsl1l"­ traversing a 4.2-mm-thick (nO) Ge crystal.
the statistics are poor for small incident angles to the (UO) axis. Most of the increase in yield for 11"- is most likely coming from an increased probability of close encounters with target electrons (see below). From this the precise structure, form, and magnitude is uncertain for the 11"- case.
Since the prospect of increasing the reaction rates for nuclear processes is a very intriguing one and since we cannot solve the problem in a decisive way from the available data up to now, we have instead performed a series of computer calculations to simulate the penetration of positive and negative pions through a germanium single crystal. The computer program, which was originally developed for electron- and positron-channeling studies at the Univer­ sity of Giessen, West Germany, simulates the motion of the projectile by means of a long series of binary collisions with the crystal atoms. As a two-body potential, we have employed a Thomas-Fermi potential, and the thermal vibrations of the crystal atoms are also taken into account by using Monte Carlo techniques. The results have been encouraging because they seem to reproduce all the observed (rather complex) angular distribution of the particles
22
ci ..... (/)
b
ci ..... (/)
(5
/
\jJ1n c ········ 0.02S\jJ1' - O.S\jJl
'---1\jJ1
/--------------------
5r-------r-------r-------r-----~
\jJin = 0.04 (jI1 --- \jJJn C 0.04 411
-'-'- \jJ,n • 1.0 4l,
ct ~
Thickness . A. 19p
DISTANCE FROM ATOMS (Angstrq,m) DISTANCE
Fig. 16. Computer simulations. Predicted impact parameter distributions for 15-GeV protons (a) and 11'- (b and c) after having penetrated 19-J.Lm- and 15Q..J.Lm­ thick (nO) Ge crystals. All shown in the figures four different incident angles have been used .
transmitted through the crystals. Consequently, we feel that we can have a good deal of confidence in the predicted impact parameter distributions, which are shown in Fig. 16. The expected impact parameter distributions of 15-GeV Ic protons are shown in Fig . 16a for two different incident angles relative to (nO) direction after the protons have passed 1,580,000 atomic layers (300 J.L) in a germanium single crystal. In Figs. 16b and 16c we show the corresponding results for 11'- mesons, when they have passed 19 J.Lm and 150 J.Lm, respectively. It is clearly seen that while the protons are kept away from the string, the 11'- particles have a much increased flux near the string, when the incoming 11'- beam is nearly parallel to the (nO) direction. For a well-aligned 11'- beam and for thin crystals we should thus expect a strongly enhanced probability for processes requiring small impact parameters.
23
Clearly the probability for 1("- small impact parameter processes decreases very fast as a function of crystal thickness and incident angle to the axis. The small impact parameter processes are especially sensitive to an increase in crystal thickness. This was to be ex­ pected because multiple scattering is strongly increased for such processes and the particles are dechanneled very fast. The same effect is found for axial channeled electrons emitting photons in the high-energy part of the radiation spectrum (see below) .
Transmission yields are illustrated for 15-Ge V /c protons (Fig. 17a) and 1("- (Fig. 17b) traversing 0.3-mm, 0.7-mm and 4.2-mm Ge crystals as a function of incident angle to the (110) axis. For protons the three crystals yield nearly identical curves which in turn shows that the dechanneling is small. The 25% reduction in channeled fraction for well-aligned particles stems from the overall angular resolution of the detection systems. This caused a minimum angular step size of 1/41#1. In general, for 1("- the dechanneling is much stronger and large differences are seen for the three different crystals. From these results it is also clear that very thin crystals should be used in order to see peaks in yield for 1("-.
From the computer simulation it is clear that negative particles undergoing larger impact parameter processes have a longer dechanneling length and could show peaks in yields for in­ cidence close to an axial direction. The results of two such processes are illustrated in Fig. 18, namely, K-shell excitations around the (110) direction in Ge and 8-electron yield [(1-5) MeV electrons] around the (UO) axis in a 630-Jlm-thick Si crystal. In the K-shell excitations around 50% of the yield comes from distant collisions for 12-Ge V / c 1("- , so there is an increase in close­ encounter yields by a factor of 2.5-3.0. For the 8 electron case the peak height is 3.5-4.0 and therefore somewhat larger than the peak for K-shell excitations.
In conclusion it seems clear that the lack of peaks in yields for 1(" - undergoing close­ encounter processes is due to a very strong dechanneling for such particles. Thus very thin crystals (~ lOJlm) should be used in order to see the effect. On the other hand, in such targets there will be very few close-encounter processes for Ge V particles and a long running time will be needed to see the effect with good statistics.
24
100
~ 80
a
b
0.5 1 1.5 0 0.5 1 1.5
INCIDENT ANGLE REL. TO STRING (1jI/1jI, I
Fig. 17. Fraction of particles still channeled after having traversed the 0.3-, 0.7-, and 4.2-mm Ge crystals, for (a) 15-Ge V /c protons and (b) 1("-. For positive well­ channeled particles, the dechanneling is seen to be very small, whereas dechanneling for negative particles is very strong.
l0r----------------------------,
a b
O~--~--~~--~--~~--~ __ ~ O. B.2 0.4 Q. 6 a.e t. 1.2 0,00 0.05 0.10 0.15 0.20 0.25 0.30
INCIDENT ANGLE TO <110>-AXIS (MRAl INCIDENT ANGLE TO <110>-AXIS (MRAD)
Fig. 18. K-shell excitation yields in Ge (a) and 8 ray yields in Si (b) from 1l.9-GeV 1["- incident along the (110) axis in a Ge and a 0.63-mm-thick Si crystal. The solid curve in (b) is a multi-string continuum calculation using the thermally averaged Doyle-Turner potential and electron density.
8. DISAGREEMENT ON DECHANNELING DATA
For MeV particles, dechanneling has been subject to rather intensive theoretical and ex­ perimental investigations. Most of the work has been concentrated on the problem of finding the distribution 9 (.&r, z) in transverse energy as a function of depth in the crystal. For clean and perfect crystals, g(.&r, 0) is sharply peaked around 1/2pv(l~ because the influence of sur­ face transmission is rather small. Here (lin is the angle of incidence to the axis. As the beam proceeds into the target, the development of g(.&r, z) will at first be dominated by electronic multiple scattering. With increasing transverse energy, nuclear multiple scattering plays a growing and, eventually, a dominant role. This increase in .&r is by nature a random walk process and can be approximately described by a diffusion equation,
8g(ET,Z) = ~ {A(E )D(.&r)_8_9(ET,Z)} 8z 8.&r T 8ET A(.&r) ,
(6)
where A(ET) is the accessible area in the transverse plane for a projectile with transverse energy ET and D(ET) is the diffusion function. This model was used by Bonderup et al.24 in their calculations of the transverse energy distribution as a function of crystal depth. The
25
model has been modified slightly to include relativistic effects in the multiple scattering depen­ dent diffusion function. This is mainly accomplished by replacing the rest mass appearing in the nonrelativistic formulas by the relativistic. mass mOl ' So far, most experimental investiga­ tions of dechanneling have been based on measurements of the minimum yield for wide-angle scattering. Few measurements of 9(ET' z} have been performed in the MeV region where perfect, thin crystals and detailed angular scans behind the crystals are required. However, for a typical high-energy experimental arrangement, such investigations are very simple and attractive. The transmission yields of 15-GeV /c protons traversing 0.3-, 0.7-, and 4.2-mm germanium crystals are shown in Fig . 19. The results are plotted as a function of t/J2, where t/J is the angle between the exit direction and the (UO) axis; hence the observed distributions can be compared directly to g(Er,z). Plots a to f correspond to a stepwise increase of 1/4t/Jl in the incident angle. Unfortunately, the overall angular resolution of the detection system was only between 1/3t/Jl and 1/2t/Jl so that for incident angles between 3/4t/Jl and t/Jl (Fig. 19d), a considerable number of random particles are present. For protons in general, it is seen that for incident angles up to - 3/4t/Jt, the transmitted yield is nearly independent of crystal thickness, showing little dechanneling even for the 4.2-mm crystal. These results have been compared to calculations based upon the diffusion model. The results of such calculations are also shown in Figs. 19b and 19f. The overall agreement is fair even for incident angles above t/J1, where the model is not expected to be particularly good. The experimental peak heights are lower than the calculated ones, which is mainly caused by angular resolution .
It is also seen that dechanneling depends very much on the incident angle . Large dechan­ neling takes place for incident angles close to t/Jl, where there is also a marked difference between the different crystal thicknesses. For incident angles larger than t/Jl , a channeled part of the transmitted beam is still present, mostly due to the rather poor angular resolution. It should be noted that generally, the influence of the increasing crystal thickness is an increase
26
::0
I
1.0 20 ! 0- 0 25 4J, 025 -050 4J, 050 - 0754J,
1.0,;
~ "U g 10
w >= z 00 2 I. 6 6 Q (f) 15 10 (f)
I 1.25 -1.50 4J, '::i' 0.7 5 -1.004J,
(f)
2
EXIT ANGLE {4J 14J,1 2
Fig . 19. Distribution in "transverse energy" (which is proportional to t/J2) of 15- Ge V /c protons transmitted through 0.3-, 0.7-, and 4.2-mm Ge crystals. Plots (a) to (f) give results for particles with increasing angle of incidence to the (UO) axis. The full curves are for the 0.3 mm, the dashed curves for the 0.7 mm, and the dot-dashed curves for the 4.2-mm crystals. In plots (b) and (f) examples of a comparison with theory (smooth curves) are shown. The agreement is satisfactory for small incidence angle, but only fair for larger angles of incidence.
008
\IJ/\IJ,
\IJ/\IJ,
Fig . 20. Emergent angular distribution of particles incident on the crystal with a uniform distribution in angular space in the range 0.8tPl < tP < tPl' tP is measured relative to (110) axis in (a), (b), (c) and relative to a random direction chosen away from axial or planar direction in (c). The smooth curves in (a)-(c) are a prediction based on a diffusion model.
d
in the average transverse energy ET • but a decrease in the most probable q, which is in agreement with general statistics. This illustrates that you can "cool" some particles but the average temperature increases.
The same type of data were published from the Fermilab experiment in 1982.25 Some of these data are shown in Fig . 20 and also compared to theoretical curves by Bonderup et al.24
From the comparison it is concluded: "that the diffusion process cannot by itself explain the observed emergent particle angular distributions" .
In order to examine this question more a new set of CERN experiments were performed using 100Ge V Ic protons transmitted along the (110) axis through 0.585-mm and 4.Q...mm-thick Ge crystals. The data are shown in Figs. 21 and 22.
The experimental data are compared to theoretical curves calculated on the basis of the diffusion model26 using a thermal average potential in calculations of the excitation function. In all cases the surface transmission on the incident and exit sides is included. Experimental angular resolution on the incident side is also folded in.
3.0
In the 0.585-mm data (Figs. 21a,b) a Gaussian angular resolution on the exit side is also folded in. When this is included there is nearly perfect agreement between experimental and calculated curves. Unfortunately the angular resolutions on the exit side (the most severe) were not included in the other data, but still the agreement is fair. In the Fermilab data there is a lack of a minimum for small exit angles although the incident angles are in the region: (0.8-1.0) tPl. This is in contradiction to calculation. In order to explain this missing minimum a broadening of 35-50 J.lrad is needed (see Ref. 25), which cannot be accounted for by "mosaic spread" or vibrations. On the other hand, in Ref. 15 the same group present a wandering of
27
the crystal axis from run to run, of as much as 30 JLrad and it is indicated that goniometer wandering and the possible motion of drift chambers could be causing these problems. The broadening due to such a wandering would explain the lack of a minimum in transmission distributions.
The same type of wandering was found in the CERN experiment. It was caused by large temperature variations from day to night. The problem was solved by a complete thermal insulation of the whole experimental set-up, by which temperature variations were reduced to less than 10 through a 24-hour period. Sudden jumps of 100-200 JLrad are still found now and then but with the broad beam technique used at CERN the crystal can be aligned using data from only part of a tape. In this way data around a jump can be discarded.
In conclusion it should be noted that data from five different CERN experiments agree with the diffusion calculations and the disagreement of the Fermilab data on the same type of crystal and with the same theory should be sought in the experimental set-up.
28
OJ
" "Vi ~
10 GeV /c p _0.585 mm (110) Ge 90° K
IjJ[X /IjJ,
Fig. 21. Comparison between measured and calculated exit angle distributions for 100GeV /c protons transmitted through a 0.585-JLm-thick Ge crystal along the (llO) axis cooled to 90 K. In the curves, different regions of incident angles are shown. The solid curves in a and b are calculated from the diffusion model of dechanneling with (2) and without (I) inclusion of the experimental angular resolution of 1/31/l1 on the exit angle side. The contribution from angular resolution on the incident side is small and has been neglected in the calculations. Solid curves in c and d are calculated without angular resolution on the exit side.
Exit angular Distribution for
10 GeV /c p _4.0 mm (110) Ge 90° K
3 a, b
2
:;::
IjJEX/IjJ,
Fig. 22. Same as Fig. 21 but for a 4.O-mm-thick (110} Ge crystal. All solid curves are calculated from the diffusion model but without angular resolution on the exit side folded in.
9. CHANNELING RADIATION FOR GeV ELECTRONS AND POSITRONS
The channeling radiation spectra was illustrated in Ph.S.L5 for 7-GeV Ie electrons and positrons traversing a 100-",m Si crystal along the (110) planes (Ph.S.L, Fig. 17 or Fig. 10, Bak, this volume). The data were compared to a classical calculation of the emitted radiation as described in Ref. 27 . The experimental energy of the first harmonics for positron radiation appeared to be 5% lower than the theoretical one. The same problem was found for other Ge y28 experiments and also for Me V experiments.29 For Ge Y electrons such a disagreement cannot be detected because the radiation spectrum is structureless. For Me V electrons discrete transitions between quantum levels are found and for this type of radiation there is, in general, good agreement with quantum calculations of the energy.
The radiation from 2-10 GeV Ie positrons channeled along the (110) plane in a 100-",m­ thick Si crystal are illustrated in Fig. 23 . For all energies the calculated energy of the first harmonic is 5-10% above the experimental one. This is worst for the 10-GeV Ie case. The same situation was found for other types of crystals. This fact led to reconsideration of both the experimental and theoretical situations. For the calculations variations of the potential were tried together with inclusion of multiple scattering and radiation from "above barrier" particles . From these calculations it developed that in order to bring the theoretical value of the first harmonic down to the experimental one, the planar potential had to be changed so much that a completely unphysical situation was created.
29
~60r_---__, 60,------, 60,------,
20
10
50
40
30
20
10
10 30 50 10 40 70 20 60 100 25 75 125 175
E.,(MeV)
Fig . 23. Photon energy spectra (for the range -t/Jp to t/Jp) and angular scans of 2-,4-,5-,7-, and 100GeV Ic positrons channeled along the (110) plane in a O.I-mm silicon crystal. Circles are experimental points while the curves are theoretical calculations.
60
~ 40
Err::::v)
Fig. 24. Photon spectra for the (110) plane in a O.l-mm-thick Si crystal. The experimental points are compared to calculated curves for 7-GeV Ic positrons (dashed) and 6.7-GeV Ic positrons (solid). With the new value of beam momentum (6.7 GeV Ic) the agreement is nearly perfect.
In the experimental set-up detectors were checked and recalibrated but without any change. Finally, the beam momentum was checked by very detailed measurements of the magnetic fields in the bending magnets. Here it was found that the apparent beam momenta in the region of 2-10 GeV Ic (for the T7 beam line at CERN) were 4% too high. The result of a 4% reduction in beam momentum is shown in Fig. 24 at 70 GeV Ic. The result of the same reduction for all the other momenta are shown in Fig. 25. Now the agreement is good
30
for practically all the CERN experiments including the Ge case. Details are given 10 the contribution of.J. Bak in this volume.
The large discrepancies in the 2-14 GeV Ic SLAC-USSR experiment on diamond have also been ameliorated. In a later SLAC-USA experiment30 the experimental energies of the first harmonics have increased considerably and are now much closer to the theoretical values. Errors were found in calibrations of the NaI detector and in the locations of the first harmonic peaks.
The influence of above-barrier particles on planar channeling radiation spectra have been studied in a SLAC-USSR experiment.31 The data are compared with calculations which take into account dechanneling, multiple scattering of above-barrier particles, and deviation from the dipole approximation. For 10-GeV Ic positrons incident on an 80-J.Lm-thick crystal it is found that good agreement between experimental and theoretical results require a multiple scattering of 1.5 x 1f;p for the above-barrier positrons . The radiation intensity from these positrons is comparable to that from channel~d particles and smears out the planar radiation spectrum. This makes it difficult to determine the energy of the first harmonic and has caused some of the disagreement between calculated and measured energies of the first harmonic in the planar case.
In general, it seems like the more serious disagreements found for planar channeling radi­ ation from Ge V positrons has disappeared for most experiments, which unfortunately is not the case for the MeV energies.
In 1982, the possibility of subharmonic peaks in planar channeling radiation was raised.32
Some indications were seen in the CERN experiments and a possible explanation was suggested. 27 Some structure was also seen in the later SLAC-USA experiment30 but the struc­ ture is extremely sensitive to angular resolution and thereby to stability of the whole set-up. This would be in agreement with the suggested explanation (see Ref. 27) where a small fraction of particles with incident angles close to 1f;p can cross one set of planes but then are reflected by the next set. The results of computer simulations are found in Ref. 27. From this model the effect is unstable and very difficult to investigate in detail.
For axial channeled Ge V electrons and positrons the emitted radiation is also dramatically enhanced as in the planar cases. In nearly all cases the radiation spectrum is structureless as with planar channeled Ge V electrons. Generally the coherent part of the radiation is enhanced by factors of 30-40 over incoherent bremsstrahlung.
1.9 GeV/e 3.8 GeV/e 4.8 GeV /e 6.7 GeV/c 9.6 GeV/e
~U\J ~~L I fl, I ~~L!LJ 1~t Alii ~~t ,J\" I -7.5 O. 7.5 - 7.5 O. 7.5 -7.5 O. 7.5 -7.5 O. 7.5 -7.5 O. 7.5
1/IN, ~60 60 c 60 60 60 Q)
E50 50 50 50 50 Q) u §40 40 40 40 40 .c ,530 30 30 30 30
20 20 20 20 20
10 10 10 10 10
0 0 0 0 0 4 12 20 10 30 50 10 40 70 20 60 100 25 75125 175
E,(MeV)
Fig. 25. As Fig. 23, but the beam energy used in the calculations has been de­ creased by 4%, as described in the text.
31
28 r 28 ~
c
d
• fflt Iftf ~ I 12 f- I I tllf fllllilltllill I II
8 ~ f li lt IlfHI
I I 1 1 III 1 1 1 1 1 1 1 o '---'-----'_..L---'_...L.--'-_-L....J 3 0 "----'-----'_..L----L_-'---'-_.l.....J 0.2 0.6 1. 1.4.10 0.2 0.6 1. 1.4'10 3
E.DN/DE-PHOTHI 1-1.5PSll E,(MeV) E.ON/DE-PHOTHI 1.5-2PSI1
28 f- 28 -
20 '-c QJ
E 16 ... /.tttlffflllttfIHI II ~ . Hllllfl1tfl I II g 12 r- ft If 1/111"11 I ~ 8 _ I 1IIIttlii Ld
4 - 4 -
E*ON/DE-PHOTHI 2-2.5PSll E,(M eV) EoON/DE-PHOTHI 2.5-3PSl l
Fig. 26. Photon spectra from 10-GeV Ic electrons traversing a 100-lJm-thick (110) Si crystal. The spectra are shown for incident angle regions of: (a) (0-0.5)fPl, (b) (0.5-1.0),/11, (c) (1.0-1.5),plJ (d) (1.5-2.0),pl, (e) (2.0-2.5),plJ and (f) (2.5-3.0),pl. All curves are normalized to the same spectra from lOO-lJm amorphous Si and given as a function of photon energy in MeV ..
Very recent experiments,33 however, show that the high-energy part of the photon spec­ trum is reduced as compared to the Bethe-Heitler intensity, which is not found in the MeV region.
The strong enhancement for the radiation is mainly produced by the very pronounced string (doughnut) scattering. These scatterings result in rotations of the transverse momentum vector PT. Thus in the transverse plane the original direction of PT will be
32
changed and after a certain number of string collisions PT may be found in any direction of the transverse plane. The net result of this equalization is that a parallel beam transmitted through the crystal will appear on the reverse side of the crystal as a ring-shaped (doughnut) distribution in angle space with a ring radius equal to the incident angle to the string. This equalization leads to a reduction in radiation intensities for low photon energies followed by a linear intensity increase (for details in calculation see the contribution by O. Pedersen, et al., this volume).
Experimental axial spectra are shown in Figs. 26, 27, and 28 for 10-Ge V /c electrons/posi­ trons and 20-GeV /c positrons transmitted through a 100-JLm Si crystal along the (110) axis. All spectra are normalized to the incoherent bremsstrahlung from 100-JLm amorphous Si. In all three figures plots a to f correspond to a stepwise increase of 1/21/11 of the incident angle to the '110) axis-starting with a circle of radius 1/21/11 around the axis. Clearly the enhancement
10 G£vjc E-t ON 110 -AXIS
28 f- 28 f-
.<:: c w
28 f- 28
24- 24- C
d ~ 2 0 rJ11fl t 20
•••• fllilllllf!! <l! Il lflll E 16 f- I \1 »tj 16 \ !Htt I <l! tH I f /I u IlIttlllfHi II II ttllll llH c 12 12 III t It 0
Illftlllt/llltlfll II III IIHlltf I t .<:: c 8 8
I III lui w
4-
\ 4-
I 3° 0.2 0.6 1. 1.4-.10 0.2 0.6 1. 1.4- .10
[.DNj D[-PHOTHI 1-1.5PSI1 [ ,(MeV) [.DNjD[ -PHOTHI 1.5-2PSI1
28 f-
24- t-
20 t-
4- t-
[.DNj DE-PHOTHI 2-2.5PSll [,(MeV) E*DNjD[-PHOTHI 2.5-3PSI1
Fig. 27 . Same as Fig. 26 but for lO-GeV /c positrons.
33
increases going from Figs. a to c,d-this is most pronounced for 20 GeV Ic. Figs. c,d correspond to incident angles (1 x ,pl - 2 x ,pI) for which the doughnut scattering is the most pronounced. The reduction in intensity for low photon energies is clearly seen for the 20-Ge Vic data and to some extent for 10-GeV Ic electrons. This reduction is mostly due to the so-called Landau­ Pomeranchuk effect, which is also discussed in the paper given by O. Pedersen et ai, ibid. For increasing particle energy the very strong enhancement is found for larger incident angles (Fig. e,f) in agreement with the fact that doughnut scattering is found to persist out to incident angles as large as (15-20) ,pl . Unfortunately the maximum photon energies measured were 1.5 Ge V so there is no chance to study the interesting high-energy part of the spectrum, for which incoherent effects are important.
Detailed investigations of axial channeling radiation and the transition to the planar case can be found in the paper by J . Bak, ibid.
34
110 -AXIS
28 r-
24- '- b
~: - ft'ffltfft/ltlllfllfIIHfllil I 11 11~fff ff f uij f U f
1 2 ,:' I f fIT f f~lffr ~
4- I-
I I I I I I I 0.2 0.6 1. 1.4 .1 0 :3
[ ,(MeV) E.DN/ DE-PHOTHI .5- 1 PSI1
16 d
o L-~~~~-L __ L--L~~ 3 0 0.2 0.6 1. 1.4. 10
I I I I I I I 0.2 0 .6 1.
QJ •
4- r-
e
28 r-
24 r- Iff 20 r- Ilitl//lfH\fllllfilltlltl\llIllIlllfHHIIIIII II I1 16 1-11 I 12 'cJ
8 -
4 -
1
o L-~-L __ i~~I __ L-I~I __ ~ 3 0 '--:-'----l __ ....I.I __ .l.I __ L--'-----:"-:-'
0.2 0.6 1. 1.4'10 0 .2 0.6 1. 1.4'10 3
E'DN/ DE-PHOTHI 2-2.5PSI1 E,(MeV) E.DN/ DE-PHOTHI 2.5-3PSI1
Fig. 28. Same as Fig. 26 but for 20-GeV Ic positrons:
10. CONCLUSIONS AND OUTLOOK
Not all the ideas that inspired the beginning of Ge V channeling have been effectuated yet. On the other hand, today it is clear that the Lindhard model for channeling developed in the mid-sixties and based on a few basic principles describes channeling very well over an energy range of more than eight orders of magnitude. The arguments for the first experiments were maybe not all too convincing but penetration phenomena for relativistic particles turned out to be a rather rich field with connections to many other subjects.
Although channeling angles are very small for GeV particles experimental techniques have been developed to such a degree of accuracy t