NATO ASI Series Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NA
TO Science Committee, which aims at the dissemination of advanced
scientific and technological knowledge , with a view to
strengthening links between scientific communities .
The series is published by an international board of publishers in
conjunction with the NATO Scientific Affairs Division
A Life Sciences B Physics
C Mathematical and Physical Sciences
o Behavioral and Social Sciences E Engineering and
Materials Sciences
F Computer and Systems Sciences G Ecological Sciences H Cell
Biology
Recent Volumes in this Series
Volume 159-Lattice Gauge Theory '86
Plenum Publishing Corporation New York and London
D. Reidel Publishing Company Dordrecht, Boston, and Lancaster
Martinus Nijhoff Publishers The Hague, Boston, Dordrecht, and
Lancaster
Springer-Verlag Berlin, Heidelberg, New York, London, Paris, and
Tokyo
edited by Helmut Satz, Isabel Harrity, and Jean Potvin
Volume 160-Super Field Theories edited by H. C. Lee, V. Elias, G.
Kunstatter, R. B. Mann, and K. S. Viswanathan
Volume 161-Quantum Measurement and Chaos edited by E. R. Pike and
Sarben Sarkar
Volume 162-Quantum Uncertainties: Recent and Future Experiments and
Interpretations edited by William M. Honig, David W. Kraft, and
Emilio Panarella
Volume 163-Thin Film Growth Techniques for Low-Dimensional
Structures edited by R. F. C. Farrow, S. S. P. Parkin, P. J.
Dobson, J. H. Neave, and A. S. Arrott
Volume 164-Techniques and Concepts of High-Energy Physics IV edited
by Thomas Ferbel
Volume 165-Relativistic Channeling edited by Richard A. Carrigan,
Jr., and James A. Ellison
Series B: Physics
Richard A. Carrigan, Jr. Fermi National Accelerator Laboratory
Batavia, Illinois
and
James A. Ellison University of New Mexico Albuquerque, New
Mexico
Springer Science+Business Media, LLC
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