Table Of ContentWELCOME ADDRESS
This Welcome Address is devoted to Schools and then to Accelerators. What does an
accelerator school, such as the one we are embarking upon, hope to achieve? The immediate
aim can perhaps be described as an exposure operation. With the help of 23 lecturers we are
going to review and expose the inner workings of high-intensity particle beams. Newcomers to
the field will find it hard to follow in detail because of the high density of information and the
short time that we have together. More experienced participants will be able to select topics of
interest, but the basic aim is to make you aware of what is known in this field.
The second aim is more durable. The lecturers will write up their talks for the
proceedings, which they will try to make as complete but as concise as possible. Several such
proceedings have been created by schools like this one and many are devoted to specialist topics
within the accelerator field. It is our hope that if you leave for a sabbatical year you will
consider it worthwhile to take the proceedings for your speciality with you.
When planning the programme for a school one discovers grey areas which may not be
adequately covered in the literature. For example, the most detailed and rigorous derivations of
the Vlasov equation are found in plasma physics books. In these derivations, it is usual to
invoke Debye shielding and a neutral plasma. Accelerator beams, however, are far from being
neutral and their dimensions are often less than the Debye shielding length. If you are interested
in the validity of the Vlasov equation when applied to accelerator beams and criteria for judging
that validity, you can now refer to one of our earlier school proceedings 1.
One also uncovers work that has never been published. This situation often comes to
light in discussions with colleagues who are about to retire. Here again the school can play a
useful role by providing the encouragement and opportunity to write up and publish this work,
which would otherwise be lost.
A school is also a good framework for preserving the history and traditions of
accelerators. Just 10 to 15 years ago, many of the founder members of the accelerator
community were still working with us. We are just now at the point where the very early years
are slipping beyond living memory. This advancing "curtain" makes it all the more important
for a school to record our history and traditions as they develop. The Prize for Achievement in
Accelerator Physics and Technology awarded by the U.S. Particle Accelerator School serves
this purpose by drawing attention to important personal contributions and main developments.
In connection with the last point, I should now like to pay a small tribute to the founders
of our community by recounting the early history of accelerators. This history can be traced
from three separate roots. Each root is based on an idea for an acceleration mechanism and all
three originated in the twenties.
The first root to be described is generally taken as the principal "history line" since it was
the logical consequence of the vigorous physics research programme in progress at the turn of
the century. Indeed, particle physics research has always been the driving force behind
accelerator development; it is therefore natural also to consider high energy physics as its
birthplace. The main events along this history line are listed in Table I. The line is started at the
end of the last century to show the natural progression through atomic physics to nuclear
physics and the inevitable need for higher-energy and higher-intensity beams of "atomic
projectiles" than those provided by natural radioactive sources. In this context, the particle
accelerator was a planned development and it fulfilled its goal of performing the first human-
controlled splitting of the atom. It was Ernest Rutherford, in the early twenties, who realised
this need, but the electrostatic machines then available were far from reaching the necessary
voltage and for a few years there was no advance. Suddenly the situation changed, in 1928,
when Gurney and Gamov independently predicted tunnelling 2, and it appeared that an energy
of 500 keV might just suffice to split the atom. This seemed technologically feasible to
Rutherford and he immediately encouraged Cockcroft and Walton to start designing a 500-kV
particle accelerator. Four years later, in 1932, they split the lithium atom with 400-keV
protons. This was the first fully human-controlled splitting of the atom 3, and it earned them
the Nobel Prize in 1951.
Table I Main History Line
1895 Lenard. Electron scattering on gases
< 100-keV electrons.
(Nobel Prize).
,~ Wimshurst-type machines
1913 Franck and Hertz excite electron shells by
electron bombardment.
1906 Rutherford bombards mica sheet with natural
alphas and develops the theory of atomic
scattering.
Natural alpha particles of
1911 Rutherford publishes theory of atomic structure. several MeV
1919 Rutherford induces a nuclear reaction with natural
alphas.
... Rutherford believes he needs a source of many MeV to continue research on
the nucleus. This is far beyond the electrostatic machines then existing, but ...
1928 Gamov predicts tunnelling and perhaps 500 keV would suffice.
1928 Cockcroft and Walton, encouraged by Rutherford, start designing an 800-kV
generator.
1932 Generator reaches 700 kV and Cockcroft and Walton split lithium atom with only
400-keV protons. They received the Nobel Prize in 1951.
Figure 1 shows the original apparatus, which is now kept in the Science Museum,
London. The top electrode contains the proton source and was held at 400 kV, the intermediate
drift tube at 200 kV, and the final drift tube and target at earth potential. This structure can be
seen inside the evacuated glass tube in Fig. 1 above the curtained booth in which the
experimenter sat while watching the evidence of nuclear disintegrations on a scintillation screen.
The voltage generator was at the limit of the in-house technology available to Cockcroft and
Walton, and the optimistic design voltage of 800 kV was never reached because of a persistent
spark discharge that occurred at just over 700 kV. However, the famous atom-splitting
experiment was carried out at 400 kV, well within the capabilities of the apparatus. The
Coekeroft-Walton generator, as it became known, was widely used for many years
afterwards as the input stage (up to 800 kV) for larger accelerators, since it could deliver a high
current.
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Fig. 1 Cockcroft and Walton's apparatus for splitting the lithium nucleus
The direct-voltage accelerators were the first to be exploited for nuclear physics research,
but they were limited to the maximum voltage that could be generated in the system (except for
the astute double use of the applied voltage in the Tandem). This limitation was too restrictive
for the requirements of high energy physics and an alternative was needed.
In fact, an alternative had already been proposed in 1924 by Ising 4 in Sweden. He
planned to apply the same voltage to the particle repeatedly, using time-varying fields. This is
known as resonant acceleration. Ising starts a new history line; its main events are given
in Table II. The difference between Cockcroft and Walton's and Ising's acceleration
mechanisms depends on whether the fields are static or time-varying. The former corresponds
to the Cockcroft-Walton machine and the latter to Ising's resonant acceleration. All present-day
high energy accelerators use the latter option of time-varying fields.
Table II The Second History Line
1924 Ising proposes time-varying fields across drift tubes. This is resonant
acceleration, which can achieve energies above that given by the highest voltage
in the system.
1928 Widerre demonstrates Ising's principle with a 1-MHz, 25-kV oscillator to make
50-keV potassium ions.
1929 Lawrence, inspired by Widerre and Ising, conceives the cyclotron.
1391 Livingston demonstrates the cyclotron by accelerating hydrogen ions to 80 keV.
1932 Lawrence's cyclotron produces 1.25-MeV protons, and he also splits the atom
just a few weeks after Cockcroft and Walton (Lawrence received the Nobel Prize
in 1939).
Ising suggested accelerating particles with a linear series of conducting drift tubes, and
Widerre built a proof-of-principle linear accelerator in 1928 5. Alternate drift tubes are
connected to the same terminal of an rf generator. The generator frequency is adjusted so that a
particle traversing a gap sees an electric field in the direction of its motion and while the particle
is inside the drift tube the field reverses so that it is again the direction of motion at the next gap.
As the particle gains energy and speed the structure periods must be made longer to maintain
synchronism (see Fig. 2).
Clearly, as the velocity increases the drift tubes become inconveniently long unless the
frequency can be increased, but at high frequencies the open drift-tube structure is lossy. This
problem is overcome by enclosing the structure to form a cavity (in a circular machine) or a
series of cavities (in a linear machine), working typically in the MHz range. The underlying
principle remains unchanged, but the accelerating structure design has several variants.
ner G
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Ising's original idea can be considered as the beginning of the true accelerator.
Indeed, the next future generation of linear colliders, which will be in the TeV range, will
probably still be applying his principle of resonant acceleration, except that the frequency is
likely to be in the range of tens of GHz.
Technologically the linear accelerator, or linac, was difficult to build and during the
1930s it was pushed into the background by a simpler idea conceived by Ernest Lawrence in
1929 6, the fixed-frequency cyclotron (see Fig. .)3 Lawrence's idea was inspired by a
written account of Wider6e's work, and M.S. Livingston demonstrated the principle by
accelerating hydrogen ions to 80 keV in 1931. Lawrence's first model worked in 1932 7. It
was less than a foot in diameter and could accelerate protons to 1.25 MeV. He split the atom
only weeks after Cockcroft and Walton. Lawrence received the Nobel Prize in 1939, and by
that year the University of California had a 5-ft-diameter cyclotron (the Crocker cyclotron)
capable of delivering protons of 20 MeV, twice the energy of the most energetic alpha particles
emitted from radioactive sources.
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Fig. 3 Schematic cyclotron
In fact two basic geometries can be used to exploit the electric field induced by a time-
varying magnetic field (Faraday's law) for acceleration. The first is the basis of Ising's idea.
The mechanism can be made more evident by enclosing the gaps in cavities. The fields in a
single rf cavity are shown schematically in Fig. 4(a). The azimuthal magnetic field is
concentrated towards the outer wall and links the beam. According to Faraday's law, the
periodic rise and fall of this magnetic field induces an electric field on the cavity axis, which can
be synchronised with the passage of the beam pulse.
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Fig. 4 Acceleration configurations
Suppose that this topology is transformed, so that the beam encircles the magnetic field as
shown in Fig. 4(b). Wider6e 8, 9 suggested this configuration and the acceleration
mechanism, now known as betatron acceleration. He called his idea a strahlung
transformator or ray transformer, because the beam effectively formed the secondary
winding of a transformer, see Figs. 4(b) and 5. As the flux through the magnet core is
increased, it induces an azimuthal emf that drives the charged beam particles to higher and
higher energies. The trick is to arrange for the increase in the magnetic field in the vicinity of
the beam to correspond to the increase in particle energy, so that the beam stays on the same
orbit*. This device, the betatron, is insensitive to relativistic effects and is therefore ideal for
accelerating electrons. The betatron also has the great advantages of being robust and simple.
The one active element is the power converter that drives the large inductive load of the main
magnet. The focusing and synchronisation of the beam energy with the field level are both
determined by the geometry of the main magnet. As noted in the third history line in Table III,
Wider6e put this idea in his laboratory notebook while he was a student but it remained
unpublished, only to re-surface many years later when Kerst 10 built the first machine of this
type.
* Known as the WiderOe condition, or 2-to-1 rule.
I
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Fig. 5 Strahlung transformator or betatron
Table III The Third History Line
1923 Wider6e, a young Norwegian student, draws in his laboratory notebook the
design of the betatron with the well-known 2-to-1 rule. Two years later he adds
the condition for radial stability but does not publish.
1927 Later, in Aachen, Wider6e makes a model betatron, but it does not work.
Discouraged, he changes course and builds the linear accelerator mentioned in
Table 2.
1940 Kerst re-invents the betatron and builds the first working machine for 2.2-MeV
electrons.
1950 Kerst builds the world's largest betatron of 300 MeV.
Thus originally three acceleration mechanisms were competing at the birth of accelerators
in the 1920s. In fact, no new acceleration mechanism entered the picture until the 1960s, when
collective acceleration 11 was proposed, and the 1980s, when a series of workshops were
specially dedicated to finding new techniques.
The name eSdrediW appears in both the second and third history lines, first for building the
first proof-of-principle linear accelerator and then for his original ideas for the strahlung
transformator. Wider6e worked in accelerators all his life and he is still alive and living in
Switzerland.
P. J. Bryant,
Head, CERN Accelerator School
REFERENCES
1 F. Ruggiero, "Kinetic theory of charged particle beams", in Proc. 3rd Advanced
Accelerator Physics Course of CERN Accelerator School, CERN 90-04 (April 1990),
52-66.
2 R.W. Gurney and E.U. Condon, Nature 122, 439 (1928); G. Gamov, Z. Phys. 51,204
(1928).
3 J.D. Crockcroft and E.T.S. Walton, "Experiments with high velocity ions", Proc. Roy.
Soc. A 136, 619-30 (1932).
4 G. Ising, Arkiv f6r Matematik, Astronomi och Fysik ,81 4-1 (1924).
5 R. Wider6e, Arch. Elektrotechnik 21,387-406 (1928).
6 E.O. Lawrence and N.E. Edlefsen, Science 72, 376-7 (1930).
7 E.O. Lawrence and M.S. Livingston, "The production of high speed light ions without
the use of high voltages", Phys. Rev. 40, 19-35 (April 1932).
81 W. Paul, "Early days in the development of accelerators", in Proc. Int. Syrup. in Honor
of Robert R. Wilson, Fermilab, 1979 (Sleepeck Printing Co. Bellwood, IL, 1979), 25-
688.
9 R. Wider6e, "Some memories and dreams from the childhood of particle accelerators",
Europhysics News, (2) 9-11 (1984).
10 D.W. Kerst, "The acceleration of electrons by magnetic induction", Phys. Rev. 60, 47-53
(July 1942).
11 G.S. James, R.H. Levy, H.A. Bethe and B.T. Fields, "On a new type of accelerator for
heavy ions", Phys. Rev. 145, 925 (1966).
INTENSITY LIMITATIONS IN CIRCULAR PARTICLE ACCELERATORS
.G .H Rees
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, UK
Abstract
First, a brief summary is given of the various intensity limitations that may
occur in circular particle accelerators. Then, as an introductory topic for a
course on coherent beam instabilities, the validity is discussed of some of the
approximations that are used in developing the existing theories.
.i INTRODUCTION
Intensity limitations in particle accelerators may be subdivided into the following
categories:
Single-particle effects,
Collective incoherent effects,
Collective coherent effects, and
Practical considerations.
A complete summary of the subject would have to cover the full range of particles,
+ -- +
e-, p, p, ions, ~- and radioactive ions, and also all the existing types of
accelerators, linear accelerators, cyclotrons, synchrotrons, storage rings, etc.
This is too large a range of topics to cover in a single fifty-minute talk, so the
lecture will concentrate on synchrotrons and storage rings, first listing the various
limiting effects and then proceeding to analyse the underlying approximations that
are made in some of the existing coherent beam instability theories.
The subject of coherent beam motion has developed progressively since the late
nineteen fifties, starting with the observation and analysis of the longitudinal
negative mass instability for a coasting beam above transition energy. There
followed the discovery of further longitudinal and transverse instabilities for both
coasting and bunched beams. Accelerator school lectures tend to follow the
historical development, starting with the longitudinal instability of a coasting
beam, even though this is not the easiest starting point for the subject. A simpler
introduction is via the longitudinal instability of a bunched beam, as the
10
instantaneous coherent synchrotron oscillation frequency is then mainly determined by
the accelerator's radio-frequency system and the presence of modulation sideband
frequencies is more readily apparent. Bunched beam theory is therefore adopted as
the starting point for the main part of the present paper, though it is developed in
a slightly different form.
A re-look at the basic theory was prompted by the discovery on RAL's high intensity
proton synchrotron, ISIS, of significant discrepancies between observation and
theory, both for coasting and bunched beams. In looking again at the theory, it was
decided to develop the beam environment interaction in a somewhat modified form to
allow for simpler physical interpretation of the coherent effects. Thus, instead of
finding the separate response for each sideband frequency in the beam current
spectrum, a simple technique has been developed to give a composite modulation
response for any pair of modulation sidebands. The advantages of doing this will
become apparent as the technique is developed in the lecture. The response is
written either in the standard form G(j~,s), where s is the Laplace Transform
Operator, or as G(j~,ip), where a dual complex number algebra is introduced to handle
separately the phase shifts of the carrier and modulation frequencies. Antidamplng
terms may then be readily identified. The response function is applicable to both
longitudinal and transverse motion and may be developed further for the special case
of coherent synchrobetatron motion. In this case, use is made of a third complex
number in the algebra.
For the longitudinal motion of a bunched beam, the usual development of the Vlasov
equation is questioned and an alternative development proposed, though a rigorous
proof of the suggested modification has not been found. The modified development is
introduced to remove two very basic anomalies that exist in the standard treatment.
Also, the modification is found to be consistent with the traditional longitudinal
envelope equation under space charge. For a more thorough analysis, it seems better
to avoid the Vlasov approach and develop directly the basic equations of longitudinal
motion. This also appears necessary to allow the correct introduction of Landau
Damping terms.
In the case of a coasting beam, the use of the G-function directs attention to a
further anomaly; the Keil-Schnell criterion I gives the threshold for longitudinal
instability in terms of the total ring impedance at a single frequency, whereas the
beam interaction function G(j~,ip) involves two separate frequencies. This
additional anomaly is also addressed in the lecture as are the implications for the
theory. Also discussed are aspects of transverse coherent motion and the topic of
beam distributions.
E~r'sno~:Th~approachisunconvcnfion~ andlacksarigorousprooL Howcv~,itisappliedin ~eproceedmgs m ~e
paper'Somelongitu~nM dynamicsofbunchedbeams'byS.R. Ko~Miak.
