12

Accelerators of Gharged Particles

12.1/ Introduction

accelerators have played the most crucial role in the development of nuclear physics. Before the advent of these machines the only sources of high energy charged particles required for the study of

and a radioactive substances emitting a

1" vfr? tod r studies ntenshv oTac li? f ® !™****'°"" the eitSify and of the intensity or the beam of particles (see § 10.7 ).

artmd^tran^.1^?’ pioneering experiment on the limiSnf SoT nil !1 ? ®‘e™e"ts. described in Ch. X, realised these Si ! !;-^i ^ students J.D. Cockloft and E.T.S. Walton to build a parti^ accelerator which would accelerate protons to high eiibugii energy to produce nuclear transmutation. ‘ S Og

At his suggestion, Cockroft and Walton built a hiafi^ voltaae generator, based on the principle of voltage multiplication, Sst^posed by H. Greinacher (1921). With this machine they^c th^^t to toS nuclear transmutation with artificially accelerated charged pr^^tiles

twra"nirfS the4isintegration of the lithium (’Li) nucleus in#

of the Cockroft-Walton voltage multiplier will be discussed in § 125. ^

12.2 Classification and performance characteristics of accelerators

class telot/Sn.'? broadly into two classes. To the first are accelerated hv accelerators m which the charged particles sTurTe andll^. Ttk"® a constant voltage difference between the ion

final LerPv^f r^ J f difference determines the genlrSLlL^^JI^- f' and the Van de Graaff fn dIfZtS ® acceleration of the charged particles m electrostatic accelerators takes place in one step

542

Accelerators of Charged Particles 543

The main limitation of these machines is the voltage breakdown due to discharge between the high voltage (HV) terminal and the walls of the accelerator chamber for potential difference greater than about a million volts. So these accelerators can accelerate the particles only upto a few

million electron volts (MeV). ' The second class of machines, known as cyclic accelerators,

accelerate the particles in multiple steps, imparting a relatively small amount of energy at each successive step. The trajectory of the particles in these machines can be curved as in the cyclotron, betatron, synchro-cyclotron and syncfirotron or straight as in-the linear accelerator. TTie acceleration at the successive steps takes place either by the repeated application of a properly phased time varying elecUic field or by time

varying magnetic induction. The performance index of the accelerators is determined by the

following considerations : (a) Maximum energy attainable ;

(b) Beam intensity ; (c) . Homogeneity of the particle energy (/.e., whether the particles

incident on the target have any energy spread) ;

(d) Energy stability ; {e) Collimation of the beam ; ^ (f) Nature of the accelerated particles ; Ion Sources

(q) Continuous or pulsed operation. g - I Discharge

12.3 Ion sources T|——voltage 12.3 Ion sources

One of the major components common to all accelerators is the ion source which supplies the charged particles to accelerated.

For electron acccler^J^ tion, the source is a heated filament from which the electrons are released by thermionic emission. They are extracted from the filament region and focused by an electrode system kept at positive potentials w.r.t. to the filament and then injected into the machine.

For positive ions» different types of ion sources are in use. The principles of positive ion

Atomic ions

Accelerating voltage

Molecular ions

Fig. 12.1. High voltage discharge type ion source.

I

IS NacUar Physics

in some of these nrc ihe same as those nsetl in mass

r Ascribed in § 8.3. However, there are major .dijercnees tn the details of the coosmiction because of the differences^ia

srr

.(«) Discharge type Ion sources :

and “““ “e*

.olU^C ,'S ^ \ <ta

betw^ ^hich there was a volugc drop of 10-20 kV (see Fig I21X

B wh« a SS

(b) Low voiiage sources :

dischIJfiT Whinf’*®!*«'“ produced by an;

which*^lteDt^at a^n^ '^ectrode on one side of fte are (see Kg. 12.^ nren is kept at a negative potential. These sources eive ion«

thatTh” ‘**®**^‘*’''*>*»*- 'The main disadvantage of these sources is

•c

0

12,2. Capilfary arc type ion souice.

(cj Electron oscillation source :

6 8 3^'* h "" of the electron impact type ion source described in

fccirtiiTo.?V from a heated filament and accelerated through a suitable potential, collides with the atoms of the

the arrangement is shown in Fig. 12.3. There are two I

t!

Acceferafors of Charged Partfefej 545

filaments Fj, F-; on the two sides of the ion box S, The electrons emitted from each filament, which are electrically heated, are forced to foHow helical paths about the lines of the force of an axial ma^etic field

g (_ 5 X lOT^ T) thereby increasing the probability of ion prodocuon became erf the longer paths of travel within the box. Since the ion box is at a posiuve potenda! (few hundred volts) wj.t the filaments, the electrons from any one ^ent, on reaching the opposite end of the box. are tum^ back by the retarding field between the box and the other filament and thus exMute damped oscillations back and forth within the box. The electron beam intensity remains constant due to space charge accumulauon near the ftlaments. There is intense ionization along the axis of Ihc box,

f3as

Probe C’i'G electrode f

Ions

Pig. 123. Electron osdJlalion type ifM sotiros.

This type of etectron oscillation discharge is also known as Pteiming

discharge after F.M, Penning- Id some forms of these sources, the electrons aie released by cold

emission fiom the cathode. These cold arc sources are ^so known as PluHips Ion Gauge (PIG) sources, because the same principle is used m

a type of Phillips vacuum gauge. , . . a ■ The drift out through a small apechne on one side of the box and

arc drawn out by an exh^^ electrode iiUq die accelerator. The so^ is simple and ragged, can'deliver large currents and can be easily pulsed.

{Pi Radicfrequency sauf^:es : ' ' ..— ^

In these soarces^ an clcctrodcless fitschaige is produced in the gas fed into a | . glass bottle at a pressure of iO-20 torr (see bL_ Fig. 12.4). The bottle is placed either j| nr

between the plates of a capacitor or within J V|t;rsoieno.d a coil which are included in an oscillator ^ circuit The electromagnetic field, which is ;y=4°=r^Lir very strong in this region, produces the /IN-(okv' discharge. The frequency of the rf field ♦ runs upto several megahertzes. F.g. I2A Radio Muency mq

t ikv

.—Ukt—.—^—

1^' lOVV

mg. 12.4 Radio &Eq«™y ^

‘loav Mognet Cqil

0 Nuclear Physics

discharge near the walls is concentrated on the axis at the ^ mouth (rf the canal through which the ions are drawn into the acceleration

ITniTpiSf- discharge is made possible by the action of an electeiG field between the electrode at the top and the tip of the canal and also by an axial magnetic field,

_ Because of the reduced recombination at the glass wall, this source gives mainly atomic iotis.

(e) Duoplasmatron sources :

These more advanced versions of the arc source were devised by M Van Ardenne and are shown in Fig. 12.5. Electron emission by a hot fil^ent creates a weak plasma discharge in the inner chamber. A strong inhomogeneous axially sym- metric magnetic field acts on ^ ^ f] (]. the plasma near the gap — kognet cqii between the conical electrode

and the flat plate which has a J ^ ^ \ V ^

small aperture ( -1 mmj in it. ^ Y J Y Y ^

This field pinches the plasma to dj/ \i / Y f\ ^ very small-dimensions. As a result, an ionized plasma, in the 7 ^ sha^ of a thin fd^ent. fills the piosmo aperture. Very high ion currents ^ -

(-^500 niA) are obtained from |[ these sources. ITiese are now ^ commejncially available.

if) Cyclotron ion sourtes : idh source, y

Ion sources for cyclotrons require special design considcratipps since these arc located at the centre of the machine where stitmg elec&jc fields arc present. ^ i

High voltage sources cannot be used in cycKrtrons because of the alternating field^between the Mees’ (see later). So only low

voltage arc sources and oscrflating electron sources are used. > __ Feeler -y Anode ^

= =n]F^ = Di Ions o* rit ^. Ions r:I

piasmo

Pinched ptasmo

Gofi

7\ f

ptiffj

Anode

^Jj^Fhamtnt

' i * \y—

nlT

FiTomsnt

(«)

Rg. 12.6

Gas (b)

(a) Si;iiple arc type ion source for qrclotron. (ft) Hooded arc ion source for cyolottion.

P

Accelerators of Charged Particles

A simple arc source is shown in Fig. 12.6fti). Its principle has

already been explained in (a) above. The only difference is that the extraction hole can be made larger in this case, to increase the intensity

of the ion-heam, the dispersion of the electron beam being limited by the magnetic field. A larger volume of gas also gets through the aperture into the accelerating chamber which however does not affect the vacuum of

the latter because of efficient differential pumping.

An improved version of the source is shown in Fig. 12.6fb). In this,

the arc is produced in arcylindrical anode with a thin vertical slit FF cut

parallel to its axis, which faces one of the ‘dees’. In contrast to a simple source in which the ions are emitted at all azimuths, in t\us hooded ton

source, the ions leave the source at an azimuth' most convenient for acceleration. To increase the electric field for extraction of the ions, special arrangements are provided in the form of metallic protruding lips

(feelers) at the entrance of die dees or two vertical bars (pullers). The

latter are perferable for fixed frequency cyclotrons.

In the case of electron oscillation sources (see h above), the required

magnetic field is provided fey the cyclotron magnet itself.

The filaments in these sources, as also die anodes of the arc, have ladier

short life times (~ 100 hr), 'fhus these require frequent replacement. So it

should be possible to remove the source at^fi?equcnt intervals without

disturbing the vacuum of the cyclotron.

The above types of sources can give ion beam current of ~ 500 |iA 3t

0,5 kW power dissipation. Arc voltages of the order of 200 volts at a few

amp^s of current are used.

(g) Negative ion sources :

Ther^ are several methods of producing negative ions. In tiw direct extraction method, the negative ions from a discharge

type source, e.g., a dua^ifcsmalron. with a high ion yield, are directly extracted by reversing the polarity of the extraction voltage. The yield is usually low and the negative ions are mixed with a, high percentage of electrons. This method is mostly used for obtaining intense beams of negative ions of hydrogen and its isotopes. Introduction of caesium vapour into the plasma discharge greatly enhances the negative ion yield.

In the charge exchange method, a beam of positive ions, with energy ranging from a fraction of a keV to several tens of keV is passed through a gas or a vapour. Due to charge exchange at these energies, a few per cent of the ions emerge as negatively charged. At the optimuin energy (depending on the ion) and with vapours of lithium, sodium an magnesium, the negative ion yield is greatly enhanced (I to 90%).

In the caesium ion sputtering method, a beam of positive.ions of caesium, falling on a solid surface, yields a high percentage of negative

<riS imoagst the sputtered particles. Highly efficient negative ion sources itHve usen developed using this method.

production, described m S 3, has also been used for negative ion production by therinaliy desorbing an clement of high electron affinity on a low work function surface- Negative ions of halogens arc produced by this method, {/i) Sp^c[£ii of iojt sources ;

For any Qrpe of ion source, the gas pressure is ~ KT^ torr while that

in the ^ekrator is much lower (10“* to torr or less). Since the gas from the ion source continuously flows into the accelerator vacuum chamber, it tends to disturb the vacuum in the latter. To avoid this, the ion extraction^ hole should be as small as possible and very efficient ditterential pumping must be provided.

The following are some other special features required in the design

(i) High intensity of the ion cuircnt *

{«) Larger percentage of atomic ions compared to molecular ions ; (iii) Uniform energy distribution ;

(iv) Low consumption of the gas being ionized ; (v) X'OW cofisumptioti of elecliic power t

(vO Long life-time of the source so that frequent repl^racnt of the source is not necessary.

12-4 Electrchstatlc accelerators

All electrostatic accelerators Have two main compon«its viz the HV ge„„.Mr ft, accelMinj ftbe, Tliough differ*,! tySV rf,

f «■= "OelFratot ft^ same m all of them. The accelerator tube is made up of a ^es of cylinders of insulating materials <e.g., glass) with a total ienSi of a

^ ^ n^tatlic j4ctrodes are vacuum tight seal^ coming out through

the insulating cylinders. These are connected to a chain of resistors with 27 r i resistances to provide a uniform potential distribution from one

d of toe tube to the other. The electrodes within the cylinders protect the'^ walls of the latter from the beam and have a focusing action on the beam. Somewhat different designs are used for some of the modem accelerators generating higher voltage,

, Two electrodes are located at the two ends of the accelerator tube

h?th‘" *'‘Shly evacuated. The target to te bombarded y the charged particle beam forms one of the electrodes wtoch is kept at ground Pokntml The other electrode is connected to the high voltage

terminal of the HV generator and the ion source is placed wittiin it. * .if V IS the voltage generated, then a particle of charge a gains a

kjiretfe ,v „„ g!,i„g ,if. e’liZfe

vokMMV)" " ''8'“" “ ™““"

549 Accelerators of Charged Particles

The limit of the voltage generated is set by the discharges between the two electrodes outside the tube or between the HV terminal and the ground or the walls of the chamber accommodating the accelerator tube. To avoid such breakdown, the accelerator tube is made as long as possible and kept as far away from the chamber walls as possible. In pr^tice, the u|^)er limit of the voltage is only a few million volts which can be increased to some extent by enclosing the machine within a closed tank containing some dry insulating gas, such as A mixture of COj and Nj at pressures upto ~ 15 atmospheres. Pure sulphur hexafluoride (SFg ) gas is also used fex- this purpose. The use of the pressure tank allows the machine to be housed in a much smaller space than would be possible in at atmospheric pressure. As an example, it has been possible to house an accelerator generating 17 MV with a clearance of less than 2 m between the HV terminal and the ground.

Two most commonly used electrostatic generators are the Cockroft- Walton voltage multiplier and the Van de Graaff generator which will be described below.

12.5 Cockroft'Walton generator

The Cockroft-Wakon generator consists of two columns of capacittMs, shown in Fig. ]2M, connected in series. A chain of diode rectifiers Rj, R2, Rj etc., connect the capacitors C„ C2 etc., in the left column to the capacitms C,',C2' etc., in flie right column. The diodes craidact in the directions of die

arrows. A traiisformer T generating an ac voltage V(t) feeds the generator. The peak ac voltage is about 100 kV.

Fig. 12.7. (a) Principle of vottiigc multi pi icaiiom. (i>> Simplified tepresentation of

voltage muMptic^on using switohes and banery.

ric Nuclear Physics

IJic basic principle of the generator can he understood by referring

“e*' are replaced by the chain of switches S, ; v, fjvfs capacitors in the two columns are all assumed to be equal!

III place of ^e transformer in the actual machine, the voltage V^is assumed ,to be supplied by the battery B. the two ends of which are connected to the two ends of the capacitor C,.

Assume the blades of the switches S,, Sj etc. to be all in the downward positions (shown by solid lines in the figure). The two capacitors Ci and C,' are connected in parallel so that each of them is

ch^ged to a potential V. The pairs of capacitors C2 and Cj', C3 and C3 etc. are also connected in pivallei in this case.

blades of the switches are moved upwards (shown by dashed lines) so that C,' and Cj, Cj' and C3 etc. are connected in parallel,

pairwise. Obviously C,' now shares its charge with Cj so that each of

them IS charged to a potential va. Hence the potential difference between the lower end of C, to upper end of Cj is now (V + V/2) or 3V/2.

The blades of the switches are again moved to* downward positions. As a result C,^ is again charged to the potential V. Cj now shares its

charge with' Cj' so that each of them is charged to the potential V/4. As

the switch-blades are again moved to upward posiUons, C,' and Cj are

connected in parallel and each attains die potential 5V/ST Similarly the capacitors Cj' and C3 being connected in parallel, each is charged to the

potential W8. Hence the potential difference between lower end of C, and the upp»- end of C3 becomes (K + + Vi%) or 7 V/4.

• If these process are continued by repeated altemtions of the jmsitions of the switch-blades, charge from the battery B moves ifp through the capacitors in the right column and is transferred to the c^acitors C,, Cj. Cj etc. in the left column to charge them to iTigher and hi^gher potenuals. Finally each of the capacitors/in th^f^ column is c^ged to the potential V of the battery, so that tile total potential across he column of capacitore on the left becomes equal to the sum of the

potentials of all the capacitors. a

generator, there are a number of dioite r^titiers Kj, R^, R3 etc. in place of the switches (see Rg. 12.7a), Instead of the battery B, a transformer T supplies the voltage, which is usually of the order of 100,000 volts. The directions of the cuircnt flow ii, the rectifiers arc shown by the arrows. During a particular half-cycle of the transformer, the alternate rectifiers R„ R3 etc. become conducting while to the next half^ycle the rectifiers Rj, etc. become conducting. When the terminal A of the transformer is positive, the rectifier R, is conducting, '

' ^ the next half-cycle, the terminal B of the transformer is positive which makes conducting while R, becomes non-conducting. If the transfonner peak voltage is V, the potential across R, varies between 0 to

Acceleraton of Charged Particles 551

2V during the cycle, Asa result C2 is charged to the potential 2V through the rectifier R2. Similarly the voltage across R2 varies between 0 to 2K The process continues. The points V2t V4 and have fixed potentials with respect to the earth. So the potentials at V2, V4 and V5 are 2V, 4V

and 6V respectively. If there n stages, the final potential is 2nV.

If a current 1 is drawn at the output stage, there is a voltage drop across the last condenser, amounting to

A Vo = — (2/3 n’ + 1/4 nV 1/12 n)

where/is the frequency of the ac voltage and C is the capacitance. There, is a ripple in the output voltage whose magnitude is

AV, = —n{n + l)

Since the reduction A Vq in the output voltage depends on n^, it is

desirable to reduce the number of stages n and increase a high peak transformer voltage. It is also convenient to use high frequency (/) since this reduces both A Vj, and A V,. In practice, frequencies above 500 Hz

are not used because of the effect of stray capacitances. In modem Cockroft-Walton generator, solid state rectifiers are used

since these do not require any power supply for their operation and can operate without dtflicuity within a pressure vessel hotasing the generator. Cockroft-Walton generators are very suitable for delivering large ion

currents (~ 1 mA) at voltages upto about I MV. Hm^ever, the voltage has relatively large fluctuations (of the order of 1%).

12.6 Van de GraalT generator

R.J. Van de Graaff developed this machine about the same time as the Cockroft-Walton generator. It uses a convective current to charge the HV terminal. IThe principle of the machine can be understood by referring of Fig. 12.8.

There are two pullt^^i and P2 ovelf which an insulating belt is made to move with the help of a motor connected to the pulley P2 which is kept at the ground potendal. Near

this pulley, a row of metal points extends across the width of the belt maintained at a potential of 20-30 kV. Assume the points to be kept at a positive potential. A corona discharge between the points and the belt ionizes the gas which helps to spray positive charge on the moving belt.

The upper pulley Pj is iriside a large

hollow metal sphere S. Hem-another set of sharp corona points connected to S removes

Rg. 12.S. Van dc Graaff generator.

Nuclear Physics

the charge from the belt to the latter. This charge gradualiy accumulates on the outer surface of S, Its^ potential thus increases gradually^ Because of the large size of the sphere S, it cm hold a large quantity of charge and its potential can rise to a very high value.

The ion source and the top end of the acceleralor tube T are located inside the HV terminal S.

Wi* the help of the first machine built by Van de Graaff in 1931, a potential of about 1.5 million volts was generated which was used to accelerate a beam of protons to L5 MeV energy. The beam current was 25 PA,

As stated in § 12.4, because of insulation breakdown at high yojtagcs, these machines are usually housed within pressure tanks either in vertical or in horizontal position.

The convective charging ciiirent is determined by the characteristics of tfie belt and is given by i^a v w where a is the surface charge densi^ on the belt which has the width w and moves with a velocity v. Usually V < 25 cm per second and w < 50 cm. The value of a is limited by gas break down, AH these ^tors limit the charging current. At atmospheric

pressure the maximum v^ue of a- which can be raised to

cy10 C/m in a high pressure tank. Charging currents upto several mi ill amperes can be produced which can be raised in various ways, by using a number of parallel belts or by charging the belt during its downward motion with opposite charge.

Van de Graaff generators cm smeh higher voltages tfian the Cockrt^- Walton generators and sibove about 2 MV, these ate file only pra<^cal electrostatic generators. The homogend^ of the partpcle ensgy is much fetter

with these machines (uplo ^ 0.01 %). Thus ftM" the proton energy qf^ MeV, Ihd energy spread is only about 1000 cV. The cunetit is however much !ow^. These hi^ preotion machines make &mn acceleratfus of oMoice in a voy wide spectrum erf sciendhc areas.

Pelletroii accelenitors j/

A piofe recently developed concept of the Van de Graaff genei^dor differs in the design of the chaiging mechanism. This is known as a pelleiiWi* in It, the charging belt is replaced by a charging chain, consisting of steel cylinders joined by links of solid insulating material such as nylon. The metal cylinders are charged as they leave a pulley at the pound potential and the charge is removed as they pass over a pulley within the high voltage terminal.

J^ktrons range tn tire operating voltage from 200 kV to 25 MV, Beam cnrreiit capabilities range from a few microamperes to

“ 0.8 mA. At voltages above ~ 1 MV, the machines are enclosed within pressure tanks filled with SFg gas at a pressure of about 8 atm.

Accelenitors of Charged Particles 553

Many of the older Van de Graaff generators are now being converted

into.pclletrons.

Tandem accelerators In these accelerators, singly charged positive ions arc accelerated to

twice the energy corresponding to the energy of the HV terminal, principle of the machine can be understood from Fig. 12.9. The accelerating tube crosses the HV terminal so that the ion source and the target at the opposite ends of it are both at the ground potential.

Pig. 12.9. Tandem Van dc Graaff airangcraenl.

The positive ions from the ion .source passes through an electron-adding canal where a flow of hydrogen adds two electrons succes.sivcly to them, so that they are trarsframed to negative ions. Hiese arc then accelerat^ to the positive HV terminal. Here the beam passes through an electron stnpp^, m which colHrfons with some ncntnii gas atoms reconverts the high negative ions into a beam of positive ions, without change of energy. Tins positive ion beam, in travelling down the remaining porUon of the accelerator tube, is further accelerated through an equal voltage when it reaches the target, so that the final energy is twice the value corresponding to the voltage produced in the mwhine.

The accelerated beam intensity is rather low in these machines, temg of the order of a few microaraperes. However, the energy available is

quite high ( ~ 12 to 20 MeV) for singly charged ions. The problems of construction and control of the machine are much simpUfied as the ion source and the target are both at the ground potential. The energy can be incrcased‘furthcr by the addition of a thirf stage. These latter machines are in operafion in se^ru^flabonitories. >

In the folded design tandem accelerator at the Oak Ridge National Laboratory in the U.s!a., two accelerator lubes are placed side by side between the ground and die HV terminals. The negative ions travel vertically upwards as they are accelerated from the ground potential to the HV terminal. A large magnet inside the HV terminaJ bends them through 180“ to travel downwards after being stripped to high positive charge states They are finally accelerated in the second accelerator tube back down to the g.reund potential to attain their final energy. They are fin^aUy directed to the experimental site by another magnet. The advantege of the folded design is that the aceeierator and the building are much smaller.

Tandem accelerators are particularly useful in the case of hewy ion acceleration, an example, suppose a beam of O ions is first converted into a beam of O’ ions, which is accelerated to 10 MeV energy m the first

554 ^celeraiors of Charged Particles

555 Nuclear Phyy

stage of a 10 MV Van de Graaff generator. These negative ions are . The principle of the cyclotron can be understood by referring to Fig^ reconverted into positively charged oxygen ions in an electron stripr^^'^Tn iq M is an electromagnet between the pole-pieces of which are placed

■ "■ _hv n. and D,. These have the appearance of two flat. The ions which are produced carry different multiples of electtdffl

charges such as 0^0^0^, 0^.0^^ etc Afi.r .L. , o ,0 etc. After going through °*^***^ accelerator tube they gain additional energy^

lOn MeV where ne is the charge on the ion. So the final energy of

positive ion beam becomes 10 (n -i-1) McV. Thus ions will attain

-mi' energy of 60 MeV in this machine.

12,7 Cyclic Accelerators ; Cyclotron

As stated in § 12.2 , in the cyclic accelerators the charged particles’; gam energy in steps, fhe voltage used is much smaller than in the i electrostatic machines and varies periodically. The path of the particle^ from the ion source to the target can be either curved (in cyclic " accelerators) or straight (m linear accelerators). In the former, a magnetic field IS used to force the particles to follow the curved path, which may be circular or spiralling. We shall discuss these machines first.

The cyclotron was first built and successfully operated by E.Q. I?M.S. Livingston at (he University of California, Berkely in .'M

the U S.A. around 1930-31, about the sametime as the Cockroft-Walton ^

and Van de Graaff generators were built. It is the most well-known of all p^icle accelerators and is the fore-runner of all cyclic^acfcelerators which have played such vital role in the devel

rk ■ cr%, I j*iL H ^ particle physics.

In the cyclotron the acceleration of the charged par^i^;les occurs in successive steps, the energy gain in each ^tep being a fraction of the fjiiarenergy achieved. Unlike in the linear accelerator to be discussed in § 12.1 a which was ii^ predecessor ^d in wh^ch acceleration ft the Successive steps takes place along a series of drift tubes ait'ariged longit-udinally, in a cyclotron the charged part¬ icles move in spiralling orbits of gradually incre¬ asing radii due to the action of a transverse magnetic field, Lawrence was awa¬ rded Noble Prize in physics Fig

for the development of the cyclotron in 1939.

lopment of nuclear physics and

/

-TO vdcuyni pump

To rodtoire- <TUtnty osci-t Ha tor

i_

Oisc-shaptd vocuum chambtr

I t_r

12*10. Principle of the fixed frequency cyclotron. Upper picture is a vertical

__!-“■ !.'■ ■ ■ * X H.

section while the tower pictiire shows die horizontal section.

4wo electrodes denoted by D, and D,. Ibcse have the appearance of two flat,

Shollow semicircular metal boxes of the same sixe and are known as ‘dees’ ":• because of their resemblance with the English letter D. The fares is somewhat less than the diameter of the magnetc pole pirees. The two -’dees are insulated from each other with small gap between their diamemcal '4anc faces as shown in the figure. An rf voltage is applied between the dees^

An. ion source is placed at the centre between the dees which are placed inside a vacuum chamber-vkept at very low pressure.

The positive ions from the ion source (such as protons a-particles) enter one of the decs (say D.) when the phase of the rf voltage

is such that they arc attracted towards Dj at the peak voltage V„, and

thereby gain the energy , being the charge of the ion. Due to the

action of the transverse magnetic field, these ions describe a semicircular path within D,. Though the rf potential goes on changing during this time,

tiiere is no further change of the ion energy because the metallic dres act as electrical shield. After describing a semicircle within D,, as the

feenter the narrow gap between the two dees, they arc the opposite dee Dj if the phase of the rf voltage changes by 180 so that

D2 attains the negative peak voltage w.r.t. D, exactly at the same instant.

thus the ions gain an additional amount of energy gV„,.

The ions now describe a semicircular phlh within Dj of radius

slightly greater than their previous path within D,. As before there is

no change of the ion energy as long as the ions move within Dj. As

ions emerge, from Dj and enter the gap between D, and Dj, they wi

gain an additional energy qV„ in going from D2 to D, if the J

the rf voltage changes again by 180“ exactly at that mstant WitMn D, they again describe a semicircular path of slightly larger radius. The

nrocess of energy gain described above fepeats itself every time the

ro"r«osI "b.'gap® bbffiS.n .h» deaa. Tbu. ^^-""7-''/ '’;/. semicircular paths of,gradually mcreasing p7„ successively so that they describe a spiralling path. 12.10 and ultimately gain quite high energy when they reach the

""‘t ordir'fflat'hi' ion, may ba accelaratad .0 high anargy by this mated, Ihela mu,, ba raaoaauc. ba.waau te bma tor de,c.jb,„g .

armiaircla withia a daa and the half-perrod of the rf ''0'“*',

curvatea on« lamiciraular p..h within .ha daa. If B ba .ha magna..a

induction field, then for an ion of mass M we can wn c

Mv^ B^v=——

i

1

3

I

^56 Nuclear Physic^

This gives ...(12.7.1)

So the time taken by an ion to describe a semicircie within the dee

icr nM

Bq .»(12,7*2)

This time t is independent of the mdius r of the ionic path or of the^ veiocity v of the ion. It depends only on the specific charge q/M of the ion and is the same for the ions of the same q/M. Obviously here we assume that there is no change in the mass M of the ion due to its increasing velocity (non-re I ati vis tic approximation). For the resonance condition mentioned above to be satisfied, we must have i — 7/2, or

itAf T 1 Bq~ l~7f ***(12.7-3)

where/= l/T is the frequency of the if voltage. We then have

/=^ ...(12.7-4)

Eq. (12.7-4) is the resonance condition which must be satisfied for the ions ^ gain the energy at each D-cro^ing,

• r If die ions cross the dee-gap n times before Teaebung the periphciy

of the dees, the final energy gained by them is Fm- If is the fin^

velocity of the ions and R is the radius of the dees, the maximum energy attained by the ionsr can be calculated ftom the condition

*

which gives ...(12.7-5)

Hence the final energy of the ions is Ji ^

= = Mf ...(12.7-6)

Bq. (12.7-6) shows that the final ion energy depends on the radius R of the decs and the strength of the magnetic induction field B (or the frequency / of the rf voltage which depends on B}. So to achieve higher energy, a magnet of larger radius is to be used.

Putting in numeiica! values, one gets for protons, deuterons and oc-paiticles (E in MeV)

■Bp = 47.9 . Ej=23.95 , E^ = 41.9

Here B is in tesla (T) and R is in meters (m). The magnetic induction B is usually of the order of 1 T or higher.

Using Eq. (12.7-4) we can calculate the frequency of the rf field. For p, d and a we get respectively

/«15.2x10*5 Hz, /j=/„ = 7.6xlO*SHz.

which gives V = ...(12.7-5)

= 2s? ...(12.7-

Accelerators of Charged Particles " '

For B= 1.6T, we get for deuterons

f^~ 12.2 MHz

This shows that the frequency lies in the, rf .region.-.. The number of steps in which the final energy is reached depends on

the amplitude of the rf voltage. Larger the amphtude smal er is the number^^f st^s required to attain the full energy and shorter is the leng* of the path travelled by the ion to reach this energy. In this case the probability of energy loss by the accelerated ions by collisions with the Virtual gas molecules in the vacuum chamber is smaliei. which is hi^h y desirable for the more’effkient operation of the cyclotron. Usually th voltage amplitude lies in the range of 100 to 200 kV. So the number of revolutions executed by an ion to attain the full energy is only a few

hund^^.^ be shown that the radius of the orbit after the nth dee crossing

is given by

^1/2 ...(12.7-7)

Thus the successive orbits get closer together as n become larpr.

Eq (12 7-6) can be used to calculate the final energy of the ions

■nius for thfe case cited above, the final ener^^of Ae from the University of Pittsburg cyclotron with a dee radius R - 0.54 m

comes out to be E^= 17.9MeV The fir^ cyclotron built by Lawrence and used for nuclear phy^^s

research had a magnet of diameter 37" (94 cm). Later, another cyclobon with a magnet of diameter 60" (152 cm) was bmlt under **is su^isiom This cvetotron could produce deuterons of energy 20 MeV and cJartSlL of energy 40 W. A large number of cyclotrons are now in

operation in different parts of the world. . . , „ ,i,,„ -rp After the ions have been accelerated to their full ^

deflected'from their orbit by the momentary applicauon of a fairly high

voltaue (~ 150 kV) between two deflection plates located near the

oerioKry of the deesl^e deflected beam can either fall on Lget M can be brqught out through a suitable port *or different

exper^ents^^^ beam current may reach values of 1 roA or more. The

external beam current is much lower (~ 200pA or less) The early cyclotrons discussed above, known as fixed frequency

cyclotrons, are suitable for operation at relatively low (non-relativistic) P

energm ^ shows that the same cyclotron can be used to accelerate

ions S^different mass and charge, provided they q/M. ITius deuterons and a-particles can be accelei ^ y cyclotron with slight changes of the operating conditions (fe «r^ich aL required due to small difference in the q /M vames of these two .ypes

Nuclear.Physics

Ofions, Most fixed frequency cyclotrons in operation arc designed for jjieiitcrons. However, to accelerate protons by the sanic cyclotron, major changes in the value of cither B or / is to be made, which is difficult to achieve in practice, H^ice a cyclotron built to accelerate deuterons cannot in general be used to accelerate protons. On the other hand, heavier ions,

^such as t , can be accelerated by the same cyclotron. In some cases harmonic operation can accelerate heavier ions by a cyclotron designed fon a lighter ion. In this case the time required for the heavier ion to describe a semicircle within a dee is an odd multiple of the half-period of the rf oscillator. The resonance condition (12,7-3) now changes to

27EAf' r = 3,5,7, etc. ,,-{12,7-8)

For example a cyclotron for protons can accelerate ^He^ ions on the third harmonic {s = 3),

The cyclotron cannot be used to accelerate electrons because the relativistic increase in the mass of the electron becomes important at

relatively much lower energies (E-- 0,01 MeV), This makes it impossible to achieve resonance condition (12,7-3) in the case of electrons.

Focusing of the acceierated beam in cycioiron :

To get a beam of charged particles of sufficient iiitensity, it is necessary to focus the beam as it emerges from the. cyclotron. The focussing is done by the fringing magnetic field as also by the electric field between the dees.

Magnetic focusing :

As shown in Fig, 12,11, near the edges of the pole-gieces and beldw (he ftiedian plane the fringing magnetic field lines have both vei^cal (V and radial components. The veirlicai cotnponent keeps the ions

' semi-ciicular orbits within the dees. Hie radial coiT(|>inent giv^ riw to a fcM’ce (Lorentz force) in the vertical direchc^, fbree pushes an

; Reflected from the median place back into the^tter, A^ually the particle 6ycrshi)^ and crosses the median plane to the opposite side. Here the radial t or^incnt is oppositely ditected (see figure) due to which the vertical fgipe iigfith pushes it back towards the median plane. Thus the ions execute vertical Mwillaljons about the median plane,

Pole Br _/

Mediori plone

ole piece > - ' , B Br I ^11 12*11, Focusing action of the magi^eiic field in a eycJotion.

559 AccelcTaloTS of Charged Panicles

To accentuate the effect, the magnetic field is deliberately disorted by shimming (with the help of a few auxiliary coils, suitably located, caiTying current) so that the field drops all the 'way from the centre outwards by about 2 to 3 per cent. However, this has the result that resonance condition cannot be satisfied at all radii, which adds to the effect of relativistic mass increase in disturbing the resonance condition (see later). Even so, the improved magnetic focussing greatly increases the emergent beam current, making shimming a practical necessity. In practice* the field at the centre is chosen slightly greater than the resonance field, while that at the periphery is somewhat lower.

Apart from the vertical oscillations discussed above, there is also radial oscillation about the equilibrium orbit An ion can move in an orbit of radius determined by its velocity and the magnetic field. If it is somehow deflected from this equilibrium orbit, the effect of the magnetic field is to push it back into that orbit If it overshoots the equilibrium orbit, it will execute radial oscillations about the latter,

' Assuming the magnetic field to vary according to the relaUon

...(12.7-9)

where B = Bq for r = J? , it can be shown that the ftequencies of vertical

and radial oscillations (betatron oscillations) are given by (see Eq.

12.12-16) ...(12.7-10)

/r=/oVnir ...(12.7-11)

where/o is the orbital ftequency (Eq. 12.7-3). Obviously n must lie between

0 Mid 1. So die field must decrease with increasing radius. Values of n which make/, and/, either equal to or small multiples of each other w of/o produce

instabitity in the orbit osdllatitais. So these must be avoided.

Electrical focusing ; The electric field between the xtees' lias a foeussitjfe effect on the accelmted ion beam. There is fringing of the field lines between the dees. Thus tire electric field off the medi an plane has a vertical component wHf^ has a net focussing action as in a proton linear accelerator to be discussed in detail in § 12.19. Actually if a particle crosses the dee gap wfien the field is rising; there is net defocussing effect while if it crossc.s the gap when the field is decreasing, there is net

focussing effect. The ion beam obtained fi-om a cyclotron has a relatively large energy

spread amounting to about 1% or more which is much higher than in the

Van de Graaff generator (see § 12.6).

12J Limitations of the fixed frequency cyclotrons

The energy attained in a cyclotron increases with the increasing

radius of the magnet used. However, there is a practical limit to the energy which can be

attained in a fixed frequency cyclotron due to the relativistic increase of

I

Nuclear Physics i

/M IP& ion mass. Since the mass appears in the denominator of Eq. {12.7-4), ^sonance condition cannot be satisfied, as the velocity of the ions become ‘high and the circulating ions fall out of step with the applied rf voltage.

Writing M-Mq /Vl-p^ where p = v/c, we can rewrite Eq. (12.7-4) as

...(12.8-1) ^ InMf,

For protons with energy E= lOMeV, the relativistic increase of maSh is about 1%. In practice^ the upper limit of the energy to which protons can be accelerated by these machines is about 20 MeV. For deuterons and a-particles the limits are about 40 and 80 MeV respectively.

To accelerate ions to relativistic energies by the cyclotron principle it is necessary either to change the mode of acceleration or the distribution of the magnetic field in such a way that the resonance condition (12.7-4) is obeyed inspite of the relativistic increase of the mass of the ion. Eq.

(12.8-1) shows that this is possible if either

(i) the magnetic induction B is increased with time in such a way

that the factor Vl cancels out, the frequency of the rf fieid

being held constant; or (ii) if the frequency/of with time in such

a way that the reasonance con3itioni(12.8«^l) holds at ail

instants, the magnetic induction B remaining constant.

The first of the above mathods forms the basis of the working principle of synchrotrons while the second method is applied in synchro-cyclotrons (or phasotrons). Both these machines (as also some others) utilize the principle of phase stability for their operation. . j

\23 Synchro-cyciotron ^ The synchro-cyclotron (also known as frequenc}^ "^modulated

cyclotron) is a cyclotron in which the frequency the rf/iScclcrating field is decreased as the particles become relativisfic. The most important conc^t upon which the operation of synchro-cyclotrons (as also of the synchrotrons and the lindar accelerators) for accelerating the ions relativistic enwgy range depends, is the principle of phase stability. Tnis principle was discovered by E.M. McMillan at the University of California, Berkeley in the U.S.A.(1945) and independently by V.I.

Veksler in Russia (1944). The principle can be stated as follows. In a phase-stable accelerator,

paiticlei are accelerated at a series of gaps by an alternating electric field. When n vtny large number of such gap-crossings occurs, the problem of keeping the Ions in step with the accelerating field becomes formidable. Howevor» with a suitable arrangement of the fields, the particles tend to droll the gaps automatically at a time when they receive the energy fiendlsd to keep them in resonance. The particles, with the correct energy irrivlog It a gap with incorrect phases in the neighbourhood of an

equilibriuo. phase. «ill experience an antomntie correction towards rhe

'‘*^'-?r»,er^:,d"iS;c-iaer a, ion cireninUnP in w. orhi.

of rndins r=— where M is Ore reladvistic mnss. If the ipn arrives at the

. rf field is zero it will not gain any energy and will s»neoH^ih provided B and/remain nnehanged.

Phase Rg 12.12. Principle of phase stability.

graph. Another ion ^i« A-g«ns some energy due „,olution (see Eq. As a result it moves more slowly dnnng tte next

,2.7-2) mid mkes aliule '""Ef.« fh; next time so lhat ns phase IM™ die dee gap exactly at

r&'S?S".d"j!'.s fans in step wiih dre oiher ions

which were originally crossing die 8**’“f^g,^,^,enlcd by A" a

Similarly an beq„„,es zero is deceletaled and litde qflex iB telLivistic mass decreases. This makes

SrSXV - Ss i.n:h™ Other ions crossing the gap at zsto p about the zero phase

^ur.':hiniS“h:Lro?7a.i“—‘" ® K may be noted .bar this type of J“Vm‘ph7s

Stable orbit occurs part of tL voltage-time graph, near the zero phase pomt m t^edecr^ gp^ ^

i.e., near about the point A • ’ j„g part of the voltage-time graph near the zero phase point in the increasing pan

Nuclear Physic^

f Sr*™ ‘’’’f' lorn falling oM of arf

- As stated, the ions crossing the gap at zero phase at A do not oairi*- any energy as long as the magnetic induction field B and the rf freque^ncv "

fomvrat *'»«"/ temased with B conaltt J

o“at't “■«

detnrJliaTbv'&'mi".? ‘”8» radio, y ( 2.7 1). This becomes possible since the freauencv f

Sa ihtolf «, ,L

Tht. Jk! continues till the ions reach the periphery of the dee '"‘greases very slowly which means that the ions must go

cyclotron Thus "T‘’r" («) than in a conventional

ge IS 10 kV, « _ 30,000 compared to only a few hundred in a

sinufe “PPh®*^ 's relatively low a

Svttrai^^^^^^^^^ -d

In the synchro-cyclotron, lighter atomic ions, such as protons deu eron, or a-particles are accelerated to ener^es ^t wWcTS i) V increase of mass becomes significant, the ions travel in rcular orbits in approximately uniform and steady magnetic field The

rr/rr' ““8y e«". The ioT^^SJ S' tT ’ '"S' »n.i^ircyler Mow

»' rf M i« e,u,l to dy ion rcvoIoM trequepcy. t|ie radius of the orbit can easily be calLlated frotfi" the foUowing re|MK»s in toms of the ion kinetic energy ?^B :

.r-A

r=-^ Bq ...(12.9-1)

L2^2 Ip C

r —

...(12.9-2)

...(12.9-3)

°q i ‘

and WJ=M^\ Hence .

'fT(T+2Wo) '^TjT+l.Wn)

Bqc ~ 300 B ...(12.9-3)

ii in T»' energy

Uelng B<), (12.7^) we get the ion revolution frequency as

...(12.9.4)

easing energy. Hence the frequency of the applied rf

MHz ...(12.9-4)

Accelerators.flf Charged Particles 563

.m-' - : Hfield has to be gradually decreased. "he^ij^e

IP rtiake several hunted which are initially spread widely in |i; oscniations discussed aw. ^ accelerated. Along with

(^y—

nddt»“«w^ -“O'"* “«

"Swd to » which is positive lies between 0 and 1. For Ae

S) ri Eqs. ( 12.7-10) and (12-7-11) if n is known. For n - 0.05, /y . /o

'""“in the synchroK^y^^n^e frequency

a frequency m^ulahon of cent ^ ^

a modul^n fL^vSSu^y^otron. was constructed atBeikelcy large scale mwhine, d^uteroM and 380 MeV a-particles of fatfly (1946) which gav^l90 MeV de^ro^^ ^

good intensities. The machin „ . Research at Dubna in Russia,, a Lnn b the Join. Antnhe. shnilur

ntwthine giving 600 MeV pn^s has been in operanon at

Labofatoiy in Geneva since 1958.

i-f ^f» roX^JSrraci.:. ““ synchrooyclotton and ,‘’“VSJ^‘**JS In some cases.

Dunng nneei-h^ ~le «

iTnct^iolt mlap voluge »ay also vary. So we can repose..

the significant field component as ( f r cOy ds 1

£ = £(s) sin M CO, dt - J -;7-+ % j

^e vSTanO «. "rSrS^flSn-^ oT^'rf^^^^

^ Nuclear Physics

Vq be the voltage amplitude ofthe rfjflel^" We shall denote the parameters

of the synchronous particle {S.P^ by the subscript ‘0’ and assume that the deviations of the non-synchronous particle parameters are small. For example the deviation in the momentum p (momentum error) will be written as Ap = /? —; A/? «Pq. Similarly the position error is given by A5^S-Sq«Sq,

The equation of motion of a non-synchronous particle revolving in an orbit of radius r under the action of the magnetic field — in the

azimuthal direction (j = r0) in cylindrical coordinates will be

\ d . ^ . rp^

( ^ \ f f CD as = ^£(5) sin J -+ CPQ +qe-^q 'rB^ ...(12.10-1)

V ^ J Here q is the charge of the particle and E represents all possible

electric fields other than the applied rf field. Thus E includes the betatron accelerating field (if any) and the decelerating field due to radiation loss. The last term gives the magnetic field contribution to the Lorentz force. The momentum p^ of a particle in an orbit of radius r is

Pe=p = qrB^ ...(12.10-2)

or. 'rp/r = q'rB^ ^ ...(12.10-3)

Eq. (12.10-1) then becomes

p = qE{s) sin in f f — V

+ (ft) -t-^iir...(12.10-4)

Eq. (12.10-4) is the general equation of motioii valid fqr both relativistic and non-relativistic particles and applies to all accelerators operating on the principle of phase stability. For a linear acp^erator the term in E will be absent.

We define the phase of a non-synchronoijt partiiife al

r c w.ds (p = <Po^Aq) = J —+ (Po ...(12.1^5)

Then A(p = J j — (a.ds

...(12.10-6)

is the phase error and is zero for an S.P.

Since the particle travels through one wavelength (v/f^) in one full

time period, where is the frequency of the rf field, we can write

2 jt

9 = %-

As f As

v/S~ V

Inf^As <Po-

OgAs

...(12.10-7)

...(12.10-8)

565 /Accelerators of Charged Particles

(12.10.*) »«1 (12-">-«) & - Wt __ .a « AV

..dm A£_ 'r dt V

...(12.10-10)

...(12.10-11) A.Ap = g£(sin<p-sin%)

Here the momentum effor is assumed to be small-

Then from the above two equations, wc ^

^^ (sin 9 - sin % )=0 -(12-10-12)

Eq (12 10-% is% equation for phase oscillation 9 about the

. K .rn ohL cL If A® is small, we can write

" sinip-sin9o = sin(%+A«P)-«n% = ‘^^«^^

So we get _•» WA CII» a-Tfc 1 to^E cos ^fa ^A9+-- dt^ Po

=0 ...(12.10-13)

E,.02,.0-13)n=p«»n«asi™ge*™onico»l^ -

neglect slow variation of £(s)/Pfr

Case of synchro-cyclotron : radius R given by

The synchronous particle has (12.10-14) R=p^/Boq

• A decreaises outwards according to the relation The magnetic induction decreases

(12.7-9) __ B=Bn(r/R)

,•1. c \A hoiiK focussine in radial and vertical Thfe shaping of the ficl 3^ n„Bt lie between 0 and 1.

directions. As will be ^ IJe J^Lating gap with a phase error (for Particles arriviijg at the acceierau g

non-synchronous particles) wi ttave i (12.10-15) r = p/qB

p = = -.-(12.10-16) so that p^qBr = qo^ry.,.^, ■*-« ^

J„ the case of 5.F. the frequency of the rf field © must be an integral

multiple of the frequency of revolution © of the partic e.

with k=\ 2, 3 etc. We then have from Eq. (12.1fT7>

fc«%A5 ...(12.10-18) Acp = -

Nuclear Physich

. '-15'*??J3i -

Since As — rA9 and v = rcOg we have I

. A<p=-jtAe A9 = -)kA0

3

and ..(12.10-19)

Accelerators of Charged Particles

Po ^ c Ho

567

'**r ?•?]« OA * '■

But A j=A(/-e)=^Af-i-v?Ae

i\

rn radial velocity of the particle to be negligible compared to its azimuthal velocity. We then have ^ ^

A * ^ 4,,,_ r

Usinn%n"?n ‘he using tq. (12.10-15), we get

Ar ^ Ap

^ Po(l-«)

To evaluate the second term on the r.h.s. of Eq. (12.12-20) we note that the momenta of a non-^./*. and of an S.P. are

^oi Po-

Ar

R As

V

\

...(12.10-20)

momentum error.

...(12.10-21)

P =

These give y Po

and Jnal

Po

VT +Po/«oC^

where we have written i1o=Po/tooc. ^ -

Also mn j=r-r~ ^

Since p and do not differ much, we get

Ap Ai 2

vr + ■00 ...(12.10-22)

..(12:10-23) * /

■ /■

A j Ap

/»o(»+Tlo) We then have from (12.10-20). using Eq. (12.10-22)

*(« + i1o)Ap

...(12.1

A (j) = /?/no(l~rt)(l +j]l) 2x3/2 ...( 12.10-25)

For t£* synchro-cyclotron and synchrotron. For the synchro-cyclotron, we usually take k = 1 and «'= 0 L that

A 9 = TloAp

/? »lo (1

In this case R is not constant. So Bq to get

...(12.10-26)

we can express R in terms of and

so that

P =

A<i)=

9^0

Bpq_

"*o‘^(l+Tlo)'^

qBo

HoAp ...(12.10-27)

The accelerating field is E=V/6k r, where V is the dee to dee ■Jv.oltage. Eq. (12.10-9) then becomes (with n = 0 . E' = 0),

. pf qV .

qBp n ^ .(12.10-28)

’n*.- For the S.P. we have Rp=p^

p qV sin tp^

9^0 IC tv So we get

d q^VBo — (qo Ap) = :73—: (sin 9 - sin <po)

7C mg c

q^VBg

K m^c A (p cos % .(12.10-29)

Combining Eqs. (12.10-27) and (12.10-29), we then get

d

dt |H-q3j A4)J =

{q/nqy VBI

KC 2.

- A q> cos % ...(12.10-30)

But 1 + T|o = 1 ““ P^. Hence we get finally

;jVV d

dt

(q/m^^ VEt^

ICC A 9 cos % ...(12.10-31)

.5-'. |A'W-

i This represents the equation for phase oscillations, provided cos is

fVnegative. We have seen before that for phase stable (^ration, the rf field should decrease through zero (see Fig. 12.12 ). This means that <p^ should

7C lie between -r and n which makes cos cp^ < 0. The oscillations are damped.

Approximate solutiflKs of Eq. (12.10-31) valid for the non-relativistic and extreme relativistic ranges can be obtained easily.

I However, these are not of fnuch use since the. synchro-cyclotroh mostly Operates outside these ranges. So the equation must be solved numerically. In most cases, the frequencies of phase oscillations lie in the tens of kilocycles per second range.

12.11 Betatron

We have seen that electrons cannot be accelerated in a cyclotron. Being very light, the relativistic increase of the mass of the electron becomes important at relatively much lower energies (> 10 keV).

to circumvent this difficulty, D.W. Kerst at the University of Illinois in the U.S.A. devised a new instrument, known as the betatron (1940) for the acceleration of electrons, which is based on an altogether new principle.

fel58 ^ Nuclear Physics"^

velockSyr/X'? ^ el^trons at I MeV or higher is close to the velocity of light c. (For kinetic energy T= 1 MeV, 6 = v/c = 0 94 So JiiK

inching energy, the electron velocity does n^t increL aplckbif. y U mass increases. Thus the electrons at the relativistic eneroies

ma^etoc field which vanes with tiine. The acceleration of the elecL„!

L^T",h I ol>™*i"8 magneiic dux thmughlStS !k„ •’f' “ojooloxy. According Co Far«lays lawTf electromagnetic induction, the changing magnefic fl,.» « oLh ^ clecoic field JT gi™ by Maxwell'n S,l2ti™® P-odnces an

Vv ^ ■% 4^ 4 « V

Vxjir=- -.(12.11-1)

This induced electric field X accelerates the electrons which en nn

7! Sol"

'™“* foofoosen and hlcnce (be S^nMic

BR=p/e (12 112)

where/>=mPc/Vr:^ is uie relativistic momentum. Eq. (12.11-1), on integration, gives the induced e.iri.f e ;

e=® Jr-rf/= f Vv V. w«r_ f e=^Z <«=J Vx;r.dS=_J e=-^

dt -(12.11-3)

wbeie 9 - J «. dS is die icncgnelic fiiix. e delenniiies thi energy gniiKd per fi * be d« rndb^ of Cbe elrc.ro. erbib Cbe ind,,^

£=2kRX / The fOTce acting on the election is # '

v__£§_g d 2jti? 2nR dt —(12.11-4)

d/9t^Jrrin(?‘£^j|^ ^»2.I1-4). Also w4ake a/oi-a/at, since the medium is not in motion. From Eq. (12.11-2) we get

#r-"5P_ d^B dt dt - (12-11-5)

Comparing Eqs. (12.11-4) and (12.11-5) we have

• -(12.11-6)

-elecJSl‘/!^n ^ condition. It shows that the electrons will revolve in an orbit of constant radius R nrovided the magnetic flux tp changes at twice the rate at which it would change if the

..(12.11-6)

569 i Accelerators of Charged Particles

magnetic field had been uniform throughout the area enclosed by the

electron orbit.

Integration of Eq. (12.11-6) gives

^-ipg = 2’%F^B . - (12.11-7)

S assuming that the magnetic induction increases from 0 to B at the electron

‘"'‘^%o7a uniform magnetic induction, on the other hand, the flux would

be nB^B So for the betatron condition (12.11-6) to told, the magnetic

, c flux must be twice tha? for a uniform magnetic induction.

.Tik vocmam pump

Electron, gun

For this to be possible, the magnetic indnetkm in of Cboir^^ by .he ebrcin. mMbobigber■•»»«« « ■!«=

orbit itself. This is made possible by shaping the pole-pieces of the magnet as shown in Fig. 12.13 (upper p^)-K <B> is the mean induction within the orbit, then the condition, <B>-2B^ must be satisfied during the / I acceleration. In addition, a few discs of j \ high permeability aic placed hetwwn / Bectroni the pole-pieces in the central region. T pumga /

In the actual instrument, the \ I electrons emitted from a heated \ ^ / filament are first accelerated to about ^g / "-K 50.000 volts and thca injected into t^ "”1 • V W9e»-^jr doughnut at the instant when B-0. V

The magnetic field then rises ^ sinusoidally. Tto acceleration tote ^

place duni^ tto Rg. 12.13.PlBiiciple of a betamin. TTie time period T of the s appcf mctiifc shows a viatic^ magnetic field. Exactly at the cn o ^ section while ihc lower ^ctore the acceleration per^Q^W when B i shows ite horizoned section.

few (he feble J.it val-bb » . .e»lt .rf .bich (be eleeboes are

Target-

!!Ooughnut<sKaped

vocuum chatnber

shows tfac horizoiribd sectiem.

deflected from stable orbit to hit a suitably located target, makes aetlecteu af enemv fiom 0 upto the maximum the latter emit x-rays with a spreto of energy electron cnerBV by bremsstrahlung process. The iHor^s r^ats itseii a Srinterval of the time period Tof the magnetic field so that the x-ray

is obtained in pulses of very short duration durmg each interval T.

The electron energy can be estimated as foUows. Suppo^ the

to.g(Sl n» 9 iaries ScsoWall, .i.b (fee e-x-fib® » q,=9„sin(at ...(12.11-8)

where co=2n/T. Then the energy gained per turn is (ne^ecting the

negative rign)

Nuclear Physics

-(12.11-9)

^^T/4 = ii/2(0, the mean

The number of turns described by the elecbon during this period is

2kR 4©/? ...(Iz.ll-L

Hence the final energy of the electrons is

^_Af-2ea)<p„ ec<p„

n 2k R —(12.11-13)

th« “ determined by the peak value (p„ of the ma^ehc flux and the radius of the election orbit.

=»^*'o?Sie If® *" maximum value R "m Of the mgnetic mduchon at the electron orbit :

B=pc=eB„Rc ...(12.11-14

which (is also obtained from Eq. (12.11-13) if >ve put ® =2n R^B

Thus fori?=0.5m,B^=lT weget I “

£=1.6xl0r'’xlxO.5x3xio®J = 150MeV /

H».l U-VleclTO^Wfer. ™chi.g Ihe final energy comes out to be W=4x 10?«and thT .u ^ *i, travelled is £= 1250 km: » ttr and the length of the path

"^±=“S)'S’ ““ •» » l» ‘U »

I JtW

AV

t i'.-

-Ato

^

K »

1. r

1

vmm

.Accelerators of Charged Particles

Vpllore in Tamil Nadu. There is a proposal for installing a betatron at the SakTJku7Snc^ Hospital near Calcutta in West Bengal (see § 12.20)

Upper limit of energy of the eiectrons accelerated by betatrons :

■ It is a well-known consequence of Maxwell’s electromagnetic theory that a charged particle with motion, accelerated or decelerated, rad ates

: energy in the form of electromagnetic waves. This type f

rrgHosftpTs^Snd^p.«icl/of Charge , aad velociry

V(0 is . .. . , ,

6n£oC (i-p;

For motion in a circular orbit with a uniform velocity, since v and v

are mutually perpendicular, we have

...(12.11-16

V* - (v X v)Vc^ ...(12.11-15)

an(i

...(12.11-16)

...(12.11-17)

Here R is the radius of the electron orbit. We then get ^ A

7 3V 1 -

^'^'■“’bJCEo (1-pV ^

...(12.11.18)

where 1-3'

W and Wo are the mass energies of the particle for the velocity v-» c and

rhrechon -

T- p Jr ^h: rSr confined within a narrow cone in the forward direction. For

^ ~ Th^waveTength distribution has a maximum at the wavelength

(in A) given by A , Y w

X = 1.76x10'“i? ...(12.11-19)

An upper limit of the energy to which the ^ ® accelerated is set by the radiation-loss discussed above. If this loss

1

572 Nuclear Physics

appnxiable compai^ to the gain of energy per turn by the betatron pnncipic, tiiOT the electron orbit will shrink and the beam will hit the.

'I!!!* energy loss per turn can easily be calculated using Eq. (12.11-lg). ^

Si^ the tune fw one complete revolution is t=2 ic R/c, we set the energy loss per turn as (writiiig q — e} ^

6 Jt CoJ c SCq/? ...(12.11-20)

OT an HK) MeV betatron, A£= 11 eV per turn which is small company to the energy gain p(»^ turn (see Eq. 12.1 l-IO) which is 420 eV for R - 0.84 m. - 0.4 T, to = 2 nf with the frequency /= 60 Hz. On the

other h^. fOT the electron energy of I GeV and with "m^ ^30 keV per turn.

12.12 Betairon osdllatimis

f .1.^** evident from Fig. 12.13, the doughnut shaped accelerator tube of the betatron is located in a region between the magnet pole-pieces where the magnetic fcld is not uniform, Init decreases slowly in the radial directioii. The field-index n is

r ^ \

>* = -r —^r-^ B dr

V * A=o ...(12.12-1)

which follows front Eq. (12.7-9)

...(12.12-2)

where 0q is the value of at the equilibrium (dr ideal) orbit fr=R). The

field index is a positive number (n > 0) and is independent of t.

.1. .•station the electrons from the hof filain^ ^ injected into tfm equihtoum mbit at low energies. The eie^ons released close to this ortit. ^ually move clj^ser to it and execute damped radial oscillatiods about It in the median plSne. The amplitude of these oscillations grad^lly decreases ^ the magnetic field builds up. TTie electrons emitted i a carefully designed electron gun will miss the gun on its second and subsequeitt revolutions.

To study the problem in detail, we write down the equations of motion of the elytron in the radial and vertical directions, using cylindrical coordinates (r. B. z). It is assumed that tlie equilibrium ortiit IS crculai lor which r = R = constant and z '= 0 (median plane). »IS the azimuthal angle. I^splacement from the equilibrium orbit takes place due to collisions with the residual gas molecules and can have

, three components along r. 8 and z. Displacement along 0 will not be considered in the case of the betatron.

r assume the displacements to be small and write 1 and p = (r-R)/R« l.

573 Accelerators of Charged Particles

Due to the action of the electric and magnetic fields, the Lorentz

force acting on a charge q is . /n n F = q{X + vxB) ...{II.U-J)

For the betatron X = 0. The m^Sneti^nduction field has the radml

and Z-components B.andS, while B^=0. The components of the Lorentz

force are a F, = g(vxB)j = q(VrBs“''e^r)-

c F, = q (v X B),. = q (vg - Vj Ba) = 7^9

/^=q(vxB)e = q(v,B,-V,B,

For the electron q=-e. Writing Ve = re the equations of motion

along r and z beconic

—p =zmr — mrO^ — erQ

—p = tni = erBBr

But r = B p so that p = r/B and z = B^.

So the equations of motion become .

itiR p = mB(l + p) 0^ ~ + p) 0

mBC = cB(l + p)0Br

...(12.12-4)

(12.12-5)

...(12.12-6)

...(12.12-7)

of p

From Eq. (12.12-2) we have, neglecting second and higher powers

B, = Bo 1- r-R

R = Bo(1 + P) " ■= ®(i(l - ”P) -.(12.12-^)

3B, 3Bz_zBq"

^^=""37"" ar " «

...(12.12-9)

last steo follows from the Maxwell’s equation v xB-0 since there are no currents in the accelerator tube except space charge curren ‘^TdSacemenL««^nt which we neglect. We thus have

mB p = mB(U p) + P) ? ^ d " "P) -(12.12-10)

...(12.12-11) mB; = -eB(l+p)9—^

For the equilibriuin orbit r.Rendp = 0. Also the derivatives of p

are zero. We have in this case

mv^_„ ! ...(12.12-12) m V = BftC V

u » m m heinu the angular velocity in the equilibrium orbit. Thus;='///ar* » valid i„ the relativistic case also.

m being the relativistic mass. Also since

V0 = re = /f(l+p)9 = B<i>

Nuclear Physics

re have A UJ

...(12.12-13)

qurfntfties of^^coid aTd” high!? ordm^" small

P=-0)\l-n)p t ..■(J2.I2-14)

^ = -a?nt -

0<«.?r?„ i!. ^ *"“®‘ positive which requires that 0<n< 1 So the field must decrease from the axis outwardsTc thl

fZTthIt Z‘r"lT' 5' «f •I’' -.achi^ Tte too repidly ' ‘ »« "I'orease

oscillations are naturally damped as the field grows the amphtude of osc.lla^ being proportional to 1/V^. For radiaJ

iations, ©^_cqa Vl _B and for vertical oscillations ©, = cObV^. The corresponding frequencies are "

fr =/oVT-n and ...(12.12-16)

w ere/o -0^/211 is the frequency of the applied magnetic field. In most modern betatrons /q = 60 cycles per second.

settle^i^Th! the damping of the oscillations, the electrons ultimately

nrnHiirr ff n » • ^ '* *' as 15^ 1 increased steadily the pr^uct If p B, remains constant so that the fractional chahge p in the orbit

em.n l instantaneous radius r approaclfes the

^. ’“^'^tion A of dunng one turn is very small.

pitch of the sn'^ this means that Ap«p so that the above condition fails if 5 =0 But^

df-iTr “ ;;s;s,r.s fulfXa orbit. In practice, these conditions are never

u -rt'i-. «■ to to decopn perfpnp,

dS™ t'r^SlSm “O »«to

575 Accelerators of Charged Particles

1213' Electron synchrotron lo, tr. that of the

synchro-cyclotron. As J" the latter, Jh y frequency of the electron.

L rf oscillator is matcto in to synchrotron, to

“^^Sacccicrationintosy—^

principle, explained • ' -igctric field along the orbit, which orbit of the electrori^in 2 MeV. At this point, the betatron accelerates the electrons up rj^ie orbit with almost constant action ceases and the electro So the angular speed

velocity v = c where c is ihe^orbit radius R remains practically 0) = c/R is also constant and hence the orou

• ■ » ,1 at fairlv high speed from the electron-gun The electrons are injected at fa ^y ^ afterwards. After the

so that the orbit radius . J-ons revolve in phase-stable orbits initial betatron acceleration, t^ el ^ is ^he relativistic

(see below) with the .“on by the rf electric field within the

i„„etod .c mainuin ^„3gn.. can be

„adc much lighter .hau i. itllx linked crirh to

in the latter is of the Magnetic field. In the synchrotron, no orbit is twice that ^ "“^Jieid is to act only in tile region iron core is needed. oelerator tube is placed, since the

primary function of the field s to keep t^^^^

phase-stable orbit. The “ ,j,g gjagnet since there is no iron in considerable saving in the cost o . The vertical section of

the core region, as^the f ^^mr Tshown in Fig. 12.146. The the magnet has Jb^^^pe o parallel with slight tapering pole-faces of ^eam The bltatron acceleration in the initia outwards for focusing the beam. i flux-bars made of steel stages is achieved with the he p . ^ iron-core of locited in the central region, i.e., {octittd. At lower field

the betatron magnet wou flux-bars increases as the ftel

strengths, the betatron accelerating field. At higher field saturated, so that the betatron

^"‘^Threlectrons revolve

toroidal section (doughnut) m outside, fhere

f •' #. / Nuclear Physics

e aris accelerated by the rf field aln axis of the doughnut. The Irequencv of the

ar frequency of revolution of the “l- eqSSTtS-Bi^ " fn.rgy. .heir feMvfciC^ 1*' *' «'«•«».» gain

in orbits of larger radiu<s 7li#»r which tfiey tend to move ort,i. mdi» c?« “f *' ”“«■»«» fiew mainairfe

Toroidal vacuum chamber

Flux bars

Accelerating ^oltoge

R.F. Cavity E

E+AE ►^•AE

Time

(o)

Coil

I4 tube

Unognen

o (b)

F.g P™^* otdacno. .ynnfcom,. ^ ^

e|ac.ron''aS“^“

higher energies is due to radiation lo« Tin **"'*^*>®a to achiey^ the region of few GeV (= lo’ eVt <:» ® **>g**ost energy attained is in

the alternating gradient (AG) princiole fiTthl incorporate

protons in the GeV (lO’ eV) ree^n p" f*^*’*'®'* acceleration of effective accelerators for ob Jninf * synchroVons are the most 'vill be described ij § 12 i ^ ^ beams. They

■Phase stability in the synchrotron :

and applini'^il “f ptiase-subili^r »as diacnsscd

Accelerators of Charged Particles 577

We rewrite the general equation (12.10-25) governing the variation of the phase-error A <p for a non-synchronous particle with the momentum error Ap :

k{n-hr\l)Ap A cp=-TTTj ,..(12.13-1)

where n is the magnetic field index (see Eq. (12,7-9). = Po/f^oC and

Jt= 1,2, 3 ... is an integer appearing in the relation (12.10-17) between 0% and co^.

We now combine Eq.(f!2.13-1) with the equation of motion (12.10-9):

p = eE sin (q>o + A (p) + ...(12,13-2)

Here we have put q^e, the electronic charge. As stated in § 12.10, E' includes all electric fields other than the rf accelerating field. Here we neglect the radiation loss field and assume E' to be the accelerating (or decelerating) field due to the changing magnetic flux (betatron field) :

Er=-r^ f ^Inrdr ...(12.13-3) 2jtr •' 3/

...(12.13-3)

The accelerating rf field is

E=V/2nr

From Eqs. (12.13-2) and (12.13-3), we get

r p=sin (qjo H-A <1>)-I-J

Frar the S.P., we get

dB ' r dr

...(12.13-4)

...(12.13-5)

eV [

We note that

rp-RpQ = A(rp)=pQAr + RAp

and Ajr^fr = rAr ^ .? 'I

Also since Po = e Bq i?-^liave

...(12.13-6)

...(12.13-7)

...(12.13-8)

Po = ^^

BB. ...(12.13-9)

We then get from the difference between the Eqs. (12-13-5) and

(12.13-6)

eV ^^0 Po A r +/? A p=—A (p cos % + e/? — A r

In view of Eq. (12.13-9) this reduces to

Using Eq. (12.13-1), we then get

...(12.13-10)

578 Nuclear Physics^,

d J(/+Tlo)2 A(p

dt n + T13

^eV'cos(po -1-A(p 2k R ahqC 1 ~n)

...(12.13-11)^

If B and the accelerating field are held constant, we get

/7q cWsin9^^

^0 m z' TT I? m^c 2nR m^c ...(12.13-12)

Writing

we get finally

d _ d ^o_ g^sintPp ^

dt dx\^ dt 2 71 R.m^ c di}^

I - n d . (i +ilo)^^- d

d% « + ilo ‘^0

where K = 2nmQC cos %

...(12.13-14)

...(12.13-15) eV sin %

Eq. (12.13-14) gives the phase oscillation for a non-synchronous particle. Since /i >0 and lies between 0 and 1, it gives stable oscillations provided cos cPq<0. For acceleration sin %> 1 and so (p^ must lie

between 7C /2 and 7i on the standing wave and hence must be in the rising part of the travelling wave.

/T is a dimensionless constant. For electron synchrotron A'^10^

while for proton synchroton A ~ 10^. It is possible to solve Eq. (12.13-14) assuming the change of

momentum in one complete period of phase oscillation to be small. At non-relativistic energies, the solution takes the form :

. (. 3n-2 2^1 1 ^ ''■'-i'—>1-

which shows that the phase oscillations are weakly damped if /i > 2/3 when the quantity withjn the first bracket on the r.h.s. is less than 1. In the extreme relativistkf range, T|o» 1 and J

A (p ~ Tj"''"'cos ri- ...(12.13-17) ^ I 1 — n ^

/ \ J A

This shows that the oscillations damp as Hq .

Along with the phase oscillations, there is a damped radial oscillation (synchrotron oscillations) which has a frequency much lower than the frequency of betatron oscillation discussed in § 12.12.

In the above analysis, radiation-loss is neglected which however becomes important at electron energies above 100 MeV. This gives damping in the phase oscillations for n < 3/4. When the energy loss per turn becomes equal to or greater than the energy gain from the acceleration process, the approximate theory discussed above will not hold. In this case V has to be increased to get acceleration so that the assumption of V = constant cannot hold.

Electrons Extractor

Resonator

Microwave power

linoc

(mac 2

(mod

Fig. 12.15. (fl) Principle of a microtron, (o (f) Double-sided microtron.

Since relativistic electrons move

cyclotron principle cannot be applie principle can be understood by reb accelerating cavity is located near the chamber placed between the pole-1 traverse the resonator and are acceb

e(]ual to the electron rest energy c

under the action of the magnetic fi approximately an integral number of i orbit is thus a series of circles of u

resonator.

Nuclear Physics'

Electrons are accelerated to a few tens of MeV in the microtron. The beam is more easily extracted than in a betatron. The beam current drops sharply at higher energies. The electrons experience variety of phase stability similar to that found in a synchrotron. Since the energy-gain-at each step in the resonator is a large fraction of the total energy, it is not possible to describe the phase-oscillations by a differential equation, as in the case of a synchrotron. The correct approach is to set up a finite difference equation which must be solved numerically. The microtron is more useful as a relatively low energy accelerator.

One important characteristic of the microtron is the very small spread in the beam energy, second only to that of the e.s generators.

Energy relations in microtrons :

Suppose the electrons are injected from the electron gun with an

energy ot^ Wq where Wo = mo c is the electron rest energy. The electrons

gain an amount of kinetic energy A W equal to a constant fraction a of their rest energy every time they pass through the resonator, so that A 1V= aWQ. So after the first crossing of the resonator, the total energy of

the electron is

W^={1 +a, + a) IVo ...(12.14.1)

The radius of the circular path of an electron of total energy

W = mc in a magnetic induction field B is given by r=mv/Be where v is the electron velocity. The time for these electrons to describe one

complete circle is x = 2 tc r/v = 2 jc W/B e c Thus the electrons after the first crossing of the resonator, describe a complete circle in time

x,= 2 7c(l +u, + a) Wq

Bec^ = *1 ...(12.14-2)

where x^ is the time period of the rf acceleratiim fieldis an integer.

After describing one complete turn, the Sectrons gain an additional amount of energy A W^olWq in the resonator for the second time so that

their total energy becomes 0

1^2 = (!+«/ +2a) Wo ...(12.14-3)

The time taken by these electrons to describe one more complete turn (second turn) is

2 71(1-f*a,-+2a) Wo ^ x^ =-^ ...(12J4-4)

^ Bec^

where ^2 is again an integer.

Wc thus get

2 K aWQ ^ ^ _

...(12J4-4)

Bee = {k2--kj)X^ = kX^ ...(12.14-5)

where k = k2 — kj is an integer {k = 1, 2, 3.).

Accelerators of Charged Particles 581

We then have

Xn — Xi aWr

T, (1-Ha, + a) Wq k^

it (1 + a,-) li gives a = ^

If X be the wavelength of the travelling wave, we have

X 2nWoa 2jiWo(l+a,)

which gives ...(12.14-6)

B ec^k B e (?' {ki—k) ...(12.14-7)

B\^ 271 Wo(l +a,)

...(12.14-8) ce{k,^k) -'

After n turns, the gain in energy by the electrons is A W„ = waWo so

that the final kinetic energy becomes

= a, Wq + naWQ=(a, + n a) Wq ...(12.14-9)

The difference in the radii of the electron orbits in successive turns can be calculated by noting that each orbit must contain an integral number of full waves. We can then write

Inry^k^X and 2 7cr2 = ik2^

So we get

kX > mm r=r,-r, —

^ * 27C ...(12.14-10)

The maximum energy of the electrons in this type of microtrons is limited by the finite size of the rf cavity. A later development, known as the race-track microtron uses a magnetic field split into two half circular sectors separated by a sufficient drift space in which a linear accelerate^' is inserted (see Fig. 12.15h). In this geometry the electrons are accelerated to a few hundred MeV energy.

To achieve higher energies, a double sided microtron with two linacs (Rg. 12.15c) or a hej(|yg[pn with three linacs have been proposed.

The earlier versions of microtrons are now used in medicine and industry as also for nuclear physics and other research fields. They are also used as injectors for larger machines, such as electron synchrotrons.

The most advanced concept in microtron known as the MAMI has been designed to furnish 100 pA of electron beam current at an energy of 175 MeV. A small 20 turn 14 MeV microtron feeds the electrons into a larger 51 turn 175 MeV machine. High efficiency rf wave guid^^^^^ makes it possible to push up the electronJcurrenrtd 1(J0 pA. With the addition of a third race-track microtron, the energy is expected to go upto 800 MeV.

The race-track microtron may be regarded as a folded linac with each traversal of the same linac section being equivalent to the passage of the beam through successive sections of a conventional linac. RF power loss is much reduced than in conventional linacs.

Nuclear Physics

.15 Sector focused cyclotron

f In § 12.8 we saw that the maximum energy attainable by a conventional fixed frequency cyclotron is limited by the relativistic increase of the mass to about 20 MeV for protons. Of the two possible methods to overcome this problem, the principle of frequency modulation (synchro-cyclotron) was discussed in § 12.9. An alternative method using an azimuthally varying magnetic field (AVF) was proposed by L.H. Thomas as early as 1938. The theory developed by Thomas forms the basis for the construction of the sectors focused cyclotrons.

It may be noted that though Thomas’s proposal was recognized as theoretically sound, the computations were rather involved. As a result, no action was taken to give practical shape to these ideas for nearly twenty years. We shall briefly discuss Thomas’s theory in Appendix A 11.

As seen in § 12.8, the method (i) of maintaining the resonance condition at relativistic energies involves the radial increase of the magnetic field. This however conflicts with the requirement of focusing the ion beam for which the magnetic field should decrease at larger radii (see § 12.7). Thomas proposed the use of alternately high and low regions of the magnetic field around the orbit which could be produced by

I attaching radial sectors of iron (ridges) to the pole-faces so that the pole-gaps were alternately wide and narrow. The average field around the orbit could then be increased radially outwards to maintain resonance. Thomas showed that an orbiting particle in such an azimuthally varying magnetic field would be subjected to an axial restoring force which would provide orbit-stability.

Tho principle of sector focusing can be understood by refitrring to Fig. l2A6a which shows a highly simplified sysjieni of three symmetrically arranged wedge-shaped magnetj^ meeting at a point in the centre. In between the magnets, the orbits are approximately straight lines. They bend>firough 120* within the magnets and emerge obliquely at the angle tp at the exit ends of the magnets. The i^i velocity has a radial component = v sin (p at the exit end. When acted

upon by the azimuthal component of the field, the ions experience

a force in the vertical direction (sec Fig. 12.166. This force pushes

back an ion straying from the median plane back into the latter and thereby helps focusing of the ion beam, overriding the defocusing action of the radially increasing field. Computer studies have shown that almost any azimuthal form of the field variation would give focusing action as discussed above.

% _

Later studies showed that Thomas’s proposal of sector-focusing was

a special case of the general theory of AG (alternating gradient) focusing

in a constant magnetic field to be discussed in § 12.17. These studies have

583 Accelerators of Charged Particles

shown that it is more advantageous to use spiral ridges on the cyclotron

pole-faces rather than 'radial sectors as proposed by Thomas. The final

outcome has been the development of a number of cyclotrons of Ais type

known as the isochronous or sector-focused or AVF. some of which use

radial and some spiral ridges.

Re 12 16 FUcusing in an aziinnWially varying m^etic field, (a) Aimgement of ® three symmetric wedge shaped irnghets; (b) Vertical force doc to

azimuthal component of the magnetic field.

Unlike in a synchro cyclotron, these machines use fixed frequency rf

field for a given energy which gives high beam intensity, as in a common

cyclotron. Another important feature is to obtain variable energy beam.

This is made possible by the use of sector-shaped coils to control the

azimuthal variation of the magnetic field. In addition, circular coils of

different radii are usedon the pole-faces to control the radial increase of

the average field. TTlSB coils maintain (the field pattern as the mam field

is varied. The radio frequency circuits are tuned over a suitable range of

frequencies to match the variation of energy.

The AVF cyclotrons with arrangement for energy variation are also

known as Variable Energy Cyclotrons (VEC).

Fig 12.17a shows the arrangement of the spiral ridges on the

pole-faces. 'Ihe nature of th6 azimuthal variation of the magnetic field at

Tgiven radius is shown graphically in Fig. 12.17h. The oscillations of the

field pattern on the median plane distorts the particle orbit from circulm

to a curved polygon. The high and low field regions at a usually termed hills and valleys. The nature of the ion orbit in such an

varving field is shown in Fig 12.17c.

i

^4 Nuclear Physics

^ A large number of AVF cyclotrons are in operation in different parts of the world including one in India located at Bidhan Nagar near Calcutta

(se6 § 12.20). South pole Hin—y .^^^alley

ions

North pole (b)

Circular orbit —■

Actual orbit-

Fig. 12.17. (a) Arrangement of the spiral-ridges on the pole-faces of the magnet; (fr) Nature of azimulthal variation of B at a particular radius ; (c) Nature of the ion orbit in an azimulthally varying magnetic field.

f ,

Vacuum chamber K—7 The first sector

focused cyclotron was built at Los Alamos in the U.S.A in 1954 by rebui¬ lding the 42'' conventional eyclotron using radial sectors and coils and with a ' variable frequency oscillator. Hiis and the one at Delft in Holland were the oiliest machines of this type built before 195^ Within a few years after that several sector-focused cyclotrons in the 50-75 MeV proton range had been built and operated successfully. Several higher energy machines have since been built.

To the machine center

Vacu!#m chamber

TO the, ► machine chnter

Mognet coils

^ To the. machine center

Vacuum chomber

Fig. 12.18. Cross section of the ring magnet in a synchrotron, (a) C section facing outwards ; (b) C section facing inwards ; (c) H section magnet.

A

i

=!-

. „ • , 5«5 Accelerators of Charged Particles

in 20,0UU. A large i m Tt uqpc <;Diral ridaes. A similar 500 MeV negative yields 100 mA proton beam at machine built at Zurich in fartnrv Arraneements are being made 500 MeV and is known “ ”'7IfnSinei

acceleration.

Superconducting cyclotron : .nm.

higher by a factor ot superconducting cyclotron is in

conventions ^ A „ State University in the U.S.A. A liffge operation at the Micnigan there which is expected to superconducting cyclotron is eing c ^ nucleon. These

acUlmte heavy io» to ^“Xuon. With high

”“!j2“onducE'&ng developed, great Unprovements are eapeeted tn

?i‘Se‘ofT.“e“' because oi uic ^ i^ize of the cyclotron is reduced by

XSTrf?'”? for tie sante energy, and hence n rednetion in the

areas by a factor of 9 or more is obtained.

heavy 1^^^

principle (see § resonance condition. This is necessary frequency are v^i^ to ma .wl velocity of the protons |s not because, unlike in the case o e c ^ ^ ^ frequency of

constant even upto P „ inf does not remain constant, "aere can revolution for the of frie rf accelerating field is increased be resonance only f the revolution frequency. At the same ttme,

re^err-b" “

itg^r:^^tSe rof aS annular magnet, thereby reducing the

cost of the machine considerably. synchrotron is the The most important component of he pro on sync

srn "rirVeCr. :ith^r-=a the outside or .he

h volts) for 10^ eV.

I

- J

1

Nuclear Physicsh

inside of the machine (see Fig.'12.18a, b). In some cases a ring magnet ^with an H*-section (Fig. 12.18c) has been used. In case a, the field : •decelerates the particles while in b it accelerates the particles. The core

, region is empty.

For magnets with C sections, as in the Brookhaven cosmotron, the vacuum chamber (ring shaped), bet ween the poles is accessible around the entire outer periphery. In this case the magnetic field at the orbit has a value of 1.4 T upto the point where the saturation effects become significant. With the H section magnet, as in the Argonne machine, the maximum field of over 2.1 T can be used which results in a smaller radius for a given energy. However, this severely restricts the accessibility to the vacuum chamber and hampers research activities.

As the magnetic Held is increased to keep the particles on track, there is some gain of energy due to betatron effect. This however is small’ compared to the energy gain per turn by the particle in crossing the accelerating system (z.e., one or sometimes more rf cavities).

The pole-faces of the magnet are tapered so that the field decreases outwards. The field index n lies between 0 and 1. This helps achieve orbit stability due to betatron oscillations. In addition there are phase oscillations due to operation of the principle of phase-stability discussed in § 12.13. These are also associated with a type of slower radial oscillations (synchrotron oscillations).

The choice of the magnet structure and the maximum field determines the orbit radius for a given energy. If T be the kinetic energy then for an induction field B, the orbit radius R is given by (see Eq. 12.4-3)

T{T+2WQ) = {BeRcf ...(12.16-1)

where'Wo = Afo c is the particle rest energy. For 7 and Wq in GeV and B in'tesla, we get ^ ^ ^

„ { 7(r+21Vo)}'^' -metres i ...(12.16-2)

In the cosmotron, the chosen energy 7 is 3 GeV. So for a maximum field of 1.4 T, ^ = 9.10 m. Actually the mean orbit radius for this machine; was chosen as 9.15 m.' . 0

The dimensions of the orbit aperture to be chosen are determined by the amplitudes of different types of particle oscillations in the orbit as discussed above. The orbit-aperture can be greatly reduced in machines operating on strong focusing principle (AG focusing) to be discussed in § 12.17.

In the cosmotron, the orbit aperture was so chosen that a 36" pole-face width with a 9" gap could be used. This accommodates a vacuum chamber with the net internal clearance of 30" x 6". This allows a geometrical factor of safety about 1.5 times the maximum oscillation amplitudes, both radial and vertical. In the 6.4 GeV Berkeley proton synchrotrtm, known as Xht bevatron, the corresponding dimensions are 40" and 10" respectively.

Accelerators‘of Charged Particles 58^7

; ^ The magnet design, both at Brookhaven and at Berkeley provides sectors in the orbit free from magnetic field (see Fig. 12.19). In both cases, the circular magnet is split into four quadrants. The four sectors of the accelerating tube within these are joined by four straight sections free from the magnetic field. The straight sections have been shown not to disturb orbit stability. They however, require a slightly larger aperture for the particle orbit within the quadrants. The cosmotron magnet is a steel

ring of 75' diameter with a cross-section 8' x 8' (2.44 x 2.44 m^) weighing about 2000 tons.

Extraction target and magnet MI

H Extraction \\ magnet Targets!

Fig. 12.19. Proton synchrotron assembly showing the four quadrants and four straight .sections of the vacuum chamber. '

The four field-free straight sections serve a number of purposes, such as housing the rf accelerator, accomodation for the injector, the target, vacuum manifolds etc. The value of the fielci index n required in the case of orbits with straight sections must lie between 0.5 to 0.8 for effective focusing of the ion beam.

The magnetic field rise-time is usually between 0.5 to 2 s. For the cosmotron, it is 1 s. The pulse repetition rate is 12 min. In the bevatron, these are Is and 10 min respectively.

The particles injected into the prot^ synchrotron are pre-accelerated to a fairly high energ^n the cosmotron, this is done by a 4 MeV Van de Graaff generator. In the bevatron, a 0.5 MeV Cockroft-Walton generator feeds a 10 MeV proton linac which was chosen as the injector.

The orbital frequency, as derived from Eq.( 12.7-4), comes out to be

27c(Wo + r) This is somewhat modified (to /') due to addition of the straight

sections such that the ratio

tL ...(12.16-4) / 2nR + 4L

where R is the quadrant radius and L is the length of the straight sections. For the cosmotron with /? = 30' and £-=10', /'//= 0.824. Actually the

I

Nuclear Physics

^^equency of revolution increased from 0.4 MHz at 5 MeV proton energy (injection energy) for B = 0.0311T to 4.2 MHz at the maximum energy of

^STCeV for B= 1.4 T in the same machine. The ion energy increases by 10^ *eV per turn at injection, decreasing to about 600 eV at the maximum energy. These requirements are met by choosing the rf peak voltage to be 2000 volts at injection, falling to about 1400 volts at the maximum energy to allow for the phase and energy oscillations.

The accelerating system in the cosmotron uses an electronically tuned oscillator which enables it to follow the frequency schedule. This includes tuning over the wide range of frequency variation. Ferromagnetic loading materials are used to give the desired high impedance in the oscillator. The frequency must be held constant to 0.2%.

The accelerating system of the cosmotron is essentially an rf transformer with a ferromagnetic tubular core to surround the particle orbit in one of the straight sections. The ferromagnetic material chosen is a ferrite tube (to eliminate eddy current losses). The primary winding is a copper rod equivalent to a one-turn coil which is fed by the high

frequency current from an oscillator. The secondary is the beam itself. In the bevatron, the accelerating system is a drift-tube of rectangular

section made of a conductor surrounding the beam aperture. The drift-tube is contained within a straight section of the-accelerating tube and its voltage is higher by a factor 5 than the energy gain per turn. As the particles enter the drift-tube they are accelerated while they are decelerated as they leave. There is net gain of energy as the particles cross the drift-tube. The drift-tube actually forms a resonant circuit with a ferrite loaded inductor. The circuit is driven by a power amplifier at the frequency of revolution of the particles. ^

A number of proton synchrotrons (with or without AG focusing) have been or are being built in different countries of the world.

0 12J.7 Altematiiig gradient focusing

The [Mfincipie of alternating gradient focusing simpl^AG focusing for restricting the radial motions of charged particles in accelerator beams was first proposed in a m^script (which was not published) by N.C. Christophilos in Athens,'Greece in 1950. The principle was independeji^y discovered by the three American scientists E.D. Courant, M.S. livingston and H.S. Snyder at the Brookhaven National laboratory in 1952. Since the method of AG focussing gives very strong restraining forces on the beam of particles, it is also known as the method of strong focusing.

In the original work of Christophilos, the proposal was for magnetic gradient focusing. In addition, electrostadc gradient focusing is also possible.

The AG focusing concept has added a new dimension to accelerator design and has become the most significant forward step in very high energy accelerator development. By the application of this principle, it is possible to ^uce die amplitudes of both betatron and synchrotron oscillations considerably. This pomits die use of a vacuum chamber of much smaller cross section, so that the magnet size can be gready reduced. Further, the quantity of copper for the

589 Accelerators of Charged Particles

cf^ and die electromagiiedc en^gy to be stored in the capacitors All these i» appeceble seving

in the cost of construction of the proton synchrotrons. m rSton synchrotrons using A.G. focusing technique are by to

% mSt important for high energies and above a certain energy me the onlv ones practically feasible. There does not s^rn to be y limit to the energy attainable with these machines in the foreseeable

the limit Sing set by radiation^ the protons, which

si: —ri? ch.n.h.1

determined by the am^Htudes of the betatron oscillations in the vertical and radial directions. Let us consider the vertical oscillations first.

It can be shown that the vertical displacement z (j) at the

the unpenurtH^d orbit P“"‘ oscillations are excited is given by (see Eq. 12.1 )

z{s) = Zsmks ...(12.1/-l)

where the wave number jt = 2«/^ is related to the parameters of the

machine as given below

(12.17-2)

V where n is the field-index. ITie amplitude Z is determined by the initial

slope z'(0)=(9z/3*)o ■ ...(12.17-3)

For a given z'(0), the amplitude of the vertical oscillation can be

^”*ShiSai!y^amplitnde of the radial oscillations X is dependent on

the initial angular deviation (0) and is given by

...(12.17-4)

To reduce X. it is thus necessary to make n negative and large in

”"^^**■^^0 requirei^ts discussed above are mutually contradictory. AB..5S rnSI d« of verticd, oscillahoo. would ruuul...

the loss of radial stability and vice-versa. . , The fundamental idea of the AG synchrotrons is to satisfy ‘he ^wo

requirements alternately, rather than nrincinie wc first consider the analogous case of the action oi a

LmbiLtion of two lenses, one convergent and of lieht If we take two thin lenses, one convex and ‘hej>oher conca . baviS i; ^ longtha/, and/, .ospectively, separated by a d.stauee d,

then the combination has the focal length F given by

1 .l+l ±. ...(12.17-5)

F ft fi f\h.

...(12.17-5)

^ Nuclear Physics

I If l/j I = 1/2 I =/ (say) then since/j and/2 have opposite signs, we get

' ' F=f^/d ' : ...(12.17-6)

F is obviously positive. Hence the combination always acts as a Converging lens, independent of the order in which the lenses are placed. For thick optical lenses, the focal lengths are to be measured from the principal planes. Since the latter have different locations for the converging and diverging systems, the location of the focal points depends on the order of the lenses.

It may be noted that the focusing action of the combination of the two lenses as above is due to the fact that the bending of a ray is proportional to its distance from the axis. Hence it is on the average, greater for the converging lens than for the diverging lens.

A pair of quadrupole lenses with opposite gradients acts on a beam of charged particles in the same way as a pair of thick lenses acting on a beam of light, as discussed above. We shall consider the mathematical theory of AG focusing using a pair of electrostatic quadrupoles with opposite gradients in Appendix AIII. The case for AG focusing using magnetic quadrupole lenses is essentially similar, except that the electric field gradient is replaced by the product of the particle velocity and the magnetic field gradient.

An electrostatic quadrupole lens consists of four-symmetrically spaced electrodes, having the shape of rectangular hyperbolas, as shown in Fig. 12.20a. The opposite electrodes have potentials of the same sign. As will be shown in the appendix, such a system focuses the beam along Z in one plane (say ) and has defocusing action in the perpendicular plane (y—z). If two such quadrupole lenses are used successively with the potentials of the electrodes reversed in the second system, there is overall focussing in both planes. ; xxpion* ^

:■ • ’I I

il A I ^

—-- A«»S Z«0 z»0

yz plone

H4 to) ; ^ (b)

Fig. 12.20. (a) Electrostatic quadrupole lens for AG focusing, (b) Focusing action of a two element AG lens.,

The displaicement of the beam from the z-axis in both and y-z planes due to the two lenses is shown in Fig. 12.20h.

In the simplest case, alternate focusing and defocusing in AG synchrotrons is achieved by using a magnet with a large number of sectors having cross sections in the shape of the letter *C\ the alternate

Accelerators of Charged Particles 591

C-sections facing inwards and outwards with the n values « 1 and

rt2» 1 respectively (see Fig. 12.18a, b). This will result in alternately

producing focusing and defocusing actions in the horizontal radial direction and defocusing and focusing actions respectively in the vertical radial direction. The overall effect on the particle trajectory is to produce

a strong focusing action in both directions.

Z{s)

Pig 12.21. Plot of the vertical displacement z as a function of the distance s along

the oibit in an AG synchrotron.

To illustrate the overall focusing effect, we have plotted in Fig. 12.21 the vertical displacement z{s) as a function of s in an AG synchrotron with alternate focusing sectors. In the focusing sectors, the particle is attracted towards the median plane (5-axis) so that the curve is concave towards the 5-axis, the force of attraction being proportional to the distance from the median plane. In the defpeusing sector, it is just the reverse, the particle being repelled from the median plane by a force proportional to die displacement from the latter. The curve is thus convex towards the 5-axis. However, since the particle enters a focusing sector after having been repelled away from the median plane in the previous defocusing sector, the distance of the particle is on the average larger

from the median plane in the focusing sectors than in the defocusing sectors. Hence the force on the particle average larger in the focusing sectors than in the defocusing sectors, resulting in overall preponderance of the focusing

action. Similar considerations

apply, in the case of radial displacement x{s) in the

4;.: ■ .

' closed orbit of reference particle

x-direction, the roles of the sectors with Wj > 0 and n2<0

being reversed. The overall effect is a trajectory oscillating

Fig. 12.22. Oscillations of the trajectory of a non-synchronous particle about the closed trajectory for a synchronous

particle.

;.r. Nuclear Physics

i cAund the instantaneous circular orbit (closed orbit) for a synchronous |liticle (see Fig. 12.22).

The wavelengths of the oscillations in both cases are much shorter than in a constant gradient (CG) synchrotron for which /i(>0) lies between 0 and 1. The complete betatron oscillation in the latter case occurs in more than one revolutions of the particle, while an the AG synchrotron, many betatron oscillations occur in each revolution. The shorter wavelength also produces, as already explained, much smaller amplitude of oscillations for a given initial slope ^'(0) or jc'(O). Since n

can be chosen to be very high (•- 300) reduction by a factor of 10 or more in the aperture over the weak focusing case is easily produced. This means a reduction in the magnet cross-section by a factor of 100 or more and similar reduction in the weight of the iron used for the magnet.

It may be noted that the amplitude of the synchrotron oscillations (see § 12.9) is also much reduced in the AG synchrotron.

Phase oscillations in AG synchrotrons can be treated in a manner similar to that in a conventional synchrotron (see § 12.10 and § 12.13). However, Eq. (12.10-21) has to be replaced by the following equation, which is due to the deviations irom the equilibrium orbit resulting from large momentum errors (see Particle Accelerators by Livingston and Blewett, Ch. XV):

^ Po ...(12.17-7)

where v is the number of betatron oscillation wavelengths per turn. This changes the phase oscillation equation (12.13-11) which now contains

v in place of (1 - n). For energies less than vlVJ), the phase oscillation is

stable if cos % in Eq. (12.13-11) is positive, whild for E > vlVo , the phase

oscillations are stable i^ cos cp is negative. Thus vIVq is th^transition

energy at which % changes discontinuousW from/^ne side of the

accelerating wave to the other. The reason for tljis is that at low energies, (W<vWo) the

momentum compaction prevents the orbit from too much ra/^al excursion, so that conventional synchrotron mechanism works v/ith the particles of too high a momentum (compared to Pq) quickly returning

to the starting point by linear accelerator mechanism. For W>v Wq ,

the particle velocity v « c so that linear accelerator mechanism is no longer strong enough and the machine suddently begins to operate like a conventional synchrotron.

A magnetic gradient lens pair has the same focal properties as a pair of thick optical lenses discussed earlier. In general such a lens is constructed using four symmetrically placed magnetic poles of alternate polarity, as shown in Fig. 12.23. The poles are shaped as rectangular hyperbolas.

Accelerators of Charged Particles

PS. 17 ri A Mir cniadnipole (*) Action of quadtopde iM^t to Hg. 12.23. of a heam of particles in two planes ; (c) Basic

triplet in AG focusing.

Use of quadrupole magnets . For producing strong focusing, most modern A use

separate luadrupoU magnets which constrain the parades to follow

narrow p^. with opposing gradients as sho^»“

^ Jrrr;:? ^ r ^^2 resultant effect is net focusing m bodi planes.

The theory of A.O. focuang is discussed in Appendix AIII.

The the<riaa mdysis the .el fociisleg ^

hit of OM polo loos. follo»ed ^ one Ml ^

Sotrrf-n I. Appeal, Am, aseoMoP

die first clement to be forcussing. ^ When a large number such basic triplets is arrange on

:i ii

Nuclear Physics

in rig. 1Z.Z4 wim a cross sectional diameter of only a few centimetres. In addition to the usual bending magnets (dipole magnets); pairs of quadrupole magnets with opposing gradients are distributed around the ring. Injection and extraction magnets are energized only at the beginning and at the end respectively of an acceleration cycle. The protons are injected into the synchrotron ring at relatively low energy when the dipole bending magnets are

adjusted to prodl^e a corres- pjg \22A. Horizontal section of the race-track ponding weak field. After accelerator tube in an AG proton injection, the magnetic field is synchiotion.

steadily increased to keep the particles in track as they gain energy while passing through the rf carity repeatedly. As already expained in § 12.16, the frequency of the rf field must be increased continuously to maintain the resonance condition. Once the desired energy is reached, the protons are allowed to hit the target in the ring (usually a jet of gas) or are extracted from the ring and directed to the experimental area, sometime more than a kilometre away.

In Table 12.1 we list some of the largest proton synchrotrons either in operations or under construction. '

strong focusing quadrupole rf cavity magnets \ /

dipole ^bending

magnets

rf oscillators

MmputersUnf97'»«: —- field

control

Injec tion 'magnets

extraction magnets

Table 12.1

^ Date cf -

commissioning Approximate radms (m)

Proton synchrotron m energy

(C5cv)

Proton syndiiDtroii * / : 1959 . .: .. W 28

(PS)it Cetn, Switzeriand.^' ; : . x .

AO Bynduotron, Brodc- 1%1 125 33 # haven, Upton. New Yofk.

lllliFi Serpukov,'Russia. • ^l%7 \ 235 76

MWiiriij, ftmiiJIib* ' 1000 400 MnVII, IIHnoto.

Niiiwi pmnm lyni'bnitnM 1976.' ' ilOO 400

— 1 - * ■ _ ^ ‘ - i» • .

levMlKiii. IVtinllsIi .920 1000

, __ J99«(D 3060 3000 ---------:_

* i*!l^ * Mslnring Puinllah aiei now used mainly as injectors for the nri| swi lovalnm rsi|iei»iivnly

Accelerators of Charged Particles 595

The main limitation of the energy that can be attained in a proton synchrotron is due to the limit in the field strength of the dipole magnets used to constrain the particles within the accelerator ring. Magnets have been designed to produce field strengths upto 9 T. These are wound with coils of superconducting alloys (usually Nb-Ti-or .Nb3Sn) and must

be cooled to 4.5 K by liquid helium. Magnets designed for mass production go upto 6.5 T when superconducting coils are used.

The enormity of the problems tackled in the construction of these machines can be gauged from the fact that the first superconducting proton synchrotron, the Fermilab tevatron, uses 774 superconducting dipole magnets, each m long, producing a maximum field strength of 4.4 T each. Highly reliable refrigeration systems are required for the large number of superconducting magnets. Appropriate couplings and insulators to withstand the effects of very low temperatures are needed. The length of the tevatron magnet decreases by about 2 cm when it is cooled to the operating temperature. The most severe problems are associated with the loss of superconductivity of the magnet windings due to the appearance of local hot spot, resulting from the tiny movement of the superconducting wires, which may result in various kinds of damages. Appropriate steps must be taken to prevent such eventualities.

Alternating gradient electron synchrotron :

Electron-synchrotrons can also be built using the AG twhnique. One such machine at Massachusetts Institute of Technology in the U.S.A. yields a 6 GeV electron beam. A similar machine is also in operation at the DESY (Deutsches Electronen Synchrotron) at Hamburg in Germany

(30 GeV). As already stated in §12.13, the main limitation for the electron

synchrotron (as in any other cyclic accelerators for electrons) is the radiation energy loss. The loss per turn is given by (see §12.11)

A W'„v=8.85xlO*(W*//f)eV/rev ...(12.17-8)

where W is the total electron energy in GeV and R is the radius of the orbit in metres. The radiation-loss can be diminished to some extent by

increasing the radius ^^owever, since A IV ^ it is not possible to

go beyond about 50 GeV. * tr. The radiation eirtitted during the proross described above is known

as the synchrotron radiation and forms an intense source of e.ni. radiation from infra-red to X-ray region.

The design of the electron synchrotron is simpler than for the proton synchrotron, since the velocity of the electron does not change much above about 1 MeV, being close to the velocity of light (v = c). Some amount of pre-acceleration either by a linac injector or by initial betatron acce¬ leration in the synchrotron field brings the electron velocity close to c.

The "problems associated with the phase transition is easy to avoid in electron synchrotron. As stated above, this occurs at an energy where v

is the number of betatron oscillation wavelengths per turn and Wo = rttfp is

Nuclear Physics

the rest-energy of the electron. For most AG_ electron synchrotrons, V ~ 10 so that the transition energy is about 5 Mev, which is lower than the injection energy (by linacs) in tfiese machines. '

Heavy-ion synchrotron :

These machines are similar to proton synchrotrons. In many cases, the same facility is used for acceleration of both protons and heavy ions, such as the Princeton and Berkeley synchrotrons. The Berkeley synchrotron is linked to a high intensity linear accelerator, known as the Super HILAC (see § 12.19) which serves as one of the injectors. The combined facility can produce beams of ions as heavy as uranium stripped of some or all of its electrons.

The principal heavy ion synchrotrons are listed in Table 12.2. When working with heavy ion accelerators, it is the energy per

.nucleon which is important. The behaviour of the ions in the

magnetic field depends on q/A. Since the ions with lower q/A require higher fields, it is desirable to have high q/A beams, which is an important determining factor in choosing the paramaters of the machine.

Table 12.2

Facility Maximum beam energy

GeV/nucleon Maximum ion mass

:#

Satume II, Saclay. France. 12 40

Bevalac, Berkeley, California, 2.1 238 U.S.A.,

Synchrophasotron.Dubna, Russia

4.6 20

AGS, Brook haven, Upton, New i

15.6 /32 York, U.S.A.

SPS, Cfem, Switzerland. 200 i 16

The ion sources^re usually much simpler in heavy ion machines.

Simple sources of ionized hydrogen or H~ ions are used to produo^ pure beams of any atomic ions with very high intensity and in the highest possible charge state

After pre-acceleration in a Cockroft-Walton or in a Van de Graaff generator, followed by a series of linacs, the beam passes through one or more strippers (carbon foils or streams of liquid or vapour) in which the particles lose electrons and are elevated to higher charge states. The beam intensity is much diminished in the process.

After selecting the proper charge state of the particles, they are allowed to enter the synchrotron in which the acceleration cycle is more or less the same as in a proton synchrotron. Very good vacuum

(10 to 11 torr) must be maintained in the accelerator to minimize the

/

597 Accelerators of Charged Particles

effect of the change of charge states of the particles due to collisions with

the residual gas molecules.

12.18 Storage rings A storage ring consists of a ring shaped vacuum chamber, placed

between the pole-faces of a ring of magnets, within which a beam of very high energy charged particles can be stored for a long time. These can thL be made to collide head-on with the particles in another similarly stored beam and the resulting interaction processes studied in detail.

The motivation behind the use of this technique in high energy collision studies 'arises from the following kinematical considerations.

When a particle of rest-mass Mq travelling with relativitic^velocity

V, = p, c in the laboratory frame (L-frame) collides with an identical

particle at rest in the L-frame, only a very small fraction of the kinetic energy of the incident particle is available for producing nuclear reactions. If Tc and T,^ represent the kinetic energies of the particles in the centre of

mass frame (C-frame) and L-frame respectively, then it can be shown that

(see Ch. XVIII) : rj, = 2rc+7|/2Wo ...(12.18-1)

where 'Wq = MqC^ is the rest-energy of the particles. Tq is equal to the

actual amount of energy available for the reactions.

On the other hand, when two identical particles, travelling with equal and opposite velocities (relativistic), collide head-on, as in the storap ring experiment, the entire energy which is equal to the suin of the kinetic ."„e,gie» of Ihe two panioles (r,= 2r) is available tor produeteg

reactions. As an example, two proton beams, each of kinetic energy r=5MoC^ = 4.69GeV will by, head-on collision, provide a total pt

10 M = 9.38 GeV for reaction, in place of a 65.66 GeV proton beam

required, if the target proton is at rest (see Ch. XVHI). Similarly two electron beams, each of energy 500 MeV will provide tht same ainount of energy (1 would be provided by a 10 GeV electron beam

striking a stationary electron. Storage rings are useful only at relativistic energies. In the

non-relativistic case Tc increases linearly with and hence most of e

energy of the incident particle in the L-system is available for producing

reaction. ^ l As seen above, the most important advantage of using beams of

charged* particles in storage rings is to produce reactions with the hig centre of mass energy available. However, the reaction rate is usually much lower than in the fixed target experiments. While the beam density in the storage rings is of the order of 10^“ particles/m . the °f nuclei or electrons in the fixed target is of the order of 10 per m . In the first case, the number of events produced by collision (reaction rate) i

V.

m Nuclear Physics

proportional to the product of the particle densities which is 10“° in the present case. On the ofter hand, if a beam of particles with the beam current

1 |iA ( ~ 10 particles incident per second), the reaction rate is proportional to the product of the beam intensity and the number of target nuclei per

in the target (10^® per m^) which is about 10^^ in the present case.

The performance of the storage ring is measured by the luminosity L given by

L = R/c=N^N2f/bS ...(12.18-2)

where R is the reaction rate and a is the reaction cross section. and N2

are the numbers of particles in the two beams ; / is the number of stored particle-revolutions per second. S is the beam cross sectional area at the point of collision and b is the number of bunches in each beam. L is of

the order of 10^^ - 10^^ m“^ . s“^ L can be increased by reducing the beam cross sectional area (5) which can be done by focusing the beams at the colliding point.

Electron positron-storage rings :

An electron-positron e~-\-e storage ring comprises a doughnut shaped vacuum chamber located between the poles of a ring of magnets. The beams of electrons and positrons rotate in opposite directions within the chamber and are stored for several hours. They are then made to collide head-on with each other about 10^ times per second within the straight sections of the chamber, which are surrounded by detectors (see Fig. 12.25). In an alternate arrangement, there are two separate but intersecting rings within which are stored the two types of colliding particles, which are made to collide at the intersection faints.

•Such collisions give deep insight into the fundamental structure of

matter. The e'^ e storage rings at the Stanford Positron-Electron Accelerating Ring (SPEAR) at the Stanford Linear Accelerator Centre (SLAC) in the U.S.A. and with DORIS at the Deutsches Electror^ SynoMIrotron (DESY) in Hamburg, Germany, provide stored beams with energies upto 5 GeV for such studies. Another facility ^DESY, known as the --v. Positron Electron Tandem Ring Accelerator tf

(PETRA), started operation in 1978 at which the / * ^ highest energy (45 GeV centre of mass energy) / interaction \ e^ e~ collision studies can be made. The Positron T ^ Points- .— I Electron-Project (PEP) at SLAG, which became I /

operational in 1980, provides 30 GeV e^ e~ \ / beams. The TRISTAN storage ring at REK in ^ Japan is designed for 60-70 GeV energy. • -/

The LEP storage ring at Cern in eA Switzerland will have circumference of 27 km injection

and' will store beams upto 180 GeV Rg. 12.25. Schematic diagram energy. ^

interaction ^ points---

• «

injection

/

111

Accelerators of Charged Particles

Some of the problems encountered in the operation of the storage rings include the beam instabilities and the emission of synchrotron radiations. Beam instabilities are caused when one of the stored beams encounters the second beam of much higher intensity. This causes defocusing of the first beam due to the e.m. field of the latter with consequent diminution in the reaction rate. Single beam instabilities are also observed, which are due to the electromagnetic interactions between the particles in the beam or by fields due to the induced currents m the chamber walls. Best results are obtained with two beams of equal intensities.

The above limitation applies equally to e* e~ and pp (or pp) storage

rings. However, forV c" storage rings, the problem of radiation-loss at ultra-relativistic energies giving rise to synchrotron-radiation (see 8 12.11) becomes prohibitively great. The radiation loss has to be counteracted by additional power supply from the rf system. At very high energies, it is a question of balance between the increased cost due to increasing sizes of the magnets and building and the decreasing cost of the rf system with increasing R which sets the limit.

The emission of synchrotron-radiation has positive aspect also, since

it dampens the oscillation-amplitudes of and e" about the equilibrium orbit. It also produces transverse polarization of the stored beams.

The magnets used for the storage rings serve the du.al purpose of keeping the particles in orbits and of focusing them at the interaction point which increases the luminosity. There are straight sections in the vacuum chamber (as in AG synchrotrons) where the rf cavities and interaction

points itte located (see Fig. 12.1^- ^

Fig. 12.26. Intersecting storage rings at Cern with the injector.

Nuclear Physics

The electrons are injected from a high energy machine, such as a ^ linac or a synchrotron, at an energy close to the desired collision energy.

The positrons are produced in a target bombarded, by high energy electrons and then accelerated. Accumulation of particles in the beam to

the desired level takes several seconds for and several minutes for e^.

The stored beams consist of bunches (^ 2 to 5 cm long) since rf power is fed continuously. For best luminosity, the number of bunches (b) in Eq. (12.18-2) should be small and about half the number of interaction regions.

Proton and pp storage rings :

Proton storage rings consist of two annular vacuum chambers in which high energy protons can be stored and made to collide head-on.

There is only one proton ring which has been constructed at Cern. This is known as the I^ intefsecting storage rings. There are two nearly circular intersecting"rings, as shown in Fig. 12.26. Each ring has 132 magnets, similar to those used in proton synchrotrons. Circumference of each ring is about 1 km. Protons from the 26 GeV proton synchrotron (PS) are injected into the rings. Protons are accumulated by stacking procedure till a current of 50 A builds up. The whole stack of protons are then further accelerated to 32 GeV. There are eight intersection points where the collisions occur. The rings have also been used for the study of

_ r

pp, dd and a a collisions. The ISR has now been decommissioned. . The program for constructing a proton-antiproton {pp) collider

system, using the 400 GeV super-proton synchrotron (SPS) at Cern has been successfully executed. The antiprotohs are produced in collisions of protons from the Cern PS with nuclei. Because of their large energy spread, they are first “cooled” to reduce the beam size slifficientiy, so that it is usable in the collider. Elaborate feed-back loops within the collider damp the average motion of the particles. It is possib}p to use these elaborate and difficult techniques to accumulate and qool about 10^^ antiprotons which are injected into the PS to b^cceleiwed to if) GeV and then injected into the SPS. A biinch of protons is also similarly injected info the SPS aftjj^ initial acceleration in the PS. The pandp beams are accelerated to 270 GeV and are made to collide within tjj^

straight section of the collider ring. A luminosity of 10^^ cm”^ s~* has been achieved. The most important result obtained from these pp collision

studies has been the discovery of and Z® intermediate bosons in 1982 and 1983 respectively (see Ch. XVIII).

At the Fermi^National Accelerator Laboratory in the U.S.A. an even more ambitious pp collider System has been constructed. It is designed to achieve an energy of 1 TeV (10^ GeV) for each beam and a luminosity of 10^^^ m“^ s"**.

* The superconducting super-collider (SPC) at the Brookhaven National Laboratory in the U.S.A. is visualized as a pp storage ring with energies upto 20 TeV in each beam. Very high luminosity is expected.

Accelerators of Charged Particles ou i

The circumference will be at least 65 km* or may be even double that figure. The problem confronting the designers is how to construct a magnet system cheap enough to make the construction of the machine

financially feasible.

12.19 Linear accelerator

Linear jKxelerators (also called linac) as the name implies, accelerate charged particles along a strmght line in multiple steps by an oscillating electric field. In ail linear accelerators, the rf field includes an e.m. wave whose phase-velocity la^ equal to the velocity of the accelerated particles. Since this latter velocity increases as the particle travels through frie accelerator, in the case of atomic ions the phase-velocity of the travelling wave roust also increase with distance along the accelerating system. In the case of electrons at extreme relativistic energies, v « c — constant, so that the phase-velocity becomes equal to the velocity of light. For; this reason, the design of the linear accelerators in this case is relatively

simpler. The first working linear accelerator was built by R. Wideroe in 1928

which is regarded as the direct ancestor of all resonance accelerators. It

used only fliiee coaxial cylindrical electrode and could accelerate and

Na^ ions to twice the energy available for a single traversal of the field. In 1931, E.O. Lawrene and D.H. Sloan at the University of California, Berkeley, built an accelerator using 10 accelerating electrodes to produce 1.25 MeV mmury ions. Since then a large number of linear accelerators, both for atomic ions and electrons have been built in different parts of the

world- . * r We fiist consider flic principle of working of a linear accelerator for

ions, such as protons. As shown in Fig. 12.27. the linew accelerator consists of a scries of coaxial cylindrical dnft-tubes along the axis of which flie charged particles travel. One set of alternate electrodes is connected to one terminal of the if supply system wiule the other set of alternate electrodes is connected to ^flic other terminal. In a linear accelerator for ionsrIBS' successive electrode tubes are of gradually

increasing lengths, p

fig. 1227. Proton lin^ accelerator.

Nuclear Physics

Suppose the rf voltage between.the first and second drift-tube is near its maximum negatiye v^ue when a positive ion^ crosses the gap between them. As a result, the ions gain an amount of energy qV where V is the amplitude of the if.yoltage and q is the charge of the ion. As the ions travel down thd second drift-tube, they do not gain any energy because of the screening action of the tube. If the frequency of the rf voltage is such that its phase changes by when the ions emerge from the second drift-tube into the gap between the second and third tubes, they gain an additional amount of energy qV so that their energy becomes 2qV, as they enter the third drift-tube. Because of the increased energy, the ions travel though the third tube with a higher constant velocity. The phase of Ae accelerating voltage again changes by n when the ions emerge into the gap between the third and the fourth drift-tubes, so that they gain an ener^ qV again as they cross the gap and the total energy becomes 3qV. In this way, the process of energy gain by qV continues at each successive gap-crossing and the ions finally emerge with an energy nqV after traversing the /ith gap. The velocity of the ions at this point is (assuming

non-relativistic relationship) = (2nqV/M)^^^ where M is the ion mass. Here we have neglected the initial energy of the ions as they enter the first stage of the accelerator.

Since the ions travel with gradually increasingfeonstant) velocity through the successive drift-tubes, it is necessary to increase the lengths of the successive tubes to maintain resonance condition which is such that the time taken by the ion to traverse through the drift-tube at any stage must be equal to the half the time period (7^ of the rf accelerating voltage.

2 v„ 2c y(12.19-l)

This gives Ji 2c 2c M

...(12.19-2) ^ V y-.

In their first linear accelerator, Lawrence and Sloan used rf voltgge of amplitude 42 kV at a fi*^uency of 10 MHz. In 1947, L.W. Alvare^'at the University of California, Berkeley, built a proton linear accelerator using 47 drift-tubes at an rf frequency of 202.5 MHz. The protons were initially accelerated to 4 MeV by a Van de Graaff^generator and then iInjected Into the first drift-tube of the accelerator. The final energy of the proton beam was 32 McV. The drift-tubes were housed within a 40' long vacuum chamber of diameter 4' which acted as the resonant cavity of the radio* frequency waves. The standing waves geh^atc^ within this reson'ant cavity prf>duced the electric field E along the axis of the drift-tubes (see idle laler). This linear accelerator gave an average proton beam current of 0.4 it A shout I mm in diameter with an energy-spread of only 0.3%. This ihilply homogeneous beam is more intense than any external t-yidojioii tieain. In f9$S« the machine was dismantled and moved to the UiiivNlsiy .Southern California.

^'Accelerators of Charged Particles 603

The development of linear accelerators received a great fillip with the development of radars during the second World War.

In addition to the work at Berkeley discussed above, a parallel ^gram of building an electron linear accelerator at Stanford University

• in the U.S.A. was starte'd by W.W Hansen to be continued after his death . by F T. Ginzton, M. Chodrow and others. The development of the

klystron amplifier greatly facilitated the program. It delivered between 10 to 20 megawatts power at 2855 MHz. The acclerating system was an iris-loaded wave-guide (see later) powered by a 3000 MHz klystron. The first machine (Mark®) was 12' in length and gave 35 MeV electron beam. This was the prototype of the much larger 1 GeV machine subsequently built by W.K.H. Panofsky and R. Hofstadter (Mark III) in 1960. It consists essentially of thirty 10 ft Mark II machines arranged in a line. The total length of the machine is 300 ft, buried underground in a tunnel

of concrete shielding. _ The largest machine of the above type, known as the SLAG, is 2

miles long, using 240 klystrons yielding a maximum energy of 20 GeV. The ultimate aim is to push up the electron energy to 45 GeV. using 960

klystrons. Superconducting linear accelerators have been built both for

electrons and heavy ions. The advantages of the linear accelerators include a relatively high

beam current and the simplicity of injection and extraction of the beam. Another great advantage is that there is practically no radiation-loss, even for the highest energy electron beam available, since they travel in straight line. This is in sharp contrast to the cyclic accelerators (betatrons and synchrotrons) in which the rMiation-loss becomes prohibiUvely large above several hundred. MeV energy for electrons.

p Field direction ot ipn ero.seing

Fig 12 28 Radial focusing in a linear accelerator, (a) Opsn-ended dnft-tubes, (b) Focusing by the use of wire grid at the entrance end; (c) Use of

quadnipole magnetic lens. ____

Nuclear Physics

r Linear accelerators, now a days, are mostly used for electron acceleration. Proton linear accelerators, in the energy range of 10 to 50 MeV have found useful application as injectors for high energy machines, such as proton synchrotrons.

Focusing in linear accelerators :

Linear accelerators give very well collimated beam with relatively small energy spread. Because of the phase-stability in their operation, the beam current can be made significantly large (see later).

Let us first consider radial focusing of the ion-beam in the lion-relativistic case. Fig. 12.28^ shows the lines of force of the accelerating electric field at the gap between two successive drift-tubes. As can be seen from the figure, an ion off the axis is acted upon by (i) m axially directed field component, which accelerates the ion and (ii) by a radial component of the field. The radial component acts inwards in the first half (region ab) of the gap, tending to push the ion towards the axis. In the second half (region be) of the gap, the radial component acts outwards, tending to push the ion away from the axis. Thus the radial component has a focusing action in the first half and a defocusing effect in the second half. However, since the ions move fasto* in the second half than in the first, they spend less time in this region (be) so that there is a net focusing effect due to the radial field.

It can be shown that the radial focusing is effective if the ions cross the gap during the quarter cycle, in which the accelerating potential V decreases from the maximum () towards zero, so that its phase lies

t*'

between 7c/2 and n. On the other hand, if the gap is crossed during the quarter cycle in which increases ftom 0 towards (^hase between 0

and K /2), there may not be any net radial foebsing. Radial focusing becomes less effective as the speed of the ion increases.

It is found that the conditions fi phase-focusing (see below) which is field lies between n /2 tojx.

To circumvent this problem in the 32 MeV proton linear acceleri^r at Berkeley, a metallic grid has been placed across the entrance cna of each drift-tube. This makes the lines of force of the electric field converging all the way from the exit-end of a tube to the entrance end I • ♦ of the next tube, as shown in Fig. 12.28i. Radial\focusing now occurs at any time during the accelerating half cycle of the rf field, i.c., for the phase of the field between 0 to n. This )automatically covers the range of phase focusing which can therefore be satisfied at the same time. Its disadvantage is that some beam is lost to the grid. “

sing go against phase of the rf

0 0.2 0.4 O.S O.t 1.0

Time Pig. 12.29. Phase focusing in a linear

accdcfatcr.

Accelerators of Charged Particles ^^5

A second method of achieving radial focusing is to use a quadrupole

masnet within each drift-tube which compensates the defocusing of the ion*” beams during the gap-crossing (see Fig. 12.28c ). Its mam disadvantage is that the high power magnets inside the vacuum chamber

require elaborate cooling.

Phase-focusing in the linear accelerators :

The problem of phase-focusing in a linear accelerator can be

understood by referring to Fig. 12.29. Suppose the accelerator is so designed as to give optimum

acceleration at the potential Vo, slightly less than V^ = V„. The potential

attains the value Vo twice during each accelerating half-cycle, represented

by the points A and A'. The first of these is in the phase-range 0 to 71/2 while the second is in the phase-range n/2 to n. Suppose there is perfect resonance between an ion crossing tne gap at the phase represented by A and the accelerating potential at the successive gaps.

Consider an ion arriving too early at the gap at the phase represented by the point B. This can happen if the ion moves faster than the synchronous particle mentioned above as it amves at the gap It is then accelerated less than the S.P. during the gap-crossing and hence falls closer in step with the S.P. as it enters the next drift-tube. Similarly an ion arriving a little too late at the phase represented by C (due to its lower velocity than the S.P.) moves faster than the S.P. during gap-crossing and

catches up with the latter as it enters the nMt tube. Thus the ions crossing the gap near about the phase ^int A

0 and 71/2) are bunched together in phase as they enter the next dnft-tube resulting in phase-focusing. This is clearly against the radial focusing condition without the special correcting devices discussed above.

There will be no phase-focusing if the ions cross flie gap at phases

near about the point A' (between 7t ./2 and 7t). The mathematical theory of phase-focusing in linear accelerators can

be developed, startingTrom Eqs. (12.ip-8) and (12.10-9), assuming the non-rf field E' = 0. ''rtise equations are'revmtten below

, (OgAX . ,, ,,

Aq> = 9-‘Po = -“V“ :>)

p = ^£sin (% +Aip) ...(12.19-4)

Writing TJo = Po for ‘*’0 synchronous particles, it is possible to obtain the following relationship between Ap and A9 , using the relations

(12-10-22) and (12.10-23) :

Ap=_^Lo(1+t)o)4a‘P + %A9[ ...(12.19-5) <0 t ^

For the S.P., Po = "H. % = 9 £« <Po ^h^re £„ is assumed constant. We \

then get

Nuclear Physics

% = q Eq t sin <Po

moc

phase equation for the linear accelerator is obtained by

Livingston and J.R Blewett)f ^ Accelerators by M.S.

moC ...(12.19-6)

Eq. (12.19-6) IS valid for both relativistic and non-relativistic cases If cos (Po IS positive It gives damped oscillations.

If we introduce the independent variable a *

« = = sin moC ...(12.19-7)

we get the following transformed equation for the phase oscillations : I 1 ’ *

^(11oAq))+ £-1 {t,„A9) = 0 ...(12.19-1

...(12.19-9)

= U ...(12.19-8)

where AT is a dimensionless constant given by

wmocco^

...(12.19-9)

If > 0, stable oscillations are maintained .till tio = A"; A 9 will be

damped in oscillations. K will be positive if cos9>0. Since sin 90 must

be positive for acceleration to take place, tpj, mpst lie between 0 and Jt/2.

/ .

or.electton accelerators K~2 which shows that ^ will execute oscillations if or the electron kin^c elQy is about

is quite Jow. Thus the electron linear accelerators mostly operafe in the regton (relativistic and ultra-relativistic) whe»e no phase-oscillations "occur.

For proton linear accelerators, AT = 1 O’. But for protons tIq " 10^

only when the proton kinetic energy reaches the value of lO’ CeV. Since the proton-accelerators operate in the low energy region (<50MeV), they show stable phase oscillations.

n ^ A* X I . ^ v«c over most of the ange of acceleration. In this case the phase-velocity v of the

" throughout the whole inachine. The electrons are then bound to the accelerating wave and continue to be accelerated indefinitely without falling out of step with the accelerating field.

: :Mr

...(12.9-10)

P = ■...(12.19-11)

Accelerators of Charged Particles

Power requirement of a linear accelerator .

Linear accelerators,require very-high power for their operation This makes these machines very costly..^The,required pqwer P for a linear

accelerator of length I delivering a proton beam of final kinetic energy

is given by

l.lxlO~^r^VX ...(12.9-10) I

Here X is the free space wavelength of the accelerating field. A more fJLh. .aldng Into account lotacs in all auxil.an^ smtctures ,a

r cf<X '...(12.19-11) I

Where C is a constant ; C = 2.5x10-* for proton accelerators and

C» 10"^ for electron accelerators.

The power necessary for a 50 MeV proton linear^acpelerator of

length 25 m with 1.5 m comes oiit to be 1.3 MW which shows the enermous power requirement in^these machines.

All linear accelerators are operated in the pulsed «"?de becau^ the very high power requirement. The duty cycle is so adjusted that actual duraLn at the full poWee averages to a reasonable value in terms

of cost and cooling requirements. ,

Accelerating system for the linear accelerators :

In the case of the electron linear accelerators, the consists of rhetal discs with small cavities, (ms-loaded cavities) operating in the TM^jo mode, modified slightly because the end wal s o

the cavities are not solid, but have openings into the adj^nt cavities (see Fig 12 30). The apertures allow the passage of the electron beam ^ provi^'coupling between the cavities,so that many cavities cm te excit^ from the sle sou«SK%he aperture ^ determines the shift hbtween the successive cavities. These phase changes determ me the

vTtoC,. Since V - c over uuuU of Ihe tagth of Sie eleei™.

aceelerator. the effective phase velocity is also ' ““f injection and where v is usually much lower (v = c/2) for an injection energy of 80 keV. The phase velocity of the wave has to be adjusted

accordingly in this region.

50 K„ n 30. Iris-loaded cavity as accelerating system in an electron liner acccleralor.

r Nuclear Physics

There is some defocusing of the beam due to the transverse field component at lower energies (upto a few Mev) which is usually compensated by using dc magnetic focusing in the initial stages.

Several parameters, such as the wavelength A. of the accelerating signal, the thickness and spacing of the irises, the hole radius and the radius of the wave guide must be chosen properly for the guide to function effectively. Once these choices have been made, the dimensions must be maintained very precisely, so that the phase velocity does not deviate from the chosen value. Otherwise the electrons will drift away from the acceleration phase and will not attain the desired energy. Mechanical tolerances of the dimensions of the parameters are usually of the order of 0.0002". Temperature must be controlled very accurately to avoid change of these dimensions.

The iris-loaded cavities are unsuitable for proton or heavy ion acceleration. The standard ion accelerator is a single standing wave cavity operating in the mode. To protect the ion beam from the

decelerating effect during the many periods of the rf field that it takes to reach the exit end, a series of drift-tubes of gradually increasing lengths (see Eq. 12.19-2) is us^.

Heavy ion linear accelerators :

Heavy ions accelerated to high energies are nowadays in routine use for production of various types of nuclear reactions. Fot these studies, a linear accelerator is a very useful source, apart from tandem Van de Graaff generators discussed earlier. ^

We describe below two identical machines built at Yale University and at Berkeley Lawrence Laboratory under a joint program (1955).

Various types of heavy ions, such as He*", C*", 0*"^, Ne*" etc. are

produced in special ion sources (see § 12.6) ^d pre^tccelerated in a Cockroft-Walton generator; These are then injected into a 15^ drift-tube type accelerator and acc^l^ated to the energy of 1 MeV per nucleon. The ion beam then passes through an electron-stripper (gas jet or carbon fj^l)

in which the ions lose more electrons, so that their specific charge

iq/M) is doubled. After stripping, the rf accelerating field doubles the acceleration. The beam now enters a post-stripper linear accelerator, 9ff in length, from which it emerges with 10 MeV per nucleon energy. This machine is known as the HILAC.

At Dubna in Russia, a heavy ion linear accelerator for ions upto argon was built in 1960. Another instrument capable of accelerating xenon ions upto 1 GeV energy was commissioned there in 1972.

In some laboratories, such as the Unilac in Dramstadt in Germany, post-stripper acceleration is achieved by the use of a number of individually controlled single cell resonant cavities, which may increase the energy upto 13 MeV per nucleon.

Accelerators of Charged Particles 609

Development of superconducting structure for heavy ions has been

in progress in some laboratories as also of spiral splitting and quarter

wave structures. Work on radio frequency quadrupole structure for heavy

L acceleration is also in progress. These structures are P/oposed as

methods of increasing the capability of tandem accelerators by as post-accelerators. The Atlas project at the Argonne NaUonal

Laboratory in the U.S.A. uses a super conducting niobium split-nng

resonator to attain energies upto 30 MeV/nucleon. n w Another machine at the University of New York at Stony Brook

tSUNYLAC) proposed to use super-conducting lead-plated copper

split-ring resonators following a tandem Van de Graaff to achieve energy

At Berkeley Lawrence Laboratory, the HILAC was upgraded to accelerate ions with A >40 by a modification of the injector. This so-called super-HILAC accelerates ions upto 8.5 MeV/nucleon. A trans er

line between the super-HILAC and the bevatron (proton synchro*™") accelerates uranium ions to 1 GeV/nucleon. (See § 12.16 an § • )•

niis facility, known as the BEVALAC may be further upgraded by addine two<-rings of 6 T superconducting magnets following the

super-HILAC. TOs projeeL kuo.n us *e “ increase the beam current by a factor between 100 to 1000. (See also

§ 12.17)

12,2» Acederators in Ind^a §§

(di Tbc first attempt to build a charged particle accelerator in India was started during the second World War at the Palit Laboratpry of Physira^^ Calcutta Univeisity. It w^ a 37" fixed frequency cyclotron milled after the similar machine at Berkeley. After some iniUal piSSL due to the technolo^cal limitations obtaining m

diat time, it was finally commissioned in ,00 proton beam which was used for rsp^troscopy for A - 80 to m Siort-Kved nuclear leveLstudies, and biomedical investigaupns. Itas now

dectHumissioned. ' . .i ‘ t.«oV...n (W A 66 cm fixed frequency cyclotron with variable energy has been

in operation at th& Physics Department cf P_unjab Un.ver«^^^^

Chandigarh since 1971. It gives a proton beam of 5 MeV- It has b^^ obtained as a gift from the University of Rochester m the .y.S.A. where

it was in operation in 1953-64. . fcl The biggest accelerator in India, which is in operation, is the

'224 cl redor!|Lused (AVF type) cyclotron at the Variable Energy

Cyclotron Centre (VECC) at Bidhan Nagar near Calcutta. J* to^deliver proton beam of 6 to 60 MeV, deuteron beam of 12 *« jS MeV and a- paiticle beam of 24 to 130 MeV. At present it gives a-parUcles of

80 MeV energy. The beam current is a few microamperes.

§§ Data supplied by N^K. Mukherjee of V.E.C.C. Bidhan Nagar. Calcutta.

rr ^ , Nuclear Physics

The following .are some- of its charactristics. The magnet, which ^eighs 262 tonne,^ produces the highest field 2.1'T. Three spiral ridges are used for azimuthal variation of the field. The rf frequency can be changed from 5.5 to 16.5 MHz. The magnetic field is increased outwards with the help of seventeen auxiliary coils, nr

{d) A K-14 tandem type pelletron has been set up at the Tata Institute of Fundamental Research in Bombay (1986). According to

definition the energy available h where q is the charge and A is the mass number of the acceletrated ion. This machine has been purchased from High Voltage Corporation, U:S.A. It gives 28 MeV protons, 42 MeV a-particles and correspondingly higher energies for heavier ions. It is used for heavy ion reaction studies. A super-conducting linac booster is expected to boost the energy considerably.

(e) An almost similar pelletron (K-15) giving the energy of 8 to 30 MeV per nucleon, obtained from the same source, has been set up (1990) at the Nuclear Science Centre at Delhi under the aegis of the University Grants Commission (UGC) of India. It is also used for heavy ion reaction; studies and caters to the University community in India.

(/) It is proposed to build a K-50 cyclotron, using super-conducting magnet at the V.E.C.C., Bidhan Nagar near Calcutta. Design studies are in progress. It will be completety indigenously built.

(g) A number of electrostatic accelerators of lower energy are in operation in different laboratories in India. These include the 5.5 MeV Van de Graaff accelerator at T.I.F.R. Bombay, a 2 MeV Tandem Van de Graaff at B.A.R.C., Bombay (and similar machines/at I.G.C.A.R., Kalpakkam, Tamil Nadu, Institute of Physics, Bhuvneshwar, Orissa and at I.I.T., Kanpur) and a number of Cockroft-Waltoh generators. The latter includes the 1 MeV rhachiiie at the University of Calicut ir/Kerala, a 400 kV machine at T.I.F.R., Bombay and a :^0 kV^achine at Bose Institute, Calcutta. /

(h) Some electron federating machines are also in operation in different parts of India, rftainly for biomedical purposes. These inclucle a 42 MeV betatron at Vellore Christian Medical Hospital in Tamil Nadii, a 20 MeV electron linear accelerator at A.I.I.M.S. New Delhi, a 12 MeV linear accelerator in the Department of Radiotherapy at Srinagar, a 10 MeV Ijnear af elerator at the Tata Memorial Hospital, Bombay. An 8 MeV microtron accelerator for electrons is in operation in the Physics Department of the University of Pune for physics research.

Besides those listed above, a number of lower energy accelerators, both for positive ions and for electrons, are in operation in different laboratories.

(0 A synchrotron radiation facility is being built at the Centre of Advanced Technology (CAT) at Indore. This will comprise a 700 MeV electron synchrotron with a 20 MeV electron beam from a microtron as the injector. Electrons accelerated to 450 MeV will then be injected into

Accelerators of Charged Particles oil

a Storage Ring (Indus-I). The synchrotron radiation coming out of this storage ring will be used for a wide spectrum of studies, including surface physics. X-ray crystallography etc. In the final stage, a larger storage ring (Indus II) will be built for a maximum energy of 2 GeV. For further details, see Indian Synchrotron Radiation facility by D.D. Bhowalkar, R.V. Nandekar, S.S. Rammurthy and G. Singh.

References

1. Particle Accelerators by M.S. Livingston and J.P. Blewett, McGraw Hill

Book Co. (1962).^ 2. Principles of Particle Accelerators by E. Perisco, E. Ferrari and S.Z. Segre,

W.A. Benjamin Inc. (1968). 3. Nuclei and Particles by E. Segre. 2nd Ed., W.A. Benjamin Inc. (1977).

4. D.W. Kerst, Phys. Rev. 58. 841 (1940). 5. E.M. McMillan, Phys. Rev. 68, 143 (1945).

6. V. Veksler, J. Phys. (U.S.S.R.) 9, 153 (1945). 7. W.M. Brobeck. E.O. Lawrence, K.R. Mackenzie. E.M. McMillian, R.

Serber, D.C. Sewell, K.M. Simpson and R.L. Thornton, Phys. Rev. 71r449

(1947). 8. E.O. Lawrence and M.S. Livingston, Phys. Rev. 37, 1707 (1931). 9. J.D. Cockroft and E.T.S. Walton, Proc. Roy Soc. (London), A 136, 619

(1932). 10. R.J. Van de Graaff. Phys. Rev. 38, 1919A (1931).

11. R. Wideroe, Ach. Electrotech 21, 387 (1928). 12. D.H. Sloan and E.O. Lawrence, Phys. Rev. 37, 2021 (1931). 13. Linear Accelerators by Lloyd Smith in Handbuch der Physik, Vol XLIV,

341, Springer Verlag (1959). 14. Introductory Nuclear Physics by David Halliday, John Wiley & Sons. Inc.

(1955). 15. Nuclear and Particle Physics by E.B. Paul, North Holland Publishing

Company. (1969). 16. Nuclear and Particle Physics Source Book, Ed. S.B. Parker, McGraw Hill

Book Co. (1987).

Problems

1. A fixed frequency cyclotron magnet of radius Im produces the maximum magnetic induction of 1.5 T. Calculate the energies of deuterons and a-particles accelerated by it. What are the frequencies of the r.f. field in the

two cases ? If it is desired to accelerate protons by the same cyclotron without changing the frequency, what is the required magnetic induction ?

(53.9 MeV ; 107.8 MeV; 11.4 MHz ; 0.75 T )

2. A 2 pA a-particle beam accelerated by a Van de Graaff generator is incident on a target of 5 cm^ area. What is the number of a- particles falling on

1 cm^ area of the target per second ? If an a active source placed at the centres of a sphere of 10 cm radius is to bombard the same target with the same number of a-particles per second,

what should be its strength ? (1.25 x lo'? ; 4.246 x 10 Ci)