Nuclear Instruments and Methods in Physics Research A 334 (1993) 562-568North-Holland

Orbit stability in the steady state MIGMAArvind JainNuclear Physics Division, Bhabha Atomic Research Centre, Bombay-400085, India

Received 25 January 1993 and in revised form 13 April 1993

The stability of the ion orbits has been investigated in a MIGMA fusion reactor for the case of a ring magnet . It is shown thatthe large number of ions which are generated off centre during the process of injection effectively execute betatron oscillationsaround the centre . An expression for the effective radial betatron frequency has been derived . The vertical stability has beenstudied by actual orbit integration in the magnetic field of a typical ring magnet generated with the magnet design code POISSON.A space charge limit, Na, = 0.64x10 8 ions/cm3 , for the average ion density, has been obtained by orbit integration for a typicalring magnet .

1 . Introduction

The MIGMA device shows the possibility to beused as a fusion reactor and has been studied well bothexperimentally [1,2] and theoretically [3,4] . The fusionreaction 3 He + D -, p + 4He + 18.35 MeV used in theMIGMA, where the end products are finally hydrogenand helium gases, offers an alternative and easily ac-ceptable route to non-radioactive aneutronic nuclearpower in the future and merits a thorough investiga-tion . It has been shown recently, that a completeburn-up of the injected beam can be achieved in theMIGMA if the radius of the magnet is properlymatched with the average injector beam current [4] .The interdependence between the various MIGMAparameters and the condition for a steady state opera-tion have also been derived [4] . The process of injec-tion and beam capture has been discussed [3] . The useof a "ring magnet" to confine the orbits of the ions wassuggested earlier by Blewett [5] . The ions make severalmillion revolutions in the MIGMA magnet before fu-sion takes place and it is essential that the orbit of theion remains stable, in both the radial and verticalplanes, during these revolutions . The radial and verti-cal stability of the ion orbits in a ring magnet has beeninvestigated in the present work .

2. The ion orbits

Typical orbits in the MIGMA magnet are shown infig . 1 . A molecular beam of DZ ions, injected from anaccelerator injector A, bends in the annular ringmag-net, passes through the centre C and, after bending

3. The radial motion

0168-9002/93/$06.00 C 1993 - Elsevier Science Publishers B.V . All rights reserved

NUCLEARINSTRUMENTS& METHODSIN PHYSICSRESEARCH

Section A

again in the magnet, emerges at point E. Due to a highdensity of ions at centre C, a fraction of this molecularbeam gets stripped when DZ - 2D ++ e-. While thecharge to mass ratio q/m of the molecular DZ ion is1/4, after stripping it becomes 1/2 for the atomic ionD+. Consequently, the radius of curvature in the mag-netic field for the D+ ion is one-half that of themolecular ion, it bends completely in the magnet pass-ing again through the centre C as shown by the dottedtrajectory in fig . 1 and this orbit pattern repeats itselffrom the centre . Thus, by virtue of having half theradius of curvature of the original molecular ion DZthe stripped D+ ion gets trapped in a "MIGMA orbit" .In each revolution the ion passes through the centre Cand the orbit precesses around this centre . Since theinjection is continuous and the orbits of all the trappedions pass through the centre, the density of ions buildsup at C and fusion conditions are created between theself colliding ions at the centre .

We define an "equilibrium orbit" for a referenceion which starts at the centre, bends in the magnet andpasses again through the centre C, thus closing theloop on itself . We consider first the radial motion ofthe ions in the plane of the MIGMA disc. Due to thefinite size of the injected beam, the stripping anddissociation of the DZ molecules during the process ofinjection does not take place exactly at the centre C,but within an impact parameter CB depending uponthe spread of the injected beam . It can easily be seenthat the D+ ions which are generated exactly at the

Fig . 1 . DZ is injected from the injector A, bends in themagnet and gets stripped at C due to the central ion density nwhen DZ --> 2D + +e . The dissociated ion bends fully in themagnetic field and passes again through the centre C, and thispattern repeats while the non-dissociated DZ beam emergesat point E. In the hardedge approximation, the magnetic fieldhas a step function value B in the magnet gap and is zero

elsewhere.

centre and enter the magnet gap radially will continueto pass through the centre in each revolution irrespec-tive of the ion energy and thus remain bounded withinthe radial plane as seen in fig. 1. The ions which aregenerated "off-centre" will enter the magnet gap non-radially and after bending in the magnetic field will notpass again through the centre C. Since most of the ionsare formed off-centre during injection, it is importantto trace the ultimate trajectories of such ions in theradial plane while they make several million revolu-tions . It is found that some general conclusions can bedrawn regarding the radial motion of an off-centre ioneven if the motion is traced through a single revolu-tion .

The hard-edge approximation has been used in thisstudy i.e ., the magnetic field has a step function valueB in the magnet gap and is zero elsewhere. As shownin fig . 2, a reference ion which leaves the centre C inthe direction CD bends in the magnetic field B in acircle with the centre at H and a radius of curvature"a". The entry and exit points in the magnet are at Dand E respectively and the lines DHX and DE are adiameter and chord on this circle respectively . Frompoint E, the ion moves radially and passes again throughcentre C. The rays CD and CE make tangents to thecircle at the points D and E respectively . The polarequation (r, 0) of the circle made by the ion orbit isgiven byr=2a cos 0,

A. Jain / Orbit stability in the steady state MIGMA

where 0 = 0 is the line DHX. The angle 0 is assumedto increase in the anticlockwise direction and the ori-

563

gin of the polar coordinate system in fig. 2 is chosen atpoint D i.e., when 0 = 0, r = 2a and when 0 = ,rr/2,r=0.

Consider now the motion of an ion which is gener-ated off-centre at point B in fig. 2. This ion, indicatedby the double arrow, moves non-radially along the pathBD and enters the magnet at an angle ß with respectto the path CD of the reference ion. This ion alsobends in a circle with the radius a but with the centreat I which lies on the diameter DIY which is normal toBD and emerges at point F. The diameter DIY of thiscircle is rotated by an angle ß with respect to thediameter DHX of the circle made by the reference ion.DKF is a chord of the second circle, and the polarequation of this circle isr=2a cos(B - ,0) .

(2)The points D, F and E in fig . 2 lie on the inner circlemade by the annular magnet ring . The equation of thiscircle, with the origin at D is given byr = 2R1 cos(B - 3ar/2),

where R1 is the radius of this circle, i.e., the innerradius of the annular magnet ring . The polar coordi-nates of the points E and F at which the two ions

Fig. 2 . The reference ion starts from the centre C, enters themagnet at point D, bends in a circle with a radius "a" andcentre at H, emerges at E and again passes through C (dottedtrajectory) . An "off-centre" ion starts at B and follows thetrajectory BDYFJLAG (solid curve). The various angles in the

figure are explained in the text.

564

originating at C and B emerge can be obtained bysolving for the points of intersection of the circles (1)and (2) with the circle (3) respectively .

After emerging at F, the off-centre ion follows astraight trajectory uptil point G. Rays BD and GFmake tangents to the second circle given by eq . (2). 0 1and 02 are the angles made by the chords DE and DFof the circles (1) and (2) with reference line 0 = 0respectively . In the triangle IDF in fig. 2, the angleIDF = IFD =,6 + 192 . Also, in the similar triangles CDKand CFK, the angles CDK= CFK =,8 + g. Since in thetriangle JDF, the segment DK = KF, angle JDF = JFK.Thus angle GFC =,ß and from the equilateral triangleGCF, angle FGC=,6 .

We draw a line CA on FG from centre C, such thatangle GCA= ,rr/2 . The ion starting at B has made onecomplete revolution and the starting point of the ionhas shifted from point B to point A. The trajectoryBDYFJAG will now repeat itself with the startingpoint at A since the triangles BCD and ACG areidentical with respect to centre C. As the off-centre ionoriginating at B makes a large number of revolutions,the starting point B of the ion moves on a circle with aradius CB around centre C since CB = CA in fig . 2.

Angle S in fig . 2 is given by

S = ,rr/2 - 2(ß + 02 ),

(4)

where angle 02 can be obtained from eqs. (2) and (3)i .e.

j o, 1 =tan-1 (a cos ß/(a sin ß+R1)) .

(5)

The number of revolutions of the off-centre ionrequired for point B to complete one revolution aroundC will be given by

dit = 2 ,rr/(rr/2 + ô) .

(6)

From eqs. (4), (5) and (6), the frequency of revolutionf ' of point B around C will be given by

f'=(f/2ar)(, r-2 [ ß+tan -r (a cos /3)/

(a sin /3+R,)]),

where f is the orbital frequency of the reference ion.The motion of point B on a circle around centre C isequivalent to a betatron oscillation around the centrein the radial plane with a frequency f ' and this beta-tron frequency is given by eq. (7) . Thus the motion ofthe off-centre ion B can be assumed to consist of twomotions, a motion similar to that of the ion passingthrough centre C, on which is superimposed a radialoscillation around C with a frequency f ' given by eq .(7) . The maximum amplitude of this oscillation will beequal to the maximum impact parameter CB made bythe ions in the injected beam with respect to thecentre .

A . Jain / Orbit stability in the steady state MIGMA

From eq . (7), when ß - 0, i.e . when the ion B isformed very close to the centre,

f'= (f/2 ,7r) [ Tr - 2 tan -1(a/R I )] .

(g)Since the half-angle subtended by the orbit at thecentre a/2 --- a/R1,

f' = (f/2,r)[,TT -a] .

(9)When the orbit subtends a very small angle, i.e ., whena - 0, f ' -~f/2. Thus for ions which are formed veryclose to the centre, point B oscillates on opposite sidesof the centre C in each revolution if the orbit angle issmall. The maximum frequency of the effective beta-tron oscillation which any ion can take is f' = f/2, i .e .,half the orbital frequency. The trajectories of the off-centre ions will pass within the area of the circle madeby the impact parameter CB . Thus the central iondensity n may be assumed to be spread over an areamade by a circle with a radius d R = CB [4] .

The analysis above shows that the large number ofions which are generated away from the centre duringinjection and dissociation effectively execute betatronoscillations around centre C and their motion is stablein the radial plane. The ring magnet thus offers a largeradial acceptance to the injected beam .

4. Magnet design for a 3 MW MIGMA

The parameters for a 3 MW power level MIGMAhave been discussed earlier [3,4] and are listed in table1 .

The magnet design for the 3 MW MIGMA, indi-cated in table 1, was studied with the magnet designcode POISSON (TRIM) [6] . A cross section of the ringmagnet, which has a cylinderical symmetry, is shown infig . 3a . The circular coil runs around the magnet be-tween the outer periphery of the annular pole pieceand the inner periphery of the magnet yoke . The axialmagnetic field B, obtained with the code POISSON is

Table 1Parameters for a 3 MW MIGMA [3,41

Central ion density nDZ beam energy ETotal injector current I

after conversion to atomic formCapture efficiency 7 with

recirculation injectionD and 3He fusion cross sectionHalf-thickness d z of

the MIGMA discTotal input beam powerTotal output fusion power(thermal) for complete burnup

2.7X1014 ions/cm3500 keV

500 mA

0.710-24 cm2

15 cm .0 .23 MW

3 .226 MW

50

25

750

500

250

Z

I - Bz

80 100

120 134

1Radius R-

64 184

t- __ ss i

i

i

i

i

i0 80 100 120 140 160 180

220

464030

15

Fig. 3 . A quarter crosssection of the 3MW MIGMA magnetwhich has a cylinderical symmetry . The MIGMA disc ofradius Rm and halfthickness zl is shown by the hatched area .In the lower part, the magnetic field Bs , obtained with

POISSON, is plotted in the median plane .

plotted as a function of the radius r for the medianplane in fig. 3b . Due to the cylinderical symmetry inthe magnet, the magnetic field BZ(r, 0) required on apolar grid (r, 0) for orbit integration in the medianplane can easily be generated from the two dimen-sional output of the code POISSON.

The principal parameters of the magnet obtainedwith the code POISSON for the 3 MW MIGMA de-sign are given in table 2. The main coil current re-quired to produce a field of 0.367 T in a half-gap of 15cm is 45 kA, and the power requirement for the roomtemperature copper coils is 37.2 kW . Due to the largemagnet gap-to-width ratio used in the design, the fring-ing field extends over a substantial distance as seen infig . 3b .

5. Vertical stability

The fringing field gives rise to a radial componentin the magnetic field B� and as mentioned earlier, dueto the cylinderical symmetry in the magnet, this fieldBr (r, 0, z) can also be constructed on a polar grid withthe aid of the two dimensional output of the codePOISSON for any value of the position z above themedian plane. The radial field component Br is plot-ted as a function of the displacement z above themedian plane for a value of the radius R = 165.5 cm in

A. Jain / Orbit stability in the steady state MIGMA

Table 2Magnet and orbit parameters for the 3 MW MIGMA. Case Ais for a smaller radius (R Z = 164 cm) and larger halfgap(dz =15 cm) magnet . Case B considers a larger radius andsmaller gap combination

fig . 4. As seen in fig . 4, the increase in the radial fieldcomponent is nearly linear with z with a deviation ofless than 10% up to a gapheight of 80%, and in furtheranalysis the following symplifying relation may be as-sumed:

Br(z) = kz .

(10)

As the ion orbits at a height z above the medianplane, it experiences an axial force eu,B� where ve is

m`

100

CaseA B

565

Z -.Fig. 4. Variation of the radial component of the magneticfield Br with the displacement z above the median plane, at

R =165.5 cm ., obtained with the code POISSON.

Magnet radius(maximum) [cm] 164 259.3(minimum) [cm] 134 249.3

Annular width [cm] 30 10Half-gap d z [cm] 15 6Magnetic field B [T] 0.367 0.7Coil dimensions,

width x height [cmz] 20x 10 20x10Main coil current,

using POISSON [kA] 45 35 .1Power (J2R) assumingCu coils [kW] 37 .2 33 .1

Angle a subtendedby the orbit atthe centre C [deg] 24 .2 6.8

Central ion density n requiredfor a 3 MW power level[ions/cm 3] 2.7 x 10 14 2.7 X 10 14

566

the component of the velocity in the 0-direction . Thisforce is defocussing and is directed away from themedian plane when the ion moves from the centre Ctowards the rising part of the fringing field and then,due to the change in the sign of Br becomes focussingwhen the ion traverses the falling edge of the magneticfield. Since the path of the ion when it enters themagnetic field is nearly radial, the azimuthal velocityv. is very small and the radial velocity ur - 1, in dimen-sionless units . The defocussing force eu,B, is thus verysmall. When the ion turns and bends in the falling partof the magnetic field on the outer edge of the ring, theradial velocity v, =0 and ue = 1 and the focussingforce eue Br is very large . Thus while the ion moves inan equilibrium orbit at a constant height z above themedian plane, the net time averaged axial force (F,)which acts on the ion is always focussing and is di-rected towards the median plane provided the ionturns and bends in the falling edge of the magneticfield. If the magnetic field is very strong for the case ofa low energy ion, and the ion is reflected in the risingpart of the field itself, it will experience a net axialdefocussing . It is for this reason that the maximummagnetic field in the design of the 3 MW MIGMAshown in tables 1 and 2 was chosen to be 0.367 T sothat the 250 keV ion turns and bends in the fallingedge of the fringing field at a radius of RT = 166.1 em .

The net time averaged axial force seen by the ion isgiven by

(F,) = (11T)feu,Br (z) dt,

(11)

where T is the time period in the equilibrium orbit .The integral in eq . (11) can be evaluated by the inte-gration of the orbit using orbit codes in the magneticfield obtained with the magnet design code POISSON.

6. Orbit integration

The time averaged axial force (Fz ) seen by the ionover the equilibrium orbit for the 3 MW MIGMA wasobtained by orbit integration using a modified form ofthe code ORBIT [7]. The input to this code is themedian plane field BZ(r, 0) . In this code, the differen-tial equation of motion is integrated on a polar grid(r, 0) using a four point Runga-Kutta method . Themagnetic field B,(r, 0), stored on a polar grid, is usedat each Runge-Kutta integration step . The magneticfield between the mesh points is obtained by a threepoint Lagrangian interpolation . The integration of theorbit is started near the centre C, where the radialmomentum pr - 1 and the azimuthal momentum pe = 0as shown in fig . 5. The integration is carried out up tothe turning point R =RT where pr = 0 and pe = 1.The value of the radius RT at which the turning is

A. Jain / Orbit stability in the steady state MIGMA

7. Space charge limit

Fig . 5 . The integration of the orbit is started near the centreC, with the conditions pe = 0, pr -- 1 at R = 0, and carried outuptil the turning point R= RT, with the condition pe = 1 and

pr -0 .

desired, i .e . the condition Pr = 0 or pe = l, at R = R-,-,can be specified as an input to the code ORBIT. Thecode first searches for the normalization factor for themagnetic field which produces these conditions at R =RT . The ion can thus be chosen to turn at any point inthe falling part of the fringing field on the outer radiusof the magnet gap to study the variation in the verticalfocussing . After the field normalization factor is ob-tained in the search, the integral in eq . (11) is obtainedby summing the product eu,Br(z) dt at each integra-tion step on the equilibrium orbit and the correspond-ing time period T obtained by summing the timeinterval dt. We note that the integral (11) need beevaluated only between the limits R = 0 and R = RT,due to the symmetry of the orbit .

The Coulomb repulsion between the ions within theMIGMA volume produces an axial defocussing forcewhich increases with the average ion density N, Alimiting density can be obtained when this repulsiveaxial force is balanced by the net axial focussing force(F,) obtained in eq . (11) . This density is the spacecharge limit for the machine. We evaluate the spacecharge limit for the 3 MW magnet of table 2. TheMIGMA disc is shown schematically with a radius Rmand a half-thickness zr in fig . 3a . The total number ofions Nz contained within a cylinderical disc of half-thickness z will be given by Nz = 27rR2 zN, where zis the height above the median plane. Using Gauss'theorem, the normal electric field Ez on the surface ofthis disc will be given by

Ez = 4 ,rr v̂ z,

(12)

which is directed away from the median plane and

produces a force F, on an ion on the surface of thedisc which is given by

F, = 4,rre 2N.,,z .

(13)

If this ion is to remain vertically bound, the timeaveraged force due to the space charge repulsion in eq .(13) must be balanced by the net time averaged fo-cussing force (F.) due to the magnetic field given ineq . (11) over one complete revolution . Equating theforces in eq . (11) and (13), the maximum ion densityNav in the space charge limit will be given by

Na, = (1/4rreT) foe[B,(z)/z] dt,

(14)

where the integral in eq . (14) and the time period Tcan be obtained by numerical integration of the orbitas discussed earlier . We note that both the defocussingforce due to the space charge repulsion in eq . (13) andthe net focussing force due to the radial field compo-nent B, in eq . (10) increase linearly with the displace-ment z above the median plane and the ratio B,(z)/zrequired in the integral in eq . (14) is independent ofthe displacement z. The resulting density Na, obtainedin eq . (14) is thus independent of the z-motion of theion, and the integral required in eq . (14) can be evalu-ated on an equilibrium orbit at any constant height zabove the median plane.

8. Results

The equilibrium orbit for a 250 keV deuteron in the3 MW MIGMA was integrated from R = 0 to RT =166.1 cm in the median plane field B,(r, 0) obtainedusing the code POISSON. A maximum field of 0.38 Twas required in the magnet gap to produce a bendingradius RT = 166.1 cm for the 250 keV deuteron . Theorbit integration for the time averaged force (Fz ) ineq . (11) was carried out at a constant height z = 10 emabove the median plane. This yielded a value N V -

0.64 aX108 ions/cm' for the space charge limit, in eq .(14).

It has been shown earlier that the central ion den-sity n is related to the volume average ion density Navthrough the relationNav=n(dR/2Rm) ln(1+2Rm/dR),

(15)where dR is the interval chosen in the averaging pro-cedure while obtaining the average density and Rm isthe maximum radius of the MIGMA disc [4] . A valuedR/Rm =0.1 had been chosen in eq . (15) in the earlierwork [4] . Using this value and the space charge limitNav = 0.64 X 108 ions/cm3 obtained by orbit integra-tion in the present work, a value n = 0.42 X 109ions/cm3 for the central ion density is obtained in thespace charge limit. We note that the average densityNav used in eq . (12) to (14) is the time averaged valueencountered by the ion over the orbit while eq . (15)

A. Jain / Orbit stability in the steady state MIGMA 567

gives the average density over the radius of the disc [4].When the bending radius of the ion in the magneticfield is very small compared to the inner radius of themagnet ring i.e ., a/R, << 1, most of the orbit is radialand both these densities become identical . Thus appli-cation of eq . (15) does not produce a serious error inthe calculations .

9. Discussion

The value for the average ion density Nav = 0.64 X108 ions/cm3 obtained for the space charge limit byorbit integration is not very large as was expected,although alternating gradient strong focussing is pre-sent in the ring magnet used in the design . This is dueto the fact that the bending radius "a" in the magneticfield is very small compared to the annular radius ofthe magnet R, . Thus, since a/R, << l, the fraction ofthe time spent in the magnetic field is very smallcompared to the time spent in the field free radial partof the orbit . This decreases the time averaged forceover the orbit in eq. (11) by a very large factor . If themagnetic field used in the gap is increased further,although the focussing field component Br(z) in eq .(14) will become larger, the bending radius "a" andhence the time averaged force will become proportion-ally smaller since the product aBz , and hence aBr,remains constant in a magnetic field for a given energy .Thus, so far as vertical focussing is concerned, no realadvantage will be gained by the use of higher magneticfields. The value Nav = 0.64 X 10 8 ions/cm3 obtainedfor the space charge limit in the present work will thusbe typical for any ring type magnet and will be nearlyindependent of the absolute value of the magnetic fieldused in the design .

The value of Nav obtained for the ring magnet ishowever sufficient, and in fact essential, for the processof "electron seeding" which is used to obtain an ulti-mately high value for the central ion density n. In thisprocess, due to the initial potential well of the positiveions, electrons will get trapped within the MIGMAdisc until the positive ion space charge limit is neutral-ized . At neutralization, by the electrons, the net fo-cussing force provided by the magnetic field in eq . (11)still exists for the positive ions and a fresh batch ofpositive ions, corresponding to a value Na, = 0.64 X 10 8ions/cm3, may be injected again. This process cancontinue until a central ion density n = 2.7 X 10 4ions/cm3 is built up . At the final equilibrium, since thepositive ion space charge is completely neutralized bythe presence of an equal number of oscillating elec-trons [1-4], the net focussing force given by eq . (11)will still exist for the positive ions, providing stablebetatron oscillations in the vertical direction as theyorbit in the MIGMA disc .

56 8

10 . Resonance conditions and orbit precession

We note from fig . 1, that if the angle a subtendedby the orbit at the centre is given bya = 2-rr/n,

(16)

where n is an integer, then, since the point of injectionis fixed, the ion will make an integral number of loopsin 2 ,rr radians and will retrace its path in the radialplane after every n revolutions . This situation willcause an undesirable bunching of the positive ion spacecharge within n loops in the radial plane. Since theorbit angle a ~-_ 2a/Rt, the magnet parameters for agiven energy E must be chosen such that a 0 2-rr/n i.e .the angles a = 90, 72, 60, etc. in degrees correspondingto the value of n = 4, 5, 6, etc. should be excluded inthe design . However, if we choose the orbit angle asuch thata - 2 ,rr/n = AO,

(17)

where AO is small, the n-looped orbit pattern will notclose on itself in 2-rr radians but will process aroundthe centre C with a precessional frequency fp given byfp= (AO/2,rr)f, (18)

where f is the orbital frequency. Such a precession isessential to distribute the space charge uniformly withinthe MIGMA disc . This precession will also minimizethe effect of small perturbations on the ion orbits . Theperturbation may arise due to a small residual techni-cal imperfection in the magnet construction or due to asmall local asymmetry in the space charge distribution .Due to the precession of the orbit, the ion will not passthrough the local perturbation again until all the ionsin the MIGMA volume have passed through the sameperturbation .

We note that due to a spread in the value of theangle ß for different groups of ions in fig . 2, a spreadwill be introduced in the effective radial betatron oscil-lation frequency f' in eq . (7) . Similarly, due to thedifferent edge focussing experienced in the verticaldirection for different entry angles /3, a spread will alsoarise in the vertical oscillation frequency. This spreadin the betatron frequencies is of advantage, since afrequency spread in a collection of oscillating ionsdampens the growth of coherent or collective oscilla-tions of the ions [8] .

11 . Conclusions

The motion of ions has been discussed in both theradial and vertical planes within the MIGMA disc . Ithas been shown that the large number of ions whichare formed "off-centre" due to the finite spread of thebeam during the process of injection, effectively exe-cute betatron oscillations around the centre of the ring

A . Jain / Orbit stability in the steady state MIGMA

magnet . An expression for the effective radial betatronfrequency has been derived and it has been shown thatthe maximum amplitude of these oscillations will beequal to the maximum impact parameter in the in-jected beam . The MIGMA disc thus offers a largeacceptance to the injected beam in the radial plane.The present description of the radial motion is ex-pected to be useful in many studies [4] .

The vertical stability of the ions in the orbit hasbeen studied by detailed orbit integration in the mag-netic field of a typical ring magnet which provides analternating gradient strong focussing. The space chargelimit for a typical 3 MW MIGMA magnet has beenobtained by orbit integration with the orbit code OR-BIT in the magnetic field, generated with the magnetdesign code POISSON. A value for the average iondensity N,v = 0.64 x 108 ions/cm3 has been obtainedin the space charge limit for a typical magnet design .Due to the strong radial magnetic field componentspresent in a ring magnet, a very large space chargelimit was expected when compared to classical "weakfocussing" magnets. But in the present work it is seenthat this advantage of the presence of large radial fieldcomponents is offset by the very small time spent bythe ion in the "ring" magnetic field and this results ina similar order for the average space charge limit. Asdiscussed earlier [1-4], if central ion densities in theregion of 10" ions/em3 are to be achieved, the posi-tive ion space charge in the MIGMA disc has to beultimately neutralized by the presence of electrons ;thus although a high value for the space charge limitthrough magnetic focussing is desirable, the exact valuedoes not appear to be critical in the design . It is seenin the present work that the magnetic focussing pro-vided by the ring magnet is sufficient, and in factessential, to provide stable vertical oscillations for thespace charge neutralized ions in the MIGMA disc .

References

[1] For a review, see for example, B.C . Maglich, Proc . Int .Symp . on the Feasibility of Aneutronic Power, Princeton,New Jersey, September 10-11, 1987, eds . B.C. Maglich, J.Norwood Jr. and A. Newman, Nod. Instr. and Meth .A271 (1988) 13 .

[2] D. AI Salameh, S. Channon, B.R . Cheo, R. Leverton, B.C .Maglich, S. Manasian, R.A. Miller, J. Nering and C.Y .Wu, Phys . Rev. Lett. 54 (1985) 796 .

[3] A. Jain, Nucl . Instr. and Meth . A316 (1992) 391.[4] A. Jain, Nucl . Instr. and Meth . A323 (1992) 671.[5] J.P . Blewett, ref. [1], p. 214.[6] J.S . Colonias, UCRL-18439, 1968, LBL, Berkeley .[7] A. Jain, Proc . Nucl . & SSP Sym., Feb. 1-4, 1972, Bombay,

Vol. 14B (1972) p. 531. (Library & Information Services,BARC, Bombay-85, India .)

[8] See for example, E.J .N . Wilson, CERN-77-07 (1977) 44 .