NUCLEARINSTRUMENTS& METHODS

North-Holland

IN PHYSICSRESEARCH

Section A

Nuclear Instruments and Methods in Physics Research A340 (1994) 605-611

The diamagnetic current limit in the steady state MIGMAArvind JainNuclear Physics Division, Bhabha Atomic Research Centre, Bombay-400085, India

(Received 28 January 1993 ; revised form received 5 October 1993)

The diamagnetic field produced by the tons circulating in the MIGMA fusion reactor has been discussed for the case of a ringmagnet . An expression for the diamagnetic current and its distribution to the magnet gap has been derived in terms of the basicMIGMA parameters It is shown, with the magnet code POISSON, that the diamagnetic field can be substantially reduced and theusable central ion density can be exceeded by one order of magnitude with the aid of correction coils which carry opposite currentsand are located in the magnet gap close to the region where the diamagnetic currents are flowing .

1 . Introduction

The MIGMA fusion device [1] has been well stud-ied experimentally [2,3] and theoretically [2,4-6] . Thefusion reaction D + 3He , p + 4He + 18.35 MeV usedin the MIGMA offers an easily acceptable route tononradioactive aneutronic nuclear power and such adevice merits a detailed study. The basic principle ofthe device has been described earlier [2-6]. A molecu-lar beam DZ is injected in the radial plane of a magnetby an accelerator injector A as shown in Fig. 1 . Due tothe central ion density n at the centre C, a fraction ofthe beam gets stripped to D+, the q/m becomesone-half, the ion bends completely in the ring magnetpassing again through the centre C as shown by thedotted trajectory in Fig. 1 . The nondissociatedmolecules in the injected beam emerge at E. Thus byvirtue of having half the radius of curvature of theoriginal molecular ion DZ , the stripped D+ ion getstrapped in a "MIGMA orbit" . In each revolution theion passes through the centre C, and the orbit pre-cesses around the centre . Since the injection is contin-uous, the density at C builds up and fusion conditionsare created at C between the self-colliding ions . Theuse of a "ring" magnet for bending the trapped ionshas been suggested by Blewett [7] . Such a ring magnetproduces strong focussing of the ions in the orbit . Thedesign for a 3 MW reactor has been discussed earlier[4,5] . It has also been shown that a complete burnup ofthe injected beam can be achieved in the MIGMAreactor if the radius of the magnet is properly matchedwith the injector beam current [5] . The interdepen-dence between the various MIGMA parameters andthe condition for a steady state operation has beenderived [5] . The stability of the orbits in both the radial

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and vertical planes has been discussed [6] . The approxi-mate parameters for a 3 MW MIGMA are given inTable 1 .

An equilibrium orbit for a reference ion may bedefined as one in which the ion starts at the centre C,bends in the magnet and again passes through C, thusclosing the loop on itself as shown in Fig. 1 . The ioncurrent flowing in the equilibrium orbit creates a mag-netic field which is opposite in sense to the bendingfield in the magnet and this "diamagnetic" field tendsto cancel the magnetic field of the main coil . As theintensity of ions in the equilibrium orbit builds up, thediamagnetic field which is proportional to the centralion density n, also builds up and would eventuallycancel the original guiding field thus destroying theequilibrium orbit itself . We may therefore define a"diamagnetic limit" such that the diamagnetic fieldproduced by the circulating ions is 10% of the bendingfield . We call the corresponding central ion density atthis limit n, .

In Fig. 2, three equilibrium orbits are depicted bythe solid curves, displaced in angle by an amount a,where a is the angle subtended by an equilibrium orbitat the centre . As seen in the figure, the inward andoutward ion currents in the radial part of the neigh-bouring orbits mutually cancel while the currents withinthe bending field for the three orbits add up in thesame sense. Thus a net diamagnetic current flowswithin the magnet gap at the outer edge of the MIGMAdisc . It is shown below that this diamagnetic current isnearly as effective in producing a magnetic field as thecurrent in the main coil, but in the opposite direction.We first derive an expression for the net diamag-

netic current flowing in the ring in terms of the basicMIGMA parameters . The maximum central ion den-

606

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 emerges

at the point E.

Table 1Parameters for a 3MW MIGMA

Central ion density

2.7 X 10 14 ions/cm3DZ beam energy E

500k_VTotal injector current 1 after

conversion to atomic form

500mACapture efficiency 77 with

recirculation injection

0.7D and 3 He fusion cross section

10-24 cm2Half-thickness dz of theMIGMA disc

3 cmTotal input beam power

0 .23 MWTotal output fusion power (thermal)

for complete burnup

3 .226 MW

Fig. 2. A schematic of three equilibrium orbits. The currentsm the radial part are in opposite directions and mutuallycancel . The currents in the bending field add up in the samesense, giving rise to a diamagnetic current flowing in the ring.Only ions having an orbit within an angle a to the left of theplane AC, drawn to bisect an equilibrium orbit, will cross to

the right of this plane .

A. Jain /Nucl. Instr. and Meth. in Phys. Res. A 340 (1994) 605-611

sity np which can be used, for a decrease in the maincoil field by 10%, is also calculated . The radial distri-bution of the diamagnetic current in the magnet gap isderived. The actual diamagnetic field produced withthis current distribution is computed with the magnetcode POISSON (TRIM) [8] . This code simulates theexperimental field to within a few percent and relativefield distributions have been obtained with this codewith accuracies upto 0.1% [8] . Finally, it is shown, withthe code POISSON, that the diamagnetic field pro-duced by the circulating ions can be cancelled with theaid of correction coils placed in the magnet gap whichcarry equal and opposite currents to the diamagneticcurrents . The diamagnetic limit np can thus be ex-ceeded by one order of magnitude with the aid of thecorrection coils.

2. Magnet design

The design of the magnet for the parameters of the3 MW MIGMA given in Table 1 was studied with themagnet design code POISSON [8] . A ring magnet, withcyliderical symmetry, has been chosen in the design .The main coil runs around the magnet between theouter periphery of the annular pole piece and theinner periphery of the magnet yoke . A cross-section ofthe magnet with the position of the coil is shown inFig. 3a . The current in the main coil obtained with thecode POISSON, which is required to produce thedesired magnetic field of 0.8 T in the magnet gap is 142kA . The full width half maximum (FWHM) of the peakin the main coil field can be made narrower with theuse of "bucking coils", not shown in the figure, whichcarry opposite currents to the main coil and cancel themain coil field at the inner radii. The mesh used in thecode is shown in Fig. 3b and typical field lines in Fig.3c . For a room temperature copper coil of cross-sec-tion 20 X 28 cm 2, the main coil power requirement forthe 3MW MIGMA magnet is 200 kW. The otherprinciple parameters obtained in the magnet designare given in Table 2.

3. Expression for the diamagnetic current

Let the equilibrium orbit subtend an angle a at thecentre C and let AC be a reference plane passingthrough the midpoint of the equilibrium orbit . FromFig. 2, we note that only the ions having an orbit withinan angle a lying to the left of the reference plane ACwill cross this plane in one revolution . The fraction ofthe ions in the MIGMA volume crossing the plane ACin one second is therefore

dNAc = (a/2ir)fN1 ,,

where NT is the total number of ions contained in theMIGMA volume and f is the revolution frequency ofthe ions, given by

f =/3c/2(RI + ,rra/2) .

(2)

Here ß = u/c, u is the velocity of the ion, c thevelocity of light, R I the inner radius of the annularmagnet ring and a the bending radius of the ion in themagnetic field. The angle a subtended by the orbit atthe centre is approximately given by

a = 2a/RI,

and the total number of ions contained in the MIGMAvolume is

N.r = 2TrR2� dz N,,, .

(4)

Here Rm and dz are the outer radius and half-thick-ness of the MIGMA disc, and Na, the average iondensity in the MIGMA volume . From Eqs. (1) to (4),the total diamagnetic current IDIA flowing in the ringwill be

,DIA =aßc dz Nav/(6.21 X 10 1s ) .

Since a <<R, we have neglected the term Tra/2 incomparision to R I in Eq . (2) and have assumed Rm =

200

250

300

346 .7 366 7

392 7

A. Jain /Nucl. Instr. and Meth . to Phys. Res. A 340 (1994) 605-611

R(cm)-

Fig. 3. (a) A cross section of the MIGMA magnet which hascylinderical symmetry . (b) and (c) show the mesh and typical

field distribution, generated by POISSON.

Table 2Magnet and orbit parameters for the 3 MW MIGMA

Nav = n(dR/2R .)ln(1 + 2Rm/dR),

'oil = 02440 X 10-12E dzn/gB

IDIA= 1 .6 X 10-12(E dz Nav)l(gB)

[A] .

607

(R I + a) =R1 in Eq . (4) . The bending radius a in themagnetic field is given by a =M&/Beq =Am&/Beq .Here q/A is the charge to mass ratio for the ion, mthe proton mass, e the elementary charge and B themagnetic field. Putting this value of a in Eq . (5), thediamagnetic current flowing in the magnet half-gap, inamperes, will be

In Eq . (6), the bending field B is in T, the ion energyE in keV and the half-thickness of the MIGMA discdz, in cm . It has been shown earlier [5], that a relationcan be derived between the central ion density n andthe volume average density Nav i .e .

where dR is the radial interval used in the averagingprocedure for obtaining the average density Nd� andRm is the radius of the MIGMA disc . A value dR/Rm= 0.1 had been chosen in the previous work [5] . Withthis value, from Eq. (6) and (7), the diamagnetic cur-rent in terms of the central ion density n will be givenby

We note from Eq . (8) that the diamagnetic currentflowing in the magnet ring is independent of the mag-net radius R 1 and depends only on the MIGMA half-thickness to magnet field ratio dz/B .

Magnet Radius (R2, maximum) 366.7 cm(R I , minimum) 346.7 cm

Annular width 20 cmMagnet half-gap 15 cmHalf-thickness of the MIGMA disc d z 3 cmMagnetic field B 08TCoil dimensions, width x height 20 X 28 cm2Main coil current, using POISSON 142 kAPower (I 2R) assuming Cu coils 200 kWAngle a subtended by the orbit at

the centre C 3.99'Diamagnetic current IDIA, Eq (8) 64.3 kADiamagnetic limit for the

central ion density nD which willproduce a field 10% of B abovea) from POISSON calculation(peaks in Fig. 5b) 3 .8X 10 13 [ions/cm3 ]b) from consideration of areasunder the magnetic field curves 1 .9 X 10 13 [ions/cm3]

Central ion density n requiredfor a 3 MW power level 2.7X 10 14 [ions/cm2]

608

4. Example

The principal parameters of the 3 MW MIGMA areshown in Tables 1 and 2. The maximum radius of themagnet in the design in Table 2 is 366.7 cm, themagnet half-gap is 15 cm, and the annular magnetwidth Rz - R, = 20 em . The hardedge value of theangle a subtended by the equilibrium orbit at thecentre is 3.99° . The diamagnetic current for the param-eters of the 3 MW MIGMA can be calculated usingEq . (8) . In this design, the parameters are: E = 250kcV, half-thickness of the MIGMA disc dz =3 cm, thecentral ion density n = 2.7 x 10'4 ions/cm3, q = 1, andthe magnetic field in the gap B = 0.768 T. With thesevalues, the diamagnetic current obtained using Eq. (8)is 64 .3 kA. As we shall see later, the diamagneticcurrent flows on the outer periphery of the magnet gapand is nearly as effective, within a factor of two, as themain coil itself in producing a magnetic field. If, forthe present purpose, the main coil current and diamag-netic current are assumed to produce approximatelysimilar but opposite field effects, then the diamagneticcurrent limit, if restricted to 10% of the main coilcurrent of 142 kA, would be only 14 .2 kA . This limit issmaller than the diamagnetic current of 64 .3 IcA calcu-lated to be flowing in the magnet half-gap by a factorof 4.5 . Using Eq. (8), the ten percent limit of 14.2 kAcan be translated into a maximum limit on the centralion density nD < 6 x 10 13 ions/cm3. Thus if the dia-magnetic field is to be restricted to 10% of the maincoil field, the central ion density np must be restrictedto this value, which must be compared to the centralion density n = 2.7 x 10 14 ions/cm3 required in thedesign . Since the power level of the reactor scales asthe second power of the central ion density i.e . as nz, acentral ion density lower by a factor of 4.5 would resultin a power level which is lower than the design value bya factor of 20.3 . Thus, in practice, the diamagnetic fieldwill set a severe restriction on the operation of theMIGMA and methods must be devised to overcomethis limit.

We note from Eq . (8) that the diamagnetic currentcan be made smaller by choosing a higher bending fieldB in the design . However, it has been shown earlier [61,that if alternating gradient focussing is to be used, thenvertical stability of the ion orbit requires the ion tobend and turn in the falling edge of the magnetic field.If the magnetic field chosen is very high and the energyof the ion is low, then the ion would get reflected inthe rising part of the field itself which would lead to anacute vertical defocussing . If the magnetic field in thegap is to be very high and the ion is to bend in thefalling edge of the field profile, then the full width halfmaximum (FWHM) of the radial field profile requiredm the gap is very "narrow" . Studies with the magnetdesign code POISSON indicate that when the median

A Jain/Nucl. Instr and Meth to Phys . Res A 340 (1994) 605-611

Fig . 4. Polar coordinates (r, 0) of the equilibrium orbit . 0increases in the anticlockwise direction. R, and Rz are theinner and outer radii of the magnet respectively, a/2 thehalf-angle subtended by the equilibrium orbit and a the

bending radius in the magnetic field

plane magnetic field chosen in the gap is very high, it isdifficult to produce a narrow FWHM field profileespecially for large gap magnets as required in theMIGMA. Thus, in practice, the magnetic field in Eq .(8) cannot be made arbitrarily large and must be re-stricted to a practical value consistent with alternatinggradient vertical focussing requirements [6] .

5. Distribution of the diamagnetic current in themagnet gap

We now investigate the distribution of the diamag-netic current as a function of the radius in the magnetgap. The radial density of the diamagnetic current willbe proportional to the radial density of the orbits p(R)crossing the plane AC in Fig. 4 or 2. The density oforbits p(R) crossing the plane AC increases from theinner radius of the magnet R, to the outer radius ofthe MIGMA disc Rm = (R, + a) . When the orbit angleis small, it can easily be seen that the density of orbitsp(R) between the radius R, and (R, + a) will be in-versely proportional to the derivative dr/d0, wherer(0) is the polar equation of the equilibrium orbit inthe magnetic field. As seen in Fig. 4, the equation ofthe equilibrium orbit in polar coordinates (r, 0) be-tween 0 = 0 and 0 = a/2 is given by

rz -2R,r cos 0+(R2 -az)=0.

The azimuth 0 increases in the anticlockwise direction.Rewriting Eq . (9), we get

r =R, cos 0 +a[ 1 - (R, sin 0/a)z 1

(10)

Since the range for the angle 0 lies between 0 anda/2, and a/2 << ,rr/2, we may assume cos 6 = l andsin 0 = 0 in Eq . (10) . Putting x = (r - R,), we get

x=a[1 - (R,0/a)z ] r/z

We note that when 0 = 0, x = a and when 0 =a/2a/R 1 , x = 0. Changing the origin from the line 0 = 0 tothe line 0 = -a/2, Eq . (11) becomes

x =all - [R1(0 -a/2)/a1211/2.

(12)

In Eq . (12), 0 increases from 0 to a/2, and x increasesfrom 0 to a. Differentiating Eq . (12) and putting thedensity of orbits p(x) as inversely proportional to thederivative dx/d0, we get,

The constant of proportionality k in Eq . (13) can beobtained by integrating p(x) in Eq . (14) over the widthx = 0 to x = a and equating this to the total diamag-netic current IDIA obtained using Eq . (8). Whence,

I = IDIA{ 1- [ I - (xla)z] I/z

} .

(15)

Eq . (15) gives the fraction of the diamagnetic currentIII,,, lying in the interval 0 to x. When the region 0to a is divided into ten equal 10% segments, thepercentage of the current flowing in each segment islisted in Table 3. We note from the table that 43.59%of the diamagnetic current flows in the last 10% radialsegment.

6. Correction coils to cancel the diamagnetic field andmagnetic field calculations

The distribution of the diamagnetic current in themagnet halfgap of the 3 MW MIGMA, as a function ofthe radius, can be obtained using Eq . (15), with Ioil =64 .3 kA . If the magnet width, Rz - Rt = 20 cm, isdivided into ten equal segments of width AR = 2 cm,then the diamagnetic current flowing in each segmentobtained using Eq . (15) is listed in Table 3. The dia-magnetic field produced by these currents was calcu-lated with the magnet design code POISSON. To cal-culate the diamagnetic field with this code, the cur-rents in Table 3 were assumed to flow in ten virtualcoils (regions) D1 to D lo placed, in the code, on themedian plane within the magnet gap as shown in Fig.5a at the positions where the diamagnetic currentswould actually be flowing. Thus the coil D 1 carries 0.32kA and the coil D to 28.03 kA. With these currents, thediamagnetic field B. in the median plane obtainedwith the code POISSON is shown by the curve (2) inFig. 5b . The distribution of the diamagnetic current inTable 3 produces, in the code POISSON, a peaknegative field of 5 .7 kG at the magnet radius R = 354.7cm as shown in Fig. 5b . In these calculations, the main

A. Jain INucl. Instr. and Meth . In Phys. Res. A 340 (1994) 605-611

Table 3Percentage of the diamagnetic current IDIA In ten equalsegments and the currents in the segments D1 to D, () for atotal diamagnetic current 64 .3 kA

609

coil current MC was "switched off" in the code . Thedotted curve (1) in Fig. 5b shows the main coil fieldalone calculated with only the main coils on in thecode and the diamagnetic currents absent . The maincoil field has a peak of 8.0 kG at a radius R = 364.7cm . If the diamagnetic field is to be limited to 10% ofthe main coil field i.e . to 0.8 kG, then the calculateddiamagnetic field exceeds this limit by a factor of 7.12.

Correction coils C1 to C1 ,1 are shown placedschematically in the magnet gap in Fig. 5a to cancel thediamagnetic field. Thus for each region in the MIGMAdisc D1 to Dlo, there is a corresponding correction coilC 1 to Cto placed directly above the region D 1 to D10 .Currents which are equal and opposite to the diamag-netic currents in the D coils are introduced in the coilsCt to Cto . Thus the pair of coils Ct and D1 carry equaland opposite currents and it is expected that the netmagnetic field produced anywhere in the magnet willbe zero due to each pair. When the diamagnetic cur-rent in coils D 1 to D10 , along with equal and oppositecurrents in C1 to C,,,, are used as input in POISSON,the net field obtained in the median plane with thecode is shown by the dashed curve (3) in Fig. 5b . Dueto the equal and opposite currents in each pair D,C 1etc., it was expected that the net field produced in thecode would be zero . This would be true if the C and Dcoils were geometrically coinciding . However, due to aseparation of 6 cm between the C and D coils, a smallsecond order residual field remains since the current inthe C coil is slightly less effective in producing amagnetic field in the median plane than the corre-sponding D coil which has a closer proximity to themedian plane. As seen in Fig. 5b, this gives a peak of-2.2 kG at the radius R = 362.7 cm . This residualdiamagnetic field in the median plane can be reducedfurther by using an opposite compensating current of 6kA in the main coils. If this is done, the residualdiamagnetic field left, after cancellation, is shown by

D-Segment % of ID)A Current [kA]1 0 .50 0 .322 1 .52 0 .983 2.59 1 .674 374 2.405 5 .05 3 .256 6.60 4 .247 8.59 5 .528 11 .41 7.349 16 .41 10.5510 43.59 28.03

p(x) =kx/(R2(0-a/2)) . (13)

From Eqs. (12) and (13),

p(x) = kx/(RI(a2- x2 )1/2 ( 14)

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A. Jain /NucL Instr. and Meth

CORRECTIONCOILS Ct TO C10IN CRYOSTAT

l

I

I3267

346.7

366 7

366.7(RADIUS) cm --.

Fig . 5 . (a) Concentric correction coils CI to C IO are shownattached schematically to the pole tip to cancel the diamag-netic current flowing m the annular ring. The C cods carry anequal and opposite current to the currents flowing in theregions D1 to Dlo . (b) The median plane magnetic fieldobtained with the code POISSON (1) guiding field due to themain coils alone, (2) diamagnetic current field only, (3) theresidual diamagnetic field when the correction coils C i to C IOcarry equal and opposite currents to the D region currents (4)the residual diamagnetic field when a current of 6 kA is used

in the main coil to further reduce the diamagnetic field .

the dash-dotted curve (4) in Fig. 5b . The peak of -2.2kG is further reduced to -1 .6 kG .

We note that during orbit integration, it is the totalarea under the guiding field upto the turning point ofthe ion rather than the peak field alone which influ-ences the orbit radius . Thus it is the areas under thevarious curves in Fig. 5b which must be comparedrather than the peak fields alone. The area under theguiding field curve (1) upto a turning point RT = 368.7em is 97 .8 kG cm while the area under the full diamag-netic field curve (2) is 137.4 kG cm . If the area underthe diamagnetic field is to be restricted to 10% of thearea under the guiding field i.e to 9.78 kG cm, then thecalculated area exceeds this limit by a factor of 14 .1 .This corresponds to a limit on the central ion density

to Phys . Res . A 340 (1994) 605-611

no -< 1.9 X 10 13 ions/cm3 . With equal and oppositecurrents in the C and D coils, the area under thediamagnetic field is reduced from 137.4 to 16 .9 kG cmwhich is 12.3% of the original value. With a furthercompensation of 6 kA in the main coil this area isfurther reduced to 3.8 kG cm i.e . to 2.8% of the fulldiamagnetic field and 3.9% of the area under the mainguiding field. Thus the correction coils aid in decreas-ing the area under the diamagnetic field by more thanone order of magnitude and the residual diamagneticfield is less than 5% of the guiding field . In practicethe C and D coils are shifted to the right by 2 cm (notshown in Fig. 5a) to coincide with the turning pointRT= 368.7 cm for the ion. We also note that thesuperconducting correction coils C 4 , CS and C6 in Fig.5a also serve as the "bucking coils" mentioned inSection 2 which are required to make the peak in themain coil field narrow in the radius .

7. Typical cryostat dimensions

Due to the large currents involved, the correctioncoils Ct to Cto are required to be superconducting andmust be contained in a cryostat as shown in the Fig. 5a .While the half-gap required in the magnet design forthe MIGMA disc is only 3 cm, the actual half-gap usedin all the POISSON calculations is 15 em . Thus aprovision of 9 cm has been provided for the liquidhelium cryostat which houses the correction coils Ct toC oo . The external dimensions of the cryostat wouldtypically be 30 cm wide and 9 cm thick . The maximumcurrent would flow in the coil C IO which from Table 3is 28 .03 kA. The normal current densities used insuperconducting coils is in the range 10 kA/cmZ, andthis would require the cross section of the coil CIO tobe 2.8 cmZ. With a width of 2 cm, this would require acoil height of 1 .4 cm . Typical dimensions of the super-conducting coil C IO would thus be 2x 1.5 cmZ (widthx height), with the other C coils, which carry lessercurrents, having smaller dimensions . No power wouldbe consumed in the correction coils themselves sincesuperconducting windings would be used ; but the liq-uid helium consumption in the cryostat would requirean estimated operating power of about 30 KW percryostat . The cryostat would run within the magnet gaparound the magnet ring, with separate cryostats for theupper and lower pole tips . During operation, the cur-rents in the correction coils would initially be zero, butas the central ion density and the diamagnetic currentslowly builds up and reaches the steady state, thecurrents in the correction coils would be required toincrease proportionally with the aid of suitably de-signed feedback signals from the main coil field whichkeep the guiding field from deviating, at any instant, bymore than a specified value.

8. Self consistent field calculations

If the main guiding field (curve (1) in Fig. 5b) bedenoted as B1(r) and the residual field (the curve (3)in Fig. 5b) as BZ (r), then the total guiding field BT(r)after cancellation of the diamagnetic field will be givenby

BT (r) =kBl(r)+BZ(r),

(16)

where we assume B2(r) to be small compared to Bl(r) .The constant k in Eq . (16) can be adjusted in thecomputer program which integrates the orbit to pro-duce the same turning point RT for the reference ionin the new total field BT(r) as in the previous fieldB,(r). The radial diamagnetic current distribution F1(r)as shown in Table 3 was determined in the hardedgeapproximation while the field BT(r) will give rise to anew diamagnetic current distribution FT(r). If theresidual field B2(r) << B1 (r), the change in the diamag-netic current distribution will be small. The effect ofthis change in the diamagnetic current distribution hasbeen calculated in a self consistent way. The newdiamagnetic current distribution FT(r) can be obtainedby integrating the ion orbit in the new total field BT(r)and setting the diamagnetic current I(r) at each radiusproportional to the derivative d0/dr, obtained numeri-cally, by orbit integration in the field BT (r), at severalvalues of the radius r, as discussed in Section 5. Thisprocedure gives a more accurate distribution of thediamagnetic current than the hard-edge approxima-tion . A new set of equal and opposite currents, accord-ing to the new distribution FT (r), are used in the Cand D coils as an input in the code POISSON which,after cancellation, gives rise to a new residual fieldB2(r) in Eq . (16) . This process is repeated several timesuntil it converges. In each cycle, the factor k is ad-justed to reproduce the same turning point RT for theion during orbit integration . The error arising in BT(r)due to the linear addition used in Eq . (16) and theactual output of the code POISSON when the calcu-lated currents are used in each cycle, are also lumpedin the residual field B2(r) and corrected by adjustingthe constant k in each cycle. At convergence, thedistribution of the diamagnetic current in the outputfield is the same, within computational errors, as thedistribution used in the previous cycle .

A. Jain INucl. Instr. and Meth. in Phys. Res . A 340 (1994) 605-61 1

9. Conclusions

An expression for the diamagnetic current due tothe ions circulating in the MIGMA magnet has beenderived in terms of the MIGMA parameters . It isshown that the central ion density nD which can beused in a practical design, due to the diamagnetic fieldlimit, is low when compared to the value of n = 2.7 X10 14 ions/cm3 required in the design for a 3 MWpower level and methods have to be devised to over-come this limit. The radial distribution of the diamag-netic current in the annular magnet gap has also beenderived. The introduction of correction coils to cancelthe effect of the diamagnetic current on the orbitgeometry has been suggested. Calculations with themagnet design code POISSON show that effect of thediamagnetic field can be reduced by one order ofmagnitude with the aid of such coils. Thus the diamag-netic limit for the central ion density no can be ex-ceeded by one order of magnitude with these coils .These correction coils are essential if the central iondensity n = 2.7 X 1014 ions/cm3, required to produce a3 MW power level in the present design, is to beachieved .

References

[1] R. Macek and B. Maglich, Part. Accel. 1 (1970) 121.[2] For a review, see for example: B.C . Maglich, Proc . Int .

Symp. on the Feasibility of Aneutronic Power, Princeton,New Jersey, September 10-I1, 1987, eds. B.C . Maglich, J.Norwood Jr . and A. Newman, Nucl . Instr . and Meth . A271 (1988) 13 .D. A] Salameh, S. Channon, B.R . Cheo, R. Leverton, B.C .

[7] J.P . Blewett, Proc. Int . Symp . on Feasibility of AneutronicPower, Princeton, New Jersey, September 10-11, 1987,eds. B.C . Maglich, J. Norwood Jr . and A. Newman, Nucl .Instr . and Meth . A 271 (1988) 214.

[8] J.S. Colonias, UCRL-18439, 1968, LBL, Berkeley.

Maglich, S. Manasian, R.A. Miller, J.Wu, Phys . Rev. Lett. 54 (1985) 796.

Nering and C.Y .

[4] Arvind Jain, Nucl . Instr . and Meth . A 316 (1992) 391.[5] Arvind Jain, Nucl . Instr . and Meth . A 323 (1992) 671.[6] Arvind Jain, Nucl. Instr . and Meth . A 334 (1993) 562.