Nuclear Instruments and Methods in Physics Research A 349 (1994) 558-562 North-Holland

NUCLEAR INSTRUMENTS

& METHODS IN PHYSICS RESEARCH

Secbon A

Self focussing of the diamagnetic currents in the

Arvind Jain Nuclear Physics DiL, ision, Bhabha Atomic Research Centre, Bombay-400085, India

Received 21 February 1994; revised form received 3 May 1994

steady state MIGMA

It is found that the diamagnetic currents flowing in the MIGMA fusion reactor produce a vertical focussing which is many times stronger than the alternating gradient (AG) focussing used in the ring magnet. Correction coils, which cancel the vertical component of the diamagnetic field but which do not affect the focussing (radial) component, can be used to enhance this source of vertical focussing by nearly one order of magnitude when compared to the AG focussing.

The MIGMA, an accelerator based fusion reactor, has been studied both experimentally [1] and theoretically [2- 5]. The principal advantage of this reactor would be the production of nuclear power without any radioactivity and it offers an alternative and attractive route to aneutronic, non-radioactive nuclear power in the future. The reactor thus merits a detailed study. Further, the clean fusion reaction 3He + d ~ 4He + p + 18.35 MeV used in the MIGMA, where the end products are finally hydrogen and helium gases, requires centre of mass energies in the region of 250 keV which is well beyond the range of tokamaks.

The basic principle of the reactor has been described earlier [2-6]. Fig. 1 shows, schematically, an accelerator injector A injecting a molecular beam of D~ into a ring magnet. Due to stripping during injection, the molecule dissociates when D ~ 2D++ e, the charge to mass ratio q/A of the molecule changes from 1 / 4 to 1 /2 , the bending radius for the D ÷ ion with q/A = 1 / 2 is one-half that of the molecule D~- with q/A = 1/4. Thus after injection, the stripped deuteron bends completely in the magnetic field and gets trapped in a " M I G M A " orbit which passes through the centre of the magnet C as shown in the figure and repeats this orbit in each revolution. The non-dissociated beam emerges at the point E. We may define an equilibrium orbit for the ion which starts at the centre C, bends in the magnet and again passes through the centre thus closing the loop on itself. Fig. 2 shows schematically three equilibrium orbits in the radial plane displaced by an angle a, where a is the angle subtended by the orbit at the centre. The ion, as it traverses the loop, produces a magnetic field which opposes the main guiding field, the so called "diamagnetic field". As observed in Fig. 2, the ion currents in the radial parts of the neighbour- ing orbits mutually cancel, while the currents in the ring

magnet add up in the same sense. This gives rise to circular currents flowing at the outer periphery of the MIGMA disc which produce a magnetic field opposite in sense to the main coil field and which tends to cancel the guiding field. An expression for the total diamagnetic current ID[A flowing in the ring in terms of the basic MIGMA parameters has been derived earlier [5]

/DIA = 0.244 × 10-12E d z n / q B , [A] (1)

where E is the energy of the ion [keV], d z the half-thick- ness of the MIGMA disc [cms], n the central ion density [cm 3], q the charge state and B the magnetic field [T].

The radial distribution of this current in the magnet gap

} has also been derived. Here a is the annular width of the magnet in the radial direction and l (x ) / Io l A is the frac- tion of the diamagnetic current flowing in the region x, where x is measured outward radially from the inner periphery of the ring [5]. In the present work we show that the diamagnetic current given by Eq. (1) produces a verti- cal focussing for the ions in the orbit which is many times larger than even the alternating gradient focussing which is used in the ring magnet. The diamagnetic currents provide a very important source of vertical focussing for the ions as they orbit in the MIGMA.

The design of a 3 MW MIGMA has been discussed earlier [3-5] and typical parameters for the magnet of a 3 MW MIGMA are given in Table 1. For the parameters of this magnet, the diamagnetic current flowing in the ring can be calculated using Eq. (1). Using the parameters: the ion energy E = 250 keV, the magnetic field in the gap B = 0.8 T, the charge state q = 1, the central ion density n = 2.7 × 1014 ions /cm 3, the half-thickness of the migma

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A. Jain /NucL Instr. and Meth. in Phys. Res. A 349 (1994) 558-562 559

Fig. 1. D~- is injected from the injector A, bends in the magnet and gets stripped at C due to the central ion density n when D~" ~ 2D ÷ +e. The dissociated ion bends fully in the magnetic field and passes again through the centre C, and this pattern repeats, while the non-dissociated D~- beam emerges at the

point E.

disc d z = 3 cm, we get, from Eq. (1), IDt A = 64.3 kA. The distribution of this current in the magnet gap, using Eq. (2), is given in Table 2 where the radial gap has been divided into ten equal segments. We note that 43.59% of the total diamagnetic current flows in the outermost 10% segment of the magnet gap.

The diamagnetic field produced by this diamagnetic current was obtained with the magnet design code POIS- SON [7]. When used in the code POISSON, the currents in Table 2 were assumed to flow in ten coils D 1 to D10, placed in the magnet gap in the median plane, where the diamagnetic currents would actually be flowing, as shown schematically in Fig. 3a. When the currents in Table 2, corrected for self-consistent field calculations [5], are used for the coils D 1 to Din in the code POISSON, the vertical

Table 1 Magnet and orbit parameters for the 3 MW MIGMA

Magnet Radius (R 2, maximum) [cm] 366.7 (R 1, minimum) [cm] 346.7

Annular width [cm] 20 Magnet half-gap [cm] 15 Half-thickness of the MIGMA disc dz [cm] 3 Magnetic field B IT] 0.8 Coil dimensions, width × height [cm 2 ] 20 x 28 Main coil current, using POISSON [kA] 142 Power (I2R) assuming Cu coils [KW] 200 Angle a subtended by the orbit at the centre C, [deg] 3.99

component of the diamagnetic field produced in the me- dian plane is shown by the curve (2) in Fig. 3b. A peak negative field of 4.4 kG is produced at a radius R = 358.7 cm. The main coil guiding field, without the diamagnetic currents, is also shown by the curve (1) in Fig. 3b. If left uncorrected, the diamagnetic field would nearly cancel the main coil field and it is necessary to limit the diamagnetic field to less than 10% of the main coil field inorder to maintain a well defined orbit geometry. It has been sug- gested in an earlier work [5] that this large diamagnetic field produced by the diamagnetic currents could be can- celled with the aid of correction coils C 1 to Clo placed on the pole tip directly above the region where the diamag- netic currents are flowing as shown schematically in the Fig. 3a. The residual diamagnetic field left when the coils C 1 to Clo carry equal and opposite currents to the regions D 1 to Din in the code POISSON is shown by the curve (3) in Fig. 3b.

The ion makes several million revolutions before fusion occurs and vertical focussing must be provided by the magnetic field to keep the ion from straying away from the median plane. The vertical focussing of the ion, as it moves in the equilibrium orbit has been discussed earlier [4]. Due to the large gap used in the ring magnet, the fringing field gives rise to a radial component in the

Fig. 2. A schematic of three equilibrium orbits. The currents in the radial part are in opposite directions and mutually cancel. The currents in the bending field add up in the same sense, giving rise to a diamagnetic current flowing in the ring. Only ions having an orbit within an angle ot to the left of the plane AC, drawn to

bisect an equilibrium orbit, will cross to the right of this plane.

Table 2 Percentage of the diamagnetic current Inl A in ten equal segments and the currents in the segments D 1 to D10 for a total diamagnetic current 64.3 kA

D-Segment % of IDI A Current [kA]

1 0.50 0.32 2 1.52 0.98 3 2.59 1.67 4 3.74 2.40 5 5.05 3.25 6 6.60 4.24 7 8.59 5.52 8 11.41 7.34 9 16.41 10.55

10 43.59 28.03

560 A. Jain /NucL Instr. and Meth. in Phys. Res. A 349 (1994) 558-562

la) -"k- Z " M A I n "'///~

/ / / / /

/ 7 7 7 / / ,o

I

6 ~ ' 11 MAiN COIL

0 3 ) RESIOUA? - ' ~uc2~ ~ r , o pposi ;E-G., j k2 , DIAMAGNETIC

- 2 -- CURRENTSIN THE C ~'O COILS l FIELD(COILS __L__ (b) ~ J O1TO ol0 0N)

-6 l I I 13727cmsi 326.7 34 62 3 66.7 386.7

( 'RADIUS) cm-.---J,,.

Fig. 3. (a) Concentric correction coils C 1 to Cl0 are shown attached schematically to the pole tip to cancel the diamagnetic current flowing in the annular ring. The C coils carry an equal and opposite current to the currents flowing in the regions DI to D~0. (b) the median plane magnetic field obtained with the code POISSON (1) guiding field due to the main coils alone, (2) diamagnetic current field only, (3) the residual diamagnetic field when the correction coils C~ to Ct0 carry equal and opposite

currents to the D region currents.

magnetic field B, which is proportional to the height z above the median plane i.e,

Or(Z ) =kz . (3)

As the ion orbits at a height z above the median plane, it experiences an axial force F z = evoB r where v o is the component of the velocity in the 0-direction. This force is defocussing and is directed away from the median plane when the ion moves from the centre C towards the rising part of the fringing field [Fig. 1 and Fig. 3b] and then, due to the change in the sign of B r becomes focussing when the ion traverses the falling edge of the magnetic field at the outer edge of the ring. Since the path of the ion when it enters the magnetic field is nearly radial, the azimuthal velocity v o is very small and the radial velocity u r ~--- 1, in dimensionless units. The defocussing force evoB r is thus very small. When the ion turns and bends in the falling part of the magnetic field on the outer edge of the ring, the radial velocity v r = 0 and v o = 1, and the focussing force

euoB r is large. Thus while the ion moves in an equilibrium orbit at a constant height above the median plane, the net time averaged axial force ( F z) which acts on the ion is always focussing and is directed towards the median plane provided the ion turns and bends in the falling edge of the magnetic field. The net time averaged alternating gradient axial force seen by the ion is given by

( F~) = ( f / f ) f euoB,( z ) dt , (4)

where T is the time period in the equilibrium orbit. The integral in Eq. (4) can be evaluated by integration of the orbit in the magnetic field with the orbit integration code ORBIT [4,8] using the computed value of Br(r, O, z ) at each integration step.

The Coulomb repulsion between the ions within the migma disc produce an axial defocussing force F" which increases with the average ion density Nav i.e using Gauss' theorem for the electric field E on the surface of the MIGMA disc of thickness z,

F" = 4"rr e2Na. z. (5)

The maximum ion density Na, within the MIOMA disc which can be compensated by magnetic focussing alone i.e. without any neutralization by electrons, is the space charge limit (SCL) and can be obtained by equating the forces in Eqs. (4) and (5),

= ( 1 / 4 ~ v e T ) f u o [ B r ( z ) / z ] dt. (6) Nav(SCL)

Evaluating the integral for the parameters of the magnet in Table 1 by orbit integration, we obtain a value for the space charge limit Na,(SCL)= 0.2 X 10 s ions /cm 3 i.e. the Coulomb repulsion between the ions at this density will be balanced by the magnetic focussing in the absence of electron neutralization.

The vertical and radial magnetic field components B z and B r of the various coils were obtained with the code POISSON and are plotted in Fig. 4 at a height z = 1.5 cm. above the median plane for various cases.

The vertical and radial field components of the main coil field are shown by the curves I(B z) and the curve I(B r) respectively in Fig. 4. The vertical component B z of the main coil field rises upto a radius R = 264.7 cm and falls thereafter. The corresponding B r component is nega- tive upto R = 264.7 cm. and becomes positive therafter providing the alternating gradient focussing.

The vertical component of the full diamagnetic field curve II(B z) in Fig. 4, is cancelled by the vertical compo- nents of the C coil fields, leaving the residual component, curve III(B,) in Fig. 4. We note that while the B z compo- nent of the C coil nearly cancels the B, component of the corresponding D coil the radial component of the diamag- netic field [curve II(B r) in Fig. 4] is not cancelled, but is in fact slightly raised by the radial component of the C coil field, leading to the curve III(B r) in Fig. 4. As pointed out

A. Jain /NucL Instr. and Meth. in Phys. Res. A 349 (1994) 558-562 561

earlier [5], during operation, the currents in the correction coils would initially be zero, but as the central ion density and the diamagnetic current slowly builds up and reaches the steady state, the currents in the correction coils would also be required to increase proportionally, with the aid of suitably designed feedback signals, which keep the guiding field from deviating, at any instant, by more than a speci- fied value [5].

The total field, with all the coils on, i.e. the main coil, the diamagnetic currents of Table 2 in the D coils along with equal and opposite currents in the C coils is shown by the curve IV(B z) and the curve IV(B r ) respectively in Fig. 4. These last two curves are almost the linear additions of the main coil field and the residual field i.e. curve IV(B z) --- curve I(Bz) + curve III(B z) and curve IV(B r) -~ curve I(B r) + curve III(Br). The dashed vertical line in Fig. 4 shows the edge of the MIGMA disc at a radius R = 372.7 c m .

The " w e l l " in the diamagnetic field and then the sharp

340.7 346.7 RADIUS(cm) 366.7 372.7 Fig. 4. Variation of the magnetic field with the radius at a height z = 1.5 cm above the median plane obtained with the code POIS- SON for various coil positions and currents. The solid curves give the vertical component Bz: I(B~), main coil field; II(Bz), full diamagnetic field; llI(Bz) , residual diamagnetic field with equal and opposite currents in the C and D coils; IV(Bz), total field. The

dotted curves give the corresponding radial field components.

median plane

median plane

~ R

Hz

median plane Br

Br

(a)

r.......lb

H~

3rBZ~B r (c) B z

Bz

Fig. 5. (a) The magnetic field produced by a solid conductor of radius R carrying a uniform current density, (b) the magnetic field increases linearly with the radius r within the conductor and then

falls as 1 / r , (c) radial and vertical components of the field H4,.

rise beyond the radius 366.7 cms. from a value - 4 . 2 kG to + 2.2 kG [curve II(B z) in Fig. 4] can be understood by considering the magnetic field inside a conductor of radius R carrying a current I with a uniform current density as shown schematically in Fig. 5a. The magnetic field H outside the conductor is obtained from Amp~re's law

~I = I / 2 ~ r . (7) Inside the conductor, the value of H at a radius r is determined solely by the current inside the radius r. Thus inside the conductor, (for r < R) H = l ' / 2~rr , where I ' = I ( r / R ) e. Thus inside the conductor,

H = I r / ( 2 7 r R 2) (8)

i.e., inside the conductor, the field increases linearly with the radius. At the surface of the conductor at r = R, Eq. (7) equals Eq. (8). The variation of the field H from the centre of the conductor as a function of the radius r is plotted in Fig. 5b. H z = H 6 in the median plane. We note that the diamagnetic current flowing in Dlo in Fig. 3a is equivalent to the conductor in Fig. 5a. except for the rectangular cross-section of the coil D10 and the non-uni- form current density of the diamagnetic current. The basic forces in both the cases will remain the same. The position of the coil D10 is also shown in Fig. 4. Qualitatively, the total diamagnetic field of the coils D~ to Dlo [curve 2 in Fig. 3b or curve II(B z) in Fig. 4] can be obtained from Fig. 5b by superimposing the field profile of the current carry- ing conductor in Fig. 5b for each D coil, displaced radially to the left by an amount DR for each D coil position and

562 A. Jain /Nucl . Instr. and Meth. in Phys. Res. A 349 (1994) 558-562

making the peak field for each of the D coil proportional to the percentage of currents in Table 2. The resultant superposition for the ten coils D a to D10 of the field profiles of the type in Fig. 5b will produce the resultant curve 2 in Fig. 3b or the curve II(B z) in Fig. 4.

The field components B z and B r within the conductor of Fig. 5a are examined in Fig. 5c. The current in the conductor produces a radial field component Br which is directed away from the centre of the magnet C above the median plane, and towards the magnet centre below the median plane. As discussed earlier, this component B r will produce an axial focussing force evoB r which will always be directed towards the median plane. Thus although the B z component of the diamagnetic field is negative, (oppo- site to the guiding field) upto the centre of the coil D10, it gives rise to radial field components B r [cf. Fig. 5c] which are always focussing in the vertical direction. The radial field component produced by the diamagnetic field [curve llI(Br)] is very large when compared to the corresponding alternating gradient component [curve I(Br)] due to the main coil in Fig. 4. Furthermore, while the B r component of the main coil field [curve I(Br) in Fig. 4] changes sign at the peak of the main coil field at the radius R = 264.7 cm, the B~ component of the diamagnetic field [curve II or III(B r) in Fig. 4] increases monotonically with the radius right from the centre of the magnet.

Thus the radially focussing field component in the total field [curve IV (Br) in Fig. 4] has contributions from two sources viz. the main coil field I(B r) and the residual diamagnetic field III(Br). When the space charge limit in Eq. (6) for the average ion density is evaluated for the total field, it increases by a very large factor due to the large radial diamagnetic component III(Br). The space charge limit for the average ion density obtained by orbit integra- tion in the total field becomes N~v = 1.3 x 10 s ions /cm 3 i.e. a factor of 6.5 more than the value 0.2 X l0 s ions /cm 3 obtained if alternating gradient focussing of the main coil field I(B r) is used alone. Thus the space charge limit is increased by a very large factor due to the vertical fo- cussing action of the diamagnetic currents. This "se l f focussing" of the ion currents in the D coils can easily be identified with the same forces which bend an electron normal to its direction of motion and set an upper limit on the intensity of an electron beam which can be propagated in vacuum, i.e. the Alfv6n current limit.

In conclusion, it has been shown that the diamagnetic current in the MIGMA disc gives rise to an additional source of vertical focussing for the ions which is many times stronger than even the alternating gradient focussing used in the ring magnet. Calculations with the code POIS- SON for a typical 3 MW design indicate that the space charge limit can be increased from a value Nay = 0.2 X 108

to 1.3 X 108 ions /cm 3 i.e. by a factor of about seven due to this additional focussing. We note that this source of focussing is distinct from both " w e a k " focussing and "alternating gradient focussing" i.e. the radial field com- ponent B r is focussing even when the vertical component B z of the total guiding field rises with the increasing radius. Due to this distinct kind of focussing, the plasma within the M1GMA disc is situated in a minimum field position upto the edge of the disc [curve IV(B z) in Fig. 4], i.e. the magnetic field strength increases outward from the centre toward the periphery of the system. The peculiar nature of the diamagnetic field, especially the sharp rise from a negative field value to a positive value beyond the last D coil, leads to a "minimum B geometry", a situation very favourable for the radial stability of the plasma.

We note that this additional source of very strong focussing can be utilized only with the aid of the "correc- tion coils" mentioned above [5]. Thus the presence of an initially large diamagnetic current given by the Eq. (1) is required in the initial design. A subsequent cancellation of the large Bz component of this diamagnetic field by the correction coils including the main coil, while the required B r component is retained intact, is required for the addi- tional focussing. It is emphasised that the large increase in the vertical focussing and the space charge limit by nearly one order of magnitude produced by the self-focussing of the diamagnetic currents discussed above can be utilized and will play a very crucial role in the confinement of the electrons and ions within the MIGMA disc and thus in the successful functioning of the MIGMA type of fusion reac- tor.

References

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[2] 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, Nucl. Instr. and Meth. A 271 (1988) 13.

[3] Arvind Jain, Nucl. Instr. and Meth. A 323 (1992) 671. [4] Arvind Jain, Nucl. Instr. and Meth. A 334 (1993) 562. [5] Arvind Jain, Nucl. Instr. and Meth. A 340 (1994) 605. [6] John P. Blewett, Proc. Int. Symp. on Feasibility of Aneutronic

Power, 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.

[7] John S. Colonias, UCRL-18439, 1968, LBL, Berkeley. [8] Arvind Jain, Proc. Nucl. & SSP Symp., Feb. 1-4, 1972,

Bombay, vol. 14 B (1972) 531, (Library and Information Division, BARC, Bombay-85, India).