A new accelerator option for driving a sub-critical reactor

Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

A new accelerator option for driving a sub-critical reactor G.H. Rees n ASTeC Division, Rutherford Appleton Laboratory, STFC, Chilton, Didcot, Oxon OX11 0QX, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 May 2012 Accepted 5 June 2012 Available online 18 June 2012

A linac and two cyclotrons, of a new type, are considered as a proton driver for an ADSR sub-critical reactor. The cyclotrons use reverse bending, for orbit length adjustments, and harmonic number jumps to achieve a synchronous acceleration. This is a different basis from that of a separated orbit cyclotron (SOC) where accelerating fields must vary over the cavity apertures in a defined manner to maintain synchronism. The new ring is called an orbit separated cyclotron (OSC) as it has the shape of an irregular SOC. An OSC design example is presented, with a six-cell superconducting cavity in each of the eight ring periods. Acceleration, over four turns, of a 10 mA (cw) proton beam from 0.5 to 1.0 GeV relates to input and output beam powers of 5 and 10 MW, respectively. A key design issue is the unusual, spiral magnet system, with all 32 superperiods requiring different combined function magnet designs. One main advantage of an OSC ring over a linac stage is the four times fewer cryogenic and cavity systems, due to the use of the four turn acceleration. Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.

Keywords: ADSR OSC Proton driver SOC Superfish

1. Introduction New challenges arise in a 3–10 MW, 1–1.5 GeV, proton accelerator for driving a sub-critical reactor (ADSR). A stable, reliable beam current is required to limit the thermal-stress damage induced in the neutron target at the reactor core. The reliability and the availability of an ADSR [1] depend on the extent of component redundancy and on the number of high power acceleration units in the proton driver, all with rapid response times for protection. A new driver option is studied to see if the number of such systems can be reduced. CW beam operation is preferred, both for lower target stress levels and to avoid switch-on power for the driver rf acceleration systems. Synchrotrons are thus not an option, but linacs, FFAGs and cyclotrons are. The most straight-forward of these, a superconducting linac, needs to have many rf cavities, so a scheme for beam re-circulation is sought. The range of the proton velocities is unsuitable for a re-circulating linac, so a new type of ring accelerator is considered, with a single, multi-cell superconducting cavity in each of the ring periods. Different energy, circulating beams have to arrive each turn with the correct rf phase at each superconducting cavity, despite their limited apertures. To achieve this, there are only four beam circulations, harmonic number jumps are used, and some reverse bending is added to expand the arc orbits (see overleaf). Orbits spiral via the common rf cavities and the separated, combined-

n

Tel.: þ44 1635200429. E-mail address: [email protected]

function magnet arcs as the beam energy rises. The length adjustments for synchronous acceleration are a basic design difference from that of a separated orbit cyclotron (SOC) [2]. The new ring is called an orbit separated cyclotron (OSC) as it has the shape of an irregular SOC. The OSC offers several benefits. The non-isochronous, separated beam orbits have both longitudinal and transverse focusing and allow direct beam injection and extraction, so beam loss is less likely than in other types of ring accelerators. The design avoids dissipative, room temperature cyclotron-type cavities, at frequencies r  100 MHz. The acceleration over four-turns leads to four-times fewer cryogenic, cavity stages than for a cw linac, and each of the four separated beam orbits in the superconducting cavities may have zero dispersion. The cavity beam currents of  40 mA are lower than in several proton linacs, while space charge forces in the arcs are only for  10 mA currents. The ring has long straights for the cavities and shorter straights in the arcs for injection, extraction and correction units. Each combined function magnet has a common yoke, but separate poles for the separated orbits. Beam dynamics is as in a linac, but new study areas are the long, beam transport in the bending-focusing sections, the interactions between the separated beams in each cavity, and the effect of phase slip in the fixed geometric-beta (bg) cavities. 2. Layout of the orbit separated cyclotron (OSC) A ring superperiod extends from one long straight section centre to the next, as shown schematically in Fig. 1. There are

0168-9002/$ - see front matter Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.06.006

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

F

D

F f

triplets d f

d cavity

B (θ) b t b

F D F

S

reverse bends

cavity

S c

s π/N

s

85

s

S

2(π/N−θ) s b t b

π/N

S

d orbit c orbit

ejected beam Fig. 3. Two adjacent orbits of an OSC ring of N superperiods.

cavity

reverse bends

injected beam

cavity

Fig. 1. Schematic drawings of the OSC ring superperiods.

Fig. 3, with straight sections 2S, s and t (t may be zero), and forward, B(y), and reverse, b(p/N  y), bend units. In expanding the arcs, the lengths, L, of the superperiods vary more than H, the end separations. A closed polygon forms if H(dd) and H(cc) remain fixed over a turn so, for orbits to spiral outwards as the beam is accelerated, H must increase by DH; e.g. in the case of N ¼8, spiral ‘‘octagons’’ require the following: Superperiods DH ¼

14 58 912 1316 1720 2124 2528 2932 0

D

2D

3D

4D

5D

6D

7D

,

ð1Þ H=2 ¼ S cos p=N þ rðsinp=N þ 2 sinðyp=NÞÞ þ s cosðyp=NÞ þ t=2, ð2Þ

Fig. 2. Pattern of straight lines between the superperiod ends.

either four separated orbits, or three together with an injection and an ejection orbit. A six-cell superconducting cavity is set centrally in each long straight section and there are separate correction units in the short straights. Some of the central magnets are sub-divided, providing further short straights. To visualize the ring, consider straight lines joining the start and end of each superperiod. A closed polygon forms if all have equal length, so lengths are varied to obtain separated, spiral orbits. The scheme chosen for an eight period per turn ring is to make the first four lengths equal, and to have increasing, equal incremental lengths for the next seven groups of four, as in Fig. 2.

3. Magnet lattice design for the orbit separated cyclotron, OSC N superperiods of the magnet lattice need to have

 a length, nbl, for an integer, n, beam velocity, bc, and rf wavelength, l

2D ¼ HðddÞHðccÞ ¼ 2ðcdÞ sinp=N,

ð3Þ

L=2 ¼ S þ rð2yp=NÞ þ s þ t=2ð  nbl=2Þ,

ð4Þ

Bend angles are large, as indicated in Fig. 1, and reverse bends are a key item, both to adjust the arcs and to reduce the lengths needed for the zero dispersion straight sections. Harmonics, n, are chosen to allow synchronous cell lengths for equal values of (H  DH). Conditions needed are most readily met if the superperiods are evaluated in sequence from the final to initial energy, as the larger angles at low energy allow more design flexibility. Parameters include lengths, harmonic numbers (n), magnet gradients, bending angles and radii (r) and superconducting cavity frequency and beam separations. Beam energies change in each superperiod, but beam arrives with the correct rf phase at the following cavity if the chosen parameters satisfy Eqs. (1–5). A linac, based on 4N, equal field level, OSC-type cavities for the OSC energy range, is used to find the beam velocities. Cavity transit time factors depend on the off-axis beam positions. The first and fourth turn beams are 9 cm off-axis and the second and third turn beams, 3 cm off-axis (see Section 4). The data tables for these, and for on-axis beams, are derived from the Superfish code, and given in Appendix A. The linac cavity velocities are then used to find each of the synchronous OSC superperiod lengths (L¼nbl) required:

 equal lateral spacing between separated adjacent long straight     

sections the centres of bends in the arcs aligned with those of the adjacent turns dispersion-free, cavity straight sections for closely spaced parallel beams increasing cavity straight section lengths for the beam being accelerated each of the combined function magnets sub-divided for triplet focusing magnets with adjustable focusing gradients, bend angle and bend radius

A four-turn ring with N superperiods per turn needs 4N, separate designs. Each one has the form S B s b t b s B S of

L ¼ nbl þ ðbg l=2Þð6

6 X 4

b=bm 

3 X

b=bn Þ,

ð5Þ

1

where b is the value in a superperiod arc, bm are average b in cells 4, 5, 6 of the upstream, six-cell cavity, and bn are the average b in the cells 1, 2, 3 of the downstream cavity. The B and the bb magnet poles in all 4N superperiods have separate correction windings. The subdivided poles give continuous triplet focusing (except when a bb is separated). A symmetric bb triplet and two symmetric B triplets in a superperiod have mirror symmetry about the arc and long straight centres, thus ensuring that the lattice a(v), a(h) and a0 (p) functions are zero at these points. Four field gradients in each superperiod are adjusted to obtain smooth b(v) and b(h) transitions at the centre

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G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

Fig. 4. Beam amplitudes for 10 mA in OSC ring (upper) and associated linac (lower).

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

of each long, non-dispersive straight section, though the lattice functions in the arcs differ from arc to arc. 4. Accelerating cavities A superconducting linac stage is usually designed with constant cavity energy gains and beam powers (though it may not operate as designed). Constant energy gains are obtained in fixed, geometric–beta (bg) cavities by increasing the fields to compensate for the phase-slips when b a bg. This is not possible in the bg cavities of the OSC ring as its four beams have different b values and phase slips. It is simpler using constant cavity fields than aiming for constant cavity beam powers. A separate klystron

87

powers each cavity, e.g. an OSC ring providing 5 MW beam power requires N, 6.5/N MW, cw klystrons, assuming 25% for control power, and 5% feeder losses. The power ratings are practical as a typical value of N is eight, or more. A six-cell, elliptical superconducting cavity is set centrally in each long straight. At an operating frequency of 324 MHz and a free space wavelength of 0.9252856 m, a bore diameter of 22 cm accommodates the 18 cm between the first and the fourth turn beams (6 cm turn separation). Appendix A gives cavity transit time factors [3] for 9 and 3 cm off-axis beams (the transit time factors for on-axis beams are lower). The shape of the cavity is optimized using the Superfish code, to obtain the values of Emax/ Eav E2.1 and Bmax/Emax E2.3 mT/ (MV/m).

Fig. 5. Beam phase space plots at 501 Mev (upper 4) and 1000 MeV (lower 4).

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G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

Table A1 Superfish input data for the 6-cell, 324 MHz superconducting cavities. Design beta Particle Super Conductor number of cells Sequence_number Beta E0T_normalization Frequency Diameter Bore_radius Second_beam_tube Second_tube_Radius Delta_frequency Mesh_size Increment Start

File name prefix Right_Beam_tube Right_Dome_B Dome_B Dome_A/B Wall_Angle Equator_flat IRIS_flat IRIS_A/B

0.82 Hþ 2 6 1 0.75–0.88 10.0 324.0 83.10778579853 11.0 0 0 0.01 0.2 2 1–5

9.2

1.00000E  08

Inner cells

End cells

82B 0

82BE 30.0 6.89068918198 7.0 and left_DOME_B 1.0 and left, right_DOME_A/ B 8.0 and left, right_Wall_angle 0 and left, right_Equator_flat 0 and left, right_IRIS_flat 1.4 and & left, right_IRIS_A/B

7.0 1.0 8.0 0 0 1.4

Table A2 SFDATAQ cavity end cell data tables for respective 9, 3, 0 cm off-axis beams. Beta

T

Tp

S

Sp

g/bl

Z

E/E0

Tave

dZct

b (g)

0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88

0.7638161 0.7667860 0.7696628 0.7724512 0.7751556 0.7777750 0.7803185 0.7827885 0.7851857 0.7875145 0.7897779 0.7919785 0.7941180 0.7962006

0.066398 0.065689 0.064999 0.064327 0.063672 0.063035 0.062413 0.061806 0.061215 0.060638 0.060076 0.059526 0.058991 0.058467

0.255162 0.253838 0.252539 0.251265 0.250015 0.248789 0.247586 0.246405 0.245246 0.244110 0.242994 0.241900 0.240826 0.239771

0.02783 0.02785 0.02787 0.02788 0.02789 0.02790 0.02780 0.02790 0.02789 0.02789 0.02788 0.02787 0.02786 0.02784

0.033646 0.030427 0.030066 0.029971 0.031979 0.031616 0.031262 0.028303 0.027995 0.027935 0.029836 0.029525 0.029222 0.026481

5831810 5813671 5792833 5771722 5749561 5728093 5706536 5686885 5665447 5643905 5621682 5600406 5579329 5560385

1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.7638161 0.7667860 0.7696628 0.7724512 0.7751556 0.7777750 0.7803185 0.7827885 0.7851857 0.7875145 0.7897779 0.7919785 0.7941180 0.7962006

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88

0.7038782 0.7076343 0.7112756 0.7148067 0.7182319 0.7215631 0.7247970 0.7279384 0.7309946 0.7339680 0.7368615 0.7396792 0.7424231 0.7450971

0.077800 0.077126 0.076462 0.075808 0.075165 0.074530 0.073905 0.073290 0.072685 0.072089 0.071503 0.070926 0.070358 0.069799

0.268145 0.267205 0.266266 0.265330 0.264397 0.263467 0.262541 0.261619 0.260703 0.259791 0.258885 0.257985 0.257090 0.256202

0.02394 0.02416 0.02436 0.02455 0.02473 0.02489 0.02505 0.02520 0.02534 0.02546 0.02559 0.02570 0.02580 0.02590

0.033646 0.030427 0.030066 0.029971 0.031979 0.031616 0.031262 0.028303 0.027995 0.027935 0.029836 0.029525 0.029222 0.026481

5831810 5813671 5792833 5771722 5749561 5728093 5706536 5686885 5665447 5643905 5621682 5600406 5579329 5560385

1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.7038782 0.7076343 0.7112756 0.7148067 0.7182319 0.7215631 0.7247970 0.7279384 0.7309946 0.7339680 0.7368615 0.7396792 0.7424231 0.7450971

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

0.75 0.76 0.78 0.79 0.81 0.82 0.84 0.85 0.87 0.88

0.6968693 0.7006414 0.7080713 0.7116597 0.7184447 0.7215945 0.7278266 0.7308519 0.7365942 0.7392704

0.079039 0.078389 0.077065 0.076409 0.075142 0.074545 0.073334 0.072734 0.071578 0.071032

0.269515 0.268642 0.266826 0.265908 0.264102 0.263235 0.261448 0.260548 0.258787 0.257943

0.02344 0.02368 0.02411 0.02432 0.02468 0.02484 0.02514 0.02528 0.02553 0.02564

0.033646 0.030427 0.029971 0.031979 0.031262 0.028303 0.027935 0.029836 0.029222 0.026481

5831810 5813671 5771722 5749561 5706536 5686885 5643905 5621682 5579329 5560385

1 1 1 1 1 1 1 1 1 1

0.6968693 0.7006414 0.7080713 0.7116597 0.7184447 0.7215945 0.7278266 0.7308519 0.7365942 0.7392704

0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

5. A 10 mA, 500–1000 MeV, OSC ring example A 500–1000 MeV, OSC ring with a cw beam current of 10 mA has a 5 MW input and a 10 MW output beam power. The rf synchronous phase angle, js, is chosen at  151 and the peak electric fields in the cavities are kept o25 MV/m. Eight, six-cell superconducting cavities are then required, each one fed from a 0.8125 MW cw klystron,when 25% is assumed for control power and 5% for the feeder losses. The off-axis beams have larger transit time factors, allowing 2% lower cavity fields to be used. The number of ring superperiods per turn is thus made eight, and harmonic numbers (n), bend angles, bend radii (r) and element lengths have to be found for the 32 superperiods. The long straight sections are made Z6.0 m, and the bending radii, r, are set for practical field levels in room temperature magnets. The parameters for the different superperiods are given in Tables B1–B3 of Appendix B. A smooth transition of focusing is needed. The FDF–fdf–FDF triplets in each superperiod have D, twice the length of F, and d twice that of f, and their field gradients are set for a smooth b(h) and b(v) transition at the centre of the non-dispersive, straight sections. The F, D, f, and d normalized field gradients, the zero current, q(h) and q(v) superperiod tunes and the b(h) and b(v) values are given in the Table B3. Superperiod tunes are 40.570 and gamma transition values are imaginary. Large differences in normalized field gradients

89

are needed, as shown in the Table. The respective maxima of b(v), b(h) and ap(h) are 14.26, 7.34 and 3.81 m at low energy and 19.78, 8.84 and 2.47 m at the top energy.

6. Beam tracking study for the 500–1000 MeV, OSC ring design A linac code has been adapted to track beam in the 8, 6-cell, superconducting cavities and the 32 matched adjacent arcs. Fig. 4 gives tracking results for 105 protons, representing 10 mA beam currents, in both the OSC ring and the associated linac (which used doublet quadrupole focusing for 32 of the same 6-cell cavities). Fig. 5 gives the beam phase space plots for the ring at 500 and at 1000 MeV. The input beam distributions are 6-D waterbag, of local Gaussian density, with normalized rms emittances of 0.397 (p) mm mr, transversely, and 0.2341 MeV, longitudinally. Beam dynamics for the linac and ring are different due to the dispersion in the ring arcs. The 10 mA beam is accelerated smoothly in both cases, but the ring has less longitudinal focusing due to its longer superperiods. Beam amplitudes in the example ring vary more in the vertical than in the horizontal plane, as seen in Fig. 4. Emittance growths are small in both linac and ring. Those for the ring at 1000 MeV are: rms and total growths, respectively, of  3.0% and 39% longitudinally,  1.5% and  7%

Table A3 SFDATAE cavity inner cell data tables for respective 9, 3, 0 cm off-axis beams. Beta

T

Tp

S

Sp

g/bl

Z

E/E0

Tave

dZct

b(g)

0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88

0.7903101 0.7922977 0.7942366 0.7961271 0.7979724 0.7997724 0.8015292 0.8032451 0.8049210 0.8065587 0.8081582 0.8097215 0.8112522 0.8127460

0.061315 0.060767 0.060231 0.059709 0.059198 0.058699 0.058212 0.057736 0.057271 0.056815 0.056370 0.055935 0.055509 0.055092

0.249335 0.248167 0.247020 0.245894 0.244788 0.243702 0.242637 0.241590 0.240562 0.239552 0.238561 0.237588 0.236630 0.235691

0.02954 0.02948 0.02943 0.02937 0.02931 0.02925 0.02920 0.02914 0.02908 0.02902 0.02897 0.02891 0.02885 0.02880

0.028053 0.027726 0.027408 0.027544 0.027237 0.026340 0.026056 0.025778 0.025508 0.025244 0.025396 0.025140 0.024349 0.024111

4605361 4633828 4659991 4683984 4705932 4725972 4744211 4760778 4775788 4789333 4801534 4812481 4822261 4830976

1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.7903101 0.7922977 0.7942366 0.7961271 0.7979724 0.7997724 0.8015292 0.8032451 0.8049210 0.8065587 0.8081582 0.8097215 0.8112522 0.8127460

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88

0.7854816 0.7862767 0.7870837 0.7879023 0.7887297 0.7895669 0.7904124 0.7912650 0.7921241 0.7929891 0.7938593 0.7947349 0.7956115 0.7964945

0.062497 0.062277 0.062054 0.061827 0.061598 0.061367 0.061133 0.060897 0.060659 0.060419 0.060178 0.059935 0.059692 0.059448

0.250247 0.249766 0.249278 0.248781 0.248278 0.247769 0.247254 0.246734 0.246209 0.245679 0.245146 0.244609 0.244069 0.243526

0.029254 0.029229 0.029203 0.029176 0.029149 0.029122 0.029094 0.029066 0.029037 0.029008 0.028978 0.028949 0.028919 0.028888

0.02805 0.02773 0.02741 0.02754 0.02724 0.02634 0.02606 0.02578 0.02551 0.02524 0.02540 0.02514 0.02435 0.02411

4605361 4633828 4659991 4683984 4705932 4725972 4744211 4760778 4775789 4789333 4801534 4812481 4822261 4830976

1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.7854816 0.7862767 0.7870837 0.7879023 0.7887297 0.7895669 0.7904124 0.7912650 0.7921241 0.7929891 0.7938593 0.7947349 0.7956115 0.7964945

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

0.75 0.76 0.78 0.79 0.81 0.82 0.84 0.85 0.87 0.88

0.7853151 0.7859986 0.7874116 0.7881390 0.7896310 0.7903937 0.7919485 0.7927389 0.7943414 0.7951531

0.062536 0.062347 0.061956 0.061755 0.061342 0.061130 0.060699 0.060480 0.060036 0.059811

0.250254 0.249838 0.248977 0.248533 0.247621 0.247154 0.246200 0.245714 0.244728 0.244228

0.02924 0.02922 0.02917 0.02915 0.02910 0.02907 0.02902 0.02899 0.02894 0.02891

0.028053 0.027726 0.027544 0.027237 0.026056 0.025778 0.025244 0.025396 0.024349 0.024111

4605361 4633828 4683984 4705932 4744211 4760778 4789333 4801534 4822261 4830976

1 1 1 1 1 1 1 1 1 1

0.7853151 0.7859986 0.7874116 0.7881390 0.7896310 0.7903937 0.7919485 0.7927389 0.7943414 0.7951531

0 0 0 0 0 0 0 0 0 0

0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

90

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

horizontally and  1.8% and  17.5% vertically. Details of the emittance growths are as follows: Plane Longitudinal Horizontal Vertical

Normalized rms 0.235–0.242 0.396–0.402 0.397–0.404

Normalized total 2.589–3.589 (p) deg MeV 4.345–4.637 (p) mm mr 4.352–5.117 (p) mm mr

7. Ring beam dynamics Transit time tables for off-axis cavity fields [3] are given in Tables A2 and A3 of Appendix A. As short bunch space charge fields are better defined in linac than in ring tracking codes, a linac code has been used to track 10 mA beams in the OSC ring and in its related linac. Emittance growths are small in both cases, but additional ring studies are now required:

beam deflections. Separate orbit correctors are planned for each superperiod to limit the coherent deflections due to the transverse space charge and cavity fields.

8. A proton driver chain for an ADSR One option for an ADSR proton driver is a 0.25 GeV linac and two OSC rings of energies 0.25–0.5 and 0.5–1.0 GeV. Each stage may have a frequency of 324 MHz. The cw linac includes a proton ion source, an RFQ and room temperature and superconducting stages. The room temperature stages include the ion source, LEBT, 3 MeV RFQ, MEBT and a 25 MeV Alvarez drift tube linac, with the complete section duplicated to allow regular maintenance. The 25–250 MeV, superconducting stage consists of only spoke cavities or spoke cavities followed by elliptical multi-cell cavities. The first OSC ring needs a larger range of beam velocities than the second, and remains to be studied.

 the effects of different types of machine errors on the ring performance

 the effects of the off-axis, transverse deflecting fields in the cavities  the effect of space charge forces changing from the arcs to the cavities  the effects of the beam phase slips in the fixed bg accelerating cavities  the effects of the dispersion in the orbit bending and focusing sections The effect of the machine errors on the ring’s performance is the major study required. Cavity transverse fields are under study using the Superfish code (3) and will be used for estimates of

9. Summary A linac and two OSC rings are considered as a new option for an ADSR proton driver. The new feature of the orbit separated cyclotron (OSC) is its use of harmonic number jumps, reverse bending and orbit length adjustments for synchronous acceleration. This is in contrast to a separated orbit cyclotron (SOC) where, to maintain the synchronism, the accelerating fields have to vary over the cavity apertures in a defined manner. The main advantage of an OSC over a linac is the four-times fewer cryogenic and cavity systems, due to the use of the

Table B1 Velocities, n values and superperiod lengths for the 500–1000 MeV OSC. Superperiod

b ¼ v/c

n

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0.76353 0.76903 0.77442 0.77969 0.78482 0.78981 0.79466 0.79937 0.80379 0.80807 0.81222 0.81625 0.82014 0.82391 0.82755 0.83108 0.83449 0.83779 0.84099 0.84408 0.84708 0.84997 0.85278 0.85550 0.85822 0.86086 0.86341 0.86588 0.86827 0.87060 0.87285 0.87503

29 29 29 29 28 28 28 28 27 27 27 27 27 27 27 26 26 26 26 26 26 26 26 25 25 25 25 25 25 25 25 Direct beam extraction is

L¼ nbl þ (bgl/2) (6  S b/bm)  Sb/bn) (5) Cell 1: n ¼29, b ¼0.76353, bg ¼0.82, l ¼ 0.9252856 m, b(500 MeV)¼ 0.75791 and b4 ¼ 0.76099, b5 ¼ 0.76204, b6 ¼ 0.76305, b1 ¼ 0.763875, b2 ¼ 0.76465, b3 ¼ 0.765565,

L (m) 20.488016  0.000511¼ 20.487505 20.635599  0.000433 ¼20.635166 20.780230  0.000381¼ 20.779849 20.921641  0.000341 ¼20.921300 20.333114  0.000288 ¼20.332826 20.462394  0.000230¼ 20.462164 20.588048  0.000193¼ 20.587855 20.710075  0.000234¼ 20.709841 20.089853  0.000121¼ 20.080732 20.187779  0.000082¼ 20.187697 20.291457  0.000051¼ 20.291406 20.392138  0.000057¼ 20.392081 20.489320  0.000027¼ 20.489293 20.583505  0.000008 ¼ 20.583497 20.674442 þ0.000017 ¼20.674459 19.993645 þ0.000024 ¼19.993668 20.075681þ 0.000049¼ 20.075730 20.155070 þ 0.000068¼ 20.155138 20.232054þ 0.000067 ¼ 20.232121 20.306391þ 0.000096¼ 20.306487 20.378564 þ0.000081 ¼20.378645 20.448090 þ 0.000103¼ 20.448193 20.515691 þ0.000098 ¼20.515789 19.789545 þ0.000145 ¼19.789690 19.852465 þ0.000115 ¼19.852580 19.913534 þ0.000115 ¼19.913649 19.972520þ 0.000115 ¼19.972635 20.029657þ 0.000121 ¼20.029778 20.084943þ 0.000143 ¼20.085086 20.138841 þ0.000125 ¼20.138966 20.190888þ 0.000125 ¼20.191013 made from the arc of the last cell.

Mean R (m) 26.085501 26.273508 26.457725 26.637826 25.888557 26.053235 26.213270 26.368588 25.567581 25.703773 25.835820 25.964003 26.087777 26.207721 26.323538 25.456729 25.561212 25.662342 25.760336 25.855021 25.946896 26.035447 26.121513 25.197015 25.277089 25.354845 25.429948 25.502705 25.573125 25.641727 25.707995

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

91

Table B2 Bend radii (r), bend angles and superperiod and ‘‘octagon’’ lengths for OSC ring. Superperiod

S (m)

r (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.450 3.450 3.450 3.450 3.450 3.450 3.450 3.450 3.900 3.900 3.900 3.900 3.900 3.900 3.900 3.900 4.350 4.350 4.350 4.350 4.350 4.350 4.350 4.350

3.5875 60.5  38.0 10.243752 3.4455 62.0  39.5 10.317583 3.3337 63.5  41.0 10.389924 3.2215 64.5  42.0 10.460650 3.5000 62.0  39.5 10.166413 3.4350 64.0  41.5 10.231082 3.4000 65.0  42.5 10.293927 3.3853 62.0  39.5 10.354920 3.8000 52.0  29.5 10.040366 3.7392 54.0  31.5 10.093848 3.5992 56.0  33.5 10.145703 3.4406 57.0  34.5 10.196040 3.3656 58.0  35.5 10.244646 3.3145 60.0  37.5 10.291748 3.22773 61.0  38.5 10.337229 3.9169 47.0  24.5 9.9968340 3.6441 48.0  25.5 10.037865 3.5084 50.0  27.5 10.077569 3.4180 52.0  29.5 10.116060 3.3546 54.0  31.5 10.153243 3.23045 54.0  31.5 10.189322 3.2110 56.0  33.5 10.224096 3.1326 57.0  34.5 10.257894 3.8800 43.0  20.5 9.8948450 3.8000 43.0  20.5 9.9262900 3.8000 45.0  22.5 9.9568245 3.8000 47.0  24.5 9.9863175 3.8574 45.5  23.0 10.014889 3.6379 45.5  23.0 10.042543 3.6012 48.0  25.5 10.069483 3.4870 49.5  27.0 10.095506 Direct injection of beam is made in the arc upstream of this final cell.

b (deg)

y (deg)

L/2 (m)

s (m)

t/2 (m)

(H  DH)/2 (m)

1.0763041 1.2138490 1.3096920 1.4726050 0.4738012 0.4899300 0.6090790 1.3578310 1.0497820 1.0640030 1.0735030 1.2514840 1.3023820 1.2014770 1.2818450 1.6588942 1.4631520 1.4320020 1.3541490 1.2473200 1.4686620 1.3082910 1.3552070 0.5184031 0.3245108 0.4284552 0.4452584 1.0531730 1.3432500 1.0998030 1.0897450

0.00000 0.00000 0.00000 0.00000 0.49233 0.41621 0.30567 0.00000 0.13530 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.17630 1.04030 0.70160 0.44900 0.00000 0.00000 0.00000 0.00000

9.410025 9.410027 9.410039 9.410026 9.410026 9.410025 9.410027 9.410025 9.410030 9.410025 9.410024 9.410021 9.410032 9.410036 9.410036 9.410028 9.410026 9.410021 9.410027 9.410028 9.410025 9.410033 9.410034 9.410023 9.410028 9.410027 9.410030 9.410024 9.410030 9.410022 9.410031

Table B3 Normalized gradients, superperiod tunes and b(h), b(v) values for OSC ring. Superperiod. F (m  2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

 0.31151164  0.29044800  0.26725408  0.24635907  0.29637837  0.27359592  0.26076703  0.28665583  0.47796087  0.45089648  0.42849287  0.41924464  0.40517238  0.37756447  0.36299691  0.55579697  0.62096526  0.59528312  0.56104340  0.53413239  0.54134699  0.50583025  0.48771294  0.68116238  0.75103073  0.70544401  0.65944341  0.68413702  0.71631073  0.66440036  0.64713858 There is no synchronous

D (m  2) 0.35829215 0.34869683 0.33600447 0.32632819 0.34480333 0.33000246 0.32247630 0.34994242 0.48602993 0.46544059 0.45295766 0.45464507 0.44717736 0.42555158 0.41897252 0.55359184 0.62055988 0.60509614 0.57724948 0.55877672 0.57518275 0.54320702 0.53441077 0.66699901 0.73129891 0.68739391 0.64512439 0.66735523 0.71004649 0.66349278 0.65421797 length requirement for

f (m  2)

d (m  2)

q (h)

q (v)

b (h)

b (v)

 0.13502710  0.12646700  0.11837200  0.11046020  0.13184060  0.12473405  0.12181730  0.12943690  0.16682660  0.15628870  0.14533280  0.13625960  0.12958360  0.12030335  0.11252155  0.20181450  0.17987610  0.16226910  0.14729840  0.13290000  0.12789240  0.11694540  0.10913290  0.26453720  0.25361700  0.21257500  0.18551040  0.19708700  0.19826060  0.17298360  0.15884470 this final superperiod of

0.285 0.286 0.287 0.288 0.289 0.290 0.291 0.292 0.293 0.294 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 0.295 the ring.

0.5893 0.5817 0.5748 0.5680 0.5922 0.5859 0.5813 0.5789 0.6516 0.6460 0.6395 0.6339 0.6302 0.6262 0.6224 0.6611 0.6842 0.6785 0.6741 0.6700 0.6668 0.6642 0.6575 0.6986 0.7166 0.7076 0.6995 0.7005 0.6964 0.6926 0.6887

0.6718 0.6808 0.6905 0.6991 0.6732 0.6902 0.6986 0.6830 0.6456 0.6584 0.6673 0.6714 0.6766 0.6862 0.6917 0.6256 0.6585 0.6686 0.6646 0.6887 0.6877 0.6945 0.6988 0.6034 0.6450 0.6662 0.6804 0.6767 0.6762 0.6904 0.6957

4.702 4.700 4.700 4.701 4.701 4.702 4.702 4.700 4.701 4.700 4.702 4.702 4.701 4.701 4.700 4.701 4.701 4.701 4.702 4.702 4.701 4.701 4.802 4.902 5.002 5.100 5.202 5.200 5.200 5.200 5.201

5.701 5.701 5.700 5.700 5.402 5.201 5.200 5.703 5.701 5.701 5.702 5.703 5.700 5.702 5.702 5.200 5.203 5.202 5.701 5.201 5.201 5.301 5.302 5.302 5.301 5.202 5.201 5.200 5.002 5.000 5.000

92

G.H. Rees / Nuclear Instruments and Methods in Physics Research A 688 (2012) 84–92

four-turn acceleration. A major design issue is the unusual, spiral magnet system, which will require careful design. A direct line drawn between the superperiod ends is a ‘‘spiral polygon’’, but each arc is different, so an OSC ring has four separated beam orbits of varying shapes. Despite the ring’s irregularity, beam is smoothly accelerated and emittance growths for beams of 10 mA are found to be small. The effect of machine errors remains to be investigated.

Appendix A See Tables A1–A3.

Appendix B See Tables B1–B3. References

Acknowledgement I am most grateful to C. Plostinar for deriving the off-axis cavity transit time factors.

[1] C. Rubbia et al., Conceptual design of a fast Neutron Operated, High Power Energy Amplifier, CERN/AT/95-44 (ET), Geneva, 1995. [2] F.M. Russell et al., Feasibility study of a 15 to 70 MeV Separated Orbit Cyclotron as a Booster Injector, RAL Report RHEL/R122, June 1966. [3] C. Plostinar, Off-axis Transit Time Factors, Rutherford Appleton Laboratory, 2012.