Superconducting proton synchrotrons

NUCLEAR

INSTRUMENTS

AND METHODS

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(I967) 298-308;

© NORTH-HOLLAND

PUBLISHING

CO.

SUPERCONDUCTING PROTON SYNCHROTRONS P. F. SMITH and J. D. LEW1N

Rutherford Laboratory, Chilton, Berkshire, England Received 13 March 1967

This paper summarizes the results of a preliminary study of the possibility of using pulsed high field superconducting magnets in place of the conventional iron-core magnets in high energy proton synchrotrons. Cost estimates are compared for alternating gradient synchrotrons using conventional, cryogenic, or superconducting magnets. The latter offer, in principle, the possibility of substantial reductions in size and cost; for example, a superconducting 1000 GeV accelerator operating at a field of 60 kG might be built at a cost no greater than that of a conventional 300 GeV machine. In-

expensive conversion of existing accelerators to higher energy might also be possible. Estimates are made of the heat dissipated in the magnet by ac losses, eddy currents, and high energy particles. Reduction of the ac loss to a tolerable level necessitates subdivision of the superconductor into insulated strands no greater than about 5/~m in dia. Although this requires a substantial advance in superconductor technology, several possibilities for producing such a conductor can be envisaged and are being investigated. A number of other aspects of the magnet and accelerator design are surveyed briefly.

1. Introduction Superconducting magnets are likely to come into widespread use in high energy physics during the next ten years, providing in particular more powerful bending and focusing magnets, and large volumes of high field for bubble chambers and spark chambers. The most spectacular development would be the use of superconducting magnets in the large particle accelerators. By this means, large reductions in size and cost might be achieved, or, alternatively, higher energies for the same cost. Inexpensive conversions of existing accelerators to higher energy might also be possible. Since the economic advantage of superconducting magnets is mainly evident in de applications (where conventional high field magnets would necessitate costly multi-megawatt power supplies), the FFAG class of accelerators would be the most natural choice for detailed study; however, this type of accelerator has not yet been operated at high energies, there still appear to be many unresolved theoretical and practical difficulties, and it seems unlikely that any large accelerator design will be based on this principle in the foreseeable future. In this paper, therefore, we have confined our attention to the well-established alternating gradient proton synchrotron, and have attempted a preliminary assessment of the economic advantages and the practical problems of using high field pulsed superconducting magnets in place of the conventional pulsed iron-cored magnets. In general, superconductors are not advantageous for pulsed magnet systems, since the cost of energy storage is usually dominant; in an accelerator, however, although the stored energy is substantially increased by

using a high field air cored magnet, this should be more than compensated by the large reduction in building, site-preparation, and other radius-dependent costs. This is examined in more detail in section 2, with the costs for the proposed European 300 GeV accelerator taken as a specific example. The optimum field for the superconducting version is shown to be about 60 kG, giving a factor 5 reduction in radius, and a possible cost saving of 20-30 million pounds. Alternatively, a superconducting 1000 GeV accelerator might be built at a capital cost no higher than that of a conventional 300 GeV machine. These are, of course, to be regarded as examples only, and not proposals, since it is unlikely that the feasibility of superconducting accelerator magnets can be established in time for the next generation of large accelerators. Perhaps of more immediate interest, therefore, is the possibility of increasing the energy of existing accelerators by a factor 5 or more, at relatively low cost. The experience gained in such projects might subsequently be used to convert a conventional 300 GeV machine to an energy of 1500 to 2000 GeV. In section 3 the alternative possibility of producing the high fields by more conventional means is discussed briefly. Pulsed coils at room temperature would have an excessive mean power consumption at the high duty cycle required in accelerator magnets. This can be overcome by the use of cryogenic magnets (using, for example, very pure aluminium cooled to about 17° K), but their higher refrigeration and power supply requirements make them economically inferior to superconducting magnets. The principal problem to be overcome before this proposal becomes technically feasible is that of the

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hysteresis loss in the superconductor (usually known as the "ac loss"), which, with conductors available at the present time, would increase the refrigeration requirements by at least two orders of magnitude. The total heat dissipated is, however, proportional to the diameter of the individual strands of superconductor; and it is shown in section 4 that, to reduce the refrigeration cost to a tolerable level, this diameter must be about 5 x 10-4cm. This presents a difficult, but not unreasonable, technical problem. Two other sources of heat dissipation are also examined in section 4. Eddy current heating in any normal metal used to stabilize or support the superconductor necessitates an order of subdivision which is in the millimetre rather than the micron region, and this should therefore present no difficulties. Heating resulting from loss of high energy particles to the magnet is greatest during beam extraction, and at high beam currents could significantly increase the refrigeration requirements unless a high extraction efficiency can be achieved, or unless the extraction geometry can be arranged to reduce the peak dose received by the magnet. In addition to increasing refrigeration costs, these sources of heat could result in an increase in superconductor temperature sufficient to lower its critical current or drive it into the resistive state. Our calculations indicate, however, that, even if the superconducting strands are embedded in material of low thermal conductivity, there would be no difficulty in providing sufficient liquid helium cooling to restrict the temperature rise to less than 1° K. In the final section a variety of other aspects of the magnet and accelerator design are reviewed briefly, none having yet been examined in detail. These include coil design, internal cooling, coil stresses and fatigue, radiation damage, operating current and power supply, factors determining size of beam aperture, rf requirements, remanent fields, and stray fields. Although a considerable amount of detailed study will be necessary before the feasibility and limitations of superconducting accelerator magnets can be established, the only serious obstacle encountered so far is the necessity to develop (at reasonable cost) a new form of superconducting cable with a low ac loss. Discussions with manufacturers of superconducting materials have indicated several possible ways in which this might be achieved, and initial small scale experiments have given encouraging results. It seems almost certain, therefore, that it will be possible to make serious proposals for superconducting proton synchrotrons within a period of 5 to 10 years.

TABLE 1

Summary of conventional 300 GeV accelerator estimates (units of £ l0 °, 1967 prices*). (a) Items affected by magnet design

Magnet Power supply Injector

(b) Items approximately proportional to R

Magnet tunnel Vacuum system Installation cost

(c) Items partly dependent on R

Site preparation Other buildings Site power and cooling Controls, etc.

(d) Items assumed independent of R

rf system Shielding Beams and data analysis equipment Staff, overheads, general expenses Total

15.8 3.0 10.9

14.8] 23.1

6.5

i:!l

33/ 38.0

4.8 '19.1

57.2

a0.0j 148.0

* Ref. 1) gives 1964 prices. Revised figures for 1967 were supplied by C. J. Zilverschoon (CERN), private communication.

2. Cost analysis It is convenient to take as a specific example the proposed European 300 GeV accelerator 1) (although it must be emphasised that superconducting magnets cannot be considered as an alternative possibility at the present time). An approximate breakdown of the estimated capital cost is shown in table I, most of this cost being spread fairly evenly over a period of at least seven years. With a superconducting magnet system, the magnet and power supply costs have to be recalculated in detail (as have the injector costs, if the proposed booster synchrotron is also made superconducting), but we make the simplifying assumption that all the other costs can be divided into a term proportional to radius R and a term independent of radius. Assuming that items (b) and, say, about 25% of items (c) are proportional to R, table 1 reduces to the first column of table 2. The superconducting magnet cost can be estimated without knowing the detailed coil design, and this is done in appendix 1. The superconductor and power supply costs are approximately proportional to the field H, whereas the refrigeration, engineering, and radiusdependent costs are inversely proportional t o / 4 , giving an optimum field in the region of 60 kG. If the aperture required for the beam is taken to be the same as in the conventional proposal, and assuming safe present-day prices for superconducting material,

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P. F. S M I T H A N D J. D. L E W l N TABLE 2

Comparison of cost estimates for conventional, superconducting and cryogenic synchrotrons (units of £ 108, rounded off to nearest [ 106). ConvenS/C, 4° K S/C, 4° K tional 300 GeV 300 GeV 300 GeV 60 kG 60 kG 12 kG air core air core iron core (estimate 1) (estimate 2)

S/C, 4°K 1000 GeV 60 kG air core

Cu, 300°K Cu, 300°K A1, 17°K 300 GeV 300 GeV 300 GeV 30 kG 22 kG 60 kG air core air core air core

AI, 17°K 300 GeV 60 kG air core

f

Magnet Power supply Refrigeration Injection Costs oc R All other costs (assumed constant) Total

16 3

II 9

11 7

9 10 19 11 13

5 19 30 11 7

-

6

11 32

11 7

9 7

13 8 12 9 20

86

86

86

86

86

86

86

86

148

130

116

148

165

200

148

158

engineering, and power supplies, the cost figures for the superconducting version are as shown in column 2 of table 2. Taking a more optimistic view, the aperture might be reduced by a factor 2 (depending on beam intensity considerations as discussed in section 5), superconductor and engineering costs might be substantially reduced for very large quantities, cheaper power supply systems may be feasible, and a superconducting booster could be used. The resulting figures are given in column 3 of table 2, and are perhaps the lowest that can be foreseen at the present time (apart from possible reductions in some of the assumed "fixed" costs which have not been investigated). The cost for other energies may be estimated by noting that the fixed cost appears to depend largely on the size and staffing of the laboratory as a whole, whereas most of the other costs will be approximately proportional to accelerator energy. The o p t i m u m operating field is independent of energy. In scaling costs up to 1000 GeV, it seems probable that items associated with the magnet can be kept to about 2.5 times their cost at 300 GeV and that the overall laboratory size and staff requirements need be no greater. We thus arrive at the fourth column of table 2, which suggests that a superconducting 1000 GeV synchrotron could, in principle, be built at a cost comparable with that of a conventional 300 GeV machine. As an alternative to the usual type of alternating gradient magnets, it is probable that serious consideration would be given to a "separated function" machine (consisting of separate bending and focusing magnets), which would simplify construction and final tuning problems at the expense of a slightly larger radius. Although a greater total length of magnet would

17 33

6 90

11 18

be required, the peak magnetic field would be lower; thus the quantity of superconducting material would probably not be increased, and the overall cost would not be significantly different. In addition to the construction of new accelerators, there is also the possibility that existing accelerators might be converted to higher energy by replacing the magnet, rf system, and power supply. In the case of constant gradient machines, it may not even be necessary to replace the power supply, since the stored energy in the large aperture magnet would be sufficient to provide a much higher magnetic field in a small ape rture alternating gradient magnet. As an example, the Rutherford Laboratory 7 GeV synchrotron (Nimrod) could, in principle, be converted to an energy of 35-40 GeV at a basic cost of perhaps 3 or 4 million pounds. In practice, most conversion proposals would probably also include a new injection system, in order to obtain an improved beam intensity in addition to the higher energy. Table 3 lists some of the parameters which might be typical of a superconducting alternating gradient magnet ring. To make the system easier to compare with other superconducting magnet projects we have imagined the ring divided into separate 200 cm long magnets. These individual magnets are smaller than m a n y superconducting magnets at present under construction, and even the complete magnet ring for a 300 GeV accelerator has no more stored energy than the large bubble chamber magnets being proposed at the present time. Such a magnet, therefore, represents an extension of existing technology not so much because of its size but because of the requirement of pulsed operation. This is examined in more detail below; first,

SUPERCONDUCTING PROTON SYNCHROTRONS TABLE 3

Typical "list of parameters" for superconducting alternating gradient magnets. Operating field Peak field on windings Length of individual magnet sections Required beam aperture Internal coil dimensions Winding thickness Coil volume (per magnet section) A" cm of superconductor (per magnet section) Maximum coil current density No. of magnets in ring Orbit radius Peak stored energy in complete magnet ring Rise time of pulse Total time for 1 complete cycle Assumed refrigeration requirement at 4° K (per magnet section): (a) Conduction and radiation (b) Current leads (c) Eddy currents (d) High energy particles (e) Superconductor ac losses

60 kG 80 kG 200 cm 8cm x 5cm 11 cm x 8 cm 3 cm 3 x 104cm3 4.3 x 108 42000 A/cm2 1.7 per GeV/c 55 cm per GeV/c 1.1 MJ per GeV/c 1 sec 3.3 sec 3W 2W 2W 3W 20 W

however, we must for completeness compare the superconducting system with alternative techniques for producing the high magnetic field. 3. Alternative pulsed high field systems

The technique, of producing very high pulsed fields using copper coils at r o o m temperature is well known, but depends on the use of a low duty cycle (e.g. a 1 msec pulse every second) to reduce the mean power dissipation to a tolerable level. In the case of a particle accelerator a high duty cycle is usually required to obtain a high output of particles, and the power requirement of the magnet will therefore approximate to that of a de magnet. Although for this reason we can immediately discount it as a competitive possibility, it is of some interest to calculate the actual figures. In appendix 2 we use approximate expressions for the resistive loss and stored energy in pulsed air cored magnets to obtain approximate optima for coil thickness and power supply cost; we then optimise the cost of the complete installation as before, arriving (for the optimum field of 22 kG) at the figures shown in column 5 of table 2. The figures for a 60 k G copper coil are also given; at 300 GeV this would have a mean power dissipation of about 1600 MW, and a peak stored energy four times as high as the thinner-walled superconducting magnet. This difficulty can be overcome by the use of cryoge-

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nic magnets, i.e. by using normal resistive conductors operated at low temperature. This possibility was examined in some detail by Post and Taylor 2) before the development of high field superconductors; for most metals the lower resisitivity is offset by the low refrigeration efficiency, but for several highly pure materials a net gain is expected. One of the best is aluminium at an optimum operating temperature of about 17°K, with which one could achieve a resistivity about 10 -3 times that of copper at 300°K and a refrigeration efficiency of about 10 -2, giving a net factor 10 gain in power. The various costs are estimated in appendix 2, and shown in the final columns of table 2 for the costo p t i m u m field of 30 kG, and for a field of 60 k G (for convenience of comparison with the superconducting case). These figures are very approximate, but superconducting magnets appear to have a clear potential economic advantage in view of their much lower refrigeration and stored energy requirements. This advantage would be lost if materials could be developed with a factor 5 lower resisitivity (including magnetoresistivity), but there appears.to be no prospect of doing this in the foreseeable future. The preceding considerations do not mean that we can rule out completely the use of conventional high field magnets in accelerators. In particular, combined superconducting-plus-conventional magnet systems might be considered, in conjunction with a shorter rise time and lower duty cycle. With faster magnet pulses, however, the rf acceleration problems become more formidable, and at high beam currents (where beam loading becomes dominant) an order of magnitude reduction in rise time would in most cases necessitate a prohibitively complex and costly rf system. Furthermore, "flat t o p " operation would not be possible. 4. Heating effects

In addition to the usual heat influx through the cryogenic enclosure by radiation and conduction (about 10 -4 W/cm 2 of magnet surface area) and down the current leads (about 1 W/1000 A per lead), there are three sources of heat in the magnet itself, each of which might seriously increase therefrigerationrequirements. These are the ac loss in the superconducting material, the eddy current loss in any normal metal present, and the heating caused by high energy particles scattered on to the magnet during injection, acceleration, and extraction. We now give the order of magnitude of each of these, and discuss their practical implications. 4.1. AC LOSS Circulating currents

are induced in the super-

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conducting material by the changing external field. The material is thus taken around a magnetization loop during each cycle, with consequent dissipation of heat. This is usually referred to as the "ac loss", and should not be confused with the frequency-dependent ac resistance skin effect which occurs in normal metals. Since there is complete penetration of flux into the superconducting strands during most of the cycle, the effect is essentially free from any theoretical uncertainty and its order of magnitude can be calculated without difficulty, as shown in appendix 3. More precise estimates can also be made, taking into account the field distribution in the coil, the current-field characteristic of the material, the interaction between transport and magnetization currents, and the shape of the superconductor cross section. The total heat released is directly proportional to the diameter of the individual strands of superconductor. With the typical commercially-produced 0.25 mm dia. wire there would be a dissipation of about 1 kW in every 200 cm magnet section; to keep the refrigeration cost to a tolerable level this must be reduced to something like the 20 W figure assumed in section 2, thus necessitating an average strand diameter no greater than about 5 x 10 -4 cm. Some comments are necessary on the reliability of this estimate. Experimental techniques for the measurement of ac losses are still in a relatively early stage of development, and agreement with theory to better than a factor 2 has not yet been reached3). It is necessary, therefore, to treat any precise estimate with some reservation at this stage, particularly as it is not yet clear whether the occurrence of magnetization discontinuities, known as flux jumping, could significantly increase the loss. It is expected, however, that flux jumping will be less likely, and perhaps completely absent, in finely divided material, so that we can be reasonably confident that the theoretical model used is satisfactory, and provides a realistic estimate of the degree of subdivision which will be necessary. Typically, therefore, a synchrotron magnet might require a 2000 A, 1.5 mm dia. conductor containing 100000 individual filaments. This presents some formidable but by no means insuperable problems. The optimum operating field is low enough to allow the use ofeitherthe ductile alloys such as niobium-titanium, or the brittle compounds such as niobium tin. It is important, however, to note that in the final product the individual strands must be insulated from one another, or at least in very poor electrical contact. If the strands are connected together by low resistance metallic paths the magnetization currents will flow round the con-

J. D. L E W I N

ductor as a whole rather than being confined to the individual wires, and the ac loss will not be reduced. It appears difficult to estimate even approximately the condition for independence of the superconducting strands in a composite conductor, but it seems likely that no metallic substrate can be used. In the case of ductile alloys, therefore, if the drawingdown process is most readily achieved in conjunction with another material, either this must be an insulator, or alternatively any metal used must be subsequently replaced by (or converted to) an insulator. The insulation thickness can be very small, since voltages between strands will in general be no greater than a few mV. Small scale investigations of the possibility of manufacturing the ductile alloys in filamentary form have been initiated, although, for commercial reasons, we cannot at this stage enlarge on the remarks made in section 1. Niobium-tin is already produced as a vapour deposited or diffusion layer only a few kLm thick on a supporting tape substrate. In principle, one could use such a conductor without modification by designing the coil so that the tape width is always parallel to the field lines. This idea, however, seems completely impracticable on examination, and it would therefore be necessary to subdivide the tape into strips 5-10 #m wide. As before, the supporting material cannot be metallic, but in view of the successful development of processes for the fabrication of micro-circuits it seems quite likely that similar techniques could be used to deposit fine parallel lines of niobium-tin onto an insulatingtape. Other suggested possibilities for producing finely divided niobium-tin include vapour deposition on very fine wires or fibres. Having specified that the individual strands in the conductor must be separated by insulation, there remains the problem that at the end of a length of conductor all the strands must be connected to a current junction, and, therefore, to each other. An order of magnitude calculation suggests that there would easily be sufficient voltage within the conductor to drive the magnetization currents from one strand to another across such a junction. In the case of stranded conductors it should be possible to overcome this by the use of a "litz wire" configuration, or simply by twisting, which would reduce the internal voltage by many orders of magnitude. In the case of subdivided tapes we cannot see a correspondingly simple solution, unless one twist per turn can be shown to be sufficient. An important factor in the development of these conductors is their eventual cost, since we have assumed that the cost of the magnet material will be no higher

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than at present. This appears to favour the use of niobium-titanium rather than niobium-tin, although in very large quantities both may be competitive; also niobium-tin has the advantage of being usable at higher fields if necessary. The use of normal metal for stabilization has not been mentioned, since it will probably be unnecessary in this type of conductor. At present, composite conductors of superconducting material and normal metal constitute the only means of ensuring reliable nondegraded performance of superconducting coils. However, it is very probable that in a well subdivided conductor the heat released by flux movement will be lowered sufficiently for degradation to be completely absent, without the need for any normal metal or local helium cooling4); if, at 5 #m, some normal metal is still necessary, this can of course be included provided that the individuality of the strands is preserved. 4.2. EDDY CURRENTHEATING It is evident from the preceding discussion that it is not clear how much, if any, normal metal will be present in the magnet. However, allowing for the possibility that some may be required for support or stabilization, we derive an approximate expression for the eddy current heating in appendix 3. As with the ac loss, this heating can be reduced by subdivision, and we find that, even assuming the presence of a substantial proportion of good conducting material such as copper, subdivision into strands about 1 mm dia. is sufficientto reduce the heating to the same order as the other heat sources, as assumed in section 2. There should obviously be no difficulty in meeting this requirement. 4.3. HEATINGBY HIGH ENERGYPARTICLES Particles will be lost to the magnet during injection, acceleration, and extraction. The resulting radiation dose rates have been estimated by LewinS), the relevant figures being summarized in appendix 3. The heat dissipation will be no higher than the other sources of heat provided the average dose rate is less than about l04 rad/h. The dose resulting from injection and acceleration losses should be considerably less than this, but a 50% extraction loss could give a mean dose as high as 250 E 0'6 rad/h, where E is the accelerator energy in GeV. This assumes that the particle beam intensity is proportional to accelerator energy, and normalized to 3 x 1013 protons per 3 sec pulse at 300 GeV and 60 kG. This figure indicates that the average radiation heating should be just acceptable at energies up to 1000 GeV. A complicating factor, however, is that the extraction

303

loss will not be distributed uniformly, but concentrated into perhaps only 4% of the circumference (in one of several possible extraction regions), so that an increase in refrigeration capacity might be necessary to cope with the high local heating. The actual dose may, in fact, be considerably lower, since a high proportion of the particles may not have a long enough path in the magnet to permit full build up of the nuclear cascade. Nevertheless it is evident that there is a strong incentive to achieve an extraction efficiency of at least 90%, reducing the above estimate by a factor 5. Some other possibilities are the use of conventional magnets in the extraction regions, or the provision of metal absorbers to limit the dose reaching the superconducting magnets.

5. Miscellaneous design problems In this section we discuss very briefly a number of other aspects of the magnet and accelerator design. This is intended to form a starting point for further work, none of these problems having yet been assessed in detail. Considering first the coil design, the choice of a sinusoidal distribution of current around a circular or elliptical aperture would appear to be the most desirable from almost all viewpoints; it enables high uniformity of field and field gradient to be achieved with greatest economy in superconducting material and stored energy, and probably also minimizes the ac loss. The theory of the required current distributions has been analysed in some detail by Beth6), and good practical approximations to sinusoidal distributions have been achieved in the superconducting quadrupoles constructed at the Brookhaven National Laboratory7,8). To achieve the conventional combined focusing and bending field a current distribution o f the form (Acos0 +Bcos20) is necessary; this unsymmetric arrangement could be disadvantageous from the viewpoints of construction and mechanical support, and it may therefore be preferable to use separate bending and focusing magnets, giving the further advantage of allowing adjustment of the focusing properties of the accelerator without the necessity for correction windings in the magnets. The production of alternating gradient fields with sets of rectangular coils has also been considered at Brookhaven9), but this appears to result in a larger and more expensive magnet. The nature of the conductor has already been discussed in some detail in the previous section. It is desirable to operate at a current in the region of several thousand A, both to ease conductor handling and winding, and to keep the voltage applied to the magnets to a

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reasonable level. Even higher currents might be preferred but for the increased heat leak down the current leads, which must, of course, be permanently in position. (The alternative possibility could be envisaged of inductive excitation of closed superconducting loops, but this would require completely different power supply concepts and is beyond the scope of this paper). The power supply itself, therefore, would not be materially affected by the use of a superconducting magnet, and could be a completely conventional system, with the advantage that it would not have to supply the usual ohmic loss in the magnet (e.g. 26 MW for the conventional 300 GeV machine). On the other hand, the superconducting magnet system is sufficiently flexible to accommodate any new power supply developments (remembering that the development of low ac loss conductors could also widen interest in superconducting energy storage systems). The possibility exists, for example, of a dc biased magnet system, gaining a factor 2 in the peak stored energy supplied. The internal structure of the coil should present fewer problems than in the case of stabilized superconducting magnets being designed at the present time, since we can reasonably hope that any finely divided conductor will be self-stabilizing, and it will not therefore be necessary to ensure local cooling of the individual turns. Some cooling channels may, however, be required to prevent an excessive temperature rise in the conductor resulting from the sources of heat discussed in the previous section; if d(cm) is the distance between cooling channels, k (W, cm, ° K) the mean thermal conductivity of the windings, and P(W/cm 3) the heat dissipation in the coil material, the maximum temperature rise will be about ~PdZ/k( ° K). Taking P ~ 10- 3 and assuming k to be in the region 10-3, T will be less than I°K provided d < 3 cm; since this is of the same order as the coil thickness, itis probable that it will be sufficient to cool the surface of the coil only. Nevertheless, it is conceivable that localized resistive regions might be formed which would propagate in the presence of the transport current, and experimental tests are necessary to establish with certainty the internal thermal environment compatible with satisfactory pulsed operation. Containment of the electromagnetic forces on the coil should present a substantial but not severe problem. For a sinusoidal current sheet there is a total outward force of ½ztaH 2 dyne per cm length of coil, which is in the region 3 ton/cm length for H ~ 60 kG and a ,-~7 cm; this is less than the forces encountered in many superconducting magnets at present in the design and construction stage. Forces at the ends of the coil are no greater, and are partly resolved into hoop stress

J. D. L E W I N

in the conductor itself. Perhaps of greater concern is the pulsed nature of the force, which could give rise to small movements of the strands in the conductor, resulting in possible fatigue effects over a period of time (remembering that there would be perhaps l0 T pulses per year). This problem is complicated by the low temperature, and the possible high peak radiation environment, and may be another reason for preferring the ductile alloy Nb Ti to the brittle Nb3Sn. However, although it is important to answer these questions with complete certainty, it should be possible to provide a conductor and coil structure which is sufficiently rigid to eliminate the possibility of fatigue. The radiation dose, arising mainly from extraction losses, could present a number of problems. In the previous section it was estimated that (at high energies and high intensities) a 50% loss of particles at extraction could give a mean dose of 104 rad/h, with some magnet sections receiving as much as 105-106 rad/h. In addition to the high local heat dissipation, this is a level which could result in radiation damage over a period of years. The superconductor itself is unlikely to be affected adversely (the critical current is, in fact, generally improved by irradiation) but some care may be necessary in selecting the insulating materials to be used in conjunction with the conductor and (because of eddy 'currents) to construct the cryostat. In most cases the dose may be considerably lower, for the reasons indicated in the previous section. The magnitudes of several of the key items in the magnet (amount of superconductor, ac loss, stored energy, etc.) are approximately proportional to the magnet aperture, which will be a few cms larger than the aperture required for the beam. For most of the estimates in this paper we have assumed a beam aperture of about 10 crux6 cm. i.e. the same as that proposed for the conventional 300 GeV synchrotron. This may appear to be pessimistic, since all fields and field gradients have been increased by a factor five, reducing all orbit parameters by the same factor. Assuming, therefore, that the percentage errors in magnet alignment and field uniformity would be no greater than for conventional magnets, one would expect, from these considerations alone, to be able to use a very much smaller aperture. However, since the use of conventional injection, extraction, and rf acceleration techniques will probably preclude any appreciable reduction in straight section length, the latter will constitute a much larger fraction of the circumference; this results in an increased amplitude of particle oscillations, which can only be partly compensated by the use of additional quadrupoles in the straight section1).

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More general considerations of achievable beam intensity could also make a large reduction in aperture undesirable; firstly since the transverse space charge limit is proportional to (aperture radius)2/(machine radius), and secondly because the reduced circumference may necessitate multi-turn injection to obtain an intensity comparable with that of a larger radius machine. These considerations are intimately linked with the choice of injection system; without detailed study of specific accelerator designs, therefore, it cannot be assumed that any reduction in aperture would be possible. Other factors affecting beam behaviour after injection, and choice of injection energy, are gas scattering and remanent fields. Gas scattering would become negligible at the lower pressures which could more readily be achieved in the low temperature environment. Remanent fields (due to residual magnetization currents in the superconducting material) will have to be checked experimentally, but an approximate calculation indicates that they should be less than HMq/a, where Hu is the maximum field, a the aperture radius, and q the individual superconducting strand diameter. With H M,,~60kG, q ~ 5 x 1 0 -4 cm, and a,-~5cm, therefore, we expect remanent fields of only a few G. This is somewhat smaller in absolute value than the values characteristic of iron cored magnets, and when expressed as a percentage error in the injection field it is at least five Limes less significant. These two gains may not be significant in the design of accelerators in the 300 GeV region, but could be relevant to the conversion of lower energy machines. Problems associated with the rf acceleration system have not yet been studied in detail. The voltage required per turn is reduced in proportion to the circumference, allowing a simpler rf system. On the other hand the total beam loading (which will predominate at high beam intensities) is unaltered; this may prevent any reduction in the number of cavities, which may therefore have to occupy a somewhat larger fraction of the circumference. One possible problem is the effect of the large stray magnetic fields on the rf cavities and other equipment. To eliminate these, the use of iron in conjunction with the superconducting magnet is probably more attractive than either the provison of compensating superconducting windings, which could increase the cost, or the use of a bulk superconducting shield, which would increase cost and have an unacceptably large ac loss. The use of iron could also lower the magnet cost, adding up to 20 kG to the field relatively inexpensively. The shield would probably be in the form of a concentric

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cylinder situated either outside the cryostat at room temperature, or adjacent to the coils at 4 ° K, where it would be more effective in increasing the field but would have to be finely laminated to reduce eddy current and magnetic hysteresis losses. The possible advantages of iron would also have to be balanced against the loss of the ideal field distribution readily obtainable with air cored magnets, perhaps requiring extensive model work to establish a satisfactory design. However, the development of field mapping computer programmes which take into account regions of field-dependent permeability may well make this a relatively minor design problem. The authors benefitted in particular from discussions with L. C. W. Hobbis and D. A. Gray on general problems of accelerator design and conversion; with G. H. Rees and R. Billinge on injection, rf and beam intensity problems; with H. London, R. Hancox, and M. N. Wilson on ac losses and stability of subdivided conductors; and with many commercial organisations on the possibility of fabricating filamentary superconductors.

Appendix 1 COST ESTIMATES AND OPTIMIZATION FOR SUPERCONDUCTING SYNCHROTRON MAGNETS

We first determine the cost per GeV of each item as a function of magnetic field H [remembering that the product HR (G.cm) is equal to (107/3) x particle momentum in GeV/c], and then determine the value of H which minimizes the total cost.

a. Superconductor To a first approximation we can neglect the effect of the field gradient, and assume a current distribution Ilsin0 (A/cm) around a circular aperture radius a (cm) to give a uniform field H G, where H = 0.2hi I. The total amount of superconductor in a magnet ring of radius R cm is then 40 HRa (A-cm), or 1.3 x 108 a (A. cm/GeV). The lowest cost of superconducting material at the time of writing is about 4x 10 -1° H (£/ A. cm). We take a = 7 cm to allow for particle aperture, vacuum chamber and coil thickness, and multiply by further factors of 1.3 (ratio peak field to mean field), 0.7 (assuming elliptical instead of circular aperture) and 1.15 (to allow for coil ends). The result is: total cost of superconductor = 0.4/-/(£/GeV). b. Engineering Estimates of the cryogenic engineering costs for various superconducting magnets indicate a figure of

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about £ 0.3S, where S (cm z) is the magnet surface area. To this we add a coil fabrication cost in the region 0.005 £/cm length of conductor, giving (at an operating current of about 2000 A): total engineering cost = 2000 + 6 x 108/H (£/GeV). c. Power supply cost The stored energy E in the magnet will be given by E ,~ (2nR)HZa2/(4 x 107) ,~ ~rta 2H

[J]

[J/GeV].

The additional contribution resulting from the field gradient is small in this case6). For a thick coil the effective value of a is the inner coil radius plus one third of the coil thickness; we take a = 7 cm. We also multiply by 0.75, assuming an elliptical aperture6), obtaining for the total stored energy 18H J/GeV. Power supply costs using a motor alternator system will be approximately proportional to stored energy (giving a term proportional to H), together with costs proportional to both stored energy and radius (giving a term independent of H). From the 300 GeV estimates we obtain the approximate expression:

total field-dependent cost = 0.85H+3.1 x 109/H (£/GeV). This is a minimum at H ~ 60 kG, although the field can be chosen anywhere within a factor 1.5 of this without increasing the cost by as much as 10%. 60 kG is, however, an attractive choice, giving a large reduction in radius, while remaining within the economic range of the ductile alloys. At optimum, all five items in the cost are of the same order, so that estimating errors are unlikely to affect appreciably the optimum field or the general conclusions of this paper.

Appendix 2 COST ESTIMATES AND OPTIMIZATION FOR ROOM TEMPERATURE AND CRYOGENIC AIR CORE MAGNETS

The energy requirements of pulsed magnets can be conveniently analysed by considering the general problem of producing a long length of transverse field H(G) over an aperture diameter d (cm) with a current density J (A/cm2). The coil thickness must then be of order H/J (cm) and the coil height of order (d+ H/J). A good approximation to the peak stored energy will be

total power supply cost = 3000+0.45 H (£/GeV). d. Refrigeration A comparison of cost estimates for a variety of refrigeration and distribution schemes for a system of superconducting magnets has been given by Strobridge et alJ°), together with estimated refrigerator costs as a function of power. We assume that a typical arrangement might consist of 1 refrigerator every 400 ft of magnet circumference, serving 60 magnet sections, with 14 000 ft of transfer line. Assuming first of all a figure of 10 W per magnet (i.e. without the ac loss) and a further 600 W in the transfer lines, a capacity of 1200 W would be required, giving a total cost in the region of £500000 per 400 ft of circumference, or about 8 x 108/H(£/GeV). If the ac loss is P W/magnet, the additional cost would be (at £ 200/W): 0.2 P x 108[H(£/GeV), so that e can be as high as 20 W before the total refrigeration cost is increased by 50%. Assuming, then, a total of 30 W per magnet, we conclude: total refrigeration cost = 12 x 108/H (£/GeV).

E n ~ H2(d+½H/j)2/(4n × 107) [J/cm length], (1) and in a pulse of rise time Tsec, the resistive loss will be E R ~ j 2 T ( p / 2 ) ( H / J ) ( d + H / J ) [J/cm length], (2) where p is the resistivity, and 2 is the space factor. Define a length 61 (cm), by the expression 61 = (4nlOVTp/2) ~

(3)

and let the actual coil thickness (H/J) be k61. We then have the expressions E n = {HZ/(8n x 107)}[261d/k+26~] J/cm,

(4)

EH={H2/(8nxlOT)}[Zd2+2k6~d+lk26~] J/cm, (5) and En + En = {2H2d2 /(Sn x 107)} • • [1 + (61/d) (k + k - 1) + (6Z/d2) (1 + ¼k2)] J/cm.

(6)

total radius-dependent cost = 13 x 108/H (£/GeV).

These expressions illustrate the well known fact that there is an optimum thickness for a pulsed coill~), which usually occurs when k ~ 1, i.e. when the thickness is given by eq. (3)*. A thinner coil will have a high resistive loss, and a thicker coil will have a high peak stored energy.

Collecting together the terms in items a. to e. above we obtain:

• When 61/d is large, the minimum in (ER+EH) occurs when k ~ (d/61)~.

e. Radius-dependent costs Assuming that about £ 32 million of the 300 GeV estimate would scale linearly with radius we obtain:

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For conventional pulsed high fields coils, using copper, and rise times in the millisecond region, 61 is in the region 0.5 to 1 cm, but for a 1 sec rise time the optimum thickness increases to about 20 cm. This is also an approximate cost optimum, since, for a rise time of 1 sec (and, of course, continuous operation), the multiplyingfactorsconverting eqs. (4) and (5) to capital cost are of the same order (£ 0.04/W and £ 0.025/J), and give, for d = 8 cm and 61 = 20 cm, a total power supply cost ,-~ 5H£/GeV. To this we must add the magnet construction cost, which, at say 0.015 £/cm 3 (for Cu), will be in the region 1.2 x 109/H(£/GeV). Adding the radius dependent cost as in appendix 1, the total field dependent cost becomes 5 H + 2 . 5 x 109/H (£/GeV), which is a minimum at 22 kG. The resulting cost estimates are summarized in table2 and confirm that high field magnets of this type have unreasonably high power requirements. This can be overcome by utilizing very pure metals at low temperature, where, despite the low refrigeration efficiency, a factor 10 gain in power can be achieved. Assuming the use of pure aluminium (total resistivity ~ 10-912 • cm) at an optimum operating temperature of about 17° K2), we can again calculate a cost optimum with the aid of expressions (4) and (5). The quantity ~t is about 0.6 cm but this time the optimum k is not 1, because the unit cost of providing refrigeration for the resistive loss is very much greater than the unit cost of providing a power supply for the stored energy. Leaving k undetermined for the moment, we arrive at the following cost items (in £/GeV, and assuming a 10 cm coil i.d.): 1. Radius dependent costs, as before: 1.3 x 109/H; 2. Refrigeration distribution, as before: 5 x 108/H; 3. Refrigerator cost (at, say, £ 15/W at 17°K): 0.8H[1 + 18k-1]; 4. Coil cost (at 0.015 £ / c m 3 ) : 107[0.7k+O.O4k2]/H; 5. Cryogenics (at 0.3 £/cm2): 107124+ 1.4k]/H; 6. Power supply (at 0.025 £/J): HI400 + 24k + 0.4k2]/103 . The sum of these has a rather flat minimum at k ,-~ 18, H ~ 30 kG, giving the costs shown in table 2. The cost figures for H = 60 kG are also calculated, showing the much greater refrigeration and stored energy requirements for the cryogenic magnet than for the superconducting magnet. In view of the uncertainty both in the resistivity of the conductor material and in the cost of refrigeration on this scale, the precise figures arrived at must not be taken too seriously. Nevertheless we cannot at present envisage a reduction sufficient to compete with the superconducting system (nor is the cryogenic system necessarily potentially simpler from a technological viewpoint).

Appendix 3 CALCULATION OF HEATING EFFECTS

a. The ac loss in the superconductor Heat dissipation occurs in hard superconductors owing to the irreversible penetration of flux into the material. At low frequencies and high magnetic fields the calculation is relatively straightforward 3' 12) since for most of the cycle the conductor cross section is completely filled with a dipole current density __+Jc. To obtain the order of magnitude, consider a long strip of rectangular cross section pq c m 2 with the field perpendicular to side p changing at /2/(G/sec). Assume the critical current density Jc to be independent of field, assume the opposing field produced by the induced currents to be negligible, and ignore any transport current. If E is the local voltage gradient associated with the changing flux, the power dissipation, given by the volume integral of JE, is P = p2qI21Jc/(4 x 108) [W/cm length of conductor] = ~IJ~p/(4 x 108)

[ W / c m 3 o f conductor]

= 12Ip/(4 x 108)

[W/A.cm],

since the product (Jc x volume) is identically equal to the number of A. cm of conductor. Thus, if the peak field is HM, with a rise time T sec, and the averages over the coil volume of the field components perpendicular to sides p and q are f p H u and fqH M, the power dissipation will be

P = HM[fpp+fqq]/(4 x 108 T) [W/A.cm]. In any practical coil design it appears likely that fp and fq will be of the same order. For example, in the design assumed in this paper, i.e. a sinusoidal distribution of current around a circular aperture (or its elliptical equivalent), we estimate thatf~ andf~ will each be in the region 0.4 to 0.5, so both dimensions of the strip are of equal significance. Taking circular or square section conductor, with an estimated average resultant field in the windings of ,-~ 0.6 H M, with H u = 60 kG and T = 1 sec, and using the previously calculated figure of 4.3 x l0 s A. cm for a 200 cm magnet section, we obtain an ac loss per magnet section of 3.9 x 104 p W. This estimate must be corrected to take into account the variation of current density with field, the presence of the transport current, and the shape of the conductor cross section. The use of a circular section conductor decreases the loss by a factor ~ 0.85; the effect of the transport current alone is to increase the loss by a factor 1.3, and this factor increases to ,-~2 when a typical variation of critical current density with field is included3).

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We thus arrive at a corrected estimate of about 6.6 x 104p W/magnet section for a 2 sec cycle. For comparison with the 300 GeV proposal we can assume the 3.3 sec cycle of the latter; this reduces the previous figure to 4 x 104 p W, which, for 5 #m dia. wire, gives a dissipation of about 20 W/magnet section. ,k

b. Eddy current losses The ohmic dissipation in a strip of rectangular cross section rs cm 2, resistivity p (ohm. cm), when the field perpendicular to side r changes at H (G/sec), is easily shown to be P = r3s/~/2 [(12 x 1016p) ~ W / c m l e n g t h ] .

Thus for a peak field HM, rise time T sec, writing the averages over the coil volume of the components of H 2 perpendicular to r and s asfrH~ a n d f s H 2, and writing the volume of metal asfM times the coil volume V, the total power dissipation is P(W) = fMVH~[f~r 2 +As 2]/(12 x 10'6T2p). With r = s, HM = 60 kG, T = 1 sec, fr = f s = 0.2, and V = 27000 cm 3, this gives P(W/magnet section) = 3 x lO-4fMr2/p. It is difficult to say how much (if any) normal metal might be included in a coil of this type. Consider two examples: l. fM = 0.1, p ( C u ) = 3 x 10-s; requires r <0.05 cm for P < 2 W/magnet. II. fM = 0.5, p (stainless steel) = 4 x 10-6; requires r < 0.25 cm for P < 2 W/magnet. These examples show that subdivision of any normal metal associated with the conductor is a relatively simple problem compared with the subdivision of the superconductor itself. The other obvious source of eddy current losses is the containing vessel; using the above formula, a continuous 1 mm thick cylinder of high resistivity stainless steel surrounding the magnet would give an eddy current loss in the region of 1000 W/magnet section. The vessel must therefore either be of a non-metallic material or be fabricated from millimetre long laminations.

c. Radiation dose A dose of D rad/h is equivalent to D/(4 x l0 B) W/g, which (for a coil volume Vcm a at an average density of,

say, 4 g/cm 3) is equivalent to a power dissipation of DV/IO 8 W. To keep this below, say, 3 W for a 200 cm magnet section (V,~ 30000 cm 3) we therefore require D < 104 rad/h. Dose rates for accelerator magnets have been estimated by LewinS). The results, assuming a beam intensity proportional to accelerator energy and normalized to an accelerated beam intensity of 3 x 1013 protons per 3 sec pulse at a radius of ,~ 165 m (300 GeV at 60 kG), are as follows: 1. Injection losses: 500 to 500E °'6 rad/h (where E i GeV is the injection energy), averaged over the whole circumference, and assuming 50% loss of particles. 2. Scatter during acceleration: ranges from less than 1 rad/h at 1 GeV to 4-100 rad/h at 1000 GeV, assuming a residual gas pressure ,-~ 1 0 - 6 Torr. 3. Extraction losses: 250 to 2 5 0 E ° ' 6 rad/h (where Ea GeV is the accelerated particle energy), averaged over the whole circumference, and assuming 50% loss of particles. In the above estimates, the lower limit is that due to the incident flux, and applies to the first 10 cm or so of the magnet. The energy-dependent upper limit applies if the particles pass through sufficient length of magnet (say 100 cm on average) for a fully developed cascade of secondary particles to be built up. If the extraction losses are concentrated into a region of circumference about -4tRin length, the peak dose will be about 25 times higher than the above figures. References 1) Design Study of a 300 GeV proton synchrotron. CERN (Geneva) report 863 (1964). 2) C. E. Taylor and R. F. Post, Proc. Intern. Conf. High Magnetic Fields, (M.I.T. press) (1961) 101. a) R. Hancox, Proc. IEE 113 (1966) 1221. 4) R. Hancox, Culham Lab. report CLM-P 121 (1966). 5) j. D. Lewin, Rutherford Lab. report RHEL/R 118 (1966). 6) R. A. Beth, Brookhaven Nat. Lab. int. reports AADD 102, 103, 106, 107, ll0, 112, 119 (1966). 0 P. G. Kruger, W. B. Sampson and R. B. Britton, Brookhaven Nat. Lab. int. report AADD-104-R (1966). 8) p. G. Kruger, J. N. Snyder and W. B. Sampson, Brookhaven Nat. Lab. int. report AADD-113-R (1966). 9) p. F. Dahl, K. Jellett and G. Parzen, Brookhaven Nat. Lab. int. report AADD-108 (1966). 10) T. R. Strobridge, D. B. Mann and D. B. Chelton, National Bureau of Standards (Boulder) report NBS 9259 (1966). n) j. D. Lewin, Rutherford Lab. report NIRL/R/19 (1962). 1~) H. London, Phys. Lett. 6 (1963) 162.