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To my wonderful daughter Hilda and my lovely wife Frida List of Papers This doctoral thesis is based on the following papers, which are referred to in the text by their Roman numerals. Reviewed journal papers and manuscripts: I Hagnestål, A., Ågren, O., Moiseenko, V.E., Field and Coil Design for a Quadrupolar Mirror Hybrid Reactor, Journal of Fusion Energy 30, 144 (2011). II Hagnestål, A., Ågren, O., Moiseenko, V.E., A Compact NonPlanar Coil Design for the SFLM Hybrid, Journal of Fusion Energy 31, 379 (2012). III Hagnestål, A., Ågren, O., Vacuum Field Ellipticity Dependence on Radius in Quadrupolar Mirror Machines, Journal of Fusion Energy 31, 448 (2012). IV Hagnestål, A., Ågren, O., Moiseenko, V.E., Radial Confinement in Non-Symmetric Quadrupolar Mirrors, Journal of fusion energy: DOI: 10.1007/s10894-012-9573-x (2012). V Hagnestål, A., Ågren, O., Moiseenko, V.E., Finite corrections to the magnetic field in the SFLM Hybrid, Manuscript (2012). VI Ågren, O., Moiseenko, V.E., Noack, K., Hagnestål, A. Studies of a Straight Field Line Mirror with Emphasis on FusionFission Hybrids, Fusion Science and Technology 57, 326 (2010). VII Ågren, O., Moiseenko, V.E., Noack, K., Hagnestål, A., Radial Drift Invariant in Long-Thin Mirrors, The European Physical Journal D 66, 28 (2012). VIII Noack, K., Moiseenko, V.E., Ågren, O., Hagnestål, A., Neutronic model of a mirror based fusion-fission hybrid for the incineration of the transuranic elements from spent nuclear fuel and energy amplification, Annals of Nuclear Energy 38, 578 (2010). Conference papers: IX X XI XII Hagnestål, A., Ågren, O., Moiseenko, V.E., Coil design for the Straight Field Line Mirror, Presented as a poster presentation at the OS-2008 conference at Daejon, Korea, published in the conference proceedings in Transactions of Fusion Science and Technology, 55 (2T), 127 (2009). Hagnestål, A., Ågren, O., Moiseenko, V.E., Theoretical field and coil design for a single cell minimum-B mirror hybrid reactor, Presented as a poster presentation at the OS-2010 conference at Novosibirsk, Russia in July 2010, published in the peer-previewed conference proceedings in Transactions of Fusion Science and Technology 59 (2T), 217 (2011). Hagnestål, A., Ågren, O., Moiseenko, V.E., Coil System for a Mirror-Based Hybrid Reactor, Presented at the FUNFI conference at Varenna 2011 as a poster presentation and published in the conference proceedings “Fusion for Neutrons and Subcritical Nuclear Fission”, AIP Conference Proceedings 1442, 217 (2012). Noack, K., Ågren, O., Källne, J., Hagnestål, A., Moiseenko, V.E., Safety and Power Multiplication Aspects of Mirror Fusion-Fission Hybrids, Presented at the FUNFI conference at Varenna 2011 by Klaus Noack and published in the conference proceedings “Fusion for Neutrons and Subcritical Nuclear Fission”, AIP Conference Proceedings 1442, 186 (2012). The author has also contributed to the following work, not included in the thesis. A. B. C. Hagnestål, A., Ågren, O., Moiseenko V. E., Coil design for the SFLM Hybrid, Proceedings of the EPS conference 2012. Noack, K., Ågren, O., Moiseenko, V. E., Hagnestål, A., Comments on the power amplification factor of a driven subcritical system, Annals of Nuclear Energy: DOI: 10.1016/ j.anucene.2012.06.020 (2012). Ekergård, B., Boström, C., Hagnestål, A., Rafael Waters and Mats Leijon, Experimental results from a linear wave power generator connected to a resonance circuit, Wiley Interdisciplinary Reviews: Energy and Environment: DOI 10.1002/wene.19 (2012). There are also a number of conference papers from the Open Systems, Alushta, FUNFI and EPS conferences to which the Author’s contribution is small. Contents 1. 2. 2.1. 2.1.1. 2.1.2. 2.2. 2.3. 2.4. 2.5. 2.6. 2.6.1. 2.6.2. 2.7. 2.8. 2.9. 2.9.1. 2.9.2. 3. 3.1. 3.1.1. 3.1.2. 3.1.3. 3.1.4. 3.2. 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. 3.2.8. 3.3. 3.3.1. 3.3.2. 3.3.3. Introduction ...................................................................................13 Fission power ................................................................................15 Fission power today.......................................................................15 Neutron multiplication...................................................................16 Negative feedback factors and delayed neutrons ..........................16 Resources.......................................................................................17 Front end........................................................................................19 Back end ........................................................................................19 Nuclear non-proliferation ..............................................................20 Transmutation................................................................................21 Transuranics ..................................................................................21 Long-Lived Fission Products ........................................................21 Breeding ........................................................................................22 Critical fast reactors.......................................................................23 Driven systems ..............................................................................23 Accelerator-driven systems ...........................................................25 Fusion-driven systems ...................................................................25 Fusion energy and plasma physics ................................................29 Basics in fusion .............................................................................29 Plasmas..........................................................................................29 Fusion reactions.............................................................................29 Fusion and confinement ................................................................30 Resources.......................................................................................31 Plasma physics ..............................................................................31 Magnetic confinement ...................................................................31 Particle drifts .................................................................................32 Collisions in fusion plasmas..........................................................34 Radial transport .............................................................................35 Kinetic theory: the Vlasov equation ..............................................35 MHD Equations.............................................................................36 Diamagnetism, MHD equilibrium and the concept of ................38 Plasma instabilites .........................................................................39 Mirror machines ............................................................................39 End confinement and electron temperature ...................................39 Mirror geometries..........................................................................42 The flute instability in magnetic mirrors .......................................44 3.4. 3.4.1. 3.4.2. 3.4.3. 4. 4.1. 4.2. 4.3. 4.4. 4.5. 4.5.1. 4.5.2. 4.6. 5. 5.1. 5.2. 5.3. 5.3.1. 5.3.2. 6. 6.1. 6.1.1. 6.1.2. 6.1.3. 6.1.4. 6.1.5. 6.1.6. 6.2. 6.3. 7. 7.1. 7.2. 7.3. 7.4. 7.4.1. 7.4.2. 7.4.3. 7.5. 7.6. 8. 8.1. 8.1.1. 8.1.2. 8.1.3. Other devices for magnetic confinement.......................................46 Tokamaks ......................................................................................46 Stellarators.....................................................................................47 Other schemes for magnetic confinements....................................47 Theory for the magnetic mirror vacuum field ...............................49 The long-thin approximation.........................................................49 Flux coordinates ............................................................................50 Flux tube ellipticity........................................................................53 Flute stability.................................................................................55 Drifts and neoclassical transport in mirror machines ....................56 Particle drifts .................................................................................56 Neoclassical transport....................................................................58 The Straight Field Line Mirror field..............................................60 Finite ß effects on the mirror magnetic field .................................63 The equilibrium .............................................................................63 The plasma currents.......................................................................65 Magnetic field from plasma currents.............................................66 The method from Paper V .............................................................66 The method from Paper VII ..........................................................67 Superconducting coils in fusion ....................................................69 Properties of superconducting coils...............................................69 Superconductivity..........................................................................69 Superconducing magnets...............................................................70 Cooling and shielding....................................................................71 Power supply and stability.............................................................72 Internal structure............................................................................72 Use of copper coils in reactor scenarios ........................................73 Mirror machine coil types .............................................................73 Existing or earlier mirror machine coil systems ............................74 The SFLM Hybrid project .............................................................75 Geometry .......................................................................................75 Plasma parameters and magnetic field properties .........................76 Radio frequency heating................................................................78 Fission mantle and shielding .........................................................79 Geometric design...........................................................................79 Reactor safety ................................................................................81 Cooling ..........................................................................................83 Electron temperature .....................................................................84 “Divertor plates” and heat load .....................................................85 Summary of results and discussion ...............................................87 Design of coils and magnetic fields...............................................87 Optimization methods ...................................................................87 Optimization of vacuum magnetic fields.......................................89 Function modelling with splines ...................................................90 8.1.4. 8.1.5. 8.1.6. 8.1.7. 8.1.8. 8.1.9. 8.1.10. 8.2. 8.2.1. 8.2.2. 8.3. 8.3.1. 8.3.2. 8.3.3. 8.3.4. 8.3.5. 9. 10. 11. 12. 13. 14. Superconducting coil modelling....................................................91 Results from Paper IX ...................................................................98 Results from Paper X.....................................................................98 Results from Paper I ......................................................................99 Results from Paper II (and XI) ....................................................104 Results from Paper III .................................................................107 Results from Paper V...................................................................109 Radial transport and radial invariant ...........................................111 Results from Paper VII................................................................111 Results from Paper IV .................................................................113 Discussion ...................................................................................116 Discussion on coil accuracy requirements...................................116 Discussion on coil calculation limitations ...................................117 Discussion on finite ß issues........................................................117 Discussion on radial invariant, E and low ß limit .......................117 Comparing axisymmetric and quadrupolar drivers .....................119 Conclusions .................................................................................121 Future studies ..............................................................................123 Summary of papers......................................................................125 Sammanfattning...........................................................................129 Acknowledgements .....................................................................131 References ...................................................................................133 Abbreviations and nomenclature ADS BOC BWR CTD D ELM EOC FDS FTWR GDT IAEA ICRH ITER LBE LLFP LLNL LOCA LWR MHD NEA PWR RW SABR SFLM SKB T TRU WR Accelerator-Driven System Beginning Of fuel Cycle Boiling Water Reactor Coolant Temperature Density (effect) Deuterium Edge-Localized Mode End Of fuel Cycle Fusion-Driven System Fusion Transmutation of Waste Reactor Gas Dynamic Trap International Atomic Energy Agency Ion Cyclotron Radio frequency Heating International Thermonuclear Experimental Reactor Lead-Bismuth Eutectic Long-Lived Fission Products Lawrence Livermore National Laboratory Loss Of Coolant Accident Light Water Reactor MagnetoHydroDynamic Nuclear Energy Agency Pressurized Water Reactor React & Wind Subcritical Advanced Burner Reactor Straight Field Line Mirror Svensk KärnBränslehantering Tritium TRansUranics Wind & React B B̂ E F j R T V/m N A/m2 m Magnetic field Unit vector parallel to B Electric field Force Current density Position vector u v xc = (xc,yc,zc) A a B c D f g I keff M m m n n p p p P Pfiss Pfus Q Q Qr q Rm rg r0 s s T kBT/e v v W x0 y0 Z 0 m/s m/s m m-1 m T m T/m A kg kg kg 1/m3 N/m2 N/m2 N/m2 N/m2 W W N/m2 C m m m m K eV m/s m/s J m m - Mass velocity in MHD Velocity of a particle Gyro center position Curvature vector Number of proton masses in a nucleus Plasma radius in a mirror machine Magnetic field modulus on the z axis Axial scale length of a mirror machine Diffusion constant Distribution function Quadrupolar field contribution Current Effective neutron multiplication Ion mass Electron mass Particle mass Neutron or number of neutrons Particle density Scalar pressure Parallel pressure (vs magnetic field) Perpendicular pressure (vs magnetic field) Total perpendicular pressure (incl. B) Fission power Fusion power Total parallel pressure (incl. B) Fusion Q, produced power/input power Fission to fusion power ratio Electric charge Mirror ratio Gyro radius of a particle Radial Clebsch coordinate Arc length coordinate along B Arc length-like coordinate Temperature Temperature (thermal energy) Velocity component perpendicular to B Velocity component parallel to B Energy x-like Clebsch coordinate y-like Clebsch coordinate Atomic charge Fraction of delayed neutrons Plasma pressure/magnetic pressure at the midplane 0 m m 0 c mfp eV Rad/s Rad/s J/T V Tm C/m3 kg/m3 m2 rad m2 s 1/s m Energy of a particle Gyro angular frequency at the midplane Gyro angular frequency Magnetic moment Electric scalar potential Magnetic scalar potential Charge density Mass density Cross section Angular-like flux coordinate Normal (Radial-like) flux coordinate B0r02/2 Collision time (time between collisions) Average number of neutrons per fission Collision frequency a/c, used for ordering in paraxial approx. Mean free path 1. Introduction The world’s demand for energy is increasing and the increase is likely to continue for many years to come. The main resource for energy production today is fossil fuels such as coal, oil and gas. Since these resources are limited and prices are rising, other energy sources ought to replace them. The environmental impact of the fossil fuels is also a large concern targeted by many governments all over the world and the need for a new clean energy source is rather urgent. A few alternative energy sources are available. Renewables can give a significant contribution to the energy production. There is definitely enough renewable energy sources to fulfill today’s energy needs, but a problem is how to harvest the often intermittent renewable energy resources in an economically and an environmentally (often referring to acceptable by the public) viable way. It is far from obvious and perhaps not even likely that renewables can provide all energy demanded by the worlds growing population within the next 100 years. The remaining candidates are few, and fission energy is likely to play a significant role in the future energy production. Fission energy is a stable base energy supplier in many countries including Sweden. Fission energy faces a rather massive resistance from a large part of the world population due to the accident risk, the waste management problem, problems with past and present uranium mining and the risk of nuclear proliferation. Recent events in Fukushima have further diminished the public trust in fission power, and Germany has now decided to decommission all their fission plants. Advantages of fission are that it is almost free of CO2 emissions, that the energy production cost is quite low and currently lower than for most renewables (except hydropower) and that the energy production is independent of weather and time of day. In addition, the technology to burn U-235 in light water reactors (LWRs) and some other reactor types is known and has been well tested during the last 50 years of commercial operation. As will be pointed out in section 2, the available resources for fission are enormous, taking into account future breeding technologies not existing commercially today. Another possible future source of energy is fusion energy. The available fuel sources for fusion are enormous. Just to illustrate how energy dense fusion fuel is, a comparison can be made with a coal power plant. A 1 GW coalfired power plant consumes about 2.7 million metric tons of coal a year, while a 1 GW fusion device would consume about 250 kilograms of 13 deuterium and tritium [1]. The nuclear waste problem from fusion would be almost negligible (there would be some low-active waste from activated reactor parts) and the overall environmental impact would be very low. Fusion power would therefore be an excellent solution for energy supply, but a serious problem with fusion energy is that it is hard to find a reactor configuration in which the energy gain, i.e. produced energy/consumed energy (the fusion Q factor), is sufficiently large. Despite the worldwide fusion research efforts since the 1950ies, commercialization of fusion power is still very far from realization. The complexity of the problem has proved to be greater than first anticipated, and there is still no commercial reactor scenario identified today. This implies that there is still at least 50 years before commercialization may become possible, in any case for magnetic fusion. There may however be another way for fusion research to contribute to society, which also probably could be realized in a shorter time scale. To increase fast fission reactor safety, subcritical reactors have been proposed (see for example Ref. [2]). Subcritical reactors are not self-sufficient in neutrons, and are driven by a neutron source. Fusion devices can be developed to excellent neutron souces, and it seems possible to combine a fusion reactor with a subcritical fission reactor into a fusion-fission (or hybrid) reactor. The fusion device in a hybrid reactor would be much less complicated to accomplish than a pure fusion device. Several fusion devices available today can with moderate extrapolation reach a sufficient fusion Q for becoming drivers for hybrid reactors. The immediate aim of such a device is transmutation of transuranics in combination with energy production. Another aim for long term sustainability is also breeding of fissile material. This doctoral thesis is about a fusion-fission reactor concept called the SFLM Hybrid which is based on a single cell magnetic mirror fusion device. The main work in the thesis is about the magnetic coil system and the magnetic field for that concept. Work has also been done on a radial invariant and effects of asymmetry in quadrupolar mirrors. The outline of this doctoral thesis is as follows. Section 2-6 contains theory and background information. In section 2, basics of fission power is described as well as the possible role for fusion-fission devices. Section 3 introduces the reader to fusion and plasma physics. Section 4 gives some theory for the magnetic mirror vacuum field and section 5 incorporates the modification of the magnetic field from the plasma currents. Section 6 gives a brief introduction to superconducting coils. Section 7 describes the SFLM Hybrid project, and section 8 gives a summary of the results in the thesis. Section 9 summarizes the conclusions made, and section 10 gives suggestions for future work. Section 11 gives a short summary of the papers in this doctoral thesis, section 12 gives a short summary in Swedish, section 13 contains acknowledgements and section 14 contains the references. The papers included in this doctoral thesis can be found after section 14. 14 2. Fission power A fusion-fission reactor gains the vast majority of its energy from fission. However, fission power is already a widely used source of electric power in the world. Thus, a question naturally arises: what is the point with a fusionfission reactor? A fusion-fission device is a more complex and expensive reactor than an ordinary fission Light Water Reactor (LWR). Also, the LWR technology is well tested during many years of operation (50 years or so), although the risk of nuclear accidents can never be completely eliminated. The point is that a fusion-fission reactor has the possibility to operate with a fast neutron spectrum with better safety margins than critical fast reactors have. A fast spectrum is required for transmutation of minor actinides and is desirable for breeding of fertile nuclear fuel. In order to understand the possibilities with fusion-fission reactors, some basic knowledge of fission power, resources and the problems associated with fission power is needed. In this section, background information about fission is provided and the possible future role of fusion-fission devices is described in the end of the section. 2.1. Fission power today Fission power is a well-known subject that can be learned from standard text books; see for example Ref. [2]. A brief overview is given here. Commercial reactors today are thermal reactors, which means that the neutrons are slowed down (moderated) to thermal energies. With thermal neutrons, reactor safety is improved and the probability of fission in U-235 and plutonium is increased. Most thermal fission reactors are LWRs, where the water is an efficient moderator. LWRs work in the following way: The fuel rods that form the core are arranged in a water tank. To control power, control rods that absorb neutrons are used as regulators. Water is either boiled at the fuel rods (Boiling Water Reactor, BWR) or heated by the fuel rods and boiled in a separate system (Pressurized Water Reactor, PWR). The steam goes through a turbine generating electricity, condensates and is returned to the system. 15 2.1.1. Neutron multiplication The effective neutron multiplication constant keff is a key parameter in fission. The parameter keff can be interpreted as the ratio of neutrons in any generation to the number of neutrons in the next generation. Todays fission reactors are critical reactors, which means that they operate with keff = 1. The parameter keff depends on many factors and can for a LWR be expressed in the so-called 4, 5, or 6-factor formulas where different effects are extracted into a number of factors. For example, the 6-factor formula is keff fp PFNL PTNL (2.1.1) where is number of fission neutrons produced per absorption in the fuel, f is the absorption in fuel probability, p is slowing down without being absorbed probability, is the fast fission factor and the last two are correction factors for neutron leakage of thermal and fast neutrons [2]. By following a neutron from its creation, the cascade of neutrons that will follow from fission reactions triggered by this neutron or later generations of neutrons in the cascade can be viewed. For critical reactors, the average ratio of the number of neutrons in the next generation to the number of neutrons in the present generation in this cascade remains at a nearly constant value keff for many generations, since all neutrons have the same source (fission) as the first neutron and thereby the same energy spectrum. For driven systems which have an external neutron source, this is not the case. The source neutrons in driven systems have a different source than the neutrons in subsequent generations which are fission neutrons, and thus have a different energy spectrum. This means that the first generation (and to some extent even the second) will have a different neutron multiplication than the subsequent ones due to the energy dependence in different cross sections and since the number of neutrons produced per fission reaction increases with neutron energy [3]. Also, the location of the neutrons has a different distribution. 2.1.2. Negative feedback factors and delayed neutrons Since LWRs operate at keff = 1 it may be questioned how the reactor can remain stable. The time between two generations of neutrons is in the order of 10 5 seconds. For a slightly supercritical keff = 1.0001 this corresponds to a neutron multiplication (and hence power increase) of 1.0001100 000 after one second. This is a large number (about 20 000). What saves the situation is a combination of two effects. The first effect is that keff is lowered by an increased temperature. The negative feedback factors on keff however need some time to become effective due to the need for heat conduction and boiling gas expansion (in the order of a second), and this effect alone is not sufficient for reactor stability. The second effect is that about 0.65% of the 16 neutrons are delayed in LWRs, and come from decay processes in the fuel on average 15 seconds or so after the fission reaction. This makes the changes in the neutron flux much slower as long as keff 1.0065 1 , where is the fraction of delayed neutrons. Due to the fraction of delayed neutrons, a moderate increase i keff only causes a slow increase in power and neutron flux. The negative feedback factors are effective at this time scale and respond to such a moderate increase in keff by reducing keff to unity at a new power balance point. Thereby, the reactor stays critical (or subcritical) if changes in keff are sufficiently slow and within some range. In practice, operation at LWRs follow predefined schemes for moving control rods etc. to ensure that supercriticality accidents should not happen as long as those rules are followed. There are two main negative feedback effects on keff that arise from higher temperature. One comes from the density decrease in water (or in a BWR from a higher steam percentage) which deteriorates the moderation of neutrons. The other effect is that resonance cross sections for neutron capture for epithermal neutrons in U-238 are broadened by the Doppler effect. This causes a larger fraction of the neutrons to be captured in U-238 when the temperature is increased. 2.2. Resources The fuel used today in fission comes from uranium ore. The natural uranium extracted from the ore consists of 99.3 % U-238, 0.7 % U-235 and negligible amounts of other isotopes. Most reactor types do not use natural uranium directly, but instead uranium enriched in U-235 up to a level of typically 4 % for LWRs. This enrichment process is similar to that which is used for producing nuclear weapons, where an enrichment level of 20 % is enough to build a bomb but 90 % or more would be desirable [4]. The reported amounts of available uranium for power production varies a lot depending on information source, partly since the amounts of uranium considered available varies with the uranium price and different prices are used in different information sources. The price depends on the percentage of uranium in the ore. A rough scaling is given in Ref. [5], where uranium ore is assumed to have a log-normal distribution over the world. This means that a tenfold decrease in ore grade would correspond to a 300-fold increase in amount of recoverable uranium from that ore in the Earth’s crust [5]. The Red Book [6] by IAEA has been produced in 24 editions since 1965 and is here regarded as a reasonably reliable source. The avaliable resources are sufficient for at least 100 years with the current uranium consumption at a fuel price of 260 USD/kg [6]. If the price is increased, there will be a lot more uranium available. Since the price of fission fuel is in the order of 17 0.005 EUR/kWh and the uranium cost is about 50% of the fuel cost [7] (0.0025 EUR/kWh) there is a large margin to increase the uranium price without having a dramatic increase in electricity price. Another possibility is to extract uranium from seawater. Seawater contains 3-4 ppb uranium, and the total amount of uranium in the seas is estimated to 4 gigatons, corresponding to about 100 000 years with the consumption rate of the world today [6]. There is no technique available today for extracting uranium from the sea at a competitive cost, but research is going on in Japan [6]. Currently the extraction cost is about 700 USD/kg [6] compared to todays market price of 100 USD/kg, which does not seem to be an unreasonable price in the future. When discussing resources, the possibility of breeding should also be taken into account. Today, only about 1% of the energy in the original uranium ore is consumed (converted) in the fission plants (see breeding, section 2.7). If future technology could solve the safety problems with breeding and TRU burning, there is a factor of 100 times more energy resources available from uranium. Also, then the mining costs per kWh will be much lower, enabling mining of lower-grade ore to be commercially feasible and thus increasing the amount of available uranium with a factor of 90 000 according to the very rough log-normal distribution [5], giving in total 9 000 000 times more fuel. There are however other problems associated with mining (such as difficulties of extracting uranium from very low-grade ore), and it is probably overoptimistic to believe that such a large portion of the earths crust will be available for mining purposes. This is however a rough indicator of how much fuel there could be. If the safety problems with breeding are solved, also thorium can be used for energy production. Thorium is about 3 times as abundant as uranium [8]. There are no facilities today that use thorium for commercial energy production, but a few test reactors have been built and some are under construction [9]. Specifically, India is aiming for a thorium fast breeder reactor that is supposed to be operational in 2013-2014 [10]. China has also started a program for thorium reactors [9]. For comparison with fusion fuel, C. Rubbia has claimed that the availability of fission fuel is about the same as the availability for D-T fusion fuel [11]. Lithium is 7 times as abundant as thorium in the earths crust, but only 7.5% of the lithium is Li-6 which is the primary isotope used. With this in regard, thorium is 4 times more energy dense than lithium (per unit mass) and the available energy resources for fission and fusion is about the same [11]. To summarize, the resources for fission power are vast. If breeding technology becomes commercial, the available resources are likely to be 18 immense. Although breeding is not the main target in this project, fusionfission hybrids seem well suited also for breeding, although critical fast breeder reactors are likely to be considerably cheaper since they do not need a fusion driver. 2.3. Front end The uranium used today originates from uranium mines. Uranium mining has caused environmental problems during the early stages of nuclear power and continues to do so today in some development countries, where environmental regulations are weak and risks of prosecution are small. Numerous examples are presented by Greenpeace [12] and are used as an argument against fission power. It is however evident that uranium mining can be done in ways that are safe for the personnel and has a similar environmental impact as other metalliferous mining activities [13]. Such mining has taken place in Canada and Australia where most mines have ISO 14001 certification [13]. Thus, this is not an argument against uranium mining in general but against uranium mining using environmentally benign techniques with lack of control. The mining capacity in the world today is lower than the consumption. The reason for this is that 25-50% of the uranium supply the last years has come from stockpiles of uranium and from downblending of weapons-grade uranium from nuclear weapons [6]. Since this source soon will diminish, the mining capacity needs to increase. 2.4. Back end The fission power industry must be able to handle the whole life cycle of the fission fuel which implies that the spent nuclear waste must be taken care of. There are two main ways to handle the problem. One is to get rid of some of the environmentally benign isotopes in the fuel by transmutation and fission, and store the remaining waste. This is described in section 2.6. The other method is to create a geological repository for the fuel such that the fuel remains safely stored for 100 000 years. After 100 000 years, the strongest radiating radioactive isotopes has decayed and the rest is regarded as fairly safe to leave in the ground. Sweden is in the front line of the development of geological repositories, and a repository large enough to store the Swedish nuclear waste is planned to be built near the fission plant Forsmark about 75 km outside Uppsala by Svensk KärnBränslehantering AB (SKB) [14]. The repository is to be built 500 m below the surface where it is assumed to be below the permafrost during an ice age (assumed to maximally reach somewhere around 400 m). The fuel rods are placed in thick copper canisters with cast iron inserts. The copper canisters are put in bentonite clay in 19 prepared caverns in the bedrock [14]. The Forsmark bedrock is considered to be very stable. A lot of research has already been made on this SKB model called KBS-3, but still some issues are debated, for instance the corrosion rate of the copper canisters. Natural questions also arises concerning nuclear non-proliferation, since the plutonium weapon-grade quality will increase in time due to the shorter half-life of Pu-240 (6 500 years) compared to Pu-239 (24 100 years). Although such patient terrorist organizations are unlikely to exist, the political situation 20000 years from now is hard to predict and it is inconvenient that high-quality weapons-grade plutonium will lie buried in the ground for tens of thousands of years. Also, there is always a possibility to use the spent fuel as a dirty bomb, where the explosion is created with conventional explosives. Another risk is treasure hunting, since some of the fission products are rare and valuable. The cost of the geological deposit in Sweden is estimated to about 12 billion euros in current monetary value [14]. This is financed by a nuclear waste fund, which receives 0.001 euro for every kWh fission energy sold. 2.5. Nuclear non-proliferation Nuclear non-proliferation, to prevent nations and groups to develop nuclear weapons, is probably the largest concern for nuclear power. The strategy is to have a sophisticated control of weapons-grade fissile material such as enriched uranium and weapons-grade plutonium, and to prevent the spread of knowledge and technology required to produce such materials, in particular enrichment facilities for uranium. Uranium with more than 20 % U-235 is considered weapons-grade uranium. Plutonium is considered weapons-grade if it contains less than 7 % Pu-240. This is due to a considerable rate of spontaneous fission in Pu-240, which could cause predetonation in a bomb before sufficient plutonium mass is assembled. Even with weapons-grade plutonium, a plutonium bomb must be of implosion type to give a massive explosion, which is considered more complicated to build than the simpler uranium bomb. LWRs can be used to produce weapons-grade plutonium if the fuel is removed from the reactor shortly after refueling (1-3 months). Pu-240 is formed from neutron capture in Pu-239, and therefore the plutonium is weapon-grade in the beginning of the fuel cycle and the amount of Pu-240 increases continuously towards the end of the fuel cycle. 20 2.6. Transmutation Transmutation is the process of changing one nucleus to another (or two others) by neutron capture or fission. In fission power, transmutation can be used to get rid of undesired radiating species in the nuclear waste in order to reduce the geological storage time [15]. There are two groups of elements that are targets of transmutation: transuranics (TRU) and long-lived fission products (LLFP). It should be pointed out that transmutation is not recommended by all sources, see Ref. [16]. A list of the most important targets of transmutation is given in Ref. [17]. Complications with transmutation are the reprocessing of the waste necessary to separate the different species [16] and to find safe reactors that can handle sufficient densities of the transmutation targets. Another concern is that industrial transmutation may be connected with nuclear proliferation. 2.6.1. Transuranics Transuranics are created from neutron capture in U-238 followed by subsequent neutron captures and decays. The main component of the transuranics is Pu-239, which is the main component responsible for the long geological storage times for the nuclear waste with a halflife decay time of 24 100 years. Radiotoxicity is a measure of how dangerous a radionuclide is for the human body. After 200 years of storage, TRU contributes with the major part of the radiotoxicity of the nuclear waste [16]. Transuranics are much more radiotoxic than fission products since they typically are emitters (emits helium nuclei) and fission products typically are -emitters (emits electrons) [17]. Pu-239 is produced in LWRs but also to some extent fissioned. To transmute minor actinides, the transuranics except plutonium, a fast neutron spectrum is required [18]. The options to produce this fast neutron spectrum are fast reactors and driven systems, but there seems to be a consensus that driven systems are required for the transmutation of minor actinides for reactor safety reasons [18]. Fission is always the goal of transmutation of transuranics, since the decay chain to stable isotopes is long for the elements in this group and since neutron capture will only result in another transuranic isotope [16]. This implies that transmutation of transuranics produces a lot of thermal energy which could be possible to utilize in a power plant. 2.6.2. Long-Lived Fission Products Fission products are the rest products from fission and elements created by subsequent neutron capture and decay from these. Fission of one isotope (for example U-235) can result in a lot of different combinations of fission products. Some are stable, some have short half-times and 7 are long-lived. 21 Fission products with a half-life of less than 90 years are classified as medium-lived or short-lived. The largest radiation emitters are Sr-90 and Cs137, with a half-life of about 30 years [16]. These cannot be transmuted due to a low cross section for neutron capture and must be stored until they decay, which takes up to 500 years [16]. Since Sr-90 and Cs-137 anyway need to decay, there is no point in transmuting any medium-lived or shortlived isotopes. Targets for transmutation have therefore been the 7 long-lived fission products (LLFP). Of those, Tc-99 and I-129 are the isotopes that have been targeted in most studies [16], since they give the largest contribution to radiation [17], are possible to (slowly) transmute with thermal neutrons and since they are mobile enough in the environment (soluble in water) [17] to pose a large threat for geological storage [15]. Cs-135 is hard to transmute without isotope separation [16]. The other 4 isotopes are either in smaller amounts (at least for LWR waste from fissioning U-235), or considered immobile in the environment (like the noble metal isotope Pa-107). Transmutation of Tc-99 and I-129 can be done in LWRs [16] or possibly with faster neutron spectras [19]. Transmutation of fission products does not produce significant amounts of energy, and consumes neutrons. Although it may be possible to transmute some of the LLFP, there is now a general consensus in the fission community that transmutation of LLFP is not necessary. Studies have shown that the radiation doses given from LLFP leaking from a geological repository to the most exposed groups of humans in any of the investigated scenarios are several orders of magnitude smaller than the background radiation [20]. Thereby, transmutation efforts are nowadays focused on transuranics, in particular plutonium and americium. 2.7. Breeding Fertile nuclei are nuclei which are not fissile in a self-sustainable way, but may be transformed into fissile nuclei by neutron capture. Breeding is the process of turning fertile nuclei into fissile ones. The two species mainly considered for breeding are U-238 and Th-232. U-238 is transmuted into Pu239 and Th-232 into U-233 by neutron capture and subsequent decay processes. The point with breeding is to produce more fuel, where there is a factor of 100 to gain only on the uranium. At a first glance, this may seem to be in contrast to transmutation, where the aim is to get rid of plutonium. However, both can be used in combination, and the final waste may still be nearly cleansed on plutonium if a working transmutation scheme can be implemented. Breeding is already present to some extent in todays LWRs, but the gain of fissile fuel is less than the consumption. To have a net gain in fissile fuel, which is the goal of a breeder reactor, a fast neutron spectrum is preferred since a fast neutron spectrum produces more neutrons per fission 22 on average [16] and the capture-to-fission ratio is smaller, giving a significantly larger in Eq. (2.1.1) [2]. 2.8. Critical fast reactors Critical fast reactors are critical reactors with a fast neutron spectrum, which differ from LWRs that use thermal neutrons for fission. The goals of fast reactors are transmutation, breeding and also hydrogen production enabled by the high temperature of the coolant [2]. Fast reactor programs have however been plagued by safety problems etc. [16] and have not yet been commercialized. The main two stabilizing negative feedback factors in LWRs on keff are Doppler broadening in U-238 and effects on moderation due to water density changes (in BWR, void percentage). In fast reactors, water is not present since the water would moderate the neutrons. Instead, liquid metal (lead, sodium, lead-bismuth), molten salt or gas is used as coolant [2]. The Doppler broadening effect is much less effective since a smaller fraction of the neutrons that may undergo fission pass the neutron capture resonances and since there often is less U-238 in the fuel. Another feedback factor on keff , fuel/coolant heat expansion, plays a role in fast reactors. A safety concern is also that the fraction of delayed neutrons is less in plutonium ( Pu-239 = 0.26%) and even less in americium ( Am-241 = 0.12%) and curium ( Cm-244 = 0.13%) [21]. Loss of coolant, in particular for sodiumcooled fast reactors, is a safety concern. Replacement of the metal coolant by water around the fuel could lead to a catastrophy. Together, this makes the fast reactor concepts much less safe than thermal reactors, and breeder reactors have today not reached public acceptance. Critical fast reactors are a key area for the generation IV studies. 2.9. Driven systems Driven systems are non-critical reactors with keff less than unity. Thereby, an external source of neutrons is required to maintain the neutron flux. The reactor safety in a driven system is not dependent on negative feedback factors and delayed neutrons, although they may still add somewhat to safety. Instead, a driven system relies on the keff that keeps the reactor subcritical. In all possible scenarios, keff must be kept below unity (plus the fraction of delayed neutrons) [21]. For Accelerator Driven Systems (ADS) a typical value is keff = 0.97. This is a larger margin than the delayed neutron fraction give in LWRs (0.0065) and it may seem that driven systems are safer. However, a driven system is more complex, has much smaller negative feedback factors (if any) on temperature and is in several aspects regarded as less safe than LWRs. The goal of driven systems is primarily transmutation in combination with energy production [18]. Driven fast reactors could also be used to accomplish breeding, although competition 23 from fast critical reactors may be too strong. As mentioned earlier, Stacey claims that driven systems are necessary to accomplish transmutation [18], see also Ref. [2]. The very point of driven systems is that they can perform the tasks that require a fast neutron spectrum with a much larger safety margin than critical fast reactors have, and that they are almost independent of the fraction in the fuel. The fission core provides a rather large neutron multiplication which gives a large power multiplication. If the neutron multiplication keff is assumed to be constant for each successive neutron generation, which is not exactly true, in particular for the first generation, the total number of neutrons n produced for each source neutron on average can be calculated as a geometric series. n keff2 keff keffi 3 ... keff i 1 keff 1 keff (2.9.1) For keff = 0.96, this would increase the total number of neutrons produced by a factor of 25, and for keff = 0.98 this gives n 50. It is obvious that keff should be as high as possible constrained by the safety issues. An approximation of the total energy gain is Qr rh nc next W fis keff Wsn 1 keff (2.9.2) where W fis 195 MeV is the average energy produced in a fission reaction, Wsn is the average energy cost for each source neutron, v 2.9 is the average number of neutrons produced in each fission reaction, nc is the fraction of source neutrons that enter the fission core, next is a correction factor that takes effects of different source neutron energy into account and rh is the fraction of heat contribution that does not come directly from fission reactions (primarily decay of fission products). There are two types of neutron sources considered today. One is ADS, and the other is fusion. For fusion devices to be useful as neutron sources, they do not only need to fulfill the requirements for transmutation (concerning energy consumption, environmental impact, overall cost etc.). They must also be competitive against ADS systems. It is today not obvious which system will be superior, and at the end the overall cost and reliability are likely to be important factors. Also, the safety restrictions on keff for the different systems will strongly affect the energy efficiency, and this depends on the layout of the fission core. 24 2.9.1. Accelerator-driven systems ADS is the most developed concept for driven systems today. An extensive comparison with fast reactors can be found in Ref. [15]. The system consists of a proton accelerator that produces a proton beam. The beam is injected into a spallation target of heavy metal, typically lead, which is aimed to produce about 20-30 source neutrons for each injected proton. The source neutrons enter the fission core which surrounds the spallation target [15]. The energy of the injected neutrons is in the order of 1 GeV, and the electric energy efficiency of the accelerator is up to 50% [15]. The average energy for the spallation neutrons is about 1.6 MeV, and the average electric energy cost to produce one neutron is about 100-150 MeV. The ADS technology faces some challenges [22]. One challenge is to increase the average accelerator beam current. Another challenge is to maintain the neutron production in the spallation target, i.e. to prevent vaporization at the beam target point. Another concern is the utility, which presently is very low. 2.9.2. Fusion-driven systems Fusion-driven systems (fusion hybrids) use a fusion device as neutron source. The concept was proposed already in the early days of fusion, and was persued by H. Bethe [23] and others (see for example Ref. [24]) in the 1970ies. The Three-Mile-Island accident in 1979 led to a decline in the fusion-fission research, and especially after the Chernobyl accident in 1986 it seems that the fusion community wanted to keep a distance from fission to avoid negative publicity. It even seems that some researchers in fusionfission had problems both to publish their work and to get financed due to this policy (see the acknowledgement in [25]). Also, after the Fukushima accident, several countries have an ambition to avoid fission energy if possible. Fusion is however in several ways naturally linked to fission. A fusion neutron source (like any sufficiently intense neutron source) can be used for breeding of weapons-grade plutonium or U-233 (which also probably can be used for production of nuclear weapons) from thorium. Fusion therefore has a link to the nuclear non-proliferation problems, which was pointed out already by L. Lidsky in his criticism against fusion [26]. Also, it has been questioned if a tokamak fusion reactor can breed sufficient amounts of tritium be self-sufficient. If fusion reactors cannot, fission reactors or fissile inserts in fusion reactors are probably required for producing tritium. Fission reactors are the source of tritium today. In the new millennium there has been a renewed interest in fusion-fission, and the subject is being pursued by several groups. Some of them are (in no specific order): 25 1. The SFLM Hybrid project. This is what this thesis is about. 2. Researchers at Budker Institute who studies a mirror-based hybrid scenario using the axisymmetric Gas Dynamic Trap (GDT) and the new modified GDT as a driver [27]. 3. R.W. Moir et al. at Lawrence Livermore who recently presented an axisymmetric mirror-based concept [28]. 4. S. Taczanowski et al. has some activities in mirror-based fusionfission [3]. 5. W.M. Stacey et al. at Georgia Tech who have studied several tokamak-based concepts with downscaled ITER parameters [29]. 6. Y. Wu et al. in China who studies tokamak-based hybrids (several FDS concepts) and are putting large resources into hybrid studies [30]. 7. M. Kotschenreuther et al. of Institute for Fusion Studies who studies tokamak-based hybrids [31]. 8. M. Gryaznewich et al. at Culham Laboratory is examining the possibilities to use spherical tokamaks as neutron sources [32]. 9. A Russian program has recently been initiated to build a sperical tokamak neutron source. 10. V. E. Moiseenko et al. in Kharkiv, Ukraine have an experimental stellarator-mirror facility aimed for fusion-fission and to become a neutron source [33]. 11. H. Yapici et al. are working with fusion-fission using catalyzed fusion as a driver [34]. 12. M. Ragheb, A. N. Eldin et al. at the University of Illinois who are considering thorium breeding using hybrid reactors [35]. 13. F. Winterberg has presented ideas concerning fusion-fission reactors [36]. 14. W. Manheimer is advocating fusion-fission [25]. Several types of fusion devices may be used as neutron sources, where the tokamak is the most studied source so far. The two largest theoretical tokamak hybrid projects that exist today are the FTWR in USA [29] and the FDS [30] in China. Both are based on tokamaks with downscaled ITER parameters. The strength with the tokamak concept is the reasonably good plasma confinement, which allows for large safety margins on keff. However, since the power multiplication in the fission mantle probably can be large ( 100), it is not obvious that such a confinement is crucial for hybrid reactors. This enables the use of other concepts. Tokamaks have some major drawbacks that make them less suited for hybrid reactors if a better suited fusion device can have sufficient plasma confinement for hybrid reactor configuration. The tokamak cannot (at least presently) be run in steady-state due to the need for inductive toroidal current drive. The pulses will be in the order of 20 minutes for ITER, which would correspond to a pulsed gigawatt 26 power source for the grid and could cause material problems. There is also a lack of space in a tokamak for the fission mantle due to all instrumentation and plasma heating, which typically causes more than half of the source neutrons to escape from the reactor without having the chance to produce fission. Specifically, in SABR only 39 % of the fusion neutrons enter the fission core [37]. Large scale plasma activities (disruptions and instabilities) are also concerns for tokamak hybrid reactor scenarios. Mirror hybrid reactors have been considered by Bethe [23], Taczanowski [3], Moir [28][38] and Noack et al [27] at the Budker Institute in Novosibirsk, Russia. Mirror machines are relatively simple, can operate in steady-state and can conveniently decrease the plasma heat load on the walls by using magnetic expanders and thereby increasing the wall area. The fraction of source neutrons that escape without entering the fission mantle can be less than 10% and the first wall neutron load can be made acceptable. The major concern is the electron temperature in the plasma. For the SFLM hybrid case, the approximate formula for the energy amplification Qr Pfiss / Pfus , the produced fission power divided by the produced fusion power, is Qr where keff case. 1.2 195 0.97 150 17.6 2.9 0.03 0.97 has been selected and the product rh nc next (2.9.3) 1.2 in this The 14 MeV fusion neutrons are about ten times more energetic than the average spallation source neutrons and the fission neutrons. Source neutrons with energy larger than 6.5 MeV produce essentially more fission neutrons than strongly moderated source neutrons. In Ref [3], the high energy of the fusion neutrons gives about 50 % extra fission neutrons in the first neutron generation. A too strong moderation of the source neutrons before they reach the fission core should therefore be avoided. 27 28 3. Fusion energy and plasma physics A fusion-fission reactor needs a fusion driver, and the main work in this doctoral thesis is about the mirror fusion driver. In this section, some basics about fusion and plasma physics are given, as well as some information specific to mirror machines. 3.1. Basics in fusion 3.1.1. Plasmas When a gas is heated to a high temperature, the molecules start to break up and the atoms are ionized. An ionized gas is called a plasma. Plasmas exist in many applications, such as for example low ionized plasmas in high voltage circuit breakers. In nature, plasmas exist for example in a lightning bolt. Outside the Earth, most matter is in plasma state, including all stars and the Sun. 3.1.2. Fusion reactions Fusion energy is produced by joining two lighter nuclei into a heavier one, plus other particles in some cases, where the resulting particles have a lower total rest mass than the original ones. The rest mass difference is released as energy. The most commonly targeted reaction is D + T = He + n + 17.6 MeV (3.1.1) where D is deuterium (hydrogen with 1 neutron), T is tritium (hydrogen with 2 neutrons) and n is a neutron. The neutron receives 14.1 MeV and the alpha particle (He) 3.5 MeV [39]. There are also a number of other reactions, for example D + D reactions giving either He-3 + n + 3.2 MeV or T + p + 4.0 MeV [39]. D-D fusion requires higher colliding energies than D-T fusion and gives less energy in each reaction, but uses on the other hand only deuterium. To initiate a fusion reaction, the electric force repelling the two nuclei that should react must be overcome. When the nuclei get close enough, the attracting nuclear force become greater than the repelling electric force and the nuclei will undergo a fusion reaction. This implies that the particles must 29 have a high velocity, and also a lot of “target” particles to collide with. For thermonuclear D-T fusion, an ion temperature of about 10 keV is required [39], i.e. about 100 million °C. The densities in magnetically confined fusion plasmas are typically in the order of 1019-1020 particles per cubic meter. This is far less than in air at atmospheric pressure, having about 2 1025 molecules per cubic meter. The reason for the “low” density compared to air is that since such a high temperature is required, the pressure would be huge and the magnetic field required (> 100 T) to contain such a pressure would be too high. Fusion plasmas that are considered here are fully ionized, at least if impurities are neglected. 3.1.3. Fusion and confinement One simple idea to create fusion would be to accelerate deuterion ions and let them burst into a solid target containing tritium. This approach will however not work for energy production, since the ions will lose their energy through collisions too fast and too few fusion reactions would result per incoming ion to produce a net gain in energy [39]. Another idea is to use muons to catalyze fusion reactions, sometimes referred to as cold fusion but more often as muon catalyzed fusion [40]. Muons are subatomic particles with the same charge as an electron and a mass about 200 times larger, and a “muon hydrogen atom” (where a muon replaces the electron) is compact and has a much smaller Coulomb barrier for fusion. Muon catalyzed fusion can be accomplished at room temperature. To the author’s knowledge, there is no reactor scenario for muon catalyzed fusion yet, but studies aimed at catalyzed fusion-fission are carried out [41]. The main path of fusion research is to heat a D-T mix to a hot plasma having fusion temperature. The difficult task is to confine the plasma. A well working fusion reactor is the Sun, which confines the plasma with strong gravity forces. That option is however not available on the Earth. The fusion plasma is also far too hot for any container material to withstand, so it cannot be kept in a container without other means of confinement. Today, there are two main paths pursued in fusion research; inertial confinement and magnetic confinement. Inertial confinement is pulsed, and aims to find a way to heat fusion fuel (typically a frozen pellet) into a hot plasma very fast and keep it together as long as possible with momentum transfer from lasers or similar radiation sources. The H-bomb is an example of inertial fusion, where a fission bomb supplies both heating and compression with radiation targeting the fusion material. The National Ignition Facility in USA is a large inertial fusion experiment where lasers are used to target a frozen D-T pellet [42]. In this doctoral thesis, magnetic confinement is addressed, and henceforth only magnetic confinement will be discussed. 30 3.1.4. Resources In fusion, several fuels can be used. This doctoral thesis is focused on D-T fusion. Deuterium is widely abundant in the world. About 1/7000 of the hydrogen atoms are deuterium. There is a lot of hydrogen in the oceans, and the available amount of deuterium is huge. For D-T fusion, the deuterium resource will not be limiting. Tritium is produced from neutron radiation of lithium, mainly from Li-6, and has a lifetime of about 12 years. The resources of lithium are large, and would probably last for millions of years of intensive utilization [11]. 3.2. Plasma physics In plasma physics, three different detail levels can be used whilst performing calculations. The first is to look at single particle motion in the magnetic field. From this approach, particle drifts can for example be derived. However, in these calculations effects of the surrounding particles are not taken into account, and all the important effects related to multi particle interaction are thereby lost. It is impossible to take into account all the plasma particles as single particles, since there are typically 1019-1020 particles per m3 in the plasma and each particle gives a force on all other particles. The second approach is to look at the plasma from the statistical mechanics viewpoint and use the kinetic theory. Kinetic theory takes most effects into account but it is often a tremendous task to carry out the calculations. The third approach is to treat the plasma as an electromagnetic fluid and use the MHD (MagnetoHydroDynamic) equations. MHD theory is easier (although not necessarily easy) than the kinetic theory, but some information is lost and not all effects can be found. In this section, some basics of plasma physics are explained. 3.2.1. Magnetic confinement In a fully ionized plasma, all particles are charged. Magnetic confinement is based on the Lorentz force, F q (E v B ) (3.2.1) where v is the particle velocity. The magnetic force is perpendicular to both B and v, forcing each particle to gyrate around a certain magnetic field line (to leading order). Thereby the particles are confined, at least to some extent, in the directions perpendicular to B if the magnetic field is strong enough to give a sufficiently small gyro radius (Larmor radius). The gyro radius rg is given by 31 rg v qB mv , qB (3.2.2) m where is the gyro angular frequency. For comparable temperatures, the ions have much larger gyro radii than the electrons due to the larger ion mass. The ratio is proportional to ( M / m)1/ 2 , giving the deuterium ions a radius that is about ( M D / me )1/ 2 60 times larger than the electron radius [43]. A typical ion gyro radius in a fusion plasma is in the order of 1 cm (B =2 T, Ti = 10 keV gives for deuterium ions rL,i = 0.7 cm) while the electrons typically have a gyro radius less than 1 mm. In the direction parallel to B the particles movement is not as restricted, and the particle trajectory is locally helix-like. The problem is to find a way to confine the particles along B. There are several solutions to this problem. The most obvious choice is to make a toroidal (doughnut-shaped) magnetic surface where the flux surfaces are closed, although most of the field lines are not closed (construction of “closed”, or nested, magnetic surfaces is straightforward for axisymmetric tokamaks but a difficult task for stellarators). Tokamaks and stellarators are toroidal devices. Mirror machines, which will be considered here, have an open magnetic configuration and a straight pipe-like vacuum chamber for the plasma. The ends of the pipe are magnetically plugged for the plasma by the reflecting magnetic mirror force, which arises when particles move from weaker magnetic field to stronger, see section 3.3.1. 3.2.2. Particle drifts Particles are restricted in the perpendicular direction by the magnetic force. However, if a force perpendicular to B is applied on the particle, the gyro radius varies during the gyro period which results in a particle drift perpendicular to both the magnetic field and the force. With B B0 zˆ is constant, E = 0 and F qB0 v B F0 xˆ , the equations of motion give vx vy F0 , vy m vx , vz v 0 (3.2.3) where vx 2 vx 0 vy 2 vy qB F0 m2 (3.2.4) If F0 = 0, the solution to these equation is a helical trajectory with a circular movement in the projection on the xy-plane as described earlier with a gyro frequency given in Eq. (3.2.2).The guiding center variables 32 xc x vy , yc y vx (3.2.5) are constant (no perpendicular drift of the guiding centers). With a finite F0, a guiding center drift appears. With F0 constant, the solutions are vx v 0 cos t vy v 0 sin t (3.2.6) F0 qB0 with the chosen initial condition v (t 0) v 0 xˆ . For a more general case the velocity in vector notation can be written (apart from higher order gyro oscillations) v v Bˆ v g vd (3.2.7) where vg is the gyrating part and vd is the drift velocity averaged over the gyro motion. There are a number of forces that can cause drifts. The gravity drift is one example. Since electric fields give a force that is charge dependent, the electric drift is charge independant. Magnetic field gradients and curvature gives rise to charge dependent so-called B and centrifugal or curvature drifts. The leading order gyro-averaged total drift velocity for time-independent fields can be written vd BB B 3 q B E B B2 mv 2 B (Bˆ Bˆ ) q B2 (3.2.8) where the magnetic moment mv 2 / 2 B (3.2.9) is an approximate constant of motion and the gravitational drift has been omitted as well as the drift from a nonuniform E. The first term is the E B drift, the second is the drift from a gradient in B and the third term is the curvature drift, where (Bˆ Bˆ ) 1/ rc rˆc is the curvature [44] and rc is B 0 ) this simplifies to the curvature radius. For a vacuum field ( vd E B B2 2 1 B mv (B q B2 ) (3.2.10) which would be zero for E = 0 and 0 . The guiding center velocity v gc v and the energy conservation for the particle can be rewritten as a constant energy for the guiding center motion, m 2 q (x gc ) B (x gc ) (3.2.11) v gc const. 2 33 There are drifts for time-dependant fields as well, but those are not addressed in this text. 3.2.3. Collisions in fusion plasmas In a plasma, particles collide. To describe collisions, a few concepts are useful. A cross section is used to express the likelihood of interaction between particles. Cross sections are dependent on the particle velocity, and collision cross sections are typically denoted pq where p and q are the two colliding particles. An example of a cross section is the 90° electron scattering cross section of ion-electron collision ei Z 2 e 4 ln 4 02 me2 v 4 (3.2.12) where ln is about 20 and Z is the atomic charge [43]. The mean free path mfp is the distance particles on average can move in between collisions. The collision frequency pk describes how often particles will collide on average, and the collision time (or drag time) pk =1/ pk is the average time between collisions. We have for a nearly Maxwellian distribution v th pk mfp nk v th 1 pk , mfp nk (3.2.13) pk Only a small fraction of the collisions lead to fusion reactions. The remaining collision processes are more or less distant Coloumb collisions where the momenta of the colliding particles are changed by the localized electric forces around the charges. The main contribution of the average momentum change for a particle comes from cumulative weak distant collisions. The difference in mean square momentum impact of the collisions that change the particle velocity less than 90° compared to those that change the velocity more than 90° are about a factor of 70 [43]. This means that velocity changes of particles in a fusion plasma mainly happens in small steps distributed in time which is an important property of hot plasmas. The collision frequency for low angle scattering goes down with increasing temperature, as can be seen in Eq. (3.2.12), but does not become zero. The properties described above are for electron-ion collisions, but similar properties yields for electron-electron scattering and ion-ion scattering. An important difference is that the diffusion impact of electron-electron collisions are on a much faster time scale than for ion-ion collisions due to the higher charge to mass ratio for electrons. Another important difference is that through ion-ion collisions and electron-electron collisions, energy transfer is efficient, but for electron-ion collisions the energy transfer is slower due to the large mass difference between the colliding particles. Thereby, the electron temperature and the ion temperature may be different. 34 3.2.4. Radial transport Radial transport, i.e. particle transport perpendicular to the magnetic surfaces in the radial direction, occurs through different processes. One is through collisions. If particles collide and their momentum is changed, they move from the original encircled flux line to a new position. The step-size is in the order of the gyro radius. This collisional transport is called classical transport. However, there are other mechanisms through which radial transport occurs, and those typically deteriorate the plasma confinement considerably more than the classical collisions. The problem of constructing a working fusion energy producing test facility would probably have been much less challanging if the classical transport was the only transport mechanism present. Another mechanism for radial transport is the neoclassical transport. This transport is also based on collisions, but the impact is in many cases far worse. In a magnetic field, the guiding centers do not follow a single flux line in any fusion devices except for the SFLM vacuum field, since there is magnetic field curvature and gradients in B. These cause particle drifts that deform the particle orbits. In tokamaks, some projected guiding center orbits of mirror-trapped particles are banana-shaped and are called “banana” orbits. Those large-scale orbits have typically much larger widths than the gyro radius, and a collision for such a particle will lead to a much larger radial step than the gyro radius on average. The diffusion coefficient in “random walk” processes can be estimated from D coll where coll is the collision frequency and collision process. r2 (3.2.14) r is the typical step length in the There are also other radial transport mechanisms caused by fluctuations or formation of nonlinear structures, which are referred to as anomalous transport. In many cases, for example in tokamaks, anomalous transport gives the major contribution to radial transport. The anomalous transport is important but complicated and is not discussed any further in this text. 3.2.5. Kinetic theory: the Vlasov equation A very brief overview of kinetic theory is given in this section. In kinetic theory, the plasma is treated from a statistical mechanics viewpoint. The plasma particle densities are described by the distribution function, or which is a function probability density, f (x, v, t ) for the particle species of 6 dimensions plus time. The 6-dimensional (x,v) space is called the phase35 space. If collisions are neglected, which is a good approximation for a hot plasma, the time dependence of f (x, v, t ) in phase-space is determined by the Lorentz force and E and B fields. The Boltzmann equation describes the evolution of f (x, v, t ) with a force F and a collision term at the r.h.s. df dt f t v f F m f v f t (3.2.15) c Replacing the force F with the Lorentz force and setting the collision term to 0 gives the Vlasov equation f t v f q (E v B ) m f v 0 (3.2.16) The E and B fields should be interpreted as the smoothened fields where the fluctuating fields from the nearby individual particles do not show, and the collision term ( f / t )c represents the forces from the near fields from individual particles and plasma heating. Collision terms can be added, such as in the Fokker-Planck equation [43], but this will not be discussed in this text. The Vlasov equation takes most effects into account that can not be handled with MHD theory, but is quite difficult to use. An example of an effect where kinetic theory is required is Landau damping. For the equilibrium, an arbitrary function of constants of motion ci (x, v ) , f f (c1, ..., cn ) is a solution for the Vlasov equation. A question is how large n is. 3.2.6. MHD Equations A brief overview of the Ideal MHD equations is given in this section. The Ideal MHD equations are basically fluid equations combined with Maxwells equations. They are derived from the Boltzmann equation [45][46], f t v f q m E v B v f f t 0 (3.2.17) c and the where f is the distribution function for the particle type gravitational force has been neglected. By taking moments of the Boltzmann equation (integrating over velocity space), Qi V f t v f where the different Qi are 36 q m E v B v f f t d 3v 0 (3.2.18) c Q1 m Q2 m v Q3 ( mass ) (momentum) (3.2.19 a-c) 2 m v / 2 (energy ) the two-fluid equations can be derived for the two species ions and electrons (we assume here that there is only one species of ions). These equations can then be joined to form the single fluid equations. By invoking a number of approximations and assumptions, one arrives at the Ideal MHD Equations. The approximations are [45] 1. Quasineutrality. It is assumed that 0 E / en 1 , ne ni n . 2. The displacement current is neglected. Thereby, it is assumed that thermal velocities and phase velocity of waves is considerably lower than the speed of light. 3. The electron inertia is neglected. 4. The gyro radius is small. 5. The plasma resistivity is negligible. 6. High collisionality, which means that the pressure can be modelled as a scalar pressure. The Ideal MHD fluid equations then becomes u t j B m t ( m E u B 0 d p 0 dt m p Momentum balance equation (3.2.20) u) 0 Continuity equation (3.2.21) Ohm’s law (3.2.22) Adiabatic equation of state (3.2.23) and should be combined with the Maxwell equations B E B 0 0 j B t Amperes law (3.2.24) Faradays law (3.2.25) Gauss law for magnetic fields (3.2.26) where is the ratio of specific heats ( would correspond to incompressible flow, 5 / 3 corresponds to the adiabatic equation in Ideal MHD), m is the mass density and u is the local mass velocity. The adiabatic equation results when it is assumed that there is no heat conduction [46]. Although the high collisionality assumtion is virtually never satisfied in fusion plasmas, the Ideal MHD equations can predict many phenomena that 37 have been verified experimentally. Examples are MHD instabilities, causing dramatic plasma displacements, and the frozen in flux theorem predicting that plasma tends to be tied to the magnetic flux lines for time dependent magnetic fields if the plasma resistivity can be neglected. Stability within Ideal MHD can be analyzed by linearizing perturbations around a static (u 0) equilibrium. A selfadjoint boundary value problem arises by considering Fourier components of perturbed quantities, i.e. B B (x)e i t etc. resulting in an equation of the form 2 F( m ) (3.2.27) is a small plasma displacement and F ( ) is linear in where first derivatives. Selfadjointness leads to the energy principle W 2 2 m d 3x , Fd 3 x W and its (3.2.28) (x) , the system is stable within Ideal If 2 0 for all test functions MHD. Using simplified analythical test functions is the most common way to identify MHD instabilities. 3.2.7. Diamagnetism, MHD equilibrium and the concept of In a magnetically confined plasma, each gyrating particle creates a magnetic field in the opposite direction to the external field B, and will thus partly cancel the magnetic field. The plasma is thereby diamagnetic. The force balance for equilibrium becomes (from Eq. 3.2.20) j B p (3.2.29) and combined with the other two Ideal MHD equilibrium equations Eq. (3.2.24) and Eq. (3.2.26) this gives p j B ( B) B / 0 (B )B (B )B ( B 2 / 2) / 0 (3.2.30) This can be reorganized as p B2 2 0 (3.2.31) 0 The second term on the l.h.s. acts as a “magnetic pressure” and is called the magnetic pressure. The term on the r.h.s. is called the magnetic tension and has to do with magnetic field line curvature. If the curvature can be neglected, which is reasonable in some cases, the sum of the magnetic pressure and the plasma pressure is constant. To give a measure of how much pressure there is in the magnetic field, 38 p B /2 (3.2.32) 2 0 is defined. For the most common definition of , B is the vacuum field. A device has a certain limit for MHD stability, where the limit depends on geometry and magnetic field in particular. Mirror machines have higher limits than tokamaks and stellarators, where for mirror machines normally is defined at the midplane (x = y = z = 0) where B has a local minimum. In this thesis 0 will be the value at (x = y = z = 0) and will refer to the local . Also, for mirrors the perpendicular pressure component p should be used instead of p in the definition of . 3.2.8. Plasma instabilites To confine a plasma, it is not sufficient to find a device that gives a satisfactory equilibrium. The plasma must also be stable, at least against MHD stabilities. The consequences of MHD instabilities are normally a rapid loss of confinement, and there is a general consensus in the fusion community that MHD stabilities must be avoided in fusion devices [45]. If the plasma is slightly displaced from its equilibrium position, the system is stable if the plasma returns to the equilibrium position and unstable if the displacement continues to grow. As stated by the Energy Principle for the Ideal MHD theory [45], instabilities occur whenever the plasma can displace in such a way that it reduces the total potential energy of the system. There are many types of instabilities that can occur in a plasma, and some of them are briefly described here. Usually, the most serious instabilities are the MHD instabilities, which can be derived from the MHD equations. Other types of instabilities are tearing modes, which feed upon magnetic reconnection that decreases the magnetic field energy. There are also many small scale instabilities called microinstabilities or velocity space instabilities that have many different causes. These instabilities typically increases transport, but on a less dramatic scale than MHD instabilities. 3.3. Mirror machines 3.3.1. End confinement and electron temperature The end confinement is the weakest point of a mirror machine, and it is here for simplicity described for a single cell magnetic mirror. Both ions and electrons are confined through the magnetic mirror effect, where the confinement for each particle depends on the particles pitch angle in the midplane. If we first assume that E can be neglected, particles having high perpendicular (compared to the magnetic field lines) velocity compared to 39 parallel velocity will be confined, and particles that do not will escape the magnetic trap. Particles that are lost longitudinally are said to be in the loss cone, where the midplane pitch angle arctan(v / v ) is lower than the loss cone angle c arcsin 1/ Rm (3.3.1) where Rm Bmax / B0 is the mirror ratio, the ratio between the magnetic field modulus at the mirror end and at the midplane [47]. The reflection mechanism arise as a consequence of the conservation of the magnetic moment, mv 2 / 2 B and the energy mv 2 / 2 B q . From Eq. (3.3.1), it can be seen that some plasma particles are immediately lost upon creation (ionization). The longitudinal confinements of the remaining particles depend on how long they can stay out of the loss cone. This is determined by collisions and high frequency instabilities [47], which will isotropize the plasma. The timescale of angular scattering of particle species is much shorter for electrons than for ions due to the electron’s lower mass, where the ratio for deuterions to electrons is ( M / m)1/ 2 60 . Thus, the electrons will leak out to the loss cone much faster than the ions. However, since the plasma then will get a surplus of ions, a positive plasma electric potential called the ambipolar potential will be built up in the plasma after a very short period of time. This potential will confine all but the most energetic electrons, so that the ion loss rate is the same as the electron loss rate which ensures quasineutrality of the plasma (i.e. that there are almost equal numbers of positive and negative charges). The loss cones are thereby converted to hyperboloids of rotation, nicely illustrated in Ref. [47] p. 1634. A sketch of the loss cones for ions and electrons including the ambipolar potential is shown in figure 3.1. Figure 3.1. A sketch of the loss cones in a magnetic mirror. Typically, e , where is the ambipolar potential difference, is about 46 times kbTe, confining all electrons with lower energies and expelling all ions with lower energies (partly dependent on where E is located) [47]. This 40 implies that the electron distribution is close to a Maxwellian distribution. For high energy particles, the hyperboloids asymptotically approach the loss cone shape. A measure of the time scale on which electrons are scattered 90° on average due to Coloumb collisions with other electrons is approximately ee 5.5 108 Te3 / 2 / ne (3.3.2) for fusion plasmas where ne (cm-3) is the electron density and Te is in keV [47]. For a fusion plasma with Te = 10 keV and ne = 1014 cm-3 this corresponds to ee 0.17 ms [47], which is a short time scale. For ion-ion scattering, the time scale is approximately 1.25 1010 Ei3 / 2 A / ni ii (3.3.3) where A is the atomic number (number of proton masses) [47]. The scattering time is about 50-60 times larger for deuterium ions than for electrons, and estimates the time for pitch angle scattering loss through the mirror ends. The ion-ion scattering time is too high to allow for efficient fusion energy net production in a conventional mirror, and the theoretical limit of fusion Q is only slightly above unity in such machines [47]. In addition, high ion temperatures (50-220 keV) are required. Since the electrons are confined electrostatically, the electron temperature will be much lower than the ion temperature. This is because whenever an either cold or hot ion is lost, one of the hottest electrons will be lost together with it. Also, if neutrals (for example impurities) enter the plasma, cold electrons will replace the hot plasma electrons when the inserted ion leaks out. It is, however, primarily the ion temperature that is important for fusion reactions, and plasma heating is often designed to heat ions. The problem with a low electron temperature is that the electrons will cool the ions by Coloumb collisions, the so-called electron drag. This effect will increase the power loss but only to a minor extent increase the particle loss. The power loss from ions to electrons is for energetic ions approximately described by the ion energy density formula W where the electron drag time d d W0 e t/ d (3.3.4) is 5.5 1011 A 3/ 2 Te / ne Z2 (3.3.5) and Z is the ion charge [47]. For deuterions, this corresponds to d 11 ms at ne = 1014 cm-3 and Te = 1 keV [47]. The electron drag is today and has indeed earlier been the dominant energy loss mechanism in mirror machines, and the primary limiting mechanism for the fusion Q. There have however been successful experiments rather recently indicating that higher electron 41 temperatures can be reached. In the axisymmetric Gas Dynamic Trap (GDT) experiment, an electron temperature of 250 eV has been reached (by forcing a rotational shear by potential plates) [48], and in the multimirror GOL-3 an electron temperature of as high as 2500 eV has been measured by Thompson scattering (which is considered a reliable method) [49]. According to D.D. Ryutov et al. [50], the lower electron temperature of earlier experiments was “not due to an intrinsic failure of mirrors, but rather a result of a particular operational mode”. With proper reduction of the neutral gas in the expander tanks, the electron temperature may approach classical values as in the GDT experiment [50]. In the theoretical design for an axisymmetric fusion-fission reactor by Moir et al. [28], there are even measures taken (gas injection) to lower the electron temperature to 3000 eV to avoid microinstabilities caused by a “hole” in velocity space that would increase the ion transport to unacceptable levels. The reason is that the sloshing ion peaks are predicted to be reduced by a higher electron temperature, which would deteriorate the warm plasma stabilization of microinstabilities. The mechanisms responsible for recent increases in electron temperature vary between experiments, but rotational shear, multimirror confinement and enhanced power for plasma heating have been shown to be favorable. Further enhancement in Te may result from increased power for heating and control of neutral gas and background plasma density. 3.3.2. Mirror geometries Different mirror geometries have been explored by different groups over time. The base geometry is the axisymmetric single cell mirror. The magnetic field lines for such a field are shown in figure 3.2. Figure 3.2. Magnetic field lines in the mirror cell for a single cell axisymmetric magnetic mirror. This field is however grossly unstable to flute modes, and has a poor plasma confinement. There are a couple of ways to stabilize such a field. One is to stabilize the field by adding a quadrupolar (or other multipolar) field component. This was shown in an experiment by Ioffe in the 1960ies [51]. 42 The successful XIIB experiment in the late 1970ies used quadrupolar values approaching unity [52]. A quadrupolar stabilization to demonstrate field flattens the flux surfaces at the ends as shown in figure 3.3 where field lines for the SFLM field are shown. Figure 3.3. Magnetic field lines on a flux surface in the Straight Field Line Mirror. Another method is to have strongly stabilizing regions at some locations, where a possibly smaller but finite plasma pressure in these regions will add so much to stability that overall stability to flutes is achieved. In tandem mirrors, stabilizing anchor cells have been used. In axisymmetric mirrors, magnetic expanders (or cusps) outside the confinement region provide stability, in combination with other effects [50]. Stabilization of axisymmetric mirrors at high has been demonstrated with the Gas Dynamic Trap by the Budker Institute in Novosibirsk, Russia [48]. Since the single cell mirror concepts are not likely to have an end confinement that is good enough for pure fusion, at least not as compact devices, some improvements have been made. The main path is the tandem mirror, described in for example Ref. [47]. The basic idea with the tandem mirror is to achieve electrostatic confinement of both ions and electrons of the bulk plasma. To accomplish this, the large main mirror cell is surrounded by two smaller plug cells in which a positive hump in the electric potential is created. A first idea of the tandem mirror “plugging” was proposed by independently by Dimov et al. [53] in the Soviet Union and by Fowler and Logan [54] in the USA. The idea was to increase the plasma pressure in the plug cells with neutral beam injection. This however required high neutral beam injection particle energies of 0.5-1 MeV and a high magnetic field in the plug cells, which would be expensive but perhaps possible to obtain with a fusion plasma [47]. A modified concept with thermal barrier was then proposed by Baldwin and Logan [55]. The electric potential in this concept has a negative hump in potential between the plug cell and the central cell, separating the electrons in the central cell from those in the plug cells. The size of the positive potential hump depends on the electron temperature and by introducing this negative “thermal” potential barrier, the electrons in the 43 plug cell can be heated separately if the system is stable. This would make the cost in energy for creating the positive potential hump much lower, since one may choose to only heat the electrons in the smaller plug cells [47]. There are also other concepts. The multimirror is one idea which is tested experimentally in the GOL-3 experiment at the Budker Institute (see for example Refs. [49][56]). The basic idea is to have many aligned mirrors and a strongly collisional and high density plasma having a mean-free path for ion 90° scattering of the same order as the mirror cell length. In this way, a leaking ion only leaks one or a few mirror cells before being trapped again. An electron temperature of around 2.5 keV has been reached with a heating based on an injected relativistic electron beam [49]. At the Budker Institute, there is now a new design project, aimed to combine favorable results from GDT and GOL-3. The new device will be axisymmetric where the long central cell is followed by shorter mirror cells on each side. Electron beams in combination with Neutral Beam Injection (NBI) and Ion Cyclotron Resonance Heating (ICRH) will be used for plasma heating. 3.3.3. The flute instability in magnetic mirrors The flute instability in magnetic mirrors radically deteriorates the confinement, and was observed already in the first mirror experiments. The effect is that the plasma edge rapidly will change its shape, moving parts of or the entire plasma radially outward. The instability can look very different, depending on the mode number m that is dominant, but the instability has a long longitudinal wavelength. The perturbation can be viewed as an alternating radial disturbance of the plasma edge depending on the angle where m is the number of oscillations around the plasma edge, roughly looking like r rb ri cos(m 0 0 ) (3.3.6) where rb is the original plasma radius and ri is the instantaneous amplitude of the growing perturbation. In figure 3.4, two plasma edges are shown with growing flute instabilities having m = 6 and m = 1. The flute instability corresponds to a perturbation (or test function) that is not varying in z [57]. The flute-like instabilities that are dependent on z, which may be triggered by a finite ß, are called MHD ballooning modes. The flute instability is a Rayleigh-Taylor instability that is driven by the magnetic field curvature. A physical picture of the flute instability can be made from particle drifts. The field line curvature in the radial direction will cause a charge dependent curvature drift which will separate the charges, making positive charges drift in one azimuthal direction and negative charges in the opposite direction. If there is a small perturbation, which will always be present, the charges will be separated on the peaks of the perturbation, as shown in figure 3.5. This 44 charge separation will give rise to an electric field, which will in turn cause a radial E B drift in that peak. If the magnetic field lines bend outward (concave) from the center when moving from the midplane, the E B drift will flat out the perturbation. This is stabilizing and is referred to as “good curvature”. If the field lines instead bend inwards (convex) like in an axisymmetric mirror, the peaks will drift outwards. This causes the perturbation to grow and is referred to as “bad curvature” [43][47]. Figure 3.4. The plasma surface in an axisymmetric mirror cross section showing flute instability for m = 6 and m = 1. The dashed line shows the plasma deformation after a short time. Figure 3.5. Perturbed plasma edge showing the charge separation from magnetic field radial curvature. The view is perpendicular to the magnetic field. However, as mentioned earlier, there are ways to design a magnetic mirror to avoid flute instabilities. The first theoretical prediction was made by Rosenbluth and Longmire [58] and is called the average minimum B criterion. It is zend r0 zend dl B zend zend B dz 0 (3.3.7) is the where the integration is to be taken along a magnetic field line and normal (radial-like) component of the field line curvature. It states that on average along a magnetic field line the magnetic field strength should grow stronger outwards in the r0 direction to grant stability. This is equivalent to having more good curvature than bad on average in the low limit. However, this criterion does not take into account the varying pressure profiles along the field lines. A more genuine flute stability criterion including the effect of pressure weighting is derived by Kaiser and Pearlstein [57]. It is 45 zend pˆ zend dz B 0, pˆ ( pˆ pˆ ) / 2 (3.3.8) where p̂ is a radially independent pressure function. 3.4. Other devices for magnetic confinement 3.4.1. Tokamaks Most fusion research today is directed towards the tokamak concept. The tokamak was invented in the Soviet Union during the early days of fusion research by A. D. Sacharov and I. E. Tamm. The concept is a toroidal (“doughnut-shaped”) device. If a simple solenoid would be used to form a toroid to confine plasma, no plasma confinement would be achieved. The reason for this is that the field curvature would cause a charge-separating drift, where the positive ions drift up/down and the electrons drift in the opposite direction. This causes a vertical electric field that will give a charge independent radial drift outwards of the entire plasma which will hit the outer wall. Therefore, equilibrium is never established. This problem is solved in the tokamak by inducing a toroidal plasma current. This current gives a helical-like toroidal magnetic field which does not suffer from an average charge separation by the drifts. Examples of field lines can be seen in figure 3.6, with a normal plasma current in (a) and a heavily exaggerated plasma current in (b) to give a good view of the helical field lines. a b Figure 3.6. Tokamak orbits with a normal plasma current in (a) and a heavily exaggerated plasma current to show the helix shape in (b). Note that the magnetic field lines, and thereby the orbits, are generally not closed with the exception of rational surfaces. The ITER project is a huge international tokamak fusion project with the aim “to demonstrate that fusion is an energy source of the future” (see Ref. [59]). The strongest argument for tokamaks is the reasonably good plasma 46 confinement compared to other devices [60]. Tokamaks have however also, as all fusion devices, some drawbacks. The largest drawback is probably that there currently is no way to drive the plasma current in steady-state with fusion parameters, although some progress with steady-state tokamaks (advanced tokamaks) has been made where the bootstrap current is an important parameter [60]. Toroidal current drive is today arranged by transformer action with an increased magnetic flux generated by a large central solenoid. This magnetic flux has to be increased to induce a toroidal electric field in the plasma, and when the maximum magnetic field strength in the central solenoid is reached, the reactor must be restarted. This is a major drawback, both for cost efficiency, the electric grid and material durability reasons, and for ITER this limits the full power pulse length to 500-800 s [60]. This is also a drawback for tokamak hybrids. There are also severe material problems, where the divertor plates in ITER is predicted to have a lifetime corresponding to about 7000 pulses [61]. The high heat and neutron material load in the divertor is a severe problem for reactor scenarios in all compact fusion devices. The problem is however even larger for tokamaks unless the ELM (Edge Localized Mode) instabilities, which are present in H-mode, associated with large intermittent plasma heat loads deposited on the divertor wall, can be suppressed. The material problems (to withstand both a massive heat load from the plasma and the heavy neutron bombardment from the fusion neutrons) are a cumbersome argument against many pure fusion devices that needs a solution, but not an argument against fusion-fission where the heat and neutron loads can be made substantially lower. 3.4.2. Stellarators The stellarator concept is also a toroidal device. In a stellarator, there is no plasma current. Instead, the rotational transform is created with external windings. This eliminates the strong drawback tokamaks have with the plasma current, and allows for steady-state operation. However, as all fusion devices, the stellarator has some problems. Currently, the limit is low, 3% or so. Also, there could be some problems with stochastic field lines, and to assure that the drifting gyro centers will remain within a flux tube. However, the stellarator may prove to be the best alternative for pure fusion in the future. The Wendelstein 7x is a stellarator project in Germany that is planned to start operation in 2014 (see for example Ref. [62]). This device has complex 3D superconducting coils. 3.4.3. Other schemes for magnetic confinements During the more than 50 years of fusion research many devices have been proposed, each with its own benefits and problems. To mention some of 47 them, there are reversed field pinches (RFP), z-pinches, -pinches, FRCs, multipoles, ion rings, bumpy tori mirrors etc. Those will not be described in this text. 48 4. Theory for the magnetic mirror vacuum field The magnetic vacuum field is the magnetic field when no plasma is present, i.e. when = 0. When > 0, the magnetic field will be altered due to the diamagnetic effect of the plasma. When plasma calculations are made in the vacuum field, they are in general only valid for low plasmas. In this chapter, some expressions used in several papers in this thesis are derived, based on the magnetic vacuum field. 4.1. The long-thin approximation In order to be able to achieve analytic expressions for calculations in a mirror device, the common approach is to use the long-thin (or paraxial) approximation. In the long-thin approximation, it is assumed that the magnetic field lines make a small angle with the system axis near this axis and are parallel to the axis at the axis [63]. This is true for points where the distance to the axis is small compared to the characteristic scale of the field variation along the axis [63]. For the vacuum field, there are no currents within the confinement region. Thus, the magnetic field can be expressed as B (4.1.1) m where m is a scalar magnetic potential, and from Laplaces equation, 2 m B 0, 0 B ( z )dz m 0 x2 y2 4 B obeys (4.1.2) With the paraxial approximation and quadrupolar symmetry, expressed as z m x2 y2 4 g ( z ) o( 4 ) m can be (4.1.3) where a / c is the small parameter, i.e. in this case x or y divided by c, for the expansion and prime denotes differentiation in z. Due to the symmetry of a quadrupolar mirror, terms containing odd power of x and y are omitted. Eq. (4.1.3) obeys to leading order Laplaces equation and has two free funcions, B ( z ) and g ( z ) . B ( z ) is the magnetic field strength on 49 axis, and g ( z ) gives rise to the so-called quadrupolar field component. The quadrupolar field component can be oriented to be directed towards the z axis at the y axis and away from the z axis at the x axis. There are also other components like octopolar components etc. but those are of higher order in . The magnetic field components can be derived from Eq. (4.1.1). They are Bx Bz x g B , By 2 x2 d ( g B ) B dz 4 y g B , 2 y2 d ( g B ) dz 4 (4.1.4 a, b, c) neglecting third order terms in . As can be seen, both B and g contributes to the Bx and By components. 4.2. Flux coordinates In fusion, Clebsch coordinates are often used to hide the topology of the magnetic field in the coordinate system. The curvilinear coordinate system can be defined by B B0 x0 y0 (4.2.1) where x0 and y0 are the two Clebsch coordinates and the constant B0 is the magnetic field at x y z 0. The Clebsch coordinate x0 is an x-like coordinate and y0 is an y-like coordinate, equal to x and y at the midplane z 0 . Since B x0 0 and B y0 0, x0 and y0 are constant along B. A natural choice for the third coordinate in the vacuum field is m . It is however often more comfortable to replace m with the arc length s that is equal to z at the z axis. The x0 and y0 coordinates can also be expressed in cylinder-like Clebsch coordinates by introducing x0 r0 cos 0 , y0 r0 sin Often in literature, r0 is replaced by divided by 2 . Then, B often called flux coordinates. 0 , B B0 r0 r0 0 (4.2.2) B0 r02 / 2 which is the enclosed flux 0 , and these Clebsch coordinates are The Clebsch coordinates and for example s form a curvilinear coordinate system, where vectors can be represented in contravariant and covariant form. The contravariant (“normal”) form for an arbitrary vector F is F F i R / u i F i ei , where R is the position vector, the u i are the curvilinear coordinates and Einstein’s summation convention is implied. The covariant form is F Fi u i Fi ei , where the base vectors u i are dual vectors to R / u i . The scalar product is A B Ai Bi Ai B i and the base vectors can be converted between covariant and contravariant form using 50 e j ek ei ei e j e k e j ek ei e j e k , ei (4.2.3) where the indexes can be cyclically shifted (see Ref [44]). From these, the cross product can be easily found by solving for e j e k or e j e k . Another useful identity is that ei e j e k 1/ ei e j ek is the determinant of the Jacobian ( x, y, z ) / (u1 , u 2 , u 3 ) . The point with the Clebsch coordinates is that x0 and y0 (or alternatively r0 and 0) are constant along a magnetic field line, and thus identify or label a field line. The third coordinate s marks the position along the field line. To find an expression for the Clebsch coordinates, magnetic field lines x B [ x( z ), y ( z ), z ] can be traced by the field line equations Bx , Bz dx( z ) dz dy ( z ) dz By (4.2.4) Bz where x and y are the coordinates for the selected field line at position z. The field components in the vacuum field using the paraxial approximation are inserted from Eq. (4.1.4) giving two first order differential equations dx( z ) dz g B 2B x( z ), g dy ( z ) dz B 2B y( z) (4.2.5) to first order in . The solutions are z x( z ) x0 e 0 g B dz 2B z g x0 B0 0 2 B dz e B y0 B0 e B x0 ax ( z ) (4.2.6) and z y( z ) y0 e 0 g B dz 2B z 0 g dz 2B y0 a y ( z ) (4.2.7) where the integration constants are set to x0 and y0 which can be identified as the Clebsch coordinates, thereby stating that x = x0 and y = y0 at z = 0. To get a more intuitive understanding of the field line equations in Eq. (4.2.6) and Eq. (4.2.7), the g / B integrand is the contribution from the quadrupolar field component. The second term is just the axisymmetric resizing of a flux tube due to different field strengths. To simplify the notation in the coming expressions, we introduce 51 g h1 ( z ) h2 ( z ) B ' 1 Bx 2B B x g B ' 1 By 2B B y ( 2 ), ( 2 (4.2.8 a, b) ) Instead of using s, we introduce the arclength-like variable s ( inverse m ( s ) by m ) and its s m (s ) (4.2.9) B ( s )ds 0 and since the magnetic scalar potential along the z axis is given by z m ,0 ( z) (4.2.10) B( z )dz 0 an inverse function s (x) m1,0 [ m (x)] can be used. By treating the off-axis terms in Eq. (4.1.3) as a small parameter , a Taylor expansion gives 1 m ,0 s ( x, y , z ) z 1 d [ 1 m ,0 m ( x, y , z )] z m ,0 1 B 1 B z d m ,0 ( z )] [ x 2 y 4 1 m ,0 dz x2 2 B y2 4 g ( z) (4.2.11) dz or more compactly s ( x, y , z ) z x2 h1 ( z ) 2 y2 h2 ( z ) o( 2 4 ) (4.2.12) The magnetic field B can now be expressed using s as B B( s ) s m 1 x2 (h1 2 h12 ) B( s ) 1 x2 2 (h1 2 dh1 ) dz y2 (h2 2 h22 ) zˆ o( 3 ) dh2 ) dz o( 4 ) In Clebsch coordinates ( s , r0 , 0 ) , this becomes apart from o( 4 B ( s ) xh1xˆ yh2 yˆ (4.2.13) and the modulus is B B ( s , r0 , 0 ) 52 B(s ) r02 [u1 ( s ) cos 2 2 0 y2 2 ( h2 2 u2 ( s )sin 2 0 ] o( 4 ) (4.2.14) ) corrections (4.2.15) where u1,2 ( s ) B ( s )e 2 s 0 h1,2 ( s ') ds ' dh1,2 ( s ) 2 [h1,2 (s ) ds ] (4.2.16) The expressions derived in this chapter can be used to examine some properties of the magnetic field. 4.3. Flux tube ellipticity A vacuum field flux surface for a mirror geometry is normally defined as an extension of a circle centered on the z axis at z = 0 extended along the field lines in both directions and thereby forming a tube. It is defined by r0 const , i.e. [ x / ax ( z )]2 [ y / a y ( z )]2 const which for each z defines an elliptical cross section of the flux tube. The flux enclosed by a flux surface is constant. The flux tube ellipticity is a measure of how elliptic a flux surface is and is a function of z. The ellipticity of a flux surface is to first order in z ell a ( z) ay ( z) max x , a y ( z ) ax ( z ) ( z) max e 0 g(z) dz B( z) z ,e 0 g(z) dz B( z) (4.3.1) In Paper III, an expression for the ellipticity to 3rd order in has been derived. To derive such an expression, higher order terms of the solution to Laplace’s equation (Eq. (4.1.2)) must be included. The solution is in cylindrical coordinates Al , n I l m l n n r i(l e c n z / c) (4.3.2) where the I l are the modified Bessel functions. The odd terms in l (i.e. l 1, 3,... are omitted due to the choice of symmetry. To order ( 5 ) z B ( z )dz m 0 4 r B ( z) 16 4 r2 B ( z ) g ( z ) cos 2 4 g ( z) cos 2 3 (4.3.3) h( z ) cos 4 4 where h( z ) represents the octupolar field. This gives the B components r r3 B ( z ) g ( z )cos 2 1 ( z , ), 2 4 r r 3 g ( z) h( z ) g ( z )sin 2 sin 2 sin 4 2 8 3 2 Br B (4.3.4) , (4.3.5) 53 Bz B( z ) r2 B ( z ) g ( z )cos 2 4 r4 16 2 ( z, ) (4.3.6) where 1 d 3 B( z ) 1 d 2 g ( z ) cos 2 4 dz 3 3 dz 2 1 d 4 B( z ) 1 d 3 g ( z ) ( z , ) cos 2 2 4 dz 4 3 dz 3 1 h( z ) cos 4 , 4 h( z ) cos 4 . 4 ( z, ) (4.3.7) (4.3.8) The position vector along a field line can be parameterized using z, yielding x B rB ( z )rˆ B ( z ) ˆ zzˆ . The field lines can now be traced with the field line equation drB ( z ) dz Br Bz (4.3.9) where only the radial part is needed for determining ellipticity. Inserting the field components yields drB ( z ) dz rB ( z ) rB ( z )3 B ( z ) g ( z ) cos 2 1 ( z, ) 2 4 r ( z )2 rB ( z ) 4 B( z ) B B ( z ) g ( z ) cos 2 2 ( z, ) 4 16 to fourth order in (4.3.10) which becomes drB ( z ) dz A3 ( z , )rB3 ( z ) A1 ( z , )rB ( z ) ( 5 ) (4.3.11) where 1 B 2B A1 A3 1 B 4B 4 g cos 2 3 h cos 4 4 g cos 2 B , g cos 2 (4.3.12) A1 ( z, ) . (4.3.13) By assuming that the term A3 ( z , )rB3 ( z ) is small, the equation can be linearized. The field line radius can be written rB rB ,1 rB ,3 ( 5 ) (4.3.14) where rB ,1 is the first order solution and rB ,3 is the small contribution from the third order terms. The differential equation then becomes d rB ,1 dz rB ,3 A1 ( z , ) rB ,1 rB ,3 A3 ( z , ) rB3,1 3rB ,3 rB2,1 The first order solution is (compare Eqs. (4.2.6 – 4.2.7)) 54 o(rB2,3 ) (4.3.15) z rB ,1 r0 e 0 B ( z ) g ( z ) cos 2 dz 2B( z ) (4.3.16) and inserting this yields drB ,3 f 2 rB ,3 dz f1 , (4.3.17) where z f1 A3 ( z , )r03e z 3 A1 ( z , ) dz 0 , 3 A3 ( z , )r02 e f2 2 A1 ( z , ) dz A1 ( z , ). (4.3.18) 0 The solution to the differential equation is z rB ,3 e z z f 2 ( z , ) dz e0 0 f 2 ( z , ) dz f1 ( z )dz C (4.3.19) 0 where the constant C is zero since rB ,3 ( z z rB ( z, ) rB ,1 rB ,3 r0 e 0 0) 0 . The total result yields z A1 ( z , ) dz e z f 2 ( z , ) dz z 0 e0 f 2 ( z , ) dz f1 ( z )dz (4.3.20) 0 to fourth order in . The third order eccentricity or ellipticity (the cross section may not be exactly elliptic to this order) becomes ell ,3 max ell 1 , rB ,1 ( z ) , ell rB ,1 ( z ) ell 0 rB ,3 ( z ) /2 rB ,3 ( z ) 0 . (4.3.21) /2 4.4. Flute stability The average minimum B criterion from Eq. (3.3.7) becomes in flux coordinates using Eq. (4.2.14) zend r0 zend zend dl B zend B / r0 dl B2 0 zend r0 (u1 ( z )cos 0 u2 ( z )sin 0 ) dz B2 zend 0 (4.4.1) 0 W1RL ,2 RL 0 where zend W1RL ,2 RL zend dz 2 e B( z ) z 0 h1,2 ( z ') dz ' 2 [h1,2 ( z) dh1,2 ( z ) dz ] 0 (4.4.2) 55 It can be shown that the W1RL and W2RL expressions will give the same result by symmetry. The flute stability criterion, Eq. (3.3.8), that takes the varying pressure along a field line into account becomes zend pˆ ( z ) W1,2 zend dz 2 e B( z ) pˆ ( z ) ( pˆ ( z ) z 0 h1,2 ( z ') dz ' dh1,2 ( z ) 2 [h1,2 ( z) dz ] 0 (4.4.3) pˆ ( z )) / 2 This can be verified against Eq. (3.3.8) using the expressions given in Ref. [57]. 4.5. Drifts and neoclassical transport in mirror machines 4.5.1. Particle drifts In mirror machines, except for the exceptional case of the ideal SFLM field, the magnetic field lines are curved which gives rise to curvature drifts. There will also be gradients in the magnetic field which give rise to drifts. In the vacuum field, these gradients can be expressed in terms of curvature since 0 B BBˆ ˆ B 0 j B Bˆ j Bˆ Bˆ BBˆ Bˆ Bˆ B Bˆ B Bˆ where 0 j BBˆ Bˆ and vacuum case when j 0 gives Bˆ B Bˆ ˆ B Bˆ ( B) . (4.5.1) which in the (4.5.2) The magnetic drifts in the vacuum field can then be divided into radial drifts dr0 / dt that depend only on geodesic (angular-like) curvature and angular drifts d 0 / dt that depend only on normal (radial) curvature. Since the curvature components are important in this thesis, they are derived explicitly. We now use the radial-like flux coordinate r02 / 2 and derive the curvature components as Bˆ Bˆ s Bˆ 0 0 (4.5.3) If a field line is parameterized as R f ( x( z ), y ( z ), z ) where R f is the position vector for a field line, Bˆ R f / s which gives ˆ B s x 2 ds x 2 Rf y 2 Rf dz 2 y x x 0 56 y (4.5.4) y 0 0 The curvature components on the r.h.s. can now be evaluated in the paraxial approximation from z h1dz x x0 e y y0 e 0 z h12 h1 h2 dz z z x0 e 0 0 h2 z 2 sin 0 e 0 (4.5.6) z h1dz 1 cos 0 e 0 , 2 (4.5.7) z h2 dz 1 sin 0 e 0 , 2 h2 dz z (4.5.5) h22 , h1dz 2 cos 0 e 0 0 z h1dz 2 sin 0 e 0 h1dz (4.5.8) , (4.5.9) 0 z y y0 e 0 0 h2 dz z h1dz h12 , h1 2 cos 0 e 0 h2 dz y0 e 0 x 2 sin 0 e 0 h1dz x0 e 0 y h1dz z h22 h2 z x 2 cos 0 e 0 z h2 dz 2 sin 0 e 0 0 z h2 dz h2 dz 2 cos 0 e 0 (4.5.10) 0 which gives cos 2 0e 2 z 0 h1dz h12 h1 sin 2 0e 2 z 0 h2 dz h22 h2 (4.5.11) and 2 sin 0 0 cos e 0 2 z 0 h1dz h12 h1 e 2 z 0 h2 dz h2 h22 . (4.5.12) The magnetic drift can now with Eqs. (4.5.2, 4.5.3) be calculated as 2 1 B mv ˆ (B q B2 v d ,m 2 1 B mv q B 2 1 B mv ˆ (B q B Bˆ ) Bˆ 0 Bˆ 0 ) (4.5.13) . Since Bˆ ( m ) / B and since the cross products can be evaluated from the definition of contravariant base vectors [44], R R m 0 0 0 R m m m 0 (4.5.14) 2 B / B0 57 R R m m 0 0 R 0 m 0 m (4.5.15) 2 B / B0 0 where R is the position vector, the magnetic drift velocity becomes 2 1 B mv q B B 12 2 q B Bˆ v d ,m qB0 where R/ written as m ˆ B m 0 R B 2 0 0 0 (4.5.16) R 0 0 2 mv / 2 . Since (r0 ) r02 / 2, R / r0 ( R / )(d / dr0 ) ( R / r0 ) / r0 the magnetic drift velocities for a particle can be dr0 dt d 0 dt 2 B qB0 2 B qB0 0 r0 , (4.5.17) (4.5.18) where B B to first order in . It is from these equations clear that in the vacuum field the radial magnetic drift is caused by a geodesic curvature, and that the normal curvature gives rise to an azimuthal drift. 4.5.2. Neoclassical transport Neoclassical transport is in mirror machines caused by geodesic field line curvature, which gives radial drifts away from the flux surface. If these radial drifts are oscillatory and cancel during one bounce back and forth across the mirror, the magnetic field is said to be omnigenous (in the average sense), a concept introduced by Hall and McNamara [64]. For a quadrupolar mirror, this requires that the device is symmetric in the sense that B ( z ) and g ( z ) are even functions. Thereby, the radial drifts on opposite sides of the mirror will be equally large but with opposite signs, and will almost exactly cancel out. In such mirrors, a radial invariant I r r0 exists, which has been derived in Paper VII. If this symmetry is not present, there will be a collisionless radial transport that may be unacceptably large as pointed out in Paper IV. The particle orbits in symmetric mirrors have a similarity with the banana orbits in tokamaks and are here called banana orbits as well. These 58 gyro center orbits may have larger radial excursions than the Larmor radius, and will lead to neoclassical transport. Catto and Hazeltine [65] introduced the concept of local omnigenity as referring to magnetic fields where the radial drift off a flux surface is zero everywhere. The banana orbit widths in such fields are zero, and thereby neoclassical transport of any kind is abscent. Axisymmetic fields are locally omnigenious in the vacuum field since 0 vanishes everywhere, and even at if fluctuations and non-ideal effects are ignored. The SFLM arbitrary vacuum field is an example of a locally omnigenious quadrupolar field, and for well behaved even functions of B ( z ) there exists a non-trivial (i.e. the axisymmetric solution g ( z ) 0 z is excluded) g ( z ) which gives a locally omnigenious vacuum field in the paraxial approximation (see Paper VII). 0 everywhere. For locally This can be accomplished by solving 0 omnigenious fields, the radial transport should be small unless there are anomalous transport processes present or there for some non-ideal reasons exists azimuthal electric fields. In mirrors with finite ß, or that have a strong radial electric field, the azimuthal drift may become an important factor for radial transport by giving rise to the so-called resonant neoclassical transport. The resonant neoclassical transport is a collisionless transport that occurs for particles which have an azimuthal drift that allows them to come in “resonance” in the sense that the number of bounces back and forth the mirror it takes before the particle has travelled one lap around the z axis is a simple rational number. For some particles, this drift can be very fast. In quadrupolar mirrors that are not locally omnigenous, the two sides of the mirror causes radial drifts that are in opposite directions. However, if one looks in the xyplane, if the drift is outward for positive z in the first and third quadrant, it is opposite in the second and forth quadrant. This is illustrated in figure 4.1, where the regions with outward ion drift are white and the regions with inward ion drift are gray. The worst possible particle trajectory is the one illustrated in figure 4.1. Here, the trajectory is in perfect phase and travels half way around the z axis on one longitudinal bounce back and forth the mirror. As can be seen in figure 4.1, the particle drifts outwards constantly and will soon be lost, if d 0 / dt does not change rapidly with radius. Particles with a ratio bounces/lap close to nbounce 2 4n will suffer from neoclassical radial transport if the phase does not cancel the effect, where the effect is larger for small n and largest for n 0 . Also faster spinning particles like those with nbounce 2 /(1 2n) will experience resonant radial transport. Resonant transport is predicted to be the most serious type of neoclassical transport for at least some mirror fusion devices ([66], section 9.3). 59 Figure 4.1. This figure illustrates neoclassical resonant transport in a projection where the rolled out circumferential surface with r0 = const is shown for a quadrupolar mirror. The white regions have a negative geodesic field line curvature and a corresponding positive radial drift, and the opposite yields for the gray regions. As can be seen, the worst particle resonance has been selected for demonstration where the drift is everywhere outward. For particles that spin very fast around the z axis, a stochastic behavior is expected [67] and this is referred to as stochastic neoclassical transport. To summarize, there are three regimes of neoclassical transport that depend on the angular drift per longitudinal bounce 0 of the ions: 1. 2. 3. 0 0 0 1 : Neoclassical transport. 1: Resonant neoclassical transport. 1 : Stochastic neoclassical transport. 4.6. The Straight Field Line Mirror field The Straight Field Line Mirror (SFLM) field is a minimum B magnetic mirror field derived by Ågren and Savenko [68]. It is defined by B B0 1 ( z / c) 2 (4.6.1) 2/c [1 ( z / c) 2 ]2 (4.6.2) and g where c is the axial scale length of the system. Alternatively, the SFLM field can be written B 60 B( s ) s B0 x0 y0 (4.6.3) where B ( s ) B0 /(1 s 2 / c 2 ) by Eq. (4.6.1) and the Clebsch coordinates are to leading orders x o( 3 ), 1 z/c y y0 o( 3 ), 1 z /c x 2 / 2c y 2 / 2c s z 1 z/c 1 z/c x0 (4.6.4 - 4.6.6) o( 4 ). The field lines in the SFLM field are straight non-parallel lines, stretching between two focal lines located at c with an angle of 90° between them (see figure 3.3). This implies that in this vacuum field the gyro center drift is abscent when there is no electric field, since no B -drift is present for this particular vacuum field. For the vacuum field SFLM, B is parallel to B, B B(s) 0 (4.6.7) Therefore, the both guiding center Clebsch coordinates are constants of motion [69], and the field is locally omnigenius. Each guiding center bounces back and forth on a single flux line. The arc length turning points in the vacuum field are given by [69] sturn B0 c 1 . (4.6.8) Inserting this magnetic field into the stability criterion in Eq. (4.4.3) results in W1,2 0 , since the second factor 2 h1,2 ( z) dh1,2 ( z ) dz (4.6.9) becomes zero for all z (or alternatively since 0 ). The SFLM field is therefore marginally flute stable for all z in the low limit. Recalling that the field lines are straight, this can intuitively be seen since no “bad curvature”, being convex, or “good curvature”, being concave, exists; the field lines do not have a curvature at all. This also implies that the flute stability will be independent of the pressure profile in the low limit. However, the SFLM field modulus is monotonically increasing with increasing z , having a monotonically increasing strong gradient dB / d z . The mirror must be ended somehow, and this cannot be done with a continuous B derivative in z without concatenating the SFLM field with another field at some point. These added parts of the field which will end the mirror also have to be stable to flutes. 61 The flux tube ellipticity in SFLM is also low, even if it is possible to find solutions that are stable to the flute criterion or average minimum B criterion that have lower ellipticities, see for example Ref. [70]. For a mirror ratio of four, the maximum flux tube ellipticity of SFLM is 13.9 [68]. To leading order, the equation determining the SFLM flux surfaces is K x 1 z/c 2 2 y 1 z /c 2 (4.6.10) where K is a constant. For the plasma surface, K = a, i.e. the midplane radius. The SFLM flux tube ellipticity is ell ( z) 1 z /c 1 z /c Rm Rm 1 2 (4.6.11) For Rm >> 1, this becomes ell 62 ( z ) 4 Rm 2 o( 1 ) Rm (4.6.12) 5. Finite ß effects on the mirror magnetic field When there is a finite present, the magnetic field will be significantly modified by the associated plasma currents. In mirror machines, the limit is typically high, and 0 1 has been reached in the midplane (short-fat) single cell quadrupolar mirror 2XIIB at LLNL [52]. In this chapter, the equations required to calculate the magnetic field from the plasma in equilibrium is derived. Also, a numerical method to calculate this field, used in Paper V, is presented as well as a method described in Paper VII. 5.1. The equilibrium To find the magnetic field contribution from the plasma, the pressure distribution must be determined. From the pressure distribution, the plasma currents can be derived. To find the equilibrium, we follow Newcomb [71]. The plasma pressure tensor in flux coordinates is assumed to be of the form P p ˆˆ I BB ˆˆ p BB (5.1.1) where off-diagonal terms are neglected. By including the magnetic pressure, the total stress tensor becomes T ˆˆ P I QBB (5.1.2) P B2 2 0 (5.1.3) where p and Q B2 p p. (5.1.4) 0 The force balance can be found from T 0 . The divergence of the first ˆ ˆ . With use of the term is simply P, and the second term is a dyadic QBB vector formula ( AB) B( A ) ( A )B , the force balance then becomes 63 ˆˆ P I QBB T 0 P Bˆ QBˆ ˆ QB Bˆ (5.1.5) is the magnetic field line curvature. This equation can now where Bˆ Bˆ be divided into perpendicular and parallel force balance, yielding P Bˆ ( P) Q P QBˆ ˆ Q PB (5.1.6) and Bˆ Q B B ˆ BB Q . B (5.1.7) From the parallel force balance equation, Eq. (5.1.7), the relation between the parallel and perpendicular pressure components on a single flux line can be found. It becomes (see Paper V, Appendix A or [71]) p p B2 , B/ s s B s p B L p B/ s ds B2 (5.1.8 a, b) where s is an arc length coordinate along B and s L at the mirror throats. Thereby, if either the parallel pressure or the perpendicular pressure is specified, the other component can easily be determined. To determine the pressure distribution, the plasma feeding and heating must also be taken into account. In the SFLM Hybrid, the intention is to use ICRH (Ion Cyclotron Radiofrequency Heating) for heating, which would create a sloshing ion pressure distribution where there is one pressure peak at each side of the mirror. The plasma feeding has not yet been determined. The effects of ICRH heating could in principle be calculated with complicated ICRH-codes. However, in this thesis this is postponed to later work, and prespecified simple pressure profiles have been used. We choose to follow Newcomb [71] and Pearlstein et al. [72] and model the perpendicular pressure as p p ,0 p , r0 (r0 ) p ,B ( B) (5.1.9) where p ,0 is the perpendicular pressure at origo, r0 is a radial Clebsch coordinate equal to x at the x axis and the functions p ,r0 (r0 ) and p , B ( B ) are normalized so that they are equal to unity at r0 0 and B Bmin . Here, Bmin is the magnetic field modulus at origo. We have from Eq. (3.2.32) that p where origo. 64 0 is the plasma 0 B02 (5.1.10) 2 0 at origo and B0 is the vacuum magnetic field at ,0 5.2. The plasma currents The plasma currents are formed by gradients in the plasma pressure. Since the aim of fusion devices with magnetic confinement is to confine plasma, there must be pressure gradients since the plasma pressure (and density) should vanish (or nearly vanish) at the boundary of the confinement region. The most important contribution to the plasma currents in a magnetic mirror is the diamagnetic current (however, in tokamaks there is a toroidal current as well). Each charged particle in the plasma gyrates roughly around a magnetic field line and forms a small current loop in the movement projection on the plane locally perpendicular to B. The ions and electrons gyrate in opposite directions so that the currents from the two species add. These currents are diamagnetic, i.e. orientated so that the magnetic field is reduced by the currents. When the pressure is constant in space, all currents from the gyrations cancel. When there is a pressure gradient, the currents no longer cancel but form the diamagnetic current. This is illustrated in figure 5.1, where the pressure is strongest at the center. The gyrating currents clearly add to form the diamagnetic current (thick lines) since there will be more circular currents on the high-pressure side than on the low-pressure side. Figure 5.1. This figure illustrates how the small currents from gyrating particles add to form the diamagnetic current (thick line) when there is a negative pressure gradient from the centre and outwards. In quadrupolar mirrors, there is also a parallel current j Bˆ if the mirror field is not locally omnigenous everywhere. This current is formed by the neoclassical banana orbits, which add in a similar way as the gyro orbits above since there normally is a negative pressure gradient radially outward. This current is closely linked to the geodesic curvature of the field lines and also has a sin 2 0 angular dependence if the azimuthal drift is small, showing that the parallel current in adjacent quadrants have opposite directions. The plasma currents can be found from the equilibrium. The details of the derivations can be found in Paper V, Appendix A or to a large extent in Ref. 65 [71] and are not repeated here. By applying the operator B̂ on Eq. (5.1.6), the expression for the parallel current j becomes after some algebra B3 0Q j Bˆ s p p B2 L ds. (5.2.1) The perpendicular current j is most straight forward to derive from the usual equilibrium relation j B and by taking B̂ (5.2.2) P on both sides of the equation, this yields j 1 B B2 p p p . (5.2.3) 5.3. Magnetic field from plasma currents Due to the plasma currents, a finite ß will modify the vacuum magnetic field and it is no longer possible to represent the magnetic field with a scalar magnetic potential. The total magnetic field can be expressed as B B v B pl (5.3.1) where B v is the vacuum field and B pl is the magnetic field from the plasma currents. In Paper V, a numerical approach has been made. In Paper VII, a different method is described. 5.3.1. The method from Paper V The magnetic field from the plasma, B pl the currents by solving Poisson’s equation 2 A pl A pl , can be determined from 0 j (5.3.2) in the plasma region and the surrounding vacuum. A Dirichlet boundary condition for the magnetic vector potential A pl at any chosen boundary can in the Coulomb gauge be calculated from the Coulomb integral A pl 0 4 V j d 3x x x (5.3.3) where the integration is to be performed over the plasma region and the boundaries where A pl is calculated are well outside the plasma region. The integral can be evaluated numerically with quite large integration steps, and Eq. (5.3.2) is then solved numerically with a boundary condition determined from Eq. (5.3.3). 66 To distribute the plasma pressure, the shape of the constant perpendicular (or parallel) pressure contours in the z 0 plane must be determined. This can be done by an iterative procedure, where those contours are traced numerically. The pressure is distributed according to these profiles, and the magnetic field is calculated. Then new contours can be calculated in the next iteration step. The constant perpendicular pressure contours can be found from From Eq. (5.1.6), where we have that P Q p B2 . 2 0 Q (5.3.4) By taking B̂ on both sides of the equation and evaluating the r.h.s., the contours can be traced by a Runge-Kutta method. The field lines can also be traced by a Runge-Kutta method, and then the pressure can be distributed. 5.3.2. The method from Paper VII In Paper VII, another method is described to calculate B pl . In the long-thin approximation, the magnetic field can be expressed as B 1 2 Bv The gradient term can be determined from integral (see also [69]) m , pl 1 8 (5.3.5) m , pl dV B(x ) x x B 0, yielding the Coulomb (x ) s' (5.3.6) where the integration is to be performed over the plasma region. This is the solution of 2 m , pl 1 B s 2 (5.3.7) with a proper boundary condition. In practice, Eq. (5.3.7) can be used to solve for m, pl and the boundary condition can be calculated from Eq. (5.3.6) in a similar way as in the method of Paper V described above. 67 68 6. Superconducting coils in fusion In a fusion device based on magnetic confinement, the strong magnetic field confining the plasma must be generated by a coil system. There are two main options for coils: standard copper coils and superconducting coils. In small pulsed experimental devices, copper coils are almost always used. Typically, they however have too high resistive losses for fusion reactors, and therefore superconducting coils – which have no resistance – are used. The drawbacks with superconducting coils are the cost and the sensitivity to nuclear radiation. For ITER, about 30 % of the total cost is associated with the coil system and the cryostat [73]. In this section, basic characteristics of superconducting coils and to some extent copper coils are described to give a background to coil design. 6.1. Properties of superconducting coils 6.1.1. Superconductivity The electrical resistivity of metallic conductors (like silver or copper) is reduced gradually with decreasing temperature. At a temperature approaching 0 K, they have a low – but nonzero – resistivity (as long as they do not become superconducting). In a superconducting material the dependence on temperature is similar, but below a certain critical temperature the electrical resistivity drops to zero abruptly. There are basically three parameters that control if the superconductor is in superconducting state [74]. They are: 1. Temperature. The critical temperature is dependant upon material, but is below 140 K for known superconducting materials and often as low as a few K. 2. Current density. There is a maximum allowed current density called critical current density J c , and if the current density exceeds this value the superconductor becomes resistive. 3. Magnetic field. The maximum allowed temperature for maintaining superconductivity is lowered by magnetic fields, and the maximum allowed magnetic field strength to maintain superconductivity for a certain temperature and material is called the critical magnetic field. 69 These three limiting factors are not independent on each other. Generally, if the temperature is lowered, the critical current density and the critical magnetic field will become larger [74]. Another property of superconductivity is that magnetic fields can be expelled from superconducting materials. This is known as the Meissner effect. 6.1.2. Superconducing magnets There are many known superconducting materials, but very few of them, only a few in 10 000, qualify for making superconducting magnets [74]. These are called magnet-grade superconductors if they are available commercially. The desired properties are high critical current, very high critical magnetic field and good strain properties. The two types of superconductors that have been used in fusion coils due to their high critical magnetic field are Nb3Sn and NbTi. The advantages with NbTi are that it has good manufacturing properties and is not very sensitive to strain. NbTi is a ductile material that can be wound on a coil [75]. The drawback with NbTi is that the critical temperature is about 10 K, limiting the maximum magnetic field to about 11-11.5 T with ITER parameters even at 1.8 K operation [76]. NbTi is a commonly used superconductor and is relatively cheap. Nb3Sn is a brittle material that cannot be wound easily. To manufacture Nb3Sn coils, one can use Wind and React (WR) or React and Wind (RW) techniques, where WR seems to be the better choice [75]. In the WR manufacturing process the niobium and tin are wound separately and then the Nb3Sn is formed by a heat treatment of typically 650-700°C for about 180-200h [75][77]. After the heat treatment, the brittle conductors are carefully taped. If a conductor is broken during this process, it cannot easily be repaired. This makes the fabrication difficult and expensive, and the small global production of these coils also add to the cost. In addition, Nb3Sn strands are sensitive to strain. However, Nb3Sn has a critical temperature of 18 K, and with ITER parameters this corresponds to a maximum allowed external magnetic field of more than 14.5 T [76]. The consequence of this is that NbTi is used in low magnetic field regions while Nb3Sn is used in high magnetic field regions. Nb3Al has also been suggested as a superconducting material for fusion coil systems and has been examined for ITER [78]. It is also considered for DEMO [79]. Already in the 1970ies laboratory experiments showed that Nb3Al had excellent properties concerning critical magnetic field ( B 20T seems possible [80]), critical current density and strain. There has however been problems to find an appropriate manufacturing process [80], but it seems now that these problems have been overcome [80] and a large Nb3Al 70 coil was fabricated (with jelly-roll technique) and successfully tested in 2005 [79]. The critical magnetic field of this superconductor is even higher than for Nb3Sn, and it seems likely that the brittle Nb3Sn conductors will be obsolete in future if the manufacturing process of Nb3Al will be competitive in price. Also, HTS (High Temperature Superconductors) are under investigation for future fusion plants and show promising properties [81]. HTS are already used for the current lead to the coils, and the use of HTS here reduces the power consumption for cooling the current lead significantly, by roughly a factor of 4. This is since the heat conductivity of these materials is lower, giving less heat transfer from the concatenation point to the resistive conductor, and since there is no ohmic heating [81]. The energy consumption of the coil system is however typically a small part of the total cost, and reliability of operation may be more important [74]. 6.1.3. Cooling and shielding To maintain superconductivity, the superconductors must be cooled. This is done by using either liquid nitrogen or liquid helium. The choice of coolant depends on the critical temperature. For type II superconductors with a critical temperature above the boiling point of liquid nitrogen at 77 K at atmospheric pressure, liquid nitrogen can be used. For type I superconductors and type II superconductors with a critical temperature below 77 K, liquid helium is used. A cooling system with liquid helium is much more expensive, and could suffer from problems with solid air plugs. In fusion, liquid (sometimes in combination with compressed) helium is used to cool the coils. In the ITER cryogenic system, one system of 80 K compressed helium precooled by liquid nitrogen is used to thermally shield the coils and other systems, and the 4 K liquid helium coolant loops cools the interior of the coils [82]. The 80 K system has a cooling capacity of 1300 kW and the liquid helium 4 K system has a cooling capacity of 65 kW [59]. The internal heating in the coils comes primarily from the neutron (and gamma) radiation and for pulsed systems like tokamaks from AC heating. The AC heating comes from hysteresis effects (in the superconducting material), coupling losses (from currents between filaments) and induced eddy currents [74]. During one pulse in ITER, 13 MJ is being deposited from AC losses [82], which roughly would average about 17 kW for a 750 s pulse. The nuclear heating is about 14 kW [82], making the two effects roughly equal in magnitude for ITER. There is also external heating from heat conduction and radiation. 71 In a fusion power plant, it is important that the superconducting coils are sufficiently shielded from neutrons (and possibly some gamma radiation) to reduce cooling power and radiation damage to the materials. Detailed computations of these doses can be made with Monte Carlo simulations, as has been done in Paper XII by Prof. Klaus Noack. Both the nuclear heating and the radiation damage to the epoxy insulators can possibly set the minimum required thickness for the shields [83], and without shielding both these radiation limits would be violated by orders of magnitude. It is rather easy to stop thermal neutrons with a few cm of boron shield or similar. The 14 MeV fusion neutrons are however much more difficult to stop, and roughly 0.44-0.5 m thick shields are required in ITER on the inboard side [84]. For shielding, a mixture of water (for moderation of neutrons and coolant) and borated steel is typically used [84]. Note, however, that the ITER coils do not seem to be dimensioned for steady-state operation [85], and thicker shielding may be required for a reactor scenario. Also, note the cost for coil heating: At 4 K, 500 kWh worked by the cryogenic system is required to remove one kWh of heat [86]. The range 250 – 8000 kWh is given in [74]. 6.1.4. Power supply and stability The current feed to superconducting coils is supplied by a DC-source with high current and low voltage, since only the feeder wires will cause a voltage drop. It is of great importance to change the current slowly to prevent mechanical stresses and heating from induced eddy currents. Heating from eddy currents can cause temperatures over the critical temperature locally which in turn gives rise to ohmic heating and overall heating with loss of superconductivity in the entire coil. Loss of superconductivity in a coil is called a quench. Quenches can arise from a variety of causes, for example too high magnetic fields. The ability to avoid quenches is referred to as stability for the superconducting coil and should also take into account external events such as consequences of plasma instabilities [74]. 6.1.5. Internal structure The internal structure of a superconducting coil is very complicated. A superconducting coil consists of a large number of wound strands. Each strand consists of a large number of superconducting filaments embedded in a stabilizing material, for example copper. The current in each strand is typically of the order of a few hundred amperes [75]. There are a few different techniques that can be used in fusion coils for structuring the strands, described in Ref. [75]. Cable in conduit is the chosen design for the ITER coils. A number of strands – typically in the order of 1000 – are put into a cable and the cable is wound many laps and embedded into structure 72 material. There is a cooling channel in the center of the cable. There is also a cover of structure material outside the current-carrying region [76]. A detailed description for the ITER central solenoid is found in Ref. [87]. A limiting factor in superconducting coil design is the stesses that arise from the often very high magnetic fields. Stress can both hamper the operation capabilities of the superconductor and damage the superconductor permanently (especially Nb3Sn coils are sensitive). In general, coil curvature should be kept low and the usual axisymmetric coils have good properties for this. This is an advantage for axisymmetric mirrors compared to quadrupolar mirrors. 6.1.6. Use of copper coils in reactor scenarios Copper coils are typically used in plasma experiments to reduce cost. These experiments are almost always pulsed and stored energy from capacitor banks or flywheels can then be used to create the magnetic field for a short while. Even some large experiments have copper coils, like for example JET (Joint European Torus) in Culham, England. However, the ohmic losses are typically regarded too large for reactor scenarios. For spherical tokamaks, however, superconducting coils cannot be used at the inboard side, since there is not sufficient space for shielding. In the ARIES-ST study, a 3 GWth spherical tokamak reactor is proposed [88]. The TF (Toroidal Field) coils are made of copper, and the best option seems to be to cool them with water at room temperature [88]. Using hot water or liquid lithium to increase the power conversion efficiency does not pay off due to the increase in resistivity [88]. Neither does it pay off to have the copper coils at cryogenic temperatures even if the resistivity drops significantly, since the efficiency of the cooling at such low temperatures is too low [88]. One problem in the ARIES-ST project is that the inboard leg of the TF coils suffer from radiation damage, and need to be replaced every third year or so [88]. Copper is rather sensitive to neutron damage, and has a low DPA (Displacement Per Atom) limit. The ohmic heating in the TF coils for ARIES-ST was calculated to 329 MW. This is reasonable (although quite high) for a power plant that aims for a power production of 3 GWth. 6.2. Mirror machine coil types For quadrupolar mirror machines, the first quadrupolar coils were the “Ioffe bars”, see for example Ref [89]. These are similar to the quadrupolar coils used in Paper I. In later designs, two types of coils have primarily been used to produce the quadrupolar (and partly the axisymmetric) magnetic field component. One is the baseball coil (with the shape of a baseball seam), which is described in Ref. [90]. The other is the yin-yang coils [91], which 73 are “bent” elliptic coils ordered in pairs embracing each other. Both these coil types are shown in figure 6.1. To add contributions to the axisymmetric part of the field, circular coils are normally used, possibly combined with elliptical-like racetrack coils (formed like a racetrack) at the elliptic regions. A yin-yang pair is used to create an anchor cell of a tandem mirror, giving rise to both the axisymmetric and the quadrupolar field components. A baseball coil is typically used to produce the quadrupolar field and other coils are normally needed to give a sufficient axisymmetric field. Figure 6.1. Quadrupolar coil types for mirror machines, with a baseball coil in (a) and a yin-yang coil pair in (b). 6.3. Existing or earlier mirror machine coil systems A number of mirror machines have been built for fusion research. The coil system of the Gamma 10 device can be viewed in Ref. [92]. It consists of circular coils, racetrack coils and baseball coils. In the reactor design study MARS (Mirror Advanced Reactor Study) at Lawrence Livermoore National Laboratory (LLNL), yin-yang coils were used [93]. For the large MFTF-B tandem mirror (40 m long) at LLNL, which was ready for operation in 1986 but never used due to budget cuts, both yin-yang coils and baseball coils were used in combination [93]. 74 7. The SFLM Hybrid project Studies of a hybrid concept based on the SFLM started in 2007 at Uppsala University. The aims are to examine the possibilities of making a mirrorbased fusion-fission device for transmutation of minor actinides in combination with energy production, and to make a reactor design study. The participants are Prof. Olov Ågren, Dr. Vladimir Moiseenko, Prof. Klaus Noack, Anders Hagnestål (the author), Prof. Henryk Anglart and Prof. em. Jan Källne. The concept is described in Paper VI, the coil system is described in Papers I-II, IX-XI and the fission mantle is described in Papers VIII, XII. The goal of the project at this stage is to show that it seems probable that such a mirror-based fusion hybrid can be an alternative for combined transmutation of TRU and energy production. In this section, most parts describe results mainly produced by other members of the project group, which is here regarded as background information. 7.1. Geometry The SFLM Hybrid is an average minimum-B mirror machine with a mirror ratio of 4. The orientation of the device can be either horizontal or vertical, since both orientations allow for natural circulation of the coolant. Currently, a vertical orientation is the selected option. The length of the confinement region is 25 m. Beyond each mirror end a 6.25 m long magnetic expander including a recirculation region is added, giving a vacuum chamber length of 37.5 m. The plasma radius is 40 cm at the midplane. The vacuum chamber has a circular cross section with a 90 cm radius in the confinement region, which expands to 100 cm at the beginning of the recirculation region. Near the magnetic expanders, the vacuum chamber radius expands to about five meters. The fission mantle is located outside the vacuum chamber confinement region, having a total length of 26 m. The cross section of the fission mantle is circular and has an outer radius of 2 m. In the first design in Paper VIII, there was no neutron shield included in the fission mantle. In the second design in Paper XII, a neutron shield which protects the coil system is included. Outside the fission mantle the superconducting coil system is located, where a radial 10 cm space separates the coils from the neutron shield. The coils are described in section 8.1. Inside the vacuum vessel at the recirculation region, antennas for radiofrequency heating are located. The device stripped from coils is shown in figure 7.1. 75 Figure 7.1. The SFLM hybrid device (here horizontally orientated) where the coils have been removed. 7.2. Plasma parameters and magnetic field properties The vacuum magnetic field is described in chapter 8.1, and provides flute stability with some margin in the vacuum field. The device is aimed for a of about 0.4-0.6 with the latest coil design (Paper II), which may affect stability. Therefore, a stability margin for maintaining flute stability has been applied in Papers I-II. There are also other stabilizing effects such as line tying (see for example Ref. [94]) and finite Larmor radius effects [95]. Furthermore, the magnetic expanders add to stability, although their main purpose is to take care of plasma loss. The device is in this version aimed for an initial fusion neutron production of around 3.6·1018 neutrons per second. Each fusion reaction give one neutron and in total 17.6 MeV 2.82 10 12 J. This corresponds to a fusion power of about 10 MW and a fission power of about 1.5 GWth with a Qr 146 150. These values are for the beginning of the fuel cycle (BOC) with keff = 0.97. Control rods, suggested in Paper XII, or burning absorbers can be used to keep a fixed value of keff = 0.97, and with this option the fusion power will be held fixed. Without control of keff the fusion power needs to be gradually increased to 20 MW in the end of the fuel cycle (EOC) to maintain the power production of 1.5 GW, since keff will be reduced to 0.945 before refuelling is performed due to burnout of fissile isotopes (see Paper VIII). A 76 fuel cycle is 311 days of steady-state operation (or nearly one year), which is a standard fission reactor operation time. It is, from the fusion point of view, strongly recommended to keep the fusion power fixed for several reasons: 1. The first reason is that the device must be dimensioned for the maximum neutron (and fusion energy) production during the fuel cycle. For the fixed keff case, only half the maximum neutron production need to be achieved, which reduces the maximum or the magnetic field strength. The requirements of the plasma neutron production is approximately given by nr nT nD D T v d 3x (7.2.1) V where is the averaged reaction rate for D-T fusion D Tv reactions, nr is the number of fusion reactions and nT and nD is the number densities of tritium and deuterium respectively. If the temperature is kept constant, the pressure is proportional to the number densities nT and nD and thus the neutron production nr p 2 . With only half the required maximum nr , the pressure can thus be reduced by a factor 2 , corresponding to a reduction of the with a factor 1/ 2 for the same magnetic maximum plasma is fixed, the magnetic field can be field. If instead the plasma reduced by 2 1/ 4 0.84. This factor can be very important for the cost of the coil system, since by this perhaps all coils can be made of the considerably cheaper material NbTi instead of Nb3Sn. 2. Following the first point, the heating system (ICRH antennas) and the massive associated power feed can be dimensioned to half the size, about 70 MW. Also, the power consumption for heating is reduced to 2/3 on average, which would save in the order of 10 million euro/year. is 3. The magnetic field can be optimized for the pressure profile. If to be varied by a factor of 2 , it will probably not be possible to avoid neoclassical transport, and other field properties could also be suboptimal (perhaps higher ellipticity is needed). This is since the curvature of the field lines is very sensitive to changes in the magnetic field. The stability limits to ballooning modes also depend , although we do not expect this to be problematic (see on introduction of Ref. [95]). The drawback with operation at a fixed keff is perhaps that control rods may complicate reactor safety. The required fusion Q, the ratio of thermal power output to total electric power consumed for the fusion device, is 0.15. This is a factor of 100 lower 77 compared to a pure fusion reactor (Q = 15), but it is still not obvious that it can be achieved in a mirror device of this type. The critical parameter is the electron temperature. Note that an increase of the midplane magnetic field is presupposed when for a certain neutron production is estimated, so that the the required mirror ratio would be about 4 on average with the diamagnetic effect effects are included has included. The final coil design where finite however not been completed yet. The plasma that leaks out at the mirror ends would be removed by vacuum pumps at the expander region. The feeding of the plasma has not been fully specified yet. New plasma could be injected using pellet injection from the mirror ends, and gas feed can also be used. Another option is to use neutral beam injection from the mirror ends, although the pitch angle may become unfavourable. In the current design, there is no room for feeding with neutral beams at the central cell since this would require holes in the fission mantle. R. Moir et al. has dealt with this problem in their design by making the fission mantle a bit shorter [28]. A disadvantage with pellet injection and gas feed is that both methods may lower the electron temperature. 7.3. Radio frequency heating For plasma heating, ion cyclotron radiofrequency heating (ICRH) is planned. There are several reasons for this. 1. Calculations predict that the efficiency of the ICRH could be high, theoretically more than 90% [96]. 2. The antennas are small and can be fitted in at the elliptic regions just outside the mirror ends, where the plasma density is low. There can be two antennas at each end, giving a possibility for four antennas in total. As mentioned above, the competing neutral beam technology with midplane injection would require large holes in the fission mantle to operate where fusion neutrons could escape, and thereby such neutrons would not contribute to the fission power generation. 3. For a fusion-fission device, electric breakdown criteria for sequenced antennas admit an ICRH power of 130 MW, which would be required at EOC if no control rods are used (100 MW indicated in [96]). 4. The ICRH may improve the end confinement since perpendicular energy is added to the ions, and the ions thereby move further away from the loss cone in velocity space. 78 7.4. Fission mantle and shielding The fission mantle is thoroughly described in Papers VIII, XII. In Paper VIII, two versions of the fission mantle are outlined, and in this doctoral thesis the “near-term” option is described which is aimed for 1.5 GWth. In Paper XII, the fission mantle is modified to include neutron shielding of the coil system and to include small control rods. The calculations on the fission mantle have been made with Monte Carlo simulations using the program MCNP5. In the calculations, the geometry of the fission mantle is simplified as axisymmetric layers of a certain thickness. Each layer constitutes of a homogenous mixture of the present structure material, fuel, coolant etc. in that layer. Several of the materials and concepts from the fission mantle study are adopted from FTWR tokamak hybrid concepts (see for example Ref. [29]), which have been studied since the 1990ies by W. M. Stacey et al. at Georgia Tech in high technical detail. One important difference is that a 15 cm thick buffer region is added in the SFLM Hybrid to protect the first wall from fission neutrons. 7.4.1. Geometric design The geometric design of the fission mantle is shown in figure 7.2. The 3 cm thick first wall has an inner radius of 90 cm and is made of HT-9 steel. Outside the first wall is a 15 cm thick reflecting buffer of Lead-Bismuth Eutectic (LBE), which is the primary coolant material. It may seem strange at first to have a reflector here. The primary reason for having the buffer is to protect the first wall from fission neutron damage. Without such a buffer, the lifetime of the first wall would be too short. The neutron damage primarily comes from the fission neutrons since they constitute the majority of the neutrons, and by reflecting them the lifetime of the first wall is extended significantly to more than 30 years using a maybe somewhat high DPA-limit (Displacement Per Atom) of 200. Another benefit of the buffer is that LBE is a good neutron multiplier for fusion neutrons through (n,2n) reactions, and the number of neutrons that enter the fission core is increased. On the other hand, the buffer moderates the neutrons which gives lower average fission neutron gain per source neutron than would otherwise be expected. The combined effect is that Qr is slightly lowered by the buffer. 79 Figure 7.2. The radial structure of the fission mantle. The figure is produced by Klaus Noack. Outside the buffer is the fission core, which is about 22 cm thick in the socalled “near-term” option. The core is composed of fuel, structure/cladding and coolant. The fuel is of dispersion type containing TRU-zirconium elements enbedded in a zirconium matrix. The structure/cladding is made of HT-9 steel. Outside the fission core is a core expansion zone filled with LBE. This zone will work as a neutron reflector, and will give a negative contribution to the void effect since a loss of coolant accident (LOCA) will result in a poorer neutron confinement. In this zone, two 2.5 m long and 1 cm wide boron carbide annuli (control rods) are located which can be inserted axially all the way to the midplane. They are used to keep keff constant during a fuel cycle (see Paper XII), and can lower keff by about 0.04. Also, there is space for special irradiation assemblies and extra fuel if needed in this region. The core expansion zone is 15 cm thick. Outside the core expansion zone, which also acts as a neutron reflector, is the radial reflector. The purpose of the radial reflector is to confine the neutrons and thereby increasing keff per amount of fuel and reducing needs for neutron shielding. Also, the tritium generation is located here. There are axial reflectors (neutron end plugs) as well at the mirror ends which cover the whole fission mantle endings. The material in these has been adopted from the NEA benchmark for ADS, using 70% HT-9 steel and 30 % LBE coolant. For the radial reflector, the LBE is replaced with the lithium-lead coolant Li17Pb83 to generate tritium (T), where the lithium is 20% enriched in Li-6. Tritium is primarily produced by the reaction Li-6 + n = He + T + 4.8 MeV [11]. It is vital that the device is self-sufficient in tritium. Calculations show that tritium generation will be larger than the consumption during the whole fuel cycle, averaging 3.60 times the consumption in the first model in Paper VIII. This large tritium production implies that the enrichment of Li-6 can be substantially reduced. The tritium 80 regeneration is sufficient in the newer fission mantle version in Paper XII as well. Outside the radial reflector, the neutron shield is located as specified in Paper XII. Neutron shields for fusion is typically very thick, 0.5-1 m, since it is hard to stop the 14 MeV fusion neutrons. For the SFLM Hybrid, the shielding problem is however a fission shielding problem, since such a small fraction of the neutrons that escape through the reflector is fusion neutrons. Typical fission neutrons have energies of 1-2 MeV directly after fission. A neutron shield of 25 cm thickness is therefore sufficient to reduce the nuclear heating and damage in the coils to acceptable levels. The shield is made from a mixture of stainless steel alloy S30467 type 304B7 (60 vol%), which has 1.75 % wt% boron added, and water (40 vol%). 7.4.2. Reactor safety Reactor safety is a complicated subject, and much work is needed to analyze various safety cases in the SFLM Hybrid project. The most severe accident case to avoid is a supercriticality, preventing keff from exceeding 1 + , where is the fraction of delayed neutrons. The Chernobyl accident was partly of this type, with known disastrous results. Also, a core meltdown must be prevented. A core meltdown will probably destroy the facility and cause a lot of sanitation work, and could also lead to radioactive leakage like the (small) leakage in the TMI (Three Mile Island) accident. In a reactor with a fast neutron spectrum, the presence of coolant in the core effects keff directly in a few ways. First, the coolant acts as a reflector. How much reflection a coolant causes depend on the coolant material. A change of the coolant reflecting property will affect keff, and the amount depends on geometry and location of the coolant. In most cases, presence of the reflecting property will increase keff by confining the neutrons. Second, the coolant will moderate the neutrons. This might affect keff in either direction, depending on the neutron spectrum, the fission cross sections of the fuel and the number of neutrons per fission, but will in the SFLM Hybrid decrease keff. Third, (n, 2n) reactions in the coolant might increase keff and neutron absorption in the coolant might decrease keff. In reactors with lead coolant, the total impact of all these effects is that the presence of the coolant leads to an increase in keff. It is however not so for all regions within a reactor, especially not in the central core region where presence of coolant in leadcooled reactors cause a decrease in keff. The coolant can be voided in a reactor, for example if the coolant loop has a leakage or if the coolant is blocked by solid coolant etc. causing local boiling. If the coolant by leakage or other effects is removed from the reactor, keff will be affected. This is called the void effect. Lead-cooled cores have in most cases a negative void 81 coefficient if all coolant is lost (see for example Ref. [97]), implying that keff will decrease when the coolant is removed. However, as stated above, this is not always true for voiding subsections of the coolant by for example coolant boiling. The core must remain subcritical in all possible coolant distributions, and such safety case studies should be addressed. In Papers VIII, XII, two safety cases concerning coolant void effects are handled. The studies are made for a horizontal reactor orientation in Paper VIII and for a vertical orientation in Paper XII. Since no detailed core layout even has been done yet, the safety cases are simplified to a homogenized model. The first case is loss of coolant accidents in the LBE loop, which is investigated by calculating keff for a number of LBE coolant levels. The result is a strong decrease in keff in all cases for both orientations, being more pronounced for lower coolant levels. There is no indication that an increase in keff would occur at any coolant level. The strong decrease in keff is caused by the loss of the neutron confining coolant reflector in the core expansion region. The second safety case examined is partial voiding in the core caused by local boiling. This case is examined by voiding axisymmetric rings of different thicknesses, and here the cases are the same for both reactor orientations. The expected increase in keff is well below 0.02 and it seems unlikely that this increase will exceed 0.02 for any local boiling. It should however be noted that voiding of a small localized part, say a cooling channel, has not been checked. The calculations suggest that keff = 0.97 is an adequate reactor scenario value concerning these studied safety criteria. In Paper XII, a continuation of a LOCA accident is examined where the core is flooded with water to remove the residual heat. The case reveals that if only the core is filled with water, the reactor becomes supercritical. If the whole coolant region is flooded, the reactor remains deeply subcritical. Since this is a possible case, there should be large and many connection channels between the different coolant loops. Safety cases concerning supercriticality where parts of the core are moved/compacted due to core meltdown have not yet been adressed. Although core meltdown should be avoided at all costs, a core meltdown should not result in supercriticality. A drastic increase of keff (from 0.84 to 0.96) is pointed out in Ref. [98] in a horizontally oriented mirror machine when the different materials (fuel, structure/cladding, coolant etc.) melts and ends up in a sandwich structure on the ground. The reason for the large increase in keff is that the fuel becomes more compactly distributed when the large empty space in the vacuum chamber compacts. The SFLM Hybrid can be oriented vertically, but this does not guarantee that such scenarios cannot be found, for example if a part of the core falls into the vacuum chamber. The described increase in keff may be a critical safety issue for magnetic 82 mirror hybrids, and should be studied for the SFLM hybrid concept. However, there are most likely ways of handling such a scenario. The criticality keff is also affected by the temperature. In a fast reactor, the change in keff comes from the Doppler broadening effect and the CTD (Coolant Temperature Density) effect. In the SFLM Hybrid case, the Doppler broadening effect is small due to the fast neutron spectrum and the very low content of U-238 in the fuel. For Doppler broadening, keff / T was calculated to about 1.1 10 6 for the SFLM Hybrid where keff was calculated at two different temperatures and the temperature dependence was approximated as linear between those two temperatures. For the CTD effect in the central core keff / T was calculated to about 4.27 10 5. for the LBE. This reflects the void effects in the central core, and is not negligible. If the entire buffer and expansion zone is added, the CTD effect reduces somewhat to 3.37 10 5 , but there is still a positive increase in keff with increasing temperature. Further analysis of the CTD effect has not yet been made. 7.4.3. Cooling The cooling has been considered by Prof. Henryk Anglart at KTH for the vertically orientated case, and some results are presented in Ref. [99]. The LBE and Li17Pb83 coolants are pumped during normal operation, and the pumping power can be well below 50 MW. The MHD forces have not been included in this analysis, and they may add to the pumping power. However, the cooling channels would be almost parallel to the magnetic field except at the influx/outflux, which should make the MHD forces less significant than would otherwise be the case. It may also be possible to reduce them with dielectric pipe coating. When the reactor is stopped, the residual heat (or decay heat) must be removed which became evident to the broad public during the Fukushima accident. The residual (or decay) heat comes from decay of fission products and gradually fades away after shutdown. It however constitutes of more than 5 % of the nominal power at shutdown, and cooling is needed for a rather long time (at least weeks) afterwards. In the SFLM Hybrid, passive cooling is planned for removal of residual heat. This removes dependence on external pumps after shutdown. With vertical orientation of the SFLM Hybrid, it is shown in Ref. [99] that it seems possible to have natural circulation of the coolant that is sufficient for removal of the residual heat. The circulation is driven by the heat differences in the core and the outer loop. It may however probably also be possible to achieve this for the horizontal case. 83 It seems sufficient to have only one point of influx and one point of outflux of coolant along the z axis. Since the influx/outflux of coolant must be regarded during coil design so that there is sufficient space available for the pipes, this means that the coil system only needs to have gaps at the recirculation regions. A vertical orientation also simplifies the refuelling process, since this can be done from the top. For a horizontal refuelling process, the coolant must be prevented from leaking out. 7.5. Electron temperature The weakest point in the fusion driver is probably the electron temperature, which has been a major roadblock for fusion mirror machines since too much input energy is lost due to electron drag. A typical requirement for mirror fusion devices is Te = 10 keV [47]. In an experiment in the Gamma 10 tandem mirror device in Japan an electron temperature of about 650 eV has been reported, indicated in Ref. [100]. For the axisymmetric Gas Dynamic Trap (GDT) device an electron temperature of 250 eV [48][101] has been achieved, measured with Thompson scattering. The multimirror device GOL-3 has achieved an electron temperature of 2.5 keV. In Paper VI, an estimate of the required electron temperature for the SFLM Hybrid has been made. The requirement on Te is substantially relaxed due to the strong fission to fusion power amplification Qr, and is estimated to 500 eV for a power producing device. The estimate is based on two assumptions. First, the major power loss in the mirror is assumed to be due to electron drag. Second, the requirement for a fusion device with Q = 15 would be 10 keV or higher. The strong Qr relaxes the requirement on Q to about 0.15 with Qr 100. Assuming that the electron drag time increases as Te3/ 2 from Eq. (3.3.5), that d is proportional to the power loss and that the densities ne are the same, the result is Te ,hybrid Te, fusion 100 2/3 10 keV 1002 / 3 0.5 keV (7.5.1) Although this is a large relaxation of the required electron temperature, it is not obvious that such a temperature can be reached in the SFLM hybrid. Therefore, a scenario for increasing the electron temperature is outlined in Paper VI. The scenario is based on reducing the plasma and neutral gas density in the expander region by pumping. Thereby, the electrons would lose less energy in that region from collisions (where line radiation, secondary emission and other effects can be important). 84 7.6. Divertor plates and heat load A critical point for fusion reactors is the divertor plates. The divertor plates are the material surfaces that are designed to receive the majority of the leaking plasma. The requirements on the divertor plates for toroidal systems for pure fusion are very high, and the best material so far tested for ITER would only last for about 7000 pulses (i.e. about 3 months of steady-state operation) [61]. The divertor plates for ITER should be able to withstand both a very high heat load from the plasma of about 5 MW/m2 (requiring good heat conduction), the strong neutron load from fusion neutrons and the pulsed mode of operation with varying heat loads which causes material stresses. The maximum acceptable upper heat load limit of 10 MW/m2 is set in Ref. [102], but the 5 MW/m2 is assumed to be more correct. For the SFLM Hybrid, the situation is much better. Due to the strong Qr, the fusion power is strongly reduced. The input power (at BOC) is about 70 MW. In such a fusion device, the radial losses are expected to be small compared to the axial losses, not more than 10% of the total losses. It should however be mentioned that this has not been checked for the SFLM Hybrid yet, and some experimental work suggests otherwise [103], although this may be due to electron cyclotron resonance heating. For the first wall, a limiter would receive most of this load of about 7 MW. It is not expected to be a problem to distribute this load on a surface large enough to withstand the heat load, say 10 m2 or so, and it is expected that a larger radial heat flux can be dealt with. The rest of the heat load comes from the axial losses and will reach the expander. The expander can – and should – be made sufficiently wide to distribute the heat load on a large area. For example, if 100 m2. This the radius is 4 m, the total expander area will be 2 42 2 would give an average heat load of about 0.7 MW/m , which is a tolerable heat load if it would be evenly distributed on the surface. 85 86 8. Summary of results and discussion The results presented in this thesis can be grouped in the following way. In Papers I-III, V, IX-XI, the results are associated with magnetic field and coil design and those results will be summarized in section 8.1. In Papers VI, VIII, XII the results are associated with the fission mantle and the overall concept. These results are presented in section 7 and will not be repeated here. In Papers IV, VII, the results are associated with radial transport and constants of motion. These results are given in section 8.2. An overall discussion of the results is given in section 8.3. 8.1. Design of coils and magnetic fields To understand the results from these papers, it is necessary to present them in cronological order, which is done in sections 8.1.5 - 8.1.10. The field and coil design has been a process over 3-5 years, resulting in some intermediate stages and a final vacuum design. The focus in this work has not been to reproduce a magnetic field with as high accuracy as possible, but rather to show that it is possible to find a coil set that reproduces the magnetic fields within the given geometric constraints with a tolerable accuracy. First, the optimization methods and coils used are described, and then the specific results from the papers on coil and field designs are given. 8.1.1. Optimization methods The basic task for an optimization algorithm is to select the best element or value from a number of available elements or values according to some criteria. In this doctoral thesis the available values are function values of some n-dimensional continuous cost function, where the cost is to be minimized. It is here important to realize the difference between global and local optimization. Local optimization, where a local minimum – not necessarily the best minimum available – in the cost function is found, is rather straightforward. To illustrate this, consider a 2-dimensional cost function and interpret it as the altitude of some terrain. If an initial starting point is given, it is simply just to walk downhill until you have uphill in all directions to find a minimum. This can be done in numerous ways, where the result will be the same but the processing time different. This is the principle of local optimization. Global optimization, on the other hand, is much more 87 difficult. A global optimizer tries to find the best minimum of all available minima. For a cost function with many parameters, say 30, it is often in reality not possible to guarantee that all minima have been checked. Global optimizers are therefore often heuristic and look for an as good minimum as possible, but do not guarantee that it is the best minimum available. There are numerous optimization methods available for both global and local optimization. At this stage of the project, time has not been spent to identify and implement/use a global optimizer. Instead, simple local optimizers have been used. After trying some different local optimizers, an implementation of the Nelder-Mead algorithm was selected. This method is heuristic and does not always converge, but in practice this showed to be no problem in the cases studied. The results were comparable to the results achieved with the other methods, and the implementation worked well. For coil optimization, there are two magnetic field components that are to be reproduced, the quadrupolar field represented by g ( z ) and the axisymmetric field B ( z ) . In Papers I, IX-X coils types that only give contribution to one of the field components have been selected. Thereby, the coil optimization problem is separated into two independent optimization problems, one for g ( z ) and one for B ( z ) which simplifies the problem. The cost function (in this case for B ( z ) ) typically punishes deviances from an ideal field and power use and a typical cost function could be zmax fz dz 0 B ( z ) Bc ( z ) 2 B ( z) 2 ri I i 2 Ki (8.1.1) i where Bc ( z ) is the B ( z ) contribution from the coil system and K i is a weight factor for the power term. In some cases, more terms were added. The power term is not at all correct for superconducting coils but serves the purpose of restricting a waste of current to gain accuracy. It is important that the power term includes all conductors to get an appropriate and practical result. Also, it proved useful to do the coil optimization by hand, which means that the input parameters were changed manually and the result was observed between each change. This was done in Paper II which at first may sound reckless, but almost as accurate results were found with this method compared to using an optimizer and it takes only an hour or so to do the optimization. The numerical optimization process produces a lot of impractical and strange solutions, and it is sometimes difficult to find a way to avoid these and find a proper solution. For the coils, there are some auxillary constraints that are not so easy to implement in a practical way in the cost function. One is that the coils should not intersect, and should be 88 spaced with some margin (which though in principle can be implemented). Another is engineering simplicity. Also, for the 3D-coils, no analythical expression has been found for the contribution to B( z ) and g ( z ) , which makes an optimization process very slow. 8.1.2. Optimization of vacuum magnetic fields The magnetic field can be optimized in many different ways. A list of properties that should be addressed for coil design in tandem mirrors has been given by Baldwin [93]. He listed 8 properties to be targeted. Many of them refer to problems specific to tandem mirrors. In Papers I-II, IX-XI the vacuum fields were optimized for field smoothness, flute stability with a margin and low ellipticity. There are two of the properties relevant for single-cell mirrors listed by Baldwin that are not fully targeted in the work in Papers I-II, X-XI. The first is neoclassical radial transport, which is associated with geodesic curvature causing radial particle drifts that creates “banana” particle orbits [93]. This is not targeted in the field design in Paper X and only to a certain extent targeted in Papers I-II, XI. It was argued in those papers that in the case of a fusion hybrid, this property is not as important as it would be for a pure fusion device, since the required particle confinement time is considerably shorter and the majority of the losses still will be end losses. The author has however after these papers were written understood that they may be quite important, and that they should be into account. targeted in a future coil design that takes effects from finite The other property is stability to MHD ballooning modes. The reason for not targeting MHD ballooning modes (yet) is the later publication by Newins and Pearlstein [95], where the authors expected that ballooning modes would not be limiting in any tandem mirror device due to the strong stabilization of finite larmor radius (FLR) effects. At this stage of the project, ballooning mode analysis is therefore postponed to future studies. The properties that are addressed are important. The field must be stable to flute modes. This is the very point of having a quadrupolar field. Two different methods have been used to determine flute stability. In Paper X, the average minimum-B criterion from Eq. (3.3.7) has been used. However, this criterion does not take into account the effects of a varying pressure along the field line. In Papers I-II, this is corrected by instead using the flute stability criterion given in Eq. (3.3.8). It is also important to obtain a fairly low flux tube ellipticity. A too high ellipticity gives an impractically shaped plasma which could hit the first wall in the strongly elliptic regions. In practice, this reduces the plasma volume, and for the SFLM Hybrid it would increase the required magnetic field strength to get a sufficient neutron production. If the fans get too thin, plasma feeding may get complicated. Also, a slight argument is that classical transport may increase somewhat in 89 the thin fans. It is also crucial to avoid too sharp gradients in the magnetic field, since it will not be possible to create sharp gradients when the thick fission mantle is present. It should be noted here that the stability requirement and the low ellipticity requirement are contradicting, and the task is to find a field that is suitable in both aspects. 8.1.3. Function modelling with splines One way to optimize the vacuum magnetic field is by modelling the B ( z ) and g ( z ) functions with equidistant cubic clamped B-splines, which has been done in Paper X. Splines are chosen since they are fairly easy to implement, have continuous derivatives up to second order and since the control points only affects the function in a region close to itself. Clamped splines are chosen, since with this representation the end derivatives can be prescribed. Typically, between 10 and 20 control points are used for each function. The control points are then used as variables in a cost function which is optimized by a local optimization function (Nelder-Mead). Since both B ( z ) and g ( z ) are even functions, it is sufficient to model half of those functions. The boundary conditions for the spline representation of the B function are B (0) B0 , B ( L ) Rm B0 , B (0) 0, B ( L) 0 (8.1.2) where prime denotes differentiation in z and L is half the length of the confinement region. For the g function, the only fixed boundary condition is g (0) 0 (8.1.3) due to symmetry and the rest of the boundary conditions g (0), g ( L) and g ( L) can be chosen freely. The boundary conditions for B are at least nearly fulfilled by locking the first and last spline control points to the boundary value and placing them at z = 0 and z = L. The derivatives of B can automatically be prescribed with the clamped spline representation. For g, all control points are free and g ( L) is either prescribed to some value or used as a variable in the optimization. A cost functional to optimize has been chosen and the optimization performed. The optimization has been made with the average minimum-B criterion as stability criterion in Paper X. An attempt has also been made to optimize the field using the flute stability criterion and a prespecified sloshing ion distribution to model the pressure. It proved however hard to do such an implementation, since the optimizer always selected a profile that disqualified the pressure profile. It was obvious that such a simple method was not sufficient, and that an equilibrium calculation was required to get even close to appropriate results. 90 8.1.4. Superconducting coil modelling The superconducting coils need to be modelled somehow to determine their size and to find the magnetic field that is generated from them. The size is required both since there is an obvious auxillary condition that the coils are not allowed to intersect and since the current distribution in a coil is affected by its size. The widths of the coils are determined from several aspects. The minimum width is obviously dependent on the coil current. The dependence is three-fold. First, the magnetic field strength should not exceed the critical magnetic field for the superconducting material anywhere. If so, a superconductor quench will occur. Second, the coil must be strong enough to withstand the huge magnetic forces. Third, the critical current density for the superconductor must not be violated. Also, space is required for cooling channels within the coils. To determine the dimensions for a coil set is therefore a complicated task. For instance, the cooling requirements need to be determined for the expected maximum heat load (which itself is not very easy to determine, see section 6.1.3), and after this the cooling channels must be dimensioned. However, what can be done conveniently and probably with a fairly realistic result is to use an existing similar coil system as a reference. The method employed is to calculate the average current density in the winding pack for a coil system producing a similar magnetic field strength as the device that should be modelled. Two existing or planned systems were analyzed: ITER and the Japanese tokamak JT-60. In the (somewhat old) specification for the ITER toroidal coils (Nb3Sn) they have a current carrying region of about 0.85 m2, and carry a current of 11.5MA [76]. This gives an approximate average current density of 1.35 kA/cm2. In the remodelling of JT-60 for superconducting coils (NbTi), approximately the double current density was used [104]. In Paper I, X, a current density of 1.5 kA/cm2 is assumed for the Nb3Sn coils, and in the lower field solution in Paper II, where we hope that NbTi coils can be used, 2.6 kA/cm2 was directly adopted from the JT-60SA NbTi coils [105]. The size of the stabilizing structure material outside the current carrying region is approximated by again looking at ITER and JT-60 toroidal field coils. Both have a cover of structure material with a thickness of about 10 % of the total coil width. Thereby, in total 20% extra width is added to the coils for structure material. The cross sections of the coils are selected to be quadratic for the circular coils, and quadratic-like for the quadrupolar and fishbone coils. The magnetic field is then calculated from Biot-Savarts law, where the winding pack has been divided into m m equidistant filamentary line currents. With m 10 the solution becomes very accurate, but already m 1 gives a fairly good approximate solution in many cases. 91 Four different coil types have mainly been used in this thesis. The simplest coil type is the circular coil, which has been used to produce the axisymmetric field in Papers I, IX-XI. It only contributes to B ( z ) (in ideal cases) and it is straight forward to calculate the contribution from one filament as B( z ) 0 2 I r r 2 z 2 z 2 3/ 2 (8.1.4) where r is the filament radius and z is the filament z position. To produce the quadrupolar field component, the simplest coil geometry consists of four parallel conductors with quadrupolar symmetry. Such a configuration is shown in figure 8.1, where also the quadrupolar field is illustrated with arrows. Figure 8.1. Quadrupolar coil configuration where the arrows show the direction of the quadrupolar component of the magnetic field. Two different types of coils have been used that only contributes to the quadrupolar field. One coil type that was evaluated in Paper IX was long conductors like those shown in figure 8.1 but where the distance from the z axis to the coil was a function of z. Since they proved not to be very useful, they are not further commented here. Another type of quadrupolar coil that has been used in Papers I, X look like symmetric Ioffe bars which are interconnected at certain positions to modify the current. A series of connected such coils are shown in figure 8.2. Note here that the current does not form circles at the connection circles, since the direction of the current shifts at each quarter-circle connection. 92 Figure 8.2. A 3D-image of three connected quadrupolar Ioffe bar-like coils. The currents follow the black arrows. The contributions to the quadrupolar field from such a coil can be grouped as contributions from sets of four straight conductors g c , s and contributions from sets of four quarter-circle segments g c ,c . The contributions to zeroth order in r are gc,s 3 0 I1 z2 z1 [( z r '2 dz ' z )2 r '2 ]5 / 2 3 0 I1 K (z z2 ) K ( z z1 ) (8.1.5) where K (z ) [3r 2 2 z 2 ]z 3r 2 [r 2 z 2 ]3/ 2 (8.1.6) r 2 (z [( z z )2 (8.1.7) and g c ,c 3 0 I1 / 2 z) . r 2 ]5 / 2 In the equations, I1 is the current in the straight conductors, with a positive sign for currents that give positive contributions to g. For gc,c in Eq. (8.1.7), a positive sign shall be taken for coils with positive, inner z and negative, outer z . The other two rings have a negative sign. The parameter r is the coil radius. When making filamentary line currents of the straight segments, the angular distance of the filamentary current from the center, q , changes the contribution to gc,s in Eq. (8.1.5) with a factor of cos(2 q ) when the symmetry is regarded. The g ( z ) field for such a coil can be seen in figure 8.3. 93 Figure 8.3. The g(z) field from one quadrupolar coil with end points at z 0.5 and radius of 0.4, where the contributions from the different parts of the coil are shown. These coils are good for optimization since the contribution to the g ( z ) function is well localized in z and since they only contributes to g ( z ) . Compared to the baseball coil, they however waste current since baseball coils contribute to B ( z ) as well. All coil types described so far have only contributed to one of the functions g ( z ) or B ( z ) . This is beneficial for the optimization process since it can be divided into two separate optimization problems, one for B ( z ) and one for g ( z ). However, this creates two layers of coils, and the size and cost of the coils will be unneccesarily high. Also, the chain of quadrupolar coils has some drawbacks. They are interconnected in a rather complicated way and they have sharp corners that must be rounded off significantly. Even if they are rounded off, the curvature of the conductors will be quite high. This is a bad trait for superconducting coils since the strain in the condutors will get high. To deal with these drawbacks, a new type of coil was invented. We called it fishbone coil (not connected to fishbones in tokamaks) due to its resemblence of a fish skeleton when several coils are put together. This type of coil has a 3D-geometry and give contributions to both the g ( z ) and B ( z ) fields. A fishbone coil is shown in 3D in figure 8.4. Figure 8.4. A fishbone coil shown from two different angles. The coil type is a 3D coil with quadrupolar symmetry that give field contributions to both g(z) and B( z ) . 94 The coil geometry is best described by looking at the circumferential surface that the coil is oriented on. The parameterized curve that describes a current thread in the coil is at a constant distance from the z axis. Thus, the curve is well described using cylindrical coordinates (r , , z ) with r kept constant for each filament. The coil layout on that circumferential surface is shown in figure 8.5. Figure 8.5. The fishbone coil layout on the circumferential surface that wraps the fission mantle. Only half of the circumference is shown. Half the coil is shown, and the curve consists of straight lines and circle segments in that plane. Such a curve can be parameterized using H, L and a from figure 8.5, where a is the radius of the circle segments in that plane and a negative L wraps the coil so that a positive coil contribution to B ( z ) gives a negative coil contribution to g ( z ) . From the symmetry, we get H r / 2 since H is one quarter of the circumference. To parameterize the curve, it is convenient to express the distances c and d in figure 8.5 in L, H and a and the angle in d and a. Within reasonable values of H and a ( H 2a ), d aH 2 aL2 a2 H 2 (H 2 4a 2 H2 L2 L2 4a L ) 4a L 2a 2 L , (8.1.8) 95 a2 c (a d )2 (8.1.9) a d . a (8.1.10) and cos 1 Each coil is divided into (m m) current filaments. To parameterize the filament curve it is divided into 8 subsections, 4 straight lines and 4 circle segments (on the flattened out circumferential surface). For two filaments that have different z positions the radii of the circle segments differ (see figure 8.5). Also, for all filaments except those that are in the coil midline in z, the radii of the circle segments varies along the curve where 2 circle segments have radii a1 and 2 circle segments have radii a2 (see figure 8.5). In the parameterization, n 0,1, 2,3 where each number represents one quadrant of the coil and to simplify we define 1 for even n n (8.1.11) 1 for odd n and L . L L (8.1.12) The filament circle segments are parameterized with the curve parameter t as aeff a z, n x(t ) r cos z (t ) n (t ) n (t ) , a L eff aeff 2 r y (t ) r sin 1 cos sin t tmax / 2 tmax / 2 (t ) , t t tmax / 2 tmax / 2 , (8.1.13 - 8.1.14) 0, tmax , (8.1.15 - 8.1.17) L 2 L z z0 (8.1.18) and the filament line segments are parameterized as z sin , (t ) n 2 1 c r x(t ) r cos z (t ) 96 n zl t d H tmax (t ) , L z cos , L 2 2c n y (t ) r sin t tmax L (8.1.19 - 8.1.20) L 2d , (t ) , (8.1.21) t L zl 0, tmax , (8.1.22 - 8.1.24) z0 (8.1.25) where z a represents the filament displacement in z from the coil central filament, z0 is the coil center z position and r is the radial distance to the filament from the z axis. An analytic expression for the magnetic field components for such a coil has not been found. However, it is straight forward to calculate them numerically using the parameterization above. The contribution to B( z ) is roughly comparable to that for the circular coil, at least when the L parameter is small. The contribution to the g ( z ) function is localized in roughly the same way as the contribution for the quadrupolar coil shown in figure 8.3. In figure 8.6, the contributions to B ( z ) and g ( z ) for a fishbone coil is shown. Figure 8.6. The contribution to the functions B( z ) (a) and g ( z ) (b) for one fishbone coil with parameters a 1 m, r 2.3 m, z 0 m, L 2 m and I 100 kA. The field is calculated with a single filament at the center of the coil. There are many advantages with these coils compared to other coil combinations: 1. There will be only one layer of coils, which reduces the size of the coil system significantly. Also, this most likely reduces the maximum magnetic field in the coils. The cost of the coils is also likely to be reduced, although the more complicated 3D geometry may add to cost. However, 3D coils with more complicated geometry have been manufactured for the Wendelstein 7-X stellarator [106], and 3D coil manufacturing is here considered to be a known technology. 2. Compared to baseball coils, these coils have softer curvature. Also, they can be inserted into each other like a pile of drinking glasses, which allows for more freedom during the design of a coil system. Equally sized baseball coils cannot overlap. In the case of the SFLM Hybrid, the ability to overlap is crucial. 3. The coils give contributions to both B ( z ) and g ( z ), and thus the optimization problem is not separated as it was for earlier coil sets. 97 However, it is almost separated, since the current roughly controls B ( z ) and the L parameter can afterwards be used to control g ( z ) when the current is determined. Changing L only has a very slight effect on B ( z ) . 4. The coils are separate, and not connected to each other. This should simplify engineering in the whole device. 8.1.5. Results from Paper IX This paper presents the first attempt in coil design, where the possibilities of making coil systems for the SFLM vacuum field were explored. The main result of this paper is the knowledge gained about the restrictions on the fields imposed by the coils. Effort was in the beginning of this work put into creating a coil set for the SFLM field up to a mirror ratio of four. It turned out to be hard (or not even possible with some geometrical constraints) to create the sharp gradients in z direction near the mirror ends in the axisymmetric field (the B ( z ) function) and especially in the quadrupolar field (the g(z) function). The difficulty to create such gradients is caused by the minimum allowed radial distance to the coil system. Also, when trying to produce those gradients, the field outside the planned confinement region was not regarded. This was a mistake, which was not yet realized in Paper IX. Since the gradients dB / dz and dg / dz are so strong, the field components (the B ( z ) and the g(z) functions) form a huge overshoot outside the SFLM region. Thus, the mirror ratio will be much larger than planned, and the field will be grossly unstable to flute modes or have extremely elliptical flux tubes. The actual results from these coils are rather uninteresting, and will not be repeated here. 8.1.6. Results from Paper X At the time when this paper was produced, it became obvious that another field than the SFLM field had to end the mirror, concatenated with the SFLM field at some points before the mirror ends. The requirements selected for such a field was that the whole field should be stable to flutes, that the flux tube ellipticity should be minimized and that too sharp field gradients in B ( z ) and g(z) should be avoided. It is however not obvious that such a concatenated field would be optimal according to those critera, i.e. it is not obvious that the best field for these criteria would partly consist of an SFLM region. Therefore the problem of designing the whole field was addressed. The field functions B ( z ) and g ( z ) was modelled using a spline representation, and an optimization was made that targeted flute stability (not taking the pressure distribution into account), low ellipticity and low field gradients. A coil set was then created for this field. The coil set was 98 similar to the coils in Paper I (see below), and reproduced the field with sufficient accuracy. These results are now obsolete, since the solution is superseeded by the results in Paper I (which also is obsolete). 8.1.7. Results from Paper I In Paper I, the average minimum-B criterion was replaced by the flute stability criterion given in Eq (3.3.8), which takes a varying pressure profile into account. The SFLM field was used concatenated with an ending field at z 8.75 . This ending field was finally modelled with splines and set up by hand to avoid ripple, when some optimized solutions had been observed. A margin was requested for the flute stability criterion to assure stability also when the pressure profile would deviate somewhat from the pressure profile selected for the computations. The resulting field is flute stable with some margin, and is probably stable for all reasonable pressure profiles in the low limit. The flux tube ellipticity of 20 is fairly low, and considered low enough. The gradients of the field are acceptable. An advantage of using the SFLM field at the center is also that there will be almost no neoclassical transport from particles that stays within z 8.75 due to the omnigenuity property of this field, and the overall neoclassical radial transport will probably be quite low in the low limit. The field characteristics are shown in figure 8.7, where the midplane B is 2 T. 99 Figure 8.7. The selected field for the fusion-fission device. The figures show B in (a), g in (b), W1 for both a constant pressure profile and a sloshing ion distribution in (c), the flux tube ellipticity in (d) and the representative sloshing ion distribution in (e). A coil set was then optimized for this field. In this case, a current density of 1.5 kA/cm2 was used to dimension the coils. The resulting coil system reproduces the magnetic field with satisfying accuracy for the SFLM Hybrid. The field errors for the circular coils are shown in figure 8.8 and the field errors for the quadrupolar field are shown in figure 8.9. In the confinement region, the field errors are about 0.2% in B in the SFLM region. In the spline region the errors in B are up to 1.5%, but this is expected to be of less importance since it is not a ripple in the ordinary sense but rather a change of the profile shape. For the g function, the relative errors are about 1-1.5% with a maximum of up to 10% in g at the end of the confinement region, 100 cutting off the top of the pointed peak. In the expander region the errors are larger, but this is expected to be of less importance. The coil set is visualized in 3D in figure 8.10 and figure 8.11, where figure 8.10 shows the mirror machine with the entire coil set and figure 8.11 shows the mirror machine with the quadrupolar coils only. The axisymmetric coils are listed in table 8.1 and the quadrupolar coils are listed in table 8.2. Figure 8.8. The relative error in the B( z ) function (a, b) and the B( z ) function (c). Figure 8.9. The relative error in the g(z) function (a,b) and the g(z) function (c). The reason for the sharp relative error peak at z = 13m in (b) is that g(z) changes sign. 101 Table 8.1. The circular coils on the positive z side for Paper I, defined by inner radius, cross section center z coordinate, cross section width/height and current. Inner radius (m) 2.45 2.45 2.45 2.50 2.50 2.57 2.66 2.89 3.09 3.09 3.09 2.87 2.77 2.55 z (m) 0.900 1.833 2.701 3.604 4.769 6.497 7.391 9.381 10.441 11.470 12.650 14.000 17.289 18.125 Coil width (m) 0.473 0.184 0.393 0.323 0.445 0.513 0.286 0.209 0.750 1.023 1.258 1.384 0.241 0.789 Current (kA) 2335 352 1609 1086 2061 2740 852 456 5852 10905 16497 19956 -606 -6482 Table 8.2. The quadrupolar coils on the positive z side for Paper I, defined by end z coordinate, cross section width/height of the straight bars, cross section width/height of the quarter-circle segments, current in the straight bars and current in the quartercircle segments. The inner radius for all segments is 2.1 m. The array of coils is cut in two at the magnetic expander, indicated by zero current and zero width for the straight segments. End z (m) 2.000 3.781 5.416 7.000 8.700 9.602 10.957 12.500 13.800 16.250 16.900 17.600 19.375 21.005 102 Width 0.315 0.332 0.365 0.420 0.519 0.684 0.972 0.978 0.392 0.753 0.666 0.581 0 0.232 Width c. (m) 0.074 0.108 0.147 0.216 0.314 0.488 0.074 0.633 0.600 0.248 0.230 0.411 0.164 0.164 Current (kA) 1032 1147 1390 1840 2809 4869 9840 9954 1597 -5905 -4617 -3519 0 559 Current c. (kA) 58 121 225 484 1030 2485 57 -4178 -3751 644 549 1760 280 -280 Figure 8.10. The mirror machine with the entire coil set. Figure 8.11. The mirror machine with the quadrupolar coils, where the axisymmetric coils have been removed. 103 8.1.8. Results from Paper II (and XI) In Paper II, fishbone coils have been used to create the magnetic field. The magnetic field has the same geometry as in Paper I, but is downscaled to 1.25 T at the midplane. A number of properties of the magnetic field that would be produced by the coil set are shown in figure 8.12. The deviation from the ideal field (see figure 8.7) in the confinement region is shown in figure 8.12 (a) for B ( z ) and (b) for g ( z ). The maximum deviation in B ( z ) is about 2.5 %, which partly arises from a smoothening of a somewhat poor concatenation point between the SFLM field and the ending field at z 8.75 m. For g ( z ), the typical deviation is below 1 % except near the mirror end where the quadrupolar field deliberately has been made weaker to lower the flux tube ellipticity. This does not have a significant effect on the flute stability function W1, since the plasma pressure is low in this region. Figure 8.12 (c) shows the B ( z ) function and (d) shows g ( z ). In the recirculation region and the magnetic expander, the ideal field profile is only very roughly followed. The field that would be produced by the coils has however similar properties in this region. In figure 8.12 (e), the pressure weighted stability function W1 in the confinement region is shown and compared with that of the ideal field. There is a stability margin to flutes in the low limit, since W1 ( z 12.5) 0 . In figure 8.12 (f), the flux tube ellipticity is shown. The maximum ellipticity is about 19.4, which is somewhat lower than in Paper I since g ( z ) has been slightly modified. The recirculation region reduces the ellipticity to about 1 at the magnetic expanders and makes the plasma receiving “divertor plates” circular. With a midplane plasma edge radius of a 40 cm, the outermost plasma edge rvc ,min ( z ) is calculated for the coil set to first order. The result is shown in figure 8.12 (g), and a magnification of the mirror end region for rvc ,min ( z ) is shown in figure 8.12 (h). At the mirror end ( z 12.5 ), the outermost plasma edge radius is 87 cm. At about z 13.5 in the recirculation region a maximum of about 93 cm is reached, which illustrates the need to expand the vacuum chamber radius to 1 m beyond the mirror ends. At the end of the magnetic expander, rvc ,min (18.75) 4 m. The resulting coil set consists of 30 coils. Since the coil set is symmetric, only the 15 coils on the positive z side needs to be specified. Those coils are described in table 8.3. The C-14 coils are the recirculation coils and have a negative L which gives a wrapped coil that produces a negative contribution to g ( z ) for a positive contribution in B ( z ) . The C-15 coils are the cusp coils which have a negative current that gives a negative contribution to B ( z ) . The C-15 coils are circular ( L 0 ). The coils do not intersect other coils or other parts of the device, at least not significantly (only possibly very slightly in the structure material, which can be handled in a detailed design). 104 Figure 8.12. The components and some properties of the magnetic field that would be generated by the coil set. The figures show the deviation from the ideal field of B( z ) in the confinement region in (a) and g ( z ) in (b), B( z ) in (c), g ( z ) in (d), the W1 ( z ) stability function in the confinement region in (e), the flux tube ellipticity in (f) and the outermost plasma edge radius rvc ,min ( z ) in (g) and (h). 105 Table 8.3. The 3D coil parameters on the z > 0 side of the midplane. Coil name C-01 C-02 C-03 C-04 C-05 C-06 C-07 C-08 C-09 C-10 C-11 C-12 C-13 C-14 (recirculation) C-15 (cusp) z(m) 0.4 1.25 2 2.75 3.5 4.25 5 5.75 6.5 7.7 9.5 11 12.45 15.05 17.4 I(kA) 787 700 700 710 762 720 680 750 1100 1640 2750 5600 11150 11200 -5070 L(m) 0.915 1.085 0.905 1.05 1.035 1.06 1.20 1.68 0.88 2 2.33 2.32 0.71 -2.98 0 r(m) 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 3 a(m) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Coil width(m) 0.209 0.197 0.197 0.198 0.205 0.200 0.194 0.204 0.247 0.301 0.390 0.557 0.786 0.788 0.530 The resulting coil system is also shown in 3D in figure 8.13, where only the coils are shown in (a) and the vacuum chamber, fission mantle and the coolant influx/outflux has been added in (b). Figure 8.13. The resulting coil system in 3D, where the coils are shown in (a) and the coils with the vacuum chamber, fission mantle and coolant influx/outflux are shown in (b). The significant size reduction of this coil system compared to the coil system in Paper I is interesting to note, and is shown in figure 8.14. It is due to the following three causes: 106 1. The overall magnetic field has been lowered. 2. The goal is to use NbTi coils, at least for most of the coils, which in combination with the weaker magnetic field implies that a higher current density of 2.6 kA/cm2 can be used instead of 1.5 kA/cm2. 3. The fishbone coils produce both the B ( z ) and the g ( z ) components, which makes the overall coil size smaller and allows the coils to be organized in one single layer. Figure 8.14. A cross sectional comparison between the size of the old coil system and the new coil system, where the cross sectional area of the largest coil of the new coil system is about 27% of the combined cross sectional area (quadrupolar coil + circular coil) of the largest old coils. 8.1.9. Results from Paper III In Paper III, the radial dependence of the vacuum field flux tube ellipticity (or excentricity) is examined. To first order in the long-thin approximation the flux tube ellipticity is independent of r0 . An expression to third order is derived in Paper III, and is given in section 4.3. These solutions are compared to a more or less exact solution where the field lines are followed numerically from the midplane using the fourth order Runge-Kutta method. Two devices are examined, the SFLM Hybrid and a hypothetical device where the g ( z ) function is much more localized. The results for the SFLM Hybrid can be seen in figure 8.15, where the traced outermost plasma field line is shown in (a, b) along with the vacuum chamber wall, and the ellipticity is shown in (c, d). In general, the 3rd order solution mitigates most of the errors from the first order approximation. For the SFLM Hybrid, the outermost field line is 3.5 cm further out in radius than would have been 107 expected from the first order approximation (97 cm compared to 93.5 cm). The corresponding values for the ellipticity are 19.9 instead of 19.4. Figure 8.15. The vacuum magnetic field properties for the SFLM Hybrid. In (a, b) the radial dependence in z of the field line that intersects the midplane at (r 0.4, 0) is shown and compared with the vacuum chamber first wall. In (c, d) the ellipticity for r0 40 cm is shown. In (a–d), a first order paraxial approximation, a third order paraxial approximation and an almost exact numerical Runge-Kutta solution is compared. The radial dependence of the maximum ellipticity is also examined for the SFLM Hybrid, and is shown in figure 8.16. As expected, the ellipticity increases with radius for the examined interval, and the 3rd order solution seems to be fairly accurate up to about r0 0.4 m. We suspected that the ellipticity dependence on radius would go up if the g ( z ) function was more localized in z, like for example in a tandem mirror. Therefore, such a case was examined. For this case, much to our surprize, the ellipticity dependence on radius was weaker than in the SFLM Hybrid. 108 Figure 8.16. The maximum ellipticity as a function of the radial Clebsch coordinate r0 in the SFLM Hybrid. A first order paraxial approximation, a third order paraxial approximation and a more or less exact numerical Runge-Kutta solution is compared. 8.1.10. Results from Paper V In this paper, the finite corrections to the magnetic field have been calculated for a prespecified sloshing ion distribution function in the SFLM Hybrid. The constant pressure profiles at the midplane have been determined. It was found that they had a square-like octupolar distortion near the plasma edge if no specific boundary conditions were applied for the pressure, and that they were almost circular within most part of the plasma region. These contours are shown in figure 8.17 for different values, and the shape of the contours is not sensitive to although a slight increase in the octupolar distortion can be observed with increasing . Figure 8.17. Contours of constant pressure at the midplane cross section for different ß values. 109 case, and The flux tube ellipticity has also been calculated for the finite the results are shown in figure 8.18. It was found that the ellipticity was increased by the plasma currents. However, the increase in ellipticity was more pronounced near the z axis than at the plasma edge, and the total ellipticity decreases with radius in the plasma region. Also, although the plasma presence increases ellipticity, the outermost field line radius is even slightly decreased with finite , which is a beneficial result. Figure 8.18. Flux tube ellipticities for different ß values and radii in (a-d), and the corresponding outermost plasma radius in (e-f). The radial dependence of the ellipticity is shown in (g). 110 8.2. Radial transport and radial invariant In Paper VII, a radial invariant that is valid for fields with small drifts (typically vacuum fields) has been found for particle orbits in quadrupolar mirrors that exhibit a certain symmetry. The geodesic curvature in quadrupolar mirrors give rise to gyro center radial drifts. It is therefore here required that the geodesic curvature along a field line is an odd function of z, so that the radial drifts on each side of the midplane cancel. This is fulfilled if both functions B ( z ) and g ( z ) are even functions. If this is not the case, a collisionless radial transport will be present, which is shown in Paper IV. 8.2.1. Results from Paper VII The main result in this paper is the derived radial invariant I r , which is valid in fields where the drifts are small (typically vacuum fields). To sketch the derivation (see Paper VII for details), we first define the field line equations to first order as xB ( z ) 1 1 ( z ) x0 (8.2.1) yB ( z ) 1 2 ( z ) y0 . (8.2.2) and The functions B ( z ) and g ( z ) are related to B( z ) 1 B0 z ( ) 1 1 1 ( z ) and 2 2 ( z ) by (8.2.3) ( z) and g ( z) 1 1 ( z) 1 1 ( z) 2 B0 1 ( z) 1 (z) 1 2 ( z) 2 ( z) (8.2.4) where prime denotes differentiation with respect to z. We now specialize the calculation to the case where B ( z ) and g ( z ) are even functions and B ( z ) has a minimum at z 0. Then 1 ( z ) 2 ( z ) ( z ) and the function ( z ) can be divided into its even and odd components e ( z ) and o ( z ) . We write ( z) 1 where we choose e (0) center can be written o e ( z) o ( z) (8.2.5) (0) 0 . The equations of motion for the gyro dx0 dt y0 ( s ), (8.2.6) 111 dy0 dt x0 ( s ), (8.2.7) M ds 2 dt B( s ) 2 (8.2.8) where 2 B m 1 (8.2.9) 0 can also be divided and bars denote guiding center values. The function into even and odd components, yielding 2 B e m 1 e e (8.2.10) o o 0 and 2 o B o e m 1 e o . (8.2.11) 0 With cylindrical Clebsch coordinates, the equations of motion become dr0 dt r0 o ( s )sin(2 0 ) (8.2.12) and d 0 dt e (s ) o ( s ) cos(2 0 ). (8.2.13) With these equations and the assumption that e ( s ) and o ( s ) are small (which imply small drifts), the radial invariant can be defined as I rN (x, v ) r0 2 x0 y0 r0 N hn ( , ) cos n n cos /4 n 1 (8.2.14) where hn ( , ) 2 n /4 d 0 o s ( ) sin n /4 , (8.2.15) is a bounce time variable which satisfies ds / d s, is the time for one bounce back and forth the mirror and N is the number of Fourier components to include in the invariant. Eqs. (8.2.11 - 8.2.12) give a condition for locally omnigenius equilibria ( r0 const ), yielding 112 o e 1 e o 0 (8.2.16) which can be satisfied for any choice of B ( z ) . To illustrate the invariance of the radial invariant, it is shown in figure 8.19 for different N. As can be seen, as N become large, the invariant is almost constant. Figure 8.19. Variations of Nth order radial invariants versus the arc length s(t) over a longitudinal bounce for N = 0, 1, 2 and 500. The curves are for a long-thin system with some geodesic curvature. The plot for the guiding center radial coordinate r0 (t ) reveals a 2.5 cm banana width. The variations of the successive approximations for the radial invariant decreases with N, approaching diminishingly small values for large N. An observation in Paper VII is also that the parallel invariant J to leading and . Therefore, it is not sufficient to use J to order is a function of model a radial pressure profile. For this, the radial invariant can be used. 8.2.2. Results from Paper IV The purpose of Paper IV was to point out that quadrupolar mirror machines must be symmetric in the sense that B( z ) and g ( z ) are even functions for a radial invariant to exist. This is in some sense known for a restricted group of mirror researchers, see for example Ref. [71] (Appendix D), but perhaps not widely known. If this symmetry is not present, there will be a collisionless radial transport (or leakage). This can intuitively be seen from the geodesic field line curvature. If there is a geodesic field line curvature, indicating that the magnetic field is not omnigenius in the local sense, there will be radial drifts. If the symmetry is fulfilled, the radial drift on one side of the mirror will be cancelled by an opposite radial drift on the other side of the mirror, if the azimuthal drift in one bounce is small. To calculate the orbits in this paper, two methods are applied. The main method is to just follow the particle with a fourth order Runge-Kutta method and calculate the Lorentz force. The magnetic field was calculated in a grid of (100 100 300) points and quadratic interpolation was used to calculate 113 B at a specific point. The method turned out to be very accurate but fairly slow. The second method was to use the equations of motion from Paper VII, and this was used to check the first method. Due to the sensitivity at the turning points, this method was actually (to our surprise) not much faster than the first one if a good accuracy was required and equidistant steps were used. A simple coil configuration with 8 fishbone coils was varied to create fields with fluctuating field line curvatures. Four different configurations were tested: 1. A fully asymmetric mirror where one side was axisymmetric and the other side quadrupolar. 2. A symmetric quadrupolar mirror. 3. A mirror with about 15% asymmetry in g ( z ) . 4. A mirror with about 1.5% asymmetry in g ( z ) . For the asymmetric cases, there was a slight asymmetry in B( z ) as well. The results for particle orbits can be seen in figure 8.20. The particle orbits can be divided into quadrant locked orbits, that tend to stay in their quadrant, and encirculating orbits, that encircle the z axis. This is determined by the azimuthal drift. The encirculating orbits in asymmetric fields will be oscillatory in r0 . For quadrant locked particles in asymmetric fields, the radial drift will be either inwards or outwards dependent on which quadrant they are in. For quadrant locked particles, there is a risk for a net radial drift and thereby a collisionless radial transport. 114 Figure 8.20. In (a), drift trajectories in the non-symmetric mirror field for 16 deuterium ions that are launched at the midplane with different initial positions in the x0 y0 plane. The small circles mark the initial positions for each ion in (a-c). The ions are tracked for about 100 bounces back and forth the mirror in (a) and the two particles that are still confined after 100 bounces are traced for up to 400 bounces in (b). As can be seen, all of these particles are lost in less than 400 bounces for the non-symmetric mirror. In (c), an ion with identical velocity and position as in (a) was launched in the symmetric mirror and traced for about 200 bounces. As seen in the magnification in (d), the trajectory is almost closed and the radial drift after one lap around the z axis is only about 0-0.1 mm. In (e) and (f), the orbit is shown for the error case where the coil parameter L is lowered by 20% on one side of the mirror, and in (g) and (h) the corresponding orbit for the case where L is lowered by 2% on one side is shown. In (e-h), the orbit appears to be drawn with a thick line, which is due to the neoclassical “banana” orbits that the particle gyro center follows. 115 In general, one can see that for the fully assymetric mirror, the radial drifts are totally unacceptable. For the 15% asymmetry, they seem to be too large (unless a fast azimuthal particle drift is introduced, by for example finite or a strong electric field). For the 1.5 % asymmetry case, the drift is small and most likely acceptable. Mirrors with asymmetry smaller than this will most likely not suffer from problems with radial transport of this type. For the symmetric case, the radial drift is very small or totally abscent as expected. The calculations in this paper have been made on a vacuum field and the electric field has been omitted. With a non-neglible finite or a strong radial electric field, they would no longer be valid, since both these effects will give a fast azimuthal drift. A radial electric field is always present (if the electron temperature is not zero) due to the ambipolar potential. This electric field can be further increased with potential plates at the end plates. 8.3. Discussion 8.3.1. Discussion on coil accuracy requirements When attempting to find an accurate solution for a coil design, the question of how accurate the magnetic field needs to be arises. For a tokamak, the ripple caused by the discrete toroidal coils give rise to increased neoclassical-type radial ion losses. These are claimed to be (at that time) 1-2 orders of magnitude greater than the ordinary neoclassical losses in tokamak fusion D-T plasmas in Ref. [107]. However, no such effects are expected in a mirror. A heavy ripple of more than 1 % (top to bottom) in the axisymmetric field is claimed to give rise to ballooning modes in tandem mirrors in Ref. [108]. Later theoretical achievements by Newins and Pearlstein [95], suggest that the FLR effects are so strong that theoretical MHD ballooning instabilities would not be important for tandem mirror experiments. In any case, ballooning modes are expected to be less important for a single cell minimum-B mirror of the SFLM Hybrid type than for a tandem mirror, since the regions of bad curvature are smaller. Also, the ripple can be substantially reduced by using ferromagnetic inserts if such a measure would be required [108]. Evidence of any serious problems with a ripple of the size presented in this paper for a mirror machine has not been found. Concerning the flute stability, the impact of the ripple is small and the resulting field flute stability is checked in Paper II. Ripple effects on the flux tube ellipticity are negligible. The ripple will cause particle drifts, but they will be of oscillatory nature as long as the field is symmetric in the sense of Papers IV, VII, and the banana widths from the ripple is expected to be small. 116 8.3.2. Discussion on coil calculation limitations The coil calculations have been made using a rough scaling law for the coil dimensions. This is not fully accurate. To produce real superconducting coils, a considerably deeper analysis must be done. The coils must be properly dimensioned to hold (both the whole coils and the internal structure) for the very strong magnetic forces. This requires numerical modelling of the coils. Also, the magnetic field must be kept below the critical magnetic field for the superconductors everywhere within the superconducting coils. This has not been checked. A detailed check would require that the internal structure of the coils must be decided. Implications of this are that there is an uncertainty of the position of the filamentary line currents due to the uncertainty in coil sizes. The consequence will be a need for recalculation of the currents, which is straightforward. There is also an uncertainty if the geometric constraint that coils shall not intersect will be fulfilled when the structure material is properly dimensioned. This can be handled by for example changing the coil cross section shapes or moving some coils. Also, the proper material of each coil has to be decided from the maximum magnetic field strength. The aim in this study was to show that it seems realistic that a coil set can be created which generates the magnetic field with satisfying accuracy without violating the geometric constraints. If problems should arise, there are ways to handle them. A somewhat lower mirror ratio or alternatively a somewhat more long-thin configuration would allow for coils for the mirror field to be constructed. In summary, coil construction for the fusion-fission mirror system seems realistic. 8.3.3. Discussion on finite ß issues The constant pressure contours of the midplane become square-shaped with finite ß in the SFLM Hybrid. This is in contrast to the result for tandem mirrors calculated by Pearlstein et al. [72], where they found a diamondshaped distortion. Their distortion was however caused by the parallel current. In the case with the SFLM Hybrid, the parallel current is small and does not have a noticeable impact on the flux surface shapes. Other effects, such as a small octupolar component in the vacuum magnetic field that come from the coil system, contribute in the SFLM Hybrid case. 8.3.4. Discussion on radial invariant, E and low ß limit The derivation of the radial invariant is only valid when the azimuthal drift is so small that the gyro center roughly stays at the same 0 value during one bounce back and forth the mirror. With a finite , or strong electric field, 117 this is no longer the case. For finite , the drift from the gradient in B is typically much larger than the curvature drift. The magnetic drift is BB B 3 q B vd ,m mv 2 B q B2 (8.3.1) /(2 0.4) which is If B is assumed to be perpendicular to B, B . Preliminary numerical calculations for a 7 keV considerably larger than deuterium ion in the SFLM Hybrid with 0 0.4 show that this drift represents a change in 0 of the particles position in the order of 15° to 60° in one single bounce back and forth the mirror, depending on r0 for the particles position. For such high values, the derivation of the radial invariant is no longer accurate (although there may still exist a radial invariant). The drift is almost proportional to , so typical drift velocities in the SFLM Hybrid can be approximated from this value. For a radial electric field, an approximation can be made. For the simulated particle above, the time for one bounce back and forth the mirror is about 8 10 5 s. If we assume that the ambipolar potential is 5Te (where Te is in eV) at the center and drops to zero just outside the plasma edge, the radial electric field roughly becomes Er 5Te r 5Te 12.5Te 0.4 (8.3.2) for the SFLM Hybrid where it is assumed for simplicity that Er is constant in the plasma region (which is not true). The electric drift is vd , E E B B2 E B 12.5Te 2 6Te (8.3.3) where it is assumed that E B and B 2T. If we assume that Te 500 eV, the drift during one bounce is 6 500 8 10 5 24 cm. At a radius of 10 cm, this represents an angular displacement of about 140° in one single bounce. From this approximation, a reasonable validity limit for the derivation of the radial invariant would be about Te 50 eV in the SFLM Hybrid. The drift from finite and from the electric field are in different directions for ions and will to a large extent cancel if they are in similar magnitude. Then, the validity must be approximated from the sum of these drifts. The E B drift is independent of a particle’s speed if E remains unchanged, and the drift in one bounce will decrease with increasing energy of the particle. For the magnetic drift, the situation is opposite and the drift during one bounce will increase linearly with the velocity of the particle. 118 8.3.5. Comparing axisymmetric and quadrupolar drivers The other mirror-based hybrid systems that have been proposed recently use axisymmetric mirrors as drivers [27][28], see also [109]. There are several advantages with axisymmetric systems: 1. The coil system is considerably simpler. All coils are circular, which also means that the magnetic field can be made higher since the strain in the coils would be smaller. This would also reduce the cost of the coil system. 2. The mirror ratio can be made higher, which is beneficial for axial confinement. 3. The flux tube have a more convenient shape (axisymmetric), which in the SFLM Hybrid would implicate that the plasma radius could be about 80-85 centimeters (since there are no wide quadrupolar fans) and the plasma volume would roughly be quadrupled. This means that the magnetic field or the plasma can be lowered while the neutron production would remain the same, compared to the SFLM case. This would also reduce radial transport, since each particle has a longer radial distance to pass before reaching the limiters where the particles are lost. 4. Axisymmetric devices are locally omnigenius, even to arbitrary , which eliminates all types of neoclassical transport. This may be hard to accomplish with finite in a quadrupolar device. Stabilization of axisymmetric devices is however not as well explored as for quadrupolar devices, and for the time being we avait more results from these experiments. Explicity, there might be problems of getting a high electron temperature since axisymmetric devices are partly stabilized by having plasma in the magnetic expanders. In the SFLM Hybrid, the density in the expander should be as low as possible to increase the electron temperature. The electron temperature is probably the single most important parameter for a mirror fusion driver, since it at least for the low electron temperatures reached so far in mirrors directly controls the fusion Q and thereby the amount of heating required. 119 120 9. Conclusions Conclusions from the work in this doctoral thesis are: 1. A steady-state fusion-fission hybrid reactor with a single cell minimumB mirror as fusion driver has been proposed in Paper VI. The fusion device is based on the SFLM magnetic field. There are currently no major showstoppers known for this project, although a lot of issues have not been examined in detail yet. A concern is the electron temperature, and reactor safety needs to be addressed more in detail, especially the consequences of a core meltdown. 2. The fission to fusion power multiplication in a mirror hybrid can be made very high, about 150 with an assumed keff of 0.97 since the geometry can be arranged so that almost all fusion neutrons enter the fission mantle. The power multiplication for a mirror hybrid could be substantially higher than what can be achieved for tokamak hybrids. A mirror machine with a power multiplication of 150 have a strongly relaxed requirement for the plasma confinement and even a thermal fusion Q = 0.15 is expected to be good enough for energy production compared to Q = 15 for a pure fusion reactor. 3. The SFLM magnetic field cannot constitute the entire mirror field in a mirror machine, and has to be concatenated with some other field before the end of the mirrors. This is necessary, since the z derivative of the field on the z axis dB / d z must be zero at the mirror ends, and dB / d z in SFLM is monotonically increasing with z . Such a concatenated field has been selected in Papers I-II that has margins to flute stability in the low limit, a maximum flux tube ellipticity of about 20 and sufficiently soft gradients for a mirror ratio of four. The concatenation point is at z 0.7c where c is the axial scale length of the device. 4. The further away the coil system is from the plasma, the harder it is to produce large relative gradients ln B / z in the magnetic field. This limits the fields that could be realized for a fusion-fission reactor, since the fission mantle including shielding is in the order of 1.2 m thick (in principle the fusion neutron shields in a fusion reactor would be of almost the same thickness). This property also sets a minimum length of the machine for a specific magnetic field. 5. A new type of coil has been invented, which we call the fishbone coil. They have practical geometrical shapes and can be inserted into each 121 6. 7. 8. 9. 10. 11. 12. 13. 14. 122 other like a pile of glasses which offers great flexibility in coil design. They contribute to both the B ( z ) and g ( z ) functions, which means that only one layer of coils is required, and the contributions to B ( z ) and g ( z ) can almost independently be controlled by the coil parameters. They also have soft curvature which is beneficial for superconducting coil design. Two different coil systems have been found that reproduces the vacuum magnetic field with satisfying accuracy and within the geometric constraints for the SFLM Hybrid. The preferred version is the version based on the fishbone coils. There is sufficient space available inside the coil system to shield the coils from neutrons. The vacuum field ellipticity increases slightly with radius. For the SFLM Hybrid, the ellipticity increases from 19.4 to 19.9, which increases the outermost field line maximum radius by 3.5 cm. A magnetic expander region with a recirculation region has been added to the mirror machine at each end. By this, several advantages are gained. The heat load from end losses can be distributed over an almost arbitrarily large area, some extra stability to flutes is gained and the electron temperature may rise due to density depletion in the ends. The heat and neutron loads on both the first wall and the “divertor plates” at the magnetic expanders are predicted to be tolerable. This is in contrast to ITER (and supposedly most compact fusion devices), where the divertor plates is only expected to last for less than 3 months of steady-state operation or so. A radial invariant in symmetric quadrupolar mirrors has been identified that can be used to model radial pressure density variations. It has been emphasized that assymetric quadrupolar mirrors have a collisionless radial transport. For mirrors with a pronounced asymmetry, this radial transport will be unacceptably large. If a finite is present in the SFLM Hybrid, the flux tube ellipticity is increased. However, this increase in ellipticity is more pronounced near the z axis, and at the plasma edge the increase in ellipticity is moderate. A beneficial property is that even though the ellipticity is increased at the plasma edge, the outermost field line on a flux surface have a smaller maximum radial extension in the finite case compared to the vacuum field. At the plasma lateral boundary (the plasma edge), the flux surface cross sections have a square-shaped distortion at the midplane with finite . Near the z axis, they are circular. 10. Future studies There are a lot of studies remaining in the SFLM Hybrid project and related research areas, and some of them can be done by the author, others have to be done by people with other competences. There is currently no long-term project plan on which a decision has been made. However, the following tasks could be done (in no specific priority order): 1. Find a proper financiation for the project. This is the single most important point. 2. Specify how the plasma should be fed with deuterium and tritium, and investigate the impact on the electron temperature etc. of such feeding. 3. Predict the electron temperature and the overall fusion Q. 4. Try to determine the proper equilibrium and magnetic field with ICRH heating and optimize the coil system for such a realistic case. Also, here the neutron production should be calculated more accurately and the required 0 to produce the neutrons should be determined. 5. Investigate the need of and maybe design a (hopefully weak) correction coil system that can modify the magnetic field profile. It is not expected that the equilibrium can be determined with such accuracy that the optimal field can be entirely determined in advance. 6. Write an overall project specification that includes all required systems. Such papers can be published in for instance Journal of Fusion Energy. 7. Extend reactor safety studies. 8. Perform burnout studies for the fission core. So far, impact on burnout has only been extrapolated from other systems (FTWR). 9. Examine the impact of the reactor poison Xe-135. This impact is expected to be low for a fast neutron spectrum, but may still be finite. 10. Perform a detailed internal coil modelling, taking strain, cooling and maximum magnetic field into account. 11. Continously evaluate the progress with axisymmetric mirrors. If axisymmetric mirrors would prove to be well working with respect to stability, limits and electron temperature, they would most likely be superior to quadrupolar devices. 12. Analyse ballooning stability properly. 13. Verify that fusion hybrids can be competitive against ADS systems and critical fast fission reactors for burning actinides. 123 14. Examine the particle orbits for symmetric quadrupolar fields with a finite , and investigate the impact of resonant and possibly stochastic neoclassical transport. 15. Design of the magnetic field and overall geometry of the magnetic expanders. 16. If the theoretical results seem promising, plans for building an experiment should be made. 124 11. Summary of papers Paper I Field and coil design for a quadrupolar mirror hybrid reactor This paper describes a version of the magnetic field and coil design for the SFLM Hybrid project, where the midplane magnetic field was 2 T. Magnetic field properties are derived using the paraxial approximation. A magnetic field, based on the SFLM field concatenated with another field to end the mirrors, is constructed, optimized for flute stability (using the flute stability criterion), ellipticity and field smoothness. A coil set is modelled theoretically to produce the constructed magnetic field. The author has made most of the work in this article. The paraxial formulas are derived by O. Ågren. The paper is published in Journal of Fusion Energy 2011. Paper II A Compact Non-Planar Coil Design for the SFLM Hybrid This paper describes the version of the magnetic field and coil design for the SFLM Hybrid project, where the midplane magnetic field was set to 1.25 T. The magnetic field is in other aspects similar to that in Paper I. Magnetic field properties are derived using the paraxial approximation. A new 3D coil type, the fishbone coil, has been designed and the vacuum field can be generated with satisfying accuracy with 28 fishbone coils and 2 circular cusp coils at the magnetic expanders. The fish-bone coil has soft curvature and is believed to be well suited for superconducting coils. The author has made most of the work in this article. The paper is published in Journal of Fusion Energy 2012. Paper III Vacuum Field Ellipticity Dependence on Radius in Quadrupolar Mirror Machines This paper investigates the radial dependence of the vacuum flux tube excentricity (ellipticity). To first order in the long-thin approximation, the ellipticity is independent of radius. A third order expression is derived and is compared with numerical “exact” solutions. For the SFLM hybrid vacuum field, the ellipticity increases slightly with radius and the outermost field line 125 is 3.5 cm further out than expected from the first order calculation. The author has made most of the work in this article. The paper is published in Journal of Fusion Energy 2012. Paper IV Radial Confinement in Non-Symmetric Quadrupolar Mirrors This paper investigates the effects of non-symmetric quadrupolar and axisymmetric fields in quadrupolar mirrors. The aim with the paper is to clearly point out that if there is a non-symmetry of the geodesic curvature with respect to the midplane, particles are not confined. The author has done most of the work in this paper. The paper is published in Journal of Fusion Energy 2012. Paper V Finite ß corrections to the magnetic field in the SFLM Hybrid In this paper the finite effects on the magnetic field is investigated for the SFLM Hybrid. The flux tube cross sections at the midplane have been calculated as well as the modification of the flux tube ellipticity from the finite . The parallel current has also been investigated. The author has done most of the work in this paper. The paper is a manuscript intended for journal publication. Paper VI Studies of a straight field line mirror with emphasis on fusion-fission hybrids This paper is a concept article, proposing a fusion-fission minimum B mirror hybrid reactor using the Straight Field Line Mirror (SFLM) as a fusion driver. In the paper, the strong fusion-fission power amplification is pointed out. The paper reviews the SFLM concept, discusses radio frequency heating, discusses Monte Carlo simulations and discusses results from coil calculations. Also, a scenario for increased electron temperature is outlined and advantages with mirror geometry compared to toroidal geometry are emphasized. The article is written mainly by Olov Ågren with assistance from Vladimir Moiseenko and Klaus Noack. The author has contributed with the coil calculations. Since this is a concept article, the author has also been involved in overall discussions of the project, especially regarding the size of the device. The paper is published in Fusion Science and Technology in 2010. 126 Paper VII Radial Drift Invariant in Long-Thin Mirrors In this paper, gyro center orbits for long-thin quadrupolar mirrors are examined and a radial invariant is derived. It is also shown that the parallel invariant J is insufficient to model radial variations in pressure. The radial invariant can be used for this purpose. Most of the work in this article is made by Prof. Olov Ågren. The author has contributed with numerical calculations of particle orbits. The paper is published in The European Physical Journal D in 2012. Paper VIII Neutronic model of a mirror based fusion-fission hybrid for the incineration of the transuranic elements from spent nuclear fuel and energy amplification This paper presents the first fission mantle design for the SFLM Hybrid project. Monte Carlo simulations suggest that the energy amplification of the fission mantle would be about 150 at BOC. Reactor safety for LOCA and local boiling of coolant is also addressed. The major contribution was done by Klaus Noack (in particular all Monte Carlo simulations) with minor contributions and proposals from Olov Ågren and Vladimir Moiseenko (on some overall parameters and plasma questions). The author’s contribution to this paper is small. The author has been involved in the axial sizing of the device and has been working with the text. The paper has been published in Annals of Nuclear Energy in 2010. Paper IX Coil design for the Straight Field Line Mirror This paper presents early work in coil design, and is superseded by Paper II. Two applications of mirror machines are addressed, a small material science experiment and a short-fat fusion-fission device for transmutation. For the material science experiment, a simple coil set is found, although with serious drawbacks not realized when the article was written. For the transmutation device, a coil set could not be found. The author has done most of the work in this paper. Some of the formulas are derived by O. Ågren. The paper was presented by the author as a poster presentation at the conference Open Systems 2008 in Daejon, South Korea. The paper is published in the (non-peer-reviewed) conference proceedings in Fusion Science and Technology 2009. 127 Paper X Theoretical field and coil design for a single cell minimum-B mirror hybrid reactor This paper describes a magnetic field and coil design, which is now obsolete and superseded by the magnetic field and coil set given in Paper II. The magnetic field is optimized for flute stability (using the average minimum-B criterion), ellipticity and field smoothness using a spline representation for the magnetic field components. A coil set is then produced theoretically that reproduces that field. The author has made most of the work in this article. The paraxial formulas are derived by O. Ågren. The paper was presented by the author as a poster presentation at the conference Open Systems 2010 in Novosibirsk, Russia. The conference papers were peer-reviewed and this paper is published in the conference proceedings in Fusion Science and Technology 2011. Paper XI Coil System for a Mirror-Based Hybrid Reactor In this paper, the coil sets in Paper I and Paper II are compared and discussed. The author has made most of the work in this article. The paper was presented by the author as a poster presentation at the FUNFI conference 2011 in Varenna, Italy. The conference paper was not peer-reviewed and this paper is published in the conference proceedings in AIP conference proceedings in 2012. Paper XII Safety and Power Multiplication Aspects of Mirror Fusion-Fission Hybrids In this paper, the fission mantle design is modified in two ways: shielding for the coils is added and small annuli (control rods) to be able to keep keff at constant level are added. Also, safety cases are examined for a vertical orientation of the device and a safety case with water cooling is examined. Klaus Noack has made most of the work in this article. The author has assisted in specifying coil data and material. The author’s contribution to this paper is small. The paper was presented by Klaus Noack at the FUNFI conference 2011 in Varenna, Italy. The conference paper was not peer-reviewed and this paper is published in the conference proceedings in AIP conference proceedings in 2012. 128 12. Sammanfattning För att förhindra att en ny energikris uppstår och för att minska skadeverkningarna på vår miljö, behöver de fossila bränslena – som idag står för mer än 80 % av världens energiproduktion – ersättas med alternativ. De alternativ som finns tillgängliga idag är förnybara energikällor och kärnkraft. Kärnkraften ger en stabil basproduktion av el, vilken har stora möjligheter att stå för en större del av den framtida elproduktionen än vad som sker idag. Kärnkraften är dock – såsom andra energislag – behäftad med ett antal problem av mer eller mindre allvarlig karaktär. Allvarligast är troligen sambandet med kärnvapenspridning samt kanske risken för reaktorolyckor. Dock utgör även förvaringen av uttjänt kärnbränsle ett problem. Det finns ett antal radioaktiva ämnen i kärnbränslet som måste slutförvaras på ett säkert sätt, och en förvaringstid på 100 000 år eller mer krävs för att radioaktiviteten i bränslet skall sjunka till sådana nivåer att det kan betraktas som säkert att lämna det i berggrunden. De ämnen som ger allvarligast konsekvenser för levande varelser efter några hundra års förvaring, transuranerna, går dock att bränna bort genom s.k. transmutation, vilket för transuraner innebär fission. För att transmutation skall fungera krävs dock en annan typ av reaktor än dagens lättvattenreaktorer. För att bränna americium krävs med stor sannolikhet s.k. drivna system vilka måste drivas av en yttre neutronkälla. Möjliga sådana neutronkällor är partikelacceleratorer och fusionsreaktorer. Med drivna system finns också möjligheter att skapa fissilt kärnbränsle ur fertila isotoper, s.k. breeding. Genom sådan teknik kan bränsletillgången för kärnkraft bli mycket stor, och räcka för världsproduktion av elektricitet i 10000-tals år genom att breeda U-238 och thorium. Även s.k. snabba reaktorer eller breederreaktorer är tänkbara för detta ändamål, och det är möjligt att drivna system får svårt att konkurrera med dessa för breeding. Fusionskraften har länge setts som ett möjligt alternativ för framtidens energiproduktion. Tillgångarna på energiråvarorna är mycket stora. Dock har problemet med att få fusionsreaktorer att leverera mer el än de konsumerar visat sig vara mycket svårare än väntat, och det kommer inte att finnas någon fusionskraft kommersiellt tillgänglig på åtminstone 30 år (förmodligen betydligt längre). Det är dock möjligt att fusionsforskningen kan lösa ett annat problem än det i förväg tänkta genom att fungera som neutronkälla till drivna system och därmed möjliggöra transmutation och breeding med goda 129 reaktorsäkerhetsmarginaler. I denna doktorsavhandling presenteras ett koncept på en fusion-fissionsreaktor baserat på en s.k. spegelmaskin. I den rörformade spegelmaskinen innesluts fusionsplasmat m.h.a. den magnetiska spegeleffekten, och neutroner skapas genom fusionsreaktioner. Spegelmaskinen omges av en fissionsmantel vilken har en neutronmultipliceringsfaktor som är mindre än ett, typiskt 0,97. Fusionsneutronerna som skapas åker in i den omgivande fissionsmanteln och genererar en kaskad av fissionsreaktioner. Energimultipliceringen som fås med fissionsmanteln blir i storleksordningen 100-150 ggr. Därför blir energiproduktionskravet för fusionsreaktorn bara 1/100 jämfört med en ren fusionsreaktor. Detta krav kan flera olika typer av fusionsapparater komma att kunna klara av. En av dessa är spegelmaskinen. Projektet SFLM hybrid bedrivs för att teoretiskt påvisa att det verkar vara möjligt att bygga en hybridreaktor baserad på en encellig spegelmaskin. I denna doktorsavhandling har det huvudsakliga arbetet varit att beräkna strömspolar för att skapa magnetfältet för SFLM hybrid. Olika magnetspolsdesigner med supraledande magnetspolar har tagits fram. En ny typ av spole har uppfunnits, vilken kallas fishbone coil och är lämplig att använda för att producera magnetfältet till enkelcellsspegelmaskiner av kvadrupoltyp. De beräknade spoluppsättningarna återskapar magnetfältet med tillräckligt god noggrannhet och uppfyller de preliminära utrymmeskrav som angetts. Den senaste designen är med stor sannolikhet möjlig att realisera med befintlig teknik. Det radiella beroendet av ellipticiteten för vakuumfluxrören i spegelfältet har också undersökts, och magnetfältet från plasmat har beräknats. En viktig slutsats som kan dras beträffande magnetfältet från plasmat är att även om plasmaströmmarna ökar ellipticiteten för fluxrören i spegelmaskinen, så kommer fluxrören inte att få större radiell utsträckning. I övrigt har rörelsekonstanter och partikelbanor i spegelmaskiner behandlats, där en radiell invariant har identifierats. 130 13. Acknowledgements First of all, I would like to thank my supervisor Prof. Olov Ågren for letting me do PhD studies as Uppsala University, and for the support and guidance in my work. Without him, this work would not have been possible. He is also greatly acknowledged for his cheerful company during these years, and for sharing his both broad and deep knowledge in physics and other various subjects. I would also like to thank my assistant supervisor Dr. Vladimir Moiseenko for the support, although the physical distance to Ukraine makes the contacts less frequent. Vladimir is also greatly acknowledged for his contributions to the project with his expertise in ICRH heating and fusion in general. Prof. Klaus Noack is also acknowledged. He has a great expertise in fission and hybrids, and the project would be less deep without his contribution. Henryk Anglart is also acknowledged for providing knowledge on liquid metal cooling. Jan Källne is acknowledged for arranging and taking initiative to the successful FUNFI conference, the first fusion-fission conference in a long time in Europe, in Varenna, Italy. I would also like to thank my 3:rd supervisor Ladislav Bardos. The head of the department, Prof. Mats Leijon, is acknowledged for support. The administrative staff at the department is also acknowledged. The Swedish institute is acknowledged for financing the two conference visits to OS-2008 and OS-2010, and for financing my assistant supervisor Dr. Vladimir Moiseenko with a grant. Liljewalchs resestipendium is acknowledged for travel funds. My office room mates Johan Abrahamsson and Emilia Lalander, as well as my former office room mates Martin Ranlöf and Johan Lidenholm are acknowledged for the great company. Johan Abrahamsson is especially acknowledged for countless long discussions on electromagnetism, pedagogic issues and a variety of other subjects. He is also acknowledged for helping out with 3D images for this thesis. Mårten Grabbe, Linnea Sjökvist and Kiran Kumar Kovi are acknowledged for the joyful teaching experiences that we have had together. The rest of the staff on the division for electricity is also acknowledged for their company during these years. Last but definitely not least my family is greatly acknowledged. My wonderful wife Frida should be greatly acknowledged for all love and 131 support and for taking the watch shift during the early mornings, late evenings and weekends in which large portions of this doctoral thesis were produced. I would also like to thank my soon 4 year old daughter Hilda for letting her father sleep much more during the production of this thesis than during the production of the licentiate thesis. My twin brother Johan should be greatly acknowledged for the enthusiasm he induces, and my mother Inger for taking a few child watches and providing support despite of the relatively large distance to Gothenburg. Acknowledgements also go to the rest of the family, my sisters Annica and Ulrika and my father Gary. My parents-in-law, Lennart and Kerstin, are also greatly acknowledged for all the support that they provide despite the long distance to Ömmesala city. 132 14. References In the reference list, all web references are dated 2012-10-25. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] See https://www.iter.org/sci/fusionfuels. W.M. Stacey, Nuclear Reactor Physics second edition, Wiley VCH Verlag 2007. S. Taczanowski, G. Domanska, J. 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