Coil Design and Related Studies for the Fusion-Fission Reactor Concept SFLM Hybrid

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To my wonderful daughter Hilda and my lovely wife Frida

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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 Non- Planar 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 Fusion- Fission 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).

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Conference papers:

IX 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).

X 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).

XI 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).

XII 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. Hagnestål, A., Ågren, O., Moiseenko V. E., Coil design for the SFLM Hybrid, Proceedings of the EPS conference 2012.

B. 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).

C. 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.

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Abbreviations and nomenclature

ADS Accelerator-Driven System

BOC Beginning Of fuel Cycle BWR Boiling Water Reactor

CTD Coolant Temperature Density (effect) D Deuterium

ELM Edge-Localized Mode

EOC End Of fuel Cycle

FDS Fusion-Driven System

FTWR Fusion Transmutation of Waste Reactor GDT Gas Dynamic Trap

IAEA International Atomic Energy Agency ICRH Ion Cyclotron Radio frequency Heating

ITER International Thermonuclear Experimental Reactor

LBE Lead-Bismuth Eutectic

LLFP Long-Lived Fission Products

LLNL Lawrence Livermore National Laboratory LOCA Loss Of Coolant Accident

LWR Light Water Reactor

MHD MagnetoHydroDynamic NEA Nuclear Energy Agency

PWR Pressurized Water Reactor

RW React & Wind

SABR Subcritical Advanced Burner Reactor SFLM Straight Field Line Mirror

SKB Svensk KärnBränslehantering

T Tritium TRU TRansUranics WR Wind & React

B T Magnetic field

ˆB - Unit vector parallel to B

E V/m Electric field

F N Force

j A/m² Current density

R m Position vector

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u m/s Mass velocity in MHD v m/s Velocity of a particle xc = (xc,yc,zc) m Gyro center position

m^-1 Curvature vector

A - Number of proton masses in a nucleus a m Plasma radius in a mirror machine

B T Magnetic field modulus on the z axis c m Axial scale length of a mirror machine

D - Diffusion constant

f - Distribution function

g T/m Quadrupolar field contribution

I A Current

keff - Effective neutron multiplication

M kg Ion mass

m kg Electron mass

m kg Particle mass

n - Neutron or number of neutrons

n 1/m³ Particle density

p N/m² Scalar pressure

p N/m² Parallel pressure (vs magnetic field) p N/m² Perpendicular pressure (vs magnetic field) P N/m² Total perpendicular pressure (incl. B)

Pfiss W Fission power

Pfus W Fusion power

Q N/m² Total parallel pressure (incl. B)

Q - Fusion Q, produced power/input power Qr - Fission to fusion power ratio

q C Electric charge

Rm - Mirror ratio

rg m Gyro radius of a particle

r0 m Radial Clebsch coordinate

s m Arc length coordinate along B

s m Arc length-like coordinate

T K Temperature

kBT/e eV Temperature (thermal energy)

v m/s Velocity component perpendicular to B v m/s Velocity component parallel to B

W J Energy

x0 m x-like Clebsch coordinate y0 m y-like Clebsch coordinate

Z - Atomic charge

- Fraction of delayed neutrons

- Plasma pressure/magnetic pressure

0 - at the midplane

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eV Energy of a particle

0 Rad/s Gyro angular frequency at the midplane Rad/s Gyro angular frequency

J/T Magnetic moment

V Electric scalar potential

m Tm Magnetic scalar potential C/m³ Charge density

m kg/m³ Mass density

m² Cross section

0 rad Angular-like flux coordinate

m² Normal (Radial-like) flux coordinate B0r02/2 s Collision time (time between collisions) - Average number of neutrons per fission

c 1/s Collision frequency

- a/c, used for ordering in paraxial approx.

mfp m Mean free path

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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 coal- fired 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

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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.

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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 fusion- fission 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.

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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

eff FNL TNL

k fp P P (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 ⁵seconds. For a slightly supercritical keff = 1.0001 this corresponds to a neutron multiplication (and hence power increase) of 1.0001^{100 000} 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

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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 k_eff 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 k_eff 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

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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

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immense. Although breeding is not the main target in this project, fusion- fission 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

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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.

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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.

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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 Cs- 137, 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 short- lived 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 Pu- 239 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

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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 k_eff, 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 sodium- cooled 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 k_eff 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

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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 k_eff 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.

2 3

1

... 1

i eff

eff eff eff eff

i eff

n k k k k k

k ^(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 k_eff should be as high as possible constrained by the safety issues. An approximation of the total energy gain is

1

fis eff

r h c ext

sn eff

W k

Q r n n

W k ^(2.9.2)

where 195W_fis MeV is the average energy produced in a fission reaction, Wsnis the average energy cost for each source neutron, v 2.9 is the average number of neutrons produced in each fission reaction, n_c is the fraction of source neutrons that enter the fission core, n_ext 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 k_eff for the different systems will strongly affect the energy efficiency, and this depends on the layout of the fission core.

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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 fusion- fission 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):

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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 fusion- fission [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

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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

r fiss/ fus

Q P P , the produced fission power divided by the produced fusion power, is

195 0.97

1.2 150

17.6 2.9 0.03

Qr (2.9.3)

where 0.97k_eff has been selected and the product r n n_{h c ext} 1.2 in this case.

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.

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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

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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 10 molecules ²⁵ 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.

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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,

( )

q

F 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

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v v

g ,

m q B

r q B m (3.2.2)

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/m_e)^{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 B₀zˆ is constant, E = 0 and F qB₀v B F₀xˆ, the equations of motion give

v_x v_y F0, v_y v , v_x _z v 0

m (3.2.3)

where

2

2 2 0

v v 0

v v

x x

y y

qBF m

(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

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v v

y, x

c c

x x y y (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

0

0 0

0

v v cos v v sin

x

y

t t F

qB

(3.2.6)

with the chosen initial condition v(t 0) v₀xˆ . For a more general case the velocity in vector notation can be written (apart from higher order gyro oscillations)

v ˆ _g _d

v B v v (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

2

2 3 2

ˆ ˆ

v ( )

d

B B m

B q B q B

E B B B B B

v (3.2.8)

where the magnetic moment

v / 22

m 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 ˆ(B Bˆ) 1/r_{c c}r is the curvature [44] and rˆ c is the curvature radius. For a vacuum field ( B 0) this simplifies to

2

2 2

1 v

( )

d

B m

B q B

v E B B (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,

( ) ( ) v2 .

gc gc 2 gc

q x B x m const (3.2.11)

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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

2 4

2 2 4

0

ln

4 v

ei

e

Z e

m (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

th

th ,

v 1

pk kv pk mfp

mfp k pk

n n (3.2.13)

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.