Hans-Jurgen Hartfuß and Thomas Geist

Fusion Plasma Diagnostics with mm-Waves

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Hans-Jurgen Hartfuß and Thomas Geist

Fusion Plasma Diagnostics with mm-Waves

An Introduction

The Authors

Prof. Dr. Hans-Jurgen HartfußMax-Planck-Institutfur Plasmaphysik (IPP)Wendelsteinstr. 117491 GreifswaldGermany

Dr.-Ing. Thomas GeistIsarstr. 189250 SendenGermany

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V

Contents

Preface XIII

1 Fusion Research 11.1 Reaction Scheme 11.2 Magnetic Plasma Confinement 41.2.1 Tokamak 51.2.2 Stellarator 61.2.3 Physics Issues of Magnetic Confinement 71.2.4 Plasma Heating 101.3 Plasma Diagnostic 111.3.1 Generic Arrangements 121.3.2 Microwave Diagnostics 15

References 17

2 Millimeter-Waves in Plasmas 192.1 Basic Equations 202.2 Plasma Dielectric Tensor, General Properties 232.3 Dielectric Tensor from Kinetic Theory 252.4 Cold-Plasma Limit 292.5 Derivation within Fluid Description 322.6 Discussion of Cold-Plasma Dispersion Relations 342.6.1 Nonmagnetized Plasma, �B0 = 0 342.6.2 Magnetized Plasma, Parallel Propagation, �k‖�B0 372.6.3 Magnetized Plasma, Perpendicular Propagation, �k ⊥ �B0 392.6.4 Slightly Oblique Propagation 412.7 Finite-Temperature Correction to Cold-Plasma Dielectric Tensor 422.7.1 Finite Larmor Radius Expansion 422.7.2 Warm-Plasma Approximation 442.7.3 Relativistic Corrections 462.8 Inhomogeneous Plasma 482.8.1 WKB Approximation 492.8.2 Refraction 51

VI Contents

2.8.3 Ray Tracing 532.9 Finite-Size Probing Beam 542.9.1 Gaussian Beam Description 542.10 Radiation Transfer 582.10.1 Transparent Plasma 582.10.2 Plasma Emitting and Absorbing 602.10.3 Multiple Chords, Imaging 61

References 62

3 Active Diagnostics 653.1 Interferometry 653.1.1 Single-Chord Interferometry 683.1.2 Multiple Chords 693.2 Polarimetry 703.2.1 Faraday Effect 713.2.2 Cotton–Mouton Effect 753.2.3 Common Generalized Description 773.3 Reflectometry 833.3.1 Time Delay Measurement 863.3.2 Phase Change at Cutoff 893.3.3 Profile Reconstruction 923.3.4 Localization of Reflecting Layer 933.3.5 Relativistic Corrections 953.3.6 Influence of Density Fluctuations 953.4 Scattering 1003.4.1 Single-Particle Thomson Scattering 1013.4.2 Doppler Shift 1023.4.3 Incoherent Scattering 1043.4.4 Relativistic Incoherent Scattering Spectrum 1063.4.5 Role of Density Fluctuations 1083.4.6 Coherent Scattering 1083.4.7 Electron and Ion Feature 1103.4.8 Summarizing Comments 113

References 115

4 Passive Diagnostics 1174.1 Bremsstrahlung 1184.2 Electron Cyclotron Emission 1224.2.1 Electron Motion in a Static �B-Field 1224.2.2 Electric Field and Spectrum, Single Electron 1234.2.3 Perpendicular Observation, Characteristic Modes 1264.2.4 Spectrum, Electron Ensemble 1284.2.5 Absorption Coefficient 1304.2.6 Emission Profile 1324.2.7 �B0-Field Varying along Sightline 135

Contents VII

4.2.8 Optical Depth of Most Relevant Modes 1374.2.9 Visibility Depth and Localization 1394.2.10 Electron Cyclotron Absorption Measurement 1424.3 Electron Bernstein Wave Emission 1434.3.1 Electron Bernstein Waves 1444.3.2 Mode Conversion 146

References 149

5 Guided Waves 1515.1 Transmission Line Properties 1515.1.1 Waves on a Lossy Transmission Line 1515.1.2 Terminated Transmission Line 1535.1.3 Classification of Transmission Lines 1575.1.4 Surface Currents 1605.2 Coaxial Transmission Line 1615.2.1 Characteristic Properties 1625.2.2 Losses and Limits of Coaxial Lines 1625.3 Rectangular Waveguides 1635.3.1 TE Waves 1645.3.2 TM Waves 1665.3.3 Attenuation in Rectangular Waveguides 1665.3.4 Fundamental TE10 Wave 1675.4 Circular Waveguides 1705.4.1 Fields in Circular Waveguides 1715.4.2 TM Waves 1725.4.3 TE Waves 1735.4.4 Loss in Circular Waveguides 1755.5 Multimode Waveguides 1765.5.1 Number of Modes Propagating 1765.5.2 Multimode Propagation 1785.5.3 TE11 Mode in Overmoded Circular Waveguides 1795.6 Corrugated Circular Waveguides 1825.6.1 Fields of Corrugated Circular Waveguides 1835.6.2 Characteristics of HE11 Hybrid Mode 1855.7 Gaussian Beams 1855.7.1 Solution of Approximate Wave Equation 1855.7.2 Transformation of Gaussian Beams 1865.7.3 Lenses and Curved Mirrors 1915.7.4 Truncation of Gaussian Beams 1935.7.5 Coupling Coefficient for Fundamental Gaussian Beams 1945.8 Vacuum Windows 1965.8.1 Single-Disk Window 1965.8.2 Half-Wave Window 1975.8.3 Thin Window 198

VIII Contents

5.8.4 Antireflection Coating 198References 199

6 Radiation Generation and Detection 2016.1 Signal Sources 2016.1.1 Backward-Wave Oscillator 2016.1.2 Solid-State Oscillators 2036.1.2.1 Gunn Oscillator 2036.1.2.2 IMPATT Oscillator 2056.1.3 Multiplier Chain 2066.2 Antennas 2086.2.1 Basic Definitions 2086.2.2 Antenna Temperature 2116.2.3 Pyramidal Horn 2126.2.4 Conical Horn 2146.2.5 Excitation of Gaussian Beams 2156.2.6 Antenna Arrays 2176.3 Detection 2216.3.1 Overview and Classification 2216.3.2 Bolometer 2236.3.3 Hot Electron Bolometer 2256.3.4 Noise Equivalent Power, NEP 2266.3.5 Schottky Diode 2276.3.6 Schottky Diode Frequency Multiplier 2296.3.7 Diode Direct Detector 2316.3.8 Schottky Detector Noise 2336.4 Heterodyne Detection 2366.4.1 Square-Law Mixer 2376.4.2 Diode Mixer 2396.4.3 Two-Port Mixer 2416.4.4 Mixer Construction 2456.5 Thermal Noise 2466.5.1 Noise Temperature 2476.5.2 Noise Figure 2496.5.3 Noise Temperature of Cascaded Systems 2506.5.4 Mixer Noise Temperature 2516.5.5 Noise Temperature of Heterodyne Receiver 2536.5.6 Measurement of Noise Temperature 2556.6 Sensitivity Limits 2566.6.1 Shot Noise Term 2566.6.2 Thermal Radiation Term 2586.6.3 Influence of Bandwidth 2596.6.4 Noise-Equivalent Power, Incoherent Detection 2606.6.5 Noise-Equivalent Power, Coherent Detection 2616.6.6 Minimum Detectable Temperature 263

Contents IX

6.7 Correlation Radiometry 2646.7.1 Intensity Fluctuations 2646.7.2 Cross-Correlation Function 2656.7.3 Intensity Fluctuations and Coherence 2666.7.4 van Cittert-Zernike Theorem 2686.7.5 Intensity Interferometer 2696.7.6 Accuracy of Cross-Correlation Measurements 2706.7.7 Alternative Decorrelation 271

References 273

7 Components and Subsystems 2757.1 Two-Port Characterization 2757.1.1 Scattering Parameters 2757.1.2 Transmission and Reflection 2787.1.3 Directional Coupler 2817.1.4 Nonreciprocal Devices 2837.2 Network-Analysis Measuring Techniques 2867.2.1 Transmission Measurement 2867.2.2 Reflection Measurement 2877.2.3 Substitution Measurement 2887.2.4 Measurements Using Noise Sources 2897.3 Frequency- and Polarization-Selective Filters 2907.3.1 General Definitions 2917.3.2 Waveguide Band-Stop Filter 2927.3.3 Band-Pass Filter in Overmoded Waveguide 2937.3.4 Metallic Meshes 2967.3.5 Polarization Filters 2987.4 Phase Measurement 2997.4.1 Phase Measurements with Analog Output 2997.4.2 All-Digital Phase Measurement 3017.4.3 Phase Determination by Software 3037.5 Signal Linearity 3047.5.1 Gain Compression 3047.5.2 Intermodulation 3057.6 Frequency Stability 3087.6.1 Control Loop Components 3087.6.2 PLL Circuits in the Millimeter-Wave Range 3097.6.3 Comments on the Theoretical Concept 310

References 313

8 Architecture of Realized Millimeter-Wave Diagnostic Systems 3158.1 Interferometer 3158.1.1 Comments on Wavelength 3168.1.2 Mach–Zehnder Interferometer 3188.1.3 Mach–Zehnder Heterodyne Interferometer 319

X Contents

8.1.4 Frequency Stability 3208.1.5 Path Length Variations 3228.1.6 Swept Frequency Interferometer 3248.1.7 Multichannel Interferometer 3248.2 Polarimeter 3268.2.1 Evolution of the Polarization State 3268.2.2 Modulation Techniques 3278.2.2.1 Modulation Scheme 1 3278.2.2.2 Modulation Scheme 2 3288.2.2.3 Modulation Scheme 3 3298.2.3 Faraday Polarimeter 3298.2.4 Cotton–Mouton Polarimeter 3308.3 Reflectometer 3328.3.1 Swept Single-Frequency System 3338.3.2 Multifrequency Systems 3378.3.3 Pulse Radar Technique 3398.3.4 Ultrashort Pulse Radar 3428.3.5 Distance Calibration and Spurious Reflections 3448.3.6 Comments on Fluctuation Measurements 3458.3.7 Doppler Reflectometry 3468.3.8 Imaging Reflectometry 3488.4 Radiometry of Electron Cyclotron Emission 3498.4.1 General Requirements 3508.4.2 Michelson Interferometer 3528.4.3 Martin–Puplett Polarizing Interferometer 3548.4.4 Grating Spectrometer 3568.4.5 Heterodyne Radiometers 3578.4.6 ECE Imaging 3628.4.7 System Parameters 3638.4.8 Calibration 3668.5 Detection of Electron Bernstein Wave Emission 3708.6 Coherent Scattering 3738.7 Summarizing Comments 375

References 378

Appendix A: Symbols and Constants 381

Appendix B: Formulas and Calculations 387B.1 Functions Qij 387B.2 Cold-Plasma Limit 388B.3 FLR Approximation 388B.4 Warm-Plasma Approximation 390B.5 Waveguide Attenuation 391B.6 Metallic Mesh Transmission 393

References 393

Contents XI

Appendix C: Tables and Material Constants 395C.1 Waveguides, Technical Data 395C.2 Waveguides, Theoretical Relations 396C.3 Dielectric Materials, Electrical Data 396C.4 Dielectric Materials, Mechanical Data 397C.5 Dielectric Materials, Names 397C.6 Gunn Oscillators 398

References 398

Index 401

XIII

Preface

This book deals with the standard microwave diagnostics established on magneticplasma confinement experiments in modern nuclear fusion research, giving anintroduction to the field by introducing the physics principles behind the diag-nostic methods as well as the experimental techniques applied. Since the latterare belonging to the field of microwave engineering, which is not a part of thecurriculum of students interested in plasma physics, in particular in experimentalplasma and fusion physics, broad room was given to the most important meth-ods, the instruments, and the measuring techniques established and applied inmicrowave experiments. The text evolved from lectures on plasma diagnostics and,in particular, also on microwave diagnostics given for many years at the Universityof Greifswald. The outline of this book follows the outline of the lectures.

Within the complete diagnostic system of a fusion experiment, microwave diag-nostics can be categorized as wave diagnostics both actively and passively probingthe plasma. The introduction therefore starts with the propagation of waves inplasmas. On the basis of kinetic theory, in Chapter 2, wave propagation in a hotplasma is treated and various approximations are given, sufficient to describe theplasma conditions envisaged. The dielectric properties of the magnetized plasmadetermine cutoffs and resonances, thus determining the frequency range withstrongest dispersion changes, offering optimum diagnostic capabilities. With thedensities and the magnetic induction of modern fusion experiments, this rangeextends from about 30 to a few hundreds of gigahertz corresponding to millimeterwavelengths—the range we are exclusively concentrating on. Thus, the subjectwaves in plasmas is strongly restricted to that particular range and to those wavesand modes that are of significance for the standard microwave diagnostics. On thisbasis, the active diagnostic methods interferometry, polarimetry, reflectometry, andscattering are introduced on an elementary level. No details of the fusion experi-ments are given, neither on details of the fusion device, nor on special topics infusion research. The ideas behind and the aims of fusion research are introduced inChapter 1 and the geometry of the diagnostic probing scenarios is being sketched.In the frame of the introduction of the various diagnostic methods, hot and dense fu-sion plasmas are considered with parameters as typical in modern fusion research.

The methods are discussed in a simplified geometry. The torus geometry isapproximated by a straight cylinder, thus with circular cross section of the plasma.

XIV Preface

However, the confining magnetic field forms nested flux surfaces as necessary formagnetic confinement. It has twisted field lines and a field gradient as typical intorus geometry. All diagnostic methods are discussed in this simplified geometry.After the discussion of the active probing diagnostics in Chapter 3, the followingchapter deals with the emission of the magnetized plasma in the millimeter-waverange. It concentrates on electron cyclotron emission, with a brief discussion of theapplicability of the radiometry of the emission that is generated by mode conversionfrom electron Bernstein waves.

The first four chapters are not going into any details of the experimental real-ization of the diagnostic methods described. Before this is possible, the specialtechniques of generating, guiding, and detection of microwaves need to be intro-duced. Thus, Chapter 5 deals with the methods of guiding waves within metallictubes, along wires, and as Gaussian beams, covering also the case of interruptionof the path by a vacuum window. Chapter 6 introduces signal sources, antennas,and detection systems and defines the figures of merit and the ultimate sensitivityof detectors in general. Chapter 7 finally introduces measuring techniques, thecharacterization of components and devices, and briefly introduces stabilizingtechniques of importance in microwave diagnostic installations.

On this basis, the various realizations of microwave diagnostic systems areintroduced in Chapter 8. The standard active and passive diagnostic systemsare discussed again, however, now concentrating on the microwave aspects: theinfluence of instable waveguide runs, instabilities in probing frequency, ways tomeasure the reflectometry time delays, to measure polarization states, to resolveemission spectra, and to measure the plasma radiation temperature. It is thearchitecture of the systems that concludes the introduction into the field. Thearchitecture is governed by microwave technology, which thus determines, to alarge extent, the progress in microwave diagnostic possibilities.

With the interested student in mind, the authors have assembled what theythink is of importance for the design, the construction, and the operation ofmicrowave diagnostics for fusion research. Of course, parts of the various fieldsthey are covering are treated, often in much more depth, in well-establishedexcellent textbooks, which had formed the vast chest of knowledge for the authorsin their own laboratory work as experimentalists. The books of M.A. Heald andC.B. Wharton, Plasma Diagnostics with Microwaves, of G. Bekefi, Radiation Processesin Plasmas, and of I.H. Hutchinson, Principles of Plasma Diagnostics need to bementioned, representative for many others referenced in the course of the book.

The authors thank their colleagues Klaus Fesser, Henry Greve, Matthias Hirsch,Eberhard Holzhauer, Walter Kasparek, Fritz Leuterer, Stefan Schmuck, TorstenStange, and Friedrich Wagner for their helpful critical comments. Finally, wewould like to thank the staff of Wiley-VCH, in particular the Project Editor, AnjaTschortner, for their friendly collaboration throughout the project.

February 2013 Hans-Jurgen Hartfuß and Thomas Geist

1

1Fusion Research

This chapter provides a brief overview of the physics basis and the aims offusion research and of the types of experimental devices used for the magneticconfinement of hot plasmas. It sketches the geometry in which plasma diagnosticsystems are operated and gives one possibility to order and categorize, from anexperimental viewpoint, the large number of diagnostic systems in use at modernfusion experiments.

The diagnostic systems collect the experimental data, thus providing the basisfor fusion research aiming at understanding the complex behavior of the hotmagnetized plasma, which is considered as necessary for the development of theoptimum confinement device and optimal scenarios for a burning fusion plasma.

1.1Reaction Scheme

Fusion research is the long-term effort to develop an almost inexhaustible energysource, based on fusion reactions among light atomic nuclei similar to thosepresent in the interior of stars. The physics basis for these burning processes is thefact that the binding energy per nucleon in an atomic nucleus is a function of itsmass number Am, increasing by about one order of magnitude from about 1 MeVper nucleon at Am = 2, deuterium (2D), to the maximum at Am = 56, iron (56Fe).Beyond iron, the binding energy per nucleon decreases. Therefore, energy can begained by the fusion of light elements as well as by the fission of heavier ones.The fusion-based energy production is connected with the formation of heavierelements. The stars create in this way the elements beyond hydrogen and helium.The young universe consisted of only light elements, about 75% hydrogen (1Hand 2D), about 25% helium (4He), and a very small amount of lithium (7Li) andberyllium (9Be). The first stars are formed out of this mixture. All elements withmass numbers up to 56 are produced by nuclear synthesis within the stars. Theelements beyond are mainly produced by neutron capture and subsequent decayprocesses when the stars are collapsing in a supernova [1].

Taking the Sun as an example of a typical star in the stable, longest lasting periodof its life, most of the power is generated by burning hydrogen into helium in a

Fusion Plasma Diagnostics with mm-Waves: An Introduction, First Edition.Hans-Jurgen Hartfuß and Thomas Geist.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Fusion Research

process called proton–proton-chain (pp-chain). This process involves a three-stepreaction: (i) two protons are combined to form first deuterium, p(p, e+νe)d; (ii)after this, the deuterium incorporates with another proton, forming helium-3,d(p, γ )3He; (iii) and then two helium-3 nuclei are merged together, finally forminghelium-4, 3He(3He, 2p)4He, releasing two protons [2]. Altogether, four protonsare combined into one α-particle, the helium nucleus: 4p → 4He + 2e+ + 2νe. Byalmost 1038 fusion reactions per second, a mass of 567 × 109 kg hydrogen is burnedinto 563 × 109 kg of helium, releasing a total power of about 1026 W, equivalent tothe mass loss of 4 × 109 kg each second. The generated fusion power of the Sunis dissipated mainly as electromagnetic radiation with a near-blackbody spectrumof 5800 K radiation temperature, corresponding to the physical temperature of theSun’s photosphere.

Energy production is concentrated in the very center of the Sun (<0.2 of the Sunradius) and it is taking place under conditions of extreme pressure (about 1016 Pa)and high temperature (1.5 × 107 K) caused by the contracting gravitational forcesof the huge mass concentration.

The high temperature is necessary for the burning process to occur, as it isneeded to overcome the repelling Coulomb forces between the equally chargedions.

The high kinetic energy enables the fusion partners to come close enoughtogether (10− 15 m) that the attracting strong but short-range nuclear forces areoutbalancing the repelling Coulomb forces, combining the two into a stable heaviernucleus. The fusion power density Pfus generated depends on the densities n1 andn2 of the reaction partners, the energy W fus released per reaction, and the stronglytemperature-dependent velocity-averaged reaction rate coefficient 〈σ v〉v,

Pfus = n1n2〈σ v〉vWfus (1.1)

Since weak interaction is involved in the first step of the pp-chain (e+-decay), therate coefficient is extremely small and the fusion power density in the Sun centeris only of the order 100 W m−3, despite the extreme density of reaction partners.Thus, the large total power released is attributable to the size of the Sun and notconnected with a large reaction rate per volume.

Copying this reaction scheme for energy production on the Earth has thereforelittle chance of success. Fortunately, more promising reaction schemes exist. Theone envisaged for controlled thermonuclear fusion on the Earth is the d(t,n)αreaction:

2D + 3T → 4He [3.5MeV] + 1n [14.1MeV] (1.2)

It is characterized by a rate coefficient higher than that within the Sun by about 27orders of magnitude, as shown in Figure 1.1.

However, the temperature needed is about 108 K, higher than the temperaturein the Sun center by almost one order of magnitude. The reaction envisaged isthe reaction between the hydrogen isotopes deuterium and tritium resulting in anα-particle (4He) plus a neutron (1n), releasing in total 17.59 MeV of energy.This energy is distributed as kinetic energy within the reaction products. Due

1.1 Reaction Scheme 3

1 10 100

E (keV)

1000

pp

1027

DT

1E−20

1E−24

1E−28

1E−32

1E−36

1E−40

1E−44

1E−48

1E−52

Rat

e co

effic

ient

(m

3s−1

)

Figure 1.1 The velocity-averaged rate coefficients 〈σ v〉 for the pp-chain of the Sun at15 million K and the DT-reaction at 100 million K envisaged for fusion energy productionon the Earth differ by about 27 orders of magnitude.

to the conservation of momentum, it is distributed inversely proportional totheir mass, mα/mn = Wn/Wα . The kinetic energies of the α-particle and theneutron are given in square brackets in Equation 1.2. Applying Equation 1.1with typical densities of the reaction partners deuterium and tritium of modernfusion experiments, which are envisaged as well for the future fusion reactor,nD = nT = 0.6 × 1020 m− 3, with the rate coefficient 〈σ v〉v ≈ 10− 22 m2 ms− 1 and thefusion energy per reaction, W fus = 2.8 × 10− 12 W s, the resulting power densityshows promising Pfus ≈ 1 MW m− 3.

While the fuel element deuterium is present in the oceans, the hydrogen isotopetritium is unstable, decaying into helium plus an electron and an electron neutrino,with a half-life of 12.3 years, according to

3T → 4He + e− [20keV] + νe (1.3)

Thus, in the Earth’s atmosphere, tritium is present only in very small quantitiesas a result of cosmic radiation interaction or imported with the solar wind. Theestimated total equilibrium tritium mass in the atmosphere is only about 3 kg.

For large-scale industrial applications in a fusion power plant, tritium needs tobe generated by neutron impact from lithium isotopes, according to

6Li + 1n → 4He + 3T + 4.8MeV7Li + 1n → 4He + 3T + 1n − 2.5MeV (1.4)

It is aimed at using fusion neutrons for that purpose. The fuel elements for apower plant based on the deuterium–tritium (DT) fusion reaction are thereforedeuterium from the oceans and lithium occurring in the Earth’s crust as well asin the oceans. They are almost uniformly distributed on the Earth. Along with theexpected safety and environment-friendly properties, fusion power might thereforebe called sustainable [3].

4 1 Fusion Research

1.2Magnetic Plasma Confinement

The hydrogen isotopes deuterium and tritium heated up to temperatures of theorder 108 K are in the fully ionized plasma state. They must be confined withoutany material contact for a sufficiently long time that fusion reactions can occurat an adequate rate. The storage of the plasma in a vacuum chamber enclosedby a magnetic configuration with torus shape has turned out to be a promisingconfinement concept [4, 5]. Owing to the Lorentz force, �F = q(�v × �B), a chargedparticle with charge q and velocity �v can move freely along the magnetic field �B,but it is forced to gyrate around the �B-field line in the case where it has a velocitycomponent perpendicular to it, in this way being bound to the field line. Thetoroidal �B-field applied in magnetic confinement devices is of the order of a fewtesla, resulting in gyro-radii of several millimeters for the plasma ions and about atenth of a millimeter for the electrons at the envisaged temperature. The pure torusfield, however, has, in addition to curvature with radius �Rc, also a radially inwarddirected �B-field gradient �∇B.

Field curvature and field gradient cause charge-dependent particle drifts pro-portional to �B × �Rc and �B × �∇B, respectively, leading to a separation of ions andelectrons perpendicular to �B. The resulting electric field �E gives rise to a radialoutward drift of ions and electrons of the plasma proportional to �E × �B. Thus,no force equilibrium is established. Since for particles on the inner edge of thetorus, this outward drift is directed to the torus axis, and for those closer to theouter edge the drift direction is further out, the drift can be avoided on average bytwisting the field lines to which the charged particles are bound. To accomplishthis, it is necessary to superimpose a poloidal field �Bθ to the pure toroidal field �Bφ ,in order to cause the curvature – and the �∇B – drift to cancel on average. Thus, thevertical charge separation is avoided, since short-circuited by the helical field lines(HFLs). The resulting net field is helical and might be expressed with unit vectorsin toroidal and poloidal directions, nφ and nθ , by �B = �Bφ nφ + �Bθ nθ with �Bφ �Bθ .

As depicted in Figure 1.2, the pitch of the resulting field line is determinedby the ratio of the poloidal and toroidal field components and characterized bythe rotational transform ι, which is the poloidal angle θ a field line is turnedwhen performing a full revolution in toroidal direction φ= 2π . The rotationaltransform is not constant along r. With ι0 = ι(r = 0), the value on the plasma axis,the dependence can be described by ι(r) = ι0 + (∂ ι/∂ r)δr = ι0 + 2π • Sδr, with thequantity S called shear.

The total field must be shaped and adjusted such that field lines never cross andthat they form toroidal nested surfaces. Only in this case, particles confined to afield line lying further in would stay further in; those farther out would stay fartherout. The existence of nested flux surfaces is a necessary condition for magneticconfinement.

The flux surfaces can be labelled by the magnetic flux they are enclosing. Theinnermost flux surface encloses zero volume. It is called magnetic axis. In the idealcase, field lines forming a magnetic surface never close on itself. Their rotational

1.2 Magnetic Plasma Confinement 5

Torus axis

Magnetic axisV = const

R

B

Φ θ

Figure 1.2 Magnetic confinement in torus geometry demands for twisted field lines thatbuild up nested magnetic surfaces. The total �B-field is composed of a toroidal, B , and asmaller poloidal component, B�. The nested magnetic surfaces show up as nested circlesin a poloidal cross section, as indicated with the grey circles.

transform is therefore an irrational number. The radial range with nested magneticsurfaces is limited. A last closed magnetic surface exists. Farther out, field linesend on material boundaries, intersecting the vacuum chamber walls. Along thoselines, particles leave the plasma, and no confinement is possible any longer. Inthis sense, the last closed surface defines the outer plasma edge in a magneticconfinement device. Two concepts have been developed differing in the way thefield line twist is generated: the tokamak and the stellarator.

1.2.1Tokamak

In the tokamak, a strong current of the order 106 A is induced in the toroidalplasma column generating the poloidal field, twisting the field lines and buildingup the nested magnetic surfaces as deemed necessary for confinement [4, 6].

The primary coil of this transformer-like arrangement, in which the plasmaforms the secondary, is a solenoid coil along the center of the torus axis. Togenerate a constant plasma current, the flux in the primary solenoid coil mustchange at a constant rate to keep the induced toroidal loop voltage constant, whichdrives the plasma current. Since the swing in the primary transformer windingsis finite, a classical/conventional tokamak is necessarily a pulsed device, althoughoperation in modern devices can extend to several tens of minutes. The toroidal�B-field is produced by typically 12–20 planar equidistant discrete coils along thetoroidal circumference. The radial position of the plasma is controlled by a verticalfield generated by a pair of coils, one above and the other below the plane of thetorus (Figure 1.3).

The torus geometry, as sketched in Figure 1.2 and Figure 1.6, is described by themajor radius R0 and the minor plasma radius a.

The radial coordinate r varies between the plasma axis at r = 0 and the plasmaedge at the last closed flux surface at r = a. The ratio A = R0/a is called aspectratio, typically lying between 2 and 6. The rotational transform expressed by the

6 1 Fusion Research

Transformer

TFC

HFL

VFC

IPlasma

Figure 1.3 In a tokamak, the poloidalfield component is generated by the strongplasma current Ip induced by magneticflux changes in the central solenoid wind-ing acting as primary winding of a trans-former arrangement, and the toroidal plasmacolumn forming the secondary one. Themain toroidal field is built up by planar

field coils (TFC). A pair of windings, thevertical field coils (VFC), one above andone below the torus plane, produce a ver-tical field that allows to shift the plasmacolumn radially. A helical field line (HFL)demonstrates the twist as a result of thesuperposition of toroidal and poloidal fieldcontributions.

field component ratio and the geometry parameters is ι/2π = R0B�/rB . The valueq = (ι/2π )− 1 is called safety factor.

The tokamak is axisymmetric, which means that arbitrary poloidal plasmacross sections are equivalent with respect to the plasma parameters. In cylindricalcoordinates, r,�, , with� the poloidal angle and the toroidal angle, it is thereforesufficient to label any volume element of the plasma column by r,�. In cases wherethe plasma has circular cross section, the poloidal angle �, when consideringphysical quantities that are constant on flux surfaces, is negligible too. The toroidal�B-field, however, depends on the poloidal angle, Bφ = Bφ(�). In modern tokamaks,the poloidal plasma cross section is noncircular, typically vertically elongated, andD-shaped, allowing for equilibria at higher plasma currents [6].

1.2.2Stellarator

In a stellarator, the whole confining field is produced by currents flowing outsidethe plasma. No induced plasma current is needed to build up the confining �B-field[4, 7]. Nevertheless, pressure-driven currents are present also in the stellarator;however, they are significantly smaller than the plasma current in tokamaks.

The coil system of a classical stellarator is composed of toroidal field coils similarto those of a tokamak, and pairs of helical windings with opposite currents within

1.2 Magnetic Plasma Confinement 7

HFC TFCHFL

Figure 1.4 In the classical stellarator, thetoroidal field is generated as in the toka-mak by a set of planar toroidal field coils(TFC). The poloidal field component, how-ever, is generated by currents exclusivelyoutside the plasma. In the l = 2 stellarator,a pair of conductors with opposite currents

are wound helically around the torus (HFC,helical field coil) generating the poloidalfield component, resulting in helical fieldlines (HFL) necessary for plasma confine-ment. The plasma cross section is ellipticalshaped, with the ellipse orientation varyingwith toroidal position.

the conductors of the pair, generating the rotational transform. Owing to thehelical windings, the stellarator has no axial symmetry. With two pairs of helicalwindings (l = 2), the poloidal plasma cross section is elliptical; with three (l = 3), it istriangular, rotating around the plasma axis with toroidal coordinate . The aspectratio of stellarators is larger than that of tokamaks, typically around R0/a ≈ 10.Figure 1.4 shows a classical l = 2 stellarator.

Modern stellarators use modular nonplanar field coils that are able to generatearbitrary superpositions of classical stellarator fields, allowing for the optimizationof the confining field configuration that is necessarily three-dimensional. It is opti-mized in various respects, considering the technical feasibility and, in particular, ac-counting for physics aspects, that is, improving the stability of the confined plasmaas well as minimizing particle and energy transport across the magnetic surfaces [7].

Stellarators are intrinsically steady-state devices, and are highly advantageous inview of the applicability as power reactor [8]. However, the experimental databaseof tokamaks is by far larger. The next-step device, the International ThermonuclearExperimental Reactor (ITER), is therefore based on the tokamak principle [9].

1.2.3Physics Issues of Magnetic Confinement

The hot plasma is confined in the magnetic torus configuration with nestedmagnetic surfaces. Assuming the electrons and ions of the plasma in Maxwellian

8 1 Fusion Research

energy distributions, the plasma pressure p (p = nkBT) is similar to the pressure ofan ideal gas given by the product of particle density, in the plasma composed ofelectron and ion density, n = ne + ni, and the temperature T = T i = Te.

Stable operation is obtained if the plasma pressure is kept smaller than that of themagnetic field; the ratio of the two, the normalized plasma pressure, β = 2μ0p/B2,is therefore typically below 0.1. Since the plasma is confined to a bounded region,pressure gradients �∇p evolve, balanced by the electromagnetic forces generated bycurrents with density�j in the plasma and the external �B-field, �∇p = �j × �B. The forcebalance implies �B • �∇p = 0. Consequently, field lines lie in a constant-pressuresurface and accordingly magnetic surfaces are surfaces of constant pressure.

Along with the pressure gradients, radially directed gradients of temperatureand density exist, which are driving energy transport, QE ∝ −n �∇T , and particletransport, �P ∝ −�∇n, across the flux surfaces from the hot and dense plasmacenter to its edge [10]. However, it turned out that collisional energy transport ofthe electrons is small compared to the transport driven by microturbulence in theplasma. Thus, small-scale turbulence of density and temperature correlated withelectric and magnetic field fluctuations within the plasma is forming the main losschannel in magnetic confinement devices.

In addition to these main loss processes, energy is lost because of the emission ofelectromagnetic radiation from the plasma. The most effective radiation processesof the electrons are bremsstrahlung (due to their acceleration in the field of ions) andcyclotron radiation (due to the gyration around the field lines). Not fully ionized im-purity ions in the plasma can give rise to atomic line emission after excitation by elec-tron impact. Although the radiative losses due to the accelerated motion of the elec-trons are unavoidable, certain effort must be undertaken to keep the impurity levelsmall enough that impurity radiation stays below a maximum acceptable level [11].

The necessary physics conditions of a fusion reactor based on the DT fusionreaction can be formulated by balancing the energy gain and loss processes. Ina burning DT-plasma, the generated energetic α-particles stay confined. Theyheat the plasma when they slow down. Only their contribution enters the energybalance as gain, as the generated energetic neutrons leave the plasma, providingtheir kinetic energy to external systems.

Energy loss is caused by turbulent transport, diffusion, convection, and theradiative losses mentioned earlier. The various processes can be combined andglobally be described by the quantity energy confinement time, τE. Its size is ameasure of the energy insulation quality of the confinement device.

Positive energy balance is obtained in the case where the triple product of tem-perature (in energy units) and density of the reaction partners and the confinementtime exceed a certain value called Lawson criterion: nkBTτE ≥ 5 × 1021 keVsm−3.With temperature T , with kBT ≈ 10 keV, and at particle densities of 1020 m− 3, theconfinement time must amount to a few seconds [12].

The energy confinement time in modern laboratory fusion experiments isexperimentally determined from the ratio of total stored energy W in the wholeplasma volume, W = 3

2

∫kB(neTe + niTi) dV , to the net heating power PH = Pext +

Pα − Prad, which is composed of external plasma heating with power Pext, the

1.2 Magnetic Plasma Confinement 9

heating Pα by the fusion generated α-particles, and the radiation losses Prad

under steady-state conditions: τE = W/PH. Defining the Q-factor as the ratio of thefusion power output to the input power necessary to sustain the fusion reaction,the so-called break-even condition, Q = 1, corresponds to the minimum conditionto sustain burning given by the Lawson criterion.

It turned out that the energy confinement time τE depends on a number of phys-ical parameters as well as on the geometry of the confinement device. The energyconfinement time improves with major and minor radii of the device, the plasmadensity, the main toroidal magnetic field, and the plasma current in the case of atokamak, and it degrades with increasing heating power, recalling only the mostimportant parameters. The dependencies explain the need for large devices tofulfill the Lawson condition. They are explored by deriving empirical scaling lawsbased upon the experimental results of many different devices of largely varyingparameters (Figure 1.5).

Energy confinement time scaling is known accurate enough to allow for theextrapolation to reactor-like conditions, although the physics behind is still notunderstood in every detail. This is true especially for the turbulent transport. Thetriple product obtained in the most advanced fusion experiments is within a factorof 5 of that necessary in a fusion reactor [13].

10−4 10−3

10−3

10−2

10−2

Calculated τE (s)

Mea

sure

dτ E

(s)

10−1

10−1

100

100

101

101

ITER

Figure 1.5 The energy confinement timeτE is a global measure of the confinementquality of the device. It depends on ma-chine parameters such as size and B-fieldas well as on the operation scenario, thatis, heating power and plasma density. Com-parison of experimental results from many

different machines allows for the formula-tion of scaling laws that enables to extrapo-late to next-step devices (here ITER), evenif the physics behind it is still not under-stood in every detail. Data given in the figureare gained in both tokamak and stellaratorexperiments [13].

10 1 Fusion Research

1.2.4Plasma Heating

In a fusion reactor based on the DT-reaction, the plasma will be heated by the α-particles. However, DT-operation with fusion power gain has so far been conductedonly in two major experiments, the Joint European Torus (JET) and the TokamakFusion Test Reactor (TFTR), with a strongly limited number of experiments [14, 15].The typical fusion plasma experiments are conducted with hydrogen or deuteriumor with mixtures of both. Therefore no internal energy gain from fusion reactionsoccurs; thus, heating is continuously necessary to study the plasma behavior atfusion-relevant temperatures and densities.

Basically three different heating schemes are possible and in use: (i) Joule orohmic heating, (ii) particle heating by injected energetic particles, and (iii) heatingby electromagnetic waves launched into the plasma.

Ohmic heating in tokamaks by the electron-carried induced plasma current Ip

is based on the fact that the plasma column has a finite resistance Rp. Thus, thepower PJ = RpI2

p is dissipated. The resistance is caused by electron–ion collisions.

Since the resistance decreases with increasing electron temperature, Rp ∝ T−3/2e ,

the heating efficiency decreases as well. Ohmic heating is therefore restricted tothe very start-up phase of tokamak operation. It is of course not used at all instellarators.

The energy content of the plasma can efficiently be increased by neutral beaminjection (NBI) of H- or D-particles with high energy (50–500 keV), which areionized and slowed down and finally thermalized in collisional processes withthe plasma electrons and ions, thus increasing the plasma energy content W.NBI heating affects the particle balance because, for example, some 1019 energeticparticles with an energy of 100 keV need to be injected per second for generating1 MW of heating power [16].

Wave heating is done at frequencies resonant with the gyration motion ofelectrons or ions called electron cyclotron resonance heating (ECRH) and corre-spondingly ion cyclotron resonance heating (ICRH). Since, at high temperatures,the plasma is almost collisionless, electromagnetic waves outside these resonancesare not absorbed at all or not dissipated efficiently enough for heating purposes.Wave heating is therefore possible only if resonance conditions are fulfilled. Theresonant frequencies depend on the B-field of the device they are applied, andare typically in the range 50–200 GHz in the case of ECRH, and 30–100 MHz forICRH. Since the B-field varies with location within the plasma, resonance becomesa local phenomenon. Wave heating methods ECRH and ICRH allow therefore forlocalized heating of electrons and ions separately, as well as for current drive andshaping of the current profile, providing wide experimental fields of operation. Theparticle and wave heating methods are experimentally tested and technologicallydeveloped to provide heating powers of the order of several tens of megawatts evenunder steady-state conditions [17].

1.3 Plasma Diagnostic 11

1.3Plasma Diagnostic

Plasma diagnostic provides the experimental database for fusion research. De-pending on the scientific problem and the related experimental program, a largenumber of plasma parameters needs to be known simultaneously. Among thoseparameters, the most important ones are the density and temperature of theplasma-forming constituents, electrons, ions, neutrals, and impurities, the totalenergy content of the plasma, the plasma pressure, plasma currents, local fields,plasma drift motions, and electromagnetic radiation of various origins. Most ofthem are time-dependent local quantities that must be measured with sufficientspatial and temporal resolutions.

The large variety of diagnostic methods applied originated from all areas ofphysics. Two general issues to be considered are redundancy and complementarity.Redundancy means, to determine the same physical quantity, different methods areto be applied to avoid or to detect systematic errors. Complementarity is necessary,on the one hand, to cover the full dynamic range of a certain plasma parameterthat might range over orders of magnitude, demanding for different methods tocover that range. It is necessary, on the other hand, to provide information by onediagnostic system needed for the interpretation of another one, to complement oneanother.

These demands affect the various systems based on different physical methodsapplied, for example, to measure the electron temperature and density.

To optimally combine their results, integrated data analysis (IDA) is advanta-geous. In this attempt, the raw data of several diagnostic systems are combined toform a common physical picture as complete as possible, from which the quantityof interest is derived, instead of deriving it individually from each of the diagnosticsystems, comparing and discussing possible discrepancies [18].

Generally, the physical quantities are time and space dependent. Owing to thefast equalization processes, however, they are generally constant on a flux surface.As we have seen, the pressure, as an equilibrium property, is constant on aflux surface. This means that measurements undertaken at different positions ofthe toroidal plasma are identical in the case where they are made at the sameflux surface. Therefore, for comparison, the laboratory coordinates defining themeasurements need to be transformed to flux coordinates or to the effective radialcoordinate of an equivalent axisymmetric plasma with cylindrical cross section.

If not explicitly mentioned otherwise, we will assume that this is possible underthe conditions discussed in this book. Figure 1.6 shows the geometry we arereferring to with R0 a, thus treating the torus in the limit of a straight cylinder.

The only local coordinate will then be the radial position r, ranging from theplasma axis to the plasma edge, 0 ≤ r ≤ a. To stay descriptively connected with theexperimental arrangement, profiles are often given, despite they are symmetric inthis representation, across the full plasma column, − a ≤ r ≤ a. Profile maxima,either peaked or broad, of the most important quantities (pressure, density, andtemperature) are located in the plasma center near the axis, with the quantities

12 1 Fusion Research

Plasma axis

Poloidalfield

Toroidal field

Plasma currentMinor radius a

Major radius

R0 r

Figure 1.6 The geometry used to describethe plasma in the frame of this book. Axialsymmetry is assumed with circular poloidalplasma cross section. Other symmetriescan be transformed to this by comparison

of the volumes enclosed by flux surfaces.In the geometry shown, the radial coor-dinate r is sufficient to describe densityand temperature profiles of the confinedplasma.

approaching zero at the edge. Typical scale lengths are of the order of centimeters(in large devices, tens of centimeters), which need to be resolved by the diagnosticsystems. All quantities are time dependent, varying on a time scale of the order ofthe confinement time. Many diagnostic systems should be able to resolve the muchfaster magneto-hydrodynamic (MHD) phenomena, occurring on a millisecondtime scale as well. Special fluctuation diagnostic systems dedicated to turbulencestudies, however, need sub-microsecond time resolution.

1.3.1Generic Arrangements

To avoid perturbation of the plasma by the measuring diagnostic instruments and toavoid destruction of their detectors, probing of the hot fusion plasma must be con-ducted without any material contact between the detection system and the plasma.The only exceptions are Langmuir probes applied for short time intervals at the lesshot very plasma edge. All other diagnostic systems are either based on the analysisof waves or particles emitted by the plasma or involve passing waves or particlebeams through the plasma and analyzing the result of their interaction with it.

From the experimental viewpoint, the large variety of different diagnostic systemspresent on modern fusion experiments can be arranged into four groups: composedof wave or particle diagnostics, and either active or passive.

In addition to these four groups, we have Langmuir probes and magneticdiagnostics, which do not fit unconstrained into that scheme. Probes eitherinject electrons into the plasma or extract them out of it. Magnetic diagnosticsmeasure magnetic flux changes caused by the plasma diamagnetism as well as byinduced and pressure-driven currents. Besides this more experimentally orientedordering scheme, the variety of diagnostic systems can be distinguished withrespect to the physical processes [19] or by the experimental methods involved [20].Figures 1.7–1.11 show the generic arrangements for active and passive probing

1.3 Plasma Diagnostic 13

Transmission

(A, ω, Φ, k, p)final(A, ω, Φ, k, p)initial

Figure 1.7 Active probing of the plasma bylaunching a wave and measuring changesin the wave’s characterizing quantities am-plitude, frequency, phase wave vector, andpolarization state. Conclusions on the kindand strength of the plasma–wave interactioncan be drawn, from which plasma param-eters can be determined. The arrangement

shown in this figure is used to measurechanges in phase and polarization statesof the wave in interferometry and polarime-try diagnostic systems (Section 1.3.1 andSection 1.3.2). The single chord arrange-ment gives line-integrated information. Toobtain local information, multiple chords areneeded.

Scattering

(ω, k)final(ω, k)initial

Figure 1.8 The electric field of the wavepassing the plasma accelerates individualplasma electrons, thus becoming them-selves emitters of electromagnetic radiation(Thomson scattering, Section 3.4). Sincethe scattering electrons are moving corre-sponding to their temperature, their emis-sion is Doppler shifted with respect to theprobing wave frequency. At fusion-relevanttemperatures, the emission is relativistically

blue-shifted, in addition. The width of thespectrum reflects the velocity distributionalong the scattering vector. Depending onthe scattering geometry and the wavelengthof the primary wave, scattering is caused byindividual electrons (incoherent scattering)or can as well be caused by collective actionof the large number of electrons in a De-bye cloud, reflecting the motion of the ions(coherent scattering).

Reflection

(ω, Φ)final

(ω, Φ)initial

Figure 1.9 The wave launched into theplasma can be reflected back when reach-ing a cutoff layer where the refractive indexapproaches zero. The conditions are mainlydetermined by the space-dependent elec-tron density in the plasma. By measuring

the round trip phase delay of the wave inthis RADAR-like arrangement, the locationof the cutoff layer can be determined. Thus,local plasma parameters determining thewave cutoff can be derived with the method(Section 3.3).

14 1 Fusion Research

Emission of waves

Ejections of particles

Figure 1.10 The plasma emits electro-magnetic radiation in a wide spectralrange extending from the gyration fre-quency of the ions at tens of megahertzto the X-ray region. Passive spectroscopyof the emission is the classical way togain information about the plasma con-stitution and state. The physical mecha-nisms causing the emission range fromgyration motion of the charged particlesaround the field lines and bremsstrahlungof the electrons in the fields of the ionsto line emission of not fully ionized impu-rity atoms within the plasma. The presenceand the concentration of the impurities

can basically be derived from the line in-tensity. The line width of the emission iscarrying information on the velocity distri-bution. All processes are strongly dependenton temperature and density of the plasma.Neutralized particles are leaving the plasmaas well. These particles are recombinedplasma ions undergoing charge exchangewith neutral particles from the NBI heatingsystem, or with neutrons from fusion reac-tions. All particles escaping from the plasmareflect the ion energy distribution in theplasma, which can be determined by mea-suring the energy distribution of the leakingparticles.

(I, E, Z n+)initial

Interaction region:

collisional excitationcharge exchange

fluorescenceparticles

Observation:

Figure 1.11 Atomic beams are injectedinto the plasma in arrangements of ac-tive particle probing diagnostics. The beamatoms are excited in electronic collisions.The subsequent emission is analyzed spec-troscopically, giving local information ondensity and temperature within the plasma

volume defined by the crossing of parti-cle and observation beams. The neutralatomic beam also provides electrons forcharge exchange processes with the plasmaions. Their broadened and shifted emissionis carrying information on the ion velocitydistribution.