Magnetic Fusion Energy Formulary

MAGNETIC FUSION ENERGY FORMULARY

Zachary S Hartwig MIT Plasma Science and Fusion Center

Yuri A Podpaly National Institute of Standards and Technology

Revision : February 2014 Preface

The guiding principle behind the MFE formulary (the “Formulary” here- after) is to provide a comprehensive reference for students and researchers working in the ﬁeld of magnetic conﬁnement fusion. It is a product of the authors’ frustration with searching dozens of textbooks while studying for the MIT doctoral qualifying exams.

The Formulary consists of three broad sections. Chapters 1–2 cover the mathematics, fundamental units, and physical constants relevent to magnetic fusion. Chapters 3–9 cover the basic physics of thermonuclear fusion plasmas, beginning with electrodynamics as a foundation and developing single particle physics, plasma parameters, plasma models, plasma transport, plasma waves, and nuclear physics. Chapters 10–13 cover the physics of toroidally conﬁned plasmas, the fundamentals of magnetic fusion energy, and the parameters for the major magnetic fusion devices of the world.

Much of the content of the Formulary has been derived from an original source, such as peer-reviewed literature, evaluated nuclear data tables, or the pantheon of “standard” mathematics and physics textbooks commonly used in magnetic fusion energy. References are given immediately following the cited item in superscript form as “a:b”, where “a” is the citation number of the reference and “b” is the page number. Full bibliographic entries for all references may be found at the end of the formulary. In addition to providing transparency, this unique feature transforms the Formulary into a gateway to a deeper understanding of the critical equations, derivations, and physics for magnetic fusion energy.

Ultimately, we hope that this work is useful to all those trying to make magnetic fusion energy a reality.

Z & Y

1 2 Information

The following sections contain important information regarding the devel- opment, use, and distribution of the MFE Formulary. The most up-to-date contact information for the authors of the MFE Formulary is:

Email: mfe [email protected] • Web: http://www.psfc.mit.edu/research/MFEFormulary • GitHub: https://github.com/MFEFormulary/MFEFormulary • Licensing

The Formulary has been released under open source licenses in order to maximize its distribution and utility to the plasma physics and magnetic fusion communities. The hardcopy printed book and digital PDF format have been released under the Creative Commons SA 4.0 license. The LaTex source code and build system has been released under the GNU GPL v3.0 and is freely available from our GitHub repository on the web. Both licenses permit copying, redistributing, modifying, and deriving new works under the aforementioned license. You are encouraged to distribute this document as widely as possible under the terms provided in the license above. You are also encouraged to fork or clone our GitHub repository, make your own contributions to the LaTeX codebase, and submit a pull request for inclusion of your contribution to the Formulary. (Email to works as well.) Finally, you are welcome to use any part of the hardcopy or electronic copy of the Formulary or its LaTeX code in your own projects under the terms of the licenses above.

Contributors

The following is a list of people who have contributed their time, eﬀort, and expertise. Their contributions have been critical to continually improving the accuracy and content of the Formulary. Please consider contributing your expertise either through our GitHub repository or email!

Michael Bongard Mark Chilenski • •

Dan Brunner Samual Cohen • • 3 4

Luis Delgado-Aparicio Alex Tronchin-James • • Chi Gao Anne White • • Greg Hammett Dennis Whyte • • Disclaimer

The MFE formulary is by no means complete or error-free. In fact, the sta- tistical laws of the universe essentially guarantee that it has grievous errors and glaring omissions. Thus, we welcome suggestions for additional material, improvements in layout or usability, and corrections to the pesky errors and typos we have tried so hard to eliminate. Even better, we encourage you to obtain the source code from our GitHub repository (see above), make the corrections, and then submit them to us via pull request. You can also reach us by email (see above).

Despite vigorous proofreading, no guarantee is provided by the authors as to the acuracy of the material in the MFE Formulary. The authors shall have no liability for direct, indirect, or other undesirable consequences of any character that may result from the use of the material in this work. This may include, but is not limited to, your analysis code not working or your tokamak not igniting. By your use of the Formulary and its contents, you implicitly agree to absolve the authors of liability. The reader is encouraged to consult the original references in all cases and is advised that the use of the materials in this work is at his or her own risk.

Acknowledgments

The authors would like to thank Dan Brunner, Chi Gao, Christian Haakon- sen, Dr. John Rice, Prof. Anne White, Prof. Dennis Whyte (all of MIT), and Dr. Samuel Cohen (PPPL) for their encouragement, feedback, and proofreading. We would also like thank Heather Barry, Prof. Richard Lester, and Prof. Miklos Porkolab of MIT for their support.

During the writing of the Formulary, the authors were at various times sup- ported by the following funding agencies, to which we would like to extend our gratitude: the MIT Department of Nuclear Science and Engineering, the MIT Plasma Science and Fusion Center, U.S DoE Grant DE-FG02- 94ER54235, U.S. DoE Cooperative Agreement DE-FC02-99ER54512, and U.S. ORISE Fusion Energy Sciences Program. Contents

Preface 1

Information 3

1 Mathematics 9 1.1 Vector Identities ...... 9 1.2 CurvilinearCoordinateSystems...... 10 1.3 Integral Relations of Vector Calculus ...... 12 1.4 LegendrePolynomials ...... 13 1.5 BesselFunctions ...... 13 1.6 ModiﬁedBesselFunctions ...... 14 1.7 Partial Diﬀerential Equations ...... 16 1.8 GaussianIntegrals ...... 18 1.9 Errorfunctions ...... 18

2 Fundamental Constants and SI Units 21

3 Electricity and Magnetism 25 3.1 Electromagnetic in Vacuum ...... 25 3.2 Electromagnetics in Matter ...... 27 3.3 Dipoles ...... 27 3.4 CircuitElectrodynamics ...... 28 3.5 ConservationLaws ...... 29 3.6 Electromagnetic Waves ...... 29 3.7 Electrodynamics ...... 32

4 Single Particle Physics 35 4.1 Single Particle Motion in E and B Fields ...... 35 4.2 Binary Coulomb Collisions ...... 37 4.3 Single Particle Collisions with Plasma ...... 38 4.4 Particle Beam Collisions with Plasmas ...... 40

5 Plasma Parameters and Definitions 43 5.1 Single Particle Parameters ...... 43 5.2 PlasmaParameters...... 44 5.3 PlasmaSpeeds ...... 45 5.4 Fundamentals of Maxwellian Plasmas ...... 45 5.5 Definition of a Magnetic Fusion Plasma ...... 46 5.6 Fundamental Plasma Definitions ...... 47

5 6 CONTENTS

6 Plasma Models 49 6.1 Kinetic...... 49 6.2 TwoFluids ...... 50 6.3 OneFluid...... 52 6.4 Magnetohydrodynamics (MHD) ...... 53 6.5 MHDEquilibria...... 54 6.6 Grad-ShafranovEquation ...... 54 6.7 MHDStability ...... 54 6.8 StabilityoftheScrewPinch ...... 56

7 Transport 59 7.1 ClassicalTransport...... 59 7.2 NeoclassicalTransport ...... 62

8 Plasma Waves 65 8.1 Cold Plasma Electromagnetic Waves ...... 65 8.2 Electrostatic Waves ...... 67 8.3 MHDWaves...... 68 8.4 HotPlasma ...... 68

9 Nuclear Physics 73 9.1 Fundamental Deﬁnitions ...... 73 9.2 Nuclear Interactions ...... 74 9.3 CrossSectionTheory...... 76 9.4 ReactionRateTheory ...... 77 9.5 Nuclear Reactions for Fusion Plasmas ...... 78 9.6 Nuclear Reactions for Fusion Energy ...... 78 9.7 Fusion Cross Section Parametrization ...... 79 9.8 Fusion Reaction Rate Parametrization ...... 80 9.9 Cross Section and Reaction Rate Plots ...... 81

10 Tokamak Physics 83 10.1 Fundamental Deﬁnitions ...... 83 10.2 Magnetic Topology ...... 85 10.3 MagneticInductance ...... 86 10.4 Toroidal Force Balance ...... 86 10.5 Plasma Para- and Dia-Magnetism ...... 87 10.6 MHD Stability Limits ...... 88 10.7 Tokamak Heating and Current Drive ...... 88 10.8 EmpiricalScalingLaws ...... 90 10.9Turbulence ...... 91

11 Tokamak Edge Physics 97 11.1 TheSimpleScrapeOﬀLayer(SOL) ...... 97 11.2 BohmCriterion...... 98 11.3 A Simple Two Point Model For Diverted SOLs ...... 99

12 Tokamak Fusion Power 101 12.1 Deﬁnitions...... 101 CONTENTS 7

12.2 Power Balance in a D-T Fusion Reactor ...... 102 12.3 The Ignition Condition (or Lawson Criterion) ...... 103

13 Tokamaks of the World 105

References 110

Index 111 8 CONTENTS Chapter 1

Mathematics

A,B, ..., are vector functions ←→T is a tensor ψ and ξ are scalar functions σ and τ refer to surfaces and volumes, respectively dσ is a differential surface element pointing away from the volume dτ is a differential volume element dr is a differential line element

1.1 Vector Identities

1.1.1 Identities Involving Only Vectors 12: 4 (a) A B C = A B C = B C A · × × · · × (b) A (B C) = (C B) A = B(A C) C(A B) × × × × · − · (c) (A B) (C D) = (A C)(B D) (A D)(B C) × · × · · − · · (d) (A B) (C D)= C(A B D) D(A B C) × × × × · − × · 1.1.2 Identities Involving 12: 4 10 ∇ (a) (ψA)= ψ( A)+ A ( ψ) ∇ · ∇ · · ∇ (b) (ψA)= ψ( A) A ( ψ) ∇× ∇× − × ∇ (c) (A B)= B ( A) A ( B) ∇ · × · ∇× − · ∇× (d) (A B) = (B )A (A )B + A( B) B( A) ∇× × ·∇ − ·∇ ∇ · − ∇ · (e) (A B)= A ( B)+ B ( A) + (A )B + (B )A ∇ · × ∇× × ∇× ·∇ ·∇ (f) (AB) = ( A)B + (A )B ∇ · ∇ · ·∇ (g) (ψ←→T )= ψ ←→T + ψ ←→T ∇ · ∇ · ∇ · (h) ( A) = 0 ∇ · ∇× 9 10 CHAPTER 1. MATHEMATICS

(i) 2A = ( A) A ∇ ∇ ∇ · −∇×∇× (j) (ψξ)= (φξ)= ψ ξ + ξ ψ ∇ ∇ ∇ ∇ (k) ( ψ ξ) = 0 ∇ · ∇ ×∇ (l) ψ = 2ψ ∇·∇ ∇ (m) ψ = 0 ∇×∇ 1.1.3 Identities Involving 12: 5 R (a) ψ dτ = ψ dσ ∇ volumeZ surfaceZ

(b) A dτ = dσ A ∇× × volumeZ surfaceI

(d) dr A = (dσ ) A × ×∇ × boundaryI surfaceZ

1.2 Curvilinear Coordinate Systems

1.2.1 Cylindrical Coordinates (r, θ, z) 12: 6−7 10 Diﬀerential volume: dτ = r dr dθ dz

Relation to cartesian coordinates:

x = r cos θ xˆ = cos φ rˆ sin φ φˆ − y = r sin θ yˆ = sin φ rˆ + cos φ φˆ z = z zˆ = zˆ

Unit vector diﬀerentials

drˆ dθ dθˆ dθ = θˆ = rˆ dt dt dt − dt Gradient ∂ψ 1 ∂ψ ∂ψ ψ = rˆ + θˆ + zˆ ∇ ∂r r ∂θ ∂z 1.2. CURVILINEAR COORDINATE SYSTEMS 11

Divergence

1 ∂ 1 ∂A ∂A A = (rA )+ θ + z ∇ · r ∂r r r ∂θ ∂z Curl 1 ∂A ∂A A = z θ rˆ ∇× r ∂θ − ∂z ∂A ∂A + r z θˆ ∂z − ∂r 1 ∂ 1 ∂A + (rA ) r zˆ r ∂r θ − r ∂θ Laplacian

1 ∂ ∂ψ 1 ∂2ψ ∂2ψ 2ψ = r + + ∇ r ∂r ∂r r2 ∂θ2 ∂z2 Vector-dot-grad

∂B A ∂B ∂B A B (A ) B = A r + θ r + A r θ θ rˆ ·∇ r ∂r r ∂θ z ∂z − r ∂B A ∂B ∂B A B + A θ + θ θ + A θ + θ r θˆ r ∂r r ∂θ z ∂z r ∂B A ∂B ∂B + A z + θ z + A z zˆ r ∂r r ∂θ z ∂z

1.2.2 Spherical Coordinates (r,θ,φ) 12: 8−9 10 Diﬀerential volume: dτ = r2 sin θ dr dθ dφ

Relation to cartesian coordinates

x = r sin θ cos φ xˆ = sin θ cos φ rˆ + cos θ cos φ θˆ sin φ φˆ − y = r sin θ sin φ yˆ = sin θ sin φ rˆ + cos θ sin φ θˆ + cos φ φˆ z = r cos θ zˆ = cos θ rˆ sin θ θˆ −

Gradient ∂ψ 1 ∂ψ 1 ∂ψ ψ = rˆ + θˆ + φˆ ∇ ∂r r ∂θ r sin θ ∂φ

Divergence

1 ∂ 1 ∂ 1 ∂A A = r2A + (sin θA )+ φ ∇ · r2 ∂r r r sin θ ∂θ θ r sin θ ∂φ 12 CHAPTER 1. MATHEMATICS

Curl 1 ∂ 1 ∂A A = (sin θA ) θ rˆ ∇× r sin θ ∂θ φ − r sin θ ∂φ 1 ∂A 1 ∂ + r (rA ) θˆ r sin θ ∂φ − r ∂r φ 1 ∂ 1 ∂A + (rA ) r φˆ r ∂r θ − r ∂θ Laplacian 1 ∂ ∂ψ 1 ∂ ∂ψ 1 ∂2ψ 2ψ = r2 + sin θ + ∇ r2 ∂r ∂r r2 sin θ ∂θ dθ r2 sin2 θ ∂φ2 Vector-dot-grad ∂B A ∂B A ∂B A B + A B (A ) B = A r + θ r + φ r θ θ φ φ rˆ ·∇ r ∂r r ∂θ r sin θ ∂φ − r ∂B A ∂B A ∂B A B cot θA B + A θ + θ θ + φ θ + θ r φ φ θˆ r ∂r r ∂θ r sin θ ∂φ r − r ∂B A ∂B A ∂B A B cot θA B + A φ + θ φ + φ φ + φ r + φ θ φˆ r ∂r r ∂θ r sin θ ∂φ r r 1.3 Integral Relations of Vector Calculus

In this section, let A A(x ,x ,x ) be a vector function that deﬁnes a vector ≡ i j k ﬁeld.

1.3.1 The Fundamental Theorem of Calculus If f(x) is a single-valued function on the interval [a,b] 14: 88 b f(x)dx = F (b) F (a) − Za

1.3.2 Gauss’s (or the Divergence) Theorem If τ is a volume enclosed by a surface σ, where dσ = nˆdσ and nˆ is a unit vector pointing away from τ 10: 31

( A) dτ = A dσ ∇ · · volumeZ surfaceI

1.3.3 Stoke’s (or the Curl) Theorem If σ is an open surface deﬁned by a boundary contour at the surface edge 10: 34

( A) dσ = A dr ∇× · · surfaceZ contourI 1.4. LEGENDRE POLYNOMIALS 13

1.4 Legendre Polynomials

Legendre’s equation 14: 337

d2y dy (1 x2) 2x + l(l + 1)y = 0 1 x 1 and l = 0, 1, 2, − dx2 − dx − ≤ ≤ · · ·

Legendre Polynomials 14: 289 Order Corresponding polynomial

l = 0 P0(x) = 1

l = 1 P1(x)= x l = 2 P (x)= 1 (3x2 1) 2 2 − l = 3 P (x)= 1 (5x3 3x) 3 2 − l = 4 P (x)= 1 (35x4 30x2 + 3) 4 8 − l = 5 P (x)= 1 (63x5 70x3 + 15x) 5 8 − ··· ···

Rodrigues’ formula 14: 286

1 dl P (x)= (x2 1)l l 2ll! dxl − Orthonormality 14: 286

1 π 2 P (x)P (x) dx = P (cos θ)P (cos θ)sin θ dθ = δ l m l m 2l + 1 lm Z−1 Z0 where δ is the Kronecker delta: l = m, δ = 1; l = m, δ = 0. lm lm 6 lm 1.5 Bessel Functions

1.5.1 Bessel’s Equation The most general form of Bessel’s equation is 14: 269

d2y 1 dy p2 + + λ2 y = 0 dx2 x dx − x2 which has the general solution 14: 270

y = AJp(λx)+ BYp(λx) where Jp are Bessel functions of the first kind and Yp are Bessel functions of the second kind (also known as Neumann Functions Np), both of order p. Bessel functions of the first kind have no closed form representation; 14 CHAPTER 1. MATHEMATICS however, they can be used to define Bessel functions of the second kind: 14: 270

J (x) cos(pπ) J (x) Y (x)= p − −p p sin(pπ)

1.5.2 Bessel Function Relations

The following relationships are also valid for Yp(x) by replacing Jp(x) with 14: 278−279 Yp(x)

2 (a) J (x)= J (x) J (x) 2 x 1 − 0 d (b) [J (x)] = J (x) dx 0 − 1 d (c) [xpJ (x)] = xpJ (x) dx p p−1 d (d) [x−pJ (x)] = x−pJ (x) dx p − p+1 2p (e) J (x)+ J (x)= J (x) p−1 p+1 x p d (f) J (x) J (x) = 2 J (x) p−1 − p+1 dx p d p p (g) J (x)= J (x)+ J (x)= J (x) J (x) dx p −x p p−1 x p − p+1

1.5.3 Asymptotic forms of Bessel Functions For x 14: 273 →∞ 2 1 1 J (x) cos x pπ π p ≈ πx − 2 − 4 r 2 1 1 Y (x) sin x pπ π p ≈ πx − 2 − 4 r For p 14: 273 →∞ 1 ex p 2 ex −p J (x) Y (x) p ≈ √2πp 2p p ≈− πp 2p r

1.6 Modiﬁed Bessel Functions

1.6.1 Bessel’s Modiﬁed Equation The most general form of Bessel’s modiﬁed equation is14: 274 d2y 1 dy p2 + λ2 + y = 0 dx2 x dx − x2 1.6. MODIFIED BESSEL FUNCTIONS 15

1 0

1 2 0.5 3 4

-0.5 0 1 2 3 4 5 6 7 8 9 10

Plots of Jp(x)

5 4 3 2 1 0 1 2 3 4 0 -1 -2 -3 -4 -5 0 1 2 3 4 5 6 7 8 9 10

Plots of Yp(x)

which has the general solution 14: 275

y = AIp(λx)+ BKp(λx)

where Ip are Modified Bessel functions of the first kind and Kp are modified Bessel functions of the second kind. Modified Bessel functions of the first kind have no closed form representation; however, they can be used to define Bessel functions of the second kind: 14: 275

π I (x) I (x) K (x)= −p − p p 2 sin(pπ) 16 CHAPTER 1. MATHEMATICS

1.6.2 Modiﬁed Bessel Functions Relations 14: 280 Relations involving Ip(x)

(a) xI (x) xI (x) = 2pI (x) p−1 − p+1 p d (b) I (x) I (x) = 2 I (x) p−1 − p+1 dx p d (c) x [I (x)] + pI (x)= xI (x) dx p p p−1 d (d) x [I (x)] pI (x)= xI (x) dx p − p p+1 d (e) [I (x)] = I (x) dx 0 1 2 (f) I (x)= I (x)+ I (x) 2 −x 1 0

14: 280 Relations involving Kp(x)

(g) xK (x) xK (x)= 2pK (x) p−1 − p+1 − p d (h) K (x)+ K (x)= 2 K (x) p−1 p+1 − dx p d (i) x [K (x)] + pK (x)= xK (x) dx p p − p−1 d (j) x [K (x)] pK (x)= xK (x) dx p − p − p+1 d (k) [K (x)] = K (x) dx 0 − 1 2 (l) K (x)= K (x)+ K (x) 2 x 1 0

1.6.3 Asymptotic Forms of Modiﬁed Bessel Functions For x 14: 278 →∞ ex 4p2 1 Ip(x) 1 − ≈ √2πx − 8x π 4p2 1 K (x) e−x 1+ − p ≈ 2x 8x r

1.7 Partial Diﬀerential Equations

1.7.1 Basis Functions for Laplace’s Equation Basis functions are the most general solutions to 2ψ = 0.18 ∇ ∗ If m is an integer, J−m → Ym. If k is imaginary, Jm(kr) → Im(|k| r) and Ym(kr) → Km(|k| r) † Ylm(θ,φ) is the spherical harmonic function 1.7. PARTIAL DIFFERENTIAL EQUATIONS 17

10 9 8 7 6 5 4 3 0 1 2 2 3 1 4 0 0 1 2 3 4 5 6

Plots of Ip(x)

5 4.5 4 3.5 3 2.5 2 4 1.5 3 2 1 0 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Plots of Kp(x)

Geometry Basis Function ψ

2D Cartesian ψ = (A sin kx + B cos kx) Ceky + De−ky n −n 2D Cylindrical ψ = A0 + B0 ln r + (A sin nθ + B cos nθ) (Cr + Dr ) × 3D Cartesian ψ = Aeikxxeikyyekzz with k2 + k2 k2 = 0 x y − z ±imθ ±kz ∗ 3D Cylindrical ψ = Ae e J±m(kr) l −(l+1) 2D Spherical ψ = Ar + Br Pl(cos θ) l l −(l+1) † 3D Spherical ψ = Almr + Blmr Ylm(θ, φ) m=−l P 18 CHAPTER 1. MATHEMATICS

1.8 Gaussian Integrals

Deﬁnite integral relations of Gaussian integrals 14: 255

∞ 2 1 π 1/2 (a) e−ax dx = 0 2 a Z ∞ 2 π 1/2 (b) e−ax dx = a Z−∞ ∞ 1/2 2 −ax2 −2bx π b (c) e e dx = e a for a>0 −∞ a Z ∞ 2 π 1/2 (d) xe−a(x−b)x dx = b a Z−∞ ∞ 1/2 2 −ax2 1 π (e) x e dx = 3 −∞ 2 a Z 1 n + 1 Γ /a(n+1)/2 a>0 2 2 ∞  n −ax2  (2k 1)!! π (f) x e dx =  − n=2k,a>0 0  2k+1ak a Z  r k! n=2k+1,a>0  k+1  2a  

Deﬁnite integrals of common Gaussian relations 21: 65 ∞ ∞ ∞ n −ax2 n −ax2 1 (n−1) −ax n x e dx x e dx x 2 e dx 0 −∞ 0

R 1/2 R 1/2 R 1/2 0 π π π 2a1/2 a1/2 a1/2 1 1 1 2a 0 a 1/2 1/2 1/2 2 π π π 4a3/2 2a3/2 2a3/2 1 1 3 2a2 0 a2 1/2 1/2 1/2 4 3π 3π 3π 8a5/2 4a5/2 4a5/2 1 2 5 a3 0 a3 1/2 1/2 1/2 6 15π 15π 15π 16a7/2 8a7/2 8a7/2

1.9 Error functions

The error function 14: 242

x 2 2 erf (x)= e−t dt √π Z0 1.9. ERROR FUNCTIONS 19

Taylor expansion of the error function 14: 242

∞ 2 ( 1)nx2n+1 2 x3 x5 x7 erf (x)= − = x + + . . . √π n!(2n + 1) √π − 3 10 − 42 Xn=0 The complimentary error function 14: 242

∞ 2 2 erfc (x)= e−t dt = 1 erf (x) √π − Zx

Taylor expansion of the complimentary error function 1: 469

2 e−x 1 3 15 erfc (x) 1 + + . . . ≈ x√π − 2x2 2 2 − 2 3 (2x ) (2x ) 20 CHAPTER 1. MATHEMATICS Chapter 2

Fundamental Constants and SI Units

Numerical values for all constants taken from the 2006 CODATA Interna- tionally Recommended Values of the Fundamental Physical Constants.

Universal Constants Constant Symbol Value Unit

Avagadro’s Constant N 6.022 141 79(30) 1023 mol−1 A × Boltzmann’s Constant k 1.380 650 4(24) 10−23 J/K B × 8.617 343 1(15) 10−5 eV/K × Elementary Charge e 1.602 176 487(40) 10−19 C × Impedance of Vacuum Z0 376.730 313 461 Ω µ0/ǫ0 Permittivity of Vacuum ǫ 8.854 187 817 10−12 F/m p 0 × Permeability of Vacuum µ 4π 10−7 N/A2 0 × Planck’s Constant h 6.626 068 96(33) 10−34 J s × · 4.135 667 33(10) 10−15 eV s × · H-bar (h/2π) ~ 1.054 571 628(53) 10−34 J s × · 6.582 118 99(16) 10−16 eV s × · Speed of Light (vacuum) c 2.997 924 58 108 m/s ×

21 22 CHAPTER 2. FUNDAMENTAL CONSTANTS AND SI UNITS

Atomic and Nuclear Constants Constant Symbol Value Unit

Electron Rest Mass m 9.109 382 15(45) 10−31 kg e × 5.485 799 0943(23) 10−4 u × 0.510 998 910(13) MeV/c2 Proton Rest Mass m 1.672 621 637(83) 10−27 kg p × 1.007 276 466 77(10) u 938.272 013(23) MeV/c2 Neutron Rest Mass m 1.674 927 211(84) 10−27 kg n × 1.008 664 915 97(43) u 939.565 346(23) MeV/c2 Deuteron (2H) Rest Mass m 3.343 583 20(17) 10−27 kg d × 2.013 553 212 724(78) u 1875.612 793(47) MeV/c2 Triton (3H) Rest Mass m 5.007 355 88(25) 10−27 kg t × 3.015 500 7134(25) u 2808.9209 06(70) MeV/c2 Helion (3He) Rest Mass m 5.006 411 92(25) 10−27 kg h × 3.014 932 2473(26) u 2808.391 383(70) MeV/c2 Alpha (4He) Rest Mass m 6.644 656 20(33) 10−27 kg α × 4.001 506 179 127(62) u 3727.379 109(93) MeV/c2 Proton to Electron m /m 1836.152 672 47(80) 6π5 p e ≈ Mass Ratio −10 Bohr Radius a0 0.529 177 208 59(36) 10 m 2 2 × (a0 = 4πǫ0~ /mee ) −15 Classical Electron Radius re 2.817 940 289 4(58) 10 m 2 2 × (re = e /4πǫ0mec ) Inverse Fine 1/α 137.035 999 679(94) Structure Constant Rydberg Constant R 1.097 373 156 852 7(73) 107 m−1 ∞ × R∞hc 13.605 691 93(34) eV Stefan-Boltzmann σ 5.670 400(40) 10−8 W/m2/K4 × Constant −28 2 Thomson Cross Section σTh 0.665 245 855 8(27) 10 m 2 × (σTh = (8π/3) re ) 0.665 245 855 8(27) barns 23

The System of International Units Quantity Symbol SI Unit Dimensions

Activity Becquerel s−1 A s2·C2 Capacitance C farad (F) kg·m2 Charge q *coulomb (C) C s·C2 Conductance siemens (S) kg·m2 s·C2 Conductivity σ siemens/meter (S/m) kg·m3 C Current I amp´ere (A) s 2 C Displacement D coulomb/m m2 kg·m Electric Field E volt/meter s2·C kg·m2 Electromotance ε volt (V) s2·C kg·m2 Energy W joule (J) s2 kg·m Force F newton (N) s2 Frequency ν hertz (Hz) s−1 kg·m2 Impedance Z ohm (Ω) s·C2 kg·m2 Inductance L henry (H) C2 Length l *meter (m) m kg·m2 Magnetic Flux Φ weber (Wb) s·C kg Magnetic Flux Density B tesla (T) s·C 2 m2·C Magnetic Moment µ ampere-m s C Magnetization M ampere-turn/m s·m kg·m Permeability µ henry/meter C2 s2·C2 Permittivity ǫ farad/meter kg·m3 2 C Polarization P coulomb/m m2 kg·m2 Electric Potential V volt (V) s2·C kg·m2 Power P watt (W) s3 kg Pressure p pascal (Pa) m·s2 kg·m2 Resistance R ohm (Ω) s·C2 kg·m3 Resistivity η ohm-meter s·C2 Temperature T *kelvin (K) K kg·m Thermal Conductivity κ watt/meter/kelvin s3 Time s *second s m Velocity v meter/second s

* denotes a fundamental SI base unit 24 CHAPTER 2. FUNDAMENTAL CONSTANTS AND SI UNITS

Energy Conversion Factors

Energy Temperature 1 eV = 1.602189 10−19 J ↔ × 1 eV = 1.1604505 104 K × Energy Mass 1 u = 931.501 MeV/c2 = 1.660566 10−27 kg ↔ × Energy Wavelength hc = 1239.8419 MeV fm = 1239.8419 eV nm ↔ · · ~c = 197.329 MeV fm = 197.329 eV nm · · e2/4πǫ = 1.439976 MeV fm 0 · Chapter 3

Electricity and Magnetism

In this chapter, all units are SI. e is the elementary electric charge q is the total particle charge Z is the particle atomic (proton) number n is the particle density U is energy r is the particle position v is the particle velocity ρ is volumetric charge density σ is surface charge density J is volumetric current density K is surface current density τ and σ are the volume and surface, respectively dτ, dσ, and dl are the volume, surface, and line elements, respectively b and f subscripts refer to bound and free charges

3.1 Electromagnetic in Vacuum

3.1.1 Fundamental Equations

Maxwell’s equations 10: 326

E = ρ/ǫ B = 0 ∇ · 0 ∇ · ∂B ∂E E = B = µ J + µ ǫ ∇× − ∂t ∇× 0 0 0 ∂t

Electrostatic scalar potential relations 10: 87

E = VV = E dl −∇ − · Z ρ 1 ρ 2V = V = dτ ∇ −ǫ 4πǫ r 0 0 Z

25 26 CHAPTER 3. ELECTRICITY AND MAGNETISM

Electrostatic vector potential relations 10: 240

µ J B = A 2A = µ JA = 0 dτ ∇× ∇ − 0 4π r Z Electromagnetic energy stored in the ﬁelds 10: 348

1 1 U = ǫ E2 + B2 dτ em 2 0 µ Z 0

Coulomb Force 10: 59

q q r r F = 1 2 1 − 2 4πǫ r r 2 r r 0| 1 − 2| | 1 − 2| Lorentz force law 10: 204

F = q (E + v B) ×

Biot-Savart law 10: 215

µ I ˆr B = 0 × dl 4π r2 Z

3.1.2 Boundary Conditions

For given surface , + and - refer to above and below , respectively. nˆ is S S a unit vector perpendicular to . S Electrostatic boundary conditions on E 10: 179

E⊥ E⊥ = σ/ǫ + − − 0

E|| E|| = 0 + − −

Magnetostatic boundary conditions on B 10: 241

B⊥ B⊥ = 0 + − −

B|| B|| = µ K + − − 0

B B = µ (K nˆ) + − − 0 × 3.2. ELECTROMAGNETICS IN MATTER 27

3.2 Electromagnetics in Matter

3.2.1 Fundamental Equations Maxwell’s equations in matter 10: 330

D = ρ B = 0 ∇ · f ∇ · ∂B ∂D E = H = J + ∇× − ∂t ∇× f ∂t 10: 179 The polarization in linear media (χe is the polarizability)

−3 P = ǫ0χeE [electric dipole moments per m ]

10: 274 The magnetization in linear media (χm is the magnetization)

−3 M = χmH [electric dipole moments per m ]

The displacement ﬁeld 10: 175,180

D = ǫ0E + P = ǫE (linear media only where ǫ ǫ (1 + χ )) ≡ 0 e The H-ﬁeld (Magnetic ﬁeld) 10: 269,275 1 H = B M µ0 − 1 = B (linear media only where µ µ (1 + χ )) µ ≡ 0 m

10: 167,168,267,268 Associated bound charges (σb, ρb) and currents (Kb, Jb)

σ = P nˆ ρ = P b · b −∇ · K = M nˆ J = M b × b ∇×

3.2.2 Boundary Conditions For given surface , + and - refer to above and below , respectively. nˆ is S S a unit vector perpendicular to . 10: 178,273 S

D⊥ D⊥ = σ D|| D|| = P|| P|| + − − f + − − + − − H⊥ H⊥ = M ⊥ M ⊥ H|| H|| = K nˆ + − − − + − − + − − f × 3.3 Dipoles

In this section, p and m are electric and magnetic dipoles, respectively. N is the torque and F is the force generated by the dipole. 28 CHAPTER 3. ELECTRICITY AND MAGNETISM

Deﬁnition 10: 149,244 Fields10: 153,155,246 Potentials10: 166,244

1 E (r)= 3 [3(p ˆr)ˆr p] p r dip 4πǫ0r 1 ·ˆ p rρ(r)dτ V = 2 · − dip 4πǫ0 r ≡ p ˆ Edip(r, θ)= 3 2 cos θˆr + sin θθ R 4πǫ0r B µ0 m r r m dip = 4πr3 [3( ˆ)ˆ ] µ0 m×ˆr m I dσ · − A (r)= 2 ≡ dip 4π r B (r, θ)= µ0m 2 cos θˆr + sin θθˆ R dip 4πr3

Electric 10: 164,165 Magnetic10: 257,258,281

F = (p ) E F = (m B) ·∇ ∇ · N = p E N = m B × × U= p E U= m B − · − ·

3.4 Circuit Electrodynamics

Microscopic Ohm’s law 10: 285

J = σcE

where σc is the conductivity. Resistivity, ρr, is deﬁned as ρr = 1/σc.

Macroscopic Ohm’s law 10: 287

V = IR

where V is the voltage, I is the current, and R is the resistance.

The voltage due to a changing magnetic ﬁeld (Faraday’s Law) 10: 295,296 ,

dΦ V = Φ= B dA − dt · surfaceZ

Capacitance is written as C, and inductance is written as L.18

Q = CV Φ= LI dV dI I = C V = L − dt − dt

Energy stored in capacitance and inductance 10: 106,317

1 1 U = LI2 + CV 2 2 2 3.5. CONSERVATION LAWS 29

3.5 Conservation Laws

Conservation of charge 10: 214 ∂ρ = J ∂t −∇ · Poynting vector 10: 347 1 S (E B) ≡ µ0 × Poynting’s theorem (integral form) 10: 347

dU d 1 2 1 2 1 = ǫ0E + B dτ (E B) dσ dt −dt 2 µ0 − µ0 × · volumeZ surfaceI

Poynting’s theorem (diﬀerential form) 10: 348

∂ (U + U )= S ∂t mechanical em −∇ · Maxwell’s stress tensor 10: 352 1 1 1 T ǫ E E δ E2 + B B δ B2 ij ≡ 0 i j − 2 ij µ i j − 2 ij 0 Electromagnetic force density on collection of charges 10: 352 ∂S f = ←→T ǫ µ ∇ · − 0 0 ∂t Total electromagnetic force on collection of charges 10: 353 d F = ←→T dσ ǫ µ S dτ · − 0 0 dt surfaceI volumeZ

Momentum density in electromagnetic ﬁelds 10: 355

pem = µ0ǫ0S

Conservation of momentum in electromagnetic ﬁelds 10: 356 ∂ (p + p )= ←→T ∂t mech em ∇ · 3.6 Electromagnetic Waves

In this section,

λ is the wavelength k = 2π/λ is the wave number ν is the frequency 30 CHAPTER 3. ELECTRICITY AND MAGNETISM

ω = 2πν is the angular frequency T = 1/ν is the period k = kkˆ is the wave number vector nˆ is the polarization vector in the direction of electric ﬁeld X˜ is a complex vector

The wave equation in three dimensions 10: 376

1 ∂2f 2f = ∇ v2 ∂t2 is satisﬁed by two transformations of Maxwell’s equations in vacuum 10: 376

∂2E ∂2B 2E = µ ǫ 2B = µ ǫ ∇ 0 0 ∂t2 ∇ 0 0 ∂t2 These have sinusoidal solutions, but it is more convenient to work with imaginary exponentials and take the real parts 10: 379

i(k·r−ωt) E˜ (r,t)= E˜0e nˆ

1 1 B˜ (r,t)= E˜ ei(k·r−ωt)(kˆ nˆ)= kˆ E˜ c 0 × c ×

EM Wave Relations 10: 381−382 Parameter Symbol Equation

Averaged energy per unit volume u 1 ǫ E2 h i 2 0 0 Averaged energy ﬂux density S 1 cǫ E2kˆ h i 2 0 0 Averaged momentum density P 1 ǫ E2kˆ h i 2c 0 0 Intensity I S h i I Radiation pressure P c

3.6.1 EM Waves in Matter In this section, θ is measured from the normal to the surface

Assuming no free charge or current in a linear media, the EM wave equations become 10: 383 ∂B E = 0 E = ∇ · ∇× − ∂t ∂E B = 0 B = µǫ ∇ · ∇× ∂t 3.6. ELECTROMAGNETIC WAVES 31

Speed of light in a material 10: 383 1 c v = = √ǫµ n

Index of refraction 10: 383 ǫµ n √ǫ ≡ ǫ µ ≈ r r 0 0 Intensity 10: 383 1 I = ǫvE2 2 0 Boundary conditions at a material surface 10: 384

⊥ ⊥ || || ǫ1E1 = ǫ2E2 E1 = E2 ⊥ ⊥ 1 || 1 || B1 = B2 B1 = B2 µ1 µ2 Reﬂection and transmission coeﬃcients 10: 386 I n n 2 I 4n n R ref = 1 − 2 T trans = 1 2 R + T = 1 ≡ I n + n ≡ I 2 inc 1 2 inc (n1 + n2) Snell’s Laws for oblique incidence on material surface 10: 388

kinc sin θinc = kref sin θref = ktrans sin θtrans

θinc = θref

sin θ n trans = 1 sin θinc n2

Fresnel Equations 13: 305−306 Polarization to E /E E /E incident plane trans inc ref inc

2 2 2 2n1 cos θinc n1 cos θinc−(µ1/µ2)√n2−n1 sin θinc Perpendicular 2 2 2 2 2 2 n1 cos θinc+(µ1/µ2)√n2−n1 sin θinc n1 cos θinc+(µ1/µ2)√n2−n1 sin θinc

2 2 2 2 2n1n2 cos θinc (µ1/µ2)n2 cos θinc−n1√n2−n1 sin θinc Parallel 2 2 2 2 2 2 2 2 (µ1/µ2)n2 cos θinc+n1√n2−n1 sin θinc (µ1/µ2)n2 cos θinc+n1√n2−n1 sin θinc

Brewster’s angle (no reﬂection of perpendicular incidence wave) 10: 390

2 2 1 β sin θB = − (n /n )2 β2 1 2 − 32 CHAPTER 3. ELECTRICITY AND MAGNETISM

where β = µ1n2/µ2n1

If the wave is in a conductor, it will experience damping due to the presence of free charges, subject to Jf = σE. Solving Maxwell’s equations gives 10: 394

k˜2 = µǫω2 + iµσω

Decomposing gives real and imaginary parts of the wave vector k˜ = k + iκ

1/2 1/2 ǫµ σ 2 ǫµ σ 2 k ω 1+ + 1 κ ω 1+ 1 ≡ 2 " ǫω # ≡ 2 " ǫω − # r r r r Knowing the imaginary part of the wave number, allows to know the damping of the wave which is characterized by a skin depth or the e-folding length d 1/κ. 10: 394 ≡ 3.7 Electrodynamics

We are allowed to choose A; two common gauges used are the Lorentz ∇ · and Coulomb gauge. 10: 421−422

Gauge A V Equation A Equation ∇ ·

∂V 2 ∂2V ρ 2 ∂2A Lorentz µ0ǫ0 V µ0ǫ0 2 = A µ0ǫ0 2 = µ0J − ∂t ∇ − ∂t − ǫ0 ∇ − ∂t − 2 ρ 2 ∂2A ∂V Coulomb 0 V = A µ0ǫ0 2 = µ0J + µ0ǫ0 ∇ − ǫ0 ∇ − ∂t − ∇ ∂t For any scalar function λ, any potential formulation is valid if 10: 420

∂λ A′ = A + λ V ′ = V ∇ − ∂t

3.7.1 Fields of Moving Charges In this section, A evaluates A at the retarded time18 || || Deﬁnition of retarded time 10: 423 r(t) r(t ) t t | − ret | ret ≡ − c The Lienard-Wiechert potentials 10: 432−433

q 1 µ q v V = A = 0 4πǫ rκ 4π rκ 0

3.7. ELECTRODYNAMICS 33

Electric ﬁeld of a moving point charge 10: 438

q 1 1 E = (ˆr v/c) 1 v2/c2 ˆr ((ˆr v/c) a/c) 4πǫ − − κ3r2 − × − × κ3rc 0 10: 438 Magnetic ﬁeld of a moving point charge

B = ˆr E/c || || × where κ = 1 rˆ v . − · c 3.7.2 Radiation by Charges In this section, u cˆr v, γ 1/ 1 v2/c2 is the relativistic gamma ≡ − ≡ − factor, and a is the acceleration. p Poynting vector associated with a moving charge 10: 460 1 S = E2ˆr (ˆr E) E µ0c − · Non-relativistic power radiated by a moving charge 10: 462

µ q2a2 P = 0 6πc Relativistic power radiated by a moving charge per solid angle 10: 463

dP q2 ˆr (u a) 2 = 2 | × ×5 | dΩ 16π ǫ0 (ˆr u) · Relativistic power radiated by a moving charge10: 463

µ q2γ6 v a 2 P = 0 a2 × 6πc − c !

Relativistic force of a moving charge 10: 467

µ q2 da F = 0 rad 6πc dt 34 CHAPTER 3. ELECTRICITY AND MAGNETISM Chapter 4

Single Particle Physics

In this chapter, all units are SI with the exception of temperature, which is deﬁned in the historical units of eV (electron-volts). e is the elementary electric charge q is the total particle charge Z is the particle atomic (proton) number m is the particle mass r is the particle position v is the particle velocity U is energy T is temperature; TkeV = T in units of kiloelectron-volts E and B are the electric and magnetic ﬁelds bˆ is a unit vector in the direction of B and indicate parallel and perpendicular to bˆ || ⊥

4.1 Single Particle Motion in E and B Fields

4.1.1 General Formulation Single particle trajectories result from solving Newton’s second law for a particle with charge q and mass m in electric and magnetic ﬁelds: 8: 141 dv dr m = q (E + v B) = v dt × dt If E and B are independent of time, the particle’s kinetic and potential energy is conserved 8: 142 1 mv2 + qV = constant 2 where V is the scalar potential (E = V ). −∇

4.1.2 Gyro Motion Solutions for B = B0zˆ;E=0

Particle initially has r = (x0,y0, z0), v = vk + v⊥, and arbitrary phase, φ.

35 36 CHAPTER 4. SINGLE PARTICLE PHYSICS

Parallel to the ﬁeld: 8: 143

z(t)= z0 + vkt

Perpendicular to the ﬁeld: 8: 144

x(t)= x + ρ sin(Ω t φ) x x + ρ sin φ g L c − g ≡ 0 L y(t)= y + ρ cos(Ω t φ) y y ρ cos φ g L c − g ≡ 0 − L

The guiding center position is (xg,yg); the larmor (or gyro) radius is ρL = v⊥/Ωc = mv⊥/qB; the larmor (or gyro) frequency is Ωc

4.1.3 Single Particle Drifts

In this section, Rc is the particle’s radius of curvature in a magnetic ﬁeld and is deﬁned as b b = R /R2 ·∇ − c c E B drift 8: 149 × E B v = × E B2

B drift 8: 153 ∇ mv2 B B v = ⊥ ×∇ ∇B 2qB B2

Curvature drift 8: 159

mv2 R B v k c κ = 2× qB Rc B

Polarization drift 8: 162

m dv v = ˆb E p qB × dt

Vacuum ﬁeld only B = 0 8: 160 →∇× m v2 R B v v 2 ⊥ c ∇B + κ = (vk + ) 2× qB 2 Rc B

Particle drift velocity for a general force F 8: 153

F B v = × F qB2 4.2. BINARY COULOMB COLLISIONS 37

4.1.4 Magnetic Moment And Mirroring In this section, i refers to the initial point, and f stand for the ﬁnal, or mirror, point.

Magnetic moment (the ﬁrst adiabatic invariant) 8: 167 mv2 (t) µ = ⊥ = constant 2B(t) Force on particle in magnetic ﬁelds where B/B 1 8: 171 ∇ ≪ F µ B k ≈− ∇k Velocity in terms of velocity space pitch angle 8: 174

v⊥i = v0 sin θ vki = v0 cos θ Conservation of energy 8: 174 1 1 m v2 + v2 = mv2 = U = constant 2 ⊥i ki 2 ⊥f total Mirroring condition 8: 175 2 2 U⊥i v⊥i Bmin sin θc = = 2 = Utotal v⊥f Bmax Fraction of trapped particles (Maxwellian distribution) 8: 176

1 π−θc 2π ∞ = sin θdθ dφ (v)v2dv Ftrapped n FMaxwellian Zθc Z0 Z0 where n is the total number of particles in the distrubition function

4.2 Binary Coulomb Collisions r is the relative distance v1 and v2 are particle velocities in the lab frame V and v are the center of mass and relative velocities v0 and b0 are the initial relative velocity and impact parameter χ is the scattering angle in the center of mass frame x˙ is the time derivative of quantity x

Force between 2 charged particles q q F = 1 2 −∇ 4πǫ r 0 Transformation to center of mass frame 8: 186

m1v1 + m2v2 V = v = v1 v2 m1 + m2 − m2v m1v v1 = V + v2 = V m1 + m2 − m1 + m2 38 CHAPTER 4. SINGLE PARTICLE PHYSICS

µ vo

rmin θmin b χ

Schematic of two body collision in the reduced mass frame.

Reduced mass 8: 186

m1m2 mµ = m1 + m2 Conservation of energy 8: 187

1 2 q1q2 1 2 mµv + = E0 = mµv0 = constant 2 4πǫ0r 2 Conservation of angular momentum 8: 187

m r v = L = m bv = constant µ × 0 − µ 0 Transformation to cylindrical coordinates 8: 188

r = rrˆ v =r ˙rˆ + rθ˙θˆ

Ordinary diﬀerential equation for unknown r(t) 8: 188

b b2 1/2 r˙ = v 1 2 90 ∓ 0 − r − r2 Solution in terms of scattering angle χ 8: 190

χ b90 q1q2 tan( )= = 2 2 b 4πǫ0mµv0b Impact parameter for 90 degree collision 8: 188

q1q2 b90 = 2 4πǫ0mµv0

4.3 Single Particle Collisions with Plasma a is an approaching test particle; t is a target plasma particle. Qat are quantities depending on approaching particle a incident upon target particle t. 4.3. SINGLE PARTICLE COLLISIONS WITH PLASMA 39

Total loss in test particle linear momentum 8: 192 d (m v )= (∆m v )n σv dt a a − a a t a = (∆m v )f (v ) v v b db dα d3v − a a a t | a − t| Z Deﬁnition of test particle collision frequency 8: 193 d (m v ) ν (m v ) dt a a ≡− at a a Test particle collision frequency 8: 193 1 ν (v )= (∆m v )f (v ) v v b db dα d3v at a m v a a t t | a − t| a a Z 4.3.1 Collision Frequencies Approximated expressions for frequencies hold only for v v v e ∼ te ≫ ti Electron-ion 8: 197 e4n ln Λ 1 e4n ln Λ ν = i i ei 4πǫ2m m v3 + 1.3v3 ≈ 4πǫ2m2v3 0 e µ e ti 0 e e 5 ni ln Λ −1 8.06 10 3 [s ] ≈ × ve Electron-electron 8: 197 e4n ln Λ 1 ν = e [s−1] ee 2πǫ2m2 v3 + 1.3v3 0 e e te Ion-ion 8: 197 e4n ln Λ 1 ν = i [s−1] ii 2πǫ2m2 v3 + 1.3v3 0 i i ti Ion-electron 8: 197 e4n ln Λ 1 ν = e [s−1] ie 4πǫ2m m v3 + 1.3v3 0 e i i te Collision frequency scalings 8: 197

m 1/2 m ν ν ν e ν ν e ν ee ∼ ei ii ∼ m ei ie ∼ m ei i i The Coulomb logarithm 8: 194

12πǫ3/2T 3/2 T 3/2 Λ 0 e = 4.9 107 kev ≈ 1/2 3 × 1/2 ne e n20 where ln Λ 15 20 for most plasmas. ≈ − 40 CHAPTER 4. SINGLE PARTICLE PHYSICS

4.3.2 Collision Times Electron-ion collision time 11: 5

12π3/2ǫ2m1/2T 3/2 τ = 0 e e ei 2 4 √2niZi e ln Λ T 3/2 16 e, keV = 1.09 10 2 [s] × Zi ni ln Λ

Ion-ion collision time 11: 5

12π3/2 ǫ2m1/2T 3/2 τ = 0 i i ii 1/2 4 4 2 niZi e ln Λi 3/2 m 1/2 T = 4.67 1017 i i, keV [s] × m Z4n ln Λ p i i i

An ion collision time is sometimes deﬁned to be τi = √2τii Ion-impurity collision time 11: 177

12π3/2 m1/2T 3/2ǫ2 τ = i i 0 iI 1/2 2 2 4 2 nI Zi ZI e ln Λ 3/2 m 1/2 T = 4.67 1017 i i, keV [s] × m Z2Z2n ln Λ p i I i Electron-to-ion energy transfer time

3 2 n (Te Ti) Rei = − mi τ 2me e Thermal equilibration frequency (rate of species a equilibrating to species b) 12: 34

(m m )1/2Z2Z2n ln Λ νt = 1.8 10−19 a b a b b [s−1] ab 3/2 × (maTb + mbTa)

For ions and electrons with T T = T , ν n = ν n 12: 34 e ≈ i ei e ie i

Z2 ln Λ νt = 3.2 10−15 [m3s−1] ei 3/2 × (mi/mpT )

4.4 Particle Beam Collisions with Plasmas

20 In this section, plasma density is written as n20 = n/10 4.4. PARTICLE BEAM COLLISIONS WITH PLASMAS 41

Beam-electron collision frequency 8: 202−203

Z2e4n ln Λ 1 ν (v )= b e be b 4πǫ2m m v3 + 1.3v3 0 e b b te 2 4 1/2 1 Zb e me ne ln Λ 3 3 (vte v only) ≈ 3(2π)3/2 2 3/2 ≫ b ǫ0mbTe n20 −1 = 100 3/2 [s ] (alpha heating only) TkeV Beam-ion collision frequency 8: 203

Z2e4n ln Λ 1 ν (v )= b i bi b 4πǫ2m m v3 + 1.3v3 0 µ b b ti 2 4 1 Zb e ni ln Λ 3 3 2 3 (vti vb only) ≈ 4π ǫ0mµmbvb ≪ n = 0.94 20 [s−1] (alpha heating only) 3/2 (Ebeam)MeV

When νbe = νbi, vb = vcrit and the beam changes the plasma particle it preferentially damps upon. For vb > vcrit, the beam damps on plasma electrons; for vb < vcrit, the beam damps on plasma ions. The critical 2 2/3 beam energy is calculated as Ec = 1/2mbeamvbeam = 14.8(mbeam/mi )Te. For a 15 keV plasma, the critical energy corresponds to 660 keV.

A high energy beam entering a plasma has energy behavior (purely slowing down on electrons) 26: 249

3/2 2/3 −3t/τse Ec −3t/τse Eb = Eb,0 e 1 e " − Eb,0 − # where Eb,0 is the initial beam energy and τse is the slowing down time 26: 246 assuming vb < vte

3/2 2 3/2 3(2π) ǫ0mbTe τse = 1/2 4 me ne ln Λ 42 CHAPTER 4. SINGLE PARTICLE PHYSICS Chapter 5

Plasma Parameters and Deﬁnitions

In this chapter, all units are SI with the exception of temperature, which is deﬁned in the historical units of eV (electron-volts). e is the elementary electric charge Z is atomic (proton) number m is the particle mass e and i subscripts refer to electrons and ions, respectively B is the magnetic ﬁeld 20 n is the particle density; n20 = n/10 T is temperature; TkeV = T in units of kiloelectron-volts

5.1 Single Particle Parameters

Thermal speed1

T 1/2 T 1/2 v = e v = i te m ti m e i Plasma frequencies [radians/s] 8: 135

n e2 1/2 n (Z e)2 1/2 ω = e ω = i i pe m ǫ pi m ǫ e 0 i 0 Cyclotron frequencies [radians/s] 8: 134

eB ZieB Ωe = Ωi = me mi 1Some references deﬁne thermal speed with an additional factor of √2. The deﬁnition given here results in vt being the standard deviation of a Gaussian distribution, which is the widely used convention in statistics and mathematical physics. In either case, it’s just a question of how the factor of “1/2” in the exponential term of the Gaussian distribution is treated.

43 44 CHAPTER 5. PLASMA PARAMETERS AND DEFINITIONS

Gyro radii [m] 8: 134

1/2 1/2 (2meTe) (2miTi) ρLe = ρLi = eB ZieB

Single particle parameters as functions of magnetic ﬁeld B [T], −3 3 density n20 [m ], and temperature T [keV] ω ω Particle Plasma frequencies f Cyclotron frequencies f Gyro radii [m]

h i h i 1/2 11 1/2 11 −4 TkeV electron 5.641 10 n20 rad/s 1.759 10 B rad/s 1.066 10 B × 1/2 × × 89.779 n20 GHz 28.000 B GHz 1/2 10 1/2 7 −3 TkeV proton 1.316 10 n20 rad/s 9.579 10 B rad/s 4.570 10 B × 1/2 × × 2.094 n20 GHz 15.241 B MHz 1/2 9 1/2 7 −3 TkeV deuteron 9.312 10 n20 rad/s 4.791 10 B rad/s 6.461 10 B × 1/2 × × 1.482 n20 GHz 7.626 B MHz 1/2 9 1/2 7 −3 TkeV triton 7.609 10 n20 rad/s 3.200 10 B rad/s 7.906 10 B × 1/2 × × 1.211 n20 GHz 5.092 B MHz 1/2 9 1/2 7 −3 TkeV helion 7.610 10 n20 rad/s 6.400 10 B rad/s 3.952 10 B × 1/2 × × 1.211 n20 GHz 10.186 B MHz 1/2 9 1/2 7 −3 TkeV alpha 6.605 10 n20 rad/s 4.822 10 B rad/s 4.554 10 B × 1/2 × × 1.051 n20 GHz 7.675 B MHz

5.2 Plasma Parameters

Debye length 8: 125

2 2 1 1 1 e n0 e n0 2 = 2 + 2 = + λD λDe λDi ǫ0Te ǫ0Ti ǫ T 1/2 T 1/2 λD 0 e = 2.35 10−5 keV [m] e ≈ e2n × n 0 20 Debye-shield ion potential (spherical coordinates) 26: 37 e V = e−r/λD 4πǫ0r

Volume of a Debye sphere 3 4 = πλ3 VD 3 D 5.3. PLASMA SPEEDS 45

Plasma parameter 8: 133

3/2 4 ǫ T 3/2 T Λ = n = π 0 e n 5.453 106 keV D D 0 2 0 1/2 V 3 e n0 ≈ × n20 Eﬀective plasma charge 8: 56

2 njZj all ions 1 2 Zeﬀ = = njZj P njZj ne all ions all ions X P 5.3 Plasma Speeds

In this section, ρ0 is the mass density of the plasma, p0 is the plasma pressure, and γ is the adiabatic index.

Alfv´en speed 8: 314

2 1/2 va = (B0 /µ0ρ0) Sound speed 5: 96,67

ZT + γT 1/2 c e i s ≈ m i where

γ = 1 (isothermal ﬂow) 2+ N γ = (N is the number of degrees of freedom) N

Plasma mach number 5: 298 v M plasma ≡ cs

5.4 Fundamentals of Maxwellian Plasmas

General Maxwellian velocity distribution function 21: 64−65 bm (v , v , v )= (v)= C exp (v a )2 + (v a )2 + (v a )2 FM x y z FM − 2 x − x y − y z − z where ax, ay, az, b, and c are constants. If ax = ay = az = 0 we have an (ordinary) Maxwellian; otherwise we have a drifting Maxwellian where the drift (or mean) velocity is vdr = (ax, ay, az).

Ordinary Maxwellian velocity distribution function for a plasma 21: 64−65 m 3/2 m (v)= n exp v2 + v2 + v2 FM 2πT −2T x y z 46 CHAPTER 5. PLASMA PARAMETERS AND DEFINITIONS

Deﬁnition of temperature in a Maxwellian plasma 21: 66

3 1 1 nT n m(v2 + v2 + v2) (v) v2 + v2 + v2 dv 2 ≡ 2 x y z ≡ FM 2 x y z Z Total number density of particles in a Maxwellian plasma 21: 66

∞ n = (v) dv = 4π w2 (w) dw FM FM allZ Z0 velocity space

where we have transformed to spherical coordinates such that dv = 2 2 2 2 1/2 w sinθ dw dθ dφ and w = (vx + vy + vz ) .

Average (thermal) particle speed in a Maxwellian plasma 21: 67

∞ 1 8T 1/2 c¯ w (w) dw = ≡ n FM πm Z0

Thermal particle ﬂux in a single dimension x for a Maxwellian plasma 21: 67

∞ ∞ ∞ 1 Γ v (v) dv dv dv = nc¯ ≡ x FM x y z 4 Z0 −∞Z −∞Z

5.5 Deﬁnition of a Magnetic Fusion Plasma

A fusion plasma is defined as an electrically conducting ionized gas that is dominated by collective effects and that magnetically confines its composing particles. If L is the macroscopic length scale of the plasma, ωtransit is 1 over the time required for a particle to cross the plasma, and vT is the thermal particle velocity, the criteria to be a fusion plasma are: 8: 136

Required condition Physical consequence

λ L Shielding of DC electric fields D ≪ ω ω = v /L Shielding of AC electric fields pe ≫ transit te Λ 1 Collective effects dominate D ≫ ρ L Magnetic confinment of particle orbits Li ≪ Ω v /L Particle gyro orbits dominate free streaming i ≫ ti 5.6. FUNDAMENTAL PLASMA DEFINITIONS 47

5.6 Fundamental Plasma Deﬁnitions

5.6.1 Resistivity Plasma resistivity of an unmagnetized plasma 5: 179,183 m Z ln Λ η = e 5.2 10−5 eﬀ [Ω m] 2 3/2 nee τc ≈ × · TeV Spitzer resistivity of a singly charged unmagnetized plasma 26: 71 m m1/2e2 ln Λ η = 0.51 e = 0.51 e s 2 2 3/2 nee τe 3ǫ0(2πTe) ln Λ 1.65 10−9 [Ω m] ≈ × 3/2 · Te, keV Spitzer resistivity of a singly charged magnetized plasma 26: 71

ηs,|| to B = ηs ηs,⊥ to B = 1.96ηs Spitzer resistivity of a plasma with impurities 26: 72

ηs = Zeﬀ ηs Spitzer resistivity of a pure non-hydrogenic plasma of charge Z 26: 72 N = 0.85 for Z = 2 η(Z)= N(Z)Zηs where ( N = 0.74 for Z = 4

5.6.2 Runaway Electrons Volumetric runaway electron production rate 26: 74 2 n E 1/2 E 2E 1/2 R = exp D D √π τ E − 4E − E se D " # 26: 74 where the Driecer electric ﬁeld, ED is ne3 ln Λ ED = 2 2 4πǫ0mevte 6 n ln Λ 4.582 10 2 [V/m] ≈ × vte and the electron slowing down time, τ , for v v is 26: 74 se e ≫ te 4πǫ2m2v3 τ = 0 e e se ne4 ln Λ v3 1.241 10−6 e [s] ≈ × n ln Λ Relativistic runaway electron limit (Connor-Hastie limit) 26: 74 ne3 ln Λ E < 2 2 4πǫ0mec n ln Λ 4.645 10−53 ≈ × me 48 CHAPTER 5. PLASMA PARAMETERS AND DEFINITIONS Chapter 6

Plasma Models

In this chapter, all units are SI with the exception of temperature, which is deﬁned in the historical units of eV (electron-volts). u is the plasma ﬂow velocity v is the particle velocity vector a is the particle acceleration vector m is the particle mass α is a general particle nα is the number density of particle α ρ is the plasma mass density p is the plasma pressure a and R0 are the minor and major radius of a toroidal plasma κ is the plasma elongation e is the fundamental charge unit i and e subscripts refer to the ions and electrons, respectively

6.1 Kinetic

The kinetic Vlasov equation7: 10

∂f + v f + a f = (f) ∂t ·∇ ·∇v C where (f) is the integro-diﬀerential collision operator. C The collisionless kinetic Vlasov equation

∂f q + v f + (E + v B) f = 0 ∂t ·∇ m × ·∇v where in a Maxwellian distribution8: 46

1 2 2 f = 3/2 exp v /2vt (πvt) −

49 50 CHAPTER 6. PLASMA MODELS

6.2 Two Fluids

Continuity equation 7: 15

dn α + n v = 0 dt α∇ · α α Momentum equation 7: 15

dv n m q n (E + v B)+ ←→P = R α α dt − α α α × ∇ · α α α Energy equation 7: 15

3 dT n α + ←→P : v + h = Q 2 α dt α ∇ α ∇ · α α α Maxwell’s equations 7: 15

∂B E = ∇× − ∂t

1 ∂E B = µ (Zen v en v )+ ∇× 0 i i − e e c2 ∂t

e E = (ni ne) ∇ · ǫ0 −

B = 0 ∇ · where

Convective derivative 7: 14 d ∂ + v dt ≡ ∂t α ·∇ Heat generated by unlike collision 7: 13 1 Q m w2 C dw α ≡ 2 α α αβ Z Mean momentum transfer from unlike particles 7: 14

R m wC dw α ≡ α αβ Z Heat ﬂux due to random motion 7: 14 1 h n m w2w α ≡ 2 α α

6.2. TWO FLUIDS 51

Temperature 7: 14

T p /n α ≡ α α Anisotropic part of pressure tensor 7: 14

←→Π ←→P p ←→I α ≡ α − α Pressure tensor 7: 14

←→P n m ww α ≡ α α h i Scalar pressure 7: 14

1 p n m w2 α ≡ 3 α α

Collision operator 7: 11

(f)= C C αβ Xβ Macroscopic number density

n (r,t) f du α ≡ α Z Macroscopic ﬂuid velocity

1 v (r,t) uf du α ≡ n α α Z Velocity representing thermal motion of the particles

w = u v (r,t) − α Relations involving the collision operators 7: 11 Cij

(a) du = du = du = du = 0 Cee Cii Cei Cie Z Z Z Z (b) m u du = m u du = 0 e Cee i Cii Z Z 1 1 (c) m u2 du = m u2 du = 0 2 e Cee 2 i Cii Z Z (d) (m u + m u ) du = 0 e Cei i Cie Z 1 (e) m u2 + m u2 du = 0 2 e Cei i Cie Z 52 CHAPTER 6. PLASMA MODELS

6.3 One Fluid

Taking the two ﬂuid equation, we assume me 0 and ni = ne n so that 7: 17 → ≡

ρ = min

v = vi v = v J/en e −

Additional simpliﬁcations p = nT = pe + pi and T = Te + Ti lead to the one-ﬂuid equations 7: 18

Continuity of mass equation 7: 18

∂ρ + ρv = 0 ∂t ∇ ·

Continuty of charge equation 7: 18

J = 0 ∇ · Momentum equation 7: 18

dv ρ J B + p = (←→Π + ←→Π ) dt − × ∇ −∇ · i e

Force balance 7: 19 1 E + v B = J B p ←→Π R × en × −∇ e − ∇ · e e Equations of state 7: 19

d p 2 i = Q h ←→Π : v dt ργ 3ργ i − ∇ · i − i ∇

d p 2 J 1 p e = Q h ←→Π : v + J e dt ργ 3ργ e − ∇ · e − e ∇ − en en ·∇ ργ Maxwell’s equation (low frequency limit) 7: 19

B = µ J ∇× 0

B = 0 ∇ ·

∂B E = ∇× − ∂t 6.4. MAGNETOHYDRODYNAMICS (MHD) 53

6.4 Magnetohydrodynamics (MHD)

MHD scalings 8: 247

length : a >> ρ >> [ρ λ ] Li Le ∼ De frequency :ν ¯ << v /a << ω << [ω ω ] e1 ti ci ce ∼ pe velocity : v v << v << c ti ∼ a te where a is the tokomak minor radius.

The MHD equations

Continuity of mass8: 252 dρ + ρ v = 0 dt ∇ · Momentum equation8: 252 dv ρ = J B p dt × −∇ Ohm’s law 8: 252

E + v B = 0 (ideal MHD) ×

E + v B = η J (resistive MHD) × || Equation of state8: 252 d p = 0 dt ργ Maxwell’s equation (low frequency limit)8: 252 ∂B E = ∇× − ∂t

B = µ J ∇× 0

B = 0 ∇ ·

where η|| is the parallel resistivity.

6.4.1 Frozen-in Magnetic Field In ideal MHD, it is found that 8: 300 dΦ = 0 dt where Φ = B dS is the magnetic ﬂux. · R 54 CHAPTER 6. PLASMA MODELS

MHD relations for linear plasma devices Case Current Relation Equilibrium Relation

2 8: 265 dBz d Bz θ-pinch µ0Jθ = p + = 0 − dr dr 2µ0 B2 B2 z-pinch 8: 267 µ J = 1 d(rBθ ) d p + θ + θ = 0 0 z r dr dr 2µ0 µ0r 2 2 2 8: 269 dBz 1 d(rBθ ) d Bθ Bz Bθ screw pinch µ0J = θˆ + ˆz p + + + = 0 − dr r dr dr 2µ0 2µ0 µ0r 6.5 MHD Equilibria

The generalized MHD equilibrium equations 8: 261

J B = p B = µ J B = 0 × ∇ ∇× 0 ∇ ·

6.6 Grad-Shafranov Equation

The Grad-Shafranov equation is the solution of the ideal MHD equations in two dimensions. In this section, Aφ is the toroidal component of the vector potential. 7: 110−111 dp dF ∆∗ψ = µ R2 F − 0 dψ − dψ where

1 B = ψ φˆ + F/Rφˆ R∇ ×

1 dF 1 µ J = ψ φˆ ∆∗ψφˆ 0 R dψ ∇ × − R where

∂ 1 ∂ψ ∂2ψ ∆∗ψ R + ≡ ∂R R ∂R ∂Z2 ψ 1 ψ = p = RA = B dA 2π φ 2π p · Z F (ψ)= RBφ

6.7 MHD Stability

This section deals with how a plasma behaves when it is perturbed slightly from equilibrium. Therefore, most physical parameters have an equilibrium value (subscript 0) with an added perturbed value (subscript 1). We also 8: 311 deﬁne the displacement vector ξ = v˜1. R 6.7. MHD STABILITY 55

Full Solutions to the Grad-Shafranov Equation Geometry Solution

∞ 7: 148 A 2 C 3 m High β Noncircular Tokamak 4 r + 8 r cos(θ)+ Hmr cos(mθ) m=0 Spherical Tokamak 7: 163 A Z2 + C R4 + c + c RP2 + c R4 4R2Z2 − 2 8 1 2 3 −

ρ˜ = ρ ξ˜ 1 −∇ · 0 p˜ = ξ˜ p γp ξ˜ 1 − ·∇ 0 − 0∇ · Q˜ B˜ = ξ˜ B ≡ 1 ∇× × 0 ∂2ξ˜ ρ = F(ξ˜) ∂t2 F(ξ˜)= J B˜ + J˜ B p˜ × 1 1 × −∇ 1 1 1 F(ξ˜)= ( B) Q˜ + ( Q˜ ) B + (ξ˜ p + γp ξ˜) µ0 ∇× × µ0 ∇× × ∇ ·∇ ∇ · Then assuming all pertubed quantities Q = Q exp [ i (ωt k r)], we can 1 1 − − · find expressions for the first order terms by letting ∂ ω and k. ∂t → ∇→ 6.7.1 Variational Formulation A different way of looking at the stability problem is given by 7: 250

δW (ξ∗, ξ) ω2 = K(ξ∗, ξ) where

1 δW (ξ∗, ξ)= ξ∗ F(ξ)dr −2 · Z 1 1 1 = ξ∗ [ ( Q) B + ( B) −2 · µ ∇× × µ ∇× Z 0 0 Q + (γp ξ + ξ p)] dr × ∇ ∇ · ·∇ 1 K(ξ∗, ξ)= ρ ξ 2dr 2 | | Z

An equilibrium is stable if δW 0. The energy principle can be evaluated ≥ simply in two cases: a conducting wall directly in contact to the plasma (n ξ (r ) = 0) or with a vacuum region next to the plasma. A vacuum · ⊥ wall region is more realistic than an adjacent conducting wall, so the variational principle becomes δW = δWF + δWS + δWV , where F, S, V refer to ﬂuid, surface, and vacuum, respectively. 7: 261 56 CHAPTER 6. PLASMA MODELS

The general screw pinch stability with a vacuum region is given by two diﬀerent expressions. For internal modes, 7: 293

a 2π2R δW = 0 fξ′2 + gξ2 dr µ 0 Z 0 with ξ(a) = 0. For external modes, 7: 293

a 2π2R krB mB r2ΛF 2 δW = 0 fξ′2 + gξ2 dr + z − θ rF + ξ2 µ  k2r2 m a 0 Z 0 | | a 0 

where ξa is arbitrary and 

rF 2 f = 2 k0

k2 k2r2 1 k2 mB g = 2 (µ p)′ + 0 − rF 2 + 2 kB θ F k2 0 k2r2 rk4 z − r 0 0 0

′ ′ m K 1 (K Ia)/(I Ka) Λ= | | a − b b − kaK′ 1 (K′ I′ )/(I′ K′ ) a − b a b a

mB F = k B = θ + kB · r z 6.8. STABILITY OF THE SCREW PINCH 57

6.8.1 Suydam’s Criterion For stability in a screw pinch 7: 298

rB2 q′ 2 z + 8p′ > 0 µ q 0 58 CHAPTER 6. PLASMA MODELS Chapter 7

Transport

In this chapter, all units are SI with the exception of temperature, which is defined in the historical units of eV (electron-volts). e is the elementary electric charge Z is the atomic (proton) number m is the particle mass 20 n is the number density; n20 = n/10 v is the particle velocity vector ρ is the plasma mass density T is the plasma temperature χ is the thermal diffusivity q is the heat flux and indicate parallel and perpendicular to B || ⊥ a and R0 are the minor and major radii of a toroidal plasma S is a general source term

7.1 Classical Transport

7.1.1 Diﬀusion Equations in a 1D Cylindrical Plasma If the following approximations are made to the MHD single ﬂuid equations 8: 451−453

∂v (a) neglect inertial terms: ρ ∂t = 0

(b) split resistivity into components: ηJ η J + η J → ⊥ ⊥ k k (c) equate electron and ion temperature: T T T e ≈ i ≡

(d) introduce the low-β tokamak expansion: Bz(r, t) = B0 + δBz(r, t) where B is a constant and δB 1 0 z ≪ then a short calculation leads a set of diﬀusion-like transport equations

59 60 CHAPTER 7. TRANSPORT

Particles 8: 453

∂n 1 ∂ ∂n n ∂T 2ηk n ∂(rB ) 2nT η = rD + + θ D = ⊥ ∂t r ∂r n ∂r T ∂r β η rB ∂r n B2 p ⊥ θ 0 where the poloidal beta is β = 4µ nT/B2 1 p 0 θ ∼ Temperature 8: 453 ∂T 1 ∂ ∂T 3n = rnχ + S ∂t r ∂r ∂r Magnetic ﬁeld 8: 453

∂(rB ) ∂ D ∂(rB ) ηk θ = r B θ D = ∂t ∂r r ∂r B µ 0

7.1.2 Classical Particle Diffusion Coefficients Classical particle diffusion results from coulomb collisions. No shift in the center of mass occurs for like-particle collisions; therefore, D = 0 for like- particles and only unlike-particle collisions lead to particle diffusion, imply- electrons ions ing Dn = Dn .

Net momentum electron-ion exchange collision frequency 8: 217

2 ω n ν¯ = pe ln Λ 1.8 105 20 ei π Λ ≈ × 3/2 r TkeV Random walk diffusion coefficient 8: 462 2 ν¯eimeTe rLe Drw = 4 2 2 e B0 ∼ τ¯ei Fluid model diffusion coefficient 8: 463

2nT η⊥ ν¯eimeTe Dfm = 2 = 2 2 2 B0 e B0 Braginskii diﬀusion coeﬃcient 8: 463

−3 n20 2 DBrag = 2.0 10 [m /s] × 2 1/2 B0 TkeV

7.1.3 The Collision Operator

In this section, Einstein summation notation (AaBa = AaBa) is used, a and u v v′, and d3v′ is a diﬀerential volume elementP in velocity space. ≡ − The Fokker-Planck form of the collision operator 11: 29 ∂ ∂ C (f ,f )= Aabf + (Dabf ) ab a b ∂v k a ∂v kl a k l 7.1. CLASSICAL TRANSPORT 61 where 11: 28−29 m ∂φ Aab 1+ a Lab b k ≡ m ∂v b k

2 ab ab ∂ ψb Dkl L ≡− ∂νk∂vl

Z Z e2 Lab a b ln Λ ≡ m ǫ a 0

1 1 φ (v) f (v′)d3v′ b ≡−4π u b Z

1 ψ (v) uf (v′)d3v′ b ≡−8π b Z Rosenbluth form of the collision operator (general) 11: 30

Z2Z2e4 ln Λ ∂ f (v) ∂f (v′) f (v′) ∂f (v) C (f ,f )= a b U a b b a d3v′ ab a b − 8πǫ2m ∂v kl m ∂v′ − m ∂v 0 a k Z b l a l where 11: 29 u2δ u u U kl − k l kl ≡ u3 Rosenbluth form of the collision operator (Maxwellian plasma) 11: 37

1 ∂ m 1 ∂f C (f ,f )= νab (f )+ v3 a νabf + νabv a ab a b0 D L a v2 ∂v m + m s a 2 || ∂v a b 11: 38 where xα = v/vT hα and 1 1 ∂ ∂f 1 ∂f (f ) sin θ a + a L a ≡ 2 sin θ ∂θ ∂θ sin2θ ∂φ

2x x 0 φ(x) xφ′(x) 3√π → G(x) − = ≡ 2x2  1  x 2x2 →∞  ab erf(xb) G(xb) νD (v) =ν ˆab −3 xa

2T m G(x ) νab(v) =ν ˆ a 1+ b b s ab T m x b a a 62 CHAPTER 7. TRANSPORT

ab G(xb) ν|| (v)=2ˆνab 3 xa

2 2 4 nbZa Zb e ln Λ νˆab = 2 2 3 4πǫ0mavT ha Krook collision operator 11: 84

C(f)= ν(f f) 0 −

where ν is chosen as some characteristic collision time and f0 is a base distribution frequency, which is often chosen to be a Maxwellian.

7.1.4 Classical Thermal Diﬀusivities Random walk thermal diﬀusivities 8: 464 2 2 1 vti rLi χi = 2 4 Ωi τ¯ii ∼ τ¯ii

2 2 1 vte rLe χe = 2 4 Ωeτ¯ee ∼ τ¯ee Braginskii Thermal Diﬀusivity Coeﬃcients (50%-50% D-T plasma) 8: 465

n20 2 χi = 0.10 [m /s] 2 1/2 B0 Tk

−3 n20 2 χe = 4.8 10 [m /s] × 2 1/2 B0 TkeV

7.2 Neoclassical Transport

The following formulae are only valid in the so-called “banana regime” of tokamak confinement devices, where a significant fraction of confined particles undergo magnetic mirroring due to inhomogeneity of the magnetic fields.

7.2.1 Passing Particles Half-transit time 8: 480

l πR0q τ1/2 = = vk vk

Drift Velocity 8: 481 m v R B 1 v v i 2 ⊥ c 2 ⊥ rˆ θˆ D = vk + 2× vk + ( sin θ + cos θ) eB 2 Rc B ≈ ΩiR0 2 7.2. NEOCLASSICAL TRANSPORT 63

7.2.2 Trapped Particles In this section, q is the tokamak safety factor.

The minimum (inboard) and maximum (outboard) magnetic ﬁelds 8: 484 R R B = B 0 B = B 0 min 0 R + a max 0 R a 0 0 − Trapped particle condition 8: 484

2 vk Bmin R0 a a 2 < 1 = 1 − 2 v − Bmax − R0 + a ≈ R0 Fraction of trapped particles (maxwellian distribution ) 8: 485 FM 1 π−θc 2π ∞ 2a 1/2 = sin θdθ dφ (v)v2dv = cos θ Ftrapped n FM c ≈ R Zθc Z0 Z0 0 Half-bounce time8: 487

l 2l 2πR0q τ1/2 ≈ vk ≈ vk ≈ vk Full-bounce frequency 8: 487

vk ωB = 2R0q Mean square step size (random walk model) 8: 488

2 2 4 2 2 vD 2 q v (∆l) = (∆r) = 4| 2| cos θ0 = 2 2 2 h i ωB h i Ωi vk Mean step size (averaged over velocity) 8: 488 R v2 R (∆l)2 3 q2 0 ti q2 0 ρ2 ≈ a Ω2 ∼ a Li i where it has been assumed that

2 3T 3 2 2 a 2 a 2 v = vti vk v 3vti ∼ m 2 ≈ R0 ∼ 2R0

7.2.3 Trapped Particle Neoclassical Transport Coefficients Random walk model neoclassical diffusion coefficient 8: 489 (∆r)2 R 3/2 2m T R 3/2 DNC = f h i = 5.2q2 0 e e = 5.2q2 0 DCL [m2/s] n τ a e2B2τ¯ a n eff 0 ei Neoclassical diffusion coefficient (Rosenbluth, Hazeltine, and Hinton) 8: 489

R 3/2 DNC = 2.2q2 0 DCL n a n 64 CHAPTER 7. TRANSPORT

Thermal diﬀusivities (Rosenbluth, Hazeltine, and Hinton) 8: 489

3/2 3/2 NC 2 R0 CL 2 R0 n20 2 χi = 0.68q χi = 0.068q [m /s] a a 2 1/2 B0 TkeV !

3/2 3/2 NC 2 R0 CL −3 2 R0 n20 2 χe = 0.89q χe = 4.3 10 q [m /s] a × a 2 1/2 B0TkeV !

7.2.4 Transport regime criteria Deﬁnition of collisionality 11: 149

ν∗ νqR/v ǫ3/2 ≡ t The banana regime 11: 149

νqR/v ǫ3/2 T ≪ The plateau regime 11: 149

ǫ3/2 νqR/v 1 ≪ T ≪ The Pﬁrsch-Schluter regime 11: 149

νqR/v 1 T ≫ Chapter 8

Plasma Waves

In this chapter, all units are SI with the exception of temperature, which is deﬁned in the historical units of eV (electron-volts).

E and B are the electric and magnetic ﬁelds, respectively bˆ is a unit vector in the direction of B and indicate parallel and perpendicular to bˆ || ⊥ k is the wave vector ωpi is the plasma frequency for particle i Ωi is the cyclotron frequency for particle i X˜ implies that X can be a complex number

8.1 Cold Plasma Electromagnetic Waves

Starting from Maxwell’s equation, linearize Q = Q exp(ik r iωt), J = ←→σ E 22: 8 · − · e i σ n n E + ←→I + ←→ E = 0 × × ǫ ω · o

where n = ck/ω

By combining this equation with the momentum conservation equations,

J = j njqjvj, setting p = 0, and setting collisionality to 0, it is possible to solve for σ. Plugging in σ and then writing in tensor form with Stix P notation,

2 S n|| iD n⊥n|| Ex − − 2 iD S n 0 Ey = 0  − 2    n⊥n|| 0 P n⊥ Ez  −    65 66 CHAPTER 8. PLASMA WAVES

where 22: 7 2 ωpj S = 1 2 2 − ω Ωj Xj − 2 Ωj ωpj D = 2 2 ω ω Ωj Xj − ω2 P = 1 pj − ω2 Xj ω2 R = S + D = 1 pj − ω (ω +Ωj) Xj ω2 L = S D = 1 pj − − ω (ω Ωj) Xj −

For a wave propagating at an angle θ to bˆ 22: 8

An4 Bn2 + C = 0 − A = S sin2 θ + P cos2 θ B = RL sin2 θ + PS 1 + cos2 θ C = PRL The equation can be rearranged, conveniently obtaining a handy mnemonic (“P Nar Nal Snarl Nap”) 22: 9

P n2 R n2 L tan2(θ)= − − −(Sn2 RL) (n2 P ) − − For a wave traveling at an angle θ to bˆ, the dispersion relation is given by the Appleton-Hartree equation 22: 38

2ω2 ω2 ω2 n2 = 1 pe − pe − 2ω2 ω2 ω2 Ω2ω2 sin2 θ Ω ω2Σ − pe − e ± e 2 1/2 Σ= Ω2 sin4 θ + 4ω4 1 w2 /ω2 cos2 θ e − pe h i Polarization of a wave 22: 10 iE n2 S x = − Ey D

8.1.1 Common Cold Plasma Waves Deﬁning special frequencies 5: 127 22: 29

1Cutoﬀ/Resonances listed for single ion species plasma 8.2. ELECTROSTATIC WAVES 67

Name Dispersion Relation5: 145 Type Resonance Cutoﬀ 1

ω2 Light Wave n2 = 1 pj B = 0 NA ω − j ω2 0 pe O wave n2 = P n B NA ω P ⊥ 0 pe X wave n2 = RL/S n B ω ,ω ω ,ω ⊥ 0 uh lh r l R wave n2 = R n B , right-handed Ω ω || 0 e r L wave n2 = L n B , left-handed Ω ω || 0 i l

2 2 Ωe + Ωe + 4ωpe ω = R q 2

2 2 Ωe + Ωe + 4ωpe ω = − L q2

2 2 2 ωuh = ωp +Ωe

2 2 2 ωpi ωlh =Ωi + 2 2 1+ ωpe/Ωe For waves that are almost one of the O, X, L, R waves, with a small angle with respect to the proper propogation direction, we get the following 2 2 19 dispersion relations. In these equations, α = ωpe/ω .

Wave Name Dispersion Relation

QT-O n2 1−α ≃ 1−α cos2 θ 2 2 2 2 2 (1−α) ω −Ωe sin θ QT-X n 2 2 2 ≃ (1−α)ω −Ωe sin θ QL-L n2 1 αω ≃ − ω+Ωe cos θ QL-R n2 1 αω ≃ − ω−Ωe cos θ

8.2 Electrostatic Waves

These are waves with no perturbed magnetic ﬁeld. The dispersion relation can be derived by combining Gauss’s law, momentum conservation, and continuity equations. It is important to note that k E. Let k lie in the (x,z) || plane, where z points the direction of bˆ.19

2 2 2 2 ωpj kx ωpj kz 1 2 2 2 + 2 2 = 0 − ω Ωj k ω k ! Xj − 68 CHAPTER 8. PLASMA WAVES

Electrostatic wave solutions Solution for k Dispersion relation Solution Type

2 ωpj 2 2 kx = 0 1 ω2 = 0 ω = ωpj Plasma oscillations − j j ω2 P pj ω =Pωuh Upper hybrid kz = 0 1 ω2−Ω2 = 0 − j j ω = ω Lower hybrid P lh kx = 0 ω2 ω2 4Ω2ω2 cos2 θ 1/2 6 2 uh uh e pe Trivelpiece-Gould ω = 2 2 1 ω4 kz = 0 ± − uh 6

8.3 MHD Waves

Using the MHD formulation describing plasmas, one can deﬁne a perturba- tion and analyze what waves propogate in a plasma. The matrix equation 8: 314 is given below, where va = B0/√µ0ρ0 and vs = γp0/ρ0. p 2 2 2 ω k||va 0 0 ξx − 2 2 2 2 2 2  0 ω k va k⊥vs k⊥k||vs  ξy = 0 − −2 −2 2 2   0 k⊥k||vs ω k vs ξz  − − ||     

MHD wave solutions Solution Solution (β 1) Wave type ≪

2 2 2 8: 314 ω = k||va Shear Alfv´en 2 k2 2 2 1/2 2 2 2 2 8: 315 ω = 2 va + vs 1 (1 α) ω (k⊥ + k||)va Compressional ± 2 2− ≈ k|| vs va where α = 4 2 2 2 2 Alfv´en k (vs +va) ω2 k2v2 Sound8: 316 ≈ || s

8.4 Hot Plasma

The hot plasma dispersion relations are calculated using the Vlasov equation and including finite Larmor orbit effects. Hot plasma effects are also characterized by electrostatic and electromagnetic waves.

8.4.1 Electrostatic The dispersion relation (general plasmas)19

ǫ(ω, k)=1+ χj (ω, k) = 0 Xj 8.4. HOT PLASMA 69

∞ ∞ ∂f mΩ ∂f 2 J 2 ( k⊥v⊥ ) k 0j + j 0j ω 2π m Ωj || ∂v v ∂v χ = pj dv v dv || ⊥ ⊥ j k2 n || ⊥ ⊥ ω hmΩ k v i 0j m j || || −∞Z Z0 X − −

where the number subscripts refer to the linearization order.

The dispersion relation (Maxwellian plasmas)

m=∞ 1 ǫ =1+ 1+ ζ Γ (b )Z(ζ ) k2λ2 0j m j mj Dj " m=−∞ # Xj X

where

∞ 2 2 v||/vT hj 1 dv||e 2 2 Z(ζmj)= bj = k⊥ρLj √π ω−mΩj v|| −∞Z − k|| ζ = (w mΩ )/(k v ) Γ (b ) I (b ) exp( b ) mj − j || Thj m j ≡ m j − j

This function can be evaluated as a power or an asymptotic series.

ζ 1 (kinetic limit) | m|≪ 2 4 ReZ(ζ ) 2ζ 1 ζ2 + ζ4 + ... (power series) m ≈− m − 3 m 15 m

ζ 1 (ﬂuid limit) | m|≫ 1 1 3 ReZ(ζ ) 1+ + + ... (asymptotic) m ≈−ζ 2ζ2 4ζ4 m m m

Γ can be expanded as well as

3 Γ 1 b + b2 + O(b3) 0 ≈ − 4 b Γ (1 b)+ O(b3) 1 ≈ 2 − Γ b2/8+ O(b3) 2 ≈ Γ O(b3) 3 ≈ 70 CHAPTER 8. PLASMA WAVES

8.4.2 Electromagnetic The dispersion relation in the electromagnetic limit19

K n2 K K + n n E xx − z xy xz x z x K K n2 n2 K E = 0  yz yy − x − z yz   y  K + n n K K n2 E zx x z zy zz − z z    

∞ 2 3 l Ω 2 2 Kxx =1+ 2 dv Jl (λ)Pl k⊥ jX=i,e l=X−∞ Z ∞ 2 lΩ 2 ′ Kxy = Kyx = i dv v⊥Jl(λ)Jl (λ)Pl − k⊥ Xj l=X−∞ Z ∞ 2 lΩ 2 2 Kxz = dv v||Jl (λ)Ql k⊥ Xj l=X−∞ Z ∞ 2 ′ 2 Kyy =1+ Ω dv v⊥Jl (λ) Pl Xj l=X−∞ Z ∞ K = i Ω dv2v v J (λ)J ′(λ)Q yz − ⊥ || l l l Xj l=X−∞ Z ∞ 2 lΩ 2 2 Kzx = dv v||Jl (λ)Pl k⊥ Xj l=X−∞ Z ∞ 2 ′ Kzy = i Ω dv v⊥v||Jl(λ)Jl (λ)Pl Xj l=X−∞ Z ∞ 2 2 2 Kzz =1+ Ω dv v||Jl (λ)Ql Xj l=X−∞ Z

∞ ∞ dv2 2 dv v dv ≡ || ⊥ ⊥ Z −∞Z Z0

∂f0j k||v|| ∂f0j ∂f0j 2 ∂v2 + ω ∂v2 ∂v2 ω ⊥ || − ⊥ P = 2π pj l ωΩ ω lΩ k v j − j − z ||

∂f0j lΩj ∂f0j ∂f0j 2 ∂v2 ω ∂v2 ∂v2 ω || − || − ⊥ Q = 2π pj l ωΩ ω lΩ k v j − j − z || where λ = k v /Ω and ∂f/∂v2 ∂f/∂ v2 | ⊥ ⊥ | ≡ 8.4. HOT PLASMA 71

Dispersion relation evaluation for an isotropic Maxwellian plasma 5: 276

2 ∞ ω e−bj K =1+ pj ζ n2I (b )Z(ζ ) xx ω2 b 0 n j n j n=−∞ Xj X 2 ∞ ωp K = K = i e−bj ζ n I (b ) I′ (b ) Z(ζ ) xy − yz ±ω2 0 n j − n j n n=−∞ Xj X 2 ∞ ω e−bj K = K = pj ζ nI (b )Z′(ζ ) xz zx ω2 1/2 0 n j n (2bj) n=−∞ Xj X 2 ∞ ω e−bj K =1+ pj ζ n2I (b ) + 2b2 I (b ) I′ (b ) Z(ζ ) yy ω2 b 0 n j j n j − n j n j j n=−∞ X X 2 1/2 ∞ ωpj bj K = K = i e−bj ζ I (b ) I′ (b ) Z′(ζ ) yz − zy − ± ω2 2 0 n j − n j n j n=−∞ X X ω2 ∞ K = 1 pj e−bζ I (b )ζ Z′(ζ ) zz − ω2 0 n j n n n=−∞ Xj X 72 CHAPTER 8. PLASMA WAVES Chapter 9

Nuclear Physics

In this chapter, all units are SI with the exception of: temperature and energy, which are deﬁned in the historical units of eV (electron-volts); cross sections, which are deﬁned in the historical units of barn; and Bosch-Hale reaction rates, which are given in cubic centimeters per second. e is the elementary electric charge Z is the number of nuclear protons N is the number of nuclear neutrons A is the number of nucleons (N+Z) m is the particle mass n is the particle number density v is the particle velocity vector E is the particle kinetic energy

9.1 Fundamental Deﬁnitions

Nuclear reaction notation 15: 378−381

a = bombarding particle entrance channel a + X b + Y ⇒ X = target nucleus ) or where   b = ejected particle(s) exit channel X(a, b)Y  Y = product nucleus )   X(a,b)Y general nuclear reaction X(a,a)X elastic scattering X(a,a’)X* inelastic scattering X(n,n)X neutron elastic scattering X(n,n’)X* neutron inelastic scattering X(n,2n)Y neutron multiplication X(n,γ)Y neutron capture X(n,f)Y neutron-induced ﬁssion

73 74 CHAPTER 9. NUCLEAR PHYSICS

Nuclear mass 15: 65

m = Zm + Nm E /c2 p n − B Nuclear binding energy 15: 68

2 2/3 Z(Z 1) (A 2Z) EB(A, Z)= avA asA ac − aa − + δ(A, Z) − − A1/3 − A where the value of the coeﬃcients a in MeV is 15: 68

av = 15.5 as = 16.8 ac = 0.72 aa = 23 ap = 34 MeV

−3/4 apA (Z,N even) δ(A, Z)=  0 (Aodd)  a A−3/4 (Z,N odd) − p  Nuclear reaction Q-value 15: 381

Q = ((m + m ) (m + m )) c2 = E E a X − b Y f − i Reaction threshold energy (Q< 0) 15: 382

m + m m E = Y b Q 1+ a Q thresh − m + m m ≈− m y b − a X

9.2 Nuclear Interactions

A m /m is approximately the nucleus atomic mass number ≡ A N i and f refer to initial and ﬁnal, respectively θ is the angle between the a and b trajectory in the lab frame θcm is the angle between the a and b trajectory in the center of mass frame

Reaction Q-value from kinematics 15: 384

m m m m 1/2 Q = E 1+ b E 1 a 2 a b E E cos θ b m − a − m − m2 a b Y Y Y Ejected particle energy: X(a,b)Y 15: 382

1/2 2 1/2 1/2 (mambEa) mambEaζ + (mY + mb)[mY Q + (mY ma)Ea] Ey = ζ − mY + mb ± mY + mb Maximum kinetic energy transfer fraction: X(a,a)X 3

E 4A A f = 1 2 E 2 i θ=0 (A1 + A2)

9.2. NUCLEAR INTERACTIONS 75

9.2.1 Charged Particle Interactions

Stopping power of heavy charged particles (Bethe formula) 15: 194

dE e2 2 4πz2N Zρ 2m c2β2 = A ln e ln 1 β2 β2 dx 4πǫ m c2β2A I − − − 0 e where βc = v and ze are the incident particle’s speed and charge; ρ is the mass density of the stopping medium; NA is Avogadro’s number; I is mean excitation of atomic electrons, typically taken as I 10Z. ≈ Collisional stopping power of electrons 15: 196

dE e2 2 2πN Zρ E (E + m c2)2β2 = A ln i i e + 1 β2 dx 4πǫ m c2β2A 2I2m c2 − c 0 e e 1 2 2 1 β2 1+ β2 ln2+ 1 1 β2 − − − 8 − − p p Radiative stopping power of electrons (& 1 MeV) 15: 196

2 2 2 2 2 dE e Z NA Ei + mec ρ 2 Ei + mec 4 = 4 ln dx 4πǫ 137m2c4A m c2 − 3 r 0 e " e #

9.2.2 Neutron Interactions Ejected neutron energy: X(n,n)X 15: 448

2 A + 2A cos θcm + 1 En,f = (A + 1)2

Maximum kinetic energy transfer fraction: X(n,n)X 15: 448

E A 1 2 n,f = − E A + 1 n,i θ=0

15: 450 Neutron lethargy (En . 10 MeV)

(A 1)2 A 1 ξ =1+ − ln − 2A A + 1 Number of collisions required to change neutron energies 15: 450

ln(E /E ) N = i f ξ 76 CHAPTER 9. NUCLEAR PHYSICS

9.2.3 Gamma Interactions Klein-Nishina cross section for Compton scattering 15: 201

dσ 1 3 1 + cos2 θ α2 (1 cos θ) = r2 1+ − dΩ 0 1+ α (1 cos θ) 2 (1 + cos2 θ)[1+ α (1 cos θ)] − − 2 2 2 where (α = Eγ /mc ) and (r0 = e /4πǫ0mc = 2.818 fm) is the classical electron radius.

Energy shift from Compton scattering 15: 201

Ei Ef = 1+ Ei (1 cos θ) mc2 − 9.3 Cross Section Theory

A reaction cross section σ(Ea), or more simply σ, is a measure of the probability that reaction X(a, b)Y will occur. Consider a beam of a particle with current Ia directed onto X target nuclei with an areal density of NX per unit area. By observing the energy and angular distribution, e(Eb) and r(θ, φ) respectively, of the ejected particle b into a solid angle dΩ then the doubly diﬀerential cross section can be determined, which is the probability 15: 392−394 of observing particle b at in solid angle dΩ with energy Eb.

Doubly differential cross section 15: 392−393 dσ r(θ, φ) e(E ) = b dΩ dEb 4πIaNX Differential energy cross section 15: 393 dσ dσ = dΩ dE dΩ dE b Z b Differential angular cross section 15: 393 dσ dσ = dE dΩ dΩ dE b Z b Reaction cross section 15: 393

2π π dσ dσ σ = dΩ= sin θ dθ dφ dΩ dΩ Z Z0 Z0

Total cross section 15: 393

all reactions σT = σi Xi=0 9.4. REACTION RATE THEORY 77

The Sommerfeld parameter 23: 5

Z Z e2 η = a X ~v Astrophysical S-factor 23: 5

S(E)= σ(E)E exp (2πη)

9.4 Reaction Rate Theory

For a thermonuclear plasma, the volumetric reaction rate (also known R as thermal reactivity) describes the number of reactions occuring per unit volume per unit time. In thermonuclear fusion plasmas, is obtained by R integrating the energy-dependent cross section, σ(v), over the distribution functions of the participating species. 26: 5−8

Partial volumetric reaction rate 26: 5−6

= σ(v)vf (v )f (v ) v = v v Rp 1 1 2 2 1 − 2 Total volumetric reaction rate for general f(v) 26: 6

1 = σ(v)vf (v )f (v ) d3v d3v R 1+ δ 1 1 2 2 1 2 12 ZZ where δ12 is the Kronecker delta for particle species 1 and 2 that prevents double counting for like-species plasmas.

Total volumetric reaction rate for Maxwellian f(v)) 2: 624 26: 6

3 1/2 8µ 1 −µE/m1 T 3 −1 σv = σ(E)Ee dE [m s ] n1n2 h i πT 3 m2 = σv where  1 Z R 1+ δ12 h i  m m 1  µ = 1 2 E = mv2 m1 + m2 2   78 CHAPTER 9. NUCLEAR PHYSICS

9.5 Nuclear Reactions for Fusion Plasmas

All data from the ENDF/B-VII nuclear data libraries.4

Abbreviations: n=neutron, p=1H, d=2H, t=3H, h=3He, α =4He Reactants Products Branching Q-value (kinetic energy in MeV) ratio (MeV)

1. d+t α(3.52) + n(14.07) 1.00 17.59 −→ 2. d+d t(1.01) + p(3.02) 0.50 4.03 −→ h(0.82) + n(2.45) 0.50 3.27 −→ 3. d+h α(3.67) + p(14.68) 1.00 18.35 −→ 4. t+t α + 2n 1.00 11.33 −→ 5. h+t α + p + n 0.51 12.10 −→ α(4.77) + d(9.54) 0.43 14.32 −→ 5He(1.87) + p(9.34) 0.06 11.21 −→ 6. p+ 6Li α(1.72) + h(2.30) 1.00 4.02 −→ 7. p+ 7Li 2 α 0.20 17.35 −→ 7Be + n 0.80 -1.64 −→ 8. d+ 6Li 2 α 1.00 22.37 −→ 9. p+ 11B 3 α 1.00 8.62 −→

9.6 Nuclear Reactions for Fusion Energy

All data from the ENDF/B-VII nuclear data libraries.4

Abbreviations: n=neutron, t=3H, B=tritium breeding, M=neutron multiplication Reaction Q-value Purpose σ(0.025 eV) σ(14.1 MeV) [MeV] [barn] [barn]

1. 6Li(n,t)4He 4.78 B 978 0.03 2. 6Li(n,2nα)6Li -3.96 M - 0.08 3. 7Li(n,2n)6Li -7.25 M - 0.03 4. 7Li(n,2nα)3H -8.72 B/M - 0.02 5. 9Be(n,2n)8Be -1.57 M - 0.48 6. 204Pb(n,2n)203Pb -8.39 M - 2.22 7. 206Pb(n,2n)205Pb -8.09 M - 2.22 8. 207Pb(n,2n)206Pb -6.74 M - 2.29 9. 208Pb(n,2n)207Pb -7.37 M - 2.30 9.7. FUSION CROSS SECTION PARAMETRIZATION 79

9.7 Fusion Cross Section Parametrization

The Bosch-Hale parametrization of the fusion reaction cross section 2

S(E) σ(EkeV)= [millibarn] E exp(BG/√E) where

A1 + E(A2 + E(A3 + E(A4 + EA5))) S(EkeV)= 1+ E(B1 + E(B2 + E(B3 + EB4)))

Bosch-Hale parametrization coeﬃcients for several fusion reactions 2 2H(d,n)3He 2H(d,p)3H 3H(d,n)4He 3He(d,p)4He

BG [keV] 31.3970 31.3970 34.3827 68.7508

4 4 4 6 A1 5.3701 10 5.5576 10 6.927 10 5.7501 10 × 2 × 2 × 8 × 3 A2 3.3027 10 2.1054 10 7.454 10 2.5226 10 × −1 × −2 × 6 × 1 A3 -1.2706 10 -3.2638 10 2.050 10 4.5566 10 × −5 × −6 × 4 × A4 2.9327 10 1.4987 10 5.200 10 0.0 × −9 × −10 × A5 -2.5151 10 1.8181 10 0.0 0.0 × × 1 −3 B1 0.0 0.0 6.380 10 -3.1995 10 × −1 × −6 B2 0.0 0.0 -9.950 10 -8.5530 10 × −5 × −8 B3 0.0 0.0 6.981 10 5.9014 10 × −4 × B4 0.0 0.0 1.728 10 0.0 × Valid Range [keV] 0.5

Tabulated Bosch-Hale cross sections [millibarns] 2 E (keV) 2H(d,n)3He 2H(d,p)3H 3H(d,n)4He 3He(d,p)4He

3 2.445 10−4 2.513 10−4 9.808 10−3 1.119 10−11 4 2.093×10−3 2.146×10−3 1.073×10−1 1.718× 10−9 5 8.834×10−3 9.038×10−3 5.383×10−1 5.199×10−8 6 2.517×10−2 2.569×10−2 1.749× 100 6.336×10−7 7 5.616×10−2 5.720×10−2 4.335×100 4.373×10−6 8 1.064×10−1 1.081×10−1 8.968×100 2.058×10−5 9 1.794×10−1 1.820×10−1 1.632×101 7.374×10−5 10 2.779×10−1 2.812×10−1 2.702×101 2.160×10−4 12 5.563×10−1 5.607×10−1 6.065×102 1.206×10−3 15 1.178× 100 1.180× 100 1.479×102 7.944×10−3 20 2.691×100 2.670×100 4.077×102 6.568×10−2 × × × × 80 CHAPTER 9. NUCLEAR PHYSICS

9.8 Fusion Reaction Rate Parametrization

The Bosch-Hale parametrization of the volumetric reaction rates 2

[cm3 s−1] ξ −3ξ or σv = C1 θ 2 3 e h i · · smµc Ti, keV 10−6 [m3 s−1] × where

T (C + T (C + TC )) B2 1/3 θ = T/ 1 2 4 6 ξ = G − 1+ T (C + T (C + TC )) 4θ 3 5 7

Bosch-Hale parametrization coeﬃcients for volumetric reaction rates 2 2H(d,n)3He 2H(d,p)3H 3H(d,n)4He 3He(d,p)4He

1/2 BG [keV ] 31.3970 31.3970 34.3827 68.7508 2 mµc [keV] 937814 937814 1124656 1124572

−12 −12 −9 −10 C1 5.43360 10 5.65718 10 1.17302 10 5.51036 10 × −3 × −3 × −2 × −3 C2 5.85778 10 3.41267 10 1.51361 10 6.41918 10 × −3 × −3 × −2 × −3 C3 7.68222 10 1.99167 10 7.51886 10 -2.02896 10 × × × −3 × −5 C4 0.0 0.0 4.60643 10 -1.91080 10 −6 −5 × −2 × −4 C5 -2.96400 10 1.05060 10 1.35000 10 1.35776 10 × × × −4 × C6 0.0 0.0 -1.06750 10 0.0 × −5 C7 0.0 0.0 1.36600 10 0.0 ×

Valid range (keV) 0.2

Tabulated Bosch-Hale reaction rates [m3 s−1] 2 T (keV) 2H(d,n)3He 2H(d,p)3H 3H(d,n)4He 3He(d,p)4He

1.0 9.933 10−29 1.017 10−28 6.857 10−27 3.057 10−32 1.5 8.284×10−28 8.431×10−28 6.923×10−26 1.317×10−30 2.0 3.110×10−27 3.150×10−27 2.977×10−25 1.399×10−29 3.0 1.602×10−26 1.608×10−26 1.867×10−24 2.676×10−28 4.0 4.447×10−26 4.428×10−26 5.974×10−24 1.710×10−27 5.0 9.128×10−26 9.024×10−26 1.366×10−23 6.377×10−27 8.0 3.457×10−25 3.354×10−25 6.222×10−23 7.504×10−26 10.0 6.023×10−25 5.781×10−25 1.136×10−22 2.126×10−25 12.0 9.175×10−25 8.723×10−25 1.747×10−22 4.715×10−25 15.0 1.481×10−24 1.390×10−24 2.740×10−22 1.175×10−24 20.0 2.603×10−24 2.399×10−24 4.330×10−22 3.482×10−24 × × × ×

Approximate DT volumetric reaction rate (10 . T [keV] . 20) 26: 7

σv = 1.1 10−24T 2 [m3 s−1] h iDT × keV 9.9. CROSS SECTION AND REACTION RATE PLOTS 81

9.9 Cross Section and Reaction Rate Plots

10-27 Bosch-Hale Nuclear Reaction Cross Sections 10-28

] 10-29 2 10-30 10-31 2H(d,n)3He 10-32 2H(d,p)3H -33 10 3 4 Crossm section [ H(d,n) He 10-34 3He(d,p)4He 10-35 100 101 102 103 Energy [keV]

-20 Bosch-Hale Volumetric Reaction Rates ] 10 1 −

s -22 3 10 −

10-24

10-26 2H(d,n)3He -28 10 2H(d,p)3H

3 4 10-30 H(d,n) He 3He(d,p)4He 10-32 Volumetricreactionm [ rate 100 101 102 Temperature [keV] 82 CHAPTER 9. NUCLEAR PHYSICS Chapter 10

Tokamak Physics

In this chapter, all units are SI with the exception of temperature, which is defined in the historical units of eV (electron-volts). e is the fundamental charge unit Z is the number of nuclear protons R0 is the major radius of a toroidal plasma a and b are the horizontal and vertical minor radii of a toroidal plasma Note: a = b for circular cross sections, and a is used by convention R and r denote lengths in the major and minor radii, respectively θ and φ are the poloidal and toroidal angular coordinates, respectively Bx is the magnetic field in direction x Bxa is the magnetic field evaluated at the plasma edge in direction x Ip is the toroidal plasma current p is the plasma pressure v is velocity of the plasma

10.1 Fundamental Deﬁnitions

Inverse aspect ratio 26: 117 a ǫ = R0 Plasma elongation 26: 741 b κ = a Plasma triangularity 26: 741

(c + d)/2 δ = a where c, d are distances to the top of the plasma and the x-point, respectively, from the plasma center.

83 84 CHAPTER 10. TOKAMAK PHYSICS

Large aspect ratio expansion (ǫ 1) 8: 280 ≪ 1 1 r 1 cos θ R ≈ R − R 0 0 Surface area of a torus

2 14: 24 Sc−torus = 4π aR0 (Circular cross section) 1+ κ2 1/2 S = 8πaR E(k) 4π2aR (Elliptical cross section)27 e−torus 0 ≈ 0 2 Volume of a torus

2 2 14: 24 Vc−torus = 2π a R0 (Circular cross section) 2 2 27 Ve−torus = 2π a κR0 (Elliptical cross section)

MHD toroidal plasma volume 7: 112

2π 2π rˆ V (ψ)= Rrdrdθdφ Z0 Z0 Z0 2π 2 2 rˆ = πR0 dθrˆ 1+ cos θ (low aspect ratio) 3 R0 Z0

Volume averaged plasma pressure 3

1 p = p dτ h i V volumeZ

Toroidal plasma beta 7: 71

2µ0 p βt = 2h i Bφa

Poloidal plasma beta 7: 71

2 2 2 2µ0 p 8π a κ p βp = 2h i = 2 h i Bθa µ0Ip

Radial electric ﬁeld in a rotating toroidal plasma 3

1 Er vφBθ vθBφ + p ≈ − Zien∇ 10.2. MAGNETIC TOPOLOGY 85

10.2 Magnetic Topology

Toroidal magnetic field for plasma confinement 3 B (r)R B φ 0 φ ≈ R B (r) φ (valid for ǫ 1) ≈ 1 ǫ cos θ ≪ − Poloidal magnetic field for plasma confinement 3 µ I (r) B 0 p θ ≈ 2πr Safety factor (general) 26: 111 # of toroidal field line orbits at r q(r)= # of poloidal field line orbits at r

Safety factor for cylindrical plasma (r, θ, z) 26: 112

2 rBφ(r) 2πr Bφ(r) q(r)cyl = = RBθ(r) µ0Ip(r)R Safety factor for toroidal plasma (R,θ,φ) 8: 288

2π ∗ 1 rBφ q(r )tor = dθ 2π RBθ Z0 r B (r ) = 0 φ 0 R B (r ) 1 r2/R2 1/2 0 θ 0 − 0 0 ∗ where the ﬂux surfaces r = r0 are circles.

Safety factor at the edge for toroidal plasma (R,θ,φ) 8: 387 2πa2B πk q 0 = ∗ 1/2 ≡ µ0R0Ip 4E(k)(β/ǫ) where E(k) k2 + π2/4(1 k2) is the complete elliptic integral of the ≈ − second kind and the deﬁnition of k is given by 2 Bi 2µ0p 4ǫµ0p 2 2 1 2 + 2 2 2 k B0 ≡ − B0 k B0 − Approximate edge safety factor for a large aspect ratio toroidal plasma 8: 414 2πB a2 1+ κ2 q 0 ∗ ≈ µ R I 2 0 0 p Magnetic shear 8: 408 r dq s = q dr 86 CHAPTER 10. TOKAMAK PHYSICS

10.3 Magnetic Inductance

Deﬁnition of magnetic inductance 8: 281 1 B2 LI2 dτ 2 ≡ 2µ0 volumeZ Normalized inductance per unit length [dimensionless] 8: 281 L/2πR 2L ℓ 0 = ≡ µ0/4π µ0R0 Internal inductance of a toroidal plasma 8: 281 a 2 2 8πR0 Bθ µ0R0 Bθ Li = 2 rdr = h2 i Ip 2µ0 2Bθa Z0 2 Bθ ℓi = h 2 i Bθ (a) External inductance of a toroidal plasma 8: 281 ∞ 2 8πR0 Bθ 8R0 Le = 2 rdr = µ0R0 ln 2 Ip 2µ0 a − Za 8R ℓ = 2ln 0 4 e a −

10.4 Toroidal Force Balance

Equation of toroidal force balance 8: 279

Rˆ (J B p) dτ = 0 · × −∇ Z where R ∂B 1 ∂ R µ J = B = 0 φ θˆ 0 rB φˆ 0 ∇× R ∂r − r ∂r R θ

Rˆ J B = · × R2 ∂ B2 R B ∂ R B ∂ R cos θ 0 φ + 0 θ 0 rB vert 0 rB − R2 ∂r 2µ µ rR ∂r R θ − µ r ∂r R θ " 0 ! 0 # 0 The hoop force8: 280 I2 2 p ∂ ˆ 2 2 Bθa ˆ Fhoop = (Li + Le) R = 2π a (ℓi + ℓe + 2) R 2 ∂R 2µ0 2 µ0I 8R ℓ = p ln 1+ i Rˆ 2 a − 2 10.5. PLASMA PARA- AND DIA-MAGNETISM 87

The tire tube force 8: 280

F = 2π2a2 p Rˆ tire h i 2 µ0I βp = p Rˆ 4 The 1/R force 8: 280

B2 B2 2 2 φa φ ˆ F1/R = 2π a h i R 2µ0 − 2µ0 ! 2 µ0I = p (β 1) Rˆ 4 p −

where p = 1 B2 B2 + B2 h i 2µ0 φa − h φi θa have been used.

The vertical ﬁeld force on toroidal plasma ring 8: 282

F = 2πR B I Rˆ vert − 0 vert p The vertical magnetic ﬁeld required to balance toroidal forces 8: 282

Fhoop + Ftire + F1/R Bvert = 2πR0Ip ǫ 2µ p B2 B2 0 φa − h φi = Bθa ℓe + ℓi +2+ 2h i + 2 4 Bφa Bθa ! µ I 8R 3 ℓ = 0 p ln + i + β 4πR a − 2 2 p 0 Shafranov shift of the plasma center 7: 2129

b2 l 1 a2 b B ∆= β + i − 1 + ln vert 2R p 2 − b2 a − B (b) 0 θ1

10.5 Plasma Para- and Dia-Magnetism

Global pressure balance equation in a screw pinch 8: 270

1 2 2 2 p = Bza Bz + Bθa h i 2µ0 − h i can be rearranged to give

2 2 Bza Bz βp = −2 h i + 1 Bθa

Diamagnetic : βp < 1 Paramagnetic : βp > 1 88 CHAPTER 10. TOKAMAK PHYSICS

10.6 MHD Stability Limits

Using experimental data from a wide variety of tokamaks, empirical scalings for critical tokamak instabilities have been constructed. Units: current in 20 MA, length in m, magnetic ﬁeld in T, and density in n20 = n/10 .

(a) Beta limits (no plasma shaping)

I βt βL ≤ aBφ 24 βL = 0.028 Troyon kink limit - no wall

βL = 0.044 Sykes ballooning limit - no wall

βL = 0.06 kink - ideal conducting wall

26: 347 (b) Deﬁnition of βN

aBφ βN βt[%] ≡ Ip[MA]

I [MA] n n = p 20 ≤ G πa2 10.7 Tokamak Heating and Current Drive

(a) Ohmic plasma heating The neo-classical resistivity approximation is 8: 538

1 Spitzer η|| = η 1 (r/R )1/2 2 || − 0

8: 539 Plugging in for the current as J|| = E0/η||

−2 2 5.6 10 R0 (I[MA]) PΩ = × [MW] 1 1.31ǫ1/2 + 0.46ǫ 2 3/2 − a κTkeV !

(b) Neutral beam plasma heating 26: 246−248

4 1/2 3/2 ne lnΛ 2me Eb mb P = mb + 2πǫ2m2 1/2 3/2 3/2 1/2 0 b 3(2π) Te 2 miEb ! 10.7. TOKAMAK HEATING AND CURRENT DRIVE 89

where Eb is the energy of the beam. The critical beam energy when the ions and the electrons are heated equally by the beam is

A E = 14.8 b T c 2/3 e Ai

10.7.1 Current Drive (a) Inductive current This current is driven via the central solenoid. The current distribution is calculated through the use of Faraday’s law and J|| = E0/η||. Total current normally has to be measured in order to normalize the distrubition of current density.

(b) Bootstrap current Bootstrap current is the self-generated current drive in the plasma from trapped and passing electrons in the plasma. The exact form of the bootstrap current density is given by 8: 496

R 1/2 T ∂n n ∂T j = 4.71q 0 + 0.04 B − r B ∂r T ∂r 0

The total bootstrap fraction is given by 8: 496

∂ ∂ r 1/2 f 1.18 (lnn + 0.04lnT )/ (lnrB ) β ǫ1/2β B ≈− ∂r ∂r θ R p ∼ p 0 (c) Neutral beam current drive By positioning a neutral beam in the tangential direction, it is possible to drive both rotation and current. Neutral beam current drive 6 eﬃciency scales as (at Eb = 40AbTe)

0.06Te I[A]/P [W] (1 Zb/Zeﬀ ) ≈ n20RZb −

(d) Lower hybrid current drive Currently one of the most used current drive mechanisms is the lower hybrid system. It launches a wave that Landau damps on the fast electron population and preferentially drives electrons in one direction. 8: 623 90 CHAPTER 10. TOKAMAK PHYSICS

2 I[A]/P [W] = 1.17/ n||R0n20 There exists an accessibility condition for the waves which forces an 22: 100 increase in the launched n||

2 D2 1/2 n2 > S1/2 + || P !

where S, P, and D are defined in Chapter 8. Because LHCD relies on Landau damping, there is an additional con- c 1/2 9 straint on the n||: Landau damping dominates at n|| & 7.0/TkeV (e) Fast Magnetosonic wave current drive Allows peaked on-axis profiles and has the following current drive ef- ficiency6

T I[A]/P [W] = 0.025 keV n20R0

10.8 Empirical Scaling Laws

10.8.1 Energy Conﬁnement Time Scalings Goldston auxiliary heated tokamak scaling (l refers to the plasma size a) 26: 152 ∼

τ B2l1.8/nT E ∼ p The ITER-89 L-Mode (ITER89-P) 26: 740

0.85 1.2 0.3 0.5 0.1 0.2 0.5 −0.5 τE = 0.048IM R0 a κ n¯20 B0 A PM [s] The ITER-98 L-Mode17

0.96 0.03 0.40 0.20 1.83 −0.06 0.64 −0.73 τE = 0.023IM BT n19 M R ǫ κ PMW [s] The ITER-98 (IPB98[y,2]); ELMy H-mode17

0.93 0.15 0.41 0.19 1.97 0.58 0.78 −0.69 τE = 0.0562IM BT n19 M R ǫ κ PMW [s] Scaling for linear regime energy transport17

0.5 2 τE = 0.07n20qκ aR [s] Critical density of linear to saturated regime17

0.5 −1 −1 n20 = 0.65Ai BT q R 10.9. TURBULENCE 91

10.8.2 Plasma Toroidal Rotation Scaling Plasma toroidal rotation (Rice) scaling20

∆V ∆W/I tor ∝ p

The 2010 multi-machine scaling database found that (with va being the Alfv´en speed)20

1.4 2.3 v/va = 0.65βT qj

2 where qj = 2πκa B/µ0RIp

10.8.3 L-H Mode Power Scalings The ITPA empirical scaling law for the L to H mode transition power thresh- old16

±0.107 0.782±0.037 0.772±0.031 0.975±0.08 0.999±0.101 PL-H[MW] = 2.15e n20 BT a R

10.9 Turbulence

Fundamental deﬁnitions 26: 422−424

L = n/ n L = T/ T n ∇ T ∇

2 2 b = kθ ρi ηj = Lnj/LTj

2 ǫ = mjv /2Tj

The diamagnetic drift velocity 26: 420

B p v j dj = ×∇ 2 qjnjB

Diamagnetic frequency 26: 421

k T dn ω = y j ∗j − eBn dr

Ion Larmor radius evaluated at the sound speed25

ρ c /Ω S ≡ s i Normalized Larmor radius

ρ ρ /a ∗ ≡ S 92 CHAPTER 10. TOKAMAK PHYSICS

10.9.1 General Drift Wave Turbulence Mixing length estimate25

n˜rms/n 1/k L 0 ∼ ⊥ n Density ﬂuctuations and plasma potential correlation25

n/n˜ eφ/kT˜ (1 iδ) 0 ≈ e − where δ is the dissipation of the electron momentum to the background plasma.

˜ Time averaged electrostatic turbulent ﬂux of particles Γ, momentum ←→µ , and heat Q˜ 25

n˜ φ˜ B¯ Γ=˜ h ∇ i× + n˜v˜ B/B − B2 h ||i

φ˜ B¯ φ˜ B B B ←→µ = ∇ ×2 +v ˜|| /B ∇ ×2 +v ˜|| /B * − B ! − B !+

5 1 T˜ φ˜ B¯ 1 n˜ φ˜ B¯ Q˜ n¯T¯ + T˜v˜ B/B + + n˜v˜ B/B ¯ h ∇ i×2 || h ∇ i×2 || ≡ 2 "T − B h i ! n¯ − B h i !# where fluctuating values are marked by a tilde and is a time average. h i Time averaged momentum and and energy fluxes due to fluctuating magnetic fields25

˜ ˜ EM BB ←→µ = h i µ0nM¯ i

˜ q˜||eB Q˜EM = h i B¯

10.9.2 General Drift Turbulence Characteristics Perpendicular drift wave turbulence is characterized by ρ , with k k S || ≪ ⊥ and k⊥ρS depending on dissipative mechanism, linear free energy source, and nonlinear energy transfer.

Ion thermal gradient (ITG) turbulence occurs when η > η 1 and has i crit ∼ the following approximate characteristics25

k ρ 0.1 0.5 ⊥ s ∼ − 10.9. TURBULENCE 93

R/L > R/L 3 5 Ti Ti |crit ∼ −

v v ph ∼ di Trapped electron mode (TEM) instabilities occurs at approximately k ρ ⊥ S ∼ 1. At higher wavenumbers the TEM transitions into the electron thermal gradient (ETG) instability with η > η 1 and the following approximate e crit ∼ characteristics25

k ρ 1 10 ⊥ s ∼ −

R/L > R/L 3 5 Te Te |crit ∼ −

10.9.3 Passing Particle Instabilities

In this section, it is assumed that k||vte >>ω >>k||vti, such that electrons respond to the electrostatic potential. Also, the frequency of the magnetic curvature drifts is assumed to be 26: 422

ω = 2L ω /R ω di n ∗i ≪ 94 CHAPTER 10. TOKAMAK PHYSICS

The passing particle dispersion relation 26: 424

2 2 2 ∂ Ln/R ∂ 2R/Ln i sin θ ∂ ρ + ikθsx cos θ + i ∂x2 − b1/2(T /T )qΩ ∂θ − (T /T )Ω k ∂x − " e i e i θ Ω 1 − + b φ˜ = 0 (T /T )Ω + (1 + η ) e i i where x is the distance from the reference mode rational surface m = nq(r) and φ˜ is the perturbed electrostatic potential.

26: 428 Ion thermal gradient (ITG, eta-i, ηi) toroidal frequency

ω (η ω ω )1/2 ITG ≈ i ∗i di ITG critical instability limit 26: 429

1.2 R/Ln < (R/Ln)crit ηic =  4 Ti 1+ (1+2s/q) R/Ln R/Ln > (R/Ln)crit  3 T e where 0.9 (R/Ln)crit = (1 + Ti/Te)(1 + 2s/q)

Electron thermal gradient (ETG, ηe mode) dispersion relation with Ti Te 26: 429 ≈

2 2 k||vte ω ω 1 ∗e (1 + η ) +1+ ∗e = 0 − ω2 − ω e ω If η 1 then there is an unstable mode with 26: 429 ω ( k2v2 η ω )1/3 e ≫ ≈ − || te e ∗e 10.9.4 Trapped Particle Modes The collisionless trapped particle dispersion relation 26: 432

1 1 1 1 ω ω 1 ω ω + = − ∗i + − ∗e √2ǫ Ti Te Ti ω ω¯di Te ω ω¯de − − where

2 2 ω v|| v k ω¯ = dj + ⊥ cos θ + r sin θ dj 2 v 2v { k } " Thj Thj # θ

This dispersion relation gives rise to the trapped ion mode if νeﬀ = νj/ǫ>ωdj and has growth/frequency 26: 433−434

2 2 √2ǫ νi ǫ ω∗e ω = ω∗e i + i 2 1+ Te/Ti − ǫ (1 + Te/Ti) νe 10.9. TURBULENCE 95

Parameter Approximate range Approximate length Mode −1 in kθ (cm ) scale(cm)

< 1 60 ITG < 2 1 ITG,TEM Density fluctuations (ñ) < 7 1-10 ITG, TEM 3-12 1 TEM > 20 1, 20 ETG Temperature fluctuations (T˜e) < 1 1 ITG Flows, GAMs, ZF < 1 1 ITG n˜eT˜e cross phase < 1 1 ITG

This mode has the largest imaginary part if ν ǫ3/2ω . e ≈ ∗e The TEM can be calculated due to the trapped particle dispersion relation. The mode is driven by trapped electron collisions and electron temperature gradients. 26: 434−435

If ν ω then the growth rate is eﬀ ≫ ∗e

2 3/2 ω∗e γ ǫ ηe ≈ νe 96 CHAPTER 10. TOKAMAK PHYSICS Chapter 11

Tokamak Edge Physics

In this chapter, all units are SI with the exception of temperature, which is deﬁned in the historical units of eV (electron-volts). e is the elementary electric charge q is the total particle charge Z is the particle atomic (proton) number T is the plasma temperature n is the plasma number density p is the plasma pressure e and i refer to electrons and ions, respectively a and R0 are the minor and major radii of a toroidal plasma

11.1 The Simple Scrape Oﬀ Layer (SOL)

The simple SOL model describes 1D plasma flow from the core plasma to material boundary surfaces for limited or diverted plasma along the toroidal magnetic topology. By assuming a high degree of collisionality (ν∗), fluid approximations for plasma flow are valid and the neoclassical effects on particle orbits due to toroidal magnetic topology can be safely ignored.

Parallel SOL connection length (rail/belt limiters, poloidal divertors) 21: 17

L πRq || ≈ Particle time in the SOL (simple 1D model) 21: 20

t L /c dwell ≈ k s SOL width (simple 1D model) 21: 23

λ D L /c 1/2 SOL ≈ ⊥ k s where D⊥ is the anomalous diﬀusion coeﬃcient.

97 98 CHAPTER 11. TOKAMAK EDGE PHYSICS

Force balance / pressure conservation in the SOL 21: 47

2 pe + pi + mnv = constant

Plasma density variation along the SOL 21: 47 n n(x)= 0 1+ M(x)2

where n0 is the density at the ’top’ of the SOL and M = v/cs is the plasma mach number.

Electrons follow a Boltzmann distribution in the SOL 21: 28

n = n0exp (eV/Te)

SOL particle sources: ionization (i) and cross-ﬁeld transport (t) 21: 35−40

S = S + S = n n σv + D n/λ2 p p,i p,t plasma neutralsh ii ⊥ SOL where σv σv (T ,Z) is the ionization rate coeﬃcient. h ii ≡ h ii e Particle ﬂux density in the SOL at the sheath edge (se) 21: 47

1 Γ = n c se 2 0 s

where n0 is the density outside the pre-sheath.

Electric ﬁeld through SOL required to satisfy the Bohm Criterion 21: 48 T V = 0.7 e se − e Floating sheath voltage 21: 79

T m T V = 0.5 e ln 2π e Z + i s e m T i e Debye sheath width 21: 27

ǫ T 1/2 λ 0 e Debye ≈ n e2 e

11.2 Bohm Criterion

2 The Bohm Criterion is derived from conservation of energy (1/2miv = eV ) and particle conservation (n v = constant). In an unmagnetized − i plasma it sets the SOL plasma exit velocity into the sheath edge (se). In magnetized plasma, it sets the SOL plasma exit velocity parallel to the magnetic ﬁeld, after which the ions become demagnetized and perpendicularly 11.3. A SIMPLE TWO POINT MODEL FOR DIVERTED SOLS 99 enter the sheath; electrons remain magnetized. 21: 61−98 .

21: 73 Bohm Criterion (assuming Ti = 0)

T 1/2 v e = c se ≥ m s i Bohm Criterion (general form) 21: 76

∞ i fse(v) dv mi 2 v ≤ Te Z0

11.3 A Simple Two Point Model For Diverted SOLs

Diverted plasmas can obtain signiﬁcant ∆T along the SOL, resulting in divertor temperatures less than 10 eV. The SOL can be approximated using a two point model: point 1 is the outboard midplane entrance to the SOL (“upstream” or “u”) and point 2 is the divertor terminus of the SOL (“target” or “t”). It is assumed that upstream density, nu, and the heat ﬂux into the SOL, q||, are control parameters; upstream and target temperatures, Tu and Tt, as well as plasma density in front of the target, nt, are subsequently determined.

11.3.1 Deﬁnitions

Dynamic and static pressure 21: 435

M 2 1 2 u ≪ p = nT 1+ M where 2 ( Mt 1 (Bohm Criterion) ≈ Heat conduction parallel to magnetic ﬁeld 21: 187

−1 7/2 dT ke,0 2000 [W m eV ] q = k T 5/2 where ≈ k, cond 0 −1 7/2 − dx ( ki,0 60 [W m eV ] ≈ Sheath heat ﬂux transmission coeﬃcient at a biased surface 21: 652

T eV m T −1/2 eV γ = 2.5 i + 2 2π e Z + i exp + χ ZT − T m T T i e e i e e where T = T , χ is the electron-ion recombination energy, e 6 i i and no secondary electrons emitted. 100 CHAPTER 11. TOKAMAK EDGE PHYSICS

11.3.2 Fundamental Relations SOL pressure conservation 21: 224

2ntTt = nuTu

SOL power balance 21: 224

7/2 7/2 7q||L Tu = Tt + 2k0 SOL heat ﬂux limited to sheath heat ﬂux 21: 224

q = γn T c 7 (D-D plasma, ﬂoating surface) || t t st ≈ 11.3.3 Consequences Upstream SOL temperature 21: 226

2/7 7q||L T assuming that T 7/2 T 7/2 u ≈ 2k t ≪ u 0 T is independent of n → u u T is insensitive to parameter changes due to the 2/7 power → u q is extremely sensitive to T due to the 7/2 power → k u Target SOL temperature 21: 227

10/7 2m qk T i t 2 2 4/7 2 ≈ γ e (Lk0) nu

1 Tt is proportional to 2 → nu Target SOL density 21: 227

3 6/7 2 2 n 7q||L γ e n = u T q2 2k 4m || 0 i n is proportional to n3 → T u Chapter 12

Tokamak Fusion Power

In this chapter, all units are SI with the exception of temperature and energy, which are deﬁned in the historical units of eV (electron-volts). n is the plasma density; n = n/1020; n = n n 20 0 e ≈ i T is the plasma temperature; TkeV = T in units of kiloelectron-volts p is the plasma pressure ν is the radialproﬁle peaking factor P is a power density S is a total power U is a total energy E is the nuclear reaction energy gain e is the elementary electric charge q is the total particle charge Z is the particle atomic (proton) number e and i subscripts refer to electrons and ions, respectively D and T refer to deuterium and tritium, respectively κ is the plasma elongation; κ=b/a a,b, and R0 are the 2 minor and major radii of a toroidal plasma V is the volume of the plasma

12.1 Deﬁnitions

Fusion power density 26: 8 1 P = n n σv E = n2 σv E fusion D T h iDT fusion 4 eh iDT fusion Fusion power 26: 22 π S = E n2 σv R dS total 2 fusion h iDT plasmaZ cross section 0.15 n 2 = Rab T 2 [MW] 2ν + 1 1020 keV 101 102 CHAPTER 12. TOKAMAK FUSION POWER

where it has been assumed that: r2 ν Pressure proﬁle: nT =n ˆTˆ 1 − a˜2 Reaction rate: σv 1.1 10−24T 2 h iDT ≈ × keV Plasma cross section:a ˜ = (ab)1/2 Alpha power density 26: 10 1 P = n n σv E = n2 σv E (1/5)P α D T h iDT α 4 eh iDT α ≈ fusion Neutron power density 26: 10 1 P = n n σv E = n2 σv E (4/5)P neutron D T h iDT neutron 4 eh iDT neutron ≈ fusion Ohmic heating power density 26: 240

P ηJ 2 ohmic ≈ plasma Stored energy in conﬁned plasma 26: 9

W = 3nT dτ = 3 nT V h i Z Definition of energy confinement time 26: 9 Stored energy in the confined plasma W τE = = Power lost from the confined plasma Sloss Power loss from a confined plasma due to conduction 26: 9−10 3nT Pconduction = τE Power loss from a confined plasma due to bremsstrahlung radiation 26: 227−228

P (5.35 10−37)Z2n n T 1/2 [W m−3] bremsstrahlung ≈ × e i keV

12.2 Power Balance in a D-T Fusion Reactor

Conﬁned fusion plasmas are not in thermal equilibrium, and, therefore, power must be balanced in a steady-state tokamak reactor. Power that is lost from the conﬁned plasma due to conduction, radiation and other mechanisms must be continuously replenished by alpha particle and auxilliary heating mechanisms.

0 = (P + P ) (P + P + . . .) alpha auxilliary − conduction bremsstrahlung where

Pauxilliary = Pohmic + PICH + PECH + Pneutral beam + . . . 12.3. THE IGNITION CONDITION (OR LAWSON CRITERION) 103

12.2.1 Impurity Eﬀects on Power Balance

The fractional impurity densities fj = nj/n0 in the plasma core cause:

(a) Modiﬁed quasi-neutrality balance 26: 36

ne = nD + nT + Znj Xj

(b) Increased radiated power loss 3

P (5.35 10−37)n2T 1/2Z [W m−3] bremsstrahlung ≈ × e keV eﬀ

1 P = n2(1 f Z )2 σv E alpha 4 e − j j h i α Xj

12.2.2 Metrics of Power Balance The physics gain factor for D-T plasma 26: 12

1 2 4 ne σv Efusion Vplasma 5Pα Qphys = h i · = Pheating Pheating

where

(a) Qphys=1 is break even

(b) Qphys >5 is a burning plasma

out Pelectricity Qeng = in Pelectricity

12.3 The Ignition Condition (or Lawson Criterion)

The ignition condition describes the minimum values for density (n), temperature (T ), and energy confinement time (τE) that are required for a confined plasma to reach ignition. Ignition is defined as Palpha > Ploss, where 26: 10−15 Pauxilliary = 0. For a given temperature, T , the following equations describe the minimum nτE required to reach ignition under different assumptions: 104 CHAPTER 12. TOKAMAK FUSION POWER

26: 10−11 (a) Palpha = Pconduction

12kT nτ = E σv E h i α Using σv 1.1 10−24T 2 and E = 3.5 MeV 3: h iDT ≈ × keV α nTτ & 3 1021 m−3 keV s E ×

3 (b) Palpha = Pconduction + Pbremsstrahlung : 12kT nτE = σv E 2.14 10−36T 1/2 h i α − × keV

3 (c) Palpha = Pconduction+Pbremsstrahlung with alpha impurities fα = nα/ne :

12kT nτE = (1 2f )2 σv E (1 + 2f )2.14 10−36T 1/2 − α h i α − α × keV

3 (d) Palpha = Pconduction+Pbremsstrahlung with impurity densities fj = nj/ne :

12kT nτE = 2 −36 1/2 (1 fjZj) σv Eα (2.14 10 )Zeﬀ TkeV − j h i − × P Chapter 13

Tokamaks of the World

Comprehensive list of all major tokamaks with parameters • www.tokamak.info

ASDEX-U (Garching, Germany) • http://www.ipp.mpg.de/ippcms/eng/for/projekte/asdex/techdata.html

Alcator C-Mod (Cambridge, USA) • http://www.psfc.mit.edu/research/alcator/

DIII-D (San Diego, USA) • https://fusion.gat.com/global/Home

EAST (Hefei, China) • http://english.hf.cas.cn/ic/ip/east

FTU (Frescati, Italy) • http://www.efda.org/eu_fusion_programme/machines-ftu_i.htm

Ignitor (Kurchatov, Russia) • http://www.frascati.enea.it/ignitor

ITER (Cadarache, France) • http://www.iter.org/mach

JET (Culham, United Kingdom) • http://www.jet.efda.org/jet/jets-main-features

J-TEXT (Wuhan, China) • http://www.jtextlab.com/EN/SortInfo.aspx?sid=43

JT-60SA (Naka, Japan) • http://www-jt60.naka.jaea.go.jp/english/figure-E/html/figureE_jt60sa_5.html

KSTAR (Daejeon, S. Korea) • http://www.pppl.gov/kstar/html/about_kstar.html

NSTX-U (Princeton, USA) • http://nstx-u.pppl.gov/

105 106 CHAPTER 13. TOKAMAKS OF THE WORLD

SST-1 (Gandhinagar, India) • http://www.ipr.res.in/sst1/SST1parameters.html

T-10 (Kurchatov, Russia) •

T-15U (Kurchatov, Russia) • www.toodlepip.com/tokamak/t15-ft_p7-3.pdf

TCV (Lausanne, Switzerland) • https://crppwww.epfl.ch/tcv

TEXTOR (J¨ulich, Germany) • http://www2.fz-juelich.de/ief/ief-4//textor_en

TFTR (Princeton, USA) • http://w3.pppl.gov/tftr/info/tftrparams.html

Tore Supra (Cadarache, France) • http://www-fusion-magnetique.cea.fr/gb/index.html 107 ------34 10 39.5 ------1 4 - - - - 7 2 0.5 4.5 0.35 n - - - - 6 3 3 12.5 ICRH ECRH LHCD Beam (18-24) c C C C M M C/Li SS/C SS/C PFC Be/W b L L L DN/SN SN/DN L/SN/DN L/SN/DN L/SN/DN L/SN/DN 5 1 4 10 92 10 2.5 0.3 1000 ) (s) (MW) (MW) (MW) (MW) 3 − 4 3 6 13 11 15 1.0 1.4 Vp Pulse Config 200 - - - 0.6 0.5 0.4 0.3 0.33 -0.7–1 Li = lithium; M = molybdenum; SS = stainless steel; W = tungste 2 1 1 1 DN = diverted double null κ δ 1.7 1.7 2.5 2.8 1.83 ing (SC) p 2 5 I 11 0.5 0.4 2.0 0.8 1.2 2.7 a φ 8 3 5 13 3.8 1.0 1.4 5.6 B 3.5 SC ǫ 0.33 0.24 0.36 0.32 0.25 0.61 0.24 0.28 0.35 0.22 0.40 0.47 0.96 0.26 0.57 0.36 0.25 0.87 6.2 2 0.32 5.3 SC 17 1.86 0.51.1 837 0.21.5 400 0.18 3C SN 0.22 Be/C/W 1.9 20 0.8 20 2 1000 - DN 33 C 1.5 0.2 1 0.8 (m) (m) (T) (MA) (m 0.67 1.70 1.32 2.96 3.16 1.021.05 0.32 2.7 SC 5.50.93 1.83 0.57 132 1000.88 DN2.52 C - 7 - 34 f d h g e Tokamak RASDEX-U a 1.65C-Mod 0.5 0.30EAST 3.1 1.4Ignitor 1.8 0.4JET 14 10J-TEXT SNNSTX-U C/WT-10 6 4TCV -TFTR 20 DIII-D 1.66FTU 0.67 0.40 2.2 0.93ITER 0.3 3 0.32JT-60SA 2.6 8 1.0 1.6KSTAR 12 1.7 1.8 0.55 5SST-1 0.5 3 SN/DN 0.28 3.5 1.5T-15 C 2.0 L 2.0 2.43Textor 5 0.42 0.8 SS/M/W 0.17 1.75 3.5 SC 6 0.5Tore 18 0.47 Supra 0.27 1 2.25 1.3 20 - 0.7 2.8 1.47 0.31 2.5 0.25 4.5 0.8 DN SC 20 12 1.7 1 - 1000 C 1 - SN - 6 8 22 C 10 - 390 L/SN 1.5 - L SS/C 8 7 4 C 4 1 9 9 - 2.4 4 5 1.7 Magnet coils are conducting unless denoted as superconduct Plasma configuration: L = limited; SN = diverted single null; Proposed; construction not begun Construction begun; first plasmaConstruction predicted begun: 2019 first plasmaConstruction predicted begun: 2014 first plasmaDecommissioned predicted in 2012 1997 Plasma Facing Components: Be = beryllium; C = CFC/graphite; f b e g c a d h 108 CHAPTER 13. TOKAMAKS OF THE WORLD References

[1] M. Boas. Mathematical Methods in the Physical Sciences. John Wiley & Sons, 1983.

[2] H.-S. Bosch and G.M. Hale. Improved formulas for fusion cross- sections and thermal reactivities. Nuclear Fusion, 32(4):611, 1992. URL http://stacks.iop.org/0029-5515/32/i=4/a=I07.

[3] Calculated, derived or produced by the authors of the Magnetic Fusion Energy Formulary explicitly for this work.

[4] M.B. Chadwick, P Oblo˘zeinsk´y, et al. ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Sci- ence and Technology. Nuclear Data Sheets, 107(12):2931 – 3060, 2006. ISSN 0090-3752. doi: 10.1016/j.nds.2006.11.001. URL http://www.sciencedirect.com/science/article/pii/S0090375206000871. Evaluated Nuclear Data File ENDF/B-VII.0.

[5] F. F. Chen. Introduction to Plasma Physics and Controlled Fusion. Springer Science, 2006.

[6] J G Cordey. A review of non-inductive current drive theory. Plasma Physics and Controlled Fusion, 26(1A):123, 1984. URL http://stacks.iop.org/0741-3335/26/i=1A/a=311.

[7] J. P. Freidberg. Ideal Magnetohydrodynamics (Out of Print). Plenum Press, 1987.

[8] J. P. Freidberg. Plasma Physics and Controlled Fusion. Cambridge University Press, 2007.

[9] W. L. Granatstein and P. L. Colestock. Wave Heating and Current Drive in Plasmas. Gordon and Breach Science, 1985.

[10] D. J. Griﬃths. Introduction to Electrodynamics. Prentice Hall, 1999.

[11] P. Helander and D. Sigmar. Collisional Transport in Magnetized Plas- mas. Cambridge University Press, 2002.

[12] J.D. Huda. NRL Plasma Formulary. The Oﬃce of Naval Research, 2007.

[13] J. D. Jackson. Classical Electrodynamics. John Wiley & Sons, 1999.

109 110 REFERENCES

[14] A. Jeﬀrey. Handbook of Mathematical Formulas and Integrals. Aca- demic Press, 2008.

[15] K. S. Krane. Introductory Nuclear Physics. John Wiley & Sons, 1988.

[16] Y R Martin, T Takizuka, and the ITPA CDBM H-mode Threshold Database Working Group. Power requirement for accessing the H-mode in ITER. Journal of Physics: Conference Series, 123(1):012033, 2008. URL http://stacks.iop.org/1742-6596/123/i=1/a=012033.

[17] ITER Physics Expert Group on Confinement, Transport, ITER Physics Expert Group on Confinement Modelling, Database, and ITER Physics Basis Editors. Chapter 2: Plasma confinement and transport. Nuclear Fusion, 39(12):2175, 1999. URL http://stacks.iop.org/0029-5515/39/i=12/a=302.

[18] R. R. Parker. Course Notes (MIT Course 22.105 Electromagnetic In- teractions. 2007.

[19] M. Porkolab. Course Notes (MIT Course 8.624 Plasma Waves. 2008.

[20] J.E. Rice, A. Ince-Cushman, et al. Inter-machine comparison of intrinsic toroidal rotation in tokamaks. Nuclear Fusion, 47(11):1618, 2007. URL http://stacks.iop.org/0029-5515/47/i=11/a=025.

[21] P. C. Stangeby. The Boundary of Magnetic Fusion Devices. Taylor & Francis, 2000.

[22] T. H. Stix. Waves in Plasma. Springer Science, 1992.

[23] I. J. Thompson and F. M. Nunes. Nuclear Reactions for Astrophysics. Cambridge University Press, 2009.

[24] F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato, and S. Succi. Mhd-limits to plasma conﬁnement. Plasma Physics and Controlled Fusion, 26(1A):209, 1984. URL http://stacks.iop.org/0741-3335/26/i=1A/a=319.

[25] G. R. Tynan, A. Fujisawa, and G. McKee. A review of experimental drift turbulence studies. Plasma Physics and Controlled Fusion, 51(11):113001, 2009. URL http://stacks.iop.org/0741-3335/51/i=11/a=113001.

[26] J. Wesson. Tokamaks (3rd Edition). Oxford University Press, 2004.

[27] Wolfram Mathworld, 2011. Elliptic Torus. Available at http://mathworld.wolfram.com/EllipticTorus.html. Index

Γ factor, 69 Braginskii thermal diffusivity coeffi- βN , 88 cients, 62 ǫ, 83 Bremsstrahlung power losses, 102 ν∗, 64 Brewster’s angle, 31 ρ∗, 91 ρS, 91 Capacitance, 28 ζ function, 69 Center of mass transformation, 37 q, q∗, 85 Charged particle interactions, 74 1/R force, 87 Circuit electrodynamics, 27 Classical electron radius, 76 Alcator C-Mod, 105 Classical transport, 59 Alfvén speed, 45 Cold plasma dispersion relation, 65 Ampere’s law, 25 Cold plasma waves Appleton-Hartree, 66 L-wave, 66 ASDEX-U, 105 Light wave, 66 Astrophysical S-factor, 77 O-wave, 66 Atomic constants, 21 R-wave, 66 X-wave, 66 Banana regime, 64 Collision frequencies Basis functions of Laplace’s equation, Definition, 39 16 Evaluation, 39 Beam-electron collision frequency, 40 Collision frequency scalings, 39 Beam-ion collision frequency, 41 Collision operator relations, 51 Bessel functions Collision operators, 60 Asymptotic forms, 14 Collisionality, 64 equation, 13 Complimentary error function, 19 Modified asymptotic forms, 16 Compressional Alfvén wave, 68 Modified equation, 14 Compton scattering, 76 Modified useful relations, 15 Conduction power losses, 102 Plots, 14 Connection length, 97 Useful relations, 14 Connor-Hastie limit, 47 Beta limit, see Troyon limit Conservation laws, 28 Bethe formula, 75 Conservation of charge, 29 Binary coulomb collisions, 37 Coulomb force, 26 Biot-Savart law, 26 Coulomb logarithm, 39 Bohm criterion, 98 Couloumb gauge, 32 Bootstrap current, 89 Crook collision operator, 62 Bounce frequency, 63 Cross sections Bound electric charge, 27 Definitions, 76 Braginskii diffusion coefficient, 60 Parametrization, 79

111 112 INDEX

Plotted data, 81 Equations Tabulated values, 79 1D cylindrical diffusion, 59 Curl theorem, see Stoke’s theorem Bessel’s equation, 13 Curvature drift, 36 Grad-Shafranov, 54 Cyclotron frequency Legendre’s equation, 13 in plasmas, 43 Maxwell’s, 25 Cylindrical coordinate systems, 10 Vlasov, 49 Wave, 30 Debye length, 44 Error function, 18 Debye sphere, 44 ETG, see Electron thermal gradient Density limit, see Greenwald limit instability Diamagnetic drift frequency, 91 External inductance, 86 Diamagnetic drift velocity, 91 Diamagnetic plasmas, 87 Faraday’s law, 25 DIII-D, 105 Fast Magnetosonic wave CD, 90 Dipoles, 27 First adiabatic invariant, 37 Displacement electric field, 27 Fokker-Planck collision operator, 60 Divergence theorem, see Gauss’s the- Fresnel equations, 31 orem Frozen-in law, 53 Driecer electric field, 47 FTU, 105 Drifts Fundamental theorem of calculus, 12 Curvature, 36 Gamma interactions, 75 E B, 36 × Gauss’s law, 25 grad-B, 36 Gauss’s theorem, 12 Polarization, 36 Gaussian integrals, 16 E B drift, 36 Goldston scaling, see Energy confine- × EAST, 105 ment scalings Electric field of a moving charge, 32 Grad-B drift, 36 Electric potential, 25 Grad-Shafranov equation, 54 Electromagnetic force density, 29 Greenwald limit, 88 Electron collision time, 40 Gryo motion in E and B fields, 35 Electron thermal gradient instability, Gyro radius 93, 94 in plasmas, 44 Electron-electron collision frequency, H-field, 27 39 Heat conduction, 99 Electron-ion collision frequency, 39 Heat flux tranmission coeficient, 99 Electrostatic boundary conditions Hoop force, 86 In matter, 27 Hot plasma waves, 68 In vacuum, 26 Electromagnetic, 70 Electrostatic waves, 67 Electromagnetic (maxwellian), 71 Electrostatics, 25 Electrostatic, 68 Elongation, 83 Energy confinement scalings, 90 Ignition condition, 103 Energy confinement time, 102 Ignitor, 105 Energy stored in EM fields, 26 Index of refraction, 31 Energy transfer time, 40 Inductance, 28 Engineering gain factor, 103 Inductance of plasma, 86 INDEX 113

Inductive current, 89 Magnetohydrodyamics, 52 Intensity, 30, 31 Maxwell’s equations Internal inductance, 86 In matter, 27 Inverse aspect ratio, 83 In vacuum, 25 Ion collision time, 40 Maxwell’s stress tensor, 29 Ion thermal gradient instability, 92, Maxwellian distribution function, 45 94 MHD Equations, 53 Ion-electron collision frequency, 39 MHD equilibrium equations, 54 Ion-impurity collision time, 40 MHD stability, 54 Ion-ion collision frequency, 39 MHD waves, 68 ITER, 105 Mirroring condition, 37 ITG , see Ion thermal gradient instability Neoclassical transport, 62 Neutral beam current drive, 89 J-TEXT, 105 Neutral beam energy loss, 41 JET, 105 Neutral beam heating, 88 JT-60SA, 105 Neutron interactions, 75 Neutron lethargy, 75 Klein-Nishina cross section, 76 NSTX-U, 105 KSTAR, 105 Nuclear binding energy, 74 Nuclear constants, 21 L-H mode power scalings, 91 Nuclear Interactions, 74 Large aspect ratio expansion, 83 Nuclear mass, 73 Lawson criterion, see Ignition condition Ohm’s law Legendre polynomials, 12 Macroscopic, 28 Legendre’s equation, 13 Microscopic, 28 LHCD, 89 Ohmic heating, 88 Lienard-Wiechert potentials, 32 One fluid model of plasma, 51 Lindear regime scaling, 90 One-fluid equations, 52 Lorentz force law, 26 Lorentz gauge, 32 Paramagnetic plasmas, 87 Lower hybrid oscillations, 67 Partial differential equations, 16 Particle transport coefficients Mach number, 45 Classical, 60 Magnetc field of a moving charge, 33 Neoclassical, 63 Magnetic boundary conditions Passing particle dispersion relation, In matter, 27 93 In vacuum, 26 Passing particle modes, 93 Magnetic inductance, 86 Pfirsch-Schluter regime, 64 Magnetic mirroring, 36 Physics gain factor, 103 Magnetic moment, 36 Plasma distribution function, 45 Magnetic shear, 85 Plasma frequencies, 43 Magnetic turbulence flux of energy, Plasma oscillations, 67 92 Plasma parameter, 44 Magnetic turbulence flux of momen- Plasma pressure, 84 tum, 92 Plasma resistivity, 47 Magnetic vector potential, 25 Plasma temperature definition, 46 Magnetization, 27 Plateau regime, 64 114 INDEX

Polarization, 27 Stopping power, 75 Polarization drift, 36 Stored plasma energy, 102 Poloidal beta, 84 Surface area of a torus, 84 Poloidal magnetic field, 85 Suydam’s criterion, 56 Power balance, 102 Power densities, 101 T-10, 105 Poynting vector, 29 T-15U, 105 Poynting’s theorem, 29 TCV, 105 TEM, see Trapped electron mode Q TEXTOR, 105 Fusion, 103 TFTR, 105 Nuclear reactions, 74 Thermal diffusivities (Classical), 62 Quasi-transverse waves, 67 Thermal equilibration, 40 Radial electric field, 84 Thermal speed, 43 Radiation pressure, 30 Times Radius of curvature, 36 electron collision, 40 Random walk diffusion coefficient, 60 electron slowing down, 47 Reaction rates energy tranfer, 40 Definitions in plasma, 77 ion collision, 40 Parametrization, 80 ion-impurity collision, 40 Plotted data, 81 Tire tube force, 87 Tabulated values, 80 Tore Supra, 105 Reaction threshold energy, 74 Toroidal beta, 84 Reduced mass, 38 Toroidal force balance, 86 Reflection coefficient, 31 Toroidal magnetic field, 85 Retarded time, 32 Toroidal plasma volume, 84 Rice scaling, see Toroidal rotation scal- Toroidal rotation scalings, 91 ing Transmission coefficient, 31 Rodrigues’ formula, 13 Trapped electron mode, 93, 95 Rosenbluth collision operator, 61 Trapped ion mode, 94 Runaway electrons, 47 Trapped particle condition, 63 Trapped particle dispersion relation, Safety factor, 85 94 Scrape off layer (SOL), 97 Trapped particle fraction, 37 Screw pinch stability, 56 Trapped particle modes, 94 Semi-empirical mass formula, 73 triangularity, 83 Shafranov shift, 87 Trivelpiece-Gould, 67 Shear Alfvén wave, 68 Troyon limit, 88 Single particle drifts, 36 Turbulence, 91 Snell’s law, 31 Turbulent flux of energy, 92 Sommerfeld parameter, 77 Turbulent flux of momentum, 92 Sound speed, 45 Turbulent flux of particles, 92 Sound wave, 68 Two fluid model of plasma, 49 Speed of light in material, 30 Two point model, 99 Spherical coordinate systems, 11 Spitzer resistivity, 47 Unit definitions, 21 SST-1, 105 Universal constants, 21 Stoke’s theorem, 12 Upper hybrid oscillations, 67 INDEX 115

Vector identities, 9 Vertical magnetic ﬁeld, 87 Vlasov equation, 49 Volume of a torus, 84

Wave equation, 30 Wave polarization, 66

Zeﬀ , 45