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The Physics of Inertial FusionBeamPlasma Interaction, Hydrodynamics, Hot Dense Matter$

Stefano Atzeni and Jürgen Meyer-ter-Vehn

Print publication date: 2004

Print ISBN-13: 9780198562641

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198562641.001.0001

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(p.429) Appendix

(p.429) Appendix

Source:
The Physics of Inertial Fusion
Publisher:
Oxford University Press

A. Units and conversion of units

In this book, we use mainly Gaussian units and measure temperature and atomic transition energies in eV. In some cases, we use SI units, too. Concerning advantages and disadvantages of Gaussian versus SI units and also for how to convert them, we refer the reader to the discussion by Jackson (1999) in his book on classical electrodynamics. Here we give only a few conversion relations needed to evaluate formulas in this book.

In the Gaussian system, the basic mechanical units are cm, g, s, while they are kg, m, s, in the SI system, and the conversion factors involve only powers of 10. Some representative conversion relations are:

pressure 1 bar = 10 5 Pa = 10 6 erg/cm 3 , energy 1 J = 10 7 erg, power 1 TW = 10 12 W= 10 19 erg/s .

For the quantities of electrodynamics, the basic Gaussian units are statcoulomb (statC) for the electric charge, statvolt (statV) for the electric potential, and gauss (G) for the magnetic field. The conversion factors to the corresponding SI units of coulomb (C), volt (V), and tesla (T) involve the velocity of light and are given by

electric charge 1 C = 2 .9979 × 10 9 statC, electric pontential 1V = 1/299 .79 statV, magnetic field 1T = 10 4 G .

The unit charge (electron charge) in the two systems is

(A.1)
e = 4.8033 × 10 10 statC = 1 .6022 × 10 10 C .

In this book, we always take the electric charge in Gaussian units. In using them, one should keep in mind the dimensional relations statC · statV = erg and (statC)2 = erg cm. They correspond to the expression E = q 1 U 2 = q 1 q 2/r, written in Gaussian units and giving the Coulomb energy E of charge q 1 in the potential U 2 of charge q 2 at distance r.

(p.430) For conversion of energy units, the central relation is

(A.2)
1 eV = 1 .6022 × 10 -12 erg = 1 .6022 × 10 19 J,
where we have used the SI relations 1 C = 1 As and 1 AV s = 1Ws = 1J. Temperatures are expressed in energy units (electron volt, eV) by using the relation E = k B T, with k B the Boltzmann constant. Therefore
(A.3)
1 eV= (1 .6022 × 10 -12 / 1.3807 × 10 16 ) K=11605 K .

Within this book, electric quantities occur mostly in the context of atomic collision and radiation physics, where Gaussian units are commonly used. In particular, one should notice that most expressions, including also the plasma frequency, contain the unit charge only as e 2, which is obtained from eqn (A.1) in pure mechanical units:

(A.4)
e 2 = 2.307 × 10 19 erg cm = 1 .440 × 10 7 eV cm .

Furthermore, the fine structure constant in its canonical form, αf = e 2/ℏc = 1/137.04 (Gaussian units!), may be conveniently used to evaluate e 2 in many formulas of this book.

B. Physical constants

velocity of light

c = 2.9979 × 1010 cm/s

Planck's constant

h = 6.6261 × 10−27 erg s

ℏ = h/2π = 1.0546 × 10−27erg s

c = 1.9732 × 10−5 eV cm

electron charge

e = 1.6022 × 10−19C

e = 4.8033 × 10−10 statC

e 2 = 1.4400 × 10−7 eV cm

fine structure constant

αf = e 2/ℏc = 1/137.04

electron mass

m e = 9.1096 × 10−28 g

m e c 2 = 0.51100 MeV

proton mass

m p = 1.6726 × 10−24g

m p c 2 = 938.24 MeV

proton–electron mass ratio

m p/m e = 1836.1

atomic mass unit

𝓂 p = 1.6605 × 10−24g

atomic energy unit

EA = e 2/a B = 27.20 eV

Stefan-Boltzmann constant

σ B = 2 π k B 4 / 15 c 2 h 3

= 5.6703 × 10−5 erg s−1 cm−2 K−4

= 1.0285 × 1012 erg s−1 cm−2 eV−4

Bohr radius

a B = ℏ2/m e e 2 = 0.52918 × 10−8 cm

classical electron radius

r 0 = e 2/m e c 2 = 2.8179 × 10−13 cm

Compton wavelength

r C = ℏ/m e c = 3.8616 × 10−11 cm

Boltzmann constant

k B = 1.3807 × 10−16 erg/K

= 1.6022 × 10−12 erg/eV

SI vacuum dielectric constant

ɛ0 = 8.8542 × 10−12 F/m

SI vacuum magnetic permittivity

μ0 = 4π × 10−7 H/m

power unit

P 0 = m e 2 c 5 / e 2 = 8.7 GW

Alfvén current unit

I 0 = m e 2 c 5 / e 2 = 17 kA

(p.431) In this book, we have written nuclear masses in the form Am p as the product of mass number A and proton mass m p. More precisely, when evaluating numerical factors, one should use the atomic mass unit ¯ p instead of m p in order to account for the average mass defect of nucleons bound in nuclei.

C. Frequently used symbols

We list symbols used frequently in this book. Unavoidably a few symbols are used with different meanings at different places. This applies in particular, to the symbol e, which stands for unit charge, specific energy, and Euler constant (e = 2.7182818 …). In these cases, the meaning should be clear from the context.

The list below also includes a number of subscripted variables. Many others appear in the text. For convenience, a list of the most common subscripts is printed following the list of the main symbols.

Italics symbols

a

acceleration Lagrange coordinate

a B

Bohr radius

A

mass number

A if

in-flight aspect ratio

A t

Atwood number

𝒜

entropy constant

B

magnetic field

c

velocity of light sound velocity (chapter 6)

c s

sound velocity

c T

isothermal sound velocity

e

electron charge specific internal energy Euler constant

e c

cold fuel specific energy

e h

hot spot specific energy

e x, e y, e z

unit vector in x, y, z direction

E

electric field energy

E c

cold fuel energy

E d

driver energy

E f

fuel energy

E h

hot spot energy

E n

n-th energy level

ɛ

particle energy

F

(Helmholtz) free energy

Froude number

g n

degeneracy factor of level n

G

target energy gain

G f

fuel energy gain

G P

Planck weighting function

G R

Rosseland weighting function

h

Planck constant specific enthalpy

h/2π

H B

burn parameter

H c

cold fuel confinement parameter

H f

fuel confinement parameter

i

imaginary unit

j

mass flow

I L

laser intensity

I ν

spectral intensity

I νP

Planck spectral intensity

k

wave number

k B

Boltzmann constant

l

angular momentum quantum number spherical mode number

L, L 0, L min

characteristic lengths

In Λ, In Λe, In Λi, In Λαe, In Λfe

Coulomb logarithms

m

Lagrange mass coordinate

m e

electron mass

m p

proton mass

m f

mass of average fuel ion

m i

ion mass

m r

reduced mass

M

mass

M c

cold fuel mass

M h

hot spot mass

M f

fuel mass

Mach number

n

number density geometry index

n e

electron number density

n i

ion number density

p

pressure

p a

ablation pressure

p c

cold fuel pressure

p e

electron pressure

p h

hot spot pressure

p i

ion pressure

p L

light pressure

P b

beam power

P d

driver power

q DT

DT fuel specific yield

q

heat flux

Q

fusion reaction quality factor Q

r

radial coordinate reflectivity coefficient

R

radial coordinate, sphere radius

reaction characteristic range (mass/area)

R h

hot spot radius

R f

fuel radius

s

specific entropy

S

entropy surface area

S r

radiation flux

t

time

T

temperature

T e

electron temperature

T i

ion temperature

T F

Fermi temperature

T h

hot spot temperature

T r

radiation temperature

T id

ideal ignition temperature

𝒯

barrier transparency surface tension

u, u

hydrodynamic velocity

u a

ablation velocity

u imp

implosion velocity

U ν

spectral radiation energy density

U ν p

Planck spectral energy density

υ

particle velocity velocity relative to shock front

V

volume specific volume

x,y,z

Cartesian coordinates

W b

bremsstrahlung power density

W e

thermal conduction average power density loss

W fus

fusion power density

Z

atomic number

Z i

ion charge number

(p.432)

Greek symbols

α

isentrope parameter

αif

in-flight-isentrope parameter

αf

fine structure constant

γ

adiabatic exponent

Γ

plasma parameter

ΓB

gas constant

ε

particle energy

εF

Fermi energy

ɛ

dielectric function

ζ

perturbation amplitude

η

overall coupling efficiency

ηh

hydrodynamic efficiency

ηcon

conversion efficiency

ηtran

transfer efficiency

θ

angle

κa

absorption coefficient

κP

Planck mean opacity

λ

wavelength of perturbation

λD

Debye length

λL

wavelength of laser, in vacuum

ν

frequency

νe

electron-ion collision frequency

ξ

dimensionless similarity variable

ρ

mass density

ρc

cold fuel density critical density

ρDT

density of solid DT

ρh

hot spot density

ρR

confinement parameter

σ

instability growth rate cross section

σB

Stefan-Boltzmann constant

σRT

classical RTI growth rate

τe

electron collision time

τE

energy confinement time

τei

electron-ion energy exchange time

τi

ion collision time

Φ

burn efficiency

χ

conductivity

χe

electron thermal conductivity

χR

radiation heat conductivity

ω

(circular) frequency

ωp

plasma frequency

Ω

solid angle

(p.433)

Frequently used indices

c

cold fuel

DT

deuterium-tritium

e

electron

f

fuel

h

hot spot

i

ion

l

l-th angular momentum component order of spherical mode

n

n-th energy level

p

proton

r

radiation

x,y,z

cartesian components

α

alpha particle

ν

photon of frequency ν

(p.434) D. Acronyms

AIM

average ion model

CPA

chirped pulse amplification

DT

deuterium-tritium

ENEA

Ente per le Nuove Tecnologie, l'Energia e l'Ambiente, Italy

EOS

equation of state

ICF

inertial confinement fusion

IFE

inertial fusion energy

ILE

Institute of Laser Engineering, Osaka, Japan

ISI

induced spatial incoherence

KHI

Kelvin-Helmholtz instability

LLE

Laboratory for Laser Energetics, Rochester, NY, U.S.A.

LLNL

Lawrence Livermore National Laboratory, Livermore, CA, U.S.A.

LMJ

Laser MegaJoule

LTE

local thermodynamic equilibrium

LWFA

laser wake-field acceleration

MCF

magnetic confinement fusion

MFE

magnetic fusion energy

MPQ

Max-Planck Institute für Quantenoptik, Garching, Germany

NIF

National Ignition Facility

PIC

Particle-in-cell

RMI

Richtmyer-Meshkov instability

RPP

random phase plate

RTI

Rayleigh-Taylor instability

SBS

stimulated Brillouin scattering

SRS

stimulated Raman scattering

SSD

smoothing by spectral dispersion

STA

super transition array

TF

Thomas-Fermi

UTA

unresolved transition array

WKB

Wentzel-Kramers-Brillouin approximation