A First-Principles Model for Estimating Atmospheric ...janow/Vitae/NWM200606-1.pdf · This paper develops a simple first-principles model for estimating the fireball temperature,

Copyright 2006 by Rich Janow Page 1

A First-Principles Model for Estimating Atmospheric Nuclear Weapons Effects on Command, Control, and Communication - I

(Unclassified)

Dr. Rich Janow

May 30, 2006

Applied Physics Department New Jersey Institute of Technology

Newark, NJ 07102-1982 [email protected]

The Analytic Solutions Group, LLC South Orange NJ 07079

[email protected] 973-762-4987

ABSTRACT

This paper develops a simple, first principles model for nuclear weapon effects pertinent to Command, Control, and Communications. It describes the fireball temperature, size, electron density, and radio frequency thermal radiation at early times after an atmospheric nuclear explosion. The goal is to provide order of magnitude estimates of communications interference and qualitative insight into the mechanisms underlying nuclear weapons effects.

The first law of thermodynamics was applied to the energy balance within an expanding, radiating mass of ionized air, using methods of near-equilibrium statistical physics augmented by phenomenological arguments. Numerical evaluation is deferred to a follow-on paper. All sources and work are unclassified.

Copyright 2006 by Rich Janow Rev. May 26, 2006 Page 2

CONTENTS

1. Introduction .......................................................... 3

2. Analytic Model for an Air Burst Fireball .............................. 3

2.1. Assumptions 3

2.2. First Law Formulation 4

2.3. Constitutive Equations for the Plasma 5

2.4. Constitutive Equations for the Radiation 7

2.5. Plasma Frequency Effects on Fireball Emissions 8

2.6. Solution for the Fireball Temperature Dependence 9

2.6.1 Equation for the Temperature Dependence 10

2.6.2 Solution for phase I temperature dependence 11

2.6.3 Fireball temperature and radius as phase I ends 12

2.6.4 Solution for phase II temperature dependence 13

2.6.5 Fireball temperature and radius as phase II ends 14

2.6.6 Solution for phase III temperature dependence 15

3. Model for temperature dependence of the electron density and ionization energy in Air ........................................................ 16

3.1. Problem Formulation 16

3.2. The Partition Function: 18

3.3. The Average Electron Density 18

3.4. The Ionization Energy 20

3.5. The Electron Density and Ionization Energy in Air 20


REFERENCES

1. Glasstone, S. and P. Dolan, eds., 'The Effects of Nuclear Weapons', 3rd edition,

USDOD/USDOE (1977).

2. Brode, H., 'A Review of Nuclear Weapons Effects', Ann. Rev. Nuc. Sci. 18, p.

153ff (1968).

3. Brode, H. L., W. Asano, A. Stevenson, M. Plemmons, L. Scanlon, 'A Program for

Calculating Radiation Flow and Hydrodynamic Motion, RAND Corporation RM-5187-PR

(1967).

4. Hillendahl, R. W., 'Theoretical Models for Nuclear Fireballs', Lockheed Missiles

and Space Company, LMSC-B006750 (1965).

5. Zemansky, M., 'Heat and Thermodynamics', 5th Ed., McGraw-Hill Book Co. (1968).

6. Landau, L. and E. M. Lifshitz, 'Statistical Physics', Addison Wesley/Pergamon

Press (1958).

7. Huang, K., 'Statistical Physics', John Wiley & Sons (1963).

8. Kittel, C., 'Elementary Statistical Physics', John Wiley & Sons (1958).

9. Bekefi, G., 'Radiation Processes in Plasmas', John Wiley (1966).

10. Akhiezer, A., I. Akhiezer, R. Polovin, A. Sitenko, and K. Stepanov, 'Plasma

Electrodynamics', Vol. I, Pergamon Press (1975).

11. Bethe, H., K. Fuchs, J. Hirschfelder, J. Magee, R. Peierls, and J. von Neumann,

'Blast Wave', LASL Report LA-2000 (1947, Declassified 1958.

12. Ziman, J., 'Principles of the Theory of Solids', Cambridge University Press

(1964).

13. Platzman, P., and P. Wolff, 'Waves and Interactions in Solid State Plasmas',

Academic Press (1973).

14. Ecker, G., 'Theory of Fully Ionized Plasmas', Academic Press (1972).

15. Pines, D., 'Elementary Excitations in Solids', W. A. Benjamin, Inc. (1964).

16. Jackson, J., 'Classical Electrodynamics', John Wiley & Sons (1962).

17. Feynman, R., R. Leighton, and M. Sands, 'The Feynman Lectures on Physics', Vols.

I, II, Addison Wesley Publishing Co. (1964).

18. Uman, M., 'Introduction to Plasma Physics', McGraw-Hill Book Company (1964).

19. Wu, T., 'Kinetic Equations of Gases and Plasmas', Addison-Wesley Publishing Co.

(1966).

20. Rishbeth, H. and O. Garriott, 'Introduction to Ionospheric Physics', Academic

Press (1969).

21. Davies, K., 'Ionospheric Radio Propagation', National Bureau of Standards,

Monograph 80 (1965).

22. Tolman, R., 'The Principles of Statistical Mechanics', Oxford University Press

(1938).

23. Tzoar, N., 'Many Body Theory', Lectures given at CUNY (l968).

24. Raimes, S., 'Many Electron Theory', North-Holland Publishing Co. (1972).

25. Schiff, L., 'Quantum Mechanics', McGraw-Hill Book Company (1955).

26. Abramowitz, M. and I. Stegun, 'Handbook of Mathematical Functions', National

Bureau of Standards (1964).

27. Douglas, B., D. McDaniel, 'Concepts and Models of Inorganic Chemistry', Blaisdell

Publishing Company (1965).

28. Kam, G., Private Communication.


1. Introduction

This paper develops a simple first-principles model for estimating the fireball

temperature, size, electron density, and radio frequency emissions produced within the

first few seconds after an atmospheric nuclear explosion. The work offers qualitative

understanding of nuclear effects on Command, Control, and Communications of Missile

Defense.

The model provides physical insight applicable to communications and radar. It can be

solved analytically and it qualitatively shows many features of fireball evolution that

are revealed in much more detailed calculations [1,2,11]. Agreement with standard

weapon codes has not been audited, as the simplicity of this model precludes

quantitatively accurate results.

The first section describes the approach and presents an intuitive picture of fireball

evolution which emerged after initial calculations. The next section develops

analytical solutions for the fireball temperature and radius as a function of time.

Section (3) develops analytical expressions for the fireball ionization as a function

of temperature.

A follow-on paper in preparation will describe the numerical methods used for

evaluation and present results.

This work was begun under a Federal contract when the author was at Bell Laboratories.

All sources and work are unclassified.

2. Analytic Model for an Air Burst Fireball

This section develops analytical expressions for the evolution of a nuclear air burst

fireball using many concepts embedded in the standard unclassified weapons effects

treatments [1,2,3,4] and basic equilibrium statistical physics [5,6,7,8].

An energy balance equation based on the First Law of Thermodynamics was used to

determine the fireball history for a hot isothermal sphere of a constant mass of air.

The rate of change of the internal energy function was equated to the energy loss

terms, and the resulting expression solved for the temperature as a function of the

radius in three solution regimes defining successive phases of the explosion’s history.

The solutions incorporate constitutive equations for the fireball material.

A deficiency of this model is its failure to automatically account for radiative

fireball growth; that is, growth in the mass of heated and ionized air as cold air

captures outbound gamma and X radiation. Theoretical treatments of radiation transport

reveal this growth but are generally not analytically solvable [2,3,9,10]. One of the

first applications of early electronic computers was numerical modeling of such growth

[11].

Blast hydrodynamic effects were summarized via a phenomenological radial expansion

velocity related to the local speed of sound. The shock, winds, and convection, are of

only peripheral interest; the intent is to describe only the nuclear weapon effects

which inhibit electromagnetic wave propagation. Elaborate computer codes are

universally used to describe the detailed hydrodynamic effects in conjunction with

radiative transport.

2.1. Assumptions

The weapon fireball was assumed to be a sphere of hot ionized air of uniform

temperature expanding at a constant radial velocity on the order of the local speed of

sound. The matter and radiation gases inside the evolving fireball were assumed to be

in 'local thermal equilibrium'.

Each small fireball volume element at all stages of fireball development is assumed to

be in 'local thermodynamic equilibrium' assumption [2,3,9,11], meaning that strong,


fast relaxation processes are assumed to be at work. The radiation, electrons, ions,

and neutral particles in each volume cell of the fireball plasma are all assigned the

same temperature. The temperature still varies from point to point and as a function of

time. Additionally, the spatial variation of the temperature across the entire

fireball region is neglected to make the problem tractable.

One should expect this simple model to give poor predictions for high altitudes burst,

where the radiation and particle mean free paths are long due to reduced air density

[1]. The local equilibrium picture also fails for the fireball immediately after

detonation; a burst initially creates distinctly non-equilibrium populations of heated,

ionized air and weapon debris molecules which can not be approximated by equilibrium

models. In the ionosphere and above, these populations persist for appreciable times

due to slow relaxation of the free electron density, while local equilibrium comes much

more quickly where the air is dense.

2.2. First Law Formulation

The early nuclear fireball is represented by a constant-temperature volume V of heated

air, expanding at a known rate dV/dt.

The total energy U trapped within the volume V includes three contributions:

radpk UUUU ++++++++==== (2.1a)

In the above:

Uk represents the energy density of an ideal gas plasma containing a mixture of

molecules, ions, and free electrons. The specific heat of the electrons dominates

when the gas is highly ionized, which is the case immediately after the weapon

detonates.

Up is the mutual electrostatic potential energy of all these particles, and consists

principally of ionization energy Ui, which can be the dominant term in U. The size

of the early energy deposition region and the rate of energy release depend

sensitively on the electron-ion recapture time.

Urad is the internal energy of the electromagnetic radiation within V, represented by

a gas of photons having a Black Body distribution at the same temperature as the

plasma.

Two competing energy loss mechanisms draw energy from the fireball:

Radiative loss of Black Body radiation through the surface bounding V.

Work done quasi-statically by the expanding fireball gasses against the surrounding

air - like a gas expanding within a cylinder capped by a massless piston. The

constant expansion velocity dr/dt related to the volume expansion rate dV/dt is an

adjustable parameter, taken to be approximately the speed of sound.

The following equation expresses energy conservation applied to the 'fireball' volume:

−−−−

−−−−====

dt

dVP

dt

dU

dt

dU

rad

(2.2)

The explicit forms for the terms above depend on the constitutive equations for matter

and radiation that are developed in the following section. The resulting equation is

then solved for the temperature T as a function of the radius, which depends on time.

The pressure P contains two contributions in principle: one representing the kinetic

(gas) pressure and a second due to radiation pressure of the photons, which can be

neglected.

The equilibrium assumptions allow the contributions to U and to the energy loss to be

expressed as functions of the temperature and other state variables alone, rather than

as complex path integrals following the evolution of the system. As a result, we are


able to obtain approximate but analytic solutions for the temperature T(r) with r(t)

describing the radial fireball expansion at an externally imposed rate.

2.3. Constitutive Equations for the Plasma

The Ideal Gas model is used to describe the fireball plasma within V at all

temperatures. The Ideal Gas equation of state

)t(V

)t(kT)t(N)t(P ==== (2.3)

will be used to find the pressure P appearing in the mechanical work expression for

energy loss. N(t) is the number of idealized particles including ions and free

electrons in V; it is not in general a constant but depends on the temperature and past

history of the fireball. The mean kinetic energy per particle per degree of freedom is

kT/2, where k is Boltzmann’s constant. The particles are regarded as non-interacting

and governed by Maxwellian statistics.

The Ideal Gas model adequately represents cold air in thermodynamic calculations;

molecular interactions would be needed mainly to describe hydrodynamic behavior related

to the viscosity. However, the high temperature early fireball is a multi-component

plasma containing free electrons, wholly and partially dissociated ions, and excited

atomic fragments. Radioactive nuclei that are present need no special recognition

below.

The Ideal Gas can also be treated classically - non-relativistic and non-quantum-

mechanical - according to following brief discussions [12,13,14,15].

Classical rather that relativistic dynamics apply to the particles, except for a

negligible fraction of the population in the high velocity Maxwellian tail of the

distribution. The mean thermal velocity for an ideal electron gas is given by:

213

/

ethermal

m

kTv

====

Even if the initial weapon debris region reaches temperatures exceeding 107 oK, the

relativistic effects are small, i.e.,

005

2

.c

v thermal ≈≈≈≈

The electron gas is non-degenerate as well, meaning that Maxwellian rather than Fermi-

Dirac statistics may be applied. The Fermi Energy defined by

[[[[ ]]]] 32

22

32

m

E ee

fermi >>>>ρρρρ<<<<ππππ≡≡≡≡ℏ

where 3

e cmelectrons/ x. 201093≈≈≈≈>>>>ρρρρ<<<<

corresponds to complete electron stripping in air at sea level density. The resulting

degeneracy temperature is

.Kk

ET ofermi

fermi 2200≈≈≈≈≡≡≡≡

confirming the applicability of Maxwellian statistics early in the fireball history.

At later times, the electron density will have relaxed, reducing the degeneracy

temperature Tfermi even further. In order to justify neglecting inter-electron

repulsion, the mean kinetic energy must dominate the mean coulomb repulsion energy,

viz.:


>>>><<<<>>>>>>>>

sr

ekT

2

2

3

where <rs> is the following measure of the average inter-electron separation:

31

4

3

>>>>ρρρρ<<<<ππππ≈≈≈≈>>>><<<<

e

s r

The preceding inequality is satisfied for temperatures above about 13,000o Kelvin when

air is at the sea-level density <ρρρρe> as above. As the altitude increases the density

<ρρρρe> falls off exponentially and the inequality is satisfied for lower temperatures. As the fireball temperatures drops, the electron-ion density falls as well due to

recombination.

The kinetic energy contribution Uk to the internal energy is approximated by:

>>>><<<<>>>>>>>>====

sk

r

eNkTU

2

2

3 (2.4)

which assumes 3 translational degrees of freedom per particle and neglects rotational

or vibrational states.

The expected number of particles N is

] )t( )t( [ )t(N >>>>ββββ<<<<++++>>>>γγγγ<<<<αααα≡≡≡≡ (2.5)

For the above,

ρρρρmol is the underlying ambient density of diatomic (cold) air molecules.

αααα = 2 ρ ρ ρ ρmolV is the number of atomic cores inside V and is taken to be a constant.

For sea level air, ρρρρmol ~ 2.7x1019 cm-3. The variation of density ρρρρmol with elevated

altitude is represented by a simple exponential decay factor incorporating a scale

height in the range of seven kilometers.

<γγγγ> is the average number of dissociated molecules or ions for each atomic core

present. If the gas is cold and diatomic <γγγγ> = 1/2; if the molecules are fully

dissociated, <γγγγ> = 1.

<ββββ> is the average number of free electrons per atomic core. The cores are called 'quasi-ions' in Section (3) of this paper which develops the electron density model.

The total particle density ρρρρtot = 2ρ2ρ2ρ2ρmol[ <γγγγ> + <ββββ> ].

Equilibrium expressions for <γγγγ> and <ββββ> are instantaneous quantities that depend on the

temperature: they may underestimate the gas specific heat. <γγγγ> and <ββββ> are assumed to be constant to facilitate analytic solutions below. For the early blast history,

fireball gases are represented as fully dissociated and ionized, i.e.:

<γγγγ> = 1 ; < ββββ> = 7.2

The potential energy Up due to dissociation is discussed in detail prior to equations

(2.18) and (2.19). We define it by:

>>>>εεεε<<<<αααα>≡>≡>≡>≡<<<< )t()t(U Ip (2.6)

in which <εεεεIt)> is the average ionization/dissociation energy per 'quasi-ion'(accounting for the mixture of gases).


2.4. Constitutive Equations for the Radiation

The early fireball contains a gas of photons that is assumed to be Black Body radiation

in equilibrium with matter. Urad represents the radiation gas component of the internal

energy.

The Planck formula for the spectral electromagnetic energy density [6] in vacuum is:

1

1

22

3

0

−−−−

ππππ

ωωωω====ΩΩΩΩωωωω

ωωωω

kTe

c)T,,(e

ℏℏ (2.7)

Here dV d d )T,,(e ΩΩΩΩωωωωΩΩΩΩωωωω0 is the number of ergs associated with photons in volume dV, with

angular frequencies in the range [ωωωω, ω ω ω ω +dωωωω] propagating into a cone of solid angle dΩΩΩΩ. The 'fireball' is assumed to be isotropic with constant temperature over volume V. Both

photon polarizations are included. The internal energy density urad is obtained by

integrating:

∫∫∫∫ ∫∫∫∫∞∞∞∞ ππππ

ΩΩΩΩωωωωΩΩΩΩωωωω====0

4

0)T,,(e d d u 0rad (2.8a)

which can be readily integrated by noting that the solid angle integration is just

4ππππ and the frequency integral is:

15

1

4

0

3 ππππ====

−−−−

∫∫∫∫∞∞∞∞

ωωωω

e

dxx

kT

ℏ

Multiplying urad by the volume yields the Black Body internal energy term:

4

4 T)c/(V VuU radrad σσσσ======== (2.8b)

The Stefan-Boltzmann constant σσσσ is

units) (cgs x.c

k 5

23

42

10675

60

−−−−====ππππ

≡≡≡≡σσσσℏ

Radiation escaping from the volume V is the dominant loss mechanism shortly after the

burst. The differential radiation flux d d S ΩΩΩΩωωωω0 is the power per unit surface area

emitted into solid angle dΩΩΩΩ with frequencies in the range [ωωωω, ωωωω+dωωωω].

)cos()T,,(ce)T,,(S θθθθΩΩΩΩωωωω====ΩΩΩΩωωωω 00

The power emitted per unit area is the integral of the above over all frequencies and

over a hemisphere into which it can be emitted (rather than reabsorbed), i.e.:

4

00

2

0

2

0T)T,,(e c dd)cos()sin(d)T(J

/rad σσσσ====ΩΩΩΩωωωωωωωωθθθθθθθθθθθθφφφφ==== ∫∫∫∫∫∫∫∫ ∫∫∫∫

∞∞∞∞ππππ ππππ

The angular integrations yielded simply 2π and ½ respectively. The total radiative

power is just:

4TA A)T(J dt

dUrad

rad

σσσσ========

(2.9)

In the above, A is the surface area of the spherical fireball volume V.


The fireball surface is assumed to be transparent to photons of all wavelengths; that

is, energy crossing toward the outside leaves V with no inbound flow. We are assuming

in effect that V is surrounded by a perfect radiation absorber maintained at zero

temperature. The radiation cannot interact with the surface in this picture and

consequently exerts no pressure. As a result, the time scale for fireball cooling will

be artificially compressed, compared with treatments that handle radiation trapping and

diffusion self-consistently. The radiative growth stage for the fireball must also be

externally imposed in order to entrain growing masses of air.

If instead we were to treat the surface enveloping the Black Body Radiation as a

perfect reflector of outbound radiation of all wavelengths, all radiant energy loss

would be prohibited, contradicting the known radiative growth process [1,2,3].

Radiation pressure would be given by the usual formula

∫∫∫∫ ∫∫∫∫∞∞∞∞

ππππ

ΩΩΩΩωωωωΩΩΩΩωωωω====0

0

43

2)T,,(e ddPrad

and would contribute strongly to the mechanical work, instead of being negligible.

The following formula may be used at low frequencies to estimate the spectral power

density per unit area received at a distance R from an extended Black body source of

radius r:

kTR

r1 )T,(

2

2

P

λλλλ====λλλλ

It is derived by expanding the Planck formula in the frequency range kT<<<<<<<<ωωωωℏ

appropriate to radio and radar communications bands, after equating the total power

exiting a sphere of radius r to the power intercepted by a larger sphere with radius R.

The wavelength ωωωωππππ≡≡≡≡λλλλ /c2 .

2.5. Plasma Frequency Effects on Fireball Emissions

The preceding treatment of the radiation does not consider the plasma character of the

fireball, which may strongly modify the amount of radio and radar frequency emission to

be expected shortly after the blast. This section suggests that this omission has only

a small impact on the fireball evolution, but may strongly modify the early-time

emissions from the fireball at radio frequencies.

The standard treatment of an electrically neutral free electron gas using classical

electromagnetic theory invokes the index of refraction, given by the following

expression [9,16,17]:

2

2

2

1

ωωωω

ωωωω−−−−====

pn

in which

e

ep

m

e >>>>ρρρρ<<<<ππππ====ωωωω

22 4

is the Plasma frequency. When ω ω ω ω is below the plasma frequency ωωωωp the index of

refraction related to vphase = c/n becomes complex. Electromagnetic waves are strongly

attenuated as they enter or propagate within conducting media and there is strong

reflection at the boundary.

This suggests a ‘plasma quieting’ effect: Urad may be depleted in photons with

frequencies less than ωωωωp and fireball emission coming from (dU/dt)rad may be


substantially cutoff below the plasma frequency. 'Plasma quieting' would turn on

emission of progressively lower frequencies during a blast, as the electron density

decreases and ωωωωp falls corresponding. At that time, the fireball's Black Body

temperature (and consequently the emitted spectral intensity) would be small compared

to the initial values. The early plasma frequency can be well above the radio/radar

region; in completely ionized sea level air it is close to 0.7 electron volts (about

150,000 GHz. and in the infra-red.

Plasma quieting should depend strongly on burst altitude, as the decreased air density

competes with longer relaxation times. The time after burstfor noise emission would

depend on the electron recapture rate and the rate at which fireball gases are diluted

by expansion.

If the suggested quieting is weak, intense radio noise should be emitted corresponding

to all fireball temperatures from the initial 107 K. through ~ 10,000 K., which is the

temperature at which rough calculations suggest plasma quieting would become

ineffective. If the quieting is strong, the peak noise received by a device pointed

toward the explosion would be reduced by a factor somewhat smaller than the 103 (30 db)

implied by the temperature ratio, since growth of the emitting fireball surface offsets

the intensity decrease attributable to declining temperature. Plasma quieting would

provide a brief time estimated to be on the order of several hundred milli-seconds

during which communications facilities might be available.

Fireball evolution might be made to incorporate ‘quieting’ by using the following

prescription in our expressions [9]:

)T,,(Sn)T,,(S ΩΩΩΩωωωω⇒⇒⇒⇒ΩΩΩΩωωωω 02

0

The approximate effect would be to replace Equations (2.8) and (2.9) with the following

expressions, that can not be readily integrated:

∫∫∫∫ ∫∫∫∫∞∞∞∞

ωωωω

ππππΩΩΩΩωωωωΩΩΩΩωωωω====

p

)T,,(e d d u 0rad4

0

∫∫∫∫∞∞∞∞

ωωωωΩΩΩΩωωωωωωωωππππ====

p

)T,,(e d c Adt

dU0

rad

A small radiation pressure term of the form

∫∫∫∫ ∫∫∫∫ωωωω

ππππ

ΩΩΩΩωωωωΩΩΩΩωωωω====p )T,,(e ddPrad

00

43

2

should also be added to the mechanical work due to reflection of the radiation below

the plasma frequency.

The fireball evolution itself is not sensitive to our choice of Equations (2.8) and

(2.9) rather than those above. Until the gas cools so that its peak Black Body

frequency of 2.822kT is on the order of pωωωωℏ , most of the intensity in the integrals

controlling the energy balance comes from the x-ray or visible range well above the

plasma frequency.

2.6. Solution for the Fireball Temperature Dependence

First we will derive a simplified differential equation for the temperature T(r) as a

function of the radius of a spherical fireball whose volume is V. This equation is

then solved in three regimes, corresponding to successive phases of fireball

development, with some discussion of the emerging picture.


2.6.1 Equation for the Temperature Dependence

The internal energy U is given by:

4

2

3T

c4V )T( VkT U Itot

σσσσ++++>>>>εεεε<<<<αααα++++ρρρρ==== (2.1b)

The above makes use of Equations (2.1a), (2.4), (2.5), (2.6), and (2.8b).

To get an analytic solution for the temperature, assume as remarked earlier that <γγγγ>,

<ββββ>, and <εεεεI> are constants. The energy U becomes a function of T and r.

The left side of Equation (2.2) is obtained by formal differentiation:

r

U

dr

dT

T

Uv

dt

dr

r

U

dt

dT

T

U

dt

dU

s

∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂====

(2.12a)

The fireball expansion speed vs = dr/dt pertains to the shock front created by the

expanding gases. The fireball is assigned a radius that expands according to:

r(t) = ro + vs t

The shock velocity vs is on the order of the speed of sound once the radiation diffusion

period in fireball evolution has ended (see phase III below). Early fireball expansion

(phases I and II) occurs via radiation diffusion, decoupling the expansion velocity

from a mechanical shock front.

The partial derivatives of the internal energy, term by term, are:

30 VT

c16 kV

2

3

T

Utot

σσσσ++++ρρρρ≈≈≈≈

∂∂∂∂

∂∂∂∂

42Trc

16 r

U σσσσππππ≈≈≈≈

∂∂∂∂

∂∂∂∂

The ionization energy derivatives are assumed to be zero. The volume V = 4ππππr2/3. V0 is the (constant) initial fireball radius, neglecting changes to the specific heat due to

expansion. In the above, the total number of particles in the plasma (see Equation

2.5) N = ρρρρtotV0.

The following coupling constants are defined to simplify the equations. C measures

kinetic effects while B marks terms involving Black Body radiation.

1 10 x 3.75 v

c

B

kVNkC

5

s

3

16

0tot

>>>>>>>>≈≈≈≈≡≡≡≡ηηηη

πσπσπσπσ≡≡≡≡

ρρρρ≈≈≈≈≡≡≡≡

8

3

2

3

2

3

(2.11)

In the ratio c/vs, c is the speed of light (3 x 108 m/sec) while the shock speed vs is

300 m/s. So the ratio is enormous. The plasma is non-relativistic even at temperatures

on the order of 107 K.,

Equation (2.12) becomes


++++++++==== 4233 Tr

c

B3

dr

dTTr

c

B4

dr

dTCv

dt

dUs (2.12b)

The loss terms contributing to the right hand side of Equation (2.2) are:

r

TCv - Tr 4-

dt

dUs

2

losses

24σσσσππππ====

(2.10)

After some rearrangement of terms, the following first order non-linear differential

equation with non-constant coefficients is found:

33

33

2

Trc

B4 C

Trv

B

4

3 C

r

T

dr

dT s

++++

++++−−−−

==== (2.13)

The terms in the numerator measure energy loss via mechanical work as the fireball

expands and fireball radiation at all wavelengths. In the denominator the right hand

term governs the rate of decrease in Urad; the left term governs the decrease in Uk.

The relaxation of Up = UI is unrepresented, as we have chosen UI to be constant.

Equation (2.13) can be solved analytically for the dominant dependence in three

distinct time/temperature regimes (temperature is monotonically decreasing). Note in

Equation (2.13) that the second term in the numerator is always larger than the second

term in the denominator by a factor on the order of c/vs, hence the inequality:

3Tr

c

B4 Tr

v

B

4

3 333

s

>>>>>>>>

The three solution regimes corresponding to successive phases of fireball evolution are

defined by:

PHASE I. C Trc

B4 33 ≥≥≥≥ (2.14a)

PHASE II. 3Tr

c

B4 C C Tr

v

B

4

3 3

and33

s

≥≥≥≥≥≥≥≥ (2.14b)

PHASE III. Trv

B

4

3C 33

s

≥≥≥≥ (2.14c)

2.6.2 Solution for phase I temperature dependence

Here the radiation terms proportional to B dominate both the numerator and denominator

in Equation (2.13). Energy is lost principally through Black Body radiation at the

expense of Urad within V. This is the earliest fireball development phase lasting as

long as Equation (2.14a) can be satisfied, which is certainly the case in high yield

(say ~ 1 Megaton) blasts within the weapon vapor region. The duration of phase I is

brief, lasting on the order of 10-8 seconds which is comparable to that for breaking-up

the weapon casing.

We simplify Equation (2.13) by retaining only the dominant terms; i.e., those

proportional to B:

r

T

dr

dTηηηη−−−−≈≈≈≈

This can be immediately integrated, yielding a power law solution:


ηηηη

====

r

rTT 0

0 (2.15a)

Here, r0 and T0 = T(r0)are the initial conditions of the blast applied to the volume V.

The solution is valid as an approximation until the following condition

3

1

14Br

cC T 3 ≈≈≈≈ (2.16)

is satisfied, with T1 and r1 marking the end of this (phase I) period. The exponent is

enormous for most plausible choices of vs ~ 300 m/s << c, and quickly damps the

fireball temperature.

An approximation to Equation (2.15a) provides some insight into early temperature

behavior. First write the right hand side as T0expηηηηlog(r0/r, note that r = r0+vst and that log(r0/r) = - log(1+vst/r0), and use the expansion:

1 x for x ... x

x)x( Ln <<<<<<<<≈≈≈≈++++−−−−====++++2

1

To lowest order, the result is:

r

ct expT T

0

08

3−−−−==== (2.15b)

The temperature drops to e-1 of its initial value in approximately the time it takes

light to cross the fireball volume.

2.6.3 Fireball temperature and radius as phase I ends

We can estimate T1 and the expanded radius r1 of the weapons volume by first invoking

Equation (2.15a) evaluated at T1, and raising both sides to the power 1/η:η:η:η:

1 T

T

r

r≈≈≈≈

====

ηηηη

1

0

1

1

0

Inasmuch as η η η η is enormous, the right hand side above is unity, and

01 rr ≈≈≈≈ (2.19)

Mechanical expansion of the original fireball during the brief duration of phase I is

negligible. However, the fireball grows by radiative coupling during this period and an

estimate (described below) of r1 based on that process is more meaningful.

Equation (16) yields a temperature estimate by direct evaluation, i.e.,

21

3

4Br

cC T ≈≈≈≈1

The ratio of coupling constants is gotten from Equations (2.11) and (2.5) plus the

discussion that follows it:

3

03

4

r k

C

B

molρρρρλλλλ

σσσσ==== (2.18a)

The parameter λλλλ measures dissociation. Full ionization is assumed, corresponding to:


8.2 ≅≅≅≅>>>>ββββ<<<<++++>>>>γγγγ<<<<≡≡≡≡λλλλ

The estimated temperature T1 assuming complete ionization during phase I is:

level) (sea6

/

mol

/

K. 10 x 1.45 6

k cT ≈≈≈≈ρρρρ

σσσσ

λλλλ≅≅≅≅ 31

31

11

3 (2.18b)

The sea level density ρρρρmol = 2.7x1019 molecules/cm3; it declines exponentially with

altitude causing T1 to fall accordingly with altitude. This prescription for

estimating the final phase I temperature is independent of initial blast parameters

such as the yield and initial volume and temperature. It depends only on the air

density as a function of the burst height and the constitutive assumptions made about

the matter inside the fireball volume V. The rate of ionization energy collapse is

omitted but the error due to this is probably small during the phase 1 early history of

blasts when the photon gas contains most of the blast energy.

By the time phase I ends, Stefan’s Law emission has drawn energy primarily from the

initial photon gas and distributed it over a volume with radius r1 via radiative

transport and absorption by the plasma. The earliest meaningful temperature is the

end-phase temperature T1, which is rapidly reached over both the expanding original

weapon region (r1’ = r0 + vst1) and also over a significantly enlarged air volume V1

(radius r1) that is formed by the direct deposition of energy from the original

radiation gas. Additional matter is ionized and entrained into the growing fireball by

the massive outpouring of X-ray and higher energy radiation from the original weapon

region.

The properties of radiation and of ionized air at temperature T1 determine the

numerical value for the internal energy density u(T1) related to Equation (2.1b).

This allows us to assign a final phase I fireball volume and radius by assuming that

the blast yield assumes a uniform energy density over the fireball volume by the end of

phase I, viz:

)T(u

YV

1

1 ≡≡≡≡ (2.17)

Y is the weapon yield in ergs, u(T1) is the energy density related to Equation 2.1b),

and V1 is the fireball volume. The exploding nuclear device is modeled as a sphere of

hot air still containing the entire yield.

The radiation component of the internal energy escapes the fireball as Black Body

Radiation, but if the ambient air density is high only a small fraction of the high

temperature Black Body Radiation gets very far. The fireball mass and volume grow

radiatively as escaping X-rays and Gamma rays are absorbed by, ionize, and heat

surrounding air which is entrained into the fireball.

Mechanical expansion of the fireball during phase I is negligible for any reasonable

mechanical velocity; fireball growth occurs by capture of the outbound radiation.

Radiative transfer is so rapid that the end of phase I is the earliest time at which

traditional thermodynamic parameters for the system become meaningful. We initialized

our machine computations at that time.

2.6.4 Solution for phase II temperature dependence

Next we simplify and solve Equation (2.13) for phase II, where condition (2.14b)

applies. Phase II is defined by the intervals T1 > T > T2 and r1 < r < r2. As

indicated in Equation (2.14b) the end of phase II corresponds to:

32

s32

Br

Cv

3

4 T ==== (2.20)


Black Body radiation continues to be the dominant energy loss mechanism rather than

mechanical work, hence the term in the numerator proportional to C is neglected. The

power is drawn principally from the plasma rather than from the photon gas. Hence the

internal energy terms representing the ions and electrons is retained and the photon

gas term in the denominator (proportional to B) is neglected. With only the dominant

terms retained, Equation (2.13) becomes:

sv

Tr

C

B

dr

dT 42

2

3−−−−≈≈≈≈

This simplified equation is separable and can be easily integrated from T1 to T and r1

to r, yielding the solution:

T

T

r

r

−−−−ηηηη++++

====

113

1

3

313

(2.21a)

The terms proportional to ηηηη dominate the denominator, in which we neglect 1 << η. η. η. η. The temperature dependence for phase II is given by:

31

3

1

331

1 1

/

r

r/ TT

−−−−

−−−−

−−−−ηηηη≈≈≈≈ (2.21b)

2.6.5 Fireball temperature and radius as phase II ends

Equations (2.21b) is evaluated for the end of phase II and solved simultaneously with

equation (2.16) for r2 – the radius of the shock front for fireball expansion:

1

312 r r /

2 ≈≈≈≈ (2.22a)

The final phase II fireball temperature is estimated by using Equations (2.20) and

(2.16), yielding:

T T /1

312

−−−−ηηηη≈≈≈≈ (2.22b)

A numerical estimate for T2 uses the numerical values of T1 and ηηηη (Equations 2.11 and 2.18b)finding that the sea level fireball is still significantly hotter than the Sun:

K. 10 x 2.00 T level) (sea 4≈≈≈≈2

During this second phase of fireball development the temperature falls to about 1% of

its value through Black Body radiation, while the initial volume V1 merely doubles

through mechanical expansion.

But radiation from the fireball is still absorbed before it has gone very far (at low

altitudes). The fireball continues to cool and grow radiatively during at least the

early portion of phase II by entraining more cold air until the peak Black Body

frequency falls below the transparency threshold for cold air. Once this radiative

growth limit is reached during phase II, the fireball simply radiates energy and

expands at velocity vs until Equation (2.20) is satisfied.


2.6.6 Solution for phase III temperature dependence

Finally, we solve an approximate version of Equation (2.13) for phase III, where

condition (2.14c) is applicable.

Radiation effects become small in both the internal energy and the loss mechanisms.

Mechanical work done by expansion is the important energy loss. Fireball growth through

radiation diffusion has basically ended, and the (low altitude)fireball has cooled

enough during phase II so that further radiative losses are neglected.

The dominant terms in the numerator and denominator of Equation (2.13) are now those

proportional to C, and so Equation 2.13 reduces to:

r

T

dr

dT2−−−−≈≈≈≈

The above is simply the equation for quasi-static expansion of an Ideal Gas doing work

on its surrounding spherical volume. Its integral from T2 to T is

2

22

====

r

rTT (2.23)

This inverse square law solution is valid for T2 > T and r > r2.

During the final stage of development the ionized fireball expands at essentially

constant velocity vs. Mechanical work is done against the surrounding air due to

pressure exerted by the heated plasma within the fireball; this reduces the internal

energy, cooling the fireball. Late in the development, the internal pressure and

density drop below the ambient values. The transfer of energy into the shock front

ceases, the fireball ‘lifts-off’ due to buoyancy begins. This model ends its coverage

with the onset of these effects.


3. Model for temperature dependence of the electron density and ionization energy in Air

This section produces approximations to the electron density and ionization energy

density of air inside the fireball of a nuclear explosion. A model for these

quantities in an arbitrary gas is first developed, and then applied using the

composition of air.

Two limiting case models applicable to signal propagation are examined: the local

thermal equilibrium case (zero relaxation time) and the fully ionized gas case

(infinite relaxation time). The equilibrium view is most applicable to low altitude

bursts for which the free electron lifetime is comparatively short. At high

temperature, stimulated emission will further hasten free electron recapture. The fully

(or partially) ionized model is useful [1] at high altitudes where the electron mean

free time can be as long as several minutes.

The nuclear fireball is a non-equilibrium system. Its ionization should properly be

approached using a set of coupled Boltzmann equations, which would include radiative

and non-radiative excitation, decay, and collision contributions for each species of

particle [3,9,10,18,19].

The simplest non-equilibrium approximation is to include only the lowest order

departure of the electron distribution function from the equilibrium solution. The

simple rate equations describing steady state electron densities derived in the

ionospheric effects literature are a special case of this approach, and they are

applicable principally to weakly perturbed systems in which the depletion of the

neutral atom reservoir may be neglected [20,21].

3.1. Problem Formulation

The Canonical Ensemble [6,7,8,21,22] was applied to a small electrically neutral volume

of air at a well-defined constant temperature. Each air parcel was assumed to be in

local thermodynamic equilibrium. The temperatures characterizing the energy level

populations, black-body radiation, and translational kinetic energy are all assumed to

be the same.

The volume of gas is defined to contain N indistinguishable quantum mechanical

particles which will be called 'quasi-ions'. Each quasi-ion in the neutral condition

contains M electrons, and possesses M + 1 states characterized by the quantum numbers:

m = 0, 1, 2, ... M.

The m'th state corresponds to m-fold ionization of the atom, with m = 0 corresponding

to a neutral. Excited neutral states and states involving partial ionization together

with excited bound electrons will occur in real systems, but are ignored in this

treatment.

In the occupation number representation [23,24,25], one state of a many-particle system

is characterized by a vector

| n0, n1,...nm.....nM >

in which nm represents the population of state m. The quasi-ions are assumed to be

non-interacting: tacitly equivalent to assuming that the many-body Hamiltonian for the

system can be well-approximated by a sum of single quasi-ion Hamiltonians. The wave

functions for the system will then be approximated by products of single quasi-ion wave

functions and the N-particle energy states for the system can be approximated by simple

sums of the single quasi-ion energies, viz:


m

M

mmm n n E εεεε==== ∑∑∑∑

==== 0

(3.1)

In the above, εεεεm is the single particle energy eigenvalue for an m-fold ionized quasi-ion. The notation nm stands for one set of M numbers defining an occupation state.

In principle, ranges of final state kinetic energies for the ionic fragments and

liberated electrons should be included in the summation. However, we consider only

ionization events at their thresholds in order to simplify the treatment, and identify:

mm ∆∆∆∆++++εεεε====εεεε 0 (3.2)

∆∆∆∆m is the ionization threshold energy - a constant. For numerical calculations it

is set equal to the sum of the 1st through mth ionization potentials. It is obvious

that ∆∆∆∆0 = 0.

The partition function Z allows one to calculate the thermodynamic functions and

statistical weights. It is defined in the Canonical Ensemble by the following

expression:

∑∑∑∑ ττττ≡≡≡≡s all

/-EseZ

In the above, kT≡≡≡≡ττττ , where k is Boltzmann's constant. Each state 's' of the system

corresponds to a particular assignment of the N particles to individual states; i.e.,

's' corresponds to a vector | m1, m2, ......., mN > in which mi is the state of particle

i. There are M**N such vectors to be summed over. Es is the total system energy for state 's', with many system states potentially yielding the same energy.

In order to convert to occupation number language we introduce the symbol g(N:nm) to represent the energy degeneracy of a particular choice of the set nm of occupation

numbers. The explicit form for it is:

(((( ))))!n!.....n!n

! Nn :Ng

m0m

1

≡≡≡≡ (3.3)

In effect g(N:nm) is the number of ways one can assign N particles to M states so

that the mth state contains nm occupants. It is also the energy degeneracy – the number of system states 's' corresponding to a particular energy eigenvalue Enm.

The general expression for the partition function in the occupation representation

becomes:

]/[-E exp ) n :g(N Zn sets all

nm

m

mττττ≡≡≡≡ ∑∑∑∑ (3.4)

Each term above includes contributions from all states 's' corresponding to a

particular energy and hence to a particular set nm of state occupations.

Each term in the partition function is the probability of finding the system in a

particular state nm is:

]/[-E exp ) n :g(N )n :N(W nmmm

ττττ==== (3.5)

A quantity is thermally averaged by calculating it’s expected value; that is, by

evaluating it for the choice nm, weighting it with the occurrence probability

W(N:nm), and then summing over all sets nm. In particular, the expected value for the fraction fm of m-fold ionized 'quasi-ions' is given by:


)n :N(WnN

N

n )(f m

nm

mm

m

∑∑∑∑====>>>><<<<

====ττττ1

(3.6)

The average number β(τ)β(τ)β(τ)β(τ) of free-electrons per 'quasi-ion' at equilibrium is the sum of the fractions fm with each weighted by the number of free electrons it yields, viz:

∑∑∑∑====

ττττ≡≡≡≡ττττββββM

0mm )(f m )( (3.7)

The quantity β(τ)β(τ)β(τ)β(τ) is the basis for charge density calculations, and is a function of temperature alone in this equilibrium treatment.

3.2. The Partition Function:

The Partition Function can be easily evaluated. First, we make substitutions in

Equation (3.4) using the rules below:

∑∑∑∑====

∆∆∆∆++++εεεε====M

0mmm0n n N E

m (3.8a)

1≡≡≡≡ΧΧΧΧ≡≡≡≡ΧΧΧΧ ττττ∆∆∆∆−−−−0

/m e m (3.8b)

after which the expression for Z is:

M

m

nM

nn

n allm0 ...... ) n :g(N ]/[-N exp Z ΧΧΧΧΧΧΧΧΧΧΧΧττττεεεε==== ∑∑∑∑ 10

10

The sum in the expression above is precisely the multinomial expansion [26]

(((( )))) M

m

nM

nn

n all M10

N M x....xx

!n!...n!n

N!x...xx 10

1010 ∑∑∑∑====++++++++++++

Hence the Partition Function is simply:

(((( )))) N M0 ... ]/[-N exp Z ΧΧΧΧ++++++++ΧΧΧΧ++++ΧΧΧΧττττεεεε==== 10

which is a product of single particle partition functions. This might have been

expected as a consequence of the non-interacting 'quasi-ion' assumption made for the

system Hamiltonian.

We define the single particle partition function Z0 by:

]/exp[ ]/[- exp ZM

mm0 ττττ∆∆∆∆−−−−++++ττττεεεε==== ∑∑∑∑

====1

0 1 (3.9)

This is itself the product of a ground state factor and a factor involving only the

'quasi-ion' excitations. The Partition Function is now simply given by:

N0Z Z ==== (3.10)

3.3. The Average Electron Density

With the Partition function known it is straightforward to calculate fm, the expected

fraction of m-fold ionized 'quasi-ions' defined in Equation (3.6):


(((( ))))∑∑∑∑ ∏∏∏∏====

ττττ∆∆∆∆ττττεεεε

====ττττn

M

0m'mmmmN

0m

m

]/'exp[-n' n )n :N(gZ N

]/[-N exp)(f

0

The above used Equations (3.5) and (3.8a). After making substitution (3.8b) and

canceling some common factors:

(((( ))))M

m

nM

nn

nN

M

mmm ......

...

n )n :N(g

N)(f ΧΧΧΧΧΧΧΧΧΧΧΧ

ΧΧΧΧ++++++++ΧΧΧΧ++++ΧΧΧΧ====ττττ ∑∑∑∑ 10

10

10

1

The ground state energy εεεε0000 has dropped out. The numerator above is a form of

multinomial expansion. Using the multinomial theorem above it is straightforward to

show that:

M

m

nM

nnm

nm

NMM

NM

mm

... n )n :N(g

)...( N)...(dx

d

ΧΧΧΧΧΧΧΧΧΧΧΧ====

ΧΧΧΧ++++++++ΧΧΧΧ++++ΧΧΧΧΧΧΧΧ====ΧΧΧΧ++++++++ΧΧΧΧ++++ΧΧΧΧΧΧΧΧ

∑∑∑∑

−−−−

10

10

11010

After some cancellation:

∑∑∑∑====

ττττ∆∆∆∆−−−−++++

ττττ∆∆∆∆====

ΧΧΧΧ++++++++ΧΧΧΧ++++ΧΧΧΧ

ΧΧΧΧ====ττττ

M

jj

m

M

mm

]/exp[

]/exp[-

... )(f

1

10

1

(3.13)

The form defined for ΧΧΧΧm was restored and we used ∆∆∆∆0 = 0 in the lower line above.

By inspection, it is clear that the following sum rule holds:

1

0

)(fM

jm ====ττττ∑∑∑∑

====

The fm are thus normalized in the manner befitting probabilities. A pair of states m

and m' are populated in the ratio:

]/)(exp[ f

f'mm

'm

m ττττ∆∆∆∆−−−−∆∆∆∆====

In the low temperature limit ττττ -> 0, the ionization disappears as it must and all the population is in the neutral ground state:

100

========ττττ ≠≠≠≠→→→→ττττ

0 0m all form f )(f Lim

In the high temperature limit ∞∞∞∞→→→→ττττ , all of the exponential factors approach unity

leaving the states uniformly populated as a result:

m all form M1

1 )(f Lim

++++====ττττ

∞∞∞∞→→→→ττττ

This regime maximizes the entropy. It places an upper limit on the number of free

electrons per quasi-ion which can be liberated in an equilibrium system:


2 / M )( ====∞∞∞∞====ττττββββ

wherein we used Equation (3.7) and the series result:

2

1)M(M M . . . . . .

++++====++++++++++++++++ 321

In our model, the most complete ionization via thermal means implies that half of the

available electrons are liberated. The initial energy release in a nuclear detonation

is sometimes assumed to completely ionize all matter in the weapons region at very

early times after detonation.

A non-thermal distribution that describes complete ionization can be expressed using

the formalism established above by taking:

M M,j0 for f ,f jM ====ββββ<<<<≤≤≤≤======== 01 (3.14)

Single-fold ionization of one electron per atomic core would be represented by setting

f1 = 1, with all others zero.

3.4. The Ionization Energy

The preceding work can likewise be used to evaluate the expectation value of the energy

bound up in ionization. Using the partition function and the prescription described

above, first find the expectation value of nm

E :

∑∑∑∑====>>>><<<<n

nmI

m

mE )n :N(W U

Next, insert the explicit form for nm

E using Equation (3.8a), after dropping the

arbitrary ground state energy εεεε0000 = 0. Finally, use Equation (3.6) after exchanging the

order of summation. The average ionization energy per quasi-ion is :

∑∑∑∑====

∆∆∆∆ττττ====>>>><<<<

====>>>>εεεε<<<<M

1jjj

II )(f

N

U (3.15)

This result can also be obtained by inspection, using the interpretation of fm as a

probability.

3.5. The Electron Density and Ionization Energy in Air

For the diatomic molecules comprising air the quasi-ion number density is 2ρρρρmol, where

ρρρρmol is the molecular density for the cold gas at altitude h. At sea level:

319

10720 cm/moleculesmol x. )h( ≈≈≈≈====ρρρρ

We correct for the N2/O2 mix of gases (80%/20%) by using partial densities for each and

assigning M = 7 for Nitrogen, M = 8 for Oxygen. The equilibrium electron density as a

function of temperature and ambient density may be evaluated using:

]0.2 0.8 [ (h) 2 )(22

ONmole ββββ++++ββββρρρρ====>>>>ττττρρρρ<<<< (3.16)

Table (3.1) shows the values used for approximating the ionization levels [27]. We

assigned values to the quasi-ion energies

∑∑∑∑====

≡≡≡≡∆∆∆∆m

1jjm V


where Vj is the j'th experimentally determined atomic ionization potential. For the

non-thermal distribution defined in Equation (3.14), corresponding to complete

stripping of the electrons, Equation (3.16) is replaced by:

molxxmole 14.4 ] 0.2 70.8 [ 2 )( ρρρρ====++++ρρρρ====>>>>∞∞∞∞====ττττρρρρ<<<< 8 (3.17)

At sea level, the maximum electron density is

3.9x10 )( 3

cmelectrons/20

e ≈≈≈≈>>>>∞∞∞∞====ττττρρρρ<<<<

The equilibrium ionization energy within a volume V can likewise be evaluated as a

function of ambient density and temperature:

] )O,( 0.2 )N,(0.8 [ V (h)2 )(U IImolI >>>>ττττεεεε<<<<++++>>>>ττττεεεε<<<<ρρρρ====>>>>ττττ<<<< 22 (3.18)

in which Equation (3.15) must be used to evaluate the average energies for Nitrogen and

Oxygen separately.

_______________________________________________________________________________________

TABLE 3.1: QUASI-ION ENERGIES and IONIZATION POTENTIALS

States for Nitrogen: States for Oxygen:

m ∆∆∆∆m(ev) Vm (ev) ∆ ∆ ∆ ∆m(ev) Vm (ev)

1 14.53 14.53 13.61 13.61

2 44.12 29.59 48.72 35.11

3 91.55 47.43 103.60 54.89

4 169.00 77.45 181.00 77.39

5 266.86 97.86 294.87 113.87

6 818.79 551.93 432.95 138.08

7 1485.62 666.83 1172.07 739.11

8 N/A N/A 2043.19 871.12

Table 3.1: Each 'quasi-ion' energy is the sum of the 1st through m'th ionization

potentials Vj. The numerical values are taken from reference [27].

_______________________________________________________________________________________

The ionization energy for complete electron stripping can also be evaluated by using

Equation (3.14) to estimate the non-thermal distribution:

] )O,( 0.2 )N,(0.8 [ V (h)2 )(U IImolI >>>>ττττεεεε<<<<++++>>>>ττττεεεε<<<<ρρρρ====>>>>∞∞∞∞====ττττ<<<< 22 (3.19)

The maximum attainable ionization energy density At sea level is approximately:

eev/molecul

3

erg/cm

3

ev/cm

3200

1.4x10

8.6x10 V

)(U

11

22I

≈≈≈≈

≈≈≈≈

≈≈≈≈>>>>∞∞∞∞====ττττ<<<<

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