NBS MONOGRAPH 120

Unified Tlieory Calculations

I of Stark Broadened

Hydrogen Lines Including

Lower State Interactions

U.S.

DEPARTMENT

OF

COMMERCE

National

Bureau

of

Standards

UNITED STATES DEPARTMENT OF COMMERCE • Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Dinctor imOHAl Bmm OF STANDMD3

AUG i 6 1371

- . 0

I Unified Theory Calculations of Stark Broadened

Hydrogen Lines Including Lower State Interactions

C. R. Vidal, J. Cooper, and E. W. Smith

Radio Standards Physics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Lj , " National Bureau of Standards Monograph 120

« tr * Nat. Bur. Stand. (U.S.), Monogr. 120, 45 pages (January 1971) CODEN: NBSMA

Issued January 1971

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C 13.44: 120), Price SO cents
Library of Congress Catalog Card Number: 79-608598

II
Table of Contents

Page

1. Introduction 1 2. Basic Relations 1 3. Properties Of The ^ And K Matrix 4

4. The Influence Of Lower State Interactions and The Static Limit 5

5. Comparison With Experiments And Other Theories 7

6. Discussion 11

7. References 14

8. Appendix 14

rrogram 17

III

Unified Theory Calculations of Stark Broadened Hydrogen Lines

Including Lower State Interactions*

C. R. Vidal, J. Cooper,** and E. W. Smith

National Bureau of Standards, Boulder, Colorado 80302

Recently published calculations of hydrogen Stark broadening on the basis of the unified classical path theory have been extended to include lower state interactions in the final line profile. A detailed comparison with experiments in the density range lO'^-lO'^ cm"'' is given.

Key words: Classical path; hydrogen lines; line wings; Stark broadening; unified theory.

1. Introduction

In a recent paper [1],' henceforth referred to as paper I, the unified classical path theory [2] was generalized for the case of upper and lower state interactions and was applied to the Stark

broadening of hydrogen. The thermal average of the time development operator and the final intensity profile of any hydrogen line were derived for the general case including lower state interactions. Numerical calculations for the thermal average including lower state interactions

have been presented. The calculations of the final line profiles, however, have so far been restricted

to the Lyman lines. In this paper we discuss calculations of the intensity profile including lower

state interactions which are more involved because they require the evaluation of tetradic operators

and contain more extensive sumations over vector coupling coefficients. The influence of lower state interactions is demonstrated for the first few lines of the Balmer series and possible simplifications

for the higher series members are pointed out. A detailed comparison with various experiments

covering the electron density range from about lO'^-lO''' cm"^ is given requiring for low electron

densities a convolution of the first series members with the Doppler profile. The unified theory

calculations are also compared with static calculations. The applied computer program is presented

in the appendix.

In this section we summarize briefly the basic relations, which have been derived in paper I and are used for the unified theory calculations presented in this paper. With the static ion field approximation the shape I{(o) of a Stark broadened line is given by

where the normalized distribution function ) is the low frequency component of the fluctuating

electric microfields. In this manner we regard the radiator as being an atom subjected to a static

field and perturbed by the electrons. In the unified classical path theory [2] the ion field de- pendent line shape I{o), Wi) is obtained from

d is the dipole moment. The matrix elements of AcOop specify the distance Aco from a particular Stark

component shifted by the static field ^, and =2^ is essentially the Fourier transform of the thermal

*This research was supported in part by the Advanced Research Projects Agency of the Department of Defense, monitored by Army Research Office— Durham

jr Contract No. DA-31-124-AR0-D-139.

"'Also at the Joint Institute for Laboratory Astrophysics and Department of Physics and Astrophysics. University of Colorado. Boulder. Colo. 80302.

' Figures in brackets refer to the literature references on p. 14.

2. Basic Relations

(1)

(2)

1
average. Within the classical path approximation and the impact approximation =2'(Aa)op) is given by

^{^(x)op)=-i^o)op exp i+it^o)op)^^^Ht)dt^o)op (3)

where the thermal average

^('>(0=Aie| c;x:6?v,r(v,)[^i(R, Xi,v,,0-1]. (4)

He denotes the electron density and W the velocity distribution function. The tetradic time development operator i is defined by the time ordered expression

^i(R, xi,vi,0 = ^exp|-| |Vi(R, xi,vi,t')c?«'j (5)

with the binary interaction

TiiR, xi, vi, t) -=exp {it^olfi}Ti{R, xi, vi, t) exp {-it^olfi}. (6)

f^i is the Coulomb interaction potential between the radiator and a single perturbing electron. The HamUtonian ^o ^o = ^a + eZ^i (7)

consists of the Hamiltonian ^^^a of the unperturbed radiator and the static ion part eZWu (For details see eqs aV.16)-(IV.18),(III.14)-(III.16) and ai.2) of paper I.) In evaluating the preceding equations for well isolated hydrogen lines we have to consider

matrix elements only between states with the same principal quantum number n (no-quenching

approximation) and it then is most convenient to work with parabolic states \nqm} . m is the magnetic

quantum number and the quantum number q is defined to be

q = ni — n2 (8)

with Hi and ^2 being the usual parabolic quantum numbers which have to satisfy the relation n — ni + n2+\m\ + l. (9)

Furthermore, we distinguish quantum numbers which refer to the lower state from the upper state quantum numbers by a prime.

In taking matrix elements of the operators Awop, d and if (Awop) between parabolic states it was shown in paper I that for the general case of upper and lower state interactions we have the

following relations. For A(Oop, which is diagonal in parabolic states, one obtains

(n'q'm'; nqm\ AcOop\n'q' m' ; nqm) — Aoj — ^coiin, q, n' , q')^ (10)

where A<u is the frequency perturbation from the position of the unperturbed line, /8 the normalized

field strength in units of the Holtsmark field strength

and

/4_\2/3

)8 = ^i/g'o;g'o=[Y) ^^^^

/4_\2/3 3 n A(Oi{n,q,n\q')^ i^^j y(ng - )- n|/3. (12)

Acoi is the frequency shift of a particular Stark component characterized by the quantum numbers

n,q, n' and q' due to the Holtsmark field strength ^o- The matrix elements of =2'(Awop) are given by

{n'qlml; nqbiub]^ {tiOiop) I n'q'am'a ; nqania)—— i7r[Acu — Aw, (n, qb,n',q'h )/3]- ^ (n'qbml,; nqbmb\K{qc, q'c) \ n'q'am'a; nqama}i{^io), fi,n,n', qb, qL qc, q'c) (13)
where

{n'qiml\nqbmb\K{qc,ql.)\n'q'am'a;nqama)= i-l)"'^^"'!' (2L + 1)

{n'q'aTn'a \ n'l'ain'n) {n'l(,m^ \ n'ql.m'(. ) {n'q[.mf. \n'll,m'. ) {n'llml \ n'qlml,

)

(nqama \ nlaTUa) (nlaTric I nqciric) {nqciric \ nlbfiic) {nlbmi, \ nqi,mi,)

I'a la L \/ II, /„ L \ / II lb L \/ II lb L

^—ml-mcM'' ^—m'umaM' ^—m^nicM'^ ^— mlmbM

(14)

The unitary transformation {nlm \ nqm) from parabolic to spherical states can be expressed in terms of S^-symbols [3]

(n/m|rao/n) = (-l)i/2(i+m-9-n) V2/+1 I , I (15)

* m — qm + q

'

2

Introducing the normalized time

s = (i}pt (16)

and the normalized frequency

A(x)R=[/:^(o — A.(Oi{n, qb, n' , q'„)(i]j(bp (17)

with (bp = VSTm^e^Jm being the plasma frequency, the Fourier transform of the thermal average

is defined to be

1 f i{A(i)R, ft, n, n\ qb, q'b, qc, q'c) =^lim — exp [ (— e + iAa>/j)s]F(5, n, qc, n\ q[)ds (18)

The thermal average F{s) was evaluated in paper I for the general case of upper and lower state interactions and it was shown that F{s) may be approximated by a function G{s) = 1.kGk(s) whose

Fourier transform can be given analytically such that

i(AcoR)=2^ i(A;, Ato«). (19)

k

For most practical situations it turns out to be sufficient to consider only the first term in the series

i(A: = l, Aa)«)-a,6fe-«i|i//(«(Zi)+//</)(^i) 1-3^ | (20)

where H^^^ and are Hankel functions and

Zi = 6,Aco«, (21)

because the higher order terms usually affect the final Une profile by not more than 2 percent

around Aw — oip. The constants Oi and 61 are given by

ai = - 4 UeD^C^ [B - In (4C2 ) ] (22)

and

6, = ^C[B-hi(4C^)]2,

3

(23)
where D is the Debye length, B a constant of the order of unity (see table II in the appendix of

paper I) and

C = \{nq-n'q')^^. (24)

Finally we give for completeness the matrix elements of d®d

{n'q'aTn'a, n'qlml \ d®d \ nqaTUa; nqbirii,) = (— l)^^^ {n'q'„m'„ \ n' I'^m'^) {nlama \ nqaTUa)

Inlh

^" ^ )P"^)[(2/« + l)(2/;+l)]'/^(«/„|r|n7^)

^m„ — ma/x/ VOO 0/

(-l)'";^; {n'q',ml\n'llml){nkmb\nqmb)( ^* ^

[ (2/^ + 1) (2/^+ 1) | r | n'/^ ) f^^ \mf, — mbH./\yjy)\J/

(25)

where the radial matrix elements are given in eq (63.2) of Bethe and Salpeter [4]. In case of no lower state interaction where the time development operator between lower

states is replaced by a unit operator the matrix elements of =2^(Aa>op) in eq (13) and (14) simplify

significantly and we have

(n'q'm' ; nqbin |o^u(AaJop) \ n'q'm'; nqatn) = — t7r[Aa» — ^(Oi{n, qb, n' , q')(3]'^

^{nqbm\Ku{qc)\nqam)i{^(xi, n, n' , qb, q', qc) (26)

<ic where

{nqbm\Ku{qc)\nqam) = ^ -^^^^^^{nqnm\nlam) {nqbm\nlam){{nlamc\nqcmc)y- (27)

Equation (26) simplifies even further for the Lyman lines with | n'g'm' ) = 1 100) . For this case numerical calculations were given in paper I. '

3. Properties of the^ and K Matrix

In eqs (13) and (26) we have split the =?'-matrix into two parts. One part, the X-matrix, contains

all the vector coupling coefficients or 3y-symbols and is independent of the plasma parameters,

while the other part, the Fourier transform of the thermal average i(Aw), contains all the broadening

parameters. The A^-matrix is completely specified by the upper and lower state principal quantum

numbers n and n' and needs to be calculated only once for every hydrogen line. From eq (14) we realize first of all that the Al-matrix is symmetric to the diagonal

{n'q'bin'b; nqbmb\K{qc q'c) \ n' q'aTn'a; nq„ma) = {n' q'aTn'a\ nqama\K{qc,q'c) \n' qiml; nqbiUb) (28)

while the o^-matrix is not, due to the factor [Aw — A(o,(n, qb, n\ q'b)fi\'^- Next we see from the

3y-symbols that K is diagonal in the quantum number M.

M= m'a — ma = m'b — nib (29)

Hence we may arrange the /C-matrix and also- the =Sf-matrix in such a manner that they are block

diagonal in M, where Af = 0,±l,±2, . . . , n + n' —2. We also notice that matrix elements which

differ only in the sign of M are identical.

(n'q^ — m'b; nqb — nib \ K(qc, q'c) \ n'q'a — m'a; nqa — nia) = { n' q'^ml; nqbtrib \ K{qc, q'c) \ n' q'^m'^; nqaTHa) (30) 4
Consequently, blocks which differ in the sign of M, can also be made identical. This greatly simplifies

the matrix inversion required by eq (2). The problem is further simplified by the fact that also the fl?(8>flf-matrix can be made block diagonal in

ti. = ma — m'„ = mh — ml,, (31)

where fx can only take on the values 0 and ±1. As a resuh we finally have to evaluate only the

blocks with M^O and M= 1 of the K and if'-matrix, respectively, because the block with M= - 1 gives identically the same contribution to the final lifte profile as the block with M=l and all the

other blocks with \M\ > 1 do not contribute due to the d!^d matrix. Further useful symmetry

properties of the A^-matrix are given in the following equations:

{n'q'^ml ; nqbmb \ K{qc, q'c) \ n'qam^; nqatna)

^{n' -ql-ml\ n-qb-mt>\ K{qc, q'c) \ n' - q'„ — m'„; n-qa-jUa) (32) = {n'qlml; nqbtrib \ K{— qc, —q[.) \ n'q'^m'^; nqam,,}

.

The latter relation simphfies our summation over qc and q'c in eq (13) because also for the

Fourier transform /(Aw) we have

i(Aaj, 13, n, n' , qh, q^, qc, qc)^i{^co, (3, n, n' , q^, qi, -qc, -qc) (33)

In calculating /(w, ^,0 according to eq (2) we always have to invert complex matrices. Since

for computational purposes it is more convenient to work with real matrices and since we need

only the imaginary part of the inverted matrix we make use of the following relations

X+iY= [A + iB]-'

where X-= [A + BA-'B]-^

(34)

Y=- [B+AB-'A]-'

with X, Y, A and B being real matrices. From eqs (2), (13) and (18) it is clear that A contains the

diagonal matrix of Awop and the sine transform of the thermal average and B the cosine transform

of the thermal average. In the limit of large Aw we have

A = B + Ao}-I (35)

(see eq (X.18) of paper I) which simplifies to lowest order the F-matrix to y--5/Aa>2 (36)

This is the one electron limit, which as pointed out already in section IV. of paper I, does not

require a matrix inversion.

4. The Influence of Lower State Interactions and the Static Limit

In the figures 1, 2, 3, 7 and 8 the intensity of the line is plotted versus the frequency perturbation Aw in units of the plasma frequency (Op= vSTrne^Jm. The frequency scale is preferred,

because it represents essentially the energy perturbation and is therefore more meaningful for the

discussion of Stark broadening than, for example, the wavelength scale, which is usually more

convenient in the measurement. Furthermore, Aa»/wp < 1 is essentially the domain of the unmodified

impact theory.

We first demonstrate the influence of lower state interaction on the final line profile in figures

1 and 2 and compare three different methods of evaluation. In the first, most general method lower

state interactions are taken into account and the line profiles are calculated on the basis of eqs (13) and (14) using the correct Stark effect for electrons and ions (solid curves in fig. 1). In the second

method based on eqs (26) and (27) we neglect lower state interactions for the electrons, i.e.. we do

5
not allow for perturbations of the lower state sublevels by electrons (dashed curves in fig. 1, solid curves in fig. 2). However, we still use the correct Stark effect for the static ions. In the third and

simplest method the line profiles are calculated like in case of the Lyman lines with no influence of

the lower state for the electrons as well as the static ions (dashed curves in fig. 2). These Lyman

profiles have been presented previously in Paper L

We realize first of all from figure 1 that in view of the accuracy to be expected from the unified

classical path theory we may neglect lower state interaction due to the electrons for Hg and therefore also for all the higher Balmer lines. For ne = 1.3 • 10'^ cm"^ and T= 1850 K the biggest calculated difference between the cases with and without lower state interaction is 43 percent for Ha,

10 percent for H^, 6 percent for Hy and 4 percent for Hg. These differences have been obtained

for the distant, almost quasistatic wing. They become slightly smaller in the purely static wing and

are significantly smaller in the line center (14% for Ha, 9% for H/3, and 3% for Hy and 2% for Hg).

Similar differences may be obtained at any electron density and temperature'. In the distant, purely

static wing the difference will always be identical (39.2% for Ha, 9.3% for H/j, 5.7% for Hy and 3.6%

for Hg), because the static ions are treated in both cases with the same, correct Stark effect, while

the static electrons are treated in the second method with the Stark effect of the corresponding

Lyman line. In figure 2 the Balmer lines and the Lyman lines with the upper state principal quantum numbers n = 6, 8 and 10 are compared. In the line center we recognize the fact that the even Balmer lines have no unshifted Stark component, while the even Lyman lines do. The difference in the wings

is also clear from the Stark effect of the Balmer and Lyman lines (see also table I of the paper by

Vidal [5]). It therefore becomes apparent that for principal quantum numbers of about n>\Q one may neglect the influence of the lower states on the final Une profile altogether and use the Lyman

profiles throughout.

Next we compare in figure 3 the results of the unified classical path theory (solid curves)

with quasistatic calculations (dashed curves) in order to show to what extent quasistatic calculations

may be useful. The even Balmer lines up to 7i= 14 are plotted. The quasistatic profiles are based

6
10-2 I0-' I 10

Figure 3. Comparison of the unified theory and the quasistatic theory for the even Balmer lines with n = 4 ton = 14.

on the low frequency component of the electric microfield distribution functions [6, 7J with a

shielding parameter

ro/Z) = 0.0898^1/6/ y/f ^^^^

and the total density A^^ being N=2ne. In addition, short vertical lines mark the position of the

average Weisskopf frequency

(38)

as defined by Unsold [8], where the average splitting

nk=-J^ n,f,l^fk (39)

k k with fk being the oscillator strength of the Ath Stark component is frequently approximated by

fik — n{n— 1) 12 (see table I of the paper by Edmonds, Schliiter and Wells [9]). The Weisskopf

frequency indicates the domain of the quasistatic theory and there have been various estimates,

which all agree within a factor of 2 to 3. From figure 3 it is apparent that with increasing principal

quantum numbers the profiles based on the unified classical path theory approach more and more

the static profiles and that the Weisskopf frequency turns out to be a rather conservative estimate

for the useful range of the quasistatic theory. In the example presented we recognize that except

for the very line center one may use quasistatic calculations almost throughout the entire profile for principal quantum numbers of n > 10. For practical purposes the hole in the center of the

static profile and the structure in the line center, which for the higher series members appears

to oscillate around the profile based on the unified classical path theory, will be smeared out by a convolution with the Doppler profile.

5. Comparison With Experiments and Other Theories

We start our comparison of the unified theory with experiments in the high and low electron

density range with recent measurements by Wiese, KeUeher, and Paquette [10] performed on a

7
high current, wall-stabilized arc. The experimental setup and the method of evaluation is similar

to the one described by Wiese, Paquette, and Solarski [11]. However, a number of refinements are incorporated in the experimental setup and the achievable accuracy has been greatly improved.

In figure 4 and 5 the red and blue wings of the measured H/3- and Hy-profiles are plotted (solid

lines), which have been obtained in the same run assuring the same electron density and temperature for both profiles. For the moment we do not consider the asymmetries of the profile, since the

theory is not yet arranged to describe them. The electron density and temperature for this particular run have been determined in several ways from the absolute intensity of H/s with

ne = 7.5 • IQifi cm-3 r= 12570 K,

from the absolute intensity of Hy with

ne = 7.7 • 1016 cm-3 7= 12640 K

and from the absolute intensity of the continuum in the visible and in the UV, which differ by the

contribution from the Balmer continuum, with

n<, = 8.0- 10'« cm-3 r= 12730 K.

0.025

1

1 \ \ \ 1 r —^

1

Ql \ \ 1

\ \> ' 3 0 10 20 30 40 50 60 TO AX [a]

Figure 4. Comparison between the unified theory for Hn and the experimental

profile as measured by Wiese, Kelleher and Paquette.

\

\

\

\

AX [A]

Figure 5. Comparison between the unified theory for Hy and

the experimental profile as measured by Wiese, Kelleher and

Paquette. 8
The three values of the electron density differ in all runs by similar amounts. In particular the

difference between the values from the and the continuum intensity is very reproducible and

gives rise to a 6 percent difference in the electron density. There are indications (private communication of Dr. Wiese) that these differences may be due to small non LTE (local thermodynamic

equilibrium) effects in the arc because of its rather small dimension. This would be consistent

with the validity criteria for LTE in inhomogeneous stationary plasmas (see section 6-10 of Griem [12]). Since the continuum intensity is least affected, the electron density and temperature

as obtained from the continuum have been adopted as the best values, in particular, since the same

values have been obtained from the continuum in the visible and the UV. For these parameter

values the profiles as calculated with the unified theory including lower state interactions have

r+oo been plotted in figures 4 and 5 with the same normalization /(AA.)rfA= 1 as the experimental

J —CO profiles. We realize that the calculated profiles are shghtly higher and narrower than the measured

profiles. In both cases, however, the best agreement between theory and experiment in a least squares sense has been obtained with 8.5- 10^^ cm~^, meaning that the theoretical value appears

to be 6 percent larger than the experimental value. Applying recent calculations of Kepple and Griem [13], one .obtains from the half-, quarter-,

and eighth-width ne = 7.38-10^^ cm~^ for Hn and /?e = 8.05 • 10'^ cm"^ for Hy (as evaluated by

Dr. Wiese), which reveals an intrinsic inconsistency because the two electron densities differ by about 10 percent.

We also notice that the worst agreement is in the very line center, a feature which is so far common to all impact theories. This fact makes the determination of the electron density on the

basis of fractional widths rather questionable because its definition may be ambiguous and it effectively normafizes wing intensities with respect to the intensity in the very line center, where

the theory seems to be least reliable. Although this is the most convenient and most widely used

method, it is definitely preferable to determine the electron density from a least squares fit of the

experimental and theoretical profile using the same normalization for both profiles. In figure 6 the calculations of Kepple and Griem [13] are compared with our calculations for

H/3 and ne = 6A 10^^ cm"^ and 7=12.200 K indicating that our profile is narrower and higher.

AX [a]

Figure 6. Comparison between the unified theory with B as obtained for Pmi«=^ -I- fn^ao, with 8=1.27 as used by Kepple and Griem and the modified impact theory of Kepple and Griem.

hence giving rise to larger electron densities. In the very line center the most important difference

appears to be due to the constant B in eqs (22) and (23). While our calculations are based on a

3 quantum number dependent constant B for a lower cutoff Pniin= ^+2 ^'^''^^^

selected according to the vahdity criterion of the classical path theory, Kepple and Griem use a 9
larger value B = 1.27 for all the Stark components. We have repeated our calculations with their

value of the constant B. The profile is included in figure 6 and does not quite agree with their calculations. It should be pointed out that the normahzation of the profile of Kepple and Griem is shghtly

too small, which may account for part of the remaining difference in the very line center, where

the unified classical path calculations should go over to the results of the impact theory, if the ion

field dependent cutoff (see eq. VI.4 of paper I) is neglected.

From the preceding comparison we realize that at this stage the most important problem seems

to be to obtain better values of the constant B whose dependence on various cutoff procedures was

discussed in the appendix of paper I for classical path theories neglecting time ordering. We note

that in case of the H/g and Hy profiles in figures 4 and 5, better agreement between theory and ex- periment may be obtained with a constant B larger than that used in this paper, and that in case of

the Lyman-a experiment of Boldt and W. S. Cooper [14] discussed in paper I, better agreement is obtained with a smaller constant B. For the moment we postpone the discussion of the various possibihties, which may affect the constant B, until we also have compared our results with measurements of the Balmer lines H3-H14

and the Paschen lines Pe-Pis performed by Vidal [5, 15]. Unfortunately, these experiments do not provide an independent electron density measurement, which is as accurate as the measured

profiles. Since, however, the measurements revealed a Aw^/^-wing, which for most of the fines extended over two orders of magnitude in intensity, the electron density was determined assuming

that these measured Aa;''^/--wings were identical to the asymptotic Holtsmark wings. In this manner

the same electron density of 1.3 • lO^^ cm~^ was obtained within ± 4 percent for all the Balmer

Unes from H4-H14 (see Vidal [5]). In paper I it was pointed out (on the basis of the Lyman fine calculations) that the unified theory calculations actually give the Aw^/- -wings which were measured

in the experiment and which extend much further into the line center than a quasistatic theory

would predict. However, over the intensity range measured, these Aw^'^-wings do not necessarily

coincide with the asymptotic Holtsmark wing.

In the following we reevaluate the measured line profiles employing now the complete line

profile from the Line center to the wings. We concentrate primarily on the Balmer fines, which

have been measured more accurately than the Paschen lines and where we also have more series

members available. The measurements were performed on a stationary radio frequency discharge

within a magnetic bottle as described by Schliiter [16]. The electron temperature, which within

the error limits is equal to the ion temperature, was measured to be T— 1850 K. Consequently, we

have to consider first of all to what extent Doppler broadening and the Zeeman effect may influence

the first series members. In figure 7 the theoretical fine profiles are shown for Ug— 1.3 • lO^^ cm~^

and T= 1850 K before (dashed fines) and after (soHd fines) the convolution with the Doppler profile.

It shows that up to around Hs Doppler broadening has to be taken into account. In estimating the

influence of the Zeeman effect we notice that for a magnetic field of 1800 gauss (0.18 tesla) as

typically used in the experiments, the separation of the outer components of the Lorentz triplet

amounts to 0.168 cm"'. Neglecting complications due to the combined Zeeman and Stark effect

this indicates that from around H7 (full half width= 1.76 cm"') we may neglect the Zeeman effect for all the higher series members.

A comparison of the experimental and theoretical profiles for the Balmer fines H7-H14 gives

the best agreement for ne=1.15 • 10'^ cm"^, a value which is slightly smaller than the value of

Tig = 1.3 • 10'3 cm"3 given by Vidal [5]. This is not surprising because within the measured intensity

range the apparent Aw^'^-wings of the unified theory calculations are all lying above the asymptotic

Holtsmark wings on which the electron density value of Vidal [5] was based. For this new value the

measured and calculated Balmer fines H7-H14 all agree within 5 percent over the entire fine

profile. These maximum deviations are comparable to the achieved accuracies of the measurement

and are so small that they hardly show up in a plot like in the figures 1, 2, 3, 7, and 8. For the lower

Balmer fines only the fine wings can be evaluated since the line center is noticeably affected by the

Zeeman effect not included in the calculations. The agreement is not quite as good. However, the

differences do not exceed 10 percent for the wings of H^g to Hg, which appears to be partially due

to the growing influence of the apparatus profile for the narrower profiles of the lower series

members.

In figure 8 our calculations are compared with the calculations of Kepple and Griem [13],

which have been extended to H12 by Bengtson, Kepple and Tannich [17]. We see that their profiles are wider and that the higher series members do not show the measured Aco^^'^.^vings. From a comparison with figure 3 we also realize that with increasing principal quantum number n our

profiles approach the static profiles while the'profiles of Kepple and Griem do not. Their profiles

actually appear to differ more and more from the static profiles with increasing principal quantum

number n, which manifests the fact that the modified impact theory is not able to describe the

static electrons. 10
Aoj/5p

Aa,/5p

Figure 7. The Balmer line profiles as calculated FIGURE 8. Comparison between the unified theory and

with the unified theory for n^. = l.S-W cm"^ and the modified impact theory of Kepple and Griem for the T= 1850 K with and without the convolution of even Balmer lines Hn to H\2. the Doppler profile. As pointed out already in paper I, we emphasize again that contrary to the H;3- and Hy-profiles

discussed above, the higher series members are insensitive to the exact value of the constant B. The reason is that these profiles are predominantly static in nature. In other words it means that the average Stark effect splitting, which is approximately equal to the half width of the higher

series members, is significantly larger than the constants ai for the different Stark components

(see eq (22)), which determine essentially the electron impact broadening. Since the hue centers

of the first series members are predominantly Doppler broadened, these low density experiments

give no detailed information on the constant B. Hence, as one would expect, we also obtain a better

agreement between theory and experiment than we did for the H^- and Hy-profiles discussed above,

where the agreement could have been worse due to the uncertainty of the constant B. This fact

is particularly important for astrophysical appUcations where one is mainly interested in low density

profiles, because these profiles are insignificantly affected by the remaining uncertainty of the constant B. One may, therefore, calculate low density profiles with rather large confidence.

6. Discussion

Since the value of the constant B appears to be the most vague and restricting quantity in the current classical path theories, we will now summarize the effects which may influence the constant B and are not included in the present calculations. First of all, we can state in general

that the exact value of the constant B is least important if either the static broadening exceeds

the electron impact broadening, as discussed above for the low density profiles, or if according to eq (22) the value of In (4C2) is much larger than the uncertainty of the constant B which means that -ln(4C2)>l. (40)

According to eq (24) the latter situation is most hkely to occur for the innermost Stark components

of any hydrogen fine. The constant B is sometimes misleadingly referred to as the strong coUision parameter. This

is only partially correct, because its value depends not only on the lower cut off" parameter pmm

11
but also on the upper cut off parameter aD , where D is the Debye length and the constant a has

been varied in the Hterature from 1.1 (Griem, Kolb and Shen [18]) down to 0.6 (Chappell, J. Cooper

and E. Smith [19]). As discussed in the appendix of paper I. this parameter a will also influence the

Hmits on the time integral J F(f')c/f' in the time development operator^, which for the S-matrix

limit are extended from — °o to +«>. In the unified theory calculations these hmits have been ex- tended from — r to + 7", where

T= VaW^-p^lv, (41)

in order to make the time limits consistent with the upper cut off parameter aD. We also saw from

the appendix of paper I that, as a function of the parameter a, the constant B varies as 5 = 5(a=l)+2 In a. (42)

The correct value of the constant a has not yet been determined conclusively. Since for large

impact parameters one is dealing only with weak colhsions, it has to be possible to determine the

parameter a from a second order classical path theory, which means essentially within the frame

work of the ordinary impact theory.

The influence of small impact parameter collisions on the constant is a much more involved

problem. First of all, it is clear that one has to worry about collisions with impact parameters

P<Pmin=^+ t «'ao=A (1 + 3.78- 10-3 «2 (43)

where classical path theories start to break down. As discussed for example, by E. W. Smith,

C. R. Vidal and J. Cooper [20] we have to distinguish in this range between "completed" strong

collisions, which contribute primarily to the impact limit, and "incompleted" strong collisions,

which mostly contribute to the static wings. While the latter collisions are properly treated in a

classical path theory and have no influence on the value of the constant 5, the "completed" strong

collisions can only be treated correctly by a quantum mechanical calculation, which has not yet

been done.

Besides the limitation imposed by a classical path approach, we also have to consider the

influence of time ordering which has been neglected in our time development operator of eq (5). We notice that time ordering will become important if Ft/^>1. With a typical collision time

T — p/f , this is the case for impact parameters

P ~Pord=t ("g-"'?')^- (44)

For most practical cases, where the value of 5 becomes critical, Pord is larger than Pmin- Hence,

there will be a domain, in which time ordering is important even within the regiorr of vahdity of a classical path theory. In the S-matrix hmit the effect of time ordering has been investigated within the classical path

theory for Lya (Bacon, Shen, and J. Cooper [21]) and for Ha (Bacon [22]) by solving the complete

set of coupled differential equations, which define all the matrix elements of the time development

operator. These calculations have demonstrated that the effect of time ordering may change the

value of the constant B by as much as ± 1 and that the effect on the final line profile may amount to a few percent in intensity. These calculations have also included higher multipole terms. If one examines the effect of the higher multipole terms with respect to the case, where one considers

only the dipole terms, one finds out that in case of Lyman-a as well as in case of Ha the higher

multipole terms lower the value of all the different constants B. However, at the electron density

and temperature of these experiments (which are essentially the same), the change in the constant B due to the higher order multipole terms affects the final line profile only very Uttle. For the more general case of the time development operator required by the unified theory,

the effect of time ordering has recently been examined for Lyman-a [23]. These calculations show

the results of the S-matrix hmit in the line center and the decreasing influence of time ordering

with increasing frequency pertubation Aw.

Due to the various outlined reasons which may affect the value of the constant B, it is rather

difficult to make a quantitative statement of the possible error. If we assume that for impact parameters p > Pord all the collisions which contribute to the value of B, have been treated correctly, one may give an upper bound on the possible error in B based on the unitarity condition of the time

12
development operator. Since the time development operator may only oscillate within the limits of ± 1, the integral over impact parameters from 0 to pord, which is weighted hy the impact parameter p, will change the constant B at the most by an amount of the order of unity. In addition to this the influence of the upper cut off parameter according to eq (42) has to be kept in mind. We also

recall that in order to obtain the best agreement between theory and experiment we have to change

the constant B for the Lyman a profile [14] by an amount which is sHghtly larger than 1, and for the

H;8 and Hy profile by an amount, which is smaller than 1. It should also be noted that the ion field splitting is not fully taken into account in the time development operator as discussed in section VI

of paper I. It was shown that this may lead to sHght modifications in the Hne center but not in the

line wings. A full treatment would be rather difficult because the ion field exponentials in eq (VI. 4) of paper I remove the spherical symmetry of the problem and a simple cut off procedure as sug- gested by Kepple and Griem [13] does not appear to be completely adequate since it affects the

normalization of the line profile. In addition, near the line center we would expect the quasistatic

description for the ions to be invalid. This however, is practically unimportant because the Weisskopf frequency for the ions, which gives the range of validity for a quasistatic approach, is for low

densities well inside the Doppler width and at high densities well inside the half width. Notice also that the breakdown of the no-quenching approximation may cause a small effect on the strong col- lision term (see ref. [20], p. 415).

So far we have ignored profile asymmetries which have been observed, for example, in Ly-a,

and Hy. In the following we give a brief discussion in which we do not consider the well known

asymmetries due to the <y^- factor in the expression for the power spectrum, the frequency to wavelength conversion, and the Boltzmann factor [24], which was neglected by assuming the

elements of the atomic density matrix to be constant for all initial states. Asymmetries due to these

effects will grow with increasing frequency perturbation Aw and are always negligible in the line center. Hence, we are mainly concerned with the higher order multipole terms and the higher order

Stark effect terms due to the electrons and ions. In section VII of paper I it was shown how the higher order multipole terms due to the electron

perturbers can be included in the interaction potential of the time development operator and it was pointed out that within the no-quenching assumption only a finite number of multipole terms

exist. The main aggravation is that the unitary transformation which diagonalizes the time development operator after the spherical average is no longer a simple rotation. In the impact limit this

will affect primarily the constant B as noted above and cause only negligible asymmetries like in case of the time ordered solutions [21, 22]. The main reason for asymmetries especially in the line center has been shown to be due to the ions and we refer to a recent paper of Sholin [25] which supersedes the earlier papers by Griem

[26] and Nguyen-Hoe, Drawin and Herman [27]. In this paper, it has been successfully demonstrated

that the observed asymmetries of the Ly-a, H/j- and Hy-lines can all be explained within a static ion approach, where the main effect is caused by the quadrupole term of the ions and to a smaller

extent by the quadratic Stark effect and the ion field dependent transition probability corrections

of the individual Stark components. The electrons were assumed to cause only collision broadening

and to introduce no further asymmetries.

It should be pointed out that the results presented by Sholin can easily be incorporated into the calculations presented here and work on this is in process. Like Sholin, we regard the radiator as being an atom in a static ion field and perturbed by the electrons. Hence, one only has to modify the matrix elements of AoJqp in eq (10) and of d(^d in eq (25) according to the relations given

by Sholin, in order to include the higher order multipole terms and the higher order Stark effect terms due to the static ions.

If we incorporate the higher multipole terms due to the electrons in the time development

operator together with those due to the ions, another interesting feature of the unified theory is that a number of the higher order multipole terms due to the electrons and ions Hke, for example,

the most important quadrupole terms, will cancel each other in the static limit of the electrons. This

effect was already suspected by Griem [12] (p. 94). However, this sort of mutual compensation

occurs only if electrons and ions are treated within the framework of the same approximations

(for example, quasi-static or impact) as pointed out by Sholin [25]. For this reason the occurrence

of asymmetries gives important information on the type of electron broadening. It is also clear

why asymmetries have been observed in the high density profiles of Lya, H^ and Hy, where the ions

may be treated quasi- statically and the electrons are over most of the measured line profile in the

domain of the impact theory, but not in the low density profiles of the higher Balmer and Paschen

lines, where ions and electrons are both in the quasi-static domain over almost the entire line

profile.

Finally, we may conclude that, at this stage, the accuracy of the unified theory calculations

depends primarily on the extent to which the final line profile is affected by the constant B and

13

406-848 0 - 71 -2
to a lesser extent on the asymmetries not yet included. Both effects are in turn determined by the

electron density and temperature. It is probably safe to say that in general the electron density

obtained with the unified theory in its present form will differ at the most by 10 percent from its true value. In the impact limit significantly better agreements between theory and experiment,

which have been reported in the hterature, have to be regarded as fortuitous. However, better

results are definitely obtained for the higher, still well isolated series members and/or at low

electron densities (n? ^ 10'^ cm"^), which are of particular astrophysical interest.

7. References

[1] Vidal, C. R., Cooper, J., and Smith. E. W., J. Quant. Spectr. Radiative Transfer, 10, 1011 (1970).

[2] Smith, E. W., Cooper, J., and Vidal, C. R., Phys. Rev. 185, 140 (1969).

[3] Hughes, J. W. B., Proc. Phil. Soc. 91, 810 (1967).

[4] Bethe, H. A., and Salpeter, E. E.. Quantum Merchanics of One- and Two-Electron Atoms (Springer Verlag, Berlin 1957).

[5] Vidal, C. R.. Proc. of the 7th Intern. Conf. on Phenomena in Ionized Gases, p. 168 (1966).

[6] Baranger. M., and Mozer, B., Phys. Rev. 115, 521 (1959); 118, 626 (1960).

[7] Hooper, C. F., Phys. Rev. 165, 215 (1968).

[8] Unsold, A., Physik der Sternatmospharen (Springer Verlag, Berlin 1955).

[9] Edmonds, F. N., Schluter, M.. and Weils, D. C, Mem. Roy. Astr. Soc. 71, 271 (1967).

[10] Wiese, W. L., Kelleher, D. E., and Paquette. D. R., to be published. (The authors would like to thank Dr. Wiese for making his measurements accessible to us prior to publication).

[11] Wiese, W. L., Paquette, D. R.. and Solarski, J. E.. Phys. Rev. 129, 1225 (1963).

[12] Griem, H. R., Plasma Spectroscopy (McGraw-Hill Book Co.. Inc.. New York, 1964).

[13] Kepple, P., and Griem, H. R., Phys. Rev. 17.3, 317 (1968). see also University of Maryland, Report 831.

[14] Boldt, G., and Cooper. W. S., Z. Naturforschung 19a, 968 (1964).

[15] Vidal, C. R.. Z. Naturforschung 19a, 947 (1964).

[16] Schluter, H., Z. Naturforschung 15a, 281 (1960).

[17] Bengtson, R. D., Kepple, P.. and Tannich, J. D.. Phys. Rev. Al, 532 (1970).

[18] Griem. H. R.. Kolb. A. C, and Shen. K. Y., Astrophys. J. 135, 272 (1962).

[19] Chappell. W. R., Cooper, J., and Smith, E. W., J. Quant. Spectr. Radiative Transfer 9, 149 (1969).

[20] Smith. E. W.. Vidal, C. R., and Cooper. J., J. Res. Nat. Bur. Stand. (U.S.), 73A (Phys. and Chem.), No. 4, 405 (1969).

[21] Bacon. M., Shen, K. Y., and Cooper, J., Phys. Rev. 188, 50 (1969).

[22] Bacon, M.. to be published.

[23] Godfrey. T., Vidal, C. R., Cooper. J., and Smith, E. W.. Phys. Rev., to be published.

[24] Huber, D. L., and VanVleck, J. H., Rev. Mod. Phys. 38, 187 (1966).

[25] Sholin, G. V., Opt. Spectrosc. 26, 275 (1969).

[26] Griem. H. R., Phys. Rev. 140, A1140 (1965).

[27] Nguyen-Hoe. Drawin, H. W.. and Herman, L.. J. Quant. Spectr. Radiative Transfer 4, 847 (1964).

[28] Vidal, C. R., Cooper, J., and Smith, E. W., Nat. Bur. Stand. (U.S.), Monogr. 116, 143 pages (May 1970).

8. Appendix. Program for Calculating the Line Profile / (Ag>)

The program presented has been arranged in a very similar manner to the program given in Appendix C of the NBS IMonograph [28], which was only able to calculate the Lyman hues. The

first general version (program version A) is in principle able to compute any hydrogen line and is only limited by the available memory space of the computer. It has been written to calculate the

Lyman lines up to n= 16, the Balmer Hues up to /i = 8 and the Paschen lines up to « = 5. By replacing the main program STBRHY and the function AIUVI by the program STBRHY and the function

AIIM Hsted in the end, the program may be simphfied to calculate any hydrogen line neglecting

lower state interaction (program version B). The arrays have been dimensioned such that the pro- gram is able to calculate all the Lyman, Balmer, and Paschen lines up to n= 16, As demonstrated

in figure 1, this simplified version is completely adequate for all the Balmer fines beyond Hg. For

all the Lyman lines it is also noticeably faster than the first general version and the program given

in Appendix C of the NBS IMonograph.

1. The Fourier transform of the thermal average.

The function FOUTR calculates essentially the Fourier transform of the thermal average such

that CCREAL=Re {zAaj|i(Awfi, /3, /I. n', 96, 9;,9c, 9c)} (Al)

and

CCIMAG^ Im \iAa)%i(\coR, (3, n, n' , qt,, q'l,, Qc, q'c) } (A2)

where

DOM = Aaj/j= [ Aw — Aa;i(n, qi,, n' , q',,) jS ]l(x),j. 14

(A3)
In evaluating i(Aajw) according to eq (19) it computes iik—1, Acu«) as defined in eq (20) and

i{k = 2, Aojr) as defined in eq (X.22) of paper I. The required Bessel functions Jd, J\, Ko and Y\ are obtained from the subroutine BSJYOl. For large and small arguments Aaj/( the \no\i^r expansions

are applied as discussed in paper I. The constants P\, Pi and b\ {Pl,P2, Bl ) specifying i(k= 1 , Aw/t)

and 02 and 62(^2, B2) specifying i{k = 2, Aojk) are set in the function AIIM and are calculated once for all the Stark components in the main program STBRHY.

2. The profile /(Aw, jS).

The function AIIM evaluates /(Ao), /3) as required by eq (1). The program is executed twice

to obtain the contribution to /(Aw, jS) from the blocks of the K and og'-matrix with M= 0 {MBL=1)

and M— ±l {MBL — 2). It establishes first of all the ^^'-matrix according to eq (13). The necessary

^-matrix (three dimensional array SLOP) as defined in eq (14) is read in or calculated once in the

main program, while the Fourier transform i(Aw/;) is obtained from the subroutine FOUTR de- scribed above. In order to reduce the size of the array SLOP the symmetry of the X-matrix with

respect to the diagonal (see eq (28)) is used.

The matrix oraering is established once in the main program to allow for the partitioning

discussed in section 3. The quantum numbers qn- m,,, q'„ and m'„ are specified by the running index

NRA, Qb, mb, q't) and ml by NRB and qc and q',, by NRC. Having set up the real and imaginary

matrices of the operator [ Awop — =^(Awop) ] the imaginary part of the invers is determined ac- cording to eq (34) performing the matrix inversion by means of the subroutine ENVERS. The re- sulting matrix is then multiplied by the dipole matrix of eq (25), which is also calculated once in the

main program such that DIPOL= ( - l)i(.+n'-,-,'-M-2) (2/+ 1) {21' + 1) {nl\r\n'r)

1 n-1

m q^ m + q

2 — m

I

m —m

(A4)

The function AIIM listed at the end for the case of no lower state interactions is arranged almost

identically to the function AIIM just described. The only difference is that now the ^- and ^- matrices are block diagonal in m, m' and q' according to eq (26) which allows a partitioning into smaller submatrices whose contribution to /(Aw, j8) is then again evaluated successively.

3. The final profile /(Aw).

The main program STBRHY calculates the final line profile by performing the ion field average

according to eq (1). It reads in first the low frequency component of the electric microfields [6, 7] for 0 < /3 ^ 30 in steps of 0.1 for 5 different values of the shielding parameter ro/D = 0., 0.2, 0.4, 0.6 and 0.8. A listing of the distribution functions is given at the end of the program. (The deck of the

distribution function was kindly supplied by Dr. C. F. Hooper.) The program then reads in the

density n^, the temperature T, the upper and lower state principal quantum numbers n and n', the initial value Aw, for which the line intensity is calculated, the logarithmic step width, the total number of points to be calculated, a parameter, which specifies the number of ion field integration

points and finally 6 numbers, which specify the characteristic constants and 62 of the G2-function

for all Stark components. These 6 numbers may be obtained from the calculations of the thermal

average described in appendix A of the NBS Monograph. As shown, however, in paper I the contribution of the G2-function can in practically all cases by neglected and hence one may set all 6 numbers to zero as done in this paper. The program starts out to calculate the microfield distribution function for the particular shield- ing parameter as defined in eq (37) by using the 5 point interpolation subroutine POLYS and stores the values in the array FIELD. It then calculates the average ion field

i8av = |ji8r(/3)rf/3 (^5)

by means of Weddle's rule (subroutine WEDDLE).

Next the program reads in a characteristic number NPUNCH. If this number is larger than 1

15
the program reads in the complete /^-matrix (array SLOP) for the particular hydrogen line, which

may have been obtained from another run of the program. Some numbers, which specify the size of the /^-matrix, are read together with NPUNCH. If NPUNCH is smaller than 1 the program cal- culates the required /^-matrix and if NPUNCH is equal to 1 it also punches the /^-matrix out on

cards. In the latter two cases the numbers NUR(l), NUR(2), IPI and NNCCQ, which specify the

size of the /^-matrix, may be set to any arbitrary value. The described feature was built into the

program because the computation of the A[-matrix may be very time consuming and is required only once for every hydrogen line as explained in section 3. It took, for example, on the CDC 3800 3

minutes to calculate the K-matrix of Ha, 8 minutes for H/j, 40 minutes for Hy and 100 minutes for Hg.

As a next step the program calculates the wavelength in standard air of the particular hydrogen line. It then calculates all the necessary radial matrix elements {ni\r\n'r) by means of the

function RADMAT, which in turn obtains the factorials from the function FCTRL. The values are stored in the array RDM according to

'I r 1

and the total line strength

STOT = 2:(2/+l)(2/' + l)(^ ^^'\{nl\r\n'l')\^ (A7)

is determined. The program then calculates all the necessary unitary transformations {nlm\nqm)

as defined in eq (15) using the 3; symbol function S3J and stores them for the upper and lower states

in the arrays SJUU and SJLL. With these transformations the dipol matrix elements between

parabolic states are calculated and stored in the array DIPOL according to eq (A4). Simultaneously,

the matrix ordering of the K- and ^-maXnx is established such that the quantum numbers q, q' , m

and m' of a particular state \nqm;n' q m') are replaced by one running index NRUN and the block

number MBLOCK, which may be 1 or 2 for M=0 andM = ± 1 (see eq (29)). The values of the indi- vidual quantum numbers are stored in the arrays NQUU, NQLL, MMUU and MMLL.

The program then proceeds to calculate the X-matrix (array SLOP) according to eq (14) if NPUNCH is smaller or equal to 1 as explained above. Although the values for the eight different unitary transformations in eq (14) are stored, the computer time may stiU become rather long for the higher series members due to the extensive sumation over the quantum numbers /„, /„, /j,, L, nic and m'c. As a last preliminary calculation the constants pi, p2, ^i, a-i-, and bz for all the Stark components

as required by the function AIIM for the subroutine FOUTR are determined and stored in the array FPAR. The constants p2, ai, and b\ are evaluated for a lower cutoff pniin = ^ + t"^ao. As explained in the appendix of paper I the constant B in eqs (22) and (23) is obtained from

5 = 2-3y-2

where

1 — cos (z) , sin (z) „. , , ; \ Li{z) (A8)

Srik^lprr^in (A9)

The necessary cosine integral is calculated by the function COSINT.

Finally the program is prepared to actually perform the ion field average. The distribution

function /*(/3) is obtained from the function WFLD, which uses a 5 point interpolation on the

values initially stored in the array FIELD. For ^ > 30 the function WFLD uses an asymptotic

expansion. The ion field integral is performed by means of Weddle's rule (subroutine WEDDLE).

The integrand is subdivided into several intervals depending on the spacing and the width of the

individual Stark components. A convenient change of variables is used in every interval. For

Aco >5iSavAw,[(/J9-n'9')n,ax] (AlO)

the ion field integral is replaced by /(Aw, ^ = y3av) as discussed in paper I. Besides the ion field integral the program calculates the wavelength perturbation AX (DLAM), which corresponds to Atu/wp, the static profile for N=ne (FHOLTS), the asymptotic Aco-^/^.^jng (WING52), the one

electron limit for j8 = 0 (AWING), /(Aco, ^3 = 0) and /(Aw, ;8 = /3av). The latter three values are printed out normalized with respect to the asymptotic Aai ^'^-wing.

As explained above the program STBRHY may be replaced by the program STBRHY listed

at the end for the case of no lower state interactions. Both programs are written in identically the same manner with the only difference that the program STBRHY listed at the end has a simpler

matrix ordering and a much simpler and faster calculation of the /(^-matrix according to eq (27), which makes the provision to read in the X-matrix no longer desirable.

16
PROGRAM STBRHY

C

C PROGRAM FOR CALCULATING THE STARKBROADENING OF HYDROGEN ON THE

C BASIS OF THE UNIFIED THEORY INCLUDING LOWER STATE INTERACTION

DIMENSION DQQ(30) ,PFAC(6) , STRONG (34) ,FF(1100) ,SJUU(136,31)

,

1 SJLL (6,5), RDM (16,3), SLUU (16 ) , SLLL ( 3 ) , NQUU ( 34 , 2 )

, NQLL ( 34 , 2 )

,

2 MMUU ( 34 , 2 ) , MMLL ( 34 , 2 ) , NQCU ( 30 ) , NQCL ( 30 ) , NXX ( 30 )

, NST ( 30 )

,

3 DST(30) ,W(300,5)

COMMON/FDAT/Pl , P2 , Bl , A2 , B2 , PPFF , CCREAL , CCIMAG

COMMON/PFW/FIELD ( 301

)

COMMON/1 1/SLOP ( 595 , 22 , 2

)

C0MM0N/22/DIP0L ( 34 , 2 )

, NUR ( 2 ) , IPI , NCQ , BET , FPAR (6,30), ADIFF ( 34 , 2 )

,

1 NSTEP

C0MM0N/33/AMATR ( 34 , 68 ) , BMATR ( 34 , 34 ) , CMATR ( 34 , 34

)

EQUIVALENCE ( SJUU , AMATR ) , ( FF ( 601 ) , NQUU ) , ( FF ( 669 )

, NQLL )

,

1 (FF(737) ,MMUU) , (FF(805) ,MMLL) , (FF(873) , NQCU ) , (FF(903) ,NQCL)

,

2 ( FF ( 933 ) , SJLL ) , ( FF ( 963 ) , RDM ) , ( FF ( 1011 ) , SLUU ) , ( FF ( 1027 ) , STRONG

)

FIELD(l) =0.0

READ 100, ((W(I,J), J = 1,5), I = 1,300)

100 FORMAT (5E12.4)

120 READ 150, DEN , TEMP , NNUU , NNLL , GIN , DGG , NTOT , NFAC ,( PFAC ( I ) , 1=1 , 6

)

150 FORMAT ( 2E10 . 2 , 13 , 12 , 2F10 . 2 , 2I5/6F10 . 5

)

IF (EOF, 60) 577, 170

C CALCULATION OF MICROFIELD DISTRIBUTION FUNCTION

170 SHIELD = 0.0898 * DEN** ( 1 . /6 .) /SQRT ( TEMP

)

DO 161 J = 1,5

161 DST(J) = 0.2 * FLOAT (J-1)

DO 163 I = 1,300

DO 162 J = 1,5

162 DQQ(J) = W(I,J)

CALL P0LY5 ( DST , DQQ , 5 , STRONG , SLUU , SHIELD , 0 . 1 , 1

)

163 FIELD (1+1) = SLUU(l)

C CALCULATION OF AVERAGE lONFIELD

DO 111 I = 1,300

AM = I 111 FF(I) = FIELD (I+l) * AM/10.

CALL WEDDLE ( 0 . 1 , 300 , FF , BAV , 0 .

)

DY = 1./300. % Y = 0. DO 113 I = 1,10

Y=Y+DY $ B=l./Y

113 FF(I) = WFLD(B) * B**3

CALL WEDDLE (DY , 10 , FF , DB , 0 .

)

BAV = BAV + DB

READ 175, NPUNCH, (NUR(I) ,1=1,2) , IPI, NNCCQ

175 FORMAT (5110)

C FOR NPUNCH GT 1 K-MATRIX IS READ IN

C FOR NPUNCH LT 1 K-MATRIX IS CALCULATED

C FOR NPUNCH EQ 1 K-MATRIX IS CALCULATED AND PUNCHED OUT

IF (NPUNCH. LE. 1) GO TO 178

DO 173 MBLOCK =1,2

NUPP = NUR (MBLOCK)

NRRAB = (NUPP+l)*NUPP/2

DO 174 J = IPI, NNCCQ

READ 172 , ( FF ( I ) , 1=1 , NRRAB

)

172 FORMAT (5E16.9)

DO 171 I = 1, NRRAB

171 SLOP (I, J, MBLOCK) = FF(I)

174 CONTINUE

173 CONTINUE

C CALCULATION OF WAVELENGTH IN STANDARD AIR

178 AUU = NNUU

ALL = NNLL

SLl = 109678.758 * ( 1 . / ( ALL*ALL ) - 1 . / ( AUU*AUU )

)

17
SL2 = SLl * SLl

ALDD = 1. 000064328+2949810. /(1-46E+10-SL2) +25540. /(4.1E+9-SL2)

ALAM = l.E+8/(ALDD * SLl)

PRINT 180, DEN, TEMP, NNUU , NNLL , ALAM, SHIELD

180 FORMAT ( IHl , * DENSITY =* ElO . 2* TEMPERATURE =*E10.2,

1 * QUANTUMNUMBERS N UPPER =*I3* N LOWER =*I2* WAVELENGTH =*

2 F8.2* ANGSTR0M*/10X, *SHIELDING PARAMETER =*F6.3,10X,

3 * INCLUDING LOWER STATE INTERACTION *//

IF (NNUU.LE.5.AND.NNLL.LE.3) GO TO 200

IF (NNUU.LE.8.AND.NNLL.LE.2) GO TO 200

IF ( NNUU. LE. 16. AND. NNLL. EQ. 1) GO TO 200

PRINT 190

190 FORMAT (* OVERFLOW OF MATRICES, PROGRAM NOT EXECUTED*)

577 CALL EXIT

CC RADIAL MATRIXELEMENTS AND TOTAL LINESTRENGTH

200 IPI = 2 STOT = 0. DO 330 KLL = 1 , NNLL

LLL = KLL - 1 $ ALL = LLL

DO 310 KUU = 1,NNUU

LUU = KUU - 1 $ AUU = LUU $ RRMM = 0. IF (IABS(KUU - KLL).NE.l) GO TO 310

RRMM = RADMAT ( NNUU , LUU , NNLL , LLL ) * S3J ( ALL , AUU ,1.,0.,0.,0.)

STOT = STOT + (2.*ALL+1. )*(2.*AUU+1. ) * RRMM * RRMM

310 RDM (KUU, KLL) = RRMM

330 CONTINUE

SQST = SQRT(STOT)

C

C TRANSFORMATION MATRICES

DO 370 NKK =1,2

NN = NNLL

IF (NKK.EQ.2) NN = NNUU

ANN = NN $ ANl = 0.5 * (ANN - 1.

KQQ = 2 * NN - 1 $ NRUN = 0

DO 370 KL = 1,NN

AL = KL - 1 FACC = SQRT(2 . *AL+1 .

IF(NKK.EQ.l) SLLL(KL) = FACC

IF (NKK.EQ.2) SLUU(KL) = FACC

DO 370 KM = 1,KL

AM = KM - 1 $ NRUN = NRUN + 1 QQ = -ANN $ ALIM = ANN - 0.9 - AM

DO 370 KQ = 1,KQQ

QQ = QQ + 1. $ SIIJ =0. $ ABQQ = ABS(QQ)

IF ( ABQQ. GT. ALIM) GO TO 360

NQ = ABQQ +0.1

IF (M0D(NN+NQ+KM+1,2) .NE.l) GO TO 360

AMQ = 0.5 * (AM - QQ) $ APQ 0.5 * (AM + QQ)

SIIJ = FACC * S3J(AN1 ,AN1,AL,AMQ,APQ,-AM)

360 IF (NKK.EQ.l) SJLL ( NRUN , KQ ) = SIIJ

IF (NKK.EQ.2) SJUU ( NRUN , KQ ) = SIIJ

370 CONTINUE

CC DIPOLMATRIXELEMENTS IN PARABOLIC STATES AND MATRIX ORDERING

KQUU = 2 * NNUU - 1 $ KQLL = 2 * NNLL - 1 DO 500 MBLOCK =1,2

AMLCK = MBLOCK $ MBL = MBLOCK - 1 $ AMBL = MBL

NRUN = 0 $ NCQ = 0 $ NAQU = -NNUU

DO 480 KUU = 1,KQUU

NAQU = NAQU + 1 S KQU = NNUU + NAQU $ NAQL = -NNLL

DO 480 KLL = 1 , KQLL

NAQL = NAQL + 1 18
KQL = NNLL + NAQL

NXDIF ^ NNUU * NAQU - NNLL * NAQL

AXDIF = NXDIF

IF ( NXDIF. LT.O) GO TO 400

NCQ = NCQ + 1 IF (MBL0CK.EQ.2) GO TO 400

NQCU(NCQ) = NAQU $ NQCL(NCQ) = NAQL

NXX(NCQ) = NXDIF $ DQQ(NCQ) =-0.

400 NPHAS = (NNUU + NNLL + MBL - NAQU - NAQL - 2 ) /2

PHASE = (-1 . )**M0D(NPHAS,2)

KMINL = NNLL - lABS(NAQL) $ KMINU = NNUU - lABS(NAQU)

KML = -KMINL $ MKML = 2 * KMINL - 1 DO 470 KAML = 1 , MKML

KML = KML + 1 IF (M0D(NNLL+NAQL+KML,2) .NE. 1) GO TO 470

KMU = KML + MBL

IF ( lABS ( KMU ) .GE. KMINU) GO TO 470

IF (M0D(NNUU+NAQU+KMU,2) .NE. 1) GO TO 470

NRUN = NRUN + 1 NQUU(NRUN,MBLOCK) = NAQU $ NQLL ( NRUN , MBLOCK ) = NAQL

MMUU(NRUN,MBLOCK) = KMU $ MMLL ( NRUN , MBLOCK ) = KML

ADIFF( NRUN, MBLOCK) = AXDIF

AMU = KMU $ AML = KML

KBBU = lABS(KMU) + I $ KBBL = lABS(KML) + I RESULT = 0. DO 450 KKLU = KBBU , NNUU

ALU = KKLU - 1 LLU = (KKLU-1) * KKLU / 2 + KBBU

FACl = SLUU(KKLU) * SJUU ( LLU , KQU

)

DO 450 KKLL = KBBL, NNLL

IF (IABS(KKLU - KKLL).NE.l) GO TO 450

ALL = KKLL - I LLL = (KKLL - I) * KKLL / 2 + KBBL

FAC2 = SLLL(KKLL) * SJLL ( LLL , KQL

)

FACC = FAC2 * FACl * S3J ( ALL . ALU , I . , AML , -AMU , AMBL

)

RESULT = RESULT + FACC * RDM ( KKLU , KKLL

)

450 CONTINUE

SIIJ = RESULT * PHASE / SQST

DIPOL( NRUN, MBLOCK) = SIIJ

IF ( NXDIF. GE.O) DQQ(NCQ) = DQQ(NCQ) + AMLCK*RESULT*RESULT/STOT

PRINT 425, NAQU, KMU, NAQL, KML, SIIJ

425 FORMAT ( 16X , *

( * . 213 , * I D I*, 213,* ) =*EI6.8)

470 CONTINUE

480 CONTINUE

NUR( MBLOCK) = NRUN

PRINT 393

393 FORMAT (/)

500 CONTINUE

: REORDERING OF THE STARK COMPONENTS

NCMl = NCQ - 1 DO 333 II = 1,NCM1

IP? =11+1

DO 333 JJ = IPP.NCQ

IF (NXX(II) .LE.NXX(JJ) ) GO TO 333

TEMPP = NXX(II) $ NXX(II) = NXX(JJ) $ NXX(JJ) = TEMPP

TEMPP = DQQ(II) $ DQQ(II) = DQQ(JJ) $ DQQ(JJ) = TEMPP

TEMPP = NQCU(II) $ NQCU(II) = NQCU(JJ) $ NQCU(JJ) = TEMPP

TEMPP = NQCL(II) $ NQCL(II) = NQCL(JJ) $ NQCL(JJ) = TEMPP

333 CONTINUE

PRINT 410, (NQCU(I) ,NQCL(I) ,NXX(I) ,DQQ(I) , 1=1, NCQ)

410 FORMAT (15X,* Q UPPER =*I3,* Q LOWER =*I3,* X =*I5,* DIP0L=*E16 . 8

]

C

19
PRINT 393

JJ = 1 $ KMST ^ 0 315 KMST = KMST + 1 DST(KMST) = 0. DO 325 II = JJ.NCQ

IF (NXX(II) .NE.NXX(JJ) ) GO TO 325

NST(KMST) = NXX(II) $ DST(KMST) = DST(KMST) + DQQ(II)

J = II 325 CONTINUE

JJ = J + 1 IF (JJ.GT.NCQ) GO TO 345

GO TO 315

345 PRINT 355, ( NST ( I ) , DST ( I ), I = l.KMST)

355 FORMAT (16X,*X =*I3,F20.10)

PRINT 393

C

C K OPERATOR MATRIX

IF (NPUNCH-1) 445,455,650

455 IPP = 2 PUNCH 465 , IPP . ( NUR ( I ) , 1=1 , 2 ) , IPI , NCQ , NNUU , NNLL

465 FORMAT ( 5110 . 15X , 215

445 NTIME = KLOCK(O)

DO 600 MBLOCK =1,2

MBL = 1 - MBLOCK $ AMBL = MBL $ NUPP = NUR (MBLOCK)

DO 600 NRC = IPI, NCQ

NCQU = NQCU(NRC) $ NCQL = NQCL(NRC)

KCQU = NNUU + NCQU $ KCQL = NNLL + NCQL

IMULIM = NNUU - lABS(NCQU) $ IMLLIM = NNLL - lABS(NCQL)

DO 590 NRA = l.NUPP

NAQU = NQUU( NRA, MBLOCK) $ KAMU = MMUU ( NRA , MBLOCK

AMU = KAMU $ lABAMU = lABS(KAMU)

NAQL = NQLL( NRA, MBLOCK) $ KAML = MMLL ( NRA , MBLOCK

AML = KAML $ lABAML = IABS( KAML)

KAQU = NNUU + NAQU $ KAQL = NNLL + NAQL

KAAU = lABS(KAMU) + 1 $ KAAL = lABS(KAML) + 1 DO 580 NRB = l.NRA

NRRAB = (NRA - 1) * NRA / 2 + NRB

NBQU = NQUU( NRB, MBLOCK) $ KBMU = MMUU (NRB , MBLOCK)

BMU = KBMU $ lABBMU = lABS(KBMU)

NBQL = NQLL( NRB, MBLOCK) $ KBML = MMLL ( NRB , MBLOCK

BML = KBML $ lABBML = lABS(KBML)

KBQU = NNUU + NBQU $ KBQL = NNLL + NBQL

KBBU = lABS(KBMU) + 1 $ KBBL = lABS(KBML) + 1 NPHAS = MBL - NCQU - NCQL - ( NAQU+NAQL+NBQU+NBQL ) /2

SIIJ = 0. DO 540 JLAL = KAAL, NNLL

ALL = JLAL - 1 LALL = (JLAL - 1) * JLAL / 2 + lABAML + 1 FACl = SJLL( LALL, KAQL)

DO 540 JLAU = KAAU, NNUU

ALU = JLAU - 1 LALU = (JLAU - 1) * JLAU / 2 + lABAMU + 1 FAC2 = SJUU ( LALU , KAQU ) $ FAC12 = FACl * FAC2

DO 540 JLBL = KBBL , NNLL

BLL = JLBL - 1 LBLL = (JLBL - 1) * JLBL / 2 + lABBML + 1 FAC3 = SJLL( LBLL, KBQL) $ FAC13 = FAC12 * FAC3

MCLUP = MINO( JLBL, JLAL) - 1 DO 540 JLBU = KBBU, NNUU

BLU = JLBU - 1 LBLU = (JLBU - 1) * JLBU / 2 + lABBMU + 1 FAC4 = SJUU (LBLU, KBQU) $ FAC14 = FAC13 * FAC4

MCUUP = MINO( JLBU, JLAU) - 1 20
lUPP = MINO(IABS(JLAU+JLAL) , IABS( JLBU+JLBL) ) - 1 MCL = -IMLLIM - 1 DO 540 JMCLL = 1 , IMLLIM

MCL = MCL + 2 $ lABCML = lABS(MCL)

IF (lABCML.GT.MCLUP) GO TO 540

AMCL = MCL

LALL = (JLAL - 1) * JLAL / 2 + lABCML + 1 LBLL = (JLBL - 1) * JLBL / 2 + lABCML + 1 FAC5 = SJLL(LALL,KCQL) * SJLL ( LBLL , KCQL

)

FAC15 = FAC14 * FAC5

IF (ABS(FAC15) .LT. 1 .E-8) GO TO 540

MCU = -IMULIM - 1 DO 535 JMCUU = 1 , IMULIM

MCU = MCU + 2 $ lABCMU = lABS(MCU)

IF (lABCMU.GT.MCUUP) GO TO 535

AMCU = MCU

LALU = (JLAU - 1) * JLAU / 2 + lABCMU + 1 LBLU = (JLBU - 1) * JLBU / 2 + lABCMU + 1 FAC6 = SJUU(LALU,KCQU) * SJUU ( LBLU , KCQU

)

IF (ABS(FAC6) .LT. 1 .E-8) GO TO 535

MGGL = MCL - MCU $ AMGL = MGGL

ILOW = MAXO(MBL,IABS(MGGL) ,IABS(JLAU-JLAL) .lABS(JLBU-JLBL) ) + 1 IF (ILOW.GT. lUPP) GO TO 535

FAC16 = FAC15 * FAC6 * (

( -1 .

) **MOD ( NPHAS-MGGL , 2 )

)

DO 530 JLDD = ILOW,IUPP

IF (M0D(JLAL+JLAU+JLDD,2) .NE.O) GO TO 515

IF (KAMU.EQ.O. AND.KAML.EQ.O) GO TO 530

IF (MCL.EQ.O.AND.MCU.EQ.O) GO TO 530

515 IF (MOD( JLBL+JLBU+JLDD,2) .NE.O) GO TO 520

IF (KBMU.EQ.O.AND.KBML.EQ.O) GO TO 530

IF (MCL.EQ.O.AND.MCU.EQ.O) GO TO 530

520 ALDD = JLDD - 1 FAC7 = S3J( ALL, ALU, ALDD, -AMCL, AMCU, AMGL) * (2.*ALDD +1.)

FAC7 = FAC7 * S3J ( ALL , ALU , ALDD , -AML , AMU , AMBL

)

FAC7 = FAC7 * S3J ( BLL , BLU . ALDD , -AMCL . AMCU , AMGL

)

FACC = FAC7 * S3J( BLL, BLU, ALDD, -BML.BMU, AMBL) * FAC16

SIIJ = SIIJ + FACC

530 CONTINUE

535 CONTINUE

540 CONTINUE

SLOP(NRRAB,NRC,MBLOCK) = SIIJ

580 FF(NRRAB) = SIIJ

590 CONTINUE

IF (NPUNCH.EQ.l) PUNCH 630 , ( FF ( I ), 1=1 , NRRAB

)

630 FORMAT (5E16.9)

600 CONTINUE

NTIME = -NTIME + KLOCK(O)

PRINT 617, NTIME

617 FORMAT ( 16X , *COMPUTERTIME FOR CALCULATING K-OPERATOR MATRIX =*I8/)

C

C BASIC CONSTANTS AND ARRAY FOR G-FUNCTION CONSTANTS

650 SDEN = SQRT(DEN)

FAC = 2064.936 * TEMP * SQRT ( TEMP/DEN

)

CFAC = 4.5645E-7 * SDEN/TEMP

DEBROG = 2. 1027E-6/SQRT(TEMP)

ANN = NNUU

RMIN = DEBROG + ANN * ANN * 7 . 9376E-9

BET = 5.6558E-5 * DEN**(l./6.)

ASY =0.0

DO 270 NRC = IPI,NCQ

QC = NXX(NRC) $ C = CFAC * QC

PI = -1.671086 * FAC * C * SQRT(C)

BS = 3. * QC * DEBROG / RMIN

21
STRONG ( NRG ) = 0 . 269-2 .*(((!. -GOSF ( BS ) ) /BS+SINF ( BS ) ) /BS-GOSINT ( BS )

PPFF = -1.128379 * FAC * C * G P2 = PPFF * (STR0NG(NRC)-2.*L0GF(2.*G)

FPAR(1,NRG) = PI $ FPAR(2,NRC) = P2

FPAR(3,NRG) = 0,5 * (P2/P1)**2 $ FIN = LOGF ( QG

A2 = P2*( (PFAG(3)*FIN+PFAC(2) )*FIN+PFAG(1)

B2 = (PFAG(6)*FIN+PFAG(5) )*FIN + PFAC(4)

IF (B2.LT.0. ) A2 = 0. FPAR(4,NRG) = A2 $ FPAR(5,NRC) = B2 $ FPAR(6,NRC) = PPFF

270 ASY = ASY +2. * PI * DQQ ( NRG

PRINT 220

220 FORMAT ( /13X*P1*18X*P2*18X*B1*18X*A2*18X*B2*17X*STR0NG*/

PRINT 240, ( (FPAR(K,I) ,K = 1 , 5 ) , STRONG ( I ) , I = IPI.NGQ)

240 FORMAT (6E20.4)

PRINT 280, FAG, GFAC , BET, ASY, DEBROG, BAV , NFAG

280 FORMAT (/* FAG =*E11.4,* CFAG =*E11.4,* BET =*E11.4,* ASY =*E11.4,

1 * DEBROG =*E11.4,* BAV =*F7.4,* INTEGRATIONFAGTOR =*I2//

2 5X*D0M*8X*DLAM*8X*IT0T*6X*IH0LTS*8X*ASY*10X*WING*7X*WH0LTS*7X,

3 *WW00*8X*WWBB*8X*WWNG*8X*TIME STEPS*/)

ADLFAC = 4.23538E-15 * SDEN * ALAM * ALAM

C

C GALGULATION OF THE lONFIELD INTEGRAL

N12 = 12 * NFAG $ AN12 = N12 $ N30 = 30 * NFAG $ AN30 = N30

DMGRT = BET * BAV * NXX(NGQ) * 5. G = GIN - DGG

DO 950 MM = 1,NT0T

NTIME = KLOGK(O)

NSTEP =0 $ G = G + DGG

DOM = 10. ** G $ DLAM = ADLFAG * DOM

WING52 = -0.2992067103 * ASY/ ( SQRT ( DOM ) * DOM * DOM)

FHOLTS =0. $ AWING = 0. DO 815 NRG = IPI.NGQ

PI = FPAR(1,NRG) $ P2 = FPAR(2,NRG) $ Bl = FPAR(3,NRC)

A2 = FPAR(4,NRG) $ B2 = FPAR(5,NRG) $ PPFF = FPAR(6,NRG)

GALL FOUTR(DOM)

AWING = AWING + DQQ ( NRG ) * GCIMAG * 2./(D0M*D0M)

QG = NXX(NRG) $ BETFAC ^ BET * QG $ BGRIT = DOM/BETFAG

815 FHOLTS = FHOLTS + DQQ ( NRG ) * WFLD( BGRIT) / BETFAG

IF ( DOM. GT. DMGRT) GO TO 985

AIRES = 0. IF(D0M.GT. (-3. *P2) ) GO TO 840

ANQl = NXX(NCQ)

BGRIT = (DOM - P2)/(ANQ1*BET)

DB = BGRIT/AN12 $ B = 0. DO 820 J = 1,N12

B = B + DB

820 FF(J) = AIIM(DOM,B) * WFLD(B)

GALL WEDDLE (DB , N12 , FF , AIII , 0 .

AIRES = AIII

DY = 1 ./(BCRIT*AN30) $ Y = 0. DO 830 J = 1,N30

Y=Y+DY $ B=l./Y

830 FF(J) = B * B * AIIM(D0M,B) * WFLD(B)

GALL WEDDLE ( DY , N30 , FF , AIII , 0 .

AIRES = AIRES + AIII

GO TO 980

840 BGRGR = DOM/BET

EPSPS = -P2/BET

DO 957 NQ = IPI,KMST

ANQ = NST(NQ)

BGR = BGRCR/ANQ $ EPS = EPSPS/ANQ

IF (NQ.EQ.IPI) GO TO 907

SLl = l./(GAM - BGR)

22
GO TO 908

907 SLl = 0. 908 SL2 = l./EPS

SL3 = l./(BCR + EPS) $ SL4 = l./(BCR - EPS)

IF (NQ.EQ.KMST) GO TO 911

ANQl = NST(NQ+1)

GAM = 0.5 * (BCR-EPS+(BCRCR+EPSPS)/ANQ1

)

GO TO 912

911 GAM = 0.5 * (BCR-EPS)

912 SL5 = l./(BCR-GAM)

GRIT = SL2 - SL5

904 Y = SLl

IF (NQ.EQ.IPI) GO TO 913

B = BCR + l./Y

FA = AIIM(DOM,B) * WFLD(B)/(Y * Y)

GO TO 914

913 FA = 0. 914 DY = (SL2 - SL1)/AN12

DO 917 J = 1,N12

Y=Y+DY $ Yl=l./Y $ B= BCR + Yl

917 FF(J) = Yl * Yl * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DY , N12 , FF , AIII , FA

)

AIRES = AIRES + AIII

Y = SL3 $ B = l./Y

FA = B * B * AIIM(DOM,B) * WFLD(B)

DY = (SL4 - SL3)/AN12

DO 927 J = 1,N12

Y=Y+DY $ B=l./Y

927 FF(J) = B * B * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DY , N12 , FF , AIII , FA

)

AIRES = AIRES + AIII

IF(CRIT.LE.O. ) GO TO 977

Y = SL5 $ B = BCR -l./Y

FA = AIIM(DOM,B) * WFLD(B)/(Y * Y)

DY = CRIT/AN12

DO 937 J = 1 ,N12

Y=Y+DY $ Yl=l./Y $ B= BCR - Yl

937 FF(J) = Yl * Yl * AIIM(DOM,B) * WFLD ( B

)

CALL WEDDLE ( DY , N12 , FF . AIII , FA

)

AIRES = AIRES + AIII

957 CONTINUE

IF(GAM.LT. 5. ) GO TO 968

Y = l./GAM $ DY = (0.2 - Y)/AN12

FA = GAM * GAM * AIIM ( DOM , GAM ) * WFLD (GAM)

DO 967 J = 1,N12

Y=Y+DY $ B=l./Y

967 FF(J) = B * B * AIIM ( DOM, B) * WFLD(B)

CALL WEDDLE ( DY , N12 , FF , AIII , FA

)

AIRES = AIRES + AIII

SL4 =0.2

GO TO 977

968 SL4 = l./GAM

977 B = 0. DB = l./(SL4 * AN30)

DO 947 J = 1,N30

B = B + DB

947 FF(J) = AIIM ( DOM, B) * WFLD(B)

CALL WEDDLE ( DB , N30 , FF , AIII , 0 .

)

AIRES = AIRES + AIII

980 WWBB = ( AIIM ( DOM, BAV) + FHOLTS ) /WING52

GO TO 990

985 AIRES = AIIM (DOM, BAV) + FHOLTS

WWBB = 0. 23
990 WING = AIRES/WING52

WINHOL = FH0LTS/WING52

WWOO = (AIIM(DOM, 0.) + FHOLTS ) /WING52

WWNG = (AWING + FHOLTS )/WING52

NTIME = -NTIME + KLOCK(O)

950 PRINT 978 , DOM , DLAM , AIRES , FHOLTS , WING52 , WING , WINHOL , WWOO , WWBB

1 WWNG, NTIME, NSTEP

978 FORMAT ( 10E12 . 4 , 110 , 16

GO TO 120

END

C

C

FUNCTION AIIM(DOM,B)

C

C CALCULATION OF I(DOM,B) FOR THE GENERAL CASE INCLUDING LOWER

C STATE INTERACTION

DIMENSION DRRF( 34) , DIIF(34), FMATR(34,34)

COMMON/FDAT/Pl , P2 , Bl , A2 , B2 , PPFF , CCREAL , CCIMAG

COMMON/1 1/SLOP ( 595 , 22 , 2

C0MM0N/22/DIP0L ( 34 , 2 ) , NUR ( 2 ), IPI , NCQ . BET , FPAR ( 6 , 30 )

, ADIFF ( 34 , 2 )

1 NSTEP

C0MM0N/33/AMATR ( 34 , 68 ) , BMATR ( 34 , 34 ) , CMATR ( 34 , 34

AIIM =0. $ NSTEP = NSTEP + 1 DO 800 MBL =1,2

AML = MBL $ NUPP = NUR (MBL)

DO 750 NRB = l.NUPP

DOMRB = DOM - BET * B * ADIFF ( NRB , MBL

DO 220 NRC = IPI, NCQ

PI = FPAR (1, NRC) $ P2 = FPAR (2, NRC) $ Bl = FPAR (3, NRC)

A2 = FPAR (4, NRC) $ B2 = FPAR (5, NRC) $ PPFF = FPAR (6, NRC)

CALL FOUTR( DOMRB)

DRRF(NRC) = CCREAL

220 DIIF(NRC) = CCIMAG

DO 700 NRA = l.NUPP

IF (NRA.LE.NRB) NRRAB = (NRB - 1) * NRB / 2 + NRA

IF (NRA.GT.NRB) NRRAB = (NRA - 1) * NRA / 2 + NRB

ARRR =0. $ AIII = 0. DO 600 NRC = IPI, NCQ

TEMP = SLOP (NRRAB, NRC, MBL)

ARRR = ARRR + TEMP * DRRF ( NRC

600 AIII = AIII + TEMP * DIIF(NRC)

AIII = AIII * 6.2831853072

AMATR(NRB,NRA) = AIII $ CMATR ( NRB , NRA ) = AIII

700 BMATR (NRB, NRA) = ARRR * 6.2831853072

750 BMATR ( NRB , NRB ) = BMATR ( NRB , NRB ) + DOMRB

CALL ENVERS (NUPP)

DO 320 NRB = 1 , NUPP

DO 320 NRA = 1 , NUPP

TEMP = 0. DO 300 NRC = l.NUPP

300 TEMP = TEMP + BMATR ( NRB , NRC ) * AMATR ( NRC , NRA+NUPP

320 FMATR ( NRB , NRA ) = TEMP

DO 420 NRB = 1 , NUPP

DO 420 NRA = 1,NUPP

TEMP = 0. DO 400 NRC = 1,NUPP

400 TEMP = TEMP + FMATR ( NRB , NRC ) * BMATR ( NRC , NRA

420 AMATR (NRB, NRA) = TEMP + CMATR ( NRB , NRA

CALL ENVERS (NUPP)

DO 795 NRB = 1,NUPP

DO 795 NRA = l.NUPP

795 AIIM = AIIM+AML*DIPOL(NRB,MBL)*DIPOL(NRA,MBL)*AMATR(NRB, NRA+NUPP)

800 CONTINUE

24
AIIM = AIIM * 0.3183099 $ RETURN

END

CC

SUBROUTINE POLYS ( X , Y , NUMX , XN , YN , XNI , DX , NXNUM

)

C A 5 POINT POLYNOMIAL INTERPOLATION ROUTINE

C FOR EACH POINT TO BE INTERP0LATED8 THE 5 NEAREST KNOWN POINTS

C ARE CHOSEN, AND A 5TH DEGREE POLYNOMIAL IS FITTED TO THESE POINTS

C X AND Y ARE THE ARRAYS OF KNOWN POINTS ON THE CURVE

C NUMX IS THE NUMBER OF KNOWN POINTS

C TWO NEW ARRAYS, XN AND YN WILL BE GENERATED FOR THE NXNUM VALUES

C OF X AND F(X) STARTING WITH X=XNI IN INCREMENTS OF DX

DIMENSION X(5) ,Y(5) ,XN(5) ,YN(5)

J=2

DO 8 1=1, NXNUM

EYE=I-1

XN(I)=XNI-fDX*EYE

IF (XN(I)-X(l) ) 1,2,3

2 YN(I)=Y(1)

GO TO 8

3 IF (XN(I) .LE.X(J) ) GO TO 4

IF (J.GE.NUMX) GO TO 7

J = J + 1 GO TO 3

1 L = 1 GO TO 70

7 L = NUMX - 4

GO TO 70

4 L = J - 1 IF(J.GT.2)L=J-2

IF(J.GT.3)L=J-3

IF ( ( J+1) .GT.NUMX) L = J - 4

70 YN(I)=0.0

LLL=L+4

DO 75 K=L,LLL

TERM=1 .

0

DO 74 M=L,LLL

IF (K.EQ.M) GO TO "^4 TDEN=X(K)-X(M)

TNUM=XN ( I

) -X ( M

)

TERM=TERM*TNUM/TDEN

74 CONTINUE

TERM=Y(K)*TERM

75 YN ( I

) =YN ( I ) +TERM

8 CONTINUE

RETURN

END

C

C

SUBROUTINE FOUTR (DOM)

CC FOURIERTRANSFORM OF THERMAL AVERAGE FOR UNIFIED THEORY

COMMON/FDAT/Pl , P2 , Bl , A2 , B2 , PPFF , CCREAL , CCIMAG

ARG = ABSF(DOM)

Z = Bl * ARG

IF (Z.LE. 0.001) GO TO 600

AR2 = ARG * ARG

IF ( Z.LE. 40. ) GO TO 300

FACl = -0.2992067103 * PI/ ( SQRT ( ARG ) * AR2

)

CC = FACl * ((1. - 1.3125/Z)*0.625/Z +1.)

SS = CC - FACl * 1.25/Z

GO TO 500

300 CALL BSJYOl (Z, AJO, YO , AJl, Yl

)

25
FACl = Y1/(2.*Z) + AJl - YO

FAC2 = AJO + Yl - AJ1/(2.*Z)

CINE = COSF(Z)

SINE = SINF(Z)

FCC = P2 * Bl * Bl

CC = FCC * (CINE*FACH-SINE*FAC2)

SS = FCC * (CINE*FAC2-SINE*FAC1)

IF (A2.EQ.0. ) GO TO 500

Z = B2 * ARC

IF (Z.GT. 10. ) GO TO 400

CALL BSJYOl (Z, AJO, YO , AJl, Yl

FACl = ( (AJ1-Y0)*16.*Z-36.*AJ0-28.*Y1)*Z+15.*Y0-3.*AJ1

FAC2 = ( (AJ0+Y1)*16.*Z-36.*Y0+28.*AJ1)*Z-15.*AJ0-3.*Y1

CINE = COSF(Z)

SINE = SINF(Z)

FCC = A2 * B2/6.

CC = CC + FCC * (CINE*FAC1+SINE*FAC2)

SS = SS + FCC * (CINE*FAC2-SINE*FAC1

GO TO 500

400 FACl = 0.1322319336 * A2 * B2 * Z**(-3.5)

FCC = FACl * (1. - (3.9375/Z + l.)*4.375/Z)

CC = CC + FCC

SS = SS - FCC - FAC1*8.75/Z

500 IF (DOM.LT.O.) SS = -SS

CCREAL = -AR2 * SS

CCIMAG = AR2 * CC

RETURN

600 CCREAL = 0.3183099 * (P2*B1-A2) * DOM

CCIMAG = -0.3183099 * P2

RETURN

END

C

C SUBROUTINE ENVERS (NN)

C C0MM0N/33/C ( 34 , 68 ) , BMATR ( 34 , 34 )

, CMATR ( 34 , 34

N = NN

IF (N.GT.l) GO TO 10

C(1.2) = l./C(l,l)

RETURN

10 DO 300 K = 1 ,N

KN = K + N NP = KN - 1 KPl = K + 1 TEMP = l./C(K,K)

DO 100 I = KP1,NP

100 C(K,I)=C(K,I)* TEMP

C(K,KN) = TEMP

DO 300 L = 1,N

IF (K.EQ.L) GO TO 300

TEMP = C(L,K)

DO 200 I = KP1,NP

200 C(L,I) =C(L,I) -C(K,I) * TEMP

C(L,KN) = -C(K,KN) * TEMP

300 CONTINUE

RETURN

END

CC FUNCTION RADMAT(N1 ,L1 ,N2,L2)

C DIMENSION F(2)

RADMAT =0.0

26
IF(IABS(L1 - L2).NE.l) RETURN

NA = Nl $ NB ^ N2 $ L = LI IF(L1.GT.L2) GO TO 10

NA = N2 $ NB = Nl $ L = L2

10 ANA = NA

DNB = NB $ AL = L IF(N1 .EQ.N2) GO TO 60

RADMAT = FCTRL(ANA + AL) * FCTRL(BNB + AL - 1 .

)

RADMAT = RADMAT/ (FCTRL( ANA - AL - 1 . ) * FCTRL(BNB - AL))

RADMAT = SQRT( RADMAT)

I = BNB + AL $ AL = 2. * AL

RADMAT = RADMAT * (

( -1 . 0

) **XMODF ( I , 2 ) ) / ( 4

. *FCTRL ( AL - 1.))

ABAB = (ANA - BNB) /(ANA + BNB)

AB2 = l./(ANA - BNB)**2

RADMAT = RADMAT * ABAB** ( NA+NB ) * ( 4

. *ANA*BNB*AB2 ) ** ( L+1

)

Z = -4. * ANA * BNB * AB2

NR = NA - L - 1 $ NRP = NB - L $ ANRP = NRP

DO 30 MM = 1,2

F(MM) = 1.0

IF(MM.EQ.2) NR = NR + 2 K = MINO (NR,NRP)

IF(K.EQ.O) GO TO 30

ANR = NR $ PROD =1.0

DO 40 M = 1,K

AM = M - 1 PROD = PROD * (AM-ANR)*(AM-ANRP)*Z/( (AL+AM)*(AM+1. )

)

40 F(MM) = F(MM) + PROD

30 CONTINUE

RADMAT = RADMAT * ( F ( 1 ) - ABAB * ABAB * F ( 2 )

)

RETURN

60 RADMAT = -1.5 * ANA * SQRT (

( ANA+AL ) * ( ANA-AL )

)

RETURN

END

FUNCTION FCTRL(A)

CALCULATION OF FACTORIALS

DIMENSION FCTI(20)

DATA ( (FCTI(I) ,1=1 ,20) =1.0,2.0,6.0,24.0,120.0,720.0,5040.0,

1 40320 . 0 , 362880 . 0 , 3628800 . 0 , 39916800 . 0 , 479001600 . 0

,

2 6227020800 . 0 , 87178291200 . 0 , 1307674368000 . 0

,

3 2.0922789888E13, 3 . 55687428096E14 , 6 . 402373705728E15

,

4 1.2164510040883 E17, 2.4329020081766 E18)

I = A + 0.1

IF(I) 50,60,70

50 FCTRL =0.0

RETURN

60 FCTRL =1.0

RETURN

70 IF (I.GT.20) GO TO 130

FCTRL=FCTI ( I

)

RETURN

130 F=20.0

FCTRL=FCTI (20)

DO 131 J=21,I

F=F+1 .

0

131 FCTRL=FCTRL*F

RETURN

END

FUNCTION S3J (FJl, FJ2 , FJ3 , FMl , FM2 , FM3

)

27
C CALCULATION OF 3J-SYMB0L. FJ1+FJ2+FJ3 HAS TO BE LESS THAN 38

DIMENSION GAM (39)

DATA ( (GAM( I) ,1=1.39) =1. ,1. ,2. ,6. ,24. ,120. ,720. ,5040. ,40320.

1 362880 . , 3628800 . , 39916800 . , 479001600 . , 6227020800 . , 87178291200 .

2 1307674368000 . , 2 . 0922789888E+13 , 3 . 5568742810E+14

3 6.4023737057E+15,1 .2164510041E+17,2.4329020082E+18,

4 5 . 1090942172E+19 , 1 . 1240007278E+21 , 2 . 5852016739E+22

5 6 . 2044840173E+23 , 1 . 5511210043E+25 , 4 . 0329146113E+26

6 1 . 0888869450E+28 , 3 . 0488834461E+29 , 8 . 8417619937E+30

7 2 . 6525285981E+32 , 8 . 2228386542E+33 , 2 . 6313083693E+35

8 8 . 6833176188E+36 , 2 . 9523279904E+38 , 1 . 0333147966E+40

9 3.7199332679E+41,1.3763753091E+43.5.2302261747E+44)

S3J=0 .

FJA = FJl+O.OOl $ FJB = FJ2+0.001 $ FJC = FJ3+0.001

IF (FJA.LT.O.O.OR.FJB.LT.O.O.OR.FJC.LT.0.0) RETURN

IF (AM0D(FJA,0.5) .GT.0.002) RETURN

IF (AMOD(FJB, 0.5) .GT.0.002) RETURN

IF (AMOD (FJC, 0.5) .GT.0.002) RETURN

FMA = FMl $ FMB = FM2 $ FMC = FM3

IF ( ABS ( FMA ) . GT . FJA . OR . ABS ( FMB ) . GT . FJB . OR . ABS ( FMC ) . GT . FJC ) RETURN

B = FJA-FMA $ P = FJB-FMB $ D = FJC-FMC $ G = FJA+FJB-FJC

IF ( (ABS(FMA+FMB+FMC)+AM0D(B,1. )+AMOD(P,l. )+AMOD(D,l. )+DIM(0. ,G)

1 +DIM(ABS(FJA-FJB) ,FJC) ) .GT.0.01) RETURN

IHH = FJA + FJB + FJC +2.

lAA = FJB+FJC+FMA+1 . $ ICC = -FJA+FJB+FJC+1

IFF = FJA-FJB+FJC+1 . $ IGG = G+1

JIM = B+1. $ J2M = P+1. $ J3M = D+1.

JIP = FJA+FMA+1. $ J2P = FJB+FMB+1. $ J3P = FJC+FMC+1

E2 = GAM(J1M)*GAM( J1P)*GAM(J2M)*GAM(J2P)

El = (GAM(ICC)*GAM(IFF)/GAM(IHH) )*GAM(IGG)*GAM(J3M)*GAM(J3P)

El = SQRT(E2/E1)

E = FJA - FJB + FMC

lAEE = ABS(E) + 0.001

lEE = lAEE +1 $ II = 0

IF (E.GE.O. ) GO TO 100

lEE = -lAEE +1 $ II = lAEE

100 12 = ICC - 1 IF (J3M.lt. ICC) 12 = J3M - 1 DO 150 1=11,12

E2 = GAM(H-1)*GAM(ICC-I)*GAM(J3M-I)/GAM(IAA-I)

150 S3J = S3J+( ( (-1. )**(I-(I/2)*2) )/E2)*GAM(J1M+I)/GAM(IEE+I)

I = ABS(FJA+FMB-FMC) +0.01

S3J = S3J * ( (-1. )**(I-(I/2)*2) )/El

RETURN

END

C

c SUBROUTINE WEDDLE (DX, N, F, A, FO

C

C INTEGRATION SUBROUTINE

DIMENSION F(N)

A = 0.0 $ K = N - 1 DO 15 I = 1,6

SUM =0.0

DO 6 J = I, K, 6

6 SUM = SUM + F(J)

GO TO (8, 10, 12, 10, 8, 14) , I 8A=A+5.0* SUM

GO TO 15

10 A = A + SUM

GO TO 15

12A=A+6.0* SUM

28
14

15

200

70

74

75

GO TO 15

A = A + 2,

CONTINUE

A = 0.3 * RETURN

END

SUM

DX ;a + Fo + f(n;

function wfld(b)

calculation of the ion microfield distribution function using a 5P0INT interpolation

COMMON/PFW/FIELD ( 301

WFLD =0.0

IF (B.LE.30.0) GO TO

SBS = l./(B * SQRT(B)

WFLD = ( (21 .6 * SBS + RETURN

FOR THE DATA READ INTO THE MAINPROGRAM

200

7.639) SBS + 1 .496) SBS/B

;b.le.

(B +

- 1 GT.2)

GT.3)

GT.300

L + 4

K = L

IF

J =

L =

IFIF

IF

LLL =

DO 75

AK = K - 1 TERM =1.0

DO 74 M = L IF (K.EQ.M)

AM = M - 1 TERM = TERM

CONTINUE

TERM = TERM

WFLD = WFLD

RETURN

END

RETURN

* 10.0

- 2 - 3

297

LLL

LLL

GO TO 74

(10. *B - AM)/(AK - AM)

FIELD (K)

TERM

FUNCTION COSINT(X)

CALCULATION OF THE COSINE INTEGRAL

TYPE DOUBLE Y2 , PROD , SUM , PT , DK

IF(X.LE.O. ) GO TO 50

X2

IF I = X * XX.GT.20,

Y2 = DBLE(X2

GO TO 30

I $ PROD -Y2 0.5 $ SUM = PROD * 0.5

10

20

30

DO 10 K = 2,50

DK = 2 * K PROD = -PROD * Y2/(DK*(DK - 1.))

SUM = SUM + PROD/DK

PT = ABS(PROD * l.D+10)

IF(ABS(SUM) .GT.PT) GO TO 20

CONTINUE

SS = SNGL(SUM)

COSINT = SS + 0.5772156649 + LOGF(X)

RETURN

FA = 1 . $ FB = 1 . $ PO = 1

.

X2 = 1./X2

DO 40 K = 1,10

AK = 2 * K $ PO = -PO * AK * X2 $

PO = PO * (AK + 1 . ) $ FB = FB + PO

IF(PA.LE.FB) GO TO 45

FA = FA + PO

S PA = ABS(PO * i.E+io:

29

406-848 0 - 71 -3
40 CONTINUE

45 FX = FA/X

GX = FB * X2

COSINT

RETURN

50 WRITE I 100 FORMAT

RETURN

END

= FX * SIN(X) GX C0S(X1

61,100) X

(* X LESS OR EQUAL TO ZERO, X = *E17.9'

10

20

50

60

SUBROUTINE BSJYOl (X, AJO, YO , AJl , Yl

YO, Jl, AND Yl FOR AN

(71

CALCULATION OF THE BESSEL FUNCTIONS JO,

ARGUMENT X

B(7), C(7), D(7), E(7), F(7), G(7), H(7) = 1,7) = 0.00021, -0.0039444, 0.0444479,

1.2656208, -2.2499997, 1.0) = 1,7) = -0.00024846, 0.00427916, -0.04261214,

-0.74350384, 0.60559366, 0.36746691)

= 1,7) = 0.00014476, -0.00072805, 0.00137237,

-0.0055274, -0.00000077, 0.79788456)

= 1,7) = 0.00013558, -0.00029333, -0.00054125.

-0.00003954, -0.04166397, -0.78539816)

= 1,7) = 0.00001109, -0.00031761, 0.00443319,

0.21093573, -0.56249985, 0.5) = 1,7) = 0.0027873, -0.0400976, 0.3123951,

2.1682709, 0.2212091, -0.6366198)

= 1,7) = -0.00020033, 0.00113653, -0.00249511,

0.01659667, 0.00000156, 0.79788456)

= 1,7) = -0.00029166, 0.00079824, 0.00074348,

DIMENSION A DATA ( (A(I) , I

1 -0.3163866,

DATA ( (B(I) , I

1 0.25300117,

DATA ( (C(I) , I

1 -0.00009512

DATA ( (D(I) , I

1 0.00262573.

DATA (

( E ( I ) , I

1 -0.03954289

DATA ( (F( I ) , I

1 -1.3164827,

DATA ( (G(I) , I

1 0.00017105,

DATA ( (H(I) , I

1 -0.00637879

AX = ABSF(X)

IF (AX.GT.0.0)

0.0000565, 0.12499612.

GO TO 10

AJO = 1 RETURN

IF (AX.GT.3.0

XX = (AX/3.0)

AJO = A(l) $

DO 20 M = AJO = AJO

YO = YO * AJl = AJl

Yl = Yl * AJl = AJl

YO = -l.E+30

GO TO

'* 2 YO = B

50

AJl = 0,

AJl = E(l

-2.35619449;

Yl = -l.E+30

Yl = F(i:

2,7

* XX + A(M)

XX + B(M)

* XX + E(M)

XX + F(M)

* X

ALF = 0.6366197724 * YO = YO + ALF * AJO

Yl = Yl/X + ALF * AJl

RETURN

X3 = 3.0/AX

L0GF(0.5 AXl

FO = C(l) $ THO = D(l) Fl = G(l

DO 60 M = 2,7

FO = FO * X3 + C(M)

THO = THO * X3 + D(M)

Fl = Fl * X3 + G(M)

THl = THl * X3 + H(M)

THO = THO + AX $ THl = THl + AX

XS = l./SQRT(AX)

AJO= XS * FO * COSF(THO) $ YO = XS

AJ1= XS * Fl * COSF(THl) $ Yl

RETURN

END

THl = H(l

XS

FO

Fl

SINF(THO;

SINFITHl

30
cc PROGRAM STBRHY

C

C PROGRAM FOR CALCULATING THE STARKBROADENING OF HYDROGEN ON THE

C BASIS OF THE UNIFIED THEORY WITHOUT LOWER STATE INTERACTION

DIMENSION DQQ ( 247 ) , PFAC ( 6 ) , STRONG ( 50 ) , FF ( 1100 )

, SJUU ( 136 , 31 )

,

1 SJLL (6,5), RDM (16,3), SLUU ( 16 ) , SLLL ( 3 ) , NST ( 50 ), DST ( 50 )

, W ( 300 , 5 )

,

2 DIP(16,3,20) ,FPAR(6,50)

COMMON/FDAT/Pl , P2 , Bl , A2 , B2 , PPFF , CCREAL , CCIMAG

COMMON/PFW/FIELD ( 301

)

COMMON/1 1/SLOP ( 16 , 16 , 60

)

C0MM0N/22/DIP0L ( 16 , 16 , 20 ), NSTEP , FPPP (6,15), NNUU , NNLL , BET

C0MM0N/33/AMATR ( 34 , 68 ) , BMATR ( 34 , 34 ) , CMATR ( 34 , 34

)

EQUIVALENCE (SJUU,AMATR)

FIELD(l) =0.0

READ 100, ((W(I,J), J = 1,5). I = 1,300)

100 FORMAT (5E12.4)

120 READ 150, DEN , TEMP , NNUU , NNLL , GIN , DGG , NTOT , NFAC , ( PFAC ( I ), 1=1 , 6

)

150 FORMAT ( 2E10 . 2 , 13 , 12 , 2F10 . 2 , 2I5/6F10 . 5

)

IF (EOF, 60) 577, 170

C CALCULATION OF MICROFIELD DISTRIBUTION FUNCTION

170 SHIELD = 0.0898 * DEN** ( 1 . /6 .) /SQRT ( TEMP

)

DO 161 J = 1,5

161 DST(J) = 0.2 * FLOAT (J-1)

DO 163 I = 1 ,300

DO 162 J = 1,5

162 DQQ(J) = W(I,J)

CALL P0LY5 ( DST , DQQ , 5 , STRONG , SLUU , SHIELD , 0 . 1 , 1

)

163 FIELD (1+1) = SLUU(l)

C CALCULATION OF AVERAGE lONFIELD

DO 111 I = 1,300

AM = I 111 FF(I) = FIELD (I+l) * AM/10.

CALL WEDDLE ( 0 . 1 , 300 , FF , BAV , 0 .

)

DY = 1./300. $ Y = 0. DO 113 I = 1,10

Y=Y+DY $ B=l./Y

113 FF(I) = WFLD(B) * B**3

CALL WEDDLE ( DY , 10 , FF , DB , 0 .

)

BAV = BAV + DB

C CALCULATION OF WAVELENGTH IN STANDARD AIR

178 AUU = NNUU

ALL = NNLL

SLl = 109678.758 * ( 1 . / ( ALL*ALL ) - 1 . / ( AUU*AUU )

)

SL2 = SLl * SLl

ALDD = 1. 000064328+2949810. /(1.46E+10-SL2) +25540. /( 4. 1E+9-SL2)

ALAM = l.E+8/(ALDD * SLl)

PRINT 180, DEN, TEMP, NNUU, NNLL, ALAM, SHIELD

180 FORMAT ( IHl , * DENSITY =* ElO . 2* TEMPERATURE =*E10.2,

1 * QUANTUMNUMBERS N UPPER =*I3* N LOWER =*I2* WAVELENGTH =*

2 F8.2* ANGSTR0M*/10X, *SHIELDING PARAMETER =*F6.3,10X,

3 * NO LOWER STATE INTERACTION *//)

IF (NNUU.LE.16.AND.NNLL.LE.3) GO TO 200

PRINT 190

190 FORMAT (* OVERFLOW OF MATRICES, PROGRAM NOT EXECUTED*)

577 CALL EXIT

C

C RADIAL MATRIXELEMENTS AND TOTAL LINESTRENGTH

200 IPI = M0D(NNUU-NNLL,2) + 1 STOT = 0. DO 330 KLL = 1 , NNLL

LLL = KLL - 1 $ ALL = LLL

31
DO 310 KUU = 1 ,NNUU

LUU = KUU - 1 $ AUU = LUU $ RRMM = 0. IF (IABS(KUU - KLL).NE.l) GO TO 310

RRMM = RADMAT ( NNUU , LUU , NNLL , LLL ) * S3J ( ALL , AUU ,1.,0.,0.,0.)

STOT = STOT + (2.*ALL+1. )*(2.*AUU+1. ) * RRMM * RRMM

310 RDM(KUU,KLL) = RRMM

330 CONTINUE

SQST = SQRT(STOT)

CC TRANSFORMATION MATRICES

DO 370 NKK =1,2

NN = NNLL

IF (NKK.EQ.2) NN = NNUU

ANN = NN $ ANl = 0.5 * (ANN - 1.)

KQQ = 2 * NN - 1 $ NRUN = 0 DO 370 KL = l.NN

AL = KL - 1 FACC = SQRT(2 . *AL+1 .

IF(NKK.EQ.l) SLLL(KL) = FACC

IF (NKK.EQ.2) SLUU(KL) = FACC

DO 370 KM = 1,KL

AM = KM - 1 $ NRUN = NRUN + 1 QQ = -ANN $ ALIM = ANN - 0.9 - AM

DO 370 KQ = l.KQQ

- QQ = QQ + 1. $ SIIJ =0. $ ABQQ = ABS(QQ)

IF (ABQQ. GT. ALIM) GO TO 360

NQ = ABQQ +0.1

IF (M0D(NN+NQ+KM+1,2) .NE.l) GO TO 360

AMQ = 0.5 * (AM - QQ) $ APQ = 0.5 * (AM + QQ)

SIIJ = FACC * S3J(AN1 , ANl ,AL,AMQ,APQ,-AM)

360 IF (NKK.EQ.l) SJLL ( NRUN , KQ ) = SIIJ

IF (NKK.EQ.2) SJUU ( NRUN , KQ ) = SIIJ

370 CONTINUE

CC DIPOLMATRIXELEMENTS IN PARABOLIC STATES

NQMAX = NNUU * (NNUU-1) + NNLL * (NNLL-1) + 1 DO 490 MM = 1 , NQMAX

490 DQQ(MM) = 0. MMAX = NNLL + 1 $ KQLL = 2 * NNLL - 1 DO 500 MM = l.MMAX

MMU = MM - 1 $ AMU = MMU $ AMLCK = 1. IF (MM.GT.l) AMLCK = 2. NAQU = -NNUU + MMU - 1 $ KQUU = NNUU - MMU

DO 500 KUU = 1 , KQUU

NAQU = NAQU + 2 $ KQU = NNUU + NAQU $ NAQL = -NNLL

DO 500 KLL = 1 , KQLL

NRUN = MMU * KQLL + KLL $ NAQL = NAQL + 1 $ KQL = NNLL + NAQL

NXDIF = NNUU * NAQU - NNLL * NAQL +1

MML = MMU - 2 S KMINL = NNLL - lABS(NAQL) - 1 DO 470 KAML =1,3

SIIJ =0. $ MML = MML + 1 IF (M0D(NNLL+NAQL+MML,2) .NE.l) GO TO 470

IF ( lABS ( MML ) .GT. KMINL) GO TO 470

AML = MML $ AMBL = AMU - AML

NPHAS = (NNUU + MMU - NAQU + NNLL - MML - NAQL - 2)/2

PHASE = (-1 . )**M0D(NPHAS,2)'

RESULT =0. $ KBBU = lABS(MMU) + 1 $ KBBL = lABS(MML) + 1 DO 450 KKLU = KBBU , NNUU

ALU = KKLU - 1 LLU = (KKLU-1) * KKLU / 2 + KBBU

FACl = SLUU(KKLU) * SJUU ( LLU , KQU

DO 450 KKLL = KBBL, NNLL

IF (IABS(KKLU - KKLL). NE.l) GO TO 450

32
ALL = KKLL - 1 LLL = (KKLL - 1) * KKLL / 2 + KBBL

FAC2 = SLLL(KKLL) * SJLL ( LLL , KQL

)

FACC = FAC2 * FACl * S3J ( ALL , ALU , 1 . , AML , -AMU , AMBL

)

RESULT = RESULT + FACC * RDM ( KKLU , KKLL

)

450 CONTINUE

SIIJ = RESULT * PHASE / SQST

PRINT 425, NAQU,MMU,NAQL,MML, SIIJ

425 FORMAT (16X , *

( * , 213 , * I D I*, 213,* ) =*E16.8)

IF (NXDIF.GT.O) DQQ(NXDIFl = DQQ(NXDIF) + AMLCK*RESULT*RESULT/STOT

470 DIP(KUU,KAML,NRUN) = SIIJ

500 CONTINUE

PRINT 393

393 FORMAT ( /

)

DO 440 MM= I.MMAX

MMU = MM - I $ KQUU = NNUU - MMU $ AMLCK = 1. IF (MMU.GT.O) AMLCK =2.

DO 440 KLL = 1 , KQLL

NRUN = MMU * KQLL + KLL

DO 440 NRA = 1 , KQUU

DO 440 NRB = 1 , KQUU

SIIJ = 0. DO 420 KAML =1,3

420 SIIJ = SIIJ + AMLCK*DIP (NRA, KAML, NRUN )*DIP( NRB, KAML, NRUN)

440 DIPOL (NRA, NRB , NRUN) = SIIJ

DST(l) = DQQ(l) $ NST(l) = 0 $ KMST = IPI - 1 DO 430 NXDIF = 2 , NQMAX

IF ( DQQ ( NXDIF ) .LT.1.E-20) GO TO 430

KMST = KMST + 1 $ DST ( KMST ) = DQQ (NXDIF) S NST(KMST) = NXDIF-1

430 CONTINUE

PRINT 355, (NST( I ) ,DST( I )

, I = 1 , KMST

)

355 FORMAT (16X,*X =*I3,F20.10)

PRINT 393

CC K OPERATOR MATRIX

NTIME = KLOCK(O) $ NCMI = NNUU - 1 DO 600 MM = l.MMAX

MMU = MM - 1 8 AMU = MMU

DO 600 NRC = 1 .NCMI

NRUN = MMU * NCMI + NRC $ KCQU = NNUU + NRC

NAQU = -NNUU + MMU - 1 $ KQUU = NNUU - MMU

DO 600 NRA = I , KQUU

NAQU = NAQU + 2 $ KAQU = NNUU + NAQU S NBQU

DO 600 NRB = I, NRA

NBQU = NBQU + 2 $ KBQU = NNUU + NBQU $ SIIJ

DO 540 JLAU = MM, NNUU

ALU = JLAU - 1 LALU = ( JLAU-1 ) *JLAU/2 + MM

FACl = SJUU( LALU, KAQU) * SJUU ( LALU , KBQU

)

FAC2 =0. $ KMINL = MINO ( JLAU , NNUU-NRC

)

DO 520 JMCU = 1 , KMINL

IF (M0D(NNUU+NRC+JMCU,2) .EQ.l) GO TO 520

LCLU = LALU - MM + JMCU $ AMC = 1

.

IF (JMCU.GT.l) AMC = 2. NPHAS = MM + JMCU + NRC - ( NAQU+NBQU ) /2

PHASE = (-1 .

) **MOD(NPHAS , 2

)

FAC2 = FAC2 + PHASE*AMC* ( SJUU { LCLU . KCQU )) **2

520 CONTINUE

540 SIIJ = SIIJ + FACl * FAC2/(2.*ALU+1.

I

SLOP (NRA, NRB, NRUN) = SIIJ

600 SLOP (NRB, NRA, NRUN) = SIIJ

NTIME = -NTIME + KLOCK(O)

PRINT 617, NTIME

= -NNUU + MMU - 1

= 0.

33
617 FORMAT ( 16X , *COMPUTERTIME FOR CALCULATING K-OPERATOR MATRIX =*I8/1

C

C BASIC CONSTANTS AND ARRAY FOR G-FUNCTION CONSTANTS

SDEN = SQRT(DEN)

FAC = 2064.936 * TEMP * SQRT ( TEMP/DEN

CFAC = 4.5645E-7 * SDEN/TEMP

DEBROG = 2 . 1027E-6/SQRT(TEMP)

ANN = NNUU

RMIN = DEBROG + ANN * ANN * 7 . 9376E-9

BET = 5.6558E-5 * DEN**(l./6.)

ASY = 0.0

DO 270 NRC = IPI.KMST

QC = NST(NRC) $ C = CFAC * QC

PI = -1.671086 * FAC * C * SQRT(C)

BS = 3. * QC * DEBROG / RMIN

STRONG ( NRC ) = 0 . 269-2 .*(((!. -COSF ( BS ) ) /BS+SINF ( BS ) ) /BS-COSINT ( BS )

PPFF = -1.128379 * FAC * C * C P2 = PPFF * (STR0NG(NRC)-2.*L0GF(2.*C)

FPAR(1,NRC) = PI $ FPAR(2,NRC) = P2

FPAR(3,NRC) = 0.5 * (P2/P1)**2 $ FIN = LOGF ( QC

A2 = P2*( (PFAC(3)*FIN+PFAC(2) )*FIN+PFAC(1)

B2 = (PFAC(6)*FIN+PFAC(5) )*FIN + PFAC(4)

IF (B2.LT.0. ) A2 = 0. FPAR(4,NRC) = A2 $ FPAR(5,NRC) = B2 $ FPAR(6,NRC) = PPFF

270 ASY ^ ASY +2. * PI * DST(NRC)

DO 275 NRC = l.NCMl

NCC = NRC * NNUU

DO 278 NRA = IPI.KMST

IF(NST(NRA) .EQ.NCC) GO TO 279

278 CONTINUE

279 FPPP(1,NRC) = FPAR(1,NRA) S FPPP(2,NRC) = FPAR(2,NRA)

FPPP(3,NRC) = FPAR(3,NRA) $ FPPP(4,NRC) = FPAR(4,NRA)

FPPP(5,NRC) = FPAR(5,NRA)

275 FPPP(6,NRC) = FPAR(6,NRA)

PRINT 220

220 FORMAT ( /13X*P1*18X*P2*18X*B1*18X*A2*18X*B2*17X*STR0NG*/

PRINT 240 , ( (FPAR(K, I ) ,K = 1 , 5 ) , STRONG ( I ) , I = IPI.KMST)

240 FORMAT (6E20.4)

PRINT 280, FAC, CFAC, BET, ASY, DEBROG, BAV , NFAC

280 FORMAT!/* FAC =*E11.4,* CFAC =*E11.4,* BET =*E11.4,* ASY =*E11.4,

1 * DEBROG =*E11.4,* BAV =*F7.4,* INTEGRATIONFACTOR =*I2//

2 5X*D0M*8X*DLAM*8X*IT0T*6X*IH0LTS*8X*ASY*10X*WING*7X*WH0LTS*7X,

3 *WW00*8X*WWBB*8X*WWNG*8X*TIME STEPS*/)

ADLFAC = 4.23538E-15 * SDEN * ALAM * ALAM

C

C CALCULATION OF THE lONFIELD INTEGRAL

N12 = 12 * NFAC $ AN12 = N12 $ N30 = 30 * NFAC $ AN30 = N30

DMCRT = BET * BAV * NST(KMST) * 5. G = GIN - DGG

DO 950 MM = 1 ,NTOT

NTIME = KLOCK(O)

NSTEP =0 $ G = G + DGG

DOM = 10. ** G $ DLAM = ADLFAC * DOM

WING52 = -0.2992067103 * ASY/ ( SQRT ( DOM ) * DOM * DOM)

FHOLTS =0. $ AWING = 0. DO 815 NRC = IPI.KMST

PI = FPAR(1,NRC) $ P2 = FPAR(2,NRC) $ Bl = FPAR(3,NRC)

A2 = FPAR(4,NRC) $ B2 = FPAR(5,NRC) $ PPFF = FPAR(6,NRC)

CALL FOUTR(DOM)

AWING = AWING + DST(NRC) * CCIMAG * 2./(D0M*D0M)

QC = NST(NRC) $ BETFAC = BET * QC $ BCRIT = DOM/BETFAC

815 FHOLTS = FHOLTS + DST(NRC) * WFLD( BCRIT) / BETFAC

IF ( DOM. GT. DMCRT) GO TO 985

34
AIRES = 0. IF(DOM.GT. (-3. *P2) ) GO TO 840

ANQl = NST(KMST)

BCRIT = (DOM - P2)/(ANQ1*BET)

DB = BCRIT/AN12 $ 8=0.

DO 820 J = 1,N12

B = B + DB

820 FF(J) = AIIM(D0M,B) * WFLD ( B

)

CALL WEDDLE ( DB , N12 . FF , AIII , 0 .

)

AIRES = AIII

DY = 1 ./(BCRIT*AN30) S Y = 0. DO 830 J = 1,N30

Y=Y+DY $ B=l./Y

830 FF(J) = B * B * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DY , N30 , FF , AIII . 0 .

)

AIRES = AIRES + AIII

GO TO 980

840 BCRCR = DOM/BET

EPSPS = -P2/BET

DO 957 NQ = IPI.KMST

ANQ = NST(NQ)

BCR = BCRCR/ANQ $ EPS = EPSPS/ANQ

IF (NQ.EQ.IPI) GO TO 907

SLl = l./(GAM - BCR)

GO TO 908

907 SLl = 0. 908 SL2 = l./EPS

SL3 = l./(BCR + EPS) $ SL4 = l./(BCR - EPS)

IF (NQ.EQ.KMST) GO TO 911

ANQl = NST(NQ+1)

GAM = 0.5 * (BCR-EPS+(BCRCR+EPSPS)/ANQ1)

GO TO 912

911 GAM = 0.5 * (BCR-EPS)

912 SL5 = l./(BCR-GAM)

CRIT = SL2 - SL5

904 Y = SLl

IF (NQ.EQ.IPI) GO TO 913

B = BCR + l./Y

FA = AIIM(DOM,B) * WFLD(B)/(Y * Y)

GO TO 914

913 FA = 0. 914 DY = (SL2 - SL1)/AN12

DO 917 J = 1,N12

Y=Y+DY % Yl=l./Y S B= BCR + Yl

917 FF(J) = Yl * Yl * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DY , N12 , FF , AIII , FA

)

AIRES = AIRES + AIII

Y = SL3 $ B = l./Y

FA = B * B * AIIM(DOM,B) * WFLD(B)

DY = (SL4 - SL3)/AN12

DO 927 J = 1,N12

Y=Y+DY $ B=l./Y

927 FF(J) = B * B * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE (DY , N12 , FF , AIII , FA

)

AIRES = AIRES + AIII

IF(CRIT.LE.O. ) GO TO 977

Y = SL5 $ B = BCR -l./Y

FA = AIIM(DOM,B) * WFLD(B)/(Y * Y)

DY = CRIT/AN12

DO 937 J = 1,N12

Y=Y+DY $ Yl=l./Y $ B= BCR - Yl

937 FF(J) = Yl * Yl * AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DY , N12 , FF , AIII . FA

)

35
957

220

AIRES = AIRES + AIII

CONTINUE

IF(GAM.LT.

Y = 1,/GAM

FA = GAM *

5.

967

968

977

947

980

985

990

950

978

GO TO 968

$ DY = (0.2 - Y)/AN12

GAM * AIIM(DOM,GAM) * WFLD(GAM)

DO 967 J = 1,N12

Y=Y+DY $ B-l./Y

FF(J) = B * B * AIIM(D0M,B; WFLD ( BB CALL WEDDLE ( DY , N12 , FF , AIII , FA

AIRES = AIRES + AIII

SL4 =0.2

GO TO 977

SL4 = l./GAM

B = 0. DB = 1./{SL4 * AN30)

DO 947 J = 1,N30

B = B + DB

FF(J) = AIIM(DOM,B) * WFLD(B)

CALL WEDDLE ( DB . N30 , FF , AIII , 0 .

AIRES = AIRES + AIII

WWBB = (AIIM(DOM,BAV) + FHOLTS ) /WING52

GO TO 990

AIRES = AIIM(DOM,BAV) + FHOLTS

WWBB = 0. WING = AIRES/WING52

WINHOL = FH0LTS/WING52

WWOO = (AIIM(DOM, 0.) + FHOLTS ) /WING52

WWNG = (AWING + FHOLTS )/WING52

NTIME = -NTIME + KLOCK(O)

PRINT 978 , DOM , DLAM . AIRES , FHOLTS , WING52 , WING , WINHOL , WWOO , WWBB

1 WWNG, NTIME, NSTEP

FORMAT (10E12.4,I10,I6)

GO TO 120

END

FUNCTION AIIM(DOM,B)

CALCULATION OF I(DOM,B) FOR THE CASE OF NO LOWER STATE INTERACTION

DIMENSION DRRF( 15) , DIIF(15), FMATR(15,15)

COMMON/FDAT/Pl , P2 , Bl , A2 , B2 , PPFF , CCREAL , CCIMAG

COMMON/1 1/SLOP ( 16 , 16 , 60

C0MM0N/22/DIP0L ( 16 , 16 , 20 ), NSTEP , FPPP (6,15), NNUU , NNLL , BET

C0MM0N/33/AMATR ( 34 , 68 ) , BMATR ( 34 . 34 ) , CMATR ( 34 , 34

AIIM =0. $ NSTEP = NSTEP + 1 MMAX = NNLL + 1 $ KQLL = 2 * NNLL - 1 $ NCMl = NNUU - 1 DO 800 MM = l.MMAX

MMU = MM - 1 $ NUPP = NNUU - MMU $ NQL = -NNLL' DO 800 KLL = l.KQLL

NQL = NQL + 1 NRUN = MMU * KQLL + KLL $ NBQU

DO 750 NRB = 1,NUPP

NBQU = NBQU + 2 DOMRB = DOM - BET

DO 220 NRC = l.NCMl

PI = FPPP (1, NRC) $

A2 = FPPP (4, NRC) $

F0UTR( DOMRB)

NRC) = CCREAL

NRC) = CCIMAG

DO 700 NRA = 1 ,NUPP

ARRR =0. $ AIII =

DO 600 NRC = 1 ,NCM1

-NNUU + MMU - 1

AXDIF = NNUU * NBQU - NNLL * NQL

B

P2

B2

AXDIF

= FPPP

= FPPP

2, NRC) $ Bl = FPPP (3, NRC)

5, NRC) $ PPFF = FPPP (6, NRC

CALL

DRRF

DIIF

36
MQC = MMU * NCMl + NRC

TEMP = SLOP (NRB.NRA, MQC)

ARRR = ARRR + TEMP * DRRF{NRC)

600 AIII = AIII + TEMP * DIIF{NRC)

AIII = AIII * 6.2831853072

AMATR(NRB,NRA) = AIII S CMATR { NRB , NRA ) = AIII

700 BMATR(NRB,NRA) = ARRR * 6.2831853072

750 BMATR(NRB,NRB) = BMATR ( NRB , NRB ) + DOMRB

CALL ENVERS ( NUPP

)

DO 320 NRB = l.NUPP

DO 320 NRA = 1,NUPP

TEMP = 0

.

DO 300 NRC = 1 , NUPP

300 TEMP = TEMP + BMATR ( NRB , NRC ) * AMATR ( NRC , NRA+NUPP

)

320 FMATR(NRB ,NRA ) = TEMP

DO 420 NRB = 1 , NUPP

DO 420 NRA = l.NUPP

TEMP = 0. DO 400 NRC = l.NUPP

400 TEMP = TEMP + FMATR ( NRB , NRC ) * BMATR ( NRC , NRA

)

420 AMATR ( NRB , NRA ) = TEMP + CMATR (NRB ,NRA)

CALL ENVERS ( NUPP

)

DO 795 NRB = 1 ,NUPP

DO 795 NRA = 1 ,NUPP

795 AIIM = AIIM + DIP0L(NRB ,NRA .NRUN) *AMATR ( NRB , NRA+NUPP

)

ouu CONTINUE

AIIM = AIIM * 0.3183099 $ RETURN

c

c

END

LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

ro/D = 0 =0.2 =0.4 =0.6 =0.8 lOjS 6.9547-003 1.1584-002 1.9378-002 3.3258-002 11 1.6670--002 2.7217-002 4.4735-002 7.3201-002 1.2093-001 2

3.6640-002 5.9060-002 9.5010-002 1.5018-001 2.3439-001 3 6.3080--002 9.9852-002 1.5606-001 2.3598-001 3.4416-001 4

9.4600--002 1.4638-001 2.2090-001 3.1741-001 4.3106-001 5 1.2959- 001 1.9523-001 2.8304-001 3.8512-001 4.8799-001 6

1.6636- 001 2.4312-001 3.3742-001 4.3436-001 5.1632-001 7

2.0323- 001 2.8721-001 3.8073-001 4.6430-001 5.2143-001 8

2.3864- 001 3.2532-001 4.1151-001 4.7674-001 5.0966-001 q 2.7122- 001 3.5597-001 4.2978-001 4.7485-001 4.8675-001 10

2.9987- 001 3.7845-001 4.3663-001 4.6214-001 4.5729-001 11 3.2378- 001 3.9268-001 4.3381-001 4.4193-001 4.2466-001 12 3.4246- 001 3.9910-001 4.2331-001 4.1702-001 3.9118-001 13 3.5570- 001 3.9855-001 4.0710-001 3.8961-001 3.5839-001 14 3.6357- 001 3.9206-001 3.8700-001 3.6133-001 3.2718-001 15 3.6633- 001 3.8077-001 3.6450-001 3.3333-001 2.9807-001 16 3.6445- 001 3.6584-001 3.4084-001 3.0635-001 2.7128-001 17 3.5850- 001 3.4831-001 3.1694-001 2.8086-001 2.4685-001 18 3.4911- 001 3.2911-001 2.9347-001 2.5712-001 2.2470-001 19 3.3694- 001 3.0904-001 2.7090-001 2.3522-001 2.0473-001 20

3.2265- 001 2.8872-001 2.4953-001 2.1516-001 1.8675-001 21 3.0684- 001 2.6865-001 2.2952-001 1.9689-001 1.7058-001 22

2.9005- 001 2.4918-001 2.1096-001 1.8031-001 1.5607-001 23 2.7275- 001 2.3058-001 1.9385-001 1.6529-001 1.4303-001 24

2.5537- 001 2.1300-001 1.7816-001 1.5170-001 1.3131-001 25

37
LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

ro/D = 0 = 0.2 =0.4 = 0.6 = 0.8 10/3 Z.JOZZ — UUl i.7Dj4 — UUl 1 ^ifiT nni l.OJoZ — UUl 11.3y4Z 10/17 — nmUUl 1 7n77 Am l.ZU/ / — UUl 26

2.2156-001 1.8124-001 1.5075-001 1.2832-001 1l.«J.J.X*^ 1129-001 \J\J 1. 27

2.0557-001 1.6709-001 1.3885-001 1 1829-001 1 0273-001 98

1.9037-001 1.5406-001 1.2804-001 1.0922-001 9 5010-002 79

1.7606-001 1.4211-001 1.1821-001 1.0100-001 8 8028-002 30

1 .OZOo — UUl 1 1 1 nni 1 .j1 id — UUl 1 noofi nni i.uyzo — UUl oy. i^/^n j3oU — nmUUZ Q 1 C(\C AAO 0.1 oVo "UUZ 31 1.5024-001 1.2117-001 1.0117-001 8.6805-002 7 5951-002 32

1.3873-001 1.1204-001 9.3793-002 8.0667-002 1 0731 -002 33

1.2812-001 1.0371-001 8.7081-002 7.5082-002 6 5969-002 34

1.1837-001 9.6123-002 8.0967-002 6.9993-002 6 1626-002 35

1 no/n nni l.uy^z — UUl fio.yzui Qoni —UUZ nm 7 ?iQ7 nm (,D.j34o ^iA<i — nmUUZ C ICO AA"! 36

1.0124-001 8.2886-002 7.0303-002 6 1103-002 5 4030-002 37

9.3751-002 7.7122-002 6.5651-002 5.7216-002 5 *\J0700-002 1 \J\J \J\J 38

8.6913-002 7.1857-002 6.1393-002 5.3652-002 4 7650-002 39

8.0670-002 6.7044-002 5.7491-002 5 0378-002 4 4839-002 40

/.4yoy — uuz o.zojy — UUZ j.jyiu— UUZ 4.A Txcc /3dd — nmUUZ A '^^AC AA'^ 4.ZZ4D — UUZ A 1 41 6.9762-002 5.8603-002 5.0620-002 4.4592-002 3 » 9859-002 y \j ^ 42

6.5004-002 5.4901-002 4.7591-002 4.2033-002 3 7647-002 43

6.0652-002 5.1502-002 4.4801-002 3.9668-002 3 5594-002 44

5.6670-002 4.8376-002 4.2226-002 3.7480-002 3 3696-002 45

c -3ATJ r\m j.iyjlj — \j\)l 4. j4yo — UUZ 1 QQ/17 nn7 j.yo4 / — UUZ 3.j4j4 — UUZ 1 ion nn7 3.iy3Z — UUZ 46

4.9678-002 4.2844-002 3.7647-002 3.3574-002 3 0289-002 47

4.6606-002 4.0395-002 3.5608-002 3 1828-002 2 8766-002 48

4.3781-002 3.8130-002 3.3716-002 3.0204-002 2 7348-002 49

4.1180-002 3.6034-002 3 1959-002 2 8692-002 2^ • 6021-002 V/ \J ^ X \J\J ^ 50

3.8783-002 T /JAO'^ AAO 3.4092-002 O A10C AAO 3.0325-002 O TlOO AAO 2.7282-002 2.4785-002 51

7 fiSTI —009 X 998Q — 009 9 880"?- 009 9z.j"uj ^o^;S — 009 UUZ 9Z. JUJ 'l^i'll1 —009 UUZ S9jZ

1 Ofil ^-009 9Z.. 7';8^— / JOJ 007 UUZ 9Z.t47^4-009 / UUZ 9Z.ZJtj 9S4^ — 009 UUZ

"7640 — 00? 9 Q0S4 — 009 7z.oujo ^;0^8 — 007 UUZ 7Z.-5J0Z 1^89 — 009 UUZ Z.9 11 JZS97/ — 009 UUZ j4

J 0890-007 9it, 7601 1 UUi -009 \j\J 9Z.*t01>' 4810-007 UUZ 9Z.ZJUZ 7507 — 007 UUZ 9Z.U_/ 0S78/ O — 009 UUZ J J z.yzoo— UUZ 9Z.OA'i ^;9471 — 009 UUZ 9Z.jOj^ "^^i^Q — 007 UUZ 9 1480 nn9 Z. 1407 UUZ 1 QAisi nn9 1 .7001 UUZ JO

2.7756-002 2.4981-002 2.2573-002 2.0538-002 1.8839-002 57

2.6347-002 2.3798-002 2.1554-002 1.9644-002 1.8052-002 58

2.5030-002 2.2690-002 2.0597-002 1.8803-002 1.7302-002 59

2.3800-002 2.1652-002 1.9697-002 1.8012-002 1.6593-002 60

Z.ZOJ J — UUZ 9 n*;78 nn9 Z.UO /o UUZ 11 .oojU 88^n — nmUUZ 1 77^A nn7 1. / zoo — UUZ 1 ^Q7fi nn7 i.jyzo — UUZ 1 01 2.1588-002 1.9764-002 1.8053-002 1.6562-002 1.5294-002 62

2.0592-002 1.8904-002 1.7302-002 1.5898-002 1.4693-002 63

1.9662-002 1.8095-002 1.6593-002 1.5270-002 1.4129-002 64

1.8790-002 1.7332-002 1.5923-002 1.4676-002 1.3592-002 65

1. /yby— UUZ 1 A nn') 1.0D14 — UUZ 1 ^7Qn nn7 1 . jzyu — UUZ 1 A\\i nn7 1 .41 13 — UUZ 1 in7Q nn7 i.jU/y — UUZ oo

1 7197-002 1.5935-002 1.4692-002 1.3580-002 1.2599-002 67

1.6470-002 1.5295-002 1.4125-002 1.3074-002 1.2142-002 68

1.5786-002 1.4689-002 1.3588-002 1.2594-002 1.1704-002 69

1.5140-002 1.4116-002 1.3079-002 1.2137-002 1.1294-002 70

1.4j31 — UUZ 1 1^71 nm 11 .Zjyo 7<Q/; — nmUUZ 1 1 7ni nn7 1.1 /U3 — UUZ 1 nom nn7 7/ 1 1 3956-002 1.3059-002 1 2138-002 1 1290-002 1.0525-002 72

1.3413-002 1.2571-002 1 1702-002 1 0897-002 1.0168-002 73

1 2898-002 1 2108-002 1 1288-002 1X t\J0523-002 ^ .J \J\J 9 8296-003 74

1 2410-002 1.1669-002 1 0894-002 1X'V/l-V/ 0167-002 \J\J 9.5003-003 75

1.1947-002 1.1251-002 1.0519-002 9.8275-003 9.1900-003 76

1.1508-002 1.0854-002 1.0161-002 9.5035-003 8.8965-003 77

1.1092-002 1.0477-002 9.8208-003 9.1941-003 8.6096-003 78

1.0696-002 1.0117-002 9.4959-003 8.8986-003 8.3371-003 79

1.0320-002 9.7745-003 9.1858-003 8.6162-003 8.0805-003 80

38
LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

ro/D = 0 =0.2 =0.4 =0.6 =0.8 10/3 9.9640-003 9.4481-003 8.8896 -003 8.3461 -003 7.8286 -003 81 9.6250-003 9.1370-003 8.6066 -003 8.0876 -003 7.5880 -003 82

9.3020-003 8.8401-003 8.3360 -003 7.8402 -003 7.3633 -003 83

8.9940-003 8.5568-003 8.0772 -003 7.6032 -003 7.1431 -003 84

8.7000-003 8.2863-003 7.8295 -003 7.3763 -003 6.9313 -003 85

8.4180-003 8.0279-003 7.5925 -003 7.1588 -003 6.7349 -003 86

8.1490-003 7.7809-003 7.3654 -003 6.9504 -003 6.5428 -003 87

7.8920-003 7.5447-003 7.1480 -003 6.7505 -003 6.3557 -003 88

7.6450-003 7.3188-003 6.9395 -003 6.5589 -003 6.1826 -003 89

7.4100-003 7.1025-003 6.7397 -003 6.3749 -003 6.0139 -003 90

7.1850-003 6.8955-003 6.5481 -003 6.1982 -003 5.8473 -003 91 6.9700-003 6.6971-003 6.3641 -003 6.0284 -003 5.6932 -003 92

6.7640-003 6.5070-003 6.1876 -003 5.8652 -003 5.5444 -003 93

6.5680-003 6.3246-003 6.0179 -003 5.7081 -003 5.3958 -003 94

6.3800-003 6.1496-003 5.8548 -003 5.5569 -003 5.2582 -003 95

6.2000-003 5.9814-003 5.6979 -003 5.4113 -003 5.1267 -003 96

6.0270-003 5.8199-003 5.5469 -003 5.2708 -003 4.9939 -003 97

5.8600-003 5.6645-003 5.4014 -003 5.1353 -003 4.8699 -003 98

5.6980-003 5.5149-003 5.2611 -003 5.0046 -003 4.7530 -003 99

5.5400-003 5.3709-003 5.1258 -003 4.8783 -003 4.6335 -003 100

5.4133-003 5.2320-003 4.9952 -003 4.7563 -003 4.5206 -003 101 5.2692-003 5.0980-003 4.8690 -003 4.6382 -003 4.4154 -003 102

5.1306-003 4.9686-003 4.7470 -003 4.5240 -003 4.3074 -003 103

4.9970-003 4.8435-003 4.6290 -003 4.4133 -003 4.2035 -003 104

4.8682-003 4.7227-003 4.5148 -003 4.3062 -003 4.1081 -003 105

4.7441-003 4.6057-003 4.4043 -003 4.2024 -003 4.0101 -003 106

4.6244-003 4.4925-003 4.2973 -003 4.1017 -003 3.9140 -003 107

4.5090-003 4.3828-003 4.1936 -003 4.0042 -003 3.8269 -003 108

4.3976-003 4.2765-003 4.0932 -003 3.9097 -003 3.7380 -003 109

4.2900-003 4.1736-003 3.9960 -003 3.8181 -003 3.6490 -003 110

4.1861-003 4.0737-003 3.9018--003 3.7294--003 3.5690--003 111 4.0857-003 3.9770-003 3.8106--003 3.6435--003 3.4886--003 112

3.9887-003 3.8832-003 3.7224--003 3.5603--003 3.4065--003 113

3.8948-003 3.7923-003 3.6370--003 3.4800--003 3.3331--003 114

3.8041-003 3.7041-003 3.5545--003 3.4024--003 3.2610--003 115

3.7164-003 3.6188-003 3.4747--003 3.3274--003 3.1860--003 116

3.6314-003 3.5361-003 3.3977--003 3.2551--003 3.1188--003 117

3.5492-003 3.4560-003 3.3233--003 3.1854--003 3.0549--003 118

3.4696-003 3.3785-003 3.2515--003 3.1182--003 2.9870--003 119

3.3925-003 3.3034-003 3.1822--003 3.0534--003 2.9259--003 120

3.3177-003 3.2307-003 3.1153--003 2.9911--003 2.8694--003 121 3.2453-003 3.1603-003 3.0507--003 2.9309--003 2.8086--003 122

3.1751-003 3.0922-003 2.9883--003 2.8728--003 2.7527--003 123

3.1070-003 3.0262-003 2.9279--003 2.8167--003 2.7027--003 124

3.0410-003 2.9623-003 2.8694--003 2.7623--003 2.6481--003 125

2.9769-003 2.9003-003 2.8127--003 2.7096--003 2.5961--003 126

2.9147-003 2.8402-003 2.7576--003 2.6583--003 2.5509--003 127

2.8543-003 2.7819-003 2.7040--003 2.6083--003 2.5013--003 128

2.7957-003 2.7253-003 2.6519--003 2.5594--003 2.4521--003 129

2.7387-003 2.6702-003 2.6009--003 2.5116--003 2.4100--003 130

2 6834-003 2 6167-003 2 551?--003 7 4646--003 2.3643--003 131

2.6296-003 2.5647-003 2.5025--003 2.4185--003 23170--003 132

2.5774-003 2.5141-003 2.4549--003 2.3732--003 2.2769--003 133

2.5266-003 2.4648-003 2.4082--003 2.3286--003 2.2346--003 134

2.4772-003 2.4169-003 2.3625--003 2.2848--003 2.1890--003 135

39
LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

r„/D = 0 =0.2 = 0.4 = 0.6 =0.8 10/3 2.4291- 003 2.3703 -003 2.3178-003 2.2417 -003 2.1505-003 136

2.3824- 003 2.3249 -003 2.2740-003 2.1994 -003 2.1115-003 137

2.3369- 003 2.2807 -003 2.2311-003 2.1578 -003 2.0682-003 138

2.2926- 003 2.2378 -003 2.1892-003 2.1171 -003 2.0313-003 139

2.2495- 003 2.1961 -003 2.1482-003 2.0773 -003 1.9960-003 140

2.2075- 003 2.1555 -003 2.1083-003 2.0384 -003 1.9555-003 141 2.1667- 003 2.1162 -003 2.0694-003 2.0004 -003 1.9205-003 142

2.1268- 003 -003 2.0314-003 1.9634 -003 1.8890-003 143

2.0880- 003 2.0408 -003 1.9945-003 1.9274 -003 1.8519-003 144

2.0502- 003 2.0048 -003 1.9587-003 1.8924 -003 1.8189-003 145

2.0134- 003 1.9698 -003 1.9238-003 1.8585 -003 1.7909-003 146

1.9775- 003 1.9358 -003 1.8899-003 1.8255 -003 1.7575-003 147

1.9424- 003 1.9029 -003 1.8571-003 1.7935 -003 1.7264-003 148

1.9083- 003 1.8708 -003 1.8251-003 1.7625 -003 1.7015-003 149

1.8750- 003 1.8397 -003 1.7941-003 1.7325 -003 1.6717-003 150

1.8425- 003 1.8094 -003 1.7640-003 1.7033 -003 1.6424-003 151

1.8108- 003 1.7799 -003 1.7347-003 1.6751 -003 1.6199-003 152

1.7798- 003 1.7511 -003 1.7062-003 1.6476 -003 1.5936-003 153

1.7496- 003 1.7230 -003 1.6785-003 1.6209 -003 1.5657-003 154

1.7201- 003 1.6955 -003 1.6515-003 1.5951 -003 1.5449-003 155

1.6913- 003 1.6687 -003 1.6252-003 1.5699 -003 1.5216-003 156

1.6632- 003 1.6424 -003 1.5995-003 1.5453 -003 1.4949-003 157

1.6357- 003 1.6166 -003 1.5744-003 1.5214 -003 1.4750-003 158

1.6089- 003 1.5913 -003 1.5498-003 1.4981 -003 1.4541-003 159

1.5826- 003 1.5665 -003 1.5257-003 1.4753 -003 1.4284-003 160

1.5570- 003 1.5421 -003 1.5021-003 1.4530 -003 1.4089-003 161

1.5320- 003 1.5181 -003 1.4789-003 1.4312 -003 1.3903-003 162

1.5075- 003 1.4945 -003 1.4562-003 1.4098 -003 1.3660-003 163

1.4836- 003 1.4713 -003 1.4338-003 1.3888 -003 1.3467-003 164

1.4602- 003 1.4485 -003 1.4118-003 1.3682 -003 1.3301-003 165

1.4373--003 1.4260 -003 1.3903-003 1.3479 -003 1.3072-003 166

1.4149--003 1.4039 -003 1.3691-003 1.3280 -003 1.2877-003 167

1.3930--003 1.3822 -003 i O AC%^ AA-^ 1.3482-003 1.3084 -003 1.2722-003 168

1.3716- 003 1.3608 -003 \ 'I'^TA AA'^ 1.3278-003 1.2892 -003 1.2506-003 169

1.3506- 003 1.3398 -003 1.3078-003 1.2703 -003 1.2307-003 170

1.3301- 003 1.3192 -003 1.2881-003 1.2518 -003 1.2162-003 171 1.3100- 003 1.2990 -003 i ^^C%C\ AA*^ 1.2688-003 1.2336 -003 1.1963-003 172

1.2904- 003 1.2791 -003 1.2499-003 1.2158 -003 1.1766-003 173

1.2711- 003 1.2597 -003 1 -X A AA-^ 1.2314-003 1.1983 -003 1.1633-003 174

1.2523- 003 1.2406 -003 1.2132-003 1.1812 -003 1.1459-003 175

1.2339- 003 1.2219 -003 1.1953-003 1.1643 -003 ,1.1266-003 176

1.2158- 003 1.2036 -003 1.1778-003 1.1478--003 1.1141-003 177

1.1981 - 003 1.1857--003 1.1607-003 1.1315--003 1.0987-003 178

1.1808- 003 1.1681 -003 1.1438-003 1.1155--003 1.0798-003 179

1.1638- 003 1.1510 -003 1.1273-003 1.0997 -003 1.0677-003 180

1.1471- 003 1.1342--003 1.1111-003 1.0843--003 1.0546-003 181

1.1308- 003 1.1178--003 1.0952-003 1.0691--003 1.0367-003 182

1.1148- 003 1.1017 -003 1.0796-003 1.0542--003 1.0250-003 183

1.0992- 003 1.0860--003 1.0643-003 1.0396--003 1.0137-003 184

1.0838- 003 1.0707--003 1.0494-003 1.0253--003 9.9633-004 185

1.0687- 003 1.0558--003 1.0347-003 1.0114--003 9.8381-004 186

1.0540- 003 1.0411 -003 1.0202-003 9.9779--004 9.7351-004 187

1.0395- 003 1.0268--003 1.0061-003 9.8441--004 9.5693-004 188

1.0253- 003 1.0128--003 9.9230-004 9.7131--004 9.4391-004 189

1.0113- 003 9.9909--004 9.7875-004 9.5849--004 9.3483-004 190

40
LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

ro/D = 0 =0.2 9.9766-•004 9.8567--004

9.8425-004 9.7253--004

9.7109-004 9.5966--004

9.5817-004 9.4703- 004

9.4550-004 9.3465--004

9.3306- 004 9.2251--004

9.2085- 004 9.1060-004

9.0887- 004 O AO A-^ 8.9892--004

8.9710- 004 8.8745- AA ,* -004

8.8554- 004 8.7620--004

8.7420- 004 8.6516- 004

8.6305- 004 8.5433--004

8.521 1 - 004 8.4370- AA A •004

8.4136- 004 8.3328--004

8.3080- 004 8.2306- 004

8.2042- 004 8.1304- 004

8.1023- 004 8.0321- 004

8.0021 - 004 7.9359- 004

7.9037- 004 7.8417- 004

7.8069- 004 7.7495- 004

7.7118- 004 7.6593- 004

7.6184- 004 7.5711- 004

7.5265- 004 7.4850- 004

7.4361- 004 7.4008- 004

7.3473- 004 7.3186- 004

7.2600- 004 7.2384- 004

7.1741- 004 7.1600- 004

7.0896- 004 7.0834- 004

7.0066- 004 7.0084- 004

6.9249- 004 6.9351- 004

6.8445--004 6.8631- 004

6.7654-004 6.7925- 004

6.6876- f\(\ A 004 0. 1 Lvi- UU4

^ 1 1 1 6.611 1 - 004 0.dj43-

6.5358- AA A 004 6.5o64- AA A 004

6.4617- 004 6.5191- 004

6.3887- 004 6.4523- 004

6.3169- AA A 004 r -5 o CO AA A 004

6.2463- AA A 004 a 1A ClA o.jiy4- UU4

6.1767- AA A 004 6.2530- AA A 004

6.1082- 004 6.1867- 004

6.0408- 004 6.1202- 004

5.9745- AA A 004 £ AC "5 ^ 6.0536- AA A 004

5.9091 - AA A 004 5.9o70- AA A 004

C O A AO AA A 004 C AO A^ 5.9203- UU4

5.7814- 004 5.8536- 004

5.7190- 004 5.7871- 004

5.6575- 004 5.7209- 004

5.5970- AA A 004 5.6551 - AA A 004

5.5373- AA A 004 C C O AO 5.5898- AA A 004

5.4786- 004 5.5253- 004

5.4207- 004 5.4617- 004

5.3637- 004 5.3991- 004

5.3075- 004 5.3377- 004

5.2522- 004 5.2775- 004

=0.4 =0.6 9.6548 -004 9.4582--004

9.5249 -004 9.3337--004

9.3977 -004 9.2114--004

9.2732 -004 9.0905--004

9.1513 -004 8.9714--004

9.0321 -004 8.8547--004

8.9154 -004 8.7397--004

8.8012 -004 8.6268--004

O /' CiC\C 8.6895 -004 8.5167--004

8.5801 -004 8.4087--004

8.4729 -004 8.3025--004

8.3680 -004 8.1991--004

8.2653 -004 8.0975--004

8.1645 -004 7.9974--004

8.0657 -004 7.8996-004

7.9688 -004 7.8035-004

7.8738 -004 7.7086-004

7.7805 -004 7.6159-004

7.6890 -004 7.5251- 004

7.5993 -004 7.4357-004

7.5114 -004 7.3485- 004

7.4254--004 7.2636- 004

7.3413 -004 7.1801 - 004

7.2591 -004 7.0988- 004

7.1787--004 7.0196- 004

7.1001 -004 6.9417- 004

7.0233 -004 6.8654- 004

6.9480 -004 6.7910- 004

6.8741 -004 6.7174- 004

6.8016 -004 6.6447- 004

6.7301 -004 6.5735-004

6.6596 -004 6.5028- 004

Z" C O AO 6.5898 AA A -004 6.4325- AA A 004

6.5206 AA A -004 6.3632- AA A 004

ZT A C A O 6.4518 AA A -004 zT 'in /1

1

6.2941 - AA A 004

6.3833 -004 6.2248- 004

6.3149 -004 6.1562- 004

Z" A ^ C 6.2465 AA A -004 Z' AOTZ!" 6.0876- AA A 004

1 no 1 6.1781 AA A -004 ^ A 1 6.0186- AA A 004

Z" 1 AA C 6.1095 AA A -004 5.9498- AA A 004

6.0409 -004 5.8812- 004

5.9721 -004 5.8121- 004

C AA1 O 5.9033 AA A -004 5.7434- AA A 004

5.8346 AA A -004 5.6752- AA A 004

5.7661 AA A -004 5.6069- AA A 004

5.6981 -004 5.5395- 004

5.6306 -004 5.4733- 004

5.5640--004 5.4080- 004

5.4985--004 C A A -\ 5.3441 - AA A 004

5.4343--004 5.2825- AA A 004

5.3716--004 5.2226- 004

5.3107--004 5.1649- 004

5.2517--004 5.1103- 004

5.1948 -004 5.0580- 004

5.1400 -004 5.0082- 004

=0.8 10/3 9.1956 -004 191 9.0622 -004 192

8.9832 -004 193

8.8498 -004 194

8.7181 -004 195

8.6485 -004 196

8.5314 -004 197

8.3942 -004 198

8.3222 -004 199

8.2177 -004 200

8.0788 -004 201

8.0060 -004 202

7.9175 -004 203

7.7797 -004 204

7.7011 -004 205

7.6243 -004 206

7.4890 -004 207

7.4041 -004 208

7.3411 -004 209

7.2164 -004 210

7.1292 -004 211

7.0807 -004 212

6.9702 -004 213

6.8795 -004 214

6.8401 -004 215

6.7430 -004 216

6.6469 -004 217

6.6118 -004 218

6.5294 -004 219

6.4302 -004 220

6.3973 -004 221

6.3308 -004 222

6.2301 -004 223

6.1949 -004 224

6.1419 -004 225

6.0400 -004 226

5.9978 -004 227

5.9553 -004 228

5.8535 -004 229

5.8011 -004 230

5.7664 -004 231

5.6671 -004 232

5.6033 -004 233

5.5742 -004 234

5.4806 -004 235

5.4062 -004 236

5.3810^-004 237

5.2970^-004 238

5.2149^-004 239

5.1931--004 240

5.1230--004 241

5.0377--004 242

5.0189--004 243

4.9664--004 244

4.8824--004 245

41
LOW FREQUENCY COMPONENT OF THE ELECTRIC MICROFIELDS

ro/D = 0 = 0.2 =0.4 =0.6 =0.8 10/3 5.1976--004 5.2185--004 5.0873 -004 4.9617 -004 4.8657 -004 246

C 1 AIR _

J . 1 1JO -004 ^ 1 60Q--004 UU4 J.UjDD ^ 0166 —— 004 UU4 44.7101 76/O — 004 uU4 4 811 0 4.OJI7 — 004 UU4 7/17 z4 / js .yjyyjy nofiQ--004 J^ . 1046- 1U4D -004 UU4 44. Q878 J70 / 0 — 004 UU4 4 87*^4 4.0 / J4 — 004 UU4 44. 7^77 / JZZ 004 UU4 7/1

z4o

^ 0187--004 -004 UU4 44.74UO Q408 — 004 UU4 44.081S8 J JO — 004 UU4 44. 7146

j4D — 004 UU4 74Q Z47

-004 44.77JD QQS6--004 UU4 44.0780^7JZ —— 004 UU4 44. 7070 ly ly —— 004 UU4 4 7171 4. / 1 / 1 — 004 UU4 7 ^O zjU

4.9364--004 4.9364--004 4.9364 -004 4.7607 -004 4.6423 -004 251

A 8864--004 4 8864--004 UU4 44.00D4 8864 — 004 UU4 44. 10/ Z4Al/ — 004 UU4 44.DI0 61 87/ 004 UU4 7^7 ZjZ

A 8'^71 - t.O J /

-004 4 8171 - t.OJ / 1 -004 uU4 44.0J8171/ 1 — 004 UU4 44.DO7U 6800 — 004 UU4 44.D16110ly — 004 UU4 7^1 ZJ J A 788S

't. / 00 J -004 4 788S- / OOJ -004 4t. 788S / OOJ -004 UU4 44.DJZ 6577/ -004 UU4 A4. J4U0 5408 — 004 UU4 7S4 Zj4

A 7406--004 4H. 7406- /4UO -004 UU4 44. 7406 / 4VJD -004 UU4 44.D161 60uu — 004 UU4 44. JUJO 5058 — 004 Uu4 7SS Zj J 4.6934--004 4.6934--004 4.6934 -004 4.5786 -004 4.5034 -004 256

At.utuo 6468--004 4 6468--004 UU4 44.D4D0 6468 -004 UU4 44. J51QS J7J -004 UU4 44.44150 J JU — 004 UU4 757 Z J / A 6008-- 004 4 6008--004 UU4 44.UUU0 6008 -004 UU4 44.477Z 4QQ7 — 004 UU4 44. JO1857J / —— 004 UU4 7sa Z JO

H.J J J J

- 004 4. J J J J -004 UU4 4. J J J J — 004 UU4 44.44S7Q- J / y — 004 UU4 A4. 1870 J0Z7 — 004 UU4 7^0 ZJ7

At. J^11 08- uo -004 44. J

'^11 08- UO -004 UU4 44. J^11 08UO -004 UU4 A4,4140 4148 -004 UU4 44. 1188 J 1 00 — 004 UU4 760 ZDu

4.4667--004 4.4667--004 4.4667 -004 4.3704 -004 4.2559 -004 261

4 491'^- 't.'tiij J -004 4 4711

4.4ZJ J -004 UU4 44.4Z4711J J -004 UU4 44. JZ1754J4 -004 UU4 44.ZJUJ 7 505 —— 004 UU4 767 ZDZ

4t. jOUt 1804--004 44. J0U4 1804--004 Uu4 44. JOUH 1804 -004 UU4 44.Z77Q1

yj -004 UU4 44.11Q18 7JO — 004 UU4 761 zdj

4 1181 - f . J jO

-004 4 1181 - 4. J jO 1 -004 UU4 44, J1181 JOI -004 UU4 A4.Z7174- JZ4 -004 UU4 44.11 700 L\jy —— 004 UU4 764 ZD4

4 ')Q64-. 004 44.Z^D4 7064--004 UU4 44.Z704 7Q64 — 004 UU4 4 1 858. 4. 1 0 JO -004 UU4 A4.11 177 IZZ — 004 Uu4 76^ ZDJ 4.2552-•004 4.2552--004 4.2552--004 4.1390 -004 4.0666 -004 266

4 71 46-. 004 44.Z140 71 46 004 UU4 44.Z14D 71 46 004 UU4 A 0071 -004 UU4 1J. 0887 700

004 uU4 767 ZD /

y1 1 74< 4.1 Z'+J — 004 44.1/1 74S4J 004 WU4 44.1/1 74^4j — 004 UU4 A 0467 4.W4D / — 004 UU4 1 0766 j.y /OD — 004 UU4 768 zDo

4 1 1^0 4. 1 J jU 004 4 1 I^O 4.1 J JU 004 UU4 4.4 1 I^O 1 J JU — 004 UU4 4 001 0 004 UU4 1 0446 J.j'44D — 004 UU4 76Q ZOy

004 4 0Q60 - 004 UU4 4 OQ60 004 UU4 1 0^77 — 004 Uu4 1 fi66< — 004 UU4 77n z /U

4.0575-004 4.0575--004 4.0575 -004 3.9153^-004 3.8501 -004 271

4.0195- 004 4.0195--004 4.0195 -004 3.8743 -004 3.8323 -004 272

3.9820- 004 3.9820--004 3.9820 -004 3.8342--004 3.7578 -004 273

3.9450- 004 3.9450--004 3.9450 -004 3.7960 -004 3.7359 -004 274

3.9084-004 3.9084- r\f\ A -004 3.9084 -004 3.7594--004 3.7307 -004 275

3.8724- 004 3.8724- 004 3.8724--004 3.7238--004 3.6628--004 276

j.ojDO 1 8168--004 1J .O8168- JUO 004 j1 .08168- JUO -004 UUH 1J .D070 68Q8--004 uut 1 611Q--004 UUH 111 Lit

J.OUl 1 801 7-/ 004 1 8017- 004 J.OUl 1 8017-

-004 UU*T 1J.UJ6571- / J -004 uut 1J .UJOH 6184--004 UUH 778 z / 0

j

1 . 7670- / o / u 004 1 7670- 004 1J . 7670- / u / u -004 UU*T 1J .UZ 6755- J J -004 UU*T j1 . J578Q- / oy -004 UUH 77Q L 1 y

J. / ji, 1 004 004 1J. 7177- / JZ

-004 uut J1 . JJ' 5Q50- JU -004 uut 1 541 5--004 UUH 780 zou 3.6989- 004 3.6989- 004 3.6989--004 3.5657--004 3.5515--004 281

1j .DO 66^6- jO 004 J1 .OOJU 66S6- 004 1J .QUJU 6656--004 UUH 1J. J516Q- JUj' -004 UUH 1J . JUU7 500Q--004 UUH

j.OjZD 004 1J .U6176- J z,u 004 1j .06176- JZU -004 UUH 1J .JUO 5087-/ -004 UUH J1 .H451Q- J j7 -004 UUH

1 6001 — J .DUUl 004 1 6001 - 004 1 6001 - J .UUU

-004 UU*T 1 481 7- j .to 1 / -004 UUH 1J .HUHH 4644--004 UUH IRA

1 S67Q — J. jD ly 004 1 '\67Q- 004 j.1 JUS67Q- / y -004 UUH j1 .HjH 4547-

-004 UUH 1 4711 - J.HZJ 1 -004 UUH

3.5362- 004 3.5362- 004 3.5362--004 3.4279--004 3.3669--004 ^ 0 y 286

1 S04Q — 004 1 S04Q- 004 uU4 1J. JU*t!7 S04Q--004 uut 1j.Hui 401Q-y -004 UUH . J.1 J1746- / HU -004 UUH 787 zo / j,^ 1 471Q_ / jy 004 j1 .*T4710- / jy 004 J1 .H471Q- / JZ7 -004 uut 1 1757--004 UUH 1 1441 - J. JHHl -004 UUH 788 zoo

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42

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