Watronat Bureau of Standards

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NBS MONOGRAPH 121

NATL INST OF STANDARDS & TECH R.I.C.

A11 100990255

/The effect of time ordering on the Lyme QC100 .U556 V121;1971 C.I NBS-PUB-C 1971

The Effect of Time Ordering

on the Lyman a Profile
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UNITED STATES DEPARTMENT OF COMMERCE • Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M, Hranscomb, £>irec<or ATjONAL BUm:.} DF STAf«S,m

The Effect of Time Ordering On the Lyman a Profile

^

( J J. T. Godfrey, C. R. Vidal, E. W. Smith,

and J. Cooper

Quantum Electronics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

4.

(/ , S , National Bureau of Standards^ Monograph 121 '*

Nat. Bur. Stand. (U.S.), Monogr. 12L 14 pages (June 1971) CODEN: NBSMA

Issued June 1971

For sale by llie Siipt- rintendenl of Documents, U.S. Government Printing Office, Wasliinglon, D.C- 20402

(Order by SD Catalog No. C 13.44:121), Price 30 cents
Table of Contents

Page

1. Introduction L 1 2. The Unified Theory Line Shape For Lyman-Alpha 2

3. Calculation Of The Time-Ordered U, 4

4. The Fourier Transform Of C(t) And The Profile 6

5. Discussion 10

6. References 11

7. Appendix A 12

II
The Effect of Time Ordering on the Lyman ol Profile

J. T. Godfrey*, C. R. Vidal, E. W. Smith

and J. Cooper**

National Bureau of Standards, Boulder, Colorado 80302

Using a unified theory of spectral line broadening previously developed, the effects ol' time- ordering over the complete line profile are investigated. The behavior of the time-ordered thermal average and un-time-ordered thermal average are compared. The F"ourier transform of the thermal average is obtained analytically. Calculations for the line profile of the Lyman-a line of hydrogen are presented and are representative in that the full thermal average is replaced by the thermal average

vi^ith the electron velocity distribution approximated by/([))=8(t)—11^) where y,,, is the thermal velocity

for the plasma in question.

Key words: Lynian-a: .Stark-broadening: time-ordering: unified theory.

1. Introduction

Classical path calculations of Stark-broadened profiles have usually been made within the framework of two different models, valid over different regions of the profile. The impact theories

of Griem, Kolb, and Shen [1]' and Baranger [2] yield results appropriate to the line center

(Aa)5 o),,, ojp being the plasma frequency). In these theories, the electron collisions are assumed

to be completed and electron correlations are approximated by impact parameter cutoffs. Initially,

the calculations of the S-operator were made to second order only. Later Shen and Cooper [3] were

able to calculate the un-time ordered 5-operator to all orders. In the other limit, quasi-static

theories were used to describe the wings of the profile (Aai-? Aw,., Aoj,. being the Weisskopf fre- quency). Recently, a "unified theory" has been developed by .Smith, Cooper, and Vidal [4]. The

expression for the line shape derived in this theory is (under certain conditions) valid over the

entire profile and yields both the impact and single perturber quasi-static results in the center and wings respectively.

Line profile calculations on the basis of the "unified theory" have been made for hydrogen by

the same authors [5]. In describing the transition region from the line center to the line wings,

the calculations include the effects of incompleted collisions through the use of a more general

time-development operator which does not employ the usual completed collision assumption.

Further, as in the case of Shen and Cooper [3], the calculations were made to all orders for the

un-time-ordered exponential form of the time-development operator. The interaction potential was approximated to be of the dipole type and the ion broadening was treated as quasi-static.

The correlation between the plasma perturbers was considered by cutting off the interaction

potential at the Debye-length of the plasma. Additionally, for numerical convenience, a strong

collision sphere was employed, inside which the collisions were assumed strong enough to destroy the phase coherence of the radiator. The radius of this spheje was taken to be pi,=A4-«-a„

where 9c is the thermal de Broglie wavelength of the colliding electron. Time-ordering was not included in these calculations.

In the S-matrix limit. Bacon, Shen, and Cooper [6] have investigated the effect of timeordering on the Lyman-alpha profile. In addition, their interaction potential included all terms in the multipole expansion. The results of these calculations indicate that a dipole type potential

yields little deviation from the more general case including all multipole terms, until impact

parameters smaller than about 3 A: are reached. The effects of level-splitting caused by the ionfields were ignored when calculating the time-development operator (S-operator). The upper

impact parameter cutoff was fixed at the Debye length and the lower impact limit was taken to be the thermal de Broglie wavelength of the perturbing electron. The purpose of this paper is to present a model calculation of the effect of time-ordering on the resultant line shape as obtained within the framework of the "unified theory" of Smith,

Cooper, and Vidal [4]. The calculation is representative in that the full thermal average is replaced by the thermal average evaluated at an average electron velocity for the plasma. The

effects of incompleted collisions and a Debye-shielded potential, as well as a strong collision sphere cutoff on the potential were included. The results of the time-ordered calculation are compared with the un-time ordered result for Lyman-alpha. The calculations are done for an

electron density of 10'" cm"' and electron temperature of 20,000 K [6].

*l niK-isil> ..r ( ..lor^iil... iiriulili-i-. C.l,,., H().f02. "Jc.iilt In.-^liluli- I'lr l.abiirjliiry A>lni|)li>-ii >. Hc.uIiJit. ( inli.. and I)i-|)arl iiii iil "I \'\\\ -ic - ami \-lni|iliysii s, L nivtT>ily nt ( Nile. rail. lii.ulili-r. ChIm.. 80302.

ijiurt's ill lirackt'l;- ret rr It) I lie iilcraliirc rcl crcm es un p. II.

1
2. The Unified Theory Line Shape for Lyman-Alpha

The approximations inherent in a classical path treatment are well known [7, 8]. In addition

to these we will approximate the effects of the ion microfields in a quasi-static manner by regarding

their collective field as essentially constant during the times of interest, l/Aoj. The final profile

is obtained by averaging the line shape for a given static ion field over the distribution of ion fields for the plasma in question:

oc i

r((w)= P{E)I{w, E) dE. (2.1)

The distribution function, P{E), is the low frequency component of the fluctuating microfields in the plasma [9, 10]. The line shape for a given static ion field obtained from the "unified theory"

is given as [5 ]

/(w, ^j^^ lm^<a|d|/3> </3'|d|a'> <a\p\a> </8'a'| -'|/3a>, (2.2)

where the sums are over atomic states. Matrix elements of Acuop give the frequency perturbation

from the position of the Stark component, shifted due to the static ion fields. The <i:'-operator is defined within the impact approximation (not to be confused with the impact theory which makes

the completed collision assumption) by the following expression:

,1" (AcOi,p)=-fAa>op e

it^.c <Ut-l> dtAcOop. (2.3)

In general, the time-development operator is tetradic. This is necessary to include the effects of both upper and lower state perturbations; however, for the Lyman lines lower state inter- actions are negligible and the time-development operator between lower states may be replaced

by a unit operator, thus, the time-development operator becomes

f7, = 0exp V(t') dt'

(2.4)

Here O denotes the chronological ordering operator and the time-development operator has to be evaluated only between the upper states of the line. The potential operator, V in the above expression is written in an interaction representation

and is defined in terms of the classical interaction potential, V, by

~ itHJh -itHJh

V {t)—e y (t) e (2.5)

where //„ is the Hamiltonian of the atomic system in the static ion field. In the expression for the

J?-operator (eq. 2.3), the brackets around the time-development operator denote the usual thermal

average over perturber states. In the classical path approximations the thermal average becomes

Fi(f)=<C/,-l> = n Jrfx, dv, W[v,) f/i(R„ x„ v„ t) -1 (2.6)

where R denotes the position of the atomic orbital electron, the perturbing electron position and

velocity being x, and v, and the subscript one indicates a binary collision of a single electron (a consequence of the impact approximation). W(vi) is the classical electron velocity distribution

appropriate to electrons of temperature T,,. Generally, this velocity distribution of the electrons

is taken to be Maxwellian resulting in the distribution W being related to the Maxwell distribution

of speeds through W(Vi)=f{Vi)l4'TTVi~. As mentioned earlier, however, in this calculation /(fi) is

replaced by 8(i;— wnere v^, th( thermal velocity Vaw — yj^kTJm, for the plasma in question. Wav) In terms of the more convenient collision variables (see Appendix of [8]), the thermal average

reduces to

Pi>

n,. Vay

''

4tt ^tdnlpdp dt,, Ui{R, p, Vav, t, to)-l (2.7)

2
Here p points along the z-axis, Vav along the x-axis and denotes some reference time of the

perturber.

To obtain the line shape it is necessary to invert the matrix

<a/3|[Aaj„p-j:'lAw„p)] |a'/3'>= | w - i£y -£„ ) M | 8„„'8^y' -<a/3|j^(Aa;„p)|a'/3'>, (2.8)

with \a>, |a'>, |/3> and |/8'> being Hq eigenstates. For hydrogen, the indicated eigenstates are taken to be parabolic. When considering well isolated lines such as Lyman-alpha, the off-diagonal

elements in principal quantum number are generally small and the no-quenching approximation can be applied. Thus, matrix elements are taken to be non-zero only when occurring between

states with the same principal quantum number. The JT-operator matrix elements for Lymanalpha become

X <2qm\£ (Aw,p)|2g'm'> = -fAajo„„Awo„,„,.f e^'^^-'""<2qm\F,{v^,., t)\2q'm'>dt, (2.9)

j(i

where the dependence on \a> and \a'> has been dropped since these states correspond to the

single final state, |100>. for Lyman-alpha which is essentially unperturbed as previously dis- cussed. The quantum number q is defined in terms of the usual parabolic quantum numbers, n, and

n.,, according to q — n^ — n-, and m is the magnetic quantum number.

In order to obtain matrix elements of Fi[v^y, t), we require elements of f/j which in turn require matrix elements of the form

<2qm\V{t)\2q'm'>=ex^ le J- iZog-Zog') Et <2qm\V(t)\2q'm'>. (2.10)

At this point we simplify the right hand side by invoking the sudden approximation which allows

us to set the exponential equal to one in the preceding equation. This approximation has been

dealt with in some detail by \an Regemorter [11] and will not be discussed further, however,

it is made in previous Stark-broadening calculations (for a more complete discussion see refer- ence [8]). Thus, matrix elements of V{t) become simply matrix elements of the classical interaction

potential, V(t). In evaluating the thermal average of we have to perform the spherical average

which is most conveniently done in spherical eigenstates. e rewrite the J^-operator

matrix elements as <2qm\£ (Aajop)|2g'm'> = — ;Acu2Q»,Aa)2Q',),'2<2gm|2/m><2/'m'|2g'/7i'>

(2.11)

X [e'^"^'^'""<2/m|Fi(t;„,, t)\2l'm'>dt.

Jo The transformation coefficients between spherical and parabolic states, <nqm\n'rm'>. have

been obtained by Hughes [12] and are diagonal in magnetic quanjtum number.

Returning to the calculation of the thermal average, we must evaluate the following expression:

Pd

<2/m|F,(i;av, t)\2l'm'>='^^jpdp j

dt^jdn <2lm\U,\2l'm'>-l . 12)

f/, is evaluated in the collision axes. In performing the angular average, the atomic axes are rotated

through n into the space axes defined by p and v. The angular average then becomes

jdn <2lm\Ur\2l'm'>-l jdfl D (H) <2/mK', |2/'/x'> Z)|' (n)-l

(2.13)

877^ ^

"2/+l^

<2Zm|C/,|2/m>-l
where the D matrices are the conventional rotation matrices [13]. The thermal average now becomes

Pd 00 <2lm\F^(v,,y, t)\2l'm'>=2TTn^, Vas^pdp^t^^ |27+y2

(2.14)

<2lm\Ui\2lm>-l 8ii,8,

which is diagonal in spherical states. Thus, only the elements <200|Fi|200> and <21m|Fi|2lTO>

are needed for the line shape. (These elements are independent of m as a result of the spherical

average.)

3. Calculation of the Time-Ordered

To obtain the full time-ordered time development operator, one may integrate the differential equation defining this operator (the subscript one will be dropped for the remaining discussion

and a binary collision is assumed to be considered). In order to obtain the differential equation

in terms of the collision variable, we differentiate the Dyson expansion of the time-development

operator (see, for example, reference [14] page 240) with respect to t„ where this expansion

is given by

(+/„ /, („- U{t+t,, ^,)= i+2(/)'|^^^i|^^'---|^^^.'^(^.)^(^2)...J^(fJ.

(3.1) The resulting differential equation becomes

ih didt,, U{t+t,„ t„)^V{t+t„) Uit+t,„ 0-f/(f+?(„ ^,) n^o). (3.2) Up to this point no mention of the form of the interaction potential has been made. In principal, the full multipole expansion of the potential can be used; however, this would result in prohibitively long machine calculations. Further, since a dipole approximation of the interaction

yields the major contribution to time-ordering [6] this form of the potential was used. Using the

Debye-cutofif and the strong-collision cutoff, the interaction potential is given as

rO (rp>po)

^ O • {r,<Pmin) (3.3)

where C!,''"'=y 47r/(2A-l-l) F/,,,, R is the position of the orbital electron and ri,— y^p~+i?av t' is the

radial position of the perturber. In effect, the perturber is considered to remain outside the

atomic dimensions as required by the classical path theory. The hard sphere cut-off, Pmim was

taken at the thermal wavelength of the colliding electron. In terms of spherical eigenstates,

eq. (3.2) becomes for Lyman-alpha

ih dldt^, <2lm\U\2l'm'>^^ <2lm\V{t+t„)\2LM> <2LM\U\2l'm'> (3.4)

-<2lm\U\2LM> <2LM\Vm2l'm'>

The cutoff on the potential at the Debye-sphere and the strong-collision sphere make the calculations velocity dependent for all values of the collision variables. For values of p>>10^ we find that the real parts of the matrix elements in eq. (3.4) become equal to the un-time ordered values. As discussed in more detail at the end of this section, this is merely a consequence of the fact that a second order expansion of the time-development operator is valid in this region. In this region U was approximated by f/,, where

f/„=exp V(t') dt' (3.5)

4
The calculation of the un-time ordered thermal average for the "unified theory" line shape has

been discussed in detail in the reference mentioned earlier [5]. At very small impact parameters

(up to about .3A) an approximation due to Berman and Lamb [15] was employed. In this a[)proximation a unitary operator R(t) is introduced which instantaneously diagonalizes the interaction

potential. The time-ordered time-development operator is then approximately given by

U it+t,^, tJ=R^ (t+to) exp

where V,>{t)=4^ C,"' (R) C„"' (r,)

.

The region of validity of this approximation occurs for values of p<<p,^. (p„. being the Weisskopf

collision radius defined as -^2/3 A n'-). In the intermediate region (.3-^ <p<10A) eqs. (3.4) were

integrated numerically (Appendix A). For large interaction times, the time development operator

has to converge to the S-matrix limit according to [8]

00 lim c?fo [U-l]av=lim t [S-l]av (3.7)

/^OO J -X (-CO

This convergence allowed the time-development operator to be replaced by the S -operator for values of t which were found to satisfy t>>10/ajp (w^, being the plasma frequency). Physically this corresponds to times of interest large enough such that virtually all collisions are completed.

The solution of the time-ordered S-operator required a Debye cut-off and hard sphere cut-off on

the interaction potential and in this respect differed slightly from the previously mentioned calculation of Bacon, Shen and Cooper [6]. (Also, the full thermal average was replaced by the thermal

average evaluated at the thermal velocity.)

For comparison, the un-time ordered time-development operator of eq. (3.5) was calculated

in spherical states. This calculation differed from that of Vidal, Cooper, and Smith [5] only in that the thermal average was evaluated at the thermal velocity instead of making the full velocity average. From the results for the {/-operator and the un-time ordered f/o-operator, it was possible

to generate a difference function, D{t) — U{t)—U(f{t), which represents the effect of time-ordering.

This effect is illustrated by figure 1, in which we plot the ratio C{t)IF{t) as a function of time

for the (200) and (21m) matrix elements of the thermal average. Here C{t) is the thermal average

of D{t) and F{t) the thermal average of the time-ordered time-development operator U(t). The

time scale is normalized with respect to the inverse of the plasma frequency such that

s^d),, t=J2 oj,jt. (3.8)

The effect of time-ordering is negligible for small times of interest corresponding to the static limit where Aw> Aw,, and increases monotonically to a constant value for large times of interest. The largest effect on the thermal average occurs for times greater than about .l/w,, and for the

(200) element is a 14 percent decrease while for the (21m) element it is a 11 percent increase. The

accuracy of the calculations for C{t)IF{t) correspond to ~ ± .005. In order to clarify again the effect of time ordering on the time development operator, one

can rewrite eq. (3.1) in the following manner [16] :

f/(; + f,„ ?„)=exp V{t')dt' exp 2\h

X exp

3\ h

i„ r I U4\ I dt'\ dt"\ ,it"'\ V{t"'),

/ + /„ rl dt' \ dt"

V{t"), V{t')

V{t'),\{t")

V{t"\

(3.9)

V{t"'), V{t') X...

5
0.16

(1 = 1) LYMAN-a

nj= lO'

Te = 20,000K

((=0)

0

LOG|oS(S=5pt)

Figure 1. Ratio of the thermally averaged difference function to the thermal average of the time- ordered time-development operator. The time scale is normalized with respect to the inverse of the plasma frequency (s=clipt = Wpl).

Clearly, if the commutators in the above expression can be ignored, time-ordering is unimportant.

For collisions which are weak (Ft/^< 1 where t is the collision time and V the interaction potential)

it is apparent that this condition holds. It is this fact which leads to the convergence of U to the

un-time-ordered result for impact parameters greater than about 10 In the quasi-static region,

Aa)T>>l, it can also be shown [8] that the commutators are negligible. Thus, we anticipate that in the static limit, for small times of interaction, the time-ordered and the un-time-ordered results

for the time-development operator will coincide.

4. The Fourier Transform of C(t) and the Profile

The thermal average of the time-ordered time development operator is defined by

F, (O-FY (0 (4.1)

where F',' (t) is the un-time ordered thermal average. Evaluation of the,i:^-operator matrix elements

in eq. (2.9) requires the Fourier Transform of Fj. As the Fourier Transform of F" {t) has been

obtained elsewhere [5], we will concern ourselves with the transform of the difference function,

C{t). We require the transform of both elements

<200 I C(t) I 200> and <21m | C{t) | 21m>.

Den(Ming these as C*"(0 and C'-'(0^ respectively, it is now possible to proceed in a manner similar

to that used for obtaining F" (Aoj). In terms of the variable s (eq. (3.8) ) and the frequency variable

AcI,,=(Aco-^^9)/ci^., (4.2)

where Aaj=(u— cu,,. Aw is the frequency perturbation from the position of the unperturbed line

(Oq, we require

X C" (Aw„)=- f e^"^"' C"is) ds. (n = l,2) (4.3)
As in the case of F?, C"{s) diverges at large values of s because of the neglect of the finite lifetimes of the states involved in the line. We thus introduce again a convergence factor exp (-€5) and

evaluate

X C" (Aaj„)=lim - e C"(s) ds. (4.4)

In performing the transform, it is convenient to fit C"(s) with an analytic function. Unfortunately, the small 5 behavior is not known analytically; however, both Fi(s) and F"{s) are known

to go as 5'"- for small s. Consideration of the results of Berman and Lamb [15] indicate that for small times of interest C"(s) is expected to go as a half-integer power of 5. It was found that a function of the form

lim H" is)=h'l s"'-+h;i 53'2 + ... (4.5)

fit the available data. A suitable function which has both the large s asymptotic behavior and

goes as eq. (4.5) for small 5 was satisfied by a function of the general form

Aai's 3 k+3

(4.6) 2A:+l/2'

The transform C"(Aaj„) is then given by

C«(Aw,)=y lim - f Y Jo (4.7) d\

(

2''^T (2 k+i) \dej \dKl p^"*' '^^"\k. b'l, (e-iAa>,)

where /C„ is the modified Bessel function of zero order. The number of terms included above is determined by the desired accuracy of the fit. In the calculation presented here it was possible to

fit C"{s) down to small values of 5 with the first two terms of equation (4.6) (A'=l and k=2 terms).

Writing these terms out explicitly, we have the following:

//'/(AwJ-a'; 6',' e \6Z\

/32

1' J, [Z'l +[f [Z'l -91 Fo [Z'l -3j [Z'l 32

^80 1 +1 i6Z';Fo(zn-(f (^i') -9]Jo[Z'{

32 80

(4.8)

m (Aw,)=-

210

^ (4200 Zi'-2112 (Zll) ) y„ (zn-(256 (Z'i) -5136 fZi') +945) F,\ [Z'i^

+ ^256 (Z'l^'-^llb (zi'^ + lOs) J, (Zi')-(l984 (zi'J'-2328 Z?j Y, (zs)

256 (z'l^-hU6 {Z'?j +9^b^ A {z'i) +

(^ 4200 Zi'-2112 [Zll] Fo U"

+ (256 te'j-4176 (zi'V+lOSj F, (^Zi'j + n984 (z'^j -2328 Zi'j 7, [z'l
where Z|'=6j' A&i,, and Z'i=b'i Aci},. The functions Jq, Y^, Ji, and Fj are Bessel functions of zero and first order. For large arguments of Z" or Z!l the following asymptotic expansions are useful:

lim H'{ (AwJ=

315 a';b'l "6^ r Z'{

(l+l)+45(1 + 0 15-21-44 (1 + i) 8 Z'l 83 Z'/ + . (4.9)

(193.8) (i_,)_

(2398.3)(l + 0_

(21434.5)(l-0 ^^

^9 9

In addition, for small values of Aw,,, C" (Awg) is approximated by

lim C" {^a)g)=-

TT Aoiq-

. 6a'i'6 i'^ 77Aa>,, (4.10)

In order to compare the results of the transform of C"(t) with the thermal average of the

time-ordered time-development operator, it was necessary to obtain F? (AtOg). In order to get this function, we approximate F? (5) with a function G(5) =G, (5) +6^9(5) for both the (200) and (21m)

elements. The functions Gi{s) and ^2(5) are the same as those defined by Vidal, Cooper, and

Smith [5] and are given in terms of the normalized time variable as

G" {s)=G'l (s)+G!! is)-

a' s- {s-+2b'{ s)"-'^\s'+2b!I sV'

The transform of the time-ordered thermal average now satisfies

F'{ (Aaj,)=G" (Aaj,,)+C" (Aw,),

(4.11)

(4.12)

where G" (Aw,,) denotes the transform of G"(s) defined in eq. (4.11). In figure 2 we have plotted the real part of C" (Aaj,,)/F" (Aw,,) versus the logarithm of Aw, for the (200) and (21m) elements. Realizing that tor a frequency perturbation Aw we are mainly inter- ested in times of the order or smaller than 1/Aw, such that AwAf> 1 (4.13) we see that in agreement with figure 1, the largest contribution to the transformed thermal average

occurs for small Aw. Unlike the effects of electron correlations on the transform, which are important for values of Aw< w,„ the effects of time-ordering extend somewhat further into the wings. As expected, however, in the distant line wings where quasistatic collisions occur, (Aw> Aw,.) the effects of time-ordering are negligible. The accuracy of the calculation of

C" (Aw„)F',' l2kT

(Aw„) is ~ ± .01. The Weisskopf frequency, defined for Ly-a as Aw,.=r;^ /

is also marked.

3h

m

with

^ The Lyman-alpha profile is obtained by evaluation of eq. (2.2) followed by averaging over the

distribution of ion fields appropriate to the plasma in question. This distribution, P{E), is taken

to be the one given by Hooper as previously mentioned [9, 10]. In order to see the effect of timeordering on the full profile, the calculation of the un-time-ordered profile was obtained with

the thermal average approximated by the G function of eq. (4.11). The time-ordered result was

obtained by calculating the profile with the thermal average of eq. (4.12). In figure 3 we have

plotted the ratio A^/3^, versus Aw where

(4.14)

A.'^l is the difference between the two line profiles based on the ordered and un-ordered timedevelopment operators. Jf, is the time-ordered profile.

8
0.14 —

0.12 —

-0.16 —

-0.18 —

u = LOG ^{Au)

FiGL RE 2. The ratio C"/F" versus Acug. C"(Acuq) is the Fourier transform of the thermally averaged

difference function between the ordered and the un-ordered tim^development operators. Fr (A(o,,) /5 the Fourier transform of the time-ordered thermal average.

-O.I4|

i 1 1 -0.12

-0.10

-0.08

LYMAN - a

-ao6

Tj = 20,000 K

-0.04

"3 -0.02

0

0.02

3<

7» ' ^

/

ao4

0.06

0.08 \\\

/

/ / —

E,o^ = 0

0.10

1 1 1 1 3 -2 -1 0 1 2

u = LOG,o(A;:n

FiGL RE .3. The ratio A."!/-' versus Ao) where A .'Us the difference between the two line profiles based on the ordered and un-ordered time-development operators. Tt ;5 the time-ordered profile.

The "unified theory"' yields a normalized profile; consequently, as expected the contribution

is normalized to zero. In the same plot we show the results with the ion microfield average

removed.

It is clear from figure 3 that the effect of time-ordering in the line center is to decrease the

intensity. The profile increases in intensity as we go towards the wings until we reach roughly

one-tenth the plasma frequency. It then decreases followed by a smaller increase near ten times

the plasma frequency. This latter increase is attributable to the effects of time-ordering near the

shifted Stark component. To demonstrate this we have included in figure 3 the case where the

Stark splitting due to the ions has been neglected by setting £'ion=0. To summarize, it appears

that the effect of time-ordering on the final line profile tends to smear out the structure near

the line center.

9
In the results of Bacon, Shen, and Cooper [6] the final profiles were normalized to one in the

line center. A plot of the data from this calculation yields an increase in the half-width of the line due to time ordering by about 14 percent (this is in close agreement with the results of the Bacon,

Shen, and Cooper calculation versus that for the Shen-Cooper case [3]).

As one goes towards larger values of u (where u=logi,i (Aw) in figs. 2 and .3), we realize that the influence of time-ordering decreases. Since the time-development operator has been calculated with the same accuracy over the entire range, the numerical accuracy of C"{Aa),,)/F"{^0}„)

decreases as one goes to large u; however, for values of u larger than one, the profile is already

down by a factor of 10*^ from its value at the line center. Thus, we have discontinued the plots

at u = l.6 (where the profile is about 10*^ below the value at line center). The effects below this point reflect primarily the error introduced in fitting the C"(s) at very small times of interest

(see eq. (4.5) ) causing the ratio ATi/tT, to go slightly negative before returning to zero.

The calculation presented here was made within the framework of the classical path model,

the effects of incompleted collisions and the full time-ordering. It was representative in that the

full thermal average was replaced by the thermal average evaluated at the thermal velocity

{i'iiv='\J "ikTi.lm). The case considered corresponds to an electron density of 10'^ electrons per

cubic centimeter and an electron temperature of 20,000 K with the interaction potential taken to be of the dipole type. The calculations were made with the "unified theory" line shape of Smith,

Cooper, and Vidal [4] and for comparison the un-time-ordered time-development operator was

also calculated at The effect of time-ordering on the thermal average was found to increase monotonically with

time. For the Lyman-alpha line (the case considered) the correction to the (200) matrix element

approaches a constant value of about —14 percent for large values of time while for the (21m)

matrix element the correction approaches +11 percent. The direction in which the correction occurs for these elements (decreases the (200) and increases the (21m) with respect to the un- time-ordered case) is in agreement with the impact calculation of Bacon, Shen, and Cooper [6]. Tlie influence of time-ordering is still important at times of interest somewhat smaller than ?=l/(t>,,; however, it eventually drops off to zero at values of t — lj^U),. (A(t)c=2kT,,l3fi. This behavior is reflected in the Fourier transform where we find an inverse behavior in frequency space.

5. Discussion

2.8

2.6

2.4 TIME-ORDERED (Gl) _

JJN-TIME-ORDERED(GI)

2.2

FULL THERMAL

AVERAGE (Gl)

1.0 10 100

o AX(A)

Kk.I HK 4. Tlie jhiid Lvniiin-a prujile nurnidlized with respect to the asymptotic Holtsmarks

W~''''-witig (iotis ottly) Jar the time-ordered and un-time-ordered (including the case where

a full thermal average is made) time-development operators (AX,. = 137.A|.

10
The largest correction takes place for small values of Aw with the effects extenflirifi past the plasma frequency and finally dropping off to zero in the wings (Aa»> Aoi,.). The influence on the profile was the greatest in the line center where the intensity was found

to decrease by about 12 percent. The half-width increases over the un-time-ordcred resuh by

about 14 percent. This is in agreement with a comparison of the impact calculation [6] and the

results of Shen and Cooper [3]. As one goes towards the wings of the profile the effect of timeordering eventually causes an increase in the intensity of the profile. This increase is smoothed

out over a fairly large region with a distinct peak occurring at slightly less than an order of magnitude past the plasma frequency. This increase results from the effects of time-ordering

in the vicinity of the Stark component shifted due to the ion fields. Beyond about ten times the plasma frequency time-ordering ceases to be important. In figure 4 we have plotted the ratio of the profile to the Holtsmark Ao) "''^ wing (resulting from considering only the ions). The un-timc

ordered result is also plotted along with the un-time-ordered result obtained using a full tlicrmal average (dotted line). In this plot only the Gi term of eq. (4.11) has been considered. As mentioned above, the effect of time ordering is to lower the Ly-a profile in the center and to increase it in the

wings which effectively reduces the structure of the line profile. In co*nparing the results presented

with various experiments in the high electron density regime we observe the following facts. For

the experiment of Elton and Griem [17] (Ly-a, n,.=3.6xl0'^ cm"'', T,,=20 400 K) the correction

from time ordering improves agreement between the theoretical profile and the experimental

data, while in case of the measurements of Boldt and Cooper [18] (Ly-a, ;ip=8.4xl0"' cm"', T,, = 12200 K) the agreement becomes worse. We also recall that in the S-matrix limit it was found

[6, 19] for Ly-a, as well as H„, that the strong collision parameter is enlarged due to the effects of time ordering. In view of these results it is therefore reasonable to expect that also in the case

of the recent measurements by Wiese, Kelleher, and Paquette [20] //^ and Hy, ?i,,=8.0x 10"" cm~^

7",,= 12700 K), which at the moment may be regarded as among the best Stark broadening measure- ments in the high electron density region, the correction due to time ordering improves the

agreement between the theoretical and experimental profiles. (For more details see also Vidal,

Cooper, and Smith [21].)

The authors would like to thank Dr. D. N. Stacey for providing the predictor-corrector

subroutine used to solve the differential equations.

6. References

[1] Griem, H. R., kolb, A. C, and Shen, K. Y., Phys. Rev. 116. 4 (1959). [2] Baranger. M.. Phys. Rev. Ill, 481 (1958). [3] Shen, K. Y., and Cooper, J., Astrophysical Journal 155, 37 (1969).

[4] Smith, E. W., Cooper. J., and Vidal, C. R., Phys. Rev. 185, 140 (1969). [5] Vidal, C. R., Cooper, J., and Smith, E. W., JQSRT, 10, 1011 (1970). [6] Bacon, M., Shen, K. Y., and Cooper, J., Phys. Rev. 188, 50 (1969). [7] Smith, E. W., Vidal, C. R., and Cooper, J., J. Res. Nat. Bur. Stand. (U.S.) 73A (Phys. and Chem.) No. 4, 389-404

(July-Aug. 1969). [8] Smith, E. W.. Vidal, C. R., and Cooper, J., J. Res. Nat. Bur. Stand. (U.S.) 73A (Phys. and Chem.) No. 4. 405-420

(July-Aug. 1969).

[9] Hooper, C. F., Phys. Rev. 165, 215 (1968). [10] Hooper, C. F., Phys. Rev. 169, 193 (1968). [11] Van Regemorter, H., Comptes Rendus 2 59, 3979 (1964). [12] Hughes, J. W. B., Proc. Phys. Soc. 91, 810 (1967). [13] Rose, M. E., Elementary Theory of Angular Momentum (John Wiley and Sons, New York. 1957). [14] Gottfried, K., Quantum Mechanics. (Benjamin, Inc., New York, 1966). [15] Berman, P. R., and Lamb, W. E., Phys. Rev. 187, 187 (1969). [16] Wilcox, R. M., J. Math. Phys. 8, 962 (1967). [17] Elton, R. C, and Griem, H. R., Phys. Rev. 135, A1550 (1964). [18] Boldt, G., and Cooper, W. S., Z. Naturforschung, 19a. 968 (1964). [19] Bacon, M., to be published.

[20] Wiese, W. L., Kelleher, D. E., and Paquette, D. R., to be published.

[21] Vidal, C. R., Cooper, J., and Smith. E. W., JQSRT, 11, 263 (1971). [22] Hamming, R. W.. J. Assoc. Comp. Mach. 6, 37 (1959).

11
7. Appendix A

The time-consuming portion of the calculation involved obtaining the time-ordered timedevelopment operator in the intermediate (.3A:<p<l(Wc) region where numerical solution of the

differential equations was required. In order to efficiently solve these equations a predictor- corrector method known as Hamming's method was used [22]. The advantage of this method is that the value of the function and its derivative at the point in question are obtained in two steps

rather than by an iterative process. The form of the differential equation solved is given as ^=/U, y). {A2.1)

The basic equations utilized in the method are given as

Predictor: p„+i=y„-;j+y (2y'„-y'„-,+2y'„_2)

Modifier:

Corrector: '^"+'"'| (^-^" ^3'"~-'+3/! (m'„+,-f 2y'„-2y'„_i)

j

9

Final Value: y„+i=c„+,+— (p„+,-c„+,).

Here p„+i is the predicted value at the point n + 1 in terms of the value of the function, y, and its first derivative, y', at the preceding four points, /ra„+i is a modification of the predicted value by

taking into consideration an estimation of the truncation error (avoiding an iterative type calculation), m'„+| is an estimate of the first derivative based on the modified predictor, c„+i is a correction term which must be applied to Pi,+i to obtain the final result, y„+i. In the above equations,

h is the step size and is determined by the desired accuracy of the calculation. A variable step

size was used which insured the calculation remained within the specified accuracy. As these

formulas are not self-starting (the first four values for y are needed any time h changes) it was

necessary to use a Runge-Kutta procedure to obtain these initial values.

U.S. GOVERNMENT PRINTING OFFICE: 1971 0-416-882

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4. TITLE AND SUBTITLE

The Effect of Time Ordering on the Lyman a Profile

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J. T. Godfrey, C. R. Vidal, E. W. Smith & J. Cooper

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Using a unified theory of spectral line broadening previously developed, the

effects of time-ordering over the complete line profile are investigated. The behavior of the time-ordered thermal average and un-time-ordered thermal average

is obtained analytically. Calculations for the line profile of the Lyman-a line of

hydrogen are presented and are representative in that the full thermal average is replaced by the thermal average with the electron velocity distribution approximated

by f (v) = 0 (v - V ) vv/here v is the thermal velocity for the plasma in question.

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