Collective coordinates for ion dynamics

Sptwrosc. Rodiar.

J. Quanf.

Tnmfer

Vol.

54, No.

112, pp. l-26,

1995

Elsevier Science Ltd. Printed in Great Britain

00224073(%)00036-4

COLLECTIVE

COORDINATES

FOR ION DYNAMICS

SPIROS ALEXIOUT Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel Abstract-Utilising a link between microfield smoothness and ion dynamics, a method for calculating the autocorrelation function of spectral line shapes in a plasma, including ion dynamics, is presented. The method is based on a time-independentsimulation, in that the dynamics are handled semi-analytically via a separable kernel approximation in Dyson’s equation. For each perturber configuration this reduces to the solution of a linear algebraic system. In addition this method naturally leads to a set of (approximate) relevant collective coordinates. It is shown that the microfield must be smooth and this implies the existence of new collective coordinates, namely the microfield values at a few time points. Nonsmooth components are separated out and treated analytically by the ion impact theory. This approach is valid for all parameter ranges and may be utilized either as an alternative to solving the Schriidinger equation in a simulation-like manner, or by using the collective coordinate approach. In the later case, knowledge of the many-times joint probability density is desirable.

1. INTRODUCTION

of ion dynamics, i.e. the often dominant collisions between the radiator and plasma ions, has attracted a great deal of attention since the first experimental demonstrations of this effect.‘,’ With hindsight, there is nothing strange about ion dynamics: Intuitively one knows that the larger the perturbation, the larger the width. Ions come as close to the emitter as the electrons (for a straight path trajectory) but stay there longer, hence the net perturbation (j [ V(t) dt]/h) is stronger. There are a number of refinements to this naive picture, for example adiabaticity should not be overlooked for nonhydrogenic radiators, but basically this intuitive picture is often not too far off from reality. The usual quasistatic (stochastically static) theory is only valid when all (including quasistatic ions) broadening mechanisms have produced memory loss before ions have time to move (and thus change their field) appreciably and is often not valid over the entire profile. Analytic theories,%’ in attempting to do the (many-body) perturber average, end up with an expression of the form6 [Go’ - xc]-’ with Go the unperturbed Green’s function and Z the self-energy, which is an infinite-body operator with respect to the atom-plasma interaction. Hence it must be replaced by some perturbation expansion, any finite truncation of which will break down for the strongly overlapping ionic collisions. One method for dealing with the ion-dynamical problem is simulation,‘-” in which case one chooses a configuration, solves the Schrijdinger equation for this configuration and averages over configurations. In other words, the autocorrelation function C(t), which is the Fourier transform of the lineshape, is obtained as the average of a function (depending on the specific line in question) of various U-matrix elements, which are in turn obtained by solving the Schrijdinger equation. Simulation methods have the best theoretical basis, but tend to be slow, especially as one approaches the ion impact regime. By far most of the CPU time in a simulation calculation is spent in solving Schriidinger equations,*O*‘obat least if one uses an independent particle model. If one obtains the microfield from a molecular dynamics (MD) simulation, then the MD calculation can also be very expensive. Other methods have met with some success, e.g. the Model Microfield (MMM),‘2-‘s and Frequency FluctuationI Model (FFM) methods and the semianalytical method of Greene.” The MMM and FFM, which also adopt the “field” point of view to be discussed later on, are fast methods and it may not be possible to improve on them, especially on the FFM, from a speed point The problem

TPresent address: Physique Atomique, Dans les Plasmas Denses, Bat 22, Universite Paris VI, 4 Place Jussieu, 75252 Paris Cedex 5, France.

2

Spiros Alexiou

of view. The MMM, and by analogy the FFM, has been criticized ‘Oafor being “a phenomenological approach, not a microscopic theory”, but for a very application-oriented discipline, such as Stark broadening this may not be an important consideration. If there is something to be desired in these methods, it is the fact that their theoretical basis is not as solid as for simulations, in spite of some work on this subject:‘4J8 studies on the foundations of the MMM have found,‘9*20for example, that the two stochastic properties of the true microfield preserved by the MMM are not sufficient. Furthermore, it has been shown*’ that the conditional probability function of the MMM is a poor approximation to the actual one. However, a major surprise, such as a case of these methods giving seriously wrong results has yet to be found and therefore these methods give very good results for their cost, especially the FFM. In addition, these methods have an important advantage in cases where the functional form of the time-dependent microfield is poorly known, as, for example, in turbulent plasmas. In such cases, it is more realistic to seek expressions for the probability distribution function and covariance than for the functional dependence of the microfield on time and relevant distribution functions. Summing up, the ion dynamic effect is well established and may result in as much as an order of magnitude increase over the quasistatic prediction in the line width for hot dense plasmas.8b Ion dynamics is important for hydrogenic neutral and ion emitters, high temperatures, “low” densities, highly charged emitters (if Doppler does not dominate) and merging of initially distinct lines. Further, it is difficult to treat analytically and bounds obtained from ion impact calculations, which is the largest possible ionic contribution, are often too large to be acceptable for hydrogenic emitters. 2.

KEY

CONCEPTS

In this section, we would like to briefly review some key concepts. 2.1. Autocorrelation function We define the autocorrelation function C(t) as the Fourier transform of the line profile, normalized to be unity at t = 0. C(t) involves linear combinations of products Ua,,(t)UBB’(t) of U-matrix elements between upper (a, a’) and lower (/3,#l’) level states. In the one state case explicitly considered here, C(t) = (Cl (t)), with Cl(t) = &ak uk(t), with ak numerical Coefficients. C,(t) is the autocorrelation function before averaging. 2.2. Particle us jield point of view There are two ways of viewing collisional broadening. One may look at this process as the radiator being perturbed by collisions, that is, particle point of view, or by a time-dependent stochastic external microfield, that is, field point of view. Clearly then, the statistical properties of this field must be related to the statistics of the plasma particles. This second point of view is taken here, as in most modern methods, and has distinct advantages. 2.3. C(t) ‘formation” The diagonal elements of the U-matrix and, hence, C,(t) start as unity at t = 0 and drop due to interaction with the field, or equivalently collisionally induced transitions. The rate of decay is faster for a stronger interaction, Each radiator sees a different time-dependent field, hence the Cl(t) for this radiator (or, equivalently, for this configuration) starts decaying with a different decay rate for each radiator. However, the decay rate does not translate into a width. The width is intrinsically an averaging effect and comes about because after long times, the Cl (t) for different radiators are out of phase and their average dies for long times, resulting in a net C(t) with a finite extent and hence, upon Fourier transforming, a width. Figure 1 illustrates for a few configurations, this random phase effect. We define the relevant (memory loss) time T as the time at which C(t) has dropped to a negligibly small value, say 1%. Practically speaking, bounds for T are easy to get: a maximum and minimum value is obtained by taking the product of the quasistatic ion C(t) and ion impact C(t) respectively with the product of the electron, Doppler and natural autocorrelation functions and observing where the total product has dropped to a negligibly small value.

Collective coordinates for ion dynamics I.0 0.8 0.6 0.4 0.2

= u

0 -0.2 -0.4 -0.6 -0.8 -1.0

0

I

I

I

I

I

I

2

3

4

5

t (ps) Fig. I. Illustration of the “formation ” of the autocorrelation function. The dashed line is the average of 7 configurations, four of which are shown as solid lines. For long times, their phases are essentially random and this gives a vanishing C(r). This graph was produced by a simulation of purely ionic electric fields for 7 configurations in an Argon plasma of temperature 16,000 K and density 2 x lO%/cm’.

3.

THE

DIFFERENT

DYNAMICAL

REGIMES

In this section we wish to look at the different dynamical regimes from the field point of view. We thus need to compare the memory loss time T to the field fluctuation frequency. Taking these regimes in turn we have: 3.1. Quasistatic In theory, this is the regime where the microfield is constant during T. In practice, it is the regime where we can decompose the microfield into a large constant field plus a small amplitude (much smaller than the constant field) fluctuating one. Although of no practical consequence, it is important conceptually to keep in mind that even a slight deviation from the static case changes the broadening picture from an inhomogeneous broadening picture, with a continuously distributed set of J-functions to the uncertainty-type broadening of the homogeneous picture. From a point of view of smoothness, the microfield is extremely smooth. As we discuss later, it is not always possible to tell apriori if this approximation is satisfied, hence any ion dynamics theory must be able to handle this case. 3.2. Impact

In this regime the microfield is produced by weak and nonoverlapping strong collisions. There are frequent sharp features. However, we can cheaply and reliably tell if this approximation is valid. Therefore, although clearly desirable, it is not imperative for an ion dynamics theory to include this case. However, we will see that it is possible, and, in fact, advantageous, to do so and doing so in fact facilitates our task. The impact approximation is further discussed from the field point of view in Appendix A. At this point one should note that for the two regimes discussed analytic solutions exist. This will be exploited later on. 3.3. Strong dynamic (ion dynamic)

The intermediate regime between quasistatic and ion impact is called ion dynamic, although “strong dynamic” may be more appropriate from our point of view, for two reasons: One may wish to consider, as the field point of view is in effect doing, an external stochastic field, such as a turbulent field. In addition, such a regime exists for electrons also, at higher densities, although at those densities other effects also become important. zz The ion dynamic regime is character&d by overlapping strong collisions. The microfield is fairly smooth on the T time scale, but not quite

Spiros Alexiou 176.0

-260.0

-323.0 0

2

I

3

4

5

t (ps) Fig. 2. Ionic electric fields for one configuration in an Argon plasma of temperature 16,000 K and density 2 x lO”e/ctn’. The solid, dashed-dotted and dashed lines represent the x, y and z-components of the electric field.

constant. Figure 2 shows a typical purely ionic microfield for the parameters of the Grtitzmacher and Wende experiments.’ The electron density is n = 2 x 10” e/cc and the temperature is 0 = 16,000 K, with Argon perturbers. These are the parameters used for every numerical result in this work. Figure 2 is to illustrate what we mean by “smooth” in this case. 4.

THE

MAIN

IDEA

The key quantity is the microfield: for each configuration, its time evolution, determines via the Schriidinger equation the U-matrix and hence the contribution of this configuration to the autocorrelation function. Hence, the microfield determines C,(t). However, if the microfield is smooth, then knowledge of it at just a few time points fixes the microfield for all times. Consequently, it should be possible to express C,(t) in terms of an integral over the (stochastic) values of the microfield at a few time points. This is very similar to the point of view taken by Dufty,23 according to whom, “An accurate determination of the time dependent electric field for given initial conditions would determine the influence of the plasma environment on the radiator, with a self-consistent treatment of both Stark broadening and Dicke narrowing. There are two main problems in implementing this point of view. The first is an adequate theory for the dynamics of the electric field; the second problem is to solve the internal atomic equations of motion for a given field history and average over initial conditions.” It is the second problem, i.e. solving the internal equations of motion and expressing C,(t) in terms of these few microfield values, that we will be mainly concerned with here. We note that the simple idea of fitting the microfield by a polynomial does not work due to the time ordering problem. In the present method, solving the Schriidinger equation is replaced by solving a linear algebraic system; in principle this may be done in closed form, in which case we have an explicit solution to the problem. 5.

THE

METHOD

Whereas until now we have been referring to the (electric) microfield, we will formulate the method in terms of the interaction V(t). This enables us to go beyond the dipole approximation. The physical model is that we have an optically active electron perturbed by a time-dependent perturbation Hamiltonian (in the interaction picture) V(t). We are not interested in the effects of the atom on the plasma field. This is certainly justified for say the experiments of Refs. 1 and 2,

Collective coordinates for ion dynamics

5

but not for strongly coupled plasmas. This is the only restriction on the interaction, i.e. that it does not depend on the state of the emitter. Other than this, the method makes no assumption on how this V(t) is specified, i.e. via an independent particle model7-10 or via MD.” It only requires the interaction at a specified set of points and this is, as far as the method is concerned, practically an external input. In the calculations used in this work, the independent particle model” has been employed, which is perfectly satisfactory for the parameters examined. For simplicity, we will restrict ourselves to the l-state case (Lyman lines), which show the largest ion-dynamical effects. We also take V(t) to vanish outside (0, T). This is not an approximation: If we are interested in the U-matrix U(r) only up to a time T, we may take V(t) to be zero outside the interval (0, T). Since U(t) = v*(-t)

(1)

we only need U for positive times. [The fact that (U(t, t,)) = (U(t - t,)) enables us to consider only U(t, 0) which we write as U(f)]. The lineshape is given by the Fourier transform of the correlation function (2) /I is the 1s state, a and u’ are upper level states, U is the usual U matrix and (. . - ) denotes average

over the plasma. If we let V(r) be the plasma-emitter U satisfies the (Schriidinger) equation

interaction in the interaction picture, then

s I

U(t) = I - f

0

with Z the unit matrix.

We therefore

W 1u(f, 1dt,

introduce a “retarded”

Green’s function

G(t)= T C(t)U(t). Note that in that case Go(t)

=

T

(5)

e(t)z

G [in view of (3)] satisfies the Dyson equation , G(t) = Go(t) + Go(t)

= Go(t) +

f -m

W, )G(b) dt,

m df,Go(r -rt)W,)G@,). I -a,

(6)

In Fourier space this reads GW=GoW+%

1

*

s

_m dw’G,(o)V(o

- o’)G(o’).

(7)

Equation (7) is exact. All we have done thus far is to give up the Schradinger equation in favor of the equivalent Dyson integral equation, which is a rather uncommon practice.24 A possible problem with Eq. (7) is the long tails of (the diagonal elements of) G and G,(o). Since in the high frequency limit G and Go have the same l/o behavior, we define quite generally

g(o) = G(o) - Gas(o)

(8)

where G,, is a truncated asymptotic expansion for G. If one wishes to work with G directly (i.e. is not bothered by the long tail), one may set G, = 0, but more to the point is to set G,,(o) = G,(w); it is possible to use even more general G,,, i.e. a part of the asymptotic expansion for G(w). Note that in general G,, involves only V(t = 0) and V(T) and its derivatives. V(T) arises only because in our model V(t > T) vanishes and this in general requires discontinuous derivatives. All order partial resummations are possible, as in the quasistatic solution, which is obtained by an all order

Spiros Alexiou

6

resummation of the terms in the asymptotic expansion that do not contain derivatives of V(t). The use of Ga8= GqUPIT is essentially the same idea as Dufty’s subtraction of the static part.25 Equation (8) correctly handles the off-diagonal elements too, since these have no long tails. Their asymptotic behavior is at worst (i.e. if we set G,, = G,) l/o*, because the off-diagonal elements of G(t = 0) vanish and hence in the asymptotic expansion derived by integration by parts we get the aforementioned leading order. Now Eq. (7) reads

m do’ _-oo271 f’(o - oQg(w’) + G,(o)

g(o) = G,(o) - G,,(w) + G,(o)

O” do’ F’(o - o~‘)G,,(o’). -m 2A s (9)

We now introduce a yet unspecified positive R, split the first r.h.s. term and write

s

n do’ _n 211 J% - cM0’)

g(o) = G,(a) - G,,(o) + G,(o)

-

V(w - o’)G,(o’)

+ E(o)

(10)

where E(o) is the error term

snm~[Y(w--w’)g(w’)+Y(w+w/)g(-w.)]

E(o) = Go(o)

G,(w)[OrdrY(t)eti’{:z

=

[g(o)‘e-‘““+g(-o’)e’~].

(11)

Equation (10) is still equivalent to the Schriidinger equation. Clearly, as R-P co, E(w)+O. We now define a truncated g

* I

s i-l

g(o) &-&J = -9 - 2n e ‘ofi dw = B(ti) + ,?(ti) + -9

B(b) = and

s*y s

By Fourier-expanding

G,,(o) +

-R

n

JW =

[G,(o) -

do G,,(o)e-‘“‘i n do’ -9% Vu - ~‘)S(~‘) 211 s

do

- o’)G,(w’)]

(13)

+ V(0 + o’)g( -o’)].

(14)

e-‘W’i

Go(w)

2n

-l-l

do’V(o Im -m

F

(12)

- o’)g(o’)

we see that the Error term is

ss m

E(t,)

=

-

f

00

dt,

dt,M(ti, tl ;fl)e-"zV(t,)g(t2)

0

0

sidn(h -

w,

-

t2)l

t2) Ml

-

t2)

1

(1%

where the 6 is infinitesmal and M(t,,

r,; Q) =

1

5 -t

%a -

h)

II

Wit, - 4 I)

(16)

with Si the sine integral Si(x) =

‘dtT s0

sin

t

(17)

I

Collective coordinates for ion dynamics

and sgn(x) the sign of x. The effect of M is to ensure that g(r) is determined primarily (i.e. if R is sufficiently large) by earlier times. It is, of course, undesirable that we have some information propagating backwards, i.e. gn at earlier times is partially determined by later times, but of course this effect diminishes as R increases. This effect arises from the sharpness of our cutoff procedure. We will discuss the error term later. To proceed, we need to specify G,, and for concreteness we adopt the simplest possible nontrivial model, namely G,,(w) = G,(o). Equation (10) must be solved nonperturbatiuely and we therefore approximate the kernel of Eq. (10) by a separable Pincherle-Goursatz6 one. Equivalently, we approximate V(o - 0’) by a quadrature rule2’ 03 T dt V(t )e’(” - w’)’ V(0 -co’)= dtV(t)e I(0- 00 = s0 s --1o =;

(18)

~k(W)Yk(~‘)

where Yk(w) = e-‘“‘k

(19)

X,(w) = wkV(tk)e’o’k

(20)

and

wk and tk are weights and abscissas respectively in some quadrature rule, which (in principle, since we do not wish to use an adaptive version) may be specified after we have drawn the configuration and have some information on V(t). Clearly such a quadrature rule cannot be good for arbitrarily large jw - 0’1. In other words, large frequency transfers are not “damped”. The solution to this problem is that we will use a quadrature rule such that V(o - o’) is accurately represented for (w - w’\ d 20. We thus have V(0 - 0’) = 1 w, V(tk)e”t(“-“‘),

[CO- ~‘1 < 20.

(21)

k

Equation (12) is now separable and we can solve it as an algebraic system. With these choices we have cc e do B(ti) = _* 211 Go(m)e-‘“‘l dtV(t)e’“‘G,(t) s -m f

We also define

If E(ti) can be either neglected or approximated analytically, in which case we can absorb E(t,) in B(ti) and A, Eq. (12) becomes the linear system (Z+A)g,=B

(24)

where the matrices act on combinations of atomic states and time indices. We will use greek letters to denote atomic state indices and latin to denote the abscissas explained above. Equation (24) thus reads (6,,..6ik+ A;“)gr(f,)

= Baa’&) = By’.

(25)

Let Np be the number of quadrature points. Since A does not depend on tl’, our problem reduces to the solution of in principle A’,, where iv, is the number of states contributing to the broadening of the level, linear systems of dimension N, x No, with the same matrix A and different Bs. For our specific case of L - a one can exploit the sparsity of the matrix and one only has to solve an

Spiros Alexiou

8

No x No system plus a few more operations, provided the Error term can be neglected. Having calculated gn(ti), we then obtain G(w)=G,(o)+g(o)[8(w

so

= s

-fi)+e(-(O

m dt e’“‘g,(t) s -m

+a))]+


(26)

if we are only interested in frequencies (01 < R, g*(t) is all we need. However, gc(,) is only determined in (0, T). One may show however, that (assuming that (G(lwI 2 Q)) = 0) (G(w))=K(o)+S:dre’“((g.(r))_-~) where K(o)

(27)

is the conjiguration-independent quantity elT(+N_

l(W - 0’)

=

e’wJJ+“)

1

G&o’) + Got--o’)

_

1

I(W + 0’)

-G,(w) Si((Q + o)T) - Si((n - o)T) II

2 a+w Ci((Q + w)T) - Ci((n - o)T) -In R-o

=-_I

-

2

. )I1

(

Ci(RT)-Ci((R-w)T)-ln&

(28)

Note that in the limit of o +O, K(w)-r(sin QT)/(hxR). The meaning of 0 and the approximation in Eq. (21) are now clear: we are interested in a separable kernel approximation to the second integral of the last term in Eq. (12). Since in this term o ranges from -0 to n and so does w’, we need a quadrature rule for V(w - o’) valid for 10 - w’l < 2Q. As for R, since we compute g*(t) rather than g(t), we want the transform of gn to be simply related to the transform of g, i.e. essentially the transform of the U-matrix. This is where the requirement that the line profile is negligible for frequency separations larger than R comes in and this fixes, at least for now, the choice for R. 6.

THE

ERROR

TERM

Both B(ti) and E(ti) can be written as

s T

dtF0(ti

0

3 t,

a)J(tv Q)

(29)

where FO(tj, tg f2) =

-$ V(t)M(tip

t; 0)

(30)

and J(t, 0) is the unit matrix for B(ti), while for E(ti) we have J(t, 0) =

=

ss 03da’

n

q-

m

dt’e-“‘[U(t’)

- I]cos[w’(t - t’)]

0

dt’e-“‘[U(t’)

-I]

s(t’ - t) - “~~~~,‘))

1 .

(31)

9

Collective coordinates for ion dynamics

A sufficient condition for neglecting the error term (which would be highly desirable) is then that .Z & I. One idea is simply to use a large R, so that the Error term will be very small and thus negligible. This however will require more quadrature points to preserve the quality of the quadrature approximation, at least theoretically speaking. Another idea is to substitute g(o’) in Eq. (14) by some asymptotic expansion. The problem with that is that the theory of asymptotic expansions has concentrated on what the asymptotic expansion is and not when it is “reached”, whereas we need to know for a given finite R how good this expansion is. Since g(o) is an oscillatory integral, it will be small due to substantial cancellation if a-’ is small enough compared to the time scale of variation of g(t). This observation brings us to the issue of the quasistatic time scale, discussed in the next section. For our purposes we note that when the asymptotic expansion is “reached” is determined by the largest V(t) giving rise to the fastest variation of U(t), as seen from the Schriidinger equation dU - -$ V(t)U(t). dtHence the largest V(t) will determine when the asymptotic expansion is reached, although the expansion itself is determined by the endpoints. This way one finds the criterion hC2B Vmax(f), which is also not too helpful. Therefore another approach must be chosen. First, it is trivial to show that

s

mdw‘

n

z

[g(o')e-'""

+

g( -o’)e’m’l

= g(t) -g*(t).

(33)

This may be expressed as

s m

g(t) -&-2(t) =

dt, e-“,g(t,)

t, ))

sin(n(t -

s(t - t,) -

nc(t -

0

4)

1

This equation means that g(t) and gn(t) are “close” if g(t) only rarely varies appreciably on a time scale of R-l. These rare occasions cannot give rise to a significant discrepancy between g(t) and g,(z), since both g and the integral of s(t - I,) -

sin(Q(t - t,)) n(t - t,)

[

1

are bounded. The idea now is to write the term in brackets as a correction from the vanishing infinite R result. Since the 6 - function is not an ordinary function, we must bring the test function into play to work out the difference for finite R. We thus substitute the Taylor expansion g(t,) = g(t) + 0, - t)g’(t) + . . .

(35)

in Eq. (34) and obtain by keeping only the first two terms (36) An improved version of Eq. (35) is to write

gOI) = e-riq)(t,

- “lhg(t)

_

;

[e.-rtw(~l

-w _

I],

(37)

which assumes that the field is quasistatic over (t, t,), which must be of the order of a-, or smaller. The improvement over Eq. (35) is that Eq. (35) was valid for a given (t, t,) interval only for V(t) small enough, whereas Eq. (37) is valid for arbitrarily large V(t), as long as it is reasonably constant over (t, t,). The validity of neglecting the variation of V(t) on a time scale of R-’ is examined in the next section. Note that this approach is applied here over a shorter time scale R-, rather than T. However, one should note that, at least for a dipole interaction, there is a heavy price to pay for including even the linear term in Eq. (35): the resulting (modified) A -matrix is no longer sparse

Spiros Alexiou

10

and inversions of an No x N, matrix instead of just No are required. One may therefore ask whether it’s worth using Eq. (37) or Eq. (35), or whether it might be preferable to use a larger 0, such that the Error term may be either neglected or approximated by the zeroth order term in Eq. (35). For the case of L - a, this approach will be preferable if one needs to increase R by a factor less than about 4 in order to obtain equivalent accuracy by neglecting the error term or using just the zeroth term in Eq. (35), since the required number of points scale (theoretically speaking) linearly with R. Based on Eq. (35) or Eq. (37), we can obtain an expression for the Error term E(fi) =

T

s T

0

df, UL TV; Q>W, ) k@,>- gn0, )I

(38)

in terms of g,. Hence we have an explicit evaluation of the Error term, which results in a modification of A and B in Eq. (24) and Eq. (25). The actual expressions are given in Appendix B. R is thus seen to play a double role: on the one hand it is the frequency separation at which the lineshape has decayed to a negligibly small value and on the other the inverse of a time scale where the fields are constant. Hence, for now, it is taken to be the largest of the two (i.e. inverse quasistatic time scale, maximum energy separation of interest). The quadrature rule depends on the product W’, which is approximately constant if R is the maximum energy separation. However, in the impact regime, the inverse quasistatic time scale becomes very large and the quadrature very inefficient; of course, it is well-known that ordinary simulation methods are also inefficient (at least compared to the analytic impact solution) in the impact regime. In the next sections we will discuss a way to get around this problem. 7. THE

QUASISTATIC

TIME

SCALE

The

question that will be addressed in this section is: on what time scale may ions be regarded as quasistatic? There are at least two reasons for attempting to establish what the quasistatic time scale is. For standard impact electron/quasistatic ion Stark-broadening calculations, it is desirable to be able to simply and reliably check the validity of the quasistatic approximation, without doing an ion-dynamical calculation (this would be desirable, even for methods such/as the MMM and Line Mixing Model, which can perform the ion dynamical calculation fairly cheaply and reliably), which would only be done if deemed necessary. In this respect the usual criterion** HWHM

$f$

(39)

i.e. R $ uo/(p) with u. = (2k@/p)“*, CLthe reduced emitter-perturber mass, 0 the temperature and (p) the mean ion separation, is not too helpful because of the 9 sign. Simulations show that even a factor of 10 is not always sufficient. This is illustrated in Fig. 3 for the x, y and z-components from 10 configurations, drawn according to the Hegerfeld-Kesting” algorithm. The percent deviations from quasistatic behavior (i.e. from a horizontal line) are respectively 6.7, 193, 15, 14, 29, 129, 9, 15, 28 and 7.4, resulting in an average deviation of 45%, for the x-component, 21 .l, 35, 3.3, 6.5, 56.6, 11.7, 4, 7.6, 10 and 5.8, resulting in an average deviation of 16%, for the y-component and 4.8, 30, 4.8, 76.2, 10, 11.8, 2, 39, 34 and 0.5, resulting in an average deviation of 21%, for the z-component. These numbers are slightly misleading, since the largest components change most slowly, and for the largest field component, the average percent deviation is 12%. The fact remains, however, that even for the largest component, the deviation was larger than 30% for 2 out of 10 configurations. Secondly, in conjunction with the present work, the quasistatic time scale comes up in the evaluation of the error term. In the calculation of the Error term, the only problems can arise from a large V(t) that will force the U-matrix to oscillate on a R-’ scale or even faster. This may be easily seen from Eq. (34), since due to the

I et-4)1

S(l_t)_sin(Q(r-t,))

II

Collective coordinates for ion dynamics

factor in the integrand, only times t, within R-’ oft are important. A large but constant l’(t) over this time scale is correctly handled by Eq. (37), i.e. the r.h.s. of Eq. (34) may be expressed in terms of g(t), yielding an explicit relation between g(t) and go(t). This in turn allows us to express g(t) -g*(t) as a function of g, in Eq. (38), so that the error term may be expressed in terms of a linear function of g,, leading to the modification of the A and B matrices. Hence it is important to determine the quasistatic time scale R-‘, over which we can use Eq. (37), since we start having problems when V’(t) 2 al’(t). Intuitively speaking, this can only be a frequent situation in the impact regime, where the fluctuation frequency is larger than the width. The point is that in the ion dynamical regime, 0-l is of the order of T, but it is important to be precise. The problem that must be addressed is the following: Large fields are usually the result of a single particle coming close to the emitter, so we must calculate how many such events we have per configuration on the average. Using a shielded interaction, one obtains for the electric field components parallel to the impact parameter E, and velocity E, due to a single perturber dEJdt

~

E

= JP’+;2(l

_Jl-3(w)‘-(‘“‘:,‘J)‘(l

+Er(t),i,)‘]-$

(40)

as t +ti with t, the time of closest approach r(r) = [p’ + v2(t - tJ*]“*

and 5 a shielding parameter”.

dE,ldt=_ E

(41)

Also

Jp2+~2(~_r,)2[3v2~2~~)+(~~~-~i)(~+Sr(~)/~D)~1]_0

(42)

as t-+ti. We should be able to neglect fluctuations on the Q-’ time scale if we have on the average one perturber with v/p 2 C2in a large number N of configurations. However, only perturbers with times of closest approach in (0, T) are relevant here. Following the Hegerfeld-Kesting” model, this leads to the following equation, which determines Q 2,/kNTR2nv,S(R)

< 1

(43)

where S(Q) is the probability of finding a perturber with v/p larger than R. Thus, if we have 1 perturber in N = 100 configurations with v/p 2 R, we may expect roughly a 1% accuracy, or better. There remains to compute S(w), defined as S(o)=

ss R2pdp -

o

R2

mdv

zv3 ,-+6

4

op

(44)

within the (straight line trajectory and Debye-shielded field) Hegerfeld-Kesting” model, which for the purposes of this analysis is equivalent to Seidel’s improvement,2g S is simply S(w) = S,(x) =

2 - (2 + x2)e-X2 x2

where x = (oR/v,)

(4’5)

with R x 3AD and In is the Debye length. Figure 4 shows So vs x. Figure 4 is not too encouraging, since according to it, in order to have only one perturber in 100 configurations with v/p > R and time of closest approach in (0, T), R must be larger than 5 x lOI Hz, whereas the line profile is practically negligible for R an order of magnitude less! In the above analysis we did not distinguish between weak and strong fields. Hence, if for a particular R, we have No perturbers per configuration with v/p, larger than R, a fraction of them

12

Spires Alexiou

will give rise to weak fields, for example less than or equal to a less than average field, such as the Holtzmark field E,, defined quite generally as (47) where nk and zk are respectively the density and charge of the kth ionic species and e is the electron charge. Such weak fields will, on the one hand, be overshadowed by the total field due to the action of all perturbers and, on the other hand, not contribute appreciably to the error term integral. Hence we obtain a modified S-function by allowing the upper limit of the p-integration to be the

(a)

-I

9-

-

I 0

1

2

0

I

2

3

4

s x 10-2

3

4

5 x 10-2

-1

-2

t (PS)

Fig. 3(a) and (b). Legend on opposite page.

I3

Collective coordinates for ion dynamics

I-------

L 9

2 d

4

10

3

3

2 t

4

5 x lo-’

(ps)

Fig. 3(c) Fig. 3. Plots of E,)E, (a), EJE, (b) and &,/I$, (c) For the parameters of Fig. 2 up to times a factor of 10 smaller than the inverse experimenta Stark HWHM due to ekctrons and ions for 10 configurations. Each line is iabelfed by the configuration number.

impact parameter corresponding to EOIWe thus obtain the probability S, to have a v/p larger than Qand E>E, (48) with pH the average distance (distance corresponding to Hoitzmark field). As expected (and also shown in Fig. S), this leads to a dramatic difference for small Zz,but makes no difference for large iz: large fields are well known to result from close collisions and these are the ones responsible for the high frequency components. 1.0

r

0.9 0.8 0.7 0.6 2 w. VI

0.s 0.4 0.3 0.2 0.1

0

Fig. 4. Probability to fmd a pertutber with u/p 2, w vs x = wR/u,, For the parameters of the numerical example, x 5 0.356 x [a-'J 0.

14

Spiros Alexiou

0

I

I

I

I

2

4

6

8

R (x

lOI Hz)

Fig. 5. In(S,,) (solid line) and ln(S,) (dashed line) vs R for an argon plasma with an electron density of 2 x 10”e/cm3 and temperature of 16,000 K.

One may argue that Eq. (43) is rather pessimistic because T is chosen large, so that C(t) has effectively “died” at t = T. However, a collision with large u/p occurring at times larger than about the inverse HWHM should not have much of an effect: cancellation due to averaging of out-of-phase U-matrices will be strong with or without such events. This argument may save a factor of two or three at the most, which is still not quite good enough. The fact remains that the quasistatic approximation is hard to justify statistically, even when applied only to the error term. However, it is hard to accept that the presence of a few close perturbers in a huge number of more distant ones should force us to use the quasistatic time scale rather than the lineshape decay scale for R. This is the object of the next section. 8. SEPARATION

OF SMOOTH

AND NONSMOOTH

COMPONENTS

As we saw in the last section, the obvious brute force answer is to just increase 0, but this increases the work needed for the system solution. We here propose a different idea. Remember how we treat dynamic line broadening: The net interaction is split into electronic and ionic interactions. We solve each one separately and convolve the profiles [multiply C(t)]. The usual justification for this procedure3’ is that normally one or the other species dominates. There is an exception to this, namely a narrow region exists close to the quasistatic ion regime, where ion and electron broadening are equally important. For example, in this region, using an impact operator for electrons misses out processes as la)+(ion

collision)lcr”)+(electron

collision)(cc”‘)-+(ion collision)lo!‘)+.

..

Even there, not much difference is expected. This separation of the net interaction into an electronic and ionic part is standard in Stark broadening and employed in virtually every method: in the standard theory3’.32(q uasistatic ions and impact electrons), in the MMM ‘*Jo(“Mother and Baby”), in simulations,‘-” in the line mixing methodI and in the semianalytic method.” Checks of this approximation (by Kesting’O via simulations and by Frerichs33 via the MMM) support this approximation. However, these checks were made for conditions where ion broadening was dominant. Gigosos and Cardenoso34 and Olchawa, Halenka and Grabowski3’ have also performed such joint electron-ion simulations, but their aim was not to check and they have not checked the validity of the separation approach. In addition, the last group seems to have missed

Collective

coordinatesfor ion dynamics

15

the very important Collision-Time Statistics method of Ref. 10. The point here is that separation is the rule and these four works are the exceptions. From the field point of view, there is no difference whether the interaction is due to electrons or ions. The standard separation discussed above in effect separates the net interaction into two components. We now proceed to do the Same with the ionic V(t) alone, namely separate it into smooth and nonsmooth parts. In other words, we split V(t) as V(u/p < 0) and V(v/p > Q). The first part is smooth, and is correctly and efficiently handled by our quadrature. For the second part, we have sharp, isolated collisions, so that the Impact approximation is applicable. The physical picture is clear, but the above statement may be made completely rigorous: If the fluctuation frequency Q > HWHM, the impact approximation is valid 36. For our purposes, we note that the total HWHM is also due to the smooth component, while n should be compared to the HWHM due to the nonsmooth component alone, which is much less, if, we are far from the ion impact regime, as in the numerical example considered. One may thus calculate the nonsmooth width as3’*32 cc PH Impact HWHM = 2nn uf(o)du... P dp s0 s nP (49) which may be expressed in terms of exponential integrals. However, one should note that it is especially important to enforce unitarity when using such a formula, as we are integrating over only small impact parameters, which would give rise to strong collisions, for which perturbation theory is not valid and other treatments” must be employed. In the ideal, “textbook”, impact case, where weak collisions dominate, this part of phase space [i.e. (v, p) space] gives a small relative contribution and its accurate evaluation may be less of a concern. For our purposes, which include computing the nonsmooth contribution, this is not necessarily the case. However, to the extent that we are far from the ion impact regime, the nonsmooth contribution should be smaller than the smooth one, hence its accurate evaluation is not a big concern. In other words, for a given R, the relative contribution of the nonsmooth (impact) component increases as we move towards the impact regime. Provided the impact contribution may be accurately computed, it is then necessary for an e#icient implementation, i.e. in order to minimize the dimension of the algebraic system to be solved, to include as large a part of the phase space as possible into the nonsmooth component. It should be emphasized that this is a question of efficiency, not a limitation of the method, which works with any R large enough that the lineshape has decayed to a negligible level. With this separation, our approach is valid from the quasistatic to the ion impact regime. With regard to the separability assumption, Fig. 6 shows a comparison between the exact C(t) and the product of the smooth and nonsmooth autocorrelation functions, illustrating the fact that this approximation is an excellent one, at least for the specific case of Fig. 6, which is, once again, a case where the smooth component is dominant. 9. NUMERICAL

RESULTS

All calculations were done in the dipole approximation, for purely ionic fields and, as already mentioned, for an electron density of n = 2 x 10” e/cc, Temperature 0 = 16,000 K, and with Argon perturbers. Figures 7-8 show a comparison between the exact result (obtained by solving the Schrcdinger equation) and the present method for the configuration shown in Fig. 2. For this particular run, the present method was faster than an Adams predictor-corrector’Ob Schrhdinger solver within the fastest method of Ref. 9, which we point out is only valid for hydrogen, but no general claim of this kind is made in this paper, since much depends on the accuracy requested and the present method is not optimized. Figure 9 shows a lOO-configuration average. With regard to convergence, this was checked by comparing the 25, 50, 75 and 100 configuration results. Convergence was excellent for t < 3psec. and afterwards, when C(t) had dropped to small values it was still good. One may see from this graph that good agreement is obtained between essentially all forms of this method and the exact

Spiros Alexiou

16

c u

0.4 -

0.2 -

O-0.1

0

I

I

I

I

I

1

2

3

4

5

t (PS)

Fig. 6. 100~configuration C(r). The solid line is an exact calculation and the dashed line is calculated by solving for the smooth component exactly, solving for the nonsmooth component exactly, and multiplying the resulting autocorrelation functions.

calculation. It is clear, however, that for 100 configurations, the noise level is unacceptably high (i.e. leads to a large relative error) for the values of C(t) close to T. To make the picture clearer, Figs. 10 and 11 show the same lOO-configuration C(t) for the total (smooth and nonsmooth) field and smooth component only, respectively, while Figs. 12 and 13 show the respective line profiles, normalized to unit peak height. These results show quite good agreement, except perhaps for long times, for which convergence was not as well achieved as for shorter times. For these long times, one might expect that the out-of-phase U-matrices will give cancellation essentially independently of the method of calculation. The differences in the autocorrelation function for large times also make their presence felt in the profile. If, as conjectured above, more configurations lead to the minimization of the long times differences, then one would obtain better agreement for the final profiles. With regard to the line profiles, they have

0.6 T= u

0.4 -

0.2 -

O.-

1 0

I I

I

I

2

3

\‘.;.I.’ 11 4

I 5

t (es)

Fig. 7. C(t) for the configuration of Fig. 2. The solid line is exact and the dotted and dashed lines are calculated according to the present method by using the quasistatic approximation for the error term and by neglecting the error term altogether, respectively

Collective

coordinates

17

for ion dynamics

18.10

-I .92 I 0

i 2

I 4

I 6

Aw (x 10” Fig. 8. ],TdrC(r)cos(Awt)

vs Ao for the configuration

1 8

I 10

Hz)

of Fig. 2. The solid line is the exact calculation and

the dotted and dashed lines are calculated according to the present method by using the quasistatic approximation for the error term and by neglecting the error term altogether, respectively.

been calculated as the integral from 0 to T of C(r)cos(Aot), since our interest here is to test the method. This, as is well-known [and may be easily verified by using a Lorentzian C(t) with a finite and infinite integration domain respectively], results in “ripples” in the profiles. The problem of a finite T is one that every simulation must deal with. Methods to overcome this problem for the Fourier transform are discussed in the literature.‘Ob Before leaving this section, we give some numerical details to help the reader gain a better understanding. Since the corresponding numerical analysis is still in progress, these details are preliminary and are meant to give some idea as to how the parameters of this work are to be calculated.

--

--

0.2 -

oI

I

I

0

2

3

I 4

I 5

Time (ps) Fig. 9. lOOconfiguration ionic C(t). The double dash-double dot line is the nonsmooth component, the solid line is the exact C(f) and the dash-dotted line is the exact C(t) for the smooth component. In the calculation according to the present method for both (smooth plus nonsmooth) components, the double dashed-dotted line neglects the error term and the dotted one uses the quasistatic approximation for the error term, Eiq. (37). In the calculation with only the smooth component, the dash-double dotted line neglects the error term, while the dashed line uses the quasistatic approximation.

18

Spiros Alexiou

0.8

0.6

l-

0

I

2

3

4

5

t (PS)

Fig. 10. 100-configuration ionic C(I). The calculations are done for the total interaction (smooth plus nonsmooth). The solid line is the exact calculation and the dashed and dotted lines are the results of the present method, neglecting the error term and treating it with the quasistatic approximation, Eq. (37), respectively. The dash-dotted line is the nonsmooth component.

The memory loss time T was taken to be Sps since, from the Hegerfeldt and at this time the (purely ionic) autocorrelation function has dropped to 1%. parameter and in any case, it must also be specified in any simulation. If one T should be, one can get bounds as explained earlier. To get a better estimate, a calculation with few configurations to see what T to use.

Kesting’O results, T is not really a

has no idea what one usually does

0.8

0.6 = u 0.4

0.2

0

0

1

3

2

4

5

t (PS)

Fig. 11. lOO-configuration ionic C(r). The calculations are done for the smooth component of V(r) only. The solid line is the exact calculation and the dashed and dotted lines are the results of the present method, neglecting the error term and treating it with the quasistatic approximation, Eq. (37), respectively.

Collective coordinates for ion dynamics

19

1.0

0.8

0.6

3 .t: T d z? x .Z

0.4

i z -

0.2

0

0

4

2

6

8

10

Ao (x IO’* Hz) Fig. 12. IOO-configuration ionic line profile. The calculations are done for the total interaction (smooth plus nonsmooth). The solid line is the exact calculation and the dashed and dotted lines are the results of the present method, neglecting the error term and treating it with the quasistatic approximation, Eq. (37), respectively.

Regarding Q, if one assumes an approximately

Lorentzian profile, with HWHM @, and defines z 4.6. Hence the line profile has dropped to, say, the 1% level when T to be the time at which C(t) has dropped to, say, the 1% level, then @T = -ln(l%)

4p

-=_ @*+w

0

2

1

loo@

4

6

8

10

Aw (x lo’* Hz) Fig. 13. 1OOtonfiguration ionic line profile. The calculations are done for the smooth component of V(t) only. The solid line is the exact calculation and the dashed and dotted lines are the results of the present method, neglecting the error term and treating it with the quasistatic approximation, Eq. (37), respectively.

20

Spiros Alexiou

i.e. R x lo@ = 46/T. Based on these estimates, n was chosen to be lOI Hz. Indications are that the profile is not perfectly Lorentzian and the 1% level is reached at smaller R. With regard to the quadrature rule, an N,-point Gauss-Legendre or Radau-Legendre rule may be used. Typically, N-point Gauss-Legendre rules work well” (say up to 2-3 significant digits, see pp. 100, 153-155 of Ref. 27) for oscillatory functions of the type exp(rot) over an interval (0, T) provided that N 2 (N, To)/(27r) where N, is the number of points needed for such accuracy over the interval (0, 1) for the integration of exp(2rcrr). In practice N, is 2 or 3 and we estimate the number of points needed to represent V(o) to the above accuracy for lo]< 20 as (N,RT)/(n), i.e. from 30 to 45 points, according to our previous QT = 46 estimate. Since V(r) is also oscillatory, perhaps more points are needed. This remains to be investigated in detail and it appears that the number of points is not as crucial as the choice of R. In numerical work, 64 points were used, confirming, for a few configurations, that the quadrature rule gives excellent approximations to the exact V(o - w’) for Iw - w’] not exceeding 2R and that the number of points used was more than enough. Preliminary tests indicate that for R sufficiently large, the number of points is of minor importance. For example, with 50,64 or 80 points, and fl twice as large as in these runs, one obtains the same results for the configuration of Fig. 2 and, incidentally, perfect agreement with the exact calculation. 10. COLLECTIVE

COORDINATES

A quest in problems such as line broadening with overlapping collisions is to find a smaller set of variables (collective coordinates) on which the solution depends, rather than all phase space variables of all the particles. An existing method that may be formulated in this way is the Model Microfield Method (MMM)‘2*‘3 dE,,dE,*..dE,dt,

C(t)=

**.dt,C,(t,

V(EO,...,

E,,t ,,...,

fN))

s

PN(Eo...EN,t,,...,t,)

(50)

where V(t) is a step (“digitized”) potential of the form V =

1 d . E[(8(ti+, - t,) - e(t - ti)]

(51)

I

and P, is the joint probability of having an electric field E,, at t = 0, E, at t = t,, . . . , E, at fN. In the MMM, a Markovian approximation is used for P,. One may thus view the MMM as a collective coordinate method in which the average is done as an integration over the collective coordinates Ei and ti. In our case =

s

dV,...dV,,P,(V,,V,,...,

b&V + A([U))-‘BWI)

=((Z+A)-‘B)

(52)

B, A and Z + A are functions of the plasma microfield V, evaluated at these quadrature denoted by V,, Vz, . . . , V,,, while

P&V,

9

vz,. . .

f

3

VN~)=

JWWW(W,

t,) - VI). . . d(V(R tNQ)- V,ve)

with [R] denoting configuration and P(R) the probability autocorrelation function is given by a similar expression C(t) =

s

points,

dV, . . .dVN,P,,(V,,

density for this configuration.

Vz,. . . , VNe)G(t,

,

V,, VI,. . . > VNv).

(53) The

(54)

Of course, since we have an initial value problem, C, (2, V, . . . , V,,,a) should depend on K. only if ti < t.

Collective coordinates for ion dynamics

21

Whether this approach will prove practical to carry out is, of course, another question, although the dimensionality of the problem has been reduced. There are remaining problems. First, the C, is, in the language of many-body physics, an infinite-body operator, since it involves a matrix inversion of the one-body operator V(t), and the usual many-body methods do not apply. In addition, only Pz, i.e. the two-times probability distribution to find a field E, at t, and E, at tZ has been studied.38,39On the other hand, Markovian models (i.e. assuming that the probability of finding an electric field E, at t,, E2 at f2, . . . , E, at t, is a product) may naturally make our task much easier, as in the MMM. In fact, there should be no reason for a Markovian approximation to be any worse here than in the MMM. 11. CONCLUSIONS

Taking the field point of view, we have analysed the ion dynamical problem and by splitting the interaction into a smooth and a nonsmooth component, we can obtain in principle analytic solutions for the entire parameters range (quasistatic to ion impact), by using a quadrature rule for the smooth and an impact approximation for the nonsmooth component. Thus, for the smooth component, instead of solving the SchrBdinger equation, we solve a linear system of dimension (number of states) x (order of some quadrature rule). The method appears to be competitive with ordinary differential equation (ODE) solvers and also works well for the (single configuration) ODE solution for fields varying on a slower than R-’ scale. Any differential equation solver can eventually run into this problem that is, if we have a sharp enough spike - for a given tolerance - on an otherwise very smooth r.h.s. of an equation dx/dt =f(x, t), we can fool the usual integration methods, but it will be harder to diagnose. We have also outlined the approach and problems to a collective coordinates attempt at an analytic solution. The many-times probability distribution P,(E, , t, , . . . , E,, fN) is, rigorously speaking, required; however, simple Markovian methods should be no worse than in the MMM and FFM. The method, being essentially as exact us simukztions and also (in some sense) “analytic”, should be useful as an alternative to ODE solvers in simulations and also in the checking of the approximations in the other models (MMM, FFM). Whether it can become competitive with them from a speed point of view, will very much depend on the success of its optimization. Thus, the choice of Q, use of quasistatic approximation or neglect of the error term, optimization of the matrix solution, number of quadrature points and especially the definition of the borderline between smooth and nonsmooth components will all have an impact on optimization. Note that the definition of the borderline can be improved, since we can include in the nonsmooth impact part more of the phase space than we are doing at present. The issue is then estimating the width produced by a nonsmooth component with u/p > R as a function of R, in order to determine R as the root of the equation HWHM(R) = R, provided this is larger than the Q at which point the lineshape has decayed to a negligible level. Work on optimization is currently underway. Acknowledgements-The conference.

author would like to thank the organizers and Y. Maron for their help with attending the

REFERENCES D. E. Kelleher and W. L. Wiese, Phys. Rev. Lett. 31, 1431 (1973). K. Gfutzmacher and B. Wende, Phys. Rev. A 16, 243 (1977); Phys. Rev. A 18, 2140 (1978). E. W. Smith, Phys. Rev. 166, 102 (1968). D. Voslamber, Z. Nuturfosch, 24a, 1458 (1969). E. W. Smith, J. Cooper, and C. R. Vidal, Phys. Rev. 185, 140 (1969). U. Fano, Phys. Rev. 131, 259 (1963). R. Stamm and D. Voslamber, JQSRT 22, 599 (1979). (a) R. Stamm, E. W. Smith, and B. Talin, Phys. Rev. A 30, 2039 (1984); (b) R. Stamm et al, Phys. Rev. Left. 52, 2217 (1984). 9. M. A. Gigosos, J. Fraile, and F. Torres, Phys. Rev. A 31, 3509 (1985); M. A. Gigosos, V. Cardenoso and F. Torres, J. Phys. B 19, 3027 (1986); V. Cardenoso and M. A. Gigosos, Phys. Rev. A 39, 5258 (1989). 10. (a) G. C. Hegerfeldt and V. Kesting, Phys. Rev. A 37, 1488 (1988); (b) V. Kesting, Ph.D. thesis, Gijttingen (1987). 1. 2. 3. 4. 5. 6. 7. 8.

22

Spiros Alexiou

11. R. Stamm, B. Talin, E. L. Pollock, and C. A. Iglesias, Phys. Rev. A 34, 4144 (1986); F. Khelfaoui, A. Calisti, R. Stamm, and B. Talin, 10th lnt. Conf on Spectral Line shapes Austin, Texas, L. Frommhold and J. Keto, eds. (1990). 12. A. Brissaud and U. Frisch, JQSRT 11, 1767 (1971); J. Math. Phys. 15, 524 (1974). 13. J. Seidel, Z. Naturforsch. 32a, 1195, 1207 (1977). 14. J. W. Dufty, in Spectral Line Shapes, B. Wende, (ed. de Gruyter, New York (1981). 15. D. B. Boercker, C. A. Iglesias, and J. W. Dufty, Phys. Rev. A 36,2254 (1987); D. B. Boercker, in Spectral Line Shapes (9), J. Szudy, ed., Ossolineum (1988) and Spectral Line Shapes (II), (1992). R. Stamm and B. Talin, eds, Vol. 7, Nova, New York. 16. A. Calisti et al, JQSRT 51, 59 (1994). 17. R. L. Greene, Phys. Rev. A 19, 2002 (1979); JQSRT 27, 639 (1982); J. Phys. B. 15, 1831 (1982); R. L. Greene, D. H. Oza, and D. E. Kelleher, in Spectral Line Shapes (9), J. Szudy, ed., Ossolineum (1988). 18. A. Calisti et al, 10th lnt. Conf on Spectral Line Shapes, Austin, Texas, L. Frommohold and J. Keto, eds., Vol. 6, APS, New York (1990). 19. E. W. Smith, B. Talin, and J. Cooper, JQSRT 26, 229 (1981). 20. R. L. Greene, JQSRT 27, 185 (1982). 21. E. W. Smith, R. Stamm, and J. Cooper, Phys. Rev. A 30, 454 (1984). 22. S. Gtinter, in Spectral Line Shapes, R. Stamm and B. Talin, eds., Vol. 7, Nova, New York (1992). 23. J. W. Dufty, in Spectral Line Shapes, R. Stamm and B. Talin, eds., Vol. 7, Nova, New York (1992). 24. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge Univ. Press (1985). 25. J. Dufty, Phys. Rev. 187, 305 (1969); Phys. Rev. A. 2, 534 (1970). 26. F. Tricomi, Integral Equations, Interscience, New York (1957). 27. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd edn., Academic Press, New York (1984). 28. D. H. Oza, R. L. Greene, and D. E. Kelleher, Phys. Rev. A 38, 2544 (1988). 29. J. Seidel, 10th lnt. Conf on Spectral Line Shapes, Austin, Texas, L. Frommhold and J. Keto, eds.; Verhandl (1990) DPG(VI), 25, 372 (1990). 30. J. Seidel, private communication. 31. H. Griem, Plasma Spectroscopy, McGraw-Hill, New York (1964). 32. H. Griem, Spectral Line Broadening in Plasmas, Academic Press, New York (1974). 33. M. Frerichs, Z. Phys. D 11, 315 (1989). 34. M. Gigosos and V. Cardenoso, J. Phys. B 20, 6005 (1987). 35. W. Olchawa, J. Halenka, and B. Grabowski, in Spectral Line Shapes, R. Stamm and B. Talin, eds., Vol. 7. Nova, New York (1992). 36. S. Sahal-Brechot, Astron. Astrophys. 1, 91 (1969). 37. M. E. Bacon, K. Y. Shen, and J. Cooper, Phys. Rev. 188, 50 (1969); M. E. Bacon, Phys. Rev. A 3, 825 (1971); M. E. Bacon, JQSRT 12, 519 (1972). 38. A. Alastuey, J. Lebowitz, and D. Levesque, Phys. Rev. A 43, 2673 (1991). 39. J. W. Dufty and L. Zogaib, Phys. Rev. A 44,2612 (1991); J. W. Dufty and L. Zogaib, Phys. Rev. E 47, 2958 (1993).

APPENDIX A Impact

Approximation

It is worth reformulating the impact approximation (IA) in terms of a field point of view (no “collisions”) in order to develop a better intuition for the applicability of this approximation. Let * be the exact propagator (U-matrix) and {a* a} denote a perturber average, so that 6’~ is the quantity of interest. The essential ingredients of the IA are the following: 1. We are interested in the not too short-time behavior of C(t). This automatically means that the impact approximation is inapplicable for short times (wings). For such (not too short) times t, we can always pick a At 4 t. 2. We assume a statistical independence of averages of U over At, i.e. {*}={*} 0 t 0 Since At $ t, t = N At, with N+co.

{*}......{*}={*}~A’ At At 2At

Thus,

{*}=l-OAt=l-@t/N 0

At

t -At

t

0

t.

Collective coordinates for ion dynamics

23

and { * } = (1 - @t/N)N = exp( - @t). 0

t

3. If r is a (strong) collision duration, then to ensure that collisions occurring closer to At than a collision duration t are negligibly rare events, one requires that t + At. Assuming that the equivalent of one (strong) collision is enough for memory loss, we estimate the half-life, or memory loss time T 2 (HWHM))’ from the following qualitative equation

s

T&Vl(t)

0

x

h

]

(AlI

where F’(t) is the interaction and V’(t) the interaction in the interaction representation. This says that the total perturbation is strong and the expansion parameter is of the order of 1 and cannot be treated by perturbation theory. We must emphasize that Eq. (Al) should not be taken as a quantitative estimate, since it neglects the matrix nature of the problem and may be wrong by as much as two orders of magnitude. Rather, Eq. (Al) must be understood as simply stating that no signz@cant memory loss is possible if the 1.h.s. of Eq. (Al) is G 1. A more common form of (Al) is

642) Eq. (A2) is, in fact, used in unitary checks and minimum impact parameter determinations in standard impact calculations (often erroneously for small impact parameters, since a small integral does not necessarily mean a small integrand). Of course, in the impact theory, perturbation theory is never applicable over time T due to Eq. (Al), but the interaction is weak and a great number of collisions must occur for any appreciable memory loss in the sense of Eq. (Al). Hence, T is much larger than a collision time and one can find an intermediate time scale At which includes many collisions, while being much smaller than T, so that perturbation theory is valid over this intermediate time scale. One then calculates the average perturbation per unit time in action units - i.e. the 1.h.s. of Eq. (A2) over T - by calculating in perturbation theory the average over this intermediate time scale and arguing that the two averages must be the same, since we have many collisions. The reason why the above reasoning does not carry over to the ion dynamical case is that the “many collisions” and memory loss time scales are similar and no such intermediate time scale exists. Naturally, Eq. (Al) or Eq. (A2) as they stand are meaningless, in the sense that memory loss is an averaging effect and exactly what z means cannot be clarified without reference to the statistics. They may be understood to refer to a “typical” interaction strength V(t). From a practical point of view, Eq. (Al) is also useful, in that it shows that our matrices in the quadrature method are well-conditioned. We now wish to clarify the issue of statistical independence. To do this, we need to distinguish between weak and strong “collisions”. For a weak collision completed in (t, , tz), & (V(t)1 dt/h 4 1, by definition. Of course then the integration limits may be extended to the entire real t-axis. Schematically, we have the following cases: Case 1. We have only weak collisions in each At and their cumulative effect is weak. If the second condition is not the case, one may choose a smaller At. This is the ideal case, since we can evaluate the averge over At by perturbation theory. Case 2. We have weak (and with a weak cumulative effect) plus only one strong collision in At. In this case, the weak collisions may be neglected. If the collision duration of the strong collision is much shorter than At, we can replace the U-matrix between the limits of our interval of length At by an S-matrix and we clearly have statistical independence, since the perturbers effective in one At interval are not effective in the next. Note however, that one cannot use the usual perturbative impact formulas and must try the approach of Ref. 37.

24

Spiros Alexiou

Case 3. We may have strong nonoverlapping collisions in At. In this case, simply choose a smaller At that will only include one strong collision and we recover case 2. The fact that we have at least two nonoverlapping strong collisions in At means that we can decrease At without seriously violating r <
B

Reduction to Computational Form In this Appendix we give the formulas for the modification of the A matrix and B vectors in Eqs. (24-25) for the L - a line, if one does not neglect the error term, but instead evaluates it using the all-order quasistatic approximation of Eq. (37). We thus write B+B + B’

(Bl)

A+A-A’

032)

where the A-matrix and B-vector on the r.h.s. are defined by Eqs. (24-25). We employ spherical states (nlm) and denote 11) = 1210), 12) = (21l), 13) = (21 - l), 14) = 1200). With these conventions we first define the matrix a(&), with the matrix elements

‘11= ’ + [EI(ti)+ E;(tj)lf:(ti)/D(ti)

IIWi) -E(h) + iEy(ti) A,, =-;i,, =qtiylf:tti) d IID(ti) a

12 =

Ex(ti)

-;i31

=

+

iEy(ti)

Ez(tiY:(ti)

(I33) (B4)

G

I.14 = A,, = iE,(ti&(ti)/D(ti)

(I361

A,, = --A,,

Y422 = a,,= 1 + a,, = a:, =

A, = -i-4,, =

Ei(ti)

E:(ti)+

(B7) +

E:(ti)

Z(ti)/D(ti)

2

Ez(ti) - E;(ti) - 2iEx(ti)Ey(ti) :(ti)/D(ti) 2 lf

-a,: = A$ =

-fi(ti)

(B5)

Ey(ti)

+ Jz

iE*(ti)

IID(ti)

(Bg)

(I391

(BlO)

A, = l/D(ti)

(Bll)

D(t) = 1 +f:(tW2(tlf(t)

(B12)

where

with

r,(+;+y?

(Bl3)

and 3u, cos(cu) h(f)

= fi(t)hRn

.

(Bl4)

Collective

coordinates

25

for ion dynamics

E,(t), E,.(t), E(t) and E(t) are the electric field components in the x, y and z directions and the electric held magnitude respectively. With these definitions, we have

A, (t, t ) = _3&M(b II 19 k

O%~z(~k)&(fk)

A,

rkhEz(tk);i42(fk)

(t, 12

19

1 ) = k

_3iWkM(li?

tk)

-

A,

ct, 14

19

=

1 ) = k

rkhE:(tk)a43(fk)

(li,

fk)

(Bl7)

h

_3iWkMtri,

rkbOE:(rk)(;i44(rk)

-

l>

U318)

h

3wkM(&y A;,

0316)

h

3iwkMb ff;3(fi,

@15)

h

tkbO@y(tk)

+

i&(fk));i41(fk)

+

iE,(tk));i42(tk)

(Bl9)

= &h 3wkM(b

-4;20,,

fk)

fkh(Ey(rk)

0320)

= Ah

@21) 3wkM(ti, A$(ti,

tk)

rkh(EJ(tk)

+

i’%(tk))(;i44(rk)

-

1)

(B22)

= @

(~23) 3wkM(ti, A&(ti,

tk)

tkh(E,.(tk)

-

i&x(rk));i42(tk)

-

i&r(rk));i43(tk)

-

iE.r(tk))(;i44(rk)

(B24)

= JZh

A,

(t, 33

t 17

1 =

3wkM(li,

tkbO(Ey(rk)

0325)

k JZh

A,

(t, 34

A;,(ri, fk) =

3wkM(ti,

tk)% h

fk~

=

3wkM(riv

tk>“O(Ey(tk)

i&(&)(1 - A,, (fk)) + i

Ex(rk)(a2,

(rk)

-

3wk(ti, =

rkh

-i&(tk)&,(tk)

+

i

‘%(fk)(;i22(fk)

-

3wkM(fi, tk)% =

1)

(B26)

(fk))

A32(rk)

(lk)

-

+

I,,

1

@k))

l>

h

-

fk)

I,,

Ey(fk)(~21

-

?k)

-

I9

h

-i&(tk)a!3(fk)

+

i

Ex(tk>(a23(1k>

E,.(tk)(a22(tk)

-

_E,.(tk)(a23(tk)

a33(fk)

+

+

a32(tk)

-

l)

-

I)

1

l>

+a33(rk)

1

26

Spiros Alexiou

Ah4(ti,

lk)

3vM(ri,

rk)&l

h

=

-iEz(r,);i,,(rk))

+ i

~x(rk)Urk)

-L(h))

& _qrk)(&drk)

Jz

+&ok))

.1 (B30)

For the B-vector, we only need to solve the systems with B,, and Baz, since the autocorrelation function only involves the diagonal matrix elements of the first two states. Defining f3(r) = cos(nr)/(hRlr)

U331)

we obtain B;,(ti) = -i C WkM(ti, fk~(tk)a44(tk)(3aoE,(t,)/h)’

(~32)

k

2 Ey(tk) B;I

(ti)

=

1

+

@&k)

(B33)

rk~V;(tk);i44(rk)Er(fk)(3a,/h)

wk”(ri, k

d

k

B;I

(Ii)

=

-i

c k

Wk”(ti,

Ez(rk);i14(tk)

fkV;(rk)Ez(tk)(3aO/h)2

Ex(fk)(a24(rk)

-

-

aM(rk))

-

iEy(tk)@24(tk)

-‘!$@k) B;2(t/)

=

(3%/h)‘c

wkM(li,

+

+

%&k))

&tfk)

1

(B35)

(B36)

rk&(tk)a44(tk)Ez(rk)

k

‘%@k) B;2(ti)

=

-i(%/h)2

c

WkM(tiT

fk&f)rk);i44(fk)

+ 2

E;(fk)

(B37)

k

2Ex(fk)‘f$(fk) B;2(fi)

=

-(3%/h)2

C k

WkM(fi,

-

:(E:(t,)

-

‘$ttk))

cB38j

fk&f3(fk)&4(rk)

Ez(rk)~,4(fk) B;2(fi)

=

(3&/h)2

c

Wk”(ti,

fk&V;(rk)(-Ey(tk)

+

iEx(rk))

k

fi

-

Ex(tk)(a24(rk)

-

&@k))

+

2

iEy(rk)(a24(tk)

+

h&k))

1

cB39j