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Overview of plasma line broadening
High Energy Density Physics 5 (2009) 225–233
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High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Review
Overview of plasma line broadening S. Alexiou University of Crete, TETY, 71409 Heraklion, TK 2208, Greece
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 June 2009 Accepted 3 June 2009 Available online 12 June 2009
We review the basics of line broadening, its relation to fluctuations and disorder, what causes broadening, the memory loss mechanism and the Standard Theory of line broadening developed by H.R. Griem and others from a modern viewpoint. This modern view benefits from many years of progress and includes a coherent theoretical perspective without the need for a conceptually different view of electrons and ions. Both electrons and ions are described in terms of their random fields. This modern and unified view allows, among other things, extending the range of validity of line profile calculations to complex situations. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Stark broadening Fluctuations Impact theory
1. Introduction: broadening and fluctuations For any atomic system embedded in a medium, the interactions between the atomic system and the medium result in a modification of the energies and lifetimes of the atomic system. From this point of view a plasma is just another medium where the interactions (usually Coulombic) are well-known, where simple approximations (e.g. a static picture as in the solid state) often fail and in which a large range of essentially very different medium behavior is spanned. Broadening is in general associated with fluctuations and randomness is essential. Conversely, broadening effectively ‘‘measures’’ fluctuations. By ‘‘fluctuations’’ we simply mean, for example, that different atoms in a plasma see a different interaction. Thus, randomness is a requirement for broadening; it is a general feature and not specific to plasmas. The above remark can be illustrated by two unrelated, but rather well-known examples: The first is Doppler broadening, where we need a distribution of velocities: A single velocity yields a Doppler shift, but no Doppler broadening. The second example is powder diffraction in crystals: The Bragg condition is nl ¼ 2dsinq with n an integer, l the radiation wavelength, d the plane spacing and q the angle of coherent scattering. Imperfections, i.e. variations in d, will result in both stress and strain. This means we will have some d that are larger and some that are smaller than their perfect crystal counterparts, resulting in shifts of q both left and right and hence broadening of the peaks. The width of these peaks therefore is a measure of crystal perfection, or equivalently disorder. This is true in general, i.e. the larger the fluctuations, the larger the widths and may be understood as follows: we can decompose the interaction into a mean term plus a fluctuating one.
The entire Hamiltonian with the mean, nonrandom, term (but not the fluctuating part) in principle may be solved to give a set of infinitely sharp energies, while the fluctuating part is a measure of disorder and will give broadening, i.e. a lifetime. Historically the view taken was that slow ions provide a static, but random, Stark field that gives rise to an inhomogeneous broadening, just like Doppler broadening: each emitter would see a different Stark field and as a result would experience a shift, just as in Doppler broadening; because these Stark fields are random, one would have a distribution of such shifts and integrating over those random fields or equivalently, shifts would result in a broadened profile. Electrons, on the other hand, would interact via fast collisions, giving rise to homogeneous broadening. Although this view enabled significant progress and actual calculations, it hampers a coherent theoretical construct as it separates from the beginning the interactions. In the modern view the effect of the plasma on the atom is coherently described for all perturbers, electrons, ions, but also collective fields, by putting the emphasis on the fields. This view, presented below, allows the ion and electron generated fields to be treated consistently, includes the older view and allows us to treat more complex cases that could not be treated by the older view. 2. The lineshape and autocorrelation function For dipole radiation, the line profileL(u) is the interesting part of the intensity of the emitted radiation I(u) [1–3]:
IðuÞ ¼
1574-1818/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.hedp.2009.06.003
(1)
where L(u) is the lineshape:
LðuÞ ¼ E-mail address: [email protected]
4u4 LðuÞ 3c3 X
d u uif hf jdjiij2 ri
if
(2)
Here the states i and f in the above equation refer to states of the entire system (medium plus atomic system and d and r are defined below). In the modern framework we often work with the Fourier Transform of the lineshape, which is the autocorrelation function C(t). Fourier-expanding the d-function, we see that the transform of L(u), C(t) is defined by:
CðtÞ ¼
hdð0Þ$dðtÞi jdj2
¼ cðtÞ
(3)
where C.D indicates the quantum and statistical averages and d(t) is the atomic dipole operator in the Heisenberg representation. The c(t) refers to the contribution to C(t) from a single random realization of the medium, i.e. the time history of that part of the medium that affects a single emitter. Equivalently, one may consider a single emitter in the medium, pick a distribution of initial plasma particle positions and velocities relative to the emitter and, in general, let them evolve; this is one realization or configuration. As shown in Fig. 1, by this term we understand an entire history of the interaction between the plasma and a radiating electron (emitter), as seen by a single emitter (different emitters see different histories, of course). The idea is that if we have M uncorrelated emitters, the intensity of the radiated light will be multiplied by M, but the lineshape will not change [4]. A broad lineshape in frequency means a narrow C(t) in time and vice versa. It is important to note that in order to get a finite width in the line profile (i.e. not a d-function), C(t) must decay to 0 for long times. This is obvious since the lineshape is a Fourier transform. The decay of C(t) to 0 is only possible over time scales for which we have very small or negative contribution for some individual c(t)’s. We define T as the time after which C(t) is negligible, i.e. the time scale of C(t). Technically c(t) is a linear combination of time evolution operators for the upper and lower levels. If jaD, ja0 D and jbD, jb0 D are atomic eigenstates (in practice jaD, ja0 D are eigenstates of the upper level and jbD, jb0 D of the lower level) for the transition in question, we have: y
cðtÞ ¼ Uaa0 ðtÞda0 b0 Ub0 b ðtÞ$dba ra
(4)
where U(t) is the time evolution operator in the interaction picture, d is the dipole moment and ra the probability of finding an emitter in state a. Given this definition of c(t) and Eq. (3) it is clear that the time dependence of C(t) is determined by the average time dependence of the U-matrix product. This seemingly simple representation of the lineshape is all we need in order to compute the lineshape exactly. Hence the central problem is the calculation of the U-matrix product and the average over the plasma.
3. Qualitative discussion of line broadening: the memory loss mechanism Fig. 2 shows the c(t)’s from different configurations for a transition with a strong unshifted component. The initial state of the atom or ion emitter is defined by a set of electrons populating the different orbitals. The transition of interest involves one of these electrons changing or ‘‘jumping’’ to a different orbital. This electron is the optically active electron and the one whose wave functions we are concerned within the line broadening problem. Essentially U(t) is the identity matrix at t ¼ 0 and its diagonal elements start decaying, meaning that the interaction with the plasma produces transitions and the optically active electron need not be found in the same state it was at t ¼ 0, and the final state may no longer be empty. At short times we have from the Schro¨dinger equation:
Electric field components (arb.units)
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
time Fig. 1. Example of a single field realization of particle configuration, i.e. the time history of the interaction between plasma and emitter as seen by a single emitter. In this example a dipole interaction has been assumed and hence the electric field components Ex, Ey and Ez are shown as solid, dotted and dashed-dotted lines, respectively.
dU ı ¼ VðtÞUðtÞ dt Z
(5)
a drop in U(t) DU z iV(t ¼ 0), with V(t) the interaction, which indicates that larger interactions result in faster decays. C(t) is the average of these c(t), so that for a given time t one needs to sum the c(t)’s and divide by their number. As mentioned above, to avoid d-functions, C(t) must decay to 0 for long times. Obviously no decay is possible until most of the individual c(t)’s have themselves decayed to values close to 0. But once U(t) and c(t) has dropped to essentially 0, we have random phases from the different realizations c(t) so that the average of c(t) over these realizations vanishes. Random phases mean that the optically active electron is uncorrelated at time t to its state at t ¼ 0. Thus, the probability amplitude of finding it at time t in the same state it was at t ¼ 0 depends critically on the interactions to which it has been subjected and that probability amplitude is quite small when averaged over different field configurations. This is referred to as ‘‘memory loss’’ and explains why C(t) is a measure of memory
1 0.9
Autocorrelation Function C(t)
226
short times, drop is proportional to the interaction
0.8 0.7 0.6 0.5
long times, random phases
0.4 0.3 0.2 0.1 0 -0.1 -0.2
Time(t) Fig. 2. Examples of c(t) (vertical axis) vs. time(horizontal axis) from different configurations. From its definition, c(t) starts as unity for t ¼ 0, can only start decreasing for times longer than 0 and displays oscillatory behavior for long times. It is the random phases of the different c(t)’s that produce the decay in time of C(t).
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
may have a ‘‘spiky’’ field, which corresponds to a high temperature, low density case which translates in the particle picture to having very short collision durations compared to T, or a much smoother field, which corresponds to a high density, low temperature case in which the collisions are ‘‘on" all the time. If T is short enough we may have a situation where the field essentially does not change on the T time scale. The picture just described is essentially correct, except for one complication: To produce memory loss a field must be ‘‘sufficiently’’ random. For a dipole interaction, a one-dimensional field is not sufficiently random for a transition with an unshifted component, because such components are completely unaffected by a one-dimensional field. This can be illustrated in the following manner: in the spherical basis a state jn l ¼ n 1m ¼ (n 1)D is modified under the interaction with a one-dimensional dipole field V(t) ¼ E(t)z as
1 0.9 0.8
C(t)
0.7 0.6 0.5 0.4 0.3
t
Fig. 3. Typical C(t) for a transition with a strong unshifted component under the action of a one-dimensional dipole field. This particular transition is the hydrogen Balmera(Ha) i.e. the transition between the principal quantum numbers n ¼ 3 and n ¼ 2. This is a transition with unshifted components, because for a transition between states n ¼ 3, l ¼ 2, m ¼ 2 to n ¼ 2, l ¼ 1, m ¼ 1 neither the upper, nor the lower level is affected by a constant field in the z-direction, i.e. these two states are not perturbed by any plasma field if we only allow the perturbation of a state by other states with the same principal quantum number. Notice therefore that C(t) cannot decay past the intensity of its unshifted component, which is about 38% in this particular case.
loss. On the contrary, at short times c(t) and C(t) are close to unity and the atomic system has lost little memory of its initial state. For completeness one should point out that the natural lifetime is typically much larger than the C(t) time scale, T or any time shown in Fig. 2. In order to gain a better understanding of time scales, one may note that the vast majority of configurations have a c(t) that approach very small values, i.e. z0, before some time T. When this happens, details of further time development are largely irrelevant (we will shortly discuss the exception to this rule), because random phases will give a vanishing C(t). One may very roughly estimate T as the time scale where U(t) becomes approximately 0 for the first time:
UðtÞz0
(6)
or equivalently
ZT
V 0 ðtÞdt z1 Z
227
dhnl ¼ n 1m ¼ ðn 1ÞjUðtÞjnl ¼ n 1m ¼ ðn 1Þi dt z nn 1 ðn 1ÞEðtÞznl0 m0 nl0 m0 UðtÞnn 1 ðn 1Þ (8) However, m-selection rules for the first term on the right hand side require m0 ¼ (n 1). But it is only the highest l, n 1 which can have such an m-state and l-selection rules, jDlj ¼ 1 do not allow l0 to take such a value. Hence the derivative is 0 and the state in question is unaffected by the external field. So for example if we have a dipole transition between an upper and a lower maximum m states, this component is unshifted and unaffected by a onedimensional field. Therefore, c(t) and C(t) cannot decay to values lower than the relative intensity of these components and one cannot avoid d-functions, unless some other broadening mechanism helps. No matter what its shape, no matter what its statistics and distribution function, a one-dimensional field is not ‘‘sufficiently’’ random in this case. This is manifested by a C(t) that does not decay past a certain level, e.g. 66% for the La(n ¼ 2 / n ¼ 1), 38% for Ha (n ¼ 3 / n ¼ 2) etc. This means that for such a field we would obtain d-function lineshapes upon Fourier transforming. Fig. 3 shows a typical C(t) for a line with a strong unshifted component if the perturbation is a one-dimensional field. This specific graph was obtained with frozen ions alone, i.e. ions that were fixed in their starting positions for the course of the calculation.
(7)
0
(b)
Electric field at emitter
with V0 the interaction in the interaction picture. This is in fact the lowest term in the Dyson [5] expansion and is to be interpreted in the sense that for times much shorter than T perturbation theory is valid and hence the deviation from the initial value of the evolution operator, U(t), is much less than 1. Hence for such times, U(t) z I and consequently c(t) z 1. This means that for such times, C(t) will be the average of quantities that are z1 and hence will itself be not much different than 1, which means practically no memory loss for such times. Therefore, one qualitatively sees that it is not the peak intensity of the interaction, but its action integral, Eq. (7), that counts and we do not have significant memory loss as long as that action is small compared to -. So it becomes clear that the action of V0 (t) determines the relevant time scale T. The main issue in plasma line broadening has thus been redefined into what is the behavior of the interaction V0 (t) on this memory loss time scale T, which in terms of the line profile is roughly of the order of the inverse of the halwidth at half maximum intensity (HWHM). This question is crucial in deciding how to treat the problem at hand. Essentially we
0
(a)
Time Fig. 4. Different temporal behavior of V(t) over the T time scale. We may have a spiky field (a), a smooth field (b) or even an essentially constant field (b, if the time scale is the shaded area).
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
When we do have the case with an unshifted component we get, of course, a finite width. But this is only because there are always broadening mechanisms, such as multi-dimensional fields, Doppler broadening, natural broadening etc. Often these mechanisms are less effective, which means they produce memory loss on longer time scales. This will be discussed below in more detail where we will show how to generalize the previous qualitative picture to include such cases. For the moment we merely note that any static field is one-dimensional and falls in the above category, while the converse is not true. As will be seen shortly, the behavior of V(t) over the T time scale is of paramount importance for treating the broadening: Certainly, the action of V0 (t) over T must be of the order of -, but there are different ways of achieving this, as shown in Fig. 4. It could be achieved by: 1) a fairly constant V0 (t), presented as field (b) when taken over the time scale indicated by the shaded area; 2) by a highly ‘‘spiky’’ field, (a) in Fig. 4 that consists of sharp features over an essentially zero background; or by field (b) that is something in between.
0,5
0,4
Intensity
228
shifted line δ(ω−CΕ)
0,3
0,2 Probability for field E
0,1
0
0
2
4
6
10
8
Δω Fig. 5. Quasistatic profile as an integral of d-functions over the microfield distribution.
4. The quasistatic approximation Like most disciplines, one normally starts with the simplest possible approximations and refines them as they start to be challenged by experiment. In case of line broadening the simplest approximation is the quasistatic approximation: The idea is that C(t) drops to very small values on such a short time scale that a certain field component, normally associated with the ions, have no time to move and hence appreciably modify their field. Therefore the time evolution happens during a static field, which is different for each configuration, but still static over T. Later we will see how this statement holds in practice. The quasistatic approximation assumes stochastically static ions, that interact with the atom via their static electric, Stark, fields. In other words we describe the medium in terms of configurations, each of which corresponds to a net ionic field that is static on the autocorrelation time scale. Since the ionic positions are random, the total (static) field is a continuous random variable E. The only quantity one needs then is the probability density Q(E) to find a field between E and E D dE. In usual isotropic situations we are only interested in the probability density W(E) ¼ 4pE2Q(E) (so R R that dEQ ðEÞ ¼ dEWðEÞ) to find a field of magnitude between E and E þ dE. If we have the W(E), then as shown in Fig. 5 each field produces a shift and the profile is just the integral of shifted d-functions over all fields, each d-function weighted by the probability density for the corresponding field.1 For example for linear shift, the otherwise unbroadened transition in a single constant field E is just d(u CE) with C a constant as shown in Fig. 5 and the profile is2:
LðuÞ ¼
ZN
dEWðEÞdðu CEÞ ¼
1 u W C C
(10)
0
Thus, in the case of a linear shift the quasistatic line profile has the shape of the distribution W(E) itself. Note that this vanishes at the
1 However, this conceptual picture does not carry over to the other regimes: it is only applicable for quasistatic behavior and unable to account for possible nonstatic field components. 2 More generally, if J(u,E) is the lineshape for a static Stark field with magnitude E, then. Z dEWðEÞJðu; EÞ (9) LðuÞ ¼
center (u = 0) because the probability of finding a zero field is vanishingly small, reflecting the improbability of having the ion charges arranged so as to exactly balance each other at the emitter. It is clear, therefore, that all that is required to calculate the line profile in the quasistatic approach is to compute the plasma-related quantity Q(E). The quantity Q(E) can be calculated using:
Q ðEÞ ¼
Z
3
3
3
d r1 d r2 .d rN Pðr1 ; r2 ; .; rN Þd E
N X
! Ei
(11)
i¼1
with P the probability density to find ion 1 in position r1, ion 2 in r2 etc. In Eq. (11) the d-function ensures that only those arrangements contribute that give a net field equal to E. So computing Q reduces to answering the questions: What is P and what are the Ei, i.e. what is the field experienced by the emitter due to the presence of the ith perturber. The initial, and simplest, answer was given by Holtsmark [6], who assumed pure Coulomb fields Ei and uncorrelated perturbers, i.e.
P ¼ V N
(12)
with V the volume, which is a statement that all configurations are equally likely. For concreteness, this calculation of the quasistatic microfield distribution is provided in Appendix A. Eventually, at a time when both theoretical and experimental advances enabled reliable tests of the theoretical predictions, screened fields were used and correlations were accounted for [7,8], which for most plasma conditions of interest very satisfactorily resolved the issue. The net result of these improved calculations is, qualitatively speaking when compared to using the Holtsmark approach, that large electric fields are less probable due to both ionic repulsion and the effect of screening by the surrounding electron cloud. This leads to microfield distributions that have peaks which occur at lower field values than those of the Holtsmark distribution, which is a high temperature limit. Currently [9] such calculations are extremely fast and accurate, so that the computation of the distribution function is considered a practically closed subject for all cases, including extreme matter conditions [10,11]. However, in many experimental situations large gradients are encountered and more generalized distributions are required [12]. The main problem with the quasistatic approximation has already been described in the previous section: Static fields are one-dimensional and by themselves cannot broaden transitions with an unshifted central component. This implies that one has to
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
ZT 0
V0 dt[
ZT 0
Vel ðtÞdt[
ZT
dVðtÞdt
(13)
0
with Vel the electron-emitter interaction. This situation is depicted in Fig. 6. In this case we may as well neglect ion motion, even for a line with an unshifted component. This will not be possible if the action of the fluctuating ion component is comparable to or even dominates the electronic action (as is the case for low densities). So we improve our qualitative understanding provided in the previous section by stating that for lines with a strong unshifted component that are unaffected by static fields, it is the action of non onedimensional fields that matters for achieving a decay of C(t) beyond the relative intensity of the unshifted component. In contrast, for lines without such an unshifted component (for example Lb, i.e. n ¼ 3 / n ¼ 1 and Hb, i.e. n ¼ 4 / n ¼ 2), the static picture for ions normally works well when one is interested in estimates of the width and not the details of the shape. This is due to the fact that the microfield distribution is 0 for E ¼ 0, so that these lines would have 0 intensity at the line center according to a quasistatic RN calculation. Clearly then for u ¼ 0, Lðu ¼ 0Þz 0 CðtÞdt ¼ 0 which means that C(t) cannot be positive for all times. In Fig. 7 it is shown that because there is no unshifted component, the action of the static field may be exploited in full to give a fast drop in C(t). The effect of the dynamics is to make the negative region shallower, and hence fill in the central dip because now the integral of the positive region is larger than the one of the negative region. Because the action of the static field is very effective in this case, dynamical effects that occur at longer times are normally relatively unimportant for the bulk of the line. In short because of the dynamics the oscillations at long times produce a damping of the long time behavior of C(t); since dynamics sets in at the times for which C(t) is negative, this results in a less negative C(t) and a filling of the central dip.
Fig. 6. Comparison of the ionic and electronic field actions: While the ionic action dominates the electronic one as it is stronger, if we decompose the ion field in a static (dotted line) and a fluctuating part, the electronic action dominates the fluctuating ionic action.
5. Impact approximation 5.1. Introduction: what an impact field looks like The impact approximation is applicable when memory loss is the result of collisions with duration much less than the memory loss time T. It is then straightforward to see that such collisions must be mostly weak. Recalling our earlier remarks that the action of the interaction, Eq. (7), is the determining factor in the effectiveness of the broadening, the weakness of collisions can be defined by the following condition:
Zti þs
V 0 ðtÞdt 1 Z
(14)
ti
Here ti the start of the collision and s its duration. Clearly this is a necessary condition because otherwise memory would be lost on the time scale of one or a few collisions and then the above relation would read z1, which is by definition excluded. Therefore memory loss is a ‘‘team effort’’ instead of arising from the interaction of a single collision. Nevertheless, there will be configurations dominated by a single collision, which, although rare, have a significant effect because then c(t) drops on a much faster time scale. Indeed, the treatment of these rare events has been an issue. On the other hand, whether we have a rare strong collision or not, the action of the impact perturbers is additive. This is clear because collisions do
1 0,9 0,8
L-beta C(t)
0,7 0,6 0,5
C(t)
rely on other broadening mechanisms and the total time scale will be determined predominantly by these other mechanisms in these cases. For example if electron broadening is strong enough to make C(t) decay before ions have time to move appreciably, then the static picture works fine. If not, the quasistatic approximation may underestimate the width by as much as an order of magnitude [13]. So, whether the quasistatic theory is valid in any given case does not depend on ions alone; it also depends on electrons. In fact it depends on all broadening mechanisms, electrons, (quasistatic) ions, Doppler and natural broadening. For the quasistatic approximation to hold, all these mechanisms must have reduced C(t) to very small values before the ions have the time to move appreciably. Coming back to the discussion of the previous section, here we have a case where the action of ions may well dominate the action of electrons, because of the longer duration of the ionemitter interactions. Because it is the total action that counts for memory loss, a constant field will normally produce more action than a spiky field that is zero, or very small, most of the time. Nevertheless, if we write the ion field as Vion(t) ¼ V0 þ dV(t), with V0 the static part and jV0 j[jdVðtÞj, the action integral3 of the electrons (or, more generally any other broadening mechanism, for instance turbulent fields) may still dominate the action integral of dV(t):
229
0,4 0,3 0,2 difference
0,1 0
dynamic
-0,1 -0,2
quasistatic 0
0,05
0,1
0,15
0,2
t(ps) 3
For simplicity we consider only hydrogen-like lines here, where V(t) is the same in the interaction and Schro¨dinger pictures.
Fig. 7. Differences between a quasistatic (dashed) and an exact dynamical calculation for a line without a central component, in this case, Lyman-b. Note that the differences are small and only nonzero for times at which C(t) has dropped to small levels.
230
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
not overlap in time, so that we effectively have a one-body problem instead of a many-body problem. For strong collisions we may effectively neglect all other collisions. Fig. 8 shows a typical ‘‘impact’’ field. Displayed is c(t) (top curve) as well as the electric field components Ex, Ey, Ez in arbitrary units. Note the sharp drop in c(t) where the field is significantly larger than the background. These ‘‘spikes’’ correspond to collisions with a particle, an electron in our case. However, over the duration of such collisions the drop in c(t) is always much less than 1, e.g. even the largest drop at t z 600 ps is only about 0.2. In other words, collisions are weak in a perturbative Dyson expansion sense. Therefore, in the impact regime memory loss is produced incrementally, each collision contributing its (normally small) share. However, the impact theory is not simple perturbation theory; it usually employs perturbation theory for what is known in manybody physics as the self-energy: it simply says that the effect of a single collision is typically small, not that the effect of all collisions is small. In fact it is precisely because the effect of a single collision typically produces very little drop in c(t) that we need to wait for many collisions and a long time T to have memory loss. However, perturbation theory is not a necessary ingredient and in some cases one can solve analytically for the U-matrix [14,15]. Nevertheless, the impact approximation is a low density, high temperature approximation that is usually valid for electrons and is also valid for ions at low densities and high temperatures. For example, for the Balmer-a (n ¼ 3 / n ¼ 2) transition, ions are impact for electron densities below 1014 e/cm3 at a temperature of 1 eV. Increasing the temperature or decreasing the density will make the impact approximation more valid; decreasing the temperature or increasing the density in this regime will make ions less impact. Using a higher line (such as Balmer-g, n ¼ 5 / n ¼ 2) at the same plasma conditions will make ions less impact because the perturbation is larger due to the larger dipole matrix elements, resulting in shorter memory loss time scales T as well as less intense unshifted components.
propagator (U-matrix) and {.} denote a perturber average, so that
is the quantity of interest. The key is to identify an intermediate time scale Dt between the collision duration time scale s and the memory loss time scale T such that:
s Dt T
(15)
This time scale is guaranteed to exist because the impact regime is precisely the regime where T[s. So we can break T into a large number N of intervals of duration Dt. Each such Dt includes many collisions and therefore statistical averages are well defined on the Dt time scale. If we denote U(0,t) the evolution from t ¼ 0 to t ¼ T, as
Uð0; tÞ ¼ Uð0; DtÞUðDt; 2DtÞ.UðT Dt; TÞ
(16)
We can, in the impact approximation, take averages over each such Dt: ð17Þ because in each Dt, U is in principle a function of the coordinates of all particles (electrons), but in reality it is only a function of the particles that collide with the atom/ion within that interval, the field due to the remaining particles being negligible. More details are given in Appendix B. Since for times of interest t not much less than the memory loss time scale T, Dt t, t ¼ NDt, with N/N. Thus, since Dt is small in a Dyson series sense, (because memory loss happens on a time scale T[Dt) the U-matrix drop is small, so that:
and for N/N we recognize the definition of the exponential: ð18Þ
5.2. Main idea and mathematics Perhaps the clearest derivation of the impact approximation is as described in H.R. Griem’s books [1,2]: Here we will use less mathematics and instead let the symbol 0 represent the exact 1 0.9
c(t). Note the large drop when there is a spike
0.8 0.7
c(t)
0.6
5.3. Usefulness of the impact approximation outside its range of validity The impact approximation is useful even when not strictly valid. The reason is that the impact approximation represents the maximum possible broadening for a given set of plasma parameters: As an example, the impact approximation normally predicts a linear density scaling, while in all other regimes density scaling is less than linear. Therefore when the impact approximation is not applicable, its predictions can still be used to obtain an upper bound on the broadening, given its validity at low enough densities.
0.5
6. The different time domains
0.4 0.3 0.2 Ex
0.1
Ey Ez
0 0
500
1000
1500
2000
2500
3000
3500
4000
t(ps) Fig. 8. Typical c(t) for an impact case. In this case we show one realization of the pure ionic field, which is impact, for the n ¼ 9 / n ¼ 2 hydrogen line in a pure hydrogen plasma with an electronic density 1.2 1012 e/cm3 and temperature 1.16 eV. The field components Ex, Ey and Ez are shown as solid, dashed and dotted lines, respectively. Also shown as a dash-dotted line is the corresponding c(t).
Having explained the two basic approximations for broadening, we must now make it clear that these approximations are never valid for all times. For any given C(t), there is always a time domain where each approximation is valid, as well as a domain where neither is valid. We are happy to use one of these two approximations if the bulk of broadening comes from times over which the approximation in question is valid. For example, it is obvious that the quasistatic approximation cannot be valid for sufficiently long times such that the ions will move appreciably and thus evolution of the U(t) no longer occurs while the field remains unchanged, or static. Similarly, to derive the impact approximation in the previous section, we had to invoke that T[s, yet for the derivation we used
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
[13] and is characterized by overlapping collisions, which is inherently a many-body problem [17]. Of course, the same arguments apply to the electrons, so that an intermediate regime also exists for electrons. In the intermediate regime, the field evolution may look as in Fig. 10, which depicts a smooth field on the memory loss time scale, which is the entire x-axis shown. The field has progressed so that the intermediate regime may also be treated and the two basic approximations discussed here are very important in many methods employed to treat this regime too [18–20]. Excluding discussions of the opacity effect on the observed line profiles the issue as far as the line width is concerned is then which of these time scales determines the bulk of C(t). In summary, short times are important for line wings where the quasistatic approximation is valid and long times are important for the profile near the line center, where the impact approximation is valid.
1 0.9 0.8 n=6.2x1018 e/cm3
0.7
T=0.86 eV
C(t)
0.6 0.5 0.4 0.3 0.2 0.1 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15
t(ps)
7. Synthesis: the standard theory
Fig. 9. Plots of C(t) and -lnC(t)/t vs. time at a high density [16]. The impact approximation is only valid for the times where -ln C(t)/t is constant. The solid line is C(t), while the dashed line is X(t) as defined in Eq. (20). Applying the impact approximation for the entire time domain is only justified if the decrease in C(t) is n1 for the short, non-impact times.
U(t) and t can range from 0 to T. Clearly this derivation is invalid for t s, which means that the impact approximation is only valid for long times. As an illustration, consider the impact approximation for electrons in a fairly dense plasma, where the electron density is 6.2 1018 cm3 and the temperature is 0.86 eV [16]. In this example C(t) was computed exactly. If the impact approximation is valid, then
In many cases the electrons may be treated in the impact approximation because they interact with the emitter mostly via short duration collisions, while the ions may be treated in the quasistatic approximation in view of their larger mass that leads to much smaller velocities. When this is the case, and noting that C(t) is a Trace and hence basis-independent, one may effectively use the Stark basis, i.e. the basis that diagonalizes the emitter plus static ion field Hamiltonian and then treat the electrons in the impact approximation. This procedure is often referred to as ‘‘Standard Theory’’ and results in an expression for the lineshape:
LðuÞ ¼ CðtÞzeFt
(19)
and hence we can define a quantity
XðtÞ ¼
ln CðtÞ t
(20)
that should be constant, independent of t. So in Fig. 9 we plot X(t) vs. time and, indeed, see a saturation for long times.4 So for the times for which X(t) is flat, the impact approximation is valid, while for the remaining times it is not. In this particular example we see that by the time that saturation occurs, which is at t z 0.02 ps, C(t) has decayed to about 0.5. This means that the contribution of a significant part of C(t) is computed incorrectly in the impact approximation because the impact approximation is not valid for C(t) between 1 and 0.5, hence non-impact effects are expected and the width will be an overestimate. The impact approximation would have been much more valid if saturation had occurred before C(t) had dropped to say the 95% mark, i.e. when C(t) ¼ 0.95. So the quasistatic approximation is valid for short enough times, when the species in question, here the ions,5 do not have the time to move appreciably and hence the field is static. While for long enough times the impact approximation is valid because we can always find a large enough Dt t where we have many collisions. Note that the times to which we refer here may be [T, and if that is the case we are only affecting the part near the line center where Du is small enough and eıDut in the Fourier transform does not oscillate too wildly. It is clear from this discussion that at intermediate times neither approximation is valid, which is the regime referred to as ‘‘ion dynamics’’ in the literature of line broadening
4
Due to the noise in the simulation we will not get a perfectly flat line. For lines with an unshifted component electrons can only be static if another broadening mechanism can cause C(t) to decay before electrons have time to move. 5
231
1
ZN
p
dEWðEÞdba $da0 b0 Re ab ı u uab ðEÞ
0
1 0 0 a b IF
ð21Þ
where I is the identity operator, W(E) is the quasistatic microfield distribution, Re means real part and jabD denotes the index of a matrix whose rows and columns include all the upper level (a) and lower level (b) combinations. Finally, the notation with the double indices should be understood in the sense:
0 0 if Q i f ¼
Z
d3 r
Z
d3 r0 j*i ðrÞjf ðr0 ÞQ ðr; r0 Þji0 ðrÞj*f 0 ðr0 Þ
(22)
Conceptually the lineshape expression means that ions have produced a random static field E and electrons add homogeneous broadening for each such field, with the integration over all static fields given the appropriate microfield distribution providing the ionic broadening. The Stark shifts are included in the new energies, uab(E). 8. Summary In summary the quasistatic and impact approximations represent important theoretical limits that are in many cases sufficient for practical purposes and have been used to guide and develop new methods that are more generally applicable [18,19] and, in fact, satisfactorily solve the line broadening problem in practically all cases. Appendix A. The simplest quasistatic microfield (Holtsmark) As mentioned above, this approximation, historically the first and the one that permits an analytical solution dates back to Holtsmark [6]. Below we derive this distribution for a plasma with 2 ionic species of charges Z1 and Z2 and densities n1 and n2, respectively, which allows us to illustrate how the Holtsmark normal field should be defined in such a case. We assume that there are N
232
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
Electric Field (arb.units)
2=3 h i2=3 4 3=2 3=2 E0 ¼ 2p jej Z1 n1 þ Z2 n2 15
(30)
The extension to more than two species is obvious. Note that we do not need to convolve the two distributions. In the more common case of a single species with charge Z and density n, 0
2=3 4 E0 ¼ 2p jejZn2=3 15
(31)
Instead of Q(E) usually the Holtsmark distribution is expressed in terms of the dimensionless quantity
HðbÞ ¼ E0 WðEÞ ¼
2
p
b
ZN
3=2
dxxex sinbx
(32)
0
Time
with
Fig. 10. A typical configuration in the ion-dynamical regime. The electric field components Ex, Ey and Ez are shown as solid, dashed and dotted lines, respectively.
plasma ions in a volume V, we label the first N1 as the ions of the first kind and ions N1 þ1 to N as ions of the second kind. Quantities relating to the ith kind have a subscript (i). The probability density to find a net electric field E is:
Q ðEÞ ¼
Z
d3 r1 .d3 rN Pðr1 ; .; rN Þd E S Ei ðri Þ i
(23)
where r1, ., rN are the coordinates of the N plasma ions in the volume V and P is the probability density for configuration (r1, ., rN). In the Holtsmark case,
Pðr1 ; .; rN Þ ¼
1 VN
Ei ¼ Zi eri3 ri Fourier transforming Eq. (23) we obtain
In an average as in Eq. (17) with M particles (1,2, ., M) we have an integral
Z
d1d2.dMUð0; Dt; 1; 2; .MÞ.Uðt Dt; t; 1; 2; .MÞV M (34)
with
Z
i
h
i d3 rN1 þ1 .d3 rN exp k: S Ej rj ¼ Q1 ðkÞQ2 ðkÞ j
(26)
Z
d1d2.dM
(35)
The integral may be written first as a product of integrals
V p
d3 Eeik:E Q ðEÞ Z h i d3 r1 .d3 rN1 exp k: S Ei ðri Þ V N1 N ¼ V N1
Z
Appendix B. Details in the impact average
VM ¼ (25)
(33)
The Holtsmark distribution is clearly very simplistic. In addition, the conditions that allow one to neglect the interionic correlations, namely low densities and/or high temperatures, are also quite unfavorable to the quasistatic picture as a whole.
(24)
The individual fields are pure Coulomb fields, i.e.,
Q ðkÞ ¼
b ¼ E=E0
Z
dk1 dk2 .dkp UðjDt; ðj þ 1Þ Dt; k1 ; k2 ; .kp
(36)
where only the particles k1, k2.kp interact with the atom/ion in the interval (jDt, (j þ 1)Dt). At this point we can integrate over the remaining redundant variables in both numerator and denominator. Then, as the integrand does not depend of these variables because they do not interact in that interval, each such integral contributes an average evolution operator:
with the sums in the exponents running over the integrated position variables. Here
h
i ð1Þ 3=2 Q1 ðkÞ ¼ exp E0 k
(27) References
with ð1Þ E0
¼
2=3 2pZ1 jejn1
4 2=3 15
(28)
with a similar relation for the second species. Thus, we are able to write
Q ðkÞ ¼ eðE0 kÞ with
3=2
(29)
[1] H.R. Griem, Plasma Spectroscopy, McGraw-Hill, New York, 1964. [2] H.R. Griem, Spectral Line Broadening by Plasmas, Academic, New York, 1974. [3] H.R. .Griem, Principles of Plasma Spectroscopy, Cambridge University Press, Cambridge, 1997. [4] M. Baranger, in: D.R. Bates (Ed.), Atomic and Molecular Processes, Academic Press, 1962. [5] F.J. Dyson, Phys. Rev. 75 (1949) 486. [6] J. Holtzmark, Ann. Phys. 58 (1919) 577. [7] M. Baranger, B. Mozer, Phys. Rev. 115 (1959) 521; B. Mozer, M. Baranger, Phys. Rev. 118 (1960) 626. [8] C.F. Hooper Jr., Phys. Rev. 165 (1968) 215.
S. Alexiou / High Energy Density Physics 5 (2009) 225–233 [9] C. Iglesias, J. Lebowitz, O. MacGowan, Phys. Rev. A28 (1983) 1667. [10] C.A. Iglesias, et al., JQSRT 65 (2000) 253. [11] C.A. Iglesias, F.J. Rogers, R. Shepherd, A. Bar-Shalom, M.S. Murillo, D.P. Kilcrease, A. Calisti, R.W. Lee, JQSRT 65 (2000) 303. [12] D.P. Kilcrease, R.C. Mancini, C.F. Hooper Jr., Phys. Rev. E 48 (1993) 3901; D.P. Kilcrease, M.S. Murillo, L.A. Collins, JQSRT 58 (1997) 677; D.P. Kilcrease, M.S. Murillo, JQSRT 65 (2000) 343. [13] R. Stamm, et al., Phys. Rev. Lett. 52 (1984) 2217; R. Stamm, B. Talin, E.L. Pollock, C.A. Iglesias, Phys. Rev. A 34 (1986) 4144. [14] H. Pfennig, Z. Naturforsch. 26a (1971) 1071; H. Pfennig, J. Quant. Spectrosc. Radiat. Transf. 12 (1972) 821.
233
[15] V.S. Lisitsa, G.V. Sholin, Sov. Phys. JETP 34 (1972) 484. [16] S. Alexiou, Phys. Rev. E 71 (2005) 066403. [17] E. Stambulchik, Y. Maron, ‘‘Plasma line broadening and computer simulations: a mini-review’’ this volume. [18] B. Talin, A. Calisti, L. Godbert, R. Stamm, R.W. Lee, L. Klein, Phys. Rev. A 51 (1995) 1918. [19] S. Alexiou, Phys. Rev. Lett. 76 (1996) 1836. [20] A. Brissaud, U. Frisch, JQSRT 11 (524) (1971) 1767; A. Brissaud, U. Frisch, JQSRT 11 (524) (1971) 1767; J. Math, Phys. 15 (1207) (1974); J. Seidel, Z. Naturforsch. 32a (1207) (1977).
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Review
Overview of plasma line broadening S. Alexiou University of Crete, TETY, 71409 Heraklion, TK 2208, Greece
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 June 2009 Accepted 3 June 2009 Available online 12 June 2009
We review the basics of line broadening, its relation to fluctuations and disorder, what causes broadening, the memory loss mechanism and the Standard Theory of line broadening developed by H.R. Griem and others from a modern viewpoint. This modern view benefits from many years of progress and includes a coherent theoretical perspective without the need for a conceptually different view of electrons and ions. Both electrons and ions are described in terms of their random fields. This modern and unified view allows, among other things, extending the range of validity of line profile calculations to complex situations. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Stark broadening Fluctuations Impact theory
1. Introduction: broadening and fluctuations For any atomic system embedded in a medium, the interactions between the atomic system and the medium result in a modification of the energies and lifetimes of the atomic system. From this point of view a plasma is just another medium where the interactions (usually Coulombic) are well-known, where simple approximations (e.g. a static picture as in the solid state) often fail and in which a large range of essentially very different medium behavior is spanned. Broadening is in general associated with fluctuations and randomness is essential. Conversely, broadening effectively ‘‘measures’’ fluctuations. By ‘‘fluctuations’’ we simply mean, for example, that different atoms in a plasma see a different interaction. Thus, randomness is a requirement for broadening; it is a general feature and not specific to plasmas. The above remark can be illustrated by two unrelated, but rather well-known examples: The first is Doppler broadening, where we need a distribution of velocities: A single velocity yields a Doppler shift, but no Doppler broadening. The second example is powder diffraction in crystals: The Bragg condition is nl ¼ 2dsinq with n an integer, l the radiation wavelength, d the plane spacing and q the angle of coherent scattering. Imperfections, i.e. variations in d, will result in both stress and strain. This means we will have some d that are larger and some that are smaller than their perfect crystal counterparts, resulting in shifts of q both left and right and hence broadening of the peaks. The width of these peaks therefore is a measure of crystal perfection, or equivalently disorder. This is true in general, i.e. the larger the fluctuations, the larger the widths and may be understood as follows: we can decompose the interaction into a mean term plus a fluctuating one.
The entire Hamiltonian with the mean, nonrandom, term (but not the fluctuating part) in principle may be solved to give a set of infinitely sharp energies, while the fluctuating part is a measure of disorder and will give broadening, i.e. a lifetime. Historically the view taken was that slow ions provide a static, but random, Stark field that gives rise to an inhomogeneous broadening, just like Doppler broadening: each emitter would see a different Stark field and as a result would experience a shift, just as in Doppler broadening; because these Stark fields are random, one would have a distribution of such shifts and integrating over those random fields or equivalently, shifts would result in a broadened profile. Electrons, on the other hand, would interact via fast collisions, giving rise to homogeneous broadening. Although this view enabled significant progress and actual calculations, it hampers a coherent theoretical construct as it separates from the beginning the interactions. In the modern view the effect of the plasma on the atom is coherently described for all perturbers, electrons, ions, but also collective fields, by putting the emphasis on the fields. This view, presented below, allows the ion and electron generated fields to be treated consistently, includes the older view and allows us to treat more complex cases that could not be treated by the older view. 2. The lineshape and autocorrelation function For dipole radiation, the line profileL(u) is the interesting part of the intensity of the emitted radiation I(u) [1–3]:
IðuÞ ¼
1574-1818/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.hedp.2009.06.003
(1)
where L(u) is the lineshape:
LðuÞ ¼ E-mail address: [email protected]
4u4 LðuÞ 3c3 X
d u uif hf jdjiij2 ri
if
(2)
Here the states i and f in the above equation refer to states of the entire system (medium plus atomic system and d and r are defined below). In the modern framework we often work with the Fourier Transform of the lineshape, which is the autocorrelation function C(t). Fourier-expanding the d-function, we see that the transform of L(u), C(t) is defined by:
CðtÞ ¼
hdð0Þ$dðtÞi jdj2
¼ cðtÞ
(3)
where C.D indicates the quantum and statistical averages and d(t) is the atomic dipole operator in the Heisenberg representation. The c(t) refers to the contribution to C(t) from a single random realization of the medium, i.e. the time history of that part of the medium that affects a single emitter. Equivalently, one may consider a single emitter in the medium, pick a distribution of initial plasma particle positions and velocities relative to the emitter and, in general, let them evolve; this is one realization or configuration. As shown in Fig. 1, by this term we understand an entire history of the interaction between the plasma and a radiating electron (emitter), as seen by a single emitter (different emitters see different histories, of course). The idea is that if we have M uncorrelated emitters, the intensity of the radiated light will be multiplied by M, but the lineshape will not change [4]. A broad lineshape in frequency means a narrow C(t) in time and vice versa. It is important to note that in order to get a finite width in the line profile (i.e. not a d-function), C(t) must decay to 0 for long times. This is obvious since the lineshape is a Fourier transform. The decay of C(t) to 0 is only possible over time scales for which we have very small or negative contribution for some individual c(t)’s. We define T as the time after which C(t) is negligible, i.e. the time scale of C(t). Technically c(t) is a linear combination of time evolution operators for the upper and lower levels. If jaD, ja0 D and jbD, jb0 D are atomic eigenstates (in practice jaD, ja0 D are eigenstates of the upper level and jbD, jb0 D of the lower level) for the transition in question, we have: y
cðtÞ ¼ Uaa0 ðtÞda0 b0 Ub0 b ðtÞ$dba ra
(4)
where U(t) is the time evolution operator in the interaction picture, d is the dipole moment and ra the probability of finding an emitter in state a. Given this definition of c(t) and Eq. (3) it is clear that the time dependence of C(t) is determined by the average time dependence of the U-matrix product. This seemingly simple representation of the lineshape is all we need in order to compute the lineshape exactly. Hence the central problem is the calculation of the U-matrix product and the average over the plasma.
3. Qualitative discussion of line broadening: the memory loss mechanism Fig. 2 shows the c(t)’s from different configurations for a transition with a strong unshifted component. The initial state of the atom or ion emitter is defined by a set of electrons populating the different orbitals. The transition of interest involves one of these electrons changing or ‘‘jumping’’ to a different orbital. This electron is the optically active electron and the one whose wave functions we are concerned within the line broadening problem. Essentially U(t) is the identity matrix at t ¼ 0 and its diagonal elements start decaying, meaning that the interaction with the plasma produces transitions and the optically active electron need not be found in the same state it was at t ¼ 0, and the final state may no longer be empty. At short times we have from the Schro¨dinger equation:
Electric field components (arb.units)
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
time Fig. 1. Example of a single field realization of particle configuration, i.e. the time history of the interaction between plasma and emitter as seen by a single emitter. In this example a dipole interaction has been assumed and hence the electric field components Ex, Ey and Ez are shown as solid, dotted and dashed-dotted lines, respectively.
dU ı ¼ VðtÞUðtÞ dt Z
(5)
a drop in U(t) DU z iV(t ¼ 0), with V(t) the interaction, which indicates that larger interactions result in faster decays. C(t) is the average of these c(t), so that for a given time t one needs to sum the c(t)’s and divide by their number. As mentioned above, to avoid d-functions, C(t) must decay to 0 for long times. Obviously no decay is possible until most of the individual c(t)’s have themselves decayed to values close to 0. But once U(t) and c(t) has dropped to essentially 0, we have random phases from the different realizations c(t) so that the average of c(t) over these realizations vanishes. Random phases mean that the optically active electron is uncorrelated at time t to its state at t ¼ 0. Thus, the probability amplitude of finding it at time t in the same state it was at t ¼ 0 depends critically on the interactions to which it has been subjected and that probability amplitude is quite small when averaged over different field configurations. This is referred to as ‘‘memory loss’’ and explains why C(t) is a measure of memory
1 0.9
Autocorrelation Function C(t)
226
short times, drop is proportional to the interaction
0.8 0.7 0.6 0.5
long times, random phases
0.4 0.3 0.2 0.1 0 -0.1 -0.2
Time(t) Fig. 2. Examples of c(t) (vertical axis) vs. time(horizontal axis) from different configurations. From its definition, c(t) starts as unity for t ¼ 0, can only start decreasing for times longer than 0 and displays oscillatory behavior for long times. It is the random phases of the different c(t)’s that produce the decay in time of C(t).
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
may have a ‘‘spiky’’ field, which corresponds to a high temperature, low density case which translates in the particle picture to having very short collision durations compared to T, or a much smoother field, which corresponds to a high density, low temperature case in which the collisions are ‘‘on" all the time. If T is short enough we may have a situation where the field essentially does not change on the T time scale. The picture just described is essentially correct, except for one complication: To produce memory loss a field must be ‘‘sufficiently’’ random. For a dipole interaction, a one-dimensional field is not sufficiently random for a transition with an unshifted component, because such components are completely unaffected by a one-dimensional field. This can be illustrated in the following manner: in the spherical basis a state jn l ¼ n 1m ¼ (n 1)D is modified under the interaction with a one-dimensional dipole field V(t) ¼ E(t)z as
1 0.9 0.8
C(t)
0.7 0.6 0.5 0.4 0.3
t
Fig. 3. Typical C(t) for a transition with a strong unshifted component under the action of a one-dimensional dipole field. This particular transition is the hydrogen Balmera(Ha) i.e. the transition between the principal quantum numbers n ¼ 3 and n ¼ 2. This is a transition with unshifted components, because for a transition between states n ¼ 3, l ¼ 2, m ¼ 2 to n ¼ 2, l ¼ 1, m ¼ 1 neither the upper, nor the lower level is affected by a constant field in the z-direction, i.e. these two states are not perturbed by any plasma field if we only allow the perturbation of a state by other states with the same principal quantum number. Notice therefore that C(t) cannot decay past the intensity of its unshifted component, which is about 38% in this particular case.
loss. On the contrary, at short times c(t) and C(t) are close to unity and the atomic system has lost little memory of its initial state. For completeness one should point out that the natural lifetime is typically much larger than the C(t) time scale, T or any time shown in Fig. 2. In order to gain a better understanding of time scales, one may note that the vast majority of configurations have a c(t) that approach very small values, i.e. z0, before some time T. When this happens, details of further time development are largely irrelevant (we will shortly discuss the exception to this rule), because random phases will give a vanishing C(t). One may very roughly estimate T as the time scale where U(t) becomes approximately 0 for the first time:
UðtÞz0
(6)
or equivalently
ZT
V 0 ðtÞdt z1 Z
227
dhnl ¼ n 1m ¼ ðn 1ÞjUðtÞjnl ¼ n 1m ¼ ðn 1Þi dt z nn 1 ðn 1ÞEðtÞznl0 m0 nl0 m0 UðtÞnn 1 ðn 1Þ (8) However, m-selection rules for the first term on the right hand side require m0 ¼ (n 1). But it is only the highest l, n 1 which can have such an m-state and l-selection rules, jDlj ¼ 1 do not allow l0 to take such a value. Hence the derivative is 0 and the state in question is unaffected by the external field. So for example if we have a dipole transition between an upper and a lower maximum m states, this component is unshifted and unaffected by a onedimensional field. Therefore, c(t) and C(t) cannot decay to values lower than the relative intensity of these components and one cannot avoid d-functions, unless some other broadening mechanism helps. No matter what its shape, no matter what its statistics and distribution function, a one-dimensional field is not ‘‘sufficiently’’ random in this case. This is manifested by a C(t) that does not decay past a certain level, e.g. 66% for the La(n ¼ 2 / n ¼ 1), 38% for Ha (n ¼ 3 / n ¼ 2) etc. This means that for such a field we would obtain d-function lineshapes upon Fourier transforming. Fig. 3 shows a typical C(t) for a line with a strong unshifted component if the perturbation is a one-dimensional field. This specific graph was obtained with frozen ions alone, i.e. ions that were fixed in their starting positions for the course of the calculation.
(7)
0
(b)
Electric field at emitter
with V0 the interaction in the interaction picture. This is in fact the lowest term in the Dyson [5] expansion and is to be interpreted in the sense that for times much shorter than T perturbation theory is valid and hence the deviation from the initial value of the evolution operator, U(t), is much less than 1. Hence for such times, U(t) z I and consequently c(t) z 1. This means that for such times, C(t) will be the average of quantities that are z1 and hence will itself be not much different than 1, which means practically no memory loss for such times. Therefore, one qualitatively sees that it is not the peak intensity of the interaction, but its action integral, Eq. (7), that counts and we do not have significant memory loss as long as that action is small compared to -. So it becomes clear that the action of V0 (t) determines the relevant time scale T. The main issue in plasma line broadening has thus been redefined into what is the behavior of the interaction V0 (t) on this memory loss time scale T, which in terms of the line profile is roughly of the order of the inverse of the halwidth at half maximum intensity (HWHM). This question is crucial in deciding how to treat the problem at hand. Essentially we
0
(a)
Time Fig. 4. Different temporal behavior of V(t) over the T time scale. We may have a spiky field (a), a smooth field (b) or even an essentially constant field (b, if the time scale is the shaded area).
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
When we do have the case with an unshifted component we get, of course, a finite width. But this is only because there are always broadening mechanisms, such as multi-dimensional fields, Doppler broadening, natural broadening etc. Often these mechanisms are less effective, which means they produce memory loss on longer time scales. This will be discussed below in more detail where we will show how to generalize the previous qualitative picture to include such cases. For the moment we merely note that any static field is one-dimensional and falls in the above category, while the converse is not true. As will be seen shortly, the behavior of V(t) over the T time scale is of paramount importance for treating the broadening: Certainly, the action of V0 (t) over T must be of the order of -, but there are different ways of achieving this, as shown in Fig. 4. It could be achieved by: 1) a fairly constant V0 (t), presented as field (b) when taken over the time scale indicated by the shaded area; 2) by a highly ‘‘spiky’’ field, (a) in Fig. 4 that consists of sharp features over an essentially zero background; or by field (b) that is something in between.
0,5
0,4
Intensity
228
shifted line δ(ω−CΕ)
0,3
0,2 Probability for field E
0,1
0
0
2
4
6
10
8
Δω Fig. 5. Quasistatic profile as an integral of d-functions over the microfield distribution.
4. The quasistatic approximation Like most disciplines, one normally starts with the simplest possible approximations and refines them as they start to be challenged by experiment. In case of line broadening the simplest approximation is the quasistatic approximation: The idea is that C(t) drops to very small values on such a short time scale that a certain field component, normally associated with the ions, have no time to move and hence appreciably modify their field. Therefore the time evolution happens during a static field, which is different for each configuration, but still static over T. Later we will see how this statement holds in practice. The quasistatic approximation assumes stochastically static ions, that interact with the atom via their static electric, Stark, fields. In other words we describe the medium in terms of configurations, each of which corresponds to a net ionic field that is static on the autocorrelation time scale. Since the ionic positions are random, the total (static) field is a continuous random variable E. The only quantity one needs then is the probability density Q(E) to find a field between E and E D dE. In usual isotropic situations we are only interested in the probability density W(E) ¼ 4pE2Q(E) (so R R that dEQ ðEÞ ¼ dEWðEÞ) to find a field of magnitude between E and E þ dE. If we have the W(E), then as shown in Fig. 5 each field produces a shift and the profile is just the integral of shifted d-functions over all fields, each d-function weighted by the probability density for the corresponding field.1 For example for linear shift, the otherwise unbroadened transition in a single constant field E is just d(u CE) with C a constant as shown in Fig. 5 and the profile is2:
LðuÞ ¼
ZN
dEWðEÞdðu CEÞ ¼
1 u W C C
(10)
0
Thus, in the case of a linear shift the quasistatic line profile has the shape of the distribution W(E) itself. Note that this vanishes at the
1 However, this conceptual picture does not carry over to the other regimes: it is only applicable for quasistatic behavior and unable to account for possible nonstatic field components. 2 More generally, if J(u,E) is the lineshape for a static Stark field with magnitude E, then. Z dEWðEÞJðu; EÞ (9) LðuÞ ¼
center (u = 0) because the probability of finding a zero field is vanishingly small, reflecting the improbability of having the ion charges arranged so as to exactly balance each other at the emitter. It is clear, therefore, that all that is required to calculate the line profile in the quasistatic approach is to compute the plasma-related quantity Q(E). The quantity Q(E) can be calculated using:
Q ðEÞ ¼
Z
3
3
3
d r1 d r2 .d rN Pðr1 ; r2 ; .; rN Þd E
N X
! Ei
(11)
i¼1
with P the probability density to find ion 1 in position r1, ion 2 in r2 etc. In Eq. (11) the d-function ensures that only those arrangements contribute that give a net field equal to E. So computing Q reduces to answering the questions: What is P and what are the Ei, i.e. what is the field experienced by the emitter due to the presence of the ith perturber. The initial, and simplest, answer was given by Holtsmark [6], who assumed pure Coulomb fields Ei and uncorrelated perturbers, i.e.
P ¼ V N
(12)
with V the volume, which is a statement that all configurations are equally likely. For concreteness, this calculation of the quasistatic microfield distribution is provided in Appendix A. Eventually, at a time when both theoretical and experimental advances enabled reliable tests of the theoretical predictions, screened fields were used and correlations were accounted for [7,8], which for most plasma conditions of interest very satisfactorily resolved the issue. The net result of these improved calculations is, qualitatively speaking when compared to using the Holtsmark approach, that large electric fields are less probable due to both ionic repulsion and the effect of screening by the surrounding electron cloud. This leads to microfield distributions that have peaks which occur at lower field values than those of the Holtsmark distribution, which is a high temperature limit. Currently [9] such calculations are extremely fast and accurate, so that the computation of the distribution function is considered a practically closed subject for all cases, including extreme matter conditions [10,11]. However, in many experimental situations large gradients are encountered and more generalized distributions are required [12]. The main problem with the quasistatic approximation has already been described in the previous section: Static fields are one-dimensional and by themselves cannot broaden transitions with an unshifted central component. This implies that one has to
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
ZT 0
V0 dt[
ZT 0
Vel ðtÞdt[
ZT
dVðtÞdt
(13)
0
with Vel the electron-emitter interaction. This situation is depicted in Fig. 6. In this case we may as well neglect ion motion, even for a line with an unshifted component. This will not be possible if the action of the fluctuating ion component is comparable to or even dominates the electronic action (as is the case for low densities). So we improve our qualitative understanding provided in the previous section by stating that for lines with a strong unshifted component that are unaffected by static fields, it is the action of non onedimensional fields that matters for achieving a decay of C(t) beyond the relative intensity of the unshifted component. In contrast, for lines without such an unshifted component (for example Lb, i.e. n ¼ 3 / n ¼ 1 and Hb, i.e. n ¼ 4 / n ¼ 2), the static picture for ions normally works well when one is interested in estimates of the width and not the details of the shape. This is due to the fact that the microfield distribution is 0 for E ¼ 0, so that these lines would have 0 intensity at the line center according to a quasistatic RN calculation. Clearly then for u ¼ 0, Lðu ¼ 0Þz 0 CðtÞdt ¼ 0 which means that C(t) cannot be positive for all times. In Fig. 7 it is shown that because there is no unshifted component, the action of the static field may be exploited in full to give a fast drop in C(t). The effect of the dynamics is to make the negative region shallower, and hence fill in the central dip because now the integral of the positive region is larger than the one of the negative region. Because the action of the static field is very effective in this case, dynamical effects that occur at longer times are normally relatively unimportant for the bulk of the line. In short because of the dynamics the oscillations at long times produce a damping of the long time behavior of C(t); since dynamics sets in at the times for which C(t) is negative, this results in a less negative C(t) and a filling of the central dip.
Fig. 6. Comparison of the ionic and electronic field actions: While the ionic action dominates the electronic one as it is stronger, if we decompose the ion field in a static (dotted line) and a fluctuating part, the electronic action dominates the fluctuating ionic action.
5. Impact approximation 5.1. Introduction: what an impact field looks like The impact approximation is applicable when memory loss is the result of collisions with duration much less than the memory loss time T. It is then straightforward to see that such collisions must be mostly weak. Recalling our earlier remarks that the action of the interaction, Eq. (7), is the determining factor in the effectiveness of the broadening, the weakness of collisions can be defined by the following condition:
Zti þs
V 0 ðtÞdt 1 Z
(14)
ti
Here ti the start of the collision and s its duration. Clearly this is a necessary condition because otherwise memory would be lost on the time scale of one or a few collisions and then the above relation would read z1, which is by definition excluded. Therefore memory loss is a ‘‘team effort’’ instead of arising from the interaction of a single collision. Nevertheless, there will be configurations dominated by a single collision, which, although rare, have a significant effect because then c(t) drops on a much faster time scale. Indeed, the treatment of these rare events has been an issue. On the other hand, whether we have a rare strong collision or not, the action of the impact perturbers is additive. This is clear because collisions do
1 0,9 0,8
L-beta C(t)
0,7 0,6 0,5
C(t)
rely on other broadening mechanisms and the total time scale will be determined predominantly by these other mechanisms in these cases. For example if electron broadening is strong enough to make C(t) decay before ions have time to move appreciably, then the static picture works fine. If not, the quasistatic approximation may underestimate the width by as much as an order of magnitude [13]. So, whether the quasistatic theory is valid in any given case does not depend on ions alone; it also depends on electrons. In fact it depends on all broadening mechanisms, electrons, (quasistatic) ions, Doppler and natural broadening. For the quasistatic approximation to hold, all these mechanisms must have reduced C(t) to very small values before the ions have the time to move appreciably. Coming back to the discussion of the previous section, here we have a case where the action of ions may well dominate the action of electrons, because of the longer duration of the ionemitter interactions. Because it is the total action that counts for memory loss, a constant field will normally produce more action than a spiky field that is zero, or very small, most of the time. Nevertheless, if we write the ion field as Vion(t) ¼ V0 þ dV(t), with V0 the static part and jV0 j[jdVðtÞj, the action integral3 of the electrons (or, more generally any other broadening mechanism, for instance turbulent fields) may still dominate the action integral of dV(t):
229
0,4 0,3 0,2 difference
0,1 0
dynamic
-0,1 -0,2
quasistatic 0
0,05
0,1
0,15
0,2
t(ps) 3
For simplicity we consider only hydrogen-like lines here, where V(t) is the same in the interaction and Schro¨dinger pictures.
Fig. 7. Differences between a quasistatic (dashed) and an exact dynamical calculation for a line without a central component, in this case, Lyman-b. Note that the differences are small and only nonzero for times at which C(t) has dropped to small levels.
230
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
not overlap in time, so that we effectively have a one-body problem instead of a many-body problem. For strong collisions we may effectively neglect all other collisions. Fig. 8 shows a typical ‘‘impact’’ field. Displayed is c(t) (top curve) as well as the electric field components Ex, Ey, Ez in arbitrary units. Note the sharp drop in c(t) where the field is significantly larger than the background. These ‘‘spikes’’ correspond to collisions with a particle, an electron in our case. However, over the duration of such collisions the drop in c(t) is always much less than 1, e.g. even the largest drop at t z 600 ps is only about 0.2. In other words, collisions are weak in a perturbative Dyson expansion sense. Therefore, in the impact regime memory loss is produced incrementally, each collision contributing its (normally small) share. However, the impact theory is not simple perturbation theory; it usually employs perturbation theory for what is known in manybody physics as the self-energy: it simply says that the effect of a single collision is typically small, not that the effect of all collisions is small. In fact it is precisely because the effect of a single collision typically produces very little drop in c(t) that we need to wait for many collisions and a long time T to have memory loss. However, perturbation theory is not a necessary ingredient and in some cases one can solve analytically for the U-matrix [14,15]. Nevertheless, the impact approximation is a low density, high temperature approximation that is usually valid for electrons and is also valid for ions at low densities and high temperatures. For example, for the Balmer-a (n ¼ 3 / n ¼ 2) transition, ions are impact for electron densities below 1014 e/cm3 at a temperature of 1 eV. Increasing the temperature or decreasing the density will make the impact approximation more valid; decreasing the temperature or increasing the density in this regime will make ions less impact. Using a higher line (such as Balmer-g, n ¼ 5 / n ¼ 2) at the same plasma conditions will make ions less impact because the perturbation is larger due to the larger dipole matrix elements, resulting in shorter memory loss time scales T as well as less intense unshifted components.
propagator (U-matrix) and {.} denote a perturber average, so that
is the quantity of interest. The key is to identify an intermediate time scale Dt between the collision duration time scale s and the memory loss time scale T such that:
s Dt T
(15)
This time scale is guaranteed to exist because the impact regime is precisely the regime where T[s. So we can break T into a large number N of intervals of duration Dt. Each such Dt includes many collisions and therefore statistical averages are well defined on the Dt time scale. If we denote U(0,t) the evolution from t ¼ 0 to t ¼ T, as
Uð0; tÞ ¼ Uð0; DtÞUðDt; 2DtÞ.UðT Dt; TÞ
(16)
We can, in the impact approximation, take averages over each such Dt: ð17Þ because in each Dt, U is in principle a function of the coordinates of all particles (electrons), but in reality it is only a function of the particles that collide with the atom/ion within that interval, the field due to the remaining particles being negligible. More details are given in Appendix B. Since for times of interest t not much less than the memory loss time scale T, Dt t, t ¼ NDt, with N/N. Thus, since Dt is small in a Dyson series sense, (because memory loss happens on a time scale T[Dt) the U-matrix drop is small, so that:
and for N/N we recognize the definition of the exponential: ð18Þ
5.2. Main idea and mathematics Perhaps the clearest derivation of the impact approximation is as described in H.R. Griem’s books [1,2]: Here we will use less mathematics and instead let the symbol 0 represent the exact 1 0.9
c(t). Note the large drop when there is a spike
0.8 0.7
c(t)
0.6
5.3. Usefulness of the impact approximation outside its range of validity The impact approximation is useful even when not strictly valid. The reason is that the impact approximation represents the maximum possible broadening for a given set of plasma parameters: As an example, the impact approximation normally predicts a linear density scaling, while in all other regimes density scaling is less than linear. Therefore when the impact approximation is not applicable, its predictions can still be used to obtain an upper bound on the broadening, given its validity at low enough densities.
0.5
6. The different time domains
0.4 0.3 0.2 Ex
0.1
Ey Ez
0 0
500
1000
1500
2000
2500
3000
3500
4000
t(ps) Fig. 8. Typical c(t) for an impact case. In this case we show one realization of the pure ionic field, which is impact, for the n ¼ 9 / n ¼ 2 hydrogen line in a pure hydrogen plasma with an electronic density 1.2 1012 e/cm3 and temperature 1.16 eV. The field components Ex, Ey and Ez are shown as solid, dashed and dotted lines, respectively. Also shown as a dash-dotted line is the corresponding c(t).
Having explained the two basic approximations for broadening, we must now make it clear that these approximations are never valid for all times. For any given C(t), there is always a time domain where each approximation is valid, as well as a domain where neither is valid. We are happy to use one of these two approximations if the bulk of broadening comes from times over which the approximation in question is valid. For example, it is obvious that the quasistatic approximation cannot be valid for sufficiently long times such that the ions will move appreciably and thus evolution of the U(t) no longer occurs while the field remains unchanged, or static. Similarly, to derive the impact approximation in the previous section, we had to invoke that T[s, yet for the derivation we used
S. Alexiou / High Energy Density Physics 5 (2009) 225–233
[13] and is characterized by overlapping collisions, which is inherently a many-body problem [17]. Of course, the same arguments apply to the electrons, so that an intermediate regime also exists for electrons. In the intermediate regime, the field evolution may look as in Fig. 10, which depicts a smooth field on the memory loss time scale, which is the entire x-axis shown. The field has progressed so that the intermediate regime may also be treated and the two basic approximations discussed here are very important in many methods employed to treat this regime too [18–20]. Excluding discussions of the opacity effect on the observed line profiles the issue as far as the line width is concerned is then which of these time scales determines the bulk of C(t). In summary, short times are important for line wings where the quasistatic approximation is valid and long times are important for the profile near the line center, where the impact approximation is valid.
1 0.9 0.8 n=6.2x1018 e/cm3
0.7
T=0.86 eV
C(t)
0.6 0.5 0.4 0.3 0.2 0.1 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15
t(ps)
7. Synthesis: the standard theory
Fig. 9. Plots of C(t) and -lnC(t)/t vs. time at a high density [16]. The impact approximation is only valid for the times where -ln C(t)/t is constant. The solid line is C(t), while the dashed line is X(t) as defined in Eq. (20). Applying the impact approximation for the entire time domain is only justified if the decrease in C(t) is n1 for the short, non-impact times.
U(t) and t can range from 0 to T. Clearly this derivation is invalid for t s, which means that the impact approximation is only valid for long times. As an illustration, consider the impact approximation for electrons in a fairly dense plasma, where the electron density is 6.2 1018 cm3 and the temperature is 0.86 eV [16]. In this example C(t) was computed exactly. If the impact approximation is valid, then
In many cases the electrons may be treated in the impact approximation because they interact with the emitter mostly via short duration collisions, while the ions may be treated in the quasistatic approximation in view of their larger mass that leads to much smaller velocities. When this is the case, and noting that C(t) is a Trace and hence basis-independent, one may effectively use the Stark basis, i.e. the basis that diagonalizes the emitter plus static ion field Hamiltonian and then treat the electrons in the impact approximation. This procedure is often referred to as ‘‘Standard Theory’’ and results in an expression for the lineshape:
LðuÞ ¼ CðtÞzeFt
(19)
and hence we can define a quantity
XðtÞ ¼
ln CðtÞ t
(20)
that should be constant, independent of t. So in Fig. 9 we plot X(t) vs. time and, indeed, see a saturation for long times.4 So for the times for which X(t) is flat, the impact approximation is valid, while for the remaining times it is not. In this particular example we see that by the time that saturation occurs, which is at t z 0.02 ps, C(t) has decayed to about 0.5. This means that the contribution of a significant part of C(t) is computed incorrectly in the impact approximation because the impact approximation is not valid for C(t) between 1 and 0.5, hence non-impact effects are expected and the width will be an overestimate. The impact approximation would have been much more valid if saturation had occurred before C(t) had dropped to say the 95% mark, i.e. when C(t) ¼ 0.95. So the quasistatic approximation is valid for short enough times, when the species in question, here the ions,5 do not have the time to move appreciably and hence the field is static. While for long enough times the impact approximation is valid because we can always find a large enough Dt t where we have many collisions. Note that the times to which we refer here may be [T, and if that is the case we are only affecting the part near the line center where Du is small enough and eıDut in the Fourier transform does not oscillate too wildly. It is clear from this discussion that at intermediate times neither approximation is valid, which is the regime referred to as ‘‘ion dynamics’’ in the literature of line broadening
4
Due to the noise in the simulation we will not get a perfectly flat line. For lines with an unshifted component electrons can only be static if another broadening mechanism can cause C(t) to decay before electrons have time to move. 5
231
1
ZN
p
dEWðEÞdba $da0 b0 Re ab ı u uab ðEÞ
0
1 0 0 a b IF
ð21Þ
where I is the identity operator, W(E) is the quasistatic microfield distribution, Re means real part and jabD denotes the index of a matrix whose rows and columns include all the upper level (a) and lower level (b) combinations. Finally, the notation with the double indices should be understood in the sense:
0 0 if Q i f ¼
Z
d3 r
Z
d3 r0 j*i ðrÞjf ðr0 ÞQ ðr; r0 Þji0 ðrÞj*f 0 ðr0 Þ
(22)
Conceptually the lineshape expression means that ions have produced a random static field E and electrons add homogeneous broadening for each such field, with the integration over all static fields given the appropriate microfield distribution providing the ionic broadening. The Stark shifts are included in the new energies, uab(E). 8. Summary In summary the quasistatic and impact approximations represent important theoretical limits that are in many cases sufficient for practical purposes and have been used to guide and develop new methods that are more generally applicable [18,19] and, in fact, satisfactorily solve the line broadening problem in practically all cases. Appendix A. The simplest quasistatic microfield (Holtsmark) As mentioned above, this approximation, historically the first and the one that permits an analytical solution dates back to Holtsmark [6]. Below we derive this distribution for a plasma with 2 ionic species of charges Z1 and Z2 and densities n1 and n2, respectively, which allows us to illustrate how the Holtsmark normal field should be defined in such a case. We assume that there are N
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S. Alexiou / High Energy Density Physics 5 (2009) 225–233
Electric Field (arb.units)
2=3 h i2=3 4 3=2 3=2 E0 ¼ 2p jej Z1 n1 þ Z2 n2 15
(30)
The extension to more than two species is obvious. Note that we do not need to convolve the two distributions. In the more common case of a single species with charge Z and density n, 0
2=3 4 E0 ¼ 2p jejZn2=3 15
(31)
Instead of Q(E) usually the Holtsmark distribution is expressed in terms of the dimensionless quantity
HðbÞ ¼ E0 WðEÞ ¼
2
p
b
ZN
3=2
dxxex sinbx
(32)
0
Time
with
Fig. 10. A typical configuration in the ion-dynamical regime. The electric field components Ex, Ey and Ez are shown as solid, dashed and dotted lines, respectively.
plasma ions in a volume V, we label the first N1 as the ions of the first kind and ions N1 þ1 to N as ions of the second kind. Quantities relating to the ith kind have a subscript (i). The probability density to find a net electric field E is:
Q ðEÞ ¼
Z
d3 r1 .d3 rN Pðr1 ; .; rN Þd E S Ei ðri Þ i
(23)
where r1, ., rN are the coordinates of the N plasma ions in the volume V and P is the probability density for configuration (r1, ., rN). In the Holtsmark case,
Pðr1 ; .; rN Þ ¼
1 VN
Ei ¼ Zi eri3 ri Fourier transforming Eq. (23) we obtain
In an average as in Eq. (17) with M particles (1,2, ., M) we have an integral
Z
d1d2.dMUð0; Dt; 1; 2; .MÞ.Uðt Dt; t; 1; 2; .MÞV M (34)
with
Z
i
h
i d3 rN1 þ1 .d3 rN exp k: S Ej rj ¼ Q1 ðkÞQ2 ðkÞ j
(26)
Z
d1d2.dM
(35)
The integral may be written first as a product of integrals
V p
d3 Eeik:E Q ðEÞ Z h i d3 r1 .d3 rN1 exp k: S Ei ðri Þ V N1 N ¼ V N1
Z
Appendix B. Details in the impact average
VM ¼ (25)
(33)
The Holtsmark distribution is clearly very simplistic. In addition, the conditions that allow one to neglect the interionic correlations, namely low densities and/or high temperatures, are also quite unfavorable to the quasistatic picture as a whole.
(24)
The individual fields are pure Coulomb fields, i.e.,
Q ðkÞ ¼
b ¼ E=E0
Z
dk1 dk2 .dkp UðjDt; ðj þ 1Þ Dt; k1 ; k2 ; .kp
(36)
where only the particles k1, k2.kp interact with the atom/ion in the interval (jDt, (j þ 1)Dt). At this point we can integrate over the remaining redundant variables in both numerator and denominator. Then, as the integrand does not depend of these variables because they do not interact in that interval, each such integral contributes an average evolution operator:
with the sums in the exponents running over the integrated position variables. Here
h
i ð1Þ 3=2 Q1 ðkÞ ¼ exp E0 k
(27) References
with ð1Þ E0
¼
2=3 2pZ1 jejn1
4 2=3 15
(28)
with a similar relation for the second species. Thus, we are able to write
Q ðkÞ ¼ eðE0 kÞ with
3=2
(29)
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[15] V.S. Lisitsa, G.V. Sholin, Sov. Phys. JETP 34 (1972) 484. [16] S. Alexiou, Phys. Rev. E 71 (2005) 066403. [17] E. Stambulchik, Y. Maron, ‘‘Plasma line broadening and computer simulations: a mini-review’’ this volume. [18] B. Talin, A. Calisti, L. Godbert, R. Stamm, R.W. Lee, L. Klein, Phys. Rev. A 51 (1995) 1918. [19] S. Alexiou, Phys. Rev. Lett. 76 (1996) 1836. [20] A. Brissaud, U. Frisch, JQSRT 11 (524) (1971) 1767; A. Brissaud, U. Frisch, JQSRT 11 (524) (1971) 1767; J. Math, Phys. 15 (1207) (1974); J. Seidel, Z. Naturforsch. 32a (1207) (1977).