- Email: [email protected]
Hydrogenic spectroscopy for various unusual plasmas
J. Quanf. .Specrrosc. Radiar. Transfer Vol. 4, No. I, pp. 93400, Printed in Great Britain
HYDROGENIC
1990
O&22-4073/90 $3.00 + 0.00 Pergamon Press plc
SPECTROSCOPY FOR VARIOUS UNUSUAL PLASMAS
P. BAKSHI and G. KALMAN Department of Physics, Boston College, Chestnut Hill, MA 02167-3811,U.S.A. Abstract-The theory of hydrogenic spectra for combined static and dynamic electric fields is reviewed. The important spectral modifications arise when the level of the dynamical field is high and its frequency is nearly resonant with the static Stark split sublevels of hydrogen. Observable effects include line broadening, satellite formation and polarization changes. Possible diagnostic applications include fusion related, solar and dense plasmas.
I. INTRODUCTION
The importance of hydrogenic spectroscopy for plasma diagnostics has been recognized for some time. The electric fields in plasmas affect the line emission through the Stark effect and this imprint on the observed spectra, properly analyzed, can provide information about the characteristics of the prevailing plasma fields. Slowly varying fields such as the ionic Holtsmark fields and low frequency oscillations will appear “static” to the atom. High frequency plasma oscillations and turbulent electric fields with fast time variation, on the other hand, constitute a time dependent perturbation and require a “dynamical” treatment. While the theories of the static Stark effect’ and the dynamical Stark effect2 for a monochromatic field were developed soon after the advent of quantum mechanics, the effects of combined static and dynamic fields were considered relatively recently.‘” It should be noted that the plasma electric fields can be quite strong in some situations and a non-perturbative treatment3-5 is thus required. Different atoms in the plasma see different electric fields. The observed line profiles reflect the cumulative intensity from all the emitters and consequently are averaged over the prevailing field distributions. The observed profiles, however, also reflect the effects of various broadening mechanisms such as Doppler broadening and instrumental broadening. Radiative transport, if relevant, further modifies the observed profiles. Thus the problem of inferring the plasma electric fields from the observed line shapes is a rather difficult “inverse” problem. The important question in any specific situation is to determine which features of the source line profile survive or remain noticeable in the final, observed profiles. The first step, in any case, is to determine the line spectrum for a single atom with prescribed static and dynamic electric fields. Our approach to this problem is briefly reviewed in Sec. 2. Averaging over the plasma field distributions and taking into account the broadening mechanisms are described in Sec. 3. Characteristic features of field distributions in some unusual plasmas are discussed in Sec. 4, along with comments on plasma diagnostics. Suggestions for further developments of theory and diagnostics are given in Sec. 5. 2. COMBINED
STATIC
AND
PERIODIC
FIELDS
For a hydrogen atom in the presence of an electric field E(t), the Hamiltonian H=H,+H,(t)=H,+r.eE(t),
is given by: (1)
where Ho is the usual Hamiltonian
of the unperturbed atom. In general, H, has inter- as well as intra-n multiplet matrix elements, n being the principal quantum number. The dynamics of the system are, however, primarily governed by the latter as long as the splittings induced by the external fields and the basic frequency of the dynamical field remain small compared to the inter-n separations. Familiar special cases of Eq. (1) include the static Stark effect’ (E = constant) and the purely dynamical Stark effect’ E = E, cos wt. It is a simple matter to extend the latter theory to include a parallel static field, and the resultant spectrum is a superposition of the static and dynamic 93
94
P. BAKSHI and G.
KALMAN
effects, acting independently. When the static and dynamic fields are not parallel, their influence on the hydrogenic spectrum is quite complex, and the solution to this problem was obtained relatively recently.334 The specific case of the Lyman-a transition for perpendicular static and dynamic fields was given in Ref. 3. A complete theory of the intra-n dynamics for combined static and periodic electric fields (including magnetic fields as well) was developed in Ref. 4. The salient features of this approach are summarized here. The O(4) symmetry of the hydrogen atom 6.7leads to the additional constant of motion, the Lenz-Runge vector M (besides the angular momentum L), and on the n-multiplet it can be identified’ with the positron vector r, r = (3/2)na0K,
K = M(me4/2 1E,,
(2)
lijkLk,
(4)
where eijkis the antisymmetric symbol and we have set h = 1. Since H, commutes with K, the intra-n dynamics is decoupled and is governed only by H, . In the general form, Eq. (3) when all the three components of K are involved, the full six-element Lie-algebra of K and L is required. On the other hand if the electric field E(t) remains confined to some fixed plane, one can simplify the problem by choosing two axes (x and z) in that plane, with the resultant Hamiltonian involving only K, and Kz. Noting from Eq. (4) that J = (K,, L,, Kz) satisfies the standard angular momentum commutation relations [Ji, J,] = i E~,~Jk, the Hamiltonian can be re-expressed as H, = J.c,(t).
(5)
Since components of J form a three-dimensional Lie algebra, the resulting spectrum for the n-multiplet is considerably simplified. Now consider a periodic electric field, E(t) = E(t + r). The Schrodinger equation for the evolution operator U,(t) for the n-multiplet constitutes a set of nz linear equations with periodic coefficients and consequently its solution has the Floquet structure4 U,(t) = g exp( - ik,t)P,(t),
(6)
y=l
where k, are constants to be determined and p,(t) are periodic matrices. The nz eigenvalues of U,,(r), where r is the period of the imposed field, form the set (exp(-i&z)}. These can be determined as follows. The solution for U,,(r) can also be written in the Magnus form8 u,(t) = exp(-Q(t)),
(7)
where Q,(t) is expressible in terms of quadratures of repeated commutators of the Hamiltonian H, at different times. Since the components of J form a closed Lie-algebra, the Magnus form reduces to u,,(t) =
ew(-iJ~/%(t>),
(8)
AWI = /-WI.
(9)
where b,,(t) is a unique functional fi of c,(t), B,(f) =
The entire spectrum can now be inferred from the structure of U,(T), U,,(r) = exp(-iJ.fi(~))
= exp(-iJBpn(T)),
(10)
where J, is the component of J in the direction of /In(t) and j?,,(z) = I/.?,,(T)~. U,,(r) is clearly diagonal in the nz (angular momentum like) basis states 1njp ) whose eigenvalues are J’+j(j + l),
Hydrogenic spectroscopy for various unusual plasmas
95
J,-+,u. The eigenvalues of U,(r) are then exp( -i&(r)). The set of n2 Floquet constants {k,} is thusgivenby{~rc,},K,=/?,(r)/z,~= -(n-l),...O,. . . (n - 1). The quasi-energies {pro,} depend only on ,u, and are degenerate with respect to j. Thus for the planar electric field, there are only 2n - 1 distinct values of {k,} = {PK,}, equally spaced with the level spacing characterized by K,. We call this the Type I spectrum.4 The periodic functions P,(t) give rise to a harmonic structure around the quasi-energies at multiples of the fundamental frequency w = 27c/r. The ensuing line spectrum for the transition n +n’ is given by the difference of the corresponding quasi-energy levels, (E,, + TIC,)- (E,, + p’rc,,,), at the (2n - 1)(2n’ - 1) basic positions {prcn- 11’~) around the unperturbed line. Each of these line components is accompanied by harmonic satellites at integral multiples ro of the basic frequency o. Thus the full line splitting due to combined static and dynamic planar electric fields is given by a,,.+~,,,
+ (PK, - ~‘rc,,) + ro.
(11)
The basic quasi-energy spacings rc, and K,, are related, but not necessarily in the ratio n/n’ as in the static Stark effect. From Eqs. (9) and (3) we note that /In(t), and thus the spacings K, for different n-multiplets obey the scaling relations
BnWI= Bn,Knln?El, K,[E] = K,,[(n/n ‘)E].
(12)
This led to the important inference4 that the full spectrum as well as the dynamics of an arbitrary n multiplet can be obtained from the solution of the (more easily solved) Lyman-a problem3, e.g., K,[E] = x,[(n/2)E]. Equation (11) can now be rewritten explicitly in terms of the basic Lyman-a spacing rc2, q,,,-co,,,,, + pc2[(n/2)E] - p’rc2[(n’/2)E] + ro.
(13)
We have shown in Ref. 3 how rc2can be obtained for any given fields, thus providing through Eq. (13) a complete scheme for determining the hydrogenic line structure for arbitrary planar static and periodic electric fields. As an example, Fig. 1 provides rc2 for a simple field configurationE(t) = Es + En cos or, where the angle between the static and dynamic field amplitudes is 19.The dimensionless variables S = 3ea,E,/o and D = 3ea, En/o characterize the respective field strengths, and K = rcZ/o is the measure of the basic spacing for the Lyman-a problem. Because of the harmonic structure, x2 is only defined modulo o, and it suffices to give K in the range O-l. Three different values of D are considered, and K vs S plots given for five different angles 0. As D increases, K(S, 0) acquires a progressively complex structure. Other illustrative plots for perpendicular static and dynamic fields have been given in Ref. 3. Figure 2 provides equi-K lines for perpendicular static and dynamic fields. This graph can be used to obtain K, for the higher levels by the scaling relations. The intensities in the line spectrum depend on the detailed dynamics of the n, n’ multiplets, governed by /l,,(t) and /I,,(t) and the radiation dipole matrix elements between the initial and final states. One can obtain /I,(t) through Eq. (12) by determining /3*(t) for an appropriately scaled electric field vector, and Eq. (8) then provides the full dynamics. A more direct method for determining the time evolution is as follows, Since [J’, Z-Z]= 0 the quantum number j (which we call the “planarity” quantum number) is conserved,5 and the n2 states of the n multiplet decouple into n independent completely non-degenerate j multiplets4 (14) Each j multiplet (consisting of 2j - 1 states) evolves independently in time according to UC’. This dynamic is conveniently studied by introducing’ “planarity states”. Different choices of the azimuthal direction lead to different representations of the planarity states. A computer program based on planarity states was developed’ to determine hydrogenic spectra up to Hs. If the field vector E(t) does not remain confined to a plane, the planarity conservation is broken and the n* states separate into as many distinct quasi-levels, leading to a Type II spectrum.4 The problem is still exactly soluble, even though all the generators of O(4) are now involved in a
P.
96
BAKSHIand
G. KALMAN
La)
(b)
2.0
2.0
K I.0
K 1.0
K 1.0
0.0 0.o
1.0
2.0
3.0 S
4.0
6.0
6.0
Fig. 1. K as a function of S, for three cases: (a) D = 0.3, (b) D = 1.5, (c) D = 2.7. In each case the solid line refers to 13= 90”. The other lines successively represent 0 = 70, 50,30, and IO”.The straight line K = S corresponds to fI = 0”.
six-dimensional Lie algebra. The problem remains soluble4 even if a constant or periodic magnetic field, B(r) = B(t + z) is introduced in an arbitrary direction. By introducing the two commuting angular momenta G, = (1/2)(L f K), one obtains a factorization U,,= U,+U; =exp(-iG;/?,+)exp(-iG_*/I;)
(15)
with B: = for a non-planar
B[k4
(16)
electric field, and
B’=B[bfd
Fig. 2. Contours of constant K for perpendicular static and dynamic fields for various Sand D. The value of K for a given contour can be read from the S scale at D = 0.
(17)
Hydrogenic spectroscopy for various unusual plasmas
91
if a magnetic field B is included in the Hamiltonian, H, = K-L,, + L *b, with b = (e/2mc)B. It should be noted that there are some special cases4 where a Type I spectrum is restored, even when a magnetic field is present or when the electric field is non-planar. The Type II spectrum is characterized by n2 quasi-levels E,+ E,, + p +K: + P-K; for the n-multiplet, and a line splitting for the transition n+n’ with n’n” basic positions (apart from harmonics), w,,,-+o,,, + (P+K,+ + P-K;)
- (j.4”~:
+p-‘K;)
+ ro,
(18)
where j.4+= -(f)(n
- l),
-(i)(n
- 3). . . ,
(i)(n - l),
K: = fi’(r)/r.
A similar definition holds for p * ’ in terms of n’. Again, scaling arguments apply through Eqs. (16) or (17) and the level spacings and dynamics for the n th level can be determined from a study of the Lyman-a problem with appropriately scaled fields. We note that the magnetic field parameter b remains the same for all levels. In summary, our formalism provides an exact solution for the n-multiplet splittings and dynamics for arbitrary static and periodic electric and magnetic fields. The time dependent fields have to be periodic but not necessarily monochromatic. Such a non-perturbative approach is useful, and even necessary, in many situations. Multiphoton transitions are obviously included here. The only limitation of the approach comes from the requirement that the field-Hamiltonian should not induce significant inter-n mixing. This requires E ~7 x 106/n5kV/cm and o K lo-” see-‘, conditions usually well met in plasmas. We have also ignored the fine structure splittings. The field induced effects far exceed the fine-structure if E >>9 l/n4 kV/cm and B >>124/n 3 kG. Finally, ignoring the finite level widths amounts to requiring o > lo9 set-’ the inverse lifetime of the fastest decaying state within a multiplet. Again, these are conditions satisfied for a large class of plasmas. 3. PLASMA
FIELD
DISTRIBUTIONS
Different atoms in a plasma see different electric fields. An individual atom responds to the prevailing fields in its neighborhood at the time of emission. The random ionic motion gives rise to Holtsmark fields which vary from one atom to another, and are to be taken into account statistically through their characteristic field distribution. Their time variation for a given atom is too slow on the emission time scale and thus they are essentially static. Any low frequency (ionic) oscillations will also appear to be static from this point of view. Only the high frequency electron oscillations, whether due to some coherent plasma mode or due to turbulence, will produce a dynamical perturbation of the atomic level structure. All the quasi-state fields can be combined to provide the effective static field Es seen by a given atom at the time of emission. Similarly, all the dynamic fields can be combined into a single effective dynamic field E,(t). For a well defined plasma mode, this will have a monochromatic character and also space coherence, provided the dispersion is weak. For such a situation one can ascribe a unidirectional monochromatic dynamical field to all the atoms on a large enough sample, and the theory of the previous section can be applied directly to obtain the emission line structure of each individual atom in this sample. Due to the variation of E, from one atom to another, we have to average over the individual line structure with a distribution in angles and magnitude of the static field strength to obtain the resultant line profile from the whole sample. It should be noted that since the theory of the previous section is based on periodicity of the field rather than monochromaticity, it can be applied even for large amplitude nonlinear plasma waves, which develop nonsinusoidal, but periodic behavior. If the dynamical fields arise due to turbulence, but are monochromatic as in Langmuir turbulence, the individual atom line emission structure is still obtained the same way. Determination of the line profile from the sample, however, now requires averaging over the distribution in angles and magnitudes of the dynamical fields as well. The observed line profiles also reflect the effects of various broadening mechanisms such as Doppler broadening and instrumental broadening. In an optically thick medium, the radiative transport also becomes significant and may further modify the observed profiles.
98
P. BAKSHIand G. KALMAN
The field averagings lead to relatively smooth line profiles instead of the discrete line structure characteristic of single fields, We have developed a program 9~‘ofor the angle and magnitude averaging of the static fields for given unidirectional dynamic fields. Intensities of radiation polarized along and perpendicular to the direction of the dynamical field were obtained’ for various dynamical field strengths, for isotropic static fields with spherical, Gaussian and shell distributions ‘O Effects of Doppler broadening were simulated by introducing a finite window averaging.‘*‘O’Diagnostic considerations are discussed in the next section. 4. PLASMA
DIAGNOSTICS
The formal problem of plasma diagnostics in this context may be stated as the determination of the properties of the electric (and magnetic) fields in the plasma by an analysis of the observed spectral line profiles. In view of the various field averagings and broadening mechanisms mentioned in the previous section, this is clearly a rather difficult “inverse problem” in general. When the fields are strong, and lead to sharp features in the line profiles in spite of the relevant broadening mechanisms, at least some of the properties of the plasma fields can be discovered by spectral line analysis. Unusual plasmas are likely to carry some corresponding unusual signatures of their properties in these line profiles. The theoretical considerations suggest at least three types of observable consequences’ where static and dynamic fields play a combined role. Magnitude of broadening
For a high level of turbulence the overall broadening of the line shape may be enhanced even when the field distributions do not lead to the more specific signatures mentioned below. Such an enhanced broadening would provide an estimate of electron density since electric fields at high level of turbulence9 scale as na’2. Solar flare plasmas are amenable to this approach. Position spectroscopy
If the dynamical field is essentially monochromatic and unidirectional (as can be the case in many fusion related plasmas), one can observe a line spectrum in which the stronger components are recognizable as humps or peaks on a broadened background. The positions of these components are often sufficient to determine the underlying static and dynamical field strengths, by recognizing the strengths K, which would produce the observed peaks. Correlating spectra of different hydrogenic lines (H, , H, . . .) is very helpful in identifying K, and establishing the correctness of the inferred field strengths. An example where such an approach was feasible is given by the hydrogenic spectra (up to Hd) for a fusion related plasma given in Ref. 11. The H, line shows a pronounced peak corresponding to a 28~ Stark-like component, which however is due to an interplay of quasi-energy spacings ICYand rc2. Other independent diagnostics, based on helium spectroscopy confirm these results.” Quite often, the finer harmonic satellite structure rw is also visible,” and it then provides the electron density through a determination of the plasma frequency. A recent example of plasma diagnostics based on the harmonic satellite structure is provided by Ref. 12. Satellites were observed on the line profile of a hydrogen-like ionized helium line from a high density theta pinch plasma. In this case the inferred dynamical field frequency w was not the electron plasma frequency (which is very high), but seemed to be related to a current driven ion mode.12 This is an example of the possibility of diagnosing plasma instability mechanisms through hydrogenic spectroscopy. Polarization
A sensitive measure of anisotropy in the system is the observable difference in linear polarization in two orthogonal directions. Various plasma instabilities have a directional character and are likely to lead to dynamical fields which are polarized along some preferred direction. By measuring the difference in polarization in orthogonal directions, one can infer the underlying directional properties of the fields. An illustration of some expected features of the polarization differences for turbulence broadened profiles was provided in Ref. 9 through computed Lyman-a profiles for given isotropic static and unidirectional dynamic field distributions.
Hydrogenic spectroscopy for various unusual plasmas
99
It has been suggestedI that a combination of a dynamic field and broadly distributed static fields will lead to special features, dips or valleys, on the observed line profiles. The resonant interactions in Figs. 1 and 2 show that some K values (near integers) are not realized for small D and moderate variations of S; the quasi-energies move away from exact resonance with integral multiples of o. These gaps in the quasi-energies may lead to gaps in the combined field spectra, which in the absence of the dynamical field would have been continuous. While this may indeed be the case in some situations, it is necessary to examine carefully the role of various broadening mechanisms as well as consider the possibility that harmonic satellites of other K values may fill these gaps. Our analysis for solar flare plasma conditions is given in Ref. 9. For strongly coupled plasma, various special effects arise which have been discussed in Ref. 14 and in another paper in this issue.” 5.
CONCLUDING
REMARKS
Hydrogenic spectroscopy is a powerful and versatile technique for plasma diagnostics. The sensitive and multifaceted response of the hydrogenic line profiles to various plasma electric fields provides a convenient, non-invasive method for the determination of the state of the plasma. Our non-perturbative, exact theory for the spectra due to combined static and dynamic periodic fields provides a comprehensive framework (Sec. 2) for the determination of the precise line structure. The ensuing field averaged and otherwise broadened observed line profiles present an inverse problem for the determination of the underlying field distributions. Depending on the particular characteristics of a given plasma, various distinguishing features remain noticeable (Sec. 4) in the observed profiles and provide information about the plasma fields. As with any complex inverse problem, the diagnosis improves when the data base allows cross correlations; in this instance, observing a whole set of spectral lines (H,, H,, . . .) and demanding internal consistency in the observed features, that is to be expected on the basis of the full theory, eliminates the danger of spurious identification. Further correlations with other diagnostic techniques, e.g., helium spectroscopy, should also be included in the observational program. There are two important limitations of our approach. First, the inter-n dynamics is not included; the conditions for the validity of our aproach were quantitatively stated in Sec. 2. With the advent of high powered lasers, inter-n dynamics becomes more important. Even otherwise, it is relevant for (high n) Rydberg states. The identification of the position operator with the Lenz vector is valid only within an n-multiplet. For inter-n matrix elements one needs to enlarge the group from SO(4) to the non-compact SO(4,l) and SO(4,2) with all the attendant complications. Such an extension can be carried out, but the results will not be as simple. The second limitation is the assumed periodicity of the dynamical field. Our approach is certainly adequate for external field driven experiments where the frequency of the applied field is monochromatic (or includes harmonics of a fundamental frequency). Some internal fields of plasmas will also qualify, e.g., coherent oscillations at the plasma frequency, or at any other (high enough) frequency due to other plasma modes. Even multimode systems where all the frequencies can be expressed as simple integral multiples of some (lower) fundamental frequency will be amenable to this approach. Yet this leaves out many realistic systems which have two or more non-commensurate frequencies, or which possess broadband turbulence. The other central feature of our approach, the O(4) symmetry, of course, is an integral property of the hydrogenic Coulomb systems. But for dense plasmas, the screening effects due to many body effects can alter the Coulomb potential and destroy the O(4) symmetry. Small departures can be treated perturbatively, but a different approach may be needed for very strongly coupled plasma systems. On the experimental side, it would be worthwhile to set up external field studies, where the field parameters can be controlled, to test or confirm the discrete line structure predictions of the theory. Observed spectral line profile for various plasma systems (solar, fusion-related, strongly coupled, etc.) can be analyzed in the light of the structural and other predictions of the combined fields theory; this may lead to possible reinterpretations of the plasma parameters in some situations. Finally the polarization studies should be emphasized in observational programs; due to their sensitivity to plasma field effects it may be possible to discern the directional properties of the underlying fields.
loo
P. BAKS~and G. KALMAN
wish to thank the U.S. National Science Foundation, Division of International Programs, for support for our participation at this conference. G.K. also wishes to acknowledge partial support from NSF Grant ECS 871337.
Acknowledgements-We
REFERENCES 1. E. Schrodinger, Ann. Phys. 80, 457 (1926). 2. D. Blochintsev, Phys. Z. SowjUn. 4, 501 (1933) 3. A. Cohn, P. Bakshi, and G. Kalman, Phys. Rev. Lett. 29, 324 (1972). 4. P. Bakshi, G. Kalman, and A. Cohn, Phys. Rev. Lett. 31, 1576 (1973). 5. P. Bakshi, G. Kalman, and A. Cohn, in Fourth Znt. Conf on Atomic Physics, Abstr. pp. 114-117, J. Kowalski and H. G. Weber eds., University of Heidelberg, Heidelberg (1974). 6. H. Kleinert, in Lectures in Theoretical Physics, Vol. XB, p. 427, A. 0. Barut and W. E. Brittain eds., Gordon & Breach, New York, NY (1968). 7. G. Flamand, J. Math. Phys. 7, 1924 (1966). 8. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954). 9. P. Bakshi and G. Kalman, Sol. Phys. 47, 307 (1976). 10. R. Hogaboom, Computer generated spectra of atomic hydrogen under certain kinds of periodic electric fields, M.S. Thesis, Boston College, MA (1975). 11. C. C. Gallagher and M. A. Levine, Phys. Rev. Lett. 30, 897 (1973). 12. B. Amini, Phys. Fluids 29, 3775 (1986). 13. A. I. Zhuzhunashvali and E. A. Oks, Soviet Phys. JETP 46, 1122 (1976). V. P. Gavrilenko and E. A. Oks, Soviet Phys. Plasma Phys. 13, 22 (1987). 14. P. Bakshi and G. Kalman, Phys. Rev. A 30, 613 (1984). 15. G. Kalman and P. Bakshi, JQSRT 44, 1 (1990).
HYDROGENIC
1990
O&22-4073/90 $3.00 + 0.00 Pergamon Press plc
SPECTROSCOPY FOR VARIOUS UNUSUAL PLASMAS
P. BAKSHI and G. KALMAN Department of Physics, Boston College, Chestnut Hill, MA 02167-3811,U.S.A. Abstract-The theory of hydrogenic spectra for combined static and dynamic electric fields is reviewed. The important spectral modifications arise when the level of the dynamical field is high and its frequency is nearly resonant with the static Stark split sublevels of hydrogen. Observable effects include line broadening, satellite formation and polarization changes. Possible diagnostic applications include fusion related, solar and dense plasmas.
I. INTRODUCTION
The importance of hydrogenic spectroscopy for plasma diagnostics has been recognized for some time. The electric fields in plasmas affect the line emission through the Stark effect and this imprint on the observed spectra, properly analyzed, can provide information about the characteristics of the prevailing plasma fields. Slowly varying fields such as the ionic Holtsmark fields and low frequency oscillations will appear “static” to the atom. High frequency plasma oscillations and turbulent electric fields with fast time variation, on the other hand, constitute a time dependent perturbation and require a “dynamical” treatment. While the theories of the static Stark effect’ and the dynamical Stark effect2 for a monochromatic field were developed soon after the advent of quantum mechanics, the effects of combined static and dynamic fields were considered relatively recently.‘” It should be noted that the plasma electric fields can be quite strong in some situations and a non-perturbative treatment3-5 is thus required. Different atoms in the plasma see different electric fields. The observed line profiles reflect the cumulative intensity from all the emitters and consequently are averaged over the prevailing field distributions. The observed profiles, however, also reflect the effects of various broadening mechanisms such as Doppler broadening and instrumental broadening. Radiative transport, if relevant, further modifies the observed profiles. Thus the problem of inferring the plasma electric fields from the observed line shapes is a rather difficult “inverse” problem. The important question in any specific situation is to determine which features of the source line profile survive or remain noticeable in the final, observed profiles. The first step, in any case, is to determine the line spectrum for a single atom with prescribed static and dynamic electric fields. Our approach to this problem is briefly reviewed in Sec. 2. Averaging over the plasma field distributions and taking into account the broadening mechanisms are described in Sec. 3. Characteristic features of field distributions in some unusual plasmas are discussed in Sec. 4, along with comments on plasma diagnostics. Suggestions for further developments of theory and diagnostics are given in Sec. 5. 2. COMBINED
STATIC
AND
PERIODIC
FIELDS
For a hydrogen atom in the presence of an electric field E(t), the Hamiltonian H=H,+H,(t)=H,+r.eE(t),
is given by: (1)
where Ho is the usual Hamiltonian
of the unperturbed atom. In general, H, has inter- as well as intra-n multiplet matrix elements, n being the principal quantum number. The dynamics of the system are, however, primarily governed by the latter as long as the splittings induced by the external fields and the basic frequency of the dynamical field remain small compared to the inter-n separations. Familiar special cases of Eq. (1) include the static Stark effect’ (E = constant) and the purely dynamical Stark effect’ E = E, cos wt. It is a simple matter to extend the latter theory to include a parallel static field, and the resultant spectrum is a superposition of the static and dynamic 93
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P. BAKSHI and G.
KALMAN
effects, acting independently. When the static and dynamic fields are not parallel, their influence on the hydrogenic spectrum is quite complex, and the solution to this problem was obtained relatively recently.334 The specific case of the Lyman-a transition for perpendicular static and dynamic fields was given in Ref. 3. A complete theory of the intra-n dynamics for combined static and periodic electric fields (including magnetic fields as well) was developed in Ref. 4. The salient features of this approach are summarized here. The O(4) symmetry of the hydrogen atom 6.7leads to the additional constant of motion, the Lenz-Runge vector M (besides the angular momentum L), and on the n-multiplet it can be identified’ with the positron vector r, r = (3/2)na0K,
K = M(me4/2 1E,,
(2)
lijkLk,
(4)
where eijkis the antisymmetric symbol and we have set h = 1. Since H, commutes with K, the intra-n dynamics is decoupled and is governed only by H, . In the general form, Eq. (3) when all the three components of K are involved, the full six-element Lie-algebra of K and L is required. On the other hand if the electric field E(t) remains confined to some fixed plane, one can simplify the problem by choosing two axes (x and z) in that plane, with the resultant Hamiltonian involving only K, and Kz. Noting from Eq. (4) that J = (K,, L,, Kz) satisfies the standard angular momentum commutation relations [Ji, J,] = i E~,~Jk, the Hamiltonian can be re-expressed as H, = J.c,(t).
(5)
Since components of J form a three-dimensional Lie algebra, the resulting spectrum for the n-multiplet is considerably simplified. Now consider a periodic electric field, E(t) = E(t + r). The Schrodinger equation for the evolution operator U,(t) for the n-multiplet constitutes a set of nz linear equations with periodic coefficients and consequently its solution has the Floquet structure4 U,(t) = g exp( - ik,t)P,(t),
(6)
y=l
where k, are constants to be determined and p,(t) are periodic matrices. The nz eigenvalues of U,,(r), where r is the period of the imposed field, form the set (exp(-i&z)}. These can be determined as follows. The solution for U,,(r) can also be written in the Magnus form8 u,(t) = exp(-Q(t)),
(7)
where Q,(t) is expressible in terms of quadratures of repeated commutators of the Hamiltonian H, at different times. Since the components of J form a closed Lie-algebra, the Magnus form reduces to u,,(t) =
ew(-iJ~/%(t>),
(8)
AWI = /-WI.
(9)
where b,,(t) is a unique functional fi of c,(t), B,(f) =
The entire spectrum can now be inferred from the structure of U,(T), U,,(r) = exp(-iJ.fi(~))
= exp(-iJBpn(T)),
(10)
where J, is the component of J in the direction of /In(t) and j?,,(z) = I/.?,,(T)~. U,,(r) is clearly diagonal in the nz (angular momentum like) basis states 1njp ) whose eigenvalues are J’+j(j + l),
Hydrogenic spectroscopy for various unusual plasmas
95
J,-+,u. The eigenvalues of U,(r) are then exp( -i&(r)). The set of n2 Floquet constants {k,} is thusgivenby{~rc,},K,=/?,(r)/z,~= -(n-l),...O,. . . (n - 1). The quasi-energies {pro,} depend only on ,u, and are degenerate with respect to j. Thus for the planar electric field, there are only 2n - 1 distinct values of {k,} = {PK,}, equally spaced with the level spacing characterized by K,. We call this the Type I spectrum.4 The periodic functions P,(t) give rise to a harmonic structure around the quasi-energies at multiples of the fundamental frequency w = 27c/r. The ensuing line spectrum for the transition n +n’ is given by the difference of the corresponding quasi-energy levels, (E,, + TIC,)- (E,, + p’rc,,,), at the (2n - 1)(2n’ - 1) basic positions {prcn- 11’~) around the unperturbed line. Each of these line components is accompanied by harmonic satellites at integral multiples ro of the basic frequency o. Thus the full line splitting due to combined static and dynamic planar electric fields is given by a,,.+~,,,
+ (PK, - ~‘rc,,) + ro.
(11)
The basic quasi-energy spacings rc, and K,, are related, but not necessarily in the ratio n/n’ as in the static Stark effect. From Eqs. (9) and (3) we note that /In(t), and thus the spacings K, for different n-multiplets obey the scaling relations
BnWI= Bn,Knln?El, K,[E] = K,,[(n/n ‘)E].
(12)
This led to the important inference4 that the full spectrum as well as the dynamics of an arbitrary n multiplet can be obtained from the solution of the (more easily solved) Lyman-a problem3, e.g., K,[E] = x,[(n/2)E]. Equation (11) can now be rewritten explicitly in terms of the basic Lyman-a spacing rc2, q,,,-co,,,,, + pc2[(n/2)E] - p’rc2[(n’/2)E] + ro.
(13)
We have shown in Ref. 3 how rc2can be obtained for any given fields, thus providing through Eq. (13) a complete scheme for determining the hydrogenic line structure for arbitrary planar static and periodic electric fields. As an example, Fig. 1 provides rc2 for a simple field configurationE(t) = Es + En cos or, where the angle between the static and dynamic field amplitudes is 19.The dimensionless variables S = 3ea,E,/o and D = 3ea, En/o characterize the respective field strengths, and K = rcZ/o is the measure of the basic spacing for the Lyman-a problem. Because of the harmonic structure, x2 is only defined modulo o, and it suffices to give K in the range O-l. Three different values of D are considered, and K vs S plots given for five different angles 0. As D increases, K(S, 0) acquires a progressively complex structure. Other illustrative plots for perpendicular static and dynamic fields have been given in Ref. 3. Figure 2 provides equi-K lines for perpendicular static and dynamic fields. This graph can be used to obtain K, for the higher levels by the scaling relations. The intensities in the line spectrum depend on the detailed dynamics of the n, n’ multiplets, governed by /l,,(t) and /I,,(t) and the radiation dipole matrix elements between the initial and final states. One can obtain /I,(t) through Eq. (12) by determining /3*(t) for an appropriately scaled electric field vector, and Eq. (8) then provides the full dynamics. A more direct method for determining the time evolution is as follows, Since [J’, Z-Z]= 0 the quantum number j (which we call the “planarity” quantum number) is conserved,5 and the n2 states of the n multiplet decouple into n independent completely non-degenerate j multiplets4 (14) Each j multiplet (consisting of 2j - 1 states) evolves independently in time according to UC’. This dynamic is conveniently studied by introducing’ “planarity states”. Different choices of the azimuthal direction lead to different representations of the planarity states. A computer program based on planarity states was developed’ to determine hydrogenic spectra up to Hs. If the field vector E(t) does not remain confined to a plane, the planarity conservation is broken and the n* states separate into as many distinct quasi-levels, leading to a Type II spectrum.4 The problem is still exactly soluble, even though all the generators of O(4) are now involved in a
P.
96
BAKSHIand
G. KALMAN
La)
(b)
2.0
2.0
K I.0
K 1.0
K 1.0
0.0 0.o
1.0
2.0
3.0 S
4.0
6.0
6.0
Fig. 1. K as a function of S, for three cases: (a) D = 0.3, (b) D = 1.5, (c) D = 2.7. In each case the solid line refers to 13= 90”. The other lines successively represent 0 = 70, 50,30, and IO”.The straight line K = S corresponds to fI = 0”.
six-dimensional Lie algebra. The problem remains soluble4 even if a constant or periodic magnetic field, B(r) = B(t + z) is introduced in an arbitrary direction. By introducing the two commuting angular momenta G, = (1/2)(L f K), one obtains a factorization U,,= U,+U; =exp(-iG;/?,+)exp(-iG_*/I;)
(15)
with B: = for a non-planar
B[k4
(16)
electric field, and
B’=B[bfd
Fig. 2. Contours of constant K for perpendicular static and dynamic fields for various Sand D. The value of K for a given contour can be read from the S scale at D = 0.
(17)
Hydrogenic spectroscopy for various unusual plasmas
91
if a magnetic field B is included in the Hamiltonian, H, = K-L,, + L *b, with b = (e/2mc)B. It should be noted that there are some special cases4 where a Type I spectrum is restored, even when a magnetic field is present or when the electric field is non-planar. The Type II spectrum is characterized by n2 quasi-levels E,+ E,, + p +K: + P-K; for the n-multiplet, and a line splitting for the transition n+n’ with n’n” basic positions (apart from harmonics), w,,,-+o,,, + (P+K,+ + P-K;)
- (j.4”~:
+p-‘K;)
+ ro,
(18)
where j.4+= -(f)(n
- l),
-(i)(n
- 3). . . ,
(i)(n - l),
K: = fi’(r)/r.
A similar definition holds for p * ’ in terms of n’. Again, scaling arguments apply through Eqs. (16) or (17) and the level spacings and dynamics for the n th level can be determined from a study of the Lyman-a problem with appropriately scaled fields. We note that the magnetic field parameter b remains the same for all levels. In summary, our formalism provides an exact solution for the n-multiplet splittings and dynamics for arbitrary static and periodic electric and magnetic fields. The time dependent fields have to be periodic but not necessarily monochromatic. Such a non-perturbative approach is useful, and even necessary, in many situations. Multiphoton transitions are obviously included here. The only limitation of the approach comes from the requirement that the field-Hamiltonian should not induce significant inter-n mixing. This requires E ~7 x 106/n5kV/cm and o K lo-” see-‘, conditions usually well met in plasmas. We have also ignored the fine structure splittings. The field induced effects far exceed the fine-structure if E >>9 l/n4 kV/cm and B >>124/n 3 kG. Finally, ignoring the finite level widths amounts to requiring o > lo9 set-’ the inverse lifetime of the fastest decaying state within a multiplet. Again, these are conditions satisfied for a large class of plasmas. 3. PLASMA
FIELD
DISTRIBUTIONS
Different atoms in a plasma see different electric fields. An individual atom responds to the prevailing fields in its neighborhood at the time of emission. The random ionic motion gives rise to Holtsmark fields which vary from one atom to another, and are to be taken into account statistically through their characteristic field distribution. Their time variation for a given atom is too slow on the emission time scale and thus they are essentially static. Any low frequency (ionic) oscillations will also appear to be static from this point of view. Only the high frequency electron oscillations, whether due to some coherent plasma mode or due to turbulence, will produce a dynamical perturbation of the atomic level structure. All the quasi-state fields can be combined to provide the effective static field Es seen by a given atom at the time of emission. Similarly, all the dynamic fields can be combined into a single effective dynamic field E,(t). For a well defined plasma mode, this will have a monochromatic character and also space coherence, provided the dispersion is weak. For such a situation one can ascribe a unidirectional monochromatic dynamical field to all the atoms on a large enough sample, and the theory of the previous section can be applied directly to obtain the emission line structure of each individual atom in this sample. Due to the variation of E, from one atom to another, we have to average over the individual line structure with a distribution in angles and magnitude of the static field strength to obtain the resultant line profile from the whole sample. It should be noted that since the theory of the previous section is based on periodicity of the field rather than monochromaticity, it can be applied even for large amplitude nonlinear plasma waves, which develop nonsinusoidal, but periodic behavior. If the dynamical fields arise due to turbulence, but are monochromatic as in Langmuir turbulence, the individual atom line emission structure is still obtained the same way. Determination of the line profile from the sample, however, now requires averaging over the distribution in angles and magnitudes of the dynamical fields as well. The observed line profiles also reflect the effects of various broadening mechanisms such as Doppler broadening and instrumental broadening. In an optically thick medium, the radiative transport also becomes significant and may further modify the observed profiles.
98
P. BAKSHIand G. KALMAN
The field averagings lead to relatively smooth line profiles instead of the discrete line structure characteristic of single fields, We have developed a program 9~‘ofor the angle and magnitude averaging of the static fields for given unidirectional dynamic fields. Intensities of radiation polarized along and perpendicular to the direction of the dynamical field were obtained’ for various dynamical field strengths, for isotropic static fields with spherical, Gaussian and shell distributions ‘O Effects of Doppler broadening were simulated by introducing a finite window averaging.‘*‘O’Diagnostic considerations are discussed in the next section. 4. PLASMA
DIAGNOSTICS
The formal problem of plasma diagnostics in this context may be stated as the determination of the properties of the electric (and magnetic) fields in the plasma by an analysis of the observed spectral line profiles. In view of the various field averagings and broadening mechanisms mentioned in the previous section, this is clearly a rather difficult “inverse problem” in general. When the fields are strong, and lead to sharp features in the line profiles in spite of the relevant broadening mechanisms, at least some of the properties of the plasma fields can be discovered by spectral line analysis. Unusual plasmas are likely to carry some corresponding unusual signatures of their properties in these line profiles. The theoretical considerations suggest at least three types of observable consequences’ where static and dynamic fields play a combined role. Magnitude of broadening
For a high level of turbulence the overall broadening of the line shape may be enhanced even when the field distributions do not lead to the more specific signatures mentioned below. Such an enhanced broadening would provide an estimate of electron density since electric fields at high level of turbulence9 scale as na’2. Solar flare plasmas are amenable to this approach. Position spectroscopy
If the dynamical field is essentially monochromatic and unidirectional (as can be the case in many fusion related plasmas), one can observe a line spectrum in which the stronger components are recognizable as humps or peaks on a broadened background. The positions of these components are often sufficient to determine the underlying static and dynamical field strengths, by recognizing the strengths K, which would produce the observed peaks. Correlating spectra of different hydrogenic lines (H, , H, . . .) is very helpful in identifying K, and establishing the correctness of the inferred field strengths. An example where such an approach was feasible is given by the hydrogenic spectra (up to Hd) for a fusion related plasma given in Ref. 11. The H, line shows a pronounced peak corresponding to a 28~ Stark-like component, which however is due to an interplay of quasi-energy spacings ICYand rc2. Other independent diagnostics, based on helium spectroscopy confirm these results.” Quite often, the finer harmonic satellite structure rw is also visible,” and it then provides the electron density through a determination of the plasma frequency. A recent example of plasma diagnostics based on the harmonic satellite structure is provided by Ref. 12. Satellites were observed on the line profile of a hydrogen-like ionized helium line from a high density theta pinch plasma. In this case the inferred dynamical field frequency w was not the electron plasma frequency (which is very high), but seemed to be related to a current driven ion mode.12 This is an example of the possibility of diagnosing plasma instability mechanisms through hydrogenic spectroscopy. Polarization
A sensitive measure of anisotropy in the system is the observable difference in linear polarization in two orthogonal directions. Various plasma instabilities have a directional character and are likely to lead to dynamical fields which are polarized along some preferred direction. By measuring the difference in polarization in orthogonal directions, one can infer the underlying directional properties of the fields. An illustration of some expected features of the polarization differences for turbulence broadened profiles was provided in Ref. 9 through computed Lyman-a profiles for given isotropic static and unidirectional dynamic field distributions.
Hydrogenic spectroscopy for various unusual plasmas
99
It has been suggestedI that a combination of a dynamic field and broadly distributed static fields will lead to special features, dips or valleys, on the observed line profiles. The resonant interactions in Figs. 1 and 2 show that some K values (near integers) are not realized for small D and moderate variations of S; the quasi-energies move away from exact resonance with integral multiples of o. These gaps in the quasi-energies may lead to gaps in the combined field spectra, which in the absence of the dynamical field would have been continuous. While this may indeed be the case in some situations, it is necessary to examine carefully the role of various broadening mechanisms as well as consider the possibility that harmonic satellites of other K values may fill these gaps. Our analysis for solar flare plasma conditions is given in Ref. 9. For strongly coupled plasma, various special effects arise which have been discussed in Ref. 14 and in another paper in this issue.” 5.
CONCLUDING
REMARKS
Hydrogenic spectroscopy is a powerful and versatile technique for plasma diagnostics. The sensitive and multifaceted response of the hydrogenic line profiles to various plasma electric fields provides a convenient, non-invasive method for the determination of the state of the plasma. Our non-perturbative, exact theory for the spectra due to combined static and dynamic periodic fields provides a comprehensive framework (Sec. 2) for the determination of the precise line structure. The ensuing field averaged and otherwise broadened observed line profiles present an inverse problem for the determination of the underlying field distributions. Depending on the particular characteristics of a given plasma, various distinguishing features remain noticeable (Sec. 4) in the observed profiles and provide information about the plasma fields. As with any complex inverse problem, the diagnosis improves when the data base allows cross correlations; in this instance, observing a whole set of spectral lines (H,, H,, . . .) and demanding internal consistency in the observed features, that is to be expected on the basis of the full theory, eliminates the danger of spurious identification. Further correlations with other diagnostic techniques, e.g., helium spectroscopy, should also be included in the observational program. There are two important limitations of our approach. First, the inter-n dynamics is not included; the conditions for the validity of our aproach were quantitatively stated in Sec. 2. With the advent of high powered lasers, inter-n dynamics becomes more important. Even otherwise, it is relevant for (high n) Rydberg states. The identification of the position operator with the Lenz vector is valid only within an n-multiplet. For inter-n matrix elements one needs to enlarge the group from SO(4) to the non-compact SO(4,l) and SO(4,2) with all the attendant complications. Such an extension can be carried out, but the results will not be as simple. The second limitation is the assumed periodicity of the dynamical field. Our approach is certainly adequate for external field driven experiments where the frequency of the applied field is monochromatic (or includes harmonics of a fundamental frequency). Some internal fields of plasmas will also qualify, e.g., coherent oscillations at the plasma frequency, or at any other (high enough) frequency due to other plasma modes. Even multimode systems where all the frequencies can be expressed as simple integral multiples of some (lower) fundamental frequency will be amenable to this approach. Yet this leaves out many realistic systems which have two or more non-commensurate frequencies, or which possess broadband turbulence. The other central feature of our approach, the O(4) symmetry, of course, is an integral property of the hydrogenic Coulomb systems. But for dense plasmas, the screening effects due to many body effects can alter the Coulomb potential and destroy the O(4) symmetry. Small departures can be treated perturbatively, but a different approach may be needed for very strongly coupled plasma systems. On the experimental side, it would be worthwhile to set up external field studies, where the field parameters can be controlled, to test or confirm the discrete line structure predictions of the theory. Observed spectral line profile for various plasma systems (solar, fusion-related, strongly coupled, etc.) can be analyzed in the light of the structural and other predictions of the combined fields theory; this may lead to possible reinterpretations of the plasma parameters in some situations. Finally the polarization studies should be emphasized in observational programs; due to their sensitivity to plasma field effects it may be possible to discern the directional properties of the underlying fields.
loo
P. BAKS~and G. KALMAN
wish to thank the U.S. National Science Foundation, Division of International Programs, for support for our participation at this conference. G.K. also wishes to acknowledge partial support from NSF Grant ECS 871337.
Acknowledgements-We
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