Dielectronic recombination cross sections for fusion plasmas

Nuclear Instruments and Methods in Physics Research A240 (1985) 549-551 North-Holland, Amsterdam

549

DIELECTRONIC R E C O M B I N A T I O N C R O S S S E C T I O N S FOR F U S I O N P L A S M A S K.J. L A G A T T U T A Physics Department, University of Connecticut, Storrs, CT 06268, USA

The almost simultaneous appearance of calculated and measured dielectronic recombination cross sections for singly and few-times charged target ions has produced an interesting situation. The large disparity between theory and experiment has provoked speculation about a possible "electric field effect". The implications of this idea are examined, in the context of distorted-wave theory, and using the isolated-resonance-approximation.

1. Introduction Dielectronic recombination (DR) has been found to be a process of significance for both astrophysical and for fusion plasmas. The coronal density of multiply charged metal ions is greatly affected by DR, while the ionization balance of lighter elements in cool nebular plasmas is also altered substantially by this process. The energy losses experienced by present day controlled fusion device plasmas, running near maximum power input, are due largely to radiation from multiply charged impurity ions. The intensity of this radiation is proportional to the 4th power of the impurity charge (for An ~ 0 excitations). The impurity charge state is determined, in part, by the DR rate coefficients (aDR), at a given plasma electron temperature. DR theory was first propounded in a simple form by Burgess [1]. More exhaustive treatments are attributable to Seaton [2], Fano [3], and to Hahn [4]. The earliest rough calculations of OtDR may be credited to Burgess [5]. More precise work was performed next by Dubau [6], although this was limited to He-like ions, for the most part. In the last five years Hahn and coworkers, at the University of Connecticut, have performed a large number of DR calculations for a wide range of ions. Both a D R and the DR cross section (O DR) have been computed. The University of Connecticut group has completed calculations of a DR for the isoelectronic sequences of H, He [11], Li [12], Ne [13] and Na [14] ions, and is presently computing a DR for the Mg [15] and Ar sequences. Additionally, o DR has been determined for a variety of ions; e.g., C1 VIII [16], C II [17], Mg II [18], and Li sequence ions [19]. Because of technical problems, the first direct measurements of O"DR have been made only recently. Crossed electron-ion beam experiments have been run at JILA [7] for Mg II targets, and at Perth [8], for Ca II. Merged electron-ion beam experiments have been per0168-9002/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

formed at W. Ontario [9], for C II, and at ORNL [10], for B III, C IV, and O VI. These experimental results were compared with detailed calculations of o °R, performed by the University of Connecticut group. The results of this interesting comparison will be discussed in this talk.

2. General theory In its simplest form, the DR process is described by ( Il BIB)i -F- k c l c ~ (( n ala )(Ill)) d

-~ ( ( I l d ~ ) ( I l O ) / +

V.

(1)

In the initial state, i, the target ion is struck by a continuum electron of orbital angular momentum l~, and energy k 2. During the collision, an nBl B target electron is excited to h a l a, and the continuum electron is captured resonantly to nl. The intermediate state, d, is metastable, and may either Auger decay back to the state i, or decay radiatively to f + 7. The f-state represents a state of the recombined ion. Our usual approach makes use of distorted-wave (DW) theory, and employs the isolated-resonance-approximation (IRA). Matrix elements for capture-excitation are determined numerically. Bound state wavefunctions are single-configuration, nonrelativistic, Hartree-Fock functions. Continuum wave-functions are computed in the field of the frozen initial Hartree-Fock state, i, with nonlocal exchange included. LS coupling is invoked throughout, for i, d, and f states. Occasional add-ons to this basic method of calculation include relativistic corrections to wave-functions, configuration interaction in i, d, or f states, and intermediate coupling. The effect of coupling order [20], as well as the " n o coupling" approximation (all azimuthal quantum numbers treated as good quantum numbers) has been considered. The correct treatment of overlapping resonances has recently been investigated. Channel coupling

K.J. LaGattuta / Dielectronicrecombinationcross sections

550

effects are a subject for future study [21]. The amplitude for D R is, to lowest order in the electron-electron interaction, V, and the electron photon coupling, D,

M/ =~a ( Y + f L D ] d ) ( d l V l i + k c l ~ ) [~, + k3 - ~. + ir'(d)/2] where the fwhm of the resonance state, cross section is

(2)

d,

is F ( d ) . The

oDR = ~ o D R ( d ) ,

(3)

d

where o D R ( d ) = 2~rZa02"r0 k~

Y~ g a A , ( d ) w ( d ) f ( d ,

k~).

(4)

l~ gi

In (4), ga and g, are the statistical weights of the intermediate and initial states, respectively, ~ ( d ) is the fluorescence yield, A~(d) is the Auger probability (for d ~ i + k~l~) and f(d,

k~)- C(d) 2~r

+

1

(5)

The fluorescence yield is

~(d)=Ar/[Ar+~Aa], where A r is the radiative probability fwhm is

F( d ) = Ar + Y~A a.

(6)

(d~f+y).

The

case of LS coupling, and - 3.5 × below experiment if the " n o coupling" approximation is invoked. A similar, if not more pronounced, discrepancy was seen to exist between the LS coupling calculation [17] and the W. Ontario measurement [9]. However, the calculations for B III and C IV [19], in LS coupling, disagree at most by - 50% with the O R N L experiments [10]. The situation for the singly charged ions is clearly unsatisfactory, and a failure of either D W theory, the IRA, or both is possible for these cases. There is one other possibility which comes to mind. Past work of Burgess [22], Jacobs [23], and Bottcher [24] had suggested that large enhancements of o DR might be induced by electric fields, if these were present within the interaction region. In the context of the IRA, these enhancements are easy to understand. Using the Mg II target as an example, we consider the capture of a continuum electron to a high Rydberg state (HRS), hi, of Mg I. Keeping in mind that in the J I L A experiment only n ~ 64 contribute to the measured cross section, we note that when n _< 100, and l_< 4, then A a >> A r, for Mg II. Consequently, the D R cross section is proportional to

AaAr EAo,~=~2 ---At, 1, :~ [Ar+ Y'A,] t< j

and depends upon n only through a weight factor. The weight factor is given, for wave-functions written in spherical coordinates, as

(7)

I

/max

E= Individual resonances, d, have a shape described by the Lorentzian function, f(d, k~), of eq. (5). It is assumed that the resonances are entirely nonoverlapping. Consequently, upon squaring the amplitude (eq. (2)) the incoherent sum of eq. (3) appears. Experience suggests that this assumption is almost always correct, especially for multiply charged ions. For some time now it has been widely accepted that D W theory should give a good description of collisions involving multiply charged ionic targets. It has been suspected, however, that the quality of this description might decline when singly, doubly, or few-times charged ions are involved. With the advent of experiments geared to measurements of o DR for just such ions, we may be able to test these notions.

(S)

E

d

Ima×

E

/=0

m-

= Y~ ( 2 / + 1 ) = ( l l

.... + 1 ) 2,

where lm~× = 4 (actually, lm~x depends weakly upon n). In the presence of an external electric field (static and homogeneous) states of different l will " m i x " . To a lesser extent, states of different n will mix too. If the nl states are purely Coulombic, then full mixing of states of different l, for fixed n, is achieved for fields of arbitrarily small, but nonvanishing, field strength, F. The resulting mixed states are the so-called Stark states, which are Coulombic wave-functions written in parabolic coordinates [25]. For such wave-functions, the weight function becomes / ......

g = d

n - 1 -

Im I

E

E

m = -/max

n 2 -- 0

= n (2/ma x + 1) --/ma× (/rnax+ 1), 3. Results

A calculation of o Iz'R for Mg II by the University of Connecticut group [18] was c o m p a r e d with the recent measurement at J I L A [7]. The calculated value of o DR was found to lie - 5.5 × below the experiment, for the

(9)

/=0

(10)

where again, l .... -- 4. Contrasting eqs. (9) and (10), we see that the Stark states, yielding larger weights, should lead to larger cross sections. Allowing for the fact that the HRS's of Mg I are not precisely Coulombic, we need to answer the following questions: What is the actual dependence of o DR upon F for Mg 11? What is

K.J. LaGattuta / Dielectronic recombination cross sections

the magnitude of F in the J I L A experiment? To answer the second question first, the JILA a p p a r a t u s included a field of F = 24 V / c m in the interaction region, in the ion b e a m rest frame. A n initial estimate [26] of o DR for Mg 1I, when F = 24 V / c m , showed the peak value of the cross section to be - 1.6 × 1 0 - 1 7 c m 2. This should be c o m p a r e d with the experimental value, which was - 30% lower. A recent more sophisticated calculation by the University of C o n n e c t i c u t group bears out this estimate. In this newest calculation, the Mg I Hamiltonian, including external electric field, is diagonalized in a basis of functions of fixed n, m, m', m s, and m~. Here, m a n d m~ are the orbital a n d spin azimuthal q u a n t u m n u m b e r s of the H R S electron, respectively, while m ' a n d m ; are the c o r r e s p o n d i n g quantities for the excited 3p electron. Exchange is included explicitly. The H R S electron is coupled to a polarization potential, which incorporates an a p p r o x i m a t e value for the dipole polarizability. The fwhm of d is increased by the fieldionization width, A v, which depends solely, but explicitly, upon all of the spatial q u a n t u m n u m b e r s of the H R S electron. In evaluating o DR, we used the no-coupling approximation. In this approach, the Auger probability, A,, which includes exchange, has two forms for each n, m, m', c o m b i n a t i o n , d e p e n d i n g upon whether the ( 3 p ) ( n l ) electron spins are parallel or antiparallel. The radiative probability, A r, for 3p ~ 3s + y, is assumed constant, i n d e p e n d e n t of n , m , etc. Similar calculations are now being performed for Li sequence ions. Naturally, for doubly or few-times charged targets, larger electric field strengths will be required to produce a given increase in o DR. Finally, we m e n t i o n that the newest JILA results [27] for Mg II have included m e a s u r e m e n t s of the distrib u t i o n o ~ R vs n'. Here, n ' =- (3.2 × I O S / F ' ( V / c m ) ) , where F ' is the strength of an analyzing electric field, used to strip recombined H R S electrons from Mg I. We have simulated this experiment, using our knowledge of the d e p e n d e n c e of o DR upon all of the q u a n t u m numbers of the intermediate state, and of the d e p e n d e n c e of A v upon all of the q u a n t u m n u m b e r s of the H R S electron. The agreement between calculated and measured o,~ R values is not impressive. This leads us to speculate that transitions between states of the recomb i n e d ion may have been induced, during its passage through the field-filled post-interaction region. Simulation shows that if all H R S electrons suffer displacements of the form, H 2 ---->0, for fixed n a n d m, then the calculated o ~ R resembles the measured value very closely.

4. Conclusion The long awaited c o n j u n c t i o n of theory with experim e n t for D R has presented us with a very interesting set

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of discrepancies a n d puzzlements. The questions stirred by this controversy are: C a n D W theory be applied successfully to singly charged ions? Is the IRA reliable for singly or few-times charged ions? Is the electric field effect real? T o d a y we are aware of almost one dozen different groups making c o n t r i b u t i o n s in this area of physics. We thank the Center for Energy and Mineral Resources, of Texas A & M University, for its generous support. This work is funded by the US D e p a r t m e n t of Energy.

References [1] A. Burgess, Astrophys. J. 139 (1964) 776. {2] M. Seaton and P. Storey, Atomic Processes and Applications, eds., P. Burke and B. Moiseiwitsch, (North-Holland, 1976) p. 133. [3] U. Fano, Phys. Rev. 142 (1961) 1866. [4] J. Gau and Y. Hahn, J. Spectrosc. Radiat. Transfer 23 (1980) 121. [5] A. Burgess, Astrophys. J. 141 (1965) 1588. [6] F. Bely-Dubau, A. Gabriel and S. Volonte, Mon. Not. Roy. Astron. Soc. 189 (1979) 801. [7] D. BeliC G. Dunn, T. Morgan, D. Mueller and C. Timmer, Phys. Rev. Lett. 50 (1983) 339. [8] J. Williams, Phys. Rev. A29 (1984) 2936. [9] J. Mitchell et al., Phys. Rev. Left. 50 (1983) 335. [10] P. Dittner et aL Phys. Rev. Lett. 51 (1983) 31. [11] I. Nasser and Y. Hahn, J. Quant. Spectrosc. Radiat. Transfer 29 (1983) 1. [12] D. McLaughlin and Y. Hahn, Phys. Rev. A29 (1984) 712. [13] Y. Hahn, J. Gau, R. Luddy and J. Retter, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 65. [14] K. LaGattuta and Y. Hahn, Phys. Rev. A30 (1984) 316. [15] M. Dube and Y. Hahn, unpublished. [16] K. LaGattuta and Y. Hahn, Phys. Rev. A26 (1982) 1125. [17] K. LaGattuta and Y. Hahn, Phys. Rev. Lett. 50 (1983) 668. [18] K. LaGattuta and Y. Hahn, J. Phys. B 15 (1982) 2101. [19] D. McLaughlin and Y. Hahn, J. Phys. B 16 (1983) L739. [20] K. LaGattuta, Phys. Rev. A30 (1984) 3072. [21] Y. Hahn, K. LaGattuta and D. McLaughlin, Phys. Rev. A26 (1982) 1385. [22] A. Burgess and H. Summers, Astrophys. J. 157 (1969) 1007. [23] V. Jacobs, J. Davis and P. Kepple, Phys. Rev. Lett. 37 (1976) 1390. [24] W. Huber and C. Bottcher, J. Phys. B 13 (1980) L399. [25] H. Bethe and E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer, Berlin, 1957) p. 29. [26] K. LaGattuta and Y. Hanh, Phys. Rev. Lett. 51 (1983) 558. [27] G. Dunn, D. Belid, N. Djurid and D. Mueller, JILA preprint (1984).