Tài liệu Plane wave born collision strengths for electron ion excitation comparison with other theoretical methods

. .. . —. 7-. - .A — .-A.. : .., . . ., .,,—. AISAffiitfve Action/Equal @pOCtUSdty EIlS@3yCS Thisworkwassupported bytheUS Department ofEnergy, OffIce ofMagnetic Fusion Energy. DISCLAIMER TM report waspepared as an account of work sponsoredby an agencyof the UnitedStztti Covcrnment. Neitherthe UrdtedStatcaGovernmentnor any agencythermf, nor any of their employea, maka any warranty,expre= or implied,or assumesany legalliabifityor responsibilityfor the accuracy,completeness, or uscfu!oesaof any information,apparatus, product, or procmadisclosed,or reprcaentsthat its use would not infringeprivatelyowned rights. Reference hereinto any speciIiccomrciaf product, process,or serviceby trade mme, trademark, manufacturer,or otherwise,docs not newsarily constitute or imply ita endorscmeru,recommendation,or favoringby the United States Governmentor any agencythereof. The viewsand ophdonaof authors expressedherefndo not necessarilystate or reflect those of the United Statca Governmentor any agencythweof. LA-10192-MS UC-34a Issued:September1984 Plane-Wave BornCollisionStrengths for Electron-Ion Excitation: Comparisonwith Other Theoretical Methods R. E. H.Clark L. A. Collins , —. .- . . . . ,. . -. . ., . . . . . . . . . -. A m— ..- —. ..- — —. . . . -. . ,m - — ~~~~k)~~~ LosAlamos National Laboratory LosAlamos,NewMexico875.5 . . +-—- - PLANE-WAVE BORN COLLISION STRENGTHS FOR ELECTRON-IONEXCITATION:COMPARISON WITH OTHER THEORETICALMETHODS by R. E. H. Clark and L. A. Collins ABSTRACT Collision rates and strengths for electron-impact excitation of atomic ions are calculated in the approximation using Born plane-wave (PWB) Cowan and the programs of Robb. Two modifications of the PWB, which correct for the ionic investigated. distorted threshold Comparison is and wave behavior, made with are the Coulcanb-Born-exchange techniques. I. INTRODUCTION The plane-wave Born (PWB) approximation~ provides a simple, economical means of generating collision strengths for electronic excitationsin atoms and ions due to electron impact. Since plane waves are employed and exchange effects are neglected, the method is strictly applicable only at high energies for spin-allowedtransitions. However, in practice, the PWB collision strengths are in reasonable agreement (<30%) with methods such as the distortedwave those of other, more sophisticated (DW)2 down to energies of a few times threshold. The prohibitionagainst spin-forbiddentransitionsis valid only for pure-LS coupling. In general, the states consist of a mixture of LS terms, and thus the collision strength will not be zero due to the presence of a spin-allowedcomponent. Since no explicit account is taken of the long-range Coulcmb field, the threshold behavior for ionic targets is incorrect--goingto 1 zero rather than a finite value. near-threshold behavior of the PWB Through simple modifications of collision strength, this defect can be rectified. In many cases, these modified PWB strengths agree to collision better than a factor of 2 with the DW results even in the region near In this report, we the threshold. calculate PWB and modified PWB excitation collision strengths and rates for a variety of ionic targets using a program developed by Cowan and Robb. This program calculates the atomic wave generalized oscillator strengths using the Hartree-Fock and functions and configuration interaction codes of Cowan and determines the PWB collision strength from a subroutineof Robb. These results are compared with the distorted wave and the hydrogenic Coulomb-Born exchange (CBX). 3 In Section II, we give a brief review of the calculationsof Sampson —. et al. PWB method while in Section 111 we describe the various calculations,listing calculations of Mann2 the species, the transitions,and, where appropriate,the mixing coefficientsof the CI calculation, the scaling parameters of spin-orbitcomponent,and notes on any the Coulomb integralsand special features of the calculation. This section is followed by a brief discussionof the results (IV) and a series of the graphs, which gives the collision strengthsfor the various species and transitions under considerationas a function of the ratio (X) of the incident electron energy (kz) to the threshold energy (AE). In transitions, we present the ratio of the PWB addition, for selected and CBX to the DW collision strengths as well as the rates as a function of temperature. 11. METHODS The plane-wave Born (PWB) collision strength S+=rt(x) for a 1 between initial and final levels J7and r’ respectivelyis given by PWB Qrr,(x) = 8 ‘max gfrr~(K) d(ln K) ~ i min , transition (1) where k2 is the energy of the incident electron in Rydbergs, (k’)* is the energy of the outgoing electron [(k’)* = k2 - AE], AE is the thresholdenergy, gfrr’ is the generalized oscillator strength (see Ref. 1, Sees. 18-12), and X is the ratio of the incident and thresholdenergies (X = kz/AE). The momentum transfer K over which the integrationis performed is defined by 2 ++ K= k’ -: , with the limits given by Kmin =kTk’ max The . label of the level consists of the total angular momentum quantum number J plus a designationa describingall other quantum numbers and the configuration that specify the level. For ions, the PWB form has been modified in order to give a more realistic thresholdbehavior. The two modified forms employed in this study are given by~ S-P(x) = Q‘B(3) F(X) , and PWB SF(x) =$2 (x + 3/(1 + x)) (2) , (3) where F(X) s 1. - 0.2 exp (0.07702 (1 - X)) Most of . the collision strengths considered in Section III involve transitions between single values of J in the initial and final levels. For cases involving several J values, we present the summed collision strengthQaa~ given by S-2 aadx) = ~ %J,a0J8 ● (4) JJ’ The PWB collision strengthswere obtained from the structure programs of Cowan (RCN31, RCN2, RCG8), which have been modified byRobb to produce the scattering information. For some of the transitionsunder consideration,we also calculate the rate coefficientsas a function of electron temperatureT. The rates are determinedby integrating the cross section over a Boltzmann distribution. 3 III. DESCRIPTION—OF CALCULATIONS In this section, we give a calculationswere performed. For description of all cases, we the systems for which give the transitions and configurationsemployed. For certain transitions,we also include the CI mixing coefficientsgenerated by RCG8 as well as the scaling parameters of the Coulunb integrals. (Fk(li,2i), Fk(.tijlj),Gk(Ai,lj), and spin-orbit term (~i).l Where appropriate,additional comments are supplied to clarify the precise nature of the calculation. The figure numbers associated with each transition are also given. A. Lithium-like CIV Transitions: 2s + 2p Fig. 1 Configurations: [He] 2s ; 2p Note: An atomic symbol in brackets is used to denote the closed-shellcore of the ion under consideration.For example [He] 2s implies a full configuration of 1s22s. Si XII Transitions: 2s + 2p Fig. 2 Configurations: [He] 2s ; 2p Ar XVI Transitions: 2s + 2p Fig. 3 Configurations:[He] 2s ; 2p Fe XXIV Transitions: 4 a) 2s+ 2p Fig. 4a,b b) 2s+3s c) 2s+ 3p Fig. 5a,b,c d) Fig. 7a,b,c 2s+3d Fig. 6a,b,c Configurations: a) [He] 2s ; 2p b) [He] 2s ; 3s c) [He] 2s ; 3p d) [He] 2s ; 3d Notes: For purposes of comparison,the collision strengths for the 2s + 2p transitionare summed at the same value of X even though the Pi/2 and P3/2 levels have different thresholds, AE(2sl/2 - 2P1/2) = 3.574 Ry; AE(2S1/2 - 3P3/2) = 4.752 Ry. Mo XL Transitions:2s + 2p Fig. 8 Configurations:[He] 2s , 2p Notes: For purposes of comparison,the collision strengths for t~e 2Pi/2 and 2 ‘3/2 transitionsare summed at the same value of X [AE(2SIJ2 - 2P1/2) = 6.184 Ry; AE(2S1/2 - 2P3/2) = 15.605 Ry]. B. Beryllium-like C III Transitions: a) 2s2 + 2s2p 1P Fig. 9 b) 2s2 + 2s2p 3P Fig. 10 Configurations: a-b) [He] 2s2 ; 2s2p ; 2p2 Mixing Coefficients: a) initial state 2s2 1s 2.P23p b) final state J=O 1s -0.96819 -0.00014 J=l 2s2p 1P 0.99999 2s2p 3P 0.00075 lp Scaling Coefficients:0.85, 0.85, 0.85, 0.85, 1.00 5 Fe XXIII Transitions: a) 2s2 + 2s2p 1P Fig. lla,b b) 2s2 + 2s2p 3P Fig. 12a,b 2s2 + 2s3s 1S Fig. 13a,b,c d) 2s2 + 2s3p 1P Fig. 14a,b,c e) 2s2 + 2s3p 3P Fig. 15a,b f) 2s2 + 2s3d ID Fig. 16a,b,c C) Configurations: a-b) [He] 2s2 ; 2s2p ; 2p2 c) [He] 2s2 ; 2s3s ; 2p3p ; 2p2 d-e) [He] 2s2 ; 2s3p ; 2p3s ; 2p3d ; 2p2 f) [He] 2s2 ; 2s3d ; 2p3p ; 2p2 Mixing Coefficients: a) initial state b) J=O 2s2 IS 0.97940 2p2 3P 0.02390 2p2 1s 0.20052 fiml state 1s J=l 2s2p 3P 0.98691 2s2p 1P 0.16126 3p Scaling Coefficients: a-b) 0.95, 0.95, 0.95, 0.95, 1.00 c-d) 0.87, 0.87, 0.87, 0.87, 1.00 Notes: Since the PWB formulation does not contain exchange effects, the collision strength for spin-forbiddentransitionsbetween unmixed states is zero. This is not the case for the IS+ 3P transition inFe XXIII due to the mixing of the 3P and 1P levels. c. Neon-like Al IV Transitions: 6 a) 2p6 + 2p53s 1P Fig. 17 b) 2p6 + 2P53S 3P Fig. 18 Configurations: a-b) [He] 2s22p6 ; 2s22p53s Mixing Coefficients: J=l final state 2p53s 3P 0.31083 2p53s 1P -0.95047 1P Scaling Coefficients:0.80, 0.80, 0.80, 0.80, 1.00 Fe XVII Transitions: a) 2p6 + 2P53s 1P Fig. 19 a,b b) 2p6 + 2P53S 3P Fig. 20 a,b Configurations: a-b) [He] 2s22p6 ; 2s22p53s ; 2s22p53d ; 2s12p63p Mixing Coefficients: J=l final state 2p53s 3P 0.66556 2p53s 1P 0.74545 2p53d 3P -0.00745 2p53d 1P -0.02442 2p53d 3D 0.00174 2s2p63p 3P -0.02125 2s2p63p 1P -0.01496 1P Scaling Coefficients:a) 0.90, 0.90, 0.90, 0.80, 1.00 Notes: The spin-forbidden PWB collision strength is nonzero due to triplet-singletmixing in the final state wave function. D. Sodium-like Fe XVI Transitions: a) 3s + 3p Fig. 21 b) 3s + 4s Fig. 22 Configurations: a) [Ne] 3s ; 3p b) [Ne] 3s ; 4s the E. Aluminum-like Ti X Transitions: a) 3S23P + 3S3P2 2D Fig. 23 b) 3s23p + 3s23d1 2D Fig. 24 Configurations: a-b) [Ne] 3s23p ; 3S3P2, 3s23d Mixing Coefficients: final state J=-3/2 3s23d1 2D 0.95040 3s13p2 2D 0.30797 3s13p2 2P 3s13p2 4P 0.03773 0.02181 Scaling Coefficients: 0.90, 0.90, 0.90, 0.90, 1.00 Iv. DISCUSSION The figures, which compare the various calculational methods, are reasonably self-explanatory and present a broad comparison for different types of systems and transitions. Therefore, we are relieved of presenting a detailed discussion of the results and instead concentrateon the mjor conclusionsthat can be drawn from this comparison. These conclusionsare as follows: (1) The M2 modified PWB prescription generally gives the best agreement with the DW results. The exceptionsto this rule involve s- to d-type transitions [Fe XXIV 2s + 3d and Fe XXIII 2s2 + 2s3d ID] although these differencesdisappear by energies of 10 times threshold (X = 10). (2) The M2-PWB agrees with the DW to within better than thirty percent (30%) for energies above a few times threshold (x = 2- 3). The exceptions to this condition are the spin-forbidden transitions for weakly coupled systems [C 111 2s2 - 2s2p 3P and Fe XXIII 2s2 - 2s2p 3P]. (3) For energies near threshold (X< 1 - 2), the M2-PWB results are usually within better than a factor of 2 of the DW. exception to The most pronounced this rule comes from the spin-forbidden transition in A IV (2p6 + 2p53s 3P) in which the exchange contributiondominates that of the mixing at low energies. (4) Spin-forbiddentransitionsare given reasonably well by the M2-PWB provided that the states are sufficientlymixed. We can see this by viewing the progressionfrom C 111 to Fe XXIII to Fe XVII. triplet-singlet mixing is negligible, and 8 For C 111 the the spin-forbiddentransition is ~ practically zero. The amount of singlet component in function is the Fe XXIII 3P wave about 3% while for Fe XVII this value has risen to nearly 45%. As expected, the agreement improves as the mixing becomes stronger. (5) The M2-PWB rates in general agree with those of the DW to better than 30% over a range of temperaturesfrom O tO 104 K. The exception again occurs for transitions involving a d-type configurationwith the largest difference being on the order of 50%. 9 1 ~0 DO Cowan, ~ Theory Qf_ Ato~c ‘tructure —and Spectra (University of California Press, Berkeley, 1981). 2 J. B. Mann, At. Data Nucl. Data Tables 29, 409 (1983); A. L. Merts, J. B. Mann, W. D. Robb, and N. H. Magee, “Electron Excitation Collision Strengths for Positive Atomic Ions: A Collectionof TheoreticalData,” Los Alamos ScientificLaboratory report LA-8267-Ms (1980). 3 D. H. Sampson, S. J. Goett, R. E. H. Clark, At. Data Nucl. Data Tables 30, 125 (1984); L. B. Golden, R. E. H. Clark, S. J. Goett, and D. H. Sampson, Astrophys. J. Suppl. 45, 603 (1981). 10 FIGURES The basic figure is a plot of the collision strength!2rr’as a function of x, the ratio of the incident energy of the electron to the thresholdenergy. The species and transitionare given at the top of the graph. The PWB, modified PWB (Ml, M2), DW, and CBX collision strengths are plotted ina consistent notation throughoutthe set. For some transitions,we present a second graph which gives the ratio of the PWB and CBX collision strengths to that of the DW as a function of X. In addition, the rate coefficients are present for a selected set of transitions. The notation is as follows: --— Plane– Wave Born 00 PWB (Ml) ++ PWB (M2) AA Coulomb–Born Distorted (PWB) Exchange (CBX) Wave (DW) 11 CIV 2s– 2p 1 I I t # 1 1I 1 30.0 I 1 1 1 1 1 11 1 I I I t 1 I I 1 1 11 1 1 I 1 1 1 1b lcf 25.0 20.0 c 15.0 / 1 ,1’ !5.0 I If 0.0 ~ Id’ 1 I I 1 I I Id x I 1 I I 1 I l(f I 1 1 Fig. 1. SixIr29– 2p I 1 1 1 1 I I1 1 4.0! c 1.o- / // / If o.5- / I I 0.0 Id I ! 1 1 1 I I II lxd Fig. 2. 12 ArXVI t 2.25 2s – 2p 1 I I 1 1 1 I 1 t I I I I I 1 L 1 1 I 1 1 2.00 1.75 1.50 c / 1 0.50 o “ It I 0.25 ; I 0.00 / I Id’ 1 1 i I I 1 1 I Id 1 I I 1 x Fig. 3. FeXXIV 2s 1 1.0 1 2p – 1 t 1 I I 1 I 1 1 I 1 1 1 t , .4) .s)O o.9- .4!” . “8”4 0.8+“ 0.7- c N“ ,9”” +,0’ 0.8+ o.5- + /’ + /’ ,’ +,’ y’: ,+ “ ~ o >0 0 o T 0.4o.30.2 Id’ / / ~ o ~’ o 0 / / / / / // / / I I 1 1 1 I 1I I Id x 1 1 1 I I 1I 1 Id Fig. 4a. 13 FeXXIV + 1 1.1- in ..” 0.9 1 1 1.2 1 + 9’ I 00 0.8 I t 1 1 1 I I 1 x1 = + +. --‘- +- * +/ ‘+. + ,d’ 00 -a’w. e ,~ 0° / o 1 + + + 2s – 2p /’ / I 1 1 11 ‘-e-. L o / o 0 o I 1’ { 1’ 1’ I I 0.7 i o.6- 1 I I t o.5- k /’ 0.4 r I I 1 1 I I 1I I 1 1 1 I I 1I l(f 1. ld’ Fig. 4b. FeXXIV 2s – 3s I 1 0.018 I AA AAA 0.o17- A ++. A + O 0 +’ A o + o b + 0.014- 0 + + t d, m 0 A 0.013- AA a 0 A c 1 1 11I -d + + A 0.015- t AAA AA A 0.016- I t 1 I ! 1 1I + 0 + 0 0 0.0120 0.011{) 0.010 ld 00 00 1 1 I i I r 1 1I ld x Fig. 5a. 14 1 I I I I I I I Id FeXXIV I 1.05; # 1 1 1 1 1 1I A AA AA 1 AA AA AA 2s – 3s A 1.00 + + o + + o + 0.90 o + g 4 + + 0.95 E- + 1 # 1 I 1 11 A + +++ 0.85 0 o 0 0.80 o 0.75 0 o 0 0.70 0 0.65 ~ 1 0 0 1 1 I ~T 1 ! 1 1I Id Id x l(f Fig. 5b. FeXXIV 2s – 3s RATE COEFFICIENT N ‘o -( ● 9.0 8.0 A 7.0 A + + + 6.0 El 5.0 2 G 4.0 3.0 2.0 1.0 0.0 I .0 I 1.0 I 2.0 I 3.0 I 4.0 I 5.0 T(eV) I 6.0 I 7.0 1 8.0 1 9.0 10.0 “Id Fig. 5c. 15 I 0.16 I FeXXIV 2s – 3p 1 1 1 I 1 11 1 1 1 1 I 1 11 0.14- h e @ o.12- al o.1oc 0.060.060.040.02-~. 0.00 Id 1 1 1 1 1 1 I II 1(Y x 1 I I I 1 I 11 Id 1 1 1 1 1 I 11 I I 1 Fig. 6a. FeXXIV 1 1.8 1 2s – 3p 1 1 1 1 1 11 1.6 1.4 6 0 i? 4 u 1.2 + A 1.0- o+ A + A A o 0 0.8- 0 0 0 0 0.6 1 Id’ 1 I 1 1 1 1 1I 10’ x Fig. 6b. I 1 1 1I 1 + + (y N . ‘o + ● 1 20.a 17.5 FeXXIV 2s – 3p RATE COEFFICIENT I 1 1 1 1 1 1 1 + 15.a + . 4. 12.5 la 10.0 ‘a a 7.5 5.0 +/ L- 2.5 0.0 .0 1.0 2.0 I 3.0 , 4.0 , 5.0 6.0 , , 7.0 1- 8.0 9.0 1 .*12 T (eV) Fig. 6c. FeXXIV 2s – 3d I 0.055 I 1 1 1 1 1 11 1 t I t AAA AA $++ 0.050+ +A + A 0 1 1 1 1 1 AA 1 ,&~ee o + + 0 A + 0.045 + + A + + c 0.040 0.035 000 A A 0.030 A 0.025 I Id 1 1 [ 1 1 I 1 1I 1(Y x Fig. 7a. I 1 I 1 I I v 1 I Id