ABSTRACT Electron collision excitation strengths of inelastic transitions among the 52 Ðne-structure levels of the

THE ASTROPHYSICAL JOURNAL, 530:1091È1104, 2000 February 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. ELECTRON COLLISION EXCITATION OF FINE-STRUCTURE LEVELS IN S IV S. S. TAYAL Department of Physics, Clark Atlanta University, Atlanta, GA 30314 Received 1999 June 28; accepted 1999 September 30 ABSTRACT Electron collision excitation strengths of inelastic transitions among the 52 Ðne-structure levels of the ground 3s23p and excited 3s3p2, 3s23d, 3s24s, 3p3, 3s24p, 3s3p3d, 3s24d, 3s24f, and 3s3p4s conðgurations in S IV are calculated using full Breit-Pauli R-matrix method. The target levels are represented by conðguration-interaction wave functions, and the mass, Darwin, and spin-orbit relativistic terms are included in the scattering equations. The collision strengths in the threshold energy regions are dominated by the complicated resonance structures. Our collision strengths are compared with earlier available calculations and signiðcant di erences are noted. The collision strengths are integrated over a Maxwellian distribution of electron energies to obtain e ective collision strengths over a wide temperature range. Subject headings: atomic data È atomic processes 1. INTRODUCTION The International Ultraviolet Explorer and the Goddard High-Resolution Spectrograph (GHRS) on board Hubble Space T elescope have observed strong S IV emission features at 657, 745, 816, 1070, and 1406 A in the spectra of the Io plasma torus (Durrence, Feldman, & Weaver 1983; Moos et al. 1991). The spin-forbidden emission lines of S IV from the multiplet 3s23p 2PoÈ3s3p24P are observed in the spectra of the Sun by the Skylab experiments (Feldman 1981; Dere 1982; Bhatia, Doschek, & Feldman 1980) and in the spectra of the transition region of Capella observed by the GHRS (Linsky et al. 1995). The line intensity ratios within the multiplet 3s23p 2PoÈ3s3p2 4P can provide excellent density diagnostics in the electron density range 109È1012 cm~3. The ratio of emission lines at 1062.7 and 1073.0 A from the multiplet 3s23p 2PoÈ3s3p2 2D are also density sensitive. These lines are observed in the far ultraviolet spectra of the Sun by the Naval Research Laboratory SO82-B slit spectrograph on Skylab (Feldman & Doschek 1991). The relative intensities of the S IV lines together with the strong emission lines of the abundant ions of carbon, nitrogen, and silicon can be used to determine relative element abundances. The transition probabilities and electron collision strengths for Ðne-structure levels are needed to determine theoretical line-intensity ratios that eventually may be used to infer electron temperature, density, and elemental abundances in astrophysical plasmas. Electron impact excitation collision strengths of electricdipole, intercombination, and forbidden transitions among the Ðne-structure levels of the 3s23p, 3s23d, 3s24s, 3p3, 3s3p3d, 3s24p, 3s24d, 3s24f, and 3s3p4s conðgurations of S IV are calculated using a full Breit-Pauli R-matrix approach. We considered 52 Ðne-structure levels in our calculation. These levels are represented by conðguration-interaction (CI) wave functions. The relativistic e ects are included in the Breit-Pauli approximation via one-body mass correction, Darwin, and spin-orbit interaction terms in the scattering equations (Scott & Burke 1980). Rydberg series of resonances converging to excited Ðne-structure levels are explicitly included in the scattering calculations. These resonance structures make substantial contributions to collision strengths for many transitions. 1091 Recently Saraph & Storey (1999) reported collision data for the 2Po È2Po Ðne-structure transition in P III, SIV, 1@2 3@2 and Cl V. They included 21 LS terms of S IV in the closecoupling (CC) expansion in their elaborate R-matrix calculations. They included mass and Darwin terms of the Breit-Pauli operator explicitly in their LS R-matrix calculation and considered the spin-orbit interaction as a perturbation by the use of term-coupling coefficients (Saraph 1978). Johnson, Kingston, & Dufton (1986) reported e ective collision strengths for the ground state Ðne-structure transition 2Po È2Po in S IV and noted that the reso- 1@2 3@2 nance e ects enhance the e ective collision strengths by a factor of 4 compared to distorted wave calculation (Brocklehurst 1972). Saraph & Storey (1999) demonstrated that the positions of the resonances depend on the coupling to higher excited states. They found that the positions of resonances shift to lower energy side when channels coupled to higher excited states are included in the CC expansion. This translates to increase in e ective collision strengths at lower temperatures. The e ective collision strengths reported by Saraph & Storey (1999) are 30% higher than those of Johnson et al. (1986) at 10,000 K. Bhatia et al. (1980) reported collision strengths at 2.0, 4.0, and 6.0 ryd for Ðne-structure transitions among the levels of the 3s23p, 3s3p2, and 3s23d conðgurations that were calculated in a distorted wave approximation (Eissner & Seaton 1972). Bhadra & Henry (1980) used two-state and Ðve-state CC approximations to calculate collision strengths among the Ðne-structure levels of the 3s23p and 3s3p2 conðgurations. Bhatia et al. (1980) and Bhadra & Henry (1980) carried out their calculations in LS coupling and obtained collision strengths for Ðne-structure levels in intermediate coupling by the use of transformation code of Saraph (1978) and did not include important resonance e ects in their calculations. Electron impact excitation collision strengths of the spin-forbidden transition 3s23p 2PoÈ 3s3p2 4P were calculated by Dufton & Kingston (1980) using R-matrix method in LScoupling, while Dufton et al. (1982) reported transition probabilities and e ective collision strengths for Ðne-structure levels of the 3s23p 2Po and 3s3p2 4P terms. We recently reported oscillator strengths of electric dipole transitions in S IV (Tayal 1999).

1092 TAYAL TABLE 1 CALCULATED AND EXPERIMENTAL ENERGY LEVELS (IN ryd) OF S IV PRESENT CALCULATION INDEX LEVEL J Ab Initio Adjusted EXPERIMENT 1... 3s23p 2Po 1/2 0.0000 0.0000 0.0000 2... 3/2 0.0078 0.0088 0.0087 3... 3s3p2 4P 1/2 0.6282 0.6487 0.6487 4... 3/2 0.6315 0.6519 0.6578 5... 5/2 0.6370 0.6575 0.6568 6... 3s3p2 2D 3/2 0.8662 0.8575 0.8575 7... 5/2 0.8668 0.8579 0.8580 8... 3s3p2 2S 1/2 1.1616 1.1255 1.1255 9... 3s3p2 2P 1/2 1.2589 1.2178 1.2176 10... 3/2 1.2651 1.2234 1.2233 11... 3s23d 2D 3/2 1.4440 1.3870 1.3863 12... 5/2 1.4447 1.3878 1.3865 13... 3s24s 2S 1/2 1.6643 1.6537 1.6535 14... 3p3 2Do 3/2 1.6814 1.6840 1.6863 15... 5/2 1.6824 1.6850 1.6872 16... 3p3 4So 3/2 1.7901 1.7901 1.7902 17... 3s3p(3Po)3d 4Fo 3/2 1.8649 1.8539 1.8539 18... 5/2 1.8670 1.8557 1.8581 19... 7/2 1.8699 1.8582 1.8581 20... 9/2 1.8737 1.8614 1.8614 21... 3p3 2Po 1/2 1.9468 1.9217 1.9261 22... 3/2 1.9703 1.9392 1.9262 23... 3s24p 2Po 1/2 1.9434 1.9489 1.9457 24... 3/2 1.9707 1.9494 1.9476 25... 3s3p(3Po)3d 4Po 5/2 2.0343 2.0247 2.0248 26... 3/2 2.0366 2.0274 2.0275 27... 1/2 2.0356 2.0293 2.0293 28... 3s3p(3Po)3d 4Do 1/2 2.0563 2.0443 2.0444 29... 3/2 2.0598 2.0451 2.0452 30... 5/2 2.0601 2.0461 2.0462 31... 7/2 2.0624 2.0469 2.0469 32... 3s3p(3Po)3d 2Do 5/2 2.1635 2.1179 2.1288 33... 3/2 2.1634 2.1123 2.1291 34... 3s3p(3Po)3d 2Fo 5/2 2.2654 2.2025 2.2020 35... 7/2 2.2730 2.2120 2.2091 36... 3s24d 2D 3/2 2.3423 2.3279 2.3273 37... 5/2 2.3426 2.3281 2.3274 38... 3s24f 2Fo 7/2 2.3989 2.3481 2.3475 39... 5/2 2.4057 2.3494 2.3487 40... 3s3p(3Po)4s 4Po 1/2 2.3886 2.3973 2.3975 41... 3/2 2.3944 2.4006 2.4007 42... 5/2 2.4001 2.4061 2.4064 43... 3s3p(3Po)3d 2Po 3/2 2.4712 2.4168 2.4134 44... 1/2 2.4656 2.4181 2.4154 45... 3s3p(3Po)4s 2Po 1/2 2.4994 2.4668 2.4679 46... 3/2 2.5014 2.4727 2.4735 47... 3s3p(1Po)3d 2Po 3/2 2.6444 2.5407 2.5392 48... 1/2 2.6471 2.5418 2.5395 49... 3s3p(1Po)3d 2Do 3/2 2.6510 2.5725 2.5615 50... 5/2 2.6472 2.5739 2.5628 51... 3s3p(1Po)4s 2Po 1/2 2.9366 2.8152 2.8153 52... 3/2 2.9396 2.8159 2.8158 2. COLLISION CALCULATION The one-electron orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, and 4f are used in our scattering calculation. The 1s, 2s, 2p, 3s, and 3p radial functions are chosen as the Hartree-Fock functions (Clementi & Roetti 1974) for the ground state 3s23p 2Po, and the parameters of the 3d, 4s, 4p, 4d, and 4f orbitals are optimized on various states using atomic structure code CIV3 (Hibbert 1975). The details of the wave TABLE 2 COLLISION STRENGTH FOR TRANSITIONS AMONG FINE-STRUCTURE LEVELS IN S IV PRESENT BH (1980) BDF (1980) TRANSITION 4.0 6.0 4.0 6.0 4.0 6.0 1È2... 0.890 0.889 1.184 1.084 0.858 0.804 1È3... 0.049 0.034 0.044 0.029 0.042 0.027 1È4... 0.071 0.049 0.066 0.043 0.059 0.038 1È5... 0.043 0.030 0.041 0.026 0.029 0.019 1È6... 1.335 1.349 1.236 1.677 3.004 3.367 1È7... 0.195 0.151 0.219 0.194 0.129 0.093 1È8... 1.492 1.850 1.587 2.089 1.611 1.851 1È9... 5.055 6.309 6.029 6.928 7.448 8.699 1È10... 2.778 3.449 3.184 3.652 3.767 4.343 2È3... 0.031 0.022 0.031 0.020 0.026 0.017 2È4... 0.088 0.061 0.084 0.054 0.068 0.044 2È5... 0.196 0.136 0.184 0.119 0.129 0.086 2È6... 0.462 0.411 0.542 0.603 0.692 0.713 2È7... 2.433 2.585 2.447 3.225 5.002 5.614 2È8... 2.147 2.690 2.730 3.676 2.673 3.061 2È9... 2.991 3.728 3.305 3.790 3.991 4.634 2È10... 13.270 16.568 15.491 17.791 18.432 21.487 3È4... 0.401 0.281 0.387 0.281 0.395 0.261 3È5... 0.964 0.993 1.124 1.096 0.882 0.831 3È6... 0.072 0.043 0.067 0.042 0.078 0.042 3È7... 0.046 0.029 0.046 0.029 0.046 0.026 3È8... 0.018 0.010 0.011 0.005 0.013 0.006 3È9... 0.011 0.004 0.006 0.003 0.010 0.005 3È10... 0.023 0.011 0.019 0.011 0.015 0.008 4È5... 1.670 1.582 1.856 1.706 1.459 1.272 4È6... 0.115 0.070 0.108 0.067 0.118 0.065 4È7... 0.125 0.077 0.119 0.074 0.108 0.061 4È8... 0.029 0.012 0.021 0.009 0.024 0.011 4È9... 0.023 0.009 0.015 0.008 0.018 0.010 4È10... 0.047 0.022 0.034 0.019 0.032 0.017 5È6... 0.102 0.063 0.098 0.062 0.095 0.053 5È7... 0.306 0.197 0.247 0.154 0.228 0.127 5È8... 0.042 0.017 0.032 0.014 0.032 0.016 5È9... 0.039 0.019 0.029 0.016 0.024 0.013 5È10... 0.069 0.029 0.044 0.024 0.049 0.026 6È7... 0.990 0.910 1.134 0.948 0.885 0.776 6È8... 1.066 1.077 1.096 1.150 1.222 1.212 6È9... 0.128 0.093 0.080 0.065 0.065 0.043 6È10... 0.200 0.123 0.137 0.091 0.125 0.072 7È8... 1.550 1.654 1.661 1.744 1.774 1.763 7È9... 0.130 0.074 0.098 0.062 0.094 0.054 7È10... 0.310 0.204 0.215 0.157 0.183 0.113 8È9... 0.109 0.080 0.043 0.033 0.046 0.032 8È10... 0.189 0.137 0.080 0.062 0.086 0.059 9È10... 0.749 0.761 0.996 0.884 1.076 0.979 functions can be found in Tayal (1999), where orbital parameters, energy levels, transition probabilities and lifetimes are presented. The 24 LSstates are represented by 114 conðgurations and 257 conðgurations are used to represent 52 Ðne-structure levels. The calculated excitation energies of the Ðne-structure levels relative to the ground level 3s23p 2Po are presented in Table 1, where these are compared with 1@2 experimental results (Martin, Zalubas, & Musgrove 1990). The present ab initio results are within 4% of the experimental results except for the 3s23p 2Po level, at which theory and experimental results di er by 3@2 10%. The relativistic e ects in intermediate coupling are incorporated through the Breit-Pauli Hamiltonian. We have made small adjustments to the diagonal elements of the Hamiltonian

TABLE 3 EFFECTIVE COLLISION STRENGTHS FOR TRANSITIONS IN S IV 1È2... 8.536 8.473 8.307 8.082 7.499 6.899 5.949 4.915 3.943 3.042 1È3... 0.596 0.544 0.523 0.502 0.453 0.403 0.332 0.265 0.207 0.156 1È4... 1.051 0.968 0.933 0.897 0.806 0.712 0.579 0.457 0.353 0.261 1È5... 1.168 1.099 1.063 1.020 0.908 0.790 0.629 0.485 0.365 0.262 1È6... 2.055 2.040 2.017 1.988 1.926 1.906 1.919 1.898 1.825 1.710 1È7... 1.777 1.732 1.703 1.666 1.546 1.399 1.171 0.949 0.752 0.574 1È8... 1.297 1.277 1.260 1.241 1.201 1.178 1.173 1.206 1.279 1.395 1È9... 2.856 2.853 2.869 2.896 2.990 3.108 3.320 3.617 4.016 4.523 1È10... 1.934 1.937 1.940 1.945 1.965 1.996 2.061 2.170 2.350 2.578 1È11... 4.164 4.370 4.468 4.564 4.764 4.965 5.341 5.909 6.727 7.728 1È12... 0.822 0.798 0.786 0.774 0.738 0.691 0.614 0.535 0.462 0.391 1È13... 0.458 0.492 0.528 0.577 0.689 0.735 0.696 0.608 0.523 0.462 1È14... 0.357 0.373 0.378 0.382 0.383 0.380 0.379 0.381 0.381 0.380 1È15... 0.350 0.358 0.359 0.357 0.346 0.332 0.313 0.299 0.292 0.288 1È16... 0.069 0.065 0.062 0.058 0.048 0.039 0.028 0.021 0.015 0.010 1È17... 0.230 0.216 0.209 0.202 0.185 0.167 0.144 0.122 0.102 0.082 1È18... 0.299 0.280 0.271 0.261 0.237 0.213 0.180 0.151 0.124 0.099 1È19... 0.305 0.281 0.269 0.256 0.228 0.201 0.166 0.135 0.109 0.085 1È20... 0.192 0.171 0.160 0.149 0.124 0.102 0.075 0.055 0.039 0.026 1È21... 0.553 0.792 0.894 0.966 0.988 0.897 0.725 0.573 0.461 0.379 1È22... 0.392 0.490 0.530 0.555 0.559 0.522 0.456 0.401 0.363 0.338 1È23... 1.918 3.748 4.438 4.874 4.930 4.321 3.263 2.362 1.692 1.180 1È24... 1.385 2.454 2.851 3.096 3.096 2.706 2.043 1.477 1.052 0.725 1È25... 0.177 0.174 0.171 0.168 0.159 0.148 0.131 0.115 0.098 0.081 1È26... 0.101 0.095 0.092 0.089 0.080 0.072 0.061 0.051 0.043 0.034 1È27... 0.040 0.037 0.035 0.033 0.028 0.024 0.019 0.015 0.012 0.009 1È28... 0.047 0.046 0.046 0.045 0.043 0.041 0.037 0.033 0.028 0.023 1È29... 0.081 0.079 0.077 0.076 0.072 0.067 0.060 0.053 0.045 0.036 1È30... 0.102 0.099 0.097 0.095 0.089 0.082 0.072 0.062 0.052 0.042 1È31... 0.122 0.121 0.119 0.116 0.108 0.100 0.087 0.074 0.062 0.050 1È32... 0.228 0.214 0.205 0.195 0.172 0.151 0.128 0.111 0.097 0.085 1È33... 0.150 0.147 0.144 0.140 0.129 0.118 0.104 0.094 0.086 0.080 1È34... 0.881 0.810 0.777 0.746 0.688 0.655 0.642 0.658 0.682 0.676 1È35... 0.336 0.305 0.287 0.268 0.224 0.187 0.143 0.108 0.080 0.057 1È36... 0.260 0.254 0.252 0.249 0.243 0.239 0.235 0.234 0.237 0.239 1È37... 0.204 0.193 0.187 0.180 0.166 0.152 0.133 0.115 0.098 0.081 1È38... 0.174 0.174 0.172 0.169 0.161 0.151 0.137 0.122 0.106 0.090 1È39... 0.266 0.267 0.267 0.266 0.267 0.267 0.275 0.282 0.282 0.267 1È40... 0.042 0.039 0.038 0.036 0.034 0.031 0.027 0.023 0.019 0.015 1È41... 0.063 0.058 0.056 0.053 0.048 0.043 0.037 0.031 0.025 0.020 1È42... 0.044 0.038 0.036 0.033 0.027 0.022 0.016 0.011 0.008 0.005 1È43... 0.200 0.189 0.185 0.182 0.178 0.178 0.182 0.190 0.194 0.188 1È44... 0.149 0.137 0.132 0.128 0.122 0.121 0.121 0.120 0.116 0.109 1È45... 0.500 0.518 0.526 0.536 0.554 0.567 0.585 0.602 0.610 0.584 1È46... 0.098 0.084 0.078 0.073 0.064 0.057 0.049 0.042 0.035 0.029 1È47... 0.093 0.091 0.091 0.092 0.093 0.097 0.106 0.116 0.123 0.122 1È48... 0.105 0.101 0.099 0.097 0.093 0.090 0.088 0.086 0.085 0.081 1È49... 0.257 0.261 0.265 0.268 0.276 0.282 0.289 0.296 0.301 0.299 1È50... 0.233 0.238 0.241 0.245 0.252 0.257 0.263 0.268 0.272 0.268 1È51... 0.113 0.114 0.115 0.116 0.119 0.123 0.130 0.136 0.142 0.144 1È52... 0.047 0.047 0.047 0.046 0.046 0.044 0.042 0.039 0.036 0.032 2È3... 1.031 0.907 0.854 0.803 0.690 0.589 0.462 0.353 0.264 0.189 2È4... 1.908 1.754 1.683 1.608 1.424 1.239 0.990 0.768 0.582 0.423 2È5... 2.733 2.601 2.533 2.453 2.230 1.979 1.616 1.277 0.987 0.731 2È6... 2.532 2.467 2.424 2.369 2.207 2.022 1.744 1.469 1.217 0.981 2È7... 5.020 4.976 4.917 4.833 4.579 4.286 3.867 3.497 3.212 2.946 2È8... 2.304 2.235 2.184 2.128 2.007 1.919 1.849 1.850 1.925 2.059 2È9... 2.118 2.087 2.078 2.076 2.086 2.112 2.180 2.304 2.497 2.742 2È10... 7.718 7.730 7.775 7.849 8.088 8.377 8.898 9.639 10.65 11.96 2È11... 1.884 1.897 1.904 1.908 1.906 1.893 1.885 1.920 2.011 2.155 2È12... 8.251 8.613 8.784 8.950 9.285 9.613 10.25 11.23 12.59 14.51 2È13... 0.912 0.924 0.922 0.915 0.880 0.825 0.738 0.668 0.634 0.641 2È14... 0.479 0.495 0.497 0.495 0.480 0.461 0.436 0.420 0.413 0.411

TABLE 3ÈContinued 2È15... 0.904 0.936 0.945 0.948 0.940 0.924 0.905 0.900 0.898 0.863 2È16... 0.135 0.128 0.121 0.114 0.095 0.077 0.056 0.041 0.029 0.020 2È17... 0.189 0.174 0.166 0.157 0.136 0.116 0.092 0.071 0.055 0.040 2È18... 0.340 0.316 0.303 0.289 0.255 0.222 0.180 0.144 0.114 0.087 2È19... 0.570 0.530 0.509 0.487 0.435 0.384 0.318 0.260 0.210 0.164 2È20... 0.896 0.828 0.796 0.763 0.687 0.615 0.519 0.432 0.355 0.281 2È21... 1.070 1.088 1.096 1.095 1.061 1.002 0.919 0.857 0.818 0.795 2È22... 0.424 0.423 0.419 0.411 0.380 0.345 0.302 0.271 0.253 0.245 2È23... 0.658 0.972 1.085 1.151 1.132 0.997 0.775 0.586 0.442 0.328 2È24... 1.160 1.357 1.419 1.445 1.388 1.260 1.077 0.936 0.837 0.762 2È25... 0.260 0.241 0.231 0.220 0.194 0.168 0.137 0.110 0.088 0.069 2È26... 0.194 0.185 0.180 0.174 0.158 0.142 0.120 0.100 0.082 0.066 2È27... 0.102 0.100 0.098 0.096 0.089 0.082 0.071 0.060 0.051 0.041 2È28... 0.063 0.063 0.063 0.062 0.059 0.055 0.049 0.042 0.036 0.029 2È29... 0.140 0.140 0.139 0.138 0.132 0.124 0.111 0.097 0.083 0.067 2È30... 0.229 0.229 0.228 0.226 0.216 0.204 0.184 0.162 0.139 0.114 2È31... 0.328 0.325 0.322 0.317 0.304 0.286 0.258 0.227 0.195 0.159 2È32... 0.411 0.407 0.400 0.390 0.358 0.327 0.290 0.260 0.237 0.213 2È33... 0.237 0.236 0.233 0.228 0.210 0.190 0.165 0.145 0.129 0.113 2È34... 0.597 0.535 0.506 0.478 0.423 0.382 0.343 0.321 0.312 0.312 2È35... 1.798 1.677 1.612 1.546 1.415 1.331 1.285 1.304 1.348 1.337 2È36... 0.300 0.285 0.277 0.268 0.249 0.231 0.207 0.185 0.165 0.145 2È37... 0.632 0.613 0.604 0.594 0.573 0.554 0.533 0.517 0.507 0.498 2È38... 0.593 0.594 0.592 0.588 0.580 0.574 0.575 0.582 0.584 0.556 2È39... 0.305 0.305 0.302 0.298 0.287 0.275 0.258 0.241 0.226 0.210 2È40... 0.040 0.034 0.032 0.029 0.023 0.019 0.014 0.010 0.008 0.005 2È41... 0.097 0.086 0.081 0.075 0.064 0.055 0.043 0.035 0.027 0.021 2È42... 0.188 0.170 0.162 0.153 0.135 0.118 0.097 0.079 0.064 0.050 2È43... 0.481 0.444 0.428 0.415 0.394 0.385 0.381 0.381 0.382 0.386 2È44... 0.199 0.186 0.181 0.177 0.172 0.171 0.174 0.182 0.191 0.200 2È45... 0.110 0.093 0.086 0.080 0.069 0.061 0.052 0.044 0.037 0.030 2È46... 1.121 1.138 1.147 1.156 1.176 1.195 1.223 1.252 1.265 1.209 2È47... 0.295 0.286 0.282 0.280 0.280 0.283 0.291 0.296 0.293 0.273 2È48... 0.083 0.081 0.080 0.080 0.081 0.083 0.090 0.101 0.110 0.112 2È49... 0.333 0.338 0.341 0.346 0.356 0.365 0.377 0.389 0.393 0.374 2È50... 0.653 0.665 0.674 0.684 0.704 0.721 0.743 0.764 0.770 0.732 2È51... 0.048 0.048 0.048 0.048 0.047 0.045 0.043 0.040 0.037 0.034 2È52... 0.282 0.283 0.284 0.285 0.291 0.298 0.308 0.318 0.322 0.305 3È4... 3.366 3.298 3.238 3.167 3.010 2.857 2.569 2.191 1.784 1.372 3È5... 2.924 2.852 2.813 2.777 2.709 2.635 2.453 2.191 1.907 1.650 3È6... 0.843 0.749 0.710 0.677 0.622 0.578 0.508 0.424 0.340 0.258 3È7... 0.678 0.625 0.601 0.579 0.541 0.507 0.446 0.371 0.294 0.219 3È8... 0.192 0.186 0.187 0.190 0.196 0.193 0.172 0.141 0.111 0.082 3È9... 0.455 0.432 0.421 0.408 0.373 0.329 0.263 0.201 0.148 0.104 3È10... 0.395 0.419 0.426 0.428 0.411 0.374 0.307 0.238 0.179 0.127 3È11... 0.914 0.776 0.719 0.665 0.560 0.470 0.360 0.268 0.196 0.137 3È12... 0.651 0.567 0.534 0.504 0.441 0.380 0.296 0.222 0.163 0.113 3È13... 0.369 0.315 0.286 0.255 0.194 0.147 0.100 0.068 0.046 0.030 3È14... 0.740 0.715 0.694 0.668 0.607 0.550 0.473 0.399 0.330 0.261 3È15... 0.530 0.503 0.480 0.453 0.387 0.328 0.256 0.198 0.151 0.110 3È16... 2.108 2.100 2.100 2.105 2.143 2.214 2.374 2.596 2.857 3.091 3È17... 0.650 0.570 0.528 0.487 0.402 0.333 0.256 0.194 0.146 0.106 3È18... 0.882 0.790 0.740 0.688 0.579 0.489 0.388 0.311 0.253 0.205 3È19... 0.904 0.830 0.785 0.736 0.629 0.539 0.436 0.359 0.304 0.261 3È20... 0.634 0.560 0.518 0.474 0.383 0.310 0.231 0.172 0.127 0.090 3È21... 0.216 0.200 0.190 0.180 0.155 0.133 0.106 0.083 0.064 0.047 3È22... 0.147 0.135 0.129 0.123 0.110 0.098 0.081 0.066 0.053 0.040 3È23... 0.120 0.107 0.101 0.095 0.083 0.074 0.061 0.050 0.040 0.030 3È24... 0.224 0.191 0.176 0.161 0.133 0.110 0.084 0.064 0.048 0.034 3È25... 0.279 0.271 0.265 0.259 0.240 0.219 0.188 0.157 0.129 0.102 3È26... 1.998 2.043 2.068 2.095 2.168 2.267 2.478 2.784 3.171 3.597 3È27... 0.490 0.501 0.507 0.513 0.530 0.553 0.601 0.673 0.764 0.867 3È28... 2.097 2.126 2.144 2.167 2.240 2.350 2.587 2.924 3.349 3.819 3È29... 1.805 1.830 1.845 1.863 1.920 2.004 2.185 2.443 2.767 3.119 3È30... 0.234 0.243 0.245 0.246 0.242 0.230 0.207 0.182 0.156 0.129 1094

TABLE 3ÈContinued 3È31... 0.264 0.270 0.270 0.268 0.259 0.246 0.225 0.205 0.189 0.174 3È32... 0.154 0.139 0.132 0.125 0.111 0.097 0.080 0.065 0.052 0.039 3È33... 0.174 0.160 0.151 0.141 0.119 0.100 0.077 0.059 0.045 0.033 3È34... 0.237 0.220 0.211 0.203 0.184 0.167 0.145 0.122 0.101 0.079 3È35... 0.133 0.136 0.136 0.134 0.128 0.118 0.102 0.084 0.070 0.054 3È36... 0.062 0.053 0.048 0.044 0.037 0.031 0.025 0.019 0.014 0.010 3È37... 0.055 0.047 0.043 0.039 0.033 0.028 0.023 0.018 0.013 0.010 3È38... 0.053 0.051 0.049 0.047 0.042 0.037 0.031 0.025 0.020 0.014 3È39... 0.067 0.066 0.064 0.062 0.057 0.052 0.045 0.037 0.029 0.021 3È40... 0.062 0.059 0.058 0.058 0.058 0.059 0.059 0.060 0.064 0.070 3È41... 0.224 0.213 0.211 0.211 0.215 0.221 0.232 0.249 0.277 0.317 3È42... 0.154 0.127 0.117 0.108 0.094 0.082 0.066 0.051 0.038 0.027 3È43... 0.079 0.072 0.069 0.066 0.060 0.056 0.049 0.042 0.035 0.027 3È44... 0.047 0.041 0.039 0.036 0.031 0.027 0.023 0.019 0.016 0.012 3È45... 0.052 0.049 0.048 0.047 0.044 0.041 0.035 0.029 0.023 0.018 3È46... 0.055 0.051 0.049 0.047 0.043 0.038 0.032 0.026 0.021 0.015 3È47... 0.050 0.047 0.045 0.043 0.037 0.031 0.024 0.018 0.014 0.010 3È48... 0.024 0.025 0.025 0.024 0.021 0.018 0.014 0.011 0.008 0.006 3È49... 0.044 0.045 0.045 0.044 0.039 0.034 0.027 0.021 0.016 0.012 3È50... 0.053 0.056 0.057 0.057 0.054 0.051 0.044 0.036 0.029 0.022 3È51... 0.004 0.004 0.004 0.004 0.004 0.004 0.003 0.003 0.002 0.001 3È52... 0.008 0.008 0.007 0.007 0.007 0.006 0.005 0.004 0.003 0.002 4È5... 7.233 7.086 6.977 6.862 6.630 6.395 5.880 5.151 4.382 3.593 4È6... 1.438 1.285 1.221 1.165 1.071 0.995 0.871 0.726 0.579 0.437 4È7... 1.628 1.492 1.432 1.378 1.286 1.205 1.062 0.885 0.705 0.529 4È8... 0.369 0.355 0.357 0.362 0.371 0.363 0.322 0.264 0.206 0.150 4È9... 0.708 0.689 0.679 0.666 0.616 0.549 0.441 0.339 0.252 0.177 4È10... 0.992 1.016 1.021 1.015 0.959 0.864 0.701 0.541 0.404 0.286 4È11... 1.533 1.305 1.211 1.123 0.949 0.799 0.612 0.456 0.333 0.232 4È12... 1.669 1.444 1.352 1.267 1.091 0.928 0.716 0.535 0.390 0.272 4È13... 0.679 0.585 0.532 0.477 0.364 0.278 0.190 0.129 0.088 0.057 4È14... 1.183 1.142 1.105 1.060 0.952 0.852 0.721 0.600 0.490 0.383 4È15... 1.331 1.272 1.222 1.163 1.020 0.889 0.726 0.585 0.464 0.354 4È16... 4.238 4.206 4.205 4.215 4.288 4.430 4.751 5.200 5.736 6.229 4È17... 1.077 0.977 0.922 0.864 0.739 0.635 0.515 0.424 0.357 0.303 4È18... 1.378 1.233 1.155 1.074 0.903 0.761 0.598 0.471 0.374 0.294 4È19... 1.650 1.455 1.351 1.245 1.024 0.845 0.643 0.486 0.363 0.261 4È20... 1.686 1.563 1.484 1.395 1.195 1.021 0.818 0.663 0.548 0.456 4È21... 0.481 0.446 0.425 0.402 0.349 0.303 0.244 0.193 0.151 0.113 4È22... 0.290 0.259 0.245 0.230 0.201 0.175 0.143 0.115 0.091 0.068 4È23... 0.236 0.208 0.195 0.182 0.157 0.137 0.111 0.089 0.070 0.052 4È24... 0.467 0.401 0.371 0.341 0.283 0.237 0.184 0.142 0.108 0.079 4È25... 2.858 2.890 2.911 2.935 3.002 3.101 3.332 3.681 4.129 4.619 4È26... 0.462 0.450 0.444 0.438 0.423 0.408 0.391 0.400 0.599 0.812 4È27... 1.286 1.311 1.326 1.342 1.384 1.440 1.562 1.745 1.989 2.268 4È28... 0.784 0.799 0.806 0.813 0.832 0.858 0.914 0.999 1.111 1.238 4È29... 3.393 3.447 3.479 3.515 3.625 3.785 4.131 4.630 5.262 5.959 4È30... 5.016 5.088 5.130 5.180 5.340 5.580 6.094 6.825 7.736 8.712 4È31... 0.618 0.631 0.633 0.632 0.614 0.583 0.527 0.469 0.413 0.358 4È32... 0.371 0.334 0.316 0.298 0.258 0.224 0.182 0.146 0.117 0.090 4È33... 0.300 0.275 0.260 0.244 0.208 0.177 0.139 0.108 0.084 0.062 4È34... 0.404 0.377 0.363 0.348 0.315 0.285 0.245 0.206 0.169 0.132 4È35... 0.362 0.361 0.357 0.350 0.328 0.301 0.260 0.218 0.178 0.139 4È36... 0.114 0.096 0.087 0.080 0.066 0.056 0.044 0.034 0.025 0.018 4È37... 0.122 0.105 0.097 0.089 0.073 0.062 0.049 0.038 0.028 0.020 4È38... 0.122 0.119 0.115 0.110 0.098 0.087 0.072 0.058 0.045 0.033 4È39... 0.117 0.113 0.109 0.105 0.093 0.082 0.068 0.054 0.042 0.031 4È40... 0.224 0.214 0.212 0.212 0.217 0.222 0.233 0.251 0.282 0.327 4È41... 0.226 0.198 0.189 0.181 0.168 0.158 0.145 0.136 0.133 0.136 4È42... 0.414 0.374 0.361 0.352 0.340 0.330 0.322 0.323 0.338 0.368 4È43... 0.165 0.149 0.141 0.134 0.121 0.111 0.097 0.082 0.067 0.052 4È44... 0.086 0.076 0.072 0.068 0.061 0.055 0.048 0.041 0.034 0.026 4È45... 0.084 0.079 0.077 0.075 0.069 0.064 0.055 0.045 0.036 0.027 4È46... 0.131 0.122 0.118 0.114 0.105 0.096 0.081 0.066 0.052 0.039 4È47... 0.097 0.094 0.091 0.087 0.076 0.064 0.050 0.038 0.029 0.021 1095

TABLE 3ÈContinued 4È48... 0.047 0.047 0.047 0.045 0.040 0.034 0.026 0.020 0.015 0.011 4È49... 0.082 0.085 0.085 0.083 0.076 0.067 0.055 0.043 0.033 0.024 4È50... 0.112 0.118 0.119 0.117 0.110 0.101 0.086 0.070 0.055 0.042 4È51... 0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.007 0.006 0.004 4È52... 0.015 0.015 0.015 0.014 0.014 0.013 0.011 0.009 0.007 0.005 5È6... 1.455 1.341 1.288 1.239 1.148 1.067 0.929 0.768 0.608 0.453 5È7... 3.196 2.865 2.725 2.601 2.392 2.224 1.951 1.630 1.306 1.066 5È8... 0.496 0.485 0.490 0.499 0.515 0.505 0.450 0.370 0.288 0.210 5È9... 0.567 0.595 0.605 0.609 0.587 0.535 0.440 0.344 0.260 0.186 5È10... 1.864 1.865 1.856 1.830 1.710 1.530 1.236 0.952 0.709 0.500 5È11... 1.453 1.248 1.166 1.092 0.939 0.800 0.619 0.463 0.338 0.236 5È12... 3.431 2.918 2.709 2.516 2.131 1.797 1.377 1.027 0.750 0.523 5È13... 1.030 0.886 0.805 0.721 0.549 0.418 0.285 0.194 0.131 0.086 5È14... 1.030 0.976 0.931 0.878 0.752 0.640 0.505 0.394 0.303 0.224 5È15... 2.640 2.541 2.456 2.356 2.117 1.896 1.606 1.337 1.092 0.854 5È16... 6.364 6.310 6.306 6.320 6.428 6.642 7.129 7.818 8.652 9.444 5È17... 0.749 0.673 0.631 0.587 0.493 0.414 0.323 0.252 0.197 0.153 5È18... 1.487 1.336 1.254 1.169 0.986 0.834 0.660 0.526 0.425 0.344 5È19... 2.527 2.273 2.136 1.994 1.690 1.435 1.142 0.915 0.744 0.604 5È20... 3.583 3.287 3.108 2.914 2.486 2.116 1.679 1.333 1.065 0.844 5È21... 0.809 0.751 0.718 0.682 0.604 0.533 0.441 0.357 0.284 0.217 5È22... 0.300 0.274 0.259 0.244 0.210 0.180 0.142 0.110 0.084 0.061 5È23... 0.325 0.280 0.260 0.239 0.199 0.167 0.129 0.098 0.073 0.052 5È24... 0.711 0.617 0.574 0.532 0.452 0.387 0.310 0.245 0.191 0.143 5È25... 3.133 3.146 3.164 3.186 3.247 3.342 3.570 3.925 4.397 4.929 5È26... 1.655 1.666 1.675 1.685 1.711 1.750 1.849 2.012 2.237 2.499 5È27... 0.241 0.232 0.227 0.222 0.207 0.190 0.162 0.136 0.112 0.089 5È28... 0.221 0.224 0.223 0.221 0.213 0.201 0.183 0.165 0.150 0.135 5È29... 0.964 0.977 0.981 0.983 0.984 0.987 1.006 1.050 1.119 1.202 5È30... 3.957 4.030 4.065 4.103 4.205 4.351 4.676 5.184 6.168 8.787 5È31... 11.50 11.70 11.80 11.93 12.30 12.84 13.99 15.62 17.57 19.37 5È32... 0.661 0.601 0.568 0.533 0.455 0.387 0.305 0.239 0.184 0.136 5È33... 0.313 0.284 0.270 0.257 0.226 0.199 0.163 0.131 0.104 0.079 5È34... 0.392 0.381 0.372 0.360 0.329 0.297 0.252 0.210 0.171 0.133 5È35... 0.823 0.793 0.772 0.747 0.684 0.620 0.532 0.447 0.366 0.286 5È36... 0.124 0.104 0.095 0.086 0.070 0.058 0.045 0.035 0.026 0.018 5È37... 0.237 0.203 0.186 0.170 0.141 0.119 0.094 0.072 0.054 0.038 5È38... 0.229 0.226 0.219 0.210 0.187 0.165 0.136 0.109 0.084 0.062 5È39... 0.133 0.127 0.122 0.116 0.103 0.091 0.076 0.061 0.047 0.035 5È40... 0.160 0.132 0.121 0.112 0.096 0.084 0.068 0.053 0.040 0.029 5È41... 0.429 0.385 0.370 0.359 0.343 0.332 0.324 0.328 0.348 0.386 5È42... 0.665 0.631 0.625 0.624 0.632 0.641 0.656 0.689 0.761 0.916 5È43... 0.259 0.230 0.217 0.205 0.182 0.164 0.142 0.120 0.098 0.076 5È44... 0.108 0.100 0.097 0.093 0.087 0.082 0.073 0.062 0.051 0.039 5È45... 0.082 0.075 0.072 0.069 0.063 0.057 0.048 0.039 0.031 0.023 5È46... 0.245 0.231 0.225 0.219 0.204 0.187 0.159 0.131 0.104 0.079 5È47... 0.138 0.138 0.136 0.132 0.117 0.101 0.080 0.062 0.047 0.034 5È48... 0.071 0.068 0.066 0.064 0.058 0.052 0.044 0.036 0.028 0.021 5È49... 0.110 0.116 0.117 0.117 0.113 0.106 0.093 0.077 0.062 0.047 5È50... 0.179 0.186 0.186 0.183 0.169 0.151 0.127 0.104 0.083 0.062 5È51... 0.011 0.010 0.010 0.010 0.010 0.009 0.007 0.006 0.005 0.003 5È52... 0.027 0.027 0.026 0.026 0.024 0.022 0.018 0.015 0.011 0.008 6È7... 6.288 6.324 6.300 6.249 6.038 5.704 5.029 4.222 3.429 2.705 6È8... 1.408 1.431 1.441 1.446 1.440 1.411 1.342 1.259 1.182 1.106 6È9... 1.040 1.062 1.064 1.058 1.021 0.959 0.835 0.691 0.550 0.417 6È10... 1.581 1.600 1.599 1.591 1.538 1.443 1.252 1.031 0.818 0.617 6È11... 7.779 8.284 8.451 8.532 8.318 7.685 6.591 5.843 5.740 6.205 6È12... 1.694 1.826 1.867 1.886 1.831 1.679 1.386 1.082 0.815 0.585 6È13... 1.541 1.649 1.815 2.066 2.669 2.933 2.740 2.238 1.703 1.215 6È14... 2.615 2.542 2.501 2.458 2.367 2.297 2.238 2.226 2.256 2.310 6È15... 2.155 2.062 2.001 1.931 1.751 1.572 1.336 1.131 0.960 0.811 6È16... 0.509 0.455 0.427 0.397 0.325 0.262 0.190 0.136 0.096 0.065 6È17... 1.218 1.143 1.102 1.057 0.947 0.840 0.700 0.576 0.469 0.367 6È18... 1.527 1.428 1.374 1.315 1.171 1.033 0.854 0.697 0.563 0.438 6È19... 1.469 1.361 1.302 1.238 1.084 0.942 0.763 0.613 0.487 0.374 1096

TABLE 3ÈContinued 6È20... 0.909 0.811 0.759 0.703 0.578 0.470 0.347 0.254 0.184 0.128 6È21... 2.088 3.447 4.039 4.457 4.631 4.167 3.247 2.416 1.779 1.287 6È22... 2.044 2.419 2.575 2.666 2.616 2.384 2.024 1.743 1.563 1.459 6È23... 1.936 1.930 1.924 1.917 1.906 1.925 2.028 2.226 2.509 2.865 6È24... 1.301 1.236 1.190 1.137 1.008 0.896 0.778 0.707 0.677 0.676 6È25... 0.646 0.604 0.578 0.549 0.483 0.426 0.358 0.299 0.245 0.193 6È26... 0.373 0.341 0.324 0.306 0.268 0.235 0.195 0.161 0.131 0.103 6È27... 0.158 0.143 0.135 0.127 0.111 0.096 0.078 0.063 0.050 0.039 6È28... 0.205 0.195 0.186 0.177 0.153 0.132 0.106 0.083 0.065 0.048 6È29... 0.362 0.340 0.325 0.306 0.264 0.226 0.180 0.142 0.110 0.082 6È30... 0.454 0.424 0.402 0.378 0.324 0.277 0.221 0.173 0.134 0.099 6È31... 0.477 0.444 0.423 0.399 0.346 0.300 0.246 0.200 0.159 0.121 6È32... 1.663 1.579 1.533 1.487 1.401 1.351 1.329 1.355 1.416 1.484 6È33... 6.038 6.202 6.279 6.363 6.603 6.933 7.596 8.505 9.661 11.04 6È34... 4.842 4.675 4.633 4.618 4.690 4.875 5.305 5.922 6.705 7.617 6È35... 2.838 2.397 2.177 1.961 1.542 1.233 0.913 0.678 0.503 0.362 6È36... 0.703 0.641 0.614 0.586 0.530 0.484 0.433 0.395 0.368 0.343 6È37... 0.533 0.465 0.431 0.397 0.327 0.271 0.210 0.163 0.126 0.095 6È38... 0.512 0.472 0.448 0.423 0.366 0.318 0.261 0.213 0.173 0.138 6È39... 1.841 1.918 1.947 1.970 2.006 2.042 2.136 2.298 2.516 2.747 6È40... 0.101 0.098 0.097 0.095 0.089 0.081 0.068 0.055 0.044 0.033 6È41... 0.160 0.155 0.153 0.150 0.140 0.126 0.105 0.085 0.067 0.050 6È42... 0.115 0.111 0.109 0.106 0.096 0.083 0.065 0.049 0.036 0.026 6È43... 0.338 0.318 0.308 0.299 0.277 0.258 0.237 0.223 0.232 0.308 6È44... 0.204 0.193 0.188 0.182 0.169 0.157 0.143 0.137 0.140 0.150 6È45... 0.251 0.242 0.240 0.239 0.242 0.249 0.263 0.286 0.321 0.368 6È46... 0.237 0.222 0.214 0.204 0.184 0.168 0.151 0.139 0.134 0.132 6È47... 0.162 0.170 0.173 0.174 0.168 0.155 0.136 0.119 0.105 0.094 6È48... 0.109 0.113 0.115 0.117 0.119 0.120 0.120 0.125 0.135 0.148 6È49... 0.271 0.273 0.275 0.276 0.281 0.289 0.310 0.341 0.378 0.412 6È50... 0.228 0.225 0.222 0.218 0.205 0.192 0.173 0.154 0.137 0.118 6È51... 0.080 0.080 0.080 0.080 0.081 0.082 0.084 0.087 0.092 0.100 6È52... 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.033 0.034 7È8... 2.137 2.173 2.189 2.198 2.186 2.132 2.013 1.884 1.771 1.665 7È9... 1.229 1.198 1.180 1.159 1.092 0.999 0.840 0.677 0.529 0.394 7È10... 2.700 2.761 2.768 2.756 2.642 2.431 2.047 1.648 1.287 0.964 7È11... 1.686 1.821 1.863 1.881 1.823 1.665 1.367 1.064 0.800 0.573 7È12... 12.40 13.19 13.44 13.55 13.18 12.15 10.41 9.152 8.704 8.586 7È13... 2.347 2.239 2.140 2.018 1.715 1.428 1.078 0.804 0.602 0.448 7È14... 2.034 1.963 1.912 1.851 1.692 1.534 1.326 1.144 0.992 0.856 7È15... 4.897 4.757 4.674 4.582 4.377 4.205 4.027 3.932 3.928 4.030 7È16... 0.741 0.667 0.627 0.582 0.477 0.384 0.280 0.202 0.144 0.098 7È17... 0.623 0.561 0.527 0.490 0.407 0.336 0.254 0.190 0.142 0.102 7È18... 1.209 1.111 1.056 0.995 0.854 0.727 0.575 0.451 0.351 0.264 7È19... 2.136 1.986 1.901 1.808 1.588 1.382 1.124 0.906 0.722 0.555 7È20... 3.557 3.335 3.209 3.070 2.736 2.414 1.999 1.636 1.323 1.030 7È21... 3.144 3.028 2.965 2.894 2.726 2.585 2.455 2.407 2.422 2.448 7È22... 0.824 0.745 0.706 0.662 0.561 0.468 0.358 0.270 0.203 0.148 7È23... 1.019 1.243 1.313 1.340 1.252 1.072 0.806 0.586 0.420 0.292 7È24... 4.125 4.245 4.266 4.252 4.134 4.013 3.973 4.129 4.456 4.834 7È25... 0.777 0.697 0.657 0.617 0.532 0.461 0.378 0.308 0.249 0.195 7È26... 0.514 0.477 0.456 0.433 0.381 0.336 0.280 0.231 0.188 0.147 7È27... 0.276 0.259 0.249 0.237 0.211 0.187 0.158 0.132 0.108 0.085 7È28... 0.176 0.165 0.158 0.150 0.131 0.115 0.095 0.078 0.062 0.048 7È29... 0.422 0.395 0.378 0.358 0.313 0.274 0.227 0.185 0.149 0.114 7È30... 0.724 0.685 0.656 0.623 0.546 0.477 0.393 0.320 0.256 0.197 7È31... 1.055 1.009 0.969 0.922 0.806 0.699 0.567 0.453 0.357 0.269 7È32... 9.737 9.960 10.06 10.17 10.51 10.98 11.97 13.33 14.97 16.51 7È33... 1.257 1.244 1.233 1.220 1.192 1.176 1.180 1.219 1.288 1.358 7È34... 3.300 2.988 2.774 2.546 2.080 1.733 1.392 1.174 1.043 0.968 7È35... 10.24 9.479 9.121 8.795 8.288 8.111 8.295 8.891 9.798 10.87 7È36... 0.533 0.464 0.430 0.395 0.326 0.271 0.210 0.163 0.126 0.096 7È37... 1.376 1.233 1.168 1.104 0.977 0.876 0.765 0.681 0.614 0.546 7È38... 2.781 2.879 2.914 2.943 2.990 3.040 3.169 3.411 3.775 4.189 7È39... 0.669 0.632 0.609 0.583 0.524 0.475 0.422 0.387 0.368 0.364 1097

TABLE 3ÈContinued 7È40... 0.066 0.060 0.058 0.055 0.048 0.041 0.031 0.023 0.017 0.012 7È41... 0.161 0.153 0.150 0.145 0.132 0.116 0.093 0.073 0.055 0.040 7È42... 0.333 0.330 0.327 0.322 0.303 0.274 0.229 0.184 0.145 0.108 7È43... 0.588 0.551 0.533 0.516 0.482 0.455 0.428 0.413 0.411 0.425 7È44... 0.246 0.231 0.224 0.217 0.203 0.189 0.174 0.163 0.157 0.153 7È45... 0.171 0.157 0.149 0.141 0.122 0.106 0.087 0.072 0.060 0.049 7È46... 0.579 0.558 0.550 0.543 0.535 0.534 0.544 0.574 0.629 0.713 7È47... 0.305 0.318 0.326 0.333 0.344 0.349 0.353 0.360 0.373 0.394 7È48... 0.119 0.124 0.126 0.127 0.122 0.112 0.096 0.081 0.067 0.055 7È49... 0.222 0.220 0.217 0.213 0.201 0.188 0.170 0.151 0.132 0.111 7È50... 0.506 0.510 0.511 0.512 0.510 0.510 0.523 0.554 0.600 0.657 7È51... 0.096 0.095 0.093 0.091 0.084 0.075 0.062 0.050 0.039 0.030 7È52... 0.182 0.182 0.182 0.183 0.184 0.187 0.190 0.194 0.201 0.213 8È9... 0.629 0.647 0.645 0.635 0.592 0.541 0.465 0.388 0.315 0.247 8È10... 1.235 1.290 1.289 1.269 1.178 1.068 0.904 0.744 0.597 0.462 8È11... 0.756 0.791 0.801 0.806 0.792 0.752 0.673 0.592 0.524 0.469 8È12... 1.087 1.145 1.164 1.174 1.161 1.106 0.992 0.873 0.774 0.693 8È13... 1.053 0.991 0.947 0.897 0.777 0.664 0.521 0.405 0.316 0.244 8È14... 0.345 0.382 0.392 0.393 0.373 0.335 0.271 0.210 0.159 0.116 8È15... 0.464 0.528 0.546 0.553 0.532 0.481 0.391 0.303 0.228 0.163 8È16... 0.071 0.070 0.067 0.064 0.054 0.044 0.032 0.023 0.016 0.011 8È17... 0.358 0.331 0.315 0.298 0.260 0.226 0.182 0.145 0.113 0.085 8È18... 0.530 0.491 0.468 0.443 0.388 0.337 0.271 0.215 0.168 0.126 8È19... 0.716 0.660 0.628 0.594 0.519 0.450 0.361 0.285 0.222 0.166 8È20... 0.879 0.807 0.768 0.728 0.639 0.557 0.450 0.356 0.278 0.208 8È21... 0.928 0.912 0.905 0.897 0.865 0.828 0.779 0.734 0.692 0.649 8È22... 0.449 0.427 0.418 0.408 0.383 0.357 0.324 0.294 0.269 0.245 8È23... 0.626 0.747 0.792 0.818 0.813 0.762 0.682 0.615 0.566 0.527 8È24... 1.127 1.187 1.206 1.214 1.192 1.148 1.091 1.051 1.027 1.008 8È25... 0.382 0.428 0.444 0.452 0.446 0.420 0.369 0.314 0.260 0.205 8È26... 0.261 0.292 0.301 0.306 0.299 0.280 0.244 0.207 0.170 0.134 8È27... 0.130 0.145 0.150 0.152 0.149 0.139 0.120 0.102 0.084 0.066 8È28... 0.074 0.078 0.078 0.077 0.071 0.063 0.052 0.040 0.031 0.022 8È29... 0.146 0.153 0.153 0.151 0.140 0.125 0.103 0.082 0.063 0.046 8È30... 0.206 0.218 0.220 0.217 0.202 0.180 0.146 0.115 0.088 0.064 8È31... 0.268 0.284 0.285 0.282 0.262 0.231 0.186 0.145 0.110 0.079 8È32... 0.594 0.560 0.541 0.518 0.456 0.394 0.317 0.254 0.204 0.159 8È33... 0.399 0.378 0.366 0.350 0.309 0.267 0.215 0.172 0.138 0.108 8È34... 0.539 0.529 0.513 0.491 0.432 0.378 0.311 0.256 0.213 0.176 8È35... 0.467 0.509 0.521 0.524 0.501 0.458 0.388 0.323 0.267 0.218 8È36... 0.256 0.222 0.204 0.185 0.147 0.117 0.086 0.064 0.048 0.035 8È37... 0.386 0.334 0.307 0.279 0.220 0.176 0.129 0.096 0.071 0.052 8È38... 0.246 0.240 0.232 0.221 0.191 0.163 0.128 0.100 0.078 0.060 8È39... 0.182 0.178 0.172 0.164 0.143 0.122 0.096 0.075 0.059 0.045 8È40... 0.086 0.072 0.067 0.062 0.052 0.044 0.034 0.027 0.020 0.015 8È41... 0.169 0.144 0.133 0.123 0.103 0.087 0.068 0.053 0.040 0.029 8È42... 0.220 0.191 0.178 0.166 0.139 0.117 0.091 0.071 0.054 0.039 8È43... 4.283 4.342 4.401 4.482 4.733 5.049 5.627 6.393 7.372 8.563 8È44... 2.006 2.002 2.017 2.045 2.148 2.286 2.541 2.882 3.323 3.869 8È45... 0.621 0.630 0.638 0.649 0.684 0.729 0.812 0.926 1.075 1.261 8È46... 1.055 1.084 1.105 1.131 1.207 1.293 1.434 1.614 1.845 2.131 8È47... 0.637 0.616 0.599 0.580 0.532 0.490 0.443 0.409 0.381 0.353 8È48... 0.342 0.335 0.328 0.319 0.297 0.277 0.255 0.239 0.226 0.211 8È49... 0.096 0.090 0.087 0.084 0.078 0.073 0.065 0.058 0.051 0.044 8È50... 0.131 0.123 0.120 0.116 0.110 0.103 0.093 0.080 0.067 0.054 8È51... 0.043 0.043 0.043 0.043 0.044 0.044 0.046 0.048 0.051 0.055 8È52... 0.102 0.102 0.102 0.102 0.104 0.105 0.108 0.113 0.120 0.130 9È10... 3.726 3.768 3.715 3.622 3.336 3.027 2.585 2.158 1.778 1.419 9È11... 1.532 1.399 1.339 1.281 1.164 1.066 0.934 0.801 0.674 0.545 9È12... 1.259 1.152 1.109 1.068 0.981 0.892 0.753 0.611 0.483 0.365 9È13... 0.359 0.335 0.324 0.313 0.287 0.256 0.205 0.157 0.116 0.082 9È14... 1.531 1.601 1.636 1.667 1.714 1.728 1.726 1.736 1.778 1.845 9È15... 0.474 0.490 0.498 0.503 0.494 0.461 0.392 0.316 0.248 0.187 9È16... 0.373 0.399 0.406 0.408 0.394 0.364 0.312 0.260 0.211 0.163 9È17... 0.382 0.379 0.371 0.359 0.320 0.276 0.214 0.161 0.119 0.083 1098

TABLE 3ÈContinued 9È18... 0.490 0.484 0.474 0.459 0.411 0.355 0.277 0.209 0.154 0.108 9È19... 0.514 0.501 0.489 0.472 0.423 0.367 0.289 0.220 0.163 0.115 9È20... 0.416 0.393 0.380 0.364 0.325 0.284 0.226 0.174 0.130 0.093 9È21... 0.682 0.682 0.676 0.664 0.621 0.570 0.503 0.449 0.418 0.397 9È22... 0.766 0.690 0.662 0.636 0.593 0.569 0.567 0.593 0.635 0.685 9È23... 0.850 1.084 1.170 1.223 1.225 1.152 1.043 0.967 0.924 0.895 9È24... 0.801 0.906 0.936 0.944 0.890 0.795 0.659 0.552 0.474 0.412 9È25... 0.331 0.320 0.315 0.308 0.281 0.247 0.196 0.150 0.111 0.079 9È26... 0.230 0.216 0.209 0.202 0.182 0.160 0.129 0.100 0.076 0.055 9È27... 0.111 0.103 0.099 0.096 0.086 0.075 0.061 0.048 0.037 0.027 9È28... 0.157 0.161 0.162 0.161 0.150 0.134 0.108 0.084 0.064 0.047 9È29... 0.282 0.285 0.285 0.282 0.262 0.232 0.187 0.145 0.110 0.080 9È30... 0.340 0.343 0.342 0.338 0.312 0.276 0.221 0.171 0.129 0.093 9È31... 0.319 0.324 0.322 0.317 0.292 0.258 0.208 0.163 0.125 0.090 9È32... 0.525 0.581 0.597 0.603 0.587 0.546 0.473 0.396 0.322 0.248 9È33... 0.903 0.999 1.045 1.082 1.134 1.150 1.151 1.152 1.161 1.171 9È34... 0.917 0.839 0.794 0.745 0.636 0.542 0.436 0.353 0.287 0.228 9È35... 0.760 0.772 0.764 0.746 0.675 0.590 0.475 0.377 0.299 0.229 9È36... 0.345 0.287 0.262 0.241 0.204 0.175 0.138 0.105 0.077 0.053 9È37... 0.345 0.298 0.274 0.250 0.202 0.163 0.121 0.088 0.063 0.043 9È38... 0.291 0.291 0.285 0.276 0.247 0.217 0.177 0.141 0.111 0.083 9È39... 0.294 0.284 0.275 0.263 0.233 0.205 0.168 0.135 0.106 0.080 9È40... 0.077 0.067 0.062 0.057 0.048 0.040 0.030 0.022 0.017 0.012 9È41... 0.148 0.127 0.117 0.108 0.089 0.074 0.056 0.042 0.031 0.022 9È42... 0.177 0.154 0.143 0.132 0.108 0.088 0.066 0.048 0.034 0.024 9È43... 0.759 0.709 0.682 0.655 0.603 0.568 0.541 0.535 0.548 0.576 9È44... 1.086 1.081 1.074 1.067 1.056 1.059 1.095 1.171 1.287 1.440 9È45... 0.336 0.275 0.248 0.223 0.182 0.159 0.143 0.142 0.150 0.169 9È46... 0.410 0.335 0.303 0.275 0.229 0.203 0.185 0.180 0.187 0.202 9È47... 1.146 1.174 1.202 1.238 1.327 1.414 1.546 1.716 1.938 2.210 9È48... 1.179 1.218 1.250 1.290 1.394 1.505 1.690 1.936 2.258 2.655 9È49... 4.618 4.820 4.948 5.093 5.454 5.853 6.578 7.575 8.867 10.42 9È50... 0.404 0.398 0.397 0.397 0.394 0.388 0.370 0.343 0.308 0.261 9È51... 0.104 0.104 0.105 0.106 0.109 0.113 0.122 0.135 0.147 0.150 9È52... 0.100 0.101 0.101 0.101 0.101 0.102 0.104 0.105 0.105 0.106 10È11... 1.810 1.660 1.599 1.542 1.420 1.300 1.114 0.921 0.741 0.571 10È12... 3.695 3.401 3.271 3.142 2.866 2.614 2.256 1.898 1.566 1.243 10È13... 0.752 0.694 0.663 0.629 0.550 0.471 0.366 0.275 0.203 0.143 10È14... 0.843 0.878 0.894 0.905 0.900 0.861 0.776 0.687 0.613 0.553 10È15... 3.104 3.237 3.304 3.364 3.441 3.441 3.387 3.346 3.363 3.429 10È16... 0.739 0.796 0.810 0.814 0.786 0.728 0.628 0.525 0.428 0.332 10È17... 0.402 0.384 0.373 0.358 0.320 0.277 0.218 0.166 0.123 0.087 10È18... 0.663 0.644 0.628 0.607 0.544 0.472 0.371 0.282 0.209 0.147 10È19... 1.023 1.002 0.979 0.947 0.848 0.735 0.579 0.441 0.327 0.231 10È20... 1.505 1.481 1.449 1.402 1.254 1.086 0.852 0.648 0.481 0.340 10È21... 1.972 1.911 1.877 1.839 1.753 1.692 1.664 1.697 1.773 1.866 10È22... 0.785 0.716 0.685 0.654 0.588 0.533 0.475 0.438 0.420 0.413 10È23... 0.898 1.104 1.174 1.208 1.166 1.045 0.864 0.718 0.613 0.532 10È24... 1.986 2.066 2.083 2.079 2.012 1.931 1.858 1.835 1.841 1.850 10È25... 0.635 0.592 0.572 0.550 0.492 0.428 0.339 0.261 0.196 0.140 10È26... 0.393 0.379 0.372 0.362 0.329 0.288 0.229 0.176 0.131 0.093 10È27... 0.186 0.185 0.184 0.180 0.166 0.146 0.117 0.089 0.066 0.047 10È28... 0.173 0.174 0.174 0.171 0.158 0.140 0.112 0.088 0.067 0.048 10È29... 0.392 0.395 0.393 0.387 0.358 0.317 0.256 0.201 0.153 0.111 10È30... 0.669 0.677 0.676 0.667 0.618 0.549 0.443 0.347 0.265 0.193 10È31... 1.041 1.049 1.047 1.033 0.957 0.846 0.679 0.527 0.400 0.290 10È32... 1.960 2.200 2.300 2.379 2.463 2.456 2.398 2.346 2.321 2.309 10È33... 0.868 0.936 0.960 0.973 0.962 0.918 0.835 0.750 0.667 0.577 10È34... 1.408 1.323 1.264 1.195 1.027 0.876 0.700 0.559 0.447 0.346 10È35... 1.955 1.921 1.878 1.814 1.621 1.418 1.155 0.936 0.756 0.592 10È36... 0.509 0.434 0.397 0.362 0.292 0.237 0.175 0.128 0.091 0.063 10È37... 0.874 0.734 0.670 0.608 0.494 0.407 0.308 0.229 0.166 0.115 10È38... 0.722 0.705 0.686 0.661 0.593 0.524 0.429 0.343 0.269 0.202 10È39... 0.468 0.463 0.452 0.435 0.389 0.342 0.280 0.225 0.178 0.134 10È40... 0.143 0.120 0.110 0.101 0.081 0.066 0.049 0.036 0.026 0.017 1099

1100 TAYAL Vol. 530 TABLE 3ÈContinued 10È41... 0.287 0.244 0.224 0.205 0.168 0.137 0.102 0.075 0.055 0.038 10È42... 0.415 0.365 0.340 0.316 0.263 0.217 0.164 0.122 0.088 0.061 10È43... 2.745 2.675 2.631 2.586 2.502 2.461 2.480 2.588 2.787 3.064 10È44... 0.721 0.691 0.673 0.655 0.620 0.598 0.584 0.589 0.614 0.657 10È45... 0.362 0.296 0.267 0.240 0.194 0.166 0.144 0.133 0.133 0.138 10È46... 0.978 0.813 0.743 0.683 0.595 0.557 0.544 0.560 0.610 0.683 10È47... 2.870 2.941 3.010 3.097 3.318 3.546 3.919 4.416 5.074 5.891 10È48... 0.844 0.858 0.877 0.900 0.959 1.014 1.098 1.206 1.352 1.536 10È49... 1.803 1.859 1.898 1.945 2.060 2.181 2.379 2.638 2.977 3.391 10È50... 9.359 9.764 10.02 10.32 11.04 11.83 13.24 15.16 17.62 20.54 10È51... 0.111 0.111 0.111 0.111 0.112 0.114 0.116 0.119 0.119 0.119 10È52... 0.357 0.357 0.358 0.358 0.359 0.362 0.371 0.389 0.409 0.417 matrices calculated in the ab initio calculation to bring the calculated energies as close as possible to the experimental results. These adjusted energies are also listed in Table 1. These adjustments also caused small changes in the weights of some eigenvectors. We have used adjusted energies and eigenvectors in the scattering calculation. The total wave function representing the scattering of electrons by S IV ion for each total angular momentum J and parity n is expanded in the inner region (r ¹ a) inthe R-matrix basis as (Scott & Taylor 1982; Berrington et al. 1987) ( (Jn) \ A ; c ' J(x, x,..., x ; x, p ) k ijk i 1 2 N N`1 N`1 ij ] u (x ) ] ; d / J(x, x,..., x ), (1) ij N`1 jk j 1 2 N`1 j where A is the antisymmetrization operator; ' J are channel i functions representing the 52 Ðne-structure target levels coupled with angular and spin functions of the scattered electron to form channel functions of J and n; / J are j (N ] 1) electron conðgurations formed from the atomic orbitals and are included to ensure completeness of the total wave function expansions and to allow for short-range correlations; and u are the orthogonal set of continuum ij basis. The coefficients c and d are obtained by diagonal- ijk jk izing the (N ] 1) electron Breit-Pauli Hamiltonian in the inner region. We have chosen the boundary radius a \ 15.9 AU and included 22 continuum orbitals for each angular momentum to obtain convergence in the energy range up to 8.5 ryd. The maximum number of channels retained in our calculation is 241. The coupled equations are solved in the asymptotic region using a perturbation method to yield K-matrices and then the collision strengths. The Breit-Pauli R-matrix method is used to calculate partial collision strengths from J \ 0toJ \ 20. The higher partial waves contributions for the dipole-allowed transitions are calculated using the Bethe approximation (Burgess & Sheorey 1974), and top-up ÏÏ is used for other transitions. The electron excitation rates are obtained by averaging total collision strengths over a Maxwellian distribution of electron energies. The excitation rate coefficients for a transition from state i to a state f at electron temperature T is e given by 8.63 ] 10~6 B C(i ] f ) \ c(i ] f )exp A[*E if cm3 s~1, g T 1@2 KT i e e (2) where g \ (2j ] 1) is the statistical weight of the lower level i i i, *E \ E [ E is the excitation energy, and c(i ] f ) is the if f i e ective collision strength given by c(i ] f ) \ P 0= )(i ] f ) exp A[v f kt e B. (3) Here )(i ] f ) is the total collision strength for transition between levels i and f, v is the energy of incident electron f TABLE 4 COMPARISON OF EFFECTIVE COLLISION STRENGTHS ELECTRON TEMPERATURE 10,000 K 40,000 K 100,000 K 400,000 K TRANSITION Present Dufton Present Dufton Present Dufton Present Dufton 1È2... 8.54 6.42 7.50 6.35 5.95 5.60 3.04 2.96 1È3... 0.60 0.51 0.45 0.41 0.33 0.31 0.16 0.15 1È4... 1.05 0.87 0.81 0.71 0.58 0.52 0.26 0.24 1È5... 1.17 0.95 0.91 0.77 0.63 0.53 0.26 0.21 2È3... 1.03 0.66 0.69 0.54 0.46 0.37 0.19 0.15 2È4... 1.91 1.47 1.42 1.18 0.99 0.84 0.42 0.37 2È5... 2.73 2.53 2.23 2.08 1.62 1.50 0.73 0.69 3È4... 3.37 3.04 3.01 2.30 2.57 1.68 1.37 0.86 3È5... 2.92 2.92 2.71 2.11 2.45 1.56 1.65 1.05 4È5... 7.23 7.01 6.63 5.16 5.88 3.77 3.59 2.24

No. 2, 2000 ELECTRON COLLISION EXCITATION IN S IV 1101 with respect to the upper level f, T is the electron temperature, and k is the Boltzmann constant. e 3. RESULTS AND DISCUSSION We chose a Ðne energy mesh (0.001 ryd) for collision strengths calculation in the thresholds energy region. This allowed us to delineate the resonance structures in the collision strengths. In Table 2 we present total collision strengths at incident electron energies 4 and 6 ryd and compare these with the calculations of Bhadra & Henry (1980) and Bhatia et al. (1980), which are denoted BH ÏÏ and BDF,ÏÏ respectively. These energies are above the highest excitation threshold in our calculation. It may be noted that the incident electron energy 2 ryd lies in the resonance region in our calculation, which is why we did not compare our results with the available earlier calculations at this energy in Table 2. The keys to the lower and upper levels involved in a transition are given in Table 1. The agreement between the present results and those of Bhadra & Henry (1980) is 40% or better except for a few transitions with smaller collision strengths. There are larger di erences (up to a factor of 2) with the distorted wave calculation of Bhatia et al. (1980). The main reason for the discrepancies at these energies with the calculation of Bhatia et al. (1980) may be the use of limited correlation e ects in their calculation. We have plotted total collision strengths as a function of electron energy in the region of thresholds from the threshold to 0.7 ryd in Figure 1 for the ground state Ðne-structure transition 2Po È2Po and in Figures 2, 3, and 4 for the 1@2 3@2 transitions 2Po È3s3p2 4P, 2Po È3s3p2 4P, and 1@2 1@2 1@2 3@2 2Po È3s3p2 4P, respectively, from the threshold to 1.2 ryd. 1@2 The results 5@2 of our full Breit-Pauli R-matrix calculation are shown by solid curves. It is clear from these Ðgures that there are complicated resonance structures in collision strengths. Since all the previous calculations were performed in the LS coupling scheme and the transformation code JAJOM (Saraph 1978) is then used to obtain collision strengths of Ðne-structure levels, we also carried out LS R-matrix plus JAJOM calculation, and these results are shown by dashed curves in these Ðgures. The mass and Darwin terms of the Breit-Pauli operator are included in our LScalculation. We used the same wave functions in the two calculations, but the energies in our LS calculation were adjusted in order to experiment. Some signiðcant differences in the resonance structures from the two calculations may be noted from these Ðgures. It is well known that the transformation of LS K-matrices to intermediate coupling using the JAJOM code (Saraph 1978) has two limitations. In this method the Ðne-structure splittings of the target terms are neglected, and in the resonance region, where there are both open and closed channels, only those components of term-coupling coefficients are used for which channels are open. Both of these approximations may lead to some inaccuracies in collision strengths depending upon the spin-orbit interaction. Recently Griffin, Badnell, & Pindzola (1998) studied electron impact excitation of Fe XV and presented detailed comparison of the collision strengths obtained in full Breit-Pauli R-matrix and pure LScoupling plus transformation to intermediate coupling using JAJOM and found signiðcant discrepancies between the two sets of results. 20 18 16 14 COLLISION STRENGTH 12 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ELECTRON ENERGY IN RYDBERGS FIG. 1.ÈCollision strength for the 3s23p 2Po È3s23p 2Po transition in S IV as a function of electron energy in rydbergs: present full Breit-Pauli R-matrix calculation (solid curve) and LSR-matrix plus JAJOM 1@2 (dashed 3@2 curve).