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Electron-impact cross sections for dipole- and spin-allowed excitations of hydrogen, helium, and lithium.

Electron-impact excitation cross sections are presented for the dipole- and spin allowed transitions from the ground states to the tip [P.sup.2] states for hydrogen and lithium, and to the 1snp [P.sup.1] P states for helium, n = 2 through 10. Two sealing formulas developed earlier by Kim [Phys. Rev. A 64, 032713 (2001)] for plane-wave Born cross sections are used. The scaled Born cross sections are in excellent agreement with available theoretical and experimental data.

Key words: electron-impact; excitation cross section; hydrogen; helium; lithium.

1. Introduction

We have scaled Plane-Wave Born (PWB) cross sections to calculate dipole- and spin-allowed excitation cross sections from the ground state of neutral hydrogen, helium, and lithium. The scaling method was developed by one of us (1), and uses two simple scaling formulas to convert PWB excitation cross sections into reliable cross sections comparable to the most accurate theoretical or experimental data available for dipole-allowed transitions. The PWB cross sections are calculated from uncorrelated wave functions, and the scaling requires only the binding energy B of the electron being excited, the excitation energy E, and an accurate dipole oscillator strength for the transition. The oscillator strength is needed only if electron correlation strongly affects the f value, i.e., when the wave functions used to calculate the PWB cross section are not accurate. Simplicity of the method to scale PWB cross sections allows us to generate a large number of cross sections reliably and quickly.

In this paper, we present calculated excitation cross sections for hydrogen from the 1s [S.sup.2] ground state to the tip [P.sup.2] excited states. For helium, the cross sections are given for excitations from the [1s.sup.2] [S.sup.1] ground state to the 1snp [P.sup.1] excited states. For lithium, the results are given for excitations from the [1s.sup.2]2s [S.sup.2] ground state to the [1s.sup.2]np [P.sup.2] states. In all cases, the values of n are from n = 2 through 10.

2. Outline of Theory

A PWB cross section for electron-impact excitation, [[sigma].sub.PWB], has the form

[[sigma].sub.PWB] 4[pi][a.sup.2.sub.0]R/[[sigma].sup.PWB] = T [F.sub.PWB](T), (1)

where T is the incident electron energy, [a.sub.0] is the Bohr radius (0.529 A), and R is the Rydberg energy (13.61 eV). The [F.sub.PWB](T) is the collision strength (different from the standard definition by a multiplicative constant).

The first scaling method, BE scaling, replaces T in the denominator of Eq. (1) by T + B + E, i.e.,

[[sigma].sub.BE] = [[sigma].sub.PWB] [T/(T+B+E)] (2)

This scaling is similar to a scaling for ionization cross sections used earlier by Burgess (2), who shifted the incident energy T by B+U, where U is the kinetic energy of the target electron. However, in the BE scaling adopted by Kim (1) for excitation cross sections, T is shifted by B+E. The BE scaling not only changes the magnitude but also the shape of the original PWB cross sections. The BE scaling corrects the deficiency in the collision theory; i.e., the use of the PWB approximation.

The second scaling formula, the f scaling, multiplies the entire cross section by the ratio of an accurate f value to the less accurate f value calculated by the actual wave functions used to generate the unscaled PWB cross sections:

[[sigma].sub.f] = ([f.sub.accu]/[f.sub.sc]) [[sigma].sub.PWsc] (3)

where [f.sub.sc] is the single configuration (or uncorrelated) f value and [f.sub.accu] is the more accurate value obtained from correlated (or multiconfiguration) wave functions or from a reliable experiment. Accurate f values are frequently available (3). The f scaling compensates for the inadequacy of the wave functions when electron correlation effect is significant. The BE and f scalings may be applied consecutively, i.e.,

[[sigma].sub.BEf] = ([f.sub.accu]/[f.sub.sc])[[sigma].sub.BEsc], (4)

where [[sigma].sub.BEsc] is the BE-scaled PWB cross section calculated from single-configuration wave functions.

Kim has shown many examples (1) in which the BE scaling alone or in combination with the f scaling transformed PWB cross sections for dipole-allowed and spin-allowed excitations into reliable cross sections comparable to the convergent close coupling (CCC) method (4) or accurate experiments.

In reality, electron-impact excitation cross sections of atoms have resonances in the vicinity of the excitation thresholds caused by the formation of transient compound states between the incident electron and the target atom. First-order perturbation theories such as the PWB approximation cannot account for such compound states, and hence the present scaled cross sections do not exhibit any resonances.

The numerical data in Tables 1, 2, and 3 can easily be extended to higher incident energies by using the well known Bethe formula (5) for the plane-wave Born approximation for fast (but nonrelativistic) incident electrons. In our notation, the asymptotic expression becomes:

[[sigma].sub.asympt](T) = 4[pi][a.sub.0.sup.2]R/T+B+E [a ln (T/R)+b+cR/T] ([f.sub.accu]/[f.sub.sc]), (5)

where a, b, and c are dimensionless constants. Equation (5) should be used for T> 3 keV. The values of a, b, and c are included in Tables 1, 2, and 3. Note that a relativistic form (5) of Eq. (5) should be used for T> 10keV.

3. Theoretical Results

We present the calculated cross sections for hydrogen, helium, and lithium in Tables 1-3. Our PWB cross sections were generated from single configuration Dirac-Fock wave functions. The calculated cross sections are compared to other theories and experiments in Figs. 1-7.

The CCC results for these elements are from the web site of Bray (6). The experimental results by Sweeney et al. (7) for hydrogen include all dipole-allowed and dipole-forbidden states of hydrogen for each n, and hence are higher than the cross sections for just the dipole-allowed excitations.

The ionization energies B and the excitation energies E are all experimental values. Only BE scaling is needed for hydrogen as its exact wave functions are known. The accurate f values for helium have been obtained from the detailed variational calculations of Drake (8). The f value for the 2s-2p transition in lithium is from the calculations of Yan et al. (9), while the values for the 2s-np transitions, n = 3 through 7, are from the non-relativistic multiconfiguration calculations including core polarization by Qu et al. (10). For the 2s-8p, 9p, and 10p excitations of lithium, we extrapolated f[([n.sup.*]).sup.3] of Qu et al. (10) from n = 5 through n = 7, where [n.sup.*] is the experimental effective principal quantum number of quantum defect theory. We had found that the f values by Qu et al. for the 8p and 9p transitions were inconsistent with their values for n < 8. The extrapolation of f[([n.sup.*]).sup.3] is shown in Fig. 8 through [n.sup.*] = 17 and is given by the expression:

F[([n.sup.*]).sup.3] = 0.343 + 0.0283/([n.sup.*]) + 0.533/[([n.sup.*]).sup.2] - 6.289/[([n.sup.*]).sup.3] (6)

Beyond [n.sup.*] [approximatly equal to] 17, the formula begins to break down but the actual curve should remain flat. At the ionization limit ([n.sup.*] [right arrow] [infinity]), the value of f[([n.sup.*]).sup.3] is extrapolated to be 0.345.

It is apparent that for all cases where experimental data and CCC results are available, the scaled PWB cross sections give values that are in good agreement.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]
Table 1

Hydrogen. Excitation energies E in eV, dipole f values, and BE-scaled
excitation cross sections [[sigma].sub.BE] in [[Angstrom].sup.2] as
functions of incident electron energy T in eV. The experimental
ionization energy B = 13.5984 eV has been used in the scaling. The
constants a, b, and c of Eq. (5) are included

Excitation 1s-2p 1s-3p 1s-4p 1s-5p 1s-6p

E 10.204 12.094 12.755 13.061 13.228
f 0.4164 0.0791 0.0290 0.0139 0.00780
Const. a 0.555512 0.089083 0.030956 0.014534 0.008031
Const. b 0.271785 0.060202 0.022984 0.011243 0.006348
Const. c 0.000112 -0.019775 -0.009279 -0.004880 -0.002853

Excitation 1s-7p 1s-8p 1s-9p 1s-10p

E 13.328 13.393 13.438 13.470
f 0.004816 0.003185 0.002217 0.001606
Const. a 0.004919 0.003237 0.002246 0.001623
Const. b 0.003939 0.002550 0.001824 0.001323
Const. c -0.001806 -0.001213 -0.000854 -0.000623
 T [[sigma].sub.BE] [[sigma].sub.BE] [[sigma].sub.BE]

 11 0.15876
 12 0.24099
 13 0.30186 0.03033 0.00573
 14 0.35119 0.04382 0.01274
 15 0.39256 0.05372 0.01697
 16 0.42786 0.06166 0.02018
 17 0.45828 0.06827 0.02278
 18 0.48468 0.07387 0.02496
 19 0.50768 0.07867 0.02681
 20 0.52779 0.08282 0.02840
 21 0.54540 0.08642 0.02977
 22 0.56084 0.08957 0.03096
 23 0.57439 0.09231 0.03200
 24 0.58627 0.09472 0.03291
 26 0.60580 0.09867 0.03440
 28 0.62069 0.10169 0.03554
 30 0.63189 0.10398 0.03641
 32 0.64013 0.10569 0.03706
 34 0.64597 0.10695 0.03754
 36 0.64986 0.10782 0.03788
 38 0.65216 0.10840 0.03811
 40 0.65315 0.10872 0.03825
 45 0.65130 0.10870 0.03828
 50 0.64520 0.10787 0.03801
 55 0.63647 0.10654 0.03756
 60 0.62615 0.10490 0.03700
 65 0.61489 0.10308 0.03636
 70 0.60315 0.10116 0.03569
 75 0.59121 0.09919 0.03500
 80 0.57929 0.09721 0.03431
 85 0.56750 0.09524 0.03362
 90 0.55593 0.09331 0.03294
 95 0.54465 0.09142 0.03227
 100 0.53369 0.08959 0.03162
 110 0.51276 0.08607 0.03038
 120 0.49322 0.08278 0.02922
 130 0.47500 0.07972 0.02814
 140 0.45805 0.07686 0.02713
 150 0.44227 0.07420 0.02619
 160 0.42756 0.07171 0.02531
 170 0.41383 0.06940 0.02449
 180 0.40100 0.06723 0.02372
 190 0.38898 0.06520 0.02301
 200 0.37771 0.06330 0.02233
 250 0.33045 0.05533 0.01952
 300 0.29445 0.04926 0.01737
 350 0.26607 0.04447 0.01568
 400 0.24309 0.04060 0.01431
 450 0.22408 0.03741 0.01318
 500 0.20807 0.03471 0.01223
 600 0.18254 0.03042 0.01071
 700 0.16303 0.02715 0.00956
 800 0.14760 0.02456 0.00865
 900 0.13505 0.02246 0.00790
1000 0.12463 0.02071 0.00729
1500 0.09094 0.01508 0.00530
2000 0.07236 0.01198 0.00421
3000 0.05214 0.00862 0.00303

 T [[sigma].sub.BE] [[sigma].sub.BE] [[sigma].sub.BE]

 11
 12
 13
 14 0.00528 0.00267 0.00154
 15 0.00752 0.00401 0.00240
 16 0.00916 0.00496 0.00300
 17 0.01046 0.00570 0.00346
 18 0.01154 0.00632 0.00385
 19 0.01245 0.00684 0.00417
 20 0.01323 0.00728 0.00445
 21 0.01391 0.00766 0.00468
 22 0.01449 0.00799 0.00489
 23 0.01500 0.00828 0.00507
 24 0.01544 0.00853 0.00522
 26 0.01617 0.00894 0.00548
 28 0.01673 0.00925 0.00567
 30 0.01715 0.00949 0.00582
 32 0.01747 0.00967 0.00593
 34 0.01771 0.00981 0.00602
 36 0.01787 0.00990 0.00608
 38 0.01799 0.00997 0.00612
 40 0.01806 0.01001 0.00614
 45 0.01808 0.01002 0.00615
 50 0.01796 0.00996 0.00611
 55 0.01775 0.00984 0.00604
 60 0.01749 0.00970 0.00595
 65 0.01719 0.00953 0.00585
 70 0.01688 0.00936 0.00575
 75 0.01655 0.00918 0.00564
 80 0.01622 0.00900 0.00552
 85 0.01590 0.00882 0.00541
 90 0.01557 0.00864 0.00530
 95 0.01526 0.00846 0.00520
 100 0.01495 0.00829 0.00509
 110 0.01437 0.00797 0.00489
 120 0.01382 0.00766 0.00471
 130 0.01331 0.00738 0.00453
 140 0.01283 0.00712 0.00437
 150 0.01238 0.00687 0.00422
 160 0.01197 0.00664 0.00408
 170 0.01158 0.00642 0.00394
 180 0.01122 0.00622 0.00382
 190 0.01088 0.00603 0.00370
 200 0.01056 0.00586 0.00360
 250 0.00923 0.00512 0.00314
 300 0.00821 0.00455 0.00280
 350 0.00741 0.00411 0.00252
 400 0.00676 0.00375 0.00230
 450 0.00623 0.00345 0.00212
 500 0.00578 0.00320 0.00197
 600 0.00506 0.00281 0.00172
 700 0.00452 0.00250 0.00154
 800 0.00408 0.00226 0.00139
 900 0.00373 0.00207 0.00127
1000 0.00344 0.00191 0.00117
1500 0.00250 0.00139 0.000851
2000 0.00199 0.00110 0.000676
3000 0.00143 0.000791 0.000486

 T [[sigma].sub.BE] [[sigma].sub.BE] [[sigma].sub.BE]

 11
 12
 13
 14 0.000965 0.000646 0.000455
 15 0.00155 0.00106 0.000763
 16 0.00195 0.00135 0.000969
 17 0.00227 0.00157 0.00113
 18 0.00252 0.00175 0.00126
 19 0.00274 0.00190 0.00137
 20 0.00292 0.00202 0.00146
 21 0.00308 0.00213 0.00154
 22 0.00321 0.00223 0.00161
 23 0.00333 0.00231 0.00167
 24 0.00344 0.00238 0.00172
 26 0.00361 0.00250 0.00181
 28 0.00373 0.00259 0.00187
 30 0.00383 0.00266 0.00192
 32 0.00391 0.00271 0.00196
 34 0.00396 0.00275 0.00199
 36 0.00400 0.00278 0.00201
 38 0.00403 0.00280 0.00202
 40 0.00405 0.00281 0.00203
 45 0.00405 0.00281 0.00204
 50 0.00403 0.00280 0.00202
 55 0.00398 0.00277 0.00200
 60 0.00392 0.00273 0.00197
 65 0.00386 0.00268 0.00194
 70 0.00379 0.00263 0.00190
 75 0.00371 0.00258 0.00187
 80 0.00364 0.00253 0.00183
 85 0.00357 0.00248 0.00179
 90 0.00350 0.00243 0.00176
 95 0.00343 0.00238 0.00172
 100 0.00336 0.00233 0.00169
 110 0.00323 0.00224 0.00162
 120 0.00310 0.00215 0.00156
 130 0.00299 0.00207 0.00150
 140 0.00288 0.00200 0.00145
 150 0.00278 0.00193 0.00140
 160 0.00269 0.00187 0.00135
 170 0.00260 0.00181 0.00131
 180 0.00252 0.00175 0.00126
 190 0.00244 0.00170 0.00123
 200 0.00237 0.00165 0.00119
 250 0.00207 0.00144 0.00104
 300 0.00184 0.00128 0.000925
 350 0.00166 0.00115 0.000835
 400 0.00152 0.00105 0.000762
 450 0.00140 0.000970 0.000702
 500 0.00130 0.000900 0.000651
 600 0.00114 0.000788 0.000570
 700 0.00101 0.000703 0.000508
 800 0.000915 0.000636 0.000460
 900 0.000837 0.000581 0.000420
1000 0.000771 0.000536 0.000387
1500 0.000561 0.000390 0.000282
2000 0.000445 0.000309 0.000224
3000 0.000320 0.000222 0.000161
Table 2

Helium. Excitation energies E in eV, dipole f values from uncorrelated
wave functions ([f.sub.sc]), f values from correlated wave functions
([f.sub.accu]) by Drake (8), and BEf-scaled excitation cross sections
[[sigma].sub.BEf] in [[Angstrom].sup.2] as functions of incident
electron energy T in eV. The initial state is [1s.sup.2] [S.sup.1].
The experimental ionization energy B = 24.5874 eV has been used in the
scaling. The constants a, b, and c of Eq. (5) are included

Final state 1s2p [P.sup.1] 1s3p [P.sup.1] 1s4p [P.sup.1]

E 21.218 23.087 23.742
[f.sub.sc] 0.2583 0.07061 0.02899
[f.sub.accu] 0.2762 0.07343 0.02986
Const. a 0.165601 0.041611 0.016111
Const. b -0.076942 -0.018087 -0.007040
Const. c 0.033306 0.002104 -0.000045

Final state 1s5p [P.sup.1] 1s6p [P.sup.1] 1s7p [P.sup.1]

E 24.046 24.211 24.311
[f.sub.sc] 0.01466 0.00844 0.00529
[f.sub.accu] 0.01504 0.00863 0.00541
Const. a 0.008298 0.004740 0.002963
Const. b -0.003475 -0.001972 -0.001227
Const. c -0.000228 -0.000194 -0.000146

Final state 1s8p [P.sup.1] 1s9p [P.sup.1] 1s10p [P.sup.1]

E 24.375 24.420 24.452
[f.sub.sc] 0.00354 0.00248 0.00181
[f.sub.accu] 0.00361 0.00253 0.00184
Const. a 0.001975 0.001383 0.001006
Const. b -0.000816 -0.000570 -0.000414
Const. c -0.000108 -0.000080 -0.000061
 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]

 23 0.01939
 24 0.02474 0.00337
 25 0.02933 0.00498 0.00159
 26 0.03343 0.00625 0.00216
 27 0.03716 0.00735 0.00264
 28 0.04058 0.00832 0.00305
 29 0.04375 0.00921 0.00342
 30 0.04669 0.01002 0.00376
 35 0.05875 0.01329 0.00510
 40 0.06757 0.01565 0.00607
 45 0.07413 0.01740 0.00678
 50 0.07903 0.01871 0.00731
 60 0.08542 0.02043 0.00802
 70 0.08883 0.02138 0.00841
 80 0.09043 0.02185 0.00861
 90 0.09088 0.02202 0.00868
 100 0.09060 0.02199 0.00868
 110 0.08983 0.02184 0.00862
 120 0.08876 0.02161 0.00853
 130 0.08748 0.02132 0.00842
 140 0.08609 0.02100 0.00830
 150 0.08462 0.02066 0.00817
 160 0.08313 0.02030 0.00803
 170 0.08162 0.01995 0.00789
 180 0.08012 0.01959 0.00775
 190 0.07864 0.01923 0.00761
 200 0.07718 0.01888 0.00747
 225 0.07370 0.01805 0.00714
 250 0.07046 0.01726 0.00683
 275 0.06747 0.01654 0.00655
 300 0.06471 0.01587 0.00628
 350 0.05982 0.01468 0.00581
 400 0.05564 0.01365 0.00541
 450 0.05203 0.01277 0.00506
 500 0.04890 0.01201 0.00476
 600 0.04372 0.01074 0.00425
 700 0.03962 0.00973 0.00386
 800 0.03628 0.00891 0.00353
 900 0.03350 0.00823 0.00326
1000 0.03116 0.00766 0.00303
1500 0.02333 0.00573 0.00227
2000 0.01885 0.00463 0.00183
3000 0.01383 0.00340 0.00135

 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]

 23
 24
 25 0.000684 0.000354 0.000206
 26 0.000997 0.000543 0.000329
 27 0.00125 0.000688 0.000421
 28 0.00146 0.000812 0.000500
 29 0.00165 0.000922 0.000569
 30 0.00182 0.00102 0.000631
 35 0.00250 0.00141 0.000878
 40 0.00299 0.00169 0.00105
 45 0.00335 0.00190 0.00118
 50 0.00362 0.00206 0.00128
 60 0.00397 0.00226 0.00141
 70 0.00417 0.00237 0.00148
 80 0.00427 0.00243 0.00152
 90 0.00431 0.00245 0.00153
 100 0.00431 0.00245 0.00153
 110 0.00428 0.00244 0.00152
 120 0.00424 0.00242 0.00151
 130 0.00419 0.00239 0.00149
 140 0.00413 0.00235 0.00147
 150 0.00406 0.00231 0.00144
 160 0.00399 0.00227 0.00142
 170 0.00392 0.00224 0.00139
 180 0.00385 0.00220 0.00137
 190 0.00378 0.00216 0.00135
 200 0.00372 0.00212 0.00132
 225 0.00355 0.00202 0.00126
 250 0.00340 0.00194 0.00121
 275 0.00326 0.00186 0.00116
 300 0.00313 0.00178 0.00111
 350 0.00289 0.00165 0.00103
 400 0.00269 0.00153 0.000957
 450 0.00252 0.00144 0.000896
 500 0.00237 0.00135 0.000842
 600 0.00212 0.00121 0.000753
 700 0.00192 0.00109 0.000683
 800 0.00176 0.00100 0.000625
 900 0.00162 0.000925 0.000578
1000 0.00151 0.000861 0.000537
1500 0.00113 0.000645 0.000402
2000 0.000913 0.000521 0.000325
3000 0.000670 0.000382 0.000238

 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]

 23
 24
 25 0.000130 0.0000879 0.0000621
 26 0.000214 0.000148 0.000106
 27 0.000277 0.000192 0.000139
 28 0.000330 0.000229 0.000166
 29 0.000376 0.000262 0.000190
 30 0.000418 0.000291 0.000211
 35 0.000583 0.000407 0.000295
 40 0.000700 0.000489 0.000355
 45 0.000787 0.000550 0.000399
 50 0.000851 0.000595 0.000432
 60 0.000937 0.000655 0.000476
 70 0.000985 0.000689 0.000501
 80 0.00101 0.000706 0.000513
 90 0.00102 0.000713 0.000518
 100 0.00102 0.000713 0.000518
 110 0.00101 0.000709 0.000515
 120 0.00100 0.000702 0.000510
 130 0.000991 0.000693 0.000504
 140 0.000976 0.000683 0.000497
 150 0.000961 0.000672 0.000489
 160 0.000945 0.000661 0.000481
 170 0.000929 0.000650 0.000472
 180 0.000912 0.000638 0.000464
 190 0.000896 0.000627 0.000456
 200 0.000880 0.000616 0.000448
 225 0.000841 0.000589 0.000428
 250 0.000805 0.000563 0.000410
 275 0.000772 0.000540 0.000393
 300 0.000740 0.000518 0.000377
 350 0.000685 0.000479 0.000349
 400 0.000638 0.000446 0.000324
 450 0.000597 0.000417 0.000304
 500 0.000561 0.000392 0.000285
 600 0.000502 0.000351 0.000255
 700 0.000455 0.000318 0.000231
 800 0.000416 0.000291 0.000212
 900 0.000385 0.000269 0.000196
1000 0.000358 0.000250 0.000182
1500 0.000268 0.000187 0.000136
2000 0.000216 0.000151 0.000110
3000 0.000159 0.000111 0.0000807
Table 3

Lithium. Excitation energies E in eV, dipole f values calculated from
uncorrelated wave functions ([f.sub.sc]), f values calculated from
correlated wave functions ([f.sub.accu]) as explained in the text, and
BEf-scaled excitation cross sections [[sigma].sub.BEf] in
[[Angstrom].sup.2] as functions of incident electron energy T in eV.
The experimental ionization energy B = 5.3917 eV has been used in the
scaling. The constants a, b, and c of Eq. (5) are included


Excitation 2s-2p 2s-3p 2s-4p 2s-5p 2s-6p

E 1.848 3.834 4.522 4.837 5.008
[f.sub.sc] 0.7685 0.00340 0.00353 0.00217 0.00135
[f.sub.accu] 0.7468 0.00483 0.00428 0.00260 0.00158
Const. a 5.658148 0.012056 0.010631 0.006113 0.003666
Const. b 17.288057 0.219978 0.047005 0.018340 0.009244
Const. c -0.226058 0.0.22768 0.011300 0.005517 0.003062


Excitation 2s-7p 2s-8p 2s-9p 2s-10

E 5.110 5.177 5.222 5.254
[f.sub.sc] 0.000880 0.000601 0.000427 0.000314
[f.sub.accu] 0.00101 0.000683 0.000482 0.000353
Const. a 0.002342 0.001580 0.001113 0.000813
Const. b 0.005372 0.003419 0.002319 0.001650
Const. c 0.001872 0.001229 0.000850 0.000613
 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]
 2 14.59367
 2.5 27.24288
 3 33.14707
 3.5 36.69134
 4 38.97838 0.53291
 4.5 40.48711 0.87436
 5 41.47440 0.97533 0.15740
 5.5 42.09531 1.00527 0.18668
 6 42.45067 1.00592 0.19520
 8 42.33337 0.92889 0.18523
 10 41.07186 0.83871 0.16733
 15 36.87740 0.66670 0.13449
 20 32.98349 0.55296 0.11354
 25 29.73768 0.47298 0.09886
 30 27.06106 0.41366 0.08789
 35 24.83566 0.36787 0.07932
 40 22.96265 0.33143 0.07241
 45 21.36655 0.30171 0.06671
 50 19.99071 0.27700 0.06190
 60 17.73927 0.23823 0.05425
 70 15.97298 0.20918 0.04839
 80 14.54840 0.18657 0.04375
 90 13.37361 0.16847 0.03998
 100 12.38704 0.15364 0.03684
 110 11.54596 0.14126 0.03420
 120 10.81975 0.13076 0.03193
 130 10.18588 0.12175 0.02996
 140 9.62741 0.11393 0.02823
 150 9.13135 0.10707 0.02671
 160 8.68756 0.10101 0.02535
 170 8.28799 0.09561 0.02413
 180 7.92620 0.09077 0.02303
 190 7.59694 0.08641 0.02203
 200 7.29592 0.08246 0.02112
 225 6.64488 0.07403 0.01917
 250 6.10761 0.06719 0.01756
 275 5.65611 0.06153 0.01621
 300 5.27096 0.05677 0.01507
 350 4.64769 0.04920 0.01323
 400 3.77721 0.03892 0.01068
 450 3.77721 0.03892 0.01068
 500 3.46013 0.03526 0.00976
 600 2.97030 0.02972 0.00834
 700 2.60850 0.02570 0.00730
 800 2.32962 0.02267 0.00650
 900 2.10765 0.02028 0.00587
1000 1.92652 0.01836 0.00535
1500 1.36023 0.01251 0.00375
2000 1.06067 0.00953 0.00291
3000 0.74561 0.00649 0.00203


 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]
 2
 2.5
 3
 3.5
 4
 4.5
 5 0.04132
 5.5 0.06775 0.03110 0.01653
 6 0.07506 0.03657 0.02057
 8 0.07364 0.03691 0.02123
 10 0.06671 0.03351 0.01930
 15 0.05414 0.02736 0.01582
 20 0.04625 0.02354 0.01367
 25 0.04071 0.02085 0.01215
 30 0.03654 0.01881 0.01099
 35 0.03324 0.10719 0.01007
 40 0.03056 0.01586 0.00932
 45 0.02832 0.01475 0.00868
 50 0.02643 0.01380 0.00813
 60 0.02337 0.01226 0.00725
 70 0.02100 0.01106 0.00655
 80 0.01911 0.01009 0.00599
 90 0.01755 0.00930 0.00553
 100 0.01625 0.00863 0.00513
 110 0.01515 0.00806 0.00480
 120 0.01419 0.00756 0.00451
 130 0.01336 0.00713 0.00426
 140 0.01263 0.00675 0.00403
 150 0.01198 0.00641 0.00383
 160 0.01140 0.00611 0.00365
 170 0.01087 0.00583 0.00349
 180 0.01040 0.00558 0.00334
 190 0.00997 0.00536 0.00321
 200 0.00958 0.00515 0.00309
 225 0.00872 0.00470 0.00282
 250 0.00802 0.00433 0.00260
 275 0.00743 0.00402 0.00241
 300 0.00693 0.00375 0.00225
 350 0.00611 0.00331 0.00200
 400 0.00497 0.00270 0.00163
 450 0.00497 0.00270 0.00163
 500 0.00455 0.00248 0.00150
 600 0.00391 0.00213 0.00129
 700 0.00344 0.00188 0.00114
 800 0.00307 0.00168 0.00102
 900 0.00278 0.00152 0.000923
1000 0.00254 0.00139 0.000845
1500 0.00179 0.000988 0.000600
2000 0.00140 0.000773 0.000470
3000 0.000984 0.000545 0.000332


 T [[sigma].sub.BEf] [[sigma].sub.BEf] [[sigma].sub.BEf]
 2
 2.5
 3
 3.5
 4
 4.5
 5
 5.5 0.00976 0.00622 0.00421
 6 0.01277 0.00851 0.00598
 8 0.01341 0.00905 0.00643
 10 0.01221 0.00825 0.00586
 15 0.01003 0.00679 0.00483
 20 0.00869 0.00589 0.00420
 25 0.00775 0.00526 0.00375
 30 0.00702 0.00478 0.00341
 35 0.00645 0.00439 0.00314
 40 0.00597 0.00407 0.00291
 45 0.00557 0.00380 0.00272
 50 0.00522 0.00357 0.00255
 60 0.00466 0.00319 0.00228
 70 0.00422 0.00289 0.00207
 80 0.00386 0.00264 0.00190
 90 0.00357 0.00244 0.00175
 100 0.00332 0.00227 0.00163
 110 0.00310 0.00213 0.00153
 120 0.00292 0.00200 0.00144
 130 0.00276 0.00189 0.00136
 140 0.00261 0.00179 0.00129
 150 0.00248 0.00170 0.00123
 160 0.00237 0.00163 0.00117
 170 0.00226 0.00155 0.00112
 180 0.00217 0.00149 0.00107
 190 0.00208 0.00143 0.00103
 200 0.00200 0.00138 0.000991
 225 0.00183 0.00126 0.000907
 250 0.00169 0.00116 0.000837
 275 0.00157 0.00108 0.000778
 300 0.00147 0.00101 0.000727
 350 0.00130 0.00894 0.000644
 400 0.00106 0.000732 0.000580
 450 0.00106 0.000732 0.000528
 500 0.000976 0.000673 0.000485
 600 0.000842 0.000580 0.000419
 700 0.000742 0.000512 0.000369
 800 0.000664 0.000458 0.000331
 900 0.000603 0.000416 0.000300
1000 0.000522 0.000381 0.000275
1500 0.000393 0.000271 0.000196
2000 0.000308 0.000213 0.000154
3000 0.000218 0.000151 0.000109


Acknowledgements

We gratefully acknowledge partial financial support by the Office of Fusion Energy Sciences of the U.S. Department of Energy.

Accepted: June 3, 2002

Available online: http://www.nist.gov/jres

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(12.) H. Merabet, M. Bailey, R. Bruch, J. Hanni, S. Bliman, D. V. Fursa, I. Bray, K. Bartschat, H. C. Tang and C. D. Lin, Phys. Rev. A 64, 012712 (2001).

(13.) F. J. de Heer, International Nuclear Data Committee Report IDC(NDS)-385, Vienna, Austria (1998). Unpublished--Recommended values.

(14.) W. B. Westerveld, H. G. M. Heideman, and J. van Eck, J. Phys. B 12, 115 (1979).

(15.) D. E. Shemansky, J. M. Ajello, D. T. Hall, and B. Franklin, Astrophy. J. 296, 774 (1985).

(16.) S. Trajmar, J.M. Ratliff, G. Csanak, and D. C. Cartwright, Z. Phys. D 22, 457 (1992).

(17.) D. C. Cartwright, G. Csanak, S. Trajmar, and D. F. Register, Phys. Rev. A 45, 1602 (1992).

(18.) D. Leep and A. Gallagher, Phys. Rev. A 10, 1082 (1974).

(19.) W. Williams, S. Trajmar, and D. Bozinis, J. Phys. B 9, 1529 (1976).

(20.) L. Vuskovic, S. Trajmar, and D. F. Register, J. Phys. B 15, 2517 (1982).

(21.) I. Bray, D. Fursa, and I. E. McCarthy, Phys. Rev. A 47, 1101 (1993).

(22.) J. Schweinzer, R. Brandenburg, I. Bray, R. Hoekstra, F. Aumayer, R. K. Janev, and H. P. Winter, At. Data Nucl, Data Tables 72, 239 (1999).

About the authors: Philip Stone is a guest researcher and Yong-Ki Kim is a physicist, both in the Atomic Physics Division of the NIST Physics Laboratory. Jean-Paul Desclaux has recently retired from the French Atomic Energy Commission. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce.
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Author:Desclaux, J.P.
Publication:Journal of Research of the National Institute of Standards and Technology
Geographic Code:1USA
Date:Jul 1, 2002
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