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Convergent calculations for electron impact broadening and shift of neutral helium lines
Z Quant. Spectrosc. Radiat. Transfer Vol. 28, No. 2, pp. 75-80, 1982 Printed in Great Britain.
CONVERGENT BROADENING
0022--40731821080075-06503.0010 © 1982Pergamon Press Ltd.
CALCULATIONS FOR ELECTRON IMPACT AND SHIFT OF NEUTRAL HELIUM LINES J. M. BASSALO
Departamento de Ffsica, Universidade Federal do Par:i, ParL Brazil
and M . CATI'ANI
and V. S.
WALDER
Instituto de Fisica, Universidade de Silo Paulo, Caixa Postal 20516, Silo Paulo, Brazil
(Received 22 September 1981) Abstract--Using the convergent semiclassical method, we have calculated the electronic widths and shifts of 42 neutral He lines for an electronic density of 10~6/cm3 and T = 5000, 10,000, 20,000, and 40,000 K. To account of ions effects, we have calculated the Stark parameter A and parameter R.
1. INTRODUCTION
The broadening and shift of spectral lines produced by electronic collisions can be estimated by using a semiclassical formalism with convenient approximations to determine the effects of strong collisions, 1'2 for which perturbation theory breaks down. There are two slightly different semiclassical impact theories I to evaluate the effects of electronic collisions in plasmas: an impact-parameter cutoff theory and a convergent theory. The cutoff theory has been adopted by , many authors 1 and has been used extensively by Griem and collaborators t'3'4 and by SahalBrechot 5-7 to study the line shapes of neutral atoms and ions. The convergent theory, developed by Vainshtein and Sobel'mana in a two-level approximation, was applied with some modifications by Dyne and O'Mara9 to calculate the widths and shifts of neutral He lines. For He(I) lines, it is necessary, in principle, to use a many-levelt'x5 instead of a two-level approximation, as is done in Ref. 2. We have verified2 that the cutoff and convergent methods are about equally successful in describing the experimental results. Within the experimental errors, the agreement between theoretical and experimental results is good for the widths and reasonably good for the shifts./° The calculations have been performed with and without Debye shielding and it has been verified, according to Griem, ! that somewhat better agreement with the experimental results is found if the Debye screening effects are neglected. In our preceding paper, 2 we have not analysed the validity of the straight-path approximation for the electron trajectory to evaluate the widths and shifts of He lines. We verified that, within the framework of the convergent approach used in the cutoff formalism,l'u it is a very good approximation since deviations of the straight-line trajectory are of minor importance and can be neglected. We have now calculated, with the convergent method, the electronic widths toe and de of 42 neutral He lines, for an electron density of N = 10~6/cm3 at temperatures T = 5000, 10,000, 20,000, and 40,000 K. For these conditions, we have also calculated the Stark parameter A and the parameter R defined by Griem) Our results for toe, de, A, and R are listed in the next section. 2. CALCULATION OF
fJ0e
AND d, FOR He(I) LINES
According to the convergent method in a many-level approximation,2 the half half-width ~, and the shift de caused by electron collisions for an isolated Lorentzian line (measured in Hertz) are given by
toe = N fo dr vF(v)
db b{1 - cos[~b1~(b, v)] × exp[- Flu(b, v)/21} 75
(1)
76
J. M, BASSAL~)et
al.
and
de = - N
dv vF(v)
Jo
db b sin[~t~(b, v)] x exp[-FiF(b, v)/2],
(2)
where the indices I and F refer to the initial and final states of the line, respectively, N is the density of the perturbing electrons, v the electron velocity, F(v) the Maxwell-Boltzmann velocity distribution, b the impact parameter, FIF(b, v)= Ft(b, v)+ Fv(b, v), and ~n:(b, v)= ~f(b, v) - ~bF(b, v). The functions Fr(b, v) and ~br(b, v), with K = I or F, have been calculated by assuming that the electron describes a straight path with constant velocity during the collision and that its interaction with this atom is dipolar; these are shown in our preceding paper. 2 As is easily verified, the contributions of the quadrupole interaction to the broadening and shift of He(I) lines are negligible. The He(I) states are indicated by InK'~lKmr), where a = 1 for para-helium and a = 3 for ortho-helium; the final state F has been chosen as the lowest energy level. In all cases, we have taken into account the contributions of both initial and final states to the widths and shifts. In some cases, since the lowest energy levels are much less polarizable than the initial states, their contributions to the widths and shifts could be neglected compared with those of the initial states. The reduced dipole matrix elements for n _<-4 have been calculated by using the oscillator strength of Wiese et al. j2 For n _->5, we have used hydrogen-like wavefunctions with principal quantum numbers adjusted to give the measured bound states energies. The energy differences htonol,nol between the states Inal) and [n~l') are given by Moore. 13 Using Eqs. (1)-(4), we have calculated the electronic half half-widths toe and shifts de. We have also calculated the Stark parameter A and the parameter R defined by 1 A
=
(47r/3)(C4/toe)3/4N I/4
and R = 61/3"n"I/6(e2/KT)I/2N
I/6.
These parameters have been introduced by Griem et al. 1"3to obtain the contributions of the ions to the widths and shifts of the lines. The total half half-widths to and shift d, due to electrons and ions, are given by to = toe[1 + 1.75A(1 - 0.75R)] and d = de -+2.0A(1 - 0.75R)toe. Our results for toe and de (measured in/~) and A are shown in Table 1. The R values for N = 10~6/cm3 and T = 5000, 10,000, 20,000, and 40,000 K are 0.5893, 0.4167, 0.2946, and 0.2083, respectively. The electron impact widths and shifts are linear in N, A scales as N TM, and R as N 1/6. The dependence of toe and de and, consequently, of A on temperature is not straightforward and an interpolation is necessary to obtain values for temperatures between 5000 and 40,000 K, which are not listed in Table 1. Comparing our electronic widths and shifts (see Table 1) with the corresponding values of Griem, 1 we see that our widths are smaller. These characteristics can be understood in view of Fig. 3 of Dyne and O'Mara, 9 who show the differences between the cutoff and convergent methods in calculating the widths and shifts of the lines.
Convergent calculations for electron impact broadening
77
Table 1. Electron impact widths (toe), shifts (de), and the Stark parameter A for He(I) for N = 1016/cm3 and T = 5000 10,000, 20,000, and 40,000 K.
L(.~) 584
11s ÷ 21p
537
11s ÷
31p
522
11s + 41p
20581
r (z)
21s ~ 31p 3965
21s + 41p 7281
21p ÷ 31s 5048
-0.1371E-03
0.1290E-03
-0.1328E-03
0.0112
20000
0.1472E-03
-0.1202E~03
0.0102
40000
0.1709E-03
-0.1019E-03
0.0091
5000
0.4066E-02
-0.2887E-02
0.1604
10000
0.3863E-02
-0.2383E-02
0.1667
20000
-0.1904E-02
0.1777
40000
0.3548E-02 0.3169E-02
-0.1474E-02
0.1934
5000
0.1624E-01
-0.1089E-01
0.2965
10000
0.1519E-01
-0.8826E-02
0.3118
20000
0.1373E-01
-0.6924E-02
0.3363
40000
0.1205E-01
-0.5269E-02
0.3710
5000
0.3066E+00 0.3724E+00 0.4524E+00 0.5427E+00
-0.4131E+00
0.0428
40000
21p -* 51s 6678
21p ÷ 31d 4922
21p ÷ 41d 4388
21p ÷ 51d
-0.4121E+00
0.0370
-0.3812E+00
0.0320
-0.3258E+00
0.0279
5000
0.3562E+00
-0.2587E+00
0.1610
10000
0.3394E+00
-0.2168E+00
0.1670
20000
0.3138E+00
-0.1770E+00
0.1771
40000
0.2832E+00
-0.1406E+00
0.1913
5000
0.9373E+00
-0.6303E+00
0.2958
10000
0.8769E+00
-0.5122E+00
0.3110
20000
0.7932E+00
-0.4035E+00
0.3353
40000
0.6968E+00
-0.3090E+00
0.3695
5000
0.2754E+00
0.3595E+00
0.0905
10000
0.3178E+00
0.3546E+00
0.0813
20000
0.3519E+00
0.3278E+00
0.0753
40000
0.3698E+00
0.2838E+00
0.0725
5000
0.5347E+00 0.6064E+00
0.6515E+00
0.1517
0.6235E+00
0.1380
0.5578E+00
0.1306
40000
0.6528E+00 0.6622E+00
0.4684E+00
0.1292
20000
4438
0.0126
0.1108E-03
10000
21p + 4 1 s
A
5000
20000
5016
d e (~)
10000
10000 21s + 21p
toe ( ~ )
5000
0.1265E+01
0.1460E+01
0.2171
10000
0.1406E+01
0.1371E+01
0.1995
20000
0.1488E+01
0.1203E+01
0.1912
40000
0.1481E+01
0.9927E+00
0.1918
5000
0.3954E+00
O.2332E+00
0.1539
10000
0.3576E+00
0.1801E+00
0.1659
20000
0.3128E+00
0.1362E+00
0.1835
40000
0.2683E+00
0.1014E+00
0.2059
5000
0.2254E+01
0.8776E+00
0.6947
10000
0.1923E+01
0.6398E+00
0.7827
20000
0.1598E+01
0.4526E+00
0.8991
40000
0.1300E+01
0.3120E+00
1.0497 1.2667
5000
0.5443E+01
0.1915E+01
10000
0.4620E+01
0.1350E+01
1.4325
20000
0.3810E+01
0.9213E+00
1.6552
40000
0.3069E+01
0.6133E+00
1.9468
J . M . BASSALO et aL
78
Table 1 (Contd) L([~) 15084
31s + 41p 11013
31s + 51p 9603
31s + 61p 19089
31p ÷ 41d 12968
31p ~ 51d 11045
31p ~ 61d 18697
31d + 41f 10830
23s ~ 23p 3889
23s + 33p 3188
23s + 43p 2945
23s + 53p 2829
23s ÷ 63p 7065
23p ~ 33s
T (K)
I
~e ( ~ )
d e (~)
5000
0.1375E+02
-0.9741E+01
0.2973
10000
0.1300E+02
-O.8162E+01
0.3~O1
20000
0.1196E+02
-0.6658E+01
0,3301
40000
0.IO73E+O2
-0.5272E+01
0,3581
5000
O.2121E+02
-0.134OE+02
O.4571
10000
0.197OE+O2
-0.1091E+02
0.4830
20000
0.1773E+O2
-0,8620E+01
0.5227
40000
0.1552E+O2
-0,6613E+01
0.5775
-O.2138E+02
0.6497
5000
0.3777E+02
10000
0.3471E+02
-0,1707E+02
{).6922
20000
0.3086E+02
-O.1321E+O2
O. 7560
40000
0.2666E+O2
-O,9903E+01
0.8436
0.6707
5000
0.3630E+O2
O.1588E+02
10000
O.3156E+02
O.12OOE+02
O.7449
20000
O.2675E+O2
O.8817E+01
0.8434
40000
0.2217E+02
O.6311E+01
~.9709 1,2556
5000
0.4826E+02
0.1774E+02
I0000
0.4122E+O2
O.1273E+02
1,4132
20000
0.3423E+02
0.8873E+01
1.6240
40000
0.2777E+02
0.6044E+01
1,9006
5000
0.8128E+02
O.2582E÷02
,.9498
10000
0.6844E+02
O.1763E+02
*.2182 2 .5744
20000
O.5611E+02
0.1178E+02
40000
O.4502E+02
O.7753E+01
3 .0369
5000
0.1873E+02
-O.5803E+O1
0.7524
10000
O.1555E+02
-0,4125E+01
O.8651
20000
0.1266E+O2
-0.2834E÷01
~.O094
40000
0,IO13E+02
-0.1882E+01
!.1928 O.0321
5000
0.4046E-01
-0.5627E-01
10000
O.5128E-01
-0,5718E-01
0.0269
20000
0.6718E-01
-0.5449E-O1
6,O219
40000
O.8896E-01
-O.4849E-01
O.0178
0.0848
5000
0.8537E-01
0.5968E-O1
10000
0.9425E-01
0.4610E-01
0,0788
20000
0.9863E-01
0,3230E-O1
0,O76~
40000
0.9836E-01
0.2075E-O!
0.0763
5000
0.2672E+00
0.1983E+00
0,1511
10000
0.2891E+OO
0.1620E+00
O~!424
20000
0.2959E+00
O.1261E+00
0,!400
40000
0.2871E+00
0.9512E-01
O.1431
5000
0.6823E+00
O.4261E+00
10000
(].7357E÷00
0.3268E+0(]
20000
0.7458E+00
(I.2355E+OO
40000
0.7139E+00
0.163OE+00
I
5000
0.1563E+01
0.9212E+00
10000
O.1649E+O1
O.6955E+O0
20000
0.1639E+01
0.4868E+00
0.
40000
O.1542E+01
O.3433E+0!)
0.3188
,
(). 3032 3046
5000
0,1532E+00
0.2 I7OE+OO
0 . 0758
10000
0.1798E+00
{I. 2 2 2 2
; , 0B ] ?
20000
0.2055E+OO
O ,2144E+I)i
40000
0.2253E+00
i). 1935E+00
E ÷ (}(I
i,05~7
79
Convergent calculations for electron impact broadening Table i (Contd) L(~)
4713
23p .
43s
4121
23p + 53s
5876
T (K)
4472
23p ÷ 43d 4026
23p ÷ 53d 12528
33S ÷ 43p 9464
33s ~ 53p
5000
0.2923E+00
0.3904E+00
0.1284
0.3391E+00
0.3888E+00
0.1149
20000
0.3780E+00
0.3630E+00
0.1059
40000
0.3989E+O0
0.3170E+00
0.1017
5000 10000 20000
0.7728E+00
0.9994E+00
0.1726
0.8905E+00
0.9817E+00
0.9807E+00
0.9020E+00
40000
0.I018E+01
0.7755E+00
0.1552 0.1444 0.1404
5000
0.1343E+00
-0.1157E+00
0.0722
10000
-0.9678E-01
0.0709
-0.7615Er01 -0.5715E-01
0.0725
40000
0.1377E+00 0.1338E+00 0.1253E+00
5000
0.1360E+01
0.7900E-01 0.6972E-02 -0.2754E-O1 -0.4055E-01
0.6144 0.6802 0.7653 0.8746
0.2518E÷00 0.4626E-01 -0.6152E-01 -0.1074E+00
1.0564
10000
0.1188E+0t
20000
0.1015E+01
40000
0.8495E÷OO
5000
0.3605E+01
10000
0.3129E+01
20000
0.2649E+01
40000
0.2187E+01
5000
o.411oz+oi
10000
0.4495E÷01
20000
0.4657E+O1
40000
o.4585z+oi
0.7501E+01
5000
0.8007E+01 0.6021E+01
0.3026
0.4266E+01
0.3038
0.2904E+01
0.3177
0.6114E+01
0.1009
0.6236E+01 0.6011E+01 0.5416E+01
0.0774
0.6076E+01 0.7482E+01 0.8654E+01 0.9331E+01
33p + 53d
0.2199 0.3151
0.0864 0.0732
5000
0.7423E+01
0.9304E+01
0.1682
0.8674E+01 0.9679E+01 0.1016E+02
0.9117E+01
0.1496
O.8383E+01
0.1378
0.7229E+01
0.1329
0.1528E+02
20000
0.1243E+02 0.1422E+02 0.1549E+02
40000
0.1587E+02
0.1665E+02
0.2454 0.2217 0.2080 0.2042
40000
5000 10000
11969
0.2153 0.2132
10000 20000
33p * 43d
0.2274
0.1337 0.1353
40000
5000 10000 20000 40000
17002
0.4400E+01 0.3319E+01 0.2324E+01 0.1539E+01
0.1373
0.7818E÷01
21120
33p + 63s
0.]468
20000
40000
10668
0.2645E÷01 0.1980E÷01 0.1333E+01 0.8161E+00
0.7716E+01
20000
33p * 53s
1.5370
0.7176E÷01
33s * 63p
12846
1.1749 1.3312
5000
10000
33p + 43s
0.0761
10000
0.1365E+02 0.1441E+02 0.1433E+02 0.1350E+02
8362
o de(A)
10000
20000
23p ~ 33d
~e(~)
0.1472E+02 0.1325E+02
5000
0.2007E+02
0.4487E+00
0.6020
10000
0.1778E+02
-0.5750E+00
0.6593
20000
0.1548E+02
-0.9799E+00
0.7313
40000
0.1323E+02
-0.I040E+01
0.8228
0.1987E+01 0.1455E+00
1.0529
-0.7953E+00 -0.1160E+01
1.3191
5000
0.3195E+02
10000
0.2781E+02
20000
0.2366E+02
40000
0.1965E+02
1.1683 1.5161
80
J, M. BASSAU) et aL
Table 1 (Contd)
L(~) 19543
33d ~ 43p
12985
33d + 53p
10997
33d -~ 63p 18686
33d + 43f
T (K)
We ( ,~ )
de(X)
5000
0.I034E+02
0.7777E+01
0.1546
10000
0.1126E+02
0.63,17E+01
0.1451
20000
0.1157E+02
0.4818E+01
0.1422
40000
0.1123E+02
0.3518E+01
0.1454
5000
0.1357E+02
0.8659E+01
0.2295
I0000
0.1460E+02
0.6698E+O1
0.2172
20000
0.1480E+02
0.4883E+O1
0.2151
40000
0.1418E+02
0.3430E+01
0.2221
0.3162
5000
0.2363E+02
0.1403E+02
10000
0.2496E+02
O.I064E+02
0.3035
20000
0.2483E+02
O.7652E+01
0.3046
40000
0.2340E+02
0.5329E+01
0.3186
5000
0.1102E+02
-0.1924E+01
0.8650
10000
0.9227E+01
-0.1120E+01
0.9884
20000
0.7622E+01
-0.609OE+00
1.1408
40000
0.6219E+01
-0.2951E+00
1.3289
REFERENCES H. R. Griem, Plasma Spectroscopy. Academic Press, New York (1974). J. M. Bassalo, M. Cattani, and V. S. Walder, Phys. Rev. A 22, 1194 (1980). H. R. Griem, M. Baranger, A. C. Kolb, and G. Oertel, Phys. Rev. 1.7,5, 117 (1%2). H. R. Griem, Phys. Rev. 128, 515 (1%2). S. Sahal-Brechot, Astron. Astrophys. I, 91 (1969). S. Sahal-Brechot, Astron. Astrophys. 2, 322 (1%9). C. Fleurier, S. Sahal-Brechot, and J. Chapelle, JQSRT 17, 595 (1977). L. A. Vainshtein and 1. I. Sobel'man, Opt. Spectrosc. 5, 279 (1959). R. J. Dyne and B. J. O'Mara, Astron. Astrophys. 18, 363 (1972). H. R. Griem and C. S. Shen, Phys. Rev. 145, 1% (1%2). M. S. Dimitrijevic and P. Grujic, JQSRT 19, 407 (1978). W. L Wiese, M. W. Smith, and B. M. Glennon, "Atomic Transition Probabilities", U.S. National Bureau of Standards, National Standard Reference Data Series-4. U.S. GPO, Washington D.C. (1966). 13. C. E. Moore, "Atomic Energy Levels". U.S. National Bureau of Standards Circ. Vol. I. No. 467. U.S. GPO, Washington D.C. (1949). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
CONVERGENT BROADENING
0022--40731821080075-06503.0010 © 1982Pergamon Press Ltd.
CALCULATIONS FOR ELECTRON IMPACT AND SHIFT OF NEUTRAL HELIUM LINES J. M. BASSALO
Departamento de Ffsica, Universidade Federal do Par:i, ParL Brazil
and M . CATI'ANI
and V. S.
WALDER
Instituto de Fisica, Universidade de Silo Paulo, Caixa Postal 20516, Silo Paulo, Brazil
(Received 22 September 1981) Abstract--Using the convergent semiclassical method, we have calculated the electronic widths and shifts of 42 neutral He lines for an electronic density of 10~6/cm3 and T = 5000, 10,000, 20,000, and 40,000 K. To account of ions effects, we have calculated the Stark parameter A and parameter R.
1. INTRODUCTION
The broadening and shift of spectral lines produced by electronic collisions can be estimated by using a semiclassical formalism with convenient approximations to determine the effects of strong collisions, 1'2 for which perturbation theory breaks down. There are two slightly different semiclassical impact theories I to evaluate the effects of electronic collisions in plasmas: an impact-parameter cutoff theory and a convergent theory. The cutoff theory has been adopted by , many authors 1 and has been used extensively by Griem and collaborators t'3'4 and by SahalBrechot 5-7 to study the line shapes of neutral atoms and ions. The convergent theory, developed by Vainshtein and Sobel'mana in a two-level approximation, was applied with some modifications by Dyne and O'Mara9 to calculate the widths and shifts of neutral He lines. For He(I) lines, it is necessary, in principle, to use a many-levelt'x5 instead of a two-level approximation, as is done in Ref. 2. We have verified2 that the cutoff and convergent methods are about equally successful in describing the experimental results. Within the experimental errors, the agreement between theoretical and experimental results is good for the widths and reasonably good for the shifts./° The calculations have been performed with and without Debye shielding and it has been verified, according to Griem, ! that somewhat better agreement with the experimental results is found if the Debye screening effects are neglected. In our preceding paper, 2 we have not analysed the validity of the straight-path approximation for the electron trajectory to evaluate the widths and shifts of He lines. We verified that, within the framework of the convergent approach used in the cutoff formalism,l'u it is a very good approximation since deviations of the straight-line trajectory are of minor importance and can be neglected. We have now calculated, with the convergent method, the electronic widths toe and de of 42 neutral He lines, for an electron density of N = 10~6/cm3 at temperatures T = 5000, 10,000, 20,000, and 40,000 K. For these conditions, we have also calculated the Stark parameter A and the parameter R defined by Griem) Our results for toe, de, A, and R are listed in the next section. 2. CALCULATION OF
fJ0e
AND d, FOR He(I) LINES
According to the convergent method in a many-level approximation,2 the half half-width ~, and the shift de caused by electron collisions for an isolated Lorentzian line (measured in Hertz) are given by
toe = N fo dr vF(v)
db b{1 - cos[~b1~(b, v)] × exp[- Flu(b, v)/21} 75
(1)
76
J. M, BASSAL~)et
al.
and
de = - N
dv vF(v)
Jo
db b sin[~t~(b, v)] x exp[-FiF(b, v)/2],
(2)
where the indices I and F refer to the initial and final states of the line, respectively, N is the density of the perturbing electrons, v the electron velocity, F(v) the Maxwell-Boltzmann velocity distribution, b the impact parameter, FIF(b, v)= Ft(b, v)+ Fv(b, v), and ~n:(b, v)= ~f(b, v) - ~bF(b, v). The functions Fr(b, v) and ~br(b, v), with K = I or F, have been calculated by assuming that the electron describes a straight path with constant velocity during the collision and that its interaction with this atom is dipolar; these are shown in our preceding paper. 2 As is easily verified, the contributions of the quadrupole interaction to the broadening and shift of He(I) lines are negligible. The He(I) states are indicated by InK'~lKmr), where a = 1 for para-helium and a = 3 for ortho-helium; the final state F has been chosen as the lowest energy level. In all cases, we have taken into account the contributions of both initial and final states to the widths and shifts. In some cases, since the lowest energy levels are much less polarizable than the initial states, their contributions to the widths and shifts could be neglected compared with those of the initial states. The reduced dipole matrix elements for n _<-4 have been calculated by using the oscillator strength of Wiese et al. j2 For n _->5, we have used hydrogen-like wavefunctions with principal quantum numbers adjusted to give the measured bound states energies. The energy differences htonol,nol between the states Inal) and [n~l') are given by Moore. 13 Using Eqs. (1)-(4), we have calculated the electronic half half-widths toe and shifts de. We have also calculated the Stark parameter A and the parameter R defined by 1 A
=
(47r/3)(C4/toe)3/4N I/4
and R = 61/3"n"I/6(e2/KT)I/2N
I/6.
These parameters have been introduced by Griem et al. 1"3to obtain the contributions of the ions to the widths and shifts of the lines. The total half half-widths to and shift d, due to electrons and ions, are given by to = toe[1 + 1.75A(1 - 0.75R)] and d = de -+2.0A(1 - 0.75R)toe. Our results for toe and de (measured in/~) and A are shown in Table 1. The R values for N = 10~6/cm3 and T = 5000, 10,000, 20,000, and 40,000 K are 0.5893, 0.4167, 0.2946, and 0.2083, respectively. The electron impact widths and shifts are linear in N, A scales as N TM, and R as N 1/6. The dependence of toe and de and, consequently, of A on temperature is not straightforward and an interpolation is necessary to obtain values for temperatures between 5000 and 40,000 K, which are not listed in Table 1. Comparing our electronic widths and shifts (see Table 1) with the corresponding values of Griem, 1 we see that our widths are smaller. These characteristics can be understood in view of Fig. 3 of Dyne and O'Mara, 9 who show the differences between the cutoff and convergent methods in calculating the widths and shifts of the lines.
Convergent calculations for electron impact broadening
77
Table 1. Electron impact widths (toe), shifts (de), and the Stark parameter A for He(I) for N = 1016/cm3 and T = 5000 10,000, 20,000, and 40,000 K.
L(.~) 584
11s ÷ 21p
537
11s ÷
31p
522
11s + 41p
20581
r (z)
21s ~ 31p 3965
21s + 41p 7281
21p ÷ 31s 5048
-0.1371E-03
0.1290E-03
-0.1328E-03
0.0112
20000
0.1472E-03
-0.1202E~03
0.0102
40000
0.1709E-03
-0.1019E-03
0.0091
5000
0.4066E-02
-0.2887E-02
0.1604
10000
0.3863E-02
-0.2383E-02
0.1667
20000
-0.1904E-02
0.1777
40000
0.3548E-02 0.3169E-02
-0.1474E-02
0.1934
5000
0.1624E-01
-0.1089E-01
0.2965
10000
0.1519E-01
-0.8826E-02
0.3118
20000
0.1373E-01
-0.6924E-02
0.3363
40000
0.1205E-01
-0.5269E-02
0.3710
5000
0.3066E+00 0.3724E+00 0.4524E+00 0.5427E+00
-0.4131E+00
0.0428
40000
21p -* 51s 6678
21p ÷ 31d 4922
21p ÷ 41d 4388
21p ÷ 51d
-0.4121E+00
0.0370
-0.3812E+00
0.0320
-0.3258E+00
0.0279
5000
0.3562E+00
-0.2587E+00
0.1610
10000
0.3394E+00
-0.2168E+00
0.1670
20000
0.3138E+00
-0.1770E+00
0.1771
40000
0.2832E+00
-0.1406E+00
0.1913
5000
0.9373E+00
-0.6303E+00
0.2958
10000
0.8769E+00
-0.5122E+00
0.3110
20000
0.7932E+00
-0.4035E+00
0.3353
40000
0.6968E+00
-0.3090E+00
0.3695
5000
0.2754E+00
0.3595E+00
0.0905
10000
0.3178E+00
0.3546E+00
0.0813
20000
0.3519E+00
0.3278E+00
0.0753
40000
0.3698E+00
0.2838E+00
0.0725
5000
0.5347E+00 0.6064E+00
0.6515E+00
0.1517
0.6235E+00
0.1380
0.5578E+00
0.1306
40000
0.6528E+00 0.6622E+00
0.4684E+00
0.1292
20000
4438
0.0126
0.1108E-03
10000
21p + 4 1 s
A
5000
20000
5016
d e (~)
10000
10000 21s + 21p
toe ( ~ )
5000
0.1265E+01
0.1460E+01
0.2171
10000
0.1406E+01
0.1371E+01
0.1995
20000
0.1488E+01
0.1203E+01
0.1912
40000
0.1481E+01
0.9927E+00
0.1918
5000
0.3954E+00
O.2332E+00
0.1539
10000
0.3576E+00
0.1801E+00
0.1659
20000
0.3128E+00
0.1362E+00
0.1835
40000
0.2683E+00
0.1014E+00
0.2059
5000
0.2254E+01
0.8776E+00
0.6947
10000
0.1923E+01
0.6398E+00
0.7827
20000
0.1598E+01
0.4526E+00
0.8991
40000
0.1300E+01
0.3120E+00
1.0497 1.2667
5000
0.5443E+01
0.1915E+01
10000
0.4620E+01
0.1350E+01
1.4325
20000
0.3810E+01
0.9213E+00
1.6552
40000
0.3069E+01
0.6133E+00
1.9468
J . M . BASSALO et aL
78
Table 1 (Contd) L([~) 15084
31s + 41p 11013
31s + 51p 9603
31s + 61p 19089
31p ÷ 41d 12968
31p ~ 51d 11045
31p ~ 61d 18697
31d + 41f 10830
23s ~ 23p 3889
23s + 33p 3188
23s + 43p 2945
23s + 53p 2829
23s ÷ 63p 7065
23p ~ 33s
T (K)
I
~e ( ~ )
d e (~)
5000
0.1375E+02
-0.9741E+01
0.2973
10000
0.1300E+02
-O.8162E+01
0.3~O1
20000
0.1196E+02
-0.6658E+01
0,3301
40000
0.IO73E+O2
-0.5272E+01
0,3581
5000
O.2121E+02
-0.134OE+02
O.4571
10000
0.197OE+O2
-0.1091E+02
0.4830
20000
0.1773E+O2
-0,8620E+01
0.5227
40000
0.1552E+O2
-0,6613E+01
0.5775
-O.2138E+02
0.6497
5000
0.3777E+02
10000
0.3471E+02
-0,1707E+02
{).6922
20000
0.3086E+02
-O.1321E+O2
O. 7560
40000
0.2666E+O2
-O,9903E+01
0.8436
0.6707
5000
0.3630E+O2
O.1588E+02
10000
O.3156E+02
O.12OOE+02
O.7449
20000
O.2675E+O2
O.8817E+01
0.8434
40000
0.2217E+02
O.6311E+01
~.9709 1,2556
5000
0.4826E+02
0.1774E+02
I0000
0.4122E+O2
O.1273E+02
1,4132
20000
0.3423E+02
0.8873E+01
1.6240
40000
0.2777E+02
0.6044E+01
1,9006
5000
0.8128E+02
O.2582E÷02
,.9498
10000
0.6844E+02
O.1763E+02
*.2182 2 .5744
20000
O.5611E+02
0.1178E+02
40000
O.4502E+02
O.7753E+01
3 .0369
5000
0.1873E+02
-O.5803E+O1
0.7524
10000
O.1555E+02
-0,4125E+01
O.8651
20000
0.1266E+O2
-0.2834E÷01
~.O094
40000
0,IO13E+02
-0.1882E+01
!.1928 O.0321
5000
0.4046E-01
-0.5627E-01
10000
O.5128E-01
-0,5718E-01
0.0269
20000
0.6718E-01
-0.5449E-O1
6,O219
40000
O.8896E-01
-O.4849E-01
O.0178
0.0848
5000
0.8537E-01
0.5968E-O1
10000
0.9425E-01
0.4610E-01
0,0788
20000
0.9863E-01
0,3230E-O1
0,O76~
40000
0.9836E-01
0.2075E-O!
0.0763
5000
0.2672E+00
0.1983E+00
0,1511
10000
0.2891E+OO
0.1620E+00
O~!424
20000
0.2959E+00
O.1261E+00
0,!400
40000
0.2871E+00
0.9512E-01
O.1431
5000
0.6823E+00
O.4261E+00
10000
(].7357E÷00
0.3268E+0(]
20000
0.7458E+00
(I.2355E+OO
40000
0.7139E+00
0.163OE+00
I
5000
0.1563E+01
0.9212E+00
10000
O.1649E+O1
O.6955E+O0
20000
0.1639E+01
0.4868E+00
0.
40000
O.1542E+01
O.3433E+0!)
0.3188
,
(). 3032 3046
5000
0,1532E+00
0.2 I7OE+OO
0 . 0758
10000
0.1798E+00
{I. 2 2 2 2
; , 0B ] ?
20000
0.2055E+OO
O ,2144E+I)i
40000
0.2253E+00
i). 1935E+00
E ÷ (}(I
i,05~7
79
Convergent calculations for electron impact broadening Table i (Contd) L(~)
4713
23p .
43s
4121
23p + 53s
5876
T (K)
4472
23p ÷ 43d 4026
23p ÷ 53d 12528
33S ÷ 43p 9464
33s ~ 53p
5000
0.2923E+00
0.3904E+00
0.1284
0.3391E+00
0.3888E+00
0.1149
20000
0.3780E+00
0.3630E+00
0.1059
40000
0.3989E+O0
0.3170E+00
0.1017
5000 10000 20000
0.7728E+00
0.9994E+00
0.1726
0.8905E+00
0.9817E+00
0.9807E+00
0.9020E+00
40000
0.I018E+01
0.7755E+00
0.1552 0.1444 0.1404
5000
0.1343E+00
-0.1157E+00
0.0722
10000
-0.9678E-01
0.0709
-0.7615Er01 -0.5715E-01
0.0725
40000
0.1377E+00 0.1338E+00 0.1253E+00
5000
0.1360E+01
0.7900E-01 0.6972E-02 -0.2754E-O1 -0.4055E-01
0.6144 0.6802 0.7653 0.8746
0.2518E÷00 0.4626E-01 -0.6152E-01 -0.1074E+00
1.0564
10000
0.1188E+0t
20000
0.1015E+01
40000
0.8495E÷OO
5000
0.3605E+01
10000
0.3129E+01
20000
0.2649E+01
40000
0.2187E+01
5000
o.411oz+oi
10000
0.4495E÷01
20000
0.4657E+O1
40000
o.4585z+oi
0.7501E+01
5000
0.8007E+01 0.6021E+01
0.3026
0.4266E+01
0.3038
0.2904E+01
0.3177
0.6114E+01
0.1009
0.6236E+01 0.6011E+01 0.5416E+01
0.0774
0.6076E+01 0.7482E+01 0.8654E+01 0.9331E+01
33p + 53d
0.2199 0.3151
0.0864 0.0732
5000
0.7423E+01
0.9304E+01
0.1682
0.8674E+01 0.9679E+01 0.1016E+02
0.9117E+01
0.1496
O.8383E+01
0.1378
0.7229E+01
0.1329
0.1528E+02
20000
0.1243E+02 0.1422E+02 0.1549E+02
40000
0.1587E+02
0.1665E+02
0.2454 0.2217 0.2080 0.2042
40000
5000 10000
11969
0.2153 0.2132
10000 20000
33p * 43d
0.2274
0.1337 0.1353
40000
5000 10000 20000 40000
17002
0.4400E+01 0.3319E+01 0.2324E+01 0.1539E+01
0.1373
0.7818E÷01
21120
33p + 63s
0.]468
20000
40000
10668
0.2645E÷01 0.1980E÷01 0.1333E+01 0.8161E+00
0.7716E+01
20000
33p * 53s
1.5370
0.7176E÷01
33s * 63p
12846
1.1749 1.3312
5000
10000
33p + 43s
0.0761
10000
0.1365E+02 0.1441E+02 0.1433E+02 0.1350E+02
8362
o de(A)
10000
20000
23p ~ 33d
~e(~)
0.1472E+02 0.1325E+02
5000
0.2007E+02
0.4487E+00
0.6020
10000
0.1778E+02
-0.5750E+00
0.6593
20000
0.1548E+02
-0.9799E+00
0.7313
40000
0.1323E+02
-0.I040E+01
0.8228
0.1987E+01 0.1455E+00
1.0529
-0.7953E+00 -0.1160E+01
1.3191
5000
0.3195E+02
10000
0.2781E+02
20000
0.2366E+02
40000
0.1965E+02
1.1683 1.5161
80
J, M. BASSAU) et aL
Table 1 (Contd)
L(~) 19543
33d ~ 43p
12985
33d + 53p
10997
33d -~ 63p 18686
33d + 43f
T (K)
We ( ,~ )
de(X)
5000
0.I034E+02
0.7777E+01
0.1546
10000
0.1126E+02
0.63,17E+01
0.1451
20000
0.1157E+02
0.4818E+01
0.1422
40000
0.1123E+02
0.3518E+01
0.1454
5000
0.1357E+02
0.8659E+01
0.2295
I0000
0.1460E+02
0.6698E+O1
0.2172
20000
0.1480E+02
0.4883E+O1
0.2151
40000
0.1418E+02
0.3430E+01
0.2221
0.3162
5000
0.2363E+02
0.1403E+02
10000
0.2496E+02
O.I064E+02
0.3035
20000
0.2483E+02
O.7652E+01
0.3046
40000
0.2340E+02
0.5329E+01
0.3186
5000
0.1102E+02
-0.1924E+01
0.8650
10000
0.9227E+01
-0.1120E+01
0.9884
20000
0.7622E+01
-0.609OE+00
1.1408
40000
0.6219E+01
-0.2951E+00
1.3289
REFERENCES H. R. Griem, Plasma Spectroscopy. Academic Press, New York (1974). J. M. Bassalo, M. Cattani, and V. S. Walder, Phys. Rev. A 22, 1194 (1980). H. R. Griem, M. Baranger, A. C. Kolb, and G. Oertel, Phys. Rev. 1.7,5, 117 (1%2). H. R. Griem, Phys. Rev. 128, 515 (1%2). S. Sahal-Brechot, Astron. Astrophys. I, 91 (1969). S. Sahal-Brechot, Astron. Astrophys. 2, 322 (1%9). C. Fleurier, S. Sahal-Brechot, and J. Chapelle, JQSRT 17, 595 (1977). L. A. Vainshtein and 1. I. Sobel'man, Opt. Spectrosc. 5, 279 (1959). R. J. Dyne and B. J. O'Mara, Astron. Astrophys. 18, 363 (1972). H. R. Griem and C. S. Shen, Phys. Rev. 145, 1% (1%2). M. S. Dimitrijevic and P. Grujic, JQSRT 19, 407 (1978). W. L Wiese, M. W. Smith, and B. M. Glennon, "Atomic Transition Probabilities", U.S. National Bureau of Standards, National Standard Reference Data Series-4. U.S. GPO, Washington D.C. (1966). 13. C. E. Moore, "Atomic Energy Levels". U.S. National Bureau of Standards Circ. Vol. I. No. 467. U.S. GPO, Washington D.C. (1949). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.