In our investigation "CMBR distortion concerned with recombination of the primordial hydrogen plasma" we used hydrogen dipole transition values calculated by exact and approximated formulae. The exact Einstein coefficients given via hypergeometric functions (see Berestetzkii et al) are complicated and demand much computer time to calculate. That is why several approximations have been developed (see Kaplan, Pikelner; Johnson). We used exact formulae and Kramers's approximation with Johnson's Gaunt factor.
The comparison of exact values with approximated values of spontaneous transitions with principal quantum number smaller than 6 are presented below.
To control our calculation, the data from Bethe & Salpeter are also presented.
fprintf(fout,"%-20.8g",A_Kramers(i,f));
fprintf(fout,"%-20.8g",A_exact(i,f));
Download of the numeric code
f \ i | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|
1 | 4.6966948 | 0.55727384 | 0.12779603 | 0.041232986 | 0.01643335 |
2 | - | 0.44082910 | 0.084157168 | 0.025293477 | 0.0097278027 |
3 | - | - | 0.089822794 | 0.021998225 | 0.0077795773 |
4 | - | - | - | 0.026981317 | 0.0077077669 |
5 | - | - | - | - | 0.010249685 |
f \ i | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|
1 | 4.69624 | 0.55722784 | 0.12778672 | 0.04122999 | 0.016432142 |
2 | - | 0.44078884 | 0.084152447 | 0.025294876 | 0.009728285 |
3 | - | - | 0.089809845 | 0.021996677 | 0.0077821864 |
4 | - | - | - | 0.026948936 | 0.0077061764 |
5 | - | - | - | - | 0.010243881 |
State | n | Sum | Lifetime, 10-8s | |||||
initial | final | 1 | 2 | 3 | 4 | 5 | ||
2s | np | - | - | - | - | - | 0 | \infty | 2p | ns | 6.25 | - | - | - | - | 6.25 | 0.16 |
2 | average | 4.69 | - | - | - | - | 4.69 | 0.21 |
3s | np | - | 0.063 | - | - | - | 0.063 | 16 |
3p | ns | 1.64 | 0.22 | - | - | - | 1.86 | 0.54 |
3d | np | - | 0.64 | - | - | - | 0.64 | 1.56 |
3 | average | 0.55 | 0.43 | - | - | - | 0.98 | 1.02 |
4s | np | - | 0.025 | 0.018 | - | - | 0.043 | 23 |
4p | ns | 0.68 | 0.095 | 0.030 | - | - | 0.81 | 1.24 |
nd | - | - | 0.003 | - | - | |||
4d | np | - | 0.204 | 0.070 | - | - | 0.274 | 3.65 |
4f | nd | - | - | 0.137 | - | - | 0.137 | 7.3 |
4 | average | 0.128 | 0.083 | 0.089 | - | - | 0.299 | 3.35 |
5s | np | - | 0.0127 | 0.0085 | 0.0065 | - | 0.0277 | 36 |
5p | ns | 0.34 | 0.049 | 0.016 | 0.0075 | - | 0.415 | 2.40 |
nd | - | - | 0.0015 | 0.002 | - | |||
5d | np | - | 0.094 | 0.034 | 0.014 | - | 0.142 | 7.0 |
nf | - | - | - | 0.0005 | - | |||
5f | nd | - | - | 0.0045 | 0.026 | - | 0.071 | 14 |
5g | nf | - | - | - | 0.0425 | - | 0.0425 | 23.5 |
5 | average | 0.040 | 0.025 | 0.022 | 0.027 | - | 0.114 | 8.8 |
6s | np | - | 0.0073 | 0.0051 | 0.0035 | 0.0017 | 0.0176 | 57 |
6p | ns | 0.195 | 0.029 | 0.0096 | 0.0045 | 0.0021 | 0.243 | 4.1 |
nd | - | - | 0.0007 | 0.0009 | 0.0010 | |||
6d | np | - | 0.048 | 0.0187 | 0.0086 | 0.0040 | 0.080 | 12.6 |
nf | - | - | - | 0.0002 | 0.0004 | |||
6f | nd | - | - | 0.0210 | 0.0129 | 0.0072 | 0.0412 | 24.3 |
ng | - | - | - | - | 0.0001 | |||
6g | nf | - | - | - | 0.0137 | 0.0110 | 0.0247 | 40.5 |
6n | ng | - | - | - | - | 0.0164 | 0.0164 | 61 |
6 | average | 0.0162 | 0.0092 | 0.0077 | 0.0077 | 0.0101 | 0.0510 | 19.6 |
webmaster
Last update: Jan. 30, 2005 |
Supported by RFBR
grant 03-07-90200 |