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.

The numeric values of the hydrogen dipole transition probabilities averaged over angular momentum are presented in ASCII-format by using C-function

```
fprintf(fout,"%-20.8g",A_Kramers(i,f));
```

fprintf(fout,"%-20.8g",A_exact(i,f));

Here

A_Kramers(i,f) are the values calculated by the formulae presented in the paper L. C. Johnson, AJ.

A_exact(i,f) are the values calculated by the formulae presented in the work V.B. Berestetzkii et al., "Quantum electrodynamics".

The database of transitions with principal quantum numbers n<=200

The database of transitions with principal quantum numbers n<=100

The database of transitions with principal quantum numbers n<=50

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^{-8}s |
|||||

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 |

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