Spontaneous transitions of hydrogen atom

Coefficients of spontaneous transitions of hydrogen atom

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.

ASCII-database comment
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 i is the principal quantum number of initial state, f is the principal quantum number of final state
A_Kramers(i,f) are the values calculated by the formulae presented in the paper L. C. Johnson, AJ. 174, 227 (1972)
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

Hydrogen dipole transition probabilities averaged over angular momentum, in 10⁸s^-1 units

The values calculated by the formulae presented in the work
V.B. Berestetzkii, E.M. Lifshitz, and L.P. Pitaevskii, "Quantum electrodynamics".
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

The values calculated by the formulae presented in
the paper L. C. Johnson, AJ. **174**, 227 (1972).
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

Probabilities of dipole transitions of hydrogen atom, 10⁸s^-1

H.A. Bethe and E.E. Salpeter, "Quantum mechanics of one- and two- electron atoms", 1957
State		n					Sum	Lifetime, 10^-8s
initial	final	1	2	3	4	5	Sum	Lifetime, 10^-8s
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
4p	nd	-	-	0.003	-	-	0.81	1.24
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
5p	nd	-	-	0.0015	0.002	-	0.415	2.40
5d	np	-	0.094	0.034	0.014	-	0.142	7.0
5d	nf	-	-	-	0.0005	-	0.142	7.0
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
6p	nd	-	-	0.0007	0.0009	0.0010	0.243	4.1
6d	np	-	0.048	0.0187	0.0086	0.0040	0.080	12.6
6d	nf	-	-	-	0.0002	0.0004	0.080	12.6
6f	nd	-	-	0.0210	0.0129	0.0072	0.0412	24.3
6f	ng	-	-	-	-	0.0001	0.0412	24.3
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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Last update: Jan. 30, 2005

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Coefficients of spontaneous transitions of hydrogen atom

Hydrogen dipole transition probabilities averaged over angular momentum, in 108s-1 units

Probabilities of dipole transitions of hydrogen atom, 108s-1

Hydrogen dipole transition probabilities averaged over angular momentum, in 10⁸s^-1 units

Probabilities of dipole transitions of hydrogen atom, 10⁸s^-1