bassi_2000 ========== M\. Bassi and K\. L\. Baluja, "Transition probabilities for two\-photon H\(1s\-2s\) and He\(1 1s\-2 1s\) transitions\: A partial\-closure approach", Pramana \- J\. Phys\. 54, 377\-384 \(2000\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1007/s12043\-000\-0129\-7 `__ **Source kind:** theory **Recorded technique:** Second\-order \(lowest\-order\) perturbation theory for the two\-photon 1s\-2s transition of hydrogen and 1 1S \- 2 1S of helium in the length form, by partial closure\: the 2p \(2 1P\) intermediate state's term computed exactly and the rest of the sum over intermediate states with one mean excitation energy Delta \(Eqs\. 1\-6, 21\), the amplitude D\(omega\) in atomic units of the radial dipole elements with Rydberg energy denominators\. Hydrogen with exact hydrogenic matrix elements; helium with analytic Hartree\-Fock\-type wave functions \(1 1S\: two Slater exponents, E = \-2\.861 67 au; 2 1S and 2 1P\: a 1s\(Z = 2\) core and one outer orbital, E = \-2\.143 and \-2\.123 au; Eqs\. 10\-20\)\. Delta chosen five ways \(the excitation energy, the ionisation energy, a 'partial closure value' from the static transition polarisability \(H 0\.9522 Ryd, scaled by effective charges to He 2\.131 Ryd, Eqs\. 25\-28\), a value 'tuned' to the known rate, and the mean energy reproducing the ground\-state dipole polarisability\)\. The spontaneous two\-photon decay rate W of 2s \(2 1S\) from the Shapiro\-Breit formula \(Eqs\. 7, 23\) and the photon energy distribution A\(y\), W = \(1/2\) integral A\(y\) dy \(Eq\. 24\)\. **Materials in structured entries:** None recorded. **Dataset quantities:** None recorded. **Contents:** 0 datasets; 0 formulas. Curation notes -------------- Pramana \- J\. Phys\. 54\(3\), 377\-384 \(March 2000\) = PDF pages 1\-8 \(journal page = PDF page \+ 376\)\. Identity checked on the first page \(PDF p\. 1\: 'PRAMANA \- journal of physics, Vol\. 54, No\. 3, March 2000, pp\. 377\-384', Monika Bassi and K L Baluja, University of Delhi; MS received 14 December 1998, revised 22 October 1999\) on 2026\-10\-09\. Nothing is taken\. The paper's numbers are spontaneous two\-photon decay rates W of the metastable 2s and 2 1S levels \(s\-1, Table 1\), the emitted\-photon energy distribution A\(y\) of helium \(s\-1 per unit y, Table 3\) and the hydrogen 1s\-2s amplitude D\(omega\) \(Table 2\)\. No quantity of the vocabulary holds a two\-photon decay rate or its spectral distribution, and D\(omega\) is not the rotationally averaged strength delta^TP that two\_photon\_transition\_strength holds \(TC\-872\)\: it is a radial amplitude, defined with a factor 1/2 and with energy denominators in Rydberg \(Eqs\. 1\-6\)\. The conversion is a finding \(below\), not a value of the paper\. CHECKED \(2026\-10\-09, by direct computation from the printed equations and numbers\)\. Table 2 \(H, Delta = 1 Ryd, p\. 383, image checked\)\: all nine rows of D1 = \-10\.0623 / \[omega \(3 \- 4 omega\)\] \(Eq\. 5\), D2 = 1\.86423 \(8 Delta \- 3\) / \[\(Delta \- omega\)\(4 Delta \- 3 \+ 4 omega\)\] \(Eq\. 6\) and D = D1 \+ D2 reproduced from the matrix elements of p\. 379 within four units of the fourth decimal \(largest relative difference 3e\-7\)\. Eq\. \(5\)'s 10\.0623 and Eq\. \(9\)'s 6\.7082 use R\(1s\-2p\) = \(5/3\)^\(1/2\) = 1\.290 99 \(stated, p\. 379\); the hydrogenic integral of P\_1s r P\_2p is 2^7 6^\(1/2\) / 3^5 = 1\.290 27 \(0\.06 % smaller; R\(2s\-2p\) = \-27^\(1/2\) and \<1s\|r^2\|2s\> = \-\(512/243\) 2^\(1/2\) are exact\)\. Helium's delta\_1 = 1\.027 829 and delta\_2 = 0\.027 829 are the 2 1P energies relative to the printed E\(1 1S\), E\(2 1S\), E\(2 1P\); f1 = \<1 1s\|r\|2 1p\>\<2 1s\|r\|2 1p\> / 6\.7082 \(0\.394 57\) and 2\.979 f2 = \|\<1 1s\|r^2\|2 1s\>\| \(0\.969 36\)\. Eq\. \(23\)'s 18\.415 is twice Eq\. \(7\)'s alpha^7 c / \(216 pi a0\) \(9\.1997 s\-1 with CODATA 2018 constants; 2 x 9\.1997 = 18\.399, 0\.09 % below 18\.415\), the 'spin factor of 2'\. Table 1 \(p\. 382, image checked\) against Eqs\. \(7\)\-\(9\) and \(21\)\-\(23\) integrated numerically\: helium Delta\_2 to Delta\_5, 38\.981, 53\.653, 51\.290, 58\.472 s\-1 against the printed 38\.9940, 53\.6668, 51\.3, 58\.4864 \(all within 0\.04 %\); hydrogen Delta\_2 \(1 Ryd\) 8\.488 against 8\.4975 \(\-0\.1 %\)\. Table 3 \(He, Delta\_3 = 2\.131 Ryd, p\. 383, image checked\)\: A\(y\) = 2 x 18\.415 dE^5 d\(y\)^2 y^3 \(1 \- y\)^3 reproduces all 21 rows of 'our results' within the printing except y = 0\.200 \(110\.86 against '110'; 111 by rounding\), and \(1/2\) integral A dy = 53\.65 s\-1 is Table 1's Delta\_3 value\. The 'Exact' column of Table 3 \(Ref\. 18, Drake, Victor and Dalgarno, Phys\. Rev\. 180, 25 \(1969\), Drake\_1969\_TwoPhotonDecay in the library, uncurated\) is that paper's Table II He I column at all 21 values of y, and 51\.3 s\-1 its He I decay rate\. FINDING\: HYDROGEN'S TABLE 1 IS NOT REPRODUCED\. With Eqs\. \(7\)\-\(9\) as printed, Delta\_3 = 0\.9522, Delta\_4 = 0\.988118 and Delta\_5 = 8/9 Ryd give 7\.873, 8\.344 and 6\.878 s\-1 against the printed 7\.3240, 8\.2260 and 5\.3918 \(\+7\.5 %, \+1\.4 %, \+28 %\); the printed rates follow from Delta = 0\.9154, 0\.9787 and 0\.8181 Ryd\. Delta\_2 \(the abstract's 8\.50 s\-1\) is reproduced, and the helium rows are\. So the 'tuned value that reproduces W' \(8\.226 s\-1, the 'known value' of the abstract, no reference given\) is 0\.9787 Ryd by the paper's own equations, not 0\.988118, and the comparison of the choices of Delta for hydrogen in the text rests on rows that do not follow from its formulas\. Not resolved\. FINDING\: THE Delta\_1 ROWS\. For helium, Delta\_1 = dE \(1\.437 34 Ryd\) puts the zero of the closure term's denominator \(Delta/dE \- delta\_1 \+ y\) at y = 0\.0278, inside the range, where the integrand of Eq\. \(23\) is not integrable\: the printed 20\.2606 s\-1 cannot be reproduced\. For hydrogen, Delta\_1 = 3/4 Ryd makes both terms of Eq\. \(9\) proportional to 1/\[y\(1 \- y\)\] and the integral elementary\: W = 9\.1997 \(3/4\)^5 \(3\.7292 \- 6\.7082\)^2 / 6 = 3\.229 s\-1, not the printed 0\.0888\. The text calls these the 'worst results'\. FINDING\: THE CONVERSION OF D\(omega\) TO delta^TP, FOR HYDROGEN \(derived, not taken\)\. For 1s \-\> 2s with both photons polarised along z, the Cartesian amplitude S\_zz in atomic units \(Hartree denominators, both time orderings\) is sum\_n \(R\_1n R\_2n / 3\) \[1/\(E\_n \- E\_1s \- omega\) \+ 1/\(E\_n \- E\_2s \+ omega\)\] = \(4/3\) D\(omega\) \(the 1/3 the s\-p\-s angular factor, 2 from the Rydberg denominators, 2 from the 1/2 of Eq\. 1\), and for an S \-\> S transition delta^TP = S\_zz^2 for parallel linear polarisation \(F = G = H = 2 with 1/30, hattig\_1998b's convention, TC\-887\)\. At the degenerate frequency \(omega = 0\.375 Ryd, Table 2's first row\) this gives delta^TP = 252\.7 au \(partial closure, D = \-11\.9230\) and 246\.7 au \(Table 2's 'Exact', \-11\.7805, whose source the paper does not name\)\. Helium's D\(y\) \(Eq\. 21\) is not converted\: the normalisation of its two\-electron matrix elements and the 'spin factor of 2' are not stated precisely enough\. NOT TAKEN\. Table 1 \(W, s\-1\: no quantity; for He the 51\.3 is Drake et al\.'s\), Table 2 \(D\(omega\), above; the 'Exact' column reproduced from an unnamed source\), Table 3 \(A\(y\)\: no quantity; 'Exact' is Drake et al\.'s\), the matrix elements of Eqs\. \(18\)\-\(20\), the model energies \(Hartree\-Fock\-type, not levels of the atom\), the static transition polarisability of hydrogen 8\.939 23 au \(Ref\. 21, V\. Chandrasekharan and B\. Silvi, J\. Phys\. B 14, 4327 \(1981\), not in the library; no quantity\)\. The paper states no uncertainty; it compares with the 'exact' values only \(W within 2\-5 %, D within 2 %\)\. TWO\-PHOTON DATA CITED, NOT IN THE LIBRARY\: A\. Quattropani, F\. Bassani and S\. Carillo, Phys\. Rev\. A 25, 3079 \(1982\) and F\. Bassani, A\. Quattropani and J\. J\. Forney, Phys\. Rev\. Lett\. 39, 1070 \(1977\) \(H two\-photon amplitudes\); R\. J\. Drachman, A\. K\. Bhatia and A\. A\. Shabazz, Phys\. Rev\. A 42, 6333 \(1990\) \(pseudostates\); R\. B\. Thayyullathil and K\. J\. Prasanna, Phys\. Rev\. A 49, 2432 \(1994\) \(H 1s\-3s\); J\. Shapiro and G\. Breit, Phys\. Rev\. 113, 179 \(1959\) \(the rate formula\); A\. Dalgarno and G\. A\. Victor, Proc\. Phys\. Soc\. 87, 371 \(1966\) and G\. A\. Victor, Proc\. Phys\. Soc\. 91, 825 \(1967\) \(He 2 1S decay\); T\. W\. Hänsch, S\. A\. Lee, R\. Wallenstein and C\. Wieman, Phys\. Rev\. Lett\. 34, 307 \(1975\) \(H 1s\-2s experiment\)\. The paper's Ref\. 6 \('S A Lee, R Wallenstein and T W Hansch, Phys\. Rev\. A35, 1262 \(1980\)'\) has inconsistent volume and year as printed\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- No datasets are recorded. Formulas -------- No formulas are recorded. Corrections ----------- No dataset errata are recorded in this source record. Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register ` for the checks and decisions.