quattropani_1982 ================ A\. Quattropani, F\. Bassani and S\. Carillo, "Two\-photon transitions to excited states in atomic hydrogen", Phys\. Rev\. A 25, 3079\-3089 \(1982\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/PhysRevA\.25\.3079 `__ **Source kind:** theory **Recorded technique:** Second\-order perturbation theory of the two\-photon transition 1s \-\> ns of non\-relativistic hydrogen \(infinite nuclear mass, dipole approximation\), with the complete set of exact intermediate states\: the discrete np states summed term by term up to an n\-dependent N and by integration beyond \(to 0\.1 %\), the continuum by numerical integration, the radial matrix elements from Gordon's closed form \(Eq\. 4a, 4b\) and from the Sommerfeld\-Schur continuum form \(Eq\. 5\); in the length gauge J0 \(E\.x, Eqs\. 3a, 3b\) and the velocity gauge J \(A\.p, Eqs\. 3c, 3d\)\. Both beams polarised along z\. Energies in rydberg units\: E\_mu = \-1/mu^2, omega\_1 = R nu\_1, nu\_2 = 1 \- 1/n^2 \- nu\_1 \(Eq\. 6\)\. **Materials in structured entries:** H **Dataset quantities:** two\_photon\_transition\_amplitude **Contents:** 6 datasets; 0 formulas. Curation notes -------------- Journal pp\. 3079\-3089 = PDF pages 1\-11 \(journal page = PDF page \+ 3078\)\. Identity checked on the first page \(PDF p\. 1\: 'PHYSICAL REVIEW A VOLUME 25, NUMBER 6 JUNE 1982', A\. Quattropani \(EPF Lausanne\), F\. Bassani and Sandra Carillo \(Roma\), received 1 October 1981; p\. 1 footer '3079'\) on 2026\-10\-09\. Atomic hydrogen is the optmat material H; Table I's amplitudes are taken under two\_photon\_transition\_amplitude \(approved for bound final levels, TC\-1040\)\. THE AMPLITUDE \(pp\. 3080\-3081\)\. Eq\. \(3a\), length gauge\: D\_1^n\[J0\] = \(1/2\) sum\_mu \[\(1 \- 1/mu^2 \- nu\_1\)^\-1 \+ \(1/n^2 \- 1/mu^2 \+ nu\_1\)^\-1\] R\_10^{mu1} R\_n0^{mu1} \(\+ the continuum, Eq\. 3b\), with R the radial matrix elements of r/a0 \('the dimensionless transition amplitude', p\. 3080\)\. The denominators are in rydberg, the angular factor 1/3 of \^2 is not in D, and both photon orderings are\. Hence, in atomic units \(hartree denominators\), the zz element of OEP 8's kernel \(Eq\. 2\) is M\_zz = \(4/3\) D, and for an ns level \(rank 0, isotropic tensor\) delta^TP = \|M\_zz\|^2 = \(16/9\) D^2 at the degenerate point nu\_1 = nu\_2 = \(1 \- 1/n^2\)/2, while D\(nu\_1\)/D\(nu\_deg\) is OEP 8's two\-colour kernel M\(omega\_a, omega\_b\)/M\(Omega/2, Omega/2\) with omega\_a = R nu\_1 and omega\_b = R nu\_2\. Exact values \(a Dalgarno\-Lewis solution in a Sturmian basis, 80\-digit arithmetic, converged to 1e\-9\)\: 1s\-2s D = \-11\.780483, delta^TP = 246\.7196 au, sigma\_0 = 2\.246e\-35 cm4 \(TC\-952's convention, photon 5\.102 eV, infinite nuclear mass\); 1s\-3s D\(4/9\) = \-3\.210927, delta^TP = 18\.329 au, sigma\_0 = 2\.344e\-36 cm4; 1s\-6s D\(35/72\) = \-0\.752341, delta^TP = 1\.0063 au, sigma\_0 = 1\.539e\-37 cm4\. Eq\. \(1\)'s rate prefactor as printed, e^4 a0^4 \|E1 E2\|^2 / \(36 \(2 pi hbar^2\)^2 R^3\), is 1/\(128 pi^3\) of the golden\-rule rate that Eqs\. \(2a\) and \(3a\) give with delta\(Delta nu\) in rydberg units; the relation above rests on Eq\. \(3a\), not on Eq\. \(1\)\. OEP 8'S KAPPA \(Eq\. 4\) from the exact amplitude, M\(nu\) = M\(0\)\[1 \+ kappa nu^2\], nu the half\-split in hartree\: 1s\-2s kappa = 36\.83, D\_eff = kappa^\-1/2 = 0\.1648 hartree = 0\.879 D\_1 \(D\_1 = E\(2p\) \- Omega/2 = 0\.1875 hartree\), above OEP 8's rule 0\.75 D\_1; 1s\-3s kappa = 7\.47, D\_eff = 2\.39 D\_1 \(the mu = n term has the opposite sign, so the paths cancel\); 1s\-6s kappa = \-16\.3, negative \(\|kappa\|^\-1/2 = 1\.88 D\_1\)\: with mixed signs the bandwidth lowers the probability\. TAKEN \(tools/extract/quattropani\_1982\.py, from the library's JSON cells, every cell checked against the table images\)\: Table I \(PDF pp\. 4\-5, tbl\_bd158100b034 and tbl\_84d4824d7da0\), the totals D\_1^n\[J0\] \(gauge length\) and D\_1^n\[J\] \(gauge velocity\) against nu\_1 \(abscissa in rydberg, as printed\), for n = 3 \(14 rows, nu\_1 = 0\.3750\-0\.8860\), n = 6 \(20 rows, 0\.4860\-0\.9650\) and n = 20 \(19 rows, 0\.4950\-0\.9800\)\: six datasets, use = primary \(the paper's own calculation, exact to its printing\), m\_zz\_per\_unit = 4/3 by the derivation above, two misprints corrected by errata\. Not taken\: the discrete \(d\) and continuum \(c\) parts, which are gauge\-dependent and not observables \(they enter the script's checks\)\. The amplitude is not a delta^TP\: signed, a dimensionless multiple of M\_zz, and a function of the split\. CHECKED by the script against the exact amplitude \(above; Decimal arithmetic, 36\-70 basis functions\), every total of Table I\: all agree to the printed 4 decimals or within 0\.0055 \(n = 6 and 20 near the poles, both gauges; relative error below 1e\-5 except near the transparency zeros\), apart from two misprints confirmed on the table image\: \(a\) nu\_1 = 0\.6875, both totals printed '0\.6963' \(exact \-0\.69628; the printed parts sum to \-0\.6963 in both gauges\); \(b\) nu\_1 = 0\.9400, D\_1^6\[J\] printed '\-115\.1149' \(exact \-151\.11567; J0 prints \-151\.1172 and the J parts sum to \-151\.1150\)\. The J parts of three rows do not sum to the printed total\: \(a\) 0\.8860, \-669\.9164 \+ \-448\.0724 = \-1117\.9888 against \-1117\.2361 \(the total is right\); \(b\) 0\.9000, \-14\.2202 \+ \-9\.6863 = \-23\.9065 against \-23\.4888; \(b\) 0\.9625, \-574\.9131 \+ \-96\.7605 = \-671\.6736 against \-644\.6736 \('\-547\.9131' would fit\)\. The library's text of \(c\)'s last heading, 'D\_1^{120}\[J\]', is D\_1^{20}\[J\] in the image\. Table I\(a\)'s first row, nu\_1 = 0\.3750, lies outside 4/9 \<= nu\_1 \<= 8/9 and equals D at nu\_1 = 8/9 \- 0\.375 = 0\.5139 \(D is symmetric in nu\_1 and nu\_2\); its nu\_1 grid from 0\.6750 to 0\.7475 is that of Bassani et al\.'s 1s\-2s table, which drachman\_1990 reproduces\. TABLE II \(p\. 3085, tbl\_a5221758c076\), the transparency frequencies nu\_0^mu \(zeros of D\) against the exact zeros\: 1s\-3s 0\.6935 \(exact 0\.69357\); 1s\-20s 0\.6895, 0\.8703, 0\.9293, 0\.9551, 0\.9697 \(exact 0\.68946, 0\.87034, 0\.92934, 0\.95570, 0\.96968\); 1s\-6s 0\.6882, 0\.8675, 0\.9292, 0\.9556 against 0\.69013, 0\.87049, 0\.92952, 0\.95600, so the 1s\-6s row is off by up to 0\.0030 \(at nu\_1 = 0\.6882 the exact D is \-0\.044\) although Table I's 1s\-6s amplitudes agree with the exact ones\. The exact mu = 2 zeros fall with n \(0\.69357, 0\.69013, 0\.68946\), as the text says \('The two\-photon transparency frequency shifts towards lower values with increasing final state principal quantum numbers', p\. 3087\); the printed 1s\-6s value breaks that order\. NOT TAKEN\: no quantity\. TABLE III \(p\. 3086, tbl\_4879369ee0d8\), headed \|\\|^2 and \|\\|^2 \(a\.u\.\)\: the squares of the radial integrals \(1\.665, 0\.881, 0\.267, 162\.0 against the exact 1\.66479, 0\.88060, 0\.26697, 162\.000 for 2p\-1s, 2p\-3s, 3p\-1s, 3p\-3s\), not of the z or x matrix elements \(1/3 of those\)\. NOT TAKEN\: transition\_dipole\_moment holds a moment, not its square, and these are textbook values\. TABLE IV \(p\. 3087, tbl\_56a19e15a624\), the dimensionless threshold probability w\_1^t = lim \|D\_1^n\|^2 n^3 \(13 values of nu\_1, 0\.50\-0\.7475, the n = 45 column and the extrapolated limit\)\. The n = 45 column against the exact \|D\|^2 n^3\: within 0\.3 % except near the zero \(0\.68\: 5\.1047 against 5\.253; 0\.69\: 2\.6907e\-2 against 2\.936e\-2\)\. The text's 'two\-photon transparency occurs at nu\_1 ~ 0\.68' \(p\. 3087\) is nearer 0\.69 by the table itself\. NOT TAKEN\: a threshold two\-photon ionisation strength needs Eq\. \(11\), whose prefactor shares Eq\. \(1\)'s doubt, and a conversion to multiphoton\_ionization\_cross\_section; no quantity holds w\_1^t\. Section VI's light\-shift estimate \(Eq\. 10\: 35 MHz at 2 kW/cm2 and 10 GHz detuning, a near\-resonant single\-level term\) and the hyperfine scheme of Fig\. 5 are not taken\. Figs\. 1\-4, 6, 7 draw D, \|D\|^2 and w\_1^t; the tables print the values\. CITED, NOT IN THE LIBRARY\: F\. Bassani, J\.\-J\. Forney and A\. Quattropani, Phys\. Rev\. Lett\. 39, 1070 \(1977\) \(the 1s\-2s amplitude, both gauges\); N\. K\. Rahman, J\. Phys\. B 12, 3229 \(1979\) \(threshold two\-photon ionisation of hydrogen\); P\. F\. Liao and J\. E\. Bjorkholm, Phys\. Rev\. Lett\. 34, 1 \(1975\) \(light shifts in resonant two\-photon absorption\)\. Its Ref\. 21, M\. P\. Pindzola and H\. P\. Kelly, Phys\. Rev\. A 11, 1543 \(1975\), is pindzola\_1975\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- quattropani\_1982/D\_3s\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_3s_length", "material": "H", "file": "H/quattropani_1982__D_3s_length.csv", "table": "tbl_bd158100b034", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "3s", "gauge": "length", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(a\), column D\_1^3\[J\_0\] \(the total of the discrete and continuum parts\), 14 rows, nu\_1 = 0\.3750\-0\.8860; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/9\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. quattropani\_1982/D\_3s\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_3s_velocity", "material": "H", "file": "H/quattropani_1982__D_3s_velocity.csv", "table": "tbl_bd158100b034", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "3s", "gauge": "velocity", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(a\), column D\_1^3\[J\] \(the total of the discrete and continuum parts\), 14 rows, nu\_1 = 0\.3750\-0\.8860; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/9\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. quattropani\_1982/D\_6s\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_6s_length", "material": "H", "file": "H/quattropani_1982__D_6s_length.csv", "table": "tbl_bd158100b034", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "6s", "gauge": "length", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(b\), column D\_1^6\[J\_0\] \(the total of the discrete and continuum parts\), 20 rows, nu\_1 = 0\.4860\-0\.9650; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/36\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. quattropani\_1982/D\_6s\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_6s_velocity", "material": "H", "file": "H/quattropani_1982__D_6s_velocity.csv", "table": "tbl_bd158100b034", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "6s", "gauge": "velocity", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(b\), column D\_1^6\[J\] \(the total of the discrete and continuum parts\), 20 rows, nu\_1 = 0\.4860\-0\.9650; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/36\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. quattropani\_1982/D\_20s\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_20s_length", "material": "H", "file": "H/quattropani_1982__D_20s_length.csv", "table": "tbl_84d4824d7da0", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "20s", "gauge": "length", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(c\), column D\_1^20\[J\_0\] \(the total of the discrete and continuum parts\), 19 rows, nu\_1 = 0\.4950\-0\.9800; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/400\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. quattropani\_1982/D\_20s\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "D_20s_velocity", "material": "H", "file": "H/quattropani_1982__D_20s_velocity.csv", "table": "tbl_84d4824d7da0", "quantity": "two_photon_transition_amplitude", "unit": "1", "abscissa": "wavelength", "abscissa_unit": "rydberg", "use": "primary", "parameters": { "final_level": "20s", "gauge": "velocity", "polarization": "linear_parallel", "definition": "D_1^n[J0] = (1/2) sum_mu [(1 - 1/mu^2 - nu_1)^-1 + (1/n^2 - 1/mu^2 + nu_1)^-1] R_10^{mu1} R_n0^{mu1} + continuum (Eqs. 3a, 3b); D_1^n[J] (Eqs. 3c, 3d) equal by gauge invariance; R the radial matrix elements of r/a0, energies in rydberg, nu_2 = 1 - 1/n^2 - nu_1", "m_zz_per_unit": 1.3333333333333333, "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed and integrated, continuum integrated numerically" } } Table I\(c\), column D\_1^20\[J\] \(the total of the discrete and continuum parts\), 19 rows, nu\_1 = 0\.4950\-0\.9800; x and row\: nu\_1 = omega\_1 / R as printed, omega\_2 = R \(1 \- 1/400\) \- omega\_1 \(Eq\. 6\)\. No uncertainty stated; the exact amplitude agrees within 0\.006 \(tools/extract/quattropani\_1982\.py\)\. Formulas -------- No formulas are recorded. Corrections ----------- .. code-block:: text [ { "dataset": "D_3s_length", "row": "0.6875", "column": "value", "printed": "0.6963", "corrected": "-0.6963", "reason": "Sign lost: the printed parts sum to -0.6963 (-3.2446 + 2.5483), the table's own sign change lies between 0.6875 (negative) and 0.7000 (+0.9847), at Table II's transparency frequency 0.6935, and the exact amplitude is -0.69628.", "evidence": "Table I(a) image (tbl_bd158100b034.jpg, PDF p. 4) prints 0.6963 without a sign in both gauges; tools/extract/quattropani_1982.py." }, { "dataset": "D_3s_velocity", "row": "0.6875", "column": "value", "printed": "0.6963", "corrected": "-0.6963", "reason": "Sign lost: the printed parts sum to -0.6963 (5.1183 - 5.8146), the length gauge has the same parts' sum, and the exact amplitude is -0.69628.", "evidence": "Table I(a) image (tbl_bd158100b034.jpg, PDF p. 4); tools/extract/quattropani_1982.py." }, { "dataset": "D_6s_velocity", "row": "0.9400", "column": "value", "printed": "-115.1149", "corrected": "-151.1149", "reason": "Digits transposed: the printed velocity-gauge parts sum to -151.1150 (-126.4391 - 24.6759), the length gauge prints -151.1172, and the exact amplitude is -151.11567.", "evidence": "Table I(b) image (tbl_bd158100b034.jpg, PDF p. 4) prints -115.1149; tools/extract/quattropani_1982.py." } ] 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.