10:54:59 Apr 26, 2026 *** A program to calculate hydrogen atom hyperfine levels using the new photonic toroidal vortex (PTV) model *** written in Liberty BASIC by Dr Barry R. Clarke, completed 26 April 2026 All equations and tables refer to the following papers. PTV1: Clarke, Barry R., A photonic toroidal vortex model of the hydrogen atom fine structure, Quantum Studies: Mathematics and Foundations, 12, article 18 (2025) PTV2: Clarke, Barry R., The hydrogen atom hyperfine structure from a photonic toroidal vortex model, in peer-review (26 April 2026) EXPERIMENTAL DATA Horbatsch, M. and E. A. Hessels, Tabulation of the bound-state energies of atomic hydrogen, Physical Review A 93, 022513 (2016) _____________________________________________________________________________________________________ 1S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 1 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 1.7324035537303600 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 2.0000332836086264 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 3289885758.535002 (5) New PTV f-s energy no adjustments = 3289885758.260552 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 3288095006.026450 (7) Relativistic reduced mass multiplier = 0.9994556721741642 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 3288095029.923980 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.0010971 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 3291702389.145564 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 3288087212.213100 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3615176.932463 (13) Magnetic potential adjustment = -3615176.932463 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 7230353.864927 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 3288087212.213100 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 3288086502.010200 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 3288086502.193267 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 17961171874.889158 (19) H-f difference taken from experiment = 710.202900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 710.019833 PTV2 Eq. (35) (21) H-f error magnitude = 0.183066 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 2S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8732511083690200 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001231494850968 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822474177.055886 (5) New PTV f-s energy no adjustments = 822474177.045523 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822026487.438713 (7) Relativistic reduced mass multiplier = 0.9994556776131092 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822026493.441384 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00054847 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 822477353.040722 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822025488.313800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 451864.726922 (13) Magnetic potential adjustment = -451864.726922 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 903729.453844 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 822025488.313800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 822025399.535400 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 822025399.565678 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 27149399551.735912 (19) H-f difference taken from experiment = 88.778400 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 88.748122 PTV2 Eq. (35) (21) H-f error magnitude = 0.030278 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 3S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9163991001174104 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002241103507824 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365542862.051824 (5) New PTV f-s energy no adjustments = 365542862.054546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365343889.550878 (7) Relativistic reduced mass multiplier = 0.9994556786197338 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365343891.037741 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00036563 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365477472.518026 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365343591.599600 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 133880.918426 (13) Magnetic potential adjustment = -133880.918426 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 267761.836853 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 365343591.599600 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 365343565.294900 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 365343565.304514 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 38001239393.181736 (19) H-f difference taken from experiment = 26.304700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 26.295086 PTV2 Eq. (35) (21) H-f error magnitude = 0.009614 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 4S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9376529632206608 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003278449755088 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205617346.644788 (5) New PTV f-s energy no adjustments = 205617346.640405 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205505424.892383 (7) Relativistic reduced mass multiplier = 0.9994556789719766 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205505425.404479 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00027422 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205561778.663135 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205505298.855200 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 56479.807935 (13) Magnetic potential adjustment = -56479.712139 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 112959.520073 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 205505298.950996 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 205505287.757900 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 205505287.857912 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 2054802537.094273 (19) H-f difference taken from experiment = 11.097300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 11.093084 PTV2 Eq. (35) (21) H-f error magnitude = 0.004216 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 5S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9503639585056512 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004326891036560 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594869.718334 (5) New PTV f-s energy no adjustments = 131594869.717838 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131523239.923151 (7) Relativistic reduced mass multiplier = 0.9994556791349960 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131523240.145725 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00021937 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131552092.567694 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131523175.306400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 28917.261294 (13) Magnetic potential adjustment = -28917.141892 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 57834.403186 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 131523175.425801 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 131523169.624600 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 131523169.746193 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 1081665040.215909 (19) H-f difference taken from experiment = 5.681800 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 5.679608 PTV2 Eq. (35) (21) H-f error magnitude = 0.002192 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 6S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588462377676944 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005380879831920 PTV1 d2-bar Eq (86) gamma = 1.0069427000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91385208.595505 (5) New PTV f-s energy no adjustments = 91385208.586172 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335465.746194 (7) Relativistic reduced mass multiplier = 0.9994556792235434 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335465.848147 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00018281 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91352162.677503 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335428.313650 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 16734.363853 (13) Magnetic potential adjustment = -16734.249498 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 33468.613352 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 91335428.428005 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 91335425.025600 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 91335425.141215 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 789995643.194370 (19) H-f difference taken from experiment = 3.288050 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.286790 PTV2 Eq. (35) (21) H-f error magnitude = 0.001260 (22) A/gamma = -28724281.0992422912 _____________________________________________________________________________________________________ 2P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8732511083690200 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001231494850968 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822474177.055886 (5) New PTV f-s energy no adjustments = 822474177.045523 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822026487.438713 (7) Relativistic reduced mass multiplier = 0.9994556776135524 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822026490.094327 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00030378 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 822276202.972314 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822026516.551150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 249686.421163 (13) Magnetic potential adjustment = -249686.421164 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 499372.842327 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 822026516.551150 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 822026486.966400 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 822026486.968443 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 402478139542.300800 (19) H-f difference taken from experiment = 29.584750 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 29.582707 PTV2 Eq. (35) (21) H-f error magnitude = 0.002043 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 3P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9163991001174104 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002241103507824 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365542862.051824 (5) New PTV f-s energy no adjustments = 365542862.054546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365343889.550878 (7) Relativistic reduced mass multiplier = 0.9994556786198650 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365343890.244386 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00020251 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365417877.126191 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365343897.705100 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 73979.421091 (13) Magnetic potential adjustment = -73979.421091 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 147958.842183 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 365343897.705100 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 365343888.939200 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 365343888.940071 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 419308615338.142656 (19) H-f difference taken from experiment = 8.765900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 8.765029 PTV2 Eq. (35) (21) H-f error magnitude = 0.000871 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 4P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9376529632206608 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003278449755088 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205617346.644788 (5) New PTV f-s energy no adjustments = 205617346.640405 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205505424.892383 (7) Relativistic reduced mass multiplier = 0.9994556789720318 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205505425.132543 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00015188 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205536637.875342 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205505428.231800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 31209.643542 (13) Magnetic potential adjustment = -31209.643542 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 62419.287084 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 205505428.231800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 205505424.533700 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 205505424.534105 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 506956167706.746688 (19) H-f difference taken from experiment = 3.698100 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.697695 PTV2 Eq. (35) (21) H-f error magnitude = 0.000405 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 5P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9503639585056512 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004326891036560 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594869.718334 (5) New PTV f-s energy no adjustments = 131594869.717838 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131523239.923151 (7) Relativistic reduced mass multiplier = 0.9994556791350242 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131523240.029201 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00012151 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131539220.803966 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131523241.607050 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 15979.196916 (13) Magnetic potential adjustment = -15979.194057 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 31958.390973 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 131523241.609909 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 131523239.713600 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 131523239.716706 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 42346139338.012904 (19) H-f difference taken from experiment = 1.893450 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.893203 PTV2 Eq. (35) (21) H-f error magnitude = 0.000247 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 6P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588462377676944 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005380879831920 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91385208.595505 (5) New PTV f-s energy no adjustments = 91385208.586172 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335465.746194 (7) Relativistic reduced mass multiplier = 0.9994556792235596 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335465.790301 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00010125 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91344713.865123 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335466.701500 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 9247.163623 (13) Magnetic potential adjustment = -9247.159226 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 18494.322850 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 91335466.705897 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 91335465.605800 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 91335465.610300 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 20294948098.008756 (19) H-f difference taken from experiment = 1.095700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.095597 PTV2 Eq. (35) (21) H-f error magnitude = 0.000103 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 7P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 6 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649244845825792 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006438038634048 PTV1 d2-bar Eq (86) gamma = 0.5577389026100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67140087.867507 (5) New PTV f-s energy no adjustments = 67140087.880927 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103542.136257 (7) Relativistic reduced mass multiplier = 0.9994556792769424 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103542.179303 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00008679 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67109366.013103 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103542.752550 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 5823.260553 (13) Magnetic potential adjustment = -5823.255441 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 11646.515994 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 67103542.757662 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 67103542.062800 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 67103542.067727 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 13620660907.392432 (19) H-f difference taken from experiment = 0.689750 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.689935 PTV2 Eq. (35) (21) H-f error magnitude = 0.000185 (22) A/gamma = -28654297.8123137568 _____________________________________________________________________________________________________ 2P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8733061076133600 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001764043401488 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822463227.394777 (5) New PTV f-s energy no adjustments = 822463227.397312 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822015543.737730 (7) Relativistic reduced mass multiplier = 0.9994556776136446 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822015542.979021 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00026622 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 822234378.763348 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822015535.670900 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 218843.092448 (13) Magnetic potential adjustment = -218843.092447 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 437686.184894 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 822015535.670901 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 822015523.845100 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 822015523.838448 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 123574237011.006912 (19) H-f difference taken from experiment = 11.825800 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 11.832453 PTV2 Eq. (35) (21) H-f error magnitude = 0.006653 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 3P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164531096983408 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002773654076936 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365539617.722525 (5) New PTV f-s energy no adjustments = 365539617.710736 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365340646.987534 (7) Relativistic reduced mass multiplier = 0.9994556786198924 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365340646.969272 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00017748 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365405485.835376 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365340644.107950 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 64841.727426 (13) Magnetic potential adjustment = -64841.727426 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 129683.454852 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 365340644.107950 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 365340640.604000 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 365340640.602063 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 188620102196.937408 (19) H-f difference taken from experiment = 3.503950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.505887 PTV2 Eq. (35) (21) H-f error magnitude = 0.001937 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 4P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377066388691312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003811001296544 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615977.950867 (5) New PTV f-s energy no adjustments = 205615977.934378 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205504056.943471 (7) Relativistic reduced mass multiplier = 0.9994556789720434 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205504056.947704 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.0001331 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205531410.533370 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205504055.622150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 27354.911220 (13) Magnetic potential adjustment = -27354.911220 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 54709.822440 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 205504055.622150 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 205504054.143900 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 205504054.143112 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 260643022773.503616 (19) H-f difference taken from experiment = 1.478250 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.479038 PTV2 Eq. (35) (21) H-f error magnitude = 0.000788 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 5P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504174816877120 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004859443149632 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594168.930585 (5) New PTV f-s energy no adjustments = 131594168.941192 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522539.516855 (7) Relativistic reduced mass multiplier = 0.9994556791350302 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522539.541534 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00010648 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131536544.478135 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522538.831750 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 14005.646385 (13) Magnetic potential adjustment = -14005.643826 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 28011.290211 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 131522538.834309 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 131522538.074900 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 131522538.077045 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 61330147104.911136 (19) H-f difference taken from experiment = 0.756850 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.757265 PTV2 Eq. (35) (21) H-f error magnitude = 0.000415 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 6P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588996786783616 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005913432321440 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384803.042340 (5) New PTV f-s energy no adjustments = 91384803.044546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335060.413780 (7) Relativistic reduced mass multiplier = 0.9994556792235632 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335060.424572 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00008874 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91343165.094059 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335060.003300 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 8105.090759 (13) Magnetic potential adjustment = -8105.086892 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 16210.177651 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 91335060.007168 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 91335059.565300 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 91335059.568937 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 25114283744.979172 (19) H-f difference taken from experiment = 0.438000 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.438231 PTV2 Eq. (35) (21) H-f error magnitude = 0.000231 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 7P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649778760733312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006970591389616 PTV1 d2-bar Eq (86) gamma = 0.4887853699500001 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139832.496862 (5) New PTV f-s energy no adjustments = 67139832.496326 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103286.904616 (7) Relativistic reduced mass multiplier = 0.9994556792769444 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103286.909409 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00007606 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67108390.706689 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103286.639850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 5104.066839 (13) Magnetic potential adjustment = -5104.062531 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 10208.129370 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 67103286.644158 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 67103286.364000 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 67103286.368188 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 16022861834.387940 (19) H-f difference taken from experiment = 0.275850 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.275970 PTV2 Eq. (35) (21) H-f error magnitude = 0.000120 (22) A/gamma = -28658796.8865208416 _____________________________________________________________________________________________________ 3D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164531096983408 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002773654076936 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365539617.722525 (5) New PTV f-s energy no adjustments = 365539617.710736 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365340646.987534 (7) Relativistic reduced mass multiplier = 0.9994556786199196 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365340646.914528 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.0001437 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365393147.295243 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365340649.089800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 52498.205443 (13) Magnetic potential adjustment = -52498.205443 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 104996.410886 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 365340649.089800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 365340646.986500 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 365340646.986267 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 1570432730738.472960 (19) H-f difference taken from experiment = 2.103300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 2.103532 PTV2 Eq. (35) (21) H-f error magnitude = 0.000232 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 4D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377066388691312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003811001296544 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615977.950867 (5) New PTV f-s energy no adjustments = 205615977.934378 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205504056.943471 (7) Relativistic reduced mass multiplier = 0.9994556789720550 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205504056.926051 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00010778 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205526205.319114 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205504057.762150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 22147.556964 (13) Magnetic potential adjustment = -22147.556964 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 44295.113929 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 205504057.762150 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 205504056.874800 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 205504056.874727 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 2805358788498.620928 (19) H-f difference taken from experiment = 0.887350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.887423 PTV2 Eq. (35) (21) H-f error magnitude = 0.000073 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 5D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504174816877120 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004859443149632 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594168.930585 (5) New PTV f-s energy no adjustments = 131594168.941192 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522539.516855 (7) Relativistic reduced mass multiplier = 0.9994556791350358 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522539.531601 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00008622 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131533879.442838 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522539.938050 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 11339.504788 (13) Magnetic potential adjustment = -11339.504788 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 22679.009575 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 131522539.938050 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 131522539.483700 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 131522539.483691 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 14493149778565.277696 (19) H-f difference taken from experiment = 0.454350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.454359 PTV2 Eq. (35) (21) H-f error magnitude = 0.000009 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 6D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588996786783616 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005913432321440 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384803.042340 (5) New PTV f-s energy no adjustments = 91384803.044546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335060.413780 (7) Relativistic reduced mass multiplier = 0.9994556792235666 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335060.419440 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00007185 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91341622.841876 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335060.647200 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 6562.194676 (13) Magnetic potential adjustment = -6562.194358 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 13124.389035 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 91335060.647518 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 91335060.384300 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 91335060.384580 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 326709245016.884864 (19) H-f difference taken from experiment = 0.262900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.262939 PTV2 Eq. (35) (21) H-f error magnitude = 0.000039 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 7D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649778760733312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006970591389616 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139832.496862 (5) New PTV f-s energy no adjustments = 67139832.496326 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103286.904616 (7) Relativistic reduced mass multiplier = 0.9994556792769464 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103286.906507 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00006159 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67107419.498716 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103287.046850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 4132.451866 (13) Magnetic potential adjustment = -4132.451271 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 8264.903137 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 67103287.047445 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 67103286.881300 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 67103286.881863 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 119153435201.030496 (19) H-f difference taken from experiment = 0.165550 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.165582 PTV2 Eq. (35) (21) H-f error magnitude = 0.000032 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 8D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 8 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 6 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 15.9695580628726016 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 16.0008029731643200 PTV1 d2-bar Eq (86) gamma = 0.3957755000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 51403919.639947 (5) New PTV f-s energy no adjustments = 51403919.634302 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 51375939.428839 (7) Relativistic reduced mass multiplier = 0.9994556793115918 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 51375939.424866 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00005389 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 51378707.932672 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 51375939.517400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 2768.415272 (13) Magnetic potential adjustment = -2768.414488 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 5536.829761 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 51375939.518184 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 51375939.406500 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 51375939.407257 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 67824976255.875536 (19) H-f difference taken from experiment = 0.110900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.110927 PTV2 Eq. (35) (21) H-f error magnitude = 0.000027 (22) A/gamma = -28655841.6011097152 _____________________________________________________________________________________________________ 3D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164711126492712 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002951168570048 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365538536.306861 (5) New PTV f-s energy no adjustments = 365538536.289994 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365339566.160507 (7) Relativistic reduced mass multiplier = 0.9994556786199280 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365339565.922954 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00013601 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365389256.699024 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365339565.452000 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 49691.247024 (13) Magnetic potential adjustment = -49691.247025 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 99382.494049 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 365339565.452000 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 365339564.100300 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 365339564.099436 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 422715915879.764096 (19) H-f difference taken from experiment = 1.351700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.352564 PTV2 Eq. (35) (21) H-f error magnitude = 0.000864 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 4D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377245304992704 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003988515982480 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615521.724133 (5) New PTV f-s energy no adjustments = 205615521.709497 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205503600.965070 (7) Relativistic reduced mass multiplier = 0.9994556789720584 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205503600.898058 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00010201 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205524564.025718 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205503600.601850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 20963.423868 (13) Magnetic potential adjustment = -20963.423868 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 41926.847736 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 205503600.601850 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 205503600.031500 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 205503600.031238 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 784923912693.473536 (19) H-f difference taken from experiment = 0.570350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.570612 PTV2 Eq. (35) (21) H-f error magnitude = 0.000262 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 5D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504353224891328 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0005036957947360 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131593935.343814 (5) New PTV f-s energy no adjustments = 131593935.354022 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522306.057230 (7) Relativistic reduced mass multiplier = 0.9994556791350378 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522306.050263 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00008161 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131533039.117004 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522305.871950 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 10733.245054 (13) Magnetic potential adjustment = -10733.245054 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 21466.490109 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 131522305.871950 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 131522305.580000 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 131522305.579797 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 648898141312.649728 (19) H-f difference taken from experiment = 0.291950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.292153 PTV2 Eq. (35) (21) H-f error magnitude = 0.000203 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 6D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9589174920513680 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0006090947191888 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384667.867097 (5) New PTV f-s energy no adjustments = 91384667.866803 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91334925.312115 (7) Relativistic reduced mass multiplier = 0.9994556792235676 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91334925.304920 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00006801 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91341136.548634 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91334925.192250 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 6211.356384 (13) Magnetic potential adjustment = -6211.356019 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 12422.712403 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 91334925.192615 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 91334925.023300 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 91334925.023545 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 372357879948.899648 (19) H-f difference taken from experiment = 0.168950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.169070 PTV2 Eq. (35) (21) H-f error magnitude = 0.000120 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 7D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649956729696720 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0007148106311456 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139747.364169 (5) New PTV f-s energy no adjustments = 67139747.369835 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103201.818263 (7) Relativistic reduced mass multiplier = 0.9994556792769472 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103201.820732 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.00005829 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67107113.264497 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103201.745800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3911.518697 (13) Magnetic potential adjustment = -3911.518089 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 7823.036786 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 67103201.746409 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 67103201.639400 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 67103201.639939 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 124443021880.318128 (19) H-f difference taken from experiment = 0.106400 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.106469 PTV2 Eq. (35) (21) H-f error magnitude = 0.000069 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 8D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 8 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 15.9695758490998720 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 16.0008207246602976 PTV1 d2-bar Eq (86) gamma = 0.3745970500000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 51403862.601317 (5) New PTV f-s energy no adjustments = 51403862.606216 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 51375882.421256 (7) Relativistic reduced mass multiplier = 0.9994556793115924 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 51375882.424514 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 1.000051 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 51378502.781152 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 51375882.372400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 2620.408752 (13) Magnetic potential adjustment = -2620.407994 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 5240.816746 PTV2 Eq (24) & Table 2 (15) H-f mid-point PTV = 51375882.373158 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Lower h-f energy from experiment = 51375882.301100 Horbatsch and Hessels (2016) (17) Lower h-f energy calculated from PTV = 51375882.301832 PTV2 Eqs (12)/(13) plus Eq. (24) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 70162334111.203232 (19) H-f difference taken from experiment = 0.071300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.071326 PTV2 Eq. (35) (21) H-f error magnitude = 0.000026 (22) A/gamma = -28657420.3795785376 _____________________________________________________________________________________________________ 1S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 1 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 1.7324035537303600 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 2.0000332836086264 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 3289885758.535002 (5) New PTV f-s energy no adjustments = 3289885758.260552 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 3288095006.026450 (7) Relativistic reduced mass multiplier = 0.9994556721864408 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 3288094955.765224 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99940209 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 3286128960.524324 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 3288087212.213100 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 1958251.688776 (13) Magnetic potential adjustment = 1958251.688776 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 3288087212.213100 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 3288087922.416000 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 3288087922.232933 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 17961132850.081810 (19) H-f difference taken from experiment = 710.202900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 710.019833 PTV2 Eq. (35) (21) H-f error magnitude = 0.183066 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 2S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8732511083690200 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001231494850968 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822474177.055886 (5) New PTV f-s energy no adjustments = 822474177.045523 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822026487.438713 (7) Relativistic reduced mass multiplier = 0.9994556776146438 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822026481.849575 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99970102 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 821780713.251798 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822025488.313800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 244775.062003 (13) Magnetic potential adjustment = 244775.062003 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 822025488.313800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 822025577.092200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 822025577.061922 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 27149405415.983552 (19) H-f difference taken from experiment = 88.778400 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 88.748122 PTV2 Eq. (35) (21) H-f error magnitude = 0.030278 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 3S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9163991001174104 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002241103507824 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365542862.051824 (5) New PTV f-s energy no adjustments = 365542862.054546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365343889.550878 (7) Relativistic reduced mass multiplier = 0.9994556786201884 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365343888.289990 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99980068 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365271066.392270 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365343591.599600 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 72525.207330 (13) Magnetic potential adjustment = 72525.207330 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 365343591.599600 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 365343617.904300 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 365343617.894686 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 38001009267.388160 (19) H-f difference taken from experiment = 26.304700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 26.295086 PTV2 Eq. (35) (21) H-f error magnitude = 0.009614 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 4S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9376529632206608 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003278449755088 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205617346.644788 (5) New PTV f-s energy no adjustments = 205617346.640405 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205505424.892383 (7) Relativistic reduced mass multiplier = 0.9994556789721682 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205505424.462619 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.9998505 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205474702.418402 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205505298.855200 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 30596.436798 (13) Magnetic potential adjustment = 30596.445663 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.008865 (15) H-f mid-point PTV = 205505298.864065 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 205505309.952500 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 205505309.957148 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 44210872203.422968 (19) H-f difference taken from experiment = 11.097300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 11.093084 PTV2 Eq. (35) (21) H-f error magnitude = 0.004216 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 5S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9503639585056512 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004326891036560 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594869.718334 (5) New PTV f-s energy no adjustments = 131594869.717838 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131523239.923151 (7) Relativistic reduced mass multiplier = 0.9994556791350940 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131523239.742134 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.9998804 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131507509.965646 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131523175.306400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 15665.340754 (13) Magnetic potential adjustment = 15665.354811 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.014058 (15) H-f mid-point PTV = 131523175.320458 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 131523180.988200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 131523181.000066 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 11084061606.235370 (19) H-f difference taken from experiment = 5.681800 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 5.679608 PTV2 Eq. (35) (21) H-f error magnitude = 0.002192 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 6S1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 0 Sp = 0.5 Sommerfeld: n(phi) = 1 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588462377676944 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005380879831920 PTV1 d2-bar Eq (86) gamma = -0.5490120000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91385208.595505 (5) New PTV f-s energy no adjustments = 91385208.586172 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335465.746194 (7) Relativistic reduced mass multiplier = 0.9994556792236000 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335465.647789 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99990034 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91326362.734225 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335428.313650 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 9065.579425 (13) Magnetic potential adjustment = 9065.594956 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.015531 (15) H-f mid-point PTV = 91335428.329181 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 91335431.601700 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 91335431.615971 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 6400143111.349888 (19) H-f difference taken from experiment = 3.288050 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.286790 PTV2 Eq. (35) (21) H-f error magnitude = 0.001260 (22) A/gamma = -28535879.7131210304 _____________________________________________________________________________________________________ 2P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8732511083690200 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001231494850968 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822474177.055886 (5) New PTV f-s energy no adjustments = 822474177.045523 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822026487.438713 (7) Relativistic reduced mass multiplier = 0.9994556776143044 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822026484.413397 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99988846 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 821934792.777153 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822026516.551150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 91723.773997 (13) Magnetic potential adjustment = 91723.773996 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000001 (15) H-f mid-point PTV = 822026516.551149 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 822026546.135900 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 822026546.133857 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 402313795865.109696 (19) H-f difference taken from experiment = 29.584750 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 29.582707 PTV2 Eq. (35) (21) H-f error magnitude = 0.002043 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 3P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9163991001174104 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002241103507824 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365542862.051824 (5) New PTV f-s energy no adjustments = 365542862.054546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365343889.550878 (7) Relativistic reduced mass multiplier = 0.9994556786200880 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365343888.897758 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99992564 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365316720.843363 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365343897.705100 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 27176.861737 (13) Magnetic potential adjustment = 27176.861737 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 365343897.705100 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 365343906.471000 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 365343906.470128 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 419165262473.347776 (19) H-f difference taken from experiment = 8.765900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 8.765029 PTV2 Eq. (35) (21) H-f error magnitude = 0.000871 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 4P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9376529632206608 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003278449755088 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205617346.644788 (5) New PTV f-s energy no adjustments = 205617346.640405 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205505424.892383 (7) Relativistic reduced mass multiplier = 0.9994556789721258 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205505424.670952 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99994423 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205493963.123335 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205505428.231800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 11465.108465 (13) Magnetic potential adjustment = 11465.108465 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 205505428.231800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 205505431.929900 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 205505431.929495 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 506807146944.176000 (19) H-f difference taken from experiment = 3.698100 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.697695 PTV2 Eq. (35) (21) H-f error magnitude = 0.000405 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 5P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9503639585056512 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004326891036560 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594869.718334 (5) New PTV f-s energy no adjustments = 131594869.717838 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131523239.923151 (7) Relativistic reduced mass multiplier = 0.9994556791350722 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131523239.831407 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99995538 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131517371.513011 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131523241.607050 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 5870.094039 (13) Magnetic potential adjustment = 5870.093147 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000892 (15) H-f mid-point PTV = 131523241.606158 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 131523243.500500 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 131523243.499361 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 115428758675.933600 (19) H-f difference taken from experiment = 1.893450 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.893203 PTV2 Eq. (35) (21) H-f error magnitude = 0.000247 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 6P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588462377676944 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005380879831920 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91385208.595505 (5) New PTV f-s energy no adjustments = 91385208.586172 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335465.746194 (7) Relativistic reduced mass multiplier = 0.9994556792235876 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335465.692108 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996282 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91332069.672574 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335466.701500 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3397.028926 (13) Magnetic potential adjustment = 3397.027639 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.001287 (15) H-f mid-point PTV = 91335466.700213 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 91335467.797200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 91335467.795810 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 65690885858.281528 (19) H-f difference taken from experiment = 1.095700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.095597 PTV2 Eq. (35) (21) H-f error magnitude = 0.000103 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 7P1/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 1 Sp = -0.5 Sommerfeld: n(phi) = 1 n(r) = 6 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649244845825792 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006438038634048 PTV1 d2-bar Eq (86) gamma = -0.2048164400200000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67140087.867507 (5) New PTV f-s energy no adjustments = 67140087.880927 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103542.136257 (7) Relativistic reduced mass multiplier = 0.9994556792769598 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103542.125220 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996813 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67101403.523239 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103542.752550 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 2139.229311 (13) Magnetic potential adjustment = 2139.228187 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.001123 (15) H-f mid-point PTV = 67103542.751427 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 67103543.442300 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 67103543.441362 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 71530566364.930816 (19) H-f difference taken from experiment = 0.689750 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.689935 PTV2 Eq. (35) (21) H-f error magnitude = 0.000185 (22) A/gamma = -28664089.3886580480 _____________________________________________________________________________________________________ 2P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 2 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 3.8733061076133600 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 4.0001764043401488 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 822463227.394777 (5) New PTV f-s energy no adjustments = 822463227.397312 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 822015543.737730 (7) Relativistic reduced mass multiplier = 0.9994556776143570 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 822015541.903204 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99987278 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 821910961.787048 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 822015535.670900 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 104573.883853 (13) Magnetic potential adjustment = 104573.883853 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 822015535.670900 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 822015547.496700 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 822015547.503353 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 123556526686.649088 (19) H-f difference taken from experiment = 11.825800 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 11.832453 PTV2 Eq. (35) (21) H-f error magnitude = 0.006653 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 3P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164531096983408 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002773654076936 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365539617.722525 (5) New PTV f-s energy no adjustments = 365539617.710736 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365340646.987534 (7) Relativistic reduced mass multiplier = 0.9994556786201034 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365340646.544107 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99991518 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365309659.480071 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365340644.107950 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 30984.627879 (13) Magnetic potential adjustment = 30984.627879 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 365340644.107950 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 365340647.611900 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 365340647.613837 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 188614301583.676352 (19) H-f difference taken from experiment = 3.503950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 3.505887 PTV2 Eq. (35) (21) H-f error magnitude = 0.001937 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 4P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377066388691312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003811001296544 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615977.950867 (5) New PTV f-s energy no adjustments = 205615977.934378 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205504056.943471 (7) Relativistic reduced mass multiplier = 0.9994556789721326 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205504056.779536 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99993639 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205490984.046214 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205504055.622150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 13071.575936 (13) Magnetic potential adjustment = 13071.575936 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 205504055.622150 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 205504057.100400 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 205504057.101188 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 260643026523.264640 (19) H-f difference taken from experiment = 1.478250 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.479038 PTV2 Eq. (35) (21) H-f error magnitude = 0.000788 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 5P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504174816877120 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004859443149632 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594168.930585 (5) New PTV f-s energy no adjustments = 131594168.941192 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522539.516855 (7) Relativistic reduced mass multiplier = 0.9994556791350758 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522539.464385 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99994911 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131515846.206529 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522538.831750 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 6692.625221 (13) Magnetic potential adjustment = 6692.624227 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000994 (15) H-f mid-point PTV = 131522538.830756 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 131522539.588600 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 131522539.588021 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 227119762806.494080 (19) H-f difference taken from experiment = 0.756850 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.757265 PTV2 Eq. (35) (21) H-f error magnitude = 0.000415 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 6P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 1 Sp = 0.5 Sommerfeld: n(phi) = 2 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588996786783616 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005913432321440 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384803.042340 (5) New PTV f-s energy no adjustments = 91384803.044546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335060.413780 (7) Relativistic reduced mass multiplier = 0.9994556792235896 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335060.384709 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99995759 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91331186.964930 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335060.003300 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3873.038370 (13) Magnetic potential adjustment = 3873.036852 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.001518 (15) H-f mid-point PTV = 91335060.001782 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 91335060.441300 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 91335060.440013 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 70984761078.276064 (19) H-f difference taken from experiment = 0.438000 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.438231 PTV2 Eq. (35) (21) H-f error magnitude = 0.000231 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 7P3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649778760733312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006970591389616 PTV1 d2-bar Eq (86) gamma = -0.2336101424100000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139832.496862 (5) New PTV f-s energy no adjustments = 67139832.496326 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103286.904616 (7) Relativistic reduced mass multiplier = 0.9994556792769612 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103286.886867 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996365 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67100847.645890 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103286.639850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 2438.993960 (13) Magnetic potential adjustment = 2438.992361 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.001600 (15) H-f mid-point PTV = 67103286.638250 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 67103286.915700 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 67103286.914220 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 45348559760.113704 (19) H-f difference taken from experiment = 0.275850 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.275970 PTV2 Eq. (35) (21) H-f error magnitude = 0.000120 (22) A/gamma = -28653017.2960224576 _____________________________________________________________________________________________________ 3D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164531096983408 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002773654076936 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365539617.722525 (5) New PTV f-s energy no adjustments = 365539617.710736 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365340646.987534 (7) Relativistic reduced mass multiplier = 0.9994556786200890 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365340646.573495 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99993331 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365316283.212186 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365340649.089800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 24365.877614 (13) Magnetic potential adjustment = 24365.877614 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 365340649.089800 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 365340651.193100 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 365340651.193332 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 1574466739955.637248 (19) H-f difference taken from experiment = 2.103300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 2.103532 PTV2 Eq. (35) (21) H-f error magnitude = 0.000232 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 4D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377066388691312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003811001296544 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615977.950867 (5) New PTV f-s energy no adjustments = 205615977.934378 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205504056.943471 (7) Relativistic reduced mass multiplier = 0.9994556789721264 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205504056.791161 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99994998 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205493778.458114 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205504057.762150 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 10279.304036 (13) Magnetic potential adjustment = 10279.304036 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 205504057.762150 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 205504058.649500 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 205504058.649573 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 2816818611796.838400 (19) H-f difference taken from experiment = 0.887350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.887423 PTV2 Eq. (35) (21) H-f error magnitude = 0.000073 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 5D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504174816877120 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0004859443149632 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131594168.930585 (5) New PTV f-s energy no adjustments = 131594168.941192 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522539.516855 (7) Relativistic reduced mass multiplier = 0.9994556791350726 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522539.469718 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99995999 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131517276.952313 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522539.938050 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 5262.985737 (13) Magnetic potential adjustment = 5262.985737 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 131522539.938050 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 131522540.392400 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 131522540.392409 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 14710547126880.131072 (19) H-f difference taken from experiment = 0.454350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.454359 PTV2 Eq. (35) (21) H-f error magnitude = 0.000009 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 6D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9588996786783616 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0005913432321440 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384803.042340 (5) New PTV f-s energy no adjustments = 91384803.044546 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91335060.413780 (7) Relativistic reduced mass multiplier = 0.9994556792235876 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91335060.387464 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996666 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91332014.945290 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91335060.647200 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3045.701910 (13) Magnetic potential adjustment = 3045.701753 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000157 (15) H-f mid-point PTV = 91335060.647043 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 91335060.910100 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 91335060.909982 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 771672186963.063936 (19) H-f difference taken from experiment = 0.262900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.262939 PTV2 Eq. (35) (21) H-f error magnitude = 0.000039 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 7D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649778760733312 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0006970591389616 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139832.496862 (5) New PTV f-s energy no adjustments = 67139832.496326 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103286.904616 (7) Relativistic reduced mass multiplier = 0.9994556792769598 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103286.888425 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99997142 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67101369.057318 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103287.046850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 1917.989532 (13) Magnetic potential adjustment = 1917.989289 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000243 (15) H-f mid-point PTV = 67103287.046607 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 67103287.212400 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 67103287.212189 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 317710270600.387392 (19) H-f difference taken from experiment = 0.165550 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.165582 PTV2 Eq. (35) (21) H-f error magnitude = 0.000032 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 8D3/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 8 L = 2 Sp = -0.5 Sommerfeld: n(phi) = 2 n(r) = 6 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 15.9695580628726016 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 16.0008029731643200 PTV1 d2-bar Eq (86) gamma = -0.1836735000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 51403919.639947 (5) New PTV f-s energy no adjustments = 51403919.634302 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 51375939.428839 (7) Relativistic reduced mass multiplier = 0.9994556793116008 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 51375939.413906 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99997499 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 51374654.616113 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 51375939.517400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 1284.901287 (13) Magnetic potential adjustment = 1284.901042 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000245 (15) H-f mid-point PTV = 51375939.517155 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 51375939.628300 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 51375939.628082 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 235399648065.257600 (19) H-f difference taken from experiment = 0.110900 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.110927 PTV2 Eq. (35) (21) H-f error magnitude = 0.000027 (22) A/gamma = -28658364.7014947744 _____________________________________________________________________________________________________ 3D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 3 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 0 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 5.9164711126492712 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 6.0002951168570048 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 365538536.306861 (5) New PTV f-s energy no adjustments = 365538536.289994 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 365339566.160507 (7) Relativistic reduced mass multiplier = 0.9994556786200946 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 365339565.811235 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99992917 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 365313689.392794 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 365339565.452000 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 25876.059206 (13) Magnetic potential adjustment = 25876.059206 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 365339565.452000 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 365339566.803700 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 365339566.804564 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 422803395572.332480 (19) H-f difference taken from experiment = 1.351700 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 1.352564 PTV2 Eq. (35) (21) H-f error magnitude = 0.000864 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ 4D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 4 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 1 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 7.9377245304992704 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 8.0003988515982480 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 205615521.724133 (5) New PTV f-s energy no adjustments = 205615521.709497 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 205503600.965070 (7) Relativistic reduced mass multiplier = 0.9994556789721288 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 205503600.836179 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99994688 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 205492684.168772 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 205503600.601850 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 10916.433078 (13) Magnetic potential adjustment = 10916.433078 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 205503600.601850 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 205503601.172200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 205503601.172462 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 785013275419.820672 (19) H-f difference taken from experiment = 0.570350 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.570612 PTV2 Eq. (35) (21) H-f error magnitude = 0.000262 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ 5D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 5 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 2 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 9.9504353224891328 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 10.0005036957947360 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 131593935.343814 (5) New PTV f-s energy no adjustments = 131593935.354022 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 131522306.057230 (7) Relativistic reduced mass multiplier = 0.9994556791350738 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 131522306.018397 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.9999575 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 131516716.669776 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 131522305.871950 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 5589.202174 (13) Magnetic potential adjustment = 5589.202175 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000000 (15) H-f mid-point PTV = 131522305.871950 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 131522306.163900 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 131522306.164103 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 648898144193.466240 (19) H-f difference taken from experiment = 0.291950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.292153 PTV2 Eq. (35) (21) H-f error magnitude = 0.000203 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ 6D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 6 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 3 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 11.9589174920513680 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 12.0006090947191888 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 91384667.867097 (5) New PTV f-s energy no adjustments = 91384667.866803 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 91334925.312115 (7) Relativistic reduced mass multiplier = 0.9994556792235884 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 91334925.287457 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996459 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 91331690.705315 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 91334925.192250 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 3234.486935 (13) Magnetic potential adjustment = 3234.486833 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000102 (15) H-f mid-point PTV = 91334925.192148 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 91334925.361200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 91334925.361217 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 5243270388806.605824 (19) H-f difference taken from experiment = 0.168950 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.169070 PTV2 Eq. (35) (21) H-f error magnitude = 0.000120 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ 7D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 7 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 4 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 13.9649956729696720 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 14.0007148106311456 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 67139747.364169 (5) New PTV f-s energy no adjustments = 67139747.369835 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 67103201.818263 (7) Relativistic reduced mass multiplier = 0.9994556792769602 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 67103201.810497 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99996964 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 67101164.870157 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 67103201.745800 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 2036.875643 (13) Magnetic potential adjustment = 2036.875428 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000216 (15) H-f mid-point PTV = 67103201.745584 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 67103201.852200 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 67103201.852054 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 458529645358.297344 (19) H-f difference taken from experiment = 0.106400 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.106469 PTV2 Eq. (35) (21) H-f error magnitude = 0.000069 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ 8D5/2 QUANTUM NUMBERS (1) Quantum Mechanics: n = 8 L = 2 Sp = 0.5 Sommerfeld: n(phi) = 3 n(r) = 5 BOUND STATE DISTANCES (as a muliplier of a fixed radius for all states) (2) Horizontal distance between p-e Sp-2 centers = 15.9695758490998720 PTV1 x-bar Eqs (84) (3) Shortest distance between p-e Sp-2 centers = 16.0008207246602976 PTV1 d2-bar Eq (86) gamma = -0.1950811000000000 FINE STRUCTURE (ALL ENERGIES IN MHz) (4) Sommerfeld f-s energy no adjustments = 51403862.601317 (5) New PTV f-s energy no adjustments = 51403862.606216 PTV1 Eq (54) (6) Sommerfeld f-s energy reduced mass = 51375882.421256 (7) Relativistic reduced mass multiplier = 0.9994556793116010 PTV2 (1 + Mr)^(-1) Eq (13) (8) New PTV f-s energy relativ. reduced mass = 51375882.418159 PTV2 (D = 1) Eq (13) (9) Velocity D multiplier = 0.99997344 PTV2 Eq (13) (10) New PTV f-s energy with relativistic reduced mass and adjusted velocity = 51374517.826145 PTV2 Eqs (12)/(13) MAGNETIC POTENTIAL ADJUSTMENT (ALL ENERGIES IN MHz) (11) Average h-f energy = 51375882.372400 Horbatsch and Hessels (2016) (12) Difference between PTV f-s with corrections and average h-f energy (magnitude) = 1364.546255 (13) Magnetic potential adjustment = 1364.546002 PTV2 Eq (22) & Tables 2 and 3 (14) H-f mid-point error magnitude = 0.000254 (15) H-f mid-point PTV = 51375882.372146 HYPERFINE SHIFT (ALL ENERGIES IN MHz) (16) Higher h-f energy from experiment = 51375882.443700 Horbatsch and Hessels (2016) (17) Higher h-f energy calculated from PTV = 51375882.443472 PTV2 Eqs (12)/(13) plus Eq. (22) minus Eq. (35) (18) Deviation of PTV from experiment is one part in 225669400955.246208 (19) H-f difference taken from experiment = 0.071300 Horbatsch and Hessels (2016) (20) H-f difference calculated from PTV = 0.071326 PTV2 Eq. (35) (21) H-f error magnitude = 0.000026 (22) A/gamma = -28655145.5912438432 _____________________________________________________________________________________________________ *** Free license, Barry R. Clarke (aleteller@barryispuzzled.com), 26 April 2026 ***