Mathematical verification: F = hbar^2/(gamma*m*r^3) vs k*e^2/r^2 Repository: https://git.esus.name/esus/spin_paper/ Bohr radius calculation: Force balance: hbar^2/(m*r^3) = k*e^2/r^2 Solving for r: r = hbar^2/(m*k*e^2) a0 calculated = 5.29177210255e-11 m a0 defined = 5.29177210903e-11 m ratio = 0.999999998776072 Hydrogen (Z = 1): Z_eff = 1.000000 radius = 52.918 pm = 5.291772e-11 m gamma = 1.000027 F_centripetal = 8.238504e-08 N F_coulomb = 8.238504e-08 N ratio = 0.999999998776072 agreement = 99.9999998776072% deviation = 1.224 ppb Carbon (Z = 6): Z_eff = 5.689918 radius = 9.300 pm = 9.300260e-12 m gamma = 1.000959 F_centripetal = 1.516219e-05 N F_coulomb = 1.516219e-05 N ratio = 0.999999998776072 agreement = 99.9999998776072% deviation = 1.224 ppb Gold (Z = 79): Z_eff = 78.688429 radius = 0.672 pm = 6.724969e-13 m gamma = 1.223838 F_centripetal = 3.279961e-02 N F_coulomb = 3.279961e-02 N ratio = 0.999999998776072 agreement = 99.9999998776072% deviation = 1.224 ppb Verification across periodic table: Formula: F = hbar^2/(gamma*m*r^3) vs F = k*e^2/r^2 Z Elem Z_eff gamma F_ratio Dev(ppb) ------------------------------------------------------------ 1 H 1.000 1.0000 0.999999998776 1.224 2 He 1.690 1.0001 0.999999998776 1.224 3 Li 2.690 1.0002 0.999999998776 1.224 4 Be 3.690 1.0004 0.999999998776 1.224 5 B 4.690 1.0007 0.999999998776 1.224 6 C 5.690 1.0010 0.999999998776 1.224 7 N 6.690 1.0013 0.999999998776 1.224 8 O 7.690 1.0017 0.999999998776 1.224 9 F 8.690 1.0022 0.999999998776 1.224 10 Ne 9.690 1.0027 0.999999998776 1.224 11 Na 10.690 1.0032 0.999999998776 1.224 12 Mg 11.690 1.0038 0.999999998776 1.224 13 Al 12.690 1.0045 0.999999998776 1.224 14 Si 13.690 1.0053 0.999999998776 1.224 15 P 14.690 1.0060 0.999999998776 1.224 16 S 15.690 1.0069 0.999999998776 1.224 17 Cl 16.690 1.0078 0.999999998776 1.224 18 Ar 17.690 1.0087 0.999999998776 1.224 19 K 18.690 1.0098 0.999999998776 1.224 20 Ca 19.690 1.0108 0.999999998776 1.224 26 Fe 25.690 1.0185 0.999999998776 1.224 47 Ag 46.689 1.0646 0.999999998776 1.224 79 Au 78.688 1.2238 0.999999998776 1.224 92 U 91.688 1.3493 0.999999998776 1.224 ------------------------------------------------------------ Statistical results: Elements calculated: 100 Mean ratio: 0.999999998776072 Mean agreement: 99.99999987761% Mean deviation: 1.223928 ppb Std deviation: 0.000000 ppb Min deviation: 1.223928 ppb Max deviation: 1.223928 ppb Range of deviations: 0.000001 ppb Deviation range < 1 ppb: True All ratios > 0.999: True All ratios < 1.001: True Unit verification: F = hbar^2/(gamma*m*r^3) [F] = [J*s]^2 / ([kg][m^3]) = [kg*m*s^-2] = [N] F = k*e^2/r^2 [F] = [N*m^2*C^-2][C^2] / [m^2] = [N] Key numbers: Mean force ratio across 100 elements: 0.999999998776072 Deviation from unity: 1.224 ppb Expected if identical: 0.000 ppb CODATA electron mass uncertainty: ~300 ppb