Engineering & Science
Hydrogenic Bohr Radius (an) & Fine-Structure Relativistic Velocity (vn/c)
Calculate the quantized orbital radius a_n = a_0 * (n^2 / Z) in picometers (CODATA a_0 = 52.9177 pm) and relativistic electron velocity ratio v_n / c = (Z * alpha) / n (fine-structure constant alpha = 1 / 137.036) for hydrogen-like ions.
See when high-Z inner-shell electrons approach relativistic speeds!
In Niels Bohr and Arnold Sommerfeld’s atomic scaling laws for any one-electron hydrogen-like ion (H, He+, Li2+, up to hydrogen-like Uranium U91+ with nuclear charge Z and principal quantum number n >= 1), quantized orbital radius expands quadratically with n^2 and shrinks inversely with Z relative to the fundamental CODATA Bohr radius a_0 = 4*pi*eps_0*hbar^2 / (m_e * e^2) = 52.9177 pm: a_n = 52.9177 * (n^2 / Z) pm!
Meanwhile, orbital speed relative to the speed of light c is governed directly by Sommerfeld’s dimensionless Fine-Structure Constant alpha = e^2 / (4*pi*eps_0*hbar*c) = 0.00729735 (~1 / 137.036): v_n / c = (Z * alpha) / n! For ground-state Hydrogen (Z = 1, n = 1), a_1 = 52.918 pm and v_1 / c = 0.007297 (0.73% of light speed—weakly relativistic). Jump to He+ ground state (Z = 2, n = 1 -> a_1 = 26.459 pm, v_1 / c = 0.014595) or heavy ions where Z * alpha approaches 1 (requiring Dirac relativistic wavefunctions for Z > 137)!
| Hydrogenic Ion State (Z, n) | Quantized Bohr Radius an (pm) | Velocity Ratio vn / c = Z * alpha / n | Relativistic Regime |
|---|---|---|---|
| Ground-State Hydrogen H (Z = 1, n = 1) | 52.918 pm (= a0) | 0.007297 (~1 / 137 c) | Non-relativistic Schrödinger |
| Excited Hydrogen n = 2 (Z = 1, n = 2) | 211.671 pm (= 4 * a0) | 0.003649 | Expanded 2s/2p shell |
| Singly Ionized Helium He+ (Z = 2, n = 1) | 26.459 pm (= a0 / 2) | 0.014595 (~1.46% c) | Tightly bound inner shell |
Hydrogenic Quantized Bohr Orbital Radius an (pm) = 52.9177 * (n^2 / Z)
Validates integer-like positive nuclear charge Z >= 1 (B2) and principal quantum shell n >= 1 (B3).
=IF(OR(B2 < 1, B3 < 1), "Invalid Quantum Numbers", ROUND(52.9177 * (B3 ^ 2) / B2, 3))Relativistic Orbital Speed Fraction vn / c = (Z * 0.00729735) / n
Multiplies Z / n by the CODATA fine-structure constant alpha = 0.0072973525693 (guarding against Dirac limit Z > 137).
=IF(OR(B2 < 1, B2 > 137, B3 < 1), "Invalid Quantum Numbers", ROUND((B2 * 0.00729735) / B3, 6))Try the example
Use the original values or change them to test your understanding.