When the ground-state electron of hydrogen is ionised, the emitted photon lies at the Lyman limit. For infinite nuclear mass, this threshold wavelength is λ Ly ≡ λ Ly ( ∞ ) = 1 / R ∞ = 91.1267 nm [1], where R ∞ is the Rydberg constant in the infinite-nuclear-mass limit; the finite-proton-mass (real hydrogen) threshold is denoted λ Ly ( real ) and treated separately in Section 6. The size of the atom is set by the Bohr radius a 0 = 52.9177 pm [1]. Their ratio is

Standard treatments note only that this is “of order 1 / α 2 ” without making its exact value or geometric content explicit. Here we show that the ratio is not approximately 1 / α 2 but exactly 4 π / α , with every factor traceable to a distinct physical origin. This exactness holds within the idealised, infinite-nuclear-mass, non-relativistic Coulomb model in which R ∞ and a 0 are themselves defined; the measured hydrogen threshold additionally contains reduced-mass, relativistic, recoil, radiative, and finite-nuclear-size corrections not captured by 4 π / α alone (Section 6 treats the leading, reduced-mass term quantitatively).

The fine-structure constant, introduced by Sommerfeld [2], is

determined most precisely by atom-recoil interferometry with rubidium [3] and cesium [4], both agreeing with the CODATA 2022 adjustment to within 0.3 ppb—negligible next to the 10⁻⁸ precision of the identity below.

The relation λ Ly / a 0 = 4 π / α follows directly from the standard definitions a 0 = ℏ / ( m e c α ) and R ∞ = m e c α 2 / ( 2 h ) : substituting into R ∞ a 0 gives α / ( 4 π ) immediately, so the identity is implicit in every standard reference [5] [6], even though it is not stated explicitly there. Our contribution is 1) the observation that 4 π = 2 π phase × 2 virial , where each factor has a distinct physical origin; 2) the recognition that, in the h -based convention for photon energy, these two factors map onto the azimuthal and polar integrals of the sphere, so that λ Ly / a 0 = Ω sph / α —an interpretive, not derivational, correspondence made precise in Section 4; 3) a numerical verification to 10⁻⁸ including the reduced-mass correction; and 4) the hierarchy of hydrogen’s three electromagnetic length scales and its extension to hydrogenic ions and atomic units. We derive no new physics and make no claim to obtain α from first principles.

Table 1 lists the symbols used, with CODATA 2022 values [1] where applicable; SI units throughout.

Table 1.Notation, definitions, and numerical values. Measured constants use CODATA 2022 [1]. ^†Mathematical constant; not a measured quantity.

This section derives the exact identity between λ Ly , a 0 , α , and the number 4π; its geometric reading as a solid angle (Section 4) is an interpretation added afterward and does not affect the derivation.

Bohr radius. From Equation (2) and a 0 = λ ¯ C / α :

Rydberg energy. The ground-state ionisation energy (infinite nuclear mass) is

where 1 2 is fixed by the virial theorem for the Coulomb potential, 〈 T 〉 = − E total (Section 4).

Lyman-limitwavelength. The photon emitted for n = 1 → ∞ has energy E γ = E R , so

The 4π here has two independent origins: h = 2 π ℏ contributes a factor 2π, and the 1 2 in the denominator of E R , moved to the numerator, contributes a factor 2.

Main identity. Forming λ Ly / a 0 from Equations (3) and (5):

m e , c , and ℏ cancel exactly: the ratio of a macroscopic photon wavelength to a sub-angstrom radius depends only on α and the pure number 4π. Equivalently, using R ∞ = 1 / λ Ly ,

Semiclassical cross-check. The hydrogen 1s state is spherically symmetric and has no definite orbit or speed; the following is Bohr-model shorthand, retained only because it is pedagogically standard. With the characteristic speed scale v 1 = α c set by 〈 T 〉 = 1 2 m e v 1 2 = E R , the semiclassical de Broglie wavelength is

exactly one Bohr-orbit circumference. This is simply the Bohr quantization condition ( n λ dB = 2 π a 0 at n = 1 ) from which a 0 is itself derived, so it is a self-consistency check on a 0 ’s definition rather than an independent verification of Equation (6).

The correspondence developed here—between the 2π and 2 of Equation (5) and the azimuthal and polar integrals of the sphere—is an interpretive analogy, not a new derivation: it illuminates the origin of the prefactor 4π but does not alter Section 3.

Three electromagnetic length scales appear naturally in hydrogen physics, each expressible via λ ¯ C = ℏ / ( m e c ) :

From r e to a 0 the scale grows by 1 / α 2 ≈ 1.88 × 10 4 ; from a 0 to λ Ly by a further 4 π / α ≈ 1722 . The three scales span about seven and a half orders of magnitude ( λ Ly / r e = 4 π / α 3 ≈ 3.2 × 10 7 ), yet every ratio is a pure number built from α and 4π alone (Table 2, Figure 2—bar lengths in the figure are illustrative only and do not reproduce this true separation).

Table 2.The three characteristic electromagnetic length scales of hydrogen. CODATA 2022 values [1].

Physically: r e = α 2 a 0 is where the electron’s electrostatic potential energy equals its rest-mass energy, e 2 / ( 4 π ε 0 r e ) = m e c 2 , setting the scale of Thomson scattering [7]. a 0 = λ ¯ C / α is where potential and kinetic energy balance via the virial theorem, fixing the ground-state radius [5]. λ Ly = ( 4 π / α ) a 0 is the wavelength of the photon that frees the ground-state electron; its ratio to a 0 is exactly Ω sph / α .

Since α = v 1 / c is the characteristic ground-state electron speed relative to c (semiclassically, Section 3; [5]), the identity reads λ Ly = 4 π ( c / v 1 ) a 0 : the photon’s wavelength exceeds the Bohr radius by 4π times the ratio of characteristic speeds, with 4 π the geometric factor converting a speed ratio into a length ratio.

Standard textbooks [5] [6] introduce α , a 0 , R ∞ , E R separately and compute λ Ly numerically, so λ Ly / a 0 = 4 π / α is implicit but never stated. It operates at O ( α 2 ) (the order of E R and a 0 themselves), well below the O ( α 4 ) fine-structure splitting [7]; relativistic corrections to E R , of order α 2 = 5.3 × 10 − 5 , are negligible here. By the Buckingham Π theorem, any dimensionless λ Ly / a 0 in non-relativistic hydrogen must be a function of α alone (the small m e / m p enters only as the separate reduced-mass correction above); Equation (6) shows that function to be precisely 4 π / α , verifiable to ten significant figures given CODATA’s δ α / α = 1.5 × 10 − 10 [1].

The applications below follow from λ Ly / a 0 = 4 π / α ; any solid-angle language remains the interpretive shorthand of Section 4.

Atomic units. R ∞ a 0 = α / ( 4 π ) (Equation (7)) bridges Hartree units ( ℏ = m e = e = 1 ) and Rydberg units ( 2 m e = ℏ = e 2 / ( 2 ⋅ 4 π ε 0 ) = 1 ) directly, with 4π acting as a universal bridge constant between length and energy scales whenever α is the coupling.

Hydrogenic ions. For nuclear charge Z , E 1 ( Z ) = Z 2 E R and a 0 ( Z ) = a 0 / Z [5], giving

This is exact within the same non-relativistic, point-nucleus model used throughout, but should not be read as exact across the full isoelectronic sequence: as Z grows, ( Z α ) 2 relativistic corrections and finite-nuclear-size effects become essential (e.g. for He⁺ through U⁹¹⁺), as treated in the standard reference on relativistic hydrogenic ions [9]. Equation (16) is best understood as the leading-order term of an expansion in Z α , valid for low- to moderate- Z systems.

Two-photon spectroscopy. The 1S - 2S transition, the most precisely measured atomic transition (4 × 10⁻¹⁵ [1]), has 1S ionisation energy E R and 1S - 2S interval 3 4 E R ; both scale as E R = 1 2 m e c 2 α 2 , so any future revision of α propagates directly into λ Ly / a 0 via 4 π / α with no free parameters, making Equation (10) a useful diagnostic in global CODATA adjustments.

Multi-electron atoms(speculative). An effective-charge analogue, λ ion / a 0 * ≈ Ω sph / ( Z eff α ) with Z eff ≈ 1.34 for helium, is suggested only as a possible direction for future work; no calculation establishing its validity is given here.

Pedagogy. λ Ly / a 0 = Ω sph / α packages four separate textbook definitions ( α , a 0 , R ∞ , E R ) into one relation, making explicit that a single coupling α and the geometry of three-dimensional space ( Ω sph = 4 π ) suffice to connect hydrogen’s two most fundamental length scales—a convenient entry point for discussions of naturalness and dimensional analysis [6].

The ratio of the hydrogen Lyman-limit wavelength (infinite nuclear mass) to the Bohr radius satisfies the exact identity

where Ω sph = 4 π sr is the solid angle of a sphere. In summary:

1) The identity follows from four lines of algebra using only the standard definitions of α , a 0 , R ∞ ; no model beyond textbook quantum mechanics is needed.

2) 4 π = 2 π × 2 , with 2π from h = 2 π ℏ (one full phase cycle in the h -convention) and 2 from the virial theorem, 〈 T 〉 = − E total .

3) These two factors are structurally identical to the sphere’s azimuthal and polar integrals when photon energy is written as h ν . This correspondence is interpretive and convention-dependent—it does not imply angular integration over atomic wavefunctions or emission patterns—whereas the identity λ Ly / a 0 = Ω sph / α itself is exact and convention-independent.

4) r e , a 0 , λ Ly form a hierarchy governed by specific powers of α and 4π.

5) The identity extends to leading order in Z α , to hydrogenic ions as λ Ly ( Z ) / a 0 ( Z ) = Ω sph / ( Z α ) .

6) The finite proton mass shifts the ratio by +0.054%, from 1722.045 to 1722.983, matching the measured Lyman limit.

7) The three numerical routes in Section 6 are consistency checks among algebraically linked CODATA constants, confirming internal consistency to 10⁻⁸, not an independent experimental test.

The exact ratio of hydrogen’s two most fundamental length scales reduces to the fine-structure constant and the pure number 4π. We offer this as a bridge constant linking Hartree and Rydberg atomic units, an exact leading-order scaling law across the hydrogenic isoelectronic sequence, and a zero-free-parameter diagnostic for tracking future revisions of α . The solid-angle correspondence behind its 4π prefactor is a formal, convention-dependent one, not evidence of a physical angular integration—but within that stated scope it gives hydrogen’s length-scale hierarchy a geometric reading left implicit in standard textbook treatments.

All numerical results and figures were produced with the Python script plots.py (Supplemental Material). The key check:

Figure S1.Ratio Vs Alpha.

Figure S2.Sensitivity.