Find the fundamental length scale of atoms. Adjust charge, level, and reduced mass corrections instantly. Get clear unit conversions, plus downloads for reporting fast.
The Bohr radius is the characteristic length scale of the hydrogen atom. The base value is:
a₀ = 4π ε₀ ħ² / (mₑ e²)a₀ = ħ / (mₑ c α)For a hydrogen-like ion with atomic number Z and principal quantum number n:
rₙ = (n² / Z) · a₀mₑ with
μ = (mₑ M)/(mₑ + M), where M is nucleus mass.
| Case | n | Z | Reduced mass | Approx. rₙ (Å) | Use case |
|---|---|---|---|---|---|
| Hydrogen ground | 1 | 1 | No | 0.529 | Baseline atomic scale reference. |
| He⁺ ground | 1 | 2 | No | 0.265 | Radius halves for double nuclear charge. |
| Hydrogen n=3 | 3 | 1 | No | 4.761 | Excited-state orbit grows with n². |
The Bohr radius a₀ sets the natural length scale for hydrogen-like atoms and ions. It is the reference distance that appears when balancing Coulomb attraction with quantized angular momentum in the Bohr model. Even when you use modern quantum mechanics, a₀ remains a convenient unit for atomic-size comparisons, orbital scales, and simple estimates.
With standard physical constants, the calculator returns a₀ ≈ 5.29×10−11 m, which is 0.0529 nm or 0.529 Å. Angstrom and picometer units are common in chemistry and solid-state physics, while meters help when connecting to macroscopic models and simulations.
For hydrogen-like systems, the model radius is rn = (n²/Z)·a₀. This means doubling the principal quantum number roughly quadruples the radius, while doubling nuclear charge halves it. For example, for H with n=2, r2 ≈ 2.12 Å. For He+ with Z=2 and n=1, r1 ≈ 0.265 Å.
Real atoms do not have an infinitely heavy nucleus. The reduced mass μ = (meM)/(me+M) slightly changes the effective radius. Because μ is just below me, the corrected radius is slightly smaller than the base value. The effect is tiny for hydrogen (parts per thousand), but it is useful for isotope comparisons and high-precision work.
Radius scaling is tied to energy scaling in the same model: larger orbits correspond to less tightly bound states. Since the radius grows like n², the typical binding energy decreases like 1/n². This relationship underpins simple estimates of spectral line spacing in hydrogen-like ions and helps interpret why excited states are more weakly bound.
Bohr-scale estimates show up in plasma physics (screening length comparisons), materials science (atomic spacing context), and spectroscopy (order-of-magnitude orbital sizes). When teaching, it is also a fast way to connect quantum numbers and nuclear charge to a physical size students can visualize.
The calculator includes a custom-constants mode so you can see how a₀ depends on inputs such as ε₀, ħ, and e. Because the base formula contains ħ² and e², small relative changes in those terms produce amplified changes in the computed radius. This is helpful when checking unit consistency in derivations.
The Bohr model is most accurate for hydrogen and hydrogen-like ions. For multi-electron atoms, electron shielding, relativistic effects, and quantum wavefunctions matter. Use the output as a physically meaningful scale and a comparison tool, not a full electronic-structure prediction.
The Bohr radius a₀ is the characteristic atomic length for hydrogen. It sets the size scale of the ground-state orbit in the Bohr model and is widely used as a reference unit in atomic physics.
In the Bohr model, angular momentum is quantized and increases with n. Balancing centripetal motion with Coulomb attraction leads to an orbital radius proportional to n², so higher energy levels are much larger.
Z increases the Coulomb attraction. For a single electron around charge Z, the radius scales as 1/Z. A He⁺ ion (Z=2) has about half the ground-state radius of hydrogen.
Enable it when you care about isotope-level accuracy or precision comparisons. Reduced mass accounts for finite nuclear mass, slightly changing the effective radius, especially for lighter nuclei.
They are equivalent: a₀ can be written using ε₀, ħ, and e, or using c and the fine-structure constant α. With consistent constants, both forms return the same numerical result.
Angstrom (Å) and picometer (pm) are common in atomic and molecular contexts, while nanometer (nm) is common in materials. Use meters for cross-domain calculations, simulations, or when combining with SI-based formulas.
It provides a useful scale but not a full prediction. Multi-electron atoms have shielding, electron–electron interactions, and relativistic corrections. For detailed sizes, use quantum chemistry or experimental data.
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