AT.3 · Thermal spectra and one-electron scaling

A spectral peak, a total radiation flux and an atomic transition energy are three different observables.

Prerequisites: 7, 9, 34.

• Distinguish wavelength spectral peak, integrated radiation and photon energy
• Derive Bohr radius and energy scaling under one-electron assumptions
• Calculate Coulomb-ion transitions, series limits and ionisation thresholds

For ideal thermal equilibrium radiation, Planck’s wavelength spectral radiance 光谱辐亮度 is B_λ=2hc²/{λ⁵[exp(hc/(λk_B T))−1]}. B_λ is per wavelength interval and per solid angle, not a total power. A blackbody’s hemispheric surface flux spectrum is M_λ=πB_λ; integrating over all wavelengths gives σ_SB T⁴. All temperatures are absolute Kelvin. The wavelength peak obeys λ_max T≈2.898×10⁻³ m·K. A spectrum per frequency has a different peak because B_ν dν and B_λ dλ include a Jacobian; c/λ_max is not the peak frequency of B_ν. Increasing T moves the wavelength peak shorter and increases the integrated flux by T⁴, not by the peak-position ratio alone.

For an original uniform grey surface of area A and wavelength-independent emissivity ε, net radiative power to a large uniform environment is εσ_SB A(T⁴−T_env⁴) under the stated view-factor assumptions. Use σ_SB≈5.670×10⁻⁸ W·m⁻²·K⁻⁴. Spectrally varying emissivity or incomplete surroundings requires a more detailed model. A photon at a specified wavelength has E=hc/λ, conveniently about 1240 eV·nm/λ_nm. A thermal spectrum contains many photon energies; a photon at the wavelength peak is not the mean energy of every photon. For T=3000 K, λ_max≈966 nm; its photon energy is about 1.284 eV. A surface with A=2×10⁻⁴ m² and ε=0.5 emits 459.27 W to a negligibly cold environment.

In the historical Bohr one-electron model with a heavy nucleus of charge Ze, Coulomb force m_e v²/r=Ze²/(4πε₀r²) and angular momentum m_e vr=n$\hbar$ lead to r_n=a₀n²/Z, v_n=Zαc/n and E_n≈−13.6Z²/n² eV. The radius scales with n²/Z while binding energy scales with Z²/n²; do not use the same charge power in both. These circular-orbit assumptions are a historical scaling model. Wave-mechanical angular momentum is √[l(l+1)]$\hbar$ with l=0,…,n−1, so a 1s orbital has zero orbital angular momentum rather than the Bohr n$\hbar$ value. Reduced-mass corrections replace m_e with μ: Coulomb radius scales as 1/μ and energy as μ. Multielectron screening, large-Z relativistic corrections and fine structure are outside the simple model.

For an emission ni→nf with ni>nf, E_γ≈13.6Z²(1/nf²−1/ni²) eV. Convert this positive difference to λ≈1240/E_γ nm. Ionisation from n requires energy 13.6Z²/n² eV to reach the continuum zero. Absorption reverses the level ordering and needs the corresponding positive incoming photon energy. For a series ending at fixed nf, the largest bound-bound photon energy occurs as ni→∞, giving E_limit=13.6Z²/nf² and the shortest series wavelength. For one-electron helium Z=2, n=3→2 gives 7.5556 eV and λ≈164.12 nm. Its Bohr n=3 radius is 4.5a₀; this radius describes the historical orbit, not a universal radial mode for every l at n=3.

Wien's law 维恩定律 refers to the wavelength-density peak of a blackbody spectrum.

A photon at that wavelength has energy The source emits a distribution of photon energies. Its integrated surface flux is $\sigma T^4$, and the frequency-density peak is not obtained by simply using c divided by this peak wavelength.

Declare a wavelength or frequency density, use Kelvin for fourth powers, and distinguish transitions from ionisation. A Bohr circular radius and angular momentum are not every wave orbital’s radius and L.

spectral radiance: Radiation intensity per projected area, solid angle and stated spectral interval.

series limit 谱线系限: Limiting bound-bound photon energy or wavelength as the initial level approaches the continuum for a fixed final level.