Thermal spectra and one-electron scaling
| English | 中文 | Pinyin |
|---|---|---|
| spectral radiance | 光谱辐亮度 | guāng pǔ fú liàng dù |
| series limit | 谱线系限 | pǔ xiàn xì xiàn |
A decision before an answer
- A spectral peak, a total radiation flux and an atomic transition energy are three different observables.
- Your goal: Distinguish wavelength spectral peak, integrated radiation and photon energy.
Identify the spectral measure
- For ideal thermal equilibrium radiation, Planck’s wavelength spectral radiance 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.
An ideal blackbody temperature doubles. Its wavelength peak and integrated emitted surface flux change by:
Wien gives inverse-T wavelength; Stefan–Boltzmann gives T⁴ integrated flux.
Separate power from one photon
- 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 heavy-nucleus Bohr one-electron model, r_(n=2) for Z=2 is:
r=a₀ n²/Z=a₀·4/2=2a₀.
Derive one-electron scaling
- 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=nℏ 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)]ℏ with l=0,…,n−1, so a 1s orbital has zero orbital angular momentum rather than the Bohr nℏ 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.
A 3000 K blackbody has wavelength peak 966 nm and peak-wavelength photon energy 1240/966≈1.284 eV. For ε=0.5,A=2×10⁻⁴ m²,T_env=300 K, net grey-surface power is 459.224073 W; the cold-environment approximation gives 459.27 W. Separately, a Z=2 one-electron ion emits n=3→2: 13.6·4(1/4−1/9)=7.5556 eV, λ≈164.12 nm. Its n=3 binding threshold is 54.4/9≈6.0444 eV, smaller than this photon energy because that photon ends at a lower n=2 level.
Using λ_max T=2.898×10⁻³ m·K, the wavelength peak at 6000 K is ____ nm.
2.898×10⁻³/6000 m=483 nm.
Subtract levels and locate limits
- 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.
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.
Which answer fits this case?
Distinguish wavelength spectral peak, integrated radiation and photon energy
The wavelength-density Planck peak and frequency-density Planck peak are related simply by ν_peak=c/λ_peak.
Changing the spectral interval introduces a Jacobian and changes the location of the maximum.
Keep the distinctions
- spectral radiance 光谱辐亮度 — 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.
- 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.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.