熱スペクトルと一電子スケール
導入済み| English | 意味 |
|---|---|
| spectral radiance/ˈspektrəl ˈreɪdɪəns/ | スペクトル放射亮度 |
| series limit/ˈsɪəriːz ˈlɪmɪt/ | スペクトル系列の極限 |
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:
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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:
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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.
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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.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.