AT.1 · Photoelectrons, reduced mass and atomic excitation

Repeated equally spaced current dips can represent repeated losses of the same excitation energy, rather than a sequence of different atomic levels.

Prerequisites: 7.

• Distinguish photoelectric thresholds and characteristic X rays
• Apply reduced-mass spectral scaling and many-electron spin filling
• Interpret Franck–Hertz energy spacing without inventing new levels

Characteristic X rays result when an electron fills an inner-shell vacancy, emitting a photon equal to the shell energy difference. Their sharp lines depend on target atoms. Bremsstrahlung arises from deceleration of energetic electrons in nuclear electric fields and produces a continuous background, with an energy endpoint set by the incident electron energy. A continuous background and discrete lines can appear together; the existence of one does not exclude the other. These mechanisms differ from visible fluorescence, phonon scattering and particle capture.

For a hydrogen-like two-body atom, replace electron mass by reduced mass μ=m_eM/(m_e+M). In the simple Coulomb model, level energies scale as −μZ²/n², so spectral frequencies scale as μZ² and wavelengths inversely. For positronium M=m_e, μ=m_e/2 and its Rydberg constant is half the infinite-nuclear-mass value. For a heavy nucleus μ approaches m_e. In many-electron atoms, use orbital filling and Hund’s rule rather than the hydrogen model: degenerate orbitals are occupied singly with parallel spins before pairing. Carbon’s 2p² gives two unpaired electrons and total spin S=1; oxygen’s 2p⁴ has two unpaired electrons and also S=1. Filled pairs contribute zero net spin.

In Franck–Hertz experiments, accelerated electrons lose energy through inelastic excitation once they reach an atomic threshold. Repeated current-dip or peak spacing in accelerating voltage can therefore identify the same excitation energy lost one, two or more times. A spacing ΔV corresponds to energy eΔV; peaks at 4,8,12 V need not represent three separate excited-level energies. Contact potentials and retarding fields can shift absolute peak positions, so use the spacing and stated apparatus conditions. If the excited atom returns by one photon of that energy, wavelength is hc/(eΔV).

Current intensity and stopping voltage answer different questions. Include reduced mass and orbital degeneracy, and use Franck–Hertz spacing rather than treating every peak as a new energy level.

work function: Minimum energy required to remove an electron from the specified surface.

reduced mass: Two-body effective mass m1m2/(m1+m2) for relative motion.