Element Z has four bound electronic energy levels relative to the ionized state (0 eV):
• E4 = −4 eV
• E3 = −7 eV
• E2 = −10 eV
• E1 = −13 eV (ground state)
In an experiment electrons are first excited to E4. Two distinct photon energies are subsequently recorded for relaxation events that end in the ground state:
• Photon X: 3.0 eV
• Photon Y: 6.0 eV
Assume no other radiative transitions are possible for this set of levels.
(a) Show quantitatively that a sequential two-step cascade can account for both recorded photon energies. Specify the order of level transitions in your explanation.
(b) For each recorded photon energy, calculate the corresponding wavelength and state whether it lies in the infrared, visible, or ultraviolet portion of the electromagnetic spectrum.
(c) Determine the wavelength of the single photon that would be produced if the electron were instead to drop directly from E4 to the ground state. Comment on whether this photon would have been detected if the experimental apparatus were limited to visible wavelengths (400 nm – 700 nm).
(d) Discuss how simultaneous (two-photon) emission from E4 to the ground state could conserve energy even though neither individual photon has energy equal to 9 eV. Include in your discussion the role of momentum conservation in such a process.