What Are the Work Function Limits of the Metal in Photon Emission?

[ProPatto16]

Homework Statement

hypothetical one electron searsium element. n=-20eV, n2=-10eV, n3=-5eV, n4=-2eV. photon emission n3>n2 and n3>n1 will eject photoelectrons from unkown metal, but photon emitted from n4>n3 will not. what are the limits (max and min values) of work function of the metal.

The Attempt at a Solution

photon emitted will have energy of 3eV. which is 4.806*10^-19 J. which equates to frequency of 7.25*10¹⁴ Hz.

work function = hf₀ where f₀ is minimum freq required to emit electron which is what we found.
so work function = 7.25*10¹⁴ x 6.626*10^-34
= 4.806*10^-19

so the work function of the metal is 3eV. but how is there a max and a min?

 

[ProPatto16]

since the minimum is 3eV cause that's what's required to emit a photo and drop from n4 to n3. and the max must be the same cause there's only the one possible energy it can be?