Step 1: The threshold frequency (\( u_{\text{th}} \)) is the minimum incident radiation frequency needed to eject photoelectrons from a surface. This frequency represents the minimum energy required to overcome the material's work function (\( \phi \)).
Step 2: Einstein’s photoelectric equation states:
\[E_{\text{photon}} = h u\]Here, \( E_{\text{photon}} \) is the energy of the incident photon. For photoelectron emission, the photon energy must meet or exceed the work function (\( \phi \)):
\[h u \geq \phi\]Step 3: The threshold frequency \( u_{\text{th}} \) is the lowest frequency where this condition holds true, meaning \( h u_{\text{th}} = \phi \). Therefore, the threshold frequency is calculated as:
\[u_{\text{th}} = \frac{\phi}{h}\]Step 4: When the incident radiation frequency is below the threshold frequency, the photons lack sufficient energy to overcome the work function, resulting in no photoelectron emission.