Question #44698

In a high-energy spectroscopic experiment, a hydrogen-like ion $X^{Z+}$ undergoes an electronic transition from $n=n_2$ to $n=n_1$, emitting a photon of energy $E_\gamma = h c R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$. This photon is incident on a metallic target with a work function $\Phi$, ejecting a photoelectron. If the atomic number $Z$ is sufficiently large such that the maximum kinetic energy of the ejected photoelectron is a significant fraction of its rest mass energy ($m_0 c^2$), which of the following expressions accurately represents the exact de Broglie wavelength ($\lambda_e$) of the fastest ejected photoelectron?