Question #43629

A beam of electrons is accelerated from rest and directed normally onto the surface of a simple cubic crystal. The surface atoms form a 2D square lattice with a lattice constant $D = \sqrt{2} \text{ \AA}$. The lowest-order non-zero diffraction maximum in the azimuthal direction $\phi = 45^\circ$ (measured from a principal lattice axis) is observed at a polar angle $\theta = 60^\circ$ (measured from the surface normal).

The kinetic energy of these incident electrons exactly matches the energy of a photon emitted during the $n=3 \to n=1$ transition of a hydrogen-like ion embedded in a dense plasma. Due to plasma screening, the effective Rydberg energy of this ion is modified to $E_R^*$. Assuming the electron diffraction follows the 2D surface Laue condition and taking $\frac{h^2}{2m_e} = 150 \text{ eV \AA}^2$, what is the value of $E_R^*$?