Energy Level Diagram of 3d2 Configuration in Tetrahedral Crystal Field and Its Applications to Cr4 + /Mn5 + ‐Doped Luminescent Materials

Abstract Tanabe–Sugano diagram is a successful guideline for the energy levels of 3d n electrons for half a century. However, it is developed only under the regular octahedral crystal field. With the surging research interests in the near‐infra‐red luminescence of 3d block transition‐metal elements, it becomes an urgent demand to know the energy level diagram of Cr 4 + and Mn 5 + with 3d 2 configuration, which are usually embedded in the tetrahedral site due to the smaller ionic size. Although the Tanabe–Sugano diagram of 3d 8 configuration in octahedron is usually used as a substitution for that of 3d 2 configuration in tetrahedron, the result mismatches with the crystal field strength and emission profile. In this work, irreducible tensor operator method is applied to derive the full analytical expressions of 3d 2 energy levels in the tetrahedral crystal field, based on which the Tanabe–Sugano diagram of 3d 2 configuration in tetrahedron is built for the first time. Also provided is a set of formulas for fast estimation of values, which shows consistency between the crystal field strength and the emission profile among Cr 4+ , Mn 5+ ‐doped luminescent materials. As it is hoped, this work could provide more deep insights into the luminescence of 3d 2 elements.