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Color centers envisioned as confined quantum systems: the case of F, F' and F⁺₂ centers

J.L. Marín^a,*, R. Acevesª, R.A. Rosas^b and S. Grijalva^b

ª Departamento de Investigación en Física, Universidad de Sonora, Apartado Postal 5–088, 83190, Hermosillo, Son. MÉXICO.

^b Departamento de Física, Universidad de Sonora, Apartado Postal 1626, 83000, Hermosillo, Son. MÉXICO.

* Corresponding author: ]]> e–mail: jmarin@cajeme.cifus.uson.mx

Recibido el 9 de mayo de 2007
Aceptado el 18 de septiembre de 2007

Abstract

Color centers in alkali halides, as well as point defects with dimensions of a few nanometers, have been considered to be confined systems and were studied with a variational formalism within a semi–continuum model. This new approach was applied to the well–known F, F' and F⁺₂ centers, which are assumed to be cavities of a determined shape that can trap one or two electrons. Inside of the cavity, the electron is subject to a constant potential (V₀) related to the Madelung energy and outside of it, the potential is Coulomb type due to a continuum polarizable medium. Because the F, F' and F⁺₂confined systems were considered to be hydrogen–like, helium–like and F⁺₂–like molecular ion systems, respectively, the ansatz functions were constructed from wave functions corresponding to these kinds of systems. For these systems, the energy transition (ΔΕ) from the ground state to the first excited state in KCl crystals was calculated and compared with experimental and calculated values obtained from the literature. The ΔΕ behavior is shown for different values of V₀. It is worth mentioning that the formalism presented in this work would be useful for both graduate and undergraduate students embarking on the study of some properties of confined quantum systems, or some simple nanostructures as well.

Keywords: Color centers; semi–continuum model; variational method; confined systems.

Resumen

]]> Se presenta un estudio con un formalismo variacional dentro del modelo semicontinuo de centros de color en halogenuros alcalinos así como defectos puntuales con dimensiones nanométricas, los cuales fueron considerados como sistemas confinados. Este enfoque fue aplicado para los bien conocidos centros F, F' y F⁺₂, los cuales fueron asumidos como una cavidad de forma específica que puede atrapar uno o dos electrones. Dentro de la cavidad el electrón "siente" un potencial constante V₀ relacionado a la energía de Madelung, mientras que en la region exterior el potencial es de tipo Coulombiano debido al medio continuo polarizable. Las funciones correspondientes a los centros F, F' y F⁺₂ fueron construidas tomando en cuenta que estos sistemas fueron considerados como sistemas tipo hidrogenoide, helioide y ion molecular H⁺₂, respectivamente. Así, para estos sistemas se calculó la energía de transición ΔΕ del estado base al primer estado excitado en cristales KCl y se comparó con los valores experimentales, así como los calculados en otros trabajos. Se presenta también el comportamiento de ΔΕ para diferentes valores de V₀. Es importante mencionar que el formalismo presentado en este trabajo puede ser útil para estudiantes de licenciatura o posgrado que deseen incursionar en el estudio de algunas propiedades de sistemas cuánticos confinados o algunas nanoestructuras sencillas.

Descriptores: Centros de color; modelo semicontinuo; método variacional; sistemas confinados.

PACS: 61.72.Bb; 61.72.Ji; 68.65.–k

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Acknowledgments

One of the authors (JLM) would like to dedicate this work to the memory of Professor Carlos Ruiz–Mejía, his former teacher and very close friend and colleague.

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