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Below shown image is a cut out from the NCERT book.

F-centre

Referring to the highlighted sentences: I know that an $\ce{e^-}$ absorbs and emits light of a particular wavelength when bound in an atom and corresponding to that wavelength a particular colour is visible.

But in the case of F-centres as NCERT explains..... that an $\ce{e^-}$ in the F-centre when excites(i.e. absorbs a photon of a particular wavelength) and then de-excites, emits that photon and due to that ionic crystals possess some color.

But here comes the problem, how can an free $\ce{e^-}$ (present in the F-centre) excite? What's the phenomenon here which is going on? I am not able to understand that how will an $\ce{e^-}$ present in the anionic vacancies excite? Overall how F-centres impart color?

Link to the book.

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    $\begingroup$ It isn’t a free electron - it is bound to the point defect. $\endgroup$
    – Jon Custer
    Commented Dec 25, 2022 at 17:29

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A "sea" of free electrons looks silvery, i.e., reflective, as in solid and liquid metals, such as aluminum or gallium, and even in ionized dissolved metals, such as sodium in $\ce{NH3}$ (although the $\ce{Na+ in NH3}$ looks blue, at first, as electrons are interspersed by $\ce{NH3}$ molecules, as it becomes more saturated, it appears metallic, i.e., specularly reflective). The key thing to understand is that delocalized electrons can reflect EMR.

However, in the F-center, the electrons are not free, but are bound by the surrounding ions. These point defects not only stay in place, but can even be used for long-term data storage, as in nitrogen vacancies in diamond.

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You can explain the phenomenon using both crystal field theory and band theory.

Crystal field theory: The electron occupies the atomic site of a vacant Cl-; it will occupy the position that minimizes the repulsion of the surrounding Cl- ions. When you excite it, the electron will move and different higher energy levels will be available, dependent on the symmetry of the anion's environment and the strength of the crystal field.

Band theory: In NaCl there's wide gap between valence and the conduction band, so no visible radiation is absorbed. Vacancies in the Cl-substructure (filled by electrons) forms new levels in the band-gap; hence lower-energy excitations become allowed and some radiation in the visible spectrum can be absorbed.

see K. Nassau: The origins of color in minerals. American Mineralogist, Volume 63, pages 219-229, 1978

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