Brus equation emission energy Δ"E" = "hc"/λ, where c is the speed of light). This is useful for calculating the radius of a quantum dot from experimentally determined parameters. The overall equation is "E", "m"*, and "m"* are unique for each nanocrystal composition. For example, with CdSe nanocrystals: Brus equation The Brus equation can be used to describe the emission energy of quantum dot semiconductor nanocrystals (such as CdSe nanocrystals) in terms of the band gap energy "E", Planck's constant "h", the radius of the quantum dot "r", as well as the effective mass of the excited electron "m"* and of the
Lane–Emden equation series solution: This leads to a recursive relationship for the expansion coefficients: This relation can be solved leading to the general solution: The boundary condition for a physical polytrope demands that formula_67 as formula_68. This requires that formula_69, thus leading to the solution: We start from with the Lane–Emden equation: Rewriting for formula_72 produces: Differentiating with respect to "ξ" leads to: Reduced, we come by: Therefore, the Lane–Emden equation has the solution when formula_77. This solution is finite in mass but infinite in radial extent, and therefore the complete polytrope does not represent a physical solution. Chandrasekhar believed for a