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Calculate the iterval (in eV units) between neighbouring levelsof free electrons in a metal at T=0 near the Fermi level, if the concentration of free electrons is n= 2.0.10^(22)cm^(-3) and the volume of the metal is V=1.0cm^(3). |
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Answer» Solution :We WRITE expression for the NUMBER of electrons as `dN=(Vsqrt(2)m^(3//2))/(pi^(2)ħ^(3))E^(1//2)dE` Hence if `DeltaE` is the spacing between neighbouring levels near the FERMI level we must have `2=(Vsqrt(2)m^(3//2))/(pi^(2)ħ^(3))E_(F)^(1//2)DeltaE` (`2` on the `RHS` is to take care of both spins `f` electrons). Thus `DeltaE=(sqrt(2)pi^(2)ħ^(3))/(Vm^(3//2)E_(F)^(1//2))` But `E_(F)^(1//2)=(ħ)/(sqrt(2)m^(1//2))(3 pi^(2)n)^(1//3)` So `DeltaE=(2 pi^(2)ħ^(2))/(MV(3PI^(2)n)^(1//3))` Substituting the data we get `DeltaE= 1.79xx10^(-22)eV` |
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