Question

why is that in a solid the allowed bands become wider as one proceeds from
the inner to the outer atomic electrons.​

Answer 1

Answer:

In a single isolated atom, the electrons in each orbit have definite energy associated with it. But in case of solids all the atoms are close to each other, so the energy levels of outermost orbit electrons are affected by the neighboring atoms.  

When two single or isolated atoms are bring close to each other then the outermost orbit electrons of two atoms are interact or shared with each other. i.e, the electrons in the outermost orbit of one atom experience a attractive force from the nearest or neighboring atomic nucleus.  Due to this the energies of the electrons will not be in same level, the energy levels of electrons are changed to a value which is higher or lower than that of the original energy level of the electron.

The electrons in same orbit exhibits different energy levels. The grouping of this different energy levels is called energy band.

However, the energy levels of inner orbit electrons are not much affected by the presence of neighboring atoms.