Question

19. [3 pts.] Explain the effect on energy bands in the Kronig-Penney model when the barrier strength approaches (i) infinity; (ii) zero.

Answer 1

19. The most desirable outcome of laronig- Penney model is p Sin da + cas da = coska ----- I da VO Pem is sometimes known as the scattering power of the potential barrjen ano Vob is known as the bannien Strengthe. m=imass of electron { X= 2ME per Et energy of electron a Lattice constant Vo =+Hight of periodic potential bannien b = Width of potential barrier (i) When the potential barrier strength /vob) approaches in binity ie as P tends to infinite, the allowed energy "bands are compressed info energy levels and line spectrum is resulted. In such a case the equation o las solution only if Sin da=0 "Xa=nn - In is any integer) q, xa non & 2ME annan " Iwa 2. E como na This expression shows that energy is quantised.

W when potential banruen strength (Vob) approaches zero ise when Pto, equation becomes Cosna = Coska i da= ka => " KW , =) QME In - kn MW K > E= ? 4) аз емегеу еxр пиязои іѕ аррliсаn btn соmрладеlу finee edle/tto Bore which all the energies are allowed. These results are summarized in Figure below. On the extreme left it for P=0 the energy-spectrum is quasi- continuous and on the extreme rughtlié Bure -too, a line spectrum is resulted . P=0 P=0