ABSTRACT
The additional influence of neighboring atoms other than the nearest neighbors is studied in Exs. 3-7. 2. Lattices with More Than One Atom per Unit Cell Consider a 1D crystal with two kinds of atoms of mass M and m where the distance between the nearest neighbors is “a” and the force constant is b in the Hooke’s approximation. Limiting the interactions to the nearest neighbors only, the equations of motion for these two types of atoms can be written as
mu u u u Mu u u un n n n n n n n 2 2 1 2 1 2 2 1 2 2 2 2 12 2= + - = + -+ - + + +b b( ), ( )From solutions of the form u A i t k nan2 2= - ◊exp ( )w and u B i t k n an2 1 2 1+ = - +exp [ ( ) ]wthe dispersion relation, w versus k, is w b b2 2 2 1 21 1 1 4= + 1ÊËÁ ˆ¯˜ ± +ÊËÁ ˆ¯˜ -ÈÎÍÍ ˘˚˙˙M m M m Mm kasin
The dispersion curves have two branches with a second branch called “optical branch,” which extends to a domain of high frequencies compared to the acoustic branch. The optical branch corresponds to the vibrations of two consecutive atoms in opposite direction, A/B = −1, that can be excited by electromagnetic (EM) radiation in the infrared region for ionic crystals thus leading to selective absorption of this EM radiation (see Ex. 10). This optical branch can exist even if the atoms of the basis are chemically identical but differ in their crystallographic environment and then on the inter-atomic force constant (see Ex. 1). Again in a 1D situation with two types of atoms at a distance of “a,” the lattice parameter is 2a and the total length of the first BZ is p/a. As a result the number of modes for a given polarization (L or T) corresponds again to the number of atoms able to move. For linear crystal consisting of a basis of p atoms, one obtains three acoustic branches (1L and 2T) and 3(p − 1) optical branches (see Ex. 2b and Pb. 8). The main results are summarized in Fig. 1.
