Question

In a linear atomic chain where each atom and bond are identical, and the distance between atoms is a.

A) Why are the vibrations with wave vectors k and k + 2π/a identical? What is the reciprocal lattice of this atomic chain?

B) If the strength of the atomic bonds begins to slowly grow in every other bond, what happens to the dispersion relation of the vibrations? How does the general description of the vibrations change?

Answer 1

a) as = 2sin(ka/2) i.e dispersion of normal waves , here if we change k - k+ , then also we get same result as sin((k+)*a/2) = sin ( ka/2 + ) = sin(ka/2) . i.e identical ( herre is stiffness constant)

b) if we consider our lattice as system of masses and spring of stiffness ( represent strength of atomic bond) , then if stregth of bond grows dispersion relation will chnange and the normal waves are not plane . and in the region of defect ,stiffness varies than those of ideal crystals callled "standard of matrix" . as only impurity atom take part in the vibration called local vibration . changes so as our dispersion relation changes