Question

A sin(nkoa), where A and na Let us suppose that the interplanar force constant K is of the form, K = ko are constants and p runs over all integers. Such a form is usually expected in metals. Use this in the Force equation du(na) Fn = M t2 du(na) -K (2u(na) – u((n + 1)a) – u((n − 1)a)) = Mºu dt2 (1) Find oin terms of k and ko (ii) Find out (iii) Plot o’ vs k (iv) So far in this problem, we have considered the role of displacements of atoms immediately prior and next to the u(n). However, this can be extended to the other planes n' and the total force on the atom in the plane where we located u(n) can be calculated. In that case, the force will be: Fn = > Kn (Un+ni - Un) Obtain (i) and (ii) for the above force with Kn =- A sin(nkoa) na (v) (for the 2 bonus points) If you apply Lagrange's trigonometric identity you can show that the derivative goes to infinity at k=ko

Answer 1

Let up=uo expi (wt - k'na) Unt=uo exp [i (wl-k(111) a)] Up-1= 40 exp {i (cut-k(n-1) a)] K = A sin(nkoa) k is force constant and K' is ware vector. 1 - is wave vector - na Fn = M du (na) dir dt? na Mna -k[au(na) - ui(co-)a) – u (mn)] marum - Asin (ntoa [a - eik'a _ eikla] = - Mw2 w? Asinots) ( 2- 2 (eilla tells)] 22 - Asinchko) (2-2058)*k'a (02- 4 silnice) sin kla Faso - 2shty] 0w2 Za sin (nkoa) cos klasycoskla sin (nik, e) δ' AMOR 2 M 2 . Mna 1 4 Mna