Phonon modes in one-dimensional monoatomic chain

Question

I am familirazing myself with the lattice dynamics reading Ashkroft-Mermin (p2 ch 22). My question is what it the mathematics behind the deriviation

$U^{harm}=1/2 K \sum_{n} [u(na)-u([n+1]a)]^2$ (22.22)

$\frac{\delta U^{harm}}{\delta u(na)}=-K[2u(na)-u([n-1]a)-u([n+1]a)]$ (22.23)

I feel like it is very basic and I fail to see it or I totally misunderstand the notation. ( I get something like $-K(u(na)-u[(n+1)a]$).

How it comes, that one of the displacements is for (n-1)a atom? Is the 22.23 has this form to fit the concept of this lattice to be a system of balls connected by spring?

Answer 1

Make sure when you vary your first equation that you use a different index instead of $n$. Do the following:

$$\frac{\delta U^{harm}}{\delta u(ma)}$$

this gives

$$ \frac{1}{2}K \sum_{n} 2 [u(na) - u([n+1]a)]\left[\frac{\delta u(na)}{\delta u(ma)} - \frac{\delta u([n+1]a)}{\delta u(ma)} \right] $$ by the chain rule. But $$ \frac{\delta u(na)}{\delta u(ma)} = \delta_{n,m} \text{ and } \frac{\delta u([n+1]a)}{\delta u(ma)} = \delta_{n+1,m} $$ Now we have four terms: \begin{equation} K\sum_{n}[u(na)\delta_{n,m} - u(na)\delta_{n+1,m} - u([n+1]a)\delta_{n,m} + u([n+1]a)\delta_{n+1,m}] \end{equation} and performing the sum for each term forces $n$ to take on the necessary values. This gives $$ K[u(ma) - u([m-1]a) - u([m+1]a) + u(ma)] = K[2u(ma) - u([m-1]a) - u([m+1]a)] $$