= Solution
The small parameter controlling the local <Taylor expansion> is the typical bond displacement divided by $\xi$. In the classical regime, the root-mean-square displacement is $x_{\mathrm{rms}}\simeq\sqrt{k_BT/\kappa}$. Thus
$$
\boxed{\frac{x_{\mathrm{rms}}}{\xi}\ll1\quad\Longleftrightarrow\quad
T\ll\frac{\kappa\xi^2}{k_B}=5.80\times10^3\,\mathrm K.}
$$
For example, the magnitude of the cubic term divided by the quadratic term is $|x|/\xi$, and subsequent terms carry further powers of this ratio. A physically safe bound-bond regime also has $k_BT\ll D$, or $T\ll2.90\times10^3\,\mathrm K$. These are the same parametric scale, differing only by a factor two. The inequalities are strong: they do not imply that a truncated local expansion is accurate close to several thousand kelvin.
At very low <temperature>, replace the classical fluctuation estimate by the <zero-point motion> as discussed in part ii. Its bond displacement is small compared with $\xi$ for the supplied parameters, so low <temperature> does not itself invalidate the local expansion. Actual melting, other neighbours and bond breaking are outside this one-dimensional bond model.
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