Theory of domain growth in an order-disorder transition

We consider the growth of order in a two-dimensional Ising system with spin-flip dynamics (no conservation laws) quenched from above to below $T_c$. We use an iterative method, which allows for the interaction among many length and time scales, to calculate correlation functions in coordinate and Fourier space as a function of the time after quench. The method which we use to derive recursion relations is a reformulation of our previous work. We derive analytically scaling forms for the "Bragg peak" part of dynamic structure factor $\tilde{C}(\overrightarrow{\mathrm{q}},t)$, the nearest-neighbor correlation function, the width of the central peak, $q_w$, and the maximum value of the peak. We evaluate the corresponding scaling functions. These scaling laws for the width and peak height are new and elucidate the role of the final temperature in the problem. In particular, we find that the peak height grows as $t^{\frac{7}{8}}$ for quenches to precisely $T_c$. Our results also show that the Cahn-Allen curvature-driven growth law, $q_w\sim t^{\frac{-1\mathrm{}}{2}}$, is valid after relatively short times in this system. Our results agree quantitatively with Monte Carlo calculations in direct comparisons.