(a) Compute the surface energy cost for a flat domain wall in three dimensions in the Ising model.

(b) Suppose that an Ising system at low temperature has nearly all spins aligned up. An external magnetic field is then applied, so that the lowest-energy state corresponds to all spins aligned down. To evolve to a state with all spins down, however, requires fluctuations that create domains large enough to overcome the surface energy cost. This is called nucleation. Assuming a spherical domain with radius large enough to make using the surface energy of part (a) reasonable, calculate the total energy cost of a fluctuation that creates a magnetic domain of the opposite spin with radius R. What is the critical radius for the nucleated domain to be stable?