The Avrami law for the fraction transformed, y(t)=1exp(ktn), can be (sort of) justified using various different growth models. (a) For small t, show that y(t)=ktn. (b) Suppose that all the growing nuclei are spherical and form very early in the phase transformation (near t=0 ), and that their radius grows linearly with time; r=vt, where v is the growth velocity. Determine, at short times t, how the total volume of the nuclei scales with t, and thus how y(t) scales with t. What is the Avrami exponent n in this case? (c) Suppose that all the growing nuclei are spherical and form very early in the phase transformation (near t=0 ), but that their volume grows linearly with time; r=vt1/3. What is the Avrami exponent n in this case? (d) Suppose that the radius of a growing nucleus increases linearly with time (r=vt), but now the nuclei appear throughout the phase transformation at a constant rate of N nuclei per unit time. Calculate the Avrami exponent in this case. Hint: at a given time, nuclei will be all different sizes, depending on when they first appeared. You need to integrate over this distribution