text: The simplest derivation of the Avrami equation makes a number of significant assumptions and simplifications:
  * Nucleation occurs randomly and homogeneously over the entire untransformed portion of the material.
  * The growth rate does not depend on the extent of transformation.
  * Growth occurs at the same rate in all directions.
  If these conditions are met, then a transformation of  into  will proceed by the nucleation of new particles at a rate  per unit volume, which grow at a rate  into spherical particles and only stop growing when they impinge upon each other. During a time interval , nucleation and growth can only take place in untransformed material. However, the problem is more easily solved by applying the concept of an extended volume – the volume of the new phase that would form if the entire sample was still untransformed. During the time interval  to  the number of nuclei N that appear in a sample of volume V will be given by
  where  is one of two parameters in this simple model: the nucleation rate per unit volume, which is assumed to be constant. Since growth is isotropic, constant and unhindered by previously transformed material, each nucleus will grow into a sphere of radius , and so the extended volume of  due to nuclei appearing in the time interval will be
  where  is the second of the two parameters in this simple model: the growth velocity of a crystal, which is also assumed constant. The integration of this equation between  and  will yield the total extended volume that appears in the time interval:
  Only a fraction of this extended volume is real; some portion of it lies on previously transformed material and is virtual. Since nucleation occurs randomly, the fraction of the extended volume that forms during each time increment that is real will be proportional to the volume fraction of untransformed . Thus
  rearranged
  and upon integration:
  where Y is the volume fraction of  ().
  Given the previous equations, this can be reduced to the more familiar form of the Avrami (JMAK) equation, which gives the fraction of transformed material after a hold time at a given temperature:
  where , and .
  This can be rewritten as
  which allows the determination of the constants n and k from a plot of  vs . If the transformation follows the Avrami equation, this yields a straight line with slope n and intercept .