is the average grain size of the columnar grains, 
is the diffusional flux along the grain boundaries,
![$ \partial c^B/\partial x
= \left[ c^B(0) - c^B(l) \right]/l$](img140.png){kind=small}
, where 
and 
are the diffusant concentrations in the boundaries at the source
surface and accumulation surface, respectively, and 
is the specimen thickness. In the early stages,
and, therefore, to a good approximation, ![$\displaystyle \rho = \frac{4}{\pi \delta^2} \frac{^\star \! D^{XL}}{^\star \!
D^D}
\left[ \frac{\ave {^\star \! D}}{^\star \! D^{XL}} -1
\right]
$](img145.png)
