Volume and area of intersection of a ball and an infinite parallelepiped

In the paper we study the solid being a model of the new phase nucleus for a phase change reaction. The solid is the intersection of the ball of a given radius $R$ and an infinite parallelepiped, i.e. the cartesian product of the square with a given side $a$ and a line. Such model appears, e.g., when describing dehydriding of activated alane: numerous nuclei of new metal phase appear and grow as hemispheres, but later they intersect being cut off by planes. Their total surface increases, reaching the maximal value that exceeds the initial total surface area $S_{0}$ of the old phase, then reduces, asymptotically tending to $S_{0}$. This property can explain the higher dehydriding rate (which depends on the surface area of the new phase) in the middle of the dehydriding reaction. We calculate volume, surface area, and some other quantities for this solid as functions of $R$ and $a$. They are expressed via non-trivial integrals as elementary functions. Using these quantities, we present the conservative mathematical model describing the dehydriding reaction. Also we discuss properties of the obtained functions and the constructed model.