Beraha numbers and zeros of the chromatic polynomial of planar graphs

I don't think that there are known families of graphs with (real) chromatic roots converging to particular Beraha numbers.

Alan Sokal and Jesus Salas wrote a series of papers, the first of which is titled "Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial" (http://arxiv.org/abs/cond-mat/0004330) where they claim evidence that certain families have roots converging to $B_2$, $B_3$, $\ldots$, $B_5$.

One intriguing leftover from their paper is that they can show that despite the likelihood that they are accumulation points of roots, no Beraha numbers are actually chromatic roots themselves - except possibly $B_{10}$. (I've looked quite hard to find a chromatic polynomial with $B_{10}$ as a root, but with no luck.)

The graphs that I used in the paper of mine that you quoted were actually just modifications of the original family of graphs used by Beraha-Kahane in their famous paper "Is the four-color conjecture almost false?" So while I am not sure what you mean by periodic, I'm pretty sure that if the graphs they used are periodic, then so are the graphs that I used.

(One problem is that statistical physicists working with strips of lattices have to make some decision about the "edges" of their lattice; if they "wrap-round" so that the left-hand edge is identified with the right-hand edge, then they call that "periodic boundary conditions." But I don't think that that is the meaning you had in mind?)