  The diffusionless martensitic phase transition from a hexagonal close-packed
(hcp) arrangement to the face-centered close-packed (fcc) and subsequently the
body-centered cubic (bcc) lattice is discussed for a Lennard-Jones solid. The
associated lattice vectors to construct the underlying bi-lattice for a
Burgers-Bain-type of transformation require a minimum of four parameters
$(a,\alpha,\beta,\gamma=c/a)$ describing, beside the change in the base lattice
parameters $a$ and $c$, the shear force acting on the hexagonal base plane
through the parameter $\alpha$, and the sliding force of the middle layer in
the original AB hexagonal packing arrangement through the parameter $\beta$. By
optimizing the lattice parameters $a$ and $\beta$ for a $(n,m)$-Lennard-Jones
potential, we obtain a simple two-dimensional picture for the complete
Burgers-Bain-type hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition.
From the generalized lattice vectors we were able to construct the
corresponding lattice sums in terms of inverse power potentials applying fast
converging Bessel function expansions using a Terras decomposition of the
Epstein zeta function combined with a Van der Hoff-Benson expansion for the
lattice sums. This allows the cohesive energy to be determined to computer
precision for a Lennard-Jones solid. For six different combinations of
$(n,m)$-Lennard-Jones potentials the energy $(\alpha,\gamma)$ hypersurface was
then mapped out and studied in more detail. We show that for a Lennard-Jones
model the minimum energy path is found to be a two-step
hcp$\rightarrow$fcc$\rightarrow$bcc transition process. The lowest transition
state in each case can be regarded as an upper limit to a hypothetical true
minimum energy path out of the many possibilities in a
hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition.
