  The elastic constant tensors for the hcp phases of three transition metals
(Co, Re, and Fe) are computed as functions of pressure using the Linearized
Augmented Plane Wave method with both the local density and generalized
gradient approximations. Spin-polarized states are found to be stable for Co
(ferromagnetic) and Fe (antiferromagnetic at low pressure). The elastic
constants of Co and Re are compared to experimental measurements near ambient
conditions and excellent agreement is found. Recent measurements of the lattice
strain in high pressure experiments when interpreted in terms of elastic
constants for Re and Fe are inconsistent with the calculated moduli.
