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A continuous tetragonal deformation connecting FCC and BCC geometries. In a simple uniaxial version, retain basal conventional edges $a,a$ while reducing the third edge to $a/\sqrt{2}$. The alternate vectors ${\mathbf{b}}_1=(a/2)(1,1,0)$, ${\mathbf{b}}_2=(a/2)(-1,1,0)$ and ${\mathbf{b}}_3=(0,0,a/\sqrt{2})$ form a BCC conventional cube, of half the volume of the deformed face-centered tetragonal cell. The name refers to Edgar Bain. Real metallurgical transformations can also change the other edges; this geometric path alone does not specify the energetic or kinetic transformation mechanism.

After compression along $z$, let the original conventional edges be $a,a,c$ with $c=a/\sqrt{2}$. This gives exactly the specified volume reduction. The compressed FCC points admit the alternative translation vectorsThey are mutually perpendicular, have equal length $b=a/\sqrt{2}$, and form a right-handed cube. Besides its corners, the lattice has the pointwhich is the compressed image of an original FCC face centre. All compressed FCC sites are generated by the cube's integer translations and this body-centre coset. ThereforeThe old compressed tetragonal conventional cell has volume $a^3/\sqrt{2}$ and four sites; the new BCC cube has half that volume and two sites. This verifies the same physical site density rather than silently losing atoms during the change of cell. The continuous uniaxial deformation is the Bain path, named for Edgar Bain.