Equation (1) is reduced to the dimensionless form using the reciprocal lattice vector
where the neutron-electron coupling constant in parenthesis is -0.27 X [10.sup.-12] cm, [tau] and M are unit vectors in the direction of the reciprocal lattice vector [tau] and the spin direction, respectively, and the orientation factor <1-[([tau] * M).sup.2]> must be calculated for all possible domains.
The scattering for a nuclear Bragg peak always preserves the spin alignment of the neutron (non-spin-flip scattering), while the magnetic cross sections depend on the relative orientation of the neutron polarization P and the reciprocal lattice vector [tau].
Note that the [a.sup.*] axis of the reciprocal lattice vector is not identical to the crystal a-axis of a mono-clinic crystal; e.g., it is tilted about 9 [degrees] away from the crystal a-axis of the monoclinic crystal of PP.
It was found that all reciprocal lattice vectors of (110), (040), and (130) planes aligned perpendicular to the direction of rolling with preferential orientation in the film normal direction.
In Equations (3) and (4) [beta] = 2[pi][[epsilon].sup.1/2.sub.0]/[lambda] where [[epsilon].sub.0] = [epsilon](G = 0) is the averaged dielectric permittivity ([[epsilon].sup.1/2.sub.0] corresponds to averaged refractive index n), [alpha] an averaged gain in the medium, K(G) the coupling constant, [lambda] the Bragg wavelength, G = (m[b.sub.1], n[b.sub.2]) the reciprocal lattice vector
, and m and n the arbitrary integers.
If b is rational, f is periodic, the two delta functions can be indexed with a single reciprocal lattice vector
. If b is irrational, f is quasiperiodic; there are two incommensurate lengths in the Fourier transform; D = 2.
One more difference from strictly periodic PhC is the definition of the reciprocal lattice vectors
If we assume the basis functions are those of an empty lattice with [[epsilon]bb](r) = 1, then the basis functions are given by plane waves with wavevectors k + G, where k and G are the Bloch and reciprocal lattice vectors
where [k.sup.m.sub.T] are the reciprocal lattice vectors
, depending on [m.sub.1] and [m.sub.2] and [r.sub.T] = x[u.sub.x] + y[u.sub.y].