The dynamical or wave-mechanical theory of electron diffraction is extended to include several diffracted beams. In the Brillouin zone scheme this is equivalent to terminating the incident crystal wave vector at or near a zone edge or corner. The problem is then one of determining the energy levels and wave functions in the neighborhood of a corner. The solution of the Schrödinger equation near a zone corner is a linear combination of Bloch functions in which the wave vectors are determined by the boundary conditions and the requirement that the total energy be fixed. This leads to a multiplicity of wave vectors for each diffracted beam giving rise to interference phenomena and is an essential feature of the dynamical theory.

At a Brillouin zone edge formed by boundaries associated with reciprocal lattice points $S$ and $G$, the orthogonality of the unperturbed wave functions in conjunction with the periodic potential requires that another reciprocal lattice point $\lambda$ be included in the calculation. The indices of $\lambda$ must be such that $\left({\lambda }_1{\lambda }_2{\lambda }_3\right)=\left(s_1{s}_2{s}_3\right)-\left(g_1{g}_2{g}_3\right)$. The perturbation at the zone edge results in non-zero amplitude coefficients $C_g$, $C_s$, and $C_{\lambda }$ for the diffracted waves irrespective of whether or not the structure factor for $\lambda$, $s$ or $g$ vanishes. This is the basis of the explanation of the (222) reflection and since it arises through perturbation at a Brillouin zone edge or corner, the term "perturbation reflection" is advanced to replace the commonly used "forbidden reflection."

The octahedron formed by the (222) Brillouin zone boundaries exhibits an array of lines due to intersections with other boundaries to form edges. This array of lines is called a "perturbation grid" and the condition for the occurrence of a (222) reflection is simply that the incident wave vector terminate on or near a grid line. Numerical intensity calculations are presented which show that a strong (222) can be accounted for by the dynamical theory.

An impedance network model is briefly discussed which may aid in qualitative considerations of the dynamical theory for the case of several diffracted waves.

• Received 16 September 1949

DOI:https://doi.org/10.1103/PhysRev.77.271