Elastic deformation in a crystal plate where lattice–parameter mismatch is present between adjacent growth sectors. I. Anisotropic elasticity theory, with application to lattice-parameter measurements

Abstract

A method is described for analysing strain in a plate–shaped crystal divided into growth sectors differing in lattice parameter and joined together at coherent dislocation–free sector boundaries running between the surfaces of the plate. The specimen studied was a large synthetic diamond, grown by the high–pressure hightemperature method, from which a near–central slice had been cut and polished parallel to (110). It contained dilatation–producing atomically dispersed substitutional nitrogen impurity, with adjacent growth sectors alternating in nitrogen concentration. In order that stress functions could be used to approximate the exact solution of the three–dimensional problem, the configuration of crystal growth sectors was modelled by an assembly of elongated rectangular parallelepipeds of alternating lattice parameter. Such a model is considered to be a good representation of the sector configuration of special interest in the real crystal. Using this model, the strains induced by lattice–parameter mismatch were calculated by adapting the classical theory of initial stresses due to Timoshenko and Goodier. In the present application, this involved four stages of imaginary operations: disassembly of the crystal model into stress–free sectors, deforming them by uniform lattice–matching strains, in planes parallel to the sector boundaries, ‘welding’ them together, and restoration of the initial conditions by application of ‘annulling’ stresses to cancel those applied in the deformations. Because uniform strains were applied in the matching procedure, only forces on the free surfaces of the crystal were required in this final stage, and it is argued that these should be applied in plane strain in planes orthogonal both to the sector boundaries and to the crystal–plate surface. Stress functions for orthorhombic symmetry are derived from first principles and applied to the geometry of the present problem, namely that of a cubic crystal under plane strain in which the cubic axes are rotated by 45 ° about the axis determined by the sector–boundary normal, [001]. Displacements and strains have been calculated at all points within the sector cross–section of interest, and extended to include wide variations in ratios between sector widths, and between sector width and plate thickness; the physical reasons for the resultant changes in strain distributions are discussed. Convergence of the Fourier–series solution under this variation of parameters is examined in detail. Representative deformation profiles of (110) planes in the real crystal are exhibited. The calculation has enabled a close limit to be placed on systematic error due to coherency strains in the findings of earlier X–ray–diffractometric experiments on the specimen discussed in the present study. Those experiments compared the symmetrical 440 surface Bragg reflection from a large (11 bar 1) growth sector fairly rich in substitutional nitrogen impurity with that from an adjacent, smaller, (11 bar 0) sector, virtually nitrogen free. The measurements aimed to establish the proportionality factor between substitutional nitrogen concentration and dilatation. The present analysis supports the earlier X–ray diffractometry: it changes the finding for Δa 0 /ā 0 quite insignificantly from (1.18 ±) × 10 −5 to (1.20 ± 0.07) × 10 −5 . On the other hand, revision of the nitrogen–concentration difference between the two growth sectors concerned, resulting from a more recent conversion factor between infrared absorption and nitrogen content (independent of the present study), raises that difference from ca .88 to ca .100 ppm atomic. Combining this large change with the small change in Δa 0 /ā 0 reduces the earlier finding for the ratio of the effective volume of a single substitutional nitrogen atom to the volume of the carbon atom it replaces from 1.41 ± 0.06 to 1.36 ± 0.1.