The Geometry of Lattice Planes

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To date, such ball models have been used only for cubic and hexagonal crystals, but they can be constructed for many other structures.
In papers with the above title, Jaswon & Dove (1955) and Bevis (1969) have presented systematic methods for mapping the projection of a lattice on to a plane of given Miller indices (hkl). The aim of th.is paper is to draw attention to another method, that of constructing a ball model of a crystal divided parallel to (hkl), and to note an error in the earlier papers. The essential problem is that of finding lattice vectors u,v,w such that u,v define a primitive mesh. in an to define a unit mesh in (hkl) and then searching out all extra lattice points within this mesh in order to determine a primitive mesh. They then look for a suitable w = wl, w2, u'3 by solving hwl + kw2 + lw3 = 1 . (2) On the other hand, Bevis (1969) chooses where d is the highest common factor of h and k, looks for an integral solution ml,m2 of and can then write down v=k(1 nt-m3)-hn2, -h(1 +m3)+hrll, -d, where m3 (or, more simply, l+m3) is an arbitrary integer. Equation (2) is then solved to give w.
Both methods calculate the shift vector t, which is the projection of w onto (hkl), as t=w-d, where d is a vector perpendicular to (hkl) and of magnitude equal to the spacing between (hkl) planes. The projections of successive planes are then plotted by calculating the magnitudes of and angles between u,v and t. Since w is not uniquely defined by equation (2), there are many possibilities for t and Jaswon & Dove specify that t shall be as short as possible; even this restriction often leaves t ambiguous (see, for example, Fig. 2). It may be noted that Bevis uses h to denote the vector d in his equations (4) and (14) although in his Introduction and in the preamble to equation (14) he has used h in the more usual sense as the vector with components (hkl) relative to the reciprocal axes, i.e. as the vector parallel to d but of length equal to the reciprocal of the interplanar spacing.
Equations (2) and (4) can normally be solved very quickly by a trial-and-error process but, since the necessary algorithms exist, it is of course possible to carry out the whole process on a computer (e.g. Bacigalupi, 1964). This only gives immediate and complete results for those fortunate enough to have direct access to a terminal with plotting facilities.
A completely different approach is to construct a ball model of a half-crystal which terminates on an (hkl) surface. Methods of doing this for a variety of cubic and hexagonal structures are given in Moore & Nicholas (1961) and Nicholas (1961Nicholas ( , 1962, while a selection of results appears in Nicholas (1965). From such a model, a plot of the unit mesh and of the stacking of planes can be produced photographically or, if greater accuracy is required, suitable vectors u, v, w can easily be selected and indexed on the model and the plot carried out as in the other methods. Thus, the ball model can be considered as a simple and economical analogue computer for solving Diophantine equations such as (2) and (4) and for producing a plot. As an example, Fig. 1 shows a model of a (5,8,11) surface in a body-centred cubic crystal (the example chosen by Bevis), together with_ a plot showing the unit mesh in the surface and the stacking vector w as derived from the photograph.
The advantages of modelling are that a visual impression of the stacking over several layers is available, a variety of planes can be considered in quick succession, the simplest unit mesh (e.g. the rectangular mesh in Fig. 1, which is not described by Bevis) is obvious, and any gross errors in calculating a plot of mesh shape can be eliminated by comparison with the model. The indexing of the vectors depends on prior indexing of a basis such as OABC in Fig. l(a), but this is necessarily done when the model is being set up. It is worth noting that the identification of equivalent vectors is always simpler on the model itself than on a photograph.
Although surface models have only been built for cubic and hexagonal structures, the theory in Nicholas (1961) describes a method for their construction in any structure for which a 'ball' model of the bulk crystal can be made. No general rules can be laid down for the construction of such bulk models but a large range of structures can be modelled by using base plates to force the first layer of balls into a predetermined pattern and/or by using balls having a degree of asymmetry.
The error referred to in the opening paragraph arises when the analysis is applied to lattices indexed relative to a centred unit cell and (hkl) is such that hWl +kw2+lw3 =½ (7) has solutions for w1,w2,w3 equal to a lattice vector (with at least two of the wl non-integral). This implies that there are other lattice planes between those whose maps are separated by t. Bevis (1969) tabulates the conditions when this occurs for various centrings of the cell and then asserts that the extra planes are to be plotted at ½t from the original ones. Jaswon & Dove (1955) and Jaswon (1965) make equivalent assertions. In fact, as can be seen from Fig. 2, which shows a plot of (111) planes in a body-centred cubic lattice, the displacement need not be ½t. However, it must be one of ±t2 ,2\ I(u -t-t), ½(v + t), ½(u + v + t), the selection depend-