Spiral steps

These three features correspond to the two-dimensional morphology of a crystal, and are directly related to the problems of the three-dimensional morphology of polyhedral crystals. Habitus and Tracht. This is because the normal growth rate R which determines Habitus and Tracht is related in the following way to the height of a step, h, the advancing rate of the step, v, and the step separation, A.  

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The seeond-order dependenee of the growth rate on the supersaturation ean be explained by a number of growth theories. The most eonvineing, however, is that of Burton etal. (1951). In their BCF theory about the serew disloeation eentred surfaee spiral step, it is assumed that growth units enter at kinks with a rate proportional to cr and that the kink density is also proportional to cr whieh gives the faetor cr in the rate expression. 

Surface Spiral Step Control. Many crystals grow faster at small supersaturation than allowed by Equation 7. This lead Frank (17) to suggest that steps may also originate from the presence of a screw dislocation, and that this kind of steps is not destroyed by spreading to the crystal edge, but continues infinitely. The rate law according to this theory is parabolic (7). We shall use the following version of the kinetic equation (10)... 

The concept of dislocations was theoretically introduced in the 1930s by E. Orowan and G. I. Taylor, and it immediately played an essential role in the understanding of the plastic properties of crystalline materials, but it took a further twenty years to understand fully the importance of dislocations in crystal growth. As will be described in Section 3.9, it was only in 1949 that the spiral growth theory, in which the growth of a smooth interface is assumed to proceed in a spiral step manner, with the step serving as a self-perpetuating step source, was put forward [7]. 

Thus, spiral step patterns served as excellent subjects, and many observations were reported using these new techniques [3], [4]. It was also around this time that the movement of spiral growth layers spreading on the (0001) face of Cdl growing in aqueous solution was first observed in situ. By using these optical techniques, spiral growth layers with monomolecular height (0.23 nm) were observed and measured on natural hematite crystals [5]. 

Figure 5.3. Spiral steps observed on the mineral kaolin using transmission electron microscopy with the decoration method [7].

On quartz crystals synthesized in a hydrothermal solution of NaOH, KOH series, 1010 faces show polygonal spiral steps, whereas rounded step patterns are observed on 1011 and 0111. ... 

In crystals containing no zigzag stacking, the orientations of the polygonal elemental spiral steps may be used as a criterion to identify whether the crystal is twinned or contains stacking faults. 

Figure 5.7. Phase contrast photomicrographs depicting interlaced spiral steps observed on (0001) faces of (a) magnetoplumbite and (b) SiC 6H polytype, (c) Schematic figure. The very narrow step separations observed at the center of the spiral in (b) are due to a sharp increment of supersaturation at the final stage due to the discontinued electrical supply.

Elemental growth spiral layers originating from an isolated dislocation can advance, keeping the step separation constant, unless factors which affect the advancing rate of the spiral steps, such as a local fluctuation in driving force or impurity adsorption, takes place. The step separation of a spiral, A, is related to the critical radius of two-dimensional nuclei, r, in the following manner (see ref. [11], Chapter 3) ... 

When the step separation is wide enough, typical spiral step patterns observable by optical microscopy may appear, but if the separation becomes narrower than the resolution power of the optical microscope, the spirals appear in the forms of polygonal pyramids or conical growth hillocks. Even if spiral patterns are not directly observable, we may assume that these growth hillocks are formed by the spiral growth mechanism. Examples representing the two cases are compared in Fig. 5.8. 

That a hollow core is formed by the creation of a free surface along a dislocation core implies that the curvature of the spiral step is reversed due to the strain field along the dislocation core. The effect of a strain field upon the advancement of a step was theoretically treated by Cabrera and Levine [14], [15],... 

In the case of growth spirals originating from dislocations with large b, hollow cores with diameters of micrometer order are observed at the spiral center however, when a number of dislocations with small b concentrate in a narrow area, a basin-like depression appears at the central area of the composite spirals, since the curvature of advancement of the spiral steps is reversed near the center. A straight step may appear near the spiral center as an intermediate state in the reversal of step curvature. Several examples are shown in Fig. 5.11. 

Elemental spiral steps originating from dislocations form various composite step patterns through cooperation or repulsion depending on the sign of dislocations and the distance between neighboring dislocations. 

It is easily understood that the rotation of a spiral step created by a dislocation is reversed depending on the sign of the dislocation. If two dislocations with the... 

If a new dislocation is introduced while a spiral step is advancing, the new step from the new dislocation will be affected by the advancement of the earlier spiral step, and can have only one-half or a couple of turns. This results in the coexistence of spirals with a small number of turns with those covering a wide area Fig. 5.14 is such an example. 

When we observe the process of advancement of elemental spiral steps in situ, it is often noticed that two steps bunch together to form a step with the height of two layers as they advance. The advancing rate of the bunched layer is retarded... 

Figure 5.17. Phase contrast photomicrograph showing bunching of elemental spiral steps (arrows) in hematite, (0001) face.

Figure 12.1. Elemental spiral step patterns observed on the 0001 face of beryl, (a) Low magnification, reflection (b) high magnification, differential interference contrast photomicrograph.

Elemental growth spiral step patterns are observed on all (0001), 1011, and 1010 faces of hematite crystals grown by post-volcanic action. 

Figure 12.7. Spiral step patterns observed on (a) 0001) and (b) 1011) faces of hematite. In (a), smooth growth steps are retarded two-dimensionally by etching, producing rough dissolution steps [3]-[6].

Figure 12.9. Spiral step pattern observed on both sides of a twin boundary on a 0001 face of a hematite crystal [7]. T.B. is the twin boundary, J. and T indicate orientations. Phase contrast photomicrograph.

Figure 12.12. Five-sided spiral steps observed on 001 face of phlogopite [10].

Solution. Before any growth, the two dislocations are associated with steps that may be as indicated in Fig. 12.9a. During growth, each dislocation rotates about its point of intersection to produce a spiral step, as in Fig. 12.5. However, the spirals will rotate in opposite directions, and sections will annihilate one another when they meet as in (a) and (b). The process will then continue as in (c)-(f) generating a potentially unlimited series of concentric steps. 

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