Crystal twinned

Some crystals have two parts symmetrically related to one another. These, called twinned crystals, are fairly common both in minerals and in metals and alloys. For a detailed discussion of twinning, see Kelly and Groves [G.33] and Barrett and Massalski [G.25]. 

The relationship between the two parts of a twinned crystal is described by the symmetry operation which will bring one part into coincidence with the other or with an extension of the other. Two main kinds of twinning are distinguished, depending on whether the symmetry operation is 180° rotation about an axis, called the twin axis, or reflection across a plane, called the twin plane. The plane on which the two parts of a twinned crystal are united is called the composition plane. In the case of a reflection twin, the composition plane may or may not coincide with the twin plane. 

Of most interest to metallurgists, who deal mainly with FCC, BCC, and HCP structures, are the following kinds of twins  

Annealing twins, such as occur in FCC metals and alloys (Cu, Ni, a-b ass, Al, etc.), which have been cold-worked and then annealed to cause recrystallization. 

Annealing twins in FCC metals are rotation twins, in which the two parts are related by a 180° rotation about a twin axis of the form 111 . Because of the high symmetry of the cubic lattice, this orientation relationship is also given by a 60° rotation about the twin axis or by reflection across the 111 plane normal to the twin axis. In other words, FCC annealing twins may also be classified as reflection twins. The twin plane is also the composition plane. 

Like a grain boundary, the twin boundary is a higher energy state, relative to the crystal. However, because a twin boundary is highly ordered, it is of lower energy than 

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Although specific calculations for i and g are not made until Sect. 3.5 onwards, the mere postulate of nucleation controlled growth predicts certain qualitative features of behaviour, which we now investigate further. First the effect of the concentration of the polymer in solution is addressed - apparently the theory above fails to predict the observed concentration dependence. Several modifications of the model allow agreement to be reached. There should also be some effect of the crystal size on the observed growth rates because of the factor L in Eq. (3.17). This size dependence is not seen and we discuss the validity of the explanations to account for this defect. Next we look at twin crystals and any implications that their behaviour contain for the applicability of nucleation theories. Finally we briefly discuss the role of fluctuations in the spreading process which, as mentioned above, are neglected by the present treatment. 

The importance of twinned crystals in demonstrating that nucleation is the relevant growth mechanism has been realized since 1949 [64, 99]6. They were first investigated extensively in polymer crystals by Blundell and Keller [82] and they have recently received increased attention as a means of establishing, or otherwise, the nucleation postulate for lamellar growth [90, 91, 95,100-102]. The diversity of opinion in the literature shows that it is very difficult to draw definite conclusions from the experimental evidence, and the calculations are often founded upon implicit assumptions which may or may not be justified. We therefore restrict our discussion to an introduction to the problem, the complicating features which make any a priori assumptions difficult, and the remaining information which may be fairly confidently deduced. 

If the stacking faults occur only rarely (say, every 105 layers on average), the result is a polysynthetic twinned crystal (cf. Fig. 18.8, p. 223). Depending on the frequency of the stacking faults, there is a smooth transition between crystals with stacking faults and poly synthetic twinning. 

Symmetry Relations at Phase Transitions. Twinned Crystals... 

The lattice parameters a and b become different during the transition from the tetragonal to the orthorhombic structure. Either a > b or a < b results, depending on the direction of the rotation of the octahedra strands. During the phase transition both directions of rotation occur at random domains appear, in which either a > b or a < b. The obtained crystal is a twinned crystal or twin, consisting of intergrown domains. The twin domains are... 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

Before going further, it may be noted that the flipping ratio does not depend either on the Lorentz factor or on absorption in the sample. Certain instrumental parameters such as the polarisation of the neutron beam for the two spin states, the half wavelength contamination of the neutron beam and the dead-time detector can readily be taken into account when analysing the data. On the other hand, the extinction which may occur in the scattering process is not so easy to assess, but must also be included [14]. Sometimes, it is even possible to determine the magnetisation density of twinned crystals [15]. 

Figure 1 HRSEM images (A, C + E) and AFMerror images (B, D + F) of silicalite 2. A +B is of twinned crystals and C+D of single crystals with E+F at higher magnification...

Initiation of growth may also proceed by formation of metastable structures when nucleation is inhibited. Multiply twinned structures have been observed for a number of metals. Their presence indicates an icosohedral or decahedral precursor cluster which has decomposed to a multiply twinned crystal at a critical size [117, 118], Another example of metastable intermediate structures was reported by Dietterle et al. 

Twins are intergrown crystals such that the crystallographic directions in one part are related to those in another part by reflection, rotation, or inversion through a center of symmetry across a twin boundary. Twinned crystals are often prized mineralogical specimens. When twins are in contact across a well-defined plane (which is not always so), the boundary is generally called the composition plane. The only twins that are considered here will be reflection twins, where the two related parts of the crystal are mirror images (Fig. 3.22). The mirror plane that relates the two components is called the twin plane. This is frequently, but not always, identical to the plane along which the two mirror-related parts of the crystal join, that is, the composition plane. Repeated parallel composition planes make up a polysynthetic twin (Fig. 3.23). 

Figure 1. Cross section of single and twin crystal of sucrose a) single crystal view along 001 direction b) single crystal view along 010 direction c) twin crystal view along I 001 I direction.

As another example of utilizing platy forms of twinned crystals, we should mention a recent remarkable increase in the photosensitivity of photographic film. Photographic emulsion film consists of minute AgBr crystallites, with octahedral or cubo-octahedral Habitus. The photosensitivity is determined by the area... 

M. Kitamura, S. Hosoya, and 1. Sunagawa, Re-investigation of the re-entrant corner effect in twinned crystals. J. Crystal Growth, 47.1979,93-9... 

The crystal figures shown in Fig. 9.2 were selected from sketches that appeared in a series of books published between 1913 and 1923, and all represent forms of single or twinned crystals larger than a few millimeters. They are mostly gem-quality diamonds. Other than these forms, there are translucent or opaque... 

As already explained in Section 7.2, the re-entrant corner effect in its original sense can be expected only when two individuals are perfect crystals, containing no dislocations either in the individual crystals or in the composition plane. In real crystals, the pseudo re-entrant corner effect, by the mediation of dislocations, creates changes in the morphology of the twinned crystal. 

Sunagawa and T. Yasuda, Apparent re-entrant corner effect upon the morphologies of twinned crystals A case study of quartz twinned according to Japanese twin law,... 

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