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7.1 STRESS AND STRAIN IN BULK CRYSTALS A mechanical force acting on a crystal lattice changes the relative positions of the lattice points (sites) i.e. the positions of the atoms forming the crystal structure. This can be characterised by a vector u, which defines the relative displacement of an atom into a new position r1 = r + u in some arbitrary Cartesian coordinate system. Different crystal lattice points can have different relative displacements making the vector u coordinate dependent i.e. u = u(r). If the lattice points return to their original positions after the force is removed then the deformation is described as elastic. Knowing the relative displacements of each lattice site, the state of the crystal deformation can be described by strain components defining a second-rank tensor

"Jfe,p Joif \ }. kjjyj \

(for more about application of tensors in crystals see Nye [152]) as:

'large'

where u\, U2, and 113 are the relative displacements of the crystal lattice points along the xi, x-2 and #3 axes, respectively. The diagonal components represent extensions per unit length along the x1, X2 and x3 directions and are usually referred as 'stretches' (see Fig. 7.1), while the off-diagonal components eij are related to 'rotations' e.g. the term 6 u i / 6 x j represents a rotation about the x3 axis toward x\ of a line element parallel to x2. As such the angle between the two line elements parallel to x1 and X2 changes from r/2 before deformation to r/2 eij after *. The strain tensor is symmetrical (eij = eji) and can be written as:

'small'

(3.13)

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If the shear strain components are zero, the diagonal elements also determine the change in the crystal volume (also known as 'dilation') as:

'medium < X'

This can be explained in terms of the unity cube strained by the diagonal components. Assuming the strains are small, which is true in the limits of linear strain theory as well as in most applications, the distorted volume will then be (1 + en) (1 + e22) (1 + e33) giving a total change in the cube volume (to first-order) of e11+e22 +e33 (see Fig. 7.1). It is also important to define the properties of the force causing the strain in the crystal lattice. If a crystal is acted on by an external force or if part of a crystal is applying a force on a neighbouring part then the crystal is said to be in the state of 'stress', where the stress is usually defined as the force per unit area of the crystal. If a unit cube is considered then normal stress components can be defined as an e.g. o11> 022, 033 etc. and the shear stress components as oij e.g. o12, o21, o13 etc. as shown in Fig. 7.2. In the homogeneous case when the force is independent of the position on the crystal surface, the stress components form a symmetric second-rank tensor (cr^ = &ji) as:

'medium = X' 'medium > X'

v"--'-^/

The diagonal elements of the stress tensor are of special importance in applications. If only o11 is a non-zero stress component then the stress is said to be uniaxial, while if both o11 and o22 are not equal to zero the stress is defined as 'biaxial'. The latter is the case common in quantum well structures as elaborated on in the following section.

'large'

*Note that in the literature an intuitive notation is occasionally seen replacing the indices 1, 2 and 3 with x, y and z, e.g. en > exx.

be non-empty, since it includes the array element previously stored at index K-l. Figure 14.5 depicts three possible outcomes, depending on the relationship of K to m and n. In each of the three possible outcomes, the five segments are ordered in the sense that the values stored in the different segments are strictly increasing as one proceeds from left to right. (That is, each value in a segment is at most each value in the segment to its right.) In the first case depicted in Figure 14.5,

Simple recurrent neural networks (SRNN) have feedback connections which add the ability to also learn the temporal characteristics of the data set. Several different types of SRNNs have been developed, of which the Elman and Jordan SRNNs are simple extensions of FFNNs. The Elman SRNN, as illustrated in Figure 3.3, makes a copy of the hidden layer, which is referred to as the context layer. The purpose of the context layer is to store the previous state of the hidden layer, i.e. the state of the hidden layer at the previous pattern presentation. The context layer serves as an extension of the input layer, feeding signals representing previous network states, to the hidden layer. The input vector is therefore (3.15)

Figure 7.1 Schematic illustration of the influence of the diagonal strain components or stretches (en, 22, 33) on the crystal lattice.

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