## 4 Discussion
This total energy can therefore be written as \(E=k\cdot t\cdot E_{d}+2\cdot t\cdot(D-2\tau_{0})\times\gamma\), where \(\gamma\) is the energy per unit surface of the subgrain boundary-like volume. The expressions of \(\gamma\) and \(E_{d}\) are known [2]: \(\gamma=G\cdot w\cdot\alpha^{2}\), and \(E_{d}=G\cdot b^{2}\), where \(G\) is the elastic shear modulus, and \(w\) is the width of the region where the plane orientation varies. The parameter \(k\) is a function of the angle \(\alpha\): we know that \(k=2\) for \(\alpha=0\) and that \(k\) diminishes if the modulus of \(\alpha\) increases. The experiment shows that \(\alpha\) is small, i.e., \(<\)-\(10^{-5}\) radians for all the cases that can be observed in Figure 3. This suggests a phenomenological approach that assumes a linear variation of \(k=k(\alpha)\) as a function of the modulus of \(\alpha\), thus writing \(k=2-z\mid\alpha\mid\), where \(z\) is a proportionality factor for the linearization. We will consider, for simplicity in this discussion, that \(\alpha\) is positive. The shortest distance \(D\) observed in Figure 3 is 50 \(\upmu\)m, and it corresponds to \(\alpha=10\) uradians. Let us assume that this pair of dislocations and its (\(D\), \(\alpha\)) couple can be considered an \"experimental minimum\". By writing \(\mathrm{d}E/\mathrm{d}\alpha=0\) for this configuration, we can deduce that \(z=4\) (\(D-2\) r0).\(w.\alpha/\mathrm{d}^{2}\). The effective misorientation \(\Delta\omega_{\mathrm{exp}}\) of an edge dislocation lying perpendicular to the diffracting planes is expected to vary as a function of the angle between the diffraction and Burgers vectors and to display a maximum given by \(\Delta\omega=1.75\)\(b/\pi\tau_{0}\)[16]. Using this formula for the isolated dislocation indicated by the black circle in Figure 3, we can estimate, with \(\Delta\omega\)-18 \(\upmu\)radian and \(\tau_{0}\)-5 \(\upmu\)m, that the modulus of the Burgers vector b is \(\sim\)0.2 nm. This, with \(w\)-1.5 \(\upmu\)m, allows a rough estimation of \(z\) and \(k\) that leads to \(1<k<2\), which is consistent with the model. We will consider, for the \(D\) larger distances case, (i) that the value of \(z\) remains constant when \(D\) varies and (ii) that \(D>>r_{0}\). With these approximations, we can write \(E=(2-z\cdot\alpha)G\cdot b^{2}\cdot t+2D\cdot G\cdot w\cdot t\cdot\alpha^{2}\) and, from \(\mathrm{d}E/\mathrm{d}\alpha=0\), deduce that the \(\alpha\) that minimises the energy for a given \(D\) is \(\alpha\)-(1/D) (\(z\)\(b^{2}/4w\)). This means that \(\alpha\) should be nearly proportional to \(1/D\). This statement allows estimation of the expected \(\alpha\) for the different  
Figure 6: Schematic representation of the distortion of the Bragg reflecting planes associated with two parallel dislocations with opposite Burgers vectors and distant from \(D\). A zone in between the two dislocations retains a misorientation \(\alpha\) instead of going down to zero (red straight line instead of continuation of the decreasing curve). The energy of the dislocations is associated with the zones indicated by the green rectangles and reduces to \(k\cdot t\cdot E_{d}\) instead of \(2\cdot t\cdot E_{d}\), with \(k<2\). \(D\) observed in Figure 3 and comparison between them with the experiment. The value of \(\alpha=10\)\(\upmu\)radians, corresponding to \(D=50\)\(\upmu\)m, is the biggest we measured. It is therefore the one where the error is the smallest. We will use this point as a reference for all the other measured (\(\alpha,D\)) pairs. The experimental and calculated (\(\alpha,D\)) pairs are given in Table 1.  
Figure 7 graphically visualises the comparison between the calculated curve and the experiment. It shows that the experimental points (measured in the same way as for the first point on Figure 4) are in fair agreement with the curve resulting from our very simple model. Let us mention again that this phenomenological approach only applies when dislocations are such that they cannot move easily, i.e., are sessile.