Note that these points represent the coefficients of the point locations in terms of the unit cell dimensions (Van Vlack, 1989). They do not represent the absolute positions of the points (Van Vlack, 1989). Thus, these point coefficients can be multiplied by the side length, a, of the unit cell in order to obtain the absolute locations of each of the points (Van Vlack, 1989). This is why the symbol a is used to denote the lengths of the sides of the unit cell in the above diagram, as well as in each of the following diagrams. Note also that this question has been answered on the assumption that the unit cell structure under consideration is cubic. A non-cubic unit cell structure would yield precisely the same point coefficients for the axial intercepts (and thus the same Miller indices) (Van Vlack, 1989). However, the side lengths would not all be equal to a. This would cause the absolute locations of each of the axial intercepts to be different (Van Vlack, 1989).
This plane passes through the origin. However, the origin can be shifted. The Miller index of the plane will depend on where the origin is shifted. However, the Miller index that is obtained from shifting the origin in one direction will be equivalent to any other Miller index that is obtained from shifting the origin in any other direction. This is because all the Miller indices obtained through this technique of shifting the origin will represent parallel (and equivalent) planes (Van Vlack, 1989).
Let us consider the situation of shifting the origin one unit in the positive x-direction. The plane in question has been resketched below with the origin shifted one unit in the positive x-direction. The plane now includes the points (-1,0,0), (0,0,1), and (0,0.5,0.5). From these points, we can see that the intercept with the x-axis is -1, and the intercept with the z-axis is 1. Note again that these intercept numbers represent the coefficients of the intercepts’ positions in terms of the unit cell dimensions, rather than the absolute locations of the intercepts (Van Vlack, 1989).
The absolute distance from the new origin to the intercept with the x-axis is therefore equal to -1a. The absolute distance from the new origin to the intercept with the z-axis is equal to 1a. To determine the location of the intercept with the y-axis, consider a vector on the plane that intercepts the y-axis. From vector algebra, we know that any two points on the plane will define a vector that lies on the plane (The Vector Equation of a Plane, 2003). Therefore, let us consider the vector v that connects the two points (0,0,1) and (0,0.5,0.5). This vector can be defined as the difference between the two points: Any scalar multiple of this vector will yield a parallel vector (Young-Hoo, 1998). Therefore, any scalar multiple of the vector v can be defined as: where s is a scalar multiplier.
By applying the principles of vector algebra, we can determine the point at which this vector intersects the y-axis. Any point on the y-axis, py, is defined by: where y1 is the y component of the point py, and can be any number. Let us consider the relationship between point py and the point (0,0,1). The scalar-multiplied vector sv, point py, and the point (0,0,1) all lie in the y-z plane. We can therefore define the scalar-multiplied vector sv as the difference between the point py and the point (0,0,1): Since we have already determined the value of s*v in a previous calculation, we can substitute this value into the above equation.
This equation can now be rearranged to solve for y1 in the following fashion: The x, y and z components on the left side of the above equation must be equal to the x, y and z components on the right side of the equation. The equation therefore yields the following three relationships: The last of the above three relationships can be rearranged to solve for s. Substituting this value of s into the previous relationship for y1 allows us to solve for y1: This indicates that the plane under consideration must intersect the y-axis at the point (0,1,0).
As a result, the intercept with the y-axis is 1. Note again that this intercept number represents the coefficient of the intercept’s position in terms of the unit cell dimensions, rather than the absolute position of the intercept (Van Vlack, 1989). The absolute distance from the new origin to the intercept with the y-axis is therefore equal to 1a. In summary, therefore, the coefficients of the three axial intercepts occur at the points: The absolute locations of the three axial intercepts occur at the points: In question #3 above, we had defined the Miller indices to be “the reciprocals of the three intercepts that the plane makes with each of the axes, cleared of fractions and common multipliers” (Van Vlack, 1989).
This indicates that the Miller Indices of the plane in question are: This plane is again displayed in the following sketch: 5). Chain addition polymerization is a highly exothermic process because heat is volumetrically created with the addition of each monomer to the polymer (Polymerization Processes, 2003) – a pi-bond in the monomer converts into a sigma-bond in the polymer – typically resulting in the generation of 8 to 20 kcal/mol (Polymers, 2003).
Heat removal normally occurs by transferring heat through the wall of the reactor – which requires at least one dimension of the reactor to be small – or through solvent or gas transport media, or by monomer evaporation and condensation, so the reactor and processes should be very well designed, controlled and monitored (Polymerization Processes, 2003) because explosions and fires can result if process control is lost and if polymerization occurs when not desired (Polymerization, 2002). Additionally, excess heat energy can cause polymers to be converted back into monomers (Polymerization Processes, 2003).