A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure a. The atoms reside at the corners of cubes of side L = 0.250 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal, as shown in Figure b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves. nm
![Imaging with a Thick Lens. Consider a glass lens of refractive index n , the z axis and thickness d , and two spherical surfaces of equal radii R as shown in Figure 3.1. Determine the ray-transfer matrix of the system between the two planes at distances d 1 and d 2 from the vertices of the lens. The lens is placed in air ( n = 1 ) . Show that the system is an imaging system (i.e., the input and output planes are conjugate) if 1 z 1 + 1 z 2 = 1 f or s 1 s 2 = f 2 , where z 1 = d 1 + h s 1 = z 1 − f z 2 = d 2 + h s 2 = z 2 − f and h = ( n − 1 ) f d n R 1 f = n − 1 R [ 2 − n − 1 n d R ] The points F 1 and F 2 are known as the front and back focal points, respectively. The points P 1 and P 2 are known as the first and second principal points, respectively. Show the importance of these points by tracing the trajectories of rays that are incident parallel to the optical axis. Figure 3.1: Imaging with a thick lens. P 1 and P 2 are the principal points and F 1 and F 2 are the focal points. Imaging with a Thick Lens. Consider a glass lens of refractive index n , the z axis and thickness d , and two spherical surfaces of equal radii R as shown in Figure 3.1. Determine the ray-transfer matrix of the system between the two planes at distances d 1 and d 2 from the vertices of the lens. The lens is placed in air ( n = 1 ) . Show that the system is an imaging system (i.e., the input and output planes are conjugate) if 1 z 1 + 1 z 2 = 1 f or s 1 s 2 = f 2 , where z 1 = d 1 + h s 1 = z 1 − f z 2 = d 2 + h s 2 = z 2 − f and h = ( n − 1 ) f d n R 1 f = n − 1 R [ 2 − n − 1 n d R ] The points F 1 and F 2 are known as the front and back focal points, respectively. The points P 1 and P 2 are known as the first and second principal points, respectively. Show the importance of these points by tracing the trajectories of rays that are incident parallel to the optical axis. Figure 3.1: Imaging with a thick lens. P 1 and P 2 are the principal points and F 1 and F 2 are the focal points.](https://www.doubtrix.com/uploads/editor/9885508202VQeDlDyKSs.png)
