9780030839931, Chapter 4, Problem 8P

Prévia do material em texto

Chapter 4, Problem 8P Problem (a) Given a Bravais lattice, let a1 be a vector joining a particular point P to one of its nearest neighbors. Let P' be a lattice point not on the line through P in the direction of a1 that is as close to the line as any other lattice point, and let a2 join P to Let P" be a lattice point not on the plane through P determined by a1 and a2 that is as close to the plane as any other lattice point, and let a3 join to P". Prove that a1, a2, and a3 are a set of primitive vectors for the Bravais lattice. (b) Prove that a Bravais lattice can be defined as a discrete set of vectors, not all in a plane, closed under addition and subtraction (as described on page 70). Step-by-step solution Step of 4 (a) The numbers of point groups in two and three dimensions are 10 and 32, respectively. These point groups form the basis for construction of different types of lattices. Only those lattices are permissible which are consistent with the point group operations. Such lattices are called Bravais lattices. Let the vectors and represent the primitive translations along the axes x, y, and Let R denotes a vector from the origin to the reference corner of a unit cell in the crystal. The R is related to the primitive translation vector by the relation Here, and n3 are integers. Step 2 of 4 Let P is an arbitrary point on the lattice. The vectors connect P to one of its nearest neighbors. Again, P' be a lattice point that is not on the line passing through P and parallel to but which is as close to this lines as any other point. Join P and P' by the vector Lastly, P" is the lattice point, defined by the vectors and but which is not passing through the point P. P and P" lattice points connected by the vector So, R will be, R = The above equation is the set of equations for the Bravais lattice and is called the lattice B As only those lattices are permissible which are consistent with the point group operations are called Bravais lattice, therefore one thing one has to notice, there are any points in lattice A which are not in lattice B. Step of 4 Let us consider there is such a point, call it Q. So, there is at least one point on Q' in lattice B Thus, the vector joining Q and Q' has the form = + + having