9780030839931, Chapter 9, Problem 4P

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Chapter Problem 4P Problem Alternative Definition Brillouin Zones Let space res are drawn about every reciprocal lattice except the origin. Show the interior spheres and on the surface then the zone. Show that the spheres and on the then point the boundaries Step solution Step Solution We have seen that Kroning Penny mode that one dimensional lattice the energy discontinues occur when wave number satisfies the condition k= Here positive negative and at consider line (representing the values) divided discontinuities into segments of length a then the segment and similarly the segment second Step 2 Construction zones:- many reduced zone sufficien but the higher zones are The Brillouin zones are onstructed from the planes which are the perpendicular bisectors of all reciprocal draw vectors from origin other lattice points draw planes bisect these vectors one is the smallest volume about origin by these planes similarly second zone volume between the zone next the procedure consider the two as hown 1. 1(a) VP H E S -π/a G R Third zone construction Second Third First zones Step Consider figure the different cons stru follows Take the origin and join to nearest attice points and the vectors OR and bisectors square EFGH which the first For Zone origin next points Take and draw these These perpendic enclose square PQRS The area between and EFGH the second zone Similarly the other zones are constructed theory know that the form equation 2K (1) propagation vector the direction rays reciprocal vector Now consider simple square lattice two its lattice vectors will be and b=ay The reciprocal lattice vectors the above vectors B. and - ay Here and are the unit vectors along the edges Therefore reciproca vector So, (2) (3) measured from the origin of reciprocal Substitute equation (2) and equation (1) This and and Thus, vectors originating from one lines will produce Bragg's reflection the area enclosed perpendicular bisectors of G, three reciprocal vectors symmetry shown below figure second zone constructed from the three vectors equivalent by and similar for third zone Figure: ky a O Step The first zone -space that be reached from the origin without crossing any Bragg The second zone points which can be reached crossing only one Bragg the first zone by The zone set of points that the (n- zone, and reached from the zone by crossing only one Bragg's plane the zone can be defined set points which can reached by crossing Bragg from the origin no smaller quantity From the above observation conclude spheres and surface " additional spheres then common the boundaries