1. (a) Write down a definition of the term Bravais Lattice, making reference to primitive lattice vectors. Give an example of a regular array that is a 2D Bravais lattice and a 2D non-Bravais lattice. [3 marks] (b) Derive the relationship between a set of points (vec{R}) constituting a Bravais lattice and the reciprocal lattice vectors that all reciprocal lattice vectors (vec{k}) satisfy. [5 marks] (c) State the relationship between the primitive vectors in the reciprocal lattice, (vec{b}_i), and the primitive vectors of the real space Bravais lattice, (vec{a}_i). Using this, prove that the reciprocal lattice is itself a Bravais-lattice. [5 marks] (d) What is meant by term Brillouin Zone and what is the relationship between the volume of the primitive cell in the real space and that in the reciprocal space? [5 marks] (e) Draw a sketch of a tetragonal unit cell, marking in the various side lengths of the cell and calculate the distances between the (211) planes of this tetragonal structure, in terms of the side lengths indicated in your diagram. [7 marks]
Added by Donald W.
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It is defined by a set of primitive lattice vectors, which are the shortest vectors that can be used to generate the entire lattice by translation. Example of a 2D Bravais lattice: A square lattice, where the primitive lattice vectors are a1 = a(1,0) and a2 = Show more…
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1. (30 pts) Consider a Bravais lattice whose primitive unit cell is shown below. (a) Calculate the three primitive vectors of the Reciprocal Lattice (represent them in Cartesian Coordinates). Draw the primitive unit cell of the reciprocal lattice. (b) Direct lattice is composed of families of lattice planes and the separation in each family of lattice planes is related to the magnitude of the associate reciprocal lattice vector. (Work on the following problems using the reciprocal lattice rather than the direct lattice) a. Calculate the largest separation in the family of lattice planes. What is the corresponding reciprocal lattice vector? b. What is the 2nd largest separation in the family of lattice planes. What is the corresponding reciprocal lattice vector? c. What is the 3rd largest separation in the family of lattice planes. What is the corresponding reciprocal lattice vector?
Madhur L.
Name the Bravais lattices encountered in cubic system. Sketch their unit cells and find for each type unit cell (a) the number of atoms per unit cell. (b) co-ordination number for each atom in the lattice.
Hexagonal space lattice. The primitive translation vectors of the hexagonal space lattice may be taken as $$ \mathbf{a}_{1}=\left(3^{1 / 2} a / 2\right) \hat{\mathbf{x}}+(a / 2) \hat{\mathbf{y}} ; \quad \mathbf{a}_{2}=-\left(3^{1 / 2} a / 2\right) \hat{\mathbf{x}}+(a / 2) \hat{\mathbf{y}} ; \quad \mathbf{a}_{3}=c \hat{\mathbf{z}} $$ (a) Show that the volume of the primitive cell is $\left(3^{1 / 2} / 2\right) a^{2} c$. (b) Show the primitive translations of the reciprocal lattice are $\mathbf{b}_{1}=\left(2 \pi / 3^{1 / 2} a\right) \hat{\mathbf{x}}+(2 \pi / a) \hat{\mathbf{y}} ; \quad \mathbf{b}_{2}=-\left(2 \pi / 3^{1 / 2} a\right) \hat{\mathbf{x}}+(2 \pi / a) \hat{\mathbf{y}} ; \quad \mathbf{b}_{3}=(2 \pi / c) \hat{\mathbf{z}}$ so that the lattice is its own reciprocal, but with a rotation of axes. (c) Describe and sketch the first Brillouin zone of the hexagonal space lattice.
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