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Section 01_Crystal Structure

Section 01_Crystal Structure - Physics 927 E.Y.Tsymbal...

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Physics 927 E.Y.Tsymbal 1 Section 1: Crystal Structure A solid is said to be a crystal if atoms are arranged in such a way that their positions are exactly periodic . This concept is illustrated in Fig.1 using a two-dimensional (2D) structure. Fig.1 A perfect crystal maintains this periodicity in both the x and y directions from - to + . As follows from this periodicity, the atoms A, B, C, etc. are equivalent . In other words, for an observer located at any of these atomic sites, the crystal appears exactly the same. The same idea can be expressed by saying that a crystal possesses a translational symmetry . The translational symmetry means that if the crystal is translated by any vector joining two atoms, say T in Fig.1, the crystal appears exactly the same as it did before the translation. In other words the crystal remains invariant under any such translation. The structure of all crystals can be described in terms of a lattice , with a group of atoms attached to every lattice point. For example, in the case of structure shown in Fig.1, if we replace each atom by a geometrical point located at the equilibrium position of that atom, we obtain a crystal lattice. The crystal lattice has the same geometrical properties as the crystal, but it is devoid of any physical contents. There are two classes of lattices: the Bravais and the non-Bravais . In a Bravais lattice all lattice points are equivalent and hence by necessity all atoms in the crystal are of the same kind. On the other hand, in a non-Bravais lattice, some of the lattice points are non-equivalent. Fig.2 In Fig.2 the lattice sites A, B, C are equivalent to each other. Also the sites A 1 , B 1 , C 1 , are equivalent among themselves. However, sites A and A 1 are not equivalent: the lattice is not invariant under translation AA 1 . A B C A 1 B 1 C 1 y x A B C a 1 T
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Physics 927 E.Y.Tsymbal 2 Non-Bravais lattices are often referred to as a lattice with a basis . The basis is a set of atoms which is located near each site of a Bravais lattice. Thus, in Fig.2 the basis is represented by the two atoms A and A 1 . In a general case crystal structure can be considered as crystal structure = lattice + basis. The lattice is defined by fundamental translation vectors. For example, the position vector of any lattice site of the two dimensional lattice in Fig.3 can be written as T =n 1 a 1 +n 2 a 2 , (1.1) where a 1 and a 2 are the two vectors shown in Fig.3, and n 1 ,n 2 is a pair of integers whose values depend on the lattice site. Fig.3 So, the two non-collinear vectors a 1 and a 2 can be used to obtain the positions of all lattice points which are expressed by Eq.(1). The set of all vectors T expressed by this equation is called the lattice vectors . Therefore, the lattice has a translational symmetry under displacements specified by the lattice vectors T . In this sense the vectors a 1 and a 2 can be called the primitive translation vectors . The choice of the primitive translations vectors is not unique. One could equally well take the vectors
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Section 01_Crystal Structure - Physics 927 E.Y.Tsymbal...

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