6.2: Section The Crystal, Lattice, Atomic Basis and Miller Notation A crystal consists of a single atom or a group of atoms arranged as a periodic array. The mathematical construction, the lattice, gives the array its periodic nature. Bravais lattices have special symmetry properties. Attaching an atom or a group of atoms to each lattice point produces the crystal. Topic 6.2.1: Lattice A physical crystal receives its periodic structure from a lattice, which is a mathematical object. We now supply two equivalent definitions for the lattice. ! ! ! Defintion 1: Given three noncoplanar vectors a1 , a 2 , a 3 , we define the lattice to be the collection of points given by ! ! ! ! r = n 1a 1 + n 2 a 2 + n 3 a 3 (6.2.1) for all integers n i (positive and negative). The lattice Figure 6.2.1: A 2-D lattice with ! ! ! consists of a set of points r not atoms! The primitive primitive vectors a1 , a 2 . vectors are the shortest vectors that span the lattice (in ! the sense of Equation 6.2.1). The primitive vectors a i generate (span) the lattice but a ! given lattice does not uniquely define the primitive vectors. For example, the vector a1 ! can be replaced by a1 and still generate the same lattice. As an important note, the ! ! ! vectors a1 , a 2 , a 3 in the definition need be neither unit vectors nor orthogonal. For this reason, the primitive vectors should not be called basis vectors in order to avoid any possible confusion (although some authors do call them basis vectors).
! ! ! Definition 2: If there exists three noncoplanar vectors a1 , a 2 , a 3 such that the point ! ! ! ! ! ! r = r '+ n1a 1 + n 2 a 2 + n 3a 3 is equivalent to the point r ' for all integers n1 , n 2 , n 3 , then the array of points forms a lattice. Equivalent means that the arrangement of atoms looks ! ! ! ! the same from point r as it does from point r ' . The points r, r ' do not necessarily coincide with lattice points or with atoms. The two definitions can be easily seen to be ! ! ! ! ! ! equivalent by noting that r = r r ' = n1a1 + n 2 a 2 + n 3 a 3 must be the lattice points themselves.
Example 6.2.1: The primitive vectors for Figure 6.2.1 can be written as ! ! " " " a1 = 2x and a 2 = x + y " " where x, y represent orthogonal unit vectors. Example 6.2.2: A finite array of points does not form a lattice because you can find ! ! integers such that points entirely surround r ' but not r .
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Topic 6.2.2: The Translation Operator We can translate the lattice by any combination of primitive vectors and still end up ! with the same lattice. In other words, if the symbol R represents a specific vector in the ! ! ! ! lattice (i.e., there exists integers n1 , n 2 , n 3 such that R = n1a1 + n 2 a 2 + n 3a 3 is a lattice ! point) then translations by R leave the lattice unchanged. Equivalently stated, translations through lattice vectors leave the lattice invariant. We define the translation ! operator TR by ! ! ! ! (6.2.2a) TR V = V + R ! where V represents an arbitrary vector that does not necessarily correspond to a lattice point. The translation operator represents vector addition. Notice that the subscript on the ! ! operator TR gives the vector R through which all other vectors must be translated. We ! ! will generally use the equivalent ( R R ) but more convenient definition for the translation as ! ! ! ! TR V = V R (6.2.2b) The definition given in Equation 6.2.2b is a special case of a more general one. The ! translation operator can be defined for functions. Let f ( r ) be a function of the position ! ! vector r . We define the translation operator TR by Put trans ! ! ! ! (6.2.3) TR f ( r ) = f r R
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Notice the use of the minus sign to match the convention in Section ! here 3.15. The translation operator assigns a new value to f ( r ) , namely the Fig 6.2.2 ! ! value it would have at the position r R . A moments thought shows the translation moves the function to the right (see Figure 6.2.2) while the vector moves to the left. The ideas can be best illustrated with a couple of examples.
Example 6.2.3:
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Consider a one-dimensional crystal of atoms with ! spacing a as shown in Figure 6.2.3. Let f ( r ) be the electrostatic potential as illustrated in the figure. We expect the electrostatic potential to be periodic along ! the chain of atoms. Let R be one of the Bravais lattice Figure 6.2.3: The electrostatic ! ! ! vectors given by R = n1a1 where a1 denotes the potential is periodic. primitive vector with magnitude a and n1 is an ! arbitrary integer. The translation operator T R produces ! ! ! ! TR f ( r ) = f r R
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! ! ! ! We know that f r R = f ( r ) because the electrostatic potential f ( r ) must be periodic ! ! ! in the lattice. So in this case, TR f ( r ) = f ( r ) and the function must f be invariant under ! translations by a lattice vector R .
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Example 6.2.4: Show that the definition of the translation operator in Equation 6.2.3 leads to the definition in Equation 6.2.2b. # # # # # ! # ! ! ! Solution: Let f ( V ) = V then TR V = TR f ( V ) = f V R = V R as required.
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Topic 6.2.3: Atomic Basis
The crystal consists of an atomic basis (or atomic cluster) attached to the lattice points. The basis can be a single atom or a group of atoms attached to each lattice point. Each lattice point receives an identical basis (or cluster). The (infinite) crystal consists of the collection of these regularly arranged clusters.
Topic 6.2.4: Unit Cells
Figure 6.2.4: A crystal is a lattice with an attached atomic basis.
Unit cells consist of small regions of space that, when duplicated, can be translated to fill the entire volume of the crystal. We briefly consider the primitive unit cell and the conventional unit cell. The primitive unit cell contains exactly a single lattice point and a single cluster. The primitive cell has boundaries made of the primitive vectors, which are the shortest vectors that span the lattice. Therefore, translating the primitive unit cell through every possible integer combination of primitive vectors covers the entire crystal. Figure ! ! 6.2.4 shows two equally valid primitive unit cells. In both cases, the two vectors a1 , a 2 span a region of space that contains only one lattice point or atomic cluster as is obviously true for the primitive cell in the bottom of the figure. The upper unit cell contains exactly one point and one unit cluster since the sides of the parallelepiped cut through the points and clusters in such a way that the sum of the pieces adds up to a single unit. Although primitive unit cells might appear to be the simplest for calculation purposes, it is sometime more convenient to work with non-primitive unit cells. The conventional unit cell does not necessarily contain exactly one lattice point and one atomic cluster. For calculational convenience, we usually choose orthogonal spanning vectors to define this unit cell. Translating the conventional unit cell by all integer combinations of spanning vectors covers the entire crystal. The next section lists the most typical examples of conventional unit cells. Example 6.2.5: Consider the vectors described in the previous Example 6.2.1 ! ! " " " a1 = 2x and a 2 = x + y A non-primitive unit cell can be defined by
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! " a1 = 2x and The spanned volume contains two points. Topic 6.2.5: Miller Indices
! ! ! b = 2a 2 a1 2
The points of intersection of a plane with the primitive or non-primitive spanning vectors can be used to specify a crystal plane. Figure 6.2.5 shows an example of an infinite plane intersecting the axes defined by three spanning vectors. In this example, the intersect...