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The scalar product and special bases

In order to express the angle between two vectors the scalar product is now introduced. In this way also the bases can be characterized by their lattice constants.

Definition (D 1.5.3) The scalar product (x,y) between two vectors x and y is defined by (x,y) = $\vert\textbf{x}\vert\,\vert\textbf{y}\vert \,\cos\,(\textbf{x},\textbf{y}).$

For the scalar product the following rules hold:

$$\left. \begin{array}{rrcll} \rule{0em}{3ex}1.&(\mathbf{x}\,,... ...(\mathbf{x}\,,\,\lambda\, \mathbf{y}). & \end{array} \right\}$$ (1.5.1)

Special cases.

$(i)$

Because of $\cos \, 90^{\circ}=0$ the scalar product is zero if its vectors are perpendicular to each other. Therefore, a scalar product may be zero even if none of the vectors is the o vector.

$(ii)$

If x = y, then because of $\cos\, 0^{\circ}=1$ the scalar product is the square of the absolute value of x: $(\mathbf{x}\,,\,\mathbf{x}) = \vert\mathbf{x}\vert^2$.

Two types of special bases shall be considered in this section.

The first one is the basis underlying the Cartesian coordinate system, see Section 1.2. It has the property that the scalar products between different basis vectors are always zero: $(\mathbf{a}_i\,,\,\mathbf{a}_k)=0$ for $i, k=1, 2, 3$, $i\ne k$, because the basis vectors are perpendicular to each other. In addition, $\vert\mathbf{a}_i\vert=1$ for any $i$ because the basis vectors have unit length. Such a basis is called an orthonormal basis. An orthonormal basis allows simple calculations of distances and bonding angles, see the next section.

The other bases are those which are mostly used in crystallography. Real crystals in the physical space may be idealized by crystal patterns which are 3-dimensional periodic sets of points representing, e.g., the centers of the atoms of the crystal. Because of the periodicity of the crystal pattern there are translations which map the crystal pattern onto itself (often expressed by `the crystal pattern is left invariant under the translation'). We consider these translations. If each of successive translations leaves the crystal pattern invariant, then so does that translation which results from the combination of the successive translations.

To each translation there belongs a translation vector. To the resulting translation belongs that vector which is the sum of the vectors of the performed successive translations. This means that for any set of translation vectors, all their integer linear combinations are translation vectors of symmetry translations of the crystal pattern as well.

Due to the finite size of the atoms the symmetry translations of a crystal pattern can not be arbitrarily short, there must be a minimum length (of a few Å). We choose three shortest translation vectors a$_1$, a$_2$, and a$_3$ which do not lie in a plane, i.e. which are linearly independent. Then any integer linear combination , $v_1, v_2, v_3$ integer, of a$_1$, a$_2$, and a$_3$ is a translation vector of a symmetry translation as well. One can show that no other translation vector may belong to a symmetry translation.

Definition (D 1.5.3) The set of all translation vectors belonging to symmetry translations of a crystal pattern is called the vector lattice of the crystal pattern (and of the real crystal). Its vectors are called lattice vectors. A basis of 3 linearly independent lattice vectors is called a lattice basis. If all lattice vectors are integer linear combinations of the basis vectors, then the basis is called a primitive lattice basis or a primitive basis.

Fig. 1.5.1 Finite part of a planar `crystal structure' (left) with the corresponding vector lattice (right). The dots mark the end points of the vectors.

Several bases are drawn in the right part of Fig. 1.5.1. Four of them are primitive, among them the one which consists of the two shortest linearly independent lattice vectors (lower left corner). The upper right basis is not primitive.

Fig. 1.5.2 c-centered lattice (net) in the plane with conventional a, b and primitive a', b' bases.

Remarks.

1. If the vectors a$_1$, a$_2$, and a$_3$ or a, b, and c form a lattice basis, then any integer linear combination of the basis vectors is a lattice vector as well. However, there may be other vectors with rational non-integer coefficients which are also lattice vectors. In this case crystallographers speak of a centered lattice although not the lattice is centered but only the basis is chosen such that the lattice appears to be centered.

   Example, see Fig.1.5.2. The lattice type c in the plane with conventional basis a, b consists of all vectors v = $n_1$a + $n_2$b and v = ($n_1+1/2$)a + ($n_2+1/2$)b, $n_1,\ n_2$ integer. This basis is a lattice basis but not a primitive one.

   The basis a' = a/2 - b/2, b' = a/2 + b/2 would be a primitive basis. Referred to this basis all lattice vectors have integer coefficients.

2. For any lattice a primitive basis may be chosen (for each lattice in the plane or in the space there even exists an infinite number of primitive bases). The basis chosen in IT A for the description of a lattice is called the conventional basis. If the conventional basis is primitive, then also the lattice is called primitive. For other reasons, the conventional basis is frequently non-primitive such that the lattice appears to be centered. The conventional centerings are c in the plane and C, A, B, I, F, or $R$ in the space.
3. In higher dimensions (dimension $n>3$) the condition that the basis vectors are shortest is no longer sufficient to guarantee a primitive basis.

Let a$_i$ be a basis. Then one can form the scalar products (a$_i$,a$_k$) between the basis vectors, $i$, $k$ = 1, 2, 3. Because (a$_i$,a$_k$) = (a$_k$,a$_i$), there are only 6 different scalar products.

Definition (D 1.5.3) The quantities

a $_1=\vert\mathbf{a}_1\vert=+\sqrt{(\mathbf{a}_1\,,\, \mathbf{a}_1)}$, a $_2=\vert\mathbf{a}_2\vert=+ \sqrt{(\mathbf{a}_2\,,\,\mathbf{a}_2)}$, a $_3\rule{0em}{3ex}=\vert\mathbf{a}_3\vert=+ \sqrt{(\mathbf{a}_3\,,\,\mathbf{a}_3)}$,

$\alpha_1= \arccos\,(\vert\mathbf{a}_2\vert^{-1}\vert\mathbf{a}_3\vert^{-1}(\mathbf{a}_2\,,\, \mathbf{a}_3))$, $\alpha_2=\arccos\,(\vert\mathbf{a}_3\vert^{-1}\vert \mathbf{a}_1\vert^{-1}(\mathbf{a}_3\,,\,\mathbf{a}_1))$,

$$\mbox{and \ \ }\alpha_3=\arccos\,(\vert\mathbf{a}_1\vert^{-1}\vert\mathbf{a}_2\vert^{-1} (\mathbf{a}_1\,,\,\mathbf{a}_2))$$

are called the lattice constants of the lattice.

The lengths of the basis vectors are mostly measured in Å (1 Å= 10$^{-10}$m), sometimes in pm (1 pm = 10$^{-12}$m) or nm (1 nm = 10$^{-9}$m). The lattice constants of a crystal are given by its translations, more exactly, by the translation vectors of the crystal pattern, they can not be chosen arbitrarily. They may be further restricted by the symmetry of the crystal.

Normally the conventional crystallographic bases are chosen when describing a crystal structure. Referred to them the lattice of a crystal pattern may be primitive or centered. If it is advantageous in exceptional cases to describe the crystal with respect to another basis then this choice should be carefully stated in order to avoid misunderstandings.

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