Chapter 1
Unit Cell Composition
1.1 Translation Vectors in Plane and Space Lattices
(a) When an object (say the number 7) is repeatedly translated through an interval
“a,” we obtain a one-dimensional array of the number 7, as shown in Fig. 1.1a.
When each object in the array is replaced with a point, a collection of points is
obtained as shown in Fig. 1.1b. This is known as a linear lattice. Since, a
geometrical point has no (or zero) dimension, therefore the lattice point is an
imaginary concept but the array of the number “7” is real.
(b) When we add another non-collinear translation “b” to the entire lattice array
due to the translation “a,” a two-dimensional array of objects is obtained
(Fig. 1.2a). The corresponding collection of two-dimensional points in
Fig. 1.2b is called a plane lattice.
The characteristic feature of a plane lattice is that the environment around
any one point is identical to the environment around any other point in the
lattice. In a plane lattice, any two points connected through a translation vector ~ t
is given by
~ t ¼ n 1 ~ a þ n 2 ~ b
ð1:1Þ
where n 1 and n 2 are arbitrary integers, when putting together in the form [n 1 n 2 ]
gives the direction of the translation vector (Fig. 1.3) in the given lattice, ~ a and
~ b are the primitive translation vectors along x and y axes, respectively.
The magnitude (say R) of the resultant translation vector between any two
points (in a plane lattice) can be determined by the formula
R = P
2 + Q
2 + 2PQ cos c
À
Á 1=2
ð1:2Þ
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_1
1
Unit Cell Composition
1.1 Translation Vectors in Plane and Space Lattices
(a) When an object (say the number 7) is repeatedly translated through an interval
“a,” we obtain a one-dimensional array of the number 7, as shown in Fig. 1.1a.
When each object in the array is replaced with a point, a collection of points is
obtained as shown in Fig. 1.1b. This is known as a linear lattice. Since, a
geometrical point has no (or zero) dimension, therefore the lattice point is an
imaginary concept but the array of the number “7” is real.
(b) When we add another non-collinear translation “b” to the entire lattice array
due to the translation “a,” a two-dimensional array of objects is obtained
(Fig. 1.2a). The corresponding collection of two-dimensional points in
Fig. 1.2b is called a plane lattice.
The characteristic feature of a plane lattice is that the environment around
any one point is identical to the environment around any other point in the
lattice. In a plane lattice, any two points connected through a translation vector ~ t
is given by
~ t ¼ n 1 ~ a þ n 2 ~ b
ð1:1Þ
where n 1 and n 2 are arbitrary integers, when putting together in the form [n 1 n 2 ]
gives the direction of the translation vector (Fig. 1.3) in the given lattice, ~ a and
~ b are the primitive translation vectors along x and y axes, respectively.
The magnitude (say R) of the resultant translation vector between any two
points (in a plane lattice) can be determined by the formula
R = P
2 + Q
2 + 2PQ cos c
À
Á 1=2
ð1:2Þ
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_1
1