Question

If the lattice parameter for a crystalline structure is , then the atomic radius in fcc crystal is:
(A) $$7.20$$
(B) $$1.80$$
(C) $1.27$
(D) $2.90$

Answer 1

Hint:A Crystal lattice is defined as a three-dimensional arrangement of atoms, ions, or any constituent particle in space. These atoms, ions, or particles that comprise the lattice are called lattice points. These points are joined together by a straight line.

Complete step by step answer:
Every crystal lattice is formed from a unit cell. A unit cell is the basic entity of a lattice. A unit cell is the smallest repeating unit of the cell which when repeated over and over gives the crystal lattice. A lattice constant or parameter is used to define the physical dimension of unit cells in a crystal lattice. Generally, there are six lattice parameters of a unit cell, three are edges and three are the angles between them. The edges are termed as $a,b\,and\,c$ and the angles are $$\alpha ,\beta \,and\,\gamma $$.
Here, we are given an FCC lattice. In a Face-centred cubic unit cell, atoms are placed at each corner and the centre of all the faces of the cell. In FCC there are total $8$ corners and $6$ face centre. So the total Number of atoms in an FCC structure is $4$. The relation between lattice parameter and the radius is given by the formula:
$a = \dfrac{{4r}}{{\sqrt 2 }}$
Where $a = $ edge or lattice parameter
$r = $ radius of the lattice
Now, we will calculate the radius using the value of the lattice parameter. Here, we are given $a = 3.6$. We will put this value in the formula and we will get;
$a = \dfrac{{4r}}{{\sqrt 2 }}$
$r = \dfrac{{\sqrt {2a} }}{4} = \dfrac{{\sqrt 2 \times 3.6}}{4} = 1.27$
Hence, the value of Radius of the given FCC lattice will be $1.27\,{A^0}$.

Therefore, option (C) is correct.

Note:
Unit cells are divided into two categories: (a) Primitive unit cell (B) Centred Unit cell. In the primitive unit cell, the constituent atom or particle occupy the positions of the corners. Centred Unit cells are divided into three types:
Body Centred Unit cell
Face centred Unit cell
Edge Centred Unit cell