A hypothetical 2D solid forms a square lattice. Atoms of type X occupy the corners of the squares, and an atom of type Y occupies the center of each square. The atoms touch each other along the diagonal of the square. If the radius of atom Y is twice the radius of atom X, what is the 2D packing fraction (ratio of the area occupied by atoms to the total area of the square unit cell)?

Step-by-step solution

Let the radius of atom X be r . Then the radius of atom Y is 2r . The atoms touch along the diagonal of the square unit cell. The length of the diagonal is equal to the sum of the diameters of the atoms along it: 2 a = 2r_X + 2r_Y = 2r + 2(2r) = 6r a = 6r 2 = 3 2 r The total area of the square unit cell is: A_ cell = a^2 = (3 2 r)^2 = 18r^2 The effective number of atoms per 2D unit cell is: For X (corners): 4 1 4 = 1 For Y (center): 1 The total area occupied by the atoms is: A_ occupied = r_X^2 + r_Y^2 = r^2 + (2r)