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How many atoms are there in a face-centered cubic unit cell of an atomic crystal in which all atoms are at lattice points?
The first thing to do here is to determine how many lattice points you get for a face-centered cubic unit cell.
For starters, all three types of cubic lattice unit cells, i.e. simple cubic, body-centered cubic, and face-centered cubic, have lattice points in the corners of the cubic unit cell.
So right from the start you know that the face-centered cubic cell has at least ##8## lattice points, one for each corner of the cube.
Now, the characteristic of a face-centered cubic cell is that it also contains lattice points in the centers of the cube's faces.
Since a cube has a total of six faces, it follows that the total number of lattice points you get per unit cell is
##"no. of lattice points" = overbrace("8 lat. pts.")^(color(blue)("corners")) + overbrace("6 lat. pts.")^(color(red)("faces")) = "14 lat. pts."##
To determine the number of atoms that you can fit into a face-centered cubic unit cell, you need to look at how the lattice structure is arranged.
For a given unit cell, pick one of its corners. This corner is being shared by a total of eight unit cells. This means that each corner in a unit cell will contain ##1/8"th"## of an atom.
Simply put, eight unit cells share one atom per corner.
For a given cell, pick one of its faces. This face is being shared by a total of two unit cells, which means that each face in a unit cell will contain ##1/2## of an atom.
In other words, two unit cells share one atom per face.
So, the total number of atoms that can fit into a face-centered cubic unit cell will be
##"no. of atoms" = overbrace( 1/8 xx 8)^(color(blue)("8 corners")) + overbrace(1/2 xx 6)^(color(red)("6 faces")) = color(green)("4 atoms")##