Plains dan Direction in Crystal Many properties of materials depend on the backing of the atom But properties like density depend on the backing in three dimension However, other properties depend on the backing in a particular plane or along particular directions For example, the stiffness of a perfect single crystal of a copper depend on which directions the modulus is measured If single crystal copper is full in the 100 110 direction The stiffness is about 67 gigapascal If it is full in 110 directions, the stiffness is about 130 gigapascal The stiffness value change again when pulling direction is changed in 111 direction The stiffness change the way we pull on The backing of the atom in a direction is called linear density The backing of the atom on a plane is a planar density Linear density Look at this simple cubic itself The length 110 direction is set to A or 2 set to R in terms of atomic rays There are two atomic radiations along 111 directions The linear density is defined as LD, length of atom in terms of R divided by length of line The length of the 110 direction in simple cubic unit service is 0.71 In 111 directions in simple cubic has the highest linear density with the LD value is 1 The linear density is 1.0 plus the back direction All crystal structures have closed back directions You should be able to determine which directions are closed back for simple cubic, and body center cubic by looking at the hot sphere model Linear density Look again at this simple cubic unit The area of 110 plane in terms of the lattice parameter is set to A square And the area of 110 plane in terms of atomic radius is A equal to R So, 4 R square is set to total area of atom that lie on the 110 plane is 4 atoms So, the 110 plane is 4 atoms times 1 over 4 times pi R square equal to pi R square The linear density is defined as PD equal area of atom on a plane divided by area of plane So, the PD for the 110 plane is 0.56 At this point we have considered only individual directions and planes Look at the structure of simple cubic unit service All the edges have the same linear density even though they have different mirror indices Similarly, all the faces of the unit same have the same plane density Even though there are planes with different mirror indices Because this direction and planes have the same linear density and plane density And thus the same properties we grouped them together and say they are same family Things like mirror indices linear density and plane density will let the benefit you When you want to predict properties such as mechanical properties based on crystal structures