Question

Calculate the linear atomic density for Cu in the [110] and [111] directions in terms atomic radius R
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Answer 1

For this [110] direction there is one atom at each of the two unit cell corners, and, thus, there is the equivalence of 1 atom that is centered on the direction vector. The length of this direction vector is denoted by x in this figure, which is equal to

x = z2 ! y2

where y is the unit cell edge length, which, from Equation 3.3 is equal to 4 R3 . Furthermore, z is the length of the

unit cell diagonal, which is equal to 4R Thus, using the above equation, the length x may be calculated as follows: 2 "4R%2 32R2 2

     x=(4R)!$3'= 3=4R3 #&

Therefore, the expression for the linear density of this direction is

LD110 = number of atoms centered on [110] direction vector length of [110] direction vector

= 1atom = 3 4R24R2

3

A BCC unit cell within which is drawn a [111] direction is shown below.

For although the [111] direction vector shown passes through the centers of three atoms, there is an equivalence of only two atoms associated with this unit cell—one-half of each of the two atoms at the end of the vector, in addition to the center atom belongs entirely to the unit cell. Furthermore, the length of the vector shown is equal to 4R, since all of the atoms whose centers the vector passes through touch one another. Therefore, the linear density is equal to

LD111 = number of atoms centered on [111] direction vector length of [111] direction vector

       

=2atoms= 1 4R 2R

(b) From the table inside the front cover, the atomic radius for tungsten is 0.137 nm. Therefore, the linear density for the [110] direction is

   LD110 (W) = 3 = 4 R 2

While for the [111] direction

3 (4)(0.137 nm)

= 2.23 nm!1 = 2.23 " 109 m!1 2

=3.65nm!1 =3.65"109 m!1

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