D = Γα²/6 applies to any diffusing species in any cubic lattice. Show that this is true for vacancy diffusion in a pure FCC metal. (hint: consider two adjacent {111} planes and determine what fraction of all possible jumps result in the transfer of a vacancy between the two planes. Is the same result obtained by considering adjacent {100} planes?

The explanation is shown below

Explanation:

The equation is by a three dimensional analysis. This yields the equation:

In cubic lattices, the jump distance is a fraction of the lattice constant represented as:

the total jump frequency is given by:

Question 1

f(x) represents the distance of the cannon from the ground

Question 2

When f(x) = 0, there will two values of 'x'. The point on the positive x-axis where the graph crosses represent the point where the cannon hits the ground.

Question 3

Yes, it would. By knowing the spot where the cannon will hit the ground, we can set the net at the spot.

Question 4

f(x) = 0
-0.05 (x² - 26x -120) = 0
x² - 26x - 120 = 0

Question 5

Please refer to the table attached below

Question 6

The value of p and q that gives the correct factors are -30 and 4 since it gives p+q = -26

Question 7

Factorising the equation completely

-0.05 (x² - 26x - 120) = 0
-0.05 (x+4) (x-30) = 0 
(x+4) (x-30) = 0

Question 8
 
Solving the equation

x+4 = 0 and x-30=0
x=-4 and x=30 

Question 9 

The roots of the equation is x = -4 and x = 30

Question 10

Please refer to the third graph

Question 11

Yes

Question 12

The negative zero means the initial distance of the cannon, where it was fired from

Question 13

The distance between x = -4 and x = 30 is 34 units. If the cannon was fired from the point when x = -4, the cannon will hit the ground again 34 units from the point it was fired from. If Nik put a net 30 units from the firing point, the cannon will fly pass it.