The relationship between D and diffusion distance

Suppose an atom is moving from one stable site to the next in the x direction by way of a random walk, as in self-diffusion. The net displacement of a diffusing atom after N jumps will be the algebraic sum of the individual jumps. If x,- is the distance moved along the x axis in the th jump, the distance moved after a total of N jumps, x, will simply be the sum of all the individual steps that is, 

In a linear crystal in which the sites are separated by a distance a, each individual value ofx,- can be - -a or —a. 

As the jumps take place with an equal probability in both directions, after N jumps the total displacement may have any value between zero and -iiNa. In order to proceed, use a mathematical shortcut. If the jump distances are squared, the negative quantities are removed. Thus  

In the limit of a large number of jumps, knowing that each jump may be either positive or negative, the double sum terms average to zero. The equation therefore reduces to the manageable form  

The result is called the mean square displacement, (x. As each jump, x can be equal to +a or —a, 

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