(5-13)   ①   We have our scaled Gaussian function, with a mean (corresponding to the top of the peak) at zero:

If x=0, we simply get G'(x) =  e0  = 1.  

②   So at the top of the peak, G'(x) = 1 and hence we want the width at half-height, where G'(x) = 0.5.

Taking the natural log of both sides, this becomes:

③  This result applies when x is the positive value of x at half height.  This means that x =  Δx½ /2 

where Δx½ is the peak width at half height.  So we end up with:

④  In Question 5-11, we derived - by measuring the linewidth from the graph - the relationship  Δx½   =   1.4 / kmax .

Substituting this in, and crunching the numbers a bit, we then get: