(5-2)    ① Bragg’s law states that nλ  =  2d sinθ.  Since we are dealing with the smallest angle at which a maximum is observed, and therefore the smallest value of sinθ, we can say that n = 1, and hence:

           λ  =  2d sinθ  =      2 x 9.1 x 10-11 x sin(65o)  =  1. 65 x 10-10 m

②  We know that voltage = energy / charge.  So the energy (E)  picked up by an electron (in J) when it is subject to a voltage of 54 V is given by:

           E   =  voltage x charge  =  54 V x  1.602 x 10-19 C  =  8.65 x 10-18 J

③     Provided that v is much less than the speed of light, we can calculate Ek  =  ½ mv2.

So velocity of the electron, v = √(2 x Ek / m)  =  √(2 x 8.65 x 10-18  / 9.109 x 10-31) = 4.36 x 106 ms-1

This is much less than the speed of light, so the non-relativistic expression is appropriate.

④   Momentum  p  =  mv  =  9.109 x 10-31  x  4.36 x 106  =  3.97 x 10-24 kgms-1

So now we come to the crunch. The de Broglie equation gives the wavelength of the electron’s associated “matter wave” as:

           λ  =  h/p  =  6.626 x 10-34 / 3.97 x 10-24   =  1.67 x 10-10 m

Therefore, to 2 significant figures, which is the limit of precision of our lattice spacing and reflection angle data, we agree perfectly with the wavelength derived from Bragg’s law.