Question

A crystal diffracts a beam of electrons, like a diffraction grating, as they hit it perpendicular to its surface. The crystal spacing is 0.18 nm, and the first maximum scattering occurs at 80° relative to the normal to the surface. (e= 1.60 × 10-19C, melectron= 9.11 × 10-13kg, h= 6.626 × 10-34J · s)(a) What is the wavelength of the electrons?(b) What potential difference accelerated the electrons if they started from rest? Answer: (a)0.18 nm(b) 48 V

Answer 1

part a:

using the formula d*sin(theta)=m*lambda

where d=crystal spacing=0.18 nm

theta=angle w.r.t. normal=80 degrees

m=order of the maximum =1

then wavelength=lambda=d*sin(theta)/m=0.17727 nm ~ 0.18 nm

part b:

from de-broglie's equation:

momentum p=h/lambda

where h=planck's constant

as p=m*v where m=mass of electron and v=speed achieved at the end of acceleration

then m*v=h/lambda

==>v=h/(m*lambda)

and q*V=0.5*m*v^2

where V=potential difference, q=charge of electron

==>V=0.5*m*v^2/q

=0.5*m*(h/(m*lambda))^2/q

using the given values,

V=47.928 volts ~ 48 volts