Electron Diffraction: Calculate Crystal Plane Spacing

[throneoo]

Homework Statement

a beam of electrons with a kinetic energy of 50eV is incident on a powdered crystal sample. A series of bright and dark rings is observed on a screen behind the sample, as shown below (a picture which shows two concentric circles with the beam passing through the center . ) If bright rings are observed at angles of theta =20 degrees and theta =40.7 degrees to the beam direction , calculate the spacing of the planes of the crystal.

Homework Equations

bragg diffraction , p=h/lambda ,E=p^2/2m

The Attempt at a Solution

I can find the wavelength using a combination of the 2nd and 3rd equation (E=(h/lambda)^2 / 2m )

but since no further information is given regarding the order of the bright fringes at those 2 angles (e.g. the relationship between the two maxima , in which order they are in etc.) I assumed they correspond to the 1st and 2nd maxima .
using 2d sin theta=n lambda , I can get two different values of d with the two angles .

However , the discrepancy between the two values is quite huge (about 10%) . therefore I don't know how to get a consistent value .

so I decided to find two integers n and m such that n/m is approximately equal to the ratio of the sines of the angles ~ 1.906608182 (which is just from the formula) , and minimize the discrepancy of d .

but I'm afraid I would get large values of n and m just to satisfy the condition , so large they are not even physically feasible .

 

[throneoo]

just realized I need to divide the angle by 2 before applying the formula . the values are much more consistent now .