Davison-Germer experiment - Bragg condition

[Strides]

Hey,

I'm trying to adapt the Bragg condition for the David-Germer experiment, so I can then use their experimental results to show that the measured wavelength is compatible to the wavelength theorized by De Broglie. However I'm having issue with the calculation, any help would be much appreciated.

Starting with the Bragg condition for constructive interference:

$$nλ = 2d sin(θ)$$

If D denotes the spacing of the atoms in the crystal, where:

$$d = Dsinα$$

with: α = π/2 - θ_B

where the scattering angle is:

θ = 2α

Then, somehow the Bragg condition becomes:

$$nλ = Dsin(θ) $$

Maybe I'm just forgetting some obscure trig identities but I just can't seem to get the final result.

 

[Charles Link]

For the Bragg scattering, there are two conditions that get satisfied: 1) From the individual crystal planes, angle of incidence =angle of reflection 2) The reflected wave from adjacent crystal planes constructively interferes, so that $n \lambda=2D \sin{\theta}$ , where $\theta$ is the angle of incidence measured from the plane of these crystal planes, (and not from the normal to these planes). $\\$ The $d$ you have is $d=D \sin{\theta}$, is the extra path distance the beam travels on just one side of being incident and then reflected from the adjacent crystal plane. Total extra path distance is $2 d$.If you do a little trigonometry, and compute how much the scattered peak gets deflected relative to the incident beam, that angle is $\theta_B=2 \theta$. It would be easier to show with a diagram, but I don't know how to use Powerpoint very well. Here's a "link" that did it for me: https://www.google.com/imgres?imgurl=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/bragglaw.gif&imgrefurl=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html&h=205&w=431&tbnid=OthoXYdSpI9wWM:&tbnh=100&tbnw=211&usg=__zzkvTLwMgFGUOjWDHBA7QGbRrxQ=&vet=10ahUKEwi71uCri-DYAhUDMqwKHRBmC2AQ9QEIKzAA..i&docid=fznSJo6pR1Vl4M&sa=X&ved=0ahUKEwi71uCri-DYAhUDMqwKHRBmC2AQ9QEIKzAA (The small $d$ in the "link" is your large $D$).

 

[Charles Link]

It should be mentioned that the first condition, that angle of incidence is equal to the reflected angle, makes it so that the scattered light off of each of the atoms in a single plane makes them all constructively interfere with each other by making the path difference equal to zero for the scattered light from each of those atoms of the same plane.