Bragg Diffraction by synthetic Opal of a pen-style 650 nm red Laser (Part II)

In Part I of this investigation into laser-based Bragg diffraction the synthetic Opal cube showed strong Bragg diffraction, using a 650 nm laser.

In Part II I'll attempt to measure the characteristic d value.

Immediately a problem becomes apparent:

In the schematic, θ is the angle of incidence and α' ('alpha prime') the angle between the purple line and the 'pseudo-crystal's' centre line (green). That blue line represents the orientation of the molecular planes (against which diffraction takes place). Without a value for α' it is impossible to determine α and β and thus d (recall the Bragg condition here.

One solution (among others) is to orient the cube so that:

and:

α = β = π/2

In that case there's no diffraction ('zero order diffraction'), only non-interactive reflection and the angle α' is bound by:

α' + θ = π/2

This however is easier said than done: at high angles of incidence the diffracted ray becomes weak (because travelling distance and thus absorption become high?) Furthermore, suspected higher order diffraction (n > 1) and/or unwanted internal refections make it hard to discern signal from noise.

To avoid this it was decided to determine the critical angle θ_c by means of correlation and subsequent extrapolation. 4 data points, each with a duplicate, were generated as below. The angles were determined with a small protractor (goniometer) and by simple Pythagorean trigonometry. The raw data were:

Correlation of the data:

I'm no longer a student or an 'institutionalised' scientist so don't have access to near-automated ANOVA but the data as well as intuition prompted me to believe the (slightly) quadratic model is the preferred one here, which yields:

γ = 0.04916 + 1.658 θ + 0.8458 θ²

Solved for γ = π/2 we then obtain θ_c:

θ_c = 0.681

and with:

α' = π/2 - θ_c = 0.8898 (51.0 deg)

This value of 51.0 deg 'feels' right and seems in accordance with some preliminary estimates (not reported here). But caution needs to be sounded, as the value is an extrapolated one, not a directly measured value.

(The meaning of γ should be clear from the diagram below:)

Determination of d:

Using the diagram for γ (above) another additional 5 data points, using the schematic below, were generated:

Measurable angles were determined by Pythagorean trigonometry and the angles α and β with some angle algebra. A total of 13 data points where thus obtained and processed (including the 8 previously obtained points):

Which gives me an average μ = 404 nm +/- 40.2 (95 % confidence)

While there's considerable spread on the results, one has to bear in mind the experiment was executed on a dining room table, with a mini protactor, some blue tack, a tape measure and two pen-type lasers.

Testing the model's predictive power:

Next week I'll test the model with 2 other lasers because:

sin β = n λ / d - sin α

Here's the result for 3 data points with the 532 nm green laser:

So the predictions for d = 404 nm aren't perfect but they're not bad either.

Here's the set up again:

The experiment specifically with the green laser also solved another mystery: where are these other diffractions that can be expected from a number of almost random micro-globule planes? Well, with the green laser, which is much more powerful (a range of a couple of kilometer) than the red and purple pen-lasers, they reveal themselves (more about that later) clearly (potentially, it the completest darkness, the red laser would also work to show the additional diffractions)

Here's the result for 3 data points with the 432 nm purple laser:

Although the predictions with the 432 nm laser are slightly less good, the trend prediction is still there.

Overall conclusions so far:

1. This synthetic Opal cube shows clear and strong Bragg diffraction for an inter-molecular plane with d = 404 nm, oriented at 51.0 deg w.r.t. the central line of symmetry of the cube.

2. Experiments with the powerful green laser (532 nm) shows multiple Bragg diffractions, corresponding to multiple intermolecular planes at various angles w.r.t. the central line of symmetry of the cube. This requires further investigation.