# Diffraction Grating: Can We Detect the Doublet?

## Introduction

Even toys can provide interesting insight into the world of physics. Newton's Cradle, for example, is a common desktop knick-knack that demonstrates basic conservation of momentum and energy. Another example of is a pair of glasses Dr. Noe showed me during my first visit to the LTC. These glasses, priced roughly 40 cents a pop, are simply cardboard frames housing slides of diffraction grating. The novelty of this toy is that, when peered through, it gives each light source a halo of rainbows. But, this toy does much more than just make pretty colors. Diffraction grating is a fundamental tool in the world of optics, and it can be used in many different applications, one example being in spectroscopy.

A primary goal of spectroscopy is to analyze radiated energy in order to understand the nature of the object radiating it, as in the identification of elements by their emission spectra. All elements, when put into a gaseous state and energized, will emit characteristic wavelengths of light as electrons fall from high to low energy levels. In sodium, there are two wavelengths of light that are quite famous, and they are jointly known as the "sodium doublet." When electrons fall from the 3p to the 3s orbital, they can either emit light with a wavelength of 589.0 nm or 589.6 nm. These wavelengths of light are remarkably close, and can provide unique opportunities for research.

Diffraction grating, in a very basic sense, spreads out the wavelengths of light comprising a light source. So when light from sodium is sent through a diffraction grating with a high enough resolving power, we are able to distinguish these two D-lines. Given this knowledge, we then ask the question: is it possible to resolve the famous sodium doublet using our toy diffraction grating?

## More on the Sodium Doublet

The sodium doublet is a product of a phenomenon known as "spin-orbit interaction." Like all elements, an atom of sodium has a nucleus and an electron cloud surrounding it. The electrons around the nucleus are not stationary, and since they are charged, produce their own magnetic field. Another important characteristic of these electrons is their "spin magnetic moment." This arrises from the fact that an individual electron has intrinsic angular momentum, entirely separate from its orbital angular momentum. It is important to note that, while a rotating, charged sphere can produce a magnetic moment, this does not describe how the effect arrises for an electron.

The magnetic moment of an electron, which can either be either 3/2 or 1/2, then interacts with the magnetic field generated by the electrons orbit. Thus, two electrons of different spins in the same electron orbital have different energies associated with them. So, in sodium, when the electron with a higher spin falls from the 3p to the 3s energy level, it will emit a photon with more energy than one emitted by its less energetic counterpart.

## Diffraction Grating

Diffraction grating relies on the same principles that are important to the double slit experiment. When a light beam is sent through multiple slits, light going through one slit will interfere with light from another slit. This produces an interference pattern that manifests as a sequence of high and low intensities. Diffraction gratings take this to an extreme, fitting as many slits together as possible. For each slit that the light travels through, any area that does not coincide with 100% constructive interference decreases in intensity, causing very narrow peaks of intensity.

The locations of these areas of high intensity correspond to the diffraction equation,

where d is the separation between the slits, \theta is the is the angle between the central high intensity point and its diffracted image, m is the diffraction order, and \lambda is the wavelength of the light. As we fit these slits closer and closer together, each wavelength of light will correspond to a narrower peak, allowing us to resolve different wavelengths of light.

The toy glasses we used had the slit distance printed on the side, 500 lines/mm or 2 microns, but we first had to verify this value. I found a green laser pointer with a known wavelength of 532 nm, directed it through the glasses, and recorded the distance from the grating to the wall and from the central bright spot to the first order. To verify the consistency of the glasses, I made sure to repeat the process multiple times using different spots on the lens. Running these data through the diffraction equation, the resulting slit spacing was actually closer to 1.98 microns, with a 0.012 micron tolerance for measurement errors. The nominal 2 micron value is within this range, so we can conclude that this is the average slit spacing.

## Resolution Limitations

Now, with an understanding of what causes the sodium doublet to arise, and the parameters of our diffraction grating, we began our attempt to resolve the D-lines. When doing this, we need to take into account a few limiting factors.

First, the intensity of the D-lines must be considered. If they are too dim, we might not be able to see them with either our eyes or the camera. If this is the case, we could put a convex-cylindrical lens in front of the sodium lamp. This lens is essentially a clear dowel that has been split down the middle. If we place this perpendicular to the slit we will allow the sodium light to travel through, it will focus the light that is directed away from our camera toward it, making the lines more visible.

Second, we must be aware of the resolution of our camera. A camera's sensor is an array of pixels, so our ability to record data is limited to the size of these pixels. Because the wavelengths of the D-lines differs by 1 part in 1000, we need to have a camera that can accomplish this. We verified this by assuring we could detect the 1 mm lines on a meter stick from 1 m away, as shown below.

Finally, the resolving power of the grating must be taken into account. If the image of the D-lines are not fine enough, the two will blur together and be indistinguishable. So, to check if this will be an issue, we look at the equation

where \lambda is the average wavelength of light, \Delta\lambda is the difference in the wavelengths, \m is the diffraction order, and N is the amount of lines we'd need to illuminate to be able to resolve two wavelengths of light \Delta\lambda apart. In our setup, the wavelengths of light are 1 in 1000 apart, so we'd need to illuminate 1000 slits. Since our grating has 500 lines/mm, all we would need is 2 mm worth to get the needed resolving power.

## Results

Armed with an understanding of the factors that affect our resolving ability, we warmed the sodium lamp up and set about attempting to image the D-lines. To make the two lines easier to distinguish, we placed a slit, roughly 0.5 mm wide, in front of the sodium lamp's opening, as shown here:

To first see if it was even remotely possible to resolve them, we tried first to simply see the D-lines. The setup was a small handheld telescope, aimed through the diffraction glasses, at the spot where the first diffraction order of the lamp's image would appear.

This preliminary test was successful, and we could see the two D-lines, clear as day.

Next, we set about attempting to image them using a digital camera. The first camera gave us a little hope that we would be successful, but was still a little blurry. To minimize blurring caused by bends in the grating, we apertured the lens to the calculated minimum of 2 mm. We got some promising shots, but nothing conclusive. It appeared as thought the camera's inconsistent zoom was going to provide us with some issues.

We then tried with a different, slightly older camera. The Sony Mavica had a bit more reliable focus, but did not have the resolution needed to distinguish the two lines, even when set to a black and white setting to maximize contrast.