Atomic Spectra

In this lab you will observe the visible spectrum of Hydrogen and use it to measure the Rydberg constant. Then you will use the spectrum of two unknown sources to determine their atomic content.

Background

The quantum explanation for atomic structure was developed in the early 1900’s based largely on measurements of the wavelengths of light emitted by atoms in gas-discharge lamps.
These measurements, first made in the late 1800’s, provided evidence that the energy levels of atoms are quantized (i.e. have a discrete set of possible values). In particular, in 1885, Johann Balmer measured the visible spectrum of Hydrogen and found the wavelengths of light that were present were related according to

The ultraviolet and infrared portions of the Hydrogen spectrum were later measured (by Lyman, Paschen, Brackett and Pfund) and found to fit similar formulas but with integers other than 2 in the first term (and in the inequality). They all measured the same coefficient, $R$, called the Rydberg constant.

The origin of this formula was eventually understood within the Bohr model of the atom, which posited that the light from Hydrogen consists of photons of energy $E = h\nu = hc/\lambda$ that are emitted by atoms transitioning between electronic states that have discrete energies of $E_n = -hcR/n^2$, where $c$ is the speed of light, $h$ is Planck's constant, $R$ is the Rydberg constant and $n$ is a positive integer (Fig. 1). Transitions from excited states to the $n = 2$ state produce visible light (red to violet, left to right for the four lines shown) that you will observe in this lab. Transitions to the $n = 1$ state produce ultraviolet light, and those to the $n = 3$ state produce infrared light, neither of which can be seen by eye.

This understanding allowed the Rydberg constant to be formulated in terms of purely fundamental constants like the mass, $m$, and charge, $e$, of the electron and the permittivity of free space, $\epsilon_0$:

whereas previously it had been only an empirically determined value. [1]

In this lab, you will measure the visible spectrum of Hydrogen as Balmer originally did and, from it, obtain a measurement of the Rydberg constant.

Note that the Rydberg constant applies only to Hydrogen. Other atoms have a similar $1/n^2$ structure to the equation that describes their spectra, but with a different coefficient. Each element’s spectrum is uniquely characteristic, so spectra are often used to identify the elemental composition of a sample (or a star! [2]).

After you completing your measurements of the Hydrogen spectra, you will measure the visible spectra of two unknown samples and use them to identify each sample's elemental composition.

Instruments

The apparatus used in this lab consists of

• A spectrum tube - a special kind of gas-discharge lamp that shapes the light emitted by a gas to optimally illuminate a narrow slit

• A carousel - that rotates to power your choice of three spectrum tubes. One contains Hydrogen gas and the other two contain gases you will identify based on their spectra.

• A diffraction grating - an optical element that uses the phenomenon of interference to send light of different wavelengths in different directions.

You have learned about interference in the context of collimated light passing through a double slit. In another lab in this course, you may have experimented with passing light through an array of three, four or five slits. When collimated light passes through multiple, equally spaced slits, the result is an intensity pattern with maxima and minima at the same positions as in the pattern created by two slits with the same spacing, but with maxima that are much more narrowly peaked, (i.e., the contrast is much improved).

The diffraction grating you will use was made from an etching of thin parallel lines on a piece of glass. Etched regions scatter light instead of transmitting it, so the non-etched regions between the etched lines act as slits. The etching process allows hundreds or thousands of lines per mm, so this type of diffraction grating has slits that are not only plentiful, but closely spaced as well.

Because of its very large number of slits, the diffraction grating creates very sharp maxima. Because of their close spacing, the slits impose a large "lever arm" on the pattern (i.e. a greater angular separation between maxima) making it possible to distinguish maxima of subtly different wavelengths. This can be seen from the equation for constructive interference. Light passing through slits separated by a distance $d$ will have intensity maxima at angles $\theta_m$ for which

where $m$ is any positive integer. The smaller $d$ is, the larger $\theta_m$ will be for the same $\lambda$.

The $m = 0$ case is the central maximum. By measuring $\theta_m$ for $m = 1$ maxima for each of the different wavelengths, you can precisely determine those wavelengths. [3]

Note that this equation applies only if the collimated light is "normally incident" on the grating. You may have to adjust the grating to make this be the case.[^4]

• A spectroscope - which sends collimated light onto the diffraction grating and enables you to measure the various angles at which the light that passes through the grating constructively interferes.

Two cylinders define the light path: the collimator (C); and the telescope (E). The collimator has an opening at its far end that only admits light through a vertical slit (B). The width of the slit is narrow but adjustable. It shapes the light beam into a vertical line. Letting the light spread vertically doesn’t degrade the resolution in $\theta$ because $\theta$ is an angle in the horizontal plane. Because a line allows more light to hit the detector, it is much easier to determine the location of a line than a point. The use of vertical slits in spectral measurements is so common that the observed maxima are called spectral lines.

The telescope swings to different angles around the grating (A) and images the light that comes through the grating onto a detector. The lens at the far end of the telescope is called an "eyepiece". If this were an in person lab, you would look through the eyepiece and your eye would be the detector. To allow you to view the lines remotely, we have put a webcam behind the eyepiece and made what it sees available to you by clickling on the "eyepiece" button.

The angular position of the telescope is read off of scales (I,J) situated on either side of the spectrometer. The scales measure degrees down to one arc minute with Vernier markings.

Using a Vernier scale takes a bit of explanation and some practice. There are some good extended descriptions online (e.g. in Wikipedia), so here we’ll introduce it with a few examples specific to the scales you will be using. The scales on the spectroscope consist of two sets of uniformly spaced lines that are arranged so that they slide past each other. The lower set's lines mark each half degree, while the upper one's (which are very slightly more closely spaced) have markings corresponding to 0-30 arc minutes. The two scales are used together.

Cartoons of two different measurements are shown in Figure 7 to illustrate the procedure. First you look where the zero mark on the inner scale is located. In the left hand cartoon that is to the right of $210^\circ$; in the right hand cartoon it is to the right of $229.5^\circ$. Then you look where the lines on the inner and outer scales align most closely. In the left hand cartoon that happens for the $15^\prime$ mark; on the right side it happens at the $16^\prime$ mark. These measure the number of arc minutes to add to the values obtained from the first step. So, the measurement on the left side is $210^\circ + 15^\prime = 210^\circ + (15/60)^\circ = 210.250^\circ$, and the measurement on the right is $229.5^\circ + 16^\prime = 229.5^\circ + (16/60)^\circ = 229.767^\circ$.

The precision of these $\theta$ readings is limited by the one arc minute granularity of the Vernier scale, but it is possible to make mistakes in reading or calculating the values. Here are a few things you can do to reduce the chance of such mistakes and their impact:

1. Have someone else (e.g. your lab partner) make the same reading (without telling them what you got) and compare.
2. Recall that the position of the telescope can be read off of scales situated on either side of the spectrometer. The angles recorded by the two scales should differ by exactly $180^\circ$. Read and record both. If, after correcting for the $180^\circ$ difference, you find that their values differ by more than the $0.005^\circ$ precision of the scales, something is wrong - if not with your readings, then with the spectrometer itself (*e.g., warping of the scales). This approach of making multiple, independent measurements of the same quantity is always a good idea and helps validate your results.
3. Record your raw readings in degrees and arc minutes, and then calculate the corresponding decimal degree values. This will help you disentangle any calculation errors that may occur. While the calculations in this case are not complicated, and the consequences of a mistake are small, this sort of fastidious data recording is a good habit to develop for professional laboratory work, when both the chances and consequences of mistakes increase.

Finally, be aware that the Verniers read angular position but the location of $0^\circ$ is arbitrary.

In order to determine the wavelength using $d\sin\theta = \lambda$, where $\theta$ is the angle between maxima, you need to measure two angles and subtract, just like measuring a length requires two position measurements. You could do this by measuring the angular position of the central maximum, which we can call $\theta_0$ , and the angular position of the $m = +1$ maximum, which we can call $\theta_{+1}$. Then $\lambda = d\sin(\theta_{+1}-\theta_0)$. Similarly, you could measure the angular position of the $m = +2$ maximum and use $2\lambda = d \sin (\theta_{+2}-\theta_0)$ or $\lambda = d \sin (\theta_{+2}-\theta_{+1})$. Finally, note that the direction in which the angles increase or decrease on the scale is arbitrary, so you should use the absolute value of the differences.

Approach

Using the instruments above, you will determine the wavelengths of light emitted by three different gasses.

The web-portal for this experiment enables you to manipulate the spectroscope, see the lines and read the verniers just as you would in person.

Measurement

Now that you have familiarized yourself with the spectroscope and have it aligned on the central maximum, you are ready to measure the Hydrogen spectrum!

You may see a background of colors in addition to the main bright lines. This background is typically due to the ultraviolet light emitted by the hydrogen lamp exciting other atoms to emit. In fact, that is how fluorescent lights work. Ultraviolet emission produced by running a current through the gas in a long tube excites atoms in the white fluorescent material that coats the inside of the tube. That material then emits a large number of different wavelengths that we perceive as white light.

[1]: Interestingly, the measured value of $R$ did not match the prediction of the simplest Bohr model which uses $m = m_e$ because the electron and proton together lead to an effective “reduced mass”, $m$, that is very slightly below the electron mass. Furthermore, the spectrum of hydrogen was found to contain a set of wavelengths with a slightly different $R$ value. These come from deuterium, which is hydrogen with a neutron in the nucleus, the mass of which further alters $m$). Both effects are smaller than 0.1% and only revealed by exquisitely precise measurements.

[2]: It is amusing to note how often dialogue in Star Trek episodes involves the captain asking for a “multi-spectral analysis” of some anomaly. Analyzing the spectrum of astrophysical objects is common, in fact so common that it would be done automatically, without the captain’s specific request! But it does provide tech-y sounding, and in this case not incorrect, dialogue.

[3]: To validate your results, it is good practice to measure $\theta_m$ values for other values of $m$, but be aware that the grating is optimized for $m=+1$, so the other maxima can be hard to see.

[4]: If the collimated light hits the diffraction grating at some other angle $\theta_i \neq 0$, the equation becomes $d (\sin \theta_i \mp \sin \theta_{\pm m}) = \pm m\lambda$, where the angles are measured from the grating's normal. If $\theta_i \gg 0^\circ$ and you neglect the $\theta_i$-term, you will have a systematic error in your wavelength measurements. Fortunately, it is easy to either (i) determine $\theta_i$ directly using the overview camera, which is positioned directly above the grating's axis of rotation, or (ii) cancel out its effect by measuring $\theta$ for two different values of $m \neq 0$ for the same wavelength and substracting the corresponding equations to cancel out the $\theta_i$-term.

[5]: This value takes into account the effective reduced mass of the electron in the Hydrogen atom. See Wikipedia for elaboration.