Most hydrogen atoms have only one proton in their nucleus, but some are composed of one proton and one neutron. These atoms are called deuteron, but are simply an isotope of hydrogen. Neutrons are neutrally charged atoms, but have about the same mass as a proton. Thus the ratio of the weight of a hydrogen atom to a deuteron atom is 1:2. I was able to measure the ratio of hydrogen mass to deuteron mass experimentally by looking at the light emitted by these atoms as electrons move to lower energy states.
The experimental setup consisted of a lamp which excited the electrons in hydrogen or deuteron atoms via an electric current. This lamp was placed so that the light entered a darkened chamber through a slit. By refracting the light, we were able to spatially separate light of different wavelengths. Since an atom has defined energy states, there are specific wavelengths of light that may be emitted. These wavelengths are different for every atom, since they depend on the atomic number and reduced mass of the particular particle. Thus, since hydrogen and deuteron atoms have different masses, the wavelengths of light they emit will differ. The energy levels of an atom can be described by
where Z is the nuclear charge of the atom, α is the fine structure constant, μ is the reduced mass, c is the speed of light in a vacuum, and n is the principle quantum number in the Bohr theory. Thus the wavelength of light emitted by a change in energy levels is
I took four spectra of these emissions around four of the peaks of deuteron. These pictures are shown below. Each photo has two peaks. The larger one is from deuteron atoms, and the smaller one is from hydrogen atoms. Because it is so hard to separate deuteron from hydrogen, each deuteron sample contained some hydrogen. These pictures have good resolution, so it is easy to estimate their centers and their widths. The resolution was determined by the width of the slits, such that a good resolution featured two well defined, separate peaks. I fit each spectra to two Gaussians (one for each peak), to get numerical values for the mean and the standard deviation of each peak.
Figure 1: The first transition state. This spectrum is not a good fit for the double Gaussian because it has plateaus at both peaks. However, the centers of the peaks are easy to find, so it was possible to get a rough estimate for the ratio of md/mp.
Figure 2: Light emitted from electrons going from n=4 to n=2. Again, there are clearly defined peaks that allowed me to calculate the difference in wavelength of the emissions.
Figure 3: The transition from n=5 to n=2. In order to get a clear peak, it was necessary to increase the integration times to get more data. This happens because the n=5 is so energetically unfavorable that few electrons begin here.
Figure 4: The transition from n=6 to n=2. When the light emitted is from a large transition, there seems to be a smaller change in wavelength than from a smaller transition. In calculations of md/mp, this difference also appears in the change of Aair for different Δλs.
In order to find the ratio of md/mp, I derived a relationship between the mass of a hydrogen atom and the mass of a deuteron atom.
where Aair= Δλh-Δλ∞ and Δλair is the separation of the peaks shown by my spectra. Aair was different for each emission, since each λ depends on the energy of its energy state. I found that my measured ratio was very close to the expected value of 2. For the first spectrum, the fit was not very close because each peak had a plateau, but by doing a preliminary calculation I found md/mp=2. In the other cases I was able to find more precise values. For the n=4 to n=2 case, md/mp=2.0±0.1, for the n=5 to n=2 case, md/mp=2.03±0.05, and for the n=6 to n=2 case md/mp=1.94±0.05. This led to a weighted average of md/mp=1.97. It is thus evident that my measurements for md/mp adhered to what was theoretically expected.
This lab was very finicky, but overall I enjoyed it. Finding a slit width that provided me with a good resolution of the peaks was tricky, and required many tries. However, spending this time to get a good spectra paid off in the accuracy of my calculations. Deriving the relationship between md and mp was also a bit tricky, but I got it to work eventually!