Atomic optical spectroscopy can study the electromagnetic spectrum of elements and determine the elemental composition. In atoms, electrons reside in different levels of well defined energies. In order to move from lower to higher energy level, an electron has to absorb energy, and to move from higher to lower energy level, an electron will emit energy. This energy is in the form of light. Every element has its unique electronic structure, and the emitted or absorbed electromagnetic radiation has wavelength determined by the electronic transition. Therefore, the study of the wavelength of this spectral line leads to the study of the unique element.
In this experiment, ionized gas of hydrogen and deuterium in gas filled tube was provided with electric discharge, and the free electrons accelerated under the effect of electric field collide with the electrons in the atomic orbitals, which move to higher energy levels. Radiations of characteristic energies were emitted as excited electrons decay to lower energy levels. The emission spectra of hydrogen and deuterium may consist of various possible emission lines, and in this experiment the Balmer α, β, γ, and δ transition lines were measured.
The spectra of deuterium and hydrogen are the subject of study in this experiment. Hydrogen has atomic number of one, and is the lightest element. Deuterium is one of the two stable isotopes of hydrogen. While hydrogen has one proton in its nucleus, deuterium contains one proton and one neutron. Deuterium has an abundance of 1.5 x 10^−4 compared to 0.99985 for ordinary hydrogen.
The goal of this experiment is to calculate the mass ratio between hydrogen and deuterium by measuring the wavelength splitting between the hydrogen and deuterium emission lines. The energy levels of a hydrogen atom are given by
E_n = −(Z^2)(α^2)μ(c^2)/2(n^2)
where Z is the nuclear charge; α is the fine structure constant; μ is the reduced mass defined by μ=(m_e)(m_N)/(m_e+m_N), in which m_e is the mass of electron and m_N is the mass of the relevant nucleus; c is the speed of light in vacuum, and n is the principle quantum number in the Bohr theory. The energies of corresponding levels in hydrogen and deuterium differ by approximately 0.25% due to the different reduced masses of these two atomic systems.
A schematic diagram of energy levels associated with the Balmer α, β, γ, and δ transitions for atomic hydrogen is shown below.
As seen in the diagram, emission line from energy level 6 to 2 is the Balmer δ emission line, from energy level 5 to 2 is Balmer γ emission line, from energy level 4 to 2 is Balmer β emission line and from energy level 3 to 2 is Balmer α emission line. The vacuum wavelengths λ_(Hvac) of the Balmer α, β, γ, and δ lines for hydrogen as calculated from the equation above are given in the following table. Since the equation calculates transition wavelengths in vacuum, a correction must be applied to obtain wavelengths as measured by a diffraction spectrometer in air at standard pressure and temperature. The formula for this correction is:
λ_air = λ_vac/n_air
where n_air is the index of refraction of air. We will be using it as 1.000277 at STP.
All experimental parameters used were recorded including slit widths, integration times, incremental steps, and ranges of wavelength scan. The emission peaks look Gaussian in shape. In the lamp discharges, far fewer atoms are produced in the high atomic states so the Balmer δ lines were much less intense than Balmer α lines. And as mentioned before, slit widths needed slight adjustments when scanning for different Balmer lines in order to obtain well resolved spectra for all four transition lines.
The measured values of spectra wavelengths will be used to calculate the wavelength separation, (λ_Hair – λ_Dair), between the hydrogen and deuterium peaks of the four Balmer spectra. The deuteron-to-proton mass ratio can be determined from single Balmer transition
E = hc/λ
λ_D/λ_H = (1/μ_D)/(1/μ_H) = (1+m_e/m_d)/(1 + m_e/m_p)
The equation can be re-written as:
m_d/m_p = (λ_H m_e/m_p)/(λ_D m_e/m_p + λ_D − λ_H)
where the electron-proton mass ratio m_e/m_d is known to be 5.446 x 10^−4. By method of error propagation, the uncertainty in m_d/m_p is:
δ(m_d/m_p) = ((∂(m_d/m_p)/∂_λH)2δ_λH+(∂(m_d/m_p)/∂_λD)2δ_λD)^(1/2)
This experiment studies the emission spectra of light output of several gas discharge lamps. The light of the lamp is generated via electrical discharge through an ionized gas. To do this, one need to separate the light by its wavelength in order to measure the spectra. A grating can be used to achieve this purpose. It can diffract light into separate beams traveling in different directions which depend on the spacing of the rulings on the grating and incident angles of the light. In this experiment, a spectrometer with such a grating was used to obtain dispersed emission spectra of the light output of several gas discharge lamps.
The spectrometer consists of two mirrors and a grating, which direct and collimate incident light from an entrance slit. Connected to the spectrometer at its exit slit are a photomultiplier (PMT) tube and a motor. The PMT is a vacuum tube and a very sensitive detector of electromagnetic spectrum, which amplifies the light signal. The motor scans the angle of the grating to send light of various wavelengths to the PMT. A schematic diagram of the instrument is shown.
Spectra of three sources were measured here: mercury, hydrogen, and deuterium. The mercury lamp was used to verify the calibration of the spectrometer by looking for a line at wavelength 5460.7 angstroms and a doublet at 5769.6 and 5789.7 angstroms. A spectrum of mercury is shown in, in which a high peak at 5472.0 angstroms and a doublet at 5781.0 and 5802.0 angstroms were detected. This measurements show that the measurements by the spectrometer are approximately 11 to 13 angstroms higher than the actual values. Therefore, when scanning for spectral peaks in deuterium and hydrogen transition lines, wavelength values of slightly higher than theoretically calculated values are expected. Adjustments were also performed to test the optimization of the slit widths and integration time. Generally, if slit widths of the entrance and exit were too large, spectra will too noisy. But different elements and different transition lines in the same elements may need to be individually adjusted to achieve the best looking spectra. A reasonably long integration time would be sufficient to obtain a good spectrum, such as 50 or 100 ms.
Spectra of both hydrogen and deuterium can be measured by using the deuterium gas filled tube, because the deuterium tube contains not purely deuterium but also hydrogen. Spectra were taken several times with different slit widths, integration times, scan ranges, and increments, and the best resolved spectra are presented here. The calculation of the exact energy of the Balmer α, β, γ, and δ transitions in vacuum of hydrogen and deuterium can be done, in which Z=1 for both deuterium and hydrogen, α = 7.72973525689 x 10^−3, m_e=9.10938291 x 10^−31 kg, m_p=3.34358348 x 10^−27 kg, m_d=1.672621777 x 10^−27 kg, c=299792458 m/s, n=2, 3, 4, 5, 6. Therefore for hydrogen and deuterium.
Using these calculations, wavelengths of transition lines can be obtained through:
λ = hc/(E_n1 − E_n2)
where h = 6.62606957 x 10^−34 J s. So the wavelengths of hydrogen and deuterium in the four Balmer transitions in vacuum and air (at STP) can be calculated.
Spectra and the Gaussian fit of the Balmer α, β, γ, and δ transitions are shown. In each of the graphs, the larger peak is the deuterium peak and the smaller peak is the hydrogen peak. Values of the mean peak positions and their associated uncertainties were obtained from the gaussian fit of the spectra and presented in Table 2. The Balmer α, β, γ, and δ lines were obtained with same entrance slit of 5 mm, exit slits of 35, 23, 23, and 26 mm, respectively, wavelength range of 10, 10, 6 and 7 angstroms, increment of 0.2, 0.1, 0.1, and 0.05 angstrom, and same integration time of 100 ms.
The deuteron-proton mass ratio and associated uncertainties were calculated from each of the spectra for the Balmer α, β, γ, and δ lines and were shown in the table below:
The average value of the deuteron-proton mass ratio based on the four calculated means and four uncertainties is 1.98 ± 1.11.