Department of Physics, Middlebury College | 1992-93 |

Modern Physics Laboratory |

The energy levels of a hydrogenic atom are given by

^{1}

E _{n}= -^{1}/_{2}Z^{2}^{2}µ c^{2}^{1}/_{n2}(1)where Z is the nuclear charge (Z = 1 for H and D atoms), is the fine structure constant, µ is the reduced mass µ = m

_{e}m_{N}/(m_{e}+ m_{N}) (m_{N}is the mass of the relevant nucleus), c is the speed of light in vacuum, and n is the principal quantum number in the Bohr theory. The most recent adjusted values of the fundamental constants may be found in Appendix A of this manual.The energies of corresponding levels in hydrogen and deuterium differ by approximately 0.25% because of the different reduced masses of these two atomic systems. Precise measurements of Balmer emission spectra for these atoms permit a determination of the deuteron-to-proton mass ratio m

_{d}/m_{p}to an uncertainty of 1 - 2%.In this experiment you will assume that Eq. (1) predicts the spectrum of hydrogen exactly and you will use your experimental spectra to determine the mass of the deuteron through its effect on the deuterium spectrum. Do not assume the deuteron mass or even look it up in tables during the course of this experiment; it is to be determined from your experimental spectra.

The energy levels involved in the Balmer , , , and transitions of atomic hydrogen are shown in Fig. 1. The vacuum wavelengths

_{H vacuum}of the Balmer H_{}, H_{}, H_{}, and H_{}lines for hydrogen, as calculated from Eq. (1), are given in Table I. To obtain these wavelengths to six significant figures, the full precision of the fundamental constants is necessary. It is essential throughout this experiment to maintain the precision of all numerical calculations to six or seven significant figures.Eq. (1) determines transition wavelengths in vacuum, so a correction must be applied to obtain wavelengths as measured by a diffraction spectrometer in air at STP. This correction is given by

_{air}=_{vacuum}/n_{air}, (2)where n

_{air}the index of refraction of air. A useful tabulation of the index of refraction of air for visible light can be found in Ref. 2. Throughout your analysis treat this index of refraction correction as exact and do not estimate its (negligible) contribution to the uncertainty in your final experimental result for m_{d}/m_{p}.

TABLE I. Wavelengths of Balmer H_{}, H_{}, H_{}, and H_{} lines in vacuum and in air at STP. | ||

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Transition | _{H vacuuma} | _{H airb} |

Name | (Å) | (Å) |

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H_{} (n = 3 --> n = 2) | 6564.70 | 6562.89 |

H_{} (n = 4 --> n = 2) | 4862.74 | 4861.38 |

H_{} (n = 5 --> n = 2) | 4341.73 | 4340.51 |

H (n = 6 --> n = 2) | 4102.94 | 4101.78 |

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**Procedure**

Obtain high resolution spectra of the Balmer , , , and lines of atomic hydrogen and deuterium using the SPEX Model 1704 spectrometer and the McSPEX program

^{3}for the IBM PC/XT. A diagram of the experimental apparatus is given in Fig. 2. Spectra for the Balmer H_{}/D_{}and H_{}/D_{}are given in Fig. 3(a) and 3(b). You may notice that the lineshapes of the Balmer H_{}and D_{}lines of Fig. 3(a) appear to be asymmetrical. The small splitting of the H_{}and D_{}lines is due to the fine structure splitting of the 2p state into 2p_{1/2}and 2p_{3/2}states. Fine structure effects have not been considered in Eq. (1) and will not be considered in this experiment.Record slit and lock-in amplifier settings in your laboratory notebook. Store your four Balmer spectra on disk and plot them on 11" x 17" graph paper using the IBM 7372 plotter.

During data acquisition be mindful of the following.

(1) It is essential to avoid high photon counting rates at the lock-in amplifier. A sure sign of excessive counting rate is "flat-topping" of the peaks in your spectra. Peaks should have a generally Gaussian or Lorentzian shape. Decreasing the slit widths or moving the H and D sources away from the beam splitter will eliminate the problem.

(2) The H and D sources produce far fewer atoms in high n states, thus the Balmer lines are far less intense than the Balmer lines. It will therefore be necessary to adjust source positions, slit widths and lock-in amplifier sensitivity to obtain useful spectra for all four Balmer lines.

(3) The H and D lamps use 5000 V power supplies to which the experimenter is dangerously exposed near the metal caps at the ends of the lamps. Please turn off the H and D sources when adjusting their position so as to avoid a potentially fatal shock. Students receiving fatal shocks will not be permitted to work in the laboratory for the remainder of the term. Please take your time and be careful.

Additional Comments:Follow the sequence of steps outlined below.

- (1)From your raw experimental spectra, record in a table entitled "Raw Experimental Data" the following quantities for each H and D peak in your four Balmer , , , and spectra.
- (a) Peak centroid and uncertainty in its determination (you should be able to locate centroids to within ± 0.01 to ± 0.02 Å from your spectra),
- (b) The full width at half maximum (FWHM) of each peak (you need not state an uncertainty in this determination),
- (c) The slit setting used to obtain the appropriate spectrum.
- (2) Compare your raw peak centroids for H
_{}, H_{}, H_{}, and H_{}to the_{H air}values given in Table I. Is your agreement satisfactory, given SPEX's claim of a calibration accuracy of ± 1.0 Å for the SPEX Model 1704 spectrometer?- (3) Use your "Raw Experimental Data" to calculate the wavelength separation,
_{air}=_{H air}-_{D air}, between the H and D peaks of the four Balmer spectra. Also calculate the associated uncertainties (_{air}).- (4) In your laboratory notebook show that the deuteron-to-proton mass ratio can be determined from a single Balmer transition using

^{md}/_{mp}= A_{air}/ A_{air}-_{air}) (3)

where A_{air}=_{H air}-_{air}is the theoretical wavelength separation (for air) between the hydrogen peak and the peak corresponding to the hypothetical, infinitely heavy nucleus case, and_{air}=_{H air}-_{D air}is the experimental H-D peak separation from part (3). Show also that the uncertainty in a m_{d}/m_{p}determination is given by

d( ^{md}/_{mp}) = A_{air}(_{air})/(A_{air}-_{air})^{2}. (4)

You will receive considerable help with the derivation of these results during the class sessions.- (5) Calculate four values of
^{md}/_{mp}± (^{md}/_{mp}) from your four Balmer spectra. Put your results from steps (3) and (5) in a table entitled "Final Deuteron-to-Proton Mass Ratio Results". Compute a weighted mean of your m_{d}/m_{p}results and its associated uncertainty. Compare your final value for m_{d}/m_{p}with the currently accepted value.

Begin working on wavelength calculations from Eq. (1) immediately. Be sure you can calculate the numbers of Table I right down to the last significant figure. You should agree with every number in those values if you maintain the required number of significant figures.

Use a programmable calculator or computer to make your analysis less time-consuming. If you use a computer, you must be absolutely certain that it can perform the necessary calculations with the required precision.

1. R. Eisberg and R. Resnick,

2.

3. E.B. Anthony,