### Rydberg Analysis of Emission Spectra for Atoms Beyond Hydrogen

In this laboratory-computation exercise, we will collect emission spectra data for helium in the visible range of the spectrum, and then use that information to examine the effect of "core" electrons upon the effective nuclear charge for the spectroscopic transitions. We will use emission spectra data for the lithium atom to extend the analysis. To provide a basis for understanding the analysis, we will begin with a brief presentation of the Lyman and Balmer series spectra for the hydrogen atom.

The Lyman series Excitation of the hydrogen atom by high voltage discharge sets up a steady state condition in which several series of spectral emission lines can be observed by spectroscopy. The Lyman series, in which electrons fall from principal quantum number states 2,3, ... to the ground state, is a group of lines in the far ultraviolet. The following presentation should be read with reference to the accompanying Lyman series Mathcad document (g:\classes\chapman\ch445\hspec.xmcd). The data array z contains the initial quantum number for several transitions, and the wavelength (in Angstrom) for the energy radiated. We will use equation 1 as the basis for analysis.

DE = Z2RH(1/nf2 - 1/ni2) (1)

Eqn. 1 expresses the relationship between the energy of the radiation and the change in principal quantum number from the initial (n=2,3,...) to the final (n=1) state of each transition. RH is the Rydberg constant, whose value was accurately known well before it was first calculated by Bohr using his model of the hydrogen atom. Z is the nuclear charge. Because the final state for all the transitions in this series is the same, we can treat its value as unknown, combining it with Z, which is known in this case, to define the intercept of the linear plot. From this point of view, Eqn. 1 can be re-written as

DE = Z2RH(1/nf2) - Z2RH(1/ni2), Z = 1, nf = 1 (2)

Linear regression with Eqn. 2 and the Lyman series data is shown on the Lyman document. From the slope we find RH, and by combining the slope and intercept we are able to calculate the quantum number for the common final quantum state in the Lyman series. From the standpoint of the regression line, the "x" axis intercept is the initial state from which a transition of zero energy difference occurs. The energy required to separate the proton and the electron in the hydrogen atom can be calculated from the regression result by allowing the initial quantum number in Eqn. 1 to take on very large values. The document shows that the Lyman series data lead to a dissociation (ionization) energy of 13.60 eV.

The Balmer series In the Balmer series of spectroscopic lines, some of which fall within the visible spectrum, a group of transitions from n=3, 4,... to n=2 are observed. The Balmer Mathcad document (same document as above) should be used to follow this discussion. The array z contains the initial quantum number and the associated wavelength (in Angstrom) in the first and second columns, respectively. Equation 1 again serves to define the relationship between the energy of each transition and its change in quantum state. Following the same analysis as applied to the Lyman series, we obtain from this spectral series a value of the Rydberg constant and the common final state of the Balmer transitions. We find 3.40 eV for the dissociation energy from the quantum state with n = 2. Having found the dissociation energy from the ground state to be 13.60 eV, we conclude that the state n = 2 in the hydrogen atom lies 10.2 eV above the ground state. The self-consistent results support our underlying assumption that the Bohr energy relation is valid for any series of hydrogen atom spectral lines, and is not restricted to the Lyman series in which the lowest quantum state in the spectral series is also the ground state of the atom.

#### Helium Atom Spectra

The presence of a second electron gives rise to a more complex set of transition probabilities whose explication requires quantum considerations beyond that represented by Eqn. 1. Nonetheless, a modification of the Lyman and Balmer hydrogen emission spectral analysis presented above can be used to learn some interesting things about multi-electron atoms.

We will use the form of Eqn. 2, but with these modifications of interpretation:

1. We accept the Rydberg constant as applicable to any spectral series,
2. The nuclear charge Z is replaced by an "effective" value, to be determined from the spectroscopic data by regression analysis,
3. Subject to the validity of the preceding approximations, a value for the ionization from the first state of a spectral series can also be calculated from the regression.

To illustrate the modified analysis, we use the spectroscopic data for helium, using the energy level diagram taken from Herzberg, "Atomic Spectra and Atomic Structure". The helium energy level diagram shows that this emission spectrum consists of two groups of intermixed lines, one group for singlet - singlet and the other for triplet - triplet transitions. The numbers written on the diagonal lines connecting energy levels are the wavelengths of the radiation associated with that transition.

Refer now to the Mathcad document for the helium 1S-1P series (g:\classes\chapman\ch445\hespec.xmcd), which falls in the far ultraviolet range; these data were taken from the Herzberg diagram. The independent variable is the reciprocal of the quantum numbers, with the lowest or series "ground" state implied to be one less than that of the lowest "excited" state of the series. We are treating the spectral series as arising from a one electron hydrogen-like atom. The extent to which the spectral series deviates from that of a one electron atom is reflected in the apparent nuclear charge and in the ionization energy from the lowest state in the spectral series. The calculations assume that the Rydberg number is a universal constant, also applicable to atoms having more than one electron; we must also know the assignment of the observed spectral lines to the electronic transitions in the atom. Consideration of the helium 1S-1P document shows an effective nuclear charge of 0.996; the 1s electron "screens" the "valence" electron from the nuclear charge by essentially one full unit of charge. The dissociation energy of the electron from the ground electronic state is found to be 24.6 ev, in good agreement with the accepted value.

#### Computations and Report

To complete this exercise in atomic spectra, we will include an examination of the emission spectrum of the lithium atom; energy levels and transition wavelengths are shown in the accompanying diagram, again taken from Herzberg. Using the document provided for analysis of the helium 1S-1P transitions as a guide in your calculations, find the effective nuclear charge and the dissociation energy for each of the three spectroscopic series shown in the lithium atom energy level diagram (1S-2P, 2P-2S, and 2P-2D). Calculate and record in separate tables for each spectral series the energies of all the levels in the series, using the regression-derived dissociation energies and the ground state energy of the lithium atom.

Consider the differences in the effective nuclear charge among the series, and discuss in terms of the properties of hydrogen-like wave functions.

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