The Spectra of Atoms

Calculate the wavelength of the light absorbed when an electron in a box of length 1.000 nm makes the transition from n = 1 to n = 2. In what region of the electromagnetic spectrum (X-ray, ultraviolet, visible, infrared, microwave) does this light lie  

Argue that the transition dipole moment for this transition is nonzero for all transitions. Hint Consider graphs of the factors in the integrand function in the integral used to calculate the transition dipole moment. 

The spectra of atoms are due to electronic transitions. The following selection rules are derived when the hydrogen atom orbitals are substituted in the integral of Eq. (23.1-8)  

From Rydberg s formula in Eq. (14.4-10), find the wavelength and freqnency of the photons emitted by a hydrogen atom nndergoing the n = 2 n = 1 transition. 

---

High-resolution spectroscopy used to observe hyperfme structure in the spectra of atoms or rotational stnicture in electronic spectra of gaseous molecules connnonly must contend with the widths of the spectral lines and how that compares with the separations between lines. Tln-ee contributions to the linewidth will be mentioned here tlie natural line width due to tlie finite lifetime of the excited state, collisional broadening of lines, and the Doppler effect. 

Spectroscopy The science of analyzing the spectra of atoms and molecules. Emission spectroscopy deals with exciting atoms or molecules and measuring the wavelength of the emitted electromagnetic radiation. Absorption spectroscopy measures the wavelengths of absorbed radiation. 

The intensity of an absorption line in the spectra of atoms or molecules can be measured and it follows the empirical law called the Beer-Lambert law. Consider the amount of light entering a cylinder full of hydrogen gas, as shown in Figure 3.1. If the radiation is absorbed by the hydrogen gas, the amount of light emerging from the opposite end of the cylinder is reduced. 

Astronomical measurements have some limitations not present in laboratory investigations but a detailed knowledge of the spectra of atoms and molecules can be used to overcome the restrictions of resolution and atmospheric windows. The detailed knowledge is the key to this success, however, and confidence in the conclusions of astrochemical observations must come from the understanding in the laboratory. 

Labzowsky, L. N., Khmchitskaya, G. L., and Dmitriev, Y. Y. Relativistic Effects in the Spectra of Atomic Systems (Institute of Physics Publishing, Bristol and Philadelphia, 1993). 340 pp. 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

Let us discuss now some aspects of practical applications of this method again following [296]. Explicit expressions for the lowest moments of the spectrum may be employed not only for an approximate description of the distribution function without carrying out the detailed calculations of separate levels, but also for studies of general properties of spectra, for evaluation of coupling schemes and the contributions of various interactions to the main spectral characteristics, for estimation of the accuracy of the approximation used, etc. Let us illustrate these statements using the example of the spectra of atoms and ions with one open shell. 

The late 30 s brought a further important step in the investigation of the interstellar medium — the discovery of the first molecular species. In the optical region, the electronic spectra of the diatomic radicals CH, CH+, and CN, seen in absorption against the continuum spectra of bright background stars, furnished the first evidence that the interstellar medium was not devoid of molecules but contained at least some simple ones. However, the intensities of the molecular spectral peaks seen via optical absorption studies were quite weak compared with the spectra of atoms, indicating that the sources observed in these early studies were not rich in molecules. These sources, now labeled diffuse interstellar clouds, possess very low gas densities (n 102 cm-3) and are of limited interest chemically. 

A note of caution The Bohr theory, even when improved and amplified, applies only to hydrogen and hydrogen-like species, such as He+ and Li+. The theory explains neither the spectra of atoms containing even as few as two electrons, nor the existence and stability of chemical compounds. The next advance in the understanding of atoms requires an understanding of the wave nature of matter. 

The spectra of atoms are often called line spectra, for it will be recalled that they consist of a relatively small number of discrete wavelengths, represented as thin lines on the traditional spectrograms. On the other hand, the spectra of molecules are termed band spectra, for they consist... 

This interaction leads to "fine-structure" splittings in the spectra of atoms and molecules. For atoms and molecules in the S = 1 triplet state, the electron spin-electron spin dipolar interaction leads to the "D and E" fine-structure Hamiltonian. 

In general, a simple oscillator model of classical electrodynamics that ceases to be valid in some stage of the analysis of the spectra of atoms and simple molecules, can describe, at least formally, all the phenomena in the polarized liminescence of complex molecules in solution (Ref. p. 121). 

Indications of the feasibility of this approach are to be found in the work of Bowers et al. (2, 3) on the photolysis of iodine vapor in the cavity of an EPR spectrometer, and in the theoretical investigations of Beltran-Lopez et al. on the microwave Zeeman spectra of atomic fluorine and chlorine 1, 8). However, these studies were mainly concerned with the precise determination of atomic g factors to verify the Zeeman theory and involved specialized spectral equipment. The present investigation demonstrates that the spectra of atomic fluorine, chlorine, and bromine are readily observable in commercially available EPR equipment and that reasonable estimates may be made of their concentrations. 

However, the theory presented several limitations and could not explain the spectra of atoms other than monoelectronic atoms. Bohr himself would recognize the need for a new theory and, indeed, would contribute to it. In particular, it is noted that, although Eq. (1.8) is reproduced by quantum mechanics, the angular momentum of the electron is not given by Eq. (1.9) in particular, the minimum value possible is zero and not hjl-K as required by Eq. (1.9). 

The spectra of atomic absorption are obtained with instruments called atomic absorption spectrometers. These instruments, as already described for other types of spectrometers, consist of the light source, monochromator and detector. However, the atomic absorption spectrometers and atomic emission spectrometers differ from all other spectral spectrometers by the absence of the sample chamber. Instead of the sample chamber, they contain a burner. A schematic of the atomic absorption spectrometer is shown in Figure 2.54. 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

The spectra of atoms are characterized by a set of bound Rydberg states below the photoionization threshold for excitations in the continuum " The Rydberg states become very weak in molecules and disappear in condensed systems. Review papers on x-ray absorption spectra for atoms and molecules are available and these spectra will not be discussed here. 

Many industrially important selective oxidation reactions are catalyzed by transition metal oxides. The activity of such catalysts is related to the reducibility of the transition metal ion, which enables the bulk oxide lattice to participate actively in the redox processes present in the Mars van Krevelen mechanism. Unfortunately, NMR spectroscopic investigations are severely limited by the occurrence of paramagnetic oxidation states. As a general rule, NMR signals from atoms bearing unpaired electron spins cannot be detected by conventional methtxls, and the spectra of atoms nearby are often severely broadened. For this reason, most of the work published in this area has dealt with diamagnetic vanadium(V) oxide-based catalysts. 

The main focus in this chapter is on two themes On the one hand, the discussion aims at presenting, without many formulae, certain critical characteristics and properties of unstable states in general, and, in particular, of states that are normally represented by multi-particle wavefunctions, such as the ones that are found in the spectra of atoms and small molecules. On the other hand, the discussion aims at demonstrating why and how it is possible to tackle, from first principles and from a many-electron point of view, a large number of problems that have to do with field-free and field-induced unstable states, for which reliable electronic structure calculations are, in principle, feasible. 

The spectra of atomic absorption of an element are made up of a series of lines of resonance from the fundamental state to different excited states. The transition between the fundamental state and the first excited state is known as the first line of resonance, being that of greatest absorption, and is the one used for analysis. 

Stark effect The splitting of lines in the spectra of atoms due to the presence of a strong electric field. It is named after the German physicist Johannes Stark (1874-1957), who discovered it in 1913. Like the normal Zeeman effect, the Stark effect can be understood in terms of the classical electron theory of Lorentz. The Stark effect for hydrogen atoms was also described by the Bohr theory of the atom. In terms of quantum mechanics, the Stark effect is described by regarding the electric field as a perturbation on the quantum states and energy levels of an atom in the absence of an electric field. This application of perturbation theory was its first use in quantum mechanics. 

---