3. Theory
3.4 Spectroscopy
‘An optical spectrum consists of radiative transitions between stationary states
of an atom or molecule.’101
When an electron within an atom undergoes a transition to a state of lesser energy it can emit a photon. These photons produce spectral lines of the atom, ion or molecule, known as an emission spectrum.
Optical spectroscopy is the analysis of spectral lines emitted or absorbed from atoms, ions or molecules and the use of this information to ascertain the identity, structure and/or environment of such species. These lines represent the energy levels of the emitting species.
Spectrochemical analysis is the search for characteristic emission line patterns from an atom, molecule or ion. Analysis of their wavelengths, intensities, widths, shifts and spectral distribution can provide further information. Analysis of the emission spectra can be made quantative if one can determine the relative intensities of such spectra to determine the abundance of that species in the substrate.
Plasma spectroscopy can be used to analyse electron densities, temperatures, pressures, velocities and relative abundances. A plasma’s physical properties can be shown to affect the emission spectra in many ways. Studying these effects has been covered in section 3.3, many of these effects are produced by line shifts and broadening mechanisms as a result of plasma properties. A plasma’s properties, such as temperature and electron density, can be determined from the emission lines themselves.
The link between line wavelength and individual atom emissions/absorptions was first discovered in 1860 by Kirchhoff and Bunsen. The foundations of spectroscopy and atomic structure are covered in quantum mechanics.
E ne rgy a bo ve gr oun d s ta te E n 4 3 2 1 continuum Ionisation Ground state
Quantum theory states that energy is quantised; it can only exist in discrete packets of energy which satisfy the relation:
υ
nh
E= (3.15)
Where E = total energy n = quantum number h = Planck’s constant
υ = frequency of oscillations
The energy quantisation can be found for the one electron atom from the time- independent Schroedinger equation. An energy level diagram for such a system is shown below:
Although this theory has been developed for a one-electron atom, and as such an energy level is only dependent on the quantum number n, the theory also holds when applied to multielectron atoms but the energy of the system, due to the levels having sub-levels and the electrons having spin, then depend on three quantum numbers n, l and ml.
It can be seen that there are situations where a single energy value En can
actually depend on a few different configurations of energy in the system, this
Figure 3.7: Energy level diagram for one electron atom. (composite drawn from many sources)
corresponds to atomic states that have different behaviour but the same total energy, known as degeneracy.
The energy level diagram above should actually be a lot more complicated than shown due to the degeneracy mentioned and also due to perturbations of the energy levels from interactions such as electrostatic interactions, magnetic interactions, nuclear mass/volume, and spin. This results in energy level splitting as shown in figure 3.8.
A suitable simplified version of a multielectron atom can be explained by the Hartree102 theory, predicting that:
‘the total energy of an electron in the outermost populated shell of any atom is comparable to that of an electron in the ground state of hydrogen.’102
This prediction is based on the fact that the outer electrons of a multi-electron atom are shielded from the strong nuclear charge by the inner shell electrons. This theory holds well but corrections are needed to allow for the weaker interactions mentioned above. These weaker interactions result in the fine structure of the energy levels of atoms. Such interactions are:
Splitting:
• Residual Coulomb interaction, (Spin coupling and Orbit coupling) o Adds corrections for electron spin-orbit interactions and
relativistic dependences of mass on velocity, the Dirac theory.
Fine-structure splitting:
• Spin-Orbit interaction, the fine-structure splitting of degenerate levels,
o known as LS coupling and JJ coupling for atoms with large Z, Quantum electrodynamics and the Lamb Shift.
Hyperfine splitting:
• Interaction of the intrinsic magnetic dipole moment of the nucleus and the magnetic field produced by atomic electrons.
o smaller than spin-orbit interactions by 3 orders of magnitude
Figure 3.8: Influences on atomic energy levels103.
The Hartree theory yields information on ordering, according to energy, of the outer filled subshells of multi-electron atoms.
Spectroscopic notation can be explained using standard notation, as shown: For a boron atom: 5B: 1s22s22p1
The principle quantum number n is represented by the integer before the letter; the azimuthal quantum number, or subshell, is represented by the spectroscopic notation in table 3.2; the superscript on the subshell designation specifies the number of electrons which it contains; the superscript on the chemical symbol specifies the number of electrons in the atom.
The filling of subshells is governed by the Pauli exclusion principle, whereby: ‘In a multielectron atom there can never be more than one electron in the same
quantum state’
The first subshell may contain two electrons and not violate the exclusion principle as one electron may have spin ‘up’ and one spin ‘down’.
Energy levels are often listed as wavenumbers (cm-1) where the ground state is zero, allowing the wavenumber of a transition to be interpreted from the energy level difference:
( )
hc E cm wavenumber, σ −1 =∆ (3.16) Derived from: υ h E= ∆ (3.17) υλ = c (3.18) λ σ = 1 (3.19)Where: E = Energy difference of two levels (J)
υ = frequency (Hz)
c = speed of light (ms-2)
λ = wavelength (nm)
The ionization energy, the energy required to remove an electron from the atom, varies with the number electrons in the outer shells of each atom.
For example a noble gas, in which the p subshell is completed, is much harder to ionize than an alkali which has a single weakly bound electron in an s subshell. These alkali elements have correspondingly high chemical activity due to their energetic favourability to interact with other elements to produce a more stable arrangement.
An element’s chemical properties and its ability to interact with other atoms depend on the number of electrons in the outer subshell of the atom, as these are the electrons that govern the electric and magnetic fields of that element.
Modern quantum theories are also able to give very satisfactory treatments of the transition rates and selection rules observed in the measurements of the spectra emitted by atoms.