Spectroscopists of the late 19^{th} and early 20^{th} century created a system of spectroscopic notation to describe the observed line spectra. Quantum numbers were invented to to provide an quantitative description of observed (and unobserved) transitions. These together provide a short-hand description of the state of the electrons in an atom or ion (I will use the terms interchangeably). The notation is confusing, is case-specific, and sometimes ambiguous.
The Pauli exclusion principle requires that no two electrons have the same set of the 4 quantum numbers n, l, m,s, therefore there are 2n^{2} possible states for an electron with principal quantum number n. The n=1 levels can contain only 2 electrons. This level is called the 1s orbit or the K shell (shells with n=1,2,3,4,5,6,7 are called the K, L, M, N, O, P, Q shells, respectively). An orbit, or shell, containing the maximum number 2n^{2} electrons forms a closed shell.
For example, when n=1, l=m=0 and m=+/- 1/2. The possible combinations of quantum numbers are nlms = 1, 0, 0, +1/2 and 1, 0, 0, -1/2. When n=3, the 18 possible states are:
3, 0, 0, +1/2 and 3, 0, 0, -1/2 3, 1, -1, +1/2 and 3, 1, -1, -1/2 3, 1, 0, +1/2 and 3, 1, 0, -1/2 3, 1, 1, +1/2 and 3, 1, 1, -1/2 3, 2, -2, +1/2 and 3, 2, -2, -1/2 3, 2, -1, +1/2 and 3, 2, -1, -1/2 3, 2, 0, +1/2 and 3, 2, 0, -1/2 3, 2, 1, +1/2 and 3, 2, 1, -1/2 3, 2, 2, +1/2 and 3, 2, 2, -1/2
Consequently, the statistical weight g_{n} = 2n^{2} for the level characterized by principal quantum number n.
In a hydrogenic atom or ion (H, He II, Li III, ... [The roman numeral refers to the ionization state. C I is neutral; C IV is triply ionized. C IV is equivalent to C^{+++}.]), the energy levels are fully described by the principle quantum number n (aside from the fine structure terms, which modify the levels by terms of order α, where the fine structure constant α is 1/137, and relativistic corrections needed for heavy ions). In a hydrogenic atom or ion, the frequency of a transition between upper state n=U and lower state n=L is
L is the total orbital angular momentum. For hydrogenic ions and alkalis, with a single electron in the outer shell, L=l. L corresponds to the term of the ion (S terms have L=0, P terms have L=1, etc.). In the case of more than one electron in the outer shell, the value of L takes on all possible values of Σl_{i} (see Table 1, which is Table 5 from Herzberg).
Table 1 | ||
Electron | L | Term |
Configuration | ||
s | 0 | S |
sp | 1 | P |
pp | 0 1 2 | S P D |
pd | 1 2 3 | P D F |
dd | 0 1 2 3 4 | S P D F G |
ppp | 0 1 1 1 2 2 3 | S P P P D D F |
The quantum number S is the absolute value of the total electron spin abs(Σs_{i}). Note: this S is not the same as the term S). Each electron has a spin of +/- 1/2. S is integral for an even number of electrons, and half integral for an odd number. S=0 for a closed shell. J represents the total angular momentum of the atom of ion. It is the vector sum of L and S. For a hydrogenic ion, L=0, S=1/2, and J=1/2. For more complex atom, J takes on the values L+S, L+S-1, ..., abs(L-S), where abs is the absolute magnitude. For a given L, there are 2S+1 possible values of J, unless L<S, in which case there are 2L+1 possible values of J. M, the Magnetic quantum number, takes on values of J, J-1, ..., 0, ..., -J-1, -J.
You may also see the level described as
The configuration describes the n and l values for all the electrons in an atom. For example, the ground state of Boron has a 1s^{2}2s^{2}2p configuration, with 2 electrons filling the n=1 level (l=0), 2 electrons in the n=2, l=0 s orbital, and the fifth electron beginning to populate the 2p orbital.
The level is the set of 2J+1 states with specific values of L, S, and J. The difference in the energy between two levels gives the wavelength or frequency of an atomic transition.
The term is the set of levels characterized by a specific S and L. The ground state of Boron has a ^{2}P_{1/2} term. Closed shells always have a ^{1}S_{0} term.
Atoms whose outer electrons have l=0,1,2,3,4 are referred to as S, P, D, F, G terms, respectively ( Note that an electron with l=0 is called an s electron; lower case terms refer to individual electrons. For example, In the ground state, Boron has 4 s electrons (2 in the n=1 level and 2 in the n=2 level) and one p electron. The ground state term of the atom is P. Warning: the s in an s electron has nothing to do with the quantum number s.). This is a carryover from early spectroscopic nomenclature (sharp, principal, diffuse, and fundamental bands, with G following F alphabetically) for alkali atoms, those with a closed shell of electrons plus a single valence electron, such as Li, Na, K, Mg II, Ca II.
As the atoms become more complex, strict L-S coupling fails to hold, and these selection effects become weaker.
Permitted lines are those whose transition probability is high, i.e., whose Einstein A value is small, of order 10^{8}s^{-1}. Forbidden lines have small A_{UL} values, of order 1, because they cannot radiate in a dipole transition. Forbidden transitions are often possible through quadrupole or octopole transitions, which have low transition probabilities. Metastable lines have intermediate A values. A^{-1} gives the radiative lifetime of the excited state. Forbidden lines are characteristic of low density media, because at high densities the time between collision is short compared to the radiative lifetime, so collisional deexcitation is the dominant process.
The multiplicity of a term is given by the value of 2S+1. A term with S=0 is a singlet term; S=1/2 is a doublet term; S=1 is a triplet term; S=3/2 is a quartet term, etc.
Alkali metals (S=1/2) form doublets. Ions with 2 electrons in the outer shell, like He, Ca I or Mg I, form singlets or triplets.
Multiplet splitting increases with the degree of departure from strict L-S coupling.
The normal Zeeman effect operates in a singlet state, and results in 3 lines. The lines with ΔM = 0, the π components, are unshifted, and are polarized parallel to the magnetic field; the lines with ΔM = +/- 1, the σ components, are shifted by +/- 4.7 X 10^{-13}g λ^{2}B, where g is the Lande g factor, λ is the wavelength of the unshifted line in Angstroms, and B is the strength of the magnetic field in Gauss. The Lande g factor is
The anomalous Zeeman effect operates on lines that are not singlets, and produces more than 3 components.
The ^{2}S_{1/2} ground state of Hydrogen has J=1/2, I=1/2 (because the spin of the proton is 1/2), and F can take on the values 0 or 1. F=1 corresponds to parallel spins for the proton and electron; F=0 corresponds to anti-parallel spins, and is the lower energy configuration. The energy difference corresponds to a frequency of 1420 Mhz, or a wavelength of 21 cm. This is a very important line astrophysically, for it permits us to map the distribution of cold Hydrogen in our galaxy and the universe.
Astrophysics of the Sun, by Harold Zirin (Cambridge University Press, 1989) contains a good discussion of this matter in Chapter 5.
See also chapters 2-4 of Optical Astronomical Spectroscopy by C. R. Kitchin (Institute of Physics Publishing).
Frederick M. Walter
2 October 1998; updated 26 September 2000