A Primer on Quantum Numbers and Spectroscopic Notation

Contents

Regular patterns in the spectra of hydrogen (e.g., the Balmer series) and alkali metals lead Bohr to propose a model for the atom consisting of a positively charged nucleus orbited by electrons with discrete, quantized values of energy. While the Bohr model is strictly applicable only to the simplest ions, those having a single electron, the Bohr model provided enormous insights into the structure of atoms, and lead to the development of quantum mechanics.

Spectroscopists of the late 19th and early 20th 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.

Quantum Numbers

4 quantum numbers suffice to describe any electron in an atom. These are:
• n, the principal quantum number. This is also known as the radial quantum number, and defines the distance of the electron from the nucleus in the Bohr model. n also describes the azimuthal angular momentum. n takes on integral values 1, 2, 3, ... .
• l, the azimuthal quantum number. In Sommerfeld's generalization of the Bohr model, the circular orbits of the electrons are replaced by elliptical orbits, and l describes the shape of the orbit. l takes on the integral values 0, 1, 2, ... , n-2, n-1. If n=1, l=0. l is sometimes called the reduced azimuthal quantum number, because the Sommerfeld formulation used a quantum number k, which equals l+1. k=0 corresponds to no angular momentum, or a radial orbit which takes the electron through the nucleus. This is unphysical, and is forbidden.
• m, the magnetic quantum number. m takes on the integral values -l , -(l-1), ..., -1, 0, 1, ..., (l-1), l. In Sommerfeld's formulation, m described the orientation of the ellipse. This is known as the magnetic quantum number because its effects are generally observed only under the influence of strong magnetic fields (which set a preferred spatial orientation).
• s, the spin quantum number. This describes the spin of the electron, and is either +1/2 or -1/2.

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 2n2 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 2n2 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 gn = 2n2 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

ν = R Z2(1/(U2) - 1/(L2))
where R is the Rydberg constant, 2π2με4/h2, where μ is the reduced mass Mme/(M+me), M is the mass of the nucleus, me is the mass of the electron, ε is the unit of electric charge, and h is Planck's constant. R=13.598 eV. In the Bohr atom, all electrons with common n are degenerate, i.e., they all lie at the same energy.

Quantum Numbers for Atoms

As with electrons, 4 quantum numbers suffice to describe the electronic state of an atom or ion.

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 Σli (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(Σsi). 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.

Spectroscopic Notation

The atomic level is described as
n 2S+1LJ
where S, n, and J are the quantum numbers defined above, and L is the term (S,P,D,F,G, etc). 2S+1 is the multiplicity.

You may also see the level described as

n lx 2S+1LJ
where l is the orbital of electron (s, p, d, etc.) and x is the number of electrons in that orbital (e.g., 1 or 2 for an s orbital, 1 to 6 for a p orbital. n lx is the configuration of the outermost electrons.

Terms, Configurations, and Levels

The outermost electron in an atom or ion is the one that usually undergoes transitions, and so the state of that electron describes the state of the atom or ion.

The configuration describes the n and l values for all the electrons in an atom. For example, the ground state of Boron has a 1s22s22p 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 2P1/2 term. Closed shells always have a 1S0 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.

Selection Rules

In complex ions, there are an enormous number of possible transitions. Not all of these possible transitions are observed. This is because some transitions are more likely than others. Selection rules were arrived at empirically to describe those changes in quantum numbers that were observed (permitted transitions) and those which did not (forbidden transitions). The basic selection rules, which strictly apply only to simple configurations which obey strict L-S coupling (In a simple atom or ion, L and S vector-sum to J. The levels of L and S do not affect each other. This lack of interaction is known as L-S coupling. In complex atoms or ions, levels of L and S can interact, leading to a breakdown in L-S coupling (physicists can be just as illogical as atronomers in their nomenclature!). When this happens, L and S are no longer interpretable in terms of angular momenta.), are:
• ΔL = 0, +/- 1
• Δl = 1
• ΔJ = 0, +/- 1, except that J=0 -> J=0 is forbidden.
• ΔS = 0
• ΔM = 0, +/- 1, except that M=0 -> J=0 is forbidden if ΔJ=0.

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 108s-1. Forbidden lines have small AUL 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.

Multiplets

Although the energy levels in the Bohr model of Hydrogen depend only on the principal quantum number n, in reality the degeneracy of the levels are broken by various interactions between the electron(s) and the nucleus. Transitions arising from a specific nL term (with a number of values of J) to another term (which has multiple values for J) give rise to a multiplet. For example, the Na I doublet (the Fraunhofer D lines) at 5890,5896 Angstroms arise from the 32P - 32S transition. The 32P term is split into 32P1/2 and 32P3/2 levels. The energy difference between these levels with different J values amounts to about 5.9 Angstroms.

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.

M Degeneracy and the Zeeman Effect

The M degeneracy can be broken by application of a magnetic field B. This is the Zeeman effect.

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 10-13g λ2B, 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

g = 1 + (J(J+1) + S(S+1) - L(L+1))/2J(J+1)

The anomalous Zeeman effect operates on lines that are not singlets, and produces more than 3 components.

Hyperfine Structure

Hyperfine structure arises from the coupling between the magnetic moment of the electron and the nuclear magnetic moment. For the quantum number I defining the net nuclear spin (analagous to the net electron spin S), you can construct another quantum number F=I+J, which takes on values J-I,J-I+1, ... J+I-1, J+I.

The 2S1/2 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.

References

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