Glow-Sticks are Forbidden Zachary G Wolfe Department of Physics and Physical Oceanography University of North Carolina Wilmington Abstract: Physics students learn early that certain electronic transitions are forbidden under selection rules. This misnomer in terminology is the result of a more rigid approach required to learn quantum mechanics. This paper describes the source of the ambiguity in forbidden transitions and describes an example of such a transition. It is also discuss transition principles and what is defined as a forbidden transition in an atomic orbital and how it relates to the quantum state description.
Transition Principles: In quantum mechanics, the relaxation of an electron from one quantum state to another releases energy in the form of a photon with a quantized energy. 1 The Lyman series of the hydrogen atom is an example of such a transition. This transition takes place from a higher energy state ( n 2 ) and relaxes to the ground state (n = 1). The lyman series 1 is based upon the Rydberg formula shown in equation (1). 1 λ = RZ 2 ( 1 n' 1 2 n ) 2 (1) In the Rydberg formula, R is the rydberg constant, and Z is the atomic number of the atom in equation, in this case 1 for hydrogen. If the relaxation takes place from the energy state represented by the principle quantum number 2, the wavelength of this transition takes place at 121.57 nm and has an equivalent energy of 10.199 ev. These transitions are described by selection rules. These selection rules describe which optical transitions are allowed. 1 The selection rules for Hydrogen are shown in equations (2) and (3). 2 Δ l = ±1 (2) Δ m l = 0,±1 (3) This indicates that for every transition that takes place in a hydrogen atom, the orbital quantum number and the magnetic quantum number must differ by one of these values as the principle quantum number changes. In the Lyman series, this means that the 2 1 transition must occur from the 2p orbital and not the 2s orbital as the orbital quantum number, l, has no change of ±1. This is represented in figure 1. Figure 1: Example Transitions of Atom Hydrogen
The Forbidden Transitions: Despite it s name, the transitions that are forbidden under the selection rules do in fact happen. If an allowed transition can happen then the relaxation to a lower energy state will occur via that route. Otherwise, these forbidden transitions have a probability of occurring, albeit at a much lower probability. Allowed transitions occur on the microsecond scale (10-6 seconds) whereas forbidden transitions have lifetimes on the order of mili-seconds to seconds. 3 In fact, selection rules are derived from time dependent perturbation theory and are inherently time-dependent as implied by the transition lifetimes. This also means that when looking at any transition, stationary states cannot be considered. These selection rules are then derived from the timedependent Schrodinger equation seen in equation (4). ^H ψ = iħ ψ t (4) The Hamiltonian in this case is not the traditional total energy operator introduced in Serway 1 but in fact is represented by the Hamilitonian shown in equation (5). ^H = μ E o cos(2 π ν t) (5) Where ν is the frequency of the radiation and μ is the dipole moment. Solving equation (4) for various scenarios like the rigid rotor model or the harmonic oscillator results in the various selection rules for atoms and molecules. It is generally too complicated to derive rules explicitly from the Schrodinger equation, just as any system larger than a helium atom cannot be solved for exactly. It is then we are required to use approximations for various transitions. Transitions may be forbidden under approximations like the electric-dipole approximation, which accounts for the atomic interaction of light, but may be allowed under higher-level approximations like the magnetic dipole or electric quadrupole approximation where the restrictive approximation is not considered. 4 This, in essence, is the source of the forbidden transitions and is the result of quantum uncertainty. They do, in point of fact, happen but are far less likely to occur if more efficient transitions are possible. The Glow-stick: Glow sticks emit light when two compounds are mixed together starting an energy exchange. They also contain a suitable dye for color. Such an example of this reaction is the oxidation between diphenyl oxalate and hydrogen peroxide seen figure 2. 5
Diphenyl oxalate is oxidized into a phenol and peroxyacid seen first in figure (2). The peroxyacid then spontaneously decomposes into carbon dioxide which energetically excites the dye, color depending on the structure of the dye compound. It is this energy input that causes the transitions that result in photon emission. The exact transition that causes these emissions are from molecular orbitals, not atomic. Such orbital representations are generally more complicated than the atomic representations such as hydrogen. The mechanism used to describe the chemi-luminescence is the proposed [2+2] photocyclization mechanism seen in figure 3. In order for the dye compound to energetically stabilize, an electron must be excited from a singlet state to a triplet state via intersystem crossing. A singlet state is a molecular electronic state where all electron spins are paired via Pauli's exclusion principle. A triplet state is where those excited electrons are no longer paired and violate the exclusion principle. The excitation from a singlet to a triplet state, unpairing the electrons under a non-radiative process is called an intersystem crossing and this is the forbidden transition. To stabilize the phenol, an electron must be excited to a triplet state so that it may bond with an alkene and after the bonding orbitals relax, an emission is observed. Eventually the phenol compound is stabilized through this complex relaxation mechanism that results the phosphorescence observed when a glow-stick is activated. The relaxation from the excited triplet state to the ground-singlet state results in chemi-luminescence, called phosphorescence and has a time scale between 10-3 and 10-8 seconds, one of the slowest forms of electronic relaxation. Under general selection rules, these transitions are forbidden and by proxy, glow-sticks described in restrictive rules should not function, an obvious contradiction. However it is these approximations of these absorbtion and radiative processes that are the source of the ambiguity of a forbidden transition.
References: 1. R.A. Serway, C.J. Moses, and C.A. Moyer, Modern Physics, 3 rd ed. (Thomson Brooks/Cole, Belmont, CA, 2005), pp 132, pp 131, pp 281, pp 222. 2. P. Atkins and J. De Paula, Physical Chemistry, 8 th ed. (Oxford University Press, 2006), pp 335. 3. D.A. McQuarrie and J. D. Simon, Physical Chemistry A molecular approach, 1 st ed. (University Science Books, Herndon VA, 1997), pp 584. 4. P.R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2 st Ed. (NRC Research Press, 2006) 5. Kuntzleman, Thomas Scott; Rohrer, Kristen; Schultz, Emeric (2012-06-12). The Chemistry of Lightsticks: Demonstrations To Illustrate Chemical Processes. Journal of Chemical Education. 89 (7): 910 916.