Hyperfine structure

Angular momenta

Both iodine nuclei have spin, i = 5/2 for ¹²⁷I and i = 7/2 for ¹²⁹I, and an electric quadrupole moment. The nuclear spins couple to the total nuclear spin I = I₁ + I₂, and I couples to the rotational angular momentum J to give the total angular momentum F = J + I. The hyperfine Hamiltonian is set up in the coupled basis |J, (i₁ i₂) I; F⟩ and diagonalised separately for each F. For ¹²⁷I₂ each rovibrational level has 15 or 21 hyperfine sublevels, depending on the parity of J (see below).

Effective hyperfine Hamiltonian

The effective Hamiltonian of Broyer, Vigué & Lehmann (1978) is

\[ H_\text{hfs} = eQq\,H_\text{EQ} + C\,H_\text{SR} + d\,H_\text{TSS} + \delta\,H_\text{SSS}, \tag{1} \]

with the electric quadrupole interaction (eQq), the nuclear spin–rotation interaction (C), and the tensor (d) and scalar (δ) nuclear spin–spin interactions. Within one value of J the operators are

\[ H_\text{EQ} = -\sum_{n=1,2}\frac{3(\mathbf I_n\cdot\mathbf J)^2 + \tfrac32(\mathbf I_n\cdot\mathbf J) - \mathbf I_n^2\mathbf J^2}{2i_n(2i_n-1)(2J-1)(2J+3)},\qquad H_\text{SR} = \mathbf I_1\cdot\mathbf J + \frac{C_2}{C_1}\,\mathbf I_2\cdot\mathbf J, \tag{2} \]

\[ H_\text{TSS} = \frac{3(\mathbf I_1\cdot\mathbf J)(\mathbf I_2\cdot\mathbf J) + 3(\mathbf I_2\cdot\mathbf J)(\mathbf I_1\cdot\mathbf J) - 2(\mathbf I_1\cdot\mathbf I_2)\,J(J+1)}{(2J-1)(2J+3)},\qquad H_\text{SSS} = \mathbf I_1\cdot\mathbf I_2 . \tag{3} \]

The quadrupole and tensor spin–spin operators are second-rank tensors with respect to the molecular axis. They therefore also have matrix elements between J and J ± 2 (and between I and I ± 2). These matrix elements shift the components by up to a few tens of kHz at high J and are included; the energies of the neighbouring rotational levels are taken from the level model. All matrix elements are evaluated with Wigner 3j, 6j and 9j symbols. For ¹²⁷I¹²⁹I the two nuclei have different eQq and C, which gives additional matrix elements with ΔI = ±1.

Nuclear-spin statistics

In ¹²⁷I₂ and ¹²⁹I₂ the two nuclei are identical fermions. The total wavefunction must be antisymmetric under their exchange, so in X (0g⁺) only values of I with the same parity as J occur, and in B (0u⁺) only values of the opposite parity. For ¹²⁷I₂ (I = 0 … 5) the nuclear-spin statistical weight is therefore 15 for even J″ (I = 0, 2, 4) and 21 for odd J″ (I = 1, 3, 5); a line with even J″ has 15 main components and a line with odd J″ has 21.

Figure 1. Hyperfine sublevels of the lower level X(v″ = 0, J″ = 56) and the upper level B(v′ = 32, J′ = 57) of R(56) 32–0, and the 15 main transitions a1–a15. Energies are relative to the centre of each group. The upper group is drawn with a three times larger vertical scale; the scale bars give 100 MHz (upper) and 500 MHz (lower), and the separation of the two groups is not to scale. The splitting of both levels is dominated by the quadrupole interaction; eQq is −2453 MHz for the lower level and −545 MHz for the upper level, so the lower level is split about 4.5 times more widely (1.1 GHz against 0.25 GHz).

Parameters

The parameters eQq, C, d and δ of each level are taken from interpolation formulae in the vibrational energy: those of Bodermann, Knöckel & Tiemann (2002) for ¹²⁷I₂ up to v′ = 43, and those of Salumbides et al. (2006) above v′ = 43 and for the other isotopologues, scaled by the ratios of the nuclear moments. For ¹²⁷I₂ the formulae are supplemented by a table of B-state corrections fitted to the measured hyperfine spectra at each v′ for which measurements exist.

Components and their intensities

The relative intensity of a component is the square of the electric-dipole matrix element between the upper and lower hyperfine eigenvectors, normalised to unity over the line. The dipole operator acts on the rotational coordinates only, so ΔF = 0, ±1 and ΔI = 0 in the uncoupled limit. The main components have ΔF = ΔJ and carry almost all of the intensity at high J. They are labelled a1, a2, … in order of increasing frequency, as in the BIPM tables.

Figure 2. Main hyperfine components of R(56) 32–0 (even J″, 15 components) and P(53) 32–0 (odd J″, 21 components), with the Doppler profile at 20 °C. The full width of the pattern, about 1 GHz, is comparable to the Doppler width, so the components are resolved only by sub-Doppler methods.

Accuracy of the hyperfine offsets

The component offsets relative to the hyperfine-free position are accurate to a few tens of kHz for ¹²⁷I₂ with v′ ≤ 43, and to about 1 MHz for ¹²⁹I₂ and ¹²⁷I¹²⁹I.