## 3 Calculation and Discussion  
The measured intervals of the hyperfine transitions were fitted to a four-term Hamiltonian [14] to calculate the hyperfine constants of the iodine lines. The four-term Hamiltonian includes the electric quadrupole, spin-rotation, tensor spin-spin, and scalar spin-spin interactions, where \(e\)Q\(q\), \(C\), \(d\), and \(\delta\) are the respective hyperfine constants corresponding to these interactions [14]. In the present calculation, a nonlinear least-squares fit was performed using the ROOT analysis libraries of CERN [17]. The detailed procedure and parameters of the fitting are described elsewhere [18; 19]. The hyperfine splitting calculated from the fit was compared with the measured hyperfine splitting. The differences between the measurement and calculation are listed in the columns \"Obs.-Cal.\" in Tables 2 and 3. The measured frequencies of the a\({}_{2}\) transition of the R(100)49-0 line, the b\({}_{7}\) and b\({}_{8}\) transitions of the R(59)45-0 line, and the a\({}_{6}\) and a\({}_{7}\) transitions of the P(82)47-0 line were excluded from the fitting because they were too close to each other or to other hyperfine transitions. The residual slope on the error signal from adjacent transitions will introduce an offset in the signal and hence shift the line center when the lock point is set to the averaged level of the off-resonance spectral baseline. The standard deviation of the theoretical fit was approximately 2 or 3 kHz, except for the R(106)50-0, R(100)49-0 lines. The larger standard deviation of the fit for the R(106)50-0, R(100)49-0 lines (8.6 and 5.0 kHz, respectively) is caused by overlapping with other relatively weak lines, as shown in Figure 2a,b. Through the theoretical fit, hyperfine constants \(\Delta e\)Q\(q\), \(\Delta C\), \(\Delta d\), and \(\Delta\delta\), which are the differences in the hyperfine constants between the upper and lower states, were obtained. The calculated hyperfine constants of the six iodine lines are listed in Table 4.  
Figure 2: Observed hyperfine structures of the \({}^{127}\)I\({}_{2}\) lines near 514 nm. The vertical scales are the same for all graphs. The R(84)47-0 (“a” series) and R(59)45-0 (“b” series) lines overlap and are shown in the same graph. The uncertainties in the values for the main hyperfine constant, \(\Delta\)_eQq_, were \(<\)10 kHz, except for the R(106)50-0 line. The slightly larger uncertainty of \(\Delta\)_eQq_ (17 kHz) of the R(106)50-0 line obtained from fitting is due to slightly larger measurement uncertainties caused by the overlapping with other lines. Furthermore, the intensity of the R(106)50-0 line was the lowest among the observed lines. Small uncertainties were also observed for \(\Delta\)_C_, \(\Delta\)_d_, and \(\Delta\delta\). Accurate hyperfine constants were used to reproduce the hyperfine spectra of iodine. Therefore, formulae for predicting hyperfine constants are essential for spectroscopy [20]. In our previous work, we derived formulae to express the rotational dependence of the ground state _eQq\({}^{\prime\prime}\)_ (\(v^{\prime\prime}=0\)) [21] and excited states _eQq\({}^{\prime}\)_ (\(v^{\prime}=32\) and 44) [10,19]. The hyperfine constants obtained in this study can be used to improve our understanding of the rotational dependence of the excited-state hyperfine constants for \(v^{\prime}=45\), 46, 47, 49, and 50. We observed 97 hyperfine transitions in six lines with an uncertainty of 5.6 kHz over a range greater than 64 GHz. By combining the absolute frequencies presented in Table 1 and the hyperfine splittings in Tables 2 and 3, we can determine the frequencies of all the hyperfine transitions, except for the unresolved and excluded transitions. Such a grid of iodine reference lines was already investigated in the 571-596 nm wavelength region, with an uncertainty of approximately 2 MHz in an early stage of Doppler-free spectroscopy [22]. However, the present study has achieved a measurement uncertainty approximately 400 times smaller than that in Ref. [22].