8 Using the standard deviation of the points shown in Fig. 7 as an estimate of the uncertainty we assign a 40 kHz uncertainty to the A constant due to possible magnetic fields. For an analysis of the shifts due to collisions, the pres- sure of the Cs vapor was changed by heating the vapor cell as described above. Data were taken at tempera- tures of 297, 318.6, and 345.5 K, corresponding to Cs vapor pressures of 0.15, 1.2, and 11 mPa. As described above, we determine the center-of-gravity frequency with a fixed hyperfine A constant for each pressure. We see no indication of any variation of the data with the Cs vapor pressure (Fig. 7). The standard deviation of the frequen- cies extracted at the different temperatures is 31 kHz, which we take as the uncertainty in the center-of-gravity frequency due to temperature or pressure effects. For the hyperfine A constant, the scatter in the data taken at different temperatures gives a standard deviation of 18 kHz, which we adopt as the uncertainty. The final systematic considered was possible error due to imperfect alignment of the counter-propagating laser beams. The effect of misalignment of the two beams is to broaden and shift the peaks through a first-order Doppler shift. As can be seen from Eq. (2), the position of the peaks depends on the difference in the magnitude and direction of the two vectors ⃗ k$_{1}$ and ⃗ k$_{2}$ as well as the velocity class excited. Additional first-order Doppler shifts come into play if the wavefronts of ⃗ k$_{1}$ and ⃗ k$_{2}$ are not precisely counter-propagating, which can result from divergence of the counter-propagating beams and mis- match of the spatial modes. For a spectrum that contains many peaks coming from many different velocity classes the shift due to the misalignment largely averages out. This effect was investigated theoretically by modifying the program to include the effect of misalignment of the two beams. This was done by changing the ⃗ k$_{2}$ · ⃗ v term in Eq. (2) to include an additional perpendicular velocity component. For the 8S state three spectra were calcu- lated with different angles of misalignment. Analysis of the calculated spectrum and an estimate of the maximum possible misalignment of the beams gives a conservative limit on possible effects from angular misalignment of 22 kHz for the center-of-gravity frequency and 20 kHz for the hyperfine A constant. If we attribute the scatter in the experimental results to angular misalignment, we conclude that for all the spectra, the effective angular de- viation due to either misalignment or to divergence and mode mismatch of the beams is less than 0.04 radians. We note that for the calculated data the misalignment of the counter-propagating laser beams leads to no sig- nificant shift in the extracted frequency, it does lead to a larger spread of the data and an increased value of χ $^{2}$, as observed. This additional spread of the data beyond what is expected from the statistical noise is taken into account by conservatively increasing the uncertainties to achieve the expected χ $^{2}$, as described in Section IV A. In addition, the effect of misalignment was investigated experimentally on the 6S$_{1}$$_{/}$$_{2}$ → 6P$_{1}$$_{/}$$_{2}$ → 7D$_{3}$$_{/}$$_{2}$ transition. These results are in good agreement with previous mea- surements of the 8S$_{1}$$_{/}$$_{2}$ state, summarized in Table II. Table I summarizes the corrections and associated un- certainties on the extracted center-of-gravity and hyper- fine A constants. We see no evidence for any shift that is larger than the uncertainty associated with evaluat- ing the effect. As our final value we take the mean of the 8S$_{1}$$_{/}$$_{2}$ frequencies determined from the 6P$_{1}$$_{/}$$_{2}$ and 6P$_{3}$$_{/}$$_{2}$ intermediate states, weighted by their respec- tive statistical uncertainties, and apply the corrections and systematic uncertainties (propagated in quadrature) listed in Table I. We find final values for the center-of- gravity frequency and hyperfine A constant of 8S$_{1}$$_{/}$$_{2}$ state of The non-zero second-order Doppler shift ( v/c ) 2 ν$_{0}$ is about 300 Hz for this transition and is negligible com- pared to the other uncertainties. The study of this transition confirmed that while the in- dividual peaks do shift, there was no shift in the mean value of the transition frequency, within the uncertainty of our measurement. FIG. 7: The dependence of the center-of-gravity frequencies for the 6S 1 / 2 → 6P 3 / 2 → 8S 1 / 2 state on the power (upper left), the magnetic field (upper right), and the Cs vapor pressure (bottom). The straight line for the power dependence is a weighted linear fit to the data. The vertical axis has been offset and centered so that the nominal operating point is at zero for the power-dependence plot. The magnetic field and pressure data are shown scattered about the weighted mean of the data. The horizontal axis for the power dependence is plotted as a function of the fraction of the maximum power. The horizontal axis for the magnetic field dependence is plot- ted as a function of the residual magnetic field, ≈ 50 µT . -0.1 0.0 0.1 1.0 0.5 0.0 Normalized Power Center-of-Gravity Frequency Shift [MHz] -0.1 0.0 0.1 2 1 0 Normalized Magnetic Field Center-of-Gravity Frequency Shift [MHz] -0.1 0.0 0.1 10 5 0 Pressure [mPa] Center-of-Gravity Frequency Shift [MHz] ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:8S}$$_{1}$$_{/}$$_{2}$ = 729 009 798 . 86(19) MHz A$_{8S}$$_{1}$$_{/}$$_{2}$ = 219 . 14(11) MHz .