## Abstract

A sign error in the equations for the mixing correction factors *η* and *ζ* from [Phys. Rev. A **77**, 012512 (2008)] was identified. These equations were used in our article [Opt. Express **20,** 21379 (2012)], resulting in a miscalculation of the corrected hyperfine constants and nuclear octupole moment. The revised values are presented.

© 2013 OSA

We recently became aware that Eq. (4) from [1] has an additional (−1)^{J−J′} factor in front compared to the corresponding equation from [2]. This equation is used for the calculation of the second-order energy corrections to a *J* level due to the mixing with a *J*′ level. For a typical *J* − *J*′ = 1 difference (mixing with the nearest *J* level), this leads to an additional (−1) factor which follows through to the values of the mixing correction factors *η* and *ζ*. As a result, *η* and *ζ* (Eqs. (5) and (6)) from [1] have the wrong sign. Similarly, we believe that the equations for *η* and *ζ* (Eqs. (6) and (7) from [2]) have the wrong sign. These were used in this form in our article [3] and yielded wrong results for the corrected values of the hyperfine constants A, B and C, as well as the calculated nuclear octupole moment.

In order to avoid further confusion, we use Eq. (4) from [1] for writing the two correction factors in terms of any (*γJ*,*γ* ′*J*′) electronic states

Consequently, with the changed sign for the *η* and *ζ* correction factors, the corrected hyper-fine constants in Table 3 from our article [3] should read as in Table 1.

The recalculated octupole moment becomes

The sign error is confirmed by our similar recent measurements on the *D*
_{5/2} manifold of the same ion, which yield the value of the octupole moment independent from the *D*
_{3/2} measurements. Despite this sign error which affected some of the presented values, all qualitative statements made in our article [3] are unaffected.

## References and links

**1. **K. Beloy and A. Derevianko, “Second-order effects on the hyperfine structure of *P* states of alkali-metal atoms,” Phys. Rev. A **78**, 032519 (2008) [CrossRef] .

**2. **K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P_{2} state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A **77**, 012512 (2008) [CrossRef] .

**3. **Nicholas C. Lewty, Boon Leng Chuah, Radu Cazan, B. K. Sahoo, and M. D. Barrett, “Spectroscopy on a single trapped ^{137}Ba^{+} ion for nuclear magnetic octupole moment determination,” Opt. Express **20**, 21379–21384 (2012) [CrossRef] [PubMed] .