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.

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We recently became aware that Eq. (4) from [1] has an additional (−1)JJ 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 JJ′ = 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,γJ′) electronic states

η=(1)JJEγJEγJ(I+1)(2I+1)Iμ2[γJT1eγJ]2
ζ=(1)JJEγJEγJ(I+1)(2I+1)I2I+32I1μQγJT1eγJγJT2eγJ

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.

Tables Icon

Table 1. Hyperfine coupling constants.

The recalculated octupole moment becomes

Ω(Ba+137)=0.04885(53)(μN×b).

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 3P2 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 137Ba+ ion for nuclear magnetic octupole moment determination,” Opt. Express 20, 21379–21384 (2012) [CrossRef]   [PubMed]  .

References

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  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 3P2 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 137Ba+ ion for nuclear magnetic octupole moment determination,” Opt. Express 20, 21379–21384 (2012).
    [Crossref] [PubMed]

2012 (1)

2008 (2)

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]

K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P2 state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
[Crossref]

Barrett, M. D.

Beloy, K.

K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P2 state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
[Crossref]

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]

Cazan, Radu

Chuah, Boon Leng

Derevianko, A.

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]

K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P2 state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
[Crossref]

Johnson, W. R.

K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P2 state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
[Crossref]

Lewty, Nicholas C.

Sahoo, B. K.

Opt. Express (1)

Phys. Rev. A (2)

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]

K. Beloy, A. Derevianko, and W. R. Johnson, “Hyperfine structure of the metastable 3P2 state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
[Crossref]

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Tables (1)

Tables Icon

Table 1 Hyperfine coupling constants.

Equations (3)

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η = ( 1 ) J J E γ J E γ J ( I + 1 ) ( 2 I + 1 ) I μ 2 [ γ J T 1 e γ J ] 2
ζ = ( 1 ) J J E γ J E γ J ( I + 1 ) ( 2 I + 1 ) I 2 I + 3 2 I 1 μ Q γ J T 1 e γ J γ J T 2 e γ J
Ω ( Ba + 137 ) = 0.04885 ( 53 ) ( μ N × b ) .

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