Ionic dipole and quadrupole matrix elements from nonadiabatic core polarization

E. S. Shuman and T. F. Gallagher

Phys. Rev. A 74, 022502 – Published 2 August 2006

Abstract

The radial matrix elements connecting the ionic ${Ba}^{+} 6 s$ ground state to low-lying excited $6 p$ and $5 d$ states can be extracted from the $K$ splittings of the bound $6 s n ℓ$ states in much the same way that ionic polarizabilities are extracted from the separations between $ℓ$ states. We develop an expression for the $K$ splitting by a pair of expansions which allows us to compare the contributions of different ionic states. This comparison confirms that all but the lowest two may be safely ignored. Finally, we extract the radial ${Ba}^{+}$ matrix elements $⟨ 6 s ∣ r ∣ 6 p ⟩ = 4.03 (12)$ and $⟨ 6 s ∣ r^{2} ∣ 5 d ⟩ = 9.76 (29)$ from the experimentally obtained $K$ splittings.

Received 8 March 2006

DOI:https://doi.org/10.1103/PhysRevA.74.022502

Authors & Affiliations

E. S. Shuman and T. F. Gallagher

Department of Physics, University of Virginia, Charlottesville, Virginia 22904-0714, USA

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Vol. 74, Iss. 2 — August 2006

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Figure 1
Energy distribution of the squared matrix element per unit energy ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{2}} ∣ n ℓ ⟩ ∣}^{2}$ which enters into the dipole interaction between the $6 s n h K$ states and $6 p n ℓ K$ states. The ${Ba}^{+}$ $6 s_{1 ∕ 2}$ energy is taken to be zero. The matrix elements are plotted per unit energy so that the area of the curve is the squared matrix element. Since there are no $6 p_{1 ∕ 2} n^{'} i K = 9 ∕ 2$ states, the $6 s n h K = 9 ∕ 2$ state is coupled only to the $6 p_{1 ∕ 2} n^{'} g$ states. In contrast, the $6 s_{1 ∕ 2} n h K = 11 ∕ 2$ state is only coupled to the $6 p_{1 ∕ 2} n^{'} i$ states. This difference causes the $K$ splitting. The matrix elements plotted here are ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{2}} ∣ n g ⟩ ∣}^{2}$ (dashed line) and ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{2}} ∣ n i ⟩ ∣}^{2}$ (solid line). The arrows represent ${\bar{W}}_{1 n ℓ ℓ^{'}}$ , the average shift of the matrix elements from the energy of the $6 p_{1 ∕ 2} n ℓ$ state for $n = 18$ and $ℓ = 5$ .Reuse & Permissions
Figure 2
Energy distribution of the squared matrix element per unit energy ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{3}} ∣ n ℓ ⟩ ∣}^{2}$ which enters into the quadrupole interaction between the $6 s n h K$ states and $5 d n ℓ K$ states. The ${Ba}^{+}$ $6 s_{1 ∕ 2}$ energy is taken to be zero. The matrix elements are plotted per unit energy so that the area of the curve is the squared matrix element. Since there are no $5 d_{3 ∕ 2} n^{'} k K = 9 ∕ 2$ states, the $6 s n h K = 9 ∕ 2$ state is coupled only to the $5 d_{3 ∕ 2} n^{'} h$ and $5 d_{3 ∕ 2} n^{'} f$ states. In contrast, the $6 s_{1 ∕ 2} n h K = 11 ∕ 2$ state is only coupled to the $5 d_{3 ∕ 2} n^{'} h$ and $5 d_{3 ∕ 2} n^{'} i$ states. This difference causes the $K$ splitting. The matrix elements plotted here are ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{3}} ∣ n f ⟩ ∣}^{2}$ (solid line), ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{3}} ∣ n h ⟩ ∣}^{2}$ (dotted line), and ${∣ ⟨ 18 h ∣ \frac{1}{r_{o}^{3}} ∣ n k ⟩ ∣}^{2}$ (dashed line). The arrows represent ${\bar{W}}_{2 n ℓ ℓ^{'}}$ , the average shift of the matrix elements from the energy of the $5 d_{3 ∕ 2} n ℓ$ state for $n = 18$ and $ℓ = 5$ .Reuse & Permissions
Figure 3
The dipole (open markers) and quadrupole (solid markers) contributions to the average polarization energy shift for the $6 s 18 ℓ$ states, $P_{18 ℓ}$ , without inclusion of $K$ splitting for $ℓ = 5$ , $ℓ = 6$ , and $ℓ = 7$ . The shifts have been calculated using numerical evaluation of Eq. (7) (●), the Van Vleck model (▵) from Eq. (42), and the adiabatic model (엯) from Eq. (17). It is apparent that the dipole shift is represented to better than 15% in all cases by any of the methods. However, the quadrupole shift is not represented accurately by the approximations except for $ℓ = 7$ .Reuse & Permissions
Figure 4
Dipole (marker), quadrupole (엯), and total (◻) $K$ splitting terms for the $6 s 18 ℓ$ states. The splittings have been calculated using numerical evaluation of Eq. (7) (●), the Van Vleck model (×) from Eq. (42), and the adiabatic model (+) from Eq. (27). As in Fig. 3, any of the methods produce an accurate estimation of the splitting in the dipole case for any $ℓ$ state. However, the quadrupole splitting is only represented accurately by the approximations for $ℓ = 7$ . The solid horizontal lines represent the experimentally observed $K$ splitting.Reuse & Permissions
Figure 5
The measured $K$ splittings using Eq. (48) with $γ$ calculated numerically. The line fitting the data yields $⟨ 6 s ∣ r ∣ 6 p ⟩ = 4.03$ from the $y$ intercept and $⟨ 6 s ∣ r^{2} ∣ 5 d ⟩ = 9.76$ form the slope. The $n k$ experimental splittings observed by Gallagher et al. have been given twice the uncertainty of the other measurements because they appear to be mutually inconsistent.Reuse & Permissions

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