Infrared extrapolations of quadrupole moments and transitions

We study the convergence of bound-state quadrupole moments in finite harmonic oscillator spaces. We derive an expression for the infrared extrapolation for the quadrupole moment of a nucleus and benchmark our results using different model interactions for the deuteron. We find good agreement between the analytically derived and numerically obtained convergence behavior. We also derive an extrapolation formula for electric quadrupole transitions and find good agreement with the numerical calculation of a simple system.

Figure 1

Convergence of the quadrupole moment for the deuteron modeled by a simple square well potential as a function of the infrared length $L$ (given in units of the oscillator length $b$). The yellow circles are the data from the computation. The thin blue solid line is the fit to Eq. (30). The thin, orange dashes are the fit to Eq. (22). The thick blue solid line is $Q_{\infty }$ from the fit to Eq. (30). The thick orange dashes are $Q_{\infty }$ from the fit to Eq. (22).

Figure 2

Extrapolation of the deuteron quadrupole moment computed from a chiral potential. The yellow circles are the data from the calculation. The thin blue solid line is the fit to Eq. (30). The thick blue solid line is $Q_{\infty }$ from the fit to Eq. (30). The thin red dashes are the fit to Eq. (22), and the thick red dashes are $Q_{\infty }$ from the fit to Eq. (22).

Figure 3

Difference between the $L$-dependent quadrupole moment and its asymptotic value from an extrapolation based on Eq. (30) (blue circles are data and the solid blue line is the fit) and from an extrapolation based on the more precise Eq. (22) (orange squares are data and the dashed orange line is the fit).

Figure 4

The yellow circles represent $\delta Q_1$ as defined by Eq. (41). The blue triangles represent $\delta Q_0$. The orange crosses represent $\delta Q_{-1}$. The red line is proportional to $e^{-2k_{\infty }L}$.

Figure 5

Convergence of the $E2$ transition amplitude as a function of $L$ for a simple Gaussian well model. The yellow dots are the results of the numerical calculation. The thin light blue dashes are the fit to the leading order result in Eq. (35). The thick red dashes correspond to a fit including the ${(1/k_{2}L)}^2$ term in Eq. (35). And the solid orange line includes the ${(1/k_{2}L)}^3$ term in the fit.

Figure 6

The residual transition amplitude from fitting to increasing powers of $(1/k_{2}L)$ in Eq. (35). The yellow dots represent a leading order fit ($m=0$), the light blue squares include also the $(1/k_{2}L)$ term ($m=-1$), the dark blue diamonds include also the ${(1/k_{2}L)}^2$ term ($m=-2$), and the orange triangles include also the ${(1/k_{2}L)}^3$ term ($m=-3$). The red solid line represents $e^{-2k_{2}L}$ where $k_2$ is determined from a $d$-wave energy fit.