Effect of finite nuclear size on the electric quadrupole hyperfine operator

The study of the hyperfine structure (hfs) in heavy and superheavy atomic systems is a valuable tool for obtaining the information about nuclei Sobelman ; Johnson . Comparing experimental hfs with theoretical calculations of the magnetic dipole hfs constants $A$ and electric quadrupole hfs constant $B$ allows extraction of the nuclear magnetic moment $\mu$ and nuclear electric quadrupole moment $Q$. Electric quadrupole $Q$ is strongly enhanced in deformed nuclei. This may serve as a guidance in the search for the nuclear stability island since the nuclei in the vicinity of the island are expected to be spherical (see, e.g. Oganessian ).

Effect of the finite nuclear size for the magnetic hfs constant $A$ has been extensively studied in numerous publications. It is sufficient to mention the Bohr-Weisscopf effect where constant $A$ is not exactly proportional to the nuclear magnetic moment since it also depends on the distribution of magnetization inside the nucleus - see e.g. BW1 ; BW2 ; Orozco ; BW3 ; BW4 ; BW5 ; BW6 ; BW7 ; Roberts and references therein. On the other hand, we are not aware of such study for the electric quadrupole hfs constant.

Standard operator of the electric quadrupole hfs interaction (the $Q_{20}$ component) has the form Sobelman ; Johnson

$$\hat{Q}=Y_{20}/r^{3},$$

(1)

where $Y_{20}$ is the spherical function, $r$ is the distance to the nucleus. We omit here a coefficient which does not play any role in further discussion since we will discuss relative corrections to the hfs constant $B$, i.e. $\delta B/B$. From the symmetry of the problem we conclude that the quadrupole electric field in the centre of the nucleus vanishes, ${\bf E}=-{\bf\nabla}\phi=0$. The vanishing gradient means that the quadrupole electrostatic potential near $r=0$ is $\phi\propto r^{2}Y_{20}$. This leads us to a simple analytical form of the quadrupole operator which takes into account the finite nuclear radius $R$:

	$\displaystyle\hat{Q}=F(r)Y_{20},$		(2)
	$\displaystyle F(r)=\left\{\begin{array}[]{ll}r^{2}/R^{5},&r\leq R\\ 1/r^{3},&r>R\end{array}\right.$		(5)

Let us start from a qualitative consideration of the dependence of $B$ on the nuclear radius $R$. Integrals in the matrix elements of the singular quadrupole operator (1) are dominated by small distances $r$ from the nucleus where we can neglect energy of an electron compare to the Coulomb potential and screening of the nuclear Coulomb potential by electrons. Solution of the radial Dirac equation in the Coulomb field for zero energy is expressed in terms of the Bessel functions $J_{\gamma_{j}}(x)$ and $J_{\gamma_{j}-1}(x)$, where $\gamma_{j}=\sqrt{(j+1/2)^{2}-Z^{2}\alpha^{2}}$, $j$ is the electron angular momentum, $Z$ is the nuclear charge, $\alpha$ is the fine structure constant, $x=(8Zr/a)^{1/2}$ is the dimensionless distance variable, $a$ is the Bohr radius- see e.g. Refs. Khriplovich ; FG . Therefore, the radial dependence of the charge density of an electron may be presented as

$$\rho(r)=\frac{C_{nlj}}{r^{2}}f(x),$$

(6)

where the normalisation constant $C_{nlj}$ may be omitted since it cancels out in the ratio $\delta B/B$, dimensionless function $f(x)$ is expressed as products of Bessel functions. We can present matrix element of the operator (1) as

	$\displaystyle B\sim\int_{0}^{\infty}\frac{\rho(r)}{r^{3}}f(x)r^{2}dr=C_{nlj}\left(8Z/a\right)^{2}I,$		(7)
	$\displaystyle I=2\int_{0}^{\infty}\frac{f(x)}{x^{3}}dx\sim 1$

Near the nucleus Bessel functions $J_{\gamma_{j}}(x)$ may be expanded for $x\ll 1$, and we have Khriplovich ; FG

$$\rho(r)=C_{nlj}\left(8Z/a\right)^{2\gamma_{j}}r^{2(\gamma_{j}-1)}.$$

(8)

If we use operator (2) instead of the singular operator (1), the contribution of the area inside the nucleus is significantly suppressed and this effect produces the change in the hfs constant $B$,

	$\displaystyle\delta B\sim-C_{nlj}\int_{0}^{R}\frac{\rho(r)}{r^{3}}f(x)r^{2}dr,$
	$\displaystyle\frac{\delta B}{B}\sim-\left(\frac{RZ}{a}\right)^{2(\gamma_{j}-1)}.$		(9)

We should note that Eq. (8) for the density $\rho(r)$ is valid outside the nucleus where the nuclear Coulomb potential is equal to $V(r)=-Ze^{2}/r$. However, practically all numerical calculations of hfs have actually taken into account finite size of the nucleus in the electron wave functions. Analytical calculations of the electron wave function in the finite size nucleus potential $V(r)$ (instead of the point-like potential $V(r)=-Ze^{2}/r$) have been done in Refs. Khriplovich ; FG . The main difference is that in the leading term $\rho(r)\propto r^{2j-1}$ (instead of $r^{2(\gamma_{j}-1)}$; the difference in power of $r$ is $\sim Z^{2}\alpha^{2}/(j+1/2)$). Such modification of $\rho(r)$ inside the nucleus produces coefficient $\sim 1$ in the estimate of $\delta B/B$ and does not change any conclusions. This is easy to explain since the Coulomb wave function for $r>R$ provides boundary condition at $r=R$ for the solution inside the nucleus, therefore, the factor $\left(\frac{RZ}{a}\right)^{2(\gamma_{j}-1)}$ in the estimate of $\delta B/B$ appears in any case.

The $s_{1/2}$ and $p_{1/2}$ electronic states have zero value of $B$. Simple estimates for the states with total angular momentum $j>1/2$ gives $\delta B/B$ equal to a small fraction of per cent. Indeed, power of the small parameter $RZ/a$ in Eq. (9) is positive, from 2 for small $Z\alpha$ to 1.46 for $Z$=137. However, this naive estimate is only valid for the hydrogen-like single electron atoms.

In many-electron atoms the core polarization corrections and other correlation corrections contain large non-diagonal matrix elements of the hyperfine interaction such as $\langle s_{1/2}|\hat{Q}|d_{3/2}\rangle$, $\langle s_{1/2}|\hat{Q}|d_{5/2}\rangle$ and $\langle p_{1/2}|\hat{Q}|p_{3/2}\rangle$. These large non-diagonal matrix elements may strongly enhance effects of configuration mixing on $B$. They are also responsible for the transition amplitudes induced by electric quadrupole hyperfine interaction - see, for example Ref. hfs-tr , where probabilities of E3 and M2 atomic clock transitions, which are transformed to E1 by the hfs operators, have been calculated. Electron wave functions $s_{1/2}$ and $p_{1/2}$ tend to infinity for point-like nucleus and this significantly increases the sensitivity to the nuclear size:

$$\frac{\delta B}{B}\sim\left(\frac{RZ}{a}\right)^{\gamma_{1/2}+\gamma_{3/2}-2}$$

(10)

Power of the small parameter $RZ/a$ becomes negative for $Z>132$ (this means ”infinite” $\delta B/B$ for $R=0$). However, the ratio $\delta B/B$ may exceed 1% already for $Z>80$. Therefore, we should use a more accurate electric quadrupole operator (2) inside the nucleus. Below we complement our rough estimates by the accurate numerical calculations.

We start our study from the hydrogen-like systems. We use the Fermi distribution of the electric charge over the nuclear volume with $R=1.2A^{1/3}$, here $A$ is the number of nucleons in the nucleus. The same nuclear radius is used in (2). We perform calculations of the allowed diagonal and non-diagonal matrix elements of operator $\hat{Q}$ for the $3s$, $3p_{1/2}$, $3p_{3/2}$, $3d_{3/2}$ and $3d_{5/2}$ states. Note that all single-electron states of the same symmetry are proportional to each other on short distances, therefore, states with any principal quantum number $n$ can be used in the study. We choose $n=3$ just for the convenience. The calculations are done for a set of different values of nuclear charge $Z$ and for two forms of the operator $\hat{Q}$, (1) and (2). The results are compared in Table 1. The results are presented as a difference in per cent between the two forms of the operator. One can see that the difference is small for the diagonal matrix elements. It reaches $\sim 10^{-3}$ for $Z=120$. Note also that the effect is practically zero for states with $j>3/2$. However, the effect is much larger for the off-diagonal matrix elements involving $s_{1/2}$ or $p_{1/2}$ states. This is because these states penetrate inside the nuclei. The effect reaches $\sim$ 1% for $Z=83$ (Bi atom) and becomes even larger for higher $Z$ (see Table 1).

Next, we study the effect of changing the electric quadrupole operator $\hat{Q}$ from (1) to (2) on the diagonal matrix elements (i.e. on the electric quadrupole constants $B$) in many-electron atoms. As examples, we consider heavy atoms or ions with a relatively simple electronic structure, one electron above closed shells. In atoms with several valence electrons the effect may be even bigger due to the large configuration mixing, which involves the $s_{1/2}-d_{3/2},d_{5/2}$ and $p_{1/2}-p_{3/2}$ matrix elements of $\hat{Q}$ . The calculations are done in the $V^{N-1}$ approximation, which means that the initial relativistic Hartree-Fock (RHF) calculations are performed for the closed-shell core, the states of external electron are calculated in the field of the frozen core. To calculate matrix elements of the $\hat{Q}$ operator, we use the time-dependent Hartree-Fock method which is equivalent to the well-known random-phase approximation (RPA) - see e.g.  Johnson . The RPA equations can be written as (see e.g  TDHF )

$$(\hat{H}_{0}-\epsilon_{c})\delta\psi_{c}=-(\hat{Q}+\delta V)\psi_{c}.$$

(11)

Here $\hat{H}_{0}$ is the RHF operator, index $c$ numerates states in the core, $\psi_{c}$ is a single-electron wave function for a particular state in the core, $\delta\psi_{c}$ is the correction to $\psi_{c}$ caused by external field $\hat{Q}$, $\delta V$ is the correction to the self-consisted RHF potential caused by the change in all core wave functions. The RPA equations (11) are solved self-consistently for all states in the core to find $\delta V$. Matrix elements for valence states $v$ are then calculated as
$\langle v|\hat{Q}+\delta V|v\rangle$.

The results of calculations are presented in Table 2. We included Fr and U as heavy atoms of a broad experimental interest. We also included Fm and No as heaviest atoms for which atomic spectra measurements are in progress Fm1 ; Fm2 ; No1 ; No2 ; No3 ; No4 . We included E120 for illustration on how big the effect could be for very high $Z$. We see from the table that the effect on $B$ in many-electron atoms is significantly larger than that for the hydrogen-like systems (see Table 1). This is due to the contribution of the $s_{1/2}-d_{3/2},d_{5/2}$ and $p_{1/2}-p_{3/2}$ off-diagonal matrix elements into the core polarisation correction (see Fig. 1).

For better understanding of the role of the off-diagonal matrix elements in the core polarisation correction we present in Table 3 the decomposition of the corrections to matrix element of different states of E120⁺ over different channels in the core. One channel is the sum over all core states $c$ of particular symmetry and all possible states $m$ above the core. For example, $s$-channel contains terms with the $\langle 1s|\hat{Q}+\delta V|nd_{3/2}\rangle$, $\langle 1s|\hat{Q}+\delta V|nd_{5/2}\rangle$, $\langle 2s|\hat{Q}+\delta V|nd_{3/2}\rangle$, etc. The $s_{1/2}$ and $p_{1/2}$ channels give non-zero contribution due to off-diagonal matrix elements only. The off-diagonal matrix elements contribute to other channels too. For example, the $\langle 2p_{3/2}|\hat{Q}+\delta V|np_{1/2}\rangle$ matrix elements contribute to the $p_{3/2}$ channel.

As can be seen from Table 3, the contribution of the off-diagonal matrix elements is huge (mostly, in channels $s_{1/2}$ and $p_{1/2}$) Some contributions of the off-diagonal matrix elements exceed many times the final answer since there are partial cancellations of these big contributions. Large off-diagonal matrix elements are more sensitive to the form of the operator $\hat{Q}$, see Table 1. These two facts lead to significant enhancement of the effect in many-electron atoms.

Calculations of the probabilities of clock transitions induced by the hyperfine interaction (see, for example Ref. hfs-tr ) can also benefit from considering correct form of the electric quadrupole operator $\hat{Q}$. These transitions, forbidden as electric dipole transitions, are open by off-diagonal mixing of states with different electron angular momentum by the magnetic dipole or electric quadrupole interaction hfs-tr .

Acknowledgements — The work was supported by the Australian Research Council Grants No. DP230101058 and DP200100150.