Finite Nuclear Size Effect on the Relativistic Hyperfine Splittings of 2s and 2p Excited States of Hydrogen-like Atoms

Abstract

Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the $1s$ ground state and $2s$ excited state of the one-electron systems ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}^{+}$, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic $2p_{3/2}$ orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.

1. Introduction

The solution of Dirac’s equation for finite-size potentials has been investigated in detail in the last century [1,2,3,4], originally motivated by finding electronic eigenenergies for nuclear charge numbers $Z>137$, where Sommerfeld’s fine structure formula fails [5] and by removing the $r\to 0$ divergency of the relativistic wavefunctions of the s states [5,6]. A further motivation comes from the calculation of hyperfine splittings, i.e., small energy splittings arising from the interaction of the electron spin and the nuclear magnetic moments. Since half a century, these quantities have been used to explore atomic structures in magnetic resonance spectroscopy [7]. Nowadays, they start to play a central role in quantum information and spintronics applications [8]. They allow for readout of spin qubits while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact as possible knowledge is thus of prior relevance. Since the hf splittings are determined in the region close to the nuclei, nuclear-structure and relativistic effects play a decisive role [6,9]. Numerical methods in the framework of density functional theory (DFT) are able to provide estimates for hyperfine splittings within complex systems containing several hundreds or even thousands of atoms [7,10], but the methods still require improvement with respect to accuracy. Modern combinations of quantum electrodynamics (QED) and two-body relativistic quantum mechanics (RQM) provide excellent accuracy in the <1 kHz [11,12] regime, but are often limited to simple systems (isolated atoms) [13,14]. Besides native QED-related effects like radiative, recoil, and self-energy corrections, also the effect of reduced mass and finite-size nuclear structure are taken into account. Recent calculations of the hf splitting in hydrogen and helium eliminate the uncertainty of nuclear structure effects and electron-electron correlations by using precise experimental hf values measured for the $1s$ state [15,16,17,18], and improve the accuracy of theoretical prediction for the $2s$ state to 1 Hz or even below. However, without use of experimental reference or restriction to normalized differences between $ns$ and $1s$ hf splittings [15,18], the accuracy is limited by the still incomplete knowledge of the nuclear charge and nuclear magnetization distributions, which in the case of hydrogen is expected to contribute about 50 kHz [14,15,19].

Several studies have already investigated analytically the influence of structural finite nuclear-size (FNS) effects onto relativistic hyperfine splittings. i.e., effects depending on the spatial distribution of nuclear charge and nuclear magnetic moments. A finite-size effects covering solution of Dirac’s equation for the Yukawa-potential together with a calculation of the hyperfine parameters for proton-neutron interaction based on a non-relativistic formula for the Fermi contact term have been reported by Azizi et al. [20], but does not provide the required accuracy. Recently, Kuzmenko et al. [9] present a comprehensive study of FNS effects on the hyperfine splittings of $1s$ ground states of isolated atoms. Using a rigorous reduced-mass approach [21], they provide hf values based on the exact solution of Dirac’s equation for the nuclear charge distributed in a spherical shell, as well as based on a numerical solution of Dirac’s equations for full-sphere distributed nuclear charge and nuclear magnetic moments. For ${}^1\mathrm{H}$, a very good agreement with experiment is obtained, but with FNS related hf shift much larger than previously expected, suggesting that contributions of recoil energy are already included [9].

In this work, we aim to identify an accurate treatment of elastic FNS effects that can be applied in DFT calculation for complex structures. For this purpose, we do not focus on absolute hf values but on the shifts induced by given finite-size nuclear structure models compared to the simple point-like case. Our analytical study is not only restricted to the $1s$ ground states but also includes the $2s$ and $2p$ excited states. In contrast to the isotropic s states, the latter provide essentially anisotropic hyperfine splittings, which in the non-relativistic limit, are determined by dipole-dipole interactions exclusively. To investigate the FNS-induced effect in the general case, we apply the method of Blügel [6] to different nuclear structure models. Based on the relativistic form function $S(r)$, which enters RQM if the small components are analytically substituted by the large components, the concept yields a set of parameters for the isotropic and angular-dependent dipolar splittings, which can be directly compared with the experiment. As in Ref. [9], the influence of the nuclear structure effects are not only taken into account for the nuclear charges but also regarding the spatial distribution of the nuclear magnetic moment. We do not, however, restrict our study on models with unique treatment but also investigate mixed models, where nuclear charge and magnetization are distributed differently. Based on a comparison with residual differences between accurate QED and the experiment, the differences in the finite-size-related hf shifts predicted for different models are discussed in order to identify simplified but still accurate calculation schemes, which can be applied to complex many-particle systems. In addition, by applying our fully relativistic approach to excited hydrogen ${}^1\mathrm{H}$ $2p$ states, we reveal that these orbitals are affected by surprisingly large FNS effects.

Our paper is organized as follows: We first revise the exact analytical solution of Dirac’s equation for different nuclei models (Section 2) before deriving additional inside-core contributions to the relativistic hyperfine splittings, which enter theory if Blügels approach via the relativistic form function is applied onto finite-size nuclei (Section 3). In Section 4, the approach is then applied to the $1s$ ground state and $2s$, $2p$ excited states of the real one-electron systems ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and${}^3\mathrm{He}^{+}$, whereby the complex interplay of inside-core and outside-core contributions balanced by the relativistic form function $S(r)$ and its derivative $dS(r)/dr$ is demonstrated in Section 4.1; the influence of the reduced mass and the comparison with QED-calculated values are discussed in Section 4.2.

2. Exact Solution of Dirac’s Equation for Finite-Size Potentials

In order to solve Dirac’s equation $[c\overrightarrow{\alpha }·\overrightarrow{p}+\beta m{c}^2+V(r)]\Psi =E\Psi$ for a single electron in a radialsymmetric potential, the following product approach for the four-component wave function $\Psi (\overrightarrow{r})=\left(\begin{array}{c}{\psi }_L\\ {\psi }_S\end{array}\right)$ consisting of two-dimensional, relativistic spherical harmonics (see Appendix A), and radial wave functions is expedient [22]:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\psi }_L& =g(r){\Omega }_{1,j,l,m_j}\hfill \\ \hfill {\psi }_S& =if(r){\Omega }_{2,j,l,m_j}.\hfill \end{array}\end{array}$$

(1)

This leads to a system of coupled differential equations for the radial wave functions $f(r)={\displaystyle \frac{F(r)}{r}}$, $g(r)={\displaystyle \frac{G(r)}{r}}$ [22]:

$$\left(\begin{array}{c}\hfill {\displaystyle \frac{dG(r)}{dr}}\\ \hfill {\displaystyle \frac{dF(r)}{dr}}\end{array}\right)=\left(\begin{array}{cc}-{\displaystyle \frac{\kappa }{r}}& {\displaystyle \frac{E+m{c}^2-V(r)}{\hslash c}}\\ -{\displaystyle \frac{E-m{c}^2-V(r)}{\hslash c}}& {\displaystyle \frac{\kappa }{r}}\end{array}\right)\left(\begin{array}{c}\hfill G(r)\\ \hfill F(r)\end{array}\right).$$

(2)

with relativistic quantum number $\kappa =\mp (j+1/2)$ for $j=l\pm 1/2$. The solution for a pure Coulomb potential $V(r)=-{\displaystyle \frac{Z\alpha \hslash c}{r}}$, i.e., a nuclear point charge, is well known (cf. e.g., Ref. [22]). With $\lambda ={\displaystyle \frac{\sqrt{{(m{c}^2)}^2-E^2}}{\hslash c}}$ and $\gamma =\sqrt{{\kappa }^2-Z^2{\alpha }^2}$ as parameters describing the asymtotic behaviour of the radial wave function for $r\to \infty$ and $r\to 0$, respectively, the general solution can be written as

$$\begin{array}{cc}\hfill & G(r)=C\sqrt{1+{\displaystyle \frac{E}{m{c}^2}}}\left[{\varphi }_1(r)+{\varphi }_2(r)\right]·e^{-\lambda r}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & F(r)=C\sqrt{1-{\displaystyle \frac{E}{m{c}^2}}}\left[-{\varphi }_1(r)+{\varphi }_2(r)\right]·e^{-\lambda r}\hfill \end{array}$$

(4)

with

$$\begin{array}{cc}\hfill {\varphi }_1(r)& ={\displaystyle \frac{\gamma -\frac{EZ\alpha }{\hslash c\lambda }}{-\kappa +\sqrt{n^2-2(n-|\kappa |)(|\kappa |-\gamma )}}}·F_1(\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }}+1,1+2\gamma ,2\lambda r)·{(2\lambda r)}^{\gamma }\hfill \\ \hfill {\varphi }_2(r)& =F_1(\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }},1+2\gamma ,2\lambda r)·{(2\lambda r)}^{\gamma },\hfill \end{array}$$

where $F_1(a,b,x)$ denotes the bridging polynom given by a confluent hypergeometric series, and

$$C={\displaystyle \frac{\sqrt{\lambda }}{\sqrt{2}}}{\displaystyle \frac{\sqrt{\Gamma (2\gamma +1+n-|\kappa |)}}{\Gamma (2\gamma +1)\sqrt{(n-|\kappa |)!}}}\sqrt{1-{\displaystyle \frac{\kappa }{\sqrt{n^2-2(n-|\kappa |)(|\kappa |-\gamma )}}}}$$

ensures normalization of the wave function.

The electronic eigenenergies are given by Sommerfeld’s fine structure formula

$$E_{n,\kappa }={\displaystyle \frac{m{c}^2}{\sqrt{1+{(\frac{Z\alpha }{n-|\kappa |+\gamma })}^2}}},$$

(5)

which is a direct consequence of the normalizability of the wave functions, which in $F_1(a,b,x)$ requires $a=\tilde{n}\in {\mathbb{N}}_0$ denoting the radial quantum number entering the principal quantum number $n=\tilde{n}+|\kappa |$ in Equation (5) [22].

Two types of finite-size potentials are used in our calculations (see also Figure 1):

(i)

The nucleus is considered to be a full sphere with radius R, i.e., with the potential [23]

$$V(r)=\left\{\begin{array}{cc}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{Z\alpha \hslash c}{R}}\left(3-{\displaystyle \frac{r^2}{R^2}}\right),& \mathrm{if} r\le R\hfill \\ -{\displaystyle \frac{Z\alpha \hslash c}{r}},& \mathrm{if} r\ge R\hfill \end{array}\right.$$

(6)

(ii)

The nucleus is considered to be a spherical shell with radius R, i.e., with the potential

$$V(r)=\left\{\begin{array}{cc}-{\displaystyle \frac{Z\alpha \hslash c}{R}},& \mathrm{if} r\le R\hfill \\ -{\displaystyle \frac{Z\alpha \hslash c}{r}},& \mathrm{if} r\ge R\hfill \end{array}\right..$$

(7)

For finite-size nuclear potentials, the system of differential Equations (2) needs to be solved in and outside the core. The solution needs to be continuous and differentiable at $r=R$. For finite-size potentials that coincide with the Coulomb potential outside the core (i.e., for $r\ge R$), the solution of Equation (2) for $r>R$ thus has to be built up by partial waves with $\gamma$ and $-\gamma$ (and $C\in \mathbb{C}$ used for normalization of the wave function):

$$\begin{array}{cc}\hfill & G(r)=C\sqrt{1+{\displaystyle \frac{E}{m{c}^2}}}\left[{\varphi }_1(r)+{\varphi }_2(r)\right]·e^{-\lambda r}\hfill \\ \hfill & F(r)=C\sqrt{1-{\displaystyle \frac{E}{m{c}^2}}}\left[({\varphi }_1(r)-{\varphi }_2(r)\right]·e^{-\lambda r}\hfill \end{array}$$

for both finite-size potentials [13,22], where

$$\begin{array}{cc}\hfill {\varphi }_1(r)& ={\displaystyle \frac{1}{-\kappa +\frac{m{c}^{2}Z\alpha }{\hslash c\lambda }}}\left[\left(\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }}\right)F_1(\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }}+1,1+2\gamma ,2\lambda r)·{(2\lambda r)}^{\gamma }\right.\hfill \\ \hfill & +\left.\left(\gamma +{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }}\right)s{F}_1(-\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }}+1,1-2\gamma ,2\lambda r)·{(2\lambda r)}^{-\gamma }\right],\hfill \\ \hfill {\varphi }_2(r)& =F_1(\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }},1+2\gamma ,2\lambda r)·{(2\lambda r)}^{\gamma }-s{F}_1(-\gamma -{\displaystyle \frac{EZ\alpha }{\hslash c\lambda }},1-2\gamma ,2\lambda r)·{(2\lambda r)}^{-\gamma },\hfill \\ \hfill s& ={\displaystyle \frac{\Gamma (1+2\gamma )·\Gamma (-\gamma -\frac{EZ\alpha }{\hslash c\lambda })}{\Gamma (1-2\gamma )·\Gamma (\gamma -\frac{EZ\alpha }{\hslash c\lambda })}}\hfill \end{array}$$

The solution of Dirac’s equation for the potential of a full sphere has been investigated by Pieper and Greiner [1]. The wave functions inside the core have been found to be (For the calculation of the hyperfine splitting, we aim to obtain an easy power series representation of G; unlike in [1], we thus derive a second-order differential equation for G and calculated F based on the solution for G):

$$\begin{array}{cc}\hfill G(r)& =A\sqrt{d-b{r}^2}{r}^{\mu }\sum _{i=0}^{\infty }{c}_i{r}^i\hfill \\ \hfill F(r)& =A{\displaystyle \frac{r^{\mu -1}}{{\sqrt{d-b{r}^2}}^3}}\sum _{i=0}^{\infty }{c}_{f,i}{r}^i.\hfill \end{array}$$

The coefficients a, b, d and the $c_i$ and $c_{f,i}$ as well as the characteristic exponent $\mu$ can be found in Appendix A, $A\in \mathbb{C}$ is defined by the normalization of the wave function.

The exact solution of Dirac’s equation for the potential of a spherical shell has been investigated by Pomeranchuk and Smorodynskii; the resulting wave functions are spherical Bessel functions [2]. They are—in our notation—given by:

$$\begin{array}{cc}\hfill G(r)& =A\sqrt{r}{J}_{\mu -\frac{1}{2}}(\sqrt{v}r)\hfill \\ \hfill F(r)& ={\displaystyle \frac{1}{u}}A[(\kappa +{\displaystyle \frac{1}{2}}){\displaystyle \frac{1}{\sqrt{r}}}{J}_{\mu -\frac{1}{2}}(\sqrt{v}r)+{\displaystyle \frac{1}{2}}\sqrt{vr}(J_{\mu -\frac{3}{2}}(\sqrt{v}r)-J_{\mu +\frac{1}{2}}(\sqrt{v}r))],\hfill \end{array}$$

where the coefficients u and v as well as the characteristic exponent $\mu$ are listed in Appendix A and $A\in \mathbb{C}$ is defined by the normalization of the wave function.

Figure 1. Left: Potential $V(r)$ for ${}^1\mathrm{H}$ generated by a point charge $Z=1$ and by homogeneously charged full spheres and spherical shells, both with $R_{\mathrm{H}}=0.83$ fm (radius taken from [24]). All three potentials match outside the nucleus, i.e., for $r\ge R_{\mathrm{H}}$. Right: Radial wave function $g(r)$ for the $1s$ state of the three hydrogen isotopes for the three potentials. Note that the difference due to the different nuclear charge models is similar to that caused by the different isotopes (isotope shift).

Figure 1. Left: Potential $V(r)$ for ${}^1\mathrm{H}$ generated by a point charge $Z=1$ and by homogeneously charged full spheres and spherical shells, both with $R_{\mathrm{H}}=0.83$ fm (radius taken from [24]). All three potentials match outside the nucleus, i.e., for $r\ge R_{\mathrm{H}}$. Right: Radial wave function $g(r)$ for the $1s$ state of the three hydrogen isotopes for the three potentials. Note that the difference due to the different nuclear charge models is similar to that caused by the different isotopes (isotope shift).

In order to construct the wave function for the finite-size potentials, it has to be ensured that $F(r)$ and $G(r)$ are continuous and differentiable at $r=R$. This leads to the condition

$${\displaystyle \frac{F_{r\le R}(R)}{F_{r\ge R}(R)}}={\displaystyle \frac{G_{r\le R}(R)}{G_{r\ge R}(R)}},$$

(8)

which allows us to determine the electronic eigenenergies numerically via bisection [1] for a given nuclear radius R. As expected, the eigenenergies obtained for a point-charge model (Coulomb potential) coincide for $Z=1$ (hydrogen) with the eigenenergies provided by Sommerfeld’s fine structure formula Equation (5) (see also

Table A3. Values of $E-m{c}^2$ (in eV) calculated with Sommerfeld’s fine structure formula for a point-like nuclear charge (reflecting the j degeneracy) and the numerically determined energy eigenvalues for the finite-size nuclear models for hydrogen ${}^1\mathrm{H}$.

State	Point Charge (Sommerfeld’s Formula)	Full Sphere	Spherical Shell
$1s_{1/2}$	−13.605874258219	−13.605874255666	−13.605874253623
$2s_{1/2}$	−3.401479885623	−3.401479885112	−3.401479885112
$2p_{1/2}$	−3.401479885623	−3.401479885623	−3.401479885623
$2p_{3/2}$	−3.401446016289	−3.401446016289	−3.401446016289
$3s_{1/2}$	−1.511763806324	−1.511763806324	−1.511763806324
$3p_{1/2}$	−1.511763806324	−1.511763806324	−1.511763806324
$3p_{3/2}$	−1.511750389026	−1.511750389026	−1.511750389026
$3d_{3/2}$	−1.511750389026	−1.511750389026	−1.511750389026
$3d_{5/2}$	−1.511745916689	−1.511745916689	−1.511745916689

in Appendix C.1) including degenerate $2s$ and $2p_{1/2}$ eigenenergies. Similar to the mass effect and the here not covered Lamb-shift [5,25] the introduction of finite-size nuclei is able to lift this degeneracy, but to much smaller extend. The resulting eigenenergies show a minor dependence on the nuclear radius ($<2·{10}^{-2}\mu$eV) restricted to the $1s$ ground state and $2s$ excited state (see also

Table A4. Calculated energy shift (in ${10}^{-9}$ eV) for hydrogen ${}^1\mathrm{H}$, deuterium ${}^2\mathrm{H}$ and tritium ${}^3\mathrm{H}$ for the different finite-size nuclear charge models compared with the values from Sommerfeld’s formula (point charge nucleus). Note that we focus on the effect of finite nuclear structure; the much larger mass effect ($\approx {10}^{-4}$ Ry) is not included here.

	State	Point Charge	Hydrogen ${}^1\mathrm{H}$	Deuterium ${}^2\mathrm{H}$	Tritium ${}^3\mathrm{H}$
R [fm]			0.83	2.13	1.76
full sphere	$1s$	−13605874258.219	+2.533	+17.360	+11.744
	$2s$	−3401479885.623	+0.511	+2.042	+1.532
spherical shell	$1s$	−13605874258.219	+4.595	+29.614	+19.913
	$2s$	−3401479885.623	+0.511	+3.574	+2.553

in Appendix C.1). Nevertheless, it allows us to calculate estimates for the isotope shifts of the eigenvalues for ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, and ${}^3\mathrm{H}$. which are in good agreement with values from numerical calculations reported in Ref. [4].

Figure 1, right shows the radial wave function $g(r)$ of the $1s$ state for hydrogen, deuterium and tritium for the three different core models near $r=0$. A pure Coulomb potential does not allow to distinguish between the wave functions of the different hydrogen isotopes, whereas the finite-size potentials show small, but clearly visible differences between the three hydrogen isotopes due to the different nuclear radii. The use of finite-size potentials leads to non-divergent wave functions for $r\to 0$, which have a finite value at $r=0$. For larger r, the wave functions become similar to each other.

3. Calculation of Relativistic Magnetic Dipole Hyperfine Splittings

After presenting the exact analytical solution of Dirac’s equation for three different core models, we aim to use the resulting wave functions to calculate fully analytically the relativistic hf splittings due to the interaction of the electron spin with that of the nuclei. The nuclei’s magnetic moment ${\mu }_I$ leads to a vector potential $\overrightarrow{A}(\overrightarrow{r})$. Instead of solving Dirac’s equation including this vector potential explicitly

$$[c\overrightarrow{\alpha }·\overrightarrow{p}-ec\overrightarrow{\alpha }·\overrightarrow{A}(\overrightarrow{r})+\beta m{c}^2+V(r)]\Psi =E\Psi ,$$

one can use first-order relativistic perturbation theory in order to calculate the hf splittings:

$$\Delta E_{\mathrm{hyperfine}}=-ec\langle \Psi |\overrightarrow{\alpha }·\overrightarrow{A}(\overrightarrow{r})|\Psi \rangle$$

(9)

For some finite-size models an exact treatment has been recently reported in literature [9]. For the model of a full sphere, i.e., a homogeneously charged ball as suggested by the liquid drop model, however, calculated hf splittings are only available based on numerically determined wave functions. Here we will show that the approach of Blügel et al. [6], allows for a straightforward extension of the fully analytical treatment also onto this model.

3.1. Breit’s Formula for the Relativistic Hyperfine Splitting of Point-Charge Nuclei

In case of a point-like nucleus, i.e., for a Coulomb potential and a vector potential $\overrightarrow{A}(\overrightarrow{r})={\displaystyle \frac{{\mu }_0}{4\pi }}\nabla \times {\displaystyle \frac{{\overrightarrow{\mu }}_I}{r}}={\displaystyle \frac{{\mu }_0}{4\pi }}{\overrightarrow{\mu }}_I\times {\displaystyle \frac{\overrightarrow{r}}{r^3}}$, large and small component from Equations (3) and (4) may be directly used to evaluate the hf splitting of relativistic atomic orbitals analytically [26,27]. Breit has derived simple formula for $1s$ and $2s$ orbitals of hydrogen [26], which can be used to evaluate alternative calculation schemes [9]. With ${\Delta }_0:={\displaystyle \frac{4}{3}}m{c}^2\alpha {(Z\alpha )}^3{g}_I{\displaystyle \frac{m}{m_p}}$ describing the non-relativistic limit for $1s$ electrons, the relativistic hyperfine splitting of the $1s$ state may be calculated via

$${\Delta }_{1s}^{\mathrm{B}}={\displaystyle \frac{1}{\gamma (2\gamma -1)}}·{\Delta }_0,$$

(10)

that of the 2s state via

$${\Delta }_{2s}^{\mathrm{B}}={\displaystyle \frac{1}{8}}{\sqrt{{\displaystyle \frac{2}{1+\gamma }}}}^3{\displaystyle \frac{1+\sqrt{2+2\gamma }}{\gamma ·(4{\gamma }^2-1)}}·{\Delta }_0,$$

(11)

which leads to a hf splitting of 1421.273 MHz and 177.665 MHz, respectively.

According to Ref. [27], the formula can be generalized; the hf splitting of any relativistic orbital of a hydrogen-like isotope with atomic number Z and nuclear spin I is given by

$$\Delta E_{\mathrm{hf}}/{\Delta }_0={\displaystyle \frac{3}{8}}{\displaystyle \frac{\kappa }{j(j+1)}}{\displaystyle \frac{2\kappa (\gamma +(n-|\kappa |))-N}{N^4\gamma (4{\gamma }^2-1)}}·[F(F+1)-I(I+1)-j(j+1)]$$

(12)

with $N=\sqrt{n^2-2(n-|\kappa |)(|\kappa |-\gamma )}$, whereby F denotes the total (sum of nuclear and electronic) angular momentum. From this, one can obtain exact reference data for the $2p$ orbitals of hydrogen. We find that the hf splitting of a $2p_{\frac{1}{2}}$ state is given by:

$${\Delta }_{2p_{\frac{1}{2}}}={\displaystyle \frac{1}{8}}{\sqrt{{\displaystyle \frac{2}{1+\gamma }}}}^3{\displaystyle \frac{1-\sqrt{2+2\gamma }}{\gamma (4{\gamma }^2-1)}}·{\Delta }_0,$$

(13)

which leads to ${\Delta }_{2p_{\frac{1}{2}}}=59.221$ MHz. For the $2p_{\frac{3}{2}}$ orbitals, we obtain

$${\Delta }_{2p_{\frac{3}{2},m_j}}={\displaystyle \frac{|m_j|}{5}}{\displaystyle \frac{1}{\gamma (2\gamma -1)}}·{\Delta }_0,$$

(14)

yielding 11.843 MHz and 35.530 MHz for $m_j=\pm$1/2 and $m_j=\pm$3/2, respectively, whereby these values describe the hf splitting along the symmetry axis of the p orbitals.

Notably, for non-point-like nuclear charge and magnetization densities $\rho (r)$ and $m(r)$, a direct evaluation of Equation (9) is no longer feasible. For s states, the FNS effects can be discussed by means of the non-relativistic Zemach formula [28]:

$$\Delta E_{\mathrm{hf}}^{\mathrm{Zemach}}=-{\Delta }_0·2Z\alpha m\int \int \rho ({\overrightarrow{r}}^{\prime })·|\overrightarrow{r}-{\overrightarrow{r}}^{\prime }|·m(\overrightarrow{r})d^{3}r{d}^3{r}^{\prime }$$

(15)

where the double-integral is the Zemach radius $r_{\mathrm{Z}}$ representing the size of the nucleus. Relativistic formulas require perturbation theory with respect to the external potential [29,30]. Assuming a homogeneously charged sphere model, FNS corrections of about 45 kHz for the nuclear charge and 15 kHz for the magnetization are reported for the ${}^1\mathrm{H}$ $1s$ state. However, since the FNS effect also alters the wave functions, it remains unclear if the two effects can actually be treated separately or provide a rather delicate mutual interaction, as also indicated by the Zemach formula Equation (15). In the following, based on relativistic wave functions obtained from the exact solution of Dirac’s equation, we will directly evaluate the influence of finite-size effects by calculating the hf splittings for different nuclear structure models.

3.2. Modelling Finite-Size Effects via the Relativistic form Function $S(r)$

In the general case, the hyperfine splitting due to a specific nucleus within a complex microscopic structure defines a tensor with isotropic contributions (given by the Fermi contact) and anisotropic terms, i.e., angular-dependent contributions, predominantly given by magnetic dipole-dipole interaction. In the experiment, the hf splittings are thus discussed with the help of a phenomenological spin hamiltonian containing several contributions [7]. By introducing the relativistic form function ${\displaystyle S(r)=\frac{2m{c}^2}{E+m{c}^2-V(r)}}$, which in general couples the large and small component via ${\psi }_S=S(r){\displaystyle \frac{\widehat{\overrightarrow{\sigma }}·\widehat{\overrightarrow{p}}}{2mc}}{\psi }_L$, Blügel et al. have shown that the hyperfine splitting Equation (9) for a radialsymmetric potential $V(r)$ consists of three terms [6],

$$\begin{array}{c}\hfill \Delta E_{\mathrm{hyperfine}}=\Delta E_{\mathrm{contact}}+\Delta E_{\mathrm{dipolar}}+\Delta E_{\mathrm{orbital}}\end{array}$$

In the case of point-like nuclear magnetic moments, they are given by

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{orbital}}& =-{\displaystyle \frac{e}{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\overrightarrow{\mu }}_I·\langle {\psi }_L|{\displaystyle \frac{S(r)}{r^3}}\overrightarrow{L}|{\psi }_L\rangle \hfill \\ \hfill \Delta E_{\mathrm{dipolar}}& ={\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}\langle {\psi }_L|{\displaystyle \frac{S(r)}{r^3}}[\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I-3(\overrightarrow{\sigma }·{\overrightarrow{e}}_r)({\overrightarrow{\mu }}_I·{\overrightarrow{e}}_r)]|{\psi }_L\rangle \hfill \\ \hfill \Delta E_{\mathrm{contact}}& =-{\displaystyle \frac{8\pi }{3}}{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}\langle {\psi }_L|S(r)\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I\delta (\overrightarrow{r})|{\psi }_L\rangle -{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}\langle {\psi }_L|{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial S}{\partial r}}[\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I-(\overrightarrow{\sigma }·\overrightarrow{e_r})(\overrightarrow{{\mu }_I}·\overrightarrow{e_r})]|{\psi }_L\rangle \hfill \\ \hfill & :=\Delta E_{\mathrm{contact},\delta }+\Delta {\overline{E}}_{\mathrm{contact}}\hfill \end{array}$$

(16)

and the relativistic form function $S(r)$ depends on the distribution of the nuclear charge, i.e., on the external potential $V(r)$. For a Coulomb potential $S(r)$ and its radial derivative read

$$\begin{array}{cc}\hfill S(r)& ={\displaystyle \frac{2m{c}^2}{m{c}^2+E+\frac{Z\alpha \hslash c}{r}}}={\displaystyle \frac{2r}{ϵr+r_{Th}}}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial S}{\partial r}}(r)& ={\displaystyle \frac{2r_{Th}}{{\left(r_{Th}+ϵr\right)}^2}}\hfill \end{array}$$

(18)

with $ϵ=ϵ(E)={\displaystyle \frac{E+m{c}^2}{m{c}^2}}$ and $r_{Th}={\displaystyle \frac{Z\alpha \hslash c}{m{c}^2}}$ being the Thomas radius, for hydrogen about the nuclear radius R (see also Figure 2); for large Z about ten times R [6].

Figure 2. $S(r)$ (top) for hydrogen ${}^1\mathrm{H}$, deuterium ${}^2\mathrm{H}$, and tritium ${}^3\mathrm{H}$ for all three nuclei models. $S(r)$ matches for all three potentials if $r\ge R$. The inset shows the non-relativistic case ($S(r)=1$) together with the relativistic form function of a pure Coulomb potential. The derivative $S^{\prime }(r)={\displaystyle \frac{ds}{dr}}$ of the relativistic form function is shown for hydrogen, deuterium, and tritium for all three nuclei models (bottom); ${\displaystyle \frac{\partial S}{\partial r}}$ matches for all three potentials if $r\ge R$.

Figure 2. $S(r)$ (top) for hydrogen ${}^1\mathrm{H}$, deuterium ${}^2\mathrm{H}$, and tritium ${}^3\mathrm{H}$ for all three nuclei models. $S(r)$ matches for all three potentials if $r\ge R$. The inset shows the non-relativistic case ($S(r)=1$) together with the relativistic form function of a pure Coulomb potential. The derivative $S^{\prime }(r)={\displaystyle \frac{ds}{dr}}$ of the relativistic form function is shown for hydrogen, deuterium, and tritium for all three nuclei models (bottom); ${\displaystyle \frac{\partial S}{\partial r}}$ matches for all three potentials if $r\ge R$.

The fact that Equation (16) depends on the large component exclusively allows us to evaluate it with scalar wave functions from the solution of Schrödinger’s equation (non-relativistic treatment) or from more elaborate equations, like the scalar relativistic approximation [31]. This does not mean, however, that the approach Equation (16) itself relies on a reduced Hamiltonian or any other kind of approximation. By construction, it is mathematically equivalent to the direct approach, where the hf interaction is determined by evaluating the expectation value Equation (9) with the help of the large and small components. This becomes clear if evaluating the resulting formula for the hf splitting analytically. For a $1s_{m_j=\mp \frac{1}{2}}$ state of hydrogen, e.g., the angular part of the contact contribution in Equation (16) becomes equal to $\pm {\displaystyle \frac{2}{3}}$ so that only a radial integral remains:

$$\begin{array}{cc}\hfill {\Delta }_{1s}=& {\displaystyle \frac{4}{3}}{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}{\int }_0^{\infty }{\displaystyle \frac{\partial S}{\partial r}}{|g(r)|}^{2}dr\hfill \\ \hfill & ={\displaystyle \frac{4(1-\gamma )}{\Gamma (2\gamma +1)}}\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{e^{-x}{x}^{2\gamma -2}}{{(x+2(1-\gamma ))}^2}}dx·{\Delta }_0\hfill \\ \hfill & ={\displaystyle \frac{1}{\gamma (2\gamma -1)}}·{\Delta }_0\hfill \end{array}$$

In the last step, we used partial integration and the identity $\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{e^{-x}{x}^{-\kappa }}{{(x+\kappa )}^2}}dx={\displaystyle \frac{1}{\kappa }}\Gamma (1-\kappa )$, yielding again Equation (10) and highlighting the equivalence of the two methods. The main difference between them is the following: The formalism via $S(r)$ allows a distinction between the orbital, dipolar, and contact contribution, while the direct evaluation of Equation (9) leads to a formula for the entire hyperfine splitting, i.e., the sum of all contributions.

In the following, we show that Blügel’s calculation scheme for the relativistic hyperfine splittings can be extended to finite-sizes nuclei, both with respect to the nuclear charge (i.e., the potential V(r)), as well as the related nuclear magnetic dipole distribution. For a homogeneously charged full sphere with radius R, we obtain

$$\begin{array}{c}\hfill S(r)=\left\{\begin{array}{cc}{\displaystyle \frac{2R}{ϵR+\left(\frac{3}{2}-\frac{r^2}{2R^2}\right)r_{Th}}}& ,\mathrm{if}r\le R\hfill \\ {\displaystyle \frac{2r}{ϵr+r_{Th}}}& ,\mathrm{if}r\ge R\hfill \end{array}\right.\end{array}$$

(19)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial r}}(r)=\left\{\begin{array}{cc}{\displaystyle \frac{2r_{Th}·r/R}{{\left(ϵR+\left(\frac{3}{2}-\frac{r^2}{2R^2}\right)r_{Th}\right)}^2}}& ,\mathrm{if}r\le R\hfill \\ {\displaystyle \frac{2r_{Th}}{{\left(r_{Th}+ϵr\right)}^2}}& ,\mathrm{if}r\ge R\hfill \end{array}\right.,\end{array}$$

(20)

whereas the potential of a charged spherical shell with radius R yields

$$\begin{array}{c}\hfill S(r)=\left\{\begin{array}{cc}{\displaystyle \frac{2R}{ϵR+r_{Th}}}& ,\mathrm{if}r\le R\hfill \\ {\displaystyle \frac{2r}{ϵr+r_{Th}}}& ,\mathrm{if}r\ge R\hfill \end{array}\right.\end{array}$$

(21)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial S}{\partial r}}(r)=\left\{\begin{array}{cc}0& ,\mathrm{if}r\le R\hfill \\ {\displaystyle \frac{2r_{Th}}{{\left(r_{Th}+ϵr\right)}^2}}& ,\mathrm{if}r\ge R\hfill \end{array}\right.\end{array}$$

(22)

The function $S(r)$ is shown in Figure 2 top for the $1s$ state for all three nuclei models (point-charge, full sphere, and spherical shell), and the three hydrogen isotopes ${}^1\mathrm{H}$, ${}^2\mathrm{H}$ und ${}^3\mathrm{H}$. Due to the fact that the potential $V(r)$ matches for all three core models outside the nucleus and thanks to almost identical ground state energies for all isotopes, there is no difference for $r\ge R$. For the point-like nucleus, $S(r)$ matches for all isotopes, whereas the finite-size potentials show differences for $r<R$ due to the different nuclear radii of the isotopes. Figure 2 bottom shows ${\displaystyle \frac{\partial S}{\partial r}}$ for all three hydrogen isotopes and core models for the $1s$ state. Again, the functions $S(r)$ match outside the core; differences between core models and isotopes are of the same size and restricted to the region inside the cores, $r<R$.

The effect of finite-size distribution of the nuclear magnetic moment is more evolved, as it requires explicit case distinction with respect to the vector potentials $\overrightarrow{A}(\overrightarrow{r})$ and derivation of additional inside-core terms not available from Ref. [6]. Apart from a point-like magnetic moment, two finite-size models shall be considered:

1.

In model 1, the magnetization density $m(r)$ is distributed homogeneously over a surface, more precisely over a spherical shell with nuclear radius R, resulting in the following vector potential [32]:

$$\overrightarrow{A}(\overrightarrow{r})={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{{\overrightarrow{\mu }}_I\times \overrightarrow{r}}{r^3}}\Theta (r-R)=\left\{\begin{array}{c}{\displaystyle 0,\mathrm{if}r<R}\\ {\displaystyle \frac{{\mu }_0}{4\pi }\frac{{\overrightarrow{\mu }}_I\times \overrightarrow{r}}{r^3},\mathrm{if}r\ge R}\end{array}\right.$$

(23)

2.

In model 2, the magnetization density is distributed homogeneously over a full sphere with nuclear radius R [32,33] yielding the vector potential:

$$\overrightarrow{A}(\overrightarrow{r})=\left\{\begin{array}{c}{\displaystyle \frac{{\mu }_0}{4\pi }\frac{{\overrightarrow{\mu }}_I\times \overrightarrow{r}}{R^3},\mathrm{if}r<R}\\ {\displaystyle \frac{{\mu }_0}{4\pi }\frac{{\overrightarrow{\mu }}_I\times \overrightarrow{r}}{r^3},\mathrm{if}r\ge R}\end{array}\right.$$

(24)

Note that both models are reasonable. Real distribution of the nuclear magnetization $m(r)$ is assumed to be somewhere intermediate between the two models [29,34,35].

The hyperfine splitting from interactions outside the nuclear radius can be calculated by reusing Equation (16), i.e., from Ref. [6], whereby the radial integrals have to be restricted to ${\langle \rangle }_R^{\infty }$. The inside-core contributions, however, have to be re-examined explicitly via ${\langle \rangle }_0^R$ from

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{hyperfine}}& =-ec\langle \Psi |\overrightarrow{\alpha }·\overrightarrow{A}(\overrightarrow{r})|\Psi \rangle \hfill \\ \hfill =& -{\displaystyle \frac{e}{m}}\langle {\psi }_L|S(r)\overrightarrow{A}(\overrightarrow{r})·\overrightarrow{p}|{\psi }_L\rangle \hfill \\ \hfill & -{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|S(r)\overrightarrow{\sigma }·(\nabla \times \overrightarrow{A}(\overrightarrow{r}))|{\psi }_L\rangle -{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|\overrightarrow{\sigma }·(\nabla S(r)\times \overrightarrow{A}(\overrightarrow{r}))|{\psi }_L\rangle \hfill \\ \hfill & -{\displaystyle \frac{e\hslash }{2mi}}\langle {\psi }_L|(\nabla ·\overrightarrow{A}(\overrightarrow{r}))S(r)|{\psi }_L\rangle -{\displaystyle \frac{e\hslash }{2mi}}\langle {\psi }_L|\overrightarrow{A}(\overrightarrow{r})·\nabla S(r)|{\psi }_L\rangle \hfill \end{array}$$

(25)

Like in the case of a point-like nuclear magnetic moment, the last two terms of Equation (25) vanish thanks to the definition of the vector potential ($\nabla ·\overrightarrow{A}=0$) and the spherical symmetry of the potential so that $\nabla S(r)\perp \overrightarrow{A}$.

In the case of model 1, the vector potential is zero within the sphere, so that almost all contributions are vanishing. Only the term with $\nabla \times \overrightarrow{A}(\overrightarrow{r})$ contributes a surface term. In contrast to the point-like case, however, it does not give rise to an anisotropic dipolar term but contributes to the contact term exclusively:

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{hyperfine}}^{\mathrm{model}1}(r<R)& =-{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|S(r)\overrightarrow{\sigma }·(\nabla \times \overrightarrow{A}(\overrightarrow{r}))|{\psi }_L\rangle \hfill \\ \hfill & =-{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}\langle {\psi }_L|S(r){\displaystyle \frac{1}{r^2}}\delta (r-R)[\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I-({\overrightarrow{e}}_r·{\overrightarrow{\mu }}_I)(\overrightarrow{\sigma }·{\overrightarrow{e}}_r)]|{\psi }_L\rangle \hfill \\ \hfill & =\Delta E_{\mathrm{contact}}^d\hfill \end{array}$$

(26)

For model 2, the vector potential inside the sphere is non-vanishing and depends on r so that more terms contribute. The first term of Equation (25) provides a modified orbital term:

$$\Delta E_{\mathrm{orbital}}=-{\displaystyle \frac{e}{m}}\langle {\psi }_L|S(r)\overrightarrow{A}(\overrightarrow{r})·\overrightarrow{p}|{\psi }_L\rangle =-{\displaystyle \frac{e}{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{1}{R^3}}{\overrightarrow{\mu }}_I·\langle {\psi }_L|S(r)\overrightarrow{L}|{\psi }_L\rangle$$

(27)

The remaining two terms can be summarized as the inside-sphere contact term. Again, $-{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|S(r)\overrightarrow{\sigma }·(\nabla \times \overrightarrow{A})|{\psi }_L\rangle$ does not give rise to an anisotropic dipolar term but contributes by $\Delta E_{\mathrm{contact}}^{\mathrm{d}}$ to the contact term exclusively

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{contact}}& =-{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|\overrightarrow{\sigma }·(\nabla S(r)\times \overrightarrow{A}(\overrightarrow{r}))|{\psi }_L\rangle -{\displaystyle \frac{e\hslash }{2m}}\langle {\psi }_L|S(r)\overrightarrow{\sigma }·(\nabla \times \overrightarrow{A}(\overrightarrow{r}))|{\psi }_L\rangle \hfill \\ \hfill & =-{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{1}{R^3}}\langle {\psi }_L|r{\displaystyle \frac{\partial S}{\partial r}}[\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I-({\overrightarrow{e}}_r·{\overrightarrow{\mu }}_I)(\overrightarrow{\sigma }·{\overrightarrow{e}}_r)]|{\psi }_L\rangle -{\displaystyle \frac{e\hslash }{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{1}{R^3}}\langle {\psi }_L|S\overrightarrow{\sigma }·{\overrightarrow{\mu }}_I|{\psi }_L\rangle \hfill \\ \hfill & =\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}+\Delta E_{\mathrm{contact}}^d\hfill \end{array}$$

(28)

and in total: $\Delta E_{\mathrm{hyperfine}}^{\mathrm{model}2}(r<R)=\Delta E_{\mathrm{orbital}}+{\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}+\Delta E_{\mathrm{contact}}^d$.

In this form, the formula appears to be applicable in density functional theory (DFT), allowing us to determine the hyperfine fields in complex microscopic structures with several hundred or even thousands of atoms. In the following, we will evaluate the formula analytically for one-electron systems, i.e., for hydrogen-like atoms. For this purpose, we use the exact solutions of Dirac’s equation for finite-size nuclear potentials presented in Section 2. By inserting Ansatz 1 for the wave functions into the integrals for Equation (16), we obtain for the region outside the nuclear core ($r>R$, whereby $R=0$ yields the case of point-like magnetic dipoles)

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{orbital}}=& -{\displaystyle \frac{e\hslash }{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[(m_j-{\displaystyle \frac{1}{2}})N_1^2+(m_j+{\displaystyle \frac{1}{2}})N_2^2\right]{\int }_R^{\infty }{\displaystyle \frac{S(r)}{r}}{|g(r)|}^{2}dr\hfill \\ \hfill \Delta E_{\mathrm{dipolar}}=& {\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2-N_2^2-3N_1^2{N}_{l,m_j-\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}{(x)}^2{x}^{2}dx\right.\hfill \\ \hfill & +3N_2^2{N}_{l,m_j+\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j+\frac{1}{2}}{(x)}^2{x}^{2}dx\hfill \\ \hfill & -\left.6N_1{N}_2{N}_{l,m_j-\frac{1}{2}}{N}_{l,m_j+\frac{1}{2}}{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}(x)P_{l,m_j+\frac{1}{2}}(x)x\sqrt{1-x^2}dx\right]{\int }_R^{\infty }{\displaystyle \frac{S(r)}{r}}{|g(r)|}^{2}dr\hfill \\ \hfill \Delta E_{\mathrm{contact},\delta }=& -{\displaystyle \frac{4}{3}}{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2{N}_{l,m_j-\frac{1}{2}}{\delta }_{m_j,\frac{1}{2}}-N_2^2{N}_{l,m_j+\frac{1}{2}}{\delta }_{m_j,-\frac{1}{2}}\right]{\int }_R^{\infty }{|g(r)|}^{2}S(r)\delta (r)dr\hfill \\ \hfill \Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}=& -{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2-N_2^2-N_1^2{N}_{l,m_j-\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}{(x)}^2{x}^{2}dx\right.\hfill \\ \hfill & +N_2^2{N}_{l,m_j+\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j+\frac{1}{2}}{(x)}^2{x}^{2}dx\hfill \\ \hfill & -\left.2N_1{N}_2{N}_{l,m_j-\frac{1}{2}}{N}_{l,m_j+\frac{1}{2}}{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}(x)P_{l,m_j+\frac{1}{2}}(x)x\sqrt{1-x^2}dx\right]{\int }_R^{\infty }{\displaystyle \frac{\partial S}{\partial r}}{|g(r)|}^{2}dr\hfill \end{array}$$

(29)

Here, $\Delta E_{\mathrm{orbital}}$ and $\Delta E_{\mathrm{dipolar}}$ contain the same radial integral, allowing the derivation of analytically exact rules for the ratio between orbital and dipolar hyperfine splittings for purely p-, d-, and f-like atomic states (see also [36]).

For finite-size distributed nuclear magnetic dipoles, the contributions for Equation (29) have to be accomplished by additional ones from inside the nuclear core ($r<R$, Equations (26)–(28)).

The magnetization density of a spherical shell (model 1) yields

$$\begin{array}{c}\hfill \Delta E_{\mathrm{orbital}}=0\\ \hfill \Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}=0\\ \hfill \Delta E_{\mathrm{contact}}^{\mathrm{d}}=& -{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2-N_2^2-N_1^2{N}_{l,m_j-\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}{(x)}^2{x}^{2}dx\right.\hfill \\ \hfill & +N_2^2{N}_{l,m_j+{\displaystyle \frac{1}{2}}}^2{\int }_{-1}^1{P}_{l,m_j+{\displaystyle \frac{1}{2}}}{(x)}^2{x}^{2}dx\hfill \\ \hfill & -\left.2N_1{N}_2{N}_{l,m_j-\frac{1}{2}}{N}_{l,m_j+\frac{1}{2}}{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}(x)P_{l,m_j+\frac{1}{2}}(x)x\sqrt{1-x^2}dx\right]S(R){|g(R)|}^2,\hfill \end{array}$$

whereby $P_{l,m}$ denote the associated Legendre-polynomials, and the factors $N_{l,m}$, as well as $N_1$, $N_2$, $N_3$, and $N_4$, are listed in

Table A1. Prefactors in the definition of the spherical harmonics.

$\mathit{j}$	$\mathit{\kappa }$	$\tilde{\mathit{l}}$	${\mathit{N}}_{\mathbf{1}}$	${\mathit{N}}_{\mathbf{2}}$	${\mathit{N}}_{\mathbf{3}}$	${\mathit{N}}_{\mathbf{4}}$
$l+{\displaystyle \frac{1}{2}}$	$-l-1$	$l+1$	$\sqrt{{\displaystyle \frac{l+m_j+\frac{1}{2}}{2l+1}}}$	$-\sqrt{{\displaystyle \frac{l-m_j+\frac{1}{2}}{2l+1}}}$	$\sqrt{{\displaystyle \frac{l-m_j+\frac{3}{2}}{2l+3}}}$	$\sqrt{{\displaystyle \frac{l+m_j+\frac{3}{2}}{2l+3}}}$
$l-{\displaystyle \frac{1}{2}}$	l	$l-1$	$\sqrt{{\displaystyle \frac{l-m_j+\frac{1}{2}}{2l+1}}}$	$\sqrt{{\displaystyle \frac{l+m_j+\frac{1}{2}}{2l+1}}}$	$\sqrt{{\displaystyle \frac{l+m_j-\frac{1}{2}}{2l-1}}}$	$-\sqrt{{\displaystyle \frac{l-m_j-\frac{1}{2}}{2l-1}}}$

in Appendix A.

The magnetization density of a full sphere (model 2) leads to:

$$\begin{array}{cc}\hfill \Delta E_{\mathrm{orbital}}=& -{\displaystyle \frac{e\hslash }{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[(m_j-\frac{1}{2})N_1^2+(m_j+{\displaystyle \frac{1}{2}})N_2^2\right]{\displaystyle \frac{1}{R^3}}{\int }_0^{R}S(r)r^2{|g(r)|}^{2}dr\hfill \\ \hfill \Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}=& -{\displaystyle \frac{e\hslash }{2m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2-N_2^2-N_1^2{N}_{l,m_j-\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}{(x)}^2{x}^{2}dx\right.\hfill \\ \hfill & +N_2^2{N}_{l,m_j+\frac{1}{2}}^2{\int }_{-1}^1{P}_{l,m_j+\frac{1}{2}}{(x)}^2{x}^{2}dx\hfill \\ \hfill & -\left.2N_1{N}_2{N}_{l,m_j-\frac{1}{2}}{N}_{l,m_j+\frac{1}{2}}{\int }_{-1}^1{P}_{l,m_j-\frac{1}{2}}(x)P_{l,m_j+\frac{1}{2}}(x)x\sqrt{1-x^2}dx\right]{\displaystyle \frac{1}{R^3}}{\int }_0^R{r}^3{\displaystyle \frac{\partial S}{\partial r}}{|g(r)|}^{2}dr\hfill \\ \hfill \Delta E_{\mathrm{contact}}^{\mathrm{d}}=& -{\displaystyle \frac{e\hslash }{m}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\mu }_{I,z}\left[N_1^2-N_2^2\right]{\displaystyle \frac{1}{R^3}}{\int }_0^R{r}^{2}S(r){|g(r)|}^{2}dr\hfill \end{array}$$

It is important to note that in the case of the finite nuclear radius $R>0$, the derivation of a simple analytical formula for the hyperfine splitting, as in the case of point-charge nuclei [26,27], is no longer possible. The remaining radial integrals are evaluated numerically by using special functions and integrational routines provided by gnu scientific library [37], whereby via $S(r)$, the eigenenergies of the investigative states enter the formula. The values for the Coulomb potential give Sommerfeld’s fine structure formula. The eigenvalues for finite nuclear-charge models differ significantly only in the case of $1s$ and $2s$ by ${10}^{-9}$ eV and by ${10}^{-10}$ eV, respectively (see also Table A3 and Table A4). For the sake of numerical stability, we use Sommerfeld’s formula in our calculations.

4. Results and Discussion

4.1. FNS Effects onto the Hyperfine Fields of s States

For s states, the angular integrals for the orbital and dipolar terms vanish, so that the hyperfine splittings of s-electrons is determined by the contact terms exclusively. The corresponding energy correction. $\Delta E_{\mathrm{contact}}$ can be split up according to

$$\Delta E_{\mathrm{hyperfine}}\stackrel{s\mathrm{state}}{=}\Delta E_{\mathrm{contact}}=\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}+\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}+\Delta E_{\mathrm{contact}}^{\mathrm{d}}+\Delta E_{\mathrm{contact},\delta }$$

(30)

where $\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}$, $\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}$ refer to the radial integrals evaluated in- and outside the nuclear core region, defined by the nuclear radius R, for better comparison also in the case of point-like nuclei. $\Delta E_{\mathrm{contact}}^{\mathrm{d}}$ provides a core contribution that appears in the case of finite-size nuclear magnetic dipole distribution exclusively. $\Delta E_{\mathrm{contact},\delta }$, in contrast, only contributes if finite-size nuclear charges are combined with point-like nuclear magnetic dipoles.

Table 1 lists the contributions of $\Delta E_{\mathrm{contact}}$ for the $1s$ state of hydrogen ${}^1\mathrm{H}$ for any combination of the three different nuclear charge and three different nuclear magnetic moment models. Corresponding data for the ${}^1\mathrm{H}$ $2s$ state (see also

Table A5. Contributions to the hyperfine splitting (in MHz) of the $2s$ state of hydrogen ${}^1\mathrm{H}$ for the different nuclear charge ($\rho$) and nuclear magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$
	point charge	177.665	65.899	111.766	-	-
point-like	full sphere	177.662	15.771	111.766	-	50.125
	spherical shell	177.661	0	111.766	-	65.896
	point charge	177.661	12.268	111.766	53.627	-
full sphere	full sphere	177.660	7.057	111.766	58.837	-
	spherical shell	177.659	0	111.766	65.894	-
	point charge	177.659	-	111.766	65.894	-
spherical shell	full sphere	177.659	-	111.766	65.894	-
	spherical shell	177.659	-	111.766	65.894	-

) as well as for the $1s$ and $2s$ states of the other hydrogen isotopes (${}^2\mathrm{H}$, ${}^3\mathrm{H}$) and ionized helium atoms (${}^3\mathrm{He}^{+}$) are compiled in

Table A8. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $1s$ state of hydrogen ${}^1\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	9.417	3.493	5.924	0	-
point-like	full sphere	9.417	0.836	5.924	2.657	-
	spherical shell	9.417	0	5.924	3.493	-
	point charge	9.417	0.650	5.924	-	2.843
full sphere	full sphere	9.417	0.374	5.924	-	3.119
	spherical shell	9.417	0	5.924	-	3.493

,

Table A9. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $2s$ state of hydrogen ${}^1\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	1.178	0.437	0.741	0	-
point-like	full sphere	1.178	0.105	0.741	0.332	-
	spherical shell	1.178	0	0.741	0.437	-
	point charge	1.178	0.081	0.741	-	0.355
full sphere	full sphere	1.178	0.047	0.741	-	0.390
	spherical shell	1.178	0	0.741	-	0.437

,

Table A10. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $1s$ state of deuterium ${}^2\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	1.446	0.871	0.575	0	-
point-like	full sphere	1.446	0.145	0.575	0.726	-
	spherical shell	1.446	0	0.575	0.870	-
	point charge	1.446	0.117	0.575	-	0.753
full sphere	full sphere	1.446	0.062	0.575	-	0.808
	spherical shell	1.446	0	0.575	-	0.870

,

Table A11. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $2s$ state of deuterium ${}^2\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	0.181	0.109	0.072	0	-
point-like	full sphere	0.181	0.018	0.072	0.091	-
	spherical shell	0.181	0	0.072	0.109	-
	point charge	0.181	0.015	0.072	-	0.094
full sphere	full sphere	0.181	0.008	0.072	-	0.101
	spherical shell	0.181	0	0.072	-	0.109

,

Table A12. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $1s$ state of tritium ${}^3\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	10.045	5.582	4.463	0	-
point-like	full sphere	10.044	1.015	4.463	4.566	-
	spherical shell	10.044	0	4.463	5.581	-
	point charge	10.045	0.815	4.463	-	4.766
full sphere	full sphere	10.044	0.441	4.463	-	5.140
	spherical shell	10.044	0	4.463	-	5.581

,

Table A13. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $2s$ state of tritium ${}^3\mathrm{H}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	1.256	0.698	0.558	0	-
point-like	full sphere	1.256	0.127	0.558	0.571	-
	spherical shell	1.256	0	0.558	0.698	-
	point charge	1.256	0.102	0.558	-	0.596
full sphere	full sphere	1.256	0.055	0.558	-	0.643
	spherical shell	1.256	0	0.558	-	0.698

,

Table A14. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $1s$ state of ${}^3\mathrm{He}^{+}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	−57.405	−23.665	−33.740	0	-
point−like	full sphere	−57.399	−5.379	−33.739	−18.280	-
	spherical shell	−57.396	0	−33.739	−23.657	-
	point charge	−57.399	−4.210	−33.740	-	−19.449
full sphere	full sphere	−57.396	−2.392	−33.739	-	−21.264
	spherical shell	−57.395	0	−33.739	-	−23.656

and

Table A15. The contributions to the hyperfine splitting (in ${10}^{-25}$ J) of the $2s$ state of ${}^3\mathrm{He}^{+}$ for three different charge ($\rho$) and two different magnetization (m) density models.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}\mathbf{,}\mathit{\delta }}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$
	point charge	−7.177	−2.959	−4.218	0	-
point-like	full sphere	−7.176	−0.673	−4.218	−2.285	-
	spherical shell	−7.176	0	−4.218	−2.958	-
	point charge	−7.176	−0.526	−4.418	-	−2.432
full sphere	full sphere	−7.175	−0.299	−4.218	-	−2.659
	spherical shell	−7.175	0	−4.218	-	−2.957

, in Appendix C.3 and provide qualitatively the same results.

The contribution $\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}$ matches for all nuclear structure models, due to the fact that the wave functions are almost identical outside the core region. $\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}$ is equal to zero for the spherical shell potential since ${\displaystyle \frac{\partial S}{\partial r}}$ vanishes for $r<R$ in this model. For the full sphere potential, $\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{in}}$ is always smaller than for the point-like nucleus. For models with finite-size nuclear magnetic moment, this is compensated by the $\Delta E_{\mathrm{contact}}^d$ term. For models with spatially distributed nuclear charges but magnetic point-dipole (first three lines in Table 1), the total value is balanced by $\Delta E_{\mathrm{contact},\delta }$, which, for these mixed models, substitutes for the $\Delta E_{\mathrm{contact}}^d$ term. As $\kappa =-1$ is required for $g(0)\ne 0$, only s states are able to fulfill $\Delta E_{\mathrm{contact},\delta }\ne 0$. We obtain ${|g(0)|}^{2}S(0)=2{|A|}^2{\displaystyle \frac{m{c}^2}{\hslash c}}$ for a full-sphere distributed and ${|g(0)|}^{2}S(0)=2{|A|}^2{\displaystyle \frac{m{c}^2}{\hslash c}}({\displaystyle \frac{E-m{c}^2}{\hslash c}}+{\displaystyle \frac{Z\alpha }{R}})$ for a spherical-shell distributed nuclear charge, so that $\Delta E_{\mathrm{contact},\delta }$ is larger for the latter. This also can be rationalized by considering Figure 2 bottom, showing that $S(0)$ is by definition larger for the spherical shell than for the full sphere potential, which provides a further decrease in $S(r)$ inside the core region.

Table 1. Contributions to the hf splitting $\Delta E_{\mathrm{hyperfine}}$ (in MHz) of the $1s$ state of hydrogen ${}^1\mathrm{H}$ for the different nuclear charge ($\rho$) and nuclear magnetization (m) density models. For better comparison also in case of point-like nuclei, $\Delta E_{\mathrm{hyperfine}}$ is split up into outside- ($R>r$) and inside-core ($R<r$) contributions. $\Delta E_{\mathrm{contact}}^d$ and $\Delta E_{\mathrm{contact},\delta }$ are explicit FNS-corrections defined in Equations (30) and (29).

Model $\mathit{m}$	Model $\mathit{\rho }$	$\Delta {\mathit{E}}_{\mathrm{hyperfine}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{in}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{contact}}^{\mathit{d}}$	$\Delta {\mathit{E}}_{\mathrm{contact},\mathit{\delta }}$
	point charge	1421.273	527.173	894.100	-	-
point-like	full sphere	1421.248	126.165	894.100	-	400.983
	spherical shell	1421.243	0	894.100	-	527.143
	point charge	1421.240	98.141	894.100	428.999	-
full sphere	full sphere	1421.227	56.456	894.100	470.671	-
	spherical shell	1421.220	0	894.100	527.120	-
	point charge	1421.228	-	894.100	527.129	-
spherical shell	full sphere	1421.228	-	894.100	527.128	-
	spherical shell	1421.228	-	894.100	527.128	-

Table 1. Contributions to the hf splitting $\Delta E_{\mathrm{hyperfine}}$ (in MHz) of the $1s$ state of hydrogen ${}^1\mathrm{H}$ for the different nuclear charge ($\rho$) and nuclear magnetization (m) density models. For better comparison also in case of point-like nuclei, $\Delta E_{\mathrm{hyperfine}}$ is split up into outside- ($R>r$) and inside-core ($R<r$) contributions. $\Delta E_{\mathrm{contact}}^d$ and $\Delta E_{\mathrm{contact},\delta }$ are explicit FNS-corrections defined in Equations (30) and (29).

Model $\mathit{m}$	Model $\mathit{\rho }$	$\Delta {\mathit{E}}_{\mathrm{hyperfine}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{in}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{contact}}^{\mathit{d}}$	$\Delta {\mathit{E}}_{\mathrm{contact},\mathit{\delta }}$
	point charge	1421.273	527.173	894.100	-	-
point-like	full sphere	1421.248	126.165	894.100	-	400.983
	spherical shell	1421.243	0	894.100	-	527.143
	point charge	1421.240	98.141	894.100	428.999	-
full sphere	full sphere	1421.227	56.456	894.100	470.671	-
	spherical shell	1421.220	0	894.100	527.120	-
	point charge	1421.228	-	894.100	527.129	-
spherical shell	full sphere	1421.228	-	894.100	527.128	-
	spherical shell	1421.228	-	894.100	527.128	-

Although the total values are built up by many different contributions, their balancing is achieved to a large extent, so that the different FNS models give almost identical results. Besides the unique models with analogous distribution of nuclear charge and magnetic moment, some of the mixed models promise a reasonable description. As we will see, providing the splitting into different contributions, isotropic and angular dependent, from outside and inside the core region, the present method allows to rationalize the small, but still apparent model-dependent differences with details of the nuclear structure.

4.2. Influence of the Reduced Mass onto the FNS Effect and Comparison with QED Results

Table 2 lists the hyperfine splitting frequencies for the s-like ground states as well as for some excited states of the three hydrogen isotopes and singly positive charged helium ions calculated in a fully relativistic scheme. A quite good agreement with experimental data is already obtained for point-charge nuclei. Notably, this agreement becomes worse if the finite-size effects of nuclear charge and nuclear magnetic moment are taken into account. This observation is in-line with modern QED calculations showing that also other effects, like radiative corrections, recoil energy, and those of reduced mass, are required to reproduce the experimental data [15,16,38].

Strictly seen, electron (mass $m_e$) and nuclear core (mass $m_N$) define a relativistic two-body problem. In classical mechanics, such a problem can be straightforwardly solved by introducing the reduced mass $\mu ={\displaystyle \frac{m_e}{1+m_e/m_N}}$. Table 2 shows that following Ref. [9] the rigorous use of a reduced mass $\mu$ in Dirac’s equation is actually able to provide better agreement with experiment, but only for ${}^1\mathrm{H}$ and in connection with point-charge nuclei.

If finite-size effects of nuclear charge and nuclear magnetic moment are taken into account, the corresponding corrections appear to be overestimated, cf. Table 2. Obviously, the classical concept of the reduced mass has to be applied with care in relativistic quantum mechanics. As shown in Ref. [39], the reduced mass $\mu$ tends to contain some contributions from self-energy diagrams, making an estimate for the FNS effect alone difficult to achieve. In this work, we thus do not only follow the rigorous reduced-mass approach used in Ref. [9] but also investigate the influence of finite nuclear size effects without reduced mass, i.e., we simply keep $m=m_e$ in Dirac’s equation.

Using the reduced mass, we actually obtain FNS-induced shifts very similar to those reported recently in Kuzmenko et al. [9] for the $1s$ states of the three hydrogen isotopes and${}^3\mathrm{He}^{+}$, see also Table 3. Using the spherical shell model, we are able to reproduce the hydrogen data of Ref. [9] more or less exactly. The moderate deviations for the full shell model can be explained by the numerical solution of Dirac’s equation in Ref. [9], whereas in this work, analytically exact solutions are also used for this model.

Table 2. Calculated hyperfine splittings (in MHz) for s-like one-electron systems. For the calculated values, different charge ($\rho$) and magnetization (m) models are used with and without reduced mass (for mixed models, see

Table A7. Calculated hyperfine splittings (in MHz) for s-like one-electron systems. For the calculated values, different charge ($\rho$) and magnetization (m) models are used. Experimental data taken from [40] ($1s$ state of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$)), [41] (${}^1\mathrm{H}$ $2s$), [42] (${}^2\mathrm{H}$ $2s$), [43,44] (${}^3\mathrm{He}^{+}$$1s$) and [45] (${}^3\mathrm{He}^{+}$$2s$).

Atom	State	Exp *	$\mathit{\rho }$	Point Charge			Full Sphere			Spherical Shell
			$\mathit{m}$	Point	Sphere	Shell	Point	Sphere	Shell	Point	Sphere	Shell
${}^1\mathrm{H}$	$1s$	1420.406		1421.273	1421.240	1421.228	1421.248	1421.227	1421.228	1421.243	1421.220	1421.228
${}^1\mathrm{H}$	$2s$	177.557		177.665	177.661	177.659	177.662	177.660	177.659	177.660	177.659	177.659
${}^2\mathrm{H}$	$1s$	218.556		218.174	218.161	218.162	218.164	218.156	218.162	218.157	218.153	218.162
${}^2\mathrm{He}^{+}$	$2s$	27.283		27.273	27.271	27.271	27.271	27.270	27.271	27.270	27.270	27.271
${}^3\mathrm{H}$	$1s$	1516.701		1515.987	1515.911	1515.886	1515.931	1515.883	1515.886	1515.886	1515.865	1515.886
${}^3\mathrm{H}$	$2s$	-		189.505	189.495	189.511	189.498	189.492	189.511	189.492	189.490	189.511
${}^3\mathrm{He}^{+}$	$1s$	−8665.650		−8663.502	−8662.533	−8662.210	−8662.791	−8662.173	−8662.210	−8662.209	−8661.951	−8662.210
${}^3\mathrm{He}^{+}$	$2s$	−1083.355		−1083.082	−1082.961	−1082.920	−1082.993	−1082.916	−1082.920	−1082.920	−1082.888	−1082.920

* The experimental value is multiplied by $·{\displaystyle \frac{2}{2I+1}}$.

in Appendix C.2.) Experimental data taken from [40] ($1s$ state of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$)), [41] (${}^1\mathrm{H}$ $2s$), [42] (${}^2\mathrm{H}$ $2s$), [43,44] (${}^3\mathrm{He}^{+}$ $1s$), [45] (${}^3\mathrm{He}^{+}$$2s$), and [46] (${}^1\mathrm{H}$ $2p_{1/2}$). Using electron mass $m_e$ and assuming a point-like nuclei, the values for ${}^1\mathrm{H}$ $1s$, $2s$ as well as for the three related $2p$ states, are identical with the prediction of Breit’s relativistic formula (cf. Equations (10), (11), (13) and (14)).

Atom	State	Exp *	Model with Reduced Mass $\mathit{\mu }$			Model with Electron Mass ${\mathit{m}}_{\mathit{e}}$
			Point-like	Full Sphere	Spherical Shell	Point-like	Full Sphere	Spherical Shell
${}^1\mathrm{H}$	$1s$	1420.406	1420.499	1418.918	1418.910	1421.273	1421.227	1421.228
${}^1\mathrm{H}$	$2s$	177.557	177.558	177.389	177.388	177.665	177.659	177.659
${}^2\mathrm{H}$	$1s$	218.556	218.115	217.981	217.978	218.174	218.156	218.162
${}^2\mathrm{H}$	$2s$	27.283	27.266	27.251	27.251	27.273	27.270	27.271
${}^3\mathrm{H}$	$1s$	1516.701	1515.711	1515.076	1515.059	1515.987	1515.883	1515.886
${}^3\mathrm{H}$	$2s$	—	189.471	189.410	189.408	189.505	189.492	189.511
${}^3\mathrm{He}^{+}$	$1s$	−8665.650	−8661.926	−8657.707	−8657.485	−8663.502	−8662.173	−8662.210
${}^3\mathrm{He}^{+}$	$2s$	−1083.355	−1082.885	−1082.357	−1082.428	−1083.082	−1082.916	−1082.920
${}^1\mathrm{H}$	$2p_{1/2}$	59.22	59.189	59.203	59.201	59.221	59.244	59.241
${}^1\mathrm{H}$	$2p_{3/2,\pm 1/2}$	—	11.834	11.872	11.850	11.843	11.871	11.854
${}^1\mathrm{H}$	$2p_{3/2,\pm 3/2}$	—	35.511	35.605	35.549	35.530	35.613	35.561

* The experimental values are multiplied by $·{\displaystyle \frac{2}{2I+1}}$.

Table 2. Calculated hyperfine splittings (in MHz) for s-like one-electron systems. For the calculated values, different charge ($\rho$) and magnetization (m) models are used with and without reduced mass (for mixed models, see Table A7 in Appendix C.2.) Experimental data taken from [40] ($1s$ state of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$)), [41] (${}^1\mathrm{H}$ $2s$), [42] (${}^2\mathrm{H}$ $2s$), [43,44] (${}^3\mathrm{He}^{+}$ $1s$), [45] (${}^3\mathrm{He}^{+}$$2s$), and [46] (${}^1\mathrm{H}$ $2p_{1/2}$). Using electron mass $m_e$ and assuming a point-like nuclei, the values for ${}^1\mathrm{H}$ $1s$, $2s$ as well as for the three related $2p$ states, are identical with the prediction of Breit’s relativistic formula (cf. Equations (10), (11), (13) and (14)).

Atom	State	Exp *	Model with Reduced Mass $\mathit{\mu }$			Model with Electron Mass ${\mathit{m}}_{\mathit{e}}$
			Point-like	Full Sphere	Spherical Shell	Point-like	Full Sphere	Spherical Shell
${}^1\mathrm{H}$	$1s$	1420.406	1420.499	1418.918	1418.910	1421.273	1421.227	1421.228
${}^1\mathrm{H}$	$2s$	177.557	177.558	177.389	177.388	177.665	177.659	177.659
${}^2\mathrm{H}$	$1s$	218.556	218.115	217.981	217.978	218.174	218.156	218.162
${}^2\mathrm{H}$	$2s$	27.283	27.266	27.251	27.251	27.273	27.270	27.271
${}^3\mathrm{H}$	$1s$	1516.701	1515.711	1515.076	1515.059	1515.987	1515.883	1515.886
${}^3\mathrm{H}$	$2s$	—	189.471	189.410	189.408	189.505	189.492	189.511
${}^3\mathrm{He}^{+}$	$1s$	−8665.650	−8661.926	−8657.707	−8657.485	−8663.502	−8662.173	−8662.210
${}^3\mathrm{He}^{+}$	$2s$	−1083.355	−1082.885	−1082.357	−1082.428	−1083.082	−1082.916	−1082.920
${}^1\mathrm{H}$	$2p_{1/2}$	59.22	59.189	59.203	59.201	59.221	59.244	59.241
${}^1\mathrm{H}$	$2p_{3/2,\pm 1/2}$	—	11.834	11.872	11.850	11.843	11.871	11.854
${}^1\mathrm{H}$	$2p_{3/2,\pm 3/2}$	—	35.511	35.605	35.549	35.530	35.613	35.561

* The experimental values are multiplied by $·{\displaystyle \frac{2}{2I+1}}$.

Table 3. Residual difference (in kHz) between experiment and hf values calculated by QED compared with finite-size induced changes $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}\end{array}$ (=difference w.r.t. to the respective point-like case) calculated in this work (using electron mass $m_e$ and reduced mass $\mu$) and in Ref. [9] (reduced mass $\mu$). 4th order ${\mathrm{QED}}_{(4)}$, inelastic FNS (nuclear recoil and polarization) including corrected ${\mathrm{QED}}_{(4)}^{+}$, and experiment taken from (${}^1\mathrm{H}$) [19], (${}^2\mathrm{H}$) [47,48], (${}^3\mathrm{H}$) [14], and (${}^3\mathrm{He}^{+}$) [49]. For tritium ${}^3\mathrm{H}$, nuclear recoil is the dominating inelastic correction and has been estimated from formulas given in Ref. [49].

Atom	State			Full Sphere			Spherical Shell
				This Work		Ref. [9]	This Work		Ref. [9]
		${\mathit{E}}_{\mathbf{hf}}^{\mathbf{exp}}\mathbf{-}{\mathit{E}}_{\mathbf{hf}}^{{\mathbf{QED}}_{\mathbf{(}\mathbf{4}\mathbf{)}}}$	${\mathit{E}}_{\mathbf{hf}}^{\mathbf{exp}}\mathbf{-}{\mathit{E}}_{\mathbf{hf}}^{{\mathbf{QED}}_{\mathbf{(}\mathbf{4}\mathbf{)}}^{\mathbf{+}}}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}{\mathit{m}}_{\mathit{e}}\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}{\mathit{m}}_{\mathit{e}}\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$
${}^1\mathrm{H}$	$1s$	−47	−58	−46	−1582	−1604	−45	−1590	−1591
${}^1\mathrm{H}$	$2s$	−6	−7	−5	−169	—	−6	−170	—
$i^2$H	$1s$	+30	+1	-18	−134	−142	−12	−137	−137
${}^2\mathrm{H}$	$2s$	+4	0	−3	−15	—	−2	−15	—
${}^3\mathrm{H}$	$1s$	−59	(−85)	−104	−635	−682	−101	−652	−653
${}^3\mathrm{H}$	$2s$	—	—	−13	−61	—	−13	−63	—
${}^3\mathrm{He}^{+}$	$1s$	+1841	+1701	+1329	+4219	+5148	+1292	+4442	+4870
${}^3\mathrm{He}^{+}$	$2s$	+230	+213	+166	+528	—	+162	+447	—
${}^1\mathrm{H}$	$2p_{1/2}$	—	—	+23	+14	—	+20	+12	—
${}^1\mathrm{H}$	$2p_{3/2,\pm 1/2}$	—	—	+28	+38	—	+11	+16	—
${}^1\mathrm{H}$	$2p_{3/2,\pm 3/2}$	—	—	+83	+94	—	+31	+38	—

Table 3. Residual difference (in kHz) between experiment and hf values calculated by QED compared with finite-size induced changes $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}\end{array}$ (=difference w.r.t. to the respective point-like case) calculated in this work (using electron mass $m_e$ and reduced mass $\mu$) and in Ref. [9] (reduced mass $\mu$). 4th order ${\mathrm{QED}}_{(4)}$, inelastic FNS (nuclear recoil and polarization) including corrected ${\mathrm{QED}}_{(4)}^{+}$, and experiment taken from (${}^1\mathrm{H}$) [19], (${}^2\mathrm{H}$) [47,48], (${}^3\mathrm{H}$) [14], and (${}^3\mathrm{He}^{+}$) [49]. For tritium ${}^3\mathrm{H}$, nuclear recoil is the dominating inelastic correction and has been estimated from formulas given in Ref. [49].

Atom	State			Full Sphere			Spherical Shell
				This Work		Ref. [9]	This Work		Ref. [9]
		${\mathit{E}}_{\mathbf{hf}}^{\mathbf{exp}}\mathbf{-}{\mathit{E}}_{\mathbf{hf}}^{{\mathbf{QED}}_{\mathbf{(}\mathbf{4}\mathbf{)}}}$	${\mathit{E}}_{\mathbf{hf}}^{\mathbf{exp}}\mathbf{-}{\mathit{E}}_{\mathbf{hf}}^{{\mathbf{QED}}_{\mathbf{(}\mathbf{4}\mathbf{)}}^{\mathbf{+}}}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}{\mathit{m}}_{\mathit{e}}\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}{\mathit{m}}_{\mathit{e}}\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{\mu }\mathbf{)}\end{array}$
${}^1\mathrm{H}$	$1s$	−47	−58	−46	−1582	−1604	−45	−1590	−1591
${}^1\mathrm{H}$	$2s$	−6	−7	−5	−169	—	−6	−170	—
$i^2$H	$1s$	+30	+1	-18	−134	−142	−12	−137	−137
${}^2\mathrm{H}$	$2s$	+4	0	−3	−15	—	−2	−15	—
${}^3\mathrm{H}$	$1s$	−59	(−85)	−104	−635	−682	−101	−652	−653
${}^3\mathrm{H}$	$2s$	—	—	−13	−61	—	−13	−63	—
${}^3\mathrm{He}^{+}$	$1s$	+1841	+1701	+1329	+4219	+5148	+1292	+4442	+4870
${}^3\mathrm{He}^{+}$	$2s$	+230	+213	+166	+528	—	+162	+447	—
${}^1\mathrm{H}$	$2p_{1/2}$	—	—	+23	+14	—	+20	+12	—
${}^1\mathrm{H}$	$2p_{3/2,\pm 1/2}$	—	—	+28	+38	—	+11	+16	—
${}^1\mathrm{H}$	$2p_{3/2,\pm 3/2}$	—	—	+83	+94	—	+31	+38	—

We furthermore extend our calculation on the respective $2s$ and $2p$ excited states. For the $2p$ states, we find only moderate differences if reduced masses are taken into account in Dirac’s equation or not. For the $1s$ and $2s$ states, however, the influence of the FNS models is essentially reduced if the calculations are performed with electron mass $m_e$, for ${}^3\mathrm{He}^{+}$ by about 2 MHz, for the ${}^1\mathrm{H}$ $1s$ ground state by nearly two orders of magnitude from roughly $-1.6$ MHz to $-46$ kHz. This value coincides perfectly with the value of $-47$ kHz deduced in fourth-order QED studies [14]. The difference $E_{\mathrm{hf}}^{\exp }-E_{\mathrm{hf}}^{\mathrm{QED}}$ between precise experimental data and QED-predicted hf splittings is usually discussed as a measure for the elastic FNS-effect due to the spatial distribution of the nuclear charge and magnetic moments. For low-Z nuclei, however, the ${\mathrm{QED}}_{(4)}$ values have to be accomplished by inelastic effects of nuclear recoil and nuclear polarization [47,49,50,51]. The effect of nuclear polarization is nearly negligible (<2 kHz) for ${}^1\mathrm{H}$ and ${}^3\mathrm{H}$, but becomes maximum for deuterium ${}^2\mathrm{H}$, where it amounts to 43 kHz [47]. In Table 3 and respective references, the forth order ${\mathrm{QED}}_{(4)}$ data was corrected by these values. As a result, taking $E_{\mathrm{hf}}^{\exp }-E_{\mathrm{hf}}^{{\mathrm{QED}}_{(4)}^{+}}$ as a reference, for hydrogen the calculated $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}(m_e)\end{array}$ data is not only able to explain the size of the remaining effect of elastic FNS in the <100 kHz regime, but also reflects the trends with respect to isotopes and the $1s$ ground and $2s$ excited states even in the critical case of deuterium ${}^2\mathrm{H}$. Here, the model of a spherical shell fits better to the experimentally expected very small, nearly vanishing elastic FNS-shift, but with −12 kHz the value for the $1s$ ground state appears to be still too large. Also for ${}^3\mathrm{He}^{+}$, the reference data has to include contributions from inelastic FNS effects. Nuclear polarisation accounts for 25 kHz, whereas nuclear recoil contributes in total with 115 kHz [49]. Independent from these corrections, with $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}(m_e)\end{array}\approx 1300$ kHz most of the deviations between QED and experiment can be explained by the elastic FNS contributions of nuclear charge and nuclear magnetic moment.

Further interesting insight is obtained by extending this comparative study to the mixed models; see Table 4 and

Table A6. Contributions to the hyperfine splitting (in MHz) of the $2p_{1/2}$ state of hydrogen ${}^1\mathrm{H}$ for different nuclear charge ($\rho$) and nuclear magnetization (m) models. (≈0 denotes very small numbers below 0.1 Hz.) $\Delta E_{\mathrm{orbital}}^{\mathrm{out}}=\Delta E_{\mathrm{dipolar}}^{\mathrm{out}}$ reflects the fact that for $p_{1/2}$ orbitals these quantities are analytically identical (see also Ref. [36]). Note that $\Delta E_{\mathrm{contact},\delta }$ is zero for all models, also in case of the magnetic-dipole models.

Model$\mathit{m}$	Model$\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hyperfine}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{in}}$	$\mathbf{\Delta }{\overline{\mathit{E}}}_{\mathbf{contact}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{contact}}^{\mathit{d}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{dipolar}}^{\mathbf{in}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{dipolar}}^{\mathbf{out}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{orbital}}^{\mathbf{in}}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{orbital}}^{\mathbf{out}}$
	point charge	59.221	≈0	≈0	-	≈0	29.611	≈0	29.611
point-like	full sphere	59.281	−0.003	−0.004	-	0.021	29.623	0.021	29.623
	spherical shell	59.267	0	−0.004	-	0.013	29.623	0.013	29.623
	point charge	59.221	≈0	≈0	≈0	-	29.611	≈0	29.611
full sphere	full sphere	59.244	−0.002	−0.004	−0.004	-	29.623	0.008	29.623
	spherical shell	59.246	0	−0.004	−0.004	-	29.623	0.008	29.623
	point charge	59.221	-	≈0	≈0	-	29.611	-	29.611
spherical shell	full sphere	59.242	-	−0.004	≈0	-	29.623	-	29.623
	spherical shell	59.241	-	−0.004	≈0	-	29.623	-	29.623

. Despite apparent differences in the absolute hf values, all models reflect the 8:1 ratio of the FNS-induced hf shifts of related $1s$ and $2s$ states [15,18] and, thus, fulfill the condition of almost vanishing normalized differences $8\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}(2s)\end{array}-\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}(1s)\end{array}$ for all investigated systems even for deuterium ${}^2\mathrm{H}$ and ${}^3\mathrm{He}^{+}$, where the contributions of inelastic FNS-effects are quite large. For $1s$ and $2s$ states modelling the nuclear magnetization m localized on a spherical shell yields almost the same hf splittings independent from details of the nuclear charge distribution $\rho$. This observation is in-line with Zemach’s formula Equation (15) where $\rho$ and m enter the integral kernel in a multiplicative way. As the magnetization is zero inside the shell, there will be no extra inside-core contribution as long as the wave functions outside the nuclear core remain the same. Already the combination with a point charge (Coulomb potential) is perfectly reproducing the results of nominally better models where both magnetic moments and charges are consistently treated within finite size. This fact allows for the promising possibility of obtaining accurate relativistic hyperfine splittings including FNS-effects already based on wave functions derived using Coulomb potentials, i.e., without the need for an explicit solution of Dirac’s equation within the nuclear core region.

Table 4. Comparison of the hyperfine splitting (in MHz) of the $1s$ and $2s$ state of hydrogen ${}^1\mathrm{H}$ for the different nuclear charge ($\rho$) and magnetization (m) density models. In the non-relativistic limit, the FNS-induced hf shifts $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}\end{array}$ (in kHz) for $2s$ and $1s$ have to fulfill a 8:1 ratio [15,18]; the relativistic deviation from this 8:1 ratio is evaluated in the last row.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hf}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}\end{array}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hf}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}\end{array}$	${\displaystyle \frac{\mathit{8}\mathbf{·}\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}\end{array}}{\mathit{1}\mathbf{·}\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}\end{array}}}\mathbf{-}\mathit{1}$
	point charge	1421.273		177.665
point-like	full sphere	1421.248	−25	177.662	−3	−0.040
	spherical shell	1421.243	−30	177.661	−4	0.066
	point charge	1421.240	−33	177.661	−4	−0.030
full sphere	full sphere	1421.227	−46	177.660	−5	−0.130
	spherical shell	1421.222	−51	177.659	−6	−0.058
	point charge	1421.228	−45	177.659	−6	−0.066
spherical shell	full sphere	1421.228	−45	177.659	−6	−0.066
	spherical shell	1421.228	−45	177.659	−6	−0.066

Table 4. Comparison of the hyperfine splitting (in MHz) of the $1s$ and $2s$ state of hydrogen ${}^1\mathrm{H}$ for the different nuclear charge ($\rho$) and magnetization (m) density models. In the non-relativistic limit, the FNS-induced hf shifts $\begin{array}{c}\hfill {\delta }_{\mathrm{hf}}^{\mathrm{FS}}\end{array}$ (in kHz) for $2s$ and $1s$ have to fulfill a 8:1 ratio [15,18]; the relativistic deviation from this 8:1 ratio is evaluated in the last row.

Model $\mathit{m}$	Model $\mathit{\rho }$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hf}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}\end{array}$	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{hf}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}$	$\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}\end{array}$	${\displaystyle \frac{\mathit{8}\mathbf{·}\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{2}\mathit{s}\mathbf{)}\end{array}}{\mathit{1}\mathbf{·}\begin{array}{c}\hfill {\mathit{\delta }}_{\mathbf{hf}}^{\mathbf{FS}}\mathbf{(}\mathit{1}\mathit{s}\mathbf{)}\end{array}}}\mathbf{-}\mathit{1}$
	point charge	1421.273		177.665
point-like	full sphere	1421.248	−25	177.662	−3	−0.040
	spherical shell	1421.243	−30	177.661	−4	0.066
	point charge	1421.240	−33	177.661	−4	−0.030
full sphere	full sphere	1421.227	−46	177.660	−5	−0.130
	spherical shell	1421.222	−51	177.659	−6	−0.058
	point charge	1421.228	−45	177.659	−6	−0.066
spherical shell	full sphere	1421.228	−45	177.659	−6	−0.066
	spherical shell	1421.228	−45	177.659	−6	−0.066

4.3. FNS Effects onto the Hyperfine Fields of Hydrogen $2p$ States

Table 5 shows the nuclear-structure-dependent balancing of the hyperfine contribution for anisotropic orbitals using the 2$p_{1/2}$ and 2$p_{3/2}$ excited state of hydrogen ${}^1\mathrm{H}$ as prototype examples. They provide true one-particle system too and, thus, also allow exact analytical analysis. Within a simple non-relativistic picture, the hf splitting of pure p orbitals are determined by the dipolar contribution exclusively. In general, hyperfine splittings of p-like states are thus expected to depend to minor extend or not at all on the nuclear structure. Table 5, however, shows that for finite-size nuclear charges, there are small but non-vanishing contributions (below 10 kHz) from contact terms, suggesting a higher probability of finding $2p$ electrons inside the nucleus. In fact, as shown in Figure 3 and already reported in Ref. [1] for finite superheavy nuclei, a solution of Dirac’s equation for an extended nucleus modelled by a homogeneously charged sphere yields an accumulation of electron density around $r=R$, whereby this effect rapidly decreases with larger nuclear radius R. Actually, the non-vanishing contact terms are restricted to ${}^1\mathrm{H}$; for tritium ${}^3\mathrm{H}$, they are already below 1 kHz. Notably, the density accumulation is only slightly modified if the model of a spherical shell is used; see Figure 3. Obviously, the discontinuity of the potential for $r=R$ has only a minor influence on this polarization effect.

The relativistic treatment yields to the appearance of the orbital term $\Delta E_{\mathrm{orbital}}$. For $2p_{1/2}$ orbitals it coincides 1:1 with $\Delta E_{\mathrm{dipolar}}$; for $2p_{3/2}$ orbitals it is larger by a factor of $-5$. It is, thus, by far not negligible but leads to a factor-of-two or even factor-of-four relativistic enhancement of the hyperfine splittings [36]. In addition, inside- and outside-core contributions to $\Delta E_{\mathrm{orbital}}$ and $\Delta E_{\mathrm{dipolar}}$ slightly increase in case of finite nuclear charges and accomplish the nuclear-structure induced changes from the contact terms. Again, this increase is due to the FNS-induced accumulation peak of the wave functions around $r=R$ depicted in Figure 3.

Table 5. Contributions to the hf splitting (in MHz) of the $2p$ states of hydrogen ${}^1\mathrm{H}$ for different nuclear charge ($\rho$) and magnetization (m) models (for mixed models, see Table A6). ≈0 denotes very small numbers < 0.1 Hz. $\Delta E_{\mathrm{orbital}}^{\mathrm{out}}=\Delta E_{\mathrm{dipolar}}^{\mathrm{out}}$ for $2p_{1/2}$and $\Delta E_{\mathrm{orbital}}^{\mathrm{out}}=-5·\Delta E_{\mathrm{dipolar}}^{\mathrm{out}}$ for $2p_{3/2}$ reflect the fact that the radial part of both quantities are analytically identical (see also Equation (29) and Ref. [36]).

Model $\mathit{\rho }$ and $\mathit{m}$	$\mathit{j}$	${\mathit{m}}_{\mathit{j}}$	$\Delta {\mathit{E}}_{\mathrm{hyperfine}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{in}}$	$\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{contact}}^{\mathit{d}}$	$\Delta {\mathit{E}}_{\mathrm{dipolar}}^{\mathrm{in}}$	$\Delta {\mathit{E}}_{\mathrm{dipolar}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{orbital}}^{\mathrm{in}}$	$\Delta {\mathit{E}}_{\mathrm{orbital}}^{\mathrm{out}}$
point-like			59.221	≈0	−0.001	-	≈0	29.611	≈0	29.611
full sphere	${\displaystyle \frac{1}{2}}$	$\pm \frac{1}{2}$	59.246	−0.002	−0.004	−0.004	-	29.624	0.008	29.624
spherical shell			59.244	-	−0.004	≈0	-	29.624	-	29.624
point-like			11.843	≈0	≈0	-	≈0	−2.961	≈0	14.804
full sphere	${\displaystyle \frac{3}{2}}$	$\pm \frac{1}{2}$	11.871	0.001	0.002	0.008	-	−2.963	0.008	14.814
spherical shell			11.854	-	0.002	≈0	-	−2.963	-	14.814
point-like			35.530	≈0	≈0	-	$\approx 0$	−8.882	≈0	44.411
full sphere	${\displaystyle \frac{3}{2}}$	$\pm {\displaystyle \frac{3}{2}}$	35.613	0.004	0.007	0.020	-	−8.889	0.024	44.443
spherical shell			35.561	-	0.007	≈0	-	−8.889	-	44.443

Table 5. Contributions to the hf splitting (in MHz) of the $2p$ states of hydrogen ${}^1\mathrm{H}$ for different nuclear charge ($\rho$) and magnetization (m) models (for mixed models, see Table A6). ≈0 denotes very small numbers < 0.1 Hz. $\Delta E_{\mathrm{orbital}}^{\mathrm{out}}=\Delta E_{\mathrm{dipolar}}^{\mathrm{out}}$ for $2p_{1/2}$and $\Delta E_{\mathrm{orbital}}^{\mathrm{out}}=-5·\Delta E_{\mathrm{dipolar}}^{\mathrm{out}}$ for $2p_{3/2}$ reflect the fact that the radial part of both quantities are analytically identical (see also Equation (29) and Ref. [36]).

Model $\mathit{\rho }$ and $\mathit{m}$	$\mathit{j}$	${\mathit{m}}_{\mathit{j}}$	$\Delta {\mathit{E}}_{\mathrm{hyperfine}}$	$\Delta {\overline{\mathit{E}}}_{\mathrm{contact}}^{\mathrm{in}}$	$\Delta {\overline{E}}_{\mathrm{contact}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{contact}}^{\mathit{d}}$	$\Delta {\mathit{E}}_{\mathrm{dipolar}}^{\mathrm{in}}$	$\Delta {\mathit{E}}_{\mathrm{dipolar}}^{\mathrm{out}}$	$\Delta {\mathit{E}}_{\mathrm{orbital}}^{\mathrm{in}}$	$\Delta {\mathit{E}}_{\mathrm{orbital}}^{\mathrm{out}}$
point-like			59.221	≈0	−0.001	-	≈0	29.611	≈0	29.611
full sphere	${\displaystyle \frac{1}{2}}$	$\pm \frac{1}{2}$	59.246	−0.002	−0.004	−0.004	-	29.624	0.008	29.624
spherical shell			59.244	-	−0.004	≈0	-	29.624	-	29.624
point-like			11.843	≈0	≈0	-	≈0	−2.961	≈0	14.804
full sphere	${\displaystyle \frac{3}{2}}$	$\pm \frac{1}{2}$	11.871	0.001	0.002	0.008	-	−2.963	0.008	14.814
spherical shell			11.854	-	0.002	≈0	-	−2.963	-	14.814
point-like			35.530	≈0	≈0	-	$\approx 0$	−8.882	≈0	44.411
full sphere	${\displaystyle \frac{3}{2}}$	$\pm {\displaystyle \frac{3}{2}}$	35.613	0.004	0.007	0.020	-	−8.889	0.024	44.443
spherical shell			35.561	-	0.007	≈0	-	−8.889	-	44.443

In total, we find a remarkably large nuclear-structure-related increase in the hyperfine splitting of about $0.1\%$, up to 28 kHz for $p_{1/2}$ and up to 83 kHz for $p_{3/2}$ orbitals, respectively. In particular, the $2p_{3/2,\pm 3/2}$ orbital is thus predicted to be stronger affected by FNS effects than the respective hydrogen $1s$ and $2s$ orbitals, where the effect amounts to maximum only 40 ppm. The size of this finite-size effect considerably depends on the specific nuclear model. Whereas the model of a full sphere gives the maximum values mentioned above; the spherical-shell model yields clearly smaller FNS-induced hf shift of 11 kHz and 31 kHz, respectively. Notably these FNS shifts are vanishing completely if relativistic wave function from Dirac’s equation for a pure Coulomb potential, i.e., for a point charge are used to calculate the hf splittings, see also Table A6. Accordingly, perturbative approaches give much smaller (by four orders of magnitude) hf splittings [29,30]. In any case, the use of non-Coulomb potentials in Dirac’s equation thus appears to be essential if evaluating possible models for extended nuclear structures. In this context, the hyperfine splittings of the hydrogen ${}^1\mathrm{H}$ $2p$ excited states appear to be a promising reference system for exploring the proton structure.

Figure 3. Electron probability ${\psi }^2(r)·r^2$ (r in fm) of the excited $2p_{1/2}$ state for the hydrogen isotopes ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}^{+}$ (curve for the latter multiplied by 500). For ${}^1\mathrm{H}$, beside that for the full-sphere nucleus (solid lines) also the curve for the spherical-shell model (dashed line) is shown.

Figure 3. Electron probability ${\psi }^2(r)·r^2$ (r in fm) of the excited $2p_{1/2}$ state for the hydrogen isotopes ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}^{+}$ (curve for the latter multiplied by 500). For ${}^1\mathrm{H}$, beside that for the full-sphere nucleus (solid lines) also the curve for the spherical-shell model (dashed line) is shown.

5. Summary

In this work, we have investigated the influence of the nuclear structure on the relativistic hyperfine splittings of real one-electron systems, for which analytical solutions of Dirac’s equation are possible and where many-particle effects can be readily excluded, namely the $1s$ ground states and the $2s$, $2p_{1/2}$ and $2p_{3/2}$ excited states of hydrogen, deuterium, tritium, and helium ions ${}^3\mathrm{He}^{+}$. Based on the analytically exact relativistic wave functions, we (i) evaluate the feasibility of the different nuclear structure models and (ii) identify nuclear structure models that provide high accuracy but can also be efficiently applied in numerical many-particle schemes. The nucleus is approximated by a point charge (Coulomb potential) and by homogeneously charged full spheres and spherical shells. In addition, the nuclear magnetic moment is not only treated point-like but also distributed in a full sphere and on the surface of a spherical shell. Thereby, we do not restrict our study on models with unique treatment but also investigate mixed models, where nuclear charge and magnetization are distributed in a different way. In particular, the combination of spherical-shell distributed nuclear magnetic moments while retaining a Coulomb potential appears to be a promising simplified approach, providing an improved many-particle treatment without need for an explicit solution of Dirac’s equation within the nuclear core region. Using the electron mass $m_e$ while solving Dirac’s equation, we obtain moderate FNS-related shifts of the hf splitting, which are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This also holds for ${}^1\mathrm{H}$, where a shift of −45 kHz confirms the estimated values around −50 kHz. Larger differences between different nuclear structure models are found in the case of the anisotropic $2p$ orbitals of hydrogen ${}^1\mathrm{H}$, rendering these excited states as promising reference systems for exploring the proton structure.