Fitting for the energy levels of hydrogen

Precision measurements of atoms are used for metrological purposes and testing the theory of quantum electrodynamics (QED). This is of current interest also in the context of beyond-the-Standard-Model phenomena, as they could manifest themselves in atomic spectroscopy [1]. The theory of bound-state QED is sufficiently mature that the dominant uncertainty in its predictions for the levels of hydrogen and deuterium is due to the nuclear radius. The proton radius puzzle first appeared in 2010 [2] when muonic hydrogen measurements indicated that $r_{p}$ is 4% smaller than had been previously determined, a value near $0.88$ fm [3]. Over the last decade or so, more scattering and spectroscopic experiments have been performed that suggest a value of $r_{p}$ closer to $0.84$ fm [4]. However, discrepancies remain, such as the results of Fleurbaey et al. [5] and Brandt et al. [6], that indicate a value of $r_{p}$ larger than $0.84$ fm with substantial statistical significance. Thus, the puzzle is not entirely solved; more measurements are planned in the near future, including that of the $1S_{1/2}\to 4S_{1/2}$ interval [7].

The bound-state QED predictions for the energy levels of hydrogen involve a combination of long analytic expressions and numerical results that are cumbersome to use; see, e.g., [8]. Our goal here is to provide a fitting formula that reproduces the bound-state QED predictions for those energy levels to sufficiently high accuracy that bound-state QED need not be used directly. To this end, we use the so-called relativistic Ritz approach, which is a long-distance effective theory describing the bound states of two-particle systems whose binding potential is dominated by the Coulomb interaction [9]. In that effective theory, the energy levels of atomic hydrogen were shown to be

$$\frac{E}{c^{2}}=\sqrt{m_{e}^{2}+m_{p}^{2}+\frac{2m_{e}m_{p}}{\sqrt{1+\left(\frac{\alpha}{n_{\star}}\right)^{2}}}}-\Big{(}m_{e}+m_{p}\Big{)}\,,$$

(1)

where $\alpha$ is the fine-structure constant and the effective quantum number

$$n_{\star}=n-\delta\,.$$

(2)

The quantum defect, $\delta$, itself depends on the principal quantum number, $n$, and accounts for interactions that are shorter in range than the Coulomb interaction; it also depends on the orbital, total electronic, and total system quantum numbers, $\ell$, $j$, and $f$, respectively.

To make the numerical analyses more efficient, we Taylor expand equation (1) in small $\alpha$ up to eighth order¹¹1There is no ninth order term so this is sufficient for the accuracy required here. Furthermore, the highest computed QED terms have so far not been computed at higher precision than this [8]. and factor out the Rydberg frequency,

$$cR_{\infty}\equiv\frac{m_{e}\alpha^{2}c^{2}}{2h}\,,$$

(3)

allowing us to write

$$\frac{E}{h}=cR_{\infty}\left(\frac{A_{2}}{n_{\star}^{2}}+\frac{A_{4}}{n_{\star}^{4}}+\frac{A_{6}}{n_{\star}^{6}}+\frac{A_{8}}{n_{\star}^{8}}\right)\,,$$

(4)

where the $A_{2k}={\cal O}\!\left(\alpha^{2k-2}\right)$. For the value of the fine-structure constant we use

$$\alpha^{-1}=137.035\,999\,166(15)\,,$$

(5)

derived from a recent measurement of the electron g-factor [10]. Together with the mass ratio

$$\frac{m_{p}}{m_{e}}=1\,836.152\,673\,349(71)\,,$$

(6)

inferred from spectroscopy of $\text{HD}^{+}$ [11]²²2This value is consistent with that of Ref. [12]. Both values reported in Refs. [11] and [12] rely on the proton-to-deuteron mass ratio obtained by Fink and Myers [13]., this allows us to determine the constants

	$\displaystyle A_{2}$	$\displaystyle=$	$\displaystyle-0.999\,455\,679\,424\,739(21)$		(7)
	$\displaystyle A_{4}$	$\displaystyle=$	$\displaystyle 3.990\,953\,7921(87)\times 10^{-5}$		(8)
	$\displaystyle A_{6}$	$\displaystyle=$	$\displaystyle-1.770\,774\times 10^{-9}$		(9)
	$\displaystyle A_{8}$	$\displaystyle=$	$\displaystyle 8.25\times 10^{-14}\,.$		(10)

There are uncertainties in $A_{6}$ and $A_{8}$; however, they are irrelevant at the level of accuracy needed here.

The simplest Ritz-like expansion is posited for the quantum defect, namely a series expansion in terms of the energy eigenvalues, which are assumed to be small relative to some high-energy scale, $\Lambda$:

$$\delta=\delta_{0}+\lambda_{1}\frac{E}{\Lambda}+\lambda_{2}\left(\frac{E}{\Lambda}\right)^{2}+\dots\,,$$

(11)

where $\delta_{0}$ and the $\lambda_{i}$ are dimensionless coefficients. However, because in this form $E$ depends implicitly on $\delta$, it is impractical to use for most theoretical or empirical applications. A modified ansatz written as a series in inverse powers of $(n-\delta_{0})$ is asymptotically ($n\to\infty$) equivalent to (11) and is significantly easier to use for data fitting. Analyzing the large-$n$ behavior of (4) with (11), it may be verified that

$$\delta=\delta_{0}+\frac{\delta_{2}}{\left(n-\delta_{0}\right)^{2}}+\frac{\delta_{4}}{\left(n-\delta_{0}\right)^{4}}+\frac{2\delta_{2}^{2}}{\left(n-\delta_{0}\right)^{5}}+\frac{\delta_{6}}{\left(n-\delta_{0}\right)^{6}}+\frac{6\delta_{2}\delta_{4}}{\left(n-\delta_{0}\right)^{7}}+\frac{\delta_{8}}{\left(n-\delta_{0}\right)^{8}}+\frac{4\delta_{4}^{2}+8\delta_{2}\delta_{6}}{\left(n-\delta_{0}\right)^{9}}\\ +\frac{\delta_{10}}{\left(n-\delta_{0}\right)^{10}}+\frac{-40\delta_{2}^{4}+10\delta_{4}\delta_{6}+10\delta_{2}\delta_{8}}{\left(n-\delta_{0}\right)^{11}}+\frac{\delta_{12}}{\left(n-\delta_{0}\right)^{12}}+\frac{-296\delta_{2}^{3}\delta_{4}+6\delta_{6}^{2}+12\delta_{4}\delta_{8}+12\delta_{2}\delta_{10}}{\left(n-\delta_{0}\right)^{13}}+\dots\,,$$

(12)

where the $\delta_{i}$ are free parameters. As shown in the following section, $\delta_{0}$ is small (of order $\alpha^{2}$) and thus $1/(n-\delta_{0})$ is small for $n>1$, but we find that the modified defect expansion (12) satisfactorily reproduces energy levels even for $n=1$ with the inclusion of a sufficient number of $\delta_{i}$.

A truncation of equation (12) is required for any application and we specify the order of the analysis by the highest inverse power of $(n-\delta_{0})$ included. Actually, truncations made at each successive inverse odd power includes one additional defect parameter. Because there is no $-1$st or $-3$rd term, including terms through $(n-\delta_{0})^{-1}$ requires only $\delta_{0}$ and is considered lowest order (LO), whereas including terms through $(n-\delta_{0})^{-3}$ requires both $\delta_{0}$ and $\delta_{2}$ and is considered next-to-lowest order (NLO). At higher orders we use the abbreviation N^kLO, where $k+1$ is equal to the number of defect parameters needed. For practical purposes, as shown below, the largest expansion is needed for $S$-states, where we truncate at the level N⁶LO, thereby including defect parameters up to $\delta_{12}$. For higher angular momentum eigenstates fewer terms are required to reach the same level of accuracy.

As outlined below, we use a limited number of precisely calculated energy levels of hydrogen employing the most up-to-date bound-state QED calculations [8], and fit them with equation (4) using the defect formula in equation (12). We determine the necessary $\delta_{i}$ to reproduce all theoretical energy levels to within their uncertainties and demonstrate the power of our fits by testing our results against higher-$n$ calculated energies. Because some energies can be predicted with a relative precision that is better than $10^{-13}$, to ensure this level of reproducibility, the parameters $A_{2}$ through $A_{8}$ are given to an absolute precision of $10^{-15}$ and, likewise, we report our fit values of the $\delta_{i}$ to the same level of precision.

According to bound-state QED, the theoretical energy levels of hydrogen can be written as the sum of a gross level structure, fine-structure (FS), and hyperfine-structure (HFS) contribution,

$$E_{n\ell jf}=-\frac{cR_{\infty}}{n^{2}}\frac{m_{p}}{m_{p}+m_{e}}+E_{n\ell j}^{(\text{FS})}+E_{n\ell jf}^{(\text{HFS})}\,,$$

(13)

where we have chosen units in which Planck’s constant, $h=1$. The electron’s reduced Compton wavelength, muon-to-electron mass ratio, proton g-factor, and electron magnetic-moment anomaly taken from CODATA-18 [8] are

	$\displaystyle\lambdabar_{e}$	$\displaystyle=$	$\displaystyle 3.861\,592\,6796(12)\times 10^{-13}\,\text{m}$		(14)
	$\displaystyle\frac{m_{\mu}}{m_{e}}$	$\displaystyle=$	$\displaystyle 206.768\,2830(46)$		(15)
	$\displaystyle g_{p}$	$\displaystyle=$	$\displaystyle 5.585\,694\,6893(16)$		(16)
	$\displaystyle a_{e}$	$\displaystyle=$	$\displaystyle 1.159\,652\,181\,28(18)\times 10^{-3}\,.$		(17)

We also use the proton radius inferred from the muonic hydrogen spectroscopy of Antognini et al. 2013 [14],

$$r_{p}=0.840\,87(39)\,\text{fm}\,.$$

(18)

Following the procedure described in [15], the measured $1S_{1/2}$ [16] and $2S_{1/2}$ [17] hyperfine intervals may be used to the determine $E_{n\ell jf}^{(\text{HFS})}$ to sufficient accuracy such that $cR_{\infty}$ may be determined using the measured $1S^{(f=1)}_{1/2}\to 2S^{(f=1)}_{1/2}$ interval from [18]; this completely specifies the theory.

The theoretical uncertainty in the energy levels is dominated by the uncertainty in $E_{n\ell j}^{(\text{FS})}$, which affects the levels directly through $E_{n\ell j}^{(\text{FS})}$ itself and also indirectly through the determination of $cR_{\infty}$. There are 5 uncertainties relevant to $E_{n\ell j}^{(\text{FS})}$ at the level of precision needed in this work. Four of these are QED uncertainties taken directly from CODATA-18 [8]: the uncertainty in the two-photon correction term $B_{60}$ yields $\delta_{\ell,0}\left(0.94\,\text{kHz}\right)/n^{3}$ (a reduction by about 50% compared to CODATA-14); the uncertainty in the three-photon correction term $C_{50}$ yields $\delta_{\ell,0}\left(0.96\,\text{kHz}\right)/n^{3}$; nuclear polarizability uncertainty yields $\delta_{\ell,0}\left(0.39\,\text{kHz}\right)/n^{3}$; and a radiative recoil uncertainty yields $\delta_{\ell,0}\left(0.74\,\text{kHz}\right)/n^{3}$.

In addition to the QED uncertainties mentioned above, there is an error in $E_{n\ell j}^{(\text{FS})}$ due to the proton radius (18) which amounts to $\delta_{\ell,0}\left(1.03\,\text{kHz}\right)/n^{3}$. Adding all of these errors in quadrature, the overall uncertainty in the fine-structure correction is

$$\delta(E_{n\ell j}^{(\text{FS})})=\frac{\left(1.9\,\text{kHz}\right)}{n^{3}}\delta_{\ell,0}\,,$$

(19)

and it follows that

$$cR_{\infty}=3\,289\,84\,960\,249.1(2.2)\,\text{kHz}\,,$$

(20)

a shift upward of 0.2 kHz compared to the result reported in [15], but well within the uncertainty computed therein. Accounting for the correlated uncertainties in the QED predictions and determination of $cR_{\infty}$, the theoretical uncertainty on any given level is

$$\delta(E_{n\ell jf})=\left\lvert\frac{1.9\,\text{kHz}}{n^{3}}\delta_{\ell,0}-\frac{2.2\,\text{kHz}}{n^{2}}\right\rvert\,,$$

(21)

and is therefore below $0.6$ kHz for all levels.

Lastly, given the potential issue of a remaining proton radius puzzle, we consider the possible systematic implications of a shift away from the value quoted in equation (18). Defining the proton radius shift,

$$\Delta r_{p}=r_{p}-0.840\,87\,\text{fm}\,,$$

(22)

it follows that the Rydberg frequency shifts by

$$\Delta\left(cR_{\infty}\right)_{r_{p}}=3.1\,\text{kHz}\left(\frac{\Delta r_{p}}{0.001\,\text{fm}}\right)\,,$$

(23)

and the energy levels shift by

$$\Delta\left(E_{n\ell jf}\right)_{r_{p}}=\left(\frac{2.6\,\text{kHz}}{n^{3}}\delta_{\ell,0}-\frac{3.1\,\text{kHz}}{n^{2}}\right)\left(\frac{\Delta r_{p}}{0.001\,\text{fm}}\right)\,.$$

(24)

Equations (23) and (24) will be utilized below to cross check our results against a selection of experimental results.

Here we have presented a simple fitting formula and parameters, equation (4) and Tables 1, and 5 through 8, that are sufficient to reproduce all hyperfine energy levels of hydrogen up to $\ell=30$. The theoretical uncertainty of any level is given by equation (21) and additional systematic shifts in those levels due to a proton radius that differs from the one determined by Antognini et al. [14] is parameterized in equation (24).

Acknowledgements

We greatly appreciate the initial suggestion of Eric Hessels to pursue this work.