Chapter 1

Tests of Theory in Rydberg States of

One-electron Ions

Joseph N. Tan and Peter J. Mohr

Comparison of optical frequency measurements to predictions of quantum electrodynamics (QED) for Rydberg states of one-electron ions can test theory and allow

new determinations of constants of nature to be made. Simplifications in the QED

theory of high-angular-momentum states reduces the uncertainty in the prediction

of transition frequencies to a level where a new value of the Rydberg constant which

is independent of the proton radius can be determined. Since the energy-level spacing between neighboring Rydberg states grows as the square of the nuclear charge

number, it is possible to study transitions with optical frequencies that are accessible to femtosecond laser frequency combs. Recently at the US National Institute of

Standards and Technology (NIST), highly-charged ions (including bare nuclei) created in an Electron Beam Ion Trap (EBIT) were extracted and captured in a novel

compact Penning trap. An ongoing experiment aims to produce one-electron ions

isolated in an ion trap designed for laser spectroscopy. Tests of theory in a regime

free of nuclear effects would be valuable in shedding light on the puzzle surrounding

the large discrepancy in the value of the proton radius inferred from the observed

Lamb shift in muonic hydrogen as compared to the value deduced from hydrogen

and deuterium spectroscopy and electron scattering measurements.

1.1 Introduction

Quantum electrodynamics (QED) is the first satisfactory quantum description of the

interaction of charged particles (and antiparticles) via the exchange of photons and

Joseph N. Tan

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD 20899,

U.S.A. e-mail: joseph.tan@nist.gov

Peter J. Mohr

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899,

U.S.A. e-mail: mohr@nist.gov

1
2 Joseph N. Tan and Peter J. Mohr

of the creation and annihilation of elementary particles. QED makes precise predictions for various physical quantities, and these predictions have been tested across

a vast array of phenomena. Spectroscopic measurements in atoms have played a

crucial role in spurring the development of QED to be the most accurate physical

theory yet invented.

This success seems all the more remarkable because, as the quantum field theory of electrons and photons, QED does not remove the divergences well known in

Maxwell’s classical theory of electromagnetism; on the contrary, new infinities are

found in QED, associated with virtual processes in the vacuum. Sensible, finite results are obtainable only after renormalization; this is an art inextricably tied to the

introduction of fundamental constants, such as the electron mass and charge, into

the theory.

One-electron atomic systems are among the simplest quantum-mechanical objects that have enabled very stringent tests of QED to be made, yielding some of

the most precise measurements of fundamental constants. For example, the value

of the Rydberg constant recommended by the Committee on Data for Science and

Technology (CODATA), an interdisciplinary scientific committee of the International Council for Science (ICSU), is obtained from precise measurements of various transitions in hydrogen and its stable isotope deuterium taken together with

electron scattering measurements [1].

For over a hundred years, the study of hydrogen-like atoms has contributed to

our understanding of the quantum world. Figure 1.1 illustrates the eight ordersof-magnitude improvement in the accuracy of the Rydberg constant in a 130-year

history that includes many theoretical and experimental discoveries and developments [2, 3]. As another example, assuming the validity of quantum electrodynamics (QED) calculations at the tenth order, the fine-structure constant α (the universal

constant of electromagnetic interactions) is most precisely obtained from the electron |g|−2 experiment, which measures the anomalous magnetic moment of a single

electron stored in a Penning trap (dubbed geonium, an engineered atom consisting

of one electron bound to the earth).

The first tests of the emerging formalism of QED came shortly after World War

II. Ref. [4] reported the first measurement of the anomalous magnetic moment of the

electron; in the same year, Ref. [5] presented the first measurement of the “Lamb

shift” of the 2s level of hydrogen [6], another departure from the Dirac theory of the

hydrogen atom. These discoveries have led to more stringent tests of QED, with remarkable progress over six decades. On the one hand, with control and minimization

of cavity effects that limited early geonium |g| −2 experiments [7, 8], the magnetic

moment of the electron |g| = 2(1+ae) has been measured recently at Harvard University with a relative uncertainty of 2.8 × 10−13 using a single electron isolated

in a cylindrical Penning trap [9]; by comparison, the calculated magnetic moment

of the electron has a relative uncertainty of 5.2 × 10−12 coming mainly from the

uncertainty of the best independent determination of the fine-structure constant. On

the other hand, the hydrogen 1s− 2s transition has been measured with a relative

uncertainty of 1.4 × 10−14 [10]; here a test of theory is hampered by uncertainties

in the nuclear-size corrections.
1 Tests of Theory in Rydberg States of One-electron Ions 3

Currently the accuracy in the determination of the Rydberg constant has a relative

uncertainty of 5.0×10−12 [1]. It seems astonishing that QED should have attained

such high accuracy based on abstractions employed to represent physical objects and

measurements, particularly when one of the pioneers in its development has noted

the “mathematical inconsistencies and renormalized infinities swept under the rug”

[11].

Looking to the future, in Section 1.5 , we consider possible determinations of fundamental constants and tests of theory in Rydberg states of one-electron ions. This

has been discussed in [12], where it was pointed out that the problems that limit

the theoretical predictions in low-angular-momentum states are strongly suppressed

in Rydberg states with high angular momentum (high-l), because the electron has a

very small probability of being near the nucleus in such states. Remarkably, the simplification of higher-order QED terms in Rydberg states yields an overall theoretical

uncertainty that is smaller than the uncertainties propagated from the fundamental

constants. In particular, even with its currently-assigned uncertainty of 5.0×10−12,

the Rydberg constant is the leading source of uncertainty in this regime, larger than

the uncertainties due to other constants by a factor of 100. Hence, if precise measurements can be made for comparison with theory, one-electron ions in Rydberg

states can be useful in testing QED in a regime with negligible nuclear effects. Assuming that QED remains valid, this could enable a Rydberg constant determination

year

1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

log10(uncertainty)

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3

first authors:

1890 Rydberg 

**** Rydberg 

1914 Curtis 

1921 Birge 

1929 Birge 

1952 Cohen 

1959 Martin 

1968 Csillag 

1971 Masui 

1973 Kessler 

1973 Kibble 

1973 Cohen 

1974 Hansch 

1976 Hansch 

1978 Goldsmith 

1980 Petley 

1981 Amin 

1986 Hansch 

1986 Zhao 

1986 Hildum 

1986 Barr 

1986 Biraben 

1987 Zhao 

1987 Beausoleil 

1987 Boshier 

1992 Hansch 

1997 Hansch 

h

cm

R

e

2

2

=a

¥

proton radius 

controversy 

Rydberg 

Fig. 1.1 Relative uncertainty in the determinations of the Rydberg constant over its history. Diamonds stand for uncertainties in experimental values; circles with dashed lines stand for uncertainties in least-squares adjustments (LSA). Its current uncertainty of 5×10−12 does not take into

account the puzzling 7 σ discrepancy in the proton radius described in Sec. 1.3.
4 Joseph N. Tan and Peter J. Mohr

that is independent of the proton radius. As discussed in section 1.3, interest in such

a determination is heightened by the large discrepancy in the proton radius determined from the recent measurement of the muonic hydrogen Lamb shift compared

to the determinations of the radius by other methods, which potentially would be

impacted by an independent measurement of the Rydberg constant. In section 1.6,

we describe an experimental effort at NIST to make one-electron ions in Rydberg

states via charge transfer from highly-excited atom to bare nuclei isolated in a trap.

1.2 The Rydberg Constant

The Rydberg constant is a combination of fundamental constants:

R∞ = α

2mec/2h, (1.1)

where h is the Planck constant, c is the speed of light, me is the electron mass, and α

is the fine-structure constant. It plays a pivotal role in the CODATA least-square adjustments. Historically, it first appeared as the overall coefficient in the mathematical

formula connecting the light frequency (or wavelength) to the numbers characterizing the atomic states involved in the light emission. This relationship was first

discovered phenomenologically in the spectral line emissions of hydrogen by Johann Balmer, who showed that the visible lines in the series named after him are

related by a simple formula involving integers. Inspired by this discovery, Johannes

Rydberg showed that a more general formula which included small non-integral

offsets (now called “quantum defects”) has much broader application by accounting

for perturbations due to core electrons when studying atoms with one highly-excited

electron (Rydberg atoms) [13]. Figure 1.1 shows the improvement in the determinations of the Rydberg constant going back over one hundred years; a compilation of

the experimental values can be found in Ref. [3].

The CODATA recommended value of the Rydberg constant has been obtained

primarily by comparing theory and experiment for twenty-four transition frequencies or pairs of frequencies in hydrogen and deuterium [1]. The theoretical value for

each transition is the product of the Rydberg constant and a calculated factor based

on QED that also depends on other constants. While the most accurately measured

transition frequency in hydrogen (1S–2S) has a relative uncertainty of 1.4 × 10−14

[10], the recommended value of the Rydberg constant has a larger relative uncertainty of 5.0 × 10−12 which is essentially the uncertainty in the theoretical factor

which includes the proton radius. The main source of uncertainty in the theoretical factor is from the charge radius of the proton with additional uncertainty due to

uncalculated or partially calculated higher-order terms in the QED corrections.
1 Tests of Theory in Rydberg States of One-electron Ions 5

1.3 The Proton Radius Puzzle: Is QED in trouble?

The uncertainty introduced into the theory of hydrogen and deuterium transition frequencies due to uncertainty in the proton and deuteron charge radii can be reduced

by employing accurate independent determinations of the radii. Electron-proton and

electron-deuteron scattering experiments are important sources of information about

the radii. Potentially more accurate information is provided by comparison of theory

and experiment for muonic hydrogen. In this section, the various sources of information on the charge radii are briefly reviewed, and the discrepancy that leads to the

proton radius puzzle is described.

1.3.1 Spectroscopic data and the Rydberg constant

Values of the Rydberg constant and the proton and deuteron charge radii follow

from precise spectroscopic measurements on hydrogen and deuterium. For the 2010

CODATA determination of the constants, 24 frequencies or differences of frequencies were taken into account. These data are listed in Table 1.1. Comparison of

theory and experiment, which does not include electron scattering values for the

radii, leads to a value of the Rydberg constant given by [1]

R∞ = 10 973 731.568 521(82) m

−1

, (1.2)

which has a relative uncertainty of 7.4 × 10−12, and a value for the proton radius

given by [1]

rp = 0.8758(77) fm. (1.3)

The theoretical input used in the analysis is described in [1]. The main source of

uncertainty, as has been mentioned, is the uncertainty in the proton charge radius. A

smaller, but not negligible, uncertainty arises from uncalculated theoretical contributions.

1.3.2 Scattering determinations of the radii

The accuracy of the Rydberg constant extracted from hydrogen and deuterium spectroscopic data can be improved by including information on the nuclear charge radii

from electron scattering experiments in the analysis. Electron-proton scattering data

values of the proton radius rp and an electron-deuteron scattering data value of the

deuteron radius rd used as input data in the 2010 CODATA adjustment are

rp = 0.895(18) fm, (1.4)
6 Joseph N. Tan and Peter J. Mohr

Table 1.1 Measured frequencies ν considered in the LSA determination of the Rydberg constant

R∞ (H is hydrogen and D is deuterium).

Frequency interval(s) Reported value Rel. stand.

ν/kHz uncert. ur

MPQ 2004 (Fischer et al. [14])

1 νH(1S1/2 −2S1/2

) 2 466 061 413 187.080(34) 1.4×10−14

MPQ 1995 (Weitz et al. [15])

2 νH(2S1/2 −4S1/2

)− 1

4

νH(1S1/2 −2S1/2

) 4 797 338(10) 2.1×10−6

3 νH(2S1/2 −4D5/2)−

1

4

νH(1S1/2 −2S1/2

) 6 490 144(24) 3.7×10−6

4 νD(2S1/2 −4S1/2

)−

1

4

νD(1S1/2 −2S1/2) 4 801 693(20) 4.2×10−6

5 νD(2S1/2 −4D5/2

)−

1

4

νD(1S1/2 −2S1/2

) 6 494 841(41) 6.3×10−6

MPQ 2010 (Parthey et al. [16])

6 νD(1S1/2 −2S1/2)−νH(1S1/2 −2S1/2) 670 994 334.606(15) 2.2×10−11

LKB/SYRTE 1997 (de Beauvoir et al. [16])

7 νH(2S1/2 −8S1/2) 770 649 350 012.0(8.6) 1.1×10−11

8 νH(2S1/2 −8D3/2

) 770 649 504 450.0(8.3) 1.1×10−11

9 νH(2S1/2 −8D5/2

) 770 649 561 584.2(6.4) 8.3×10−12

10 νD(2S1/2 −8S1/2) 770 859 041 245.7(6.9) 8.9×10−12

11 νD(2S1/2 −8D3/2

) 770 859 195 701.8(6.3) 8.2×10−12

12 νD(2S1/2 −8D5/2

) 770 859 252 849.5(5.9) 7.7×10−12

LKB/SYRTE 1999 (Schwob et al. [17])

13 νH(2S1/2 −12D3/2

) 799 191 710 472.7(9.4) 1.2×10−11

14 νH(2S1/2 −12D5/2

) 799 191 727 403.7(7.0) 8.7×10−12

15 νD(2S1/2 −12D3/2

) 799 409 168 038.0(8.6) 1.1×10−11

16 νD(2S1/2 −12D5/2

) 799 409 184 966.8(6.8) 8.5×10−12

LKB 2010 (Arnoult et al. [18])

17 νH(1S1/2 −3S1/2

) 2 922 743 278 678(13) 4.4×10−12

LKB 1996 (Bourzeix et al. [19])

18 νH(2S1/2 −6S1/2

)−

1

4

νH(1S1/2 −3S1/2

) 4 197 604(21) 4.9×10−6

19 νH(2S1/2 −6D5/2

)−

1

4

νH(1S1/2 −3S1/2

) 4 699 099(10) 2.2×10−6

Yale 1995 (Berkeland et al. [20])

20 νH(2S1/2 −4P1/2)−

1

4

νH(1S1/2 −2S1/2

) 4 664 269(15) 3.2×10−6

21 νH(2S1/2 −4P3/2

)− 1

4

νH(1S1/2 −2S1/2

) 6 035 373(10) 1.7×10−6

Harvard 1994 (Hagley and Pipkin [21])

22 νH(2S1/2 −2P3/2

) 9 911 200(12) 1.2×10−6

Harvard 1986 (Lundeen and Pipkin [22])

23 νH(2P1/2 −2S1/2

) 1 057 845.0(9.0) 8.5×10−6

U. Sussex 1979 (Newton et al. [23])

24 νH(2P1/2 −2S1/2

) 1 057 862(20) 1.9×10−5

1 MPQ: Max-Planck-Institut f ̈ur Quantenoptik, Garching. LKB: Laboratoire Kastler-Brossel, Paris.

SYRTE: Syst`emes de r ́ef ́erence Temps Espace, Paris, formerly Laboratoire Primaire du Temps et

des Fr ́equences (LPTF).
1 Tests of Theory in Rydberg States of One-electron Ions 7

rp = 0.8791(79) fm, (1.5)

rd = 2.130(10) fm. (1.6)

The value of rp in Eq. (1.4) is based on Sick’s analysis of the world data available

at the time [24]. Equation (1.5) gives a value based on measurements made at the

Mainz Microtron MAMI and analyzed using a variety of form-factor models [25].

The result for rd in Eq. (1.6) is the result of an analysis of the world data by Sick

[26].

More recent values for the proton radius from scattering data have been given

since the 2010 CODATA cutoff date for input data. Precise measurement of the

ratio of the electric to magnetic form factors over a range of Q

2 was made at the

Thomas Jefferson National Accelerator facility in Virginia (JLab). These data, combined with selected earlier data, yield the value rp = 0.875(10) fm [27, 28]. Sick

has obtained rp = 0.894(8)fm [29] and rp = 0.886(8)fm [30] by supplementing the

scattering data with a calculation of the shape of the large radius proton charge distribution. The analysis leading to the latter value for rp includes data from Ref. [25].

An analysis of existing data with an analytic form-factor model has yielded a smaller

radius rp = 0.8489(69) fm [31].

The consensus of these values is consistent with the 2010 CODATA recommended value for the proton radius. Exceptions are the result of [29] which is

slightly larger and the result of [31] which is significantly smaller. The various treatments of the electron scattering data and the difficulties in arriving at a unique result

for the proton radius are discussed in Ref. [32].

1.3.3 Proton radius from muonic hydrogen

It is expected that a better value for the proton radius, which would lead to a better

value for the Rydberg constant, can be obtained from muonic hydrogen, an atom

consisting of a negative muon and a proton. Due to its larger mass, the Bohr radius

of the muon is about 207 times smaller than the electron Bohr radius. As a result,

the splitting of the 2S and 2P states in muonic hydrogen is quite sensitive to the size

of the proton, which contributes about 2 % to the total Lamb shift. (Because of the

large electron-vacuum-polarization effect, the 2S1/2level is below both the 2P3/2

and 2P1/2levels.)

In measurements carried out at the Paul Scherrer Institute (PSI), Villigen, Switzerland, the 2S1/2

(F = 1)−2P3/2(F = 2) and 2S1/2(F = 0)−2P3/2(F = 1) transitions

in muonic hydrogen have been accurately measured [33, 34, 35]. These results,

when combined with the theory for the transition, lead to [35]

rp = 0.840 87(39) fm. (1.7)

Because of the discrepancy between this value of the proton radius and the values obtained from spectroscopic data or electron scattering data, the theory for the
8 Joseph N. Tan and Peter J. Mohr

transition frequency has been reviewed many times. Recent reviews, which also list

earlier reviews, are given in Refs. [32, 36, 37, 38, 39, 40, 41, 42].

1.3.4 Comparison of the determinations of the proton radius

Values of the proton radius discussed in Secs. 1.3.1-1.3.3 are shown in Fig. 1.2.

Based on both spectroscopic data and the electron scattering data, the CODATA

recommended value of the Rydberg constant is [1]

R∞ = 10 973 731.568 539(55) m

−1

(1.8)

which has a relative uncertainty of 5.0×10−12, and the recommended value of the

proton radius is [1]

rp = 0.8775(51) fm. (1.9)

rp/fm

0.85 0.90 0.95

0.85 0.90 0.95

10−2

e-p: Sick (2003)

e-p: Bernauer, et al. (2010)

μp: Pohl, et al. (2010)

H,D: CODATA (2010)

CODATA-2010

e-p: Ron, et al. (2011)

e-p: Sick (2011)

e-p: Sick (2012)

e-p: Adamuscin, et al. (2012)

Fig. 1.2 Values of the proton radius rp given in Secs. 1.3.1-1.3.3. The entry at the top sets the

scale for the graph. The values, from top to bottom, are from Refs. [24, 43], [25], [33, 35], [1], [1],

[27, 28], [29], [30], [31]. Error bars represent 1 standard uncertainty.
1 Tests of Theory in Rydberg States of One-electron Ions 9

If the proton radius from muonic hydrogen in Eq. (1.7) is compared to the

CODATA 2010 recommended value in Eq. (1.9), there is a 7σ disagreement. If

it is compared to the value in Eq. (1.3), based on only H and D spectroscopic data,

the disagreement is 4.5σ.

Alternatively, if the proton radius derived from muonic hydrogen is included

among the input data for a least-squares adjustment of the spectroscopic data, the

result is an apparent inconsistency between the fitted values of the transition frequencies and the experimental values used as the input data, as shown in Fig. 1.3

[1]. An additional problem with including the muonic hydrogen proton radius in a

least-squares analysis of the spectroscopic data is that it leads to a value of the finestructure constant given by α

−1 = 137.035 881(35) [2.6×10−7

], which differs from

the 2010 recommended value by 3.4σ [1]. The value of R∞ from such an adjustment

is [1]

Fig. 1.3 Transition frequencies in hydrogen and deutrerium. Deviations from theory are large when

the proton radius deduced from the muonic hydrogen Lamb-shift is included among the input data.

Error bars represent 1 standard uncertainty.
10 Joseph N. Tan and Peter J. Mohr

R∞ = 10 973 731.568 175(12) m

−1

. (1.10)

This value of the Rydberg constant differs from the 2010 recommended value by

6.5σ.

These discrepancies constitute the proton radius puzzle. At this time, there is no

explanation for the disagreements between the muonic hydrogen Lamb shift data

and both the H and D spectroscopic data and the electron-proton scattering data. It

may be necessary to modify QED to properly account for the muon data, or there

still may be a contribution from conventional QED that has not been taken into

account properly.

1.3.5 Comparison of the determinations of the Rydberg constant

As shown in the previous sections, there is disagreement between the values of the

Rydberg constant determined from the H and D spectroscopic data with and without

the inclusion of the proton radius data from muonic hydrogen. The reason for this

is that S states play an important role in the determination of the Rydberg constant

in the least-squares adjustments, and these states are relatively strongly affected by

the size of the nucleus.

On the other hand, there is the possibility of using Rydberg states to determine

the Rydberg constant. Such a determination would be virtually independent of the

proton or nuclear radius, because high-l states have a negligible overlap with the

nucleus. Thus such an independent value for the Rydberg constant might favor one

of the values associated with either including or not including the muonic hydrogen

proton radius in the analysis, thereby shedding light on the proton radius puzzle.

In this regard, it is of interest to note that there is an existing unpublished measurement of the Rydberg constant made in Rydberg states [44]

R∞ = 10 973 731.56834(23) m

−1

(1.11)

Values for the Rydberg constant in Eqs. (1.8), (1.10), and (1.11) are shown in

Fig. 1.4. It is apparent from that figure that an independent determination of the Rydberg constant from Rydberg states with even a modest accuracy might distinguish

between the values involving S states with or without the muonic hydrogen data

included.

1.4 Optical transitions between Rydberg states

Optical frequency combs [45] can be used to make precise measurements of optical

transitions between Rydberg states. For example, laser spectroscopy of antiprotonic

helium together with theoretical calculations of the atomic structure have been used
1 Tests of Theory in Rydberg States of One-electron Ions 11

to weigh the antiproton [46]. On the other hand, if CPT symmetry is assumed, the

experiments and theory can be interpreted as a determination of the electron mass.

For one-electron ions in Rydberg states, the optical frequency combs have the

potential of making precise absolute frequency measurements which can lead to a

determination of the Rydberg constant. Although transitions between Rydberg states

in neutral atoms are typically in the microwave frequency range, in highly-ionized

ions, the frequencies can be in the optical range, as shown in Figure 1.5. In that

figure, the frequencies corresponding to transitions between adjacent Bohr energy

levels (n to n − 1) in hydrogen-like ions are shown as a function of the principle

quantum number of the upper level n and the charge number of the nucleus Z. The

colored bands indicate the (optical) colors corresponding to the frequencies in that

figure. The pink band is approximately the region of infra-red frequencies.

Much of this parameter space is accessible to optical frequency synthesizers

based on mode-locked femtosecond lasers, which readily provide ultra-precise reference rulers (optical frequency combs) spanning the near-infrared and visible region of the optical spectrum (530 nm–2100 nm). Even when the absolute accuracy

is limited by the primary frequency standard (a few parts in 10

16), optical frequency

(R∞ − 10 973 731.568) × 103

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−11R∞

De Vries (2001)

μH (2010)

CODATA-2010

Fig. 1.4 Values of the Rydberg constant based on microwave transitions in circular states of hydrogen, the value obtained by including the muonic hydrogen proton radius in the least-squares

adjustment of H and D data, and the 2010 recommended value. First entry sets the scale of the

graph. Error bars represent 1 standard uncertainty.
12 Joseph N. Tan and Peter J. Mohr

combs can enable relative frequency measurements with uncertainties approaching

1 part in 1019 over 100 THz of bandwidth [47]–making comb-based measurements

generally more precise than other existing technologies. The precisely controlled

pulse train from a femtosecond laser can also be used directly to probe the global

atomic structure, thus integrating the optical, tera-Hertz, and radio-frequency domains [48].

This is an advantage, because heavier hydrogen-like ions with a higher nuclear

charge Z have larger fine-structure separations, which scale as Z

4

. As a consequence,

the Rydberg states are far more robust against perturbations such as Stark mixing

than Rydberg states in the microwave regime. In addition, the charged ions repel

each other, so the effect of short range collisions can be expected to be less important.

1.5 Theory of Rydberg states

In this section, we consider the theory of one-electron ions in Rydberg states, focusing on aspects unique to the high-l case. In particular, simplifications in the theory

reduce the uncertainties and allow accurate predictions of the energy levels to be

made. Radiative transitions are also discussed.

Fig. 1.5 Iso-frequency plots showing values of Z and approximate n that give a specified value of

the frequency for transitions changing the principal quantum number n by 1 in a one-electron ion

with nuclear charge Z. Color bands represent frequencies in the near infrared and visible region.
1 Tests of Theory in Rydberg States of One-electron Ions 13

1.5.1 Simplification of nuclear size and higher-order QED effects

The largest source of uncertainty in determining the Rydberg constant from spectroscopy of H and D is the uncertainty in the nuclear radius, which translates into

uncertainty in the theoretical values of the transition frequencies. For states with

high orbital angular momenta l, this uncertainty is relatively smaller than in lower-l

states, because in the high-l case, the probability P(r) for the electron to be found

within a short distance r of the nucleus is very low. As a consequence, the nuclearsize correction to the energy level is small, and knowledge of the nuclear size is

unimportant. For a Rydberg state of a hydrogen-like ion with charge number Z, high

principal quantum number n, and angular momentum l = n−1, this probability is

P(r) = Z

|x|<r

dx|ψ(x)|

2 ≈

1

(2n+1)!



2Zr

na0

2n+1

, (1.12)

where a0 is the Bohr radius. If r is taken to be the nuclear radius, then the high-power

factor, together with the factorial in the denominator, leads to an almost complete

suppression of nuclear effects for circular or near-circular Rydberg states. For example, if n = 5 and Z = 5, then P(1 fm) ≈ 10−56

.

Another advantage of high-l states for theory is the fact that the higher-order corrections in the QED theory of the levels are relatively smaller for these states compared to low-l states. Thus, the perturbation expansion of the theoretical expressions

for the energy levels, as a function of Zα, provides relatively more accurate results

with a given number of terms.

Since these two portions of the theory have the largest associated uncertainties,

their elimination significantly reduces the relative uncertainty for the theory of the

transition frequencies to the extent that in certain cases the Rydberg constant is the

largest source of uncertainty. Then, a comparison of theory and experiment for the

transitions can provide information on the Rydberg constant and potentially improve

its accuracy.

1.5.2 High-l-state energy levels in hydrogen-like atoms

In this section, we list the known theoretical expressions for the energy levels of

hydrogen-like ions and give numerical results with estimates of the uncertainties for

special cases. Reviews of the theory and references to original work are given in

[49, 50, 1]. Here the theory is given only for l ≥ 2.

The energy levels can be written as

En = EDM +ERR +EQED , (1.13)

a sum of the Dirac energy with nuclear motion corrections EDM, relativistic recoil

corrections ERR, and radiative corrections EQED.
14 Joseph N. Tan and Peter J. Mohr

For the first term, the difference between the Dirac eigenvalue ED and the electron

rest energy, is proportional to

α

2Dmec2 = ED −mec2 =

 

1+

(Zα)

2

(n−δ)

2

−1/2

−1

!

mec

2

, (1.14)

α

2D = −

(Zα)

2

2n

2

+



3

8n

−

1

2 j +1



(Zα)

4

n

3

+... , (1.15)

where j is the total angular momentum quantum number, δ = |κ| −p

κ

2 −(Zα)

2

,

and κ = (−1)

l+j+1/2

(j +1/2) is the Dirac spin-angular quantum number. The expansion in Eq. (1.15) shows the leading Schr  ̈odinger and fine-structure terms. The

energy level, taking into account the leading nuclear motion effects, is given by [50]

EDM = 2hcR∞

"

μrD−

rNμ

3

r α

2

2

D

2 +

r

2

Nμ

3

r Z

4α2

2n

3κ(2l+1)

#

, (1.16)

where rN = me/mN is the ratio of the electron mass to the nucleus mass, and μr =

1/(1+rN) is the ratio of the reduced mass to the electron mass.

Relativistic corrections to Eq. (1.16), associated with motion of the nucleus are

ERR = 2hcR∞

rNZ

5α

3

πn

3



μ

3

r



−

8

3

lnk0(n, l)−

7

3l(l+1)(2l+1)



+πZα



3−

l(l+1)

n

2



2

(4l

2 −1)(2l+3)

+..., (1.17)

where lnk0(n, l) is the Bethe logarithm. We assume that the uncertainty due to uncalculated higher-order terms is Zα ln(Zα)

−2

times the last term in Eq. (1.17).

Quantum electrodynamics (QED) corrections for high-l states are

EQED = 2hcR∞

Z

4α

2

n

3



− μ

2

r

ae

κ(2l+1)

+ μ

3

r

α

π



−

4

3

lnk0(n, l) (1.18)

+

32

3

3n

2 −l(l+1)

n

2

×

(2l−2)!

(2l+3)!

(Zα)

2

ln

1

μr(Zα)

2



+ (Zα)

2 G(Zα)

,

where ae is the electron magnetic moment anomaly and G(Zα) is a function that

contains higher-order QED corrections. Equation (1.19) contains no explicit vacuum polarization contribution because of the damping of the wavefunction near the

origin where the polarization effect is largest. The quantity ae replaces the order-byorder terms in the theory that are equivalent to the free-electron magnetic moment

anomaly. Instead, the experimental value is used for the sum of all such terms in order to eliminate uncertainty associated with the calculated values and higher-order

omitted terms. We employ the value ae = 1.159 652 180 73(28)×10−3 obtained in

a recent experiment [51].
1 Tests of Theory in Rydberg States of One-electron Ions 15

The dominant terms in G(Zα) are expected to be of the form

G(Zα) = A60 +A81(Zα)

2

ln(Zα)

−2 +A80(Zα)2 +...

+

α

π

B60 +...+

α

π

2

C60 +... , (1.19)

where the letters A, B, and C indicate contributions arising from one-, two-, and

three-photon Feynman diagrams, respectively. The function G(Zα) = GSE(Zα) +

GVP(Zα) is the sum of contributions from self-energy (SE) diagrams and vacuumpolarization (VP) diagrams. The coefficients A60 and A81 arise from the self energy,

and A80 arises from both the self energy and the long-range component of the vacuum polarization.

The first two terms in Eq. (1.19) come from the one-loop self energy

GSE(Zα) = A60 +A81(Zα)

2

ln(Zα)

−2 +... , (1.20)

and the leading contribution is given by the term A60. Calculated values for this

coefficient are given in Table 1.2 [12, 52]. Results of a complete calculation of the

one-loop self energy, given in Table 1.3, are consistent with the calculation of the

leading term, which provides an independent check against substantial errors in either calculation [53]. The good agreement also indicates that the higher-order terms

are not significantly larger than expected. The uncertainty in GSE(Zα) is assumed to

be given by A60(Zα)

2

ln(Zα)

−2

, where the coefficient A81 in the first omitted term

is replaced by A60 for the estimate.

The vacuum-polarization contribution to A80 is extremely small [54]. The twophoton coefficient B60 has not been calculated for high-l states. However, a comparison of calculated values of B60 [55] and A60 [56] for l ≤ 5, suggests it is of the

order of 4|A60|, which is taken to be its uncertainty. The three-photon term C60 is

expected to be the next term, and is assumed to be negligible.

1.5.3 Transition frequencies and uncertainties

To put the theory presented above into perspective, we list the numerical predictions

for transition frequencies for two different ions, and indicate the various sources

of uncertainties. We consider the frequency of the transition between the state with

n = 14, l = 13, j =

27

2

and the state with n = 15, l = 14, j =

29

2

in the hydrogen-like

ions He+ and Ne9+. The constants used in the evaluation are the 2010 CODATA

recommended values [1], with the exception of the neon nucleus mass m(

20Ne10+)

which is taken from the neon atomic mass [57], corrected for the mass of the electrons and their binding energies. Values of the various contributions and the total are

given as frequencies in Table 1.4. Standard uncertainties are listed with the numbers

where they are non-negligible. The largest uncertainty arises from the Rydberg frequency cR∞, which is a common factor in all of the contributions. There is no un-
16 Joseph N. Tan and Peter J. Mohr

Table 1.2 Calculated values for the coefficient A60 from [12, 52]. The numbers in parentheses are

standard uncertainties in the last figure.

n l = n−2, j = l−

1

2

l = n−1, j = l−

1

2

9 7.018 373(5)×10−5 3.860 349(5)×10−5

10 3.655 111(5)×10−5 2.158 923(5)×10−5

11 2.008 438(5)×10−5 1.259 580(5)×10−5

12 1.019 187(5)×10−5 0.759 620(5)×10−5

13 0.679 575(5)×10−5 0.469 973(5)×10−5

14 0.410 825(5)×10−5 0.296 641(5)×10−5

15 0.252 108(5)×10−5 0.189 309(5)×10−5

16 0.155 786(5)×10−5 0.121 749(5)×10−5

n l = n−2, j = l+

1

2

l = n−1, j = l+

1

2

9 28.939 225(5)×10−5 14.918 400(5)×10−5

10 16.589 245(5)×10−5 9.141 150(5)×10−5

11 10.111 871(5)×10−5 5.882 197(5)×10−5

12 6.331 080(5)×10−5 3.940 256(5)×10−5

13 4.318 998(5)×10−5 2.729 475(5)×10−5

14 2.979 937(5)×10−5 1.945 279(5)×10−5

15 2.116 050(5)×10−5 1.420 631(5)×10−5

16 1.540 181(5)×10−5 1.059 674(5)×10−5

Table 1.3 Calculated values for the coefficient GSE(Zα) with Z = 14, 16 from [53]. The numbers

in parentheses are standard uncertainties in the last figure.

n Z l = n−2, j = l−

1

2

l = n−1, j = l−

1

2

13 14 0.676(9)×10−5 0.469(9)×10−5

13 16 0.682(4)×10−5 0.468(7)×10−5

14 14 0.403(8)×10−5 0.296(9)×10−5

14 16 0.408(6)×10−5 0.296(9)×10−5

15 14 0.243(9)×10−5 0.184(7)×10−5

15 16 0.249(3)×10−5 0.191(9)×10−5

n Z l = n−2, j = l+

1

2

l = n−1, j = l+

1

2

13 14 4.317(5)×10−5 2.728(9)×10−5

13 16 4.321(2)×10−5 2.728(5)×10−5

14 14 2.974(5)×10−5 1.944(9)×10−5

14 16 2.978(3)×10−5 1.945(9)×10−5

15 14 2.107(9)×10−5 1.415(7)×10−5

15 16 2.114(2)×10−5 1.423(9)×10−5

certainty from the Planck constant since ν = (E15 − E14)/h, and h drops out when

calculating the frequency.

Table 1.5 gives sources and estimates of the various known uncertainties in the

theory. By way of comparison, in hydrogen, the relative uncertainty from the twophoton term B60 for the 1S–2S transition is of the order of 10−12 whereas in the
1 Tests of Theory in Rydberg States of One-electron Ions 17

Table 1.4 Transition frequencies between the highest-j states with n = 14 and n = 15 in hydrogenlike helium and hydrogen-like neon.

Term 4He+ ν(THz) 20Ne9+ ν(THz)

EDM 8.652 370 766 016(43) 216.335 625 5749(11)

ERR 0.000 000 000 000 0.000 000 000 1

EQED −0.000 000 001 894 −0.000 001 184 1

Total 8.652 370 764 122(43) 216.335 624 3909(11)

n = 14 to n = 15 Rydberg transitions the QED relative uncertainties are orders of

magnitude smaller, as indicated in that table. The convergence of the expansion of

the QED corrections in powers of Zα is significantly faster for Rydberg transitions,

because of the smallness of the higher-order terms for the high-l states compared to

S states.

Table 1.5 Sources and estimated relative standard uncertainties in the theoretical value of the

transition frequency between the highest-j states with n = 14 and n = 15 in hydrogen-like helium

and hydrogen-like neon.

Source He+ Ne9+

Rydberg constant 5.0×10−12 5.0×10−12

Fine-structure constant 3.3×10−16 8.2×10−15

Electron-nucleus mass ratio 5.5×10−14 1.1×10−14

ae 5.1×10−20 1.3×10−18

Theory: ERR higher order 6.2×10−17 2.4×10−14

Theory: EQED A81 1.7×10−18 1.6×10−14

Theory: EQED B60 8.6×10−18 5.4×10−15

1.5.4 Natural line widths

The advantages of using optical transitions between Rydberg states come with experimental trade-offs associated with a large spontaneous emission rate. Natural

decay linewidths tend to be small for states from which electric dipole (E1) decay is

forbidden (as in the case of the 2S level). In contrast, the spontaneous decay rate for

a circular Rydberg state is dominated by an electric dipole E1 transition from the

highest-l value of the state n to the highest-l value of the state n−1.

Formally, the QED level shift given by Eq. (1.19) is the real part of the radiative

correction. The complete radiative correction to the level can be written as EQED =

EQED − iΓ /2 which is complex with an imaginary part proportional to the decay

rate A = Γ /h ̄ of the level. In the nonrelativistic framework, for a state with principal
18 Joseph N. Tan and Peter J. Mohr

quantum number n and angular momentum l = n−1, the dominant decay mode is

an E1 transition to state with n

′ = n − 1 and l′ = n′ − 1 [58]. Ref. [58] gives the

corresponding expression for the decay rate, which is the nonrelativistic limit of the

imaginary part of the level shift:

Γn → hcR∞Z

4α

3

4n

2n−4

(n−1)

2n−2

3



n−

1

2

4n−1

. (1.21)

The imaginary part of the level shift gives the natural width of the level, and when

this is taken into account, the resonant frequency of the transition between states

with quantum numbers n and n − 1, as a function of the frequency of the applied

laser field, has a full width at half maximum given by Γn +Γn−1. A measure of the

quality of the line for making precision measurements is the ratio of the transition

energy to the resonance width, given by

Q =

En −En−1

Γn +Γn−1

→

3n

2

4α(Zα)

2

+... , (1.22)

where the expression on the right is the asymptotic form as n → ∞ of the nonrelativistic value. Figure 1.6 gives a contour plot of the values of n and Z that give a

specified value of Q based on Eq. (1.21).

Fig. 1.6 Graph showing combinations of nuclear charge Z and approximate n values that give the

same ratio of the transition frequency to the natural linewidth of the transition resonance between

circular states of one-electron ions with principal quantum number n and n−1.
1 Tests of Theory in Rydberg States of One-electron Ions 19

In addition to a width of the resonance, there can be an asymmetry in the line

shape that affects the relationship between the measured frequency distribution and

the level splitting. Such effects have been shown to be small by Low [59], of order

α(Zα)

2 EQED. For example, for the 1S–2S transition in hydrogen, they are completely negligible at the current level of experimental accuracy [60]. However, for

Rydberg states of hydrogen-like ions, particularly at higher-Z, asymmetries in the

line shape could be more significant. Such effects, some of which might depend on

the detailed configuration of the experiment, can be calculated if necessary [12].

1.6 Experiment

As discussed in Sec. 1.5, Rydberg states of hydrogenlike ions with l > 2 essentially

avoid a number of problems associated with either higher-order binding corrections

to QED interactions or the nuclear size effect in Lamb shift predictions. In the cases

considered for planned experiments, the higher-order QED corrections for Rydberg

states are smaller by a factor of about 107compared to S states while the nuclear size

effect is completely negligible, providing significant advantages in making accurate

predictions.

Of particular interest are Rydberg states with maximum angular momentum or

circular states, so-called because the probability density is significant only in an

annular region centered on the nucleus. In addition to aforementioned advantages,

other useful features of circular states include: (1) the longest lifetime in a given

shell n, and (2) suppressed Stark effect. In fact, circular Rydberg states of hydrogen

in the microwave regime have been used in very precise measurements of transition

frequencies [61]; as a result, a determination of the Rydberg constant has been made

with a 2.1×10−11 relative uncertainty [61]. In Fig. 1.4 we compare this result based

upon circular-state transitions with more recent determinations of the Rydberg constant. Unfortunately, the precision of the hydrogen circular-state measurements is

not sufficient to help resolve the large discrepancy between the μp measurement

and the hydrogen measurements (CODATA). In retrospect, the large dipole moments in circular Rydberg states of hydrogen would significantly increase the measurement uncertainty due to the possible perturbations from the dipole-dipole interaction between Rydberg atoms [62]. In contrast, such dipole-dipole interaction

would be orders of magnitude smaller between one-electron Rydberg ions since the

wavefunction is concentrated more tightly around a highly-charged nucleus.

Experiments with cold, one-electron Rydberg ions may be possible for a wide

range of nuclear charge Z and angular momentum l. A considerable array of tools

and techniques have emerged in the last two decades which enable “engineered

atoms” to be formed in traps and tailored in specific states of experimental interest.

Cooling techniques [63] routinely used with singly- or doubly-ionized atoms, for instance, can be exploited to cool highly-charged ions. Man-made quantum systems—

for example, antihydrogen [64] or a single electron in a Penning trap [8]—have the

potential to extend the range of precision measurements that probe nature. In this
20 Joseph N. Tan and Peter J. Mohr

section, we describe an experimental effort at NIST to produce one-electron ions

in Rydberg states by isolating bare nuclei in a trap for recombination via electron

capture from laser-excited alkali atoms.

1.6.1 Source of fully-stripped ions

Various methods have been used to produce Rydberg states [13]. Generally, one (or

a combination) of the following mechanisms is used: (1) electron impact excitation,

(2) photoexcitation, or (3) charge exchange. High angular-momentum states present

some challenges when dealing with highly-charged ions. Electron impact excitation

from the ground state tends to produce low-l Rydberg states since it changes angular momentum by only one unit. Even for low-Z one-electron ions, lasers with

the proper frequency are not readily available for photoexcitation from the ground

state. On the other hand, recombination by electron capture from a highly-excited

atom produces Rydberg states with high-l; it is noteworthy that, following charge

exchange, the recombined Rydberg ion tends to evolve via radiative cascade towards

a circular state (maximum angular momentum) [67].

To use charge exchange for producing hydrogenlike ions, fully-stripped ions

(bare nuclei) must be isolated in a trap. Fully-stripped ions can be obtained from an

Fig. 1.7 Simplified schematic of the experimental set-up (not to scale). Highly stripped ions are

produced in an EBIT (right). Extracted ion pulses are transported to an analyzing magnet in the

beamline. Detailed descriptions of the extraction beamline have been published in [65, 66]. Ions

of a selected charge state are deflected by the analyzing magnet into the vertical beamline, where

a 90-degree bender deflects them into the experimental apparatus (left).
1 Tests of Theory in Rydberg States of One-electron Ions 21

electron beam ion trap (EBIT). An EBIT has been in operation at NIST since 1993

[68]. The EBIT at NIST has an existing beamline for ion extraction [66]. Figure 1.7

provides a simplified schematic diagram of the experimentalset-up. As discussed in

the next section, we recently demonstrated that a compact Penning trap [69] can be

used for capturing ions extracted from the EBIT at NIST.

Highly-charged ions (HCI) are produced in an EBIT by repeated electron impact

ionization of atoms injected as a neutral gas target via a nozzle (alternatively, low

charge-state metal ions can be injected from a metal-vapor vacuum arc [MeVVA]

source [70]). In an EBIT, a nearly-monoenergetic electron beam is accelerated

through high voltage Ue to reach high energies eUe as it travels through the axis

of a stack of three electrodes called drift tubes; for the EBIT at NIST, an electron

beam current as high as 150 mA can be used, accelerated to energies up to 30 keV.

Coaxial with the drift tubes is a strong magnetic field (≈ 3 T) which compresses the

radial extent of the electron beam, yielding very high current density. Confinement

of positively-charged ions is similar to that in a Penning trap, with the drift tubes

biased to form an electrostatic potential well along the trap symmetry axis; however,

radial confinement of ions is dominated by attraction to the tightly-compressed, axial electron beam. Confined ions collide with the intense electron beam repeatedly

and are ionized to successively higher charge states limited by the kinetic energy of

the electron beam.

A mixture of several charge states is produced in an EBIT, bounded by the ionizing energy of the electron beam, with a relative distribution determined by various

tuning parameters. For the purpose of capturing and isolating the ions in an experimental apparatus, the HCI mixture is extracted in bunches by quickly ramping the

EBIT middle drift tube up in voltage. The extracted ion pulse accelerates out of the

drift tube region, with each ion of charge number Q acquiring substantial kinetic

energy (≈ QeUe) as it departs from the potential “hill” that accelerates the electron

beam. Typical ion pulse energies for the experiments of interest here are between

2000 Q eV and 4000 Q eV. The process of slowing and capturing the extracted ions

in a compact Penning trap is discussed in detail in a forthcoming publication, and is

briefly described below in Sec. 1.6.2.

During ion extraction, an ion pulse from the EBIT is injected vertically into the

extraction beamline (described elsewhere [66]). As shown in Figure 1.7, a 90-degree

electrostatic bender turns the ion pulse into the horizontal beamline, which transports the ion bunch to an analyzing magnet located ≈ 3.8 m from the EBIT. The

analyzing magnet allows selection of one specific ion charge state by adjusting

its magnetic field to the proper value necessary for deflecting an ion of a chosen

mass/charge ratio into the vertical section of the beamline. Figure 1.8 illustrates the

selection of a specific charge state by controlling the magnetic field of the analyzing magnet. Downstream in the vertical beamline, another 90-degree electrostatic

bender is used to divert the ion pulse to the experimental apparatus as illustrated in

Figure 1.9, where ions can be detected on a Faraday cup or a microchannel plate.

The total path length from the EBIT to the compact Penning trap in the experimental

apparatus is ≈ 7.2 meters.
22 Joseph N. Tan and Peter J. Mohr

1.6.2 Capturing bare nuclei in a compact Penning trap

We have recently demonstrated that a compact Penning trap can be useful for capturing ions extracted from an EBIT [69]. The experimental apparatus shown in Figure

1.9 was designed to utilize the remaining space (≈ 1m3) at the end of the ion extraction beamline. It incorporates a compact Penning trap for capturing ion pulses

extracted from the EBIT. A Penning trap can be made very compact by embedding

annular rare-earth (NdFeB) magnets within its electrode structure. The electrode

structure uses electrical-iron to close the magnetic circuit. The unitary architecture of this two-magnet, compact Penning trap is represented in Figure 1.10, with

a three-dimensional rendering shown in the inset of Figure 1.9; a more detailed description has been presented in Ref. [69]. The endcap electrodes have holes to allow

passage of ions along the trap axis. Not shown in Fig. 1.9 are the vacuum pumps

(non-evaporable getter, turbo-molecular and ion pumps) which evacuate the roomtemperature chamber to a residual gas pressure of ≈ 1.2×10−7 Pascal (or 9×10−10

Torr).

An ion beam or pulse entering the apparatus is steered by two orthogonal pairs

of deflection plates, and focussed by an Einzel lens into the compact Penning trap.

The electrodes in the trapping region have been carefully designed to optimize ion

capture. Details on capturing ions with a compact Penning trap are presented in a

forthcoming publication [71]. Here we give a brief summary.

The axial confinement well of the compact Penning trap is formed by symmetrically biasing the endcaps to higher electrical potential than on the ring electrode,

Bending Magnet Field (Gauss)

650 700 750 800 850 900 950 1000 1050 1100

Ion Current on Faraday Cup (pA)

0

5

10

15

20

25

30

35

40

45

20Ne10+

20Ne9+

22Ne10+

20Ne8+

20Ne7+

20Ne6+

20Ne5+

22Ne5+ 22Ne6+ 22Ne7+

22Ne 22 8+ Ne9+

Neon extraction from NIST-EBIT

 I

e = 150 mA

 Ue = 4.987 kV

Fig. 1.8 Charge states of neon resolved by the analyzing magnet, detected on a Faraday cup near

the bottom of the vertical beamline. The Faraday cup signal is plotted as a function of the magnetic

field generated by the analyzing magnet. The leftmost peak corresponds to fully-stripped 20Ne10+

(bare nuclei).
1 Tests of Theory in Rydberg States of One-electron Ions 23

typically between 10V and 40V. In preparation for injecting an ion pulse into the

Penning trap, the potential of the “front” endcap facing the incoming ion pulse

is momentarily lowered below the ring electrode potential. The trap is kept open

during the transit of the ions pulse from the EBIT to the compact Penning trap.

Ions steered and focussed into the Penning trap are captured by switching the front

endcap rapidly back to its high potential setting, ideally when the ions are turning around from the “back” endcap. Simulations aid the design of trap components

to attain good capture efficiency. Various operating parameters are also important;

foremost are: (1) proper timing for closing the trap; and (2) matching the electrical

potential of the compact Penning trap to the extraction energy in order to slow the

ions as much as possible in the trapping region. Various ion species extracted from

1 

Photomultiplier 

Tube (PMT) 

TOF 

compact 

Penning trap 

vacuum window 

NdFeB magnet 

lens 

Fig. 1.9 Experimental apparatus for capturing ions extracted from an EBIT using a compact Penning trap. Quarter-cut three-dimensional representation of major components with vacuum chamber in lighter shades. Inset shows a zoomed view of the unitary Penning trap and the light collection

optics. Ions enter the apparatus from the right hand side.
24 Joseph N. Tan and Peter J. Mohr

an EBIT (including: Ne10+, Ne9+, Ne8+, Ar16+, Ar15+, Ar14+, Ar13+, and Kr17+)

have been captured and isolated in the two-magnet Penning trap [72].

The apparatus provides relatively easy access to the stored ions along several

directions for various experiments. Occupying less than 150 cm3 of volume, the

compact Penning trap is readily configured with several detectors. In particular, the

ring electrode has four equidistant holes to provide midplane line-of-sight access to

the stored ions. The trap is centered on a six-way cross with one of the ring electrode

holes aligned with the vertical axis of the vacuum cross, and an orthogonal hole

aligned with the horizontal window. As illustrated in the inset of Fig. 1.9, the top

hole in the ring electrode has an embedded aspheric lens which, together with the

lens system mounted outside the top window, collects and focusses light emitted by

the trapped ions into a photomultiplier. Very recently, highly charged ions isolated

in this unitary Penning trap have been used in experiments to observe forbidden

radiative decay [73].

Stored ions can be detected also by ejection to a retractable, time-of-flight (TOF)

micro-channel plate (MCP) detector, with ≈ 1 ns response time to resolve different charge states. Alternatively, if the TOF detector is retracted, a position-sensitive

MCP detector can be used. Details of the ion detection scheme are in [69]. These capabilities have been useful in fine-tuning the ion beam and in measuring the storage

lifetime of captured ions [72, 74].

Fully-stripped neon ions (bare Ne10+ nuclei) have been extracted from an EBIT

and captured in the unitary Penning trap. An ion cloud in a Penning trap attains

dynamical equilibrium by rotation (spin) about the trap axis to generate a v × B

compressive force which counter-balances the Coulomb repulsion (or space-charge

Fig. 1.10 A Penning trap made extremely compact with a unitary architecture. (Left) Crosssectional view showing 2 annular NdFeB magnets embedded within the electrode structure. Reentrant, soft-iron endcap electrodes are used to close the magnetic circuit to strengthen the field at the

center. (Right) Calculated magnetic field generated by the two rare-earth magnets is represented in

false color. More details are provided in [69].
1 Tests of Theory in Rydberg States of One-electron Ions 25

repulsion) between ions. For most practical cases, an ion cloud expands radially if

the rotation slows down due to some torque acting on the cloud (which may come

from trap imperfections or misalignment). Radial expansion of the ion cloud can

lead to number loss when some ions strike the wall of the ring electrode. Even in

a well-constructed trap wherein cloud expansion is very slow, an ion can change

charge state due to electron capture from residual gas atoms.

The charge-state composition of an ion cloud can be analyzed from the time-offlight of ejected ions to the fast MCP detector. The endcap nearest the TOF detector

is switched to 400 V below the ring voltage in about 50 ns, ramping out the ions

in a pulse. Figure 1.11(a) shows the TOF signal of bare Ne10+ nuclei ejected to the

detector after 1 ms of confinement in the two-magnet Penning trap. If there are multiple charge states, several peaks will be observed. Lower charge stages have later

arrival times due to their smaller charge-to-mass ratio. Figure 1.11(b) illustrates the

mixture of charge states produced from the charge exchange between a cloud of

bare nuclei (initial state) and the background gas atoms, detected after 2 s confinement. The TOF is sufficiently fast to fully resolve the lower charge states produced

Fig. 1.11 Time-of-flight detection of highly-charged ions ejected from the two-magnet Penning

trap after various storage times [72, 69], showing charge-state evolution of captured bare nuclei

due to electron capture from residual background gas. Neon ions are stored in the Penning trap for

times ranging from 1 ms to 4 s, with applied potential difference ∆V = 10 V between the ring and

endcap electrodes. In particular, output of the TOF detector is plotted versus arrival time (relative to

the ejection trigger) for storage times of (a) 1 ms and (b) 2 s. The detector signal scale is magnified

by a factor of 10 from (a) to (b). The TOF signal peak for each charge state is converted to ion

counts in order to show (c) the evolution of charge states due to charge exchange, normalized to

the initial population of Ne10+ ions observed at the 1 ms storage time. Error bars represent 1σ

uncertainty.
26 Joseph N. Tan and Peter J. Mohr

by charge exchange. Figure 1.11c shows the charge-state evolution as a function of

confinement time.

Collisions with the residual gas atoms or molecules in the vacuum chamber deplete the initial ion species captured in the Penning trap. In Fig. 1.11c, the initial

charge state (Ne10+ bare nuclei; red squares) decays exponentially with time constant 1.09(2) s, as lower charge states grow; the sum of all observed charge states

decays with a longer time constant 2.41(6)s. The variation of these decay rates with

the amount of residual gas is illustrated in Figure 1.12, showing their linear proportionality to the background gas pressure. Loss of stored bare nuclei is largely due

to charge exchange. Ion cloud expansion can explain the decay of the charge-state

sum [69]. It is noteworthy that the decay rates extrapolate to nearly zero, indicating

that the compact Penning trap imperfections do not contribute significantly to the

torque driving the ion cloud expansion. If the background gas pressure is reduced,

for example by cooling the vacuum envelope with a cryocooler, then the lifetime

of the charge state of interest can be lengthened (and the ion cloud expansion rate

reduced) considerably.

1.6.3 Charge-exchange recombination: one-electron Rydberg ions

One-electron ions in high-l Rydberg states have attractive features for testing theory. Undoubtedly challenging to realize, such elegantly-simple quantum system can

be engineered using an array of tools and techniques that have emerged in atomicmolecular-optical physics within the last two decades. For example, production of

cold antihydrogen at the European Laboratory for Particle Physics (CERN) demonstrated two ways for an antiproton to capture a positron in high-l Rydberg states:

(1) three-body recombination during positron cooling of antiprotons; or (2) positron

capture from positronium in a two-stage charge-exchange mechanism [64]. In addition, Doppler-tuned laser spectroscopy of Rydberg states has been used to study the

atomic cores of highly charged ions in crossed-beam experiments [75].

The effort underway at NIST will develop a unitary, compact Penning trap (Fig.

1.10) to facilitate the interaction of stored bare nuclei with an excited atomic beam

for the formation of one-electron ions in Rydberg states via charge exchange. In

the planned experiment, a beam of alkali atoms enters the compact Penning trap

through one of the holes in the ring electrode and they are excited by lasers to

Rydberg states. The charge-exchange reaction between a bare nucleus A of charge

Z colliding a target atom B prepared in an excited state with a principal quantum

number n is represented by the formula

A

+Z +B(n) → A+Z−1

(n ̃) +B

+ , (1.23)

where  ̃n is the principal quantum number of the one-electron ion A

+Z−1

immediately

after recombination. A classical-trajectory Monte Carlo (CTMC) method has been

used to estimate the electron capture cross-section, with the low-velocity limit given
1 Tests of Theory in Rydberg States of One-electron Ions 27

by [76]

σcx ≈ 5.5Z π n

4

a

2

o

, (1.24)

which indicates that the large geometrical cross-section for a Rydberg target atom

with radius rn = n

2ao is further enhanced by a factor 5.5Z.

CTMC simulations also showed that the excited electron in the target atom attempts to preserve its original orbital dimension and energy during the electron capture process, leading to an initial post-collisional quantum number distribution given

by

n

√

Z ≤ n ̃ ≤ nZ , (1.25)

with the most probable state being roughly  ̃n ≈ [nZ + nZ1/2]/2 [76]. Results of

crossed-beam experiments which studied ion-Rydberg atom collisions are consistent with this model [13, 77]. As an illustration, the CTMC model predicts that RyFig. 1.12 Loss rate of ions stored in the compact Penning trap. (Left) Exponential decay of detected ions as a function of confinement time. The vacuum chamber pressure for these measurements is ≈ 1.7 × 10−7 Pa (1.3 × 10−9 Torr). (Right) The ion loss rate varies linearly with the

residual gas pressure in the vacuum chamber. A voltage difference of ∆V = 10 V was applied between the ring and endcap electrodes of the Penning trap. Error bars represent 1σ uncertainty. See

Ref.[69] for more details.
28 Joseph N. Tan and Peter J. Mohr

dberg states in the range 63 < n ̃ < 200 are obtainable for Ne10+ nucleus colliding

with a target atom prepared in n = 20 excited state, with the peak near  ̃n ≈ 130.

1.7 Summary

With the assumption that the theory is correct, QED is used to determine values of

the relevant fundamental constants by adjusting their values to give the best agreement with experiments [78]. The Rydberg constant, for example, is currently determined to 5.0×10−12 uncertainty by adjusting its value to give predictions of atomic

transition frequencies that agree best with twenty-four spectroscopic measurements

in hydrogen and deutrerium. The accuracy of the Rydberg constant determination

using hdyrogen spectroscopy is limited by theoretical uncertainties associated with

the nuclear-size effects and, to a lesser extent, the two-loop QED contributions. In

such a comparison between theory and experiments, the proton radius can also be

adjusted to give the best agreement.

Surprisingly, recent measurements of the Lamb shift in muonic hydrogen, when

compared with QED calculations, require a proton radius that is almost seven standard deviations smaller than the radius inferred from hydrogen spectroscopy or

electron scattering experiments. The discrepant proton-radius value obtained from

munoic hydrogen Lamb shift measurements has a signficant impact on the determination of the Rydberg constant if used together with spectroscopic measurements

of hydrogen and deuterium. This has generated renewed interest in alternative measurements that are independent of the proton radius. Although transitions between

circular states of neutral hydrogen have been measured in the microwave regime and

yield such a value of the Rydberg constant, the uncertainty is larger than the aforementioned discrepancy, and may be dominated by large dipole-dipole interactions.

In view of the availability of optical frequency combs, earlier theoretical work

at NIST has addressed the possibility of comparing theory with measurements of

optical transitions between Rydberg states in one-electron ions isolated in a trap.

Attractive features of using one-electron ions include the robustness of Rydberg

states when the energy spacing between neighboring states is magnified by some

power of a larger nuclear charge, making the Rydberg states more immune to external perturbations. Moreover, in contrast to neutral Rydberg atoms, the dipole-dipole

interaction (which may have limited the MIT Rydberg hydrogen experiment [62])

would be orders of magnitude weaker due to the smaller orbitals around a stronger

nuclear attraction. Finally, as reviewed here, recent calculations at NIST have shown

that the theory of Rydberg states in one-electron ions is remarkably simplified for

high angular momentum (l > 2).

Acknowledgements Experimental work by S.M. Brewer and N.D. Guise has been supported by

the Chemical Physics Program of the University of Maryland, and the Research Associateship

Program of the U.S. National Research Council, respectively. Valuable contributions to the theory

of Rydberg states have been made by U.D. Jentschura and B.J. Wundt.
1 Tests of Theory in Rydberg States of One-electron Ions 29

References

1. P.J. Mohr, B.N. Taylor, D.B. Newell, Rev. Mod. Phys. 84(4), 1527 (2012)

2. G.W. Series (ed.), The Spectrum of Atomic Hydrogen: Advances (World Scientific, Singapore,

1988)

3. J.S. Rigden, Hydrogen: the essential element (Harvard University Press, Cambridege, Massachusetts, 2002)

4. P. Kusch, H.M. Foley, Phys. Rev. 72, 1256 (1947)

5. W.E. Lamb, Jr., R.C. Retherford, Phys. Rev. 72(3), 241 (1947)

6. W.E. Lamb, Jr., Rep. Prog. Phys. 14, 19 (1951)

7. R.S. Van Dyck, Jr., P.B. Schwinberg, H.G. Dehmelt, Phys. Rev. Lett. 59(1), 26 (1987)

8. L.S. Brown, G. Gabrielse, Rev. Mod. Phys. 58(1), 233 (1986)

9. D. Hanneke, S. Fogwell, G. Gabrielse, Phys. Rev. Lett. 100, 120801 (2008)

10. T. H ̈ansch, J. Alnis, P. Fendel, M. Fischer, C. Gohle, M. Herrmann, R. Holzwarth, N. Kolachevsky, T. Udem, M. Zimmermann, Philos. Trans. R. Soc. London, Ser. A 363(1834), 2155

(2005)

11. F. Dyson. letter to G. Gabrielse, July 15 (2006)

12. U.D. Jentschura, P.J. Mohr, J.N. Tan, B.J. Wundt, Phys. Rev. Lett. 100, 160404 (2008)

13. T.F. Gallagher, Rydberg Atoms, 1st edn. (Cambridge University Press, New York, 1994)

14. M. Fischer, N. Kolachevsky, M. Zimmermann, R. Holzwarth, T. Udem, T.W. H ̈ansch,

M. Abgrall, J. Gr ̈unert, I. Maksimovic, S. Bize, H. Marion, F.P. Dos Santos, P. Lemonde,

G. Santarelli, P. Laurent, A. Clairon, C. Salomon, M. Haas, U.D. Jentschura, C.H. Keitel,

Phys. Rev. Lett. 92, 230802 (2004)

15. M. Weitz, A. Huber, F. Schmidt-Kaler, D. Leibfried, W. Vassen, C. Zimmermann, K. Pachucki,

T.W. H ̈ansch, L. Julien, F. Biraben, Phys. Rev. A 52(4), 2664 (1995)

16. C.G. Parthey, A. Matveev, J. Alnis, R. Pohl, T. Udem, U.D. Jentschura, N. Kolachevsky, T.W.

H ̈ansch, Phys. Rev. Lett. 104, 233001 (2010)

17. C. Schwob, L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef,

A. Clairon, Phys. Rev. Lett. 82(25), 4960 (1999)

18. O. Arnoult, F. Nez, L. Julien, F. Biraben, Eur. Phys. J. D 60(2), 243 (2010)

19. S. Bourzeix, B. de Beauvoir, F. Nez, M.D. Plimmer, F. de Tomasi, L. Julien, F. Biraben, D.N.

Stacey, Phys. Rev. Lett. 76(3), 384 (1996)

20. D.J. Berkeland, E.A. Hinds, M.G. Boshier, Phys. Rev. Lett. 75(13), 2470 (1995)

21. E.W. Hagley, F.M. Pipkin, Phys. Rev. Lett. 72(8), 1172 (1994)

22. S.R. Lundeen, F.M. Pipkin, Metrologia 22(1), 9 (1986)

23. G. Newton, D.A. Andrews, P.J. Unsworth, Philos. Trans. R. Soc. London, Ser. A 290(1373),

373 (1979)

24. I. Sick, Can. J. Phys. 85(5), 409 (2007)

25. J.C. Bernauer et al., Phys. Rev. Lett. 105, 242001 (2010)

26. I. Sick, in Precision Physics of Simple Atoms and Molecules, ed. by S.G. Karshenboim

(Springer, Lect. Notes Phys. 745, Berlin, Heidelberg, 2008), pp. 57–77

27. Z. Zhan et al., Phys. Lett. B 705(1-2), 59 (2011)

28. G. Ron et al., Phys. Rev. C 84, 055204 (2011)

29. I. Sick, Few-Body Syst. 50(1-4), 367 (2011)

30. I. Sick, Prog. Part. Nucl. Phys. 67(2), 473 (2012)

31. C. Adamuscin, S. Dubnicka, A.Z. Dubnickova, Prog. Part. Nucl. Phys. 67(2), 479 (2012)

32. R. Pohl, R. Gilman, G.A. Miller, K. Pachucki, (2013). ArXiv:1301.0905

33. R. Pohl et al., Nature 466(7303), 213 (2010)

34. R. Pohl et al., J. Phys. Conf. Ser. 264, 012008 (2011)

35. A. Antognini et al., Science 339(6118), 417 (2013)

36. A. Antognini, F. Kottmann, F. Biraben, P. Indelicato, F. Nez, R. Pohl, Ann. Phys. (N.Y.) 331,

127 (2013)

37. P. Indelicato, Phys. Rev. A 87, 022501 (2013)

38. E. Borie, Ann. Phys. (N.Y.) 327(3), 733 (2012). ArXiv:1103.1772-v6
30 Joseph N. Tan and Peter J. Mohr

39. S.G. Karshenboim, V.G. Ivanov, E.Y. Korzinin, Phys. Rev. A 85, 032509 (2012)

40. U.D. Jentschura, Ann. Phys. (N.Y.) 326(2), 500 (2011)

41. U.D. Jentschura, Ann. Phys. (N.Y.) 326(2), 516 (2011)

42. J.D. Carroll, A.W. Thomas, J. Rafelski, G.A. Miller, Phys. Rev. A 84, 012506 (2011)

43. I. Sick, Phys. Lett. B 576(1-2), 62 (2003)

44. J.C. De Vries, A precision millimeter-wave measurement of the rydberg frequency. Ph.D.

thesis, MIT (2001)

45. T.W. H ̈ansch, Rev. Mod. Phys. 78(4), 1297 (2006)

46. M. Hori, A. Dax, J. Eades, K. Gomikawa, R. Hayano, N. Ono, W. Pirkl, E. Widmann, H.A.

Torii, B. Juh ́asz, D. Barna, D. Horv ́ath, Phys. Rev. Lett. 96, 243401 (2006)

47. L.S. Ma, Z. Bi, A. Bartels, L. Robertsson, M. Zucco, R.S. Windeler, G. Wilpers, C. Oates,

L. Hollberg, S.A. Diddams, Science 303(5665), 1843 (2004)

48. A. Marian, M.C. Stowe, J.R. Lawall, D. Felinto, J. Ye, Science 306(5704), 2063 (2004)

49. J.R. Sapirstein, D.R. Yennie, in Quantum Electrodynamics, ed. by T. Kinoshita (World Scientific, Singapore, 1990), chap. 12, pp. 560–672

50. M.I. Eides, H. Grotch, V.A. Shelyuto, Phys. Rep. 342(2-3), 63 (2001)

51. G. Gabrielse, P. Larochelle, D. Le Sage, B. Levitt, W.S. Kolthammer, R. McConnell,

P. Richerme, J. Wrubel, A. Speck, M.C. George, D. Grzonka, W. Oelert, T. Sefzick, Z. Zhang,

A. Carew, D. Comeau, E.A. Hessels, C.H. Storry, M. Weel, J. Walz, Phys. Rev. Lett. 100,

113001 (2008)

52. U.D. Jentschura, P.J. Mohr, J.N. Tan, B.J. Wundt, Can. J. Phys. 87(7), 757 (2009)

53. U.D. Jentschura, P.J. Mohr, J.N. Tan, J. Phys. B 43, 074002 (2010)

54. E.H. Wichmann, N.M. Kroll, Phys. Rev. 101(2), 843 (1956)

55. U.D. Jentschura, Phys. Rev. A 74, 062517 (2006)

56. E.O. Le Bigot, U.D. Jentschura, P.J. Mohr, P. Indelicato, G. Soff, Phys. Rev. A 68, 042101

(2003)

57. G. Audi, A.H. Wapstra, C. Thibault, Nucl. Phys. A 729(1), 337 (2003)

58. H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-electron Atoms (Academic

Press, New York, 1957)

59. F. Low, Phys. Rev. 88(1), 53 (1952)

60. U.D. Jentschura, P.J. Mohr, Can. J. Phys. 80(6), 633 (2002)

61. J.C. De Vries, A precision millimeter-wave measurement of the rydberg frequency. Ph.D.

thesis, MIT (2001)

62. D. Kleppner. private communication (2012)

63. W.M. Itano, et al., Phys. Scripta T59, 106 (1995)

64. G. Gabrielse, in Advances in Atomic Molecular, and Optical Physics,, vol. 50, ed. by B. Bederson, H. Walthers (Elsevier, San Diego, 2005), pp. 155–217

65. L.P. Ratliff, E.W. Bell, D.C. Parks, A.I. Pikin, J.D. Gillaspy, Rev. Sci. Instrum. 68, 1998 (1997)

66. L. Ratliff, J. Roberts, in Trapping Highly Charged Ions: Fundamentals & Applications, ed. by

J. Gillaspy (Nova Science Publishers, Hauppauge, 2001), p. 257

67. M.R. Flannery, D. Vrinceanu, Phys. Rev. A 68, 030502(R) (2003)

68. J.D. Gillaspy, L.P. Ratliff, J.R. Roberts, E. Tak ́acs, Highly charged ions: Publications of the

ebit project, 1993-2001. Special Publication 972, NIST (2001)

69. J.N. Tan, S.M. Brewer, N.D. Guise, Rev. Sci. Instrum. 83(2), 023103 (2012). DOI

10.1063/1.3685246. URL http://link.aip.org/link/?RSI/83/023103/1

70. G.E. Holland, C.N. Boyer, J.F. Seely, J.N. Tan, J.M. Pomeroy, J.D. Gillaspy, Rev. Sci. Instrum.

76(7), 073304 (2005)

71. S.M. Brewer, N.D. Guise, J.N. Tan. Capture and isolation of highly charged ions in a compact

penning trap. arXiv. Submitted to Phys. Rev. A

72. N.D. Guise, S.M. Brewer, J.N. Tan, ArXiv e-prints (2012)

73. S.M. Brewer, N.D. Guise, J.N. Tan. Observing forbidden radiative decay of highly charged

ions in a compact penning trap. BAPS.2012.DAMOP.U3.8 (2012)

74. N.D. Guise, S.M. Brewer, J.N. Tan. Charge exchange and spectroscopy with isolated highlycharged ions. BAPS.2012.DAMOP.U3.6 (2012)
1 Tests of Theory in Rydberg States of One-electron Ions 31

75. S.R. Lundeen, C.W. Fehrenbach, Phys. Rev. A 75(3), 032523 (2007). DOI

10.1103/PhysRevA.75.032523

76. R. Olson, J. Phys. B: Atom. Molec. Phys. 13, 483 (1980)

77. B.D. DePaola, M.T. Huang, S. Winecki, M.P. Stockli, Y. Kanai, S.R. Lundeen, C.W. Fehrenbach, S.A. Arko, Phys. Rev. A 52, 2136 (1995)

78. P.J. Mohr, B.N. Taylor, Rev. Mod. Phys. 77(1), 1 (2005)