1271

Cascade problems in precisionphysics with muonic and hadronichydrogen

V.E. Markushin

Abstract: Recent progress in the theory of atomic cascade in hydrogen-like exotic atoms isreviewed from the viewpoint of precision experiments with µ−p, π−p, K−p, and p̄p atoms.

PACS Nos.: 36.10-k, 32.30Rj, 32.70Jz, 32.80Cy

Résumé : Nous passons en revue les progrès récents de la théorie des cascades atomiquespour les atomes exotiques de type hydrogène dans une optique de mesures de précisionimpliquants des atomes µ−p, π−p, K−p et p̄p.

[Traduit par la Rédaction]

1. Introduction

Exotic atoms provide numerous opportunities to investigate nuclear properties and low-energynuclear reactions in well-defined and control conditions. The properties of atomic states can be calculatedwith high precision from QED (quantum electrodynamics), while the nuclear interaction in exoticatoms can be described with a special kind of perturbation theory based on the hierarchy of scales[1–4]: the range of the nuclear interaction is much smaller then the characteristic atomic scale andthe nuclear-energy scale is much larger than the atomic one. The nuclear interaction usually results insmall corrections to the energies of atomic states; nevertheless, due to the extremely high precision ofspectroscopic methods, some of these nuclear effects can be measured with an accuracy of 1% or better.Of particular interest are the cases where this level of accuracy, which is often considered as a precisionmeasurement in nuclear physics, cannot be reached with other methods.

There are both experimental and theoretical challenges in precision spectroscopy of exotic atoms.The proton-charge radius can be determined with an accuracy of 10−3 from the energy splitting betweenthe 2S and 2P states in muonic hydrogen. This goal is expected to be reached in a muonic-hydrogenLamb-shift experiment [5, 6] presently in progress at PSI (the Paul Scherrer Institute). The feasibilityof this experiment depends crucially on the population and the lifetime of the metastable 2S state ofµ−p, and a good understanding of the atomic cascade is essential for performing this measurement.The πN -scattering length can be determined with an accuracy better than 1% by measuring the nuclearshifts and widths of the K X-ray lines in pionic hydrogen. A significant improvement of the earliermeasurements [7] is expected in the new experiment at PSI [8,9]. The ultimate accuracy of the nuclearwidth is determined by the Doppler-broadening corrections to the line profile, that must be calculated in areliable cascade model. Similar problems exist in the spectroscopy of other hadronic atoms. To improvethe measurement of the K−p-scattering length using X-ray spectroscopy of kaonic hydrogen [10, 11],

Received 9 September 2002. Accepted 10 September 2002. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 8 November 2002.

V.E. Markushin. Paul Scherrer Institute, CH-5232Villigen PSI, Switzerland (e-mail: valeri.markushin@psi.ch).

Can. J. Phys. 80: 1271–1280 (2002) DOI: 10.1139/P02-087 © 2002 NRC Canada

1272 Can. J. Phys. Vol. 80, 2002

Table 1. Cascade processes in exotic atoms with Z = 1 and their energy dependence.

Mechanism Example E-dependence

Radiative deexcitation (µp)i → (µp)f + γ NoneExternal Auger effect (µp)i + H2 → (µp)f + e− + H+

2 WeakStark mixing (µp)nl + H → (µp)nl′ + H ModerateElastic scattering (µp)n + H → (µp)n + H StrongCoulomb transitions (µp)ni + H2 → (µp)nf + H + H, nf < ni StrongTransfer (isotope exchange) (µp)n + d → (µd)n + p StrongAbsorption (π−p)nS → π0 + n, γ + n None

one needs to use theoretical predictions for the relative yields of the individualK lines, which cannot beresolved experimentally because of the large nuclear width of the (K−p)1S state exceeding the spacingbetween the K lines. The 2P nuclear widths of antiprotonic hydrogen can be determined from the LX-ray spectra [12, 13], and the Doppler-broadening corrections turn out to be important for precisionmeasurements.

At the same time, these precision spectroscopic experiments allow one to obtain detailed informationabout the kinetics of atomic cascade that was not available in the past. On the theoretical side, theultimate goal is the ab initio cascade calculations. By confronting the theoretical predictions with theexperimental data one can verify whether the cascade model is complete and all important cascadeprocesses are adequately treated. In this paper, I shall focus my attention on the hydrogen-like exoticatoms with Z = 1, which, despite their simple structure, exhibit very complicated “life histories”reflecting a broad variety of reactions occurring during the cascade, and I will discuss the cascadefeatures that are important for the feasibility of the above-mentioned precision experiments.

2. Extended standard cascade model

Exotic atoms are usually formed in highly excited states with the principal quantum number n ∼√mx/me where mx is the reduced mass of the exotic atom x−p and me is the electron mass [14].

The exotic atoms are deexcited in a sequence of collisional and radiative processes forming a so-calledatomic cascade. The standard cascade model (SCM) for hydrogen-like exotic atoms developed in the1960s and 1980s [14, 15] was able to describe the basic features of the atomic cascade, and, with theuse of a few phenomenological parameters, provided a fair qualitative description of the X-ray yieldsand absorption fractions [15–17]. For the sake of simplicity, the SCM does not take into account theinterplay between the internal and external degrees of freedom: the cascade rates calculated for a fixedvalue of the kinetic energy T are used in the kinetics calculation, with the value of T being treated as afit parameter. This severe limitation of the SCM was removed in the extended standard cascade model(ESCM) [18, 19] where the evolution of the energy distribution of the exotic atoms during the cascadeis calculated from a master equation.

A general review of the deexcitation processes in light exotic atoms is beyond the scope of this paperand can be found in refs. 14, 15, and 19 and references therein.A brief summary of the cascade processesincluded in the ESCM is given in Table 1. A significant improvement over the previous calculations[18–21] was achieved in recent cascade studies [22–25], where a new set of differential cross sections forelastic scattering and Stark mixing was used. The cross sections were calculated using the close-couplingmethod for low-n states and the classical-trajectory Monte Carlo (CTMC) method for high-n states[24,26–28]. The inelastic collisions with hydrogen molecules were simultaneously calculated using theCTMC method, and it was found that the Coulomb transitions dominate the collisional deexcitation athigh n [24]. The Coulomb deexcitation leads to significant acceleration of exotic atoms because thetransition energy is shared by the recoiling particles having comparable masses (contrary to the radiativeor Auger deexcitation where the transition energy is mainly carried away by the light particles).

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Using the self-consistent set of collisional cross sections one can describe the competition betweenthe acceleration and deceleration mechanisms without employing any fit parameters. At present, theparameter-free cascade calculations are available for low-density targets where remaining theoreticaluncertainties, which are mainly related to the Coulomb transitions and additional cascade mechanismsat the lower stage of the cascade, play a minor role (at low density the lower part of the atomic cascade isdominated by the radiative deexcitation). By confronting the theoretical predictions with experimentaldata one can determine the domain of validity of the ESCM and identify possible significant mechanismsbeyond the ESCM.

3. Muonic hydrogen

A precision measurement of the Lamb shift (the 2S−2P energy difference) in muonic hydrogen canimprove the uncertainty in the RMS proton-charge radius rp by more than one order of magnitude [6].The proton radius is not only a fundamental quantity [29], but is also important for further precisiontests of QED in bound systems with ordinary hydrogen, which are presently limited by the experimentalerror in the rp. The QED calculations for the µ−p Lamb shift are available at a precision level of 10−6

(see refs. 30–32 and references therein)

�E2S−2P = 210.005(6) meV − 5.166r2p meV fm2 (1)

= 207.167(107) meV, r2p = 0.880(15) fm2 (see ref. 33) (2)

Since the finite-size effect is about 2%, a precise measurement of the Lamb shift can provide a value ofthe proton radius with an accuracy of 0.1%. This goal is expected to be reached by the Muonic HydrogenLamb Shift Collaboration group at PSI [6].

The feasibility of the µ−p Lamb-shift experiment depends on the population and the lifetime ofthe metastable 2S state. The metastability of the 2S state, which has a larger binding energy than the2P state, see (1), depends, in particular, on the kinetic energy. The (µ−p)2S atoms below the thresholdof the 2S → 2P Stark mixing, Tlab = 0.3 eV, can be metastable because they are “immune” todeexcitaion via the Stark transition to the 2P state followed by the fast radiative transition 2P → 1S (theradiative quenching during the Stark collisions is small). For kinetic energies above the 2P threshold,the collisions with the target molecules lead to a deceleration competing with the depletion via the2S → 2P Stark transitions. The lifetime of the metastable (µ−p)2S is determined by the muon lifetimeand the collisional (nonradiative) quenching via formation of molecular resonances [6, 34].

The existence of long-lived metastable muonic hydrogen atoms in the 2S state has been demonstratedin a series of experiments at PSI [6,34] where the initial kinetic energy distributions of muonic hydrogenatoms in the ground state were measured using a time-of-flight (TOF) method. Low-energy muons werestopped in a cylindrical low-pressure gas target, and the time difference between the muon stop and thearrival of the µ−p atom at the inner gold-coated target wall was measured. The target was located in astrong axial magnetic field that forced the muons to stop near the axis of the target, but did not affectthe neutral µ−p atom. The kinetic energy distributions were reconstructed from the TOF spectra inthe density range 0.06–16 mbar. The atomic cascade at 0.06 mbar is nearly pure radiative (the averagenumber of collisions is less than one), therefore, the kinetic energy distribution changes very little duringthe cascade. This allows one to determine the initial kinetic-energy distribution in the very beginningof the atomic cascade and use it in the cascade calculations at higher densities.

In the recent theoretical studies of the atomic cascade in µ−p [24], the evolution of the kinetic-energy distribution, for the first time, has been calculated from the very beginning of the cascade. Thecascade calculations used the new results for the collisional cross sections [27]. Figure 1a shows anexample of the n dependence of the rates of different cascade processes. Especially important wastaking into account the molecular structure of the hydrogen molecules in the Coulomb deexcitation ofthe µ−p states with high n, see Fig. 1b. The Coulomb deexcitation leads to a significant acceleration at

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Fig. 1. (a) The l-average rates for muonic hydrogen with the laboratory kinetic energy T = 1 eV in a gaseoustarget at 10−4 LHD (79 mbar). The Coulomb deexcitation (filled diamonds) and Stark mixing (filled triangles) ratescalculated in the CTMC model [24] are shown in comparison with the results of the semiclassical fixed-field modelfor Stark mixing (continuous line), Auger deexcitation (light broken line), and nuclear absorption during collisions(broken line) [24]. The Coulomb deexcitation rate from ref. 35 is shown with a broken-dotted line. The radiativerates np → 1s and n(n − 1) → (n − 1)(n − 2) are shown with dotted lines. (b) The Coulomb deexcitation crosssections (in atomic units) for (µp)ni=13 scattering from molecular and atomic hydrogen at the laboratory kineticenergy T = 1 eV as a function of the final state nf [24].

0 5 10 15 20

n

10–4

10–3

10–2

10–1

Rate

(10

12

s–

)

Radiative

Stark

Coulomb

Auger

µp, 10–4

LHD

6 10 11 12 13

nf

0

2

4

6

8

10

σ(a

0

2)

CMC, H2

CMC, H

Bracci, Fiorentini

ni=13

(a) (b)

1

7 8 9

Fig. 2. (a) The calculated cumulative energy distributionW(T ) of theµ−p atoms at the end of the cascade for initialconditions: ni = 14 and T0 = 0.5 eV [24]. The data are from ref. 34. (b) The calculated density dependence [24]of the median kinetic energy of the µ−p at the end of the cascade for different initial average kinetic energies andni = 14. The result for atomic target is shown for T0 = 0.5 eV. The data are from ref. 34.

100

101

T (eV)

0

0.2

0.4

0.6

0.8

1.0

W(T

)

0.06 mbar

0.25 mbar

1 mbar

4 mbar

16 mbar

n=140.5 eVMolecular

10–7

10–6

10–5

10–4

10–3

N/LHD

0

1

2

3

4

Tm

edia

n(e

V)

0.25 eV

0.5 eV

1.0 eV

Atomic H

n=14

(a) (b)

the very beginning of the atomic cascade due to the dominance of transitions with �n > 1. This newversion of ESCM provides a good description of many properties of the atomic cascade without usingany fitting parameters. Figure 2a shows the calculated density dependence of the cumulative energydistribution in the ground state in comparison with the experimental data [34]. The calculated increaseof the (µ−p)1s median kinetic energy with the density is in agreement with the data [34] for the initialconditions corresponding to kinetic energies about 0.5 eV and principal quantum number n ≈ 14, seeFig. 2b. The molecular structure of the target is essential for explaining the observed density dependenceof the kinetic-energy distribution at the end of the cascade.

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Markushin 1275

Fig. 3. (a) The density dependence of the relative X-ray yields, Kα , Kβ , and Kγ , and the absolute total yield Ktot

in muonic hydrogen [24]. The experimental data are from refs. 38–40. (b) The calculated 2s arrival probabilityε2s (broken line) [24], in comparison with the values derived from X-ray measurements: ref. 38 (circles), ref. 36(triangles), and ref. 37 (filled square). The continuous line shows the calculated metastable 2s fraction R2s ; thecorresponding experimental data (filled diamonds) are from ref. 34.

10–7

10–6

10–5

10–4

10–3

10–2

10–1

100

N/LHD

0

0.2

0.4

0.6

0.8

1.0

Yie

ld KαKβKγ

Kα

Kβ

K γ

K

Ktot

(a) (b)

δ

The calculated X-ray yields in muonic hydrogen are in good agreement with the data as shown inFig. 3a. The measured ratio of the Kα and Kβ yields in liquid hydrogen can be reproduced only if themetastable 2S state is depopulated via a nonradiative quenching (see ref. 34 and references therein fora discussion of the quenching mechanisms).

The calculated density dependence of the metastable 2S fraction is shown in Fig. 3b. The calculatedarrival probability (the probability of the deexcitations from the states n > 2 to the 2S state) is in goodagreement with the estimates based on the measured X-ray yields (see ref. 34 and references therein).As discussed above, only a fraction of the initial 2S population can survive to form the metastable stateif the radiative deexcitation of the 2P state is faster than the Stark mixing 2P → 2S. The relationshipbetween these two processes changes at high density when the slowing down proceeds faster than theradiative transition 2P → 1S, and more than half of all muons in liquid hydrogen go to the 2S metastablestate. However, the metastable 2S state in liquid hydrogen has a short lifetime because of collisionalquenching. Most suitable for the Lamb-shift experiment is the density range around 1 mbar where thecollisional-quenching rate is comparable with the muon lifetime [6, 34]. The calculated 2S metastablefraction at 1 mbar is about 1% in perfect agreement with the experimental data. In conclusion, thepopulation and the lifetime of the metastable (µ−p)2S state have been found to be in the range suitablefor a feasible measurement of the Lamb shift by means of laser-induced spectroscopy [6].

4. Pionic hydrogen

The pion–nucleon scattering lengths can be determined from the nuclear shift �E1S and width $1Sof the 1S state of pionic hydrogen using the Deser-type formula [1]

�E1S = −4 aπ−p→π−prB

E1S(1 + δE) (3)

$1S = 8q

rB

(1 + 1

P

)|aπ−p→π0n|2 E1S(1 + δ$) (4)

where aπ−p→π−p is the π−p scattering length, aπ−p→π0n is the S-wave amplitude of the charge-exchange reactionπ−p → π0n, rB = 222.56 fm is the Bohr radius of pionic hydrogen,E1S = 3238 eVis the point-like Coulomb binding energy of the 1S state, P = 1.55 is the Panofsky ratio taking into

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1276 Can. J. Phys. Vol. 80, 2002

Fig. 4. (a) The energy distribution of the π−p atoms at the instant of the nuclear reaction (π−p) → π0 +n in liquidhydrogen reconstructed from the neutron TOF spectra [43]. The kinetic energies corresponding to the Coulombtransitions ni → nf are shown by arrows, with horizontal bars indicating the experimental energy resolution.(b) The calculated cumulative energy distribution of the π−p atoms at the instant of the nuclear reaction [23] incomparison with the experimental data [43].

Tπp ( eV )

f (T

πp)

(eV

-1)

65

54

43

32

64

53

LH2

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

50 100 150 200 2500

0.2

0.4

0.6

0.8

1.0

1 10 102

E (eV)W

(E) theory

data PSI-2000

π-p liq.(a) (b)

account the reaction channel (π−p) → γ + n, q = 28.04 MeV/c is the CMS momentum of the π0 inthe charge-exchange reaction, δE and δ$ are small electromagnetic corrections. PrecisionπN -scatteringlength measurements are of fundamental importance for the theory of strong interaction and symmetrybreaking in hadronic systems. The goal of the new experiment at PSI [9] is to improve the presentvalues [7]

�E1S = −7.108 ± 0.013 ± 0.034 eV (5)

$1S = 0.868 ± 0.040 ± 0.038 eV (6)

by measuring the $1S with an accuracy about 1%. At this level of precision, the Doppler-broadeningcorrections determine the ultimate limits of the π−p X-ray spectroscopy.

The atomic cascades in theµ−p and π−p atoms are expected to have many similarities as they havecomparable reduced masses. The main difference in the atomic cascade is due to the nuclear absorption inπ−p that is especially important for low-lying levels (n < 6) where the absorption significantly reducesthe population of atomic states. The π−p → π0n reaction during the atomic cascade was studied usingthe neutron time-of-flight (n-TOF) method in refs. 41–43 where the kinetic-energy distribution at theinstant of nuclear reaction was investigated by measuring the Doppler broadening of the n-ToF spectra.The discovery of a high-energy component in the π−p kinetic-energy distribution in ref. 41 was thefirst direct evidence of the importance of the Coulomb-like acceleration in the atomic cascade. Thekinetic-energy distributions reconstructed from the n-TOF spectra in ref. 43 reveal important detailsconfirming the Coulomb acceleration mechanism, see Fig. 4.

The kinetic-energy distribution at the instant of the radiative transitions nP → 1S are dependenton the initial state n and the target density, and so are the profiles of the X-ray lines. A typical exampleof the cumulative kinetic-energy distribution and the corresponding Doppler broadening of the Kβ linein a gaseous target is shown in Fig. 5 [23]. If the Doppler-broadening corrections are applied properly,then the values for the nuclear width $1S obtained from measurements at different target densities mustagree with each other. This check of the self-consistency of the data analysis is crucial for a reliabledetermination of systematic errors.

Additional checks of the cascade model, which was used in the calculations of the Doppler-broadening corrections, can be done by using the same model for the muonic hydrogen and comparing

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Fig. 5. (a) The calculated cumulative energy distribution of pionic hydrogen at the instant of the radiative transition3P → 1S at 3 bar. (b) The calculated profile of theKβ line in pionic hydrogen at 3 bar with the Doppler broadeningtaken into account (continuous line) in comparison with the Lorentzian shape corresponding to the width of the 1Sstate (broken line).

∆E (eV)w

(eV

-1)

0

0.2

0.4

0.6

0.8

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1.0

10-1

1 10 102

E (eV)

W(E

)

πp

3P-1S

(a) (b)

Fig. 6. (a) The calculated cumulative energy distribution of muonic hydrogen at the instant of the radiative transition3P → 1S at 3 bar. (b) The calculated Doppler broadening of the Kβ line in muonic hydrogen at 3 bar.

0

0.2

0.4

0.6

0.8

1.0

10-1

1 10 102

E (eV)

W(E

)

µp

3P-1S

0

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

∆E (eV)

Y (

eV-1

)

µp

3P-1S

(a) (b)

the calculated X-ray profiles with measured ones. Figure 6 demonstrates the cumulative energy distribu-tion for the µ−p atom at the instant of the radiative transition 3P → 1S at 3 bar and the correspondingDoppler profile of theKβ line. There is a clear similarity between theµ−p andπ−p energy distributionsunder the same experimental conditions (compare Figs. 5a and 6a) that warrants a detailed study of theDoppler broadening of the µ−p X-ray lines. The measurements of the µ−p X-ray Doppler broadeningare planned as a part of the pionic hydrogen experiment [9].

5. Antiprotonic hydrogen

The K and L X-ray yields and cascade times in antiprotonic hydrogen calculated in the ESCM[24] are in good agreement with the data as shown in Fig. 7. For the first time, good agreement hasbeen obtained without using any phenomenological tuning parameters, which were necessary in thecalculations using the SCM [15]. Due to the fast collisional deexcitation at high n with �n > 1, the

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1278 Can. J. Phys. Vol. 80, 2002

Fig. 7. The density dependence of the absolute K and L X-ray yields in antiprotonic hydrogen [24]. The experi-mental data are from refs. 44–47.

107–

106–

105–

104–

103–

102–

N/LHD

0

0.01

Yie

ld

Ktot

KαKβ

Ktot

Kα

Kβ

107–

106–

105–

104–

103–

10–2

N/LHD

102

101

100

Yie

ld

Ltot

LαLβ

Ltot

Lα

Lβ

(a) (b)

calculated cascade times are much shorter than in the earlier calculations [48] and are in good agreementwith the measurements by the OBELIX collaboration [49]. The acceleration of the p̄p atoms duringthe cascade results in a significant Doppler broadening of the L X-ray lines. This effect, which wasneglected in the past, must be taken into account when the hadronic widths and shifts are obtained fromthe measured line profiles.

6. Conclusion

A reliable theoretical model of the atomic cascade is essential for the current generation of precisionexperiments with light exotic atoms. One of the main problems is a detailed description of the kinetic-energy distributions at the instant of the radiative transitions in pionic and antiprotonic hydrogen.The corresponding Doppler-broadening corrections determine the ultimate precision with which thestrong interaction widths can be determined from the X-ray spectra. The precision experiments withexotic atoms are typically low-count-rate measurements, therefore, they can significantly benefit fromtheoretical guidance in finding optimum experimental conditions to minimize statistical errors. In thissituation, it is not sufficient to rely anymore on the phenomenological cascade models that can be fit tothe data with some ad-hoc parameters. To meet these experimental challenges one needs the ab-initiocascade calculations.

The first systematic implementation of this approach was undertaken in the recent calculations of theatomic cascades inµ−p, π−p,K−p, and p̄p in the framework of the extended standard cascade model[23, 24]. For the first time, the acceleration and deceleration mechanism have been taken into accountfrom the very beginning of the atomic cascade, and the evolution of the kinetic-energy distributionhas been calculated in a straightforward way from the master equation. As a result, a number of long-standing problems have been solved, in particular, a long-standing puzzle of chemical deexcitation [14],which was included in many cascade calculations in a pure phenomenological way for about 40 years.The underlying dynamics actually corresponds to the Coulomb deexcitation of highly excited statesin collisions with hydrogen molecules, with the target molecular structure being important for largedecrease of the principal quantum number n in a single collision. The consequences of this mechanismcan be directly seen in the energy distributions of muonic hydrogen in the ground state at low densities[6, 34]. The ESCM also provides a good description of the data on the cascade times and the X-rayyields in antiprotonic hydrogen, with which the earlier calculations always had a problem.

The ESCM will be subjected to stringent tests in the precision experiments with pionic and muonichydrogen. The Doppler broadening of the µ−p X-ray lines will be measured by the pionic-hydrogen

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Markushin 1279

collaboration as a cross check of the atomic cascade model because the acceleration and decelerationprocesses are expected to be similar in the π−p and µ−p atoms. Another critical test of the cascadecalculations will come from the determination of the strong interaction width $1S of the (π−p)1S statefrom the radiative transitions nP → 1S at different densities. While the X-ray line shapes depend,because of the Doppler broadening, on the initial nP state and the target density, the correspondingvalues of $1S obtained by applying theoretically calculated corrections must agree with each other.

The present version of the ESCM is definitely not the final one. In particular, there are indicationsthat additional mechanisms (presently not included in the calculations), like the resonant formation ofmolecular states [50], are important at the lower stage of the cascade. These mechanisms are not veryimportant at low density, when the lower part of the cascade is dominated by the radiative deexcitation,but may not be neglected at high density. Finding disagreements between the “no-free-parameters”cascade calculations and experimental data, in particular, the dependence of these effects on the exper-imental conditions, would be important for identifying these additional cascade mechanisms. Thus, theprecision measurements with exotic atoms are mutually beneficial for both nuclear and atomic physics.

Acknowledgment

The author thanks Thomas Jensen for fruitful collaboration and numerous discussions on the prob-lems of atomic cascade. Stimulating discussions with M. Daum, D. Gotta, P. Hauser, H.J. Leisi,F. Kottmann, R. Pohl, L.M. Simons, and D. Taqqu, were very important for the development of theESCM. The author thanks S. Karshenboim for an invitation to this conference.

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