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The Energy Levels of Muonic Atoms

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The energy levels of ii-iuonic atoii-is- E. Boric Kernforschungszentrum Karlsruhe GmbH, Institut fiir Kernphysik and Institut fur Experimentelle Kernphysik der Uniuersita't Karlsruhe, 7500 Karlsruhe 1, Federal Republic of Germany~ G. A. Rinker Theoretical Di Uision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 The theory of muonic atoms is a complex and highly developed combination of nuclear physics, atomic physics, and quantum electrodynamics. Perhaps nowhere else in microscopic physics are such diverse branches so intimately intertwined and yet readily available for precise experimental verification or rejection. In the present review we summarize and discuss all of the most important components of muonic atom theory, and show in selected cases how this theory meets experimental measurements. CONTENTS I ~ INTRODUCTION I. Introduction 67 Perhaps the most interesting theoretical aspect of A. General description of a muonic atom 68 muonic atoms is that their study provides one of the B. Principal kinematics and interactions 69 and com- C. Notation and numerical values 69 most highly developed complete theories of a II. Classical Interactions and Kinematics 69 plex physical system which is amenable to both accurate A. Dirac equation 69 calculation and accurate experimental verification. There B. Zero-order approximations 70 C. Decomposition in spherical coordinates 70 are few serious outstanding difficulties in our present D. Numerical methods 71 understanding. This has not always been so. Each ad- 1. Eigenvalues 71 vance in experimental technique has opened new ques- 2. Solution of the radial Dirac equations 72 E. "Model-independent" interpretation 72 tions which were met with increasingly refined and com- F. Corrections for static nuclear moments 73 plicated theoretical calculations. Old effects have had to l. Electric multipole interactions 74 be recalculated repeatedly to higher precision, and new 2. Magnetic dipole interactions 74 G. Intrinsic nuclear dynamics 75 effects have been added periodically to the list of known 1. Formal description 75 contributions. Meaningful calculations require that some 2. Explicit formulas 77 of these effects be recalculated every time comparison is 3. Radiative transition rates 78 4. Examples 78 made with experiment. For other e6ects, such recalcula- a. General case of nuclear polarization 78 tion is impractical, owing to the complexity of the calcu- b. Helium 80 lations, and in experimental comparisons one resorts ei- c. Lead 81 d. Deformed nuclei 82 ther to published numbers or to semiempirical approxi- e. Transitional nuclei 82 mations to the more laboriously calculated values. H. Translational nuclear motion 83 The main intent of the present review is to collect in 1. Nonrelativistic corrections 83 all 2. Relativistic corrections 83 one place in logical order currently available methods, I. Electron screening 86 formulas, and results which are necessary for the accu- III. Quantum Electrodynamic (QED) Corrections 87 rate calculation of muonic atom energy levels throughout A. Vacuum polarization 87 the periodic table. Where possible, we either derive or 1. Order +ZAN 87 a. e+e pairs 87 motivate the relevant expressions. Naturally, some omis- b. p+g pairs 89 sions must be made. We hope and believe that these c. Effect on higher nuclear multipoles 89 d. Effect on the recoil correction 89 omissions will be unimportant for most cases of interest. e. Hadronic vacuum polarization 89 In selected cases we present experimental results which 2. Ordero~Za 90 bear directly upon the validity of the calculations, but we e'e pairs 90 b. Mixed p —e vacuum polarization 90 do not attempt an exhaustive review of the present exper- 3. Ordersn(Zn)", n = 3,5,7. 91 imental situation. For this the reader is referred to the 94 4. Order n'(Zn)' review by Engfer et al. (1974). B. Self-energy 99 1. Orderso, (Zn)", n ~ 1 99 A number of excellent books and review articles have 2. Ordersn~(ZO. ),n ~ 1 101 appeared on this subject in the last 15 years (Devons and C. Anomalous interactions 101 Duerdoth, 1968; Wu and Wilets, 1969; Kim, 1971; Bar- D. Experimental tests of QED with muonic atoms 102 1. Heavy elements 103 rett, 1974; Hufner et al. , 1977; Scheck, 1977; Hughes and 2. Experiments using crystal spectrometers 109 Wu, 1975), in which a wealth of information is con- = 3. Very light elements (Z 1,2) 110 tained. An additional source of information, insight, and IV. Acknowledgments 113 Appendix 113 inspiration is the classic paper of Wheeler (1949), which References 114 marked the beginning of the theory of muonic atoms and Reviews of Modern Physics, Vol. 54, No. I, January 1982 Copyright 1982 American Physical Society E. Boric and G. A. Rinker:in er: Enernergy levels of muonic atoms ~ ~ ' set forth numerous physicalca ideasi eas whichw i are still impor- transition energies comparable e too nucnuclearear excitationex ener- nsu these refer- gies (see Fi'g.. 2).. Th e capture and cascade poroc tk ences for alternaterna e disiscussions from different po s 0 e or er 10 —10 sec so er istorical details. time the muon reach es th e 1s state , it't as barely begun its life. Ther A. General description of a moonie atom ppens most of the time via deca int eu rinos. n heav nuclei ' %'hen a negative muon is stos oppeded in matter, it passes usually capturedre by th e nucleus via t e elementary reac- throuug h a variety of environments before 6na11 tion p +p~n+v&.~n Its mass a ears a vicinity of an atomic nucleus. In the earl gy or as neutrino kinetic energy. o a om as ddoes a free elec- Because of the variety of ph y sical ductse which occur tron, but graduallua y it loses energy until it is capt d n e ormation and the destruestruction of a muonic , i s stu y involves a number ofo diistinct physical a i i y or capture into various ororbitsi s is knnown to theories and types of measurement. Thhose effects con- ' e c emical corn position and crystalline cerned with thee captureca t and cascade throu gh the elect o structure oof th e -material (Fermi and T e c emica e ects. Theories of th 1973 Leon and Seki 197977; Kunselman et al., 1 ti 1 iti'veve at present, owin g to th e complexity 1976; Gershteiners tein and Ponomarev 1975 Daniel et QI. , si uation;a,we shall notno discussdis them here 1977, 1978 1979;1 Schneuwly, etal. , 1978' Leon except as theey re1ate to other topics.ics. MeasurementsM in Bergmann, et al , 197.9 Schnchneuwly, 1977; Daniel, 1979. the "quiet" zone,zone h owever, have stimulated a re u er etal. , 1979; Leisi, 1980. In~ ftho t 1 rk oover tth e last1 tenn years. Discussion of ' ia istri ution of orbits persistspersis s as tthe muon cascades thesee calculations (primaril y QED) and how th ey relate ' throuug h thee eelectron cloud, ejecting Auger electrons and to experiments formorms a su stantial parta of the present re- e ic ra iation, until it arriarrives in a view. Inn thee ssame context we discusss howow tthe muon can region inside thee innermostinn electron orbit. Th be viewed not as a known probe but as a pa rt'ic1e whose of re 1ative quiet, bein ertur & assumed propertiesies can be tested. Interactionn with the its stationary states only b its int p extended nucleus is thee oold est use of the muon as a probe gy [muonic atoms provided th e first accurate meeasurements w' ine y essentially static interactions of nuclear size (Fitchi c and Rainwater, 1953) . Wee also an t e nucleus, and by a variety of discuss the calculationon ofo tth ese nuclear eQects oth static quantum electrodynamical (QED) effects (see Fi and dynamic, inn detaile ai and relate these to someome sesel ected ansitions provide experimental measuremurements.t . F'inall one gen ests of our understanding of QED. As the 1, d h e wea k interaction and nuclearnuc ear excitation b' ' 's muon moves into lowwer or its it is affected more strong- spectra through thee destruction of the muonicmu atom by ly by th e size and sha pe of the nucleus and its ossibili re. n practice, details of the we for excitation. For all butu tth e 1'ightest nuclei, muon or- 1"' '" 'h'e details of nuclear structureuc ure, and al- bits can be comparable e too tthee nuclearn size, and muon though considerable e worwork hasas been done on the subject, O O O- O O O Z=82 CI-h2 O O ~ CV ~O U + O CQ O O O O e 1s (x10) O O CI-O O O I I I I I I 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 r(f rn) ' FIG. 1. MuMuon wave functions (solid lines) for rela FIG. 2. Muon 1 i o red o th e electronoon 1s1 wave function I, dashed line). Rev. Mod. Phys. , Vol.o . 54,54 No. 1, January 1982 E. Boric and G. A. Rinker: Energy levels of muonic atoms we shall not discuss it here. For more information, see constants which reduce the dynamic and quantum in- the reviews by Mukhopadhyay (1977, 1980). teractions among the diff'erent constituents to small per- The principal quantities to be calculated for a muonic turbations and allow the approximate isolation of many atom are the energies and relative rates for various radia- degrees of freedom. The time-averaged classical static tive transitions. To our knowledge, no quantitative mea- interactions determine nearly all of the basic structure of surements of Auger transitions exist.
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