The electric multipole hyperfine interaction in muonic atoms is discussed. In particular, the influence of the finite size of the nuclear electric multipole-charge distribution on the values of nuclear spectroscopic multipole moments that are extracted from muonic hyperfine measurements is considered. It is shown that nuclear electric quadrupole moments can be deduced from the observed hyperfine splittings of muonic M X-ray transitions with high precision and practically independently of the model nuclear quadrupole-charge density. Measurements of the ground-state electric quadrupole moments of 11 deformed nuclei in and near the rare-earth region are discussed.

Monopole and quadrupole charge distributions of $^{176}\mathrm{Hf}$, $^{177}\mathrm{Hf}$, $^{178}\mathrm{Hf}$, $^{179}\mathrm{Hf}$, and $^{180}\mathrm{Hf}$ were investigated by muonic atom $K$ and $L$ x-ray measurements. The model-independent Barrett charge radii ${R}_{k}$ and the isotope shifts $\ensuremath{\Delta}{R}_{k}$ were measured, and values of $〈{r}^{2}〉$ and $\ensuremath{\Delta}〈{r}^{2}〉$ were deduced. A weak odd-even staggering of the nuclear charge radii was observed for the series $^{176\ensuremath{-}178}\mathrm{Hf}$ and $^{178\ensuremath{-}180}\mathrm{Hf}$. A large negative isomer shift was observed in the ${2}^{+}$ state of the $^{176}\mathrm{Hf}$ nucleus, a fact that existing theories do not explain. The quadrupole moments of the first excited states of the hafnium nuclei were determined to be ${Q}^{176}({2}^{+})=\ensuremath{-}2.10(2) e$ b, ${Q}^{177}({\frac{9}{2}}^{\ensuremath{-}})=1.30(2) e$ b, ${Q}^{178}({2}^{+})=\ensuremath{-}2.02(2) e$ b, ${Q}^{179}({\frac{11}{2}}^{+})=1.88(3) e$ b, and ${Q}^{180}({2}^{+})=\ensuremath{-}2.00(2) e$ b. These quadrupole moments and the simultaneously determined $B(E2)$ values for the respective nuclei are in satisfactory agreement with the predictions of the axially symmetric rotor model.

Monopole and quadrupole charge distributions of $^{151}\mathrm{Eu}$ and $^{153}\mathrm{Eu}$ were investigated by muonic atom $K$ and $L$ x-ray measurements. The model-independent Barrett charge radii ${R}_{k}$ and the isotope shift $\ensuremath{\Delta}{R}_{k}$ were measured, and the value of $\ensuremath{\Delta}〈{r}^{2}〉=0.606(18)$ ${\mathrm{fm}}^{2}$ was deduced. This isotope shift is the largest known of all nuclear pairs. The isomer shift of the first excited state of $^{153}\mathrm{Eu}$ is found to be close to zero, in contrast to the large isomer shifts observed in its neighbors: $^{152}\mathrm{Sm}$ and $^{154}\mathrm{Gd}$. The quadrupole moments of the first excited states were determined as ${Q}^{151}({\frac{7}{2}}^{+})=1.28(2) e$ b and ${Q}^{153}({\frac{7}{2}}^{+})=0.44(2) e$ b. The value for $^{151}\mathrm{Eu}$ and its ground-state quadrupole moment of ${Q}^{151}({\frac{5}{2}}^{+})=0.90(1) e$ b reported previously are several times larger than the respective single particle units. This fact shows that a fair amount of collectivity is involved in the ${\frac{5}{2}}^{+}$ ground state and in the ${\frac{7}{2}}^{+}$ first excited state of $^{151}\mathrm{Eu}$.

The X-ray energies and widths of the pionic 4f → 3d transitions (M X-ray) of 166, 168Ev and 176Hf have been measured. The measured widths [Г(166Er) = 19.7(0.9) keV, Г168Er) = 19.4(1.0) keV, and [Г(176Hf) = 27.8(4.3) keV] are in good agreement with theoretical calculations, and suggest that anomalous widths, if they exist, begin at or above Z = 73.

The ground-state quadrupole moments of $^{151}\mathrm{Eu}$, $^{153}\mathrm{Eu}$, $^{159}\mathrm{Tb}$, $^{163}\mathrm{Dy}$, $^{167}\mathrm{Er}$, $^{177}\mathrm{Hf}$, $^{179}\mathrm{Hf}$, $^{191}\mathrm{Ir}$, and $^{193}\mathrm{Ir}$ were determined by measuring the quadrupole hyperfine-splitting energies of muonic $M$ x rays. The results are $Q=0.903(10)e$ b for $^{151}\mathrm{Eu}$, $Q=2.412(21) e$ b for $^{153}\mathrm{Eu}$, $Q=1.432(8) e$ b for $^{159}\mathrm{Tb}$, $Q=2.648(21) e$ b for $^{163}\mathrm{Dy}$, $Q=3,565(29) e$ b for $^{167}\mathrm{Er}$, $Q=3.365(29) e$ b for $^{177}\mathrm{Hf}$, $Q=3.793(33) e$ b for $^{179}\mathrm{Hf}$, $Q=0.816(9) e$ b for $^{191}\mathrm{Ir}$, and $Q=0.751(9) e$ b for $^{193}\mathrm{Ir}$. The present quadrupole moments, compared with values obtained from electronic-atom hyperfine measurements, show that the Sternheimer correction factors used in the rare-earth electronic-atom analysis are unreliable. Systematics of deformation parameters ${\ensuremath{\beta}}_{2}$ calculated from the present quadrupole moments for odd-$A$ nuclei, and from $B(E2)$ values of Coulomb excitation measurements for even-$A$ nuclei, also indicate that the largest deformation change so far known exists between $^{151}\mathrm{Eu}$ and $^{153}\mathrm{Eu}$. Except at the onset of nuclear deformation, the deformation parameters of the odd-$A$ nuclei are quite consistent with those of the even-$A$ neighbors.

Precise intrinsic quadrupole and hexadecapole moments of $^{233,\phantom{\rule{0ex}{0ex}}234,\phantom{\rule{0ex}{0ex}}235,\phantom{\rule{0ex}{0ex}}238}\mathrm{U}$ have been determined from muonic $K$, $L$, $M$, and $N$ x rays. For $^{233,\phantom{\rule{0ex}{0ex}}235}\mathrm{U}$ seven $E2$ matrix elements were independently determined. These $E2$ matrix elements are in good agreement with the adiabatic rotational model; this agreement is further improved if a correction for $\ensuremath{\Delta}K=1$ band mixing is included. The measured hexadecapole moments are in good agreement with shellcorrection calculations and Hartree-Fock calculations.

Monopole and quadrupole charge distributions of $^{154}\mathrm{Gd}$, $^{155}\mathrm{Gd}$, $^{156}\mathrm{Gd}$, $^{157}\mathrm{Gd}$, $^{158}\mathrm{Gd}$, and $^{160}\mathrm{Gd}$ were investigated by muonic-atom $K$ and $L$ x-ray measurements. The model-independent Barrett charge radii ${R}_{k}$ and the isotope shifts $\ensuremath{\Delta}{R}_{k}$ were measured, and values of $〈{r}^{2}〉$ and $\ensuremath{\Delta}〈{r}^{2}〉$ were deduced. A pronounced even-odd staggering effect of the nuclear charge radii was observed for the series $^{156\ensuremath{-}158}\mathrm{Gd}$. The quadrupole moments of the first excited states of the even-$A$ Gd nuclei were determined to be ${Q}^{154}({2}^{+})=\ensuremath{-}1.82(4) e$ b, ${Q}^{156}({2}^{+})=\ensuremath{-}1.93(4) e$ b, ${Q}^{158}({2}^{+})=\ensuremath{-}2.01(4) e$ b, and ${Q}^{160}({2}^{+})=\ensuremath{-}2.08(4) e$ b, and the quadrupole moments of the ${\frac{3}{2}}^{\ensuremath{-}}$ ground states of the odd-$A$ $^{155,157}\mathrm{Gd}$ nuclei were determined to be ${Q}^{155}({\frac{3}{2}}^{\ensuremath{-}})=1.27(3) e$ b and ${Q}^{157}({\frac{3}{2}}^{\ensuremath{-}})=1.35(3) e$ b. Comparison with a separate measurement of the odd-$A$ ground-state quadrupole moments based on the static hyperfine splitting of the muonic $M$ x rays showed that the model error in the extracted quadrupole moments of these nuclei is less than 2 percent. The quadrupole moments and the $B(E2)$ values obtained in the present experiment for the low-lying Gd states are in satisfactory agreement with the axially symmetric rotational model. However, the $^{154}\mathrm{Gd}$ nucleus exhibits a considerable softness as indicated by the isomer shift of the ${2}^{+}$ excited state and by the experimental value of the ratio $\frac{Q({2}^{+})}{B(E2:{0}^{+}\ensuremath{\rightarrow}{2}^{+})}$.NUCLEAR STRUCTURE $^{154\ensuremath{-}158,160}\mathrm{Gd}$; measured muonic x-ray spectra; deduced monopole and quadrupole charge parameters; isotope and isomer shifts.

The ground-state quadrupole moments of $^{151}\mathrm{Eu}$, $^{153}\mathrm{Eu}$, $^{155}\mathrm{Gd}$, $^{157}\mathrm{Gd}$, $^{159}\mathrm{Tb}$, $^{163}\mathrm{Dy}$, $^{167}\mathrm{Er}$, $^{177}\mathrm{Hf}$, $^{179}\mathrm{Hf}$, $^{191}\mathrm{Ir}$, and $^{193}\mathrm{Ir}$ were determined with an uncertainty of less than one percent by measuring the quadrupole hyperfine-splitting energies of muonic $M$ x rays. The results are used to determine experimentally Sternheimer shielding factors for the $4f$, $5d$, and $6p$ electronic states of the respective atoms. The deduced shielding factors for the $5d$ electronic states were found to vary considerably among these elements, presumably as a result of configuration mixing.

The decay of the 2648- and the 803-keV excited states in muonic $^{206}\mathrm{Pb}$ has been observed and the isomer shifts of these states have been measured. The isomer shift of the nuclear ${2}^{+}$ state at 803 keV is -1.45\ifmmode\pm\else\textpm\fi{}0.31 keV; the isomer shift of the nuclear ${3}^{\ensuremath{-}}$ state at 2648 keV is +6.81\ifmmode\pm\else\textpm\fi{}0.46 keV. The measured shifts in $^{206}\mathrm{Pb}$ are similar to those measured in $^{208}\mathrm{Pb}$ and support the suggestion that the corresponding states in those nuclei have similar origin.NUCLEAR STRUCTURE $^{206}\mathrm{Pb}$; measured isomer shifts of 803-keV ${2}^{+}$ and 2648-keV ${3}^{\ensuremath{-}}$ nuclear states.

The influence of the assumed model charge density distribution in deducing nuclear quadrupole parameters from muonic X-ray experiments is investigated within the framework of the Davydov model. It is shown for the case of 192Os that the extracted quadrupole parameters are not seriously affected by a triaxial deformation of the charge distribution. The model uncertainty involved in the muonic K and L X-ray analysis is estimated to be of the order of a few percent for the deduced quadrupole parameters of 192Os. The results of the calculation also preclude the possibility that muonic X-ray measurements can be used to obtain information concerning triaxial deformations of nuclei.

The ground-state quadrupole moments of 155Gd and 157Gd were determined by measuring the quadrupole hyperfine-splitting energies of the M X-rays of muonic Gd atoms. The results are Q = 1.30 ± 0.02 eb for 155Gd and Q = 1.36 ± 0.02 eb for 157Gd.

Precise $2p\ensuremath{-}1s$ muonic x-ray transition energies and energy shifts were measured for the $1{f}_{\frac{7}{2}}$-shell nuclei $^{39,41}\mathrm{K}$, $^{40,42,43,44,46,48}\mathrm{Ca}$, $^{45}\mathrm{Sc}$, $^{46,47,48,49,50}\mathrm{Ti}$, $^{51}\mathrm{V}$, $^{50,52,53,54}\mathrm{Cr}$, $^{55}\mathrm{Mn}$, and $^{56}\mathrm{Fe}$. The data were analyzed in terms of the Barrett moments $〈{r}^{k}{e}^{\ensuremath{-}ar}〉$ of the nuclear charge distribution, from which equivalent nuclear radii ${R}_{k}$ and isotopic and isotonic differences $\ensuremath{\Delta}{R}_{k}$ were computed. For those nuclei for which electron-scattering data were available, model-independent rms radii and radius differences were deduced from a combined analysis of the present data and elastic electron-scattering data. The $\ensuremath{\Delta}N=2$ isotope shifts between even-$A$ nuclei decrease nearly linearly with increasing $N$ and become negative in the second half $24\ensuremath{\le}N\ensuremath{\le}28$ of the $1{f}_{\frac{7}{2}}$ shell; a strong shell-closure effect is evident at $N=28$. The isotope shifts are independent of $Z$, suggesting that the added neutrons interact with the entire proton core rather than with the valence protons. The $\ensuremath{\Delta}Z=2$ isotone shifts between even-$A$ nuclei decrease smoothly and uniformly with increasing $Z$ for $20\ensuremath{\le}Z\ensuremath{\le}28$ and are essentially independent of $N$. Like the isotope shifts, the isotone shifts display a sudden increase at $Z=28$. Both the $\ensuremath{\Delta}N=1$ isotope shifts and the $\ensuremath{\Delta}Z=1$ isotone shifts show pronounced odd-even staggering. A strong correlation is found between nuclear deformation, as revealed by experimental $B(E2)$ values, and the measured isotope and isotone shifts. The measured isotope and isotone shifts, including those for odd-$A$ nuclei, are satisfactorily described by a simple empirical formula. The experimental results were compared with Hartree-Fock calculations; some recent calculations that include ground-state correlations display improved agreement with experiment through the inclusion of these deformation-related effects.NUCLEAR STRUCTURE $^{39,41}\mathrm{K}$, $^{40,42,43,44,46,48}\mathrm{Ca}$, $^{45}\mathrm{Sc}$, $^{46,47,48,49,50}\mathrm{Ti}$, $^{51}\mathrm{V}$, $^{50,52,53,54}\mathrm{Cr}$, $^{55}\mathrm{Mn}$, $^{56}\mathrm{Fe}$; measured muonic x-ray spectra; deduced nuclear charge radii, isotope and isotone shifts; performed model-independent combined analysis of muonic x-ray data and elastic electron scattering data; compared charge parameters with Hartree-Fock and other calculations.

In view of the higher precision of the experimental data in measurements of the energies and relative intensities of the muonic and pionic x-ray transitions, the validity of the approximations made in extracting nuclear spectroscopic values from the raw data must be scrutinized more carefully if the reliability of the extracted parameters are to approach the accuracy of the experimental data. (GHT)

Precision measurements have been made of the muonic x-ray spectra of the transitional nuclei $^{186,188,190,192}\mathrm{Os}$. Equivalent Barrett radii and isotope shifts have been determined, as have isomer shifts of the first excited ${2}^{+}$ states. These results are compared with other experiments and with theoretical calculations. The systematics of isotope shifts in the deformed nuclei are also discussed. Generalized $E2$ moments of the charge distribution have been extracted in a nearly model-independent way and conventional electromagnetic moments have been deduced by assuming a specific transition charge density model. The latter are in good agreement with recent calculations of both the interacting boson approximation and the boson expansion theory. However, a serious discrepancy in the values of the quadrupole moments determined from the muonic and Coulomb excitation experiments is apparent. The model dependence of the muonic results (including the effect of a triaxial model charge distribution) is explored as a possible cause of the discrepancy; however, no effect large enough to explain the discrepancy is found. Furthermore, no feature of the muonic spectra was found which could be used to distinguish between a triaxial and an axially symmetric charge distribution.NUCLEAR STRUCTURE $^{186,188,190,192}\mathrm{Os}$; measured muonic x-ray spectra; deduced monopole and quadrupole charge parameters, isotope and isomer shifts.