Energy levels: 1s_1/2, 2p_1/2, 2p_3/2, 3d_3/2, and 3d_5/2 of the muon in the spherical nuclei:¹²⁰Sn,¹⁹⁷Au and²⁰⁸Pb have been calculated under the assumption of harmonic oscillator potential. The levels are corrected for vacuum polarisation. The agreement with experimental values is better than 0·5%. An accurate method of solving the Dirac equation to obtain the energy eigenvalues is outlined. The importance of choosing the ‘classical turning point’ as the radius for matching the interior and exterior solutions is discussed.

Muonic x-ray transitions in various spherical nuclei in the region 13⩽Z⩽83 have been analysed and the isotope and isotone shifts in charge radiusR are investigated. AssumingR=r₀A^1/3, the isotopic and isotonic behaviour of the parameterr₀ (=RA^−1/3) is also studied. The variation ofr₀ with mass numberA reveals the variation of average nucleon density, which in turn sheds light on the compressibility of nuclear matter. The isotope and isotone shifts inR exhibit the shell effects in the vicinity of magic neutron and proton numbers: 20, 28, 50, 82 and 126. The results indicate that neutron-proton interaction is maximum at the beginning of a major neutron shell and decreases gradually as the shell gets filled up. The behaviour of parameterr₀ clearly suggests that low-Z nuclei are highly compressible while high-Z nuclei are more or less incompressible. The parameterr₀ too is observed to exhibit profound shell effects.

The variation of nuclear parameter with mass number elicits information about nuclear compressibility. Analysis of muonic x-ray transitions provides an elegant method to investigate the behaviour of the nuclear parameterr₀. It is observed from the behaviour ofr₀ that nuclei in the regionA⩽70 are highly compressible while those in the regionA∼210 are almost incompressible. The behaviour ofr₀ is incorporated into the semi-empirical mass formula through the Coulomb energy term. From the modified mass formula thus obtained binding energies of about 440 spherical nuclei have been calculated. The results suggest that nuclear compressibility imposes certain relationship between excess binding energies (E_exp−E_cal) and neutron. proton number. The present study also points out that shell effects exhibited by nuclear binding energies cannot be accounted for by simply varying the coefficients of the mass formula: on the other hand extra terms are necessary to explain them.