Mean-square nuclear radius

The relative change of the mean-square nuclear radius in going from the excited to the ground state, A r )/ r ), is positive for u. An increase in observed isomer shifts S therefore reflects an increase of the s-electron density at the Ru nucleus caused by either an increase in the number of s-valence electrons or a decrease in the number of shielding electrons, preferentially of d-character. 

On one hand, the large magnitude of the change of the mean-squared nuclear charge radius, A(r ) = (r )e — (r )g, between the excited state and the ground... 

From the effects of quasihydrostatic pressure on the transition energy of the 77.3 keV gamma rays of Au in metallic gold, a value of A(r >= +9X10" has been derived for the change of the mean-squared nuclear charge radius. The positive sign of implies an increase in p(0) with increas-... 

According to O Eq. (25.56), one obtains that the energy shift due to electric monopole interaction is proportional to the product of the s electronic density and the second moment of the nuclear charge distribution also called the mean square nuclear charge radius (see also O Sect. 2.2.3.1 in Chap. 2, Vol. 1). When the nucleus is considered to be a homogeneously charged sphere with a radius R (often called charge equivalent nuclear radius), then... 

We shall briefly review the relationship between spectroscopic observables and nuclear properties/ These properties include spins, magnetic dipole and electric quadrupole moments, as well as the variation in the mean square charge radius within a sequence of isotopes. They manifest themselves in the hyperfine structures and isotope shifts. The hyperfine energies of the different F states within a hyperfine multiplet / - / F J + I, given by the well-known formula... 

Similarly, the isotope shift of an optical transition is related to the change in the nuclear mean square charge radius between the... 

In addition to the mass variation by isotopic substitution also the nuclear size will vary slightly giving rise to small changes in the Coulomb interaction between the electrons and the nucleus. This isotope effeet which is called field shift in the theory of atomic spectra [78Hei] can be traced back to yield a very similar form as of eq. (6) where the mean square nuclear charge radius A,B is used as the expansion parameter instead of and the new molecular parameter is introduced [82Tie]. 

Let us consider the next term eq. (8.2.7). Two multipliers exist in this term (r ) is the mean-square nuclear charge radius (determined according to the general rule (eq. (3.1.6 )) r )=( p(x X2X2)r dv)l( p(x2X2X2)dv) and tA(0)p is the probability density of finding an electron at the origin. Notice that this term accounts for the nonzero nuclear size. 

The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

The root-mean-square radius Rrma for this uniform distribution is /inns = y/S/5R. A fit to nuclear radii obtained in electron-nucleus scattering experiments and muonic x-ray measurements led to the empirical formula [27]... 

The root-mean-square radius of the nuclear charge distribution is related to the 50% fall off radius C by... 

F is a quantity related to the change in the total probability density of the electrons at the nucleus and the leading term in Aaa > 6 is the change in the mean square radius of the nuclear charge... 

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