Odd-even staggering

The differential IS s (6 + or 6 + ) are measures of the radius change of the neighbouring nuclei and yield more clearly than the integral IS the effect of the addition of a neutron or a neutron pair. In this way, the increase or decrease of deformation is easily observable as a function of neutron number. A still more sensitive measure of the influence of an unpaired neutron on the charge radius is the odd-even staggering parameter y introduced by H.H. Stroke [TOM64] and given by... 

Analysis of the data obtained is under way. Preliminary results for isotope shift show the same inversion in odd-even staggering in the same neutron range as in francium and radium. This is a possible evidence for an octupole mode of deformation as suggested in [AHMAD 83], .see also [ LEA 84], [NAZ 84], and [SHE 83]. ... 

In the past ten years a huge amount of data by laser spectroscopy of short lived isotopes have been collected and the field is still developping with new proposed technics. Instead of analysis of individual data, which is not desirable, trends may be found and thus raise up further theoretical interest. In this respect, the odd-even staggering in isotope shift, a feature well known long ago and very well documented, deserves certainly a close theoretical attention. 

Figure 17. Differences of mean square charge radii and odd-even staggering in Ra. The upper curve corresponds to the plot of Figure 16. The lower curve represents the changes between neighboring isotopes with solid dots (open circles) for the steps from odd to even (even to odd) neutron number.

In our studies of the RIS (Eq. 14), our goal is to probe such nuclear interactions. Thus, while the spherical variations of the nuclear charge radius are given quite well phenomenologically in a parametrized form by the droplet model a picture in which the additional nucleon producing the isotope shift polarizes a nucleus may be more revealing. For the odd-even staggering we have thus... 

H. H. Stroke, D. Proetel, H.-J. Kluge, Odd-even staggering in mercury isotope shifts evidence for Coriolis effects in particle-core coupling, Phys. Lett. 82B 204 (1979). 

The 102-104 isotope shift is appreciably larger than the previously published values [24, 25] and the odd-even staggering of ° Pd is quite marked. 

The strong fluctuation of IP or of the mass abundance is an electronic-structure effect, reflecting the global shape of the cluster, but not necessarily its detailed ionic structure. This is demonstrated in Fig. 6, where the ionization potentials of sodium clusters obtained by the spheroidal jellium model [32] are compared with their experimental values [46]. The odd-even oscillation of IP for low N is reproduced well. The amplitude of these oscillations is exaggerated, but this is corrected by using the spin-dependent LSDA, instead of the simple LDA [47]. The same occurs for the staggering of d2 N) [48]. 

Additionally, a staggering of rotational band parameters has been found [HAG78] for 176-182pt> where the odd-mass Pt isotopes appear more deformed than the even-mass ones. 

The anti-prismatic structures ML2n are constructed from two n-planar moieties which are staggered with respect to each other. For odd n, all of the L functions generated by an anti-prism are the same as those generated by the corresponding prism because the L s of the two n-planar moieties span the same series of L° s. The situation for even n, however, is different. In this case, the expansions of the L wave functions for the upper and lower n-planar moieties are different in the final term. If the L s of the upper n-planar moiety span the S , P 1, D+2> (A - 1) (a-i), Ag (where A = L ax = n/2), the L s of the lower one span the S , P+i, D+2,..., (A - 1) (a-i), A Therefore, all the L° s for anti-prisms when n is even are ... 

Improper axes can also be associated with several symmetry operations. We noted earlier that e, applied twice in succession, results in a simple 2tt/3 rotation about the S6 axis. In other words, we can write the set 5e, 5g, Sl, S, as Se, C3, S2, C. We stop at s because S = E due to the combination of C and an even number of reflections. 5g is equivalent to S2 because it contains three rotations by 2 r/6 and an odd number of reflections, and S2 means one rotation by n and one reflection. The operation S2, however, is easily shown to be equivalent to an inversion, and so we have, using a compressed notation, 25e, 2C3, i associated with the Se axis. Since we have already explicitly listed the elements C3 and i in our set of elements for staggered ethane (or any other system containing an Se axis) only the ISe operations are unique to the Se axis. The generalization of this case is that an S2n axis with odd n implies that elements C and i are also present. Of the 2 — 1 operations associated with S2 , — 1 are preempted by the C axis and 1 by the element i leaving — 1 operations to be attributed to the S2a axis. [If = 1, we have S2, C, and i as elements. But Ci = E, so we ignore it. There is only one operation here (21 — 1 = 1) and it is preempted by i. Therefore, S2 has no unique operations and it is not listed as a symmetry element.] For S2n with n even, i is not implied and there are n unique operations. 

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