Nuclear radius, change

The difference in energy caused by the nuclear radius change <37 will then be... 

The Mossbauer isomer shift is a function of the fractional nuclear radius change on excitation ( R/R) and the change in charge density at the nucleus from source to absorber (d 11//(0) 19),... 

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. 

Next to Fe, Sn is the second best Mossbauer isotope, in the sense that the y-ray energy, Mossbauer lifetime, and parent lifetime are all satisfactory. To use isomer shift measurements eflFectively, one must know the relative change in nuclear radius (8R/R). In the case of Sn, the correct value for this quantity was indefinite, both as to sign and magnitude. 

To get a fairly good number for the change in nuclear radius, we can consider a wider class of tin-containing materials. The results in Figure 1 are the work of Lees (12). Here we have taken arbitrarily as our origin, the source, the intermetallic compound Mg2Sn, which provides a convenient cubic environment for tin. It gives a line which has natural... 

Because the interactions measured in Mossbauer experiments are products of atomic and nuclear factors, experiments on iodine isotopes have yielded values of the change of nuclear radius between the ground state and the excited state, AR/R, quadrupole moment values Q, and magnetic moment values, fi, as well as electric field gradients and internal magnetic fields. 

The observed half-life of 238U is 4.47 x 109 y, which is a factor of 25 times longer than the calculated value. Note the qualitative aspects of this calculation. The a particle must hit the border of the parent nucleus 1038 times before it can escape. Also note the extreme sensitivity of this calculation to details of the nuclear radius. A 2% change in R changes A. by a factor of 2. In our example, we approximated R as 7 xh + Ra. In reality, the a particle has not fully separated from the daughter nucleus when they exit the barrier. One can correct for this by approximating R 1.4A1 3. 

We show this curve together with a similar one for the vacuum polarization in Fig. 12. Already a change of 2 % in the nuclear radius yields a change of 0.1 eV in each of the radiative corrections. Due to the different sign of both contributions the net effect almost cancels. 

Fig. 3.1 The electrostatic potential of an electric charge of —ep dr at distance r from a point nucleus is given by V, but when the nucleus has a finite radius, the potential curve within the sphere is different. The shaded area indicates the effect of a change in the nuclear radius from to...

The fractional change in the nuclear radius dR/R can also be compared with theory. For some of the heavier elements it is also common to describe a deformation parameter which is given by [47t/(5/ )] dR/R [19]. The difficulties of estimating the electronic charge density at the nucleus result in experimental dR/R values being approximate only. Numerical values are listed under individual MSssbauer nuclides in later chapters. 

The existence of a chemical isomer shift produced by the change in nuclear radius and differing chemical environments was first demonstrated by Kistner and Sunyar [15] in 1960 for the case of a stainless steel source and an absorber of a-FeaOs. Early work quickly showed that the chemical isomer shift measured was related to the formal oxidation state of the iron. This is illus-... 

As already discussed in some detail in Chapter 3, the magnitude of the chemical isomer shift depends not only on the values of the electron density at the tin nucleus for the compounds being compared, but also on the value of the fractional change in nuclear radius on excitation to the 23-88-keV level, 6R/R. Determination of the sign and magnitude of the nuclear constant dR/R has proved more difficult for Sn than for Fe because of the initial lack of accurate electronic wavefunctions for tin compounds. The various values which have been proposed are given in Table 14.1 in approximate chronological order [1, 23-29]. 

The fractional change in the nuclear radius ARJR can be obtained from the expression of isomer shift (O Eq. (25.83)). For this reason, the accurate knowledge of the difference in the electronic density at the site of nucleus of source and absorber is necessary. Because of the difficulties in the estimation of electronic densities at the nucleus, the values of ARJR have considerable uncertainty (Shenoy and Wagner 1978). 

In the processing of nuclear materials, precipitation/coprecipitation techniques are used for the separation of the actinides from most fission products. Both fluoride and oxalate complexes of these metal ions are sufficiently insoluble to accomplish this separation (Stary 1966). Coprecipitation with bismuth phosphate has also been used for this purpose (Stary 1966). Because of their insensitivity to subtle changes induced by minor cation-radius changes, such techniques are not useful for the separation of the lanthanides from the trivalent actinide metal ions. 

AR = change in nuclear radius Rq, -Roe = Sternheimer shielding factors Rfim =distancefrommthtonthion n = radial coordinate of ith electron rtj = distance from ith to /th nucleus [Pg.389]

Accurately calibrating the chemical isomer shift scale for a particular isotope is not without its difficulties. The change in nuclear radius, SR/R, cannot be determined independently of the chemical environment, which means that the electron density 1 s(0)P must be estimated by molecular orbital methods in at least two compounds before a value of SR/R can be obtained. 

Ionic radius. Ionic radii do not change much in going across a transition row. The reason for this is essentially a balance of two effects (1) As nuclear charge increases across the row, the electrons would be pulled in, so the ions ought... 

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... 

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