Nucleus/nuclear radius

Nuclei suitable for fusion must come near each other, where near means something like the nuclear radius of 10" cm. For positively charged nuclei to make such a close approach it requires large head-on velocities, and therefore multimillion-degree Celsius temperature. In contrast, fission can occur at normal temperatures, either spontaneously or triggered by a particle, particularly an uncharged neutron, coming near a fissionable nucleus. 

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. 

The Mossbauer effect involves the resonance fluorescence of nuclear gamma radiation and can be observed during recoilless emission and absorption of radiation in solids. It can be exploited as a spectroscopic method by observing chemically dependent hyperfine interactions. The recent determination of the nuclear radius term in the isomer shift equation for shows that the isomer shift becomes more positive with increasing s electron density at the nucleus. Detailed studies of the temperature dependence of the recoil-free fraction in and labeled Sn/ show that the characteristic Mossbauer temperatures Om, are different for the two atoms. These results are typical of the kind of chemical information which can be obtained from Mossbauer spectra. 

The main difference between the quoted papers lies in the modeling of the magnetic moment distribution in the nucleus a bulk distribution is assumed in the present paper and in paper [11] and a surface distribution is adopted in ref [22]. A systematic 1% difference is observed, which cannot be explained by the uncertainty in the nuclear radius. However, it is known that variations of the nuclear size within reasonable limits can lead to variations in the value of A of several orders of magnitude [11, 14]. This question will be analysed in a separ-ate paper. 

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. 

In Rutherford s mind these results could be explained only in terms of a nuclear atom—an atom with a dense center of positive charge (the nucleus) with electrons moving around the nucleus at a distance that is large relative to the nuclear radius. 

Isomer Shift (IS). The shift observed in the Mossbauer lines with respect to zero velocity is produced by the electrostatic interaction of the nuclear and electron charge distributions inside the nuclear region. One assumes the nucleus is a uniformly charged sphere of radius R, and the electronic charge density is taken to be uniformly distributed over the nucleus. Then the difference between the electrostatic interaction of a point nucleus and a nucleus with radius R is given by... 

The most accurate calculations of the SE correction were carried out in Mohr (1974a, 1992) and in Indelicato and Mohr (1998) for the point nucleus, and in Mohr and Soff (1993) for the extended nucleus. For heavy systems (Z > 50) the dependence of the self-energy correction FSE on the nuclear radius R also Ahas to be taken into account (Soff 1993). 

The solution to this dilemma is to recognize that the nucleus has a finite size, and that this should be accounted for. Ishikawa and coworkers showed that the use of a finite nucleus instead of a point nucleus allowed for more compact basis sets [12] and also eliminated problems with basis set balance close to the nucleus [13]. Visser et al. [14] performed a full relativistic optimization of exponents for the one-electron atoms Sn and U with and without a finite nucleus, showing that the use of a finite nuclear radius significantly decreased the maximum exponent. 

The nucleus is composed of protons (charge = +1 mass = 1.007 atomic mass units ([p.]) and neutrons. The number of protons in the nucleus is called the atomic number Z and defines which chemical element the nucleus represents. The number of neutrons in the nucleus is called the neutron number N, whereas the total number of neutrons and protons in the nucleus is referred to as the mass number A, where A = N -I- Z. The neutrons and protons are referred to collectively as nucleons. A nucleus with a given N and Z is referred to as a nuclide. Nuclides with the same atomic number are isotopes, such as and whereas nuclides with the same N, such as and are called isotones. Nuclei such as " N and which have the same mass number, are isobars. Nuclides are designated by a shorthand notation in which one writes Chemical Symbol, that is, for a nucleus with 6 protons and 8 neutrons, one writes gCs, or, g C, or just The size of a nucleus is approximately 1 to 10 x 10 m, with the nuclear radius being represented more precisely as 1.2 x A x 10 m. We can roughly approximate the nucleus as a sphere and thus we can calculate its density... 

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...

Although the nuclear radius effect is the principal factor in producing a shift of the resonance line, there are two other factors, namely temperature and pressure, which are also acting, and it is often forgotten that the term chemical isomer shift is generally applied to the sum effect of all three. The temperature effect can be very important when measuring small differences in 5-electron density at the nucleus and is considered in the next section. 

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. 

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]. 

Since the nuclear radius is approximately 10 to 10 m, 1 b is approximately equal to the cross-sectional area of a nucleus. 

The decay constant can be regarded as the product of p and the frequency, /, by which the a-particle hits the barrier from inside. If we assume that the deBroglie wavelength, hipv, for an a-particle of velocity v inside the nucleus is approximately equal to the nuclear radius, R, we obtain... 

Alpha-particles from Po ( 6.0 MeV) are used to bombard a gold foil, (a) How close to the gold nucleus can these particles reach (b) What is the nuclear radius of gold according to the radius-mass relation (r = 1.3 fm) ... 

The probability for a nuclear reaction is expressed in terms of the reaction cross-section. The geometric cross-section that a nucleus presents to a beam of particles is t/. If we use 6 X 10 m as an average value for the nuclear radius, the value of becomes 3.14 (6 X 10 ) 10 m. This average geometric cross-section of nuclei is reflected in... 

Isotope shifts for the various isotopes of lithium Li to Li have been measured by the group of Kluge et al. at GSI (including a collaboration with TRIUMF for Li) and the results reported in Refs. [66,65]. The result for Li is of special interest because, like He, it is a halo nucleus with a Li core, and so the nuclear radius is very sensitive to the details of nuclear structure. All the experiments involve measurements of the isotope shift for the 2 Si/2-3 Si/2 two-photon transition. As an interesting test of the method. Table 4.9 compares theory and... 

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 radius used in penetrability calculations is not strictly the nuclear radius R[=r A ), regarded as a property of a particular nucleus and used, for instance, in simple calculations of Coulomb energy (Sect. 56). Following Christy and Latter the radius geneially used is ... 

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