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Introduction to Inorganic Chemistry
Published in Caroline Desgranges, Jerome Delhommelle, A Mole of Chemistry, 2020
Caroline Desgranges, Jerome Delhommelle
Yet the power of group theory is far more impressive! Indeed, the question of solving quintic equations (polynomial equations of degree 5) still remains a very challenging problem at the end of the 18th century. Lagrange in “Sur la résolution algébrique des équations” (1771) understands that new methods need to be developed to solve these equations. Furthermore, Abel (1802–1829) confirms: “Au lieu de demander une relation dont on ne sait pas si elle existe ou non, il faut demander si une telle relation est en effet possible […] de telle ou telle manière”. Lagrange understands that symmetry and permutation play a major role in finding the solutions of polynomial equations. A young French mathematician Galois (1811–1832) (see Figure 6.3) finally finds the solution in 1831 but dies one year later during a duel. His idea is to define a new method based on “groups”: “On peut se donner arbitrairement une première permutation, pourvu que les autres permutations s’en déduisent toujours par les mêmes substitutions de lettres. Le nouveau groupe ainsi forme jouira évidement des mêmes propriétés que le premier.” (One may take an arbitrary first permutation, if the others are deduced from the same substitutions of letters. The newly obtained group will have the same properties as the first one.) For instance, if we take the sequence “A B C D” then the permutations give “B A D C”, “C D A B” and “D C B A” giving us a group of four elements. Regarding the quintic problem, Galois theorizes that the group keeping all the qth roots p1/q, in which q is the prime of any polynomial p, invariant has at most 1/(q(q – 1)) times as many elements as the group leaving p invariant. Other mathematicians develop group theory, including Cauchy (1789–1857), Cayley (1821–1895) and Lie (1842–1899). In particular, Cauchy introduces the theory of permutation groups, as well as the theory of matrices. Klein (1849–1925) in his 1872 Erlangen program classifies geometries by their symmetry groups, linking geometry and group theory. But the most impressive work comes from Frobenius (1849–1917) and his idea of representation theory. Thanks to this new formalism, it is possible to simplify complex problems in terms of group representations and their associated characters. More specifically, the elements of a group can be described as matrices and the characters as the traces of matrices. During the 20th century, Weyl (1885–1955) and Wigner (1902–1995) make crucial contributions by linking group theory and quantum mechanics! According to Wigner, symmetry, and especially group theory, is the key: “I have come to agree […] that the recognition that almost all rules of spectroscopy follow from the symmetry of the problem is the most remarkable result”. In 1963, Wigner receives the Nobel Prize in physics “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”. Bethe (1906–2005), another Nobel Prize awardee, shows that group theory can be applied to problems concerning the nature of crystals.
Sensitivity and uncertainty analyses of fission product nuclide inventories for passive gamma spectroscopy
Published in Journal of Nuclear Science and Technology, 2020
Next sensitivities of Eu-154 number density to one-group (n,) cross-section are shown in Figure 4. Note that the present DPT calculations yield sensitivities of nuclide number densities to cross-sections in 107-group representation. To summarize this, one-group cross-section sensitivity, which is obtained by summing up sensitivities of all the energy groups, is presented here. This figure shows that (n,) cross-sections of several nuclides such as Eu-153, −154, and Sm-152 have large sensitivities. Note that among these three nuclides, covariance data are not evaluated for Eu-154 (and Sm-150 also) in ENDF/B-VIII.0. This will be discussed later. Based on these sensitivities, the generation mechanism of Eu-154 can be simply presented as shown in Figure 5. In this figure, FP nuclides which are explicitly treated in the simplified chain model are shown with bold-line boxes, and those which are neglected are shown with dashed-line boxes. Half-lives of unstable nuclides are also presented inside the boxes.
Numerical benchmark problem of solid-moderated enriched-uranium-loaded core at Kyoto university critical assembly
Published in Journal of Nuclear Science and Technology, 2020
When one begins to perform experimental analyses with his own deterministic procedure, it is mandatory to confirm the applicability of their numerical tool to these problems. A numerical benchmark problem simulating the KUCA-ADS benchmark data with reference solutions would be beneficial and helpful. In the past, Takeda and Ikeda proposed a set of neutron transport benchmark problems, and a small light water-moderated coresimulating KUCA is included [1]. This benchmark is quite helpful, but a reactor core model is extremely simplified and energy treatment is also simple: two-group representation. Nowadays, sophisticated deterministic procedures with less approximations and simplifications have been developed, but careful discretization about energy, space, and angle is necessary at the same time because of complicated behavior of neutron flux inherent to these less-approximated procedures.
A simplified Bixon–Jortner–Plotnikov method for fast calculation of radiationless transfer rates in symmetric molecules
Published in Molecular Physics, 2023
A. I. Martynov, A. S. Belov, V. K. Nevolin
The operator of the non-adiabatic coupling is totally symmetric with respect to molecular point group operations. The consequence of this is that the non-adiabatic constant belongs to the same symmetry group representation as the product of WFs: . Valiev [30] also tried to associate the rate constants with representations of modes. Since only the modes with were taken into account in that work, the modeled IC in naphthalene and anthracene proceeded only via the A and B modes. This contradicts the zero displacement of non-totally symmetric modes according to Li [36].