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Relationships between bond lengths and bond numbers and also between atomic volumes and valencies are derived and parameters for their calculation are given for the s-block, p-block, and d-block metals. From the atomic volumes under pressure, the valencies of three solid lanthanoids have been confirmed or redetermined: La 3; Ce 2. 3. and 4; Yb 2 and 3.
The volume changes of cesium under pressure are discussed with respect to the packing density of the atoms and valence. The element is univalent in densely packed Cs I and Cs II. Valence increases in Cs III (packing density q = 0.973), in Cs IV (q = 0.943), in Cs V (q ~ 0.99), and in close packed Cs VI. The diminuition of volume beyond ~ 15 GPa is caused by this increase only which implies that electrons of the fifth shell act as valence electrons.
In the body-centered cubic structure and in closest sphere-packings the atoms are arranged to give structures with equal densities. A measure of the packing density of atoms is derived. Several crystal structures of elements, including the bcc structure and the closest sphere packings, represent a state of maximum density in which the atomic volume is characteristic of each element. From any crystal structure of an element its atomic volume in this state can be calculated to a good approximation.
The volume changes of solid iodine under pressure are discussed with respect to the packing density of the atoms and to valence. The packing density of solid iodine which is 0.805 under ambient pressure increases to 0.976 in monoatomic iodine-II, 0.993 in iodine-III, and 1 in fcc iodine-IV. Simultaneously, the valence increases from 1 in the free molecule to 1.78 in the crystal structure under ambient pressure, 2.72 – 2.81 in iodine-II, 2.86 – 2.96 in iodine-III, and 3 in fcc iodine-IV. The valence then remains constant up to about 180 GPa and rises moderately to 3.15 at the highest investigated pressure of 276 GPa. Parameters for calculating bond numbers, valences and atomic volumes of densely packed halogens, hydrogen, oxygen, and nitrogen are given.
The volume changes of lithium and sodium under pressure are discussed with respect to the packing density of the atoms and their valence. In densely packed Li I (bcc), Li II (fcc), and Li III (alpha-Hg ype), valence increases from 1 at ~ 5 GPa to ~ 2.5 at 40 GPa. The maximum valence 3 is attained in Li IV (body-centered cubic, 16 atoms per cell, packing density q = 0.965) at 47 GPa. In densely packed Na I (bcc) a linear increase of valence from 1 at ~ 10 GPa to 2.9 at 65 GPa is found which continues in Na II (fcc) up to 4.1 at 103 GPa.
In recent publications Otto Hahn, last president of the Kaiser-Wilhelm-Gesellschaft, is charged with having favoured the Nazi regime, before World War II by politically purging institutes and suppressing Lise Meitner’s contribution to the discovery of nuclear fission, and during the war by contributing to the German war efforts, mainly to the development of nuclear weapons. These charges, however, which partly concern also the Kaiser-Wilhelm-Gesellschaft and some of their institutes are based on ignorance or disregard of the historical sources.
It is considered whether Fermat’s so called Last Theorem can be understood by substituting variables by polynomials and discussing their properties. The same substitution yields a survey of the Pythagorean Triples.
