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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.
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.