The Pythagorean Theorem and the Solid State

Abstract

Solid-state parameters such as radius ratios, packing efficiencies, and crystal densities may be calculated for various crystal structures from basic Euclidean geometry relating to the Pythagorean theorem of right triangles. Because simpler cases are often discussed in the standard inorganic chemistry texts, this article only presents calculations for closest-packed A-type lattices (one type of particle) and several compound AB lattices (A and B particles) including sodium chloride, cesium chloride, zinc blende (sphalerite), wurtzite, and fluorite. For A-type metallic crystals, the use of recommended values of atomic radii results in calculated densities within 1% of observed values. For AB lattices, assuming ionic crystals, the use of recommended values of ionic radii results in density determinations that are usually but not always close to observed values. When there is covalent character to the bonding, the use of covalent radii results in calculated densities that correlate well with observed values. If interionic or interatomic spacings are used, the calculated densities are always close to the observed values. As indicated by a survey of the standard inorganic texts, these calculations are generally not presented. However, as an illustration of the application of simple mathematical principles to the study of chemistry, discussion of the methods presented in this manuscript may be of value in classroom presentations pertaining to the solid state.

Improvement in the Filtration Performance of an Ultraporous Nanofiber Membrane by Atmospheric Pressure Plasma-Induced Surface Modification

“…Figure b displays the internal occupancy ratio φ for the filtered membranes calculated from eq (descripted in Section ). The PMMA particles used were spherical and uniform in size, so if the entire space inside the pore was occupied by PMMA particles in a dense state, the occupancy ratio would be 0.74 from the perspective of the hexagonal closest packing. , Therefore, 0.74 was taken as the maximum value, and the effect of filtration performance improvement was evaluated according to how close the occupancy ratio could get to it. The pores in the original membrane were utilized about a mere 10%, while those of the plasma-irradiated membrane were utilized nearly 60%, and most of the internal pores of the membrane were occupied by PMMA particles.…”

Using Latex Balls and Acrylic Resin Plates To Investigate the Stacking Arrangement and Packing Efficiency of Metal Crystals

“…It seems interesting for students to examine the packing efficiencies of various types of lattice structures experimentally and theoretically. The crystal lattice models in classroom demonstrations have been proposed. − Commercial display models are also available through companies such as Carolina Biological Supply Company and Indigo Instruments in the United States and Maruzen Company and Kenis Co. Ltd. in Japan . Although these models are very convenient to show the crystalline lattice in the classroom, it is difficult to understand the concepts of the unit cell, atomic arrangement, and coordination number of atoms using the space-filling-type models; on the other hand, the concepts of closest packing and packing efficiency in the lattice structures are hard to understand using the ball-and-stick type models.…”

A Unit Cell Laboratory Experiment: Marbles, Magnets, and Stacking Arrangements

“…Various classroom demonstrations using crystal-lattice models have been described in this Journal and others presenting face-centered, body-centered, and simple cubic unit cells for first-semester general chemistry courses. Models presented include software programs, , BBs and watch glasses, paper and glue, − Styrofoam balls, , and others. − Commercial display models are also available through companies such as Carolina Biological Supply Company (Burlington, NC) and Indigo Instruments (Waterloo, ON, Canada). A very popular commercial crystal-lattice model set is the Solid-State Model Kit distributed through the Institute for Chemical Education.…”