Determination of Crystal Structure

Solid nitrogen, N2, has a density of 1.026 g/cm at about 20 K. If solid nitrogen is known to have some cubic lattice with a length of 5.66 A, what kind of cubic lattice does solid N2 have  

From the length of the lattice, we can determine the volume of the unit cell  

From the known density (and a few unit conversions), we can now determine the amount of mass in each unit cell  

Each nitrogen molecule, with two nitrogen atoms each, has a mass of 28 u. If there are 112 u per cell, then there are 

Note that we cube the unit as well as the number. 

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H. A. Hauptman (Buffalo, NY) and J. Karle (Washington, DC) outstanding achievements in the development of direct methods for the determination of crystal structures. 

The preceding quotation serves admirably as a brief historical introduction to x-ray diffraction. This field of research has proved enormously fruitful in yielding information about crystal structure, and in providing a means of obtaining monochromatic x-rays and of measuring their wavelength. The determination of crystal structure, though important to analytical chemistry, is outside the scope of this book.31... 

The theoretically obtained electron densities of ions may be used for the calculation of the so-called F curves, which give the effective reflecting power of the ion as a function of the angle of reflection and the wave-length of X-rays, and which are of use in the determination of crystal structures. It may be mentioned that the high maximum value of the electron density at the nucleus given by our calculations provides considerable justification for the method of determining crystal structures with the aid of the relative intensities of Laue spots produced by crystal planes with complicated indices. 

X-ray crystallography is still the best technique for complete and accurate determination of crystal structures. However, electron crystallography has several important advantages over X-ray crystallography ... 

The X-ray determination of crystal structure of tetracyclopentadienyl-hafnium has established that the molecule should be represented as [(7r-Cp)2-Hf(Cp)2], and thus resembles the titanium rather than the zirconium analogue. Both [(7i-Cp)2Hf(Cp)2] and [(it-Cp)2Zr(Cp)] exchange nonequivalent cyclopentadienyl rings very easily such that, even at — 150°C, only one sharp line is observed in the H n.m.r. spectrum. ... 

Coppens, P., and M. Eibschiitz Determination of crystal structure of YFeOg and refinement of GdFeOg. Acta Cryst. 19, 524 (1965). 

HAUPTMAN, HERBERT A. (1917-). An American biophysicist who won ihe Nobel prize for chemistry in 1985 along with Jerome Karle for their outstanding achievements in the development nf direct methods for the determination of crystal structures. Hauptman s work involved developing equations that allow determination of phase information from X-ray crystallography intensity patterns. The use of computers permitted use of the equations to determine the conformation of thousands of chemicals. Haaptmao was director of research and v ice president of the Medical Foundation of Buffalo and a professor of biophysics in Buffalo at the Stale University of New York. 

The atoms that comprise a solid can be considered for many purposes to be hard balls which rest against each other in a regular repetitive pattern called the crystal structure. Most elements have relatively simple crystal structures of high symmetry, but many compounds have complex crystal structures of low symmetry. The determination of crystal structures, of atom location in the crystal, and of the dependence of many physical properties upon the inherent charactensdcs of the perfect solid is an absorbing study, one that has occupied the lives of numerous geologists, mineralogists, physicists, and other scientists for many years. 

The determination of crystal structure in synthetic polymers is often made difficult by the lack of resolution in the diffraction data. The diffuseness of the reflections observed in most x-ray fiber patterns results from the small size and imperfect lattice nature of the polymer crystallites. Resolution of individual reflections is also made difficult from misorientation of the crystallites about the fiber axis. This lack of resolution leads to poor accuracy in measurement of peak positions. In particular, this lack of accuracy makes determination of layer line heights difficult with a corresponding loss of significant figures in evaluation of the repeat distance for the molecular conformation. In the case of helical conformations, the repeat distance may be of considerable length or, as we shall show, indeterminate and, in effect, nonperiodic. This evaluation requires high accuracy in measurements of layer line heights. 

The determination of crystal structures by X-ray crystallography provides precise and unambiguous data on intermolecular interactions. Crystal engineering has been defined by Desiraju as the understanding of intermolecular interactions in the context of crystal packing and in the utilization of such knowledge in the design of new solids with desired physical and chemical properties. ... 

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

The discovery of x-rays provided crystallographers a powerful tool for the thorough determination of crystal structures and unit cell sizes [20-26], X-rays have wavelengths between 0.2 and 10 nm. As x-rays possess dimensions comparable to the interplanar distances in crystals, x-ray crystallography is an ideal nondestructive method for material characterization, since nanometer parameters as well as macroscopic properties of the tested samples can be determined from x-ray diffraction data. 

Lipson, H., and Cochran, W. The Determination of Crystal Structures, G. Bell and Sons, Ltd., London (1953). 

Hydrogen bonds are also formed in the solid state. It is true that in the determination of crystal structure by X-rays the position of the hydrogen atom can hardly be determined directly but an abnormally short distance between two oxygen atoms points to the presence of a hydrogen bond. 

Ellison RD, Johnson CK, Levy HA (1971) Glycolic acid direct neutron diffraction determination of crystal structure and thermal motion analysis. Acta Cryst B 27 333-344... 

Gu, Y.-X. Determination of crystal structure of norsesquiterpenoid lactone. Acta Physica Sinica, 1982, 31(7) 963-968. 

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