Introduction

Lindley, P. F.

doi:10.1107/97809553602060000586

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 3.2, p. 156

Section 3.2.1. Introduction

P. F. Lindley^b

3.2.1. Introduction

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The measurement of the density of a crystal has become a neglected art, and yet, in combination with an accurate knowledge of the unit-cell dimensions, it can provide vital information regarding the total molecular weight of the unit-cell contents. From this quantity, it is usually possible to determine the number of molecules in the unit cell and their individual molecular weights. The equation relating the crystal density (ρ), unit-cell volume (V), and the overall molecular weight is $[ \rho ={M}m_a/{V},]$ where m_a is the atomic mass unit (1.66057 × 10⁻²⁴ g) and V is expressed in mm³. Alternatively, $[ {M}=0.602206{V}\rho,]$ where V is in units of Å³. The mass per asymmetric unit can be determined by dividing M by the number of asymmetric units, Z (dependent on the space group), and this will normally correspond to the molecular weight. However, the quotient can either be a fraction of the molecular weight (normally 1/2) when the molecular symmetry permits the molecule to lie on a special position such as a centre of symmetry or a symmetry axis, or a multiple if the asymmetric unit contains more than one molecule. In either case, a special examination of the choice of unit cell and space group should be undertaken to ensure that the correct ones have been chosen. Normally, the measured and calculated densities should agree within at least 1.5%; discrepancies greater than this may indicate an incorrect molecular formula (not unknown in preparative chemistry) or the presence of solvent molecules or other additives. Incorrect choice of space group, inappropriate choice of unit cell, and incorrect asymmetric unit contents can all have profound effects on the success of a structure analysis and on the refinement of the resulting structure.

The classical techniques of density measurement are described by Tutton (1922) and by Reilly & Rae (1954). An excellent and detailed review of both the standard and the less common methods is given by Mason (1944), but, because this work can be difficult to obtain, some of the references compiled by this author are cited herein.

3.2.1.1. General precautions

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Meticulous temperature control is essential for the highest precision. The allowable temperature fluctuation will depend on the thermal coefficient of expansion of the material and on the required accuracy of the measurement. The utmost care must be taken to avoid air bubbles and inclusions. In those techniques that require immersion of the solid in a liquid, it is assumed that no chemical or physical interaction occurs between the liquid and the solid, and that the volume of the liquid displaced represents the true volume of the solid. For most hard crystalline materials, liquids can easily be found for which these assumptions are valid. However, for amorphous powders, porous structures such as zeolites, crystalline proteins, and natural and synthetic fibres, the measured `density' may depend markedly on the particular liquid chosen and on the details of the method applied. In these cases, penetration or swelling of the solid will depend on a variety of factors such as interfacial tension, the relation of pore size to molecular dimensions, adsorption, and electrostrictive forces. The structural unit to which the measured density applies may be very difficult to specify. Even with materials not subject to these difficulties, variability in the measured density is frequently found. Such variations may arise from differences in trace impurities or in the previous history of the sample (Johnston & Adams, 1912).

References

Johnston, J. & Adams, L. H. (1912). On the density of solid substances, with especial reference to permanent changes produced by high pressures. J. Am. Chem. Soc. 34, 563–584. Google Scholar

Mason, B. (1944). The determination of the density of solids. Geol. Foeren. Stockholm Foerh. 66, 27–51.Google Scholar

Reilly, J. & Rae, W. N. (1954). Physico-chemical methods, Vol. 1, 5th ed., pp. 577–608. New York: van Nostrand. Google Scholar

Tutton, A. E. (1922). Crystallography and practical crystal measurement, Vol. 1, pp. 625–639. London: Macmillan. Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 3.2, p. 156