A least-squares procedure is described for modeling an empirical transmission surface as sampled by multiple symmetry-equivalent and/or azimuth rotation-equivalent intensity measurements. The fitting functions are sums of real spherical harmonic functions of even order, ylm(− u0) + ylm(u1), 2 ≤ l = 2n ≤ 8. The arguments of the functions are the components of unit direction vectors, −u0 for the reverse incident beam and u1 for the scattered beam, referred to crystal-fixed Cartesian axes. The procedure has been checked by calculations against standard absorption test data.

An empirical correction for absorption anisotropy

An extension of Furnas's method is described. The variation of intensity of an axial reflection as the crystal is rotated about the goniometer axis is used to give a curve of relative transmission T against azimuthal angle ϕ for the corresponding reciprocal lattice level. Transmission coefficients for any general reflexion hkl are then given approximately by T(hkl) = [T(ϕinc) + T(ϕret)]/2 where ϕinc and ϕret are the azimuthal angles of the incident and reflected beams. Equations are derived for (ϕinc and ϕret and the accuracy of the method is discussed.

A semi-empirical method of absorption correction

Absorption effects usually present the most serious source of systematic error in the determination of structure factors from single-crystal X-ray diffraction measurements if the crystal is not ground to a sphere or cylinder. A novel method is proposed for the correction of these effects for data collected on a diffractometer. The method works from the premise that the manifestation of systematic errors due to absorption, unlike most other sources of systematic error, will not be evenly distributed through reciprocal space, but will be localized. A Fourier series in the polar angles of the incident and diffracted beam paths is used to model an absorption surface for the difference between the observed and calculated structure factors. Knowledge of crystal dimensions or linear absorption coefficient is not required, and the method does not necessitate the measurement of azimuthal scans or any extra data beyond the unique set. Moreover, application of the correction is not dependent upon the Laue symmetry of the crystal or the geometry of the diffractometer. The method is compared with other commonly used corrections and results are presented which demonstrate its potential.

An empirical method for correcting diffractometer data for absorption effects

phenix.refine is a program within the PHENIX package that supports crystallographic structure refinement against experimental data with a wide range of upper resolution limits using a large repertoire of model parameterizations. It has several automation features and is also highly flexible. Several hundred parameters enable extensive customizations for complex use cases. Multiple user-defined refinement strategies can be applied to specific parts of the model in a single refinement run. An intuitive graphical user interface is available to guide novice users and to assist advanced users in managing refinement projects. X-ray or neutron diffraction data can be used separately or jointly in refinement. phenix.refine is tightly integrated into the PHENIX suite, where it serves as a critical component in automated model building, final structure refinement, structure validation and deposition to the wwPDB. This paper presents an overview of the major phenix.refine features, with extensive literature references for readers interested in more detailed discussions of the methods.

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Towards automated crystallographic structure refinement with phenix.refine

A program for evaluating the solution scattering from macromolecules with known atomic structure is presented. The program uses multipole expansion for fast calculation of the spherically averaged scattering pattern and takes into account the hydration shell. Given the atomic coordinates (e.g. from the Brookhaven Protein Data Bank) it can either predict the solution scattering curve or fit the experimental scattering curve using only two free parameters, the average displaced solvent volume per atomic group and the contrast of the hydration layer. The program runs on IBM PCs and on the major UNIX platforms.

CRYSOL : a program to evaluate X-ray solution scattering of biological macromolecules from atomic coordinates

International tables for X-ray crystallography

AKHTAR, M. & WEEDON, B. C. L. (1959). J. Chem. Soc. 4058. BART, J. C. J. & MACGILLAVRY, C. H. Acta Cryst. B24, 1587. CRUICKSHANK, D. W. J. (1961). Computing Methods and the Phase Problem in X-ray Crystal Analysis. Oxford: Pergamon Press. pp. 45, 70. DRENTH, W. & WIEBENGA, E. H. (1955). Acta Cryst. 8, 755. HERSCHBACH, D. R. & KRISHER, L. C. (1958). J. Chem. Phys. 28, 728. INHOFFEN, H.H.,BOHLMANN, F.,BARTRAM, K., RUMMERT, G. & POMMER, H. (1950), Liebigs Ann. 570, 54. ISLER, O., LINDLAR, H., MONTAVON, M., RUEGG, R. & ZELLER, P. (1956). Helv. Chim. Acta, 39, 449. JEFFREY, G. A. & CRUrCKSHANK, D. W. J. (1953). Quart. Rev. Chem. Soc., Lond. 7, 335. KLUG, H.P. (1965). Acta Cryst. 19, 983. KOCH, B. & MACGILLAVRY, C. H. (1963). Program and Abstracts, I.U.C. Summer Meeting, Roma. LABHART, H. (1957). J. Chem. Phys. 27, 957. PAULING, L. (1960). The Nature of the Chemical Bond, p. 136. Ithaca: Cornell Univ. Press. SCHOMAKER, V., WASER, J., MARSH, R. E. & BERGMAN, G. (1959). Acta Cryst. 12, 600. SLY, W. G. (1964). Acta Cryst. 17, 511. STAM, C. H. & MACGILLAVRY, C. H. (1963). Acta Cryst. 16, 62. WASER, J. (1955). Acta Cryst. 8, 731. WILSON, A. J, C. (1949). Acta Cryst. 2, 318. ZECHMEISTER, L., LE ROSEN, A. L., SCHROEDER, W. A., POLGAR, A. t~ PAULING, L. (1943). J. Amer. Chem. Soc. 65, 1940. ZECHMEISTER, L. (1944). Chem. Rev. 34, 267. ZECHMEISTER, L. • POLGAR, A. (1954). J. Amer. Chem. Soc. 65, 1522. ZELLER, P., BADER, F., LINDLAR, L., MONTAVON, M., MULLER, P., ROEGG, R., RYSER, G., SAUCY, G., SCHAEREN, S. F., SCHWIETER, U., STRICKER, K., TAMM, R., ZORCI-IrR, P. & ISLER, O. (1959). Helv. Chim. Acta, 42, 841.

The crystal and molecular structure of canthaxanthin.

these measurements are made by a manual device with digital counters; each measurement must be recorded by hand and combined to yield orientation parameters. This procedure is tedious and subject to human error at several stages. We have devised a more efficient procedure using a system which is commonly employed for the measurement of contour lengths of DNA molecules photographed in the electron microscope. We illuminate the oscillation photograph from below or project it from above with a photographic enlarger. Each fiducial mark and reflection is located with the magnifying cursor of the digitizer and, upon command, its film coordinates are transmitted to the calculator. The resolution of each measurement is 250 #m. The calculator is programmed to list each pair of coordinates and to calculate the correction constants which describe the crystal orientation (Nyborg, Wonacott, Thierry & Champness, 1975). The reproducibility of these numbers is approximately 50 #m. All of the measurements and calculations required to process a single film can be completed in less than four minutes. Further details, including a listing of the program, are available from the authors.

Caroline H. MacGillavry

Papers

International tables for X-ray crystallography

The crystal and molecular structure of canthaxanthin.

Symmetry Aspects Of M.C. Escher's Periodic Drawings

Röntgenanalyse von Krystallen

Physical and chemical tables