The number of measurable reflections

If the sphere of reflection has a radius of 1/A, then any reciprocal-lattice point within a distance 2/X of the origin can be rotated into contact with the sphere of reflection (Fig. 4.12). 

Because there is one lattice point per reciprocal unit cell (one-eighth of each lattice point lies within each of the eight unit-cell vertices), the number of reflections within the limiting sphere is approxiniately the number of reciprocal unit cells within this sphere. So the number N of possible reflections equals the volume of the limiting sphere divided by the volume Vrecip of one reciprocal cell. The volume of a sphere of radius r is (4ir/3)r3, and r for the limiting sphere is 2/X, so 

Equation (4.7) shows that the number of available reflections depends only upon Vand A. For a modest-size protein unit cell of dimensions 40 x 60 x 80 A, 1.54-A radiation can produce 1.76 X 106 reflections, an overwhelming amount of data. Fortunately, because of cell and reciprocal-lattice symmetry, 

---

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

IR absorbance was measured with a Fourier-transform IR spectrometer. The absorbance at wave number a is defined as (1 /TV) In [F(U0)/ F(U)], where N 10 is the number of useful reflections at the electrochemical interface, F(U) the light intensity at wave number a reaching the detector at potential U, and F(U0) the same but under reference conditions at potential U0. 

Simply stated, the goal of data collection is to determine the indices and record the intensities of as many reflections as possible, as rapidly and efficiently as possible. One cause for urgency is that crystals, especially those of macromolecules, deteriorate in the beam because X rays generate heat and reactive free radicals in the crystal. Thus the crystallographer would like to capture as many reflections as possible during every moment of irradiation. Often the diffracting power of the crystal limits the number of available reflections. Protein crystals that produce measurable reflections from interplanar spacings down... 

The goal of data collection is a set of consistently measured, indexed intensities for as many of the reflections as possible. After data collection, the raw intensities must be processed to improve their consistency and to maximize the number of measurements that are sufficiently accurate to be used. 

You can see from Table 8.1 that 98% of the reflections available out to 2.7 A [those lying within a sphere of radius 1/(2.7 A) centered at the origin of the reciprocal lattice] were measured, and on the average, each reflection was measured four times. Additional reflections were measured out to 2.4 A. The number of available reflections increases with the third power of the radius of the sampled region in the reciprocal lattice (because the volume of a sphere of radius r is proportional to r3), so a seemingly small increase in resolution from 2.7 to 2.4 A requires 40% more data. [Compare (1/2.4)3 with (1/2.7)3.] For a rough calculation of the number of available reflections at specified resolution, see annotations of the 4/92 paper. 

It will now be clear that the accuracy that may be attained in crystal analysis depends on the number of observed reflections and on the precision with which their intensities can be measured. (We assume that the structure is not complicated by any randomness or disorder, and that the necessary absorption and extinction corrections can be made.) A very useful discussion of the requirements necessary for determining bond lengths to within a limit of error of 0-01 A has been given by Cruickshank (1960). This is, of course, a very ambitious limit, but if it could be achieved it would enable the predictions of the molecular-orbital and valence-bond theories in aromatic hydrocarbons to be distinguished. It is pointed out that at the 0-1% level of significance a bond length difference must be 3-3 times the standard deviation to be accepted as genuine, so the limit of error of 0-01 A would require an e.s.d. (estimated standard deviation) of 0-003 A or better in the bond difference, or a coordinate e.s.d. of 0-0015 A or better. 

Although the heavy atom method has been successful in establishing many structures, particularly those of alkaloids since alkaloids have a propensity for crystallizing as halide salts, there had been an urgent need to develop a procedure for phase determination that was not dependent on the presence of a heavy atom in a crystal. Such a procedure, now commonly called the direct method of phase determination, has been devised. Karle and Hauptman 13) recognized that the number of unique reflections measured in an X-ray pattern is 25-50 times greater than the number of unknowns in a crystal, the unknown quantities being the three coordinates... 

E.g. analytical, numerical, empirical via -scans max. and min. transmission factors T must be specified, 0 < Tmin < Tmax < 1 Total number of measured reflections no of unique (independent) reflections after merging symmetrical equivalents no of observed reflections and the criterion for observed e.g 6896 measured refls., 4323 unique, 2544 observed refls. with I > 2a (/) ... 

The number of Bragg reflections N that can be measured to any particular interplanar spacing, dmjn, depends upon the volume of the unit cell. 

FIGURE 7.3. The effects of different wavelengths. The reciprocal lattice of a crystal is shown as a grid, with a circle indicating the limit of measurement (sin 6 — 1). Note how, the longer the wavelength, the fewer the number of Bragg reflections amenable to measurement. 

The measurement of diffraction data from crystalline macromolecules presents additional problems. In the first place, the intensity of the diffraction is related to the size of the unit cell. Crystals with large unit cells diffract less strongly than do crystals with small unit cells. This is because there are fewer unit cells per unit volume for a macromolecular crystal. As a result, there is a need for very sensitive detection devices that can measure the intensities of weak Bragg reflections with high precision. A second related complication that arises with large unit cells is that the number of Bragg reflections is increased, and therefore the... 

To carry out the experiment we let a beam of electrons impinge on the crystal at a constant angle and vary F, the velocity of the electrons. The number of electrons reflected is measured by means of a collector F (fig. 1), the axis of which makes an angle with the crystal surface equal to the angle of incidence. An opposing field applied to the diaphragm N keeps off slow secondary electrons and electrons which have lost part of their velocity. 

In modern powder diffraction the measurement delivers a raw-file of some thousand step-scan data of counted X-ray photons per step. This raw file contains all the needed information to carry out a crystallographic analysis, but in a way that requires follow up. More informative is a list of distinguishable reflections that includes the position (mostly in the form of f-values) and intensity of each reflection. This dif-file (d-values and intensities) contains some tens to hundreds of reflections. The number of reflections depends on the complexity of the structure and the crystal symmetry the more atoms per cell and the lower the symmetry the more reflections can be identified. But the number of detectible reflections also depends on the resolving power of the equipment, best documented by the half-width of the reflections (more accurately half-width at half-maximum, FWHM). Reflections nearer together than this half-width (or even two half-widths) cannot be resolved. In a second step, very often the Miller indices of the originating lattice planes are added to the dif-file. For this the knowledge of the unit cell is necessary (though not of the crystal structure itself). The powder diffraction file PDF of the International Centre for Diffraction Data (ICDD) contains over 100000 such dif-files for the identification and discrimination of solid state samples. 

To visualize the difference in simple terms, consider a population of cells exposed to radiation. Imagine that an average cell dies after it has taken 100 hits. If we now double the radiation intensity, the cells will accumulate 100 hits in half the time. They age at twice the rate. Time is not an appropriate measurement of their age the number of hits is far more relevant. In this instance, the number of hits reflects the biological age. 

R-value (normally <0.07) due to good-quality crystals with no disorder and a high (usually >10) ratio of the number of observed reflections to the number of refined parameters. The R-value is a measure of the level of disagreement between the properly scaled observed structure factors (Fobs) and calculated structure factors (Fcaic)- It is usually given as %, i.e., an R-value of 0.07 is reported as 7%. 

An IRE consists of an IR-transparent material with a high refractive index, for example zinc selenide (ZnSe), a mixmre of thallium bromide and iodide (KRS-5), cadmium telluride (CdTe) or germanium (Ge) (Hind et al., 2001). Its shape is determined by functional aspects (1) the IR beam must enter and leave it with an angle providing total reflection at sample side, (2) the number of total reflections has to be defined and (3) the design is determined by the respective measuring accessory. 

Uncertainty Scientists report measured quantities so that the number of digits reflects the certainty in the measurement. Write measured quantities so that every digit is certain except the last, which is estimated. 

---