Intensities of Reflections

We have seen that the shape and size of the unit cell of a crystalline substance may be determined from the positions of the reflected beams. The number of atoms in the cell follows directly from, the size of the cell and the density of the substance for information concerning the positions of atoms, however, it is necessary to rely almost entirely on the relative intensities of the beams. There are a number of photoelectric devices known for measuring the degree of blackness on a clear film or the degree of whiteness on a black film a good measurement of the intensity of beams is obtained by using a movable Geiger counter instead of a film, for the intensity of ionization within the spectrometer chamber is proportional to the number of photons received. Reflections from some planes may be completely absent. 

On the other hand, it should be noted that the 200 type reflections should appear. In this case, since the waves from B are two lengths behind those from A, the waves from C would be one length behind those from A and one length ahead of those from Bf leading to complete reinforcement. One might alternatively say that the (200) reflections are strong because interplanar spacing associated with these reflections are half those for the (100) reflections and there are thus no extra atoms between planes in the (200) set. 

Ignoring for the present ionization and contraction of the lattice, the structure of cesium chloride may be considered similar to that of cesium metal, but with the cesium atoms removed from the body centers and chloride ions inserted. In the diffraction pattern for cesium chloride, the 100 reflections, 111 reflections, and other reflections absent from the pattern for cesium metal are present but are weak. Wave interference similar to that occuring for cesium metal must occur, but here interference is not complete. The planes of chloride ions are not as strong reflectors or scatterers as the planes of cesium ions (the scattering power of an atom rises sharply with atomic number). Thus, interference in cesium 

---

The single most severe drawback to reflectivity techniques in general is that the concentration profile in a specimen is not measured directly. Reflectivity is the optical transform of the concentration profile in the specimen. Since the reflectivity measured is an intensity of reflected neutrons, phase information is lost and one encounters the e-old inverse problem. However, the use of reflectivity with other techniques that place constraints on the concentration profiles circumvents this problem. 

Reflectance A measure of the extent to which a surface is capable of reflecting radiation, defined as the ratio of the intensity of reflected radiant flux to the intensity of the incident flux. 

The Rutile Structure.—A large number of compounds MX crystallize with the tetragonal structure of rutile, TiCfe. In this structure the position of the ion X is fixed only by the determination of a variable parameter by means of the intensity of reflection of x-rays from various crystal planes. In accordance with the discussion in a following section, we shall assume the parameter to have the value which causes the distances between X and the three ions M surrounding it to be constant. With this requirement the inter-atomic distance R and the edges a and c of the unit of structure are related by the equation R = (a/4 /2) [2 + (c/o)2]. In this way the inter-atomic distances in Table XII are obtained. In the case of magnesium fluoride the agreement is satisfactory. 

It was concluded by Zachariasen [Norsk geol. Tidsskrift, 8, 189 (1925) Z. physik. Chem., 119, 201 (1926)] from the intensities of reflection of x-rays that beryllium oxide does not contain Be++ and O" ions. However, it has since been shown by Claassen [ibid., 124, 139 (1926)] and Zachariasen himself [Z. Physik, 40, 637 (1926)] that if the electron distribution of the ions is taken into account, the x-ray measurements are compatible with an ionic structure. 

The first was not the structure of brookite. The second, however, had the same space-group symmetry as brookite (Ft,6), and the predicted dimensions of the unit of structure agreed within 0.5% with those observed. Structure factors calculated for over fifty forms with the use of the predicted values of the nine parameters determining the atomic arrangement accounted satisfactorily for the observed intensities of reflections on rotation photographs. This extensive agreement is so striking as to permit the structure proposed for brookite (shown in Fig. 3) to be accepted with confidence. 

The intensities of reflections (A00) depend only on the parameter tv, and of (OfcO) on u, so that these parameters can be easily evaluated separately. In Table VI of Hassel and Luzanski the observed intensities are reported to be ... 

Fig. 2. Curves showing square-root of intensity of reflection for (liOO) and (OH)) as functions of the parameters w and u, respectively.

We believe that our conclusions can be accepted with considerable certainty, for the agreement between the predicted structure and the experimental results in regard to space-group symmetry, size of the unit of structure, and intensities of reflections on rotation photographs is so striking as to remove nearly completely from consideration the possibility of its being accidental. 

The verification of the suggested structures by the comparison of the observed and calculated intensities of reflection of x-rays has been begun. So far the calculations have been carried out for 18 even orders of re-... 

Observed and Calculated Intensities of Reflection from (001) of Fuchsite... 

The observed small intensity of reflection of these planes in general thus rules out fie. We can arbitrarily place 6 Si in 6 and 6.4/in 6y without loss of generality. 

We have investigated this region with the aid of the observed intensities of reflection in eight orders from (110). The value of the sodium parameter u was simultaneously determined. 

The integrated intensity of reflection of an X-ray line from an extended face of a mosaic crystal is4)... 

The comparison of observed intensities of reflection in eight orders from (110) and intensities calculated for a range of parameter values suggested by assumed minimum interionic distances led to the values w=0.175, x = 0.135, y = 0.440, = 0.150. These values were verified by a large number of other reflections. The resultant structure is shown in Fig. 1 and 3. 

The reflecting powers of Mn and Fe are nearly the same, and may be taken equal without serious error. This reduces the number of distinct structures to three namely, 1 ab, %abc, and 3, of which 1 ab depends on two parameters and the others on one. It is possible to decide among them in the following way. Let us assume that the contribution of oxygen atoms to the intensity of reflection in various orders from (100) is small compared with the maximum possible contribution of the metal atoms that is, with 32M. The metal atom structure factor for structure 1 for (/a 00) is... 

All distinct structures are included in the parameter range — 0.25 u 0.25, and, moreover, positive and negative values of u give the same intensity of reflection from (A0 0). Hence we need consider only 0 u 0.25. In Figure 1 are shown values of S calculated over... 

Now there are two physically distinct arrangements of the metal atoms corresponding to w = 0.030, the first with u = 0.030, and the second with u = — 0.030 and it is not possible to distinguish between them with the aid of the intensities of reflection of X-rays which they give. Let us consider the positions 24e. The structure factor for 24e is ... 

The predicted structure has been verified by the comparison of the observed intensities of reflection for a large number of planes and those calculated with the use of Equation 1. Data for such comparisons for planes (feOO) and (hOl) reflecting on oscillation photographs are given in Tables I and V, and for other planes giving Laue reflections in... 

Each oxygen ion is then nearly equidistant from four cations. Making the four (Mn, Fe)—O distances equal, values of the parameters are predicted which lead to good agreement between observed and calculated intensities of reflection from a large number of planes. The structure found for bixbyite has... 

Observed and calculated intensities of reflections on two oscillation photographs, one of which is reproduced in Fig. 5, are given in Table III. The first number below each set of indices (hkl) is the visually estimated observed intensity, and the second the intensity calculated by the usual Bade-methode formula with the use of the Pauling-Sherman /0-values1), the Lorentz and polarization factors being included and the temperature factor omitted. No correction for position on the film has been made. It is seen that the agreement is satisfactory for most of the... 

The parameter for the remaining structure, with 3 Cm in 3 b, can be limited to a narrow range of values with the aid of intensity comparisons on oscillation and Laue photographs. In Fig. 1 there are plotted the intensities of reflection in various orders from (100) for the parameter range 0.20 to 0.30 (limitation to this region being easily made) calculated from the equation... 

On calculating intensities of reflection for these parameter values, it was found that the general agreement with observation for all except the very weak reflections was excellent, as is shown by the data in Table IV for useful reflections from 45° oscillation photographs from (100) with... 

Table IV. Observed and calculated intensities of reflection on oscillation photograph from (100) with [001] as axis. Equator...

Comparison of Calculated and Observed Intensities of Reflection for Be40-... 

---