The theory of fibre diagrams as given by P o 1 a n y i and first applied to cellulose fibres by S i s s o n and K r a t k y enables one to derive the orientation in a fibre from the intensity distribution in the diatropic interferences. Since these interferences are usually too weak in the X-ray diagrams of cellulose fibres, we must resort to the paratropic reflexions. The present paper shows how these paratropic reflexions must be evaluated to obtain the orientation factor fr as a rational measure of average orientation. It is shown on experimental grounds that the proximity of the A, interference does not invalidate the evaluation of the As sickle. A special procedure must be applied to fibres in which the orientation is not very pronounced, since here the interferences of the (021) planes partly overlap those of the paratropic A, planes.It is assumed throughout this article that in all experiments the B r a g g angle is negligible. The correction required when dealing with finite B r a g g angles is discussed in section 6.The relation between orientation and optical constants is set forth in section 5. Finally, section 8 gives an illustration along with a short summary of the method.In a previous study on the orientation of cellulose fibres, based upon optical and X-ray data, the results of the quantitative evaluation of X-ray diagrams of various fibre specimens were communicated and discussed *) . The theoretical basis of the evaluation and the experimental method applied were only briefly referred to in the publication quoted. In the present paper the underlying theory and the experiments performed in order to determine certain fundamental constants are given. W e may omit introductory remarks of a general nature, since these can be found in the publication referred to.It is assumed throughout this article that the incident beam of X-rays is perpendicular to the fibre axis. The film is a plane one at distance L from the fibre and perpendicular to the incident beam. W e have restricted the discussion to X-ray reflexions of the first order. Higher order reflexions are usually very weak in fibre diagrams: in the particular case of regenerated cellulose they play no part at all a). Besides, the extension to reflexions of higher order would be a very easy matter, if required.W e shall often make use of the fact that the B r a g g angle 6 is small. The error involved is discussed in section 6.The reader who is not interested in the mathematical details can find a summarizing description of the procedure, along with a few examples illustrating the method, in section 8. 1.1.1. Following P o 1 a n y i 4 ) we shall designate a lattice plane by its representative point on a sphere of unit radius: the tangent plane in this point Basic relations governing the fibre diagram.

A quantitative investigation of the intensity of scattered radiation in a series of native and regenerated cellulosic objects, including bacterial cellulose and fibers transformed into cellulose IV, is offered. The technique used involves the use of copper radiation monochromatized by reflection, specimens with randomized orientation of equal size and density, rotation of the plane film, and measurement of primary beam intensity by means of a device introduced by Goppel. The procedure of evaluation consists of measuring integrated intensities of the crystalline peaks and that of the diffuse background. After correction of the latter for radiation scattered by air and for the background components due to Compton radiation and thermal scattering, as derived from exposures of sugar crystals and diamond, the diffuse background is represented by a curve exhibiting a flat maximum. The height of this maximum was assumed to be a correct relative measure of the disordered portion and the total intensity of the coherently diffracted radiation of the crystalline peaks as a relative measure of the crystalline portion. Within the series of native specimens the ordered and disordered fractions are found to be equal in cotton, cotton linters, ramie, and flax fibers. In woodpulp the disordered fraction is slightly higher and in bacterial cellulose considerably higher. Within a series of rayons, including slightly and highly orientated viscose rayons, Cellophane, Lilienfeld rayon, and cuprammonium rayon, the ordered and disordered fractions were also found to be equal. Slightly less disordered substance is found in Fortisan, in mercerized ramie and in specimens where cellulose II has been transformed in cellulose IV by a heat treatment. It is shown that the relative proportions of the fractions of disordered substance, for all objects investigated, are in satisfactory quantitative agreement with those deduced from the sorptive capacity for water vapor. The absolute percentage of the crystalline portion calculated from the experimental data is 70±2 percent for the native fibers and 39±3 percent for the rayons. These figures are in reasonable agreement with those previously estimated from sorption isotherms (68 percent and 35 percent), from density determinations (60 percent and 25 percent), from birefringence and x-ray orientation measurements in regenerated fibers (≤40 percent for rayon), and from recrystallization of amorphous cellulose powder (∼35 percent for rayon). The observed relative intensities of the principal crystalline interferences give rise to a discussion of the limited scope of the concept ``crystallinity'' in cellulose fibers, and it is argued that the concept ``degree of lateral order'' may be preferable. Comparison of observed relative line intensities with the theoretically calculated ones, reveals another marked difference between native and regenerated fibers, pointing to the fact that certain states of lateral order prevalent in the latter may be absent in the former.

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A simple procedure for the determination of the crystalline fraction in isotactic polypropylene, suitable for routine work, is described. It gives lower figures for the crystalline fraction tban the method recommended by NATTA, CORRADINI, and CESARI in 1957.The correctness of the results (within the limitations inherent to the concept of crystallinity in polymers) is supported by the consistency of the results with those obtained by the recently published very remarkable though quite elaborate and sophisticated procedure of W. RULAND. ZUSAMMENFASSUNG:Ein einfaches Verfahren fiir die Bestimmung des kristallinen Anteiles in Praparaten aus isotaktischem Polypropylen wird beschrieben. Es ergibt niedrigere Zahlen als die von NATTA, CORRADINI und CESARI im Jahre 1957 empfohlene Methode.Die Richtigkeit der Ergebnisse (innerhalb der Beschrankungen, die dem iiblichen Begriff der Kristallinitat in Makropolymeren iiberhaupt innewohnen) wird gestiitzt durch die abereinstimmung mit denen, die nach dem vor kurzern von W. RULAND veroffentlichten theoretisch sehr raffbierten, aber urnstfindlichen Verfahren erhalten werden.

It is demonstrated that the various procedures recommended in the literature for the determination of the crystalline fraction in polyethylene specimens by means of X-ray diffraction lead to consistent results if certain errors made by some of the previous authors are avoided.A simple procedure which is not inferior to other more elaborate ones and does not require corrections for LORENTZ factor, polarization or incoherent radiation is presented.It is easily adaptable for routine work.Attention is drawn to the observed close correlation between crystallinity and line width of the crystalline interferences. ZUSAMMENFASSUNG:Es wird gezeigt, dal3 die verschiedenen in der Literatur empfohlenen Verfahren zur Bestimmmg des kristallinen Anteiles in Polyathylenen auf rontgenographischem Wege zu iibereinstimmenden Ergebnissen fiihren, wenn bestimmte Fehler, die von einigen der friiheren Autoren gemacht worden sind, vermieden werden.Ein einfaches Verfahren wird angegeben, das den sonstigen umsttindlicheren nicht unterlegen ist, keine Korrektur fiir LORENTZ-Faktor, Polarisation und inkoherente Strahlung erfordert und sich leicht fiir einen Routinegebrauch anpassen liiBt.Auf die Beobachtung einer engen Korrelation zwischen Kristallinitat und Lmienbreite der kristallinen Interferenzen wird hingewiesen.