The Distribution of Diffracting Power in Certain Crystals

VoL. ll,- 1925

1

PHYSICS: R. J. HA VIGHURST

507

evident in this figure. Bey itkeseturves, the points would vary about zero in haphazard- fashion. When the curves are drawn to a large scale, and the areas measured with a planimeter, the ratio of Cl to Na is found to be 1.78. The ratio oYthe numbers of electrons in the ions is 1.8, and the ratio of the nunibers of electrons of the neutral atoms is 1.55. The existence-of ions seems to be indicated. But it is apparent that in the curve for sodium, one series of points, representing p(x=y=z), falls outside of the curve. A check of the calculations disclosed no mistake. Probably the limit. of accuracy of the ratio is about 10%, and conclusions should hardly be drawn from these figures. But if equally good data could be obtained for crystals with lighter elements, e.g., NaF or LiF, it should be possible to decide definitely between the two alternatives. The problem of extinction6 must be considered in the experimental determination of these relative intensities. B., J., B. used a method of correcting for the secondary extinction, and their corrections have been applied to the data we have used, but a method of determining relative intensities 'without the necessity of making this rather uncertain correction is to be desired. It has been often pointed out that the use of powdered crystals would obviate the difficulty due to secondary extinction. Work is being done in this laboratory by this method, and it is hoped to check the results of B., J., B. on NaCl as well as to make determinations on other crystals. NATIONAL RESEARCH FELLOW. Epstein and Ehrenfest, Proc. Nat. Acad. Sci., 10, 133 (1924). 3 W. L. Bragg, James and Bosanquet, Phil. Mag., 41, 309; 42, 1 (1921). 4 James, Phil. Mag., 49, 585 (1925). 6 Debye and Scherrer, Phys. Zeit., 19, 474 (1918). 2

6 Darwin, Phil. Mag., 43, 800 (1922).

THE DISTRIBUTION OF DIFFRACTING POWER IN CERTAIN CRYSTALS By R. J. HAVIGHURST' JEFFERSON LABORATORY, HARVARD UNIVERSITY It has been shown (cf. two preceding notes, pp. 489,. 502) that the distribution of diffracting power in the unit cell of a crystal with the NaCl or CsCl structure may be represented by a Fourier's series of the form:

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2, Z2 z A5525, cos 27rnix/a cos 2in2y/a cos 2sn3z/a

(1)

where A",",, is the square root of the intensity of reflection of X-rays frotg the plane (nin2n3); nl, n2, n3 are the Miller indices of the different

508

PROC. N. A. S.

PHYSICS: R. J. HA VIGHURST

crystal planes, multiplied by the dlumber representing the order of reflection. The purpose of this paper is to demonstrate the application of this series with some qualitative data which is available. Figs. 1 and 2 represent the distribution of electron density in crystals of KI, NH4I, and NH4C1. They show (what is already well known from

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previous work) that the first two have the NaCl structure and the last has the body-cetitered CsCl structure. Values for the A's to be substituted in (1) were obtained from data taken in the course of some precision measurements on the lattice constants of these crystals.2 The intensities were estimated visually from powder photographs. Only rotigh corrections

509

PHYSICS: R. J. HA VIGHURST

VOiL. 1 1, 1925

were made for variation of intensity due to polarization, absorption, increasing radius of the diffracted cone, etc., for the data are at best only qualitative. It is important to have intensities from as large a number of planes as possible. Fortuitous errors in individual coefficients will neutralize each other, provided a large number of coefficients are used; and the curves do not begin to take their final shape until a fairly large number of terms in the series are evaluated. The data on the substances mentioned above were fairly complete, and were used for that reason. Relative intensities of the reflections from 29 planes of KI were used. For NH4I there were 22 values of the coefficients, and 28 for NH4CL.

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The width of a peak includes, of course, the amplitude of thermal agitation. This amplitude is generally considered to be much smaller than the width of these peaks. The expression for the electron density of diamond is not so simple. The structure of diamond is that of two interpenetrating face-centered cubes, the origin of one being at the point (1/4, 1/4, 1/4) with respect to the origin of the other. The origin of the unit cell is no longer a center of symmetry, and there are no axial planes of symmetry running through it. Consequently the phase constants of the terms of the Fourier's series (cf. preceding two papers) are not equal to an odd multiple of ir/2, and the series does not become a simple cosine one. However, if the two interpenetrating lattices are treated separately, each may be expressed by a simple cosine series, and the problem is to combine the two series. For a point (xyz) in the unit cell: P(Xyz) = X2 ' A'nlntn cos 27rnlx/a cos 27rn2y/a cos 2rnz3z/a +A cos 2irnix'/a cos 27rn2y'/a cos 27rn3z'/a where x' = x-a/4; y' = y-a/4; z' = z-a/4. A',,, is the coefficient for plane (nln2n3) due to the first lattice; A",M is the coefficient for the same plane due to the second lattice. Therefore:

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= Al2 A cos 2rnix/a cos 2rn2y/a cos 27rnaz/a + Affl2fl cos (27rn1x/a - n,7r/2) cos (27rn2y/a - n2lr/a) cos

(27rn3z/a - n37r/a). (2)

The second term on the right will be positive or negative, with sines or cosines, depending upon the values of nl, n2 and n3. The measured A is the algebraic sum of A' and A ". Since we are here dealing with interpenetrating lattices of carbon atoms, it might be supposed that A' equals A ". Then it would be possible to evaluate the coefficients of the series. The curves of figure 3 were obtained on this assumption. But if A' equals A ", there should be no (222) reflection, as its coefficient is (A'- A"), or zero. W. H. Bragg3 has found a very faint (222) reflection. This must mean that the distribution of reflecting power of the atoms of one lattice is not equal to that of the atoms of the other, perhaps because of a difference of orientation of electrons. If (A'-A") has a positive value, it is possible, knowing the value of (A' + A 'i) at the same angle (from interpolation on a curve as described in the preceding paper), to get the ratio of A '/A ", and to separate each measured A into two parts on the basis of this ratio. The ratio A 'A " will vary with the angle of reflection. The curves of figure 3 were obtained by substituting in (2) the values of A from Hull's data4 for reflections from powdered diamond. It will be noticed that at the half-way point along the cube edge, and at the half and

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PHYSICS: R. J. HA VIGHURST

VOL. 11, 1925

511

three-quarter way points along the diagonal, there are slight humps, although there are no atomic centers at these points. One explanation is that these represent valence electrons, another is that the assumed equality of A' and A" has produced the humps, which would disappear if

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showing the same abnormal humps when A' and A' are assumed equal. If, from the value of (A'-A ") obtained from the (222) reflection, the ratio A '/A" is calculated, and the coefficients in the series adjusted accordingly, the humps are very nearly wiped out. The data on intensity of reflection from diamond are not at all satisfactory. As Bragg indicated, there seems to be a strong effect of secondary extinction, which has been left out of consideration. This effect, or some unknown one, is illustrated clearly if Bragg's intensities are corrected by sin 20 the factor + os2 20 and plotted against 0. The curve, instead of rising

rapidly at small values of 0, reaches a maximum and then decreases as e decreases. This anomalous behavior' could be explained as a secondary extinction effect, which becomes more effective in decreasing the reflected intensity at small angles or large intensities. Some powder method data of Debye and Scherrer,5 which is presumably free from the effect of secondary extinction, shows a normal decline of the corrected intensity for the first few reflections from diamond. It is quite evident that no conclusions can safely be drawn from the curves concerning the electron density distribution in diamond. However, they serve to indicate that the more complex structure of diamond can be treated by this method, provided that accurate intensity data can be obtained. NATIONAL RESEARCH FELLOW. Havighurst, Mack and Blake, J. Amer. Chem. Soc., 46, 2368 (1924). 3 W. H. Bragg, Proc. Phys. Soc., London, 33, 304 (1921). 4 Hull, Physic. Rev., 10, 661 (1917). 5 Debye and Scherrer, Phys. Zeit., 19, 474 (1918). I

2

IONIZATION PROD UCED IN GA SEO US REA CTIONS BY A. KEITH BRBWSR' NORMAN BRIDGE LABORATORY, CALIFORNIA INSTITUTE OF TECHNOLOGY

Communicated June 22, l925

The presence of ionization in chemical reactions has been 4etected bya large number of investigators. The nature of the ionization has remained obscure due largely to the fact that most of the reactions studied were

accompanied by physical phenomena which might in themselves produce the observed ionization. The author has shown in previous papers2 3 that reacting gases become conductors of the current; and that the conductance is proportional to the number of molecules reacting, and to the potential drop between the electrodes.