The Crystal Structure of MgZn₂ Citation Friauf, James Byron (1926) The Crystal Structure of MgZn₂. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/A7KS-CZ31. https://resolver.caltech.edu/CaltechETD:etd-11112004-102634 Abstract [See Abstract in scanned thesis]. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Chemistry Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Dickinson, Roscoe Gilkey Thesis Committee: Unknown, Unknown Defense Date: 1 January 1926 Record Number: CaltechETD:etd-11112004-102634 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-11112004-102634 DOI: 10.7907/A7KS-CZ31 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4503 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 11 Nov 2004 Last Modified: 03 Oct 2019 23:35 Thesis Files Preview PDF (Friauf_jb_1926.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

The Crystal Structure of Mgzn, Thesis by James B, Friauf In partial fulfillment of the requirements for the degree of Doctor of Philosophy. California Institute of Technology Pasadena, California. 1926 Abstract Crystals of the intermetallic compound, Mg Zn, » were prepared and the crystal structure was determined from x-ray data furnished by Laue and rotation photo- graphs. The crystal was found to have hexagonal axes with a,= 5.154 and c,= 8.484. The unit cell contains four molecules. The effect of absorption in the crystal in determining the wave-length giving a maximum inten~ sity of reflection in Iaue photographs was used to confirm the dimensions of the unit cell. The atoms have the positions: Za: wu,u,t; 2u,u,t; u,2u,e; U,u,z; 2u,u,a; wu . Me: 1/3,2/3,v; 1/3,2/3,8-v; 2/3,1/3,3+v; 2/3,1/3,¥ where u = .830 and v = .062. The least distance between two magnesium atoms is 3.164, between two zine atoms, 2.524, and between a magnesium and 4 zinc atom, 3.02A. The constitution diagram for the binary system, magnesium-zinet>® has a pronounced maximum corresponding 1) Grube, 2. anorg. Chem., 49, 80 (1906). 2) Bruni and Sandonnini, Z. anorg. Chem., 78, 276 (1912). to the formation of an intermetallic compound, MgZne, which forms eutectics with both constituents, Since both magnesium and zine erystallize in the hexagonal elose packed arrangement, a determination of the crys~ tal structure of their compound was thought to be of interest. Crystals of the compound were formed by melting together the calculated amounts of magnesium and zine under a molten mixture of sodium and potassium chlo- rides to prevent oxidation. The melt was then allowed to e001 slowly in the electric furnace, about four hours being taken to cool from ten degrees above to ten de- grees below the melting point of the compound, 595 C. In this way a mass of erystals was obtained from which individual erystals were separated for the production of Laue and spectral photographs. fwo rotation photographs taken with the x-rays from a molybdenum water cooled tube on an x-ray spectrograph of the kind deseribed by Wyckoff? furnished data for the determination of the size and shape of the unit eell, As no information on the crystal class or axial ratio of crystals of this compound was found in the literature, a consideration of the secondary spectra as well as of the principal spectrum was necessary in order to obtain the quadratic form which gives the spacings of the planes, These spacings can be computed from the positions of the reflections on the plate and furnish information of the same nature as that available from a powder photograph, the difference being that in a rotation photograph taken with the crystal turning about a definite axis, the reflections occur in spots instead of in complete circles as they would in a powder photograph taken on a plate. Furthermore, due to the limited rotation of the crystal (30° in this case) certain planes will never reach a position to reflect, while in a powder photograph reflections are to be expeeted from all planes having a suitable spacing. Table I gives the data from a rotation photograph. The observed spacings are the means of those calculated from the reflections produced by the K, doublet and the 2 Kp line of molybdenum for all the planes of the same form showing on the plate. These spacings were 4 compared with the charts given by Hull and Davey, and -—— El Se eee -— mm mm eee were found to agree with the spacings for a hexagonal unit cell having a,= 5,154 and o,= 8.48\. Since the erystal was rotated about one of the a, axes for this photograph, the asSignment of indices obtained from the chart was checked by comparing the computed and observed values for the x and y co-ordinates of the spots. The fact that certain reflections could not oceur due to the limited rotation of the crystal could also be used in some cases to distinguish between planes having nearly the same spacings. On another photograph taken with the crystal rotating about the Gc, axis, only the principal spectrum was measured. This gave a,* 5.154 which is in agreement with the value just given. The third colum of the table gives the values for the spacings computed from the dimensions of the unit cell. This unit cell agrees With the data from Laue photographs taken with the white radiation from a tungsten target. When the wave-lengths of the x-rays 4 producing the spots on symmetrical and unsymmetrical Laue photographs were ealeulated on the basis of this unit cell, no values were found less than the short wave-length limit, about .244, of the x-rays used. The curves showing the intensity of reflection from different planes of the same form reflecting at dif- ferent wave-lengths in unsymmetrical Laue photographs, start from the short wave-length limit, rise to a maximum between .36A and .40A, and then decrease for Longer wave-lengths. The presence of a maximum inten- sity so far below the wave-length of the silver absorp- tion edge is due to absorption in the crystal. The photographic intensity, I, of the white radiation from a tungsten target operated at 50 kv ean be approximately represented between the short wave-length limit, ’ 5) Wyckoff, The Structure of Crystals (New York, 1924), pe 142, | and the wave-length of the silver absorption edge, .485A, by the equation T= B(A-).) where B is a constant. This must be modified, however, if the erystal is strongly absorbing as is the ease with MgZnz For a first approximation it can be assumed that 5 all the rays producing spots on a Laue photograph are absorbed for a distance equal to the thickness of the crystal. The absorption coefficient of the erystal can be computed from data given by Richtmeyer and 6 Warburton for the atomic scattering and fluorescent absorption coefficients. Since the absorption due to scattering is small and nearly independent of the wave- length® it will have no other effect than to decrease the value of the constant, B, but the fluorescent absorption, which is proportional to the cube of the wave-length, will cause greater weakening of the longer wave-lengths and the maximum intensity is accordingly shifted to the short wave-length side of the silver absorption edge. The density of the crystal, 5.16, its thickness, about .3mm, and the computed absorption eoefficient give 3 It = Bt (A-).) a7 where I' is the photographic intensity of the white radiation after passing through the crystal, and B' is the constant, B, multiplied by the factor which represents the common decrease in intensity of all waveelengths due to scattering. The curve given by 6 this equation has a maximum at .36A and agrees in form with the curves showing the intensity of reflection as -@ funetion of the wave-length, thus furnishing additional evidence for the correctness of the unit cell chosen. The density of MgZng was determined by weighing in a specifie gravity bottle after breaking the sample into small pieces in order to avoid, as far as possible, the 4nelusion of blowholes. Two determinations gave 5.164 and 5.155. Using the value 5.16 for the density, the computed number of molecules in the unit cell was found to be 3.93, the deficiency from the integral number, 4, doubtless being due to the fact that the density deter- mined by the use of a specifie gravity bottle is likely to be less than the density determined by x-ray measure~ ments unless porosity of the sample can be completely eliminated. Smaller unit cells, containing 1, 2, or 3 molecules were found to be inconsistent with the data available. A Laue photograph taken with the incident beam of x-rays parallel to the principal axis of ‘the crystal had a six-fold symmetry axis intersected by six planes of symmetry. The space-group giving the arrangement of atoms in the crystal must consequently be isomorphous with one of the point-groups 0B, Ce, De, OF Dee Refer- ence to a tabulation of the results of the theory of 5 space-groups’ shows the possible ways of arranging four 7) Wyokoff, The Analytical Expression of the Results of the Theory of Space Groups (Washington, 1922). magnesium and eight zine atoms in the unit cell. If it is assumed that the magnesium atoms are equivalent and that the zinc atoms are likewise equivalent, the poss- ible arrangements are those which can be obtained from the space-groups a and Den since these are the only space-groups considered having a group of eight equiv- alent positions. All these arrangements give zero for the amplitude factor of the first order reflection from 04.1. The data given in Table I for an unsymmetrical Laue photograph show, however, that 04.1 gives a strong first order reflection. These arrangements are consé- quently inadmissible and the assumption of equivalence of chemically like atoms must be relinquished. With the freedom of choice thus allowed there are numerous ways of arranging the atoms, The zinc atoms may be in two groups of four equivalent positions, two groups of six and two equivalent positions, or in some other com- bination giving the required number of atoms. The num- ber of possible combinations for the megnesiun atoms is somewhat less. The choice of the correct atomic Q arrangement is simplified by the observation that many of the groups of six equivalent positions lie in a plane ‘parallel toe the base of the unit cell. If, however, six zine atoms which constitute more than half the reflecting _ power of all the atoms contained in the unit cell, are arranged in such a plane, the absence of odd oréer re- flections from 00.1 and the observation that the fourth order reflection from 00.1 is stronger than the second order cannot be satisfactorily explained. Groups of Six equivalent positions having such an arrangement are consequently excluded from further consideration. No ways of arranging four magnesium and eight zinc atoms in the unit cell can be obtained from the space~ groups ws D°, me, or D Since none of these space-groups contains the requisite number of equivalent positions. All the arrangements that can be derived from the space- groups Des oe, oe, Dea and Den can be readily elim- inated since for each of these space-groups, each group of eight or less equivalent positions (and hence any com- bination ef them) gives. zero for the amplituae factor of the first order reflection from 04.1. Such arrangements are consequently inconsistent with the data, With the restriction that has been made as to the character of _ the groups ef six equivalent positions to be considered, all the arrangements that ean be derived from the 9 space- groups c2 Dee and Den give the same amplitude faetors for 34.3 and 16.3, and since the data show that the more complicated plane, 16.3, gives a stronger re= flection, such arrangements are inadmissible. A number of three or four parameter structures which cannot be so readily eliminated may be obtained from the space~ groups Den and ae but none ef these seems to offer the slightest possibility of accounting for the observed intensity relations. The only structures remaining to be considered are those arising from the space-groups Tene Oe De, and Dene Of the structures which can be obtained from the space-groups nf and Dene the only one not conflicting With the data is that in which the atoms have the following positions: a: u,B,t; 20,0,4; u,2u,4; u,u,Z; 2u,u,2; G,20,2; 0,0,0; 0,0,% Mg: 1/3,2/3,v; 1/3,2/3,h-v; 2/3,1/3,h+v; 2/3,1/3,7%. This arrangement is obtained by placing the magnesium atoms in one group of four equivalent positions and the zinc atoms in two groups of six and two equivalent positions, A consideration of the type of strusture involved shows that it is sufficient to consider only values of the parameters satisfying the conditions Os us,.5 and -.25i v2.25. If it is assumed that there 10 is a reasonable distance between the two magnesium atoms in the same vertical line, v will be restricted to the middle half of this range. . fhe amplitude factor, S, is computed from s =[a2+ 3? where A and B have their usual significance of sine and eosine summations” and is zero for first order reflections 8) Wyokorf, The Structure of Crystals (New York, 1924), pe 107. from planes of the forms hh.e2ptl irrespective of the values of u and v. No first order reflections from any such planes were found on any of the Laue photographs although planes of the forms 22.1, 33.1, 4463, 55.3, and 44.5 were in a position to give first order reflections at a favorable wave-length. Another characteristie feature of this structure is that the magnesium atons eontribute nothing to the amplitude factors for first order reflections from planes of the forms O3h,.2ptl. The intensities of such reflections are consequently useful in determining the positions of the zine atoms, Of the observed planes of this kind, 03.1 was found to give a weak first order reflection while no reflections were found from 06,5 and 09,5 although planes of both 11 these forms were in as position to reflect at favorable wavelengths and reflections were observed from more complicated planes. The value of the parameter, u, must consequently be such as to give only a small value to the amplitude factors for these planes. By plotting these amplitude factors as a function of the parameter it is readily seen that the only values of u which will reduce them all to zero are 0, 1/6, 1/3, and 1/2. While u eannot be equal to one of these values because of the weak reflection from 03.1, it can hardly be much differ- ent from them because of the rapidity with which the amplitude factors change. Consideration of the amplitude factors for other planes shows that the only values of u giving general agreement with the requirements of the Iaue data are those in the neighborhood of u = 1/6. With this restriction on the value of u, some information concerning v can be obtained from a consid- eration of the intensity relations for the planes 26.3, 26.5, and 26.7. The amplitude factors for the first order reflections from these planes are: 26.35 s= 2 2n( sin28Tu-sin8Tu-sin20tu)44Mgsin60 sinéTy 26.5 8 = 22Zn(sin28Tu-sinstu-sin20Tu)-4Mgsin60 sinloly 26.7 S = 22n(sin26Tu-sin8Tu-sin20Tu)+4Mgsin60 sinl4ty. fhe first part, due to the zine atoms, is the same for all three planes, positive for values of u near 1/6, 12 and changes only slightly for small changes in the value of ue. Ags the data show that 26.5 is stronger than 26.3 it must have a larger amplitude factor since it is a more complicated plane. Also, since 26.7 is & more complicated plane than 26.3 and gives an equally strong reflection, it must have a greater amplitude factor than 26.3. ‘These conditions are satisfied by giving v & small negative value. A consideration of the intensity relations for the planes 08.3, 08.5, and 08.7 leads to the same conclusion. With the values of u and v restricted in this way it was found by trial that satisfactory agreement with the data was obtained for u = .170 and v = -.062. The extent of the agreement is shown in Table ll which gives the data from an unsymmetrical Laue. photograph. The table shows the spacing of the plane producing the re flection in A. U., the intensity as estimated visually by comparison with a plate which had been given a Sé@Yries of graduated exposures, the product of the order of reflection by the wave-length producing the reflection, and the amplitude faetor computed for the values of the parameters given on the assumption that the reflecting powers of the Zine and magnesium atoms are proportional to their atomic numbers. Im comparing the intensities of two planes, if the plane with the smaller spacing 13 gives the greater intensity under comparable conditions of wave-length, it must have a greater amplitude factor. As previously stated, the maximum intensity falls be- tween .364 and .40A and the intensities in the table have been given in this region when possible. | This two parameter structure is the simplest which Will give agreement with the data available. The only other possible structures are a three parameter struc- ture derived from Den and a five parameter structure derived from Caps Neither of these can be eliminated sinee suitable values for the parameters reduces each to the two parameter structure which has been found to give agreement with the data. A consideration of these two more general structures indicates, however, that neither will give satisfactory agreenent with the data except for values of the parameters which reduce them to forms closely approaching that of the two parameter structure, and it is consequently concluded that this structure or a more general structure so similar as to be indistinguishable from it, represents the crystal structure of Mg2n, when u and v have the values given. This structure can be described in an alternative way Without the use of a negative parameter by setting u = .830 and v = .062. Figure 1 shows the arrangement of atoms in the unit cell. The least distance between 14 two magnesium atoms is 3,164, between two zine atoms, 2,52A, and between a magnesium and @ zine atom, 302A. fhe values computed from the atomic radii determined from the crystal structures of magnesium and zine are ‘respectively 3.224, 2.67A, and 2.954. In conclusion, the writer wishes to express his thanks to Dr. R. G Dickinson for reading the manuscript and suggesting some modifications. The Bureau of Metallurgical Research, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, April 19, 1926. 15 fable I Plane Observed Spacing Calculated Spacing , in Be Ue. 00.1(2) 4.241 4.240 10.3 20377 22588 11.2 2.204 2.201 00.1(4) 22122 20120 10.1(2) 1.966 1.974 10.4 1.916 1.915 20.3 1.746 1.751 12.61 1.658 1.653 12.3 1.449 1.448 00.1(6) 1.411 1.413 20.5 ; 1.351 1,350 12.4 1.321 1.320 11.6 1.244 1.239 2026 1.198 1.194 20.7 1.071 1.065 00.1(8) 1.062 1.060 Hexagonal axes, a,= 5.154, o,= 8.484, co, /a,= 1.646 Table iL Plane Spacing Intensity n\ Amplitude Factor 12.1 1.65 32 037 31 03,1 1.46 4 46 15 13.1 1.22 20 36 38 04.1 1.10 90 243 171 02.1(2) 1.08 70 276 132 23e1 1.01 12 41 46 2502 299 6 0357 30 14.2 95 60 206 153 05.3 285 11 58 77 3302 84 36 38 152 24el1 - 84 28 ool 170 24.5 81 25 40 116 12.2(2) 278 3 91 29 15.3 277 11 2043 83 15.4 75 12 41 98 03.1(2) 073 8 275 83 34,3 271 4.2 0359 77 25.2 70 10 36 155 03.2(2) 270 50 077 237 5444 269 8 037 111 16.3 66 5 039 98 16.4 065 3 036 72 07.3 262 5 37 108 53523 262 4.8 238 89 35.4 61 3 043 84 2665 260 5 236 114 22.23(2) 258 6 - 80 149 2665 258 10 035 191 07.6 58 2.5 41 71 13.3(2) 257 9 074 164 5567 56 1.4 044 68 2667 55 5 042 136 08.3 055 265 2350 113 17.6 054 245 256 83 08.5 053 5 035 192 04.3(2) e051 5 073 164 18.4 250 1 252 108 08.7 250 2.5 056 134 46.5 049° Ks) «56 190 27.8 248 1 258 149 46.7 047 1 054 134 @ O= Mg Figure I (Reprinted from the Journal of the American Chemical Society, Vol. XLVI, No. 11. November, 1924.] [ConrTRIBUTION FROM THE GATES CHEMICAL LABORATORY, CALIFORNIA INSTITUTE OF Tecuno.ocy, No. 45] THE CRYSTAL STRUCTURE OF TETRAGONAL LEAD MONOXIDE By Roscor G. Dickinson AND JAMES B. FRIAUF RECEIVED Ju1y 28, 1924 PusiisHep NovemMBer 5, 1924 Lead monoxide, PbO, may be prepared in two modifications, a yellow, rhombic form and a red, tetragonal form. Of these the former is the more stable!? at higher, and the latter more stable at ordinary tempera- tures. It has been suggested* that the differences between these forms are due simply to a difference in state of subdivision; but this hypothesis ? Geuther, Ann. Chem., 219, 56 (1883). 2 Ruer, Z. anorg. Chem., 50, 265 (1906). 3 Glasstone, J. Chem. Soc., 119, 1689, 1914 (1921). 2458 ROSCOE G. DICKINSON AND JAMES B. FRIAUF Vol. 46 is not regarded‘ as tenable in the face of a difference in solubility deter- mined both gravimetrically and electrometrically and in the face of crystal- lographie evidence. Moreover, the two forms have been found’ to give quite different X-ray powder diagrams. Apparently no goniometrical measurements of the red modification have been published, but the fact that it gives optically uni-axial, rectangular plates places it in the tetragonal® system. ‘The present paper describes an X-ray investigation of the structure of the tetragonal modification. Method of Experimentation One part! of lead hydroxide was melted with five parts of potassium hydroxide and the melt allowed to cool slowly in a covered iron crucible. After treatment with water, some very thin red crystals remained. Ex- amination between crossed Nicols showed these to be basal plates. The angles of reflection of the molybdenum K radiation from the base (001) and from the planes (100) and (110) were measured photographically ;? TABLE I SPECTRAL DATA FROM TETRAGONAL LEAD OXIDE Observed angle d Relative hkl Line of reflection n intensities® (001) Mo K a 4° 2! + X 5.050) mw oY 7 6 % X 5.014 w B 7 14 + X 5.018 m ay 8 7 3 X 5.013 $s Cy 8 il 3 X 5.004 s oy 12 14 % X 5.011 vw a2 12 19 4 X 5.008 vw OY 14 20 4 X 5.006 vw B 14 36 t X 5.007 mw oy 16 25 + X 5.009 ms ae 16 31 ; X 5.010 m (100) Y 8° 57’ z X 3.983 w B 9 8 3 X 3.976 ms a 10 15 3 X 3.990 8 (110) B° 6° 26’ i; X 2.816 ms e 7 18 ; X 2.826 s oY 12 438 $ X 2.815 w B 12 56 4 X 2.820 m a 14 34 3 X 2.823 ms * The abbreviations are: s, strong; ms, medium strong; m, medium; mw, medium weak; w, weak; vw, very weak. > ‘The line y slightly overlapped £. 4 Applebey and Reid, J. Chem. Soc., 121, 2129 (1922). 5 Kohlschiitter and Scherrer, Helvetica Chim. Acta, 7, 337 (1924). 6 Groth, ‘““Chemische Krystallographie,” Engelmann, Leipzig, 1906, vol. 1, p. 77. 7 The X-ray methods used have already been described; Dickinson, Tuis JOURNAL, 44, 276 (1922). Nov., 1924 CRYSTAL STRUCTURE OF LEAD MONOXIDE 2459 the reflections from the last two planes were obtained by transmission of the beam through a basal plate. The data from these photographs are given in Table I. A number of symmetrical and unsymmetrical Laue photographs were taken through (001). To secure proper codrdination of all these photographs in the absence of good face-development, a sym- metrical Laue photograph and reflections from (110) and (100) were made with the same crystal. The Unit of Structure Application of the equation nA= 2d sin @ to the data of Table I shows that don = 5.01 A. or a multiple of this value. Prior to a structure de- termination the indices of (100) and (110) may be interchanged. If this is done, then diy = 2.82 A. or a multiple of this value. The smallest possible unit of structure thus has the dimensions 2.82 X 2.82 X 5.01; taking the density® as 9.27, the number of PbO in this unit is found to be 1.003. But if Laue photographic spots are assigned indices referred to the axes of this unit, the values of md calculated for them using the 2doo1 sin On kl lower than the shortest wave length, 0.23 A., which could have been pres- ent in the spectrum (the peak voltage was about 53 kv.). This unit is therefore impossible. Any larger unit differing from this one only by having doo, a multiple of 5.01 A. is impossible for the same reason. When, however, the directions of the axes are taken in accord with the indices of Table I, dio. becomes 3.99 A. or a multiple of this value. The number of PbO in a unit 3.99 X 3.99 X 5.01 is two, and the Laue photographic data do not conflict with this unit. ‘There is thus no evidence necessi- tating the assumption of a larger unit. equation n\ = where ¢ = dou /dioo, are often found much The Arrangement of the Atoms On all the Laue photographs examined no plane having h + k + 1 odd was found to give more than a weak reflection in the first order. Since the reflecting power of the lead atoms must be far greater than that of the oxygen atoms, the weakness of this type of reflection must arise from interference between the lead atoms themselves, rather than from inter- ference between the lead and oxygen atoms. It can readily be shown that the only arrangement of two lead atoms in a unit such that the con- tribution of the lead atoms to first order reflections from planes having h + k + 1 odd is zero is the body-centered arrangement; consequently, the lead atoms must have at least approximately this arrangement. If the lead atoms have precisely the body-centered arrangement, the 8 Ref. 4, p. 2132. 2460 ROSCOE G. DICKINSON AND JAMES B. FRIAUF Vol. 46 reflections of odd order from (001) must be due to the oxygen atoms alone. But reference to a tabulation’ of the coérdinates of possible positions!® in the unit of structure shows that, if the lead atoms are given the non- variant body-centered arrangement, there is no way of placing the oxygen atoms with any tetragonal space-group symmetry in such a way as to give the odd orders from (001). Consequently the lead atoms must have an arrangement which involves one or more parameters and which is capable of approximating the body-centered arrangement when suit- able values are given the parameters. There is only one such arrangement: (0 $ u) G0 &), where! u is near 0.25. The oxygen atoms may then have any of the arrangements: (a), (00 u) (00 a); (b), (000) ($40); (c), (03 v) G0 ¥). Arrangement (a) is rendered improbable by the absence of the first- order reflection from (100) on the spectral photographs. _ The values of the structure factor, S, for first-order reflections from arrangements (b) and (c) are given by Arrangement (b) h+keven. S = 2(—1)* Pb cos 2reJu'+ 20. h+kodd. S = 2Pbhsin2r/u. Arrangement (c) h+keven. S = 2Phcos2rlu+20cos2alv. h+kodd. S = 2Pbsin2r]u+2O0sin2alv, Planes of the forms {241}, d = 0.878, and {041}, d = 0.980, were found to reflect more strongly than those of the forms {331}, d = 0.924, and {131}, d = 1.223. As all of these are planes having h + k even and the same value of 1, arrangement (c) gives them all the same structure- factor and is hence ruled out. The structure-factor for these planes for arrangement (a) is shown in Fig. 1 for values of u near 0.25. The above intensity relations evidently necessitate taking u slightly below 0.25. However, the value of u cannot be as low as 0.23 for {075}, d = 0.496, reflected considerably more strongly than {546}, d = 0.500. ‘The value of u must, therefore, be close to 0.24. It will be noted that this con- clusion is not dependent on any quantitative assumption as to the relative reflecting powers of atoms of lead and of oxygen. In Table II are given representative data from one Laue photograph; in the last column of this table are given values of the structure-factor calculated for arrangement (a) placing u equal to 0.24 and placing the reflecting powers proportional to the atomic numbers. We have found no data in conflict with this arrangement. 9 Wyckoff, “The Analytical Expression of the Results of the Theory of Space- Groups,”’ Washington, Carnegie Inst. Pub., 318, pp. 73-102. 10 The assumption is made here that all of the lead atoms are in equivalent positions and likewise all of the oxygen atoms. The value u = 0.75 also satisfies the conditions but leads to the same possible structures. Nov., 1924 Plane 131 312 410 331 241 413 431 243 510 T50 510 250 053 153 350 531 523 434 504 514 CRYSTAL STRUCTURE OF LEAD MONOXIDE 20 1.25 wt? Ss % 2461 Fig. 1.—Values of the structure factor S plotted against u; the oxygen Inter- planar distance 1.222 1.128 0.968 925 878 838 788 -787 782 - 782 782 742 .720 -709 -684 678 .678 674 674 664 Wave length 0.34 .33 -40 Al 387 36 .38 .30 34 .36 45 .39 34 39 .39 34 .30 32 38 33 tensity u=0.24 faint om SHHEOONHD ‘= OFOrRrrNNOONWOOO MW atoms have arrangement (a). - wNaea TasBiE II S for 6 179 0 6 26 161 164 15 148 148 148 161 47 - 148 161 41 41 143 Plane 154 352 352 161 621 525 631 445 071 355 Til 605 165 172 536 278 635 546 075 Inter- planar distance 0.664 -660 -660 .651 -625 -596 591 577 -567 565 .561 555 -550 550 530 522 .512 .500 497 Wave mated in- length 0.40 29 36 33 38 37 30 33 82 33 37 34 36 38 37 41 .36 81 30 Esti- LAvE PHotocrapuic Dara; INcipENT BEAM 18° FROM PERPENDICULAR TO (001) Esti- mated in- S for tensity u=0.24 0.7 CO Fee ho 143 179 179 164 26 156 164 67 . 164 35 6 67 156 179 169 161 156 60 156 2462 ROSCOE G. DICKINSON AND JAMES B. FRIAUF Vol. 46 Discussion of the Structure The structure obtained for lead oxide is shown in Fig. 2. This structure is derivable from any of the space-groups: V3, Cj, D?, Dj. In it the T i | || | O = Pb ; 0 z e=0 l we | | | | Q | 0 | | | g.---—-. LiLo Lb Loa <] A ez oe Fig. 2.—Arrangement of the atoms in tetragonal PbO. shortest distance between lead and oxygen atoms is 2.33 A. ‘The arrange- ment of layers of atoms parallel to (001) is shown in Fig. 3. The distance IK —— — dog-— — + | A B. | | | | I ! | | Ph 2-0 Pb Ph 20 Pb Fig. 3.— Arrangement of atom-planes parallel to (001) in tetragonal PbO. between the nearest lead atoms in layers A and B is 3.84 A., while the distance between the nearest lead atoms of layers A and C is 3.71 A.; the lead atoms are thus closer together when there is a layer of oxygen atoms between them than when there is not. Summary The crystal structure of red lead monoxide, PbO, has been determined, using X-ray spectral photographs and Laue photographs. It is found that there are 2 PbO in a tetragonal unit of structure having dioo = 3.99 Nov., 1924 CRYSTAL STRUCTURE OF LEAD MONOXIDE _ 2468 and do. = 5.01 A. The oxygen atoms are at (000) (3 4 0), and the lead atoms at (03 u) (0 a) with u = 0.24. PASADENA, CALIFORNIA "Reprinted. from Journat or THe “Oprican Soctery. OF Awentcd 3 ‘AND Review 0 OF. Sctentrere InstauMents, Vol. 11, No. 3, , September, 1925 | THE DESIGN OF A CAM FOR AN X-RAY SPECTROGRAPH — Jawes B. Frraur | THE DESIGN OF A CAM FOR AN X-RAY SPECTROGRAPH James B. FRIAUF In photographing x-ray spectra by the rotating crystal method it is frequently desirable that the crystal be given such an oscillatory motion that it rotates through a given angle with constant angular velocity, then reverses its direction of rotation and returns with the same constant speed. The same type of motion may be useful in other cases and can be obtained from a suitably designed cam rotating with constant angular velocity. This article contains an account of an analytical method which enables the computation of the proper profile for the cam. This could also be done by graphical construction which was the method used for the spectrograph designed by Dickinson and described by Wyckoff.! In these spectrographs the distances designated by R, and R; in this article have been made equal. In Fig. 1, O represents the axis, perpendicular to the plane of the paper, about which the cam rotates with constant angular velocity in the direction indicated by the arrow. A represents the axis of rotation of the crystal holder, likewise perpendicular to the plane of the paper, and ACB the curved lever which rests upon the cam at B and serves to rotate the crystal holder. R, is the distance AO and R, the distance AB. r, and rz are the least and greatest radii respectively of the cam, and r is the radius of the cam at an angle a with the least radius 1. The angle from OA to 7; is wt where w is the angular velocity of the cam and ¢ the time measured from the coincidence offn with OA. The meaning of the angles @ and ¢ is clear from the figure. It is evident from the figure that r? = Ry?+ Ro? —2R,Recos i] (1) from which R 24R 2__ 72 cos @=$ ——_——— (2) 2RiR2 The smallest angle 6, occurs when the cosine is greatest, that is, when ris least, and the largest angle 6. when the cosine is the smallest, that is, when r is largest (as long as @ is less than 180°). ?R. W. G. Wyckoff, The Structure of Crystals, p. 163. 289 290 JAMES B. FRIAUF [J.O.S.A. & R.S.T., 11 RP+R2P—377? cos 0, = (3) : 2RiRe R 21R 2» 2 0s 63 =— (4) 2RiRs The angle 8 through which the crystal holder rotates is then given by the relation B=02.—6; (5) When convenient values of R,, R, and 7, have been chosen, 6, can be determined from equation (3) and 72 can then be calculated from equa- tion (4) by putting 6.=0@,+8 where 8 is the angle through which it is desired to rotate the crystal holder. Fic 1. Differentiating equation (2) gives dr 2r— . ag dt —sin§— = — (6) dt 2RiRe which can be written sin ¢— = —— — — (7) dt RR da di But it is clear from the figure that a=wt—g which gives when differ- entiated da. dg — — & di” dt (8) Sept., 1925] CaM FOR AN X-Ray SPECTROGRAPH 291 so that equation (7) becomes dé r ar *) sin @— = —lo- (9) dt RR, da dt dé , . The quantity ” can be expressed in terms of a in the following manner. It is evident that R= RY+r?— 2rRicos — giving RY—-R2+r2 cos g= — 10) Ry Differentiating this gives d R?—-R-? 1 \dr sing> = (( =x)5 (11) dt 2r?Ry 2Ri/ dt But ; Re , sing = — sin# r while equation (6) gives dr RiRe . ae —= sin 6 — dt r Substitution of these values in equation (11) then gives dy Ri— R? 1\ dé we _ ‘a _.)\% (12) dt 2r? 2/ dt . dy . . . . When this value of au put into equation (9) the result is _ d0 or ad R2—R2 1\d6 sin @— = —— sea - 5S] . (13) dt R,Re da 2r? 2/ dt It is desired to have the angular velocity of the crystal holder con- stant and of the same magnitude in both directions. Consequently, the crystal holder must rotate through the angle 8 (measured in degrees) while the cam rotates through 180°. Since the two angular velocities 292 James B. Friaur [J.O.S.A.& R.S.L, 11 are constant they must be in the same ratio as the angles through which the crystal holder and the cam rotate in the same time. Hence, dé dt B w 180 and dé qo Bo (14) dt 180 Substituting this value for d6 in equation (13) and making a few simpli- fications we have (15) sin @=——— —(—- +. — — r dr {180 1 R?- “) RiRo da 2 2r? When the value of sin 6 in terms of 7 is introduced by using sin 6= V1 ~cos?@ there results 180 1 R?— R? 2 2r? 2Ri Ry as the differential equation giving the relation between r and a for the part EBD of the cam. The introduction of the substitution R?~+R2—r? OS ORR» reduces the integrals to forms which can be found in the tables. The result of the integration is 180 1 R?+ Re —r? a+C = (= + 5) cos”! —— - B 2 2Ri Re 1 Ry — Ro’)? — (Ry2+ Re?) r? gb gin BET RT AE Oe (17) 2 2R,Ror? The constant of integration C can be determined from the condition that y=7,, when a =o and is found to be 180 (RY— R?)?-— (RVR)? 1 1 c= (-—4+—)o4 ir 18 B + >) a) 2RiRor? (18) Sept., 1925] CAM FOR AN X-RAY SPECTROGRAPH 293 The constant of integration having been determined, equation (17) can be used to calculate values of the angle a corresponding to different values of r and the profile of the cam for the part EBD can be drawn. The angle a: corresponding to the greatest radius r2 is found to be (Ri2— Rs®)?—(R2+ R22) re? 180 + B 4 t int a= Pot ? 2° 2 2R,Rors? 1 RP— R?)?—-(R2+AR!? rye — Ft! - *) . yn (19) 2 2R, Ror \* Fic. 2 The profile of the remainder of the cam can be determined in a sim lar way. In Fig. 2 r’ represents the radius of the cam at an angle a’ with the greatest radius 7. The only difference with the preceding case is dé, : that 7 8 now negative ag Bo dt ‘180 When this is substituted in equation (13) the resulting differential equation is dr! (-— — — 4 a B 2 2r’2 R, Roda (20) yi- -(-— ——) 2RiRe Integration gives cr gt - (180 ‘) RYE R= 7? a =[-—— — — J cos™!——_—_—-—— B 2RiRe 1 RY— Ro?)?—-(R,2 R»2 12 — foi! 1 2?) ( v+ 2”)r (21) 2 2R Ror’? 180 1 R?- a) 294 James B. FRIAUF [J.O.S.A. & R.S.L, 11 where the constant of integration C’ determined from the condition that r’=r2 when a’ =o is found to be Cc’ (= 1 1 (RP — Re)? — (RP + Re?) 12? 9, — sin (22) B 2 2 2Ri Rope” The values of a’ for different values of-r’ can now be calculated and the complete profile of the cam drawn. - It is evident that a’ must be equal to 360 —a. when r’=7; and this is found to be the case by equa- tion (21). Re qi R, Fic. 3 The equations show clearly that the cam is unsymmetrical and that in general the angle between the least and greatest radii is not equal to 180°. The reason for this is due to the fact that as @ changes the angle y also changes and the line OB accordingly moves forward or backward. It is desired that the change from the least angle 4; to the greatest take place while the cam rotates through 180°. Consequently, Fic. 4 the point B of the lever must make contact with the greatest radius ro when the cam has turned 180° from the initial position in which B makes contact with the least radius 7. If ¢ has decreased by the amount ¢ in this half revolution of the cam, the angle a2 from the least to the greatest radius must be 180°-+¢o measured in the direction EBD, while if ¢ has increased by the amount ¢o’ the angle a, must be 180°—g,’. As shown in Figs. 3, 4 and 5 the locus of B is on a circle Sept., 1925] CaM FOR AN X-Ray SPECTROGRAPH 295 having a radius R; and a center at 4. When R2>R, as in Fig. 5, or when R= R, as in Fig. 4 it is evident from the figures that ¢ decreases as @ increases. The angle az is therefore greater than 180° for these cases. But when R:<R;, as in Fig. 3 it is clear that starting from the initial position shown in the figure, ¢ at first increases to a maximum and then decreases as @ increases. In this case aj may be either greater or less than 180° depending upon the values chosen for Ri, Ro, 7: and 8, the angle through which the crystal holder is to be rotated. The cam drawn in Figs. 1 and 2 is calculated to rotate the crystal holder through 30°. Ri, Re and 7, were chosen to be 12, 10 and 3 cm respectively. In this case az is less than 180° being equal to 166.9°. 7:=8.05 cm. The equations given make it possible to determine the proper profile to be given to the cam for general values of R, and Re. When the apparatus can be so constructed that R; and R2 can be made equal, the equations become much simpler and the computation of the profile is much facilitated. When R,=R:=R equation (17) reduces to 4c 180 4 1 ) ; 2R?-r 4 1. : ( 2R? (23) a. = {[—-+ — }cos~ — + —sin“!( — B 2 2R? 2 7) The second term on the right hand side is a constant and can be com- bined with C so that if Cy, =C ~$sin-1(— 1) we have +C ‘ee + :) aiid (24) atc, = { -— +1} cos-?———_ ; Bp ° 2R? This gives r atc, —_— — = cos 2R? 180 +4 B 296 JAMES B. FRIAUF [J.O.S.A.& R.S.L, 11 or r= 2R sin —--——- (25) C, is determined from the condition that r=7, when a=o and is found to be 180 CQ = ([-——+ 1)o (26) B The angle a: from the least to the greatest radius is found to be a2 = 18045 from equation (19). The expression (21) for the profile of the remainder of the cam simplifies to , . Cy'—a’ y’ =2R sin-————___— (27) 180, y -4) with 180 | BUREAU OF METALLURGICAL RESEARCH, CARNEGIE INSTITUTE OF TECHNOLOGY, PITTSBURGH, PENNSYLVANIA, DECEMBER 22, 1924.