X-RAY INVESTIGATION OF THE CRYSTAL STRUCTURE OF LITHIUM AND LITHIUM HYDRIDE ACADEMISCH PROEFSCHRIFT TER VERKRIJGING VAN DEN GRAAD VAN DOCTOR IN DE WIS- EN NATUURKUNDE AAN DE UNIVERSITEIT VAN AMSTERDAM OP GEZAG VAN DEN RECTORMAGNIFICUS Dr. P. ZEEMAN, HOOGLEERAAR IN DE FACULTEIT DER WIS- EN NATUURKUNDE, IN HET OPENBAAR TE VERDEDIGEN OP DONDERDAG 5 JULI 1923 DES NAMIDDAGS TEN 3 URE IN DE AULA DER UNIVERSITEIT DOOR JOHANNES MARTIN BIJVOET GEBOREN TE AMSTERDAM A. W. SIJTHOFF’S UITGEVERSMAATSCHAPPIJ LEIDEN UNIVERSITEITSBIBLIOTHEEK UTRECHT 3794 0776 X-RAY INVESTIGATION OF THE CRYSTAL STRÜCTÜRE OF LITHIUM AND LITHIUM HYDRIDE AAN MIJNE OUDERS. Zeergeleerde Hemmes, U dank ik voor Uw uitnemend H. B. S. onderwijs. Met groote dankbaarheid, Hooggeleerde Sissingh, zal ik uit mijn eerste studiejaren Uw leiding blijven gedenken, welke liefde vormt voor het experimenteeren. Bovenal dank ik U voor Uw hartelijke persoonlijke belangstelling. U, Hooggeleerde Holleman, dank ik voor het vele dat ik, inzonderheid gedurende den tijd dat ik op Uw practicum werkte, van U heb geleerd en voor de mij betoonde belangstelling. U, Hooggeleerde van der Waals, ben ik zeer erkentelijk voor Uw colleges, welke ik als de leerrijkste uit mijn'studie beschouw, en voor Uw groote bereidwilligheid mij bij moeilijkheden te helpen. Hooggeleerde Keesom, U dank ik voor den aangenamen tijd, gedurende welken ik onder Uw leiding te Utrecht het kristalonderzoek met Röntgenstralen mocht bestudeeren; voor Uw belangstelling en hulpvaardigheid ook daarna. U, Zeergeleerde Kolkmeyek, voor Uw hulp, samenwerking en vriendschap. In het bijzonder richt ik mij met gevoelens van zeer groote erkentelijkheid tot U, Hooggeleerde Smits, Hooggeschatte Promotor, om U mijn oprechten dank te brengen voor Uw belangwekkende colleges en practica; voor het vele dat ik gedurende mijn assistentenjaren van U leerde; voor Uw belangstelling, aanmoediging en leiding bij het onderzoek mijner dissertatie. Bij uitstek echter dank ik U voor Uw zoo groote en hartelijke belangstelling buiten mijn werk. Aan het einde van mijn academische studie is het mij een behoefte allen, die tot mijn wetenschappelijke vorming hebben bijgedragen, in het bijzonder U, Hoogleeraren inde faculteit der Wis- en Natuurkunde, mijn hartelijken dank te betuigen. I. INTRODUCTION The investigation of the crystal structure of lithium and lithium hydride *) was undertaken with the idea of finding out whether, in these crystals built up of atoms with a small number of electrons, something could be determined regarding the positions (orbits) of these electrons. As is always the case, here X-ray analysis cannot lead to the structure directly; it can only be used to test definite configurations. Where it concerns a determination of the arrangement of the centres of the atoms, the number of structures to be tested is limited by conditions, derived from the assumption of definite hypotheses in crystallography2). When we consider the orbits of rnoving electrons, then the difficulty is increased, and the question, which orbits must be considered to test the X-ray diffraction, is still unanswered while, also, the problems of whether, and how these rnoving particles in the crystal comply with symmetry conditions, has not yet been tested. 3) Various attempts to determine something regarding the configuration of electrons in the atoms from the intensity of the diffraction of Xrays, may be found since 1915 4). If we assume that it is the electrons outside the nucleus which scatter the X-rays 5), then the wave which is scattered through the whole atom in a definite direction, will be the resultant of the waves sent out in that direction by its electrons. If all the electrons lie so close to the nucleus that their mutua] distances are small compared with the wave length of the radiation, then this holds also for the difference in path between the rays dif *) J. M. Bijvoet and A. Karssen, Verslag Akad. Wetenschappen Amsterdam 29, 1208 (1921) ; 31, 49 (1922). *) Cf. A. Karssen this number of the Recueil. 3) Cf. N. H. Kolkmeyer, Verslag. Akad. Wetenschappen Amsterdam 28, 767 ; 29, 824, 980 (1921) ; Physik. Z. 22, 457 (1921). ‘) G. G. Darwin, Phil. Mag. 27, 315 (1914) ; W. H. Bragg Phil. Trans. Roy. Soc. A. 215, 253 (1915) ; P. Debye, Ann. Phys. 46, 809 (1915). 6) Suppositions in the calculations, e. g. the different electrons scatter independently of each other; the scattering can be calculated lrom the classical electromagnetic theory as if the eleotron were held in equilibrium with a frequency, which is small compared with that of the incident radiation. (See Darwin loc. cit. § §11-—-12). , X-RAY INVESTIGATION OF THE CRYSTAL STRUCTURE OF LITHIUM AND LITHIUM HYDRIDE fracted in a particular direction by the different electrons and the mutual phase differences of these rays will be scarcely perceptible ; thus the scattering power of the atom, independent of the angle of deviation, is proportional to the number of electrons. This must also hold for greater distances with small angles of deviation, which has been found experimentally and thus explained1). When the relative distances between the electrons are no longer small in comparison with the wave length of the radiation, then for large angles of deviation, the phase differences between rays diffracted by different electrons in the atom make themselves obvious by a decrease of the' scattering power with the angle of deviation, ö 2). This opens the possibility of testing atom models by determining this dependence of the scattering power on it from the intensity of X-ray reflections the y function of vcn Laue and comparing with the calculated dependence of different electronic configurations. Hull already deduced an electronic configuration for iron in this manner in 1917 3) ; however, no further significance can be attached to it, sincé some factors, which must be taken into account in order to arrivé at the scattering power from the observed intensities of reflection, were not considered4). Comptons) tested the possibility of atom models with electrons rotating in rings on the observed diffraction intensities, for example, for rock salt and calcite. Debye and Scherrer 8), in a well known investigation on lithium fluoride determined the number of electrons on the lithium and on the fluorine partiele from the ratio of their scattering powers, for o = O : they concluded that Li+ and F~ions occur at the points of the lattice. An attempt to calculate from the X-ray diffraction of diamond, whether rings of binding electrons between neighbouring carbon atoms can be assumed, a question of great significance for an insight into the nature of homopolar combination, led to no definite conclusion 7). Recently, Bragg, James and Bosanquet 8) in a detailed investigation on sodium chloride, determined, from accurate measurements and calculations, the absolute scattering power of the sodium and chlorine particles and their dependence on the angle of deviation (measurement of the diffraction intensity compared with that of the incident beam). They find it impossible, by extrapolation to 9= O to decide ‘) P. Debye & P. Scherrer, Physik. Z. 19, 474 (1918). _ 2) Below we shall find, that with a distanee from the nucleus such as in a H ion, the scattering power between 9 = 0° and 9 = 180°, passes several times through zero. 3) A. W. Hull, Phys. Rev. 9, 84 (1917). 4) Por instance, we miss here the factor which is introduced by the beam being not completely parallel; the continuous decrease of the intensity with 9 observed by Hull, ascribed by him to a decrease in the scattering power, appears to be sufficiently explained by these factors. Langmuir, [J. Am. Ghem. Soc. 41, 881 (1919).], claims unjust value lor Hull’s proof of the “cubical atom” model, on X-ray analysis of iron. 5) A. H. Compton, Phys. Rev. 9, 29 (1917). 6) P. Debye and Scherrer, Phys. Z. 19, (1918). !) Debye and Scherrer loc. cit., D. Coster, Verslag. Akad. Wetenschappen Amsterdam 28, 391 (1919), N. H. Kolkmeyer, ibid. 28, 767 (1920). 8) W. L. Bragg, R. W. James and G. H. Bosanquet, Phil. Mag (6) 41, 309, 42, 1 (1921), 44 433 (1922). Z. Physik 8, 77 (1922). between ions or atoms at the lattice points; they show also, that the conclusion of Debye and Scherrer on lithium fluoride is not sufficiently certain. As regards the electron configuration m the sodium and chlorine particles, they can only conclude that the electron density round the nucleus is a function of the distance from the nucleus. Through the small number of ring electrons in the constituent atoms we hoped that the investigation of lithium and its hydride, perhaps better than manv of the compounds hitherto investigated, would allow some conclusions to be drawn regarding the structure and binding of the particles and so bring this problem, so extremely important for chemistry, a step further towards solution. II RESEARCHES ON THE STRUCTURE OF LITHIUM i) Previous investigutions. Observations with X-rays on lithium, have been made by Schmidt 4) and Hull2). Using the method of Debye and Scherrer the fust named obtained no clear interference lines. From exposures by his method (identical in principle with that of Debye), Hull could not decide between cubes with two atoms at each corner of the lattice and centred cubes. Hull obtained two different photograms, that which led to the conclusion of a centred cube appears to agree with our films, on the others, on the contrary, as Hull himself considers possible, parasitic lines occur. Besides we must reject the reasomng o Hull on the last film : the structure cubes with two atoms, per pomt of the lattice, was given because the film was in agreement with a simple cubic structure, while from the dimensions found for the unit cell together with the molecular volume, it followed that the number of atoms per cell was two. Ina structure, however descnbed as cubes with two atoms at each lattice-pomt we should have to place these atoms, through their space filling 3), so far from the points of the lattice that the structure, (not sufficiently descnbed for the calculation of the diffraction effect), would give a diffraction picture completely deviating in intensity from that of a simple cubic lattice. This structure model has still found its way into the literatuie, toi example, see Borelius4). Similarly little is known of the crystal structure of the remaimng alkali metals. With sodium and potassium Schmidt also found no clear interference pattern. Hull mentions for the structure of sodium : probably a centred cube. Recently, Mc. Keehan has found a centred cube for the structure of potassium at low temperatures, he considers, however, that it is not crystalline at ordinary temperatures. 5) i) Pliysik. Z. 17, 554 (1916). Phys. Rev. 10, 661 (1917). _ 3) Bragg’s atomio domains. Phil. Mag. (6) 40, 169 (1920). al PlVmc rKEEHAN!4proc92Nat. Acad. Sci. 8, 254, 270 (1922). One must be careful about the conclusion that a substance is not crystallme smce good rnter ference ligures cannot be obtained ; sometimes a very long time of exposure appears to be neoessary. We obtained, for example, a good ülm from red phosphorus while Burger, Physica 2, 121 \,L922) finds it amorphous. After the conclusion, of this investigation we found in a communication of Haber1), the statement of the result of a still unpublished research of Debye on lithium which in some respects, (centred cubic structure ; no lattice of stationary electrons), agrees with our conclusions. 2) Outline of the investigation. In *the ordinary way it was established that lithium crystallises in centred cubes (pages s—-io).5—-io). Subsequently, the following assumptions concerning the position and orbit of the valency electron2) were tested with the intensity requirements. A) A model where the valency electron continues to revolve round its original nucleus, (p. 10—12). B) A model where the valency electron takes up a definite position in the space lattice, which is possible. crystallographically (p. 12—13) The assumption, that electrons and ions may be considered as equivalent structure elements in the lattice3), has much to be said in favour of it, e.g. the explanation of the cohesion of the crystal and of the known ratio between infra red and ultra violet frequencies, according to which -mfra 'ed— is of the same order of magnitude as vultra violet 1 / Mlon Subsequent to the lattice of stationary electrons, a structure was tested where this lattice was replaced by a moving one. Such a structure, with such large orbital motion of the electrons that neighbouring circles touch one another, serves to explain the super conductive state 5). On page 15, the model is investigated, where the electrons revolve round the points of the stationary electron lattice, in circles perpendicular to the trigonal axes. A similar model with the nuclei in the centre of the orbit is considered on page 186). Finally, (page 20), the possibility of making a choise between the two models, found to agree with the intensities of the diffracted rays, is considered. 3) Apparatus. Fora description of the X-ray apparatus, the exposure camera etc., 1) F- Habeb, Sitz. b. preuss. Akad. Wiss. 51, 990 (1919), Nachtrag. 2) Por a short review of the pros. and nons. of different electronio theories for metals, (free electrons ; lattice of stationary electrons ; “Bewegungsgitter” of electrons) see W. Meissner, Jahrb. Radioact. Elektronik. 17, 264 (1920). 3) J. Stark, Jahrb. Radioakt. Elektronik. 9, 188 (1912) ; Die Principien der Atomdynamik 111 p. 179; P. Haber, Sitz. b. preuss. Akad. Wiss. 506, (1919) ; M. Born, Dynamik der Kristallgitter p. 71 ; J. J. Thomson, Phil. Mag. 43, 721 (1921) ; 44, 657 (1922). 4) For metals of higher atomie number the testing of X-ray analysis is much more difficult (practioally impossible) ; the diffraction of the valency electrons shows itself much less in the total interference result the greater the number of atoms in the metal ion. Cf. R. W. G. Wyckoff, J. Frank. Inst. 191, 199 (1921). 6) F. Haber loc. cit. p. 1003. A. Einstein, Gedenkboek aangeboden aan H. Kamerlingh Onnes page 429 (1922). 6) The lattice of Haber, not precisely stated by him belongs to this case or that discussed on page 9. see the communication of A. Karssen in this issue of the Recueil. For some exposures the specimen was not rotated by a clock work whilst in others it was (films I and II in the plate). It is clearly seen that in the former case, dots of greater intensity occur in the interference lines ; these dots are caused by larger crystals which are rightly oriënt ated for reflection. Rotation of the specimen is desirable for good estimations of the intensity. 4) Exposures. A small rod of lithium, 1 . 5—2 mms. thick, after cleaning under paraffin oil and washing with dry ether, was covered with a thin protecting layer of molten paraffin wax and placed by means of a small glass foot in the axis of the camera. Even after some days the nietal'ic surface remained clear and shining. A chromium anticathode was used. The time of exposure varied from four to twelve hours with an average current strength of ± 12 milli amps while the parallel spark between point and plate amounted to ± 3 cms. An exposure was also made from a glass rod covered with paraffin wax in order to be able to eliminate the interference lines caused by these substances on the lithium film 1). 5) Observations. Calculation of the size of the cell from the diffraction angles. TABLE I. j • CrK„ radiation j CrK(* radiation j Distance in mm. * „ No I and estimated 103 sin3 =- 10« sin2 i 103 sin3 intensitv. 1 zh2 2 bih2h3223h32 liih2h3 calculated calculated 0|123 45 6 7 8 179 2 177 110 2 47.5 vs 214 2 213 110 3 70.4 m 428 4 427 200 4 81.0 vw 585 6 530 «11 5 91.6 s 640 6 640 211 6 116.1 ms 852 8 854 220 7 120. vw 879 10 884 310 In table 1 are given : in column 1 the distances on the film from the middle of the pattern to the middle of the interference line 2) and the estimated intensity 3). Owing to the small absorption in lithium, the correction of the distance for the thickness of the rod was unnecesary. i\ Except for the bands round about the central spot, these parasitic lines on the lithium films are so weak that they are undiscernable on the reproductions, yet on the film itsell about fifteen of these Unes, all of which comcide with lines of the glass paraffin exposure, can be seen. 2) These distances were determined as half the distance between correspondmg lines on either side, the change in the middle (zero point) caused by the excentricity of the rod amounted to only a few tenths of a millimeter over the whole fÜII3j vs = very strong ; s = strong ; ms = moderately strong ; m = moderate ; mw = moderately weak ; w = weak ; vw = very weak. 9 Column 2 gives the corresponding values of io3 sin2 (where &is the angle between the incident and refracted beam ; the radius of the camera was 50 m.ms.), It is well known that these values fulfilaquadratic relation of the Millerian indices , which, fora regular crystal takes up the simple form. sin2 |= A (h2 +h2 + h*) (1) Here, hx, h2, h3 are the indices of the lattice plane against which the rays may be considered to be reflected and A,a constant given by i2 A = – (11) 4 a2 where l is the wave length of the diffracted ray, and a, the edge of the elementary cube 1). In the first place, the numbers in column 2 were separated according to whether they belonged to the a or /3 lines and this was accomplished with the help of the following ratio derived from (I) and (II), between the angles of deviation of the a and (3 rays respectively on reflection from a definite lattice surface : ■ „ & sin2 2 -— = ~ (for the CrK rays used by us \tt = 2.284 A, lp = O 1 i2 sin2 H 2p 2 . 079 A2) hence the required ratio is o . 808). Thus it appears that 1 is the (3 line of 2 ; 4 the (3 line of 5, while after determining the factor Ap, 7 appears to be a (3 line also. By the identification of a line exclusively as a (3 line, it must be considered that its intensity must be related to that of the corresponing a line, almost as the intensity ratio of the incident a' + a and (3 rays, i.e. approximately as 4 to 5 : 1. 3) In agreement with the regular crystal form for lithium, the « lines, (and in consequence the (3 lines also), appear to conform to an equation of the form (I) with a mean factor, which by averaging for the a lines is found to be 213.4. The factor A from equation (I), therefore, is p ;s a small whole number. P To determine p we remember that the edge of the cell, a, fulfils A.2 A A Aa = " = —. It also follows from a3 =—— n, where A = the 4a2 p d N atomic weight = 6.94 ; d = density == 0.5344) > N == Avagadro !) Por a short account of the theory of the angles of diffraction and the diffraction intensity see Sommerfeld, Atombau und Spectrallinien, 2e Auflage, No. 3 der Zusatze und Erg. p. 464—471 (taken out of the 3rd edition). Marx Handbuch der Radiologie V § 165 (1919). 2) Sommerfeld, Atombau und Spectrallinien, 2e Auflage p. 172. 9) Sommerfeld, Atombau, 3e Aufl. pg. 186. 4j Landolt-Börnstein, Phys.-Chem. Tabellen, pg. 165. foCoTumns 3 and 5, 6 and 8 of table I, with the help of the values thusTund3for A, identify further the reflecting surfaces. The values of io3 sin2- calculated from equation (I) (columns 4 and 7) appeai to be in good2agreement with the observed values (column 2). From equation (II) the lattice parameter has the value 3.50 X 1 • From the deviation angles of the diffracted ray it «Hows that the elementary cell of lithium is a cube (regular crys al system) with edge of 3.50 X iO~8 cm, each cell contaimng two lithium atoms. 6) Calculation of the positions of the particles in the cell from the diffraction intensities. General. As is known the diffraction intensities dependonfhfterent fartors a) With regard to a discussion of them it will sultice to ïeie to the most important literature ; structure factor (1,2) wherei m our case, the influence of the electron configurations (1, 3,5, 7. 8 [*7) is considered; factor concerning the number of planes 9. 4 f tor which comes into consideration on account of the departure Rom parallelism and casu quo homogenity of the rays .3,4- , – 12); temperature factor (1/13,3); influence of extmction m the reflecting crystal (6, 3,4- 8) and absorption in the rod (n, 16). а) Por short review see lorexample *« sar» ftiW&fófe Pws- “'•*■ss ,m,,‘ ? S: vm W eiJ' з) C. G. Darwin, Phil. Mag. 27, 315, 675 (19 )• *) G. G. Darwin, ibld. 43, 800 (1922). f1915-) » W. H. Bragg, Phil. Trans. Roy. Soc A. 215, 253 (Ulo). б) W. H. Bragg, Phil. Mag. 27, 881 (1914). 7) P. Debye, Ann. Physik 46, 809 8) A. H. Gompton, Phys. Rev. 9> 29 ? • Göttingen 1916. 1. «) P. Debye and P. Scherrer, Nachr. Kg • • „g3 (1914). P. Debye (alter H. A. Lorentz), Ann. P Y • ’ и) p! Debye and P. Scherrer, Physik, Z. , * Mag. 41,309 (1921) W. L. Bragg, R. W. James and C. H g™ (1913). Ann. Phyi3) p. debye, Verh. deut. physik. Ges. 15, 678, ïdö , G 1 Si»i4j: j9’thomson, Conducüon of Eleotricity through Gases p. 326. 1S) A. H. Gompton, Phys. Rev. 7, 658 (19i ~ n,,,, i‘) W. Gerlach and O. Pauli, Z. J9l7|' 17) A. W, Hvll, Phys. Rev. 9, 84 (1917) 10, bbl (ui n. number = 0.6062 X io24 ; n = number of paxtides per umt successive trials of n = 1,2 etc, ït was mquired foi which of these values a whole number for p followed from both equations. appeared that n = 2, p=2 sufficed within one That the above given density for lithium is accurate to withui one per cent., is proved by the X-ray photograph; from A. = 106.7, it "lhe last four factors alter gradually for successive interference lines b These factors (among which the temperature factor and extinction factor can only be estimated very uncertainly), can be left out of consideration if only the calculated and observed intensity ratios of neighbounng lines are compared; 2) this also holds for the influence of the election configuration in atoms which are not too light. The structure factor S. This gives the composition of the amplitudes diffracted by the particles present in the cell and has the form, S = As en * (fs hi +°sh2 + rg h3) where o and r are the coordinates of the diffracting partiele with regard to the system of coordinates, for which the edges of the cell are the axes, expressed in terms of the length of the edges The quanmeasures the amplitude scattered by the partiele. If we ascribe the diffraction to the electrons outside the nucleus3) then the summation m S must be carried out for each electron separately; the proof tJle analysis is then to determine properly the positions (orbits) of all the electrons in the crystal. Only with atoms with a very small number of electrons, can any such thing be tried. (In heavier atoms, as an orientating calculation shows, the greater part o the electrons are situated so closely to the nucleus that in the formula for S, their coordinates can be taken as approximately equal. ln th*s case- the structure factor is transformed into a summation over different atoms; the scattering power of an at om appears proportional to the atomie number, which is approximately found experimentallyfor simple structures, and then applied for determining structure). Structure of Lithium ; testing the models. I. Ihe number of lithium 'particles in the elementary cell is two. Thus the crystal period becomes equal to the X-ray period4) a== 3’s° A- For the two lithium particles there exists only one two-fold position, namely o, o, o, and |, j, l Separate positions for the two valency electrons are no longer available. By supfiosition (I.), therefore, he only possible case «s that where the valency electron remains on it ’s parental nucleus (statte lattice). The orbit of the valency electron cannot have one privileged position in the atom, since then, there were at most two privileged positions (orbital plane in the atom o, o o and that in the atom mi i), which is contrary to the regular structure. hor reasons of symmetry we must assume then, that the junction lmes, nucleus-electron, occur at each moment equally distri- ~ the absorption lithium and its hvdride have the advantae-p tbat their absorption coefficients are very small. advantage that l GL?\^°}jK™eyer’M. Bijvoet and A. Karssen, Z. Physikl4,29l (1993) .T*1® electrons attain a much greater oscillation the incident rav of thT^uölus1 arf'much more strongly. That the electrons strong btadlng! noticeable by this means is explained by their 4) Cf. P. Niggli, Geometrische Kristallographie des Diskontinuums p. 480, 506. buted, in all directions on such a part of the crystal as causes the interference (‘ spherical atom”) ; or that in each atom defmite planes of these radii vectores are privileged, for example, the four planes perpendicular to the trigonal axes J). Then the atoms in o, o, o and f, J, J are equivalent for the reflection, since there exists no difference in their orientation with regard to the incident radiation. If we introducé the scattering power of this tri-electronic atom by i/mi (for definite land radius of sphere only a function of d), the structure factor becomes : S = T(,Li + e"1 (hi +h» + "J i.e. 2 n>u for planes with even 1 h : zero for those with odd £h. The absence of planes with odd £ h appears to be in agreement with observation. For the planes with even £h, taking into account only the “planes number” factor v, the factor of Lorentz and £ h2 the square of the structure factor the intensity becomes : V I— —— ip2. £h2 Table II gives the observed and calculated results in columns 3 and 4, which agree well with each other. TABLE II Intensities .... , , Calculated Plane “Number of indices P“’ °bserved p = 0.20 A | f' = 0.65 A 1 23 | 45 100 6 00 11012 vs 6.0 Vu 38 111 8 00 200 6 m 1.5 7: 210 24 00 211 24 s 4.0 14 220 12 ms 1.5 Vlj 4 221 ) 24 ) q q 300 ) 6 \ ~ u u *) That the reduction factor on these last suppositions are not much different from those fora “spherical” electron, (especially for reflections from latticc planes which are not coincident with orbital planes, cf. page 17 note 2) appears from a eomparison of the values of ip from the table XT p. 33 with those of from table XII p. 35. How Haber considered the orientation (loc. cit. 4 p. 1003) of these orbital planes is not stated; for the explanation of the super conducting state the completely regular model with one, privileged orbit in each atom, which in the super conducting state, is still more poivileged, would best suffice. Such a model would be possible after doubling the cell (case II) and is considered along withthe intensity on page 18. Lithium therefore crystallises in a centred cube. From the lattice parameter for this structure, the valuc 3.04 X io8 for the diameter of Bragg’s atomic domain, follows, in very good agreement with the value 3.00 xlO "8 given by Braggl). It remains now to investigate for the determination of the possible radii of the spheres, the dependence of i|>Li with 9. To this end we will first calculate, for the directions of the diffracted pencils, the diffraction effect of a particular electron in a spherical atom at a distance p from the nucleus. The diffraction amplitude resulting from the diffraction by the electrons, corresponding to the one considered in all the N participating cells, is independent of the position of each of the electrons at the moment; indeed, since rays diffracted from corresponding points of different cells in the direction of the diffraction, are in phase, the diffraction result is equivalent to that of N electrons, equally distributed over a sphere of radius q about one of the atomic centres. Or it is thus, the radiation per spherical electron agrees in phase with this resultant while the amplitude amounts to times the resulting amplitude. Let us determine the magnitude of the amplitude, (expressed in the amplitude per electron scattered in that direction) fora given angle of deviation, resulting from N diffracting electrons uniformely distributed over a sphere of radius p, and the phase difference of the diffracted beam with a ray diffracted at the centre of the sphere 2). Rays possessing similar phases are diffracted by electrons lying in a plane parallel to that from which the rays appear to be reflected. Rays diffracted at points a distance d below the corresponding plane through the centre of the sphere have a difference of path 2d sin – comparedwith the ray diffracted at the centre of the sphere ; i.e. a phase difference 4 jt d sin t- The number of electrons at a distance between d and d + X ö d below the plane considered is —. N, indeed, the surf ace of 2 p ■ da . a zone of height ö d of a sphere of diameter 2 p is times the 2 Q surface of the sphere. The amplitude of the scattering by the elec- M W. L. Bragg, Phil. Mag. (6) 40, 169 (1920). ») debye, [Arm Physik. 46, 609 (1915)], deduces the analogous formula lor the scattering by au amorphous body with a definite electron conliguration in the atoms ; in this formula all the mutual distances make their appearance. Glocker [Z. Physik, 5, 369 (1921)], incorrectly, took over this formula unchanged for atoms without any regular orlentation, which stand at the lattice points of a crystal. Bragg, James and Bosanquet have shown [Z. Physik 8, 77 (1922)] that in the electron configuration factor for this case, only the distances of the electrons to the centre of the atom come in, as appear? from the above deduction (after Compton loc. cit. p. 49). By the disappearance of the imaginary part of this amplitude, the rays diffracted through the sphere appear to be in phase or diner by iBo°, with the ray diffracted at the centre. By mutual mterference the resulting amplitude is decreased in the ratio compared with the result for complete intensification. Fig. I (page 33), gives the course of this function. In calculating the diffraction intensiües we can thus consider an electron of a spherical atom as if it were placed at the centre of the atom with its scattering power reduced by sin_P With the help of equations (I) and (II) on page 34, we can P ' 2 sin * H c 2 substitute for directions in which the crystal reflects, for where H = V £ h2. , The reduction factor fora spherical electron then reads o- «H bm 2 n – , a firn *e “ eH [ ' 2 TC a and the reduction factor fora spherical atom: c. Q H Sm 2n ♦ – * Vh— (III,) 2 n a whereby all electrons in the atom are summed. Let us put in (III1) the radii of the lithium atom equal to the values given previously by „ ~ nor, 1 ■ n' =o 63 A in order to calcu- BOHR -1) , pinner ring 0.20 A , Pouter ring O i) N. Bohr, Phil. Mag. (6) 26, 490 (1913). . , . 9 4 Ti ld sin— N 2 , trons lying in this zone amounts then to —e- —d a- ne amplitude diffracted by the whole sphere is represented, therefore, as : late tpu. For the reduction factors tp, tp' and ipLi respectively for the inner electron, outer electron and the whole atom we find (a = 3.50 A) : plane tp tp' i/jLI = 2tp + tp'. no 0.96 0.60 2.52 200 0.92 0.31 2.15 211 0.88 0.10 1.86 220 0.84- —0.05 1.63 Column 5 of table II is obtained from column 4 by substituting this value of tpjj. Column 5 must be considered as agreeing sufficiently well with observations. However, this also holds certainly fora 1p which alters less with F, so that smaller values of (F are not excluded; on the other hand, one would expect that larger values of p' can be excluded from the consideration that if tp' becomes too strongly negative for large 9, the intensity of the plane 220 will be calculated too small. Also a similar limiting of the possible values of g' leads to no result since tp at its (first or most strongly negative), minimum amounts to only – 0.22 4), a value which, as will be seen by substitution, must be considered as allowable. From X-ray analysis, therefore, the case of the "spherical atom” appears possible, without, in this case, anything being deduced about the orbits of the electrons. II The number of lithium particles in the elementary cell is sixteen. The crystal period a' is now taken equal to twice the X-ray period; a' = 7.00 A. A reflection in case I indicated by h1; h2, h3 (table 11 P- 9) agrees with the reflection 2hlt 2h2,2h3 in case II (table V p. 18) 2). For the sixteen lithium atoms deviations are possible from their positions given under I, without upsetting the regular symmetry.3) Other possible positions4), the number of which increases as one allows the assumption of the equivalence of all sixteen lithium atoms to fall away, owing to their physical improbability are not further discussed; besides, a connection with the observed intensities is only to be expected with values .of the parameters determined in this case, which the model would scarcely differentiate. In this case, separate positions in a regular model are available for the sixteen electrons, although not in a lattice which is congruent with that of the ions. In placing the electrons midway between the ions, an arrangement which is obtained exclusively as follows has to be considered5). ‘) E. Jahncke and F. Emde, Functionentafeln p. 3; fig. 1. 2) The doubling of the parameters oom es from a choice of the values of n and p. on page 6 : n' =l6, p' = 8. Obviously, A„ = and = iAa or aceording to equation I. page 6, for the same reflection, 2h'a = 42h2. Therefore h' = 2 h. 3) Gf. P. Niggli, loc. cit. p. 443. 4) Deduced from tables of Niggli loc. cit. p. 410—411 or bv R W G Wyckoff p. 105—106. 5) Neither the three equivalent axial directions nor the six equivalent face diagona! directions are to be considered for arrangements of the sixteen electrons, The coordinates of the electrons The lithium ions will be considered as spherical ions (for the values of ti> of these electrons for different reflections see, table XI) ; on account of the small distance of these two electrons from the nucleus, other assumptions give only slightly differing reduction factors. For the valency electron we shall test the following assumptions : 1) The electrons remain stationary at their lattice points (except for their heat motions). 2) The electrons travel round these points in circles perpendicular to the trigonal axis on which their centres lie. On the first assumption, is constant = 1. On the second assumption, for an arbitrary reflection, in general, ipQ 41 33* J 420 ; 421; 430 ; 500 ; 333 ; 5n ! 520 ; 432 ; 1. On the first supposition on page 13, (lattice of stationary valency electrons), =Vb M’o IPd —T- Table 111 gives the calculation of the intensities according to TABLE III*) Caloulated _ observed h, b2 h3 A 2 . j Li +* e y ~h 110 – 211 2 24 6 16 220 7.68 12 8 88 310 321 248 1414 400 7.36 4 6 6 4 m 330\ _ _ 411 ) 332 + 2 24 224 422 7.04 24 24 50 s 431) – __ _ 510 S 521 + 248 30 6 440 6.72 + 412 32 43 ms i) in the oorresponding table Verslag. Akad. Wetenschappen Amsterdam 29, 1212 an error occurs : the planes 521 and 440 must not be bracketed together, their intensities 6 and 54 respectively should not be added. The intensity of 400 seems to be calculated so much too weak that this model must be rejected1). Thus from X-ray analysis a lattice of stationary valency electrons aMears to be unsatisf actory. . , 2. On the second assumption of page 13, (rings of binding electrons) the diffraction result must be calculated from one electron revolving in a definitely placed orbit. . , Let us consider N electrons together revolving m circular paths (radius O in mutually parallel orbital planes about such pomts that ü the electrons stood in these centres, the rays diffracted by them, would completely reinforce one another in the reflection under consideration. Averaging over the N electrons, we assume a distribution o e radii vectores equally in all directions in the orbital plane. Then the amplitude of diffraction of these N electrons is equal to that scattered by a similarly placed ring, with N electrons about one of the pomts One may calculate again as if the ray diffracted by one such electron is in phase with the resultant and its amplitude amounts to —th part of the resultant. This is calculated in the following marnier lor reflection from thei latüce plane h, ha h, which makes an angle « with the orbit plane *). The rays rellected from points in the same plane h, h2 h3 are alike m phase . oomp the ray diffracted in the same direction by a point a distance d below t, 4 nd sin 2 plane, lias a difference of path 2 d sin. or a phase difference of – – Taking 0 the angle between the intersecting orbital plane lattice plane and the radius vector to a circle element d0 the diffracted amplitude by this ring & 4 ni d sin <5- element (containing electrons), is given ty e 1 671 & 2 sin ö" tt ... stituting in this d= q sin 0 sin « and -— = (page 11) and ïntegratmg i) That this eonclusion would be altered by bringing into the calculation, the heat motions is improbable (Cf. Bragg, James, and Bosanquet, PM. Mag. 44, 433 (1922) ; the amplitudes of these electrons oscillatmg about their equihbrium positions will be considered small in this model with regard to those of the heavler particles. Indeed, if we assume, that the forces with wlnch the electrons and ions are bound at their equilibrium positions m the ion-e ectron lattice are of the same order (Cf. Born, Dynamik der Knstalgitter p. 71), then the mucn smaller masses of the electrons will consequently have a much bigger vibrati number ; this frequency lying far in the ultraviolet has, accordmg to the quan tum theory an energy of vibration, which at ordinary temperatures, remains far below the èquipartition amount: from the small potential energy with regard to that of the ions, follows the smallness of the displacements compaxed with those of the ions. The average displacement of the ions at ordinary tempera tures is measured in hundredths of A ; thus, without doubt, the displacements of the valency electrons in this model are very small compaxed with that which would be necessary to decrease the diffraction effect sufficiently (Cf. the CMC lated value on page 17, p = ± 0.08 a = ± 0.6 A necessary for such S in the circular orbit considered). oa *) According to D. Coster, Verslag. Akad. Wetenschappen Amsterdam 28, 391 (1919). over the whole circle, we find for the amplitude of the beam diffracted by the ring, (phase compared with rays diffracted from a point in the lattice plane, e. g. the centre of the ring): were J0 (x) represents the Bessel function of the zeroth order. 1 herefore for the reduction of a ring electron we find (IV) and for the reduction factor of an atom with circular electron orbits : (IV1) The value of this function for real values of the argument is real and less than unity; in fig. Ha (p. 36), the curve for the plane 200 gives the course of this function. Comparing with the system of stationary electrons, each completely reinforcing the other, we find also for the rotation of the electrons in circles in definite positions, a weakening of the diffracted wave with conservation of or reversal of the phase. Thus, by substituting in the structure factor on page 14, according to equation (IV) we find the structure factor (V) where aA, »B, o0 and aD represent the angles between the lattice plane and the orbital planes perpendicular to the trigonal axes A, B, C and D repectively. This substitution, after putting in the values for 27t H sin a, shows that also the structure factors of the planes iio, 310» 33°. 411 and 510 are zero independent of the values of —; the ■ a' calculated non-occurrence of these reflections is in agreement with the observations. It was then investigated, for which of the —r values the calculated a and observed intensities of the remaining planes agreed. In the first place the intensities of the 321 and 400 planes were compared: from table 111 (for = 0) it follows that small values of are to be rejeca a' ted because they give too low a calculated intensity of 400 (e.g. comp- TABLE IV. Intensities , . , Calculated hj h2 h3 observed o_ yalues. 0.05/0.06/0.07/0.08/0.09/0.10 8. 00 \ 8 6 42 1 0 400 m 7 8 10 1214 16 Thus for values ol 0.08 the calculated intensity ratios appear cl to agree with the calculated. It may easily be seen that thisremams also the case for all larger values of For successive minima and maxima cl the value of function becomes continually smaller. Foi small values of the arguments, on the first descending branch of the function, the ratio was unallowably large. The limit for which this S4oq was the case is determined. For larger values of the arguments, in each case : i 5321 = —Jo (23-51 Ji) —Jo (18.50 {-,) <—2 X (value of J0 (x) in the first min.) i.e. 0.80 while i S4OO = 7.36-4 Jo (20.52 £) >7-36 -4 (value of JG (x) in the second max.) i.e. > 7.36—4 X 0.30 or > 6.1. This gives for the intensities, 1321I321 < 2.3 and 1400I400 > 14, in agreement with observation. Table V gives in column 2 the calculated intensities for binding rings with -, = o.i; planes for which the structure factor becomes zero independent of the value of —t have not been included in the table. Owing to the great weakening in the diffraction effect of the valency electrons for these and largei radius values -) the calculated intensities differ only slightly from the intensities found in case A ij The necessary Bessel functions are taken trom the Punktionentafeln ot E. Jahnke and F. Emde p. 110. , ... , 01 course this does not occur in a reflection Irom a lattice. plane whicn coincides with an orbital plane ; then, independent ol the orhit radius, no weakening occurs. In our case, however, there are lour different positions of the orbital planes. Thus the coincidence can only be fulfilled fora quarter of the electrons, so that, great weakening of the diffraction effect will occur for large orbit radii also. Besides, the octahedral planes for which sin « is zero, do not reflect here. See figs. Ila and Ilb. 2 ared with 321) Table IV gives the intensities for increasing values of T-T calculated from I = —1). a' (simple cubic lattice). These latter, given in column 4 of table 11, are inserted for comparison in column 3 of table V (after multiplication by 4 in order to be able to compare in absolute magnitudes). TABLE V. Intensities. Galculated. hit h3 observed. Lilions between which o Li atoms are binding orbits -7 = 0.1 a Q. 1 1 | 2 {_ 3 211 8 220 vs ' 71 24 321 0 400 m 16 6 V’lj 332 1 422 s 61 15 JLI 431i 2\ 510 5 os 521 0 440 ms 15 6V-Li A choice between the two suppositions in table V seems to be possible after very accurate intensity measurements of 211. Smaller intensity of 211 may, however, be obtained also on the supposition of column 2 by accepting a larger orbit radius. Meissner says (loc. cit. p. 268 ; according to a communication of Scherrer) : “Auch wenn das Elektron auf einer festen Bahn um den Gleichgewichtspunkte umliefe, müsste es sich zeigen, da es z.B. bei kreisförmigen Umlauf in erster Naherung so wirkt, alsob es im Mittelpunkt der Bahn liegt”. It is at once obvious, that such a “first approximation” is by no means allowed for orbital radii of the expected magnitude (e.g. one-quantum rings), ■ not even for the lower reflections. A model in which the electrons in the doublé cell describe paths about the ions, in planes perpendicular to the trigonal axes, (Cf p. 9 note i), will, independent of the value of the orbital radius, scarcely deviate in diffraction intensity from the intensities calculated fora spherical atom. In both cases, (in opposition to that of binding rings between the ions), small values of p suffice, owing to the valency electron working together with the electrons in the inner ring; while for values of p for which the ray diffracted by the valency electron is opposite in phase to that diffracted by the inner electrons, in both cases the diffraction effect of the valency electrons appears to be weakened to such an extent, that their influence is almost unnoticeable. From X-ray analysis, a structure for Lithium appears possible in which electron rings, with orbital planes perpendicular to the trigonal axes, are found between or round about the lithium ions. For the allowable value, of the radius in the case of the binding orbits between the ions, a lower limit was found, namely q larger than about one fifth of the distance between the two ions. Let us calculate for comparison with this lower limit, the radius p of an n-quantum ring of a single electron between the two monovalent \/ O o o Li+ions which are at a distance 2b = . 7.00 A = 3.03 A from each 4 other. Simplifying by considering only the attraction of these two neighbouring ions, we find for the determination of q . Centripetal force = electrostatic attraction : quantum condition : 2n; m « p2 = nh After elimination of co : p = 1.2 A. The radius of a one-quantum ring fulfills the condition deduced from X-ray analysis 9 > ± 0.6 A. The influence of the extinction in the reflecting crystal, of the absorption in the rod (cf. page 8 note 1), of the heat inotion (cf. page 15 note 1) and the taking into account of the polarisation factor, in all probability will not appreciably alter the above conclusions, concerning different ; suppositions as to the position and orbit of the different valency electrons. Also, on adding to the intensity formula a factor ——, as will be described on pages 27 and 38 below, & cos 2 the agreement in table V and the nonagreement in table 111 remain. 1 + cos2 tt Then, bringing in the polarisation factor also, the factor —— . _JÜ tobt added has the following values for the four calculated in» cos 2 tensities of table V, 0.75 ; 0.67 ; 0.90 and 1.96, successively. lts effect is, that the intensity ratio of the lines 422 and 440 are better reproduced*) while the ratio of the lines 321 and 400 in table 111 are not noticeably 'changed by adding the factor. Neither will the introduction of not circular i) Throush a defect in the X-ray apparatus it could not be settled if, as we expect, the 440 reflection becomes relatively weaker by exposure to Cu^-rays. orbits bring in much alteration ; in fact, from the calculatie®, of the diffraction effect of the valency electrons and by testing different suppositions, we need only the conclusion that the diffraction effect of the rotating valency electron is weakened to a relatively much greater extent than that of the inner ring electrons. On the other hand, owing to the above apparent insensibility of X-ray analysis for electrons with not very small orbital radii, along with the present day uncertainy about several intensity factors, a choice between, and further consideration of both models (ions with electron rings round them or between them), by X-ray analysis is not possible; not even after more accurate determination of the diffraction intensities. Summary. From the X-ray photographs of lithium it appears that : Lithium crystallises in centred cuhes; the lattice parameter a = 3.50 o _ o A; the diameter of the “atomic domain” =3.04 A. There is no lattice of stationary electrons. It is possible that the valency electror. rotatesround the nucleus, or also, that the valency electrons describe orbits between the lithium ions, in planes perpendicular to the trigonal axes\ the radius in the latter case being larger than about one fifth of the distance between two ions (one-quantum orbits suffice). It seeins to me desirable, that both models should be subjected to an investigation into the relevant values of the radii from the standpoint, that the cohesion of the crystals is to be explained by the electrostatic forces between the particles1). The model with binding rings between the ions is in agreement with the usual idea of the homopolar binding and explains 'the cohesion, at least, for not too great radial values. 111. INVESTIGATION OF THE CRYSTAL STRUCTURE OF LITHIUM HYDRIDE. i. Lithium hydride had not been investigated with X-rays. The investigation was taken up in order to see how much concerning the behaviour of the valency electrons could be brought to light in this simplest compound; more especially, in order to test the analogy drawn between lithium hydride and the heteropolar alkali halogenides. Moers 2) electrolysed fused lithium hydride. Besides the separation of lithium at the cathode, he found hydrogen was produced; he concluded that probably the hydrogen is present in the form of HM ons, although it was not shown, that the hydrogen is freed at the anode. Bardwell 3) afterwards so directed his experiments with other hydrides, (NaH, KH, CaH2), that it could be concluded that separation ot hydrogen takes place at the anode in the electrolysis of the fused hydrides or *) Gompare H. Thirring, Z. Physik 4, 1 (1921). W. Meissner, Jahrb. Radioakt. Elektronik. 17, 272, (1921). 2) K. Moers, Z. allg. anorg. Chem. 113, 179 (1920). W. Nernst, Z. Elektrochem. 26, 323 and 493 (1920). 3) D. G. Bardwell, J. Am. Ghem. Soc. 44, 2499 (1922). 2, Review of the research. The difficulty presented itself that after the exposures, in most cases the hydride content had been reduced by 15 —2O per cent. by weight. The lines relating to the hydride could be known with certainty in three different ways (pp. 22—23). With the hydride a great difference was found in the intensity ratios of some corresponding lines by exposures with Crk—or Cuk—rays respectively. Something similar appears from exposures of lithium hydroxide and can be found in the literature fora few substances investigated by different workers using different rays. The intensity formula in the form generally applied in the powder method, gives no explanation of this. Probably this is to be found in the addition of a factor —— to the intensitv formula, which can be deduced theorecos-2 tically and is experimentally proved in the case of lithium hydride (p. 39). First, neglecting all the continuous factors, as intensity conditions for the hydride, only those ratios of neighbouring lines were taken, which agreed on the Cr —and Cu – films and the conditions deduced were very amply set. (p. 29—30). Finally the calculation was based on the intensity formula on p. 38 The calculation of the diffraction directions showed that lithium hydride belongs to the cubic system and crystallises with four groups of LiH per unit cell (§4). This is in agreement with the supposed NaCl-structure together with the absence of the planes of mix ed indices. In absence of all further crystallographical data we have confined ourselves to the question whether, sticking to a NaCl or ZnS structure, an A, Klemenc, Z. Elektrochem, 27, 470 (1921), of their Solutions in fused alkali chlorides. Thomson already had demonstrated the occurrence of negatively charged hydrogen ions in discharge tubes. Klemenc l) , by thermochemical calculations, tested a structure for the hydride analogous to that of sodium chloride, with positive lithium and negative hydrogen ions. He came to the conclusion that two-quantum orbits in the H ion suffice. These calculations, however, lose all their value simplv because in his equation (I) l.c. page 471, one of the terms has a wrong sign. Further it was interesting to calculate the diffraction effect of a hydrogen partiele, in order to see in how far, the usual approximate supposition, reflecting power constant, can be allowed as a basis for the determination of the positions of the atoms in the crystal for orbit radii in the atom or negative ion of about the same magnitude as those calculated by Bohr for the free partiele. For the testing of different assumptions concerning the orbits of the hydrogen electron from the intensity of the crystal diffraction, after the solid element, only its simplest compounds with the lightest elements, among which lithium hydride has the first place, are suitable. electron grouping could be found which agreed with the reflection intensities found. In the suppositions tested (p. 30—31) concerning the orbits of the valency electrons (atomic lattices ; ionic lattices ; connecting rings between the ions ; the path of the electrons whether or not in a plane of definite position), all radius values for the valency electrons were systematically tried (pp. 34 40) after considering the particular structure factors (pp.3333) and reduetion factors (pp. 33—-35); the small radius of the inner ring of the lithium partiele was here taken as equal to the value given by Bohr ; the calculations remained preliminarily confined chiefly to circular orbits 1). 3. Preparation of Lithium hydride and exposures. The hydride was prepared as given by Moers 2) from lithium and carefully purified hydrogen. It was finely powdered in an atmosphere of hydrogen, and preserved in sealed tubes. For the exposure in the Debye-Scherrer method the substance was introduced into a little shell (diameter about 2 m.ms) of cigarette paper, covered with a celluloid skin. After exposure the hydride-content in most cases had been reduced by 15—20 per cent. by weight. Instead of trying to bring this decomposition to a minimum we have eliminated these parasitic lines as follows : (a) By continued decomposition in the air and exposure of preparations of decreasing contents3), one series of lines (table VI no. 5,6, 7,12, 14, 16, 17, 20 and 22) appeared to decrease in intensity and the remainder to increase. Obviously the first group must belong to the hydride. The position of these lines appears to be independent of the degree of decomposition so that lithium hydride forms no mixed crystals with the decomposition products 4). (b) On the exposure of a coarsely crystallised non rotated sample, the interference lines of the hydride by their dots of greater intensity (caused by the larger crystals) appeared clearly distinguished form those of the very finely crystallising decomposition products. 1) The introduction of elliptical orbits into the caleulation is, in principle, not difficult; in this case the reduotion factor of an electron is found by averaging in a definite manner the reduction factors for values of « between the distarices of the electron from the. nucleus at perihelion and apheiion; considering only two-quantum orbits, assuming undisturbed elliptical orbits (excen■ -4 l/3 tricity _—) and supposing all directions of the major axis in the orbital plane occur with equal probability then, with the help of tables XI and XII on p. 33 and 35, the reduction factors for these orbits can be calculated as a function of the major axis. By means of these reduction factors the exclusion of the axis values follows completely in the same way as given on p. 34—37 with those of tables XI and XII. Performing such calculations will only become of consequence when it is possible to deduce sharper conditions for the values of the structure factor for the different reflections. 2) loc. cit. 3) Specimen analysis: 0.0268 gr. hydride gave in water 71.1 ccs. of Hs at 763 m.m. of mercury 205 m.m. of water; t = 23°. Content 85.3 per cent. 4) Gf. Vegard, Z. physik, 5, 17 (1921), Distances of outside edge in m.m. and estimated intensities. LiH- LiOHlines lines Pilrn XIII. I Film XVII. Film XVIII. Film XXVII. 85% LiH. ' 30% LiH 10% LiH LiOH _Ö 1 23 1 ±2B (band) ±25 (band) ! (band) ±2B (band) 2 37 v.w. j 40 m.w. 40 m.w. 3 43.6 m.s. 43.7 s. ' 44 s. 44 v.s. 4 48.1 v.w. 48.3 w. 49 w. 49 m.w. 5 51.1 w. ! 6 54.0 v.W. 53 v.w. | 55 v.w. 7 60.0 m.s. 60.0 m.w. 59 v.w. o 62 ? 54 v.w. 64 v.w. 9 66 ? 68 w. 10 70.1 w. 70.3 m. 71 m.w. 71 m.s. 11 77.0 w. 77.1 m. 78 m. 78 s. 12 79.9 v.w. 80 ? 13 86 3 w. 87.1 in. 88 m.s. 88 s. “ 15 90-’ m!- r T >9 m. 98 w. 15 X5: X ? « t «o r,o i 109 v.w. 110 m.w. 110 m. 19 113.5 W. 113.8 s. 114 s. 114 v.s. 20 117.8 v.s. ! 118.2 s. 119 m. 2! I 124 w. 125 m.w. 125 m. 22 131.1 s. j 131.9 m.s. (c) By comparison with the exposures of a few substances which may occur by decomposition, the lines I, 2,3, 4,8, g, io, ii 13, 15, 18, 19, and 21 appear to form the X-ray photograph of lithium " Exposures were made with Crk—and Cuk—rays, of stationary and rotated samples, which were more and less finely powdered and slightly or to a larger degree decomposed; in one Cu-exposure the Kp rays were taken out by a nickel window. ' . , In using chromium rays the parallel spark gap from plate to point was ± 3 cm., for copper exposures ± 4 cm. The mean current was about 12 milliamps. The time of exposure varied from 2 to 12 houis. In some of the long exposures, the sample was renewed after a lew h°observed intensities. The intensity ratios of the diffraction lmes mutually agree completely on the different Crk—films, eight in number (Cf. also columns 1,2 and 3 in table VI)- The agreement between the seven copper films which were much weaker for the same time ol i) Lithium oxide (LisO) too was exposed ; the X'ray. to be internretable as: CaF.» structure with lattice parameter 4.62 .10 cm. Then the W is 1.99; tAe sum of the radii of the “atomic domrnns” of Lx and O 2.00 A (according to Bragg 1. c. — 1.50 A ? Rq u- ' A)- TABLE VI. exposure, was much less satisfactory : table VII gives the estirnated intensities of four different films and in column 5 the mean of these readings. Column 6 gives the chromium intensities. On the same sample Crk—Cuk—and Crk—exposures were made directly after each other. TABLE VII. Observed intensities. hlhA Cuk rays CrK ~ film film film film XVII. XXVIII. XXXI XXXVI Mean' o 1 23 45 6 111 m.w. in. m.w. v.w. m.w. w. s v.s. s. m.w. s m.s. 220 m.s. m.s. m.s. m. s. m.s. m.s. 311 m. in. m.w. m.m. v.s. 222 v.w. v.w. v. w. v.w. s. Both the Crk—.exposures agreed completely, so that the dilference in intensity ratios between the Chromium and Copper exposures cannot be caused by an alteration of the sample. Compared with the Cuk—films, the diffraction lines on the Crk—films, for large angles of deviation increase strongly in intensity with the angle of deviation. By comparing columns 5 and 6 there appears such a strong dependence of the intensity ratio on the wave length of the diffracted ray, that even the succession in intensity of the successive lines 220 and 311, although not very close together, differs and the line 220 is one of the strongest on the Cr—film but the weakest on the Cu—film. On exposures of lithium hydroxide tooit was found that the intensities of the higher lines on the Crk—films were relatively stronger than the corresponding lines on the Cuk—films. Some examples of the same phenomenon can be taken trom the literature; a small number of substances appear to have been studied by the powder rnethod by different investigators using different rays : TABLE VIII. DIAMOND , , , Observed intensities. hj li2 h3 CuK rays | MoK rays. 9on w a 406 100 i*°A 158 j 50 fnn 94 ö 40 409 Q * g 55 x 10 331 02 171 25 fable VIII gives the observations for diamond of Debye and Scher- rer l) with Cu—rays and of Hull 2) using Mok—rays ;in the former is Isii'Óssi and in the latter ïsi^ïasx- TABLE IX. IRON , . ! Observed intensities. I11 II2 H3 CuK rays | MoK rays i 2 110 Sa s j 400 200 § ’ö o m J 46 211 s 54 220 J S M 24 Table IX gives the observations for iron of Westgren and Lindh with Cuk—rays3) and of Hull4) with Mok—rays : In the former is 1200I200 1220 and 4n the latter 1200I200 1220.I220. For aluminium with Cuk—rays Scherrer 5) gives : 5420 _ (observed) = (calculated) while Hull 6) on the other 5422 4145 77 S 2*5 hand with Mok-rays gives Q 420 =—. And here it concerns successive lO neighbouring lines the intensity ratio of which is used for testing structure models. As is known 7) the intensity formula, which, according to Debye and Scherrer, has been applied in most investigations with the powder 1 + cos2- v. [S]2 . & 2 Jd sin2 method, reads : I 00 e 2 2 ü sin2 2 where S=2*e ~ ni hi +®sh» + Ts . Sometimes a correction for the absorption in the rod is also applied in the above8). For large extinction the reflection intensities will also depend on the limiting planes of the crystal and on the dimensions of the crystal fragments9). Besides, the bréadth of the interference lines and the distribution of the intensity over them should also be taken into account. In most cases the absorption factor, like the heat and polarisation *) P. Debye and P. Scherrer, Physik. Z. 19, 481 (1918). >) A. W. Hull, Phys. Rev. 10, 661 (1917). 3) A. Westgren and A. E. Lindh, Z. physik. Ghem. 98, 181 (1921). 4) A. W. Hull. Phys. Rev. 84 (1917). 5) P. Scherrer, Physik. Z. 19, 26 (1918). 6) A. W. Hull, Phys. Rev. 10, 661 (1917). 7) See e. g. P. Debye and P. Scherrer, Naehr. Kgl. Akad. Wiss. Göttingen 1, (1916). 8) P. Debye and P. Scherrer, Physik. Z. 19, 474 (1918). ») With this point in mind an investigation was begun with the heavy potassium iodide, which can be obtained crystallised in cubes or octahedra ; however, this research was abandonned for the present. factors, is left out and in testing structure factors only the intensities of neighbouring lines are compared1). Most of these factors for agiven crystal depend only on h1( h2 and h3. In the usual intensity formula, given above, occurs, independently of l, only in the polarisation and i “I- cos2 O 1 absorption factors. The first , especially for lines lying close together, can explain no large difference in the intensity ratio. Also, owing to the absorption a weakening of the forward reflected rays is to be expected ; it is not probable, in spite of the uncertainty here, that in the light lithium hydride the absorption would cause so strong an effect, and even less, that the intensity ratio of lines lying close to one another for large {l’s would be strongly influenced by this cause2). It seems best to look for the explanation in the factor, which after Debye 3) and Scherrer is usually introduced in the calculation. for the powder method as —- ». sm2 2 At the time of the deduction of this factor the powder method was not yet invented. The factor takes into account that the beams are not quite parallel and monochromatic and is most suitable for the arrangement of the Laue method. The intensity of the reflected beam is not found only from the reflection in whïch the phase differences of the wavelets diffracted at corresponding pointsfollowingin succession alongthe direction of the axis amount to a whole number of times 2n, but by integration of the intensities of the pencils for which these phase differences have somewhat deviating values. For the method of Bragg, Darwin 4), Compton 5) and Bragg, James and Bosanquet 8) deduced intensity formulae for different cases in which, for example, a factor 1 occurs fora crystal which -ft 'tf sm cos 2 2 turns during the exposure with a constant angular velocity and totally absorbs the incident radiation. Here, as is usual in the powder 1) Among others P. Scherrer, Physik. Z. 19, 26 (1918) ; N. H. Kolkmeyer, J. M. Bijvoet and A. Karssen, Z. Physik 14, 291 (1923). 2) Cf. the magnitude of the correction for the absorption in a rod of diamond powder (diameter 2 mm.) for Cuk rays: P. Debye and P. Scherrer, Physik. Z. 19, 474 (1918), Table V. Gompare, however, also the quotation from W. Gerlach and O. Pauli, Z. Physik. 7, 119 (1921) : “Die Intensitat der Linien ïst namlich, wie sich aus vielen anderen Aufnahmen herausstellte, absolut nicht nur durch Structurfactor u. s. w. bestimmt. Das Absorptionsvermögen und eine weitere Punktion des Materials, welchevorlaufig als Reflectionsvermögen bezeichnet sein moge, spielen eine hervorragende Rolle”. 3) P. Debye (after H. A. Lorentz), Ann. Phvsik. 43, 93 (1914). 4) G. G. Darwin, Phil. Mag. 27, 315, 675 (1914); 43, 800 (1922). 5) A. H. Compton, Phys. Rev. 9, 29 (1917). 6) W. L. Bragg, R. W. James and G. H. Bosanquet, Phil. Mag. 41, 309, (1921) § 13. appears to occur (loc. cit. equations io . 4 and 4.9). In this case it was taken into consideration, that incident and reflected directions which differ somewhat from the most favourable ones contribute to the reflection ; each orientation of the crystals is considered equally probable ;3) it is assumed that the crystals are so small that absorption and extinction in the reflecting crystal may be neglected. In calculating the energy absorbed along equal lengths of the interference Jines we consider that in the neighbourhood of the central strip of the film in our cylindrical camera, it is just as big as it would be in a spherical camera : in the latter case the lengths of the circular interference lines would be proportional to their sin ■&, from which it follows that in the intensities a factor ——- will appear. Tosin tt gether with the factor deduced by Darwin 4), this gives a factor lnstead of the intensity (blackening), jI ds then must be . » » ■ / sin2 cos J 2 2 measured over the total width s of the interference lme. Debye and Scherrer 5) by determining the scattering power of the carbon atoms in diamond have introduced the same factor, and promised to give the deduction. The factor did not reappear in the literature. Should, as seems to be verv probable, a factor occur in the V’ COS 2 intensities, then this would naturally explain a strong increase in the 1) The kriown falling off of the intensities of higher orders found experimentally by W. H. Bbago (I nearly co Phil. Mag. 27, 89, 468 (1914)) is then sin2 j explained by Compton by the factor mentioned and a factor (for which nearly cot. remains), ascribed to the configuration of the electrons. 2) Darwin, Phil. Mag. 43, 800. (1922) §§ 4 and 10. 3) The number of crystals for which the angle between the incident beam » + d-9 . and the normal to the lattice plane lies between 90— and 90 is therefore proportional to sin (90—g). This had never been oonsidered. 4) On account of the uncertainty caused in Bragg’s method by the extinetion the increased absorption of the rays at the reflecting angle Darwin, for quantitative determinations of the scattering power of the atoms, takes the powder method as choice with powder so fine that the extinction may be neglected. 6) P. Debye and P. Sgherrer, Physik. Z. 19, 481 (1918). method, we mean by ■Ö' the diffraction angle and not the “glancing angle”). x) Recently Darwin 2) deduced for the powder method the total energy thrown off over the cone of half angle &, in which a factor – sin 2 reflection intensities for large diffraction arigles1). It is remarkable that in the crystal investigations by the Debye-Scherrer method, the intensities of reflection have practically never been compared for different rays. Most German investigators use Cu-rays (in imitation of Debye and Scherrer) while most Americans use Mo or Wo-rays (in imitation of Hull). Yet on the usually accepted basis of the calculation, Hull’s observations for aluminium for example (5420: 5422S422 =25 : 10) would reject the structure given by Scherrer (5420 : 5422S422 =72 : 77). More experimental material must be compiled for the establishment and interpretation of the influence of the diffracted wave length on the intensity ratio of corresponding lines in the powder method*). See p.p. 38—40. hor lithium hydride we shall first try to draw only such conclusions as are not affected by the continuous factors. In this case, contrary to investigations on the scattering power of heavier atoms, this is possible to a certain extent. In the latter the reflecting power alters only slightly with & and all which is wrongly brought into the calculation in the continually altering factors, is found back in the scattering powers deduced. In lithium hydride the difference in structure factor for different suppositions concerning the positions of the electrons is so pronounced, that in spite of the uncertainty in some intensity lactors, several models could be rejected. 4. Observations; C alculation of the dimensions of the cell from the dijfraction dngles. TABLE X. Distance oi mid- CrK(I—rays Gr —rays die of interfe- 9 ' : rence lines in 103sin2 ,„s . ,«■ » m.m. and estima- 2 ih2 1(J sm hxh h. 3 W3sm2- h hïh, ted Gr-intensity. caloulated j caloulated j 1 23 45 6 7 8 49.6 w. 232 3 233 111 52.5 v.w. 257 4 257 200 58.5 m.s. 312 4 310 200 78.4 V.W. 508 8 514 220 89.2 m.s. 616 8 620 220 ll 706 311 V‘W' lil 12 770 222 116.3 v.s. 853 11 853 311 129.6 s. 934 12 930 222 x) In the case of LiII with CrK rays the ratio of this factor for the neiglibouring lincs 311 and 222 amounts to =1: 1,5 ; their #’s stiii differ considerably from 180°. 3) Should some influence of the wave length of the diffracted beam remain, ?annot be explained by the more obvious factors considered above which by the uncertainty in these factors is not likely to be expected there are sufficiënt other causes on which an explanation can be constructed: inter alia the scattering power of an electron may depend on the wave length of the inci- As appears from the occurrence of a common factor 77 • 5 ± 0 • 5 in the values of io8 sin2 for the a-lines, the crystal form of lithium 2 hydride belongs to the regular system and the edge of the elementary cell (eq.nll p. 6) is a=4. io . io—8 cm. With the aid of the density given bv Moers1), molecular weight, Avagadro’s number, and the wave length of Cii;a-rays (o . 816 ; 7.94 ;o . 6062 . io24 and 2.284. lO' 8 respectively) n, the number of particles in the elementary cell is calculated from this common factor to be 4.30. This points ton = 4 which is in agreement with the probable sodium chloride structure. Putting n = 4 the said common factor determines the density as 0.76 ± o . 01. Impurities have no influence on this value of the density as it was deduced on p. 22 sub. a), that the hydride forms no mixed crystals with the impurities2). The diffraction angles of the diffracted ray gives, therefore : the elementary cell of lithium hydride is a cube (regular system) with a lattice parameter a = 4.10.10—8 cm\ each cell containing four lithium hydride groups. 5. Calculation of the positions of the particles m the cell from the diffraction intensities. Intensity conditions. We shall first calculate the intensities according to I v S2 (1/ = number of cooperating planes). Thus, owing to the great uncertainty in most of the remaining intensity factors, we neglect all factors continually altering with Hand, limiting ourselves exclusivelv to the successive lines 111 and 200, and 311 and 222, fix such ample conditions to the intensity ratios of these lines, that at least it is not probable that structure models will be excluded incorrectly. In this connection it is important that the conditions fixed are common to the Cu—and Cr—exposures. We require 12a2>If dent radiation and be different for eleetrons not smilarly bound. See Darwin, Phil. Mag. 27, 326 equation 8 ; this formula is deduced with the help of some suppositions which may be rather doubtful (see p. 1 note 5). On the other hand the absolute measurements of W. L. Bragg'oii rook salt are in agreement with the A 02 simplified formula = -2 for the amplitude scattered by eaoh electron. *) Loc. cit. p. 194. 2) Also for other substances difficult to get pure, X-ray analysis often gives densities deviating considerably from other results ; e. g. Gerlach found for SrO 5.143 while the literature gave 3.93—4.75 Z. Physik 9, 184 (4-922). In table X the distances of the centres of the interference lines are & recalculated as io3 sin2 and the B lines separated. 2 From table X it iollows that the a reflection of m is considerably stronger than the (3 reflection from 200 lying close by; the intensity of this /3 reflection, as appears from the intensity ratio of the K * + a' and is about one fifth as large as that of the corresponding «-reflection (influence of the absorption etc. is not taken into account). Hence Sm wiJl be certainly greater than ¥ S2OO. On the other hand in spite of the great difference between the observed intensities of 111 and 200, a value \ is allowed as the limit for the calculated ratio in order to be certain not to underestimate the intluence of the absorption, which, as said above, will not be taken into account in deducing from the above equations conditions for the structure factors. As regards the inequalities (2) : the line 222 observed as strong on the Cr _—film must certainly have a calculated intensity larger than 1/3 S3U, even if one considers that the reflection of 222, lying nearer the edge of the film than 311, may be more relatively intensified *). Besides, on the Cr—films the (3 reflection from 222 like the /3-reflection from 311 is observed to be very weak and on the Cu —films the (3-reflection from 311 is never observed although the « from 222 does appear. On the other hand 5222S222 O S32i is a weak requirement for the clear difference in intensity ot these lines, especially on the Cu-films. The relations between the absolute values of the structure factors V which follow from (1) and (2) by the equation I 00 , are o. 40 S2OO Sln 00. 60 S2OO (1 ) and 1. 5° S3n 5222S222 S3IJ (2 ) Forms of the structure factor S. Diiferent possibilities for the position of the valency electrons have been investigated in an arrangement of the lithium and hydrogen nuclei analogous to that of the particles in sodium chloride or zinc sulphide 2): I. The valency electron remains with its original nucleus (three electrons round the lithium nucleus and one round the hydrogen nucleus : atomic lattice). 11. The lithium has its valency electron transferred to the hydrogen (Li + two electrons ; H~ two electrons : ionic lattice) 3). 111 The binding of the lithium and hydrogen takes place through rings of electrons perpendicular to the non-intersecting trigonal axes halfway between the lithium and hydrogen nuclei (binding rings) ; in this case we may imagine passing along a trigonal axis : (a) single-electron rings between lithium and hydrogen as well as between hydrogen and lithium. (b) two-electron rings between lithium and hydrogen (molecular lattice). ’) Cf. note 1 p. 28. 2) In order to save space the caloulations in the case of the Zinc sulphide structure which showed that it would hot fit, are not given. ?) Also the improbable case L~ and H+ was considered and excluded. As to the orbits of the electrons it has been assumed that: A. the electrons are so close to their nucleus that they may be considered as being in one point for the diffraction result (points; reflecting power proportional to number of electrons). This supposition is taken only for comparison. B. The connecting line from nucleus to electron is of definite length pand occurs equally in all orientations throughout the part of the crystal which is cooperating in the interference (spheres; the reductsin 2 rr— H a ion factor for such an electron is found on p. n, = m 2w- H a which a is the lattice parameter and H = S h2). C. The connecting lines nucleus-electron are in planes perpendicular to the non-intersecting trigonal axes and thereby again all directions are equally distributed (rings ; reduction factor for an electron moving in such a ring, (p. 16), tpQ =JO (2tt—R siny) where J0 (x) is theßessel cl function of the zeroth orderand y the angle between the plane of the orbit and the lattice plane). D. In the binding rings for the connecting lines centre-electron the same is supposed as under C. A. Poit t atoms and ions. Coordinates: Li H 000 £ i i ££l I O I iii i i o lii ó i i g__ _|_ e*i (hi + h2) ie’* +W|e ni + hl'J [fa + fce‘ill'I+UI',)J (a) in which in case I (atoms) : t|7L. = 3 Vu 1 „II ( ions ) : t|7l.+ = 2 tH- = 2 S = O for mixed indices. S=4 (ij7L. if>H) for hj, h2 and h3 odd ;S = 4 (i/'Li+ ■’f'jj) I°r hi» h2 and h3 even. B. Spherical atoms and ions. Coordinates of the centres of the particles and structure factor the • „ Prr sin 2 n— H same as equation (a) in case A but, i|> = £— mustbesummed 2*9-H a over all electrons in the partiele. Values for S for mixed, odd and even indices as under A. C. Ring atoms and ions. Coordinates: Li partiele H partiele lies on o o o i i i A axis llii o | B „ iIII i o C „ lilo | i D „ S = L1A + LiBe + h*> + fLic e« H •• • • (b)1)- Here the summations must be carried out over the four particles with different axial directions while =2 J0 (2 n-- H) is to be summed over the electrons in the partiele. D. Binding rings. a. one-electron rings. Coordinates : Li + H~ one-electron rings round: on A axis 000 lil lil fff „ B „ I| 1 1o |f | | lI i „ c „ 1 H I1 o iI -i lil „ D „ |I | o| I lil 444 Taking together electron rings on the same trigonal axial direction and limiting ourselves to planes with mixed indices. : S = syu+ +ji + (—i)11* +ha + ei’i(hi +h2 + h.) 2)° *0 for odd : S = +••••' (c) 2h even :S = j+ + 2(—1) 2 • • (^) b. two-eledron rings. Coordinates as under a taking away the system of electron rings from the last column and doubling the first. M In this form the structure factor (b) also liolds for the case that the four planes perpendicular to the trigonal axes in one and the same atom (ion) are privileged orbital planes. For even values of ha, h2 and h3 the structure factor becomes the same as in the case above (equation (d) ). Testing the models. In table XI are given the reduction factors for some values of for the cl spherical electron and in table XII for the ring electrons perpendicular to the four trigonal axes; also fora spherical electron is graphically represented in fier I as a function of for different ö a reflections, i £üil> as a function of atq a for the planes mand 200 in fig. Ua and for the planes 311 and 222 in fig. Ilb. In the tables, arranged in ascending order, the values are included which a we have need ot below forthe systematic exclusion of radius values in the different cases and also the values of of the a cases calculated in tables XIII and XIV. TABLE XI. JL = 0.05 = 0.13 =0.16 =O.lB , , , 'd 3- 9 _ q c>2 JL. _q 35 Ü1 -3 (inner ring (radius (outerring (radius .a ' a Li-atom or H-atom) Li-atom) H-ion) ion) O 1 23 45 6 111 .95 .70 .56 ’ 47 .28 —-.17 200 .94 .61 .45 34 .13 j —.22 220 .88 .32 .10 —.02 —.lB j O 311 .84 .15 —.06 —.15 —.22 j .12 222 .82 .11 —.lO —.lB —.21 j -13 ~ _ I 3 Systemotic testing of the models with the conditions (1') and (2') onp. 30. Spherical ions. Sm 0.38. This demands 0.22, as appears by sub-3. stituting the corresponding values of 200 and ip111 from table XI and considering that for larger values of the value 01 ti> decreases in a successive maxima, while already in the neighbourhood of the second maximum (i//max. = 0.13) they do not suffice. Sin>o.4oS2oo: ijkll + 0.40 ii>2''° S3n : ip222 + 41)!11 > 0.02. It appears from table XI that in the 00 remaining region 0.22 > 0.18 this inequality does not hold. ci The case of the spherical ion, therefore, is excluded. Spherical atoms and ring atoms. In this case values suffice for the radii of the outer ring of the lithium atom and q2 of the hydrogen atom, which are in the neighbourhood of those initially givenby Bohr x) *) N. Bohr, Phil. Mag. 26, 490 (1913). Pig. Ha. Pig. Ilb. A B G D J£°ve A B G D | A B G D | D°r|>e A B G D| £ Ö ï 23 4 p=O.2OA P=O.55A P = 0.65 A P = 0.73 A (outer ring Li atom) (radius Hatom) (outer ring Li) (radius H-ion) = 0.05 ' —=0.13 -f=0.16 V=o-18 Hl 100 94 94 94 3.82 100 58 58 58 2.74 100 44 44 44 2.32 100 32 32 32 1.96 200 94 94 94 94 3.76 58 58 58 58 2.32 44 44 44 44 1.76 32 32 32 32 1.28 220 94 82 82 94 3.52 58 01 01 58 1.18 44 —l9 —l9 44 50 32 —3l —3l —32 .02 311 94 77 82 82 3.35 58 —l6 01 01 .44 44 —33 —l9 —l9 —.27 32 —4O —3l —3l —.70 222 100 77 77 77 3.31 109 —lO —-16 -—-16 .52 100 —-33 —'33 —33 .01 100 —4O —4O —-40 —.20 5 6 7 8 ~= 0-38 g =0,22- (radius 1-quantum one electron (radius 1-quantum two electron "a~”°'5° binding ring) binding ring) Hl 100 08 08 08 L24 100 —34 —34— 34 0.02 100 —4O —4o 40 .20 100 —l3 —l3 —l3 +.61 200 0 8 0 8 0 8 0 8 .32 —34—34 34 34 —1.36 —4O —4O —4o 40 1.60 —l3 —l3 —l3 —l3 —.52 ,220 08 40 —4O 08 .64 —34 +O6 +o6 34 .56 —4O +29 +29— 40 .22 —l3 —o6 06 —l3 —.38 311 08 —32 —4O —4O —1.04 —34 +27 +o6+ 06 + .05 —4O +22 +29+ 29 + .40 —l3 —25—06 06 —.50 222 100 32 —32 —32 + .04 100 +27 +27 +27 + 1.81 100 +22 +22 +22 + 1.66 100 —25 —25 —25 +.25 ‘ 9 ÏÖ 1112 JL =0 56 —= 0.58 = 0.60 = 0.63 a a a a 111 100 08 08 08 1.24 100 1414 14 1.42 100 19 19 19 1.57 100 25 25 25 1.75 200 0 8 08 0 8 08 .32 1414 1414 .56 19 19 19 19 .76 25 25 25 25 1.00 920 08 —24 —24 08 —.32 14 —25 —25 14 —.22 19 —22 —22 19 —.06 25 —l5 —l5 25 .20 ng ,Q 7-24 —24 —• 47 14 03 —25 —25 -—.33 19 11 —22 —'22 —-.14 25 20 —l5 —l5 .15 222 100 —O7 —Ó7 —O7 + .79 100 03 03 03 1.09 100 11 H 11 1.33 100 20 20 20 1.60 " __ 13 =0.70 a 111 100 30 30 30 1.90 =Jo (2 n—H Sin y\ 200 30 30 30 30 1.20 0 \ a / 220 30 17 17 30 .94 311 30 11 17 17 .75 222 100 1111 11 1.33 TABLE XII. Pl = 0.65 A and p2 = 0.55 A. This appears by substituting in (i') and (2') the structure factors for these cases from pp. 31 and 32 sub Band C, and the i|> values from columns I—31—3 of tables XI and XII respectively, which nearly suffice. Also values suffice for the radius of the outer ring of the lithium atom in the neighbourhood of a two-quantum ring. Spherical or ring atoms with radius values almost equal to those calculated by Bohr are in agreement with the diffraction intensities. Binding rings. Case a. Sm o . 10 p o . 10, as appears from the tables of Jahnke and Emde (Table 111 p. in.) demands that, Higher values of may be excluded from consideration on account cl of the condition that rings in the same octahedron plane do not cut each other (in cutting, each ring would enclose six other nuclei besides the one in the centre). This condition gives – <(£ Vz. Thus the a region (b) is limited to o . 56 < < o . 71, a (a) o < – < O . 22. Sm > o 40 S20o demands, as appears by substituting equation (b) on p. 32 and (f) on p. 36, that p< o . 29 or in the region considered o. 18 (see table XII.) cl By substitution of equation (b) on p. 32, S222 S3u demands 4- £^e222>- o . 04. In the remaining region 0.18 < 9- <0.22, Ihis condition is not fulfilled as may bc scen from dable XII. (b) O . 56 < | < O . 71. The first of the conditions deduced under (a), p 6.8 (See table 111 J. and E. loc. cit.). In the remaining region 0.56 S222> demands, on substitution of equation (b) p. 32, that + 2>*>° < 1.72. This excludes in the region considered, as is seen in table XII, values of greater than 0.63. Ring ions with radü of the H~ion between the approximate limits •0-560. 63 appear to fit the conditions. a These q values (about 2.5 A) are in the neighbourhood of those of a two-quantum ring in the free H ion. Calculated intensities. In table XIII the intensities are calculated for some models from Ir=l/ ( S )2, for the sodium chloride structure. In columns I—6 are some cases which do not agree with observation. i) Similar to the calculatlon on p. 19, we find e = 1.3 A lor the radius of a 1-quantum one electron ring between the monovalent Li+ and H atoms, distant 2i) = ? x lattice parameter = 3.55 A. The radius of_ an ?i-quantic doublé electron ring between these ions can be found in a smilar manner, taking into account TABLE XIII. , Calculated intensities (NaCl structure) Observed intenSitieS VÜnUincr „ , . h>h2h3 Point Point SP*e™ Ring S^1C 4>g Ring CrK CuK atoms ions ions mnriS atoms *1 iOnS4) 12345 6 789 111 w mw 0 32 77 29 29 25 26 10 200 ms s 96 96 40 38 39 43 52 50 31 220 ms ms 192 192 36 38 26 33 51 57 36 311 vs m 0 96 94 99 68 68 52 54 73 222 s vw 128 128 14 19 45 i 22 26 4a *) In this case according to Bohr (Phil. Mag. 26, 490 (1913). Radius inner ring Li = 0.05 a. (a = 4.10 A)- „ outer ring Li = 0.16 a. „ H ring = 0.13 a. „ Haring = 0.18 a. 2) Radius Li+ ring =0.05 a ; radius of binding ring =0.32a 3) ditto ; „ = 0.38a 4) ditto ; radius of H—ring = 0.60a. Columns 7—9 show the above found cases which fulfil the intensity conditions : in columns 7 and 8 spherical and ring atoms respectively with radii as given by Bohr for the tree particles ; in column 9 ring ions with a radius for the Li + ion as given by Bohr and for the Hlon q = 0.6 a. In the' calculation of these intensities only the structure factor and the "planes number” factor are taken into account. Comparing columns 7 or 8 with column 9 we see that the intensity ratio of, for example, the 220 and 311 reflections, differ considerably. A choice between the model with atoms and that with ions will be possible, therefore, as soon as better knowledge of the intensity factors will enable us to conclude as to the allowable ratios of the values of the structure factors for these further outlying lines also. Finally let us investigate if by neglecting only the absorption, extinction and heat motion factors the intensities of both the CuK- and Cr – exposures can be interpreted by, which seems to me the best formula: now the electrostatic repulsion between the two valency electrons (we assume them diametrically opposite one another in the same circle). The equation which n2h2 81-4 determines p is : = p frorri which after substituting the numerical . eWm (b2 + e2)f values taking n = 1 and solving graphically: p = 1.6 A-4) The factor i3 has little meaning here since the intensities of the incident radiation in the Cugp and Crg;- exposures were not the same, the estimations of blackness of different films are not related to one another and the relation- Intensities 1 + Cos2 & | Calculated h, b„ h, Observed 1 2 2 sin2#/2coss/2 casecolumn7orBol:Tab.Xlll column9Tab.Xlll CrK CuK firK GuK CrK CuK CrK | CuK —Öjjï| 2 j 3 | 4 111 3.2 8.2 w. m.w. 27 21 12 9 200 2.2 5.8 m.s. s , 37 29 20 16 220 1.4 2.5 m.s. m.s. 26 1415 8 311 2.3 1.7 v.s. m. 40 10 61 14 222 3.7 1.6 s v.w. 29 4 46 6 Table XIV gives the calculated intensities for CuK-and CrK-rays : in columns i and 2 for the cases in columns 7 and 8 of table XIII, in columns 3 and 4 for the case in column 9of table XIII (in which = 0.58 is taken instead of 0.60 because on the now accepted intensity formula, taking = 0.60, 222 calculated is too strong compared 3, with 311). , I + cos2 # . , . , VT,. Also the values of the factor are given m table XIV. .. 9 tl sm2 cos 2 2 The factors supplied appear to explain the alteration of the ratio between corresponding confirmation of the intensity formula used. On this basis of calculation the connection with the observed values appears to be better in columns 1 and 2 than in 3 and 4 ; in the latter case the ratio 311 : 220 is too big. The agreement of columns 1 and 2 may even be considered very good, when it is taken into account that the lower reflections are observed as relatively weaker owing to the absorption. The conditions (i') on p. 30 :o. 40 S2OO Sm o. 60 S2OO » which are used in the exclusion ot p. 34—37, on applying the intensity formula I lo 1 + cos fr gives as intensity conditions : sin2 cos 2 2 for chromium rays :o. 30 1200I200 <( Im 0• 53 1311 for copper rays : o. 68 1311I311 » I222 >o. 30 131lI31l On this present basis of calculation the upper limit for both kinds of rays appear unnecessarily high and the lower limit for the copper film somewhat narrow. If we modify these limits somewhat by multiplying the upper limit by o. 82 and the lower limit by o. 92 (conditions then for Cr-rays :o. 77 I3u> 1222I222 >o .43 I3xi and for Cp-rays :o. 44 13ii]>1222 > 0.24 I3n) then the conditions (2') become 1.20 S3ll 5222S222 o . 90 S3jj {2 ). The exclusions made with conditions (1') and (2") are almost the same as those on pp. 34—37 except that in the case ot ring ions by conditions (2") under (b) the upper limit of the allowable yalues is changed from o . 63 to o . 58. Apart from the change of this limit the conclusions remain unaltered. Finally if the ratio 220 : 311 is taken into consideration, then, as appeared above, the model with atoms is to be preferred to that with ions. From X-ray analysis, therefore, a structure appears possible for lithium hydride, in which there is placed at the lattice points in a structure like sodium chloride : either atoms with radii of approximately the same order of magnitude as those calculated by Bohr for the free particles, or positive lithium and negative hydrogen ions, in which the electrons travel round the nucleus, in paths whose planes are perpendicular to the trigonal axes ; the possible radius value for the H“ion is ~ = about 0.58 and lies in the neighbourhood of that of the 2-quantum ring in a free H~ion. Fig. 111 gives the elementary cell with the position of the electron orbits for the case of the ring ions from column 9. Fig. IV A gives a iii plane containing H ions and Fig. IV B the parallel plane withLLions.i+ions. o 6. From the lattice parameter a = 4.10 A it follows, on the sodium chloride structure that the sum of the radii of the “atomic domains” of Bragg x) of Li+ and H~ is 2.05 A ; using the value of Bragg for lithium, namely 1.50 A, that of hydrogen follows as 0.55 A; pn the other hand taking Richard’s2) value for Li+, namely 1.18 A, we find as the radius of the atomic domain of H : 0.87 A3). 'q W. H. Bragg, Phil. Mag. (6) 40, 169 (1920). a) Th. W. Richards, J. Am. Ghem. Soo. 43, 1584 (1921). 3) W. H. Bragg (Proo. Phys. Soc. 34, 98 (1922)) finds lor the radius ol the atomic domain of hydrogen in ice : 0.73 A- SUMMARY. The X-ray photograph of lithium hydride has been taken by the method of Debye and Scherrer with Crand CuK~rays- The lines for large o on the Cr-films were relatively much stronger than the corresponding reflections on the Cu-films, so that even the suc- cession of intensities in successive lines appeared to be dependent on the wave length of the diffracted radiation. This was explained by a factor 1 in the intensity formula (which can be theoretically deduced) O O sma cos 2 2 j instead of the factor ; ■-7- usual in the powder method. 'ssa 0/2 Lithium hydride appears to crystallise regularly with four LiH groups per unit cell. Lattice parameter = 4.10 A. The density was found to be 0.76 ± 0.01. On the basis of calculation applied, there appears to be two. models having structures analogous to that of sodium chloride, which suffice . (1) with atoms at the points of the lattice with radü of the same order oj magnitude as found by Bohr for the free atoms; (2) with positive lithium ions and negative hydrogen ions at the points of the lattice ; the electrons rotate round the nuclei in planes perpendicular to the non intersecting trigonal axis ; radius of Li+ion about 0.05a, radius of H~ion about 0.58a. pig m< Fig. IV a and b. IV. Thus in lithium hydride, hy X.-ray analysis, we found shown the possihility of a heteropolar binding (exchange of electrons) and in lithium probably a homopolar (binding rings between the ions). Further from tables XI and XII it appears that the electrons of the hydrogen, the valency electron of the lithium and the electrons rotating in the binding rings of expected dimensions, have reduction factors, which even for our unusually long wave length (Crk-rays) change sign once or more times between tt =o° and = iBo°. Thus the place of a hydrogen atom cannot be approximately determined without taking into account the dependence of the scattering power on tf. At first sight, a structure for lithium hydride, analogous to that of sodium chloride, with positive Landi+and negative H ions, appears to be incompatible with the great strength of the 311 reflection. Indeed S3ll v) + -—l|; ,or “approximately” 2—2 =O. Even exclusively in the presence of so light an atom as lithium along with hydrogen, for determining the position, at least, of this atom, the diffraction of the valency electron is better completely neglected, leaving then the lowest reflections out of consideration. In somewhat heavier atoms, e.g. oxygen, the valency electrons form a predominant portion of the total number of electrons. Should they be placed between the atoms in the crystal complex, as seems not improbable 1), then for them too, such deviations from the approximate supposition, reflecting power a> atomic number, must be expected that the usual calculation of their position on the basis of this supposition is not allowable; e.g. suppose the distance of the valencj' electrons from the nucleus in an oxygen atom is approximately equal to the atomic radius == about o . 65 A ; putting the lattice parameter at, for exarnple, 5 A then becomes about o . 13. The scattering power of the cl valency electrons (calculated as spherical electrons table XI) becomes already negative for Eh2 = 16. The excellent research of Bragg, James and Bosanquet, shows, how little can be concluded at present even with the greatest care, from the diffraction intensities of rock salt with regard to the finer structure of the relatively light sodium and chlorine particles : here it is only possible to deduce the electron density as a function of the distance from the nucleus. As regards the question whether atoms orions stand at the lattice points, the X-ray analysis gives no definite answer: “it seems that crystal analysis must be pushed to a far greater degree of refinement bef ore it can settle the point” 2). Also with the lightest atoms, as appears from the above described investigation, the difficulties are so big that a definite choice appears ‘) Gf. e. g. G. N. Lewis, J. Am. Ghem. Soc. 38, 762 (1916). I. Langmuir „ 41, 868 (1919). J. J. Thomson, Phil. Mag. 41, 510 (1921) ; 43, 721 (1922) W. L. Bragg. „ „ 40, 169 (1920). 2) Phil. Mag. 44, 448 (1922). From the results of intensities the X-ray investigation is able only: 1. to determine accurately the positions of the middle points of the heavy atoms. It is thereby limited to testing suggested models which are determined by means of symmetry conditions for the arrangement of the particles borrowed under defimte suppositions trom crystallography (See A. Karssen this number of the Recueil). 2. to deduce something concerning the configuration of eiectrons in compounds containing only very light atoms. Here, the conclusions on the configuration of the eiectrons are dependent on the knowledge of the other factors. Indeed, as scattering power of the atom (by the change of which with tt one tests the configuration of the eiectrons), is considered the factor which remains unexplained after the other factors have been taken into account. That this effect actually depends on the electron configuration is only probable by plausible results1). Repeatedly it appeared in the interpretation of the diffraction intensities, that conclusions regarding one factor had to be altered on account of a change in or increase of the number of the intensity factors2). Such conclusions, therefore, are to be accepted with care. To take an electron configuration which agrees with the diffraction intensities as the only one possible is naturally excluded by such a “cut and try method” (Compton, loc. cit p. 55). In our investigation we met the difficulties of the X-ray analysis with regard to eiectrons moving in orbits with not very small radius; these hold even more when heavy atoms are present, by which the diffraction effect of the valency eiectrons is overruled by that of the large number of eiectrons near the nuclei. While Prof. Smits at the beginning of this research expressed the hope that by the investigation with X-rays more would be known of the “finer internal states of equilibrium” 3) in the crystal4), we must now expect, that if these conditions are determined chiefly by the paths of the valency eiectrons, *) In 1917 Hull deduced an electron configuration for iron which was in agreement with the model of Langmuir; yet now no more value ean be atlatched to his proof (see p. 2). 2) As an illustration of the faulty knowledge of intensity factors I will give two quotations from Darwin: the first in 1914 (1. c. pg. 600): “Gomparison with experiments suggests that the new formulae can account for the obser vod intensity as little as the old”, the second in 1922 (I. c. pg. 829): “The theoretical results are composed with experiment. The experimental data are rather inadequate and the agreement is not very good.” 3) See A. Smits, The Theory of Allotropy p. 2. *) Verslag Akad. Wetenschappen Amsterdam 29, 1208 (1921). A. Smits, The Theory of Allotropy p. 181. impossible even among the models tested : with lithium the possibilities appear of rings of valency electrons round the ions as well as hetween the ions : in lithium hydride the possibility of atomic and of ionic lattices remained. In lithium hydride the difficulty of the choice is caused chiefly by the uncertainty in the intensity factors whilst in lithium it is moreover caused by the insensibility of the diffraction result to the structure differences in question. as Prof. Smits considers possible1), at present X-ray analysis will be able to furnish information or even indications on this pomt only m very few cases. SüMMARY (see pp. 20 and 41). Through the great kindness of the Municipality and of the Amsterdamsche Universiteitsvereeniging a grant was receiveci to provide the Laboratory with an X-ray apparatus. I feel obliged to exPjess 7 hearty thanks to both bodies. Also I thank the techmcal Staff ofthis Laboratory Messrs G. J. van Deene, H. J. van der Grind and J. Gunt r for the accuracy with which they constructed several Röntgen requisites. !j verslag Akad. Wetenschappen Amsterdam April 1923. I Aan de gebruikelijke intensiteitsformule voor de door kristalpoeder gediffracteerde Röntgenbundels moet een factor (* = afbuigingshoek) worden toe-2 gevoegd. II Bij de kristalstructuurbepaling met Röntgenstralen is voor lichte atomen parameterberekening op grond der benadering: verstrooiend vermogen is evenredig aan het atoomnummer, bedenkelijk. 111 Uit de dampdrukberekeningen van Stern kan geen aanwijzing gevonden worden voor het aannemen eener nulpuntsenergie. Stern, Z. Electrochem. 25 66 (1919). Reiche, Die Quantentheorie (1921) pg. 107. STELLINGEN IV Onnauwkeurig geformuleerd is “any indefiniteness in the time in which we take a system at random from an ensemble has the practical effect of diminishing the average index of the ensemble from which the system may be supposed to be drawn, except when the given ensemble is in statistical equilibrium” Gibbs, Statistical Mechanics, laatste alinea Hoofdstuk XII. V De afhankelijkheid van de statistische entropie van het aantal moleculen is bij herhaalde besprekingen, o.a. van Planck en Ehrenfest, niet duidelijker uiteengezet dan aanvankelijk door Gibbs. Gibbs, Statistical Mechanics Hoofdstuk XV. o.a. Planck, Ann.Physik 66 365 (1921); Warmestrahlung. 4e Aufl. (1921) pg. 209. Ehrenfest en Trkal, Verslag Akad. Wetenschappen Amsterdam 28 (1921). VI Inde definitie van phasenvergelijking, gegeven in het leerboek van van der Waals-Kohnstamm, kan beter worden weggelaten de voorwaarde dat spanning, temperatuur en samenstellingen de thermodynamische potentialen eenduidig moeten bepalen. van der Waals-Kohnstamm, Lehrbuch der Thermodynamik, Deel II pg. 11. VII De berekening van Tolman over de ligging van het evenwicht He 4 H is aan bedenking onderhevig. Tolman, J. Am. Chem. Soc. 44 1902 (1922). VIII Er zijn bezwaren tegen den term „bewegelijkheid” inden door Brandsma gebruikten zin. Brandsma (naar een lezing van Sckeffer), Chem. Weekbl. 19 322 (1922). IX Bij de omzetting van Beckmann zijn het niet steeds de cis-groepen welke van plaats wisselen. Meisenheimer, Ber. 54 3206 (1921). X Dat trans-66'-dinitrodipheenzuur in optisch actieve componenten te splitsen is, wordt het eenvoudigst verklaard door aan te nemen dat de vlakken der beide benzolkernen niet samenvallen. Kristie en Kenner, J. Chem. Soc. 121 614 (1922). XI In Haas, Einführung in die Theoretische Physik, is de behandeling der statistische waarschijnlijkheid aan bedenking onderhevig. Haas, 1. c. Deel 11, Hoofdstuk XIII. VERBETERING. Pag- 25 regel 7 en 8 v. b. en pag. 28 regel 7 en 8 v. b. staat: T '-422 lees: J420. M22