I. The Crystal Structures of Certain Intermetallic Compounds: (a) The Nature of the Bonds in the Iron Silicide FeSi and Related Crystals; (b) A Study of the Structure of the Intermediate Compounds MG₂Tl, Mg₂In, and Mg₂Ga. II. Metallic Valence and X-Ray Emission Fine Structure: a Discussion. III. Crystal Structures of Certain Inorganic Compounds: (a) Crystal Structure of Sodium Borohydride; (b) A Redetermination of the Structural Parameters in the Cubic Modification of Arsenious Oxide Citation Soldate, Albert Mills (1950) I. The Crystal Structures of Certain Intermetallic Compounds: (a) The Nature of the Bonds in the Iron Silicide FeSi and Related Crystals; (b) A Study of the Structure of the Intermediate Compounds MG₂Tl, Mg₂In, and Mg₂Ga. II. Metallic Valence and X-Ray Emission Fine Structure: a Discussion. III. Crystal Structures of Certain Inorganic Compounds: (a) Crystal Structure of Sodium Borohydride; (b) A Redetermination of the Structural Parameters in the Cubic Modification of Arsenious Oxide. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/82p4-ba49. https://resolver.caltech.edu/CaltechTHESIS:06232025-183816561 Abstract Part I. Section (a) The structure of the iron silicide FeSi was determined by means of X-ray single-crystal photography. The previously reported structure was verified. The structure is based upon the space group T 4 - P2 1 3 with a o = 4.489 ± 0.005 Å. Four iron and four silicon atoms are in the positions (xxx; x + ½. ½ - x, x;) with x Fe = 0.1370 ± 0.0020 and x Si = 0.842 ± 0.004. A detailed discussion of the structure was given in terms of the resonating-valence-bond theory developed by Professor Pauling. Part I. Section (b) An investigation of the structures of the intermetallic compounds Mg 2 Tl, Mg 2 In, and Mg 2 Ga was made. The methods of X-ray single-crystal and powder photography were employed. The structures of Mg 2 Tl end Mg 2 In are based upon the space group D 3 3h - C62m with three molecules in a unit cell of dimensions a o = 8.068 ± 0.005 Å, c o = 3.457 ± 0.0007 Å; a o = 8.324 ± 0.005 Å, c o = 3.457 ± 0.0007 Å, respectively. The three thallium or indium a toms are in the positions 1: (a) 000, 2: (c) ⅓ ⅔ ½; ⅔ ⅓ ½. The six magnesium atoms are in the positions 3: (f) x 1 00; 0x 1 o; x 1 x 1 0 3: (g) x 2 0½; 0x 2 ½; x 2 x 2 ½. In the case of Mg 2 Tl x 1 = 0.3746 ± 0.0028 and x 2 = 0.2866 ± 0.0038. It was not possible to fix the parameters x 1 and x 2 accurately in the instance of Mg 2 In, but it is highly probable that they do not differ greatly from the corresponding values of Mg 2 Tl in terms of the resonating-valence-bond theory (1),(2) was given and it was found that the valences of the thallium atoms are larger than that exhibited in the elementary substance. The structure of Mg 2 Ga is apparently similar to that of Mg 2 Tl and Mg 2 In. The positions of the atoms, however, necessitate a doubling of the unit cell dimension along the c axis. It was not possible to determine the exact position of the atoms. Part II. A discussion is given of the relationship between metallic valence and the fine structure of the X-ray emission bands of metals, alloys, and intermetallic compounds. The study of fine structure seems to offer a chance to gain valuable information concerning metallic valences; there is, hov1ever, a great need for more experimental information. Part III. Section (a) The structure of sodium borohydride was investigated by means of X-ray powder photography and found to be based on a face-centered cubic lattice. The unit cell of dimensions a o = 6.151 ± 0.009 kX contains four sodium and four boron atoms arranged in the following manner: 4 Na at ooo, o½½, ½o½, ½½o 4 B at ½½½, ½oo, o½o, oo½ It is probable that the structure consists of tetrahedral BH - 4 ions and Na + ions. The dimensions of the ions would permit oscillation or rotation of the BH - 4 tetrahedra. Part III. Section (b) The structural parameters of the cubic modification of arsenious oxide have been redetermined by means of X-ray powder and single crystal photography and found to agree closely with those reported by Almin and Westgren (5) . The structure is based upon the packing of As 4 0 6 molecules which have the same dimensions as those found for the molecules of the gas (2),(12) . Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry and Mathematics) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Pauling, Linus (advisor) Sturdivant, James Holmes (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1950 Additional Information: Title varies in the 1950 Caltech commencement program: I. The Critical Structures of Certain Intermetallic Compounds. II. Metallic Valence and X-Ray Emission Fine Structure: a Discussion. III. Crystal Structures of Certain Inorganic Compounds Record Number: CaltechTHESIS:06232025-183816561 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06232025-183816561 DOI: 10.7907/82p4-ba49 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. 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I. THE CRYSTAL STRUCTURES Olil CERTAIN INTERl\1ETALLIC COMPOUNDS '!'he ·N ature of the Bonds in the Iron Silicide ( e.) FeSi and Related Crystals (b) A Study of the Structures of the !ntennete.llic Compounds Mg 2Tl, Mg 2 In, and Mg 2 Ga II. METALLIC VALC:NCE AND X-RAY 1<J.1ISSION FIWE STRUCTURE: A DISCUSSION III. CRYSTAL STRUCTURES OF CJ:i.Jt'l'AIN INORGANIC COMPOUNDS (a) Crystal Structure of Sodium Borohydride (b) A Redetermination of the Structural Parameters in the Cu-b ic Modification of Arsenious Oxide Thesis by Albert Mills Soldate In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1950 !Jae 1nvest1ga.t1of);s of th1s 'th•sis )'esul te4 fJ"om t:J14ggeti'f,1otul by P~ettUJ<i$l" Lin-u s Faul1ag to Wh'3m I e.m further indebted. f<>~ adv1(HJ1{ an'd eneouragemeat. in tlae o~urse of eomplt>\1~& ·tt'lem • J :Wl:$b. to Et1tpt1etts my -at.1tude to :Professor J. H.. Sturiiv~ni •ftr h1St geiu,:reu.s f1tpend1 ture of time and effort la oftevi~g- mueh :a ,lpftal $.dv:1-c $ a-a aoZllst.~e-t 1 ve, eJ>1 t1c1srn t~ me: 4urlwig •Y ~a.4ua.te studtes at, tne Inet.i tut;e. Ab stracts Abstract of Part I. Section {a)* The structure of the iron silicide FeSi was detennined. by means of X-ray single-crystal photography. ported structure was verified. The pre,.riously re- The structure 1s basec upon the 0 4.489 ± 0.005 A. four silicon atoms are in the positions ( :xxx; Four iron and x+½, i - x, x;)) with xlt'e: 0.1370 ± 0.0020 and ¾i : 0.842 ±0.004. A detaile<l discussion of the structure was given in terms of t he resonating-valence-bond theory developed by Professor Pauling. - .. ............. ............... * The wo:rk of this section was done in collaboration with Professor Linus Pauling. Abstract of Pa rt I. Section ( b) An investi gation o f t h e structu res of the inte;rmetallie· compounds Mg2 Tl, :Mg2In , a nd Mg 2 0a. wa s m~~de, Th e me t h od s of X-ray single-crystal an d p owde r photo graphy we re emp loyed . The stru ctu res of Mg2 Tl end Mg 2 In a:re based upon t he s p a ce group D~h - C62m with thr ee molecules in a unit cell of = 8.068 ± 0.005 A, c 0 _ = 3.457 ± 0.000'7 A; a 0 : 8.324 ± 0.005 A, c0 : 3 . 4 57 ± 0.0007 A, res pe c t i vely. dimensions a 0 The t hree t hallium or indium a toms a re in t h e position s 1: ( e. } ooo, 2: (e) 1 2 l; 3 3 2 2 1 1. 332 The six magnesium a.toms are in the positions 3: (f ) X100; Ox10; X1X10 3, ( g) l X20½; ox2-in -X~2~• In t h e case of Mg 2 'rl x 1 = 0,3746 ± 0.0028 and x 2 : 0 . 2866 ± o . 0038 . It was not possible to fix t h e pa rame ters x 1 and x 2 accurately in t he instance of Mg 2 In, but it is highly probable that they do not di f f e .r greatly- from the eor'.?'esponding ,ralues of Mg2 Tl . A di &cussion of t h e structure of Mg 2 Tl in terms of the resonating-valence-bond theory ( l), (2) was given and it was found that the valences of the thallium atoms are larger the.n that exhibited in the elementary substance •. The strueture of Mg2 Ga is appa rently simila r to that of Mg 2 Tl and Mg2 In. The positions of the a.toms, however, necessitate - a doubling of t h e unit cell dimension along the e axls. It was not possible to determine the exact position of the atoms. Abstract of Part II. A discussion is giYen of the relationship between metallic valence and the fine struct,ure of the X-ra-:{ emission bands of metals, alloys, and intermetallic compounds. The study of fine structure seems to offer a chance to g111.in valuable information concerning metallic valences; thel. .e is, hov1 ever, a ereat need for more experimental information. Abstract of Part III. Section (a) The structure of soditun borohydride was investigated by means of X-ray pow<'ler photography e.rd found to be based on a face-centered cubic lattice. mens ions a 0 : The unit cell of di~ 6 .151 ± 0.009 kX contains four sodium and four boron a toms arranged in the following manner: 4 Na 1 1 1 l l l at ooo, 0?Ii:i 1 13oz, irzo 4 B l ' at -LL.1. 00·2 22·2 I ½oo , OiJO, It is probable that the structure consists of tetrahedral BH; ions and Na'° ions. The dimensions of the ions would penni t oscillation or rotation of the BE 4 tetrahedra. Abstract of Part III. Section (b) l1he structural parwneters of the cubic modificatior of 1 arsenious oxide have been redetermined by means of X-re.y powder and single crystal photogPEtphy and found to agree closely with those reported by AJJnin and Westgren( 5 ). The sttt\cture is based upon the packiri.g of As 4 0 6 molecules v.t.tich have the same dimensions e.s those found for the molecules of the gas ( 2), \1 2 ) ~ Table of Contents Part I Title Page The Crystal Structures of Certain Intermetallic Compounds (a) 'I1he Nature of the Bonds in the Iron Silicide FeSi and Related Crystals (b) 1 A Study of the Structures of the Intermetallio Compounds 11g 2Tl, Mg 2In, and Mg 2Ga II Metallic Valence and X-ray Emission Fine Structure : III 11 A Discussion 81 Crystal Structures of Certain Inorganic Compounds (a) Crystal Structure of Sodium Borohydride (b) 118 A Redetermination of the Structural Para.meters in the Cubic Modification of Arsenious Oxide IV Propositions 121 142 I• THE CRYSTAL STRUCTURES OP CERTAIN !NTERMETALL!C COMPOUNDS (a) The Nature of the Bonds in t he Iron Silicide FeSi and Rela ted Crystals 1 R eprinted from Acta Crystallographica, VOL. 1, PART 4, September 1948 PRIN TED IN GREAT BRITAIN Acta Cryst. (1948). 1, 212 The Nature of the Bonds in the Iron Silicide FeSi and Related Crystals* BY L. PAULING AND A. M. SOLDATE Gates and Crellin Laboratories of Chemistry, California Institute of Technology, Pasadena, Gal., U.S.A . (Received 7 June 1948) The iron silicide FeSi has been reinvestigated by X-ray photography of single crystals, and the reported structure for the substance has been verified. The space group is T4-P2 1 3, with a0 = 4·489 ± 0·005 A. Four iron atoms and four silicon atoms are in positions (x, x, x; x + ½, ½- x, x; j), with Xr. = 0·1370 ± 0·0020 and Xsi = 0·842 ± 0·004. A detailed discussion of the structure and the values of the interatomic distances has been given, by application of the resonating-valence-bond theory, and it has. been shown that the interatomic distances are compatible with those found for elementary iron and elementary silicon. Introduction The structure of the iron silicide FeSi was first investigated by Phragmen (1923), who found it to be based upon the space group T4-P21 3, with a unit cell containing four iron atoms in the positions (x, x, x; x+½, ½-x, x; J) and four silicon atoms in a similar set. The edge of the cubic unit cell (a 0) was given as 4·48 A., and it is evident from Phragmen's drawing of the structure that xFe and x 8 ; were taken as approximately land!- Wever&Moeller(l930)laterstudiedthis , 1 structure by use of powder photography and reported 1the values XFe = 0·1340 ± 0·0020, x 8; = 0·8445 ± 0·0020, and a0 = 4·467 A. (It is mentioned in the Struktur1 bericht, 2, p. 241, however, that the authors have stated j in a personal communication that an error was made in their determination of the parameters.) Another powder photographic study of this substance was made by Boren (1933), who reported xFe=0·139±0·001, x 8 ;=0·845 ± 0·001, and a 0 =4·478 A. The structure found for FeSi and other silicides (CrSi, MnSi, CoSi, NiSi) (Boren, 1933) is interesting as an example of co-ordination number 7. Each iron atom is surrounded by seven silicon atoms, and each silicon 1 atom by seven iron atoms, at the distances 2·28 A. (one ! ligand), 2·35 A. (3), and 2·50 A. (3) (calculated with the parameter values of Boren; those of Wever & Moeller lead to values agreeing with these to within 0·03 A.). Each iron atom also has six iron neighbors at 2·75 A. We became interested in the structure in connection with our efforts to develop a rational theory of metals and intermetallic compounds (Pauling, 1938, 1947), and attempted to find an explanation of the choice by these substances of this unusual structure, in preference to the sodium chloride, nickel arsenide, or cesium chloride structure, and also an explanation of the * Contribution no. 1172 from the Gates and Crellin Laboratories. --, 2 unusually small interatomic distances shown by the crystal. These small distances make themselves evident when an effort is made to calculate the valences \1f the iron and silicon atoms from the bond distances;_with use of the recently proposed relation between bond distance and valence (Pauling, 1947): Rn= R 1 -0·300 log n. (1) The bond numbers obtained from the reported interatomic distances lead, when summed, to the valences 6·98 for iron and 6·93 for silicon, in obvious disagreement with those usually accepted, 5·78 and 4·00, , respectively. The seriousness of the apparent anomaly in valence and the incompleteness of the published details of the only single-crystal investigation of the structure that has been reported (Phragmen, 1923) caused us to consider the possibility of an error in the structure determination, and led us to verify the structure and to ·r edetermine the parameters by single-crystal X-ray methods. Experimental methods and results A preparation which consisted of silicide crystals embedded in a silicon-rich a-iron matrix was made by slowly cooling a mfxture of reagent-grade iroi;i and silicon containing 25 (wt.) % of silicon from the molten state, under 40 mm. of hydrogen. A residue of silicide crystal fragments and silica was obtained by the decomposition of the sample with hot aqua regia. A single-crystal fragment with longest dimension less than 0· l mm. was selected, and its individuality as a single crystal was demonstrated by the preparation of heavily exposed symmetrical Laue photographs. The Laue point group was found to be Th-2 /m3. The value of a0 was found to be 4·489 ± 0·005 A. from equatorial measurements of 15° oscillation photographs; no re- 1 1 i 3 flections requmng a larger cell were observed on the Laue or oscillation photographs.* The lattice is . primitive, restricting the space group to T 1-P23, T 4-P21 3, T1-Pm3, TJ.-Pn3, or T~-Pa3 (Internationale Tabellen ... ). T;.-Pn3 is eliminated by the observation of reflections (0kl) with k+l odd, and T~-Pa3 by (0kl) with k odd. The density 5·95 g.cm.- 3 reported for a specimen containing 33·0 (wt.) % of silicon (Landolt-Bornstein, 1931, p . . 330) leads to 3·86~4 atoms of iron and 3·86 ~ 4 atoms of silicon in the unit cell. It is assumed that the iron and silicon atoms are distributed in a definite, non-statistical fashion. The final correspondence between calculated and observed structure factors constitutes the ultimate justification of this assumption. (Furthermore, ther!:) seems to be no reasonable statistical distribution involving four iron and four silicon atoms for any one of the three possible space groups.) It is impossible to place four iron and four silicon atoms in sets of equivalent positions of the space groups T 1-P23 and T1-Pm3 which are not also equivalent positions of the space group T~-P43m (which has the Laue symmetry Oh-4/m32/m). The structure must therefore be based on the space group T 4-P2 13. The observed absence of reflections of the type (hOO) with h odd is consistent with the choice of this space group. Four iron and four silicon atoms may be placed in sets of equivalent positions in only one manner, namely, in two sets of equivalent positions of the type (x, x, x; x+½, ½-x, x; ,;). A set of 15° oscillation photographs about a twofold axis was prepared in a cylindrical camera with a radius of 5 cm. by use of Mo Ka 1 , a 2 radiation filtered by zirconium foil. Three films interleaved with 0·001 in. copper foil were used for each exposure (Hughes, 1941). Intensities of reflections observed in the photographs were estimated visually. Absorption corrections, omitted here, were estimated to be small. The atomic scattering factors for iron were taken as averages between those of Pauling & Sherman (1932) and of Thomas & Fermi (Internationale Tabellen ... ), and the atomic scattering factors for silicon as averages between those of Pauling & Sherman and of J ames & Brindley (Internationale Tabellen ... ). A temperature factor * The Mo Ka. 1 components of well-resolved doublets were used for these measurements; the value 0·70926 A . was assumed for ii.. It is probable that earlier measurements of a0 are reported in kX. units. If this is the case, the measurement of the present investigation is in good agreement with the measurements of Phragmen (1923) and Boren (1933). t Owing to the experimental arrangement the Ka. 2 component of the reflection (10,0,0) was coincident with the Ka. 1 components of the reflections (10,L0) and (10,L0) on one exposure and the Ka. 1 components of the reflections (Ll0,0) and (LlO,0) on another exposure. The well-lmown intensity ratio of the Ka. 1 to the Ka. 2 line, 2 : 1, was used to make the intensity comparisons involving these coincidences shown in Fig. 1. 4 I exp [0·5{(sin 0)/,:\.}2] was applied to calculated structurefactor values. Table 1. Intensities of X-ray reflections on oscillation photographs of FeSi (hkl) 110 020 120 210 220 130 310 230 320 040 140 410 330 240 420 340 430 150 510 250 520 440 350 530 060 160 610 260 620 450 540 360 630 170 Obs. Cale. 11 9·5 8·8 11 13 12 26 33 2·5 2·7 5·6 4·7 5·9 6·9 14 13 8·4 7·0 25 25 Abs. 0·6 8·5 10 7·4 6·3 4·6 5·1 4·8 5·1 < 1·9 2·3 3·8 3·4 5·9 6·2 3·9 4·3 18 19 6·5 7·2 16 15 2·7 3·0 3·3 3·2 11 11 5·6 5·8 8·9 9·9 2·4 2·9 2·6 2·9 7·5 6·9 <3·0 2·3 5·8 6·3 4·9 5·3 4·4 4·2 Obs. Cale. <3·0 3·1 4·5 4·6 7·0 6·6 6·8 6·6 Abs. 0·2 5·6 5·9 3·3 4· 1 4·2 4·4 (hkl) 710 550 460 640 270 720 370 730 560} 650 080 180 810 470 740 280 820 660 380 830 570 750 480 840 190 910 670 760 290 920 580 850 390 930 9·1 7·2 Abs. 6·3 Abs. Abs. Abs. Abs. 4·4 Abs. <3·0 4·5 Abs. 5·9 5·7 <3·0 3·1 Abs. 6·2 8·2 <3·0 <3·0 5·8 Abs. 4·4 9·0 6·4 1·2 6·4 0·9 0·6 0·8 0·8 4·2 0·7 2·7 4·4 0·3 5·2 5·2 2·4 3·7 l· l 5·2 7·7 2·7 2·0 5·5 0·7 4·2 (hkl) 490 940 770 680 860 0,10,0 1.10,0 10,LO 2,10,0 10,2,0 590 3,10,0 10,3,0 780 870 4,10,0 10,4,0 690 960 LlLO 11.LO 2.11.0 11.2,0 5 ,10,0 10,5,0 880 790 970 3,11.0 11.3,0 6,10,0 4,11.0 11.4,0 Obs. Cale. 3· 1 2·5 Abs. 1·6 <3·0 2·4 <3·0 1·8 <3·0 1·8 6·8 5·6 3·7 3·1 3·0 2·2 Abs. 1· 2 Abs. 1·2 <3·0 2·0 4· 1 4·0 Abs. 1·8 Abs. 1·2 Abs. 0·4 4·9 4·3 4·5 4·3 6·1 4·5 Abs. l·O 3·5 3·4 Abs. 0·7 Abs. 1·6 6·0 4·3 Abs. l ·l 2·8 2·3 4·0 2·9 Abs. 0·6 4•5 3·6 2·3 2·3 Abs. 1·3 2·8 2·3 Abs. l •l Abs. l •l 0·136 Xfe 0·140 3 0·838 0·842 0·846 Fig. 1. Determination of parameters. Curve 1 2 3 4 5 Comparison 2Ffo,o,o < 4Ff.10,o + Ffo,o,o 2F'f o,o,o > 4F'f 0,1,0 + Ff o,o,o IF' 1920> IF' lsoo I F' lsso> IF' l.. o I F' luo,o > I F' la,10,0 Structure factors calculated with XFe = 0· 1370 and x 8 i = 0·8420 show good agreement with those calculated from the observed intensities (Ta ble 1 ). Intensity comparisons between reflections having nearly identical angle-dependent factors and observed on the same side of a given film were used to establish the limits of error shown graphically in Fig. Lt Presumably reliable 5 values of XFe and Xs; are 0· 1370 ± 0·0020 and , 0·842 ± 0·004, respectively. These are in essential agreement with those of Wever & Moeller (1930) and of Boren (1933) . Discussion of the structure The interatomic distances The ligands of each iron atom in FeSi are seven silicon atoms at distances 2·29 A. (1), 2·34 A. (3) and 2·52 A. (3), and six iron atoms at 2·75 A., these distances being reliable to about 0·01 A. The corresponding bond numbers, calculated with the assumption of the normal single-bond radii 1·165 A. for iron and 1·173 A. for silicon (Pauling, 1947), are 1·21, 1·00, 0·50 and 0·20, respectively; the sum of these values for the bonds formed by an iron atom is 6·91, which differs significantly from the expected value 5·78. Each silicon atom is surrounded by seven iron atoms at 2·29 A. (1), 2·34 A. (3) and 2·52 A. (3), and six silicon atoms at 2·78 A., the corresponding bond numbers being 1·21, l ·00, 0·50 and 0· 19, and the calculated valence of silicon 6·85, which is very much greater than the normal value 4. Let us now consider the electronic structure that would be expected for the iron silicide crystal according : to the resonating--valence-bond theory of metals (Pauling, 1948). According to this theory, silicon, with only four stable orbitals in its outer shell, could not form more than four electron-pair bonds, which might, however, resonate among more than four positions. In order for metallic resonance of the valence bonds to occur, some of the atoms must possess an extra orbital, the metallic orbital; it is not necessary, however, ·that the silicon atom itself possess a metallic orbital,· inasmuch as its four bonds might undergo pivoting resonance about it, if the surrounding atoms possess the metallic orbital. Moreover, it is to be expected that the resonance of the four bonds of the silicon atom would be of such a nature as to lead to the assignment to the interatomic positions of bond numbers equal to simple fractions, such as ½, ½, i, .... We note that the observed interatomic distances 2·34 A. (for three bonds) and 2·52 A. (for three) differ by just 0· 18 A., which corresponds to a factor ½ in bond number, and that the remaining bond distance, 2·29 A., is slightly smaller (0·05 A.) than the smaller of the other _two, suggesting that its bond number is slightly greater. These distances hence suggest that the corresponding bond numbers are 1, i, and ½, their sum thus being equal to the normal valence 4 for silicon. We assume, as has been found necessary for other crystals also, that the valence of the silicon atom is concentrated entirely into the bonds with its nearest neighbors, and that the six silicon neighbors at 2·78 A. are not bonded by the silicon atoms, the bond number 0·19 given by equation (1) being illusory. The silicon atom is thus to be described as using one 6 of its four tetrahedral sp 3 bond orbitals in forming an essentially non-resonating single bond with one iron atom, at 2·29 A.; the other three orbitals resonate between the three iron atoms at 2·34 A., and the three at 2·52 A., the latter set being the less favored, by the factor ½, presumably because of their less satisfactory relation to the tetrahedral directions (the angle with the non-resonating bond being 40° less than the tetrahedral angle, as compared with 26° more for the former set). When the two longer distances are corrected by 0· 11 and 0·29 A. for bond numbers} and½, respectively, and the single-bond radius of silicon is also subtracted, there are obtained the values l •12, l ·06 and l ·06 A., respectively, for the single-bond radius of iron. Moreover, we expect that the iron atom has the valence 6 (rather than tbe valence 5·78 found in elementary iron), for reasons described below; of this valence four units are used in bonds with silicon, leaving two valences to be divided among the six iron-iron bonds. The iron-iron distance 2·75 A., when corrected by the amount 0·29 A. corresponding to bond number½, leads to l ·23 A. for the single-bond radius of iron. These four values show striking deviation from the single-bond radius l •165 A. found for elementary iron (Pauling, 1947) . However, it is known (Pauling, 1948) that elementary iron represents a resonance average between two valence forms of iron. In one of these, Fe.A, two of the eight outer electrons of the atom are atomic electrons, contributing to the ferromagnetic moment, and the other six are valence electrons. Of the nine stable outer orbitals of the iron atom, two are used for ' the atomic orbitals, and the remaining seven, d3 sp 3 , are for the six valence bonds and the metallic orbital. The metallic orbital accordingly has or 42·9 % d character, on the assumption that the hybrid character of the metallic orbital is the same as that of the bonding ; orbital. The other kind of iron atoms, FeB, contains three atomic electrons in three of the d orbitals, the other six orbitals (d 2 sp 3 ) constituting five bonding orbitals and the metallic orbital. These six orbitals have 33·3 % d character. The ferromagnetic saturation moment, 2·22 magnetons, shows that the structures A and B contribute 78 and 22 %, respectively, to elem en' tary iron, the average amount of d character thus being 39·7 %. The corresponding single-bond radius is 1·165 A. The single-bond radius for sp 3 bonds, with no d character, is found by linear extrapolation (.Pauling, 1947) to be 1·481 A. On the assumption (Pauling, 1947, 1 1948) that there is a linear relation between the singlebond radius formed by a hybrid orbital and the amount of d character, there can be derived the equation t (2) where the symbol /J means the fractional amount of d character in the bond orbital. This leads, for example, to the value l •138 A. for the single-bond radius of the d3sp 3 bonds formed by Fe-4, which is sexivalent. We may now use equation (2) in interpreting the 7 values l •12, l ·06, l ·06, and l ·23 A. found for the singlebond radius of iron for the different kinds of bonds formed in the FeSi crystal. Application of equation (2) leads to the values 45, 53, and 31 % d character, respectively, for these bonds. Since there are altogether three d orbitals available for use in the six bond orbitals and the metallic orbital, the difference between 3 and the summed amount of d character for the bond orbitals is the d character of the metallic orbital. This difference is found to be 34 %, an entirely reasonable value. The state of the iron atom in FeSi may thus be described in the following way . The iron atom is sexivalent iron, FeA. It uses four of its orbitals in forming bonds with the seven surrounding silicon atoms-one single bond, three f bonds, and three ½ bonds-and two of the remaining orbitals, and the corresponding valence electrons, in forming six ½ bonds with six neighboring iron atoms . The orbitals involved in the bonds with silicon are different in hybrid character from those involved in the bonds with iron; they have between 45 and 53 % of d character, whereas the bonds with iron have only 31 % d _character. The metallic orbital has essentially the same hybrid character, 34 %d, as the bond orbital with the smallest d character. It is interesting to note that the four iron-silicon bond orbitals make use of just two of the three available d orbitals, the third being used by the two bond orbitals involved in iron-iron bonds and the metallic orbitals. Moreover, the distribution of the d orbitals among the orbitals of each class is essentially uniform; the four orbitals involved in bonds with silicon have approximately one-half d character, and the two orbitals involved in iron-iron bonds and the metallic orbital have approximately one-third d character. If it is assumed that the amounts of d character for these classes of orbitals are precisely ½and ½, respectively, the bond numbers calculated for the four kinds of bonds become 0·85, 0·70, 0·35 and 0·33, respectively, adding up to the valence 6. It is not clear whether the treatment of the distances in which the bond numbers are supposed to be simple (1, ¾, ½) or the slightly different treatment in which the amounts of d character are supposed to be simple (½,½)is to be preferred. It has hence been found that the interatomic distances in FeSi are compatible with those in metallic iron and elementary silicon, with the principal difference that the iron atom in FeSi is · sexivalent iron, rather than iron with the average valence 5·78, and the effective radius of iron for the different non-equivalent bonds that have formed in FeSi is different because of a difference in the amount of d character. These values of the iron radius are, however, related to that shown in elementary iron, and may be derived from it. The compounds CrSi, MnSi, CoSi and NiSi were found by Boren (1933) to have the same structure as FeSi, and he reported values for a0 and for the two atomic parameters differing very little from those for FeSi. The corresponding bond distances are given in 8 Table 2, together with the single-bond radii of the metallic atoms calculated from them with the same assumption about bond numbers as made above for FeSi. The values for the amount of d character (o) are also given, as calculated by the following equation (Pauling, 1948): R 1 (o, z) = 1·825-0·043z-(1·600-0·100z) o. (3) Here z is the atomic number minus 18. It is seen that the amounts of d character lie in the neighborhood of 50 %for the bonds between the metal atom and silicon, and of 30 % for the metal-metal bonds. The values found for the d character of the metallic orbital show rather wide fluctuations, which presumably have little significance. (The magnetic data reported for MnSi, FeSi, and CoSi (Foex, 1938) are difficult to interpret because oflack of correspondence with the Weiss-Curie equation, and hence are not useful in this case as a criterion of electronic structure.) Table 2. Bond distances and bond character in FeSi and related crystals (All distances in _A.) M-Si ~-----A..---~M-M Bond distance R(l) % d character 1 bond 3 bonds 3 bonds 6 bonds n=l n=¾ n=¼ n=¼ CrSi (a 0 = 4·620) 2·32 2·43 2·58 2·83 1·15 1-15 1·12 1·27 42 42 45 30 Bond distance R(l) % d character MnSi (a0 =4·548) 2·30 2·39 2·55 1·13 1·11 1·09 43 46 48 2·79 1·25 30 57 Bond distance R(l) % d character FeSi (a0 =4·489) 2·29 2·34 2·52 1·12 1·06 1·06 45 53 53 2·75 1·23 31 34 Bond distance R(l) % d character CoSi (a 0 = 4·438) 2·28 2·33 2·47 1·11 1·05 1·01 47 56 61 3i Bond distance R(l) % d character NiSi (a0 =4·437) 2·28 2·33 2·47 1·11 1·05 1·01 47 57 64 2·73 1·22 29 Metallic orbital 69 2·73 1·22 17 The choice of the FeSi structure Some general considerations about the selection of the FeSi structure by iron silicide and related substances may now be formulated. It is clear that the sodium chloride structure would not be a satisfactory one-in this structure an atom has as its neighbors only six atoms of the other kind, all more distant neighbors being so far removed as not to permit the formation of a significant bond with them; the effective valences of the two kinds of atoms are hence the same, and the structure is accordingly an unsuitable one for two elements with such different valences as 4 and 6. On the other hand, the nickel arsenide structure, with - 9 ~ • J / small 3:xial ratio, such as shown by AuSn (Pauling, . 1947);m:ig4t be assumed by FeSi. The axial ratio could ~ adjusti~seWin such a way that the silicon atom would use its valence of 4 in forming four bonds with the surrounding six iron atoms, and the iron atom would use its extra valence 2 in forming two iron-iron single , bonds, one with each of the iron atoms above and below ' it along the c axis. Similarly, the cesium chloride structure might be assumed by F eSi, each silicon atom then forming eight half-bonds with the iron atoms surrounding it in a cubic arrangement, and each iron , atom forming eight half-bonds with silicon atoms, and six one-third bonds with the six adjacent iron atoms. , It seems not unlikely that co-ordination number 6 (bond number i ) is more suitable for silicon in an intermetallic compound than co-ordination number 8* (bond t I number ½), and that, moreover, resonating iron-iron : bonds, with bond number½, are more stable than nonresonating bonds, with bond number 1. The actual FeSi structure would thus be preferred to the cesium I * A substance in which the four bonds of the silicon atom show pivoting reson ance among eight positicms _is Mg2 Si, with the fluorite structure (Pa uling, 1948). • 7 c_l~l~fide structure for the first reason, and to the nickel arseili:aJ ·s tructure for the second reason. . .;- • • ... ·W!)·are grateful to Prof. J. H . Sturdivant for assistance'. with the experimental part of this investigation. The work reported in this paper is part of a series of studies of metals and alloys being carried on with the aid of a grant from the Carbide and Carbon Chemicals Corporation. References Ark. Kemi Min. Geol. 11 A, 1. FoEx, G. (1938). J. Phys. Radium, 9, 37. HUGHES, E . W. (1941). J. Amer. Chem. Soc. 63 , 1737. Internationale Tabellen zur B estimmung von Kristallstrukturen (1935). Berlin: Borntraeger. LANDOLT, H . & BORNSTEIN, R. (1931). Physikalischechemische Tabellen. Berlin: Springer. PAULING, L. (1938). Phys. Rev. 54, 899. PAULING, L. (1947). J. Amer. Chem. Soc. 69, 542. PAULING, L. (1948). Nature, L ond., 161, 1019. PAULING, L. & SHERMAN, J . (1932). Z. Krystallogr . 81, 1. PHRAGMEN, G. (1923) . Jernkontor. Ann. 107, 121. WEVER, F. & MOELLER, H. (1930). Z. Krystall_ogr. 75, ' 362. ,,.· • BOREN, B. (1933). _) I. THB CRYS TAL STRUCTURES OF GI~:RTAH! ! N'r BRMETALLJ:C COM.POUNDS (b) A Study of the Structures of the Intermetallic 0('.)mpounds Mg Tl, Mg 2 In, and Mg2 Ga 2 11 A S tud~{ of t h e Struc t ures of t h e I nterme t a. llic Compounds I n tr•oc1u c tion The meta llic e l ements g allium t l n d ium, and thall ii.1.m e.re members of t h e ,a scen ding b ranches of t he f irs t , se co n d , and third long periods o f t h e periodic table in t h e s en s e f i rs t d iscussed by I' au ling <1 ). E l ements of the a::;c ending •branches "' e t y of va l enc e s i n t he forma tion of typi c all y show a vari i n t erme t all i e compounc.s and it has been s hown t heoretical ly tha t the t hree clements o f Group III B ce.n exhi b i t va l enc e s rangi ng f rom f i ve down wa rds, a lth ough the elementary valences for gaJ.lhun, inditun 1 an d t h allium a re 3½, 2-k , and 2 respective( 1), ( 2) . • Inte me tallic structu re s involv i ng val ences d ifJ.y. ferin g f rom the elemen tary iralue s are ge ne r a lly o f' i n t e rest, a n d the p resen t investi gation was undertaken with the expeeta tion tha t the pha se s Mg 2 Tl, M.g2In, end Mg2 Ga ( 3) would have structure s in which such di f ferences in valence a~e exh ibite d. Thi s expe cta t i on is made reasonabl e by t he exi s tence of a numbe r o f in t e nnetallic compounds of d l ffe ren t stoichiometric propor t i ons in e a ch of the bine.ry s y stems o f ma gnesium wi t h t hal l i um , i ndium, o r ga llitun. In the magnesium-thallium s ys tem t h e phases Mg 5 Tl , Mg 2 Tl, an d MgTl have been r eported, 2 a ll with n a rrow re gions of homo geneity< 4 ); in the magnesiumindium sy stem phases o f composition Mg In , Mg 2 In, Mgin , and 5 2 Mgin 2 were found 1n X-ray studies, although the explicit details of the phase d iagrams have not been worked out; and 12 in the magnesium- ga.llitun sys tern the phase diagram ind icates the existence of the interme tall:J.c compounds Mg Ga. 2 , Mg 2 Gtit , 5 5 MgGa, and MgGa 2 ( }, in agreern.en t vd. th a.n X-ra:y inves tige.. tion of the system< 6 >. The structures of MgTl , Mgin 2 , and Mgin ( 3) (7 ) are kno?m ' ; no structural determinations have been ma.de f or the other compounds o It has been reporte d that the structures of Mg 5 Ge. 2 , M:g In .. and Mg 5 T1 2 are similar and are based on an 5 2 unit cell containing 28 atoms ( Z = 4} ( 3 ) . orthorhombic In addition a common structure based on a hexagonal unit cell with 18 atoms {Z = 6) was implied for the compounds Mg 2 Ga, Mg 2In , and Mg 2 Tl. The hexagonal structure seemed. the simpler of the two .. !!la J:2riori, and thus a structural study of the three intermetallic General Exp erimental Details (a) PreEara.tion of Gompoun.da Use of the following method was found to effect easy and rather a ceura.te synthese-s from the elementary substances. A molten mixture of KCl and LiGl (11:9 by wei Bht) was added to an alundum extraction thimble and then _heated to incipient red hea t, about 4.50° C.. A calculated a.mount of he a vy metal was then added and tbe mo lten mass wa s again brought to about 450° Ce Next., the specimen of magnesium was added to the melt, and the mixture we.s sti .r red vigorously for ten minutes with a clay rod and allowed to cool to room temperature. Heating was accomplished with an oxygen- gas flame from a hand torch clmnped into posi tlon under the alundt1m 13 thimbleo Chan.ges in composition due to heating losses during the course of the X-ray powder photography we re estimated. to be small. In v iew •Of previous reports concerning the existence of the compounds and the verification found here for the ( '6), (? ) essen ti a 1 resu_1 t so f the X-rays t u di es o_f' them • no attempt was made to check the composition by means of chemical analyses. (b) Preparation cf Specimens and Powder Photography The intermetallic compounds prepa~ed, espe.cially Mg2 Tl, we re found to corrode when left unprotected from the atmosphere, The preparation of specimens suitable for X-ray powder photography was therefore carried out in a dry box under a carbon dioxide or argon atmosphere• Filings from clean surfaces of the melts were sieved with a. 200 mesh screen and the resulting powder was loaded into thin-walled pyrex capillaries. The open end of each capillary was then capped with vase line in order to secure temporary protection from the atmosphere upon removal from the dry b ox. After removal each capillary was quickly sealed off with an oX"IJgen-gas flame. The breadth of the high angle lines in X-ray photo- graphs of specimens prepared in this manner revealed the presence of lattice distortions; annealing of the specimens for short periods at low temperatures effected a satisfactory sharpening of the high- ang le reflections. In the cases of Mg 2 Tl and Mg2 In an annealing period of one hour at. 100° C was found sufficient. A s omewhat higher temperature was apparen tly necessary for the Mg 2 Ga specimensi one hour at 200° C being 14 effective,. The e,nnealed speclmens always appeared clean and free of oxides upon microscopi c examin ation, and no X-ray evidence was found for impurities other than those resulting from the presence of other intermetallic compounds in the system concerned., such as Mg 11'l, Me;In, e tc. Powder photographs were prepared with copper radiation filtered through nickel. The cylindrical camera generally used ha.<'l a radius of 5 cm • ., and u system cf 0.6 mm . slits was used to define the incident beam. After the present investi- gation was we ll under way , t...l-J:e labor~,tor~r acquired a 57 rnm . camera employing the Straumanis mounting. Several powder photographs were ta.ken with the use of this equipment . T.ae St:ro. ctural Determinat~on of Mg 8Tl The reflections which appeared on powder photographs of the compound. Mg2 T1 and whi ch we re not due to the presence of small amounts of MgTl or Mg 5 Tl 2 were successfully indexed on the basis of a hexagonal unit cell of approximate dimensions • This cell is half as large as *r:rhe powder photo grQphs of annealed specimens prepared from a melt of calculated composition Mg 2Tl contained a few reflections of low intensity which could be indexed on the basis of the cubic unit cell of MgTl~ With the addition of a small amount of magnes ium to the melt and subseqJ en t powder phot.oe;raphy, these lines c1 lsappeared, and a small number of faint reflections in other positions appeared which presumably due to the presence of a small amount of MgRTl2 • In this way the lines which were those of Mg 2 Tl could be i dentified with certainty. 15 that proposed by Haucke( :3 ), the value for c 0 being approximately one-half his reported va~ue for Mg2 Ga. The smaller cell was accepted on a tentative basis and was confirmed in subsequent experiments~ A density determination with the use of the displacement tecbni(lue gave the value 2.75 f',, 3 for the number or molecules of Mg 2 Tl in the unit cell * • It was noticed. a.t once that reflections of the t ype hk•2n were fain t or f1bsent when h-k # 3m, but we re generally stron g when h - k = 3m. 'fuis suggested the approximate positions 000, 1/3 2/3 1/2, 2/3 1/3 l/2 for the three thallium a toms, from the following argument. In a hexagonal lattice i t is always :pos~ible to use e. triply primi ti ve hexagonal cell with a volume three times the. t of the smallest ~ - - - - - - i - , , ,· ... - - - - - - {'• 'The low value for this number i s rea.sone.ble in view of the extreme rea.ctivi ty of the substance and the resulting tendency to measure a d isplacemen t volurne wh ich is too high. 'l'hiophene free benz ene of h i qh purity was used as the li quid and was dri ed by refluxing ove~ P205 for several hours with a subsequent dis ti lla.tion, Even after thia prec.a utien was ta.ken to ensure an inert liquid, l:)ubbles of gas we:t;>e slowly evolved upon contact with the specimen. It was; of course, considered futile to attempt the removal of ocelu.ded gases by boiling. During the determin~tion the speeimen was never dire ctly in contact wi th the atmo s phe~e, but WSt.a in con tact irlith CO when not immersed in t h e purified benzene., All necessary 2 transferences were made in a dry box under CO 2 .. The surface of the specimen ,u sed was sore.pe<tl f ree of' oxide initially .. 16 hexagonal cell. - The A a~ea of t he large cell are inclined at 306 to corresponding~ axes of t he smallest hexagonal -' ~l. cell and lattice translat.1.ons of n • A,~ ; n -... 0 , 1 , 2• , •,, exist which lead to t he extinction of reflections of t he type H - Kj 3m. Since Mg2 Tl t'efleetions of the type hk•O,- h - k l 3m are weak or absent and reflecti ons of the t ype hk •O, h - k = 3m are strong; the projections of t he thallium. atoms on the base or a sui tahly oheaen primi t1ve hexagonal cell are indicated to be at the positions 00, l/5 2/3, 2/3 1/3 ( the vee tore from the origin to the ;pos.ltions a.re o, l, 2• 'l,;~2. , respeetively) . The absence or weakness of refleetions of the type hk · 2n, h - k f 3m indicates that one or two of the atoms lie ap proximately on the h0riz0ntal plane haltwa~t along· the vertical cell edge( , =½). The choice of one or two of the atoms in any eomb1nat10n to be placed in this plane is arbitrary J th~re is enly one resulting vector diagram. One convenient choice has been indicated, with the three thallium atoms at 000; l/3 2/3 l/2; choice is O O 1/2; 2/3 1/3 1/ 2; l/3 2/3 O; another convenient 2/5 1/3 o. The general intens1 ty relationships among the remaining reflections observed 7ror i.y:1.'11ch ~ is odd were e.at:tsfied by the proposed arrangement for the t h allium atoms . It was eon• sidered probable that the proposed pos1t1oi'ls were exactly correct, and the chance of determining the magnesium. positions was considered promising,. A diagram or the spatial arrangement of the thallium atoms with the use of the radius observed 1n 1'7 the elem.ent.ary substance indicated. that the large vacancies in the structure were centered about the paeit1ens x.1 O O, 1 - 0; -~ 0.• 12, O· X22; i . · 0 X10 , X1X1 X2lt!.22 vrhere X1, - ¾ > o. . .F ran eonsiderations of symmetry it seemed reasonable that these e.re the true positions of the :magnesium atoms; if the . radius of elementary magneaium is t,sed; there are no sterie hindrances for suitable values ef ~i Qlld ~• The completed arrangement is based upon the space g:roup Di~ .... S62m. Preliminary ea.1- eulations were made to see if the valuea of x 1 and x 2 could be determined from the pow-d er photog~aphie de.ta, but tt was immediately e,.ppat'ent t-he.t it is itnpessible to do so . Re- .t'leetions of the type f -o:P whieh the $Catte:ring contriblltiens or the thallium atom.1i ·cane.e l according to the proposed ar ... re.ngement weH obs·el:'iTed in one instance only, and the the.llitun eontrib.ltione were so la~ge in the remaining eases that the seatter1ng eontr1t>ut1on~ of the magne.sium e.toms could be changed ·a ppre1iabl7 without <le.terminable ehangea in intensitl.es . lt became apparent that, if the etruetttre of M.~Tl were to be determined. in the. present investigation . 1 t would be necessary to obtain. e.·d dit1onal infonnat1t:m from eingle .. eryetal X-ray photography~· It wae obvieul. that, ee~tain rather fonnidable difficulties would have to be overcome in the pl"eparation or suitable single crystals of tbe sub.s tance.. The substance was very reactive and would have to be proteeted from the atmos.phere at all times. The substance was soft and it would not be possible to pul verize a melt to small particlea which were 18 single crystals. Since the adjoining phases Mg5T1 2 and MgTl were so close in composition to that of Mg Tl, a successful 2 chemical separation of single crystals .grown from a. t wo ... phase me lt was conai de red unlikely. lowing technique was first tried. Accordingly, the fol- A melt of calculated composition Mg2 Tl was prepared in the usual way with a KCl LiCl flux in an alundum crucible.. Vilhile the mass was molten, at a temperature of about 450° C, a clay tube with a 0.5 w..m.. bore was lowerec1 into the crucible and a cylindrical casting was for.med by sueking u.p M.g2 Tl in the liquid state with a vacuum pump whi.ch was attached to the upper end of the clay tube. Gooling took place immediately and the tube was quickly and carefully demolished under a binocular microscope wi th a r-azor blade and ordinary- clissecting in- struments. The casting thus obtained was immersed in a pro- tective layer of vaseline-ligro:tn s0lution--- initially sane gas evolution was noted Whieh soon ceased without visible corrosion of the . specimen. .In this manner specimens were prepared which consisted of clumps of aoieular crystals; the long axes of' the needles were misaligned by a few degrees in each elump. The problem of isolating a single individual from such an aggregate seemed difficult, and an attempt was made to grow crystals by .a slower cooling process. Crystals were grown in a two-phase melt by slow cooling in a Swedish iron crucible under a protective atmosphere of argon. Upon cooling the crystals were again clumped together and grew 19 on the surface of the molten mixture. One promising specimen was removed from the solid mass by a systematic procedure of excavation in which selected regions of the surrounding matrix were e:xposed to moistut>e and the resulting products of the reaction were removed with a dissecting needle. At the conclusion of the procedure the specimen removed was. found to consist of the usual clump of ae1culae; it was therefore decided to attemp.t the isolatien of one individual from such a clump w1 thout further expendi tttre of time e.nd effort in trying to grow indi vidua.la. A specimen was selected which was indicated by means of Laue. phote-graphy to be nearly a single individual on one side an~ over a reasonable length. The specimen; always under a coating of vasel1ne.-ligro1n solution, was fastened t~ a thin Pyre~ fiber by dried shellac using the hot-wire te¢hnique. One side of the specimen was systematically dissolved with 20% nitric acid, and the course of the isolation was f -olloweC,, and direeted w:t th Laue photo• graphy.. The apparat:1.1.s us~d in the C¢>ntrolled dissolution was designed to aeeomplish the grinding of small crys ta.ls to cylindrical fenn; 1 t was modified he:re by replacing the abra.si ve surface of the gtttnd!.ng wheel wt th hardened filter paper which served as .a very gentle 1';tbbing surface which could bl"ing the surface of the metal :tnto contact w1 th absorbed aeid solution. At the oonelu$10n of the . proeedure the original _specimen was SMped at one end. to a cylinder 0.1 mm. in diameter e.nd o. 5 mm., long. The cylinder was alm.os t a single ind! vi dual with 1 ts long axis nearly parallel to the £_ axis of the 20 crystal. The specimen was then washed repeatedly with vaseline- ligro:in solution and, a f ter a day, a coating of material formed from the reaction of the remaining acid was removed by gentle abrasion with a dry surface. The specimen was then removed from the Pyrex rod and sealed in a thin-walled Pyrex capillary. It was held ri gidly in position by a small shellac joining between the inner surface of the capillary and the untreated end of the specimen., All photo graphs of the crystal were taken with the c axis vertical. Orientation of the crystal was accomplished with Laue photography. During ph otography the specimen was always in such a position that the incident beam of r adiation did not pass through the untreated pa.rt of the specimen, In Figure la and lb prints are shown of t wo Laue photo- graphs t aken for t wo symmetrical settings of the crystal. The c axis is vertieal in oo th cases; in Figure la the incident beam is parallel to one of the a axes and in Pigure lb the specimen has been rotated 30° about the .£ axis from this position. Although consi derable differences in the spot sh.apes a r e caused by a bsorption effects and the presence of other individti.als on one side of the specimen. it can be seen in both figures that the local intensity relationships around each spot indic~te two perpendicular mirror planes, one vertical, the other horizontal. This indicates that the Laue point group of the crystal is D6h - &mm. As a c heck to this m conclusion, s~metrical Laue photog raphs were prepared with a specimen which consisted of a clumped aggregate with a very 21 c vertical sO N ll a~ "' 22 c rv verti.cal 25 small ·tangle crystal tip• The ineident beam touched c>nly the tip and extremely long e:,iposures were neeessary. The photographs are repl'Qduced in Figures 2a and S:\-, • The set"'!' tings corre$pond · to those of Figures la alld ¾" Although the be.ckgnn.Uld 1s high and repro.duction of the pho't egr$phe . . d!ff''leult,., it :ts evident th.a.t the symmetry found in the preced.in~ photogl'aph$ is eonf1rrned. Additional Laue pho~~aphs of the • eylindviee.l ijpeeimen were prepared w1 th the ineident beam inclined at small angles to ~ • The poai t1ene or the ret1eet10ns of only one s:t de of the photographs eo\ll<.1 be aceU?'t\teli detennined in gnomonic projections J the sides PrG Jeeted •,cox-respond to the left side of the reproductioncS of Figure l • No reflections were found which required the assumption or a unit eell larger than the one proposed from th~ powder photeg:raphie data. The first order re:f'le:o tions 0b$e1"Ved tn the symmetrical photographs are 11steli in fables l and 2, ln the appllcat.lon ot th-e Laue phot.o grapbic data to the determination of .S tNeture; 1 t ts safe to assume first that the unit cell and Laue point group f~ncl holds at least for the positions of the th$111wn a tom's ; it is conceivable that the pos'itions of the magnesium atom~ permit neither the unit eell nor the Laue symmetry ft)und, but that the cc->ntributions to scattering from these atoms are negligible with r-eipect to the contributions from 1;,he thallium atoms. 'l!he correctness of the poeitions proposed f0r the thallium atans are tested byan examination of the ways in wh1eh three atoms, either cryste.llographically equivalent or not, in any com.b1na.tion, eoul.d 'be ....c vertical 25 c vertical ,,...., 26 placed in a unit cell based upon a space group of the Laue point group D h.-. 6 Thus a ll space groups were examined be- longil'lg to the point groupa, D6h - §!nnl, D6 - 62, Cov - 6mm, m and D5 h - 62m. .The only possible arrangement which does not lead to gross disagreement with the intensities of the reflection.s observed in the powder photogris.phs is the one pre ... viously proposed., thallium atoma at O O O; 1 2 z; 2 l z with 'Z '3 the value er z el.ose tQ 1. 2 '3 '3 The intensity relat1onehip$ be• tween the :reflections er the Laue photographs ehov.r that 1 t is very unlikely that t di.f'fer,,s from ½ by ae, much e.s 0,05. statement is derived from the- follow:tng argument. This If z is ½, the geometrical s t~tcture fattor _f'ormulae for the thallium positions are ). : 2n , n=: 0,, 1, 2, ,. ,; • .... S ii! O• h ""k f 3m J : 8n + 1 ·s :: -1, h ~ k -: 3ln S ;: 2, h ... k t 3m On the other hand, if. z differs fr-om ½ by 0.05 and i ; 5 S ;:1±21 • h .., k : 3m S: 1 ± 1, h ... k f 'lml. In a few instances the intensity of a reflection of the type hk•5, h - k = 3n could be compared to that of a ref_lection 8'1 of. the type hk •5., h ... k =i= 3n. !n every ease the observed in- tensity of the reflection of the first type is much smaller than that of the reflection of the second t ype. This would be expected :t.f z is precisely ½; on the other hand if z differs frcm ½ by ta.$ much as O,05, 1 t would be predicted triat the reverse of the observed relationship should hold . Ftl.rthennore, only four very faint reflections of the type hk •5, h - k : Sn were ,observed to twentr•one of the type hk • 6 1 h ... k /:- 3n, which were generally or moderate intensity . There are a.pproXim.a.tely one-half as many reflections of the fit'st type as ~r ,the second type {i.n fi'i. given r;p here .o f . reflection) :and the much greater preponderance of". reflections. of the second type WOl-tld be unexpected,. if the thallium ·.~ on• tribut1ons to reflections of the first type were greater than those to ref'lection.£? of the. .s,econd type · to the extant indicated for z : i ± 0,05, The positiqns of the • thallium atoms h$Ving been fixed with som·e degree of fil.CCuracy • it is now prot1 t .a b.le to discuss the possibility that the size of the unit cell found is not correct with respect to the positions of the magnesium atoms . It c.an be noted fr-om Tables 1 and 2 .that a. few reflections or the type hk•O, h .. k =/=. 3n are observed . There is no doubt that the contributions made to the intensities of these re• f'lections by the scattering of the thallium a tans exaetly ceneel one another, and that the intensities of the reflections are due only to seat taring by magnesium atoms . Since it 1s possible to obaerve reflectiens which depend on magnesium scattering alone, it is reasonable to expect that, if the true unit cell were larger than the one proposed., some evidence should be found to this effect from the Laue photographs. It was, therefore, considered safe to proceed with the assumption tha. t t,he unit cell found is correct. With the thallium parameter z fixed in the range ½± 0.05, the scattering contributions of the magnesilun atoms should have an appreciable effect upon the intensities of many of the reflections observed in the Laue photographs. stance if z ;; ½± 0.05 and For in- P: 1, 5,,t : -0.902 - 0.6181, h - k: 3n 5T.l ; lo 9 02 + 0.3091, h - k :/- 3n. Thus, even if z were to differ from½ by as much as 0.05, still only one thallium atom out of t hree would contribute to the intensity of the first type of ref'lection and two out of thi'ee to the secQnd type,.. !he maximum magnesium contribution in any case would be that for which six magnesium atoms would scatter in phase. of sin _fr i\ A reasonable average value tor the values observed. in Tables l 1J..nd 2 is 0.30/0.37 ; 0.81. At this value of sin .fr /'- the ratio of the atcmic scattering factor of magnesium to that of thallium is sabout 0.075. The ma:x:imwn. contribution of magnesium would therefore be 6 x 0.075 :: 0.45 compared to approximate ly l for the thallium contri-bu tion in the case h .. k;: 3m., 1 : 1. It is improbable, of course, tbat the maximum m~gnestum contribution would be realized; it is, however, reasonable to expect that the contributions would be large enough to be able to lead to contradictions o f the ob served Laue symmetry for reflections = 3m, ,/ : 1 1 at least. of the type h .. k No contra.dictions to the general symmetry relationships were observed and t he Laue point group D6 h can be a ssigned to the structure without reasonable d oub t s , I t is now possible to derive the spa ce group of the structure from the followin g considerations. If all the space groups of the Laue p oint r,roup D611 are considered in whi ch it i s poss ible to place t hree t ha llium atoms in the positions O O o, 1 2 z, 2 l z o r an equiva.J.ent set, it is 3 3 3 3 found that the only po si tions for six magnes ium atoms which are not irn.mecl i ate l y eliminated by gro sfi steric h i ndrance s are (l ) X 0 ( 2) X 0 ).,_2 ' . X 1. 2 , ( 3) X 0 z; 0 X z; (4 ) x , 2x, O; O; 0 X O; . 0 O; X 0 o; 0 X O; X X 0 • X X .1. 2, X 0 1.. 2 , 0 X ).,_f:?'. X X ·! z; X 0 z; 0 X z; X X z X X XX x O; x, 2x, O; 2x, x, O; ~,. x, 2x, .1. • 2 , 2x., x, i2,. x, 2x, z. . , 2x., x, .z ; 2x, x ., 0 ;. X 2x., x , la,• X X ],_ 2x, x; z; X X z; x X 0 (5) x., 2x., .¾ ,• .; 1 X X 1:?' (6 ) x, 2x, z; X X (7) (8) z X ,,. O; X - X .. y .12,; <J x-yp y., Y11 X - y , O; - .x x., O; V .; "It X ,t " O; x, y - x, O; Y, 0 y, X y, 2·l. -y, l 11; y - x, x, .1. 0 2 .11 1 y X z, x., y-x, .1.z;• 30 ( 9} X y O; Y, x-y, o; y-x, x, O; y X O; x, x-y, o; y -x, y, 0 {10} X . Y: x-y, , l y -21 .:;:.. 0 2 , t y -x, x, .1. 2 ,. -- 1 l • y X 2 J X; x-y, a; y-x, y, 21 ( 12 ) x OJ .,, X x 2 , ( 13 ) thraee atoms :tn ( 11) and three ( 14) (15) o, x x · O ( two sets) J. J. X 0 TI, 0 X ·2 , x x -~ .., (two sets} • ( 16 ) three e.toms in { 14) and three a toms in ( 15)., {11) X ..:±.. X x, 2x, o; 2x, x, 0 ( t y,ro x, 2x, .l.2 ,• 2x, x, -zl ( two sets) sets) atoms in ( 12) 0 o, 0 X In the discussion of t he various arrangements listed above it the positions O O o, 1 2 z, 2 1 z will be a.ssir;ned the three 3 3 3 3 thallium atoms with z in the range ·! ± 0 .. 05. The generality of the discussicm is not affected by this choice, It will be noted that the majority of the permissible arrangements ( l) to ( 16} are planar. These arrangements might be eliminated at o:nce by a consideration of the 00; 2n - 1 intens:tt.i es~, Unfortunately there were no re- flections of this type which were observed unambiguously in the powder photographs, and it \Vas not considered feasible to attempt single crystal photography of the specimen prepared in such a way as to observe t hem'" If intensities of the type hk•O e.re considered, arrangements (1), (2), (3) 9 (9) 9 (10), ( 14), ( 15), and ( 16) differ only in that one or two magnesium parameters x are possible. The various structure factor expressions can be made forme.lly identical to that of arrangement ( 14) with two para.me ters x 1 and x2 by letting x1 in certain c.s.ses. = ± x2 A c1i scus sion of these arrangements wi ll be deferred until later. It is necessary now to ha.v~ some idea of the sizes of the various a toms without making assumptions which would vitiate the e liminations of the rernaining space groups . A 0 lower limit of 2 • 50 A for both the tha llium-magne sium and magnesium-magnesitun distences seems safe. A minimum bond number of 2.6 was calcula ted for a thall:tum.-magnesturr. distru1ce of 2.50 A with the assump tion of the n ormal sin gle bond radius of magnesium a.nd the sine;le oond radius of quinquevalent thallh1m, 1.387 A(l), ( 2 >., The bond nUmber 2.4 0 was calculated for a magne sium-magne sium distance of 2.50 A. Bond numbers significantly g reater than one are rarely, if ever, observed in interme tallic compounds. By graphical means it we.s seen that arrangement ( 5) is not permissible if the lower l:tmi ts to both rna.gnes:i.um-thallium and mag~esii1m-magnesium distances a re to be maintained. In this arrangement the value of the parameter x must be between o.84 and 0.88 if the magnesium atom at xx ~ is to be fitted between the thallium atoms at O O O and 2 1 1. 332 As a result p a irs of magnesium atoms., such as those at 2x, x, ½ and x, 2x, ½ are P.t u distan ce between 1 . 8 and 2.3 A apart. A similar a rgument can be used to eliminate arrangements (8) and {12). Arrangemen ts (9) and (10) a re permissible for values of x between 0.31 and o.35 and of y between 0.03 and 32 -0.03. Be cause of t.he small allowa b le value of ~, these ar- r angements can be considered as nearly equivalent to those of arran Gements (14 ) a.."i.d (15) with x 1~ x 2 • Furth er discussion of arrangements (9) and (10 ) will therefore be postponed. Arrangements (4) md {11) are perrrd.tted only for ,,alues of x between 0.'78 and Oo83. If x or x 1 and x 2 are t a ken as o.so it i s e asy to show t:l:1a. t the gross 1ntensi ty relation ships b etween thR small an gle powder 1~ef'lections are not satisfied . For e xam.ple , the reflection f or the plEl.nes f30.01 was ob- s erved to be about t wice as strong as the adjacent reflection f or the pls.ne.s {22 . 0} • Since the thallium c·o ntributions e.nd the frequency factors for the two reflections are the s a.me, the magnesium contribution in the '.ease of {30,0] must reinforce that of thalliu1r1. much more strongly than it does in the case of f22.01 • On the other hand., the geometrical structure factor for arrangements (4 ) and (11 ) w'ith x:: 0.80 is -1 . 73 for f 30;0} and -1'!9'7 for f22.0} , the geometrical strt1.cture factor for thallium being 3.0 in each case. (4) and {11) are thus eliminated, Arrangements In f;tr r angement {7) steric consid erations show that x must &,gain be close to o~so and the ,,alue of y must not differ from the.t of x by more than about 0 . 05 .. Arrangement (7) may therefore be eliminated by the e.r•gtunent used to eliminate arrangements ( 4) and ( 11 }. In the case of arrangement (6 ) it is easy to show that x must be between Oo78 and 0.88 o At x = 0.78 the previous calculations ind icate that the calcula ted intons:i.t:i.es of 33 l30.o1 tm d f 22 ,o} fail comple tely to satisfy the observed r elationship. At x :: 0.88 the geonrntrica.l structure factors for a r rangement (6) are -1,60 and. · -1.91 for f3o,o) and f22.oi r espe c t ively; again the observed intensity relationship is not s a tisfied. Arrangement (6) is accordingly eliminated . Arrangemen t (13) may be split into t wo divisions. In the f irst , the t wo v a lues of x lea d to an a rrangement wh ich., whe n pr o j ected on t h e b a s e of the unit cell, is n ear•ly i den tica l to a rrangement (5); the observed. intensity relationship between this po s sibility. f3o.o} e.nd f22.o} may be used to eliminate In the second, one vt~lue of x lies between 0.8 4 an d 0.88 ; the othe r value lies in the nei f,hborhood of ½ within rather wide limits. The latter arran gement leads to a very unfavorable t ype of coordine. tion, one o f the t h allium 1:1. toms having nine ne&i.re st magnesium neighbors and the other t wo having only three nearest neighbors. For this reason t he a rrangement was eliminated from further consideration. All possible arrangements have now been eliminated except those which are formally equi Yalen t to the a rrangement ( 1'7) x1 O z1 ; O x1 z1 ; xx z 1 1 1 ; x2 o z2 ; O x2 z 2 ; X2 "i2 z2 • All the arrangements of this type with the exception of arran c ement (16) require that all the magnesium atoms be in one plane. Such a degree of plarrnri ty was felt to be un- likely , a.nd it was considered that hk•O dP.ta from single crystal photo graph s togetr.,er with the powder ph oto graphic data would be sufficient to ccrn.plete the determination of 34 structure. In order to determine the positions of the magnesium atoms it is necessary; as discussed previously, to observe as many hk•O reflections of the type h - k f 3n as possible. The Laue data of Tables 1 and 2 was found to be useless for this purpose; only a small number of such reflections were observed, encl the possibility of intercompar1son of intensities, which is necessary for a parameter determination, is denied by the fact that the reflections are observed at greatly different values of n ?l and 5in 8 . It was deemed impractical to attempt a systematic program of Laue photography with successive orientations of' the crystal in sueh a manner as to observe $elected pairs of the reflections at identical wave lengths• Instead a program of monochromatic photography was undertaken in order to complete the determination of structure._ - A series of oscillation photographs with thee axis vertical were prepared with the use of Cu K radia. tion filtered through nickel. I'n this manner all the hk•O., h - k j 3n reflections were obtained which it was possible to observe under the experimental condi tiens. The choice of copper K"""- radiation instead of molybdenum K c::>{ radiation, the only other feasible choice, was made for the following reasons. It was realized that heavy exposures would be re- quired in order that the reflections of the type desired could be observed with sufficient intensity to afford 35 reliable intercomparisons. Previous experience with the apparatus avRilable at t :i.e time coupled with the fact that t b.e mass absorption coefficients for both copper Ko<. and molybdentrrn Kc::,C radiation are high, 231 anci 136 respectively, led to t'ne conclusion t:r.e. t heavs:J exposures could not be ob tained in a reasonable length of time with mol-:,rbdenum radiation. Furthermore, filtered radiation of the highest pos- sible spectral purity was desired; filtered copper K raclia tion is much cleaner than is fi 1 tered molybdenum K radiation. Fina1ly, the existence of a high temperature factor WRS indicated by both the po11vder e,nd Laue photographic data, and it is doubtful whether the larger sphere of re f le ct ion gained with molybdenum Ko< radiation could be used to good advantage even under favorable e:xperimental conditions. The exposures necessar:r for the preparation of suitable 30° oscillation photographs were of the order of 15 x 168 = 2520 milliampere hours. The successive oscillation ranges over- lapped 5° and the total oscillation range was over 60°. Three films were used for each exposure in accordance with the multiple film technique ( 8 ) and the intensities were estimated visually. No reflections were observed on any of the films ·which necessitated an enle-rgement of the unit cell chosen. In Figure 3 the most heavily exposed film of one set of three is reproduced. It shows the same general asymmetry as that shown in the Laue photographs , one side of the specimen ap parently cons is ting of several l a rge individu al s inc lined 36 at small angles to one another. The equatorial reflections on one side of the film, however, are reasonably well formed and perm1 t fairly reliable intensity estimations. rrh.e powder lines of the intennetallic compound MgTl are also observed in t he photograph indicating its presence a$ a.n impurity; little difficulty was occasioned by this complication. '!'he eylindrieal absorption fa.cters of Ch\assen ( 9 ) were applied to the obse!"Ved intensities t©gether with the frequency factor and the usual angle-dependent factors• The corrected intensities so obt.a ined are listed in Table 3. If a rrangement ( 16) is chosen for the magnesium a.toms, the space g:roup is D,33h - C62m and the z parameter of the thallium atoms must be exactly½, 0 The lovrer limit of 2, 50 A for the magnesium-thallium end magnesium-magnesium distances requires that 0,312 ~ x 1 ~ 0,492 and 0,212 ~ x 2 ~ 0.3'72, This parameter region includes any of the parameter regions sterioally permissible .for the remaining possible arrangements. Complete gener-ali ty with regard to the choice of any one of the arrangements formally equivalent to arrangement ( 17) is therefore preserved, The obser,,ed relationfihips between the intensities of the reflections 21•0, s1,o, 32 •0, and 50•0 are approximately satisfied only in the sub region 0.402 ~ x 1 ~ 0,352, 0,252 ~ x 2 ~ 0.302, In this region curves were calculated for each ratio between the corrected observed intensities of adjacent reflections. From the points of intersection of 37 Figure .3 Left side of speeimen • 30? -o.sc:illation Right s id:e of specimen, 30° osci 11.ation 38 these curve s the approxim6.te parameters ~l:: 0,.187» 2 x2 -:: 0,142 were obtained. With the use of these pa.:rameters !he temperature fa.etor B in the expression ! wa s found to be approximately 1.65. ce.le. ct IFt.£.4.f-'~-.ZB~} This temperature factor wa.s used to correct the intensities listed 1n Table 3 and. the final vs.lues of the parmneters were detennined by the points of intersection of the corrected curves, calculated as before. In the final parameter determination the coordinates of the various points of intersection were given the weights o, 1, or 2 by the use of the followin g scheme. If the point of intersection does not lie within the region 0. 402 ~ xi L o.332, ~~ 0.252 ~ x 2 ,s .302, the weight is zero'. If the angle of intersection is close to 90°, both coordinates should be weighted heavily, subject to the remaining considerations of the scheme~ On the oth er hand, if' the angle of intersection is very small, but both curves are nearly perpendicular to one axis (x 1 or x 2 axis)., the coordinate e.long this axis should be determinable w:1. th some certainty• Finally the ma.gni tude of the slope o.f a given ratio along a given axis at the point of intersection should be large if the corresponding coordinate is to receive a maximum weigh tf ~~· Only t wo cu:r-ves were completely removed from t h is regions, those involving the observed. intensity of the reflection 43•0. During the estimation of intensities it was noticed that th1s spot was ill-formed , appa rently be cause of abso rpt ion effects, and some dmtbt was felt at the time ~-s to whether or not an attemyt ah0uld be made to es timate its inten sity. A number of curves lying well within the region were parallel to the extent that no included points of intersection existed. 39 The va.lues of ! i and ! 2 derived from each point of inter2 2 section, together with the assigned weights ., are given in Table 4. The final values of the parameters and t he corresponding probable errors detennined by the weighted averacing of t he data of Table 6 are ~l = 0 . 1873 ± 0.0014, x 2 = 0.1433 ± 0.0019. 2 . ~ The calculated values of the hk •O., h - k j: 3n intensiti es, corrected for thermal motion, e.re gi iren in Table 3 and gen- erally are in satisfactory agreement with the 6bserved intensities, although t he discrepanc:tes a.re e.pprecie.bly greater than those o bserved in the usua.1 structure determinations made under more favorable experimental conditions . It is felt, however, that the final parameter values are reliable to the extent indloated. Because the values of x 1 and x 2 differ greatly, it is possible to eliminate arrangements (1), (2), (3), (9), and (lO) for the magnesium a.toms. The only possible arrangements are ( 14), ( 15), and { 16), al 1 of whi ch lead to the space group D~h - C62m and the exac.t value ½ for the .:> . thallium par~neter., In order to verify the conclusion made previously that arrangernent (16) represents the actual configuration of the magnesium a toms, a set of multiple fi 1m powder photographs were prepared and the intensities of the observed reflections were estimated visually. The use of a.t>ra.ngement {16) leads to satisfactory agreement between calculated and observed intensities; on the other hand it is easy to show that g ross disagreements wou ld exist if eithe r arrangement (14) or {15) were used. For example, the reflection for the planes {00.2} 40 is observl~d to be a.bout 2½ times. e.s intense as that for the planes t 30.11 • With arrangement {14) the calculated ratio I / I becomes about 19; ,~11th arrangement ( 15) \ Oo.21 \ 30.lJ1 the ratio becomes 0.85. The corresponding ratio calculated for arrangement {16) is approximately 2.4 .. Arrangement (16) was t he refore chosen. In order to obtain the best possible agreement between calculated and observed intensities the expression was used. t his expression contains the general form of the - . tempera ture factor for a bia2eial crystal derived with the t . assumption of central linear res,..oring forces. The value of the parameter B was found to be 1.80 from a consideration of the f hk•Of powder reflections, necessarily those for whieh h - k = 3n • 'r'.ai s value 1 $ in good agreement with the va.:lue 1.65 wh ich was found for B fran the intensities of Table 3. The parameter b was {biVen the value 1.26 because of the close equality of the intens ities of the reflections {33.0J and f22.2} ; its ma.gni tude is not of such an order a.s to affect the argument used in the elimination of arrangements (14) and (15). Observed intensities., corrected with Lorentz, (q) pc lariza tion, and absorption fe-,etors are listed in 'fable 5, as a.re the corresponding calculated intensities. The structural investigation was completed with a precise determination o f the dimensions of the unit cell. 0 The ·best value for a 0 was found to be 8 .086 A from the {hk•O} powder spacings with the use of the extrapolation 41 method of Ne lson and Riley ( 10) In this procedure values of o a 0 calculated from each {:hk ~o} spacing were plotted e:.ga.inst corresponding values of the function ½{ cos %,e ,frr1 0 + cos;e }. The plot so obtained was a pproximately linear wit,h a slope which i s a measure of the magnitude of the absorption error in the values of the .observed spacings. At the position 8 ; 900 t he absorption e.f.fect vanis hes and the intercept of a straight line drawn through the points of the plot with t he .a.0 axis represents the correct value of a 0 , if shrinkage and -c·c entr-icity errors are negligible .. .rhree plots of the type described ,~ere made from three dif- 1 ferent powder pho to graphs.• Two · 6.f the photographs we r e of the usual type with both ends .of the film in the ba ck reflec• tion J'egion; the third photograph ,was prepe.red with the . mounting fir~t used by Htraumanie (11 ) • In each cas e shrinkage co rrectiona were ma.de and eccentrie:t ty errors we re assumed to be negligible •. .'fhe three values of a 0 found from the plots were 8.087 .'A, 8,086 A, and 8.085 A. A reasonable limit of 0 error was estimated to be 0 .005 A from a consi ·eration of the change s in the va.lu.e of 0. 0 on drawing other reasonably wellfl tting straight lines through the point s of each plot o The values of A for the Ko<,, and Ko< components of the copper 2 K o( line we r~ taken as 1,5405 A and 1,5443 A, respectively. 0 0 One of t he plot.,s is reproduced in Figure 4 . The extrapolation procedure o f Nelson and Riley c ould not be us e0 directly in the determination of c 0 be cause no 42 mo r e than fou r re fl e ctio n s of t h e type {oo.l.J li e •P i thin the range in which the lines of the powder photog raph could be indexed with certainty, and of these only (00.2} could be observed free from the interference of other reflections. 'I'hli.S it wa s necess a r :r to determine c 0 from general\hk • iJ data. Again three determinations were made with the use of the three photographs mentioned previously. In each deter- mination the value of a 0 was tak en from the fhk · OJ data in the manner first described. The spacings of as many -·of the \hk•21 reflections as could be measured accurately were corrected for absorption shifts by : the use of the a. 0 plots in the following way. According to Nelson and Riley(lu ) 2 and Taylor and Sinclair(l ), under certain experimental conditions ·which are commonly satisfied, a: _L ( cos i e + cos~ e) -z s,-n e e (a) where D is the true spacing and Dobs : D + dD. of hk•O reflections dD = da 0 -u (b} - obs ( a,0 a: 0 and we find that e) - Cl,) C( O.o ( cos~e + cos%. e 6 '/., In the instance S l"(l e I the proportionality constants of Equations (a) and ( b) being equal. The proportionali ty ,:constant is determined from the plot of the type shown in Figure 4 concerned, more accurately from the slope of the straight line drawn throur.,h the points of the plot, and is then used to find the correction dD to be applied to the observed spacings of the~hk •2J reflections . The value of a 0 derived from thelhk·oJspacings of the photo- ~ rtl 0 00 (A) 8.080 8.060 8.040 8.020 8.000 0 33·0 1.00 I F19vre 4 .0/sm 0+ 0 41·0 2 2(COS 2 22·0 0/0) 2.00 COS 21·0 o· 3.00 44 graph in quest ion,. together with the values of the corrected {hk•2 J spacings v.;ere then used to calculate corresponding values of c 0 • i'hese values v;rere then averaged to obtain the 1 best value of c 0 for each photograph. The probable error in each case was estimated from the internal consistency of the set. The three values of c 0 and the corresponding 0 0 probable errors are 3.6'74 :t 0.003 A, 5.670 ±0.005 A, and 3.67 2 ± 0 .. 005 A. The final value of c0 was taken as .t: 3.672 ± 0.00'7 St ru ct u:ra 1 Summarz '!'he str•ucture of the intennetallic compound Mg Tl is 2 br-tsec1 upon the BPRCe group D~h - C62rr. with a unit cell of ,) . di mensions a 0 = s . r~,.:-3 ± 0.005 A. c 0 • 6 Three thallium. e.toms and six magne sium atoms are contained in the unit c e ll. T'.!'1e thallium atoms Rre in the positions 1: (a) 2: ( C) 000 1 2 l ; 3 3 2 2 1 1 332 *The increased value of the error assigned to the final value of c 0 represents the stun of the a v eraee probable error f rom the three determinations and one-half t he error assigned to value of a. 0 o The deterrninat1.on of' c 0 is partially dependent upon the determination of a 0 ~ A ratfi.er quRli tative argument was fo rmulat ed to show tha. t t h is par tial dependen ce of e 0 upon a could be expressed reasonably by the calculation of t he er?.or for c 0 in the manne r described. 45 The magnesium atoms are in the positions 3: (f) x 1o O; 0 xl O; Xl xl 0 3: (g) x2 0-. ·""-t;'I 0 x2 .1.2,• x2 x:2 2. :l. The values 0.3746 .:t 0,0028 and 0.2866 ± 0,0038 were found for structural discussion 1'he positions found for 'the magnesium and thalli-um atoms in the stru.cture .Mg2 Tl lead to the following details of coordination. The thallium atom at 000, hereafter to be 0 denated Tl I, is surrounded by six magne sium atoms at 2.96 A which lie on the corners of a trigonal prism, and by three • which are situated on lines per• magnesium a.toms at 3,03 A pendicular to the rectangular faces of the prism and passing through the point 000. A closest thallium atom is at 001, 3.67 A dis.tant ' ; this .distance is not considered to be a. bonded di,.s ta.nee. Eltteh thallium. atom of the positions 2; ( c), hereafter Tl II; :ts surround.eel by ss.x magnesium atoms a.t, a . 0 distance of 3.14 t+. and three magnesil::lln atans at a distance • of 2,90 A., The coordination polyhedron is very similar to that of Tl I; the si.x magnesium. atoms a.t 3 .• 14 A• for m a trigonal prism, ~n<!l. the three magnesium atoms at 2.90A are again situated on lines which are perpendicular to the rectangular fa ces of the prism and which pass through its center. Ea.ch magnesium a tom of the set 3: (f), hereafter Mg I, is surrounded by .four thallium atoms at the distance 0 0 3.14 A and one thallium atom 3.03 A away. The pol~rhedron 46 is a rectangular pyr>am:td with the magnesium atom close to the base on the line drawn through tb apex and perpendicular to the base . Each magnesium atom of positions 3:(g) is surrounded 0 by two thallium atoms at a distance of 2 . 96 A and two thallium 0 atoms at a distance of 2 . 90 A. The cocordination polyhedron in this case is ·an ·~ppro;nmately regular tetrahedron. In a ddi tlon therEl en et magnesium ... magne1¼1 um d.1.'a tances . 0 of a'bout 3 .so A whi~h ·are perbap$ . rtori -bonded distance$; the decision as to the n$t,u~e et these distance$ will be attanpted subsequently~ • $ .31 A. Ef!.eh aton:i~g -1 is s:urpotmded by :six. Mg JI .· ,., . 81:x: Mg l atoma $Urr,ound each Mg Il atom, four at 0 . 0 ' ' • . 5 .30 A and two nt 3,3:L. I-,,'.,. _ The eoordtne.tien polyhedron in both cases is v-e.r y nea.l?lY. a trigonal priain . The nee.re.s t Ot' ve·r y nea.r neighbor distances found in the s true ture e.an be tt$ad to derive the valenaes exhi b1 ted by the varioU$ t,yp~e ot at<;,me w1 th the application of the rules proposed by l'·atil1n;g( l.)' ( 2 ) -i The procedure is b$-sed upon the rela.tiensh:1P 1\i ::. a ... 0.300 log n 1 which is now too well kn.Qwn to require further explanation . (c) 'l"ne principal difficulty in t he application or this relation between distance and bond number is the uncertainty as to the correet value of the single bond radius R1 . As a $tart in the discussion of va..le.n ee we ihall e.ssurne that the value or R1 for both t;,IT>es of magnesium atoms is nonnal, 1.364 A ( 2 ). 47 We shall now· seek a self-consistent solution of the thallium valences, defining a self-consistent solution as one in which the use of single bond radii derived from assumed valences in a manner to be described leads to valences equal to those assumed . As e.n example, let us etssume that the valence of Tl I is 4;00. If we assume the t the d character of all bond- ing and. metal11e orbi ta.ls. 1.s the same, we find that the use of the relationship R1 (d\ z ) : l.850 ... 0.,030 z - (J..276 ... o •.070 z)d' 0 . {2} leads to the value l.419 A for R , The use of Equation 1 { d) { c) now leads to the valence 4.20 for rrl ! • Thus 4 .,00 is a nea!"ly s elf-eonsisten t solution for the valence of Tl I• The exact self-consistent ve.lenees of Tl I and Tl II were found to be 4.14 ·and 3~63 with the respective single cond radii 1.414 A and 1.433 A,. We c0n no\".r calculate the valences of the magnesium atoms by the use of the self-consistent single bond radii of the thallium atoms. If the shor t mag- nesium distances are non-1Jonded 1 the calculated valences of Mg! and MgII are l . 45 and 2.3511 If the ma.gnesit.trn- magnesium dis ta.noes are bonded, the corresponding valences Elre 2 .10 and 3.00. The magnesium val ences in e ither case seem im- probable, e-nd thus the validity of the self-consistent solutions for the thallium vs.lences is to be questioned11 It is worthwhile a t this point to consider the effect of small variations in the distance$ upon the results of the 48 preceding ca l culations. 0 The Tl I - Mg I distance of 3.03 A is very sen sitive to the value of the parameter x 1 and is A r easonable minimum value o f x 1 is independent of x 2 • 0.3'746 - 3•0 .0028 = 0.-3662. 'fh e Tl l - Mg I dis tance is now decreased to 2.96 A; the Tl II - Mg I distance, previous l y 0 t ak en e. s 3 .14 A, is not de creas ed. appreciably . 'l.'he self .. consist,ent valence of Tl I is now 4.36, the self-consisten t valence o f Tl II being about 3 .• 63 . Corresponding valences of' Mg I and Mg II a re l,56 and 2.32 ·with the assumption that magnesium-magnesium .istunces are non-bonded. Although the maximum allowable decrease in t be value o f x 1 brings the magne sium valences closer to t he expect e d. value 2 . 00 , the shift is not nearly great enough. An increase i n the value of x 1 '1ould obviously mak e ma tters worse. I f t he value of x 1 foun d experimentally is retained, and variations in the value of x 2 are considered, we have t he following situation . If the value of x 2 is increased to 0 . 2866 + 3•0,.0038 ; 0.2980, the Tl II - Mg II distanc e is decreased. t o 2 , 85 i\. and the Tl I - Mg II dis t ance is :i.nereased 0 t o 3 .04 A. The self-consistent valences of Tl I and Tl II 0 are no··- 3 . 60 and 3 . 9 0 with t he single bond radii 1 ., 434 A and I o 1. 423 A, respectively . Co rre s ponding megnesium valen ce s are 1 . 45 (I) and 2 . 37 (II) if short magnes ium- magnesium di s tances are assumed to be non-1·;onded. Although the thallium valences are changed appreciably by an increase in the value of x 2 , the magnesium valences are essen tially unchanged. 49 If the value of x 2 is decreased to 0.2866 • 3•0.0038 = 0.2752, the Tl II ~ Mg 0 . !I distance is 2.96 A and the fl I • Mg II distance is 2.89 A, 1.'he aelf'.,.consistent valences of Tl I and Tl It are 4.71 and 3.,35 with single bond radii 0 O or l.396 A and 1.445 A. of Mg Il is 2 .34" 1.t'he valence of Mg I is 1 . 48; that It ie as s\.imed ag$in that ne magnesium• magne&J ium bonds exist• It 1$ .eV:ident therefore that wi th the la.r geat reaaonable variations of x 1 and x 2 frem the most probable values, the magnesium valences ealeulated in the manner prev10usl1 indicated e.re essentially unchanged.. It is improbable that the magnesium valences are correctly indicated in the preceding calculations, either with or wi thout the assumption that the magnesium - magnesimn d1 stances are non - bonded. One immediate objection to the method employed in the calculations is that the assumption of t he equality of the percentages of g_ character of all the bonding and metallic orbitals for a given thallium atom may well be incorrect. In order to avoid this aasumption it is necessary to attack the problem in an entirely different way . •. • first enunciated by Pauling ( l) We first assume the principle, , that bond numbers should always be equal to rati.o s of small integers 2 1 1 1 1, l; etc. 3' 2 i l r The ocour,.e nce of m .unbers gr-eater than about 5 in t he ratios seems to be unlikely; this enables us to make a decision as to the nature of the short magnes ium- magnesium distances. The bond number of a magnesium ... magnesium distance of 3 .31 A 50 is 0.10?' ,v l 'It) with the use of the single bond ratdius l.364 A. . ' It seems likely, in view of the principle just eited; tha.t the . . . bond number .l • TO' has no physieal significance and that the . . . ' . magnesium - magnesiu,m di stance& are non•bonded. It is hard to see why the valences of the magnesium atoms $hould differ from two. Electr.o n transferences ap.Pe.a r 1m .. probable because of the small difterenees between the electro .. negativities of magne$.fum. -and thallium> and the t-esul.ting . . l e of eleet~neutrality . . ( 13) difficulty 1n satisfying the pri ne1p if significant fonnal charges exist. 'l'hus it seems wholly reasonable the. t the valences of both types of magnesium a toms are 2.00. In aecordance with this valence requirement we now select bond numbers for the various distances which seem satisfactory. From the bond numbers end the single bond radius ~ of magnesitnn, l.364 A, the corresponding single bond radii of the thallium atoma is calculated with the use of Equation orbitals ~mployed in bond formation is then calculated f!'Qm the applicat:ten of Equ0.tion ( d). ( C) o The number or d The valence of a given thallium. atom is determined initially from the h:,nd numbers assumed, and must be compatible With the d orbitals . em.ployed 1n bond formation, as cal'!'> . number of . ... eulated in the manner just described. If valences are in• compatible with e.mou.nta of.~ character employed in bond formation, the initial choice or bond numbers is incorrect. A new choice of bond ·numbers must be made then and the procedure repeated. When valences are found to be compatible with the nlllrtber of !! orbitals employed, a decision is reached 51 concerning the valences of the thallium atoms, the bond numbers of the various distances, and the numbers of ~ orbitals employed in bonding, Application or the method deser!bed above» with the use • of the most probable values of x 1 and x2 led to the conclu$ion that the valence of Tl I !$. 4.40 and ·the valence o.f Tl l:t is 5.80, Each atom Mg l f-0me one bond witb 'fl I of order 2 • and fou:r bonds with fl tX. of or,<:1,e~ 1. . . I 3 •:Each atom Mg tt f«trms . . two bonds of order 2 with Tl l .a nd . twe bon~s of order $ ,vith · 5 Tl IX~ s . Every thallilnn atpm I f<>nns $ilt 2 bond$ With the ~se of 0.67 ,2 orbitals and thre.e 5 J bo31da withou t the .ua~ of !1 orbitals; 0.30 metal.lie orbit.ala are employed which have 10% t ..:t fonns .t hree .15· 3 bonds with the use of ....d character. .·. '!'hall1~ . 0.29 d orbit.al$:, Md . stn .1 bond.a w:tth .:r t the use of d - orbitals . . ., . 3 ~ fh~ll1\\ln I..l l:l.a$ o •.60 metallic orbitals Whieh have 19-% ._4 Character .• . fhfs ~~.s er:tption of vcl.encee. s.nd distances in the struetur.e of Ms2 Tl seem.a . the most reasonable. ~ne $trU,etural :o etenningtion or M&zln l'reliminacy powde?;J phctography :indicated that the structure of Mg2 In was isomorphous to that or Mg2 '!'1 . Experience gained in the determination of the structure of Mg 2 Tl led to the conclusion that, if t he sti:uctu:ral parameters of Mg2 tn were to be determined accurately, it would be nec.e ssary to obt'-1n single-et""JStal data~ Accordingly, a number of attempts to prepare single crys ta.ls were made along the lines described 58 in the preceding seotion, t,Chese experiments were unsuccessful . Additional experiments Wf3re m&de to· prepare slowly cooled cylindrical easting~ nf' t,he eornpeund w:t.th the use of the small bore elay tubing de.$ eribed. previously. Several· single crystals were pr~pared. by thia method., bl.ti all or . them were found to have the Le.ue s~'l0lnett7 . °t,_. •Sinc.e the only known pl).ase 1n th~ magn~-:t1uin- !P:diwn S'IJS t..em with the t.e.ue. $~etry ~ is that o.f lligJn2 <3 \ :tt was ass.\1med . that la~ge de~~ea13es in the magnesium content of the :melt occurred whene-v0r the molten :material was in prolonged contact with the clay tubing ♦. .Although the procedure fo llo\n1ed in the preparat~on of slowly cooled eas~inga v1as unsuccessful in the pre$ant investigation, it will be described in detail with the hope that some. future use can be me.de of it in the preparati0n or single erys tals in ays-tems of less reactive metals.. A tubular fumace was prepared by Winding a double heli..x or re~istanee wire around a piece or el.o.y tubing in sueh a way that both leads were si tua terl at one end of the tube. The inside diameter of the tube was of $UCh a size as to pe:i:mlt an easy, bUt snug fit with the small bo~e tubing in which the casting was to be prepared.. A sheath of le.rge-bore clay tubi.n g surrounded the winding and was atte.ehed to ea.eh end of it with sodium silicate . '!'he end of the intermediate tube at which the leads of the winding emerged, and which projected beyond the sheath tube, was then sealed with silver chloride 53 to the small-bore tubing on the inside~ and to a piece of Pyrex t ubing en the outside. t ight . 'fhese seals were made vacuum 'fhe Pyrex tube was connected. ta a two-way stopcock and pump. The innermost, snall bore tube pro,jeeted int() the Pyrex tube at one end imd through • the ends of the sheath and intermediate tube at~ the other; thus to.:rm:!ng a pass~wa7 from the pi1.mp to the atmosphere . The spacing et the, fum~ee winding was increased from one end to the other sc, thl\t a le..rge difference between the ~tead:y-.state tempe:ea.tur-e~ at either end of the furnace could be obtained_, ln th!~. w~y a .s udden decrease in the power inpttt pr.odueed a co0.l 1ng front which traveled along the length ef' the fmmace • . lt was hoped that coolins in thia mannel'.' wrn..lld eneoux,age ~ne r .onna,.t!on of feweii, and l a.r,ger a,ipgle 0eysta.l$ 11. Wht:m a taS:t:tns was to be prepared;, the f\u•nace r,ra~ brought. .t e. a ete~~ ,s ta.te temperature with th.e lewer- ,e ~d ef the !ttm~ee :s.omewhat above the estimated tne.lt:i.ng point. tttf the all@f and the upper end about 0 -i..-~ 100° ~- cooler-~ • The pump Wa$ tu:rned ort, btJt was isolated from the innenn0.st tllbe by means o.t the two- way stopcock . The easting 6\$$embl&ge was l.ewe~d into the orucible until the tip of the $Jnall ...bQ.:re tu.be touched the bottom; the rest of the furnace was · well above the surface of the mel ten compound . The pump wa.$. then connected to the system by a quick turn of the stopeoek .and the molten material was sucked into the small-001--e t~be., In most eases solidifieatien occurred toward the top of the tumaee before metal was sucked 54 in te .the Pyre,:: tube. '!'he assemblage V:Te,s wi thdre.wn ·from the e1'le1ble a.nd the power input to the t ubule.!' furnace was dis- ccmtinued • . After a few m.inutes of COQl.i:ng the e~e.t ing W·aB ra,u>"red by careful demo 11 tion of the entire fum,a~e assemblage a.nd quickly placed under a protective coating of .V~$eline• 11groin solution+ Very ra:r'ely did the :ea-ating eon$18t of a cGnt!nuoua rod, Usually s.eeti.on$ of alloy were .interspersed by regions of solidified :flux~ The ceystallin-1.tJ ot ,the various pa.r,tteles obtai11-ed we.s tested by meana of tau.a photography . As was stated . previously, several single cp.ystals of identiee.l stt1Uetures wi th th$ Laue aynunetey ~ were obtained; no single <;rys't.als of he.1(ago:nal syro.meti,y were found. Because of the failure to prepare single crystals, 1 t was necessary to rely en powder p:ho-tographie , dt;tta for the determination of stru.etttre. '!'wo seta of multiple .. film photo- graphs we~e prepared in the usual wf).y t o:ne of Which was an eJ(tremely heavy exposure. !ntensi ttes were estimated vlsu.e.lly; the datra is given in t.ra.b le 6. A few l.irtes or very low in• tensity which were identified as belong:tng te Mgln and Mg5 In were ob&e~red in the photographs . The presence of both the phases was attributed to the fact that .M~ln prob~bly melts ineongru.ent;ly ♦. It is well known that, 1n the case or f:ocon- g!'.'tt~nt melting, 1 t is of'ten possible to obtain three se lid ph$Se:s from the eoo11ng ot a liquid phase< 14 >. . .ln e~der to vertty the identification of the Mg5ln 2 lines• $ melt. cot Cf;\leu.lated eomposi tion Mg0In2 was prepa~ed 2 55 and photographed. During the eour.se of the a.ttempt,s to pre• pa.re single ery-stals ~ . some alloya w-er-e prepared which had large per~entages of the phase Mgln, and no difficulty was found in the iden tificat:lon of these lines in ttte final . Mg In powder photographs• 2 !be powder reflections which were those or ,Mg2:tn were indexed with a unit cell or dime,n sions ~0 .:: s.524 ± 0,005 1 .• c0 0 t:: 3.457 ± 01.1.007 A.. The dete,nninatien of precise uniti oell dimens ions was carried out by the U$e of the ext~apole.tiqn procf)<lure of .N elson .and Ri le~t( lO) ,in the mariner , previo~s ly de8.6r1 '.b.e d" !he abJerved intenai ties of Table 6 a how the same gen~ . era.l relati<:mshipe found 1n the powder patte~ of Jl!g2 'J?l , and 1 t is evident th.at the s tructi1res of Mg2In t;.n.d Mg2 Tl are itH:>ntori,h~u.s., ;tt was not pQss:lble to fix the magnesium . para ... meter$ ~,1 and l,!:2 with precislon1 because only a small ntunber of reflections ot the form hk .• 2n, h-k f 3m were observee.. 'i'he intensity :r-elationeb.1ps betw~en the reflections and fs1•i} and . tao •1} ., \72 •l } • and t81 •'1} and {22•1, [61 ~ 1J limit the parameters to the region , O. l70 ~ , l 4 0 . 2001 O,.l50 ~ fi ~ O~l.7Q~ ?n the d~rivstt~n of this region a .le.wefi limi.t ef' 2 .;.50 A . w.a~ e.sel!lmed t'(ir .magnesium .. magnes!t'Un and m.agn,~~ium - .ind:hun. d:t.s.te.nce,e ~- Intens-1 ties, e~leulate.d .w1 t.h the .pai--amet~~ ita.l.µe~ ,_. x _ o . 187, ~(;} • . o .• 143 are :t~ aatiefactqcy agreement witb ob.D1 ~~ • served in tens it1ea • ~s is tbown · :tn, table th . The . tempere,ture 9 51 factor ell!) ~4.00 ( ";,_: + 04:(!Jwal! applied to the eale\llated l 56 intensities • The struotu~i de~ail_a., or'•g2ln are, the.retore, very similar to tho~e of _Mg_2 tl' .~d ~the two ~t~ueturee are ap parently 1$(l)JnOrphoU~ •, ·,:t-t is net e.onttt dered $~fa to siate a • • .. • • ~• more . than this withQllt, -~441_t1or1al difi'raetlen data: • . . . Ai;, t};le .ot.\t~et. (;>t ;, t~t~< Jnve~t:tg~~1ori: . t t wa~ deet4e,l to - att~mpt the ¢omple~e . detenn'in$tlc:>.n ;9.f ·structure with powd~x• , • ' • ' ' t ' ''. • •• j • ., :·' photegrapht·e data. ' ~'1., J'!l".~\t-iQUS ~iffl<;~.lty teuna ' in growing . ' single crystals -etn~ the. p,te.a sur~ t~ . complete. .the wo:rk 1n a reaaon~ble length fJf t~.e. l~d t,;() th~s dee!,st&x'h A melt o.f .c al.eu~te4. •compo.$'lt..1 ~n ,M:g2 Gt\. and s.eve~l . spee!mens suitable fo.r powder .p hetogpt.phy were prepai>ed in ' . ' ' the usu~i way . • A heaV][ ln.µl~tplt - fi hn ~xposure waa me.de and the inten&i ties of the . obeerved. re.fle-cti.o• ns were estimated. ' : . ' v1-Smtllty • , •The refliiH~~lorui of · t~e -ph0togre.ph were· "indexed a. uecesat\f\ly wtth. . . .. ' ~e . ~t'sun:tptd.,czm of -a unit ¢ell ef dim.ens:tons ' . ., * '\, ~ 7 .80'1 ,A., e · ;; 6.,900.. A· , . 0 .. ' ., 0 •.. . ·•·. !he dtmel!laio,ru:i •of :tll~ tUHt eel.1 found here are in es-·_ . • ,. - . · _· ·-. . • , • _. • ae-ntial ~gr.er,me.nt .vri th., thOs~ •fet:md . by tf.aucke ( S) o , a0 :: ? . 85 A, c) e@ .= 6 . 9;4 .A. ,. 'rile !Val\le}1f t, 0 _ is approximately t wice that of those . found - for o0 in .,t.he _~f.\$.e s of M.g2Tl and Mg2 In, and the voltw.te of .the ~nit ce.11, 1$ approltimately twice as le.rge • . ,it-The detem.ln$.tion ,of the p~eise dimensions of the unit cell wae eerried out in the manner already deseri bed . f/7 The reflections for which i' is ocld, which ne-0essi t~te the doubling of the value of ¢.0 , are of low .1ntansi ty ~ Emd the possibility that the st.~c:ture was gro,s.aly the st=1me ~s that found for Mg2 Tl anq ~s2 in was auggeated•~ • . ,8 -eflections of the type h-k ;: 3n, J. ,-; 4m. ~ere $trong and. reflections h - k 1 = 4m wel.'le weak • . Thia. ean be teken . as oonflrmt;\tory . :r- 3n, evid~ce tl't..a t tb.e s tl"ucture o.f :Mg Qa. is g:re ssly tsomorpttou~ 8 to the t of Mg2Tl and Mg2 Xn ..,. . A,e . a · artart, : th.erefore .', -thEt • structure previously found was . a-ssum(l)d·· ro~ Mg2Ga and- the· values of it1 and X2 were dete·r niin~d :tn t.he ·u$ual way With the assumption of min!nrum megneeitnn .,., gallium. and magnesium ,.. . 0 magnesium di s -t ences or 2.50, Jh· ·ft+e bEr$t v$lue~ , ot. il and ~ seemed to 'be :tn the nei'gb.b(,,rh.ood. or, (hl82 a~d 0 .,1417;, . re&pe-et• ively i fhe temper~:tu?>:e t~e-t or was- e$'M.m-ate4 in the U$UA1 ,qa-y and the intensit1e$ o.r the f$:i:_-.s-t twenty Peflections of .•· 'the type hk•2m were ea.leu-l-at.e d.• . The t\greemen t between calc1..1.1ated and o b$tn1v~~ 1nts.ni1:t 1ea W~$ very goed and confirmed the gene~al de ttti ls asaumed• for the $.· true ture * !n or-der to ·d erlve- the space group of' the actual structure, it was ass1im.ed t,h~t ~e $ynll'net-ry (;}f the space group requires a truly hexagonal latti.ce ancl that the unit cell contains six gallium @, toms and twelve. magnesittm atoms in posi ttons *~'Alloys of composition lg5Ga 2 and MgGe. were prepared and photographed in order t() be · su:r-e that the reflections of type = ,E S'tl + l were not due to their presence in small amounts . The strongest lines,; of• both substances we'r e not obee·r ved in the Mg2Ga photegre.phS; 1 t seems eerta.in the doubling of the unit cell of Mg2 0a is necessary, 58 which a.re very close to those fotind fet' the stt'Ucture 1d$Q.11zed in the manner just dese.ribed·. fhe $pae.e grcup-s whieh satisfy these requirements were found t<1> :be Pfn ·- ee2c, l)~h "'\ (f62m, 1 C;,n 2 l. CS, D3 - 032; · C§v ;..· ;Qolo" C!v ~ -G~'lm, ~.nd iji"' <t'.a. or these sp.aee groupe :it l~ 1:1:aturel te tt\v.or the .tir-s't t-,we, 4 · ·. 3 ;,'.• ·. D h and D3b'. because, ot · their · re1:a. tlvely high $jtn'tmetey. 3 A· systematic • ~xtinetlE,n e::d$t'S : fo.r . , :11 ;. . c.6St, it· ' £,:, ~ + l i :reflections e>t , ~he type hh;.J'. airi.e •abff&ht;,:. • 16 ref·l ect1,;mtr ··c:;r{:' • of this type were.. ob$Ef?J'W'¢Q ,-. · a:l.th()Ugh( In : $-~·'V'.e rttl:' eates :tt would have been · po.ss1b ·te. :t a.' o~~e-" ~"' :~ $I};btguElusly .a · te~ • f lecti.0.n o :f' this typ-e if 1~ .iY't1ll"e st ,m,pd~,;rate 1:ntens1 ty . • • Violations or· th$ ,j ,~~~:t:t·Ot!Qtl l'\ll~ -.ftr· Pt ·- Cij23· were not observedJ th~re :$,s 'inO ':e~t:tnttton' n,1le t'~r. x,gh . ....-. C62m . • ' ' In v:Lew or tb.,e ' tiliWtbeti· et; N):-1!-Qn f 1. aba.ente~. . it wa.s felt ·that the v:h spc $,-Oi, g~<tUp ,11l -~ 1r;,i~ Vi~$ • 1mp'r ob,i t,le • ·a nd fro sys tema tic effo~t was, l'il~,ae:. ~ · fl~d: " ' :~ tw~~~; l1'a'$e('i. upon t h is Sp$Ce group··whien wqq.·l d ··be -e orie:1$ttni- wtth the Qbaeraved i ntensities . • '!Che •&.paee, gretip .l{~ ..:~· ·sJ;ae. w.~s :though t to be the most probable $p~ee • g~up b'e c~u:te e'!'' the: nwnber of absences. whi ch could be e~lained, by J't$. a~$tltript·1 cn,. and because of i te relatively h.ir):l. s:ymmet'I\V.,. l f the' magnesium a. tows are to be placed in the manner postul~ted~ tt ia ne~ess a~r to p1.e1t siJt magnesium a t(:)ms in the poa!ti;ons . 6 ( g)' 1 x O O; 0 >t O; xx 0 and six magnesium atoms :tn the pos i tlons 59 (h) . . The r1rst arrangement is tdent1ca1 to that · fc.,un<i in M~tl, - two atoms at 000; Oo½; two eqUi vale:nt atema at ,::,, l 2 _, l• ., 3 4 2 1 3; and two ~qui"ralent atoms at 2 l 1; 1 a. 3. 3';;4 !34 334 ;t:n. the $eCond arrangement t vr.o equivalent gallium atoms arie in the positions 00¼; ooi; and f,our equivalent e.toms are :i:n the poeitions The first arrro1gement was considere~ to be incorreet rrott.t the following .e ons:tder~:t..ion.th ):f t he first ar~angem.ent we"' correct, the corrected intensities of refleet:!ons with the same values o-f h and . k but dif'fe1;1ent odd values of ,,l be the Sfflle .. shm.,ld tb.1-s ie obvious beeaus-e there a.re no general z pa.r emetera ir1 the flr$t,, possible arrangement of gallium atom$. On the other hand, 1:f the second.· e~rangement is correct, the• intensities o:f hk~ l greatJ,y . an4 h,k• ,-t' ~an differ Fbq,erimentally-,. it w~$ ebserverl that the cor:r~ete<i 1ntenslty of the !''e t'.leetion that of the reflection \ 10•5 1 1$ nea;rly ten times f10•1} .. The $eeond arrs.ntentent was therefore chosen for the galli1..tm atoms; the value of re was presumed to be ne~rly o. If reflections of the type \ h k • 2l + 1} are con .. a tdered, the para.meter x of the magnesium pos i ti<zme e { g) 60 is eliminated., an~ it ia nec.e asary to -c onsid~r only the !t magnestu.m parameters u, y and th~ gallium pa.remeter z was assumed that the value of ll 2 •0,.182, determined initially from the lhk•4n 1 h-k ¥:Sn inten$l tie~ w~s appr~:rKim6:tely. correct. Intensity relationships between the observed re.... . 1 flections with -~ odd ~re in~ent.rit;1-v--e to large 1.::ariations in u from this value ,. end thua the ,. e;xplsi:nation of the :tntens:tM.es. of these refl~etions become.a eaeenti.ally e. two -parameter. problem. A consider$.tion qf the intenei ties of the , observed reflections of :t:rpe • [ ii..lt·• 2rn + 1 1 - -~rith the use of the p.r elimina riJ .v alue of :u , led to the pa ra:m.e-te rs - z :: o. •031, 1r = o. 022 • These :pEtrem.eters were tpen used !n-. the co:nsidEH"~tion of the 11 \ nk •4nJ intensities to oo:r,r,e:et the· Initial values found f'or and x, '.tt was now f~und fl"Qm a yonsid~:r-ation of the in- • tensity ratios 3-~ . o-, ~O •O• and 21-4 ·· that u = 0.192,. x i!";'U 32 ~0 30 •·4 1 i 1 , ·, '··, =o .. 142-1 ·, With the parameter values derived so ffiri the tempers.tu.re ' ' f'a.o tor 001?rect,,t0n waa $hewn t~ be- approximately tcale * f' ff j"vx+f "·'IZ ( ~ : e -t- 1.00 Jc;) J With the new value of;, 0.192, _the int~nsities of the 1hk•2n + 1} ·intenslttes.,11:,e~ ·nse<.1 to de'term1ne that y :: 0.022 end z == 0 . 034 • . ·T hese valtie_s -for y and z were then use<l in the final detennination of u ·and x. The. observed inter1si ty ratios 32•0, 21•-:\.,, a:t:1d _. J 40 •2 1 .. + . {32 •1J l'r;o ;;ij .. % , . ~3~hO J when u -• 0 .:387 1 x # 0.,284 . are best satisfied intensities were th-en calculated for all reflections o,bserved and compared to the ve.luea 6). estimated ,risually,. This compariscn :ts given in Table 7. In general the agreem~t between ,obs,ervecl e,nd calculated inten sities 1s rathe-r good; there a.x-.e, however, a few marked d iscre.panc1e$ ~ Th~ ~greem(:3l'lt. between cal cu.la. t ed an d observed l22 • l J + f 5 l. •O1 , t22 •4 } + lsa •31 , {.ss •l 1 + l 20 • 5 J + ' { 42 . 0 J and ( 3 l •4J in t,ensi ties r or th a ret"le c tfons {30 • 2 J + 1 is ePpepie.112.r poe~. s-t~enuous., bu\ i,nsuece~sful, e f foI"ts - were made to eJ1:pJ..a1n these diserepa.neit)S in .• number of '1$.fS ., Ern:>n ~:n the indexir,:i.g proc.e ss were n~t , fsµnei,_ e.nd there were no gross errors in the estimation er intenGi ties .. '!'here·;,$eemed to be no eystematic effect due to pre.fe!'l"ed _orie:q.t,at~on 0f. crystallites ~ Since the obt:H~rved. intene i ties in the ·ease or marked discrepancy a.rt€ a.lways too sip,all, the presence o.f 1m• . purity lines cannot be invoked. to e:&:plain the dif.fiCttlty, Accidental phyaice.1 effects, such ae the occurrence of multiple reflections< 15 ) vii. thin ea.oh mosaic block, seem iJ:n ... probable for a num.bet• of reae ons . Tne st:r,a.eture fc.,u.nd £or Mg2 tta in the . present investigat ion. 1$ not, eo.:mpletely aco.ept~l:?le 1n View ef the inelCI)lieabl.e discrepanc1ea between . tbe ca.lQ1,tle.t~d e.nd obeerve-d :tntensi ties fer a sm:all number o.f . t1e.(lect1or1s ,_ It is quite probable that the g~o$ s strtletitral cteta.fls- proposee are correct, but, that the preo1se p.o at tion-s of th~ ~toms, have not been detenni.n ed correctly .o A consideration of the other space g~eupe whieh were proposed above weuld-be extremely :in,lt')lved without adtUtion.e .l I 62 experimental in.formation. ?n pais1ng, it should be noted that tbe e~t1net.18n rule for the apaee group elv - .Q:3ie ts the same as that .for Djh - Q.62~ • . In thts. spaee greup eight pattemeters would generally be neeessary to deeeribe a structure which is grosslt the eame 8.$ that of Mg2Tl . It ia reasQnable to expect that• with single ... oryetal data. the atructure ot Jtg 2G.a could be completely determined.. A !'u.rther attaek \lpcm the s tNeture in this di reet:ton was net possible in t,he present 1nveetige.tion. tihllnmt1£,1 ~d Cmie luS31.o.ns, An investigation of the structures of the 1nt.ermetall1e eompettnds. Ms2 fl• ,Mg2 In, and Mg20a haa been made • ',l!he structure of Mg2 Tl was determined preeiaely with the use of aingle•erystal and powde?" X-ray data,. The v~e:nces of the t wo types or thallium atoms were found to be greater than the elementary valence; :t t is prob~'ble that cne ~r the th~•Efe''''th'allium atoms of each unit eell has a valen'~e•'·~.r 4.4 and the remaining thallium atoms a valenee of a.s, the magnesium valences being 2.00. No th11-lllum•th lliwn valence bonds exist in the structure. . ~, 1'he s tru:ct:ure ef Jtg2ln Wt\ta ftJund t~ be isomel1)heus t0 that of lltg2 1l . It was nQt possible to determine the positions ef the magnesium atQctns with sufficient accuracy to give a. discussion of valences . The etNcture of ·M.g2 Ga is ~pparently similar te those of Mg2Tl and Mg2In in 1 ts. maj.o,:- aA:Jpe4ts., ~e posi ti.ens or the atoms, h.Qwe.ve:r, nee.es$ttate a d$'t1bling of the length of the - unit cell dimension along the c axis. • l.t was not possible 64 , ' Laue paotQgr!\\phf.~ da,ta fer Mg2 Tl ~ bk•l 38-,4 4l'1 ,2 40,5 46~3 15'12 35•3 31•1 . 2s.1 55.5 41 •5 57.3 33.4 58•,2 4$•1 ~3•5. 51•6 J?.. •3 54,5 11 il o.,~02 0.479 0.4'74 . 0,4'74. (h471 0,4$0 · (h3® sa~o 4l;5 25,2 58,1 55,4 sine 0.248 23~3 36,2 &.3&·4 . Q. • 34~ <h-242- ., <:>.380 0~380 o. .•$39 0.339 o.·s,3a 0 .17.Q 0.292_ .o.2.90 . ·0.237 0.370 0.389 0.36€1 o.as4. 0.366 0.32'7 0..322 ().356 .o•.~ .s 15"·0 0.355 0,158 69 •2 0.353 0.384 59•1 47•3 41•~ 31•5 27•0 38•1 (hlO•l 15•1 10•3 49 •l. 52•6 ol•5 14•2 87•1 11,3 ve~tieal., 1o II ~ 0,;314 0..310 0.26'1 0.215 0,264 o.372 0.1$2 0.150 0.301 o.535 0.259 0.150 0.211 0.147 G.461 , d.. (A> -t 0.102 0.061 0111790 0,615 o.eea 0.951 0.~460 0~458 0.605 ().466- , 0.573 0.456 0~452 · ().451. 0.450 · 0·!1445 .<>-•~ o.,44a O·@S . . 0.430 e.f2s o.. t2:5 o•.424 ·<h4ll, 0.402 0.40l 0,6()3 0.678 o.•e&a · · l,32 O.fo/-o .a ♦.·7.6'6 (l.13.& o.sas . o.s58 0.594 o..754 o.&11 0.$48 o.-o:38 0~564 0.400 . 0 .:S.93 0.~564 .l~ij6 0 .393 ·o~&ll • o.sal. o.:615 o.597 0.$95 0.390 0 :. 375 0,.370 0.367 ().560 0.$8l 0,'702 o.seo 0,695 0.359 Q.493 1.18 1.20 0,596 0.358 0.353 0,682 0.367 0,360 0,359 0.350 0.350 0.348 0,534 1.17 0,828 l,.16 65 ~bl.e· 1 {oont .,) SIYI 25•3 G ni\. 0.342 Ow342 Q. 359 0,334 0~355 o ., :f2$ 0~52? 5.lO•l 48 ··3 59•3 S2 '!5 31 •3 58 •4 45•5 Oio23 . 47•4 Ot3g3 41 ·6 6.ll •l 4.10 •1 12~3 36 •4 0,322 o.~et, 0 . 3le,· · O:i:iQV •. 0 •.305 · .. (),305 0,.$-0 5 Q-.297 39 • 1 ~hll •O 49t3 26<i3 39~2 13•5 16•1 O.822 0 . 521 0.5?8 0 , 5).4 0.663 o:e:55 0-~.500 Q.~554- 0,I>i.6 G.~.5~4 :0~4621 o•.saa i •.~E> _ o.tSea 0~634: • >0~4~1 0., 541 o.~s · 0,7S.6 o,~S$l Q.• 271 0.144 0-t:~2 2S·1 cl cr-f ),;( 0.26f O~i60a 1._03. i.o.a *c~leulatef w:ith the p~$limt:naey eel1 dimensions ao : : 8.04 A, e0 : 3.,.64 A. . . 66 Table 2 Laue photographic data for Mg 2 Tl ~ vertical , L. ( ~o, ~) = 30° 10 .. 2 •1 13 •5 42 ° 5 81• 3 91 •3 11 . 2•1 10 . i •3 9 0" 4 . 32 • 5 62 "1 61 11 3 10•2 12 .2• 1 ii.1•3 21 •4 l3 • 2 •0 03 •6 61 · 5 01•3 13 .2• 1 l4 "6 22 • 5 51 "5 . nA .0 . 322 .0 +348 .0 . 372 0 . 320 .0 . 479 0 . 4'75 0. 344 o•.S40 0+364 o . 38 6 0 . 339 0 . 179 0 . 254 o. i.2s 5l•o 13 . ~•2 20 •3 52 · 6 11 .1:- 4 5g• o al •-4 so·s 11 . 2 •3 , 10+2" 3 12. 2° 3 70 °5 9 2" 3 0 +440 ·( . ) ~ o.744 0 . 682 0 . 638 0 . '735 0 - 676 ,Q. 674 0 . 626 0 . 590 0 . 664 l . 24 0 . 8'71 1 . 76 0 . 61'7 0 . 580 0.440 0 . 440 o.•a,2 0 0435 o •.se1 0 . 427 0 . 425 0 . 421 0 . 420 0 . 419 0 . 417 0 . 409 0 . 405 0 , 574 0 . 696 0 . 601 1 . 20 0 . 56'7 0 . 398 0 , 526 0 . 951 0 . 949 o.s19 0 . 25:3 <h305 o.sao 0 . 1'75 0 , 369 o. s10 0 . 322 0 . 379 o.. eos 0.200 o_.35e o..1ae 0 *358 0 . 285 41 •5 o .474 . 0 . 470 0 . 465 0 . 458 0 . 455 0.455 0 . 450 0 . 444 0 . 442 : cl -0 . 356 0 . 298 14 .2• 1 7 g-. g 12• 5 • Stn ti o.:;so 0. 115 0 , ®)84 o.327 0,. 305 0 . 255 0 . 201 0 . 32S 0,.301 0 . 254 0 . 396 0 . 394 o. ss4 0 . 686 0 . 630 0 , 390 0 . 386 0 , 548 0,.376 0 . 534 0 . 658 0 . 536 0 , 375 0 . 375 o . 368 0 . 368 0 . 366 0 . 365 o . 362 0 . 362 o . 359 0 . 3540 . 354 1. 15 l , 60 0 .. 648 0 ; 559 0 , 598 0 . 702 0 . 644 0 . 556 0 . 588 0 . 697 67 Table 2 . feont .. ) hk • 1. Sin ll "4 0 ,. 197 0 0 . 550 11 . 3 •1 12 . 3 °1 13.3 1 14 *2"3 0 . 251 0 . 295 0.354 -0 . 344 .. 10. ;&•1 71"'6 0 . 221 0 ~33:~ 11 31"5 14 . 3 •1 o . 2?5 o. 2so 0 . 333 0 . 311 82•3 . 31 · 2 . 0 . 110 12 . 3 •2 0 . 266 93"1 30 '" 4 15.3 •1 15 . 3•2 41 "·3 52 •1 13 . 8 °4 02 • 5 72°3 21 •5 93 • g 0 .• 220 o . 186 0 . 187 0 . 338 0 . 319 • o. 1s1 0 . 105 0 . 314 0 . 21'0 o . 1a-1 0 . 206 0 ,.. 34:9 0 . $48 0 , 344 0 ~34i o.~sss 0 .. 336 · 0 . 33.2 0-.330 0 . 522 0. 317 0 ~317 0 . 317 0 . 312 0~311 o.sos 0 . 505 0 . 300 o. 2tn 0 . 28$ 5l 4 • 0 . 1'7'5 Oel75 0 . 27'(; 83 •'1 0 . 142 0 , 27'2 0 0 . 274 68 Table 3 hk •,O observed intensity . ( eo rree ted ) 10 .. 0 20 . 0 21 . 0 31 . 0 40 . 0 not obs . not obs. * 32 110 50 . 0 42.0 51 . 0 43 . 0 61 . 0 70 . 0 53 . 0 62 . 0 54 .0 80.0 72 . 0 81 . 0 {i- 15 •.4 • 5 . 32 l . 68 4 . 00 1 . 79 7 . 24 6 , 69 2 . 94 2 . 29 1.18 1. 51 l . 17 2 , 81 2 ,12 2.27 1.cm calculated intensity {sealed) 0 . 06 0 . 07 10.3 5 . 30 1.76 4 . 31 1 .85 6 ,36 7.04 1.55 2.35 1.50 l.'75 1 .. 4'7 2 . 52 1 .95 1.46 0.90 *I n tl?,is . region of the .photograph there was a rather heavy background. 1 1 2 3 2 ~ 0.1885 1 3 4 6 7 0.1870 0.1890 2 0.1430 0.1870 2 2 2 8 9 10 11 0.1900 0 .185( 1 1 12 Table 4 0.1435 0.1425 0.1870 2 2 2 0.1425 4 5 0.1865 0.1865 0.1860 2 2 2 0.1865 1 0.1865 0.1910 0.1850 2 1 2 0.1870 0.1870 2 0.1915 2 1 2 2 0.1905 0.1915 1 2 ~ 0.1425 5 0.1405 1 0.1865 1 0.1475 0.1510 0.1850 0 11840 1 1 0.1470 0.1885 2 2 1 1 0.1390 0.1395 8 2 0.1425 0.1400 0.1395 2 1 2 1 0.1865 1 2 9 2 0.1435 0.1435 10 1 2 0.1450 0.1450 0.1435 0.1435 2 2 0.1450 0.1455 0.1430 0.1435 2 2 0.1450 11 2 12 2 2 2 2 0.1365 0.1385 0.1380 1 1 2 40-0:31·0 32·0:40·0 50•0:32·0 42•0:50•0 51•0:42•0 70•0:61•0 53·0:70•0 62•0:53•0 54·0:62·0 80•0:54•0 72•0:80•0 81•0:72•0 9 0.1855 0.1490 1 2 3 4 5 6 7 8 10 11 12 7 1 Intensity Ratio 0.1875 6 2 Curve No. 0.1845 70 Table 5 Mg2 Tl Powder Date. observed spacing A• calculated spacing A 7 . 004 4 . 043 3 . 992 3 . 628 3.6'72 3 . 502 3.252 2 . 718 3 . 225 2 . 696 hk•J. 2 . 618 2 . 647 2 . 516 2 . 536 not obs . 11 . 7 00 . 1 20 . 0 10 . 1 11 , l 21.0 2 . 334 2.. 143 21 . 1 1.963 l . 970 1 . 942 l .836 l . 7'76 1 /771 1 . 751 l . '715 1 . 672 l . 626 1 ~711 1 . 663 1 . 575 l .~524 22 . 0 30 . l 31 . 0 00.2 0 . 01 3 . 93 3 . 46 3 4! 40 1 . 606 32 . 0 1 . 580 1 . 528 not obs . 1.19 41 . 0 40 . 1 21 .2 52 . l 1,357 1. 344 1 . 344 1-334 1 . 823 50.0 22.2 33 . 0 31 . 2 42 . 0 1 . 309 50 . l 1 . 202 1 . 188 1.173 '?; 1 . 268 1 . 265 1 . 257 1 . 245 1 . 224 1 . 209 1 . 206 1 . 190 1 . 174 l . 171 5. 63 not obs . 31 . 1 11 . .2 1.,401 l . 359 1 . 242 6 . 15 2,73 0 . 63 0 . 05 40 . 0 22 . 1 1 . 406 1 . 261 5 . 99 '7 .21 '7 . 32 3 . 47 0 . 45 not obs . 1 ,09 0.53 30 . 2 1-306 o.oo 3 . 50 1 . 28 0 . 09 4 . 19 ~ 0 . 3'7 {~ l.4'72 o.oo 4 . 09 0 . 18 10 . 21 l.442 l . 410 l,;439 ~ 0 . 38 not obs . 6 . 10 l , 46 20.2 1. 508 1 . 468 inter:.si t y ( scaled ) 10_.o 20 . 1 30 . 0 1 . 830 l . 763 calculated intensity ( corr~cted) 11 . 0 2.:320 . 2 . 139 2 . 010 2 . 021 observed 41,1 40 .2~ 33 . .1 51 . 0 42 . l 00 . 3 32.21 10 . 3 51.l 41 . 21 11 . 3 5 . 50 not obs . 51190 1. 12 0 .73 2 . 92 o.oo 0 . 04 1 . 25 3 . 96 0 . 06 4 .72 5 . 62 0.66 ·4 .19 1 . 01 1 , 92 2. l9 not oba. 1.94 not obs. o.oi 2 . 20 o.os not obs. 0 . 07 1.75 2 . 01 < 0.43 0 . 41 l ,89 2. 38 0 . 49 0 . 02 0 . 57 2.32 3 . 36 2~66 3 . 42 not obs . 71 Table 5 (cont . ) observed s~eing c alculated SPf-Cing A A l.165 1 . 167 1. 120 1 . 110 1. 097 1.085 l . 058 1 . 155 1 . 151 1.121 1 . 113 1. 112 1 . 111 1 .. 098 l .. 087 1 . 084 1 . 072 1 . 0'71 1 . 068 1.047 1 . 034 1.03'7 1 . 024 1 .010 l . 035 1 . 025 1 . 011 1 . 002 l . 001 0 . 9835 0.9727 0 ,. 9641 0.9557 o . 9378 0 . 9265 0 . 9198 l ,. 00:3 1 . 001 0.9850 0.;9'7741: 0 ~9756 hk •.Q observed intensity calcula t ed i ntensity { eo!"reeted ) (scal ed } 60 . 0 20 . 3} o•.76 ~ 0 . 47 0 . 67 52 . 0 50 . 2l 60 . 1 2. os 2 . 22 0 . 68 1 . 06 0 . 91 . 1 . 25 1 . 80 · l . 99 < o..39 0 . 44 43 . 0 21 . 3 43 '1 1 3:3 . 2} 30 . 5 52 . l J 42 . 2 61 . 0 l . 66 22.:; 51 . 2} 31 . 3 not obs . 0 . 49 0 .77 61.l 0 . 73 1 . 18 0 . 90 <0 . 35 · 0 . 29 1 . 09 1 . 32 0 . 56 1 . 11 44 , . 0 40 . 3] 53,.0 70 ~0 60 . 2 32.31 43 . 2 0 . 09 1 . 23 0 ,; 9'7-43 44 .. l 0,9710 0 .i 9658 .6 8 . 0 53,1} 2 . 06 · 2. 11 52 . 2J 2 . 33 2.21 o ..9esa 0.9568 ?O~l 0.9552 0 , 9386 41 . 3 62 ~1 0!19274 71 . 0 0 . 9227 0 , 9214 1 . 24 l , 32 · 1 . 51 l.38 61.2} 50 . 3 0 . 48 00.4 < 0 . 25 0.9180 0 . 9098 O,&W60 o.8985 0,8991 0 . 8984 0 . 8964 0,8949 0 , 8848 0.8880 0 . 8855 o.oo 10 , 4 33.3 ?l 42 ol ~3 l 64 . 0 not oba o not obs . 0 . 06 0 . 68 1 . 11 80 . 4 } 0 . 92 0 . 85 11.4 44 . 2 ~1-The absence or this line waa,. not certain because of the pr'"esence of the l ines of either Mg2'1'1 or Mg5 T12 • 72 Table 6 . Jg2 I?il P.owder Data observed spa~!ng A 0 4·.-127 3.416 3 . 096 2,/700 2.647 2.481 2 .. 390 2f/J.31 2 . ()78 ctil,cu la. ted spa,eing ,A ' . 7.209 4,l,.62 3.6Q5 3 .. 451 3.ll3 2 ·/7:24 2.657 2,49~ 2 , 403 2.159 2_.oe1 2.0,00 1"963 l.9?2 l.802 1.?76 1. 783 1.,'725 111.1 30 1.647 1 . 678 l.684 l .590 1 . 567 1.487 l.428 1.398 l . 385 111·360 1.326- hk• i l . -?26 lr598 1~594 1.573 l.566 l.492 l,457 l . 442 l.431 1~401 1,387 l , 3£2 111330 l,328 10.;o 11,iO . • 20,0 00.1 10.1 , 21.,0. , 11.1. 20.l 5:0.0 . 21.1 22.0 . l . l68 3,67 1,2'7 0 , 60 0.53 2.33. 2 •.28 not obs . o.oo 82 , l. . ~i:~1: 10.2 . sa~o . .:::: 0 . 26 40~1) 1 ♦. 20 41.,0 20 •.a. not o~a . ll,2 . 32.t l . 21. 2 . so.o. 41.1 , 2 .. 00 3,92 not obs . not 3bSt o.oo O,,C'/7 1 . 70 1 .•88 o.oo S.28 o•.l l 0.01 0/70 3"•2 . Oe;66 ~, 90 41?.0 . < 0.29 0,1~ 2 . 49 2 930 <0.30 not obs. < 0.30 0~06 0 . 12 35,-.0 . so. ~,· 22. •. ~ltl2 , 42.l l.193 l.162 1 . 16 2 . '78 4 e05 0.54 0,01 1.267 1.2,6 l . 194 0 . 16 not e~a . l,281 1.201 4.42 l.99 o.oo 0.10 1.es 40 . Q . 51.0 . 331 .. . . . • 1.200 2 •.80 <0 .. 19 1 •.24 3.36 5 . 82 o. oo 31.o 30.l . l1t295 l .-2 87 1.212 not obs. 6.05 not obs. 0 . 33 l.:64 not obs , 0.44 • 111306 1~210 ealeulated intensity ( ee>rreeted) .{sea.led) 1 11 304 l.266, observed intensity l . 34 2.88 L,33 40 .• ~ not obs. 60118 52~2n o.78 l-.23 0.14 1.16 0.01 l,54 not obs . o. o:; 41.2 l , 29 51.l 43.0 o.•·8'7 o.es 1,60 '73 Table 6 (cont.) oh,e!"Ted calculated spacing spacing A A l.153 1.154 0 1.134 1.120 1,150 l~l36 1 .. 134 1.120 lo107 1:093 l.~096 1.057 1,-047 1.106 1 ~100 1~094 l~.081 1~069 1,-0eo l!t047 Q . 9275 43~l 11~31· 50.e:2 61~01 0,·3 8 <-0.29 0~61 0.17 0~58 0!'62 1. ·~ -05 l~l:3 20.~S 52," l $3 ·~-~ 42~2 21~5- not obs. o.. ss 0~10 0 . 66 s1;2 • 0;58 lt035 0~60 1~0'30 15'.3~.01· '70.0 2$~3 6.2~-~-. RGt ob · • o.so n9t e_ be~ Q~09 0 ,. 43 0:.4:6 1. 84 1~9$ 'Oj.99'70 o,.sa6a o.g:aso o;~'76t · 0 ♦, 94.29 0.41 44,0} 30~5 . -0 ►986S 0~9539 0~29 10, 31· 1 ' o•·ss-• 0~9968 o~-~ase eo.1 : 0;54 1,.000 o •.t1868 1.08 el~l 1., 030 l.~007 o . 9961 o.,, .0_1 oo,i 52_ ,0 ,!&38 l;,640 1 . 0:39 (corrected) Qaleulated 1.ntensi ty (scaled) intensity 0 1~109 1~079 observed bk•l. 0.-.·~ "9:69.6 ' • 0,-:9602 0. -~"". 0-594. 0;.954$ (j:i!9443 0;,9286 ' 6 :·;ft.9271'' Q~922~ ' o.~9eo1 •',,_(' :!:r : 7e_tiJ •53.1 • 00;2 . 43~2 . 4e.s n9t obs~ not obs~ O!P-'02 62~ln 5~(♦, 8 11,. 0 l~o/0 2~60 i2;. 3 4:l.~51 s1:~2 0~56 <'.' 0.19 0~58 not obs,, <= o _.19 o~,03 0~.18 so.01. 0~60 0.1a o•.ss 0,. 16 54-.1 o.eo O.f,8 33~3· 53~2] not obs~ not obs. 0.03 5~~0 0-,91~4 o.. 9078 0.8981 e..0000 ?l,-1 63,0 (>.8909 0~$915 50+3 44~2} 0-~&0)JJ 0~89~$ Q~-8915 0,8856 0 .,;:8842 .. o,.ee4s ; 70.2 0,09 0. 1, 0 . 03 '74 Table 6 (e9nt.) e, b.se?1ved spacing calculated hk.•l apaeing observed intensity calculated (. eo rrected} intensity (ace.led) A J. o ·. 8780 O.,SB07 0 .. 8?89 0.-8783. '78.0~ 42.3 63 . l o•.3o o.35 80_. l 62 .. 2 0.27 not Ob$, -0 .. 20 0 !'8650 0 0,8718 o_ .s11e 0.86$7 0.8599 00.4 not obs. u.02 0~07 o.•8567 5)..31 10., 4 0.30 0 . 3B o .8533 0.6533 72irl o.:sa 0.35 0 08438 o.•a4a7 ·o.•15 o.8~54 0.8354 0.8324 o.s310 11.· •.: 41.·· 81;.,0 ~ o.• l l 20~4 · ,1.2 o. 172 55,0~ (J.l.5 60 . 3 \ 64 ,-.01· 43,,.3 L Q.09 81 .• 4 81 .l o.:.s2 o .8 596 o .a;,25 0.8448 0 .. 8390 Q.,,8270 o .e.252 0 . 0195 0.8254 0.8225 o.el96 ·0 . 67 0 •. 24 0 . 12 01o48 '7 5 obi e rved apa~i:ng A Table 7 Mg 2 Ga Powder Data calculated hk•.2 ~bserved •s paeing intensity '( corrected) A 0 6.900 6/764 4 .. 830 4.827 3 .. 904 3.935 3it454 3«450 3.398 3 . 382 3.074 3 . 03'7 3,080, 2.586 2 .. 556 2 , 415 2. 077 2.553 9.;417 2 .. 39'7 2,300 2 ., 259 2.179 2 tc057 l,.B83 1 . 815 1,729 1.704 1,477 1 . 454 l.430 1 ., 415 ru,.t, obs.- 10. 21 l . ....18 0 .. 93 1 11 176 not .ob$. 20. 1 11 . 2 21.0 20.21 2 . 50 ·2~33 4.80 s . 22 00;3 not obs.• o,oo 21.1 0,24 o.7s not obf.J. 1.•952 4,'72 :not ebs . 1 ; ~02 20 . 3 0,38 1 .887 l .879 1 . 875 1.a10 0.40 so-.Ja~ 0 . 4.1 0.98 "" o.,0'1 o.os. 21.31 o.7a - l..15 4·0..• 0 10 .. 4 - not ob~h not obs. o_.07 0 .• 04 l .oo, 1.00 .. 1 .. a10 l . 578 l..• 558 -1 . 531 , o~oo 11 . 1 20.0 30.l 21.2 11.3 22.0 : ,:(I· .· 1 . 5HJ 1.85 0,.09 0.71 ·l 648 ·• 1., 642 ;':, 0.011 10 . 3 l , 6~1 • 2. 0, 0.11 o.oo 2 . 178 2 . 143 2.054 l.671 1 , 552, o.oo 7 , 01 l..11.699 l . 579 not obs . o.oo 30.0 l , '725 1 , 710 1 . 649 11 •.0 00,. 2 not obs . (scaled} 2 _.255 · l .• 982 1+955 1 .. 906 00 .. 1 10 .• 0 10.1 calculated :tntensit1 l . 518 l . {?13 l .488 1 . 476 l . 464 l.442 1,429 1 11 415 1.iao 22~1 31 .. 0 Sl . l 00 . 4 22 . 2 31,,,2] 4·0 .•-1 7. 5€}' ,. .:, 11,.4 1.50 1 . 1;5 6,25 0;65 Q.Ol ·h48 o.oo 1 •.e4 l,-l~ , 38,.0 20 . 4 not obs. ·O·. sa 0.00 0,.51 0,'. 16 0~16 32 .•.1 4C>.2J (J.35 o. S7 82 , 3 41 .. 0 31 .•3 41 ., l :not Qbs . 2;.41 0 •.1.s not ebs . 0 . 46 o.•oo 21 •.s 32 . 2 00.5 not obs .• 4.es not Ob$_. 2,42 0.'73 o.04 0 . 44 4 . 88 o.oo 76 fable 7 (cont._) ebeerved spicing ealeu.lated hk~l A { oor~cted) 3.l.4 3,.1'"/ 10,5 0,69 50,0 50.l ll~Sl o~e2 not obs111 Ott03 1;32 1.29 ae~4~ 0~14 0~93 20.5. 42 ,0 0!048 0.75 31114 < 0 •.12 A 1.570 1~562 1.357 1.352 l~:355 1~326 l.·3Cl 1,,301 l .. 3'72 1~356 1,302 1.293 1 .290 1,286 1.277 1.27$ 1,.269 1 . 259 1,.:2'78 l,270 1.259 J.,j,,2'79 .,,; __ ., 1._19f!,_ 1 ., 242 41.:3 33,2} 21,5 51,0 not obs. Ot02 0,9B 1,1'7 40.4 not obs. 42.21 51,l 1.09 0.11 ' ,1,.0.f:I, 30,5 50.,3 32,-4B 00,6 5lf2 • 10·;.a 33,3 not obs. .c o.1:; 0.15 c:: 0.13 0.16 1.83 1,69 1,..214 1,.214 i.207 l,l98 1.196 l.150 1.103 l.090 1.083 33.,lJ l . '71 1.156 1.113 32/(13 50 .• 2.1 1.166 L.121 33,0 ~2.1 li.166 lo128 4l!l2} 1,257 1, l.77 1 .. 146 30~4~ 4;0~3 0,30 2.08 lit:81S 1,.215 observed intensl ty calculated intensity (scaled) Sf-e:~ing l,153 -, •., 1401:': ' ~ ·, lfl35 1.155 ·1 ,121 i.127 1,121 141117 1 .. 112 1,111 1.111 ~~:~1 4l.4J 42.3 60 .. 1] 31.5 11.6 1.002 43.l 20.6 52,0] 1 .. 08$ 1.080 0,05 o.oo 0.46 0 .. 63 0.76 0.99 0.23 0 .. 4'7 0.,15 0.20 0,.04 0.20 0.30 0.87 1~43 4340 1 .. 103 1,09'7 not obs. not obs. o,oo 51.3 not obs. 77 Table 7 (eont~) observed spa~1ns c~lcula.ted ap~cing A A l . 073 1 ,. ()71 1,070 1~070 1.058 1.oro 1 •.064 1,049 1-.038 1 .031 1.027 1.049 l;,.039 1,.033 1.031 1,mn 1 ♦ 027 1.024 1 ..mm 1 .• 012 1,007 1.001 1,001 0,9932 0 .. 9933 0.,9910 0~988g 0 ,9882 Ot985'7 Q;,9804 0~9805 0.9761 Q68<' 0. ..... ·a. o.tn5s <f•.97,6() o.,9£65 0.9EH32 0.966,8 0,9660 <J.9.56'1 O~fJ5B17 0,'9560 0.,9810 ()~9469 o.943 4 0.9375 0.9295 0~9465 0.9435 0-.94017 0,9395 0,9577 0.9315 0,9346 o.9304 0.930-4 0.9297 hk ~,.f obeerved calculated intensity intensity ( co~reeted) ( S4aled) 60 . 2u 6_ 2_•••.. 1 40 •.5 ·50.4 43.8 2.l,.6 33.4 52 . 2 61.0 32,5 42.4] 30.,6 61 .l .c o.13 0 . 41 ~ 0.13 o•.16 0.62 O'i22 0;,55 0 . 30 0.38 ~ 0.17 0 . 23 nQt ebs. not ohs. not obs. 0 .. 21 0 . 24 0.01 0.02 0.01 0.29 0.16 0 !' 22 0 . 3"1 0 . 39 not ebfh not obs. 0.04 0 . 54 0 . 56 0 .. 11 0 • .22 not ~bs. 0.02 o.oo e1,s 60~41 net e'bso not obs, n:o t obs . Q.25 0.07 0 . 42 -44,.~1 42 •.5 0.29 0!'63 o~se 0.98 60.3 41 .s 43.3 51.4} 28.6 e1.2J 00 .'7 31 , 6 58,3 l0.,7 J 4i.O «.1} 70.-. "0 53 •.0 0.02 so.s 70,.lJ · Od~l 1 ,. •"':/ 40.6 51.,S 20 -./1 .. ~ .- , #, " ·6 2.,Q 43~!53.:a, } 70 . 2 62.l 0~00 78 Table 7 (cont~) observed spa~ing A 0 . 9231 0.:9165 calculated spa~ine hk•-l intensity (corrected) A o.s·239 obser,.red caleulated inten-si ty (sealed) 32!>6} 0 . 31 0.48 62"4 o•.3a 0,55 net obs. 1.ms not obs. 0 .02 o.oo 536·31 70.-3 ?J.l 0.20 0,27 0,.8887 0.8852 61.4 22/7., not obs~- 0.9220 0.91172 21.-7 0 . 911 0 C .-9 1.17 51 . 5 0.9071 41.6 0 .9045 0.9031 Q.9051 p2.21 30 .7 0.8986 44,.3 0 " 8948 0.8958 0.8908 '71.0 0'1£900 0 .. 8908 0.8798 0;26 0.65 not obs. Oo'24 0 , 80 0 ,62 0 . 04 o~oo ?9 (l) L. Pauling., Proc. Ro y . Soc . (Lon don), A 196, 343 (1949) . { 2) L . Pauling, J .A .c .s . ., .fil?., 542 ( 1947 ). (3) w. Haucke, Natur wiss., gQ_, 5?7 (1938) . (4) Q. Grube and J. Hill, z. Elek trochem ., 40, 101 (1934) . (5) N. A. Pushin and O. D. Micic, Z. anorg. Chem.,~, - 229 (1937} . t- (6 ) K. Weokerle, Disse~a.tion, Freiburg, 1935 . (7) E . Z:tntil and G. Brauer, z . phys . Chem ., IL.,g.Q, 245 (1933). (8) J. J. deLange, J . M. Rob ertson, end I. Woodward, Proc. Roy. Soc . {London ), A 171, 398 ( 1939) . (9) 0 Intemat1onal fab les for the Determination of Crystal Structures, Geb ru.der Borntraeger, Berlin, 1935, Vol . II, PP• 583-585 . ( 10) J . B. Nelson and D. P. Rile y , Pro c. Phys . Soc . (London), .§!, 160 ( 1945). (ll) M. Straumanis and o. Mellis, z . Physik, 94, 185 (1935). (12) A. Taylor and H. Sin clair, Proc. Phys . Soc . (London) , 57 , 126 ( 1945). ( 13) - -t . Pauling, J . Chem . Soc., 1461 (1948). ( 14) Handbuoh Der Meta llphys.ik 6 Vol II, Leipzig ( 1937 ), p. 2~j1 ff . ( 15) M. Renninger, z. Phys., Lpz . , 106, 141 ( 1937). - 11 . ME'!''ALLl ·O VAT.a GE AND X- RAY EMISSION FINE STRUCTURE: \ I A DlSQUS.SIOI • ' ' 81 Metallic· Valence 'imd X-ray Jinission Fine Stnietu.re't A Discussion Theories ot 1!U\>d6?'1'1 stl'Uotural chemistry tin:d m~y severe tests in the f'ie.~d of' 'J rie~als and alloy$• ?n ·add! t101f ·to • ciu.es,tion$ ~ncemin€£ ,org;inary ph:yeJcall .p poperties; d.uet111 ty • • ' ~ ' I condt1.ctivi ty, ate . , there is the central question ae t() the ree.l/H:>n fpr the existeJ;1ce of thE:! vat>idus . 0bser ved . Jtrit~turae t:,pes which can vacy in comple:x.1 ty f~an the eubi~ 9r hel¢agQna.l closest packing of iden ti.CQl spheres to the ~tructu,re ct Na.(?d21 which is reported to be based on a unit eel~ c.p ntain• ing nearly twelve hundred atoms ( l) . One necessary eubdiVision ot the existence question is concerned with valence; in other' words, how many strongly bonding electrons does a given atom contribute in any particular metallic or ii:itennetallic structure, In some eases such classical ehemieal quantities as o;xidation numbers and stoichiometrie proportic.ms seem to provide a. satiafaetery answer; for example, the metallic valences wh:tch are predicted for the alkali and alkaline earth metals from these qtumtit:tes, one and t wo respectively, aeem to be correct. On t he other hand., although in classical chem.is try the v-alenees or iron are given as t wo and three, convincing pb1sical evidence has been .r ound recently tor the belief that the metallic valences of iron ~re as high as ri ve and- s "J.:X ( 2a), (20), (2c ) • I• ron i s , o r course, no t un i.qu.e 88 ' ' in this respect; most e!' the other transition gt-oup elements * apperently have great.el' valences in the metallic state than was heretofere 1Jna.gined.~ In view of the importance, the general:t t y • and the tm .. expectedness of the recent diac~:rvel'.'ies eoneeming metallic ve.lenees it :t.s w:ortbWh1 le to se~reh fer addit,ional experimental evidence bea~tng upe>n them. !he ro llow1ng d:tsellssion is concem.ed · with a P'rQ.rn1$lng phya1 eal method f'or the deter ... m.inf;l.tion of vale.no&; namely,: the analysis or the fine structure or the X-rny em1a$lmn epeetra of' se lids. In the eases 0f such metal$ ae sod!um, e.l\in~:tnw.n, ,ind magnesium• t his method he.a gt ven the CE>'.r.l'eet ~$Wlltf the extension or the method to the mor,e !nteree't1ng trafl$1t1t\>n group t) laments :ts difficult and mus-t be .exemlned wttib ca'.t'e, I,t .1s well known. :that the gross fe.atures of the X-ray • emission speetn:un ef · $. ·given element are not affected by chemical or pby$;teal . dhMges • Closer exmninELti.on of S.peet~al details; how~ver,. :r,evee.ls the existenee of small effects• >;· :::-;-'">- .:,:•- :\: ·~':'."· ··' , ;.. ,., • --~·- ,: -· ;; especially 1rt the highest frequency line of a given •emission ff seiries . • it In th,ese tnst&1e.e s one of the states concerned in '.Ph,e use of th:ts term is that of L. Pauling ( General Chemist~,., Frf3eman and Go•, .e\fAn F~anc1soe , 1947, J>. 146). The elements of the first t~ansitien series, for example, az:ae Sc to Zn .• '"'.§.See, fer ins t~ee, Man.tie- SJ;e-gbahn, nepektroskopie der Roentgenstrahlen1t, Second Edition , Julius Springer, ('Serl1nl 1931, pp, ).60 ff•, 278 ff. • 83 a particular electronie transition is an outer state; one in which the electron 1s for the most part relatively far removed from the nucleus in que~t1on~ and it is unde~stand- able that the enePgtJ of such a at.ate shoul4 .l;>e perturbed , , , . ' : : I,. , . • , • ' ' ' , •• :·. ", '../ .' I - ' ' I , • ! appreciably b¥_ nei€#lbPr~g at~s ~- 'l'be ~er~~1rb~~10}?B . ~l"~~ ... 1ng fre1n ~djaeent ~tom$ artl . l-a~g(i) in met.$.ls . anq. the ene:rgy · ' ' ' ' • ~ · • - ' ' ' ' : ' ." 'i • I ,' • of !\ particultu:~ OUt~_r a t()m!C state -~s, given a _~earl..y continuous set or VAlues OV(U:' ·.e. range Wbi-e h 1.$ of t:n:e . (iJ1der .er . . ' : . i • • ., . . • ' • • several. ~leet~n v~l.ts,: .a~c~ a tlE>~ . ~1ng $~+l.e4 a -~and, : ~ line emitt.e d as: . a re-s ult o't tt'an.tt1tiems. rm~ €>Uter oeeup1.e~ • • • • • : • ' ' ! ~ I ' i • ' ! '• ; ' • state$ in f. m.e tal, ~h0rerQ):"8~, is, mue~ w,!de~ . tban in , tlle caae of the. . eorr~spon. <.\l:ng gatteou. s'. ep~et~ai line;• with a :· . -·· ', . .• • ' ' ';• : \ ' ' .• fine strlicture wh10b !ti depen~-ent ~:r,o:n ~be nat~e 0~ the in t~ra tom1 e• int~UiA~tt,ns ., ,. ·. , ,. ' ' The . fine strueture of a broadened:. solid spe.ctre.l line . . . ' is .related explie!t.lj tt> .th~ energy distribution of outer oecupie4 ele,ctron~~ ·s tate$ . 1:n th~ fol.lowing mat:1ner.. If the . . . .. . Cl). (B) ' . . 3 I (E) a: J/ P(E) N(E), I<.E) ~o' E / Em The number .o r Ot\ter el~ctronic etates in the energy range ls to E -\- dE is ~tven, "b.y nti.)dE; l ,(E} is the intensity er radiation emitted With spontaneous transitions from outer electronic states in the range E to E + cm; P(E) is the 84 square or the ele~trie di.p ole moment integral'; * e.nd Em ts the rnaximtun energy Q;f the ri11ed levels • If the for.m of the function P{EJ were known, Eq:uation 1 would enable N(E) , to be f ·ound direetlw ff!Om the mea,s uremen.t of l {E ) . '!'here !a; however., a theor~ti,•O al l ,i m:lt . to the resolution of the fe~tureel of the d1strfb:ut1on function wh1ch arises from the fini,te width or the lnne:r lev•l, radiation damping, and A:\iger etfeet&J teatlilres f)f the !'un-0tion whi.ch are withi:n l ev of one another cannot be ~e,solved in the X"".ray region which ex-tends trom a tew Anset~tn$ upwards ( 3 ) • the function },P {IJ 1.$ generally di ffi cult to determine, but a eonven.!ent. .recM~\1()11 of the problem can be ma.de< 4 ), lo ) . ' ' ' • ,.• ,' 2- ' • The fortn ef f(i,) ts ,.~ 'fen, . t?1 /f1!! 1/1-x,d.,,-J*-t· and it is reason- . ., E/\ble to asaume t,hat n 7~ .; Ii . ~ .t .ll.e wave function or an inne.r st,a te, ' 1s loeali£ed $.COUt ~•a.Ch r:nleleus w1 thout a.ppree:table perturb- • • .· . . ' ' .. . . ·*~ ~tions fmm ne1ghbe.r 1ng atoms • :tn order to evaluate P(E) , there.f'ore, it !$ . nec~.$ s.a cy to calculate the ei,ter wave f'u.nct.i&n, •1t ' rir , .~Jitly in the 1mroe.d1ate neighborhood of each *:tt haa been as$Ullled by most workers that the probabilities of trans1t1ona whieh \d.ielate the eleetrle dipole moment se leetion rules are negligible , ;;,.,..;} ' . ' ' If the outer ele~tronic state~ are €1,6generate, P( E l sh0uld be repr,e sented: b~cr a we(ghte,<i average 0£ term$ ot this type, ~.Jt't'his assumption mar be open to ctuestion; Ps.rra tt( 6 ), f(jr exe.mple. found. that the halt widths o f the Ko< lines of' Mn, Fe, Gr, and Ni een be va.r:te.d by as mueh es 83 pe.:reent by al1oy1nti• 85 nucleus. J.n the neighborhood of a nucleus as a series of functions 7/{., ]4:. can be expanded 7? 't ·... , the !, £, £ wave > functions of the free a tom" • Thus, 7j; == As-;j{ + Arfr +Acl~ -r (3} !f now 7/1: is $\n ! function, es it 1 s in the ins ta.nces of K or t ~• spectra,. 1 (4) (51/( r ,i; d ,,.r r,: A; (E) = = P(E). For LII Q!' LIII $pee,t ri,: ( 5) P( E) = { ~F 11; (E) +- ~p A; Since the functions P ps (E) 11. , .P . , etc~ are to be regarded as in• sp dependent of the enet•gy of' the outer wave func M,one, and ae known for each given t ype of atom, the problem of the fonn of P(E) has been reduced to the determination of the functions Alkali and Alkaline . Earth .Metals In the case of metals euch as 11 thum, sodium, beryllium., magnesittrn, and aluminu.m, low energy eleetrona in the bend fonned from the outer a torn!c ! states, the valence band, ean be regarded to move in e. field of nearly conatant potential energy within a given specimen and in a. £1eld cf very ni.u.eh higher or infinite potential energy· outside the specimen. -- .. --·- - ... -~ .... -....... %1'"'or an explanation o.f these symbols see Compton and " D. Van Nostrana • • Allison, nX-Rays •in Theory an d Experi ment, Co., (New York) l.935, P• 630. 86 The wave f'unet1ons in thi.$ instance ean be given the form J exp \ Z 1f i. ( ~ •,!) (? ) wher,e the vector £ loce. tes a parti Qular point in the metal specimen · with reference to an or•igin wbieh may be arb:t trtwily cliosen &1nd }! is a veeter having a discrete set of values which depend upon the dimensions of the metal specimen~ (For met8\ls based on a eubi c lattice the speebnen 1.s usually considered to . be a cube aS. a matteP of conveni~nce.) The distribution function hl(E) has a pai"abol1e f'enni (6) ' ~/; ½ N(E) = 1..1rt1 (zm) t. (E- E) it h'J' N • wh~re M is t .h e e. iPmie weight, p is the density, N is Avogadro's number, m· ts the e.leet~riic m.e .ss, and 8 0 is. the ener~r of the lowest, V't:ilen ee st~ t ·e . As the ld.n e,ti.c energy of the valence e le-0trons increases , heweve:r , a J?egbm 1s· ~ltlcountered in whieh the electrens, eons1{tered as de Br~gl:te. 1$;aves; can undergo e. Bragg reflection as a res.u .l t or interactions w1 th the cr:rstal lattice; in th:ta region NtEl t"luetua.tes markedly from the parabolic fonn or Eque:t1on (6) . 'l?hta 13:. ituat:!.on <Hm be treated mere preei$ely 1f' a ! ~pace i .s C<;ltlceived in which veeters drawn fr1om the origin to a set of' point.~ based upo.n an appropriate lattice are the vector$ !! . .11' tche metal ape e!men is a cube with edge length ' La, such a lattice is cubie with a period , • L~ along each al\tiaS, and the corresponding compone:nt,s of k have the fonn.- Lo.. A , where n and L are !nte"ers . t:I> We shall 87 assume that the crystal lattice is cubic; the crystal lattice is aligned with the. ! $ea,ce lattice in such a manner that, the points { n, n ,._ n3 } in !i spae~ fom the re•ciproce.l et.' a,' CL lattice of the Ct"'Jstal. An eleetron in the state exp\z 1f L('!:.!) J can be re.f l-ecte<! ac·eo r41ng to tne Br~.gg eond.1 tion fr-om the Cr'Jstallograph1c plane ( l m n) if k touches a plane which f" is at a d1at:anee W:t ~'2:t + (~tJ -t1/z from the or1gtn, wt th a · unit normal veeta?" 1s):i- + (ms)k+ (Y75)2.,]-½_. [ 1.vf + ~ + t:!.4-[( Za Za. 7-a.. Za • ~a.. ~a.. where ,2 1 , 22 , and 1? 3 are unit veetors directe~ along the lJ, f,. axes of the ..,.. k lattice. •• ' :tr the .angl~ be·t ween t.ne veet.or k .. •. ' and the unit noma.1 18.· . .veeter · . , ., 1f z _' 0 ,'. ' ; The de Broglie wave- len.gth of the e,lectz-on eonaid.ered is l ~ r' and 1 ts momentum is therefore (8) E-E = (9) 0 ~i. "= hz f ~I~ . h;z. ~l'Y'I Z.m :1- zm St1' e Therefore; s A= Za. sin 0 0 ,._ %. ( ;e.t-m+Y'l '.2.)1/.z h I~ \ • fhus [(is\2.,+(ms ).4,+ (!] f ]-=K 5 7-a. l Za., = z d. • \J.f½ 1- . zl'YI X s,n 0 -;fr>'1YI Since the crystallographic plane ( l m n) is perpendicular 1. s f. +- ~ '½. -r ~ -h ) ~a.. Z a. 2 c.. from the properties of the reciprocal lattice, Equa tion (9) to the veetor ( ia the eond:t tion for Bragg reflection from ( 1 m n). If i; (J!) i •s plotted age.inst J! I , the k' e beil1g collinear, a parabolic curve, given by »~quaticn (6); is obtained until k, A, 88 drawn from the origin in k spe.ce, is in the neighborhood of a plane correspondin g to e. erysta11ograph1c plane ( l m n) . In this neighbo~hood the curve flattens sharply a.nd has e. discont1nui ty when !f_ lie s in t h e plane . Ask increases, ~ the curve, s •t ill flat, qtl ick1y increases in slope until the parabolic law is age.in satisfied. This behavior is in ~eneral repeated for eaoh plane as !5. 1s further increased. Discontinuities along different lines in! space may overlap; that is, energies which a re forbidden an eleetron travo.ling along a certain direction may be allowed. for anotherdireetion . Mott and Jones< 8 ) have shown that in t he case of a monatoml.c solid and loos-ely bound electrons the ma.gni tude or each d1scont1nu1 t y is roughly proportional to the structure factor for Bragg reflection from the corresponding crystallographic plane"' Dis continu1tiea are non-existent when . corresponding ·$tl"l.1ct,ure factors e.:re exactly zero.,. .' In the- case of. a polyatomic sol1d this proportiona.11ty is assumed , bl.ft' ma·r 'be in serious. error( 9 >' {.lO)., Tb.e pla.l'les of energy d!soontimd.ty in _! space form a series of polyhedra, concentPic •a bout the or1g1n. For a primitive oubie latt1·c e, the firtJt of such polyhedra is generally a cube defined by t he oryS''.tallographic planes.: l a J • This polyhed ron is then called. .the first 1:3rillou1:n /0 z.one end its boundaries Bri lloutn zone boundaries. The space between the dodecahedron defined by the erystallographie f planes • 110 } and the cube 11001 forms the secemd Brillouin zone, and so on. !n t h.a pt'1m1t1,re eu.b:tc case, wi~h no vanishing strueture fact,ors, each zone contains 2N. ( twice the number of unit eells 0r the ¢'\:Jbie bl0ck) electronic states, . ·when spin degentHJaey is taken into account . . When the electrons of the valence b~d are loosely bound , theo!'y predicts e.n ~lE) t\mo.t ien of 1;,he types ehown in Figure 1. .At small kinetie energies the curves follow the parabolic law, Equation (6), As · the ki;netfe er;iergy is lne.reas,ed and ~, drawn from the origtn in ! Sjt\.~~ j appr.oe.ches the first zone boundar1r 1, the N(EJ eurv:e r!ee:s Sllal'ply t .o point i' . At a point in the ne:t.ghborhOQd o!' P • the tote! number of states with lower energ~es a~~ given by thEt ntunher or poin·ts of the k 'S£;;. !i'a:c:~ ·1atti~e whieh lie v11th1n the inscribed splre·re of the .,_ l first z.o ne poly,hed.:ron ( l.Ji)" As· the energy is increased the N{E) cur'\1e falls sha~l.y te point K because ~f the energy discontinuity •a t the zone boundary. The N{IO curve risee sharply from K because of the contri bution o"f states in the second zone. A detailed consideration of the function P(E) ind i cates that points P and K 0.r .,f:t&rur(~ 1 cannot both be found in one emission bsnd( 5 ) , . and, therefo!"e;i that the K e.nd t 111 emi~eion curves of an element for v1hich ll(E) is appreciably greater than zero at point K should differ greatly in appearance. Aluminum and. magnesiunt both have filled zones and p t ··• •• .. •· N{E) {a) .... -· . Figure 1 LiK 1 t l(E)/ v 3 Np(E) 0 I 0 I 3 4 5 6 7 8 O· 2 4 6 8 10 12 14\·olls 0 4 6 8 10 12 14 \"Oi l s 2 - - - - - -- - E - - - - - - - - - - - - ' > - Figure 2 I AIK Mg K 91 in each case the K and LIII emission bands differ in ap pearance, as oan be $een in Figure 2 . Further consider- ation of P(E) shows that, a t low energies, I{ E) should very 3/2 1/2 . • · • as (B - E0 ). and _(E ,.. E0 ) . . in the K and LIII bands, respect~'tl'aly( 3), (5 ). On the other he.nd, the curves .of Figure 2, especially the Lill curves, extend toward very low frequencies with a small but finite ordinate . Thi~ be • havior. has been explained by advanced , cons idet>ations of the radiation processes of solids which are quite beyond the scope of t.his discussim.1 ( 5 ).. The high freq/eney edges of the bands shown in Figure 2 have nearly infin1 te slopes in . , a.ccoroance with Fenni-Di:i"ae statistics . ' ' The widths of the bands of Figure 2, aJ,I mea.~ured by Skinner(B); are given in Tab le 1 togethe-r ·with v_a1,ences ~ calculated fran them with the use of Eque.tion (6)" . Table 1 Metal Band Measured Breadth ( ev) • Li Be Ma :Mg Mg Al Al ~!. 4"1 ± o·. 3 14 .a.:r o..5 3 .0 ± 0 . 2· 7 . 3 :t l 7 .6 ± 0.3 12 . 7 ± l 13 . 2 ±·0 . 5 Calculei. ted valence o ..7·7 2 . 02 0 . 88 2 . 02 2 . 14 3 . 32 3 . 52 nSkinner obtained the low fre~uency edges of the bands , that is , E. by the linear extrapolation of the LIII tails to zero; K tai 0J.S were extrapolated by the t:.se of an E2 Zorm . This seems to have been done on the basis of expediency . 92 The agreement between observed valences and those ordinarily assigned to these elements is very good, even in the eases of ber--.rllium, magnesium, and aluminum, wh ere marked d'~viations in the form or the N(Ft) cu. rve from that · of Equation { 6 ) must occur at :filled zone boundaries-. . .• 'l'he only investigation of alloys between the me tals of Table l has b~e:n made by Far:tneat1i1 who studie·d the alleys ·u - Al mS3 .- • d Al 11" .- ( 12 ) • 2 an · •. 3mg2 mt,., ,i:.u,e L IlI • b an d $ 0.1, • "" the t•WQ com:ponen • ts we re found to be : superposable ~n both eases, indicati~g that an electron of the valence band can approach either type of a.tan with nearly equal p1.. ·o bability~. The observed widths of the bands of the two alloys are iden tie•~ l, 11 •.5 :¼.1.5 ev, ,, which differs somewhat from the eorresp~:mding widths for Mg3 Al 2 and Al 3Mg 2 calculs.ted w'~ th the a~swnption of n?nnal ,raJ. ences by application of Equation (6),; 9,0 ev and 9 •.4 ev; respectively• T.r an.si tion ~roup· M.etal$ ·- The oo.ter electrons of transition group metals are more ' , tightly bound thR:n in t he case,: of the alkali or alkaline earth metals and. it ·would be unsound to ' aasume that the distributton function N(li!) follows the parabolic law of ' . : Equation (5) eve~. at low kinet~c energies . , Various theoreti • cal calculations of N(E) have ~een made for transition meta.ls with the ~tse of the mol~cular orbital approximation, and some degree or experimental verification has been claimed. 93 Prominent features of Jli(E) ¢aleulated for copper by Rudberg . and S.l a ter<l 3 ) can be placed in correspondence wi th prominent features of .K $mis$ion and absorption curves for copper, as h&s been shown by .Beemrui and Fr1edman(l4:): on the other hand, the exper-imental verification of the . •. Manning-Cbodorow distribution function . f.or tungsten not been carried through, aa suc.e essfully( l&) . (15) has The theoretical N(E ) curv~ of Ru.dberg and Slater shows the,t the energies of all the outer, . that is, 3d .... and !ls~ - electrons lie in a ccmtinuoue band . for cepper, . but thts does not mean that all the outer electrons are necessarily valence electrons . There is good evidence~ in fact, that ·of 5.,4.4 pf the eleven outer eleetro:ns of copper are ve.lenee. electron.s • .This conclusion is a re$ult of a general soheme , for the dist:ribution of the outer o lee trons of trans 1 ti.on . group m·e tale whi ch has. been propQsed by Pauling(aa). In th1:s distribution the outer ele(}trone are Q.onsidere<i to he divided into t wo overlapping ban<ls . The first band contains electronic states wbieh cover a large range of energy wtth ~ small ensity ; the second band has a small energy t'ange wi tn a high density of st~tes and overlaps the first, band at the high energies . The electrons of the broad band are valence electrons. while those of the narrow band are tightly bound to ind ividual atoms and do not contribute appreciable to binding . In fact , although Fenni- Dirac statistics governs the filling of the states of the valence band, the electrons of the narrow 94 non - bonding band tend to remain unpaired with parallel spins 1n accordance with Hund' s .rule of maximtun multiplicity and are responsible for ferromagnetic prope~ties . Now the p!'esenee of a narrow band of tightly bound electrons overlapping i:r\ energy a broad .valence band might well be ex ... pected to render the experimental determination or the distribution of.' occupie<;l states · 1.n the valence band d,lfficult or impossible • .ln _addition, the effects of gross changes in crys tal structure upon the distribution of states in the valence bend -001.1.ld be at least partially obscured by the presence of a relativellr unehE.'l'lging distribution of tightly bound s ta tea .. Furthennore; .c hanges in !.(.E ). where no over• lapping occurs mi~ht not be apparent because of the relatively low density of oec~pied states in the valence band . The nature of the emiss i on band studied is important, to be sure . Pauling has indicated that the states of the narrow • non bonding ba~d are f.onned from atomic d states< 2 a>; therefore , it might be assumed that,, if a. K errd.saion be.nd were studied ; the complications described above could be avoided because of the dipole selecti-o n rules ~ There is e. ce rtain amount of experimental evidence, however, whi ch indicates that this method o,f attack is unavailing . Bea!"den and Fried.man< l?) have s tud.i ed several alloys in the copper- zinc s:rs tern and found. only;_ small systematic eban ges in K spectral detail with eompqsiti~n_, although correspondin g structural changes are g ross 8,nd unsystematic. It is quite possible then, 95 that the quantities A8 end AP in the expansion of Equation (3) are ra.ther appreciable for the wave functions of the non-bonding bend . A certe.in percentage of the observed intensity mi ght reen.tlt from electric quadropole transitions v<ihich in some cases have a probability of a.bout one - tenth that of dipole transitions ( 18) • ; · t he selection rules for this - - tJpe of. transition of course, permit d - s tra.n si tions e Studies of transitio_n metals end alloys between transition mBtals, therefore, wotld seem riither fruitless as sou rces of information concerning the distribution of occupied states in the valence band; this conclusion may be premature because of the small amount of experimenta;t. information available, l..-ut i t is certain that such studies can be rr.ade fruitful only after formidable complications have been overcome . It is possible, however, to gain information concerning the valences of transl tion elements from spectral studies without regard for the diffi culties in the interpretation of tranei tion element emission bands. In order to explain this statement let us consider the probable nature of the emission bands of both components in an alloy of the copperalvmu-1.vm :aitokol s:1 stem . The copper emission band wot,ld be expected to result from transitions involving states of both the broad valence band and the narrow non - bonding band in accordance with the previous discussion . On the other hand , it seems plausible that there should be no appreciable 96 contributions to the alum.im.1m emission band from transitions of the copper non-bondin g states. Since there are no non- bonding states centered a bout aluminum atoms in the energy region concerned, the aluminum emission band s h ould be the result of transl tion from valence s t a. tes alone. Consider- atior1 of the observed f'onn of the aluminum emission band sh ould thus lead to some information about t he distribution of valence states in t he alloy. Fa. rinea.u ha.s investigated the fine deta ils of the outermost emission bands of aluminum-copper and e.luminum- niclrnl alloys ( 19) . His spectral curves are shovm in Figure 3. The copper emission bands shown a.re L c,(. bands; that is, the transitions in,,o lve outer occupied states and inner atomic fl. states. Accordingly, the ma.eking effect of the outer narrow non-bonding band would be expected to be large; however, considerable changes in spectral det.ail with changes in aluminum content are to be seen. " (Curves I to VII contain o, 19, 35, 49, 60, 82, arid 96 atomic percent of aluminum, respectively.} No attempt will be me.de to discuss the features of these curves. The nickel £ex. emi$Sion bar1ds, in contrast to t hose of coppe r , sh01i1.r 11 ttle detail, as Vl.r ould have been predicted in the present discussion. The aluminum K~ emission bands of Figure 3e. possess a. considerable amount of structure which suggest fluctua• tions in correspon ding :N ( E ) functions at the surfaces of important Brillouin zone boundaries . It is of interest, Al eutectic 19 at.% Al 1555 1550 eV fS4S Al Kp" 925 e V 930 Cu L°' Figure 3a: Copper-Aluminum NiAl t a I , 1555 1550 I t I p 1545 1 1 , 1 sso eV e V Al KfJK Figure 3b: Ni Loe Nickel-Aluminum 98 accordingly, to find the important zone boundaries for the valence bands of each of these alloys. following manner. This is done in the The imports.nee of a given boundary .is est ima tad from the magnitudes of' the s true ture fa.c tors of' the set of planes involved; this procedure has been discussed previously* • Next, the extent to which an important zone is filled by occupied states is estimated by calculating the number of' states in the inscribed ephere of the zone polyhedron. If the number of valence st&l.tes :ts greater than the number of states contained by the inscribed sphere» the zone is at •least approximately filled **•• At the value of the energy for which the occupied states just fill the inscribed sphere, the N(E) curve rises to a maximum, the points P of Figure l (B) ~ When the next zone overlaps appreciably end the region of occupied. states extends well into it, the points of discontinuity K of Figure 1 are contained in the filled pc{tton ◊f the N(E) curve. In general it is difficult ... ')f' In many c.a aee :1 t is el ther inconvenient or impossible to calculate structure f'actcrs. The alternate procedure !n these cases is to uae obset"Veo. intensities, if possible, corrected with frequency factor, and, in any event, corrected with the appropriate Lorentz and polarization factors • . **The number of states in e.n inscribed sphere is calculated from the volume of the sphere and the dens1 ty of ate.tee in k-ipaie 1 2V, where V is the volume of the c~Jsta.l paral• Ie ep ped with the edges case. La, Mb. and Ne in the orthorhombic 99 to calet.1la.te the function E(!f). In the discussion of the features of the aluminum bands o.f Figure 3, the free electron approximation Will be used to calculate t,he positions of" the points P, with the 1:--ecogn1 tion that this approx ... imation may be quite c:rude. In Tables .2-? the calculations oi1.tlined in the preceding paragraph a.re given for the substances Ali CugAl, CUAl, Cu Al 4 , 19% Al-Cu solid solution ( o{ ph$.se) and N! Al. In 9 the first column of each table the incUcea of the observed reflections are given; in the second.., the. uncorrected ob ... served intensities; in the third) the Lorentz and polarization factors; in the fourth• the frequency factor; in the fifth, the number of states per unit cell ( .e rystallogre.phic unit cell) in a ephere tangent to the corresponding set of planes in the first column; in the sixth, the radius of the o -I sphere in A units J and in the seventh, the value of E(g.) - E(o) calculated from the r.adius , ot the inscribed sphere with the use of the free electron app'l"OXimation. The number of occupied states 1-n the valence bands of each of the alloys is calcule. tad in two ways. , The first value is obtained with the use of 5.44 for the valence of copper and 5.76 for the valence of nickel; these are the valences proposed by Pauling (2) a ~ ThEl second value is de- rived from the assumption of. the valences l and 0.6 for 100 copper and ni ck el respectlvelir , following Mott and J ones ( 20) ~i- . I nspection of 'ra.blea 2 ...7 s hows the,t important Brillouin zones of the different s tru.ctu:res a. re filled w:t t h either set of valences. In general, howeve r, there is better co1'- respo:nden ce bet.v1een the fea tures of the emission bands and t he fi lled.-zone scheme obt,a ined with t he le.rger ,ralences . '.r'.ae form of the aluminum emission band f or the i nte:rmetallic cor:1pound Cul1.1 2 ~ugges ts ·t hat at least two, e.nd perhaps three important zones are filled . From Table 3 it is s een t hat the insc r1. bed spheres •of the two mo$ t ilnportan t sets of p lanes {22 •0J , t ll•2! , and t 3l • OJ , l_20 •01 can be filled iHith either set of valences.• 't.!1.e s.ph ere i ns cribed in the poly ... hedr-on bounded by the planes l3l •OJ $ \20 • 2) contedns 2? states per unit cell; the number of occupied states \'IJ i th the valences of Mott and Jones is 28, hardly more than enough to fill the inscribed sphere" It might be reasonably expected,. tberefore, that the emission band would ernibi t two instead o f the observed t hree maxima, if t he idea of ____________ __ ,.. .. ·*It has recently been shown by Pauling< 2c), (2d) that the valences of cop per and nickel c an as sume a variety of i.ralues , the ma."timum values being about 7 .o and 6 . 0 respec t ively . The use of the valence 5.44 for copper has been previously justified for the ~ • brass structure, C\.tgAl4( 10), and the inter- atomic distances found 1n the structure of CuA12 (21) ~re consistent, with valence 5.44 for copper., The structure of Cu.Al is unknovm(22) and a che ck of the valences from structural data is impossible.. It seems lilrn ly that t he v alence of copper in the 19% Al o<- - brass structu re is s.lso 5 . 44 . 101 monovalancy for copper is correct. !f the valence of copper is about 5½, however, the m.u nber of filled states greatly exceeds the. t contained by the second sphere, and three maxima. in the emission curve would be expected, the third maximum resulting from the abrupt decrease in the number of filled levels in t~e region of maximum energy due to Fermi-Dirac statistics~] The positions of the two maxima which a.re to correspond zto the filling of the two inscribed spheres are ~r ealculated '.,to be 8.6 and 19.1 ev; respectively, from the lovr frequency edge of the emmission band. Ctn calculating these positlona the free electroriapproximat:ton 18 assumed e,s before stated• with . the additional aS$Ulnpt1on that the low frequency edge of the em1.ss1en • bend correspond$ t_o a trans! t:!.on f:rom the lowest state o:t the valence bMd.,) The reapeetive :observed maxima seem to be about 7 •.5 and 10.0 ev from the low frequency length edge, ,1£ we may take the edge • ae· ~giV81l' .bY the long wave length · extremity of thtf "curve '·in Figure 3a*• The corre~pondence between calculated and ~This se.::ms .a re:...i! :;::nab:J.~ choice, especially from comparison with the indicated· edge .for· the band of 100~ Al, the point A in --Pigure 3. • Farineau does not report his low frequency edges exp11c1 tly" I!e does state, however, that the observed maxima in question lie betwt?en 5 and 6 ev and at 8.5 ev, respectively, from the low frequency edge . An inspection o:f' the curve foF CuA1 2 given: 1.n Figtire 3a. shows that, if the pos l tions reported by Fa.rineau are eorrect, the low frequency edge would be sit• us.tad at some distance to t he left of the indicated extrem'.t.ty • The values reporteq. by F'arinee.u thus seem in error.; one .explanation tor this discrepancy might lie in l1'arineau's. desi re to ascribe the first maximum to the presence of a filled sphere inscribed to the set {20•0} , {.00 •2, at calculated energies . 4~2 - 6.1 ev. As Table 3 shows (20•01 end \00•2} are much less l?rominent than e:tther t22•0l , tll•2} , or t3l•O} • t20•2} • 102 observed positions is as good as can be expected in view of the roughly approximate nature of the calculation. The cal- cula ted. width for 45.8 occupied states per unit cell is 14.7 ev, which seems somewhat high. It is dif'fieult to choose important planes in the ease of CUAl. For one thing, the. 1ntens1 ties of the planes have been reported. on a qua.11tat1ve basis only. Fox- anothe:r-, there are a large n'U?nber of closely spaeetl, heavily reflecting planes• An appreciable eoncentration of important planes seems to occur in the vieinit;y of th~ planes 200, however, and the first marked fluctuation in the N(E) curve may occur e.t this point, calculiated to be. 8.6 ev from the low frequency . • ~ . The observed m~ml.)m is abput 8-.0 ev from the edge, edge~ in reasonable 1:1sr~i)!nent • :Che in$eribed sphere to the planes 200 contains 49 states, and is 1"1llet1 in either valence scheme, If eopper is monovalent in this structure, however. it is hard to understand why there are two very pronounced ma.x:tma wt th one broad m1ninn1m in the observed emission curve. The free electron w:tdth for 84 ooeuptao~ states per unit cell is only 10.4 ev, which is about the po.s 1tton of the minimum of the observed cur,1e. l'he assumption of the higher valence for copper can exp la.in t h e shape of • the curve very well -- there is again a discrepancy, even rrore pronounced than in the first ease, between calculated and observed widths, 17.0 e-..r and about 14 ev, res pectively. The Brillouin zones of the Hume-Rothery o •phases are well kno'Wll ( 10) • The first important zone is formed by tb.e 103 planes t411J , t3001 , and the second by t he planes t 6001 , The inseribed sphe:re · ,,r the first ZQne would just \ 442 J • be filled, wi th the :monova lent as.s.umption; both inscribed • spheres are filled with t he use of the valence 5.44. The caleulatecl position · of the ma~irnum which corresponds to the f illing of the first $phere ill (h~i ,ev~ the valculated position of the see.ond maximum. is 17 . 8 ev. 11:he observed emi.s s!on eurve shows tvro maxima, one at a:p preximate.ly 8 .• 9 av and one near the high frequencr edge et the curve at about 13 .. 5 ev . It would again be. diffie-ult te understand the presence of twe 11u1:g:tma in the o'.bSeMed curve if copper were assumed t0 be monovalent s-ine,e the first i _n ser!bed. sphere contains 80 of the 84 ·o eeupi,e d - -s tates " A.g ain .• the diecrepancy between the obeerie<t wi.dth ,, .e .a . '1 4•6 ev, and that ee.leulated with • the 5 . 44 valency; lt ,fa', is quite pt.'Qneunced . tn· orde.l" te c.a~cy ·cut · a di$.eU.$iion of the 19~ Al-Cu solid sol:ution• it Will, be 11e.~ etsaey to assume that t he method of dete!ffi!hing t.he importance of a given plane, 1 .e . the magnit,u,de ··.of the energy di.~continuity in the region of !~:,.$12aoe a.djQining it;11 is not to be altered b~cause 0f the existenee of a disordered arrangement of a toma. in the airueture . '.Chis. as$umptien seeJ!iS plausibLe beca~se the . . ' relative magni tudea of the stru-cture factors are unchanged. by the ex1stenee of a solid so1ut1o~, 'Che only planes with' a finite structure fa~tor are \ lllJ , \ 200} , l220 l • etc . 'l?he sphere inseri bed to the {lll} planes 104 contains 5.4 of the 5.5 occupied states calculated w:tth the . Mott an<;i Jones valences. It vrould be predicted, therefore• that the observed emission curve should exhibit but one maxiinum ., actually two are observed, at 9.0 and 14.0 ev, reapeetively. The appearance of the emission curve is thus more re.asone.bly explained with the as.aUJnption c,f 5.44 ( 5½) vs.lent copper" Again, the calculated position of the first maximum, 8.4 ev, a.grees well with the observed position. As before, with the high va.lent copper, the width of the emission curve is less than the width of the filled portion of the free electron N(E) curve J the discrepancy between ee.leulated and observed widths., 20.0 e,, e.nd c.a. 15 ev, respectively, 1s very large. The general appearance of the aluminum emiasion curvee for the copper alloys . is, then, more reasonably explained if the metallic valence of copper is .a ssumed to be much gree. ter than the value l; which 1A as commonllr accepted e few 1 years ~go. It is aom.ewhat difficult to estimate the precise valences exhibited by copper in the various · etruetures, however. With 1ncreaa1ng copper contents the free-electron widths caleula.ted with the aasumpt1on or 5.44 (5½) ve.lent copper become progressively larger than the observed wHlths, which are .approximately conatant. This would indicate that the valence eXhibited 1n eaeh case is much smalle1~ than 5½, being close to 311 . On the other hand, as is indicated in a previous footnote; there is reliable evidence that the valence of copper is a.pp4'¢x1mately 5tt in the structures of 105 C u A l a n d 2 ou Al 4 • tt m a y b e t h e . t t h e f r e e electron a p ... 9 proximation is completely invalid here; this seems improbable for the low energy region of the band at lee.st, since the calculated post tion of the firs~ maximum .:tn the emission curve agrees well with the ob&e?'Ved po~li tion in every case• Even if the ft'ee electron approxlmaticm la -Vf:\lid exe~pt in regions close to zone boundaries• the large deviations from the p#Jlrabel1c N(E) ,c urve at points corresponcling. to the filling of important zones may cause the large diacrepancies found between eel~ul~te<:l e.nd observed widths. tr this v1ere the ca.sei it might tteasonably be expected that the widths of' the various e.lumimim . curves would ,racy irregu.la.rly end. markedly in view of the lat>ge st!\l.ctural differences ex• hib:tted in the series. An explanation of the dise.greement between calculated and observed widths which should be advanced at this point lies in the recognition that the one-electron approximation, which has been J.mplici tly assumed throughout the discussion, may be inadequate to ex:plair1 completely the features of the various emission cur,tes • It is possible t~at ·, although in each case the width of the filled portion of the N(E) curve is reasonably that calculated for the free--electron fem, exchange effects prevent e.leetrona in one region of the band from approaching the aluminum atoms frequently enough to give rise to much radiation of freciuencies corresponding to the energies of such e. region, Thus, if electrons 1n the 106 high energ-J region of ea.ch band were prevented from approaching-~ the aluminum a toms f1'-..equentl~t, the observed widths of . the corresponding emiseion curves would be smaller than 1.f this mechanism did not obtain. In the present explanation this is the effect postulated to exis.t. The observed widths of the emission bands indicate that no more than about three of the 5½ valence electrons of' r.ovper can e.pproe.ch the aluminum atoms frequently. The remaining 2½ valence electrons, situated in the high energy region of each b~d,. are forced to remain in the neighborhood of copper atoms. Such a sit- uation seems fairly plausible although a specific theoretical justification for it is lacking. It is re~.sone.ble t .o expect that a high energy emission cutoff due to exchanb"8 ef-feot should be less abrupt than e. cutoff of the type described by the elementary theory of the introductor'/ section. Indeed, Figure 3a shows that &s the copper content is increased,, the slope of the cutoff generally decreases, In the alu.mtnum ...hickel series the alumin:ll.rn curves are for the most part t~.ati,reless • It is likely :that many or the composi tic:ms are in two-phase regions; the only kno\\'n struct'lu•e is that of NiAl (Curve !V). W:lth either valence for nickel, the first important inscribed sphere, containing 2.8 occupied states per unit cell., is filled. The appearance of the emission curve indicates the filling o f at least one inscribed sphere• The width of the emission band is again nearly the same as that for pure aluminum. 107 Cone lus ions The study of the fine structure of X-ray emission bands of metals, alloys, and in tennete.llic ec-mpounds seems to offer a. chance to gs.in valuable info:rme.tion concerning metallic Yalences. In many eases. however, it Will be necessary to have an extremely refined theoretical description of the situation in metallic structures before the interpretation of the spectral date. can be made with confidence.. In other cases there is some hope that the roughly approximate picture, that developed in this discussion, will be effective in the interpretation of the data. In any event there 5.s a. gree~t need for more experimental informat.,ion; the study of alloys and intermetalllc compounds can be said to have just begun. a) 0 .-i intensity Table 2 Alum1num Stru-0ture: A 1 t-:,rpe Number of occupied states per un1t cell: no. of s ts.tes 12.0 radlus of E(t)-E(d (ev) sphere 6.9 hkf in inscribed sphe.re 0.21 9.2 (X•l) 5,.4 0-. 25 18.3 Prequency factor 111 8 -.4 0.35 factor zoo 2.4 Lorentz & polarization 1-zo ()) 0 rl Structure< 23) •• nl8. 4h,.' - o - 6.054 A; e0 Table 3 0 Cu. A lz a : e 4 .864 A; Z . cA- ... J 4 no of states 1rt inscribed sphere 0.12 ~ 2.4 8 .. 6 45~8 (Pauling); 28 .o ,(Mott e.nd Jones) factor 6 .• 7 0.16 0.21 0 .. 24 200 420 660 • 3.8 (ev) E(k)-E(o) Lorentz &. polarization factor 4 14 10.1 r adius of spher~ intensity oO 4 20 0.26 not oba. 66 frequency . 440 28 16 27 o.•31 not obs. 13 16 4 8 8 8 45 0.33 Number of occupied states per unit cellt hki 11.0 10•1 20•0 9.7 8 55 22•2 32•1 21•1 22•01 910 6.5 16 not obs. 11 •21 31•0} 20·2 30•1 not obs. 86 not obs. 5.5 10•3 40·0 31•2 0 rl rl hki 020 021 002 110 111 022 040 003 130 023 132 113 043 150 200 024 133 151 060 061 220 221 202 Table 4 CuAl . Str-ueture< 22): . a -~ 4~ .ps A) . b : 12.. 0 A, c :: 8 "~63 A; z =., 16 Lo?'entz & factor frequeney • no·• . of state.a in inscribed sphere s_ p - here (.A-1) re.dius of 0 -.12 0.21 0.24 0.24 0.2~ 0.24 0.25 0 .• 25 0~25 0·~27 8,.6 E(k) • E(o) (ev) Number of occupied states per' unit, cell;- • • 155 (Pauling)• 64:. 0 (Mott and Jones) • inten.s1 ty polarization factor unity throughout 4.l 0.13 0.14 0.15 s M MW MS 3.2 w s s,,, MS 49 0.17 0.17 0.19 1-.4 1 -. e 2.2 s s MW MS w s vs MS VS 5 s w s s M r-i r-i r-i hkf vv1 , f .e .ctor 1.0 . 1.2 1.3 intensity Lorent.z & polar-i~ation 222 044 M ,~, s 005 134 MW s s MW MW s w s w ~'Y 240 203 063 223 154 135 081 224 006 rec.Hus of (!-1) sphere 0.28 0.31 inscribed sphere no of states in Table 4 (cont.) .frequency factor unity . throughout 1.05 .. 0.35 •. 0.35 0.30 150 (ev) E(~) - E(o) w r-i r-i hk :R 210 211 300~t 221 222 321 4101 322 4111 330 331 420 421 332 422 510} 431 511} 333 5201 432 Lorentz & factor ...-Pre <•t· ·i l enc"· -· ,) 24 24 6 48 6 24 24 48 24 24 24 48 24 8 24 48 24 24 24 24 24 12 8 48 24 1£' 16 24 24 24 5.0 9.0 11 25 53 43 p olar•i zatlon fe.ctor Table 5 Cu ·\ 1 226 80 sphere no . of states 1n inscribed 0 radi us of s-ob.ere tti.-1) 0.24 0 . 34 17 .8 8 .9 E (k) - E {o) (ev) • 9i• -4 Structure: •r 1 ; ' e. = 8 0700 A; Z : 4 Nu.mber of occupied states per unitdcell : 2Ll4 (Pe,uling); 84 {Mott a nd Jones) in.tensity 128 26 4 2 58 43 90 110 2920 26 34 31 142 183 6t3 59 38 31 14 20 522} 441 530} 433 531 366 442 600} t') r-1 r-1 11 intensity 62 hk.£ 611~ 1.6 610 532 25 10 540 20 62lj 443 541 622 75 137 6301 542 631 444 .fe.etor Lorentz & polarization 348 sphere no. of st~tes in inscribed Table 5 (cont.) frequency fact,0r 24 24 48 48 24 24 48 24 24 8 48 48 ()\-1) rad:tus of sphere 0.40 E (k) - E(o} {ev) ~ r-i r-i hki 220 111 200 100 110 111 211 200 T.:,orentz & .frequency Table 6 no . of states in 1nseribed sphepe ( ..~-1) • 0.27 22:. 4 8.4 11.2 (ev) E{t) - E (o) 0 .·38 o•.24 radius of sphere <:t· , U~ ). Cu - Al _{ 1~ at_.. o / o Al ) ,. _ 0 • 1. } tructure 3 . A l solid solut.ion; a::3o65 A, Z. _ 4 Number of occupied states per unit cell: 1:9.9 (Pauling); 5 . 52 ( .M~tt and Jones} intensity facter factor polarization 5. 4 8.4 24 Table 7 2.8 0.35 0.25 1'7 . 2 9.0 1'11/il Structure< 24 }; L21: a : 2.88 A Number of occupied states per unit cell; 8.87 (Pauling); 5 .• 52 {M0tt and. Jone$} 8.4 · ( 11) 115 References (1) G~ B. Jordan, Unpublished Research. ( 2e.) L. Pa~ling, Phys• Rev• , §! >899 ( 1938 ) .. (2b) L, Pau_ling~ J.A.C .. S ., 69 542 (1947). .( 2c) L. Pauling; Proc • Ro:r • Soc. (London) A19 6, 343 ( 1949) • -, - • {2d) L. Pauling, Physica, 2SY, 23 (1949). (3) H. w. B. Skinr1:er., . .Reports on Progress in Ph;srsice, y, 257 (1938). (4) H. .Tones, N • £. Mott, e.nd H. W. R. Skinner, Phys , Rev, , 45 , 379 ( 1934) • (5) H. w, . B. Skinner., Roy, Soc. Phil. Trans.,~, 95 - (1940)0 • (6) L. G. Farrat.t, Pb.ye. Rev., 44, 328 (19~~3); 45, 364 (1934). - {7 l F . Seitz, t''l'he Modem Theory of Solidsn, M.cGre.w Hill, (li e\-'r -::!ork and London) 1940, p, 272. (8) N, F. Mott and H. ,Tones, ''The Theories of the Properties of Metals end Alloystt, (Oxford) 1936, p. 154, (9) F. Seitz , op• cit • , p. 29 6. ( 10) L,. Pauling and F, J. Ewing, Hev. Mod. Phy s., 20, 112 (1948). - - (11) H, Jones,. Proc. Phys,. Soc,. (London)., 49• 243 (1937). { 12) J' ~ Farine1:P.1 t< Ann u de Phys", 10, 20 ( 1938 ) • ( 13) E . Rudbe rg and J .. ( 14) W. vit . Beeman and E . Friedman , Phy s • Rev. , 56, 393 (1939). - (15) M. F. Manning t.'1.nd M. I. Chodorow, Phys~ Rev., .§§,, 78'7 c. Slater, Phy s. Rev., .§_Q; 150 (1936). ( 1939) • (16) J. A. Be~rden end J, M. Snyder, Phys , Rev. , 59, 162 (1941). - 116 References (cont.} (17) .T . A. Bearden and H. Friedman, Phys. Rev.,,§§, 387 ( 1940) • (18) A.H., Compton ands. K. Allison, "X-R~ys in Theory and Experiment 0 , D. Van Mostrand Co., (New York) 1935, P• 632. , { 19, J •• Farineau, J. Phys. et Rad., !Q, 32'7 ( 1939). (20) Mott and Jcmes,. op. cit., P• 315 rr. (21) J. B. .~"'riauf, J. A. c. s .• 49, 3107 (1927). ( 22) o. D. Preston, ,Phil. Mag., 12, 980 ( 1931). (23) A. J. Brodle:r E:.nd P. Jones, J. Inst. Mete.ls,. g, 131 - ( l933h ,. ( 24} ' A, J. Bradley and A. Taylor, Proc. Roy. Soc. (London) ~ , 56 ( 1937 ) • . III. CRYS 1.VAL S'l1HUCTURES OF CERTAIN INORGANIC COMPOUMDS (a) G!"'JSta.1 Stri..1.cture of Sodium Bo!"ohydride 118 [Reprinted from the Journal of the American Chemical Society, 69, 987 (1947). I CONTRIBUTION FROM THE GATES AND CRELLIN LABORATORIES OF CHEMISTRY, CALIFORNIA INSTITUTE OF TECHNOLOGY, No. 1086] Crystal Structure of Sodium Borohydride BY A. M,SOLDATE The unexpected properties of sodium boro- Ka radiation filtered through nickel in a cylindrihydride, such as its stability in vacuum at cal camera with a 5-cm. radius. The twenty retemperatures as high as 400°, 1 give Iinterest flections observed were indexed on the basis of a to an investigation of its structure, hitherto cubic cell with ao = 6.15 A. More precise lattice unreported. constant measurements using sodium chloride as Experimental Methods and Results.-A sam- an internal standard gave 6.151 ± 0.009kX for the ple of powdered sodium borohydride was kindly value of a 0. No reflections with mixed indices supplied by Dr. H. C. Brown of Wayne Univer- were observed, the lattice is, therefore, face-censity, who reported it as being 98-99% pure. tered. Specimens suitable for X-ray photography were _ The unit cell, being face-centered, must contain obtained by sealing powdered material in thin- a multiple of four sodium borohydride molecules. walled Pyrex capillaries about 0.5 mm. in diam- It was assumed that the correct number of moleeter. To ensure adequate protection of this cules in the unit cell is four. The density calcuhygroscopic substance, the procedure was car- lated with this assumption is 1.074 g./cc. No exried out over sulfuric acid in a sealed enclosure. perimental value is known to us, but this value Powder photographs were made with copper is a reasonable one. There are two ways in which the sodium and boron atoms can be arranged in (1) Private communication from Dr. H. I. Schlesinger, The Unithe unit cell: versity of Chicago. 119 988 A. M. SOLDATE Vol. 69 (A) 4Naat000, 0 ½ 1/z, 1/z0 1/z, 1/z 1/z0 surrounded by a tetrahedral configuration of four 4B at½ !/2 ½, 1/z00, 01/zO, 001/z hydrogen atoms and that the structure consists of BH 4- and Na+ ions, intensities were calculated (B) 4Naat000,01/z 1/z, 1/z0 1/z, 1/z½0 4B at ¼ ¼ ¼, ¼ ¼ ¼, ¼. ¼ ¼, ¼ • for two arrangements of BH 4- tetrahedra: (1) a completely random distribution of tetrahedra ¼¼in positive and negative configurations (mirror 2 By use of Pauling-Sherman f values and by images) with hydrogen atoms on cube diagonals; neglect of the scattering power of the hydrogen (2) tetrahedra all positive or all negative throughatoms, intensities were calculated for both ar- out the structure with hydrogen atoms on cube rangements of the sodium and boron atoms. Ta- diagonals. In both instances the calculated inble I shows good agreement between intensities tensities of {111) and (400) were the only calculated on the basis of arrangement A and ob- values which differed from corresponding values served intensities. The net effect of absorption of Table I by more than 20%. Agreement beand temperature corrections, omitted here, was tween observed and calculated intensities was not estimated to be small. improved over that shown in Table I with either arrangement of tetrahedra. It was decided that TABLE I it was impossible to reach any conclusion as to A COMPARISON OF OBSERVED INTENSITIES AND INTENSITIES the positions of the hydrogen atoms from the inCALCULATED WITH SODIUM AND BORON ATOMS IN ARtensity data obtaine.d here. RANGEMENT A With the assumption of a covalent boron-hy(hkl) Calqd. intensity Obs. intensity drogen bond di~tance of 1.16 A., a van der Waals 111 231 m+ radius of 1.00 A. for hydrogen, and a crystal ra200 562 vs dius of 0.95 A. for sodium, it can be seen that the 220 386 s boron-sodium distance of 3.07 A. and the boron311 138 111 boron distance of 4.35 A. are large enough to cause 222 130 mvery little steric hindrance to rotational displace400 72 ments of BH4- tetrahedra. Thus, rotating or os331 51 f cillating BH4- tetrahedra can be reasonably pre420 145 msumed to exist in the structure and statistical 422 105 f+ distributions of tetrahedra, such as the one al333-511 31 fready considered, are of relevance. 440 f33 I thank Dr. Linus Pauling for the suggestion of 531 24 vf the problem and for helpful advice during the 442-600 65 f preparation of this report. 620 f45 0 533 7 622 38 444 15 711- 551 14 640 47 642 108 as, Strong; m, medium; f, faint. vvf f- vvf+ vvf f- Intensities calculated for the second arrangement were in marked disagreement with those observed. For example, the calculated intensity of the line ( 220) is much greater than that of (200) and the calculated intensity of {331) is greater than that of {420). With the assumptions that each boron atom is (2) L. Pauling and J. Sherman, Z. Krist., 81, 1 (1932). Summary Powder photographic X-ray data have been used to show that the structure of sodium borohydride is based on a face-centered cubic lattice. The unit cell contains four sodium and four boron atoms arranged in the following manner: 4 Na at 0 0 0, 0 ½ ½, ½ 0 ½, ½ ½ 0 4B ½ ½ ½, 1/z 0 0, 0 1/z 0, 0 0 1/z. The value 6.151 ± 0.009 kX was found for a 0 . It is probable that the structure consists of tetrahedral B H4 - ions and Na+ ions. The dimensions of the ions would permit oscillation or rotation of the BH4- tetrahedra. PASADENA, CALIFORNIA RECEIVED NOVEMBER 26, 1946 III• CRYSTAL STRUCTURJ::'"!S OF CF.!RTA!M INORGAN IC COMPOUNDS ( b) A Redetermination of the Structural Parruneters in the Cubic Modifica tion of Arsenious Oxide 121 A Redetermination of the Structural Parameters in the Cubic Modification of Arsenioue Oxide In trodue tion 'l'he structure of the cubic modification of arsenious oxide, wh ich occurs naturally e.s the mineral ars.enoli te, is based upon the packing of As 4,o 6 molecules in the pos! tions of the A-4 ( diamond type) arrangement ( 1 ). tnterest in the strt1cture was stimulated by the discovery that in the ' 0 molecular state the ar~,.mic ... 0,xygen distance, l.80±0.02A, is somewhat shorter than the co!'re.s ponding distance of 2,01 A reported in Bozorth' s original ¢rye tal structure study( 2 ). Because the preeis.1cm of Bozorth' s det.e nnination c1id not exclude the possibility that this difference is only apparent, another. investigation of the crystal structure was undertaken, which led to a oonfirmation of Bozorth•s result< 3 ). Accordingl)t it was concluded that the arsenic-oxygen intramolecular distance is lengthened in the formation of the molecula.l;'l cr-Jstal; this lengthening waa attributed to the formation of weak intemolecula:t1 o~gen-arsenic bonds ( 4 ). The structure of cubic areenious oxide was therefore con - sidered to be well understood. Recently, hov,ever, Emother redete:rmination of the stru.ctural parameters of the cubic modification has been made, and it is reported t h at the intramoleoular areenie-oxygen distance is 1.ao + 0.05 A( 5 >, 182 the sani.e distance found in the molecular state * • The results of the reported ceystal structure determinations are discrepant, and 1 t wa.s considered that still another crystal structure detenninatien was worthwhile . ~pe:rm~nt,al. Methods. .and i>ar$neter Evalue. tion~ Boeo:rthtl.} found that. the structure of the cubic mod• ification of a~ium.l&U.S o"ide is based upon the space group o7 - li'd3m and that the length of the unit eell ed~e is 11 . 06 h . o k .X• !he densi tr et the substance indicates that the:re e.re 32 arsenic ~terns arid 48ox:ygen atoms in the unit cell . The arsenic a'totn$ are in the positions which may be conveniently g1ven - «Ii.Ci! ( 0 0 O; i i 0; 0 ½ O; 0 0 ½) + (;!, 1 1J +- (X X x, X X XJ ) as e ) fl Ci 1 1J + ex x x .. x x xJ ) l, 888 " l- and the oxygen ',l1:i()tl12 are in the posit.ions ( 0 :0 O; ¼} O; 0 0 ½) + (l I l) +- (O o v; JI,) ;• o o v; 8 :=;- J ). (i J; ,¾.} + ( 0 0 V; ) j. ) • ., C 8 o 8 8 • J O O V; ., The-Se. eonelusion$ were based upon Laue photographic de.ta and the pl(;).us!ble ~s.sump.tion that all the arsenic atoms and *The value for the a.rsenic ... oxygen clistanee in molecular ar- sen1ous oxtde we.s f0tmd to be l . '78 ± 0 . 02 A in the electron diffraction investigation of Lu and Donohue, J"A . c.s., ~, 818 ( 1944)., 123 all the oxygen atoms are crystallographieally equivalent. tn addition• the parameters t t1.nd v were determined with. enough precision from the Laue data to establish beyond doubt that the structure 1s based upon the packing of' As 4 o6 molecules. It was not deemed necessa.riJ to make a systematic eheek of the gross struetural details given above; the good e.gree ... ment between observed and ca.leule.ted intensities to be re .. ported subsequently ean be considered to confirm them. The ma-in purpose of the present 1:nvest1gation, therefore, was to obtain ve.lues of the parameters x and v with enough precision to decide whether or not a significant difference in the molecular dimensions is (}e:used by e:ryste.lliz.ation. Initially the problem of the determination of the parameters x and v was attacked with the use of powder photography. 6 A heavy multiple-film expoeure( ) was made with Cu K radiation filtered through niekel. 'the specimen photogrEtphed con- sisted of a thin Pyrex fiber which we.a coated w1 th the powdered substance by mefms cf a small amount of vase line. Reagent grade arsenious oxide was used and the powder was prepared by' sublimation. Absorption errors were considered to be negligible. A preliminary determination of the value of the arsenic parameter x was made by a consideration of the intensities of reflections or the type h + k + '.~ = 4n +-2, which e.re in .. dependent of the positions of the oxygen atoms.. Structure 124 factors were calculated for these reflections in the reg~on 0.100.;;;:x ~0.,110 w:tth the ti.se of Pauling'!"'She?,rtan atomic scattering fa.ctore< 7 ). The permissible range for x was found to be o.102~xE-0.105 by the observations that 2 F2662<2 F842 an<l .F 2662 > 2 F! 42 , the intensity ratio for the latter 1nequali ty being large enough to insure the. t 1 t is not reversed when corrections for Lorentz, polarization, or reasonable isotropic temperature factors are ma.de. Structure factors were then calculated 1n the region 6.102 ~ X. ~ 0.105 1 0.160 ~VL 0.230. It was found that v must be less than o.185 because of the observation that F: 20 ? Fj33 • On the other hand, if v = 0.160 the general agreement between ca.loulated and obeeri1ed intensities becomes appreciably less satisfactory. A further limitation or the permissible values of x and ..., was found to be impossible with the powder photo• graphic data. It becane necessar-.r, thet'efere, to secure single ..erystal data • .. S.ingle crystals were prepa:r:-ed by the slow cooling of a hydrochloric ,acid s.Qlution or the substance. The crystals were octahedral in habit and, e.t first, it did not seem "" . • advis~able to grind them to cylinders even though the linear absorption coefficient· for Cu Ko< radiation is rather high, 236 ern-1. Oscillation and rotation photographs were taken with er;stals of sme..11 dimensions, 0.1 mm to 0.3 mm on an edge. Selected intensity ratios were corrected for 125 the effect of Lorentz and polarization factors, and the ma.gni tude s of absorption errors were estimated b y t h e use of t h e graphical method fi r st suggested by Albre ch t< 8 ). The pennissi ble raf! ge for x was t h en mo re precisely defined as o.102Lx ~0.104 by t h e obse"Mra.tion that Ff 2 • 0 • 0 -.; ; : 3F: 80 , and the permissible range of v as o.165 ~v ~0.178 b y the ob2 serva tion that F~ 0 ~F~oo• A p recise measurement of the pos1 tions of the l a rge angle powder reflections, which were clearly resolved. into the K«1, and Ko( 2 components, led to the value of 11.04 ±O.OlkX for a 0 , the unit cell edge ·~•• The vah.1.ea of the structural pa rameters and the length of the unit cell edge reported above fix the intremolecule.r arsenic-oxygen d iate.nce as 0 1.78 :t0,04 A, e value which is in essential agreement with that of Almin and Westgren l.80 ± 0.05 A. As a final step in t h e investigation.t the intensities of the powder. reflections were estimated visually , the best value for the isotropic temperature f~ctor e.stime.ted by graphical means, and a list of calculated and observed intensities prepared. During a verbal report o·t the investigation made in a seminar on s tructtiral chemistry at the Ce.lifornia Institute ""'The value of a 0 was found to be 11.045'7 ± 0.0002 kXby F.Lihl, z. Kryst., 81, .142 (1931) and 11.05208 ±0.00010 kXby M. Straumanis _et al, z. Phys. Chem.,~~~ 143 (1937). 126, 127, 128 of Teebnology; it vras pointed out that the agreement between the cnlculated and observed lntensi ties of the po\'!rder reflections was not wholl;r satisfactory, and it was siig ... gested tha t the existence of an anisotropic temperature effect. of' the type found for crys tel line he~amethyle:n.e ... tet1')am;ine<9 ) might eXplain the discrepancies-i~. 8inee such a correction is not a smooth f\tnction of the angle of reflection~ the parameter determination reported above eoncetvabl~ could be incorrect. Ac© rdingly it beceme necessary to estimate the m.agni tude of' the anisotropic tem.pera.tur-e effeet" A · consideration of the data led to the conclusion that the anisotropic tempe1--atu.re effect was not large J however, the method used in tbe datermina tion of the parameters is in principle extremely sensitive to even small effects of this type, Fttrthemer·e, 1 t aeemed the. t the intensities of the powder reflections could not be estimated vri th enough precision to decide whether o:r not the effect really exists . * I am indebted to Profe$$Or v. F. !. Schomaker. for this sug- gestion. It ts made plausible by the fact that the cr-1ste.l strt;wture of hexsmethylenetetram1ne is based upon the packing of (CR2 ) 6 N4 molecules, the configuration of the carbon . and nitrogen atoms of ea_c h molecule being identical to that of the arsenic and OJ{ygen atoma of the Aai)6 molecule. 129 Before the single-crystal data at hand could be used it v,ras necessar'J to apply the absorption correction calculated for an octahedral shape. Since the calcule.tion of these corrections, even b~r graphical means was found to be l~borious, it was decided to obt,ain single-cr,Jstal data with the use of a cylin ,~· drieally shaped specimen. A rather large octahedron was ground to a cylinder o.•5 mm, in length and 0.2 mm. in diameter. The axis of the cylinder was parallel to one of the four-fold axes. A· heavily exposed multiple-film rotation photograph wan pre• pa~ed with the use of copper K radi~tion filtered through nickel. The cylinder axis was aligned with the axis of rotation. The inteneities of the observed refleetions wer"e estimated visually with the use of the film factor 3.7, v,hich :ls a1)propriate for ponna.l incidence upon Eastma."1 NoScreen X-~ay film(lOJ{{•• Tb..e intensities were corrected for the effect of Lorentz, pol.arizat.ion, freqieney, and absorption factors. Absorption corrections were obtained 1n the following manner. 'J,'rJ.e corrections or Claassen< 11 ) V r :; 2.4) were used for the equatorial reflections. The correction curves for the second and . fourth layer lines ·~Because of an oblique angle of incidence, the film factor for each layer line above the e<11ue.tor is somewhat greater than 3.7. T"ne film factors used for the first, second, third, and fourth layer lines were 3.7, 3.9, 4.2, and 4.9 respectively. 130 were estimated from 8. comparison of the intensities of a given reflection vrhich ap peared on the equatorial layer line and one or the other of the hi ghe~ layer lines. The curves for the first and third layer lines wer-e estimated by inter• pole.tion and a simila l" intercomparison of intensities• The • visually estima ted intensitie s corrected tn the me..nner outline d ab ove are listed in Table 1. 1':he magnitude of t h e anisotropic temperature effect was 1 estimated from a con$ideration of the intensities of type h + k + .£: 4n -t-2. First. t h e value of the arsenic p a 2"ameter x was estimated from t he observed structure factor ratios of reflections which occurred on the eeme layer line and ·~ihich were close to one another. In th1a way., en isotropic temperature correction effect eould be omitted in tbe first approximation and possible errors in absorption corrections ignored. Gal cu lated curves of four intens:1. ty ratios were found to h ave large slopes in the !'ange 0.095 ~x ~0.115. Tl1e Yal ues of x derived fran the observed ratios 442., 842, 622 ~ 10.2.2., and 10.2.,2 a;r e 0.1025, o •.1026, 0.1029., and 0.1035, 842 r ae2 • • :• re$pectively. • 'l'hese values are, of course, derived with the neglect of an anisotropic temperature factor as well as an isotropic temperature factor. Since the variations in the observed structure factor ratios due to an anisotropic temperature . effect would be expected to be more or less random, 131 the s.vera.ge v~lu,e of x should not be greatly in error. Structtire factors were then ce. lculated for all observed reflections of the ty'p.e h -r k + i the function 1n This plot i:-;; .· JIFl 1 c.a.lc • I FlobS , = 4n + 2 for x = O.J.03 e.nd plotted against ~,n;._;1.e • snown in. Figure 1- With the exception of the point for 222, the points lie very close to a straight line 0 0 • 1 o:r• $1Qpe• 2 .. 2 ,Al!:,i " ..1!'h~$ • .a r-easone.bl • e ve._ue o f t h e i• sotrop i c temperature !'actor is obtained and the magni t.ude of the anisot~opie tempe:x-;$:tui"e effect is indicated to be small . W.1 tl.'l '. the -- uae ef · tbe , :taptrop:tc ·wmperature coefficient 2 . 2 • the Values ·ot ,;. ~f ·e.tld v were found from a eompari~on of ob... , ··. 1 : se.r-veq antf eal¢"1l.te.flt· l;>,tructure factor ratios . .. .,, ' ' In. Figure 2 •cut-Ve$ a~ 61Y~t- wtifch •represent the loci of point.s for which calcula·t ed structure factor ratios are equal to observed stru.ctu1'a f'aetor i'$.tff,,0,ir~ : !he varioue points of intersection are centered abbut the: point x:; 0 . 1029, v;; 0 , 1749 with an • 1 • • • ave~age d.evie.Jt,:1on 0,f 0 ., 0003 and 0 . 0030 respectively.. The paraJ'neter values d~X"l:V<i,d here are in essential agreement with those de:riveQ ;pr,evtc;>.u al.y and w1 th those feund by Almin an~ ~Vestgren.f$) ~ ·; -6:,:;1 ~$ ± 0 . 001:, y ;: 0 . 175 ± 0 . 010 . Table 1 shmrn good general agr~el)1ent between corrected obser•v·ed in• tensities and inteni:r$.t:t.e,s, eai-~ ulated with the structural and isotropic ternpe.r ature pavemeters given above . The value of the- paramet,e r x seems particularly well fixed &.."1.d can be used · t..q gain a qu.ant1 tative estimate of- the C --.,.J LL 0.0 00.2 C, u ........ t--4 _....._flJ .0 0 ~0.4 en 0.6 0 0 0.10 sm 0 / )... 2 644 0 0 622 662 222 2 0.20 10.2.2 864 00 0.30 12.4.2 0 u..l r-..> - 13 3 6 5 0.105 0.103 X 3 0.101 2 0 .160 curve no. 1 2 3 4 5 6 7 8 9 10 0.170 V 0.180 ratio 933:953 513:533 620:800 884:10.6.4 11.5.3 :993 822=662 442=622 662=842 842=10.2.2 10.2.2:882 0.190 134 magnitude of the an:tsotropie temperature factor . '!'he anisotropic temperature factor to be considered here arises from the possibility that the As 4o6 molecules can under go small torsional vibrations a.bout the various molecular centers of gravity . These small vibrations should be veey nearly symmetric with respeet to direoti9n and would cause a smearing of the charge di$tributicn cf each a.tom in a plane normal to a 11ne drawn through the atom and the molecular center of gravity . The calculation of the effective atomic scattering factor fer a smearing of this type has been dis cussed previoue11< 9 ) and no report Will be ma.de here concerning its details . The speeifi c form of the correetlon used here v,1111 be given for convenience . in the subsequent discussion . If following Shaffer, FAis and F 0 are defined by the e:x:presaion F = e-xpf- Bs,te}[Fo+FAs], we find that F0 :\6f0 cos 2 n (h+R+.Q)lcos2.1t"...eN"' e~pf-...l2..2.(h1.+R 1) } 8 . .4a 0 4 + (OS 21T ~v e-y.p {-4b 2. ( r?+ ,l )} +cos1 lT kV eip{~ o~l.+l~Jl) ~ 4~ J fAs=· 8f"s e1p{-J'cs2. (hl.+Rl-+J.•)J [ cos 2lT{ h+a+...Q + 0 . ' . (h-\-~+ih}e~p{-.1'.L (-hR-hl-Rl)1 + 60; cos 2 Tr { h+l~.t + ( h-t ~ +.i)x}e1 p{-!! ( h ~+ hl-\:1..t)} + 1 co~ 2:rr { h+\2.-\-.R. +( h + t"'"J. )x e.) p f--J1. (-h~+\\.Q.+ R.l)l+ 8 6a; • S cos 2.1T { h+8R~.t +_C h+ R. +l >~! e.1pf 6~t( hR- hl-+ ~~)}J. 135 The value of the parameter b determines the me.gnitude of the anisotropic temperature effect. It is quite plausible that in most cases the changes in the terms of each expression due to the exponential factors tend to cancel When the value of b :is small. In a few instances, however, the various changes might reinforce eaeh other. Even so, the effect of an anisotropic temperature factor for small values of b would be slight unless the various terms of each e::xpreseion, especially that for FAs tend to cancel 'Nhen no exponential factors are applied. The agreement between ce.lculated and observed structure factors is quite appreciably improved in the case of the weak reflections 10.e.2, and 10.4.4 when b is 0.5 and B is 1.9. The positions of the curves of Figure 2 are not appreciably shifted by· .an anisotropic temperature factor of 0.5, however. It is elear that, although the final agreement between ca lcula.ted and observed structure factors may be sanewhat improved by the anisotropic temperature factor, the structural parameters will be left essentially unchanged. l)iscuasion of Structure The parameters x : 0.1029 ± 0.0003 and v : 0.1749 ± 0.0030 determine the 1ntrmnoleeular nrsenic~o:xygen distance as l.80 ±0.02 A. 3.22 A. The intramolecular arsenic•areenic distance ts The most probable O - As - O e.ngle is 99° :121 ; the 136 most probable As ... O ... .As angle is l2"1°24t • The dimensions ot the As 4o6 molecule in the cubic m.od1f1ce.tion of arsenlous oXide are trhus n-early 1dentice.1 to those found for the free O 0 AS406 molecule; As ... 0 • 1.78 ±0.02 A, AS • As :: 3,20±0.02 A L O .. ,As • 0 • 99° , L As .. 0 - AS : 128 o ( 12 ) • ' o '!he arsenie•oxygen distance of 1.80 A is appreciably 0 smaller than the sum of the single bond rad11, 0.66 A +1.21 A :;: l.87 A( 13 ). If the equation rela.t:tng distance to bond I order R( l ) ... R( n ) :: 0. 300 log n ( 13 ) is used, it is found that the bond order, n, for the arsenicC oxygen distance of 1.80 A is 1.31. !Jfue increase in bend number over the value 1.00 1s presumably due to the use of the i.mshared electron pairs of the oxygen atoms(-4) • It is of interest to calculate the bond number for the smallest intennolecular arsenic-oxygen distance. This distance is 0 3. 03' ·A 'S:nd the corresponding bond number is 0.012 wh'i"eh is 0 so small as to have no significenee ( 13) • . The structure of the cubic mod1f1ca.t1on of arsenious ox:l. de, therefore, can be deset-1:bed in terms of the packing o of undistorted As 4 6 molecules which are attracted to one another by secondary valence or Van de'r Waals forces. Summa.a The structural param.e ters of the cub1 c modification of a.rsenious oxide have been redetermined by means of X-ray powder and single-crystal photography and found to agree 137 closely with those reported by Alrn1n and Wes tgren ( 5 ); The etructure is based upon the packing of' As 4o6 molecules which have t he same dimensions as those found for the molecules of the gas ( 2) •(12) ' • 138 Table l hkl }F} cale . 111 220 311 222 400 331 422 511 ..2.0.0 2.06 26 .. 4 15.,0 . 9.47 5.62 18 . 7 :5'53 440 15.0 448 620 53.3 622 .5.·89 444 7.J. _._>_l~ 551J 642 731 553 800 733 644 882 660 751 662 3 . e1 1 . 30 Ji.37 11 , 4 l3"5 15.0 ' 3.8'7 5. .11 7 . 22 '7 • 38 1'7 . 0 3 . 88 5.66 . 1 . 20 2.49 12.1 13.0 14.7 3.03 '7' . 06 S.85 3 . 65 8 . 94 5. 29~ ll . O 3.16 2.02 10 .9 3e49 11 . 5 2 -. a4 2 . 25 11 ~9 3 . 25 5 •.86 6 . 24 15-70 5 . 4.0 , . 41 842,t···•,;, .. , 6.82 664 ~. 69 5 . 81 931 11 . 2 '1.06 . 840 911 '753 1 .. 69 2 . 13 _.18,.7 ., 6,55 6_.64 531 ll.5 l . 94 6. 98 1 ., 54 • • ·' '-"6 . 00 4.-16 5 . '70 2 . 50 2 , 44 '771 4 . 61 . 4,15 10. 2~:o 3,.63* 3 . 44 6,. 16 844 933 862 . 2,.43 1 . 30 2. 17 773 6 . 46 1 . 45 5 . 69 6.61 10. 2.2 2.07 1 . 90 9fil 953 864 2. 22 3,60 l . 97 3.30 l39 Table l (cont.) hki )Fl obs. 10.4.2 11 . 1 . 1 3 .24 6 . 29 > 1 . 10·; .(. 1 . 52 3 . 86' 880 11 .s . 1 971 10 . 4. 4 882 10.s.o 11 . 3 ,~ 973 10 . e . 2 12.0. 0 0 . 55* 7 0 . 84; -< 1.00 7 . 19' 6 . 07 6 . 75 ,;,-0 .74'; <'.'.. 0 .87 \FI eal e . 3 . 02 5 . 86 1 . 45 3 . 84 0.73 0 . 44 7 . 10 5. 2i 7.11 1 . 06 2 . 75 2 . 37 2 . 64 ~ 0 . 67 2 . 42 O. l? l .. 94 1 . 9'7 12.4 . 0 991 < 0 . 59 2. 66 2.25 12.4 . 2 3.71 < 0 . 60 e . 20 1.73 c::: 0 . 41 2.84* 884 ll . 5.l 12.242 10 . 6 . 4 11.5.3 10 .e. 2 13 .1. 11 11 . 11 . 1 993 12.4 . 4 13 . 3 . l 11 .'7 . 3 5~oa 2 . '78 2 .·56 2.68 5. 39 0 ,,30 3 .72 0.09 5 . ?6 1.56 0 . 53 2 , 74 ~.oo *'rb,ese. intensities . . . detennined by difference; that is, when crystal in reflecting posi t!on, another reflection alse ~sdbh . • 140 :Reference a. Cl) R. 1<.L, Bozorth,, J .. A. {2) G, C!lt 5., !§., 1621 (1923) . a .. Ham.peon and A. J. Stosick, :r. A. C. s . filh 1814 . ( 1938) • (5) • D4! Harker and J. Eskijian, Unpublished Re•aee.rch.- , ( 4) • L. Pauling, 11 The :Nature ·or the Chemical Bond*', Second Edition, Cornell Univer$1 ty Press, lthaea, N.,Y . , •. ( 1942) , p . 248 . • • . < ( 5) , K. E . Al.min and A . Westgren, Al"klV -- 15 B, Mo• 22, ( 1942) . ........ . • JmMI., M:tnerolGeol; . , ( ~) : J. de. Umge,. J .• M, Robertson, and l, Woodward, Proc . Roy . Soc . (Lor.tdon), A l'71; 39-8 ( l939) . ( 7) t . Pauling an4 J._ .Shem.an, Z. K:rii t, . , ll• 1 • ( 1932) • . (~) : G. Albt1~eht, Rev •. Sci • l.:n strument s, !Q, 221 (1959) • ' • { 9) • P . A. ij.h aff'er 1 _Jr . 11 .1 , A. e. S •. , 69 ,. 1557 .. ( 1941) • . (10 ) • D. • .,P . Sh.oeme.ker, Thesis;' Calif, . J:nat,,. of Tech • ., (1947) . ·, ( ll) • ftlxr~ernational Tables fer the Detenn1:nat1on of Crystal ' • Struetu~~ ,t • Gebru~der Borntl:'e.eget'; Berlin, 1935 1 Vol . 2, .pp ·• 583- 585 ii; • ( l~): Cbia- S.i tu and Jercy Donohue, J . A. C. S . ,§!_. 818 {13) t , Pauling , J. A. c. $,., §!, 542 (194'7). ( 1944) . IV. PROPOSITIONS 142 IV. l. Propositions A relationship between the band structure of nearly free electrons and diffraction patterns of liquids exists which is closely analogous to that between the band structure of nearly free electrons and crystalline diffraction patterns. The demon- stration of this relationship can be easily generalized to include the case of disorder in crystalline solids. Some systems in which the proposed relation- ship might fruitfully be applied are the alkali metal-aru.monia systems, metallic liquid systems, and ( 1) metal-molten salt systems • (2) 2. ·Mott and Jones have given a discussion of the Brillouin zones of graphite and find that the valence electrons just fill the zone bounded by the planes {11•0] and {00·2] • This zone has the form of a flattened hexag onal prism with the ratio of the perpendicular distance of the {00•2) planes to that of the l11• o} planes equal to o.362. Because ot the marked flattening it might be expected that the {00•2l zone boundaries are overlapped l:)y the Fermi-surface, with resultant vacancies at the zone vertices. This expectation is supported by the abnormally low value 143 of the specific resistivity. 2 •10- 5 ohm om. at oo C, and the high diamagnetic susceptibility 9 l 3) -3.5 • 10 6 emu • ( 4) A. F .. Wells 3. has stated that the re are no hydrogen bonds in the diaspore structure, and that one-half of the oxygen atoms exist as hydroxyl groups and the other half as oxygen atoms; accordingly, the most representative chemical formula for diaspore would be AlO(OH). As evidence for this conclusion Wells cites the existence of an absorption band at 3 }A • Cl. Duval and Jo Lecomte ( 5) indeed , however, find an OH absorption band at 1521 c.m... l {--1.5;\-:t ) . 'fha ,hcw~ver, result of Duval and Lecornte ,. is in accordance with the ( 6) conclusion reached by Pauling that there are no hydroxyl groups in the structure and that all hydrogen bonds exist in such a way as to give the structure a residual entropy of R log 2. It is probable that Well s> conolu~ion is incorrect. 4. Pha.se · transformations which are extremely sluggish at low temperatures m:i:ght be carried out by the p:re$ence of suitably unstable isotopes which by their disintegration would furnish the required energy of activation. In section 3 of this thesis, it was found that 144 the interatomie distances in the structure of sodium borohydride permit the existence of freely rotating or ,oscillating BH~ tetrabedrons o It would be of interest to carry out a determination of the C and p Cv curves from about 60° K to room temperature in order to investigate this possibility further~ 6. The famous relationship of Gibbs - r _, dfnQ d er- (1) = RT . , has general validity tor planar surfaces of two component systems, in the absence of external fields. A previous suggestion that it be modified in view of properties peculiar to one or both of the components, such as a tendency for long chain hydrophobic molecules to o~ient themselves at the (8) surface is unwarranted, 7. Slow exchange reactions mi ght be studied in an electrical mi gration apparatus with a scanning count~ ing device. In such a manner 9 complications intro- duced by preoipitation separa tions could be avoided as could the necessity for the separation of enriched {9} solutions . (10) ta.chari asen has developed a general theory for the calculation of structure factors of disordered structures. The initial model for his calculations 145 is an. assemblage of crystallites of uniform dimensions. This model seems reasonable; it is, however, immediately modified to consist of but one averag5t crystallite. Although such a .modification seems plausible and does enable the development of a rather elegant treatment, it might be thought to lead to some degree of error, possibly or serious proportions. In order to o.h eck the validity .of Zaohariasen's treatment several of his examples were reealeulated by a more direct. less elegant procedure in whioh the averaging process was ·carried out directly over the crystallites of the assemblage; this procedure led to identical results in every instance and lends confirmation to Zachariasen's treatment. The presence of a protein has been noted to markedly affect the reduction of such substanees as (11) organic dyes at the dropping mereury electrode • With referenoe t o this phenomenon the following oalculat,ion may be of interest. Th~ flux-time relationship is calcula ted for the linear flow of a. solute at initial concentration o0 to a moving, initially olea.n; plain surfaee upon whiah the solute is adsorbed instantaneously a.oeording to a lj_near isotherm, Jt == Ko ( li.mi ting form of Langmuir isotherm at low concentrations). If the flux, taken in the 146 direction toward the surface is <p , t the time, c 0 the initial concentration. and k the diffusion ~ constant, • 10. ~ ::: c RK e \ t ( 1- er f{ ~ Vii}] O It would be of great assistance and convenience to have a continuing index of all proposit.ions submitted by candidates for the degree of Doctor of Philosophy in Chemistry at the California Institute of Technology. In this nia-nner unintentional duplica- tions of propositions : and the embarrassment of an innocent plagiarism. oould be avoided. References D. D. Oubicciotti and 0, Do Thurmond, J. A. C., s., 21., 2149 ( 1949) N. J!" • Mott and H. Jones, 0 The T.heory of the Properties of :Metals and Alloys", (Oxford} 1936, p. 163 F. Seitz, '"fhe 1'.f odern Theory of Solids", McGraw Hill (New York) 1940 l p E> I A. 1f. Wells, . ''Struetural Inorga~io Chemistry", ( Oxford) 1945, P• 356 .• C.l. Duval and J. Lecomte• Bull. Soc • . frano. mineral., .!§., 284 (1943) · L. P auling , J • . A. C; s., .§1, 2680 (1935) "The Collected Works of J. Willard Gibbs", Lon{!, mans, 0-reen and Co, (New York) 1928, V-ol. I, :p. 229 ff, 8. J. w. McBain and G. F. Mills, Reports on Progress 1n Physios, Y.., 30 (1938) v. J. Linnenbom and A. o. Wahl, J. A. c. s., 11, 2589 { 1949) • 147 10. w. H. Zaohariasen, "Theory of X-ray Diffraction 11. B o Keilin, Jo A,, Co S .,, 1Q, 1984 (1948) in Crystals" , John Wiley & Sons, Inc .• (New York) 1945; p . 213 ff~