The Cleavage of Ionic Minerals. The Crystal Structure of Bixbyite and the C-Modification of the Sesquioxides Citation Shappell, Maple Delos (1933) The Cleavage of Ionic Minerals. The Crystal Structure of Bixbyite and the C-Modification of the Sesquioxides. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PTJH-EN65. https://resolver.caltech.edu/CaltechTHESIS:05122017-095401216 Abstract Mineral cleavage can be resolved into two components; cleavability and optical effect. The electrical theory of matter in the solid state leads to a quantitative expression for the cleavability of ionic minerals C {hkl} = A (hkl) /Σ í n i s i cos θ i Approximate values for s are obtainable by using the electrostatic bond strength. Systematic application to minerals whose constituent atoms have inert-gas cores gives good agreement with observation. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Geology Degree Grantor: California Institute of Technology Division: Geological and Planetary Sciences Major Option: Geology Thesis Availability: Public (worldwide access) Research Advisor(s): Pauling, Linus Thesis Committee: Unknown, Unknown Defense Date: 1 January 1933 Funders: Funding Agency Grant Number National Research Council UNSPECIFIED Record Number: CaltechTHESIS:05122017-095401216 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05122017-095401216 DOI: 10.7907/PTJH-EN65 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10168 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 27 Jun 2017 17:15 Last Modified: 16 Aug 2023 23:12 Thesis Files Preview PDF - Final Version See Usage Policy. 18MB Repository Staff Only: item control page

THE CLEAVAGE OF IONIC MINEP.ALS THE CRYSTAL STRUCTURE OF BIXBYITE AND THE C-MODIFI CATION OF THE SESQUIOXIDES Thesis by Maple D. Shappell In partial fulfilment of the requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1933 THE CLEAVAGE OF IONIC MI NERALS ABS TRACT Mineral cleavage can be resolved into two components; cleavability and optical effect. in the solid state The electrical theory of matter leads to a quantitative expression for the cleavability of ionic minerals c{ hklJ = Ei n.s.cos e 1 J. J. Approximate values for s are obtainable by using the electrostatic bond strength. Systematic application to minerals whose constituent atoms have inert-gas cores gives good agreement with observation. CONTENTS Page 1. Introduction l 2. Previous Work on Cleavage 2 3. Derivation of a Quantitative Expression for Cleavabili ty 5 4. Selection of Ionic Minerals 12 5. Structure and Cleavage 15 6. Discussion of Cleavage 38 7. Conclusions 48 1. 1. INTRODUCTION It has long been recognized that a close interrelationship :J" '." 1-.,etric Q/ C1rra,.,, e ,..,..~11f o ( t-h c co.., '!> h"ttJ e nt must exist between the ... particles of a mineral and the cleavage it exhibits; as a result the development of the theory of crystal structure has led to attempts to explain cleavage while a.t the sa.me time work on clea.va.ge has of necessity been required to gain an understanding of structure. Nevertheless in spite of its long standing the problem of cleavage has continued as one of great difficulty. 1 Cleavage has been defined as "the natural fracture of a crystallized mineral yielding more or less smooth surfaces; it is due to minimum cohesion." It is evident that the term cleavage should be resolved into two distinct parts, first a comparison of the work of separation, i.e. "cleava.bility", and second the nature o'f the surface fracture and its interaction with light. For exrunple, the goniometric signal from the cleavage form £111) of diamond is strong - hence t he cleavage is said to be perfect. '.Ihe necessary distinction between these two phases of cleavage has not ahmys been made. Cleava.bili ty is the more funda- mental and the present investigation is largely restricted to a study of this property. Experimentally cleavage is obtainable by several methods~ of which the following are the more important. 1. Blow. This is likely to result in a general shattering of the mate1~al with the process largely one of shearing. 2. Wedge-action, e.g. with a knife, It can be either static or dynamic; in the former case pressure on the wedge is increased until fracture occurs while in the latter a number of blows on the wedge is made. 2. 3. Bending. Analogous to the deflection of a berun. 4. Direct pull normal to the cleavage plane, i.e. the stress is a tension. The second method is the one usually applied. 2 Up to the present time the experimental data on cleavage have been chiefly of a qualitative nature. 2. Previous Work on Cleavage Cleavage investigations have taken in general two distinct directions; either that cleavage can be accounted for by the point geometry of the crystal or that it depends on the breaking of bonds between the atoms. The first method is a development of the fundamental researches of Bravais. 3 He held that cleavage is obtained parallel to the plane of greatest net density and that in case cleavage is obtained parallel to several forms the ease of obtaining it decreases in the order of the decreasing net densit~es. However since the net densities are inversely proportional to the lattice face areas, he used the latter for calculating relative values. Moreover since net density and interplanar distance are proportional, he gave as an alternative condition for most complete cleavage that the interplanar distance be greatest. Sohncke 4 used the interplanar distance condition but because of lattice interpenetration added to it the further condition that the cleavage planes be parallel to plane~ where the tangential cohesion is greatest. This is because interpenetration often causes in a se- quence of parallel planes several of them to be grouped closer together 3. into a layer which is repeated at regular intervals. The assumption being ma.de that such layers a.re more strongly held together the result is greater tangential cohesion. i'ihen Brave.is 1 theory was systematically applied to crystals of the trigonal and hexagonal system, Tertsch found numerous exceptions. 5 The application of Sohncke's condition requires la1owledge of the crystal structure. on x-ray determinations, Ewald and Friedrich Ba.sing their conclusions 6 independently coone to Stark 7 considering crystals in which the same condition as Sohncke. the atoms can be considered to exist as ions, related cleavage to the repulsion of like ions of adjacent net planes approaching each other during a shearing process; however the possible cleavage faces are too numerous for this to be unique condition. Scharizer 8 postulated that the adjoining planes of two layers must be similar, the sllill.e holding true for a non-layer sequence; this condition evidently permits the application of Stark's relation. 9 Niggli , by swnming the number of electrons of the atoms at the lattice points, converted Brave.is' net 10 density into planes of electron density. Beckenka.mp's treatment is essentially a combination of the conditions of Sohncke and Stark. Parker 11 has applied Niggli' s method to the structure of octa.hedri te. The concept of conditioning cleavage on the breaking of bonds between particles was used by Ba.rlow12 as a. result of his studies on the close-packing of spherical particles; the bonds broken in the process a.re not necessarily those which under static conditions have least strength. He gave as a probable condition that cleavage planes separate opposed unlike particles, a condition directly opposite to that of Stark. 13 Elvald postulated that cleavage will take place where the fewest bonds a.re broken. Rela.ti ve calculated values per unit area 4. for diamond gave agreement with observation; however discrepancies ~rising in a later study14 caused this postulate to be questioned and the condition of the previous paragraph to be set up as essential. • 15 iugg1ns Cfu~e to the conclusion that new crystal surfaces should be :r left electrically neutral, that weak bonds would be ruptured in preference to strong bonds but that where all bonds are equally strong, the cleavage plane would break fewest bonds per unit area. He considered tnat the inclination of the bond to the cleavage normal could be neglected. Tertsch; 6118 considering the problem as one of attractive and repulsive forces between ions, calculated a value for the force across various possi. ble cleavage planes, the minimum indicating the most cleavable. The inclination of the individual force directions to the cleavage normal is ta.ken into account by using its direction cosine. Pauling 17 calculated the density (bonds per unit area) of Al-0-Si bonds for cleavage in s~veral aluminosilicates; he found that ease of cle ~ decreases as bond-density i ncreases. 5. 3. Derivation of a Quantitative Expression for Cleavability. Cleavage in minerals is one phase of the phenomenon of the cohesion of matter in the solid state; it accordingly is concerned with the interactions of the constituent particles when disturbed by mechanical forces from the configuration ta.ken by them at equilibrium. The necessary and sufficient condition for equilibrium in a conservative system is tha.t 19 where ~ is the entropy and E the energy. A more workable criterion is gained however by using the free energy, F, which is a function of s. The condition then becomes in the comrn.only used notation of Lewis and Randall 20 = o. dF (1) In the wave equation of quantum mechanics, 21 v 1. ~ + TT' .u (} .Ji:-~ q ( w - v lf = 0 ) the product of the eigenfunction 'P by its complex conjugate be interpreted as the electron density ;O the con.figuration of the system. J ljl may , i.e. the probability of '.Ihe distribution of ;O has been shown to be spherically symmetrical a.bout the nucleus for a.toms and ions having completed subgroups of electrons. 22 23 ' It follows that the n electrons in an ion can be considered in effect as if located. at the center of symmetry. 24 Under the assumption that polarization can be neglected, the center of symmetry for both the resultant positive and negative charges coincide, permitting a system of ions to be treated mathematically as discrete point charges (plus a. repulsive tenn to 6. be taken into account later) of value ! z where z = Z - n, denotes the valence, the number of electrons. ( 2) Z the charge on the nucleus and n Since the atoms are in thermal agitation about a mean position it is unnecessary to refer the system to a. temperature of absolute zero and accordingly the lattice constants e.s detennined at ordinary temperatures can be used. In the equilibrium condition the quantity F F is given by = E + P V - T S, where E is the total energy, P absolute temperature and the entropy of the system. S the pressure, V the volume, T the The term con- taining the entropy drops out at the absolute zero, and if the region surrounding the system is void of matter the P V term also disappears. Hence under these conditions F = E. (3) It is convenient to consider the crystal as at absolute zero with a volume energy content E obtained by using the ordinary lattice con- stants without extrapolating to zero, since the difference of energy is small. 25 er;,; . (AF9URd ~ or Qbout 2 l&rge G~lories p8r mole for GFJr&tQl& 0£ t lce fie.lite type ) • A method of evaluating E for ionic crystals is known, 261 27 the energy expressed in either ergs or calories per mole, being designated as the lattice energy. The results of crystal structure determi- nations while giving the configuration of ions in a crystal say nothing as to the interactions between them. 28 It is knovm from Earnshaws Theorem 29 that in classical electrostatics a system of electric charges alone cannot be in equilibrium; a repulsive term is accordingly required. 7. 1he potential law between two ions is assumed as a first approximation . to be = where z1 and Equation 2, b e z2 + r b ( 4) rn are the charges on the ions obtained from the electron charge,, a proportionality constant and r n the distance between ions_, the repulsive exponent, which ordinarily can be ta.ken as ap proximately equal to nine. 1he equilibrium energy of a unit cell is given by f where b z , z2 e = A A (l _ !.) ( 5) n has been eliminated, adjacent ions, and 2 r 0 is the equilibrium distance between is the Ma.delung constant. the repulsive tenn is evidently small. Dividing 1he energy due to p by p, the number of molecules in the unit cell, and multipl ying by N, Avogadro's number, gives ( 6) E Accordingly if a system of ions, e.g. a crystallized mineral, satisfies the condition d E = 0 ( 7) the system is in equilibrium and its energy is given by Equation 6. When cleavage takes place the crystal is separated a.long a surface into two parts, each containing a volume energy, E1 and in addition if Ez, A is the area. of the cleavage plane the area of surface has been increased by energy and 2 A. On sepe.ra.tion the interaction E 12 will equal zero since the ions have only a small radius of influence; there enters 30 31. 32 33 , ' , however a surface free energy, 2 cr- A, s. ~~ose absolute specific value is given by = The value of E12 -· (8) E 12 2 A can be obtained by a method of a similar nature to that used in calcul ating E. The change in free energy then becomes (9) For a sequence of variously oriented planes through a crystal, a series of values is obtained for ~ F and the plane of easiest separation is given by D. F = minimum. Accordingly the cleava.bility (10) C of a mineral is defined as the re- ciprocal of the change in free energy c = -1 ..:'.J F ( 11} In terms of unit area Equation 11 becornes _L. (12) 2 0- If the volume remains constant which is approximately correct 34 Equati on 10 becomes for the cleavage fo:nu the surface energy law of Gibbs 35 and Curie3 6,37 L: a- s = minimum where s is the face area. (13) It may be mentioned that a fictitious "surface tension" is often conveniently used in the mathematical calculations instead of surface energy since the dimensions are the same for both. 3 s, 39 , 4o A 3-dimensicnal method to exhibit the value of for various orientations of the separation surface in a crystal is known. 4l, 4 z, 43 , 44 , 45 If nonnals to the plane are drawn from a center of' coordinates 1rithin the crystal 46 proportional to o- , their ex- tremi ties form a surface in which the minima are depressions. lhese 9. will be symmetrically placed according to the synunetry of the crystal and may consist of secondary as well as primary minima, the secondary being due to a lesser degree of cleava.bility. The calculation of the surface energy o- needed for Equation 12 is involved and has only been carried out for the most simple ionic canfigurations; moreover it is desirable to express the cleavability in terras more closely related to the mechanical properties of the crystal rather than in terms of energy. This can be done by identifying of Eque.tion 8 with the mechanical work W E12 done on the svstem 47 , 4 s, 49 , 5o, 5 l v • Equation 12 then becomes (14) w Since the ma.xbnum force per unit area. is the tensile strength, i.e. the breaking strength of the crystal for the direction of the normal to the W cleagage plane, = k S where k is a parameter which without loss of generality can be placed equal to unity. Cleava.bility is the reci- procal of the tensile strength, 1 ( 15) 1he force betvreen pairs of ions having the cleavage surface interposed betvreen them ca...n be considered as in the nature of a bond. On stressing the crystal until have a value s S is reached each of these bonds will giving as its normal component s cos e Let the number of such bonds for each ith ion be denoted by • (see fig. 1) n, then the maximum force nonna.l to the face area is given by = Since = Fw'A(hkl)' Equation 15 now becomes (16) 10. Fig. 1. The (110) plane for periclase showing the relationship of s and normal. e to the cleavage 11 . A simple estimate of the bond strength s is obtainable from the coordination theory of ionic structures. 52 . It is evident from crystal geometry that the strongest bonds are those between cations and anions of the coordinated polyhedra, Hence the electro- static valence bond strength equal to the charge on the cation z divided by the coordination number V, s ::: z v can be used a.s a first approximation. 53 The angular change in e from its equilibrium value is usually small and in general may be neglected. The expression for clea.vability, Equation lq, may now be applied to ionic minerals. 12. 4. Selection of Ionic Minerals. Minerals are a heterogeneous group of substances and a classification is accordingly necessary in order to have some basis of selection for those likely to be amenable to a simple treatment. A rational basis for grouping is to be found in an examination of the structure of the elements. If the valence electrons a.re stripped from the a.toms of the periodic table, the elements can be divided into two ma.in groups, (1) Ionic,, (2) Covalent. The vovalent group is composed largely of elements whose cores have an outer shell of eighteen electrons. The bond between such elements is predominately due to the she.ring of electrons, requiring treatment by quantum mechanical methods. Moreover due to their ease of deformation, polarization effects further complicate the cohesive phenomena of minerals composed of such a.toms. The ele- ments of this group are those ¥nth atomic numbers 26 (Fe) to 35 (Br), 44 (Ru) to 53 (K), and 76 (Os) to 84 (Po). Minerals consisting essen- tially of atoms from this group will be eliminated from this investige.tion. The ionic group have cores whose electron configuration is that of the inert gases, hence with an outer shell of eight electrons. As previously shown the bonds between such ions in the solid state can be considered as of an electrosta.tice.l nature. The superimposed effects of a dipole field can be greatly reduced by restricting the anions to those of the smallest atomic radii, namely oxygen o= and fluorine F-. The hydroxyl ion OH-, where it is a subordinate consti- tuent as in the amphiboles, will be assumed to give no appreciable polarization effect. 13. The fundamental nature of this two-fold grouping has been t•igat•ions. 54,55,56 . 1 inves . sh own b y severa1 recen t geoch enuca As a basis for systems.tic application of the clea.vabili ty expression, ionic minerals treated in this study a.re classified a.s follows: A. Simple ions I. Binary system, A - X. Class 1. A : x = 1. Villiaumite, NaF. Bromelli te, BeO. Periclase, MgO. Lime, Cao. Class 2. A : x (Halite structure) (Wurtzite structure) (Halite structure) (Halite structure) = 2 : 3. Y' Corurii.dum, Al 2 0 3 • Class 3. A : x = 1 : 2. Sellaite, MgF2 • (Rutile structure) Fluorite, CaF2 • Quartz, Si~. Cristobalite, Si02 • Tridymite, Si02 • Rutile, Ti02 • Octahedrite, Ti02 • Brooki te, Ti02 • II. Terne.ry system, A - B - X. B. Complex radical, ARX 3 • Ca.lei te, CaC0 3 • c. Silicates. 57158 I. Independent tetrahedral groups. Class 1. Single Si0 4 groups. Phena.$ite, Be2 Si0 4 • Kye.nite, A1 2 0Si0 4 • Topaz, Al 2 Si0 4 F2 • Zircon., ZrSi0 41 • Grossula.rite (garnet group), Ca. 3 Al2(Si0 4 ) 3 • 14. c. Silicates (cont) Class 2. Si2 07 groups. Me l i l i te, Ca2 MgSi 2 07. II. Tetrahedral chains. Class 1. Single chains. Diopside (pyroxene group), CaMgSi2 0 6 • Class 2. Double chains Tremolite (amphibole group), C~Mg 5 (Si 4 0,, ) 2 (0H)2. III. Tetrahedral planes. lviuscovi te, KA1 2 ( Si 3 Al )0 10 ( OH.,F) 2 • IV. '.l.hree-dimensional network o~ tetrahedra. Soda.lite, Na.g,Al 3 Si 3 0,2Cl. 15. 5. Structure and Cleavage. For most of the structural arrangements given below reference is made to Ewald, P.P. and Herman~ .. c., "Strukturbericht, 1913-1928," Z.Krist. Ergl:inzungsbe.nd (1931). In the comparison of calculated relative values with observation the work of Dana (see Ref. 1) is used. It is found that electrical neutrality of cleavage surfaces where it does not follow as a result of a plane of cleavage is obtainable by a non-planar cleavage surface; possible cleavage surfaces are considered to be electrically neutral. In cases where a cleavage form has several alternative cleavage surfaces the most probable is taken to be the one giving the highest w.lue for the cleavability. C1eavability values marked by a star denote cases where it is probable that the angular change i .n e cannot be neglected. Vil~aumite, Periclase and Lime. . r-' Cub1c, 'c Structural characteristics; , Oh5 • Z = 4. Villiaumite, NaF, a= 4.62 X; periclase, MgO, a= 4.20 !; lime, CaO, a= 4.80 K. Coordination; octahedra of anions. Areas: Villiaumite, A(lOO) = 21.15 Periclase, A(lOO) A:, A(llO) = 33.65 !~ A(lll)= 36.65 = 17.63 K A(llO) = 24.93 !, A(lll)= 30.54 A 1 A( lOO) = 23. 00 ! Lime, Bond strengths: !: Villiaupli te, sNa-F 2 , A( llO) = 32. 52 !; A( lll )= 39. 84 .!2 = 1/6. Periclase, sMg-O = 1/3. Lime, sCa-O = 1/3. Summations: Villiaumite, Z (100) = 4 x 1/6 x 1.00( e =0°) = 0.67,i:(llO) = s x 1/6 x o.n( E> =45°) = o.95, ~ (111) = 12 x 1/6 x 0.82( e =35°) = 1.64. Periclase, L (100) =:= 4 x 1/3 x 1.00( e =0°) = 1.33, z.. (llO) = ax 1/3 x o.n( e =45°) = 1.89, z: (n1) = 12 x 1/3 x o.s2(e =35°) = 3.28. Lime, same as periclase. 16. TABLE 1. Clea~abilities of Villiaumite, Periclase and Lime. Form [ 100 } { 110 J f 111 } Type clo s ed cubic closed dode c ahedral closed octahedral 6 12 8 31.5 31. 5* 22.3 complete - - Number of f a ces Villiaumite Cale. Cleave.bili ty Obs. 59 Pericle.se Cale . 13.2 13. 2* 9.3 Obs. Perfect - Less distinct Cleavabili ty Lime Cale. Cleavability Obs. 60 17. 2 17. 2* 12.l Complete Possibly - Remarks: The separation surfaces for { lOOj are coplanar ions. 17. Bromellite Structural characteristics: Hexagonal, f"h Lattice constants: a= 2.69 K, C = 4.37 !. Composition: Coordination: BeO. Be-0 tetrahedra. 2 2 = 6.24 K J A(1010)= 11.75 K J A(lOll)= 13.4 ! Areas: 2 J 2 = 20. 35 ! • Bond strengths: SUr;Jillations: sBe-O = 1/2. 2_ (lOlo) = 2 x 1/2 x 0.94( 6 =20°) = 0 94,, ~ (0001) = 0 z:: (1120) = 4 x 1/2 x 0.87( e =30°) = 1.74, z (1011) i 1x1/2 x i.oo( e =0°) + 2 x 1.2 x 0.11( e =45°) = 1.21 1x1/2 x 1.00( €> = 0°) = o.50, TABLE 2. Cleavability of Bromellite. Form tienoJ { 0001) tll20} \1011) Type open prismatic open basal open prismatic closed bi"Iframidal ,. 6 2 6 12 12. 5* 11. 7 11.1 Doubtful - - Number of faces Cleavabili ty 12.5 Cale. Obs. Remark: Distinct I According t o Groth (see Ref. 6D) there is no distinct cleavage. 18. Corundum Structure characteristics: a= 5.12 'J..., Lattice constants: Compo sition: r;.h Trigonal, 0( Al 2 0 3 • Coordination: = 55°17'. Al-0 octahedra. ·· 2 A(lOO) = 18.16 K , A(lll) = 19.20 ! Areas: A( 211 Bond strength: Summations: 2-. 2 , A(llO) = 49.28 2 'J... , 2 = 60. 57 !, A( lOl) = 104. 79 'J... • sAl-O = 1/2. (110) = 8 x 1/2 x 0 , 98( $ = 10°) = 3.92, Z (211) =ax i/2 x 0.11( e = 40°) +ax 1/2 x o.n( E> = 45°) = 5.92, Z:. (111) = 3 x i/2 x 0.11( 0 =45°) + s x i/2 x o.64( e =50°) = 2.02, ~ (10D = 6x1/2 x 0.11( 8 =40°)+6x1/2 x o.n( e =45°) + 12 x 1/2 x o. 64( C> =50°) + 12 x 1/2 x o. 50( (:) =60°) = 11. 28, z. (100) = 2 x 1/2 x 0.42( 0 =65°)+,2x1/2 0.71( 0 =45°) + % 2x1/2 x 0.98( 9 =10°) = 2.11. TABLE 3. Form Cleavability of Corundum x-ray unit i 110 1 { 2IT l i lll ) l lOl j { lOO J Dana fl oI1} [ lOlO} l OOOl} !_1120 } \ 0221 J closed rhomb6hedral Open prismatic Open basal Open prismatic Closed Dhombohedral 6 6 I 2 6 6 Cale. 12.4 10.2 I 9.5 9.2 8.6 Obs. - - - - - Type Number of f a ces Cle avability Remarks: l Parting on ~ HO ! often prominent. 19. Fluorite Cubic., f:c I Structure characteristics: Lattice constant: a z = 4. = 5.45 X. Composition: Ca.F2 • Coordination : Ca-F hexahedra. 2 Areas: A(lOO) = 29.7 K2 , A(llO) = 42.0 A2 , A(lll) = 51.4 A • sca-F = 1/4. Bond strength: Surrunations: L (110) = 8 x 1/4 x o. 71( 0 -=45°) = 1.42, ~ (n1) = s x 1/4 x l.oo( e =0°) = 2.00, ( 0 =54-0 ) z (100) - s x 1/4 x o. s11 = 1.15 TABLE 4. Form - Cleavabili ty of Fluorite. f 110 1 closed dodecahedral Type Number of faces Oa.lc. l_ 111 ~ i 100 ! closed octahedral closed cubic 12 8 6 29. 7* 25.7 25. 7* perfect - Cleava.bili ty Obs. occasional~~ . distinct 20. Quartz Areas: \0001) Si-0 tetrahedra. = 21.8 ! Si02 • High temperature modification. 2 2 , A(llZO) = 47. 6 1t2 • Bond strength: sSi-O = 1. Summations: z = 3. a = 5.01 .ll, c = 5.47 A.. Composition: Lattice constants: Coordination: 4 or n • 6 Hexagonal, Structure characteristics: A(lOlO) = 27.40 K , A(l Oll) = 35.00 !, L (1011) = 1 x l x 0.87( e =30°) + l x 1 x 1.00( e =0°) = 1.87,, L. (112'0) = 2 x 1 x o.so( t> =60°) + 2 x 1 x o.87(6 =30°) = 2.14, ;£ (0001) = 2 x 1 x o.64(6=50°)=1.28, Z... (1010) = 1x1 x o.87 ( '9 =30°) + 1 x 1 x 0.82(8 = 35°) = 1.69. TABLE 5. Cleavability of Qu artz. Form ~ lOll J ! 112'0 } l 0001 ) f lOlO } Type closed ' bipyramidal open prismatic open basal open prismatic 12 6 2 6 Cale. 18.7 17.4 17.0 16. 2 Obs. difficult and seldom observed more difficult more difficult - - Number of f a ces Clea.vabili ty Remark: According to Rogers 63 imperfect { lOll ! cleavage is rather common, especially in thin section. 21. Cf'iristobalite Structure characteristics: Lattice constant: Coordination: Cubic, a= 7.12 .K. 'c' z = 8. 7 , Oh, •~ Composition: Si-0 tetrahedral fra.r_~ework. Si02 , High temperature modification. !i'2 Areas: 87.85.a. Bond strength: sSi-O Summations: Z, (100) = 1. = 4 x 1 x 0.57(6 = 55°) = 2.28, ; 3. :l ~ Z. (llO) = 4 x 1x0,82(6 = 35°~, ~ (lll) = 4xlx1.00 ( $ = 0 °) :::: 4. 00 TABLE 6. Cleavability of C~ristobalite. Form ! 100 1 llll} Type closed cubic closed octahedral 6 a 12 Cale, 22.2 22.0 21,9 Obs. - - - Number of faces Cleavability ( 1101 closed dodecahedral 22. Tridyfllit e Structure characteristics: Latti ce constants: Coordination: fh , D:h. Hexagonal, a= 5.03 !, -· Z = L1. Co~nposition: c = 8.22 1. Framework of Si-0 tetrahedra. SiOz. High temperature modification. Areas: 2 2 = 41.35 K' A(lOll) = 46.5 K . A(OOOl) A (1120 - ) Bond strength: sSi-O = 1. Summations: ~ (1010) = 2 x 1 x 0.94( 8 = 20°) = 1.88, Z (0001) = lxl x i . oo( C> = 0°) = i.oo, Z (1120) = 4 x 1 ¢ o.87( e = 30°) = 3.48, 2.. (1011) = i + 1 x i.oo( e =0°) + 2 x i x o.64( 9 =50°) = 2. 28. TABLE 7. Cleavability of Tridymite. po11 1 Fonn t 1010 1 t 0001 } { 1120 } Type open prismatic open basal open . prismatic closed bi pyramidal 6 2 6 . 12 Cale. 22.0 21.9 20.6 20,4 Obs. not distinct - - - - Number of faces Cleavability Remark: Parting sometimes observed II {OOOlJ 23. Sellaite and Rutile Structure characteristics: Lattice constants: Tetragonal, , n14 • 411 ~ Z = 2. = 4. 64 !, c = 3.06 !; Sella.ite, MgF2 , a Rutile, Ti02 , a= 4. 58 .ft., c = 2.95 !. Areas: Sellaite, A(lOO) = 14.2 1, A(llO) = 20.1 1, A(ooi) = 21.53 !, = 29.41 !. Rutile, A(lOO) = 13.51 !, A(llO) = 19.18 !, A(OOl) = 20.94 !, A(lll) :: 28.40 !. A(lll) Bond strengths: Surrunations: sMg-F ::: 1/3, sTi-O = 2/3. Sellaite (1/2 those of rutile), :£. (100) = 0.47, L (110) = 0.67, Z (001) = 0.94, £ (lll)::: 1.76. z_ (100) = 2 x 2/3 x o.n( a =45°) = o.94, i.oo( e =0°1=1.33, Rutile, ~ (110) = 2 x 2/3 x z (001) = 4 x 2/3 x o.nc 0 = 45°) = 1.ss, r (111) = 2 x 2/3 x i.oo( a =0°) + 2 x 2/3 x o.64( e =50°) + 4 x 2/3 x o. 50( €> =60°) = 3. 52. TABLE 8. Cleavability of Sellaite and Rutile. Fonn f 100 ? Type open open prismatic prismatic Number of faces 4 { 110 } 4 i OOl ~ l lll } open basal closed bi pyramidal 2 8 Sellaite Cle avability Cale. 30.l 30.l 23.2 16.7 Obs. perfect perfect - - Ru tile Cle avability Cale. 14.4 14.4 11.1 8.1 Obs. distinct distinct - traces 24. Octahedrite Tetragonal, Structure characteristics: Lattice constants: Coordination: rt.I , n19 4h. z = 4. a. = 3. 73 K, c = 9. 37 !. Composition: Ti02 Ti-0 octahedra . = 13.94 ! 2 , A( 100 ) = 34.93 ! 2 , A( 101 ) = 37.60 K2 , A(ll O) = 49.40 !2. Bond strength: sTi-O = 2/3. Summations: ~ (101) = 4 x 2/3 x 0.82( 8 = 35°) = 2.19, Z (001) = Areas: A (001) ix 2/3 x i.oo( e =0°) + 2 x 2/3 x o.11( e =80°) = o.9o, 4 x 2/3 x o.98( e:> = io 0 ) = 2.s1, ~ (110) = 8 x 2/3 x 0.11( e =45°); 3. 84. TABLE 9. Cleavability of Octa..~edrite. x-ra.y { 101 ~ { 001} l lOO J \llO J Dana. i lll ~ f OOl) f llO } i lOOJ closed bip;>rramida.l open basal 8 2 4 4 Ca.le. 17. 2 15.5 13.1 12. 9 Obs. perfect perfect - - Fonn Type Number of faces Cleavibili ty 2. (100) = open open prismatic prismatic 25. Brookite 64 Structure characteristics: Orthorhombic, r10 , 15 Vh • Z =8 Lattice constants: a= 9.16 !, b = 5.44 A, c = 5.14 A. Composition: Coordination: Ti02 • Ti-0 octahedra. = 47.02 fl!', A(OOl) = 49.83 ! 2 , A(110) -- 54.8 0 !2 .. A(210) -- 73.25 A~2 .. A(lll) = 74.10 A~2 • Bond strength: sTi-O = 2/3. Surrunations: Z. (210) = 6 x 2/3 x 1.00( e =0°) = 4.oo, Z (110) = 3 x 2/3 x A(lOO) = 27.96 Ji.2, A(OlO) Areas: i.oo( e = 0°) + 3 x 2/3xo.10( e = 45°) = 3.40 .. z (111) = 14 x 2/3 x 0.10( e = 45°) = 6.53, z (001) =lox 2/3 x 0.10( e =45°) = 4.67, 2. (010) =ax 2/3 x o.81( e =30°) = 4.64, Z° (100) = 6 x 2/3 x o. 75( e =40°) = 3. oo. TABLE 10. Cleavability of Brookite. x-ray 1 21o l ~ 110 5 f lll J f OOl } Dana {110 } f l20 ~ l l21 ) ~ 001 ) Form Type ( lOO J t 010 ! f lOO } I open open closed open open open brismati1 prism. bipyram. basal pinacoid. pinacoidal Number of faces Cle avability II f 010 } Cale. 4 4 8 18.3 16.1 11.0 2 10. 6 I Obs. indi sti1 ct - - l more ~ iindistin t 2 2 10.1 9.3 - - 26. Spinel Lattice constant: Coordination: ~ Cubic, Structure characteristics: a= 8.09 !. 7 I I Oxi• z = 8. Composition MgA12 0 4 • Mg-0 tetrahedra, Al-0 octahedra. 2 Areas: 2 9 2 A(lOO) = 65.45 K , A(llO) = 92.28 K, A(lll) = l ~.64 K. Bond strengths: Su.rn.mations: io.39, sMg-O = 1/2, sAl-O = 1/2. L._ (111) = 8x1/2x1.00( 6 =0°) + 18 x 1/2 x 0.71( <3 = 45°) = z= (110) = 16 x 1/2 x o.n( e = _45°) + 4 x 1/2 x o.77( e =40°) = 1.22, L (100) =ax 1/2 x l.oo( e =0°) + 4x1/2 x o .57( 6 ==-55°) = 5.14. TABLE 11. Clea.vability of Spinel. Form f lll J f 110 } ~ 100 } Type closed octahedral closed do de ca.he dral closed cubic 8 12 6 Cale. 15. 3 12.8 12.7 Obs. imperfect - - Number of faces Cleavabili ty Remark: The reflection surface * for ~ lllj is poor. 27. Calcite Structure characteristics: Areas: r.:~ z = 2. a = 6. 36 !, o( = 46°7 1 • Lattice constants: Coordination: Trigonal. C-0 triangles, Composition: Ca.C0 3 • Ca-0 octahe.d ra. = 22.02 K2 , A( 2'il) = 87.54 K2 , A(lol) = 151.43 K2 , A(lll) 2 A( 2ll) = 187.52 K. Bond strengths: C-0 unbroken, sca-O = 1/3. = 16 x 1/3 x 0.94( 8 =20°) = 5.01, L (lll) = 3 x 1/3 x 0.64( c;) =50°) = 0.64, L.. (2IT) = 8x1/3=0.71(<9 ~ 45°)+4x1/3 x <).64( 8== 50°) = 2.75,, Z (101) = 24 x 1/3 x 0.77( S =40°) = 6.16. Summations: Z (211) TABLE 12. Cleava.bility of Calcite. x-ray unit { 211 1 t 111 J i 2IT ~ { lOl J Dana ~ 1011~ { 00011 { 1010 l t 1120 closed open basal open prismatic open prismatic 6 2 6 Form 1';pe rr ombohedral Number of faces I J 6 I Cale. 35.4 Obs. highly perfect Cleavability Remark: 34.4 31.8 24.6 - - - Ions of the { 2llj surface are coplanar. 28. Phenacite Structure characteistics: Trigonal, Composition: Be 2 Si0 4 • Be-0 tetrahedra, Si-0 tetrahedra. A(lOO) = 56.09 ! Areas: z = 6. = 7. 68 !, 0( = 108°1'. Lattice constants: a Coordination: r;:" 2 2 2 , A( 2ll) = 102.21 ! , A(lll) = 134.19 ! , 2 A(lOl) = 176.82 K. Bond strengths: Summations: L sSi-O = l (unbroken), sBe-O (100) = 2 x 1/2 x 0,98( $ = 1/2. =0°) + 2 x 1/2 x 0.57 (&..:- 55°) + l x 1/2 x 0.11( 6== 45°) + 2 x 1/2 x o.34( $== 70°) + l x 1/2 x o.98 ( e =10°) = 2.74, L. (111) = 16 x 1/2 :iti o.82( e;: 35°) = 6.56, .[ (101) = 12x1/2 x 0.98(61:==10°) + 12x1/2 x 0.57(C7· =55°), L. (2IT) = 4 x 1/2 x o.57( e = 55°) + 8 x 1;2 x o.81( e= 30°) + 2 x i/2 x o. 77( e =40°) TABLE 13. = 5.39. Cleavability of Phenacite. x-rtJ.y t 100 } \ 111 } f 1or 1 t 2IT } Dana f lOll } I 0001~ { 1120 l l 1010} closed 1hombohedral open--·.:. basal open prismatic open p ri srna. tic 6 2 6 6 Cale. 20.4 20.4 19. 0 18.9 Obs. imperfect - distinct - Form Type Number o:r faces Cleava.bil i ty Remark: According to Niggli 65 ~ lOO J very imperfect,, llOl } not very distinct. \lll j .perhaps, 29. Kyanite Structure characteristics: 66 , 67 , 68 =90°5.5',,,& =101°2 1 , . Composition: Areas : Al 2 0Si0 4 • A(OlO) = 38.64 ! Bond strengths: Summations: Z 2 , , Ci•l z = 4. r;~ = 5.56 !. a= 7.09 !, b = 7.72 !, c Lattice constants: o( Triclinic, ( =105°44.5' • Coordination: Al-0 octahedra, Si-0 tetrahedra. A(lOO) = 42.90 ! 2 , A(OOl) = 53.54 ! 2 • sSi-O = 1 (unbroken), sAl-O = 1/2. (100) = 8 x 1/2 x o. 71( 6::.45°) = 2.84, ;E._ (010) = 6 x 1/2 x 0.98(8~10°) + 4 x 1/2 x Q.11( e:::: so ) = 3.28io 0 z (001) = 4 x 1/2 x o.26(0"'75°) + 4 x 1/2 x o.n(e"'45°) + 6 x 1/2 x o. 87( €>:;;: 30°) = 4. 55. TABLE 14. Cleavability of Kyanite. {OlO J l 001 ! open pinacoidal open pinacoidal open basal 2 2 2 15.1 ll. 7 ll. 7 very less perfect - Form f lOO J Type Number of faces Cale. Cleava.bili ty Obs. perfect Remarks: Ions of the [ lOO J surface are coplanar. Parting/{ f OOlJ • 30. Topaz 69, 70 . Structure chara.cte ri sti cs: Orthorhombic, Areas: , 16 Vh • Z = 4. 8 a= 4.64 !, b = 8.7« A, c = 8.37 !. Lattice constants: Coordination: r J0 Si-0 tetrahedra, Al-0 octahedra. 2 2 2 A(OlO) = 38.84 K, A(OOl) = 40.74 K 1 A(lOO) = 73.48 K, 2 A(llO) = §3.12 K. Bond strengths: ssi-0 = l(unbroken); sAl - O = 1/2, sAl-F = 1/2, sAl-O = 1/4. Swnmations: 2_ (001) = 4 x 1/2 x o. 71(8 ::: 45° ) = 1.42, 2._ (100) = 8 x 1/2 x o.65( 8= 50°) + 4 x 1/4 x o.11(e =45°) = 3.21, Z.. (010) = 4 x i/2 x o.nc e =45°) + 4x1/4 x 0.11( e =45°) = 2.13, :E. (no)= 3 x 1/2 x 1.00( $ =0 °) + 8 x 1/2 x 0.11( 8 =45°) + 4 x 1/4 x 0.71(6 "'45°) = 5. 05. TABLE 15. Cleavability of Topaz. Form l 001 ! Type open basal Number of Faces Cale. 2 28.6 Cleavabili ty Obs. {100 I open fpinacoidal I I I I ! highly perfec~ Remark: 1 } f OlOJ ! ll01 open pinacoidal open prismatic 2 2 4 22.4 18.1 lo. 4 - - - Ions of { 001 ~ surface are coplanar. 31. Zircon a= 6.58 !, c = 5.93 !. Lattice constants: Coordination: Areas: 19 , D411. Tetragonal, Structure characteristics: Si-0 tetrahedra., z = 4. Composition: ZrSi0 4 • Zr-0 hexahedra.. 2 A(lOO) = 39. 02 -8.. , .A(OOl) = 43. 30 ! 2 2 , A(llO) = 55.15 f... , = 58. 30 ! A(lll) = 70.14 A Bond strengths: ssi-0 = 1 (unbroken), sz r -0 = 1/2. 2 A(lOl) Sunnnations: 2._ (100) 2 , • = 4 x 1/2 x 0.98( 8 =10°) = 1.96, L_ (llO) = 8 x 1/2 x 0.71(8:::45°) = 2.84, <s -= :ra z (101) = 4 x 1/2 x 0.71( 6 =45°) + 0 ) z (001) = 8x1.2 x 0.85( 6=32°) + 4x1/2 x 8x1/2 x 0.64.., = 3.98, o.14(ch s2°) = 3.68,, 2. (lll) = s x 1/2 x o.82( CJ =35°) + 8 x 1/2 x o. n(e =10°) + 6 x 1/2 x o.34( e =10°) = 1.14. TABLE 16. Clea.vability of Zircon x-ra.y { 100 l ~llO J po1 1 f 001 ~ f 111 ~ Dana f_ no J t lOO } I 111 } f 001 } { 221 ) Form open closed open open prismatic prismatic bipyra.m. basal Type Number of faces closed bi pyramidal 4 4 8 2 8 Cale. 19. 9 19.4 14. 6 ll. 7 9.8 Obs. imperfect less distinct - - Cleavability I - 32. Grossularite Structure characteristics: Lat tice constant: Coordination: Areas: Summations: ~ a = ll. 83 1{. Si-0 tetrahedra, A(lOO) = 139.95 ! Bond strengths: Cubic, 2 , II 1:0 , Oh • Z = 8. Composition: Ca 3 Al 2 ( Si0 4 ) 3 • Al-0 octahedra, Ca-0 hexahedra. A(llO) = 19 7.90 ! 2 , A(lll) = 242.5 ! sSi-O = 1 (unbroken), sAl•O = 1/2, sca-O 2 • = 1/4. (llO) = 4 x 1/2 x 0.98( e =10°) + 8 x 1/2 x 0.71( &:::- 45°) + L 8 x i/2 x 0.26(0 =75°) + 12 x 1/4xo.n( d= 45°)+4x1/4 x 0 0.87( $:; 30) = 8.84, z z (100) % 8 x 1/2 x 0. 94( 8 =20°) + 8 x 1/2 x 0.26(8 ::: 75°) + 8 x 1.4 x 0.87( C> =30 °) + 8 x 1/4 x 0.98( G =10°) = 8. 50. TABLE 17. Cleavability of Grossularite. i llO ~ Form Type J i lll J closed dodecahedral cl os e d cubic closed octahedral 12 6 8 Cale. 22.4 16.5 Obs. sometimes rather distinct - Number of faces Clea vabi li ty Remarks: { 100 probably less than p oo} - It is uncertain whether ll lO } is cleavage or parting. The limit of error for the calculated [ lll J cleavabili ty value is large. 33. Meli'hte . t·ics: St rue t ure ch arac t eris 71 T t e ragona1 , z = 2. a= 7.73 X, c = 5.01 !. Lattice constants: Composition: (Ca.,Na) 2 (Mg,Al) 1 (Si,Al) 2 0 7 • Coordination: Si-0 tetrahedra, Mg-0 tetrahedra, Ca-0 hexahedra. 2 2 2 Areas: A(lOO) = 38.73 K, A(llO) = 54.76 ! , A(OOl) = 59.74 K, Bond strengths: sSi-O = 1 (unbroken), sMg-O = 1/2, sca-O = 1/4. = 1,68, f (llO) Summation: Z (001) = 16 x 1/4 x 0,42( 8 =65°) =2 x 1/2 x 0.17( 6 =80°)+2x1/2 x 0.91( 0 =25°)+4x1/4 x o.64( e=- so ) + 2 x 1/4 x o.71( e =45°) = 2.01, Z 0 o.91( e = 25°) + s x i/4 x o.64( e =50°) TABLE 18. (100) = 2 x 1/ 2 x = 2.19. Cleavability of Melilite. x-ray f 001 ~ {llO J f 100} Dana. 1 0011 £100J f llOj open basal open prismatic open prismatic 2 4 4 Cale. 35.5 26. 5 17. 6 Obs. distinct indistinct - Form Type Number of faces Cleavability Remark: Ions of the { 001 ! · surface are coplanar. 34. Diop side Structure characteri sties: 72 Monoclinic,, r' l1'o\ ,, c62h Z = 4. a= 9.71 !, b = 8.89 !,, c = 5.24 !,I' = 74°10'. Lattice constants: Composition: Ca.MgSi2 0 6 • Coordination: Si-0 te;t rahedra_, Mg-0 octahedra, Ca-0 hexahedra. Areas: A(lOO) = 46.58 ! 2 , A(OlO) = 48.84 ! 2 , A(llO) = 67.50 ! 2 _, 2 A(ool) = 86.33 1 • Bond strengths: sSi-O = 1 (unbroken for t lOOJ and l llO~ ),, sMg-O = 1/3, s Ca-O = 1/4. 2_ (110) j° 8 x 1/3 x 0.64( e= 50°) + 8 x 1/4 x 0.50( e =60°) + Summations: 4 x 1/4 x o.94( e:::; 20°) = 3.64_, .Z (001) = 4 x 1 x o.87( e =30°) + 2 x 1/3 x o.98( e =10°) + 4 x 1/4 x o. n( e = 45°) = 4.85, z (100) = 6 x 1/3 x 0.11( e =45°) + 2 x 1/4 x o.64( $-== 50°) + 2 x 1/4 x o.so( e =60°) + 4 x 1/ 4 x o.n( e =45°) = 2.10, z:. (010) = 4 x 1 x o.34( GJ =70°) + 4 x 1/ 3 x o.71( e =45°) + 4 x 1/4 x o.n( e 45°) = 3. 02. Table 19. Cleavability of Diopside. Form t 1101 t_oo1J { lOO ! t OlO} Type open prismatic open basal open pinacoidal open pinacoidal 4 2 2 Number of faces 2 - Cale, 18.5 Cl eavabili ty Obs. Rema rks: rather perfect ( srnnetimes) but interrupted 17. 8 17. 2 - - I i 16.3 - ~ {1101 often only observed in thin sections ..1. C • Parting on f 0013 often very prominent, on ~ 100 } less distinct and less common. 35. Tremoli te 73 St rue t ure ch ara ct eris · t ics: · , -once 1inic, · · M r I 1""" <orien · t e d as a b o dy centered lattice in order to agree with the usual crystallographic axes), 78 Lattice constants: a= 9.~ A, b = 17.8 K, c = 5.26 &,13 = 73°58'. Composition: CEl.:;,iMg 5 (Si 4 0,,) 2 (0H) 2 • Coordination: Si-0 tetrahedra, Mg..()octahedra, Ca-0 hexa.hedra. Areas: A(OlO) = 49.38 ! 2 A(ool) = 174.08 ! Bond strengths: A(lOO) = 93.63 !~ , 2 A(l lO) = 104.60 K , 2 • sSi-O = 1 (unbroken for f llO ~ , [ lOOJ and {OlOj ) sMg-O = 1/3, sMg-OH = 1/2 sMg-O = 1/6, sCa.-O = 1/4. Surnm.ations: 2.. (110) = 12 x 1/4 x 0.64( e =so 2.37, L. (100) = 12 x 1/4 x o.64( B =so 0 ) 0 ) + 4 x 1/3 x 0.34($ =70°) + 4 x 1/3 x o.34( e =70°) = 2.37, ~ (010) • 4 x 1/3 x o.71( e =45°) + 4 x 1/4 x o.64( e =50°) = 1. 59, z (001) = s x i x i. oo( e =0°) + 8 x 1/4 x o. 64( e =50°) + 4 x 1/ 3 x o.87( e =30°) + 2 x l j 6 x o.87( e =30°) = io. 73. TABLE 20. Cleavability of Tremolite. Forrn { 110 1 l lOO J f 010 J \001} Type open prismatic open pina.coidal open pinacoida.l open basal 4 2 2 2 Ca.le. 44.1 39. 5 31.0 16.6 O:bs. highly perfect sometimes distinct sometimes distinct - Number of faces Clea.va.bili ty Remarks: See discussion. 36. Muscovite • t•ics: 74,75,76 St rue t ure c h ara.c t eris Lattice constants: • 1Mv1onoc i·inic, r::' ,.,..,., , c621'. z = 4• a= 5.19 !, b = 9.00 ~. c = 20.04 K,fi Composition: KA1 2 ( Si 3 Al )0 10 ( OH,F) 2 • Coordination: Si,Al-0 tetrahedra, Al ~O,OH) octa.hedra; VK, , Areas: 2 A (OOt) = 46.8 K , A (OtC>) = 103.0 fi 2 , A (/oo) = 95°30 1• = 12. 2 = 180.0 ft. • = l (unbroken), sA1 _ 0 (tetra.hedron) = 3/4, sAl-O(octa.hedron) = 1/2, sAl-(OH,F) = 1/2 . sAl-O = 1/4, sK-O = 1/12. Bond strengths: sSi-O (001)=12 x 1/12 x 0.64( B =50°) = 0.64, Z (100) = Summations: 2 6 x 3/4 x o.71( e =45-~) + s x 1/2 x o.n( e =45°) + s x 1/4 x = 1.1s, ~ (010) = 3 x 3/4 x i.oo( e =0°) + 4 x 1/2 x o.71( e =45°) + s x 1/ 4 x o.57( $ =55°) = 4.s1. o.57( e =55°) TABLE 21. Cleavability of Muscovite. x-ray t 001 1 t lOO } {OlO J Dana { 001 } f. 201 } t OlOJ open basal open pina.coida.l open pina.coidal 2 2 2 Ca.le. 73.1 25. 0 21.4 Obs. eminent - - Form Type Number of faces Cle a va.bi li ty Remark: Fourier analysis indicates that the ions of the separation surfaces for t OOl~ are exactly coplanar. 37. Sodalite Structure characteristics: 77 ' 78 body-centered lattice), A(lOO) = 78 68 ! 2 , 0 Bond strengths: lc (closely approximates a. 4 1 Td (possibly Td). z = 2. Si-0 tetrahedra, Al-0 tetrahedra, Na-0 hexahedra. Coordination: Areas: Cubic, sSi-O A(llO) = 108.72 ! 2 , A(lll) = 136.12 !~ = 1 (unbroken), sAl-O = 3/4, sNa-O = 1/8. Swruua.tions: 2: (110) = 4 x 3/4 x 0.71( e =45°) + 2x1/8 x 0.71( e =45°) ~ = 2.39, L. (100) = 2 x 3/4 x 0.11( e =40°) + z x 3/4 x o. 71( e =45°) + 4 x 1/8 x o.34( e =70°) = 2.40, ~ (lll) = z x 1/8 x o.34( c9 =70°) 4 x 3/4 x o. 87( e =30°) + 2 x 3/4 x o. 98( e =10°) + 1 x 1/8 x i.oo( 0 =0°) = 4.21. TABLE 22. Cleavability of Soda.lite. Form l llO ~ { lOO J t lll! Type closed dodecahedral closed cubic closed octahedral 12 6 8 Ca.le. 45.5 32.7 32.2 Obs. more or less distinct - - Number of faces Cleava.bili ty 38. 6. Discussion of Cleavage . Comparative cleavabilities and data on the cohesive properties of the ionic minerals studied are given in Table 23. '.l.'he twenty-five minerals listed are arranged in t he order of their high est cleaV8.bility. The calculated cleavabilities for the different forrns, taken from the foregoing tables,, ar e contained in the second column, observed cleavage being denoted by underlines, three for very good, two for good, and one for poor. 'Ihe table shows that the general agreement between cal- culated cleavability values and observed cleavage is good. The best observed cleavage for each species is for the form witzy the highest calculated cleavability va lue, except those of fluorite, lime, periclase and bromellite. neglect of angular change in e These anomalies are probably due to the during the process of rupture; since the treatment is in the nature of a first order a pproximation it is to be expected that such cases may occur. In the case of fluorite there may exist the additional effect of a good reflecting surface for a form of lower cleava.bility in the sequence. The lack of observed cleavage for cristobalite and corundum is discussed later (see Table 24). Second order effects due to impurities, growth conditions and mechanical strain such as gliding set up during the process of cleavage although difficult to evaluate do not seem to materially affect the results. The agreement of muscovite is excellent; its high clea.vability is due to the comparatively weak K-0 bonds which hold strong layers together; the statistical alternation of K;- ions a.cross the cleavage surface results in electrical neutrality. An extreme case of this nature is that of talc whose stntcture consists of neutral l ayers held together by second order electrical effects; since t here are no direct 39. TABLE 23. Comparative Clea.vabilities and Cohesive Properties of Ionic Minerals. Species Sequence of cleava.bilities Muscovite 73.l 25.0 Soda.lite 45.5 32.7 Tremolite 44.1 Melilite Calcite Total no. of planes Hardness Fracture Tenacity 21.4 6 2.00-2.25 - - 33.2 26 5.50- 600 uneven to subc. Brittle 39. 5 31. 0 16.6 --35. 5 26. 5 17. 6 -- - - 10 5.00-6.00 10 5.oo 35.4 20 3.00 26 < 3.00 -- - = 34.4 31.8 24.6 fl Brittle uneven to conch. conch. (with dif.) Brittle - - Villiaumi te 31. 5 31. 5* 22.3 Sellai te 30.1 30.l 16. 7 18 5.00-6.00 conchoida.l Brittle Fluorite 29. 7* 25.7 25. 7* --28.6 22.4 18.1 16.4 -22.4 16.5 -- 26 4.00 flat-conch. Brittle 10 8.00 uneven to subc. Brittle 18 6. 50-7. 50 II Brittle 26 6.00-7.00 - - 20.4 26 1.00 conchoida.1 Brittle Phena.cite 20.4 20.4 19. 0 18.9 --- 20 7.50-8.00 conchoida.l Brittle Zircon 19.9 26 7.50 conchoidal Brittle Quartz 18. 7 17.4 17.0 -17.8 l'l. 2 16. 3 -18.5 26 7.00 StlbC 0 to Brittle Topaz Grossularite Cristobe.lite Tridymite Diop side Brookite -- -- 23.2 22.2 22.0 21.9 22.0 21.9 20.6 -- 19.4 14.6 - - ? 11. 7 9.8 16.2 12 5.00-6.00 18.3 16.l 11.0 10.6 ? 22 -9.3 · 10.l 5.50-6.00 conch. uneven to conch. subconch. 5.50-6.00 subconch. Brittle - - - Brittle Octahedrite 17. 2 15. 5 Lime 17.2 17.2* 12.1 26 . Spinel 15.3 12.8 12.7 -- 26 a.oo Kyanite 15.1 6 5.00-7.25 - - 18 6. 00-6. 50 I uneven to Brittle -- -- Rutile - -- 11. 7 13.1 12.9 11. 7 ---14.4 14. 4 11.1 8.1 - - -- Pericla.se Bromellite 13.2 12.5 13. 2* 9.3 12. 5* 11. 7 Corundum 12.4 10.2 -- 9.5 ? 11.1 9.~ z 18 Brittle 26 26 6.00 26 9.00 9.oo 8.6 Note: Observed cleavage is denoted by underlining. -:= very good = good - poor. conchoida.l I i subc. uneven to conch. I Brittle Brittle 40. bonds the cleavability value is very high. In the case of diopside although there are single chains of Si-0 tetrahedra parallel to the c-axis the numerical values for the possible cleavage forms are fairly close together indicating a compact structure (see Table 24); this is confirmed by the observed frequent failure of the prismatic {110 } cleavage except in thin section. Tremolite with double chains of tetrahedra. gives an interesting contrast. The high cleavability (44.1) for { llOJ together with the low cleavability (16.6) f or (001! clearly indicates a structure capable of yielding fibres; the probable cleavage path for £110} is shown in Figure 2; a suggested 79 cleavage giving the very low value of 15.9 for the cleavability of { llOJ is erroneous, if the calculations of this paper are trustworthy. '.Ihe cleavage of that portion of the path shown in the figure parallel t o b is analogous to the mica £ 001J cleavage •rith its accompanying high cleavability; since two double Si-0 chains are held together by strong Mg-0 bonds the resulting columnar units have nearly square cross-section of greatly increased strength. '.Ihe silica modifications, namely quartz, cristobalite and tridymite, have their cleavabilities calculated from the high temperature forms since it is probable that these configurations are approximate to those of the low temperature forms; the agreement obtained is good; the compact structures indicated by their cleavabilities is reflected in the poor or no cleavage (see Table 24). and octahedrite give excellent agreement. Sellaite, rutile The form f lll} of spinel has the highest cleavs.bility but probably due to a poor reflecting surface t he observed cleavage instead of being good is imperfect; this is in marked contra.st to the effect of surface on the { 211} form of calcite (see Table 25). The agreement of kye.nite. topaz and melilite 41. t PJ70l iVl7t4~ ~ 1 tre mol ite col um ti 1 +--a'~ Figure 2. Basal projection of tremolite columns. The {1101 cleava ge path is indicated. 42. is excellent. Bodalite, a mineral with a three-dimensional network of tetrahedra., gives very good agreement. It is seen that the cleavage of a mineral is predominately determined by the relative magnitude of' the calculated cleava.bilities for the crystal. In species where pr6- nounced clea.va.ge occurs the calculated cleave.bility value for that form is appreciably greater than the values for other forms of the same crystal; an example is furnished by muscovite 'V'Ihose eminent { 001) cleavage has a cleavability 200% greater than that of any other form. That the change in observed cleavage from mineral to mineral is not determined by the relative values of the highest calculated cleava.bilities is shovm by the form { llOi of tremolite having the same clea.va.bility as {110} of soda.lite while the observed cleavage for tremolite is highly perfect in marked contrast to the more or less distinct cleave.ge of soda.lite. If the clea.vabili ties for a large number of forms were calculated, it is probable that the values would approach an asymptotic lower limit. vuch a tendency is shown by the data of Table 23 as for example in the cases of brookite whose last form calculated values a.re 11.0, 10.6, 10.1, 9.3 and corundwn whth values of 10.2, 9.5, 9.2, 8.6. Since the cleava.bili ty range of a mineral is the interval between the greatest and least of its cleavabilities, these may be replaced by the greatest and lea.st of the calculated values and the approximate range so obtained may be designated as the "re.nge of cleavabili ty values. 11 The lower asymptotic limit being unknown, the significance of this range is to some extent determined by the total nurnber of planes in the forms considered. '.1his number is given in the third col~Ttlll. There seems to be a correlation between the number of planes in a form and the degree 43. TABLE 24. Restricted Range of Clea.vability Values and Cleavage. Number of Planes Range of Clea.va.bility Values Cristoba.lite 26 22.2-21.9 None Tridymite 26 22.0-20.4 Not distinct Phena.cite 20 20. 4-18. 9 Poor Quartz 26 18.7-16.2 Difficult Diopside 12 18.5-16.3 Often only in t hin section Spinel 26 15. 3-12. 7 Poor Bromellite 26 12. 5-11.1 Doubtful Corundum 26 12. 4-8. 6 None Species Cleavage 44 . of cleavage as evidenced by the excellent cle avage of muscovite, topaz and kyanite, each having two planes in the cleavage form while sodalite and grossularite with tv,.-elve planes for f llO} have poor cleavage. It follows that minerals having a restricted range of cleavability values should not show pronounced cleavage on any .form. As shown by the minerals listed in Table 24, this is in agreement with the results of observation. 1be cleavability values given in column three for the forms considered change by only a few percent. Cristobali te and corundum show no cleavage, while for tridymite, phenacite, quartz, etc., the reported cleavage is poor. The observational data on hardness (according to the scale of Moh~ is given in colunm four of table 23. It is noteworthy that the difference in cleavability between the two end species muscovite and corundum is large, the former having six times the cleavability of the latter, and is correlative vrith the least and greatest of the hardness values, and further that in a general way the intermediate cleavability and hardness values tend to be inverse to each other. 1hese data sug- gest that hardness increases as the cleavability decreases. To establish the exact nature of this inverse relationship requires more detailed work. Departure of the rupture surface from a planar condition with its accompanying good cleavage results in the irre gular surface of the conchoidal fracture. As indicated by columns five and six of Table 23~ the observed fracture range is uneven to conchoidal with the tenacity uniformly brittle. The re l a ti vely less important eomponent of cleavage is the effect of the optical properties of the cleavage surface. There are listed in Table 25 mineral species whose highest calculated cleavabilit ies 45. TABLE 25. Effect of Plane Optical Surface on Cleavage Species Cleavage form Clea.vabili ty Surface Cleavage Muscovite { OOl J 73.1 Exactly plane Eminent Calcite f 211} 35. 4 Plane Highly perfect Villiaumi te [ 100 } 31.5 Plane Complete Topaz {OOlJ 28.6 Plane Highly perfect Lime flOOJ 17. 2 Plane Complete Kyanite {1001 15.1 Plane Very perfeet Perie lase {100} 13.2 Plane Perfect 46. range from 73.l (muscovite) to 13.2 (periclase), being the maximum and nearly the minimum of Table 23. All these cleavage forms have planar surfaces, with that of muscovite known to be exactly so. In strong contra.st to the variation in cleavability~ the degree of cleavage given in the last colunm of the table is observed to be much better than average. It is evident that the sequence of cleavabilities for a species need not be the same a.s that of observed cleavage (cleavability plus optical effect) although in general they correspond. ~~e cleavabilities of parting forms for grossularite, tridymite, diopside and corundum are given in 1'able 26. Comparison of the data of comfills three and four show that the cleavabilities for these fonns a.re at or near the maximum limit of the range of cleavability values, indicating that cleavage may frequently occur on such forms rather than parting. The use of Equation 16 gives not only calculated cleava.bilities for various forms of the same species but also permits comparing cleava.bilities of different minerals. Although restricted in this study to ionic minerals it is probable that the expression for cleavability is of more general scope. 47. TABLE 26. Cleava.bility and Parting. Species Form Cleavability Range of cleavability values of species Grossularite {1101 22.4 22.4 - 16.5 Uncertain if' cleavage or parting Tridymite [ 0001 } 21.9 22.0 - 20.4 Parting sometimes observed Diopside { 001 ! 1 7. 8 18. 5-16. 3 Parting often verly prominent Corundum [ 110 } 12.4 12.4 - 8.6 Parting often prominent Observation 48. 7. Conclusions 1. TI1e dominant component of the phenomenon of mineral cleavage is cleavability. 2. Ionic cleavability is given by the expression 3. A limited range of cJ.ea.vability values indicates absence of cleavage. 4. The higher degrees of cleavage a.re due to high relative clea.vability plus plane optical surfaces. 5. The data suggest that hardness increases as cleavability decreases. 6. High relative cleavabili ties for parting forms indicate cleavage rather than parting. Balch Graduate School of the Geolo gical Sciences, California Institute of Technology. BIBLIOGRAPHY 1. Dana, J.D., (Sixth ed. by ~.S.Dana) "System of Mineralogy" Wiley, New York (1892). 2. Tertsch, H. "Einfache KohHsionsversuche I, II, III." Z.Krist., 74, 476-500 (1930); 78, 53-75 (1931); 81, 264-274 (1932). - 3. Bra.va.is, A. "Etudes Crista.llog ra.phiques. 11 Jour.d.l'Ecole Poly., 22, 101-278 (19 51). 4. Sohncke, L. "Ueber Spa.li ungsflli.chen und naturliche Krista])fH!chen. 11 z. Krist., ..!..'.?..• 214-235 (1887). 5. Tertsch, H. "Spaltbarkeit und Struktur im trigonalen und hexagonalen ~ysteme. 11 Z.Krist., 47, 56-74 (1909 ). 6. E.'wald, P .P. and Friedrich, w. 11 R8ntgenaufna.hmen van kubischen 11 Kristallen, insbesondere Pyrit. Ann.d.Phys., 44, 1183-1184 (191 4) 7. Stark, J. "Neure Ansi ch ten Uber die zwi sch en- und innermolekulare Bindung in Krista.llen. 11 Jahrb. der Rad. u.Elek., 12, 279-296 (1915). 8. Scharizer, R. "Die Bra.gg'schen Kristallgitter und die Spaltbarkeit. 11 Z.Krist., ~· 440-443 (1916). 9. Niggli, P. 11 Geometrische Kristallo g raphie des Diskontinuums. 11 Borntraeger, Leipzig. (1919). 10. Beckenkamp, J. "Atoma.nordung und ~paltbe.rkeit. 11 Z.Krist., 58, 7-39 (1923). ll. Parker, R.L. 11 Zur Kristallogra.phie von Ane.tase und RutiU. 11 II. Teil. Die .Anatasstruktur. Z.Krist., 59, 1-54 (1923). 12. Barlow, w. "Geometrische Untersuchung Uber eine mechanische Ursuche der Hrunogenitlit der Struktur und der Syrrunetrie." Zeit.Krist., ~. 433-588 (1898). 13. Ewald, P .P. "Die Ihtensi tHt der Interferenzflecke bei Zinckblende 11 und das Gitter der Zinkblende. Ann.der Phys., 44, 257-282 (1914) 14. See Ref. 6. 1 5. Huggins, M. L. 11 Crystal Cleavage and Crystal Stru:Bture" Amer.Jour.Sci., E_, 303-313 (1923). 16. Te rt sch, H. "Bemerkungen zur Spal tbarkei t. 11 z.Krist. I 65, 712-718 (1927). Biblio graphy - 2 17. Pauling, L. "The St ructure of Some Sodium and Calcitun Alurninosilicates. 11 Proc.Nat.Acad. Sci., 16, 453-459 (1930). 18. Tertsch,, H. ''wie erfolgt der Spa.ltungsvorgang bie Kristallen?" Z.Krist., ~, 275-284 (1932). 19. Gibbs, J. w. 11 0n the Equilibrium of Heterogeneous Substances. 11 (1876) Collected Works. Longmans 1 New York (1928). 20. Lewis and Randall, "Thermodynamics." McGraw-Hill, New York (1923). 21. Condon and Morse. "Wave Mechanics. 11 McGraw-Hill, New York (192i>). 22. Uns8ld, A. "Bei trage zur Q.uantenmechanik der Atome." Ann.der Phys., ~' 355-393 (1927). 23. Pauling, L. "The Sizes of Ions and the Structure of Ionic Crystals." J .Am. Chem. Soc.,, 49,, 765-790 ( 1927). 9 24. Jeans,, J.H. "The Mathematical Theory of Electricity and Magnetism. 11 5th ed. University Press, Cambridge (1925). 25. Around 1% or about 2 large calories per mole for crystals of the halite type. 26. Born, M. "Atomtheorie des Festen zustandes. 11 Eney. d.Math. Wiss.,, ~, 527-789. Teubner, Leipzig (1923). 27. Born, M. and Bollnow, O.F. "Der Aufbau der festen Materie" Hdbk.d.Phy., 24, 370-465. Springer, Berlin (1927). 28. See Ref. 13 29. See Ref. 24. 30. Langmuir, I. "T'ne Constitution and fundamental Properties of Solids and Liquids." J.Am.Chem.Soc., 38, 2221-2295 (1916). 31. Harkins, w. D. et al. "The Structure of the Surfaces of Liquids, etc." J.Am.1,;hem.Soc., 39, 354-364 (1917). 32. Born, M. and Stern, o. "Uber die Oberflachenenergie der Kristalle und ihren Einfluss auf die Kristallgestalt. 11 Sitzber.d.Pr.Akad.d.Wiss., Berlin. ~' 910-913. 33. Harkins, w. D. and Cheng, Y. c. "The Orientation of Mo1ecules in Surfaces." J.Am.Chem.Soc., _£, 35-53 (1921). 34. Lenna.rd-Jones, J.E. In Fowler, R.H. "Statistical Mechanics. 11 Univ.Press, Cambridge (1929). 35. See Ref. 19. 36. Curie, P. "Sur la formation des cristaux et sur les constantes capillaires de leurs differentes faces." Bull.Soc. Miner.Fr., ~" 145-150 (1885). Bibliogra.phe - 3 37. Ehrenfest, P. "Zur KapillarittJ.tstheorie der Kristallgestalt." .Ann.der Phy., 48, 360-368 (1915). 38. See Ref. 19. 39. Bakker, G. "Ka.pillari t!l.t und OberfHl.chenspa nnung." Hdbk.d.Exper.Phy., Vol.VI, Akad.verlags, Leipzig (1928). 40. Adams, N.K. "The Physics and Chemistry of Surfaces. 11 Clarendon Press, Oxford ( 1930 ). 41. Wulff, G. 11 Zur Fra.ge der Geschwindigkeit der Wachst\(ums und der Aufl8sung der Kristallflll.chen." z. Krist., 34, 449-530 ( 1901). 42. Hilton, H. "Me.thematice.l Crystallography." Clarendon Press, Oxford (1903). 43. Yamada, M. "tfuer de Oberflllchenenergie der Kriste.He und die Kriste.llformen." Phy.Zeit •• 24, 364-372 (1923). z Phy.Weit. 25, 52-56 (1924). 44. Yamada, M. "Anha.ng ••• n 45. Biemuller,, J. "tlber die OberfHl.chenenergie der Alka.lihalogenide." Zeit.f.Phy., ~· 759-771 (1926). 46. Usually coinciding with the center of symmetry. 47. Griffith, A.A. "The Phenomena of Rupture and 1 Flow in Solids." Trans.Roy. Soc., London,, A 221, 163-198 (192~). 48. Polanyi, M. "Uber die Natur der Zerreissvorganges." Zeit.f.Phys., '!_, 323-327 (1921). 49. See Ref. 30. 50. See Ref. 31 . 51 0 Smekal, A. 11 Kohllsion der Fe stk8rper." In Auerbach und Hort. Hdbk.d.phy.u.tech. Mechanik, £• 1-153 (1931). 52. Pauling, L. "The Principles Determining the Structure of Complex Ionic Cryste.ls. 11 J • .Am.Chem.Soc., 51, 1010-1026 (1929). 53. For a physical justification of the electrostatic valence bond see Bragg, W.L. "The Architecture of the Solid State." (Kelvin Lecture) Nature, 128, 210-212, 248-250 (1931). 54. Washington, H. s. "Chemistry of the Earth's Crust. 11 Jour.Franklin Inst.~ 190, 757-815 (1920). 55. Niggli, P. "Ore Deposition of Mag~tic Origin . " Murby, London, (1929). Bibliography - 4 56. Goldschmidt, V. M. "Geochemische Verteilungsgesetze und kosmische HM.ufi gkeit der Elemente." Na:turw. ~, 999-1013 (1930). 57, Bragg, W.L. "The Structure of the Silicates" Z.Krist., 74,, 237-305 (1930). 58, Na.ra.,-Sza.bo,, St, " Ein auf der Kristallstruktur basierendes Silicatsystem." Z.Phy.Ghem., ~,, 356-377 (1930). 59. Hintze, 60. Groth,, P. "Chemische Kriste.llogra.phie ." Engelmann, Leipzig (1906). 61. Aminoff, G. "Uber Berylliumoxyd a.ls Mineral und des~ Kristallstruktur." Z,Krist., 62,, 113-122 (1925); 63, 175 (1926). 62. See Ref. 59, 63, Rogers, A,F. "Cleavage e.nd Parting in Quartz." ( Abstract) Amer.Miner., 18,, 111-112 (1933). 64, Pauling, L. and Sturdivant, J/H. "The Crystal Structure of Brookite. 11 Z.Krist., ~,, 239-256 (1928). 65. Niggli, P. "Lehrbuch der Minera.logie." 2nd ed. Vol. II. Borntraeger, Berlin (1926). 66, Bragg, w. L. and West , J. "The Structure of Certain Silicates. 11 Proc.Roy.Soc., A 114, 450-473 (1927). 67. See Ref. 52. 68. Ntray-Sza.bo, s., Taylor, lN,H. and Jackson, Vlf.W, "The Structure of Cyanite. 11 Z.Krist., 71, 117-130 (1929 ). 69, Pauling, L. "The Crystal Structure of Topaz. 11 Proc.Nat.Acad.Sci., 14, 003-606 (1928). 70. Alston, N. A. e.nd We st,, J. "The Structure of Topaz. 11 Z,Krist., 69, 149-167 (1928). 71. Warren, B. E. "The Structure of Meli lite Zeit.Krist., 74, 131-138 (1930). 72. Warren, B. and Bragg, w. L. "The Structure of Diop side, • • • " Z.Krist., 69, 168-193 (1928}. 73. Warren, B. E. 11 The Structure of Tremolite • • • " Z.Krist.# 72, 42-57 (1929). 74. Ma.uguin, c. 11 Etude du mica. muscovite au moyen rayons X. 11 Comp.Rend., 185, 288-291 (1927). I )I c. "Ha.ndbuch der Mineralogie. I 11 Veit, Leipzig (1915). ... II Biblioe;raphy 5 75. Pauling. L. "The Structure of the Micas and Related Minerals." Proc.Nat.Acad.Sci., 16, 123-129 (1930). 76. Jackson, W.W. and West J. 11 The Crystal Structure of Muscovite Z.Krist., 76, 211-227 (1931). 77. Pauling L. "The Structure of Soda.lite • •• II Z.Krist., 74, 213-225 (1930). 78. Barth, T.F. 11 The Structures of the Minenals of the Soda.lite Family. 11 Z.Krist., ~' 405-414 (1932). 79. See Ref. 73. ... " ACKNOWLEDGll!IENT This research was greatly aided by a Storrow Fellowship in Geology of the National Research Council. I wish to thank Professor Linus Pauling for his kindly interest and advice throughout the investigation. also indebted to the Gates Chemical Laboratory for research facilities. I am 1. Reprint from ,,Zeitschrift fiir Kristallographie". Ed. 75 , Heft 1/ 2. Akademische Verlagsgesellschaft m. b. H. in Leipzig, 1930. The Crystal Structure of Bixbyite and the C-Modification of, the Sesquioxides. By Linus Pauling and M •. D. Shappell in Pasadena. Gates Chemical Laboratory, California Institute of Technology, Pasadena, California. Communication No. 259. ('<Vith six figures.) 2. 128 1. Introduction. It was discovered in 1925 by Goldschmidt!) that an extensive series of sesquioxides form cubic crystals with the unit of structure containing 16 M2 0 3 , the value of a varying between 9.3 A and i0 .9 A. An atomfo arrangement based on the space group T5 was assigned this C-modification of the sesquioxides by Zachariasen 2J, who studied crystals of Sc2 0 3 , Mn 2 0 3 , Y2 0 3 , Jn2 0 3 , Tl 2 0 3 , Sm 2 0 3 , Eu.i.03 , Gcl,.03 , Tb 2 0 3 , Dy2 0 3 , Ho 2 0 3 , Er2 0 3, Tm 2 0 3, Yb2 0 3 , Lu2 0 3 , and the mineral bixbyite, (Fe, Mn)20 3 • Zachariasen ' s procedure was the following. Using data from powder and Laue photographs of Tl2 0 3 , and neglecting the contribution of the oxygen atoms to the reflections, he decided that the space group is T\ with the 32 Tl in 8 b with parameter t = 0.25, in 12 c with parameter u = 0.021, and in 12 c with parameter v = 0.542. These parameter values were assumed to hold for all members of the series. The consideration of intensities of reflection of Sc2 0 3 then was found to indicate the 4.s 0 to be in two groups of 24 in the general position of T5, with parameters x1 r-.J t, y 1 r-.J t, x1 r-.J !- and x2 ,--. . ,_, -}, y 2 r-..; -~ , ~ r-.J ;J-. The same structure was also assigned bixbyite, with ·16 (Mn, Fe)203 in a unit 9.35 + 0.02 A on edge. On beginning the investigation of the tetragonal pseudo-cubic mineral bra unite, 3M~ 0 3 • MnSi0 3 , we found the unit of structure to be closely related to that of bixbyite, and, indeed, to have dimensions nearly the same as those for two superimposed bixbyite cubes. This led us to 1) V. M. Go Id s chm id t, •Gcochem. Vert.-Ges. d. El. • IV, V, Videnskapsselsk. Skr., 5, 7, Oslq. ·1925. 2) W . Zachariasen, Z. Krist. 67, 455. 1928; •Untersuchungcn iiber die Kristallstruktur von Sesquioxyden und VerbindungenAB03 •, Videnskapsselsk . Skr. 4, Oslo. 1928. 3. 129 The Crystal Structure of Bixbyile etc. make a study of Zachariasen's structure, leading to the observation that not only are the interatomic distances reported abnormally small, but also the structure does not fall in line with the set of principles found to hold for coordinated structures in general 1) . It was further noted that Zachariasen's atomic arrangement, with the symmetry of space group T5, approximates very closely an arrangement with the symmetry of T1/ (of which T5 is a sub-group), and it is difficult to find a physical explanation of this distortion from a more symmetrical structure. This led to the reinvestigation of this mineral and the determination of a new and satisfactory structure for the C-modificalion of the sesquioxides. 2. The Unit of' Structure and Space-group Symmetry of Bixbyite. Bixbyite, found only in Utah, about 35 miles southwest of Simpson, is described by Penfield and Foote2) as forming shiny black cubic crystals with a trace of octahedral cleavage. The composition assigned it by them was JJ'e++ jJ!fn+40 3, with a little isomorphous replacement of Jte++ by J.lfg++ and J.lfn++ and of .ili_faH by Ti+4. lt was shown by Zachariasen that the X-ray data exclude this formulation, and indicate instead that the mineral is a solid solution of J.lfn 2 0 3 and JJ'e2 0 3 • We shall reach a similar conclusion. Table I. Spectral Data from (100) of Bixbyite (with rock salt comparison). It/cl Line Eslimated Intensity s2;1o,ooo 0.05 0.26 11 7 2'1.16 0.2 0.61 d/n I 2 00 1, 0 0 111.oICai a1 -1fx 9.r.o.A :}x 9.38 400 600 a2 :}x9.3 7 (f.1 i - X 9. 38 600 CC;? ·&x 9.36 } sbo flt ~- X 9.36 t x 9.34 } 5 20.0 9.38 0.1 0.2 3 T"u X 9. 34 T':r X 9.36 ,-':r X 9.3 7 0.4 0.3 '1.80 0. 71 0.1 0. 71 800 1 0.0 . 0 1 0.0.0 1 i.0.0 111.0. 0 a2 rt1. a2 flj fl[ 16 .0.0 -i1u x Average: a= 9 .365 ± 0.020 A. 1) Linu s P a uling, J. Am. chem. Soc, 51 , 1010 . ·1929. 2) S. L. P e nfield and H. W. Foote, Z. Krist. 28 , 592. 1897 . Zeitschr. l. Kristallogrnp hie. 75. Bd. 9a 4. 130 Linu s Paulin g and l\L D. Shappell Data from oscillation photographs of bix byite show a t o be a multiple of 4. .68 A (Table I). The Polanyi layer-line r elation applied to photographs wilh f I 00 ) as rotation axis showed that this multiple must be 2, giving a unit with a= 9. 365-+- 0.020 1. This unit sufficed to account for the occurrence of all spots observed on several Laue photographs taken with a tube operated at a pea k voltage of 54. kv. (the incident beam making small angles with [·I 00 ] or [•11 OJ), and may be accepted as the true unit. Tabl e II. Laue Data for Bixbyite. Incident beam nearly normal to (100) . Estimated Intensity . nl, = dhk l {hkl} .A ~-1 0.25;.-0.29 10.30 1.5 I 451 415 1.4 4 631 61 3 1. 38 - -- 7.0 7.0 2 7 ·1 217 '1. 2 7 65 1 61 5 us 0. ·I 0.2 1.15 0.1 0.1 0.8 1.1 5 1.0 1. 6 471 SH I 1_ _ 0. 4 1 0.0 209 1 0.0 1 0.0 157 7.0 1 0.0 - 2 3;J 4.3 7 I 2.96 ·17 2 I 2 18 1.0 3.0 9.w. \.> ,. 1117 s A0.341° . H5 A0.3 910.4 OA0.11 5 Calculated s2r1o,ooo I 4. 75 5.43 1. 42 . 1 -19 -154 0.'1 0.4 2 .46 2.37 14 2 1.93 2.02 S4 107 1.1 4 1SO 3.211 0.1 18 25 0.03 0.06 0.0 0. 2 11 0.01 30 o.ou 139 - -- - 275 257 ·1. 06 219 291 1.0 ·1 293 23 9 0.96 0. 1 0.1 0.1 0. 2 43 43 0.18 U.'18 277 .93 0.3 0.4 -10 3 1.06 837 873 .85 0.G 0.6 - ·10 2 ·1 07 '1.0 4 U 4 ·I 0.5.3 1 0. 3.5 . 8~ 0. 86 0.86 4.11. 3 4.3.11 0.0 0.7 ·1 I .77 I I I 0.1 0.05 0.1 -9 3 0 ..1 O. •I 0.1 0 ..1 -6 3 -68 93 I I I 0.110 0.46 5. 131 The Crystal Slructme of Bixbyite etc. The value 4.945 for the density of bixbyite reported by Penfield and Foote leads to 16 (1lfn, Ji'e)2 0 3 in the unit. lt was observed that the only planes giving odd-order reflections (see Table II) were those with h k l even, indicating strongly that the structure is based on the body-centered cubic lattice T'~. Moreover, a Laue photograph taken with the incident beam normal to (100 ) showed only two symmetry planes and a two-fold axis, requiring that the pointgroup symmetry of the crystal be that of T or '1'1i. The only space groups compatible with these conditions are 'J'3, 'J'5, Tf., and '1'1;. Of these Tl. requires that planes (0 !cl) with k and l odd give no odd-order renections, while 'J'3, '1'0, and T1~ allow such reflections to occur. On our photographs no such reflections were found, although a number of planes of this type were in positions favorable to reflection (Table IIl). This makes it highly probable that Tl. is the correct space group, for it would be very difficult to acco unt for Lh e absence of these reflections with an atomic arrangement derived from T3, 'J'5, or Tf. which at the same time did not come indislinguishably close to an arrangement derivable from Tl.. In view of these considerations we have assumed Tl. to be the correct space group . + + Table Ill. Data for Prism Forms from Bixbyite. A. Forms not reflecting on Laue photographs: {hkl} nl. {07 ·1} 0.35 , {701} {11 .0.3} {0 .11 .3} 0.3'•, 0.4 o, 0.41, {13 .0.3} {0.13.3} 0.30, 0.31, o. 39 0.40 .A 0.45 0. 4 4 0.33 0.32 13. Forms not reflecting on osci llation photographs: {031}, {013}, {033}, {05 ·1}, {Oi 5}, {053}, {035}, {055}, {0 7·1}, {017}, {073}, {U93}, {U37}, {039}, {U75}, {095}, {057}, {059}. {091}, {019}, This choice of space group is further substantiated by Zacharias en 's data for the other substances as well as bixbyite. His reproduced Laue photographs of 'l'l2 0 3 and of bixbyite show no spots due to {071 }, {017}, {09 •1}, {0 119}, {0.11.1} or {O.U 1}, allhough planes of these forms were in positions favorable to reflection, while the powder data show that {031 }, {013}, {073}, and {037} gave no reflections for any of the sesquioxides studied. Zachariasen's rejection of Tl arose frorri his 9* 6. 132 Linus Pauling and i\I. D. Shappell assumption that the oxygen contribution to the intensities was negligible, and his consequent inability to account for the observed inequalities in intensity of pairs of forms such as {271} and {2 ,17} with the metal atoms in positions provided by Tf.· But actually the oxygen contribution is by no means negligible. For example, the structure which we find gives for the metal contributions to the structure factor for {27 ·I} and {2'17} the values 66.2 and - 66.2, which are changed by the oxygen 119.6 and -154-.2, respectively. For Tl2 0 3 the effect contribution to of the oxygen would be only abo ut one-fourth as great, which is, however, still sufficient to account for the observed inequalities 1). + + 3. The Arrangement of the Metal Atom8. The equivalent positions provided by Ti, are: 81:: 000; i{- 0; -}0 -}; O:H-; -H-t; oot; O§O ; 1,- oo. 8 e: +-H; :4"}~- ; -Hd-; -H-L 't '}f; -Hd-; t-H-; -U d· ii; t - u, u"u; ·u, t - ii, ·u; u u u; u , u, ii+ {; ii+{-, u, ii; u, ii+ -}, u; u+{-,u+-J-,ii+-}; ii+ -},J- ii, u; u , u+t,t--'u; t-it,u,it+-§-; -~ - it, {--u, {-- ii; -} - it, u + -§-, u; ii, {--u, ii+{-; ii+-}, ii, -} - u. 24-e: u 01; ; u{-i1; }-u,0,1;; u+ -L i·,+; '16 e: u ii u; ii, u, {- - fu O; t ·u{-; -h -~- -u,O; f,it +}, {-; Otii; -§- tu; 0,-f.,-§--ii; t,i,1~+t; u Of; u-§-{; ii+-§-,0,{; j- -u,}, -~- ; "t u 0; "k u -~- ; ii t, 0; -1-, :1 - ii, t; O{ ·u; -H u ; 0,-!,ii+{-; {,i_,,{- - ii. 4-8: x y x; x, y, j- - x; t - x, y, x ; x, -}-y, x ; x x y; -~- - x, x, y ; x, t - x, y; x, x, t - y; y xx; y, t - x, x; y, x, t- x; -§- -y, x, x; xyx ; x,y,x + {-; x+~,y,x; x,y+{-,x; x x y; x -~' x, y; x, x y; x, x, y t; yxx; y,x+-§-,x; y,x,x+{-; y+-§-, x,x; x+-§-,y+~,x+-4; x+ {-,{--y, x ; x,y+ -},-} - x; {- - x,y,x+{-; x + -h x + -~-, y + } ; x, x + -§-, -} - y; t - x, x, y + -} ; x + -§-, -~- - x' y; y+ -§,x+{-,x+{-; §-- y, x, x+ -}; y+{-, t - x, x; y, x+-§-, t-x; -} - x, :4. - y, -§- - x; !-x, y+ :~, x; x, § - y, x+{-; x+t, y, {--x; t - x, ~- - x, l - y; x, ~- - x, y + -§-; x + -;-, x, -§- - y; t - x, x + t, y; { - y, -§-- x, {- -x; y+-§-, ,~, -§- - x; } - y, x +}, x ; y, {- - x, x+-§-. +, + + + {' + 1) DI'. Zacharia s e n has kin dly inform ed us that he now agrees with our choice of the space gro up Tii 7. 133 The Crysta l Structure of Bixbyite etc. Of these, all but the general pos ition may be occupied by th e metal atoms. The 32 metal atoms may have any one of the following arrangements : A: Formula, F'e111.n03 : 1 a. •16.F'e in 16 e, •16Mn in 16 e. 2a. 16F'e in 16 e, S..ilfn in 8i, SMn in Se. 2b. 1611.fn in •I 6e, Sli'e in Si, SF'e in Se. B: Formula, (Mn, J1'eh0 3 : '1b. 16(Mn, F'e) in 16 e, 16 (Mn, fl'e) in 16 e. 2c. 16 (Mn, F'e) in •16 e, S (Mn, F'e) in Si, S (Jlfn, F'e) in Se. 3. 2ft (1l'In, F'e) in 24 c, S (Mn, Fe ) in Si, or 2r~ (.Mn, f!'e) in '.H e, S(Mn, F'e) in Se. The reflecting powers of Jlin and F'e are nearly the same, and may be taken equal without serious error. This reduces the number of distinct structures to three; namely, 1 ab , 2 abc, and 3, of which 1 a b depends on two parameters and the others on one. It is possible to decide among them in the following way. Let us assume that the contribution of oxygen atoms to the intensity of reflection in various orders from (100) is small compared with the maximum possible contribution of the metal atoms; that is, with 32 Jl'f. The metal atom structure factor for structure 1 for (h 0 0) is 1 S,,00 = 16 JlI (cos 2 1rh u 1 + cos 2 rr h it 2). 1 Now (200) gave a very weak reflection, so that 8 200 must be small. This is true only for it1 + it 2 ,...__, -}, for which S,, 00 ,...__, 0 for h = 2, 6, 1 O, S,,00 ,...__,32 111cos2nhit1 for h=l+, S, 12. Now th e gradual decline in intensity for h = ti., S, ·12 (Table 1) requires that it1 = t, and hence u 2 = .g-. This puts the two sets of metal atom s in the same place, and is hen ce ruled out. It may also be mentioned tbat structure '1 would place eight metal atoms on a cube diagonal, giving a maximum metal-metal distance of 2.03 J\., which is considerably smaller than metal-metal distances observed in other crystal s. Structure 2, dependent on one parameter ii, has structure factors 81i0o = 16 111cos 21chu for h = 2, 6, . . . , 8 ,,00 = 1611{ (1 +cos 2 n hu) for h = !~, S, .... All values of the parameter u are eliminated by the comparisons 600 2 00, 4.00 200 , and 10.0.0 200. > > Zeitschr. f. Kristallographic. 75. Bd. 9b > s. Linus Pauling and M. D. Shappell 134 There accordingly remains only structure :3 . We may take S (1l1n, Fe) in Se rather than Si, which leads to the same arrangements. The structure factor for various orders from (100) is then 1) for h = 2, 6, 'IO, etc. 8 1100 = S.ilf (cos 2 1chib S1ioo =SM (cos 271:hu + 3) for h = 4, S, ·12, etc. All distinct structures are included in the parameter range - 0.25-<; u 0.25, and, moreover, positive and negative values of u give the same intensity of reflection from (hO 0). Hence we need consider only 0 t b [ -<; 0. 25. In Figure 'I are shown values of [SI calculated over < <[ .oz .oq .06 08 .10 .12 Jq .15 .t8 .20 .22 ,gq Fig. 1. Structure factor curves over the range C <: lu: <: 0.25 with .M constant. this range with a constant value for Jlif. It is seen that the observed intensity inequality 600 > 200 rules out the region 0.125-<; 'u [ -<; 0.25, and ,I 0.0.0 > 200 and 1ILO.O>200 further limit [u [ to between 0.00 and 0.06. The value of [u [ can be more closely determined by the use of atomic amplitude curves. The intensity of the diffracted beam can be taken as I = with A1i1cl = .:Jf: A; !(. A~lcll (·I) e2"i (h x; + ky ; + lz;) . (2) i In this expression Ai, the atomic amplitude function, is given by ·1 + cos220} :4 A·= { . Ji'. ' 2 sin 2 0 '' (3) in whi ch Fi is the atomic Ji'- function. Values of Are and Ao calculated for JlfoKu radiation and for an average wave-length of 0.40 A effective on Laue photographs from Bragg and West's Ji'-curves1) are given in 1) W. L. Bragg and J . W es t , Z. Kris t. 69, 118. 1928. 9. The Cry5tal Structure of Bi xbyite etc. Tab le IV. Figure 2 shows values of A1t0o for h = 2, 4. • •• '16 over the range of values 0 to 0.06 for f It is seen that the value u [ = 0.0 30 : 0.005 is indicated by the observed intensities of Table I. u:. 2'10 200 IA/ t 160 120 80 40 .01 02 .OJ M .05 .06 Fig. 2. A-curv es over the ran ge o <:: ju j <:: o.OG. Table IV. Atomic A-values for Iro n and Oxygen. d h/ii .A I.= 0.110 A I.= 0.709 A A,, A Fe A" A"'" 2~ .0 79 .0 2·1.2 59.S 2.50 1 11. 5 115.0 ·I 0.11 33 .6 1.67 Li 30 .1 5.S 2L7 1.25 !,,!, 2LS 3 .5 1 5.G 1.00 2 .7 16.8 ~u ·11 .G 0.83 1.5 12.5 '1. 2 9.0 0.72 0 .9 1 0.3 0.6 7. 3 0.63 0.G 8.4 0.11 5.9 0 .55 0.11 G.G 0.2 4.6 0.50 0. 3 5.2 0 ..1 11. 0 5.00 0.115 4.5 3 .6 0.112 3. 7 3.11 0.39 3.1 3.2 Now there are two physically distinct arrangements of the metal a toms corresponding lo [ii [ = 0.030, the first with u = 0.0 30, and the second with ii = - 0.030 ;. and it is not possibl e to distinguish between them with the aid of the intensities of reflection of X-rays which th ey give. Let us consider th e positions 24 e. The structure factor for 24 e is: 10. 136 Linus Pauling and M. D. Shappell S1iu = 8.ilf [cos 2n (hu + Z/ 4) +cos 2 n (lvil + !t/ 4) +cos 2 n; (lu + k/ 4 )] for h, k, l all even; = 8 111 cos 2 n (hil + l/ 4) for h even, k odd , l odd; = 8Mcos2n: (ku+h/ !i ) for h odd , /( even, l odd; = 8 Jlf cos 2 1r (lu + k/ 4) for h odd, k odd , l even. = 0 otherwise. It is seen that the value of the structure factor is the same for a given positive as for the same negative value of i i, except for a difference in sign in some cases. But the positive and the negative parameter values correspond to structures which are not identical, but are distinctly different, as can be seen when the attempt to bring them into coincidence is made. This is a case where two distinct structures give the same intensity of X-ray reflections from all planes, so that they could not be distinguished from one another by X-ray methods. The presence of atoms in 8 e or Si does not change this result. In the case of bixbyite a knowledge of the positions of the oxygen atoms would enable the decision between these alternatives to be made, but the rigorous evaluation of the three oxygen parameters from the X-ray data cannot be carried out. Za ch a riasen's arrangement of the metal atoms approximates the first of our two (that with the positive parameter value), and would be identical with it if his parameters were taken to be 0. 0 3 0 and 0. 53 0 rather than 0.021 and 0.54.2. 4:. 'l'he Prediction and Verification of the Atomic Arrangement. Recognizing the impracticability of determining the positions of the oxygen atoms from X-ray data, we have predicted a set of values for the oxygen parameters with the use of assumed minimum interatomic distances which is found to account satisfactorily for the observed intensities of a large number of reflections and which also leads to a structure which is physically reasonable. The Ji'e -0 distances in hematite are 1.99 and 2.06 k The (Mn, E'e)-0 distances in bixbyite are expected to be the same in case that (1lfn, Ji'e) has the coordination number 6, and slightly smaller, perhaps 1. 90 A, for coordination number 4. The radius of o= is 'J.4.0 A, and the average 0-0 distance in oxide crystals has about twice this value. When coordinated polyhedra share edges the 0-0 distance is decreased to a minimum value of 2.50 A, shown by shared edges in rutile, anatase, brookite, corundum, hydrargillite, mica, chlorite, and other crystals. Our experience with complex ionic crystals leads us to believe that we may ll. The Crystal Structure of Bixbyite etc. 137 safely assume that the (l"vfn, Fe)-0 and the 0-0 distances in bixbyite will not fall below 1.80 A and 2.4.0 A respectively. On attempting to build up a structure on the basis of the first arrangement of the metal atoms, with n = 0.030, we found that there is no way in which the oxygen atoms can be introduced without causing interatomic distances smaller than the assumed minimum ones. This arrangement (which approximates Zacharias en's) is accordingly eliminated. The second arrangement of the metal atoms, with ii = - 0.0 30, is such that satisfactory interatomic distances are obtained only when the , y ~ -$- , and x ~ %oxygen atoms are in the general position with x ~ % Each oxygen atom is then at about 2 A from four metal atoms ; if it be assumed that these four metal-oxygen distances are equal, the parameters are found to have the values y = O.H5 , x = 0.380. With this structure each metal atom is surrounded by six oxygen atoms at a distance of 2.0 '1 A, and the minimum 0-0 distance is 2.50 A. These dimensions are entirely reasonable. It is probable that the various metal-oxygen distances are not exactly equal, but show variations of possibly + 0.05 .il- . The predicted parameter values may correspondingly be assumed to be accurate to only about + 0.005. Table V. Data from an Oscillation Photograph of Bixbyite 1) . x = 0.385, {It 0 l} Es tim at ed Intensit y s211o,ooo 202 0.00 402 40 1, 0.01 0.01 0.28 10 67.00 602 0.2 0.68 6011 0.6 1.69 802 O. 'l 804 2 0.7 4 3. 311 806 0. 05 0.19 The predicted structure has been verified by the comparison of the observed intensities of r eflection for a large number of planes and those calculated with the use of Equation 1. Data for such comparisons for planes (hO 0) and (hO l) reflecting on oscillation photographs are given in Tables I and V, and for other planes giving Lau e reflections in 1) These rell ections are from th e fir st, second, and third layer lines of the same ph otograph as that from which th e data of Table I were obtained , so th a t intercomparisons between Tabl es I and V may be made. 12. 138 Linus Pauling and l\L D. Shappell Table II. It is seen that the agreement between calculated and observed intensities is almost complete; the existent discrepancies are generally explicable as resulting from small errors in the parameter values (within the limits + 0.005) or from errors in the assumed F-curves, for which an accuracy greater than + 20% is not cl aimed. o. Description of the Structure. The structure found by the methods just described agrees well with th e general principles und erlying complex ionic crystals. Th e arrangement of the metal ions is shown in Fig. 3. These ions are nearly in cubic close-packing, so that the structure gives nearly th e maximum dispersion of cations with given molal volume. Each cation is surrounded by six oxygen ion s at a distance of 2.0i 1, at t he corners of a hig hly distorted octahedron. These octahedra are of two types, corresponding to the two positions Se and 24 e. Each 8 c octahedron (with point-group symmetry C3 i) shares six edges with adjoining 24-e octahedra, and each 24c octahedron shares six edges also, two with 8 c and four with other 2 4. e octahedra. Every s har ed e dg e is 2.5 0-2 .521 long, in striking agreement with the minimum dimensions found in other crystals for shared edges and th e theoreti cal values obtained for rutile and anatase 1). These shared edges are a rranged differently for 8c and 2ke ; the distortion accompanying their shortening leads to octahedra of the shapes shown in Fig. 4. and 5. Various intera tomic distances are giv en in Table VI. Table VI. Interato mic Distances in 13ixbyite. (F e,Jlin)- 0.l =2.0 1 A (Fe,llln)- 0 1,=2.01 A O,i - 0F=2.52A OJ/ - 0 /;' =3 .29A (Fe,Mn)-011 =2.0·I (Fe, Mn)- 0 1,=2. 01 011- of;'= 2.52 OL- Oa= 3.29 (Fe,11In)- Oc = 2.0 1 (Ji'c, Mn) - OD= 2.0 ·1 (F e, llln)- OH= 2.01 (F e,Mn) - 0p=2.0 1 (Fe,1lfn) - Or,- = 2.0·I (Fe. Mn) - 0H =2.0 ·I ( F c,Mn )- 0 1 =2.0 I OJ - 0 11 =3. 1 3 0 11 - 0 c= 3.·13 OH - 0 1,-=2 .52 OH - 0 1 =2. 5 0 l Oc -0li=2.~2 OA.-01, = 2. <iO O}J- 0};'=:~ .1 3 I Oc - o,,- 01 = 3.38 01,· = 2..•2 011 - 01i· = 2.5 1 01 - 01.=2 .51 0,,-01,-=2 .9 2 Oc- 0 1, = 2.50 OJJ - Op=3. 1 3 Or;-Or=B.13 OJ-0c =3.13 o _l-OJJ = 2.52 IOll-0 u =3. ·12 OH-01 =2 .GO 01, - 0};'=3 .1 2 Each oxygen ion is common to four octahedra, and h as .)_,' s; = 2, in accordance with the electrostatic valence rule . i The structure can be instructively compared with th at of fluorite, CaF2 • In fluorite the calcium ions are arranged at face-centered lattice points, and each is surrounded by eight flu orin e ions at cube corners. 1) Linus Pauling, Z. Krist. 67, 377. 19 28. 13. 139 The Crystal Structure of Bixby ite etc. / -......,..-- -I I I -h.. 0, I I I ~ ~~=~7-/--...+- - ..- I - t- 0- I ---o-- .I- - I ..- - - _.J- / ,.. 1 --l ,.. / I I I I _)__ __ I I Fig. 3. Th e struc ture of bixbyite. The metal ions a re shown, together with one of each kind of disto r ted octahedron . I 1..-, - ..-'G - -) ---------,- -J. Fig. 4. The s e octahedron, showing its relation to a cube. Fig. 5. The 21, e octahedron, showing it s re lation to a cube. 14. 140 Linus Pauling and M. D. Shappell If one-fourth of the fluorine ions are removed and the others are re- placed by oxygen ions , calcium being replaced by (Mn, B'e) , a structure is obtained which approximates that of bix byite, which diITers from it only in small displacements of the ions. This similarity is shown by the fact that the highly distorted octahedra have corners which are nearly at six of the eight corn ers of a cube, the six being chosen differently for the 8 e and the 24 e octahedra, as is seen from Fig. 4 and 5. This analogy was, indeed, pointed out by Zachariasen for bis incorrect structure. As a matter of fact the "ideal" structure, with n = O and Fig. 6. A photograph of a model repres entin g one half of the unit cube. Th e arrangement of the si.-.: 21, e octahcd ra sharing edges with an Se octa hedron is clearl y shown. x=;~, y=~~, :;; =},corresponding to Zach ar ias e n 's original atomic arrangement also corresponds to ours. Zacharia s en very instructivel y pointed out that this id eal structure lies midway .between the fluorite and the sphalerite arrangements, being obtained either by removing certain anions from fluorite, or by adding anions to sphalerite, the positions of the other ions remai ning unchanged in either case. With the ideal struoture the coordinated polyhedra are cubes with two truncated corners; for 8 e these corners are at the ends of a bod y diagonal , fo r 24 e at the ends of a face diagon al. The actual structure is distorted from th e ideal one, which leads to too small interioni c di stances, in such a way as to give a constant metal-oxygen di stan ce and a minimum oxygenoxygen distance (for shared edges ) of 2.50 1\. In Zacha riasen ' s arrangement the distortion was in th e opposite directi on. 15. The Crystal Structure of Bixbyite etc. 141 Table VII. Interat omic Distances in Sesquioxides. Subst an ce (Fe, Jlinb Oa .il ln20a &203 Y20a In2 0a 'l'l203 Sm20a Eu20a Gd20 3 Tb20a Dy20a Ho20a IYJ·203 'l'm20a Yb20a Lu2 03 Cb 9.365 .A 9.41 9.79 ·I 0.60 '10. ·l 2 ·1 0.57 1 0.8 5 1 0.84 ·I 0. 79 ·I 0.70 ·I 0.6 3 ·1 0. 58 1 0.54 10. 52 1 0.39 10.37 M-0 2.01 A 2.02 2.10 2.27 2.17 2. 26 2.33 2.33 2.32 2.3 0 2.28 2.27 2.26 2.26 2.23 2.22 As mentioned by Zac haria sen, Goldschmidtl) found that the range of radius-ratio values leading to stability of the C-modification is about 0.60 <~::' < 0.88, which is high 2) for a structure in which the coor- dination number is 6. The explanati on of this is obvi ous; the coordinated octahedra are deformed so that the anions are nearly at six cube corners, and the radius rati o will accordingly tend to the range of values giving th e coordination number 8. A photograph of a model representing the structure is shown in Fig. 6. Zacharias en's inves tigati on makes it highl y probable that the sesquioxides forming crystals of the C-modification have the same structure as that which we have found for bixbyite, and the similarity in intensities on powder photographs of the different substan ces which be reports indicates that the parameter values do not change very much throughout the series. Thus in all these crystals the cations are attributed the coordination number 6. Values of interionic distances calculated from Zacha ria se n' s values of a with the bixbyite parameters are given in Table VII. It is probabie, however, that the oxygen parameters do change as ci increases in such a way as to keep shared edges short, for with the bixbyite parameters th e shared edges increase from 2.50 A to about 2.9 0 A in Sm 2 0 3 and Ei~03 . As a consequence the metal-oxygen dis1) V. M. Goldschmidt, •Ge och cm . Vcrt.-Ges. cl. El. «, VII, p. 76 . 2) Linu s Paulin g, J. Am. chem. Soc, 51, 1010 . 19 29. 16. 142 Linus Pauling aml M. D. Shappell, The Crystal Structure of Bixbyite etc. tances in Table Vil are probably a little larger than the true ones, the maximum error being 0.1 0 A. It is worthy of mention that the 0-structure and the corundum structure correspond to nearly the same interionic distances (1.99-2.06 A in hematite as compared with 2.0 ·1 A in bixbyite), as is to be expected from the equality in coordination number of the cation. Summary. With the use of data from oscillation and Laue photographs it is shown that the unit of structure of bixbyite has a= 9.36 5 A and contains 1 fi (jJfa, Fe)i03 . The lattice is the body-centered cubic one, T~, and the space group is Tf,. Two possibl e arrangements alone of th e metal atoms are found to be compatible with the X-ray data (oxygen atoms being neglected), the first wilh 8 (1J1n, Fe) in 8 e, <;H (Mn, Fe) in 24. e with ii = 0.030, and the second the same except with i i = -0.030. It is pointed out that these two physically distinct arrangements give the same intensities of reflection of X-rays from all planes, so that an unambiguous structure determination for a crystal containing only atoms in 24 e (or 24 e, 8 e, 8 i) could not be made with X-rny methods alon e, despite the dependence on only on e parameter. The assumption that the (Mn , Fe)-0 and 0-0 distances can nol fall below 1.80 A and 2.40 A, respectively, eliminates the first metal atom arrangement, for there are no positions for oxygen satisfying it. With the second arrangement of metal atoms this assumption requires rn 0 to be in the general position of '1'1;, with x ,....._, -~ , y ,....._, t , x ,..__, -§-. Each oxygen ion is then nearly equidistant from four cations. Making the four (Mn, Fe)-0 distances equ al, values of the parameters are predicted which lead to good agreement between observed and calculated intensities of reflection from a large number of planes. The structure found for bixbyite has S(Mn, Fe) in 8 e 24(.il!n, Fe) in 21\.e with i t = -0.0 30 + 0.005 48 0 in x , y, x, etc. with x = 0.385 + 0.005, y = 0. 1145 + 0.005, x = 0.380 + 0.00 5. A description of the structure with values of interatomic distances for bixbyite and for Sc2 0 3 , Mn 2 0 3 , Y2 0 3 , Jn2 0 3 , 'l'l2 0 3 , Sm2 0 3 , E!J,203, Gd2 0 3 , Tb 2 0 3 , Dy 2 0 3 , Ho 2 0 3 , Er2 0 3 , 'l'm 2 0 3 , Yb 2 0 3 , and Lu2 0 3 , which are shown to have the same structure by Zacharias e n ' s investigation, is given in Section 5. Received July 5th, 1930. Printed in Germany.