J. 'I'URKEVICH, P. C. STEVENSON AND J. HILT,IER 75 THE INTERPRETATION OF BROADENED X-RAY REFLECTIONS WITH SPECIAL REFERENCE TO CLAY MINERALS BY G. W. BRINDLEY Received 24th May, 1951 The problems arising in the determination of colloidal sizes from the broaden- ing of X-ray reflections are surveyed. Instrumental broadening can be taken fully into account. Diffraction broadening may be produced by lattice strain, lattice mistakes or small crystal size separately or in combination. In the domain of clay minerals, the two latter commonly occur together and this is the main complication in using X-rays for size determinations of clay colloids. In addition, the diffraction process is often 2- rather than 3-dimensional in character76 BROADENED X-RAY REFLECTIONS so that experimental studies have been concerned as much with the difiraction process as with crystal size determinations.Electron microscope data on clay particle sizes are summarized. Crystal sizes of balloysite and montmorillonite from X-ray data are of the order of 0~01-0~02p which is comparable with the smallest observed particle dimensions. Investigations of the thickness of kaolinite and dickite crystals could be usefully undertaken but similar studies of other clays are rendered difficult by the occurrence of interstratified mixed- layer structures. 1. 1ntroduction.-This contribution to the Discussion on the size and (I) a survey of the interpretation of broadened X-ray reflections, and As regards (I), the subject can be discussed broadly since comprehensive treatments of the subject up to about 1947 have been given by James and by Wilson ; 2 the latter has also summarized some of the more recent developments.8 X-ray diffraction by clay minerals has also been recently reviewed.4 Since colloidal sizes are measured with X-rays by determining the breadths of reflections, the questions involved are concerned with the interpretation of line breadths.* The observed breadth B of a re- flection arises partly from instrumental effects for which a correction must be made before the true diffraction breadth /3 can be obtained.Since diffraction broadening of lines can arise in a number of ways, i t is important to be able to distinguish them experimentally. Failure to do so may result in quite false results for the size and shape of particles. 2.Corrections for Instrumental Broadening .-Instrumental broaden- ing arises from factors such as slit widths defining the X-ray beam, size and form of powder specimen, type of X-ray camera, doublet character of the Kci radiation usually employed. The instrumental breadth b of a line is found by using a well-crystallized material such as quartz or an annealed metal wire with a crystal size N 10-4 cm. which gives negligible diffraction broadening. To obtain the diffraction breadth ,!l of a broadened line from the observed breadths B and b, various procedures axe possible, for example shape of colloidal particles aims at (11) a summary of results obtained with clay minerals. P = B - b , . ' (1) B'=Ba--bS . ' (2) Eqn. (I) was given by Scherrer,6 and recently Wood and Rachinger have shown that i t is generally valid if the line profile takes the form I / ( I + cZxS), (G, a constant and x, distance along the film).Eqn. (2) given by Warren and Biscoe was shown by Taylor * to be valid for a Gaussian intensity distribution. Graphical methods involving line pro- files have been described by Jones and widely applied. Rachinger lo and Pease 11 have described direct methods for correcting for Koc,a, separation. 1 James, The Crystalline State, Vol. 2 ; The Optical Principles of the Daflrac- tion of X-rays (Bell, London, 1948). 2 Wilson, X-ray Optics (Methuen, London, 1949). Wilson, Research, 1950, 3, 394. 4 Brindley, X-ray Identificataon and Crystal Structures of Clay Minerals (The Mineralogical Society, London, 195 I).* The breadth may be taken either as the angular width of a line a t half its maximum intensity (the so-called half width) or as the integrated intensity divided by the maximum intensity (the integral width) ; the latter is almost invariably used in theoretical treatments of the subject. 6 Scherrer, Nachr. Gottingen Gesell., 1918, 98. Zsigmondy's Kolloidchemie (3rd edn.) , p. 387. Wood and Rachinger, J . Inst. Metals, 1949, 75, 571. 7 Warren and Biscoe, J . Amer. Ceramic SOC., 1938, 21, 49. Taylor, Phil. Mag., 1941, 31, 339. Jones, Proc. Roy. SOC. A , 1938, 166, 16. lo Rachinger, J . Sci. Instr., 1948, 25, 253, 353. l1 Pease, J . Sci. Instr., 1948, 25, 353.G. W. BRINDLEY 77 Stokes la has made a notable contribution by developing a Fourier method giving the true diffraction profile of a line corrected for all forms of instrumental broadening.The profile enables more information to be obtained than can be obtained singly from the width of a line, and the form in which Stokes' results are obtained is particularly suitable for these further developments (see 3 6 ) . 3. Diffraction Broadening Due to Crystal Size .-Measurable broadening, distinguishable from instrumental effects, is obtained only with crystals of colloidal and near-colloidal size. ( A .= wavelength, L = crystal size, and K is a constant of the order of unity) gives correctly the order of magnitude of L if K is made equal to unity. The value of K depends on the relation of the reflecting planes to the shape of the crystal, and can only be calculated explicitly for relatively simple shapes.The theoretical calculation of K has been the subject of many investigations too numerous to list here. Reference may be made to the books by James 1 and by Wilson and to recent papers by Stokes and Wilson.18 The latter show that for a general reflection hkZ from a crystal of any shape whatever, The Scherrer e q ~ a t i o n , ~ j? = KA/LCOS e . (3) j? = A/L cos e, where L = V-lST(hkZ)dV, . (4) or L is the " volume average of the thickness of the crystal measured perpendicular to the reflecting planes ", T(hkZ). Only for spherical crystals will L be a constant for all reflections. In other cases, the variation of L with (hkl) will enable the crystal shape to be determined. 4. Broadening of X-Ray Reflections Due to Strain.-X-ray reflections are broadened by any departure of a crystalline structure from strict regularity.Crystal size broadening may be included in this statement if the boundary of a crystal is regarded as a discontinuity in its regularity. Strain broadening arises (a) if the strain 7 is uniform in a crystal but varies from crystal to crystal in a powder,ld (b) if the distortion in a crystal is non-uniform, so that 9 varies with direction, hkl.15 This type of broaden- ing has been investigated chiefly in relation to cold-worked metals 16 and is probably not likely to be important for free colloidal particles. Briefly, in case (a) the strain breadth & is proportional to tan 8 and in- dependent of A (19 = Bragg angle), and in case (b) Comparison of eqns. (4) and (5) suggests that strain broadening may be distinguished from crystal size broadening by (i) the variation of with sec 8 or tan 0 ; (ii) the variation of p for a particular reflection hkl with A.Of these possibilities, (i) tends to be insensitive and is likely to be com- plicated by the dependence of L or 7 on hkl ; (ii) is the more satisfactory method but is inconvenient experimentally. 5. Broadening of X-Ray Reflections Due to Lattice Mistakes.-This type of broadening is less easily summarized because i t can arise in a variety of ways. Among clay minerals and layer silicates generally, lattice mis- takes of several kinds are common (Brindley,4 especially Chap. XI and XII), and i t is principally for this reason that the X-ray determination of the size and shape of clay colloids is rendered difficult.l3 Stokes and Wilson, Proc. Carnb. PhiE. Soc., 1942, 38, 313 ; 1944, 40, 197. l4 Brindley and Ridley, Proc. Physic. Soc., 1938, 50, 501. Brindley, Proc. Physic. Soc., 1940, 52, 117. l6 Stokes and Wilson, Proc. Physic. Soc., 1944, 56, 174. 16 Smith and Stickley, Physic. Rev., 1943, 64, 191. Wood, Nature, 1943, 151, 585. Lipson and Stokes, Nature, 1943, 152, 20. Hall, PYOC. Physic. j?, oc q m . tan 8. - ( 5 ) Stokes, Proc. Physic. Soc., 1948, 61, 382. sot., 1949,162, 741.78 BROADENED X-RAY REFLECTIONS The diffraction patterns of such structures are characterized by a mixture of relatively sharp and diffuse reflections, a general account of which is given by Wilson.8 Crystal size broadening gives lines in powder diagrams having a symmetrical or largely symmetrical profile whereas lines broadened by certain types of mistakes may appear as markedly asymmetric bands.", 1% 19,20 Thus, if in layer lattices the displacements of the layers are so frequent that the structure may be treated as a random stacking of two-dimensional lattices, characteristic bands are obtained having relatively sharp low-angle terminations and spreading towards high angles ; the band profile can show considerable variation from one band to another depending on the variation of the structure factor F with angle of diffraction.% 89 When F is constant or largely constant over the range of diffraction, then a formula developed by Warren18 analogous to the Scherrer formula is applicable for the determination of the layer size : A slight modification of this procedure is required if L is very small, -20 A, as Miss Franklin 23 has recently' indicated.Other bands are largely insensitive to crystal size, so that considerable care is required in utilizing data of this kind for the determination of crystal dimensions. The theory of diffraction by two-dimensional lattices has been treated by Laue,l7 Wa,rren,lB Wilson, l 9 Brindley and MBring.20, 22 An alternative method of estimating crystal sizes from two-dimensional diffraction bands rests on the fact that the peaks are displaced from the positions of the corresponding normal reflections which can be calculated from the lattice parameters. The method is feasible only when the structure factor F is largely' constant and for low-order bands.Warren Is has given, in effect, the following approximate formula, (Ad = apparent error in lattice spacing, d = spacing calculated from the lattice parameters). Another type of lattice mistake which is also common among clay minerals arises from the interstratification of layers of different kinds, having different thicknesses and/or scattering factors. With layers of different thickness, a non-integral series of reflections is obtained of very varying widths. Hendricks and Teller 24 first treated problems of this kind theoretically assuming the crystal size to be large (effectively in- finite) and M6ring 2 5 has more recently developed the theory in a form applicable to a specified number of layers. Brown and MacEwan (see BrindleyJ4 Chap. XI) have given numerous curves illustrating these effects for mixed layer clays of several kinds. 6 .Range of Crystal Sizes in a Powder Specimen.-In any powder specimen a range of crystal sizes and shapes will be present so that L determined from the diffraction breadth /3 will represent an average value, This average is expressed by eqn. (4) for a single crystal or an assemblage of crystals. Bertaut 26 and independently Warren and Averbach 27 have @ = 1e84X/L cos 9. . - (6) Ad = o-32d2/L, . (7) 17 Laue, 2. Krist., 1932, 82, 127. l8 Warren , Physic. Rev., 1941 , 59, 693. lS Wilson, Acta Cryst., 1949, 2, 245 ; also Nature, 1948, 161. 2o Brindley and Mbring, Nature, 1948, 161, 774. 21 Brindley and Robinson, Min. Mag., 1948, 28, 393. 22 Brindley and Mering, Acta Cryst., 1951 (in press) 22a Brindley and Mering, paper in course of preparation ; a continuation of 24 Hendricks and Teller, J.Chem, Physics, 1942, KO, 147. 2sM6ring, Acta Cryst., 1949, 2, 371 ; see also Fourth Int. Cong. Soil Sci., 26 Rertaut, Acta Cryst., 1950, 3, 14 ; also Conzpt. rend., 1949, 228, 187, 492, 27 Warren and Averbach, J. Appl. Physics, 1g50,21, 595. ref. (22). Amsterdam, 1950, 3, 21. 1597- Franklin, Acta Cryst., 1950, 3, 107.G. W. BRINDLEY 79 shown that by a Fourier analysis of the profile of a diffraction-broadened line, the distribution of the particle sizes may be found. In Warren's notation, if A , is the nth Fourier coefficient, then (un/dn)n,o = - I I N , and (d2A,/dn2f = (r/N)$(n), where IN is the average number of unit cells in a crystal normal to the reflecting planes, N is the total numbers of cells and p(n) is the number of columns containing n cells.7. Combination of X-Rays and Other Techniques.-In the study of clay mineral colloids, the writer believes that the most useful results will be obtained by a combination of techniques and in particular by combining electron microscope studies with X-ray analysis and perhaps also electron diffraction analysis. A pure X-ray approach to the deter- mination of clay mineral dimensions is beset with very great difficulties on account of the frequency with which these minerals exhibit lattice imperfections. The electron microscope, by showing directly the indi- vidual particles of a clay colloid, is of very great value, but i t will not replace entirely the X-ray method of examination, since the latter is con- cerned essentially with crystal dimensions.With well crystallized clays such as many kaolinites which show clear hexagonal forms in the micro- scope there is little doubt that the individual particles are single crystals, but with poorly crystalline clays, such as the montmorillonites, the par- ticles will often be crystal aggregates. An important contribution to this question has been made by Mdring and co-workers 28 who have shown that montmorillonite crystals tend to adhere along edges and faces in a manner depending on the number and kind of exchangeable cations. Experiments such as those of Birks and Friedman29 have clearly shown that crystal sizes determined by X-rays and with the electron microscope are in close agreement ( f 10 yo) over a wide range of crystal sizes when the conditions are favourable for both methods. Electron diffraction has so far been applied very little to clay minerals, but by combining the diffraction technique with the electron microscope i t may be possible to obtain useful results for single crystals of clay colloids.Experiments of this kind on kaolinite and montmorillonite have been described by Forslind 30 while MacEwan and Finch 81 have reported diffraction experiments on montmorillonite. The accompanying table summarizes data obtained with the electron microscope and is based largely on a recent report of the American Petroleum Institute s2 which contains a valuable bibliography and an extensive series of electron micrographs (see also the book by Marshall 9.The numerical data are to be regarded as indicating orders of magnitude only. The majority of clays form thin hexagonal or pseudo-hexagonal plates but a number exist as thin laths, rods or tubes. In the former group, the plane of the flake corresponds to the plane of the Si-0 network in the crystal structure, and is the basal (001) plane. In the latter group, i t is of great interest to determine the relation of the crystal habit to the structure. Clays having flaky crystals tend to form well-ordered ag- gregates when sedimented and these can be used to obtain clear basal (ool) reflections. One would expect to be able to determine the thickness 2* MBring, Mathieu-Sicaud and Perrin-Bonnet, Fourth Int. Cong. Soil Sci., 29 Birks and Friedman, J .Appl. Physics. 1946, 17, 687. 3O Forslind, Svenska Forskningsinst. f& Cement, (1948), Bull. No. 11. 31 MacEwan and Finch, to appear in Clay Minerals Bulletin. 33 American Petroleum Institute (Project 49, Clay Mineral Standards), (Col- 33 Marshall, The Colloid Chemistry of the Silicate Minerals (Academic Press, Amsterdam, 1950, 3, 29. umbia Univ., N.Y., 1950). N.Y., 1949).SO BROADENED X-RAY REFLECTIONS of the flakes from the (004 reflections and the layer dimensions from the (hko) reflections, or from the (hk) bands when the crystalline layers are randomly disordered. SURVEY OF SIZES OF CLAY MINERAL PARTICLES FROM ELECTRON MICROGRAPHS 28, 32, 35, 34, 95 Mineral Dickite . . Kaolinite . . Kaolin mineral in many fireclays Halloysite . Montmorillonites Nontronite .Hectorite . c Hydrous mica, illite Palygorskite or Sepiolite . Attapulgite Habit I Partidesize Well-defined hexagonal plates. Hexagonal plates, often well-developed. Thick- ness, variable. Some- times elongated. Hexagonal plates, often poorly developed. Elongated forms having the appearance of rods or tube.s.85 Split and partly unrolled tubes are observed. Usually very poorly de- fined, occasionally show- ing hexagonal forms. Tendency to aggregate.28 Poorly crystallized. Lath- like or ribbon-like ap- pearance. Thin laths. Poorly defined, thin hexa- Ribbon-like particles. gonal flakes. Fibres. Rod-like and flaky forms observed. Commonly - r-rop, Plates usually 0.1-3 p. occasionally larger. Plates usually much less than ~ p . Outer diameters N 0-05-0.2~.Length - 0-1-1 p. Wall thickness N o*ozp. than ~ p . Plates much less Length N rp. Width N 0 . 1 ~ . Width - ~ p . Length +- ~ p . Length - 0-I-5p. Width N O - O I - O . I ~ . Length of rods N 0-I-5p. Flakes N 0-1-0-5p. 8. Difficulties of Determining the Thickness of Clay Mineral Particles with X-Rays .-In the montmorillonite clays, the non-integral orders of basal reflections ordinarily obtained and the marked variations in their breadths arise from the occurrence of randomly interstratified water layers or hydrated layers.24 Organic-montmorillonite complexes 8% a7 (especially with glycerol and glycol) give clearer reflections and a series of integral orders which are much less influenced by random effects. They have not been critically considered as yet from the standpoint of the thickness of the flakes.I n the mica clays also the line profiles in- dicate random interstratifications. The lines are considerably sharpened by heat treatment, but they have not been considered in relation to particle size measurements. 34 CaiUbre, Mathieu-Sicaud and Henin, Bull. SOC. Frmnq. Min. Crist., 1950, 35 Bates, Hildebrand and Swineford, Amer. Min., rgjo, 35, 463. 38 MacEwan, Nature, 1944, 154, 577 ; J . SOC. Chem. Id., 1946, 65, 298 ; 37 Bradley, J . Amer. Chem. SOG., 1945, 67, 975 ; Amev. Man., 194.5, 30, 704. 73, 193. 7’wm.s. Faraday SOC., 1948, 4, 349,G. W. BRINDLEY 81 The situation is less complex among the kaolin group of clays, apart from halloysite which must be considered separately. Kaolinite and dickite could be usefully examined from this standpoint, but no measure- ments have so far been reported.The less well-defined kaolin clays found in many fireclay deposits must be considered more cautiously. They commonly give broader basal lines than the well-crystallized kaolinites, suggesting thinner flakes, but there is evidence (admittedly rather slender a t present) that they may contain some hydrated layers.4 The electron micrographs were first interpreted as showing lath-like crystals but with improved techniques it has become apparent that at least some halloysite crystals are tube-like.ss It was inferred from the non-orientation of halloysite sedimented in water that the particles were probably not plate-like. Halloysite exists in a variety of hydrated forms. Fully hydrated halloysite, Al,Si,0,(OH)4 .2H,O, with alternate silicate and water layers, may be a regular layer structure suitable for an X-ray study of crystal thickness, but some water is lost very readily and the regularity of the layer sequence is not very certain. Fully dehydrated halloysite is difficult to obtain and seems to require heat-treatment to the point of decomposition.38, 39 Naturally occurring metahalloysite is largely but not wholly dehydrated and this applies also to the mineral heated at temperatures up to about 3ooOC. The possibility that the incompletely dehydrated mineral may contain randomly interstratified water layers requires careful consideration in any study of X-ray line breadths. The fact that some crystals are curved, even tubular, is a further complication, and Wilson’s analysis 40 of diffraction by curved layers may find an important application in this connection.To summarize : to the writer’s knowledge, no detailed measurements have been made to obtain the thickness of clay mineral particles from the broadening of basal X-ray reflections. Kaolinite and dickite axe suitable for such measurements, but with all other clay minerals serious difficulties arise from the irregularity of the layer sequences. 9. Layer Dimensions of Clay Mineral Particles .-Detailed studies have been made to interpret quantitatively the broad diffraction bands given by the clay minerals halloysite 21 and montmorillonite.22 Although the centre of interest in this work has been the process of diffraction by single silicate sheets acting as two-dimensional gratings, the determin- ation of the size of the layers is implicit in the work.The procedure adopted was to estimate L by applying eqns. (6) and (7) to suitable bands and then to calculate the distribution of intensity in the diffracted bands for a range of crystal sizes of the estimated order of magnitude. In this way the approximations involved in arriving at eqns. (6) and (7) do not enter into the final results which rest directly on comparisons of observed and calculated intensity distributions. The results obtained for halloysite ar although broadly satisfactory in that they furnish ample evidence that the mineral consists essentially of two-dimensional diffracting units and that the theory of diffraction developed by Warren l8 is essentially correct, do not give more than the order of magnitude of the crystal size, namely about 100-200 A, i.e.0~01-0~02p. The crystal size therefore appears to be of the same order of magnitude as the smallest particle dimensions seen in the electron microscope. Since the electron micrographs of halloysite show elongated forms (laths, rods or tubes) some variation of L with direction hR in the crystal lattice may exist, i.e. if the diffracting units (crystals) are similar in shape to the observed particles. While some of the anomalies may be explainable in this way, other difficulties remain, notably for the bands Halloysite presents a particularly complex problem. 38Brindley and Goodyear, Man. Mag., 1958, 28, 407. 39 Brindley, Robinson and Goodyear, Mdn.Mag., 1948, 28, 423. 4 0 Wilson, Acta Cryst., 1949, 2, 220.82 RHEOLOGICAL PHENOMENA OF CLAY SOLS hk = 02, 11 and 06, 33. These, being different orders of reflection from the same planes, should at least give the same L value, in fact they give about zoo A and 85 respectively ; this difference may be due to a small but appreciable separation of the 06 and 33 reflections, but this explana- tion is not yet fully established. Halloysite, however, is not ideal for testing the theory of diffraction by random layer structures in view of the probable curvature of the layers and for this reason no further attempt has yet been made to analyse the data in detail. Montmorillonite appeared to offer a better chance of obtaining quanti- tatively satisfactory results.The theory' of the diffraction process has been re-examined and certain approximations in the purely algebraic treatment involving the assumption of a constant or slowly varying F factor have been removed by developing a method involving partly algebraic, partly numerical integration.22 The results obtained by the more detailed calculations do not differ very greatly from those obtained from the Warren formulae and it seems probable that the anomalies in the L values obtained for halloysite cannot be attributed entirely if at all to approximations in the Warren theory. In the application of the Warren theory 18 or the more detailed analysis of Brindley and MQing 22 to the diffraction bands from montmorillonite, a further complication has arisen. In order to carry through these ana- lyses for two-dimensional diffracting units, i t is necessary to know the angular variation of F within a diBraction band, i.e., to know the struc- tural arrangement in the layer. Now with montmorillonite not only is the structure of the silicate layer still a matter of discussion (see, for example, MacEwan, in ref. (4), Chap. IV) but also the exchangeable cations and water molecules between the layers must be considered. The shape of the bands has been found to be dependent on the particular saturating cations and on the degree of hydration. The problem is still not yet fully solved, but the prospects of obtaining quantitative agreement be- tween observed and calculated intensity distributions and therefore of reliable L values from the X-ray data appear to be good. At the time of writing, the most probable value for the layer dimensions of montmorillon- ite appears to be about 250 A.aaa and this agrees with the electron micro- graphs of the same material. 28 Physics Laboratories, University of Leeds.