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Chapter5_Cullity-Stock copy 1 - Chopier5 Diffraction Ill...

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Unformatted text preview: Chopier5 Diffraction Ill: Real ‘ Samples INTRODUCTION Before turning to the practical aspects of diffraction from materials, it is valuable to consider how diffraction peaks are altered by the presence of various types of defects. Indeed, knowledge of how diffraction peaks are changed by defects under- lies many of the analyses described in the third section of the book. Defects can be small numbers of dislocations in crystals with dimensions of millimeters. At the other extreme, the dislocation density may be so high that it is difficult to imagine the existence of discrete dislocations. Small crystal or grain size can be thought of as another kind of defect and can alter diffraction peak widths. In the limit of ‘grain’ size approaching that of an atom, as in gases, liquids and amorphous solids such as glasses and many polymers, sharp diffraction peaks no longer exist, and important information about these materials can be gleaned from how they scatter x—rays. At the other limit, that of defect—free crystals with millimeter or greater dimensions, diffracted intensity must be treated in a fashion quite different from the approach in Chap. 4. 5-2 CRYSTALLITE SIZE In Chap. 4 the discussion of the structure factor revealed that destructive interfer- ence is just as much a consequence of the periodicity of atom arrangement as is con— structive interference. If the path difference between x-rays photons scattered by the first two planes of atoms differs only slightly from an integral number of wave- lengths, then the plane of atoms scattering x—rays exactly out of phase with the pho— tons from the first plane will lie deep within the crystal. If the crystal is so small that this plane of atoms does not exist, then complete cancellation of all the scattered x- rays will not result. It follows that there is a connection between the amount of “out-of-phaseness” that can be tolerated and the size of the crystal. The result is that very small crystals cause broadening (a small angular divergence) of the dif- 167 168 Chapter 5 Diffraction Ill: Real Samples fracted beam, i.e., diffraction (scattering) at angles near to but not equal to, the exact Bragg angle. Therefore consider the scattering of x-rays incident on the crys- tal at angles deviating slightly from the exact Bragg angle. Suppose, for example, that the crystal has a thickness t measured in a direction perpendicular to a particular set of Bragg plane (Fig. 5-1). Let there be (m + 1) planes in this set. Define the Bragg angle 9 as a variable and let BB be the angle which exactly satisfies Bragg’s law for the particular values of A and d involved, or A = 2d sin 63. In Fig. 5—1, rays A, D,..., M make exactly this angle 63 with the diffraction planes. Ray D”, scattered by the atoms of the first lattice plane below the surface, is there- fore one wavelength out of phase with A’; and ray M’, scattered by the mth plane of atoms below the surface, is m wavelengths out of phase with A”. Thus, at a dif- fraction angle 263, rays A’, D’,..., M’ are completely in phase and unite to form a dif— fracted beam of maximum amplitude, i.e., a beam of maximum intensity, since the intensity is proportional to the square of the amplitude. Incident X-rays that make angles only slightly different from 03, produce incom— plete destructive interference. Ray B, for example, makes a slightly larger angle (91, such that ray L’ from the mth plane below the surface is (m + 1) wavelengths out of phase with B’, the ray from the surface plane. This means that midway in the crys— tal there is a plane populated by atoms scattering x-rays which are one—half (actu- ally, an integer plus one—half) wavelength out of phase with ray B’ from the atoms of the surface plane. These rays cancel one another, and so do the other rays from similar pairs of planes throughout the crystal, the net effect being that rays scat- tered by the top half of the crystal annul those scattered by the bottom half. The intensity of the beam diffracted at an angle 201 is therefore zero. It is also zero at an angle 262 where 02 is such that ray N’ from the mth plane below the surface is Figure 5-1 Effect of crystal size on diffraction. 5-2 Crystallite Size 169 (m - 1) wavelengths out of phase with ray C’ from the surface plane. This defines, therefore, the two limiting angles, 261 and 202, at which the diffracted intensity must drop to zero. It follows that the diffracted intensities at angles near 263, but not greater than 291 or less than 262, are not zero but have values intermediate between zero and the maximum intensity of the beam diffracted at an angle 263. The curve of diffracted intensity vs. 26 will thus have the form of Fig. 5-2(a) in contrast to Fig. 5—2(b), which illustrates the hypothetical case of diffraction occurring only at the exact Bragg angle. The width of the diffraction curve of Fig. 5-2(a) increases as the thickness of the crystal decreases, because the angular range (261 — 202) increases as m decreases. The width B is usually measured, in radians, at an intensity equal to half the maxi- mum intensity, and this measure of width is termed the full-width at half maximum of FWHM. [Note that B is an angular width, in terms of 26 (not 6), and not a linear width.] A rough measure of B, is one-half the difference between the two extreme angles at which the intensity is zero, which amounts to assuming that the diffraction line is triangular in shape. Therefore, 1 B = 5(261 “ 202) = 01 _ 62. The path-difference equations for these two angles are similar to Eq. (3-1) but related to the entire thickness of the crystal rather than to the distance between adjacent planes: , 2: sin a, = (m Jr 1», 2: sin 02 = (m — 1». [max INTENSITY INTENSITY 292 293 20, 293 29—» 20————«» (a) (b) Figure 5-2 Effect of fine crystallite size on diffraction curves (schematic). u,” .. 170 Chapter 5 Diffraction HI: Real Samples By subtraction, t(sin 61 - sin 02) = A, 9 + 6 9 — 9 2t cos (3—5—3) sin (—173) = A. But 61 and 62 are both very nearly equal to 63, so that 91 + 62 = 263(approx.) m<L62>—<———61”2> s 2 2 (approx). and Therefore 2 A t = ——-——-. 5-1‘ B cos 63 ( ’ A more exact treatment of the problem gives 0% - t = ————— 5-2 B cos 63’ ( which is known as Scherrer’s formula [5.1]. It is used to estimate the size of verj small crystals from the measured width of their diffraction curves. Note tha whether a value of 0.9 or 1 is used depends on the shape(s) of the crystallite assumed to be in the sample. A detailed discussion appears elsewhere [G17]. Wha is the order of magnitude of this effect? Suppose A = 1.5 A, d = 1.0 A, an 0 = 49". Then for a crystal 1 mm in diameter the breadth B, due to the small crys tal effect alone, would be about 2 X 10—7 radian (10‘5 degree), or too small to b observable. Such a crystal would contain some 107 parallel lattice planes of th spacing assumed above. However, if the crystal were only 500 A thick, it would cor tain only 500 planes, and the diffraction curve would be relatively broad, name} about 4 X 10‘3 radian (02°), which is easily measurable. Nonparallel incident rays, such as B and C in Fig. 5—1, actually exist in any re: diffraction experiment, since the “perfectly parallel beam” assumed in Fig. 3-2 is a idealization. As will be shown in Sec. 6-4, any actual beam of X-rays contains dive gent and convergent rays as well as parallel rays, so that the phenomenon of di fraction at angles not exactly satisfying Bragg’s law actually takes place. Neither is any real beam ever strictly monochromatic. The usual “monochr' matic” beam is simply one containing the strong Ka component superimposgid C the continuous spectrum. But the Ka line itself has a width of about 0.001 A a1 5—3 Interference Fuction 171 this narrow range of wavelengths in the nominally monochromatic beam is a fur-' ther cause of line broadening, i.e., of measurable diffraction at angles close, but not equal, to 263, since for each value of A there is a corresponding value of 0. (Translated into terms of diffraction line width, a range of wavelengths extending over 0.001 A leads to an increase in line width, for )t— — 1. 5 A and 6— — 45°, of about 0.08” over the width one would expect if the incident beam were strictly mono- chromatic.) Line broadening due to this natural “spectral width” is proportional to mat? and becomes quite noticeable as 0 approaches 90". 5-3 INTERFERENCE FUNCTION The calculation of the intensity of diffraction peaks in Ch. 4 was for diffraction at the exact Bragg angle 03. At this angle, and in the absence of any defects producing displacement of the unit cells of the crystal, the total amplitude diffracted by the N unit cells of the crystal is the sum of the amplitude Fn scattered by each unit cell: ATOTAL = NFn- i (5'3) At deviations from the exact Bragg angle, the individual unit cells will scatter slight— ly out of phase. Also, the vector (S —- So)//\ no longer extends from the origin of the reciprocal lattice to a reciprocal lattice point. As was shown in the preceding sec- tion, x-rays scattered from an effectively infinite crystal at 6' ¢ 63 will be out of phase and the diffracted intensity will equal zero. If the crystal is small enough, however, the intensity will not go to zero off the exact Bragg condition, and the cal- culation which follows shows how diffracted intensity varies with angle as a func- tion of the number of unit cells along the direction of the diffraction vector (S — So), i.e., along the direction normal to the Bragg planes. Figure 5-3 shows the direct space and reciprocal space diagrams, respectively, for diffraction from a crystal at 9(1) > 93(1) for 001 and at 0(2) > 63(2) for 002, where “1” and “2” in parenthesis indicate the angle for the first and second order diffrac- tion. If a 0—29 diffractmeter is used, the portion of the reciprocal lattice sampled during a scan is indicated by the horizontal line from the origin in Fig. 5-3b (i.e., along b3 in this example). If Hhkl (or H for short) is the reciprocal lattice vector from the origin of the reciprocal lattice to the reciprocal lattice point hhe, the diffraction off the exact Bragg condition means that (S - So)/A #9 H. The difference between these two vectors along the direction of (S ~ SO)//\ will be written as / is often termed the deviation parameter and is shown in Fig. 5-3. In order to calculate the intensity diffracted from the crystal at 6 :3 63, the phase differences for scattering from different unit cells must be included. For the three unit vectors of the crystal 81, a2, and a3: N.—1N_~1N_.—1 277i ATOTAL: 2 E 2 Fepr — [(S ‘ So) ' (“131 + H232 + 11333)ls (5‘4) 111:0 m =0 113 =0 172 Chapter 5 Diffraction III: Real Samples (3) 30(1) SAMPLE 80(2) (b) 050 052 s 1 m N001 o( ) a1 0 _ O 0 30(2)” 8(1)/>\ smut s (2) S (1) 1(1) s (1)/ i (d) (C) Intensity 001 002 b3 Figure 5-3 (a) Incident and diffracted beam directions for 001 and for 002 diffraction in direct space. (b) Ewald sphere construction for incident beam directions shown in (a). (c) Deviation parameter I for the geometries in (a) and (b). (d) Diffracted intensity verses orientation given by the intereference function as a function of deviation parameter, i.e., of Ewald sphere orientation. “E” denotes the Ewald sphere, and the number in parentheses (“1” or “2”) indicates whether the quantities apply to 001 or 002 dif- fraction, respectively. the integers ni define the particular unit cells for which the phase difference is being calculated, and Ni are the total number of unit cells along a. From the definition for the reciprocal lattice vector expressing the deviation from the exact Bragg condi- tion (Fig. 5—3), i.e., ATOTAL : F2 2 2 exp[27rl(H + J) ‘ (”131 + ”232 + H333)] "1 "2 "3 = F2 2 E exp[27ri(hb1 + kbz + [b5 + J) ' (”131 + ”232 + "339] (5'5) n1 n2 n3 Applying the orthonormality conditions for reciprocal and direct space vectorS 5-3 Interference Fuction 173 b]- - a, = 8,], where is Kronecker’s delta produces ATOTAL = F2 6Xp(27Ti/" nlal) 2 exp(27r i/- nzaz) 2 exp(27ri/- n3a5) (5—6) Each sum is independent and may be evaluated separately. Converting from exp form to Sines, yields 3 sin 7T)\f - Niai , sin WA/- 3, i=1 ATOTAL = F eXP{7Tl(Ni — Dal + (N2 -1)32 + (N3 * 1)aal}, (5-7) where H denotes the product of three terms shown in Eq. 5—6. Note that thehlast term is the phase factor relating |ATl to AT, and this term is eliminated when" the intensity is calculated: sinzw J - Niai _ sinzrr J ‘ ai _ (5-8) 1 I = ArorAL ATOTAL = F2 Equation 5-8 is known as the interference function. By use of L’Hopital’s rule, the maximum intensity at the Bragg peak can be shown to equal F2N2 and, to a reasonable approximation, the width of the Bragg peak can be; calculated to be-proportionalito 1/N, where N is the number of unit cells along (S — SO). Thus, the integrated intensity increases linearly with N. Intensity, therefore, is a periodic function around each Bragg peak, a function which depends on the number of unit cells. The vector / is a three-dimensional vec- tor so that the intensity for a certain deviation along a specific direction in recipro- cal space depends on the number of unit cells lying along that direction. Another way of thinking about this is to think of the reciprocal lattice points lengthening into reciprocal lattice rods, i.e., rel rods, along the direction with the small number of unit cells. A simple way of showing this is to plot a constant contour of intensity, in reciprocal space, say one—half of the maximum. Figure 5-4(c) and (d) show such plots for the thin crystallite dimensions shown in a) and b). Vectors SO, S, and (So — S)/)t are shown for 6 < OB. In the case where the rel rod is elon- gated parallel to the reciprocal space sampling region for the 6—20 diffractometer, i.e., where the thin dimension of the crystal is parallel to the sample normal, a wider diffraction peak is observed. When the rel rod is perpendicular to the reciprocal Space sampling region, the effective of the small crystal dimension is not seen in the scan. This result, obtained from a reciprocal space perspective, is the same as that of Scherrer’s equation. Both results highlight an important property of direct and reciprocal spaces. If a feature’s dimension along a certain direct space direction is large, the feature’s size along the corresponding direction in reciprocal space is small. The converse is true as well. 174 Chapter 5 Diffraction lll: Real Samples (a) DIRECT SPACE (b) RECIPROCAL SPACE CRYSTALLITE G) <9 G) Q) Q so hkl m e— 29 S - So 000 SAMPLING S <3 G (D a (D t” e o (C) 9 So 0 0 W 0 26 l:::l ——-> SAMPLING S — So 000 CRYSTALLIT E S 0 Figure 5-4 Illustration of detectability of diffraction peak broadening for a 0—26 diffractometer for two crystallite orientations. (a) and (c) show direct space and reciprocal space, respectively, for one crystal- lite orientation relative to S — So and (b) and (d) show the two spaces for a second crystallite orienta- tion. The ellipses surrounding the reciprocal lattice points show the eleongation of the rel points into rel rods (due to small crystallite dimensions) and represent contours of constant diffracted intensity. The horizontal, solid bar represents the reciprocal space sampling region (RSSR). 5-4 STRAIN In the preceeding sections crystal size was seen as a type of defect, i,e., a deviation from the crystal of infinite extent and perfect atomic periodicty assumed in the der- ivation of Eq. 4-20 or 21. Dislocations and subgrains are another type of defect which have important consequences in diffraction. Before the existence of disloca- tions was established experimentally, considerable indirect evidence had been gath- ered showing that all real crystals possess, to a greater or lesser degree, a mosaic structure such as is illustrated in a greatly exaggerated fashion in Fig. 5-5. A crystal with mosaic structure does not have its atoms arranged on a perfectly regular lattice extending from one side of the crystal to the other; instead, the lat- tice is broken up into a number of tiny blocks, each slightly disoriented one from another.The size of these blocks is of the order of 1000 A, while the maximum angle of disorientation between them may vary from a very small value to as much as one degree, depending on the crystal. If this angle is s, then diffraction of a parallel monochromatic beam from a “single” crystal will occur not only at an angle of inCi'. dence 93 but at all angles between 63 and 63 + 8. Another effect of mosaic struC‘ Strain 175 Strained \’ \r Al x/if i t J. i ,_ ( i \/ J1} 1% \/r \X‘k \( T9» {TFTRX (:3 Ti~ L7 {X XL T/\ i . T \/ . by)» «x 1‘ J. i . T J. )3, 1 EN v4 '- ii 41 «5" Li 14+ J. v I {X (flTTTyfix/kuyy t— «"TI‘XJQ \rki 31¢? \’ ii .L 1 Unstrained (b) (C) Figure 5-5 Mosaic structure of a real crystal. (a) Rotations between adjacent domains (left), (b) dislo- cations walls separating different mosaic blocks (middle) and (e) regions corresponding to high disloca- tion densities in (b) where microstrain is significant. In (b) the symbol J. shows the positions of dislo- cation lines running through the plane of the drawing. ture is to increase the integrated intensity of the diffracted beam relative to that theoretically calculated for an ideally perfect crystal (Sec. 5—5). In the 1960s the TEM provided direct evidence of mosaic structure. It showed that real crystals, whether single crystals or individual grains in a polycrystalline aggregate, had a substructure defined by the dislocations present. The density of these dislocations is not uniform; they tend to group themselves into walls (sub- grain boundaries) surrounding small volumes having a low dislocation density (sub- grains or cells). Today the term “mosaic structure” is seldom used, but the little 176 Chapter 5 Diffraction Ill: Real Samples blocks of Fig. 5-5 are identical with sub—grains and the regions between the blocks are the dislocation walls. It is the strains and strain gradients associated with the groups of dislocations that is responsible for the increase in integrated intensity of diffraction not the fact that there are rotated domains. It is useful at this juncture to consider the effects of strain on diffraction peaks. Two types of stresses can be identified, microstresses and macrostresses. Microstresses and the corresponding microstrains vary from one grain to another, or from one part of a grain to another part, on a microscopic scale. On the other hand, the stress may be quite uniform over large distances; it is then referred to as macrostress. The effect of strain, both uniform and nonuniform, on the direction of x-ray reflection is illustrated in Fig. 5-6. A portion of an unstrained grain appears in (a) on the left, and the set of transverse diffraction planes shown has everywhere its equilibrium spacing do. The diffraction line from these planes appears on the right. If the grain is then given a uniform tensile strain at right angles to the diffraction planes, their spacing becomes larger than do, and the corresponding diffraction line shifts to lower angles but does not otherwise change, as shown in (b).This line shift CRYSTAL LATTICE DIFFRACTION LINE -ldol-— / So \ S (a) NO STRAIN \‘ (b) UNIFORM STRAIN Ah (C) NON—UNIFORM STRAIN 29 Figure 5-6 Effect of uniform and non-uniform strains (left side of the figure) on diffraction peak P05 tion and width (right side of the figure). (a) shows the unstrained sample, (b) shows uniform strain an (c) Shows non-uniform strain within the volume sampled by the x-ray beam. 5-4 Perfect Crystals 177 is the basis of the X-ray method for the measurement of macrostress, as'will be described in Chap. 15. In (c) the grain is bent and the strain is nonuniform; on the top (tension) side the Bragg plane spacing exceeds do, on the bottom (compression) side it is less than do, and somewhere in between it equals do. Thus, a single grain can ...
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