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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. 52 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 xrays 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
“outofphaseness” 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 xrays 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. 51). 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 Xrays 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 xrays 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 51 Effect of crystal size on diffraction. 52 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. 52(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. 52(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 fullwidth 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 onehalf 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 pathdifference equations for these two angles are similar to Eq. (31) 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 52 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 = ————. 51‘
B cos 63 ( ’
A more exact treatment of the problem gives 0% 
t = ————— 52 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. 32 is a
idealization. As will be shown in Sec. 64, any actual beam of Xrays 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". 53 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, xrays 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 53 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. 53b (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. 53. 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 53 (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 53 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}, (57) 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 _ (58)
1 I = ArorAL ATOTAL = F2 Equation 58 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 beproportionalito 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 threedimensional 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 54(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 54 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).
54 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. 420 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. 55. 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 55 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. 55 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 xray
reﬂection is illustrated in Fig. 56. 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 56 Effect of uniform and nonuniform 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 nonuniform strain within the volume sampled by the xray beam. 54 Perfect Crystals 177 is the basis of the Xray 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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