XRay Diffraction from Small Crystallites
Victor H. Tiensuu, Sabri Ergun, and Leroy E. Alexander
Citation: Journal of Applied Physics 35, 1718 (1964); doi: 10.1063/1.1713726
View online: http://dx.doi.org/10.1063/1.1713726
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JOURNAL
OF APPLIED
PHYSICS
YOLUME 35,
NUMBER
6
JUNE
1964
X-Ray Diffraction from Small Crystallites
VICTOR
H.
TIENSUU,* SABRI ERGUN,t AND LEROY E. ALEXANDER!
Solid State Physics, Pittsburgh Coal Research Center, U. S. Bureau of Mines, Pittsburgh, Pennsylvania
(Received 1 November 1963)
The classical crystallographic equations used in deducing structure from peak positions, peak widths, and
integrated intensities of x-ray scattering are facile enough, but they involve assumptions that are not valid
for extremely small crystallites. An alternate approach more generally valid is to use the Debye interference
function. These two different approaches have been compared, using as examples a diamond crystallite
containing three unit cells on a side ('"'-'10.7 A) and one containing ten unit cells on a side (",35 A).
in experimental x-ray diffraction techA DVANCES
niques make possible the study of the structure of
ultrafine crystallites or crystalline regions, and interest
in this field is increasing. The classical crystallographic
equations used in deducing structure from peak positions, peak widths, and integrated intensities in x-ray
diffraction patterns are facile, but they involve a lattice
sum technique which becomes questionable as the number of unit cells in the crystallite approaches one. An
alternate approach more generally valid is to use the
Debye interference function. However, the calculation
of the Debye interference function becomes laborious
when the number of atoms in the crystallite is large (in
thousands). Therefore it is desirable to obtain information about the lower limit of crystallite size to which
the classical crystallographic equations can safely be
applied. The information sought can be obtained
numerically by comparing the results of diffraction intensity distributions calculated by these two different
approaches. Although such comparisons have been
made for peak positions and peak widths for facecentered cubic structuresl and for aromatic molecules,2
the calculations have not been extended to integrated
intensities which are important in structure studies.
If absorption effects are neglected, the intensity of
x-rays scattered coherentlYaby a crystallite is generally
expressed as
1 =1 •(FF*) (GG*) ,
(1)
where 1 e is the intensity of scattering by an electron,
(FF*) is the unit cell interference function, and (GG*)
is the lattice interference function. 1 e is given by the
well-known Thomson interference function which, for
an unpolalized beam, takes the form
called the structure factor and, if vibration effects are
neglected, is given by4
n
L
F=
IS
* Physical Chemist, Pittsburgh Coal Research Center, U. S.
Department of the Interior, Pittsburgh, Pennsylvania. Present
address: Dow Chemical Corporation, Midland, Michigan.
t Project Coordinator, Solid State Physics, Pittsburgh Coal
Research Center, U. S. Department of the Interior, Pittsburgh,
Pennsylvania.
:/: Senior Fellow, Mellon Institute, Pittsburgh, Pennsylvania.
1 L. H. Germer and A. H. White, Phys. Rev. 60, 447 (1941).
2 R. Diamond, Acta Cryst. 10,359 (1957).
3 W. H. Zachariasen, Theory of X-Ray Diffraction in Crystals
(John Wiley & Sons, Inc., New York, 1945), p. 91.
(3)
where the summation is carried out over the atoms of a
unit cell, f being the atomic scattering factor, r the
position vector (relative to the origin of the unit cell)
of the atom p, and H being defined from H= (S-So)/A,
S and So being unit vectors in the directions of the diffracted and primary beams, respectively. The magnitude of H is equal to 2 sinO/A and is customarily designated by s. Designating the position vectors of the
origins of unit cells by Am, G is defined from
G= L exp(211'iH·A m ).
(4)
m
If, for convenience, it is assumed that the crystal has
the shape of a parallelepiped with edges MIaI, M2a2,
and M 3a3(al, a2, a3 being space lattice vectors) it follows
that4
3
(GG*) = II (sin211'M;H· a;/sin211'H· ai).
(5)
i=l
If M's are large, say 102 or greater, it is readily recognized that the OSCIllations of 1 are determined by (GG*).
Hence peak positions are derived from Eq. (5), d. the
three Laue equations. 4 Also, in deriving expressions relating peak broadening to crystallite dimension and in
obtaining expressions for integrated intensities, only
(GC*) is considered. Equation (1) can be written in a
more general form
1=1.(DD*),
(2)
where the symbols have their usual meanings. 3 F
fp exp(211'iH'rp),
p=I
(6)
where (DD*) is the inteference function for the crystallite. If vibration effects are neglected, D can be expressed
as
'N
D=
L
fp exp(27riH.r p)
(7)
p=I
where r is the position vector of the atom p relative to
• B. E. Warren, Mimeographed resume of lectures, cf. A. H.
Compton and S. K. Allison, 'X-Rays in Theory and Experiment
(D. Van Nostrand Company Inc., Princeton New Jersey 1954)
pp. 406-420.
'
,
"
1718
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1719
X-RAY DIFFRACTION FROM SMALL CRYSTALLITES
any arbitrary origin and the summation is carried out
over all of the atoms in the crystallite. When the
assemblage assumes all possible orientations with respect to the primary beam with equal probability, the
average value of (DD*) takes the form
(DD*) =
N
N
p~1
q~l
I: I:
fpfq(sin27rsrpq/27rsrpq).
(8)
Equation (8) is the Debye interference function. s For a
crystallite containing one kind of atom, Eq. (8) can be
written in a more convenient form, and its substitution
into Eq. (6) yields
(1) = I eN pi (s),
(9)
where i(s) is the Debye interference function per atom
and
+
i(s) = 1 l/N I: ni(sin27rsr;j27rsri),
ri~O.
(10)
In Eq. (10) 11; is the number of interatomic distances of
magnitude ri, ni being counted twice between any two
atoms. This equation has been used in computing scattering intensities of molecules containing only a few
atoms, the maximum number in one case being 359. 1
This equation gives the average profiles of diffraction
peaks in absolute units. In comparing the peak locations and peak widths obtained from Laue and Scherrer
equations, respectively, with those obtainable from the
Debye inte:ference function, i(s), not (I), should be
plotted agamst s.
As examples we have calculated i(s) for a cube of
diamond crystallite having three unit cells on an edge
(216 atoms) and one having ten unit cells on an edge
(8000 atoms). The interatomic distances and the exact
number of like distances were computed by two procedures. One procedure involved calculation of interatomic distances in a unit cell and summing over unit
cells. In the second procedure the projections of the
a~oms on the (00l) planes were considered. Four equidIstant layer sequences parallel to (001) planes describe
the positions of all atoms in the crystallite. For a spherical model t~e estimates of ni can be obtained by a procedure outlmed by Germer and White. 1 However to
avoid any uncertainties that may be associated ~ith
estimates, the exact numbers have been obtained. Calculations yielded-S8~different interatomic distances (excl~sive of the zero distance) for the crystallite having 27
umt cells and 790 for the one having 1000 unit cells.
The i(s) function was computed using an IBM-704
computer. A general survey calculation was made over
the range s=0.00-2.00 at increments Lls=O.Ol and detailed calculations were made at increments ds=O.OOl
to cover each of the first eight peaks and over the region
0.000< s <0.020. The tetrahedral C-C bond distance
was taken as 1.544426 A.
The plots of i(s) are shown in Fig. 1. All of the peaks
5
P. Debye, Ann. Physik 46, 809 (1915).
4,,-'-r-r-,,-~r-,,-,-,-r-''-'-'-~''
:5
2
I
Or-~~~--~~----~--------------~
220
9
8
III
~7
6
5
4
113
422
133
:5
2
511
333
135 206
440
o
FIG. 1. Calculated x-ray patterns for diamond crystallites. The
top pattern belongs to a cube with 3 unit cells on edge (216 atoms)
the bottom pattern to a cube with 10 unit cells on edge (8000
atoms).
seen correspond to normal crystalline reflections except
the ripples at low s values which arise from the highly
symmetrical shape of the model (i.e., from the transform
of the shape). It is readily seen that the intensities belonging to the small crystallite (top curve) are not very
sharp and some of the peaks remain unresolved. For the
large crystallite the peak positions and linewidths of the
first eight reflections, obtained from a separate enlarged
plot of each reflection, are shown in Table I. Included in
the table are also peak breadths calculated using the
Scherrer equation with K being taken as 0.89. 6 The
nominal si~es of the crystallites are 3a and lOa, where
a=3.5667 A, the side of the unit cell; therefore according to the Scherrer equation, iJ=0.89/10.7=0.0832 A-I
and iJ=O.89/35.667 =0.02495 A-I for the small and the
large crystallites, respectively. It is seen that the peak
TABLE I: Comparison of pe~k positions and peak breadths calculat~d usmg. the Laue eqUl;tlOns and the Scherrer equation, respect~vely, WIth those .obtamed graphically for the interference
functIOns calculated usmg the Debye equation.
Peak breadth
Line
index
111
220
113
400
133
422
511,333
440
«(3), l1s
Peak position, s
Large
Laue
Small
equacrystal- crystaltions·
lite
lite
Large
crystallite
Small
crystallite
0.4856
0.7930
0.9299
1.1215
1.2221
1.3735
1.4568
1.5860
0.0241
0.0236
0.0259
0.0254
0.0244
0.0252
0.0258
0.0247
0.091
0.081
0.094
0.4856
0.7931
0.9297
1.1218
1.2220
1.3734
1.4566
1.5862
0.484
0.794
0.924
1.127
1.223
1.375
.
Average =0.02489
From the Scherrer equation = 0.02495
as = (h' +k' +12)
0.089
0.0832
'Ia.
6 L. Bragg, The Crystalline State, A General Survey
and Sons, London, 1919), Vol. 1, p. 189.
(G. Bell
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1720
TIENSUU,
ERGUN,
positions agree well with those calculated using the
Laue equations; however, the Scherrer equation predicts the peak breadth accurately for the large crystalite only.
The information about the arrangement of atoms in
the unit cell is deduced from (FF*). At the peak maximum (GG*) assumes the value of M2, M being the total
number of unit cells in the crystallite. Thus, theoretically
(FF*) =1max/l eM2. For experimental reasons 1 max cannot generally be obtained, and equations have been developed for integrated intensities which are measurable.
If a crystallite assumes all possible orientations relative
to the primary beam with equal probability, the Darwin
equation 7 relating the total scattered intensity for a
given reflection, sometimes termed the power of the reflection, to the crystallite and unit cell dimensions takes
the form
P=le(FF*) (mR 2'A3N/4vn sinO),
(11)
where m is the multiplicity of the (hkl) considered, N is
the total number of the atoms in the crystal, v is the
volume of the unit cell, and n is the number of atoms in
it. Equation (11) is applicable for randomly oriented
powdered crystals except at small angles.
The applicability of Eq. (11) can be determined by
computing numerically the integrated intensities from
the i(s) function. The total scattered intensity P can
be expressed as
P=
f
leN j2i(s)27rR sin (20)Rd(20).
(12)
In Eq. (12), d(20) can be replaced by 'Ads/cosO, and
1 e, j2, and sinO can be taken outside the integral sign
whenever the integration involves a range of (I; Eq. (12)
then takes the form
f
P=1e!2JV47rR 2'A sinO
i(s)ds.
(13)
Comparison of Eq. (13) with (11) yields
(14)
where F 02 = (FF*)/ j2 and is sometimes called the
geometric structure factor.
Integrated areas under the diffraction peaks can be
obtained graphically or numerically from the plot of
i(s) vs s. From Fig. 1 it is seen that the intensities do
7
C. G. Darwin, Phil. Mag., 27, 315, 675 (1914); 43, 800 (1922).
AND
ALEXANDER
II. Comparison of the integrated intensities calculated
Darwin equ~tion with those obtained graphically from
the Interference functlOn of the large crystallite calculated using
the Debye equation.
TABLE
fro~ the
Line
index
s=2 sinll/X
m(F/f)2
411"nvs2
fi(s)dsa
111
220
113
400
133
422
511,333
440
0.4856
0.7930
0.9299
1.1215
1.2221
1.3735
1.4568
1.5860
0.239
0.269
0.196
0.0672
0.113
0.179
0.106
0.0672
0.242
0.265
0.195
0.0712
0.114
0.174
0.106
0.0715
a Integrated areas under the peaks extend to the abscissa the limits on
the abscissa being points midway between the adjacent peaks.
not fall to zero between the normal diffraction peaks.
Therefore uncertainties arise concerning the limits of
the graphical integration. It was found that only the
large crystallite yielded meaningful results. The integrated areas were obtained using the midpoints (on
the s scale) between the adjacent peaks as cutoff points.
Integrated areas and those calculated using Eq. (5) are
shown in Table II. F i was calculated from
F i= 32[1 +cos!7r(h+k+l)],
(15)
where (hkl) are either all odd or all even. It is seen that
maximum deviations (about 6%) are encountered with
the (400) and (440) reflections ...,:The latter are the two
smallest of the first eight peaks and the background intensity apparently had the greatest effect on their integrated areas. The agreement obtained with other reflections is good.
It appears that the Laue equations predict satisfactorily the peak positions of x-ray diffraction from
crystallites as small as 10 A in size (about 200 atoms).
The Scherrer equation for the peak breadth is fairly
accurate for crystallite sizes to 30 A. It is also evident
that if the crystallite size becomes about 35 A, the intensities obtaine((experimental or theoretical) will not
fall to zero between the diffraction peaks, and the smaller
the crystallite size the greater the error will be in the
integrated intensities. The calculations made here should
permit a better understanding of the relationships that
exist between the various crystallographic parameters
and give an idea about the lower limit of crystallite
size to which the classical crystallographic equations
can be applied safely.
ACKNOWLEDGMENT
We wish to thank John R. Townsend of the University of Pittsburgh for valuable discussions.
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