Introduction to Nanotechnology

3.2. STRUCTURE 41
The widths of the Bragg peaks of the X-ray scan of Fig. 3.4 can be analyzed to
provide information on the average grain size of the TiN sample. Since the widths
arise from combinations or convolutions of grain size, microcrystalline strain, and
instrumental broadening effects, it is necessary to correct for the instrumental
broadening and to sort out the strain components to determine the average grain
size. The assumption was made of spherical grains with the diameter D related to the
volume V by the expression
D=(?) 113
(3.4)
and several ways of making the linewidth corrections provided average grain size
values between 10 to 12nm, somewhat larger than expected from the TEM
histogram of Fig. 3.3. Thus X-ray diffraction can estimate average grain sizes, but
a transmission electron microscope is needed to determine the actual distribution of
grain sizes shown in Fig. 3.3.
An alternate approach for obtaining the angles 8 that satisfy the Bragg condition
(3.2) in a powder sample is the Debye-Scherrer method sketched in Fig. 3.5. It
employs a monochromatic X-ray beam incident on a powder sample generally
contained in a very fine-walled glass tube. The tube can be rotated to smooth out the
recorded diffraction pattern. The conical pattern of X rays emerging for each angle
28, with 8 satisfying the Bragg condition (3.2), is incident on the film strip in arcs, as
shown. It is clear from the figure that the Bragg angle has the value 8 = S/4R, where
S is the distance between the two corresponding reflections on the film and R is the
radius of the film cylinder. Thus a single exposure of the powder to the X-ray beam
provides all the Bragg angles at the same time. The Debye-Schemer powder
technique is often used for sample identification. To facilitate the identification,
ic .-@s O R
0 .... s c
0
diffracted rays
~~~j~~~~~ 0 11
Figure 3.5. DebyeScherrer powder diffraction technique, showing a sketch of the apparatus
(top), an X-ray beam trajectory for the Bragg angle 0 (lower left), and images of arcs of the
diffraction beam cone on the film plate (lower right). (From G. Burns, Solid State Physics,
Academic Press, Boston, 1985, p. 81 .)