When examining the incoming wave (as represented by their path in Figure 12.6),
a difference becomes apparent in the path length between the wave scattered in the
zeroth lattice plane, representing the plane directly at the surface, and the wave
scattered in the first plane. The difference in the path length d is a function of
the distance between the lattice planes d and the angle between the incoming wave
and the lattice planes q. This path difference is d ¼ d sin q. The scattered wave,
leaving the specimen under the same angle, has the same difference in path length
and therefore the total difference of the path length is d total ¼ 2d sin q. The path
difference between the zeroth and the second plane is, therefore, exactly doubled;
more generally, the difference of the path length to the nth plane is 2nd. An
interference maximum is observed when the difference in the path length is exactly
an integer multiple of the wavelength. This may be the path difference between two
parallel adjacent lattice planes or between planes in the distance nd. This leads to the
condition for an interference maximum:
nl ¼ 2d sin q; n ¼ 1; 2; 3;. . .
ð12:14Þ
where d is the distance between two arbitrary planes in the lattice and n is the order
of the diffraction line.
In order to understand the appearance of complex diffraction patterns, it is
necessary to explain the geometry and the notations in crystallography. This
explanation uses the simplest case of a primitive cubic lattice, but for more details
the use of more specialized books on diffraction techniques and crystallography
is advised. Figure 12.7 shows a cubic cell and three lattice planes indicated in blue;
the lattice planes are denoted by Miller indices.
Figure 12.6 Basic geometry of diffraction. The waves incoming under an angle of q against the
surface are reflected at the same angle. Between two consecutive lattice planes, the “reflected”
waves have a phase difference of 2d.
Figure 12.7 Lattice planes in a cubic lattice. These planes are characterized by their Miller
indices.
342j 12 Characterization of Nanomaterials
a difference becomes apparent in the path length between the wave scattered in the
zeroth lattice plane, representing the plane directly at the surface, and the wave
scattered in the first plane. The difference in the path length d is a function of
the distance between the lattice planes d and the angle between the incoming wave
and the lattice planes q. This path difference is d ¼ d sin q. The scattered wave,
leaving the specimen under the same angle, has the same difference in path length
and therefore the total difference of the path length is d total ¼ 2d sin q. The path
difference between the zeroth and the second plane is, therefore, exactly doubled;
more generally, the difference of the path length to the nth plane is 2nd. An
interference maximum is observed when the difference in the path length is exactly
an integer multiple of the wavelength. This may be the path difference between two
parallel adjacent lattice planes or between planes in the distance nd. This leads to the
condition for an interference maximum:
nl ¼ 2d sin q; n ¼ 1; 2; 3;. . .
ð12:14Þ
where d is the distance between two arbitrary planes in the lattice and n is the order
of the diffraction line.
In order to understand the appearance of complex diffraction patterns, it is
necessary to explain the geometry and the notations in crystallography. This
explanation uses the simplest case of a primitive cubic lattice, but for more details
the use of more specialized books on diffraction techniques and crystallography
is advised. Figure 12.7 shows a cubic cell and three lattice planes indicated in blue;
the lattice planes are denoted by Miller indices.
Figure 12.6 Basic geometry of diffraction. The waves incoming under an angle of q against the
surface are reflected at the same angle. Between two consecutive lattice planes, the “reflected”
waves have a phase difference of 2d.
Figure 12.7 Lattice planes in a cubic lattice. These planes are characterized by their Miller
indices.
342j 12 Characterization of Nanomaterials