5.1 Basic Considerations 83
Box 5.3 Miller Indices to Describe Crystallographic Planes and Orientation
The Miller indices, a notation system applied to characterize planes and directions in a crystallographic lattice, are, in the simplest case, explained in a cubic
lattice. Figure 5.4 shows a cubic cell and six different lattice planes indicated
in gray and the associated Miller indices.
The Miller indices of a lattice plane are the reciprocal values of the intercept
of the lattice planes with the axes of the coordinate system. The reciprocal
intercept in the x-direction is denoted with h, in the y-direction with k, and in
the z-direction with l. The coordinate system is normalized; it achieves the
value 1 at the lattice constant a; therefore, Miller indices are always integers.
Miller indices of planes are written in round brackets (hkl). The minus sign is
denoted above the index, for example, 112
( ) . The set of all planes with equivalent symmetry is denoted in curved brackets {hkl}. Crystallographic directions
(lattice vectors) are written in square brackets, they designate a direction in the
lattice from the origin to a point. Important: The vector [hkl] is perpendicular
to the plane (hkl). The set of all lattice vectors perpendicular to the set of lattice
planes {hkl} is written in arrow brackets 〈hkl〉.
sions beyond 100 nm are, strictly speaking, no longer a nanomaterial. However,
as this micrograph is so perfect, it was selected for illustration purposes.) At one
end of most of the particles a bulb is visible, this is a structure element indicating
a synthesis process via the gas phase.
By attaching surface-active molecules, it is possible to alter the surface energy.
This technology makes it possible to synthesize even nanoplates of gold, as is
depicted in Figure 5.3 [2]. The size of these platelets is around 400 nm; the thickness varies in the range from 25 to 60 nm. The shape of all platelets is hexagonal,
this indicated that the nanoplates have single orientation with (111) plane at the
surface. In the cubic system, the (111) plane has hexagonal symmetry. This is
perfectly visible in the electron diffraction pattern in Figure 5.3b. In this case, the
surface active molecules added during synthesis led to plates, the application of
other organic agents may also lead to the formation of nanorods. The nanoplates
displayed in Figure 5.3a are nearly atomically flat. Such gold platelets are applied
in nanotechnology for manufacturing of small devices.
Altering the habit of nanoparticles is possible not only with metals, but also with
ceramic materials. Such an example is displayed in Figure 5.5. Again a material,
CuFe 2 O 4 .crystallizing in a cubic structure (spinel structure) was forced by surfaceactive substances to crystallize as platelets, again with the (111) plane at the top.
The magnetic properties of materials depend strongly on the anisotropy of the
particles; therefore, to obtain very special properties one adjusts the shape of the
particle during synthesis. Adding different amounts of surface-active molecules
changes the shape from a plate to a rod [3].
Box 5.3 Miller Indices to Describe Crystallographic Planes and Orientation
The Miller indices, a notation system applied to characterize planes and directions in a crystallographic lattice, are, in the simplest case, explained in a cubic
lattice. Figure 5.4 shows a cubic cell and six different lattice planes indicated
in gray and the associated Miller indices.
The Miller indices of a lattice plane are the reciprocal values of the intercept
of the lattice planes with the axes of the coordinate system. The reciprocal
intercept in the x-direction is denoted with h, in the y-direction with k, and in
the z-direction with l. The coordinate system is normalized; it achieves the
value 1 at the lattice constant a; therefore, Miller indices are always integers.
Miller indices of planes are written in round brackets (hkl). The minus sign is
denoted above the index, for example, 112
( ) . The set of all planes with equivalent symmetry is denoted in curved brackets {hkl}. Crystallographic directions
(lattice vectors) are written in square brackets, they designate a direction in the
lattice from the origin to a point. Important: The vector [hkl] is perpendicular
to the plane (hkl). The set of all lattice vectors perpendicular to the set of lattice
planes {hkl} is written in arrow brackets 〈hkl〉.
sions beyond 100 nm are, strictly speaking, no longer a nanomaterial. However,
as this micrograph is so perfect, it was selected for illustration purposes.) At one
end of most of the particles a bulb is visible, this is a structure element indicating
a synthesis process via the gas phase.
By attaching surface-active molecules, it is possible to alter the surface energy.
This technology makes it possible to synthesize even nanoplates of gold, as is
depicted in Figure 5.3 [2]. The size of these platelets is around 400 nm; the thickness varies in the range from 25 to 60 nm. The shape of all platelets is hexagonal,
this indicated that the nanoplates have single orientation with (111) plane at the
surface. In the cubic system, the (111) plane has hexagonal symmetry. This is
perfectly visible in the electron diffraction pattern in Figure 5.3b. In this case, the
surface active molecules added during synthesis led to plates, the application of
other organic agents may also lead to the formation of nanorods. The nanoplates
displayed in Figure 5.3a are nearly atomically flat. Such gold platelets are applied
in nanotechnology for manufacturing of small devices.
Altering the habit of nanoparticles is possible not only with metals, but also with
ceramic materials. Such an example is displayed in Figure 5.5. Again a material,
CuFe 2 O 4 .crystallizing in a cubic structure (spinel structure) was forced by surfaceactive substances to crystallize as platelets, again with the (111) plane at the top.
The magnetic properties of materials depend strongly on the anisotropy of the
particles; therefore, to obtain very special properties one adjusts the shape of the
particle during synthesis. Adding different amounts of surface-active molecules
changes the shape from a plate to a rod [3].