The reason for the detection of Raman scattering over this broad region is that the Raman selection rule for crystals restricting Raman scattering to phonons at the Brillouin zone center no longer holds in amorphous Si and all of the phonons from Brillouin zone center to edge are now sampled. The Raman spectrum of amorphous Si looks very much like the calculated populations of phonon energy states within the first Brillouin zone of crystalline Si. We need to understand concepts such as wave vector, reciprocal space, and the Brillouin zone to accomplish that goal.
Therefore, we will explain each of these topics and finally bring them all together, leading to the conclusion that in a crystal only those phonons at the Brillouin zone center are Raman active. With the loss of long-range translational symmetry in an amorphous solid, the Raman selection rules for crystalline solids no longer hold. If after reading this article you would like to view a detailed lecture on this topic, please see the companion video on YouTube 2. Earlier, we defined a phonon as a lattice vibrational wave propagating through the crystal arising from repetitive and systematic atomic displacements.
It should also be understood that these displacements are quantized vibrations of the atoms in the lattice and are traveling waves. The phonon has the characteristics of a traveling wave insofar as it has a propagation velocity, wavelength, wave vector, and frequency. It is important to note that all of the peaks in a Raman spectrum of a crystalline solid are attributed to phonons and not only those at low energy or Raman shift.
Unlike the normal vibrational mode of an individual molecule, a phonon is a lattice vibrational wave traveling through the crystal. We describe the phonon as a traveling wave with one of its characteristics, the wave vector k , defined as. The wave vector is in units of wavenumbers cm -1 because the phonon wavelength is in the denominator. Perhaps you can anticipate why a reciprocal lattice with units of distance in the denominator for example, cm -1 would be compatible with our treatment of phonon wave vectors in units of cm Depictions of a direct linear monatomic lattice along with its reciprocal lattice are shown in Figure 3.
Each of the dots in the linear crystal lattice represents an atom and the distance between the atoms is given by the letter a. Without providing the derivation, we simply state that there is one solution for a wave traveling either to the right or left in a linear monatomic lattice. Progressing in our understanding of phonon propagation and dispersion curves, we now consider vibrational waves in a linear diatomic lattice.
Again without providing the derivation or the mathematical expressions, we state that there are two solutions for waves traveling either to the right or left in the linear diatomic lattice. Recall that there was only one solution for a traveling wave in a linear monatomic lattice. The derivations and analytical expressions for vibrational waves in linear monatomic and diatomic lattices can be found in all text books on solid-state physics.
A phonon dispersion curve for the two solutions of the wave equations of a linear diatomic lattice is shown in Figure 5.
The lower curve is called the acoustic branch and is similar to the phonon dispersion curve for the linear monatomic lattice.
The upper optical branch is so named because these lattice vibrational waves can be excited with light of infrared wavelengths, energies much higher than those of the acoustic branch, especially at the Brillouin zone center. The optical branch phonons are the origin of the bands in the Raman spectrum of a solid. Of course, the number of Raman bands is dictated by the symmetry of the crystal and their energies by the masses of the atoms and force constants of the chemical bonds.
The Raman bands would be very broad if all of those phonons from center to edge were sampled, and that is indeed the case for amorphous solids. The Raman intensity as a function of frequency for a given Raman band is related to populations of the phonon state as a function of wave vector from the Brillouin zone center to the edge, referred to as the phonon density of states.
In contrast with amorphous solids, crystals restrict phonon sampling to the Brillouin zone center and therefore have narrow Raman bands as we explain in the following section. Of course, the real world of crystals is not one-dimensional but is three-dimensional. A detailed description of the construction of the Brillouin zone for a three-dimensional crystal lattice is beyond the scope of this article.
That would involve a mathematical description of the construction of the reciprocal lattice from the real lattice also called a direct lattice and the subsequent construction of the Wigner-Seitz lattice from the reciprocal lattice.
A helpful description of the construction of three-dimensional reciprocal lattices, Wigner-Seitz lattices and Brillouin zones is given in the excellent book by Richard Tilley 4. Nevertheless, we can describe one example to give you a sense of how the construction of a Brillouin zone is accomplished. Consider a face-centered cubic Bravais F crystal lattice in real space. The first step is to construct the reciprocal lattice of the face-centered cubic F crystal.
Following the mathematical procedure for doing so produces a body-centered cubic I lattice. The next step is to construct a Wigner-Seitz cell from the body-centered I reciprocal lattice, thereby generating a truncated octahedron. The Wigner-Seitz truncated octahedron is the shape of the first Brillouin zone of the real face-centered cubic F lattice. Depictions of the face-centered cubic F unit cell and its corresponding first Brillouin zone are shown at the bottom of Figure 6.
The first Brillouin zones of a simple cubic lattice and a body centered cubic I lattice are also shown in Figure 6. The significance of depicting the three-dimensional Brillouin zone is that we have just made the leap from one-dimensional to three-dimensional reciprocal space, which prepares us for the typical phonon dispersion curves found in the literature.
Phonon dispersion curves generally plot the frequency of the phonon from the Brillouin zone center to the various crystal faces and points within the first Brillouin zone. You can find examples of the phonon dispersion curves and corresponding phonon density of states plots of various solids in the companion video lecture to this installment on YouTube 2. Having developed our understanding of the Brillouin zone and its basis in reciprocal space, we relate it to the wave vector in Raman scattering from a crystalline solid.
Energy, wave vector, and momentum are conserved in the Raman scattering process involving real crystals, which are anharmonic. Therefore, the incident radiation transmits momentum to the crystal through the creation of a lattice vibrational wave or phonon. A wave vector diagram of Raman scattering in a crystal is shown in Figure 7. The conservation of the wave vector is implicit in the diagram, which shows the dependence of the direction at which the Raman scattered light is collected on the direction of phonon propagation.
The most frequently cited example of an amorphous solid is glass. However, amorphous solids are common to all subsets of solids. Additional examples include thin film lubricants, metallic glasses, polymers, and gels. Samples of amorphous metallic glass are shown below. Most classes of solid can be found in an amorphous form. Amorphous solids can be prepared in a variety of ways, such as rapidly cooling from the molten state or seeding the solid with an additive that disrupts long-range order.
Privacy Policy. Skip to main content. Liquids and Solids. Search for:. Amorphous Solids Amorphous Solids Amorphous solids lack a crystalline or long-range order to their atomic structure.
If an amorphous solid is maintained at a temperature just below its melting point for long periods of time, the component molecules, atoms, or ions can gradually rearrange into a more highly ordered crystalline form.
Solids are characterized by an extended three-dimensional arrangement of atoms, ions, or molecules in which the components are generally locked into their positions. The components can be arranged in a regular repeating three-dimensional array a crystal lattice , which results in a crystalline solid, or more or less randomly to produce an amorphous solid.
Crystalline solids have well-defined edges and faces, diffract x-rays, and tend to have sharp melting points. In contrast, amorphous solids have irregular or curved surfaces, do not give well-resolved x-ray diffraction patterns, and melt over a wide range of temperatures. How do amorphous solids differ from crystalline solids in each characteristic? Which of the two types of solid is most similar to a liquid?
Why is the arrangement of the constituent atoms or molecules more important in determining the properties of a solid than a liquid or a gas? Why are the structures of solids usually described in terms of the positions of the constituent atoms rather than their motion? What physical characteristics distinguish a crystalline solid from an amorphous solid? Describe at least two ways to determine experimentally whether a material is crystalline or amorphous.
A student obtained a solid product in a laboratory synthesis. After it had cooled, she measured the melting point of the same sample again and found that this time the solid had a sharp melting point at the temperature that is characteristic of the desired product. Why were the two melting points different? What was responsible for the change in the melting point? The arrangement of the atoms or molecules is more important in determining the properties of a solid because of the greater persistent long-range order of solids.
Gases and liquids cannot readily be described by the spatial arrangement of their components because rapid molecular motion and rearrangement defines many of the properties of liquids and gases. The initial solid contained the desired compound in an amorphous state, as indicated by the wide temperature range over which melting occurred.
Slow cooling of the liquid caused it to crystallize, as evidenced by the sharp second melting point observed at the expected temperature.
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