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Section 56.2 Structure of Solids

Various types of bonds between atoms and molecules described in the previous section can continue among very large number of atoms and molecules when they form liquids and solids. Solids are fascinating materials characterized by machinable shape, strength, and stability. If a solid has a regular arrangement of atoms and molecules, it forms crystals. For instance, the crystal of sodium chloride, has sodium and chlorine ions at the corners and faces of a cubic structure such that each sodium ion is surrounded symmetrically by six chlorine ions and vice versa as illustrated in Figure 56.1.2.

Figure 56.2.1. The arrangement of atoms in the sodium chloride crystal.

Many solids are found in crystalline form in nature. Figure 56.2.2 shows three cubic crystals of pyrite (FeS\(_2\)) in a rock sample. The flat surface of the crystal forms due to arrangement of atoms in a cubic lattice.

Figure 56.2.2. A rock containing three crystals of pyrite (FeS\(_2\)). The arrangments of atoms in the crystal are in the form of a cubic lattice which over a macroscopic sample is reflected in the formation of cubic shaped crystal. Credits: Wikicommons.

Most solids in nature are not crystalline. The atoms and molecules in them are randomly placed. They are called amorphous. Glass and sand are examples of amorphous materials.

Subsection 56.2.1 X ray Diffraction from Single Crystals

X rays were discovered by William R\"ontgen in 1895 by the marks they left on photographic plates. They are produced when highly energetic electrons strike a target. X rays are very penetrating and can pass through human body and many other materials that are opaque to the visible light.

Are X rays particles or waves? Wilhelm Wien calculated that if X rays were waves their wavelength will be very small, of the order of \(10^{-11}\) m. Based on the calculation of the inter-atomic distances in crystals to be or the order of \(10^{-10}\) m, Max von Laue wondered if crystals could act as a diffraction grating for X rays. Two of his assistants, Walter Friedrich and Paul Knipping, set up the experiment to test Laue's ideas. First they used copper sulfate crystal since it was readily available in the lab. The crystal was fixed to a holder with wax which was placed in a lead box with a 3 mm hole in one side for the X rays to enter while the back end and the sides were lined with photographic plates. They found that, in addition to the central spot, there were many smaller dots on the screen in a regular pattern where diffracted beams had formed constructive interferences. These patterns are often referred to as Laue pattern. Von Laue received Nobel prize for his discovery in 1914.

Figure 56.2.3. Von Laue experiment. A continuous wavelength X rays beam is incident on a crystal and diffracts from the atoms of the crystal forming Laue pattern on the photographic plate.

Von Laue's patterns did not have all the diffraction spots expected from a source of continuous wavelengths. Von Laue took it to mean that the source he was using had only certain wavelengths. However, William Lawrence Bragg, the son of William Henry Bragg, the Cavendish Professor of Physics at the University of Leeds, thought differently. Lawrence Bragg was convinced that missing diffraction spots were due to some property of crystals themselves. He put forth a theory for diffraction of X rays based on their reflection from crystal planes.

Crystal planes are planes in space that are parallel to each other and spaced equally apart and every atom of the crystal belongs to one of these planes as illustrated for a square lattice in Figure 56.2.5. According to [William] Lawrence Bragg each crystal plane acts as a reflecting surface. The X rays reflected from these planes would interfere constructively if they were all in phase. For instance, the ray reflected from the surface plane will have traveled smaller distance than the ray that reflected off the plane a distance \(d\) from the surface just below as shown in Figure 56.2.4. The difference in the paths of the two rays will depend on the angle \(\theta\) and plane separation \(d\text{.}\)

\begin{equation*} \textrm{path difference} = 2d \sin\theta. \end{equation*}

If this distance is an integral multiple of the wavelength \(\lambda\text{,}\) then then reflected rays will interfere constructively in that direction.

\begin{equation} 2d \sin\theta = n \lambda,\ \ \ n = 1, 2, 3, \cdots. \label{eq-bragg-law}\tag{56.2.1} \end{equation}
Figure 56.2.4. The path length difference for the reflections from successive crystal planes is equal to \(2d\sin\theta\text{.}\) If the path length difference is equal to the an integral multiple of the wavelength, then there will be constructive interference of the two reflected waves in that direction.

The formula in Eq. (56.2.1) is called Bragg's law. This formula can be used to find \(\lambda\) of the X rays if we know \(d\) in a crystal and to find \(d\) between two crystal planes if we know \(\lambda\) of the X rays. By rotating the crystal, or changing the direction of the beam, or the direction of the detector, it is possible to figure out \(d\) corresponding to various crystal planes in the crystal as shown in Figure 56.2.5. From the data for \(d\) for various crystal planes, the arrangements of atoms in the crystal can be determined. The father/son team of Braggs worked on the crystal structure determination using this formula and were awarded Nobel prizes in 1915.

Figure 56.2.5. Two sets of crystal planes. The Bragg condition will occur at different angles \(\theta\) for the two sets of planes for the same \(n\) and same \(\lambda\text{.}\) Experimentally, one keeps the direction of the beam same and rotates the crystal to change \(\theta\text{.}\)

Subsection 56.2.2 The Powder X ray Diffraction

If the single crystal is not large enough, you can still gain the information about the crystalline structure from a powder of the material. If each particle of the powder itself is a tiny crystal, then each crystallite will diffract X rays independently of the way some other particle of the powder would. Since particle of the powder would be randomly oriented with respect to each other, we get circles at the photographic plate instead of sharp dots as illustrated in Figure 56.2.6. The rings are are called the Debye-Scherrer rings.

Figure 56.2.6. X ray diffraction of powder of a crystalline material. Powder particles, though small, have many thousands of atoms in a crystal lattice. Each crystallite produces diffraction peaks at particular angles. Since the crystallites are randomly oriented, the dots from different crystallite fall on circles as shown here.

When the crystallites becomes smaller each peak spreads out over larger angles. Eventually, when there is no long range order, the material is called amorphous. The X ray diffraction of amorphous materials are broad and lack the sharp peaks.

(a) If the distance between atoms in the following figure is 6.00 \AA, what is the distance between the lattice planes? (b) If X rays of wavelength 1.54 \AA are incident on the crystal, what will be the smallest angle to the plane should the beam direction be such that there will be a constructive diffraction in that direction?

Figure 56.2.8.


(a) \(4.24\:\textrm{\AA}\text{,}\) (b) \(21.3^{\circ}\text{.}\)


(a) We begin by noting that there is a right-angled triangle possible whose sides are \(a\text{,}\) \(a\text{,}\) and \(2d\) as shown in Figure 56.2.9

Figure 56.2.9.
Therefore, we will have \[ (2d)^2 = a^2 + a^2, \] which gives \[ d = \frac{a}{\sqrt{2}}. \] Plugging in the numerical value of \(a\) we find the spacing between the given lattice planes to be \[ d = \frac{6.00\:\textrm{\AA}}{\sqrt{2}} = 4.24\:\textrm{\AA}. \]

(b) Now, we have the \(d = 4.24\) \AA. We are given \(n=1\) ( which corresponds to the smallest angle) and \(\lambda = 1.54\) \AA. We plug these values in the Bragg law to obtain \[ \sin\theta = \frac{1\times 1.54\:\textrm{pm}}{4.24\:\textrm{pm}} = 0.363. \] Therefore, \[ \theta = 21.3^{\circ}. \]


\​begin{exercise} A particular set of crystal planes in a sodium chloride crystal are separated by $d = 2.52$ nm. Find the angles at which different order diffraction diffraction will occur for an X ray beam of wavelength 0.71 AA? \end{exercise}

\​begin{exercise} A crystal of copper is being analyzed by a monochromatic X rays of wavelength 0.056 nm. It is found that the smallest angle at which the diffraction occurs is $7.74^{\circ}$. (a) Find the spacing of crystal planes from which the X rays are reflecting off. (b) Find all other angles the diffraction would occur. \end{exercise}

\​begin{exercise} A crystal of copper is mounted so that the crystal planes will be separated by $d = 0.361$ nm. An X ray source has strong diffraction at $15^{\circ}$ and again at $ 18^{\circ}$. Decide whether the two diffractions correspond to different orders of the same wavelength or to two different wavelengths. If you decide the two correspond to two different wavelengths then find out the wavelength assuming they are first orders in each, and if you decide the two correspond to the same wavelength then figure out the two orders assuming the smaller angle corresponds to the first order. \end{exercise}