Powder diffraction patterns of three monatomic cubic crystals with BCC, FCC and diamond structur

Powder diffraction patterns of three monatomic cubic...

Powder diffraction patterns of three monatomic cubic crystals with $\mathrm{BCC}, \mathrm{FCC}$ and diamond structures are recorded using a Debye-Scherrer camera. The angles $2 \theta$ in degrees of the first four diffraction lines for the three samples marked as $A, B, C$ are as under: $\begin{array}{ccc}A & B & C \\ 42.2 & 28.8 & 42.8 \\ 49.2 & 41.0 & 73.2 \\ 72.0 & 50.8 & 89.0 \\ 87.3 & 59.6 & 115.0\end{array}$ (a) Determine the structure type of each sample. (b) What is the size of the cubic cell in each case? Take the wavelength of incident $x$ -rays as $1.5 \AA$.

Powder diffraction patterns of three monatomic cubic crystals with $\mathrm{BCC}, \mathrm{FCC}$ and diamond structures are recorded using a Debye-Scherrer camera. The angles $2 \theta$ in degrees of the first four diffraction lines for the three samples marked as $A, B, C$ are as under:
$\begin{array}{ccc}A & B & C \\ 42.2 & 28.8 & 42.8 \\ 49.2 & 41.0 & 73.2 \\ 72.0 & 50.8 & 89.0 \\ 87.3 & 59.6 & 115.0\end{array}$
(a) Determine the structure type of each sample.
(b) What is the size of the cubic cell in each case? Take the wavelength of incident $x$ -rays as $1.5 \AA$.

Key Concepts

X-ray Powder Diffraction

X-ray powder diffraction is a technique used to identify the crystal structure of a material by analyzing the pattern of x-rays diffracted from a powdered sample. Since powders contain many randomly oriented crystallites, all possible diffraction directions are sampled, resulting in a series of concentric diffraction rings that encode information about the spacing of lattice planes.

Debye-Scherrer Method

The Debye-Scherrer method employs a cylindrical camera to record the diffraction pattern from polycrystalline samples. This method is particularly useful for analyzing fine powders and involves the collection of diffracted x-rays on a curved photographic film, transforming the information into a pattern of rings whose radii correspond to different interplanar spacings.

Bragg’s Law

Bragg’s Law, expressed as n? = 2d sin?, is fundamental in crystallography as it relates the wavelength of incident x-rays (?), the interplanar spacing (d), and the diffraction angle (?). It is used to calculate the distances between planes in the crystal lattice and to assign Miller indices to the observed diffraction peaks.

Reciprocal Lattice and Miller Indices

The concept of the reciprocal lattice links the periodicity of a crystal lattice with its diffraction pattern, with Miller indices (h, k, l) labeling the sets of crystallographic planes. In cubic systems, the interplanar spacing is given by d = a/?(h²+k²+l²), where a is the lattice parameter, and this relationship is key for indexing diffraction peaks and determining the structure.

Cubic Crystal Structures

Cubic crystal structures, such as body?centered cubic (BCC), face?centered cubic (FCC), and diamond, have distinct arrangements of atoms that result in unique sets of allowed reflections in their diffraction patterns. Each structure possesses a unique combination of structure factors that can be used to differentiate and confirm the type of cubic arrangement present.

Structure Factor

The structure factor is a quantity that determines the intensity of a diffraction peak by accounting for the arrangement and basis of atoms within the unit cell. It plays a critical role in identifying the symmetry and atomic positions in various crystal structures, allowing for the discrimination between different types such as FCC, BCC, and diamond structures.

Lattice Parameter Calculation

The lattice parameter, which defines the size of the unit cell in a crystal, can be calculated by combining the measured diffraction angles with Bragg’s Law and the known wavelength of the incident x-rays. In a cubic system, using the relationship between the interplanar spacing and Miller indices allows for the extraction of the lattice constant, a key metric in characterizing the material’s crystal structure.

Transcript

00:01 For this question, we have an x -ray wavelength lambda of 0 .1 nanometers, which is 0 .1 times 10 to the minus 9 meters.

00:08 The sample is located 12 centimeters from the photographic film.

00:12 So l is 12 centimeters or 0 .12 meters.

00:16 And we're also told that the atomic spacing, d, is 0 .222 nanometers or 0 .22 times 10 to the minus 9 meters.

00:23 So always make sure that if you're dividing or anything or multiplying anything together that they have the same.

00:30 Units.

00:31 So if one has nanometers and one has centimeters, you need to convert both the centimeters or both meters, however you see fit.

00:38 Just be consistent.

00:40 So we're asked to figure out the radii diffraction rings corresponding to the first and second order.

00:47 Look at figure 28 and you see that deflected, the x -ray beam is deflected through an angle of 2 .5.

00:53 Okay.

00:54 So first things first, from bragg's equation, we find the angle of incident for the first and second order come from the equation that says 2 times d times the sign of the angle phi is equal to m times lambda.

01:12 Or in other words, because we're asked to first find the first thing we need to do is find the angles phi to find the two radii of curvature, we find that phi is equal to the inverse sign, sine to the minus one of m times lambda divided by two times d.

01:32 We're asked to find the first two...

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