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Chapter 3: Problem 2

Given that the lattice parameter of primitive cubic zeolite is \(12.400 \AA\), calculate the \(2 \theta\) positions of the 301,400 and the 111 reflections determined by copper radiation \((\lambda=1.54 \AA\) ).

Short Answer

Expert verified
In conclusion, the 2胃 positions for the given reflections using copper radiation are: For the 301 reflection, 2胃 is approximately \(27.59掳\). For the 400 reflection, 2胃 is approximately \(28.64掳\). For the 111 reflection, 2胃 is approximately \(20.45掳\).

Step by step solution

01

Determine the d-spacing for each reflection

Using Miller Indices (h, k, l) for each reflection, the d-spacing for a cubic lattice can be calculated by the formula: \[ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] Where \(a\) is the lattice parameter, and \(h, k, l\) are the Miller Indices of the plane. First, let's compute d-spacing for each given reflection (301, 400, and 111). For the (301) reflection: \(h = 3, k = 0, l = 1\) For the (400) reflection: \(h = 4, k = 0, l = 0\) For the (111) reflection: \(h = 1, k = 1, l = 1\)
02

Calculate the d-spacing for each reflection

310 reflection: \(d_{301} = \frac{12.400}{\sqrt{3^2 + 0^2 + 1^2}} = 3.2995 \AA\) 400 reflection: \(d_{400} = \frac{12.400}{\sqrt{4^2 + 0^2 + 0^2}} = 3.1000 \AA\) 111 reflection: \(d_{111} = \frac{12.400}{\sqrt{1^2 + 1^2 + 1^2}} = 4.3371 \AA\)
03

Use Bragg's law to calculate 2胃 for each reflection

Using Bragg's Law equation: \[ n\lambda = 2d\sin{\theta} \] Where \(n\) is an integer which is equal to 1 for all reflections in this case, \(\lambda\) is the wavelength of the radiation, and \(d\) is the d-spacing for the respective reflection. We want to find the angle 2胃, which can be obtained using the relation: \[ 2\theta = 2\arcsin{\left(\frac{\lambda}{2d}\right)} \] Now, let's calculate the 2胃 values for each reflection:
04

Calculate 2胃 for each reflection

301 reflection: \(2\theta_{301} = 2\arcsin{\left(\frac{1.54}{2 \times 3.2995}\right)} 鈮� 27.59掳\) 400 reflection: \(2\theta_{400} = 2\arcsin{\left(\frac{1.54}{2 \times 3.1000}\right)} 鈮� 28.64掳\) 111 reflection: \(2\theta_{111} = 2\arcsin{\left(\frac{1.54}{2 \times 4.3371}\right)} 鈮� 20.45掳\) In conclusion, the 2胃 positions for the given reflections using copper radiation are: For the 301 reflection, 2胃 is approximately 27.59掳. For the 400 reflection, 2胃 is approximately 28.64掳. For the 111 reflection, 2胃 is approximately 20.45掳.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

d-spacing
In crystallography, **d-spacing** refers to the distance between adjacent planes of atoms (or ions) in a crystal. This is a crucial concept when it comes to understanding X-ray diffraction, which is used to determine the arrangement of atoms in solids. The d-spacing is particularly important in calculating the diffraction angles using Bragg's Law.

For a cubic lattice, the d-spacing can be found using the formula:
  • \[ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \]
where:
  • \(a\) is the lattice parameter.
  • \(h, k, l\) represent the Miller indices of a crystal plane.
Understanding d-spacing allows for the calculation of the angle of reflection using Bragg's Law, helping to reveal information about the structure of the material being studied.
Miller Indices
**Miller Indices** are a notation system in crystallography to denote the orientation of crystal planes. These indices \((h, k, l)\) are essential for identifying and discussing the different faces of a crystal. They are expressed as a set of three integers.
  • The integers are derived from the reciprocals of the fractional intercepts that the plane makes with the crystal axes.
  • They indicate how surfaces or planes are oriented within the crystal lattice.
When calculating d-spacing or understanding reflections off crystal planes, Miller indices guide the mathematical computations and interpretations. They link directly with Bragg's Law calculations, as seen in diffraction experiments, to interpret the structure based on recorded reflections.
Cubic lattice
A **cubic lattice** is one of the simplest types of crystal structures and can be thought of as a repeating three-dimensional grid of atoms, ions, or molecules. In this structure, each unit cell is a cube, making calculations of properties like d-spacing straightforward.

Characteristics of a cubic lattice include:
  • All edges of the cube have equal length (denoted as the lattice parameter \(a\)).
  • The angles between the edges of the unit cell are all 90 degrees.
  • The repeating pattern in all three spatial directions is identical, making symmetry a key characteristic.
Cubic lattices are noted for their simplicity in mathematical analysis, especially when using formulas and laws like the d-spacing formula and Bragg's Law. This type of lattice helps streamline various calculations crucial in structural chemistry and physics.
Lattice parameter
The **lattice parameter** is a physical dimension that defines the overall size of the unit cell of a crystal lattice. It is usually denoted as \(a\) for cubic lattices and is crucial in determining the geometric structure of the crystal.
  • It represents the edge length of the cubic unit cell.
  • Plays a vital role in calculating d-spacing using the relation \[ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \].
A correct understanding of the lattice parameter is essential for interpreting X-ray diffraction patterns. The lattice parameter, along with knowledge of the wavelength used in diffraction, aids in applying Bragg's Law to find angles of reflection and gain insights into the atomic arrangement within a crystal.

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Most popular questions from this chapter

The spinel \(\mathrm{ZnFe}_{2} \mathrm{O}_{4}\) has reflections at the following \(2 \theta\) values in its neutron diffraction pattern: \(22.593,26.145,37.311,44.059\), 46.129, 53.792. Determine the lattice type, and given that the neutron wavelength was \(1.90 \AA\), calculate the lattice parameter.

\(\beta\)-Tungsten crystallizes with a body-centred cubic structure. What \(h, k, l\) values would you expect the first ten reflections to have?

On a TGA, \(24.20 \mathrm{mg}\) of \(\mathrm{SrFeO}_{3}\), decomposed in a stream of hydrogen at \(700^{\circ} \mathrm{C}\) to give \(23.34 \mathrm{mg}\) of a mixture of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) and SrO. Use these data to determine the value of \(x\).

(i) Potassium chloride crystallizes with a face-centred cubic lattice similar to sodium chloride. If the X-ray powder pattern was recorded, what restrictions, if any, would be expected on \(h, k, l\) ? (ii) If the powder neutron diffraction pattern was recorded instead, would the restrictions change?

What type of cubic lattice would have a diffraction pattern where the 110 reflection would be observed but the 234 would be absent? Give reasons for your answer.
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