Question

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

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

Step-by-step solution

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: \(h = 1, k = 1, l = 1\)

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: \(d_{111} = \frac{12.400}{\sqrt{1^2 + 1^2 + 1^2}} = 4.3371 \AA\)

Use Bragg's law to calculate 2θ 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 2θ, which can be obtained using the relation: \[ 2\theta = 2\arcsin{\left(\frac{\lambda}{2d}\right)} \] Now, let's calculate the 2θ values for each reflection:

Calculate 2θ for each reflection

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