Answered: The coordinates, in units of a, of the…

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Question

The coordinates, in units of a, of the atoms in a cubic P unit cell are (0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,0), (1,1,0), (1,0,1), and (1,1,1). Calculate the structure factor F_hkl when all the atoms are identical. Where possible, simplify your expression by using e^inπ = (-1)ⁿ, as in Example 15B.1.

Expert Solution

Step 1

Given:

Coordinates: (0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,0), (1,1,0), (1,0,1), and (1,1,1)

Step 2

The formula for the structure factor is given by,

$\Rightarrow F (h' k' l') = f S$

where,

$S = \sum_{i}^{} e^{2 πi (uh' + vk' + wl')}$

S is called as the geometrical structure factor as it depends upon the geometrical arrangements of atoms within the unit cell.

For (0,0,0):

$S = \sum_{}^{} e^{2 πi (0 h' + 0 k' + 0 l')}$

On solving,

$S = 1$

Step 3

For (0,1,0):

$S = \sum_{}^{} e^{2 πi (0 h' + 1 k' + 0 l')}$

On solving,

$S = \sum_{}^{} e^{2 πi k}$

For (0,0,1):

$S = \sum_{}^{} e^{2 πi (0 h' + 0 k' + 1 l')}$

On solving,

$S = \sum_{}^{} e^{2 πi l}$

For (0,1,1):

$S = \sum_{}^{} e^{2 πi (0 h' + 1 k' + 1 l')}$

On solving,

$S = \sum_{}^{} e^{2 πi (k + l)}$

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