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The lattice constant for the fcc structure of Platinum (Pt) at 300 K is 0.3912 nm.
A. List the Miller Indices of three diffraction planes you would be able to observe from your
selected X-ray wavelength. Discuss reasons for your choice.
B. What is your primary concern when you select the wavelength of an X-ray source to examine
the crystal structure? Show your reasoning.
C. Does increasing the temperature affect the diffraction peak position (20)? Explain how and
why
D. In order to generate an X-ray source with a wavelength of 0.12 nm, propose the minimum
acceleration voltage for the X-ray tube. Show your calculation and reasoning.

Expert Solution

Step 1: Explanation about part a and part b:

(a) For the FCC structure of Platinum (Pt) at 300K with a lattice constant of 0.3912 nm, we can use Bragg's law to determine the Miller indices (hkl) of three diffraction planes that can be observed with a selected X-ray wavelength. In an FCC structure, only those values of (hkl) are allowed where all are either even or all are odd due to the arrangement of atoms. Bragg's law, which relates diffraction angle (θ), wavelength (λ), and lattice spacing (d), is given by:

$2 dsin straight theta equals straight n straight lambda$

Where:

$Error converting from MathML to accessible text.$ $straight d space is space the space lattice space spacing comma space which space is space equal space to space fraction numerator straight a over denominator square root of straight h squared plus straight k squared plus straight l squared end root end fraction comma space with space straight a space being space the space lattice space constant.$
$Error converting from MathML to accessible text.$
$Error converting from MathML to accessible text.$

$For space straight n equals 1 comma space we space have space left parenthesis hkl right parenthesis space equals space left parenthesis 111 right parenthesis.$
$For space straight n equals 2 comma space we space have space left parenthesis hkl right parenthesis space equals space left parenthesis 002 right parenthesis.$
$For space straight n equals 3 comma space we space have space left parenthesis hkl right parenthesis space equals space left parenthesis 022 right parenthesis.$

The selection is based on the requirement that the Miller indices must have either all even or all odd values to fit the FCC structure.

(b) When selecting the wavelength of an X-ray source to examine the crystal structure, the primary concern is to ensure that the selected wavelength falls within the range that can be effectively diffracted by the crystal. X-rays have a frequency range between 0.1 Å to 10 Å. Using Bragg's law $left parenthesis 2 dsin straight theta equals straight n straight lambda right parenthesis$ the wavelength of X-rays required for diffraction can be determined based on the lattice spacing ( $d$ ) of the crystal and the desired diffraction angle ( $θ$ ). It is crucial to choose a wavelength within this range that corresponds to a measurable diffraction angle for the crystal structure of interest.

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