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Chapter 12: Problem 120

In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation \((\lambda),\) the angle at which the radiation is diffracted \((\theta),\) and the distance between planes of atoms in the crystal that cause the diffraction \((d)\) is given by \(n \lambda=2 d \sin \theta . X\) rays from a copper \(X\) -ray tube that have a wavelength of \(154 \mathrm{pm}\) are diffracted at an angle of 14.22 degrees by crystalline silicon. Using the Bragg equation, calculate the distance between the planes of atoms responsible for diffraction in this crystal, assuming \(n=1\) (first-order diffraction).

Short Answer

Expert verified
The distance between the planes of atoms responsible for diffraction in the crystalline silicon crystal can be found using the Bragg equation \(n\lambda = 2d \sin\theta\). Given a wavelength of 154 pm, an angle of diffraction of 14.22°, and first-order diffraction (n=1), the distance (d) is approximately 313.9 pm.

Step by step solution

01

Understand the Bragg Equation

The Bragg equation relates the wavelength of radiation with the angle of diffraction and the distance between planes of atoms in a crystal. It is expressed as: \[n\lambda = 2d \sin\theta\] where \(n\) is the order of diffraction, \(\lambda\) is the wavelength of the radiation, \(d\) is the distance between planes of atoms in the crystal, and \(\theta\) is the angle of diffraction.
02

Given Data

We are given the following information: - Wavelength of radiation (λ) = 154 pm - Angle of diffraction (θ) = 14.22° - Order of diffraction (n) = 1
03

Convert the Angle to Radians

The angle of diffraction is given in degrees, so we need to convert it to radians before using the Bragg equation. To convert degrees to radians, we use the following formula: \[\text{Radians} = \frac{\text{Degrees}}{180} \times \pi\] Plugging in the angle in degrees: \(\theta = \frac{14.22}{180} \times \pi = 0.2481\) radians.
04

Use the Bragg Equation

Now that we have the angle θ in radians, we can apply the Bragg equation to find the distance between planes of atoms (d). First, we need to rewrite the equation for d: \[d = \frac{n\lambda}{2 \sin\theta}\]
05

Plug in the Values and Calculate d

Insert the given values for λ, n, and the calculated value for θ in radians: \(d = \frac{(1)(154 \times 10^{-12})}{2 \sin(0.2481)}\) Now calculate the value of d: \(d = \frac{154 \times 10^{-12}}{2 \times 0.24567} = 313.9 \times 10^{-12} \: \text{meters}\)
06

Convert Meters to Picometers

Since our original wavelength was given in picometers, it's convenient to express our answer in the same units for easier comparison. To convert meters to picometers, simply multiply by \(10^{12}\): \(d = 313.9 \times 10^{-12} \times 10^{12} = 313.9 \: \text{pm}\) The distance between the planes of atoms responsible for diffraction in the crystalline silicon crystal is approximately 313.9 pm.

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