Question

An X-ray beam of unknown wavelength is diffracted from a NaCl surface. If the interplanar distance in the crystal is \(286 \mathrm{pm}\), and the angle of maximum reflection is found to be \(7.23^{\circ},\) what is the wavelength of the X-ray beam? (Assume \(n=1 .\) )

Short answer

\( \lambda = 1.786 \times 10^{-10} \text{m} \) or \( \lambda = 1.786 \text{Å} \).

Step-by-step solution

Identify the relevant equation

To find the wavelength \( \lambda \) of the X-ray beam, use Bragg's Law: \[ n\lambda = 2d\sin(\theta) \] where \( n \) is the order of reflection, \( d \) is the interplanar distance, and \( \theta \) is the angle of maximum reflection.

Insert known values into the equation

Given that \( n=1 \) (first order of reflection), \( d = 286 \text{pm} = 286 \times 10^{-12} \text{m} \) and \( \theta = 7.23^\circ \), insert these values into Bragg's Law: \[ 1 \times \lambda = 2 \times (286 \times 10^{-12} \text{m}) \times \sin(7.23^\circ) \].

Calculate the sine of the angle and the wavelength

First, calculate the sine of the angle: \[ \sin(7.23^\circ) \]. Then, multiply it by \( 2d \) to get the wavelength: \[ \lambda = 2 \times 286 \times 10^{-12} \times \sin(7.23^\circ) \].

Perform the calculation

Use a calculator to compute the value of \( \sin(7.23^\circ) \) and consequently \( \lambda \) to get the answer: \[ \lambda = 2 \times 286 \times 10^{-12} \times \sin(7.23^\circ) \approx 2 \times 286 \times 10^{-12} \times 0.1257 \].

Finish the calculation to find the wavelength

The calculation results in \( \lambda \) = \( 2 \times 286 \times 10^{-12} \times 0.1257 \) which simplifies to: \[ \lambda = 142 \times 10^{-12} \times 0.1257 \approx \lambda = 1.7857 \times 10^{-12} \text{m} \], or \( \lambda = 1.786 \text{Å} \) when converted to Angstroms with \( 1 \text{Å} = 10^{-10} \text{m} \).

Key concepts

Bragg's Law

Understanding Bragg's Law is central to the field of X-ray diffraction, which is a technique widely used to determine the structure of crystalline materials. The law was formulated by William Lawrence Bragg in 1912 and relates the wavelength of electromagnetic radiation to the crystal structure by means of diffraction.

In the simplest form, Bragg's Law is represented by the equation:
\[ n\lambda = 2d\sin(\theta) \]
Here, \( \lambda \) is the wavelength of the X-ray, \( n \) is the order of reflection which must be a positive integer (in many cases, it is sufficient to consider the first-order reflection, where \( n = 1 \)), \( d \) is the interplanar distance between crystal layers, and \( \theta \) is the angle of diffraction, typically known as the angle of maximum reflection. This law provides an essential connection between the observed diffraction pattern and the atomic structure of the crystal.

Interplanar Distance

The interplanar distance, represented by the symbol \( d \), is the distance between parallel planes of atoms or ions in a crystal. It's a significant quantity in crystallography and plays a key role in understanding X-ray diffraction patterns.

In the context of Bragg’s Law, the interplanar distance can affect the diffraction pattern significantly. Each set of crystallographic planes will diffract X-rays at specific angles, corresponding to specific interplanar distances.

Crystalline solids like sodium chloride (NaCl) have a regular arrangement of atoms, allowing for precise measurement of \( d \). In the given exercise, the interplanar distance is provided as 286 picometers (pm), where \( 1 \text{pm} = 10^{-12} \text{m} \). Calculating this distance is crucial, as it is one half of the product used to calculate the wavelength in Bragg's equation.

Angle of Reflection

In X-ray diffraction, the angle of reflection, often denoted as \( \theta \), is the angle at which X-rays are scattered upon hitting the crystal. This is not to be confused with reflection in mirrors, where the angle of reflection equals the angle of incidence. Instead, this angle in X-ray diffraction denotes the specific angle at which constructive interference of the reflected X-ray waves occurs, resulting in a detectable peak, also known as maximum reflection.

In the given exercise, the angle of maximum reflection is found to be 7.23 degrees. This value is critical when applying Bragg’s Law since it influences the path difference of the diffracted beams and, hence, determines whether constructive interference will occur.

Wavelength Calculation

Calculating the wavelength of an X-ray beam via diffraction involves rearranging Bragg's equation to solve for \( \lambda \). The wavelength can reflect information about the energy and penetrating power of the X-ray. This calculation is fundamental in the field of spectroscopy and material science, revealing insights into the atomic scale structure of materials.

The steps to calculate the wavelength typically involve identifying the order of reflection (usually \( n = 1 \)), knowing the interplanar distance (\( d \)), and the angle of reflection (\( \theta \)). The sine function is used as it relates the opposite side of the right triangle (path difference that leads to constructive interference) to the hypotenuse (distance between mirror reflection points at atom layers).

In the exercise, the value for \( d \) and \( \theta \) are plugged into Bragg's equation and computed, resulting in an X-ray wavelength measured in meters, which then can be easily converted into more conventional units like Angstroms depending on the context of use.