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Magazine Chapter 26: Problem 43
\(\bullet\) Monochromatic \(x\) rays are incident on a crystal for which the spacing of the atomic planes is 0.440 nm. The first-order maximum in the Bragg reflection occurs when the incident and reflected rays make an angle of \(39.4^{\circ}\) with the crystal planes. What is the wavelength of the rays?
Short Answer
Expert verified
The wavelength of the X-rays is approximately 0.557 nm.
Step by step solution
01
Understand the Bragg's Law
Bragg's Law is given by the formula \( n\lambda = 2d\sin(\theta) \), where \(n\) is the order of the maximum (which is 1 for first-order maximum), \(\lambda\) is the wavelength of the X-rays, \(d\) is the spacing between the atomic planes, and \(\theta\) is the angle between the incident rays and the crystal planes.
02
Identify Given Values
From the problem statement, we know the order \(n = 1\), the plane spacing \(d = 0.440\, \text{nm}\), and the angle \(\theta = 39.4^{\circ}\).
03
Convert Angle to Radians (if Necessary)
Bragg's Law calculations often use degrees directly, but it's important to check if units need conversion. In this case, we can keep \(\theta\) as \(39.4^{\circ}\) since the sine function accommodates degree input in many scientific calculators.
04
Apply Bragg's Law to Solve for Wavelength
Substitute the known values into Bragg's Law: \( \lambda = \frac{2d\sin(\theta)}{n} = \frac{2 \times 0.440\, \text{nm} \times \sin(39.4^{\circ})}{1} \). Calculate \(\sin(39.4^{\circ})\) and proceed with the calculation.
05
Perform Calculation
Calculate \(\sin(39.4^{\circ})\), which approximately equals 0.633. Therefore, \( \lambda = 2 \times 0.440 \times 0.633 \approx 0.55744\, \text{nm} \).
06
Finalize the Answer
The wavelength of the X-rays is approximately \(0.557\, \text{nm}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
X-ray diffraction
X-ray diffraction is a powerful technique used to study crystal structures by observing how x-rays are scattered by the crystal's atoms. When x-rays hit a crystal, they are diffracted in specific patterns. These patterns reveal information about the atomic arrangement within the crystal. This occurs because x-rays have wavelengths on the order of atomic distances and can thus be diffracted through the regular, repeating structures in crystals. This method is widely applied in fields such as physics, chemistry, and biology to determine the detailed atomic structure of a wide range of materials.
- Sketch the pattern: X-rays are scattered by atoms in the crystal.
- Discover atomic arrangements: The angles and intensities of the diffracted x-rays help deduce the positions of atoms.
- Utilize in research: X-ray diffraction is essential for crystallography and is frequently used to determine the structures of complex molecules like proteins.
atomic plane spacing
The spacing between atomic planes in a crystal, often denoted by the letter \(d\), is a crucial parameter in the study of crystal structures. Atomic plane spacing refers to the distance between two parallel planes of atoms in a crystal lattice. This spacing can greatly affect the diffraction pattern observed when x-rays are used to probe the crystal. The precise knowledge of this spacing allows scientists to predict and understand how x-rays will interact with the crystal, based on Bragg's Law.
- Measured in nanometers: Typical values for atomic plane spacing range in the order of angstroms or nanometers.
- Impacts diffraction: The spacing influences the angles at which constructive interference occurs.
- Links to Bragg's Law: \(d\) is a key component in the equation \(n\lambda = 2d\sin(\theta)\), governing the diffraction of x-rays.
wavelength calculation
Calculating the wavelength of x-rays involves using Bragg's Law, which is expressed as \(n\lambda = 2d\sin(\theta)\). Here, \(\lambda\) represents the wavelength of the x-rays, \(n\) is the order of reflection, \(d\) is the atomic plane spacing, and \(\theta\) is the angle of incidence with respect to the crystal planes. This formula helps find the specific wavelength that will satisfy the diffraction condition, leading to the construction of a resultant, observable maximum in the diffraction pattern. Walking through the steps of this calculation ensures that the wavelength corresponds to the physical characteristics of the crystal under examination.
- Relate to crystal structure: The wavelength calculated is intrinsically linked to the crystal's atomic arrangement.
- Use known variables: Input known values of \(d\), \(\theta\), and \(n\) to determine \(\lambda\).
- Essential for experiments: Accurate wavelength calculation is vital for scientific studies that rely on precise x-ray information.
angle of incidence
The angle of incidence, represented by \(\theta\) in the context of Bragg's Law, is the angle at which incoming x-rays strike the crystal's atomic planes. It plays a significant role in determining the diffraction pattern. When calculating the diffraction condition using Bragg's Law, this angle helps determine the specific path differences needed for constructive interference of the x-rays. The value of \(\theta\) influences whether or not the x-rays will constructively interfere and produce a prominent peak in the diffraction pattern.
- Critical for diffraction: The precise angle is crucial as it determines the setting for optimal diffraction.
- Measured in degrees: Often, the angle \(\theta\) is given in degrees, but sometimes it may be necessary to convert it to radians for calculations.
- Defines path length difference: The angle affects how x-rays traverse through different paths within the crystal structure.
