Question

X rays of wavelength $0.154\mathrm{nm}$ strike an aluminum crystal; the rays are reflected at an angle of ${19.3}^{\circ }.$ Assuming that $n=1,$ calculate the spacing between the planes of aluminum atoms (in pm) that is responsible for this angle of reflection.

Short answer

The plane spacing is approximately 233 pm.

Step-by-step solution

Understand Bragg's Law

Bragg's Law states that for constructive interference (reflection) from the crystal planes, the condition $2d\sin \theta =n\lambda$ must be fulfilled. Here, $d$ is the spacing between planes, $\theta$ is the angle of reflection, $n$ is the order of reflection (given as 1), and $\lambda$ is the wavelength of X-rays.

Convert Given Values to Desired Units

Convert the wavelength from nanometers to picometers: $$0.154\text{ nm}=154\text{ pm}$$ Keep the angle in degrees for calculation since the sine function can work with degrees directly.

Rearrange Bragg's Law to Solve for d

Rearrange the equation $2d\sin \theta =n\lambda$ to solve for $d$:$$d=\frac{n\lambda }{2\sin \theta }$$Substitute $n=1$ to simplify.

Calculate $\sin \theta$

Calculate the sine of the reflection angle $\theta ={19.3}^{\circ }$:$\sin {19.3}^{\circ }\approx 0.3305$.

Substitute Values into Bragg's Law

Substitute $n=1$, $\lambda =154$ pm, and $\sin \theta =0.3305$ into the equation from Step 3:$$d=\frac{1\times 154}{2\times 0.3305}\approx \frac{154}{0.661}\approx 233.0\text{ pm}$$.

Conclusion

The spacing between the planes of aluminum atoms is approximately $233.0$ picometers.

Key concepts

X-ray diffraction

X-ray diffraction is a powerful and widely used technique that helps scientists explore the arrangement of atoms within complex crystal structures. When X-rays are directed at a material, they can scatter in specific ways that produce patterns. These patterns offer valuable insights into the internal arrangement of atoms. This process is based on the principle of interference, where scattered X-rays can constructively or destructively interfere. X-ray diffraction is used in a variety of fields, from physics to chemistry and materials science. It can unravel intricate details about the atomic structure of different materials, including minerals, metals, and even proteins. Scientists often use X-ray diffraction to determine crystal structures, understand material imperfections, or even identify new compounds.

• Utilizes interference of X-rays by crystal planes.
• Helps to determine the internal structure and spacing between atoms.
• Important for both scientific research and industrial applications.

crystal lattice

A crystal lattice is the structured arrangement of atoms, ions, or molecules within a crystalline material. This highly ordered structure extends in multiple dimensions, forming a repeating pattern that can vary depending on the substance. In essence, a crystal lattice lays the foundation for the overall structure of the material by defining the position of each component. This specific arrangement is crucial because it affects the material's properties, from hardness to transparency. Crystal lattices can be visualized as three-dimensional grids that stretch over space, with each point corresponding to a location where an atom or molecule resides. Understanding the crystal lattice structure is fundamental in fields such as crystallography, materials science, and solid-state physics. The coordination and uniformity of atoms in a lattice contribute significantly to the properties and behavior of the crystalline substance.

• Defines how atoms are arranged in a crystal.
• Impacts many physical properties of the material.
• Essential for understanding material behavior at the atomic level.

atomic spacing

Atomic spacing refers to the distances between adjacent planes or atoms in a crystal lattice. This spacing plays a pivotal role in determining many of the material's physical properties and its interactions with external influences such as light or radiation. In the context of X-ray diffraction, atomic spacing is crucial because it determines how X-rays are scattered by the crystal. By knowing the atomic spacing, researchers can utilize Bragg's Law to understand how these reflections correlate with the arrangement of atoms inside the crystal. This concept is also essential for predicting material behavior, such as thermal expansion, electrical conductivity, and even chemical reactivity. Precise knowledge of atomic spacing can lead to innovations in material design and the development of novel materials with tailored properties.

• Important for understanding material properties and behavior.
• Facilitates the use of Bragg's Law in X-ray diffraction studies.
• Key factor in interpreting X-ray diffraction data to reveal atomic structure.

wavelength conversion

Wavelength conversion involves changing the units of wavelength from one system to another, a necessary step when working with different measurement systems. In scientific computations, it's vital to ensure consistency in units for accurate calculations. For instance, converting wavelengths from nanometers to picometers often helps match the preferred scale used in context with atomic and molecular dimensions. This practice of converting units is commonly used in many scientific fields to maintain clarity and precision in experiments. In the exercise at hand, converting the given X-ray wavelength from nanometers to picometers was crucial for using Bragg's Law accurately to solve for the atomic spacing. This step ensures that all units are consistent across the calculation, reducing the risk of errors.

• Converts between different units of wavelength.
• Ensures precision and consistency in scientific calculations.
• Critical for aligning measurements to the scale of atomic dimensions.