Question

Using the expression $2d\sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angles $\theta$ in the range 0 to ${90}^{\circ }.$ The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta .$ As $\theta$ increases from $0^{\circ }$, (A) the absolute error in $d$ remains constant. (B) the absolute error in $d$ increases. (C) the fractional error in $d$ remains constant. (D) the fractional error in $d$ decreases.

Short answer

The fractional error in $d$ decreases as $\theta$ increases from $0^{\circ }$, thus the correct answer is (D).

Step-by-step solution

Understand the problem and given equation

The expression to calculate the spacing, d, is given by Bragg's Law, which in this form is expressed as: $2d\sin \theta =\lambda$, where $\lambda$ is the known wavelength, $d$ is the spacing between planes in the crystal lattice, and $\theta$ is the angle of incidence. What varies in the problem is $\theta$, and we're interested in understanding how the error in $d$ changes as $\theta$ changes.

Identify the relationship of interest

We need to find how the error in measurement of $d$ changes with $\theta$. Errors can be expressed in two ways: absolute error, which is the difference between the measured value and the true value, and fractional (or relative) error, which is the absolute error divided by the true value.

Analyze the absolute error in $d$

The absolute error in $d$ depends on the error in $\theta$ because $d$ is calculated through $\theta$. However, since the error in $\theta$ is constant for all values of $\theta$, the sensitivity of $d$ to changes in $\theta$ is key. Differentiating both sides of the equation with respect to $\theta$, we get the relationship for the error in $d$ as $2d\cos \theta d\theta =0$, which suggests the absolute error in $d$ depends on $\cos \theta$.

Analyze the fractional error in $d$

The fractional error for $d$, by definition, is the absolute error divided by the true value of $d$. Hence, if the absolute error in $d$ increases with $\theta$ due to the $\cos \theta$ factor (whether it increases or decreases), then the fractional error in $d$ will change accordingly, since the denominator (the true value of $d$) would be constant for a given $\theta$.

Determine the trend of the absolute error as $\theta$ increases

As $\theta$ increases, $\cos \theta$ decreases because the cosine function decreases from 1 to 0 as the angle increases from $0^{\circ }$ to ${90}^{\circ }$. Thus, the absolute error in $d$, which is proportional to $\cos \theta$, would decrease.

Determine the trend of the fractional error as $\theta$ increases

Since the true value of $d$ isn't changing with $\theta$, a decrease in the absolute error in $d$ means the fractional error in $d$, which is the absolute error divided by the true value, also decreases.

Conclusion

Given that the absolute error in $d$ decreases as $\theta$ increases and since the true value of $d$ remains the same for a given $\theta$, the fractional error in $d$ decreases. Therefore, the correct option is (D).

Key concepts

X-ray Diffraction

X-ray diffraction (XRD) is a powerful tool that scientists use to study the structure of materials at the atomic level.

When X-rays are aimed at a crystalline substance, they are scattered by the atoms within the crystal lattice. This scattering follows a pattern defined by Bragg's Law, which relates the angle at which the X-rays are diffracted to the spacing between the lattice planes in the crystal.

The equation we use to describe this relationship is the famous Bragg's equation: $2d\sin \theta =\lambda$, where $d$ represents the distance between crystal planes, $\theta$ is the angle of incidence of the X-ray hitting the crystal, and $\lambda$ is the wavelength of the X-rays. By analyzing the diffraction pattern, the arrangement of atoms and the distances between them can be determined.

Crystal Lattice Spacing

The crystal lattice spacing, denoted as $d$ in Bragg's Law, is the distance between parallel planes of atoms or ions in a crystal.

This spacing is essential for determining the structure of materials because it can affect the properties of a material in significant ways, including its electronic and mechanical characteristics.

Importance of Lattice Spacing in Physics

A precise measurement of lattice spacing is important in crystallography as it helps scientists understand the fundamental properties of different materials and how they interact with various forms of energy, such as light and X-rays.

Error Analysis in Physics

Error analysis in physics is a critical process for understanding the accuracy and precision of measurements.

Every physical measurement has some degree of uncertainty, which can stem from the instruments used, the observer, or the object being measured. Understanding the types and sources of errors is crucial to interpreting results correctly.

Absolute and Fractional Errors

In the context of Bragg's Law, we focus on two types of errors: absolute error, which is the deviation from the true value, and fractional error, which is the ratio of the absolute error to the true value.

In the step-by-step solution for the exercise, we analyzed how these errors behave as the angle $\theta$ varies and found that while the absolute error in $d$ decreases with $\theta$, the fractional error also decreases.

Angular Measurement in Crystallography

Angular measurement in crystallography is fundamental, as the angles at which X-rays are diffracted from a crystal provide information about the spacing between the lattice planes.

The precision with which these angles are measured can greatly influence the accuracy of the determined lattice spacings and, consequently, the entire structural analysis of a crystalline material.

Impact of Angular Measurement Error

A constant error in the angular measurement can lead to varying impacts on the calculated lattice spacing depending on the angle of incidence. As seen in the exercise, this error can impact the absolute and fractional errors in measurements of the lattice spacing, affecting the outcomes of crystallographic studies.