# X-Ray Diffraction Homework: Find Lattice Spacing d

• ShotgunMatador
In summary, the problem deals with X-rays of wavelength λ incident on a cubic crystal with spacing d. The angle of incidence is fixed at θi degrees and a detector is rotated until the first maximum is detected at θs degrees. The lattice spacing, d, can be found using Bragg's Law, but since the first condition of angle of incidence equaling angle of reflection is not met, a modified version of the interference equation must be used. This can be achieved by considering the path difference between rays from adjacent slits/scatterers, and the condition for constructive interference between these rays. From this modified equation, the distance d between the crystal planes can be computed.
ShotgunMatador

## Homework Statement

X-rays of wavelength λ are incident on a cubic crystal with spacing d as shown in the figure. The incident angle is fixed at θi degrees. A detector is rotated from perpendicular to the 100 plane (90 degrees (grazing)) until the first maximum is detected at θs degrees due to interference from the 100 plane. What is the lattice spacing, d, of the cubic crystal?

## Homework Equations

Bragg's Law : mλ=2dsinθ

## The Attempt at a Solution

Known: m=1, λ
The angle to be used for Bragg's Law is not the incident/reflected angle because θs does not equal θi. So I figured out the angle between the incoming and reflected ray (180-θsisplit) and that would not be the angle we are looking for either. So i divided that θsplit by 2, and subtracted it from 90. To find what I thought was the correct angle but apparently not.

If this is not the angle, what is and how do i get to it?
Is m≠1 despite the problem stating "first order maximum"?
is my θs not the correct angle?

Thank you in advance. also sorry about the images, they work better if you drag them into a new tab.

#### Attachments

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This is not (IMO) characteristic of what are typically called Bragg maxima, which need to meet two conditions : 1) The angle of incidence equals angle of reflection, and 2) The reflected rays(waves) from adjacent planes must constructively interfere. Here, the first condition is not met. (You will get a peak, and it could perhaps even be called a Bragg peak, but it is not the extremely intense type of Bragg peak that you get when conditions (1) and (2) are both met). ## \\ ## For this one, simple interference principles and equations will apply (and get you the answer) with a slight modification. The simple interference/diffraction equations usually are written with zero degree angle of incidence and you get ## m \lambda=d \sin(\theta_s) ## for the interference maxima. ## \\ ## Question for you (I am not allowed to give you the answer, at least not without some effort on your part) How would the interference equation be modified for a bunch of equally spaced slits if the incident angle were ## \theta_i ## instead of zero degrees? For a couple hints on this, at normal incidence, ( ## \theta_i=0 ##), why is the path distance difference between adjacent slits (or scatterers) equal to ## d \sin(\theta_s) ## ? What is the additional path distance difference (between rays at adjacent slits) that occurs if the incident angle is some non-zero ## \theta_i ##? How would you then get the total path distance difference between rays from adjacent slits(or scatterers) in the case of a non-zero ## \theta_i ## plus a ## \theta_s ##, and what is the condition for constructive interference between the rays from adjacent slits (or scatterers)? How does your modified equation for constructive interference read for this (more general non-zero ## \theta_i ##) case? From this equation you can compute the distance ## d ## between the crystal planes.

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## What is X-Ray Diffraction?

X-Ray Diffraction is a technique used to determine the atomic and molecular structure of a crystal by measuring the diffraction patterns produced when a beam of X-rays is directed at a crystal.

## How is X-Ray Diffraction used to find lattice spacing d?

X-Ray Diffraction is used to find lattice spacing d by measuring the angle at which the diffracted X-rays intersect with a detector. This angle, along with the wavelength of the X-rays and the known properties of the crystal, can be used to calculate the lattice spacing of the crystal.

## What is the importance of finding lattice spacing d?

Finding lattice spacing d is important because it provides valuable information about the structure of a crystal. It can help determine the type of crystal structure, the number of atoms per unit cell, and the arrangement of the atoms within the crystal lattice.

## What factors can affect the accuracy of lattice spacing d calculations?

The accuracy of lattice spacing d calculations can be affected by factors such as the quality of the X-ray source, the condition of the crystal, and the accuracy of the measurements taken. Additionally, factors such as temperature, pressure, and crystal defects can also impact the accuracy of the results.

## What are some practical applications of X-Ray Diffraction in scientific research?

X-Ray Diffraction has a wide range of practical applications in scientific research, including the study of crystal structures in materials science, the determination of protein structures in biochemistry, and the analysis of pharmaceutical compounds in drug development. It is also used in fields such as geology, archaeology, and forensics to identify and analyze different types of crystals.

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