Bragg Law & Diffraction Condition

• James Marquez
In summary, the derivation in Chapter 2 of "Introduction to Solid State Physics" by Kittel uses the diffraction condition and the equation for spacing between parallel lattice planes to arrive at the Bragg law, which states that 2d sin(theta) = n lambda, where d is the spacing between lattice planes, theta is the angle of diffraction, n is an integer, and lambda is the wavelength. The derivation involves rearranging equations and using the definition of reciprocal lattice vectors.
James Marquez
Hello. I am reading "Introduction to Solid State Physics" by Kittel and there is a derivation in the textbook that I am understanding. This should be a fairly simple question but I am unable to see it.

1. Homework Statement

In Chapter 2, it derives the Bragg law using the diffraction condition and equation for spacing between parallel lattice planes. The Bragg law is given by:
$$2d\sin{\theta}=n\lambda$$

Homework Equations

The Diffraction Condition:
$$2\textbf{k}\bullet\textbf{G}=G^2$$

Spacing Between Parallel Lattice Planes:
$$d=\frac{2\pi}{G}$$

Wave Vector:
$$\textbf{k}=\frac{2\pi}{\lambda}\hat{\textbf{k}}$$

The Attempt at a Solution

I just don't see how we get the Bragg law using these two equations? I rearranged it so that
$$G=\frac{2\pi}{d}$$
But given by the diffraction condition, I see that:
$$G=2\frac{2\pi}{\lambda}\cos{\theta}$$
$$\frac{1}{d}=\frac{2\cos{\theta}}{\lambda}$$

I don't understand how the textbook arrives at that conclusion. Thank you.

I think the answer is that ## G=(\frac{2 \pi}{d}) m ##, where ## m ## is an integer, defines the reciprocal lattice vectors for a cubic lattice. ## \\ ## Editing... That will not give them all because the reciprocal lattice vectors are found from the lattice vectors for the primitive cell of the reciprocal lattice by having integer numbers of primitive basis vectors in each direction... If you look at the previous page or two in Kittel, he defines a reciprocal lattice vector ## G=h \vec{A}+k\vec{B}+l \vec{C} ## , ## (h,k,l= ## integers## ) ##.

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What is Bragg's law?

Bragg's law, named after British physicist William Henry Bragg, is a fundamental law of diffraction that describes the relationship between the angle of incidence of X-rays or neutrons and the spacing between atomic planes in a crystalline material.

What is the diffraction condition according to Bragg's law?

The diffraction condition states that for a particular set of crystal planes, the angle of incidence of X-rays or neutrons must be equal to the angle of reflection in order for constructive interference to occur and produce a diffraction pattern.

How does Bragg's law relate to crystallography?

Bragg's law is essential in crystallography as it allows scientists to determine the arrangement of atoms in a crystalline material. By measuring the angles at which X-rays or neutrons are diffracted, the spacing between atomic planes can be calculated and used to create a three-dimensional model of the crystal's atomic structure.

What are the main applications of Bragg's law?

Bragg's law is used in various fields such as materials science, chemistry, and biology to study the atomic and molecular structure of crystalline materials. It is also used in X-ray diffraction techniques for determining the atomic structure of proteins and other biological molecules.

What are the limitations of Bragg's law?

While Bragg's law is a powerful tool for studying the atomic structure of crystalline materials, it has limitations. For example, it only applies to regular crystal structures and cannot be used for amorphous materials. Additionally, it assumes that the crystal is perfect and does not account for defects or imperfections in the crystal lattice.

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