Section 38–3 Crystal diffraction

Bragg diffraction / Bragg condition / Bragg cutoff

 

In this section, Feynman discusses Bragg diffraction, Bragg condition, and Bragg cutoff. Interestingly, the section is titled as “crystal diffraction,” but he explains the phenomenon as the reflection of particle waves from a crystal. However, the term Bragg diffraction is more appropriate to acknowledge the contributions of W. H. Bragg and his son W. L. Bragg in x-ray diffraction, a discovery for which they received the 1915 Nobel Prize in Physics.

 

1. Bragg diffraction

“Next let us consider the reflection of particle waves from a crystal. A crystal is a thick thing which has a whole lot of similar atoms—we will include some complications later—in a nice array. The question is how to set the array so that we get a strong reflected maximum in a given direction for a given beam of, say, light (x-rays), electrons, neutrons, or anything else. In order to obtain a strong reflection, the scattering from all of the atoms must be in phase. There cannot be equal numbers in phase and out of phase, or the waves will cancel out. The way to arrange things is to find the regions of constant phase, as we have already explained; they are planes which make equal angles with the initial and final directions (Feynman et al., 1963).”

 

Feynman could have continued using the term wave packets or wave trains instead of particle waves to model x-rays, electrons, and neutrons. More importantly, the term reflection is a misnomer, as the underlying process is diffraction, not simple specular reflection. A more precise term is Bragg diffraction, which accurately describes the phenomenon as wave interference arising from periodic layers of atoms rather than mere bouncing off a surface. The process involves the scattering of incoming waves that interact with parallel atomic planes, leading to constructive interference among outgoing waves. The scattering phenomenon is also known as Bragg scattering or elastic scattering, as the interaction between the incoming waves and the crystal lattice does not result in an observable change in energy—only a change in direction.

 

Historically, in 1912, Max von Laue proposed that crystals act as three-dimensional diffraction gratings for x-rays. To simplify analysis, the x-ray source and detector are idealized as being far from the crystal, allowing both the incident and outgoing waves to be treated as plane waves.  Specifically, x-rays induce oscillations in the electrons within the crystal, causing them to emit secondary x-rays. These scattered waves interfere and give rise to diffraction patterns at certain angles. This process is a form of elastic scattering, meaning that while the x-rays interact with the crystal lattice, their wavelength remains constant. The experiments showed that x-rays have wavelike properties and provided insight into the periodic arrangement of atoms in crystals.

 

2. Bragg conditions:

“… the waves scattered from the two planes will be in phase provided the difference in distance travelled by a wavefront is an integral number of wavelengths. This difference can be seen to be 2dsinθ, where d is the perpendicular distance between the planes. Thus the condition for coherent reflection is 2dsinθ = nλ (n=1,2,…) (Feynman et al., 1963).”

 

Feynman states the condition for coherent reflection as 2d sin θ = nλ (n = 1, 2,…), where d is the interplanar spacing. However, instead of single condition, we may emphasize three key Bragg conditions:

(1) Bragg’s equation: For diffraction to occur, the scattered waves must interfere constructively satisfying Bragg’s equation, nλ = 2dsin θ.

(2) Angle of diffraction: The incident and diffracted waves must obey the relation: Angle of Incidence = Angle of Diffraction.

(3) Interplanar spacing: The crystal must have a regular, periodic arrangement of atoms with a well-defined interplanar spacing d.

Additionally, while Bragg’s equation provides a simplified scalar description of diffraction, the Laue condition offers a more general vector-based formulation that relates the incident and diffracted wave vectors. Bragg’s equation can be derived as a special case of the Laue condition, particularly when considering diffraction from parallel atomic planes.

 

“If, on the other hand, there are other atoms of the same nature (equal in density) halfway between, then the intermediate planes will also scatter equally strongly and will interfere with the others and produce no effect. So d in (38.9) must refer to adjacent planes; we cannot take a plane five layers farther back and use this formula! (Feynman et al., 1963).”

 

Perhaps Feynman could have clarified the distinction between intermediate planes and adjacent planes. In a crystal, multiple sets of parallel planes exist, each with its own interplanar spacing, leading to different pairs of incidence and diffraction angles (as shown below) that satisfy the conditions for constructive interference. Bragg’s law applies not only to regular lattice structures but also to specific lattice planes, such as hexagonal planes in certain crystals. Furthermore, if the diffraction conditions hold for a particular atomic layer and its neighboring layers, they can be assumed to apply consistently across all layers with identical spacing. Experimentally, x-rays penetrate deeply into the crystal, allowing diffraction to arise from thousands or even millions of layers, collectively contributing to the observed diffraction pattern.

 

Source: (Mansfield & O'sullivan, 2020)

3. Bragg cutoff:

“Incidentally, an interesting thing happens if the spacings of the nearest planes are less than λ/2. In this case (38.9) has no solution for n. Thus if λ is bigger than twice the distance between adjacent planes then there is no side diffraction pattern, and the light—or whatever it is—will go right through the material without bouncing off or getting lost. So in the case of light, where λ is much bigger than the spacing, of course it does go through and there is no pattern of reflection from the planes of the crystal (Feynman et al., 1963).”

 

Bragg cutoff refers to the wavelength λ_b beyond which Bragg diffraction cannot occur. This wavelength can be determined by substituting two extreme values, θ = 90° (maximum angle) and n =1 (minimum order) into Bragg’s equation. Mathematically, if λ > 2d​, no real angle θ satisfies Bragg’s law, making diffraction impossible. However, Feynman’s claim that “light will go right through the material without bouncing off or getting lost” oversimplifies the situation. Even if Bragg diffraction does not occur, incident waves can still interact with the crystal through scattering, absorption, or transmission. In materials with sufficient electron density, electromagnetic radiation can be significantly absorbed rather than simply passing through unaffected. Bragg cutoff represents a fundamental limit in crystal diffraction, defining the range of wavelengths that can undergo diffraction.

 

“If we take these neutrons and let them into a long block of graphite, the neutrons diffuse and work their way along (Fig. 38–7). They diffuse because they are bounced by the atoms, but strictly, in the wave theory, they are bounced by the atoms because of diffraction from the crystal planes. It turns out that if we take a very long piece of graphite, the neutrons that come out the far end are all of long wavelength!... In other words, we can get very slow neutrons that way. Only the slowest neutrons come through; they are not diffracted or scattered by the crystal planes of the graphite, but keep going right through like light through glass, and are not scattered out the sides (Feynman et al., 1963).”

 

Feynman’s statement—“if we take a very long piece of graphite, the neutrons that come out the far end are all of long wavelength!”— oversimplifies the underlying physics. As neutrons diffuse through graphite, they undergo multiple collisions with carbon atoms, losing kinetic energy in a process known as neutron moderation. This slowing-down effect is why graphite serves as a moderator in nuclear reactors, reducing neutron energy to facilitate optimal fission reactions. In a sufficiently long piece of graphite, the neutrons that emerge at the far end are mainly slower neutrons with longer de Broglie wavelengths. This occurs partly because high-energy (short-wavelength) neutrons satisfy Bragg's diffraction condition for scattering from the crystal planes and are thus deflected. In contrast, slow neutrons, which do not satisfy Bragg’s condition, pass through the lattice with minimal scattering, similar to light passing through glass.

 

Review questions:

1. What is Bragg diffraction? Is it due to the reflection of particle waves from a crystal?

2. How would you explain the Bragg condition(s)? How many are there?

3. How would you explain the Bragg cutoff? Does light simply pass through the material without bouncing off or getting lost?

 

The moral of the lesson: The wave properties of x-rays, electrons, and neutrons are revealed through Bragg diffraction, which occurs when the path difference between the adjacent waves scattered from different planes is an integer multiple of the wavelength.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Mansfield, M. M., & O'sullivan, C. (2020). Understanding physics. Hoboken, NJ: John Wiley & Sons.