Diffraction In Nature

Learning Objective

  1. Diffraction In Nature Photography
  2. Diffraction In Nature
  3. Diffraction In Nature Science
Diffraction
  • Recognize the difference between constructive and destructive interference, and between interference and diffraction

Key Points

Diffraction In Nature Photography

  • In physics, interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude.
  • Constructive interference occurs when the phase difference between the waves is a multiple of 2π, whereas destructive interference occurs when the difference is π, 3π, 5π, etc.
  • Diffraction refers to various phenomena that occur when a wave encounters an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.

Terms

Electron diffraction refers to the wave nature of electrons.However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern.

Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon is commonly known as wave–particle duality, which states that a particle of matter can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to. Diffraction is the bending of light around the sharp corner of an obstacle. When light is incident on a slit, with a size comparable to the wavelength of light, an alternating dark and bright pattern can be observed. This phenomenon is called the single slit diffraction. The behavior of a wave (or pulse) upon reaching the end of a medium is referred to as boundary behavior. There are essentially four possible behaviors that a wave could exhibit at a boundary: reflection (the bouncing off of the boundary), diffraction (the bending around the obstacle without crossing over the boundary), transmission (the crossing of the boundary into the new material. Light can bend around edges. Light bends when it passes around an edge or through a slit. This bending is called diffraction. You can easily demonstrate diffraction using a candle or a small bright flashlight bulb and a slit made with two pencils.

  • interferenceAn effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects. In physics, interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude.
  • amplitudeThe maximum absolute value of some quantity that varies, especially a wave.
  • diffractionThe breaking up of an electromagnetic wave as it passes a geometric structure (e.g., a slit), followed by reconstruction of the wave by interference.

In physics, interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same (or nearly the same) frequency. Interference effects can be observed with all types of waves, including light, radio, acoustic, and surface water waves. In chemistry, the applications of interference to light are the most relevant to the study of matter.

Mechanism of Interference

The principle of superposition of waves states that when two or more waves are incident on the same point, the total displacement at that point is equal to the vector sum of the displacements of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the magnitude of the displacement is the sum of the individual magnitudes; this is known as constructive interference. If a crest of one wave meets a trough of another wave, then the magnitude of the displacements is equal to the difference in the individual magnitudes; this is known as destructive interference.

Constructive interference occurs when the phase difference between the waves is a multiple of 2π, whereas destructive interference occurs when the difference is π, 3π, 5π, etc. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase and will produce a maximum displacement. In other places, the waves will be in anti-phase and there will be no net displacement at these points. Thus, parts of the surface will be stationary.

Diffraction

Diffraction refers to various phenomena that occur when a wave encounters an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects occur when light waves travel through a medium with a varying refractive index or a sound wave through one with varying acoustic impedance. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays, and radio waves. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1665.

The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. All these effects occur because light propagates as a wave.

Richard Feynman said, “No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.”

He suggested that when there are only a few sources, say two, we call it interference (as in Young’s slits), but with a large number of sources, the process can be labelled diffraction.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is roughly similar to the dimensions of the diffracting objects. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. This is due to the superposition, or interference, of different parts of a wave that traveled to the observer by different paths (see diffraction grating).

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Learning Objectives

By the end of this section, you will be able to:

  • Describe interference and diffraction effects exhibited by X-rays in interaction with atomic-scale structures

Since X-ray photons are very energetic, they have relatively short wavelengths, on the order of m to m. Thus, typical X-ray photons act like rays when they encounter macroscopic objects, like teeth, and produce sharp shadows. However, since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the location, shape, and size of atoms and molecules. The process is called X-ray diffraction, and it involves the interference of X-rays to produce patterns that can be analyzed for information about the structures that scattered the X-rays.

Perhaps the most famous example of X-ray diffraction is the discovery of the double-helical structure of DNA in 1953 by an international team of scientists working at England’s Cavendish Laboratory—American James Watson, Englishman Francis Crick, and New Zealand-born Maurice Wilkins. Using X-ray diffraction data produced by Rosalind Franklin, they were the first to model the double-helix structure of DNA that is so crucial to life. For this work, Watson, Crick, and Wilkins were awarded the 1962 Nobel Prize in Physiology or Medicine. (There is some debate and controversy over the issue that Rosalind Franklin was not included in the prize, although she died in 1958, before the prize was awarded.)

(Figure) shows a diffraction pattern produced by the scattering of X-rays from a crystal. This process is known as X-ray crystallography because of the information it can yield about crystal structure, and it was the type of data Rosalind Franklin supplied to Watson and Crick for DNA. Not only do X-rays confirm the size and shape of atoms, they give information about the atomic arrangements in materials. For example, more recent research in high-temperature superconductors involves complex materials whose lattice arrangements are crucial to obtaining a superconducting material. These can be studied using X-ray crystallography.

X-ray diffraction from the crystal of a protein (hen egg lysozyme) produced this interference pattern. Analysis of the pattern yields information about the structure of the protein. (credit: “Del45”/Wikimedia Commons)

Historically, the scattering of X-rays from crystals was used to prove that X-rays are energetic electromagnetic (EM) waves. This was suspected from the time of the discovery of X-rays in 1895, but it was not until 1912 that the German Max von Laue (1879–1960) convinced two of his colleagues to scatter X-rays from crystals. If a diffraction pattern is obtained, he reasoned, then the X-rays must be waves, and their wavelength could be determined. (The spacing of atoms in various crystals was reasonably well known at the time, based on good values for Avogadro’s number.) The experiments were convincing, and the 1914 Nobel Prize in Physics was given to von Laue for his suggestion leading to the proof that X-rays are EM waves. In 1915, the unique father-and-son team of Sir William Henry Bragg and his son Sir William Lawrence Bragg were awarded a joint Nobel Prize for inventing the X-ray spectrometer and the then-new science of X-ray analysis.

In ways reminiscent of thin-film interference, we consider two plane waves at X-ray wavelengths, each one reflecting off a different plane of atoms within a crystal’s lattice, as shown in (Figure). From the geometry, the difference in path lengths is . Constructive interference results when this distance is an integer multiple of the wavelength. This condition is captured by the Bragg equation,

where m is a positive integer and d is the spacing between the planes. Following the Law of Reflection, both the incident and reflected waves are described by the same angle, but unlike the general practice in geometric optics, is measured with respect to the surface itself, rather than the normal.

X-ray diffraction with a crystal. Two incident waves reflect off two planes of a crystal. The difference in path lengths is indicated by the dashed line.

X-Ray Diffraction with Salt Crystals Common table salt is composed mainly of NaCl crystals. In a NaCl crystal, there is a family of planes 0.252 nm apart. If the first-order maximum is observed at an incidence angle of , what is the wavelength of the X-ray scattering from this crystal?

Strategy Use the Bragg equation, (Figure), , to solve for .

Solution For first-order, and the plane spacing d is known. Solving the Bragg equation for wavelength yields

Significance The determined wavelength fits within the X-ray region of the electromagnetic spectrum. Once again, the wave nature of light makes itself prominent when the wavelength is comparable to the size of the physical structures it interacts with.

Check Your Understanding For the experiment described in (Figure), what are the two other angles where interference maxima may be observed? What limits the number of maxima?

and ; Between , orders 1, 2, and 3, are all that exist.

Although (Figure) depicts a crystal as a two-dimensional array of scattering centers for simplicity, real crystals are structures in three dimensions. Scattering can occur simultaneously from different families of planes at different orientations and spacing patterns known as called Bragg planes, as shown in (Figure). The resulting interference pattern can be quite complex.

Because of the regularity that makes a crystal structure, one crystal can have many families of planes within its geometry, each one giving rise to X-ray diffraction.

Summary

  • X-rays are relatively short-wavelength EM radiation and can exhibit wave characteristics such as interference when interacting with correspondingly small objects.

Conceptual Questions

Crystal lattices can be examined with X-rays but not UV. Why?

UV wavelengths are much larger than lattice spacings in crystals such that there is no diffraction. The Bragg equation implies a value for sin⁡θ greater than unity, which has no solution.

Problems

X-rays of wavelength 0.103 nm reflects off a crystal and a second-order maximum is recorded at a Bragg angle of . What is the spacing between the scattering planes in this crystal?

A first-order Bragg reflection maximum is observed when a monochromatic X-ray falls on a crystal at a angle to a reflecting plane. What is the wavelength of this X-ray?

An X-ray scattering experiment is performed on a crystal whose atoms form planes separated by 0.440 nm. Using an X-ray source of wavelength 0.548 nm, what is the angle (with respect to the planes in question) at which the experimenter needs to illuminate the crystal in order to observe a first-order maximum?

The structure of the NaCl crystal forms reflecting planes 0.541 nm apart. What is the smallest angle, measured from these planes, at which X-ray diffraction can be observed, if X-rays of wavelength 0.085 nm are used?

On a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of relative to its surface, using an X-ray source of unknown wavelength. Additionally, when illuminated with a different, this time of known wavelength 0.137 nm, a second-order maximum is detected at Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.

Diffraction In Nature

Calcite crystals contain scattering planes separated by 0.30 nm. What is the angular separation between first and second-order diffraction maxima when X-rays of 0.130 nm wavelength are used?

The first-order Bragg angle for a certain crystal is . What is the second-order angle?

Diffraction In Nature Science

Glossary

Bragg planes
families of planes within crystals that can give rise to X-ray diffraction
X-ray diffraction
technique that provides the detailed information about crystallographic structure of natural and manufactured materials