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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 [link] . From the geometry, the difference in path lengths is 2 d sin θ . Constructive interference results when this distance is an integer multiple of the wavelength. This condition is captured by the Bragg equation ,

m λ = 2 d sin θ , m = 1 , 2 , 3 .. .

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, θ , but unlike the general practice in geometric optics, θ is measured with respect to the surface itself, rather than the normal.

Figure shows atoms in a crystal as dots arranged in a grid. They are at a distance d from each other. Two parallel rays, labeled light rays in phase, strike one atom each from up and left, are deflected, and go up and right. The atoms concerned are labeled a and b, b being directly below a. The incident rays form an angle theta with the horizontal. Their extensions form an angle of 20 degrees with the deflected rays. A dotted line connects a and b. Another one connects a with the ray incident on b, making an angle theta with ab, thus forming a triangle. The side of the triangle along the ray incident on b is labeled d sine theta. The ray deflected from b is smaller than the ray deflected from a, by a distance 2d sine theta.
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 18.1 ° , what is the wavelength of the X-ray scattering from this crystal?

Strategy

Use the Bragg equation, [link] , m λ = 2 d sin θ , to solve for θ .

Solution

For first-order, m = 1 , and the plane spacing d is known. Solving the Bragg equation for wavelength yields

λ = 2 d sin θ m = 2 ( 0.252 × 10 −9 m ) sin ( 18.1 ° ) 1 = 1.57 × 10 −10 m , or 0.157 nm .

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 ( λ = 0.157 nm) is comparable to the size of the physical structures ( d = 0.252 nm ) it interacts with.

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

38.4 ° and 68.8 ° ; Between θ = 0 ° 90 ° , orders 1, 2, and 3, are all that exist.

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Although [link] 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 [link] . The resulting interference pattern can be quite complex.

Figure shows two crystal lattices, with atoms shown as small circles, connected to each other by lines. In the first lattice, flat planes formed in the lattice are highlighted. In the second, slanted planes formed in the lattice are highlighted. In each case, the planes are seen as a combination of different atoms in the same lattice.
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.

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Problems

X-rays of wavelength 0.103 nm reflects off a crystal and a second-order maximum is recorded at a Bragg angle of 25.5 ° . What is the spacing between the scattering planes in this crystal?

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A first-order Bragg reflection maximum is observed when a monochromatic X-ray falls on a crystal at a 32.3 ° angle to a reflecting plane. What is the wavelength of this X-ray?

0.120 nm

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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?

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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?

4.51 °

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On a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of 27.1 ° 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 37.3 ° . Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.

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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?

13.2 °

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The first-order Bragg angle for a certain crystal is 12.1 ° . What is the second-order angle?

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Questions & Answers

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because beta particles give the information that the electron is ejected from the nucleus with very high energy
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Practice Key Terms 2

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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