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Chemistry LibreTexts


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  • Learning Objectives

    • Explain X-rays.
    • Interpret the symbols used in the Bragg equation.

    Like light, X-rays are electromagnetic radiation with very short wavelengths. Thus, X-ray photons have high energy, and they penetrate opaque material, but are absorbed by materials containing heavy elements.

    X-ray Diffraction

    When light passes through a series of equal-spaced pinholes, it gives rise to a pattern due to wave interference, and such a phenomenon is known as diffraction. X-rays have wavelengths comparable to the interatomic distances of crystals, and the interference patterns are developed by crystals when a beam of X-rays passes a crystal or a sample of crystal powder. The phenomena are known as diffraction of X-rays by crystals. More theory is given in Introduction to X-ray Diffraction.

    X-ray diffraction, discovered by von Laue in 1912, is a well established technique for material analysis. This link is the home page of Lambda Research, which provide various services using X-ray diffractions. For example:

    • Residual Stress Measurement
    • Qualitative Phase Analysis
    • Quantitative Phase Analysis
    • Precise Lattice Parameter Determination

    In 1913, the father and son team of W.H. Bragg and W.L. Bragg gave the equation for the interpretation of X-ray diffraction, and this is known as the Bragg equation.

    \(2\, d\, \sin \theta = n\, \lambda\)

    where d is the distance between crystallographic planes, \(\theta\) is half the angle of diffraction, n is an integer, and \(\lambda\) is the wavelength of the X-ray. A set of planes gives several diffraction beams; each is known as the nth order.

    Example 1

    The X-ray wavelength from a copper X-ray is 154.2 pm. If the inter-planar distance from \(\ce{NaCl}\) is 286 pm, what is the angle \(\theta\)?


    \sin \theta &= \dfrac{\lambda}{2 d}\\
    &= \dfrac{154}{2\times282}\\
    &= 0.273

    \(\theta = 15.8^\circ\)

    Example 2

    An X-ray of unknown wavelength is used. If the inter-planar distance from \(\ce{NaCl}\) is 286 pm, and the angle \(\theta\) is found to be 7.23°, what is \(\lambda\)?


    \lambda &= 2\, d\, \sin\theta\\
    &= 2\times282\times\sin(7.23^\circ)\\
    &= \mathrm{71\: pm}

    Example 3

    The X-ray of wavelength 71 pm is used. If the inter-planar distance from \(\ce{KI}\) is 353 pm, what is the angle \(\theta\) for the second order diffracted beam?

    The calculation is shown below:

    \sin \theta &= \dfrac{\lambda}{2 d}\\
    &= \dfrac{71}{2\times353}\\
    &= 0.100\\
    \theta &= 5.8^\circ

    These examples illustrate some example of the applications of X-ray diffraction for the study of solids.


    1. Hint: 30 degrees
    2. If the wavelength is 150 pm and the interplanar distance d is 300 pm, what is the angle \(\theta\) in the Bragg equation, for n = 2?

      Hint: 30 degrees

    3. Hint: \(\ce{NaCl}\)

      Discussion -
      The larger the interplanar distance d, the smaller the angle.

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