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Variance as a Measure of Line Broadening: Particle-size Determination

Naturevolume 207pages966967 (1965) | Download Citation



THE X-ray determination of the average particle size in a polycrystalline aggregate, using the variance as a measure of line breadth, gives two independent estimates of the average size1. The variance of the line (corrected for the background2) varies linearly with the range of scan in the region where the intensity decreases as the inverse square of the range. The slope of this linear function, after subtracting the contribution from the emission-spectrum variance, is inversely proportional to the apparent particle size, provided that the specimen is well annealed and free from ‘mistakes’. The apparent size calculated from the slope is the same as that determined by the Fourier method3, and is the mean thickness of the crystallites measured perpendicular to the reflecting planes4. (The integral breadth, however, gives a slightly higher apparent size, this being the volume average of the thickness of the crystal, again in the direction normal to the reflecting planes5.) The ‘true’ size (the cube root of the volume) is the product of the apparent size and the appropriate Scherrer constant1,5.

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  1. 1

    Wilson, A. J. C., Proc. Phys. Soc., 88, 286 (1962).

  2. 2

    Langford, J. I., and Wilson, A. J. C., in Crystallography and Crystal Perfection, edit. by Ramachandran, G. N., 207 (London: Academic Press, 1963).

  3. 3

    Wilson, A. J. C., Mathematical Theory of X-ray Powder Diffractometry (Eindhoven: Centrex, 1963).

  4. 4

    Averbach, B. L., and Warren, B. E., J. App. Phys., 20, 885 (1949).

  5. 5

    Stokes, A. R., and Wilson, A. J. C., Proc. Camb. Phil. Soc., 38, 313 (1942).

  6. 6

    Wilson, A. J. C., Proc. Phys. Soc., 85, 807 (1965).

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  1. Viriamu Jones Laboratory, University College, Cardiff

    • J. I. LANGFORD


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