What is the significance of h^2+k^2+l^2 values in BCC and FCC?I have seen how h,k,l values are the ones which are relatable to the structure factor theorem and why all odd or all even and sum should be even give us the necessary variations.But how do I interpret the h^2+k^2+l^2 as the consecutive peak values?
In a cubic system a/(h^2+k^2+l^2)^0.5 gives you the planar spacing d_hkl. Otherwise, I am not sure what you mean by 'consecutive peak values'?
Those integers are Miller indices (of vital relevance to X-ray diffraction and crystallography -- they determine the distance between adjacent parallel lattice planes in crystals, a quantity that features in the Bragg law [1]). The quadratic form h2+k2+l2, corresponding to (hkl), is particularly relevant to cubic crystal systems, as up a multiplicative constant it determines the shortest distance dhkl between two adjacent parallel lattice planes (encountered in the Bragg law) in these crystal systems. Similar, but not identical, quadratic forms are relevant to tetragonal, orthorhombic, and monoclinic crystal systems. For hexagonal crystal systems the quadratic form h2 + hk + k2 is relevant. For further details, consult [2] (Chapters 5, 6 and 7 herein) and [3] (consult in particular the Appendix with the title 'Quadratic Forms of Miller Indices', pp. 627 and 628).
[1] By definition, Miller indices h, k and l, in (hkl), have no common integer factor different from 1.
[2] NW Ashcroft, and ND Mermin, Solid State Physics (Thompson Learning, 1976).
[3] BD Cullity, and SR Stock, Elements of X-Ray Diffraction, 3rd edition (Pearson, Harlow, 2014).
Dear Rajorshi,
the sum h2+k2+l2 is related to the intensity and to the scattering angle of X-rays in a crystal. In the attached file 'indizes' this sum is plotted for the 3 cubic structures primitive, BCC (basic centered cubic) and FCC (face centered cubic). The sign - means that the corresponding reflex doesn't exist. So, the different cubic structures can be differed. The normalized intensity for consecutive peaks is plotted against the scattering angle in the file 'intensities'. The absolute intensity has to be calculated using the scattering factors.
With Regard
R. Mitdank
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