Precise lattice parameters from linear least squares

2 Precise lattice parameters from linear least squares 

We now consider the most general form of Eq. 5.28, i.e. that which relates the unit cell dimensions, Miller indices, wavelength and Bragg angles in the triclinic crystal system. In a quadratic form, it can be written as 

By comparing Eq. 5.40 with Eq. 5.30, it is easy to see that the linear least squares technique is not directly applicable in this case since the former equation is clearly non linear with respect to the unknowns (a, b, c, a, p and y). If, however, Eq. 5.40 is rewritten it terms of the reciprocal lattice parameters 

So far, we considered the application of a liner least squares technique in the case when no systematic error has been present in the observed powder diffraction data. However, as we already know, in many cases the measured Bragg angles are affected by a systematic sample displacement or zero shift error. The first systematic error affects each data point differently and considering Eq. 3.4 (section 3.5.5), when a sample displacement error, s, is present in the data, Eq. 5.43 becomes 

R is the radius of the goniometer. The second systematic error adds a small constant value, 80o, to each observed Bragg angle, which results in the following equivalent of Eq. 5.43  

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