Sample displacement error

In practice, the application of x-ray measurement techniques to thin films involves some special problems. Typical films are much thinner than the penetration depth of commonly used x-rays, so the diffracted intensity is much lower than that from bulk materials. Thin films are often strongly textured this, on the other hand, results in improved intensity for suitable experimental conditions but complicates the measurement problem. Measurements at other than ambient temperature, not usually attempted with bulk materials, constitutes additional complexity. Since typical strains are on the order of 1 X 10 , measurements of interplanar spacing with a precision of the order of 1 X 10 are needed for reasonably accurate results hence, potential sources of error must be kept to a low level. In particular, the sample displacement error can be a major source of difficulty with a heated sample. The sample surface must remain accurately on the axis of the instrument during heating. 

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... 

Least squares refinement of lattice parameter (Eq. 5.39) assuming unit weights and using all 20 available Bragg peaks results in a = 4.1599(3) A. The obtained differences between the observed and calculated 26 are shown in Figure 5.19 and it is quite obvious that there is a systematic dependence of A20 on the Bragg angle. A similar behavior is always indicative of a systematic error, namely the presence of zero shift or sample displacement errors, or a combination of both. 

Figure 5.19. The differences between the observed and calculated Bragg angles after least squares refinement of the lattice parameter of LaBs without accounting for the presence of any kind of systematic error (open circles) using a = 4.1599(3) A. The dash-dotted line drawn through the data points is a guide for the eye. The solid line represents corrections of the observed Bragg angles using the refined in the next step sample displacement error (s/R = 0.00632) and the dashed line represents a similar correction by using the determined zero shift error (50o = 0.078°).

Figure 5.20. The differences between the observed and calculated Bragg angles after the least squares refinement of the lattice parameter of LaBg simultaneously with the zero shift error (open circles) or simultaneously with the sample displacement error (filled triangles).

In Table 4.13 an example is worked out for the determination of the sample-displacement error without an internal standard. Other errors should be excluded before if possible. For example, the zero-point error should be corrected by using an external standard. If the reflections can be indexed unequivocally (uncertain reflections must be omitted), the sample-displacement error - the most important error changing from sample to sample - can be easily refined with the lattice constants. The author s program, LATCO, refines the lattice constants twice, with and without sample-displacement error, and one has to decide oneself which of the two refinements is to be preferred. The main criterion for accepting the refinement with sample-position error is the size of the error in comparison with its standard deviation it must amount to at least twice the standard deviation. Furthermore jfi should be significantly... 

Cu Ka, radiation = 1.54960 A) was used. A pinhole collimator with a 0.5 mm opening was used to define the sampling volume, and seller slits were and to minimize sample displacement errors. The (1 4 6) AI2O3 reflection at -136° 26 was step-scanned using three azimuthal angles ((j) = 9, 45, and 99°) and seven tilt angles, ( / = 0, +28.2, 42, and 55 °)... 

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