Zero shift error

The flf-spacings of the first and second order Bragg peaks should be related as 2 1 in the absence of sample displacement and/or zero shift errors. If this ratio is different, the related systematic error can be computed from the Braggs law (see Eqs. 2.39 to 2.44). 

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... 

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).

Considering the resultant unit cell dimensions, it appears that the zero shift error has the largest influence on the discussed experimental data, since the refined lattice parameter a = 4.1574 A) is the closest match with the standard value of a = 4.15695 A. 

The specimen displacement parameter usually varies from sample to sample and it usually takes up some of the effects of sample transparency. For a properly aligned goniometer, the zero shift error should be negligible. Even if the goniometer is misaligned, the zero shift correction should remain sample independent. 

Table 1 is condensed from Handbook 44. It Hsts the number of divisions allowed for each class, eg, a Class III scale must have between 100 and 1,200 divisions. Also, for each class it Hsts the acceptance tolerances appHcable to test load ranges expressed in divisions (d) for example, for test loads from 0 to 5,000 d, a Class II scale has an acceptance tolerance of 0.5 d. The least ambiguous way to specify the accuracy for an industrial or retail scale is to specify an accuracy class and the number of divisions, eg. Class III, 5,000 divisions. It must be noted that this is not the same as 1 part in 5,000, which is another method commonly used to specify accuracy eg, a Class III 5,000 d scale is allowed a tolerance which varies from 0.5 d at zero to 2.5 d at 5,000 divisions. CaHbration curves are typically plotted as in Figure 12, which shows a typical 5,000-division Class III scale. The error tunnel (stepped lines, top and bottom) is defined by the acceptance tolerances Hsted in Table 1. The three caHbration curves belong to the same scale tested at three different temperatures. Performance must remain within the error tunnel under the combined effect of nonlinearity, hysteresis, and temperature effect on span. Other specifications, including those for temperature effect on zero, nonrepeatabiHty, shift error, and creep may be found in Handbook 44 (5). The acceptance tolerances in Table 1 apply to new or reconditioned equipment tested within 30 days of being put into service. After that, maintenance tolerances apply they ate twice the values Hsted in Table 1.

Flow Low mass flow indicated. Mass flow error. Transmitter zero shift. Measurement is high. Measurement error. Liquid droplets in gas. Static pressure change in gas. Free water in fluid. Pulsation in flow. Non-standard pipe runs. Install demister upstream heat gas upstream of sensor. Add pressure recording pen. Mount transmitter above taps. Add process pulsation damper. Estimate limits of error. 

It is important to check the zero setting (or the setting of the lower range value) for an instrument as a zero error will cause the whole of the instrument span to be displaced. The zero setting may drift or change over a period of time (zero shift). Such drifting is frequently due to variations in ambient conditions—most commonly temperature. In addition to zero shift, point values of the measured variable in different regions of the span may drift by different amounts. 

Using all Bragg peaks which have been indexed and the associated observed Bragg angles, more accurate unit cell dimensions and, if applicable, systematic experimental errors, e.g. sample displacement, sample transparency, or zero shift, which are described in section 2.8.2, Chapter 2, should be refined by means of a least squares technique (see section 5.13, below). 

Experimental data from the LaNi4.85Sno.15 sample are especially useful for this illustration because as established earlier, this diffraction pattern has been successfully indexed manually. We also know that the data are affected by a small systematic error, which can be eliminated by introducing a zero shift correction of 0.032°, and that there are two low intensity Bragg peaks, which belong to an impurity phase (see sections 5.4 and 5.8). 

The last (fourth) parameter line should contain three numbers (with zeros representing the selection of default values) and their meaning is as follows. The first value indicates how the errors in the experimental data are handled. By default, the measurement errors are assumed constant at 0.03° or 20. The second value specifies the minimum acceptable M v figure of merit, where N is the number of used Bragg peaks, i.e. it is identical to the first number in the first parameter line, and its default is 5. The last number in this line is the value of the zero shift to be added to the observed experimental data. 

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. 

However, if one compares the values of the lattice parameter obtained when a different kind of a systematic error was assumed and accounted for in the data, the difference between the two is statistically significant (4.1583 vs. 4.1574 A for sample displacement and zero shift effects, respectively). This is expected given the different contribution from different errors as seen in Figure 5.19. Usually, both effects are present in experimental data. The refinement of two contributions simultaneously is, however, not feasible due to strong correlations between sample displacement and zero shift parameters as shown in Figure 5.21. 

In this example, lattice parameters and the zero shift correction have a substantial impact on the quality of the fit and the weighted profile residual, Rwp, decreases nearly two-fold (from 24 to 12 %), while the refinement of peak shape parameters decreases R p by only 4 %. Therefore, in this case lattice parameters should have been refined first. However, it is not always obvious which parameters are more important and should be released at a particular stage of the least squares refinement. Because of this, in complex cases a trial-and-error approach is often employed. ... 

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