Experimental variables

In this chapter, a brief introductory discussion is given of (i) the experimental variables that influence the nature of HRTEM images, (ii) the experimental techniques involved, and (iii) the basis of image interpretation. More detailed accounts have been given by Spence (1981) and Veb-len (1985a). 

1 Crystal orientation and thickness. The way in which changes of orientation and crystal thickness influence the nature of the image can 

2 Defect of focus. In the Gaussian image plane, defined by Eq. (1.1), the image has very low contrast. The nature of the image changes above and below this plane because the waves have different relative phases. When the objective lens is defocused by an amount A/ relative 

4 Phase contrast transfer function. The phase shifts due to the combination of spherical aberration and defect of focus can be combined into a single phase factor x given by 

Sin X is called the phase contrast transfer function (PCTF) of the objective lens. 

---

The capillary rise method is generally considered to be the most accurate means to measure 7, partly because the theory has been worked out with considerable exactitude and partly because the experimental variables can be closely controlled. This is to some extent a historical accident, and other methods now rival or surpass the capillary rise one in value. 

By proper design of experiments, guided by a statistical approach, the effects of experimental variables may be found more efficiently than by the traditional approach of holding all variables constant but one and systematically investigating each variable in turn. Trends in data may be sought to track down nonrandom sources of error. 

Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. ... 

A formal mathematical analysis of the flow in the concentric cylinder viscometer yields the following relationship between the experimental variables and the viscosity ... 

A variety of experimental techniques have been employed to research the material of this chapter, many of which we shall not even mention. For example, pressure as well as temperature has been used as an experimental variable to study volume effects. Dielectric constants, indices of refraction, and nuclear magnetic resonsance (NMR) spectra are used, as well as mechanical relaxations, to monitor the onset of the glassy state. X-ray, electron, and neutron diffraction are used to elucidate structure along with electron microscopy. It would take us too far afield to trace all these different techniques and the results obtained from each, so we restrict ourselves to discussing only a few types of experimental data. Our failure to mention all sources of data does not imply that these other techniques have not been employed to good advantage in the study of the topics contained herein. 

Although the extent of crystallinity is the variable under consideration, time is the experimental variable. Accordingly, what is done is to identify the specific volume of a sample at t = 0 (subscript 0) with V, the volume at t = °° (subscript °o) with and the volume at any intermediate time (subscript t) with the composite volume. On this basis, Eq. (4.39) becomes... 

POWDERS,HANDLING - DISPERSION OF PO XTERS IN LIQUIDS] Pol 19) -among experimental variables pESIGN OF EXPERIMENTS] pol 7)... 

Flavor and Aroma Transport. Many methods ate used to characterize the transport of flavor, aroma, and solvent molecules in polymers. Each has some value, and no one method is suitable for all situations. Any experiment should obtain the permeabiUty, the diffusion coefficient, and the solubihty coefficient. Furthermore, experimental variables might include the temperature, the humidity, the flavor concentration, and the effect of competing flavors. 

What is the most meaningful way to express the controllable or independent variables For example, should current density and time be taken as the experimental variables, or are time and the product of current density and time the real variables affecting response Judicious selection of the independent variables often reduces or eliminates interactions between variables, thereby leading to a simpler experiment and analysis. Also inter-relationships among variables need be recognized. For example, in an atomic absorption analysis, there are four possible variables air-flow rate, fuel-flow rate, gas-flow rate, and air/fuel ratio, but there are really only two independent variables. 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

Change of reaction conditions to minimize kinetic complications. For example, if two parallel reactions have substantially different activation energies, their relative rates will depend upon the temperature. The reaction solvent, pH, and concentrations are other experimental variables that may be manipulated for this purpose. 

Stage in the development of a practical and selective method involved a detailed investigation of the many experimental variables and a in depth study of the nature of the species responsible for the observed selectivity. 

For a reaction as complex as catalytic enantioselective cyclopropanation with zinc carbenoids, there are many experimental variables that influence the rate, yield and selectivity of the process. From an empirical point of view, it is important to identify the optimal combination of variables that affords the best results. From a mechanistic point of view, a great deal of valuable information can be gleaned from the response of a complex reaction system to changes in, inter alia, stoichiometry, addition order, solvent, temperature etc. Each of these features provides some insight into how the reagents and substrates interact with the catalyst or even what is the true nature of the catalytic species. 

See nomenclature for definition of symbols and units. The units presented are English engineering units, unless a conversion is required. The friction factor is the only experimental variable that must be determined by reference to the above equations and it is represented by Figure 2-3. Note that this may sometimes be referred to as the Fanning formula, and may be modified to )held a fric-... 

It has been proposed that evaluation of the resistance of materials, or the study of experimental variables, should be based on the results obtained for the attenuation zone. Other methods of assessment have been proposed by Hobbs , and by Plesset and Devine . 

Using a as an experimental variable, information concerning other reaction cross-sections will also become available. Some of the values of the cross-sections obtained by this technique are summarized in Table I, clearly demonstrating that much useful information can be obtained from the detailed studies of these simple discharge systems. 

From the foregoing discussion it may seem that a complex experimental design must be carried out before any analysis is attempted. While it is certainly a necessity/advantage that the analyst has some understanding of the effect that a parameter is likely to have on the experimental outcome, many analyses, particularly those involving mixtures containing relatively few components at relatively high concentrations, will be accomplished successfully on the basis of a simple study of selected experimental variables. 

As part of their method development, these same authors studied the effect of a number of experimental variables on the HPLC separation and the mass spectral quality and it is worthwhile considering their results, reproduced in Table 5.3, in some detail. 

Experimental design A number of formal procedures whereby the effect of experimental variables on the outcome of an experiment may be assessed. These may be used to assess the optimum conditions for an experiment and to maximize the accuracy and precision obtained. 

Factor An experimental variable that has (or may have) an effect on the outcome of an experiment, e.g. temperature, concentration of reactants, presence of a catalyst, etc. 

Factorial design One method of experimental design that allows interactions between factors to be investigated, i.e. whether changing one experimental variable changes the optimum value of another. 

The various independent variables can be the actual experimental variables or transformations of them. Dilferent transformations can be used for different variables. The independent variables need not be actually independent. For example, linear regression analysis can be used to fit a cubic equation by setting X, X and Z as the independent variables. 

The effectiveness of the chlorination of mbbers with TCI solutions strongly depends on several experimental variables. 

One-dimensional data are plotted versus an experimental variable a prime example is the Lambert-Beer plot of absorbance vs. concentration, as in a calibration run. The graph is expected to be a straight line over an appreciable range of the experimental variable. This is the classical domain of linear regression analysis. 

In ellipsometry two parameters are determined. These are A, the phase angle between the leading and trailing components in Fig. 27.24, and the ratio of the electric field amplitudes E and E, which defines the second parameter, /. IE I/IEJ = tan /. A and r may be recorded as functions of other experimental variables, such as potential and time. 

---