Algebraic structure for the crossing of balanced and stair nested designs

Stair nesting allows us to work with fewer observations than the most usual form of nesting, the balanced nesting. In the case of stair nesting the amount of information for the different factors is more evenly distributed. This new design leads to greater economy, because we can work with fewer observations. In this work we present the algebraic structure of the cross of balanced nested and stair nested designs, using binary operations on commutative Jordan algebras. This new cross requires fewer observations than the usual cross balanced nested designs and it is easy to carry out inference.

analytical model and measurements of the target erosion depth profile of balanced and unbalanced planar magnetron cathodes

The erosion depth profile of planar targets in balanced and unbalanced magnetron cathodes with cylindrical symmetry is measured along the target radius. The magnetic fields have rotational symmetry. The horizontal and vertical components of the magnetic field B are measured at points above the cathode target with z = 2 x 10(-3) m. The experimental data reveal that the target erosion depth profile is a function of the angle. made by B with a horizontal line defined by z = 2 x 10(-3) m. To explain this dependence a simplified model of the discharge is developed. In the scope of the model, the pathway lengths of the secondary electrons in the pre-sheath region are calculated by analytical integration of the Lorentz differential equations. Weighting these lengths by using the distribution law of the mean free path of the secondary electrons, we estimate the densities of the ionizing events over the cathode and the relative flux of the sputtered atoms. The expression so deduced correlates for the first time the erosion depth profile of the target with the angle theta. The model shows reasonably good fittings to the experimental target erosion depth profiles confirming that ionization occurs mainly in the pre-sheath zone.

Crossing balanced and stair nested desings

Balanced nesting is the most usual form of nesting and originates, when used singly or with crossing of such sub-models, orthogonal models. In balanced nesting we are forced to divide repeatedly the plots and we have few degrees of freedom for the first levels. If we apply stair nesting we will have plots all of the same size rendering the designs easier to apply. The stair nested designs are a valid alternative for the balanced nested designs because we can work with fewer observations, the amount of information for the different factors is more evenly distributed and we obtain good results. The inference for models with balanced nesting is already well studied. For models with stair nesting it is easy to carry out inference because it is very similar to that for balanced nesting. Furthermore stair nested designs being unbalanced have an orthogonal structure. Other alternative to the balanced nesting is the staggered nesting that is the most popular unbalanced nested design which also has the advantage of requiring fewer observations. However staggered nested designs are not orthogonal, unlike the stair nested designs. In this work we start with the algebraic structure of the balanced, the stair and the staggered nested designs and we finish with the structure of the cross between balanced and stair nested designs.

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