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This thesis presents developments in both the experimental techniques and in the theoretical understanding of the helium atom scattering. These contributions open the way for more effective experimental investigations and more quantitative analysis of helium - surface scattering data. The introduction starts with an outline of the contents of the thesis, followed by a description of the diffraction geometry of the Cambridge helium scattering apparatus and a method of measuring diffraction intensities at constant incident condition. The next two chapters describe two experimental improvements in the field of helium scattering. In chapter 2 the characterisation of a molecular beam source shows that high performance (velocity spread < 1%) can be achieved with a simple and inexpensive design. Chapter 3 describes the development, testing and characterisation of a 2-dimensional multichannel scalar data acquisition device for use with two experimental methods under development, namely an imaging helium detector and time-of-flight measurements using a double chopper gate helium beam. Chapter 4 describes the investigation of helium scattering from a lattice gas of Pb adsorbed on the Cu(001) surface. Experimental helium diffraction intensities for the Pb/Cu(001) system were previously accounted for by invoking an adsorbate - induced corrugation in the Cu(001) substrate. In this work, a more detailed analysis shows that a substrate corrugation is not necessary to reproduce the experimental results. The investigation includes a review of a range of approximations and methods available for calculating scattered intensities within the hard wall approximation of the helium-surface potential. The remainder of the thesis presents a detailed experimental and theoretical investigation of elastic helium scattering from the Si(111)-(1x1)H surface.

Equality constraints are dealt with by including them directly in the inner optimization problem of the MBF method. Exact Hessian and gradient information is used throughout all implementations. The MBF, as implemented, consists of a two-stage approach: an outer cycle where the Lagrange multipliers for simple bound constraints of the variables are updated and an inner cycle, where the resulting equality-only constrained nonlinear optimization problem is solved. At present, inequalities in the problem are converted to equalities with the addition of bounded slack variables, the subsequently solved as such. In addition, sparsity is exploited in the overall problem Jacobians. The advantages of the MBF method are demonstrated with test cases coming from the standard literature of process systems engineering. It is also observed that the solutions are attained in few iterations and function evaluations. This work also presents a rigorous approach for the solution of inequality path constrained optimal control problems (OCPs). The scheme uses the MBF method to derive a globally smooth transformation for the inequality path constraint which generates the Euler-Lagrange multiplier trajectories iteratively. It also introduces into discretization methods a novel scheme which is e-convergent with respect to satisfying the path constraints globally. This results in a derived sequence of OCPs that will converge to a prescribed accuracy within a finite number of iterations. Another important development is the initialization of the Lagrange multipliers using only a sparsely discretized path constraint over a time horizon. The method is shown to be effective and promising for future applications in both discretization/collocation and control vector parameterization implementations.