Order in two-dimensional projections of thin amorphous three-dimensional structures

Two-dimensional (2D) projections underlie the technique of high resolution electron microscopy. Projections of amorphous structures are generally considered uninterpretable due to structural disorder. We have derived the relationships between the distribution functions of an amorphous three-dimensional (3D) structure and those of the corresponding 2D projection. We have confirmed the relationships using calculations on models of amorphous structures with thickness in the range from 10.6 to 43.3 Angstrom. For example the pair and triplet distribution functions of the 2D projections show peaks corresponding to nearest neighbour distances and bond angles (respectively). The degree of order in the 2D projection decreases as thickness increases. Similar results are expected for higher order correlation functions. Thus it has been demonstrated that 2D projections of amorphous 3D structures contain a great deal of valuable structural information.