Obtaining an Image

The matrix X defines a pattern P" of n points, e.g. x, in which are projected perpendicularly upon the axis v. The result, however, is a point s in the dual space S". This can be understood as follows. The matrix X is of dimension nxp and the vector V has dimensions p. The dimension of the product s is thus equal to n. This means that s can be represented as a point in S". The net result of the operation is that the axis v in 5 is imaged by the matrix X as a point s in the dual space 5". For every axis v in 5 we will obtain an image s formed by X in the dual space. In this context, we use the word image when we refer to an operation by which a point or axis is transported into another space. The word projection is reserved for operations which map points or axes in the same space [11]. The imaging of v in S into s in S" is represented geometrically in Fig. 29.9a. Note that the patterns of points P" and P are represented schematically by elliptic envelopes. 

It is the steep variation of the tunneling current with distance that enables one to obtain an image of the atoms in the surface when the tip is rastered over the surface. The common way to measure STM images is to record the position of the tip from the surface necessary for keeping the tunneling current constant. 

We wish to obtain an image of the scattering elements in three dimensions (the electron density). To do this, we perform a 3-D Fourier synthesis (summation). Fourier series are used because they can be applied to a regular periodic function crystals are regular periodic distributions of atoms. The Fourier synthesis is given in O Eq. 22.2 ... 

The relation between an HREM image and the projected crystal potential is quite complex if the crystal is thick. To obtain an image which can be directly interpreted in terms of projected potential, crystals have to be well aligned, thin enough to be close to weak-phase-objects and the defocus value for the objective lens should be optimal, i.e. at the Scherzer defocus. 

Appropriate electrophoresis systems in various sizes are available from numerous companies. More expensive are the imaging devices, usually consisting of a UV transilluminator and a CC-camera attached to a computer. There are several types of digital image analysers commercially available. Alternatively, one may use ordinary photography (use a red filter and UV-transilluminator) to obtain an image of the gel. 

From crystallography, we obtain an image of the electron clouds that surround the molecules in the average unit cell in the crystal. We hope this image will allow us to locate all atoms in the unit cell. The location of an atom is usually given by a set of three-dimensional Cartesian coordinates, x, y, and z. One of the vertices (a lattice point or any other convenient point) is used as the origin of the unit cell s coordinate system and is assigned the coordinates x = 0, y = 0, and z = 0, usually written (0,0,0). See Fig. 2.4. 

Researchers used an STM to obtain an image of the HB-DC monolayer. The image essentially matched what they had expected HB-DC molecules were arranged in a crystal-like pattern in which every molecule was held in place by other molecules surrounding it. Because of their shape, the molecules looked like a collection of hexagonal tiles neatly laid out on the copper surface. 

Obtaining an image of the first layer deposited on a foreign substrate can be rightly considered to be not completely representative of bulk crystallization. However, this is only the first layer of a thin polymer film, the structure of which can be investigated by electron diffraction. The two techniques are indeed very complementary. AFM probes the first layer, whereas electron diffraction determines the structure of the thin film as a whole - the structure of the film interior. 

In order to obtain an image of the surface of the sample, the STM tip slowly scans across the surface at a distance of only an atom s diameter. This is represented in Fig. 6. 

---