Established methods for measurement of lattice spacings and angles of crystalline materials include x-ray diffraction, microdiffraction and HREM imaging. Structural information from HREM images is normally obtained off-line with the traveling table microscope or by the optical diffractogram technique. We present a new method for precise measurement of lattice vectors from HREM images using an on-line computer connected to the electron microscope. It has already been established that an image of crystalline material can be represented by a finite number of sinusoids. The amplitude and the phase of these sinusoids are affected by the microscope transfer characteristics, which are strongly influenced by the settings of defocus, astigmatism and beam alignment. However, the frequency of each sinusoid is solely a function of overall magnification and periodicities present in the specimen. After proper calibration of the overall magnification, lattice vectors can be measured unambiguously from HREM images.Measurement of lattice vectors is a statistical parameter estimation problem which is similar to amplitude, phase and frequency estimation of sinusoids in 1-dimensional signals as encountered, for example, in radar, sonar and telecommunications. It is important to properly model the observations, the systematic errors and the non-systematic errors. The observations are modelled as a sum of (2-dimensional) sinusoids. In the present study the components of the frequency vector of the sinusoids are the only parameters of interest. Non-systematic errors in recorded electron images are described as white Gaussian noise. The most important systematic error is geometric distortion. Lattice vectors are measured using a two step procedure. First a coarse search is obtained using a Fast Fourier Transform on an image section of interest. Prior to Fourier transformation the image section is multiplied with a window, which gradually falls off to zero at the edges. The user indicates interactively the periodicities of interest by selecting spots in the digital diffractogram. A fine search for each selected frequency is implemented using a bilinear interpolation, which is dependent on the window function. It is possible to refine the estimation even further using a non-linear least squares estimation. The first two steps provide the proper starting values for the numerical minimization (e.g. Gauss-Newton). This third step increases the precision with 30% to the highest theoretically attainable (Cramer and Rao Lower Bound). In the present studies we use a Gatan 622 TV camera attached to the JEM 4000EX electron microscope. Image analysis is implemented on a Micro VAX II computer equipped with a powerful array processor and real time image processing hardware. The typical precision, as defined by the standard deviation of the distribution of measurement errors, is found to be <0.003Å measured on single crystal silicon and <0.02Å measured on small (10-30Å) specimen areas. These values are ×10 times larger than predicted by theory. Furthermore, the measured precision is observed to be independent on signal-to-noise ratio (determined by the number of averaged TV frames). Obviously, the precision is restricted by geometric distortion mainly caused by the TV camera. For this reason, we are replacing the Gatan 622 TV camera with a modern high-grade CCD-based camera system. Such a system not only has negligible geometric distortion, but also high dynamic range (>10,000) and high resolution (1024x1024 pixels). The geometric distortion of the projector lenses can be measured, and corrected through re-sampling of the digitized image.
Balancing of Coupled Piezoelectric NEMS Resonators
