The exit waves

Therefore, if 0 (f) is the exit wave-function, the image wave-function i/ (f) taking into account the microscope aberrations is given by ... 

The reliability of the exit wave reconstruction process was independently confirmed by making available to the scientific community a reconstructed focal series from the OAM. L. J. Allen et al. demonstrated striking agreement for the identical data set with an iterative method for exit wave function reconstruction. 

Figure 2. Proof of 0.78A resolution in the phase of the exit wave from silicon [112] by removing (444) image Fourier coefficients.

Figure 6 shows a part of the exit wave shown in Figure 5 but after applying three fold symmetry. The white dots in this image represent atom columns or groups of atom columns. One can observe several (almost) round white dots and several elongated ones. Since the c axis is short (0.396 nm)... 

Given the thiekness dependenee of the eontrast of the atom eolumns in the exit waves as shown in Figure 5, the eontrast of the dots in the exit waves eannot be used for a assignment of Ce, Cu or P to the various dots in the exit wave. Therefore the MSLS refinement was started by putting Cu atoms at all sites. First the seale factor, erystal (mis)orientation and thickness for each data set were refined. Nine data sets were taken to have a range of thieknesses. All subsequent refinements were done with these nine data sets simultaneously. Next a refinement was done of the atom positions but keeping the Debye Waller faetors (B) of all atoms at zero and the occupancy at 1. This resulted in an overall R value of 10.5 %. 

A kinematic refinement starting with the atoms at the positions estimated from the exit wave or at the positions given in Table 2 does not lead to R... 

In this approximation, the exit wave function can be directly interpreted in terms of the projected potential of the specimen and the imaginary part exhibits sharp maxima at the position of the atomic columns. Unfortunately, most practical specimens do not satisfy the POA or WPOA and, thus, the electron wave function is more difficult to understand. Nevertheless, we are going to work with the WPOA in this paper to simplify the explanation of the imaging process. 

Several other approaches have been followed towards quantitative HRTEM imaging. One approach is the development of new hardware to correct for or alleviate some of the aberrations in the image, e.g. spherical aberration corrector (Rose 1990, Haider et al. 1995) and three-fold astigmatism corrector (Overwijk et al. 1997). An alternative approach is the development of new methods to retrieve the exit wave function, e.g. off-axis holography (Eichte 1986, Lichte and Rau 1994) and focal-series reconstruction (FSR) (Coene et al. 1992, 1996, Thust et al. 1996a). While each approach has its distinct advantages, we are only going to discuss focal-series reconstruction in this paper. 

The reconstruction of the exit wave occurs in four steps. It starts with an alignment step where the images of the focal-series are aligned with respect to each other. In the second step, an analytical inversion of the linear imaging problem is achieved by using the paraboloid method (PAM) to generate a first approximation to the exit wave function. This approximated exit wave function is then refined in the third step by a maximum likelihood (MAE) approach that accounts for the non-linear image contributions. Finally, the exit wave is corrected for residual aberrations of the microscope. 

With the microscope aberrations eliminated, the exit wave function can be directly understood only in terms of the electron beam - specimen interaction. 

The result of the PAM reconstruction is, in general, only an approximation of the exit wave function. Some non-linear terms may be present exactly on the paraboloid surfaces, and, thus result in artifacts for the PAM reconstruction. However, the PAM result is a good approximation to the exit wave function, which, in the present implementation, is used as a starting point for a maximum likelihood (MAL) reconstruction that takes the non-linear image contributions fully into account (Coene et al. 1996, Thust etal. 1996a). 

In the MAL approach, the approximated exit wave function, F, obtained in the previous iteration step is used to simulate the images of the focal-series. These simulated images are quantitatively compared with the original HRTEM images and a correction to the exit wave function, d is calculated to minimize the difference between the experimental data and the simulation. The corrected exit wave function is then used as the basis to simulate the images of the focal series and the whole process is repeated iteratively until the difference between simulation and experiment is sufficiently small (Figure 8). 

With the paraboloid method followed by the maximum-likelihood refinement of the exit-wave function, the inherent effects of the microscope on the exit wave function due to spherical aberration and defocus are eliminated resulting in a complex-valued wave function with the delocalization removed. However, the electron wave function frequently suffers from residual aberrations due to insufficient microscope alignment. In a single image, it is not possible to remove these aberrations, but, with the reconstructed complex wave function, one can use a numerical phase plate to compensate the effect of aberrations by applying appropriate phase shifts (Thustetal. 1996b). 

The only practical difficulty comes from determining the appropriate aberration correction. In Truelmage, this has been automated, e.g. for the defocus and the 2-fold astigmatism. For a weak phase object, the real part of exit wave function should be constant (Equation 2). Thus, the defocus and 2-fold astigmatism can be determined by minimizing the contrast of the real part of the exit-wave function. 

Figure 10. Reconstructed exit wave function (phase) of the same area as shown in Figure 5. The left image shows the phase of the exit wave function before aberration correction and the right image shows the phase after correction for residual 2-, 3-fold astigmatism, and coma. The difference clearly illustrates that numerical correction of residual aberrations is crucial for...

In a single HRTEM image, it is difficult to measure atomic positions with a high accuracy especially next to an interface due to contrast delocalization. However, the delocalization is removed in the exit wave function and the maxima in the reconstructed phase directly correspond to the center of the atomic positions. The atomic positions can, therefore, be measured with a... 

As described above, when all aberrations are corrected, the contrast of an image is dominated by the exit wave function. If the sample is thin, the exit wave function is mainly determined by the diffraction channeling effect. When electrons travel though an atomic column (along a zraie axis), each atom behaves like a small lens. Different atoms have different focus power and thus electrons travel along the colunm with different beating. This beating is characterized by extinction distance D, which is inversely proportimial to Z [27]. 

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