Image wave function

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

Further degradation of the information encoded in the electron beam takes place in the recording step since the signal is proportional to the square modulus of the image wave-function, i.e. neglecting small second order terms ... 

For pure phase objects, the observed contrast results from the aberration function. For uncorrected HRTEM imaging, if the value of defocus Af equals to the Scherzer defocus, the aberration function x(u) jr/2 over as large a range Au of u. In this case, if the sample is sufficiently thin to be a weak phase object and if the phase is proportional to the projected potential, a imiform phase shift for all u (except u = 0) results in an image wave function in which the contrast is linearly proportional to the projected potential. 

Imura, K. and Okamoto, H. (2008) Development of novel near-field microspectroscopy and imaging of local excitations and wave functions of nanomaterials. Bull. Chem. Soc.Jpn., 81, 659-675. 

The purposes of this paper are to discuss the CO effect on the i (r)-function and to show that the appearance of the atom-like image can be explained by using the fact that the 5(r)-function can be described in terms of the autocorrelation function among the electron wave functions over the occupied electronic states. Autocorrelative overlap between the CO terms explicitly containing the atomic information has a possibility to enhance a specific atom on the i (r)-function map. The overlap explains why images of the other distant atoms are not pronounced. 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

The imaging process is influenced by interactions between the wave functions of the substrate and tip that result in significant deviation from the idealized square barrier profile. This is particularly true at small tip-substrate separations, where the barrier collapses below the vacuum level, as indicated in Fig. 8 for the Al-Al junction [56,64]. These calculations indicate that for typical STM imaging conditions, the top of the barrier... 

FIG. 10. Theoretical calculations reveal that in the case of adsorption of Xe on Ni the resonance associated with Xe(6s) state is broadened significantly with a long tail that extends to the Ni Fermi level. STM images are determined by the LDOS at the Fermi level. Although the contribution of Xe to the LDOS is small, it significantly extends the spatial distribution of the electronic wave function further away from the surface thereby acting as the central channel for quantum transmission to the probe tip. (From Ref. 71.)... 

Modem electron microscopes with field emission electron sources provide brighter and more coherent electrons. Images with information of crystal stmctures up to 1 A can be achieved. A through-focus exit wave reconstmction method was developed by Coene et al. (1992 1996) to retrieve the complete exit wave function of electrons at the exit surface of the crystal. This method can be applied to thicker crystals which can not be treated as weak-phase object. It is especially useful for stud5dng defects and interfaces (Zandbergen etal, 1999). 

One such approach to counter the effects of spherical aberration is to record a through-focus series, which allows reconstruction of the electron exit-wave function and thereby removal of the spherical aberration. The basic idea of this reconstruction approach, as implemented in the Truelmage software package, will be described after a simplified introduction to HRTEM imaging theory (for more details see Williams and Carter 1996, Reimer 1984, Spence 1988). Furthermore, two application examples are shown to discuss the benefits of focal-series reconstruction and illustrate the information that can be obtained. 

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. 

Figure 7. The linear image contributions of a focal-series are located on the surface of two paraboloids obtained by 3D Fourier transformation of the focal series. The two paraboloids correspond to the electron wave function and its complex conjugate.

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). 

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... 

During the second step in the image formation, which is described by the inverse Fourier transform (3 the electron beam undergoes a phase shift x(g) with respect to the central beam. The phase shift is caused by spherical aberration and defocus and damped by incoherent damping function D(a,A,g), so that the wave function /im(R) at the image plane is finally given by... 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

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