Weak phase object

In HRTEM, very thin samples can be treated as weak-phase objects (WPOs) whereby the image intensity can be correlated with the projected electrostatic potential of crystals, leading to atomic structural information. Furthermore, the detection of electron-stimulated XRE in the electron microscope (energy dispersive X-ray spectroscopy, or EDX, discussed in the following sections) permits simultaneous determination of chemical compositions of catalysts to the sub-nanometer level. Both the surface and bulk structures of catalysts can be investigated. 

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). 

Here we take U = 2meV(f) , which is treated as constant within the small volume. This is known as the weak-phase-object approximation. For a parallel beam of incident electrons, the incident wave is described by the plane wave exp n For high-energy electrons with E V, the scattering by the atom is weak and we have approximately ... 

It is well known that under the weak-phase-object approximation (WPOA) [19], the image intensity function is linear to the convolution of the projected potential distribution function cpt (x, y) and the inverse Fourier transform (FT) of the contrast transfer function (CTF) r(u) of the electron microscope ... 

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. 

For a weak-phase-object, the Fourier transform of the HREM image lim(u) is related to the structure factor F(u) by ... 

For a very thin object that is acting as a weak scatterer, the phase modulation in the exit plane is small, i.e. aVpt l. In this weak-phase object approximation (WPOA), equation I can be further simplified to... 

Equation 10 can be interpreted as the aberrations of the objective lens multiplying the intensities of the diffracted beams by a phase factor sin[2(g)], which depends on the spatial frequency. Thus, in the WPOA, the observed image is proportional to the projected potential, but is modulated by the phase factor. Without the phase shift, j, due to the lens aberrations, a weak phase object would not be visible in HRTEM (this is analogous to the interpretation of equation 6). 

Within the weak-phase object approximation, the effect of the aberrations is most conveniently described by the Contrast Transfer Function (CTF), which gives the phase factor as a function of spatial frequency (diffraction angle). 

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. 

Structural information as well as subterranean structures of solids. The surface-related properties of materials can therefore be better understood [46], There are several other advantages of surface investigation by HRTEM. For example, specimen preparation is simple. Normally, small particles with any size and any morphology can be directly used. Multiple scattering can normally be ignored, since the surface areas are often very thin and can legitimately be treated as weak phase objects, where the image intensity indicates the projected electrostatic potential. 

Therefore, a structure image can be obtained from a weak phase object first by setting the defocus value at and then by cutting out those diffracted waves whose spatial frequencies are larger than u. These observation conditions derived from eq. (1.13) are called the Scherzer imaging conditions and the defocus value is called the Scherzer focus. 

The weak phase object approximation (eq. (1.6)) is satisfied when Fp(= FoAz) <1.2 X 10 VA for E = 200 kV. This means Az < 120 40 A since Fq (mean inner potential) = 10 30 V for most inorganic crystals. In reality, however, structure images are obtained mostly for thickness between 15 and 50 A, depending on the material, crystal structure and orientation, and accelerating voltage. That is to say, the condition for the weak phase object is not satisfied in most actual cases. 

In order to overcome this contradiction we extend the theory of weak phase object approximation to slightly thicker crystals as follows from dynamical calculations on the amplitude and phase of many waves we note the existence of such a relation as... 

When the object is not thin enough, a strong scattered beam may act as a secondary incident beam. Oscillations would then be produced in the scattered amphtude this is a dynamic effect. At the hmit, the transmitted beam dis ears and the incident beam energy is entirely spent in scattering. The corresponding thickness of the object is the extinction distance. To avoid dynamic effects [18,19], the object thickness must be much smaller than the extinction distance. For gr hite [18] at 200 kV, the thickness must be smaller than 50 A to keep 90% of the incident beam in the transmitted one (40 A at 100 kV [19]) it must be inferior to 20 A to keep 95%. In the case of graphene, because it behaves as a single layer (t < 5 A), it is a weak phase object. 

Ideally, for a weak-phase object the phases of the elastically scattered beams would fix up the image through the object s coherent CTF which is supposed to be proportional to the specimen projected potential Fp(p). This is impossible, because the microscope has its own coherent CTF, h(p), which is the impulse response of the TEM (point spread of the microscope) in turn. h(p) is dne to illumination... 

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