The electron wavelength

When an electron of rest mass and charge e is accelerated through a potential difference V, the kinetic energy it acquires is 

Inserting the appropriate values of the constants gives an equation that is a good approximation to Eq. (2.4), 

However, these equations are valid only if the velocity v is small compared with the velocity of light c. This will be true for 10,000 volts, but for the higher voltages commonly used in electron microscopy (50 to 1,000 kV), /Wg must be replaced by /Wg/(1 — Values of X for some 

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The interaction will become weaker as the electron wavelength becomes greater or less than the molecular diameter with a consequential decrease in the cross section. This leads to an expression for the ionization probability as a function of electron energy, giving the shape of the ionization efficiency curve,... 

Figure 6.7 The principle of LEED is that a beam of monoenergetic electrons scatters elastically from a surface. Because of the periodic order of the surface atoms, electrons show constructive interference in directions for which the path lengths of the electrons differ by an integral number times the electron wavelength. Directions of constructive interference are made visible by collecting the scattered...

Equation (2) describes the effect of defocus f and spherical aberration Cs on the object function and X is the electron wavelength. Higher order aberrations can be considered. An attenuation of the electron wave is taken into account by a spatial damping envelope... 

Here, AT is a constant, f is the incoming intensity, R is the distance of the scattered wave from the molecule (in practical terms, it is the distance between the scattering center and the point of observation), i and j are the labels of atoms in the jV-atomic molecule, g contains the electron scattering amplitudes and phases of atoms, 5 is a simple function of the scattering angle and the electron wavelength, I is the mean vibrational amplitude of a pair of nuclei, r is the intemuclear distance r is the equilibrium intemuclear distance and is an effective intemuclear distance), and k is an asymmetry parameter related to anharmonicity of the vibration of a pair of nuclei. 

The image contrast which is predicted by the pseudo-WPOA is in agreement with that given by multislice simulation with the crystal thickness below and around the critical value [22, 23]. The critical thickness of a crystal depends on the electron wavelength and the t5 es of the atoms that constitute the crystal. 

The point resolution of a TEM, which only depends on the spherical aberration, C, and the electron wavelength. A, (which is determined by the accelerating voltage) sets the limit for a straightforward interpretation of a HRTEM image of a thin object. However, this is different from the information limit, which defines the highest frequencies that can be transferred in a microscope. 

In CBED, zone-axis patterns (ZAP) can be recorded near the relevant zone axis and the pattern may also include a higher-order Laue zone (referred to as a HOLZ). The radius of the first HOLZ ring G is related to the periodicity along the zone axis [c] and the electron wavelength, by = 2/kc. CBED can thus provide reciprocal space data in all three (x,y,z) dimensions, typically with a lateral resolution of a few nanometres. As in any application, corroborative evidence from other methods such as HRTEM and single-crystal x-ray diffraction, where possible, can be productive in an unambiguous structural determination of complex and defective materials such as catalysts. We illustrate some examples in later sections. 

The independent variable s = sind 6, where X is the electron wavelength and 20 the scattering angle. The summation indices i and j refer to each of the M atoms in the molecule. The index pair k and 1 refers to a representative atom pair of the molecule studied, chosen to obtain a convenient form for I(s) and its Fourier transformed partner. fj(s) is the complex scattering amplitude of the i-th atom in the molecule and rji(s) is the argument of fj(s), /. e. 

The results presented above indicate a non-classical scattering process which can in principle result from two different effects. One is related simply to the fact that thin layers exist with their thickness comparable to the electrons wavelength, hence interferences in the transmission result in what is known as shape resonances . These resonances are very sensitive to the thickness of the layer, but only weakly depend on the electronic properties of... 

The interference effects resulting from electron propagation are responsible for providing the desired structural information in all the techniques discussed here. The extraction of structural information, therefore, requires a knowledge of the electron wavelengths and of any phase shifts that may occur in electron emission and electron-atom scattering. Failure to understand these processes can result in quite erroneous results. 

This leads to a local Fermi wave number, ky(r), given by / k. (r) /2m = L i.(r), and from Lq. (15-4), a local electron density N(r) =/<. (r) /(37c ). This is called the Fermi-Thomas approximation. The essential assumption that is required is that the potential does not vary greatly over the distance corresponding to the electron wavelength. 

Where L is the experimental camera length (CL) and k is the electron wavelength determined by the electron accelerating voltage, O, in volts ... 

The overall resolution in HRTEM is governed partly by the electron wavelength and partly by the optical characteristics of the objective lens. The most important effect of the latter arises from spherical aberration. This aberration introduces a phase difference into the individual diffracted beams and when the real image is synthesized by the lens from these diffracted beams this can give rise to considerable confusion in the image contrast. 

Figure 2. (a) Calculated dilTraction functions for several cluster models and a given number of atoms N. Lower curve represents Ar atomic scattering s = (4n/X)sin(6/2) is the diffraction parameter with 6 the diffraction angle and X the electron wavelength, (b) Densitometer traces of photographic plates recorded for several Ar inlet pressures. 

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