Elastic scattering within the specimen

As the electrons enter the specimen they will interact with the electrostatic potential of the specimen lattice, resulting in elastic scattering of the incident electrons. This kind of scattering is associated with only a negligible loss of energy of the incident electron beam. The other kind of scattering, inelastic scattering, where there is a significant transfer of energy between the involved particles is giving another kind of information, and is treated later. The TEM has four different modes of imaging:

• Diffraction mode

• Amplitude contrast imaging (BD/DF mode)

• High resolution mode (HRTEM)

• STEM mode

In diffraction mode an image of the back-focal plane of the objective lens is produced on the imaging plane. In the case that the incident electron beam is illuminating the specimen parallel, the image produced will consist of a pattern of spots. Each such a spot corresponds to a beam that is refracted from a plane that (nearly) satisfies the Bragg condition:

n=2dsin_B (3.5)

where n is an integer, d is the spacing of the refracting lattice planes and θB is the refraction angle (know as Bragg's angle). The diffraction pattern is defined by the intersection of the reciprocal lattice and the Ewald sphere, which can be shown to be a geometrical reconstruction of the Bragg conditions.

Amplitude contrast imaging can be done by either selecting the direct beam or one of the diffracted beams by the use of an objective aperture. In the former case it is called bright field (BF) imaging, in the latter case dark field (DF) imaging. The names originate from the fact that in DF imaging, holes in the foil appear black and the picture is generally darker than for BF imaging. These two imaging techniques can be used to image strain fields within crystals. Strain is defined by the displacement of an atom from its position that would be expected from the ordinary periodicity of the crystal. Causes of such a strain field can be dislocations, planar defects or other imperfections in the crystal lattice.

The distortion of the lattice around a dislocation core may alter the lattice in such a way that diffracted beams that are in the undistorted lattice not excited, become exited in the distorted lattice. The displacement field around a dislocation core for an isotropic solid is given by (Williams and Carter 2009):

R= 1

2



^b41−¹ {b_eb×u21−2lnrcos2}



^(3.6)

where be is the edge component of the burgers vector, ν Poisson's ratio and r and φ are the spherical coordinates along the dislocation line. The Howie-Whelan (HW) equations, the equations that describe the intensity of the direct and diffracted beam and thus the contrast in amplitude contrast images, can be modified to include a lattice distortion (Williams and Carter 2009):

d_g dz =i

₀_gi

_g exp

[

^{−2iszg⋅R}

]

(3.7)

where g is the diffracted beam used to image. The only difference to the HW-equations for an undistorted lattice is the additional g∙R term, and thus only a change in contrast will be observed when the scalar product of the diffraction vector and displacement field will be non-zero. If the displacement field is the one corresponding to a pure screw dislocation R reduces to R = bφ / 2π. The dislocation for a screw dislocation will thus not be visible is the Burgers vector is perpendicular to the diffracted beam. In the case of a pure edge dislocation we need the complete form of 3.6, so the invisibility criteria become g∙b = 0 and g∙(b x u) = 0. The last term can be ascribed to the buckling of lattice planes perpendicular to both the Burgers vector and dislocation line (Williams and Carter 2009). Using dark field imaging, one can thus determine the Burgers vector and slip system of the studied phase.

Another type of defects are planar defects. Examples are stacking faults, anti-phase domain boundaries and grain boundaries. Again one can use a displacement vector R to describe the translation of the lattice over the planar defect, a point rn corresponds on the other side of the planar fault to the point rn' = rn + R. Since R on either side of the planar defect is independent of z, equation 3.7 comes down to adding a phase term to Howie-Whelan equations for the diffracted beam, which can be written as e^iα; where α = 2πg∙R. Again, no change will be visible when α = 0. When α ≠ 0, there will be a change in contrast depending on the depth of the planar defect. In the case of an inclined planar defect, light and dark fringes will be visible, since one needs to integrate 3.7 over the amount of foil under the planar defect. Similarly to dislocations, one can determine the

displacement vector by a more detailed investigation of the fringes with different g∙R conditions.

Bright field, dark field and weak beam dark field imaging are all amplitude contrast imaging techniques, which is one of the two main imaging modes of the TEM, as already explained. The other imaging mode is phase contrast imaging, where the contrast in the image is caused by interference of multiple beams. To do so, one uses an aperture if the back focal plane of the objective lens to let some of the diffracted beams pass through and block the others. When one selects a 2D array of diffracted beams in the objective aperture, a phase contrast image will be formed with a two dimensional structure. The image formed in HRTEM images can be

expressed as the product of several contributions. The first one being the specimen function describing how the specimen interacts with the incident electron beam and what wave function of the electron beam is after they emerge from the specimen. The other contributions arise because of the electron optical system and its main components are those from the apertures, attenuation of the wave in the lens and aberration of the lens from a perfect lens (Williams and Carter 2009). The image one observes in the end can be written as a convolution of these contributions:

gr=

∫

^{fr'hr−r'dr'=fr∗hr (3.8)}

where f is the specimen function and h is the function describing the electron optical system, the * denotes the convolution f with h. The convolution can be written as the product of the Fourier transform of the functions that are convolved together. If we break down h into the contributions mentioned above one can write:

Gu=AuEuBuFu (3.9)

where A is the Fourier transform of the aperture function, E the Fourier transform of the envelope (attenuation) function, B the fourier transform of the aberration function, F the Fourier transform of the specimen function an u a reciprocal lattice vector. Equation 3.9 decribes how the detail on a certain scale (u) is affected by the

electron optical system. The specimen function can be written in the phase object approximation (POA), which holds for thin specimens and when absorption can be neglected (Spence 2009, Williams and Carter 2009):

fr=exp



^−iVpr



^(3.10)

where Vp is the projected electrostatic potential of the lattice parallel to the x-y plane (perpendicular to the incident electron beam) and σ = π/λE the interaction constant. The specimen function is thus directly related to the (projected) electrostatic potential inside the specimen itself.

Figure 3.7 :Lattice fringes in enstatite.

The dominant structure visible are lattice fringes related to the 18.2 Å lattice repeat ((1 0 0) plane spacings).

Stacking faults run from the top left to the bottom right.

The final mode is the scanning transmission electron microscope (STEM) mode. In this mode the electron optical system creates a condensed probe which scans the specimen, similar to in an SEM. For every pixel the intensity of either the direct beam (bright field STEM) or diffracted beams (dark field STEM) are measured using an angular detector. In this way the image is build by scanning over the specimen with the condensed beam.

This mode has been used to measure the diffusion profiles presented in chapter 4.

Im Dokument Major element diffusion in garnet and the exsolution of majoritic garnet from aluminous enstatite in Earth's Upper Mantle (Seite 64-67)