从晶体学到高分辨电子显微图像(3)

point
f(x,y)
Optical system
Disk g(x,y) = g(r)
A fA
B fB
Optical system
gA
gB
High-Resolution TEM
The role of the optical system
The factors contributing to H(u) include:
Is necessary to think of the microscope as an optical device which transfers information from the specimen to the image. As we saw the optics consist of a series of lenses and apertures aligned along the optic axis. Our goal is to transfer ALL the information from the specimen to the image doing a process known as mapping.
So most of our task will be concerned with finding the best compromise and producing models for the real situation.
High-Resolution TEM
The role of the optical system
As it is easy to understand there are two fundamental different types of Moiré interferences: the rotational and the translational.
In TEM the Moiré patterns correspond to interference between a pair of beams g1 and g2. If g1 is generated in the upper crystal and g2 in the lower, then each reflection g1 in crystal 1 acts as an incident beam for the lower crystal and produces a “crystal-2 pattern”.
Apertures:
the aperture function A(u)
Attenuation of the wave: the envelope function E(u)
Aberration of the lens: the aberration function B(u)
So we can write H(u) as the product of these three terms:
Phase Contrast
Imaging
Good HREM images can only be obtained as follows: 1. from thin specimen, since inelastic scattering will tend to degrade the image quality 2. the microscope must be aligned as accurately as possible 3. the objective lens astigmatism must be minimized 4. the specimen must be carefully tilted so that the beam direction coincides with a
Assuming that the astigmatism can be properly corrected, the Phase-Distortion Function is the
sum of two terms (f and Cs). If the Contrast Transfer Function
2. the image should be interpreted using an atomistic model for the material that will include a full description of the atomic potential and the bonding of the atoms (impossible!)
crystallographic axis
Introduction
High-Resolution TEM
We will now think to the TEM in a way that is more suitable for HRTEM, where the purpose is to maximize the useful detail in the image.
Phase Contrast – Moiré Patterns
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Imaging
Moiré patterns can be formed by interfering two sets of lines which have nearly common periodicities. It’s easy to understand Moiré effects if you just make three transparent sheets of parallel lines (two with the same spacing and one slightly different).
T (u ) 2 A(u ) sin (u ) is now compared to the Phase Distortion Function
(u)
a number of observation can be made.
High-Resolution TEM
The Contrast Transfer Function
The CTF is oscillatory: there are “bands” of good transmission separated by “gaps” (zeros) where no transmission occurs.
The CTF shows maxima (maximum transfer of contrast) whenever the Phase-Distortion
(u) fu 2 1 / 2C s3u 4
High-Resolution TEM
The Contrast Transfer Function
T (u) 2A(u) sin (u)
where we know that A(u) is the Aperture Function and might call (u) the Phase-Distortion
The envelope function has the same effect but is a property of the lens itself.
The aberration function is usually expressed as:
B( u ) exp[ i ( u )]
where:
g( r ) f ( r1 )h( r r1 )dr1 f ( r ) h( r )
The symbol indicates that the two functions f and g are
“folded together” (multiplied and integrated).
Since h(r) describes how a point spreads into a disk, it is known as the point-spread function or smearing function, and g(r) is called the convolution of f(r) with h(r).
There are mainly two problems to overcome and we can never be completely successful in transferring ALL the information because;
1. the lens system is not perfect so the image is distorted and you loose data
Phase Contrast – HREM images
Imaging
A more common, and indeed more useful, type of phase contrast image is formed when more diffracted beams are used to form the image.
If we consider two nearby points, A and B, they will produce two
overlapping images, gA and gB. If we extend this argument, we can
see that each image point has contributions from many points in the specimen and so we express this result mathematically by:
Function.
In other words, the Phase-Distortion Function has the form of a phase shift expressed as 2
times the path difference traveled by those waves affected by spherical aberration (Cs), defocus (f), and astigmatism (Ca).
What the microscope does is to transform each specimen point into an extended region in the final image plane. Since each point on the specimen is different, is possible to describe the
specimen by a specimen function f(x,y).
The extended region in the image which corresponds to the
point (x,y) in the specimen is then described as g(x,y).
H (u) A(u)E (u)B(u)
The aperture function says that the objective aperture cuts off all values of u (spatial frequencies) greater than (higher than) some selected value governed by the radius of the aperture.
Selecting several beams allows a structure image (often called high-resolution electron microscope, or HREM, image) to be formed.
The many lattice fringes intersect and give a pattern of dark (or bright) spots corresponding to atom columns.