The shape of the beam of electrons emanating from the source can be approximated to a cone. Manipulation of the electron beam is the key to getting information from the sample. This is achieved using electromagnetic lenses. Here we shall see how the paths of electrons in the microscope can be modified by the lenses to focus the beam as required.
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Fortunately, despite operating in a very different way, we may use electron lenses in the same way as light-optical lenses. The way of describing the function of a lens in an optical system is by means of a ray diagram, which is a slight abstraction based on the thin lens approximation. This geometric construction allows us to see the behaviour of different rays incident on a lens.
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By using a small number of lenses in series we can achieve very high magnifications very quickly, since the magnifications multiply. For example, three lenses each giving a magnification of 50× give a 503 = 125000× magnification when placed in series. Any magnification may be achieved in theory. However, beyond a limit any increase in magnification becomes meaningless, as the amount of information available is limited by resolution.
The resolution of an image is the smallest distance between two points at which they may be distinguished as separate. The limit on resolution arises because by passing a beam of light through a lens of finite size it is diffracted. Due to the interference of waves coming from different parts of the lens, and also due to the fact that there are always some rays from the object which fall outside the lens, what begins as a point on the sample becomes a series of concentric circles in the image. These patterns of concentric maxima and minima are called "Airy Rings".
To reduce the effects of diffraction it is favourable to increase the size of the lens. Unfortunately, this leads to further problems.
Unfortunately, no lens is perfect. Apart from diffraction effects, resolution is limited by artefacts in the image due to the magnification process, "aberrations". These are due to the fact that rays entering the lens from different angles, or of different wavelength are bent differently. In practice, the aberrations due to the lenses become the limiting factor in the resolution of the electron microscope. We shall deal briefly with two major aberrations: spherical and chromatic aberrations.
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By using a more expensive FEG electron source, the effects of the chromatic aberrations may be reduced substantially - at least before the beam hits the specimen. Afterwards some will remain because energy is lost (and thus the wavelength changed) as the beam passes through the specimen. Unfortunately the monochromatic aberrations such as spherical aberration will almost always be present. Resolution may be increased by restricting the electrons used to those falling close to the optic axis. As seen in the simulation, an aperture placed above the lens limits the angles at which electrons are incident on the sample, reducing the spherical aberration.
Reducing the aperture size will also reduce the beam current, and increase the amount of diffraction experienced by the beam. There is, therefore, an optimum aperture size for the greatest resolution.
A typical TEM uses a system of two condenser lenses to control the beam incident on the sample. The first lens demagnetises the source, either to increase the brightness or decrease the area of the specimen that is illuminate. A second lens with an aperture above it controls the convergence angle, α, of the beam at the specimen.
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It is possible to reduce the effects of spherical aberration dramatically through the use of a large number (as many as 50) of finely adjustable lenses acting in series, much like the lenses in a camera lens are arranged to reduce chromatic aberration. With the computing power available today it is possible to adjust the lenses simultaneously to find the optimum combination of strengths. This has made it possible to construct aberration-corrected microscopes with a resolution better than 0.1 nm (1 Å).
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