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Fresnel Diffraction of Electrons
Because of the smallness of the electron wavelength, observation of their Fresnel diffraction—in particular, observation of edge diffraction—is extremely difficult. In order to resolve interference maxima, the extent of the electron source must be small in comparison with the distance between the maxima. To resolve the \(n\)-th and \(n+2\)-th maxima, one must have an electron source of size no greater than
\[ d \leq \sqrt{\lambda \cdot a}\,(\sqrt{n+2}-\sqrt{n}). \]
Börsch\(^1\) succeeded in producing an electron source of size \(140\ \text{Å}\). With the aid of this source a diffraction photograph was obtained in which, in accordance with the equation given above, 4–5 lines could be observed. The distance between neighboring interference maxima
is a quantity of the order of 20 μ! Therefore, in the photograph reproduced by Boersch, about 2 cm in size, the emulsion grain is clearly visible.
The distances between interference lines correspond to the wavelength of the electrons. One may therefore regard as experimentally proven the application of the Kirchhoff and Sommerfeld theory of diffraction to electron waves.
On the basis of the experiment described, an absolute measurement of the wavelength can be made; until now, electron diffraction by crystal lattices made it possible to determine only the relative wavelength (with respect to the lattice parameters).
Boersch observed very complex Fresnel diffraction phenomena from an aperture, a screen, etc. A whole series of features of shadow photographs made with an electron microscope are explained by these phenomena.
Let us consider, for example, the occurrence of Fresnel diffraction when the object is not sharply adjusted in an ordinary electron microscope (see the figure).
Figure labels: Edge; Object; Image plane.
The distance \(x\) of an interference maximum will be
\[ x=\sqrt{l\lambda\left(n-\frac{1}{4}\right)} \qquad (n=1,3,5). \]
Owing to the divergence of the incident beam, a series of diffraction patterns displaced relative to one another arises. As a result of their superposition, the diffraction pattern is smeared out. In order for at least the first two maxima to be observed, the condition
\[ l\leq 0.6\,\frac{\lambda}{a^{2}}, \]
where \(a\) is the angle of divergence, must be satisfied. For ordinary \(a\) and \(\lambda\), the value of \(l\) is very small. The distance of the diffraction maxima from the edge is therefore a quantity of the order of 50 Å, i.e. of the order of the resolving power of modern electron microscopes. Fresnel diffraction phenomena are therefore observed in the form of blurring of contours², doubling of the figure³, etc.
To avoid these undesirable phenomena, large apertures must be used when the specimen is out of focus.⁴
A. I. Kitaigorodskii, Moscow
LITERATURE
- H. Boersch, Naturwiss., 28, 709, 1940.
- N. Mahl, AEG-Jahrbuch, 7, 43, 1940.
- H. Boersch, Naturwiss., 27, 418, 1939.
- H. Boersch, AEG-Jahrbuch, 7, 27, 1940.