Wave-Particle Duality || Recent Contributions of Electron Interferometry to Wave—Particle Duality

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<ul><li><p>CHAPTER 6 </p><p>RECENT CONTRIBUTIONS OF ELECTRON INTERFEROMETRY TO WAVE-PARTICLE DUALITY </p><p>FRANZ HASSELBACH </p><p>1. INTRODUCTION </p><p>Louis de Broglie's wave-particle duality hypothesis, published in his famous paper(l) in 1924, was verified for electrons only 3 years later by Davisson and Germer</p></li><li><p>110 FRANZ HASSELBACH </p><p>particle duality were done with electron and neutron interferometers. However, there is of course-apart from these experiments-an overwhelming mass of evidence for de Broglie's hypothesis. </p><p>Conventional electron interferometers were constructed by electron micros-copists according to customary principles approved in electron microscopy. In fact, they were most of the time just suitably modified electron microscopes. Their sensitivity to alternating magnetic stray fields and mechanical vibrations is therefore similar or, in interferometers with wide separation of the coherent beams, even higher than that of electron microscopes with atomic resolution. Electron interferometers therefore had to be located in special laboratories far from electric cables, far from any traffic and often additionally mounted on vibration isolation systems. To put such an interferometer on a rotating table, e. g., in order to perform a Sagnac experiment with electron waves seemed to be unimaginable. A closer look at the problem fortunately revealed that the special constructional require-ments necessary for a rugged interferometer were disregarded in conventional instruments. We therefore dropped traditional constructive principles and devel-oped a totally new design. (8) The resulting instrument is many orders of magnitude more insensitive to the disturbances just mentioned. The Sagnac experiment, for which the insensitivity of the instrument to vibrations is crucial, could be per-formed successfully as we will see later. While we focus here our interest on experiments done with this new interferometer, supplementary information may be found in review papers on conventional electron interferometry. (9-12) </p><p>2. THE NOVEL ELECTRON-OPTICAL BIPRISM INTERFEROMETER </p><p>An electron optical biprism interferometer, in principle, consists of an elec-tron source which is illuminating the biprism (Figure la). The biprism is composed of a very fine metallized quartz filament (less than 1 J.Lm in diameter) held at positive potential between two grounded electrodes. The incoming wave front is split into two partial waves when passing the biprism filament and-by the positive charge-the two partial waves are deflected toward each other. In analogy to the light optical biprism the electrons (partial waves) seem to emerge from the two virtual sources (marked by crosses). Interference fringes are formed in the region of superposition only if the spatial and the temporal coherence conditions are met. Both conditions can be satisfied very easily when the electrons emerging from the very fine virtual source of a field-emitter tip are used to illuminate the biprism.(13) When a single positively charged biprism is used, the widest separa-tion of the coherent partial waves is on the order of the diameter of the biprism filament, i.e., a micrometer or less. A wider separation of the coherent electron waves is mandatory in many experiments, e.g., when a very small coil or super-conducting tube carrying magnetic flux has to be inserted between the coherent waves in order to measure the Aharonov-Bohm phase shift (14-17) or the flux quantization in superconductors. (17,18) Likewise, an enclosed area between the </p></li><li><p>REcENT CoNTRIBUTIONS OF ELECTRON INTERFERoMETRY </p><p>electron source </p><p>a </p><p>plane of observation </p><p>+ </p><p>electron source </p><p>b </p><p>III </p><p>FIGURE 1. (a) Schematic diagram showing the path of the rays in an electron-optical biprism interferometer. (b) Schematical setup for achieving wide separation of the coherent electron waves by means of a dual biprism arrangement. </p><p>coherent beams which is as large as possible is needed for the Sagnac experiment. Wide separation of the coherent beams and in turn an enclosed area can easily be accomplished by multiple biprism arrangements as shown, e.g., for the case of two biprisms in Figure lb. Here the first biprism is charged negatively and bends the partial waves apart. A second, positively charged biprism recombines them. The separation of the wave fronts depends on the negative voltage of the first biprism filament and the distance of the two filaments. It reaches its maximum in the vicinity of the second biprism filament. </p><p>A conventional electron interferometer operated under such "wide separation conditions" has an enormously increased sensitivity to vibrations and-due to the phase shifting action of magnetic fluxes enclosed by the coherent beams-to ac magnetic stray fields. The mechanical resonance frequency of a conventional in-strument is low and, since it departs only slightly from the frequencies of the vibrations coming along the floor of the building it is excited easily. As a remedy it had to be the primary goal to make the new interferometer as rigid as possible, i.e., to raise the mechanical eigenfrequency of the whole assembly to values as high as possible; then external vibrations cause the interferometer to vibrate as a whole of course, but the relative positions of its components are not influenced. </p></li><li><p>112 FRANZ HASSELBACH </p><p>In turn, the visibility of the interference fringes is not impaired. The consequence of these considerations is that the dimensions as well as the weight of the inter-ferometer have to be reduced drastically. Mechanical alignment of the interferom-eter, while operating, has to be abandoned in favor of prealigned high-precision electron optical components. Fine alignment has to be done exclusively by electro-magnetic deflection systems. </p><p>The practical realization of the interferometer, the electron-optical setup, and a beam path are given in Figure 2a,b,c. The total length of the interferometer is only 30 cm, the diameter of the electron-optical components 28 mm, and the total mass less than 1 kg. In the setup shown, up to three biprisms can be used according to the individual requirements of the experiment. Fine alignment is achieved by the deflection elements and by the coils. The homogeneous magnetic field created by the coils allows rotation of the directions of the wave fronts. Inevitable slight rotational misalignments of the biprism filaments relative to each other can be compensated in this way. The Wien filter incorporated in our instrument is a novel component in electron interferometry. It is obligatory-as we will see later-in multiple biprism interferometers working with low-energy </p><p>/ / ~ / I I I ( I I 1V,en magnlfymg fluorescent anode I Wblprlsm 12'W1blPrtSm 3,dblprtsm fllfer quadrupole enses screen 'i0&amp;~='ffid-~(!trt&amp;t~~ ~~ m </p><p>cathode deflection I" alignment j2ndallgnment I ~ systems Call Call </p><p>hortzontal channelplate Image ",tenslfler </p><p>de flecflon deflection system system </p><p>b </p><p>view along the blprtsm wtres </p><p>I U u u u u ~1 -=====- 0 &gt;-- ~ + I r r r n n area enclosed by the two coherent electron waves </p><p>FIGURE 2. (a) Technical realization of the triple biprism interferometer. (b) Electron-optical setup. (c) Beam path. </p></li><li><p>RECENT CONTRIBUTIONS OF ELECTRON INTERFEROMETRY 113 </p><p>electrons in the range of 150 e V up to a few ke V. Figure 2c gives the beam path in a three-biprism (-, + ,-) arrangement with an enclosed area. The third, nega-tively charged biprism deflects the beams so that they intersect at a smaller angle in order to increase the width of the fringes in the interference pattern. </p><p>3. ELECTRON INTERFEROMETRIC VERIFICATIONS OF WAVE-PARTICLE DUALITY </p><p>3.1. Early Experiments: Diffraction at an Edge. Electron Biprism Interferences. and Diffraction by Slits Electron diffraction on macroscopic objects was observed for the first time by </p><p>Boersch(l9--21) in 1940 in an electron microscope. He observed contour fringes on an edge in out-of-focus electron micrographs and identified these as Fresnel diffraction fringes. </p><p>As an example of electron biprism interferences, a series of interferograms taken at an energy of 2.5 ke V with our new interferometer are given in Figure 3a. The potential of the biprism filament was chosen in the range 0.0-1.6 V. In the uppermost panel of Figure 3a the shadow of the biprism filament is visible in the middle. The Fresnel diffraction fringes of both edges of the filament are clearly visible. With increasing positive voltage applied to the biprism filament the par-tial waves begin to overlap. With further increasing angle of superposition-corresponding to an increasing lateral distance of the two virtual sources-more and more fringes become visible. </p><p>Microminiaturization was launched in TIibingen at the end of the 1950s.(22,23) Slits about 0.3 fJ-m wide in a thin copper foil were produced with this new technique. Single-, double-, up to ten-slit diffraction patterns were observed by Mollenstedt and Jonsson in 1959(24,25) and diffraction by a transmission grating by Holl in 1969.(26) In Figure 3b, single-, double-, and five-slit diffraction pat-terns taken from Jonsson's Ph.D. thesis(27) are given. The single slit interference pattern is complementary to the shadow image of the biprism filament given in Figure 3a and demonstrates Babinet's theorem. </p><p>3.2. Novel Experiments </p><p>3.2.1. Buildup of an Interference Pattern out of Single Events. One of the most impressive experiments which directly shows quantum mechanics at work is to observe the buildup process of an electron biprism interference pattern by accumulating the arrival sites of single electrons on a photographic plate or, even more impressive, in the memory of an image processing system. (28) While with the photographic method the buildup process can be seen just after the developing process is finished, with the image processor the buildup process can be visualized </p></li><li><p>114 FRANZ HASSELBACH </p><p>0.0 V </p><p>0.2 </p><p>0.4 </p><p>0.6 </p><p>0.8 </p><p>1.0 </p><p>1.2 </p><p>1.4 </p><p>1.6 </p><p>FIGURE 3. (a) Electron biprism interference patterns taken at an electron energy of 2.5 keY. The voltage applied to the (single) biprism filament is given at the right. Fresnel diffraction fringes on both sides of the filament are clearly visible especially when no voltage is applied to the biprism filament. (b) Single-. double-. and five-slit electron diffraction patterns taken by C. Jonsson in 1959. The freestanding slits had a width of about 0.3 fJ.m. </p><p>in real time. The fringe pattern, which has been accumulated in the memory of the image processor, is simultaneously displayed with the electrons incoming in every moment. In order to be able to discriminate between the incoming electrons and the accumulated fringe pattern, the brightness of the dots on the cathode ray tube, corresponding to the momentarily incoming electrons, is enhanced by a suitable program routine. Unfortunately, this dynamic buildup process cannot be demon-strated in a book. We must be content here with a static demonstration as given in Figure 4. </p><p>In order to obtain these micrographs the emission current of the cathode of our interferometer was adjusted to such a low value and the gain of the image intensifier to such a high level that the sites of incidence of single electrons become visible as tiny bright spots on the fluorescent screen of the image intensifier. In the series of micrographs the exposure time has been doubled from micrograph to micrograph resulting in an increasing density of the bright spots. While fringes </p></li><li><p>REcENT CONTRIBUTIONS OF ELECfRON INTERFEROMETRY 115 </p><p>FIGURE 4. Interference patterns obtained at extremely low emission current. The exposure time (starting from 1/8 s) has been doubled from micrograph to micrograph. With increasing integration time the fringe visibility becomes better and better. The bright spots in the micrographs show the arrival sites of single electrons and demonstrate the corpuscular character of the electrons, the arrangement to fringes their simultaneously present wave character. </p><p>cannot be seen at all when one is observing the fluorescent screen and in the first micrograph which has been taken with an exposure time of 1/8 s, they are well marked in the last ones, The appearance of well-localized bright spots demon-strates the corpuscular nature of the electrons and the arrangement of the spots to fringes the simultaneous presence of their wave nature. </p><p>A prerequisite for this demonstration of particle-wave duality was the availability of image intensifiers with a gain sufficient to visualize single electrons. The first micrographs showing the statistical nature of the formation of inter-</p></li><li><p>116 FRANZ HASSELBACH </p><p>ference fringes were taken by Merli et al. in 1976(29) followed by Wohland(30) and Matteucci and Pozzi. (31) The micrographs presented in Figure 4 were taken in the first test phase of the new interferometer and presented at the 1979 meeting of the German Electron Microscopical Society. (32) </p><p>3.2.2. The Wien Filter as a Device to Shift Wave Packets Longitudinally. What is a Wien filter, and what is the salient point of a Wien filter in an electron interferometric instrument? A Wien filter consists of crossed electric and mag-netic fields (Figure 5). It is in its compensated state when the electric force on the electrons is just compensated by the magnetic force, that is, the electrons travel through the Wien filter without any deflection rectilinearly. Let us assume that the two coherent wave packets enter into the Wien filter ax apart from each other and that the condenser plates of the Wien filter are on a potential of - U and + U, respectively. The wave packet on the right-hand side travels through the Wien condenser in a region of positive potential with respect to that on the left-hand side. That is, the wave packet on the right has a higher group velocity in the Wien filter than that on the left. Consequently, the wave packets leave the Wien filter shifted longitudinally Lly relative to each other. The acceleration and deceleration of the wave packets happens in the fringing electric fields of the Wien filter. With increasing excitation of the compensated Wien filter, the longitudinal shift in-creases, and for sufficiently high excitation, the two wave packets leave the Wien filter one behind the other. They do not overlap any more, and the contrast of the interference fringes vanishes. </p><p>!J.y </p><p>~ I I </p><p>I I </p><p>wave packets </p><p>Wien-filter </p><p>:~ </p><p>JJv interference fringes </p><p>FIGURE 5. Influence of a Wien filter in its compensated state on two spatially separated electron wave packets and on the phase of the waves. The wave packets are shifted longi-tudinally, the phase velocity is not affected (see text). There-fore, the positions of the horizontal lines, which symbolize the crests of the waves, are not shifted at all by the electromagnetic fields inside the Wien filter. </p></li><li><p>RBcENr CoNTRIBUTIONS OF Eu!crRoN INTERFERoMETRY 117 </p><p>It is noteworthy that in all compensated states of the Wien filter, the electron-optical index of refraction equals one in the nonrelativistic limit....</p></li></ul>