One such discovery occurred to me on April 8, 1992, at 10.00 a.m. (Figure 4). I was looking at an electron diffraction pattern of an aluminum manganese compound that formed in a rapidly solidified alloy with composition close to Al6Mn, taken by an electron microscope. Electron diffractions contain the same information as X-ray diffractions. While looking at this pattern, I noticed two very strange things: first, Inhibitors,research,lifescience,medical this compound had a 10-fold rotational axis and, second, it had no periodicity (Figure 5). If the distance between
two spots is taken as the periodic distance and is multiplied by two, we should expect to reach the next diffraction spot. However, in this diffraction pattern (Figure 5), we reached nothing. Therefore, this LDK378 crystal had no periodicity. This crystal violated both laws of crystallography of the time: it had no periodicity, and it had a 10-fold rotational symmetry. Figure 4 Inhibitors,research,lifescience,medical Logbook of Professor D. Shechtman. Figure 5 First view of the icosahedral phase. Heavy diffraction is noted by the black crystals. However, this crystal did have quasi-periodicity. The ratio of the distances from the central spot to two spots that are Inhibitors,research,lifescience,medical adjacent to each other equals the Fibonacci number τ, which is also known as the golden mean or golden ratio. This number
is an irrational number of approximately 1.618. It is also the ratio of sequential elements of the Fibonacci sequence Inhibitors,research,lifescience,medical (0, 1, 1, 2, 3, 5, 8, 13, 21 …) which approaches the golden ratio asymptotically. The common denominator between the Fibonacci series and quasi-periodical crystals is that there is no motif of any size which repeats itself. However, they both have governing rules enabling them to continue indefinitely. The Fibonacci series is an example of quasi-periodicity in one dimension. A two-dimensional example of quasi-periodicity Inhibitors,research,lifescience,medical is Penrose tiles, named
after Professor Roger Penrose (Figure 6). If the colors are ignored, only two types of tiles remain: a thin rhombus and a thick rhombus. for A plane can be tiled according to specific rules, and the result is a quasi-periodic array. Roger Penrose proposed that this set of two tiles could only produce non-periodic tiling. Alan L. Mackay showed that the diffraction pattern of the Penrose tiling has a five-fold symmetric pattern. Figure 6 Penrose tiles. My discovery of quasi-periodical crystals occurred while I was staying at the National Bureau of Standards, where I was on sabbatical from 1981 to 1983. The reaction to my discovery was mixed. One colleague literally came to my office with a textbook proving that this pattern was just not possible. My group leader at the time called me a disgrace and asked me to leave his group.