In this article, the Wulf-Bragg formula is given, its significance for the modern world is studied. Methods for studying substances that are made possible by the discovery of electron diffraction on solids are described.
Science and conflict
The fact that different generations do not understand each other was written by Turgenev in the novel Fathers and Sons. And the truth is, it happens this way: a family lives for a hundred years, children respect their elders, all support each other, and then once - and everything changes. And it's all about science. No wonder the Catholic Church opposed the development of natural knowledge in this way: any step can lead to an uncontrolled change in the world. One discovery changes the idea of hygiene, and now the old people are surprised to see how their offspring wash their hands before eating and brush their teeth. Grandmothers shook their heads disapprovingly: “Why, they lived without it, and nothing, twenty children each. And all this your purity is only to the detriment of the evil one. ”
One assumption about the location of the planets - and now on every corner young educated people are discussing satellites and meteors, telescopes and the nature of the Milky Way, while the older generation is unhappy: “All sorts of nonsense, what is the use of space and celestial spheres, what difference does it make, how it rotates Mars and Venus, they would go better than growing potatoes, everything would be more good. "
One breakthrough in technology, which became possible due to the fact that diffraction by a spatial lattice is known, and in every second pocket lies a smartphone. At the same time, older people grumble: "There is nothing good in these quick messages, they are not that real letters." However, paradoxical as it sounds, the owners of a variety of gadgets perceive them as a given, almost like air. And few people think about the mechanisms of their work and that great path that human thought has traveled in some two hundred or three hundred years.
At the dawn of the twentieth century
At the end of the nineteenth century, humanity was faced with the problem of the knowledge of all open phenomena. It was believed that everything is already known in physics, and it remains only to find out the details. However, Planck's discovery of quanta and discreteness of the states of the microworld literally reversed previous ideas about the structure of matter.
Discoveries poured one after another, researchers grabbed ideas from each other from their hands. Hypotheses arose, verified, discussed, and rejected. One resolved question gave rise to a hundred new ones, and there were many people who were ready to seek answers.
One of the turning points that changed the world view was the discovery of the dual nature of elementary particles. Without him, the Wulf-Bragg formula would not have appeared. The so-called wave-particle duality explained why, in some cases, an electron behaves like a body with a mass (that is, a particle, particle), and in others - like an ethereal wave. Scientists argued for a long time until they came to the conclusion that the objects of the microworld possess such different properties at the same time.
This article describes the Wulf-Bragg law, which means that we are interested in the wave properties of elementary particles. For a specialist, these questions are always ambiguous, because overcoming the threshold of sizes of the order of nanometers, we lose certainty - the Heisenberg principle comes into force. However, for most problems, a rather rough approximation is enough. Therefore, it is necessary to begin to explain some features of addition and subtraction of ordinary waves, which are simple enough to imagine and understand.
Waves and Sines
Few people in childhood loved such a section of algebra as trigonometry. Sines and cosines, tangents and cotangents have their own system of addition, subtraction and other transformations. Perhaps this is incomprehensible to children, so it is not interesting to study. And many wondered why all this is needed at all, in what part of ordinary life this knowledge can be applied.
It all depends on how curious a person is. Someone lacks knowledge like: the sun shines during the day, the moon at night, the water is wet, and the stone is hard. But there are those who are interested in how everything that a person sees is arranged. For tireless researchers and explain: the greatest benefit from the study of wave properties is, oddly enough, the physics of elementary particles. For example, electron diffraction obeys precisely these laws.
First, work on your imagination: close your eyes and let the wave captivate yourself.
Imagine an infinite sinusoid: bulge, hollow, bulge, hollow. Nothing changes in it, the distance from the top of one dune to another is the same as everywhere else. The slope of the line, when it goes from maximum to minimum, is the same for each section of this curve. If there are two identical sinusoids nearby, then the task is complicated. Diffraction by a spatial lattice directly depends on the addition of several waves. The laws of their interaction depend on several factors.
The first is phase. The parts that touch these two curves. If their maxima coincide to the last millimeter, if the angles of the curves are identical, all indicators double, the humps become twice as high, and the hollows are twice as deep. If on the contrary, the maximum of one curve hits the minimum of the other, then the waves cancel each other out, all vibrations turn to zero. And if the phases do not coincide only partially — that is, the maximum of one curve falls on the rise or fall of the other, then the picture becomes quite complicated. In general, the Wulf-Bragg formula contains only an angle, as will be seen later. However, the rules for the interaction of waves will help to realize its conclusion more fully.
The second is the amplitude. This is the height of the humps and hollows. If one curve has one centimeter in height and the other has two, then they must be folded accordingly. That is, if the maximum of a wave with a height of two centimeters falls strictly on the minimum of a wave with a height of one centimeter, then they do not cancel each other, but only the height of the disturbances of the first wave decreases. For example, the diffraction of electrons depends on the amplitude of their vibrations, which determines their energy.
The third is frequency. This is the distance between two identical points of the curve, for example, highs or lows. If the frequencies are different, then at some point in the two curves the maxima coincide, respectively, completely add up. Already in the next period this does not happen, the final maximum becomes lower and lower. Then the maximum of one wave hits strictly the minimum of the other, giving the smallest result with this overlap. The result, as you know, will also be very complex, but periodic. The picture will repeat sooner or later, and again two maxima will coincide. Thus, when superimposing waves with different frequencies, a new oscillation with a variable amplitude will arise.
The fourth is direction. Usually, when two identical waves are considered (in our case, sinusoids), it is considered that they are automatically parallel to each other. However, in the real world everything is different, the direction can be any within the three-dimensional space. Thus, only waves traveling in parallel will be added or subtracted. If they move in different directions, interaction between them does not occur. The Wulf-Bragg law stands precisely on the fact that only parallel beams are added.
Interference and diffraction
However, electromagnetic radiation is not exactly a sinusoid. The Huygens principle states that each point of the medium to which the wave front (or perturbation) has reached is a source of secondary spherical waves. Thus, in every instant of propagation of, say, the light of a wave, they are constantly superimposed on each other. This is interference.
This phenomenon becomes the reason that light in particular and electromagnetic waves in general are able to go around obstacles. The latter fact is called diffraction. If the reader does not remember this from school, we will tell you that two slots in a dark screen illuminated by ordinary white light give a complex system of light maxima and minima, that is, there will be not two identical strips, but many different intensities.
If you irradiate the strips not with light, but bombard yourself with solid body electrons (or, say, alpha particles), you get exactly the same picture. Electrons interfere and diffract. It is in this that their wave nature is manifested. It should be noted that the Wulf-Bragg diffraction (most often referred to simply as Bragg) differs in the strong scattering of waves on periodic gratings when the phase of the incident and scattered waves coincides.
Solid
Everyone can have their own associations with this phrase. However, a solid is a well-defined branch of physics that studies the structure and properties of crystals, glasses, and ceramics. The foregoing is known only due to the fact that once scientists developed the basics of x-ray analysis.
So, a crystal is a state of matter when the nuclei of atoms occupy a strictly defined position in space relative to each other, and free electrons, like electron shells, are generalized. The main characteristic of a solid is periodicity. If the reader was once interested in physics or chemistry, the image of the crystal lattice of table salt (the name of the mineral is halite, the NaCl formula) probably pops up in his head.
Two types of atoms are in very close contact, forming a fairly dense structure. Sodium and chlorine alternate, forming in all three dimensions a cubic lattice, the sides of which are perpendicular to each other. Thus, a period (or unit cell) is a cube in which three vertices are atoms of one kind, the other three are another. By putting such cubes together, you can get an endless crystal. All atoms located within two dimensions periodically form crystallographic planes. That is, the unit cell is three-dimensional, but one of the sides, repeated many times (in the ideal case, an infinite number of times), forms a separate surface in the crystal. There are a lot of these surfaces, and they go parallel to each other.
Interplanar spacing is an important indicator that determines, for example, the strength of a solid. If in two dimensions this distance is small, and in the third it is large, then the substance easily exfoliates. This characterizes, for example, mica, which used to replace people with glass in windows.
Crystals and minerals
However, rock salt is a very simple example: there are only two types of atoms and understandable cubic symmetry. A branch of geology called mineralogy studies crystalline bodies. Their feature is that one chemical formula includes 10-11 types of atoms. And their structure is incredibly complex: tetrahedra, connecting with cubes with vertices at different angles, form porous channels of various shapes, islands, complex chess or zigzag joints. Such, for example, is the structure of an incredibly beautiful, rather rare and purely Russian ornamental stone of charoite. Its violet patterns are so beautiful that they can turn your head - hence the name of the mineral. But even in the most complicated structure there are crystallographic planes parallel to each other.
And this allows, due to the presence of the phenomenon of electron diffraction on the crystal lattice, to reveal their structure.
Structure and electrons
In order to adequately describe the methods of studying the structure of matter based on electron diffraction, one can imagine that the balls are thrown inside the box. And then they count how many balls bounced back and at what angles. Then, in the directions in which most of the balls bounce, they judge the shape of the box.
Of course, this is a rough idea. But according to this rough model, the direction in which the largest number of balls bounces is the diffraction maximum. So, electrons (or X-rays) bombard the surface of the crystal. Some of them get stuck in the substance, but others are reflected. Moreover, they are reflected only from crystallographic planes. Since the plane is not one, but there are many of them, only reflected waves parallel to each other are added up (we discussed this above). Thus, a signal is obtained in the form of a spectrum, where the reflection intensity depends on the angle of incidence. The diffraction maximum indicates the presence of a plane at the angle being studied. The resulting picture is analyzed to obtain an accurate crystal structure.
Formula
The analysis is carried out according to certain laws. They are based on the Wulf-Bragg formula. It looks like this:
2d sinθ = nλ, where:
- d is the interplanar distance;
- θ is the slip angle (angle additional to the angle of reflection);
- n is the order of the diffraction maximum (a positive integer, i.e. 1, 2, 3 ...);
- λ is the wavelength of the incident radiation.
As the reader sees, even the angle is not taken from that which was obtained directly during the study, but additional to it. It is worth explaining separately about the value of n, which refers to the concept of "diffraction maximum." The interference formula also contains a positive integer that determines what order the maximum is observed.
The illumination of the screen in the experiment with two slots, for example, depends on the cosine of the travel difference. Since the cosine is a periodic function, after the dark screen in this case there is not only the main maximum, but also several dimmer bands on its sides. If we were to live in an ideal world that is completely amenable to mathematical formulas, there would be an infinite number of such bands. However, in reality, the number of light regions observed is always limited, and depends on the wavelength, the width of the slits, the distance between them and the brightness of the source.
Since diffraction is a direct consequence of the wave nature of light and elementary particles, that is, the presence of interference, the Wulf-Bragg formula also contains the order of the diffraction maximum. By the way, this fact at first greatly complicated the calculations of the experimenters. At the moment, all the transformations associated with the turns of the planes and the calculation of the optimal structure from the diffraction patterns are performed by machines. They calculate exactly which peaks are independent phenomena, and which are second or third orders of the main lines in the spectra.
Before the introduction of computers with a simple interface (relatively simple, since programs for various calculations are still sophisticated tools), all this was done manually. And despite the relative conciseness that the Wulf-Bragg equation possesses, it took a lot of time and effort to verify the truth of the values obtained. Scientists checked and double-checked whether there was any out-of-place maximum that could ruin the calculations.
Theory and practice
The remarkable discovery, made both by Wulf and Bragg, has given mankind an indispensable tool for studying the previously hidden structures of solids. However, as you know, theory is a good thing, but in practice, everything always turns out to be a little different. A little higher we were talking about crystals. But any theory has in mind an ideal case. That is, an infinite defectless space in which the laws of repetition of the structure are not violated.
However, real, even very pure and laboratory-grown, crystalline substances abound with defects. Among the natural formations to meet the perfect pattern is a great success. The Wulf-Bragg condition (expressed by the above formula) in one hundred percent of cases applies to real crystals. For them, in any case, there is such a defect as the surface. And let the reader not be confused by some absurdity of this statement: the surface is not only a source of defects, but also the defect itself.
For example, the energy of bonds formed inside the crystal differs from the similar value of the boundary zones. This means that it is necessary to introduce probabilities and peculiar gaps. That is, when the experimenters take the spectrum of the reflection of electrons or x-rays from a solid, they get not just the angle, but the angle with an error. For example, θ = 25 ± 0.5 degrees. On the graph, this is expressed in the fact that the diffraction maximum (the formula of which is the Wulf-Bragg equation) has a certain width and represents a strip, and not an ideally thin line strictly in place of the obtained value.
Myths and errors
So what turns out, everything received by scientists is not true ?! To a certain degree. When you measure your temperature and find 37 on a thermometer, this is also not entirely accurate. Your body temperature is different from a strict temperature. But for you the main thing is that it is abnormal, that you are sick and it is time to be treated. Both you and your doctor do not care what the thermometer actually showed 37.029.
So it is in science - as long as the error does not interfere with making unambiguous conclusions, it is taken into account, but the emphasis is on the main value. In addition, statistics show: while the error is less than five percent, it can be neglected. The results obtained in experiments for which the Wulf-Bragg condition is met also have an error. Scientists who do the calculations usually point it out. However, for a specific application, in other words, understanding what the structure of a particular crystal is, the error is not very important (as long as it is small).
It is worth noting that every device, even the school line, always has an error. This indicator is taken into account in measurements, and, if necessary, is included in the general error of the result.