The Wolf-Bragg formula. Diffraction on a space lattice

In this article the formula of Wolf-Bragg is given, its importance for the modern world is studied. Methods for investigating the matter, which became possible due to the discovery of diffraction of electrons on solids, are described.

Science and Conflicts

The fact that different generations do not understand each other, Turgenev wrote in the novel "Fathers and Sons". And the truth is: the family lives for a hundred years, the children respect the elders, support each other, and then time and again everything changes. And it's all about science. No wonder the Catholic Church was so opposed to the development of natural knowledge: any step can lead to an uncontrolled change in the world. One discovery changes the idea of hygiene, and now the old people look with astonishment at how their offspring wash their hands before eating and brush their teeth. Grandmothers shake their heads disapprovingly: "Why, they lived without it, and nothing, they gave birth to twenty children. And all this your purity is only to the detriment of the evil one. "

One assumption about the location of the planets - and already at every corner young educated people discuss satellites and meteors, telescopes and the nature of the Milky Way, whereas the older generation is dissatisfied: "Stupidity is all that is of use from space and heavenly spheres, what's the difference, how does it rotate Mars and Venus, they would rather grow potatoes, everything would be more useful. "

One breakthrough in technology, which became possible due to the fact that diffraction is known on the lattice, - and in every second pocket is a smartphone. At the same time, elderly people grumble: "There is nothing good in these quick reports, they are not like real letters." However, paradoxically it sounds, the owners of various gadgets perceive them as a kind of reality, almost like air. And few people think about the mechanisms of their work and the huge way that a human thought has done for some two or three hundred years.

At the dawn of the twentieth century

At the end of the nineteenth century, mankind was confronted with the problem of studying 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 the quanta and discreteness of the microworld states literally reversed the old ideas about the structure of matter.

Discoveries fell one after another, researchers snatched ideas from each other from their hands. Hypotheses arose, tested, discussed, rejected. One solved question spawned a hundred new ones, and there were a lot of people ready to look for answers.

One of the turning points that changed the perception of the world was the discovery of the dual nature of elementary particles. Without him, the Wolf-Bragg formula would not have appeared. The so-called corpuscular-wave dualism explained why in some cases the electron behaves like a body that has a mass (that is, a particle, a particle), and in others it is like a disembodied wave. Scientists argued for a long time until they came to the conclusion that the objects of the microworld possess such different properties simultaneously.

In this paper we describe 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 size of the order of nanometers, we lose certainty - the Heisenberg principle comes into force. However, for most problems there is enough rough approximation. 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 sinuses

Few in his childhood loved such a section of algebra as trigonometry. Sinuses and cosines, tangents and cotangents have their own system of addition, subtraction and other transformations. Maybe it's not clear to the children, so it's not interesting to study. And many wondered about why this is all necessary, in which part of the ordinary life, this knowledge can be applied.

It all depends on how inquisitive the person is. Someone has enough knowledge of the type: the sun shines in the daytime, the moon at night, the water is wet, and the stone is solid. But there are also those who are interested in how everything is arranged that a person sees. For indefatigable researchers, we explain: the most benefit from studying wave properties is, oddly enough, the physics of elementary particles. For example, the diffraction of electrons obeys these laws.

To begin, work on the imagination: close your eyes and let the wave draw you.

Imagine an infinite sinusoid: bulge, hollow, bulge, hollow. Nothing in it is changing, the distance from the top of one barkhan to another is the same as everywhere else. The slope of the line, when it goes from the maximum to the minimum, is the same for each section of this curve. If there are two identical sinusoids next to each other, then the task becomes more complicated. Diffraction on a spatial lattice is directly dependent on the addition of several waves. The laws of their interaction depend on several factors.

The first is the phase. The parts that these two curves meet. If their maxima coincide to the last millimeter, if the slopes of the curves are identical, all the indicators are doubled, the humps become twice as high, and the hollows become twice as deep. If on the contrary - a maximum of one curve falls to a minimum of the other, then the waves cancel each other, all the oscillations 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 Wolf-Bragg formula contains only the angle, as will be seen later. However, the rules of 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 a height of one centimeter, and the other has two inches, then they should be added accordingly. That is, if a maximum of a wave two centimeters high falls strictly on a minimum of a wave with a height of one centimeter, then they do not extinguish each other, but only the height of the perturbations of the first wave decreases. For example, the diffraction of electrons depends on the amplitude of their oscillations, which determines their energy.

The third is the frequency. This is the distance between two identical points of the curve, for example, maxima or minima. If the frequencies are different, then at some point the two curves coincide, respectively, and they add up completely. Already in the next period this does not happen, the final maximum is getting lower and lower. Then the maximum of one wave falls strictly on the minimum of the other, giving the smallest result with such an overlap. The result, as you understand, will also be very complex, but periodic. The picture will be repeated sooner or later, and again two maxima will coincide. Thus, with the imposition of waves with different frequencies, a new oscillation with a variable amplitude will arise.

The fourth is the direction. Usually, when two identical waves are considered (in our case, sinusoids), it is assumed 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 moving in parallel will add up or subtract. If they move in different directions, there is no interaction between them. The law of Wulf-Bragg stands precisely on the fact that only parallel beams are added.

Interference and diffraction

However, electromagnetic radiation is not exactly a sinusoid. Huygens' principle says that every point of the medium to which the wave front (or perturbation) has reached, is a source of secondary spherical waves. Thus, at every instant of propagation, say, of light, waves are always superimposed on each other. This is interference.

This phenomenon is the reason that light in particular and electromagnetic waves are generally capable of skirting obstacles. The last fact is called diffraction. If the reader does not remember this from school, we will suggest that two slots in a dark screen, illuminated by ordinary white light, give a complex system of maxima and minima of illumination, that is, the strips will not be two identical, but of many and different intensities.

If we irradiate the strips with light, but bombard ourselves with solid electrons (or, say, alpha particles), we get exactly the same picture. The electrons interfere and diffract. This is the manifestation of their wave nature. It should be noted that the Wulf-Bragg diffraction (often referred to simply as Bragg diffraction) consists in the strong scattering of waves by periodic gratings when the phase of the incident and scattered waves coincides.

Solid

With this phrase, everyone can have their own associations. However, a solid body is a well-defined branch of physics that studies the structure and properties of crystals, glasses and ceramics. The foregoing is known only because scientists once developed the fundamentals of X-ray structural 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 its 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 - halite, the NaCl formula) is likely to pop up in his head.

Two types of atoms are very closely contiguous, 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, the period (or unit cell) is a cube in which the three vertices are atoms of one kind, the other three are the same. By placing such cubes to each other, one can obtain an infinite 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. These surfaces are very numerous, and they run parallel to each other.

Interplanar distance 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 - a large, then the substance easily breaks down. 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: only two kinds of atoms and an understandable cubic symmetry. The section of geology, which is called mineralogy, studies crystalline bodies. Their peculiarity is that one chemical formula includes 10-11 kinds of atoms. And their structure is incredibly complex: tetrahedra, connecting with cubes with vertices from different angles, form porous canals of various shapes, islets, complex chess or zigzag connections. This, for example, is the structure of an incredibly beautiful, rather rare and purely Russian ornamental stone of charoite. His violet patterns are so beautiful that they can turn a head - hence the name of the mineral. But even in the most confused structure there are parallel to each other crystallographic planes.

And this makes it possible, due to the presence of the diffraction of electrons on the crystal lattice, to reveal their structure.

Structure and electrons

To adequately describe the methods of studying the structure of matter based on electron diffraction, it can be imagined that the balls are thrown inside the box. And then calculate 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 are "stuck" in the substance, but others are reflected. And they are reflected only from the crystallographic planes. Since the plane is not one, but there are many of them, only the reflected waves parallel to each other (we discussed this above) are added. Thus, a signal is obtained in the form of a spectrum, where the intensity of reflection depends on the angle of incidence. The diffraction maximum indicates the presence of the plane at the angle under study. The resulting picture is analyzed to obtain the exact crystal structure.

Formula

The analysis is made according to certain laws. They are based on the Wolf-Bragg formula. It looks like this:

2d sinθ = nλ, where:

• D is the interplanar distance;
• Θ - slip angle (angle, additional to the reflection angle);
• N is the order of the diffraction maximum (positive integer, ie 1, 2, 3 ...);
• Λ is the wavelength of the incident radiation.

As the reader sees, even the angle is taken not the one that was obtained directly in the study, but an additional one to it. It is worthwhile to explain separately about the value of n, which refers to the concept of "diffraction maximum". The interference formula also contains a positive integer that determines which order the maximum is observed.

The illumination of the screen in an experiment with two slots, for example, depends on the cosine of the path difference. Since the cosine is a periodic function, after the dark screen in this case there is not only a main maximum, but also a few fainter bands on its sides. We live in an ideal world, which is completely amenable to mathematical formulas, such bands would be an infinite number. However, in reality, the number of observed bright areas 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 hampered the calculations of experimenters. At the moment, all the transformations associated with the reversal of planes and the calculation of the optimal structure by diffraction patterns are performed by machines. They also calculate which peaks are independent phenomena, and which ones are the second or third orders of the main lines on the spectra.

Before the introduction of computers with a simple interface (relatively simple, as programs for a variety of calculations - all the same sophisticated tools), all this was done manually. And despite the relative laconism that the Wulf-Bragg equation has, it took a lot of time and effort to ascertain the truth of the values obtained. The scientists checked and rechecked - whether there was a skewed out where some unprincipled maximum, which could spoil the calculations.

Theory and practice

A remarkable discovery, accomplished simultaneously by Wulf and Bragg, gave an indispensable tool in the hands of mankind to investigate the structures of solids hidden before. However, as you know, theory is a good thing, but in practice everything always turns out to be a little different. A little higher it was about crystals. But any theory has in mind an ideal case. That is an infinite defect-free space in which the laws of repetition of the structure are not violated.

However, real, even very pure and grown in laboratories, crystalline substances abound in defects. Among the natural formations to meet the ideal pattern is great luck. The Wolf-Bragg condition (expressed by the above formula) is applied to real crystals in a hundred percent of cases. 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 the bonds formed inside the crystal differs from the analogous value of the border zones. This means that it is necessary to introduce probabilities and peculiar gaps. That is, when the experimenters remove the spectrum of reflection of electrons or X-rays from a solid, they get not just the magnitude of the angle, but the angle with the error. For example, θ = 25 ± 0.5 degrees. On the graph this is expressed by the fact that the diffraction maximum (the formula of which lies in the Wulff-Bragg equation) has some width, and is a strip, and not an ideally thin line, strictly in place of the value obtained.

Myths and Errors

So what happens, everything received by scientists is untrue ?! In some ways. When you measure your temperature and find 37 on a thermometer, this is also not entirely accurate. The temperature of your body is different from the strict value. But for you the main thing is that it is abnormal, that you are ill and it's time to be treated. And to you and your doctor it does not matter at all what the thermometer actually showed on 37.029.

So 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 significance. 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 satisfied also have an error. Scientists who make calculations, as a rule, indicate it. However, for a specific application, in other words, understanding what the structure of a crystal is, the error is not very important (as long as it is small).

It should be noted that every device, even a school ruler, always has an error. This indicator is taken into account in the measurements, and, if necessary, enters the overall error of the result.