"RADIOCHEMISTRY Volume I RADIOACTIVITY AND RADIATION Textbook Moscow Beckman I.N.

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I. N. Beckman

RADIOCHEMISTRY

RADIOACTIVITY AND RADIATION

Tutorial

Beckman I. N.

Beckman Igor Nikolaevich -

Doctor of Chemical Sciences, Professor of the Department of Radiochemistry, Faculty of Chemistry, Moscow State University named after M.V. Lomonosov;

Honored Professor of Moscow State University.

Editors Beckman E.M. and Polonskaya-Buslaeva O.A.

Beckman I.N.

B42 Radioactivity and radiation. Radiochemistry.

Volume 1: study guide / I.N.Bekman. - Moscow Region, Shchyolkovo:

Publisher Markhotin P.Yu. 2011.- 398 p.

ISBN 978-5-905722-05-9 "Radioactivity and radiation" - the first part of the textbook "Radiochemistry". The book contains systematic information on radioactivity, radionuclides and their accompanying radiation, the kinetics of decay and accumulation of radioactive isotopes, the structure of the nucleus and nuclear processes, sources of radioactive radiation, methods for detecting radioactive radiation, the interaction of radiation with matter, methods for isotope separation, as well as methods for statistical processing of results. radiometric measurements. Information about the biological effect of radiation is given, methods of radiation dosimetry are considered, and safety rules for working with radionuclides are discussed. The properties of some radioactive isotopes are described.

The manual can be useful for students of radiochemistry, students and graduate students of chemical and polytechnic universities, researchers working with radioactive substances, and everyone interested in isotopes, radioactive radiation and methods of their use in modern science, technology and medicine.

BBC 23.1 ISBN 978-5-905722-05-9 © Beckman I.N., 2011 Radioactivity and radiation. Radiochemistry. Volume 1

FOREWORD

"Radioactivity and radiation" - the first part of the textbook "Radiochemistry", consisting of seven volumes: 1. Radioactivity and radiation (Fundamentals of radiochemistry), 2. Radioactive elements (Nuclear-physical, radiological and chemical properties of radioactive elements; methods for their production and use ), 3. Fundamental radiochemistry (Chemistry of hot atoms, state and diffusion of radionuclides in various media; methods for separating radioactive substances), 4. Nuclear industry and industrial radiochemistry (Nuclear fuel cycles; radiochemical technologies in the nuclear industry), 5. Applied radiochemistry (Method tracers, nuclear physical and radionuclide diagnostic methods), 6. Ecological radiochemistry and radioecology (State and migration of radionuclides in natural environments), 7. Radiation and nuclear medicine: physical and chemical aspects (Synthesis of radiopharmaceuticals and their use in diagnostics and therapy) .

From the point of view of the university educational process, the manual "Radioactivity and Radiation" contains the information necessary to understand the material presented in the parallel lecture courses:

“Fundamental and Applied Radiochemistry”, “Nuclear Industry”, “Nuclear Physics”, “Radioecology”, etc. However, the book can be used without any connection with any educational process, and it can be recommended to everyone who is interested in the phenomenon of radioactivity and problems, arising from work with radionuclides and ionizing radiation emitted by them.

The proposed textbook focuses on the physical foundations of radiochemistry, in particular, the elements of atomic and nuclear physics, as well as radiation chemistry. Such aspects as the properties of nuclei, the phenomenon of radioactivity, the kinetics of decay and accumulation of radionuclides, nuclear processes, sources of radioactive radiation, the properties of various types of radiation, the interaction of radiation with matter, methods for obtaining radioactive isotopes, the properties of some radioactive isotopes of stable elements, methods for detecting radioactive radiation are considered. and methods for measuring the radioactivity of solid, liquid and gaseous preparations. Information is given on the biological effect of radiation, methods of radiation dosimetry, and safety rules for working with radionuclides are discussed. In the final part of the manual, the main methods of statistical processing of the results of radiometric measurements are given.

The textbook is written based on the materials of lectures given for more than twenty years at the Department of Radiochemistry, Faculty of Chemistry, Lomonosov Moscow State University. M.V. Lomonosov.

Beckman I. N.

1. HISTORY OF THE DISCOVERY OF THE PHENOMENA OF RADIOACTIVITY

AND IONIZING RADIATION

The discovery of radioactivity occurred at the end of the 19th century, and by accident. However, it was apparently inevitable, as evidenced by the fact that the emission of mysterious radiation by uranium salts was reported independently in the middle of the same century by the Frenchmen Niepce (1858) and Saint Victor (1867). Unfortunately, their observations did not become a discovery and were forgotten. The phenomenon of radioactivity was discovered by A. Becquerel in 1896. The discovery itself was unexpected, but it happened in the framework of purposeful work on the study of the structure of the atom and the properties of radiation.

This chapter discusses the stages of development of the doctrine of radioactivity, and the experiments that led to the discovery of the nucleus, the laws of radioactive decay, nuclear processes and the properties of ionizing radiation.

1.1 Basic elementary particles Electron.

For millennia, science has been dominated by the hypothesis of the indivisibility and "structurelessness" of the atom. A departure from these ideas began in the theory of electricity. In 1749, B. Franklin suggested that electricity is a kind of material substance. In his works, for the first time, the terms appear: charge (positive and negative), particles of electricity. The term "electron" as the name of the fundamental indivisible unit of charge in electrochemistry was proposed by J. J. Stoney in 1894. The discovery of the electron as a particle belongs to J. J. Thomson, who in 1897 established that the charge-to-mass ratio for cathode rays does not depend from the source material. M. Faraday in 1833, to explain experiments on electrolysis, introduced the term "ion" for the carriers of electricity in the electrolyte and suggested that the ion has a constant charge.

The name "electron" comes from the Greek word meaning "amber": back in ancient Greece, natural scientists conducted experiments - pieces of amber were rubbed with wool, after which they began to attract small objects to themselves.

The discovery of the electron - the carrier of the negative elementary electric charge - and ions testified to the complex structure of the atom and the possibility of its disintegration into separate components.

In the discovery of the first elementary particle - the electron - cathode rays, discovered in 1859 by Yu. Plyukker, played a significant role. The name was given by E. Goldstein, who believed that cathode rays are a wave process in the ether. V. Crooks argued that cathode rays are streams of particles of matter. In 1895, J. Perrin experimentally proved that cathode rays are a stream of negatively charged particles that move in a straight line, but can be deflected by a magnetic field. Not all physicists agreed with the hypothesis of atomic electricity. Thus, J.Maxwell, who created the fundamental theory of electrical and magnetic phenomena, categorically rejected it.

Cathode rays - an electron beam in a vacuum that generates a magnetic field and deviates in magnetic and electromagnetic fields.

4 Radioactivity and radiation. Radiochemistry. Volume 1 Since 1895, J. J. Thomson at the Cavendish Laboratory of the University of Cambridge began a quantitative study of the deflection of cathode rays in electric and magnetic fields. He worked with a Geissler tube.

Thomson proved that all particles that form cathode rays are identical to each other and are part of the substance. Thomson called the particles of cathode rays "corpuscles". According to his hypothesis, cathode rays consist of particles whose charge does not exceed the elementary charge of ions e. The mass of such particles must be thousands of times less than the mass of an atom. (Indeed, as it turned out, the mass of an electron is 1/1837 of the mass of a hydrogen atom). The hypothesis of the existence of matter in a state of even finer fragmentation than atoms was presented by Thomson at a meeting of the Royal Society on April 29, 1897. However, the idea of ​​the electron was not immediately accepted.

So, M. Planck did not believe in the hypothesis about the electron. The word "electron" was originally used to denote the magnitude of the charge of the "corpuscle". Only with time did they begin to call the particle itself an electron. The electric charge of an electron was measured by R. Millikan in 1912, and only then did this first elementary particle get the right to exist. In 1923, Louis de Broglie suggested that the electron could have wave properties. In 1925, J. Uhlenbeck and S. Goudsmit postulated the electron spin. In 1927 K. Davisson, L. Germer and J. Thomson confirmed the wave nature of the electron.

Direct experimental proof of the existence of the photon was given by R. Millikan in 1915 in his studies of the photoelectric effect, and also by A. Compton in 1922, who discovered the scattering of X-rays with a change in their frequency. Since the rest mass of a photon, unlike other particles (except for neutrinos), is equal to zero, the photon was not immediately considered a particle: at first it was believed that the presence of a finite and non-zero rest mass is a mandatory feature of an elementary particle. The concept of "quantum of light" was introduced by Planck in 1901 in order to explain the laws of radiation of an absolutely black body. But then the photon was considered not a particle, but only the minimum possible "portion" of light energy of one frequency or another. Although Planck's assumption about quantizing the energy of light was absolutely contrary to all classical theory, Planck himself did not immediately understand this. He wrote: “... I tried to somehow introduce the value of h into the framework of the classical theory. However, despite all such attempts, this value turned out to be very obstinate. Subsequently, this value was called Planck's constant (h=6.626 10-34 J s).

Photons received the status of particles within the framework of A. Einstein's theory of relativity, who in 1905 showed that quanta have not only energy, but also momentum, and that they are particles (with a rest mass equal to zero, they move at the speed of light). Electromagnetic radiation (light) is a stream of individual quanta (photons), which well explains the patterns of the photoelectric effect.

The discovery of the second elementary particle, the proton, was made by Rutherford in 1919, although the H+ ion had long been known by that time.

Beckman I. N.

In 1913, E. Rutherford put forward the hypothesis that one of the particles that make up the nucleus of an atom of any chemical element must be the nucleus of a hydrogen atom, since. it was already known that the masses of atoms of chemical elements exceed the mass of a hydrogen atom by an integer number of times. Rutherford set up an experiment to study the interaction of -particles with the nuclei of the nitrogen atom.

As a result of the interaction, a particle flew out of the nucleus of the nitrogen atom, which in 1920 Rutherford called the proton (from the Greek simplest, primary) and suggested that this was the nucleus of the hydrogen atom.

The nuclear reaction of nitrogen with helium (-particles) has the form:

7 N + 2 He 8 O +1 p (1) Rutherford concluded that "the nucleus of the nitrogen atom disintegrates as a result of the enormous forces that develop in a collision with a fast particle, and that the liberated hydrogen atom forms an integral part of the nitrogen nucleus." In 1925, P. Blacket obtained the first photographs of proton traces in a cloud chamber, at the same time confirming the discovery of the artificial transformation of elements. In 1933, O. Stern measured the magnetic moment of the proton. In 1955 O. Chamberlain, E. Segre, K. Wiegand and T. Ypsilantis discovered the antiproton. In 1956, R. Hofstadter measured the electromagnetic radius of the proton for the first time.

In 1920, Rutherford suggested that there should be a particle with a mass equal to the mass of a proton, but without an electric charge. However, Rutherford failed to detect such a particle.

In 1930, V. Bothe and G. Becker irradiated lithium and beryllium

Particles and using a Geiger counter registered the resulting penetrating radiation. Since this radiation was not affected by electric and magnetic fields, and it had a high penetrating power, the authors came to the conclusion that hard radiation is emitted. In 1932, Frédéric and Irene Joliot-Curie also experimented with beryllium by passing new penetrating radiation through a paraffin block. They discovered that high-energy protons emerge from the paraffin and concluded that, as it passes through the paraffin, the γ-radiation scatters to produce protons.

J. Chadwick in 1932 repeated the experiment on the irradiation of beryllium

Particles. He also used paraffin and, using a proportional counter, which allows determining the distribution of energy between different particles, showed that the penetrating radiation consists of neutral particles with a mass close to protons - neutrons. In the case of beryllium, for example, they are formed as a result of a nuclear reaction:

4 Be+2 He=6 C+0 n. (2) When passing through matter, neutrons do not lose energy for the ionization of atoms of matter, therefore they have a huge penetrating power.

Chadwick estimated the mass of the neutron by analyzing the energy balance of nuclear reactions involving the neutron. The properties of the new particle were investigated by the Joliot-Curies, who showed that it was unstable compared to the proton; for its mass they found a value of 1.0 (at 16O=16,000). The kinetic energy of neutrons, the emission of radioactivity and radiation were estimated. Radiochemistry. Volume 1 sourced by Ro + We. These works led to the discovery of artificial radioactivity.

In 1951, J. Robson measured the half-life of the neutron.

A fairly accurate determination of the half-life of a free neutron (11.7 min) was carried out in 1959 by P.E. Spivak. In 2005 A.P. Serebrov refined this value to T = 10.14 min, and in 2010 K. Nakamura proposed a half-life value for the neutron T = 10.18 min.

Positron.

In the 30s - 50s of the 20th century, new particles were discovered mainly in cosmic rays. In 1932, in their composition, A. Anderson discovered the first antiparticle - the positron (e +) - a particle with the mass of an electron, but with a positive electric charge. The existence of the positron followed directly from the relativistic theory of the electron developed by P. Dirac (1928-31) shortly before the discovery of the positron.

The existence of the positron was confirmed by observations by Blackett and Occhialini in a cloud chamber. Then the Joliot-Curies discovered that positrons are formed during the conversion of -rays, and are also emitted by artificial radioactive isotopes. Since the photon

Radiation, being neutral, forms a pair: a positron and an electron, then from the principle of conservation of electric charge it follows that in absolute value the charge of the positron is equal to the charge of the electron.

For the first time, the mass of the positron was measured by J. Thibault, who found that the masses of the positron and electron differ by less than 15%. Later experiments confirmed that the positron and electron have equal masses.

In 1938, E. Stückelberg introduced the concept of the baryon number to explain the stability of the proton.

Neutrino.

The discovery of the neutrino, a particle that hardly interacts with matter, began with a theoretical conjecture by W. Pauli (1930), which made it possible, by assuming the appearance of such a particle, to eliminate the difficulties of applying the law of conservation of energy to the decay of radioactive nuclei. The existence of neutrinos was experimentally confirmed only in 1953 (F. Reines and K. Cowen, USA).

In 1933, a theory of -decay was created taking into account neutrinos; the concept of a new type of interaction - weak (E. Fermi) was introduced. Fermi's theory is based on the proton-neutron model of the nucleus and relies on the concept of neutrino and the laws of conservation of spin and energy.

In the 1930s, Fermi's theory was generalized to positron decay (Wick 1934) and to transitions with a change in the angular momentum of the nucleus (Gamow and Teller 1937). In 1938, A. Alikhanov and A. Alikhanyan proposed to investigate the recoil of nuclei in the process of electron capture (electronic capture of 7Be) in order to detect neutrinos. In 1943, J.S. Allen, in the process of electron capture on the 7Be nucleus, measured the recoil momentum of the final nucleus (7Li), confirming the hypothesis of the existence of neutrinos. In 1946, B. Pontecorvo proposed the "chlorine method" for detecting neutrinos.

In 1956, F. Reines and K. Cohen registered antineutrinos. In 1962, it was found that there are two different neutrinos: electronBekman I.N.

noah and muon. In 1964, nonconservation of combined parity (introduced by Li Tsung-dao and Yang Ch'en-ning and independently by L. D. Landau in 1956) was discovered in the decays of neutral K-mesons. In 1957, B. Pontecorvo put forward the idea of ​​neutrino oscillations. In 1962, L. Lederman showed that the electron neutrino differs from the muon one. In 1998, the first evidence of neutrino oscillations was obtained (during the registration of atmospheric muon neutrinos at the Super-Kamiokande facility, Japan).

1.2 X-ray radiation Experiments with the Crookes tube, gas discharge and cathode rays led to the discovery of X-rays (W. Crooks, 1890).

Cathode rays have been known since the middle of the 18th century. As early as 1748, it was noticed that in a glass tube from which air was evacuated, fires flared up when an electric spark was passed. A hundred years later, a similar phenomenon was observed by Faraday, when he brought current from an electric machine to a glass tube with rarefied air. He noted that a violet glow emanated from the positive electrode (anode), extending almost to the cathode itself, which also flickered in the dark. Twenty years later, Plüker, who had achieved a strong rarefaction in a glass tube, noticed that not only the cathode, but also the glass located near it, glowed. Ten years later, Gittorf inserted a solid object between the cathode and the phosphorescent glass and noticed that it cast a shadow. From which he concluded that the cathode emits invisible rays.

W. Crooks, who invented many cathode ray tubes of different shapes, suggested that cathode rays are a stream of some negatively charged particles. In 1891, G. Hertz discovered that cathode rays pass through thin layers of metal. In 1894, F. Lenard removed a beam of cathode rays from a tube. He made a hole at the end of it and covered it with thin aluminum foil so that the vacuum would not be broken.

Rice. 1. X-ray of the hand of Bertha Roentgen.

The German scientist V.K. Roentgen studied cathode rays, experimenting with the Gittorf tube. 11/8/1895 X-ray discovered the glow of a screen coated with barium platinum-cyanide (barium tetracyanoplatinate, Ba). Since the screen was at a considerable distance from the radiation source (cathode rays could not reach it), and the tube was covered with an opaque casing, Roentgen suggested that the screen glow was caused by high-energy invisible rays. He called them X-rays (in some countries, including Russia, they are now called X-rays). The wide recognition of Roentgen's discovery was facilitated by his obtaining images of various objects on photographic plates in x-rays.

On January 20, 1896, at a meeting of the Paris Academy, Henri Poincaré spoke about the discovery of new rays and suggested that X-rays are associated with fluorescence and, possibly, all radioactivity and radiation arise. Radiochemistry. Volume 1 where in luminescent substances and no cathode tube is needed to obtain X-rays.

In February-March 1896, Henri Becquerel tested this hypothesis.

He used photographic action through black paper of sun-activated uranium salt crystals.

Comment. For the experiment, Becquerel chose a salt from his father's extensive collection that has a high intensity of luminescence under the action of sunlight (yellow-green phosphorescence) - double uranyl and potassium sulfate (UO2SO4 K2SO4 2H2O). The choice of salt is random - he had at his disposal salts that had similar properties, but did not contain uranium. If Becquerel had taken any other salt, then the discovery of radioactivity would not have taken place. Therefore, they speak of the accidental discovery of radioactivity. However, the thoroughness and accuracy of all operations allowed Becquerel to make a great discovery.

At the first stage, experiments confirmed Poincaré's hypothesis, but soon Becquerel discovered that uranium salt, even without exposure to sunlight, has the property of emitting radiation passing through black paper. Particularly clear confirmation of the presence of highly penetrating, but not X-ray, radiation from the preparation was an experiment in which a stand, into the recesses of which uranium salt was poured, was sandwiched between two photographic plates wrapped in black paper. Both plates gave fairly clear images (Fig. 2).

Rice. 2. An imprint of uranium salt (potassium uranyl sulfate), placed in the recesses of the stand, on photographic plates applied to different sides of the stand.

A. Becquerel's discovery of the phenomenon of radioactivity (03/1/1896).

Continuing the research, Becquerel discovered that only uranium salts emit new radiation, other luminescent or phosphorescent substances do not emit radiation.

In the case of uranium salts, the radiation intensity is determined only by the amount of uranium in the preparation and does not depend at all not only on the temperature and state of aggregation, but also on which compounds the uranium enters into. Radiation is emitted not by a compound, but by a chemical element - uranium. This was finally confirmed when working with metallic uranium, which turned out to be more active than its salts.

Thus, the phenomenon of radioactivity was discovered: the property of some elements to spontaneously decay and emit radiation without introducing energy from outside. Over the next few years, it was found that the radiation power of uranium does not decrease with time.

In 1901, M. Curie introduced the concept of radioactivity. In 1902, V. Ramsay experimentally showed that the radioactive process proceeds as a monomolecular decay reaction of matter, and E. Rutherford and F. Soddy proposed the first explanation of the mechanism of the radioactive process as a phenomenon of spontaneous decay of a chemical element: atoms of radioactive elements undergo spontaneous decay, accompanied by emission - or -particles and the formation of an atom of a new element. In 1903, they also formulated the law of radioactive Beckman I.N.

transformations and gave its mathematical expression, the canonical form of which Nt=N0.e-t is generally accepted at the present time. According to the proposed scheme, radioactive decay, for example, of radium leads to its transformation into radon and helium. The formation of helium was experimentally confirmed by W. Ramsay and F. Soddy.

A quantitative interpretation of radioactive phenomena became possible after E. Schweidler proved the statistical nature of radioactive transformations in 1905 and introduced the concepts of "decay probability" and "half-life". An experimental substantiation of these ideas was given in 1906 by K. Kohlrausch. This is how the interpretation of radioactivity as a probabilistic process arose.

In 1934, the Joliot-Curies discovered artificial radioactivity. They obtained, by means of nuclear transformations, unstable isotopes of light elements, which, depending on the relative mass number, had the ability to -radiate. Isotopes with a relatively large mass number emitted electrons, so that their atomic charge increased by one, and they experienced a shift to the right by one place in the periodic table. If the mass number turned out to be relatively small, then the isotopes moved one place to the left, emitting positrons and thereby reducing their nuclear charge by one. As a result of these nuclear reactions, radioactive isotopes of known light elements are formed, for example, oxygen, carbon, nitrogen, fluorine, and others.

1.3 Radioactive elements and isotopes In 1898, M.Curie and G.Schmidt independently discovered activity in thorium.

In 1897, Becquerel turned to P. Curie with a request to find out if there were any impurities in the radiating substances that could play a special role. P. Curie recommended M. Sklodowska-Curie to work on this topic. In 1896, the Curies found that the radioactivity of uranium minerals is greater than the radioactivity of the uranium they contain. This observation led them to speculate that the uranium minerals contained some much more radioactive element than uranium.

After processing several tons of uranium ore from the Czech Jáchymov deposit, they obtained two very radioactive precipitates: barium sulfate and bismuth hydroxide. In the sediment of bismuth hydroxide in 1898, a new element, polonium, was discovered (it was not possible to isolate it in its pure form;

Polonium chloride is 900 times more active than uranium. In 1902, radium was isolated from the precipitate of barium sulfate (M. Curie, P. Curie and J. Bemont).

In these works, a specific material carrier of radioactive phenomena (atoms of radioactive elements) was indicated and such a carrier (radium) was actually discovered, in which the process under study proceeds with a much greater intensity than in uranium.

The discovery of radioactivity aroused great interest among scientists. Unfortunately, it was impossible to buy radium. The situation was saved by the German chemist Professor F. Gisel, a specialist in quinine. He isolated active substances from uranium ores at about the same time as the Curies. By modernizing the methodology of K.R. Fresenius, used in slightly modiRadioactivity and radiation. Radiochemistry. Volume 1 of Curie's official form, Gisel switched from chloride to radium bromide and obtained a fairly pure radium salt. (In 1898, M. and P. Curie still had not quite clean samples of two new radioactive emitters). He isolated the radium preparation earlier than M. Curie. In 1900, Gisel discovered a new radioactive element, although it immediately became clear that A. Debjorn had already discovered it (1899) and called it actinium. Since 1903, F. Gisel (Hininfabrik, Braunschweig) began to sell pure radium compounds at relatively moderate prices (radium bromide hydrate contained 50% of the element). Prior to this, one had to work with compounds containing at most 0.1% radium!

In 1900, E. Rutherford discovered a radioactive gas released by thorium salts and called it an emanation (now thoron, Tn, 220Rn). Dorn in the same year established that radium salts also emit emanation (radon, 222Rn), and in 1903 A. Debjorn showed that actinium salts emit actinon (An, 219Rn). In 1902, E. Rutherford and F. Soddy proved that thoron is an inert gas. In the same year, the diffusion coefficient of radium emanation in the air was measured (P.Curie, J.Dann). In 1903 radiothorium (228Th) was discovered (O.Khan). In 1906, N. Campbell and A. Wood discovered the radioactivity of potassium and rubidium. Thus, it was proved that radioactivity is not only a property of heavy atoms, but can manifest itself in any elements of the periodic table. Protactinium was discovered by O.Khan and L.Meitner in 1907. In 1909

it has been proved that various isotopes of lead are the end product of three natural radioactive families (J. Gray). In 1910, pure metallic radium was obtained (M.Curie, A.Debiern). The first international radium standard (M.Curie, A.Debjorn) was made in 1911.

In 1912, isotopes were discovered - the existence of neon atoms with a mass of 20 and 22 was discovered (J. J. Thomson).

In 1913, the concept of an isotope was introduced and the isotopy of radioactive elements was demonstrated (F. Soddy), the shift rule (displacement law) during radioactive decay was formulated - the Soddy Faience shift rule (F. Soddy and C. Faience independently of each other, as well as A .S.Russell), which made it possible to come to the idea that the charge of the nucleus of an atom is equal to the serial number of the corresponding element in the periodic system. In the same year, isotope separation was carried out by the gaseous diffusion method (F. Aston). In 1914, the separation of chemically indivisible radioelements was carried out using the diffusion method (G. Hevesy), the existence of stable isotopes of lead was proved (F. Soddy and others). In 1915, the method of labeled atoms was developed (D. Hevesy, F. Panet). In 1916, F. Panet introduced the concept of a chemical element. In 1917, higher-order isotopes were discovered - nuclear isomers (F. Soddy), and in 1918 the existence of isotopes among the products of radioactive decay was proved (J.J. Thomson).

In 1919, F. Aston built a high-resolution mass spectrograph and proposed an electromagnetic method for isotope separation (the operating principle of a mass spectrograph was proposed in 1907 by

J. J. Thomson), with the help of which isotopes were discovered in chlorine and mercury, and in 1920 the features of isotope exchange were established (G. Hevesy). In 1918, the possibility of the existence of nuclear isomerism was predicted (S. Meyer), and in 1921, using the example of 234Pa, the phenomenon of isomerism of atomic nuclei was discovered (O. Hahn). In 1923, D.Hevesy applied the method of labeled atoms to the solutionBekman I.N.

biological problems by conducting a study of the absorption of lead from solution by plants. By 1925, the phenomenon of isotopy had been proven for almost all stable elements (mainly due to the work of F. Aston). An important role in the characterization of isotopes was played by the curve of dependence of packing coefficients on mass numbers (the Aston curve).

The packing factor is a value equal to the ratio of the mass defect of the atomic nucleus to the mass number. It characterizes the value of the specific (in terms of one nucleon) binding energy of nucleons in the nucleus.

In total, by the 20s of the 20th century, 40 natural elements and isotopes were discovered, and a genetic relationship was established between them.

In 1911, F. Soddy published the book "Chemistry of Radioactive Elements", in which he described in detail a series of successive radioactive transformations of radium through radon to lead.

The first artificial transmutation of elements was carried out in 1918. By bombarding nitrogen atoms in the air with particles,

E. Rutherford was the first to carry out the artificial transformation of elements:

the nitrogen nucleus turned under the influence of -particles (nuclei of the helium atom) into an oxygen nucleus with the release of a hydrogen nucleus.

An important event was the discovery of the neutron (Chedwig, 1932) and artificial radioactivity (I. and F. Joliot-Curie, 1934).

The first radioactive isotopes discovered during the bombardment of various elements - particles were 13N, 30P and 27Si. By bombarding a sheet of aluminum with polonium particles, I. and F. Joliot-Curie observed using a Geiger-Muller counter that when the source was removed

Particles or when their energy decreases below a certain threshold, the emission of neutrons stops, however, the emission of positrons continues with a half-life of 3 min. The authors suggested that the nuclear reaction proceeds according to the scheme:

e + (stable) 13 Al+ 2 He0 n + 15 P 14 Si (3) They confirmed their assumption by dissolving irradiated aluminum in hydrochloric acid, followed by removal of the formed radioactive product (30РН3) by a gas stream. Similar results were obtained with boron, which was converted to radionitrogen, and with magnesium, which gave radioaluminum.

Before World War II, the possibility of artificial production of radioactive isotopes of almost all known stable elements was proved. Nuclear reactions were discovered, which made it possible to begin obtaining radioactive isotopes and the synthesis of new elements, including transuranic ones. In 1937, C. Perrier, E. Segre synthesized the first artificial element - technetium (by bombarding molybdenum nuclei with deuterons), E. Segre obtained astatine (1940), M. Perey discovered francium (1939), in 1940 E. Macmillan, P. Abelson synthesized 239Np (-emitter), and G. Seaborg, E. Macmillan, A. Wahl, J. Kennedy, E. Segre - plutonium (including 239 Pu.) In 1930, the isotope was discovered 238U (F. Aston), and in 1935 - 235U (A. Dempster). In 1947, one new element, promethium, was discovered in the fission products of uranium.

In 1940, the synthesis of neptunium (E. Macmillan, P. Abelson) and plutonium (G. Seaborg, A. Wahl, J. Kennedy, E. Segre) was carried out, pure 235U was isolated. Radioactivity and radiation. Radiochemistry. Volume 1 (J. Dunning, A.

Nir), it was proved that 235U is divided by slow neutrons (Y. Booth, J. Dunning, A. Gross) and the possibility of a chain nuclear fission reaction in a system with uranium and heavy water was predicted (X. Halban, L. Kovarsky). In 1944, the actinide theory was proposed, which plays an important role in the systematics and prediction of the properties of heavy transuranium elements (G. Seaborg). In 1946, the synthesis of the 95th and 96th elements - americium and curium was carried out (G. Seaborg, R. James, L. Morgan, A. Giorso), the fission constants of uranium were measured (J. Scharf-Goldhaber, J. Kleiber ).

In 1966, L. Lederman obtained antideuterium nuclei, and in 1970, Yu. Prokoshkin, antihelium nuclei.

In 1940-1953. G. Seaborg and others synthesized transuranic elements - plutonium, neptunium, americium, curium, berkelium, californium, einsteinium, fermium.

From the second half of the 20th century to the present day, the synthesis of heavy elements in the world has been and is being carried out by three research centers: in Dubna (Russia), in Berkeley (USA) and in Darmstadt (Germany). All elements, from 93rd (neptunium) to the recently discovered 117th, were obtained in these laboratories.

In 1987, the International Unions of Pure and Applied Chemistry (IUPAC) and Physics (IUPAP) created a joint international commission that considered the issue of priority in the discovery of new elements. In 2010, this commission gave names to new elements: element 104 was named rutherfordium (Rf) in honor of E. Rutherford; element 105 - dubnium (Db) in honor of the city in Russia where this and many other new elements were discovered; element 106 - seaborgium (Sg) in honor of the American physicist and radiochemist G. Seaborg, who participated in the isolation and synthesis of many new elements - from plutonium to mendelevium; element 107 - bohrium (Bh) in honor of the famous Danish physicist N. Bohr; element 108 is named hassium (Hs) in honor of Hesse in Germany, where the largest research center for the synthesis and study of new elements is located; element 109 - maitnerium (Mt) in honor of the Austrian researcher (physicist and radiochemist) Lise Meitner, who, together with O. Hahn, discovered the element protactinium and did many other important works that contributed to the establishment of the structure of the atom; element 110 - darmstadtium (Ds) in honor of the city of Darmstadt in Germany, where many new artificial elements were discovered; element 111 - roentgenium (Rg) in honor of W. Roentgen; element - copernicium (Cp) in honor of N. Copernicus. In 2004 - 2006 the fact of successful synthesis of elements with numbers 113, 114 and 116 was officially recognized, and in 2010 - elements 117 and 118. These elements do not yet have names.

1.4 Radioactive radiation After powerful sources of radiation appeared in the hands of researchers, millions of times stronger than uranium (drugs of radium, polonium, actinium), detailed studies of the properties of radioactive radiation were begun. First of all, the penetrating power of the rays was studied, as well as the effect of the magnetic field on the radiation. It turned out that the radiation is not homogeneous, but is a mixture of "rays". Gisel was the first to demonstrate the deflection of "Becquerel rays" in a magnetic field. P. Curie discovered that under the action of magnetic and electric

–  –  –

Radioactivity and radiation. Radiochemistry. Volume 1 Radium radiation also affects biological objects. In 1900

Gisel and Walhof pointed to the physiological effect of the new radiation. In 1911, Becquerel needed a radioactive substance for a lecture, he took it from the Curies, and put the test tube in his vest pocket. After giving a lecture, he returned the radioactive preparation to the owners, and the next day he discovered reddening of the skin on the body. Becquerel told P. Curie about this, he put on an experiment: for ten hours he wore a test tube with radium tied to his forearm. A few days later, he developed redness, which turned into an ulcer, from which he suffered for two months. Soon L. Matou (Becquerel's assistant) reported that radioactive radiation accelerates the germination of seeds. Then the healing properties of radiation were discovered: radium helped with cancer, lupus, and some other skin diseases. Thus, the foundations of a new method of treatment - radiation therapy were laid.

In 1906, characteristic X-ray radiation was discovered (C. Varila), and in 1908 it was shown that it is a fundamental property of the atom (C. Barkla, C. Sandler). In 1908, a device was created for registering individual charged particles (H. Geiger-W. Muller counter). In 1934, Walter Bothe developed the coincidence method.

In 1910, the first determination of the energy of -particles was carried out by their deviation in a magnetic field (O.Bayer, O.Gan). In 1911, E. Rutherford created the theory of scattering of -particles in matter. In the same year, it was shown that the decay constants of -emitters are related to the path length of -particles (Relation between the lifetime and the decay energy of radioactive nuclei - the Geiger-Natall law). In 1912, cosmic rays were discovered (V.

Geis) and invented a device for observing traces of charged particles (Ch. Wilson chamber). In 1913, a continuous spectrum of radiation energy was discovered (J. Chadwick), the identity of the X-ray spectra of isotopes was proved, and the equality of the serial numbers of the isotopes of a given element was finally confirmed (E. Rutherford, E. Andrade).

In addition to experimental advances, significant progress was made in the field of theoretical physics in the early 20th century. In 1900, M. Planck created the quantum theory. In 1903, A. Einstein introduced the concept of a quantum of light (photon) and created a special theory of relativity, in which he included the Poincaré formula: E=mc2, linking the mass (m) with the total internal energy (E) and the speed of light (c). Einstein proposed to check this law by determining the amount of energy released by radioactive substances. Experimental proof of the existence of the photon was obtained in 1923.

In 1923, the phenomenon of short-wavelength radiation scattering by a free or weakly bound electron (A.Compton effect) was discovered and a theoretical interpretation of this phenomenon was given (A.Compton, P.Debye); recoil nuclei were discovered (P. Blackett) - a photograph of the proton trace and the splitting of the nitrogen nucleus by particles was obtained. Recoil protons were identified by I. and F. Joliot-Curie in 1932. In 1929, the quantum theory of the Compton effect was created and an equation was proposed that describes the scattering of electrons in this effect (the Klein-Nishina equation). In the same year, O. Klein and I. Nishina derived a formula for the scattering of high-energy Beckman I. N.

photons on electrons, and N.Mott - the formula for the Coulomb scattering of relativistic electrons.

In 1934, the glow of pure transparent liquids under the action of gamma rays was discovered (the effect of S.I. Vavilov - P.A. Cherenkov). The theory of this effect was given by I.E. Tamm and I.M. Frank in 1937. In 1944, synchrotron radiation was predicted (D.D. Ivanenko, I.Ya. Pomeranchuk) discovered in 1946 by Blueit.

1.5 Types of decay As already mentioned, two types of decay were discovered at the turn of the century:

Decay and --decay, which are often accompanied by

radiation.

In 1911, G. Geiger and J. Nettol established a relationship between the lifetime and the decay energy of radioactive nuclei. In 1914 internal conversion was predicted (E. Rutherford), and in 1925 the Auger effect was discovered (P. Auger). In 1928, the theory of -decay as a tunneling process was developed (G. Gamow, E. Condon, R. Gurney).

In 1930, W. Pauli suggested that, during -decay, a particle is emitted that has an incomparably greater penetrating power than electrons. The walls of the calorimeter cannot stop it, and it takes some of the energy with it. This is how the concept of the neutrino was born.

The -decay theory was created in 1934 by E. Fermi, who suggested that an electron and a neutrino arise at the moment of decay of a nucleon in the nucleus, and postulated a new interaction - weak. He introduced a constant, which plays the same role for -decay as the charge for electromagnetic processes, and calculated its value on the basis of experimental data.

Fermi's theory made it possible to calculate the shape of the -spectra and relate the limiting decay energy to the lifetime of the radioactive nucleus. The neutrino in this theory had a charge equal to zero and zero mass. The modern theory of the unified weak and electromagnetic interactions includes the Fermi model as a first approximation.

In 1934, positron (+-decay) was discovered (I. and F. Joliot-Curie). In the same year, H. Bethe and R. Peierls predicted the reverse -decay. In 1934

the idea was put forward that the reverse decay is a process caused by a free neutrino (H. Bethe and R. Bacher). In 1935, double decay was predicted and its theory was developed (M. Geppert-Mayer), in 1935 - the capture of an orbital electron (H. Yukawa), and in 1936 - K-capture (X. Yukawa, S. Sakata ), which was discovered by L. Alvarez in 1937. In 1938, the conversion radiation of nuclear isomers was discovered (L. Rusinov, B. Pontecorvo), the emission of internal conversion electrons by substances that capture neutrons was discovered (J. Hoffman, R. Bacher). In 1935, L-capture was predicted, which was experimentally observed in 1949.

(B.M. Pontecorvo). In 1936, the existence of metastable states of nuclei was explained (K. Weizsacker).

One of the central events in the history of the study of radioactivity is the discovery of spontaneous and forced fission of uranium.

E. Fermi, exposing uranium to the action of slow neutrons, observed a weak activity, which he attributed to the formation of transuranium nuclei. O.Khan, L.Meitner and F. Strassmann, after conducting similar exRadioactivity and radiation. Radiochemistry. Volume 1 experiments confirmed this hypothesis and proposed several decay chains ending with ecagold. Irene Curie was also interested in the products resulting from the neutron irradiation of thorium and uranium. In collaboration with G. Halban and P. Preiswerk, she revealed the formation of two new radioactive nuclei. Then, together with P. Savich, among the products of uranium, I. Curie discovered a new emitter with a half-life of 3.5 hours, which chemically separated from elements considered as “transuranium” and exhibited the properties of lanthanide. It was not possible to identify it then (later it turned out that it was an isotope of lanthanum, a fragment of uranium fission).

To clarify the situation, O. Hahn and F. Strassman continued their experiments and discovered the formation of an alkaline earth product.

At first it was mistaken for an isotope of radium, but, in the end, it was possible to separate it from radium, but not from barium. Chemical evidence was given that neutron irradiation of uranium produces an element with an atomic number 36 units less than uranium. Thus, the completion of work on the irradiation of uranium with slow neutrons, begun in 1934 by E. Fermi, was the discovery by O. Khan and F. Strassmann in 1938 of the forced fission of uranium under the action of neutrons.

Comment. I. Noddak wrote about the possibility of nuclear fission in 1934, but her contemporaries did not pay attention to her prediction.

The results obtained by O. Hahn and F. Strassmann were interpreted by L. Meitner and O. Frisch in 1939 as the decay of a uranium nucleus into two fragments of approximately equal mass. L. Meitner introduced the concept of "nuclear fission". F. Joliot proved the division of uranium into two fragments. A.Golstein, A.Rogozinsky and R.Valen showed that fission is accompanied by neutron emission. O. Frisch, F. Joliot-Curie, G. Anderson and J. Dunning confirmed the fission of the uranium nucleus into two fragments and carried out a direct measurement of the fission energy. In the same year, based on the drop model, N. Bohr developed a qualitative (drop) theory of nuclear fission, and together with J. Wheeler gave a quantitative interpretation (introduced the parameter Z2/A) and predicted the possibility of spontaneous fission of uranium. Fission, as a type of radioactive decay, was experimentally discovered by K.A. Petrzhak and G.A. Flerov.

In 1935, nuclear isomerism was discovered in natural (O. Khan, 1921) and artificial (I.V. Kurchatov, B. Kurchatov, L. Mysovsky, L. Rusinov, 1935) isotopes, and in 1936 the reason for nuclear isomerism was explained (G. VanVurgis). In 1934, internal conversion of -beams with the formation of electron-positron pairs was discovered (A.I. Alikhanov and others). In 1939, W. Farry suggested the possibility of a neutrinoless double decay.

In 1947, G. Baldwin and G. Klaiber observed a giant resonance in nuclear reactions under the action of photons. Discovered in 1948

Neutron decay (A. Snell and L. Miller), and in 1949 the dependence of the electron capture rate on the chemical state was established (E. Segre).

In 1951, proton radioactivity was predicted (B.S. Dzhelepov). In 1952

recoil nuclei arising from electron capture in argon were registered (J. Rodebak, J. Allen), the law of conservation of momentum during neutrino emission was proved. In 1957, longitudinal Beckman I.N. was experimentally discovered.

polarization of -particles in -decay: +-particles correspond to the left screw,

--particles - right.

In 1961, the existence of two types of neutrinos - electron and muon (L. Lederman, M. Schwartz, J. Steinberger) was proved,

The decay of a positive pion, the phenomenon of the emission of delayed protons was discovered (V.A. Karnaukhov, J. Cherny, 1970, Z. Hoffman, 1982). In 1967, double -decay and double bremsstrahlung were discovered. In 1970, proton radioactivity was discovered (by J. Cherny), and in 1984, cluster decay.

The assumption about the possibility of proton emission in radioactive decay arose as early as 1915 in the laboratory of E. Rutherford. In 1951

B.S. Dzhelepov calculated the possibility of proton decay of neutron-deficient nuclei, and in 1958 V.A. Karnaukhov estimated the limits of nuclear stability with respect to proton decay. In 1962, a team of physicists (V. A. Karnaukhov, G. M. Ter-Akopyan, V. G. Subbotin and L. A. Petrov), working at the JINR Heavy Ion Accelerator (Dubna), discovered proton decay:

emission of delayed protons. This kind of radioactivity was discovered by analyzing the properties of radioactive products obtained by irradiating nickel with a beam of neon nuclei. The emission of delayed protons is a two-step process. At the first stage of this process, the proton-rich nucleus undergoes proton decay. The resulting daughter nucleus is excited and decays, emitting a proton. In 1963, R. Burton and R. MacPherson identified a delayed proton emitter using 25Si as an example. In 1970, J. Cherny (Berkeley, USA) observed proton activity - the decay of the excited (isomeric) state of the 53mCo nucleus.

A delayed proton emitter was discovered at the JINR (Dubna) by irradiating nickel with a beam of accelerated 20Ne ions (1962). Almost simultaneously, the same emitters were discovered among light nuclei. By 1991, more than 100 emitters were discovered, the lightest of which is 9C (T = 0.13 s), the heaviest l83Hg (T = 8.8 s). For the first time, weak proton activity was observed when 96Ru was irradiated with a 32S beam (JINR, 1972). In 1981, S. Hofmann (Heavy Ion Research Center, Germany) discovered the proton radioactivity of the ground state of 151Lu and 147Tm. Today, more than 25 isotopes are known to decay from the ground (or isomeric) state through this channel.

In 2002, the process of simultaneous emission of two protons (two-proton decay), predicted in 1991, was observed for the first time. It was discovered in the 45Fe isotope in experiments at GSI and GANIL (Caen, France).

In 2005, it was found that 54Zn also undergoes a two-proton decay.

If proton activity is one of the types of radioactive transformations predicted theoretically, then the discovery of spontaneously fissile isomers is an example of surprises in the history of the study of radioactivity. The phenomenon of spontaneous fission of nuclei in the isomeric state was discovered in 1961 (S.M. Pelikanov, V.A. Druin, V.A. Karnaukhov) using the 242Am isomer as an example.

In 1984, independent groups of scientists in England (H. Rose, G. Jones) and Russia (D.V. Aleksandrov) discovered the cluster radioactivity of certain heavy nuclei that spontaneously emit clusters - atomic radioactivity and radiation. Radiochemistry. Volume 1 nuclei with atomic weights from 14 to 34. At present, 25 nuclei from 114Ba to 241Аm are known to emit clusters of the 14С, 20О, 24Ne, 26Ne, 28Mg, 30Mg, 32Si and 34Si types from the ground states. The energies of the relative motion of the emitted cluster and the daughter nucleus vary from 28 to 94 MeV.

1.6 Structure of the nucleus and nuclear reactions In 1911, the atomic nucleus was discovered and a planetary model of the structure of the atom was proposed (E. Rutherford). The nuclear model arose from the experiments of H. Geiger and E. Marsden on the scattering of -particles by various substances, for the interpretation of which Rutherford derived a formula for the scattering of charged particles in a Coulomb field. The model assumes that the atom contains a central positively charged nucleus, in which almost the entire mass of the atom is concentrated, and electrons rotate around the nucleus at a considerable distance. This model served as an important prerequisite for the physical substantiation of the law of periodicity.

In 1913, Niels Bohr proposed a quantum model of the atom. In 1924 V.

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I.N.Bekman PLUTONIUM Tutorial

INTRODUCTION

Plutonium is the first element artificially obtained by man. Enough
it quickly proved to be one of the most important components of the nuclear industry. On the
all modern atomic weapons are based on it, and its isotopes are widely
used in sources of electrical energy, heat, light and ionizing
radiation. Plutonium is associated with the prospects for the development of a large nuclear
energy. He found his place in medicine. But also its name, referring
he also justifies us to the underworld and hell. And not only with their own affairs
to Nagasaki ... Many want to get rid of him as soon as possible, and -
forever and ever.
94
Pu
PLUTONIUM

5f67s2

2
8 24
32 18
82

Plutonium (lat. Plutonium), Pu, a radioactive chemical element,
group III of the periodic system, atomic number 94, atomic weight 244;
belongs to actinides, has no stable isotopes. First
artificial element obtained in available for weighing
quantities (1942). Plutonium was found in nature after it was
synthesized artificially. Currently belongs to the group
actinides of the Periodic Table of the Elements.

Periodic system of elements
H
He
Li Be
B C N O F Ne
NaMg
Al Si P S Cl Ar
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
Fr Ra ** Rf Db Sg Bh Hs Mt Ds RgUubUutUuqUupUuhUus Uuo
* La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
** AcTh Pa U Np Pu AmCmBk Cf Es Fm Md No Lr
The world learned about the discovery of plutonium after the atomic bombing of St.
Nagasaki in 1945. No other element became known with such suddenness and
under such dramatic circumstances. Moreover, none of the elements has
such exceptional properties. Suffice it to say that he has six
allotropic modifications in a relatively small temperature range - from
room temperature to a melting point of 640°. The metal also has
unique property to noticeably shrink with increasing temperature in
relatively wide temperature range. Plutonium is highly toxic. Him
many isotopes and almost all are fissile.
Plutonium isotopes are formed in nuclear explosions, but by the main methods
synthesis are two groups of techniques: the use of charged particles is large
energies, such as deuterons and helium ions accelerated in a cyclotron, and
use of nuclear reactions in self-sustaining chain nuclear
reactors.
In this review (educational material for students of radiochemistry at Moscow State University and for
all participants in the system of Internet education in the NUCLEAR SPHERE), we will consider
nuclear, physical, chemical, mechanical and toxic properties of isotopes
plutonium and briefly dwell on the methods of their production and application in
industry, energy, science and medicine, as well as discuss methods for its
qualitative and quantitative analysis in various environments. A main attention
we will focus on the uranium-plutonium cycle and the prospects for its development.
The focus will be on the following points:
- A third of the energy produced by nuclear power plants in the world comes from plutonium, which is formed in nuclear reactors as
by-product of nuclear reactions.
- Plutonium once existed in the earth's crust, but now it is practically gone.

In the biosphere now
located
some
plutonium
as a result of nuclear weapons testing in
I.N.Bekman
PLUTONIUM
Training tons
allowance
http://profbeckman.narod.ru/Pluton.htm
1950s and 1960s.
- Plutonium is radiologically dangerous, especially when inhaled, it must be handled with great care.
caution.
- Plutonium extracted from weapons and spent fuel from nuclear reactors can become
a powerful source of energy, if it can be included in the nuclear fuel cycle.
Plutonium is an element with unique nuclear, physical, chemical and radiological
properties. In this tutorial, we will try to demonstrate how these properties are used.
in the synthesis of this element, its compounds and alloys, in applications in weapons, energy and medicine, and in
its disposal. We will also discuss the current contradiction between the Concept
sustainable development, requiring the expansion of production of all types of energy, including nuclear, and,
consequently, the production of plutonium (the use of plutonium as a fuel in power nuclear
reactors increases the world's energy reserves from burning uranium by more than 100 times) and the Concept
international security, providing for the removal of plutonium from the fuel cycle and its complete
destruction.

1. HISTORY OF DISCOVERY

In 1940, E. Macmillan and P. Abelson, conducting experiments on the cyclotron of the Radiation Laboratory
Lorenz (University of California at Berkeley), discovered the formation of neptunium in uranium irradiated
neutrons generated in beryllium by deuterium ions accelerated to high energies. (For more see
textbook NEPTUNIUS). It turned out that 239Np, formed during the β-decay of 239U, in its
queue undergoes β--decay, i.e. goes to the element one cell to the right (we now
we call plutonium). However, they could not identify the new element due to its long period.
half-life and low specific activity. This was done by radiochemists from the same University under
directed by Glenn Seaborg.
The first isotope of plutonium with a mass number of 238 was identified during the study
Seaborg group of chemical properties of indicator quantities of neptunium.
In the autumn of 1940, Glenn Seaborg, as head of the Department of Chemistry
University of California (Berkeley), commissioned a recent UC alumnus Arthur Wahl (Arthur Wahl - in
In Russian literature, he is sometimes written as Val, then as Walkh, then as Volch, don’t be surprised!) as
dissertation work to consider the possibility of studying the chemical properties of traces of element 93
(neptunium), search for and identify element 94 (plutonium). Work done with John
Kennedy, who was also one of the leaders of the Department of Chemistry. During the experiment, uranium oxide was directly irradiated at the Berkeley cyclotron with accelerated deuterons.
Official discovery of the plutonium isotope 238Pu, with a half-life of ~90 years (86.4 g),
attributed to Glenn Seaborg, Edwin McMillan, John F. Kennedy
(Kennedy), and Arthur Walh (Arthur Wahl). A year later, another isotope was discovered - 239Pu with T = ~24000 years. In 1951
Seaborg and Edwin McMillan received the Nobel Prize in Chemistry "for their discoveries in the field of
chemistry of transuranium elements. (By the way, Seaborg is the only chemist
having a patent for the discovery of an element, even two: americium and curium).
A photo. Glenn T. Seaborg (04/19/1912 - 02/25/1999) American chemist and physicist -
head of the Berkeley University plutonium research team.
Nobel Prize in Chemistry (1951).
Comment. When the Swedish Academy of Sciences in 1951 announced the award
Nobel Prize in Chemistry to E. McMillan and G. Seaborg for discoveries in the field
chemistry of transuranium elements, many decided that two professors from California
worked together. However, Seaborg and Macmillan were never full time employees.
sense of this word. Moreover, Macmillan belongs to the glory of the discoverer, and
Seaborg - the successor of the work begun.

The name of the element was proposed in 1948: McMillan called
the first transuranic element is neptunium due to the fact that the planet Neptune is the first behind Uranus. By
By analogy, they decided to call element 94 plutonium, since the planet Pluto is the second planet after Uranus. Pluto,
opened in 1930, got its name from the god Pluto (aka Hades) - the ruler of the underworld
the dead in Greek mythology.
At the beginning of the 19th century, a professor from Cambridge, Clark, proposed to rename barium to plutonium,
arguing that barium is not at all heavy, as its Greek name declares, and, moreover, its
obtained by electrolysis, which means that the name must contain fire, the real fiery Hyena,
those. hell and its chief - the god Pluto. However, this proposal was not accepted. Incidentally, the element symbol
wrong - should be Pl, but Seaborg chose Pu, remembering the exclamation of a child smelling something
opposite: "Pissed!" ("Pee-yoo!"). Seaborg expected that his initiative would be met with hostility, but
The Element Naming Committee agreed without any comment. During the Manhattan
project plutonium in secret documents was called "element 49": 4 - the last
digit of the element number in the Periodic table (94), 9 is the last digit
atomic weight of weapons-grade plutonium-239.
Rice. 1. The world's first 520 milligrams of metallic plutonium,
produced by Ted Magel and Nick Dallas in
Los Alamos 03/23/1944.
For the first time bombardment of oxide 238U (U3O8) by accelerated deuterons
in a 60-inch cyclotron up to an energy of 22 MeV was carried out on December 14, 1940. Before
than to hit a uranium target, the deuterons passed through 0.002-inch aluminum foil.
A carefully isolated fraction of element 93 (neptunium) contained α-activity, the absorption curve

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Which in aluminum differed markedly from the absorption curve of the sample of the 2.3-day isotope 93238,
obtained under identical conditions. Then an increase in the number of α-particles was found, which could be caused by
element 94 (plutonium), which is a daughter product of the two-day element 93. Physical and
chemical research continued for two months, and on February 24, 1941, decisive experiments were carried out on
oxidation of the putative element 94 with peroxide disulfate ions and silver ions as
catalyst.
Identification of the isotope showed that it is the reaction 92 U 238 (d, 2n) 93 Np 239:

U+12H → 239
93 Np + 2n
with subsequent decays of the neptunium-239 isotope:
β − 2 .1 day
α , 86 , 4 years
239
⎯→238
93 Np ⎯⎯ ⎯
94 Pu ⎯⎯ ⎯⎯→
In May 1940, the properties of plutonium were predicted by Louis Turner.
In 1941 and early 1942, the chemical properties of
plutonium with indicator amounts. It was found that the highest oxidation state can be
obtained by treating the lowest oxidation state with oxidizing agents such as persulfate in the presence of
silver ions, dichromate or potassium permanganate. The lowest valency state of plutonium is obtained by
reduction with sulfur dioxide or bromide ion. Plutonium in aqueous solutions is not
is reduced to metal with zinc and that plutonium does not form volatile tetroxide.
The stable lowest state of plutonium is tetravalent, since it coprecipitates with Th(JO3)4. For
ethereal extraction was used to separate large amounts of uranyl nitrate from plutonium.
It turned out that plutonium in its highest degree of valence is similar to hexavalent uranium, and in its lowest degree to tetravalent uranium and thorium.
In 1941, by irradiating large amounts of uranium salt with fast neutrons generated by
at the cyclotron, the more important isotope of plutonium, 239Pu, was produced, with a half-life of 24,000 years.
Kennedy, Seaborg, Wahl and Segre found 239Pu as a decay product of 239Np. To obtain 239Np, we took 1.2 kg
uranyl nitrate, distributed in a large block of paraffin placed behind a beryllium target of a 60-inch cyclotron, and irradiated for two days with neutrons obtained using a deuteron beam.
The neutron-irradiated uranyl nitrate was processed in an extraction glass plant, with
using diethyl ether as an extractant. 239Np was isolated using a redox cycle. Lanthanum and cerium fluorides were used as carriers; for removing
uranium residues, the reprecipitation process was repeated six times. 03/28/1941 it was proved that 239Pu
undergoes fission by slow neutrons with a cross section exceeding the cross section for 235U, and
neutrons obtained in the fission process are suitable for obtaining the following fission events, i.e.
allow counting on the implementation of a nuclear chain reaction. Work began immediately on
building the plutonium atomic bomb.
Research conducted at the University of California in 1941-42 made it possible to accumulate
significant data on the chemical properties of plutonium, and in 1942 a pure compound was obtained
plutonium.
The next stage in the history of plutonium is associated with its production in large quantities, which became
possible after the construction and start-up on December 2, 1942 by Fermi and Szilard of the atomic uranium-graphite reactor,
which turned out to be a powerful source of thermal neutrons. For the synthesis of the 239Pu isotope, two
nuclear reactions:
235

238
U + n → 239 U → 239 Np → 239 Pu
The reactor consisted of blocks of metallic uranium, uranium oxide (all of natural isotopic composition)
and graphite. The reactor was built by employees of the metallurgical laboratory on a tennis court under
stands at the University of Chicago stadium. Since cooling and protection against
radiation, the power was limited to 0.5 watts (at times - hundreds of watts). This power is enough
for the production of significant quantities of plutonium compared to what can be obtained with
bombardment at the cyclotron. This reactor was dismantled and reassembled in Argonskaya
metallurgical laboratory, where he worked in a more intensive mode, but to develop a weapon
plutonium, it has never been used.
238
92

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Rice. 2. The first industrial uranium-graphite reactor in Hanford (Washington, USA).
The first pure chemical compounds of plutonium, free from carrier materials and other
foreign contamination, obtained on August 18, 1942 by Cunningham and R. Werner by processing plutonium
concentrate in 10 mg of rare earths (Се4+, La3+). It was possible to synthesize Pu(OH)4 hydroxide in the amount of 5
μg, double fluoride 239Pu and iodate 239Pu. The first weighing of a pure plutonium compound took place
09/10/1942, when 2.77 micrograms of plutonium oxide were weighed by Kanningen and Werner. First
large-scale operations to separate plutonium from several hundred pounds of irradiated uranium into
cyclotrons at the University of California and Washington were carried out in the summer of 1942 by Komen and
Jaffe. Plutonium was separated from uranium and fission products by extraction with diethyl ether.
The final purification of plutonium was carried out by the lanthanum-fluoride method.
As a result of these operations, several hundred micrograms of plutonium were obtained. Until autumn 1943
cyclotron bombardment was the only source of plutonium, and for the entire period of research from
At the beginning of the discovery of plutonium, about 2000 micrograms or 2 mg of plutonium were obtained. Research by radiochemists
created the basis for the further development of the process used to separate plutonium from uranium and
fission products under industrial conditions.
When gram quantities of plutonium became available, the main research was moved to Los Alamos.
The first reactor to produce plutonium was the Oak Ridge (Tennessee) reactor,
containing tons of uranium metal. Uranium could be removed and replaced with new. The reactor was cooled
air blast. It was launched in October 1943, in January 1944 it produced milligram quantities
plutonium, and in February 1944 delivered it already in grams.
Cooled
water uranium-graphite reactors. This plant began producing plutonium in early 1945. The difference in scale
between the laboratory tests and the first plant in Hanford was valued at a factor of 109.
The created production made it possible to obtain plutonium on February 2, 1945 in quantities sufficient for
making several atomic bombs.
In our country, the history of weapons-grade plutonium began in December 1946, when in Moscow on the territory
Laboratory No. 2 (now the State Scientific Center "Kurchatov Institute") in Pokrovsky-Streshnevo, created under
the leadership of I.V. Kurchatov a small nuclear reactor F-1. Chemical processing of irradiated
reactor of uranium blocks - rods 100 mm long and 32 or 35 mm in diameter in an aluminum shell -
first tested on the U-5 unit in the nearby NII-9 (now the Federal State Unitary Enterprise Research Institute of Inorganic
materials to them. A.A. Bochvara). Then, not far from the city of Kyshtym near Chelyabinsk, Combine No. 817 was launched
(now PA "Mayak"), which included three plants: "A" - a nuclear reactor, "B" - a radiochemical
plant and "B" - metallurgical plant. The first industrial nuclear reactor began to operate at full capacity
power 06/22/1948, the dissolution of irradiated blocks at plant "B" began on 12/22/1948, and the first ingot
plutonium metal - weighing only 8.7 g - was obtained by reduction of plutonium chloride at the plant
"B" 04/14/1949. Plutonium for the manufacture of the RDS-1 atomic bomb consisted of two hemispheres of a common
weighing 6 kg, covered with a thin nickel film. In the middle of 1949 they were sent to Arzamas-16 (KB-11,
now the All-Russian Research Institute of Experimental Physics), and then to the Semipalatinsk test site. Only at the training ground
the final assembly of the bomb: a polonium-beryllium source was mounted in its central part
neutrons. The first test explosion was carried out on August 29, 1949.

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Chemical processing of irradiated uranium blocks (enclosed in aluminum containers)
consisted in dissolving them, separating uranium and plutonium from the bulk of radioactive products
fission, separation of uranium and plutonium and isolation of their compounds in pure form. Content of plutonium in
irradiated blocks was 100-200 g per ton of uranium. The blocks were highly radioactive and contained
a large number of γ-emitters (the inner surface of some radiochemical devices
plant absorbed so much radioactive substances that it emitted a faint, but noticeable glow in the dark).
The dissolution of the blocks was carried out in nitric acid. The process was accompanied by the release of brown vapors of poisonous
nitrogen oxides and radioactive isotopes of iodine, krypton and xenon. To reduce the volume of solutions
first, aluminum was dissolved in a weak acid, and then the concentration was increased and transferred into a solution
Uranus. Plutonium was extracted from nitrate solutions by precipitation methods. Poskol

“1 I.N.Bekman NUCLEAR INDUSTRY Special course. Lecture 20. PRE-REACTOR PART OF THE URANIUM FUEL CYCLE Contents. 1. URANIUM ORE MINING 1 1.1 Uranium...»

I.N.Bekman

NUCLEAR INDUSTRY

Special course.

Lecture 20. PREREACTOR PART OF THE URANIUM FUEL CYCLE

1. URANIUM ORE MINING 1

1.1 Uranium mines and uranium reserves 1

1.2 Processing of uranium ore. 6

2. PRODUCTION OF METAL URANIUM 9

3. ENRICHMENT OF URANIUM. ten

3.1 Uranium hexafluoride 10

3.2 Isotope separation methods 11 3.2.1 Isotope separation 11 3.2.2 Isotope effects 13 3.3.3 Gaseous diffusion. 14 3.2.4 Diffusion in a vapor stream (countercurrent mass diffusion) 16 3.2.5 Thermal diffusion 16 3.2.6 Gas centrifugation 17 3.2.7 Electromagnetic separation. 17 3.2.8 Chemical enrichment 18 3.2.9 Aerodynamic separation 18 3.2.10 AVLIS (evaporation using laser). 18 3.2.11 Distillation 19 3.2.12 Electrolysis 19 3.2.13 Isotope exchange 19

3.3 Separation of uranium isotopes 21 The uranium fuel cycle is the main cycle of modern nuclear power engineering. It consists of three parts: pre-reactor, reactor and post-reactor.

In this lecture, we will consider the pre-reactor part of the uranium fuel and energy cycle, which includes such stages as the extraction and enrichment of uranium ore, the production of metallic uranium and its oxides, and the enrichment of uranium with the uranium-235 isotope (Fig. 1). Let us briefly dwell on the problem of isotope separation (not only uranium (fuel), but also hydrogen isotopes (as applied to neutron moderators) and boron (as applied to neutron absorbers). All issues related to reactor materials science (manufacturing of fuel elements, fuel assemblies, moderators , control rods, etc.) will be discussed in the 29th lecture.

1. URANIUM ORE MINING The initial stage of the nuclear fuel cycle (NFC) is the extraction of ore and the production of uranium concentrate. The main stages of the stage: the actual extraction of uranium-containing ore; its mechanical enrichment by removing waste rock; grinding the resulting ore mass; leaching uranium from it with sulfuric acid or sodium carbonate; obtaining uranium concentrate by extracting uranium solutions (extraction, sorption or selective precipitation); drying of uranium concentrate and its hermetic packaging.

1.1 Uranium mines and uranium reserves Uranium is a metal, about the same as tin or zinc, found in most rocks and even in sea water. Some typical concentrations of uranium in various environments are given in the table (ppm - ppm, one part per million).

Table 1 Concentration of uranium in various natural media High-grade sources 2% U or 20000 ppm U Low-grade sources 0.1% U or 1000 ppm U Granite 4 ppm U Rocks 2 ppm U Average amount in the earth's crust 1.4 ppm U Sea water 0.003 ppm U Up to World War II, uranium was considered a rare metal. It is now known that uranium is more common than mercury, cadmium, silver and is found in industrial ores in approximately the same concentrations as arsenic or molybdenum. Its average concentration in the earth's crust is about 2 parts to 1 million, which, translated into the language of weight units, is equal to billions of tons! It ranks 48th in terms of content in crystalline rocks. In the lithosphere, uranium is more abundant than such inexpensive substances as zinc and boron, which occur in concentrations of 4 g/t. The content of uranium in granite rocks is quite sufficient for the radioactive gas radon, a decay product, to pose a serious biological hazard in places where granite comes to the surface. Uranium is also found in sea water, at a concentration of 150 µg/m3.

Uranium is not found in the form of powerful deposits, but a large number of minerals containing uranium are known:

carnotite, otenitis, uraninite, torbernite, tyuyamunite. Uranium occurs in sufficient concentration in 150 different minerals, and in small amounts in another 50. It was originally found in igneous hydrothermal veins and pegmatites, including uraninite and pitch blende. These ores contain uranium in the form of dioxide, which, depending on the degree of oxidation, has an average composition from UO2 to UO2.67. Other ores of economic importance: autanite, calcium hydrate uranyl phosphate; tobernite, hydrated copper uranyl phosphate; coffinite, hydrated uranium silicate; carnotite, potassium hydrate uranyl vanadate. Uranium ores are found all over the world. Stocks and commercial transactions are expressed in equivalent mass U3O8 plant. Deposits of tar blende, the fuel elements of the ore richest in uranium, are located mainly in Canada, the Congo and the USA.

–  –  –

Fig.2 Location of uranium deposits on the territory of Russia Balance deposits. Streltsovsky uranium ore region: Streltsovskoye, Luchistoye, Shirondukuevskoye, Tulukuevskoye, Oktyabrskoye, Dalneye, New Year's, Yubileinoye, Five-year-old, Spring, Antey, Argunskoye, Martovskoye, Malotulukuevskoye, Zherlovoe.

Zauralsky uranium ore region:

Dalmatovskoe. Off-balance deposits. Ergeninsky uranium ore region: 1 - Stepnoe.

Zauralsky uranium ore region: 3 - Dobrovolnoe. Republic of Khakassia; 5 - Primorskoye. Republic of Buryatia, Vitim uranium ore region: 6 – Khiagda; 7 - Radionovskoye; 8 - Vitlauskoe; 9 - Kolichikan; 10 - Dzhilindinskoe; 11 - Tetrakhskoye; 12 - Vertex; 13 - Inaccurate; 14 - Koretkondinskoe;

15 - Namaru; 16 - Dybryn. Outside the Vitimsky district: 17 - Imskoye; 18 - Buyanovskoye. Chita region: 19 - Gornoe; 20 - Birch; 22 - Durulguevskoe. Streltsovsky uranium ore region: 23 - Tsagan-Toron; 24 - Southwestern; 25 - Shirondukuevskoe; 26 - Riverless. Republic of Sakha-Yakutia, Elkon uranium ore region: 27 - Yuzhnoe; 28 - North; 29 - Central zone; 30 - Spring zone;

31 - Agda zone; 32 – Flat zone; 33 - Neva zone; 34 For the extraction and processing of uranium, mining and processing enterprises have been built near the explored deposits: Priargunsky industrial mining and chemical association (Krasnokamensk, Chita region, Russia), Vostochny mining and processing plant (Zhovtiye vody, Ukraine), Caspian mining and smelting plant (Aktau , Kazakhstan), Tselinny Mining and Processing Plant (Stepnogorsk, Kazakhstan), Yuzhnolimal Production Association (Bishkek, Kyrgyzstan), Navoi Mining and Smelting Plant (Navoi, Uzbekistan) and Vostochny Rare Metals Industrial Plant (Chkalovsk, Tajikistan). In addition to the Priargunsky PCU in Russia, the mining and processing of uranium and thorium ores was previously carried out by the Lermontov Production Association Almaz (Stavropol Territory) and the Novotroitsky Mining Administration.

As of 01.01.1999, the state balance of uranium reserves of Russia included the reserves of 16 deposits, of which 15 are concentrated in one area - Streltsovsky in Transbaikalia (Chita region) and are suitable for mining. The Dolmatovskoye deposit with a uranium content of 0.06% in the ore, located in the Trans-Urals (Kurgan region), is suitable for extraction by the method of borehole underground leaching. In 1999, a pilot plant for in-situ leaching with a capacity of 50 tons of uranium per year was put into operation here. One of the best deposits (Tulukuevskoye) with reserves of rich ores for open-cast mining has practically been worked out. In recent years, the trend towards the redemption of the best reserves of deposits has increased dramatically. So, in 1998, reserves with an average grade of 0.419% were redeemed. The reserves remaining in the subsoil with such a content make up only 54% of those listed on the balance sheet as "active".

Today TVEL Corporation is the only company that mines natural uranium in Russia. It includes three subsidiaries: the Priargunsky Mining and Chemical Association in the city of Krasnokamensk, Chita Region (mines 3,000 tons of uranium per year), CJSC Dalur in the Kurgan Region, and OJSC Khiagda in Buryatia. The last two enterprises have not yet reached their design capacity, construction is still underway there and at the same time exploration of adjacent deposits. It is planned that each of them should produce by 2012 1,000 tons of uranium per year.

Thus, in Russia there is still the only intensively developed uranium deposit - the Krasnokamensky mine in Transbaikalia (Chita region). The average content of uranium in the ore is 0.38%, the annual productivity of the mine is 2.5 thousand tons of uranium. At the current level of production, the ore reserves at this deposit create a 20-year supply of raw materials for the operating uranium mining enterprise (Priargunskoye Industrial Mining and Chemical Association OJSC). Uranium is mined here. That is, uranium ore is mined in underground mines at a depth of up to 800 meters, then it is delivered to the plant and uranium is extracted using various chemical processes, which is then processed to obtain fuel in the form in which it is needed.

In the Kurgan region (Dolmatovskoye deposit) and at the Khiagdinskoe uranium deposit in Buryatia, underground borehole leaching of uranium is used. Special acid solutions are pumped into one well, which leach uranium, and through another well, all this is extracted upwards with a pump. The costs for this method are much less than for laying a mine, and the leaching technology ensures the environmental safety of the work. In Buryatia, now the volume of production is 1.5 thousand tons of uranium concentrate per year. The explored reserves of the deposit are calculated for 50 years.

Comment. In Canada, the concentration of uranium in the ores of deposits mined underground is 100 times higher than what is in the bowels of the Priargunsky plant. Accordingly, to get exactly the same amount, we need to extract 100 times more.

Now a project is being prepared to develop new deposits in Yakutia. There are also very large deposits in Russia. They were discovered and explored back in the 70s of the last century, but then they were put in reserve. The state balance includes reserves of 38 uranium deposits classified as off-balance (ie explored, but not developed). Among the latter, the reserves of the Elkon and Ergeninsky uranium ore regions, considered as reserves, stand out. Thus, in the Elkon region in the Republic of Sakha-Yakutia, uranium reserves (more than 200 thousand tons) quantitatively exceed all balance reserves in the country, but due to the ordinary quality of ores, they can become profitable only at a high price for uranium. Promising regions include the Onega region (Karelia), where reserves of vanadium ore containing uranium, gold and platinum have been discovered; Vitimsky region (Siberia) with explored reserves of 60 thousand tons at a concentration of uranium of 0.054% in ore with accompanying scandium, rare earth elements and lanthanides;); The West Siberian region (Malinovskoye deposit with reserves of 200 thousand tons of uranium), as well as the Yenisei-Zabaikalsky region and the Far East ore-bearing region, located in the coastal zone of the Sea of ​​Okhotsk.

Ukrainian uranium deposits are considered one of the richest. Deposits of this element are located in the Kirovograd region: Vatutinskoye, Michurinskoye, Zheltorechenskoye.

Of the CIS countries, the most promising is the recently built Kyrgyz Navoi Mining and Metallurgical Combine (NMMC), which in 2001 began developing a large uranium deposit. Ore processing is carried out by a plant in Uchkuduk. In 2005, the joint Kazakh-Kyrgyz-Russian mining enterprise Zarechnoye, which is still under construction, produced its first products. At the first stage (2003-2006) JV Zarechnoye will produce up to 500 tons of uranium per year. Subsequently, its volumes will increase to 700-800 tons. At the same time, uranium concentrate produced at our KGRK will be supplied to Russia. In Kazakhstan, uranium mining has recently begun at the South Moinkum, Akdala and South Karamurun mines. The Chu-Sarysu uranium ore province in South Kazakhstan is unique in terms of uranium reserves. The exceptional value of this region lies also in the fact that it is possible to mine uranium in the deposits of this province by the method of underground leaching through boreholes.

Fig.3. At the uranium mine

Recall that in Russia there are currently 29 nuclear reactors at nuclear power plants, which consume more than 4,000 tons of uranium per year. Most of the uranium deposits of the former Soviet Union remained on the territory of Kazakhstan and Uzbekistan. The shortage of uranium in Russia is replenished by the strategic reserves of this raw material, made back in Soviet times, and by mining at the Chita mine.

1.2 Processing of uranium ore.

Minerals from which uranium is mined always contain elements such as radium and radon.

Therefore, although uranium itself is weakly radioactive, the ore that is mined is potentially dangerous, especially if it is a high-quality ore. The radiation hazard associated with accompanying elements is typical not only for uranium-containing ores, but also for any mining industry. Often uranium is mined in an open pit, in which the quarries have good natural ventilation. The underground uranium mine is ventilated with special powerful devices.

The final products of the processing of ore raw materials are pure chemical compounds from which metallic uranium is obtained.

Uranium ores usually contain a small amount of uranium-bearing mineral (0.05-0.5% U3O8), so that preliminary extraction and enrichment is necessary. Mechanical methods of enrichment (radiometric sorting, separation in heavy suspensions, gravity, flotation, electrostatics) are not applicable for uranium, hydrometallurgical methods are used - leaching is the usual first step in ore processing (sometimes it is preceded by annealing).

In the classical acid leaching process, the ore is first crushed in a special way and roasted for dehydration. In this case, carbon-containing fractions are removed, uranium is sulfated, and reducing agents, which can be an obstacle to leaching, are oxidized. The mixture is then treated with sulfuric or nitric acids, or a mixture of these acids.

Solid particles remaining after the dissolution of uranium are removed and placed for long-term storage in special tanks. The tanks are designed to securely store these materials. Such waste contains the bulk of the radioactive substances found in the ore (such as, for example, radium).

Uranium passes into uranyl sulfate, radium and other metals in uranium pitch precipitate in the form of sulfates. With the addition of caustic soda, uranium precipitates as sodium diuranate Na2U2O7.6H2O (often uranium is precipitated as ammonium diuranate or uranyl hydroxide).

When processing ores and poor concentrates, solutions contain only 0.5 - 2 g U per l. In this case, sorption on ion-exchange resins, extraction with organic solvents (alkyl phosphoric acids, amines are used for extraction from sulfuric acid solutions), or evaporation are widely used to extract and concentrate uranium. During solvent extraction, uranium ore is removed from the acidified rock leach liquor using a mixture of solvents, such as a solution of tributyl phosphate in kerosene. In modern industrial methods, alkyl phosphoric acids (eg, di(2-ethylhexyl)-phosphoric acid) and secondary and tertiary alkylamines appear as solvents. As a general rule, solvent extraction is preferred over ion exchange methods when the uranium content in the solution after acid leaching is greater than 1 gram per liter. However, it is not applicable to the reduction of uranium from carbonate solutions. Weapon-grade uranium is usually obtained from sodium diuranate through further purification using the tributyl phosphate refining process. Initially, Na2U2O7.6H2O is dissolved in nitric acid to prepare the raw solution. Uranium is selectively removed from it by diluting the solution with tributyl phosphate in kerosene or another suitable hydrocarbon mixture. Finally, uranium passes from tributyl phosphate into acidified water to isolate highly purified uranyl nitrate.

After being removed from the solution, the precipitate containing uranium has a bright yellow color ("yellowcake"). After high-temperature drying, uranium oxide (U3O8), now green in color, is loaded into special containers with a volume of up to 200 liters. Dried or calcined precipitates are intermediate products used to obtain pure uranium compounds (UF4, U3O8 or UO2).

Comment. The radiation dose rate at a distance of one meter from such a container is approximately half of what a person receives during an airplane flight. All these operations are carried out in accordance with radiation safety standards at mining enterprises. These rules and regulations set strict standards for the control of gamma exposure, and the possible ingestion of radon and other radioactive materials. The standards apply to both the personnel of enterprises and the public. A dose of 20 mSv/year for more than five years is the maximum allowable dose for the personnel of enterprises, including exposure to radon and other radioactive substances (in addition to the natural background and excluding exposure during medical diagnostics). Gamma radiation comes mainly from isotopes of bismuth and lead. Radon gas is released from rocks in which radium decays. Due to spontaneous radioactive decay, it passes into the daughter isotopes of radon, which are effective emitters of alpha particles. Radon is found in most rocks and, as a result, is found in the air we all breathe. At high concentrations, radon poses a health risk, as its short half-life means that alpha decay can occur inside the body after it is inhaled, which can eventually cause lung cancer. (Rn-222 is usually referred to as "Radon". Another isotope, Rn-220 (comes from the decay of thorium and is known as "thoron"), is a common constituent of many mineral sands.) In uranium mining and production, various precautions are taken to protect the health of personnel: Dust levels are carefully controlled to minimize the ingestion of gamma or alpha emitting substances. Dust is the main source of radioactive exposure. It typically contributes 4 mSv/yr to the annual dose received by personnel. The external radioactive exposure of personnel in mines, factories and waste disposal sites is limited. In practice, the level of external exposure from ore and waste is usually so low that it has little effect on increasing the allowable annual dose. Natural ventilation of open deposits reduces exposure to radon and its daughter isotopes. The level of exposure from radon rarely exceeds one percent of the level allowed for continuous exposure of personnel. Underground mines are equipped with sophisticated ventilation systems to achieve the same level. In underground mines, the average radiation dose is approximately 3 mSv/year. There are strict hygiene standards for personnel working with uranium oxide concentrate because it is chemically toxic, similar to lead oxide.

In practice, precautions are taken to protect the respiratory system from the ingress of toxins, similar to those used when working in lead smelters.

Very high requirements are imposed on the purity of uranium. Thus, the content of impurities of elements with a large neutron capture cross section (B, Cd, Li, REE, etc.) should not exceed hundred-thousandths and millionths of a percent. For purification, technical products are usually dissolved in nitric acid. An effective purification method is the extraction of uranyl nitrate with organic solvents (tributyl phosphate, methyl isobutyl ketone). Uranyl nitrate UO2(NO3)2*6H2O is crystallized from purified nitric acid solutions or peroxide UO4*2H2O is precipitated, and UO3 is obtained by careful calcination (calcination). The latter is reduced with hydrogen to UO2, which is converted into UF4 by the action of dry HF at 430-600°

- the main source compound for the production of metal.

Typical procedure for separating fissile radionuclides from uranium ore. Uranium is commonly found as uranium salt ore (uranium oxide) and carnolite (complex uranium-vanadium compound). The enriched ore is treated with a mixture of nitric and sulfuric acids.

Uranium goes into solution in the form of UO2++, and metals that form insoluble sulfates (Pb, Ba, Ra, etc.), together with silicates insoluble in acid, remain in the precipitate. By adding an excess of Na2CO3 to the solution, a basic solution is obtained, in which uranium is contained in the form of a complex carbonate, and elements that form insoluble carbonates, hydroxides or basic carbonates (Fe, Al, Cr, Zn, etc.) pass into the precipitate. Adding HNO3 again to an acidic reaction, a solution of uranyl nitrate is obtained with the formula for the solid salt UO2 (NO3) 2 * 6H2O, which is soluble in diethyl ether. Extraction of uranyl nitrate with this solvent gives an exceptionally pure product, quite suitable for the manufacture of uranium for nuclear reactors. When uranyl nitrate is calcined, the oxide U3O8 is obtained. This oxide can be reduced to metal in a bomb with Al, Ca, or Mg.

Reduction with carbon gives a product heavily contaminated with uranium carbide, and reduction with hydrogen gives UO2. This oxide can be converted to UF4 or UCl4 by treatment with anhydrous HF or HCl at reduced temperatures. Tetrahalides are reduced to metal with sodium or calcium. The halide salt KUF5, obtained from UF4, produces a very pure metal by electrolysis.

For poor ores (and in Russia now the ores are rather poor), a method of extracting uranium based on ion-exchange chromatography is usually used. However, the reagents used are very expensive, and by-products pollute the natural environment. In addition, the effectiveness of the method decreases as the uranium content in the ore decreases. The use of micro-organisms can cut costs in half, since the bacteria themselves "supply" the reagents, and the development of even low-grade ores becomes justified. To extract uranium from ore, it must be placed in a dump above a layer of impermeable rock and sprayed with an aqueous solution of iron sulfide containing populations of sulfide-reducing bacteria Thiobacillus ferrooxidans. During the vital activity of microorganisms, a ferrous sulfate reagent is formed, which oxidizes tetravalent uranium, turning it into pentavalent. The resulting compound is dissolved in acid. The radioactive element is recovered by concentration and purification by precipitation and ion exchange.

The method of bacterial hydrometallurgy was tested by Canadian scientists who, when resuming uranium mining at the mine in Stanrock, where the ore reserves were considered completely exhausted, used iron bacteria of the species Thiobacillus ferrooxidans. Such bacteria inhabit mine tailings (accumulations of waste uranium ore) and acidic water pumped out of the ground. They feed on sulfur, decomposing sulfide minerals. As a result, the insoluble uranium compounds that make up the ores become soluble, and the subsequent extraction of the metal is greatly facilitated.

Comment. In the production of enriched uranium for conventional reactors, about seven times as much depleted uranium is simultaneously produced. If uranium is enriched to 93% 235U (for military purposes), then about 200 times more depleted uranium is produced. All this, given the very large amount of all uranium that has ever been mined, is a very valuable raw material and potential fuel for nuclear installations.

After the completion of uranium extraction processes in the mining industry, almost all radioactive radium, thorium and actinium is contained in dumps and, therefore, the levels of radiation and emission of radon from such wastes will, in all likelihood, be significant. However, it is unlikely that someone will build a dwelling on top of the spoils and receive an increased dose of radiation that is outside international norms. However, the waste should be covered with sufficient soil to ensure that gamma radiation levels do not exceed natural background levels. In this case, it is also possible to cover these places with vegetation.

Comment. Approximately 95% of the radioactivity in the 0.3% U3O8 ore comes from the radioactive decay of 238U, reaching approximately 450 kBq/kg. This series has 14 long-lived radioactive isotopes and thus each gives approximately 32 kBq/kg (regardless of the mass ratio). After processing, 238U and some 234U are removed from the ore (and

U) and the radioactivity is reduced to 85% of its original value. After removing most of the U, the two short-lived decay products (234Th and 234Pa) soon disappear and, after a few months, the radioactivity level drops to 70% of its original value. The main long-lived isotope then becomes 230Th (half-life 77,000 years), which turns into 226Ra with subsequent decay into 222Rn.

2. PRODUCTION OF METALLIC URANIUM

Uranium metal is produced by the reduction of uranium halides (usually uranium tetrafluoride) with magnesium in an exothermic reaction in a "bomb"—a sealed container, usually steel, a common technique known as the "thermite process". Reactions in the "bomb"

flow at temperatures exceeding 1300°C. A strong steel case is needed to withstand the high pressure inside it. The "bomb" is charged with UF4 granules and filled with finely dispersed magnesium in excess and heated to 500-700°C, from this moment a self-heating reaction begins. The heat of reaction is sufficient to melt the “bomb” filling, consisting of metallic uranium and slag - magnesium fluoride, MF2. This same slag separates and floats up. When the "bomb" is cooled, the result is an ingot of uranium metal, which, despite its hydrogen content, is the highest quality commercially available and is well suited for nuclear power plant fuel.

The metal is also obtained by reducing uranium oxides with calcium, aluminum or carbon at high temperatures; or by electrolysis of KUF5 or UF4 dissolved in CaCl2 and NaCl melt. High purity uranium can be obtained by thermal decomposition of uranium halides on the surface of a thin filament. At the end of the uranium enrichment process, usually 0.25-0.4% 235U remains in the waste, since it is economically unprofitable to extract this isotope to the end (it is cheaper to buy more raw materials). In the USA, the residual content of 235U in raw materials after production increased from 0.2531% in 1963 to 0.30% in the 70s, due to the decrease in the cost of natural uranium.

Recovered ingots The ore concentrates are melted in vacuum and billets of the desired shape are cast, which are then subjected to Sampling by pressure treatment. The general scheme for the production of metallic nitric uranium is given in Fig. 3.

U3O8+8HNO3 3UO (NO) +2NO +4HO acid

–  –  –

Rice. 6 General scheme for the production of uranium metal.

7 3. URANIUM ENRICHMENT.

3.1 Uranium hexafluoride Nuclear power and the nuclear military complex require uranium-235, which is capable of sustaining a fission chain reaction. Its concentration in natural uranium is low - 0.7% on average. Therefore, enrichment of natural uranium up to 2.4-25% is required for power nuclear reactors and higher enrichment for military purposes. The operation of additional purification of uranium (refining) is obligatory for its transformation into a nuclear-pure material, which is then converted into uranium hexafluoride (UF6). Uranium is cleaned from boron, cadmium, hafnium, which are neutron absorbing elements, as well as from rare earth elements (gadolinium, europium and samarium). Refining consists in the extraction purification of uranium with tributyl phosphate after dissolving the uranium concentrate in nitric acid.

Uranium hexafluoride is the most suitable chemical compound for isotopic enrichment in terms of properties. The fluorination technology in a vertical plasma reactor includes the production of pure fluorine, grinding tetrafluoride (UF4) or uranium oxide to a powder state, followed by its combustion in a fluorine torch. Then uranium hexafluoride is filtered and condensed in a system of cold traps. Russian enterprises for the conversion of uranium oxide into hexafluoride are located in Verkhniy Neyvinsk (Sverdlovsk region) and Angarsk (Irkutsk region). Their total productivity is 20 - 30 thousand.

tons of uranium hexafluoride per year.

On an industrial scale, the production of uranium hexafluoride, in addition to Russia, is carried out in the USA, Great Britain, France and Canada. The capacity of factories exceeds the demand for their products (approximately 85% of capacity is used). The production capacity of Russian enterprises is sufficient not only to meet domestic needs, but also to supply a significant amount of products for export.

The separating power of a concentrator is measured in units of mass of processed material (MPM) per unit of time, such as MPP-kg/year or MPP-ton/year. The output of an enriched product from an enterprise of a given capacity depends on the concentration of the desired isotope in the input rock, output waste, and the final product. The initial content of a useful isotope is determined by its natural content. But the other two parameters can be changed. If the degree of extraction of the isotope from the initial substance is reduced, the rate of its release can be increased, but the price for this will be an increase in the required mass of the raw material.

This is subject to the relation:

where P is the product yield, U is the separating power, NP, NF, NW are the molar concentrations of the isotope in the final product, raw materials and waste. V(NP),

V(NW), V(NF) separating potential functions for each concentration. They are defined as:

Assuming a residual concentration of 0.25%, a 3100 MPP-kg/yr plant would produce 15 kg of 90% U-235 annually from natural uranium. If we take 3% U-235 (fuel for nuclear power plants) and 0.7% concentration in production waste as raw material, then a capacity of 886 MPP-kg/year is sufficient for the same output.

3.2 Methods of isotope separation 3.2.1 Isotope separation Most often, the separation of isotopes into individual isotopes is reduced to the isolation of one of the isotopic substances from a mixture or simply to the concentration of this substance in the mixture. An example is the extraction of 6Li, 235U, D. The separation of isotopes is always associated with significant difficulties, because isotopes, which are variations of one element that differ slightly in mass, chemically behave almost the same. Yet the rate of some reactions is different depending on the isotope of the element, in addition, you can use the difference in their physical properties, for example, in mass. To separate isotopes, differences in the physical or chemical properties of substances due to differences in their isotopic composition are used.

Isotope separation methods are based on differences in the properties of isotopes and their compounds associated with the difference in the masses of their atoms (isotope effects). For most elements, the relative mass difference of the isotopes is very small, and the isotope effects are also small. This determines the complexity of the task.

The efficiency of isotope separation is characterized by the separation factor. For a mixture of two isotopes C" = 1 C" C "" 1 C "" where C and (1 - C") are the relative abundances of light and heavy isotopes in the enriched mixture, and C and (1 - C) are in the primary mixture. For most methods, it is only slightly more than one, therefore, to obtain a high isotopic concentration, a single isotope separation operation has to be repeated many times.Only with electromagnetic separation, it is 10-1000 per 1 separation cycle.The choice of isotope separation method depends on the properties of the substance to be separated, the required degree of separation, the required the number of isotopes, the efficiency of the process (with a significant scale of production of isotopes), etc.

Isotope effects are understood as non-identity of the isotopes of a given element, due to the difference in the masses of isotopic atoms (atomic weights). Isotope effects are manifested in the difference in any properties of isotopes, except for radioactive ones.

According to the isotope effect used, there are various methods of isotope separation: gas diffusion (differences in diffusion coefficients), liquid thermal diffusion (difference in thermal diffusion coefficients), rectification or distillation (difference in vapor pressure), chemical exchange (uneven distribution of isotopes at isotopic exchange equilibrium), kinetic method (difference in chemical reaction rate constants), gas centrifugation (difference in density), electromagnetic method (difference in specific ion charges), AVLIS (evaporation using laser) and electrolysis.

Due to the need for large quantities of isotopes such as deuterium 235U for the needs of nuclear energy, many methods of isotope separation have received industrial use:

diffusion method - to isolate 235U using gaseous UF6, rectification, chemical exchange and electrolysis methods to isolate deuterium. The separation of lithium isotopes is of industrial importance.

A single operation of isotope separation leads only to a slight enrichment of the separated mixture in the required isotope, which is associated with small values ​​of isotope effects. Therefore, for the complete isolation or significant concentration of one of the isotopic substances, the separation operation is repeated many times in a stepwise separation cascade. The stage of the cascade is one or more parallel connected separating devices; steps are connected in series. Since the initial content of the isolated isotopic substance is usually low, the flow of the initial mixture passing through the cascade is very large compared to the amount of the resulting product.

The flow of the initial mixture is fed to the first stage of the cascade. As a result of the separation operation, it is divided into two streams: depleted - removed from the cascade, and enriched - supplied to the 2nd stage.

At the 2nd stage, the enriched stream is subjected to separation for the second time:

the enriched flow of the 2nd stage enters the 3rd stage, and its depleted flow returns to the previous (1st), etc. From the last stage of the cascade, a finished product with the required concentration of a given isotope is selected. The flow of the mixture flowing through the cascade from the previous stages to the next is called direct, or enriched, and flowing in the opposite direction is called return, or depleted.

Comment. The performance of such a cascade system is affected by two factors: the degree of enrichment at each stage and the loss of the desired isotope in the waste stream. Let's explain the second factor. At each stage of enrichment, the flow is divided into two parts - enriched and depleted in the desired isotope. Since the degree of enrichment is extremely low, the total mass of the isotope in the spent rock can easily exceed its mass in the enriched part. To avoid such a loss of valuable raw materials, the depleted flow of each subsequent stage is again fed back to the input of the previous one. The source material does not enter the first stage of the cascade. It is introduced into the system immediately to some, n-th stage. Due to this, material that is highly depleted in the main isotope is scrapped from the first stage.

The amount of enriched material produced depends on the desired degree of enrichment and leanness of the output streams. If the starting material is available in large quantities and is cheap, then the performance of the cascade can be increased by discarding along with the waste a large amount of the unextracted useful element (an example is the production of deuterium from ordinary water). If necessary, a high degree of extraction of the isotope from the raw material is achieved (for example, when enriching uranium or plutonium).

The choice of an isotope separation method depends on the magnitude of the underlying isotope effect, which determines the value of the separation factor, as well as on economic indicators: energy consumption, cost of apparatus, productivity, reliability, etc. Table 3 compares separation methods using the example of three types of isotopes isotopes for hydrogen, carbon and uranium.

Tab. 3 Efficiency of various methods for the separation of hydrogen, carbon and uranium:

Separation method H/D C-12/13 U-235/238 Chemical enrichment 1.2-3 1.02 1.0015 Distillation 1.05-1.6 1.01 Gas diffusion 1.2 1.03 1.00429 Centrifugation (250 m/s) 1.01 1.01 1.026 Centrifugation (600 m/s) - - 1.233 Electrolysis 7 - Isotope effects Isotope effects are the non-identity of the properties of the isotopes of a given element, due to the difference in the masses of isotopic atoms (atomic weights).

Isotope effects are manifested in the difference in any properties of isotopes, except for radioactive ones.

However, since for the isotopes of most elements (with the exception of the lightest ones) the relative difference in the atomic weights of the isotopes is small, the isotope effect for these elements is expressed very weakly. Even for light elements of the second period of the Periodic system (Li-Ne), the relative differences in the atomic weights of isotopes do not exceed 35%; for the third period (Na-Ar) they do not exceed 20%, for the fourth (K-Kr) and fifth (Rb-Xe) periods - 15%; for heavier elements they are always less than 10%. Only for elements of the first period (H - He) the relative differences in atoms are very large - for hydrogen, the maximum difference reaches 200%, and for helium - 100%.

The unequal atomic weights of isotopes cause certain differences in such properties of isotopic compounds as density, viscosity, refractive index, diffusion coefficient, specific charge of ions, etc. In this case, the ratio of the densities of isotopic compounds quite accurately coincides with the ratio of their molecular weights, and the specific charges of isotopic ions are inversely proportional their molecular weights. In addition, the difference in the masses of isotopic atoms causes a change in the levels of translational, rotational, and vibrational energy of molecules during their isotopic substitution, which leads to a difference in the vibrational-rotational spectra of isotopic compounds.

A change in energy levels during isotopic substitution, in turn, causes a change in thermodynamic properties, such as heat capacity, thermal conductivity, heats of evaporation and melting, saturated vapor pressure, etc. For example, the ratio of vapor pressures H2 and D2 is 2.448 at - 251.1o ; the vapor pressure ratio of H2O and D2O is 1.148 at 20° and 1.052 at 100°; the corresponding ratio for H216O and H218O is 1.009 at 23° and 1.003 at 100°.

As for the chemical properties of isotopic compounds, they remain basically unchanged, because the mass of an atom does not affect its electronic configuration, which determines its chemical properties. However, the thermodynamic nonequivalence of isotopic compounds leads to an uneven distribution of isotopes at the equilibrium of isotopic exchange (the thermodynamic isotope effect), as well as to the preferential adsorption of one of the isotopic forms on the sorbent. In addition, the thermodynamic nonequivalence of the initial isotopic compounds in combination with a similar nonequivalence of transition states (active complexes) in the chemical reactions of isotopic compounds causes a difference in the rates of these reactions (kinetic isotope exchange). Since the thermodynamic and kinetic effects depend on differences in the vibrational-rotational and translational energy levels of isotopic molecules, it is possible to calculate the isotope effects by statistical methods based on the data of the vibrational spectrum of these molecules. Thermodynamic isotope effects, expressed as deviations from unity of the equilibrium isotope distribution coefficient, for hydrogen isotope exchange in the case of tritium and protium can reach a maximum value of 16-18 times at 20o, and 8-9 times in the case of deuterium and protium; for the isotopic exchange of other light elements such as lithium, boron, carbon, nitrogen, chlorine, these deviations rarely exceed 10%, and in the case of heavier elements they usually do not exceed 1%.

Kinetic isotope effects, expressed as the ratio of the rate constants of chemical reactions for various isotopic compounds, can also be very large in the case of hydrogen isotopes. For example, the ratio of the rate constants for the synthesis of hydrogen bromide and deuterium bromide is 5. For isotopes of all other elements, the deviations of this ratio from unity never exceed 50%.

The use of isotopes as labeled atoms is based on their chemical and physicochemical identity. In fact, there are always differences in the properties of isotopes, characterized by the values ​​of isotope effects. Thus, the knowledge of isotope effects makes it possible to correct for differences in the properties of isotopes when they are used as labeled atoms. Obviously, taking into account the corresponding corrections is essential when working only with isotopes of light elements and especially hydrogen.

Differences in the properties of isotopes make it possible to separate isotopes and determine their content in isotopic mixtures. Any method of isotope separation, as well as quantitative analysis of stable isotopes, is based on the presence of isotope effects (in this case, the separation method is more effective, the greater the corresponding isotope effect). For example, the distillation separation method is based on the difference in vapor pressures of isotopic compounds. The basis of the centrifugation method is the difference in densities. The diffusion method assumes a difference in diffusion coefficients. Separation methods using isotope exchange reactions are based on the thermodynamic isotope effect. There are isotope separation methods based on the isotopic kinetic effect, such as the widely used electrolytic method for producing heavy water.

3.3.3 Gas diffusion.

Diffusion of gases through porous partitions at reduced pressure is one of the most important methods for separating heavy, as well as many light, isotopes. The gas diffusion method uses the difference in the velocities of movement of gas molecules of different masses. It is clear that it will be suitable only for substances in the gaseous state.

The gaseous compound of the element to be separated at sufficiently low pressures ~ 0.1 N/m2 (~10-3 mm Hg) is "pumped" through a porous partition containing up to 106 holes per 1 cm2. Light molecules penetrate the barrier faster than heavy ones, since the velocities of the molecules are inversely proportional to the square root of their molecular weight. As a result, the gas is enriched in the light component on one side of the baffle and in the heavy component on the other.

Let us explain the principle of operation of the gas diffusion method (see Fig. 5).

At different speeds of movement of molecules, if they are forced to move through a thin tube, the faster and lighter ones will overtake the heavier ones. To do this, the tube must be so thin that the molecules move through it one by one. Thus, the key point here is the manufacture of porous membranes for separation. They must not leak, withstand excessive pressure. Since the light isotope diffuses faster than the heavy isotope, the gas passing through the porous partition is enriched in the light isotope. For some light elements, the degree of separation can be quite large, but for uranium it is only 1.00429 (the output stream of each stage is enriched by a factor of 1.00429). Therefore, gas diffusion enrichment enterprises are cyclopean in size, consisting of thousands of enrichment stages.

Rice. 5 Scheme of the gas diffusion method Based on the inversely proportional dependence of the square of the average velocity of the thermal motion of molecules on their mass, it can be shown that the maximum separation coefficient of isotopic molecules during their diffusion through porous partitions is determined by the expression M2 =, M1 where M1 and M2 are the molecular weights of light and heavy isotopic molecules.

This method of isotope separation gives low separation factors, but allows easy cascading. Thus, in general, a large separation factor can be obtained.

If the difference in molecular weights is very small, then it is necessary to repeat this process thousands of times.

The number of division operations n is determined by the relation:

q =n, where q is the required degree of separation. This method is the basis for the work of giant gaseous diffusion plants to obtain 235U from gaseous UF6 (~ 1.0043). To obtain the required concentration of 235U, about 4000 single separation operations are required.

The diffusion separating cascade consists of many cells (stages). Each cell

– the chamber is divided into two parts by a porous partition, on one side of which a gaseous isotope mixture is pumped. The pore size is of the order of the mean free path of these molecules at the applied pressure. Part of the mixture passes through the partition and is enriched in the light component, because. its diffusion rate is higher. An enriched stream is discharged from one part of the chamber, and a depleted one from the other. Both streams enter the corresponding stages of the cascade for further separation. Diffusion of the mixture to be separated is also used in some foreign gas (or preferably in steam), which is then easily separated from the mixture by condensation. The mixture is fed into the steam jet, and part of it with a high content of the light component diffuses against the steam flow.

3.2.4 Diffusion in a vapor flow (counterflow mass diffusion) Isotope separation occurs in a cylindrical vessel (column) blocked along the axis by a diaphragm containing about 103 holes per 1 cm2. The gaseous isotopic mixture moves towards the auxiliary steam flow. Due to the gradient (difference) in the concentration of gas and vapor in the cross section of the cylinder and the larger diffusion coefficient for light molecules, a part of the gas that has passed through the vapor flow to the left side of the cylinder is enriched with a light isotope. The enriched part is removed from the upper end of the cylinder along with the main steam flow, and the part of the gas remaining in the right half moves along the diaphragm and is removed from the apparatus. The steam that has entered the right side is condensed. On separating installations, consisting of several dozen series-connected diffusion columns with an evaporating liquid (mercury, xylene, etc.), isotopes of neon, argon, carbon, krypton, and sulfur are separated on a laboratory scale (up to 1 kg).

3.2.5 Thermal diffusion In this case, again, the difference in the velocities of the molecules is used. The lighter ones, in the presence of a temperature difference, tend to end up in a hotter region. The separation factor depends on the ratio of the mass difference of isotopes to the total mass and is larger for light elements. The thermal diffusion process is carried out in hollow columns with cooled walls and with a hot wire stretched in the center along the column. Such a column, depending on its height, is equivalent to many steps connected in series.

The forward and reverse flows in the column are provided by natural convection currents (the current is directed upwards along the hot wire, and downwards along the walls). Between the flows in each cross section, thermal diffusion processes occur, the successive imposition of which leads to the accumulation of a heavy isotope at the bottom of the column, and a lighter isotope at the top. Despite its simplicity, this method requires a lot of energy to create and maintain heating. Therefore, it is not widely used.

Typically, a thermal diffusion separation column consists of two coaxially arranged tubes in which different temperatures are maintained. The mixture to be separated is injected between them. The temperature difference T between the surfaces of the pipes creates a diffusion flow, which leads to the appearance of a difference in the concentration of isotopes in the cross section of the column. At the same time, the temperature difference leads to the emergence of convective vertical gas flows. As a result, lighter isotopes accumulate near the hot surface of the inner tube and move upward. Separation coefficient = 1 + T T where is the thermal diffusion constant depending on the relative mass difference of the isotopes, and T = (T1 + T2)/2. The thermal diffusion method makes it possible to separate isotopes in both gaseous and liquid phases. The possible range of isotopes to be separated is wider than in the case of separation by gaseous diffusion or diffusion in a vapor flow. However, for the liquid phase, it is small. The method is convenient for the separation of isotopes in the laboratory due to its simplicity, the absence of vacuum pumps, etc.

This method was used to obtain He with a content of 0.2% 3He (in a natural mixture 1.510-5%), isotopes O, 15N, 13C, 20Ne, 22Ne, 35Cl, 84Kr, 86Kr with a concentration of 99.5%. Thermal diffusion has been used commercially in the USA to pre-enrich 235U before final separation in an electromagnetic plant. The thermal diffusion plant consisted of 2142 columns 15 m high.

3.2.6 Gas centrifugation This technology was first developed in Germany during World War II, but was not commercially used anywhere until the early 60s. Separation is carried out due to the difference in centrifugal forces acting on molecules of different masses. (Fig.6). In a centrifuge rotating at a high circumferential speed (100 m / s), heavier molecules are concentrated at the periphery under the action of centrifugal forces, and light molecules are concentrated at the centrifuge rotor.

The vapor flow in the outer part with the heavy isotope is directed downwards, while in the inner part with the light isotope it is directed upwards. The connection of several centrifuges in a cascade provides the necessary enrichment of isotopes.

The great advantage of centrifugation is that the separation factor depends on the absolute difference in mass, and not on the mass ratio. The centrifuge works equally well with both light and heavy elements. Therefore, centrifugation is suitable for the separation of isotopes and heavy elements. The degree of separation is proportional to the square of the ratio of the speed of rotation to the speed of the molecules in the gas. From here it is very desirable to spin the centrifuge as quickly as possible. Typical linear speeds of rotating rotors are 250-350 m/s, and up to 600 m/s in advanced centrifuges. A typical separation factor is 1.01 - 1.1.

A countercurrent gas centrifuge is used, in which the mixture circulates, moving upwards along the axis of rotation in the central part, and downwards along the periphery. Such a centrifuge is a column-type apparatus with repeated repetition of the elementary separating effect (in each cross section) along the direction of the forward and return flows.

Fig.6 Scheme of the gas centrifugation method

Centrifugation has been used to separate isotopes of carbon (as CCl4), krypton, xenon, and uranium (as UF6).

Compared to gas diffusion installations, this method has a reduced energy consumption, greater ease in increasing power. The disadvantage of the method is the low productivity of centrifuges and the need to provide very high angular velocities (of the order of 60,000 rpm). At present, gas centrifugation is the main method of isotope separation.

3.2.7 Electromagnetic separation.

The method of electromagnetic separation is based on the different action of a magnetic field on charged particles of different masses. In fact, such installations, called calutrons, are huge mass spectrometers. The ions of the separated substances, moving in a strong magnetic field, twist with radii proportional to their masses and fall into the receivers, where they accumulate (Fig. 6).

Fig.7 Scheme of the method of electromagnetic separation The substance whose isotopes are to be separated is placed in the crucible of an ion source, evaporates and ionizes.

Ions are pulled out of the ionization chamber by a strong electric field, formed into an ion beam, and enter a vacuum separation chamber placed in a magnetic field H directed perpendicular to the ion motion. Under the action of a magnetic field, ions move along circles with radii of curvature proportional to the square root of the ratio of the ion mass M to its charge e. As a result, the trajectory radii of heavy and light ions differ from each other. This makes it possible to collect ions of various isotopes in receivers located in the focal plane of the setup.

This method allows you to separate any combination of isotopes, has a very high degree of separation. Two passes are usually sufficient to obtain an enrichment above 80% from a poor material (with an initial content of the desired isotope of less than 1%).

The performance of electromagnetic installations is determined by the value of the ion current and the efficiency of ion trapping. At large facilities, the ion current ranges from tens to hundreds of mA, which makes it possible to obtain up to several grams of isotopes per day (total for all isotopes). In laboratory separators, productivity is 10 to 100 times lower.

The electromagnetic method is characterized by high and the possibility of simultaneous separation of all isotopes of a given element. Usually in large industrial plants for one stage of separation a ~ 10-100, in laboratory - 10-100 times higher. In most cases, when separating by the electromagnetic method, one stage is sufficient; rarely, the repeated separation of pre-enriched isotopic materials is performed to obtain isotopes of especially high frequency. The main drawback of the method is the relatively low productivity, high operating costs, and significant losses of the substance to be separated.

Electromagnetic separation is ill-suited for industrial production:

most of the substances are deposited inside the calutron, so it has to be periodically stopped for maintenance. Other disadvantages are high power consumption, complexity and high cost of maintenance, low productivity. The main scope of the method is the production of small amounts of pure isotopes for laboratory use. They are used to obtain radioactive isotopes necessary for nuclear spectroscopy, to study the interaction of ions with a solid body (during ion implantation and for other purposes).

The electromagnetic method made it possible for the first time to obtain kilogram quantities of 235U.

The electromagnetic plant in Oak Ridge (USA) had 5184 separating chambers - "calutrons".

Due to their high versatility and flexibility, electromagnetic facilities are used to separate isotopes of ~50 elements of the periodic table in quantities from mg to hundreds of g and are the main source of isotope supply for scientific research and some practical applications of isotopes 3.2.8 Chemical enrichment Chemical enrichment uses the difference in flow rates chemical reactions with various isotopes. It works best when separating light elements, where the difference is significant. In industrial production, reactions are used that take place with two reagents in different phases (gas/liquid, liquid/solid, immiscible liquids). This makes it easy to separate rich and lean streams. Using additionally the temperature difference between the phases, an additional increase in the separation factor is achieved. Today, chemical separation is the most energy-saving technology for producing heavy water. In addition to the production of deuterium, it is used to extract Li-6. In France and Japan, methods of chemical enrichment of uranium were developed, which never reached industrial development.

3.2.9 Aerodynamic separation This method can be considered as a variant of centrifugation, but instead of swirling the gas in a centrifuge, it swirls when it exits a special nozzle, where it is supplied under high pressure. This technology was used by South Africa and Germany.

3.2.10 AVLIS (evaporation using a laser).

Different isotopes absorb light at slightly different wavelengths. A finely tuned laser can selectively ionize atoms of a particular isotope.

The resulting ions can be easily separated, for example, by a magnetic field (Fig. 8). This technology is extremely efficient, but has not yet been applied on an industrial scale.

Fig.8. Scheme of the laser evaporation method

3.2.11 Distillation Distillation (fractional distillation) exploits the difference in evaporation rates of isotopes of different masses. The smaller the mass of the atom, the faster this isotope will evaporate.

This works best again, on light elements. Distillation has been successfully used to produce heavy water.

Since, as a rule, isotopes have different saturation vapor pressures, for example p1 and p2, and different boiling points, separation of isotopes by fractional distillation is possible. Fractionating columns with a large number of separation stages are used;

depends on the ratio p1/p2 and its value decreases with increasing molecular weight and temperature. Therefore, the process is most efficient at low temperatures. Distillation was used to obtain isotopes of light elements - 10B, 11B, 18O, 15N, 13C, and on an industrial scale to produce hundreds of tons of heavy water per year.

3.2.12 Electrolysis During the electrolysis of water or aqueous solutions of electrolytes, the hydrogen released at the cathode contains a smaller amount of deuterium than the original water. As a result, the concentration of deuterium in the electrolyzer increases. The method was applied on an industrial scale to obtain heavy water. Separation of other isotopes of light elements (lithium, potassium) by electrolysis of their chloride salts is carried out only in laboratory quantities. This most efficient method of obtaining deuterium (a separation factor of more than 7) requires such an amount of energy that, for economic reasons, if it is used, then at the later stages of purification.

3.2.13 Isotope exchange Isotope exchange is a reaction, the only result of which is the redistribution of isotopes of an element between the reacting substances.

During isotopic metabolism, substances retain their elemental composition unchanged and pass only from one isotopic form to another. Such reactions can also occur between different isotopic forms of the same substance. The possibilities of carrying out isotope exchange reactions are very different: they can proceed under homogeneous conditions (between a dissolved substance and a solvent, between reacting substances in a neutral solvent, in a mixture of gases, etc.), as well as under heterogeneous conditions (between a solid and liquid substance and insoluble gas, between gases on the surface of a solid catalyst, etc.).

The equilibrium of isotopic exchange is characterized by the distribution coefficient of isotopes and the equilibrium constant of the reaction. The equilibrium coefficient is a value showing how many times the ratio of the equilibrium concentrations of isotopes in one of the reacting components is greater than the corresponding ratio in the other. The equilibrium constant is the ratio of the equilibrium concentrations of the final and initial isotopic forms of the reacting components.

A specific feature of isotopic exchange reactions, which distinguishes them from ordinary (elemental) chemical reactions, is that the concentrations of the reacting components remain unchanged, and only their isotopic composition changes. This feature leads to the fact that these reactions, regardless of their true mechanism, can practically be described by a first-order kinetic equation.

Isotopic exchange proceeds through various mechanisms, and there are all the mechanisms inherent in elemental chemical reactions, and, moreover, mechanisms that have no direct analogues in ordinary chemistry.

Isotopic exchange can be one-, two- and multi-stage, homogeneous and heterogeneous. It can be based on transitions of electrons, ions, atoms, groups of atoms and whole molecules. As intermediate stages of isotope exchange reactions, dissociation of molecules into charged or uncharged particles, associations of individual particles, and intramolecular rearrangements of atoms can be observed. In addition, the isotopic exchange for each given element has its own characteristic features.

Isotopic exchange is widely used in various research and preparative works, as well as in industry. It is used to separate natural stable isotopes by chemical methods based on the uneven equilibrium distribution of isotopes between substances.

For example, to concentrate deuterium in the industrial production of heavy water, the isotope exchange reaction is used:

HDS + H 2O H 2S + HDO and HD + H 2O H 2 + HDO

The following reaction is used to concentrate 6Li:

7 + 6 Li +7LiZ 6 Li + LiZ (the reaction is carried out on zeolite), Z is the zeolite radical.

The use of several stages makes it possible to obtain a high enrichment of hydrogen, nitrogen, sulfur, oxygen, carbon, and lithium with individual isotopes.

*-*-* In addition to these, there are a number of other methods, the use of which is limited or is under research or technical improvements. These include: obtaining 3He based on the phenomenon of 4He superfluidity;

separation by diffusion in a supersonic gas jet expanding in space with reduced pressure; chromatographic separation based on differences in isotope adsorption rates; biological methods of separation.

Isotope separation methods have features that determine the areas of their most effective application. When separating isotopes of light elements with mass numbers of about 40, distillation, isotope exchange, and electrolysis are more economically advantageous and efficient. To separate the isotopes of heavy elements, the diffusion method, centrifugation and electromagnetic separation are used. However, gaseous diffusion and centrifugation can be used if there are gaseous compounds of the elements.

Since there are few such compounds, the real possibilities of these methods are still limited.

Thermal diffusion makes it possible to separate isotopes both in the gaseous and liquid state, but it is not enough for the separation of isotopes in the liquid phase. The electromagnetic method has a large, but has a low productivity and is used mainly with a limited scale of isotope production.

To ensure scientific research and practical applications of isotopes in Russia, the State Fund for Stable Isotopes has been created, which has a reserve of isotopes of almost all elements. Significant amounts of deuterium 10B, 13C, N, 180, 22Ne and other isotopes are regularly separated. The production of various chemicals labeled with stable isotopes has also been organized.

3.3 Separation of uranium isotopes The following technologies are used to separate uranium: electromagnetic separation, gas diffusion, liquid thermal diffusion, gas centrifugation, aerodynamic separation. Some attention should be paid to the following methods, which are not yet industrially applicable: laser evaporation and chemical separation. field perpendicular to their trajectory. By placing the ion source in the center of a uniform magnetic field so as to use multiple ion beams in different directions, an efficient use of a large magnet can be achieved. The collectors are arranged in such a way that they intersect each beam and collect separately the two main isotopes 235U and U with a fairly high degree of purity. The method gives a large separation in one setup.

The separation factor approaches 100%, but the productivity of one installation is low. The overall productivity can be increased by increasing the concentration of U in the feed product.

It was historically the first technique capable of producing weapons-grade uranium. It was used in the Y-12 electromagnetic separator at Oak Ridge during World War II.

Two separation stages are enough to enrich uranium up to 80-90%. The other two methods available at that time - gaseous diffusion, liquid thermal diffusion - were used for the initial enrichment of uranium and to increase the yield of an electromagnetic separator in relation to natural uranium feedstock. All the uranium used in the Hiroshima bomb was produced using this technology. Due to high overheads, Y-12 was closed in 1946. More recently, only Iraq has attempted to industrialize this method in its nuclear program.

Gas diffusion. The first practical industrial scale separation technology for 238U and 235U. The method is based on the difference in the rates of thermal motion of molecules of isotopic substances. The only uranium compound with the properties required for gaseous diffusion is uranium hexafluoride UF6. The saturated vapor pressure of this compound reaches the atmosphere at 56°C.

Despite requiring thousands of stages for high enrichment, this is a more cost effective method than electromagnetic separation. 235U enrichment gaseous diffusion plants are huge and have a large production capacity. The main difficulty is the creation of reliable gas diffusion barriers capable of withstanding the corrosive action of UF6. There are two main types of such barriers: thin porous membranes and barriers assembled from individual tubes. The membranes are films with pores formed by etching. For example, nitric acid will pickle a 40/60 Au/Ag (Ag/Zn) alloy; or by electrolytic etching of aluminum foil, a brittle aluminum membrane can be obtained. Composite barriers are assembled from small, discrete elements packed into a relatively thick porous baffle. The technology for manufacturing diffusion barriers continues to be classified in all countries that have developed it. Built during World War II, the K-25 facility at Oak Ridge consisted of 3,024 enrichment stages and continued to operate until the late 1970s. Developing a suitable barrier material proved difficult, causing some delay in commissioning the plant after the war, although even a partially completed plant contributed to stockpiling 235U for the Little Boy atomic bomb dropped on Hiroshima. While barriers were made from sintered nickel powder, attempts to create promising membranes from electrolytically etched aluminum failed. K-25 originally contained 162,000 m2 of membrane surface. This facility, with expansions, produced the majority of all uranium for the US Army in the sixties. With the improvement of gas diffusion barriers, the productivity of the plant has increased by 23 times. Diffusion production consumes much less electricity compared to electromagnetic, but its consumption is still quite large. In 1981, after modernization, it had a specific power consumption of 2370 kWh/MPP-kg. While low-enrichment uranium is a valuable raw material for the production of highly enriched uranium, low-enrichment gaseous diffusion plants cannot easily be converted to produce high-enriched uranium. High enrichment requires many smaller stages, due to the sharp drop in enrichment factor and criticality problems (accumulation of the critical mass of uranium) in larger blocks. The huge size of the enrichment system leads to a long time of filling it with material (enriched substance) before the product exits. Typically, this equilibration time is 1-3 months. Gaseous diffusion technology has been widely used in many countries, even Argentina has established a working enrichment plant for its covert weapons program (now discontinued). In 1979, over 98% of all uranium was produced using this process. By the mid-1980s, this proportion had dropped to 95% with the introduction of the centrifugation method.

Liquid thermal diffusion, i.e. the phenomenon of changing the diffusion equilibrium of a gas in the presence of a temperature difference is also widely used in the practice of isotope separation.

Thermal diffusion separation of uranium isotopes occurs in liquid UF6, which is under high pressure between two surfaces - hot and cold. Due to the difference in masses of uranium isotopes and complex intermolecular forces, isotope separation occurs.

Liquid thermal diffusion was the first technology to produce significant amounts of low enriched uranium. It was used in the USA during the Manhattan Project to increase the efficiency of the Y-12 separator. This is the simplest of all separation methods, but the maximum degree of enrichment in 235U is only ~1% (the S-50 plant in Oak Ridge produced 0.85-0.89% uranium-235 in the final product). A serious disadvantage of this method is the high energy consumption.

Gas centrifugation. The dominant method of isotope separation for new industries, although existing facilities are mostly gaseous diffusion. Each centrifuge provides a much higher separation factor than a single gas stage.

Many fewer stages are required, only about a thousand, although the cost of each centrifuge is much higher. Gas centrifugation requires ~1/10 of the energy required for gaseous diffusion (its energy consumption is 100-250 kWh/MPH-kg) and allows for easier scaling up. Of the developing nuclear countries, this rather sophisticated technology is owned by Pakistan and India.

Aerodynamic separation. Aerodynamic separation has been developed in South Africa (UCOR process using vortex tubes at 6 bar) and Germany (using curved nozzles operating at 0.25-0.5 bar). The only country that has put this method into practice is South Africa, where 400 kg of weapons-grade uranium was produced at a plant in Valindaba that closed in the late eighties. Separation factor ~1.015, energy consumption ~3300 kWh/MPP-kg.

Evaporation using a laser. AVLIS (atomic vapor laser isotope separation). The technology was never put into production; it was developed in the USA during the 1970s and 80s. and died out due to a general overabundance of separating capacities and a reduction in the arsenal.

Chemical separation. Chemical separation of uranium was developed in Japan and France but, like AVLIS, was never used. The French Chemex method uses countercurrent in a tall column of two immiscible liquids, each containing dissolved uranium.

The Japanese Asahi method uses an exchange reaction between an aqueous solution and a finely ground resin through which the solution slowly percolates. Both methods require catalysts to speed up the concentration process. The Chemex process needs electricity at the level of 600 kWh/MPP-kg. Iraq was developing this technology (in the form of Chemex/Asahi mixed production) for U-235 enrichment up to 6-8% and subsequent enrichment in the calutron.

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1 Back to contents I.N. Beckman SYNERGETICS Lecture 2. Dynamical systems Contents 2.1 Order and chaos 2.2 Types of complex systems 2.3 Discovery of deterministic chaos 2.4 Elements of the theory of dynamical systems 2.5 Examples of dynamical systems with deterministic chaos least chaotic). There is order and ordered structures in the world, there is disorder and random phenomena, there is chaos, i.e. absolute chaos. There is also deterministic chaos, i.e. disorder, more or less ordered, with random processes that are partially predetermined and even regular. Interest in dynamic chaos is due to the fact that this phenomenon occurs in nonlinear systems of very different physical nature and finds many practical applications. Chaotic oscillations can occur in strictly deterministic systems, but they have a number of properties that make them similar to random oscillations. Forming a new class of complex, broadband signals that are easily implemented in electronic circuits, they claim in radio engineering the role of information carriers for covert communication systems. In this lecture, we will qualitatively consider the features of deterministic chaos in relation to dynamic (dissipative) systems. 2.1 Order and Chaos In nature and society, there is a continuous struggle between order and chaos. Order is the harmonious, expected, predictable state or arrangement of something. Orderliness is a characteristic of a structure, indicating the degree of mutual consistency of its elements. In this lecture, order (determinism) will mean the possibility of unambiguously predicting the state of a system at any given time, based on initial conditions. Chaos is an aperiodic deterministic behavior of a dynamical system that is very sensitive to initial conditions. An infinitely small perturbation of the boundary conditions for a chaotic dynamical system leads to a finite change in the trajectory in the phase space. We will assume that chaos is the limiting case of disorder. Further chaos for us will mean the complete unpredictability of the system, the irregularity of movement, the non-repeatability of trajectories. Usually the order is a clear, orderly change of events in the space and time around us. In the theory of dynamical systems, order is understood as a deterministic process, i.e. a process, each step of which is predetermined by some regularities that are well known, so that with 100% probability it is possible to predict the evolution of the system. A chaotic process is random and cannot be controlled. It is impossible to predict the development of such a process; one can only raise the question of the probability of one or another variant of its evolution. Examples of chaotic processes are: throwing a ball in a roulette, Brownian motion

2 particles under random impacts of "neighbors", chaotic whirlwinds of turbulence formed when fluid flows at a sufficiently high speed, trains going when they want and where they want. An important type of chaos is white noise (noise chaos or fractional noise). Noise is random fluctuations of various physical nature, characterized by the complexity of the temporal and spectral structure. It can be stationary or non-stationary. White noise is stationary noise, the spectral components of which are evenly distributed over the entire range of frequencies involved. An example of white noise is the noise of a nearby waterfall. It got its name from white light containing electromagnetic waves of frequencies of the entire visible range of electromagnetic radiation. Distinguish between random and chaotic motions. The first term refers to situations where the acting forces are unknown or some statistical characteristics of the parameters are known. The term "chaotic" is used in those deterministic problems where there are no random or unpredictable forces or parameters, and whose trajectories of motion show a strong dependence on the initial conditions. Rice. 1. a The movement of the ball after several collisions with the sides of an elliptical billiard table. This movement can be described by a discrete set of numbers (s i, j i) called a map; b motion of a particle in a pair of potential wells under the action of a periodic excitation. Under certain conditions, the particle periodically jumps from left (L) to right (R) and back: LRLR... or LLRLLR... etc. Under other conditions, jumps are chaotic; the sequence of characters L and R is unordered. Gambling is a classic example of chaos. However, gambling is not a deterministic process, since there are many chances in it. Although the theory of chaotic dynamical systems uses the methods of probability theory, it is not part of mathematical statistics. Chaos is some random process observed in dynamic systems that are not affected by noise or any random forces. It turned out that many completely deterministic systems can exhibit chaotic unpredictable behavior. A "random" process turns out to be a solution to one or more simple, differential equations. This gives rise to the problem of the unpredictability of the long-term behavior of deterministic chaotic systems and the need to use a statistical description. On fig. 1 shows two examples of mechanical systems whose dynamics are chaotic. The first example is an experiment with a ball that hits and bounces off the sides of an elliptical pool table. If the collisions are elastic, then the energy is conserved, but for elliptical tables the ball wanders around the table, never repeating its trajectory. Another experiment is a ball in a potential consisting of two wells. If the table on which the device stands does not oscillate, then such a ball has two states of equilibrium. However, if the table oscillates, making a periodic movement of a sufficiently large amplitude, the ball begins to jump randomly from one hole to another; thus, periodic exposure to it frequency causes a disordered response with a wide spectrum of frequencies. The excitation of a continuous spectrum of frequencies located below the impact frequency is one of the features of chaotic oscillations (Fig. 2). Rice. 2. Power spectrum (Fourier transform) of chaotic motion in a pair of potential wells. Another property of chaotic systems is the loss of information about the initial conditions. Let the coordinate

3 is measured with an accuracy of Dx, and the speed is measured with an accuracy of Dv. Let us divide the coordinate-velocity plane (phase plane) into cells with area DxDv (Fig. 3). If the initial conditions are specified exactly, then the system is located somewhere in the shaded area on the phase plane. But if the system is chaotic, then this uncertainty grows with time, increasing to the size of N(t) cells (Fig. 3). The increase in uncertainty described by the law ht N» N0e, (1) is the second characteristic property of chaotic systems. The constant h is related to the entropy (information theory) and the Lyapunov exponent (a measure of the speed at which close trajectories of a system diverge). Rice. 3. An illustration of the increase in uncertainty, or loss of information in a dynamic system. The shaded square at time t=t 0 shows the uncertainty of knowledge of the initial conditions. Between the extremes: order and chaos, there is a vast area of ​​deterministic (to some extent ordered) chaos. Deterministic chaos refers to limited randomness, it can be controlled and even predicted for short periods of time ahead. Recall that the principle of determinism says: if we know the current state of a system and the laws of its evolution, then we can predict the future behavior of this system. Example: the classic Newtonian "mechanical" universe, in which the position of the planets is like the movement of the hands of a multi-hand clock. Here the future is predicted unambiguously. However, in nature there are systems that are completely deterministic in the Newtonian sense, but their future in a certain range of parameters cannot be calculated in principle. This phenomenon is known as deterministic chaos, or chaos theory. By deterministic chaos, we mean a system that behaves chaotically without noise and randomness. Let us consider situations when a random process becomes deterministic, and elements of random, chaotic behavior are found in the deterministic process. Examples of such systems are the atmosphere, turbulent flows, some types of cardiac arrhythmias, biological populations, society as a communication system and its subsystems: economic, political and other social systems, partially crystalline polymers, etc. A typical example of deterministic chaos is the water of mountain streams. If you throw two leaves into this river, one after the other, then downstream they will most likely be far from each other. In a system like this, a small difference in the initial conditions (leaf positions) leads to a large discrepancy in the output. Can we predict the outcome of a billiard game? Not! Even the problem of a billiard ball bouncing off the boards on a perfectly level table dissolves into uncertainty due to inaccuracies in measuring the angle at which the ball approaches the board at the very beginning. The behavior of a deterministic system seems to be random, although it is determined by deterministic laws. The reason for the appearance of chaos is instability (sensitivity) with respect to the initial conditions and parameters: a small change in the initial condition over time leads to arbitrarily large changes in the system dynamics (Fig. 4). Since the initial state of a physical system cannot be specified absolutely exactly (for example, due to the limitations of measuring instruments), it is always necessary to consider some (albeit very small) area of ​​initial conditions. When moving in a limited region of space, the exponential divergence over time of close orbits leads to mixing of initial points throughout the region. After such mixing, it is meaningless to talk about the coordinate of the particle, but you can find the probability of its being at some point.

4 Fig. 4. Stable and unstable systems. An example of an unstable dynamical system is the two-dimensional gas of Heinrich Lorentz (1902). It consists of circles of the same radius of scatterers randomly scattered over the plane, and a material point (particle) that moves at a constant speed between them, each time experiencing a mirror reflection upon collision. One can be convinced of the instability of such a system by considering two close trajectories of a particle coming out of the same point. From fig. It can be seen from Fig. 5 that already after two scattering events, the angle between the trajectories, initially less than 1, becomes larger than π/2: initially close trajectories diverge very quickly, i.e. the particle "forgets" the initial conditions. ("forgetting" means that with a small variation of the initial conditions, the statistical properties of the trajectories do not change). At short times, predictions of the behavior of the system are still possible, however, starting from a certain moment, one has to use a statistical approach. Rice. 5. "Loss of memory" and divergence of close trajectories as a result of instability of motion in a two-dimensional gas by G. Lorenz. An important circumstance is the fact that the degree of orderliness of chaos can often be calculated. The measure is given by the geometry of fractals. We will deal with this in subsequent lectures of this course. 2.2 Types of complex systems The task of predicting the behavior of the system under study in time and space on the basis of certain knowledge about its initial state is reduced to finding a certain law that allows, using the available information about the object at the initial moment of time at some point in space, to determine its future at any next moment of time . Depending on the degree of complexity of the object itself, this law can be deterministic or probabilistic, it can describe the evolution of the object only in time, only in space, or it can describe the spatio-temporal evolution. There are different types of systems. A conservative system is a physical system, the work of conservative forces of which is equal to zero and for which the law of conservation of mechanical energy takes place, i.e. the sum of the kinetic energy and potential energy of the system is constant. The volume in the phase space is constant. Examples of a conservative system are the solar system and an oscillating pendulum (neglecting friction in the suspension axis and air resistance). A dynamic system is a mathematical abstraction designed to describe and study the evolution of systems in time. It is a stateful system. It describes the dynamics of some process, namely: the process of transition of the system from one state to another. The phase space of a system is the totality of all admissible states of a dynamical system. Thus, a dynamic system is characterized by its initial state and the law by which the system passes from the initial state to another. A dynamic system is characterized by stability (the ability of the system to remain for an arbitrarily long time near the equilibrium position or on a given manifold) and roughness (preservation of properties for small changes in the structure of the dynamic system; “a rough system is one whose qualitative character of motion does not change with a sufficiently small change in parameters.

5 A special case of a dynamic system is a dissipative system, an open dynamic system in which an increase in entropy is observed. Rice. 6. Mixing colored plasticine in a ball after successive iterations of the Smale Horseshoe display, i.e., flattening and folding in half. A dissipative system is an open system that operates far from thermodynamic equilibrium. This is a stable state that occurs in a non-equilibrium medium under the condition of dissipation (dissipation) of energy that comes from outside. It is characterized by the spontaneous appearance of a complex, often chaotic structure. A distinctive feature of such systems is the nonconservation of volume in the phase space. A dynamic system is any object or process for which the concept of a state is unambiguously defined as a set of certain quantities at a given moment of time and a law is set that describes the change (evolution) of the initial state over time. This law allows predicting the future state of a dynamical system from the initial state. The mathematical apparatus used for the quantitative description of the law of evolution of dynamical systems is based on the use of differential equations, discrete mappings, graph theory, Markov chains, etc. A mathematical model of a dynamic system is considered given if the parameters (coordinates) of the system are introduced that uniquely determine its state, and the law of evolution is indicated. Thus, a dynamical system = a set of parameters + an evolution operator. The evolution of the system can be described by both differential equations and mappings (equations with discrete time). Dynamic systems can be described by linear (linear systems) or non-linear (non-linear systems) equations. Systems with continuous and discrete (cascades) time are possible. An important group of dynamic systems is represented by systems in which oscillations are possible. There are linear and non-linear oscillatory systems, lumped and distributed, conservative and dissipative, autonomous and non-autonomous. Self-oscillating systems represent a special class. Deterministic chaos is an abstract mathematical concept denoting a deterministic process in a deterministic nonlinear system, due to the property of this system to show instability, sensitive dependence of the system dynamics on small perturbations. Comment. It is necessary to distinguish between deterministic chaos in dissipative systems (for example, an excited pendulum with friction) and in conservative systems (for example, the motion of planets obeying Hamiltonian equations). Hamiltonian, Hamiltonian operator, total energy operator, H = E + U, where E is the kinetic energy operator, U is the potential energy operator. A synonym for deterministic chaos is dynamic chaos, a phenomenon in dynamical systems theory in which the behavior of a non-linear system appears to be random despite being determined by deterministic laws. Both terms are completely equivalent and are used to indicate a significant difference between chaos as a subject of scientific study in synergetics and chaos in the ordinary sense. The reverse to dynamic chaos is dynamic equilibrium and the phenomena of homeostasis.

6 An important circumstance is the fact that in dissipative systems chaotic dynamics develops within a certain structure. This structure is difficult to study with the usual methods of studying dynamics, for example, by plotting the dependence of the response on time or obtaining the frequency spectrum. The order should be sought in the phase space (along the axes of which the coordinate and velocity are plotted). Along the way, it can be found that chaotic movements have a fractal structure. Deterministic chaos is characterized by the presence of a periodic process, the trajectory of which is reproduced, i.e. after repeating the initial state, the same trajectory is reproduced again, regardless of its complexity. This makes it possible to predict the future by the parameters of one of the trajectory repetition periods. However, it is necessary to take into account the properties of equilibrium and nonequilibrium systems. Non-equilibrium open systems allow new structural states. Dissipative systems, regardless of the type of stability, cause a decrease in the phase volume in time to zero. So a dissipative system can pass into an ordered state as a result of the instability of the previous disordered state. The initially stable dissipative structure in the course of its evolution reaches a critical state corresponding to the structure stability threshold, starts to oscillate, and fluctuations arising in it lead to the self-organization of a new, more stable structure at a given hierarchical level of evolution. At the same time, it is important that, as in biological systems, stability-instability-stability transitions are controlled by cumulative feedback. It differs from externally regulated feedback in that it allows self-organizing such an internal structure that increases the degree of its organization. Thus, cumulative feedback due to the accumulated internal energy allows the system to carry out not only feedback, taking into account the information received about the previous critical state, but also to ensure the preservation or increase in the organization of structures. Examples of chaotic dynamical systems are the Smale horseshoe and the baker's transformation. The Smale horseshoe is an example of a dynamical system proposed by Steve Smale that has an infinite number of periodic points (and chaotic dynamics), and this property does not collapse under small perturbations of the system. Rice. 7. Evolution of Smale's horseshoe. According to the Smale horseshoe algorithm, a unit square is compressed in one direction (horizontally) and stretched in another (vertically), while the area is reduced. The resulting strip is then bent into a horseshoe shape and inserted back into the original square. This procedure is repeated many times. In the limit, a set with zero area is formed, which has a Cantor structure in cross section, a special case of fractal geometry (see the course of lectures by IN Beckman "Fractals"). We will consider the form of the Smale attractor later in this lecture. Rice. 8. Smale horseshoe mapping: stretching, shrinking and folding after a large number of mapping iterations leads to a fractal structure. The baker's map is a non-linear mapping of the unit square onto itself, which exhibits chaotic behavior. The name "baker display" comes from its resemblance to kneading dough. Since the mapping consists of stretching along the x-axis and shrinking along the y-axis, close trajectories diverge exponentially in the horizontal direction.

7 direction and converge in the vertical. From a random symbolic sequence, a chaotic trajectory is constructed, which passes arbitrarily close to each point of the square (ergodicity). Under the action of the mapping, any selected area turns into a set of narrow horizontal stripes, which, after a certain number of iterations, will evenly cover the unit square (shuffling). The transformation is reversible; when iterating in the opposite direction, any area will be divided into narrow vertical stripes and will also be shuffled over the entire square. Another example of deterministic chaos is the Hadamard billiards, i.e. billiards, in which a swirling surface of negative curvature is used instead of a flat table. Calculating the trajectory of a ball on a Hadamard billiard table is "completely unusable" because the small uncertainty inherent in the initial conditions leads to a large uncertainty in the predicted trajectory if we wait long enough to render the prediction useless. Rice. 9. Baker display. The transformation consists of uniform compression of the square by 2 times in the vertical direction and stretching in the horizontal direction. Next, the right half should be cut off and put on the left. The figure shows the action of the first two iterations. Systems of deterministic chaos allow a different attitude to the use of statistical approaches to improve the reliability of the experiment. According to traditional mathematical statistics, the more parallel experiments we carry out, the more reliably the dependences under study will be established. This is absolutely not applicable to deterministic systems; here, the effect of fundamental irreproducibility of the experiment takes place. We can set up the same experiment, reproduce the initial conditions in the most accurate way, and obtain repeatable results, but at some point (we cannot predict it) the observations will begin to give completely dissimilar results. This is due to the phenomenon of orbital recession, which is illustrated by the three examples just considered. 2.3 Discovery of deterministic chaos Consideration of deterministic chaos will begin with the theory of stochastic behavior of dynamic dissipative systems. We will be interested in the random behavior of a completely deterministic system, the evolution of which in time can be accurately predicted (and this is confirmed in a wide range of parameter changes), but which, for some values ​​of the initial conditions (and very insignificant ones), begins to fluctuate randomly and its behavior becomes unpredictable, chaotic . As everyday experience shows, for many physical systems, small changes in the initial conditions lead to small changes in the result. So, for example, the path of the car will change little if the steering wheel is only slightly turned. But there are situations in which the opposite is true. The side on which a coin placed on its edge will fall depends on the weak touch. The succession of heads and tails in a coin toss exhibits irregular, or chaotic, behavior over time, since extremely small changes in the initial conditions can lead to very different results. Until relatively recently, it was believed that the random behavior of a system is an exception, and almost all systems are deterministic. However, it is now clear that high sensitivity to initial conditions, leading to chaotic behavior in time, is a typical property of many systems. Such behavior, for example, is found in periodically stimulated heart cells, in electronic circuits, when turbulence occurs in liquids and gases, in chemical reactions, in lasers, etc. From the point of view of mathematics, in all nonlinear dynamic systems with more than two degrees of freedom (especially in many biological, meteorological and economic models) one can

8 to detect chaos and, therefore, at sufficiently long times their behavior becomes unpredictable. For a physical system whose behavior is determined in time, there is a rule in the form of differential equations that determines its future based on given initial conditions. It is natural to assume that deterministic motion is fairly regular and far from being random, since successive states are continuously evolving one from the other. This means that in classical mechanics all equations must be integrable. But already in 1892 A. Poincaré knew that in some mechanical systems, whose evolution in time is determined by Hamilton's equations, unpredictable chaotic behavior is possible. An example is the non-integrable three-body problem, which under certain conditions leads to completely chaotic trajectories. A particular case of the three-body problem is the motion of a test particle in the gravitational field of two fixed point masses. Even if the motion occurs in one plane, the trajectory of the particle looks extremely complex and confusing. She either wraps around one of the masses, then suddenly jumps to another (Fig. 10). Initially close trajectories diverge very quickly. Rice. 10. Motion of a test particle near two identical masses. The top shows the initial part of the trajectory, and the bottom shows its continuation. It is now known that there are many non-integrable systems in mechanics. 60 years after Poincare Kolmogorov, 1954; Arnold, 1963 and Moser, 1967 proved that in classical mechanics the motion in phase space is neither completely regular nor completely irregular, and the type of trajectory depends on the choice of initial conditions (now this statement is called the KAM theorem). Thus, stable regular motion is an exception in classical mechanics. The American meteorologist Edward Lorenz (1961), when modeling unevenly heated atmospheric air, found that even a simple system of three coupled first-order nonlinear differential equations can lead to completely chaotic trajectories (this is the first example of deterministic chaos in dissipative systems). E. Lorenz calculated the values ​​of the solution for a long time, and then stopped the calculation. He was interested in some singularity of the solution that arose in the middle of the counting interval, and therefore he repeated the calculations from that moment. The results of the recalculation would obviously coincide with the results of the initial calculation if the initial values ​​for the recalculation were exactly equal to the values ​​obtained earlier for this point in time. Lorentz slightly changed these values, reducing the number of valid decimal places. The errors introduced in this way were extremely small. The newly calculated solution for some time agreed well with the old one. However, as the count progressed, the discrepancy increased, and the new solution looked less and less like the old one. What Lorentz observed is now called the essential dependence on initial conditions, the main feature inherent in chaotic dynamics. Substantial dependency is sometimes called the butterfly effect. This name refers to the inability to make long-range weather forecasts. Lorenz himself clarified this concept in the article "Predictability: Can the flapping of a butterfly's wings in Brazil lead to the formation of a tornado in Texas?". Maybe! Further, by deterministic chaos, we will mean irregular, or chaotic, motion generated by nonlinear systems of equations for which dynamic laws uniquely determine the time evolution of the state of the system with a known prehistory. Deterministic chaos = non-linear system of equations + instability Deterministic chaos differs from regular motion in complex, non-repeating trajectories and unpredictable behavior of the system at large times. Deterministic chaos differs from a random process in that irregularity in it comes from the system itself, and not from an external factor (noise, fluctuations).

9 Fig. 11. The emergence of chaos at long times. Examples of non-linear systems in which deterministic chaos is manifested are: a pendulum with excitation, liquids near the threshold of turbulence, lasers, non-linear optics devices, Josephson junction (Josephson effect is the phenomenon of superconducting current flowing through a thin dielectric layer separating two superconductors) chemical reactions, classical systems that include many bodies (the three-body problem), particle accelerators, interacting nonlinear waves in plasma, biological models of population dynamics, stimulated heart cells, etc. As is known, linear differential or difference equations can be solved by the Fourier transform and do not lead to chaos. And nonlinear equations can lead to chaos, but it is important to understand that nonlinearity is a necessary but not sufficient condition for the occurrence of chaotic motion. The chaotic behavior observed in time does not arise from external sources of noise, not from an infinite number of degrees of freedom, and not from the uncertainty associated with quantum mechanics (the systems under consideration are purely classical). The real root cause of irregularity is determined by the property of nonlinear systems to exponentially quickly separate initially close trajectories in a limited region of phase space (for example, three-dimensional in the Lorentz system). It is impossible to predict the long-term behavior of such systems, since the initial conditions can only be specified with finite accuracy, and the errors increase exponentially. When solving such a non-linear system of equations on a computer, the result at increasingly distant times depends on an increasing number of digits in (irrational) numbers representing the initial conditions. Since the digits in irrational numbers are distributed irregularly, the trajectory becomes chaotic. Several fundamental questions arise here: - Is it possible to predict (for example, by the form of the corresponding differential equations) whether deterministic chaos is realized in the system? - Is it possible to define the concept of chaotic motion more strictly from the point of view of mathematics and develop quantitative characteristics for it? - What is the impact of these results on different areas of physics? Does the existence of deterministic chaos mean the end of long-term predictability in physics for non-linear systems, or can one still learn something from a chaotic signal? 2.4 Elements of the theory of dynamical systems Let us now turn to the presentation of the theoretical foundations of the description of dynamical systems. However, first we recall the concepts on which the mathematical apparatus used in this area is based. The phase space is the space on which the set of all states of the system is represented, so that each possible state of the system corresponds to a point in the phase space. Phase space = space of values ​​of system parameters. Trajectory = set of points in phase space visited by the system in sequence. The peculiarity of the phase space is that the state of an arbitrarily complex system is represented in it by a single point, and the evolution of this system by the displacement of this point. When considering several identical systems, several points in the phase space are given. The totality of such systems is called a statistical ensemble. According to Liouville's theorem, a closed curve (or surface) consisting of points of the Hamiltonian phase space evolves in such a way that the area (or volume) of the phase space contained in it is preserved in time.

10 Liouville's theorem: the distribution function of a Hamiltonian system is constant along any trajectory in the phase space. The theorem asserts the conservation in time of the phase volume, or probability density in the phase space. The Hamiltonian system is a special case of a dynamical system describing physical processes without dissipation. In it, forces do not depend on speed. A dynamic system is a system that has a state. It describes the dynamics of the system's transition from one state to another. The phase space of a system is the totality of all admissible states of a dynamical system. A dynamic system is characterized by its initial state and the law by which the system passes from the initial state to another. A dynamic system is a system whose model is a system of ordinary differential equations. Stable dynamic system - a dynamic system, the state of which is completely determined by the initial conditions and external influences in the process of development. In a conservative system, an element in the phase space only changes its shape, but retains its volume (Liouville's theorem is fulfilled), which predetermines the nature of evolution and the type of chaos that occurs in conservative systems. Conservative systems are characterized by a constant supply of energy over time. In mechanics, they are called Hamiltonian. Mechanical oscillatory systems in the absence of friction are conservative systems. In conservative systems, chaotic orbits tend to uniformly fill all parts of some subspace in the phase space, i.e. they are characterized by a uniform probability density in limited regions of the phase space. Rice. 12. Preservation of the phase volume during the evolution of the Hamiltonian system. An example of a simple conservative system with one degree of freedom is a pendulum. If friction does not have a noticeable effect on the oscillations of the pendulum, then the Hamiltonian of the pendulum of length l and mass m is equal to the sum of potential Π= mglcosϕ and kinetic K=p 2 /2ml 2 energies: H=p 2 /2ml 2 mglcosj, (2) where j is the angle deviation from the vertical, and g is the acceleration due to gravity. The pendulum motion equation is: 2 d j + w 2 0 sinj = 0, (3) 2 dt g where w 0 = oscillation frequency. l Fig. 13. Phase portrait of a pendulum with a Hamiltonian (2). When the total energy H=E of the pendulum exceeds the largest value of the potential energy, E=E rot >mgl, the momentum p will always be different from zero, which leads to an unlimited increase in the angle j. This means that the pendulum will rotate. On the phase plane (Fig. 13), this behavior is depicted by the trajectories E rot corresponding to the movement of the phase point from left to right for p>0 and from right to left for p<0. Колебаниям маятника соответствует энергия E=E osc

11 of the pendulum, and the hyperbolic points corresponding to the upper equilibrium position of the pendulum are unstable. The phase curve that started in the vicinity of the hyperbolic point moves away from it, while the trajectory near the elliptic point always remains in its vicinity. Comment. The pendulum in the case of small deviations is described by linear equations: the oscillation frequency does not depend on the amplitude. The pendulum in the case of large deviations refers to a non-linear system: the oscillation frequency depends on the amplitude. Rice. 14. Phase portrait of an integrable system with two degrees of freedom. For systems with two degrees of freedom, the phase space is four-dimensional. An example is the system of two harmonic oscillators of unit mass (Fig. 14). In the case of completely integrable systems with n degrees of freedom, the phase space is 2n-dimensional and, in the action-angle variables, has the structure of a set of n-dimensional tori. Any possible trajectory is located on one of them. In this case, some trajectories may turn out to be closed, while others will densely cover the surface of the corresponding torus everywhere. A dissipative system is an open dynamic system in which an increase in entropy is observed. In a dissipative system, due to the dissipation of energy, the volume of an element of the phase space decreases with time (the Louisville theorem is not observed). Therefore, attracting sets appear in the phase space of dissipative systems, which do not exist in conservative systems attractors (attract attract). An attractor is a state of a dynamic system to which it aspires in the course of its movement (development). In the phase space, the attractor of a stable dynamical system is represented by a point (in the case of aperiodic processes) or a limit cycle (in the case of periodic processes). A strange attractor is an attractor, which in the phase space corresponds to a region that attracts all phase trajectories from the surrounding regions. These trajectories have a complex and intricate structure and are open curves. Rice. 15. To the definition of conservative (a) and dissipative (b) dynamical systems. For dissipative systems, it is typical that over time the cloud of representing points "shrinks" and concentrates on one or several attractors of subsets of the phase space, which usually have zero phase volume (Fig. 15b). From the point of view of time dynamics, this means that the regime that occurs in a system left to itself for a long time becomes independent of the initial state. In dissipative systems, there are attractors in the phase space. Rice. 16. Construction of the Poincaré mapping in the phase space of an autonomous Hamiltonian system with two degrees of freedom. In the analysis of dynamical systems, the Poincaré map is widely used. Mapping is a law according to which each element of some given set X is associated with a well-defined element of another given set Y.

12 Poincaré mapping (first return mapping) projection of some area in the phase space onto itself (or onto another area) along the trajectories (phase curves) of the system. Rice. 17. Construction of the Poincaré mapping in the phase space of an autonomous Hamiltonian system with two degrees of freedom. A. Poincaré proposed a procedure that associates some mapping with dynamics in the framework of differential equations. The idea is as follows: some surface is selected in the phase space, and the image of the phase trajectory is constructed, which is obtained when it intersects this surface. On fig. 17 shows an illustration of this method of the Poincare section of a four-turn limit cycle. It can be seen that in such a section, the representative point will sequentially occupy the positions marked by the numbers 1, 2, 3, and 4. Thus, in terms of mappings, we can say that a cycle of period 4 is realized. It is clear that certain rearrangements of the limit cycle will lead to and to perestroikas in the Poincaré section. The latter is much easier to study, which determines the importance of this method. When analyzing specific systems, the Poincaré section is constructed using a computer. Rice. 18. Qualitatively different trajectories are distinguished by Poincaré sections: a chaotic motion; b movement to a fixed point; in cycle ;, z cycle of doubled period. On fig. 18 shows four types of Puncaré section. Note that the Poincare section method is an effective, but not always reliable, method for studying periodic motion with a decrease in the order of the system. Let us illustrate the application of the Poincaré section by the example of the Henon-Heilis system of equations (1964), which describes the motion of a particle with mass m=1 in a two-dimensional potential: 2 2 x + y U (x, y) = + x y - y 2 3 (3) two identical harmonic oscillators with non-linear interaction between them. If the total energy of this mechanical system is 0

13 Fig. 19. Henon-Heiles model: a area of ​​fenite movement (dashed lines represent equipotential curves U=const, 1 U=0.01, 2 U=0.04, 3 U=0.125); Poincare section (y, P y) at particle energy E=1/10 (b) and E=1/8 (c). Dynamic systems, which are described by ordinary (linear) differential equations, have four types of solutions: equilibrium state, periodic motion, quasi-periodic motion and chaotic. Dynamic systems modeled by a finite number of ordinary differential equations are called lumped or point systems. They are described using a finite-dimensional phase space and are characterized by a finite number of degrees of freedom. One and the same system under different conditions can be considered as either concentrated or distributed. Mathematical models of distributed systems are partial differential equations, integral equations or ordinary delay equations. The number of degrees of freedom of a distributed system is infinite, and an infinite number of data are required to determine its state. Rice. 20. Scheme of possible signal transformations in linear and non-linear systems. In a linear system, the evolution operator is linear, i.e. A(x+y)=Ax+Ay, A(lx)=lAx. In such a system, there can be no chaotic oscillations. In it, periodic external influences cause, after the damping of transient processes, a periodic response of the same period (Fig. 20). As is known, there are three classical types of motion: equilibrium, periodic motion (limit cycle) and quasi-periodic motion. These states are called attractors, because in the presence of any damping, transient deviations are suppressed and the system is "attracted" to one of the three listed states. There is, however, a class of motions (nonlinear oscillations) that cannot be reduced to any of the classical attractors. Here, the movements are chaotic in the sense that if there is a small uncertainty in the initial conditions, then they are unpredictable (strange attractor). Classical attractors correspond to classical geometric objects in the phase space: an equilibrium state is a point, a periodic motion or a limit cycle is a closed curve, and a surface in a three-dimensional phase space corresponds to a quasi-periodic motion. The strange attractor is connected to the geometric object by a fractal set. In three-dimensional phase space, the fractal set of the strange attractor looks like a collection of an infinite number of layers or parallel planes, with the distance between some of them approaching infinitesimal. An example of a non-integrable system is a double flat pendulum with point masses m 1 and m 2, (Fig. 5) which has two degrees of freedom, angles φ 1 and φ 2. If the deviation from the equilibrium position is small, then the system performs regular harmonic oscillations. However

14, as the total energy increases, there comes a moment when the oscillations become chaotic, the pendulums begin to rotate, and two close initial conditions lead to completely different dynamics of this nonlinear system with two degrees of freedom. A chaotic dynamical system is a dynamical system in which processes are described by a strange attractor. In contrast to a stable dynamical system, it is impossible to determine the state of the system from given values ​​of time and initial conditions. An important characteristic feature of all systems in which deterministic chaos is observed is that they are described by nonlinear differential equations or systems of equations. The principle of superposition is not applicable to such equations, which is valid for linear systems, according to which the sum of solutions is also a solution. A non-linear system is controlled by a non-linear operator: A(a 1 x 1 +a 2 x 2) a 1 Ax 1 +aax 2. An example is the sin(x) function. The situation is further complicated by the fact that non-linear equations often have not one, but several solutions. Among them can be both chaotic and regular, periodic solutions. Which of them is carried out in practice depends on the initial conditions. Rice. 21. Double flat pendulum and its chaotic oscillations. The simplest type of dynamic chaos is chaotic dynamics in nonlinear systems with discrete time (regular dynamics is considered in this case as a stage preceding chaos). The mathematical apparatus here is simple; in fact, it is reduced to the theory of difference equations. Understanding chaos in systems with continuous time is more difficult; deep knowledge of the theory of differential equations is required. It is important to understand that for the emergence of chaos in the case of systems with continuous time, their dimension (order N of the nonlinear differential equation describing this system) must be at least 3. Such systems (3D dynamical systems) are represented by flows of trajectories in phase space, the dimension of which is 3 (or higher, according to the order of the differential equation). However, in nonlinear dynamical systems with discrete time, chaotic motions can already appear in the case of systems of the 1st order (1D discrete dynamical systems). These motions represent cascades of discrete mappings and are described by non-linear difference equations of order 1 and higher. Note that there are four criteria for the randomness of movement: the signal "looks random"; the power spectrum exhibits broadband noise at low frequencies; the autocorrelation function falls off rapidly; the Poincaré section consists of points that fill the space. Mathematical models containing 3 or more ordinary differential equations are able to demonstrate chaotic oscillation modes, which at first glance look like random processes. The transition to the phase space makes it possible to obtain visual information about the features of the complex dynamics of the corresponding systems, and, above all, about the geometry of the limiting sets of phase trajectories that correspond to steady-state regimes. E. Lorenz's strange attractor played an important role in the analysis of chaotic systems. Lorentz showed that heating the air from the side of the Earth and cooling it from the opposite side leads to convection flows, which are approximately described by a system of three first-order ordinary differential equations that do not have an exact analytical solution: dx/dt=s(y x), (4а) dy/ dt=x(r z) y, (4b) dz/dt=xy bz, (4c) where s=10, r=28, b=8/3.

15 The Lawrence model is a dynamical system in three-dimensional phase space. The variable X is proportional to the speed of the convective flow (characterizes the speed of rotation of the convection shafts), Y and Z are responsible for the temperature distribution horizontally and vertically, respectively. The parameter r is proportional to the Rayleigh number, and s and b are some dimensionless constants characterizing the system. The solution of these equations, the functions X(t), Y(t) and Z(t) determine in a parametric form the trajectory of the system in the three-dimensional "phase" space X,Y,Z. In view of the uniqueness of the functions on the right-hand sides of these equations, the trajectory never intersects itself. Lorentz studied the form of these trajectories under different initial conditions for the values ​​of the parameters r=28, s=10 and b=8/3. He discovered that in this case the trajectory wanders chaotically from the half-space x>0 to the half-space x<0, формируя две почти плоских, перепутанных сложным образом спирали. На рис. 8 показана проекция этих спиралей на плоскость XZ для некоторого начального условия. Траектория сначала делает 1 оборот справа, затем 20 слева, затем опять 1 справа, затем 4 слева и так далее. Похожее поведение имеет место и при других значениях параметров. Хаотичность решения означает, что если мы заранее выберем каким угодно способом цепочку переходов из одного полупpостpанства в другое, то у системы Лоренца найдётся решение, которое в точности эту цепочку воспроизведёт. Рис. 22. Траектория, отвечающая хаотическому решению уравнений Лоренца, с параметрами, приведенными в тексте, и начальными условиями X(0)=Y(0)=Z(0)=1. Один эллипс отражает вращение атмосферы по часовой стрелке, другой - против неё. Причина непpедсказуемости поведения этой и других подобных систем заключается в не в том, что не верна математическая теорема о существовании и единственности решения при заданных начальных условиях, а в необычайной чувствительности решения к этим начальным условиям. Близкие начальные условия со временем приводят к совершенно различному конечному состоянию системы. Причём часто различие нарастает со временем экспоненциально, то есть чрезвычайно быстро (см. рис. 23): D(t) = D(0)e ht, (5) где инкремент неустойчивости h является функцией точки в фазовом пространстве. Рис. 23. Две первоначально близкие траектории в фазовом пространстве расходятся со временем в результате локальной неустойчивости. Оказалось, что нечто похожее происходит и с системами, в которых наблюдается детеpминиpованный хаос: они движутся таким образом, что всё время находятся в неустойчивом состоянии. Иными словами, сколь угодно малые возмущения начальных условий приводят с течением времени к сильному отклонению траектории от своего невозмущенного положения. Если фазовое пространство системы является конечным, то фазовые траектории не могут разойтись из-за неустойчивости более чем на характерный размер области движения, и начинается их запутывание. Предсказать поведение такой системы тогда оказывается практически невозможным. Странный аттрактор это некоторое «сложно устроенное» множество в фазовом пространстве, к которому притягиваются почти все траектории из его некоторой окрестности, а на самом множестве движение имеет экспоненциально неустойчивый характер. Такое сочетание глобального сжатия с локальной неустойчивостью приводит к тому, что аттрактор уже не может быть гладким как, например, тор; он определенным образом расслаивается и представляет собой в некотором сечении канторово множество (фрактально). Странный аттрактор обладает двумя свойствами: траектории на странном аттракторе разбегаются друг от друга; объёмы в фазовом пространстве со временем сокращаются.

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Lecture 5 Chaotic behavior of dynamical systems. Lorentz system The variety of behavior of trajectories on the plane is limited by the Bendixon-Poincare theorem, according to which the trajectory can go to infinity,

Dynamic systems and methods of mathematical modeling Symbolic dynamics Symbolic dynamics The method of symbolic dynamics is a description of the dynamics of a system using admissible sequences

3. Types of attractors 1 3. Types of attractors It is possible to visualize the location of attractors on the phase plane in a very clear way, largely due to the fact that there are only a few of their types,

Yaroslavl State Pedagogical University. K. D. Ushinsky Department of General Physics Laboratory of Mechanics Laboratory work 7 Experimental determination of the acceleration of gravity and characteristics

FEDERAL STATE BUDGETARY EDUCATIONAL INSTITUTION OF HIGHER EDUCATION "ORENBURG STATE AGRARIAN UNIVERSITY" Department of "Mathematics and Theoretical Mechanics" Guidelines

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Nonlinear pendulum. 1 Dimensionless equation of motion of a physical pendulum with viscous friction. The equation of motion of a physical pendulum, taking into account viscous friction: I φ + b φ + mga sin(φ) =, (1) where I is the moment

DYNAMIC SYSTEMS V. S. ANISHCHENKO The mathematical definition of a dynamic system is formulated. For the dynamic systems described by ordinary differential equations, four types of

Butikov E. I. Educational Laboratory for Computer Simulation of Oscillations in Simple Nonlinear Systems St. Petersburg State University

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Seminar 4 System of two ordinary differential equations (ODE). phase plane. Phase portrait. Kinetic curves. special points. Steady State Stability. System linearization in

Maxwell's Law of Velocity Distribution 1. Maxwell's Law of Velocity Distribution. 3.Mean free path 4.Experienced

Harmonic Oscillations Oscillations are processes (movement or change of state) that are repeated to some extent in time. mechanical oscillations electromagnetic electromechanical

12 April 11 Poincaré Mean Return Time Capture Effect as a Criterion for Forced Chaos Synchronization V.S. Anishchenko, Ya.I. Boev Saratov State University E-mail: [email protected] Received

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1. INTRODUCTION Physics is the science of the most general properties and forms of motion of matter. In the mechanical picture of the world, matter was understood as a substance consisting of particles, eternal and unchanging. basic laws,

Lyapunov's theory of stability. In many problems of mechanics and technology, it is important to know not the specific values ​​of the solution for a given specific value of the argument, but the nature of the behavior of the solution when changing

Oscillations in systems with distributed parameters Lines with losses Losses in wires L L equiv i (x,t) R L equiv Ldx u(x,t) u(x+dx,t) R equiv Rdx u(x dx,t) u(x ,t) L equiv i(x,t) t R equiv i(x,t) u x dx Ldx i t

The program is compiled on the basis of the federal state educational standard of higher education (the level of training of highly qualified personnel) in the direction of training 01.06.01 "Mathematics

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Lecture 8 Wave motion Propagation of oscillations in a homogeneous elastic medium Longitudinal and transverse waves Equation of a plane harmonic traveling wave displacement, velocity and relative deformation

Mathematical foundations of chaotic dynamical systems Alexander Loskutov, Faculty of Physics, Moscow State University Abstract The dynamical approach to the description of systems of various origins has been known since the time of Newton.

5. Parametric oscillations 5. Introduction

The concept of bifurcation. Bifurcations of equilibrium positions. Differential equations of dynamic systems often depend not only on phase variables, but also on parameters, i.e. have the following structure: ẋ =

1 LECTURE 8 Random and deterministic processes. Was Laplace right? Chaos in nature and in everyday life. What is a random number? Chaotic signal as a solution to a differential equation. Opening

TOMSK STATE UNIVERSITY Faculty of Physics STUDY OF FORCED OSCILLATIONS OF A SPRING PENDULUM Guidelines for performing laboratory work Tomsk 14 Considered and approved by the methodological

Some discrete models of turbulence Akishev A.A. FGAOU HPE "Ural Federal University named after the first President of Russia B.N. Yeltsin" Ekaterinburg, Russia The paper considers the seven-dimensional

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Topic 5. Mechanical oscillations and waves. 5.1. Harmonic oscillations and their characteristics Oscillations are processes that differ in varying degrees of repeatability. Depending on the physical nature of the repeating

2. Phase space 1 2. Phase space Before moving on to the numerical methods for solving Cauchy problems for ODEs (see the following paragraphs), let's say a few words about important aspects of their visualization,

Order and disorder in nature. Synergetics. "THE WHOLE ORDERED WORLD IS CREATED FROM CHAOS" (myth) January 25, 1917 "ORDER FROM CHAOS" (I.Prigozhin) In the course of the evolution of life, energy is needed to form order,

Objective. To get acquainted with the main characteristics of undamped and damped free mechanical oscillations. Task. Determine the period of natural oscillations of the spring pendulum; check linearity

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Oscillations and waves Oscillations are processes that are characterized by a certain recurrence in time Oscillatory system (oscillator) a system that oscillates