The law of radioactive decay is the constant of radioactive decay activity. Basic law of radioactive decay

1. Radioactivity. The basic law of radioactive decay. Activity.

2. Main types of radioactive decay.

3. Quantitative characteristics of the interaction of ionizing radiation with matter.

4. Natural and artificial radioactivity. Radioactive series.

5. Use of radionuclides in medicine.

6. Accelerators of charged particles and their use in medicine.

7. Biophysical basis of the action of ionizing radiation.

8. Basic concepts and formulas.

9. Tasks.

The interest of doctors in natural and artificial radioactivity is due to the following.

Firstly, all living things are constantly exposed to natural background radiation, which constitute cosmic radiation, radiation of radioactive elements located in the surface layers of the earth's crust, and radiation of elements entering the body of animals along with air and food.

Secondly, radioactive radiation is used in medicine itself for diagnostic and therapeutic purposes.

33.1. Radioactivity. The basic law of radioactive decay. Activity

The phenomenon of radioactivity was discovered in 1896 by A. Becquerel, who observed the spontaneous emission of unknown radiation from uranium salts. Soon E. Rutherford and the Curies established that during radioactive decay He nuclei (α-particles), electrons (β-particles) and hard electromagnetic radiation (γ-rays) are emitted.

In 1934, decay with the emission of positrons (β + -decay) was discovered, and in 1940, a new type of radioactivity was discovered - spontaneous fission of nuclei: a fissile nucleus falls apart into two fragments of comparable mass with the simultaneous emission of neutrons and γ -quanta. Proton radioactivity of nuclei was observed in 1982.

Radioactivity - the ability of some atomic nuclei to spontaneously (spontaneously) transform into other nuclei with the emission of particles.

Atomic nuclei consist of protons and neutrons, which have a general name - nucleons. The number of protons in the nucleus determines Chemical properties atom and is denoted by Z (this is serial number chemical element). The number of nucleons in a nucleus is called mass number and denote A. Nuclei with the same atomic number and different mass numbers are called isotopes. All isotopes of one chemical element have the same Chemical properties. Physical properties isotopes can vary greatly. To designate isotopes, use the symbol of a chemical element with two indices: A Z X. The lower index is the serial number, the upper index is the mass number. Often the subscript is omitted because it is indicated by the element's symbol itself. For example, they write 14 C instead of 14 6 C.

The ability of a nucleus to decay depends on its composition. The same element can have both stable and radioactive isotopes. For example, the carbon isotope 12 C is stable, but the isotope 14 C is radioactive.

Radioactive decay is a statistical phenomenon. The ability of an isotope to decay characterizes decay constantλ.

Decay constant- the probability that the nucleus of a given isotope will decay per unit time.

The probability of nuclear decay in a short time dt is found by the formula

Taking into account formula (33.1), we obtain an expression that determines the number of decayed nuclei:

Formula (33.3) is called the main law of radioactive decay.

The number of radioactive nuclei decreases with time according to an exponential law.

In practice, instead decay constantλ another quantity is often used, called half-life.

Half life(T) - time during which it decays half radioactive nuclei.

The law of radioactive decay using half-life is written as follows:

The graph of dependence (33.4) is shown in Fig. 33.1.

The half-life can be very long or very short (from fractions of a second to many billions of years). In table Figure 33.1 shows the half-lives for some elements.

Rice. 33.1. Decrease in the number of nuclei of the original substance during radioactive decay

Table 33.1. Half-lives for some elements

For rate degree of radioactivity isotope use a special quantity called activity.

Activity - number of cores radioactive drug, decaying per unit time:

The SI unit of activity is becquerel(Bq), 1 Bq corresponds to one decay event per second. In practice, more

childish non-systemic unit of activity - curie(Ci), equal to the activity of 1 g 226 Ra: 1 Ci = 3.7x10 10 Bq.

Over time, activity decreases in the same way as the number of undecayed nuclei decreases:

33.2. Main types of radioactive decay

In the process of studying the phenomenon of radioactivity, 3 types of rays emitted by radioactive nuclei were discovered, which were called α-, β- and γ-rays. Later it was found that α- and β-particles are products of two various types radioactive decay, and γ-rays are a by-product of these processes. In addition, γ-rays accompany more complex nuclear transformations, which are not considered here.

Alpha decay consists in the spontaneous transformation of nuclei with the emissionα -particles (helium nuclei).

The α-decay scheme is written as

where X, Y are the symbols of the mother and daughter nuclei, respectively. When writing α-decay, you can write “He” instead of “α”.

During this decay, the atomic number Z of the element decreases by 2, and the mass number A - by 4.

During α-decay, the daughter nucleus, as a rule, is formed in an excited state and, upon transition to the ground state, emits a γ-quantum. The general property of complex microobjects is that they have discrete a set of energy states. This also applies to kernels. Therefore, γ-radiation from excited nuclei has a discrete spectrum. Consequently, the energy spectrum of α-particles is discrete.

The energy of emitted α-particles for almost all α-active isotopes lies in the range of 4-9 MeV.

Beta decay consists in the spontaneous transformation of nuclei with the emission of electrons (or positrons).

It has been established that β-decay is always accompanied by the emission of a neutral particle - a neutrino (or antineutrino). This particle practically does not interact with matter and will not be considered further. The energy released during beta decay is distributed randomly between the beta particle and the neutrino. Therefore, the energy spectrum of β-radiation is continuous (Fig. 33.2).

Rice. 33.2. Energy spectrum of β-decay

There are two types of β decay.

1. Electronicβ - -decay consists of the transformation of one nuclear neutron into a proton and an electron. In this case, another particle ν" appears - an antineutrino:

An electron and an antineutrino fly out from the nucleus. The electron β - decay scheme is written in the form

During electronic β-decay, the order number of the Z element increases by 1, but the mass number A does not change.

The energy of β-particles lies in the range of 0.002-2.3 MeV.

2. Positronicβ + -decay involves the transformation of one nuclear proton into a neutron and a positron. In this case, another particle ν appears - a neutrino:

Electron capture itself does not produce ionizing particles, but it does accompanied by X-ray radiation. This radiation occurs when the space vacated by the absorption of an internal electron is filled by an electron from the outer orbit.

Gamma radiation has an electromagnetic nature and represents photons with a wavelengthλ ≤ 10 -10 m.

Gamma radiation is not an independent type of radioactive decay. Radiation of this type almost always accompanies not only α-decay and β-decay, but also more complex nuclear reactions. It is not deflected by electric and magnetic fields, has a relatively weak ionizing and very high penetrating ability.

33.3. Quantitative characteristics of the interaction of ionizing radiation with matter

Impact radioactive radiation on living organisms is associated with ionization, which it causes in tissues. The ability of a particle to ionize depends on both its type and its energy. As a particle moves deeper into matter, it loses its energy. This process is called ionization inhibition.

To quantitatively characterize the interaction of a charged particle with matter, several quantities are used:

Once the particle's energy drops below the ionization energy, its ionizing effect ceases.

Average linear mileage(R) of a charged ionizing particle - the path traveled by it in a substance before losing its ionizing ability.

Let's look at some characteristics interactions of various types of radiation with matter.

Alpha radiation

The alpha particle practically does not deviate from the initial direction of its movement, since its mass is many times greater

Rice. 33.3. Dependence of linear ionization density on the path traveled by an α-particle in the medium

the mass of the electron with which it interacts. As it penetrates deep into the substance, the ionization density first increases, and when completion of the run (x = R) drops sharply to zero (Fig. 33.3). This is explained by the fact that as the speed of movement decreases, the time it spends near a molecule (atom) of the medium increases. The probability of ionization increases in this case. After the energy of the α particle becomes comparable to the energy of molecular thermal motion, it captures two electrons in the substance and turns into a helium atom.

Electrons formed during the ionization process, as a rule, move away from the α-particle track and cause secondary ionization.

Characteristics of the interaction of α-particles with water and soft tissues are presented in Table. 33.2.

Table 33.2. Dependence of the characteristics of interaction with matter on the energy of α-particles

Beta radiation

For movement β -particles in matter are characterized by a curvilinear unpredictable trajectory. This is due to the equality of the masses of interacting particles.

Interaction Characteristics β -particles with water and soft tissues are presented in table. 33.3.

Table 33.3. Dependence of the characteristics of interaction with matter on the energy of β-particles

Like α particles, the ionization ability of β particles increases with decreasing energy.

Gamma radiation

Absorption γ -radiation by matter obeys an exponential law similar to the law of absorption of X-ray radiation:

The main processes responsible for absorption γ -radiation are the photoelectric effect and Compton scattering. In this case, a relatively small number of free electrons are formed (primary ionization), which have a very high energy. They cause processes of secondary ionization, which is incomparably higher than the primary one.

33.4. Natural and artificial

radioactivity. Radioactive series

Terms natural And artificial radioactivity are conditional.

Natural called the radioactivity of isotopes existing in nature, or the radioactivity of isotopes formed as a result of natural processes.

For example, the radioactivity of uranium is natural. The radioactivity of carbon 14 C, which is formed in the upper layers of the atmosphere under the influence of solar radiation, is also natural.

Artificial called radioactivity of isotopes that arise as a result of human activity.

This is the radioactivity of all isotopes produced in particle accelerators. This also includes the radioactivity of soil, water and air that occurs during an atomic explosion.

Natural radioactivity

IN initial period to study radioactivity, researchers could only use natural radionuclides (radioactive isotopes) contained in earth rocks in sufficient quantities large quantities: 232 Th, 235 U, 238 U. Three radioactive series begin with these radionuclides, ending with stable isotopes Pb. Subsequently, a series was discovered starting with 237 Np, with the final stable nucleus 209 Bi. In Fig. Figure 33.4 shows the row starting with 238 U.

Rice. 33.4. Uranium-radium series

Elements of this series are the main source of internal human radiation. For example, 210 Pb and 210 Po enter the body with food - they are concentrated in fish and shellfish. Both of these isotopes accumulate in lichens and are therefore present in meat reindeer. The most significant of all natural sources of radiation is 222 Rn - a heavy inert gas resulting from the decay of 226 Ra. It accounts for about half the dose of natural radiation received by humans. Formed in the earth's crust, this gas seeps into the atmosphere and enters water (it is highly soluble).

The radioactive isotope of potassium 40 K is constantly present in the earth's crust, which is part of natural potassium (0.0119%). This element comes from the soil through the root system of plants and with plant foods (cereals, fresh vegetables and fruits, mushrooms) - into the body.

Another source of natural radiation is cosmic radiation (15%). Its intensity increases in mountainous areas due to a decrease in the protective effect of the atmosphere. Sources of natural background radiation are listed in Table. 33.4.

Table 33.4. Component of natural radioactive background

33.5. Use of radionuclides in medicine

Radionuclides called radioactive isotopes chemical elements with a short half-life. Such isotopes do not exist in nature, so they are obtained artificially. In modern medicine, radionuclides are widely used for diagnostic and therapeutic purposes.

Diagnostic Application based on the selective accumulation of certain chemical elements by individual organs. Iodine, for example, is concentrated in the thyroid gland, and calcium in the bones.

The introduction of radioisotopes of these elements into the body makes it possible to detect areas of their concentration by radioactive radiation and thus obtain important diagnostic information. This diagnostic method is called by the labeled atom method.

Therapeutic Use radionuclides is based on the destructive effect of ionizing radiation on tumor cells.

1. Gamma therapy- use of high-energy γ-radiation (60 Co source) to destroy deep-lying tumors. To prevent superficial tissues and organs from being subjected to harmful effects, exposure to ionizing radiation is carried out in different sessions in different directions.

2. Alpha therapy- therapeutic use of α-particles. These particles have a significant linear ionization density and are absorbed by even a small layer of air. Therefore therapeutic

The use of alpha rays is possible through direct contact with the surface of the organ or when administered internally (using a needle). For surface exposure, radon therapy (222 Rn) is used: exposure to the skin (baths), digestive organs (drinking), and respiratory organs (inhalation).

In some cases, medicinal use α -particles is associated with the use of neutron flux. With this method, elements are first introduced into the tissue (tumor), the nuclei of which, under the influence of neutrons, emit α -particles. After this, the diseased organ is irradiated with a stream of neutrons. In this manner α -particles are formed directly inside the organ on which they should have a destructive effect.

Table 33.5 shows the characteristics of some radionuclides used in medicine.

Table 33.5. Characteristics of isotopes

33.6. Charged particle accelerators and their use in medicine

Accelerator- an installation in which, under the influence of electric and magnetic fields, directed beams of charged particles with high energy (from hundreds of keV to hundreds of GeV) are produced.

Accelerators create narrow beams of particles with a given energy and small cross section. This allows you to provide directed impact on irradiated objects.

Use of accelerators in medicine

Electron and proton accelerators are used in medicine for radiation therapy and diagnostics. In this case, both the accelerated particles themselves and the accompanying X-ray radiation are used.

Bremsstrahlung X-rays are obtained by directing a beam of particles to a special target, which is the source of X-rays. This radiation differs from the X-ray tube by significantly higher quantum energy.

Synchrotron X-rays occurs during the acceleration of electrons in ring accelerators - synchrotrons. Such radiation has high degree direction.

The direct effect of fast particles is associated with their high penetrating ability. Such particles pass through superficial tissues without causing serious damage and have an ionizing effect at the end of their journey. By selecting the appropriate particle energy, it is possible to destroy tumors at a given depth.

The areas of application of accelerators in medicine are shown in Table. 33.6.

Table 33.6. Application of accelerators in therapy and diagnostics

33.7. Biophysical basis of the action of ionizing radiation

As noted above, the impact of radioactive radiation on biological systems is associated with ionization of molecules. The process of interaction of radiation with cells can be divided into three successive stages (stages).

1. Physical stage consists of energy transfer radiation to molecules of a biological system, resulting in their ionization and excitation. The duration of this stage is 10 -16 -10 -13 s.

2. Physico-chemical the stage consists of various types of reactions leading to the redistribution of excess energy of excited molecules and ions. As a result, highly active

products: radicals and new ions with wide range chemical properties.

The duration of this stage is 10 -13 -10 -10 s.

3. Chemical stage - this is the interaction of radicals and ions with each other and with surrounding molecules. At this stage, structural damage of various types is formed, leading to changes in biological properties: the structure and functions of membranes are disrupted; lesions occur in DNA and RNA molecules.

The duration of the chemical stage is 10 -6 -10 -3 s.

4. Biological stage. At this stage, damage to molecules and subcellular structures leads to various functional disorders, to premature cell death as a result of the action of apoptotic mechanisms or due to necrosis. Damage received at the biological stage can be inherited.

The duration of the biological stage is from several minutes to tens of years.

Let us note the general patterns of the biological stage:

Large disturbances with low absorbed energy (a lethal dose of radiation for humans causes the body to heat up by only 0.001°C);

Effect on subsequent generations through the hereditary apparatus of the cell;

Characterized by a hidden, latent period;

Different parts of cells have different sensitivity to radiation;

First of all, dividing cells are affected, which is especially dangerous for a child’s body;

Detrimental effect on tissues of an adult organism in which there is division;

Similarity of radiation changes with the pathology of early aging.

33.8. Basic concepts and formulas

Table continuation

33.9. Tasks

1. What is the activity of the drug if 10,000 nuclei of this substance decay within 10 minutes?

4. The age of ancient wood samples can be approximately determined by the specific mass activity of the 14 6 C isotope in them. How many years ago was the tree cut down that was used to make an object, if the specific mass activity of carbon in it is 75% of the specific mass activity of the growing tree? The half-life of radon is T = 5570 years.

9. After the Chernobyl accident, in some places soil contamination with radioactive cesium-137 was at the level of 45 Ci/km 2 .

After how many years will activity in these places decrease to a relatively safe level of 5 Ci/km 2? The half-life of cesium-137 is T = 30 years.

10. The permissible activity of iodine-131 in the human thyroid gland should be no more than 5 nCi. Some people who were in the zone Chernobyl disaster, the activity of iodine-131 reached 800 nCi. After how many days did activity decrease to normal? The half-life of iodine-131 is 8 days.

11. To determine the blood volume of an animal, the following method is used. A small volume of blood is taken from the animal, red blood cells are separated from the plasma and placed in a solution with radioactive phosphorus, which is assimilated by the red blood cells. The labeled red blood cells are reintroduced into the animal's circulatory system, and after some time the activity of the blood sample is determined.

ΔV = 1 ml of such a solution was injected into the blood of some animal. The initial activity of this volume was equal to A 0 = 7000 Bq. The activity of 1 ml of blood taken from the vein of an animal a day later was equal to 38 pulses per minute. Determine the animal’s blood volume if the half-life of radioactive phosphorus is T = 14.3 days.

Radioactivity

Ionizing radiation

Radiation effects

The Earth is under constant influence of a flow of fast particles and hard quanta electromagnetic radiation coming from space. This stream is called cosmic rays. Cosmic rays come from the depths of the universe and from the Sun. Part of the cosmic ray flux reaches the Earth's surface, and part is absorbed by the atmosphere, generating secondary radiation and leading to the formation of various radionuclides. The interaction of cosmic rays with matter leads to its ionization.

The flow of particles or electromagnetic quanta, the interaction of which with a medium leads to the ionization of its atoms, is called ionizing radiation.

Ionizing radiation can also be of terrestrial origin. For example, occur during radioactive decay.

The phenomenon of radioactivity was discovered in 1896 by A. Becquerel.

Radioactivity - the ability of some atomic nuclei to spontaneously (spontaneously) transform into other nuclei with the emission of particles.

There are two types of radioactivity:

Natural, which is found in natural unstable nuclei;

Artificial, which is found in radioactive nuclei formed as a result of various nuclear reactions.

Both types of radioactivity have common patterns.

Radioactive decay is a statistical phenomenon. Can be installed probability decay of one nucleus over a certain period of time. Over equal periods of time, equal shares of existing (i.e., those that have not yet decayed at the beginning of a given period of time) nuclei decay radioactive element.

Let in a short time dt disintegrates dN cores. This number is proportional to the time interval dt and the total number of radioactive nuclei N:

where λ - decay constant, proportional to the probability of decay of a radioactive nucleus and depending on the nature of the element; the "-" sign indicates decreasing number of radioactive nuclei.

By decision differential equation(12.23) is an exponential function:

Where N 0- the number of radioactive nuclei at the moment t = 0, a N- the number of undecayed nuclei at the current time t.

Formula (12.24) expresses the law of radioactive decay.

Number of radioactive nuclei decreases with time according to an exponential law.

In practice, instead of the decay constant A, another value is often used, called half-life.

Half-life (T)- this is the time during which it decays half radioactive nuclei.

The half-life can be very long or very short. For example, for uranium T = 4.5 10 9 years, and for lithium T Li = 0.89 s.

Decay characteristics T and λ are related by:

The law of radioactive decay using half-life is written as follows:

In Fig. Figure 12.7 shows the radioactive decay processes for two substances with different half-lives.

Rice. 12.7. Decrease in the number of nuclei of the original substance during radioactive decay

Lecture 2. The basic law of radioactive decay and the activity of radionuclides

The rate of decay of radionuclides is different - some decay faster, others slower. An indicator of the rate of radioactive decay is radioactive decay constant, λ [sec-1], which characterizes the probability of the decay of one atom in one second. For each radionuclide, the decay constant has its own value; the larger it is, the faster the nuclei of the substance decay.

The number of decays recorded in a radioactive sample per unit time is called activity (a ), or the radioactivity of the sample. The activity value is directly proportional to the number of atoms N radioactive substance:

a =λ· N , (3.2.1)

Where λ – radioactive decay constant, [sec-1].

Currently, according to the current International system SI units, taken as a unit of measurement of radioactivity becquerel [Bk]. This unit received its name in honor of the French scientist Henri Becquerel, who discovered the phenomenon of natural radioactivity of uranium in 1856. One becquerel equals one decay per second 1 Bk = 1 .

However, the non-system unit of activity is still often used – curie [Ki], introduced by the Curies as a measure of the decay rate of one gram of radium (in which ~3.7 1010 decays occur per second), therefore

1 Ki= 3.7·1010 Bk.

This unit is convenient for assessing the activity of large quantities of radionuclides.

The decrease in radionuclide concentration over time as a result of decay obeys an exponential relationship:

, (3.2.2)

Where N t– the number of atoms of a radioactive element remaining after time t after the start of observation; N 0 – number of atoms at the initial moment of time ( t =0 ); λ – radioactive decay constant.

The described dependence is called basic law of radioactive decay .

The time it takes for half of the total number radionuclides is called half-life T½ . After one half-life, out of 100 radionuclide atoms, only 50 remain (Fig. 2.1). Over the next similar period, only 25 of these 50 atoms remain, and so on.

The relationship between half-life and decay constant is derived from the equation of the fundamental law of radioactive decay:

at t=T½ And

we get https://pandia.ru/text/80/150/images/image006_47.gif" width="67" height="41 src="> Þ ;

https://pandia.ru/text/80/150/images/image009_37.gif" width="76" height="21">;

i.e..gif" width="81" height="41 src=">.

Therefore, the law of radioactive decay can be written as follows:

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Where at – drug activity over time t ; a0 – activity of the drug at the initial moment of observation.

It is often necessary to determine the activity of a given amount of any radioactive substance.

Remember that the unit of quantity of a substance is the mole. A mole is the amount of a substance containing the same number of atoms as are contained in 0.012 kg = 12 g of the carbon isotope 12C.

One mole of any substance contains Avogadro's number N.A. atoms:

N.A. = 6.02·1023 atoms.

For simple substances(elements) the mass of one mole corresponds numerically to atomic mass A element

1mol = A G.

For example: For magnesium: 1 mol 24Mg = 24 g.

For 226Ra: 1 mol 226Ra = 226 g, etc.

Taking into account what has been said in m grams of the substance will be N atoms:

https://pandia.ru/text/80/150/images/image015_20.gif" width="156" height="43 src="> (3.2.6)

Example: Let's calculate the activity of 1 gram of 226Ra, which λ = 1.38·10-11 sec-1.

a= 1.38·10-11·1/226·6.02·1023 = 3.66·1010 Bq.

If a radioactive element is included in the composition chemical compound, then when determining the activity of a drug, it is necessary to take into account its formula. Taking into account the composition of the substance, the mass fraction is determined χ radionuclide in a substance, which is determined by the ratio:

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Example of problem solution

Condition:

Activity A0 radioactive element 32P per day of observation is 1000 Bk. Determine the activity and number of atoms of this element after a week. Half life T½ 32P = 14.3 days.

Solution:

a) Let’s find the activity of phosphorus-32 after 7 days:

https://pandia.ru/text/80/150/images/image019_16.gif" width="57" height="41 src=">

Answer: after a week, the activity of the drug 32P will be 712 Bk, and the number of atoms of the radioactive isotope 32P is 127.14·106 atoms.

Control questions

1) What is the activity of a radionuclide?

2) Name the units of radioactivity and the relationship between them.

3) What is the radioactive decay constant?

4) Define the basic law of radioactive decay.

5) What is half-life?

6) What is the relationship between activity and mass of a radionuclide? Write the formula.

Tasks

1. Calculate activity 1 G 226Ra. T½ = 1602 years.

2. Calculate activity 1 G 60Co. T½ = 5.3 years.

3. One M-47 tank shell contains 4.3 kg 238U. Т½ = 2.5·109 years. Determine the activity of the projectile.

4. Calculate the activity of 137Cs after 10 years, if at the initial moment of observation it is equal to 1000 Bk. T½ = 30 years.

5. Calculate the activity of 90Sr a year ago if it is currently equal to 500 Bk. T½ = 29 years.

6. What kind of activity will 1 create? kg radioisotope 131I, T½ = 8.1 days?

7. Using reference data, determine activity 1 G 238U. Т½ = 2.5·109 years.

Using reference data, determine activity 1 G 232Th, Т½ = 1.4·1010 years.

8. Calculate the activity of the compound: 239Pu316O8.

9. Calculate the mass of a radionuclide with an activity of 1 Ki:

9.1. 131I, T1/2=8.1 days;

9.2. 90Sr, T1/2=29 years;

9.3. 137Cs, Т1/2=30 years;

9.4. 239Pu, Т1/2=2.4·104 years.

10. Determine mass 1 mCi radioactive carbon isotope 14C, T½ = 5560 years.

11. It is necessary to prepare a radioactive preparation of phosphorus 32P. After what period of time will 3% of the drug remain? Т½ = 14.29 days.

12. The natural potassium mixture contains 0.012% of the 40K radioactive isotope.

1) Determine the mass of natural potassium, which contains 1 Ki 40K. Т½ = 1.39·109 years = 4.4·1018 sec.

2) Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.

13. How many half-lives are required for the initial activity of a radioisotope to decrease to 0.001%?

14. To determine the effect of 238U on plants, seeds were soaked in 100 ml solution UO2(NO3)2 6H2O, in which the mass of radioactive salt was 6 G. Determine the activity and specific activity of 238U in solution. Т½ = 4.5·109 years.

15. Identify Activity 1 grams 232Th, Т½ = 1.4·1010 years.

16. Determine mass 1 Ki 137Cs, Т1/2=30 years.

17. The ratio between the content of stable and radioactive isotopes of potassium in nature is a constant value. The 40K content is 0.01%. Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.

18. Lithogenic radioactivity of the environment is formed mainly during count of three main natural radionuclides: 40K, 238U, 232Th. The proportion of radioactive isotopes in the natural sum of isotopes is 0.01, 99.3, ~100, respectively. Calculate radioactivity 1 T soil, if it is known that the relative content of potassium in the soil sample is 13600 g/t, uranium – 1·10-4 g/t, thorium – 6·10-4 g/t.

19. 23,200 were found in the shells of bivalve mollusks Bq/kg 90Sr. Determine the activity of samples after 10, 30, 50, 100 years.

20. The main pollution of closed reservoirs in the Chernobyl zone took place in the first year after the accident at the nuclear power plant. In the bottom sediments of the lake. Azbuchin in 1999 discovered 137Cs with a specific activity of 1.1·10 Bq/m2. Determine the concentration (activity) of fallen 137Cs per m2 of bottom sediments as of 1986-1987. (12 years ago).

21. 241Am (T½ = 4.32·102 years) is formed from 241Pu (T½ = 14.4 years) and is an active geochemical migrant. Taking advantage reference materials, calculate with an accuracy of 1% the decrease in the activity of plutonium-241 in time, in which year after the Chernobyl disaster the formation of 241Am in environment will be maximum.

22. Calculate the activity of 241Am in the emissions of the Chernobyl reactor as of April
2015, provided that in April 1986 the activity of 241Am was 3.82 1012 Bk,Т½ = 4.32·102 years.

23. 390 were found in soil samples nCi/kg 137Cs. Calculate the activity of samples after 10, 30, 50, 100 years.

24. Average concentration of lake bed pollution. Glubokoye, located in the Chernobyl exclusion zone, is 6.3 104 Bk 241Am and 7.4·104 238+239+240Pu per 1 m2. Calculate in what year these data were obtained.

Under radioactive decay, or simply disintegration, understand the natural radioactive transformation of nuclei, which occurs spontaneously. An atomic nucleus undergoing radioactive decay is called maternal, the emerging core - subsidiaries.

The theory of radioactive decay is based on the assumption that radioactive decay is a spontaneous process that obeys the laws of statistics. Since individual radioactive nuclei decay independently of each other, we can assume that the number of nuclei d N, decayed on average during the time interval from t before t + dt, proportional to the time period dt and number N undecayed nuclei at the time t:

where is a constant value for a given radioactive substance, called radioactive decay constant; the minus sign indicates that total number radioactive nuclei decreases during the decay process.

By separating the variables and integrating, i.e.

(256.2)

where is the initial number of undecayed nuclei (at the time t = 0), N- number of undecayed nuclei at a time t. Formula (256.2) expresses law of radioactive decay, according to which the number of undecayed nuclei decreases exponentially with time.

The intensity of the radioactive decay process is characterized by two quantities: the half-life and the average lifetime of the radioactive nucleus. Half life- the time during which the initial number of radioactive nuclei is halved on average. Then, according to (256.2),

Half-lives for naturally radioactive elements range from ten millionths of a second to many billions of years.

Total life expectancy dN cores is equal to . Having integrated this expression over all possible t(i.e. from 0 to) and dividing by the initial number of cores, we get average life time radioactive nucleus:

(taken into account (256.2)). Thus, the average lifetime of a radioactive nucleus is the reciprocal of the radioactive decay constant.

Activity A nuclide (common name atomic nuclei that differ in the number of protons Z and neutrons N) in a radioactive source is the number of decays that occur with the nuclei of a sample in 1 s:

(256.3)

The SI unit of activity is becquerel(Bq): 1 Bq - activity of a nuclide, at which one decay event occurs in 1 s. Still in nuclear physics An off-system unit of activity of a nuclide in a radioactive source is also used - curie(Ci): 1 Ci = 3.7×10 10 Bq. Radioactive decay occurs in accordance with the so-called displacement rules, allowing us to establish which nucleus arises as a result of the decay of a given parent nucleus. Offset rules:

For -decay

(256.4)

For -decay

(256.5)

where is the mother nucleus, Y is the symbol of the daughter nucleus, is the helium nucleus (-particle), is the symbolic designation of the electron (its charge is –1 and its mass number is zero). The displacement rules are nothing more than a consequence of two laws that apply during radioactive decays - the conservation of electric charge and the conservation of mass number: the sum of the charges (mass numbers) of the resulting nuclei and particles is equal to the charge (mass number) of the original nucleus.

Nuclei resulting from radioactive decay can, in turn, be radioactive. This leads to the emergence chains, or series, radioactive transformations ending with a stable element. The set of elements that form such a chain is called radioactive family.

From the displacement rules (256.4) and (256.5) it follows that the mass number during -decay decreases by 4, but does not change during -decay. Therefore, for all nuclei of the same radioactive family, the remainder when dividing the mass number by 4 is the same. Thus, there are four different radioactive families, for each of which the mass numbers are given by one of the following formulas:

A = 4n, 4n+1, 4n+2, 4n+3,

Where P is a positive integer. Families are named by the longest-lived (with the longest half-life) “ancestor”: the families of thorium (from), neptunium (from), uranium (from) and sea anemone (from). The final nuclides are respectively , , , , i.e. the only family of neptunium (artificially radioactive nuclei) ends with a nuclide Bi, and all the rest (naturally radioactive nuclei) are nuclides Pb.

§ 257. Laws of decay

Currently, more than two hundred active nuclei are known, mainly heavy ( A > 200, Z> 82). Only a small group of -active nuclei occur in areas with A= 140 ¸ 160 ( rare earths). -Decomposition obeys the displacement rule (256.4). An example of -decay is the decay of an isotope of uranium with the formation Th:

The velocities of particles emitted during decay are very high and range for different nuclei from 1.4 × 10 7 to 2 × 10 7 m/s, which corresponds to energies from 4 to 8.8 MeV. According to modern concepts, -particles are formed at the moment of radioactive decay when two protons and two neutrons moving inside the nucleus meet.

Particles emitted by a specific nucleus usually have a certain energy. More subtle measurements, however, have shown that the energy spectrum of -particles emitted by a given radioactive element exhibits a “fine structure”, i.e. several groups of -particles are emitted, and within each group their energies are practically constant. The discrete spectrum of -particles indicates that atomic nuclei have discrete energy levels.

-decay is characterized by a strong relationship between half-life and energy E flying particles. This relationship is determined empirically Geiger-Nattall law(1912) (D. Nattall (1890-1958) - English physicist, H. Geiger (1882-1945) - German physicist), which is usually expressed as a connection between mileage(the distance traveled by a particle in a substance before it comes to a complete stop) - particles in the air and the radioactive decay constant:

(257.1)

Where A And IN- empirical constants, . According to (257.1), the shorter the half-life of a radioactive element, the greater the range, and therefore the energy of the particles emitted by it. The range of particles in the air (at normal conditions) is several centimeters, more dense environments it is much smaller, amounting to hundredths of a millimeter (-particles can be detained with an ordinary sheet of paper).

Rutherford's experiments on the scattering of -particles on uranium nuclei showed that -particles up to an energy of 8.8 MeV experience Rutherford scattering on nuclei, i.e., the forces acting on -particles from the nuclei are described by Coulomb's law. This type of scattering of -particles indicates that they have not yet entered the region of action of nuclear forces, i.e., we can conclude that the nucleus is surrounded by a potential barrier, the height of which is not less than 8.8 MeV. On the other hand, -particles emitted by uranium have an energy of 4.2 MeV. Consequently, -particles fly out from the -radioactive nucleus with an energy noticeably lower than the height of the potential barrier. Classical mechanics could not explain this result.

Explanation -decay given quantum mechanics, according to which the emission of a particle from the nucleus is possible due to the tunnel effect (see §221) - the penetration of a particle through a potential barrier. There is always a non-zero probability that a particle with an energy less than the height of the potential barrier will pass through it, i.e., indeed, particles can fly out of a radioactive nucleus with an energy less than the height of the potential barrier. This effect is entirely due to the wave nature of -particles.

The probability of a particle passing through a potential barrier is determined by its shape and is calculated based on the Schrödinger equation. In the simplest case of a potential barrier with rectangular vertical walls (see Fig. 298, A) the transparency coefficient, which determines the probability of passing through it, is determined by the previously discussed formula (221.7):

Analyzing this expression, we see that the transparency coefficient D the longer (therefore, the shorter the half-life) the smaller in height ( U) and width ( l) the barrier is in the path of the -particle. In addition, with the same potential curve, the greater the energy of the particle, the smaller the barrier to its path. E. Thus, the Geiger-Nattall law is qualitatively confirmed (see (257.1)).

§ 258. -Disintegration. Neutrino

The phenomenon of -decay (in the future it will be shown that there is and (-decay) obeys the displacement rule (256.5)

and is associated with the release of an electron. We had to overcome a number of difficulties with the interpretation of decay.

First, it was necessary to substantiate the origin of the electrons emitted during the decay process. The proton-neutron structure of the nucleus excludes the possibility of an electron escaping from the nucleus, since there are no electrons in the nucleus. The assumption that electrons fly out not from the nucleus, but from the electron shell, is untenable, since then optical or X-ray radiation should be observed, which is not confirmed by experiments.

Secondly, it was necessary to explain the continuity of the energy spectrum of emitted electrons (the energy distribution curve of -particles typical for all isotopes is shown in Fig. 343).

How can active nuclei, which have well-defined energies before and after decay, eject electrons with energy values ​​from zero to a certain maximum? That is, the energy spectrum of emitted electrons is continuous? The hypothesis that during -decay electrons leave the nucleus with strictly defined energies, but as a result of some secondary interactions they lose one or another share of their energy, so that their original discrete spectrum turns into a continuous one, was refuted by direct calorimetric experiments. Since the maximum energy is determined by the difference in the masses of the mother and daughter nuclei, then decays in which the electron energy< , как бы протекают с нарушением закона сохранения энергии. Н. Бор даже пытался обосновать это нарушение, высказывая предположение, что закон сохранения энергии носит статистический характер и выполняется лишь в среднем для большого числа элементарных процессов. Отсюда видно, насколько принципиально важно было разрешить это затруднение.

Thirdly, it was necessary to deal with spin non-conservation during -decay. During -decay, the number of nucleons in the nucleus does not change (since the mass number does not change A), therefore the spin of the nucleus, which is equal to an integer for even A and half-integer for odd A. However, the release of an electron with spin /2 should change the spin of the nucleus by /2.

The last two difficulties led W. Pauli to the hypothesis (1931) that during -decay, another neutral particle is emitted along with the electron - neutrino. The neutrino has zero charge, spin /2 and zero (or rather< 10 -4 ) массу покоя; обозначается . Впоследствии оказалось, что при - decay, it is not neutrinos that are emitted, but antineutrino(antiparticle in relation to neutrinos; denoted by ).

The hypothesis of the existence of neutrinos allowed E. Fermi to create the theory of -decay (1934), which has largely retained its significance to this day, although the existence of neutrinos was experimentally proven more than 20 years later (1956). Such a long “search” for neutrinos is associated with great difficulties due to the lack of electric charge and mass in neutrinos. Neutrino is the only particle that does not participate in either strong or electromagnetic interactions; The only type of interaction in which neutrinos can take part is the weak interaction. Therefore, direct observation of neutrinos is very difficult. The ionizing ability of neutrinos is so low that one ionization event in the air occurs per 500 km of travel. The penetrating ability of neutrinos is so enormous (the range of neutrinos with an energy of 1 MeV in lead is about 1018 m!), which makes it difficult to contain these particles in devices.

For the experimental detection of neutrinos (antineutrinos), an indirect method was therefore used, based on the fact that in reactions (including those involving neutrinos) the law of conservation of momentum is satisfied. Thus, neutrinos were discovered by studying the recoil of atomic nuclei during -decay. If during the decay of a nucleus an antineutrino is ejected along with an electron, then the vector sum of three impulses - the recoil nucleus, the electron and the antineutrino - should be equal to zero. This has indeed been confirmed by experience. Direct detection of neutrinos became possible only much later, after the advent of powerful reactors that made it possible to obtain intense neutrino fluxes.

The introduction of neutrinos (antineutrinos) made it possible not only to explain the apparent non-conservation of spin, but also to understand the issue of continuity of the energy spectrum of ejected electrons. The continuous spectrum of -particles is due to the distribution of energy between electrons and antineutrinos, and the sum of the energies of both particles is equal to . In some decay events, the antineutrino receives more energy, in others - the electron; at the boundary point of the curve in Fig. 343, where the electron energy is equal to , all the decay energy is carried away by the electron, and the antineutrino energy is zero.

Finally, let us consider the question of the origin of electrons during -decay. Since the electron does not fly out of the nucleus and does not escape from the shell of the atom, it was assumed that the electron is born as a result of processes occurring inside the nucleus. Since during -decay the number of nucleons in the nucleus does not change, a Z increases by one (see (256.5)), then the only possibility of simultaneous implementation of these conditions is the transformation of one of the neutrons - the active nucleus - into a proton with the simultaneous formation of an electron and the emission of an antineutrino:

(258.1)

This process is accompanied by the fulfillment of conservation laws electric charges, momentum and mass numbers. In addition, this transformation is energetically possible, since the rest mass of a neutron exceeds the mass of a hydrogen atom, i.e., a proton and an electron combined. This difference in mass corresponds to an energy equal to 0.782 MeV. Due to this energy, spontaneous transformation of a neutron into a proton can occur; energy is distributed between the electron and the antineutrino.

If the transformation of a neutron into a proton is energetically favorable and generally possible, then radioactive decay of free neutrons (i.e., neutrons outside the nucleus) should be observed. The discovery of this phenomenon would be a confirmation of the stated theory of decay. Indeed, in 1950, in high-intensity neutron fluxes arising in nuclear reactors, the radioactive decay of free neutrons was discovered, occurring according to scheme (258.1). The energy spectrum of the resulting electrons corresponded to that shown in Fig. 343, and the upper limit of the electron energy turned out to be equal to that calculated above (0.782 MeV).

Prerequisite radioactive decay is that the mass of the original nucleus must exceed the sum of the masses of the decay products. Therefore, each radioactive decay occurs with the release of energy.

Radioactivity divided into natural and artificial. The first relates to radioactive nuclei existing in natural conditions, the second - to nuclei obtained through nuclear reactions in laboratory conditions. Fundamentally they are not different from each other.

The main types of radioactivity include α-, β- and γ-decays. Before characterizing them in more detail, let us consider the law of the occurrence of these processes over time, common to all types of radioactivity.

Identical nuclei undergo decay over different times, which cannot be predicted in advance. Therefore, we can assume that the number of nuclei decaying in a short period of time dt, proportional to the number N available cores at this moment, and dt:

Relationship (3.5) is called the basic law of radioactive decay. As you can see, the number N the number of nuclei that have not yet decayed decreases exponentially with time.

The intensity of radioactive decay is characterized by the number of nuclei decaying per unit time. From (3.4) it is clear that this quantity | dN / dt | = λN. It's called activity A. Thus the activity:

.

It is measured in becquerels (Bq), 1 Bk = 1 decay/s; and also in curies (Ci), 1 Ci = 3.7∙10 10 Bq.

The activity per unit mass of a radioactive drug is called specific activity.

Let's return to formula (3.5). Along with constant λ and activity A the process of radioactive decay is characterized by two more quantities: half-life T 1/2 and average life time τ kernels.

Half life T 1/2- time during which the initial number of radioactive nuclei will decrease by half on average:

,

Average life time τ Let's define it as follows. Number of cores δN(t), which experienced decay over a period of time ( t, t + dt), is determined by the right side of expression (3.4): δN(t) = λNdt. The lifetime of each of these nuclei is t. This means the sum of the lifetimes of everyone N 0 of the initially available nuclei is determined by integrating the expression tδN(t) in time from 0 to ∞. Dividing the sum of the lifetimes of all N 0 cores per N 0, we will find the average lifetime τ of the kernel in question:

notice, that τ is equal, as follows from (3.5), to the period of time during which the initial number of nuclei decreases by e once.

Comparing (3.8) and (3.9.2), we see that the half-life T 1/2 and average life time τ have the same order and are related to each other by the relation:

.

Complex radioactive decay can occur in two cases:

Physical meaning of these equations is that the number of nuclei 1 decreases due to their decay, and the number of nuclei 2 is replenished due to the decay of nuclei 1 and decreases due to its own decay. For example, at the initial moment of time t= 0 available N 01 cores 1 and N 02 2 cores. With such initial conditions, the solution of the system has the form:

If at the same time N 02= 0, then

.

To estimate the value N 2(t) you can use the graphical method (see Figure 3.2) to construct curves e−λt and (1 − e−λt). Moreover, due to the special properties of the function e−λt it is very convenient to construct curve ordinates for values t, corresponding T, 2T, … etc. (see table 3.1). Relationship (3.13.3) and Figure 3.2 show that the amount of radioactive daughter substance increases with time and with t >> T 2 (λ 2 t>> 1) approaches its limit value:

and is called centuries-old, or secular balance. The physical meaning of the age-old equation is obvious.

Figure 3.3. Complex radioactive decay.

Since, according to equation (3.4), λN is equal to the number of decays per unit time, then the relation λ 1 N 1 = λ 2 N 2 means that the number of decays of the daughter substance λ 2 N 2 equal to the number of decays of the parent substance, i.e. the number of nuclei of the daughter substance formed in this case λ 1 N 1. The secular equation is widely used to determine the half-lives of long-lived radioactive substances. This equation can be used when comparing two mutually converting substances, of which the second has a much shorter half-life than the first ( T 2 << T 1) provided that this comparison is made at the time t >> T 2 (T 2 << t << T 1). An example of the sequential decay of two radioactive substances is the transformation of radium Ra into radon Rn. 88 Ra 226 is known to emit with a half-life T 1 >> 1600 yearsα particles, turns into the radioactive gas radon (88 Rn 222), which is itself radioactive and emits α particles with a half-life T 2 ≈ 3.8 day. In this example, just T 1 >> T 2, so for times t << T 1 the solution to equations (3.12) can be written in the form (3.13.3).

For further simplification, it is necessary that the initial number of nuclei Rn be equal to zero ( N 02= 0 at t= 0). This is achieved by specially setting up an experiment in which the process of converting Ra into Rn is studied. In this experiment, the Ra drug is placed in a glass flask with a tube connected to a pump. During operation of the pump, the released gaseous Rn is immediately pumped out, and its concentration in the cone is zero. If at some moment, while the pump is running, the cone is isolated from the pump, then from this moment, which can be taken as t= 0, the number of nuclei Rn in the cone will begin to increase according to the law (3.13.3):N Ra and N Rn- precise weighing, and λ Rn- by determining the half-life Rn, which has a value convenient for measurements of 3.8 day. So the fourth quantity λ Ra can be calculated. This calculation gives for the half-life of radium T Ra ≈ 1600 years, which coincides with the results of the definition T Ra method of absolute counting of emitted α-particles.

The radioactivity of Ra and Rn was chosen as a standard when comparing the activities of various radioactive substances. Per unit of radioactivity - 1 Ki- accepted activity of 1 g of radium or the amount of radon in equilibrium with it. The latter can be easily found from the following reasoning.

It is known that 1 G radium undergoes ~3.7∙10 10 per second decays. Hence.