The basic law of radioactive decay has the form. Law of radioactive decay
Under radioactive decay, or simply decay, understand the natural radioactive transformation of nuclei, which occurs spontaneously. An atomic nucleus undergoing radioactive decay is called maternal, the emerging core - child.
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 over the time interval from t before t + dt, proportional to the time interval dt and number N undecayed nuclei by the time t:
where is a constant value for a given radioactive substance, called radioactive decay constant; minus sign indicates that total number radioactive nuclei in the process of decay decreases.
Separating the variables and integrating, i.e.
where is the initial number of undecayed nuclei (at the time t = 0), N- the number of undecayed nuclei at a time t. Formula (256.2) expresses radioactive decay law, according to which the number of undecayed nuclei decreases exponentially with time.
The intensity of the process of radioactive decay 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 on average is halved. Then, according to (256.2),
The 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 . By integrating 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:
(taking into account (256.2)). Thus, the average lifetime of a radioactive nucleus is the reciprocal of the radioactive decay constant.
Activity BUT nuclide(general name for 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 the sample in 1 s:
SI unit of activity - becquerel(Bq): 1 Bq is the activity of the nuclide, at which one act of decay occurs in 1 s. Until now, in nuclear physics, an off-system unit of nuclide activity in a radioactive source is also used - curie(Ki): 1 Ki = 3.7 × 10 10 Bq. Radioactive decay occurs according to the so-called displacement rules, which make it possible to establish which nucleus arises as a result of the decay of a given parent nucleus. Offset Rules:
for -decay
for -decay
where is the parent 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 the mass number is zero). Displacement rules are nothing more than a consequence of two laws that are fulfilled during radioactive decays - conservation of electric charge and conservation of mass number: the sum of charges (mass numbers) of emerging nuclei and particles is equal to the charge (mass number) of the original nucleus.
The nuclei resulting from radioactive decay can be, in turn, radioactive. This gives rise to chains, or series, radioactive transformations ending with a stable element. The set of elements that form such a chain is called radioactive family.
It follows from the displacement rules (256.4) and (256.5) that the mass number decreases by 4 during -decay, and does not change during -decay. Therefore, for all nuclei of the same radioactive family, the remainder after 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:
BUT = 4n, 4n+1, 4n+2, 4n+3,
where P- whole positive number. The families are named according to the longest-lived (with the longest half-life) "ancestor": families of thorium (from), neptunium (from), uranium (from) and actinium (from). The final nuclides are, respectively, , , , , i.e. the only family of neptunium (artificially radioactive nuclei) ends with Bi, and all the rest (naturally radioactive nuclei) - nuclides Рb.
§ 257. Regularities of -decay
Currently, more than two hundred -active nuclei are known, mainly heavy ( A > 200, Z> 82). Only a small group of -active nuclei falls on regions with BUT= 140 ¸ 160 (rare earths). - Decay 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 the -particles emitted during the decay are very high and fluctuate for different nuclei in the range from 1.4×10 7 to 2×10 7 m/s, which corresponds to energies from 4 to 8.8 MeV. According to modern ideas, -particles are formed at the moment of radioactive decay when two protons and two neutrons moving inside the nucleus meet.
Particles emitted by a particular nucleus, as a rule, have a certain energy. More subtle measurements, however, showed that the energy spectrum of the -particles emitted by a given radioactive element exhibits a "fine structure", that is, 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 emitted particles. This relationship is determined by empirical Geiger-Nattall law(1912) (D. Nettol (1890-1958) - English physicist, H. Geiger (1882-1945) - German physicist), which is usually expressed as a relationship between mileage(the distance traveled by a particle in a substance until it stops completely) - particles in the air and the constant of radioactive decay:
where BUT and AT are empirical constants, . According to (257.1), the shorter the half-life of a radioactive element, the greater the range, and, consequently, the energy of the -particles emitted by it. The range of -particles in air (under normal conditions) is a few centimeters, in denser media it is much less, amounting to hundredths of a millimeter (-particles can be detained by 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, that is, the forces acting on the -particles from the nuclei are described by Coulomb's law. The similar nature of the 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 whose height is not less than 8.8 MeV. On the other hand, the -particles emitted by uranium have an energy of 4.2 MeV. Consequently, -particles fly out of the -radioactive nucleus with an energy much less than the height of the potential barrier. classical mechanics this result could not be explained.
The explanation of -decay is given by quantum mechanics, according to which the escape of -particles from the nucleus is possible due to the tunneling effect (see §221) - the penetration of -particles 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 escape from 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 on the basis of 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 considered 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, for the same potential curve, the barrier to the path of a particle is the smaller, the greater its energy E. Thus, the Geiger-Nattall law is qualitatively confirmed (see (257.1)).
Section 258 - Decay. Neutrino
The phenomenon of -decay (later it will be shown that there is and (-decay) obeys the displacement rule (256.5)
and is associated with the ejection of an electron. I had to overcome a number of difficulties with the interpretation of -decay.
First, it was necessary to substantiate the origin of the electrons emitted in the process of -decay. 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 do not fly out of 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 the emitted electrons (the energy distribution curve of -particles typical for all isotopes is shown in Fig. 343).
How, then, can active nuclei, which have quite definite energies before and after decay, eject electrons with energies ranging from zero to a certain maximum? That is, is the energy spectrum of the emitted electrons continuous? The hypothesis that electrons leave the nucleus during -decay with strictly defined energies, but as a result of some secondary interactions lose one or another fraction of their energy, so that their initial discrete spectrum turns into a continuous one, was refuted by direct calorimetric experiments. Since the maximum energy is determined by the difference between the masses of the parent and daughter nuclei, the 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 non-conservation of spin during -decay. During -decay, the number of nucleons in the nucleus does not change (since the mass number A), therefore, the spin of the nucleus, which is equal to an integer for even BUT and half-integer for odd BUT. However, the ejection of an electron with spin /2 must change the spin of the nucleus by /2.
The last two difficulties led V. Pauli to the hypothesis (1931) that during -decay, together with an electron, one more neutral particle is emitted - neutrino. The neutrino has zero charge, spin /2 and zero (or rather< 10 -4 ) массу покоя; обозначается . Впоследствии оказалось, что при - decay emits not a neutrino, but antineutrino(antiparticle with respect to neutrino; denoted by ).
The hypothesis of the existence of the neutrino allowed E. Fermi to create the theory of -decay (1934), which has largely retained its significance at the present time, although the existence of the neutrino was experimentally proved more than 20 years later (1956). Such a long "search" for neutrinos is associated with great difficulties, due to the absence of an electric charge and mass in neutrinos. The 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 small that one act of ionization in the air falls on 500 km of the way. The penetrating power of neutrinos is so enormous (the range of a neutrino with an energy of 1 MeV in lead is about 1018 m!), which makes it difficult to keep these particles in devices.
For the experimental detection of neutrinos (antineutrinos), therefore, an indirect method was used, based on the fact that in reactions (including those involving neutrinos) the momentum conservation law is fulfilled. Thus, the neutrino was discovered in the study of the recoil of atomic nuclei during -decay. If, during the decay of the nucleus, an antineutrino is also ejected along with the electron, then the vector sum of the three impulses - the recoil nucleus, the electron and the antineutrino - must be equal to zero. This has indeed been confirmed by experience. The direct detection of neutrinos became possible only much later, after the appearance 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 deal with the question of the 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 acts of decay, the antineutrino receives more energy, in others, the electron; at the boundary point of the curve in Fig. 343, where the electron energy is , 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 in -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 the number of nucleons in the nucleus does not change during -decay, a Z increases by one (see (256.5)), then the only possibility for the 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:
This process is accompanied by the fulfillment of the conservation laws electric charges, momentum and mass numbers. In addition, this transformation is energetically possible, since the rest mass of the neutron exceeds the mass of the hydrogen atom, i.e., the proton and electron combined. This difference in masses corresponds to an energy equal to 0.782 MeV. Due to this energy, a spontaneous transformation of a neutron into a proton can occur; the 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 (ie, neutrons outside the nucleus) should be observed. The discovery of this phenomenon would be a confirmation of the expounded theory of -decay. Indeed, in 1950, in high-intensity neutron fluxes arising in nuclear reactors, radioactive decay of free neutrons was discovered, which occurs according to scheme (258.1). The energy spectrum of the electrons arising in this case 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).
Change in the number of radioactive nuclei over time. Rutherford and Soddy in 1911, summarizing the experimental results, showed that the atoms of some elements undergo successive transformations, forming radioactive families, where each member arises from the previous one and, in turn, forms the next one.
This can be conveniently illustrated by the example of the formation of radon from radium. If placed in a sealed ampoule, then an analysis of the gas after a few days will show that helium and radon appear in it. Helium is stable, and therefore it accumulates, while radon itself decays. Curve 1 in fig. 29 characterizes the decay law of radon in the absence of radium. At the same time, the ratio of the number of undecayed radon nuclei to their initial number is plotted on the y-axis. It can be seen that the content decreases exponentially. Curve 2 shows how the number of radioactive radon nuclei changes in the presence of radium.
Experiments carried out with radioactive substances showed that no external conditions (heating to high temperatures,
magnetic and electric fields, high pressures) cannot affect the nature and rate of decay.
Radioactivity is a property atomic nucleus and for of this type nuclei in a certain energy state, the probability of radioactive decay per unit time is constant.
Rice. 29. Dependence of the number of active radon nuclei on time
Since the decay process is spontaneous (spontaneous), the change in the number of nuclei due to decay over a period of time is determined only by the number of radioactive nuclei at the moment and is proportional to the time interval
where is a constant characterizing the decay rate. Integrating (37) and assuming that we get
i.e., the number of nuclei decreases exponentially.
This law applies to statistical averages and is valid only for sufficiently large numbers particles. The value of X is called the radioactive decay constant, has a dimension and characterizes the probability of the decay of one atom in one second.
To characterize radioactive elements, the concept of half-life is also introduced. It is understood as the time during which half of the available number of atoms decays. Substituting the condition into equation (38), we obtain
whence, taking logarithms, we find that
and half life
With the exponential law of radioactive decay, at any time there is a non-zero probability of finding nuclei that have not yet decayed. The lifetime of these nuclei exceeds
On the contrary, other nuclei that have decayed by this time have lived for different times, the shorter average lifetime for a given radioactive isotope is defined as
Denoting we get
Consequently, the average lifetime of a radioactive nucleus is equal to the reciprocal of the decay constant R. Over time, the initial number of nuclei decreases by a factor.
For processing experimental results it is convenient to represent equation (38) in another form:
The value is called the activity of a given radioactive preparation, it determines the number of disintegrations per second. Activity is a characteristic of the entire decaying matter, and not separate core. The practical unit of activity is the curie. 1 curie is equal to the number of decayed nuclei contained in radium in 1 sec of decays/sec). Smaller units, millicuries and microcuries, are also used. In the practice of a physical experiment, sometimes another unit of activity is used - Rutherford disintegrations/sec.
Statistical nature of radioactive decay. Radioactive decay is a fundamentally statistical phenomenon. We cannot say exactly when a given nucleus will decay, but we can only indicate with what probability it decays over a given period of time.
Radioactive nuclei do not "age" in the course of their existence. The concept of age is generally inapplicable to them, but one can only talk about the average time of their life.
It follows from the statistical nature of the law of radioactive decay that it is strictly observed when it is large, and when it is small, fluctuations should be observed. The number of decaying nuclei per unit time must fluctuate around the average value, which is characterized by the above law. This is confirmed by experimental measurements of the number of -particles emitted by a radioactive substance per unit time.
Rice. 30. Dependence of the logarithm of activity on time
Fluctuations obey Poisson's law. When making measurements with radioactive preparations, one must always take this into account and determine the statistical accuracy of the experimental results.
Determination of the decay constant X. When determining the decay constant X of a radioactive element, the experiment is reduced to registering the number of particles emitted from the drug per unit time, i.e., its activity is determined. Then a graph of the change in activity over time is plotted, usually on a semi-logarithmic scale. The form of dependences obtained in studies of a pure isotope, a mixture of isotopes, or a radioactive family turns out to be different.
Let's take a few cases as an example.
1. We study one radioactive element, the decay of which produces stable nuclei. Taking the logarithm of expression (41), we obtain
Therefore, in this case, the logarithm of activity is a linear function of time. The graph of this dependence has the form of a straight line, the slope of which (Fig. 30)
2. A radioactive family is being investigated, in which a whole chain of radioactive transformations occurs. The nuclei resulting from the decay, in turn, themselves turn out to be radioactive:
An example of such a chain is the decay:
Let us find a law describing in this case the change in the number of radioactive atoms in time. For simplicity, we single out only two elements: considering A as the initial one, and B as an intermediate one.
Then the change in the number of nuclei A and nuclei B will be determined from the system of equations
The number of nuclei A decreases due to their decay, and the number of nuclei B decreases due to the decay of nuclei B and increases due to the decay of nuclei A.
If at there are nuclei A, but there are no nuclei B, then the initial conditions will be written in the form
The solution of equations (43) has the form
and the total activity of the source, consisting of nuclei A and B:
Let us now consider the dependence of the logarithm of radioactivity on time for different ratios between and
1. The first element is short-lived, the second is long-lived, i.e. . In this case, the curve showing the change in the total activity of the source has the form shown in Fig. 31, a. At the beginning, the course of the curve is determined mainly by a rapid decrease in the number of active nuclei, the B nuclei also decay, but slowly, and therefore their decay does not greatly affect the slope of the curve in the section . In the future, there are few nuclei of type A in the mixture of isotopes, and the slope of the curve is determined by the decay constant If you need to find and then from the slope of the curve at great importance time is found (in expression (45) the first exponential term in this case can be discarded). To determine the value, it is also necessary to take into account the influence of the decay of a long-lived element on the slope of the first part of the curve. To do this, the straight line is extrapolated to the region of short times, at several points the activity determined by element B is subtracted from the total activity according to the obtained values
they build a straight line for element A and find it by the angle (in this case, it is necessary to switch from logarithms to antilogarithms and vice versa).
Rice. 31. Dependence of the logarithm of the activity of a mixture of two radioactive substances from time: a - at at
2. The first element is long-lived, and the second is short-lived: The dependence in this case has the form shown in fig. 31b. At the beginning, the activity of the drug increases due to the accumulation of B nuclei. Then a radioactive equilibrium occurs, at which the ratio of the number of A nuclei to the number of B nuclei becomes constant. This type of equilibrium is called transitional. After some time, both substances begin to decrease at the rate of decay of the parent element.
3. The half-life of the first isotope is much longer than the second (it should be noted that the half-life of some isotopes is measured in millions of years). In this case, after a while, the so-called secular equilibrium is established, in which the number of nuclei of each isotope is proportional to the half-life of this isotope. Ratio
The ability of nuclei to spontaneously decay by emitting particles is called radioactivity. Radioactive decay is a statistical process. Each radioactive nucleus can decay at any moment, and the pattern is observed only on average, in the case of the decay of a sufficiently large number of nuclei.
decay constant
λ is the probability of nuclear decay per unit time.
If there are N radioactive nuclei in the sample at time t, then the number of nuclei dN that decayed during time dt is proportional to N.
Integrating (1) we obtain the law of radioactive decay
N 0 is the number of radioactive nuclei at time t = 0.
Average life time
τ -
Activity A - the average number of nuclei decaying per unit time
Activity is measured in curies (Ci) and becquerels (Bq)
1 Ki \u003d 3.7 10 10 decays / s,
1 Bq = 1 decay/s.
The decay of the initial nucleus 1 into the nucleus 2, with its subsequent decay into the nucleus 3, is described by a system of differential equations
where N 1 (t) and N 2 (t) is the number of nuclei, and λ 1 and λ 2 are the decay constants of nuclei 1 and 2, respectively. Solution of system (6) with initial conditions N 1 (0) = N 10 ; N 2 (0) = 0 will be
, | (7a) |
. | (7b) |
The number of cores 2 reaches its maximum value at .
If λ 2< λ 1
(>), the total activity N 1 (t)λ 1 + N 2 (t)λ 2 will monotonically decrease.
If λ 2 >λ 1 (<), суммарная
активность вначале растет за счет накопления ядер 2.
If λ 2 >>λ 1, at sufficiently long times the contribution of the second exponent in (7b) becomes negligibly small, compared with the contribution of the first and the activity of the second A 2 = λ 2 N 2 and the first isotope A 1 = λ 1 N 1 are almost equal . In the future, the activities of both the first and second isotopes will change in time in the same way.
That is, the so-called secular balance , at which the number of isotope nuclei in the decay chain is related to the decay constants (half-lives) by a simple relation.
By solving system (10) for activities with initial conditions N 1 (0) = N 10 ; N i (0) = 0 will be
Necessary condition 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 refers to radioactive nuclei that exist in natural conditions, the second - to the nuclei obtained through nuclear reactions in the laboratory. Fundamentally, they do not differ 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 course of these processes in time common to all types of radioactivity.
Identical nuclei undergo decay at 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 nuclei at that moment, and dt:
Integration of equation (3.4) gives:Relation (3.5) is called the basic law of radioactive decay. As you can see, the number N of yet undecayed nuclei decreases exponentially with time.
The intensity of radioactive decay is characterized by the number of nuclei decaying per unit time. It can be seen from (3.4) that this quantity | dN / dt | = λN. It's called activity. A. Thus activity:
. |
It is measured in becquerels (Bq), 1 Bq = 1 decay / s; and also in curie (Ci), 1 Ci = 3.7∙10 10 Bq.
Activity per unit mass of a radioactive preparation is called specific activity.
Let us return to formula (3.5). Along with constant λ and activity A the process of radioactive decay is characterized by two more quantities: the half-life T 1/2 and average life time τ kernels.
Half life T 1/2- the time for which the initial number of radioactive nuclei on average will decrease by two:
, |
. |
Average life time τ we define as follows. Number of cores δN(t) that 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. So the sum of the lifetimes of all N0 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 N0 cores per N0, we will find the average lifetime τ the kernel in question:
notice, that τ equals, as follows from (3.5), the time interval during which the initial number of nuclei decreases in e once.
Comparing (3.8) and (3.9.2), we see that the half-life T 1/2 and mean lifetime τ have the same order and are related by the relation:
. |
Complex radioactive decay
Complex radioactive decay can occur in two cases:
The 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 their own decay. For example, at the initial time t= 0 available N01 cores 1 and N02 kernels 2. With such initial conditions, the solution of the system has the form:
If at the same time N02= 0, then
. |
To evaluate the value N 2(t) can be used graphic method(see figure 3.2) plotting curves e−λt and (1 − e−λt). In this case, due to the special properties of the function e−λt it is very convenient to plot the ordinates of the curve for the 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 increases with time and t >> T2 (λ 2 t>> 1) approaches its limit value:
and is called the age-old, or secular balance. The physical meaning of the secular equation is obvious.
t | e−λt | 1 − e − λt |
0 | 1 | 0 |
1T | 1/2 = 0.5 | 0.5 |
2T | (1/2) 2 = 0.25 | 0.75 |
3T | (1/2) 3 = 0.125 | 0.875 |
... | ... | ... |
10T | (1/2) 10 ≈ 0.001 | ~0.999 |
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 is 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 ( T2 << T1) provided that this comparison is made at time t >> T2 (T2 << t << T1). An example of the successive decay of two radioactive substances is the transformation of radium Ra into radon Rn. It is known that 88 Ra 226, emitting with a half-life T1 >> 1600 yearsα-particles, turns into radioactive gas radon (88 Rn 222), which is itself radioactive and emits α-particles with a half-life T2 ≈ 3.8 days. In this example just T1 >> T2, so for times t << T1 the solution of equations (3.12) can be written in the form (3.13.3). |
For further simplification, it is necessary that the initial number of cores Rn be equal to zero ( N02= 0 at t= 0). This is achieved by a special setting of the experiment, in which the process of transformation of Ra into Rn is studied. In this experiment, the Ra preparation is placed in a glass flask with a tube connected to a pump. During the operation of the pump, the released gaseous Rn is immediately pumped out, and its concentration in the cone is zero. If at some point while the pump is running, the cone is isolated from the pump, then from that 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- accurate weighing, and λRn- by determining the half-life Rn, which has a value of 3.8, convenient for measurements days. So the fourth value 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 determination T Ra by the method of absolute counting of emitted α-particles.
The radioactivity of Ra and Rn was chosen as a reference when comparing the activities of various radioactive substances. Per unit of radioactivity - 1 Key- accepted activity of 1 g of radium or an amount of radon that is 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. Consequently.
The law of radioactive decay is a physical law that describes the dependence of the intensity of radioactive decay on time and the number of radioactive atoms in the sample. Opened by Frederick Soddy and Ernest Rutherford, each of whom was later awarded the Nobel Prize. They discovered it experimentally and published in 1903 in the works "Comparative study of the radioactivity of radium and thorium" and "Radioactive transformation", stating as follows:
“In all cases when one of the radioactive products was separated and its activity was examined, regardless of the radioactivity of the substance from which it was formed, it was found that the activity in all studies decreases with time according to the law of geometric progression.”
With the help of Bernoulli's theorem, the following conclusion was obtained: the rate of transformation is always proportional to the number of systems that have not yet undergone transformation.
There are several formulations of the law, for example, in the form of a differential equation:
radioactive decay atom quantum mechanical
which means that the number of decays?dN that occurred in a short time interval dt is proportional to the number of atoms N in the sample.
Exponential Law
In the above mathematical expression, the decay constant, which characterizes the probability of radioactive decay per unit time and has the dimension c?1. The minus sign indicates a decrease in the number of radioactive nuclei over time.
The solution to this differential equation is:
where is the initial number of atoms, that is, the number of atoms for
Thus, the number of radioactive atoms decreases with time according to an exponential law. The decay rate, that is, the number of decays per unit time, also falls exponentially.
Differentiating the expression for the dependence of the number of atoms on time, we obtain:
where is the decay rate at the initial moment of time
Thus, the time dependence of the number of undecayed radioactive atoms and the decay rate is described by the same constant
Decay characteristics
In addition to the decay constant, radioactive decay is characterized by two more constants derived from it:
1. Average lifetime
The lifetime of a quantum mechanical system (particle, nucleus, atom, energy level, etc.) is the period of time during which the system decays with a probability where e = 2.71828… is the Euler number. If an ensemble of independent particles is considered, then over time the number of remaining particles decreases (on average) by a factor of e of the number of particles at the initial moment. The concept of "lifetime" is applicable in conditions where exponential decay occurs (that is, the expected number of surviving particles N depends on time t as
where N 0 is the number of particles at the initial moment). For example, this term cannot be applied to neutrino oscillations.
The lifetime is related to the half-life T 1/2 (the time during which the number of surviving particles is on average halved) by the following relation:
The reciprocal of the lifetime is called the decay constant:
Exponential decay is observed not only for quantum mechanical systems, but also in all cases when the probability of an irreversible transition of an element of the system to another state per unit time does not depend on time. Therefore, the term "lifetime" is used in areas quite far from physics, for example, in the theory of reliability, pharmacology, chemistry, etc. Processes of this kind are described by a linear differential equation
meaning that the number of elements in the initial state decreases at a rate proportional to N(t)/. The coefficient of proportionality is So, in pharmacokinetics, after a single injection of a chemical compound into the body, the compound is gradually destroyed in biochemical processes and excreted from the body, and if it does not cause significant changes in the rate of biochemical processes acting on it (i.e., the effect is linear), then the decrease its concentration in the body is described by an exponential law, and we can talk about the lifetime of a chemical compound in the body (as well as the half-life and decay constant).
2. Half-life
The half-life of a quantum mechanical system (particle, nucleus, atom, energy level, etc.) is the time T S during which the system decays with a probability of 1/2. If an ensemble of independent particles is considered, then during one half-life period the number of surviving particles will decrease on average by 2 times. The term applies only to exponentially decaying systems.
It should not be assumed that all particles taken at the initial moment will decay in two half-lives. Since each half-life halves the number of surviving particles, a quarter of the initial number of particles remains after 2T S, one eighth after 3T S, and so on. In general, the fraction of surviving particles (or, more precisely, the survival probability p for a given particle) depends on time t as follows:
The half-life, mean lifetime, and decay constant are related by the following relationships, derived from the law of radioactive decay:
Because, the half-life is about 30.7% shorter than the average lifetime.
In practice, the half-life is determined by measuring the activity of the study drug at regular intervals. Given that the activity of the drug is proportional to the number of atoms of the decaying substance, and using the law of radioactive decay, we can calculate the half-life of this substance
Partial half-life
If a system with a half-life T 1/2 can decay through several channels, for each of them it is possible to determine the partial half-life. Let the probability of decay along the i-th channel (branching factor) be equal to p i . Then the partial half-life for the i-th channel is equal to
Partial has the meaning of the half-life that the given system would have if all decay channels except the i-th one were “turned off”. Since by definition, for any decay channel.
half-life stability
In all observed cases (except for some isotopes that decay by electron capture), the half-life was constant (separate reports of a change in the period were caused by the insufficient accuracy of the experiment, in particular, incomplete purification from highly active isotopes). In this regard, the half-life is considered unchanged. On this basis, the determination of the absolute geological age of rocks, as well as the radiocarbon method for determining the age of biological remains, is built.
The assumption of the variability of the half-life is used by creationists, as well as representatives of the so-called. "alternative science" to refute the scientific dating of rocks, the remains of living beings and historical finds, in order to further refute the scientific theories built using such dating. (See, for example, articles Creationism, Scientific Creationism, Critique of Evolutionism, Shroud of Turin).
The variability of the decay constant for electron capture has been observed experimentally, but it lies within a percentage in the entire range of pressures and temperatures available in the laboratory. The half-life in this case changes due to some (rather weak) dependence of the density of the wave function of orbital electrons in the vicinity of the nucleus on pressure and temperature. Significant changes in the decay constant were also observed for highly ionized atoms (thus, in the limiting case of a fully ionized nucleus, electron capture can occur only when the nucleus interacts with free plasma electrons; in addition, decay, which is allowed for neutral atoms, in some cases for strongly ionized atoms can be prohibited kinematically). All these options for changing the decay constants, obviously, cannot be used to “refute” radiochronological dating, since the error of the radiochronometric method itself for most chronometer isotopes is more than a percent, and highly ionized atoms in natural objects on Earth cannot exist for any long time. .
The search for possible variations in the half-lives of radioactive isotopes, both at present and over billions of years, is interesting in connection with the hypothesis of variations in the values of fundamental constants in physics (fine structure constant, Fermi constant, etc.). However, careful measurements have not yet yielded results - no changes in half-lives have been found within the experimental error. Thus, it was shown that over 4.6 billion years, the b-decay constant of samarium-147 changed by no more than 0.75%, and for the b-decay of rhenium-187, the change during the same time does not exceed 0.5%; in both cases the results are consistent with no such changes at all.