Half-life
Half life quantum mechanical system (particle, nucleus, atom, energy level, etc.) - time T½ , during which the system decays with probability 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, in time 2 T½ will remain a quarter of the initial number of particles, for 3 T½ - one eighth, etc. In general, the fraction of surviving particles (or, more precisely, the probability of surviving p for a given particle) depends on time t in the following way:
The half-life, mean lifetime τ, and decay constant λ are related by the following relationships:
.
Since ln2 = 0.693… , the half-life is about 30% shorter than the lifetime.
Sometimes the half-life is also called the decay half-life.
Example
If we designate for a given moment of time the number of nuclei capable of radioactive transformation through N, and the time interval after t 2 - t 1 , where t 1 and t 2 - fairly close times ( t 1 < t 2), and the number of decaying atomic nuclei in this period of time through n, That n = KN(t 2 - t 1). Where is the coefficient of proportionality K = 0,693/T½ is called the decay constant. If we accept the difference ( t 2 - t 1) equal to one, that is, the observation time interval is equal to one, then K = n/N and, consequently, the decay constant shows the fraction of the available number of atomic nuclei that undergo decay per unit time. Consequently, the decay takes place in such a way that the same fraction of the available number of atomic nuclei decays per unit time, which determines the law of exponential decay.
The values of the half-lives for different isotopes are different; for some, especially rapidly decaying ones, the half-life can be equal to millionths of a second, and for some isotopes, like uranium 238 and thorium 232, it is respectively equal to 4.498 * 10 9 and 1.389 * 10 10 years. It is easy to count the number of uranium 238 atoms undergoing transformation in a given amount of uranium, for example, one kilogram in one second. The amount of any element in grams, numerically equal to the atomic weight, contains, as you know, 6.02 * 10 23 atoms. Therefore, according to the above formula n = KN(t 2 - t 1) find the number of uranium atoms decaying in one kilogram in one second, keeping in mind that there are 365 * 24 * 60 * 60 seconds in a year,
.Calculations lead to the fact that in one kilogram of uranium, twelve million atoms decay in one second. Despite such a huge number, the rate of transformation is still negligible. Indeed, the following part of uranium decays per second:
.
Thus, from the available amount of uranium, its fraction equal to
.
Turning again to the basic law of radioactive decay KN(t 2 - t 1), that is, to the fact that only one and the same fraction of the available number of atomic nuclei decays per unit time, and, bearing in mind the complete independence of atomic nuclei in any substance from each other, we can say that this law is statistical in the sense that it does not indicate exactly which atomic nuclei will undergo decay in a given period of time, but only tells about their number. Undoubtedly, this law remains valid only for the case when the available number of nuclei is very large. Some of the atomic nuclei will decay in the next moment, while other nuclei will undergo transformations much later, so when the available number of radioactive atomic nuclei is relatively small, the law of radioactive decay may not be fully satisfied.
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 partial half-life. Let the probability of decay by i-th channel (branching factor) is equal to pi. Then the partial half-life of i-th channel is equal to
Partial has the meaning of the half-life that a given system would have if all decay channels were "turned off" except for i th. Since by definition , then for any decay channel.
half-life stability
In all observed cases (except for some isotopes decaying by electron capture), the half-life was constant (separate reports of a change in the period were caused by insufficient experimental accuracy, 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 strongly 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 isotope-chronometers is more than a percent, and highly ionized atoms in natural objects on Earth cannot exist for any long time. .
The range of values for the half-life of radioactive substances is extremely wide, it extends from billions of years to small fractions of a second. Therefore, the methods for measuring the quantity T 1/2 should be very different from each other. Let's consider some of them.
1) Suppose, for example, it is required to determine the half-life of a long-lived substance. In this case, having chemically obtained a radioactive isotope free of foreign impurities or with a known amount of impurities, it is possible to weigh the sample and, using the Avogadro number, determine the number of atoms of the radioactive substance that are in it. By placing the sample in front of the detector of radioactive radiation and calculating the solid angle at which the detector is visible from the sample, we determine the fraction of radiation recorded by the detector. When measuring the radiation intensity, one should take into account its possible absorption on the path between the sample and the detector, as well as its absorption in the sample and the detection efficiency. Thus, the number of nuclei is determined in the experiment n decaying per unit time:
Where N is the number of radioactive nuclei present in the radioactive sample. Then 
And 
.
2) If the value is determined T 1/2 for substances that decay with a half-life of several minutes, hours or days, it is convenient to use the method of observing the change in the intensity of nuclear radiation with time. In this case, the registration of radiation is performed either using a gas-filled counter or a scintillation detector. The radioactive source is placed near the counter so that their mutual arrangement does not change during the entire experiment. In addition, it is necessary to create such conditions under which possible miscalculations of both the meter itself and the recording system would be excluded. Measurements are made as follows. The number of pulses is counted N0 for some period of time t(e.g. one minute). After a period of time t1 pulses are counted again N 1.After a period of time t2 get a new number N 2 etc.
In fact, relative measurements of the isotope activity at different points in time are made in this experiment. The result is a set of numbers , , ..., , which is used to determine the half-life T 1/2.
The obtained experimental values, after subtracting the background, are plotted on a graph (Fig. 3.3), where the time elapsed from the beginning of the measurements is plotted along the abscissa axis, and the logarithm of the number . A line is drawn along the plotted experimental points using the least squares method. If only one radioactive isotope is present in the sample to be measured, then the line will be straight. If it contains two or more radioactive isotopes that decay with different half-lives, then the line will be a curve.
With a single counter (or camera) it is difficult to measure relatively long half-lives (several months or several years). Indeed, let at the beginning of the measurements the count rate was N 1 , and at the end - N2. Then the error will be inversely proportional to ln( N 1 /N 2). This means that if during the measurement period the source activity changes insignificantly, then N 1 And N 2 will be close to each other and ln( N 1 /N 2) will be much less than unity and the error in determining T 1/2 will be great.
Thus, it is clear that half-life measurements with a single counter must be made at such a time that ln (N 1 /N 2) was greater than one. In practice, observations should be made for no more than 5T 1/2.
3) Measurements T 1/2 in a few months or years it is convenient to produce using a differential ionization chamber. It consists of two ionization chambers, switched on so that the currents in them go in the opposite direction and compensate each other (Fig. 3.4).
The half-life measurement process is as follows. In one of the chambers (for example, K 1) a radioactive isotope with a known large T 1/2(for example, 226 Ra, which has T 1/2=1600 years); over a relatively short measurement time (several hours or days), the ionization current in this chamber will hardly change. To another camera K 2) the radioactive nuclide under study is placed. With the help of an approximate selection of the values of the activities of both preparations, as well as their appropriate placement in the chambers, it is possible to ensure that at the initial moment of time the ionization currents in the chambers will be the same: I 1 \u003d I 2 \u003d I 0, i.e. residual current =0. If the measured half-life is relatively short and equal, for example, to several months or years, then after a few hours the current in the chamber K 2 decreases, a residual current will appear: 
.
The change in ionization currents will occur in accordance with the half-lives:
Hence,
For the measured half-lives, the quantity and after expansion into a series, we obtain
In the experiment, we measure I 0 And t. They are already defined and
The measured quantities can be determined with satisfactory accuracy, and, consequently, the value can be calculated with sufficient accuracy. T 1/2.
4) When measuring short half-lives (fractions of a second), the delayed coincidence method is usually used. Its essence can be shown by the example of determining the lifetime of the excited state of the nucleus.
Let the core A as a result of -decay turns into a nucleus B, which is in an excited state and emits its excitation energy in the form of two -quanta, going in series one after another. First, a quantum is emitted, then a quantum (see Fig. 3.5).
As a rule, an excited nucleus does not emit excess energy instantly, but after a certain (even if very short) time, i.e., excited states of the nucleus have some finite lifetime. In this case, it is possible to determine the lifetime of the first excited state of the nucleus. For this, a preparation containing radioactive nuclei A, is placed between two counters (it is better to use scintillation counters for this) (Fig. 3.6). It is possible to create such conditions that the left channel of the circuit will register only quanta, and the right one. A quantum is always emitted before a quantum. The time of emission of the second quantum relative to the first will not always be the same for different nuclei B. The discharge of excited states of nuclei is of a statistical nature and obeys the law of radioactive decay.
Thus, to determine the lifetime of the level , it is necessary to follow its discharge over time. To do this, in the left channel of the coincidence circuit 1, we include a variable delay line 2 , which will in each specific case delay the pulse arising in the left detector from the quantum for some time t 3 . The pulse arising in the right detector from the quantum , directly enters the coincidence block. The number of coinciding pulses is recorded by counting circuit 3. By measuring the number of coincidences as a function of the delay time, we obtain a level I discharge curve similar to the curve in Fig. 3.3. From it, the lifetime of level I is determined. Using the method of delayed coincidences, one can determine the lifetime in the range of 10 -11 -10 -6 s.
The history of the study of radioactivity began on March 1, 1896, when a famous French scientist accidentally discovered an oddity in the radiation of uranium salts. It turned out that the photographic plates located in the same box with the sample were illuminated. The strange, highly penetrating radiation that uranium had led to this. This property was found in the heaviest elements that complete the periodic table. It was given the name "radioactivity".
We introduce the characteristics of radioactivity
This process is the spontaneous transformation of an atom of an isotope of an element into another isotope with the simultaneous release of elementary particles (electrons, nuclei of helium atoms). The transformation of atoms turned out to be spontaneous, not requiring the absorption of energy from outside. The main quantity characterizing the process of energy release during the course is called activity.
The activity of a radioactive sample is the probable number of decays of a given sample per unit of time. In the international) unit of measurement, it is called the becquerel (Bq). In 1 becquerel, the activity of such a sample is taken, in which, on average, 1 decay occurs per second.
A=λN, where λ is the decay constant, N is the number of active atoms in the sample.
Allocate α, β, γ-decays. The corresponding equations are called displacement rules:
Time interval in radioactivity
The breakup moment of the particle cannot be determined for this particular atom. For him, this is more of an "accident" than a pattern. The release of energy characterizing this process is defined as the activity of the sample.
It has been observed to change over time. Although individual elements show a surprising constancy of the degree of radiation, there are substances whose activity decreases several times in a fairly short period of time. Amazing variety! Is it possible to find a pattern in these processes?
It has been established that there is a time during which exactly half of the atoms of a given sample undergo decay. This time interval is called the "half-life". What is the meaning of introducing this concept?
half-life?
It seems that in a time equal to a period, exactly half of all active atoms of a given sample decay. But does this mean that in a time of two half-lives, all active atoms will completely decay? Not at all. After a certain moment, half of the radioactive elements remain in the sample, after the same period of time, half of the remaining atoms decay, and so on. In this case, the radiation persists for a long time, significantly exceeding the half-life. This means that active atoms are retained in the sample regardless of radiation
The half-life is a value that depends solely on the properties of a given substance. The value of the quantity has been determined for many known radioactive isotopes.
Table: "The decay half-life of individual isotopes"
| Name | Designation | Type of decay | Half life |
0.001 seconds |
|||
beta, gamma | |||
alpha, gamma | |||
alpha, gamma | 4.5 billion years |
The half-life was determined experimentally. In the course of laboratory studies, activity is repeatedly measured. Since laboratory samples are of minimal size (the safety of the researcher is paramount), the experiment is carried out with different time intervals, repeating many times. It is based on the regularity of changes in the activity of substances.
In order to determine the half-life, the activity of a given sample is measured at certain time intervals. Taking into account the fact that this parameter is related to the number of decayed atoms, using the law of radioactive decay, the half-life is determined.
An example definition for an isotope
Let the number of active elements of the studied isotope at a given moment of time be equal to N, the time interval during which the observation is carried out t 2 - t 1, where the moments of the beginning and end of the observation are close enough. Let's assume that n is the number of atoms that decayed in a given time interval, then n = KN(t 2 - t 1).
In this expression, K \u003d 0.693 / T½ is the proportionality coefficient, which is called the decay constant. T½ is the half-life of the isotope.
Let's take the time interval as a unit. In this case, K = n/N indicates the fraction of the isotope nuclei present that decay per unit time.
Knowing the value of the decay constant, one can also determine the decay half-life: T½ = 0.693/K.
It follows from this that not a certain number of active atoms decays per unit of time, but a certain proportion of them.
Law of radioactive decay (LRR)
The half-life is the basis of the RRR. The pattern was derived by Frederico Soddy and Ernest Rutherford based on the results of experimental studies in 1903. It is surprising that multiple measurements made with devices that are far from perfect, in the conditions of the beginning of the 20th century, led to an accurate and justified result. It became the basis of the theory of radioactivity. Let us derive the mathematical notation of the law of radioactive decay.
Let N 0 be the number of active atoms at a given time. After the time interval t has elapsed, N elements will remain undecayed.
By the time equal to the half-life, exactly half of the active elements will remain: N=N 0 /2.
After another half-life period, the following remains in the sample: N=N 0 /4=N 0 /2 2 active atoms.
After the passage of time, equal to one more half-life, the sample will save only: N=N 0 /8=N 0 /2 3 .
By the time when n half-lives have passed, N=N 0 /2 n active particles will remain in the sample. In this expression, n=t/T½: the ratio of study time to half-life.
ZRR has a slightly different mathematical expression, more convenient in solving problems: N=N 0 2 - t/ T½.
The regularity makes it possible to determine, in addition to the half-life, the number of atoms of the active isotope that have not decayed at a given time. Knowing the number of sample atoms at the beginning of the observation, after some time it is possible to determine the lifetime of a given preparation.
The formula for the law of radioactive decay helps to determine the half-life only if certain parameters are present: the number of active isotopes in the sample, which is quite difficult to find out.
Consequences of the law
You can write down the RRR formula using the concepts of activity and mass of drug atoms.
Activity is proportional to the number of radioactive atoms: A=A 0 .2 -t/T. In this formula, A 0 is the activity of the sample at the initial time, A is the activity after t seconds, T is the half-life.
The mass of the substance can be used in the regularity: m=m 0 .2 -t/T
During any equal time intervals, an absolutely equal proportion of the radioactive atoms present in a given preparation decays.
Limits of applicability of the law
The law in every sense is statistical, determining the processes occurring in the microcosm. It is clear that the half-life of radioactive elements is a statistical value. The probabilistic nature of events in atomic nuclei suggests that an arbitrary nucleus can fall apart at any moment. It is impossible to predict an event; one can only determine its probability at a given moment in time. As a consequence, the half-life is meaningless:
- for a single atom;
- for a sample of minimum mass.
Atom lifetime
The existence of an atom in its original state can last a second, or maybe millions of years. It is also not necessary to speak about the lifetime of this particle. By introducing a value equal to the average value of the lifetime of atoms, we can talk about the existence of atoms of a radioactive isotope, the consequences of radioactive decay. The half-life of the nucleus of an atom depends on the properties of the given atom and does not depend on other quantities.
Is it possible to solve the problem: how to find the half-life, knowing the average lifetime?
To determine the half-life, the formula for the relationship between the average lifetime of an atom and the decay constant helps no less.
τ= T 1/2 / ln2= T 1/2 / 0.693=1/ λ.
In this notation, τ is the mean lifetime, λ is the decay constant.
Use of half-life
The use of ZRR to determine the age of individual samples became widespread in studies at the end of the 20th century. The accuracy of determining the age of fossil artifacts has increased so much that it can give an idea of the life time for millennia BC.
Fossil organic samples are based on changes in the activity of carbon-14 (a radioactive isotope of carbon) present in all organisms. It enters the living organism in the process of metabolism and is contained in it in a certain concentration. After death, the exchange of substances with the environment stops. The concentration of radioactive carbon falls due to natural decay, the activity decreases proportionally.
If there is such a value as the half-life, the formula for the law of radioactive decay helps to determine the time since the cessation of the life of the organism.
Chains of radioactive transformation
Radioactivity studies were carried out in laboratory conditions. The amazing ability of radioactive elements to remain active for hours, days and even years could not but surprise physicists of the early twentieth century. Studies, for example, of thorium, were accompanied by an unexpected result: in a closed ampoule, its activity was significant. At the slightest breath she fell. The conclusion turned out to be simple: the transformation of thorium is accompanied by the release of radon (gas). All elements in the process of radioactivity turn into a completely different substance, which differs in both physical and chemical properties. This substance, in turn, is also unstable. At present, three series of similar transformations are known.
Knowledge of such transformations is extremely important in determining the time of inaccessibility of zones contaminated in the course of atomic and nuclear research or catastrophes. The half-life of plutonium - depending on its isotope - ranges from 86 years (Pu 238) to 80 million years (Pu 244). The concentration of each isotope gives an idea of the period of disinfection of the territory.
most expensive metal
It is known that in our time there are metals much more expensive than gold, silver and platinum. They include plutonium. Interestingly, plutonium created in the process of evolution does not occur in nature. Most of the elements were obtained in laboratory conditions. The exploitation of plutonium-239 in nuclear reactors has enabled it to become extremely popular these days. Obtaining a sufficient amount of this isotope for use in reactors makes it practically invaluable.
Plutonium-239 is obtained in natural conditions as a result of a chain of transformations of uranium-239 into neptunium-239 (half-life - 56 hours). A similar chain makes it possible to accumulate plutonium in nuclear reactors. The rate of appearance of the required amount exceeds the natural one by a billion times.
Energy Application
You can talk a lot about the shortcomings of nuclear energy and about the "strangeness" of mankind, which uses almost any discovery to destroy its own kind. The discovery of plutonium-239, which is capable of taking part, has made it possible to use it as a source of peaceful energy. Uranium-235, which is an analogue of plutonium, is extremely rare on Earth; it is much more difficult to isolate it from it than to obtain plutonium.
Age of the Earth
Radioisotope analysis of isotopes of radioactive elements gives a more accurate idea of the lifetime of a particular sample.
Using the chain of transformations "uranium - thorium", contained in the earth's crust, makes it possible to determine the age of our planet. The percentage of these elements on average throughout the earth's crust is the basis of this method. According to the latest data, the age of the Earth is 4.6 billion years.
From Wikipedia, the free encyclopedia
Half life quantum mechanical system (particle, nucleus, atom, energy level, etc.) - time , during which the system decays in an approximate ratio 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 reduces the number of surviving particles by half, over time a quarter of the initial number of particles will remain, for - one eighth, etc. In general, the fraction of surviving particles (or, more precisely, the probability of surviving p for a given particle) depends on time in the following way:
.
Half-life, mean lifetime and decay constant are related by the following relationships, derived from the law of radioactive decay:
.
Because the , the half-life is about 30.7% shorter than the mean lifetime.
In practice, the half-life is determined by measuring 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, you can calculate the half-life of this substance.
Examples
Example 1
If we designate for a given moment of time the number of nuclei capable of radioactive transformation through , and the time interval after , Where And - fairly close times
The values of the half-lives for different isotopes are different; for some, especially rapidly decaying ones, the half-life can be equal to millionths of a second, and for some isotopes, like uranium-238 and thorium-232, it is respectively equal to 4.498 10 9 and 1.389 10 10 years. It is easy to count the number of uranium-238 atoms undergoing transformation in a given amount of uranium, for example, one kilogram in one second. The amount of any element in grams, numerically equal to the atomic weight, contains, as you know, 6.02·10 23 atoms. Therefore, according to the above formula let's find the number of uranium atoms decaying in one kilogram in one second, bearing in mind that there are 365 * 24 * 60 * 60 seconds in a year,
.
Calculations lead to the fact that in one kilogram of uranium, twelve million atoms decay in one second. Despite such a huge number, the rate of transformation is still negligible. Indeed, the following part of uranium decays per second:
.
Thus, from the available amount of uranium, its fraction equal to
.
Turning again to the basic law of radioactive decay KN(t 2 - t 1), that is, to the fact that only one and the same fraction of the available number of atomic nuclei decays per unit time, and, bearing in mind the complete independence of atomic nuclei in any substance from each other, we can say that this law is statistical in the sense that it does not indicate exactly which atomic nuclei will undergo decay in a given period of time, but only tells about their number. Undoubtedly, this law remains valid only for the case when the available number of nuclei is very large. Some of the atomic nuclei will decay in the next moment, while other nuclei will undergo transformations much later, so when the available number of radioactive atomic nuclei is relatively small, the law of radioactive decay may not be fully satisfied.
Example 2
The sample contains 10 g of the plutonium isotope Pu-239 with a half-life of 24,400 years. How many plutonium atoms decay every second?
= \frac(2.5\cdot 10^(22) \cdot 0.693)(7.7\cdot 10^(11))= 2.25\cdot 10^(10) ~s^(-1).
We calculated the instantaneous decay rate. The number of decayed atoms is calculated by the formula
The last formula is only valid when the period of time in question (in this case 1 second) is significantly less than the half-life. When the time period under consideration is comparable to the half-life, the formula should be used
This formula is suitable in any case, however, for short periods of time, it requires calculations with very high accuracy. For this task:
2.5\cdot 10^(22) \left(1-2^(-1/7.7 \cdot 10^(11)) \right) = 2.5\cdot 10^(22) \left(1-0.999999999999910 \right) = 2.25\cdot 10^(10).
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 partial half-life. Let the probability of decay by i-th channel (branching factor) is equal to pi. Then the partial half-life of i-th channel is equal to
.
Partial has the meaning of the half-life that a given system would have if all decay channels were "turned off" except for i th. Since by definition , That for any decay channel.
half-life stability
In all observed cases (except for some isotopes decaying by electron capture), the half-life was constant (separate reports of a change in the period were caused by insufficient experimental accuracy, 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 strongly 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 isotope-chronometers 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 α-decay constant of samarium-147 changed by no more than 0.75%, and for the β-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.
see also
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Notes
An excerpt characterizing the half-life
Returning from the review, Kutuzov, accompanied by an Austrian general, went to his office and, calling the adjutant, ordered to give himself some papers relating to the condition of the incoming troops, and letters received from Archduke Ferdinand, who commanded the forward army. Prince Andrei Bolkonsky with the required papers entered the office of the commander in chief. In front of the plan laid out on the table sat Kutuzov and an Austrian member of the Hofkriegsrat.“Ah ...” said Kutuzov, looking back at Bolkonsky, as if by this word inviting the adjutant to wait, and continued the conversation begun in French.
“I only say one thing, General,” Kutuzov said with a pleasant elegance of expression and intonation, forcing one to listen to every leisurely spoken word. It was evident that Kutuzov listened to himself with pleasure. - I only say one thing, General, that if the matter depended on my personal desire, then the will of His Majesty Emperor Franz would have been fulfilled long ago. I would have joined the Archduke long ago. And believe my honor, that for me personally to transfer the higher command of the army more than I am to a knowledgeable and skillful general, such as Austria is so plentiful, and to lay down all this heavy responsibility for me personally would be a joy. But circumstances are stronger than us, general.
And Kutuzov smiled with such an expression as if he were saying: “You have every right not to believe me, and even I don’t care whether you believe me or not, but you have no reason to tell me this. And that's the whole point."
The Austrian general looked dissatisfied, but could not answer Kutuzov in the same tone.
“On the contrary,” he said in a grouchy and angry tone, so contrary to the flattering meaning of the words spoken, “on the contrary, Your Excellency’s participation in the common cause is highly valued by His Majesty; but we believe that a real slowdown deprives the glorious Russian troops and their commanders of those laurels that they are accustomed to reap in battles, ”he finished the apparently prepared phrase.
Kutuzov bowed without changing his smile.
- And I am so convinced and, based on the last letter that His Highness Archduke Ferdinand honored me, I assume that the Austrian troops, under the command of such a skilled assistant as General Mack, have now already won a decisive victory and no longer need our help, - Kutuzov said.
The general frowned. Although there was no positive news of the defeat of the Austrians, there were too many circumstances that confirmed the general unfavorable rumors; and therefore Kutuzov's assumption about the victory of the Austrians was very similar to a mockery. But Kutuzov smiled meekly, still with the same expression that said that he had the right to assume this. Indeed, the last letter he received from Mack's army informed him of the victory and the most advantageous strategic position of the army.
“Give me this letter here,” said Kutuzov, turning to Prince Andrei. - Here you are, if you want to see it. - And Kutuzov, with a mocking smile on the ends of his lips, read the following passage from the letter of Archduke Ferdinand from the German-Austrian general: “Wir haben vollkommen zusammengehaltene Krafte, nahe an 70,000 Mann, um den Feind, wenn er den Lech passirte, angreifen und schlagen zu konnen. Wir konnen, da wir Meister von Ulm sind, den Vortheil, auch von beiden Uferien der Donau Meister zu bleiben, nicht verlieren; mithin auch jeden Augenblick, wenn der Feind den Lech nicht passirte, die Donau ubersetzen, uns auf seine Communikations Linie werfen, die Donau unterhalb repassiren und dem Feinde, wenn er sich gegen unsere treue Allirte mit ganzer Macht wenden wollte, seine Absicht alabald vereitel ien. Wir werden auf solche Weise den Zeitpunkt, wo die Kaiserlich Ruseische Armee ausgerustet sein wird, muthig entgegenharren, und sodann leicht gemeinschaftlich die Moglichkeit finden, dem Feinde das Schicksal zuzubereiten, so er verdient.” [We have a fully concentrated force, about 70,000 people, so that we can attack and defeat the enemy if he crosses the Lech. Since we already own Ulm, we can retain the advantage of commanding both banks of the Danube, therefore, every minute, if the enemy does not cross the Lech, cross the Danube, rush to his communication line, cross the Danube lower and the enemy, if he decides to turn all his strength on our faithful allies, to prevent his intention from being fulfilled. Thus, we will cheerfully await the time when the imperial Russian army is completely ready, and then together we will easily find an opportunity to prepare the enemy for the fate he deserves.
Kutuzov sighed heavily, having finished this period, and carefully and affectionately looked at the member of the Hofkriegsrat.
“But you know, Your Excellency, the wise rule of assuming the worst,” said the Austrian general, apparently wanting to end the jokes and get down to business.
He glanced involuntarily at the adjutant.
“Excuse me, General,” Kutuzov interrupted him and also turned to Prince Andrei. - That's what, my dear, you take all the reports from our scouts from Kozlovsky. Here are two letters from Count Nostitz, here is a letter from His Highness Archduke Ferdinand, here's another,” he said, handing him some papers. - And from all this, cleanly, in French, make a memorandum, a note, for the visibility of all the news that we had about the actions of the Austrian army. Well, then, and present to his Excellency.
Prince Andrei bowed his head as a sign that he understood from the first words not only what was said, but also what Kutuzov would like to tell him. He collected the papers, and, giving a general bow, quietly walking along the carpet, went out into the waiting room.
Despite the fact that not much time has passed since Prince Andrei left Russia, he has changed a lot during this time. In the expression of his face, in his movements, in his gait, there was almost no noticeable former pretense, fatigue and laziness; he had the appearance of a man who has no time to think about the impression he makes on others, and is busy with pleasant and interesting business. His face expressed more satisfaction with himself and those around him; his smile and look were more cheerful and attractive.
Kutuzov, whom he caught up with back in Poland, received him very affectionately, promised him not to forget him, distinguished him from other adjutants, took him with him to Vienna and gave him more serious assignments. From Vienna, Kutuzov wrote to his old comrade, the father of Prince Andrei:
“Your son,” he wrote, “gives hope to be an officer who excels in his studies, firmness and diligence. I consider myself fortunate to have such a subordinate at hand.”
At Kutuzov's headquarters, among his comrades, and in the army in general, Prince Andrei, as well as in St. Petersburg society, had two completely opposite reputations.
Some, a minority, recognized Prince Andrei as something special from themselves and from all other people, expected great success from him, listened to him, admired him and imitated him; and with these people, Prince Andrei was simple and pleasant. Others, the majority, did not like Prince Andrei, they considered him an inflated, cold and unpleasant person. But with these people, Prince Andrei knew how to position himself in such a way that he was respected and even feared.
Coming out of Kutuzov's office into the waiting room, Prince Andrei with papers approached his comrade, adjutant on duty Kozlovsky, who was sitting by the window with a book.
- Well, what, prince? Kozlovsky asked.
- Ordered to draw up a note, why not let's go forward.
- And why?
Prince Andrew shrugged his shoulders.
- No word from Mac? Kozlovsky asked.
- No.
- If it were true that he was defeated, then the news would come.
“Probably,” said Prince Andrei and went to the exit door; but at the same time to meet him, slamming the door, a tall, obviously newcomer, Austrian general in a frock coat, with his head tied with a black handkerchief and with the Order of Maria Theresa around his neck, quickly entered the waiting room. Prince Andrew stopped.
- General Anshef Kutuzov? - quickly said the visiting general with a sharp German accent, looking around on both sides and without stopping walking to the door of the office.
“The general is busy,” said Kozlovsky, hurriedly approaching the unknown general and blocking his way from the door. - How would you like to report?
The unknown general looked contemptuously down at the short Kozlovsky, as if surprised that he might not be known.
“The general chief is busy,” Kozlovsky repeated calmly.
The general's face frowned, his lips twitched and trembled. He took out a notebook, quickly drew something with a pencil, tore out a piece of paper, gave it away, went with quick steps to the window, threw his body on a chair and looked around at those in the room, as if asking: why are they looking at him? Then the general raised his head, stretched out his neck, as if intending to say something, but immediately, as if carelessly starting to hum to himself, made a strange sound, which was immediately stopped. The door of the office opened, and Kutuzov appeared on the threshold. The general with his head bandaged, as if running away from danger, bent over, with large, quick steps of thin legs, approached Kutuzov.
- Vous voyez le malheureux Mack, [You see the unfortunate Mack.] - he said in a broken voice.
The face of Kutuzov, who was standing in the doorway of the office, remained completely motionless for several moments. Then, like a wave, a wrinkle ran over his face, his forehead smoothed out; he bowed his head respectfully, closed his eyes, silently let Mack pass him, and closed the door behind him.
>> Law of radioactive decay. Half life
§ 101 LAW OF RADIOACTIVE DECAY. HALF LIFE
Radioactive decay obeys a statistical law. Rutherford, investigating the transformation of radioactive substances, established empirically that their activity decreases with time. This was discussed in the previous paragraph. Thus, the activity of radon decreases by 2 times after 1 min. The activity of elements such as uranium, thorium and radium also decreases with time, but much more slowly. For each radioactive substance, there is a certain time interval during which the activity decreases by 2 times. This interval is called the half-life. The half-life T is the time during which half of the initial number of radioactive atoms decays.
The decline in activity, i.e., the number of disintegrations per second, depending on time for one of the radioactive preparations is shown in Figure 13.8. The half-life of this substance is 5 days.
We now derive the mathematical form of the law of radioactive decay. Let the number of radioactive atoms at the initial time (t= 0) be N 0 . Then after the half-life period, this number will be equal to
After another similar time interval, this number will become equal to:
