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From the graph, you can see that the pattern is exponential and at the same time decreases each time by half of the period, followed by a decrease.
As an experimental analysis of this phenomenon, the following can be shown. 100 measurements were carried out over the same period of time and the number of decays was measured. As a result, a graph was obtained on (Figure 2.9), where the average number of decays equal to 77.47 coincided with the value in (2.12), which is a clear proof of the validity of the general pattern.
Figure 2.9. The result of the experiment
The general view of the distribution of these statistics is already presented according to a different law. That is, the probability Pn for the time t for testing the n number of decays is given by the Poisson distribution (2.13).
This conclusion is already inherent in probability theory, and if we rely on it, then also for the case when (n>> 1) Gaussian distributions (2.14) are already used.
If we express these two patterns on graphs, we can get almost identical patterns with an increase in the average number of decays. For example, if the average number of decays is 2, then there is some difference in the results of the Poisson and Gauss distribution, but when this number, for example, reaches 7 and higher values, this difference becomes less significant, as shown in (Figure 2.10).
Figure 2.10. The graph of the probability of decay according to the Poisson and Gauss distributions for the average decay number equal to 2 and 7
After it has been decided with probability at zero speed, we can pay attention to cases when the effects of the theory of relativity come into play. In the microcosm, where the sizes of the studied objects are practically invisible, for example, for atoms with their sizes of 10-8 cm, for atomic nuclei with their 10-12-10-13 cm and for other particles with 10-13-10-17 cm, the speeds are often comparable, close or even equal to the speed of light. Thanks to this, all the features and effects of the theory of relativity are clearly manifested in the microcosm.
For this reason, it is important to consider in more detail the relations and basic equations from the theory of relativity.
One of the most important elements in the theory of relativity is the Lorentz factor (2.15), which is involved in almost all formulas of the theory of relativity, which can also be derived from the kinetic energy formula (2.16).
From these relations, it can be concluded that the total energy, which is the sum of the kinetic energy and the rest energy of the particle, is determined by (2.17).
The presence of this equality leads to the fact that the problem of the lack of a formula for calculating the energy of particles without masses (for example, a photon or a gluon) is solved. And already from (2.16) it is also possible to derive a more simplified entry for kinetic energy (2.18). In the case of applying (2.15) for the momentum formula (2.19), a simplified form is also obtained.
The velocity of the particle derived from the formulas of the total energy (2.17) looks like this (2.20).
An important element also in calculations, this is also the total energy of massless particles, is the formula (2.21), where the conclusions of which are also given from the total energy ratio (2.17).
The notion of invariant also plays a role in this definition. An invariant is a constant value, regardless of the system of the report from which the observation is conducted. In this case, the invariant is the square of the mass or (2.22).
And it does not matter if it is a single particle or a system of particles, so the total energy E also refers to a particle or a system of particles, and the momentum of a particle also refers to a particle or a system of particles.
One of the most important points in the study of the physics of the atomic nucleus and elementary particles is familiarity with the system of units, which is the easiest way to perform calculations – this is the Gaussian system of units together with some non-system quantities.
Speaking of units of energy, due to the small amount of energy, it is convenient to use a unit of electron volts (eV), which is equal to 1.6 *10-19 J or 1.6 * 10-12 erg. This value represents the energy that an electron acquires by passing a potential difference of 1 Volt. Values of 1 keV (kiloelectronvolt) or 103 eV, 1 MeV or 106 eV, 1 GeV or 109 eV and 1 TeV or 1012 eV are also appropriate, which are actively used in the physics of elementary particles and the atomic nucleus.
As a unit of length or distance, it is customary to use the value of 1 Fermi (Fm) in honor of the famous scientist Enrico Fermi, which also coincides with the value of 1 femtometer (fm), where 1 Fm is 10-13 cm. As for the mass, it is expressed in energy units mc2, for example, the mass of an electron, which in the usual SI unit system is 9.11 * 10-28 grams, then in energy units is 0.511 MeV. And the mass of the proton, which is 1.6727 * 10-24 grams, in energy equivalent will be 938.27 MeV.
Special and general relativity have many effects, then 3 of them deserve more attention. The first of them is the time dilation for a relativistic particle, the second is the effect of shortening the distance in the direction of motion of a relativistic particle, and the third effect, which, by the way, comes out of general relativity – the time dilation in a gravitational field, also known as the gravitational redshift of radiation. For a better understanding of these effects, consider 3 cases.
Let some particles undergo decay according to the law of radioactive decay (2.23), where t0 is the lifetime at rest.
In this case, if the particles move at a certain speed, then their lifetime, due to time dilation, increases and becomes equal to (2.24), where the Lorentz factor is known (2.25) and from here it can already be said that the law of radioactive decay for relativistic particles is represented as (2.26).
Consider an example. Most of all the particles that fall to Earth together with cosmic radiation are protons with energies of the order of 1020 eV. At the same time, when they enter the Earth’s atmosphere, they collide with nitrogen and oxygen atoms, charged pions are born (some particles that have the designation π+ or π- for a positively and negatively charged pion, respectively), they decay in their free flight into already relativistic muons (other types of particles that are already designated as μ+ or μ- also for a positively and negatively charged muon, respectively), muon neutrinos vμ or their antineutrinos are also isolated from pions.And the lifetime of a peony is 2.6 * 10—8 seconds. Thus, a kind of «shower» of secondary particles is formed, which is born by a proton in the Earth’s atmosphere. And these reactions forming a «shower» can be written as (2.27), (2.28) and (2.29), and also illustrated in (Fig. 2.11).
