home Online services How to calculate the mass of the nucleus of an atom. Masses of atomic nuclei. Line of beta-stability and binding energy of nuclei

How to calculate the mass of the nucleus of an atom. Masses of atomic nuclei. Line of beta-stability and binding energy of nuclei

How to find the mass of the nucleus of an atom? and got the best answer

Answer from NiNa Martushova[guru]

A = number p + number n. That is, the entire mass of the atom is concentrated in the nucleus, since the electron has a negligible mass equal to 11800 AU. e. m., while the proton and neutron each have a mass of 1 atomic mass unit. Relative atomic mass because fractional number that it is the arithmetic mean of the atomic masses of all isotopes of a given chemical element, taking into account their prevalence in nature.

Answer from Yoehmet[guru]
Take the mass of the atom and subtract the mass of all the electrons.

Answer from Vladimir Sokolov[guru]
Sum the mass of all the protons and neutrons in the nucleus. You'll get a lot in em.

Answer from Dasha[newbie]
periodic table to help

Answer from Anastasia Durakova[active]
Find the value of the relative mass of an atom in the periodic table, round it up to a whole number - this will be the mass of the atom's nucleus. The mass of the nucleus, or the mass number of an atom, is made up of the number of protons and neutrons in the nucleus
A = number p + number n. That is, the entire mass of the atom is concentrated in the nucleus, since the electron has a negligible mass equal to 11800 AU. e. m., while the proton and neutron each have a mass of 1 atomic mass unit. The relative atomic mass is a fractional number because it is the arithmetic mean of the atomic masses of all isotopes of a given chemical element, taking into account their prevalence in nature. periodic table to help

Answer from 3 answers[guru]

Hello! Here is a selection of topics with answers to your question: How to find the mass of the nucleus of an atom?

Many years ago, people wondered what all substances are made of. The first who tried to answer it was the ancient Greek scientist Democritus, who believed that all substances are composed of molecules. We now know that molecules are built from atoms. Atoms are made up of even smaller particles. At the center of an atom is the nucleus, which contains protons and neutrons. The smallest particles - electrons - move in orbits around the nucleus. Their mass is negligible compared to the mass of the nucleus. But how to find the mass of the nucleus, only calculations and knowledge of chemistry will help. To do this, you need to determine the number of protons and neutrons in the nucleus. View the tabular values of the masses of one proton and one neutron and find their total mass. This will be the mass of the nucleus.

Often you can come across such a question, how to find the mass, knowing the speed. According to the classical laws of mechanics, the mass does not depend on the speed of the body. After all, if a car, moving away, begins to pick up its speed, this does not mean at all that its mass will increase. However, at the beginning of the twentieth century, Einstein presented a theory according to which this dependence exists. This effect is called the relativistic increase in body mass. And it manifests itself when the speeds of bodies approach the speed of light. Modern particle accelerators make it possible to accelerate protons and neutrons to such high speeds. And in fact, in this case, an increase in their masses was recorded.

But we still live in a world of high technology, but low speeds. Therefore, in order to know how to calculate the mass of a substance, it is not at all necessary to accelerate the body to the speed of light and learn Einstein's theory. Body weight can be measured on a scale. True, not every body can be put on the scales. Therefore, there is another way to calculate mass from its density.

The air around us, the air that is so necessary for mankind, also has its own mass. And, when solving the problem of how to determine the mass of air, for example, in a room, it is not necessary to count the number of air molecules and sum up the mass of their nuclei. You can simply determine the volume of the room and multiply it by the air density (1.9 kg / m3).

Scientists have now learned with great accuracy to calculate the masses of different bodies, from the nuclei of atoms to the mass of the globe and even stars located at a distance of several hundred light years from us. Mass like physical quantity, is a measure of the body's inertia. More massive bodies, they say, are more inert, that is, they change their speed more slowly. Therefore, after all, speed and mass are interconnected. But the main feature of this quantity is that any body or substance has mass. There is no matter in the world that does not have mass!

§1 Charge and mass, atomic nuclei

The most important characteristics of a nucleus are its charge and mass. M.

Z- the charge of the nucleus is determined by the number of positive elementary charges concentrated in the nucleus. A carrier of a positive elementary charge R= 1.6021 10 -19 C in the nucleus is a proton. The atom as a whole is neutral and the charge of the nucleus simultaneously determines the number of electrons in the atom. The distribution of electrons in an atom over energy shells and subshells essentially depends on their total number in the atom. Therefore, the charge of the nucleus largely determines the distribution of electrons over their states in the atom and the position of the element in the periodic system of Mendeleev. The nuclear charge isqI = z· e, where z- the charge number of the nucleus, equal to the ordinal number of the element in the Mendeleev system.

The mass of the atomic nucleus practically coincides with the mass of the atom, because the mass of the electrons of all atoms, except for hydrogen, is approximately 2.5 10 -4 masses of atoms. The mass of atoms is expressed in atomic mass units (a.m.u.). For a.u.m. accepted 1/12 mass of carbon atom.

1 amu \u003d 1.6605655 (86) 10 -27 kg.

mI = m a - Z me.

Isotopes are varieties of atoms of a given chemical element that have the same charge, but differ in mass.

The integer closest to the atomic mass, expressed in a.u. m . called the mass number m and denoted by the letter BUT. Designation of a chemical element: BUT- mass number, X - symbol of a chemical element,Z-charging number - serial number in the periodic table ():

Beryllium; Isotopes: , ", .

Core Radius:

where A is the mass number.

§2 Composition of the core

The nucleus of a hydrogen atomcalled proton

mproton= 1.00783 amu , .

Hydrogen atom diagram

In 1932, a particle called the neutron was discovered, which has a mass close to that of a proton (mneutron= 1.00867 amu) and not having electric charge. Then D.D. Ivanenko formulated a hypothesis about the proton-neutron structure of the nucleus: the nucleus consists of protons and neutrons and their sum is equal to the mass number BUT. 3 ordinal numberZdetermines the number of protons in the nucleus, the number of neutronsN \u003d A - Z.

Elementary particles - protons and neutrons entering into the core, are collectively known as nucleons. Nucleons of nuclei are in states, significantly different from their free states. Between nucleons there is a special i de r new interaction. They say that a nucleon can be in two "charge states" - a proton state with a charge+ e, and neutron with a charge of 0.

§3 Binding energy of the nucleus. mass defect. nuclear forces

Nuclear particles - protons and neutrons - are firmly held inside the nucleus, so very large attractive forces act between them, capable of withstanding the huge repulsive forces between like-charged protons. These special forces arising at small distances between nucleons are called nuclear forces. Nuclear forces are not electrostatic (Coulomb).

The study of the nucleus showed that the nuclear forces acting between nucleons have the following features:

a) these are short-range forces - manifested at distances of the order of 10 -15 m and sharply decreasing even with a slight increase in distance;

b) nuclear forces do not depend on whether the particle (nucleon) has a charge - charge independence of nuclear forces. The nuclear forces acting between a neutron and a proton, between two neutrons, between two protons are equal. Proton and neutron in relation to nuclear forces are the same.

The binding energy is a measure of the stability of an atomic nucleus. The binding energy of the nucleus is equal to the work that must be done to split the nucleus into its constituent nucleons without imparting kinetic energy to them

M I< Σ( m p + m n)

Me - the mass of the nucleus

Measurement of the masses of nuclei shows that the rest mass of the nucleus is less than the sum of the rest masses of its constituent nucleons.

Value

serves as a measure of the binding energy and is called the mass defect.

Einstein's equation special theory relativity relates the energy and rest mass of the particle.

In the general case, the binding energy of the nucleus can be calculated by the formula

where Z - charge number (number of protons in the nucleus);

BUT- mass number ( total number nucleons in the nucleus);

m p, , m n and M i- mass of proton, neutron and nucleus

Mass defect (Δ m) are equal to 1 a.u. m. (a.m.u. - atomic mass unit) corresponds to the binding energy (E St) equal to 1 a.u.e. (a.u.e. - atomic unit of energy) and equal to 1a.u.m. s 2 = 931 MeV.

§ 4 Nuclear reactions

Changes in nuclei during their interaction with individual particles and with each other are usually called nuclear reactions.

The following are the most common nuclear reactions.

Transformation reaction . In this case, the incident particle remains in the nucleus, but the intermediate nucleus emits some other particle, so the product nucleus differs from the target nucleus.

Radiative capture reaction . The incident particle gets stuck in the nucleus, but the excited nucleus emits excess energy, emitting a γ-photon (used in the operation of nuclear reactors)

An example of a neutron capture reaction by cadmium

or phosphorus

Scattering. The intermediate nucleus emits a particle identical to

with the flown one, and it can be:

Elastic scattering neutrons with carbon (used in reactors to moderate neutrons):

Inelastic scattering :

fission reaction. This is a reaction that always proceeds with the release of energy. It is the basis for the technical production and use of nuclear energy. During the fission reaction, the excitation of the intermediate compound nucleus is so great that it is divided into two, approximately equal fragments, with the release of several neutrons.

If the excitation energy is low, then the separation of the nucleus does not occur, and the nucleus, having lost excess energy by emitting a γ - photon or neutron, will return to its normal state (Fig. 1). But if the energy introduced by the neutron is large, then the excited nucleus begins to deform, a constriction is formed in it and as a result it is divided into two fragments that fly apart at tremendous speeds, while two neutrons are emitted
(Fig. 2).

Chain reaction- self-developing fission reaction. To implement it, it is necessary that of the secondary neutrons produced during one fission event, at least one can cause the next fission event: (since some neutrons can participate in capture reactions without causing fission). Quantitatively, the condition for the existence of a chain reaction expresses multiplication factor

k < 1 - цепная реакция невозможна, k = 1 (m = m kr ) - chain reaction with a constant number of neutrons (in a nuclear reactor),k > 1 (m > m kr ) are nuclear bombs.

RADIOACTIVITY

§1 Natural radioactivity

Radioactivity is the spontaneous transformation of unstable nuclei of one element into nuclei of another element. natural radioactivity called the radioactivity observed in the unstable isotopes that exist in nature. Artificial radioactivity is called the radioactivity of isotopes obtained as a result of nuclear reactions.

Types of radioactivity:

α-decay.

Emission by the nuclei of some chemical elements of the α-system of two protons and two neutrons connected together (a-particle - the nucleus of a helium atom)

α-decay is inherent in heavy nuclei with BUT> 200 andZ > 82. When moving in a substance, α-particles produce strong ionization of atoms on their way (ionization is the detachment of electrons from an atom), acting on them with their electric field. The distance over which an α-particle flies in matter until it stops completely is called particle range or penetrating power(denotedR, [ R ] = m, cm). . Under normal conditions, an α-particle forms in air 30,000 pairs of ions per 1 cm path. Specific ionization is the number of pairs of ions formed per 1 cm of the path length. The α-particle has a strong biological effect.

Shift rule for alpha decay:

2. β-decay.

a) electronic (β -): the nucleus emits an electron and an electron antineutrino

b) positron (β +): the nucleus emits a positron and a neutrino

These processes occur by converting one type of nucleon into a nucleus into another: a neutron into a proton or a proton into a neutron.

There are no electrons in the nucleus, they are formed as a result of the mutual transformation of nucleons.

Positron - a particle that differs from an electron only in the sign of charge (+e = 1.6 10 -19 C)

It follows from the experiment that during β - decay, isotopes lose the same amount of energy. Therefore, on the basis of the law of conservation of energy, W. Pauli predicted that another light particle, called the antineutrino, is ejected. An antineutrino has no charge or mass. Losses of energy by β - particles during their passage through matter are caused mainly by ionization processes. Part of the energy is lost to X-rays during deceleration of β-particles by the nuclei of the absorbing substance. Since β-particles have low weight, unit charge and very high speeds, then their ionizing ability is low (100 times less than that of α-particles), therefore, the penetrating ability (mileage) of β-particles is significantly greater than that of α-particles.

Rβ air =200 m, Rβ Pb ≈ 3 mm

β - - decay occurs in natural and artificial radioactive nuclei. β + - only with artificial radioactivity.

Displacement rule for β - - decay:

c) K - capture (electronic capture) - the nucleus absorbs one of the electrons located on the shell K (less oftenLor M) of its atom, as a result of which one of the protons turns into a neutron, while emitting a neutrino

Scheme K - capture:

The space in the electron shell vacated by the captured electron is filled with electrons from the overlying layers, resulting in X-rays.

γ-rays.

Usually, all types of radioactivity are accompanied by the emission of γ-rays. γ-rays are electromagnetic radiation, having wavelengths from one to hundredths of an angstrom λ’=~ 1-0.01 Å=10 -10 -10 -12 m. The energy of γ-rays reaches millions of eV.

W γ ~ MeV

1eV=1.6 10 -19 J

A nucleus undergoing radioactive decay, as a rule, turns out to be excited, and its transition to the ground state is accompanied by the emission of a γ - photon. In this case, the energy of the γ-photon is determined by the condition

where E 2 and E 1 is the energy of the nucleus.

E 2 - energy in the excited state;

E 1 - energy in the ground state.

The absorption of γ-rays by matter is due to three main processes:

photoelectric effect (with hv < l MэB);
the formation of electron-positron pairs;

scattering (Compton effect) -

Absorption of γ-rays occurs according to Bouguer's law:

where μ is a linear attenuation coefficient, depending on the energies of γ rays and the properties of the medium;

І 0 is the intensity of the incident parallel beam;

Iis the intensity of the beam after passing through a substance of thickness X cm.

γ-rays are one of the most penetrating radiations. For the hardest rays (hvmax) the thickness of the half-absorption layer is 1.6 cm in lead, 2.4 cm in iron, 12 cm in aluminum, and 15 cm in earth.

§2 Basic law of radioactive decay.

Number of decayed nucleidN proportional to the original number of cores N and decay timedt, dN~ N dt. The basic law of radioactive decay in differential form:

The coefficient λ is called the decay constant for a given type of nuclei. The "-" sign means thatdNmust be negative, since the final number of undecayed nuclei is less than the initial one.

therefore, λ characterizes the fraction of nuclei decaying per unit time, i.e., determines the rate of radioactive decay. λ does not depend on external conditions, but is determined only by the internal properties of the nuclei. [λ]=s -1 .

The basic law of radioactive decay in integral form

where N 0 - the initial number of radioactive nuclei att=0;

N- the number of non-decayed nuclei at a timet;

λ is the radioactive decay constant.

In practice, the decay rate is judged using not λ, but T 1/2 - the half-life - the time during which half of the original number of nuclei decays. Relationship T 1/2 and λ

T 1/2 U 238 = 4.5 10 6 years, T 1/2 Ra = 1590 years, T 1/2 Rn = 3.825 days The number of decays per unit time A \u003d -dN/ dtis called the activity of a given radioactive substance.

From

follows,

[A] \u003d 1 Becquerel \u003d 1 disintegration / 1 s;

[A] \u003d 1Ci \u003d 1Curie \u003d 3.7 10 10 Bq.

Law of activity change

where A 0 = λ N 0 - initial activity at timet= 0;

A - activity at a timet.

with parameters b v , b s b k , k v ,k s ,k k ,B s B k C1. which is unusual in that it contains a term with Z to a positive fractional power.
On the other hand, attempts were made to arrive at mass formulas based on the theory of nuclear matter or on the basis of the use of effective nuclear potentials. In particular, Skyrme's effective potentials were used in works where not only spherically symmetric nuclei were considered, but also deformations of the axial type were taken into account. However, the accuracy of the results of calculations for the masses of nuclei is usually lower than in the macro-macroscopic method.
All the works discussed above and the mass formulas proposed in them were oriented towards a global description of the entire system of nuclei by means of smooth functions of nuclear variables (A, Z, etc.) with an eye to predicting the properties of nuclei in distant regions (near and beyond the nucleon stability limit, and also superheavy nuclei). Global-type formulas also include shell corrections and sometimes contain a significant number of parameters, but despite this, their accuracy is relatively low (on the order of 1 MeV), and the question arises of how optimally they, and especially their macroscopic (liquid-droplet) part, reflect the requirements of the experiment.
In this regard, in the work of Kolesnikov and Vymyatnin, the inverse problem of finding the optimal mass formula was solved, based on the requirement that the structure and parameters of the formula provide the smallest standard deviation from the experiment and that this be achieved with the minimum number of parameters n, i.e. so that both and the quality index of the formula Q = (n + 1) are minimal. As a result of selection among a fairly wide class of considered functions (including those used in the published mass formulas), the formula (in MeV) was proposed as the optimal option for the binding energy:

B(A,Z) = 13.0466A - 33.46A 1/3 - (0.673+0.00029A)Z 2 /A 1/3 - (13.164 + 0.004225A)(A-2Z) 2 /A -
– (1.730- 0.00464A)|A-2Z| + P(A) + S(Z,N),

(12)

where S(Z,N) is the simplest (two-parameter) shell correction, and P(A) is the parity correction (see (6)) with a maximum deviation of ~2.5 MeV (according to the tables). At the same time, it gives a better (compared to other global type formulas) description of isobars that are far from the beta stability line and the Z*(A) line, and the Coulomb energy term is consistent with the sizes of nuclei from electron scattering experiments. Instead of the usual term proportional to A 2/3 (usually identified with the “surface” energy), the formula contains a term proportional to A 1/3 (present, by the way, under the name of the term “curvature” in many mass formulas, for example,). The calculation accuracy of B(A,Z) can be increased by introducing more parameters, but the quality of the formula deteriorates (Q increases). This may mean that the class of functions used in was not sufficiently complete, or that another (non-global) approach should be used to describe the masses of nuclei.

4. Local description of the binding energies of nuclei

Another way of constructing mass formulas is based on a local description of the nuclear energy surface. First of all, we note the difference relations that relate the masses of several (usually six) neighboring nuclei with the numbers of neutrons and protons Z, Z + 1, N, N + 1. They were originally proposed by Harvey and Kelson and further refined in the works of other authors (for example, in). The use of difference relations makes it possible to calculate the masses of unknown, but close to known, nuclei with a high accuracy of the order of 0.1 - 0.3 MeV. However, you have to enter a large number of parameters. For example, in order to calculate the masses of 1241 nuclei with an accuracy of 0.2 MeV, it was necessary to enter 535 parameters. The disadvantage is also that when crossing the magic numbers, the accuracy decreases significantly, which means that the predictive power of such formulas for any extrapolations farther away is not great.
Another version of the local description of the nuclear energy surface is based on the idea of nuclear shells. According to the many-particle model of nuclear shells, the interaction between nucleons is not entirely reduced to the creation of some average field in the nucleus. In addition to it, one should also take into account an additional (residual) interaction, which manifests itself, in particular, in the form of a spin interaction and in the parity effect. As shown by de Shalit, Talmy and Tyberger, within the filling of the same neutron (sub)shell, the neutron binding energy (B n) and similarly (within the filling of the proton (sub)shell) the proton binding energy (B p) change linearly depending on the number of neutrons and protons, and the total binding energy is quadratic function Z and N. An analysis of experimental data on the binding energies of nuclei in works leads to a similar conclusion. Moreover, it turned out that this is true not only for spherical nuclei (as was assumed by de Chalit et al.), but also for regions of deformed nuclei.
By simply partitioning the system of nuclei into regions between magic numbers, one can (as Lévy showed) describe the binding energies by quadratic functions of Z and N at least as well as by using global mass formulas. A more theoretically serious approach based on the works was taken by Zeldes. He also divided the system of nuclei into regions between the magic numbers 2, 8, 20, 28, 50, 82, 126, but the interaction energy in each of these regions included not only the pair interaction of nucleons quadratic in Z and N and the Coulomb interaction, but also called deformation interaction containing symmetric polynomials in Z and N of degree higher than the second.
This made it possible to significantly improve the description of the binding energies of nuclei, although it led to an increase in the number of parameters. Thus, to describe 1280 nuclei with = 0.278 MeV, it was necessary to introduce 178 parameters. Nevertheless, neglect of subshells led to rather significant deviations near Z = 40 (~1.5 MeV), near N = 50 (~0.6 MeV) and in the region of heavy nuclei (>0.8 MeV). In addition, difficulties arise if one wishes to harmonize the values of the formula parameters in different regions from the condition of continuity of the energy surface at the boundaries.
In this regard, it seems obvious that the effect of subshells should be taken into account. However, at a time when the main magic numbers are established reliably both theoretically and experimentally, the question of submagic numbers turns out to be very confusing. In fact, there are no reliably established universally recognized submagic numbers (although in the literature, irregularities were noted in some properties of nuclei for nucleon numbers of 40, 56.64, and others). The reasons for relatively small violations of regularities can be different. For example, as noted by Goeppert-Meier and Jensen, the reason for the violation of the normal order of filling neighboring levels can be a difference in the magnitude of their angular momenta and, as a result, in the pairing energies. Another reason is the deformation of the core. Kolesnikov combined the problem of taking into account the effect of subshells with the simultaneous search for submagic numbers based on dividing the region of nuclei between neighboring magic numbers into such parts that, within each of them, the binding energies of nucleons (B n and B p) could be described by linear functions of Z and N, and provided that the total binding energy is continuous function everywhere, including at the borders of the regions. Accounting for subshells made it possible to reduce the root-mean-square deviation from the experimental values of binding energies to = 0.1 MeV, i.e., to the level of experimental errors. The division of the system of nuclei into smaller (submagic) regions between the main magic numbers leads to an increase in the number of intermagic regions and, accordingly, to the introduction of a larger number of parameters, but at the same time, the values of the latter in different regions can be coordinated from the energy surface continuity conditions at the boundaries of the regions and thereby reducing the number of free parameters.
For example, in the region of the heaviest nuclei (Z>82, N>126), when describing ~800 nuclei with = 0.1 MeV, due to the energy continuity conditions at the boundaries, the number of parameters decreased by more than one third (it became 136 instead of 226).
In accordance with this, the proton binding energy - the energy of proton attachment to the nucleus (Z, N) - within the same intermagic region can be written as:

(13)

where the index i determines the parity of the nucleus by the number of protons: i = 2 means Z is even, and i =1 - Z is odd, a i and b i are constants common for nuclei with different indices j, which determine the parity by the number of neutrons. In this case , where pp is the proton pairing energy, and , where Δ pn is the pn-interaction energy.
Similarly, the binding (attachment) energy of a neutron is written as:

(14)

where c i and d i are constants, , where δ nn is the neutron pairing energy, and , Z k and N l are the smallest of the (sub)magic numbers of protons and, accordingly, neutrons that limit the region (k, l).
In (13) and (14) the difference between the kernels of all four types of parity is taken into account: hh, chn, lf and nn. Ultimately, with such a description of the binding energies of nuclei, the energy surface for each type of parity is divided into relatively small pieces interconnected, i.e. becomes like a mosaic surface.

5. Line of beta - stability and binding energy of nuclei

Another possibility for describing the binding energies of nuclei in the regions between the main magic numbers is based on the dependence of the beta-decay energies of nuclei on their distance from the beta-stability line. It follows from the Bethe-Weizsacker formula that the isobaric sections of the energy surface are parabolas (see (9), (10)), and the beta-stability line, leaving the origin at large A, deviates more and more towards neutron-rich nuclei. However, the real beta-stability curve is straight line segments (see Fig. 3) with discontinuities at the intersection of the magic numbers of neutrons and protons. Linear dependency Z* from A also follows from the many-particle model of nuclear shells by de Chalit et al. Experimentally, the most significant breaks in the beta stability line (Δ Z * 0.5-0.7) occur at the intersection of magic numbers N, Z = 20, N = 28, 50, Z = 50, N and Z = 82, N = 126 ). Submagic numbers are much weaker. In the interval between the main magic numbers, the Z* values for the minimum energy of the isobars lie with a fairly good accuracy on the linearly averaged (straight) line Z*(A). For the region of the heaviest nuclei (Z>82, N>136) Z* is expressed by the formula (see)

As shown in , in each of the intermagic regions (that is, between the main magic numbers), the energies of beta-plus and beta-minus decay with good accuracy turn out to be linear function Z - Z * (A) . This is demonstrated in Fig. 5 for the region Z>82, N>126, where the dependence of the value + D on Z – Z*(A) is plotted, for the sake of convenience, kernels with even Z are chosen; D is the parity correction equal to 1.9 MeV for nuclei with even N (and Z) and 0.75 MeV for nuclei with odd N (and even Z). Given that for an isobar with an odd Z, the energy of beta-minus decay - equals with a minus sign the energy of beta-plus decay of an isobar with an even charge Z + 1, and (A, Z) = - (A, Z + 1), the graph in Fig. 5 covers all, without exception, the cores of the region Z>82, N>126 with both even and odd values of Z and N. In accordance with the above

= + k(Z * (A) – Z) - D ,

(16)

where k and D are constants for the area between the main magic numbers. In addition to the area Z>82, N>126, as shown in , similar linear dependences (15) and (16) are also valid for other areas identified by the main magic numbers.
Using formulas (15) and (16), it is possible to estimate the energy of beta decay of any (even so far inaccessible for experimental study) nucleus of the submagic region under consideration, knowing only its charge Z and mass number A. At the same time, the calculation accuracy for the region Z> 82, N>126, as a comparison with ~200 experimental values of the table shows, ranges from = 0.3 MeV for odd A to 0.4 MeV for even A at maximum deviations of the order of 0.6 MeV, i.e. higher than when using mass formulas of the global type . And this is achieved by using the minimum number of parameters (four in formula (16) and two more in formula (15) for the beta stability curve). Unfortunately, for superheavy nuclei, it is currently impossible to carry out a similar comparison due to the lack of experimental data.
Knowing the beta decay energies and plus the alpha decay energies for only one isobar (A, Z) allows us to calculate the alpha decay energies of other nuclei with the same mass number A, including those far enough from the beta stability line. This is especially important for the region of the heaviest nuclei, where alpha decay is the main source of information about the energies of nuclei. In the region Z > 82, the beta stability line deviates from the N = Z line along which alpha decay occurs so that the nucleus formed after the escape of an alpha particle approaches the beta stability line. For the beta stability line of the region Z > 82 (see (15)) Z * /A = 0.356, while for alpha decay Z/A = 0.5 . As a result, the core (A-4, Z-2) in comparison with the core (A, Z) is closer to the beta stability line by (0.5 - 0.356) . 4 = 0.576, and its beta decay energy becomes 0.576 . k = 0.576 . 1.13 = 0.65 MeV less compared to the nucleus (A,Z). Hence, from the energy (,) cycle, which includes the nuclei (A,Z), (A,Z+1), (A-4,Z-2), (A-4,Z-1), it follows that the energy of alpha decay Q a of the nucleus (A, Z+1) must be 0.65 MeV greater than the isobar (A, Z). Thus, during the transition from the isobar (A, Z) to the isobar (A, Z + 1), the energy of alpha decay increases by 0.65 MeV. With Z>82, N>126, this is on average very well justified for all cores (regardless of parity). The root-mean-square deviation of the calculated Q a for 200 nuclei of the region under consideration is only 0.15 MeV (and the maximum is about 0.4 MeV), despite the fact that the submagic numbers N = 152 for neutrons and Z = 100 for protons intersect.

To complete the overall picture of the change in the energies of alpha decay of nuclei in the region heavy elements on the basis of experimental data on alpha decay energies, the value of alpha decay energy for fictitious nuclei lying on the beta stability line, Q * a , was calculated. The results are presented in Fig.6. As can be seen from fig. 6, the overall stability of nuclei with respect to alpha decay after lead rapidly increases (Q * a falls) to A235 (uranium region), after which Q * a gradually begins to increase. In this case, 5 areas of approximately linear change in Q * a can be distinguished:

Calculation of Q a by the formula

6. Heavy nuclei, superheavy elements

AT last years significant progress has been made in the study of superheavy nuclei; were synthesized isotopes of elements with serial numbers from Z = 110 to Z = 118. In this case, the experiments carried out at the JINR in Dubna played a special role, where the 48Ca isotope containing a large excess of neutrons was used as a bombarding particle. This made it possible to synthesize nuclides closer to the beta-stability line and therefore longer-lived and decaying with lower energy. The difficulty, however, is that the chain of alpha decay of the nuclei formed as a result of irradiation does not end with known nuclei, and therefore the identification of the resulting reaction products, especially their mass number, is not unambiguous. In this regard, as well as to understand the properties of superheavy nuclei located on the boundary of the existence of elements, it is necessary to compare the results of experimental measurements with theoretical models.
Orientation could be given by the systematics of the - and - decay energies, taking into account new data on transfermium elements. However, the works published so far have been based on rather old experimental data of almost twenty years ago and are therefore of little use.
As for theoretical works, it should be recognized that their conclusions are far from unambiguous. First of all, it depends on which theoretical model of the nucleus is chosen (for the region of transfermian nuclei, the macro-micro model, the Skyrme-Hartree-Fock method and the relativistic mean field model are considered the most acceptable). But even within the same model, the results depend on the choice of parameters and on the inclusion of certain correction terms. Accordingly, increased stability is predicted at (and near) different magic numbers of protons and neutrons.

So Möller and some other theorists came to the conclusion that in addition to the well-known magic numbers (Z, N = 2, 8, 20, 28, 50, 82 and N = 126), the number Z = 114 should also appear as a magic number in the region of transfermium elements, and near Z = 114 and N = 184 there should be an island of relatively stable nuclei (some exalted popularizers hastened to fantasize about new allegedly stable superheavy nuclei and new energy sources associated with them). However, in fact, in the works of other authors, the magicity of Z = 114 is rejected and instead Z = 126 or 124 are declared to be the magic numbers of protons.
On the other hand, in the works, it is stated that the magic numbers are N = 162 and Z = 108. However, the authors of the work do not agree with this. Opinions of theorists also differ as to whether nuclei with numbers Z = 114, N = 184 and with numbers Z = 108, N = 162 should be spherically symmetric or they can be deformed.
As for the experimental verification of theoretical predictions about the magic of the number of protons Z = 114, then in the experimentally achieved region with neutron numbers from 170 to 176, the isolation of the isotopes of element 114 (in the sense of their greater stability) in comparison with the isotopes of other elements is not visually observed.

The aforesaid is illustrated in 7, 8, and 9. Figures 7, 8, and 9, in addition to the experimental values of the alpha-decay energies Q a of transfermium nuclei plotted by dots, show the results of theoretical calculations in the form of curved lines. Figure 7 shows the results of calculations for the macro-micro model of work , for elements with even Z, found taking into account the multipolarity of deformations up to the eighth order.
On fig. Figures 8 and 9 show the results of Q a calculations using the optimal formula for even and odd elements, respectively. Note that parameterization in c was carried out taking into account experiments performed 5–10 years ago, while in c the parameters have not been corrected since the publication of the work.
The general nature of the description of transfermium nuclei (with Z > 100) in and is approximately the same - the root-mean-square deviation is 0.3 MeV, however, in for nuclei with N > 170, the dependence of the Q a (N) curve differs from the experimental one, while in full compliance is achieved if the existence of the N = 170 subshell is taken into account.
It should be noted that the mass formulas in a number of papers published in recent years also give a fairly good description of the energies Q a for the nuclei of the transfermium region (0.3-0.5 MeV), and in the paper the discrepancy in Q a for the chain of the heaviest nuclei 294 118 290 116 286 114 turns out to be within the limits of experimental errors (true for the entire region of transfermium nuclei 0.5 MeV, i.e. worse than, for example, in ).
Above in section 5, a simple method for calculating the alpha decay energies of nuclei with Z>82 was described, based on the use of the dependence of the alpha decay energy Q a of the nucleus (A, Z) on the distance from the beta stability line Z-Z * , which is expressed by the formulas ( 22,23). The Z * values necessary for calculating Q a (A, Z) are found by formula (15), and Q a * - from Fig. 6 or by formulas (17-21). For all nuclei with Z>82, N>126, the accuracy of calculating the alpha decay energies is 0.2 MeV, i.e. at least not worse than for mass formulas of the global type. This is illustrated in Table. 1, where the results of Q a calculation by formulas (22.23) are compared with the experimental data contained in the tables of isotopes. In addition, in Table. Table 2 shows the results of calculations of Q a for nuclei with Z > 104, the discrepancy between which and recent experiments remains within the same 0.2 MeV.
As for the magic number Z = 108, as can be seen from Figs. 7, 8 and 9, there is no significant effect of increasing the stability with this number of protons. It is currently difficult to judge how significant the N = 162 shell effect is due to the lack of reliable experimental data. True, in the work of Dvorak et al., using the radiochemical method, a product was isolated that decays by emitting alpha particles with quite big time life and a relatively low decay energy, which was identified with the 270 Hs nucleus with the number of neutrons N = 162 (the corresponding value of Q a in Figs. 7 and 8 is marked with a cross). However, the results of this work differ from the conclusions of other authors.
Thus, we can state that so far there are no serious grounds to assert the existence of new magic numbers in the region of heavy and superheavy nuclei and the increase in the stability of nuclei associated with them, except for the previously established subshells N = 152 and Z = 100. As for the magic number Z = 114, it certainly cannot be completely ruled out (although this does not seem very likely) that the effect of the shell Z = 114 near the center of the island of stability (i.e., near N = 184) could be significant. However this area is not yet available for experimental study.
To find submagic numbers and related effects of filling subshells, the method described in Section 4 seems logical. p change linearly depending on the number of neutrons N and the number of protons Z, and the whole system of nuclei is divided into intermagic regions, inside which formulas (13) and (14) are valid. The (sub)magic number is the boundary between two regions of regular (linear) variation B n and B p , and the filling effect of the neutron (proton) shell can be understood as the energy difference B n (B p) during the transition from one region to another. Submagic numbers are not preassigned, but are found as a result of agreement with the experimental data of linear formulas (11) and (12) for B n and B p when the system of nuclei is divided into regions, see Section 4, and also .

As can be seen from formulas (11) and (12), B n and B p are functions of Z and N. To get an idea of how B n changes depending on the number of neutrons and what is the effect of filling different neutron (sub)shells, it is convenient bring the binding energies of neutrons to the line of beta stability. To do this, for each fixed value of N, we found B n * B n (N,Z*(N)), where (according to (15)) Z * (N) = 0.5528Z + 14.1. The dependence of B n * on N for nuclei of all four parity types is shown in Fig. 10 for nuclei with N > 126. Each of the points in Fig. 10 corresponds to the average value of B n * values reduced to the beta-stability line for nuclei of the same parity with the same N.
As can be seen from Fig. 10, B n * experiences jumps not only at the well-known magic number N = 126 (drop by 2 MeV) and at the submagic number N = 152 (drop by 0.4 MeV for nuclei of all parity types), but also at N = 132, 136, 140, 144, 158, 162, 170. The nature of these subshells turns out to be different. The point is that the magnitude and even the sign of the shell effect turns out to be different for nuclei of different parity types. So when passing through N = 132 B n * decreases by 0.2 MeV for nuclei with odd N, but increases by the same amount for nuclei with even N . The energy C averaged over all types of parity (line C in Fig. 10) does not experience a discontinuity. Rice. 10 allows you to see what happens when the other submagic numbers listed above intersect. It is significant that the average energy C either does not experience a discontinuity or changes by ~0.1 MeV in the direction of decreasing (at N = 162) or increasing (at N = 158 and N = 170).
The general trend in the change in energies B n * is as follows: after filling the shell with N = 126, the binding energies of neutrons increase to N = 140, so that the average energy C reaches 6 MeV, after which it decreases by about 1 MeV for the heaviest nuclei.

The energies of protons reduced to the beta-stability line B p * B p (Z, N*(Z)) were found in a similar way, taking into account the formula N * (Z) = 1.809N – 25.6 (following from (15)) . The dependence of B p * on Z is shown in Fig.11. Compared to neutrons, the binding energies of protons experience sharper fluctuations when the number of protons changes. As can be seen from Fig. 11, the binding energies of protons B p * experience a break except for the main magic number Z = 82 (a decrease in B p * by 1.6 MeV) at Z = 100 , as well as at submagic numbers 88, 92, 104, 110. As in the case of neutrons, the intersection of proton submagic numbers leads to shell effects of different magnitude and sign. The average value of energy C does not change when crossing the number Z = 104, but decreases by 0.25 MeV when crossing the numbers Z = 100 and 92 and by 0.15 MeV at Z = 88 and increases by the same amount at Z = 110.
Figure 11 shows the general trend of changing B p * after filling the proton shell Z = 82 - this is an increase to uranium (Z = 92) and a gradual decrease with shell vibrations in the region of the heaviest elements. In this case, the average energy value changes from 5 MeV in the uranium region to 4 MeV for the heaviest elements, and at the same time, the proton pairing energy decreases,

Fig.12. Pairing energies nn , pp and np Z > 82, N > 126.

Rice. 13. B n as a function of Z and N.

As follows from Figs. 10 and 11, in the region of the heaviest elements, in addition to a general decrease in binding energies, there is a weakening of the binding of external nucleons to each other, which manifests itself in a decrease in the neutron pairing energy and the proton pairing energy, as well as the neutron-proton interaction. This is demonstrated explicitly in Figure 12.
For nuclei lying on the beta-stability line, the neutron pairing energy nn was determined as the difference between the energy of an even(Z)-odd(N) nucleus B n *(N) and the half-sum
(B n * (N-1) + B n * (N+1))/2 for even-even kernels; similarly, the proton pairing energy pp was found as the difference between the energy of the odd-even nucleus B p * (Z) and the half-sum (B p * (Z-1) + B p * (Z+1))/2 for even-even nuclei. Finally, the np-interaction energy np was found as the difference B n * (N) of the even-odd nucleus and B n * (N) of the even-even nucleus.
Figures 10,11 and 12, however, do not give a complete picture of how the binding energies of nucleons B n and B p (and everything connected with them) change depending on the ratio between the numbers of neutrons and protons. With this in mind, in addition to Fig. 10,11 and 12 for the sake of clarity (in accordance with formulas (13) and (14)) Fig.13, which shows a spatial picture of the binding energies of neutrons B n as a function of the number of neutrons N and protons Z, Let us note some general patterns, manifested in the analysis of the binding energies of the nuclei of the region Z>82 , N>126 , including in Fig.13 The energy surface B(Z,N) is continuous everywhere, including at the boundaries of the regions. The binding energy of neutrons B n (Z,N), which varies linearly in each of the intermagic regions, experiences a break only when crossing the boundary of the neutron (sub)shell, whereas when crossing the proton (sub)shell, only the slope B n /Z can change.
On the contrary, B p (Z,N) experiences a discontinuity only at the boundary of the proton (sub)shell, and at the boundary of the neutron (sub)shell, the slope of B p /N can only change. Within the intermagic region B, n increases with increasing Z and slowly decreases with increasing N; similarly, B p increases with increasing N and decreases with increasing Z. In this case, the change in B p occurs much faster than B n .
The numerical values of B p and B n are given in table. 3 , and the values of the parameters that determine them (see formulas (13) and (14)) - in Table 4. they are found as differences B* n for odd-even and even-even nuclei and, respectively, even-even and odd-odd nuclei in fig. 10 and as the difference B * p for even-odd and even-even and, respectively, odd-even and odd-odd nuclei in Fig.11.
The analysis of shell effects, the results of which are presented in Figs. 10-13, depend on the input experimental data - mainly on the energies of alpha decay Q a and a change in the latter could lead to a correction of the results of this analysis. This is especially true for the region Z > 110, N > 160, where conclusions were sometimes made on the basis of a single alpha decay energy. Regarding the Z area< 110, N < 160, где результаты экспериментальных измерений за последние годы практически стабилизировались, то результаты анализа, приведенные на рис. 10 и 11 практически совпадают с теми, которые были получены в двадцать и более лет назад.
This paper is a review of various approaches to the problem of nuclear binding energies with an assessment of their advantages and disadvantages. The work contains a fairly large amount of information about the works of various authors. Additional information can be obtained by reading the original papers, many of which are cited in the bibliography of this review, as well as in the materials of conferences on nuclear masses, in particular, conferences AF a MC (publications in ADNDT Nos. 13 and 17, etc.) and conferences on nuclear spectroscopy and nuclear structure, held in Russia. The tables of this work contain the results own assessments the author related to the problem of superheavy elements (SHE).
The author is deeply grateful to B.S. Ishkhanov, at whose suggestion this work was prepared, and also to Yu.Ts. experimental work held at FLNR JINR on the problem of SHE.

BIBLIOGRAPHY

By studying the composition of matter, scientists came to the conclusion that all matter consists of molecules and atoms. For a long time, the atom (translated from Greek as "indivisible") was considered the smallest structural unit of matter. However, further studies have shown that the atom has a complex structure and, in turn, includes smaller particles.

What is an atom made of?

In 1911, the scientist Rutherford suggested that the atom has a central part that has a positive charge. Thus, for the first time, the concept of the atomic nucleus appeared.

According to Rutherford's scheme, called the planetary model, an atom consists of a nucleus and elementary particles with a negative charge - electrons moving around the nucleus, just as the planets orbit around the sun.

In 1932, another scientist, Chadwick, discovered the neutron, a particle that has no electric charge.

According to modern ideas, the kernel corresponds planetary model suggested by Rutherford. The nucleus carries most of the atomic mass. It also has a positive charge. The atomic nucleus contains protons - positively charged particles and neutrons - particles that do not carry a charge. Protons and neutrons are called nucleons. Negatively charged particles - electrons - orbit around the nucleus.

The number of protons in the nucleus is equal to those moving in orbit. Therefore, the atom itself is a particle that does not carry a charge. If an atom captures foreign electrons or loses its own, then it becomes positive or negative and is called an ion.

Electrons, protons and neutrons are collectively referred to as subatomic particles.

The charge of the atomic nucleus

The nucleus has a charge number Z. It is determined by the number of protons that make up the atomic nucleus. Finding out this amount is simple: just refer to the periodic system of Mendeleev. The atomic number of the element to which an atom belongs is equal to the number of protons in the nucleus. Thus, if the chemical element oxygen corresponds to the serial number 8, then the number of protons will also be equal to eight. Since the number of protons and electrons in an atom is the same, there will also be eight electrons.

The number of neutrons is called the isotopic number and is denoted by the letter N. Their number may vary in an atom of the same chemical element.

The sum of protons and electrons in the nucleus is called the mass number of the atom and is denoted by the letter A. Thus, the formula for calculating the mass number looks like this: A \u003d Z + N.

isotopes

In the case when elements have an equal number of protons and electrons, but a different number of neutrons, they are called isotopes of a chemical element. There can be one or more isotopes. They are placed in the same cell of the periodic system.

Isotopes have great importance in chemistry and physics. For example, an isotope of hydrogen - deuterium - in combination with oxygen gives a completely new substance, which is called heavy water. It has a different boiling and freezing point than usual. And the combination of deuterium with another isotope of hydrogen - tritium leads to a thermonuclear fusion reaction and can be used to generate a huge amount of energy.

Mass of the nucleus and subatomic particles

The size and mass of atoms are negligible in the minds of man. The size of the nuclei is approximately 10 -12 cm. The mass of the atomic nucleus is measured in physics in the so-called atomic mass units - a.m.u.

For one a.m.u. take one twelfth of the mass of a carbon atom. Using the usual units of measurement (kilograms and grams), the mass can be expressed as follows: 1 a.m.u. \u003d 1.660540 10 -24 g. Expressed in this way, it is called the absolute atomic mass.

Despite the fact that the atomic nucleus is the most massive component of the atom, its dimensions relative to the electron cloud surrounding it are extremely small.

nuclear forces

Atomic nuclei are extremely stable. This means that protons and neutrons are held in the nucleus by some forces. These cannot be electromagnetic forces, since protons are like-charged particles, and it is known that particles with the same charge repel each other. The gravitational forces are too weak to hold the nucleons together. Consequently, the particles are held in the nucleus by a different interaction - nuclear forces.

Nuclear interaction is considered the strongest of all existing in nature. Therefore, this type of interaction between the elements of the atomic nucleus is called strong. It is present in many elementary particles, as well as electromagnetic forces.

Features of nuclear forces

Short action. Nuclear forces, in contrast to electromagnetic forces, manifest themselves only at very small distances comparable to the size of the nucleus.
Charge independence. This feature manifested in the fact that nuclear forces act equally on protons and neutrons.
Saturation. The nucleons of the nucleus interact only with a certain number of other nucleons.

Core binding energy

Something else is closely connected with the concept of strong interaction - the binding energy of nuclei. Nuclear binding energy is the amount of energy required to split an atomic nucleus into its constituent nucleons. It is equal to the energy required to form a nucleus from individual particles.

To calculate the binding energy of a nucleus, it is necessary to know the mass of subatomic particles. Calculations show that the mass of a nucleus is always less than the sum of its constituent nucleons. The mass defect is the difference between the mass of the nucleus and the sum of its protons and electrons. Using the relationship between mass and energy (E \u003d mc 2), you can calculate the energy generated during the formation of the nucleus.

The strength of the binding energy of the nucleus can be judged by the following example: the formation of several grams of helium produces the same amount of energy as the combustion of several tons of coal.

Nuclear reactions

The nuclei of atoms can interact with the nuclei of other atoms. Such interactions are called nuclear reactions. Reactions are of two types.

Fission reactions. They occur when heavier nuclei break down into lighter ones as a result of the interaction.
Synthesis reactions. The process is the reverse of fission: the nuclei collide, thereby forming heavier elements.

All nuclear reactions are accompanied by the release of energy, which is subsequently used in industry, in the military, in energy, and so on.

Having become acquainted with the composition of the atomic nucleus, we can draw the following conclusions.

An atom consists of a nucleus containing protons and neutrons, and electrons around it.
The mass number of an atom is equal to the sum of the nucleons of its nucleus.
Nucleons are held together by the strong force.
The enormous forces that give the atomic nucleus stability are called the binding energies of the nucleus.