How to calculate the mass of an atomic nucleus. How to find the mass of a nucleus. Heavy nuclei, superheavy elements

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 research showed that the atom has complex structure and, in turn, includes smaller particles.

What does an atom consist of?

In 1911, the scientist Rutherford suggested that the atom contains central part having a positive charge. This is how the concept of the atomic nucleus first appeared.

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

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

According to modern ideas, the core corresponds to the planetary model proposed by Rutherford. The nucleus carries most of the atomic mass. It also has 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 - move in 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 gains electrons from others or loses its own, it becomes positive or negative and is called an ion.

Electrons, protons and neutrons are collectively called subatomic particles.

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 easy: just contact periodic table Mendeleev. The atomic number of the element to which the atom belongs is equal to the number of protons in the nucleus. Thus, if the chemical element oxygen has an atomic number of 8, then the number of protons will also be 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 designated by the letter N. Their number can 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 = Z + N.

Isotopes

When elements have equal numbers of protons and electrons, but different numbers 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 table.

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

Mass of the nucleus and subatomic particles

The size and mass of atoms are negligible in human perception. The size of the nuclei is approximately 10 -12 cm. The mass of an atomic nucleus is measured in physics in the so-called atomic mass units - amu.

For one amu take one twelfth of the mass of a carbon atom. Using the usual units of measurement (kilograms and grams), mass can be expressed by the following equation: 1 amu. = 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 an atom, its size relative to the electron cloud surrounding it is extremely small.

Nuclear forces

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

Nuclear force is considered the strongest of all existing in nature. That's why this type interactions between the elements of the atomic nucleus are called strong. It is present in many elementary particles, just like electromagnetic forces.

Features of nuclear forces

  1. Short action. Nuclear forces, unlike electromagnetic ones, appear only at very small distances, comparable to the size of the nucleus.
  2. Charge independence. This feature manifests itself in the fact that nuclear forces act equally on protons and neutrons.
  3. Saturation. The nucleons of the nucleus interact only with a certain number of other nucleons.

Nuclear binding energy

With the concept strong interaction Another thing is closely related - the binding energy of nuclei. Nuclear bond energy refers to the amount of energy required to split an atomic nucleus into its constituent nucleons. It equals 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. A mass defect is the difference between the mass of a nucleus and the sum of its protons and electrons. Using the relationship between mass and energy (E=mc 2), one can calculate the energy generated during the formation of a nucleus.

The strength of the binding energy of a 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. There are two types of reactions.

  1. Fission reactions. They occur when heavier nuclei, as a result of interaction, decay into lighter ones.
  2. Synthesis reactions. The reverse process of fission: nuclei collide, thereby forming heavier elements.

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

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

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

Many years ago, people wondered what all substances were made of. The first who tried to answer it was the ancient Greek scientist Democritus, who believed that all substances consist of molecules. It is now known that molecules are built from atoms. Atoms are made up of even smaller particles. At the center of the 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 only calculations and knowledge of chemistry will help you find the mass of the nucleus. To do this, you need to determine the number of protons and neutrons in the nucleus. Look at the table values ​​of the masses of one proton and one neutron and find their total mass. This will be the mass of the core.

You can often come across the question of how to find the mass, knowing the speed. According to classical laws mechanics, mass does not depend on the speed of the body. After all, if a car starts to pick up speed as it starts moving, 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 relativistic increase in body weight. And it manifests itself when the speeds of bodies approach the speed of light. Modern charged 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 the world high technology, but at low speeds. Therefore, in order to know how to calculate the mass of matter, you do not need 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 scale. Therefore, there is another way to calculate mass from its density.

The air around us, the air that is so necessary for humanity, 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 atomic nuclei to the mass of the globe and even stars located at a distance of several hundred light years from us. Mass as physical quantity, is a measure of the inertia of a body. More massive bodies are said to be more inert, that is, they change their speed more slowly. Therefore, after all, speed and mass turn out to be interconnected. But main feature This value means that any body or substance has mass. There is no matter in the world that does not have mass!

Isogons. The nucleus of the hydrogen atom - proton (p) - is the simplest nucleus. Its positive charge is absolute value equal to the charge of the electron. The mass of a proton is 1.6726-10’2 kg. The proton as a particle that is part of atomic nuclei was discovered by Rutherford in 1919.

To experimentally determine the masses of atomic nuclei, they have been and are using mass spectrometers. The principle of mass spectrometry, first proposed by Thomson (1907), is to use the focusing properties of electric and magnetic fields in relation to beams of charged particles. The first mass spectrometers with sufficiently high resolution were designed in 1919 by F.U. Aston and A. Dempstr. The operating principle of the mass spectrometer is shown in Fig. 1.3.

Since atoms and molecules are electrically neutral, they must first be ionized. Ions are created in an ion source by bombarding vapors of the substance under study with fast electrons and then, after acceleration in an electric field (potential difference V) exit into the vacuum chamber, entering the area of ​​homogeneous magnetic field B. Under its influence, ions begin to move in a circle whose radius G can be found from the equality of the Lorentz force and centrifugal force:

Where M- ion mass. The speed of movement of ions v is determined by the relation


Rice. 1.3.

Accelerating potential difference U or magnetic field strength IN can be selected so that ions with the same masses fall into the same place on a photographic plate or other position-sensitive detector. Then, by finding the maximum of the mass spectrum signal and using formula (1.7), we can determine the mass of the ion M. 1

Excluding speed v from (1.5) and (1.6), we find that

The development of mass spectrometry technology made it possible to confirm the assumption made back in 1910 by Frederick Soddy that the fractional (in units of the mass of a hydrogen atom) atomic masses of chemical elements are explained by the existence isotopes- atoms with the same nuclear charge, but different masses. Thanks to Aston's pioneering research, it was established that most elements are indeed composed of a mixture of two or more naturally occurring isotopes. The exceptions are relatively few elements (F, Na, Al, P, Au, etc.), called monoisotopic. The number of natural isotopes of one element can reach 10 (Sn). In addition, as it turned out later, all elements without exception have isotopes that have the property of radioactivity. Majority radioactive isotopes not found in nature, they can only be obtained artificially. Elements with atomic numbers 43 (Tc), 61 (Pm), 84 (Po) and higher have only radioactive isotopes.

The international atomic mass unit (amu) accepted today in physics and chemistry is 1/12 of the mass of the most common carbon isotope in nature: 1 amu. = 1.66053873* 10 “kg. It is close to the atomic mass of hydrogen, although not equal to it. The mass of an electron is approximately 1/1800 amu. In modern mass spectromephs relative error mass measurements

AMfM= 10 -10, which makes it possible to measure mass differences at the level of 10 -10 amu.

Atomic masses of isotopes, expressed in amu, are almost exactly integers. Thus, each atomic nucleus can be assigned its mass number A(integer), for example Н-1, Н-2, Н-З, С-12, 0-16, Cl-35, С1-37, etc. The latter circumstance revived on a new basis interest in the hypothesis of W. Prout (1816), according to which all elements are built from hydrogen.

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 have been 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, effective Skyrme potentials were used in works, and not only spherically symmetric nuclei were considered, but also axial type deformations were taken into account. However, the accuracy of calculation results for nuclear masses usually turns out to be lower than in the macro-macroscopic method.
All the works discussed above and the mass formulas proposed in them were focused on the global description of the entire system of nuclei through 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 boundary, 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 (about 1 MeV), and the question arises of how optimally they, and especially their macroscopic (liquid-droplet) part, reflect the experimental requirements.
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 a minimum number of parameters n, i.e. so that both , and the quality indicator of the formula Q = (n + 1) are minimal. As a result of selection from a fairly wide class of functions considered (including those that were used in published mass formulas ah) as optimal option a formula was proposed for the binding energy (in MeV):

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)) The optimal formula (12) with 9 free parameters provides a root-mean-square deviation from the experimental values ​​= 1.07 MeV with a maximum deviation of ~2.5 MeV (according to tables). At the same time, it gives a better (compared to other global-type formulas) description of isobars distant from the beta stability line and the course of 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 “surface” energy), the formula contains a term proportional to A 1/3 (present, by the way, under the name of the “curvature” term in many mass formulas, for example in,). The accuracy of B(A,Z) calculations 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 was not complete enough, or that a different (non-global) approach should be used to describe nuclear masses.

4. Local description of nuclear binding energies

Another way to construct mass formulas is based on a local description of the nuclear energy surface. Let us first note the difference relations that connect 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 were later 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 the work, to calculate the masses of 1241 nuclei with an accuracy of 0.2 MeV, it was necessary to enter 535 parameters. Another disadvantage is that when crossing magic numbers, the accuracy decreases significantly, which means that the predictive power of such formulas for any distant extrapolations is small.
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 this, an additional (residual) interaction should be taken into account, which manifests itself in particular in the form of spin interaction and the parity effect. As shown by de Shalit, Talmy and Tiberger, 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 the works leads to a similar conclusion. Moreover, it turned out that this is true not only for spherical nuclei (as assumed by de Shalit et al.), but also for regions of deformed nuclei.
By simply partitioning a system of nuclei into regions between magic numbers, it is possible (as Levy showed) to describe binding energies by quadratic functions Z and N at least as well as by using global mass formulas. A more theoretically serious, works-based approach 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 nuclear binding energies, 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 quite 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 when one wants to coordinate 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 it is necessary to take into account the subshell effect. However, while the main magic numbers are reliably established both theoretically and experimentally, the question of submagic numbers turns out to be very confusing. In fact, there are no reliably established generally accepted submagic numbers (although irregularities in some properties of nuclei at nucleon numbers of 40, 56,64 and others have been noted in the literature). The reasons for relatively small violations of regularities can be different. For example, as noted by Geppert-Mayer and Jensen, the reason for the violation of the normal order of filling neighboring levels may be a difference in the magnitude of their angular momenta and, as a consequence, in the pairing energies. Another reason is core deformation. Kolesnikov combined the problem of taking into account the effect of subshells with the simultaneous search for submagic numbers based on partitioning 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 Z and N, and provided that the total binding energy is continuous function everywhere, including at the borders of regions. Taking into account 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 control of a larger number of parameters, but at the same time, the values ​​of the latter in different regions can be coordinated from the conditions of continuity of the energy surface 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 taking into account the conditions of energy continuity 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 the proton joining the nucleus (Z,N) - within the same intermagic region can be written in the form:

(13)

where the index i determines the parity of the nucleus by the number of protons: i = 2 means Z - even, and i =1 - Z - 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 energy of proton pairing, and , where Δ pn is the energy of pn interaction.
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 bounding the region (k, l).
(13) and (14) take into account the difference between nuclei of all four parity types: hh, hn, nh 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 connected to each other, i.e. becomes like a mosaic surface.

5. Beta line - 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. From the Bethe-Weizsäcker formula it follows that the isobaric sections of the energy surface are parabolas (see (9), (10)), and the beta stability line, leaving the origin of coordinates at large A, deviates more and more towards neutron-rich nuclei. However, the real beta stability curve is straight segments (see Fig. 3) with breaks at the intersection of the magic numbers of neutrons and protons. Linear dependence Z* of A also follows from the many-particle nuclear shell model of de Shalit 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 less pronounced. In the interval between the main magic numbers, the values ​​of Z* for the minimum energy of the isobars fall with 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 was shown in, in each of the intermagic regions (i.e. between the main magic numbers), the beta plus and beta minus decay energies turn out to be with good accuracy 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 convenience, nuclei with even Z are selected; D is a 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). Considering that for an isobar with an odd Z, the energy of beta-minus decay is equal to the minus sign of 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 nuclei of the region Z>82, N>126 without exception, 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 region enclosed between the main magic numbers. In addition to the region Z>82, N>126, as shown in , similar linear dependencies (15) and (16) are also valid for other regions identified by the main magic numbers.
Using formulas (15) and (16), it is possible to estimate the beta decay energy of any (even not yet available for experimental study) nucleus of the submagic region under consideration, knowing only its charge Z and mass number A. Moreover, the calculation accuracy for the region Z>82, N>126, as a comparison with ~200 experimental values ​​in the table shows, ranges from = 0.3 MeV for odd A and up to 0.4 MeV for even A with maximum deviations of the order of 0.6 MeV, i.e. higher than when using mass formulas of global type. And this is achieved by using a 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 make 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 you to calculate the alpha decay energies of other nuclei with the same mass number A, including those quite distant 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 nuclear energies. 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 emission of an alpha particle approaches the beta stability line. For the beta stability line of the region Z > 82 (cm (15)) Z * /A = 0.356, while for alpha decay Z/A = 0.5. As a result, the core (A-4, Z-2) compared to the core (A,Z) turns out to be closer to the beta stability line by an amount of (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, including the nuclei (A,Z), (A,Z+1), (A-4,Z-2), (A-4,Z-1) it follows that the alpha decay energy Q a of the nucleus (A,Z+1) should be 0.65 MeV greater than the isobar (A,Z). Thus, when going from isobar (A,Z) to isobar (A,Z+1), the alpha decay energy increases by 0.65 MeV. At Z>82, N>126 this is on average very well justified for all cores (regardless of parity). The standard deviation of the calculated Q a for 200 nuclei in 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 changes in alpha decay energies of nuclei in the region heavy elements Based on experimental data on alpha decay energies, the alpha decay energy value was calculated for fictitious nuclei lying on the beta stability line, Q * a. 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 increases rapidly (Q * a falls) until 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 using the formula

6. Heavy nuclei, superheavy elements

IN last years significant progress has been made in the study of superheavy nuclei; Isotopes of elements with serial numbers from Z = 110 to Z = 118 were synthesized. In this case, a special role was played by experiments carried out at JINR in Dubna, where the 48 Ca 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 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 border of the existence of elements, it is necessary to compare the results experimental measurements with theoretical models.
Orientation could be given by a systematics of - and - decay energies, taking into account new data on transfermium elements. However, the works published to date were based on rather old experimental data from almost twenty years ago and therefore turn out to be of little use.
As for theoretical works, it should be recognized that their conclusions are far from unambiguous. First of all, it depends on what theoretical model of the nucleus is chosen (for the region of transfermium 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 supposedly stable superheavy nuclei and new energy sources associated with them). However, in fact, in the works of other authors, the magic of Z = 114 is rejected and instead Z = 126 or 124 are declared to be the magic numbers of protons.
On the other hand, the works claim that the numbers N = 162 and Z = 108 are magic numbers. However, the authors of the work do not agree with this. Theorists also differ in their opinions as to whether nuclei with numbers Z = 114, N = 184 and with numbers Z = 108, N = 162 should be spherically symmetrical or whether they can be deformed.
As for the experimental verification of theoretical predictions about the magic 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) compared to the isotopes of other elements is not visually observed.

This is illustrated in 7, 8 and 9. In Figs 7, 8 and 9, in addition to the experimental values ​​of the alpha decay energies Q a of transfermium nuclei plotted as dots, the results of theoretical calculations are shown in the form of curved lines. Figure 7 shows the results of calculations using the macro-micro model of work, for elements with even Z, found taking into account the multipolarity of deformations up to the eighth order.
In Fig. 8 and 9 present the results of calculations of Q a using the optimal formula for, respectively, even and odd elements. Note that the parameterization was carried out taking into account experiments performed 5-10 years ago, while the parameters have not been adjusted since the publication of the work.
General character descriptions of transfermium nuclei (with Z > 100) in and is approximately the same - the standard deviation is 0.3 MeV, however in for nuclei with N > 170 the course of the dependence of the Q a (N) curve differs from the experimental one, while in full agreement is achieved if we take into account the existence of the subshell N = 170.
It should be noted that the mass formulas in a number of works published in recent years also give a fairly good description of the energies Q a for nuclei in the transfermium region (0.3-0.5 MeV), and in the work there is a 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 (though for the entire region of transfermium nuclei 0.5 MeV, i.e. worse than, for example, in ).
Above in Section 5, a simple method was described for calculating the alpha decay energies of nuclei with Z>82, 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 using formula (15), and Q a * from Fig. 6 or using formulas (17-21). For all nuclei with Z>82, N>126, the accuracy of calculating alpha decay energies turns out to be 0.2 MeV, i.e. at least no worse than for mass formulas of global type. This is illustrated in table. 1, where the results of calculating Q a using formulas (22,23) are compared with experimental data contained in the isotope tables. In addition, in table. Figure 2 shows the results of calculations of Q a for nuclei with Z > 104, the discrepancy with 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 stability at this number of protons. It is currently difficult to judge how significant the effect of the N = 162 shell 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 a rather long lifetime and 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 on Fig. 7 and 8 are marked with a cross). However, the results of this work differ from the conclusions of other authors.
Thus, it can be stated 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 associated increase in the stability of nuclei other than the previously established subshells N = 152 and Z = 100. As for the magic number Z = 114, then, of course, it cannot be completely ruled out (although this does not seem very likely) that the effect of the Z = 114 shell 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 the associated effects of filling subshells, the method described in Section 4 seems logical. As was shown in (see above - Section 4), it is possible to identify regions of the nuclear system within which the binding energies of neutrons B n and the binding energies of protons B p change linearly depending on the number of neutrons N and the number of protons Z, and the entire system of nuclei is divided into intermagic regions, within which formulas (13) and (14) are valid. The (sub)magic number can be called the boundary between two regions of regular (linear) change B n and B p , and the effect of filling the neutron (proton) shell is understood as the energy difference B n (B p) during the transition from one region to another. Submagic numbers are not specified in advance, but are found as a result of agreement with 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, as well as .

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 the effect of filling different neutron (sub)shells is, it turns out to be convenient bring the neutron binding energies to the beta stability line. 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 presented 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 fact is that the magnitude and even sign of the shell effect turns out to be different for nuclei various types parity. 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 average energy C for all parity types (line C in Fig. 10) does not experience a discontinuity. Rice. 10 allows us to trace 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 towards a decrease (at N = 162) or an increase (at N = 158 and N = 170).
The general trend of changes in the energies of B n * is as follows: after filling the shell 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 approximately 1 MeV for the heaviest nuclei.

In a similar way, the energies of protons reduced to the beta stability line B p * B p (Z, N*(Z)) were found taking into account (following from (15)) the formula N * (Z) = 1.809N – 25.6. The dependence of B p * on Z is presented 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 discontinuity 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 changes in 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 region of uranium 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, the bond between external nucleons weakens, which manifests itself in a decrease in the neutron pairing energy and proton pairing energy, as well as in the neutron-proton interaction. This is demonstrated explicitly in Fig. 12.
For nuclei lying on the beta stability line, the neutron pairing energy nn was determined as the difference between the energy of the even (Z)-odd (N) nucleus B n *(N) and half the sum
(B n * (N-1) + B n * (N+1))/2 for even-even nuclei; 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 between B n * (N) of the even-odd nucleus and B n * (N) of the even-even nucleus.
Figures 10, 11 and 12 do not, however, 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. Taking this into account, in addition to Fig. 10, 11 and 12, for clarity purposes, is shown (in accordance with formulas (13) and (14)) Fig. 13, which shows the 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 nuclei in 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 neutron binding energy B n (Z,N), which varies linearly in each of the intermagic regions, experiences a discontinuity only when crossing the boundary of the neutron (sub)shell, while 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)) are in Table 4. The values ​​of n 0 n 0 nn, as well as p 0 n and p 0 nn are not given in Table 1, but they are found as the differences B* n for odd-even and even-even nuclei and, accordingly, even-even and odd-odd nuclei in Fig. 10 and as the differences 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 Fig. 10-13, depends on the input experimental data - mainly on the alpha decay energies Q a and a change in the latter could lead to correction of the results of this analysis. This is especially true in the region Z > 110, N > 160, where conclusions were sometimes drawn based on 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 work 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 works, many of which are cited in the list of references of this review, as well as in the proceedings of conferences on nuclear mass, in particular the AF and MS conferences (publications in ADNDT No. 13 and 17, etc.) and conferences on nuclear spectroscopy and nuclear structure, carried out in Russia. The tables in this work contain the results of the author’s own assessments related to the problem of superheavy elements (SHE).
The author is deeply grateful to B.S. Ishkhanov, at whose suggestion this work was prepared, as well as to Yu.Ts. Oganesyan and V.K. Utenkov for the latest information about experimental work conducted at FLNR JINR on the problem of STE.

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Atomic mass is the sum of the masses of all protons, neutrons and electrons that make up an atom or molecule. Compared to protons and neutrons, the mass of electrons is very small, so it is not taken into account in calculations. Although this is not formally correct, the term is often used to refer to the average atomic mass of all isotopes of an element. It's actually relative atomic mass, also called atomic weight element. Atomic weight is the average of the atomic masses of all isotopes of an element found in nature. Chemists must differentiate between these two types of atomic mass when doing their work—an incorrect atomic mass value can, for example, result in an incorrect result for the yield of a reaction.

Steps

Finding atomic mass from the periodic table of elements

    Learn how atomic mass is written. Atomic mass, that is, the mass of a given atom or molecule, can be expressed in standard SI units - grams, kilograms, and so on. However, because atomic masses expressed in these units are extremely small, they are often written in unified atomic mass units, or amu for short. – atomic mass units. One atomic mass unit is equal to 1/12 the mass of the standard isotope carbon-12.

    • The atomic mass unit characterizes the mass one mole of a given element in grams. This value is very useful in practical calculations, since it can be used to easily convert the mass given quantity atoms or molecules of a given substance in moles, and vice versa.

  1. Find the atomic mass in the periodic table. Most standard periodic tables contain the atomic masses (atomic weights) of each element. Typically, they are listed as a number at the bottom of the element cell, below the letters representing the chemical element. Usually this is not a whole number, but a decimal fraction.

    Remember that the periodic table gives the average atomic masses of elements. As noted earlier, the relative atomic masses given for each element in the periodic table are the average of the masses of all isotopes of the atom. This average value is valuable for many practical purposes: for example, it is used in calculating the molar mass of molecules consisting of several atoms. However, when you are dealing with individual atoms, this value is usually not enough.

    • Since the average atomic mass is an average of several isotopes, the value shown in the periodic table is not accurate the value of the atomic mass of any single atom.
    • The atomic masses of individual atoms must be calculated taking into account exact number protons and neutrons in a single atom.

    Calculation of the atomic mass of an individual atom

    1. Find the atomic number of a given element or its isotope. Atomic number is the number of protons in the atoms of an element and never changes. For example, all hydrogen atoms, and only they have one proton. The atomic number of sodium is 11 because it has eleven protons in its nucleus, while the atomic number of oxygen is eight because it has eight protons in its nucleus. You can find the atomic number of any element in the periodic table - in almost all its standard versions, this number is indicated above the letter designation of the chemical element. The atomic number is always a positive integer.

      • Suppose we are interested in the carbon atom. Carbon atoms always have six protons, so we know that its atomic number is 6. In addition, we see that in the periodic table, at the top of the cell with carbon (C) is the number "6", indicating that the atomic carbon number is six.
      • Note that the atomic number of an element is not uniquely related to its relative atomic mass in the periodic table. Although, especially for the elements at the top of the table, it may appear that an element's atomic mass is twice its atomic number, it is never calculated by multiplying the atomic number by two.

    2. Find the number of neutrons in the nucleus. The number of neutrons can be different for different atoms of the same element. When two atoms of the same element with the same number of protons have different numbers of neutrons, they are different isotopes of that element. Unlike the number of protons, which never changes, the number of neutrons in the atoms of a given element can often change, so the average atomic mass of an element is written as a decimal fraction with a value lying between two adjacent whole numbers.

      Add up the number of protons and neutrons. This will be the atomic mass of this atom. Ignore the number of electrons that surround the nucleus - their total mass is extremely small, so they have virtually no effect on your calculations.

    Calculating the relative atomic mass (atomic weight) of an element

    1. Determine which isotopes are contained in the sample. Chemists often determine the isotope ratios of a particular sample using a special instrument called a mass spectrometer. However, in training, this data will be provided to you in assignments, tests, and so on in the form of values ​​​​taken from the scientific literature.

      • In our case, let's say that we are dealing with two isotopes: carbon-12 and carbon-13.

    2. Determine the relative abundance of each isotope in the sample. For each element, different isotopes occur in different ratios. These ratios are almost always expressed as percentages. Some isotopes are very common, while others are very rare—sometimes so rare that they are difficult to detect. These values ​​can be determined using mass spectrometry or found in a reference book.

      • Let's assume that the concentration of carbon-12 is 99% and carbon-13 is 1%. Other isotopes of carbon really exist, but in quantities so small that in this case they can be neglected.

    3. Multiply the atomic mass of each isotope by its concentration in the sample. Multiply the atomic mass of each isotope by its percentage abundance (expressed as a decimal). To convert interest to decimal, simply divide them by 100. The resulting concentrations should always add up to 1.

      • Our sample contains carbon-12 and carbon-13. If carbon-12 makes up 99% of the sample and carbon-13 makes up 1%, then multiply 12 (atomic mass of carbon-12) by 0.99 and 13 (atomic mass of carbon-13) by 0.01.
      • The reference books give percentages based on the known quantities of all isotopes of a particular element. Most chemistry textbooks contain this information in a table at the end of the book. For the sample being studied, the relative concentrations of isotopes can also be determined using a mass spectrometer.

    4. Add up the results. Sum up the multiplication results you got in the previous step. As a result of this operation, you will find the relative atomic mass of your element - the average value of the atomic masses of the isotopes of the element in question. When considering an element as a whole, rather than a specific isotope of a given element, this is the value used.

      • In our example, 12 x 0.99 = 11.88 for carbon-12, and 13 x 0.01 = 0.13 for carbon-13. The relative atomic mass in our case is 11.88 + 0.13 = 12,01 .
    • Some isotopes are less stable than others: they break down into atoms of elements with fewer protons and neutrons in the nucleus, releasing particles that make up the atomic nucleus. Such isotopes are called radioactive.