Masses of atomic nuclei. How to find the mass of the nucleus What is the mass of the nucleus of an atom

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) the formula (in MeV) was proposed as an 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)) 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 a continuous function everywhere, including at the boundaries 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. The linear dependence of Z* on A also follows from the many-particle model of nuclear shells by 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 of 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 of 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.
Concerning theoretical works, then 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.
The general nature of the description 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, which appear when analyzing 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 Yu.Ts. Oganesyan and V.K. Utenkov for the latest information about the experimental work carried out at FLNR JINR on the problem of SHE.

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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 a. e.m., while the proton and neutron each have a mass of 1 atomic mass unit. 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 abundance 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 up the mass of all the protons and neutrons in the nucleus. You will get a lot of money.

Answer from Dashka[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 to the nearest whole number - this will be the mass of the atomic nucleus. Nuclear mass, or 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 a. e.m., while the proton and neutron each have a mass of 1 atomic mass unit. 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 abundance 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?

Core charge

The nucleus of any atom is positively charged. The carrier of positive charge is the proton. Since the charge of a proton is numerically equal to the charge of an electron $e$, we can write that the charge of the nucleus is equal to $+Ze$ ($Z$ is an integer that indicates the serial number of a chemical element in periodic table chemical elements D.I. Mendeleev). The number $Z$ also determines the number of protons in the nucleus and the number of electrons in the atom. Therefore it is called the atomic number of the nucleus. Electric charge is one of the main characteristics atomic nucleus, on which the optical, chemical and other properties of atoms depend.

Core mass

Another important characteristic of the nucleus is its mass. The mass of atoms and nuclei is usually expressed in atomic mass units (amu). It is customary to consider $1/12$ of the mass of a carbon nuclide $^(12)_6C$ as an atomic mass unit:

where $N_A=6.022\cdot 10^(23)\ mol^-1$ is Avogadro's number.

According to Einstein's relation $E=mc^2$, the mass of atoms is also expressed in energy units. Because the:

• proton mass $m_p=1.00728\ amu=938.28\ MeV$,
• neutron mass $m_n=1.00866\ amu=939.57\ MeV$,
• electron mass $m_e=5.49\cdot 10^(-4)\ amu=0.511\ MeV$,

As you can see, the mass of the electron is negligibly small in comparison with the mass of the nucleus, then the mass of the nucleus almost coincides with the mass of the atom.

Mass is different from whole numbers. Nuclear mass, expressed in amu. and rounded to a whole number is called the mass number, denoted by the letter $A$ and determines the number of nucleons in the nucleus. The number of neutrons in the nucleus is $N=A-Z$.

To designate nuclei, the symbol $^A_ZX$ is used, where $X$ means the chemical symbol of a given element. Atomic nuclei with the same number of protons but different mass numbers are called isotopes. In some elements, the number of stable and unstable isotopes reaches tens, for example, uranium has $14$ isotopes: from $^(227)_(92)U\ $ to $^(240)_(92)U$.

Most chemical elements existing in nature are a mixture of several isotopes. It is the presence of isotopes that explains the fact that some natural elements have masses that differ from whole numbers. For example, natural chlorine consists of $75\%$ $^(35)_(17)Cl$ and $24\%$ $^(37)_(17)Cl$, and its atomic mass is $35.5$ a.u. .m. in most atoms, except hydrogen, the isotopes have almost the same physical and Chemical properties. But behind their exclusively nuclear properties, isotopes differ significantly. Some of them can be stable, others - radioactive.

Nuclei with the same mass numbers, but different $Z$ values ​​are called isobars, for example, $^(40)_(18)Ar$, $^(40)_(20)Ca$. Nuclei with the same number of neutrons are called isotones. Among light nuclei there are so-called “mirror” pairs of nuclei. These are pairs of nuclei in which the numbers $Z$ and $A-Z$ are swapped. Examples of such nuclei could be $^(13)_6C\ $ and $^(13_7)N$ or $^3_1H$ and $^3_2He$.

Atomic nucleus size

Assuming the atomic nucleus to be approximately spherical, we can introduce the concept of its radius $R$. Note that in some nuclei there is a slight deviation from symmetry in the distribution of electric charge. In addition, atomic nuclei are not static, but dynamic systems, and the concept of the radius of a nucleus cannot be represented as the radius of a ball. For this reason, the size of the nucleus must be taken as the area in which nuclear forces manifest themselves.

When creating the quantitative theory of scattering of $\alpha $ - particles, E. Rutherford proceeded from the assumptions that the atomic nucleus and $\alpha $ - particle interact according to Coulomb's law, i.e. that the electric field around the nucleus has spherical symmetry. The scattering of an $\alpha $ particle occurs in full accordance with Rutherford's formula:

This occurs for $\alpha $ - particles whose energy $E$ is quite small. In this case, the particle is not able to overcome the Coulomb potential barrier and subsequently does not reach the region of action of nuclear forces. As the particle energy increases to a certain boundary value $E_(gr)$ $\alpha $ -- the particle reaches this boundary. Then in the scattering of $\alpha $ - particles there is a deviation from the Rutherford formula. From the relation

Experiments show that the radius $R$ of the nucleus depends on the number of nucleons that enter the nucleus. This dependence can be expressed by the empirical formula:

where $R_0$ is a constant, $A$ is a mass number.

The sizes of nuclei are determined experimentally by the scattering of protons, fast neutrons or high-energy electrons. There are a number of other indirect methods for determining the size of nuclei. They are based on the connection between the lifetime of $\alpha $ -- radioactive nuclei and the energy of $\alpha $ -- particles released by them; on the optical properties of so-called mesoatoms, in which one electron is temporarily captured by a muon; by comparing the binding energy of a pair of mirror atoms. These methods confirm the empirical dependence $R=R_0A^(1/3)$, and using these measurements the value of the constant $R_0=\left(1.2-1.5\right)\cdot 10^(-15) was established \ m$.

Note also that the unit of distance in atomic physics and particle physics is taken as the “Fermi” unit of measurement, which is equal to $(10)^(-15)\ m$ (1 f=$(10)^(-15)\ m )$.

The radii of atomic nuclei depend on their mass number and are in the range from $2\cdot 10^(-15)\ m\ to\\ 10^(-14)\ m$. if we express $R_0$ from the formula $R=R_0A^(1/3)$ and write it in the form $\left(\frac(4\pi R^3)(3A)\right)=const$, then we can see that each nucleon contains approximately the same volume. This means that the density of nuclear matter is approximately the same for all nuclei. Based on the existing data on the sizes of atomic nuclei, we find the average value of the density of nuclear matter:

As we can see, the density of nuclear matter is very high. This is due to the action of nuclear forces.

Energy of communication. Nuclear mass defect

When comparing the sum of the rest masses of the nucleons that form the nucleus with the mass of the nucleus, it was noticed that for all chemical elements the following inequality is true:

where $m_p$ is the mass of the proton, $m_n$ is the mass of the neutron, $m_я$ is the mass of the nucleus. The value $\triangle m$, which expresses the mass difference between the mass of nucleons that form the nucleus and the mass of the nucleus, is called the nuclear mass defect

Important information about the properties of the nucleus can be obtained without delving into the details of the interaction between the nucleons of the nucleus, based on the law of conservation of energy and the law of proportionality of mass and energy. Depending on how much as a result of any change in mass $\triangle m$ there is a corresponding change in energy $\triangle E$ ($\triangle E=\triangle mc^2$), then during the formation of a nucleus a certain amount of energy is released. According to the law of conservation of energy, the same amount of energy is needed to divide the nucleus into its constituent particles, i.e. move nucleons one from another at the same distances at which there is no interaction between them. This energy is called the binding energy of the nucleus.

If the nucleus has $Z$ protons and mass number $A$, then the binding energy is equal to:

Note 1

Note that this formula is not entirely convenient to use, because The tables do not list the masses of nuclei, but the masses that determine the masses of neutral atoms. Therefore, for the convenience of calculations, the formula is transformed in such a way that it includes the masses of atoms, not nuclei. For this purpose, on the right side of the formula we add and subtract the mass $Z$ of electrons $(m_e)$. Then

\c^2==\leftc^2.\]

$m_(()^1_1H)$ is the mass of the hydrogen atom, $m_a$ is the mass of the atom.

In nuclear physics, energy is often expressed in megaelectron volts (MeV). If we're talking about about the practical application of nuclear energy, it is measured in joules. In the case of comparing the energy of two nuclei, the mass unit of energy is used - the ratio between mass and energy ($E=mc^2$). A mass unit of energy ($le$) equals energy, which corresponds to a mass of one amu. It is equal to $931,502$ MeV.

Picture 1.

In addition to energy, the specific binding energy is important - the binding energy that falls on one nucleon: $w=E_(st)/A$. This value changes relatively slowly compared to the change in the mass number $A$, having an almost constant value of $8.6$ MeV in the middle part of the periodic system and decreases to its edges.

As an example, let us calculate the mass defect, binding energy and specific binding energy of the nucleus of a helium atom.

Mass defect

Binding energy in MeV: $E_(bv)=\triangle m\cdot 931.502=0.030359\cdot 931.502=28.3\ MeV$;

Specific binding energy: $w=\frac(E_(st))(A)=\frac(28.3\ MeV)(4\approx 7.1\ MeV).$

Atomic nucleus is the central part of an atom, consisting of protons and neutrons (together called nucleons).

The nucleus was discovered by E. Rutherford in 1911 while studying the transmission α -particles through matter. It turned out that almost the entire mass of the atom (99.95%) is concentrated in the nucleus. The size of the atomic nucleus is of the order of magnitude 10 -1 3 -10 - 12 cm, which is 10,000 times smaller than the size of the electron shell.

The planetary model of the atom proposed by E. Rutherford and his experimental observation of hydrogen nuclei knocked out α -particles from the nuclei of other elements (1919-1920), led the scientist to the idea of proton. The term proton was introduced in the early 20s of the XX century.

Proton (from Greek. protons- first, symbol p) is a stable elementary particle, the nucleus of a hydrogen atom.

Proton- a positively charged particle whose charge is absolute value equal to the charge of the electron e= 1.6 · 10 -1 9 Cl. The mass of a proton is 1836 times greater than the mass of an electron. Proton rest mass m r= 1.6726231 · 10 -27 kg = 1.007276470 amu

The second particle included in the nucleus is neutron.

Neutron (from lat. neutral- neither one nor the other symbol n) is an elementary particle that has no charge, i.e. neutral.

The mass of a neutron is 1839 times greater than the mass of an electron. The mass of a neutron is almost equal (slightly greater) to the mass of a proton: the rest mass of a free neutron m n= 1.6749286 · 10 -27 kg = 1.0008664902 a.m.u. and exceeds the mass of a proton by 2.5 times the mass of an electron. Neutron, along with proton under the general name nucleon is part of atomic nuclei.

The neutron was discovered in 1932 by E. Rutherford's student D. Chadwig during the bombardment of beryllium α -particles. The resulting radiation with high penetrating ability (overcame a barrier made of a lead plate 10-20 cm thick) intensified its effect when passing through a paraffin plate (see figure). An assessment of the energy of these particles from tracks in a cloud chamber made by the Joliot-Curie couple and additional observations made it possible to exclude the initial assumption that this γ -quanta. The greater penetrating ability of the new particles, called neutrons, was explained by their electrical neutrality. After all, charged particles actively interact with matter and quickly lose their energy. The existence of neutrons was predicted by E. Rutherford 10 years before the experiments of D. Chadwig. When hit α -particles into beryllium nuclei the following reaction occurs:

Here is the symbol for the neutron; its charge is zero, and its relative atomic mass is approximately equal to unity. Neutron is an unstable particle: a free neutron in a time of ~ 15 minutes. decays into a proton, electron and neutrino - a particle devoid of rest mass.

After the discovery of the neutron by J. Chadwick in 1932, D. Ivanenko and V. Heisenberg independently proposed proton-neutron (nucleon) model of the nucleus. According to this model, the nucleus consists of protons and neutrons. Number of protons Z coincides with the ordinal number of the element in D.I. Mendeleev’s table.

Core charge Q determined by the number of protons Z, included in the nucleus, and is a multiple of the absolute value of the electron charge e:

Q = +Ze.

Number Z called charge number of the nucleus or atomic number.

Mass number of the nucleus A is the total number of nucleons, i.e. protons and neutrons contained in it. The number of neutrons in the nucleus is indicated by the letter N. So the mass number is:

A = Z + N.

Nucleons (proton and neutron) are assigned a mass number equal to one, and an electron is assigned a mass number of zero.

The idea of ​​the composition of the nucleus was also facilitated by the discovery isotopes.

Isotopes (from Greek. isos- equal, identical and topoa- place) are varieties of atoms of the same chemical element, the atomic nuclei of which have same number proto-nov ( Z) and different numbers of neutrons ( N).

The nuclei of such atoms are also called isotopes. Isotopes are nuclides one element. Nuclide (from lat. nucleus- nucleus) - any atomic nucleus (respectively, an atom) with given numbers Z And N. The general designation of nuclides is……. Where X- symbol of a chemical element, A = Z + N- mass number.

Isotopes occupy the same place in the Periodic Table of Elements, which is where their name comes from. According to its nuclear properties (for example, the ability to enter into nuclear reactions) isotopes, as a rule, differ significantly. The chemical (and almost to the same extent physical) properties of isotopes are the same. This is explained by the fact that the chemical properties of an element are determined by the charge of the nucleus, since it is this charge that affects the structure of the electron shell of the atom.

The exception is the isotopes of light elements. Isotopes of hydrogen 1 N — protium, 2 N— deuterium, 3 N — tritium differ so greatly in mass that their physical and chemical properties are different. Deuterium is stable (i.e. not radioactive) and is included as a small impurity (1: 4500) in ordinary hydrogen. When deuterium combines with oxygen, heavy water is formed. At normal atmospheric pressure it boils at 101.2 °C and freezes at +3.8 °C. Tritium β -radioactive with a half-life of about 12 years.

All chemical elements have isotopes. Some elements have only unstable (radioactive) isotopes. Radioactive isotopes have been artificially obtained for all elements.

Isotopes of uranium. The element uranium has two isotopes - with mass numbers 235 and 238. The isotope is only 1/140th of the more common one.

§1 Charge and mass of 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. Carrier of 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 across energy shells and subshells significantly depends on their total number in the atom. Therefore, the nuclear charge largely determines the distribution of electrons among their states in the atom and the position of the element in the Mendeleev periodic table. The nuclear charge isqI = z· e, Where z-charge number of the nucleus, equal to the atomic number of the element in the periodic 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 hydrogen, is approximately 2.5·10 -4 the mass of the atoms. The mass of atoms is expressed in atomic mass units (amu). For a.u.m. assumed to be 1/12 the mass of a carbon atom.

1 amu =1.6605655(86)·10 -27 kg.

mI = m a - Z m e.

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 mass number m and denoted by the letter A. Chemical element designation: A- mass number, X - symbol of a chemical element,Z-charge number - serial number in the periodic table ():

Beryllium; Isotopes: , ", .

Core radius:

where A is the mass number.

§2 Composition of the core

Nucleus of a hydrogen atomcalled proton

mproton= 1.00783 amu , .

Hydrogen atom diagram

In 1932, a particle called a neutron was discovered, with a mass close to the mass of a proton (mneutron= 1.00867 amu) and has no 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 A. 3rd serial numberZdetermines the number of protons in the nucleus, the number of neutronsN =A - Z.

Elementary particles - protons and neutrons included into the core, got common name nucleons. The 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 with a charge+ e, And neutron with charge 0.

§3 Nuclear binding energy. Mass defect. Nuclear forces

Nuclear particles - protons and neutrons - are firmly held inside the nucleus, so very strong attractive forces act between them, capable of resisting the enormous repulsive forces between similarly charged protons. These special forces that arise at small distances between nucleons are called nuclear forces. Nuclear forces are not electrostatic (Coulomb).

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

a) these are short-range forces - manifesting themselves 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 - over-row independence of nuclear forces. The nuclear forces acting between a neutron and a proton, between two neutrons, and between two protons are equal. The proton and neutron are the same in relation to nuclear forces.

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

M I< Σ( m p + m n)

Mya - core mass

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

Magnitude

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

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

In general, the binding energy of a nucleus can be calculated using the formula

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

A- mass number (total number of nucleons in the nucleus);

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

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

§ 4 Nuclear reactions

Changes in nuclei when they interact with individual particles and with each other are usually called nuclear reactions.

The following are the most common nuclear reactions.

1. 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.

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

An example of a neutron capture reaction by cadmium

or phosphorus

1. Scattering. The intermediate nucleus emits a particle identical

with an attack, and it could be:

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

Inelastic scattering :

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

If the excitation energy is low, then the division 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 contributed by the neutron is high, then the excited nucleus begins to deform, a waist forms in it and, as a result, it splits into two fragments that fly apart at enormous speeds, and two neutrons are emitted
(Fig. 2).

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

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

RADIOACTIVITY

§1 Natural radioactivity

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

Types of radioactivity:

1. α-decay.

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

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

Bias rule for α decay:

2. β-decay.

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

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

This process occurs by converting one type of nucleon in 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 its charge (+e = 1.6·10 -19 C)

From the experiment it follows that during β - decay, isotopes lose the same amount of energy. Consequently, based on the law of conservation of energy, W. Pauli predicted that another light particle called an antineutrino would be ejected. An antineutrino has no charge or mass. Energy losses by β-particles when passing through matter are caused mainly by ionization processes. Part of the energy is lost to X-ray radiation when β-particles are decelerated by the nuclei of the absorbing substance. Since β - particles have low mass, a single charge and very high velocities, their ionizing ability is low (100 times less than that of α - particles), therefore, the penetrating ability (range) of β - particles is significantly greater than for α-particles.

Rβ air =200 m, Rβ Pb ≈ 3 mm

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

Bias rule for β - - decay:

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

• γ-rays.

Typically, all types of radioactivity are accompanied by the emission of γ-rays. γ-rays are electromagnetic radiation with 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 ~ MeB

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 are 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;

or

• scattering (Compton effect) -

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

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

І 0 - intensity of the incident parallel beam;

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

γ-rays are one of the most penetrating radiations. For the hardest rays (hνmax) 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 The basic law of radioactive decay.

Number of decayed nucleidN proportional to the initial 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 that decay per unit time, i.e., it 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]=с -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 undecayed nuclei at a timet;

λ is the radioactive decay constant.

In practice, the rate of decay is judged using not λ, but T 1/2 - half-life - the time during which half of the original number of nuclei decays. Relationship between 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. Number of decays per unit time A = -dN/ dtis called the activity of a given radioactive substance.

From

should

[A] = 1Becquerel = 1decay/1s;

[A] = 1Ci = 1Curie = 3.7 10 10 Bq.

Law of Activity Change

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

A - activity at a timet.