Since most nuclei are stable, there is a special nuclear (strong) interaction between nucleons - attraction, which ensures the stability of nuclei, despite the repulsion of like-charged protons.
The binding energy of the nucleus is physical quantity, equal to the work that must be done to split the nucleus into its constituent nucleons, without imparting kinetic energy to them.
It follows from the law of conservation of energy that the same energy must be released during the formation of a nucleus, which must be expended in the splitting of the nucleus into its constituent nucleons. The binding energy of the nucleus is the difference between the energy of all nucleons in the nucleus and their energy in the free state.
Binding energy of nucleons in an atomic nucleus:
where, are the masses of the proton, neutron and nucleus, respectively; is the mass of a hydrogen atom; - atomic mass of this substance.
Mass corresponding to binding energy:
is called the nuclear mass defect. The mass of all nucleons decreases by this amount when a nucleus is formed from them.
The specific binding energy is the binding energy per nucleon: . It characterizes the stability (strength) of atomic nuclei, i.e. the more, the stronger the core.
The dependence of the specific binding energy on the mass number is shown in the figure. The most stable cores of the middle part periodic table (28<A<138). В этих ядрах составляет приблизительно 8,7 МэВ/нуклон (для сравнения, энергия связи валентных электронов в атоме порядка 10эВ, что в миллион раз меньше).
With the transition to heavier nuclei, the specific binding energy decreases, since with an increase in the number of protons in the nucleus, the energy of their Coulomb repulsion increases (for example, for uranium it is 7.6 MeV). Therefore, the bond between nucleons becomes less strong, the nuclei themselves become less strong.
Energetically favorable: 1) fission of heavy nuclei into lighter ones; 2) the fusion of light nuclei with each other into heavier ones. Both processes release enormous amounts of energy; these processes are currently implemented practically; nuclear fission reactions and nuclear fusion reactions.
MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION
BLAGOVESCHENSKY STATE
PEDAGOGICAL UNIVERSITY
Department of General Physics
Binding energy and mass defect
course work
Completed by: 3rd year student of FMF, group "E", Undermined by A.N.
Checked by: Associate Professor Karatsuba L.P.
Blagoveshchensk 2000
Content
§1. Mass Defect - Characteristic
atomic nucleus, binding energy .............................................. ............... 3
§ 2 Mass spectroscopic methods
mass measurements and equipment .............................................................. .............. 7
§ 3 . Semiempirical formulas for
calculation of masses of nuclei and binding energies of nuclei ................................. 12
clause 3.1. Old semi-empirical formulas.............................. 12
clause 3.2. New semi-empirical formulas
taking into account the influence of shells .............................................. ..... 16
Literature................................................. ................................................. .24
§1. The mass defect is a characteristic of the atomic nucleus, the binding energy.
The problem of the non-integer atomic weight of isotopes worried scientists for a long time, but the theory of relativity, having established a connection between the mass and energy of a body ( E=mc 2), gave the key to solving this problem, and the proton-neutron model of the atomic nucleus turned out to be the lock to which this key fit. To solve this problem, some information about the masses of elementary particles and atomic nuclei will be needed (Table 1.1).
Table 1.1
Mass and atomic weight of some particles
(The masses of nuclides and their differences are determined empirically using: mass spectroscopic measurements; measurements of the energies of various nuclear reactions; measurements of the energies of β- and α-decays; microwave measurements, giving the ratio of masses or their differences.)
Let us compare the mass of an a-particle, i.e. helium nucleus, with a mass of two protons and two neutrons, of which it is composed. To do this, we subtract the mass of the a-particle from the sum of the doubled mass of the proton and the doubled mass of the neutron and call the value obtained in this way mass defect
D m=2M p +2M n -M a =0,03037 a.u.m. (1.1)
Atomic mass unit
m a.u.m. = ( 1,6597 ± 0,0004 ) ´ 10 -27 kg. (1.2)
Using the relation formula between mass and energy made by the theory of relativity, one can determine the amount of energy that corresponds to this mass, and express it in joules or, more conveniently, in megaelectronvolts ( 1 MeV=10 6 eV). 1 MeV corresponds to the energy acquired by an electron passing through a potential difference of one million volts.
The energy corresponding to one atomic mass unit is
E=m a.u.m. × c 2 \u003d 1.6597 × 10 -27 × 8,99 × 10 16 =1,49 × 10 -10 J = 931 MeV. (1.3)
The helium atom has a mass defect ( D m = 0.03037 amu) means that energy was emitted during its formation ( E= D ms 2 = 0,03037 × 931=28 MeV). It is this energy that must be applied to the nucleus of a helium atom in order to decompose it into individual particles. Accordingly, one particle has an energy that is four times less. This energy characterizes the strength of the core and is its important characteristic. It is called the binding energy per particle or per nucleon ( R). For the nucleus of a helium atom p=28/4=7 MeV, for other nuclei it has a different value.
 |
In the 1940s, thanks to the work of Aston, Dempster and other scientists, the values of the mass defect were determined with great accuracy and the binding energies were calculated for a number of isotopes. In Fig. 1.1, these results are presented in the form of a graph, on which the atomic weight of isotopes is plotted along the abscissa, and the average binding energy of the particle in the nucleus is plotted along the ordinate.
The analysis of this curve is interesting and important, because from it, and very clearly, it is clear which nuclear processes give a large yield of energy. In essence, the nuclear energy of the Sun and stars, nuclear power plants and nuclear weapons is the realization of the possibilities inherent in the ratios that this curve shows. It has several characteristic areas. For light hydrogen, the binding energy is zero, because there is only one particle in its nucleus. For helium, the binding energy per particle is 7 MeV. Thus, the transition from hydrogen to helium is associated with a major energy jump. Isotopes of average atomic weight: iron, nickel, etc., have the highest particle binding energy in the nucleus (8.6 MeV) and, accordingly, the nuclei of these elements are the most durable. For heavier elements, the binding energy of the particle in the nucleus is less and therefore their nuclei are relatively less strong. The nucleus of the uranium-235 atom also belongs to such nuclei.
The greater the mass defect of the nucleus, the greater the energy emitted during its formation. Consequently, a nuclear transformation, in which the mass defect increases, is accompanied by an additional emission of energy. Figure 1.1 shows that there are two areas in which these conditions are met: the transition from the lightest isotopes to heavier ones, such as from hydrogen to helium, and the transition from the heaviest, such as uranium, to the nuclei of average weight atoms.
There is also a frequently used quantity that carries the same information as the mass defect - packing factor (or multiplier). The packing factor characterizes the stability of the core, its graph is shown in Figure 1.2.
 |
Rice. 1.2. The dependence of the packing factor on the mass number
§ 2. Mass spectroscopic measurement methods
masses and equipment.
The most accurate measurements of the masses of nuclides, made by the method of doublets and used to calculate the masses, were performed on mass spectroscopes with double focusing and on a dynamic device - a synchrometer.
One of the Soviet mass spectrographs with double focusing of the Bainbridge-Jordan type was built by M. Ardenne, G. Eger, R. A. Demirkhanov, T. I. Gutkin and V. V. Dorokhov. All dual focusing mass spectroscopes have three main parts: an ion source, an electrostatic analyzer, and a magnetic analyzer. An electrostatic analyzer decomposes an ion beam in energy into a spectrum, from which a slit cuts out a certain central part. A magnetic analyzer focuses ions of different energies at one point, since ions with different energies travel different paths in a sectoral magnetic field.
Mass spectra are recorded on photographic plates located in the camera. The scale of the instrument is almost exactly linear, and when determining the dispersion in the center of the plate, there is no need to apply the formula with a correction quadratic term. The average resolution is about 70,000.
Another domestic mass spectrograph was designed by V. Schütze with the participation of R. A. Demirkhanov, T. I. Gutkin, O. A. Samadashvili and I. K. Karpenko. It was used to measure the masses of tin and antimony nuclides, the results of which are used in mass tables. This instrument has a quadratic scale and provides double focusing for the entire mass scale. The average resolution of the device is about 70,000.
Of the foreign mass spectroscopes with double focusing, the most accurate is the new Nir-Roberts mass spectrometer with double focusing and a new method for detecting ions (Fig. 2.1). It has a 90-degree electrostatic analyzer with a radius of curvature Re=50.8 cm and a 60-degree magnetic analyzer with a radius of curvature of the ion beam axis
 |
R m =40.6 cm.
Rice. 2.1. Large dual-focusing Nier–Roberts mass spectrometer at the University of Minnese:
1 – ion source; 2 – electrostatic analyzer; 3 – magnetic analyzer; 4 – electronic multiplier for current registration; S 1 - entrance slot; S2 – aperture slot; S 3 - slot in the image plane of the electrostatic analyzer; S 4 is a slit in the image plane of the magnetic analyzer.
The ions produced in the source are accelerated by the potential difference U a =40 sq. and focus on the entrance slit S1 about 13 wide µm; same slot width S4 , onto which the slit image is projected S1 . aperture slit S2 has a width of about 200 micron, gap S3 , on which the image of the slot is projected by the electrostatic analyzer S1 , has a width of about 400 µm. Behind the gap S3 a probe is located to facilitate the selection of relationships U a / U d , i.e. accelerating potential U a ion source and analyzer potentials U d .
On the gap S4 a magnetic analyzer projects an image of the ion source. Ionic current with a strength of 10 - 12 - 10 - 9 A registered by an electron multiplier. You can adjust the width of all slots and move them from the outside without disturbing the vacuum, which makes it easier to align the instrument.
The essential difference between this device and the previous ones is the use of an oscilloscope and the unfolding of a section of the mass spectrum, which was first used by Smith for a synchrometer. In this case, sawtooth voltage pulses are used simultaneously to move the beam in the oscilloscope tube and to modulate the magnetic field in the analyzer. The modulation depth is chosen such that the mass spectrum unfolds at the slit approximately twice the width of one doublet line. This instantaneous deployment of the mass peak greatly facilitates focusing.
As is known, if the mass of an ion M changed to Δ M , then in order for the ion trajectory in a given electromagnetic field to remain the same, all electric potentials should be changed to Δ MM once. Thus, for the transition from one light component of the doublet with mass M to another component having a mass of Δ M large, you need the initial potential difference applied to the analyzer U d , and to the ion source U a , change accordingly to Δ U d And Δ U a so that
(2.1)
Therefore, the mass difference Δ M doublet can be measured by the potential difference Δ U d , necessary to focus instead of one component of the doublet another.
The potential difference is applied and measured according to the circuit shown in fig. 2.2. All resistances except R*,
manganin, reference, enclosed in a thermostat. R=R"
=3 371 630 ± 65 ohm.
Δ R
can vary from 0 to 100000 Om, so attitude Δ R/R
known to within 1/50000. Resistance ∆ R selected so that when the relay is in contact A
,
on the crack S4
,
it turns out that one line of the doublet is focused, and when the relay is on the contact IN
- another doublet line. The relay is fast-acting, switches after each sweep cycle in the oscilloscope, so you can see both sweeps on the screen at the same time. 
doublet lines. Potential change Δ U d
,
caused by added resistance Δ R
,
can be considered matched if both scans match. In this case, another similar circuit with a synchronized relay should provide a change in the accelerating voltage U a
on Δ U a
so that
(2.2)
Then the mass difference of the doublet Δ M can be determined by the dispersion formula
The sweep frequency is usually quite large (for example, 30 sec -1), therefore, voltage source noise should be kept to a minimum, but long-term stability is not required. Under these conditions, batteries are the ideal source.
The resolving power of the synchrometer is limited by the requirement of relatively large ion currents, since the sweep frequency is high. In this device, the largest value of the resolving power is 75000, but, as a rule, it is less; the smallest value is 30000. Such a resolving power makes it possible to separate the main ions from impurity ions in almost all cases.
During measurements, it was assumed that the error consists of a statistical error and an error caused by the inaccuracy of the resistance calibration.
Before starting the operation of the spectrometer and when determining various mass differences, a series of control measurements were carried out. Thus, control doublets were measured at certain intervals of instrument operation. O2- S And C 2 H 4 - SO, as a result of which it was found that no changes had occurred for several months.
To check the linearity of the scale, the same mass difference was determined at different mass numbers, for example, by doublets CH 4 - O , C 2 H 4 - CO And ½ (C 3 H 8 - CO 2). As a result of these control measurements, values were obtained that differ from each other only within the limits of errors. This check was made for four mass differences and the agreement was very good.
The correctness of the measurement results was also confirmed by measuring three differences in the masses of the triplets. The algebraic sum of the three mass differences in the triplet must be equal to zero. The results of such measurements for three triplets at different mass numbers, i.e., in different parts of the scale, turned out to be satisfactory.
The last and very important control measurement for checking the correctness of the dispersion formula (2.3) was the measurement of the mass of the hydrogen atom at large mass numbers. This measurement was done once for A =87, as the difference between the masses of the doublet C4H8O 2 – C 4 H 7 O2. Results 1.00816±2 A. eat. with an error up to 1/50000 are consistent with the measured mass H, equal to 1.0081442±2 A. eat., within the error of resistance measurement Δ R and resistance calibration errors for this part of the scale.
All these five series of control measurements showed that the dispersion formula is suitable for this instrument, and the measurement results are quite reliable. Data from measurements made on this instrument were used to compile the tables.
§ 3 . Semi-empirical formulas for calculating the masses of nuclei and the binding energies of nuclei .
clause 3.1. Old semi-empirical formulas.
With the development of the theory of the structure of the nucleus and the appearance of various models of the nucleus, attempts arose to create formulas for calculating the masses of nuclei and the binding energies of nuclei. These formulas are based on existing theoretical ideas about the structure of the nucleus, but the coefficients in them are calculated from the found experimental masses of the nuclei. Such formulas, partly based on theory and partly derived from experimental data, are called semi-empirical formulas .
The semi-empirical mass formula is:
M(Z, N)=Zm H + Nm n -E B (Z, N), (3.1.1)
Where M(Z, N) is the mass of the nuclide Z protons and N – neutrons; m H is the mass of the nuclide H 1 ; m n is the neutron mass; E B (Z, N) is the binding energy of the nucleus.
This formula, based on the statistical and droplet models of the nucleus, was proposed by Weizsäcker. Weizsäcker listed the laws of mass change known from experience:
1. The binding energies of the lightest nuclei increase very rapidly with mass numbers.
2. Bond energies E B of all medium and heavy nuclei increase approximately linearly with mass numbers A .
3. E B /A light nuclei increase to A ≈60.
4. Average binding energies per nucleon E B /A heavier nuclei after A ≈60 slowly decrease.
5. Nuclei with an even number of protons and an even number of neutrons have slightly higher binding energies than nuclei with an odd number of nucleons.
6. The binding energy tends to a maximum for the case when the numbers of protons and neutrons in the nucleus are equal.
Weizsacker took these regularities into account when creating a semi-empirical formula for the binding energy. Bethe and Becher simplified this formula somewhat:
E B (Z, N)=E 0 +E I +E S +E C +E P . (3.1.2)
and it is often called the Bethe-Weizsacker formula. First Member E 0 is the part of the energy proportional to the number of nucleons; E I is the isotopic or isobaric term of the binding energy, showing how the energy of the nuclei changes when deviating from the line of the most stable nuclei; E S is the surface or free energy of the nucleon liquid drop; E C is the Coulomb energy of the nucleus; E R - steam power.
The first term is
E 0 \u003d αA . (3.1.3)
Isotopic term E I is the difference function N–Z . Because the influence of the electric charge of protons is provided by the term E WITH , E I is a consequence of only nuclear forces. The charge independence of nuclear forces, which is especially strongly felt in light nuclei, leads to the fact that the nuclei are most stable at N=Z . Since the decrease in the stability of nuclei does not depend on the sign N–Z , addiction E I from N–Z must be at least quadratic. Statistical theory gives the following expression:
E I = –β( N–Z ) 2 A –1 . (3.1.4)
Surface energy of a drop with a coefficient of surface tension σ is equal to
E S =4π r 2 σ. (3.1.5)
The Coulomb term is the potential energy of a ball charged uniformly over the entire volume with a charge Ze :
(3.1.6)
Substituting into equations (3.1.5) and (3.1.6) the core radius r=r 0 A 1/3 , we get
(3 .1.7 )
(3.1.8)
and substituting (3.1.7) and (3.1.8) into (3.1.2), we obtain
. (3.1.9)
The constants α, β and γ are selected such that formula (3.1.9) best satisfies all values of binding energies calculated from experimental data.
The fifth term, representing the pair energy, depends on the parity of the number of nucleons:
(3 .1.11 )
A
Unfortunately, this formula is quite outdated: the discrepancy with the actual values of the masses can reach even 20 MeV and has an average value of about 10 MeV.
In numerous subsequent papers, initially only the coefficients were refined or some not very important additional terms were introduced. Metropolis and Reitwiesner further refined the Bethe–Weizsäcker formula:
|
+ π0.036A -3/4
(3.1.12)
For even nuclides π = –1; for nuclides with odd A pi = 0; for odd nuclides π = +1.
Wapstra proposed to take into account the influence of shells using a term of this form:
(3.1.13)
Where A i , Z i And Wi are empirical constants, selected according to experimental data for each shell.
Green and Edwards introduced the following term into the mass formula, which characterizes the effect of shells:
(3.1.14)
Where α i , α j And K ij - constants obtained from experience; and - average values N And Z in a given interval between filled shells.
clause 3.2. New semi-empirical formulas taking into account the influence of shells
Cameron proceeded from the Bethe-Weizsäcker formula and retained the first two terms of formula (3.1.9). Surface energy term E S (3.1.7) has been changed.
Rice. 3.2.1. Nuclear matter density distribution ρ according to Cameron depending on the distance to the center of the nucleus. A -average core radius; Z - half the thickness of the surface layer of the nucleus.
When considering the scattering of electrons on nuclei, we can conclude that the distribution of the density of nuclear matter in the nucleus ρ n trapezoidal (Fig. 16). For the average core radius T you can take the distance from the center to the point where the density decreases by half (see Fig. 3.2.1). As a result of the processing of Hofstadter's experiments. Cameron proposed the following formula for the average radius of nuclei:
He believes that the surface energy of the nucleus is proportional to the square of the mean radius r2 , and introduces a correction proposed by Finberg, which takes into account the symmetry of the nucleus. According to Cameron, the surface energy can be expressed as follows:
Besides. Cameron introduced the fifth Coulomb exchange term, which characterizes the correlation in the motion of protons in the nucleus and the low probability of protons approaching. exchange member
Thus, the excess of masses, according to Cameron, will be expressed as follows:
M - A \u003d 8.367A -
0.783Z
+ αА +β 
+
+ E S + E C + E α = P (Z, N). ( 3 .2.5)
Substituting the experimental values M-A using the least squares method, we obtained the following most reliable values of the empirical coefficients (in Mev):
α=-17.0354; β=-31.4506; γ=25.8357; φ=44.2355. (3.2.5a)
These coefficients were used to calculate the masses. The discrepancies between the calculated and experimental masses are shown in Figs. 3.2.2. As you can see, in some cases the discrepancies reach 8 Mev. They are especially large in nuclides with closed shells.
Cameron introduced additional terms: a term that takes into account the influence of nuclear shells S(Z, N), and member P(Z, N) , characterizing the pair energy and taking into account the change in mass depending on the parity N And Z :
M-A=P( Z , N)+S(Z, N)+P(Z, N). (3.2.6)
Rice. 3.2.2. Differences between the mass values calculated using the basic Cameron formula (3.2.5) and the experimental values of the same masses depending on the mass number A .
At the same time, since theory cannot offer a kind of terms that would reflect some spasmodic changes in the masses, he combined them into one expression
T(Z, N)=S(Z, N)+P(Z. N). (3.2.7)
T(Z, N)=T(Z) +T(N). (3.2.8)
This is a reasonable suggestion, since the experimental data confirm that the proton shells are filled independently of the neutron ones, and the pair energies for protons and neutrons in the first approximation can be considered independent.
Based on the mass tables of Wapstra and Huizeng, Cameron compiled tables of corrections T(Z ) And T(N) on parity and filling of shells.
G. F. Dranitsyna, using new measurements of the masses of Bano, R. A. Demirkhanov and numerous new measurements of β- and α-decays, refined the values of the corrections T(Z) And T(N) in the area of rare earths from Ba to Pb. She made new tables of excess masses (M-A), calculated by the corrected Cameron formula in this region. The tables also show the newly calculated energies of β-decays of nuclides in the same region (56≤ Z ≤82).
Old semi-empirical formulas covering the entire range A , turn out to be too inaccurate and give very large discrepancies with the measured masses (of the order of 10 Mev). Cameron's creation of tables with more than 300 amendments reduced the discrepancy to 1 mev, but the discrepancies are still hundreds of times greater than the errors in the measurements of the masses and their differences. Then the idea arose to divide the entire area of nuclides into sub-areas and for each of them to create semi-empirical formulas of limited application. Such a path was chosen by Levy, who, instead of one formula with universal coefficients suitable for all A And Z , proposed a formula for individual sections of the sequence of nuclides.
The presence of a parabolic dependence on Z of the binding energy of isobar nuclides requires that the formula contain terms up to the second power inclusive. So Levy proposed this function:
M(A, Z) \u003d α 0 + α 1 A+ α 2 Z+ α 3 AZ+ α 4 Z 2 + α 5 A 2 + δ; (3.2.9)
Where α 0 , α 1 , α 2 , α 3 , α 4 , α 5 are numerical coefficients found from experimental data for some intervals, and δ is a term that takes into account the pairing of nucleons and depends on the parity N And Z .
All masses of nuclides were divided into nine subregions, limited by nuclear shells and subshells, and the values of all coefficients of formula (3.2.9) were calculated from experimental data for each of these subregions. The values of the found coefficients ta and the term δ , determined by parity, are given in Table. 3.2.1 and 3.2.2. As can be seen from the tables, not only shells of 28, 50, 82, and 126 protons or neutrons were taken into account, but also subshells of 40, 64, and 140 protons or neutrons.
Table 3.2.1
The coefficients α in the Levy formula (3.2.9), ma. eat(16 O = 16)
| Z |
N |
α 0 |
α 1 |
α2 |
α 3 |
α4 |
α5 |
Table 3.2.2
The term δ in the Lévy formula (3.2.9), defined by parity, ma. eat. ( 16 O \u003d 16)
Z |
N |
δ at |
|||
| even Z and even N |
odd Z and odd N |
odd Z and even N |
even Z and odd N |
||
Using Levy's formula with these coefficients (see Tables 3.2.1 and 3.2.2), Riddell calculated a table of masses for about 4000 nuclides on an electronic calculator. Comparison of 340 experimental mass values with those calculated using formula (3.2.9) showed good agreement: in 75% of cases the discrepancy does not exceed ±0.5 ma. eat., in 86% of cases - no more ± 1,0ma.e.m. and in 95% of cases it does not go beyond ±1.5 ma. eat. For the energy of β-decays, the agreement is even better. At the same time, Levy has only 81 coefficients and constant terms, while Cameron has more than 300 of them.
Correction terms T(Z) And T(N ) in the Levy formula are replaced in separate sections between the shells by a quadratic function of Z or N . This is not surprising, since between function wrappers T(Z) And T(N) are smooth functions Z And N and do not have features that do not allow them to be represented on these sections by polynomials of the second degree.
Zeldes considers the theory of nuclear shells and applies a new quantum number s - the so-called seniority (seniority) introduced by Cancer. Quantum number " seniority " is not an exact quantum number; it coincides with the number of unpaired nucleons in the nucleus, or, otherwise, it is equal to the number of all nucleons in the nucleus minus the number of paired nucleons with zero momentum. In the ground state in all even nuclei s=0; in nuclei with odd A s=1 and in odd nuclei s= 2 . Using the quantum number “ seniority ” and extremely short-range delta forces, Zeldes showed that a formula like (3.2.9) is consistent with theoretical expectations. All the coefficients of the Levy formula were expressed by Zeldes in terms of various theoretical parameters of the kernel. Thus, although Levy's formula appeared as purely empirical, the results of Zeldes' research showed that it could well be considered semi-empirical, like all previous ones.
Levy's formula, apparently, is the best of the existing ones, but it has one significant drawback: it is poorly applicable on the boundary of the domains of the coefficients. It's about Z And N , equal to 28, 40, 50, 64, 82, 126 and 140, the Levy formula gives the largest discrepancies, especially if the energies of β-decays are calculated from it. In addition, the coefficients of the Levy formula were calculated without taking into account the latest mass values and, apparently, should be refined. According to B. S. Dzhelepov and G. F. Dranitsyna, this calculation should reduce the number of subdomains with different sets of coefficients α And δ , discarding subshells Z =64 and N =140.
Cameron's formula contains many constants. The Becker formula also suffers from the same shortcoming. In the first version of the Becker formula, based on the fact that the nuclear forces are short-range and have the property of saturation, they assumed that the nucleus should be divided into external nucleons and an internal part containing filled shells. They accepted that the outer nucleons do not interact with each other, apart from the energy released during the formation of pairs. It follows from this simple model that nucleons of the same parity have a binding energy due to binding to the core, which depends only on the excess of neutrons I=N -Z . Thus, for the binding energy, the first version of the formula is proposed
E B = b "( I) A + A" ( I) + P " (A, I)[(-1) N +(-1) Z ]+S"(A, I)+R"(A, I) , (3. 2.1 0 )
Where R" - parity-dependent pairing term N And Z ; S" - correction for shell effect; R" - small remainder.
In this formula, it is essential to assume that the binding energy per nucleon, equal to b" , depends only on the excess of neutrons I . This means that the cross sections of the energy surface along the lines I=N- Z , the longest sections containing 30-60 nuclides should have the same slope, i.e. should be a straight line. Experimental data confirm this assumption quite well. Subsequently, the Beckers supplemented this formula with one more term :
E B = b ( I) A + A( I) + c(A)+P (A, I)[(-1) N +(-1) Z ]+S(A, I)+R(A, I). ( 3. 2.1 1 )
Comparing the values obtained by this formula with the experimental values of the Wapstra and Huizeng masses and equalizing them using the least squares method, the Beckers obtained a series of coefficient values b And A for 2≤ I ≤58 and 6≤ A ≤258, i.e. more than 400 digital constants. For members R , parity N And Z , they also adopted a set of some empirical values.
To reduce the number of constants, formulas were proposed in which the coefficients a, b And With are presented as functions from I And A . However, the form of these functions is very complicated, for example, the function b( I) is a fifth degree polynomial in I and contains, in addition, two terms with a sine.
Thus, this formula turned out to be no simpler than Cameron's formula. According to the Bekers, it gives values that differ from the measured masses for light nuclides by no more than ±400 kev, and for heavy A >180) no more than ±200 kev. In shells, in some cases, the discrepancy can reach ± 1000 kev. The disadvantage of the Beckers' work is the absence of mass tables calculated using these formulas.
In conclusion, summing up, it should be noted that there is a very large number of semi-empirical formulas of different quality. Despite the fact that the first of them, the Bethe-Weizsacker formula, seems to be outdated, it continues to be included as an integral part in almost all the newest formulas, except for formulas of the Levi-Zeldes type. The new formulas are quite complex and the calculation of the masses from them is quite laborious.
Literature
1. Zavelsky F.S. Weighing of the worlds, atoms and elementary particles.–M.: Atomizdat, 1970.
2. G. Fraunfelder, E. Henley, Subatomic physics.–M.: Mir, 1979.
3. Kravtsov V.A. Mass of atoms and binding energies of nuclei.–M.: Atomizdat, 1974.
In the physical scale of atomic weights, the atomic weight of an oxygen isotope is taken to be exactly 16.0000.
Topic 16. Atomic nucleus
Section 7. Elements of nuclear physics and elementary particle physics
Questions:
1. Atomic nucleus. mass defect. The binding energy of the atomic nucleus
2. Radioactive radiation. Law of radioactive decay.
3. Nuclear reactions. Energy nuclear reaction.
4. Elementary particles. fundamental interaction.
5. Conclusion on discipline.
The atomic nucleus is the central part of the atom, in which all the positive charge and almost all the mass is concentrated.
The nuclei of all atoms are made up of particles called nucleons. Nucleons can be in two states - in an electrically charged state and in a neutral state. A nucleon in a charged state is called a proton. The proton (p) is the nucleus of the lightest chemical element, hydrogen. The proton charge is equal to the elementary positive charge, which is equal in magnitude to the elementary negative charge q e = 1.6 ∙ 10 -19 C, i.e. charge of an electron. A nucleon in a neutral (uncharged) state is called a neutron (n). The masses of nucleons in both states differ little from each other, i.e. m n ≈ m p .
Nucleons are not elementary particles. They have a complex internal structure and consist of even smaller particles of matter - quarks.
The main characteristics of an atomic nucleus are charge, mass, spin and magnetic moment.
Core charge is determined by the number of protons (z) that make up the nucleus. The nuclear charge (zq) is different for different chemical elements. The number z is called the atomic number or charge number. The atomic number is the serial number of a chemical element in D. Mendeleev's periodic system of elements. The charge of the nucleus also determines the number of electrons in the atom. The number of electrons in an atom determines their distribution over energy shells and subshells and, consequently, all the physicochemical properties of the atom. The nuclear charge determines the specifics of a given chemical element.
Core mass The mass of a nucleus is determined by the number (A) of nucleons that make up the nucleus. The number of nucleons in the nucleus (A) is called the mass number. The number of neutrons (N) in the nucleus can be found if the number of protons (z) is subtracted from the total number of nucleons (A), i.e. N=F-z. In the periodic table, up to its middle, the number of protons and neutrons in the nuclei of atoms is approximately the same, i.e. (А-z)/z= 1, by the end of the table (А-z)/z= 1.6.
The nuclei of atoms are usually denoted as follows:
X - symbol of a chemical element;
Z is the atomic number;
A is the mass number.
When measuring the masses of the nuclei of simple substances, it was found that most chemical elements consist of groups of atoms. Having the same charge, the nuclei of different groups differ in masses. The varieties of atoms of a given chemical element, which differ in the masses of nuclei, are called isotopes. Isotope nuclei have the same number of protons, but a different number of neutrons (and; , ; ,).
In addition to the nuclei of isotopes (z - the same, A - different), there are nuclei isobars(z - different, A - the same). (And).
The masses of nucleons, atomic nuclei, atoms, electrons and other particles in nuclear physics are usually measured not in “KG”, but in atomic mass units (amu - otherwise called the carbon mass unit and denoted by “e”). For the atomic mass unit (1e), 1/12 of the mass of the carbon atom is taken 1e = 1.6603 ∙ 10 -27 kg.
Nucleon masses: m p -1.00728 e, m n =1.00867 e.
We see that the mass of the nucleus expressed in "e" will be written as a number close to A.
Spin of the nucleus. The mechanical angular momentum (spin) of the nucleus is equal to the vector sum of the spins of the nucleons that make up the nucleus. The proton and neutron have a spin equal to L = ± 1/2ћ. Accordingly, the spin of nuclei with an even number of nucleons (A even) is an integer or zero. The spin of a nucleus with an odd number of nucleons (A odd) is half-integer.
The magnetic moment of the nucleus. The magnetic moment of the nucleus (P m i) of the nucleus is very small compared to the magnetic moment of the electrons filling the electron shells of the atom. The magnetic moment of the nucleus does not affect the magnetic properties of the atom. The unit of measurement of the magnetic moment of the nuclei is the nuclear magneton μ i = 5.05.38 ∙ 10 -27 J/T. It is 1836 times less than the magnetic moment of the electron - the Bohr magneton μ B = 0.927 ∙ 10 -23 J / T.
The magnetic moment of the proton is equal to 2.793 μ i and is parallel to the spin of the proton. The magnetic moment of the neutron is equal to 1.914 μ i and is antiparallel to the spin of the neutron. The magnetic moments of the nuclei are of the order of the nuclear magneton.
To split a nucleus into its constituent nucleons, a certain amount of work must be done. The value of this work is a measure of the binding energy of the nucleus.
The binding energy of the nucleus is numerically equal to the work that must be done to split the nucleus into its constituent nucleons and without imparting kinetic energy to them.
In the reverse process of the formation of a nucleus, the same energy should be released from the constituent nucleons. This follows from the law of conservation of energy. Therefore, the binding energy of the nucleus is equal to the difference between the energy of the nucleons that make up the nucleus and the energy of the nucleus:
ΔE \u003d E nuk - E i. (1)
Taking into account the relationship between mass and energy (E = m ∙ c 2) and the composition of the nucleus, we rewrite equation (1) as follows:
ΔЕ = ∙ s 2 (2)
Value
Δm \u003d zm p + (A-z) m n - M i, (3)
Equal to the difference between the masses of the nucleons that make up the nucleus and the mass of the nucleus itself, is called the mass defect.
Expression (2) can be rewritten as:
ΔЕ = Δm ∙ s 2 (4)
Those. mass defect is a measure of the binding energy of a nucleus.
In nuclear physics, the mass of nucleons and nuclei is measured in amu. (1 amu = 1.6603 ∙ 10 27 kg), and energy is usually measured in MeV.
Considering that 1 MeV = 10 6 eV = 1.6021 ∙ 10 -13 J, we find the energy value corresponding to the atomic mass unit
1.a.u.m. ∙ s 2 = 1.6603 ∙10 -27 ∙9 ∙10 16 = 14.9427 ∙ 10 -11 J = 931.48 MeV
Thus, the binding energy of the nucleus in MeV is
ΔE sv = Δm ∙931.48 MeV (5)
Taking into account that tables usually give not the mass of nuclei, but the mass of atoms, for the practical calculation of the mass defect, instead of formula (3)
enjoy another
Δm \u003d zm H + (A-z)m n - M a, (6)
That is, the mass of the proton was replaced by the mass of the light hydrogen atom, thereby adding z electron masses, and the mass of the nucleus was replaced by the mass of the atom M a, thereby subtracting these z electron masses.
The binding energy per nucleon in a nucleus is called the specific binding energy.
The dependence of the specific binding energy on the number of nucleons in the nucleus (on the mass number A) is given in Fig.1.
Graph analysis shows:
1. Nucleons are more strongly bound in the nuclei of elements in the middle part of the periodic table (30<А< 100), для этих ядер ≈8,7 МэВ.
2. For nuclei with a mass number A > 100, the specific binding energy decreases with an increase in the number of nucleons (A); for nuclei at the end of the periodic table ≈7.5 MeV.
3. For nuclei with mass number A< 30 удельная энергия связи с уменьшением числа нуклонов (А) уменьшается; для ядер начала периодической таблицы ≈ 1 - 3 МэВ.
4. For light nuclei, the specific binding energy exhibits characteristic maxima and minima: for odd-odd nuclei (,); the maximum for even-even nuclei (,).
The analysis of the graph shows that it is energetically favorable for the lightest nuclei to merge with each other into heavier ones (energy increases). The heaviest nuclei, on the contrary, are energetically favored by the process of fission into fragments (lighter nuclei)
Topic 17. radioactive radiation
