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УДК 531.9+539.12.01
ЯДЕРНЫЕ ОБОЛОЧКИ И ПЕРИОДИЧЕСКИЙ ЗАКОН Д.И.МЕНДЕЛЕЕВА. ЧАСТЬ 2.
UDC 531.9+539.12.01
NUCLEI SHELLS AND PERIODIC TRENDS. PART 2.
Трунев Александр Петрович к.ф.-м.н., Ph.D.
Директор, A&E Trounev IT Consulting, Торонто, Канада
Alexander Trunev Cand.Phys.-Math.Sci., Ph.D.
Director, A&E Trounev IT Consulting, Toronto, Canada
На основе теории ядерных взаимодействий и данных по энергии связи нуклонов для всех известных нуклидов установлены параметры, характеризующие периодические закономерности в формировании ядерных оболочек
Parameters describing periodic trends in the formation of nuclear shells have been established based on the theory of nuclear interactions and data on the binding energy of nucleons for the set of known nuclides
Ключевые слова: НЕЙТРОН, ПЕРИОДИЧЕСКИМ ЗАКОН, ПРОТОН, ЭНЕРГИЯ СВЯЗИ, ЯДРО
Keywords: BINDING ENERGY, PERIODIC TRENDS, PROTON, NEUTRON, NUCLEI
Introduction
The periodic law discovered by Mendeleev in 1869, played a huge role in the development of ideas about the structure of matter. In one of the first formulations of this law states that "the properties of simple bodies, as well as the shape and properties of the compounds of the elements, and therefore the properties of which they form simple and complex bodies are in the periodic table according to their atomic weight" [1]. Using this law, Mendeleev created the periodic table of elements, which is predicted on the basis of new elements - gallium, scandium, germanium, astatine, polonium, technetium, rhenium, and francium.
Ivanenko [2-3], evidently, was one of the first who raised the issue of expansion of the periodic law to include the periodic patterns are observed in atomic nuclei and some exotic formations, such as exotic atoms. According to the theory of nuclear shells [4-5], periodic patterns in the nuclei are explained by analogy with the electron shells, the Pauli principle, which is applied separately for protons and neutrons fill the nuclear envelope. With this expansion of the periodic law of its original wording seems quite logical, since the properties of the nuclei depend not only on the number of protons but also on the number of neutrons.
Thus, the properties of nuclei and the properties of atoms of chemical elements due to the same type on the basis of quantum mechanics and the Pauli principle.
At present, the binding energy of the nucleon measured with high accuracy for almost all known nuclides. However, in contrast to the ionization energy of the atoms, the dependence of the binding energy of the number of nucleons does not contain any explicit reference to the existence of nuclear shells. However, attempts to establish the presence of known nuclear shells and the so-called magic numbers for the deviation from the standard binding energy trend of the Weiszacker’s semi-empirical model [6] and other models [7]. In the present work we investigated the dependence of the energy of the nucleons for all known nuclides with the use of Wolfram Mathematica 8 [8] and three models of the binding energy:
1) Semi-empirical model;
2) 5D model of nuclear interactions [9-11];
3) Information model [12].
The parameters of all models were determined locally for the isotopes of each element. Found that in all three cases, model parameters have a similar behavior depending on the number of protons, which indicates the presence of the internal structure of nuclei.
5D model description
You may notice that the periodic law in the original formulation of Mendeleev is local, as relates properties of simple substances with their atomic weight, which at the time when the law was formulated, was determined by weighing in the gravitational field of the earth. Such a correlation properties of substances and their gravitational properties are reasonable. To answer the question about the fundamental causes that lead to a law of periodicity in nature, consider a general model of atomic nuclei and atoms of matter [10-11]. In this model, the properties of matter are determined by the parameters of the metric tensor in 5-
dimensional space, which depend on a combination of charge and gravitational properties of the central core in the form [9]
k — 2gM3Ac2/Q4, e2 /k — 2gMA /c2 (1)
Here 7, c, Q - the gravitational constant, the speed of light and charge of the nucleus, respectively. About the nature of the charge will be assumed that the source is an electric charge, but it can be screened in various natural fields. The mechanisms of screening and related fields are discussed below. In the case of proton and electron parameters of the metric tensor (1) are presented in Table 1.
Table 1: The metric tensor parameters
k, 1/m e rmax, m rmin, m
e- 1,703163E-28 4,799488E-43 5,87E+27 2,81799E-15
p+ 1,054395E-18 1,618178E-36 9,48E+17 1,5347E-18
Note that the maximum scale rmax -1/ k in the case of an electron exceeds the size of the observable universe, while for protons this scale is about 100 light-years. The minimum is the scale rmm — e/ k - e /m°2 corresponds to the classical radius of a charged particle, which in the case of a proton and an electron is commensurate with the scale and the weak nuclear interactions.
It is easy to see that the second parameter of the model (1) directly included in the formula for the Mendeleev’s periodic law [1]. Combining the parameters, we
find the nuclear charge in the form: Q — einc2/£^/2g. Consequently, the periodic law in its present formulation can also be expressed through the parameters of the metric tensor (1). The metric tensor can be expanded in the vicinity of a massive center of gravity in five-dimensional space in powers of the
dimensionless distance to the source, ~— kr, here r — Vx 2 + y2 +z 2 .
Consider the form of the metric tensor, which arises when holding the first three terms in the expansion of the metric in the case of central force field with the http://ej. kubagro.ru/2012/07/pdf/3 7.pdf
gravitational potential in the Newton’s form. This choice of metric is justified, primarily because of the specified building the superposition principle holds. Suppose x = ct’x2= x x3= y,x4= z, in this notation we have for the square of the interval in the 4-dimensional space:
ds2 — (1 + 2 j/c2)c2dt2 -(1 -2 j/c2)(dx2 + dy2 + dz2)
j — -M (2)
r
Assuming that e /k — 2gM /c2 we arrive at the expression of the interval depending on the parameters of the metric in the five-dimensional space:
ds2 — (1 -e2 /k)c2dt2 -(1 + £2 /k)(dx2 + dy2 + dz2) (3)
Further, we note that in this case the metric tensor in four dimensions is diagonal with components
gu — 1 -e2/ kr; g22 — g33 — g44 — -(1 +e2/ kr) (4)
We define the vector potential of the source associated with the center of
gravity in the form
g —e/^ g — gju (5)
Here u is a vector in three dimensional spaces, which we define below. Hence, we find the scalar and vector potential of electromagnetic field
Q Mc2 e . ...
je —- —---------------------------^, A — jeu (6)
r e kr
To describe the motion of matter in the light of its wave properties, we assume that the standard Hamilton-Jacobi equation in the relativistic mechanics and the Klein-Gordon equation in quantum mechanics arise as a consequence of the wave equation in five-dimensional space [9]. This equation can generally be written as:
1 3 (4-Ga1 — o (7)
Here Y - the wave function describing, according to (7), the scalar field in five-dimensional space; Gik - the contravariant metric tensor,
(8)
' 11 0 0 0 - g1 1
0 1 0 0 - g 2
Glk = h- 0 0 ^2 0 - g 3
0 0 0 1 - g 4
V- g - g2 - g 3 - g 1 J
1= (1 -£2/ kr)-1; 1 =-(1 + £2 / kr) -1
g1 =1gl, g2 =1 g 2 , g3 =1 gз, g4 =^2 g4
1 = 1 +1g12 +^2(g2 + g32 + g4 ); G = h /(ab3); h =
We further note that in the investigated metrics, depending only on the radial coordinate, is true the following relation
F m =v—U-GG mv)=-n—d U-GG mv) dxmK ’ dxm drK '
(9)
Taking into account the expressions (8) and (9), we write the wave equation
(7) as
(10)
c2 dt2 2 dp2 ~ dx1 dp dxM
Note that the last term in equation (10) is of the order h2k =k5r4 << 1. Consequently, this term can be dropped in the problems, the characteristic scale which is considerably less than the maximum scale in Table 1. Equation (10) is remarkable in that it does not contain any parameters that characterize the scalar field. The field acquires a mass and charge, not only electric, but also strong in the process of interaction with the central body, which is due only to the metric of 5dimensional space [9-11].
Consider the problem of the motion of matter around the charged center of gravity, which has an electrical charge and strong, for example, around the proton. In the process of solving this problem is necessary to define the inertial mass of
matter and energy ties. Since equation (10) is linear and homogeneous, this problem can be solved in general.
We introduce a polar coordinate system (r,f,z) with the z axis is directed along the vector potential (8), we put in equation (10)
Y = y(r)exp(ilf + ikzz -iwt -ikpp) (11)
Separating the variables, we find that the radial distribution of matter is described by the following equation (here we dropped, because of its smallness, the last term in equation (10)):
c2
( 1 l2 ^ y-|1 | yrr + -yr —2- y-kW - *kpy + 2g1c ~lokpy - 2gzkzkpy = 0 (12)
r2z V r r у
Consider the solutions (12) in the case when one can neglect the influence of gravity, i.e. 1 »-12 » 1but 1 = 1 + g2(1 -u2) *1. Under these conditions, equation
(16) reduces to
w
—-y—
r 2 ' z
v r r у
Уг + -yr —2У-kZy -^py+ 2g1c~lwkPy-2gZkzkPy = 0 (13)
In general, the solution of equation (13) can be represented in the form of a power series [10-11]
exp(-~) ^ ~ j
¥= ~ Z j (14)
r j=0
It is indicated r = r / rn. Substituting (14) in equation (13), we find
(a2 -12 + k)^ cjr j-2 +(2a -1 + Kgrn )^ j j-1 +
j=0 j=0
(1 - klrn + K2rl )Z cj~ j - Z jcj~ j-1 - 2aZ jcj~ j-2 + (15)
j=0 j=0 j=0
Z cjj(j- 1)r j-2=0
j=0
Ku = (1 - u 2) k 2pe2 / k 2 , K2 = k2 + (O2 / c2, kg = —2ekp(kzuz + o/c)/ k > 0.
Hence, equating coefficients of like powers, we obtain the equations relating the parameters of the model in the case of excited states
/72 k n +1 — 2a 1 12 , J 2 1 O f\
a = il —ku > =--------------, — — kz+ kr +— = 0 (16)
k rn C
The second equation (16) holds only for values of the exponent, for which the inequality 2a < n +1 is true. Hence, we find an equation for determining the energy levels
4e2kl r wA2 2 2 Oo
— k2 + k2 + — = 0 (17)
p c
p
k (n +1 — 2a)
Equation (17) was used to model the binding energy of nucleons in the nucleus for the entire set of known nuclides [10-11]. In the model [10-11], the core consists of "pure" proton interacting with a scalar field. Part of the "pure" proton is screened by forming N neutrons, as a result there is an atom, consisting of the electron shell and nucleus with electric charge eZ, number of nucleons A = Z + N , mass excess ME = M — A , and the binding energy Eb = Z (me + mp ) + Nm n — (ME + A ■ mu), mu = m (12 C) /12 » 931 .494028 MeV
Note that we using the standard expression of the mass excess in atomic units. Since two types of charges - scalar and vector, appear in this problem the effect of screening manifests itself not only with respect to the scalar charge (which leads to the formation of neutrons), but also in terms of the vector of the charge, which leads to the formation of the nucleons.
It should be noted that the original metric in the five-dimensional space defined by the metric tensor, which depends only on the parameters of the central
body, i.e. of the total charge and total mass of the nucleons. Different shells can be formed depending on the combination of the charge and mass of the nucleus:
1) Nucleon shell, in which all charges are screened, therefore^/k == A2e2 / AmpC2 = Ae2 /mpC2;
2) Neutron shell, in which we have e/k = Ne 2 / mpc 2;
3) Proton shell, in which e/ k = Ze 2 / mpc 2.
Using the electron mass and Planck's constant, we define the dimensionless parameters of the model in the form
a=£l, s=<ik4,p=, E =
hC (mec)2’ mec’ mec
bX =
tX = ___________4 x (^me / mp )________________________
nl (n + 1 — 2^112 — (1 — u2)SX 2(ame / mp )2 ) (18)
Here X = A, N, Z in the case of the nucleon, neutron and proton shells, respectively.
Solving equation (17) with respect to energy, we find
„X — SbX,Pu ± ¿V— (SbX,Pu)2 + (SbX + 1)(S — P1 + SbXP1 uJ)
En = ^ (19)
Note that the parameter in the energy equation (19) can be both real and complex values, which correspond to states with finite lifetime. Given that for most nuclides the decay time is large enough quantity; it can be assumed that the imaginary part of the right-hand side of equation (19) is a small value, which corresponds to a small value of the radicand. Hence we find that for these states the following relation between the parameters
S (SbX +1)
p 2 =
1 + SbX (1 — u2)
Substituting in the momentum equation (19), we have
о3/2т X ,
T?X = S b„lU
Enl =
J(Sb'X +1)(1 + SbX (1 - «'))
(21)
Hence, we find the dependence of the binding energy per nucleon in the ground state
S3/2bnuX2 /A
, „ -__________________________________'0
J0a
Ex / A =
7(Sb0 X2 +1)(1 + Sb0 X 2(1 - u2))
(22)
It is indicated b0 = (2ame/mp0 — 2a))2. Thus, we have established a link the energy of the state parameters with the interaction parameter. Note that the energy of ground state (22) depends on the magnitude of the vector charge, which appears in equations (5) - (6). In [11] have shown that this shows the difference between the interaction of nucleons in nuclei, where the parameter u ^ 0 , and the interaction between electrons and atomic nuclei, in which u = 0 .
Equation (22) allows us to describe the dependence of the binding energy of the number of nucleons for all nuclides. The computational model is constructed as follows. Suppose that, based on equation (22) was able to accurately determine the binding energy of one of the isotopes of an element. Without loss of generality we can assume that this is isotope, which contains the minimum number of neutrons. Then the binding energy of all other isotopes of element is defined by E(N, Z) = Ei (Nmm, Z) , EN (N, Z) EN (Nmm, Z)
■ + -
A Z + Nmin Z + N Z + Nmm
(23)
Model (23) contains the arbitrary choice of the interaction parameterSb 0. Further, without loss of generality we assume that Sb 0 = 1, therefore a momentum scale in the fifth dimension appearing in equations (11) - (17) is established. Computation of the binding energy of nucleons
We consider three models of the binding energy of nucleons. Standard Weiszacker semi-empirical model has the form [13]:
Eb = avA — asA2/3 — acZ 2A~1/3 — aA (N — Z )2 A~1 + a5A~3/4 (24)
The first term on the right side of (24) describes the increase in binding energy due to the increase in the volume of the system, the second term is due to the contribution of surface energy, the third term describes the contribution of the electric charge of protons, the fourth term due to the contribution of the Fermi energy of nucleons, and finally, the fifth term describes the pairing energy. Since the model (24) depends on five parameters, and model (23), only three, we fix two parameters in equation (24). First, we assume as = I7 23 that is consistent with
the known data [6]. Second, we assume a5 = 0 that due to the specifics of the problem, in which the model parameters are defined locally for a given value of the nuclear charge, and in this case there is no sufficient data to determine this parameter. Consequently, it is necessary to determine the three parameters of the
model av, ac, a a , depending on the number of protons Z.
5D model of the binding energy (23) depends on three parameters. For a given number of protons can be represented as
bN2 / A
Eb / A = a +
V( N2 +1)(1 + k (gN )2)
(25)
+1)(1 + t
The problem is to find the values of model parameters (25) a, b, g , depending on the number of protons Z.
The information model is based on the binding energy changes in terms of energy of a thermodynamic system
dE = TdS — PdV
We can assume that in the case of core contributions of pressure and volume is described by the first term on the right side of equation (24), and entropy of the system varies with the number of neutrons, like the entropy of a discrete set [12], thus
Eb / A = a + b^N / A)(- ln(N / A) + &) (26)
Note that in the system Mathematica 8 [8] has built a database of isotopes IsotopeData [], and the procedure for finding the parameters of linear and nonlinear models - Fit, FindFit, NonlinearModelFit. The coefficients of the three models (24)
- (26) were calculated in the system [8] for the isotopes of chemical elements (Appendix B shows an example for the model (24)). Model (25) is rigid, so in the calculation of its parameters is introduced numerical coefficient k, which provides the convergence of the solution (see example below in Appendix section). Model (26) can be used without change, but in some cases, a sign in front of the logarithm should be replaced with the opposite sign on the initial iteration. Three models can be compared with experimental data (the corresponding code shows in the Appendix). Results comparing the three models are shown in Figure 1 for isotopes of O, F, Fe, Ni, Pt, Au curves of different colors - green, red and blue for models (24), (25), (26), respectively.
Parameters of the three models, calculated for the number of protons from 4 to 94 are summarized in Table 2.
-1 i I i i i i I i i i i I i i i i I i i 4 i 1 i 1 1—1—I i—i—i—i—|—iiii= i i i i i i i i i i i i i i i i i i i i i i j i j, i i i i i i i i i i i i
Pt ^4 7.90 ■ Au '
/ > 7.S5 /
■ / > ■ /
/ s f
§ 7. B0 ■ / -
: / : /
: / 7.75 / ;
£ ■ ■ . . .1 7.70 LJ — I il.. I . . i I .iii _i i i i . . . . i l:
90 95 100 105 110 115 120
JV
90 95 100 105 110 115 120 125
N
Figure 1: Comparison of three models for the isotopes O, F, Fe, Ni, Pt, Au: the green lines - Weiszacker’s model (24), red lines - 5D model (25), blue lines -an information model.
Table 2: The calculated parameters of models (24) (25) and (26).
Z a b g al b1 g1 av ac aA
4 -3.62481 27.327 0.0422609 -20.4234 51.3442 0.355978 14.7091 -0.449354 15.4203
5 -2.52089 23.6371 0.0247448 -14.204 39.0915 0.361751 13.8749 -0.548336 12.7719
6 -2.68681 25.3876 0.0165681 -19.5433 50.4696 0.366144 14.4986 -0.1836 14.5216
7 -3.85908 28.2342 0.0114133 -26.5208 63.2546 0.378508 14.8815 0.262532 16.6353
8 -3.79443 28.7792 0.0086387 -29.0956 68.485 0.382158 15.1319 0.377706 17.331
9 -4.02073 29.1897 0.0067492 -31.347 72.4102 0.38568 15.0451 0.461668 17.8339
10 -3.90832 29.5351 0.0055812 -32.327 74.7037 0.384548 15.066 0.447392 18.1195
11 -4.34471 30.6186 0.0045588 -34.9852 79.4571 0.387794 15.1316 0.538501 18.9135
12 -4.75363 32.0398 0.0037947 -34.9742 79.891 0.386432 15.2473 0.556375 19.3312
13 -4.49257 31.4826 0.0032596 -34.4678 78.9237 0.387226 15.0955 0.520236 19.0377
14 -4.86299 32.9161 0.0028544 -33.7932 78.1938 0.38417 15.1093 0.489994 19.3225
15 -4.92823 32.9845 0.0024462 -34.6704 79.514 0.38844 15.1402 0.532441 19.4656
16 -4.57793 32.2025 0.0021184 -34.6231 79.3658 0.391189 15.1662 0.53731 19.208
17 -4.4887 31.7784 0.0018284 -34.6721 79.0368 0.396352 15.1746 0.568718 19.024
18 -4.20688 31.1873 0.0016155 -34.0293 77.7887 0.398887 15.1715 0.559632 18.6967
19 -4.56505 32.023 0.0014452 -35.2885 79.9604 0.400381 15.1961 0.58579 19.1674
20 -4.64648 32.4197 0.0013115 -35.6665 80.765 0.400448 15.2171 0.580203 19.3655
21 -5.26814 33.8858 0.00119 -38.196 85.348 0.400375 15.2774 0.617602 20.2984
22 -5.67551 35.0752 0.0010958 -40.1961 89.2201 0.398872 15.3419 0.628125 21.0693
23 -6.05237 35.9337 0.0009984 -41.5921 91.6283 0.400196 15.3843 0.651503 21.5696
24 -6.11985 36.2127 0.0009177 -41.99 92.3902 0.400701 15.4094 0.648387 21.7216
25 -6.24784 36.4664 0.0008418 -42.4793 93.1508 0.401959 15.4076 0.653399 21.8633
26 -6.28165 36.6336 0.0007779 -41.7491 91.9181 0.401584 15.3744 0.633746 21.7385
27 -5.91711 35.5786 0.0007103 -40.9674 90.2611 0.403841 15.3089 0.622372 21.2347
28 -5.78781 35.2243 0.0006501 -39.7398 87.903 0.405936 15.3041 0.614025 20.8196
29 -4.99568 33.1639 0.0006016 -39.4301 87.1016 0.408116 15.1827 0.590204 20.1834
30 -4.57451 32.0833 0.0005569 -38.1885 84.7455 0.409463 15.0929 0.566158 19.6017
31 -4.63175 32.0402 0.0005134 -38.5366 85.1258 0.411775 15.099 0.577839 19.6222
32 -4.63363 32.0235 0.0004795 -38.6924 85.3594 0.412437 15.0928 0.575169 19.6399
33 -4.89913 32.5176 0.0004462 -39.7378 87.0587 0.413959 15.1314 0.593458 19.9647
34 -4.92286 32.5903 0.0004204 -42.1626 91.2954 0.415514 15.2229 0.61443 20.4833
35 -5.32932 33.4376 0.0003932 -43.7459 93.9747 0.416771 15.2993 0.639397 21.0178
36 -5.73261 34.4072 0.0003708 -45.5038 97.1115 0.417121 15.4002 0.659816 21.6395
37 -6.02828 35.0052 0.0003487 -46.3959 98.5568 0.418232 15.4387 0.673052 21.9657
38 -6.30199 35.6495 0.0003294 -47.6848 100.826 0.418783 15.5158 0.686486 22.3972
39 -5.99714 34.8385 0.0003117 -47.4196 100.218 0.419675 15.4344 0.670048 22.0897
40 -5.80949 34.4269 0.0002976 -46.5407 98.6955 0.419209 15.3424 0.644418 21.7953
41 -5.29964 33.138 0.0002831 -45.367 96.4927 0.41975 15.1905 0.61341 21.1702
42 -5.18736 32.8917 0.0002706 -44.7795 95.4757 0.419392 15.1213 0.594908 20.9812
43 -4.72927 31.7071 0.0002573 -43.0526 92.2911 0.419885 14.9618 0.56367 20.276
44 -4.68112 31.5894 0.0002456 -42.7163 91.6774 0.420027 14.925 0.553454 20.1688
45 -4.30091 30.5779 0.0002333 -40.9962 88.4722 0.420954 14.787 0.528102 19.5122
46 -3.98847 29.8081 0.0002226 -39.6453 86.0108 0.421472 14.688 0.506142 19.0082
47 -3.01173 27.3123 0.0002095 -36.5769 80.3064 0.423684 14.4568 0.463584 17.6412
48 -2.82675 26.8101 0.0001988 -35.6621 78.5572 0.425169 14.4103 0.454071 17.2941
49 -2.64241 26.2272 0.0001878 -34.8193 76.8483 0.427347 14.3553 0.447902 16.9268
50 -2.75243 26.3929 0.0001774 -35.0131 77.0089 0.429706 14.4058 0.458403 16.982
51 -3.10147 27.0323 0.0001675 -36.9114 80.0647 0.432777 14.5294 0.489927 17.5059
52 -4.1087 29.2927 0.0001595 -40.8443 86.8587 0.434208 14.8168 0.548111 18.9091
53 -4.88685 30.9368 0.0001512 -44.1034 92.3213 0.436673 15.0542 0.599221 19.986
54 -5.77758 32.9822 0.0001456 -47.6512 98.5393 0.436678 15.2876 0.642985 21.2657
55 -6.49219 34.5295 0.0001395 -50.793 103.932 0.437431 15.4808 0.68223 22.3091
56 -6.55646 34.6463 0.0001344 -50.8494 103.993 0.437599 15.4654 0.677452 22.3435
57 -6.71471 34.8831 0.0001287 -51.7677 105.403 0.439024 15.5096 0.688154 22.5647
58 -6.77462 34.9917 0.0001243 -51.8113 105.447 0.43912 15.4913 0.682866 22.5949
59 -6.61779 34.5162 0.0001194 -50.9097 103.723 0.439992 15.4053 0.669129 22.2552
60 -6.64243 34.5196 0.0001151 -51.3111 104.33 0.440713 15.414 0.669337 22.3221
61 -6.5384 34.155 0.0001105 -50.6191 102.949 0.441853 15.3515 0.660245 22.0511
62 -6.571 34.1944 0.0001069 -50.5155 102.727 0.442029 15.3256 0.654196 22.0297
63 -6.54571 34.0061 0.0001026 -50.1268 101.863 0.44335 15.2915 0.650338 21.8645
64 -6.60085 34.0906 9.923E-05 -51.0797 103.449 0.443885 15.3239 0.654754 22.0742
65 -6.50471 33.7317 9.524E-05 -50.3913 102.047 0.445361 15.272 0.647954 21.799
66 -6.38636 33.4135 9.218E-05 -49.6857 100.758 0.445726 15.2046 0.635322 21.5537
67 -6.25463 32.9776 8.857E-05 -48.8482 99.1038 0.447197 15.1396 0.626537 21.2265
68 -5.86922 32.0605 8.596E-05 -47.5027 96.7233 0.447365 15.0027 0.602623 20.6984
69 -5.54572 31.1761 8.249E-05 -45.8781 93.681 0.449166 14.8826 0.585267 20.0847
70 -5.1355 30.2033 8.013E-05 -44.3569 91.0016 0.449317 14.7324 0.560059 19.5073
71 -5.13124 30.0637 7.707E-05 -44.2486 90.6327 0.450782 14.7158 0.559955 19.4139
72 -5.1398 30.0561 7.505E-05 -44.8237 91.6306 0.45047 14.7088 0.558424 19.5368
73 -5.08636 29.7848 7.199E-05 -44.3017 90.492 0.452533 14.6794 0.556499 19.3111
74 -5.15849 29.8641 6.949E-05 -44.7317 91.095 0.4538 14.7085 0.56144 19.3941
75 -5.18135 29.7528 6.652E-05 -44.5443 90.5021 0.456279 14.7143 0.56513 19.2703
76 -4.81392 28.8112 6.396E-05 -42.837 87.3653 0.458217 14.5937 0.546876 18.628
77 -4.739 28.4772 6.119E-05 -42.4773 86.4801 0.46079 14.5782 0.547303 18.4114
78 -4.3432 27.458 5.861E-05 -40.8502 83.4595 0.463247 14.4649 0.530748 17.7622
79 -4.11711 26.7572 5.566E-05 -40.1393 81.9009 0.46695 14.4327 0.528879 17.367
80 -5.44848 29.7541 5.483E-05 -45.8294 91.7441 0.465145 14.85 0.591188 19.3432
81 -6.3089 31.5529 5.29E-05 -50.0175 98.6963 0.466802 15.1852 0.643356 20.6374
82 -7.35854 33.897 5.192E-05 -54.2297 105.945 0.465925 15.4981 0.688685 22.1181
83 -10.6838 41.2669 5.081E-05 -69.5929 132.156 0.465622 16.6923 0.86676 27.1828
84 -12.0013 44.1892 4.977E-05 -75.7497 142.678 0.465302 17.1429 0.930859 29.2048
85 -14.341 49.3045 4.842E-05 -87.2432 162.142 0.465986 18.0281 1.06011 32.8569
86 -13.1541 46.6637 4.754E-05 -57.8229 112.377 0.461763 17.588 0.991697 31.3175
87 -12.0067 44.0411 4.647E-05 -10.9011 33.4322 0.416778 17.1993 0.933971 29.8626
88 -10.7392 41.227 4.573E-05 -33.0918 70.7649 0.449922 16.6866 0.857065 28.0244
89 -9.3985 38.2087 4.493E-05 -0.240897 15.592 0.333918 16.162 0.781219 26.1047
90 -7.66393 34.3582 4.437E-05 1.39374 12.7593 0.305706 15.468 0.680812 23.5375
91 -6.60717 31.912 4.33E-05 1.25935 12.7706 0.319387 15.048 0.622713 21.8927
92 -5.29195 28.9599 4.261E-05 4.6173 7.29986 0.171406 14.5418 0.55151 19.9757
93 -10.3571 39.5927 3.857E-05 5.70092 5.94733 0.0177703 16.745 0.867284 27.4873
94 -10.6928 40.2478 3.754E-05 6.13997 5.31963 -0.0623392 16.8843 0.884977 27.9953
The Appendix provides the text of programs to calculate and plot the model parameters on the number of protons - Fig. 2-4. The parameters of all three models
vary with the number of protons. Since the 5D model is rigid, it uses the aboveintroduced coefficient k, which provides the convergence of solutions depending on the number of protons in the form k = 0.9592/Z2209. Shown in Fig. 3 parameter g is calculated with respect to this factor.
Analyzing the data given in Table 2 and Fig. 2-4, we can conclude that there is no universal model that describes the entire set of nuclides. From the data presented in Fig. 1, it follows that all three models describe equally well the binding energy of the isotopes of individual elements. In this sense, the model of Weiszacker cannot be regarded as a universal model, even with the term describing the pairing energy.
Figure2: The dependence of Weiszacker model parameters on the number of protons, the lower figures show the value of standard deviation RSquared and maximum absolute prediction error of the binding energy - MaxError.
Note the similarity in the behavior of parameters of three models: the parameters reach extreme values at the same or similar values of Z - Table 3. These results indicate the presence of nuclear structure, but the point of extremes do not coincide with the magic numbers of protons - 2, 8, 20, 28, 50, 82, as defined in the standard nuclear shell model [3-6]. A similar result was obtained in [11], in which the local parameters 5D model depending on the number of neutrons have been calculated. As it turned out, the number of neutrons corresponding to the extreme values of the parameters of the model 5D, close to the magic numbers, but nowhere with them do not match.
Figure 3: The dependence of the parameters of the 5D model on the number of protons.
In this regard, we note in Table 3 and Fig. 2-4 three points that fall on the elements Z = 31,32,85 - Ga (Gallium), Ge (germanium), At (astatine). Gallium and germanium were predicted by Mendeleev in 1870 and discovered in 1875 and 1885, respectively. Astatine predicted by Mendeleev was artificially synthesized only in 1940. Note three extreme coinciding with the Z = 26, 79, 92 - Fe (iron), Au (gold) and U (uranium). There is no doubt that the iron is clearly identified in nature and has long been used in human practice. The role of gold and uranium in human history cannot be overestimated. It is also interesting that only in the 5D model, the binding energy of one of the extremes have the element with proton number Z = 26 - iron.
Figure 4: The dependence of the parameters of the information model on the number of protons.
Table 3: Extreme values of the model parameters
Z a b a1 b1 av aA
5 -2.52089 23.6371 -14.204 39.0915 13.8749 12.7719
18 -4.20688 31.1873 -34.0293 77.7887 15.1715 18.6967
25 -42.4793 93.1508 15.4076 21.8633
26 -6.28165 36.6336
31 -4.63175 -38.5366 85.1258 19.6222
32 32.0235 15.0928
38 -6.30199 35.6495 -47.6848 100.826 15.5158 22.3972
49 -2.64241 26.2272 -34.8193 76.8483 14.3553 16.9268
58 -6.77462 34.9917 -51.8113 105.447 15.4913 22.5949
79 -4.11711 26.7572 -40.1393 81.9009 14.4327 17.367
85 -14.341 49.3045 -87.2432 162.142 18.0281 32.8569
92 -5.29195 28.9599 4.6173 7.29986 14.5418 19.9757
We can assume that there is a version of the periodic table, in which periods are associated with the trend shown in Fig. 2-4 and in Table 2. These results suggest that the periodic properties of the nuclei of atomic elements depend on the number of protons (charge), in line with the modern formulation of the periodic law [14]. It has been previously established [11] that the periodic properties of nuclei depend on the number of neutrons, which is reflected in the original formulation of Mendeleev's periodic law. The Appendix gives the texts of programs to calculate the model parameters depending on the number of neutrons - Fig. 5-6.
Figure 5: The dependence of Weiszacker model parameters on the number of neutrons, the lower figures show the value of standard deviation and maximum absolute prediction error of the binding energy.
Model (24) used in this case without change and 5D model takes the form
bZ2 / A
E / A = a + -
(27)
V(Z2 +1)(1 + k ( gZ )2)
Since the 5D model is rigid, it uses a numerical coefficient k, which provides the convergence of solutions depending on the number of neutrons in the form
k = 0.0025 - 0.0003ln N
Figure 6: The dependence of the parameters of the 5D model on the number of neutrons.
The data presented in Fig. 5 that the Weiszacker model parameters depend on the number of neutrons, and these dependencies are not monotonic, which indicates the presence of nuclear structure. Thus, we have shown that the binding energy of all known nuclides can be described approximately with the same accuracy by any of three models (24)-(26). This means that the nucleus can be regarded as a charged liquid drop (Weiszacker model), and as a set of shielded "clean" protons in the five-dimensional space [10-11], and as a statistical (information) system [12].
Note that the droplet model of the nucleus had a large development in the 3050s of last century. On the other hand, 5D model is theoretically justified by Kaluza [15], Einstein [16-19], Pauli and Einstein [20], Rumer [21], Dzhunushaliev [22], and in our papers [9-11] as well. The information model of the nucleus also has a
great potential in terms of its expansion, taking into account the spin angular momentum and other quantum numbers, as well as quantum chaos [6-7, 23-24].
The author expresses his gratitude to Professor VD Dzhunushaliev and Professor EV Lutsenko for useful discussions.
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Appendix
Source code for calculating the Weiszacker model parameters in Table 2:
Do[ model = av - 17.23*(Z + x)A(-1/3) + ac*(Z*Z)*(Z + x)A(-4/3) + aA*((x - Z)A2)*(Z + x)A(-2);
Eb = Table[IsotopeData[#, prop], {prop, {"NeutronNumber", "BindingEnergy"}}] & /@ IsotopeData[Z]; nlm = FindFit[Eb, model, {{ av, 15.5}, {ac, -.628528}, {aA, -22.03}}, x]; Print[Z, nlm], {Z, 1, 118}]
Source code for the comparison of three models - Fig. 1:
Z = 78; k = 0.000049;
Eb = Table[IsotopeData[#, prop], {prop, {"NeutronNumber", "BindingEnergy"}}] & /@ IsotopeData[Z]; nlm = NonlinearModelFit[Eb, a + b*(x*x/(Z*1. + x))*((xA2 + 1)*(1 + k*(g*x)A2))A(-.5) , {a, b, g}, x]; nlm1 = NonlinearModelFit[Eb, av - 17.23*(Z + x)A(-1/3) - ac*(Z*Z)*(Z + x)A(-4/3) - aA*((x - Z)A2)*(Z + x)A(-2) , {av, ac, aA}, x]; nlm2 = NonlinearModelFit[Eb, a2 + b2*(x/(Z*1. + x))*(-Log[x/(Z*1. + x)] + g2) , {a2, b2, g2}, x];
Show[ListPlot[Eb], Plot[{nlm[x], nlm1[x], nlm2[x]}, {x, 1., 180.}, PlotStyle -> {Red, Green, Blue}], Frame -> True,
FrameLabel -> {N, "Eb/A, MeV"}]
Source code for calculating the Weiszacker model parameters depending on the number of protons (Fig. 2):
par = {0.}; para = {0.}; parc = {0.}; RSq = {1.}; MaxEr = {0.};
Do[ Eb = DeleteCases[Table[IsotopeData[#,prop], {prop, {"NeutronNumber", "BindingEnergy"}}] & /@
IsotopeData[Z], {_, Missing["Unknown"]}];
nlm = NonlinearModelFit[Eb, av - 17.23*(Z + x)A(-1/3) - ac*(Z*Z)*(Z + x)A(-4/3) - aA*((x - Z)A2)*(Z + x)A(-2) , {av, ac, aA}, x];
RSq = {RSq, nlm["RSquared"]} // Flatten;
MaxEr = {MaxEr, Last[Sort[nlm["MeanPredictionErrors"]]]} // Flatten;
para = {para, av /. nlm["BestFitParameters"]} // Flatten;
parc = {parc, ac /. nlm["BestFitParameters"]} // Flatten;
par = {par, aA /. nlm["BestFitParameters"]} // Flatten, {Z, 2, 112}]
ListPlot[par, Filling -> Axis, AxesLabel -> {Z, aA},
ImageSize -> {200, 200}] ListPlot[para, Filling -> Axis,
AxesLabel -> {Z, av}, ImageSize -> {200, 200}] ListPlot[parc,
Filling -> Axis, AxesLabel -> {Z, ac}, ImageSize -> {200, 200}]
ListPlot[RSq, Filling -> Axis, AxesLabel -> {Z, "RSquared"},
ImageSize -> {300, 300}, DataRange -> Automatic] ListPlot[MaxEr,
Filling -> Axis, AxesLabel -> {Z, "MaxError"},
ImageSize -> {300, 300}, DataRange -> Automatic]
Source code for calculating the Weiszacker model parameters depending on the number of neutrons (Fig. 5):
par = {.0}; para = {.0}; parc = {.0};
Do[model = av - 17.23*(nn + x)A(-1/3) - ac*(x*x)*(nn + x)A(-4/3) -aA*((x - nn)A2)*(x + nn)A(-2) ;
Eb = Drop[
Cases[DeleteCases[
Table[{a - z, z, IsotopeData[{z, a}, "BindingEnergy"]}, {z, 1,
118}, {a,
IsotopeData[#, "MassNumber"] & /@ IsotopeData[z]}], {_,
Missing["Unknown"]}] // Flatten[#, 1] &, {nn, _, _}],
None, {1}];
nlm = FindFit[Eb, model, {{ av, 15.5}, {ac, 0.628528}, {aA, 22.03}}, x]; para = {para, av /. nlm} // Flatten; parc = {parc, ac /. nlm} // Flatten; par = {par, aA /. nlm} // Flatten, {nn, 2, 175}]
ListPlot[par, Filling -> Axis, AxesLabel -> {N, aA}]
ListPlot[para, Filling -> Axis, AxesLabel -> {N, av}]
ListPlot[parc, Filling -> Axis, AxesLabel -> {N, ac}]
Source code for calculating the dependence of 5D model parameters on the number of protons (Fig. 3):
par = {0.}; para = {0.}; parc = {0.}; RSq = {1.}; MaxEr = {0.};
Do[ Eb = DeleteCases[Table[IsotopeData[#, prop], {prop, {"NeutronNumber", "BindingEnergy"}}] & /@
IsotopeData[Z], {_, Missing["Unknown"]}]; nlm = NonlinearModelFit[Eb, a + b*(x*x/(Z*1. + x))*((xA2 +1)*(1 + (0.9592/ZA2.209)*(g*x)A2))A(-.5), {a, b, g}, x]; para = {para, -a /. nlm["BestFitParameters"]} // Flatten;
RSq = {RSq, nlm["RSquared"]} // Flatten;
MaxEr = {MaxEr, Last[Sort[nlm["MeanPredictionErrors"]]]} // Flatten;
parc = {parc, b /. nlm["BestFitParameters"]} // Flatten;
par = {par, gA2 /. nlm["BestFitParameters"]} // Flatten, {Z, 2, 110}]
ListPlot[par, Filling -> Axis, AxesLabel -> {Z, "g"}, ImageSize -> {200, 200}, PlotRange -> {0.8, 1.2}] ListPlot[para,
Filling -> Axis, AxesLabel -> {Z, "a"}, ImageSize -> {200, 200}] ListPlot[parc, Filling -> Axis,
AxesLabel -> {Z, "b"}, ImageSize -> {200, 200}]
ListPlot[RSq, Filling -> Axis, AxesLabel -> {Z, "RSquared"}, ImageSize -> {200, 200}, DataRange -> Automatic] ListPlot[MaxEr,
Filling -> Axis, AxesLabel -> {Z, "MaxError"},
ImageSize -> {200, 200}, DataRange -> Automatic]
Source code for calculating the dependence of 5D model parameters on the number of neutrons (Fig. 6):
par = {0.};para = {0.};parc = {0.};RSq = {0.};MaxEr = {0.};
Do[ Eb = Drop[ Cases[DeleteCases[
Table[{a - z, z, IsotopeData[{z, a}, "BindingEnergy"]}, {z, 1,118}, {a, IsotopeData[#, "MassNumber"] & /@ IsotopeData[z]}], {_,Missing["Unknown"]}] // Flatten[#, 1] &, {nn, _, _}], None, {1}];
nlm = NonlinearModelFit[Eb, a + b*(x* x/(nn*1. + x))*((xA2 + 1)*(1 + (0.0025 - 0.0003*Log[nn])*(g*x)A2))A(-.5), {a, b, g}, x]; para = {para, -a /. nlm["BestFitParameters"]} // Flatten;
RSq = {RSq, nlm["RSquared"]} // Flatten;
MaxEr = {MaxEr, Last[Sort[nlm["MeanPredictionErrors"]]]} // Flatten;
parc = {parc, b /. nlm["BestFitParameters"]} // Flatten;
par = {par, g /. nlm["BestFitParameters"]} // Flatten, {nn, 2, 102}]
ListPlot[par, Filling -> Axis, AxesLabel -> {N, "g"},
ImageSize -> {200, 200}] ListPlot[para, Filling -> Axis,
AxesLabel -> {N, "a"}, ImageSize -> {200, 200}] ListPlot[parc,
Filling -> Axis, AxesLabel -> {N, "b"}, ImageSize -> {200, 200}]
ListPlot[RSq, Filling -> Axis, AxesLabel -> {N, "RSquared"},
ImageSize -> {200, 200}, DataRange -> Automatic] ListPlot[MaxEr,
Filling -> Axis, AxesLabel -> {N, "MaxError"},
ImageSize -> {200, 200}, DataRange -> Automatic]
