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УДК 531.9+539.12.01
ХАОС И ФАЗОВЫЕ ПЕРЕХОДЫ В АТОМНЫХ ЯДРАХ
Трунев Александр Петрович к.ф.-м.н., Ph.D.
Директор, A&E Trounev IT Consulting, Торонто, Канада
В работе рассмотрена модель хаотического поведения нуклонов в атомных ядрах, построенная на основе модели ядерных взаимодействий и статистики Ферми-Дирака
Ключевые слова: НЕЙТРОН, ПРОТОН, ФЕРМИ-ДИРАКА СТАТИСТИКА, ХАОС, ЭНЕРГИЯ СВЯЗИ, ЯДРО
UDC 531.9+539.12.01
CHAOS AND PHASE TRANSITION IN ATOMIC NUCLEI
Alexander Trunev Cand.Phys.-Math.Sci., Ph.D.
Director, A&E Trounev IT Consulting, Toronto, Canada
The model of chaotic behavior of nucleons in nuclei, based on the model of nuclear interactions and the Fermi-Dirac statistics is discussed
Keywords: BINDING ENERGY, CHAOS, FERMI-DIRAC STATISTICS, PROTON, NEUTRON, NUCLEUS
It is known that the binding energy of nucleons in nuclei depends on the availability of a regular motion of protons and neutrons in the nuclear shells, and on the chaotic behavior of the nucleon, which introduces uncertainty in the measurement of the mass of the nuclides [1-3]. Models of chaotic behavior of the nucleon are based on an analogy with the chaos in classical dynamical systems, as well as on the concept of quantum chaos [4-5]. In [6] developed a model of the bifurcation of the binding energy in atomic nuclei, based on the theory of nuclear interactions [7] and on the generalized dynamics of the Verhulst-Ricker-Planck equation [8]. To derive the equations of the model using the relationship between the size of the nucleus, binding energy and the number of neutrons and protons, this relation can be represented as follows (see [6-7])
rE = B (A, Z) (1)
Here A = Z + N, N, Z are the number of nucleons, neutrons and protons, respectively.
Using the experimental data [9] and the standard expression of the nuclear
radius, reflecting the weak compressibility of nuclear matter, i.e. r (A) = r°A , we can define the left-hand side of equation (1). As a result, we find the radius
of the core product of the energy due to the number of nucleons. For consistency with the data [9], we set
B / r = a o + a A + a 2 A + a 3 Z + a 4( N — Z) A (2)
a 0 = - 14438.078 ; a, = - 15418.779; a 2 = 15181.734;
a3 = - 687.601; a4 = - 22502.817
Here are the values of the coefficients derived from the data [9] for the
12
binding energy calculated relative to the carbon isotope C. All the coefficients are given in keV. Note that expression (2) has a high degree of accuracy for all nuclides with the number of nucleons A - 20 - Fig. 1.
A
Figure 1: The dependence of the product of nuclei size and binding energy due to the number of nucleons (top) and for number of isotopes (bottom) calculated according to equation (2) and according to [9] are shown.
Construct on the basis of equation (1) a discrete model of the energy levels in nuclei as follows:
BA +1 BA = A 4prA BA + 1 BA
л +1 л r r2 4pr3 / 3 3 r A
гл+1 гл чл'л / j j гл+1 A (3)
On the other hand, the density of nucleons in the nucleus can be related to energy using the Fermi-Dirac statistics, we have
n = A = gzZ / A + gNN / A
•ІЛ . st?.. \ / n I
A 4prA / 3 e(Ez —mz)/0 + 1 e(En —mN)/q + 1 (4)
g E m 0
Here i i i are the weight factors, energy and chemical potential of protons and neutrons, and the statistical temperature of the nucleon, respectively. Consider the results obtained in the simplified model under the condition of equality of chemical potentials and binding energies of the two kinds of nucleons:
m N = m Z m A = — 0b , E Z = E N = — E A / A
In this case, the model can be written as follows
2 K
( Xa+1 + b)( Xa + b )2
e~*A + 1
_ 4 pg л *
A в " 3в3 A
xA =-4^ - b, K = 4pg3'' BaBaa+1 (1 + 1/A )2/3
(5)
gA = gN + gZ
Model (5) differs from similar models developed in [6-7] in that it has no
singularity at the point Xa = 0 . For a fixed number of nucleons the first equation (5) can be regarded as a model of equilibrium in the system of nucleons at nonzero temperature [6]. In this case we have
x +1 =---------------------------K---------------------------b
1+1 (xj + b )2( e - Xi + 1)
The main properties of the model (6) coincide with the properties of the model discussed in [6, 8]. In particular, the bifurcation diagram of model (6) has a characteristic form of "four rats", and reproduces the transition to chaotic behavior in the parameter values b > ln 137 - Fig. 2.
Consider the two-dimensional generalization of the model (6), which occurs when the chemical potential deviation from the set value is proportional to the binding energy, we have:
K u P
X,+1 = -----------n--------------b + y,, y,+1 = bx,
i+1 ( x, + b )2( e -Xi + 1) 1 ^+1 (7)
Log(x+5)
S 9 10 11 12
Figure 2: Bifurcation diagram "four rats", which contains specific zones of chaotic behavior. Calculations made with model (6) at b = ln 137 035999 .
Model (7) has a number of interesting properties. In the range
bi = b = ln 137 ; b = — 0 63 ; 800 < K < 4840 model has a solution, similar to a strange attractor, as described in [10-11] and others - see Fig. 3. In the
parameter range b = ln 137 ; b = b /137 ; b = — 1 0001 ; K > 0 the solution have a form of some geometric figure, apparently indicating the phase transitions in the system of nucleons - Fig. 4.
X
Figure 3: The strange attractor arising in the model (7) in the range of parametersbi = b = ln 137 ; b = — 0.63 ; 800 < K < 4840 .
Finally, we consider the solution of equation (7) in the one dimension case,
i.e. with parameters b = 0; y‘ = 0 , b = ln 137 ; b1 = b /137 , for which the solutions shown in Fig. 4 have been calculated. In this case, equation (6) and (7) differ only in the magnitude of the constant on the right side of these equations. Nevertheless, their bifurcation diagrams differ quite significantly - compare Fig.
3 and 5. For the solutions of equation (7) for the indicated values of the
parameters of the bifurcation diagram has only two branches. In the vicinity of the bifurcation point there is thickening of the solutions that form the line spectrum of energy - Figure 5.
Figure 4: The characteristic shapes that form in the plane of the system (7), describing the energy and chemical potential of nucleons at a constant temperature in the range of parameters
b = ln 137 ; b1 = b /137 ; b = —1.0001 ;1 < K < 200
515.0 515.2 515.4 515.6 515.8 516.0 0 100 200 300 400 500 6С0 *00
Figure 5: The bifurcation diagram with the line spectrum of energy.
Calculations made with the model (7) at b = 0; y‘ = 0 and at
b = ln 137 ; b1 = b /137
Model (6) also can be used to predict the binding energy of nuclei for isotopes of some elements and for isotones as well - see Figure 6. In general a correlation with experimental data is good. Note. That model (6) is not so differ from the information model of the binding energy discussed in our paper [12].
Figure 6: Binding energy per nucleon as a function of neutron number
computed for the gold isotopes (left), and as a function of proton numbers
computed for isotone N=90 (right).
Thus, we have shown that there are phase transitions due to the mutual influence of changes in energy and chemical potential, as well as a line spectrum of energy and chaos in the system of nucleons in nuclei at finite temperature that
previously observed in the model [6].
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