Atomic nuclei even relatively light elements are

Теоретическая физика

Fractals features of the atomic nuclei

D. K. Ershov

Physics Departament, Smolensk State University,

Smolensk, Russia

ABSTRACT

Have been calculated of the exponents Gelders / Herst/ α, β, γ: ΔZ=(ΔN)^α, Δ Z =(Δ A)^γ, Δ N = (Δ A)^β , which is connected Δ Z, Δ N and ΔA in

chains of the even ΔZ families of stable isotopes. Are observed periodical occurrence of the sets of values (α, β, γ). This is clear illustration of the fractals feature of the nuclear structures.

Keywords: stable isotopes, Gelders exponents, fractals, multifractals.

Atomic nuclei even relatively light elements are complex systems consisting of nucleons with many degrees of freedom and participating in different interactions [1] .

Unified strict theory of such systems is still not built, so it seems reasonable use of the phenomenological approaches to the study of various properties of atomic nuclei.

In modern physics is actively and successfully uses the methods of fractal analysis of various systems [2-4].

In this paper we study the fractal properties of the nuclei of stable isotopes of elements.

Connection between the number of neutrons (N) and protons (Z) in the nuclei of stable isotopes has a nonlocal nature of the nuclei of one element () have several isotopes (), and kernel - isotones with the given number of neutrons () may be owned by different elements ().

Nonlocality of mathematically reflected in the fractional value exponents Gelders [5,6]. It is almost evident, for example, if x<<1:

, (1)

while

(endless series). (2)

Therefore integer derivatives are local values, and the fractional - nonlocal (integral), i.e. not defined in point, and only in a variable values region.

Accordingly, on the plane (Z, N) has a region of stability, not a line graph stability.

In this paper we calculate the indicators holder Gelders (Herst) ():

, (3)

, (4)

. (5)

Where - the change in the number of neutrons, Δ Z - number of protons, ∆ A- change the mass number (A = N + Z).

We consider consistently neighboring even-Z values, because only even the charge on the nucleus have several stable isotopes.

And for those of neighbouring even nuclei, we compare the most light isotopes from and most heavy - with :

(6)

(7)

(8)

(9)

Δ Z = 2 (all compared pairs of adjacent elements).

It is easy to show that

(10)

(11)

(12)

It is easy to see that , but for completeness, we give the values of all three indices.

For each pair of adjacent elements we have two sets of values (max and min).

We begin with neon , settings , which determine the oxygen- and end lead . We use data [7, 8].

Just have 36x2 = 72 set of values .

Quite surprising that there are only 4 different set of values : ( 1.0, 0.5, 0.5 ); ( 0.5, 0.774, 0.387); ( 0.387, 0.861, 0.333 ); ( ∞, 1.0, ∞).

Moreover, the uncertainty ∞ are available in 5 cases, and they correspond to the magical N cores .

Another ( the only hand ) anomaly ( 1.0 , 0.387, 0.387 ) was observed in the magic nucleus , i.e. magical N kernel stable isotopes match a kind of “critical” nuclei.

The magic number N = 126 in a stable region to show themselves not “time”.

Sequential sets of values () means something like self-similarity of nuclear structures, i.e. fractal nature of nuclei (more precisely - multifractal) [9].

Recently extensively studied cluster models of atomic nuclei [10, 11].

The clustering properties of nuclei, apparently, also characterize the fractal nature of cores, but, perhaps, in a somewhat different aspect.

It is known that the definition of fractals and fractal dimensions, allows a certain variability [12].

Summing up, we can say that the structure of the nuclei of stable isotopes of elements demonstrates multifractal nature. The phenomenon of scaling /self-similarity, fractal / long been well known in the physics of high energies / z - scaling / [12].

As you can see, this manifests itself in nuclear physics medium energies. Perhaps the main role is played not the value of the energy, and the intensity of the interaction constant of the strong interaction.

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