Fibonacci numbers and the Pascal Triangle

 

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Atomic Structure and the Pascal Triangle

        

        

hydrogen

Atoms are made of 3 types of particles : electrons, protons and neutrons.The atom of hydrogen has one proton and one electron. The proton and electron stay together because the opposite electrical charges attract each other. The electron is constantly spinning around the center of the atom ( called the nucleus ).

The familiar model of an atom is that of a small nucleus composed of protons and neutrons surrounded by rapidly moving electrons. Typically, the atomic diameter is on the order of l0-10 m while that of the nucleus is on the order of 10-15 m. Protons and neutrons have about the same mass (1.00728 and 1.00867 amu respectively) and each is about 1800 times as heavy as an electron. A neutron is electrically neutral, but a proton has a positive charge (+1.6 x 10-19 coulomb*) which is exactly the opposite of the negative charge of an electron. In a neutral atom, the number of electrons around the nucleus equals the number of protons in the nucleus.
The number of protons in the nucleus (the "atomic number", Z) characterizes a chemical element. Atoms of a given element all have the same number of protons, yet may have different masses. The atomic mass number of an atom, A, is given by A = Z + N, where N is the number of neutrons in the nucleus. Since an element is characterized solely by Z, it follows that atoms of a given chemical element may have a varying number of neutrons. Subspecies of chemical elements with the same Z but differing N and A are called isotopes. The atomic weight of an element is the weighted average of the atomic masses of the various naturally occurring isotopes of the element, and the atomic weight scale is based on a value of exactly 12, after the carbon isotope that has an atomic mass number of 12.
The nucleus of an atom weighs less than the sum of the weights of its isolated component particles. The difference between the actual mass and that of the components is called the mass defect. The nuclear forces which bind protons and neutrons together are strong, and the binding energy per nuclear particle (nucleon) amounts to about 1.4 x 10-12 Joules. The greatest nuclear binding energy is found in nuclei of medium atomic number (such as Fe) where N is approximately equal to Z. For nuclei of larger atomic number, such as uranium, N is about equal to 1.5 Z, and the binding energy per nucleon is less. As a consequence of this decreased nuclear stability, some isotopes (of uranium, for example) are unstable. That is, if the uranium isotope,92U235 (Z = 92, A = 235), is bombarded with neutrons, the following reaction can take place:

92U235 + n = 39Y94 + 53I140 + 2n

Here the reaction products are smaller nuclei of increased stability. (In the above convention, subscripts indicate the atomic number and the superscripts the mass number.) Notice that one incident neutron generates fission products including two neutrons - the basis for chain reactions in nuclear reactors and nuclear explosions. Heavy nuclei (even light nuclei) which have an unfavorable ratio in the number of protons and neutrons can spontaneously decay by the emission of particles (helium ions) or particles (electrons). These nuclei are referred to as radioactive.

Numbers, known as figurate or polygonal numbers, appeared in 15th-century arithmetic books and were probably known to the ancient Chinese; but they were of especial interest to the ancient Greek mathematicians. To the Pythagoreans (c. 500 BC), numbers were of paramount significance; everything could be explained by numbers, and numbers were invested with specific characteristics and personalities. Pascal`s Triangle is an arrangement of numbers such that each number is the sum of two numbers immediately above it in the previous row.Pascal simply discovered one of nature`s structural formulas.
The Electronic Configuration Pattern can be found in Pascal`s Triangle of the Second Kind (squar numbers).

The elements of Group 18, the noble gases, are:

  symbol electron configuration
helium(#2) He 1s2
neon(#10) Ne [He]2s22p6
argon(#18) Ar [Ne]3s23p6
krypton(#36) Kr [Ar]3d104s24p6
xenon(#54) Xe [Kr]4d105s25p6
radon(#86) Rn [Xe]4f145d106s26p6

The connection between magic numbers of noble gases on one side and squar numbers, on the other,

     

1
 
 
 
 
 
 
=
1
 
2
4
1
 
 
 
 
 
=
5
 
10
4
4
1
 
 
 
 
=
9
 
18
9
4
4
1
 
 
 
=
18
 
36
9
9
4
4
1
 
 
=
27
 
54
16
9
9
4
4
1
 
=
43
 
86
16
16
9
9
4
4
1
=
59
 
118

     

is highly amusing.

Natural materials are made of natural existing elements or nuclides. Some of these elements have more than one stable isotope, and some may have long-live isotopes. Stable nuclides are the stable isotopes of all elements. Elements with Z = 2, 8, 20, 28, 50, and 82 have many isotopes. Thus, these numbers are called magic numbers. For example, the stable isotopes of calcium (Z = 20) have mass numbers 40, 42, 43, 44, 46, and 48, a total of six. Double-magic-number nuclides :Nuclides 4He2, 16O8, 40Ca20, 48Ca20, and 208Pb82 have magic numbers of protons and magic number of neutrons.Nuclei with magic numbers of protons or neutrons exhibit certain properties, which are analogous to closed shell properties in atoms, including anomalously low masses, high natural abundances and high energy first excited states. Even Z elements are more abundant than odd Z ones of comparable mass.



Also, there is a remarkable connection between magic numbers of nucleons on one side and tetrahedral numbers and triangular numbers, on the other.



This discovery will lead to a better understanding of the periodic atoms and even quantum physic.

     



Belorussian translation
German translation




     



References

Linus Pauling. Research Notebooks  http://osulibrary.orst.edu/specialcollections/rnb/index.html Research Notebook 25. 26.

The MIT Laboratory for Nuclear Science, USA

Lawrence Berkeley National Laboratory, USA

R.A.Brightsen  The Nucleon Cluster Model

The collected works of Bert Schreiber

The double tetrahedron structure of the nucleus

Pythagorean approach to problems of periodicity in fermionic system

Mayer M G, Jensen J H D 1955 Elementary Theory of Nuclear Shell Structure

Cook N D 1999 Proceedings of the St. Andrews Conf. on Fission (Singapore: World Scientific)


     

        

  2001-2005 Radoslav Jovanovic                 updated:  September 2003.