Physical Model Explaining the Periodic Pattern of the Chemical Elements

8/13/2019 Physical Model Explaining the Periodic Pattern of the Chemical Elements

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Physical Model Explaining the Periodic Pattern of the Chemical Elements

Jozsef Garai

Abstract: The fundamental organizing principle resulting in the periodic table is the nuclear

charge. Arranging the chemical elements in an increasing atomic number order, a symmetry pattern known as the Periodic Table is detectable. The correlation between nuclear charge and

the Periodic System of the Chemical Elements (PSCE) indicates that the symmetry emerges from

the nucleus. Nuclear symmetry can only be developed if the positions of the nucleons are

preserved. Thus the phase of the nucleus must be solid where the positions of the nucleons are

preserved in a lattice. A lattice model, representing the protons and the neutrons by equal

spheres and arranging them alternately in a face centered cubic structure forming a double

tetrahedron, is able to reproduce all of the properties of the nucleus including the quantum

numbers and the periodicity of the elements. Using this nuclear structure model, an attempt ismade here to give a physical explanation for the periodicity of the chemical elements.

Keywords: Periodic table; sequencies in the periodic table; atomic quantum numbers;

symmetry; nuclear structure; nuclear model

Introduction

By 1860 about 60 elements were identified, and this initiated a quest to find their systematic

arrangement. Based on similarities, Döbereiner in 1829 suggested grouping the elements intotriads [1]. Newlands (1864) in England arranged the elements in order of increasing atomic

weights and, based on the repetition of chemical properties, proposed the “Law of Octaves” [2].

Listing the elements also by mass, Mendeleev (1869) in Russia [3] and Meyer (1870) in

Germany [4] simultaneously proposed a 17-column arrangement with two partial periods of

seven elements each (Li-F and Na-Cl) followed by nearly complete periods (K-Br and Rb-I).

Mendeleev gets higher credit for this discovery because he published the results first; he also

rearranged a few elements out of strict mass sequence in order to fit better to the properties of

their neighbors and corrected mistakes in the values of several atomic masses. Additionally, he predicted the existence and the properties of a few new elements by leaving empty cells in his

table. Mendeleev’s periodic table did not include the noble gasses, which were discovered later.

Argon was identified by Rayleigh in 1895 [5]. The remainder of the noble gasses were

discovered by Ramsey (1897) who positioned them in the periodic table in a new colum [6].

Anton van den Broek (1911) suggested that the fundamental organizing principle of the table is

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not the weight but rather the nuclear charge [7], which is equivalent with the atomic number.

The extended 18-column table was slightly modified based on Moseley’s experiments (1913),

and he rearranged the table according to atomic number [8]. The discovery of the transuranium

elements from 94-102 by Seaborg (1951) expanded the table [9]. He also reconfigured the table

by placing the lanthanide/actinide series at the bottom. There is no “standard” or approved

periodic table. The only specific recommendation of the International Union of Pure and

Applied Chemistry (IUPAC), which is the governing body in Chemistry, is that the Periodic

System of the Chemical Elements (PSCE) should follow the 1 to 18 group numbering [10].

The majority of general and physical chemistry books indicates that the symmetry of PSCE is

satisfactorily explained by either the electronic structure of the elements [11] or by quantum

mechanics [12]. This presentation is misleading [13] because fundamental questions regarding

to PSCE are still under discussion. One example of the unsettled questions is the positions of

Hydrogen and Helium. Hydrogen has one 1s electron but also one electron is needed to attain

inert configuration. Thus it can be either placed in the 1st group or in the 17th group. Based on

chemical behavior, Hydrogen neither halogen nor alkali metal but rather both. Thus the position

of Hydrogen in the PSCE is uncertain [14]. Helium with its 1s2 electron configuration is the

other element with uncertain position. Helium should be in the 2nd group. However, based on

its chemical properties, equivalent to inert gas, is placed into group 18. Besides Helium, the

outermost electron configuration of group 18th is 2p6. No theoretical explanation for the shift in

the electron configuration from 2p6 to 1s2 or vice versa is offered. The substantial number of

articles in the current literature [e.g. 15] discussing the fundamental problems of PSCE nearly

one-and-a-half centuries after its invention, clearly indicates that the complete understanding of

the symmetry experessed by PSCE is still lacking.

Despite the remaining outstanding questions, there is a general agreement that the elements

should be arranged in an increasing atomic number order in PSCE. The correlaton between

nuclear charge and symmetry in PSCE indicates that nuclear structure might provide a clue for

the periodicity of the elements. This possibility is investigated here.

Nuclear Lattice Model

The most widely accepted models for the nucleus are the shell [16], the liquid drop [17] and

the cluster [18]. The shell model assumes a gas phase (Fermi) for the nucleus and is able to

explain the independent quantum characteristics of the nucleons. The liquid drop model is able

to explain the observed saturation properties of the nuclear forces, the low compressibility of the

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nucleus and the well-defined nuclear surface. The clustering of alpha particles can be explained

by a semi-solid phase. These models are able to describe successfully certain selected properties

of the nucleus; however, none of them able to give a comprehensive description. The basic

assumptions of the models are contradictory; therefore, it is impossible combine them and

develop a hybrid model.

The assumed phase of the nucleus in these models are gas, liquid or semi-solid. None of

these phases can maintain nucleon order; therefore, these models cannot explain the symmetry in

the PSCE. The preservation of symmetry requires a solid phase nucleus, in which the nucleons

are arranged in a lattice. Solid phase or lattice models have not been considered for many

decades as a viable option because of the uncertainty principle and the lack of diffraction. In the

1960s, the discovery of quarks and neutron star research satisfactorily answered these objections

and opened the door for lattice models. The first nuclear lattice models were suggested by Linus

Pauling in 1965 [19].

The lattice models can easily reproduce the various shell, liquid-drop, and cluster properties

[20]. Asymmetric fission and heavy-ion multi-fragmentation are some of the phenomena that

the traditional models of nuclear structure theory cannot explain, yet can be reproduced by lattice

models [21]. Significant effort has been made to find correlation between lattice positions and

quantum numbers with partial success for FCC structure [22]. The symmetries of the

Schrodinger’s equation also correspond to FCC geometry [23]

The common features of the developed lattice models are that the protons and the neutrons

have the same size and they are alternately arranged in a closest packing crystal structure [24].

These assumptions are reasonable. The radii of protons and the neutrons differ only slightly

[25]. The same proton and neutron magic numbers indicate same structural development for

both protons and neutrons in an alternating proton and neutron array. The equal spheres will

most likely utilize the available space in the most efficient way, which is a closest packing

arrangement. Previous investigations [22], which expanded the FCC structure spherically, had

partial success in finding correspondence between lattice positions and quantum states. The

structure of FCC consists of only tetrahedron and octahedron units. The expansion of these units

should be investigated first if someone wants to look for structural symmetry patterns in the FCC

lattice [26].

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Double Tetrahedron FCC Lattice Model

The first sequence of the nuclear structural development is completed by the nucleus of

, which contains four nucleons. These four nucleons in an FCC lattice should form a

tetrahedron. Developing a tetrahedron in the first period of the PSCE indicates that the seed for

the nuclear lattice most likely is a tetrahedron instead of an octahedron. Calculations of potential

models, constrained by the hadron spectrum for the confinement of the relativistic quark [27] and

colored quark exchange model [28], are also consistent with a tetrahedron forming the

nucleus. Expanding this tetrahedron seed of four nucleons, the nucleus of He, is investigated

here.

The tetrahedron shape of equal spheres arranged in FCC packing can be formed from layers

of equilateral triangles packed in two dimensional closest packing arrangement as shown in Fig.

1/a. Starting with one sphere and increasing the length of the side of the triangles by one

additional sphere, the number of nucleons in each triangle plane will be 1, 3, 6, 10, 15, 21, 28,

36... (Fig. 1/a). Stacking these layers, the number of spheres in two consecutive layers are 4, 16,

36, 54... Assuming that protons and neutrons are arranged alternately in the lattice; therefore,

dividing the number of spheres by two gives the proton numbers to 2, 8, 18, 32... respectively

(Fig. 1/b). These numbers are identical with the number of possible states of the principle

quantum numbers. If the tetrahedron is expended on two faces then a shell-like structure can be

formed (Fig. 1/c), which is consistent with the physics of the principle quantum number.

Investigating how many spheres can be accommodated in one row of the outer shell gives the

total number of spheres in one row to 4, 12, 20, 28..., which corresponds to 2, 6, 10, 14... proton

number (Fig. 2/a). These proton numbers are identical with the number of states determined by

the angular momentum quantum numbers corresponding to s, p, d and f orbitals. The rows in the

outer layers of the tetrahedron are a unit distance away from each other; thus the identical

agreement of the number of nucleons with the angular momentum quantum numbers is not only

in numerical agreement but also bears the same physical meaning defined by quantum theory.

The number of different positions of protons in one row of the outer shell is the same as the

number of magnetic quantum numbers (Fig. 2/b). The lattice positions also identically

reproducing the multiplicities. Thus developing a tetrahedron in an FCC lattice is able to

reproduce all of the quantum numbers. The lattice positions not only reproduce the quantum

numbers but also bear the same physical meaning.

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Fig. 1. Representing both protons and neutrons with equal spheres and arranging them in FCC

structure, the number of protons in the outer layers of a tetrahedron formation is the same as the

number of possible states of the principle quantum numbers.

(a) The number of spheres in a two dimensional closed packing arrangement in equilateral

triangles.

(b) The number of spheres in two consecutive layers of the tetrahedron formation. Assuming a

proton-neutron ratio of one, the outer layers of the tetrahedron contain the same number of protons

as predicted by quantum theory.

(c) The same tetrahedron formation can be developed by adding the new layers to alternate sides.

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Fig. 2. If a double tetrahedron has been developed from a core tetrahedron, which contains four

spheres, then the number of protons in one line of the outer layer (shell) of the tetrahedron is

equivalent with the number of states of the angular momentum quantum number or the

corresponding subshells. The number of different positions of the protons in one line of the shell

is the same as the number of magnetic quantum numbers. The red circles represent the outer shellnucleons. The tetrahedron has four vertexes; therefore, the number of spheres is multiplied by

four.

(a) Number of spheres in one layer of a vertex of the double tetrahedron.

(b) Number of the different proton positions in one layer of a vertex of a double tetrahedron.

The periodic table reveals three sequences: fundamental, periodic, and atomic number [29].

The fundamental sequence of the table is:

(1)

The number of elements within the period has the sequence of

.(2)

The atomic number or the nuclear charge of the elements in a completely developed

period follows the sequence

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.(3)

The relationship between the periods (n) and the fundamental sequence numbers (m) can be

described as:

(4)

The sequence of the number of the elements can be given as:

(5)

and the atomic number sequence as [29]:

(6)

The sequence of the principle quantum number [PQN(n)] is

(7)

which can be described as:

(8)

The sequence of the principle quantum numbers [Eq. 7] shows some similarity to the sequence of

the elements [Eq. 2] but the two sequences are not identical as can be seen from Eq. 5 and Eq. 8.

Quantum theory does not offer an explanation for the fundamental and atomic number sequences

[Eqs. 1 and 3] (Fig. 3).

Fig. 3. The sequences of quantum mechanics and the periodic table. It can be seen that quantum

theory is unable to reproduce and explain any of the three sequences of PSCE; therefore, it can

not be considered as an adequate physical explanation for the periodicity.

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Based on these deficiencies, it has been suggested that quantum mechanics is unable to explain

the most important aspects of the periodic table [13].

The difference between the principle quantum number and the periodic sequences is that the

periodic sequences are repeated following the first period. This sequence can be reproduced by

developing a double tetrahedron shape instead of a single one. Thus, a double tetrahedron

nuclear FCC lattice structure is able to reproduce both the quantum numbers and the periodicity

of the elements. Based on these agreements, it has been proposed that the nucleus of the

elements is arranged in an FCC lattice forming a double tetrahedron [26]. Three dimensional

images of the fully-developed double tetrahedrons corresponding to the elements Ne, Ar, Xe and

Ra are shown in Fig. 4. The mathematical equations [Eqs. 4-6] describing the three sequences of

the periodic table [Eq. 1-3] have been derived from this double tetrahedron nuclear lattice model

[29].

Fig. 4. 3D images of the structures of the fully developed nuclear structures of the noble gases

and the structures of C and O in the second period are shown. The red and blue spheres represent

protons and neutrons respectively. In structures, where the charge distributions are not predicted, a

grey sphere has been used to represent a nucleon

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Physical Explanation for the Periodicity of the Elements

It has been shown that the double tetrahedron nuclear lattice model succesfully reproduces

not only all the quantum numbers but also the sequences of PSCE. The physical explanation

behind this coincidence is investigated.Electrons are bounded to protons by electrostatic attraction described by the Coulomb’s law.

The electrostatic force, in scalar from, between a proton and an electron pair (F p-e) can be

calculated as:

(9)

where !o is the electric constant (8.854187817..."10-12 Fm -1), qe is the elementary charge of the

electron (1.602"10#19 C) [30], which is equal with opposite sign to the charge of the proton, and r

is the distance between the proton and the electron. In order to make a distinction among the protons in the nuclear structure, the protons are labeled by their atomic number (Z) in accordance

with the development of the nuclear structure. The first proton in the Hydrogen nucleus is then P

(1), the next proton added into the structure to form the nucleus of Helium is P(2) and the last

proton added to the nuclear structure of element Z-1 to form the nucleus of element Z is P(Z).

The electrons are labeled accordingly. Thus electron(1) is captured by P(1) and electron(Z) is

captured by P(Z). Equation 9 can be used to calculate the attraction of protons for elements

higher then Hydrogen by assuming that the rest of the charge of the nucleus [qe"(Z-1)] is

shielded by Z-1 electrons. Thus the equation is valid for any element if r > r atom-ionized, wherer atom-ionized is the atomic radius of the element ionized due to one missing electron.

The charge distribution of the nuclear lattice is not continous; therefore, the nucleus can not

be considered as a point charge. When the attraction of a single proton outside of the valence

shell is calcualted, then the lattice position of the given proton must be taken into considertion.

The strongest attraction of a given proton in the direction of P-NCC then can be calculated as:

(10)

where r e-NCC is the distance between the nuclear charge center and the electron and dP-NCC is the

distance between the charge center of the nucleus and the given proton. Depending on the lattice

positions of the protons, inequality between the attractive force of two elements occurs as:

or (11)

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In Eq. 10 dP-NCC is constant; therefore, the inequality between the forces is independent of r and

the inequality remains as long as the condition

(12)

is satisfied, where h is the Planck constant. The relative attraction of proton Z-1 and Z can be

described by the distance from the charge center of the nucleus because if:

then (13)

or vica versa. In order to overcome the repulsion force of the existing valence electron shell, the

attractive force on the captured electron(Z) must be sronger than the attraction force on any other

electrons in the valence shell. Thus the following unequality must be satisfied:

or (14)

When this condition [Eq. 14] is not fulfilled, electron(Z) is unable to join to the valence shell and

starts a new shell outside of the existing one.

Investigating the distances of the nucleons arranged in an FCC lattice forming a tetrahedron,

the hights of a tetrahedron [h$] with unit length is calculated first as:

(15)

Assuming a unit mass for the nucleons the distance between the vertex of the tetrahedron and

the mass center is:

(16)

The distance between the face of the tetrahedron and the mass center is then:

(17)

The tetrahedron is expanded by one-one layers on both sides; thus the length of the side of

the tetrahedron is increased by two units [Fig. 1]. The distance between the center of a sphere

(nucleon lattice position), placed on the surface of the tetrahedron, and the mass center in period

n+1 is then

(18)

where n is the number of periods. The distance between a sphere placed at the vertex of the

tetrahedron in period n is then:

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(19)

It can be seen that

(20)

for any n. Thus, the attraction of the proton which starts to form the new layer on the surface of

the tetrahedron is always smaller than the attraction of the proton completing the previous layer

(Fig. 5).

Fig. 5. Physical model proposed to explain the periodicity of PSCE. When a new layer starts to

build up, the proton is closer towards to the nuclear charge center (NCC) than the proton

completing the previous shell. Thus proton [P(n+1)] starting the period of n+1 has weaker

attraction for the captured electron than the proton [P(n)] completing the period n. The captured

electron(n+1) is unable to overcome the repulsion of the existing shell because of its weaker

attraction and starts to built up a new shell outside of the exising one.

The substitution of the charge center of the nucleus with the mass center does not change the

outcome of the conclusion. For the fully-developed structures of the double tetrahedron nucleus

in He, Ne, Ar, Xe and Ra, the mass and charge centers are coincident. When a new nuclear shell

is started by adding a proton, then the charge center shifts towards the proton resulting in smaller

distance between the charge center and the proton than the distance between the proton and the

mass center. Thus using the mass center instead of the charge center is a conservative estimate.

Based on the presented geometry it is concluded that arranging the nucleons in an FCC lattice

building up a double tetrahedron results in the following inequalities

(21)

therefore,

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(22)

Based on this weaker attraction it is suggested that a new electron shell should be formed

when a new layer starts to built up in the nucleus. This prediction is consistent with the sequence

of the electronic structure of the elements.

Conclusions

Investigating the symmetry expressed by PSCE it is shown that this symmetry is identical

with the symmetry of a double tetrahedron nuclear lattice arranged in face centered cubic

structure. Based on this symmetry agreement, it is suggested that the patterns present in PSCE

are generated by the lattice positions of the nucleons arranged in FCC structure forming a double

tetrahedron shape. A physical model consistent with the periodic pattern of PSCE is also proposed following the stepwise logical reasoning given below.

1./ The underlying organizing principle of the PSCE is that the elements are arranged in an

increasing nuclear charge order. The symmetry patterns emerging from this order; therefore,

should originate from or relate to the nucleus.

2./ The preservation of symmetry requires a solid phase nucleus arranged in a lattice.

3./ The possibility of a nuclear lattice structure is a viable option. Long held objections, the

uncertainty principle and the lack of diffraction, have been satisfactorily answered in the 1960s

by the discovery of quarks and through neutron star research.

4./ Lattice models have three reasonable assumptions: one, the size of protons and neutrons

are the same, which is known from experiments; two, protons and neutrons are arranged

alternately in the lattice, which is supported by the same magic numbers of protons and neutrons;

three, under the strong nuclear force, these equal spheres should occupy the available space in

the most efficient way. Thus the nucleons in the lattice should be arranged in a closest packing

arrangement, either FCC or HCP, which makes common sense.

5./ The fundamental building blocks of these closest packing lattices are tetrahedron and

octahedron. The “crystal growth” of the nucleus should start from either of these basic “seeds”.

6./ The first period is completed by Helium, with a nucleus that contains four nucleons. The

closest packing of four spheres forms a tetrahedron; therefore, the expansion of tetrahedron is

investigated.

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7./ Expanding a tetrahedron containing four spheres by adding one-one extra layers on two

sides reproduces the quantum numbers. The number of protons are calculated by adding up the

number of spheres contained in the outer two layers of the tetrahedron and dividing this number

by two based on the assumption that the neutrons and protons alternate in the lattice.

8./ If in FCC lattice arrangement a double tetrahedron is developed instead of a single one,

then the expansion of this double tetrahedron reproduces all of the sequences present in PSCE.

Based on the identical agreement with the quantum numbers and with PSCE, the double

tetrahedron FCC lattice model is suggested for the nucleus of the elements with completely

developed nuclear shells.

9./ In a nuclear lattice the protons have well-defined position; therefore, the attraction of the

protons on the electrons varies.

10./ The relative attractions of the protons can be described by their distance from the charge

center of the nucleus. Calculating these distances reveals that when a new layer of the double

tetrahedron starts to build up, the distance of the first proton in the new layer from the charge

center of the nucleus is smaller than the distance of the proton completing the preceding layer.

The shorter distance between the charge center of the nucleus and the proton results in a smaller

attraction on the captured electron.

11./ It is suggested that this weaker attraction force results in the formation of a new electron

shell because the newly captured electron is unable to overcome the repulsion of the electrons

comprising the existing outer shell.

Acknowledgement: I thank Michael Sukop for reading and commenting on the manuscript.

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Physical Model Explaining the Periodic Pattern of the Chemical Elements