Mathematical Formulas Describing the Sequences of the Periodic Table

Journal reference: International Journal of Quantum Chemistry, 108, 667-670 (2008)

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Interactive page calculating the sequences of the periodic table

ABSTRACT:

Mathematical formulas describing all of the sequences of the chemical elements are derived from double tetrahedron face centered cubic lattice model.

CONTENTS

1. Introduction

2. Sequences of the Periodic Table

3. Double Tetrahedron Nuclear Structure

4. Analytical Description of the Sequences

5. Conclusion

Figure 1.

Introduction

The modern version of periodic table was developed by Mendeleyev [1, 2] in 1869. The original table relied on the detected the relationship between the properties of the elements and the atomic weights of the elements. Anton van den Broek suggested that the fundamental organizing principle of the table is not the weight but rather the nuclear charge [3, 4]. Charge distribution of the nucleus affects the electron density distribution of the atoms [5] thus the sequence of the nuclear charge distribution might show resemblances to the periodicity of the elements. Investigating the periodicity of the nuclear charge occurring in the structural development of double tetrahedron nucleus reveals the periodicity of the elements. The analytical solution describing this periodicity is derived here.

2. Sequences of the Periodic Table

There are three sequences in the periodic table (Fig. 1-e). Following the first period each of the periods is repeated in the periodic table. This general or fundamental sequence is described as:

S _Fundamental = { 1, 2, 2, 3, 3, 4, 4 . . . }       (1)

I will call the numbers in Eq. (1) to sequence numbers of the periods. The number of elements within the period [Δ Z(n)] has the sequence of

S _{Δ Z(n)} = { 2, 8, 8, 18, 18, 32, 32 . . . }       (2)

The atomic number or the nuclear charge of the elements [Z(n)] in a completely developed period follows the sequence

S _Z(n) = { 2, 10, 18, 36, 54, 86, 118 . . . }       (3)

These sequences bear physical meaning only for the known periods and their extension is only numerical. I will call the sequences in Eqs. (1), (2), and (3) to fundamental, periodic, and atomic number sequence of the periodic table respectively. Formula only for the periodic sequence is known [8, 9] and the number of elements in a period is determined as:

       (4)

where n is the number of the period. Recently Kryachko suggested a new formula [10] for the periodic sequence:

       (5)

No numerical solutions describing the fundamental and the atomic number sequences of the periodic table are known [11].

3. Double Tetrahedron Nuclear Structure

Double tetrahedron shape with alternately arranged protons and neutrons in face-centered cubic lattice has been proposed for the structure of the nucleus [12]. This structure reproduces the symmetry of both quantum mechanics and the periodic system with no discrepancy. The structure is developed from a core tetrahedron (four nucleons) by expanding with one extra layer at each period. The number of charge in the outer shell and the nucleus are identical with the periodicity of the elements in the periodic system (Fig. 1). Deriving analytical solution for the number of charges in the shell and the nucleus describes the sequences of the period table.

4. Analytical Description of the Sequences

The equations describing the sequences of the periodic table are derived here.

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4.1 Fundamental sequence

The relationship between the periods [n] and sequence numbers [m] can be described as:

       where       n ∈ N*        (6)

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4.2 Periodic sequence

The number of protons in the k^th layer of a tetrahedron can be calculated by the triangular number [Tr(k)] [13, 14] (Fig. 3. a)

       (7)

In each structural step of the development the tetrahedron is expanded by one layer in two direction (Fig. 1 b-c) giving the relationship between the tetrahedron layers and the sequence numbers as:

k = 2m       (8)

The number of nucleons in the outer shell of the tetrahedron [Tr(m)] is the sum of the two consecutive triangular numbers.

Tr(m) = Tr(k) + Tr(k-1) = Tr(2m) + Tr(2m-1) = 4m²       (9)

The number of charge in the completely developed shell is

       (10)

Substituting the sequence number from Eq. (6) recovers the Tomkeieff formula Eq. (4)

       (11)

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4.3 Atomic number sequence

Formula giving the total number of charge in the nucleus with completely developed shells can be derived in a similar manner. The total number of nucleons in a tetrahedron with k layers can be determined by its tetrahedral number [Th(k)] [13, 15]

       (12)

Substituting the sequence number from Eq. (8) gives the number of nucleons in a tetrahedron for sequence (m) as:

       (13)

The double tetrahedron nucleus is developed by alternately expanding the tetrahedrons (Fig. 1-d). The number of nucleons in the double tetrahedron is

Th^double(n) = 2Th(m) - Tr^even-period(m) - 4       (14)

The formula

       (15)

can be used to generate 0 for add periods and 1 for even number periods, and

       (16)

Equation (14) can be rewritten then as

       (17)

The number of charge in the nucleus in a completely developed sequence is

       (18)

Combining Eqs. (6), (9), (13), (14), (17), and (18) gives the number of nuclear charge for any period. The atomic number sequence of the periodic table is described as:

       (19)

Substituting m from Eq. (6) gives the atomic number sequence

       (20)

5. Conclusion

The symmetry equivalence between the charge distribution of fcc double tetrahedron lattice and the periodicity of the elements is used to derive analytical solutions for the sequences present in the periodic table. The derived equations (6), (11), and (19) reproduce the fundamental, periodic, and atomic number sequences of the periodic table respectively.

Acknowledgement:

I thank to Xavier Borg for his encouragement and to Vadym Drozd for reading and commenting the manuscript.

References

1. Mendelejeff, D. Zeitschrift für Chemie 1869, 12, 405.

2. Mendeleyev, D. Journal of the Chemical Society 1889, 55, 634.

3. Broek, A. Nature 1911, 87, 78.

4. Broek, A. Nature 1913, 92, 372.

5. Andrae, D. Physics Reports 2000, 336, 6, 413.

6. Scerri, E.R. Scientific American 1998, 279, 78.

7. Karol, P.J.; Nakahara, H.; Petley, B.W.; Vogt, E. Pure Appl. Chem. 2003, 75, 1601.

8. Tomkeieff, M.V. Nature 1951, 167, 954.

9. Tomkeieff, M.V. Nature 1954, 173, 393.

10. Kryachko, E.S. Int. J. Quantum Chem. 2007, 107, 372.

11. On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/Seis.html

12. Garai, J. Physics Archive, 2003, arXiv:nucl-th/0309035v2

13. Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions; National Bureau of Standards Applied Math. Series 1964, 55, 828.

14. Beiler, A.H. Recreations in the Theory of Numbers; Dover, NY, 1964, p. 189.

15. Conway, J.H.; Guy, R.K. The Book of Numbers; Copernicus Press, NY, 1996, p. 83.

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