A pythagorean approach of periodicity in chemical and nuclear physics

PERIODICITY IN CHEMICAL AND NUCLEAR PHYSICS D. WEISE International Society for the Interdisciplinary Study of Symmetry (ISIS), Belovejskaya, Complex 39 - Building 2 - Suite 133, 121353 Moscow, Russia tel.: ++ 7-095-423-8979, e-mail: dweise@gol.ru Abstract. A Pythagorean approach to numerical sequences in both chemical and nuclear physics has allowed us to show geometrical analogies based on figurate numbers (three-dimensional forms of Mendeleev's periodic table and packing models for nuclei). Pascal’s triangle is used to deduce analytical equations, from which magic numbers through 12,360 for atoms and through 21.400 for nuclei are calculated. 1. Introduction Structural similarities exist between atomic nuclei and other fermionic systems such as metal clusters. In particular both of these systems exhibit specific magic numbers [e.g., 1]. In this section we briefly recall some relevant mathematical concepts. 1.1. Figurate numbers Figurate or polygonal numbers appeared in 15th-century arithmetic books. and were probably known to ancient Chinese: but they were of particular interest to ancient Greek mathematicians [2]. To the Pythagoreans (c. 500 BC). numbers were of paramount significance: they believed everything could be explained by numbers, and numbers were invested with specific characteristics and "personalities". Among the properties of numbers the Pythagoreans endowed them with "shapes" (2, 3]. 1.2. Pascal's triangle As it is well known Pascal's triangle is an arrangement of numbers such that each number is the sum of the two immediately above it in the previous row [4]. 4. Maruani et al, (eds.), Advanced Topics in Theoretical Chemical Physics, 459-474. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

461 l. natural numbers appear in the second row and second column of the table Table 1 - Pascal's triangle: each number is the sum of the two numbers above it. (shaded in Fig. 1); they are given by the formula: n=C,': (2) Pan ABACO [3 [6 [10 [15 |. 1|| 4 "006, :[10![20 [35 |... 1154115 1135 1170 |. 1 [6 [21 [56 261. Fig. 1 - Natural numbers. The 'gnomon' (generator) of the n-th number, in its familiar rows-of-balls manifestation, is a single ball. The conotation of the term gromon is that originally given by Hero of Alexandria: 'A gnomon is thatform which, when added to some other form, results in a new form similar to the original' [5, 6]. Although named after the French mathematician Blaise Pascal (1623-1662), this triangle appears as early as the 10th century in Chinese mathematical scripts 2. figurate triangular numbers (shaded in Fig. 2) appear in the third row and third column, and are given by the formula: and it may be even older. The entries in Pascal's triangle are also known as the nh = C2 = (n° + n)/2; binomial coefficients and are given by: C=(mk=n/fkm-W (1) Caen! /kifnk)! : (3) C'=n(n+1)/2 Table 2 - Binomial coefficients in Pascal's triangle (revolved counter-clockwise on 45 degrees). HOCICICE CICIFICICIA da ala al. aia la aaa ial. 4 |[10:[20] 5 1115 [35 (0: 6 |} 21 [56[126i dia ele. Fig. 2 - Triangular numbers. The gnomon of the n-th triangular number is a row of n balls. Free) Very interesting entries can be found in the columns and rows of this 3. figurate tetrahedral numbers appear in the fourth row and column, and table. are given by the formula:

n& =C = (n’ + 3n? + 2n)/6. 463 (4) 2. Periodicity of atomic properties 2.1. Atomic magic numbers In the atomic shell model, the shells are filled with electrons in order of increasing energy until they fill a closed shell, producing the inert core of noble gases. The atomic magic numbers 2, 10, 18, 36, 54, 86, correspond to the total number of electrons in filled shells [3, 8, 9]. 2.1.1. Pascal's triangle modification “A” If the one's in the first column are replaced with two's, and if the 2 and 3% columns are filled according to the rule by which Pascal’s triangle is built up, then we find: Fig. 3 - Tetrahedral numbers. The gnomon of the -th tetrahedral number is a layer of n& balls. l. odd numbers, given by the formula: n&P = 2n - 1, In the following we shall use the signs CDI, A, A, EP. etc., for the designation of figurate numbers. 4. the sequence of Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13,... , where each number is equal to the sum of the preceding two. (5) where the sign F is chosen because of its similarity to the gnomon of the square [5. 6]: U =[((1 + VB) (2) - ((1-V3)/2)0] / vB Fig. 5 - Odd numbers have the form of the gnomon of a square. The most familiar form of a gnomon is the L-shaped object of that name that serves as a sundial pointer. 2. square numbers, given by the formula: Fig. 4 - Fibonacci numbers appear as the sums of the numbers located along the slanting strips. Fibonacci numbers occur in various natural patterns, including living beings such as plants (Fig. 4) [7]. Thus, Pascal's triangle appears as a cognitive bridge between microcosm and wildlife.

465 Fig. 6 - Square numbers composed from L-shaped gnomons. After modification: - blank lines are inserted between the lines of the table: - a duplicate square is placed above each entry in the third column; - the fourth column E is filled according to Pascal's triangle rule (1, 4+1=5, 4+5=9, 9+9=18, 9+ 18=27, and so on); - doubles of the fourth column entries are entered in column 2E. It is significant that in the fourth column are written the numbers of electron pairs in inert-gas atoms. Accordingly, in the fifth column are the charges of the- Fig. 8 - 3D Mendeleev's periodic system. Two monochromatic building blocks posed above one another in each layer correspond to an electron orbital. se atoms, or the magic numbers for atoms. e A period corresponds to a horizontal layer. Thickness of each period is two building-blocks. e The number of building-blocks in such a twofold layer corresponds to the number of elements in the period and then to the principal quantum number n. e e The top stratum with the thickness of 1 in each twofold layer corresponds generally to elements with uncoupled electrons. The bottom stratum corresponds to elements for which coupled electrons exist. Two monochromatic building-blocks posed one above another in every layer correspond to one electron orbital. The elements, the properties of which are determined by outside electron subshells: s, p, d, f, g, are grouped together in modules of individual colour. Fig. 7 - Magic numbers for atoms in fifth column of Pascal’s triangle modification. 2.1.2. 3D periodic system The manipulations performed so far result in a 3D model for Mendeleev's periodic system, to which one may give the following interpretation. 2.1.3. Analytical representation of atomic magic numbers The figurate-numerical approach yields an algebraic expression for the atomic numbers of inert gases: Z=[(-1)" Bn + 6) + Zn’ + 12n° + 25n-6 J / 12.

467 where n = 1, 2, 3, ... is the period number (compare with [10]). The values of all numbers in Table 3 correspond to doubling the numbers in the conventional Pascal's triangle. From this modified table it is possible to obtain: The recurrence relation for the inert gases is: Zn+6 = 2Zn+5 + Zns4 > 423 + Zn+2 + 2Zn+1 — Ln, 2.1.4. (3) a. oblong plane numbers, given by the formula: nN = 2nA =n’ + n; Predicted atomic magic numbers (9) Formula 7 generates the following atomic magic numbers: 2, 10, 18, 36, 54, 86, 118/ 168, 218, 290, 362, 460, 558, 686, 814, 976, 1338, 1538, 1780, 2022, 2310, 2598, 2936, 3274, 3666, 4058, 4508, 4938, 5470, 5982, 6560, 7138, 7786, 8434, 9156, 9878, 10678, 11478, 12360, .... 3. Periodicity of nuclear properties Since Pascal”s triangle has proven successful in determining atomic magic numbers, the question arises as to whether it can also help to identify nuclear magic numbers — can it provide a geometrical image and an analytical formula just as for atoms? Fig. 9 - Oblong plane numbers. 3.1. Nuclear magic numbers The magic numbers for nuclei are: 2, 8, 20, 28, 50, 82, 126, corresponding to the total numbers of protons and neutrons in filled nuclear shells. The magic b. oblong pyramidal numbers, given by the formula: numbers here are nuclear equivalents to the atomic numbers of the inert gases: nl = 2n& = (n’+3n?+2n)/3. (10) 2, 10, 18, 36, 54, 86. The nuclear "valley of stability" is the major feature in nuclear stability. However, if this valley is subtracted out, then nuclei with magic numbers of protons and neutrons are seen to be unusually stable [8-11]. They have especially large gaps between their ground and excited states. 3.1.1. Pascal's triangle modification “B” Table 3 - Doubled binomial coefficients in a modification of Pascal's triangle. 8 et cS ft Oe FIA Ac ol act ac hotoe. Ic pot bc} aoosc ict fed ac? Po od cs ig. Ac Be beth bot be. dal Pp Pe Pe Pape 3.1.2. The periodic system for nuclei This system is shown in Figures 11 and 12. The figures for numbers 28, 50, 82, 126 correspond to the same central potential but with strong spin-orbit coupling

D. WEISE [11]. Similar diagrams and formulas have been found on a few pages of Linus where m = 4, 5, 6, 7, .... Pauling's notebooks [12] and in some other works [13, 14]. Analytical representations of nuclear magic numbers can be found in the literature (13, 14]. A formula valid for all nuclear magic numbers is: m = 3; MN = 20 MNm = k* (m? - m) + (mì + Sm)/3, (13) where m = 1, 2, 3, ...andk = 1 if m = 2 or 3. k = 0 otherwise. 30 42156 |. 40 [70 fuites. Here is a recurrence relation for the (11) and (12) series: 20 [40 [52/40]. 112:2521[504j924...|| + — Ma, 54 = 4MN a3 - OMN +2 + 4MN MN [56 166 fa20/9241 ... If. 3.1.4. (14) Predicted nuclear magic numbers Formula 13 generates the nuclear magic numbers: 2, 8, 20, 28, 50, 82, 126, 184, 258, 350, 462, 596, 754, 938, 1150, 1392, 1666, 1974, 2318, 2700, 3122, 3586, 4094, 4648, 5250, 5902, 6606, 7364, 8178, 9050, 9982, 10976, 12034, 13158, 14350, 15612, 16946, 18354, 19838, 21400, .... MN, = OO, ¿92 m) + (mie 5m) / 3 m=7; MN, = 126 2 E) GEE 214; el: 69 Le Lolita 4. S Periodicity of cluster properties It is possible to show that our Pythagorean approach can also be used to rationalize the magic numbers discovered in the analysis of the stability properties of clusters formed by either inert-gas [16] or alkali-metal [17] atoms. Let us now consider the magic number sequences: (a) 7, 29, 66, 118, 185; (b) 7, 19, 37, 61 [18]: (c) 2, 8, 20, 40, 58, 92, 138, 196, 260, 344, 440, 558 [19]. 28 fe 20712 feat [2 [107300 [wops2/fzo 456; 168! 420924]... |... 4.1. Sequence 7, 29, 66, 118, 185 4.1.1. Method of finite differences Fig. 12 - 3D representation of nuclear magic numbers 28, 50, 82, and 126. The number of spheres changing to each other is equal to the magic numbers 2, 8, 20, 28, 50, 82, and 126. The segments of dark blue lines bridge the numbers whose sum is equal to the next magic number. 3.1.3. Analytical representation of nuclear magic numbers The method of finite differences, which is involved in the building of Pascal’s triangle, can sometimes be used to conjecture a formula /(r). Table 4 shows the first and second differences obtained from sequence (a). The lowest values (2, 8, 20) correspond to the independent nucleon motion in a Table 4 - Finite differences (a). single particle harmonic oscillator potential [15]: MNm = (m + 3m? + 2m)/3, (11) where m = 1, 2, 3. The numbers 28, 50, 82, 126 are given by the formula: MNm = (m? + 5m)/3, (12) Magic numbers First differences 7 22 15 29 37 15 66 52 15 118 Second differences

A PYTHAGOREAN APPROACH TO ... Note that the second differences are constant (i.e., 15). If and when we reach a 471 4.2.2. Analytical representation of series (b) row that contains a constant value, we can write an explicit expression for f(n). In Pythagorean terms one would say that if the method of finite differences is Explicit form: applicable, each difference can be considered as a gnomon for a previous series. We can consider numbers in Table 4 as Pascal’s triangle modification “C”. C,=61nA+1=3n + 3n+ 1. (17) Cn+3 = 3Cn+2 T 3Cn+1 + Cr (18) Recurrence relation: 4.1.2. Analytical representation of series (a) Explicit form: C, = (I5n?-n)J/2, (15) 4.3. Sequence 2, 8, 20, 40, 58, 92, 138, 196, 260, 344, 440, 558 4.3.1. Recurrence relation: Cn+3 = 3Cy 42 - 301 + Ch (16) Method of finite differences The method of finite differences yields Table 7. Table 7 - Finite differences (c). 4.2. Sequence 7, 19, 37, 61 4.2.1. Method of finite differences Magic numbers | First differences | Second differences The method of finite differences yields Table 5. Table 5 - Finite differences (b). Magic numbers | First differences | Second differences 2 6 8 12 6 8 20 20 -2 40 18 16 58 34 12 12 92 46 7 12 6 138 58 6 19 13 6 196 64 20 37 24 260 84 12 344 96 22 440 118 61 The resulting modified Pascal’s triangle allows prediction of heavier clusters of this kind, as shown below. Table 6 - Pascal's triangle modification “D”. 6 558 Here second differences are unequal. One may think that some clusters are combinations of others: e.g., 58+138=196. On the other hand, numbers 2, 8, 20 are doubled pyramids (magic numbers for nuclei). 6 6 7 6 12 18 19 One may assume that, first, the series could involve approximate values and, 6 18 36 37 second, the series could be non-uniform. Using a modification of Pascal’s train- 6 24 60 61 6 30 90 91 6 36 126 127 6 42 168 169 6 48 216 217 gle that we will call “E” let us then construct an “ideal” series of magic numbers based on the second difference 12. It can be seen on Table 8 that the resulting ideal numbers are usually tdentical or very close to the observed numbers (usually within less than 2 %). Again this procedure can be used to predict magic numbers for heavier clusters of this kind. 4.3.2. Analytical representation of series (c)

473 ment is ever made. It could be the ideal gas “Olesium”! I do not know whether the scientific community would accept this idea. The explicit form for the “ideal” numbers given in Table 8 is: C, = 2 (3n*+ 2n + 13). (19) Acknowledgments Special thanks are due to Pr Ray Hefferlin of Southern Adventist University (Collegedale, TN, USA) for useful advice about this work, and to a referee for This equation is a second-degree polynomial corresponding to 2D forms, while the numbers 2, 8, 20, as well as 40 (badly described by the above formula), are pointing out the relevance of atomic clusters. represented by a third-degree polynomial corresponding to 3D forms: (20) MNn = (n° + In? + 2n) /3. References 1. D. Bonatsos, N. Karoussos, C. Daskaloyannis, S. B. Drenska, N. Minkov, P. P. Raychev, R. P. Roussev, and J. Maruani: "Symmetries in nuclei, molecules and Table 8 - Pascal's triangle modification “E”. atomic clusters", XIIIth International School on Nuclear Physics, Neutron Physics and Nuclear Energy (Varna, Bulgaria, September 1999). 2. The Encyclopedia Britannica (CD ROM 2000 Deluxe Edition & www pages). +26 10 10 36 40 12 22 32 58 58 12 34 66 92 92 12 46 112 138 138 12 58 170 196 196 12 70 240 266 260 12 82 322 348 344 12 94 416 442 440 12 106 522 548 558 12 118 640 666 4. http://www,math.tamu.edu/-don.allen/history/pythag/pythag.htm] Pascal's triangle http://mathforum.org/dr.math/fag/fag.pascal.triangle.htm] 5. Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, 1999, 259 pages. FDEIRiSETlEf)“iLe.i The recurrence relation corresponding to Eq. (19) is: Crea = 3Cn+2-3Cnel + Cn Pythagoras and the Pythagoreans 3. (21) It then appears that clusters at the beginning and in the continuation of the series TL 6. J. Schombert, Gnomones 7. D. Weise, Principle of minimax and rising phyllotaxis 8. 9. http://zebu.uoregon.edu/~js/glossary/gnomones.html http://members.tripod.com/vismath/dima Magic numbers http://www.ph.surrey.ac.uk/~phs Lsv/thesis/node28.html The shell model http://www.physics.carleton.ca/courses/75.364/np-2html/node2.html 10. Atomic nucleus and atomic shells http://www.apsidium.com/number/number.htm 11. M. G. Mayer and J. H. D. Jensen, Elementary Theory of Nuclear Shell Structure, correspond to different gnomons. John Wiley, New York, 1955, 269 pages. 12. Linus Pauling, Research notebooks Conclusion 13. V. Ladma, Magic numbers In this paper we have attempted to couple modern atomic theories with an ancient guiding principle. In particular, a parallel between atomic, nuclear and cluster shells is carried out using a fundamental concept of the Pythagorean school known as the gromon. Dedication | This work is dedicated to an astonishing person, Abolina Olesia. I would like to see her name given to element 118 if a valid discovery of such an elehttp://osulibrary.orst.edw/specialcollections/rnb/index.html Research Notebook 25 http://www.sweb.cz/vladimir ladma/english/notes/texts/magien.htm 14. R. Jovanovic, Atomic structure and Pascal's triangle http:/milan.milanovic.org/math/english/atom/atom.html 15. S. A. Moszkowski, Maria Goeppert Mayer Talk presented at APS meeting at Indianapolis on May 4, 1996, UCLA, USA 16. Experiment: O. Echt, K. Sattler and E. Recknagel, Phys. Rev. Lett. 47 (1981) 1121; A. Ding and J. Hesslich, Chem. Phys. Lett. 94 (1983) 54; Theory: G. S. Anagnostatos, Phys. Lett. A 124 (1987) 85; ibid. 133 (1988) 419; ibid. 150 (1990) 303. 17. Experiment: W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou and M. L. Cohen, Phys. Rev. Lett. 52 (1984) 2141; C. Brechignac, Ph. Cahuzac and J.-Ph. Roux, Chem. Phys. Lett. 127 (1986) 445; Theory: D. Bonatsos, N.

COLOUR FIGURES Karoussos, P. P. Raychev, R. P. Roussev and P. A. Terziev, Chem. Phys. Lett. 302 (1999) 392; A. I. Kuleff, J. Maruani and P. P. Raychev, Adv. Quantum Chem. 40 (2001) 279; D. Bonatsos, A. I. Kuleff, J. Maruani, P. P. Raychev and P. A. Terziev, Int. J. Quantum Chem. 89 (2002) 377. 18. J. Wang and L. Holmlid, Chem. Phys. Lett. 295 (1998) 500. 19. S. Bj@rnholm, J. Borggreen, O. Hecht, K. Hansen, J. Pedersen and A. D. Rasmussen, The Influence of shells ... on the abundance spectra of sodium metal clusters http://pubpages.unh.edu/~oee/abstracts.htm 20. Periodic Tables http://www.chemistrycoach.com/periodic_tables.htm 21. Extended Periodic Table http://www.sweethaven.com/chemele/pertab0 | html 22. Dimitri Mendeleev’s Original Periodic Table (and other tables) http://chemlab.pc.maricopa.edu/periodic/foldedtable.html 23. Jeries A. Rihani’s Extended Periodic Table of the Elements http://jeries.rihani.com/ 24. Dr Timmothy Stowe's Physicists Periodic Table http://140.198.18.108/periodic/stowetable.html SeEaS 25. Emil Zmaczynski’s Triangular Long Form Periodic Table http://140.198.18.108/periodic/triangletable.html 26. D. Weise, ISIS Symmetry congress & exhibition http://www. isis-s.unsw.edu.au/interact/gallery/image_files/wiese/weise.htm| o li] C=nt/klnk)t : G3=n(n+1)(n4+2)

COLOUR FIGURES COLOUR FIGURES 2 Sì _o | | lvol È [Ue160[|E:N0 LS) dd 00 real. D. Weise, Fig. 4. = CRE HSTCS afele | |

COLOUR FIGURES J. Maruani, R. Lefebvre, and M. Rantanen, Fig. 3 - Theodor Benfey's periodic table.