Quantum Vectors and Atomic Stack Symmetry©

The Janet Periodic Table of Elements (1929) may be re-arranged as a series of square matrices. The matrices are of different sizes and each matrix organizes the atomic orbitals into square concentric rings. Each cell may be assigned an atomic number which also identifies a “most significant electron”. The matrices may be stacked vertically to form “The Periodic Stack of Elements” as shown below.

The sub-atomic particles may also be arranged as square matrices. These matrices may be stacked to form “The Periodic Stack of Particles”.

Please send your comments to; doulting@shaw.ca Last Revision 25 August 2008.

Contents

1 Introduction

2 The Janet Periodic Table of Elements

3 Most Significant Electron

4 The Periodic Stack of Elements

5 The Quantum Numbers

6 Orbital Angular Momentum

7 Principal Angular Momentum

8 The Quantum Matrix

9 Couples

10 The Aufbau Couple

11 Quantum Forces

12 Uncertainty

13 Fundamental Uncertainty

14 Fundamental Energy

15 Fundamental Force

16 Centripetal Forces

17 Electro-magnetic Forces

18 Quantization of Energies

19 Quantization of Forces

20 Force Balance

21 Calculations

22 Dynamic Matrices

23 Fundamental Principles of Chemistry

24 The Periodic Stack of Particles

25 Particle Quantum Numbers

26 The Particle Number

27 Nomenclature

28 References

1. Introduction

Various forms of the Periodic Table of Elements are popular. One interesting arrangement is “The Periodic Table in Flatland”³. Another interesting table is derived from atomic ionization potentials¹. The Janet Periodic Table² (1929) is an interesting 2D arrangement of the natural elements. Further information on the Janet table may be found at;

http://www.ipgp.jussieu.fr/~tarantola/

The Janet table can be arranged as a series of four square matrices. Each matrix is a different size. Each matrix arranges the atomic orbitals (s,p,d,f) into a series of concentric rings. The “s” ring is the core and the other orbitals form concentric rings around the core. The atomic number is used to identify the most energetic electron of an element in the ground state. Elements which are located on a major diagonal have the quantum number for orbital magnetic moment (m_ℓ) equal to zero.

The matrices may be stacked vertically with the core elements aligned vertically. This is a 3D arrangement of the natural elements called “The Periodic Stack of Elements”. Vertical sections through the stack give interesting groupings of elements.

If a quantum number ‘m_n’ is admitted, then it may represent a magnetic moment associated with the principal quantum number (n). From observation it is possible to deduce a 2x3 matrix of quantum numbers.

The top row is; n , ℓ , s

The bottom row is; m_n , m_ℓ , m_s

A new set of quantum numbers (a, m_a) may be derived from quantum numbers in the matrix. This new set is called the “aufbau pair” and it quantifies the aufbau principle. The forces acting on a bound electron within an atom induce an “energy halo” around the electron.

The sub-atomic particles may also be arranged as a series of square matrices.

2. The Janet Periodic Table of Elements

Charles Janet first proposed this form of the Periodic Table in 1929, since then others have independently rediscovered it. For more information on the Janet Periodic Table see;

http:// www.ipgp.jussieu.fr/~tarantola

The Janet form of the Periodic Table has been proposed from time to time by various persons⁴. Acceptance of this table requires minor renumeration of the periods⁷. According to Winter⁸, the Janet table is preferred by some persons to the standard form. The Janet table as displayed below has been separated into three parts for convenience. Part C contains the Lanthanides and the Actinides. Most elements remain in the same period and group as they are traditionally placed.

Janet Periodic Table of Elements - Part A ;  Periods are in rows, groups are in columns. The orbital of the most significant electron is shown at the top of each column. Columns are numbered from right to left, and rows from top to bottom.

Orb	p	p	p	p	p	p	s	s
P\G	13	14	15	16	17	18	1	2	row
0							1 H	2 He	1
I							3 Li	4 Be	2
II	5 B	6 C	7 N	8 O	9 F	10 Ne	11 Na	12 Mg	3
III	13 Al	14 Si	15 P	16 S	17 Cl	18 Ar	19 K	20 Ca	4
IV	31 Ga	32 Ge	33 As	34 Se	35 Br	36 Kr	37 Rb	38 Sr	5
V	49 In	50 Sn	51 Sb	52 Te	53 I	54 Xe	55 Cs	56 Ba	6
VI	81 Tl	82 Pb	83 Bi	84 Po	85 At	86 Rn	87 Fr	88 Ra	7
VII	113	114	115	116	117	118	119	120	8
Col	8	7	6	5	4	3	2	1

Janet Periodic Table of Elements - Part B ;

Orb	d	d	d	d	d	d	d	d	d	d
P\G	3	4	5	6	7	8	9	10	11	12	row
IV	21 Sc	22 Ti	23 V	24 Cr	25 Mn	26 Fe	27 Co	28 Ni	29 Cu	30 Zn	5
V	39 Y	40 Zr	41 Nb	42 Mo	43 Tc	44 Ru	45 Rh	46 Pd	47 Ag	48 Cd	6
VI	71 Lu	72 Hf	73 Ta	74 W	75 Re	76 Os	77 Ir	78 Pt	79 Au	80 Hg	7
VII	103 Lr	104 Rf	105 Db	106 Sg	107 Bh	108 Hs	109 Mt	110 Ds	111	112	8
Col	18	17	16	15	14	13	12	11	10	9

Janet Periodic Table of Elements - Part C ;

Orb

f

f

f

f

f

f

f

f

f

f

f

f

f

f

P\G

19

20

21

22

23

24

25

26

27

28

29

30

31

32

row

VI

57 La

58 Ce

59 Pr

60 Nd

61 Pm

62 Sm

63 Eu

64 Gd

65 Tb

66 Dy

67 Ho

68 Dr

69 Tm

70 Yb

7

VII

89 Ac

90 Th

91 Pa

92 U

93 Np

94 Pu

95 Am

96 Cm

97 Bk

98 Cf

99 Es

100 Fm

101 Md

102 No

8

col

32

31

30

29

28

27

26

25

24

23

22

21

20

19

Mapping Algorithms;

The Janet Periodic Table has a consistent distribution of atomic numbers. The quantum numbers of the most significant electron relate to the location of an element within the table (Row and Column).

Row Number (R) ; R = n + ℓ

Column Number (C) ; C = 2(ℓ +1)² + 2(m_s – ½)(ℓ +½) - (ℓ + m_ℓ)

The atomic number of an element also relates to the row and column numbers as follows;

For even numbered rows; Z = R(R+1)(R+2)/6 - C + 1

For odd numbered rows; Z = (R+1)(R+2)(R+3)/6 - ½(R+1)² - C + 1

Other mappings of the Periodic Table are described by Kibler³.

The atomic number may be simply related if a new quantum number for magnetic moment is introduced. Permit a quantum number (m_n) representing magnetic moment associated with the principal quantum number (n) to take values as follows;

m_n = +½ (odd numbered rows)

m_n = -½ (even numbered rows)

The atomic number of any element may now be defined as;

Z = (R + m_n + 1/2)(R + m_n + 3/2)(R + m_n + 5/2) / 6 - ½(m_n+½ )(R+1)² - C + 1

3. Most Significant Electron

All electrons of an atom in the ground state may be identified by a number from 1 to Z, with 1 being the least energetic and Z the most energetic. The atomic number (Z) of any element may have a dual purpose. In addition to identifying the number of protons in the nucleus, the atomic number may also be used to identify the “most significant electron” (MSE) of an atom in the ground state. This is usually the most energetic electron, which is normally the first electron to ionize the atom. The characteristics of this electron are represented by its quantum numbers. It is possible to represent the MSE number (Z) of any element as a function of its quantum numbers.

4. The Periodic Stack of Chemical Elements

The Janet periodic table may be re-arranged into a series of square matrices. Each matrix is a different size. The matrices arrange the atomic orbitals as square concentric rings. The core is the ‘s’ orbital and the remaining orbits (p, d, f) are arranged concentrically around the core. The quantum numbers associated with the “most significant electron” of any element (in the ground state) determine the position of the element within the matrix. The matrices may be stacked vertically with the core elements in alignment. This is the “Periodic Stack of Elements” which resembles a stepped pyramid. Various pyramidal forms of the PT have been suggested in the past⁵. Any element positioned on a major diagonal of a matrix has the quantum number for angular momentum of its most significant electron equal to zero.

Matrix Identification (a) ;

Each matrix is identified by a matrix number (a) where; a = 1,2,3,4

Half-Matrix Identification (m_n , m_s) ;

One half of each matrix is identified by a number (m_n). The upper half of each matrix is defined by ; m_n = ½.  The lower half of each matrix is defined by ; m_n = -½. This number may be a new quantum number for magnetic moment associated with the principal quantum number (n). One half of each matrix is identified by the quantum number for spin magnetic moment (m_s).  The right half of each matrix is defined by ; m_s = ½.   The left half of each matrix is defined by ; m_s = -½.

Together m_n and m_S define a quadrant of any matrix.

Concentric Rings (ℓ);

Each matrix may be viewed as a set of concentric square rings arranged around a core. The core is the inner four cells. Each ring is identified by the quantum number for angular momentum (ℓ) of orbital rotation.

The core of each matrix is defined by ; ℓ = 0

The outermost ring of each matrix is defined by ; ℓ = a - 1

Electrons with quantum number "ℓ" greater than three are not known to be of any significance in chemical processes.  This implies that the matrix number (a) may have a limiting value of four.

Displacement from Diagonal (m_ℓ);

Displacement from a major diagonal of a matrix is identified by the quantum number for magnetic moment (m_ℓ). 

A cell on the diagonal is defined by ; m_ℓ = 0

A column displacement is defined by ; m_ℓ = positive

A row displacement is defined by ; m_ℓ = negative

The Periodic Stack labelled as Atomic Numbers (Z);

a = 1

2	1
4	3

a = 2

9	8	5	6
10	12	11	7
18	20	19	15
17	16	13	14

a = 3

28	27	26	21	22	23
29	35	34	31	32	24
30	36	38	37	33	25
48	54	56	55	51	43
47	53	52	49	50	42
46	45	44	39	40	41

a = 4

67	66	65	64	57	58	59	60
68	78	77	76	71	72	73	61
69	79	85	84	81	82	74	62
70	80	86	88	87	83	75	63
102	112	118	120	119	115	107	95
101	111	117	116	113	114	106	94
100	110	109	108	103	104	105	93
99	98	97	96	89	90	91	92

The Periodic Stack labelled as Atomic Orbitals;

a = 1

1s	1s
2s	2s

a = 2

2p	2p	2p	2p
2p	3s	3s	2p
3p	4s	4s	3p
3p	3p	3p	3p

 a = 3

3d	3d	3d	3d	3d	3d
3d	4p	4p	4p	4p	3d
3d	4p	5s	5s	4p	3d
4d	5p	6s	6s	5p	4d
4d	5p	5p	5p	5p	4d
4d	4d	4d	4d	4d	4d

a = 4

4f	4f	4f	4f	4f	4f	4f	4f
4f	5d	5d	5d	5d	5d	5d	4f
4f	5d	6p	6p	6p	6p	5d	4f
4f	5d	6p	7s	7s	6p	5d	4f
5f	6d	7p	8s	8s	7p	6d	5f
5f	6d	7p	7p	7p	7p	6d	5f
5f	6d	6d	6d	6d	6d	6d	5f
5f	5f	5f	5f	5f	5f	5f	5f

The Periodic Stack labelled as Chemical Elements;

3D Periodic Table of Elements - Matrix 1 ;

He	H
Be	Li

3D Periodic Table of Elements - Matrix 2 ;

F	O	B	C
Ne	Mg	Na	N
Ar	Ca	K	P
Cl	S	Al	Si

3D Periodic Table of Elements - Matrix 3 ;

Ni	Co	Fe	Sc	Ti	V
Cu	Br	Se	Ga	Ge	Cr
Zn	Kr	Sr	Rb	As	Mn
Cd	Xe	Ba	Cs	Sb	Tc
Ag	I	Te	In	Sn	Mo
Pd	Rh	Ru	Y	Zr	Nb

3D Periodic Table of Elements - Matrix 4 ;

Ho	Dy	Tb	Gd	La	Ce	Pr	Nd
Dr	Pt	Ir	Os	Lu	Hf	Ta	Pm
Tm	Au	At	Po	Tl	Pb	W	Sm
Yb	Hg	Rn	Ra	Fr	Bi	Re	Eu
No	Uub					Bh	Am
Md	Uuu				Uuq	Sg	Pu
Fm	Ds	Mt	Hs	Lr	Rf	Db	Np
Es	Cf	Bk	Cm	Ac	Th	Pa	U

Inert Gases ;

The inert gases form two vertical columns within the stack.  The location of one vertical column (18, 54, 118) is given by ; 

ℓ = 1 ,   m_n = -½ , m_ℓ = 1,  m_s = -½.

The location of the other vertical column (10, 36, 86) is given by ;   

ℓ = 1 ,  m_n = +½ ,   m_ℓ = 1 ,  m_s = -½. 

The inert gases are represented by blocks with red labels in the illustration below. The heavy score line separates quarters of the stack.

Other chemical commonalities may be viewed in vertical sections of the stack.  Diagonal sections are also interesting.

5. The Quantum Numbers

The quantum numbers (n , ℓ , m_{ℓ ,} m_s) are associated with electron motion and are defined as follows.

n is the principal quantum number

ℓ is the azimuthal quantum number for orbital angular momentum

m_ℓ is orbital magnetic moment

m_s is spin magnetic moment ( ms = ± ½ ) (spin up, spin down)

s is the quantum number for spin angular momentum (s = ½ ).

Z is the atomic number and the MSE identifier for a ground state atom

The spin quantum number is usually omitted as it is the same value for all leptons.

6. Orbital Angular Momentum

The azimuthal quantum number is associated with orbital motion relative to the nucleus⁶. The orbital angular momentum (L_ℓ) of the electron is quantized to ‘ℓ’ as follows;

L_ℓ² = ℓ(ℓ+1)ħ²

Where; ħ = h/2π

h is Plank’s constant (a fundamental unit of angular momentum)

7. Principal Angular Momentum

The principal quantum number (n) may also be associated with some kind of angular momentum. The principal quantum number defines the discreet number of electron wave-lengths which may occupy a rotational path. For a circular rotation;

2πr = nλ

Where; r is the mean radius of the surface of motion

λ is the wave length of an electron

A “principal angular momentum” (L_n) is quantized to ‘n’ as follows;

L_n = hr/λ = nh/2π = nħ

8. The Quantum Matrix

If it is assumed that the principal quantum number (n) is associated with some kind of angular momentum, then a magnetic moment (m_n) may be associated with the angular momentum.

m_n = ± ½ (rotation up, rotation down)

It may also be conjectured that ‘m_n’ represents a magnetic dipole with orbital orientation described as “dipole north” or “dipole south”.

If a quantum number for principal magnetic moment (m_n) is admitted, then a quantum matrix (M_Z) may be constructed as follows;

M_Z = n , ℓ , s

m_n , m_ℓ , m_s

The top row represents various forms of angular momentum. The sum of all quantum numbers for angular momentum (L_T) is;

L_T = n + ℓ + s

The bottom row represents various forms of magnetic moment. The sum of all quantum numbers for magnetic moment (m_T) is;

m_T = m_n + m_ℓ + m_s

9. Couples

The members of the quantum matrix may be arranged into couples of momentum and magnetic moment which correspond to the columns of M_Z.

Principal couple (N¢); N¢ = n + m_n

Orbital couple (L¢); L¢ = ℓ + m_ℓ

Spin couple (S¢); S¢ = s + m_s

10. The Aufbau Couple

It shall be assumed that an electron’s basic region of motion precesses. The momentum associated with precession will have a quantum number (a) which shall be called the “Aufbau Number”. An associated magnetic moment (m_a) completes the Aufbau Couple (A¢).

A¢ = a + m_a

Where; 'a' takes values 1, 2, 3, 4

The aufbau principle may be quantified using two simple averages.

A¢ = ½( L_T + m_T)

m_a = ½( m_ℓ + m_s)

Substitution gives; 2a = n + m_n + ℓ + s

Atomic orbitals (s, p, d, f ) are defined as;

s: ℓ = 0,

p: ℓ = 1,

d: ℓ = 2,

f: ℓ = 3

The aufbau number (a) and the magnetic number (m_n) summarize the electronic filling sequence of an atom. Groupings of atomic orbitals (n, ℓ) corresponding to the aufbau-magnetic pair (a, m_n) are;

(a, m_n) = (n, ℓ) Groups

(1, +½) = 1s

(1, -½)    = 2s

(2, +½) = 3s, 2p

(2, -½)    = 4s, 3p

(3, +½) = 5s, 4p, 3d

(3, -½)    = 6s, 5p, 4d

(4, +½) = 7s, 6p, 5d, 4f

(4, -½)    = 8s, 7p, 6d, 5f

11. Quantum Forces

Forces acting upon an electron may be represented as quantum vectors. These are vectors which are quantized. They may be separated into electric and magnetic parts and they may associate with different types of motion (spin, orbital, and precession). Fundamental force is derived from fundamental energy which relates to the principle of uncertainty. The following sections will develop this concept.

12. Uncertainty

The principle of uncertainty is normally written as;

∆x∆p ≥ h/4p

Where; ∆p is a change in momentum

∆x is a change in position

h is Plank’s constant (a fundamental unit of angular momentum)

It may also be written as ;

∆x∆p = Uh/4p

Where; U is an uncertainty ratio

It is convenient to use a compact uncertainty ratio (u) where; u = U/4p

Giving; ∆x∆p = uh

13. Fundamental Uncertainty

The principle of uncertainty may be applied to fundamental changes (∆x₀ ,∆p₀).

∆x₀∆p₀ = u₀h

If; ∆x₀ = a₀

∆p₀ = h/λ₀

Where; λ₀ is a fundamental wavelength

a₀ is the Bohr radius

u₀ is a ratio of uncertainty for fundamental changes

Then fundamental uncertainty is; u₀ = a₀/λ₀

14. Fundamental Energy

Fundamental energy (E₀) is uncertain (it is defined using the uncertainty ratio) ;

E₀ = u₀²h²/m₀a₀²