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 07 October 2009.

Contents

1 Introduction

2 Most Significant Electron

3 The Periodic Stack of Elements

4 The Quantum Numbers

5 Orbital Angular Momentum

6 Spin Angular Momentum

7 Trajectory Angular Momentum

8 The Quantum Matrix

9 Couples

10 Quantization

11 Quantum Vectors

12 Single Electron Systems

13 Calculations for Multiple Electron Systems

14 Dynamic Matrices

15 Fundamental Principles of Chemistry

16 The Periodic Stack of Particles

17 Particle Quantum Numbers

18 The Particle Number

19 Nomenclature

20 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. Eric Scerri (UCLA) has written extensively on variations of the Periodic Table. His table may be found at;

http://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=20

Another form of the PT is based upon tetrahedral sphere packing.

http://www.perfectperiodictable.com/

The Janet Periodic Table² is a 2D arrangement of the natural elements. Charles Janet first proposed this form of the Periodic Table in 1929. The Janet form of the Periodic Table has been proposed from time to time by various persons⁴. Acceptance of this table requires minor modification of the periods⁷. According to Winter⁸, the Janet table is preferred by some persons to the standard form. Further information on the Janet table may be found at;

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

and at; http://www.meta-synthesis.com/webbook/35_pt/pt.html#j

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 significant electron (MSE) of an element in the ground state. Elements (actually MSEs) 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 orbital rings aligned vertically. This is a 3D visualization 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 sub-atomic particles may also be arranged as a series of square matrices.

2. 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 (Z) of any element as a function of its quantum numbers.

3. 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 and orbital rings in alignment. This is the “Periodic Stack of Elements” which resembles a stepped pyramid. Various pyramidal and tetrahedral forms of the PT have been postulated⁵.

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) ;

The upper or lower 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.

The right or left 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 orbital angular momentum (ℓ).

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, and that the atomic number (Z) may have a limiting value of 120.

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 with cells 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 with cells 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 with cells 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.

4. 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 electrons.

5. Orbital Angular Momentum

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

L² = ℓ(ℓ+1)ħ²

Where; ħ = h/2π

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

A magnetic moment (L_z) is associated with orbital motion; L_z = m_ℓ ħ

6. Spin Angular Momentum

The spin angular momentum (S) of the electron is quantized to ‘s’ as follows;

S² = s(s+1)ħ² = ¾ ħ²

S = ½√3ħ

A magnetic moment (S_z) is associated with spin motion; S_z = m_s ħ

7. Trajectory Spin Angular Momentum

It is convenient to assume that orbital motion is confined to a single spatial plane. The trajectory of the orbit surrounds the nucleus and is a closed path such as a circle, ellipse, or lobe. A surface of electron motion is generated by rotating the plane trajectory around an axis of rotation. This shall be called “trajectory spin”. It shall be assumed that the spin axis is confined to the trajectory plane and that it passes through the nucleus. If the axis of spin is aligned with the major axis of the trajectory a simple surface is generated. If the spin axis is tilted by some angle from the major axis, then the surface has a self intersecting region.

The angular momentum (N) associated with trajectory spin is quantized to ‘n’ as follows;

N² = C_N²n(n+1)ħ²

Where; C_N is a constant depending upon trajectory shape and the tilt of the axis of rotation.

A magnetic moment (N_z) is associated with trajectory spin; N_z = C_Nm_n ħ

Where; m_n is a quantum number for the magnetic moment of trajectory spin.

m_n = ± ½ (rotation up, rotation down)

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

8. The Quantum Matrix

If a quantum number for trajectory 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

It shall be assumed that an electron’s surface 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) The couple average

m_a = ½( m_ℓ + m_s) The magnetic average

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

If ; m_n = - ½

Then the Madelung rule is formed; 2a = n + ℓ

If ; m_n = + ½

Then the Madelung rule is modified; 2a = n + ℓ + 1

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

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;

(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

10. Quantization

The quantised rotational energy of a rigid rotor (E_R) is ; E_R = q(q+1)E₀

Where; q is a generic quantum number

E₀ is a fundamental energy

The vibrational energy of a simple harmonic oscillator (E_H) is quantised; E_H = (q+½)E₀

The energy ratio (rotation/vibration) is; E_R / E_H = q(q+1) / (q+½) = tan(f_q)

q/(q+1) = tan(½f_q)

Where; f_q is the angle of quantization of rotation and vibration to quantum number q.

The total energy (E_T) is; E_T² = E_R² + E_H²

E_T - E_H = q²E₀

E_T - E_H - E_R = -qE₀

A particle energy ratio is; E_P /E_R

Where; E_P is particle energy

E_P = L_P/t

Where; L_P is angular momentum of a particle

t is a time interval

A wave energy ratio is; hω /E_H

Where; ω is frequency of vibration

h is plank’s constant

It shall be assumed that a condition for steady state motion is;

Particle ratio = wave ratio

E_P /E_R = hω /E_H

E_P /hω = E_R/E_H = tan(f_q)

L_P/thω = tan(f_q)

If; L_P = mvr

h = m₀cλ₀

Then; mvr/tm₀cλ₀ω = tan(f_q)

If; r/t = λ₀ω

and; sin(f_q) = v/c

Then; cos(f_q) = m₀/m

Giving; 1 = (m₀/m)² + (v/c)²

This is the definition of relativistic mass (m).

11. Quantum Vectors

A quantum vector has quantized magnitudes which are functions of the quantum numbers. The unit vectors (i , j , k) form the basis of an orthogonal co-ordinate system. The complex constant (i) may identify an unobservable characteristic; i² + 1 = 0

A fundamental magnitude of momentum (p₀) is; p₀ = h/λ₀

Two momentum vectors will be assumed to act upon each MSE as follows;

Electric momentum (p₁) is;

p₁ = 2[a(a+1)(a+½)/3]^½ p₀ i + i[2ℓ(ℓ+½)]^½ p₀ j + i[s(s+½)]^½ p₀ k

Magnetic momentum (p₂) is;

p₂ = i[2a²(m_n+½)]^½ p₀ i + i[2(ℓ+½)(m_s+½)]^½ p₀ j + [(s+½)(m_ℓ+½)]^½ p₀ k

The electro-magnetic momentum (p₃) is; p₃ = p₁ + p₂

If; p₁ and p₂ act orthogonally, then; | p₃|² = | p₁|² + | p₂|²

| p₁|² = (4/3)a(a+1)(a+½)p₀² - 2ℓ(ℓ+½)p₀² - s(s+½)p₀²

| p₂|² = -2a²(m_n+½)p₀² - 2(ℓ+½)(m_s+½)p₀² + (s+½)(m_ℓ+½)p₀²

| p₃|² = Zp₀²

Giving;

Z = 4/3a(a+1)(a+½) - 2ℓ(ℓ+½) - s(s+½) - 2a²(m_n+½) - 2(ℓ+½)(m_S+½) + (s+½)(m_ℓ+½)

Where the Aufbau principle gives; 2a = n + ℓ + s + m_n

12. Single Electron Systems

Electro-magnetic energy (E₃) is; E₃ = | p₃|² /m₀ = Zp₀²/m₀ = ZE₀

For a single electron system the fundamental energy (E₀) is; E₀ = -e²/2ε₀r

Where; e is the charge of a proton

-e is the charge of an electron

r is the average distance between an electron and the nucleus

The energy balance for a single electron system shall be;

(E_T)(E_ω)² + (E_m)(E₃)² = 0

Where; E_T is the total energy of a single electron system

E_ω is the wave energy

E_m is the particle energy

E₃ is the electro-magnetic energy (assuming the magnetic part is minimal)

The energies have values as follows;

E_ω = nhω

E_m = ½ mv²

E₃ = -Ze²/2ε₀r

Giving total energy; E_T = -(½ mv²)( -Ze²/2ε₀r) 2/( nhω)²

If; v = rω

Then; E_T = -mZ²e⁴/ 8ε₀²n²h²

13. Calculations for Multiple Electron Systems

The following calculations demonstrate that the atomic number of an element is related to its location within the stack. This may also be restated as follows; the MSE identification number is a function of the quantum numbers of the MSE.

Example 1;

Titanium has atomic number ; Z = 22.

The Quantum Matrix for Titanium is;

M₂₂ = 3 , 2 , ½

+½ , -1 , +½

The aufbau number is;

2a = n + ℓ + s + m_n

2a = 3 + 2 + ½ + ½

a = 3

The MSE number is;

Z = 4/3a(a+1)(a+½) - 2ℓ(ℓ+½) - s(s+½) - 2a²(m_n+½) - 2(ℓ+½)(m_S+½) + (s+½)(m_ℓ+½)

Z = (4/3)3(3+1)(3+½) - 2(2)(2+½) - ½(½+½) - 2(9)( ½+½) - 2(2+½)(½+½) + (½+½)(-1+½)

Z = 56 - 10 - >½ - 18 - 5 + - ½

Z = 22

Example 2;

Bismuth has atomic number ; Z = 83.

The Quantum Matrix for Bismuth is;

M₈₃ = 6 , 1 , +½

+½ , +1 , +½

The aufbau number is;

2a = n + ℓ + s + m_n

2a = 6 + 1 + ½ + ½

a = 4

The MSE number is;

Z = 4/3a(a+1)(a+½) - 2ℓ(ℓ+½) - s(s+½) - 2a²(m_n+½) - 2(ℓ+½)(m_S+½) + (s+½)(m_ℓ+½)

Z = (4/3)4(4+1)(4+½) - 2(1)(1+½) - ½(½+½) - 2(16)( ½+½) - 2(1+½)(½+½) + (½+½)(1+½)

Z = 120 - 3 - >½ - 32 - 3 + 1 ½

Z = 83

14. Dynamic Matrices

The Atomic Stack is a 3D Periodic Table. It places the MSE of an element in a cell determined by quantum numbers. For an element in the ground state the atomic number is the MSE number.

For any element not in the ground state, or for any ion, or for an atom bound within a molecule, the matrix may be used to show the “location” (condition) of each electron. It will be observed that some type of symmetry with respect to major diagonals is required. This means that orbital magnetic moment (m_ℓ) is a determining factor when locating dynamic electrons.

Dynamic matrices may be a useful technique to represent molecular interaction, they will not be discussed in detail.

15. Fundamental Principles of Chemistry

The following fundamental principles of chemistry have an application to the periodic stack.

The Aufbau Principle;

The Aufbau principle states that generally an electron will occupy the lowest energy state available.  This principle also indicates the order in which atomic orbitals are occupied.  It is possible to quantify the Aufbau principle using the matrix numbers (a, m_n), which indicate the filling order of atomic orbitals.

The Exclusion Principle;

The Pauli Exclusion Principle states that no two electrons within the same atom may have the same set of quantum numbers. This principle may be restated as no two electrons of the same atomic system can have the same quantum matrix.

The Hund Principle;

The Hund principle states that atomic orbitals must be half filled by electrons of the same spin before an electron of opposite spin enters the orbital. This is easily observed in the atomic stack which has the right side of an orbital with positive spin and the left side with negative spin. The right side of a square ring must be completely labeled before the left side can be labeled.

Madelung’s Rule;

Madelung’s rule states that the orbitals fill with electrons as; n+ ℓ .

This rule may be expressed generally as; 2a = n + ℓ + s + m_n

If m_n = - ½ ; Then ; 2a = n + ℓ

If m_n = + ½ ; Then ; 2a = n + ℓ + 1

Atomic Limit ;

There is no known theoretical limit to the size of an atom.  Chemical reactions with  ℓ > 3 have not been observed.  If we assume that ℓ = 3  is the limiting value of the azimuthal quantum number, then the limiting value of the aufbau number is forced to be a = 4.  This suggests an upper limit on the size of an atom of  Z = 120.  Heavy elements degenerate quickly making it unlikely that element 120 could be synthesized.

16. The Periodic Stack of Particles

The sub-atomic particles may also be arranged as square matrices. The particles within a matrix are arranged by “family” into “square rings” surrounding a central “core”. The core contains only force carrier particles. The lepton and quark families (including anti-particles) are arranged as square rings surrounding their associated bosons. The matrices may be stacked vertically to form the Periodic Stack of Particles. Some cells within a matrix may represent unknown particles.

The Periodic Stack of Particles labelled by symbol;

Matrix 1 ; This matrix contains four gluons only.

g2	g1
g4	g3

Matrix 2 ; The core is four gluons which are surrounded by quarks and anti-quarks. Anti-particles are underlined and occupy the lower half of the matrix.

s	d	u	c
b	g6	g5	t
b	g8	g7	t
s	d	u	c

Matrix 3 ; The core contains bosons which are surrounded by leptons and anti-leptons. The outer ring may represent unknown leptonic matter. Anti-matter is confined to the lower half of the matrix.

?	?	?	?	?	?
?	Vu	Ve	e-	µ-	?
?	Vt	W-	p	τ-	?
?	Vt	W+	z	τ+	?
?	Vu	Ve	e+	µ+	?
?	?	?	?	?	?

Matrix 4 ; This 8x8 matrix is not shown. The core shall be assumed to be four gravitons surrounded entirely by unknown matter particles. If Higgs particles exist, they may be located in this matrix.

17. Particle Quantum Numbers

The numbers used to locate a particle within a matrix may be called “particle quantum numbers” (pqn). Unlike the chemical quantum numbers, no known physical characteristic may be associated with a pqn. The pqn's may also be arranged as a matrix (M_P) as follows;

M_P = q₁ , q₂ , q₃

q_1- , q_2- , q_3-

Matrix Identification (q₁) ;

Each particle matrix is identified by a pqm (q₁) where; q₁ = 1,2,3,4

The concept of spin does not apply to a pqn, therefore the corresponding pqn is arbitrary; q₃ = ½

Quadrant Identification (q_1- , q_3- ) ;

One half of each matrix is identified by a pqn (q_1- ). The upper half of each matrix is defined by ; q_1- = ½. The lower half of each matrix is defined by ; q_1- = -½.

One half of each matrix is identified by a pqn (q_3-). The right half of each matrix is defined by ; q_3- = ½. The left half of each matrix is defined by ; q_3- = -½.

Together q_1- and q_3- define a quadrant of any particle matrix.

Concentric Rings (q₂);

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 a pqn (q₂).

The core of each particle matrix is defined by ; q₂ = 0

The outermost ring of each matrix is defined by ; q₂ = q₁ - 1

Displacement from Diagonal (q_2- );

Displacement from a major diagonal of a particle matrix is identified by a pqn (q_2- ).

A cell on the diagonal is defined by ; q_2- = 0

A column displacement is defined by ; q_2- = positive

A row displacement is defined by ; q_2- = negative

18. The Particle Number

Each sub-atomic particle may be assigned a “particle number” (P) as an identifier, similar in concept to the use of an atomic number (Z) as the identifier of a chemical element. Some particle numbers are;

P = 5 represents the up quark

P = 6 represents the charm quark

P = 15 represents the anti-top quark

P = 31 represents the electron

P = 37 represents the photon

P = 49 represents the positron

The P number is related to a particle's location within the stack, which is determined by its pqn matrix (M_P). The P number is related to a particles pqn's as follows;

P = (4/3)q₁(q₁+1)(q₁+½) - 2q₁²(q_1- +½) - 2q₂(q₂ +½) - 2(q₂ +½)(q_3- +½) + (q_2- +½) - ½

The Particle Stack with cells labelled as Particle Numbers (P);

q₁ = 1

2	1
4	3

q₁ = 2

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

q₁ = 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

19. Nomenclature

Constants;

h Plank’s constant (fundamental angular momentum)

ħ h barred (ħ = h/2π)

i complex constant

-e dynamic charge of an electron

e dynamic charge of a proton

ε₀ permitivity of free space

μ₀ permeability of free space

c speed of light in vacuum

m dynamic mass of an electron

Quantum Numbers;

q generic quantum number

n principal quantum number

ℓ  orbital angular momentum (azimuthal number)

s spin angular momentum

m_n principal magnetic moment

m_ℓ orbital magnetic moment

m_s spin magnetic moment

a aufbau quantum number

m_a aufbau magnetic moment

Z   atomic number and MSE identifier for ground state

(s,p,d,f ) atomic orbitals (conforming to ℓ)

Matrices;

M_Z quantum matrix

a Aufbau number, identifier of a matrix of the atomic stack

Variables;

l wavelength

w frequency

v velocity

p linear momentum

L angular momentum

F force

E energy

V potential energy

20. References

1. Hervé Le Cornec

THE DISTRIBUTION OF ATOMIC IONIZATION POTENTIALS REVEALS AN UNEXPECTED PERIODIC TABLE

Chemistry Preprint Server, Item 0201007, 9 January 2002

2. Charles Janet,

CONSIDERATIONS SUR LA STRUCTURE DU NOYAU DE L’ATOME,

Beauvais, December 1929

3. Maurice Kibler, T. Negadi

THE PERIODIC TABLE IN FLATLAND

Intl. Journal of Quantum Chemistry, Article 57, 53-61, 1996

4. Albert Tarantola

PERIODIC TABLE OF THE ELEMENTS (JANET FORM)

Chemistry Preprint Server, Item 0009002, 9 September 2000

5. Eric R. Scerri

THE EVOLUTION OF THE PERIODIC SYSTEM

Scientific American, September 1998

6. Phillip S.C. Matthews

QUANTUM CHEMISTRY OF ATOMS AND MOLECULES

Cambridge University Press, Cambridge UK, 1989

7. Edward G. Mazurs

GRAPHIC REPRESENTATIONS OF THE PERIODIC SYSTEM DURING ONE HUNDRED YEARS

University of Alabama Press, 1974

8. Mark J. Winter

CHEMICAL BONDING

Oxford University Press, 1994