Z = D1 + D2 + D3
The Quantum Numbers;
The quantum numbers of chemistry
(n , l
, mL , mS) are well documented. The quantum number for
spin (s) is constant (for a lepton); s =
½
A “quantum matrix” is;
|
n
|
l
|
s
|
|
mn
|
mL
|
mS
|
Where; mn may
represent a magnetic dipole moment; mn
= ±½
mS
= ±½
The row sums are;
A
= n + l
+ s
B = mn
+ mL + mS
Quantum Averages;
The quantum averages (a,ma)
are; a + ma = ½(A+B)
ma
= ½(mL + mS) (spin-orbit average magnetic moment)
a
= ½(A+ mn) = ½(n + l ) or
½[(n + ½) + (l + ½ )]
Where; n+ ½ is associated with
the energy of radial vibration
l
+ ½ is associated with the energy of orbital vibration
Giving; a = 1,2,3,4
(“a” is
the “step number”, also a matrix identifier, also the “Hund Number”)
Each “step” of the JPT
(identified by “a”) consists of two rows. An individual row within a step is
identified by “mn”.
Where; mn = +½
identifies the upper row within a step
mn
= -½ identifies the lower row within a step
The pair (a,mn) will
identify any row (period).
The Determinants;
The first determinant (D1) is;
|
4a(a+1)
|
mn + ½
|
|
2a2
|
(a+ ½)/3
|
Giving; D1
= 4a(a+1)(a+ ½)/3 – 2a2(mn + ½)
Where; a(a+1) is
associated with rotation
(a+½)
is associated with vibration
(mn
+½) is associated with magnetic
vibration
The second determinant (D2)
is;
|
-2l
|
mS + ½
|
|
2(l+ ½)
|
l+ ½
|
Giving; D2
= -2l(l+
½) - 2(l+ ½)(mS
+ ½)
Where; (l+
½) is associated with orbital vibration
(mS
+ ½) is associated with magnetic
vibration
The third determinant (D3)
is;
|
-s
|
ml + ½
|
|
-(s + ½)
|
s + ½
|
Giving; D3
= -s(s + ½) + (s + ½)(ml
+ ½)
Where; (s+ ½) is
associated with spin vibration
(ml + ½)
is associated with magnetic vibration
Atomic Forces;
The structure of an atom is dependant upon electric and
magnetic forces, which are associated with dimensions (r,θ,φ,t). An atom in the
ground state is time independent (or time cyclical).
The magnitude of the electric force vector (Fe)
is; |Fe|=
Fe
Fe2
= Fer2 + Feθ2 + Feφ2
The magnitude of the magnetic force vector (FB)
is; |FB|= FB
FB2
= FBr2 + FBθ2 + FBφ2
The magnitude of the “total atomic force” (FZ)
is; FZ2
= Fe2 + FB2 = ZEZ02/λZ02 = ZFZ02
FZ = Z½FZ0
Where; Z is the atomic
number
EZ0
is the atomic energy (average energy of all bound particles in the atom
including the nucleus)
λZ0
is the atomic wavelength (wavelength representing the complete atom)
FZ0 is the average
atomic force
If the atom is electrically neutral the atomic number
applies to protons and electrons (hence Z½)
The electric components of force are;
Fer2
= 4a(a+1)EZ0/λZ0 * ⅓(a + ½)EZ0/λZ0 (dependant
only on ‘a’)
Feθ2 = -2l EZ0/λZ0
* (l
+ ½)EZ0/λZ0
(dependant
only on ‘l
’)
Feφ2 = -sEZ0/λZ0 * (s + ½)EZ0/λZ0 (dependant
only on ‘s’)
The electro-magnetic components of force are;
FBr2
= – 2a2EZ0/λZ0 * (mn + ½)EZ0/λZ0
FBθ2 = - 2(l+
½)EZ0/λZ0 * (mS + ½)EZ0/λZ0
FBφ2 = (s + ½)EZ0/λZ0 * (ml
+ ½)EZ0/λZ0
The JPT is displayed in two parts
(A,B). Each cell represents a chemical element represented by the atomic number
(Z) shown as the lower number. A cell also contains the orbital (n,l)
of the most significant electron, shown as the upper number. Please note that
the periods (rows) of the JPT do not always agree with the Standard Periodic
Table (for “s” orbitals).
JPT (Part A);
R
|
a
|
mn
|
|||||||||||||||||||||||||
1s
1
|
1s
2
|
1
|
1
|
+½
|
|||||||||||||||||||||||
2s
3
|
2s
4
|
2
|
-½
|
||||||||||||||||||||||||
2p
5
|
2p
6
|
2p
7
|
2p
8
|
2p
9
|
2p
10
|
3s
11
|
3s
12
|
3
|
2
|
+½
|
|||||||||||||||||
3p
13
|
3p
14
|
3p
15
|
3p
16
|
3p
17
|
3p
18
|
4s
19
|
4s
20
|
4
|
-½
|
||||||||||||||||||
3d
21
|
3d
22
|
3d
23
|
3d
24
|
3d
25
|
3d
26
|
3d
27
|
3d
28
|
3d
29
|
3d
30
|
4p
31
|
4p
32
|
4p
33
|
4p
34
|
4p
35
|
4p
36
|
5s
37
|
5s
38
|
5
|
3
|
+½
|
|||||||
4d
39
|
4d
40
|
4d
41
|
4d
42
|
4d
43
|
4d
44
|
4d
45
|
4d
46
|
4d
47
|
4d
48
|
5p
49
|
5p
50
|
5p
51
|
5p
52
|
5p
53
|
5p
54
|
6s
55
|
6s
56
|
6
|
-½
|
||||||||
5d
71
|
5d
72
|
5d
73
|
5d
74
|
5d
75
|
5d
76
|
5d
77
|
5d
78
|
5d
79
|
5d
80
|
6p
81
|
6p
82
|
6p
83
|
6p
84
|
6p
85
|
6p
86
|
7s
87
|
7s
88
|
7
|
4
|
+½
|
|||||||
6d
103
|
6d
104
|
6d
105
|
6d
106
|
6d
107
|
6d
108
|
6d
109
|
6d
110
|
6d
111
|
6d
112
|
7p
113
|
7p
114
|
7p
115
|
7p
116
|
7p
117
|
7p
118
|
8s
119
|
8s
120
|
8
|
-½
|
||||||||
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
31
|
32
|
C
|
|||||||||
2 (d)
|
1(p)
|
0 (s)
|
l
|
||||||||||||||||||||||||
-2
|
-1
|
0
|
+1
|
+2
|
-2
|
-1
|
0
|
+1
|
+2
|
-1
|
0
|
+1
|
-1
|
0
|
+1
|
0
|
0
|
ml
|
|||||||||
+½
|
-½
|
+½
|
-½
|
+½
|
-½
|
ms
|
|||||||||||||||||||||
R
|
a
|
mn
|
||||||||||||||||
4f
57
|
4f
58
|
4f
59
|
4f
60
|
4f
61
|
4f
62
|
4f
63
|
4f
64
|
4f
65
|
4f
66
|
4f
67
|
4f
68
|
4f
69
|
4f
70
|
7
|
4
|
+½
|
||
5f
89
|
5f
90
|
5f
91
|
5f
92
|
5f
93
|
5f
94
|
5f
95
|
5f
96
|
5f
97
|
5f
98
|
5f
99
|
5f
100
|
5f
101
|
5f
102
|
8
|
-½
|
|||
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
C
|
||||
3(f)
|
l
|
|||||||||||||||||
-3
|
-2
|
-1
|
0
|
+1
|
+2
|
+3
|
-3
|
-2
|
-1
|
0
|
+1
|
+2
|
+3
|
ml
|
||||
+½
|
-½
|
ms
|
||||||||||||||||
A column number (C) is; C = 32 - 2l
(l + ½)
- 2(l + ½)(mS +
½) +
mL
C = 32 + Z - D1
C = 32 + Z - D1
Please note that the quantum number “n” is distributed throughout the JPT, therefore it does not correspond to any particular row or column. It is included in the average value (a) which corresponds to a “step” (two rows) which is also the “Hund Number”.
The magnetic quantum numbers (mn
, ms) become more meaningful if the JPT is re-arranged as a
series of four square matrices.
Conclusion;
The Janet Periodic Table (JPT),
aka “Left Step Periodic Table” may be constructed from three determinants (D1,D2,D3)
composed of quantum numbers.
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