7. The Isotope Balance Formula
(Discovered by Tom Gilmore)

by Tom Gilmore
Copyright 2018
All graphics by Tom Gilmore

Intro to Isotopes

Atomic Number and Elements

There are 3 atomic particles (formed from Quarks); Protons, Electrons, and Neutrons.  Protons and Electrons occur in pairs, although the Electron particle itself can be absent, in which case the atom is referred to as "ionized".  For the purpose of this presentation the Proton/Electron pair is simply termed a Proton.  The integer number of Protons in an atom is called its "atomic number", and these integer-based atoms are termed "Elements".  One Proton is the Element Hydrogen (H), two Protons is Helium (He), three Protons is Lithium (Li), and so on.

Atomic Weight and Isotopes

Both Protons and Neutrons have an "atomic weight" of 1.  The atomic weight (Z) of an atom is the sum of Protons (P) and Neutrons (N), or (Z=P+N).  Many Elements occur with variable numbers of Neutrons, and these are called "Isotopes" of the Element. 

Part I – Exact Conformities

In this presentation the syntax for representing an Isotope is to prefix an Element with the atomic weight (Z) and suffix with the atomic number (P). 
For example (19)Fluorine(9), and has 9 Protons and 10 Neutrons.

The derivation of the formula influencing the Isotopes is detailed further on. 
P is the atomic number (number of Protons)
Z is the atomic weight (total of Protons and Neutrons)
(z is the atomic weight derived from the formula). 

Exact Conformities with the Formula

Isotope Balance z = (2P + P2/156) Rounded

100% in one Isotope:

 (19)Fluorine(9) -- (2P + P2/156) = (2x9 + 9x9/156) = 18.52 rounds to 19
(23)Sodium(11) --  z = (2x11 + 11x11/156_= 22.78 rounds to 23
(27)Aluminum(13) – z = 27.08 rounds to 27
(31)Phosphorus(15) – z = 31.44 rounds to 31
(45)Scandium(21) – z = 44.83 rounds to 45
(59)Cobalt(27) – z = 58.67 rounds to 59
(93)Niobium(41) – z = 92.78 rounds to 93
(103)Rhodium(45) – z = 102.98 rounds to 103
(169)Thulium(69) – z = 168.52 rounds to 169

99% in one Isotope:

(12)Carbon(6) – z = 12.23 rounds to 12
(14)Nitrogen(7) – z = 14.31 rounds to 14
(16)Oxygen(8) – z = 16.41 rounds to 16
(238)Uranium(92) – z = 238.26 rounds to 238

Notice that the Elements with 100% have odd atomic weight and those with 99% have even atomic weight. 

Part II – The Derivation of the Balance Formula

The “n-Value”

With the exception of (1)Hydrogen(1), which has no Neutrons, every Element has at least as many Neutrons as Protons.  Consequently the distinguishing factor of an Isotope can be expressed as the number of excess Neutrons over the number of Protons.  For purposes of the Isotope balance equation the excess of Neutrons over Protons is called the ”n-value” and by definition it is the atomic weight less twice the number of Protons or Z-2P.

(n = Z – 2P)

Deriving the Balance Formula

The Isotopes of the Elements show a general increase of excess Neutrons as atomic number increases.  At Platinum(78) a proportional relationship of 1/2 between Protons and excess Neutrons occurs.  The Isotope balance equation was derived by applying the balance ratio of Platinum(78) to all of the atomic numbers (P).

There are multiple Isotopes of Platinum(78) but the average atomic weight of the Isotopes of Platinum(78) is 195.  The equation for excess Neutrons is (n=Z-2P), so

For (195)Platinum(78):  n = (Z-2P) = (195 - 2x78) = 39, and 39/78=1/2

The balance is 1/2 of P (or P/2) when P is 78 (or P/78). 

(P/78 x P/2 = P2/156)

This is verified by putting P=78 in the equation (P2/156) as follows.
For P=78, (P2/156) = (78x78/156) = 39.

The n-Gradient and z-Gradient

P2/156 (rounded) is termed the "n-gradient" of the Elements because it represents a gradient of exponential increase of excess Neutrons.
The calculated n-gradient is only an integer for P=78.  All other calculated values of P result in a decimal fraction (which must be rounded to an integer).  

The atomic weight (Z) equals twice the Proton count plus the excess Neutrons (Z=2P+n).  As applied to the n-gradient, the Isotope balance point is represented by z.

z= (2P + n-gradient) = (2P + P2/156) = "z-gradient"

n-gradient = P2/156
z-gradient = (2P + P2/156)

Example: z-gradient calculation for Mn(25) is:
z =(2P + P2/156) = (25+25 +(25x25/156)) = (50 + 4.00641) = 54

The z-gradient only represents where the Isotope is in balance, not where it must absolutely reside, but the “pull” of the balance induces Beta decay (discussed further on) toward the balance.

The n-gradient plots as a geometric curve, and the Beta decay gradient is tangent to the curve at P=78, the balance point.

Part III – The Non-Parity Caused Variances

Parity Demand

The odd/even (or even/odd) non-parity between the atomic number and the atomic weight is a strong factor in causing a variance in Isotopes from the calculated balance equation (z-gradient).  There is a strong demand for the odd atomic number Elements to have an odd-number atomic weight, and a less strong but noticeable preference for even atomic number Elements to have an even-number atomic weight.

Small Deviations Induced by Non-Parity

(Odd atomic number, even z-gradient, Isotope prefers odd atomic weight).

Following is a list of 6 non-parity Elements that have 100% (or nearly) in one Isotope, and exhibit a variance of 1 to the z-gradient.

Isotope     z-gradient   variance

(55)Mn(25)     54        +1        100%
(89)Y(39)        88        +1        100%
(141)Pr(59)     140      +1        100%
(175)Lo(71)    174      +1        97%
(181)Ta(73)    180      +1        99%
(197)Au(79)    198      -1         100%

Notice that Gold(79) is on the other side of the balance at Platinum(78), shifting the variance from +1 to -1.

Non-Parity Splitting

Six Elements of odd atomic number split their Isotopes around their even z-gradient.  Their percentage split exhibits a graduated percentage change centered around the Element Silver(47), which nearly equally splits around the z-gradient (see list below).  All 6 Elements have averaged atomic weights (average Z) that round to the z-gradient.  The reason the percent split increases is that the number of excess Neutrons (n-value) increases with the increase in atomic number. 

69% -- (63)Copper(29)
31% -- (65)Cu(29)      average Z = 63.62     z-gradient = 64 

60% -- (67)Gallium(31)
40% -- (69)Ga(31)      average Z = 68.16     z-gradient = 68 

51% -- (107)Silver(47)
49% -- (109)Ag(47)    average Z = 108.16   z-gradient =108

37% -- (185)Rhenium(75)
63% -- (187)Re(75)    average Z = 186.06   z-gradient = 186

37% -- (191)Iridium(77)
63% -- (193)Ir(77)      average Z = 192.01   z-gradient = 192 

30% -- (203)Thallium(81)
70% -- (205)Tl(81)     average Z = 204.06   z-gradient = 204 

Notice that the percentage split is nearly exactly reversed between Copper(29) and Thallium(81).

Part IV – Even-Atomic-Number Isotope Spreads

The chart of abundances by n-value shown below is a stripped down version of the Isotope Abundance Charts displayed in Part VI.
A section of Elements from 39 to 46 are shown, illustrating how some of the even-atomic-number Elements are spread across multiple Isotopes.


The spread of Isotopes of these even-numbered Elements average-out near the z-gradient. 

Average atomic weights (Ave-Z) are the sum of the isotope percentages times their atomic weights, divided by 100. 

Ave-Z    z    variance             
Zr(40)    91      90        +1
Mo(42)  86      85        +1
Ru(44)   101    100      +1
Pd(46)   106    106      zero

Note: Atomic weight percentages on the chart are individually rounded to an integer, so the percentages for an Element will not always sum to 100.  Percentages to 2 decimals are used to calculate Ave-Z.  The “atomic mass” is not the average atomic weight of an Element because the mass is missing the nucleic binding energy.

Ave-Z = Sum [ Isotope% x Z ] / 100

Example: Ave-Z (of Zr(40))

= ((51.45x90)+(11.32x91)+(17.19x92)+(17.28x94)+(2.76x96)) /100
= (4630.50 + 1030.12 + 1581.48 + 1624.32 + 264.96) /100
= 9131.38/100 = 91.3138 (rounded to 91)

Notice that the spread prefers Isotopes in parity.  For example in Zr(40) the percentages in parity are 51%+17%+17% = 85%.
Zr(40) is 85% in parity
Mo(42) is 74% in parity.
Ru(44) is 72% in parity.
Pd(46) is 77%) in parity.

The even numbered Elementa from 72 to 82 have Isotope spread averages that are also only off 1 or equal to the z-gradient.
The percentage spreads can be seen on the Isotope Abundance charts in Part VI.

Av- Z     z    variance 
Hf(72)   178    177      +1
W(74)    184    183      +1
Os(76)   190    189      +1
Pt(78)    195    195      zero
Hg(80)   201    201      zero
Pb(82)   207    207      zero

Part V – Radioactive Decay

Alpha Radioactive Decay

Alpha radioactive decay is when an Alpha Particle (of 2 ionized Protons and 2 Neutrons) splits off of the atom, and becomes an ionized Helium atom, which due to the positive charge (from being ionized) flies off at great speed.  (In the chart above Alpha decay is shown by a vertical arrow upward by 2 elements).

Alpha: à -2P  -2N  = -4Z

The Element transmutes to the Element with 2 less Protons, retaining the parity or non-parity of the Isotopes.  The atomic weight (Z) is reduced by 4.

Beta Radioactive Decay

Beta decay changes the number of Protons, by either converting a Proton into a Neutron (Beta+), or by converting a Neutron into a Proton (Beta-).  Since either a Proton changes to a Neutron, or a Neutron changes to a Proton, the atomic weight (Z=P+N) is unaltered.  Transmutation changes the Element, and also increases (when Beta+) or decreases (when Beta-) the number of excess Neutrons (n) by 2. 

Beta+ à -1P à +2n
à +1P à -2n

Note: Beta decay involves conversions between mass and gravity.  Refer to article on Gravity for more detail.

Part VI – The Isotope Abundance Charts

The isotope abundance charts to follow show Isotope abundance percentages by the n-value.  The advantage of this is it compacts the chart, moving each successive Element left by 2 positions from where they would display if the chart was by atomic weight.

To calculate atomic weight (Z) from the chart, use Z=2P+n.

The n-gradient is shown as a blue line (with blue boxes marking the integer jumps).  The Isotope Beta-decay target-Element is pointed to with arrows. 

Shaded boxes are in parity.  The charts show that with few exceptions Beta decays are going from non-parity to parity.  The variant decays are shown in red, and are explained in Part VII – The Major Variances.

Dotted lines plot the converging meeting points of the Beta decay arrows (the lines either follow the 45 degree angle of the linear Beta-decay gradient, or drop straight down). 

The Complete Isotope Abundance Charts
(Data from “The Elements” by John Emsley, Oxford Press)

The chart above shows how the beta-decay-gradient is tangent to the n-gradient at Element 78.  The beta-decay dotted lines are split around the blue n-gradient line because the n-gradient is out of parity (passes through the un-shaded out-of-parity boxes).

The chart below shows the multiple reductive paths (from 92 to 82) of the radioactive group of Elements

The Cosmos of Flux

Even as the solar bodies create the natural Elements out of clouds of Hydrogen and Helium (see article on fusion), Alpha and Beta decay is ever so gradually, but actively, reducing the Elements back down to Hydrogen and Helium.  All Isotopes of atomic number >2 are alpha radioactive, but to varying degrees.  Their rate of decomposition is regulated by the half-life law, where the decay rate is not linear, but proportional to the count of atoms of that Isotope in matter.  It is the inverse of exponential doubling: 1,2,4,8,16,32,64, meaning that in a given period of time 64 will reduce to 32, and in the same length of time 32 will reduce to 16 (and not to zero as would occur if it were to be a linear decay).

Part VII – The Major Variances

The major variances from the z-balance has hidden the balance formula from conventional academic chemists, and these variances begin very early in atomic number. 

The Dip in Binding Energy

Atoms are thought to be held together by nucleonic binding energy (taken from the mass of the constituent nucleons (Protons and Neutrons).  The binding energy is small because it only needs to hold the nucleons together for the brief interval when the cubic matrix dematerializes.  The binding energy generally increases with increased atomic number up to Iron(26) and then gradually decreases, but there is a dip in binding energy between Helium(2) and Oxygen(8) that results in an aberration in the Isotopes of Beryllium(4) and Nitrogen(7), where high-abundance non-parity occurs. 


In the following, non-parity is highlighted in red.

What happens is that (12)Carbon(6) captures a free neutron and then may Alpha decay to (9)Beryllium(4), or capture another free neutron and Beta- decay to (14)Nitrogen(7)

(12)C(6) + n = (13)C(6); aà [(4)He(2)) + (9)Be(4)], or

(12)C(6) + n = (13)C(6); + n = (14)C(6); b-à (14)N(7)

It is the beta-decay of carbon-14 that is used in carbon-dating. Such dating assumes that the percentage of carbon-14 in the air is relatively constant through time, and uses the fact that the decay of carbon-14 works at a constant rate to calculate how long the decay has been taking place.  By measuring the residual percentage of carbon-14 in organic matter, a rough date the carbon was metabolized can be calculated.

The Potassium Beta split

There is only one Isotope (the non-parity Isotope (40)Potassium(19)) that splits Beta decay of a single Isotope between both Beta+ and Beta-.  This out-of-parity Isotope Beta split results in the aberrations at (40)Argon(18) and (40)Calcium(20), shown in green below.

(40)K(19)  b+ à (40)Ar(18), or

(40)K(19)  b- à (40)Ca(20).

The (58)Nickel(28) Variance

As shown below, the final phase of fusion in our Sun combines Silicon(14) and Sulphur(16) into Zinc(30), which drops down to Nickel(28) due to the Isotope Balance, which also causes some of the Nickel to drop to Iron(26) where the n-gradient is 4.  The remaining (58)Nickel(28) is the source of the -3 variance to the Nickel n-gradient of 5.

(60)Zn(30) à 2H + (58)Ni(28) à 2H + (56)Fe(26) or (n=4)
             or  à 2H + (58)Ni(28) or (n=2).

The Inert Bulges

Since all Elements >2 Alpha decay, to avoid clutter the arrows are not usually shown except where the Alpha decay is significant.

The cluttered chart below shows the pattern of Alpha decay without compensating Beta- decay, that causes the bulge around the inert Element Krypton(36).  Notice that Beta+ decays (to the far left) cross the n-gradient at Elements Se(34), Kr(36), Sr(38), and Zr(40.  


There is a common pattern concerning the bulges around the inert Elements Krypton(36) and Xenon(54). 


In both bulges, there is a “ledge” (highlighted yellow) 2 Elements prior to the inert Elements (highlighted red), where the successive alpha decays (green arrows), without adjusting beta decays, terminate.  Also, in both bulges Elements with no natural isotopes (labeled “Null”) occur 8 Elements down from the inert Elements,

In the radioactive Element chart below, the bulge can be seen around the inert Element Radon(86). 

The inert Elements have no interior Bias energy planes drawing-out energy from the Protons aligned on the planes, and this may be a factor causing the bulges.

Index of all Articles by Tom Gilmore