Atomic Number Equation Based on Larson's Triplets

December 31, 1980

Dear Mr. Larson

I trust this letter finds you in good health.

After many a summer, I have taken up my ‘pen’ again, to write to you. I decided years ago, that until I has something better top do than just ask questions, I would not bother you directly. So, as you know, I have been writing to N.S.A. members, in particular Frank Meyer.

I have been doing some of my own in-depth study of R.S. and have appeared to come up with something of interest to R.S. IN GENERAL and you in particular. Please make allowance for my lack of COMPLETE understanding, which is a goal to work towards, but difficult to achieve.

As you know, my interest in anything at all is mainly in mathematical aspects, where possible. So, since you started the ball rolling with number triplets, where better for me to start also, and hopefully make my own discoveries, where possible? This renewed vigour, on my part, has come about because I have been invited to submit a paper on R.S. to the science editor of a journal, Helix. Because I have no guarantee that it will be accepted, I felt that I had to do as complete an introduction as I was capable, both philosophically and mathematically.

I have come across problems, some of which I have not solved, some solved, and yet others with theoretical developments to be presented to you for an opinion etc..

 

1) The algebraic expression for atomic number, I have found to be

ATOMIC NUMBER EQUATION BASED ON LARSON’S TRIPLETS (a,b,c,)

If a = b then this reduces to

If a = b + 1 then it reduces to

a=b		a = b+1		Range of c	Z	Range of Z
a	b	a	b
		2	1	-1 to 4	c + 2	1 to 6
2	2			-4 to 4	c + 10	6 to 14
		3	2	-4 to 9	c + 18	14 to 27
3	3			-8 to 9	c + 36	28 to 45
		4	3	-8 to 16	c + 54	46 to 70
4	4			-15 to 16	c + 86	71 to 102
		5	4	-15 to -1	c + 118	103 to 117

 

 

2) The a,b,c represent displacements, as originally defined in your books.

a) However, note well, that the sum of 3 algebraic expressions in a,b,c resp. results in (Z+2) (see later) or maybe, yet to be investigated, since only a,b are included in cubic expressions. We should especially note well (Z+2-c)

b) a,b,c have been used later, in a Reciprocity journal, representing the magnetic and electric rotations, confusing to the readers of the books and the journal.

 

3) This brought me to the full realisation, that I could not come to grips with the problem, “How does anyone, but yourself, know with certainty, the algorithm to determine the rotations for all elements?”

Frank Meyer wrote in Reciprocity a few years ago, that he, his son and Ronald Satz were working on a basic computer-program to not merely work out these rotations, but go further and work out crystal structures.

I have made a chart of all elements with their respective displacements and rotations, but I could not detect any mathematical pattern, furthermore the elements with more than one triplet of rotations were not obviously so to me, as to why they were the ones as distinct from any others, despite rereading the book. I would appreciate such an algorithm from you.

 

4) I believe that I understand the concept of Ionization level. I see it as an indication of the age of the element in question. I presume that at creation, the elements have atomic mass A = 2.Z

The gravitational charge is, from one point of view, understandable, since I see it as the ‘make-up’ to bring the element in question to the atomic mass, as it would be measured today, this value being determined by its admixture od primary and secondary isotopes, hence its non-integer value. This, of course, does not apply to those rarer elements, whose stated atomic mass is an integer, because, presumably, there is no naturally occurring isotope, so that an admixture is out of the question, and as a consequence the formula A = 2.Z + G does not apply.

I see that the Ionization Level increases with time due to bombardment, followed by some absorption of neutrinos into the lower isotopes in particular.

However, this does not explain tha rarity of some lower isotopes, where the most abundant isotope is not at the bottom of the range.

So I see how the neutrinos build up the lower isotopes atom by atom into the higher isotopes. If there is a higher probability for the neutrinos to be absorbed by the higher isotopes in some cases, and the lower isotopes in other cases, then this needs a good theoretical basis, without any appearance of assumption.

This build up is the increase of Ionization Level, I believe. However I now see an apparent contradiction, since I think that Ionization Level, by definition, can only start at zero, and rise by integral values, in units. Your statement that the terrestrial level for `I' being just over unity is an averaged situation. Then I ask:-

A) Why then are some elements at a very high Ionization level today?

Is it because of probability considerations, in which case I feel the need to be shown these considerations, not only for myself, but for others?

B) Why are some elements largely negative, and some just negative, when negativity for Ionization Level is, by definition, impossible?

This brings the definition of Gravitational Charge, ‘G’, into doubt and that it needs alteration to avoid these contradictions (See charts later)

 

5) My answer is that there is something lacking in the formula, and I have taken the liberty to alter, as a proposal for serious consideration and evaluation, so see whether it can be considered as a logically correct deduction from the original premises, as much as the other formula for `G'.

 

IONIZATION LEVELS IN THE RECIPROCAL SYSTEM OF THEORY

It is stated in the texts how to calculate atomic mass from atomic number A = 2.Z + G where G represents gravitational charge and can be represented by

where the inter-regional ratio 156.4 = 1408/9 and where k = 0

The only alteration, (at this stage, anyhow), being `k' in the Z term under the square, since:-

A) `k', I believe, has a meaning within R.S.; I am sure that you could explain it in detail; I have only an intuition about it.

B) I stressed earlier that (Z+2) = f(a,b,c) OR (Z+2-c) = g(a,b) and therefore I suspected that `k' would turn out to be equal to 2 in many instances, in fact in about 33% more cases than k = 0, as you will see in the accompanying charts.

C) The artifice of using `k' is also mathematical, since it is another way to increase, or decrease, the numerator, thus in one foul swoop explaining away an apparent negative `I' value in one instance, and avoiding a non-integral `I' value in another instance.

 

6) At last to the charts:-

The column headings are fairly explicit, however to elaborate:-

The first two columns after Atomic Mass (Nos. 3 & 4) show the apparent I-values, derived from your equation for theoretical Atomic Mass, where actual Atomic Mass is used, and `k' is taken as 0 & 2 resp. and the equation is solved for ‘I’.

In the next 8 columns, I = 1 throughout and I compare your theoretical Atomic Mass with actual Atomic Mass, on a percentage basis, taking a succession of k-values from -3 thru 4.

The results from comparing firstly those first two columns with each other, then the next 8 columns with each other are both identical and interesting.

But firstly, I opine that out of 103 elements with stated atomic masses, we have to subtract 2 groups of them, in order to find proper meaning for worthwhile deductions.

The first group are those elements, only known from the table to have an integral mass, despite their many unstable isotopes, and therefore cannot be expected to conform to your equation.

Atomic Nos.:-84,85,86,87,89,94,95,96,97,98,99,100,101,102 & 103 (Total=15)

The second group are those elements, whose non-conformity with your equation is so obvious, that it is unreasonable to include them, since we are only comparing those elements, which conform to that equation:

Atomic Nos.:-1,2,3,4,5,6,7,8,10,12,14,15,16,18,19,20 & 28 (Total=17)

RESULTS

Where the percentage figure in the columns is close to 100, they fall into two main groups.

k = 0 :- 13,17,21,26,27,29,30,41,45,47,59,61,63,69,71,

73,74,75,77,78,79,80,81,82,83,88,90,91,92,93.

Total = 30

k = 2 :- 9,11,22,23,24,25,31,32,33,34,35,36,37,38,39,40,42,43,44,

46,48,49,50,51,52,53,54,55,56,57,58,60,62,64,65,66,67,68,70,72,76

Total = 41

So, as you can see, this analysis finds that k = 2 is the hot favorite in the ratio 41:30 i.e. 57.75% of the considered selection, which represents 36.67% more elements favor k = 2

 

7) Finally, there has to be a precise reason given for non-conformity to your equation. We see, overall, a parabolic type of curve, when plotting `Z' against `A', but not the same parabola fits them all.

k = 0 goes well for atomic numbers in the 70's and 80's

k = 2 goes well for atomic numbers in the 30's, 40's, 50's & 60's

However the first 20 are the worst drop-outs, and I feel intuitively that the explanation lies, mathematically speaking, in the function (Z+2-c) or perhaps (Z+k-c), as suggested earlier.

 

8) Also I am not clear, when you discuss c-matter arriving in this sector, and taking variously 1 or 2 Gravitational Charges, that you don't appear, likewise, to allow for more than one charge in your equation for Atomic mass, as used above.

My regards to your wife and yourself,

Sincerely

David Halprin

This equation covers columns 3 & 4

These 2 equations cover columns 5 to 12

ATOMIC NUMBER	ATOMIC MASS	APPARENT IONIZATION   LEVEL ASSUMING   AL = A		PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1
1	2	3	4	5	6	7	8	9	10	11	12
Z	A	k = 0	k = 2	k=-3	k=-2	k=-1	k=0	k=1	k=2	k=3	k=4
1: HYDROGEN	1.008	-155.19	-17.24	200.95	199.04	198.41	199.04	200.95	204.12	208.55	214.26
2: HELIUM	4.0026	0.1017	0.0254	100.09	99.94	100.09	100.57	101.37	102.49	103.93	105.68
3: LITHIUM	6.941	16.357	5.8886	86.44	86.53	86.81	87.27	87.91	88.75	89.75	90.96
4: BERYLLIUM	9.01504	8.9248	4.4110	88.81	89.02	89.37	89.87	90.51	91.29	92.21	93.27
5: BORON	10.81	5.0688	2.5861	92.74	93.04	93.45	93.98	94.63	95.40	96.29	97.30
6: CARBON	12.0111	0.0482	0.0271	100.39	100.76	101.24	101.82	102.52	103.31	104.22	105.23
7: NITROGEN	14.0067	0.0214	0.0164	100.68	101.09	101.59	102.19	102.87	103.64	104.52	105.47
8: OXYGEN	15.9994	-0.0014	-0.00094	101.00	101.44	101.96	102.56	103.24	104.00	104.84	105.76
9: FLUORINE	18.9984	1.9283	1.2909	95.96	96.39	96.90	97.47	98.11	98.81	99.59	100.43
10: NEON	20.182	0.2847	0.1977	100.65	101.12	101.66	102.26	102.93	103.66	104.45	105.31
11: SODIUM	22.9898	1.2797	0.9163	97.47	97.95	98.47	99.06	99.70	100.39	101.14	101.95
12: MAGNESIUM	24.312	0.3390	0.2490	100.85	101.35	101.90	102.50	103.16	103.87	104.63	105.45
13: ALUMINIUM	26.98	0.9072	0.6814	98.74	99.23	99.77	100.37	101.01	101.70	102.43	103.21
14: SILICON	28.086	0.0686	0.0526	102.45	102.97	103.54	104.15	104.81	105.52	106.27	107.07
15: PHOSPHORUS	30.9738	0.6771	0.5271	99.83	100.34	100.90	101.50	102.14	102.82	103.54	104.31
16: SULPHUR	32.064	0.0391	0.0309	103.17	103.71	104.29	104.90	105.56	106.26	107.00	107.77
17: CHLORINE	35.453	0.8353	0.6687	99.43	99.96	100.52	101.11	101.74	102.41	103.11	103.85
18: ARGON	39.958	1.9063	1.5441	93.72	94.21	94.74	95.30	95.89	96.51	97.17	97.86
19: POTASSIUM	39.10	0.4767	0.3902	101.37	101.91	102.48	103.09	103.72	104.40	105.10	105.83
20: CALCIUM	40.08	0.0313	0.0259	102.56	103.11	103.90	104.31	104.95	105.62	106.33	107.06
21: SCANDIUM	44.956	1.0486	0.8742	98.03	98.56	99.11	99.70	100.31	100.95	101.61	102.31
22: TITANIUM	47.90	1.2606	1.0593	96.68	97.20	97.74	98.32	98.92	99.54	100.20	100.89
23: VANADIUM	50.941	1.4612	1.2368	95.32	95.83	96.37	96.94	97.52	98.14	98.78	99.45
24: CHROMIUM	51.996	1.0853	0.9248	97.74	98.26	98.82	99.40	100.00	100.63	101.28	101.95
25: MANGANESE	54.9380	1.2360	1.0597	96.64	97.17	97.71	98.28	98.88	99.49	100.13	100.80
26: IRON	55.847	0.8903	0.7677	99.17	99.70	100.27	100.85	101.46	102.08	102.74	103.41
27: COBALT	58.9332	1.0587	0.9177	97.88	98.41	98.96	99.53	100.13	100.75	101.39	102.05
28: NICKEL	58.71	0.5408	0.4711	102.19	102.74	103.32	103.92	104.54	105.18	105.85	106.53
29: COPPER	63.545	1.0315	0.9027	98.07	98.61	99.16	99.73	100.33	100.94	101.57	102.23
30: ZINC	65.38	0.9352	0.8219	98.90	99.44	99.99	100.57	101.17	101.78	102.42	103.07
31: GALLIUM	69.72	1.2568	1.1090	96.11	96.64	97.18	97.74	98.32	98.91	99.53	100.16
32: GERMANIUM	72.59	1.3124	1.1625	95.57	96.09	96.63	97.18	97.75	98.35	98.95	99.58
33: ARSENIC	74.9216	1.2817	1.1394	93.77	96.29	96.83	97.38	97.95	98.54	99.14	99.77
34: SELENIUM	78.96	1.4832	1.3230	95.90	94.41	94.94	95.48	96.04	96.61	97.20	97.81
35: BROMINE	79.903	1.2647	1.1317	95.80	96.32	96.85	97.41	97.97	98.55	99.16	99.77
36: KRYPTON	83.80	1.4244	1.2784	94.23	94.74	95.26	95.80	96.36	96.93	97.52	98.12
37: RUBIDIUM	85.466	1.3103	1.1794	95.23	95.75	96.28	96.82	97.38	97.96	98.55	99.16
38: STRONTIUM	87.62	1.2589	1.1362	95.67	96.19	96.73	97.27	97.83	98.41	99.00	99.61
39: YTTRIUM	88.905	1.1216	1.0149	97.05	97.57	98.12	98.67	99.24	99.82	100.42	101.03
40: ZIRCONIUM	91.22	1.0971	0.9951	97.29	97.82	98.36	98.91	99.47	100.06	100.66	101.27
41: NIOBIUM	92.9066	1.0150	0.9228	98.20	98.73	99.27	99.82	100.40	100.98	101.58	102.19
42: MOLYBDENUM	95.94	1.0589	0.9648	97.69	98.21	98.75	98.31	99.31	100.48	101.04	101.65
43: TECHNETIUM	99	1.0999	1.0043	97.20	97.72	98.26	98.81	99.37	99.94	100.53	101.13
44: RUTHENIUM	101.1	1.0586	0.9685	97.67	98.20	98.73	99.28	99.84	100.42	101.01	101.61
45: RHODIUM	102.906	0.9971	0.9140	98.42	98.94	99.48	100.04	100.60	101.18	101.77	102.37

 

ATOMIC NUMBER	ATOMIC MASS	APPARENT IONIZATION   LEVEL ASSUMING   AL = A		PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1
1	2	3	4	5	6	7	8	9	10	11	12
Z	A	k = 0	k = 2	k=-3	k=-2	k=-1	k=0	k=1	k=2	k=3	k=4
46: PALLADIUM	106.4	1.0647	0.9778	97.57	98.10	98.63	99.18	99.73	100.30	100.89	101.49
47: SILVER	107.870	0.9823	0.9037	98.61	99.14	99.68	100.23	100.79	101.37	101.96	102.55
48: CADMIUM	112.40	1.1136	1.0263	96.93	97.44	97.97	98.51	99.06	99.62	100.20	100.78
49: INDIUM	114.82	1.0960	1.0117	97.13	97.65	98.18	98.72	99.27	99.83	100.40	100.99
50: TIN	118.69	1.1696	1.0813	96.15	96.66	97.18	97.72	98.26	98.82	99.38	99.96
51: ANTIMONY	121.75	1.1879	1.1000	95.87	96.38	96.90	97.43	97.97	98.52	99.08	99.66
52: TELLURIUM	127.6	1.3654	1.2661	93.53	94.03	94.53	95.05	95.57	96.11	96.66	97.21
53: IODINE	126.9044	1.1642	1.0811	96.12	96.63	97.15	97.67	98.22	98.76	99.32	99.89
54: XENON	131.30	1.2501	1.1624	94.92	95.42	95.93	96.45	96.98	97.52	98.07	98.63
55: CESIUM	132.906	1.1846	1.1030	95.77	96.27	96.79	97.31	97.85	98.39	98.94	99.51
56: BARIUM	137.34	1.2641	1.1784	94.62	95.12	95.63	96.14	96.67	97.20	97.75	98.30
57: LANTHANUM	138.91	1.1995	1.1195	95.49	95.99	96.50	97.02	97.55	98.09	98.63	99.19
58: CERIUM	140.12	1.1217	1.0482	96.59	97.09	97.61	98.13	98.67	99.21	99.76	100.32
59: PRASEODYMIUM	140.91	1.0296	0.9632	97.97	98.48	99.00	99.53	100.07	100.62	101.18	101.75
60: NEODYMIUM	144.24	1.0534	0.9865	97.59	98.10	98.62	99.14	99.68	100.23	100.78	101.34
61: PROMETHIUM	145	0.9670	0.9066	98.97	99.48	100.01	101.54	101.03	101.63	102.19	102.76
62: SAMARIUM	150.35	1.0724	1.0064	97.27	97.78	98.29	98.82	99.35	99.89	100.44	100.99
63: EUROPIUM	151.96	1.0233	0.9613	98.06	98.57	99.09	99.61	100.15	100.69	101.24	101.80
64: GADOLINIUM	157.25	1.1162	1.0496	96.52	97.02	97.53	98.05	98.57	99.10	99.65	100.20
65: TERBIUM	158.92	1.0709	1.0079	97.26	97.77	98.28	98.80	99.32	99.86	100.40	100.95
66: DYSPROSIUM	162.50	1.0954	1.0319	96.84	97.34	97.85	98.36	98.88	99.40	99.95	100.50
67: HOLMIUM	164.93	1.0779	1.0163	97.12	97.62	98.13	98.64	99.17	99.70	100.24	100.78
68: ERBIUM	167.26	1.0576	0.9980	97.46	97.96	98.47	98.98	99.51	100.04	100.57	101.12
69: THULIUM	168.934	1.0165	0.9600	98.17	98.67	99.18	99.70	100.22	100.76	101.30	101.85
70: YTTERBIUM	173.04	1.0549	0.9971	97.49	97.99	98.49	99.00	99.55	100.05	100.59	101.13
71: LUTETIUM	174.97	1.0232	0.9679	98.05	98.55	99.06	99.57	100.10	100.62	101.16	101.71
72: HAFNIUM	178.49	1.0409	0.9853	97.73	98.22	98.73	99.24	99.76	100.29	100.82	101.36
73: TANTALUM	180.948	1.0260	0.9720	98.00	98.49	99.00	99.51	100.03	100.56	101.09	101.63
74: TUNGSTEN	183.85	1.0242	0.9710	98.03	98.52	99.03	99.54	100.06	100.58	101.11	101.65
75: RHENIUM	186.2	1.0068	0.9552	98.35	98.85	99.36	99.86	100.38	100.91	101.44	101.98
76: OSMIUM	190.2	1.0347	0.9823	97.83	98.32	98.82	99.33	99.84	100.36	100.89	101.42
77: IRIDIUM	192.22	1.0085	0.9581	98.33	98.82	99.32	99.83	100.35	100.87	101.40	101.93
78: PLATINUM	195.09	1.0051	0.9555	98.39	98.89	99.39	99.90	100.41	100.93	101.46	101.99
79: GOLD	196.967	0.9768	0.9292	98.96	99.48	99.96	100.47	100.99	101.51	102.04	102.57
80: MERCURY	200.59	0.9922	0.9444	98.66	99.15	99.65	100.16	100.67	101.19	101.72	102.25
81: THALLIUM	204.39	1.0108	0.9626	98.29	98.78	99.28	99.78	100.29	100.80	101.33	101.86
82: LEAD	207.19	1.0049	0.9576	98.41	98.90	99.40	99.90	100.41	100.92	101.44	101.97
83: BISMUTH	208.980	0.9760	0.9307	98.41	98.90	99.40	99.90	100.41	100.92	101.44	101.97
84: POLONIUM	210	0.9312	0.8884	99.97	100.47	100.97	101.48	101.99	102.51	103.04	103.57
85: ASTATINE	210	0.8661	0.8268	101.42	101.92	102.42	102.94	103.46	103.99	104.52	105.06
86: RADON	222	1.0576	1.0101	97.31	97.79	98.28	98.77	99.27	99.77	100.28	100.80
87: FRANCIUM	223	1.0128	0.9678	98.25	98.74	99.23	99.72	100.22	100.73	101.24	101.76
88: RADIUM	226.026	1.0106	0.9662	98.30	98.78	99.27	99.77	100.27	100.77	101.29	101.80
89: ACTINIUM	227.03	0.9678	0.9257	99.20	99.70	100.22	100.72	101.22	101.73	102.24	102.76
90: THORIUM	232.038	1.0051	0.9618	98.42	98.91	99.39	99.89	100.39	100.89	101.40	101.91

 

ATOMIC NUMBER	ATOMIC MASS	APPARENT IONIZATION   LEVEL ASSUMING   AL = A		PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1
1	2	3	4	5	6	7	8	9	10	11	12
Z	A	k = 0	k = 2	k=-3	k=-2	k=-1	k=0	k=1	k=2	k=3	k=4
91: PROTACTINIUM	231.036	0.9264	0.8870	100.20	100.69	101.18	101.68	102.19	102.70	103.22	103.74
92: URANIUM	238.04	0.9988	0.9568	98.57	99.05	99.53	100.03	100.52	101.02	101.53	102.05
93: NEPTUNIUM	237.04	0.9234	0.8849	100.31	100.79	101.29	101.79	102.29	102.80	103.32	103.84
94: PLUTONIUM	244	0.9915	0.9506	98.74	99.22	99.71	100.20	100.69	101.19	101.70	102.20
95: AMERICIUM	243	0.9187	0.8812	100.45	100.94	101.43	101.92	102.43	102.93	103.45	103.97
96: CURIUM	247	0.9336	0.8959	100.12	100.60	101.09	101.50	102.08	102.58	103.10	103.61
97: BERKELIUM	247	0.8812	0.8460	101.41	101.90	102.39	102.89	103.39	103.90	104.42	104.92
98: CALIFORNIUM	251	0.8959	0.8604	101.07	101.56	102.05	102.55	103.05	103.55	104.07	104.58
99: EINSTEINIUM	254	0.8939	0.8588	101.14	101.63	102.12	102.62	103.12	103.62	104.13	104.65
100: FERMIUM	257	0.8917	0.8571	101.22	101.71	102.20	102.69	103.19	103.69	104.21	104.72
101: MENDELEVIUM	256	0.8282	0.7963	102.88	103.38	103.88	104.38	104.88	105.40	105.91	106.43
102: NOBELIUM	254	0.7518	0.7232	104.98	105.48	105.98	106.50	107.01	107.53	108.06	108.59
103: LAWRENCIUM	257	0.7520	0.7237	105.03	105.52	106.03	106.54	107.05	107.57	108.10	108.63
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117: LARSONIUM	est.???