## APPENDIX 4: Radiometric Decay Processes

### (i) Alpha decay processes

In discussing radiometric decay, there are two quantities of importance, namely the decay constant, λ*, and the half-life, τ, of the element. They are linked as follows [176]:

τ = ln 2/λ* = 0.69315/λ*                                               (107)

The frequency of escape of alpha particles from the nucleus is equal to the decay constant, λ*, in units of seconds-1 [177]. So, a greater escape frequency of alpha particles from the nucleus results in a higher decay constant, λ*, and a shorter half-life, τ, in inverse proportion. As (16) and (17) apply to nucleons as well as electrons, the velocity, v, at which nucleons move in their orbitals within the nuclear potential well is proportional to c, and inversely proportional to h and U. As the radius, r, of atomic nuclei is essentially constant, then the alpha particle escape frequency, λ*, the decay constant defined by Glasstone [177] and Von Buttlar [178], is

λ* = Pv/r                                                             (108)

P is the probability of alpha particle escape from a nuclear potential well by the tunnelling process, and is a function of the electrostatic or Coulomb energy of the nucleus, which is constant from (7A). Radius, r, is constant, while velocity, v, is proportional to c. But a higher velocity means more hits per second against the potential well “wall”, with an unchanged escape probability. So the alpha particle tunnels out sooner since the number of hits required before it could statistically escape occurred more quickly. Thus (107), (108), (57) show that

λ* ~ c ~ 1/τ ~ 1/U                                       (109)

So the alpha decay constant is proportional to c and 1/U, with half-lives proportional to U.

### (ii) Beta decay processes

For β decay processes, Von Buttlar defines the decay constant as [179]:

λ* = Gf = mc2g2M2f/(π2h)                                   (110)

Here, M is the nuclear matrix element dependent upon the electrostatic energy of the nucleus, which is unchanged with varying ZPE, as is the constant g. The term (mc2) is a constant from (16). The term f is a function of atomic number Z, and emission energy (related to the energy of the nuclear potential well), and so is unchanged. Therefore, for β decay processes

λ* ~ c ~ 1/τ ~ 1/U                          (111)

Another approach by Burcham [180] leads to the same result.

### (iii) Electron capture processes

For electron capture, the relevant equation is given by Burcham as [181]

λ* = K2M2f(2π3)                                               (112)

Here, M is the same quantity as in (110) and remains unchanged. Also, f is a function of the fine structure constant, the atomic number, Z, and Coulomb energy, all of which remain unchanged with varying ZPE. The quantity K2 is defined by Burcham as [182]:

K2 = g2m5c4/(h/2π)7                                               (113)

where g is independent of ZPE strength. Proportionalities in (57) for m, h, and c, then give us:

K2 ~ c ~ 1/U ~ λ* ~ 1/τ                                         (114)

So electron capture has decay constants proportional to c, with half-lives proportional to h and U. All three types of radioactive decay therefore give concordant results.  Note for the decay of potassium-40, that a gamma ray is emitted. The energy density of that ray is proportional to 1/c from (77). But more rays are emitted per unit time proportional to c. Thus the intensity of this radiation remains unchanged as in (86A). So damage from a higher number of rays in the past was the same as fewer rays today since radiation energy densities remain constant.

### (iv) Coupling constants

The weak coupling constant, gw, which is involved in decay processes, is a dimensionless number and so remains invariant with changes in the ZPE strength. This is generally true for all dimensionless coupling constants. In this case, Wesson defines gw as follows [183]:

gw = [gm2c/(h/2π)3]2                                               (115)

Here, m is the pion mass and g is the constant that appears in (110) and (113) above.  It is called the Fermi interaction constant, and has units of energy density of a Coulomb field, and is thus constant. If the proportionalities of (57) are applied to the terms m, c, and h in (115), gw remains constant when the strength of the ZPE varies. This is verified by data since any variation in gw results in a discrepancy between the radiometric ages for alpha and beta decay processes [183]. This is not usually observed. Similar theoretical and experimental evidence also shows that the strong coupling constant, gs, has been invariant over cosmic time [183]. We conclude that radiometric clocks tick with the same dependence on the ZPE as other atomic clocks do. But pleochroic haloes are said to prove radioactive decay rates are constant.

### (v) Pleochroic haloes

Pleochroic haloes form in mica and other minerals which contain minute quantities of uranium etc. Alpha particles emitted by the decaying elements interact, because of their charge, with electrons of the surrounding atoms, which slows them down until they stop, producing a discoloration. The distance traveled depends first on their initial kinetic energy, and second on the composition and number of atoms per unit volume of the host material. These latter properties remain unchanged with ZPE variation, but the kinetic energy has two aspects needing scrutiny. At the moment of emission from the nucleus, the particle’s kinetic energy remains unchanged, since the Coulomb energy of both the emitting nucleus and the alpha particle are constant from (7A) with fixed nuclear distances. But a lower ZPE strength means particle masses, m, are lower from (16) with velocities, v, higher from (17), so particle kinetic energies (½ mv2) inside the nucleus remain unchanged. Since the Coulomb energies of nucleus and particle are also unchanged, particle kinetic energies at emission are constant.

The second aspect is the interaction of the particle with its host material that diminishes its initially unchanged kinetic energy. As the particle moves through its host material, it interacts with the same number of energy potential wells as before. The electrostatic energy of both the potential wells and the particle are unchanged at a given distance from (7A), so the particle’s unchanged kinetic energy carries it the same distance as before. The higher particle velocity will not carry it further as its mass will be lower. The interaction of the same charges on this lower mass is thus correspondingly greater. This can be expressed mathematically. Let a force of electrostatic attraction, F, produce a deceleration, a, on an alpha particle of mass, m. Let the force of electrostatic attraction between charges be proportional to (e1e2/ε), where e1 is the particle charge and e2 is an interacting host material charge [184]. From (7A), F must remain unchanged at a given distance. But m is proportional to 1/c2 so the deceleration, a, is proportional to c2. So the same charge interaction per particle mass produces a deceleration increased by a factor of c2, exactly counteracting the effects of mass decrease in proportion to 1/c2. Thus pleochroic halo radii will remain fixed with varying ZPE.

(vi). Spontaneous fission and the Oklo natural reactor

It has been objected that the Oklo natural reactor in the Republic of Gabon in West Africa proves fission rates are fixed. Geological conditions 1.8 billion atomic years ago allowed a chain reaction to cycle on and off repeatedly for some time in that uranium ore deposit. The cyclical nature of the process was revealed by isotopes left from reactions which were induced by the action of a geyser [185]. The water slowed down neutrons that were produced by spontaneous fission of uranium nuclei. These slow neutrons were then absorbed by nearby uranium nuclei and, in so doing, provided the activation energy needed to allow the fission of those nuclei. The reactor shut off when the water evaporated and the emitted neutrons were no longer slow enough to be absorbed. The cycle repeated when the water returned.

Isotopes from this reaction placed limits on any fine structure constant variability [186]. John Barrow then claimed c could not have differed from today’s value by more than one part in ten million [187].  Barrow’s statement, which used E = mc2 as its basis, is correct only if c alone varied. But Barrow's problem disappears if other physical quantities, such as m, varied so energy was conserved, or ZPE dependent data were allowed to dictate the scenario.

At Oklo, the spontaneous fission of U238 with a current half-life of 1016 years was the main source of neutrons [188].  This reaction’s fission tracks acting over geological time are used for atomic dating. It is true that the strong, weak and electromagnetic forces were unchanged in this fission process. But sub-atomic particle masses were lower when the ZPE was lower. The velocity of particles in the nucleus was thereby higher, just as for alpha particles, so important stable configurations of nucleons formed by chance more quickly, and the nucleus split more rapidly. Thus spontaneous fission half-lives are proportional to U, and h. This also means there will be a greater flux of neutrons in the past than there is now. But neutron capture by a nucleus to form an activated complex only occurs if a neutron is near a nucleus for a relatively long time. Short-range nuclear attractive forces then have a better chance to take effect. Such nuclei are called 1/v absorbers, where v is the neutron velocity [189].

To model a neutron flux, we note from (16) and (17), that the number of approaches per second by neutrons to a nucleus is proportional to c. Because the velocity of neutrons is faster when c is higher, the time each one is in the vicinity of a nucleus is then proportional to 1/c. So the number of neutrons approaching the nucleus per second multiplied by the time they spend near the nucleus is unchanged for all U, h, and c. So, like (90), the number of interactions to form the activated complex at any instant will also remain unchanged. As a consequence, the rate of nuclear reactions at Oklo will be unaltered as the ZPE varies.

Radiation energy densities also influence the heat from radioactive decay processes. With lower ZPE, and higher decay rates, came a greater number of gamma and X-rays. But this was moderated by the lower vacuum permittivity and permeability so radiation intensities were the same as today. Now radioactive decay processes produce heat. This applies to gamma and X-radiation, which are electromagnetic in character, as well as alpha and beta radiation, which are sub-atomic particles. Gamma and X-radiation can often accompany alpha and beta emission also. There is a reason why heat is produced by all these forms of radioactive decay.

The high energy forms of radiation ionize and/or excite the atoms of the substances through which they pass. The ionization process strips an electron(s) off those atoms with which the alpha, beta or gamma radiation interacts, while the excitation process shifts an electron(s) to a higher orbit in the host atom. In the case of excited atoms, the electrons return to lower orbits, emitting low energy photons which appear as heat, until the atom is again in its ground state. In the case of ionization, where the electrons are stripped off, these electrons cause the secondary excitation of atoms with which they interact. The process continues until all the kinetic energy originally imparted to these electrons is used up by the excitation process. As the excited atoms return to their ground state, they emit low energy photons that again result in heat. Gamma radiation produces ionization and the excitation of atoms over a relatively large distance, while alpha and beta particles only produce significant results over a short distance.

It can therefore be seen that the majority of heat from radioactive decay is generated by the lower energy photons. Since they are electromagnetic in character, they are subject to the moderating effects of the permittivity and permeability of space as shown by (76) and (77). Thus, even though a given radioactive source emitted more electromagnetic waves per unit time of both high and low energy, their effects would have been the same as the fewer number of waves emitted by that same source today. Thus heat production was similarly moderated.

### (viii) Radiometric dates & carbon 14 data

Equations (107) to (114) reveal that radiometric decay processes are affected by variations in the ZPE strength in the same way that lightspeed is. Therefore the radiometric clock ticks at a rate that is inversely proportional to U and proportional to c. In other words, atomic time intervals were shorter when the speed of light was higher and the strength of the ZPE was less. This means that equation (58) describes radiometric clock behavior in the same way that it does for other atomic ‘clocks.’ Therefore, equation (59) can be used to correct all atomic and radiometric dates to orbital dates. The results of this are the subject of separate analyses for astronomy, geology and archaeology.

What is important is that this process can be used in reverse to determine the behavior of the ZPE in the past by comparing radiometric dates obtained for objects of known or approximately known orbital dates. The difference between the two dates for the same object reveals the disparity for that particular point in time.  When a number of these objects of known dates are used, a curve begins to emerge which shows us how fast or slow the atomic clock was ticking at that time compared to our normal, orbital clock.. It is thus a direct measure of how the speed of light was behaving or an inverse measure of the ZPE behavior. Figure 8 is a graph of the results of just such a comparison between orbital and/or historical dates and the resultant carbon 14 dates from Reimer et al., published in Radiocarbon 46 (3), 1029-1058, (2004).

The graph in Figure 8 illustrates the oscillation which shows up as part of the measurements of many atomic processes as well as the strength of the ZPE. The evidence for this oscillation seems to point to its appearance starting about 2600 BC, as mentioned in the main text in section 3.17. Figure 8 also illustrates the effect of the rampant industrialization that occurred in the 20th century as the burning of fossil fuels put large amounts of extra carbon into the atmosphere. As a consequence, the final data points on that graph from 1910 to 1950 are extremely high, even though the general trend of the graph at that point is downwards. If these data points are ignored, then the zero deviation line is in the position shown for 1950. As we go back from 1950 AD we see a general rise in the trend of the graph which peaks between 500 and 800 AD. From that point there is a steady, general decline as we continue going back in time. By 500 BC it is again on the zero deviation line and continues dropping from there. Since the speed of light follows this same trend, it might be anticipated that c reached an oscillation maximum somewhere between 500 - 800 AD also, after which it dropped to its minimum about 1970 AD. However, in order to discuss the results prior to 500 BC, additional information is needed.

The radiocarbon or C-14 dates require standardized and known corrections in order for them to be converted to orbital or historical dates. The correction standard is often derived from tree-rings and dendro-chronological techniques. The reasons given that explain why this correction is necessary have been varied, but it has become customary to invoke changes in solar activity or the earth’s magnetic field. While it is true that the Sun has cycles of 11 years, 22 years, and about 440 years, this is already apparent in the C-14 data and its short-term variations. However, the long-term overall trend appears to have some other overarching cause, as there is no really satisfying solar or geomagnetic explanation for this need to consistently correct radiocarbon dates.  This correction is, however, clearly and easily explained by variation in the ZPE strength due to cosmological factors, as is evidenced in Figure 8.

The 2004 data list and their corrections is taken as being approximately correct back to about 1550 BC. This is in basic agreement with R.H. Brown [Origins 21 (2):66-79, Figure 1 (1994)] who considered dates back to 3500 BP to be valid. Here, the designation BP means Before Present where 1950 marks the zero point. More recent developments reported in Science, 312, p. 548 (28th April 2006) indicated that this method of C-14 dating, with the standard corrections, yields dates for the Santorini (Thera) eruption between 1627 and 1600 BC. But the closing sentence by Friedrich et al. in their Abstract indicates a long-standing problem. They say "Our result is in the range of previous, less precise, and less direct results of several scientific dating methods, but it is a century earlierthan the date derived from traditional Egyptian chronologies." The conclusion is, therefore, that earlier than about 1650 BC discrepancies appear in the data that are the cause for some concern. [See for example “Dating the Pyramids” in Archaeology Abstracts 52 (5) September/October 1999, or Bonani et al., in Radiocarbon 43 (3) pp.1147 ff (May 2001)].

### (ix) Tracing back the ZPE oscillation

Prior to 1650 BC the correction supplied by dendrochronology gives results that do not accord with archaeological data. There are several aspects to this. First, the oldest living tree began growing in 2766 BC. This bristlecone pine, the “Methuselah Tree,” is nearly 1000 years older than the next living specimen. This second oldest tree thus began growing around 1700 BC or a little earlier. These two living trees provide the absolute maximum to which any reliable dendrochronology correction can be applied to C-14 dates. Yet even here, the longest continuous tree-ring sequence, starting from near the present, that the Methuselah Tree has produced only goes back as far as 1203 BC. This is given in the sample labelled MWK975. The extension of the record from this important tree back into the past beyond 1203 BC is itself dependent upon the cross-matching of rings from other, older, parts of the tree. Since a similar situation may apply to the second oldest tree, the actual date at which these two trees really began growing may be open to question. The raw data itself in 10 year intervals for the Methuselah Tree can be viewed here.

Second, these tree-ring sequences need to be anchored by historical events. Indeed, Brown, op. cit., has stated that “The validity of tree-ring master sequences on which the data is based is controlled by the availability of material which can be C-14 dated and also has an unquestioned historical age. The time range from which such samples are available extends to the vicinity of 3500 BP (about 1500 BC).” It therefore becomes apparent why this system of correction breaks down for dates earlier than about 1650 BC. The Santorini explosion is the last historical event that shows up in the tree-ring sequences. Baillie and Munro have pointed out that it does so as frost-rings in Irish (bog) oaks [Nature, 332 (1988), pp.344-346]. The bristlecone pine data also have a significant frost-ring event near 1626 BC [V.C. LaMarche and K.K. Hirschboeck, Nature, 307 (1984), pp. 121-126].

Third, as recently as 1626 BC, C-14 results seem to show divergence from historical dates, and, as mentioned above, the problem appears to get worse the further back we go. Therefore 1650 BC is taken as the limit to which reliable corrections can be applied. The ‘wiggle matching’ techniques used to extend the tree-ring chronology further back into the past can only do so after the sample has first been C-14 dated. This is an extension based on a seemingly circular argument and the results obtained from this must therefore be viewed with extreme caution.  This is emphasized by the comments of D.K. Yamaguchi, who, when trying to match a single tree to the whole sequence of rings, found 113 matches that were false. Yet each of these matches had a high t-statistic score [D.K. Yamaguchi, Tree Ring Bulletin 46: 47-54 (1986)]. Nor is this a problem of the past. Girssino-Mayer et. al. have stated in 2004 that "crossdating autocorrelated tree-ring series against a reference chronology can result in many 'false positives,' i.e., a placement may be found for the chronology being dated that is temporally [time] incorrect." [Journal of Archaeological Science, 31: 167-174 (2004)]. Indeed, the statistical techniques used for tree-ring matching have produced some degree of embarrassment. On three occasions the published matches were found to be in error, each time after strong assertions of reliability [M.G.L. Baillie, "A Slice Through Time: dendrochronology and precision dating," Chapter 2, London: Batsford (1995), and M. Spurk et. al., Radiocarbon 40: 1107-1116 (1998)]. Because of these factors, 1650 BC is taken as the reliable limit to which C-14 corrections are based on dendrochronology can be taken.

However, we do have a comparison with historical objects of closely known orbital age and C-14 data in one period prior to 1650 BC. Dynasty 4 in Egypt produced pyramids over a span of 85 years centering on 2550 BC. The standardized correction to C-14 dates around 2550 BC requires the addition of about 300 years to get the historical date. In 1984 and 1995 the members of the David H. Koch Pyramids Radiocarbon Project took many C-14 samples and published the Abstract containing their results in Archaeology, 52 (5), Sept/Oct 1999. The 1984 results gave a carbon 14 age that averaged 374 years older than the recommended Cambridge Ancient History historical dates. The 1995 C-14 results ranged from 100 to 350 years older than the historical data. Data from Bonani et al. (op. cit., 2001) support this contention throughout most of the period covered by the Old Kingdom. These two discrepancies are shown as the two points on the far left of Figure 8.

Now approximately 300 years had been added to the C-14 date to get the historical date. But the resultant date is averaging about 300 years too old. This means that the uncorrected C-14 date and the historical date were the same and the dendro-correction from wiggle-matching is inaccurate. In other words, a zero correction applies as the C-14 clock is, at that point, ticking at the same rate as it is today. Thus, going backwards in time, the actual curve prior to 1650 BC rose from near a minimum then to reach a near-zero-correction level again near 2550 BC. If we allow for about 50 years of error in the data, then 2600 BC is a new origin point. Going back in time from there means an increased rate of ticking of atomic clocks (and hence an increase in c) compared with today’s values. This implies that the actual origin of the ZPE curve without the oscillation occurred sometime before that.

References

[176] A.P. French, op.cit., p.298. (see ref. #73)
[177] S. Glasstone, Sourcebook on Atomic Energy, 1st Edition, Macmillan, London, (1950) 158.
[178] H. Von Buttlar, “Nuclear Physics” Academic Press, New York, (1968) 448-449.
[179] Ibid, pp. 485, 492.
[180] W.E. Burcham, “Nuclear Physics”, McGraw-Hill, New York, (1963) 606.
[181] Ibid, p.609.
[182] Ibid, p.604.
[183] P.S. Wesson, op. cit., pp.65-66. (see ref. #168)
[184] A.P. French, op.cit., p.31. (see ref. #73, linked above)
[185] A.P. Meshik, C.M. Hohenberg, and O.V. Pravdivtseva, Phys. Rev. Lett. 93, (2004), p.182302.
[186] F. Dyson and T. Damour, Nuclear Physics B, 480 (19970, 37.
[187] J.D. Barrow, “Impossibility: the limits of science and the science of limits,” pp.186-187 (Oxford University
Press, 1998).
[188] M.R. Wehr and J.A. Richards, “Physics of the Atom,” p.311 (Addison-Wesley, 1960).

[189] Ibid, p.321.

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