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Journal of Radioanalytical and Nuclear Chemistry, Vol. 257, No. 1 (2003) 33–43
0236–5731/2003/USD 20.00 Akadémiai Kiadó, Budapest
© 2003 Akadémiai Kiadó, Budapest Kluwer Academic Publishers, Dordrecht
Periodicity of the stable isotopes
J. C. A. Boeyens
Department of Chemistry, University of Pretoria 0002 Pretoria, South Africa
(Received November 6, 2002)
It is demonstrated that all stable (non-radioactive) isotopes are formally interrelated as the products of systematically adding alpha particles to four
elementary units. The region of stability against radioactive decay is shown to obey a general trend based on number theory and contains the
periodic law of the elements as a special case. This general law restricts the number of what may be considered as natural elements to 100 and is
based on a proton:neutron ratio that matches the golden ratio, characteristic of biological and crystal growth structures. Different forms of the
periodic table inferred at other proton:neutron ratios indicate that the electronic configuration of atoms is variable and may be a function of
environmental pressure. Cosmic consequences of this postulate are examined.
Introduction
It is generally accepted1 that the stability of nuclei is
dictated by the ratio between the numbers of protons and
neutrons in the atomic nucleus. The observed trend is
summarized by the well-known Segré chart1 which
however, has never been explained in any other than
qualitative detail. The genesis of chemical elements is
assumed to relate to the Segré line of stability and
different mechanisms to account for the synthesis of
elements within stars have been conjectured.1 Elements
with atomic number less than 26 are thought to be
produced by reactions grouped together as hydrogen,
helium, oxygen and silicon burning, typically
represented by processes such as:
1H+1H 2H+e++
3 12C+
2(16O) 28Si+4He
28Si+4He 32S+
In the final phase, nuclei are assumed to exist in
equilibrium with particles, but on the basis of
observed abundances and thermonuclear arguments,
equilibrium processes are considered incapable of
producing nuclei more massive than those around iron.
Heavier nuclei are stated1 to be generated predominantly
by the process of neutron capture and beta decay.
The Segré chart does not reflect the special status of
iron, nor does it exhibit any discontinuity in the vicinity
of iron, nor at any other point. A single function seems
to characterize the proton to neutron ratio for the entire
range of nuclear stability. An analysis of isotope
stability should logically be based on this relationship.
Isotope families
Cursory examination of an isotope table2 imme-
diately reveals sequences of isotopes that differ in
composition by a constant factor, corresponding to an
a-particle, between consecutive members. The
abundance of isotopes in sequences such as:
12C 16O 20Ne
24Mg 28Si 32S, etc.
is particularly striking.
All stable (i.e., non-radioactive) isotopes can be
grouped into four sequences or families (4n, 4n+1, 4n+2,
4n+3), of the same type, as shown in Fig. 1 for the
isotopes of mass number 4n that occur between 2
4He and
56
128Ba. Each diagonal arrow in the diagram represents
the addition or removal of an -particle. The series does
not terminate at Ba, but continues without interruption to
82
208Pb. Apart from 4
8Be which is unstable (radioactive),
the addition of successive particles produces a
sequence of stable isotopes until it reaches 22
44Ti which
decays by electron capture to 21
44Sc, that decays by
positron emission to 20
44Ca. Starting from 20
44Ca a reverse
arm is generated by the removal of particles, ending in
the decay series:
SPSi 32
16
32
15
32
14
to rejoin the main sequence. The forward arm continues
to 32
68Ge that decays by electron capture and positron
emission to 30
68Zn.
The reverse arm emanating from 30
68Zn proceeds via
three -emitters towards the stable 20
48Ca. The forward
arm breaks again at 40
88Zr and/or 46
100Pd. The remainder of
the sequence that ends at 82
208Pb develops naturally on the
basis of the same principles outlined above. Four more
unstable isotopes that decay by -emission occur in the
4n family: 60
144Nd, 62
148Sm, 64
152Gd and 78
192Pt.
The number of non-radioctive or stable isotopes
generated in the 4n series totals 81, identical to the
number of stable elements.
A more convenient mapping of the isotope distribution
is obtained by plotting proton:neutron ratio as a function
of atomic number, as in Fig. 2. This diagram shows
development of both even series, characterized
by mass numbers of 4n and 4n+2, respectively.
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
34
Fig. 1. Alpha-particle relationship between isotopes of mass number 4n
Fig. 2. The even-mass-number series of stable isotopes
The two different patterns used to represent each of
the two series serve to distinguish between forward and
reverse progressions as discussed. Like the 4n series, the
second series also consists of 81 stable isotopes. Open
circles identify those isotopes unstable to or + decay
and have been included as a guide to draw limiting
straight-line segments that define the region of stability.
The turning points of the limiting profile bear some
resemblance to the magic numbers of nuclear stability,3
but correlate more strongly with features of the periodic
table of the elements.
The two series of odd mass-number isotopes are
shown in Fig. 3. The sequences start at 4
9Be (2
5He) and
2
3He respectively and proceed by forward progression
only, to terminate at 83
209Bi and 82
207Pb. Breaks at unstable
isotopes result in single steps of radioactive decay,
rather than two consecutive processes as in the even
series. The effect of this difference is that atomic
numbers switch between odd and even at each break in
the sequence. A new feature is that, at one point in each
series the sequence proceeds via – decay, e.g.:
PrCeBa 141
59
141
58
137
56
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
35
Table 1. The isotope sequences
n4n–2 4n–1 4n4n+1 n4n–2 4n–1 4n4n+1
1H He He* (He) 27 Cd Pd Ag Cd Pd Ag
2 Li Li Be* Be 28 Cd Pd Cd Sn Cd In Cd
3 B B C C 29 Sn Cd Sn Sn Cd Sn
4 N N O O 30 Sn Sn Te Sn Sb
5 O F Ne Ne 31 Te Sn Sb Xe Te Sn Te
6 Ne Na Mg Mg 32 Xe Te I Xe Te Xe
7 Mg Al Si Si 33 Ba Xe Te Xe Ba Xe Cs
8 Si P S S 34 Ba Xe Ba Ce Ba Xe Ba#
9 S Cl Ar SCl 35 Ce Ba La# Ce Pr
10 Ar K Ca Ar K 36 Nd Ce Nd Nd* Nd
11 Ca Ca Ca Ca Sc 37 Nd Sm* Sm* Nd Sm*
12 Ti Ca Ti Ti Ca Ti 38 Sm Nd Eu Gd* Sm Eu
13 Cr Ti V Cr Cr 39 Gd Sm Gd Dy Gd Gd
14 Fe Cr Mn Fe Fe 40 Dy Gd Tb Dy Gd Dy
15 Ni Fe Co Ni Ni 41 Er Dy Dy Er Dy Ho
16 Ni Cu Zn Ni Cu 42 Er Er Yb Er Tm
17 Zn Zn Zn Ga 43 Yb Er Yb Yb Yb
18 Ge Zn Ga Ge Ge 44 Hf* Yb Lu Hf Yb Hf
19 Se Ge As Se Ge Se 45 Hf Hf W Hf Ta
20 Kr Se Br Kr Se Br 46 W W Os WRe
21 Kr Se Kr Sr Kr Rb 47 Os WOs Os Os
22 Sr Kr Sr Sr Y 48 Pt* Os Ir Pt* Os Ir
23 Zr Zr Mo Zr Nb 49 Pt Pt Hg Pt Au
24 Mo Zr Mo Ru Mo Zr Mo 50 Hg Pt Hg Hg Hg
25 Ru Mo Ru Ru Mo Ru 51 Hg Tl Pb Hg Tl
26 Pd Ru Rh Pd Ru Pd 52 Pb Pb Pb Bi
Fig. 3. The odd-mass-number series of stable isotopes
In most respects the odd and even series however,
develop on the basis of equivalent principles. The region
of stability for the odd series, as demarcated by straight
line segments, is smaller than that of the even series and
completely enclosed within the latter. An immediate
conclusion is that isotopes with odd mass number are
intrinsically less stable than their even-numbered
counterparts. Each of the odd series consists of 51 stable
isotopes. Despite its different size, the limiting profile of
the odd series reveals the same periodic structure
observed with the even series (see vertical dotted lines).
The build up of isotopes is shown in tabular form in
Table 1. Asterisks indicate alpha instability. 2
4He is, by
definition, included with the -emitters. The main
sequences of the four series are shown in bold type.
Sequence breaks are marked by underlines. The symbol
# indicates that the next break occurs by –-decay. To
achieve uniformity the series are relabeled as 4n-2, 4n-1,
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
36
4n and 4n+1, such that each main sequence consists of
consecutive addition of 51 alpha-particles to the n = 1
starting particles.
Periodic trends
The stability profiles of the even and odd series of
stable isotopes reveal periodic characteristics, highly
reminiscent of the periodic table of the elements. It is
therefore interesting to note that an 8-column
arrangement of the elements, grouped into sets of two
and six elements per period, has exactly the same
periodic structure as the isotope boundaries in Figs 2 and
3. The periodic points are identified by sets of stipled
vertical lines. Later arguments will make it clear why
emphasis is placed on these atomic numbers. The table
is shown in Fig. 4. Based on the demonstrated
correspondence the hypothesis is made that isotopic
periodicities and elemental periodicities arise from a
common underlying principle or model.
This compact form of the periodic table has, to my
knowledge, not been proposed before. However, it may
be related to the prime-number cross, postulated by
PLICHTA4 as the fundamental basis of atomic structure.
Fig. 4. Compact form of the periodic table of the elements
Fig. 5. Number spiral to show the prime-number cross at 6n±1
This construct, with the natural numbers in a spiral
array, as shown in Fig. 5, may serve as a simple model
that, with appropriate identification of parameters, could
explain the observed periodicities and account for the
observed number of elements and isotopes. It requires a
periodicity of 24 to project all prime numbers, excluding
2 and 3, and their products on eight arms at 6n±1, that
form the cross.
All the numbers in periods 024, 2448, etc. add
up to multiples of 300, i.e.:
+
=
=×+=
)1(24
24
,2,1,0,300)12(
j
jn
jjjn K(1)
The accumulated sums over the first n periods, with
= 300, are:
=
=+++==
n
i
i
nanan
1
2
)]12(31[ K(2)
Each of the coefficients is a square number5 that
relates to the preceding square number by:
sn = sn–1+2n–1, (s0 = 0)
It is well known that the distribution of electron pairs
within atomic energy shells obeys the same rule, i.e.:
K shell: 1 pair = 2 electrons (1s)
L shell: (1+3) pairs = 8 electrons (2s2p)
M shell: (1+3+5) pairs = 18 electrons (3s3p3d)
N shell: (1+3+5+7) pairs = 32 electrons (4s4p4d4f)
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
37
Fig. 6. The 264 isotopes that are stable against radioactive decay arranged in an 11×24 matrix
Fig. 7. Triangle of stability with isotopes close to the periodic limits
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
38
It appears that a formal analogy exists between
atomic structure and the number spiral, as if each cycle
along the spiral covers 8 atoms (on the cross) and 24
isotopes. This analogy implies an overall periodicity of
24 among the stable isotopes, with an embedded
elemental period of 8. Remarkably, the number of stable
isotopes is 2×81+2×51 = 264 = 11×24. An 11×24 matrix
of stable isotopes is shown in Fig. 6, arranged in order of
increasing mass number. Within a group of constant
mass number the final order is decided by atomic
number. Only those isotopes generated by alpha-particle
addition are shown. This definition excludes entities
such as 1H, n and 4He that may, more appropriately be
considered as building blocks. If these are included,
together with the nine -unstable isotopes the total
number of possible isotopes, stable against beta-type
decay, becomes 11×25 = 275.
If the coefficients of a in the summands i of Eqs (1)
and (2) are consistently interpreted as sums over
individual electronic sub-shells, the constancy of a
implies that the same number of atoms or isotopes must
feature in each summation. One interpretation could be
that the number of natural isotopes corresponds to a,
which restricts the number of natural elements with
stable isotopes to a/3 = 100. However, there are two
elements, 43 and 61, without any stable isotopes. Two
additional elements with atomic numbers beyond 100
are therefore allowed, giving a maximum natural atomic
number of 102. The distinction between natural and
stable isotopes should become clear later on.
There is no immediately apparent periodicity in the
table of Fig. 6. The first period ends with the first of the
three isotopes of Si, the second ends with the third of the
five isotopes of Ti and the third ends with the fourth of
the five isotopes of Zn. The isotope at the end of each
period is invariably an apparently arbitrary one within
the range for that element. A more precise definition of
each period is obtained by using the coordinate axes of
Fig. 2. Instead of linear arrays, the sets of 24 isotopes in
each period now occur in well-defined blocks, shown in
Fig. 7. The positions of straight lines that separate the
isotopes belonging to contiguous blocks, are fixed
within very narrow limits. The procedure is illustrated
by drawing a set of slanted lines to separate the
hypothetical numbered sequence of Fig. 8 into groups of
8. In Fig. 7, the positions of isotopes not close to the
dividing lines are not shown; only those positions that
have an effect on the slope of the lines are shown. It is
important to note that the lines are not arbitrary, but
fixed by the assumed periodicity of 24. Since these end-
of-period lines are in general not parallel with the axis
that defines the proton:neutron ratio, a common base
line on which to compare these lines needs to be
established.
Fig. 8. Numbered array to show how groups of 8 consecutive points
are separated by slanted straight lines
It was noted that the straight line AC provides an
acceptable approximation to the stability profile of the
even mass-number isotopes. It extends to element 102 at
a proton:neutron ratio, conspicuously close to 0.618. The
famous Fibonacci series5 converges to this value, known
as the golden ratio. The lines that coincide with the
completion of the 11 periods of 24 extrapolate to the
familiar atomic numbers of the compact periodic table,
at about this ratio, represented in Fig. 7 by the line CD.
This important observation confirms that the periodic
table of the elements is a subset of the isotopic table.
Extrapolation of the limiting lines in the other
direction to intersect AB, defines a new form of
elemental periodicity that would apply under conditions
of a 1 : 1 proton : neutron ratio.
The triangle of stability
The connecting line between p/n = 1 at Z = 0 and
p/n = 0.618 at Z = 102 (AC) defines, to fair
approximation, the discontinuous stability curve
empirically inferred before (Fig. 2). If points on this line
represent the maximum p/n ratio allowed for stable even
mass-number isotopes, the minimum number of
neutrons that stabilize an isotope of given atomic
number p, is given by:
=
tan1
min p
p
n(3)
where
00384.0
102
618.01
tan =
=
An average line that represents minimum p/n ratios
for stable isotopes with p protons extends from
p/n = 0.74 at Z = 0 to p/n = 0.618 at Z = 102. The maximum
number of neutrons allowed for a stable isotope with p
protons follows as:
=
tan74.0
max p
p
n(4)
where
0012.0
102
618.074.0
tan =
=
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
39
Fig. 9. Modified Segré chart with lines defining 1 : 1 and golden ratios
The two simple formulae, Eqs (3) and (4) may be
used to calculate the range of stability for isotopes of
each element in the triangle of stability defined by
ACD’.
Contained within this triangle is another that defines
the region of stability for isotopes with odd mass
number. The intercepts of the lines that define nmin and
nmax for these isotopes, on the p/n axis, are 0.92 and
0.84, respectively, derived graphically from Fig. 3. To
calculate neutron numbers for the odd series of isotopes
these numbers are used in Eqs (3) and (4).
Since nuclear particle numbers must be discrete the
various lines of stability cannot be continuous. For each
atomic number the rounded minimum and maximum
number of associated neutrons are obtained by
discarding fractions. The combined stability profiles for
all isotopes, obtained by this procedure are shown in
Fig. 9. This plot is a modified Segré chart. All isotopes
of elements with odd atomic number are shown as filled
circles (isotopes with odd mass number but even atomic
number are not shown). The open circles identify only
the lowest and highest mass number isotope for each
element of even atomic number. It is important to note
that all stable isotopes are correctly predicted to lie
between the saw-tooth profiles, and hence that Eqs (3)
and (4) correctly predict the stable isotopes for any
atomic number. The two limiting lines represent p/n
ratios of 1.0 and 0.618, respectively.
The nuclear stability region starts out at the ratio
p/n = 1 and gradually bends back to p/n = 0.618, at
Z = 102. The 11×24 isotope periods are shown again, as
before with the intercepts on both the golden and 1 : 1
lines identified.
Elemental synthesis
The numerical regularities observed in this analysis
of isotopic periodicity provide fertile ground for
speculation. It is probably appropriate however, to leave
most of that to the imagination of the reader.
Nevertheless, a few pointers pertaining to the genesis of
the chemical elements are thought to be in order. Figure
7 is the most informative as a guide to isotope stability
against ±-decay.
The relationship between consecutive isotopes in
each of the four series firmly identifies particles as the
common building block that produces complex atoms. It
is argued that such build-up occurs within massive stars
under conditions of enormous pressure. Without any
attempt to identify the type of star or to anticipate the
exact conditions, it is simply assumed that the
appropriate environment allows free equilibrium
between particles and all possible isotopes that can be
obtained by adding particles to the fundamental
particles, protons and neutrons, e.g.:
p+n 2H 6Li (4n+2)
2H+ 2H 8Be 12C (4n)
n+ 5He 9Be (4n+1)
n + 2H 3H 7Li (4n–1)
It is proposed that conditions in the star be identified
with the line AB of Fig. 7, with all isotopes stable
against -decay. With increasing mass number the p/n
ratio of all isotopes approaches 1, characteristic of the
-particle. The observed periodicity under these
conditions, as derived from Fig. 7, corresponds to the
occupation of electronic energy levels in the sequence:
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
40
4f 3d 2p 1s 3p 2s 5f 4p 4d 5p 3s 6p 5d 7p 4s
for elements 1 to 102. The number of isotopic periods is
inferred to increase from 11 to 12, such that the total
number of isotopes now amounts to 12×24+12 = 300, as
predicted by the number spiral.
To arrive at the familiar condition of moderate
pressure, for instance on planets, the isotope equilibrium
has to change dramatically. The required phase
transition may be brought about by an event such as a
supernova. The effect, as many isotopes are rendered +
unstable, is that AB changes slope, moving to AC. Other
isotopes become unstable against -decay, represented
here by BC that slopes to BC', cutting off the triangle
A'CC'. The perpendicular C'D' on BC' now defines the
line of – instability.
The position of C at Z = 102 is determined by local
conditions of pressure. This pressure has an effect, not
only on isotope stability, but on the nature of all matter
and on all growth mechanisms such as crystallization
and biological growth. These ideas may cast some light
on the mysterious mathematics of biological spirals6 and
their relationship to the Fibonacci series7 and on scale-
rotational symmetries observed in snow crystals and
nucleic acids with helical structure.8
Generalized periodic tables
The straight lines that separate consecutive periods
of the isotope matrix have been redrawn as the thick
oblique lines in Fig. 10. Since these lines are not
perpendicular to the atomic-number axis any elemental
periodicity embedded in the isotope table will have a
different form depending on the proton : neutron ratio at
which the sampling is done. Extrapolation of the line
that divides the first two periods, to ratios of 1.0 and 0.6,
for instance, suggests that the first period ends at either
14Si or 10Ne. Such a conclusion can only imply that,
provided a periodic law applies, the elements of the first
period have totally different electronic configurations in
situations that correspond to these different respective
ratios. A logical further assumption is that different p/n
ratios in the context of Fig. 10 imply different
thermodynamic conditions. This crucial assumption, that
environmental conditions determine the allowed p/n
ratios of stable isotopes, leads to the novel conclusion
that the electronic configuration of an atom must,
likewise be a function of external factors.
It was noted that the observed periodicity of the
elements, which is adequately approximated by the
occupation of energy levels in the order of:
1s 2s 2p 3s 3p 4s 3d(4s) 4p 5s 4d(5s)
5p 6s 4f 5d(6s) 6p 7s 5f,
was reproduced well at a p/n ratio of 0.618.
Fig. 10. Diagram to illustrate the relationship between isotopic periodicity and four limiting cases of the elemental periodic table
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
41
Fig. 11. Periodic tables based on (a) the Schrödinger model, (b) 1 : 1 p/n ratio, and (c) extreme pressure
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
42
It is generally known that this observed order of sub-
level occupation does not exactly follow the sequence
predicted by solution of Schrödinger’s equation for the
hydrogen atom. It is, therefore, of interest to note that
the predicted periodicity at about p/n 0.585 is
compatible with the Schrödinger sequence of:
1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p
These hypothetical and actual energy-level schemes
result in periodic tables that differ in minor, but non-
trivial detail, as shown in Figs 4 and 10.
The conjecture that the electronic configuration of an
atom could vary as a function of thermodynamic
conditions finds theoretical support in the known
response of atomic energy levels to compression. The
compression of a hydrogen atom has been studied
theoretically by many authors. The major effect of
compression is a general elevation of all energy levels.9
Furthermore, the accidental degeneracy of levels with a
common principal quantum number was found lifted by
compression of the free hydrogen atom.10,11 The
calculated shifts decrease in the order d<p<s, predicting
sub-level occupation of compressed atoms in the order
nf nd np ns. The compression and subsequent ionization
of more-electron atoms has also been studied.12
Although orbital energy spectra were not analyzed
directly, some evidence of inter-level crossing was
recorded.
The evidence is considered sufficient for assuming
that the inverted ground-state configurations:
4f1143d1102p161s12
inferred from the extrapolated atomic numbers of 14, 24
and 32 for the first three isotopic periods at a ratio of
p/n = 1 (Fig. 9), could be the result of increased pressure.
These are the conditions under which the synthesis of
elements by alpha-particle addition is envisaged to
occur.
Theory, therefore, predicts that the ground-state
electronic configuration of atoms is a function of the
environment. Since atomic number is not affected, the
periodic arrangement, defined to reflect electronic
configuration, is predicted to change, unless a common
structure that fits all arrangements can be identified. It
turns out that the 8-group compact form of the periodic
table of the elements represents such a structure.
This compact form of the periodic table is based on
the distribution of prime numbers on a number spiral
with periodicity of 24, interpreted to reflect the
periodicity among the stable isotopes. To fit sub-sets of
2, 6, 10 and 14 into the 8 allowed sites per cycle, it is
necessary to make combinations of the type 2+6, 2+2+6
and 2+6+6, which can be arranged in such a way that the
closure of all sub-levels coincides with either groups 2
or 8. It is recommended that this generalized form of the
periodic table may be used to represent the electronic
configuration of the elements under all situations in the
cosmos. Four special cases are shown in Figs 4 and 11.
Cosmic states of matter
The familiar forms of matter, observed in the local
region of space-time known as the solar system, are
made up of atoms with electronic configurations
corresponding to Fig. 4. In other domains different
forms of matter, characteristic of different local
conditions, occur. The periodic table that represents
local matter is expressed at the ratio p/n 0.618.
An inverted form of the periodic table is obtained by
extrapolating the period boundaries to p/n = 1. Assuming
the points of intersection to define closed-shell
configurations, as before, the periodic law shown in Fig.
10b is inferred. As before, the closed-shell
configurations coincide with periodic groups 2 and 8.
The first f-level (4f) is lowest in energy, followed by the
first d, p and s levels. Although this pattern is not
continued through higher levels, the observed sequence
is plausible.
If the extrapolation is continued to a ratio of
p/n 1.03 the fully inverted arrangement is observed,
i.e.:
4f 3d 2p 1s 5f 4d 3p 2s 6f 5d 4p 3s.
The four periodic arrangements are shown in Figs 4
and 11.
An obvious consequence of variable electronic
configurations concerns the spectroscopic analysis of
galactic light, and especially the Doppler interpretation
of red shifts. An immediate solution to the paradox of
high red-shift quasi stellar objects (QSO) physically
associated with low red-shift galaxies13 is indicated. It is
only necessary to interpret the postulated13 intrinsic red
shifts in terms of configurational changes in the field of
the QSO. The apparent frequency shift is the effect of
different energy levels being involved in the transition.
The detection of large amounts of matter with very
different red shifts all lying near the QSO could simply
be a function of the non-uniform gravitational field in
this region.
J. C. A. BOEYENS: PERIODICITY OF THE STABLE ISOTOPES
43
References
1. P. A. COX, The Elements: Their Origin, Abundance and
Distribution, University Press, Oxford, 1989.
2. R. L. HEATH, in: CRC Handbook of Chemistry and Physics,
R. C. WEAST (Ed.), 60th ed., or any later edition, 1981.
3. W. N. COTTINGHAM, D. A. GREENWOOD, An Introduction to
Nuclear Physics, University Press, Cambridge, 1986.
4. P. PLICHTA, God’s Secret Formula, Element Books, Boston, 1998,
Translated from German.
5. J. E. MAXFIELD, M. W. MAXFIELD, Discovering Number Theory,
Saunders, Philadelphia, 1972.
6. D. BERGAMINI, Mathematics, Time-life, USA, 1970.
7. R. DIXON, Fibronacci phyllotaxis: mathematically speaking, in:
Symmetry 2000, I. HARGITTAI and T. C. LAURENT (Eds), Portland
Press, London, 2002.
8. A. JANNER, Crystal Eng., 4 (2001) 119.
9. A. SOMMERFELD, H. WELKER, Ann. Phys., 32 (1938) 56.
10. S. R. DE GROOT, C. A. TEN SELDAM, Physica, 12 (1946) 669.
11. P. PLATH, Spin-Bahn Wechselwirkung und oktaedrische Ligand-
feldaufspaltung eines räumlich beschränkten Einelektronen-
systems, Dissertation, Technische Universität, Berlin, 1972.
12. J. C. A. BOEYENS, J. Chem. Soc., Faraday Trans., 90 (1994) 3377.
13. H. C. ARP, G. BURBIDGE, F. HOYLE, J. V. NARLIKAR,
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