European Journal of Scientific Research
ISSN 1450-216X / 1450-202X Vol. 152 No 4 April, 2019, pp. 418-445
http://www. europeanjournalofscientificresearch.com
The Quantized Atomic Masses of the Elements:
Part-6; Z=100-107 (Fm-Bh)
Bahjat R. J. Muhyedeen
College of Science, Baghdad University, Iraq
E-mail: [email protected]
Abstract
This paper is the sixth part of a series of eleven of QAM-UQAM-NMT-2017-Ver-3.
An innovative semi-empirical atomic mass formula has been derived to calculate quantized
atomic masses more precisely than macro-micro formula and purely microscopic HF-self-
consistent methods. It is based on the novel mass quantization and the variable neutron
mass concepts of new nuclear theory, the Nuclear Magneton Theory of Mass Quantization,
NMT. It can calculate the atomic masses of non-existent isotopes based on the existing
experimentally measured nuclides. The discrepancy (RMS) of the mass model is less than
335 keV for UQAM and 884 keV for QAM for the entire region of ground-state masses of
3160 nuclei ranging from 1H to 118Og. The quantized atomic masses of 16000 nuclei
ranging from Z=1 to Z=226 have been calculated, 800 nuclides of them belong to Z=100-
107 (Fm-Bh). The results are compared with those of other recent macroscopic–
microscopic. Sn, Sp, β-, β+ and α decay energies are also given.
Keyword: nuclear mass formula, neutron mass, atomic masses, new isotopes, super-
heavy nuclei, alpha decay.
1. Introduction A. Background
The methodology’s history of the evaluation of the atomic masses during the last seven decades has
been discussed in details in the previuos five articles [1-5] of this series of eleven papers. Several
various microscopic, macro-micro, shell and phenomenological theoretical models have been
published [6-52]. All these theoretical methods calculate the binding energy first, and then evaluate the
neutral atomic mass. The binding energy are calculated from two parts which may be written as:
B.E. = Emac (refined version of liquid drop) + Emic (shell+paring+Wigner terms) (1)
And the theoretical neutral atomic masses are derived from the following formula:
MA = ZMH+NMN-B.E. (2)
But none of these models can be used with total confidence due several deficiencies.
Consequently, the accurate estimation of the atomic masses of the existent and non-existent isotopes in
astrophysics is considered as unsolved problem. The output of most theoretical calculations cannot
predict the atomic masses precisely which lead to improper alpha energies and half-lives. The atomic
masses calculated by Wang-Audi-Wapstra, WAW et al. [25,26], Moller et al. [31,32] and Duflo-
Zucker [40] failed to give the positive incremental difference in alpha energies between two sequential
isotopes as we will see later. The mass evaluations usually use short-range connection between close
lying neighbors isotopes. The researchers are looking for an extended complicate connectivity between
multiple isotopes.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 419
An immense project was setup in 2010 to solve this problem [1-5]. In present project a new
semi-empirical polynomial 2nd
strategy is setup to adopt the measured atomic masses, inserting them
into special quadratic equations and then extrapolate them to predict the accurate atomic masses for the
non-existent element’ isotopes. In this manner, the prodecure is avoiding any theoretical treatment
which lead to inaccuracy in the prediction processes and also use simple long-range relations among
the element’s multiple isotopes. The new ansatz depends on the pure nuclear property i.e. atomic
masses of the nuclei and avoids the external corrections used in the main approaches in the theoretical
description of nuclei.
This article is part 6 of the immense project. The aim of the project is to find out 1- a quantized
mass formula QMF to the elements that have enough isotopes to setup a polynomial equation 2nd
based
on the available isotope masses (i.e. Z=1-107), it will be called isotopic quantized mass formula IQMF,
2- two quantized mass formula QMF to the elements that do not have enough isotopes (i.e. Z=108-118)
to setup a polynomial equation that will create atomic masses for non-existent isotopes, they will be
called analytical AQMF, numerical quantized mass formula NQMF and total variable neutron mass
TVNM, to generate a database for alpha energies for more than 1140 isotopes for Z=100-118, and 3- a
quantized mass formula QMF to the non-existent elements that will create atomic masses for more than
9800 new non-existent isotopes, from the alpha energies database, it will be called energetic quantized
mass formula EQMF (i.e. Period-8, Z=119-172 (54 elements) and period-9, Z=173-226 (54 elements)).
The whole project is accomplished successfully in 2017. The polynomial equation of the 2nd
power
generates the quantized atomic masses QAM while the polynomial equation of 3rd
up to 6th
power
generates the unquantized atomic masses UQAM with RMS=335 keV. A novel Total Variable Neutron
Mass TVNM method also used for calculation of QAM and UQAM. Both QAM and UQAM of the
elements Z=1-118 have been calculated. The QAM of the isotopes of the Period-8 Z=119-172 and the
Period-9 Z=173-226 have been calculated.
B. Theory
Although the standard model theory SMT essential concepts of the mass formula are based on the
theory that both masses of protons and neutrons are invariant inside and outside of the nuclei and the
energy has an equivalent mass based on mass-energy equivalence E=mc2. But, it shows a vague idea
about this invariance when it talks about the binding energy. SMT thinks the binding energy of the
nucleus is given by the mass formula; B.E = ZMH+NMN-MA. This concept referes hiddenly that some
of the mass of p and n converted into binding energy which breaks the invariance masses of p and n.
The conversion of mass into energy is still a matter of dispute among the scientists but Einstein
confirmed this concept in 1932 after the Cockcroft and Walton experiments. Einstein was very
persuaded in the conversion of the mass to energy and vice versa where Einstein said:
“It followed from the special theory of relativity that mass and energy are both but different
manifestations of the same thing—a somewhat unfamiliar conception for the average mind.
Furthermore, the equation E is equal to mc2, in which energy is put equal to mass, multiplied with the
square of the velocity of light, showed that very small amounts of mass may be converted into a very
large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the
formula mentioned before. This was demonstrated by Cockcroft and Walton in 1932, experimentally”.
Let me explain what happened in this funny experiment. The story is that in 1932, the English physicist
John Cockcroft and the Irish physicist Ernest Walton produced a nuclear disintegration by bombarding
lithium with artificially accelerated protons. The following reaction took place: Li�� + H�
� → Be [T1/2,
8.19x10-17
s] → 2 He + energy [QT=17.254 MeV]. The product of the reaction created Be-8 nuclide
which has a very short time and fissioned into two He-4 nuclides. When the helium gas atoms escaped,
Cockcroft and Walton wrongly thought that the mass has converted into energy of 17 MeV. They said,
“The evolution of energy on this view is about sixteen million electron volts per disintegration,
agreeing approximately with that to be expected from the decrease of atomic mass involved in such a
disintegration”, Cavendish Laboratory, Cambridge, April 16. Cockcroft, J. D. & Walton, E. T. S.
420 Bahjat R. J. Muhyedeen
Nature 129, 242 (1932). This event misled Einstein to think that the mass can be converted to energy.
At that time the neutron was still not discovered and the element transmutation was not well proved.
NMT has a different thought about the source of this energy as we will see in the forthcoming text.
In my previous articles [53-55], I explained that; in quantum mechanics, regarding the E=mc2 (
or E=mc1c2, or E=pc2), the correct visualization and understanding to this equation supposed be
realized as a simple mathematical function to describe the kinetic momentum of the mass in unit of
energy and it may be written as a modest multiplication of momentum p times the speed c2 where c2 is
a unit conversion factor but not c2 as Einstein perceived. This means that the mass-energy equality
supposed be interpreted as p times c and not m times c2.
NMT [55] investigated intensively the actual meaning of the atomic mass formula; B.E =
ZMH+NMN-MA, after reviewing all the principles and facts of physics and SMT theory since 1900. It
concluded that the mass formula is an abstract, and simply a hypothetical mass equality states the
mass difference ∆m (or mass defect) between atomic mass MA and the nucleons; ZMP+NMN can be
expressed in unit of energy.
The hypothetical mass equality does not simulate the mechanism of the nucleosynthesis
process in the stars. The nucleosynthesis process in the stars is based on p-p chain, CNO and s- and p-
processes and not based on this putative equality. The concept is inherited from the wrong
interpretation of E=mc2. Nuclear scientists cannot create nuclides in their laboratories from adding
ZMP+NMN to get B.E and MA. NMT scrutinized deeply the B.E term in this virtual equation and found
out it refers to the mathematical ∆m energy and it does not reflect the actual concept of binding
energy. There is neither physical condition, nor guarantee to confine the ∆m energy (mass defect
energy) to stay in the nuclei to act as binding energy. It is supposed that the binding energy BE
explains the easy formation of; nucleosynthesis or nucleogenesis, and the degree of the stability of the
nuclide, but it seems not (see item 2.4C).
NMT believes the fermions are constructed from two building blocks; the magneton and its
anti-magneton, while the SMT believes that the fermions are constructed from quarks. The author has
explained previously that the source of the binding energy does not come from mass defect but from
the spinning magnetic dipoles of the two building blocks as explained below. NMT considers the B.E
as a misleading concepts in nuclear science. The most accepted Standard Model-Extended Theory
(SMET) is the quark-model, which was independently proposed by physicists Murray Gell-Mann and
George Zweig in 1964. However the quark theory has short comings in that it cannot explain the
nuclear transformation of the proton, and neutron in a clear way. The quark theory tells us that the
neutron is created from three quarks (two down and one up) and the proton is created from three quarks
(two up and one down). The quark theory proposes that a decrease in neutron and proton mass is
explained by the binding energy for the host nucleus. If someone were to ask “which part of the three
quarks of the neutron and the proton converts into binding energy when the neutron and the proton
form the deuterium nucleus?” the answer is unclear. Another problem occurs, when we assume that the
proton (938.272 MeV/c2) is composed of two up quarks (each quarks is 2.4 MeV/c
2) and one down
quark (4.8 MeV/c2). They consists of 1.0232 % (9.6 MeV/c
2 of 938.272 MeV/c
2), the remainder of the
proton mass, due to the kinetic energy of the quarks inside the nucleus, and the energy of the gluon
fields that bind them together. If we acquiesce that these quarks has kinetic energy of 928.672 MeV/c2,
then a reasonable question would arise: what is the source of this everlasting energy? Suppose we
consent that the energy has mass and the total mass of proton is 9.6 MeV/c2 due to quarks and 928.672
MeV/c2 due to kinetic energy (total is 938.272 MeV/c
2), and we then apply this concept to the binding
energy calculation. Does this mean that the binding energy also has mass (as SMT theory says)? Is that
correct? If it is correct, it implies that the real or actual mass of each nuclide will come from (ZMH +
NMn) but this has not been proven experimentally.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 421
C. Solutions from NMT
I. Fermion New Structure
In 2008, 2009, and 2013; the author proposed a new nuclear theory which is founded on the concept of
quantized elementary discrete mass particles, called herein the Magneton (m-1
) and Antimagneton
(m+1
). The particles are conceived to be spinning magnetic dipoles with sufficient mass to produce the
dipole-dipole interaction sufficient to act at ultra-short range – the source of the Nuclear Force Field
(NFF) - which now has a gravitational component. Since the NFF contains this component it can be
thought of as the long search for Unified Field (Electro-gravito-magnetic Field). The theory is termed
the Nuclear Magneton Theory of Mass Quantization-Unified Field, or NMT. It includes many novel
nuclear concepts such as the mass quantization principle, MQP, the mass-energy conformity MEC
rather than mass-energy equivalence, the universal particle speed constant b
(=0.6037970064x108m/s=0.20141c). The NMT concepts [53-55] considers all other heavier neutrinos
as a multiple package of electron neutrino (i.e. magnetons, m-1
and m+1
). The reasons that NMT
changed the “electron-neutrinos” name to “magnetons” name is that the magnetons are building blocks
that form all the fermionic particles, and their anti-particles (quarks and leptons) i.e. proton, neutrons,
electrons, muons, tauons etc.
In the big bang and then in the stars during the nucleosynthesis, these two building block
particles m-1
and m+1
are thought to be compacted in matrix to build a two types of a closed quantized
circular packages rings of neutrinos (CQ-CPRN) of quantized mass. The dipole-dipole interaction of
the high gravity spinning and rotating magnetic dipoles results in generating a magnetic field which
generates an electric field i.e. Electro-Gravito-Magnetic field (EGM) in the CQ-CPRN. The generated
EGM field is creating the partial negative charge from the m-1
in small CQ-CPRN (m-1
-CQ-CPRN) and
partial positive charges from m+1
in the large CQ-CPRN (m+1
-CQ-CPRN). Then these CQ-CPRN’s
merge together further vertically in parallel mode into multi-layer peripheral circles to form spherical
shells. The structure of the fermion particles is composing under drastic circumstances in stars from
these spherical shells (nmtionic shells) that finally give the shape of the fermion. The superposition of
three spherical shells of small partially negative m-1
-CQ-CPRN causes the formation of the electron of
invariant quantized mass with unitary elementary negative charge. Therefore, the electron has not a
dipole structure and its magnetic field is an intrinsic property and not due to the motion of electron.
The superposition of three spherical shells of the large partially positive m+1
-CQ-CPRN causes the
formation of the proton of invariant quantized mass with unitary elementary positive charge. While the
superposition of 1st core spherical shell of small partially negative m
-1-CQ-CPRN followed by two 2
nd
and 3rd
large partially positive m+1
-CQ-CPRN and 4th
outer small partially negative m-1
-CQ-CPRN
causes the formation of the free neutron of invariant unquantized and unstable mass with nearly zero
charge. When the proton crushed into bound neutron with variant mass they will bound each other
inside to create the nuclei. The bound neutron has the same structure of the free neutron but with lesser
mass. Because the free neutron has unquantized mass; it will disintegrate mostly into muon and tau
after a mean lifetime of 881.5s, while the bound neutron will undergo a beta decay in unstable nuclei.
The bound neutrons are stable in stable nuclei and unstable in unstable nuclei. The structure of all
particles follow this superposition process. When the two building block m-1
and m+1
change their
spinning and rotating motion in the nmtionic shells of the fermion they will create the unstable closed
unquantized circular packages rings of neutrinos CU-CPRN of unquantized mass. The structure of the
fermion particles grows hierarchically from these high gravity CQ-CPRN and their antiparticles grows
from CU-CPRN. There is a big difference between NMT and SMT regarding the building blocks.
SMT believes anti quarks build anti-particles, and the charge depends on the type of quark while NMT
believes that the change of spinning and rotating motion of m-1
and m+1
build the anti-particles and the
charge comes from the type of the magneton m-1
and m+1
. NMT believes that no tiny mass from any
nucleon could be converted to any energy or to create the binding energy or vice versa, but the mass
create energy due to the spinning-rotations of the magnetons. It confirmed that the reaction of particle
and its anti-particle lead to disintegration to the two basic building blocks rather than annihilation.
422 Bahjat R. J. Muhyedeen
Diagram 4: The Nmtionic Shells of the Electron, Proton and Neutron
II. Standard Energy’s Formation of Nuclide in the Stars
The proton-proton chains consist of thermonuclear reactions by which hydrogen is transformed into
helium in stars. The net energy emitted from these thermonuclear reactions is given by:
4p+2e → He + 2e
+ + 2νe + 24.688 MeV (3)
The two positrons eventually interact with two free electrons and neutrinos in the stellar plasma
leading to their annihilation. An amount of energy of 4me c2 = 2.044 MeV is released in the form
of gamma rays and must be added to the energy gain of this thermonuclear reaction giving a total
energy of 26.732 MeV minus the energy carried away by the neutrinos.
NMT called the energy released or consumed from nucleogenesis process of the nucleus as the
standard energy of formation of nuclide, SEFN ∆E�� (similar to standard enthalpy of formation ∆H�
� in
chemistry) [1-5,55]. The SEFN concept does not focus on the number of the protons and electrons of
the created nuclide. SEFN indicates the amount of the energy released in the stars from the
nucleosynthesis. SEFN does not refer to the stability of the nuclide as binding energy does. The higher
the SEFN value of the nuclide the more exothermic in the star and the hotter the area is. The general
formula for calculation of SEFN ∆E�� is given by the following equation;
∆E��= (ZMH + 1.00672787241186N – MA) x 931.5 MeV (4)
Where N=neutron number, Z= atomic number, MA the atomic mass of the nuclide and
MH=1.0078250322323 u. In the next article I will explain in details how we can use SEFN values in
calculating the atomic masses of the unknown isotopes of known element Z=1-118.
The relation between MA and the SEFN can be written as follow;
MA= ZMP+NMN-SEFN-NF (5)
The NF term is the amount of energy used inside the nuclei which represents one type of the
nuclear forces. The NF is a function of neutron numbers N, NF=0.001937043438N u, as we will see in
the forthcoming item 2.4C. The SEFN can be expressed mathematically in terms of Z, N and A. For
example, the intital mathematical function of SEFN for the Fm, Z=100 is given by EXP(Z*N/Ax) x
EXP(-Z/Ny), where x=1.734 and y=0.76 for Z=100. It gives a good predicted atomic masses with
RMS=1.876 MeV.
III. Application of the MQP and MEC Concept on Single and Total Variable Neutron Masses
The details of this procedure is explained underneath.
2. Method The general procedures and policies of calculating the quantized atomic masses QAM are fully
explicated here below:
NMT called the mass of the single bound neutron in the nuclide as single variable neutron
masses SVNM; M�∗ = (MA-ZMH)/N and the whole bound neutrons in the nuclide as the total variable
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 423
neutron masses TVNM; NM�∗ = (MA-ZMH). In this article, the MQP and the MEC concepts are
applied on the measured atomic masses MA (from IAEA) to amend and extrapolate them. The
application of MQP and the MEC concepts on the single / total variable neutron mass M�∗ / NM�
∗ inside
the nuclei helps in calculating the quantized atomic masses MA of existent and non-existent isotopes
which gives excellent results far from using B.E concept. The variable neutron mass is denoted by M�∗
to differentiate it from the mass of the free neutron MN. The adoption of single / total variable neutron
mass M�∗ / NM�
∗ inside the nuclei will not show mass defect because the change in the neutron mass
will equate the MA. NMT shows that the atomic masses MA of the neutral nuclei can be calculated
from summation of the total mass of proton and electrons (MH) and total mass of the corrected variable
neutron masses M�∗ (i.e. MA=ZMH+NM�
∗ ) after manipulation to give positive incremental difference in
alpha energies between two sequential even-A & even-A and odd-A & odd-A mass number A; ∆α
(ee,oo) and two sequential even-A & odd-A and odd-A & even-A mass number A; ∆α (eo,oe). NMT
believes that the nuclear forces is represented by the Electro-Gravito-Magnetic field (EGM) i.e.
strong nuclear forces, weak nuclear forces, electromagnetic forces and gravitational forces and can not
be described by the binding energy term. The atomic mass of the hydrogen MH will include the ZMe
and ZMP.
2.1. Application of Mass Quantization Principle MQP on the Neutron
NMT thinks that; contrary to the mass of the electron QMe and the proton QMp which they have a fixed
quantized mass inside and outside the atom, the SVNM has several stable variable masses M�∗ in the
stable nuclei, which are called quantized QNM and unstable variable masses M�∗ in the unstable poor
and rich nuclei, which are called unquantized UQNM/UQNM respectively. NMT believes that there
are no absolute stable nuclides at all, and they possibly live with t1/2 ca 1020
-1040
y due to the unstable
structure of the bound neutron resulted from the unstable charge over mass [1-5, 55]. In previous
articles [1-5], many concepts and definitions were explained such as Neutron Mass Plateau, Neutron
Mass Quadratic Equation, NMQE, and Isotopic Quantized Mass Formula IQMF. NMT sorts out the
SVNM M�∗ values and nominates the M�
∗ values based on their R2 values formed by the NMQE as
dissonant neutron mass DNM (i.e. DNM, DNM and DNM) if they give M�∗ -ln(A) graph with poor R
2
values ((R2<0.9999) and consonant neutron mass CNM (i.e. CNM, CNM and CNM) if they give M�
∗ -
ln(A) graph with high R2 values (R
2≥0.9999). NMT called the consonant CNM hermetic values as
Generalized Neutron Masses values and denoted by GNM (i.e. GSVNM and GTVNM, while it called
the consonant CNM non-hermetic values as Un-Generalized Neutron Masses values and denoted by
UGNM (i.e. UGSVNM and UGTVNM). The concept of GNM and UGMN are also applied on non-
alpha emitters of light nuclides. The values of UQNM, QNM and UQNM of the available isotopes
from IAEA website (more than 3300 nuclides) of the elements Z=1-118 have been studied carefully
and sorted as UGNM that generate unquantized atomic masses UQAM.
The main criteria that bound the calculated GNM are;
i. The even and odd curves of each element inside the graph have to show the parallelism
with gap. Usually, inside the graph, the gap between isotopes of small A is larger than that
in large A. The gap incraeses with an increase in Z. This gap deforms the parallelism in
the element of Z>130 and convert it to twisting in higher Z. Therefore, NMT replaces the
treatment of SVNM by TVNM to avoid the deformation of the graphs.
ii. The even curve should be lower than the odd curve in even Z but vice versa for odd Z (see
Fig. 4),
iii. The calculated GNM (i.e. GSVNM and GTVNM) should give harmonization curves of
the sequential elements (see Fig. 8A & B),
iv. The GNM values have to generate quantized atomic masses QAM that give the proper
sequential values for β-, β
+, α-decay energies without deviation and discontinuities,
v. The calculated QAM should lead to the positive incremental difference in alpha energies
between two sequential even-even and odd-odd mass number A; which denoted by the
424 Bahjat R. J. Muhyedeen
term ∆α (ee,oo). The ∆α (ee,oo) values should be positive if the atomic masses are
quantized. Otherwise, the atomic masses are not quantized. The second term is also
considered during conversion CNM to GNM that the differences between each two
sequential α-energy values of even-A and odd-A or vice versa ∆α (eo,oe) has to be
positive incremental values. All IAEA and the theoretical atomic masses, such as Moller
and Duflo-Zucker etc., in the literature failed to give positive ∆α (ee,oo) and ∆α (eo,oe),
vi. The calculated QAM values have to be very close to the existing of IAEA,
vii. The calculated QAM values should give the proper β- and β+ values in the element Z+1
and Z-1 respectively and proper α-energies values in Z and be suitable for α-decay
energies in the next element Z+2. Normally, NMT avoids the full modification of the
QNM of the stable and relatively stable nuclides of Z=1-82 to decrease the RMS values.
Achieving the seven criteria is a very challenging job because they conflict each other,
consequently, this work requires 6 years of continuous work where more than 14000
isotopes are treated.
A. Quantized Atomic Masses Calculation
Two methods were used to calculate the quantized atomic masses QAM. NMT usually correlates the
generalized single variable neutron masses GSVNM M�∗ , of the element’ isotopes versus their natural
logarithm of mass number ln(A) for light and heavy elements, M�∗ -ln(A) graph; to generate the Neutron
Mass Quadratic Equation, NMQE (2nd
power), while it correlates the generalized total variable neutron
masses, GTVNM NM�∗ , of the element’ isotopes versus their mass number (A) for Superheavy
elements, NM�∗ -(A) graph; to generate Neutron Mass Quadratic Equation, NMQE with correlation
coefficient R-squared, R2. Each element has its two equations of neutron mass quadratic equations
NMQE, one for even and the other for odd isotopes. The polynomial equations of the 2nd
power of
GSVNM lead to the prediction of the quantized atomic masses QAM of light and heavy elements
(Z=1-107) and the polynomial equation of the 2nd
power of GTVNM lead to the prediction of the
quantized atomic masses QAM of Superheavy elements (Z=100-226).
B. Uquantized Atomic Masses Calculation
Two methods were used to calculate the unquantized atomic masses UQAM. First method for Z=1-
107; uses the polynomial equation 3rd
up to 6th
power of UGSVNM with ln(A) (i.e. the M�∗ from
IAEA) which lead to the prediction of the UQAM. Second method for Z≥93, uses the 2nd
power of
UGTVNM with (A) which (i.e. the NM�∗ from IAEA) lead to the prediction of the UQAM for Z=94-
107. The first method has short range limit or extension in estimation of UQAM with a little bit lower
RMS. The second method has long range limit or extension in estimation of UQAM a little bit higher
RMS.
The calculated quantized atomic masses QAM will be used to calculate the β-, β
+, α-decay
energies and will be listed in the tables. The calculated UQAM will be listed in the tables and also will
compare them with literature values. The UQAM are very close to the IAEA and the RMS values
become 9.3 keV for Z=100-118.
The single variable neutron mass SVNM values are functions of Z, N, A, Eb/A, masses,
half-lives, decay energies, magic numbers, deformations, shapes, sizes, shell structure and other
properties of the nucleus. Therefore, NMT coins them as “Nucleus Master Key”. The Mass
Quantization Principle MQP believes that the single variable neutron masses inside the nucleus have
to increase regularly with a fixed number of magneton packages in the nucleus with increasing the
neutron number N. The regular increment in the neutron masses should create the DNM or CNM in
different nuclides with increasing Z and N. That means there is a systemic increment in the neutron
mass starting from neutron poor to neutron rich. NMT refers to the fixed increment in the neutron
masses due to N increment as quantization process.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 425
The fixed increment in neutron masses in the isotopes of the element results in giving the
positive incremental values in ∆α (ee,oo) and ∆α (eo,oe). Furthermore, the NMT entitled the isotopic
quadratic equations as Isotopic Quantized Mass Formula IQMF for the same reason and the atomic
masses generated from GNM as quantized atomic masses QAM due to the quantization process.
Fig. 8A shows how the IAEA values of DNM of neutron poor have higher values than the
DNM of the neutron rich which give arbitrary β-, β
+, α decay energies for the element with Z=80-99.
Fig. 8B shows the corresponding NMT treated consonant and hermetic GNM variable neutron masses
values.
NMT checked the neutron masses of more than 3430 isotopes of the elements Z=1-118 of
IAEA {and also JAEA (Audi et al., Private Communication (April 2011))} and found out that all of
them are dissonant and non-hermitic neutron mass DNM (i.e. DNM, DNM and DNM) except seven
elements; Z=92-99 (of 99 elements) show a consonant neutron masses CNM. They do not give
sequential α energies. For example, the DNM, DNM and DNM of the isotopes of Z=33-72 and Z=79-
92 displayed deviation in the parabolic curves that result in discontinuities in α energies.
Fig. 9A shows the deviation (curvature, marked by a circle) of the IAEA neutron mass values
which give discontinuities in α-decay energies for their isotopes. Fig. 9B shows the generalized
neutron masses of isotopes of the elements Z=81-99. The incorrect values of α-decay energies due to
the dissonant and non-hermetic neutron masses span from neutron poor to neutron rich passing the
plateau of the stable isotopes.
The discontinuities in α-decay energies of IAEA lead to curling curves of Qα vs N as seen in
Fig. 26 in Appendix-1.
Figure 8: A-The consonant neutron masses from IAEA and B- The generalized neutron masses of the
elements’ isotopes (after correction); Z=100-107
A- The IAEA consonant neutron masses CNM (SVNM)
of the elements; Z=100-107 isotopes.
B- The corresponding NMT generalized neutron masses
GNM (GSVNM) of the elements; Z=100-107
isotopes.
426 Bahjat R. J. Muhyedeen
Figure 9: The SVNM ��∗ -ln(A) graph of isotopes of the elements Z=81-99 of IAEA is sensitive to the magic
number 126. The circle in Figure-9A shows the curvature of the IAEA neutron mass values. Figure-
9B shows the corresponding TVNM ���∗ -A graph which is insensitive
A- The single neutron masses of the isotopes of the
elements; Z=81-99 of IAEA.
B- The total neutron masses values of the isotopes of the
elements; Z=81-99 of IAEA.
The correction process for these neutron masses values will affect the values of the atomic
masses of these isotopes of IAEA, therefore, NMT avoids a complete correction for Z=81-83 only to
keep the difference between IAEA atomic masses of stable nuclides, and NMT atomic masses as
minimum as possible. The curvature in Fig.8A and in Fig.9A are due to underestimation of the IAEA
atomic masses at the magic numbers 82 and 126 respectively. The M�∗ -ln(A) graph (Fig.9A) is sensitive
to the magic number while NM�∗ -A graph (Fig.9B) is insensitive. The NM�
∗ -A graph will generate the
variable neutron mass matrix that can evaluate the atomic masses of non-existent isotopes as we will
see in the forthcoming articles.
The NMT called the calculated atomic masses which are very close to the IAEA as UQAM
because they fail to give the positive incremental values in ∆α (ee,oo) and ∆α (eo,oe). While NMT
called the calculated atomic which are a little bit far of the IAEA as QAM because they succeeded to
give the positive incremental values in ∆α (ee,oo) and ∆α (eo,oe). In item 2.7 we will see how the
predicted atomic masses of the theoretical models fail to give the positive incremental values due to
their reliance on IAEA or WAW database.
Magic Numbers
NMT scrutinized the neutron magic numbers NMN of the neutrons in details from Z=2 up to 118, and
found out they are active only in few nuclei (of small Z) but they are mostly inactive in the other
nuclei. They seem as if they are stochastic complementary number with some selective proton
numbers in some nuclei to give stable nuclide more than phenomenological numbers. They do not
have any influence at nuclei with Z greater than 90. For NMN; N=8, which is available in 13 elements,
Z=2-14, has two stable nuclides only N��� , O
�� . For NMN; N=20, which is available in 19 elements,
Z=9-28, has five stable nuclides only S���� , Cl��
�� , Ar�� , K�#
�# , Ca % % . For NMN; N=28, which is available in
21 elements, Z=12-32, has four stable nuclides only Ti �% , V �
�� , Cr � , Fe �
� . For NMN; N=50, which is
available in 24 elements, Z=27-50, has five stable nuclides only Kr��� , Sr�
, Y�## , Zr%
#% , Mo # . For
NMN; N=82, which is available in 29 elements, Z=45-73, has six stable nuclides only Ba���� , La��
��# ,
Ce��% , Pr�#
�� , Nd�%� , and Sm�
� . For NMN; N=126, which is available in 18 elements, Z=76-93, has
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 427
one stable nuclide only Pb % . For higher NMN; neither Z=114 nor N=184 shows stability effect in the
superheavy isotopes of Z=100-200.
NMT found similar results for the known proton magic numbers PMN; Z=2, 8, 20, 28, 50, 82,
108, 114, 124, 126 and 164. Elements with Z = 2, 8, 20, 28, 50, and 82 have several isotopes. For
example, the stable isotopes of, helium (Z=2) have A; 3 and 4, oxygen (Z=8) have A; 16, 17 and 18,
calcium (Z=20) have A; 40, 42, 43, 44, 46, and 48, tin (Z=50) have A; 112, 114-120, 122 and 124, lead
(Z=82) have A; 206, 207 and 208. The total stable nuclides from P and N magic are 44 nuclides of 280
stable nuclides. We have to notice that the most stable nuclide in the universe C�� does not have any
magic number, while the isotope C�� with NMN=8 is unstable. What does that mean? Can we say that
the magic numbers select arbitrarily some nuclides to make them stable and select the other to be
unstable? Why NMN=8 makes O�� nuclide high stable and makes C�
� nuclide unstable? Why
NMN=20 makes 36
S, 37
Cl, 38
Ar, 39
K, and 40
Ca stable while makes 35
P (47.3s) and 41
Sc (596ms)
unstable nuclides? And why the NMN 126 makes 82
Pb nuclide one of the most stable in the universe
and makes its neighbors nuclides 207
Tl (4.8m) and 210
Po (137.37d) and 218
U (0.51ms) totally instable?
Does the magic number make themselves active in some nuclides and inactive in the other? There are
more than 70 examples.
It is supposed that when P and N magic numbers combined in any nuclide they have to give the
highest stable nuclides, but in reality this did not happen. Only five nuclei with doubly MN are stable
such 4He2,
16O8,
40Ca20,
48Ca28 and
208Pb126 while other nuclei are unstable such Ni %
(2.1ms), Ni ��
(6.07d), Ni�% � (0.11s), Sn�%�%
�%% (1.16s), Sn �%�� (39.7s). Table-1 shows the doubly magic nuclei
which is generated from combinations of the Z and N magic numbers. Only five stable isotopes are
stable which explain the suspicion of their concepts. The microscopic-macroscopic models expected
Z=114 and N=184 for the next doubly magic nucleus. NMT calculated the T1/2 of this isotope Fl�� # -
184 to be 3.01d, while next isotope Fl�� # -185 to be with T1/2=102.1d. The non-relativistic mean-field
models predict these numbers at Z=124 and 126 and N=184 are magic numbers. NMT calculated the
T1/2 of this isotope 124� �% -184 to be unstable and decay with α=13144 MeV and β
+=4659 MeV, while
the other isotope 126� ���% -184 to be unstable and decay with α=14152 MeV and β
+=6076 MeV. The
relativistic mean-field models predict these numbers at Z=120 and N=172 are magic numbers. NMT
calculated the T1/2 of this isotope 120� % # -172 to be unstable and decay with α=13602 MeV and
β+=6277 MeV, while the other isotope 120� %
��# -199 to be with T1/2=6158y. The theoretical studies
predict the deformed nucleus at Z=108 and N=162 to be more stable. IAEA calculated the T1/2 of this
isotope Hs�% �% -162 to be 3.1s, while NMT the isotope Hs�%
-174 to be 13.37d.
The most important question is that why the graphs of Eα vs N (neutron numbers) do not show
a sharp peak after the NMN; 28, 50, 82, and 126 but they show a gradual peaks that the graph raise
with 4-6 points (i.e. NMN+4-6) then comes down. This gradual peak clearly comes from two factors:
1- the overestimation of the atomic masses, 2- the neutron mass becomes closer to the quantized mass
in the stable nuclei, rather than a magic number phenomena. If the magic numbers are really come out
of the shell phenomenon then they have to be effective in that nuclei.
Although the results of nuclear shell model of spherical nuclei has been proved by many
experimental facts such as quadrupole moments, isotope abundances, binding energies, separation,
pairing energies, nuclear radii and β, α-decay energies, but it does not mean these magic numbers grant
the nuclei the stability.
NMT again confirms that the magic P and N are a complementary numbers (i.e. Proton-
Neutron integral numbers) to achieve the stable variable neutron masses and not phenomenological
numbers.
428 Bahjat R. J. Muhyedeen
Table 1: The experimental doubly magic nuclei and the expected from theoretical studies
Z,N Z=2 Z=8 Z=20 Z=28 Z=50 Z=82 Z=108 Z=114 Z=120 Z=124 Z=126 Z=164
N=2 He-4
Stable
N=8 He-8
119ms
O-16
Stable
Ca-28
β+ 14012
N=20 O-28
β- 19160
Ca-40
Stable
Ni-48,
2.1 ms
N=28 O-36
β- 42789
Ca-48
1.9x1019y
Ni-56
6.08 d
Sn-78
β+17959
N=50 Ca-70
β- 20317
Ni-78
0.11 s
Sn-100
1.16 s
Pb-132
β+25351
N=82 Ni-110
β-38261
Sn-132
39.7 s
Pb-164
NE
N=114 Sn-164
160 ns
Pb-196
37 m
Fl-228
β+25694
N=126 Sn-176
β-29345
Pb-208
Stable
Hs-234
β+15373
Fl-240
β+20401
N=162
Pb-234
β-20317
Hs-270
3.6 s
Fl-276
1.97 us 120X-282
α=15340
β+=10311
124X-286
α=17348
β+=13870
N=172
Hs-280
11.04h
Fl-286
0.16 s 120X-292
α=13602
β+=6277
124X-296
α=15588
β+=9221
126X-298
α=16603
β+=11072
N=184
Hs-292
β- 4146
Fl-298
3.01 d 120X-304
α=11166
β+=1894
124X-308
α=13144
β+=4659
126X-310
α=14147
β+=6076
164X-348
α=35871
β+=64799
N=196
Hs-304
β- 9922
Fl-310
α=5481
β=4765
120X-316
α=8348
247.89 d
124X-320
α=10316
2 m
126X-322
α=11318
β+=1652
164X-360
α=32808
β+=45980
N=236
Hs-344
β- 32981
Fl-350
β-27535 120X-356
β-22204
124X-360
β-18492
126X-362
β-16614
164X-400
α=20190
β+=13494
N=318 124X-442
β-99393
126X-444
β-55081
164X-482
β,β+emitter
2.2. The General Procedure for the Calculation of the Quantized Atomic Masses [52-55]
1. Collecting the atomic masses and other properties values of the isotopes from IAEA for
element Z=1-118 and from others such as JAEA, Wang-Audi-Wapstra (WAW) [25,26], and
from Moller et al [31,32] for comparison.
2. Deriving the single variable neutron mass SVNM M�∗ from IAEA atomic masses.
3. Using the polynomial functions to link M�∗ with ln(A) to setup the NMQE (see Eq. 4A) and the
isotopic quantized mass formula IQMF for the element Z=1-107 to calculate the QAM. The
elements from Z=108-118 require different methods. NMT uses AQMF and NQMF as
explained below.
4. Calculating the alpha energies of the isotopes for the element Z=1 to Z=99 consecutively.
5. Moving back from Z=99 to Z=1 sequentially to calculate the positive ∆α (ee,oo) values. The
aim of this step is to depend on alpha energies of the appropriate alpha emitter element Z=84-
99.
6. Moving again from Z=1 to Z=99 to calculate the positive ∆α (eo,oe) values.
7. Deriving and using the analytical mass formula AQMF and the numerical quantized mass
NQMF formula to create the atomic masses for more than 700 isotopes belong to the element
Z=108-118 as these elements don’t have enough isotopes as refference points, RP (4RP for
even and/or 4RP for odd) to setup NMQE and IQMF. Each isotope can be created from its four
ancestors. The alpha energies of the 700 isotopes have been calculated. The details of these
procedures will be published soon.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 429
8. The alpha energies of more than 1250 isotopes of the elements Z=100-118 with positive values
for ∆α (ee,oo) and ∆α (eo,oe) used to setup the database for the element Z=119-200 of Periods-
8 & 9.
9. Deriving and using the energetic quantized mass formula EQMF, which derive the SVNM &
TVNM from the daughter to the mother keeping the positive values of ∆α (ee,oo) and ∆α
(eo,oe) to secure the quantization of the atomic masses. EQMF succeeded in calculation of the
quantized atomic masses QAM for Z=119-200. The results of items 7, 8 and 9 will be published
in the next articles. The results were compared with Moller and DZ calculations.
2.3. The Special Isotopic Quantized Mass Formula for the Elements Z=1-107
As we stated above that NMT’s procedure of calculation of the atomic masses is completely different
from the SMT methods of determination or prediction of the atomic masses. NMT apply the Mass
Quantization Principle MQP to the neutron mass which lead to generation of the DNM and CNM
which they have to be converted to the generalized (consonant hermetic) neutron masses GNM. The
generated neutron masses GNM will be substituted into the isotopic quantized mass formula IQMF
to calculate the quantized atomic masses QAM. The quantized atomic masses of Einsteinium Es are
calculated below as an example below.
I. Calculation of the Quantized Atomic Masses from IQMF
NMT converts the CNM of 100Fm isotopes to GNM in order to calculate the quantized atomic masses.
The precise quantized atomic masses QAM of the radioisotopes of 100Fm element is calculated in two
steps.
First, we have to calculate the generalized neutron masses GNM/GNM/GNM values as
explained in the previous paragraph (see Scheme -1) to get two neutron mass quadratic equations
NMQE; Even-Even (Eq. 6) and Even-Odd (Eq. 7) which are generated from Fig. 10 A&B, as follow:
GNM = 0.022527785024067 Ln(A)2 - 0.238454593170083 Ln(A) + 1.62513842710033, R² = 1.000 (6)
GNM = 0.022238716748689 Ln(A)2 - 0.235311087540118 Ln(A) + 1.6165972411304, R² = 1.000 (7)
The calculated values GNM/GNM/GNM of 50 isotopes are listed in Table-2. Column 4, shows
the difference between the values of the dissonant neutron masses DNM of IAEA and the generalized
neutron masses GNM of NMT. The red values GNM refer to neutron-rich equation, the green values
GNM refer to neutron-poor.
Second, we insert the GNM/GNM/GNM values into the isotopic quantized mass formula
IQMF; QAM= ZMH + N(α*ln(A)2 - β*ln(A)+ γ) as follow:
For example, the quantized atomic masses QAM of the NR 100Fm-275 and NP 100Fm-226 and
are calculated as follow:
QAM= 100MH + 175(0.996498376125361, from Table-2) (8)
= 275.16971904494 u
QAM= 100MH + 126(0.994502763187517, from Table-2) (9)
= 226.08985138463 u
430 Bahjat R. J. Muhyedeen
Figure 10A: The generalized neutron mass of Fermium (even)
Figure 10B: The generalized neutron mass of Fermium (odd
The calculated quantized atomic masses QAM values of 50 isotopes are listed in Column 4 in
Table-4. The quantized atomic masses QAM for new non-existent such as 226
Fm -240
Fm and 260
Fm -275
Fm etc. have been calculated using equations 6-9 as seen in the Table-4. Column 2, ∆MIAEA-QAM,
shows the difference between the unquantized masses of the IAEA and the quantized masses of NMT.
Generally, the differences become larger when we go up or down far from the quantized masses of
stable isotopes. NMT discovered that most of IAEA atomic masses of far neutron-poor and far
neutron-rich isotopes are overestimated by nuclear workers which are listed by the tables of IAEA
website and also in JAEA and NNDC websites.
2.4. Calculation of the Theoretical β-, β
-, EC Energies
The beta energies are calculated from SMT (based on mass defect MD, ∆MA), NMT (based on neutron
mass defect NMD, ∆Mn) and standard energy of formation of nuclide E��, SEFN (based on a specific
perception that in severe circumstances such as in the stars, the bound neutrons are usually generated
from fusion of protons during the creation of nuclei releasing two positrons and neutrinos). The three
methods give the same results. The beta, EC-decay energy values are listed in the Tables 1-20 in the
appendix-1.
A SMT-MD; (based on mass defect MD, ∆MA) The Q-value of β-, β
-, EC-decay can be calculated
from the mass defect.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 431
SMT-MD; β-: Q
- = M(A,Z)-M(A,Z+1); β
+: Q
+ = M(A,Z)-M(A,Z-1) (10)
B NMT-NMD; (based on neutron mass defect NMD, ∆Mn)
1- β--decay:
The general formula of NMD (∆Mn) for β--decay is given by:
∆Mn = [(MA-ZMH)M – (MA-Z1MH)D] (11)
The term (MA-ZMH)M and (MA-Z1MH)D will give the total mass of neutrons in the mother M
and the daughter D respectively [55].
2- β+-decay:
The general formula of ∆Mn for β+-decay is given by:
∆Mn = [(MA-Z1MH)M – (MA-ZMH)D] (12)
3- EC-decay:
The general formula of ∆Mn for EC-decay is given by:
∆Mn = [(MA-Z1MH)M) - (MA-ZMH)D] (13)
Thus, both positron and EC decay follow that same equation. Both SMT - MD and NMT -
NMD could not establish a sharp criterion to differentiate between β+ and EC. In the following item,
we will see and understand how the new nuclear concept will sort out this issue.
Table 2: The CNM (from IAEA) and GNM (from NMT) of Fermium’ isotopes
No Nuclide
CNM/CNM
variable neutron
mass, CM�∗ (u)
GNM/GNM
variable neutron
mass, GM�∗ (u)*
The differences
between 8M�∗ and
GM�∗
∆Mn=CM�∗ - GM�
∗
Half-lives A N
1 100Fm-226 0.994502763187517 226 126
2 100Fm-227 0.994533625547725 227 127
3 100Fm-228 0.994555366703745 228 128
4 100Fm-229 0.994587764875610 229 129
5 100Fm-230 0.994610961745754 230 130
6 100Fm-231 0.994644811347080 231 131
7 100Fm-232 0.994669466761673 232 132
8 100Fm-233 0.994704685842952 233 133
9 100Fm-234 0.994730802808274 234 134
10 100Fm-235 0.994767311766329 235 135
11 100Fm-236 0.994794893449242 236 136
12 100Fm-237 0.994832614944602 237 137
13 100Fm-238 0.994861664658161 238 138
14 100Fm-239 0.994900523535987 239 139
15 100Fm-240 0.994931044725977 240 140
16 100Fm-241 0.994976643808511 0.994970967940353 0.00080030 0.73ms 241 141
17 100Fm-242 0.995006526598592 0.995002964172692 0.00050586 0.8ms 242 142
18 100Fm-243 0.995048683755245 0.995043880714114 0.00068683 231ms 243 143
19 100Fm-244 0.995080102618056 0.995077355663068 0.00039556 3.3ms 244 144
20 100Fm-245 0.995123074324138 0.995119196488972 0.00056229 4.2s 245 145
21 100Fm-246 0.995156487993151 0.995154153926120 0.00034077 1.54s 246 146
22 100Fm-247 0.995200284197279 0.995196851894294 0.00050455 31s 247 147
23 100Fm-248 0.995234346216216 0.995233295678218 0.00015548 34.5s 248 148
24 100Fm-249 0.995278015617450 0.995276785482958 0.00018329 2.6min 249 149
25 100Fm-250 0.995313452073333 0.995314719549582 0.00019012 30min 250 150
26 100Fm-251 0.995357858721854 0.995358937660465 0.00016292 5.3h 251 151
27 100Fm-252 0.995394498927632 0.995398366014029 0.00058780 25.39h 252 152
28 100Fm-253 0.995442361751634 0.995443250617163 0.00013600 3d 253 153
29 100Fm-254 0.995482799831169 0.995484177321776 0.00021213 3.24h 254 154
30 100Fm-255 0.995532005258065 0.995529668263423 0.00036223 20.07h 255 155
31 100Fm-256 0.995572251576923 0.995572097435165 0.00002405 157.6min 256 156
32 100Fm-257 0.995621674235669 0.995618136167607 0.00055548 100.5d 257 157
33 100Fm-258 0.995661859348101 0.995662071967143 0.00003359 370μs 258 158
432 Bahjat R. J. Muhyedeen
No Nuclide
CNM/CNM
variable neutron
mass, CM�∗ (u)
GNM/GNM
variable neutron
mass, GM�∗ (u)*
The differences
between 8M�∗ and
GM�∗
∆Mn=CM�∗ - GM�
∗
Half-lives A N
34 100Fm-259 0.995711281616352 0.995708601496709 0.00042614 1.5s 259 159
35 100Fm-260 0.995754048122386 260 160
36 100Fm-261 0.995801012959508 261 161
37 100Fm-262 0.995847974640906 262 162
38 100Fm-263 0.995895320752135 263 163
39 100Fm-264 0.995943801744036 264 164
40 100Fm-265 0.995991476505923 265 165
41 100Fm-266 0.996041481082681 266 166
42 100Fm-267 0.996089433237418 267 167
43 100Fm-268 0.996140965687706 268 168
44 100Fm-269 0.996189145300476 269 169
45 100Fm-270 0.996242209922376 270 170
46 100Fm-271 0.996290568340312 271 171
47 100Fm-272 0.996345169436744 272 172
48 100Fm-273 0.996393659249435 273 173
49 100Fm-274 0.996449801123883 274 174
50 100Fm-275 0.996498376125361 275 175
In column 3, GNM red color values are neutron rich isotopes, GNM green color are neutron
poor isotopes.
C. NMT- SEFN: Standard Energy of Formation of Nuclide 9:;
As I stated in Item-1C-II, NMT proposed a standard energy of formation of nuclide,
SEFN, ∆E�� (nucleosynthesis) in the stars as a new concept which is based on neutron mass
quantization [1-5,55]. In severe circumstances in Sun and stars, the bound neutrons are usually
generated from fusion of protons during the creation of nuclei releasing two positrons and neutrinos.
NMT called the mass of the bound neutron as variable neutron mass M�∗ . The energy released or
consumed from nucleogenesis process of the nucleus is called the standard energy of formation of
nuclide, SEFN ∆E�� (similar to standard enthalpy of formation ∆H�
� in chemistry).
SEFN indicates the amount of the energy released in the stars from the nucleosynthesis. SEFN
does not refer to the stability of the nuclide, unlike binding energy. The higher the SEFN value of the
nuclide the more exothermic and hotter the area in the star is. The SEFN concept does not focus on the
number of the protons and electrons of the created nuclide. It is different from binding energy concept
B.E and it has lower values than B.E by a fixed factor of 0.001937043438N as seen in Figure-12. It is
supposed that the binding energy BE explains the easy of formation; nucleosynthesis or nucleogenesis,
and the degree of the stability of the nuclide, but it seems not. The well-known statement “the energy
liberated in the formation of nucleus from its component nucleon is a measure of stability of that
nucleus” is incorrect. In other words, the higher the value of B.E, does not acoount for a higher stable
nuclide . Both the B.E and SEFN give higher energies values to unstable nuclides (which form 50% of
total nuclides) that belong to Z=1-99 rather than to stable nuclides. For example, both indicate that the 8Be and
218U nuclides have the highest B.E and SEFN among Be and U isotopes respectively. SEFN
has two advantage over B.E that it can calculate the Q-values for all nuclear processes and it can give a
sharp criterion to EC-decay, while B.E failed.
The fact that the SEFN values are similar to BE value sequence up to Mg only as seen in
Figure-12 except for 2H,
3H and
3He. SEFN shows that the E�
� value of 3H (T1/2=12.32 y) is lower than
that of 3He (stable), while the SMT shows the binding energy of
3H (T1/2=12.32 y) is higher than that of
3He (stable). Both show; the E�
� value and the B.E. value for 3H is higher than
2H which is incorrect
result. Although both values are not a measure of stability, but still we can say that SEFN values for 3H
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 433
and 3He is more reasonable than B.E due to their half-lives winging the range Z=1-12. This result
submits a muddled idea about the nuclear forces between the protons and neutrons in the nuclei.
The comparison between binding energies B.E and SEFN energies for Ra and U isotopes are
illustrated in Figures 13&14. The application of SEFN on all nuclear process calculation gives
identical results to SMT calculations while the binding energies cannot give the correct Q-values for β-,
β-, EC-decay processes.
The general formula for ∆E<= is given by Equation (4) using the atomic mass of the nuclide or it
may be given by the following equation using the single variable neutron masses SVNM; M�∗
E��= N(1.00672787241186-M�
∗ ) x 931.4940023 MeV (14)
Therefore, any nuclide with variable neutron mass M�∗ exceeds 1.00672787241186 will have a
negative E�� value. The relation between the binding energy and E�
� is expressed in Equation (5).
Table 3: ∆9>
? values (MeV/c2) for EC-decay of the neutron poor isotopes of the elements Z=100-107
Electron Capture Criteria
No Nuclear Process: EC-decay IAEA SEFN ∆E<=
1 Es## �� + e → Cf#
�� + υB 99.5% Yes
2 Fm�%% �� + e → Es##
�� + υB 88% Yes
3 Md�%� + e → Fm�%%
+ υB 80% No*
4 Md�%� �% + e → Fm�%%
�% + υB 93% No*
5 Md�%� � + e → Fm�%%
� + υB 100% No*
6 Md�%� � + e → Fm�%%
� + υB 100% No*
7 Md�%� �� + e → Fm�%%
�� + υB 93% No*
8 Md�%� �� + e → Fm�%%
�� + υB 90.8% No*
9 Md�%� �� + e → Fm�%%
�� + υB 85% Yes
10 No�% �� + e → Md�%�
�� + υB 30% No*
11 Lr�%� �% + e → No�%
�% + υB 40% No*
12 Rf�% �� + e → Lr�%�
�� + υB 19.4% No*
13 Db�%� �� + e → Rf�%
�� + υB 30% No*
14 Db�%� � + e → Rf�%
� + υB 33% No*
15 Sg�%� �# + e → Db�%�
�# + υB 13% No*
16 Bh�%� �% + e → Sg�%�
�% + υB 18% No* *EC cannot happen because E<
= for mother is less than for daughter.
Figure 12: Comparison of binding energy of IAEA with SEFN of NMT values of Z=1-100
434 Bahjat R. J. Muhyedeen
Figure 13: Comparison of binding energy of IAEA with SEFN of NMT values of Fm isotopes
Figure 14: Comparison of binding energy of IAEA with SEFN of NMT values of Lr isotopes
For any nuclear process A+B→C+D, the energy change for the nuclear reaction can be
calculated as follow:
∆E��
= ∑ ∆E��(I + J)LM� -∑ ∆E�
�(N + O)MPQ (15)
The SEFN concept is completely different from SMT’s ∆MA and NMT’s ∆Mn in two points:
first, it uses the standard energy of formation E�� of the nuclides in the calculation rather than the mass
defect in SMT or the neutron mass defect in NMT; second, it deducts the reactants from the products
unlike SMT and NMT which deducts the products from the reactants.
1. In negatron decays, the Q-value of β--decay can be calculated from ∆E<
=, where ∆E�,S�TUPM� <
∆E�,WQXYUTPM� .
∆E<=
= ∆E<=(B)Z -∆E<
=(A)[ - 1.0219978922 (16)
2. In positron decay, Q-value of β+-decay can be calculated, where ∆E�,S�TUPM
� < ∆E�,WQXYUTPM� .
The ∆E�� of the mother in β
- and β
+ decay should be smaller than the ∆E�
� of the daughter.
∆E<=
= ∆E<=(B)Z -∆E<
=(A)[+ 1.0219978922 (17)
3. In EC-decay, the ∆E�� of the mother should be larger than the daughter which is a sharp
criterion for EC-decay, where ∆E�,S�TUPM� > ∆E�,WQXYUTPM
� . In EC decay we add 2Me to the right
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 435
side (products) to get QEC value (same as in b). It is a different procedure from SMT and NMT.
SEFN add 1.0219978922 MeV to the daughter because the ∆E<=of the mother is larger than
∆E<= of the daughter. EC cannot happen because E<
= for mother is less than for daughter.
∆E<=
= ∆E<=(B)Z -∆E<
=(A)[ + 1.0219978922 (18)
The QEC
energy values for isotopes of the elements Z=81-99 are listed in the Tables-3.
2.5. Proton and Neutron Separation Energies
The proton Sp and neutron Sn separation energies are calculated from SMT (based on mass defect MD,
∆MA), NMT (based on neutron mass defect NMD, ∆Mn) and standard energy of formation of
nuclide E��, SEFN [55]. The three methods give the same results:
A. SMT-MD; the proton Sp and neutron Sn separation energies can be derived from the following
combinations of atomic masses as follow:
Sn= -M(A,Z)+M(A-1,Z)+N (19)
Sp= -M(A,Z)+M(A-1,Z-1)+H (20)
where H and N are the mass of the hydrogen atom, and the neutron respectively. The proton separation
energies (Sp) generally increase with increasing neutron number, (N) and neutron separation energies
(Sn) increase with increasing proton number (Z). For a given Z, Sn is larger for even N compared to
that for odd N. Similarly, for a given N, Sp is larger for even Z compared to that for odd Z. This pairing
of like nucleons causes e-e nuclei to be more stable than e-o or o-e nuclides which, in turn, are more
stable than o-o nuclei. The pairing energies of P and N are usually calculated from proton Sp and
neutron Sn separation energies. The values of the proton Sp and neutron Sn separation energies are
presented in the Figures 21-24 in Appendix-1, where they show that higher energy is required to
remove neutron when N is even than odd.
B. NMT-NMD: The general formula of ∆Mn for the neutron Sn separation energies and proton Sp
(MeV) are given by:
Sn=∆Mn = [-(MA-ZMH)(Z,A) + (MA-ZMH)Z,(A-1)+MN] (21)
SP=∆Mn = [-(MA-ZMH)(Z,A) + (MA-ZMH)(Z-1),(A-1)] (22)
C. NMT-SEFN: The general formula of ∆Mn for the neutron Sn separation energies and proton Sp
(MeV) are given by:
∆E�,]^�
= [(∆E��)M,(Z,A) - (∆E�
�)D,(Z,A-1)]+1.80435] (23)
∆E�,]L� = [(∆E�
�)M,(Z,A) - (∆E��)D,(Z-1,A-1)] (24)
2.6. Calculation of _`ab Energies and ∆α (ee,oo) and ∆α (eo,oe) of Z=81-99 Isotopes
The alpha energy cdeU are calculated from SMT (based on mass defect MD, ∆MA), NMT (based on
neutron mass defect NMD, ∆Mn) and standard energy of formation of nuclide, E��, SEFN. The three
methods give the same results:
SMT-MD; The Q-value of β-, β
--decay (MeV) can be calculated from the mass defect
SMT-MD; α: cdeU = M(A,Z)-M(A,Z-2)-Mα (25)
NMT-NMD;cdeU = ∆Mn = [(MA-ZMH)M – {(MA-ZMH)D +(MA-ZMH)α}] (26)
NMT-SEFN; ∆E��
= ∑ ∆E��(I + J)LM� -∑ ∆E�
�(N + O)MPQ, O =0 (27)
Where M=mother, D=daughter, α=alpha
Following the steps of the main criteria from i to vii the theoretical alpha energies of Es were
calculated with positive values of the two terms ∆α (ee,oo) and ∆α (eo,oe). The calculated QAM from
GNM values result in the positive values in ∆α (ee,oo) and ∆α (eo,oe) as seen in the Table-4 (see 8th
and 9th
columns) if the atomic masses are quantized. Otherwise, the atomic masses are not quantized.
436 Bahjat R. J. Muhyedeen
The 7th
and 10th
columns show that there are few alpha energy cdeU in red color that are unquantized in
NMT and IAEA.
Table 4: The Quantized Atomic Masses QAM of Fermium Isotopes
NMT-MQ, QAM-Z-100
QAM= ZMH + N(α*ln(A)2 - β*ln(A)+ γ) IAEA NMT
Isotope IAEA NMT-QAM SVNM ∆M= IAEA-
QAM
T1/2 β- β+ Α ∆α
(ee,o
o)
β- β+ α ∆α
(ee,o
o)
1 100Fm-226 226.08985138463 8886 9915 116
2 100Fm-227 227.08827366756 9961 9858 122
3 100Fm-228 228.08559016108 8234 9799 127
4 100Fm-229 229.08432489195 9371 9736 133
5 100Fm-230 230.08192824995 -11136 7574 9671 138
6 100Fm-231 231.08097350947 -9459 8760 9603 144
7 100Fm-232 232.07887283554 -10526 6906 9533 150
8 100Fm-233 233.07822644011 -8780 8130 9459 155
9 100Fm-234 234.07643079931 -9900 6231 9384 161
10 100Fm-235 235.07609031145 -8097 7481 9304 167
11 100Fm-236 236.07460873210 -9258 5549 9223 172
12 100Fm-237 237.07457147041 -7411 6811 9137 178
13 100Fm-238 238.07341294583 -8599 4861 9050 184
14 100Fm-239 239.07367599450 -6722 6123 8959 189
15 100Fm-240 240.07284948464 -7925 4167 8867 195
16 100Fm-241 241.0742100000 241.07340970259 0.00080030 0.73ms 5263 8764 71 -6031 5415 8769 201
17 100Fm-242 242.0734300000 242.07292413552 0.00050586 0.8ms 3598 8697 147 -7235 3466 8672 206
18 100Fm-243 243.0744650000 243.07377816512 0.00068683 231ms 4616 8693 258 -5337 4688 8569 212
19 100Fm-244 244.0740380000 244.07364243848 0.00039556 3.3ms 2939 8550 173 -6530 2760 8466 218
20 100Fm-245 245.0753490000 245.07478671390 0.00056229 4.2s -5085 3819 8435 178 -4643 3941 8356 224
21 100Fm-246 246.0753504700 246.07500969621 0.00034077 1.54s -5927 2286 8377 382 -5809 2049 8248 229
22 100Fm-247 247.0769450000 247.07644045146 0.00050455 31s -4263 3095 8257 549 -3947 3176 8133 235
23 100Fm-248 248.0771864630 248.07703098338 0.00015548 34.5s -5250 1598 7994 438 -5073 1333 8019 240
24 100Fm-249 249.0789275500 249.07874425996 0.00018329 2.6min -3713 2344 7709 284 -3250 2391 7897 247
25 100Fm-250 250.0795210340 250.07971115544 0.00019012 30min -4559 847 7557 404 -4322 612 7779 252
26 100Fm-251 251.0815398900 251.08170280973 0.00016292 5.3h -3013 1440 7425 227 -2552 1588 7651 258
27 100Fm-252 252.0824670600 252.08305485713 0.00058780 25.39h -3693 -478 7153 -155 -3556 -113 7527 263
28 100Fm-253 253.0851845710 253.08532056743 0.00013600 3d -1825 334 7198 -42 -1855 765 7393 269
29 100Fm-254 254.0868543970 254.08706653055 0.00021213 3.24h -2550 -1088 7307 280 -2775 -843 7264 274
30 100Fm-255 255.0899640380 255.08960180383 0.00036223 20.07h -1043 -290 7240 376 -1158 -76 7124 281
31 100Fm-256 256.0917744690 256.09175042289 0.00002405 157.6min -1970 -1700 7027 368 -1980 -1577 6990 286
32 100Fm-257 257.0951060780 257.09455060131 0.00055548 100.5d -406 -813 6864 394 -461 -936 6843 292
33 100Fm-258 258.0970770000 258.09711059381 0.00003359 370μs -1262 -2276 6660 -1169 -2314 6704 297
34 100Fm-259 259.1005970000 259.10017086098 0.00042614 1.5s 80 6470 235 -1814 6551 303
35 100Fm-260 260.10315092258 6y -345 -3055 6407 308
36 100Fm-261 261.10646630948 931 -2711 6248 314
37 100Fm-262 262.10987511483 494 -3800 6099 319
38 100Fm-263 263.11344050560 1624 -3627 5933 326
39 100Fm-264 264.11728670902 1347 -4547 5779 331
40 100Fm-265 265.12109684648 2316 -4561 5608 337
41 100Fm-266 266.12538908273 2214 -5297 5449 342
42 100Fm-267 267.12943857365 3007 -5513 5271 348
43 100Fm-268 268.13418545854 3095 -6050 5107 353
44 100Fm-269 269.13846877878 3695 -6484 4923 359
45 100Fm-270 270.14367890980 3990 -6806 4754 364
46 100Fm-271 271.14819040919 4380 -7473 4564 370
47 100Fm-272 272.15387236612 4899 -7563 4391 375
48 100Fm-273 273.15860627315 5063 -8480 4194 381
49 100Fm-274 274.16476861856 5821 4016
50 100Fm-275 275.16971904494 5743 3813
*T1/2 is calculated from Royer Formula, G. Royer, J. Phys. G: Nucl. Part. Phys. 26 (2000) 1149-1170.
In column 4, red color QAM values are neutron rich isotopes, green color QAM are neutron poor isotopes. All the atomic
masses of einsteinium are QAM. The calculated QAM values finally give the proper sequential values for β-, β+, cdeU
energies without deviation and discontinuities as seen in Table-4 and illustrated in Fig. 13, 14 in comparison with WAW Eα
(see Fig.15) in the appendix-1.
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 437
2.7. Comparison of Atomic Masses with Literature
The atomic masses calculated by IAEA, JAEA, Wang-Audi-Wapstra et al., WAW [26], and by Moller
et al. [32] have been investigated and compared with the present work of NMT calculation. The WAW
values are very close to IAEA values and both of them give limited isotopes to each element. WAW
calculations are highly corrected based on experimental nuclear data such as atomic masses and
nuclear reactions etc. Although Moller et al values are based on theoretical Macro-Micro MM
equations but also, they did several corrections and amendment during three decades. The neutron
masses values derived from IAEA, WAW and Moller are not able to give to the positive incremental
difference in alpha energies i.e. ∆α (ee,oo) and ∆α (eo,oe) as seen in 11th
and 12th
columns of Table-4
which are responsible for generating the proper data base Z=1-118 for atomic calculation of isotopes
with Z=119-200. The neutron masses values of IAEA and Moller show high deviation from NMT
values especially in the far neutron poor and neutron rich regions (see Fig. 15). NMT attributed the
deviation of Moller values to the improper parameters of macro-micro formula. Figures 16 and 17
shows the accuracy of the current calculations in comparison with the literature.
The MM describes the nuclei as macro model (such as liquid drop etc.) from standpoint and the
micro model (such as shell structure etc.) point of view and thus MM will give a confused model rather
than hybrid model. Moller model gives overestimated atomic mass with increasing Z. For example,
NMT calculated the quantized atomic mass QAM for g���� #� ; MA= 297.24405819090 u, while Moller
calculation gives 297.24797796322 with ∆M=0.00391977232 u and Duflo-Zucker, DZ [40] calculation
gives 297.23962686328 with ∆M=0.00443132762 u. The maximum mass number A calculated by
Moller was A=339 and A=297 by Duflo-Zucker while NMT can reach above 560 and no limit. Table-5
shows a sample for NMT calculation for the accurate atomic masses, decay energies, and their
estimated α partial half-lives hdeU for few isotopes belong to Z=118-147. Koura-Tachibana et al. derived
a nuclear mass formula and improved in 2005 and termed it as “KTUY” [42] and estimated the Qα for
Ji���% # , jk���
# , g�,� ���% g�%��
� 6.84, 8.12, 14.45, and 14.12 MeV respectively compared with
present work calculation 6.429, 8.289, 14.152, and -11.845 MeV. DZ calculation gives 7.386 MeV for
Ji� ��% # only and Moller calculation gives 6.915, 8.995 and 11.275 MeV for Ji���%
# , jk��� # ,
g�,� ���% only. KTUY is close to NMT only in the first three isotopes but there is a large descriptancy in
the fourth isotope. All theoretical calculations give inconsecutive alpha energies for Z>118.
Table 5: The NMT atomic masses, decay energies and their estimated α partial half-lives a`
ab of few isotopes
from Period-8
Isotope Atomic masses Z A N β
- β+ _`
ab SP a`ab
RF a`ab VSS a`
ab
118-Og-309 309.24639815408 118 309 191 -1060 -2321 8608 2.64E-01 8.48E-02 1.34E-01
120-VIII-12-314 314.26253061903 120 314 194 -1297 -1611 8844 4.78E-02 1.00E-02 3.29E-02
122-VIII-17-322 322.28817353226 122 322 200 -340 -2250 8270 3.50E+01 7.45E+00 2.88E+01
133-VIII-19-254 354.38579392079 133 354 221 12 -1455 7823 1.55E+08 4.31E+07 1.02E+08
139-VIII-21-368 368.42541026097 139 368 229 -2197 -79 8344 1.19E+08 3.38E+07 6.86E+07
147-VIII-26-397 397.49187641021 147 397 250 -1520 -1806 4563 3.57E+41 1.72E+43 5.25E+39
SP=Sobiczewski-Parkhomenko Formula [56], RF=Royer Formula [57], VSS=Viola-Seaborg-Sobiczewski Formula [58]
Table 6: The theoretical alpha decay energies _
`
ab of few isotopes from Z=119 and Z=120
Isotope A D-Z Munt
ian
Molle
r NMT Isotope
A D-Z Muntian Moller
Smola
nczuk NMT
119X-291 291 13434 12890 13235 13102 120X-292 292 13787 13460 13775 13590 13602
119X-292 292 13188 12730 13075 12917 120X-293 293 13530 13340 13645 13432
119X-293 293 12943 12620 12915 12736 120X-294 294 13281 13240 13485 13420 13223
119X-294 294 12706 12380 12845 12549 120X-295 295 13033 13010 13455 13047
119X-295 295 12472 12550 12935 12358 120X-296 296 12794 13230 13585 13400 12833
119X-296 296 12245 12650 12975 12168 120X-297 297 12555 13490 13645 12651
119X-297 297 12020 12860 12895 11968 120X-298 298 13440 13235 13360 12429
119X-298 298 12590 13085 11773 120X-299 299 13230 13725 12241
438 Bahjat R. J. Muhyedeen
Isotope A D-Z Munt
ian
Molle
r NMT Isotope
A D-Z Muntian Moller
Smola
nczuk NMT
119X-299 299 12630 13075 11564 120X-300 300 13110 13695 13090 12018
119X-300 300 12600 13025 11367 120X-301 301 13110 13615 11824
119X-301 301 12590 13075 11149 120X-302 302 13080 13545 13070 11596
119X-302 302 12560 13045 10948 120X-303 303 13050 13505 11396
119X-303 303 12560 13105 10722 120X-304 304 13070 13545 13120 11164
119X-304 304 13510 13855 10518 120X-305 305 13990 14255 10957
119X-305 305 13270 13855 10283 120X-306 306 13730 14275 13530 10721
119X-306 306 13160 13925 10074 120X-307 307 13650 13615 10508
119X-307 307 13090 13385 9832 120X-308 308 13560 12955 10268
Another example, Muntian et al [43] calculated the alpha energies for 119 starting from A=291
up to 307, the values of Alpha for nuclides with A=291-297 are 12.89, 12.73, 12.62, 12.38, 12.55,
12.65, and 12.86 MeV respectively. It is very clear that they decreased to the minimum at 12.38 then
increased up to 12.86 MeV. Always Moller atomic masses are higher than Muntian et al and both are
higher than the QAM of NMT. For example, the atomic masses for 292
120 and 297
120 are calculated as
292.22865416852, 292.22746253451, 292.22576296088 and 297.23150979596, 297.23071537329,
297.22834165491 by Moller, Muntian and NMT respectively. Table-6 lists alpha energies for Z=119
and Z=120 of the theoretical models of Duflo-Zucker, Muntian et al, Moller et al, and Smolanczuk et
al.
It is supposed that the values of the alpha energies of the isotopes of the same element decrease
with increasing mass number such as in the calculations of Duflo-Zucker, lim-Oh[59] and present
work, while the calculation of Muntian et al, Moller et al, Smolanczuk et al [60] and Rather et al [61]
showed an arbitrary alpha values as seen in Figures 18 and 19.
Summary Three methods were used to calculate the unquantized atomic masses UQAM. The first method for
Z=1-107; uses the polynomial equation 3rd
up to 6th
power of SVNM with ln(A); M�∗ vs ln(A) to
generate M�∗ which lead to the prediction of the UQAM. In case of UQAM, the Neutron Mass
Quadratic Equation is changed to Neutron Mass Polynomial Equation NMPE, because it is non-
quadratic. The second method for Z≥93, uses the 2nd
power of TVNM with (A); NM�∗ -A graph; to
generate NM�∗ which lead to the estimation of the UQAM for Z=94-107. The third method for Z=1-107
uses the SEFN of the known isotopes to setup a standard graphs between function of (SEFN) versus
mass number A. It gives a good estimated atomic masses with RMS=1.876 MeV for Z=100, Fm
element and with RMS=1.811 MeV for Z=98, Cf element. The third method requires very accurate
function to describe SEFN values of the existent isotopes against mass number A to generate a graph
of excellent pearson correlation coefficient formula which pridects the best SEFN values to the non-
existent isoptops. The UQAM can be calculated from these predicted SEFN.
The first method has short range limit or extension in estimation of UQAM with a little bit
lower RMS. The second and third methods have long range limit or extension in estimation of UQAM
a little bit higher RMS. The UQAM of more than 6000 isotopes of the elements Z=1-107 have been
calculated.
Only one methods was used to calculate the quantized atomic masses UQAM which uses
GSVNM.
The calculated QAM for 800 isotopes belong to Z=100-107 from GNM values result in the
positive difference in alpha energies between two sequential even-even and odd-odd mass number A
(i.e. ∆α (ee,oo)) and even-odd and odd-even mass number A (i.e. ∆α (eo,oe)) for each element. The
calculated QAM values give the proper sequential values for β-, β
+, α-decay energies without deviation
and discontinuities. Only the quantized atomic masses QAM can give the positive values for ∆α (ee,oo)
and ∆α (eo,oe) which should be started from Z=1 up to Z=200. These positive values have been
calculated for more than 7900 isotopes belong to Z=1-118. With help from these positive values, NMT
succeeded in calculation of the QAM of more than 8100 isotopes belong to period-8, Z=119-172, and
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 439
Z=173-226 of period-9 that they will be submitted in the subsequent articles. NMT also succeeded in
finding out the Island of the Stability. In the present article, we discussed the QAM of the isotopes of
the elements Z=100-107 (Fm-Bh). The UQAM are very close to the IAEA. The QAM of some neutron
poor or rich can be calculated from the isobaric equation ISQMF. Although NMT has its powerful
NMQE and IQMF to calculate the QAM for any isotope to any existent element but some time it uses
its ISQMF to re-evaluate some disputed QAM for double checking. The Q-values of β-, β
+, EC, α-
decay energies have been calculated. IAEA showed in their website that Md�%� , Md�%�
�% , Md�%� � ,
Md�%� � , Md�%�
�� , Md�%� �� , No�%
�� , Lr�%� �% , Rf�%
�� , Db�%� �� , Db�%�
� , Sg�%� �# and Bh�%�
�% undergo EC, but SEFN
calculations showed they cannot decay via EC.
Figure 15: Comparison of the neutron masses among IAEA, Moller and present work NMT, Z=100-107
440
440
Figure
Figure-16:
H. Geissel,
Figure 17: Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
Figure-
16: Comparison of the NMT
H. Geissel, Yu.A. Litvinov et al, Nucl. Phys. A746 (2004) 150c
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
-18: Comparison of the NMT QAM’E
Comparison of the NMT
Yu.A. Litvinov et al, Nucl. Phys. A746 (2004) 150c
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
Comparison of the NMT QAM’E
Comparison of the NMT- UQAM and QAM with other models
Yu.A. Litvinov et al, Nucl. Phys. A746 (2004) 150c
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
Comparison of the NMT QAM’Eα with other models, Z=119
UQAM and QAM with other models
Yu.A. Litvinov et al, Nucl. Phys. A746 (2004) 150c
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
α with other models, Z=119
Bahjat R. J. Muhyedeen
UQAM and QAM with other models
Yu.A. Litvinov et al, Nucl. Phys. A746 (2004) 150c
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100
with other models, Z=119
Bahjat R. J. Muhyedeen
UQAM and QAM with other models
Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100-107
with other models, Z=119
Bahjat R. J. Muhyedeen
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 441
Figure 19: Comparison of the NMT QAM’Eα with other models, Z=120
3. Results and Discussion The Quantized and Unquantized Atomic Masses (QAM and UQAM) - NMT- 2017 version 3, of
Z=100-107
Figures and Tables are in the Appendix-1. The polynomial equations, of the 2nd
power, were used to
generate the quantized atomic masses QAM as listed in Table-9. The mass formula has RMS 0.37
MeV for quantized atomic masses QAM for the isotopes corresponding to the IAEA peers. Actually,
the values of RMS of QAM does not reflect the accuracy of the calculated QAM values rather than
they show the discrepancies with IAEA values. The polynomial equations, of the 3rd
up to 6th
power,
were used to generate the unquantized atomic masses UQAM with RMS of 0.057 MeV. Another
method was used to calculate the UQAM of the elements Z=100-107 using total variable neutron mass
TVNM as explained in item 2.1 (which uses the polynomial equation of the 2nd
power of NM�∗ -A
graph). Table-10 lists the values of the UQAM which calculated by TVNM and compared with
literature values of DZ, WAW and Moller. The UQAM are very close to the IAEA. The NMT values
of the proton Sp and neutron Sn separation energies are plotted in the Figures 9-12. The NMT QAM
give a smooth curve for cdeU vs N while IAEA and WAW’ Eα show curling curves (see Fig. 13-15).
The smooth and straight curves of cdeU confirm the mass quantization concept. The UQAM, Moller
and IAEA failed to give the positive values for the two terms ∆α (ee,oo) and ∆α (eo,oe).
1. Fermium
The fermium element has 9 even isotopes, and 10 odd isotopes from IAEA241
Fm – 259
Fm. They were
used as reference points RP in the polynomial of Neutron Mass Quadratic Equation NMQE (see Fig.
1A). Most of RP values showed deviation in the polynomial of even and odd isotopes, and both curves
give DNM. NMT generated the GNM after several modifications to the DNM (see Fig. 1B). Table-1
shows the difference between beta energies of IAEA and NMT. NMT added 16 new neutron poor NP
isotopes 226
Fm – 240
Fm, 260
Fm and 15 new neutron rich NR isotopes 261
Fm – 275
Fm to the literature. All
the calculated atomic masses of fermium are QAM. The RMS in calculation of the QAM is 0.39 MeV
for all isotopes. The RMS in calculation of the UQAM is 0.057 MeV for all isotopes.
2. Mendelevium
The mendelevium element has 8 even isotopes and 8 odd isotopes from IAEA 245
Md – 260
Md. They
were used as reference points RP in NMQE (see Fig. 2A). Most of RP values showed deviation in the
polynomial of even and odd isotopes, and both curves give DNM. NMT generated the GNM after
442 Bahjat R. J. Muhyedeen
several modifications to the DNM (see Fig. 2B). Table-2 shows the difference between beta energies of
IAEA and NMT. NMT added 15 new neutron poor NP isotopes 230
Md – 244
Md and 15 new neutron
rich NR isotopes 261
Md – 275
Md to the literature. All the calculated atomic masses of mendelevium are
QAM. The RMS in calculation of the QAM is 0.45 MeV for all isotopes. The RMS in calculation of
the UQAM is 0.075 MeV for all isotopes.
3. Nobelium
The nobelium element has 8 even isotopes and 5 odd isotopes from IAEA 248
No, 250
No – 260
No, 262
No.
They were used as reference points RP in the NMQE (see Fig. 3A). Most of RP values showed
deviation in the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the
GNM after several modifications to the DNM (see Fig. 3B). Table-3 shows the difference between beta
energies of IAEA and NMT. NMT added 21 new neutron poor NP isotopes 232
No - 247
No, 249
No, 261
No, 263
No, 264
No, 266
No and 12 new neutron rich NR isotopes 265
No, 267
No - 277
No to the literature. All the
calculated atomic masses of nobelium are QAM. The RMS in calculation of the QAM is 0.29 MeV for
all isotopes. The RMS in calculation of the UQAM is 0.038 MeV for all isotopes.
4. Lawrencium
The lawrencium element has 7 even isotopes and 5 odd isotopes from IAEA 252
Lr – 262
Lr, 266
Lr. They
were used as reference points RP in the NMQE (see Fig. 4A). Most of RP values showed deviation in
the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the GNM after
several modifications to the DNM (see Fig. 4B). Table-4 shows the difference between beta energies of
IAEA and NMT. NMT added 18 new neutron poor NP isotopes 236
Lr – 251
Lr, 263
Lr, 265
Lr and 17 new
neutron rich NR isotopes 264
Lr, 267
Lr – 281
Lr to the literature. All the calculated atomic masses of
lawrencium are QAM. The RMS in calculation of the QAM is 0.32 MeV for all isotopes. The RMS in
calculation of the UQAM is 0.026 MeV for all isotopes.
5. Rutherfordium
The rutherfordium element has 5 even isotopes and 8 odd isotopes from IAEA 253
Rf - 263
Rf, 265
Rf, 267
Rf. They were used as reference points RP in the NMQE (see Fig. 5A). Most of RP values showed
deviation in the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the
GNM after several modifications to the DNM (see Fig. 5B). Table-5 shows the difference between beta
energies of IAEA and NMT. NMT added 20 new neutron poor NP isotopes 238
Rf - 252
Rf, 264
Rf, 266
Rf, 268
Rf - 270
Rf and 13 new neutron rich NR isotopes 271
Rf - 283
Rf to the literature. All the calculated
atomic masses of rutherfordium are QAM. The RMS in calculation of the QAM is 0.29 MeV for all
isotopes. The RMS in calculation of the UQAM is 0.2 MeV for all isotopes.
6. Dubnium
The dubnium element has 7 even isotopes and 6 odd isotopes from IAEA 255
Db – 263
Db, 266
Db, 268
Db - 270
Db. They were used as reference points RP in the NMQE (see Fig. 6A). Most of RP values showed
deviation in the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the
GNM after several modifications to the DNM (see Fig. 6B). Table-6 shows the difference between beta
energies of IAEA and NMT. NMT added 19 new neutron poor NP isotopes 240
Db - 254
Db, 264
Db, 265
Db, 269
Db, 271
Db and 14 new neutron rich NR isotopes 271
Db - 285
Db to the literature. All the calculated
atomic masses of dubnium are QAM. The RMS in calculation of the QAM is 0.29 MeV for all
isotopes. The RMS in calculation of the UQAM is 0.02 MeV for all isotopes.
7. Seaborgium
The seaborgium element has 5 even isotopes and 6 odd isotopes from IAEA 258
Sg – 266
Sg, 269
Sg, 271
Sg.
They were used as reference points RP in the NMQE (see Fig. 7A). Most of RP values showed
deviation in the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the
The Quantized Atomic Masses of the Elements: Part-6; Z=100-107 (Fm-Bh) 443
GNM after several modifications to the DNM (see Fig. 7B). Table-7 shows the difference between beta
energies of IAEA and NMT. NMT added 24 new neutron poor NP isotopes 242
Sg - 257
Sg, 267
Sg, 268
Sg, 270
Sg, 272
Sg - 276
Sg and 11 new neutron rich NR isotopes 277
Sg - 287
Sg to the literature. All the
calculated atomic masses of seaborgium are QAM. The RMS in calculation of the QAM is 0.42 MeV
for all isotopes. The RMS in calculation of the UQAM is 0.009 MeV for all isotopes.
8. Bohrium
The bohrium element has 7 even isotopes and 5 odd isotopes from IAEA 260
Bh - 267
Bh, 270
Bh - 272
Bh, 274
Bh. They were used as reference points RP in the NMQE (see Fig. 8A). Most of RP values showed a
deviation in the polynomial of even and odd isotopes, and both curves give DNM. NMT generated the
GNM after several modifications to the DNM (see Fig. 8B). Table-8 shows the difference between beta
energies of IAEA and NMT. NMT added 20 new neutron poor NP isotopes 244
Bh - 259
Bh, 268
Bh, 269
Bh, 273
Bh, 275
Bh and 12 new neutron rich NR isotopes 276
Bh - 289
Bh to the literature. All the calculated
atomic masses of bohrium are QAM. The RMS in calculation of the QAM is 0.49 MeV for all
isotopes. The RMS in calculation of the UQAM is 0.02 MeV for all isotopes.
4. Conclusion In present work, the QAM’s are calculated from the isotopic quantized mass formula IQMF of 2
nd
power. The UQAM’s are calculated from three methods applying SVNM, TVNM and SEFN values.
The first method for Z=1-107; uses the polynomial equation 3rd
up to 6th
power of SVNM with ln(A);
M�∗ vs ln(A) While the second method for Z≥93, uses the 2
nd power of TVNM with (A); NM�
∗ -A graph.
The third method for Z=1-107; uses the SEFN of the known isotopes to setup a standard graphs
between function of (SEFN) versus mass number A.
The discrepancy (RMS) of the mass model is less than 335 keV for UQAM and 884 keV for
QAM for the entire region of ground-state masses of 3160 nuclei ranging from 1H to
118Og.
The quantized atomic masses of more than 16000 nuclei ranging from Z=1 to Z=226 have been
calculated, 800 nuclides of them belong to Z=100-107 (Fm-Bh). These new atomic masses will be
submitted in the forthcoming articles. The QAM of some neutron poor or rich can be calculated from
the isobaric equation ISQMF. Although NMT has its powerful NMQE and IQMF to calculate the
QAM for any isotope to any existent element but some time it uses its ISQMF to reevaluate some
disputed QAM for double checking. The Q-values of β-, β
+, EC, α-decay energies have been calculated
through three different nuclear methods; SMT which based on mass defect MD (in proton and neutron
mass), NMT which based on neutron mass defect NMD ( in neutron mass only), and SEFN (standard
energy of formation of nuclide E��). SEFN calculations showed the Md�%�
, Md�%� �% , Md�%�
� , Md�%� � ,
Md�%� �� , Md�%�
�� , No�% �� , Lr�%�
�% , Rf�% �� , Db�%�
�� , Db�%� � , Sg�%�
�# and Bh�%� �% cannot undergo EC-decay.
The novel theory NMT examined all experimental and theoretical atomic masses values of
IAEA, JAEA, WAW, Moller and DZ which failed to give positive incremental difference in alpha
energies between two sequential even-even and odd-odd mass number A; ∆α (ee,oo) and two
sequential even-odd and odd-even mass number A; ∆α (eo,oe). NMT called these values as
unquantized atomic masses UQAM. NMT calculation proved that the prediction of the nuclear
theoretical models for magic and doubly magic numbers for Z>82 were incorrect due to their reliance
on the UQAM of IAEA and WAW.
NMT not only succeeded in evaluation of the quantized atomic masses but also it found out
several proton and neutron magic numbers in 2012 [55] such 6,14, 16 and 34 which are identical to the
literature [62-64].
444 Bahjat R. J. Muhyedeen
Acknowledgement
A great thanks to Najah Noori Alshams and Hedeer Jawad Alshams for their huge work in collecting
data from international nuclear websites, compiling nuclear data, managing excel and word files to
prepare a hundred of tables, graphs and figures for the element Z-1-228.
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