the quantized atomic masses of the elements: part-6; z=100 ... · atomic masses more precisely than...

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.


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.


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


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


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 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.


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


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.


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.


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


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.


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


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


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


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


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.


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.


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


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


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


Comparison of the NMT QAM’E

Comparison of the NMT- UQAM and QAM with other models



Comparison of the NMT QAM’Eα with other models, Z=119

UQAM and QAM with other models



α with other models, Z=119





with other models, Z=119



Mass Difference among DZ, Moller and NMT in relation to WAW, Z=100-107

with other models, Z=119



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


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




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




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


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





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




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


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].


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.

References [1] B. R. J. Muhyedeen, Euro. J. Sci. Res.144, 4 (2017) 1-23.

[2] B. R. J. Muhyedeen, Euro. J. Sci. Res.145, 2 (2017) 1-32.




[6] S.G. Nilsson, C.F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I.L. Lamm,

B. Nilsson, Nucl. Phys. A 131 (1969) 1.

[7] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, C.Y. Wong, Rev. Mod. Phys.

44 (1972) 320.

[8] M. Bolsterli, E.O. Fiset, J.R. Nix, J.L. Norton, Phys. Rev. C 5 (1972) 1050.

[9] A. Sobiczewski, A. Gyurkovich, M. Brack, Nucl. Phys. A 289 (1977) 346.

[10] Von Weizsäcker, C.F.: Zur Theorie der Kernmassen. Z. Phys. 96, (1935) 431–458.

[11] H. A. Bethe and R. F. Bacher, Rev. Mod. Phys. 8, (1936) 82-229.

[12] G. Gamow, Pro. Roy. Soc. A 136 (1929) 386-387.

[13] N. Bohr, Nature 137, (1936) 344-348.

[14] L. Spanier, S.A.E. Johansson, Atomic Data and Nucl. Data Tables, (1988) 259-264.

[15] I. Angeli, J. Phys. G: Nucl. and Particle Phys., 17, 4 (1991).

[16] D. Vautherin, D. M. Brink, Phys. Rev. C 5 (1972) 626.

[17] J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 422 (1983) 103.

[18] J. F. Berger, M. Girod, D. Gogny, Nucl. Phys. A 428 (1984) 23c.

[19] J.P. Delaroche. M. Girod, H. Goutte, J. Libert, Nucl. Phys. A 771 (2006) 103.

[20] P.-G. Reinhard, M. Rufa, J. Maruhn, W. Greiner, J. Friedrich, Z. Phys. A 323 (1986) 13.

[21] P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193.

[22] M. Bender, P.-H. Heenen, P.-G. Reihard, Rev. Mod. Phys. 75 (2003) 121.

[23] M.S. Livingston and H.A. Bethe, Rev. Mod. Phys. 9 (1937) 245, XVIII.

[24] G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A 729, (2003)1 EX1-EX2, 3-676.

[25] M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer,

Chinese Physics C 36 (12), (2012) 1603–2014.

[26] M. Wang, G. Audi, F.G. Kondev, W.J. Huang, S. Naimi and Xing Xu, Chinese Physics C 41, 3

(2017) 030003.

[27] Yu. A. Litvinov et al., Nucl. Phys. A 756 (2005) 3-38.

[28] K. Blaum et al., Nucl. Phys. A 752 (2005) 317c.

[29] D. Lunney, J.M. Pearson and C. Thibault, Rev. Mod. Phys.75 (2003) 1021.

[30] P. Möller and J. R. Nix. Nucl. Phy. A, 361, 1, (1981) 117–146.

[31] P. Möller and J. R. Nix et al, Atomic Data Nucl. Data Tables. 59, (1995) 185.

[32] P. Möller, A. J. Sierka, T. Ichikawab, H. Sagawa. Atomic Data Nucl. Data Tables, 109–110,

(2016), 1–204; arXiv:1508.06294v1 [nucl-th] 25 Aug 2015.

[33] S. Goriely, N. Chamel, J. M. Pearson, Phys. Rev. C 82 (2010) 035804. (HFB-19 - HBF-21)

[34] S. Goriely, S. Hilaire, M. Girod, and S. Peru, Phys. Rev. Lett. 102 (2009) 242501. (Gogny

HFB).

[35] S. Goriely, N. Chamel, J. M. Pearson, Phys. Rev. Lett. 102 (2009) 152503. (Skyrme HFB).

[36] W. D. Myers. W. J. Swiatecki, Nuclear Physics, 81, 1, 1966, 1-60

[37] R. C. Nayak, L. Satpathy, At. Data Nucl. Data Tables, 98, 4, 2012, 616-719


[38] L. Spanier and S. A. E. Johansson, At. Data Nucl. Data Tables, 39, 2, 1988, 259-264

[39] J. Jänecke. P. J. Masson, At. Data Nucl. Data Tables, 39, 2, 1988, 265-271

[40] J. Duflo, A.P. Zuker, Phys. Rev. C 5 2 (1995) R23.

[41] H. Koura, T. Tachibana, M. Uno and M. Yamada, Nucl. Phys. A 674 (2000) 47

[42] H. Koura, T. Tachibana, M. Uno and M. Yamada, Progr. Theor. Phys. 113 (2005) 305

[43] I. Muntian, Z. Patyk, and A. Sobiczewski, Phys. Atomic Nuclei 66, (2003) 1015.

[44] I. Muntian, S. Hofmann, Z. Patyk, and A. Sobiczewski, Acta Phys. Pol. B 34, (2003) 2073.

[45] I. Muntian, Z. Patyk, and A. Sobiczewski, Acta Phys. Pol. B 32, (2001) 691.

[46] D. Vautherin, D. M. Brink, Phys. Rev. C 5 (1972) 626.

[47] J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 422 (1983) 103.

[48] J. F. Berger, M. Girod, D. Gogny, Nucl. Phys. A 428 (1984) 23c.

[49] J.P. Delaroche. M. Girod, H. Goutte, J. Libert, Nucl. Phys. A 771 (2006) 103.

[50] P.-G. Reinhard, M. Rufa, J. Maruhn, W. Greiner, J. Friedrich, Z. Phys. A 323 (1986) 13.

[51] P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193.

[52] M. Bender, P.-H. Heenen, P.-G. Reihard, Rev. Mod. Phys. 75 (2003) 121.

[53] B. R. J. Muhyedeen, Euro. J. Sci. Res. 22, 4 (2008) 584-601.

[54] B. R. J. Muhyedeen, Euro. J. Sci. Res. 26, 2 (2009) 161-175.


[56] A. Parkhomenko and A. Sobiczewski, Acta Phys. Polonica B 36, (2005) 3095.

[57] G. Royer, J Phys G: Nucl. Part. Phys. 26 (2000) 1149-1170.

[58] V.E. Viola Jr and G.T. Seaborg, J. Inorg. Nucl. Chem. 28, (1966) 741.

[59] Y. Lim and Y. Oh, arXiv:1701.05491v2 [nucl-th] 24 Feb 2017

[60] R. Smolańczuk, J. Skalski, and A. Sobiczewski. Phys. Rev. C 52, (1995) 1871.

[61] A. A. Rather, M. Ikram, and S. K. Patra, Proceedings of the DAE Symp. On Nucl. Phys. 62

(2107) 170.

[62] O. Haxel, J. Hans, D. Jensen, and H. E. Suess, H. E., Phys. Rev. 75, (1949) 1766.

[63] A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. let. 84, (2000)

5493.

[64] D. Steppenbeck, et al., Nature, 502, (2013) 207–210., doi:10.1038/nature12522, 109–110,

2016, Pages 1–204;

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