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Article
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Physics
Nuclear Force Constants Mapped by Yukawa Potential
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
The nuclear meson theory evolved from Yukawa potential regards the observed mesons as its ex-
p erimental test, which is far fetched. It goes without saying that there are no mesons in nuclei such
as deuterons. However, it is the lack of scientific basis to construct the wave equation by using the
quantum mechanical operator to act on the unknown function, so as to piece up the exact solution of
the potential function. The relevant calculation results do not form an inevitable causal relationship
with the existence of mesons. Even if a celebrity patchwork theory is maintained, meson theory should
have at least one observable quantitative inference from the perspective of experimental test theory.
Here, the ground state energy corresponding to the Yukawa potential expressed by the nuclear force
constant and the mass of the PI meson is calculated by the variational method, and then the average
binding energy of the atomic nucleus is connected with the ground state energy corresponding to the
Yukawa potential to calculate the average nuclear force constant. From this result, it is estimated
that the distribution range of the average nuclear force constant of 118 atomic nuclei in the periodic
table of elements is from 1.093992 to 1.413981. The results also show that the estimated value of
nuclear force constant up to 15 is unreliable. However, the nuclear force constant can not be predicted
by other methods. Calculating the average nuclear force constant by using the ground state energy
corresp onding to Yukawa potential is only a dead cycle, and the desire to test meson theory through
exp erimental observation can not be realized. This unexpected result shows that modern physical
theory has long fallen into an unrealistically strange circle of formal logic shrouded in the halo of
mathematics: although the conditions and conclusions of the theory are far from the true description
of natural phenomena, the theory can be deduced infinitely, and is constantly beautified and advocat-
ed as a profound revelation of natural laws. The reason why this paper introduces the average nuclear
force constant mapped by the unreasonable Yukawa potential is to disclose some unique calculation
metho ds that need to be applied in establishing a reasonable nuclear force model in the future.
Keywords: Yukawa potential; Meson theory; nuclear force constant; ground energy.
PACS number(s): 02.30.Xx—Calculus of variations; 03.65.Ta—Foundations of quantum me-
chanics; measurement theory; 13.75.Cs—NucleonCnucleon interactions; 21.30.-x—Nuclear forces;
21.30.Cb— Nuclear forces in vacuum; 42.50.Xa—Optical tests of quantum theory.
1 Introduction
The Yukawa potential
[1, 2]
is regarded as a phenomeno-
logical central potential between two nucleons
[3]
in theo-
retical nuclear physics
[4]
. Since its birth in 1935, Yukawa
potential has been widely used, becoming an impor-
tant model
[5-7]
for establishing the theoretical frame-
work of nuclear force meson
[8-16]
and describing the in-
tuitive physical image
[17-19]
, and deriving many form-
s similar in form but different in essence. For exam-
ples, the Reid potential
[20]
, the Green’s form
[21]
, the for-
m with poles by Roriz-Delfino
[22]
, the inverse Fourier
transform by Garavelli-Oliveira
[23, 24]
, the convergence
form at the coordinate origin by Calvin Stubbins
[25]
,
the effective potential with the coupling constant ”am-
plified:” by Stefano De Leo
[26]
, Yukawa-like potential-
s by Flambaum-Shuryak
[27]
and Cordon-Arriola
[28]
re-
spectively. Donoghue also gave an integrated form
[29]
.
There are some more complex expressions of Yukawa
potential, like the high-quality nucleon-nucleon(NN)
potentials
[30, 31]
and Yukawa-Like tensor interactions
[32]
and so on. Most of these Yukawa-like potentials retain
the features of central symmetry and simple form.
In fact, there has never been a meson in the nucle-
us. However, the interaction between nucleons is de-
scribed as exchange through mesons, which is expressed
by Yukawa potential. The meson theory established
from this is still developing vigorously. This should be
attributed to the material benefits and religiousness of
modern physics. The fanatical belief and the eager pur-
suit of interests have led the theoretical creators to blind-
ly follow the leader’s singing to get more good things
without asking whether what they say conforms to the
natural law. We know that if a potential function is
used to describe the bound system, there must be the
corresponding ground state energy, which is the most im-
portant physical quantity. Is the result consistent with
the nuclear structure? The original Yukawa potential is
Citation: Dongfang, X. D. Nuclear Force Constants Mapped by Yukawa Potential. Mathematics & Nature 1, 202109 (2021).
2 X. D. Dongfang Nuclear Force Constants Mapped by Yukawa Potential
thought to come from a kind of pending particle rela-
tivistic Klein Gordon like equation:
∇
2
Φ =
m
2
c
2
~
2
Φquad (r > 0)
However, this equation is not actually the Klein Gordon
equation. Its derivation and hidden mathematical signif-
icance are intriguing. It can be said that it has enjoyed
the glory of the theory of relativity that made physicist-
s lose their minds at that time. The particle with the
mass of m is identified as π meson, which is defined as
the medium of strong interaction between nucleons.
It is generally believed that when the atomic nucleus is
in a stable state, the nuclear forces b etween protons and
protons, protons and neutrons, and neutrons and neu-
trons are equal: f
pp
= f
pn
= f
nn
[48]
. However, masses
of π
0
meson and π
±
meson are different, and the cor-
responding Yukawa potentials are not exactly the same.
Meson theory does not know how much and how to dis-
tribute pi0 and π
±
mesons in an atomic nucleus, but
there are endless fairy tales of nucleon meson exchange.
A question that readers often think of is that theories
should be tested by experiments. Then, Yukawa poten-
tial and its countless derived potentials are regarded as
the equivalent potentials of strong interaction forces be-
tween nucleons. Which form is consistent with the fact?
Is there any inference in the meson theory consistent
with the experimental observation?
In this paper, the average nuclear force constant σ
is introduced to represent the average interaction in-
tensity between nucleons in the nucleus described by
Yukawa potential, and the average ground state energy
of Yukawa potential mapping is calculated by the varia-
tional method. Then the average nuclear force constants
of 118 nuclei in the periodic table of elements are calcu-
lated. Among them, the average meson force constant
of
2
H nucleus with the lowest average binding energy is
1.093992, the average meson force constant of
62
28
Ni nucle-
us with the highest average binding energy is 1.413981,
and the average meson force constant of
295
Ei nucleus
with the heaviest average binding energy is 1.360918.
This new quantitative inference is the most realistic cal-
culation result of Yukawa potential, and it seems to be
a rare quantitative standard for experimental testing of
meson theory. Then, can these results be verified exper-
imentally to support the nuclear meson theory triggered
by the Yukawa potential?
2 Variational parameters
The energy eigenvalue of quantum systems in bound
states is one of the most important quantities of atomic
and nuclear physics. They are often derived by solving
the wave equation with a boundary condition. Yukawa
potential includes an exponential function, solving pre-
cisely the Yukawa wave equation is difficult. It usual-
ly gives a rough result
[33-36]
. Experimental observations
confirmed the existence of the nuclear energy levels
[37]
.
There is warrantable supposition, if a changed Yukawa
potential deviates from the original form not to o far
away, the approximate calculation should be within the
range of error. It is unique for the most stable structure
of any nucleus. Now that Yukawa potential is used to
describe the nuclear force, the ground energy of Yukawa
field
[12, 38, 39]
must uniquely exist. According to quantum
mechanics, if the trial wave function of the ground state
is appropriate, using the variational method to calculate
the ground energy of a quantum system in the bound
state will give accurate results.
As it is well known to us all that the fine-structure
constant α = e
2
(4πε
0
~c) represents the electromag-
netic interaction strength, and the original Coulomb
potential U
e
= ±e
2
(4πε
0
r) can be expressed as the
form of U
e
= ±α~c/r. Where e is the elementary
charge, ~ = h/2π is the Planck constant, and c is the
speed of light in a vacuum
[40]
. Similarly, according to
the gravitational potential energy U
m
= −Gm
′
m/r(in
which G is the gravitational constant.) for two bodies
of mass m and m
′
respectively, we can define a con-
stant A = Gm
′
m/~c to represent the gravity strength,
and express the Newton gravity potential as the form
U
m
= −A~c/r. As a corollary, we introduce the di-
mensionless physical constant σ to represent the nuclear
force strength between two nucleons, thus the Yukawa
potential can be expressed as the following unified form
Φ
π
= −
σ~c
r
e
−
m
π
c
~
r
(1)
The nuclear force has b oth attractive and repulsive force.
In the above equation, the minus sign expresses the at-
tractive force.
A wave function must satisfy the boundary condi-
tion ψ (r → ∞) = 0. According to the characteristics
of the equation (1), we choose the trial wave function
of the ground state ψ
0
(r) = A exp (−λmcr/~), where
λ > 0 is the variational parameters. Using the normal-
ized conditions
∞
0
ψ
0
(r) ψ
∗
0
(r) r
2
dr
π
0
sin θdθr
2π
0
dφ = 1
to calculate the normalization coefficients gives A =
λ
3
m
3
π
c
3
π~
3
. So the trial wave function takes the for-
m
ψ
0
(r) =
λ
3
m
3
π
c
3
π~
3
e
−
λm
π
cr
~
Hamilton without the rest energy is
H
=
E
−
m
π
c
2
+Φ
π
,
where E −m
π
c
2
≈ p
2
2m
π
, p and E are the momentum
and energy of the π meson respectively. From this it
gives Hamilton with a Yukawa potential,
ˆ
H = −
~
2
2m
π
∇
2
−
σ~c
r
e
−
m
π
c
~
r
In the state described by wave function ψ
0
(r),
the ground energy of the Yukawa field is
¯
H =
∞
0
ψ
∗
0
ˆ
Hψ
0
r
2
dr
π
0
sin θdθ
2π
0
dφ, namely
Mathematics & Nature (2021) Vol. 1 3
¯
H =
∞
0
λ
3
m
3
π
c
3
π~
3
e
−
λm
π
c
~
r
−
~
2
2m
π
∇
2
−
σ~c
r
e
−
m
π
c
~
r
e
−
λm
π
c
~
r
r
2
dr
π
0
sin θdθ
2π
0
dφ
the results of the above integration are as follows
¯
H =
λ
2
2
−
4σλ
3
(2λ + 1)
2
m
π
c
2
(2)
As r → ∞, the potential energy of the system is zero,
and the kinetic energy of the particles constituting the
system is also zero, so the total energy without the rest
energy of the system is zero. Since the nucleons com-
bine to form a nucleus will release energy, the ground
energy of Yukawa field must a negative number. By the
equation (2), one obtains
λ
2
2
−
4σλ
3
(2λ + 1)
2
m
π
c
2
< 0
namely,
σ −
1
2
−
σ (σ − 1) < λ < σ −
1
2
+
σ (σ − 1)
Notice that σ > 0, the quadratic radial
σ (σ − 1) of
physically significant reads σ > 1. However, making use
of the squeeze theorem
[41]
, the condition of the inequality
is
σ > 1
so the nuclear force constant must be greater than 1.
This result agrees with the experiments.
The exact ground wave function requires the varia-
tional parameters λ in the trial ground wave function
making
¯
H the minimum energy. Let d
¯
H
dλ = 0, one
obtains a cubic equation with one unknown,
8λ
3
+ 4 (3 − 2σ) λ
2
+ 6 (1 − 2σ) λ + 1 = 0
Since the ground energy of a nucleus is the only one that
exists, this equation has a unique positive root. From the
solution of cubic equation x
3
+ bx
2
+ sx + d = 0 in one
variable, mark
p = s −
b
2
3
, q = d −
bs
3
+
2b
3
27
the discriminant of the cubic equation with one
unknown
[42]
is
∆ = −108
p
3
27
+
q
2
4
=
1
4
σ
2
9σ
2
+ 20σ − 27
> 0
for σ > 1. So the cubic equation has three real roots,
λ
k
=
−
4
3
p cos
u +
2
3
kπ
−
b
3
where k = 0, 1, 2 and
cos 3u = −
q
2
−
p
3
−
3
2
Therefore, the variational parameters is determined by
the following expressions,
k = 0
k = 2
k = 1
-1
1
2
3
4
5
6
Σ
2
4
6
Λ
8 Λ
3
+ 4 Λ
2
H3 - 2 ΣL + 6 Λ H1 - 2 ΣL + 1 0
Figure 1 Three roots of the variational parameters
k = 2
200000
400000
600000
800000
Σ
-6.´ 10
-6
-4.´ 10
-6
-2.´ 10
-6
2.´ 10
-6
4.´ 10
-6
6.´ 10
-6
Λ
Figure 2 Formal root of the variational parameters with k = 2
λ
k
=
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ (2σ + 3)
3
+
2kπ
3
+
2σ − 3
6
(3)
The above results can also be obtained by solving equation 8λ
3
+ 4 (3 − 2σ) λ
2
+ 6 (1 − 2σ) λ + 1 = 0 with scientific
calculation software such as Mathematica. In order to find the variational parameters of physical meaning, we draw
the images of three solutions of the variational parameters, see the Fig.1, in which σ > 1. k = 1 reading λ < 0 does
4 X. D. Dongfang Nuclear Force Constants Mapped by Yukawa Potential
not agree with the precondition, so it should be discarded. k = 2 reads that the variational parameters λ tends to
zero rapidly with the increase of constant σ and then oscillates in the neighborhood of zero, resulting in that the
total energy tends to zero rapidly then oscillates, neither is it in agreement with facts nor self-consistent, so it should
be discarded too. λ > 0 only holds for k = 0, it determines the unique variational parameters as following,
λ =
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ (2σ + 3)
3
+
2σ − 3
6
(4)
Therefore, the approximate ground radial wave function takes the form
R
0
(r) =
m
3
π
c
3
π~
3
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ (2σ + 3)
3
+
2σ − 3
6
3
·exp
−
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ (2σ + 3)
3
+
2σ − 3
6
m
π
c
~
r
(5)
3 Ground energy distribution
The stable structure of nuclei corresponds to a min-
imum energy state. Using the Yukawa potential to de-
scribe the forces between nucleons, the minimum energy
of the nucleus is just the ground energy. Substituting
equation (4) into the equation (2) gives the relationship
that the ground energy E
0
changes with the nuclear force
constant σ, which is called the nuclear Yuakwa ground
energy distribution function,
E
0
= −
4σ
√
4σ
2
+6σ
3
cos
1
3
arccos
4σ
2
+9σ−27
√
2σ(2σ+3)
3
+
2σ−3
6
3
2
√
4σ
2
+6σ
3
cos
1
3
arccos
4σ
2
+9σ −27
√
2σ (2σ+3)
3
+
2σ −3
6
+ 1
2
−
1
2
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ (2σ + 3)
3
+
2σ − 3
6
2
m
π
c
2
(6)
This energy of the ground state does not contain the
static energy of the π meson, so it is negative. Yukawa
ground state energy (6) is the exact result of calculat-
ing the binding energy of atomic nuclei according to the
first principle
[43]
, and is an inevitable quantitative in-
ference of meson theory. The variational method is an
approximate method but also a reliable metho d for cal-
culating the ground energy. Although choosing different
trial ground wave function will cause the different cal-
culation of the ground energy, there should be no ob-
vious errors. The Yukawa potential describes forces in
short distances within 1.7fm or so, and its accurate ex-
perimental verification is difficult. However, observing
outside the nucleus, the interactions between nucleons
can always be treated as an equivalent of force that has
a center of symmetry. This is the meaning of Yukawa
potential.
Linked to the binding energies of the nucleus, the
Yukawa ground energy should have the experimental ob-
servation effect. It is the basis for experimental tests of
meson theory. Let A be the mass number and B the
binding energy of a nucleus, Yukawa ground state en-
ergy describes the minimum energy of the interaction
between two nucleons, and its absolute value is equiva-
lent to twice the average binding energy of nucleons, so
2B/A = E
∞
−E
0
. Where E
∞
= 0. So the nuclear force
constant σof nucleus with multiple nucleons implies the
average value. It can be calculated by the following e-
quation
2B
A
=
4σ
√
4σ
2
+6σ
3
cos
1
3
arccos
4σ
2
+9σ−27
√
2σ(2σ+3)
3
+
2σ−3
6
3
2
√
4σ
2
+6σ
3
cos
1
3
arccos
4σ
2
+9σ −27
√
2σ (2σ+3)
3
+
2σ −3
6
+ 1
2
−
1
2
√
4σ
2
+ 6σ
3
cos
1
3
arccos
4σ
2
+ 9σ − 27
2σ(2σ + 3)
3
+
2σ − 3
6
m
π
c
2
(7)
Mathematics & Nature (2021) Vol. 1 5
In principle, from the expressions (6) and ( 7), one can
get the average binding energies E
0
B = A |E
0
| (8)
How to obtain the average nuclear force constant
[44]
in
theory may be a puzzle. In the standard literature, the
nuclear force constant is usually expressed as a dimen-
sional constant g
2
4π, and some estimations achieve up
to g
2
4π > 16.6
[44]
. If we use the π
0
meson to calculate,
the binding energy of this nucleus is at least 16453.897
MeV. There are other mesons participating in nuclear
force, and the masses of different mesons are differen-
t. Since any ground energy corresponds to the binding
energy and has a certain value, the calculated average
force constants are not the same. The π meson is the
lightest meson, switching over to other heavier meson-
s in the Yukawa potential to describe the nuclear force
will result in that the corresponding nuclear force con-
stants become small. There is some important difference
between many reported
[30, 44, 45]
nuclear force constants,
and there seems to be no relation to each other. An
accurate nuclear force constant should ensure that the
absolute value of the average ground energy is just the
average binding energy of the nucleus. Now, as one of
the quantitative standards, the Yukawa Nuclear Ground
Energy distribution function can unify the various de-
ductions of the meson theory.
The exact binding energies of almost all nucleuses
have been confirmed by experiments. There is one-
to-one correspondence between the ground energy and
the binding energy. According to the equation (7), one
can work out conversely the average nuclear force con-
stants of all nucleuses by numerical computation. To
make use of the masses m
π
0
c
2
= 134.9766MeV and
m
π
±
= 139.57018MeV
[46]
, we calculated the average
nuclear force constants of the lightest
2
H with aver-
age binding energy 1112.283Mev, and the results are
σ
D
= 1.054487 and 1.052947 respectively. In Table 1,
it is listed for the average nuclear force constants cor-
responding to π
0
and π
±
respectively for the 118 stable
nucleuses in the periodic table of the elements. Its range
of distribution is 1.052947 to 1.264019. In calculation,
the values of the binding energy of the nucleuses are
taken from the 2012 atomic mass table
[47]
. These av-
erage nuclear force constants are obtained one by one
by image solution metho d. The specific calculation is
very time-consuming. Readers can check the accuracy
of the calculation results. However, the accuracy of cal-
culation does not mean that the conclusion is scientific,
because the construction of Yukawa potential function
is not strict scientific logic, and because the existence of
meson and the construction of Yukawa potential are not
necessarily causal, but just a coincidence. So can the ex-
perimental test these nuclear force constants to confirm
the meson theory evolved by Yukawa potential? Let’s
look forward to the answer.
Table 1: Average nuclear force constants of nuclei mapped by Yukawa potential
Z Nuclide B/A (MeV) σ
π
0
σ
π
±
Z Nuclide B/A (MeV) σ
π
0
σ
π
±
1
2
H 1.112283 1.096527 1.093992 59
141
Pr 8.353998 1.400850 1.392488
1
3
H 2.827266 1.194528 1.189953 60
144
Nd 8.326922 1.400032 1.248724
2
4
He 7.073915 1.360720 1.353066 61
145
Pm 8.302656 1.399298 1.390964
3
7
Li 5.606439 1.310294 1.303546 62
150
Sm 8.261617 1.398055 1.389742
4
9
Be 6.462668 1.340364 1.333074 63
152
Eu 8.226577 1.396991 1.388697
5
11
B 6.927716 1.355928 1.348360 64
157
Gd 8.2035 1.396289 1.388088
6
12
C 7.680144 1.380115 1.372118 65
159
Tb 8.188796 1.395842 1.387568
7
14
N 7.475614 1.373654 1.365772 66
162
Dy 8.173449 1.395374 1.387108
8
16
O 7.976206 1.389573 1.381459 67
165
Ho 8.14696 1.394566 1.386315
9
19
F 7.779018 1.383210 1.365772 68
167
Er 8.131743 1.394101 1.385858
10
20
Ne 8.03224 1.391051 1.382868 69
169
Tm 8.114475 1.393573 1.385339
11
23
Na 8.111493 1.385250 1.393482 70
173
Yb 8.087433 1.392745 1.384526
12
24
Mg 8.260709 1.398227 1.389960 71
175
Lu 8.069151 1.392185 1.383975
13
27
Al 8.331548 1.400180 1.391822 72
178
Hf 8.049456 1.391580 1.383381
14
28
Si 8.447744 1.403671 1.395260 73
181
Ta 8.023418 1.390780 1.382595
15
31
P 8.481167 1.404673 1.396245 74
184
W 8.005088 1.390216 1.382041
16
32
S 8.493129 1.405031 1.396597 75
186
Re 7.981288 1.389483 1.381321
17
36
Cl 8.521932 1.405893 1.397443 76
190
Os 7.962112 1.388891 1.380740
18
40
Ar 8.595259 1.408080 1.399592 77
192
Ir 7.93901 1.388178 1.380039
19
39
K 8.557025 1.406941 1.398473 78
195
Pt 7.926565 1.387793 1.379661
20
40
Ca 8.551304 1.406770 1.398305 79
197
Au 7.91566 1.387456 1.379330
21
45
Sc 8.618915 1.408784 1.400284 80
201
Hg 7.897561 1.386895 1.378779
22
48
Ti 8.722986 1.411870 1.403315 81
204
Tl 7.880021 1.386352 1.378245
23
51
V 8.742096 1.412434 1.403870 82
207
Pb 7.869864 1.386036 1.377935
6 X. D. Dongfang Nuclear Force Constants Mapped by Yukawa Potential
Z Nuclide B/A (MeV) σ
π
0
σ
π
±
Z Nuclide B/A (MeV) σ
π
0
σ
π
±
24
52
Cr 8.775967 1.414434 1.404852 83
209
Bi 7.847984 1.385357 1.377268
25
55
Mn 8.765009 1.413110 1.404534 84
210
Po 7.834344 1.384933 1.376851
26
56
Fe 8.790342 1.413857 1.405268 85
210
At 7.811661 1.384227 1.376158
27
59
Co 8.768025 1.413200 1.404622 86
222
Rn 7.694489 1.380565 1.372560
28
59
Ni 8.736578 1.412271 1.403710 87
223
Fr 7.683657 1.380225 1.372227
28
62
Ni 8.794546 1.413981 1.405390 88
226
Ra 7.661954 1.379544 1.371557
29
64
Cu 8.739068 1.412345 1.403782 89
227
Ac 7.650701 1.379190 1.371210
30
66
Zn 8.75963 1.412952 1.404378 90
232
Th 7.615024 1.378067 1.370106
31
70
Ga 8.70928 1.411464 1.402917 91
231
Pa 7.618419 1.378174 1.370211
32
73
Ge 8.705049 1.411339 1.402794 92
238
U 7.57012 1.376650 1.368714
33
75
As 8.700874 1.411216 1.402672 93
237
Np 7.574981 1.376803 1.368865
34
79
Se 8.695591 1.411059 1.402519 94
239
Pu 7.56031 1.368409 1.376340
35
80
Br 8.677653 1.410528 1.402519 95
243
Am 7.530168 1.375386 1.367472
36
84
Kr 8.717446 1.411706 1.403154 96
247
Cm 7.501926 1.374490 1.366593
37
86
Rb 8.696901 1.411098 1.402557 97
247
Bk 7.498935 1.374395 1.366499
38
88
Sr 8.732592 1.412154 1.403594 98
251
Cf 7.470495 1.373491 1.365612
39
89
Y 8.713987 1.411604 1.403054 99
252
Es 7.45724 1.373070 1.365197
40
91
Zr 8.69332 1.410992 1.402453 100
257
Fm 7.422191 1.371953 1.364100
41
93
Nb 8.664186 1.410129 1.402453 101
258
Md 7.409668 1.371553 1.363708
42
96
Mo 8.653974 1.409826 1.401307 102
259
No 7.400 1.371245 1.363405
43
99
Tc 8.613599 1.400128 1.408626 103
262
Lr 7.374 1.370414 1.362588
44
102
Ru 8.607392 1.408442 1.399947 104
261
Rf 7.371372 1.370329 1.362506
45
103
Rh 8.584157 1.407750 1.399267 105
262
Db 7.352 1.369709 1.361896
46
107
Pd 8.560893 1.407056 1.398586 106
266
Sg 7.332 1.369068 1.361267
47
108
Ag 8.542025 1.406493 1.398033 107
264
Bh 7.315 1.368522 1.360731
48
112
Cd 8.544738 1.406574 1.398112 108
277
Hs 7.255 1.366592 1.358834
49
115
In 8.516546 1.405732 1.397285 109
268
Mt 7.271 1.367107 1.359341
50
119
Sn 8.499449 1.405221 1.396782 110
281
Ds 7.220 1.365462 1.357725
51
122
Sb 8.468317 1.404288 1.395866 111
272
Rg 7.227 1.365688 1.357967
52
128
Te 8.448752 1.403701 1.395290 112
285
Cn 7.185 1.364330 1.356613
53
127
I 8.445487 1.403603 1.395193 113
284
Ed 7.174 1.363974 1.356263
54
131
Xe 8.423736 1.402950 1.394551 114
289
Fl 7.149 1.363163 1.355466
55
133
Cs 8.409978 1.402536 1.394145 115
288
Ef 7.136 1.362741 1.355052
56
137
Ba 8.391828 1.401990 1.393608 116
292
Lv 7.117 1.362123 1.354445
57
139
La 8.378043 1.401575 1.393200 117
283
Eh 7.097 1.361472 1.353805
58
140
Ce 8.376339 1.401523 1.393150 118
294
Ei 7.080 1.360918 1.353261
4 Conclusions and conments
As we all know, there is no meson in the nucleus, so
how can the nuclear force maintain the stable structure
of the nucleus through the meson? On the other hand,
to establish the potential energy function of nucleon in-
teraction through the solution of a differential equation,
it is necessary to demonstrate the scientific rationality of
this differential equation. The purpose of my calculation
of Yukawa’s ground state energy here does not mean to
support Yukawa’s theory, but only to supplement rele-
vant calculation methods and explain the basic content
of this theory so as to facilitate testing. Whether we es-
tablish or develop the meson theory of nuclear force, we
should always consider that since the Yukawa potential
is adopted, there is a ground state energy corresponding
to this theory, and the ground state energy calculated
by other methods must have the same accuracy as the
result of the variational metho d (6).
According to the ground state energy and the binding
energy of the atomic nucleus, the average interaction in-
tensity of two adjacent nucleons in 118 stable elements
is estimated, which is only the result corresponding to
the meson theory. The gap between the results and the
experiment shows the limitations of the meson theory
For the derived Yukawa like potential, the corresponding
ground state energy can be derived theoretically Howev-
er, the average ground state energy derived from differ-
ent nuclear force potential energy functions is different
Since the average binding energy of atomic nucleus is u-
nique, the results of calculating the average nuclear force
constant are different by changing the form of Yukawa
potential energy function An accurate average nuclear
force constant must ensure that the ground state energy
is consistent with the average binding energy of atoms
The ground state energy of Yukawa is a standard for the
Mathematics & Nature (2021) Vol. 1 7
unification of meson theory, and should be the object of
experimental testing of Yukawa meson theory.
However, the average nuclear force constant of various
nuclei can not be measured by other methods, or there
is no other reasonable definition of the average nuclear
force constant. The experiment cannot verify whether
the ground state energy corresp onding to Yukawa p o-
tential is true. From a theoretical point of view, Yukawa
constructed a wave equation by applying an operator of
quantum mechanics to an unknown function, and de-
fined the exact solution of such a wave equation as a
potential function. This operation is extremely lacking
in scientific basis. The nuclear force constant is defined
by the Yukawa potential, and the average nuclear force
constant calculated by the ground state energy of the
Yukawa potential belongs to a dead cycle. It should be
pointed out that the construction of Yukawa potential
belongs to far fetched mathematical calculation, and its
infinite deduction and development just shows the spiri-
tual essence of formal modern physics under the cover of
gorgeous mathematics. The factual basis that best illus-
trates this view is that there are no so-called mesons in
the light nucleus, otherwise the mass of the atomic nu-
cleus is greater than known, or the mass of the nucleon
is less than known.
There are no mesons in the nucleus. It can be seen
that the so-called meson theory of nuclear force is just
imagination and patchwork. Mesons have been vivid-
ly described as the medium of interaction between nu-
cleons for nearly 100 years. Yukawa’s meson theory of
nuclear force is a microcosm of modern physical theo-
ry. The mass production method of modern physical
theory, which has almost become witchcraft, has even
been applied to the production of famous experimental
reports
[49-51]
. It takes a lot of energy and time to expose
such theories and experimental reports. However, for
the reasonable nuclear force model to be established in
the future, the relevant calculation methods introduced
in this paper are applicable, which is another reason why
I write this article carefully.
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Latest findings
The End of Yukawa Meson Theory of Nuclear Forces
Yukawa meson theory of nuclear force originates from the fiction of Yukawa potential function. Its logic is that there is a mass factor in the fictitious potential function. After the mass factor is artificially given the meaning of meson mass, it imagines that the Nucleon in the atomic nucleus depends on the meson to generate exchange force, so the meson theory of nuclear force is established. However, there has never been a meson in the nucleus. It is obviously unreasonable to say that the interaction between nucleons is dominated by other substances that are not in the nucleus. Here, I have reviewed in detail Yukawa's thesis on the creation of the meson theory of nuclear forces, and solemnly pointed out that the fiction of Yukawa potential function seriously distorts the basic mathematical rules and cannot be modified. There is no scientific logic or experimental basis to designate the mass in Yukawa potential function as the meson mass. Specifically, Yukawa's meson theory of nuclear forces introduces a delta function without causality into the operator equation of relativistic momentum energy relationship, thus very covertly deleting the energy term, which essentially belongs to the operator equation corresponding to tearing and dismembering the relativistic momentum energy relationship, so it is completely illogical; A mutilated equation that meets the expectation but has no physical meaning is given for the tearing and dismembering of the relativistic momentum energy operator equation, and then Yukawa introduces the irrelevant Compton wavelength to knead, explains that the mass in the exponential factor of the exact solution of the mutilated equation is the mass of the meson that does not exist in the nucleus, and claims that the meson is the medium of nucleon interaction, thus creating a nuclear force meson theory, which is not a scientific theory; the steady-state solution of the operator equation of the relationship between momentum and energy of free particles is Bessel function, but the meson theory of nuclear force deletes the energy operator in the operator equation of the relationship between momentum and energy and then solves the mutilated equation, which is equivalent to strongly twisting the Bessel function into an exponential function that meets the expectation, and this subversive distortion of mathematics does not constitute a new theory; the calculation results for the binding energy of atomic nuclei show that nucleons cannot absorb other particles such as mesons when they combine to form atomic nuclei, and it is just a fairy tale to insist that mesons are the medium of interaction between nucleons. All these truths mean that Yukawa's meson theory of nuclear forces based on the fictitious potential function is a wrong theory and must be completely terminated.