We proposed an empirical equation for a fine-structure constant:
The symbol list is shown in Section 2. We discovered Equation 1 [
G m p 2 h c = 4.5 2 × k T c 1 kg × c 2 (1)
G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π × m e e × h c × ( 1 C J ⋅ m × 1 1 kg ) (2)
m e c 2 e × ( e 2 4 π ε 0 ) = π k T c × ( 1 J ⋅ m C ) (3)
These equations have small errors of approximately 10−3 and 10−4 [
Then, 4.48870 and 3.13189 Ω are connected as follows.
m p m e = 1 4.4887 × 3.13189 Ω q m e (4)
Next, we discovered the empirical equation for a fine-structure constant [
137.0359991 = 136.0113077 + 1 3 × 13.5 + 1 (5)
13.5 × 136.0113077 = 1836.152654 = m p m e (6)
To explain 136.0113077, we proposed the following values.
3.131777037 ( Ω ) = R k ( 3 × 136.0113077 ) 1.5 (7)
4.488855463 = 136.0113077 × 4 27 (8)
However, Equations (7) and (8) cannot be compatible with Equations (5) and (6). Main purpose of this report is to improve the compatibility between these equations.
The remainder of the paper is organized as follows. In Section 2, we show the symbol list. In Section 3, we reconsider the deviation from the factors of 9/2 and π. In Section 4, using the electrochemical method, we propose the equivalent circuit for the fine structure constant. In Section 5, we refine our three equations. In Section 6, general discussions are presented, which are mainly about the UNIT.
G gravitational constant: 6.6743 × 10−11 (m3∙kg−1∙s−2)
(we used the compensated value 6.673778 × 10−11 in this report)
Tc temperature of the cosmic microwave background: 2.72548 (K)
(we used the compensated value 2.726312 K in this report)
k Boltzmann constant: 1.380649 × 10−23 (J K−1)
c speed of light: 299,792,458 (m/s)
h Planck constant: 6.62607015 × 10−34 (Js)
ε0 electric constant: 8.8541878128 × 10−12 (N∙m2∙C−2)
μ0 magnetic constant: 1.25663706212 × 10−6 (N∙A−2)
e electric charge of one electron: −1.602176634 × 10−19 (C)
qm magnetic charge of one magnetic monopole: 4.13566770 × 10−15 (Wb)
(this value is only a theoretical value, qm = h/e)
mp rest mass of a proton: 1.6726219059 × 10−27 (kg)
(we used the compensated value 1.672621923 × 10−27 kg in this report)
me rest mass of an electron: 9.1093837 × 10−31 (kg)
Rk von Klitzing constant: 25812.80745 (Ω)
Z0 wave impedance in free space: 376.730313668 (Ω)
α fine-structure constant: 1/137.035999081
In this section, we reconsider the deviation from 4.5 and π. We notice that 4.48870 and 3.1319 can be rewritten as follows.
4.48870 = q m c ( m p m e + 1 + 1 3.854377987 ) m p c 2 (9)
For example, Equation (9) can be made sure as follows.
4.48870 = 4.13567 × 10 − 15 × 299792458 ( 1836.15 + 1 + 1 3.854377987 ) × 1.50328 × 10 − 10 (10)
3.13189 Ω = ( m p m e + 1 + 1 3.854377987 ) m e c 2 e c (11)
Next, the deviation from 4.5 and π can be explained as follows.
4.5 π × 3.13189 4.48870 = 0.999421207 ≐ 1 (12)
Regarding the values for 3.131777037 and 4.88855463,
4.488855463 = q m c ( m p m e + 1 + 1 5.106991198 ) m p c 2 (13)
3.1317770 Ω = ( m p m e + 1 + 1 5.106991198 ) × m e c 2 e c (14)
Next, the deviation from 4.5 and π can be explained as follows.
4.5 π × 3.1317770 4.488855463 = 0.999350548 ≐ 1 (15)
Therefore, using X, the deviation should be rewritten as follows.
3.1317770 = ( m p m e + 1 + 1 X ) × m e c 2 e c (16)
4.488855463 = q m c ( m p m e + 1 + 1 X ) m p c 2 (17)
The value of 4.5 is from the degree of freedom as 9/2. Therefore, 4.488855 should be dimensionless.
( m p m e + 1 + 1 X ) = q m c 4.488855463 × m p c 2 = Wb ⋅ m / s J = Wb ⋅ m J ⋅ s = m C (18)
The correct value of X and the UNIT will be discussed in detail in a later section.
For convenience, Equations (5) and (6) are rewritten as follows:
137.0359991 = 136.0113077 + 1 + 1 3 × 13.5 (19)
13.5 × 136.0113077 = 1836.152654 = m p m e (20)
We strongly believe that the fine structure constant should be explained by the transference number [
J 1 = L 11 g r a d η 1 T + L 12 g r a d η 2 T (21)
J 2 = L 21 g r a d η 1 T + L 22 g r a d η 2 T (22)
whereJ1 and J2 are the current densities of two different carriers; η1and η2are the electrochemical potentials of the two different carriers; L11, L12, L21, and L22 are Onsagar coefficients.
In the area of solid-state ionics, Rickert proposed the following equation.
L 12 = L 21 = 0 (23)
Then, the transference number (t1 and t2) can be explained as follows.
t 1 = R 2 R 1 + R 2 (24)
t 2 = R 1 R 1 + R 2 (25)
where R1 and R2 are different resistance values. Next, we consider the following equation.
( V 1 V 2 ) = Z 1 × ( 1 − 1 81 − 1 81 136.0113077 ) ( I 1 I 2 ) (26)
where V1 and V2 are the voltage losses due to different carriers; I1 and I2 are the currents due to different carriers; Z1 is the resistance, which will be explained later. In Equation (26), using an inverse of the matrix, Onsager coefficients can be obtained. Because the derivation is too complex to show here, we have:
V 1 = Z 1 × I 1 − Z 1 81 × I 2 (27)
V 2 = − Z 1 81 × I 1 + Z 1 × 136.0113077 × I 2 (28)
Next, we consider the following situation, which implies the open circuit condition.
I 1 = − I 2 (29)
Therefore, the theoretical voltage (Vth) is,
V t h = V 1 − V 2 = Z 1 × ( 136.0113077 + 1 + 1 3 × 13.5 ) × I 1 (30)
In Equation (30), −V2 is the voltage loss due to the opposite drift (not diffusion) current (I2). Thus, in Equation (30), we obtained the value of a fine-structure constant. For convenience, Equation (5) is rewritten as follows.
137.0359991 = 136.0113077 + 1 + 1 3 × 13.5 (31)
Next, we define the interaction voltage (V3) as follows.
V 3 = Z 1 81 × I 1 (32)
The transference number for the small voltage loss (V1) is
t 1 = V 3 − V 2 V 1 − V 2 = Z 1 × ( 136.0113 + 1 3 × 13.5 ) × I 1 Z 1 × ( 136.0113 + 1 + 1 3 × 13.5 ) × I 1 = 136.0359991 137.0359991 (33)
Therefore, the interaction coefficient is
1 − t 1 = 1 − 136.0359991 137.0359991 = 1 137.0359991 = 1 α (34)
The transference number for the large voltage loss (V2) is
t 2 = V 1 − V 3 V 1 − V 2 = Z 1 × I 1 Z 1 × ( 136.0113 + 1 + 1 3 × 13.5 ) × I 1 = 1 137.0359991 (35)
Equation 35 means the strong interaction. From Equations 33 and 35, V3 should be transferred from the carriers with a large voltage loss (V2) to those with the small voltage loss (V1). From Equations (33) and (35),
t 1 + t 2 = 136.0359991 137.0359991 + 1 137.0359991 = 1 (36)
The total resistance is Z0, so the large resistance (Z2) should be
Z 2 = Z 0 × 137.035999081 = 2 R k (37)
The small resistance is
Z 1 = Z 0 × 137.035999081 136.0113077 = 2 R k 136.0113077 = 379.5685505 Ω (38)
We discover Z1 as follows:
Z 1 = 3 3 × q m e × m e m p = 27 × 25812.807459 1836.152654 = 379.5685505 Ω (39)
Therefore, our argument is not a coincidence. From Equation (32), the interaction resistance (Z3) should be
Z 3 = Z 1 81 = 1 3 × q m e × m e m p = 4.686031487 Ω (40)
Consequently, Equation (26) can be rewritten as follows,
( V 1 V 2 ) = ( Z 1 − Z 3 − Z 3 Z 2 ) ( I 1 I 2 ) (26b)
We are discussing the equivalent circuit at the quantum level. Clearly, one charge is an electron. However, it is difficult to search for the other charge. The suitable charge is
q m Z 1 = 1.08957070 × 10 − 17 ( C ) = 68.00565 × e (41)
Because qm has never been observed, the charge + q m Z 1 should be the set of an antiparticle − q m Z 1 , which may be related to quarks. Then, we propose the equivalent circuit in
2 q m Z 1 = 136.0113 × e (42)
When the charge + q m Z 1 cannot be realized at the low energy level, it cannot be observed as the mass.
The direction of + q m Z 1 is opposite to the electrical field, which may prevent the increase of the electrical field.
For convenience, Equations (5) and (6) are rewritten as follows:
137.0359991 = 136.0113077 + 1 + 1 3 × 13.5 (43)
13.5 × 136.0113077 = 1836.152654 = m p m e (44)
For convenience, Equations (16) and (17) are rewritten as follows:
3.1317770 Ω = ( m p m e + 1 + 1 X ) × m e c 2 e c (45)
4.488855463 = q m c ( m p m e + 1 + 1 X ) m p c 2 (46)
Then, we notice that X should be 3. Therefore,
3.132011447 Ω = ( m p m e + 1 + 1 3 ) m e c 2 e c = ( m p m e + 4 3 ) m e c 2 e c = 1837.48599 × m e c 2 e c (47)
4.488519503 = q m c ( m p m e + 1 + 1 3 ) m p c 2 = q m c ( m p m e + 4 3 ) m p c 2 = q m c 1837.48599 × m p c 2 (48)
Here, 4/3mec2 is well known and has been discussed by Feynman. Next, the deviation from 4.5 and π can be explained as follows.
4.5 π × 3.132011447 4.488519503 = 0.999500154 ≐ 1 (49)
1 m p m e + 1 + 1 3 = 1 1837.485988 appears to be a transference number. The total resistance (Z5), small resistance (Z6), large resistance (Z7) and interaction resistance (Z8) can be defined.
Z 5 = 2 × q m e × 1 m p m e + 4 3 = 28.095787 Ω (50)
Z 6 = 2 × q m e × m e m p = 2 R k 1836.152654 = 28.11618892 Ω (51)
Z 7 = 2 × q m e = 2 R k (52)
Z 8 = 13.5 81 × Z 6 = 13.5 81 × 2 × q m e × m e m p = 1 3 × q m e × m e m p = 4 .686031487 Ω = Z 3 (53)
Consequently, Z8 is equal to Z3.
We use 2.726312143 K instead of 2.72548 K. We use 6.6737778665 × 10−11 m3∙kg−1∙s−2 instead of 6.6743 × 10−11 m3∙kg−1∙s−2. Equation (1) is refined as follows:
G m p 2 h c = 4.488519503 2 k T c 1 kg × c 2 (54)
G m p 2 h c = 6.6737778665 × 10 − 11 × ( 1.6726219 × 10 − 27 ) 2 6 .626070 × 10 − 34 × 2.9979246 × 10 8 = 9.39919318 × 10 − 40 (55)
4.488519503 2 k T c 1 kg × c 2 = 4.488519503 2 × 1.3806490 × 10 − 23 × 2.726312143 ( 2.9979246 × 10 8 ) 2 = 9.39919318 × 10 − 40 (56)
Equation (55) is equal to Equation (56). Therefore, the compensation method is perfect. Next, Equation (2) is refined as follows:
G m p 2 ( e 2 4 π ε 0 ) = 4.488519503 2 × 3.132011447 Ω × m e e × h c × ( 1 C ⋅ Ω J ⋅ m × 1 1 kg ) (57)
G m p 2 e 2 4 π ε 0 = 6.6737778665 × 10 − 11 × ( 1.672621923 × 10 − 27 ) 2 ( 1.60217663 × 10 − 19 ) 2 4 π × 8.8541878 × 10 − 12 = 8.0929175 × 10 − 37 (58)
4.488519503 2 × 3.132011447 × m e e × h c = 4.488519503 × 9.10938 × 10 − 31 × 1.986446 × 10 − 25 2 × 3.132011447 × 1.602177 × 10 − 19 = 8.0929175 × 10 − 37 (59)
Equation (58) is equal to Equation (59). Therefore, the compensation method is perfect. Next, Equation (3) is refined as follows:
m e c 2 e × ( e 2 4 π ε 0 ) = 3.132011447 Ω × k T c × ( 1 J ⋅ m C ⋅ Ω ) (60)
m e c 2 e × ( e 2 4 π ε 0 ) = 9.10938 × 10 − 31 × ( 2.9979246 × 10 8 ) 2 × ( 1.60217663 × 10 − 19 ) 2 1.60217663 × 10 − 19 × 4 π × 8.8541878 × 10 − 12 = 1.1789142 × 10 − 22 (61)
3.132011447 × k T c = 3.132011447 × 1.3806490 × 10 − 23 × 2.726312143 = 1.1789142 × 10 − 22 (62)
Equation (61) is equal to Equation (62). Therefore, the compensation method is perfect.
For convenience, Equation (54) is rewritten as follows:
G m p 2 h c = 4.488519503 2 k T c 1 kg × c 2 (63)
The value of 4.488519503 is the deviation from the degree of freedom 9/2, which is dimensionless.
Therefore, there are no dimension mismatch problems. For convenience, Equation (56) is rewritten as follows:
m e c 2 e × ( e 2 4 π ε 0 ) = 3.132011447 Ω × k T c × ( J ⋅ m C ⋅ Ω = A m ) (64)
In Equation (64), there remains the unexplained UNIT as “Am”.
m e c 2 4 π ε 0 c = 3.132011447 Ω × k T c × A m e c (65)
Therefore,
Z 0 × m e c 2 3.132011447 Ω × 4 π = k T c × A m e c (66)
In Equation (66), the UNITs of 1 J and 1 C can be separately defined. However,
1 J = 6.241509 × 10 18 eV = 6.241509 × 10 18 e c × V c (67)
where V c is the unit of the electromagnetic four potential. Therefore, A m e c may be Faraday constant at the quantum level. The proof is difficult and will be published in a future report. From Equations (47) and (65), we have
k T c e 2 c 4 π ε 0 × ( A m ) = 1 ( m p m e + 4 3 ) ( C m ) = 1 1837.485988 ( C m ) (68)
Next, the fine structure constant (α) is
e 2 4 π ε 0 = h c × α 2 π (69)
From Equations (68) and (69),
k T c h c 2 = 1 ( m p m e + 4 3 ) × α 2 π × ( 1 A m ) = 1 1837.485988 × α 2 π × ( s m 2 ) = 6.3206454 E − 07 (70)
In Equations (68) and (70), from different coordinate systems, kTc should be changed because the unit C/m is not Lorentz invariant. Therefore, from Equation (63), G is not Lorentz invariant.
According to the advanced Wagner model, the diffusion time for mixed electronic and ionic currents should exponentially decrease with distance [
We have never discussed the spin of electrons. In Equation (64), the number 3.132011447 Ω is the deviation from π. We believe that π is related to the spin of the electron. Angrick et al. have confirmed that the spin of electrons cannot be thermodynamically ignored [
We proposed an empirical equation for a fine-structure constant:
137.0359991 = 136.0113077 + 1 + 1 3 × 13.5 . We proposed several empirical equations to describe the relationship between an electromagnetic force and Tc.
Three equations were explained by the factors of 9/2 and π. We attempted to improve the accuracies by changing the values of 9/2 and π. For this purpose, using the electrochemical method, we proposed the equivalent circuit. Then, we proposed the following two values,
3.132011447 Ω = ( m p m e + 1 + 1 3 ) m e c 2 e c , 4.488519503 = q m c ( m p m e + 1 + 1 3 ) m p c 2
The calculated values of Tc and G are 2.726312 K and 6.673778 × 10−11 (m3∙kg−1∙s−2). ( m p m e + 1 + 1 3 ) appears to be related to the transference number. However, there should be the UNIT (m/C) in ( m p m e + 1 + 1 3 ) . Therefore, the values of Tc and G should not be Lorentz invariant.
The author declares no conflicts of interest regarding the publication of this paper.
Miyashita, T. (2023) A Fine-Structure Constant Can Be Explained Using the Electrochemical Method. Journal of Modern Physics, 14, 160-170. https://doi.org/10.4236/jmp.2023.142011