Mathematical electron model and the SI unit 2017 Special Adjustment Malcolm J. Macleod maclem@platoscode.com Following the 26th General Conference on Weights and Measures are fixed the numerical values of the 4 physical constants (h, c, e, kB). This is premised on the independence of these constants. This article discusses a model of a mathematical electron from which can be defined the Planck units as geometrical objects (mass M=1, time T=2π ...). In this model these objects are interrelated via this electron geometry such that once we have assigned values to 2 Planck units then we have fixed the values for all Planck units. As all constants can then be defined using geometrical forms (in terms of 2 fixed mathematical constants, 2 unit-specific scalars and a defined relationship between the units kg,m, s, A), the least precise CODATA 2014 constants (G, h, e,me, kB...) can then be solved via the most precise (c, μ0, α,R∞), with numerical precision limited by the precision of the fine structure constant α. In terms of this model we now for example have 2 separate values for elementary charge, calculated from (c, α,R∞) and the 2017 revision. Table 1 Calculated using (c∗, μ∗0, α ∗,R∗∞) CODATA 2014 values Fine structure constant α∗ = 137.035999139 α = 137.035999139(31) Speed of light c∗ = 299792458 u17 c = 299792458 Permeability μ0∗ = 4π/107 u56 μ0 = 4π/107 Rydberg constant R∞∗ = 10973731.568 508 u13 R∞ = 10973731.568 508(65) Planck constant h∗ = 6.626 069 134 e-34 u19 h = 6.626 070 040(81) e-34 Elementary charge e∗ = 1.602 176 511 30 e-19 u−27 e = 1.602 176 6208(98) e-19 von Klitzing (h/e2) R∗K = 25812.807 45559 u 73 RK = 25812.807 4555(59) Electron mass m∗e = 9.109 382 312 56 e-31 u 15 me = 9.109 383 56(11) e-31 Electron wavelength λ∗e = 2.426 310 2366 e-12 u −13 λe = 2.426 310 2367(11) e-12 Boltzmann's constant k∗B = 1.379 510 147 52 e-23 u 29 kB = 1.380 648 52(79) e-23 Gravitation constant G∗ = 6.672 497 192 29 e-11 u6 G = 6.674 08(31) e-11 Planck length l∗p = .161 603 660 096 e-34 u −13 lp = .161 6229(38) e-34 Planck mass m∗P = .217 672 817 580 e-7 u 15 mP = .217 6470(51) e-7 Gyromagnetic ratio γe/2π∗ = 28024.953 55 u−42 γe/2π = 28024.951 64(17)e-7 1 Background The mathematical electron model (see [1]) uses a unit-less (units = 1) formula for an electron fe (eqs. 1, 20) to construct the Planck units as geometrical objects. In this model MLTA are interrelated (see units, eq.1), thus we need only assign numerical values to any 2 Planck units, for example M = mP and T = tp, in order to fix the numerical values for all Planck units and from these the dimensioned constants (G, h, c, e,me, kB...) where α (fine structure constant) and Ω = 2.0071349496... are dimensionless mathematical constants. fe = 4π2(263π2αΩ5)3 = .23895453...x1023 (1) units = (AL)3 T = √ L15 M9T 11 = 1 We find this set of geometries for MLTA; M = (1) (2) T = (2π) (3) L = (2π2Ω2) (4) A = ( 26π3Ω3 α ) (5) To convert to CODATA 2014 values requires 2 dimensioned scalars. In this example I use a mass scalar k and time scalar t with associated units u15 = mass, u−30 = time. Assigning appropriate numerical values to (k, t) k = mP = .217672817... x 10−7, unit = u15 (kg) t = tp/2π = .171585512... x 10−43, unit = u−30 (s) M = mP = (1)k, unit = u15 (kg) (6) T = tp = (2π)t, unit = u−30 (s) (7) We can now calculate scalars (l, a) and the Planck units LA l = k9/15t11/15, unit = u9/15∗15+11/15∗(−30)=−13 (eq.1) (8) L = lp = (2π2Ω2)l, unit = u−13 (m) (9) l15 k9t11 = (.203...x10−36)15 (.217...x10−7)9(.171...x10−43)11 . u−13∗15 u15∗9u−30∗11 = 1 (10) a = 1 k9/15t6/15 , unit = u9/15∗(−15)+6/15∗30=3 (11) 1 1 Background A = ( 64π3Ω3 α ) a, unit = u3 (ampere) (12) a3l3 t = (.126...x1023)3(.203...x10−36)3 (.171...x10−43) . u3∗3u−13∗3 u−30 = 1 (13) The physical constants in terms of MLTA (V = 2L/T); c∗ = V = (2πΩ2) k3/5 t4/15 , u9/15∗(15)+8/30∗(−30)=17 (14) G∗ = V2L M = (8π4Ω6) k4/5t1/5, u12−6=6 (15) e∗ = AT = ( 128π4Ω3 α ) t3/5 k3/5 , u−18−−9=−27 (16) h∗ = 2πLV M = (8π4Ω4) k11/5t7/15, u33−14=19 (17) k∗B = πV M A = ( α 32πΩ ) k11/5t2/15, u33−4=29 (18) σe = 3α2AL π2 = 273π3αΩ5 t1/3, u−10 (19) fe = σ3e T = (273π3αΩ5t1/3)3 2πt , units = (u−10)3 u−30 = 1 (20) m∗e = M fe , units = u15 (21) λ∗e = 2πL fe, units = u −13 (22) μ∗0 = πV2M αLA2 = ( α 211π5Ω4 ) k14/5 t7/15 , u42+14=56 (23) R∗∞ = me 4πlpα2mP = 1 (22333π11α5Ω17) 1 k3/5t11/15 , u13 (24) For example this common equation for alpha reduces to; α = 2h∗ μ∗0(e ∗)2c∗ = α, u19−56+54−17=0 (25) Inserting the values for (α,Ω, k, t) in the above will give the calculated results listed in table 1. 2 CODATA 2014 By defining all the dimensioned constants in terms of (α,Ω), 2 scalars and the unit rule set u, we can define constants in terms of each other. In the following the geometries of the least precise constants (G∗, h∗, e∗,m∗e, k ∗ B) can be constructed using the geometries of the most precise (c∗, μ∗0,R ∗ ∞, α ∗); (h∗)3 = (8π4Ω4k11/5t7/15u19)3 = 2π10(μ∗0) 3 36(c∗)5α13(R∗∞)2 , unit = u57 (26) (e∗)3 = 4π5 33(c∗)4α8(R∗∞) , unit = u−81 (27) (k∗B) 3 = π5(μ∗0) 3 332(c∗)4α5(R∗∞) , unit = u87 (28) (G∗)5 = π3(μ∗0) 22036α11(R∗∞)2 , unit = u30 (29) (m∗e) 3 = 16π10(R∗∞)(μ ∗ 0) 3 36(c∗)8α7 , unit = u45 (30) (l∗p) 15 = π22(μ∗0) 9 235324α49(c∗)35(R∗∞)8 , unit = (u−13)15 (31) (m∗P) 15 = 225π13(μ∗0) 6 36(c∗)5α16(R∗∞)2 , unit = (u15)15 (32) γe/2π = gl∗pm ∗ P 2k∗Bm ∗ e , unit = u−13−29=3−30−15=−42 (33) (γe/2π)3 = g333(c∗)4 28π8α(μ∗0) 3(R∗∞)2 (34) Inserting the above in the alpha formula α3 = 8(h∗)3 (μ∗0) 3(e∗)6(c∗)3 = α3, units = 1 (35) We can then replace (c∗, μ∗0,R ∗ ∞, α ∗) with the numerical CODATA 2014 values for (c, μ0, α,R∞) giving the calculated results listed in table 1. 3 Revision 2017 As eq.27 uses only (c∗,R∗∞, α) and as e = 1.602176634x10 –19 is the most precise of the revised values I shall discuss it here in terms of this model. α8 = 4π5 33(c∗)4(e)3(R∗∞) , unit = 1 (36) Using the CODATA 2014 R∞ gives α = 137.03599520 R∗∞ = 4π5 33(c∗)4(e)3(α8) , unit = 1 (37) Using the CODATA 2014 α gives R∞ = 10973729.04723. In other words we now have 2 separate values for elementary charge, from (c∗,R∗∞, α) and from the revision. Notes: kB does not agree with CODATA, however it can be used in eq.33 to solve the gyro-magnetic ratio. G agrees with Rosi et al G = 6.67191(77)(62) x 10−11 [3]. A complete discussion of this model can be found here [1], the formulas used in this article downloaded in maple format [2], for convenience I use the commonly recognized value for alpha as α ∼ 137; 2 3 Revision 2017 References 1. Macleod, Malcolm J. "Programming Planck units from a virtual electron; a Simulation Hypothesis" Eur. Phys. J. Plus (2018) 133: 278 2. planckmomentum.com/rev2017.zip (formulas in maple and pdf format) 3. Rosi, G.; Sorrentino, F.; Cacciapuoti, L.; Prevedelli, M.; Tino, G. M. (26 June 2014). "Precision measurement of the Newtonian gravitational constant using cold atoms", Nature. 510: 518–521. www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/ Rosi.pdf 3 3 Revision