CALCULATION OF THE MASS OF THE NEUTRON AS AN ELECTRON AND A PROTON IN A SUBSTATE. James Keele, MSEE. Abstract

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1 HADRONIC JOURNAL CALCULATION OF THE MASS OF THE NEUTRON AS AN ELECTRON AND A PROTON IN A SUBSTATE James Keele, MSEE 3313 Camino Cielo Vista, Santa Fe, NM jkeele9@sisna.com Abstract The mass-energy states of some Hadrons are calculated as the mass of a proton and an orbiting electron at the subatomic level and in quantum state regions with n = 1, 2,3, etc., or n = 1/1,1/2,1/3, etc. The ability to calculate these states with a computer is mainly based on the following concepts: SRT, the equivalence of the electric field and rest mass of the electron, the extinction of rest mass of electrons and protons at subatomic distances, fractional energy quantum states, and Bohr Atom Theory. The mass of the neutron is shown to have a multiplicity of fractional mass-energy quantum states at the subatomic level. Support is presented for these concepts that are new and non-traditional physics. Also this paper presents the formulae and some output listings from computer calculations. INTRODUCTION This paper is a response to the call from the guest editor saying, People sometimes casually suggest that the neutron is a bound quantum state of an electron with a proton. But this idea does not seem to make sense, at least in concert with other familiar ideas. First of all, traditional quantum mechanics doesn t offer any states for electrons and protons other than Hydrogenatom states. Furthermore, the neutron mass is larger ( ) than the sum ( m e ) of the electron and proton masses. Since mass is equivalent to energy, the conventional thought is that because the neutron has more energy than its constituents, those constituents cannot be bound because that condition implies negative orbit energy. [1] The purpose of this paper is to show more than casually how the subatomic fractional energy states of Hydrogen provide a multiplicity of bound states of the proton and electron that have the mass of the neutron. m e

2 2 J. KEELE Discussions of new concepts for physics are presented; namely, the equivalence of the mass-energy of the electric field of the electron with its rest mass, and the extinction of rest mass of the electron at a finite distance from a proton, and fractional energy states. These concepts are necessary for use in the calculations, as presented in this paper, of hadronic masses such as the neutron. Discussions of the uses of SRT and the Bohr Atom Models with circular orbits are also presented. Detailed formulae that model these states, and computer-calculated values of these states, are also presented. Known masses of some hadronic particles are compared with the calculated values to provide some support and validation of the formulae presented. 1. EQUIVALENCE OF THE ELECTRIC FIELD AND THE REST MASS OF THE ELECTRON The suppositions that the rest mass of the electron is the electromagnetic mass of its E-field, and that it extends throughout space, are supported by the following arguments: 1. Logic: In electron-positron pair annihilation, there appear to be just two gamma rays produced, together equal in energy to the rest mass of the pair. If the rest masses of the pair were separate from the electromagnetic masses of the pair, and all masses were annihilated, then the total energy of annihilation should equal four m e c 2 instead of just two m e c 2, as appears in the two gamma rays. 2. An analysis of the Bohr Atom [2] shows that the mass lost by the Hydrogen atom due to photoemission equals the mass corresponding to the energy of the emitted photon. This can be true if the energy in the electric field of the electron is the rest mass of the electron. The energy in the electric fields of the proton and electron, and thus their rest masses, diminishes as the distance between them decreases. And the mass of electron has added relativistic mass in a lower orbit due to its increased speed. Also, the analysis shows that if the rest mass of the electron remains constant, as is currently believed by the physics community, then when the electron moved to a lower orbit, the atom would gain mass when the photon was emitted. This is contrary to conservation of energy. 3. The E-field of the electron can extend throughout all space. The E-field of the electron is usually terminated on nearby protons. But the E-field of an isolated electron can be thought of as extended throughout space. This can m e

3 3 MASS OF NEUTRON AS HYDROGEN SUBSTATE explain instantaneous action at a distance if, for instance, another charge acts on the E-field that is already there. Also, the mach principle of all the matter in the universe affecting the inertia of an accelerating particle is somewhat explained by the equivalence of e-field and rest mass. 4. The E-field energy of the electron is associated with kinetic energy momentum hc E e by the following formula: de e / dv = h αc /8 πr 4. (1) where V is the volume of space occupied by the E-field, h is Planck s constant normalized by 2π, α is the fine structure constant, and r is the distance over which the volume is integrated; namely, from a sphere of radius R k (defined below) to a sphere of infinite radius. Formula (1) is derived from Coulombs Law employing two charges of magnitude equal to that of the electron. However, this formula applies only to one charge, that of a single electron, and the formula remains numerically correct. As such, it implies that kinetic energy moment, hc, is inherent in the E- field of the electron. It suggests that the electron may not need spin to obtain its observed angular momentum. 5. It makes just as much sense to suppose that the charge of an electron is in one little ball as it does to speculate that the electron's charge is distributed in a toroid or a ring. In either case, one has the problem of keeping the charge together. This implies that we still do not know the true nature of charge. 6. If an integration of the energy in the electric field of an electron is performed, it is found that the energy is equal to m e c 2 providing the lower limit of the integration is a sphere of radius, a constant that is explained below. Applying SRT to the electric field, it is found that the integration of the energy in the electric field as seen by an observer moving relative to the electron is still m e c 2 providing the integration is done over the same volume of space as the former integration. This also conforms to the idea that the energy of the rest mass of the electron is the energy in the E-field of the electron. A question remains about this integration procedure regarding whether the lower limit of the integration sphere of radius should be length contracted. R k R k

4 4 J. KEELE 2. THE EXTINCTION OF REST MASS OF THE ELECTRON AT SUBATOMIC DISTANCES FROM A PROTON If it is assumed that the rest masses for both the electron and the positron are in their electric fields and equal to m e c 2, then one can determine the distance between them where the E-field energy is totally dissipated or zero. The author determines [2] this distance to be: R k ke 2 2m e c 2 (2) where k is 1/4πε 0 and ε 0 is the permittivity of free space. After all the values are substituted into (2), R k = m. Also, this author defines E k to be: E k 2m e c 2. (3) After values are substituted into (3), E k = J. e Substituting (2) and (3) into Coulomb s Law for charges of magnitude equal to that of the electron: f 12 =±E k R k r 2 (4) Also: E ep = E k ( 1 R k /r) (5) Our first observation about (5) is that the energy is positive for all r R k. Also, E ee is positive for all r. The author defines the energy, in (5) to be zero for r < R k (negative energy does not exist, except mathematically). Eq. (5) describes the stored potential energy in the electronpositron system. Dividing (5) by 2c 2, the potential mass of one of the particles in the electron-positron system becomes: E ep E ep

5 5 MASS OF NEUTRON AS HYDROGEN SUBSTATE m e0 = m e ( 1 R k /r) (6) where subscript 0 means without motion, or at rest. We see from (6) that the mass of the electron and positron becomes extinct or zero at r = R k. Appling this reasoning to the larger particle of the proton: m = m m / r (7) P0 P e R k This extinction process when combined with SRT allows, in this theory, for calculation of quantum mass-energy states at the subatomic level. 3. FRACTIONAL ENERGY STATES The mass-energy states of Hydrogen with n = 1,2,3, etc. are well known. What is not so well known are the fractional energy states of Hydrogen below the ground level. These correspond to radial quantum number n < 1. Hydrogen principal fractional quantum states are claimed to exist by Dr. Randall Mills, president of Blacklight Power, Inc. and by its employees. [3]. The company has been dedicated to establishing the validity of fractional energy states for Hydrogen and exploiting them in new technology. Dr. Mills claims to have validated their existence. Principal mass-energy states, both whole numbers and fractional, are also claimed to exist at the subatomic level by the present author. The existence of fractional energy states is crucial to the purpose of this paper. The graph in Figure 1, taken from a presentation by Dr. Randall Mills, shows the spectral emission of the electron transition from ground state to the first fractional mass-energy state. The spectral emission at 30.4 nm on the graph corresponds with a computer calculated fractional transition. The computer program that calculated this fractional state transition energy uses the formulae and concepts presented in this paper. This agreement is too close to be coincidental, and, therefore, is taken by the present author to be proof of the existence of fractional mass-energy states of Hydrogen. More spectral emission peaks would be desirable, but the difficulty of obtaining them prevents its presentation here. This one was obtained when He (helium) was added to H 2 as a chemical catalyst to cause the electron transition.

6 6 J. KEELE Figure 1. Emission Spectra of Hydrogen showing transition from ground state to 1/2 fractional state (30.4 nm). From Blacklight Power, Inc. 4. SPECIAL RELATIVITY THEORY (SRT) MODEL The basic equation from SRT [4] for the electric field E about a moving electron is: E e = ke e r γ 2 r 3 1 (v 2 / c 2 )sin 2 θ 3/2 e (8) where v = magnitude of relative velocity v, γ=1 1 v 2 / c 2, θ= angle between r and v, and e e = charge of electron. The constants are k = 1/4πε 0 ( ε 0 = permittivity of free space) and c = speed of light. We note that if v = 0, then

7 7 MASS OF NEUTRON AS HYDROGEN SUBSTATE E e = ke e r / r 3 (9) The SRT model provides a mathematical basis for the masses associated with hadronic particles. For instance, some of the masses of mesons range from m e to m e. These masses may be composed mostly of kinetic energy according to the relativistic formula m =γm0, where m 0 γ=1/ 1 v 2 / c 2, is the rest mass, v is the relative speed of the particle, and c is the speed of light. The author has shown that the electron and positron have a special place in the creation of matter because their mass is principally composed of potential energy [2]. Thus, the rest mass of the electron or positron, that can vary, constitutes the in the relativistic formula for m. In order for the mass to have the high values associated with a hadronic particle, its velocity must be that approaching the speed of light c. Table 1 illustrates the mass obtainable when the mass of an electron travels by only a very small amount to cre- at speed approaching c, differing from c ate a multitude of high mass values. TABLE 1. m 0 m e m =γm v = factor c factor = γ=1 1 v 2 / c 2 = Such a particle orbiting around another particle at near speed c would have very high centripetal acceleration or very high centrifugal force, quantified by the classical formula:

8 8 J. KEELE f c = mv 2 / r (10) where m is the inertial mass, v is the tangential speed of the particle which is rotating at radius r from a central point. It is important to emphasize here that m includes not only the rest mass of the particle, but also the mass increase due to its kinetic energy. The force between two orbiting particles must be high to hold a massive particle orbiting at a velocity near c. Again SRT provides the force for this formula: r f 12 = 12 kq 1 q 2 / r 3 (11) γ 2 1 (v 2 / c 2 )sin 2 3/2 θ f 12 is the force vector of q 1 acting on q 2, and is in a direction of the radial vector r 12 from q 1 to q 2, θ is the angle between the relative velocity vector v of charge 2 with respect to charge 1, and the vector r 12. The γ is defined as above, and here k = 1/4πε0, where ε 0 is the permittivity of free space. Eq. (11) is taken quite literally by the author to be good for all γ s and all speeds up to c. It also includes the orbital magnetic force [4]. It is assumed there are no time delays in the force interaction. Since we are considering two large particle speeds with at least one approaching c, then we must employ the SRT formula for velocity addition [5] to determine the relative speed v and the γ in (13): γ=γ 1 γ 2 ( 1+ v 1 v 2 /c 2 ) (12) γ 1, γ 2, v 1, and v 2 are the gammas and velocities of q 1 and q 2 with respect to the center-of- mass (CM) of the two orbiting charged particles. When θ=90 Eq. (11) becomes: f ep =γe k R k r 12 r 3. (13)

9 9 MASS OF NEUTRON AS HYDROGEN SUBSTATE Eq. (13) uses the new constants, and, for Coulomb s Law when the E k charge values are e. For two circular orbiting particles, θ = 90. The γ in (13) must be calculated with (12). 5. BOHR ATOM THEORY A very important assumption that Bohr made for his analysis of the spectral content of the Hydrogen atom (Bohr atom) is that the total orbital angular momentum of the orbiting masses is quantized by Plank s constant, h [6]. For the case of the electron and proton orbiting around each other, this assumption is expressed mathematically by: R k m e v e r e + m P v P r P = nh or m e v e r e a = nh (14) where n = 1, 2,3, etc., h = h / 2π, and a = 1+ m e / m P. The r s are the radii of the respective orbits about the CM. The mass m e in the computer program that follows is taken to be smaller than or equal to m. For normal quantum states v e << c, but for fractional states can become significant. When is near c, the quantized mass-energy states of me may initially be approximated by the formula: v e m e = nh / cr e a P v e (15) The classical formulae related to the electron and proton orbiting about each other and their CM are employed as necessary: m P r P = m e r e, m P v P = m e v e (16) Definitions and Constants Symbols and subscripts are employed to precisely define what is referred to. Also, values of constants are given so further referral is not necessary. All calculations are done for circular orbits, instead of elliptical orbits, to emphasis the important mass-energy states without undue mathematical complexity.

10 10 Symbols: m mass m P rest mass of isolated proton m P0 variable rest mass of proton J. KEELE m Pt total mass γ m P0 of proton with respect to Center-of-Mass (CM) m e rest mass of isolated electron m e0 variable rest mass of electron m et total mass γm e0 of electron with respect to CM v speed with respect to CM γ - SRT gamma factor with respect to CM 1 1 v 2 / c 2 n principal quantum enery level ( n = 1, 2,3, etc., n = 1/ p, p = 1, 2,3, etc.) r radius of orbit from CM r t total distance between electron and proton orbiting each other e magnitude of electrical charge of electron or proton k = 1/4πε 0, where is the permittivity of free space ε 0 Subscripts: e electron; P proton; 0 rest mass; t total; n energy level Constants: E k 2m e c 2 = J R k ke 2 /2m e c 2 = m c = m/s = speed of light h = J-s, Planck s Constant/2π m e = kg = mass of electron m P = kg = mass of proton n = 1, 2,3, etc., for normal atomic energy states n = 1/ p where p = 1, 2,3, etc., for fractional subatomic energy states. Formulae: m e0n = m e 1 R k a n r en ( ) (17)

11 11 MASS OF NEUTRON AS HYDROGEN SUBSTATE m P0n = m p R k m e / a n r en (18) (19) a n = 1+ m etn / m Ptn (20) r Pn = r en m etn / m Ptn (21) v Pn = v en r Pn / r en (22) m etn = nh / a n f n cr en v en = f n c (23) γ en = m et n / m e0n γ Pn = 1 1 v Pn 2 / c 2 f tn = 1+ v Pn / c (24) (25) (26) (27) γ tn =γ en γ Pn f tn (28) m Ptn =γ Pn m P0n r en = 2γ tn m e R k a n 2 metn f n 2 f n = 1 1 γ en 2 (29) (30) 6. HADRONIC MASSES WITH NORMAL ENERGY STATES: n = 1, 2, 3, etc. Equations (17) and (18) mathematically allow stable mass-energy states for Hydrogen and other hadronic particles [7] at the subatomic or nuclear level. The term mass-energy states is used instead of energy levels because mass of the particles is calculated for the various states instead of energy levels. Quantum theory will work just as well with mass-energy states as it will with energy levels and light spectra. As support for the existence of the mass-energy states at the subatomic level, a computer program was written to compute the mass of the created particles and matched them to within 0.1 percent of some known existing hadronic particles. This procedure is similar to using light spectra for verification, except that masses are used instead. The computer program was written in Pascal. It is not included in this paper, but is available from the author. It has two realizations, Program A and Program B, detailed below. Variables common to both program realizations are: mept is calculated total mass of the proton and electron.

12 12 J. KEELE mpt is calculated total mass of the proton. met is calculated total mass of the electron. vp is calculated velocity (m/sec) of the proton about the CM. rp is calculated distance of the proton from the CM. re is calculated distance of the electron from the CM. rt is calculated total distance between the electron and proton. Program A addresses subatomic mass-energy states of Hydrogen, n = 1, 2, 3. The program calculates for successive n, then searches for matches with mept, then mpt, then met. Figure 2 is the output listing of Program A. The results show mass matches within 0.1 percent of some known hadronic particles [8]. (Charge or other properties of the hadronic particles are not considered in this paper.) At the bottom of the output list are the matches. The column on the left shows the calculated values and the column on the right shows the measured masses of known hadronic particles. Note that the radius of the orbiting electron varies from about 3.2 fermi to about 7.6 fermi. Around 6 to 7 fermi seems to be the dividing line between atomic and subatomic particles. Figure 2. Program A Output Nuc_Nrm2 1/18/08 is the name of this Pascal program and its date. Initial center mass = ; Mass unit is m ; Length unit is one meter. e

13 13 MASS OF NEUTRON AS HYDROGEN SUBSTATE n mept mpt met vp rp re rt Matches that are within 0.1%: mept[ n = 7 ] : meson #22: mept[ n = 8 ] : meson #24: mept[n = 13] : meson #30: mpt[ n = 9 ] : meson #13: mept[n = 10] : baryon #13: mept[n = 11] : baryon #16: mept[n = 14] : baryon #26: mept[n = 14] : baryon #27: mept[n = 16] : baryon #34: mept[n = 17] : baryon #43: mpt[ n = 1 ] : baryon #2: mpt[n = 16] : baryon #5:

14 14 J. KEELE 7. FRACTIONAL MASS-ENERGY STATES FOR HYDROGEN AT THE SUBATOMIC LEVEL If fractional mass-energy states exist for Hydrogen at the atomic level, there is a possibility that fractional mass-energy states exist for Hydrogen at the subatomic level. It is shown above with Program A that normal massenergy states exist for Hydrogen at the subatomic level. Fractional massenergy states, n = 1/p where p = 1, 2, 3, Hydrogen, were inserted in Program A creating Program B. Program B is essentially the same as Program A above except that n is placed in the denominator of the third equation in the main routine instead of the numerator. Figure 3. Program B Output (fractional mass-energy states of Hydrogen at subatomic level) Nuc_Frc2 1/18/08 is the name of this Pascal program and its date. Initial center mass = ; Mass unit is m ; Length unit is one meter p mept mpt met vp rp e re rt

15 15 MASS OF NEUTRON AS HYDROGEN SUBSTATE p mept mpt met vp rp re rt Matches that are within 0.02% mept[p = 29] baryon #2: mept[p = 30] baryon #2: mept[p = 31] baryon #2: mept[p = 32] baryon #2: mept[p = 33] baryon #2: mept[p = 34] baryon #2: mept[p = 35] baryon #2: mept[p = 36] baryon #2: is close to neutron mass.

16 16 J. KEELE mpt[p = 2] baryon #1: mpt[p = 3] baryon #1: mpt[p = 4] baryon #1: mpt[p = 5] 7 baryon #1: mpt[p = 6] 4 baryon #1: mpt[p = 7] 2 baryon #1: mpt[p = 8] 1 baryon #1: mpt[p = 9] 0 baryon #1: is close to proton mass. 8. DISCUSSION OF RESULTS r t The output listing of Program A shows several matches with some measured masses of mesons and baryons. This gives support and credence to the employed theories and formulae. Further credence can be realized if in place of the mass of the proton, masses of m e and m e are substituted in this program. These masses are measured masses of other hadronic particles. The program then calculates masses of many other hadronic particles [7]. It appears that all it takes to create a hadronic particle is a heavy seed particle of positive charge, an electron, and energy. The radii of the orbiting electron vary from about 3.2 fermi to about 7.6 fermi. The total separation,, between the central proton and the orbiting electron increases with increasing n. Also the energy state and the masses of the resulting particles increase with increasing n. Figure 4.

17 17 MASS OF NEUTRON AS HYDROGEN SUBSTATE Program B shows that mass-energy states for the neutron are predicted with fractional energy levels at the subatomic level. The radii are approximately constant for all fractional energy states: 4.4 fermi. The velocity of the electron approaches c for all the fractional energy states. Figure 4 is a copy of a proton and neutron charge distribution figure From Elementary Particles, By David H. Frisch and Alan M. Thorndike, Van Nostrand Company, Inc, 1964, p. 56. Referring to Figure 4, one can see the calculated radius (from Program B) of the neutron, 4.4 x m, is within the correct order of magnitude. A fermi is 1 x cm or 1 x m. 9. ENERGY OF THE NEUTRON Neutrons have the lowest mass-energy level of the hadronic particles created from a proton and an electron (Hydrogen). They were probably created with compression in stars and exploding stars. Their long time existences are mainly due to the simple fact they are bound to the proton in normal matter. On being free of the protons, they decay into protons and electrons and release their excess energy in the form of kinetic energy of the protons and electrons or in the form of light type energy or both types of energy. A Neutrino or Etherino need not be postulated to account for this excess energy. According to the formulae used in the program for neutron mass calculation, the proton and the electron each loose about 0.34 m e of their rest mass for the synthesis of the neutron. On decaying the neutron gives this rest mass energy back to the proton and electron plus the energy difference in the masses: ( ) = 1.53 m e. This extra energy is mainly due to the relativistic mass increase of the fast orbiting electron about the proton. This energy may be available as an alternative energy source if a way is found to use neutrons as an energy source. R. M. Santilli describes a possible way in which this energy may be tapped as an alternative energy source [9]. 10. CONCLUSIONS 1. Hydrogen has far more principal mass-energy states than is generally recognized. The ones not recognized deal with higher energy levels than the Hydrogen atom exhibits and are more difficult to experiment with. 2. The formulae and methodology of this paper appear to be useful in predicting the major property of Hydrogen at the subatomic level, i.e. its mass under differing principal quantum numbers. 3. Several mass-energy states predict the mass of the neutron in the fractional subatomic region.

18 18 J. KEELE 4. The techniques employed in this paper are useful for calculating masses associated with other hadronic particles [7]. 5. This is not a complete theory of hadronic particles; it is a place to begin a more comprehensive understanding of hadronic particles. 6. Neutrinos and Etherinos need not be postulated to explain the mass increase of the neutron over the sum of the proton and electron masses. The theories outlined in this paper explain the excess energy. ACKNOWLEDGMENT The author is very grateful to Dr. Cynthia Whitney for her encouragement to write this paper, and for her suggestions about editing it. REFERENCES [1] Dr. Cynthia Whitney, Editor of Galilean Electrodynamics, Nov. 23, [2] J. Keele, Mass Changes and Potential-Energy Changes Unified, Galilean Electrodynamics 14 (5), 62 (2004). [3] R. Mills, Blacklight Power, Inc., [4] W. Rindler, Essential Relativity, Revised Second Edition, p. 101 (Springer-Verlag, New York, Heidelberg, Berlin, 1977). [5] ibid. p. 47. [6] R. Eisberg, R. Resnick, Quantum Physics, Second Edition, p. 105 (John Wiley & Sons, New York, 1974, 1985) [7] J. Keele, A Different Approach to Elementary Particles, Galilean Electrodynamics, 17 (5), 88 (2006) [8] CRC Handbook of Chemistry and Physics, 52 nd edition, The Chemical Rubber Co., Cranwood Parkway, Cleveland, Ohio 44128, 1971, p. F-211 to F217. [9] R. M. Santilli, Neutrino and/or Etherino?, Contribution to the Proceedings of IARD 2006, revised October 26, 2006, Institute for Basic Research, P.O. Box 1577, Palm Harbor, FL 34682, U.S.A.