Atonic Mechanics Part Two: The Hydrogen and Helium Atoms

Transcription

1 Atonic Mechanics Part Two: The Hydrogen and Helium Atoms Alfred Phillips, Jr. Source Institute Ithaca, NY Alfred Phillips, Jr. and Cosmic Positive Interactions (Ken-Eth) Abstract We have created a particle-base model, Atonic Mechanics, for calculating the spectra of the hydrogen and helium atoms. We make the argument that the Schrödinger theory is un-physical for one of the simplest atomic configurations, the ground state of the hydrogen atom. Although the mathematical procedures in Atonic Mechanics are relatively simple, the accuracy of the ground state helium atom calculations is comparable to the accuracy of the Schrödinger Equation using perturbation techniques. Our mathematical procedure is energy minimization with the introduction of a term Δn, the precession fraction, in the angular momentum. The precession fraction is a real quantity in the helium atom energy levels shown on the NIST website with the Atonic Mechanics interpretation of that data. We have found curve-fitted expressions for Δn for the helium singlet and triplet cases. The curvefitted equations seem to suggest a coupling scheme between the angular momentum of the electrons in an atom. We obtain good agreement between the Atonic Mechanics calculation for helium spectra and the more than 400 energy levels given on the NIST website. We present a time-independent model for calculating the energy levels for the hydrogen and helium atoms. We do this by the minimization of the energy. For helium we postulate that the angular momentum of the excited electron is an integer plus a fraction of ħ. (We note, in passing, that there is nothing sacred about an integral number of ħ for angular momentum. The other Planck units are the well known c, G, ε 0, and k. One is not compelled to drive a car at multiples of c even if it is a red Cadillac CTS.) In a sense, this model is a generalization of Bohr s approach. We do not here show a more general time-dependent schema for treating atomic phenomena. We intend, in later work, to describe the physics of time dependence for atomic systems, as well as other important atomic and relativistic physics. Hydrogen Atom Following Bohr, we assert that the angular momentum is given by ħ. 1

2 The electron kinetic energy, T, potential energy, V, and total energy, E, are given by The terms have their usual meanings. From we find that 1 2,. 0, ħ., Putting this expression for r back into that for E, we obtain 2 ħ which is the expression for the hydrogen atom energy levels, apart from the Lamb shift. Note that we did not set the Coulomb force equal to the centrifugal force as did Bohr. We know of no derivation of the energy for the hydrogen atom done in this manner, but likely others have done this heretofore. Physical inadequacy of Schrödinger Theory In our Atonic Mechanics model of atoms, the electron is not right at the proton because its angular momentum is quantized. We submit that the Schrödinger Equation does not treat the important ground state of the hydrogen atom in a manner consistent with what we have learned experimentally since he published his work in We use Feynman s work to illustrate this. Feynman, in describing the Schrödinger Equation, writes 1, For the lowest energy state, n = 1, and Ψ 1 (ρ) = e -ρ. For a hydrogen atom in its ground (lowest energy) state, the amplitude to find the electron at any point drops off exponentially with the distance from the proton. It is most likely to be found right at the proton, and the characteristic spreading distance is about one unit in ρ, or about one Bohr radius, r B. 2

3 Does it make sense that the electron is coincident in space with the proton? Classically it may be there, but Atonic Mechanics insistence on the quantization of angular momentum prevents this coincidence. We again quote Feynman 2,... From the measurements in nuclear physics it is found that there are electrostatic forces at typical nuclear distances at about centimeter and that they still vary approximately as the inverse square law. This experimental fact was not demonstrated when Schrödinger created his wave mechanics. The electron would not loiter around the proton; the electron would stay coincident with the proton from the Coulombic attraction. Guess what would happen in the inverse situation if there are two entities of the same charge within nuclear dimensions? Feynman also writes 3, If such a nucleus (i.e., U 235 ) is just tapped lightly (as can be done by sending in a slow neutron), it breaks into two pieces, each with a positive charge, and these pieces fly apart by electrical repulsion. The energy which is liberated is the energy of the atomic bomb. The energy is usually called nuclear energy, but it is really electrical energy released when electrical forces have overcome the attractive nuclear forces. Feynman pointed out in the actual lecture given at Caltech on September 27, 1962 that he had given careful though for that lecture on the subject of the 20 K ton explosion. (This lecture is on Audible.com.) We assert that it is virtually impossible for the electron to be found right at the proton because (as sophomore physics student can show) the energy required to pull the electron from the proton would be 1.2 MeV not the experimental 13.6 ev. We take this to mean that the Schrödinger Equation is not even correct for a common state in the universe, the ground state of the hydrogen atom. Schrödinger originally conceived the wave function as a charge density. Born and Bohr later worked out the commonly accepted interpretations of the wave function as probabilistic and complementary. We believe that these imaginative stretches are unnecessary. Atonic Mechanics Calculation procedure for NIST He I, II The NIST website has listed two sets of data relevant to this work: NIST Atomic Spectra Database Levels Data He I 4, and He II. 5 (We usually contract the name of the data found on that website as NIST He I and NIST He II.) The website address for the NIST He I data in the HTML format was, as of 31 March 2008, It gave nine significant-figures accuracy. When this physicist retrieved the data used in this paper, on or around 12 April 2003, only seven significant-figure accuracy was given. (The calculations presented here will not be re-done by this physicist. We invite others to do this work, or with suitable funding, this physicist would direct calculations to be done by students or employees.) Reference made to NIST data in this paper means the information given on that website around 12 April We put the first pages of the 31 March 2008 data and the entire vintage (~12 April 2003) data for NIST He I and NIST He II in Appendix I. In the dataset for NIST He I, both electrons start in the lowest level (configuration 1 Term 1S) and one electron is always kept there. (Although we give the HTML format 3

4 designation, we use the ASCII nomenclature for term designation rather than the HTML format designation. The ASCII designations are easier to use in WORD. Clearly the information is identical for HTML and ASCII formats. Both options were given at the NIST website.) One hundred ninety-one energy values were listed in 2003 for which the second electron was in states ranging from 1s to 22p. The first electron was always in the 1s state. The dataset for NIST He II is singly ionized helium atom (hydrogen atomlike) for which 243 lines of data are given in Atonic Mechanics Model for Helium The electrons kinetic energy, T, is given by We have assumed that the helium nucleus is stationary (infinite mass) and the electrons of mass, m e, have velocities v 1 and v 2. The potential energy, V, is written as r 1, and r 2 are the distances from the nucleus to the respective electrons and r 12 is the distance between electrons given by 2 cos where θ is the angle between the two electrons with the nucleus at the vertex, with Finally, newton meter.. The very robust NIST Atomic Spectra Database Levels Data website allows presentation of the data in many formats. Using the NIST website, we express energy values given in ev, and the ground state set at ev. As we calculate the extent to which the electrons are bound, there is an offset of ev. We label as electron 1(one) the electron which is always at the 1s level. Similar to Bohr, we assert that the angular momentum of that electron is given by ħ. We posit that the angular momentum of the second electron for NIST He I (which we label as electron 2) is given by 4

5 D ħ. Bohr assumed that n is an integer, that is, angular momentum is an integral multiple of the Planck constant. We posit that Dn, named the precession fraction, is a real number less than one. The velocities v 1 and v 2 obtained from the above two equations are substituted into the expression for the kinetic energy, T. We set to zero the partial derivatives of E with respect to r 1, r 2, and θ. Solving the resulting three equations for r 1, r 2, and θ, we then substitute these values into the equation for E. The energy of NIST He I for values of m and (n +Dn) is thereby obtained. For NIST He II there is no second electron. Mathematically, is infinite and v 2 is zero, that is, we have a Bohr-style hydrogenic atom. We will discuss the NIST He I case, as modeling NIST He II is straight forward. Analysis of NIST He I The NIST data given on their website is in the standard spectroscopic nomenclature which is described in a comprehensive and easy to read manner with many informative help documents. One such is NIST Atomic Spectra Database Help Page. We will not elaborate on what is given on that website. We do, however, take some definitions from the NIST website given as Helium and Helium-like Ions; LS Coupling. 6 (a) The orbital angular momenta of the electrons are coupled to give a total orbital angular momentum L =. (b) The spins of the electrons are coupled to give a total spin S =. The combination of a particular S value with a particular L value comprises a spectroscopic term, the notation for which is. The quantum numbers 2S + 1 is the Multiplicity of the term. The S and L vectors are coupled to obtain the total angular momentum, J = S + L, for a level of the term; the level is denoted as J. By introducing D, we have found redundancy in the spectroscopic nomenclature for NIST He I. We need only be concerned about the singlet (electrons anti-parallel) and triplet (electrons parallel) as a function of L- value (S, P, D, F, etc.). We also designate the configuration (n-value, and l-value) for each electron as done by NIST. Surprisingly, there is no J-value dependence using the D concept for NIST He I and He II. For example, for the 3D term, the vectors S and L can be added in three ways giving values for J of 1,2, and 3 each corresponding to D = We will show that we need only to designate the configuration, the L-value, and Dn to specify the electron parameters for NIST He I; the designation of NIST He II is simpler still, as it is hydrogenic. More surprisingly, the calculated 1S energy values were very close to measured values which we show in Table 1. 5

6 Table 1 Term 1S, Dn = 0 Configuration En NIST (ev) En Aton (ev) % Difference r 2 (Bohr radii) 1s s.2s s.3s s.4s s.5s s.6s s.7s s.8s s.9s s.10s s.11s s.12s s.13s s.14s s.15s For the helium ground state (1s2) there is a 5.48% difference between calculated and measured values in Atonic Mechanics. Using first order perturbation techniques, the authors Eyring, Walter, and Kimball 7 as well as Pauling and Wilson 8 gave values of ev and ev for the ground state of helium. The percent differences are 5.32 and 5.33 respectively. These lowest energy values, obtained with more mathematical effort than with Atonic Mechanics, are essentially of the same accuracy the same as those of Atonic Mechanics. We realize that the these two text were written shortly after Quantum Mechanics was invented and that more sophisticated analytical techniques implemented on electronics computers may be able to render highly accurate values for the energy of the ground state of helium. On the other hand, people may be able to develop mathematical refinement suitable for use with Atonic Mechanics that may give more accurate results than presented here. For example, this could be done by treating non-circular orbits and/or non-planar orbits. Note that Atonic Mechanics renders a ground state that is more bound ( ev) than the measured value ( ev). This could result from the inappropriateness of assumed circular co-planar orbits. More sophisticated calculations are not the concern here at this time. What we demonstrate is an approach to calculating helium energy levels that is conceptually simpler than the Schrödinger equation but with comparable accuracy. This is done without assuming that particles are wave-like. Einstein may have said, Make 6

7 everything as simple as possible, but not simpler. Einstein reinforced what Occam may have said, One should not increase, beyond what is necessary, the number of entities required to explain anything. Mass Multiplier In addition to improving on simple circular planar orbits, we think influences such as the finite mass of the helium nucleus, three-body considerations, and perhaps even relativistic influences should be included in a more rigorous Atonic Mechanics treatment of the helium atom. Let us, however, assume in this approximation that we can keep the helium nucleus stationary and only adjust the electron mass. Table 2 shows the electron mass (adjusted for both electrons) that renders the energy levels in Atonic Mechanics identical with those of NIST for the Term 1S energy levels. Table 2 Mass Multipliers, Term 1S Configuration E NIST (ev) Mass Multiplier 1s s.2s s.3s s.4s s.5s s.6s s.7s s.8s s.9s s.10s s.11s s.12s s.13s s.14s s.15s The mass adjustment (the multiplication of m e by the mass multiplier) was appreciable for the ground state ( ). It was close to one for all other n-values of the second electron. The further the second electron was from the nucleus, the closer the mass multiplier was to one. These results may suggest that some influences, especially threebody, may be studied in the helium atom with spectral accuracy. We made mass adjustments identical to these for the other terms (singlet and triplet S, P, D, F) as a function of the principal quantum number, n. In the results given so far, Dn, the precession fraction, has been zero. The values of Dn corresponding to the Singlet and Triplet S, P, D, and F configurations are listed in Table 3. 7

8 Dn Values for NIST Helium I L-Designation Singlet Triplet S P D F Table 3 These Dn values follow directly from the NIST He I dataset as interpreted by Atonic Mechanics, thus they have a real physical meaning. The Dn concept allows organization of the NIST He I dataset into a much more concise manner than does conventional spectral designation. It is likely that Dn can be modeled in more than one way; nevertheless, we model it as discussed in the next section. Electron Precession in Atonic Mechanics By precession we mean a subsequent rotational motion, caused by an external influence such as torque or energy, which is added to a more fundamental rotational motion. We compared our Atonic Mechanics model with data available on the NIST website. NIST He II has one ionized electron, so it is hydrogenic and we do not consider precession. NIST He I, the more interesting case, has one excited electron and one electron always in the ground state. There are two numbers for the excited electron which characterize the angular momentum, the principal quantum number n, and the precession fraction, Δn. The total the total angular momentum for the excited electron is (n + Δn) ħ in NIST He I. The angular momentum of the ground state electron is m ħ with m = 1. Below we give the principal quantum number n and the precession fraction Δn values for electrons of the first three atoms, and the P th atom, along with NIST He I and NIST He II. The subscripts designate the electron number. Hydrogen n 1 Helium n 1 + Δn 1, n 2 + Δn 2 Lithium n 1 + Δn 1, n 2 + Δn 2, n 3 + Δn 3 Atom # P n 1 + Δn 1, n 2 + Δn 2, n 3 + Δn 3,..., n P + Δn P He I (NIST) n 1, n 2 + Δn 2 He II (NIST) n 1 Curve Fitting Δn for NIST He I 8

9 The NIST website lists helium electron energy values for manifold singlet and triplet states (see Appendix I). In Atonic Mechanics we have found that the energy is primarily a function of the principal quantum number, n, and secondarily a function of the precession fraction, Δn. If we think of n representing orbital properties, we note that the values of Δn are independent of n. Physically, when additional energy is supplied to the helium atom such that it is less than that which would cause the electron to jump from one orbital level to another, the energies corresponding to the various Δn jumps are independent of n. We have found that each precession fraction corresponds to a specific so-called orbital angular momentum quantum number, l, i.e., for the socalled p, d, f, etc. states. Unlike l-values which are integers, Δn-values are different for singlet states (electron spins anti-parallel) and for triplet states (electron spins parallel). Both l and Δn would be independent of the principal quantum number, n. (The maximum value of l is, however, proscribed by n-1 in the Schrödinger theory.) We have fitted the singlet and triplet precession fractions as shown in the equations below. D 1 D 1 Listed in the Table 4 below are values of the parameters a, b, c, and d.that give a reasonable fit to the values of the precession fraction given in Table 3. Table 4 parameter Value a b c 2.5 d 0.95 Figures 1 and 2 show the comparison of the actual and the fitted precession fractions, Δn, for the Singlet and the Triplet cases. (The y-value is labeled Delta n as this author is not sufficiently familiar with Xcel to put Δn there.) 9

10 Singlet Precession Fraction Delta n l- Value Measured Fitted Figure 1 Measured and fitted values of the Singlet precession fraction Δn called Delta n on this figure. Triplet Precession Fraction Delta n l- Value Measured Fitted Figure 2 Measured and fitted values of the Triplet precession fraction Δn called Delta n on this figure. The He I NIST data can be correctly characterized by precession fractions. Our curve fitting, however, does not reflect an understanding of the physics of the precession fractions. More work can be done here. Nevertheless, the equations of Δn suggest that we may think of the quantities in the equations above as couplings. A possible interpretation for the first term, b, is a self coupling in the presence of other electrons, the second term, be -cl, coupling to the electron s orbital angular momentum, the third term, ae -dl, coupling to the spin angular momentum of the ground state electron, and the fourth term, ae -cl, coupling to the orbital angular momentum of the ground state electron. A generalization of this interpretation is that each electron is coupled to the spin angular momentum of all electrons in its atom (including itself as self-coupling) and to the orbital 10

11 angular momentum of all electrons in its atom. The mathematical form of the general coupling could be like that given above for NIST He I. Centrifugal Quantities The Schrödin ger s Equation of radial motion for rotating system is given by ħ 2 ħ where the quantity 1 ħ 2 is related to a centrifugal term in classical mechanics. The Atonic Mechanics expression for the helium atom energy is given by ħ 2 ħ D 2 D ħ 2 2, where the centrifugal quantity is D 2 D ħ 2. If n = ½ and Dn = l, the centrifugal quantities are identical. Tabulated Results for NIST He I with Atonic Mechanics Table 5 through Table 7 give the NIST He I data along with the Atonic Mechanics calculations for the singlet 1P, 1D, and 1F Terms. Table 8 through Table 11 give similar results for the triplet 3S, 3P, 3D, and 3F Terms. Of the 191 line of NIST He I data, there were 25 lines that we did not model. These lines are for 3G, 1G, 3H, 1H, 3I, 1I, 3K, and 1K Terms. The energy differences between configurations in the closest modeled Term, 1F, and configurations in these terms ranged from ev to ev. The limited accuracy of our Dn-values was not sufficiently good to resolve such small energy levels. Singlet cases Table 5 Term 1P, Dn = Configuration En NIST (ev) En Aton (ev) % Difference 1s.2p E-01 1s.3p E-02 1s.4p E-03 1s.5p E-03 1s.6p E-04 1s.7p E+00 11

12 1s.8p E-04 1s.9p E-04 1s.10p E-04 1s.11p E-04 1s.12p E-04 1s.13p E-04 1s.14p E-04 1s.15p E-04 Three place accuracy below 1s.16p s.17p s.18p s.19p s.20p Table 6 Term 1D, n = Configuration E n NIST (ev) E n Aton (ev) % Difference 1s.3d s.4d s.5d s.6d s.7d s.8d s.9d s.10d s.11d s.12d s.13d s.14d s.15d E-05 Three Place Accuracy 1s.16d s.17d s.18d We have usually chosen to show only the tables which give both the NIST value of energy and that calculated in Atonic Mechanics so that the reader can make a directly comparison. The corresponding figures form a graphical family which looks boringly similar. For the 1D Term, however, we make an exception. NIST seems to have a typo for the 1s.11d configuration. It shows up in Table 6, but it stands out in the Figure shown below. The NIST value should be close to the calculated value for this configuration, ev. 12

13 Percent Difference in Energy Term 1D: NIST, Aton Percent Difference n Value Table 7 Term 1F, Dn = Configuration E n NIST (ev) E n Aton (ev) % Difference 1s.4f s.5f s.6f s.7f s.8f s.9f s.10f s.11f Triplet Cases Table 8 Term 3S, Dn = Configuration En NIST (ev) En Aton (ev) % Difference 1s.2s s.3s s.4s s.5s s.6s s.7s s.8s s.9s

14 1s.10s s.11s s.12s s.13s E-05 1s.14s E-05 1s.15s E-05 Table 9 Term 3P, Dn = Configuration En NIST (ev) En Aton (ev) % Difference 1s.2p s.3p s.4p s.5p s.6p s.7p s.8p s.9p s.10p s.11p E-05 1s.12p E-05 1s.13p E-05 1s.14p E-05 1s.15p E-05 Three Significant-Places 1s.16p s.17p s.18p s.19p s.20p s.21p s.22p Table 10 Term 3D, Dn = Configuration En NIST (ev) En Aton (ev) % Difference 1s.3d s.4d s.5d s.6d s.7d s.8d

15 1s.9d s.10d s.11d s.12d s.13d s.14d s.15d Three Significant-Places 1s.16d s.17d s.18d s.19d s.20d s.21d Table 11 Term 3F, Dn = Configuration 1s.4f E n NIST (ev) E n Aton (ev) % Difference s.5f s.6f s.7f s.8f s.9f s.10f s.11f s.12f s.13f s.14f s.15f E-05 Tabulated Results for NIST He II with Atonic Mechanics As mentioned previously, NIST He II is spectral energy data for singly ionized helium. In the 2003 time frame this data was entitled NIST Atomic Spectra Database Levels Data He II 243 Lines of Data Found Note: Unpublished data of R.L. Kelly (not evaluated by NIST). This dataset gave seven significant-figure accuracy. On March 2008 it was referred to as NIST Atomic Spectra Database Levels Data, He II 240 Levels Found. Just as with the NIST He I March 2008 dataset, the NIST He II dataset also has nine significant-figures. (Again, the first pages of the NIST datasets copied in 2003 and in 2008 are included in Appendix II. And again, we invite others to test Atonic Mechanics calculations using the nine significant-figure NIST datasets.) Comparisons of the Atonic Mechanics calculations and NIST He II energy levels for 15

16 principal quantum number up to n = 50 are given in Table 12. Only for n < 5 was there any significant difference between calculations and measurements. Table 12 - Helium II n - Value Configuration Term J Kelly Level Atonic Calc'n Shifted Atonic Calc'n Difference Abs. Pcnt. Diff. (ev) (ev) (ev) (ev) (Percent) 50 50p; 50s 2P*; 2S 1/2, 3/2; 1/ p; 49s 2P*; 2S 1/2, 3/2; 1/ E E p; 48s 2P*; 2S 1/2, 1/2, 3/2; 1/2, 1/2, 3/ p; 47s 2P*; 2S 1/2, 3/2; 1/ E E p; 46s 2P*; 2S 1/2, 3/2; 1/ p; 45s 2P*; 2S 1/2, 3/2; 1/ p; 44s 2P*; 2S 1/2, 3/2; 1/ p; 43s 2P*; 2S 1/2, 3/2; 1/ p; 42s 2P*; 2S 1/2, 3/2; 1/ p; 41s 2P*; 2S 1/2, 3/2; 1/ p; 40s 2P*; 2S 1/2, 3/2; 1/ p; 39s 2P*; 2S 1/2, 3/2; 1/ p; 38s 2P*; 2S 1/2, 3/2; 1/ E E p; 37s 2P*; 2S 1/2, 3/2; 1/ p; 36s 2P*; 2S 1/2, 3/2; 1/ p; 35s 2P*; 2S 1/2, 3/2; 1/ p; 34s 2P*; 2S 1/2; 1/ E E p; 2P*; 3/2; p; 33s 2P*; 2S 1/2, 3/2; 1/ p; 32s 2P*; 2S 1/2, 3/2; 1/ p; 31s 2P*; 2S 1/2, 3/2; 1/ E E p; 30s 2P*; 2S 1/2, 3/2; 1/ p; 29s 2P*; 2S 1/2, 3/2; 1/ E E p; 28s 2P*; 2S 1/2, 3/2; 1/ p; 27s 2P*; 2S 1/2, 3/2; 1/ E E p; 26s 2P*; 2S 1/2, 3/2; 1/ p; 25s 2P*; 2S 1/2, 3/2; 1/ E E p; 24s 2P*; 2S 1/2, 3/2; 1/ p; 23s 2P*; 2S 1/2, 3/2; 1/ p; 22s 2P*; 2S 1/2, 3/2; 1/ p; 21s 2P*; 2S 1/2, 3/2; 1/ p; 20s; 20y 2P*; 2S; 2Y 1/2; 1/2; 37/2, 39/ p; 19s 2P*; 2S 1/2; 1/ x 2X 35/2; 37/ E E p; 18s; 18w 2P*; 2S; 2W 1/2; 1/2; 33/2, 35/ p; 17s; 17v 2P*; 2S; 2V 1/2; 1/2; 31/2, 33/ p; 16s; 16u 2P*; 2S; 2U 1/2; 1/2; 29/2, 31/

17 15 15p; 15s 2P*; 2S 1/2; 1/ t 2T 27/2, 29/ E E-05 2P*; 2S; 1/2; 1/2; 25/2, 14 14p;14s; 14r 2R 27/ p; 13s 2P*; 2S 1/2; 1/ E E q 2Q 23/2; 25/ E E p; 12s 2P*; 2S 1/2; 1/ E E p; 12o 2P*; 2O 3/2; 21/2, 23/ E E p; 11s; 11d 2P*; 2S; 2D 1/2, 3/2; 1/2; 3/ E E d; 11f; 11g; 11h; 11i; 11k; 11l; 11m; 11n 2D; 2F*; 2G; 2H*; 2I; 2K*; 2L; 2M*; 2N 5/2; 5/2, 7/2; 7/2, 9/2; 9/2, 11/2; 11/2, 13/2; 13/2, 15/2; 15/2, 17/2; 17/2, 19/2; 19/2, 21/ E E p; 10s; 10d; 10f 2P*; 2S; 2D; 2F* 1/2, 3/2; 1/2; 3/2, 5/2; 5/ E E-06 2F*; 2G; 2H*; 2I; 2K*; 2L; 2M* 7/2; 7/2, 9/2; 9/2, 11/2; 11/2, 13/2; 13/2, 15/2; 15/2, 17/2; 17/ E E-06 10f; 10g; 10h; 10i; 10k; 10l; 10 10m 9 9p; 9s 2P*; 2S 1/2; 1/ E E-05 2P*; 2D; 3/2; 3/2, 5/2; 9 9p; 9d; 9F; 9g 2F*; 2G 5/2, 7/2; 7/ E E-06 2G; 2H*; 2I; 2K*; 2L 9/2; 9/2, 11/2; 11/2, 13/2; 13/2, 15/2; 15/2, 17/ E E-05 9g; 9h; 9i; 9k; 9 9l; 8 8p; 8s 2P*; 2S 1/2; 1/ E E p; 8d 2P*; 2D 3/2; 3/ E E-05 2D; 2F*; 2G; 2H*; 2I; 2K* 5/2; 5/2, 7/2; 7/2, 9/2; 9/2, 11/2; 11/2, 13/2; 13/2, 15/ E E-06 8d; 8f; 8g; 8h; 8 8i; 8k 7 7p 2P* 1/2; E E s 2S 1/2; E E p; 7d 2P*; 2D 3/2; E E-05 2D; 2F*; 2G; 2H*; 2I 5/2; 5/2, 7/2; 7/2, 9/2; 9/2, 11/2; 11/2, 13/ E E-06 7d; 7f; 7g; 7h; 7 7i 6 6p; 6s 2P*; 2S 1/2; 1/ E E p ; 6d 2P*; 2D 3/2; 3/ E E d; 6f 2D; 2F* 5/2; 5/ E E f; 6g; 6h 2F*; 2G; 2H* 7/2; 7/2, 9/2; 9/2, 11/ E E p; 5s 2P*; 2S 1/2; 1/ E p; 5d 2P*; 2D 3/2; 3/ E E d; 5f 2D; 2F* 5/2; 5/ E E f; 5g 2F*; 2G 7/2; 7/2, 9/ E E p; 4s 2P*; 2S 1/2; 1/ p; 4d 2P*; 2D 3/2; 3/ E E d; 4f 2D; 2F* 5/2; 5/ E E f 2F* 7/2; E E p 2P* 1/2; s 2S 1/2; p; 3d 2P*; 2D 3/2; 3/ E E d 2D 5/2; E

18 2 2p 2P* 1/2; s 2S 1/2; p 2P* 3/2; s 2S 1/2; #DIV/0! Summary Calculations using Atonic Mechanics agree substantially with values of the helium atom energy levels given on the website, NIST Atomic Spectra Database Levels Data He I, and He II for the April 2003 time frame which published seven significantfigure accuracy. (The March 2008 NIST website gave nine seven significant-figure accuracy.) NIST He II is simply singly ionized helium for which the Bohr model gives reasonable accuracy. NIST He I has one electron in the 1s state and the other in states from the 1s to 22p (191 lines). Using energy minimization, our calculated ground state energy was more bound than the measured value. Yet, the accuracy of this model has the same magnitude as has been obtained with the Schrödinger Equation using perturbation theory. The excessive binding may result from the circular co-planar orbits used. This may imply that the actual electron orbits are curvilinear, but not as simple as co-planar circular orbits. Apart from the ground state, there is good agreement between NIST measurements and Atonic Mechanics calculations for the other fourteen spectral lines of 1S Term. We made electron mass adjustment to make the Atonic Mechanics calculations and NIST He I data agree exactly for 1S Term which we used throughout this work. Mass adjustments were only significant for the ground state. We introduced a quantity we call Dn, named the precession fraction, for the other terms: 1P, 1D, 1F, 3S, 3P, 3D, and 3F. The Dn quantity (called precession fraction) which is independent of the principal quantum number, n, is a fraction of ħ. We add to the angular momentum of the electron. This quantity, Dn, is in the NIST He I data when it is organized using Atonic Mechanics. We were able to obtain a reasonable curve fits to the Singlet and Triplet Dn. Various couplings of electrons were suggested from the curve fitting of Δn. The nomenclature used by NIST for electron parameters that of standard QM, seems redundant. We found that only the configuration, the L-value, and Dn are required to specify the electron parameters for NIST He I; the designation of NIST He II is simpler still, as it is hydrogenic. Atonic Mechanics assume particles only, that is, it is done without re-course to wave notions. A good test of an expanded Atonic Mechanics would be calculations of the energy levels of atoms of the complexity of lithium and greater using the Schrödinger Equation and using Atonic Mechanics. 18

19 References 1.Feynman, R. P., Leighton, R. B., Sands, M. (1965). The Feynman Lectures on Physics: Quantum Mechanics, Vol III., p Addison-Wesley Publishing Company. Reading, MA, etc. 2. Feynman, R. P., Leighton, R. B., Sands, M. (1964). The Feynman Lectures on Physics: Quantum Mechanics, Vol II., p 5-7. Addison-Wesley Publishing Company. Reading, MA, etc. 3. Feynman, R. P., Leighton, R. B., Sands, M. (1964). The Feynman Lectures on Physics: Quantum Mechanics, Vol II., p 1-2. Addison-Wesley Publishing Company. Reading, MA, etc. 4. NIST He I, as given on the NIST website around 12 April NIST He II, as given on the NIST website around 12 April Atomic Spectroscopy A Compendium of Basic Ideas, Notations, Data, and Formulas as given on the NIST website on 15 April Eyring, H., Walter, J., Kimball, G. E. (1944). Quantum Chemistry. John Wiley & Sons, Inc., New York and London. 8. Pauling, L. and Wilson, E. B. (1935). Introduction to Quantum Mechanics: With Applications to Chemistry. McGraw-Hill Book Company, Inc. New York and London. 9. Feynman, R. P., Leighton, R. B., Sands, M. (1965). The Feynman Lectures on Physics: Quantum Mechanics, Vol III., pp 20-7, Addison-Wesley Publishing Company. Reading, MA, etc. 19

20 Appendix Part 1. NIST He I data vintage 12 April 2003 (HTML format), pp Part 2. NIST He II data vintage 12 April 2003 (ASCII format), pp Part 3. NIST He I data 31 March 2008, first page (HTML format), p 26 Part 4. NIST He II data 31 March 2008, first page (HTML format), p 27 20

21 21

22 22

23 23

24 24

25 25

26 26

27 27