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Big 3-Dimensional Atoms

Outline

Atoms are objectified from eight-part compund events like this. Click for more about space-time events.

In accordance with conventional chemistry various atoms are understood to be combinations of protons, electrons and neutrons. These atomic components have all been specified from quarks and reference sensations. So for EthnoPhysics, atoms are ultimately defined from sensations too. They are generically represented using bold, serified upper-case letters like  \mathbf{A} or  \mathbf{B}.

The smallest atom is hydrogen, noted by  \mathbf{H}. Hydrogen is also the smallest stable particle defined from a fully three-dimensional array of orbital components. Other bigger atoms can meet this requirement for tridimensionality even more easily. So atomic happenings are all space-time events. That is, atomic events have positions and occurrence times that are well-defined and potentially measurable.

Quantum Numbers for Atoms

Here are some more quantum numbers for use in atomic descriptions that gloss over visual sensation. To objectively discuss atoms we adopt  \mathrm{n},  \ell,  s and  j instead of the four coefficients of rotating quarks  n^{\mathsf{u}},  n^{\mathsf{\overline{u}}},  n^{\mathsf{d}} and  n^{\mathsf{\overline{d}}} that previously accounted for achromatic visual sensations. Taken together, they are said to specify an atomic state. Note that  \mathrm{n} and  s are defined from simple sums and differences of quark coefficients, so they are always conserved. The principal quantum number is given by

\mathrm{n} \equiv  n^{\mathsf{d}}  / 4

This principal number was defined and discussed earlier. It is used to describe how an atom is excited by interactions with other particles. The azimuthal quantum number is also known as the orbital quantum number. It is defined by

\ell \equiv \dfrac{ \; n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} - 3n^{\mathsf{d}} + n^{\mathsf{\overline{d}}} + \left| \, n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} - n^{\mathsf{d}} - n^{\mathsf{\overline{d}}} \rule{0px}{9px} \; \right| \, }{8}

But recall that N^{\mathsf{Z}} \equiv n^{\mathsf{z}} + n^{\mathsf{\overline{z}}} so the azimuthal number can also be written as

\ell = \dfrac{ \; N^{\mathsf{U}} + N^{\mathsf{D}} + \left| N^{\mathsf{U}} - N^{\mathsf{D}} \rule{0px}{10px} \right| - 16\mathrm{n} \; }{8}

The spin magnetic quantum number for atomic hydrogen is

m_{S} \equiv \dfrac{ \;  n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} - 3 n^{\mathsf{d}}+ n^{\mathsf{\overline{d}}} \; }{8}

which may also be written as

m_{S} = \dfrac{ \; N^{\mathsf{U}} + N^{\mathsf{D}} - 16\mathrm{n} \; }{8}

And finally, the total atomic angular momentum quantum number is given by

j \equiv \dfrac{ \; \left| \, n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} - n^{\mathsf{d}} - n^{\mathsf{\overline{d}}} \rule{0px}{10px} \, \right| \; }{8} = \dfrac{ \; \left|  N^{\mathsf{U}} - N^{\mathsf{D}} \rule{0px}{10px} \right| \; }{8}

This is the same functional relationship used earlier to define \textsl{\textsf{J}}, the total angular momentum quantum number for nuclear particles. The duplication is so that we may use the lower case letter  j to identify the atom in problems that also involve an electron.

Chemical Quarks

The union of a conjugate seed and a chemical seed is called a chemical quark. Chemical quarks are symbolized using bold lower-case Roman letters with serifs like a b and d. (This is different from a b and d, the sans-serif font used for thermodynamic quarks.) So chemical quarks are defined by objectifying pairs of Anaxagorean sensations and named after their chemical seeds. Objectification changes narrative forms of description from using adjectives to identify sensations, to using nouns for identifying particles. Click on any icon in the table below for more detail.

Chemical Quark Definitions
tart taste on the right
A seed icon.

+

A seed icon.

A seed icon.
acidic quark
tart taste on the left
A seed icon.

+

A seed icon.

A seed icon.
acidic anti-quark
soapy taste on the right
A seed icon.

+

A seed icon.

A seed icon.
basic quark
soapy taste on the left
A seed icon.

+

A seed icon.

A seed icon.
basic anti-quark
brackish taste on the right
A seed icon.

+

A seed icon.

A seed icon.
ionic quark
brackish taste on the left
A seed icon.

+

A seed icon.

A seed icon.
ionic anti-quark
potable taste on the right
A seed icon.

+

A seed icon.

A seed icon.
aqueous quark
potable taste on the left
A seed icon.

+

A seed icon.

A seed icon.
aqueous anti-quark
sweet taste on the right
A seed icon.

+

A seed icon.

A seed icon.
dextro quark
sweet taste on the left
A seed icon.

+

A seed icon.

A seed icon.
dextro anti-quark
savory taste on the right
A seed icon.

+

A seed icon.

A seed icon.
levo quark
savory taste on the left
A seed icon.

+

A seed icon.

A seed icon.
levo anti-quark
Anaxagorean sensations are like these painted building-blocks.

Chemical quarks are building-blocks that we can combine to model more complicated particles. We use them to define atomic bonds and molecules. There are 6 different chemical seeds, so there are six different chemical quark types. Each type includes an ordinary-quark and an anti-quark, for a total of 12 particles. Chemical quarks are described by their internal energy  U as shown in the accompanying table. Note that this internal energy is reported in (eV) not (Mev), the chemical quarks are much smaller than thermodynamic quarks.

Chemical Quark Traits

Quark Type

Quark IndexInternal Energy

\mathsf{Z}

 \zeta

U (eV)

11-2.216
12-1.801
13-2.114
14-2.549
15-0.02875
16-0.04894

Atomic Bonds

Following custom, EthnoPhysics portrays molecules as compound atoms that are held together by atomic bonds. The bonds that we consider here are defined from pairs of electrons. We intend to count these atomic bonds, as for example in Lewis dot diagrams or VSEPR theory . So we need to follow some rules of arithmetic and logic. Specifically, by Anaxagorean narrative conventions, Cantor’s definition of a set, and Pauli’s exclusion principle, we cannot have two identical electrons in the same description. So we need to distinguish electrons from each other. And, for molecules with more than one bond, we need to distinguish atomic bonds from each other too.

Seeds are not enough, so Mr. Quirky is here to remind us about other sensations.
Don’t forget about other sensations. Click for more.

Traditionally, we meet this requirement by saying that various bonds and electrons are distinct because they are in different places. But, by the premise of EthnoPhysics, we cannot innocently resort to any sort of spatial explanation.

So instead we invoke the first hypothesis and assert that bonds are distinguished by association with various taste sensations. That is, we presume that bonds and electrons are logically distinct due to their union with various chemical quarks. Then subsequently, we use atomic bonds to define a Cartesian system for describing spatial relationships. The order of definition is reversed. And for EthnoPhysics, empirical analysis comes before dogmatic ideas about space.

The letter  \mathbb{B} is used to identify specific bonds in the following discussion. Bonds are characterized by  D_{\mathsf{o}} a dissociation energy given by

\displaystyle D^{\mathbb{B}}_{\mathsf{o}} \, \equiv H^{\mathbb{B}} - N_{\!\mathsf{e}} \, H^{\mathsf{e}} = \sum_{\mathsf{q} \in \mathbb{B}} \Delta n^{\mathsf{q}} U^{\mathsf{q}}

where H is the enthalpy and  U marks the internal energy. The coefficients of quark  \mathsf{q} are noted by   \Delta n^{\mathsf{q}} \,  , and the number of electrons in the bond is written as  N_{\!\mathsf{e}} \, . Experimental observations report  D_{ \mathsf{o}} for the gaseous state.1Bond Dissociation Energies in Simple Molecules National Standard Reference Data Series Number 31, B. deB. Darwent, U.S. National Bureau of Standards, 1970., 2Bond Dissociation Energies of Organic Molecules Stephen J. Blanksby and G. Barney Ellison, Accounts of Chemical Research 36 (4), 2003., 3Strengths of Chemical Bonds J. Alistair Kerr and David W. Stocker, CRC Handbook of Chemistry and Physics 81st Edition, Lide, D.R. (Editor), Boca Raton, Florida, 2000.

Single Bonds

When  N_{\!\mathsf{e}} \! = \! 2 \, , links are called single bonds and depicted by a short line segment. For example H–H indicates the tie between hydrogen atoms in a diatomic molecule of hydrogen gas.

Some single bonds are modeled from groups of quarks that are all the same quark-type, they are unusually homogeneous. For a few of these bonds, quark character can be experienced directly as the taste of an associated molecule. Thus basic quarks taste like lye-soap, and ionic quarks taste like table-salt, etc. These models are called archetypical bonds. But more often quark-types are varied, and the taste of a molecule is not specifically related to its chemical quarks.

To start a quantitative analysis of single bonds, we associate one acidic quark with one of the electrons of a bonding pair. This arrangement is called \mathbb{B} \small{\mathsf{(anti} \, \mathsf{acidic)}} . It is the only ligature defined from just one chemical quark

\mathbb{B} \small{\mathsf{(anti} \, \mathsf{acidic)}} \equiv { { \mathsf{e^{-}}, }, \mathsf{e^{-}} }

This bond correctly models the dissociation energy of Au–Au, a diatomic gold molecule. And here is another bond with two acidic quarks

\mathbb{B} \mathsf{(acidic)} \equiv { { \mathsf{e^{-}}, }, \mathsf{e^{-}}, }

This arrangement accurately represents the bond in H–Cl, a strong acid. So \mathbb{B} \mathsf{(acidic)} is an archetypical bond. The two acidic quarks are distinct from each other because one of them is logically associated with an electron, and the other is not. In the following models, we make extensive use of similar nested sets to satisfy Pauli’s exclusion principle. But no more acidic quarks can be added to this bond. So next we consider a pair of basic quarks

\mathbb{B} \mathsf{(basic)} \equiv { { \mathsf{e^{-}}, }, \mathsf{e^{-}}, }

This collection of quarks and electrons accurately represents the bond in sodium hydroxide, Na–OH, also known as lye. More archetypical bonds can be defined using ionic and aqueous quarks to model molecules of table-salt and water, as shown in the following table.

A table of single molecular bonds.

Hydrides

Acidic quarks are featured prominently in models of binary hydrogen compounds. For example, this collection accurately represents the atomic bond in hydrogen gas

\mathbb{B} ( \mathrm{H}_{\mathsf{2}} )  \equiv { { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, , }, }

The following table shows models that all work within experimental uncertainty, even though observations of these molecules are remarkably precise.

A table of bonds for binary compounds of hydrogen.

Salts and Halogens

It is possible to construct models of salt molecules and halogen gases that make extensive use of ionic quarks. But laboratory observations that report only two or three significant figures do not constrain the theory very much. Models that do not include ionic quarks are also possible.4For details about calculating bond strengths, please see the spreadsheets titled Atoms & Photons and Bond Data. So the following table is not exhaustive, especially for the bonding in salts.

A table of single bonds for salts and halogens.

Double Bonds

The single bonds discussed above all involve one pair of electrons. But we may also include more electrons to define double bonds that involve four electrons. For example, here is a double acidic bond

\mathbb{B} \mathsf{(double \ acidic)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, , } }

This model correctly yields the dissociation energy of sulphur dimers, S=S. And here is a double basic bond

\mathbb{B} \mathsf{(double \ basic)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, } }

The strength of the double-bond in carbon dioxide, O=CO, is correctly represented by this arrangement. For another example, the link between oxygen atoms in O2 is accurately given by this double aqueous bond

\mathbb{B} \mathsf{(double \ aqueous)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, }, \mathsf{e^{-}} }

These examples do not specify how pairs of electrons are associated to form a specific single-bond within the molecule’s double-bond. For that, we depend on dextro and levo-quarks. These stereochemical quarks only have about 1% of the internal-energy of other chemical quarks, but nonetheless, they still play an important logical role by distinguishing between similar arrangements. For example consider this double-bond

\mathbb{B} \mathsf{(double \ wet)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, } }

which accurately yields the dissociation energy of diazine, HN=NH. If we imagine that each single-bond of the pair is formed from one wet-quark and one stereochemical-quark, then there are two distinct possibilities. They can be written as

\mathbb{B} \mathsf{(double \ wet \ type1)} \equiv { { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, } }, { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, } } }

\mathbb{B} \mathsf{(double \ wet \ type2)} \equiv { { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, } }, { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, } } }

Both of these bonds contain the same quarks, but they are still logically different from each other. And in the laboratory, chemists do indeed find two different forms of diazine The distribution of stereochemical quarks is thus associated with geometric isomerism.

Ball-and-stick model of the cis-diazene molecule, N2H2
cis-diazene
Ball-and-stick model of the trans-diazene molecule, N2H2
trans-diazene

Triple Bonds

The forgoing double-bonds all contain just four electrons, but we may also include another pair of electrons to define triple bonds such as

\mathbb{B} \mathsf{(triple \ wet \ acidic)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, \mathsf{e^{-}} }

The very strong bonding in carbon monoxide, C≡O, is correctly represented by this arrangement. And here is another triple-bond that accurately models the link in N≡N, a molecule of nitrogen gas

\mathbb{B} \mathsf{(triple \ aqueous)} \equiv { { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, , }, { \mathsf{e^{-}}, }, { \mathsf{e^{-}}, }, \mathsf{e^{-}}, }

The following table summarizes some of these double and triple bonds

Sensory interpretation: All of the bonds discussed above are defined from three distinct classes of sensation; sour, salty and sweet tastes. They may vary independently from each other. So, in an upcoming article, we use these bonds to define a three-dimensional Cartesian coordinate system for making space-time descriptions of molecules. And after that, we stop worrying about Pauli’s principle.

Atomic Clocks

Atoms are stable, three-dimensional particles. So a position  \bar{r}, and a time of occurrence  t, can be associated with anything that happens to an atom. Assigning a position is discussed later in an article about length. But we can readily assign a time-of-occurrence to atomic events just by thinking of atoms as little clocks.

Atoms oscillate and jostle about in complex ways. So many different modes of atomic vibration have been examined in the laboratory, compared with historical standards of timekeeping, and assessed for their practical use as clocks. Some are excellent. Atomic fountain clocks can make time measurements that are good to one part in 1014, and they are still being improved. High precision laboratory work often uses atoms of caesium. But here is a generic way to tell time using any atom,  \mathbf{A} . Let  \mathbf{A} be described by a repetitive chain of events

\Psi^{\mathbf{A}} = \left(   \mathsf{\Omega}_{0}, \, \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2} \, \ldots \, \mathsf{\Omega}_{k} \, \ldots \right)

Since  \mathbf{A} is being used as a clock, we assume it has been sufficiently stabilized and isolated so that its vibrations are steady and regular. We assume that the period of the atom  \widehat{\tau}^{\, \mathbf{A}} has been measured, so that the clock is calibrated. Then we can determine some elapsed time after the initial event just by counting atomic vibrations to determine k. We write

\Delta t = t_{k} - t_{0} = k \widehat{\tau}

Without loss of generality, let t_{0} \! = \! 0 so that t_{k} = k \widehat{\tau}. And recall that the angular speed is given by \omega = 4\pi / \widehat{\tau} \, . And so event  k occurs at a time given by t_{k} = 4\pi k / \omega \, . Also remember that the phase angle  \theta is defined by \theta_{k} \equiv \theta_{0} + 4\pi k \, . So we can write

t_{k} = \dfrac{\, \theta_{k} - \theta_{0} \, }{  \omega}

This relationship expresses the time-of-occurrence of the  kth event as a function of the phase-angle. Time is thus told. However, we often think of time as the independent parameter and write the phase-angle as a function of time; the  k-subscript is dropped, and   \theta ( t ) is substituted for   \theta_{ k}. Then rearranging gives

\theta (t) =    \theta_{0} + \omega t

This form is good for describing the rotating motion of particles when they are framed in a Cartesian view. It can also be used to eliminate  \theta from other expressions. For example the displacement previously stated as

d \! \bar{r} = R \! \left(\rule{0px}{11px} - \! \sin{\theta}, \; \cos{\theta}, \; \lambda / 4\pi \! R \, \right) d \! \theta

can also be written

d \! \bar{r} = \omega R  \left(\rule{0px}{11px} - \! \sin{\omega t}, \; \cos{\omega t}, \; \lambda / 4\pi \! R \, \right) d \! t

Atoms are objectified from eight-fold patterns of sensation, similar to the anthropomorphic motifs in this Indonesian textile
Two by Four Tampan, Paminggir people. Lampung region of Sumatra, 19th century, 55 x 59 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.

an icon showing tools to indicate work in progress

Orbital Angular Momentum

The orbital angular momentum vector

\mathrm{\overline{L}} = \overline{r} \times \overline{p}

Total Angular Momentum

Here are some total angular momentum quantum numbers. Recall that \delta_{\hat{e}} is the electric polarity, \delta_{\hat{m}} is the magnetic polarity and \delta_{\hat{z}} is the helicity

{\textsl{\textsf{J}}} \equiv \dfrac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \, }{8} = \delta_{z} \! \dfrac{ \,  N^{\mathsf{U}} - N^{\mathsf{D}} \, }{8}

{\textsl{\textsf{J}}}_{m} \equiv \dfrac{ \, \left| \, N^{\mathsf{A}} - N^{\mathsf{M}} \, \right| \, }{8} = \delta_{m} \! \dfrac{ \, N^{\mathsf{A}} - N^{\mathsf{M}} \, }{8}

{\textsl{\textsf{J}}}_{e} \equiv \dfrac{ \, \left| \, N^{\mathsf{G}} - N^{\mathsf{E}} \, \right| \, }{8} = \delta_{e} \! \dfrac{ \, N^{\mathsf{G}} - N^{\mathsf{E}} \, }{8}

The total angular momentum vector is written as  {\mathrm{\overline{J}}} = \left( {\mathrm{J}}_{m} , \ {\mathrm{J}}_{e} , \ {\mathrm{J}}_{z} \right)

\overline{\mathrm{J}} \equiv \overline{\mathrm{S}} + \overline{\mathrm{L}}

Exchanging quarks for anti-quarks switches conjugate seeds but does not change dynamic seeds, so  {\mathrm{\overline{J}}} ( \mathsf{P} ) = {\mathrm{\overline{J}}} ( \mathsf{\overline{P}} ).

In general, the components  {\mathrm{J}}_{m},  {\mathrm{J}}_{e} and  {\mathrm{J}}_{z} have non-zero values, and P’s motion is complicated. But for a particle that is not electrically or magnetically polarized N^{\mathsf{A}} = N^{\mathsf{M}} and N^{\mathsf{G}} = N^{\mathsf{E}} . Then \delta_{\hat{e}} = 0 and \delta_{\hat{m}} = 0 and P is centered around the polar axis. So the angular momentum vector is aligned with the polar-axis.

get

\left\| \, {\mathrm{\overline{J}}} \, \right\| = \dfrac{h}{\rm{2} \pi} \sqrt{ \, {\textsl{\textsf{J}}} \, \left( {\textsl{\textsf{J}}} + 1 \right) \; \rule{0px}{10px} }

Next

EthnoPhysics faviconAtomic Hydrogen

The ground-state of atomic hydrogen is defined. Twenty eight excited states are also specified. Energy levels are compared with experiments.
References
1Bond Dissociation Energies in Simple Molecules National Standard Reference Data Series Number 31, B. deB. Darwent, U.S. National Bureau of Standards, 1970.
2Bond Dissociation Energies of Organic Molecules Stephen J. Blanksby and G. Barney Ellison, Accounts of Chemical Research 36 (4), 2003.
3Strengths of Chemical Bonds J. Alistair Kerr and David W. Stocker, CRC Handbook of Chemistry and Physics 81st Edition, Lide, D.R. (Editor), Boca Raton, Florida, 2000.
4For details about calculating bond strengths, please see the spreadsheets titled Atoms & Photons and Bond Data.