Summarizing Atom and Ion Structure: The Periodic Table of the … · In the later 1800s, Dmitri...

Summarizing Atom and Ion Structure: The Periodic Table of theElements

Lawrence E. Murr*Metallurgical and Materials Engineering, The University of Texas at El Paso, El Paso, TX, USA

Abstract

Having illustrated symbolically and mathematically the wave-particle dualism applied to electronsin atoms in chapter “A Brief Introduction to Quantum Mechanics”, the symmetrical and asymmet-rical behavior of quantized electron energy states in composing or characterizing atoms in thedevelopment and prediction of atomic chronology implicit in the periodic table of the elements isexplored. The systematic or asystematic addition and subtraction of electrons from the quantumstates or substates of atoms is discussed as this creates ions, both positive (cations) and negative(anions) valence states. This serves as a precursor to understanding change balance and atomic orionic binding in the creation of solid or condensed matter to be discussed in chapter “ChemicalForces: Molecules and Nanoparticles” following.

Introduction

The concept of atoms as fundamental building blocks for solids (or condensed matter) dates to atleast 440 B.C., but the Greeks, in particular Aristotle, were generally opposed to the concept. Plato(circa 330 B.C.) envisioned fundamental categories of “elements” as air, fire, earth, and water, alongwith the ether or the “universe” to be represented as five polyhedral forms respectively: tetrahedron,octahedron, cube (or hexahedron), icosahedron, and dodecahedron, referred to as the Platonic solidsillustrated in Fig. 1. These characterize the initial 5 polyhedral forms for the regular-faced, convexpolyhedra as described in the lifework of Berman (1971). Polyhedra characterize the growth ofcrystals as nanocrystals or nanoparticles, where the icosahedron and faceting of the cube corners toform a cube octahedron or a truncated octahedron shown in Fig. 2 dominate. The cube octahedronand truncated octahedron shown in Fig. 2a, b represent progressive polyhedral in the sequence ofregular-faced convex polyhedral discussed on detail by Berman (1971). The Platonic solids in Fig. 1represent the first 5 of these while the cube octahedron and truncated octahedron are ranked number6 and 9, respectively. The regular faces or polygons composing these polyhedral are represented inthe associated nets in Fig. 2a, b.

As noted in Part I, metal extraction, pottery making, glass working, pigment development, etc.,were all related to chemistry, but early metallurgy as described in De Re Metallica and Alchemy, theprotoscience of chemistry, was not able to explain the nature of matter and materials in relation totheir processing or fabrication. Robert Boyle in 1661 presented a clear distinction between alchemyand chemistry, whereupon chemistry emerged as a science with the development of the chemicalfundamentals such as the conservation of mass by A. Lavoisier (1743–1794), along with thermo-dynamic principles established later in the work of J. W. Gibbs (1839–1903). Lavoisier and JosephPriestly, around the same time, almost simultaneously discovered oxygen, and Priestly was also

*Email: [email protected]

Handbook of Materials Structures, Properties, Processing and PerformanceDOI 10.1007/978-3-319-01905-5_6-1# Springer International Publishing Switzerland 2014

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involved in a wide range of both electrical and chemical experiments – a precursor to electrochem-istry. His work in fact influenced M. Faraday and Maxwell. Sir Humphry Davy also influenced thework of Faraday in the early 1800s by his publication, “On Some Chemical Agencies of Electricity.”Davy’s pioneering work in electrolysis allowed him to discover Na, K, Mg, Ba, and B. He also

1 Tetrahedron

3 Cubo

2 Octahodron

5 Dodecahedron

4 Icosahedron

Fig. 1 The five Platonic solids

Fig. 2 Nanocrystal growth as cuboctahedra (a) or truncated octahedra (b). The nets composing the polygons or faces areshown. Note triangles in the cuboctahedron become hexagons in the truncated octahedron. (c) Projections of Al2O3

particles showing octahedral facets in the transmission electron microscope


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discovered nitrous oxide and its effect as a “laughing gas,” but he never considered its use as ananesthetic.

In the later 1800s, Dmitri Mendeleev, a Russian chemistry professor, and J. Lothar Meyer,a German chemist, independently published periodic tables of the elements in rows or columnsaccording to their atomic weight, starting new rows or columns when certain characteristicsrepeated. Mendeleev left gaps in his table where it seemed elements had not been discovered, andhe was the first to use trends in his chart to predict properties of the missing elements. Gallium andgermanium were among the notable elements to be predicted. Mendeleev also occasionally ignoredthe order suggested by the element’s atomic weight. It became apparent that Mendeleev hadorganized the elements in order of increasing atomic number (Z). Because of the importance ofthe periodic arrangement of the elements and the ability to predict new elements and their positionsin the periodic chart, it might be assumed that Mendeleev would have received a Nobel Prize. Whilehe was in fact nominated in 1906 and 1907 for the Nobel Prize in chemistry, he was rejected onpersonal bases, in particular his criticism of the work of S. Arrhenius, who pressed for his rejectionby the Nobel Committee. In addition, this ordering was consistent with the number of protons whichequaled the number of electrons (Z).

Antonius Van der Broek, almost simultaneous with an idea by Ernest Rutherford in 1911,suggested that the central charge and number of electrons in an atom was exactly equal to itsplace in the periodic table. Consequently, by the time Schrödinger developed his quantum waveequation (Eq. 13 of chapter “▶A Brief Introduction to Quantum Mechanics”), it was well recog-nized that atoms were characterized by their number of electrons or atomic number, Z, whichbalanced the number of protons in the nucleus. Using this hypothesis, Henry Moseley, in 1913,measured the wavelength of X-ray photos for electron transitions produced by elements rangingfrom Al (Z ¼ 13) to Au (Z ¼ 79) using X-ray excitation and concluded that the atomic number wasrelated to the electron quantum level energy transition, or

DE0 / 1

l0¼ C Z� sð Þ2, (1)

where c is a constant and s is a nuclear screening factor which increases from ~3 to ~5 for elementswith Z ¼ 30 (copper, Cu) to Z ¼ 90 (thorium, Th). This accounts for the increasing shielding orscreening provided as the number of electrons and the corresponding electron cloud surrounding thenucleus increases. Equation 1 has become known asMoseley’s law and forms the basis for elementalanalysis in matter using X-ray energy dispersive spectrometry (EDS) which measures characteristicX-ray energy, E0 (or DE0), or wavelength, l0 according to Eq. 1, quantifying elemental or atomicconcentrations in real-time analyses.

Physical Meaning of the Wave Function: Wave Model of AtomsWhile we have dealt with the energy of atoms in relation to atomic energy levels and quantumnumbers, we have not specifically considered the physical meaning of the wave function, c, whichrepresents the wave amplitude. Correspondingly, the notion that an electron beam is considered to bea wave with amplitude, c, the beam intensity will be represented by c2 or c*c ¼ |c|2 where c* isthe complex conjugate of c. Assuming dV to be an element of intra-atomic volume, we can observethat

e

ðcj j2dV ¼ e or

ðcj j2dV ¼ 1 (2)


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http://dx.doi.org/SpringerLink:ChapterTarget

which expresses the total charge of the electron cloud is equal to e. Therefore, e |c|2 may beconsidered the charge density of the atom. According to M. Born, the quantity |c|2 dV is theprobability of finding an electron in a given volume element, dV. This means that the quantity |c|2

becomes the probability density, and in this contextc is the amplitude of the probability. In this case,Eq. 1 indicates that the probability of finding an electron inside an atom is unity, and calculating thequantity |c|2 from the wave equation only allows for the probability of finding an electron at a givenpoint in space. This is expressed by the Heisenberg uncertainty principle:

Dx � Dp ¼ ћ (3)

where Dx is the measure of the uncertainty of the geometric position on an electron (in reference toFig. 4 of chapter “▶A Brief Introduction to Quantum Mechanics”) and Dp is the momentumuncertainty (p ¼ mv). The significance of Eq. 3 is that the more precisely we can determine theelectron momentum or energy, the less precisely its position can be determined. Correspondingly,since the energy of an electron in the steady state of an atom is fixed as given in Eq. 20 of chapter“▶A Brief Introduction to Quantum Mechanics.” it follows that the “orbit,” implicit in Fig. 4 ofchapter “▶A Brief Introduction to Quantum Mechanics,” is impossible to define.

In retrospect, it might be recalled that with reference to Fig. 4 (chapter “▶A Brief Introduction toQuantum Mechanics”) (ideally representing a one-electron atom, the hydrogen atom), the quantumnumbers n, l, and ml are directly related to the positional coordinates describing the orbital positionof the electron. The radial position, r, is dependent on n, while l and ml relate generally to sphericalcoordinates y and f. If we imagine a normalized wave function having arbitrarily fixed angles y andf for a momentarily fixed orbital position for the electron (Fig. 4 of chapter “▶ABrief Introductionto Quantum Mechanics”), then the r-dependent wave function, c(r), for the solution of theSchrödinger equation in spherical coordinates represented by Eq. 21 of chapter “▶A Brief Intro-duction to Quantum Mechanics,” can be plotted simply as a function of r. The electron orbit can beconsidered to be established when |c(r)|2 is unity for each quantum state (n). Figure 3 illustrates thisargument in representing probability distributions or atomic orbital probabilities for helium (Z ¼ 2),beryllium (Z ¼ 4), and carbon (Z ¼ 6) for electrons at a distance r from the nucleus (Fig. 4 ofchapter “▶A Brief Introduction to Quantum Mechanics”).

To conceptualize a multi-electron atom requires the superposition of probability distributionsshown in Fig. 3 visualized in 3 dimensions. As electrons are added, this complex distributionexpands, creating a dense “cloud” which increasingly shields the nucleus (Eq. 1). The importantfeatures of the atoms or elements are therefore represented by the outer reaches of this complex, 3-D,probabilistic electron cloud. When viewed, the atom or atoms composing solid matter, especiallycrystals composed of regular arrays of atoms, will appear as arrays of “cotton balls.” This is in factwhat is observed in various modes of visualization or atomic imaging shown in Fig. 4 for chargedvolumes on a crystal surface viewed in the field ion microscope in Fig. 4a in contrast to electronclouds imaged in the transmission electron microscope (TEM)where an electron beam interacts withthe electron cloud characterizing each atom in Fig. 4b. The TEM image in Fig. 4a highlights rows ofatoms composing specific crystal planes, specifically the {002} planes in fcc gold.

Actually, the notion of a dense electron cloud of probabilistic electron shells surrounding thenucleus implied in the atom images shown in Fig. 4 is somewhat misleading since from a spatial


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perspective, the actual size of an electron and the orbital radii, rK, rL, etc., in contrast to thecorresponding size of the nucleus represents this regime as mostly empty space, just as the milkyway galaxy and the universe in total represent mainly empty space relative to the sizes of thecomponent planets, moons, galaxies, stars, etc. Collectively these complex concepts providea physical model of a quantum-mechanical atom.

Fig. 4 Images of atoms. (a) High-resolution image of gold atoms or atom planes viewed edge-on in a thin gold film inthe transmission electron microscope. (b) Atom sites observed as helium ions on electric field lines from surface atoms atthe edges of atomic planes of tungsten (Z ¼ 74) in a field ion microscope (FIM)

Fig. 3 Atomic orbital probabilities represented only as radial functions, rK, rL. (not to any scale)


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The Periodic System of the Elements: Periodic TableThe representation of electrons in atoms as quantized states, along with the Pauli exclusion principleand the fact that each atom is characterized by a least-energy ground state represented by the K-shellor quantum number n ¼ 1, provides proof that atoms or elements follow a periodic systemrepresented ideally in Fig. 6 of chapter “▶Electromagnetic Fundamentals.” As shown in Eq. 24of chapter “▶A Brief Introduction to Quantum Mechanics,” the maximum number of electrons inthe n-quantum number electron groups is 2n2. These electrons are distributed among n-subgroupswith maximum number of electrons in each subgroup given by 2 (2 l + 1). This number isdetermined by the possible values of the quantum number ml (for given values of l) and bythe � 1/2 for Ms. This diversity of electron states and substates (as represented graphically inFig. 6 in chapter ▶A Brief Introduction to Quantum Mechanics) can be written as:

K n ¼ 1ð Þ L n ¼ 2ð Þ M n ¼ 3ð Þ N n ¼ 4ð Þ O n ¼ 5ð Þ1s 2s 3s 4s 5s

2p 3p 4p 5p3d 4d 5d

4f 5f5g

In state n ¼ 1 (1s) there are 2s electrons (since l ¼ 0) forming the two-electron K-shell or groupof the atom (1s2). In state n ¼ 2, we have 2s electrons and 2 (2 � 1 + 1) ¼ 6 p electrons (2p6) sincel ¼ 1 in Eq. 24 of “▶ABrief Introduction to QuantumMechanics.” This produces 8 electrons in theL (n ¼ 2) group, consisting of the subgroups 2s and 2p as shown above. For the M-shell (n ¼ 3),there are three subgroups which fill as follows: 3s23p63d10 since the total number of electrons in theM-group is 18. However, experiment shows that the maximum number of electrons for the M-group(18) is only found in heavier elements beginning with copper, Z ¼ 29. In elements following Ca(Z ¼ 20), the 4s and 3d quantum substates overlap, and this produces a quantum state fillinganomaly beginning with scandium (Sc) (Z ¼ 21) and ending with Cu (Z ¼ 29). Elements in thisanomalous series are referred to as transition elements and possess unique properties such asmagnetism (especially Fe (Z ¼ 26). After copper, the systematic electron filling of subgroupscontinues up to Z ¼ 36 (Kr). For krypton (Kr) and the next two elements (Rb: Z ¼ 37 and Sr:Z ¼ 38), the N-group stops at 8 electrons because their outer electrons occupy the 5s substate andnot the 4s substate which continues another anomalous series up to palladium (Pd: Z ¼ 46) wherethe number of N-state electrons reaches 18. Other anomalous groups include elements identified asrare earths (cerium, Ce: Z ¼ 58 to lutetium, Lu: Z ¼ 71) and the actinides: thorium, Th (Z ¼ 90) tocalifornium, Cf (Z ¼ 98). These features are represented in the periodic table of Fig. 5 whichillustrates valence states, or the charge of the atom depending upon the loss or gain of electrons (+or,respectively). Atoms (or elements) with the same valence number or charge are generally arranged incolumns beginning with positive ions (or cations) with charge +1, beginning with hydrogen,followed by the next column of positive ions with charge +2, beginning with beryllium at the leftof the table. The extreme right column contains the rare gas elements which ideally have zerovalence, beginning with helium (Z ¼ 2).

An examination of the electronic structure represented in the periodic table of Fig. 5 can providea predictive matrix for compound formation (chemistry) especially for ionic or electrovalent solids


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or ionic crystals represented ideally in Fig. 10 of chapter “▶Electromagnetic Color and Color inMaterials” for NaCl (Na+Cl�) and Fig. 21 of chapter “▶Examples of Materials Science andEngineering in Antiquity” for CaF2 (Ca

2+ F2�). Note in Fig. 5 that elements to the left of the rare

earth column usually represent anions (having a negative charge) since these elements formingionic molecules accept electrons. Those columns headed by F (Z ¼ 9) and O (Z ¼ 8) have a valenceof 1- and 2-, respectively, although other possible valences are noted. Ionic or electrovalentcompounds are therefore formed by the transfer of electrons from elements in columns to the leftto element in columns to the right in the periodic table. In the case of covalent (shared electron)bonds, the theory of electron pairs represents the most important generalization of the results ofquantum-mechanical analysis of simple molecules such as C Cl4 which can also be representedelectrovalently as C4 + Cl4

�. Similarly as a covalent compound, CH4 can also be representedelectrovalently as C4� H4

+ or CO (C2+O2�) as illustrated in Fig. 5 where carbon is represented bya valence of 2 or 4. However, for covalent binding which prevails, the 4 electrons in 2p for carbon areshared with 3p in Cl for C Cl4, stabilizing 2s (2s

2) for carbon and 3p (3p6) for chlorine. In this case,we can examine the carbon outer electron configuration: 2s22p2. Carbon can also stabilize itselectronic structure in the 2s2 substate by giving 2 electrons in the 2p2 substate filling. Thiscorresponds to a valence or charge of +2. Correspondingly, carbon can receive 4 electrons in thep-substate to stabilize (or fill) 2p6, having an electrovalence of �4. In contrast, carbon can give upthe 2s22p2 electrons to stabilize the ion in the 1s2 substate, corresponding to a valence of +4. It is thisquantum state or substate filling which forms stable and even energetically favorable pairing ofelements forming molecules. This is demonstrated by the diamond structure in Fig. 7 of chapter“▶Electromagnetic Color and Color in Materials” where the strong covalent bond is formed onsharing 4 electrons with 4 carbon atoms, but two carbon atoms can be ionically stabilized by 2s22p2

going to stabilize the 2p substate with 4 electrons. In this case the valence becomes 4+ and 4�.CaCO3 (Ca

2+C4+O2�) is a more specific ionic or electrovalent compound example.The most stable electronic configurations are observed to be the filling of the nsnp quantum

substates: n ¼ 1 (2 electrons in 1s2), n ¼ 2 (10 electrons in 1s22s22p6), n ¼ 3 (18 electrons in1s22s22p63s23p6), etc., which is represented by the rare (or inert) gases in the extreme right columnin the periodic table of Fig. 5. Considering the nsnp filling or stabilization rule can also determine thenormal or preferred valence states. For example, for the column to the left of the inert gases, thevalence is 1-, while for the next column it is 2-. But, as noted, the column represented by C and Si canhave mixed valence since fewer electrons are involved.

The gain or loss of electrons by an atom creating positive (+) or negative (�) valence(or electrovalence) must correspondingly cause an increase or decrease in the resulting ion sizewith respect to the neutral atom having 2 electrons. This feature is illustrated in the correspondingperiodic table shown in Fig. 6. The atom sizes for neutral atoms usually imply metal (metallic)elements which are characterized by metallic binding (M in Fig. 6). Covalently bound atom sizes arealso noted in Fig. 6 by C as noted. Weak binding atom sizes are also denoted V representing van der


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Waals binding. We will discuss binding in more detail in Chapter 7 following. Figure 6 presentsa scaled representation for atom or ion (+or �) sizes (radii) corresponding to Fig. 7. It can beobserved in this relative size representation of the periodic table of the elements that the columns ofelements at the left and right of the table show increasing size with increasing Z in each column(going down the columns) while the sizes also generally increase in elemental rows from left to right,also representing increasing Z from left to right in each row.

Figure 8 illustrates size and density variations detected in an Si–Fe compound by atomic forcemicroscopy. Compare Si and Fe radii as represented in Fig. 6, for example.

It is of interest to point out that the representation of “elements” invested in the Platonic solids(more than 2300 years ago) shown in Fig. 1 have been configured in more recent times to representelements from Z ¼ 1 to Z ¼ 92 by nesting, or systematically and geometrically placing one or moreof the Platonic polyhedra within one another. Simple Platonic duals, where one polyhedron can beplaced within the other and vice versa, include the cube and octahedron and the tetrahedron and theicosahedron and dodecahedron. Four nested Platonic polyhedra including the icosahedron inside the

Fig. 5 Periodic table of the elements showing valence and electron configurations (From Staudhammer and Murr(1973))


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dodecahedron, the octahedron inside the icosahedron, and the cube (or hexahedron) inside theoctahedron contain 46 vertices which form half of what is referred to as the “Keplerian atom” andillustrated in Fig. 9. This concept was originated by Robert Moon in the mid-twentieth century. Itforms a geometric basis for the periodicity of the elements to create an electron shell model usingPlatonic solids and their duals, commonly referred to as the Moon Model (Hecht 2000; Hecht andStevens 2004). In some respects, this polyhedral nesting also emulates shell and cluster formation tobe discussed in chapter ▶Chemical Forces: Nanoparticles.

Fig. 6 Periodic table of the elements showing ionic and atomic radii (From Staudhammer and Murr (1973))


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Fig.7

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theelem

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s)andions.S

izes

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References

Berman M (1971) Regular-faced convex polyhedra. J Frankl Inst 291(5):329–352Hecht L (2000) Advances in developing the moon nuclear model. 21st Century Fall:5–12Hecht L, Stevens CB (2004) The moon model. 21st Century Fall:58–74Smith WF (1990) Principles of materials science and engineering. McGraw-Hill, New YorkStaudhammer KP, Murr LE (1973) Atlas of binary alloys: a periodic index. Marcel Dekker,

NewYork

Cube

Icosahedron Octahedron

Dodecahedron

Fig. 9 Sequence of four Platonic polyhedral whose 46 vertices form half of the “Keplerian atom”

Fig. 8 Atomic force microscope image differentiating Si and Fe atoms in a mixed compound (Courtesy of OscarCustance, University College, London)


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Index Terms:

Atomic force 11Atomic orbital probabilities 5Gold atoms 5Keplerian atom 9, 11Moon Model 9Nanocrystal growth 2Periodic system of the elements 6Platonic solids 1–2Wave model of atoms 3


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