The Periodic System and Atomic Structure ZIZ. Wave Mechanical Interpretatiom WILLIAM J. WISWESSER* Cooper Union, New Y o r k City
I
N A prevlous paper the intellectual, practical, and teachmg . 'values of wave mechanics were noted, and a simplified wave-mechanical approach to the Periodic System was summarized (1). This summary should be consulted as a necessary introduction for the following more detailed discussions and supplementary notes on the remaining eight sections of the summary. SECTION 111.. BEHAVIOR WAVES AND PATTERNS
A. The Electron's Matter Wave Heisenberg's uncertainty principle (Section 11, 13) showed that, in the atomic extremities of nature, statistical limitations govern the meaning of "points" and "instants" just as they do the meaning of pressure, difiusion, or temperature. Newton's corpuscular mechanics (and force analysis) fails with small, high-speed particles just as his corpuscular or geometric optics fails for fine shadow details. The uncertainty about the atom-bound electron's Newtonian behavior was evident in the blurred motions previously described (2). However, this dilemma is merely a "tempest in a teapot," as Swann has insisted (3), for the electron's Hamiltonian behavior (with action analysis) is described with far more certainty than the best insurance tables can claim for vital statistics. The wave analysis indicates, as Hartree said (4),that the atom's electric charge distribution is static, even though the charge itself (matter wave phases) may not be. Quite recently, Menius and Rosen (5) proposed a classical field theory of elementary particles, in which a matter function $, formally similar to the Schrodinger function, was found to change with time though fixed in space. Wave functions (more recently called statefunctions) are, in the final analysis, statistically precise descriptions of natural states of behavior. Hence it is appropriate to call them behavior patterns, and compare them with other patterns (cycles, fields) in the macroworld.
ganisms; in high-speed machinery, music, light, and electricity. Our living bodies give immediate evidence for the great variety of cyclic phenomena peculiarly associated with l i e : there is the rhythm of breathing, the faster pulse of the heartbeat, the still faster alpha, beta, and delta brain wave rhythms. Less regular rhythms are evident in the blinking of our eyelids, in the peristaltic digestive pulses, in the incessant sweeping beats of respiratory cilia, and in the complex oscillations of nerve impulses. A striking chemical example of wave action is found in the mysterious rhythmic precipitation of Liesegang rings, recently given a wave-mechanical treatment (6). The property cycles of the elements in the Periodic System make this famous example of recurrent behavior stand out in its spaceless, timeless uniqueness. Gradually the enormous practical value of knowledge about recurrent phenomena is being recognized, and this greater understanding is bound to bring greater human benefits (7) John J. Grebe, Director of the Physical Research Laboratory of the Dow Chemical Company, showed that the practical interests in wave behavior are fully as great as the theoretical. When he was awarded the 1943Medal of the Society of Chemical Industry, he said in his acceptance address (8): "The major portion of all our technical and scientific activities has to do with cycling of all kinds. . . AU the natural sciences deal with cycles of various frequencies which, if undesirable, should be understood and controlled at their inception, rather than Fought with violent action after they arein full swing. . . ."
.
Dr. Grebe then illustrated the amazing number of correlations in afrequacy chart: (1) binding energies, from molecular to nuclear; (2) electron beam energies; (3) the electromagnetic spectrnm from magnetic suuspot disturbances to cosmic rays; (4) wave lengths of de Broglie particle waves; (5) mechanical and thermal vibrations, from those of constellations to those of B. Waves, Cycles, and Vibra#ions atoms; (6) vital cycles, from civilizations to cell divisions; and (7) radioactive half-lives, from Uranium I1 The evidence reviewed in the previous paper (2) for the wave nature of matter should not have been start- to Thorium C' (Po 212). So seriously does Dr. Grebe ling. Everyday experience shows that there are natural regard the industrial implications of cycles that he convibrations and cycles of all kinds, associated with all siders comprehensive unifications of this kind "the forms of matter and energy. Recurrent behavior is logical starting point for all our education in the scievident in the heavens, cloud formations, weather, and ences." sunspot activity; in civilizations, business, and individual moods; in living plants, animals, and microor- C. The Fine-grained Unity i n Nature Optics and mechanics became more perfectly unified * Present address: Willson Products, Inc., Reading. Pennin Rohr's theory (Section I, D) which recalled Hamilsvlvania.
ton's Principle of Least Action (1835). This principle, equal in analytical power to Newton's Principia of forces, interrelated the optical path of least time (Fermat's Principle, 1660) and the mechanical path of least distance (d'Alembert, 1742; de Maupertuis, 1744). Maxwell unified electricity and magnetism, and Poynting's quietly famous theory showed that the vector product of the electric and magnetic field vectors determines the intensity and direction of electromagnetic radiation. Now in the same way, the electron's distribution intensity is determined by the product of two conjugate expressions, the Schrodinger wave equations. The wave equations themselves have a deeply penetrating symmetry and unity; if any two except the complex conjugates are multiplied together, the integral of the product is zero, instead of one. Any wave equation can be represented by some combination of the others; the diierentials and integrals are symmetrically related, somewhat like the sine and cosine functions. Not only are positive numbers matched by negative ones, but real numbers by equally probable complex nnes, giving the four unit alternates:
+.\/Ti, -.\/Ti.
+a. -1/=i
Dirac's four wave equations (see Section V) r d e c t further unity in the four dimensions of space-time, and in the four quantum numbers. His comprehensive theory (9, 10) also accounted for spin and magnetism and predicted the positive counterpart of the electron. Hill and Landshoff (9) showed how Dirac's theory unified 16 mathematical operators: 3 each for the space coordinates of electric field,magnetic field,current density, and angular momentum; and 1 each for charge density, rest mass, spin, and an unnamed measure. In Dirac's more recent field quantization theory (11) he proposed still greater convergence of theories by including "negative probabilities" and "negative energy photons" as well as positive ones. Unfortunately, these profound unifications are far from simple and self-evident (aside from growing complications in other fields such as cosmic rays, nuclear properties, and atomic bombs). They are cloistered in highly technical articles such as those just cited (9,10, 11, 12); lost in a veritable jungle of spinors, tensors, and matrices; sealed with hermitian, unitary, and adjoint functions; and carried by space operators that commute and anticommute with one another. No wonder these fearful accounts are unpopular, even to the curious chemist! There are some simpler hints that further unifications remain to be discovered or proved. I t has been suggested (13) that there may be an ultimate unit to probability itself, marking off the "highly improbables" from the "utter impossibles." The Sommerfeld fine structure reciprocal (137), which interrelates three natural constants (Section II), is a new fundamental number, (1 4' - 5!) according to Eddingtou (14). The two roots of Eddington's related relativity equation :
+
10m2 - 136mm0
+ m;
= 0
give the proton-electron mass ratio ml/ml
=
curiously close to a recent (1942) "best" experimental value of 1837.5. (The uncertain mass of the meson, around 110 to 180 electronmasses, also is curiously close to this 136 or 137.) Eddington mixed his cosmological constants with Fermi-Dirac statistics to suggest that the number of elementary particles in the universe equals the square of, (a).the universe-to-electron radius ratio, and (b) the electrostatic-to-gravitational force ratio between electron and positron. How very interesting this would be if true! IV.
WAVE P A m R N S AND NODES
A . Mechanical Analogies Mechanical motion really is not necessary to explain the origin of atomic wave or field patterns, but the Bohr and vector models are so suggestive, and the mechanical illustrations of standing waves so convincing, that these dynamic interpretations will be retained for many years. Nevertheless, it is well to recognize beforehand the prevailing contemporary view most recently stated by Gustaf Stromberg (15): "It cannot he too strongly emphasized that this vibrating ~~robability] system does not consist of moving particles or of a v~hratingbody, but is simply a convenient mathematical representation of a probability structure of the same type as that needed to characterize electron beams."
This does not deny the reality of the static electric charge patterns, nor the possibility of vibrations in their wave components (complex imaginaries). Stromberg, in fact, maintains that the pattern has a more fundamental reality than the particle, being a self-contained field unit that even transcends the physical world of space-time (see Section X). Margenau (16) sharply criticized the view of "many competent physicists, some of them brilliant leaders in research, who while regarding the electron as an ultimate eonstituent of some unchanging reality, look upon the +-function of quantum mechanics as a formal and useful artifice having no 'real' existence."
Margenau pointed out that realists build sensible world descriptions "from perceived colors, sounds, aromas, and tactual impressions," and the action patterns art the same sort of physical realities. The only kind of mechanical motion necessary to illustrate (by analogy) the behavior of electrons is wave motion; and for atom-bound electrons, standing waves are the only kind necessary to consider.
B. Standing Waves
Stationary or standing waves are so called because the vibrating system producing the wave pattern contains regions which do not move. These fixed region* points on a vibrating string, lines on a vibrating membrane, or surfaces in a vibrating chamberall are called nodes (L., nodus = knot). The restricting boundary, such as the end of a whirling string or the rim of a drum, usually is regarded as a single node. Successive intermediate nodes define successive harmonics of the simp1847.51)9, lest or fundizmatnl pattern.
Standing waves have four basic characteristics pertinent to these discussions:
difficult task to show the exact intensities in the iutermediate regions.
1. There is no transfer of energy to or from the vibrating system. 2. Every part of the vibrating body pass- through theequilib. rium position at the same inatant. This property gives rise to new nodes of a higher dimension when the pattern isrotated, as shoran by previous illustrations (1). ' 3. The vibration pattern has as many types of nodes as it has dimensions. Thus, three-dimensional vibrations require three nodal numbers for their description. 4. The vibration pattern can be represented equally well by a motionless field having an intensity variable such as a gravitational, electrical, or magnetic potential gradient.
C. Simple Dynamic Models H. E. White, in making his intriguing "electroncloud" photographs (IS), used a symmetrical spindle, turned on a lathe to give the radial distribution curve; it was centerpivoted and conically rotated while constrained to the proper 8 angles b y a vertically adjusted line. While his resulting photographs cannot be surpassed for accuracy, the apparatus is not ideally simple. Spinning cardboard models that make direct use of the mathematical curves are more desirable for their instructive value and simplicity, even though the results may be less rdned.
The simplest wave patterns are illustrated by the vibrating string, symmetrically fixed at both ends. A whirling string, with one free end, shows more complex wave forms, illustrated by Morse (17). Those of vibrating bars or beams are still more complex, due to the material rigidity and variable cross section. Two-dimensional wave forms, in the simplest cases, are described by mutually perpendicular circular nodes on a spherical membrane, or by circular and diametrical nodes on an infinitely large planar membrane (17). Three-dimensional patterns, such as sound waves originating from a point source and confined in a spherical enclosure (limiting node), may be considered a combination of the preceding two types. These various types of standing wave patterns all are described most easily in terms of the constantly fixed nodes, or loci of zero vibration, rather than the more complex and changing regions of maximum vibration. As in cutting pies or melons, it is much easier to describe the number and types of slices rather than the more variable number and shapes of the resulting pieces. I t is very easy to make spinning disc models showing the origin and exact location of the nodes, but a much more
PAPER CLIP
3" DISC
The equipment needed to demonstrate radial wave patterns is even simpler than that previously described for the vector model (Z), and the method is identical with that of White for these primitive patterns. A button, two paper clips, and a piece of thread are all that are necessary to spin 3-inch discs cut from noncnrling index cards. These parts are assembled as described in Figure 1, utilizing an old parkr trick. The disc must he carefully cut, and the button carefully centered when cemented fast. The "motor" string is wound by wheeling the disc, and then if teusian is applied by hand "power" through the end clips, the disc spins rapidly. If the pull is relaxed harmoniously after an initial flexingand some practicethe thread continuaUy rewinds itself. This entertaining device easily develops over 1000 =.p.m. in rapidly reversing whirls of 3 or 4 seconds' duration. The high speed can be verified from sound effects. Besides this siren demonstration, the device incidentally illustrates gyroscopic stabilization, and additive color effects if the curve areas are tinted. The thread itself demonstrates the existence of rapidly changing , nodes in complexly blended linear vibration. The only wearing part is this severely twisted suspension, but surely the maintenance of this highly educational toy is worth a spool of thread!
Three-dimensional views are illustrated by the use of photographic technique. A rotating mask, such. as
CUT FROM NONCURLING INDEX C A W
\
HOLDER
/
YE%
4 FI: LOOP O F
FIGURE1.6RdPtE SPINNING MODEL FOR SBOWING PATTBRNS WITH RADIAL NODESONLY
The radial curve is plotted in polar coordinates (arbitrary
+
units of angle) so that the m tegration of the intensity.
e area represents the correct circular in-
the disc in Figure 1, is made by plotting all radial curves abwe the x-axis. Then these areas are cut out to make exposure windows. A negative is centerpinned underneath, and the exposure is varied (by time or diaphragm) according to the lateral wave intensity as the mask is rotated. Demonstration discs printed from this negative are spun on their vertical axis diameter (polar axis), conveniently with a laboratory stirier having a slotted shaft. Meridian nodes may be shown with stroboscopic illumination; or, in static models, by exposing a set of disc prints according to the meridian variation, then cutting into halves and slotting them into a common equatorial disc. Simpler disc patterns may be made by repeating the radial curve a large number of times in the manner suggested by Figure 3 of the e s t paper (1). While this model is more approximate, it probably is more instructive, as the same pattern-repeating principle may be applied to discs to illustrate the meridian nodes (uniformly black discs.)
+
+
Dirac W * = hh* f hh* $&* Vhh* Also, in this Dirac model as in the vector model, the spin quantum number s has a bite numerical magnitude: unity in a very strong magnetic field, degenerating to '/2 in weak fields (normal states). As a result, the spin-corrected distribution patterns remain essentially as previously described (with finite minima a t the nodes) only if s has the same direction as I and the same sign as m. Otherwise they have one less h e a r node if sJ opposes 1 ( j = 1 - s), or if s, does not have the same s). When the latter is true the sign as m (mJ = m loss is a meridian node. Thus, in the normal weak field 1is displaced by j, and m by m,, as with the vector model axes (Section I, E). These differences will be evident from a study of Table 1, and from comparison with the vector model illustrations (Part 11, Figures 2 and 4).
-
D. Color Effects Multicolor schemes can be employed to distinguish the magnetic alternates, spin alternates, and complex conjugates in spinning models. The "resolved" and "merged" relations also can be shown in such models. For example, the "resolved" positive areas may be distinguished by blue (+s) or green (-s) color while all minus areas retain the red color of the previously illustrated (resolved) patterns, these same colors showing clear through the disc. Then when spun, the "merged" pattern is revealed in lilac (+) and tan (-). Patterns without meridian nodes have opposite colors on the corresponding reverse side of the disc, still alternating with blue or green to show plus or minus spins, respectively. This coloring system is explained in Figure 2, and other variatiofis may be employed to emphasize diierent properties. These color-mixing effects are most striking if the nodal background is blacked out with an air brush, after the regions are flat colored as indicated. SECTION V.
ATOM-BOUND ELECTRONS
The need for a fourth quantum number becomes evident in designing the simple disc models (with resolved patterns). Each region in the pattern marked off by spherical, conical, and meridian nodes has a characteristic algebraic sign. Thus for each pattern, a second alternate with over-all opposite sign, should exist in the same phase time. This distinction is indicated by the fourth and h a 1 sign or spin quantum number. The Dirac theory ( 9 , l O ) (Section 11, l l ) , which gave theoretical justfication for this fourth number, introduced a welcome reiinement to explain how the electron passes the nodal regions; but this relativistic improved e n t also introduced some complications and changes. Using C . G. Darwin's formulation, the new probability distribution is a sum of four Scbrodinger-like products:
Back Front Back Front Resolved (static) colors Merged colors Nodes are shown with solid lines Frcung %-COLORSCHEME FOR D I ~ T I N ~ U RESOLVED I S H ~ ~ vs. MERGEDPAmEUTs R=Red
G=Green B=Blue
T=Tan L=Lilac
For an elementary understanding,. these relativitycorrected patterns are unnecessary and (like relativity itself) tend to be confusing. For example, pattern (nlllm~+sj)is identical except in size with (&ma-sj). However, these corrected patterns more closely resemble the vector model dynamic views (Section 11, E) and thus give a better accounting for the spectroscopic "doublet" grouping of energy levels. SECTION Vl.
THE PLAN OF THE PERIODIC SYSTEM
The introductory summary ( I ) showed how the simple Schr6dinger wave patterns account for every atom in the Periodic System. It must be emphasized that these patterns do not embody the complicating relativistic spin corrections, nor the distorting effect of inner layers of electrons. They are hydrogen-like (pointnucleus), spin-free, or nonrelativistic patterns, correctly scaled to a uniform radial maximum. Hartree's famous calculations for real atoms (4) showed that the distorting effect merely pulls the inner spherical nodes closer to the nucleus; the atom's total charge patternshowing the layer maxima-is simply the sum of individual
electron distribution patterns, each adjusted to the varying effective nuclear charge. The empirical sequence measure n 1 - 1/(1 1) showed the correct order of the subgroups, but not the pattern sequence within any subgroup. Table 1, column 1, shows this pattern sequence, as deduced by j
+
+
TABLE 1
Wcok Field Po11c.n No.
Pallam no.
-
SYn-frcc Nodes I m s.
Spin-corrected nodes 0 rn' I' j
A . The Schrddinger Wave Eguntions De Broglie initiated the matter-wave concept by combining the relations of Planck and Einstein as indicated in Section 11, 9. Then Schrodinger put de Broglie's definition of electronic wave length into the classical wave equation, in order to discover the atomic energy-space relations (Section 11, 10). The classical differential equation for wave motion of any kind is a v e q fundamental expression. It is used not only in the analysis of true waves of all kinds (mechanical vibrations, sound, heat, light, and electricity), but also in single pulses, in fluid flow, heat flow, and the basic "equation of continuity." (When written as shown below, it even suggests the relativity definition of time as part of a fourth dimension of length. The space-time "distance interval" or "event length" is: iads' = dxP
+ dya + dzs + i V d t 2
where the imaginary i marks off the distinction that we cannot "go backwards" in time.) The wave equation states that any kind of intensity variable or field potential 11. (such as displacement, pressure, temperature, concentration, density, electric or magnetic field strength) must have its second partial derivatives with respect to the space-time coordinates related as follows: and mj for the normally weak or negligible tllagnetic field. Alternatively, column 2 shows the strong field order (m 2 4 , applicable when the spin-orbital interaction is broken and the spin quantum number has its limitine unit value ( 2 ~ ~ ) .
+
Usually the terms involving the space coordinates x, y, z are abbreviated as V 211., called the Laplacian (of $). The operator V ("del") symbolizes the gradient (of a scalar 11.) or the divergence (of a vector $1; that is, . .
SECTION VU.
THE MATHEMATICAL EQUATIONS
Students are not expected to perform atomic weight determinations before thev first use the atomic weight -. -. ...- table, nor evaluate trigonometric and logarithmic functions before they use those tables. Nor need they derive the ~chrijd:mgerequation in order to understand its surprisingly simple solutions. The sfherical hurmonk kquations and solutions were developed long avo-introduced bv Laolace in 1785 and extended by Legendre from then on until 1827. They are discussed in great detail in many source books, with (18, 19, 20, 21, 22) and without (23, 24) reference to the atomic structure applications. The problem for the chemical reader is to make a satisfvinn condensation rather than - -~ -.further elaboration, and (he blowing paragraphs represent an effort in this direction. Glockler (25) gave an interesting introduction to the Schrodinger wave equation, stressing familiar mechanical applications and similarities in the analytical methods. In the appended discussion by W. A. Noyes, Jr., the revealing point was stressed that students should first have "a real appreciation and feel for physical phenomena" before the rigorous mathematics. The illustrations and sequences in this presentation (1) Were drawn with this foremost in mind.
-
~~0~
Now standing waves are a periodic function of time = ar-2""') so this t variable c a n be eliminated by the followinn identitv:
-
- ha* At?
=
""
[?g$, ,,. ,
Hence for standing Waves of all kinds,
. .
Schrijdinger replaced the wave length h for stationary, atom-bound electrons by its de Broglie equivalent in terms of electric charget Q& radius r, and total energy e, employing Hamilton's action analysis :
h' - e,) - 2m(e + Qa/rr)
h' 2mer
2m(e
substitution gives the usual form of the general schro. dinger equation: 8a2m
v P * + - ,z
(
e +
$)*=o
The tedious tasks only begin with this-transforming to polar spherical coordi>~ltcs(Cartesian solutions are T h e dielectric should not be confused e, mentioned elsewhere
the Naperian
almost impossibly difficult), factoring into functions of the separate variables (+, 8, and r ) , and finding the "finite, continuous, and single-valued" proper or characteristic or eigen functions with corresponding eigenvalues. Once obtained, the final results are surprisingly simple and easy to understand. It is something like discovering that the awesome-looking quantity (with coefficientsadding to unity) 32 case 4
- 48 cos4 4
+ 18 cosP0 - 1
is simply cos G+, defining a meridian pattern with six nodes.
The Meridian Function of the Equatorial Angle or Azimuth The meridian variations in the wave intensity J., are easiest of all to describe, being determined by a single quantum number m; showing simple sine or cosine variations between equidistant nodes which disappear in the resulting distribution function &bf. This last property is evident from the mathematical definitions: $6 = d m 4 cos m4 + i sin M * - 1 H.
+
++* =
