(12)
where ne is a new quantum number which varies between −(n − 1) and (n − 1). The motion itself can be described in the following way. If we calculate the electric “center of gravity” S of the electron in its orbit, that is, the average value of its coördinates over one revolution, we find that it is on the major axis at a distance from the nucleus O in the direction towards the aphelion. On account of secular perturbations, this point moves with simple harmonic motion in a plane perpendicular to the field E, whence it follows that ne can only change by ±1. The split of spectral lines is completely determined hereby, in good agreement with the results of experiment, also as regards the intensities calculated by Kramers.
For the Zeeman effect the calculation is still simpler and more-over can be carried out for atoms with an arbitrary number of electrons. The expression given earlier for the energy in a magnetic field is, neglecting terms containing the square of the field strength (Equation (14), Lecture 2):
(13)
where H0 is the energy of the unperturbed system. The Vector potential of a homogeneous field is
Therefore
where Pφ is the component of angular momentum P = Σr × p parallel to the field. For the unperturbed system the angular momentum is constant in magnitude |P| and direction. It is easy to see that 2 π |P| is an action integral. We place therefore
(14)
The components of P are also constant, but they are evidently conjugate to degenerate angle variables. In the magnetic field the degeneration of the angle φ, which fixes the position of the plane determined by the field and the angular momentum with respect to a fixed plane parallel to the direction of the field, is removed and the system pre-cesses around the direction of the field. Pφ is conjugate to φ, as easily seen. We have therefore the new quantum condition
(15)
If α is the angle between the angular momentum and the direction of the field, then evidently
FIG. 8
The axis of angular momentum can therefore only take 2j + 1 different directions (m = − j , ···+ j) with respect to the direction of the field axis. We shall call this result, following Sommerfeld, “directional quantization.”
The energy is
(16)
whence the number of revolutions of the axis of angular momentum, the so-called “Larmor frequency,” is
(17)
Precession does not influence the components of the motion of the electrons in the direction parallel to the field. Therefore there is no additional term in the z-component of the electric moment and light oscillating parallel to z corresponds to jumps for which m does not change. The components of motion perpendicular to the field are altered however by simple rotations in one direction or the other, hence the emitted light must be decomposed into two circularly polarized waves in opposite directions, to which correspond the jumpsm → m ± 1.
We obtain therefore the classical Zeeman triplet without any change. This contradicts experiment, however, for in most cases spectral lines are split up in a much more complicated way. Bohr’s theory in its present form gives no explanation of this more complicated effect. According to it we should expect in all cases and for every atom normal Larmor precession and the normal spectral triplet. At this point many attempts have been made to change the theory. Starting from Sommerfeld’s researches, Landé has succeeded in decomposing the observed Zeeman separation of most spectral lines into terms and discovered their relation to the periodic system of the elements. Heisenberg, Pauli and many others have investigated this problem further. The essential result of all these investigations is that the so-called “abnormal” Zeeman effect—which is, however, certainly the normal case—finds no place in the semi-classic theory which we have developed here.
The positive result is that the Zeeman effect is closely connected with the construction of atoms out of electrons describing orbits to which correspond fixed quantum numbers. We shall now treat this problem of the arrangement of electronic orbits in the atom, following the method of Bohr, who considered the series spectra as modifications of the hydrogen spectrum.
LECTURE 7
Attempts towards a theory of the helium atom and reasons for their failure—Bohr’s semi-empirical theory of the structure of higher atoms—The optical electron and the Rydberg-Ritz formula for spectral series—The classification of series—The main quantum numbers of the alkali atoms in the unexcited state.
The most obvious way of finding an exact theory of atomic structure would be to consider successively the simplest atoms, helium, lithium, etc., following hydrogen in the series of the elements. This has been tried, but even the first step from the hydrogen to the helium atom proved unsuccessful. The helium atom is an instance of the three-body problem: one nucleus and two electrons. It is well known that the three-body problem has greatly perplexed astronomers and that it has not been possible to represent the motion by analytical expressions (series) which really permit a survey of the motion at all times. In the case of atomic structure, conditions are even less favorable, for in celestial mechanics there is at least the advantage that the attraction towards the central body is much greater than the other attractions on account of the preponderant mass of the sun, so that all these other attractions can be looked upon as small “perturbations.” In atomic mechanics, however, all the attractions and repulsions of electric charges are of the same order of magnitude. On the other hand, the atomic problem has an advantage of a different kind, precisely on account of the postulate of the quantum theory, that only certain “stationary” orbits come into consideration. It has been shown that the quantum conditions allow only very simple types of orbits, because they exclude certain librations (oscillations).
Based on this result, attempts have been made to find the stationary orbits for the helium atom, and calculate its energy levels. The line of attack has been along two directions: Some investigators have considered the normal state of the helium atom (Bohr, Kramers, van Vleck), others the excited state, in which one electron is in the nearest orbit to the nucleus and the other revolves in a very distant orbit (van Vleck, Born and Heisenberg). Both calculations give incorrect results: The calculated energy of the normal state does not agree with experimental results (ionization energy of the normal helium atom), and the calculated term system for the excited states is different from that observed, qualitatively as well as quantitatively.
After all, no other result could be expected, for the validity of the frequency condition is sufficient to show conclusively that in the realm of atomic processes the laws of classical theories (geometry, kinematics or mechanics, electrodynamics) are not right. That in certain simple cases, as for a single electron, they give partially correct results is, in fact, more astonishing than that they fail in the more complicated cases of several electrons. This failure of the theory in the case of interactions among several electrons is evidently connected with the following fact: We know that electrons react quite unclassically to light waves, because the latter produce quantum jumps. In a system made up of several electrons, each electron is in the oscillating field due to all other electrons and the periods of these fields are of the same order of magnitude as those of light waves, therefore we have no reason to expect that the electron should react classically to this oscillating field. This point of view gives grounds for understanding why we obtain, by the classical theory, correct results in many cases of the one-electron problem.
Appreciating these difficulties, Bohr has given up the attempt to construct a truly deductive theory, and, instead, has endeavored, with the greatest success, to discover, by interpretation of facts, above all of the facts relating to the spectra and the chemical and magnetic properties of the atoms, something about the arrangement of the electrons. The starting point was the observation of the fact that the spectra of certain atoms are of a type quite similar to the hydrogen spectrum. The lines, or better, the terms form series quite similar to the series of terms of the H-atom. Rydberg, for instance, s
howed that in many cases expressions of the form with δ =constant, are sufficient for expressing the terms. This is the case for the alkali metals, for part of the lines of Cu, Ag, Au, and for other similar cases, all of which share chemical properties which indicate the easy detachment of one electron. From this Bohr concludes that all these spectra, as that of hydrogen, are produced by the jumps of one electron, the “optical electron.” This electron, however, does not move around a simple nucleus, but around a core consisting of the nucleus and all the remaining electrons. If the optical electron is more and more strongly excited—that is, brought up to levels of higher energy—the state of total separation called ionization is gradually reached and then the core is left as an “ion.” This argument agrees with the results of the chemists, as formulated by Lewis, Langmuir and Kossel. According to these theories the ions of the alkali metals have the same structure as the atoms of the neighboring inert gases: The latter are the most stable closed electron configurations.
Now it can be shown that the orbit of the optical electron in the lower stationary states must penetrate into the core, for otherwise the terms would differ but little from those of the H-atom. More-over, the radii of the ions are fairly well known from the theory of electrolytes and of polar crystals and it can be estimated, by a method to be given shortly, that the orbit of the optical electron must go through the core (Schrödinger, Bohr).
In postulating such “penetrating orbits,” a step is taken which is incompatible with ordinary mechanics, for following our quantization rules we must assume that the orbit of the optical electron is exactly periodic, but this cannot be understood from the standpoint of mechanics because of the intensive interaction with the inner electrons. It would be necessary to assume that the whole electronic structure is rigorously periodic in every quantized state and it is very questionable whether there exist such solutions of the mechanical equations. If in spite of this we wish to describe the paths of the optical electron within the realm of our theory and more or less approximately, this may be done following Bohr and in a purely formal way, by replacing the action of the core on the electron by a central force and neglecting altogether the reaction of the electron on the atomic core. Then the conservation of energy is always satisfied for the electron alone and we have to do, as hitherto, with a one-body problem. Conservation of angular momentum also holds and the orbit is plane.
We can show, following Bohr, that the terms must be approximately expressible by formulas of Rydberg type or more accurately of Ritz type, assuming that the core is small compared with the size of the orbit of the optical electron. The outer part of the orbit differs only slightly from a Keplerian ellipse, while the inner part is an oval of small radius of curvature, because there the electron comes in a region of strong nuclear attraction (Fig. 9).
If we replace the outer part of the orbit by an ellipse, then its energy becomes
(1)
FIG. 9
In this formula n× depends on the aphelion distance 2a×, in the same way as was given above for other quantized orbits, i.e.
(2)
and Zx is the “effective nuclear charge,” that is the difference between the charge of the nucleus and of the screening electrons of the ion. n× need not be an integer, for it is not the main quantum number of the whole orbit; if we call the latter n then the frequency of the motion from one aphelion to the next is given by
(3)
On account of our assumption that the core is small, the time of revolution the whole orbit differs slightly from the time of revolution of the ellipse replacing the actual path. The latter is given by
(4)
We therefore place
and consider b to be approximately constant; then we have
which integrated gives
and solving approximately for n×
(5)
where δ1 is an integration constant. δ2 = RbZ ×2 is determined by the mechanical system. Of course δ1 depends also on the second quantum number of the system, for the motion in a central field is double-periodic; one of the periods is the one which has been already considered, namely the motion of the electron from perihelion to perihelion with the main quantum number n; the second is that of the revolution of the perihelion itself with the quantum number k; hk is therefore the angular momentum of the electron and as the rotation of the perihelion is simple-periodic therefore k can change only by ±1, as in the case of the relativity correction for the H-atom. δ1 is a function of k; this can also be approximately found by means of relatively simple considerations.
The expression which we have found in this way for the value of the term
(6)
agrees exactly with the term formulas found empirically by Rydberg (δ1-term) and Ritz (δ1-and δ2-terms).
Since k can have different values, every atom has several series of terms. In fact we should expect, on account of the selection rule for k (k → k ± 1), that the latter can be so classified that a term of a series can be combined only with the terms of neighboring series. This indeed is the case. It is usual to classify the terms in series according to the following scheme:
where an s-term can be combined only with a p-term, a p-term with s- and d-terms, etc. From this we conclude with Sommerfeld that the following correspondence holds:
We shall now proceed to determine the main quantum numbers for all the observed terms. For this purpose we must above all determine whether the path in question is a penetrating orbit. We calculate from the observed term the effective quantum number n× in accordance with the formula
The aphelion distance then is known, that is the major axis 2 a× of the ellipse replacing the path. Moreover the parameter 2 P of this ellipse is known. This parameter depends on the value of k as shown by the formula
Therefore, the whole equivalent ellipse is known approximately and it is possible to determine whether it penetrates into the atomic core, the size of which is known from the ionic volume. If in this way the conclusion is reached that the path is wholly outside the core, then the Rydberg correction δ1 is small, that is n× is nearly an integer. If such is the case then n can be chosen as the next integer to n×. In fact all the terms corresponding to exterior paths d(k = 3), f(k = 4) . . . behave in this manner.
FIG. 10
On the other hand the s-terms (k = 1) and the p-terms (k = 2) correspond in general to penetrating orbits. Here n× departs considerably from integral values. δ1 (k) is then quite large, frequently larger than 1 or 2. The actual determination of δ1 requires approximation formulas, for the derivation of which fairly rough assumptions are sufficient. In every case the main quantum number n can be determined with fair certainty.
The main quantum number of the normal state is thus of the greatest interest. The most important result can be stated as follows: For every alkali atom (hydrogen included) the main quantum number of the normal state of the optical electron is increased by 1:
LECTURE 8
Bohr’s principle of successive building of atoms—Arc and spark spectra—X-ray spectra—Bohr’s table of the completed numbers of electrons in the stationary states.
Bohr’s construction of the periodic system is based on the supposition that every atom can be derived by the addition of one electron to an ion which is constructed essentially as the previous atom and has the same number of electrons. On this depends the possibility of deriving the structure of one atom from that of the previous one. It is first assumed that the core of the second atom has the same structure as that of the first atom and then, on the basis of a simple estimate of the Rydberg constants, it is seen whether the spectrum is not in contradiction with their value. In many cases we also know the spark spectrum, that is the spectrum of the ionized atom, which is considered as produced by one optical electron rotating around a core of a structure similar to that of the second previous atom having the same number of electrons. We understand from this the so-called “spectroscopic displacement law” given by Sommerfeld and Kossel. The structure of the s
pectrum of a neutral atom (often called “arc spectrum” because of the most convenient means for its production) resembles the first spark spectrum of the next higher atom, the second spark spectrum of the following atom and so on; except that Rydberg’s constant R must be replaced by 4 R, 9 R, . . . or generally Z×2 R. We have already used the simplest example of this rule, where the correspondence of the spectra is quite exact, when we spoke of the spectra of H, He+, Li++, ... together by introducing an arbitrary nuclear charge Z.
An electronic configuration once formed is buried more and more deeply inside the atom as it proceeds along the periodic system of the elements. Now the X-ray spectra furnish means of examining the inner parts of the atom. The production of these spectra depends, according to Kossel, on the following process: As all the quantum orbits are, so to speak, full, it is impossible for an electron to jump from one orbit to another. It is necessary that an electron be previously removed by supplying energy (electron impact or absorption of X-rays). Then other electrons may fall from higher orbits into the gaps left free and in this way the emission lines of the X-ray spectrum are produced. According to whether the removed electron had the main quantum number n = 1, 2, 3 . . . we name the line emitted when this electron is replaced, a K,L,M... line; and according to the origin of the substituting electron we indicate the line by indices Kα , Kβ . . . Lα , L β . . . or by new quantum numbers. The correctness of this conception can be tested by observing that for the X-ray lines, the Ritz combination principle must hold. Of course the energy values on the differences of which the frequencies depend are directly given by the so-called absorption limits. In the spectrum of absorption of an atom there must exist sharp limits or “edges” which separate the frequencies the energy-quanta hν of which are greater or less than the work necessary to remove to infinity the electron describing the orbit which is responsible for the absorption. In this way the system of X-ray terms is determined as exactly as that of the optical spectra.
The Dreams That Stuff is Made of Page 81