Optical Spectra and the Periodic System of Elements

S. Frisch, Leningrad

Since the time of Rydberg, who established the existence of spectral series, numerous attempts have been made to clarify the connection between the quantities characterizing these series and the position of the elements in the periodic system. For a long time, however, no such connection could be found, whereas in the younger field of X-ray spectroscopy the very first works of Moseley revealed the presence of a strict regularity linking the X-ray spectra of the elements with their atomic numbers, i.e., with their position in Mendeleev’s system. But now, when, thanks to the work of Millikan, it has proved possible to establish an analogous connection for optical spectra in the extreme ultraviolet region, it has turned out that it is precisely the study of optical spectra that makes it possible to penetrate more deeply into the essence of the periodic law; at the same time, the new data of spectroscopy compel us to modify to a considerable degree our views on the laws governing intra-atomic processes. Bohr’s original theory used ordinary mechanics and electrodynamics to determine the orbits of electrons inside the atom and only then, from among all possible orbits, selected—by means of the rules of quantization—a discrete set of stable orbits. In the case of a one-electron system, i.e., an atom consisting of a nucleus and one electron, elliptical orbits were obtained, characterized by two quantum numbers: the principal quan-

quantum number \(n\), which may take the values \(n = 1, 2, 3, \ldots\), and the azimuthal quantum number \(k\), which for a given \(n\) takes the values \(1, 2, \ldots n\). Later, however, Bohr’s theory encountered very substantial difficulties, especially in the interpretation of complex spectra and of the complex Zeeman effect. Only the hypothesis of Uhlenbeck and Goudsmit concerning the “rotating” electron enabled Goudsmit to construct a new scheme which not only does not contradict experiment, but also makes it possible to analyze those numerous complex spectra which until then had seemed completely incomprehensible.1

According to the hypothesis of Uhlenbeck and Goudsmit, each electron, along with its charge \(e\) and mass \(m\), is assigned a constant mechanical angular momentum \(\sigma\),2 equal to \(1/2\) in units of \(h/2\pi\), and a constant magnetic moment \(\mu\), equal to one Bohr magneton:

\[ \sigma = \frac{1}{2}\left(\frac{h}{2\pi}\right) \]

\[ \mu = \frac{e}{2mc}\left(\frac{h}{2\pi}\right) \]

According to the scheme developed by Goudsmit, the orbit of an individual electron in an atom is characterized by the following quantities:

a) the principal quantum number \(n\); \(n = 1, 2, 3, \ldots\)

b) the vector \(\mathbf{l}\)

c) the vector \(\mathbf{j}\)

For a given $n$, the vector $\mathbf{l}$ numerically assumes, in units of $h/2\pi$, the following values:

$$ l = 0,\ 1,\ 2,\ \ldots\ (n - 1) \tag{1} $$

The quantity $l$ replaces the former azimuthal quantum number $k$, which gave the angular momentum of the given electron orbit, but differs from it in that it takes values smaller by one. Because of this, the concrete representation of the form of the electronic orbits inside the atom is lost, and the whole scheme acquires, to a considerable degree, a formal character.

The vector $\mathbf{j}$ is the geometrical sum of the vectors $\boldsymbol{\sigma}$ and $\mathbf{l}$, and it is assumed that the vector $\boldsymbol{\sigma}$ can be situated only parallel or antiparallel to the vector $\mathbf{l}$; whence it follows that the numerical values which the vector $\mathbf{j}$ can assume for a given $l$ will be:

$$ j = l + \tfrac{1}{2}, \quad \left|l - \tfrac{1}{2}\right| \tag{2} $$

Thus, according to Hund’s scheme, the orbit of an electron inside the atom is uniquely determined by three quantities: $n$, $l$, $j$. It is customary to denote symbolically the orbits and the corresponding energies (or spectral terms) by means of letters, where

The principal quantum number $n$ is written before the letter, while the quantity $j$ is written at the lower right, as a subscript; thus, for example, an orbit characterized by $n = 2$, $l = 1$, $j = \tfrac{3}{2}$ is denoted as $2p_{\frac{3}{2}}$.

The state of the atom as a whole is denoted by capital Latin letters. For the simplest atom, possessing a single electron, capital and lowercase Latin letters have one and the same meaning, since its states are wholly determined by the motion of its single electron. For complex atoms, whose state can be uniquely determined only when the motions of all the electrons are given, lowercase and capital letters obviously have different meanings.

Using formulas (1) and (2), one can directly find the number of possible orbits, and consequently the number of possible energy levels in an atom with one electron. Scheme I is constructed so that it gives all possible numerical values of the vectors \(l\) and \(j\) for each given principal quantum number \(n\), and gives the symbolic designations of the corresponding orbits.

Scheme I

and so on.

Scheme I coincides with the well-known scheme of terms of the so-called doublet spectra, which are observed in alkali metals and certain other elements. According to what was said above, doublet spectra should be emitted by the simplest one-electron atoms, i.e. by hydrogen and ions similar to it (for example, \(\mathrm{He}^{+}\)). And indeed, even before the appearance of Hund’s scheme, Sommerfeld and Unsöld had shown that the hydrogen spectrum, if the fine structure of the lines is taken into account, must be regarded precisely as a doublet spectrum. Below we shall show that this scheme also explains the doublet character of the spectra of the alkali metals. For the moment, however, let us pass on to more complex atoms, consisting of a nucleus and two electrons, whose analysis will lead us to consequences very important for understanding the periodic system of the elements.

In a complex atom each electron moves in its own special orbit, which is characterized by its quantities \(n_i, l_i, j_i\). In addition, each electron has its inherent moment \(\sigma_i\) (where for every electron \(\sigma_i = \dfrac{1}{2}\)). Co-

the state of the entire atom as a whole is characterized by certain resultant vectors \(l\), \(\sigma\), and \(j\).

In the case of the so-called normal coupling between vectors, which, as Hund showed, is justified for a large number of elements, the resultant vector \(l\) is formed geometrically from the individual vectors \(l_i\):

\[ l = \Sigma l_i \tag{3} \]

In this case the individual vectors \(l_i\) may be arranged either parallel (or antiparallel), or at such angles to one another that the individual values of their geometrical sum differ from one another by unity. Thus, for example, for \(l_1 = 2\), \(l_2 = 3\), the vectors \(l_1\) and \(l_2\) can form only such angles with one another for which the resultant vector \(l\) will assume the values \(l = 5, 4, 3, 2, 1\) (see Fig. 1). In exactly the same way the resultant vector \(\sigma\) is the geometrical sum of the individual vectors \(\sigma_i\):

\[ \sigma = \Sigma \sigma_i \tag{4} \]

Fig. 1

The vectors \(\sigma_i\) may likewise be arranged either parallel (or antiparallel) to one another, or at such angles that the individual values of \(\sigma\) differ by unity.

Finally, the resultant vectors \(l\) and \(\sigma\) combine geometrically into the vector \(j\):

\[ l + \sigma = j \tag{5} \]

Here again \(l\) and \(\sigma\) are arranged either parallel (or antiparallel) to one another, or form between themselves

only such angles for which the separate values of their geometric sum differ from one another by one. Hence it follows that the vector \(j\), for given \(l\) and \(\sigma\), can assume the following numerical values:

\[ j = l+\sigma,\ l+\sigma-1,\ \ldots\ |l-\sigma+1|,\ |l-\sigma| \tag{6} \]

For \(l>\sigma\), the number of different values which \(j\) assumes is obviously \(\varkappa=2\sigma+1\).

Symbolically, as was indicated, the state of the atom as a whole is denoted by capital Latin letters, the letters \(S, P, D, F, \ldots\) being written respectively instead of \(l=0, 1, 2, 3, \ldots\). The numerical value of the vector \(j\) is written beside the letter in the form of a subscript; to the left of the letter, above, is written the value of the quantity \(\varkappa=2\sigma+1\). Thus, for example, the state of the atom characterized by the quantities \(l=2\), \(\sigma=1\) (therefore \(\varkappa=2\sigma+1=3\)) and \(j=2\), is symbolically denoted by \({}^{3}D_{2}\). No principal quantum number can be ascribed to the whole atom as a whole. For a completely exact designation of the state of the atom, strictly speaking, it is also necessary to indicate separately the orbits of all the electrons entering into its composition.

Fig. 2

Passing to the special consideration of an atom consisting of a nucleus and two electrons, we shall confine ourselves to the particular case in which one of the electrons always remains in the \(1s\) orbit, while the other electron is located in any possible orbit. Then for the first electron \(l_{1}=0\); for the second, \(l_{2}\) assumes any admissible values. Since \(l_{1}=0\), the resultant vector \(l\) coincides with the vector \(l_{2}\). The resultant vector \(\sigma=\sigma_{1}+\sigma_{2}\) (where \(\sigma_{1}=\sigma_{2}=\tfrac12\)) can have two values, \(\sigma=0\) and \(\sigma=1\), according as the vectors \(\sigma_{1}\) and \(\sigma_{2}\) are arranged parallel or antiparallel to one another (see Fig. 2). Considering these two cases separately, we obtain, using formula (6), two schemes IIa and IIb, giving the possible values of \(j\) for each given \(l\):

Scheme IIa

\[ \sigma = 0 \quad (\varkappa = 1) \]

Scheme IIb

\[ \sigma = 1 \quad (\varkappa = 3) \]

In the first case, to each \(l\) there corresponds one possible state of the atom—this coincides with the scheme of terms of the so-called singlet spectral series. In the second case, to each \(l\) (except \(l=0\)) there correspond three possible states of the atom, which coincides with the scheme of terms of triplet series. Their number gives the order of the “multiplicity” of the spectral terms.

Thus it turns out that an atom with two electrons must possess two different sets of series: a set of singlet series and a set of triplet series. And indeed helium has two such different sets of series that it was formerly assumed that helium was a mixture of two different elements, to which the names parhelium and orthohelium were assigned. The parhelium series are singlets; the orthohelium series were initially considered doublets, but the high-resolving-power instruments available to modern spectroscopy have revealed that they are very narrow triplets. A simultaneous set of singlet and triplet series is also observed in many other elements, which will be considered in more detail below.

The scheme obtained for the possible states of an atom with two electrons, however, disagrees with experiment at one point. When both electrons are situated in \(1s\) orbits, according to the scheme two different states of the atom are possible, \({}^{1}S_{0}\) and \({}^{3}S_{1}\), corresponding to whether the vectors \(\sigma_i\) of both electrons are directed parallel or antiparallel to one another (see Fig. 2). Study of the helium spectrum, however, indicates that in reality only one state, \({}^{1}S_{0}\), is realized. Likewise, in other more complicated cases the theory

sometimes gives superfluous states, not observed experimentally. This shortcoming of the theory is eliminated by means of Pauli’s “exclusion” rule, which is connected with the general considerations that underlie modern physical statistics and explain such diverse phenomena as, for example, the degeneracy of gases at low temperatures or the electrical conductivity of metals.

To formulate Pauli’s “exclusion” rule, a new quantity \(m_i\) is introduced, taking the values:

\[ m_i = j_i,\; j_i - 1 \ldots -|j_i - 1|,\; -j_i \tag{7} \]

Fig. 3

This quantity \(m_i\) may be interpreted as the projection of the vector \(\mathbf{j}_i\) onto some external preferred direction (for example, the direction of an external magnetic or electric field). It is assumed here that the vector \(\mathbf{j}_i\) can make with this preferred direction only such angles that the separate values of its projection onto this direction differ from one another by unity (see Fig. 3).

Then the “exclusion” rule is formulated as follows: there cannot exist in an atom several electrons characterized by identical numbers \(n_i, l_i, j_i, m_i\). Or: to each group of numbers \(n_i, l_i, j_i, m_i\) in an atom there can correspond no more than one electron. Or, in still other words: if in an atom there are several electrons with identical \(n_i, l_i, j_i\), then, according to Pauli’s principle, they must differ from one another in the values of the projections of the vectors \(\mathbf{j}_i\) onto the external preferred direction, i.e. in the different orientation of the vectors \(\mathbf{j}_i\) in space. As a consequence of Pauli’s “exclusion” rule we obtain:

a) since for a given \(j_i\) the quantity \(m_i\) cannot have more than \(2j_i + 1\) different values [see formula (7)], then

in an atom there cannot exist more than \(2j_i+1\) electrons having identical \(n_i, l_i, j_i\).

b) since \(j_i=l_i\pm \dfrac{1}{2}\) [see formula (2)], there cannot exist in an atom more than \(2(2l_i+1)\) electrons with identical \(n_i\) and \(l_i\) (“equivalent electrons”).

c) since for a given \(n_i\) the vector \(\mathbf{l}\) can numerically take the values: \(l_i=0, 1, 2, \ldots (n_i-1)\) (see formula (1)), the greatest possible number \(Z_n\) of electrons with identical \(n_i\) will be:

\[ Z_n=\sum_{l_i=0}^{n_i-1} 2(2l_i+1)=2n_i^2 \]

i.e. in an atom there cannot exist more than \(2n_i^2\) electrons with identical principal quantum numbers \(n_i\).

Applying the “exclusion” rule to two electrons situated in identical orbits \(1s\), we find that, since in this case both electrons are characterized by identical \(n_i\) \((n_1=n_2=1)\), identical \(l_i\) \((l_1=l_2=0)\), and identical \(j_i\) \((j_1=j_2=1/2)\), they must differ in their \(m_i\). Since in the present case \(|m_1|=|m_2|=1/2\), this is possible only when \(m_1\) and \(m_2\) have different signs, i.e. when \(j_1\) and \(j_2\), and consequently also the vectors \(\sigma_1\) and \(\sigma_2\), which in this case coincide with \(\mathbf{j}\), are antiparallel. In other words: according to the Pauli rule, a parallel arrangement of the vectors \(\sigma_1\) and \(\sigma_2\) in this case cannot occur (only the left arrangement of the vectors is possible, Fig. 2). It follows from this that the resultant vector \(\sigma\) can then have only one value, \(\sigma=0\), and that the atom can be only in one state, \({}^{1}S_0\); the second state \({}^{3}S_1\) is impossible. If, however, the second electron is situated not in the \(1s\) orbit but in a higher \(s\) orbit, e.g. \(2s\), then it will have a principal quantum number different from that of the first electron, and according to the Pauli rule both arrangements of the vectors \(\sigma_1\)—parallel and antiparallel—will become possible, and consequently both states of the atom, \({}^{1}S_0\) and \({}^{3}S_1\), will also be possible. Finally, we obtain the following scheme of possible states of an atom with

by two electrons, under the assumption that one electron is always in the \(1s\) orbit:

Scheme III

The symbolic designations given for the resulting states of the atom are not unambiguous. For example, the state of an atom denoted as \({}^{3}P_{1}\) may arise both as the result of the motion of the second electron in the orbit \(2p\), and as the result of motion in the orbit \(3p\) (it is assumed here that the first electron remains all the time in the orbit \(1s\)). Energetically these two states differ greatly from one another, and it is necessary somehow to distinguish them also in their symbolic designations. Spectroscopists usually denote the first of these states as \(2^{3}P_{1}\), and the second as \(3^{3}P_{1}\). In view, however, of the fact that such a notation is not suitable for more complicated cases, another notation is sometimes used, although somewhat cumbersome, but quite unambiguous: before the capital letter referring to the state of the whole atom, the symbols of the orbits of the individual electrons constituting this atom are written. For example, the state of the atom \({}^{3}P_{1}\), arising as the result of the motion of the 1st electron in the orbit \(1s\), and of the second in the orbit \(2p\), is written as \(1s \cdot 2p\ {}^{3}P_{1}\).

Consequence c) of the Pauli “exclusion” rule (see p. 119) asserts that there cannot exist in an atom more than two electrons with principal quantum numbers \(n_i = 1\), more than 8—with principal quantum numbers \(n_i = 2\), etc. Adding to this also consequence b) of the same page, we obtain the following table, giving the greatest possible number \(Z_n\) of electrons with identical \(n\) and their distribution over orbits with identical \(l\):

Table I

This table, together with the data of spectroscopy, makes it possible to trace the arrangement of electrons in orbits in individual atoms of the Mendeleev periodic system.

In the normal state, both electrons in the helium atom are situated in $1s$ orbits. According to the Pauli principle, these two electrons constitute the greatest possible number of electrons with principal quantum numbers $n_i = 1$. Moreover, they lead to the single resultant state $^1S_0$, which is characterized by the fact that all three resultant vectors $\sigma$, $l$, $j$ are equal to zero (see scheme IIa). In view of this it is customary to say that the two electrons in the helium atom constitute a closed shell. The two electrons in the positive lithium ion in the normal state likewise form a closed shell, as follows from the similarity, established by Schuler, of the spectrum of ionized lithium with the spectrum of helium.

In order to establish the distribution of electrons in the neutral lithium atom, one should suppose that the third electron is brought up infinitely slowly from infinity to the positive lithium ion, which is in the normal state. Then, by virtue of the principle of “adiabatic invariance,” the orbits of both inner electrons must retain the quantum quantities characterizing them, although they may undergo certain perturbations. Thus we arrive at the important conclusion that in the neutral lithium atom the two innermost electrons also constitute a closed shell. This closed shell of two “one-quantum” ($n_i = 1$) electrons is preserved also in all other atoms. The third electron in the neutral lithium atom ...

cannot, by the Pauli principle, have the principal quantum number \(n_i = 1\). Normally it is located in the \(2s\) orbit; in the case of excitation of the atom it may pass to energetically higher orbits, e.g. \(2p, 3s, 3p, 3d\), etc. The resultant vectors \(\sigma, \mathbf{l}, \mathbf{j}\) for the entire neutral lithium atom will be determined by the motion of only the third electron, since the two inner ones form a closed shell. Hence it follows that the number of possible energy states of the neutral lithium atom is determined by Scheme I, and thus the doublet character of the arc1 spectrum of lithium is explained.

The energy that the simplest atom, consisting of a nucleus and one electron, may assume in stationary states is, in the first approximation, expressed by the simple formula:

\[ W=\frac{Rh}{n^2}Z^2, \tag{8} \]

where \(R\) is the Rydberg constant, \(n\) is the principal quantum number, and \(Z\) is the charge of the nucleus in units of the electron charge (atomic number).

In the neutral lithium atom, the distant orbits of the third electron are in conditions close to those of the hydrogen atom, since at large distances the group consisting of the nucleus and two electrons is almost equivalent to a point charge \(Z=+1\). This is confirmed by Table II, in which the high spectral terms2 of lithium are compared with the corresponding terms of hydrogen.

Table II

Hence, in general, the formula expressing the energy of high

states of non-hydrogen atoms, can be written in a form close to formula (8), differing from it only by insignificant correction terms. Spectroscopists, for reasons chiefly of a historical nature, replacing the group consisting of the nucleus and all inner electrons by a charge exactly equal to unity \((Z = +1)\), introduce a correction into the denominator:

\[ W=\frac{Rh}{n^{*2}}, \]

where \(n^*\)—the effective quantum number—is no longer an integer and is a quantity determined empirically. For ions one has to introduce a factor \(z_{eff}\), equal to 2 for single ionization, 3 for double ionization, etc. Finally, the magnitude of the spectral term is expressed by the formula:

\[ \nu=\frac{R}{n^{*2}} Z_{eff}^{2} \tag{8a} \]

For deep orbits the correction terms become significant, but nevertheless in spectroscopy the terms are usually expressed by formula (8a); only the integer part of the effective quantum number in these cases cannot be identified with the principal quantum number of the outermost electron.

X-ray spectroscopists introduce a correction into the numerator of formula (8), referring it to \(Z\):

\[ W=\frac{Rh}{n^{2}}(Z-a)^2 \tag{8b} \]

The empirically determined quantity \(a\) proves, for a given \(n\), to be approximately constant (Moseley’s law), owing to which the use of formula (8b) becomes especially convenient.^1

Millikan and Bowen, who first obtained in the extreme ultraviolet region the spectra of multiply ionized

^1 The quantity \(a\) is called the “screening” quantity. We denote it by the letter “\(a\),” and not by the letter \(\sigma\), as is customary, so that no confusion may arise with the electron moment \(\sigma\).

atoms, showed that for optical spectra as well it is advantageous to use a formula of the form (8b).

Denoting the neutral state of an atom by the Roman numeral I, single ionization by the numeral II, double ionization by III, etc., let us consider the isoelectronic series of ions Be\(_{\mathrm{II}}\), B\(_{\mathrm{III}}\), C\(_{\mathrm{IV}}\), N\(_{\mathrm{V}}\), … . All these ions have three electrons and differ from one another and from the neutral lithium atom only in the charges of their nuclei. The doublet character, found experimentally, of the spectra of all these ions indicates that the electrons are arranged in them in exactly the same way as in the neutral lithium atom: two electrons in \(1s\) orbits, forming a closed shell, and the third normally in the \(2s\) orbit, and in an excited state in any possible orbit: \(2p\), \(3s\), \(3p\), \(3d\), etc. Applying to the expression of the serial terms the formula in the form (8b), i.e. putting

Fig. 4

\[ \nu=\frac{R}{n^{2}}(Z-a)^{2} \tag{8c} \]

Millikan and Bowen found that for analogous terms of the isoelectronic series the screening quantity \(a\) remains approximately constant, and thus Moseley’s law proves applicable also in the optical region.

Denoting the screening quantity for the normal state \({}^{2}S\) of the indicated ions by \(a_{1}\), let us rewrite formula (8c) in the form:

\[ \sqrt{\frac{\nu_s}{R}}=\frac{Z-a_{1}}{n} \]

whence it follows that the quantity \(\sqrt{\nu_s/R}\) must be a linear function of the atomic number \(Z\), if \(a_{1}\) is indeed constant. Fig. 4 (see the straight line \({}^{2}S\)), giving graphi-

dependence of \(\sqrt{\nu_s/R}\) on \(Z\) shows that, for the isoelectronic series \(\mathrm{Li_I}, \mathrm{Be_{II}}, \mathrm{B_{III}}, \mathrm{C_{IV}}, \mathrm{N_V}\), the constancy of the quantity \(a_1\) is indeed fulfilled; the dependence is expressed by an almost exact straight line, entirely analogous to Moseley’s straight lines in the X-ray region.

The constancy of the screening quantity \(a\) proves to be fulfilled also for other energy states, corresponding to the motion of the third electron in higher orbits. In the same Fig. 4 the dependence \(\sqrt{\nu_p/R}\) on \(Z_p\) is given, where \(\nu_p\) denotes the magnitude of the spectral terms \({}^2P\), corresponding to the motion of the third electron in the orbit \(2_p\). Denoting by \(a_2\) the screening quantity for these states \({}^2P\), we obtain the following expression for the frequencies of the head lines of the principal series (resonance lines) of lithium and of ions isoelectronic with it:

\[ \nu_{sp}=\nu_s-\nu_p=\frac{R}{2^2}(Z-a_1)^2-\frac{R}{2^2}(Z-a_2)^2, \]

whence

\[ \nu_{sp}=\frac{R}{2}(a_2-a_1)Z-\frac{R}{4}(a_2^2-a_1^2), \tag{9} \]

since \(a_1\) and \(a_2\) do not depend on \(Z\), consequently \((a_2-a_1)\) and \((a_2^2-a_1^2)\) also do not depend on \(Z\), whence:

\[ \nu_{sp}=AZ+B, \tag{9a} \]

where \(A\) and \(B\) are empirical constants independent of \(Z\). In other words: the frequency of the resonance line \(\nu_{sp}\) is a linear function of the atomic number \(Z\). In Fig. 5 the dependence of the frequency \(\nu_{sp}\) of the resonance lines \(\mathrm{Li_I}, \mathrm{Be_{II}}, \mathrm{B_{III}}, \mathrm{C_{IV}}, \mathrm{N_V}, \ldots\) on their atomic numbers is represented graphically.

Fig. 5

a simple regularity, analogous to the law of the so-called irregular doublets in the X-ray region, renders invaluable services in attempts to analyze the newly obtained spectra of multiply ionized atoms.

In considering the dependence of terms on the atomic numbers \(z\), we have, for simplicity, omitted their doublet character. In reality, however, only the terms \({}^2S\) are single; all the other terms \({}^2P\), \({}^2D\), etc., are double, corresponding to two different possible values of the quantity \(j\) for each given \(n\) and \(l\) (except \(l=0\)). In order to bring doublet-ness into the range of our considerations, one must turn to a more exact formula giving the dependence of the magnitude of the terms not only on \(n\), but also on \(j\). Theoretically such a formula can be derived only for the simplest atoms, consisting of a nucleus and one electron (\(\mathrm{H}, \mathrm{He}^+\)), and has the form:

\[ \nu=\frac{RZ^2}{n^2}+\frac{R\alpha^2Z^4}{n^3}\left(\frac{1}{j+\frac12}-\frac{3}{4n}\right). \tag{10} \]

The quantity \(\alpha=\frac{2\pi l^2}{ch}\) is small in comparison with \(R\), so that the whole second term represents a correction to the first term, which was given above as a first approximation (see formula (8)). For complex spectra, where theory is not yet able to give definite results, one has to introduce still empirical corrections into formula (10), and again it proves most convenient to introduce these corrections in the numerator, relating them to \(z\):

\[ \nu=\frac{R}{n^2}(Z-a)^2+\frac{R\alpha^2(Z-a^1)^4}{n^3}\left(\frac{1}{j+\frac12}-\frac{3}{4n}\right). \tag{10a} \]

Since doublet terms with identical \(n\) and \(l\), for example \({}^2P_{1/2}\) and \({}^2P_{3/2}\), differ from one another in that for one of them \(j=l+{}^1/{}_2\), and for the other \(j=l-{}^1/{}_2\) (see formula (2)), the frequency difference \(\Delta\nu\) of two such terms will be:

\[ \Delta\nu=\frac{R\alpha^2}{n^3}\frac{(Z-a^1)^4}{l(l+1)}. \tag{11} \]

Millikan and Bowen succeeded in showing how this

it is seen from Table III that the quantity \(a'\) also remains approximately constant for isoelectronic series.

Table III

\[ \Delta \nu = 2^2P_{1/2} - 2^2P_{3/2} \]

If \(a'\) were quite exactly independent of \(z\), then on a graph on one of whose axes the values of the quantities \(a'\) are plotted, and on the other the atomic numbers \(z\), a horizontal straight line should result. In fact (see Fig. 6), a slightly inclined line is obtained. For \(z\) large in comparison with \(a'\), the width of the optical doublets according to formula (11) proves to be proportional to the fourth power of the atomic numbers \(Z\), analogously to the law of regular doublets in the X-ray region.

Fig. 6.

The considerations presented above make it possible to trace the arrangement of electrons also in the subsequent atoms of the periodic system of elements. Three electrons in the positive beryllium ion \((\mathrm{Be}_{\mathrm{III}})\) are arranged in exactly the same way as the electrons in the neutral lithium atom. In the neutral beryllium atom the fourth electron may be arranged, like the third, on one of the two-quantum orbits, since, according to the Pauli principle, up to 8 electrons may be arranged on two-quantum orbits (see Table I). These two two-quantum electrons will determine the resultant vectors \(\sigma\), \(\mathbf{l}\), \(\mathbf{j}\), and consequently also the character of the spectrum of the beryllium atom, since the two inner electrons constitute a closed shell. We have seen that a system of two

of electrons must lead to sets of singlet and triplet series, and indeed the arc spectrum of beryllium consists of singlets and triplets. A more detailed study of the spectrum of beryllium shows that the normal state of the beryllium atom is the state \({}^{1}S_{0}\), whence it may be concluded that normally the fourth electron, like the third, is located on the \(2s\) orbit.

Fig. 7

The ions \(\mathrm{B}_{\mathrm{II}},\ \mathrm{C}_{\mathrm{III}},\ \mathrm{N}_{\mathrm{IV}},\ \mathrm{O}_{\mathrm{V}}\) have spectra analogous to the spectrum of \(\mathrm{Be}_{\mathrm{I}}\), whence it follows that the electrons in them are arranged in the same way as in the neutral beryllium atom. For analogous terms of this isoelectronic series as well, the screening quantities \(a\) remain approximately independent of \(Z\), so that here too Moseley’s law proves to be satisfied (see Fig. 7).

The fifth electron in the neutral boron atom can no longer be a \(2s\) electron, since according to the Pauli principle (see Table I) there cannot exist in the atom more than two electrons with \(n=2,\ l=0\); it must have \(l=1\), i.e. normally be located on the \(2p\) orbit. Since the normal state of the boron ion is the state \({}^{1}S_{0}\) (as also for \(\mathrm{Be}_{\mathrm{I}}\)) and consequently is characterized by the fact that for it all three resultant vectors \(\sigma, l, j\) are equal to zero, the resultant states of the neutral boron atom must be determined solely by the motions of its very last, fifth, electron. It follows from this that \(\mathrm{B}_{\mathrm{I}}\) must possess a simple doublet spectrum; this is also observed experimentally. The deepest term of this spectrum is the term \({}^{2}P\), which precisely corresponds to the fact that normally the fifth electron is on the \(2p\) orbit; thus all conclusions about the distribution of electrons inside atoms find direct confirmation in spectroscopic data. Discovered in the extreme ultraviolet part of the spectrum of \(\mathrm{C}_{\mathrm{II}}\),

N$_{\mathrm{III}}$, O$_{\mathrm{IV}}$ show that these ions are constructed analogously to the neutral boron atom.

In the subsequent neutral atoms: C, N, O, . . . there proceeds the further filling of the two-quantum shell with electrons, arranged normally in the $2p$ orbits. Since, according to Table I, atoms cannot contain more than 6 $2p$ electrons,

Table IV

the filling of the shell is completed at neon. The study of the spectrum and other physical properties of neon indicates that the two-quantum shell of 8 electrons is closed; the resultant moments $s$, $l$, $j$ corresponding to it are equal to zero (the normal term Ne$_{\mathrm{I}}$ is the term ${}^1S_0$). Table IV once again gives the arrangement of electrons in orbits in the first ten elements of the periodic system. The numbers in the horizontal rows of Table IV give the number of electrons in the given orbit.

With the eleventh element of the periodic system—sodium—the filling of the three-quantum shell begins. Thus this element has, outside the closed shells, one electron, which also determines the doublet character

its spectrum, analogous to the spectrum of lithium, as well as the similarity to lithium in the other physicochemical properties.

From what has been said above it is clear that the periodicity of the physical and chemical properties of the elements finds its explanation in the existence of closed electron shells. This explanation, first proposed by Kossel soon after the appearance of Bohr’s theory, now proves to be connected with Pauli’s principle. If one looks at Table I, it is easy to see that the number of elements in the first row of Mendeleev’s system—two (H, He)—coincides with the greatest possible number of electrons on a one-quantum shell, while the number of elements in the second row—8 (Li, Be, B, C, N, O, F, Ne)—coincides with the greatest possible number of electrons on a two-quantum shell. From this one might have expected that the third period of Mendeleev’s system should have contained 18 elements, since, according to Table I, the greatest possible number of electrons on three-quantum orbits is 18. In fact, however, the third period again contains 8 elements.

Let us see what a more detailed study of the spectra of the corresponding elements gives in this regard. The 18th element of the periodic system—argon—has, outside the closed one-quantum and two-quantum shells, another 8 electrons: two $3s$-electrons and six $3p$-electrons. These 8 electrons, leading to resultant moments $\sigma$, $l$, $j$ equal to zero, determine, as was to be expected, the complete similarity of the spectrum and other physicochemical properties of argon to the properties of neon. The 18 electrons of the positive potassium ion are arranged in exactly the same way as in argon, as follows from the similarity of the spectrum $K_{\mathrm{II}}$ to the spectrum $A_{\mathrm{I}}$. According to Table I, the nineteenth electron in the neutral potassium atom should have been located on the $3d$ orbit ($l = 2$). However, the doublet character of the spectrum $K_{\mathrm{I}}$, entirely analogous to the spectra $Li_{\mathrm{I}}$ and $Na_{\mathrm{I}}$, with the normal term ${}^{2}S_{1/2}$, convinces us that the nineteenth electron in the neutral potassium atom is normally located on the $4s$ orbit. Thus, here for the first time energetic considerations, not yet amenable to theoretical analysis, come to the fore: the $4s$ orbit

turns out to be energetically deeper than the \(3d\) orbit; therefore normally the 19th electron of potassium is located in the \(4s\) orbit; upon excitation it can pass to energetically higher orbits: \(4p\), \(3d\) (!), \(5s\), \(5p\), etc.

Very interesting results are yielded by the study of the spectra of the ions \(\mathrm{Ca}_{\mathrm{II}}\), \(\mathrm{Sc}_{\mathrm{III}}\), \(\mathrm{Ti}_{\mathrm{IV}}\), isoelectronic with \(\mathrm{K}_{\mathrm{I}}\). In Fig. 8 the Moseley graphs are given for the terms \({}^2S\), \({}^2P\), and \({}^2D\) of this series. As can be seen, the straight line corresponding to the \({}^2D\) terms intersects the straight lines \({}^2S\) and \({}^2P\) between \(Z=20\) and \(Z=21\). Hence

Fig. 8

it follows that the deepest (normal) term of \(\mathrm{Sc}_{\mathrm{III}}\) is the term \({}^2D\), whence in turn it follows that the 19th electron of scandium is normally located no longer in the \(4s\) orbit, as in potassium, but in the \(3d\) orbit; it can also be established that in the following element—titanium—not only the 19th, but also the 20th electron is located in the \(3d\) orbit. Once again, the results obtained are summarized in Table V.

In an analogous way, although in many cases less confidently for lack of experimental spectroscopic data, the distribution of electrons can be traced for the other elements as well. If the study of X-ray spectra once indicated that the innermost shells of all elements are constructed in the same way, and if this fact served as the starting point for the entire doctrine of the periodic system, then at the present time, with the po-

attempts to understand in greater detail the structure of the various elements, the study of optical spectra comes to the fore, providing information about the outer, not yet completed electron shells. Without dwelling on this in greater detail, since nothing fundamentally new is encountered here, we shall only show how, from the point of view of the scheme presented, the occurrence of X-ray spectra is explained.

Table V

X-ray lines arise when, from one of the inner shells of the atom, one of the electrons is torn out by some external action and its place is occupied by another, more external electron. Our scheme directly gives how many different energy states can arise when one of the electrons is knocked out of a closed shell. A closed shell is characterized by the fact that for it the resultant vectors $\sigma$, $l$, $j$ are equal to zero. If one electron is removed from a closed shell, then obviously the electron configuration remaining from the closed shell will be characterized by vectors whose numerical values coincide exactly with the numerical values of the vectors $\sigma_i$, $l_i$, $j_i$ of the removed electron, since only in this case, before the removal of the electron, could all resultant vectors be equal to zero. Since the scheme of vectors belonging to a single electron corresponds to doublet terms, it follows that the scheme of X-ray levels must correspond to the scheme of optical doublets.

In fact: both electrons of the one-quantum shell are \(1s\)-electrons, for each of which \(\sigma = 1/2\), \(l_i = 0\), \(j_i = 1/2\); whichever of these two electrons is torn out, the remaining part of the shell will be characterized by the resultant quantities \(\sigma = 1/2\), \(l = 0\), \(j = 1/2\), or by the symbol \({}^{2}S_{1/2}\). This corresponds precisely to the fact that in X-ray spectroscopy there exists only one identical \(K\) level; it now turns out that this level \(K\) corresponds to the optical level \({}^{2}S_{1/2}\).

Table VI

The two-quantum shell consists of two \(2s\)-electrons and six \(2p\)-electrons; according to scheme I, a part of these \(2p\)-electrons must have \(j_i = 1/2\), and a part \(j_i = 3/2\). If one of the \(2s\)-electrons is torn out of the two-quantum shell, then the state \({}^{2}S_{1/2}\) arises, while if one of the \(2p\)-electrons is torn out, then either the state \({}^{2}P_{1/2}\) \((\sigma = 1/2,\ l = 1,\ j = 1/2)\), or \({}^{2}P_{1/2}\) \((\sigma = 1/2,\ l = 1,\ j = 3/2)\), arises.

Thus, in all, three different states are possible here, which corresponds precisely to the presence, established in X-ray spectroscopy, of three \(L\)-levels: \(L_1\), \(L_2\), and \(L_3\).

In exactly the same way we arrive at the conclusion that, when one of the electrons is torn out of the three-quantum shell, one of five states arises: \({}^{2}S_{1/2}\), \({}^{2}P_{1/2}\), \({}^{2}P_{3/2}\), \({}^{2}D_{3/2}\), \({}^{2}D_{5/2}\) (see Table VI).

We have seen that in the optical region the difference \(\Delta \nu = {}^{2}P_{1/2} - {}^{2}P_{3/2}\) obeys the law of regular doublets —

This is in complete agreement with Table VI, since in the X-ray region the law of regular doublets refers precisely to the difference between the levels \(L_2\) and \(L_1\) (or \(M_4\) and \(M_3\), etc.). In exactly the same way, the law of irregular doublets, to which in the X-ray region the difference \(L_3 - L_2\) (or \(M_5 - M_4\), etc.) is subject, is justified in the optical region, in accordance with Table VI, by the difference of the levels \(^{2}S_{1/2} - {}^{2}P_{1/2,3/2}\).

1. In spectroscopy, an “arc” spectrum means a spectrum emitted by neutral atoms. The spectrum of ions is called a “spark” spectrum. ↩↩

2. A spectral term is a quantity proportional to the energy of an atom: \(\nu = W/h\); spectral terms are usually given in wave numbers \((\mathrm{cm}^{-1})\). ↩↩