ISOTOPIC EFFECT IN ATOMIC SPECTRA

A. R. Striganov and Yu. P. Dontsov

1. INTRODUCTION

The first attempts to detect isotopic structure in a line spectrum were undertaken by a number of investigators immediately after the discovery of the isotopy of radioactive (Soddy, 1910) and nonradioactive (J. Thomson, 1912) elements. However, up to 1918 no measurements in this direction had been published. The first experimental data on the measurement of isotopic shift in a line spectrum belong to Aronberg^212, who found the effect by comparing the spectra of two samples of lead of different origin. He found that the wavelength of the Pb 4057.8 Å line in the spectrum of ordinary lead is 0.0043 Å smaller than the wavelength of this same line in the spectrum of lead that is a product of radioactive decay. However, he was unable to observe the complete structure of the line, which is explained by the large broadening of the spectral lines in his light source, and also by the insufficiently high resolving power of the interferometer.

Beginning in 1919, mass-spectrographic investigations of the isotopic composition of nonradioactive elements acquired broad scope. The work of Aston, J. Thomson, Dempster, Nier, and many other investigators made it possible by 1935 to analyze, in the main, almost all the elements of the periodic table of Mendeleev known at that time. It is quite natural that during this period the attention of a number of spectroscopists was concentrated on investigations of the isotopic structure of atomic and molecular spectra. In 1927 Hansen^213 found an isotopic shift in the spectrum of neon. In the same year Schüler and Wurm^58 discovered and measured the isotopic structure of the resonance line of lithium. In 1931 and 1932 Schüler and co-workers^166,167 studied in detail the hyperfine and isotopic structure of 19 mercury lines, with 7 of them being

A. R. STRIGANOV AND Yu. P. DONTSOV

a measurable isotopic shift was detected. In this same period the isotopic shift in the spectrum of lead was investigated. Konferman \(^{180}\) compared ordinary lead with monoisotopic Pb \(^{206}\), which was a product of the decay of the radioactive uranium series. On the Pb I \(4057.8\,\text{Å}\) line he obtained a distinct pattern of the shift. In 1932 Schuler and Jones \(^{214}\) analyzed in detail the structure of the Pb II \(5372.5\,\text{Å}\) line. As a result they discovered the fourth isotope of lead (Pb \(^{204}\)), which had not been noted by the first mass-spectrographic investigations. A year earlier, while studying the hyperfine structure of thallium lines, Schuler and Keyston \(^{176}\) discovered two isotopes of thallium*). Using the optical method, Urey, Brickwedde, and Murphy \(^{46}\) discovered heavy hydrogen (deuterium) in enriched water in 1932. They carried out the first measurements of the isotopic structure in the hydrogen spectrum on the \(H_\beta\), \(H_\gamma\), \(H_\delta\) lines of the Balmer series. Simultaneously with the development of experimental work on the isotopic effect in atomic spectra, the theory of this phenomenon was also being developed.

Intensive work on the investigation of the isotopic effect in atomic spectra continued until 1940. During the war its scope decreased somewhat. However, beginning in 1945, and especially in recent years, this work again assumed a fairly broad scale. This was facilitated by the new possibilities that had opened up for obtaining enriched stable and radioactive isotopes, which in many cases expanded the possibilities of the spectroscopic method. As a result, in recent years the isotopic effect has been investigated in He, C, N, O, Ca, Cd, Sn, Te, Ba, Ce, Sm, Dy, Er, W, Hg, Pb, Th, U, Pu, Am.

At present more than 200 works on the investigation of the isotopic effect can be counted. Of these, approximately 30 works concern the theory of this phenomenon, while the rest are devoted to the study of isotopic shifts in the spectra of various elements. Of all 98 known elements, the isotopic effect has now been studied to one degree or another for 54 elements. If monoisotopic stable elements are excluded from consideration, then for the remaining 57 stable elements consisting of more than one stable isotope, the degree of study of the isotopic effect is presented in Table I. From the table it is evident that, of the 57 stable elements, the isotopic effect has been studied fairly fully (on many or on several spectral lines) for 33 elements, insufficiently studied (on 1–2 lines) for 17

*) Let us note that, by the optical method, with the aid of molecular spectra, the heavy isotopes of carbon (C\(^{13}\)), nitrogen (N\(^{15}\)), and oxygen (O\(^{17}\) and O\(^{18}\)) were also discovered in 1927–1929.

elements and has not been studied at all for 7 elements. It is further seen from the table that, in the case of light elements, the isotope effect has been incompletely studied for two elements (Cl and K) and has likewise not been studied at all for two elements (Si, S). Among heavy elements the isotope effect has been incompletely studied for five elements (Dy, Hf, Re, Os, Ir). In the medium elements the isotope effect has been studied much less extensively than in the light and heavy elements. Indeed, of the 25 elements of this group the effect mentioned has been studied (in each case only for a few lines) for only 10 elements. In the case of 10 elements it proved either very small in magnitude or was not detected at all (Ga, Ge, Se, Br, Kr, Rb, Sr, Zr, Mo, Sb), and for 5 elements it has not yet been studied (Ti, Cr, Fe, Ni, In).

Table I

Degree to which the isotope effect has been studied in various elements

Group of elements	Number of elements	Studied fully	Studied incompletely	Not studied
Stable elements with a number of stable isotopes greater than two	57	33	17	7
Of these:
1) light elements from \(Z = 1\) to \(Z = 20\)	15	11	2	2
2) medium elements from \(Z = 21\) to \(Z = 56\)	25	10	10	5
3) heavy elements from \(Z = 57\) to \(Z = 83\)	17	12	5	—
Radioactive elements	17	4	—	13

It follows from the table that, of the 17 radioactive elements, the isotope effect has been studied for only four elements (Th, U, Pu, Am), which is explained by the specific difficulties associated with obtaining isotopes in the necessary quantities.

The isotopic displacement of spectral lines arises as a result of the shift, relative to one another, of the energy levels of atoms belonging to different isotopes of one and the same element. This displacement is the result of the interaction of the electron shell with the nucleus. Therefore both the properties of the nucleus and the characteristic features of the electron shell must manifest themselves in it. Hence it follows that the study of the isotope effect

is of twofold interest. First, this effect to some extent makes it possible to judge the change in the interaction between the nucleus and the electron shell when neutrons are added to a nucleus with one and the same charge; it makes it possible to obtain certain information about atomic nuclei (for example, data on nuclear radii, on the character of the distribution of nucleons over the volume of nuclei, on the structure of nuclear shells, on the nonsphericity of nuclei, etc.). Second, the isotope effect in atomic spectra, as has recently been established, can in some cases be used as an auxiliary means for classifying spectra, especially in establishing and refining data concerning electron configurations for various transitions.

It should be noted that the structure of spectral lines is determined not only by the isotope shift, but also by the hyperfine splitting of the lines of individual isotopes. The nuclei of all isotopes, depending on their structure, may be divided into four groups:

1. Even-even nuclei, consisting of an even number of protons and an even number of neutrons (even $Z$, $N$, and $A$). The mechanical, magnetic, and quadrupole moments of all these nuclei are equal to zero.

2. Even-odd nuclei, consisting of an even number of protons and an odd number of neutrons (even $Z$, odd $N$ and $A$). The spins of these nuclei take only half-integral values from $1/2$ to $9/2$. The magnetic moments lie within the limits from $-2$ to $+1$ nuclear magneton. Very few quadrupole moments have been measured; their values lie in the range from $-0.08 \cdot 10^{-24}$ to $+3.9 \cdot 10^{-24}\ \mathrm{cm}^2$. For nuclei with spin $I=1/2$, the quadrupole moments are equal to zero.

3. Odd-even nuclei, consisting of an odd number of protons and an even number of neutrons (odd $Z$ and $A$, even $N$). The spins of these nuclei take half-integral values from $1/2$ to $9/2$. The magnetic moments lie within the limits from $-0.28$ to $+6.2$ nuclear magnetons. The quadrupole moments are in the range from $-1.2 \cdot 10^{-24}$ to $+6 \cdot 10^{-24}\ \mathrm{cm}^2$, and for nuclei with $I=1/2$ the quadrupole moments are equal to zero.

4. Odd-odd nuclei, consisting of an odd number of protons and an odd number of neutrons (odd $Z$ and $N$, even $A$). The spins of these nuclei take integral values from 1 to 7. The magnetic moments lie within the limits from $-1.7$ to $+3.8$ nuclear magnetons. The measured quadrupole moments are very small, with the exception of the nucleus $\mathrm{Lu}^{176}$ ($Q=+7 \cdot 10^{-24}\ \mathrm{cm}^2$).

The spectral lines of isotopes of the first group have no hyperfine structure, i.e. they consist of a single component. The spectral lines of isotopes belonging to the other three groups possess hyperfine structure and therefore consist of several

ISOTOPIC EFFECT IN ATOMIC SPECTRA

components. Thus, when analyzing isotopic displacement it is necessary, in the case of some isotopes, to take into account the presence of hyperfine structure of the lines. In these cases the isotopic effect is determined from the position of the center of gravity of the hyperfine-structure components of the given isotope*).

Let us note that in the case of light elements the fine structure of the lines must also be taken into account. For these elements the fine structure is close in width to the isotopic displacement. In hydrogen the fine doublet structure of the \(H_\alpha\) line is approximately 13 times smaller than the isotopic displacement between the \(H_\alpha\) and \(D_\alpha\) lines. In lithium the fine doublet structure of the line \(6707.84\ \text{Å}\) is, in width, of the same order as the isotopic displacement \(^{28}\) experienced by each component of this doublet, splitting into two lines \(Li^6\) and \(Li^7\). In boron \(^{29}\) the distance between the doublet fine structure (the lines \(2497.73\) and \(2496.78\ \text{Å}\)) is already 90 times greater than the isotopic displacement of the lines \(B^{10}\) and \(B^{11}\). We see, therefore, that when considering the complex structure of lines, say, of the first five elements of the Mendeleev periodic system, one must take into account, in addition to the isotopic structure, also the multiplet splitting or fine structure of the lines.

For the theoretical interpretation of the isotopic effect, two theories were proposed as early as 20 years ago. One of them, taking into account the mass of the nuclei, explained the isotopic displacement in the spectra of light elements. The other, based on accounting for the volume of the nucleus, explained the displacement in the spectra of heavy elements. Both of these theories, as further testing showed, were not entirely accurate; in a number of cases they corresponded only roughly to experiment and did not explain a number of anomalous phenomena discovered later in isotopic displacement. Proceeding from this, in order to interpret certain anomalies, additional effects were used that appear in the interaction between the nucleus and the electron shell. However, even up to the present time, it seems, a sufficiently complete and coherent theory of the isotopic effect in atomic spectra has still not been developed that would make it possible to understand this phenomenon in all its diversity.

Below we shall briefly set forth the theory of the isotopic effect, as well as the regularities found in isotopic displacement in atomic spectra according to literature data published up to 1955. At the end of the present review, for the first time, a complete systematized bibliography on the isotopic effect is given.

*) The center of gravity determines the position of a line consisting of separate components, and is found from the condition \(\sum a_i l_i = 0\), where \(a_i\) is the distance of the corresponding component from the center of gravity, and \(l_i\) is its intensity.

2. ISOTOPIC EFFECT IN THE SPECTRA OF LIGHT ELEMENTS

a) Hydrogen and hydrogen-like ions

The isotopic effect in the spectrum of hydrogen and hydrogen-like ions is readily explained on the basis of Bohr’s theory, if the mass of the nucleus is not regarded as infinitely large in comparison with the mass of the electron. In this case it is necessary to take into account the motion of the nucleus, which leads to a dependence of the spectral terms on the nuclear mass. For the value of the term one obtains the following expression:

\[ T_M = T_\infty \left(1-\frac{m_e}{M}\right), \tag{1} \]

where \(T_\infty\) is the value of the term for an atom with an infinitely heavy nucleus. The displacement of the term relative to \(T_\infty\) is equal to:

\[ \Delta T_M = T_M - T_\infty = -\frac{m_e}{M}T_\infty . \tag{2} \]

It is evident from the formula that the smaller the mass \(M\), the greater the displacement of the term will be; moreover, the minus sign indicates that the displacement occurs toward smaller numerical values of \(T\). It follows from this that the terms of the lighter isotope are situated closer to the series limit, i.e., the binding of the electron to the nucleus in the lighter isotope is weaker than in the heavier one. This latter circumstance is explained by the fact that the effect of the motion reduces the binding energy more strongly for the lighter nucleus, since, roughly speaking, the amplitude of oscillation of the light nucleus is greater than that of the heavy one.

For the magnitude of the isotopic displacement between the terms of two isotopes, on the basis of formula (1), one obtains the expression

\[ \Delta T = -m_e \frac{M_2-M_1}{M_1M_2}T_\infty . \tag{3} \]

Since \(T_\infty\) is inversely proportional to \(n^2\), the absolute value of the displacement between the terms of two isotopes will rapidly decrease with increasing principal quantum number \(n\), and at the series limit the displacement will be equal to zero. It is customary to assign a negative sign to the displacement in the case when the level of the heavier isotope lies deeper than the level of the lighter isotope.

If the finiteness of the nuclear mass is taken into account, then the wave numbers of the spectral lines of hydrogen will be expressed by the formula

\[ \nu=\left(1-\frac{m_e}{M}\right)\nu_\infty , \tag{4} \]

where \(\nu_\infty\) is the wave number of the line if the mass of the nucleus is taken to be infinitely large in comparison with the mass of the electron. From this formula it is seen that each isotope corresponds to its own spectral line, and with increasing nuclear mass \(M\) the wave number of the line \(\nu\) also increases, i.e., the lines of the heavier isotopes are shifted toward larger wave numbers. This direction of displacement is conventionally regarded as positive\({}^{12}\). In order to obtain, from the shift in terms, the isotopic shift in lines with the correct sign, it is necessary to subtract the shift of the lower term \((\Delta T)\) from the shift of the upper term \((\Delta T')\):

\[ \Delta\nu = \Delta T' - \Delta T. \]

From formula (3) or (4) it is easy to obtain the magnitude of the isotopic shift between the lines of two isotopes:

\[ \Delta\nu = \nu_2 - \nu_1 = m_e \frac{M_2 - M_1}{M_1 M_2}\,\nu_\infty . \tag{5} \]

In Fig. 1 we give the transition scheme for the \(H_\alpha\) line between the shifted terms of three isotopes (H, D, and T). On the left are given the spectroscopic symbols of the terms and the intervals between the shifted terms in \(\mathrm{cm}^{-1}\). Below is given the isotopic structure of the \(H_\alpha\) line. The intervals between the components H and D, D and T are respectively \(4.14\ \mathrm{cm}^{-1}\) (1.79 Å) and \(1.38\ \mathrm{cm}^{-1}\) (0.60 Å). They can be found directly from the transition scheme. The scheme presented does not take into account the fine structure of the hydrogen terms, since the width of this structure even for the lower term is approximately 20 times smaller than the width of the isotopic shift between H and D.

Fig. 1. Transition scheme and isotopic structure for the \(H_\alpha\) line.

The isotopic shift between the H and D lines in the hydrogen spectrum has been thoroughly measured for 20 members of the Balmer series\({}^{46,48,49,50}\), for 6 members of the Lyman series\({}^{47}\), and for several members of the Paschen series\({}^{49,50}\). For the isotopic shift of tritium there are calculated data\({}^{13}\). If the experimental data on the isotopic shift in the hydrogen spectrum are compared with the calculated data, very good agreement is obtained. The theory of the isotopic shift in the hydrogen spectrum is also valid for hydrogen-like ions. The formulas given above remain the same in this case, since the atomic number \(Z\) (or the number of protons in the nucleus) enters into the values \(T_\infty\) and \(\nu_\infty\). For hydrogen-like ions there are calculated data on the isotopic shift in spectra up to O VIII\({}^{13}\).

b) Atoms and ions with several electrons

The isotope effect considered in one-electron atoms, caused by the joint motion of the nucleus, is customarily called the “normal” mass effect. For heavier atoms this effect rapidly decreases, approximately inversely proportional to the square of the atomic mass. However, if an atom has two or a larger number of electrons, then the isotope shift in the spectrum cannot be explained solely by the normal mass effect. In atoms of this type it is necessary to take into account the exchange interaction of the electrons with one another. The total kinetic energy of an atom with several electrons in a coordinate system connected with the center of inertia is expressed as follows:

\[ E=\frac{1}{2}\left(\frac{1}{m_e}-\frac{1}{M}\right)\sum p_i^2+\frac{1}{M}\sum_{i\ne j} p_i p_j, \]

where \(p_i\) and \(p_j\) are the momenta of the electrons. The first term of this expression represents the normal shift; the second term gives a certain additional shift. In the case of independent motion of the electrons, the mean value of the second term becomes zero. If, however, the orbital motions of the electrons are in definite phase relations, then the mean value of the second term will have a finite value. It follows from this that, for a given atomic state, some additional quantity, also depending on the nuclear mass, will either be added to or subtracted from the normal mass effect, which is caused by the simple joint motion of the nucleus when the electrons move independently relative to one another. In contrast to the normal mass effect, this additional effect, characteristic of atoms with several electrons, is customarily called the “specific” mass effect \(^{1,2}\).

The specific effect, being an effect of an essentially quantum nature, is a consequence of the identity of the electrons. The identity of the electrons, in combination with the Pauli principle, requires antisymmetry of the complete wave function of the electron shell with respect to the interchange of any pair of electrons. As a result, for any specified total spin of the electron system their orbital motions are not independent but, on the contrary, are in definite phase relations. It is obvious that for different phase relations the joint motion of the nucleus will also be different. From the point of view of classical mechanics this may be represented roughly as follows: if the predominant number of electrons moves in one direction, then the joint motion of the nucleus will be intensified; conversely, when the electrons move in opposite directions, the joint motion will be weakened, owing to the fact that the center of inertia of the entire atom remains immobile.

Let us explain what has been said using the simplest example of a two-electron shell. The total spin of such a shell may be equal either to zero (spins antiparallel—paraterms), or to unity (spins parallel—orthoterms). In the first case the spin function of the two-electron system is antisymmetric, and therefore the coordinate function will be symmetric (since the complete wave function, which is the product of the spin and coordinate functions, must be antisymmetric). In the second case everything will be the reverse. Using the classical analogy, one may say that in a parastate the electrons move predominantly in the same direction, and in an orthostate—in opposite directions. Since the center of the entire atom (nucleus + shell) is at rest, this means that for paraterms (singlet states) the co-motion of the nucleus is intensified, while for orthoterms (triplet states) it is weakened. This means that the specific effect for singlet terms coincides in sign with the normal effect; for triplet terms the specific effect is opposite in sign to the normal one.

If the exchange interaction of the electrons is not taken into account, then for atoms with several electrons the normal effect will give, for each atomic term \(T_\infty\) (corresponding to the mass \(M_\infty\)), exactly the same displacement as in the case of a one-electron system\(^{1,2,3}\)

\[ \Delta T_{\mathrm{n}}=-\frac{m_e}{M}T_\infty . \tag{6} \]

The specific effect for systems of two and three electrons was first considered theoretically by Hughes and Eckart\(^{14}\). They showed that the specific shift is different from zero only for terms belonging to configurations in which the orbital quantum numbers of both electrons differ by one \((\Delta l=\pm 1)\). Since in helium-like atoms or ions one of the electrons is practically always in the \(1s\) state, this means that the specific shift will have a finite value only for the \(P\)-term, i.e., when the second electron is excited and is in the \(np\) state. The same is also true for three-electron atoms (for example, LiI), when two electrons are in the \(1s\) state.

According to Hughes and Eckart, the specific shift for the terms \(1snpP\) of two-electron atoms is expressed as follows:

\[ \Delta T_{\mathrm{c}}=\pm\frac{m_e}{M}\tau \,[\mathrm{cm}^{-1}], \tag{7} \]

where

\[ \tau=\frac{128}{3}(Z_s Z_p)^3 \left(\frac{Z_s n-Z_p}{Z_s n+Z_p}\right)^{2n-4} n^3(n^2-1)R_\infty . \tag{8} \]

Here \(n\) is the principal quantum number of the \(p\)-electron, \(R_\infty\) is the Rydberg constant for \(M=\infty\), and \(Z_s\) and \(Z_p\) are the effective nuclear charges for the \(s\)- and \(p\)-electrons. Since \(\Delta T_{\mathrm{n}}\) is negative, this negative sign means that the shift \(\Delta T_c\) has the same direction as \(\Delta T_{\mathrm{n}}\); the positive sign shows that \(\Delta T_c\) has the direction opposite to that of \(\Delta T_{\mathrm{n}}\). From what has been said above it follows that the negative sign in formula (7) refers to singlet terms, and the positive sign to triplet terms. Formula (7) is also valid for the terms \(1s^2np\,^2P\) of three-electron atoms and ions. The quantity \(\Delta T_c\) in this case has a positive value, i.e., it decreases the normal shift.

The total shift of the term \(T_\infty\) due to the normal and specific effects will be expressed by the algebraic sum:

\[ \Delta T_{\mathrm{m}}=\Delta T_{\mathrm{n}}+\Delta T_c . \tag{9} \]

The isotope shift for a given term is easily obtained by taking the difference of the corresponding shifts for two isotopes with masses \(M_1\) and \(M_2\). Using formulas (9), (6), and (7), we obtain:

\[ \Delta T=m_e\frac{M_2-M_1}{M_1M_2}\left[-T_\infty \pm \tau\right]. \tag{10} \]

In order to obtain the magnitude of the isotope shift for a line, one must take the difference of the isotope shifts for the upper and lower levels. On the basis of formula (10) it is easy to find the complete isotope shift for lines:

\[ \Delta\nu=m_e\frac{M_2-M_1}{M_1M_2}\left[\nu_0 \pm(\tau'-\tau)\right], \tag{11} \]

where \(\nu_0\) is the wave number of the given line; the values \(\tau\) and \(\tau'\) are expressed by formula (8) and refer respectively to the lower and upper levels between which the transition corresponding to the given line occurs. The positive sign, as already indicated, refers to triplet terms of two-electron atoms, and also to terms of three-electron atoms. The negative sign refers to singlet terms of two-electron atoms.

The first term in formula (11) expresses the normal isotope shift of the line \((\Delta\nu_{\mathrm{n}})\), while the second term represents the specific shift \((\Delta\nu_c)\). Since only \(P\)-terms undergo the specific shift, and since transitions between \(P\)-terms are usually forbidden by the selection rules, for any line the specific shift will be absent either in the lower or in the upper term, or in both together, i.e., in formula (11) either \(\tau\), or \(\tau'\), or both quantities simultaneously will be equal to zero. It then follows from formula (11) that for transi-

for \(P \to S\) transitions, in the case of singlet terms the total displacement of the line is
\(\Delta \nu = \Delta \nu_{\mathrm{n}} - \Delta \nu_{\mathrm{s}}\), while in the case of triplet terms
\(\Delta \nu = \Delta \nu_{\mathrm{n}} + \Delta \nu_{\mathrm{s}}\). For \(S \to P\) or \(D \to P\) transitions the reverse is obtained: in the case of singlet terms \(\Delta \nu = \Delta \nu_{\mathrm{n}} + \Delta \nu_{\mathrm{s}}\), and in the case of triplet terms \(\Delta \nu = \Delta \nu_{\mathrm{n}} - \Delta \nu_{\mathrm{s}}\). In transitions in which \(P\)-terms do not participate, the isotopic displacement is determined entirely by the normal mass effect.

From formula (11) it is seen that, like the normal isotopic displacement, the specific displacement varies inversely proportionally to the square of the mass. Calculations of the specific displacement carried out for light elements show that this effect is, in order of magnitude, equal to the normal displacement. However, unlike the normal effect, the specific effect also depends on the number of electrons in the atom. A characteristic feature of both the normal and the specific displacement is the exact proportionality between the displacement of the lines (or the centers of gravity of the hyperfine-structure components) \(\Delta \nu_1\) and \(\Delta \nu_2\) and the difference of the reciprocal mass numbers of the isotopes \(A_1\), \(A_2\), \(A_3\) (Fig. 2)

\[ \Delta \nu_1 : \Delta \nu_2 = \left( \frac{1}{A_1} - \frac{1}{A_2} \right) : \left( \frac{1}{A_2} - \frac{1}{A_3} \right). \]

Fig. 2. Isotopic structure of a line.

For sufficiently large isotope mass numbers (for elements with \(Z > 10\)) the displacements will be approximately proportional to the difference between the mass numbers of the isotopes:

\[ \Delta \nu_1 : \Delta \nu_2 = (A_2 - A_1) : (A_3 - A_2). \]

In other words, if the mass numbers of the isotopes differ by one and the same amount, then, within the limits of experimental error, the intervals between the displaced isotope lines (or the centers of gravity of the hyperfine-structure components) will be equal.

In calculating the specific displacement it is necessary to know the wave functions of the atom, which are usually found on the basis of “hydrogen-like” functions of individual electrons. Each of these functions is the solution of a one-electron problem at \(M = \infty\) with its own “effective charge,” which is usually determined by the variational method.

The most interesting object for the investigation of isotopic displacement in the spectra of two-electron atoms is helium. Although in the natural state this monatomic gas consists of two isotopes (\(\mathrm{He}^3\) and \(\mathrm{H}^4\)), the isotope \(\mathrm{He}^3\) is present in it in concentrations reaching only about \(0.0001\%\), which is quite insufficient for spectroscopic detection. Therefore investigations of the isotopic displacement in the spectrum of helium were carried out only recently, after possibilities had been found for obtaining sufficient quantities of enriched

of helium. During 1948–1951, six papers devoted to this question were published in succession \(^{52-57}\). As a result of these works, the helium spectrum is now the only spectrum in which the isotope shift has both been measured and theoretically examined quite fully. The most extensive data are presented in the papers of Bradley and Kuhn \(^{56}\), and also of Fred, Tomkins, Brode, and Hamermesh \(^{57}\), who measured the isotope shift for 16 singlet and 17 triplet lines of six series. From the shift in the lines, Bradley and Kuhn \(^{56}\) determined the shift in terms.

Table II gives a complete summary of the experimental data on the isotope shift in the helium spectrum \((\Delta \lambda_{\mathrm{exp}}, \Delta \nu_{\mathrm{exp}})\). The results of Fred, Tomkins, et al. \(^{57}\) are used mainly, as the most reliable, since for the triplet lines they took into account, to some extent, the influence of the fine and hyperfine structure*). For comparison, the same table presents calculated data for the normal \((\Delta \nu_{n})\), specific \((\Delta \nu_{c})\), and total \((\Delta \nu_{\mathrm{calc}})\) shifts. In the 8th column the relative deviation of the experimental data from the theoretical data, expressed in percent, is given.

It is evident from Table II that the shift for all lines, with the exception of one \((7065.2\,\text{\AA})\), is positive, i.e., the line of the lighter isotope \((\mathrm{He}^{3})\) is shifted toward smaller wave numbers.

The shift increases with increasing principal quantum number of the variable term, both for the lines of the singlet and for the lines of the triplet system of terms. This is explained by the decrease of the normal and specific shifts of the variable term with increasing principal quantum number (see Tables III and IV). A comparison of the experimental data \((\Delta \nu_{\mathrm{exp}})\) with the theoretical data \((\Delta \nu_{\mathrm{calc}})\) shows, in general, fairly good agreement. However, attention should be paid to the fact that for all series a systematic discrepancy is observed between the theoretical and experimental data, and the difference between them in many cases exceeds the experimental error of the measurements. If the unaccounted-for part of the isotope shift, represented by the difference \(\Delta \nu_{\mathrm{exp}} - \Delta \nu_{\mathrm{calc}}\), is compared with the magnitude of the specific shift, then for many lines it reaches 50–100% or more of the magnitude of the specific shift. This shows that the theory of the isotope shift for two-electron atoms presented above is far from covering the entire phenomenon.

*) For singlet terms, in the presence of strict Russell–Saunders coupling, the hyperfine structure is absent, since the constant of the hyperfine structure \(A\) in the formula expressing the magnitude of the hyperfine splitting is equal to zero \(^{3}\).

Table II

Isotopic shift between the lines of the isotopes He³ and He⁴

$\lambda$ (Å)	Transition	$\Delta\lambda_{\mathrm{exp}}$ (Å)	$\Delta\nu_{\mathrm{exp}}$ (cm$^{-1}$)	$\Delta\nu_{\mathrm{n}}$ (cm$^{-1}$)	$\Delta\nu_{\mathrm{c}}$ (cm$^{-1}$)	$\Delta\nu_{\mathrm{calc}}$ (cm$^{-1}$)	$\dfrac{\Delta\nu_{\mathrm{exp}}-\Delta\nu_{\mathrm{calc}}}{\Delta\nu_{\mathrm{exp}}}$ (%)
5015,7	$2s\,{}^1S — 3p\,{}^1P$	0,213	0,849	0,893	0,117	0,776	8,6
3964,5	$2s\,{}^1S — 4p\,{}^1P$	0,183	1,165	1,130	0,050	1,080	7,3
3613,6	$2s\,{}^1S — 5p\,{}^1P$	0,170	1,304	1,240	0,026	1,214	6,9
3447,6	$2s\,{}^1S — 6p\,{}^1P$	0,159	1,344	1,300	0,015	1,285	4,4
3354,5	$2s\,{}^1S — 7p\,{}^1P$	0,155	1,375	1,336	0,009	1,327	3,5
7281,3	$2p\,{}^1P — 3s\,{}^1S$	0,554	1,046	0,615	0,372	0,987	5,6
5047,7	$2p\,{}^1P — 4s\,{}^1S$	0,339	1,333	0,888	0,372	1,260	5,5
4437,5	$2p\,{}^1P — 5s\,{}^1S$	0,287	1,461	1,010	0,372	1,382	5,4
4168,9	$2p\,{}^1P — 6s\,{}^1S$	0,266	1,529	1,075	0,372	1,447	5,4
3935,9	$2p\,{}^1P — 8s\,{}^1S$	0,249	1,606	1,138	0,372	1,510	6,0
6678,1	$2p\,{}^1P — 3d\,{}^1D$	0,501	1,124	0,671	0,372	1,043	7,2
4921,9	$2p\,{}^1P — 4d\,{}^1D$	0,329	1,358	0,910	0,372	1,282	5,6
4387,9	$2p\,{}^1P — 5d\,{}^1D$	0,281	1,462	1,021	0,372	1,393	4,7
4143,7	$2p\,{}^1P — 6d\,{}^1D$	0,259	1,512	1,081	0,372	1,453	3,9
4009,2	$2p\,{}^1P — 7d\,{}^1D$	0,246	1,532	1,118	0,372	1,490	3,4
3926,5	$2p\,{}^1P — 8d\,{}^1D$	0,238	1,543	1,141	0,372	1,513	1,9
10830	$2s\,{}^3S — 2p\,{}^3P$	1,385	1,181	0,414	0,565	0,979	17,1
3888,6	$2s\,{}^3S — 3p\,{}^3P$	0,212	1,404	1,153	0,175	1,328	5,4
3187,7	$2s\,{}^3S — 4p\,{}^3P$	0,156	1,535	1,406	0,072	1,478	3,7
2945,1	$2s\,{}^3S — 5p\,{}^3P$	0,142	1,645	1,522	0,037	1,559	5,2
2829,1	$2s\,{}^3S — 6p\,{}^3P$	0,136	1,700	1,584	0,015	0,599	5,9
7065,2	$2p\,{}^3P — 3s\,{}^3S$	−0,018	−0,036	0,634	0,565	0,069	—
4713,4	$2p\,{}^3P — 4s\,{}^3S$	0,066	0,297	0,951	0,565	0,386	−29,9
4120,8	$2p\,{}^3P — 5s\,{}^3S$	0,074	0,438	1,087	0,565	0,522	−19,2
3867,5	$2p\,{}^3P — 6s\,{}^3S$	0,076	0,508	1,159	0,565	0,594	−16,9
3732,9	$2p\,{}^3P — 7s\,{}^3S$	0,076	0,548	1,209	0,565	0,644	−17,5
3652,1	$2p\,{}^3P — 8s\,{}^3S$	0,078	0,586	1,227	0,565	0,662	−12,9
5875,6	$2p\,{}^3P — 3d\,{}^3D$	0,042	0,123	0,763	0,565	0,198	−60,9
4471,5	$2p\,{}^3P — 4d\,{}^3D$	0,074	0,360	1,002	0,565	0,437	−21,3
4026,2	$2p\,{}^3P — 5d\,{}^3D$	0,077	0,474	1,113	0,565	0,548	−15,6
3819,8	$2p\,{}^3P — 6d\,{}^3D$	0,080	0,547	1,173	0,565	0,608	−11,1
3705,1	$2p\,{}^3P — 7d\,{}^3D$	0,081	0,592	1,210	0,565	0,645	−8,9
3634,2	$2p\,{}^3P — 8d\,{}^3D$	0,081	0,617	1,233	0,565	0,668	−8,2

Shown in Fig. 3 are transition schemes for two singlet and two triplet helium lines between mixed terms of the isotopes He\(^3\) and He\(^4\), without allowance for hyperfine structure. In each scheme, on the left is shown the arrangement of the terms in the presence of one normal

Fig. 3. Transition scheme and isotopic structure of helium lines:
a) 5015.7 Å \((2s\,{}^1S — 3p\,{}^1P)\), b) 3888.6 Å \((2s\,{}^3S — 3p\,{}^3P)\),
c) 7281.3 Å \((3p\,{}^1P — 3s\,{}^1S)\), d) 4713.1 Å \((2p\,{}^3P — 4s\,{}^3S)\).

mixing. On the right is given the actual arrangement of the terms when normal and specific mixing are taken into account. At the bottom the isotopic structure is shown schematically in both cases. In the case of \(P \to S\) transitions (Fig. 3a, b) the lower term, like

this follows from the theory, undergoes only the normal displacement ($\tau = 0$), whereas the upper term, along with the normal displacement, also undergoes a specific displacement ($\tau \ne 0$). In this case, as also follows from the theory, in the case of singlet terms the specific displacement is directed in the same direction as the normal one and thus increases the displacement of the term, while in the case of triplet terms the opposite picture occurs. Hence the displacement of the lines is determined by the normal displacement of the lower term ($-\Delta T_{\mathrm{n}}$) and by the total displacement of the upper term ($-\Delta T'_{\mathrm{n}} \pm \Delta T'_{\mathrm{s}}$). For singlet lines the total isotopic displacement is $\Delta \nu = -\Delta T'_{\mathrm{n}} - \Delta T'_{\mathrm{s}} + \Delta T_{\mathrm{n}} = \Delta \nu_{\mathrm{n}} - \Delta \nu_{\mathrm{s}}$, and for triplet lines $\Delta \nu = -\Delta T'_{\mathrm{n}} + \Delta T'_{\mathrm{s}} + \Delta T_{\mathrm{n}} = \Delta \nu_{\mathrm{n}} + \Delta \nu_{\mathrm{s}}$. In the case of $S \to P$ transitions (Fig. 3b, c) the lower term undergoes, along with the normal displacement, also a specific displacement. Here, for singlet lines, the displacement is $\Delta \nu = -\Delta T'_{\mathrm{n}} + \Delta T_{\mathrm{n}} + \Delta T_{\mathrm{s}} = \Delta \nu_{\mathrm{n}} + \Delta \nu_{\mathrm{s}}$, and for triplet lines $\Delta \nu = -\Delta T'_{\mathrm{n}} + \Delta T_{\mathrm{n}} - \Delta T_{\mathrm{s}} = \Delta \nu_{\mathrm{n}} - \Delta \nu_{\mathrm{s}}$. It is easy to see that the displacement for triplet lines in $P \to S$ transitions, as well as for singlet lines in $S \to P$ transitions, is always positive. At the same time, the displacement for singlet lines in $P \to S$ transitions and triplet lines in $S \to P$ transitions may be negative, as is observed for the He 7065.2 Å line ($2p\,^3P — 3s\,^3S$). A negative displacement is obtained in this case as a result of the fact that the upper term in this transition undergoes a greater displacement than the lower one (see Table IV).

On the basis of the experimental data obtained on the isotopic displacement in the helium spectrum, it proved possible to pass from displacements in lines to displacements in terms. Bradley and Kuhn used the most accurate method for this purpose. From the isotopic-displacement data for the lines of each series they constructed a graph of the dependence of the line displacement on the value of the variable term (the upper one). By extrapolating the resulting straight line to its intersection with the axis on which the displacements were plotted, they determined the isotopic displacement for the constant (lower) term ($\Delta T$). Hence, knowing the line displacement ($\Delta \nu$), it is easy to find the absolute displacement of all variable terms: $\Delta T' = \Delta T + \Delta \nu$.

Tables III and IV give generalized data on the isotopic displacement of the terms of the helium atom, supplemented by us on the basis of the experimental results of Fred, Tomkins, and others. The first two columns give the symbols and numerical values of the terms; the 3rd column gives experimental data on the displacement of the terms for He$^3$ and He$^4$. The subsequent columns present calculated data on the normal ($\Delta T_{\mathrm{n}}$), specific ($\Delta T_{\mathrm{s}}$), and total ($\Delta T_{\mathrm{calc}}$) displacement of the terms; the last column gives the relative deviation of the experimental data from the theoretical, expressed in percent.

From Tables III and IV it is seen that, despite the general agreement of the experimental and theoretically calculated data on the isotopic shift of singlet and triplet terms, nevertheless in the overwhelming majority of cases the theoretical data are smaller than

Table III

Isotopic shift between the singlet terms of the isotopes He³ and He⁴

Term symbol	$T\ (\mathrm{cm}^{-1})$	$\Delta T_{\mathrm{exp}}\ (\mathrm{cm}^{-1})$	$\Delta T_{\mathrm{n}}\ (\mathrm{cm}^{-1})$	$\Delta T_{\mathrm{s}}\ (\mathrm{cm}^{-1})$	$\Delta T_{\mathrm{calc}}\ (\mathrm{cm}^{-1})$	$\dfrac{\Delta T_{\mathrm{exp}}-\Delta T_{\mathrm{calc}}}{\Delta T_{\mathrm{exp}}}\ (\%)$
1	2	3	4	5	6	7
$2s\,{}^{1}S$	32 033	−1,560	−1,436	0	−1,436	7,9
$3s\,{}^{1}S$	15 074	−0,635	−0,603	0	−0,603	5,5
$4s\,{}^{1}S$	7 371	−0,355	−0,330	0	−0,330	7,0
$5s\,{}^{1}S$	4 647	−0,245	−0,208	0	−0,208	15,1
$6s\,{}^{1}S$	3 196	−0,155	−0,143	0	−0,143	7,7
$7s\,{}^{1}S$	2 332	−0,110	−0,105	0	−0,105	4,5
$8s\,{}^{1}S$	1 776	−0,085	−0,080	0	−0,080	6,2
$2p\,{}^{1}P$	27 176	−1,680	−1,218	−0,372	−1,590	4,7
$3p\,{}^{1}P$	12 101	−0,710	−0,543	−0,117	−0,660	7,0
$4p\,{}^{1}P$	6 818	−0,400	−0,306	−0,050	−0,356	11,0
$5p\,{}^{1}P$	4 368	−0,252	−0,196	−0,026	−0,222	12,0
$6p\,{}^{1}P$	3 036	−0,177	−0,136	−0,015	−0,151	15,2
$7p\,{}^{1}P$	2 232	−0,130	−0,100	−0,009	−0,109	16,1
$3d\,{}^{1}D$	12 206	−0,565	−0,547	0	−0,547	3,2
$4d\,{}^{1}D$	6 864	−0,328	−0,308	0	−0,308	6,1
$5d\,{}^{1}D$	4 392	−0,225	−0,197	0	−0,197	12,4
$6d\,{}^{1}D$	3 050	−0,190	−0,137	0	−0,137	27,9
$7d\,{}^{1}D$	2 241	−0,150	−0,100	0	−0,100	33,3
$8d\,{}^{1}D$	1 715	−0,140	−0,077	0	−0,077	45,0

the experimental ones, and in many cases the difference lies outside the limits of experimental error. Even for such terms as ${}^{1}S$ and ${}^{3}S$ the magnitude of the isotopic shift ($\Delta T_{\mathrm{exp}}$) is on the average 7–8% greater than the normal shift ($\Delta T_{\mathrm{n}}$), calculated by formula (6). For the ${}^{1}P$- and ${}^{3}P$-terms the observed shift, as was to be expected, differs strongly from the normal shift. Taking into account the specific shift ($\Delta T_{\mathrm{s}}$), the agreement between the experimental

and theoretical data is noticeably improved; however, the observed shift for \({}^1P\)-terms is on average \(11\%\) greater than the calculated data \((\Delta T_{\mathrm{calc}})\). The agreement between experimental and calculated data for \({}^1D\)-terms at \(n=3, 4\) and \({}^3D\)-terms at \(n=3, 4, 5\) is quite good, whereas for these same terms with higher values of the principal quantum number \(n\)

Table IV

Isotopic shift between the triplet terms of the isotopes He\(^3\) and He\(^4\)

Term symbol	\(T\) \((\mathrm{cm}^{-1})\)	\(\Delta T_{\mathrm{exp}}\) \((\mathrm{cm}^{-1})\)	\(\Delta T_{\mathrm{n}}\) \((\mathrm{cm}^{-1})\)	\(\Delta T_{\mathrm{s}}\) \((\mathrm{cm}^{-1})\)	\(\Delta T_{\mathrm{calc}}\) \((\mathrm{cm}^{-1})\)	\(\dfrac{\Delta T_{\mathrm{exp}}-\Delta T_{\mathrm{calc}}}{\Delta T_{\mathrm{exp}}}\) \((\%)\)
1	2	3	4	5	6	7
\(2s\,{}^3S\)	38 445	−1,856	−1,724	0	−1,724	7,1
\(3s\,{}^3S\)	15 074	−0,71	−0,676	0	−0,676	4,8
\(4s\,{}^3S\)	8 013	−0,38	−0,359	0	−0,359	5,5
\(5s\,{}^3S\)	4 964	−0,24	−0,223	0	−0,223	7,1
\(6s\,{}^3S\)	3 375	−0,17	−0,151	0	−0,151	11,2
\(7s\,{}^3S\)	2 242	−0,11	−0,101	0	−0,101	8,2
\(8s\,{}^3S\)	1 849	−0,09	−0,083	0	−0,083	7,8
\(2p\,{}^3P\)	29 223	−0,675	−1,310	+0,565	−0,745	−10,3
\(3p\,{}^3P\)	12 746	−0,409	−0,571	+0,175	−0,396	3,2
\(4p\,{}^3P\)	7 094	−0,260	−0,318	+0,072	−0,246	5,4
\(5p\,{}^3P\)	4 510	−0,17	−0,202	+0,037	−0,165	2,9
\(6p\,{}^3P\)	3 118	−0,12	−0,138	+0,015	−0,123	−2,5
\(3d\,{}^3D\)	12 209	−0,535	−0,547	0	−0,547	−2,2
\(4d\,{}^3D\)	6 866	−0,304	−0,308	0	−0,308	−1,3
\(5d\,{}^3D\)	4 394	−0,192	−0,197	0	−0,197	−2,6
\(6d\,{}^3D\)	3 051	−0,119	−0,137	0	−0,137	−15,1
\(7d\,{}^3D\)	2 241	−0,074	−0,100	0	−0,100	−35,2
\(8d\,{}^3D\)	1 716	−0,049	−0,077	0	−0,077	−46,8

the discrepancy increases greatly, which may be attributed to perturbations of the \({}^1D\) and \({}^3D\) terms.

For many-electron atoms of light elements the isotopic effect has been studied to one degree or another in the spectra of lithium \(^{58-65}\), boron \(^{66,67}\), carbon \(^{68-70}\), nitrogen \(^{71-73}\), oxygen \(^{74}\), neon \(^{75-79}\), magnesium \(^{80-87}\), chlorine \(^{88}\), argon \(^{89,90}\), potassium \(^{91}\), and calcium \(^{92}\). Systematized experimental data on the isotopic shift

in the spectra of these elements (with the exception of oxygen and calcium) can be found in the tables of Brix and Kopfermann[^12].

It may be pointed out that, in the investigation of isotope shifts in the spectra of boron, carbon, nitrogen, and other elements, a large negative shift was found for some lines. Such a shift usually arises as the result of a transition from a more shifted upper term to a less shifted lower one. Sometimes, however, a negative shift occurs because of an “inverted” lower term. An explanation of this is given in Fig. 4, where the scheme of transitions for the line B I 2496.8 Å \((2p\ ^2P_{1/2}—3s\ ^2S_{1/2})\) is presented. Here, as before, on the left only the normal shift of the terms is taken into account; on the right, in addition, the specific shift for the \(P\)-term is taken into account \((\Delta T_c = 0.373\ \mathrm{cm}^{-1})\), which is directed in the opposite direction relative to the normal one and, in magnitude, exceeds the normal shift. As a result of this the term becomes “inverted,” i.e., the level of the lighter isotope lies below the level of the heavier isotope (positive shift).

Fig. 4. Scheme of transitions and isotopic structure of the line B I 2496.8 Å.

It follows directly from the transition scheme given that in this case a negative shift in the line is indeed obtained. Such “inverted” terms were also found in the case of nitrogen and magnesium.

In recent years the isotope shift in the spectra of oxygen and calcium has been studied, and the data on isotope shift in the spectrum of argon have been substantially supplemented. In the spectrum of oxygen the isotope shift was measured on 20 lines of the arc spectrum[^74]. For these purposes an enriched sample containing 29% \(O^{18}\) was used. For all the lines, with the exception of one, the shift proved to be positive. In magnitude it ranges from 0 to \(0.50\ \mathrm{cm}^{-1}\). The greatest shift is observed for the lines 4233.3 and 2883.8 Å. The data on isotope shift in the spectrum of argon were recently expanded by means of a study of the spectra of enriched samples[^90]. For 26 lines of the arc spectrum a positive shift was found; for 9 lines of the spark spectrum there is both a positive and a negative shift. Careful measurements of the shifts between the components of the isotopes \(Ar^{36}\),

\( \mathrm{Ar}^{38} \) and \( \mathrm{Ar}^{40} \) on 25 lines show exact agreement between the ratio of the intervals \(\Delta \nu(38-36):\Delta \nu(40-36)=0.532\) and the ratio of the mass numbers

\[ \frac{38-36}{36\cdot 38}:\frac{40-36}{36\cdot 40}=0.526 . \]

In the spectrum of calcium, the isotope shift between \(\mathrm{Ca}^{40}\) and \(\mathrm{Ca}^{48}\) was also studied with the aid of enriched samples\({}^{92}\). For the resonance arc line 4226.7 Å, and also for two spark lines 3933.7 and 3968.5 Å, the specific shift proved to be very small and directed in the same way as the normal shift. The triplet arc lines 6102.7, 6122.2, 6162.2 Å, on the contrary, give a large, oppositely directed, specific shift.

Table V brings together, for comparison, literature data on isotope shifts in certain lines of light elements; where experimental material is available, not only lines of the neutral atom but also lines of the ionized atom are presented. For comparison, the table includes lines belonging to transitions into one of the lowest states, i.e. chiefly resonance lines have been taken. In the 1st column the chemical symbol of the element is shown; in the 2nd, the wavelength of the spectral line; in the 3rd column it is indicated whether the given line belongs to the neutral or to the ionized atom; in the 4th column the terms between which the given transition occurs are given. Next, in the 5th column the symbols of the isotopes between which the shift was measured are presented in sequence. In the two following columns the magnitude of the isotope shift is given, in wavelengths \((\Delta \lambda_{\mathrm{exp}})\) and in wave numbers \((\Delta \nu_{\mathrm{exp}})\), according to the latest literature data. In the 8th column of Table V the normal isotope shift \((\Delta \nu_n)\), calculated by us from formula (5), is indicated. In the next column, literature data on the specific shift \((\Delta \nu_s)\), obtained on the basis of theoretical calculations for certain elements, are given. In the 10th column the total isotope shift obtained on the basis of the calculations is given. Finally, in the last two columns the deviations between the experimental and theoretical values of the isotope shift \((\Delta \nu_{\mathrm{exp}}-\Delta \nu_{\mathrm{calc}})\) relative to \(\Delta \nu_{\mathrm{exp}}\) and \(\Delta \nu_s\), expressed in percent, are given.

From Table V it is seen that the magnitude of the isotope shift in the spectra of neutral and ionized atoms decreases rapidly with increasing atomic number of the element, reaching thousandths of a \(\mathrm{cm}^{-1}\) per atomic mass unit by the end of the third period of the Mendeleev system. For each element it is characteristic that the isotope shift for spark lines exceeds, on average by a factor of 2–4, the shift for arc lines. This is due to the fact that the share of the specific shift in the total isotope shift of lines of the ionized atom is considerably larger than the corresponding share in the isotope shift

Comparative data on the isotopic

Element	Wavelength (Å)	Atom type	Transition	Isotopes studied	$\Delta\lambda_{\mathrm{exc}}$ (Å)
1	2	3	4	5	6
H	1215,7	I	$1s\,{}^{2}S — 2p\,{}^{2}P$	$\mathrm{H}^{1} — \mathrm{H}^{2}$	0,329
H	6562,8	I	$2p\,{}^{2}P — 3d\,{}^{2}D$	$\mathrm{H}^{1} — \mathrm{H}^{2}$	1,784
He	5015,7	I	$2s\,{}^{1}S — 3p\,{}^{1}P$	$\mathrm{He}^{3} — \mathrm{He}^{4}$	0,213
He	3888,6	I	$2s\,{}^{3}S — 3p\,{}^{3}P$	$\mathrm{He}^{3} — \mathrm{He}^{4}$	0,212
Li	6707,8	I	$2s\,{}^{2}S — 2p\,{}^{2}P$	$\mathrm{Li}^{6} — \mathrm{Li}^{7}$	0,160
Li	5484,7	II	$2s\,{}^{3}S — 2p\,{}^{3}P$	$\mathrm{Li}^{6} — \mathrm{Li}^{7}$	0,342
B	2497,7	I	$2p\,{}^{2}P_{3/2} — 3s\,{}^{2}S_{1/2}$	$\mathrm{B}^{10} — \mathrm{B}^{11}$	−0,010
B	3451,4	II	$2s\,2p\,{}^{1}P_{1} — 2p^{2}\,{}^{1}D_{2}$	$\mathrm{B}^{10} — \mathrm{B}^{11}$	0,104
C	2478,5	I	$2p^{2}\,{}^{1}S_{0} — 2p\,3s\,{}^{1}P_{1}$	$\mathrm{C}^{12} — \mathrm{C}^{13}$	−0,009
C	2836,7	II	$2s\,2p^{2}\,{}^{2}S_{1/2} — 2s^{2}\,3p\,{}^{2}P_{3/2}$	$\mathrm{C}^{12} — \mathrm{C}^{13}$	−0,049
N	9629,6	I	$3s\,{}^{2}P_{3/2} — 3p\,{}^{2}P_{3/2}$	$\mathrm{N}^{14} — \mathrm{N}^{15}$	0,052
N	8242,5	I	$3s\,{}^{4}P_{5/2} — 3p\,{}^{4}P_{3/2}$	$\mathrm{N}^{14} — \mathrm{N}^{15}$	0,041
O	8446,4	I	$3s\,{}^{3}S_{1} — 3p\,{}^{3}P_{1}$	$\mathrm{O}^{16} — \mathrm{O}^{18}$	0,100
O	7157,4	I	$3s\,{}^{1}D_{2} — 3p\,{}^{1}D_{2}$	$\mathrm{O}^{16} — \mathrm{O}^{18}$	0,056
O	4233,3	I	$4p\,{}^{3}P_{2} — 3d\,{}^{3}P_{2}$	$\mathrm{O}^{16} — \mathrm{O}^{18}$	0,084
Ne	7173,9	I	$3p\,2p_{8} — 3s\,1s_{2}$	$\mathrm{Ne}^{20} — \mathrm{Ne}^{22}$	0,035
Ne	7032,4	I	$3p\,2p_{10} — 3s\,1s_{5}$	$\mathrm{Ne}^{20} — \mathrm{Ne}^{22}$	0,026
Ne	3323,8	II	$3s\,{}^{2}P_{3/2} — 3p\,{}^{2}P_{3/2}$	$\mathrm{Ne}^{20} — \mathrm{Ne}^{22}$	0,029
Mg	2852,1	I	$3s^{2}\,{}^{1}S_{0} — 3s\,3p\,{}^{1}P_{1}$	$\mathrm{Mg}^{24} — \mathrm{Mg}^{26}$	0,005
Mg	4571,1	I	$3s^{2}\,{}^{1}S_{0} — 3s\,3p\,{}^{3}P_{1}$	$\mathrm{Mg}^{24} — \mathrm{Mg}^{26}$	0,017
Mg	8806,8	I	$3s\,3p\,{}^{1}P_{1} — 3s\,3d\,{}^{1}D_{2}$	$\mathrm{Mg}^{24} — \mathrm{Mg}^{26}$	0,066
Mg	2795,5	II	$3s\,{}^{2}S_{1/2} — 3p\,{}^{2}P_{3/2}$	$\mathrm{Mg}^{24} — \mathrm{Mg}^{26}$	0,008
Cl	4810,1	II	$4s\,{}^{5}S_{2} — 4p\,{}^{5}P_{2}$	$\mathrm{Cl}^{35} — \mathrm{Cl}^{37}$	0,008
Ar	7147,0	I	$4s\,{}^{1}S_{5} — 4p\,{}^{2}P_{4}$	$\mathrm{Ar}^{36} — \mathrm{Ar}^{40}$	0,009
Ar	4510,7	I	$4s\,{}^{1}S_{2} — 5p\,{}^{3}P_{5}$	$\mathrm{Ar}^{36} — \mathrm{Ar}^{40}$	0,010
Ar	4579,4	II	$4s\,{}^{2}P_{1/2} — 4p\,{}^{2}S_{1/2}$	$\mathrm{Ar}^{36} — \mathrm{Ar}^{40}$	0,021
K	7699,0	I	$4s\,{}^{2}S_{1/2} — 4p\,{}^{2}P_{1/2}$	$\mathrm{K}^{39} — \mathrm{K}^{41}$	0,005

shifts in the spectra of light elements

Table V

\(\Delta\nu_{\mathrm{экс}}\) \((\mathrm{cm}^{-1})\) 7	\(\Delta\nu_{\mathrm{н}}\) \((\mathrm{cm}^{-1})\) 8	\(\Delta\nu_{\mathrm{с}}\) \((\mathrm{cm}^{-1})\) 9	\(\Delta\nu_{\mathrm{рас}}\) \((\mathrm{cm}^{-1})\) 10	\(\dfrac{\Delta\nu_{\mathrm{экс}}-\Delta\nu_{\mathrm{рас}}}{\Delta\nu_{\mathrm{экс}}}\) (%) 11	\(\dfrac{\Delta\nu_{\mathrm{экс}}-\Delta\nu_{\mathrm{рас}}}{\Delta\nu_{\mathrm{с}}}\) (%) 12
22,300 4,144	22,365 4,147	— —	22,365 4,147	— —	— —
0,849 1,404	0,893 1,153	−0,117 0,175	0,776 1,328	8,6 5,4	62,3 43,4
0,350 1,14	0,194 0,24	0,08 0,85	0,27 1,09	22,8 4,4	100,0 5,9
−0,168 0,877	0,198 0,143	−0,366 0,566	−0,168 0,709	0,0 19,1	0,0 29,6
−0,156 −0,612	0,142 0,124	−0,295 —	−0,153 —	1,9 —	1,0 —
0,070 −0,060	0,030 0,032	— —	— —	— —	— —
0,14 0,11 0,47	0,05 0,05 0,09	0,08 0,06 —	0,13 0,11 —	7,1 7,0 —	12,5 0,0 —
0,068 0,052 0,260	0,034 0,035 0,075	−0,0038 0,0157 —	0,030 0,051 —	56 1,9 —	1000 5,1 —
0,061 0,083 0,085 0,102	0,062 0,038 0,020 0,063	−0,0094 0,012 0,024 —	0,053 0,050 0,044 —	13,1 39,7 48,2 —	85,1 275 171 —
0,035	0,018	—	—	—	—
0,018 0,048 0,100	0,021 0,033 0,033	— — —	— — —	— — —	— — —
0,008	0,009	—	—	—	—

lines of a neutral atom, which is explained by the increase in the effective nuclear charge and in the Rydberg constant for ions.

Calculations of the isotope shift in the spectra of many-electron atoms involve great difficulties, since for these atoms there are no exact values of the wave functions needed for a quantitative estimate of the specific shift. However, in a number of cases these difficulties have been overcome, and for the spectra of some of the elements mentioned calculations have been carried out, the results of which are given in Table V. The most detailed calculations were made by Bredy and Kuhn[^56], as well as by Fred, Tomkins, and others[^57] for the isotope shift in the helium spectrum, which we considered above. In the lithium spectrum, calculations for three lines were also carried out in 1930–1931 by Hughes and Eckart[^14,^62] with the aim of testing the theory of the specific effect developed by them. Calculations of the isotope shift in the boron spectrum were performed by Opechowski and de Vries[^21], and later more thoroughly by Vinti[^23]. A theoretical determination of the isotope shift in the spectrum of the neutral carbon atom by means of one-electron wave functions of the Hartree self-consistent field was recently made by A. P. Yutsis, A. S. Nakechnis, and G. K. Tsunaitis[^44]. A quantitative estimate of the specific shift in the oxygen spectrum was given by Parker and Holmæss[^74]. In the neon spectrum the isotope shift was calculated by Bartlett and Gibbons[^19] and, finally, in the magnesium spectrum the calculation was carried out by Vinti[^22].

Comparison of the experimental data with the theoretical ones (see Table V, columns 7, 10) shows, at first glance, fairly good agreement for almost all elements for which calculations of the isotope shift have been made. The exceptions are the arc lines of neon and magnesium, for which the deviations of the experimental data from the theoretical reach 50% (Table V, column 11). At the same time it should be noted that, for the spectra of all elements, the theoretical data are somewhat lower than the experimental results, and in most cases the discrepancy exceeds the experimental error obtained in measuring the intervals between the components of the isotope structure. It is of interest to compare the difference between the experimental and theoretical shift with the magnitude of the specific shift obtained by calculation (Table V, column 12). It turns out that the unaccounted-for part of the isotope shift in the spectra of He I, Li I, B II, Ne I, and Mg I amounts to from 30 to 1000% of the specific shift. As was already pointed out in the discussion of the isotope shift of helium terms, a systematic difference between the experimental and theoretical data on the isotope shift occurs not only for $P$-terms, but also for $S$- and $D$-terms, which, as is usually assumed, should undergo only one normal shift.

At present, it is perhaps difficult to say whether the above-mentioned discrepancies between the experimental and theoretical data are explained by the inaccuracy of the wave functions used in the theoretical calculations and by the approximate nature of the calculations themselves, or whether they are the result of some other effects that make their own contribution to the isotopic shift of terms and lines. Recently this question was considered by I. I. Goldman \(^{41}\), who believes that, in calculating the isotopic shift in light elements, in addition to the normal mass effect and the specific effect caused by the exchange interaction of the valence electrons with one another, one must also take into account the exchange interaction between the valence electrons and the electrons of filled shells, as well as the exchange interaction between the electrons of filled shells. In earlier theoretical calculations the isotopic shift due to closed shells was not taken into account, since it was assumed that this part of the shift is the same in the initial and final states of the atom. In fact, however, in optical transitions the field in which the inner electrons move changes and, consequently, the wave functions and matrix elements of the momentum change; therefore the shift caused by a closed shell will be different in the initial and final states of the atom. In atoms with fewer than 10 electrons, the fraction of the isotopic shift under consideration goes to zero, since in such atoms there is only one filled shell. In atoms whose number of electrons is greater than or equal to 10, this fraction of the isotopic shift is not equal to zero. In such atoms there are at least two filled shells, whose orbital angular momenta differ by unity. This may explain the fact that theoretical calculations of the isotopic shift in the spectra of lithium, boron, carbon, nitrogen, and oxygen agree fairly well with the experimental data, whereas for neon and magnesium the magnitude of the isotopic shift calculated without taking into account the influence of closed shells lies far outside the limits of experimental error. Unfortunately, the considerations developed by I. I. Goldman have not been carried through, for neon and magnesium, to quantitative calculations.

It should also be pointed out that the discrepancies noted above between the experimental and theoretical data on the isotopic shift may possibly be explained by polarization of the atomic shell. This phenomenon is due to the electrostatic interaction of electrons which, repelling one another, tend to arrange themselves on different sides of the nucleus. Polarization of the atomic shell leads to a decrease in the motion accompanying the nucleus and thus acts in the opposite direction as compared with the normal mass effect. An attempt to estimate quantitatively the isotopic shift caused by the polarization effect was made for some terms of the helium atom \(^{57}\). However

the estimate showed that the share of the isotopic shift due to this effect is very small and cannot fully explain the discrepancy existing for helium between theory and experiment.

Summing up, it should be said that the isotopic effect in the spectra of light elements (from \(Z=1\) to \(Z=20\)) has been studied rather thoroughly. The most complete experimental data have been collected for 11 elements (H, He, Li, B, C, N, O, Ne, Mg, Ar, Ca). For most of these elements (with the exception of N and O), the isotopic shift has been investigated on arc and spark lines. For a number of elements a shift has been found in terms (H, He, Li, N, Ne, Mg, Ar). The isotopic shift has been insufficiently fully investigated for Cl and K, and has not been studied at all for Si and S. To explain the isotopic effect in the atomic spectra of light elements, as we have seen, a sufficiently complete theory has been developed, according to which the isotopic shift of terms is due to displacement of the nucleus as a result of the presence of rotational electronic motions and the correlation of these motions resulting from exchange interaction between electrons. However, detailed investigations show that the theory of the isotopic effect mentioned, in the case of many-electron atoms, does not completely explain the observed isotopic shift. Attempts to take other effects into account have not yet received complete quantitative confirmation.

3. ISOTOPIC EFFECT IN THE SPECTRA OF MEDIUM ELEMENTS

In the spectra of medium elements the isotopic shift is very small. For some elements the isotopic shift in lines reaches \(0.001\ \mathrm{cm}^{-1}\) per 2 mass units and, in its magnitude, lies at the limit of the experimental capabilities of spectroscopy. In the isotopic shift for the spectra of medium elements, both the mass effect (normal and specific) and the volume effect (see below) play a role. The shifts due to both effects for the spectra of medium elements are small and, moreover, the normal shift and the shift caused by the volume effect are always directed in opposite directions. For elements with atomic number from \(Z=20\) to \(Z=30\), the predominant share in the isotopic shift is due to the mass effect, whereas for elements from \(Z=40\) to \(Z=55\) the volume effect is of primary importance.

The most detailed data on the isotopic shift in the spectra of medium elements have at present been obtained for copper, zinc, zirconium, molybdenum, ruthenium, palladium, silver, cadmium, tin, and tellurium.

In the spectrum of copper, the isotopic shift has been investigated between the centers of gravity of the hyperfine-structure components of the isotopes

Cu\(^{65}\) and Cu\(^{63}\) on two resonance lines of Cu I

\[ 3247.5\,\text{\AA}\;(4s\,{}^{1}S_{1/2} — 4p\,{}^{2}P_{3/2}),\qquad 3274.0\,\text{\AA}\;(4s\,{}^{2}S_{1/2} — 4p\,{}^{2}P_{1/2}) \]

and on three Cu I lines with metastable levels

\[ 5105.5\,\text{\AA}\;(4s^{2}\,{}^{2}D_{5/2} — 4p\,{}^{2}D_{3/2}),\qquad 5700.2\,\text{\AA}\;(4s^{2}\,{}^{2}D_{3/2} — 4p\,{}^{2}P_{3/2}), \]

\[ 5782.1\,\text{\AA}\;(4s^{2}\,{}^{2}D_{3/2} — 4p\,{}^{2}P_{1/2})^{93—96}. \]

For all lines the shift proved to be positive. On the basis of measurements of the hyperfine and isotope structure, the isotope shift was found for the ground state \(3d^{9}4s^{2}\,{}^{2}S_{1/2}\) (it proved equal to \(-0.018\ \text{cm}^{-1}\)), as well as the shifts for two metastable levels \(3d^{9}4s^{2}\,{}^{2}D_{5/2}\), \(3d^{9}4s^{2}\,{}^{2}D_{3/2}\) (respectively equal to \(-0.085\) and \(-0.074\ \text{cm}^{-1}\))\(^{12}\). The negative sign of the shift means that the center of gravity of the term of the heavier isotope (Cu\(^{65}\)) lies deeper. If the normal shift, calculated by formula (3), is subtracted from the magnitude of the shift of these terms, then the remainder for the term \({}^{2}S_{1/2}\) will be \(-0.001\ \text{cm}^{-1}\), while the remainders for the terms \({}^{2}D_{5/2}\) and \({}^{2}D_{3/2}\) will be, respectively, \(-0.071\) and \(-0.061\ \text{cm}^{-1}\). Such a large difference between the \({}^{2}S\) and \({}^{2}D\) terms cannot be explained by the volume effect, since the latter is only about twice as large for electronic configurations with \(s^{2}\)-electrons\(^{120}\). Hence it follows, as was already noted by Bartlett and Gibbons\(^{19}\), that apparently the absence of one electron in the \(3d\)-shell in the case of the complex configuration \(3d^{9}4s^{2}\) leads to a large specific shift, which exceeds the normal shift several times and is directed in the same direction with respect to the latter.

The isotope shift in the spectrum of zinc was measured between the components of the isotopes Zn\(^{64}\), Zn\(^{66}\), Zn\(^{68}\) on the lines: Zn I 2138.6; 3075.9 Å\(^{98}\) and Zn II 5894.4; 6214.6; 6471.0; 7478.8 Å\(^{97}\). For arc lines the shift proved to be positive, for spark lines negative and very large. Table VI gives the experimental data on the isotope shift of arc lines, as well as the normal shift and the difference \(\Delta\nu_{\text{exp}}-\Delta\nu_{\text{n}}\).

Table VI

Isotope shift between the lines of the isotopes Zn\(^{64}\) and Zn\(^{68}\)

\(\lambda\) (Å)	Transition	\(\Delta\nu_{\text{exp}}\) \((\text{cm}^{-1})\)	\(\Delta\nu_{\text{n}}\) \((\text{cm}^{-1})\)	\(\Delta\nu_{\text{exp}}-\Delta\nu_{\text{n}}\) \((\text{cm}^{-1})\)
2138.6	\(4s^{2}\,{}^{1}S_{0} — 4s\,4p\,{}^{1}P_{1}\)	\(+0.033\)	\(+0.023\)	\(+0.010\)
3075.9	\(4s^{2}\,{}^{1}S_{0} — 4s\,4p\,{}^{3}P_{1}\)	\(+0.046\)	\(+0.016\)	\(+0.030\)

Comparing the shifts of this pair of lines, one can, without any calculations, roughly estimate the magnitude of the specific shift for these lines. Since the upper levels for the lines 2138.6 and 3075.9 Å belong to one and the same electron configuration, while the lower level is common, the isotope shift due to the volume effect will be approximately the same for these lines. The specific shift, however, will be different, since the line 3075.9 Å is an intercombination line with an upper level belonging to the triplet system of terms. According to Table VI, the difference \(\Delta \nu_{\mathrm{exp}}-\Delta \nu_{\mathrm{n}}\) for the two lines mentioned differs by \(0.020\ \mathrm{cm}^{-1}\). Taking the foregoing into account, this difference must be attributed to the specific shift. Thus it is already evident from this that the specific shift plays a considerable role, for in order of magnitude it approaches the experimental value of the isotope shift for the lines under consideration. If, moreover, one takes into account that the theoretical estimate of the isotope shift due to the volume effect gives, for these zinc lines, approximately \(-0.03\ \mathrm{cm}^{-1}\) (apparently an upper limit), while the average difference \(\Delta \nu_{\mathrm{exp}}-\Delta \nu_{\mathrm{n}}\) is equal to \(+0.020\ \mathrm{cm}^{-1}\), then the mean value of the specific shift must be approximately \(+0.05\ \mathrm{cm}^{-1}\). In the spark lines of zinc, as already noted, a large negative shift has been found between the components of the isotopes \(\mathrm{Zn}^{64}\) and \(\mathrm{Zn}^{68}\), reaching \(-0.189\ \mathrm{cm}^{-1}\). These zinc lines belong to the transitions \(3d^{10}4p\ ^2P — 3d^94s^2\ ^2D\). By analogy with the isotope shift in the spectrum of copper, the presence of a large negative shift in the Zn II lines can be explained by a large specific shift of the upper term, belonging to the same configuration \(3d^94s^2\).

Thus, the anomalously large isotope shift in the spectra of copper and zinc (if it is compared with the isotope shift of neighboring elements: Ar, K, Ga, Se, Br, Kr), which was noted at the time by Schüler and Westmeyer[^119] and was attributed by them to the volume effect with the opposite direction of the shift (it was assumed that nuclei with larger mass numbers have smaller radii), is due, as has now been established, to an anomalously large specific shift. From this it is clear that the specific effect is of great importance when considering the isotope shift for medium elements. Therefore, in order to isolate in the spectra of medium elements the isotope shift due to the volume effect, which makes it possible to judge certain properties of nuclei (for example, their radii), it is necessary first to take into account the mass effect, including the specific shift.

Of the heavier elements belonging to the middle part of the periodic table of elements, we shall first consider zirconium, molybdenum, ruthenium, and palladium. The isotope shift

in the spectra of these four elements was studied recently, and since these new data were not included in the summary tables of Briks and Kopfermann^12, they are given by us in Table VII.

It is characteristic that for the lines of these four elements the shift proved to be negative^109–114, which corresponds to the volume effect. If one calculates the normal shift for all the spectral lines listed in Table VII and roughly takes account of the specific shift by doubling the normal shift, then, subtracting the obtained values from the experimental magnitudes of the isotope shift, one can roughly estimate the shift due to the volume effect. A comparison of the data obtained in this way shows that for zirconium and molybdenum the volume effect exceeds the mass effect by approximately a factor of three; for ruthenium and palladium (the lines 3489.8 and 3404.6 Å), by approximately a factor of four. For the Pd I lines 8132.8 and 7961.1 Å the volume effect is apparently still larger, which is due to the electron configuration \(s^2\), to which the lower term belongs (see above).

Table VII

Isotope shift in the spectra of Zr, Mo, Ru, Pd

Element and atom type	\(\lambda\) (Å)	Transition	Isotopes	\(\Delta\nu_{\mathrm{exp}}\) \((\mathrm{cm}^{-1})\)	\(\Delta\nu_{\mathrm{n}}\) \((\mathrm{cm}^{-1})\)
Zr I	4081.2	\(4d^25s\ ^5F_5 — 4d^35p\ ^5D_4\)	\(\mathrm{Zr}^{90}—\mathrm{Zr}^{94}\)	\(-0.020\)	\(+0.0063\)
Mo I	5650.1	\(4d^45s^2\ ^5D_1 — 4d^55p\ ^5P_2\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Mo I	5689.1	\(4d^45s^2\ ^5D_1 — 4d^55p\ ^5P_1\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Mo I	5791.8	\(4d^45s^2\ ^5D_2 — 4d^55p\ ^5P_1\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Mo I	5858.3	\(4d^45s^2\ ^5D_3 — 4d^55p\ ^5P_3\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Mo I	5888.3	\(4d^45s^2\ ^5D_3 — 4d^55p\ ^5P_2\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Mo I	6030.7	\(4d^45s^2\ ^5D_4 — 4d^55p\ ^5P_3\)	\(\mathrm{Mo}^{95}—\mathrm{Mo}^{97}\)	on average \(-0.0126\)	\(+0.0020\)
Ru I	4554.5	\(4d^75s\ ^3F_4 — 4d^75p\ ^3G_5\)	\(\mathrm{Ru}^{96}—\mathrm{Ru}^{100}\)	\(-0.030\)	\(+0.0024\)
Ru I	4554.5	\(4d^75s\ ^3F_4 — 4d^75p\ ^3G_5\)	\(\mathrm{Ru}^{100}—\mathrm{Ru}^{102}\)	\(-0.015\)	\(+0.0024\)
Ru I	4554.5	\(4d^75s\ ^3F_4 — 4d^75p\ ^3G_5\)	\(\mathrm{Ru}^{102}—\mathrm{Ru}^{104}\)	\(-0.016\)	\(+0.0024\)
Pd I	8132.8	\(4d^85s^2\ ^3F_4 — 4d^95p\ ^3D_3\)	\(\mathrm{Pd}^{106}—\mathrm{Pd}^{110}\)	\(-0.0652\)	—
Pd I	7961.1	\(4d^85s^2\ ^3F_3 — 4d^95p\ ^1D_2\)	\(\mathrm{Pd}^{108}—\mathrm{Pd}^{110}\)	\(-0.0298\)	\(+0.0012\)
Pd I	3489.8	\(4d^95s\ ^1D_2 — 4d^95p\ ^3D_1\)	\(\mathrm{Pd}^{108}—\mathrm{Pd}^{110}\)	\(-0.016\)	\(+0.0027\)
Pd I	3404.6	\(4d^95s\ ^3D_3 — 4d^95p\ ^3F_4\)	\(\mathrm{Pd}^{108}—\mathrm{Pd}^{110}\)	\(-0.016\)	\(+0.0027\)

The isotopic shift in the spectrum of silver was measured on several arc and spark lines. This was done most carefully by Bricks, Kopfermann, Martin, and Walcher[^118] on separated isotopes Ag\(^{107}\) and Ag\(^{109}\). For the spark lines they succeeded in establishing the sign of the isotopic shift (it proved to be negative), which had not previously been done by Rasmussen[^116]. In addition, in the earlier interpretation of the components of the hyperfine structure of the resonance lines Ag I 3280.7 Å \((5s\,{}^2S_{1/2} — 5p\,{}^2P_{3/2})\) and 3382.9 Å \((5s\,{}^2S_{1/2} — 5p\,{}^2P_{1/2})\)[^115][^117], they uncovered an error, as a result of which the data on the magnetic moments and on the isotopic shift of these lines were substantially corrected. The shift between the centers of gravity of the hyperfine-structure components of the isotopes Ag\(^{107}\) and Ag\(^{109}\) proved to be equal to \(-0.015\ \text{cm}^{-1}\) and, in its sign, to correspond to the volume effect. The normal shift for the mentioned Ag lines is approximately \(+0.0028\ \text{cm}^{-1}\). If one takes into account that the experimental value of the shift for the resonance lines of the alkali elements (Li I, Mg II, K I), and also for Cu I, exceeds, on the average, the normal shift by no more than a factor of two, then the total mass shift for the resonance lines of silver may roughly be taken as equal to \(+0.005\ \text{cm}^{-1}\). This means that the shift due to the volume effect in the case of these lines reaches approximately \(-0.020\ \text{cm}^{-1}\), i.e., in absolute value it is approximately 4 times greater than the total mass (normal and specific) shift.

For cadmium, the isotopic shift was investigated between the even isotopes with mass numbers 110, 112, 114 on the resonance line Cd I 3261.1 Å \((5s^2\,{}^1S_0 — 5s5p\,{}^3P_1)\)[^120] and on the spark line Cd II 4415.6 Å \((4d^{10}5p\,{}^2P_{1/2} — 4d^95s^2\,{}^2D_{5/2})\)[^119][^120]. For both lines the shift proved to be negative, which again indicates the presence of a volume effect. Calculation shows that for the terms \({}^1S_0\) and \({}^3P_1\) the normal shift between Cd\(^{110}\)—Cd\(^{112}\) is respectively \(-0.0065\) and \(-0.0037\ \text{cm}^{-1}\). If one takes into account, by analogy with the spectra He I, Li II, and Mg I, that the specific shift for the triplet term \({}^3P_1\) approximately halves the normal shift of this term, then for the line Cd I 3261.1 Å the total mass effect will be approximately \(+0.0046\ \text{cm}^{-1}\). Subtracting this value from the experimentally found shift \((-0.016\ \text{cm}^{-1})\), one can roughly determine the magnitude of the volume effect, which proves to be approximately equal to \(-0.021\ \text{cm}^{-1}\). Hence it is seen that here, as for silver, the isotopic shift due to the volume effect is, in absolute value, approximately 4.5 times greater than the total mass shift.

A quantitative estimate of the volume effect in the spectra of some atoms with one and two valence electrons has recently been made by I. I. Gol’dman[^41]. For this purpose he calculated the total mass effect (normal and specific, ...

displacements), taking into account the interaction between valence electrons and the electrons of filled shells, according to the theory he developed. The volume effect \((\Delta \nu_0)\) was obtained by subtracting the theoretically calculated total mass shift \((\Delta \nu_{\mathrm{m}})\) from the experimental value of the shift \((\Delta \nu_{\mathrm{exp}})\). The results for the resonance lines of Cu, Zn, Rb, Ag, Cd, and Ba are given in Table VIII, which contains the isotope shifts for a change of the mass number by two units. For Zn, Cd, and Ba, where the shift was measured for several pairs of even isotopes, mean values were taken. It is seen from Table VIII that the total mass shift in the spectra of Cu and Zn is, in absolute magnitude, approximately \(1.5\)–\(2\) times larger than the shift due to the volume effect. In the spectra of Ag, Cd, and Ba, on the contrary, the volume effect is approximately the same factor larger than the mass shift. It should be noted that the isotope shifts for Ag and Cd calculated by I. I. Goldman and caused by the volume effect are \(1.5\)–\(2\) times larger than the empirical estimates of Briks, Kopferman, and others.\(^{118,120}\)

Table VIII

Volume effect for \(\Delta A = 2\) in the spectra of medium elements

Element and atom type	\(\lambda\) (Å)	Transition	\(\Delta \nu_{\mathrm{exp}}\) (cm\(^{-1}\))	\(\Delta \nu_{\mathrm{m}}\) (cm\(^{-1}\))	\(\Delta \nu_0\) (cm\(^{-1}\))
Cu I	3274.0	\(4s\,{}^{2}S_{1/2} — 4p\,{}^{2}P_{1/2}\)	0.018	0.0358	−0.018
Zn I	2138.6	\(4s^{2}\,{}^{1}S_{0} — 4s4p\,{}^{1}P_{1}\)	0.0165	0.0476	−0.031
Zn I	3075.9	\(4s^{2}\,{}^{1}S_{0} — 4s4p\,{}^{3}P_{1}\)	0.023	0.055	−0.032
Rb I	7800.2	\(5s\,{}^{2}S_{1/2} — 5p\,{}^{2}P_{1/2}\)	0.003	0.0093	−0.010
Ag I	3280.7	\(5s\,{}^{2}S_{1/2} — 5p\,{}^{2}P_{3/2}\)	−0.015	0.0182	−0.033
Ag I	3382.9	\(5s\,{}^{2}S_{1/2} — 5p\,{}^{2}P_{1/2}\)	−0.015	0.018	−0.033
Cd I	3261.1	\(5s^{2}\,{}^{1}S_{0} — 5s5p\,{}^{3}P_{1}\)	−0.0147	0.0241	−0.039
Ba I	5535.5	\(6s^{2}\,{}^{1}S_{0} — 6s6p\,{}^{1}P_{1}\)	−0.0022	0.0063	−0.0085
Ba II	4554.0	\(6s\,{}^{2}S_{1/2} — 6p\,{}^{2}P_{3/2}\)	−0.0054	0.0094	−0.0148
Ba II	4934.1	\(6s\,{}^{2}S_{1/2} — 6p\,{}^{2}P_{1/2}\)	−0.0048	0.0093	−0.0141

Recently the isotope shift in the spectrum of cadmium and tin was measured with the aid of enriched samples on the lines Cd II 4415.6 Å and Sn II 6453.6 Å[^120a]. Before this work, the isotope shift in the spectra of Sn II\({}^{123}\) and Sn II\({}^{121,122}\), although detected, had not been measured. The newly obtained results are presented in Table IX, where the shift between even isotopes is given,

Table IX

\(A\)	110—112	112—114	114—116	116—118	118—120	120—122	122—124
Cd	−0.0533	−0.0479	−0.0344	—	—	—	—
Sn	—	−0.0067	−0.0061	−0.0044	−0.0044	−0.0012	−0.0017

expressed in \(\mathrm{cm}^{-1}\). If from the data obtained one subtracts the normal shift and takes as the unit of shift the intervals between the isotopes Cd\({}^{110}\)—Cd\({}^{112}\) and Sn\({}^{112}\)—Sn\({}^{114}\) with isotone nuclei, then the relative shift between other isotone pairs of isotopes can be represented as follows:

\(N\)	62—64	64—66	66—68	68—70	70—72	72—74
Cd	1.00	0.90	0.63	—	—	—
Sn	1.00	0.93	0.72	0.72	0.32	0.39

These data show that there is a similarity in the relative shift between isotone pairs of isotopes of cadmium and tin. It might seem possible to note that such a similarity is absent between the isotone isotopes of tin and tellurium[^126] in the region of \(N\) from 68 to 74. However, it is not yet possible to draw such a conclusion, since there is no complete confidence in the reliable measurement of the intervals between the tin isotopes \(\Delta \nu\) (120—122) and \(\Delta \nu\) (122—124).

The isotope shift in the spectrum of tellurium was studied on two spark lines, 4006.5 Å \((5s5p^4\,{}^4P_{5/2} - 5p^2({}^3P_0)6p_{3/2})\) and 4048.9 Å \((5s5p^4\,{}^4P_{3/2} - [5p^2({}^3P_1)6p]_{3/2})\), with the aid of enriched samples[^126].

The structure of both lines and the intervals between components were found, within the experimental errors, to be the same. As an example, Fig. 5 gives schematically the isotopic

Fig. 5. Isotopic structure of the
Te II 4006.5 Å line.

structure of the 4006.5 Å line. Above, opposite the vertical lines representing the isotope components, the mass numbers are indicated. Below are given the intervals between components in wave numbers \((10^{-3}\ \mathrm{cm}^{-1})\), measured with an accuracy up to \(\pm 0.003\ \mathrm{cm}^{-1}\). The dashed lines mark the centers of gravity of the hyperfine-structure components for the odd isotopes. The displacement in the two mentioned tellurium lines is positive. In addition, it turns out that the centers of gravity of the odd isotopes are additionally shifted toward the lighter even isotopes. It may also be noted that the intervals between the components Te\(^{124}\)—Te\(^{126}\) on both lines are on average approximately 1.7 times greater than the intervals between the other even isotopes. This anomalously large shift is found for isotopes whose nuclei consist of 72 and 74 neutrons.

Barium may also be classed among the medium elements. The isotopic effect in the barium spectrum has been studied on enriched isotopes with mass numbers 134, 135, 136, 137, 138 [130]. Table X gives the experimental results on the isotopic shift of three lines relative to the component of the isotope Ba\(^{138}\) (accuracy of measurements \(\pm 0.0007\ \mathrm{cm}^{-1}\)). The shift on all lines is negative.

Table X

Isotopic shift in the Ba spectrum

Element and atom type	\(\lambda\) (Å)	138	137	136	135	134
Ba I	5535.5	0.0	−0.0052	−0.0022	−0.0074	−0.0044
Ba II	4554.0	0.0	−0.0064	−0.0054	−0.0126	−0.0108
Ba II	4934.1	0.0	−0.0060	−0.0048	−0.0111	−0.0096

Of some interest is the rather unusual arrangement of the components (for odd isotopes the centers of gravity are meant) for the lines Ba I (138, 136, 134, 137, 135) and Ba II (138, 136, 137, 134, 135). However, I. I. Gol’dman \(^{41}\) showed that, if the mass effect is excluded, the remaining displacement, due to the volume effect, gives the same sequence of isotope positions for the Ba I and Ba II lines, with the additional displacement of the odd isotopes toward the lighter even isotopes, characteristic of heavy elements.

The isotopic displacement in the spectra of the other middle elements has been studied very incompletely. The effect has been found in the spectra Ca II \(^{99}\), Ge I \(^{99a}\), Kr I \(^{103,104}\), Rb I \(^{106,107}\), Rb II \(^{105}\), Sb I \(^{124}\), Sb II \(^{125}\), Xe I \(^{127,128}\). It turned out that the isotopic displacement between components of isotopes with a mass difference \(\Delta M = 2\) varies within the limits \(0.001—0.005\ \mathrm{cm}^{-1}\) \(^{12}\). In the spectra Se II \(^{100}\), Br I \(^{101}\), Br II \(^{102}\), Sr II \(^{108}\) the displacement lies beyond the limits of experimental possibilities and therefore was not detected. For 5 elements the isotopic displacement has still not been measured (Ti, Cr, Fe, Ni, In).

In conclusion we note that the isotopic displacement in the spectra of krypton and xenon is, in absolute magnitude, approximately the same; as to sign, the displacement in the krypton spectrum is positive, while in the xenon spectrum it is negative. Hence, taking into account the analysis of isotopic displacement in the spectra of zirconium and molybdenum (see above), one may say that, apparently, in the region between krypton and zirconium the mass and volume effects produce approximately equal displacements of the energy levels.

4. ISOTOPIC EFFECT IN THE SPECTRA OF HEAVY ELEMENTS

a) Theory of the volume effect

The experimental material on the isotopic effect in the spectra of light and middle elements shows that the isotopic displacement in terms, and also in lines, decreases rapidly from element to element as the atomic number \(Z\) increases. In the region of rubidium the displacement approaches zero and, subsequently, having changed sign, begins to increase with increasing \(Z\), reaching for the actinide elements the same values as in the spectra of helium and lithium. The theory of the mass effect, which quite satisfactorily explains the isotopic displacement in the spectra of light elements, becomes, as we have seen, insufficient for explaining the isotopic displacement in the spectra of middle elements.

In the spectra of heavy elements the mass effect is immeasurably small, and with its help one cannot explain either the magnitude or the sign

of the isotope shift. Therefore, already in the initial period of development of spectroscopic work on the investigation of the isotope effect it was clear that, for the isotope shift in the spectra of heavy elements, the principal importance belongs not to the mass but to certain other properties of the nucleus. In 1931 Bartlett15 pointed out that, since the heavier isotopes of one and the same element have a larger radius, the electric field inside such nuclei must be weaker and the binding energy of the electrons must be smaller than in the case of light isotopes. In 1932 Racah16, Rosenthal and Breit17 developed a theory of the volume effect to explain the isotope shift in the spectra of heavy elements. This theory was tested by Breit18 on previously published experimental material concerning the isotope shift in the spectra of Hg, Tl, and Pb. Later the theory of the volume effect was refined by Broch24 and Ya. A. Smorodinsky25 and was examined in detail by Crawford and Schawlow29, Bricks and Kopfermann30, D. D. Ivanenko6, Humbach38, Bodmer42, 45b, Wilets, Hill and Ford49, and others.

The volume effect in the isotope shift, which has a specifically quantum-mechanical character, is caused by the deviation of the field near the nucleus from a purely Coulomb field owing to the finite dimensions of the nucleus. If one imagines the nucleus in the form of a point charge, then the electrostatic potential of the nucleus right up to distances \(r = 0\) will be Coulombic. In this case the energy of interaction of the electron with the nucleus is determined only by the charge of the nucleus, and therefore the energy levels of different isotopes of one and the same element will not undergo a shift. In the case of nuclei of finite size, outside the nucleus the Coulomb field predominates, whose potential is equal to

\[ V = - \frac{Ze^2}{r}, \]

where \(r > R_0\) (\(R_0\) is the radius of the nucleus). Inside the nucleus, i.e., for \(r < R_0\), the potential will be somewhat diminished in comparison with the rapid increase of the absolute value of the Coulomb potential as \(r\) decreases; i.e., the Coulomb interaction of the nucleus with the electron will be somewhat weakened.

In a rough approximation one may assume that, for heavy nuclei, the charge is distributed uniformly over the surface of the spherical nucleus. In this case the electrostatic potential inside the nucleus will be constant:

\[ V_i = - \frac{Ze^2}{R_0}. \]

The distance from the center of the nucleus at which the above-mentioned deviation from the Coulomb field begins to manifest itself is customarily taken as the radius of the nucleus. According to the “drop” model of the nucleus, this distance

A. R. STRIGANOV AND Yu. P. DONTSOV

proportional to the cube root of the mass number:

\[ R_0 = r_0 A^{1/3}, \tag{12} \]

where for heavy nuclei the factor \(r_0 = 1.2 \cdot 10^{-13}\) cm. From this it is seen that the radius of the nucleus increases with increasing \(A\). This change in the nuclear radius with the mass number leads to the fact that, for different isotopes of one and the same element, the deviation from the Coulomb field begins at different distances from the center of the nucleus, namely, for light isotopes at smaller distances than for heavy isotopes. As a result of this, for electrons that penetrate into the nuclear region, the binding to the nucleus will be stronger for the lightest isotopes. This means that the energy levels (or the centers of gravity of the hyperfine-structure sublevels) of the lightest isotope will lie lower, while the levels of the heavier isotopes will be located higher, in accordance with their mass numbers.

The decrease in the binding energy of the optical electron, or the shift of levels in the case of a nucleus of finite dimensions, as compared with a point nucleus, is calculated by means of perturbation theory. Below is given the final result obtained by Racah, Rosenthal, and Breit \(^{16,17}\) for an \(s\)-electron*):

\[ \Delta W = \frac{8\pi(1+\gamma)}{[\Gamma(2\gamma+1)]^2} \left(\frac{2Z}{a_{\mathrm H}}\right)^{2\gamma-2} \psi^2(0) \frac{Ze^2 R_0^2\gamma}{2\gamma(2\gamma+1)}, \tag{13} \]

where \(a_{\mathrm H}=\dfrac{h^2}{4\pi^2me^2}\) is the radius of the first Bohr orbit for hydrogen, \(\psi^2(0)\) is the square of the nonrelativistic wave function at \(r=0\) for the state of the \(s\)-electron under consideration, expressing the probability of finding the electron at the place where the nucleus is located, \(\gamma=(1-Z^2\alpha^2)^{1/2}\), where \(\alpha=\dfrac{2\pi e^2}{hc}\) is the fine-structure constant. From the formula it is seen that the displacement of the energy levels in the direction of a decrease in the binding of the electron to the nucleus depends on the radius of the nucleus. For a given \(Z\), for isotopes with a smaller mass number, i.e., with smaller \(R_0\), the displacement will be smaller than for heavier isotopes. Hence it is clear that in the case of the volume effect the levels of the lighter isotopes lie deeper.

In order to find the isotope shift in the levels between isotopes with a difference of radii \(\delta R_0\) in the interaction of an \(s\)-electron with the nucleus, it is necessary to differentiate expression (13) with respect to \(R_0\). Passing to terms and introducing the notation

\[ \frac{2ZR_0}{a_{\mathrm H}} = y_0,\qquad \frac{e^2}{2hc} = R_\infty a_{\mathrm H}, \]

*) The derivation of this formula was later reproduced in a number of works by other authors \(^{3,6,8}\).

we obtain:

\[ \Delta T=\frac{\delta \Delta W}{hc}=\frac{4\pi R_{\infty}a_H^3\psi^2(0)}{Z}\, \frac{1+\gamma}{[\Gamma(2\gamma+1)]^2}\,B\,y_0^{2\gamma}\frac{\delta R_0}{R_0}, \tag{14} \]

where \(R_{\infty}=\dfrac{2\pi^2me^4}{h^3c}\) is the Rydberg constant, and the factor \(B=\dfrac{1}{2\gamma+1}\) characterizes the charge distribution in the nucleus. Formula (14) does not change its form when the form of the charge distribution changes; only the factor \(B\) changes. Thus, if one assumes that the charge is uniformly distributed throughout the entire volume of the nucleus, then the potential inside the nucleus can be represented in the following form:

\[ V_i=\left[-\frac{3}{2}+\frac{1}{2}\left(\frac{r}{R_0}\right)^2\right]\frac{Ze^2}{R_0}; \]

in this case

\[ B=\frac{3}{(2\gamma+1)(2\gamma+3)}. \]

The most accurate value of the nonrelativistic electron density \(\psi^2(0)\) is given by the expression often used in calculating magnetic moments from the hyperfine structure of spectra\(^3\)

\[ \psi^2(0)=\frac{Z_iZ_0^2}{\pi a_H^3 n^{*3}}\left(1-\frac{d\sigma}{dn}\right), \tag{15} \]

where \(Z_i\) is the effective nuclear charge for the inner region (\(Z_i\simeq Z\) for an \(s\)-electron), \(Z_0\) is the effective nuclear charge in the outer region (for a neutral atom \(Z_0=1\), for a singly ionized atom \(Z_0=2\), etc.), \(n^*\) is the effective quantum number

\[ \left(n^{*2}=\frac{R_{\infty}Z_0^2}{T}\right), \]

and \(\sigma\) is the quantum defect. Substituting expression (15) into formula (14), we finally obtain the isotope shift in terms for an \(s\)-electron relative to the series limit:

\[ \Delta T=\frac{4R_{\infty}Z_0^2}{n^{*3}}\left(1-\frac{d\sigma}{dn}\right) \frac{1+\gamma}{[\Gamma(2\gamma+1)]^2}\,B\,y_0^{2\gamma}\frac{\delta R_0}{R_0}, \tag{16} \]

where \(\dfrac{d\sigma}{dn}\) is the derivative of the quantum defect with respect to the corresponding quantum numbers, representing the difference of the Rydberg corrections for the given term and the subsequent term of the given series, the numerical value of which can be found\(^ {29,30}\); \(\dfrac{\delta R_0}{R_0}\) is the relative change of the nuclear radius, characterizing the change in the distribution of protons in the transition from one isotope to another.

On the basis of formula (16), expressing the volume effect in the isotope shift, the following conclusions may be drawn:

1. The shift increases with increasing charge and radius of the nucleus, i.e., for the heaviest elements the isotope shift will be the greatest.

2. The isotope shift in the spectra of heavy elements is proportional to the increment of the mass numbers, since, according to formula (12), the increment of the radii is equal to:

\[ \frac{\delta R_{0}}{R_{0}}=\frac{\delta M}{3M}. \tag{17} \]

1. Within one and the same term series of a given element, the isotope shift tends to zero with increasing effective quantum number, since \(\Delta T\) is inversely proportional to \(n^{*3}\).

Proceeding from the theory of the volume effect, it is seen that the isotope shift should increase with an increase in the number of electrons penetrating to the nucleus, and also with the degree of their penetration. According to the laws of quantum mechanics, the most penetrating are \(s\)-electrons; therefore for them the volume effect will be greatest. For other shells with \(p\)-, \(d\)-, and \(f\)-electrons, at the same principal quantum number, the isotope shift is incomparably smaller than the shift due to \(s\)-electrons. Of these, only in the case of \(p\)-electrons does the shift attain noticeable magnitudes. Theoretically, for example, it has been established that the isotope shift in terms for \(6p_{1/2}\)-electrons is approximately \(1/20\) of the shift for \(6s\)-electrons\(^{13}\).

A more rigorous derivation of the isotope-shift formula by the method of perturbation of boundary conditions\(^{24,25}\), based on taking into account the change of the wave function in the region of the nucleus, shows that in formula (16) it is necessary to add, according to Ya. A. Smorodinsky\(^{25}\), a new factor

\[ \xi=\frac{2\gamma^{2}(2-\gamma)(1+2\gamma)}{(1+\gamma)(2+\gamma)}. \tag{18} \]

For light nuclei, when \(\gamma=1\), \(\xi\) tends to 1. For heavy elements the factor \(\xi\) differs appreciably from 1; thus, for example, for mercury \(\xi=0.8\). It may also be noted that, according to Crawford and Schawlow\(^{29}\), in the case of thallium the isotope shift calculated by formula (16) exceeds by approximately 35% the shift obtained by the method of Broch\(^{24}\). Hence it is clear that the refinement mentioned in calculating the isotope shift is of substantial importance for heavy elements.

In this connection one should also mention the derivation of the isotope-shift formula for the spectra of heavy elements

D. D. Ivanenko and A. F. Tsander[^27][^6], who abandoned the approximate perturbation method and based the derivation on “refined” electron wave functions that take into account the finite size of the nucleus. The corrections obtained by them can also be expressed in the form of a special factor in the formula for the isotope shift, which for mercury reaches 0.9.

In addition to these corrections, connected with a more rigorous theoretical derivation of the formula, when estimating the magnitude of the isotope shift it is necessary to take into account the mutual screening of the optical electrons and the screening of the inner electrons by the optical ones, as well as the mutual perturbation of the external electron configurations. If these are not taken into account, all this introduces noticeable distortions into the theoretical results when calculating the isotope shift on the basis of the volume theory.

b) Experimental data

The group of heavy elements, in which the isotope shift in the spectra is due mainly to the volume effect, may include elements beginning with atomic number $Z = 58$ and higher. The mass shift in the spectra of heavy elements is apparently very small. This follows from the fact that the isotope shift in them is determined by the number of $s$-electrons, and not by the magnitude and character of the terms, and that triplet and singlet terms give approximately equal shifts. However, calculations estimating the mass shift in the spectra of heavy elements have not yet been made, and therefore it is impossible to say with certainty whether the mass effect can be completely neglected in this case.

It should be noted that fairly extensive experimental data have been collected on the isotope shift in the spectra of heavy elements, both in the number of isotopes investigated and in the number of spectral lines studied. However, the material is still insufficient for a critical comparison with theory. First of all, for many elements the isotope shift measured on lines cannot always be transferred to terms by normalizing (or referring) them to the series limit, where the shift is equal to zero. The latter is explained by the fact that for any term, when the optical electron has a large value of the principal quantum number, the difference in the binding energy of this electron with the nucleus for different isotopes must be small and tend to zero near the series limit. If the shifts in the terms are found in this way, then they are approximately equal to the shifts of the terms caused by the optical electron. In normalization with respect to the series limit difficulties arise, since lines belonging to the series limit are usually not observed. Therefore, wherever possible, extrapolation is used.

by the measured shifts of lines of one and the same series. Such an extrapolation gives fairly accurate results if it is possible to trace the course of the isotopic shift for several members of one series with respect to a limiting value, which is then taken as the isotopic shift of the constant (unchanging) term (see, for example, helium 56, 57). It should be borne in mind, however, that quite often some of the terms of a given series are perturbed by a “complex” term not belonging to that series. In this case the perturbed terms undergo an anomalous isotopic shift, and the extrapolation mentioned above becomes difficult.

It is much simpler to pass from the shift in lines to the shift in terms if one takes some term that is known not to undergo a shift. Then, considering the lines connected with this term by the corresponding transitions, one can find, for other terms, their shifts relative to the term chosen as the initial one. Such a normalization can be continued further, using spectral lines with a measured isotopic shift in which one of the terms has a known shift on the basis of the preceding normalization. Terms belonging to configurations with \(p_{1/2}\)-, \(d_{3/2}\)- or \(d_{5/2}\)-electrons, which possess an insignificantly small shift, can be used as the initial term for normalization. For the same purpose any term belonging to configurations with a \(p_{1/2}\)- and even an \(s\)-electron at a high value of the principal quantum number \(n\) may be used. It should nevertheless be noted that such a normalization leads to a certain constant error, which is determined by the error in estimating the shift of the initial term.

The isotopic shift has been measured, to one degree or another, for the spectra of all heavy elements that consist of two or more stable isotopes, and also for some elements with radioactive isotopes. The most complete data have been obtained for W, Os, Pt, Hg, Tl, Pb, the rare-earth elements: Ce, Nd, Sm, Eu, Gd, Er, Yb, and some actinide elements: Th, U, Pu, Am. The isotopic shift in the spectra of Dy, Hf, Re, Ir, Bi has been studied poorly. In the tables of Brix and Kopfermann \(^{12}\) all data on isotopic shifts in spectra obtained up to 1952 are collected. Without going into a detailed description of the experimental material for each element separately, we shall dwell chiefly on more recent results, as well as on the characteristic experimental features of the isotopic shift in the spectra of heavy elements.

The isotopic shift in the arc spectrum of tungsten has been studied by many investigators \(^{153—158}\). The most reliable and complete results were recently obtained by Murakawa \(^{158}\), who measured

shift for 55 lines between isotopes with mass numbers 182, 184, 186 and determined the shift for 29 terms. In the case of the tungsten spectrum it has been established that on all lines the shift \(\Delta \nu(182—184)\) is somewhat larger than \(\Delta \nu(184—186)\). According to Murakawa

\[ \frac{\Delta \nu(182—184)}{\Delta \nu(184—186)}=1.206 \pm 0.010 . \]

If the interval \(\Delta \nu(182—184)\) is taken as unity, then the relative position of the even isotopes may be represented as follows:

\(A\)	180	182	184	186
Relative position . .	0.90	0	1	1.83

It has been established that the center of gravity of the components of the odd isotope \(W^{183}\) is located midway between \(W^{182}\) and \(W^{184}\). The shift in the terms is positive, i.e. the term of the lighter isotope lies lower.

The isotopic shift in the osmium spectrum has been studied on 8 arc lines between the even isotopes 186, 188, 190 and 192\(^{160,161}\). The shift for all lines proved to be negative. The relative position of the components of the even isotopes is as follows:

\(A\)	186	188	190	192
Relative position	0	1	1.90	2.72

The isotopic shift in the platinum spectrum has been found on many arc lines\(^{163,164,165}\). Below is given the relative position of the isotope components on the basis of four lines (3408.1, 4442.5, 5369.0 and 5390.8 Å):

\(A\)	194	193	196	198
Relative position	0	0.44	1	2.04

In Fig. 6, as an example, the schematic structure of two arc lines of platinum is shown, where, along with the even isotopes, the position of the center of gravity of the components of the hyperfine structure of the odd isotope is marked by a dashed line. From the data presented it is seen that, whereas the components of the even isotopes are located at equal distances from one another, the center of gravity of the odd isotope falls not in the middle of the interval \(\Delta\nu\) \((194—196)\), but is shifted somewhat toward the isotope with the smaller mass number.

\[ \lambda\,5390.8\,\text{\AA}\ \left(5d^{9}6p\,{}^{3}F_{3}-5d7s\,{}^{3}D_{2}\right) \]

\[ \lambda\,5369.9\,\text{\AA}\ \left(5d^{8}6s^{2}\,{}^{3}F_{3}-5d^{9}6p\,{}^{3}F_{3}\right) \]

Fig. 6. Isotopic structure of two Pt lines.

In the platinum spectrum, isotope shifts of 34 terms have been established for the neutral atom relative to the level \(5d^{10}\,{}^{1}S_{0}\). In all these terms the shift proved to be positive. Table XI gives the data obtained for several characteristic terms belonging to various electron configurations\({}^{12}\). It is easy to see that the greatest shift is experienced by the terms belonging to the configuration \(5d^{8}6s^{2}\), which contains two penetrating \(6s\)-electrons. The shift in the terms of the configurations \(5d^{8}6s7s\), \(5d^{8}6s6p\), which have only one \(6s\)-electron, is approximately two times smaller. The shift in the terms of the configuration \(5d^{9}6p\), in which the \(s\)-electron is absent, is still smaller. In the terms of the more complex configurations \(5d^{8}6s6p\), including \(s\)- and \(p\)-electrons, considerable fluctuations of the isotope shift are observed, which is due to the noticeable share of the \(p_{1/2}\)-electron in the total isotope shift. From the data on the isotope effect in the terms of con-

configurations \(5d^9 6s\) and \(5d^9 7s\), it is seen that the shift decreases strongly with increasing principal quantum number. It may be noted that within one and the same multiplet the magnitude of the isotope shift in the terms varies within very small limits.

Table XI

Isotope shift \(\Delta T(194—196)\) in Pt terms

Electron configuration	Term symbol	\(\Delta T\) \((\mathrm{cm}^{-1})\)	Electron configuration	Term symbol	\(\Delta T\) \((\mathrm{cm}^{-1})\)
\(5d^8 6s^2\)	\({}^3F_4\)	\(+0.203\)	\(5d^9 6s\)	\({}^3D_3\)	\(+0.082\)
\(5d^8 6s^2\)	\({}^3F_3\)	\(+0.202\)	\(5d^9 6s\)	\({}^3D_2\)	\(+0.117\)
\(5d^8 6s^2\)	\({}^3F_2\)	\(+0.200\)	\(5d^9 6s\)	\({}^3D_1\)	\(+0.090\)
\(5d^8 6s7s\)	\({}^5F_5\)	\(+0.107\)	\(5d^9 6p\)	\({}^3F_3\)	\(+0.056\)
\(5d^8 6s7s\)	\({}^5F_4\)	\(+0.107\)	\(5d^9 6p\)	\({}^3F_4\)	\(+0.030\)
\(5d^8 6s6p\)	\({}^5F_5\)	\(+0.112\)	\(5d^9 7s\)	\({}^3D_3\)	\(+0.020\)
\(5d^8 6s6p\)	\({}^5F_4\)	\(+0.107\)	\(5d^9 7s\)	\({}^3D_2\)	\(+0.012\)
\(5d^8 6s6p\)	\({}^5D_3\)	\(+0.085\)	\(5d^{10}\)	\({}^1S_0\)	\(0\)
\(5d^8 6s6p\)	\({}^5D_2\)	\(+0.050\)	\(5d^{10}\)	\({}^1S_0\)	\(0\)

Much attention was devoted from the very beginning to the study of the isotope effect in the mercury spectrum. As a result, extensive data were obtained on the spectra Hg I\(^{166,167,168,174}\), Hg II\(^{168,171}\), and Hg III\(^{171,175}\), which are collected in the tables of Briks and Kopfermann\({}^{12}\). In the Hg I spectrum a noticeable isotope shift was found in 7 lines (2536.5; 4358.3; 5460.7; 5675.9; 6072.6; 6234.4; 6716.4 Å). The relative position of the isotope components is determined as follows:

\(\cdot\)	\(A\)	198	199	200	201	202	204
Relative position . .		\(-0.94\)	\(-0.80\)	\(0\)	\(0.33\)	\(1\)	\(1.98\)

As an example, Fig. 7 shows the isotope structure of two mercury lines with different directions of shift. Vertical lines mark the positions of the components for the even isotopes and the positions of the centers of gravity for the odd isotopes. Just as in the platinum spectrum, on all mercury lines the centers of gravity of the odd isotopes preserve the sequence in mass numbers of the even isotopes; however, the odd isotopes possess an additional displacement toward the lighter even isotope.

On the basis of isotope shifts in the lines, the shift for several terms has been found\(^{12}\), with normalization to the term \(5d^{10}6s6p\ ^3P_1\). Figure 8 shows the scheme of term shifts for the neutral mercury atom\(^{166,4}\). For all terms the shift is positive. The largest shift is experienced by the terms \(5d^{10}6s^2\ ^1S_0\), \(5d^96s^26p\ ^1P_1\), whose electronic configurations include the group of \(s^2\)-electrons. In the case of the series of \(^1S_0\)-terms, the largest shift, as is usually observed, is experienced by the deepest term. Here the shift decreases rapidly upward along the series, so that for the term \(5d^{10}6s8s\ ^1S_0\) the shift can no longer be measured. The shift in the term \(5d^{10}6s7s\ ^3S_1\) is so small and decreases so rapidly upward along the series that in higher terms it can no longer be detected. In the case of the series \(5d^{10}6smp\ ^1P_1\), the opposite is obtained: the higher term \(8p\ ^1P_1\) undergoes a large shift, whereas the term \(6p\ ^1P_1\) is not shifted. This departure from the general rule is explained by perturbations of closely spaced energy levels of similar quantum nature. From the scheme of isotope shifts of the energy levels (Fig. 8), it is easy to understand that, in lines which correspond to transitions from an unshifted term to a shifted one (or from a term with a small shift to a term with a large isotope shift), a negative shift will appear (line \(2536.5\) Å, Fig. 7), whereas in the reverse case the lines will show a positive shift (line \(6072.6\) Å, Fig. 7).

Fig. 7. Isotopic structure of two Hg I lines.

On the lines and terms of Hg II and Hg III the same regularities are observed as are characteristic of the spectrum of Hg I. It should be noted, however, that the isotopic displacement increases with increasing multiplicity of ionization of the atom. This can be seen from the data of Table XII, where the displacement is presented for terms of analogous configurations of different degrees of ionization[^12].

Fig. 8. Isotopic structure of the terms of Hg I (in all terms the order of arrangement of the isotopes is the same as for \(6\,{}^{1}S_{0}\)).

Let us point out that the displacement for the Hg II term is almost 3 times greater than the displacement for Hg I.

Table XII

Isotopic displacement \(\Delta T\) (200—202)
in Hg terms

Degree of ionization	Electronic configuration	Term symbol	\(\Delta T\) \((\mathrm{cm}^{-1})\)
Hg I	\(5d^{10}6s^{2}\)	\({}^{1}S_{0}\)	\(+0.179\)
Hg II	\(5d^{9}6s^{2}\)	\({}^{2}D_{5/2}\)	\(+0.508\)
Hg III	\(5d^{8}6s^{2}\)	\(G_{4}\)	\(+0.600\)

An interesting phenomenon was discovered by Mrozowski[^169,^172] on the forbidden lines of Hg, \(2269.8\ \text{Å}\) \((6s^{2}\,{}^{1}S_{0} — 6s6p\,{}^{3}P_{2})\), \(2655.8\ \text{Å}\)

\((6s^2\,{}^1S_0 - 6s6p\,{}^3P_0)\) and 2967.5 Å \((6s6p\,{}^3P_0 - 6s6d\,{}^1D_2)\). Each of these lines corresponds to a transition between the singlet and triplet systems of terms, i.e., it is an intercombination line. Although these lines also contradict the selection rule \(\Delta S = 0\) (intercombinations are forbidden), nevertheless in the atoms of many elements such transitions occur (for example, Zn, Cd, Hg). The number and intensity of intercombination lines increase, as is known, with increasing atomic number of the element. In the spectrum of mercury the brightest intercombination line is the resonance line 2536.5 Å \((6s^2\,{}^1S_0 - 6s6p\,{}^3P_1)\). The hyperfine and isotopic structure of this line is due to magnetic splitting and isotopic shift. In contrast to it, the three intercombination lines of mercury considered here—2269.8, 2655.8, 2967.5 Å—are forbidden, since they contradict another selection rule: \(\Delta J = 0, \pm 1\) (except \(J_1 = 0 \to J_2 = 0\)). However, these three lines were discovered as early as 1927 by Wood and Reilly. Studying the structure of the 2655.8 Å line, Mrozowski found that it consists of only two components, which belong to the odd isotopes Hg\({}^{199}\) and Hg\({}^{201}\). Likewise, the lines 2269.8 and 2967.5 Å consist of only four components, of which one belongs to the isotope Hg\({}^{199}\), and the other three to the isotope Hg\({}^{201}\). The structure of these lines does not include components of the even isotopes at all.

Fig. 9. Scheme of transitions and structure of the Hg I 2655.8 Å line.

To explain this phenomenon, let us consider, proceeding from the scheme of transitions, the structure of the 2655.8 Å line, which we have given in Fig. 9. On the left is shown the transition for the unsplit line of mercury, which appears in the spectrum despite the fact that the transition \(J_1 = 0 \to J_2 = 0\) is forbidden. This prohibition is intermediate and not

always holds, since it is also necessary to take into account the influence of the magnetic moment of the nucleus. The prohibition is valid if the magnetic moment of the nucleus is equal to zero, and may fail to hold if the magnetic moment is not equal to zero. Therefore, if for the line 2655.8 Å one considers the hyperfine structure, applying the selection rule \(\Delta F=0,\pm 1\) (except \(F_1=0 \to F_2=0\)), then in the case of even isotopes the prohibition is confirmed, since for the upper and lower levels \(F=J+I=0\). For odd isotopes, on the contrary, the prohibition will be lifted, since in the case of \(\mathrm{Hg}^{199}\), \(I=\tfrac{1}{2}\), as a result of which for both levels \(F=\tfrac{1}{2}\), while in the case of \(\mathrm{Hg}^{201}\), \(I=\tfrac{3}{2}\) and, consequently, for both levels \(F=\tfrac{3}{2}\). Hence, according to the transition scheme shown in Fig. 9 on the right, a structure consisting of two components is obtained only for the odd isotopes, which was also found experimentally. Opechovskii \(^{170}\) calculated the intensity ratio of these two components and found good agreement with experiment.

Figure 10 presents the transition scheme for the line 2269.8 Å, which fully explains the structure of this line, consisting,

Fig. 10. Transition scheme and structure of the Hg I 2269.8 Å line.

according to Mrozowski, of one component of the isotope \(\mathrm{Hg}^{199}\) and three components of the isotope \(\mathrm{Hg}^{201}\). Mrozowski’s results for the line 2967.5 Å were later confirmed on mercury obtained as a result of a nuclear reaction in the bombardment of gold by neutrons \(^{175}\).

The isotope shift in the spectrum of thallium was found from the hyperfine structure of the isotopes \(\mathrm{Tl}^{203}\) and \(\mathrm{Tl}^{205}\) for spectral lines of the neutral atom, as well as of singly and doubly ionized

atoms \(^{176-179}\). On the basis of these data, the shift in the terms Tl I, Tl II, Tl III was obtained\(^{12}\).

In Table XIII below, data on the isotopic shift in terms are compared for atoms of different degrees of ionization, as well as data on the shift of terms with increasing principal quantum number.

Table XIII

Isotopic shift \(\Delta T\) (203—205) in Tl terms

Degree of ionization	Electron configuration	Term symbol	\(\Delta T\) \((\mathrm{cm}^{-1})\)
Tl I	\(5d^{10}6s^26p\)	\({}^{2}P_{1/2}\)	0
Tl I	\(5d^{10}6s^28p\)	\({}^{2}P_{1/2}\)	\(+0.060\)
Tl II	\(5d^96s^26p\)	\(1_2\)	\(+0.350\)
Tl II	\(5d^{10}6s7s\)	\({}^{1}S_0\)	\(+0.060\)
Tl II	\(5d^{10}6s8s\)	\({}^{1}S_0\)	\(+0.015\)
Tl II	\(5d^{10}6s9s\)	\({}^{1}S_0\)	\(0.000\)
Tl III	\(5d^96s^2\)	\({}^{2}_{3/2}\)	\(+0.710\)
Tl III	\(5d^{10}6s\)	\({}^{2}S_{1/2}\)	\(+0.380\)
Tl III	\(5d^{10}7s\)	\({}^{2}S_{1/2}\)	\(+0.090\)
Tl III	\(5d^{10}8s\)	\({}^{2}S_{1/2}\)	\(+0.046\)
Tl III	\(5d^{10}9s\)	\({}^{2}S_{1/2}\)	\(+0.000\)

Here the same regularities in isotopic shift appear as those already noted by us in the case of platinum and mercury. It should be especially noted that in the terms Tl II there are negative shifts\(^{12}\), which are characteristic of the configurations \(5d^{10}6p^2\). In these terms the level of the lighter isotope lies above the level of the heavier isotope, which is in contradiction with the volume theory of isotopic shift. There is as yet no sufficiently reliable explanation of this phenomenon; however, it may be assumed that such an abnormal arrangement of terms is explained by perturbations of closely lying levels.

Extensive data on the isotopic shift in the lines and terms of lead have been collected for Pb I and Pb II\(^{180-184}\), Pb III\(^{185}\) and Pb IV\(^{186}\), which in generalized form are given in the tables of Bricks and Kopfermann\(^{12}\). In the terms of lead the same regularities manifest themselves—

nesses in isotopic mixing that we had noted in the case of platinum, mercury, and thallium. We shall mention only that, just as in the case of Tl II, in the terms Pb III of the configurations \(5d^{10}6p^2\) and \(5d^{10}6s6d\) there is negative mixing.

Much attention in recent years has been devoted to measuring the intervals in isotopic mixing between the even isotopes of lead*). As a result, on the basis of experimental material in the spectra Pb I and Pb II the following arrangement of isotope components has been found, in relative units:

\(A\)	204	206	207	208	210
Relative position	\(-0.90\)	0	0.38	1	2.74

In Fig. 11, as an example, the structure of the line Pb I \(4057.8\,\text{Å}\) \((6s^26p^2\,{}^3P_2 — 6s^26p7s\,{}^3P_1)\) is shown, with carefully measured distances between the components. Along with the characteristic

Fig. 11. Isotopic and hyperfine structure of the line Pb I \(4057.8\,\text{Å}\) (the dotted line marks the center of gravity of the hyperfine structure of the isotope \(\mathrm{Pb}^{207}\)).

additional shift of the center of gravity of the even–odd isotope toward the isotopes of smaller mass numbers, in the lead lines there appears a strong deviation from equidistance in the intervals between the even isotopes. Indeed, by careful measurements of many lines of Pb I and Pb II, using enriched samples \(^{187-191,193}\), it has been established that

\[ \frac{\Delta \nu(204—206)}{\Delta \nu(206—208)} = 0.90 \pm 0.01. \]

*) Beginning in 1949, 7 papers were published that were carried out on enriched lead isotopes \(^{187-193}\).

An even larger “jump” in the isotopic shift for lead was recently found between the stable isotope Pb\(^{208}\) and radioactive lead RaD (Pb\(^{210}\)) on the line 4057.8 Å (Fig. 11). According to the data of Briks, Kopfermann, et al.\(^{192}\), it turned out that:

\[ \Delta \nu (208-210) = 0.143 \pm 0.003\ \mathrm{cm}^{-1}, \]

\[ \frac{\Delta \nu (208-210)}{\Delta \nu (206-208)} = 1.74 \pm 0.05. \]

It should be noted that the indicated anomaly in the isotopic shift in the spectrum occurs when a new pair of neutrons is added to a filled shell with 126 neutrons.

In conclusion, let us note that within the elements of this group some data on the isotopic shift have long since been obtained for the arc lines of hafnium\(^{152}\) and rhenium\(^{159}\). In the spectrum of iridium, an isotopic shift was found between the isotopes Ir\(^{191}\)—Ir\(^{193}\) for the lines Ir I 3513.6 and 3800.1 Å; it proved to be respectively \(-0.071\) and \(-0.063\ \mathrm{cm}^{-1}\)\(^{162}\). Recently an isotopic shift (\(+0.12\ \mathrm{cm}^{-1}\)) was found\(^{194}\) in the spectrum of bismuth on the line Bi I 3067.7 Å between the stable isotope Bi\(^{209}\) and radioactive bismuth RaE (Bi\(^{210}\)).

Much attention in recent years has been devoted to the study of the isotopic shift in the spectra of the rare-earth elements. It is sufficient to note that of the 26 works devoted to this question, 18 were carried out during 1949—1954. The increased interest in this problem is explained by the fact that anomalous phenomena in the isotopic shift were discovered in the spectra of the rare-earth elements, phenomena of great importance for the theory of the isotopic shift as well as for the theory of nuclear structure. The isotopic shift has been studied to one degree or another in the spectra of all rare-earth elements that consist of more than one stable isotope (Ce, Nd, Sm, Gd, Dy, Er, Yb).

In the spectrum of cerium the isotopic shift was studied\(^{131}\) on 9 spark lines between the components of the isotopes Ce\(^{140}\) and Ce\(^{142}\). On average the shift was found to be \(-0.054\ \mathrm{cm}^{-1}\). If this value is compared with the isotopic shift in the spectrum Ba II, which reaches only \(+0.005\ \mathrm{cm}^{-1}\), then, already from this, it is evident that in the spectrum of cerium there is a “jump” in the isotopic shift, which occurs when a new pair of neutrons is added to a filled shell with 82 neutrons. However, the question of the presence of an anomaly in the isotopic shift for cerium lines remained open, since to prove it it was necessary to compare the found shift \(\Delta \nu (140—142)\) with the shift between the components of isotopes

Ce$^{136}$—Ce$^{138}$—Ce$^{140}$. An attempt to prove this anomaly had already been undertaken by Murakawa and Ross$^{132}$ on an enriched cerium sample in which the content of the isotope Ce$^{138}$ had been brought up to 4.4% (instead of 0.25% in the natural sample). The maximum possible distance between the isotopes Ce$^{138}$ and Ce$^{140}$ was roughly estimated from the line Ce II 4628.2 Å as equal to 0.029 cm$^{-1}$, i.e. it amounts to approximately half the interval between the components of the isotopes Ce$^{140}$ and Ce$^{142}$. The same question is the subject of a recently published work by Arpoe$^{133}$, who, with the aid of two enriched samples (Ce$^{138}$—13.1%, Ce$^{136}$—22.3% and 30%), found that the isotopic structure Ce$^{136}$—Ce$^{138}$—Ce$^{140}$ is very narrow. For many lines this structure does not even produce a noticeable broadening of the lines and, according to the author’s rough estimate, amounts to only 6% of the shift Ce$^{140}$—Ce$^{142}$. It should be noted that these rather strange results require verification. At the same time, apparently, the “jump” in the isotopic shift between isotopes whose nuclei consist of 82 and 84 neutrons may be considered established.

Of special interest in the group of rare-earth elements is the isotopic shift in the spectra of samarium and neodymium. As early as 1934, while investigating the line Sm I 5320.6 Å, Schuler and Schmidt$^{135}$ discovered an anomalous shift between the components of the even-even isotopes Sm$^{150}$—Sm$^{152}$. The anomaly consisted in the fact that the distance between the components Sm$^{150}$—Sm$^{152}$ was approximately twice as large as the distances between any other pair of even-even isotopes, which among themselves were approximately equal. Subsequently, the same results were obtained by S. E. Frisch and M. P. Vanyukov$^{137}$ on the lines Sm I 5320.6 and 5251.9 Å, and then were studied in detail by Brick and Kopfermann$^{138}$ on the lines Sm I 5251.9, 5979.4, 6588.9 and 6671.5 Å.

The relative position of the components in the isotopic structure of the above-mentioned lines may be represented in the form of the following table:

A	144	147	148	149	150	152	154
Relative position . .	0	1.42	2	2.27	3.14	4.81	5.72

As an example, Fig. 12 gives the isotopic structure of the lines Sm 5320.6 Å, where the intervals between the components of the isotopes are indicated. From the table it is seen that the ratio

\[ \frac{\Delta \nu(150—152)}{\frac{1}{3}\Delta \nu(144—150)} = 1.59 \]

and that in the spectrum of samarium there occurs an even-odd additional

shift. The same phenomenon was discovered by Klinkenberg \(^{134}\) in the investigation of the isotopic shift in the spectrum of neodymium. The rela-

Fig. 12. Isotopic structure of the line
SmI \(5320.6\) Å.

tive positions of the components of the neodymium isotopes, obtained on the basis of the lines NdI \(4569.7\), \(4618.1\), and \(4678.5\), are given in the following table:

\(A\)	142	144	146	148	150
Relative position . . . . .	0	1.03	2	3.08	4.73

An anomalously large shift is also observed between the isotopes \(\mathrm{Nd}^{148}\)—\(\mathrm{Nd}^{150}\). The ratio between the intervals

\[ \frac{\Delta\nu(148-150)}{\frac{1}{3}\Delta\nu(142-148)} = \frac{1.65}{1.02} = 1.62 \]

agrees in magnitude with the value obtained for samarium.

Comparing the anomalous shift in the spectra of samarium and neodymium, it is easy to see that in both cases it occurs between isotopes whose nuclei consist of 88 and 90 neutrons. This means that the “jump” in the isotopic shift is connected here with the addition, to 88 neutrons, of the next, 45th pair of neutrons.

It should be noted that a comparatively large shift is observed \(^{141,142}\) in the spectrum of europium between the isotopes \(\mathrm{Eu}^{151}\)—\(\mathrm{Eu}^{153}\). This is undoubtedly connected with the presence in the nuclei of the two mentioned isotopes of the same number of neutrons (88 and 90), which is characteristic of the anomalous shift in the spectrum of neodymium and samarium. Although the anomaly of the isotopic shift on the europium lines cannot be shown, since europium has only two stable isotopes, the anomaly can nevertheless be verified by comparing the shift in two isoelectronic spectra EuI and GdII. According to experimental data, the shift of the ground level

$4f^7 6s^2\,{}^8S_{7/2}$ Eu I between the isotopes Eu$^{151}$—Eu$^{153}$ reaches$^{12}$ $-0.250\ \text{cm}^{-1}$, whereas for the ground level $4f^7 6s^2\,{}^8S_{7/2}$ Gd II the shift between the isotopes Gd$^{158}$—Gd$^{160}$ is equal$^{143}$ to $+0.115\ \text{cm}^{-1}$. Since the electron shells of Eu I and Gd II atoms are identical, the large isotope shift in the europium spectrum is connected with certain peculiarities in the structure of the europium nucleus. An anomalous shift should also appear in the gadolinium spectrum between the isotopes Gd$^{152}$—Gd$^{154}$, whose nuclei have $N=88$ and $90$. In natural gadolinium, however, these isotopes are present in concentrations of 0.2% and 2.16%, which is insufficient for their spectroscopic determination. Enriched isotopes are needed to solve this problem.

The most detailed data on the isotope effect in the spectra of rare-earth elements are available for samarium$^{135—140}$. New results have recently been obtained for europium$^{142}$. General material over the last two years has been collected for gadolinium$^{144—146}$ and erbium$^{148,149}$. There are some data on ytterbium$^{150,151}$ and dysprosium$^{147}$. Analysis of all these data shows that in the spectra of rare-earth elements the same regularities appear in the isotope shift as are characteristic of heavy elements. For samarium, for example, the presence of an additional displacement of the odd isotopes relative to the even ones has been established. In the spectra of samarium, europium, and gadolinium a noticeably larger shift has been found for spark lines than for arc lines. It turned out that the greatest shift in these elements is experienced by terms belonging to configurations with $s^2$ electrons; approximately a twofold smaller shift is observed for terms of $sp$ configurations, and negligibly small shifts are obtained for terms of $d^2$ and $pd$ configurations.

From the existence of a connection between the electronic configurations of the atom and the displacement of spectral lines, it is clear that data on isotope shifts can serve as an auxiliary means for the classification of spectra$^{10}$. As an example we shall cite the work of Bricks$^{139}$, who compared his measurements of the isotope shift in the Sm I spectrum with the term scheme, dividing all samarium lines into several groups according to the magnitude and direction of the shift. On the basis of these data Bricks confirmed, in the main, the previously established classification of the arc spectrum of samarium. In addition, he found 4 new odd terms, newly classified 5 lines, and corrected the previous classification for four lines. The same aim was pursued by Macknelly and Smith$^{140}$, who measured the isotope shift for 400 samarium lines between the components Sm$^{144}$ and Sm$^{154}$ with the aid of two enriched samples (Sm$^{144}$—72.13%, Sm$^{154}$—92.1%). Let us also note the attempt to decipher to some extent, on the basis of the isotope shift, the completely unclassified spectrum of erbium$^{148—149}$,

which made it possible to obtain some, though far from complete, but quite convincing data on the electronic configurations for various transitions.

The isotope effect in the spectra of the actinide elements has been studied for thorium, uranium, plutonium, and americium. In the spectrum of thorium the isotope shift has been measured on approximately 250 arc and spark lines[^195] between the components of the principal isotope Th\(^{232}\) and the isotope Th\(^{230}\) (ion), formed as a result of the radioactive decay of uranium. On the basis of these data the relative shift in the terms Th I and Th II was obtained. The shift for all terms proved to be positive.

Figure 13 shows the limits of the isotope shift in the terms for various electronic configurations of Th II. From

Fig. 13. Isotope shift in terms of various electronic configurations of Th.

these data it is seen that here, too, the greatest shift is undergone by terms belonging to the configurations \(5f7s^2\) and \(6d7s^2\); the smallest shift is observed for terms of the configuration \(5f6d^2\). It should be noted that in heavy atoms with a complex electron shell, neighboring levels interact. As a result of this, the position of each level is determined not only by its own function, but also depends on the eigenfunctions of neighboring levels of other configurations. Such configuration mixing not only complicates the interpretation of isotope-shift data in terms, but is also the cause of difficulties in establishing the type of coupling between electrons and the type of electronic configurations for the levels of heavy atoms.

The isotope shift in the spectrum of uranium was first discovered by Anderson and White^196, and later studied in more detail by other investigators^197–199. From the literature data, isotope shifts are now known for three arc and 12 spark lines; on this basis the relative shift has been found for 9 terms of the singly ionized atom^12. The experimental material shows that ion lines undergo larger shifts than lines of the neutral atom. As in the spectra of mercury and platinum, the isotope shift depends on the electron configurations. All lines whose lower terms belong to configurations with penetrating \(s^2\)- and \(s\)-electrons have a large negative shift. Lines whose lower terms belong to higher electron configurations with nonpenetrating \(f\)- and \(d\)-electrons undergo a smaller and only positive shift. As in mercury, the presence of perturbations between levels has been established in uranium. In the U II 4244.4 Å line an even–odd displacement has been found, which is shown in Fig. 14.

Fig. 14. Isotopic structure of the U II 4244.4 Å line.

Fig. 15. Isotopic structure of the Pu 4021.4 Å line.

The isotope shift in the spectrum of plutonium has been measured^200 in the region 3370–4020 Å for 20 lines between the isotopes \(\mathrm{Pu}^{238}\)—\(\mathrm{Pu}^{240}\). In four of these lines a shift was found between the isotopes \(\mathrm{Pu}^{239}\)—\(\mathrm{Pu}^{240}\). Figure 15 shows the isotopic structure of the \(\mathrm{Pu}\) 4021.4 Å line with four isotopes. Here, too, an additional displacement of the component of the odd isotope is observed toward the isotopes of smaller mass numbers.

In the spectrum of americium the isotope shift has been measured^201 for 28 lines between the isotopes \(\mathrm{Am}^{241}\) and \(\mathrm{Am}^{243}\). Both negative and positive shifts were found. The largest shifts are undergone by the lines 2938.96 and 3258.62 Å, for which \(\Delta\nu(241—243)\) are respectively 0.80 and 0.75 \(\mathrm{cm}^{-1}\).

The data presented above show that, in the isotope effect of the actinide elements, the same regularities appear as those found for other heavy elements. From the experimental data it is evident that the isotope shift in the spectra of the actinides greatly exceeds in magnitude the isotope shift in the spectra of the preceding heavy elements (Hg, Tl, Pb). It may therefore be considered that, just as in the case of the rare-earth

elements; in the actinide region there is a “jump” in the isotope shift.

Considering, on the basis of experimental materials, the isotope shift in the spectra of heavy elements, one may arrive at the following conclusions:

1. The components of isotopes, or their centers of gravity, are arranged in the isotope structure of lines in the order of mass numbers.

2. For most elements with even \(Z\), the components of even-even isotopes are situated (within 10% accuracy) at equal distances from one another.

3. The centers of gravity of the components of even-odd isotopes undergo an additional displacement (relative to the even-even ones) toward isotopes of smaller mass numbers.

4. The relative distances between the components of isotopes (or their centers of gravity) are more or less constant and characterize the element rather than a particular line of the neutral atom or of one of its ions.

5. In the spectra of Ce, Nd, Sm, and Pb there is an anomalously large shift between the even isotopes \(Ce^{140} — Ce^{142}\), \(Nd^{148} — Nd^{150}\), \(Sm^{150} — Sm^{152}\), \(Pb^{208} — Pb^{210}\), which is approximately twice as large as the shift between other pairs of even isotopes of the same element. A noticeable increase of the isotope shift in comparison with neighboring elements is observed in the spectra of europium, thorium, uranium, plutonium, and americium.

6. For odd isotopes, changes are observed in the relative distances between the centers of gravity of their components. The same fluctuations also occur for even isotopes, although they do not go beyond the limits of experimental errors.

7. The isotope shift for spark lines is greater than for arc lines; moreover, the shift in lines and terms increases with the order of ionization of the atom.

8. The greatest shift is experienced by terms belonging to configurations with penetrating \(s^2\)-electrons. Terms of configurations with one \(s\)-electron have an approximately two times smaller shift. Terms of configurations in which penetrating \(s\)-electrons are absent usually experience a very small shift.

9. The isotope shift in terms decreases with increasing principal quantum number: for the lower term of a given series it has the largest value and then rapidly decreases upward along the series.

10. For terms of one and the same multiplet, the isotope shift remains approximately constant, i.e. it does not depend on the quantum number \(J\).

It should be borne in mind that some of the above-mentioned regularities in the isotope shift are sometimes distorted by the mutual perturbation of terms and by the effect of electron shielding.

c) Comparison of experimental and theoretical data

A systematic comparison of experimental data on the isotopic shift of terms in Pb IV, Tl II, Hg II, Pt I, Re I, Yb I for \(s\)-electrons with theoretical data was first carried out by Kopfermann\(^3\). He showed that the isotopic shift in terms, calculated on the basis of the volume effect, exceeds the experimental values by several times.

Later, more accurate data in this direction were obtained by Crawford and Schawlow\(^ {29}\) for the isoelectronic spectra Hg II, Tl III, Pb IV. They took into account the correction associated with a more accurate derivation of the formula for the isotopic shift\(^ {24,25}\), and also allowed for the effect of electron screening. Assuming a uniform charge distribution over the volume of the nucleus, they calculated the isotopic shift between even isotopes from formulas (16), (17) with \(R_0 = 1.5 \cdot 10^{-13} A^{1/3}\)*. It turned out that the theoretical data on the isotopic shift in terms are twice as large as the experimental data. This discrepancy increases still further if it is assumed that the nuclear charge is distributed uniformly over the surface of the nucleus.

For many elements, detailed data comparing theory with experiment have in recent years been obtained by Brix, Kopfermann, and their co-workers\(^ {9,30,36,38,114,118,120,131}\). In order to compare comparable quantities characterizing the isotopic shift independently of the external structure of the atom, Brix and Kopfermann\(^ {30}\) presented formula (16) in the following form:

\[ \Delta T = - \frac{Z_0^2}{n^{*3}} \left( 1 - \frac{d\sigma}{dn} \right) C, \tag{19} \]

where the quantity

\[ C = 4R_\infty \frac{1+\gamma}{[\Gamma(2\gamma+1)]^2} \, By_0^{2\gamma} \frac{\delta R_0}{R_0} \tag{20} \]

is the isotopic-shift constant, depending only on the model of the nucleus and its charge. The theoretical value of this quantity \(C_{\mathrm{T}}\) can be calculated from formula (20), if the radius of the nucleus is taken to be

\[ R_0 = r_0 A^{1/3}. \]

The experimental value of the isotopic-shift constant \(C_{\mathrm{e}}\) can be found from formula (19), if \(\Delta T\), found from experiment, is substituted into it.

* At present it has been established that \(R_0 = 1.2 \cdot 10^{-13} A^{1/3}\) cm (see below).

In Fig. 16 the dependence of the constant of isotopic mixing between two isotopes with \(\Delta A = 2\) on the number of protons in the nucleus is given\(^{8,45,138}\). Here the theoretical dependence is shown by straight lines for \(r_0 = 1.5 \cdot 10^{-13}\) and \(r_0 = 1.1 \cdot 10^{-13}\), while the experimental data are represented by separate points expressing the constant of isotopic mixing (with error limits) for one or several pairs of isotopes of a number of elements. In Fig. 17 the same data on the comparison of theory with experiment\(^{36,43}\) are presented somewhat differently.

Fig. 16. Variation of the constants of isotopic mixing \(C_S\) and \(C_T\) as a function of \(Z\).

Here the ratio is given of the experimental value of the mixing for \(\Delta A = \Delta N = 2\) to the theoretical value obtained for the case of a charge uniformly distributed over the volume of the nucleus with \(R_0 = 1.4 \cdot 10^{-13} A^{1/3}\). This ratio is plotted for a number of elements as separate points as a function of the number of neutrons in the nucleus.

The data presented in Figs. 16 and 17 show that isotopic mixing increases with increasing atomic number of the element. If the errors of the calculation and the screening effect are estimated, then for most elements the isotopic mixing found experimentally turns out to be half as large as the theoretical value. In the region of the rare-earth elements, namely in the spectra of cerium, neodymium, samarium, and europium, a sharp increase in isotopic mixing takes place. Such “jumps,” although less pronounced, are observed in the region between rubidium and palladium, and also in the spectrum of lead.

Thus, from a comparison of the experimental data with the theory of the volume effect there follow two features of the isotopic shift in the spectra of heavy elements:

1. The observed isotopic shift in the spectra of heavy elements is, on the average, approximately half as large as is predicted by the theory of the volume effect.

2. The isotopic shift changes with increasing number of protons and neutrons in the nucleus not everywhere uniformly; in some regions rather sharp “jumps” of the isotopic shift are observed.

Fig. 17. Comparison of experimental and theoretical data on the isotopic shift in the spectra of heavy elements (circles correspond to elements with even \(Z\); dots—to elements with odd \(Z\)).

From what has been said it is clear that, although the theory of the volume effect does explain in its main features the course of the isotopic shift, it nevertheless becomes insufficient for interpreting certain reliably established details. In addition to the systematic discrepancy between experimental and theoretical data, the theory does not explain the “jumps” in the isotopic shift and the small changes in the shift of isotopes of one and the same element, as well as the additional displacement of even-odd isotopes relative to even-even isotopes. These features in the isotopic shift, it would seem, may be ascribed: 1) either to inaccuracies in the calculations, 2) or to additional effects manifested in the interaction of the electrons with the nucleus, 3) or to the imperfection of that model of the nucleus which is taken as the basis of the volume theory (sphericity, uniform change of the radius with increasing \(A\), etc.).

As for the inaccuracy of the calculations, it should first of all be noted that errors in the calculations may be due to the inaccuracy of the perturbation theory that underlies the derivation of formula (16) for the volume effect. As was already noted, a more rigorous derivation of this formula shows that, in the case of the mercury spectrum,^28 the shift calculated on the basis of perturbation theory must be multiplied by the coefficient \(\xi = 0.80\), while in the case of the thallium spectrum^29 by \(\xi = 0.75\). Some error may also be introduced into the calculations by the function \(\psi^2(0)\), which expresses the electron density. Judging from spectroscopic data on the determination of magnetic moments, the error associated with the use of this function may be estimated at approximately \(10\text{--}15\%\). Next, one should note the shielding effect, which consists in the fact that the penetrating \(s\)-electrons, being for some time closer than all other electrons to the nucleus, shield or reduce the effective nuclear charge acting on the inner electrons. This leads to a decrease in the binding energy of the inner electrons with the nucleus and, consequently, to a decrease in the contribution of these electrons to the isotopic shift. The valence \(s\)-electrons also shield the inner \(p\)- and \(d\)-electrons, but since the latter give a very small isotopic shift, the shielding effect in this case will constitute a negligible fraction of the total isotopic shift. For a direct calculation of the shielding effect, exact values of the function \(\psi^2(0)\) for the inner electrons in the presence and in the absence of valence electrons are necessary. Calculations carried out by an indirect method^29,38 show that shielding of the inner electrons by valence \(s\)-electrons reduces, in some cases, the magnitude of the theoretically calculated isotopic shift by approximately \(20\%\).

Even if one takes into account, to some extent, the errors of the calculations and the shielding effect, the discrepancy between the experimental and theoretical data, by no less than a factor of two, still remains.

In order to refine the theory of the isotopic shift, Breit, Arfken, and Clendenin^31 theoretically considered the influence of nuclear polarization on the isotopic shift. The effect of nuclear polarization consists in the action of atomic electrons on nuclear levels. Theory shows that polarization increases the binding of electrons with the nucleus, i.e., it acts in the opposite direction compared with the volume effect. This, apparently, can partially explain the discrepancy between the experimental shift and that calculated on the basis of the volume theory, which was discussed above. From studies of radioactive decay it is known that the low energy levels of the nucleus are situated more closely in nuclei with odd mass number \(A\) than in nuclei with even \(A\). Hence it is clear that atomic electrons produce a larger polarization effect in odd nuclei than in even ones, as a result of which the energy

of the electrons with the nucleus increases more in the former than in the latter. This leads to an additional shift of the center of gravity of the sublevels of odd isotopes relative to the levels of even isotopes toward smaller mass numbers. However, for a complete explanation of the even-odd shift it is necessary that the polarization effect be, in magnitude, approximately one half of the volume effect. Reasonable calculations show that the polarization effect gives only 20% of the even-odd shift. From what has been said, it is evident that nuclear polarization can explain the even-odd shift only to some extent. At the same time it should be noted that the significance of this effect in explaining the discrepancy between experimental and theoretical data is apparently small.

A second additional effect that may influence the isotope shift is the interaction between electrons and neutrons. This question was first examined in detail by I. E. Tamm \(^{20}\), and then by D. D. Ivanenko and V. Rodichev \(^{33,6}\). Just as polarization does, this effect increases the coupling of the electron with the nucleus and is directed opposite to the volume effect. A quantitative estimate of the electron-neutron interaction can be obtained by assuming that the additional neutrons of the heavier isotopes are located near the surface of the nucleus. Then the ratio of the shift caused by the interaction of the electron with the neutron to the shift due to the volume effect can be expressed by the formula \(^{34}\)

\[ \frac{\varepsilon}{\delta \Delta \omega} \simeq 0.0122(2\gamma+1)(2\gamma+3)\frac{A^{1/3}}{Z}, \tag{21} \]

where

\[ \gamma=(1-Z^2\alpha^2)\quad \text{and} \quad \alpha=\frac{2\pi e^2}{hc}. \]

For zinc \((Z=30,\ A=64)\) this ratio is approximately equal to 0.024, i.e., the contribution of the electron-neutron interaction to the isotope shift amounts to only 2.4% of the volume effect. It follows from formula (21) that the ratio mentioned decreases with increasing atomic number of the element. Thus, allowance for the electron-neutron interaction can explain only a very small part of the isotope shift in heavy elements*).

Thus, more accurate calculations and the additional effects manifested in the interaction of electrons with the nucleus do not give a complete explanation of all the experimentally observed features in the isotope shift. Therefore doubts naturally arose as to the perfection of that model of the nucleus according to which it is represented as a uniformly charged sphere, with a constantly increasing

*) See also on this question \(^{202}\).

with radius \(R_0=1.5\cdot 10^{-13} A^{1/3}\). Above all, the discrepancy between the experimental and theoretical data (see Figs. 16 and 17) shows that the adopted value of the radius is too large. Recent investigations of the scattering of high-energy electrons by nuclei of various elements give\(^{203}\) for the nuclear radius, on the average, the value \(R_0=1.1\cdot 10^{-13} A^{1/3}\). This is also confirmed by the results of investigations of \(\gamma\)-quanta in the capture of \(\mu\)-mesons by nuclei\(^{204}\) and by data on \(\beta\)- and \(\alpha\)-decay\(^{39}\). If the nuclear radius is taken equal to \(1.1\cdot 10^{-13} A^{1/3}\) cm, then, as follows from Fig. 16, the theoretical straight line expressing the course of the isotope shift for heavy elements gives, on the average, already a fairly good agreement with the experimental data. In this case, for heavy elements the magnitude of the isotope shift decreases by the ratio

\[ \left(\frac{1.15}{1.45}\right)^{2\gamma} \simeq 0.7. \]

However, this alone still does not solve the question of the agreement between theory and experiment, since the “jumps” and anomalies in the isotope shift still remain unexplained. Evidently, upon the addition of neutrons there occurs some redistribution of the charge of the atomic nucleus, and this redistribution is more complex than a gradual increase of the nuclear radius. Briks and Kopfermann\(^{26,30}\) were the first to express this point of view; they pointed out that, in calculating the isotope shift, it is necessary to take into account the deviation in the distribution of charge from spherical symmetry, which affects the isotope shift in the same direction as the increase of the nuclear volume. On this basis, Briks and Kopfermann, by analogy with Eu\(^{151}\) and Eu\(^{153}\), connect the anomalous shift between Sm\(^{150}\)—Sm\(^{152}\) with a large increase of the quadrupole moment*). The isotopes Eu\(^{151}\)—Eu\(^{153}\) give, like the isotopes Sm\(^{150}\) and Sm\(^{152}\), an anomalously large isotope shift. The nuclei of both pairs of isotopes consist of 88 and 90 neutrons, respectively. The difference consists only in the number of protons, namely, the nuclei Eu\(^{151}\) and Eu\(^{153}\) differ from the nuclei Sm\(^{150}\) and Sm\(^{152}\) by only one odd proton. The quadrupole moments of europium are too large to be explained by just one single excess particle. Hence the residues of the nuclei of europium isotopes, and the complete nuclei of samarium analogous to these residues, must deviate, in the sense of the electrostatic charge distribution, from spherical symmetry. A rough calculation shows that if this additional quadrupole contribution to the isotope shift is taken into account, using the quadrupole moments of europium (\(+1.2\) and \(+2.5\cdot 10^{-24}\ \mathrm{cm}^2\)) for the isotopes Sm\(^{150}\)

*) The attempt by a number of authors\(^{135,28}\) to explain the anomalous shift in the samarium spectrum by the \(\alpha\)-activity of the isotope Sm\(^{152}\) proved untenable, since according to the latest data\(^{205}\) the isotope Sm\(^{147}\) turned out to be \(\alpha\)-active.

and Sm\(^{152}\), then the anomalous shift between these isotopes finds a more or less correct explanation.

It should be noted that in the spectrum of neodymium, just as in the case of the spectra of samarium and europium, the anomalous shift occurs between the isotopes Nd\(^{148}\) and Nd\(^{150}\), with the same number of neutrons (88 and 90). This shows that in the three cases mentioned the principal role is played not by protons, but by neutrons*). Proceeding from this, Brix and Kopfermann\(^{9, 30, 36}\) drew attention to the connection of “jumps” in the isotope shift with the shell structure of the nucleus and, in particular, with the “magic” numbers of neutrons: 50, 82, 126. In recent years the attention of many investigators\(^{37, 49, 132, 133, 192}\) has been focused on clarifying this question. As a result, as we have already noted above, “jumps” in the isotope shift were found in the spectra of cerium and lead, and they were associated with the numbers \(N = 82\) and 126, which, according to the shell model, correspond to closed neutron shells.

It is appropriate to note that similar periodicities were discovered even earlier in graphs of the variation of nuclear moments throughout the whole system of isotopes, in curves of effective cross sections for the scattering of fast neutrons, in the periods and maximum energies of \(\beta\)-decay, in curves of mass defects, and in some other properties of nuclei. One of the most convincing examples of the manifestation of nuclear periodicity connected with the “magic” numbers of nucleons is provided by the quadrupole moments of nuclei, which characterize the deviation from spherical symmetry in the charge distribution. Graphs of nuclear quadrupole moments plotted as functions of \(Z\) and \(N\) give, for nuclei with 50 and 82 protons, and also with 50, 82, and 126 neutrons, characteristic minima close to zero\(^{207}\). Some of the periodicities mentioned here are easily interpreted as manifestations of periodicities in the values of the volumes and radii of nuclei connected with the “magic” numbers 50, 82, and 126.

Thus, in the theoretical consideration of the isotope effect in the spectra of heavy elements it is necessary to take into account the nonsphericity of nuclei and the periodic change of their shape (nuclear deformation) as a function of the filling of nuclear shells.

c) Influence of nuclear deformation on the isotope shift

The absence of spherical symmetry in nuclei is confirmed above all by the presence of rotational excitation levels in nuclei, which are possible only in the case of deformed nuclei. The nonsphericity

*) Let us also note Wefelmeier’s\(^{206}\) attempt to connect the anomalous isotope shift in the spectrum of samarium with the number of neutrons, proceeding from the \(\alpha\)-particle model of the nucleus.

the presence in many nuclei of quadrupole moments, with positive quadrupole moments characterizing an elongated shape of the nucleus, and negative ones—a flattened shape. In addition, it has been shown experimentally that nuclear deformation affects the energies of the first excited states of odd-odd nuclei and the rates of transitions from the first excited level to the ground state in these nuclei.

The question of the influence of the nonsphericity of nuclei on the isotopic shift was investigated in detail by Wilets, Hill, and Ford \(^{43}\), as well as by Bodmer \(^{456}\), who carried out a theoretical calculation of the isotopic effect taking nuclear deformation into account, comparing the results obtained for heavy elements with experimental data. The dependence of the isotopic shift on the distortion or deformation of the shape of the nucleus is explained by a change in the mean distribution of charge in the nucleus. As an illustration, Fig. 18 presents the charge distribution of a deformed nucleus for three different nuclear shapes with uniform charge density throughout the volume of the nucleus.

Fig. 18. Mean charge density for deformed nuclei.

The dashed line \((1)\) refers to a spherical nucleus with radius \(a_0\); curves \((2)\) and \((3)\) represent charge densities averaged over all angles, respectively for an elongated and a flattened nucleus. It is clear from the figure that deformation of the nucleus (both longitudinal and transverse) leads, as it were, to an increase in the nuclear radius, which should manifest itself in an additional isotopic shift. This additional shift may be very large, since the deformation of the nucleus upon the addition of a pair of neutrons may change greatly, whereas the volume of the nucleus changes only slightly.

Let us consider the simplest case, when the deformed nucleus has the shape of an ellipsoid of revolution. In this case the nuclear radius is expressed as

\[ R(\theta)=a_0[1+\alpha P_2(\cos\theta)], \tag{22} \]

where \(P_2\) is the second Legendre polynomial, \(\alpha\) is the nuclear deformation parameter, with \(\alpha>0\) for elongated and \(\alpha<0\) for compressed ellipsoids. For convenience of consideration we shall assume that the nucleus has a constant volume

\[ \frac{4}{3}\pi R_0^3 \]

and a uniform charge distribution over the volume

\[ \left. \begin{aligned} \rho_0(r)&=Ze\left(\frac{4}{3}\pi R_0^3\right)^{-1}, \qquad r<R(\theta),\\ \rho_0(r)&=0, \qquad r>R(\theta). \end{aligned} \right\} \tag{23} \]

The assumption of constancy of volume relates the constants \(a_0\) and \(R_0\) in equalities (22) and (23):

\[ a_0=R_0\left(1+\frac{3}{5}\alpha^2+\frac{2}{35}\alpha^3\right)^{-\frac{1}{3}} . \tag{24} \]

If \(m\) is the minor axis of the ellipsoid and \(M\) its major axis, then for the charge density averaged over all angles in the region \(m<r<M\), we obtain the expressions:

elongated nucleus:

\[ \rho(r)=\rho_0\{1-[(r-m)/(M-m)]^{1/2}\}, \tag{25a} \]

compressed nucleus:

\[ \rho(r)=\rho_0\{(M-r)/(M-m)\}^{1/2}. \tag{25b} \]

The distribution of charge density in the nucleus shown in Fig. 18 is obtained from these formulas.

If one uses perturbation theory and takes the electron density inside the nucleus to be

\[ \bar{\psi}\psi=\frac{2(\gamma+1)}{[\Gamma(2\gamma+1)]^2}\psi^2(0) \left[\frac{2Zr}{a_H}\right]^{2\gamma-2}, \tag{26} \]

then the isotope shift of the level due to the deformation of the nucleus can be expressed as follows:

\[ \Delta w_\alpha= \frac{8\pi Z e^2(1+\gamma)}{[\Gamma(2\gamma+1)]^2} \left(\frac{2Z}{a_H}\right)^{2\gamma-2} \psi^2(0)R_0^{2\gamma}\frac{3}{10} \times \]

\[ \times \frac{\alpha^2}{(2\gamma+1)} \left[1+\frac{2}{21}(2\gamma+3)\alpha+\cdots\right]. \tag{27} \]

Here the same notation is adopted as was used in formula (16). The ratio of this shift to the shift due to the ordinary volume effect, to accuracy up to \(\alpha^2\), will be equal to:

\[ \frac{\Delta w_\alpha}{\Delta w_{\mathrm{ob}}} = \frac{\gamma(2\gamma+3)}{5}\alpha^2 . \tag{28} \]

The ratio of the isotope shift between the levels of two isotopes is obtained as:

\[ \frac{\delta\Delta w_\alpha}{\delta\Delta w_{\mathrm{ob}}} = \frac{3}{10}(2\gamma+3)A \left\{\frac{\delta(\alpha^2)}{\delta N}\right\}_{Z}. \tag{29} \]

Using data on quadrupole moments, it can be shown that deformation of nuclei has a considerable influence on the isotope shift; thus, for example, for samarium \(Z=62\), \(A=150\), \(\delta\alpha^2=0.005\) and \(\delta N=2\), whence \(\dfrac{\delta\Delta w_\alpha}{\delta\Delta w_{\mathrm{ob}}}=0.54\). It should be noted,

that the quantity \(\alpha\), occurring in the formulas given above, is a characteristic of the deformation of the nucleus; it is not always determined by the quadrupole moment (for example, in the case of nuclei with \(I=0,\,1/2\)). Therefore this effect will occur both in even-even and in even-odd nuclei.

We see that, in order to determine the magnitude of the isotopic shift due to deformation of the nucleus, it is necessary to know the quantity \(\alpha\). Its value can be determined by various methods. First of all, we note that one could calculate the equilibrium shape of the nucleus on the basis of the known configuration of nucleons in an unclosed shell. However, for heavy nuclei there are no exact data on nucleon configurations; therefore this method of determining \(\alpha\) has no practical significance. The value of the deformation coefficient \(\alpha\) may also be found with the aid of quadrupole moments, which, however, have not been measured with especially high accuracy. For a given quadrupole moment \(Q\) it is determined by the formula\({}^{208}\):

\[ \alpha=\frac{53.5}{Z A^{2/3}}\,\frac{(I+1)(2I+3)}{I(2I-1)}\,Q_{\mathrm{exp}}. \tag{30} \]

This formula is derived on the basis of the assumption that the nuclear spin \(I\) is directed along its axis and that the total spin vector is rigidly connected with the axis\({}^{209}\). In view of this, quadrupole moments give a lower limit for the value of \(\alpha\). Another orientation, or a weak coupling of the spin vector with the axis, leads to an overestimate of the value of \(\alpha\). It follows directly from formula (30) that even if \(Q_{\mathrm{exp}}\) is equal to zero, for nuclei with \(I=0,\,1/2\) the deformation coefficient \(\alpha\) may have a finite value. Thus, even-even nuclei and nuclei with \(I=1/2\) may be nonspherical. The second method of determining the deformation is based on the use of a formula connecting the deformation of the nucleus with the first excited state of an even-even nucleus\({}^{208,210}\):

\[ \alpha=10.4\,A^{-5/6}E_1^{-1/2}, \tag{31} \]

where \(E_1\) is expressed in MeV. This simple formula gives overestimated values of \(\alpha\)\({}^{208}\). The error is connected with the particular configuration of the nucleons, and therefore it is difficult to estimate it. It is consequently necessary to normalize this formula so that the results of its application coincide with the values of nuclear deformation obtained on the basis of quadrupole moments. The normalization consists in choosing a factor by which \(\alpha\) must be reduced. This factor turns out to be equal to 1.7 for \(\alpha\), i.e. 3 for \(\alpha^2\). This method has a number of advantages in comparison with finding deformations from quadrupole-moment data. First, the data on the first excited states of even-even nuclei for \(50<N<126\) are more accurate and more extensive; second, the majority

of the measured isotope shifts pertain to even-even isotopes, while the quadrupole moments of these nuclei are equal to zero; third, the isotope shifts and excited states of even-even nuclei depend on the deformation of the nuclei, irrespective of how the nuclear spin is oriented—along the axis of deformation or not. The third method of determining deformation is based on an approximate formula that relates the deformation of a nucleus to the intensity of quadrupole electric transitions from the first excited state to the ground state in even-even nuclei\(^{210}\). However, this formula gives somewhat underestimated values of \(\alpha\) in comparison with the quantities obtained on the basis of quadrupole moments.

Figure 19 shows the change in the ratio of the total isotope shift for \(\Delta A = 2\), with allowance for nuclear deformation, to the shift due to the simple volume effect, as a function of the number of neutrons in the nucleus. The data shown by crosses were obtained on the basis of measured quadrupole moments, while the data shown by circles were obtained from information on the first excited states of even-even nuclei. Near each point the chemical symbol of the given element is indicated.

Fig. 19. Theoretical curve of the isotope shift with allowance for deformation, in units of the ordinary volume effect.

It should be noted, however, that since the deformation of the nucleus depends both on \(N\) and on \(Z\), one cannot expect all points to fall on a single curve that is a function of only one quantity, \(N\).

Comparing the experimental curve (Fig. 17) with the theoretical one (Fig. 19), it is easy to see that the latter lies, as was already noted, 0.5 above the experimental curve. Now, however, with allowance for nuclear deformation, both curves correspond surprisingly well to one another in their shape. For clarity, in Fig. 17 the theoretical curve has been plotted shifted downward by 0.5 relative to the curve presented in Fig. 19. This shifted curve coincides completely with the experimental points, which indicates that all deviations of the isotopic shift from the mean value are fully explained by the distortion of the spherical shape of the nucleus, or, in other words, by its deformation.

The presence of nonsphericity of the nucleus also easily explains the phenomenon of the even–odd shift in the isotopic shift. From the data on the first excited states it follows that even–even nuclei are deformed somewhat more strongly than even–odd nuclei. The same conclusion follows from shell theory \(^{456}\). This causes an additional shift of the even–odd isotopes in the normally observed direction. In addition, as already noted above, the polarization of the nucleus by atomic electrons makes a small contribution to the even–odd shift.

Summing up, it should be noted that the theory of the simple volume effect, supplemented by allowance for the influence of nuclear deformation, agrees rather well with experiment, explaining the “jumps” in the isotopic shift and the even–odd shift. Nevertheless, a discrepancy between the theoretical and experimental data remains, with the experimental values being on average 0.5 times smaller than the shifts determined by the simple volume effect. It is possible that this discrepancy is explained by an overestimate of the value of the nuclear radius, which, as already mentioned, is now estimated as \(1.1 \cdot 10^{-13}\cdot A^{1/3}\ \mathrm{cm}\). At the same time it should be noted that, according to Wilets, Hill, and Ford \(^{43}\), the effect of nuclear “compressibility” may be used to explain the indicated discrepancy; this also leads to a decrease in the mean value of the isotopic shift. The term “compressibility” of the nucleus means that the radius of the nucleus, upon the addition of two neutrons, increases considerably less than follows from the law \(R_0 = r_0 A^{1/3}\). In addition, it is quite possible that the discrepancy between the experimental and theoretical data is the result of an unaccounted-for part of the isotopic shift caused by deformation of the atomic core by the valence electrons. The effect of deformation of the inner shells, theoretically considered for the first time by I. I. Gol’d-

element \(^{41}\), leads to a noticeable additional shift in the \(p\)-, \(d\)- and \(f\)-states of the valence electron and gives a significant correction to the previous formulas for an \(s\)-electron. However, this correction, unfortunately, has not yet been quantitatively evaluated.

d) Perturbation of configurations

In addition to the effects considered above, which determine the total isotopic shift in spectra, let us briefly discuss the phenomenon of mutual perturbation of electronic configurations, which may distort, to one degree or another, the true magnitude of the shift. This phenomenon is analogous to the perturbation of hyperfine-structure levels of the same parity and the same resultant angular momentum \(J\), belonging to different electronic configurations. The perturbation amounts, as it were, to an interaction of levels which either move closer together or farther apart. As one of the known examples of perturbation in the case of isotope structure, one may cite the shift of the term \(8^{1}P_{1}\) Hg I (see Fig. 8), which is many times larger than the shifts of the deeper terms \(6^{1}P_{1}\) and \(7^{1}P_{1}\). This is explained by the fact that the term \(5d^{10}6s8p\,{}^{1}P_{1}^{0}\) is perturbed by a term of the more complex configuration \(5d^{9}6s^{2}6p\,{}^{1}P_{1}^{0}\); both of these terms are odd and have identical values of \(J\). Perturbations between closely lying levels of similar quantum nature have also been observed in the case of other heavy elements (Pt, Pb, U). It should be noted that the question of perturbations manifested in the isotope structure of terms has been studied very little. One can cite only a few works that deal with this phenomenon \(^{4,8,21}\). However, in the study of the isotope shift in the spectra of heavy elements, this phenomenon must always be taken into account.

5. CONCLUSION

Theoretical and experimental investigations of nuclear effects in atomic spectra, including the isotope shift caused by a whole range of properties of the nucleus, are of very great interest. These investigations make it possible not only to interpret the structure of lines, but also to obtain very important information about atomic nuclei. An analysis of the theoretical data shows that in recent years substantial results have been obtained in the study of the isotope effect, thanks to which it has been possible to find the right approaches for a reasonable explanation of many previously discovered regularities in the isotope shift. However, the work on investigating the isotope structure of lines is not complete, since there still remain many elements and isotopes for which the isotope shift has not been studied. A number of theoretical

questions also still remains unresolved. As a result, there is a very considerable, still unexplained, fraction of the shift both in the case of light and in the case of heavy elements.

SYSTEMATIZED SUMMARY OF THE LITERATURE ON THE ISOTOPE EFFECT IN ATOMIC SPECTRA

1. GENERAL LITERATURE

1. G. A. Bethe and R. F. Bacher, Nuclear Physics, part I, GNTI, Kharkov, 1938, pp. 242—246.
2. E. Condon and G. Shortley, The Theory of Atomic Spectra, IL, Moscow, 1949, pp. 398—400, 432—433.
3. H. Kopfermann, Kernmomente, Leipzig, 1940, pp. 68—78, 116—136.
4. S. Tolansky, Hyperfine Structure in Line Spectra and Nuclear Spin, London, 1946, pp. 69—86.
5. S. E. Frish, Spectroscopic Determination of Nuclear Moments, Gostekhizdat, 1948, pp. 90—100.
6. A. Sokolov and D. Ivanenko, Quantum Field Theory, Gostekhizdat, Moscow, 1952, pp. 728—780.
7. W. Walcher, Nucleonics 6, No. 6, 28 (1950).
8. E. W. Foster, Reports on Progress in Physics 14, 288—308 (1951).
9. P. Brix und H. Kopfermann, Festschrift der Akademie der Wissenschaften in Göttingen, Math.-Phys. Klasse, 1951, p. 17.
10. J. P. McNally, Am. Jour. Phys. 20, 152 (1952).
11. G. H. Dikke, Annuals Review of Nuclear Science 1, 382—391 (1952).
12. P. Brix und H. Kopfermann — Landolt—Börnstein, Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 6th ed., vol. 1, part 5, Berlin, 1952, pp. 1—69.
13. C. E. Moore, Atomic Energy Levels, vol. 1, Washington, 1949.

2. THEORY

1. D. S. Hughes and Eckart G., Phys. Rev. 36, 694 (1930) (isotopic shift in the Li spectrum considered).
2. J. H. Bartlett, Nature 128, 408 (1931).
3. G. Racah, Nature 129, 723 (1932).
4. J. E. Rosenthal and G. Breit, Phys. Rev. 41, 451 (1932); 41, 459 (1932).
5. G. Breit, Phys. Rev. 42, 348 (1932) (shift in the spectra of Hg, Tl, Pb).
6. J. Bartlett and J. Gibbons, Phys. Rev. 44, 538 (1933) (shift in the Ne spectrum).
7. I. E. Tamm, DAN, vol. XXI, No. 3, 106 (1938).
8. W. Opechovsky and de Vries, Physica 6, 913 (1939) (shift in the B spectrum).
9. J. P. Vinti, Phys. Rev. 56, 1120 (1939) (shift in the Mg spectrum).
10. J. P. Vinti, Phys. Rev. 58, 879 (1940) (shift in the B spectrum).
11. E. K. Broch, Arch. Math. Naturwidensk. 48, 25 (1945).
12. Ya. A. Smorodinskii, ZhETF 27, 1034 (1947).
13. P. Brix und H. Kopfermann, Nachrichten Akad. Wiss. Göttingen, Math.-Phys. Kl. 2, 31 (1947).
14. D. D. Ivanenko and A. F. Tsander, ZhETF 18, 434 (1948).
15. N. Feather, Nature 162, 412 (1948) (shift in the spectra of Sm and Nd).
16. M. F. Grawford and A. L. Schawlow, Phys. Rev. 76, 1310 (1949).

1. P. Brix und H. Kopfermann, Zeits. f. Phys. 126, 344 (1949). (shift in the spectrum of Sm).

2. G. Breit, G. B. Arfken and W. W. Clendenin, Phys. Rev. 77, 569 (1950); 78, 390 (1950).

3. G. Breit, Phys. Rev. 78, 470 (1950); 79, 891 (1950).

4. D. Ivanenko and V. Rodichev, DAN 70, 801 (1950).

5. L. Wilets and L. C. Bradley, Phys. Rev. 82, 285 (1951).

6. G. Breit and W. W. Clendenin, Phys. Rev. 85, 689 (1952).

7. P. Brix and H. Kopfermann, Phys. Rev. 85, 1050 (1952).

8. G. Breit, Phys. Rev. 86, 254 (1952).

9. W. Humbach, Zeits. f. Phys. 133, 589 (1952).

10. D. Ivanenko and S. Larin, JETP 24, 359 (1953).

11. D. Ivanenko and N. Kolesnikov, DAN 39, 253 (1953).

12. I. I. Goldman, JETP 24, 177 (1953).

13. A. R. Bodmer, Proc. Phys. Soc. (A) 66, 1041 (1953).

14. L. Willets, D. L. Hill and W. Ford, Phys. Rev. 91, 1488 (1953).

15. A. P. Yutsis, A. S. Janouchanis and G. K. Tsionaitis, JETP 25, 683 (1953) (shift in the spectrum of C).

16. F. Bitter and H. Feshbach, Phys. Rev. 92, 837 (1953).

45a. W. R. Hindmarsh, Proc. Phys. Soc. (A) 67, 393 (1954).

45b. A. R. Bodmer, Proc. Phys. Soc. (A) 67, 622 (1954).

3. Hydrogen (H, Z = 1)

1. H. C. Urey, F. G. Brickwedde and G. M. Murphy, Phys. Rev. 39, 164 (1932); 40, 464 (1932) (lines Hα, Hβ, Hγ).

2. S. S. Ballard and H. E. White, Phys. Rev. 43, 941 (1933) (Lyman series).

3. H. Hagenbach und H. Gärtner, Helv. Phys. Acta 8, 34 (1953) (lines Hα, Hβ, Hγ).

48a. V. I. Chernyaev, DAN 19, No. 4, 254 (1938); 20, No. 5, 374 (1939).

1. H. Nagaoka and T. Mishima, Sci. Pap. Phys. Chem. Res. Tokyo 34, 931 (1938) (higher members of the Balmer series and the Paschen series).

2. H. Nagaoka and T. Mishima, Proc. Imp. Acad. Tokyo 13, 97 (1937); 14, 53 (1938) (Paschen series).

3. J. W. Drinkwater, S. O. Richardson and W. E. Williams, Proc. Phys. Soc. (A) 174, 164 (1940); R. T. Birge, Phys. Rev. 60, 766 (1941); G. W. Hsuch, Phys. Rev. 67, 66 (1945) (Hα line).

4. Helium (He, Z = 2)

1. L. C. Bradley and H. Kuhn, Nature 162, 412 (1948).

2. A. Andrew and W. W. Carter, Phys. Rev. 74, 838 (1948).

3. T. P. Manning, Phys. Rev. 76, 173, 1949.

4. M. Fred, F. S. Tomkins and J. K. Brody, Phys. Rev. 75, 1772 (1949); 79, 212 (1950).

5. L. C. Bradley and H. Kuhn, Proc. Roy. Soc., London (A) 209, 325 (1951).

6. M. Fred, F. S. Tomkins, J. K. Brody and M. Hamermesh, Phys. Rev. 82, 406 (1951).

5. Lithium (Li, Z = 3)

1. H. Schüler und K. Wurm, Naturwiss. 15, 971 (1927).

2. H. Schüler, Zeits. f. Phys. 42, 487 (1927).

3. H. Schüler und H. Brück, Zeits. f. Phys. 58, 735 (1929).

1. H. Schuler, Zeits. f. Phys. 66, 431 (1930).
2. D. S. Hughes, Phys. Rev. 38, 857 (1931).
3. A. Bogros, Ann. d. Physique 17, 199 (1932).
4. D. A. Jackson and H. Kuhn, Proc. Roy. Soc. (A) 173, 278 (1939).
5. K. W. Meissner, L. G. Mundie and P. H. Stelson, Phys. Rev. 74, 932 (1948).
   65a. R. Hughes, Phys. Rev. 95, 621 (1954).

6. Boron (B, \(Z = 5\))

1. L. S. Ornstein und J. A. Vreeswijk, Zeits. f. Phys. 80, 57 (1933).
2. S. Mrozowski, Zeits. f. Phys. 112, 223 (1939).

7. Carbon (C, \(Z = 6\))

1. J. R. Holmes, Phys. Rev. 77, 745 (1950).
2. C. R. Burnett, Phys. Rev. 80, 494 (1950).
3. J. R. Holmes, J. Opt. Soc. Am. 41, 360 (1951).
   70a. J. P. Nikolas, Phys. Rev. 95, 1469 (1954).

8. Nitrogen (N, \(Z = 7\))

1. J. R. Holmes, Phys. Rev. 63, 41 (1943).
2. J. R. Holmes, Phys. Rev. 73, 539 (1948).
3. J. R. Holmes, J. Opt. Soc. Am. 41, 360 (1951).

9. Oxygen (O, \(Z = 8\))

1. L. M. Parker and J. R. Holmes, Phys. Rev. 83, 888 (1951); J. Opt. Soc. Am. 43, 103 (1953).
   74a. C. E. Treanor, Phys. Rev. 95, 1472 (1954).

10. Neon (Ne, \(Z = 10\))

1. E. Thomas and E. J. Evans, Phil. Mag. 10, 128 (1930).
2. H. Nagaoka and T. Mishima, Sci. Pap. Inst. Chem. Res., Tokyo 13, 293 (1930); 25, 223 (1934).
3. R. Ritschl und H. Schober, Phys. Zeits. 38, 6 (1937).
4. H. Schober, Phys. Zeits. 40, 77 (1939).
5. K. Murakawa and S. Suwa, Phys. Rev. 74, 1535 (1948).

11. Magnesium (Mg, \(Z = 12\))

1. R. F. Bacher and R. A. Sawger, Phys. Rev. 47, 587 (1935).
2. D. A. Jackson and H. Kuhn, Proc. Roy. Soc. (A) 154, 679 (1936).
3. R. A. Fischer, Phys. Rev. 51, 381 (1937).
4. K. Meissner, Phys. Rev. 53, 931 (1938); Ann. de Phys. 31, 505 (1938).
5. R. A. Fischer, Rev. Mod. Phys. 14, 79 (1942).
6. L. G. Mundie and K. W. Meissner, Phys. Rev. 65, 265 (1944).
7. M. F. Crawford, F. M. Kelly, A. L. Schawlow and W. M. Gray, Phys. Rev. 76, 1527 (1949).
8. K. Murakawa, J. Phys. Soc., Japan 8, 213 (1953).

12. CHLORINE (Cl, \(Z=17\))

1. S. Tolansky, Zeits. f. Phys. 79, 470 (1931).

13. ARGON (Ar, \(Z=18\))

1. H. Kopfermann und H. Krüger, Zeits. f. Phys. 105, 389 (1937).
2. H. Meyer, Helv. Phys. Acta 26, 811 (1953).

14. POTASSIUM (K, \(Z=19\))

1. D. A. Jackson und H. Kuhn, Proc. Roy. Soc. (A) 165, 303 (1938).

15. CALCIUM (Ca, \(Z=20\))

1. A. Pery, Proc. Phys. Soc. (A) 67, 181 (1954).

16. COPPER (Cu, \(Z=29\))

1. R. Ritschl, Physik 79, 1 (1932).
2. H. Schüler und T. Schmidt, Zeits. f. Phys. 100, 113 (1936).
3. P. Brix, Zeits. f. Phys. 126, 725 (1949).
4. P. Brix und W. Humbach, Zeits. f. Phys. 128, 506 (1950).

17. ZINC (Zn, \(Z=30\))

1. H. Schüler und H. Westmeyer, Zeits. f. Phys. 81, 565 (1933).
2. M. F. Crawford, W. M. Gray, F. M. Kelly and A. L. Schawlow, Canad. J. Res. (A) 28, 138 (1950).

18. GALLIUM (Ga, \(Z=31\))

1. H. Schüler und H. Korsching, Zeits. f. Phys. 103, 434 (1936).

19. GERMANIUM (Ge, \(Z=32\))

99a. G. V. Deverall, K. W. Meissner, G. J. Zissis, Phys. Rev. 95, 1463 (1954).

20. SELENIUM (Se, \(Z=34\))

1. J. E. Mack and O. H. Arroc, Phys. Rev. 76, 173 (1949).

21. BROMINE (Br, \(Z=35\))

1. S. Tolansky and S. A. Trivedi, Proc. Roy. Soc. A 175, 336 (1940).
2. J. D. Ranade, Phil. Mag. 42, 284 (1951).

22. KRYPTON (Kr, \(Z=36\))

1. H. Kopfermann und N. Weith-Knudsen, Zeits. f. Phys. 85, 353 (1933).
2. J. Koch and E. Rasmussen, Phys. Rev. 76, 1417 (1949).

23. RUBIDIUM (Rb, Z=37)

1. H. Kopfermann, Zeits. f. Phys. 83, 417 (1933).
2. H. Kopfermann und H. Krüger, Zeits. f. Phys. 103, 486 (1936).
3. A. V. Hollenberg, Phys. Rev. 52, 139 (1937).

24. STRONTIUM (Sr, Z=38)

1. M. Heyden und H. Kopfermann, Zeits. f. Phys. 108, 232 (1938).

25. ZIRCONIUM (Zr, Z=40)

1. S. Suwa, Phys. Rev. 86, 247 (1952).

26. MOLYBDENUM (Mo, Z=42)

1. N. S. Grace und K. R. More, Phys. Rev. 45, 166 (1934).
2. A. Steudel, Zeits. f. Phys. 132, 429 (1952).

27. RUTHENIUM (Ru, Z=44)

1. K. Murakawa, J. Phys. Soc., Japan 8, 535 (1953); 9, 427 (1954).
   112a. H. Kopfermann, A. Steudel und H. Thulke, Zeits. f. Phys. 138, 309 (1954).
   112b. A. Steudel und H. Thulke, Zeits. f. Phys. 139, 239 (1954).

28. PALLADIUM (Pd, Z=46)

1. P. Brix und A. Steudel, Naturwiss. 38, 431 (1952).
2. A. Steudel, Zeits. f. Phys. 132, 429 (1952).

29. SILVER (Ag, Z=47)

1. D. A. Jackson and H. Kuhn, Proc. Roy. Soc. A 158, 372 (1937).
2. E. Rasmussen, Medd. Danske Vid. Selskad. 18 (1940).
3. M. F. Crawford, A. L. Schawlow, W. M. Grey and F. M. Kelly, Phys. Rev. 75, 1112 (1949).
4. P. Brix, H. Kopfermann, R. Martin und W. Walcher, Zeits. f. Phys. 130, 88 (1951).

30. CADMIUM (Cd, Z=48)

1. H. Schüler und H. Westmeyer, Zeits. f. Phys. 82, 685 (1933).
2. P. Brix und A. Steudel, Zeits. f. Phys. 128, 260 (1950).
   120a. W. R. Hindmarsh, H. Kuhn, S. A. Ramsden, Proc. Phys. Soc. A 63, 478 (1954).
   120b. E. C. Woodward, D. R. Speck, Phys. Rev. 96, 529 (1954).

31. TIN (Sn, Z=50)

1. H. Schüler und H. Westmeyer, Naturwiss. 21, 660 (1933).
2. S. Tolansky, Proc. Roy. Soc. A 144, 574 (1934).

1. S. Tolansky and G. O. Forester, Phil. Mag. 32, 315 (1941).
   123a. W. R. Hindmarsh, H. Kuhn, S. A. Ramsden, Proc. Phys. Soc. A 63, 478 (1954).

32. ANTIMONY (Sb, Z=51)

1. J. S. Badami, Zeits. f. Phys. 79, 224 (1932); Journ. Univ. of Bombay, 4, 86 (1935).
2. D. H. Tomboulian and R. F. Bacher, Phys. Rev. 58, 52 (1940)

33. TELLURIUM (Te, Z=52)

1. J. S. Ross and K. Murakawa, Phys. Rev. 83, 559 (1952).

34. XENON (Xe, Z=54)

1. H. Kopfermann und E. Rindal, Zeits. f. Phys. 87, 460 (1934).
2. J. Koch and E. Rasmussen, Phys. Rev. 77, 722 (1950).

35. BARIUM (Ba, Z=56)

1. H. Kopfermann und G. Wessel, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., p. 58 (1948).
2. O. H. Arroe, Phys. Rev. 79, 836 (1950).

36. CERIUM (Ce, Z=58)

1. P. Brix und Frank, Zeits. f. Phys. 127, 289 (1950).
2. K. Murakawa and J. S. Ross, Phys. Rev. 83, 1272 (1951).
3. H. Arroe, Phys. Rev. 93, 94 (1954).

37. NEODYMIUM (Nd, Z=60)

1. P. F. A. Klinkenberg, Physika 11, 327 (1945).
   134a. G. Nöldeke und A. Steudel, Zeits. f. Phys. 137, 632 (1954).
   134b. K. Murakawa, Phys. Rev. 96, 1543 (1954).

38. SAMARIUM (Sm, Z=62)

1. H. Schüler und Th. Schmidt, Zeits. f. Phys. 92, 148 (1934).
2. A. S. King, Astrophys. Journ. 82, 140 (1935).
3. M. N. Banyukov and S. E. Frish, DAN 23, 39 (1939).
4. P. Brix und H. Kopfermann, Zeits. f. Phys. 126, 344 (1949).
5. P. Brix, Zeits. f. Phys. 126, 431 (1949).
6. J. R. McNally and D. D. Smith, J. Opt. Soc. Am. 40, 878 (1950).

39. EUROPIUM (Eu, Z=63)

1. H. Schüler und Th. Schmidt, Zeits. f. Phys. 94, 457 (1935).
2. P. Brix, Zeits. f. Phys. 132, 579 (1952).

40. GADOLINIUM (Gd, Z=64)

1. P. F. A. Klinkenberg, Physika 12, 33 (1946).
2. P. Brix und H. D. Endler, Naturwiss. 38, 214 (1951); Zeits. f. Phys. 133, 362 (1952).
   144a. K. Murakawa, Phys. Rev. 96, 1543 (1954).

1. P. Brix, Zeits. f. Phys. 132, 579 (1952).
2. S. Suwa, Phys. Rev. 86, 247 (1952); J. Phys. Soc., Japan 8, 377 (1953).

41. Dysprosium (Dy, Z=66)

1. K. Murakawa und Kameit, Phys. Rev. 92, 325 (1954).

42. Erbium (Er, Z=68)

1. L. Wilets and L. C. Bradley, Phys. Rev. 84, 1055 (1951); 87, 1018 (1952).
2. K. Murakawa and S. Suwa, Phys. Rev. 85, 683 (1952).

43. Ytterbium (Yb, Z=70)

1. H. Schüler, J. Roign und Korsehing, Zeits. f. Phys. 111, 165 (1938); 111, 386 (1938).
2. P. Brix, Zeits. f. Phys. 132, 579 (1952).

44. Hafnium (Hf, Z=72)

1. E. Rasmussen, Naturwiss. 23, 69 (1935).
   152a. W. W. Watson and J. A. Collinson, Phys. Rev. 95, 621 (1954); 96, 949 (1954).

45. Tungsten (W, Z=74)

1. N. S. Grage and H. E. White, Phys. Rev. 43, 1039 (1933).
2. N. S. Grage and K. R. More, Phys. Rev. 45, 166 (1934).
3. M. Kintl, H. Hasunuma and T. Kawada, Proc. Phys.-Math. Soc., Japan 19, 1019 (1937).
4. H. Kopfermann und D. Meyer, Zeis. f. Phys. 124, 685 (1948).
5. J. A. Vreeland and K. Murakawa, Phys. Rev. 83, 229 (1951).
6. K. Murakawa, J. Phys. Soc., Japan 8, 215 (1953).

46. Rhenium (Re, Z=75)

1. H. Schüler und H. Korsching, Zeits. f. Phys. 105, 168 (1937).

47. Osmium (Os, Z=76)

1. T. Kawada, Proc. Phys.-Math. Soc., Japan 20, 653 (1938).
2. S. Suwa, Phys. Rev. 83, 1258 (1951).

48. Iridium (Ir, Z=77)

1. K. Murakawa and S. Suwa, Phys. Rev. 87, 1048 (1952).

49. Platinum (Pt, Z=78)

1. B. Jaeckel und H. Kopfermann, Zeits. f. Phys. 99, 492 (1936).
2. B. Jaeckel, Zeits. f. Phys. 100, 513 (1936).
3. S. Tolansky and E. Lee, Proc. Roy. Soc. A 158, 110 (1937).

ISOTOPE EFFECT IN ATOMIC SPECTRA

50. MERCURY (Hg, \(Z=80\))

1. H. Schüler und J. E. Keyston, Zeits. f. Phys. 72, 423 (1931).
2. H. Schüler und E. G. Jones, Zeits. f. Phys. 74, 631 (1932).
3. H. Schüler und E. G. Jones, Zeits. f. Phys. 76, 14 (1932).
4. S. Mrozowski, Zeits. f. Phys. 108, 204 (1938).
5. W. Opechowski, Zeits. f. Phys. 109, 485 (1938).
6. S. Mrozowski, Phys. Rev. 57, 207 (1940); 61, 605 (1942).
7. S. Mrozowski, Phys. Rev. 67, 161 (1945).
8. K. G. Kessler, Phys. Rev. 77, 559 (1950).
9. K. Murakawa, Phys. Rev. 78, 480 (1950); 79, 536 (1950).
10. E. W. Foster, Proc. Roy. Soc. A 200, 429 (1950); G. R. Fowles, J. Opt. Soc. Am. 44, 85 (1954).

51. THALLIUM (Tl, \(Z=81\))

1. H. Schüler und J. E. Keyston, Zeits. f. Phys. 70, 1 (1931).
2. D. A. Jackson, Zets. f. Phys. 75, 293 (1932).
3. P. Köhler, Zeits. f. Phys. 113, 306 (1939).
4. M. F. Crawford and A. L. Schawlow, Phys. Rev. 76, 1310 (1949).

52. LEAD (Pb, \(Z=82\))

1. H. Kopfermann, Zeits. f. Phys. 75, 363 (1932).
2. H. Schüler und E. G. Jones, Zeits. f. Phys. 75, 563 (1932).
3. H. Schüler und E. G. Jones, Zeits. f. Phys. 76, 14 (1932).
4. J. L. Rose and L. P. Granath, Phys. Rev. 40, 760 (1932).
5. J. L. Rose, Phys. Rev. 47, 122 (1935).
6. A. M. Grooker, Canad. J. Res. A 14, 115 (1936).
7. M. F. Crawford, A. B. Melay and A. M. Crooker, Proc. Roy. Soc. A 158, 455 (1937).
8. T. E. Manning, Phys. Rev. 76, 464 (1949).
9. T. E. Manning, C. E. Anderson and W. W. Watson, Phys. Rev. 78, 417 (1950).
10. K. Murakawa and S. Suwa, J. Phys. Soc., Japan 5, 382 (1950) 5, 429 (1950).
11. F. E. Geiger, Phys. Rev. 79, 212 (1950).
12. A. Stendel, Zeits. f. Phys. 113, 438 (1952).
13. P. Brix, H. Van-Butlar, F. G. Houtermans und H. Kopfermann, Zeits. f. Phys. 133, 192 (1952).
14. K. Murakawa, J. Phys. Soc., Japan 8, 382 (1953).

53. BISMUTH (Bi, \(Z=83\))

1. M. Fred, F. S. Tomkins, R. F. Barnes, Phys. Rev. 92, 1324 (1953).

54. THORIUM (Th, \(Z=90\))

1. G. L. Stukenbrocker and J. R. McNally, J. Opt. Soc. Am. 43, 36 (1953).

55. URANIUM (U, \(Z=92\))

1. O. E. Anderson and H. E. White, Phys. Rev. 71, 911 (1947).
2. L. E. Burkhart, G. L. Stukenbrocker and S. Adams, Phys. Rev. 1, 83 (1949).

1. J. R. McNally, J. Opt. Soc. Am. 39, 271 (1949).
2. D. D. Smith, G. L. Stukenbrocker and J. R. McNally, Phys. Rev. 84, 383 (1951).

56. Plutonium (Pu, Z=94)

1. J. G. Conway, M. Fred, J. Opt. Soc. Am. 43, 216 (1953).

57. Americium (Am, Z=95)

1. J. G. Conway and R. D. McLaughlin, Phys. Rev. 94, 498 (1954).

58. Miscellaneous Questions

1. E. U. Condon, Phys. Rev. 49, 459 (1936); E. Fermi and L. Marshall, Phys. Rev. 72, 1139 (1947); L. J. Rainwater and W. W. Havens, Phys. Rev. 75, 1295 (1949); L. L. Foldy, Phys. Rev. 78, 693 (1952).
2. Lyman, Hanson and Scott, Phys. Rev. 84, 626 (1951); Hammer, Raka and Pidd, Phys. Rev. 90, 341 (1953); Hofstadter, Fechter and McIntyre, Phys. Rev. 91, 439 (1953).
3. L. N. Cooper and F. M. Henley, Phys. Rev. 91, 480 (1953).
4. G. T. Sborg, Rev. Mod. Phys. 25, 469 (1953).
5. W. Wefelmeir, Ann. d. Phys. 36, 373 (1939).
6. W. Gordy, Phys. Rev. 76, 139 (1949); R. D. Hill, Phys. Rev. 76, 998 (1949); C. H. Townes, H. M. Foley and W. Löw, Phys. Rev. 76, 1415 (1949).
7. K. W. Ford, Phys. Rev. 90, 29 (1953).
8. A. Bohr, Phys. Rev. 81, 134 (1951).
9. A. Bohr and B. R. Mottelson, Phys. Rev. 89, 316 (1953).
10. J. H. P. Hume and M. F. Crawford, Phys. Rev. 78, 343 (1950).
11. Aronberg, Astrophys. Journ. 47, 96 (1918).
12. G. Hansen, Naturwiss. 15, 163 (1927).
13. H. Schüler und E. G. Jones, Naturwiss. 20, 171 (1932).