METHODS AND RESULTS OF ISOTOPE RESEARCH

I. Mattauch

III. Spectroscopic Methods *

Isotopes of one and the same element have the same nuclear charge, and therefore also the same number of outer electrons. They differ from one another in the number of elements of nuclear structure and in their arrangement in the nucleus. This gives rise, first, to differences in the masses of the nuclei, second, to differences in the nuclear spin, and third, to small differences in the field of the nucleus. For methods based on the analysis of canal rays, only the first of these differences is of significance.

Thanks to the extremely high accuracy of spectroscopic measurements, it has now been possible, for many elements, to establish and measure the difference in the spectra of isotopes (isotopic shift).

For explaining the differences in the spectra of isotopes, all three of the above-mentioned differences are essential, although, as a rule, the influence of the difference in nuclear masses predominates. Since in band spectra the isotope effect is in general much more sharply expressed and, moreover, is more fully clarified theoretically than is the case for line spectra, we shall begin with an exposition of the question of isotopic shift in band spectra.

1. Shifts in Band Spectra Caused by Isotopy

a) General Remarks (Diatomic Molecules)

Since molecular lines owe their origin not only to electronic transitions but also to changes in the vibrational and rotational state of the nuclei, it is obvious that the difference in nuclear mass among isotopes must show itself much more sharply in band spectra than in line spectra.

The theory of shifts caused by isotopy was developed almost simultaneously for infrared and rotational-vibrational bands by Loomis^93, Kratzer^94, and Haas^95, and for visible and ultraviolet bands by R. S. Mulliken. The details of the theory were developed by Gibson^97, Beardsley^98, and Patkowski and Curtis^99.

* Continuation. See Uspekhi fizicheskikh nauk, vol. 15, issue 1, 1935.

In addition to Mecke’s articles in Handbuch der Physik, vol. XXI, and Weizel’s in Handbuch der Experimentalphysik, supplementary volume, vol. I, the most recent, very detailed review of the question is contained in Jevons’ Report on Band-Spectra of Diatomic Molecules. The following exposition is based mainly on this monograph.

The energy \(E\) of a diatomic molecule is, as a rule, additively composed of three parts:

\[ E = E_e + E_v + E_r . \tag{23} \]

The electronic energy \(E_e\) is the energy which the molecule would have if the nuclei were fixed in their equilibrium positions; it consists of the kinetic and potential energy of the outer electrons and the potential energy of the nuclei. Just as in the case of line spectra, it is determined by a certain group of quantum numbers. The vibrational energy \(E_v\) is the additional energy which the molecule would have if, without rotating, it underwent vibrations along the line connecting the two nuclei. This energy is determined by one quantum number \(v\) and increases with increasing \(v\). At \(v=0\) it is different from zero. The rotational energy \(E_r\) is the energy present when the nuclei, in addition to other motions, rotate about the common center of gravity. Basically it is determined by the rotational quantum number \(J\); therefore the terms \(T\left(=\dfrac{E}{hc}\right)\) are given by the expression

\[ T = T_e + G(v) + F(v,J), \tag{24} \]

and the wave numbers of the lines of a band spectrum, as differences between terms, by the expression

\[ \nu = T' - T'' = (T'_e - T''_e) + (G' - G'') + (F' - F'') = \]
\[ = \nu_e + \nu_v + \nu_r, \tag{25} \]

where, as a rule, \(\nu_e \gg \nu_v \gg \nu_r\). The prime sign denotes, as usual, the upper state, and the double prime the lower state.

All lines corresponding to transitions between the same electronic and vibrational terms, but different rotational terms, form one band. For a given band, \(\nu_e\), \(\nu_v\), \(v'\), and \(v''\) are constant, while \(J'\), \(J''\), and \(\nu_r\) vary from line to line. The place on the band for which \(\nu_r=0\) is denoted as the zero position. It is determined by the equality \(\nu_0=\nu_e+\nu_v\). As a general rule, the zero position is not occupied by any spectral line.

All bands corresponding to transitions between the same electronic terms, but different vibrational and rotational terms, form a system of bands. This system is equivalent to a single line of a line spectrum. For a given system, \(v'\), \(v''\), and \(\nu_v\) vary from band to band, while \(\nu_e\) remains constant. The place in the system for which \(\nu_v=\nu_r=0\) is called the principal line. As a rule, this place is not occu-

is not any spectral line and does not coincide with any of the zeros belonging to the band system. However, the origin is important in the study of shifts caused by isotopy. Band systems corresponding to transitions between different electronic terms are combined, by analogy with line series, into a series of systems.

If the molecule only rotates, then according to wave mechanics the terms have the form \(F(J)=B\cdot J(J+1)\), where \(B=\dfrac{h}{8\pi^{2}cI}\) and the moment of inertia \(I=\mu r^{2}\). The reduced mass \(\mu\) is related to the masses of both partners forming the molecule by the following equality: \(\dfrac{1}{\mu}=\dfrac{1}{M_{1}}+\dfrac{1}{M_{2}}\); \(r\) is the invariable distance between the nuclei. If one takes into account the stretching caused by the centrifugal force, then:

\[ F(J)=B_{e}J(J+1)+D_{e}J^{2}(J+1)^{2} +\text{[terms containing higher powers of }J(J+1)\text{]}; \tag{26} \]

\(B_{e}\) has the same value as before, but it contains \(I_{e}=\mu r_{e}^{2}\), where \(r_{e}\) denotes the distance between the nuclei in the case when they are in the position of equilibrium, i.e. when the molecule can be regarded as a rigid rotor, with moment of inertia \(I_{e}\); \(D_{e}\) is a small negative constant, which can be expressed in terms of \(I_{e}\) and the coefficients of the expansion in a series of the potential energy of both nuclei. The coefficients of the higher-order terms are still smaller. As in simple periodic motion, there must also be \(J'-J''=\Delta J=1\). The differences of two terms then give, if we write \(J\) instead of \(J''\):

\[ \nu=\nu_{r}=2B(J+1)+4D(J+1)^{3}+\cdots, \tag{27} \]

i.e. on the \(\nu\) scale one obtains a series of lines situated at almost equal distances from one another. Since \(\nu_{e}=\nu_{v}=0\), the origin line obviously corresponds to a frequency equal to zero. Moreover, since the rotational energies and their differences are extremely small, these purely rotational bands lie in the far infrared region. They were detected in the region from \(\lambda=100\) to \(\lambda=200\,\mu\) for hydrogen halide molecules; however, owing to very great experimental difficulties, the isotopic effect could not be established.

If, in addition, the nuclei oscillated about the equilibrium position along the line joining their centers, then for a harmonic oscillator according to wave mechanics we would have:
\(G(v)=\omega_{e}\left(v+\dfrac{1}{2}\right)\), where \(\omega_{e}\) is proportional to \(\dfrac{1}{I_{e}}\) and denotes the frequency of vibration of the nuclei. Since molecules are anharmonic oscillators, then

\[ \left. \begin{aligned} G(v)&=\omega_{e}\left(v+\frac{1}{2}\right)-x_{e}\omega_{e}\left(v+\frac{1}{2}\right)^{2}\\ &\quad +y_{e}\omega_{e}\left(v+\frac{1}{2}\right)^{3}+\cdots \end{aligned} \right\} \tag{28} \]

where \(x_e, y_e\) are small constants: \(1 \gg x_e \gg y_e \gg \ldots\), \(G(v)\) has the smallest value at \(v=0\), but does not vanish completely. For a harmonic oscillator the distance between two neighboring terms would be constant and equal to \(\omega_e\). In the case of an anharmonic oscillator one may, by analogy, designate the mean interval between two levels \((v+1)\) and \((v-1)\), which in this case depends on \(v\), by \(\omega_v\). Thus

\[ \omega_v=\frac{1}{2}\,[G(v+1)-G(v-1)] = \]

\[ = \omega_e - 2x_e\omega_e\left(v+\frac{1}{2}\right) +3y_e\omega_e\left(v+\frac{1}{2}\right)^2+\ldots \tag{29} \]

This quantity may also be regarded as the distance between two imaginary levels \(\left(v+\frac{1}{2}\right)\) and \(\left(v-\frac{1}{2}\right)\), or simply as \(dG(v)/dv\). \(\omega_v\) decreases with increasing \(v\), since \(x_e\) is positive and \(\gg y_e\).

The change of the rotational term which is obtained owing to the simultaneous vibration of the nuclei may be taken into account by replacing the coefficients \(B_e, D_e\) by \(B_v, D_v\), depending on \(v\). Then we obtain

\[ F(v,J)=B_vJ(J+1)+D_vJ^2(J+1)^2+\ldots, \tag{30} \]

where \(B_v=B_e-\alpha\left(v+\frac{1}{2}\right)+\ldots\), and \(D_v=D_e+\beta\left(v+\frac{1}{2}\right)+\ldots\). \(\alpha\) and \(\beta\) are positive constants which are much smaller than \(B_e\), or \(D_e\). Here \(B_v=\dfrac{h}{8\pi^2 c I_v}\), where \(I_v\) represents the effective moment of inertia of the molecule in the vibrational state, which is determined by the value of \(v\). Since \(G(0)\ne0\), strictly speaking, even for a purely rotational spectrum in equality (27) one should also write the coefficients with the index \(v\).

Further, if \(\nu_e=0\), but \(\nu_v\ne0\), then emission (or absorption) of a rotational-vibrational band will take place. Since the vibrations are anharmonic, \(\Delta v\) may take the values 1, 2, 3, but, as a rule, not larger. The bands therefore lie not in the visible, but in the near infrared region. Since \(\nu=\nu_v+\nu_r\) and \(\nu_v\gg\nu_r\), \(\nu\) remains positive also for transitions \(\Delta J=-1\). We consequently now obtain two branches—the positive, or \(R\)-branch \((\Delta J=+1)\), and the negative, or \(P\)-branch \((\Delta J=-1)\). We shall obtain, if we again write \(J\) instead of \(J''\):

\[ \nu=\nu_v+\nu_r=G(v')-G(v'')+F(v',J\pm1)-F(v'',J). \tag{31} \]

For constructing the term differences it should be taken into account that the differences of the coefficients \(B\) and \(D\) are now no longer equal to zero, but \(B_{v'}-B_{v''}=-\alpha(v'-v'')\) and \(D_{v'}-D_{v''}=\beta(v'-v'')\), where \(\alpha\) and \(\beta\) are constants, since the electronic energy does not change. Since \(\beta \ll \alpha\), for \(\nu\) as a function of \(J\) two curves are obtained, similar to parabolas (Fig. 28), which, however, owing to the fact that \(\alpha \ll B_{v'}+B_{v''}\), almost degenerate into straight lines. The line \(P(0)\), co-

which would have to coincide with the zero position, is absent, since for it one must have \(J'=-1\). The intensity of the lines, represented in the figure by the length of the strokes, will in both branches be governed by a law determined by the Boltzmann factor. Such bands have been observed for HF, HCl, HBr, CO, and NO in the wavelength region from 1 to 5 \(\mu\). In this case they were observed as absorption bands, for which (at room temperature) for the most part \(v''=0\). The band having the indices \((1,0)\) is called the fundamental band, and the bands \((2,0)\), \((3,0)\), etc., respectively the first, second, etc., since their zero positions lie near \(\omega_e, 2\omega_e, 3\omega_e\).

The preceding considerations are fully rigorous only in the case when the influence of the electronic energy on the rotational terms may be neglected, or when the resultant moment, composed of the orbital moments of the electrons and their spin moments, is equal to zero for the molecule as a whole (i.e., in the case when one is dealing with a \({}^{1}\Sigma\)-term). If, besides jumps in \(v\) and \(J\), there also occurs a jump in the electronic shell (the simultaneity of these three jumps, at least for nonpolar molecules, is required by the correspondence principle), then \(\nu_e\) also \(\ne 0\) and the line passes into the visible or ultraviolet region. The simplest situation is in the case of the transition \({}^{1}\Sigma \rightarrow {}^{1}\Sigma\). The wave number of the line will then be equal to

\[ \nu=\nu_e+\nu_v+\nu_r=\nu_0+F'(v', J\pm 1)-F''(v'', J). \tag{32} \]

In this connection it should be borne in mind that each electronic term has its own system of constants \(B_e, \alpha, I_e, r_e, D_e, \beta,\ldots\). In order to mark the difference between the constants corresponding to the two terms, they should be supplied with the signs \('\) and \(''\). The same signs should also be supplied to the designations of the terms \(T_e, G\), and \(F\). For brevity, instead of \(B'_{v'}\) and \(B''_{v''}\), we shall also write \(B'\), \(B''\ldots\).

Despite the similarity of equations (31) and (32), the bands determined by them differ greatly in appearance. Since \((B'-B'')\) may in the second (more general) case assume values many times larger, the deviation of the curves \(J/\nu\) from straightness (parabolic character) will in this case be manifested much more strongly (see Fig. 27). Even at small \(J\) the vertex of the parabola may be reached and the band forms a “head.” Further, \((B'-B'')\) may assume both positive \((I'<I'')\) and negative \((I'>I'')\) values. Therefore the bands are shaded both toward the red and toward the violet sides. Finally, since in electronic bands the transition \(\Delta J=0\) is often encountered, there must also exist a third branch (the zero branch, or \(Q\)-branch). In rotational-vibrational bands this branch, owing to the smallness of \(F'-F''\), would have to be wholly drawn together toward the zero position.

If the molecule has a rotational moment arising from the orbital moments or spins of the electrons, then this moment must, together with the rotational moment of the nuclei, give a resultant

the angular momentum, which is determined by the quantum number \(J\). Depending on the strength of the interaction, one must distinguish different types of coupling (Kopplung) of moments, of which the most important are the cases \(a\) and \(b\) considered by Hund.

The coupling of moments leads to the fact that in place of \(\sqrt{J(J+1)}\) in the equation determining the rotational term there appears the number \(N\), which determines the angular momentum of the nuclei. \(N\) is no longer a quantum number. In the general case \(N\) is composed vectorially (in different ways, depending on the type of coupling) from the quantum numbers \(L\) and \(S\) for the resultant orbital and spin moments, \(\Lambda\) and \(\Sigma\) for the components of \(L\) and \(S\) parallel to the direction of the line joining the nuclei, \(\Omega=|\Lambda+\Sigma|\)—the projection of the resultant electronic moment on the line joining the nuclei, the quantum number \(K\), which determines the resultant moment of the orbital motion of the electrons and the rotation of the nuclei \(\sqrt{K(K+1)}\), and the quantum number \(J\), which determines the total angular momentum

\[ \sqrt{J(J+1)}, \]

which is composed of the angular momentum of the nuclei and of the orbital and spin moments of the electrons.

In the cases \(a\) and \(b\), analyzed by Hund, we have:

\[ \left. \begin{aligned} N^2 &= J(J+1)-\Omega^2+S(S+1)-\Sigma^2 \\ \text{or, respectively}\qquad N^2 &= K(K+1)-\Lambda^2 . \end{aligned} \right\} \tag{33} \]

The coupling of moments causes the splitting of a line into multiplets, so that in the general case several \(P\)-, \(G\)-, and \(R\)-branches with common or separate zero positions must appear. Electronic terms are denoted by the symbols \(\Sigma,\Pi,\Delta,\Phi\) depending on the value of \(\Lambda(=0,1,2,3,\ldots)\). To account for the remaining quantum numbers, each term is supplied with an index denoting the multiplicity of the term \((2S+1)\), placed at the upper left, and an index placed at the lower right, which indicates the value of \(\Omega\).

Thus, for example, \({}^{2}\Pi_{1/2}\) denotes a doublet \(\Pi\)-term \(\left(\Lambda=1,\ S=\frac12\right)\), for which \(\Omega=\frac12\) and, consequently, \(\Sigma=-\frac12\).

In this case also, the course of the line intensities within a band is determined by Boltzmann’s distribution law (Fig. 27). However, some lines may be strengthened owing to the coincidence of lines of different branches. According to wave mechanics, in molecules with identical atoms, such as for example \({}^{16}\mathrm{O}\,{}^{16}\mathrm{O}\) or \({}^{35}\mathrm{Cl}\,{}^{35}\mathrm{Cl}\), peculiar anomalies must exist in the course of line intensities, called “intensity alternation.” If the nuclear spin is zero, then every second line is weakened. From the alternation of intensities one can infer the magnitude of the nuclear spin.

When calculating \(\nu_\upsilon\) or \(\nu_0\) in equation (32) and when distributing the bands into a system, it should be borne in mind that each electronic term has its own function \(G(\upsilon)\), with coefficients \(\omega_e, x_e, y_e,\ldots\), which therefore, in order to distinguish the upper and lower terms, must be supplied with primes \('\) and \(''\). For the zero point

\[ \nu_0=\nu_e+\nu_\upsilon=\nu_e+G'(\upsilon')-G''(\upsilon''). \tag{32a} \]

Since the vibrations are anharmonic, \(\upsilon'-\upsilon''=\Delta\upsilon\) can take all integral values; moreover, in contrast to rotational-vibrational bands, both positive and negative values are possible here. (For even for negative \(\Delta\upsilon\) the frequency \(\nu\) remains positive owing to the fact that \(\nu_e\gg\nu_\upsilon\).) Bands with the same final vibrational state \(\upsilon''\) (or, correspondingly, with the same initial state \(\upsilon'\)) form a series of bands. Bands with the same \(\Delta\upsilon\) form a group of bands (Fig. 24).

Between the molecular constants there exist theoretical relations by means of which all these constants can be expressed in terms of the two most important ones, \(\omega_e\) and \(B_e\). For the constants \(x_e, \alpha\), and \(D_e\), Kratzer found the expressions

\[ x_e=\frac{3B_e}{\omega_e},\quad \alpha=\frac{6B_e^2}{\omega_e},\quad D_e=-\frac{4B_e^3}{\omega_e^2}. \tag{34a} \]

Berdge calculated the coefficients \(F_e\) and \(H_e\) of the next two terms of equations (26) and (30), containing \(J^3(J+1)^3\) and \(J^4(J+1)^4\):

\[ \left. \begin{aligned} F_e&=\frac{2D_e}{B_e}-\frac{\alpha\omega_e D_e^2}{6B_e^3}\\ H_e&=\frac{3D_eF_e}{B_e}-\frac{5D_e^3}{B_e}+\frac{F_e^2}{D_e}-\frac{8D_e^2x_e}{3\omega_e}. \end{aligned} \right\} \tag{34b} \]

Berdge and Gayman found for \(\beta\):

\[ \frac{\beta}{D_e}=\frac{\omega_e}{24B_e}\cdot\left(\frac{\alpha}{B_e}\right)^2+5\left(\frac{\alpha}{B_e}\right)-8x_e. \tag{34c} \]

Jevons, on the basis of considerations following from the correspondence principle, deduced that \(y_e, \omega_e\), etc. in equation (28) must be proportional to \(I_e^{3/2}, I_e^2\), etc.

Let us pass to a consideration of the isotope effect. To simplify somewhat the form of the equations, put \(u=(\upsilon+\tfrac{1}{2})\). Then the vibrational and rotational terms will take the form

\[ G(\upsilon)=\omega_eu-x_e\omega_eu^2+y_e\omega_eu^3+\cdots \tag{35} \]

\[ F(J)=(B_e-\alpha u)N^2+(D_e+\beta u)N^4+F_eN^6+H_eN^8+\cdots \tag{36} \]

Let us now consider an isotopic pair of diatomic molecules \(MM'\) and \(M^iM'\). The atom with mass \(M'\) enters into both molecules, while the atoms with masses \(M\) and \(M^i\) belong to two different isotopes of one and

of the same element. Let, for example, \(M\) correspond to the more abundant isotope, and \(M^i\) to the less abundant isotope. The reduced masses of both molecules will be

\[ \mu=\frac{MM'}{(M+M')} \quad \text{and} \quad \mu^i=\frac{M^iM'}{(M^i+M')}. \tag{37} \]

The dependence of the coefficients of equations (35) and (36) on the moment of inertia \((I_e)\), i.e., on the reduced mass \((\mu=I_e/r_e^2)\), is obtained for \(\omega_e\) and \(B_e\) from the definition of these constants, and for the remaining ones from equations (34). Since \(r_e\) is determined by the charge of the nucleus and therefore should be considered identical for both molecules, the molecules \(MM'\) and \(M^iM'\) differ only in the magnitude of the reduced mass \(\mu\).

For convenience let us introduce the notation \(\sqrt{\frac{\mu}{\mu^i}}=\rho\). The ratios of the constants of the two molecules for a given electronic term will be, as is easy to verify, simple powers of \(\rho\). For example, since \(B_e\) varies proportionally to \(\mu^{-1}\), then

\[ \frac{B_e^i}{B_e}=\frac{\mu}{\mu^i}=\rho^2. \]

The results for the remaining coefficients may be arranged as follows:

\[ \left. \begin{array}{llllllllll} \text{coefficient:} & \omega_e, & x_e\omega_e, & y_e\omega_e\ldots & B_e, & \alpha, & D_e, & \beta, & F_e, & H_e \\[2mm] \text{proportional to } & \mu^{-\frac12}, & \mu^{-1}, & \mu^{-\frac32}, \ldots & \mu^{-1}, & \mu^{-\frac32}, & \mu^{-2}, & \mu^{-\frac52}, & \mu^{-3}, & \mu^{-4} \\[2mm] \dfrac{\text{coeff.}^i}{\text{coeff.}} \text{ proportional to } \rho, & \rho^2, & \rho^3, \ldots & \rho^2, & \rho^3, & \rho^4, & \rho^5, & \rho^6, & \rho^8 \end{array} \right\} \tag{38} \]

From the equalities (37) and the definition there follows the relation

\[ 2(\rho-1)\simeq \rho^2-1=\frac{M'(M-M^i)}{M^i(M+M')}. \tag{39} \]

Therefore \(\rho\) will be greater than 1 when \(M>M^i\) and less than 1 when \(M<M^i\).

a) The isotope effect in electronic terms. If this effect can be observed at all, then, just as in linear spectra, it must be very small. Only if one takes into account the “correction for the proper motion of the nucleus” can a weak splitting of terms be expected. Theoretically this question has not yet been considered. Jenkins and McKellar\(^{142}\) state that they succeeded in detecting a splitting of electronic terms for \({}^{10}\mathrm{B}^{16}\mathrm{O}\) and \({}^{11}\mathrm{B}^{16}\mathrm{O}\) from the displacement of the band origin. In most cases, however, this displacement can be completely neglected. Only for the isotopes of H does it play a known role, owing to the large relative mass difference. Recently Johnston and Dawson\(^{133}\) reported that they had discovered in the OH spectrum an effect of another type. Jeppson\(^{134c}\) found that changes in 12 bands, corresponding to the transitions \({}^{1}\Pi \to {}^{1}\Sigma\), in the spectrum of \({}^{1}\mathrm{H}{}^{2}\mathrm{H}\) reveal

effect of isotopy in electronic terms, the splitting being equal to \(\Delta \nu = 135\ \text{cm}^{-1}\). Jeppson indicates that in the line spectrum the elementary (Bohr) effect would be equal to \(9\ \text{cm}^{-1}\) if the nuclear masses were related as the masses \({}^{1}\mathrm{H}{}^{1}\mathrm{H}\) and \({}^{1}\mathrm{H}{}^{2}\mathrm{H}\). The Urey–Eckart effect gives for \({}^{6}\mathrm{Li}\) and \({}^{7}\mathrm{Li}\) a value 3.6 times larger, consequently \(42\ \text{cm}^{-1}\), i.e., a quantity of the same order.

Beitler and Mi\({}^{134d}\), however, find that for the same electronic transition \(({}^{1}\Pi \to {}^{1}\Sigma)\), \(\Delta \nu\) is equal to only \(30\ \text{cm}^{-1}\). They obtained the same value also in measurements of the lines corresponding to the transition \(({}^{1}\Sigma \to {}^{1}\Sigma)\).

\(\beta)\) The effect of isotopy of vibrational terms. From equation (35) and the first three equalities (38), it follows for the vibrational term of the rarer molecule that

\[ G^{i}(v)=\rho\omega_{e}u-\rho^{2}x_{e}\omega_{e}u^{2}+\rho^{3}y_{e}\omega_{e}u^{3}+\cdots \]

For the displacement of terms caused by isotopy, the completely rigorous equality will hold:

\[ \left. \begin{aligned} G^{i}(v)-G(v) &= (\rho-1)\omega_{e}u-(\rho^{2}-1)x_{e}\omega_{e}u^{2} \\ &\quad +(\rho^{3}-1)y_{e}\omega_{e}u^{3}+\cdots \end{aligned} \right\} \tag{39a} \]

If one replaces \((\rho^{2}-1)\), \((\rho^{3}-1)\) by \(2(\rho-1)\), \(3(\rho-1)\), then approximately:

\[ \left. \begin{aligned} G^{i}(v)-G(v) &= (\rho-1)u\left(\omega_{e}-2x_{e}\omega_{e}u+3y_{e}\omega_{e}u^{2}+\cdots\right) \end{aligned} \right\} \tag{39b} \]

and, according to equation (29),

\[ G^{i}(v)-G(v)=(\rho-1)\left(v+\frac{1}{2}\right)\omega_{v}. \tag{39c} \]

This equation, in addition to containing a finite number of terms, has the further advantage that in it the displacement is expressed in such a way that, for its determination, it is not necessary to know \(G(v)\) in advance, since the quantity \(\omega_{v}\) (half the interval between two consecutive vibrational terms) is directly accessible to measurement.

The isotopic displacement of the band \((v',v'')\)

\[ \nu_{v}^{i}-\nu_{v}= \left[G^{i'}(v')-G^{i''}(v'')\right] -\left[G'(v')-G''(v'')\right] \]

can be expressed rigorously according to equation (39a) as

\[ \left. \begin{aligned} \nu_{v}^{i}-\nu_{v} &=(\rho-1)(\omega_{e}'u'-\omega_{e}''u'') \\ &\quad -(\rho^{2}-1)(x_{e}'\omega_{e}'u'^{2}-x_{e}''\omega_{e}''u''^{2}) \\ &\quad +(\rho^{3}-1)(\ldots)+\cdots \end{aligned} \right\} \tag{40a} \]

or approximately, according to equation (39b),

\[ \nu_{v}^{i}-\nu_{v} =(\rho-1)\left[\nu_{v}-(x_{e}'\omega_{e}'u'^{2}-x_{e}''\omega_{e}''u''^{2})+\cdots\right] \tag{40b} \]

or, by equation (39c),

\[ \nu_{v}^{i}-\nu_{v} =(\rho-1)\left[\left(v'+\frac{1}{2}\right)\omega_{v}'- \left(v''+\frac{1}{2}\right)\omega_{v}''\right]. \tag{40c} \]

For bands with low values of \(v'\) and \(v''\), the difference of the quadratic terms in equation (40b) may be neglected. We then obtain:

\[ \nu'_v-\nu_v=(\rho-1)\nu_v . \tag{41} \]

In this case the shift caused by isotopy will amount to \(100(\rho-1)\) percent of the wave number corresponding to the more abundant molecule.

For the vibrational isotope effect, the following will take place: the shift for all lines of the given band \((v', v'')\) is the same and therefore has one and the same value both for the band head and for the zero point. The magnitude of the shift increases from band to band together with the distance \(\nu_v\) of the zero point of the band from the origin of the system \(\nu_e\), and depends almost linearly on \(\nu_v\).

Fig. 24

It may assume large values, since the coefficient \((\rho-1)\) in many cases is of the order of 0.01. For the origin of the system the shift should be equal to zero, but it is not equal to zero for the \((0,0)\) band, since here \(u'=u''=\frac{1}{2}\). These relations are illustrated by Fig. 24 (after Mulliken and Jevons), which shows the vibrational isotope effect in two systems of bands. In case (a) the groups of bands are clearly separated from one another (\(\omega'_e\) and \(\omega''_e\) differ very little from each other), and \((\rho-1)\) is negative, i.e., the more abundant of the two molecules is at the same time the lighter one, as, for example, in the case of \({}^{63}\mathrm{CuJ}\), \({}^{65}\mathrm{CuJ}\). In case (b) the groups of bands interlock with one another (\(\omega'_e\) differs strongly from \(\omega''_e\)); \((\rho-1)\) is considerably greater than in case (a) and has a positive sign, i.e., the more abundant molecule is the heavier one, as, for example, in the case of \({}^{11}\mathrm{BO}\), \({}^{10}\mathrm{BO}\). Each line represents one band; the long one is for the more abundant and the short one for the less abundant molecule. Oblique strokes indicate the direction of shading (in both cases toward the red-

the direction that corresponds to the case when \(\omega'_e < \omega''_e\). The dotted lines connect bands with identical \(v'\), \(v''\), the distance between which determines the isotopic shift \(\nu^{i}_{v'}-\nu_{v'}\). The signs of the shift on the two sides of the origin are different. The band of the lighter molecule always lies farther from \(\nu_e\) than the corresponding band of the heavier molecule; this means that the mutually corresponding systems of bands of two (or many) isotopic molecules are constructed in exactly the same way and differ only in scale. For the lighter molecule the scale is larger, approximately in the ratio \(1:\rho\) or \(\rho:1\).

The dependence of the magnitude of the shift on the masses can be clarified by using the approximate equality (39), which determines \((\rho-1)\), the most important coefficient in the expression for the shift. First of all, we see that the shift increases almost proportionally to the mass difference of the two isotopes \(M-M^i\). Thus, for example, the shift for the same bands will be almost twice as large for \({}^{30}\mathrm{Si}^{14}\mathrm{N}-{}^{28}\mathrm{Si}^{14}\mathrm{N}\) as for \({}^{29}\mathrm{Si}^{14}\mathrm{N}-{}^{28}\mathrm{Si}^{14}\mathrm{N}\). Secondly, the shift, expressed in percent of \(\nu_v\), increases almost proportionally to the mass \(M'\) of the second atom in the case when \(M' < M\). For example, the shifts (expressed in percent) for \({}^{65}\mathrm{Cu}^{1}\mathrm{H}-{}^{63}\mathrm{Cu}^{1}\mathrm{H}\), \({}^{65}\mathrm{Cu}^{19}\mathrm{F}-{}^{63}\mathrm{Cu}^{19}\mathrm{F}\), \({}^{65}\mathrm{Cu}^{35}\mathrm{Cl}-{}^{63}\mathrm{Cu}^{35}\mathrm{Cl}\) are in the ratio \(1:19:35\) (Fig. 25). Thirdly, the shift, expressed in percent, decreases rapidly with increasing \(M\), if \(M-M^i\) and \(M'\) remain constant; for example, for \({}^{63}\mathrm{Cu}^{37}\mathrm{Cl}-{}^{63}\mathrm{Cu}^{35}\mathrm{Cl}\) it will be much greater than for \({}^{63}\mathrm{Cu}^{81}\mathrm{Br}-{}^{63}\mathrm{Cu}^{79}\mathrm{Br}\).

For the vibrational effect in bands with large values of \(v'\) and \(v''\), especially in a system with large values of \(x'_e\omega'_e\) or \(x''_e\omega''_e\), the approximate equation (41) is no longer sufficient, and the more exact equation (40a) should be used. This applies especially to systems with long band progressions. In this case neglect of the quadratic terms in equality (40b) is no longer permissible, since one quantum number (\(v'\) or \(v''\)) reaches very large values while the other (\(v''\) or \(v'\)) is very small.

The change of the isotopic shift from band to band in a band progression is best expressed by equality (40c).

The curve that is obtained if the shift is plotted as a function of \(v'\) has the form of a parabola (Fig. 26). For bands belonging to a progression with given \(v''\), the shift first increases almost linearly as \(v'\) increases, then passes through a maximum, and subsequently, with increasing distance from the origin, falls again. In all progressions the maximum lies at one and the same value of \(v'\). We further note that the shift may change sign twice within a single band progression.

All these properties of the vibrational isotope effect were first considered by Patkowski and Curtis, \(^{99}\) who also succeeded in detecting them experimentally in the investigation of the absorption bands of JCl.

*

γ) The isotope effect in rotational terms. From equation (36) and the last six equalities (38) we obtain, for the rotational term of the molecule that occurs more rarely:

\[ F^{i}(J)=\rho^{2}B_{e}N^{2}-\rho^{3}\alpha u N^{2}+\rho^{4}D_{e}N^{4}+ \]
\[ +\rho^{5}\alpha uN^{4}+\rho^{6}E_{e}N^{6}+\cdots \]

and for the displacement of the terms:

\[ \left. \begin{aligned} F^{i}(J)-F(J)&=(\rho^{2}-1)B_{e}N^{2}-\\ &\quad-(\rho^{3}-1)\alpha uN^{2}+(\rho^{4}-1)D_{e}N^{4}+\cdots \end{aligned} \right\} \tag{42} \]

The rotational part of the wave number of the line corresponding to the transition \(v', J'\to v'', J''\) can therefore be written as

\[ \nu_{r}+(B'_{e}N'^{2}-B''_{e}N''^{2})-(\alpha'u'N'^{2}-\alpha''u''N''^{2})+ \]
\[ +(D'_{e}N'^{4}-D''_{e}N''^{4})+\cdots, \]

Fig. 25. After Mulliken

Fig. 26. After Patkowski and Curtis

and therefore the isotopic displacement of a line due to the rotational effect will be determined quite rigorously by the expression:

\[ \left. \begin{aligned} \nu^{i}_{r}-\nu_{r}&=(\rho^{2}-1)(B'_{e}N'^{2}-B''_{e}N''^{2})+\\ &\quad+(\rho^{3}-1)(\alpha'u'N'^{2}-\alpha''u''N''^{2})+\\ &\quad+(\rho^{4}-1)(D'_{e}N'^{4}-D''_{e}N''^{4})+\cdots \end{aligned} \right\} \tag{43a} \]

If \((\rho^{3}-1)\), \((\rho^{4}-1)\) are replaced by \(\dfrac{3}{2}(\rho^{2}-1)\), \(2(\rho^{2}-1)\), then approximately one obtains:

\[ \left. \begin{aligned} \nu^{i}_{r}-\nu_{r}&=(\rho^{2}-1)\left[\nu_{r}-\frac{1}{2}(\alpha'u'N'^{2}-\alpha''u''N''^{2})+\right.\\ &\quad\left.+(D'_{e}N'^{4}-D''_{e}N''^{4})+\cdots\right] \end{aligned} \right\} \tag{43b} \]

or, if the small differences in equation (43) are neglected, one obtains:

\[ \nu^{i}_{r}-\nu_{r}=(\rho^{2}-1)\nu_{r}. \]

This equation, which is constructed analogously to equation (41), contains the most important facts relating to the isotope effect in rotational spectra. This effect is illustrated by Fig. 27. The branches \(P, Q, R\) (transition \({}^{1}\Pi \to {}^{1}\Sigma\)) belong to the more common, and \(P^i, Q^i, R^i\) to the rarer, molecules. In the present case the lighter molecule is at the same time the less common one; \((\rho^2 - 1)\) is taken to be equal to \(+0.2\). The crosses in the diagram denote the actually existing spectral lines, while the dotted curves are obtained by extrapolation. The rotational displacement of the lines is approximately proportional to the difference between the wave numbers of the given line \((\nu_r)\) and of the zero point \((\nu_0)\), and vanishes at \(\nu_0\). Consequently, the splitting of a line caused by isotopy is symmetric with respect to the zero point and thereby differs substantially from the entirely analogous splitting of lines caused by the interaction of nuclear rotation and electron motion. The band structure of both isotopic molecules is the same and differs (as also in the vibrational effect) only by scale, which for the lighter molecule is greater than for the heavier one in the ratio

\[ \frac{\rho^2}{1}. \]

Although the coefficient \((\rho^2 - 1)\) in formula (44), which determines the magnitude of the rotational displacement, is approximately twice as large as the coefficient \((\rho - 1)\) in equation (41), nevertheless in general the rotational isotope effect is considerably smaller than the vibrational one, since \(\nu_r \ll \nu_v\). For very extended bands it may still reach several Å. The dependence of the coefficient of the rotational isotope effect on the masses of the atoms \(M, M^i, M'\) is practically the same as that of the coefficient \((\rho - 1)\).

Fig. 27

The rotational effect in the form in which it is depicted in Fig. 27 can be observed only for that band for which the vibrational effect is equal to zero. For any other band the observed displacement is the algebraic sum of a constant (within the band) vibrational effect and a varying rotational one (if one disregards the negligibly small effect of isotopy of the electronic terms). Therefore the parabolas of Fig. 27 must be shifted horizontally by the distance \(\nu_v^i - \nu_v\) (in the proper direction).

The total displacement caused by isotopy may be greater or less than that part of the displacement which is associated only with vibrational terms, and moreover by an amount determined by equations (43) and (44).

b) Measurements of isotopes by the method of band spectra

α) General considerations. The observation of shifts caused by isotopy is important not only because it can confirm the existence of isotopes found with the aid of the mass spectrograph, but also because, in those cases in which it is difficult to verify the existence of a new isotope with the aid of the mass spectrograph—as, for example, in the case of hydride formation—the study of band spectra can give a definite answer, since this method is free from a number of shortcomings characteristic of the mass-spectrographic one.

In the search for rare isotopes by the canal-ray method it may happen that the mass numbers corresponding to the sought isotopes are occupied by molecular lines, which makes it extremely difficult to isolate the true isotopes. Thus, for example, the detection of weak isotopes of C, N, and O is greatly hindered by the presence of the lines C, CH, CH₂, CH₃, CH₄, O, OH, OH₂, and a number of others arising from impurities. In the analysis of band spectra this difficulty is entirely absent. However, the analysis of band spectra has become one of the important means of investigating isotopy not only because it can serve to confirm doubtful isotopes and to find new ones. The high precision of the most recent measurements of the shift caused by isotopy also makes it possible to determine the value of ρ with great accuracy.

For this it is necessary to use the exact equalities (40) or (43), including, if necessary, terms of higher order. However, the measurement gives directly the mass of an isotope only in the case of isotopes of the standard element (for example, when observing shifts in \({}^{17}\mathrm{O}{}^{16}\mathrm{O}\), \({}^{18}\mathrm{O}{}^{16}\mathrm{O}\)). In all other cases the value of ρ makes it possible to calculate only the ratio of the isotope masses, and even that only when observing the band spectrum of a molecule of the type \(M_2\), for if \(M' = M\), then

\[ \frac{M}{M^2} = 2\rho^2 - 1. \]

In the general case, in order to determine the mass of an isotope from data obtained in the study of spectral bands, it is necessary to know both the mass of the principal isotope \(M\) and the mass of the partner entering into the molecule \(M'\). The values \(M\) and \(M'\) can, obviously, be obtained only with the aid of the mass spectrograph. As Mecke and Vurm have indicated¹¹, the possibility of accurate direct measurement of isotope masses by this method depends on how well justified is the assumption of equality of the internuclear distances in both molecules containing different isotopes.

Indeed, the measured shift gives directly the ratio of the two moments of inertia, and the values of the isotope masses only under the condition that the distances are equal. The distance between the nuclei depends primarily on the external field of the nucleus. This field may be non-identical for two isotopes, as shown by attempts

for explaining the displacement caused by isotope effects in the line spectra of elements. It is also possible that the distance between nuclei depends on their masses. However, as Mecke and Wurm^111 indicate, the absence of an isotope effect in electronic terms excludes this possibility.

Further, it is quite clear that, by measuring intensities in band spectra, one can also determine the relative abundance (r. a.) of isotopes. If the r. a. of isotopes \(M\) and \(M^i\) are in the ratio \(a : 1\), then the relative abundance of the molecules \(MM'\) and \(M^iM'\), corresponding to both isotopes, will also be \(a : 1\). For the molecules \(MM\), \(M^iM\), \(M^iM^i\), composed of atoms of the given element, the relative abundances will be:

\[ MM : M^iM : M^iM^i = a^2 : 2a : 1, \]

i.e.

\[ MM : M^iM = \frac{1}{2}a : 1. \]

Let us take Br as an example. Since both isotopes \({}^{79}\mathrm{Br}\) and \({}^{81}\mathrm{Br}\) have the same abundance, in the spectrum of the \(\mathrm{Br}_2\) molecule the intensities of the lines corresponding to \({}^{79}\mathrm{Br}{}^{79}\mathrm{Br}\), \({}^{81}\mathrm{Br}{}^{79}\mathrm{Br}\), and \({}^{81}\mathrm{Br}{}^{81}\mathrm{Br}\) will be in the ratio \(1 : 2 : 1\). This ratio was indeed found by Brown^163. In determining the r. a. of isotopes on the basis of intensity measurements, it is necessary to take into account the “alternation of intensities” and always to compare the sum of the intensities of an even number of successive lines belonging to one branch \(M^iM\) with the sum of the intensities of the corresponding lines \(MM\) or \(M^iM^i\). Strictly speaking, only absorption bands of unexcited molecules* (for which \(v'' = 0\)) can serve for comparison of intensities. In bands with large \(v''\), the intensity ratio gives the r. a. of the molecule with the given value of \(v''\), and this ratio is not exactly equal to the r. a. of the molecule with \(v'' = 0\), since in thermodynamic equilibrium the relative fraction of molecules with smaller \(\omega_e\), i.e. of the heavier ones, will be increased. The correction that must be introduced in order to take this circumstance into account will be rather large, as shown by Stenvinkel^101 and Elliott^102,^103. Thus, for example, Elliott^102 found that the ratio of the intensities of the lines corresponding to the molecules \({}^{35}\mathrm{Cl}{}^{35}\mathrm{Cl}\) and \({}^{35}\mathrm{Cl}{}^{37}\mathrm{Cl}\), calculated as the mean from numerous measurements with the bands \((12,1)\), \((6,2)\), and \((12,2)\), is equal to 1.35. After introducing a correction for the Boltzmann factor, which at \(v'' = 2\) is equal to 9%, the relative abundance of the two molecules proves to be 1.46. If one also takes into account the correction for mutual overlap of lines, this ratio must be increased a little more. The final value of the r. a. will be 1.58. This ratio can also be calculated from the chemical atomic weight of chlorine and the weights of isotopes 35 and 37 measured by Aston. The ratio computed in this way is 1.59. The situation is somewhat more complicated for emission bands. Here it is no longer possible to make the supp—

* According to Dungum^100, even in this case the transition probabilities for different isotopes are not identical. However, this effect is insignificant.

the assumption of thermodynamic equilibrium, and, moreover, in this case even with a constant light source the ratio of intensities changes from band to band. Thus, for example, Elliott\(^{103}\) obtains for the relative abundance of the molecules \(^{11}\mathrm{B}^{16}\mathrm{O}\) and \(^{10}\mathrm{B}^{16}\mathrm{O}\), from measurements of the intensity of the lines of the emission bands (2,6) and (3,7), two sharply different values, 3.50 and 4.37. However, after correction for the excitation function these numbers become equal to 3.68 and 3.61, i.e., good agreement is obtained.

\(\beta)\) Discovery of new isotopes. Analysis of band spectra led to the discovery of the isotopes \(^{30}\mathrm{Si}\), \(^{18}\mathrm{O}\), \(^{17}\mathrm{O}\), \(^{13}\mathrm{C}\), \(^{15}\mathrm{N}\), \((^{8}\mathrm{Be})\), \((^{118}\mathrm{Cd})\), and \((^{108}\mathrm{Cd})\). Mulliken\(^{104}\) detected in the spectrum of SiN the isotope \(^{30}\mathrm{Si}\), which had earlier been assigned by Aston\(^{25}\) to the group of doubtful isotopes and whose existence was later\(^{30}\) confirmed mass-spectrographically. At the same time Mulliken confirmed the existence of \(^{29}\mathrm{Si}\). For the relative abundance of \(^{30}\mathrm{Si}\), Mulliken finds a value somewhat smaller than for \(^{29}\mathrm{Si}\); together they amount to approximately \(1\%\) of \(^{28}\mathrm{Si}\). Recently MacKellar\(^{104a}\) found from intensity measurements in the SiN spectrum that \(^{28}\mathrm{Si} : {}^{29}\mathrm{Si} : {}^{30}\mathrm{Si} = 89.6 : 6.2 : 4.2\) (preliminary data).

According to Dieke and Babcock\(^{105}\), the red atmospheric bands of \(\mathrm{O}_{2}\) consist of a single chain of \(v'\)-bands (according to Mulliken, the transition \(^{1}\Sigma \to {}^{3}\Sigma\)) with \(v'' = 0\), of which (0,0) and (0,1) are usually denoted by the letters \(A\) and \(B\). Near the lines of band \(A\), Dieke and Babcock observed 26 lines of the weak band \(A'\), which apparently has the same structure but cannot be included in the same system. This fact received a brilliant interpretation in the work of Giauque and Johnston\(^{107}\), who showed that the weak satellites of the lines of the \(\mathrm{O}_{2}\) spectrum discovered by Dieke and Babcock belong to the (0,0) band of the molecule \(^{18}\mathrm{O}^{16}\mathrm{O}\). Giauque and Johnston established that the calculated shifts of the lines (the vibrational and rotational effects) coincide completely with the distances between the satellites and the principal lines of band \(A\). Babcock, who after this rechecked the entire experimental material obtained, found another 34 lines. These results were not published because there was no complete certainty that the lines found really belonged to the \(\mathrm{O}_{2}\) spectrum. However, Giauque and Johnston, having the opportunity to make use of Babcock’s new data, showed that of the 34 lines, 27 belong to the rotational spectrum of the molecule \(^{18}\mathrm{O}^{16}\mathrm{O}\). This interpretation is confirmed by the fact that the lines of band \(A'\) do not give the alternation of intensities characteristic of the spectrum of a molecule consisting of atoms with identical nuclei, such as, for example, \(^{16}\mathrm{O}^{16}\mathrm{O}\). Babcock\(^{108}\) confirmed this interpretation by detecting a weak band \(B'\), which is the (1,0) band of the molecule \(^{18}\mathrm{O}^{16}\mathrm{O}\). Further, Giauque and Johnston showed that 19 of the next 22 still weaker lines found by Babcock (band \(A''\)) should be ascribed to the molecule \(^{17}\mathrm{O}^{16}\mathrm{O}\). Babcock\(^{108}\) confirmed this interpretation by measuring a series of other lines; moreover, he was able to estimate the relative abundance of the isotopes \(^{17}\mathrm{O}\)

and $^{18}\mathrm{O}$. It turned out that $^{18}\mathrm{O}:^{16}\mathrm{O}=1:1250$ and $^{17}\mathrm{O}:^{18}\mathrm{O}=1:10^4$. After Bärdzh found and eliminated the error that had crept into the determination of the zero point of band $A$, previously made by Dicke and Babcock, it turned out that the agreement between the calculated and measured positions of the $^{18}\mathrm{O}$ and $^{17}\mathrm{O}$ lines is practically complete.

Hode$^{118}$ found that the structure of the NO absorption spectrum can be explained only if it is assumed that, in addition to $^{14}\mathrm{N}^{16}\mathrm{O}$, there also exist $^{14}\mathrm{N}^{18}\mathrm{O}$ and $^{14}\mathrm{N}^{17}\mathrm{O}$. He found the following values for the relative abundances: $^{16}\mathrm{O}:^{18}\mathrm{O}:^{17}\mathrm{O}=1075\pm100:1:0.12$. For the mass of $^{18}\mathrm{O}$ an exact value was very soon found by a precision measurement of $\rho$. The first determination of the mass was made by Mekke and Burm$^{111}$, who used for this purpose the rotational isotope effect in the $(0,0)$ band. However, in Vandziels’s opinion they obtained an erroneous value. After Babcock and Hoh had measured another series of lines of the $(0,0)$ band corresponding to $^{17}\mathrm{O}^{16}\mathrm{O}$ and the lines of the bands $(0,0)$, $(1,0)$ and $(2,0)$ belonging to the molecule $^{18}\mathrm{O}^{16}\mathrm{O}$, Babcock and Bärdzh$^{112}$ calculated, using the method developed by Bärdzh, a precise value for $^{18}\mathrm{O}$ ($18.0026$ with an error not exceeding $10^{-5}$). This value was obtained from the vibrational isotope effect and agrees very well with the much less exact value found from the rotational effect. For the mass of $^{17}\mathrm{O}$ there are as yet no spectroscopic determinations. Dzhyok$^{220}$ obtained for it a precise value ($17.0029$), using data from experiments with the disintegration of atoms carried out by Kirsch. The method used by Dzhyok was subsequently employed by Chadwick, Constable, and Pollard$^{221}$ for determining the masses of a number of other isotopes. However, since these values were obtained under certain assumptions concerning the structure of the nuclei under investigation, we shall not cite them here.

The relative abundances of the oxygen isotopes were found by Mekke and Childs$^{113}$ by means of especially careful measurements and proved to be $^{16}\mathrm{O}:^{18}\mathrm{O}:^{17}\mathrm{O}=630\pm20:1:0.2$. The intensities of the lines, measured by the zero method, after allowance for anomalies in the intensity progression in the symmetrical molecule $^{16}\mathrm{O}^{16}\mathrm{O}$, directly give the ratio in which the isotopes are mixed with one another. In doing this, it is, of course, necessary to take into account the correction for the law of intensity distribution, first applied by Elliott in his measurements of the spectra of $\mathrm{Cl}_2$ and $\mathrm{BO}$. To determine this correction, Mekke and Childs measured, with an accuracy of up to $1$–$2\%$, the intensity distribution in the atmospheric bands and took into account the influence of temperature.

The unexpected fact of the high abundance of $^{18}\mathrm{O}$, discovered by Babcock and Hode, must lead to a large difference between the standards used in chemistry ($\mathrm{O}=16$) and in mass spectroscopy ($^{16}\mathrm{O}=16$). Atomic weights found with the aid of the mass spectrograph must still be multiplied by $0.99978$ in order to compare them with atomic weights measured chemi-

chemical methods. According to new mass-spectrographic measurements by Smythe\({}^{83a}\), the relative abundance of the isotope \({}^{18}\mathrm{O}\) is still higher and amounts to \(1/500\); therefore the conversion factor is equal to 0.99973.

King and Birge\({}^{114}\) showed that, in the spectrum of the \(\mathrm{C}_2\) molecule, the strongest bands \((1,0)\) and \((2,0)\), corresponding to the transitions \(({}^{3}\Pi \rightarrow {}^{3}\Pi)\), have weak satellites, whose lines are displaced from the corresponding lines of the main band of \({}^{12}\mathrm{C}\,{}^{12}\mathrm{C}\) by distances equal to the isotopic shifts calculated for the molecule \({}^{13}\mathrm{C}\,{}^{12}\mathrm{C}\). The presence of \({}^{13}\mathrm{C}\,{}^{12}\mathrm{C}\) was also established by Birge\({}^{115}\) in the absorption spectrum of CO (from Hopfield’s observations) and by King and Birge\({}^{116}\) in the “violet” bands of CN. Since for \({}^{12}\mathrm{C}\) the nuclear spin is zero, in the symmetric molecule \({}^{12}\mathrm{C}\,{}^{12}\mathrm{C}\) an alternation of lines is observed in the main band, whereas \({}^{13}\mathrm{C}\,{}^{12}\mathrm{C}\) gives no “alternation.” Measurement of the relative abundance in this case is very difficult, since sharply different values are obtained at different temperatures of the source radiation. King and Birge give the approximate value \({}^{13}\mathrm{C}:{}^{12}\mathrm{C}=1:400\). The mass ratio of \({}^{13}\mathrm{C}\) to \({}^{12}\mathrm{C}\), according to these authors, is \(13:12\) with an accuracy up to \(10^{-4}\). The relative mass defect of \({}^{13}\mathrm{C}\) must therefore be equal to \(+3.0 \pm \sqrt{0.3^2+\left(\frac{13}{12}\right)^2}=+3.0\pm1.1\) (if for \({}^{12}\mathrm{C}\) one takes Aston’s value of the mass). For the mass we find \({}^{13}\mathrm{C}=13.0039\). The relative abundance, according to the new data of Jenkins and Ornstein, is approximately \({}^{13}\mathrm{C}:{}^{12}\mathrm{C}=1:106\). This value agrees well with the mass-spectroscopic measurements of Tate, Smythe and Vaughn\({}^{59a}\), Vaughn, Williams and Tate\({}^{59b}\), and with the chemical weight, for which Wudhard and Whitelaw-Gray\({}^{228}\) found the value 12.011, but is incompatible with the international atomic weight of C.

Nodé\({}^{118}\) found in the already mentioned bands of NO clear indications of the presence of the molecule \({}^{15}\mathrm{N}\,{}^{16}\mathrm{O}\). The existence of \({}^{15}\mathrm{N}\) was firmly established by Herzberg\({}^{119}\) from measurements in the spectrum of \(\mathrm{N}_2\), on lines of bands belonging to the transition \({}^{3}\Pi \rightarrow {}^{3}\Pi\). The best spectroscopic determination of this isotope belongs to Murphy and Urey\({}^{110}\), who, from measurements of intensities in the NO spectrum, found: \({}^{15}\mathrm{N}:{}^{14}\mathrm{N}=1:346\). This ratio was obtained on the assumption that the value \({}^{18}\mathrm{O}:{}^{16}\mathrm{O}\) given by Mecke and Childs is correct. The relative abundance of the isotope \({}^{15}\mathrm{N}\), found by Murphy and Urey, differs hardly at all from the value obtained by Birge and Menzel\({}^{227}\) from Aston’s value for the mass of \({}^{14}\mathrm{N}\) and the chemical atomic weight of N, namely \(1:320\). With the aid of a mass spectrograph, for the relative abundance one obtains the value \(1:265\pm8\). For the mass of \({}^{15}\mathrm{N}\), Birge\({}^{121}\) finds from Herzberg’s measurements the value 15.0027, whence it follows that the relative mass defect is equal to \(+2.0\).

Watson and Parker found in the \((0,0)\) band of the \(({}^{2}\Pi \rightarrow {}^{2}\Sigma)\) system of the BeH molecule weak lines which correspond to the isotopic shift for \({}^{8}\mathrm{Be}\,{}^{1}\mathrm{H}\). According to the authors’ estimate, the relative abundance \({}^{8}\mathrm{Be}:{}^{9}\mathrm{Be}\) is \(1:2000\).

However, Olson, who investigated the same spectrum, doubts

in the existence of $^{8}\mathrm{Be}$ and interprets part of the corresponding lines as the result of interference.

According to a preliminary communication by Svensson$^{124}$, in the spectrum of CdH it is possible, from the isotope shift, not only to confirm the presence of the known isotopes $^{114}\mathrm{Cd}$, $^{112}\mathrm{Cd}$, $^{110}\mathrm{Cd}$ and $^{116}\mathrm{Cd}$, but also to detect new isotopes $^{118}\mathrm{Cd}$ and $^{108}\mathrm{Cd}$. We have as yet no confirmation of this.

$\gamma$) Measurement of known isotopes. The shift caused by isotopy in rotational-vibrational spectra was first discovered by Loomis$^{93}$ and Kratzer in the analysis of Aymès’ observations$^{125}$ relating to the absorption band $(2,0)$ of the HCl molecule. Loomis and Kratzer succeeded in proving that both isotopes $^{35}\mathrm{Cl}$ and $^{37}\mathrm{Cl}$ are present in the HCl spectrum. The components $^{1}\mathrm{H}^{35}\mathrm{Cl}$ and $^{1}\mathrm{H}^{37}\mathrm{Cl}$ (Fig. 28) were subsequently completely separated

Fig. 28. After Jevons

by Meyer and Levin$^{126}$ not only in the $(2,0)$ band at $1.75$, but also in the fundamental band $(1,0)$ at $\lambda = 3.46\ \mu$. Becker$^{127}$, having carefully measured the line $P(3)$ of the $(2,0)$ band, suggested the existence of $^{39}\mathrm{Cl}$ and pointed out that the presence of this isotope is also confirmed by the data of Meyer and Levin$^{126}$. Gettner and Bome$^{128}$, who photographed the $(2,0)$ bands at higher resolving power, found, near each doublet of the third satellite corresponding to $^{39}\mathrm{Cl}$, and for some lines—indications of the presence of the isotope $^{40}\mathrm{Cl}$. However, Hardy and Sutherland$^{129}$, who photographed the same band with still greater dispersion, were unable to find any traces of $^{39}\mathrm{Cl}$ and $^{40}\mathrm{Cl}$, although they could have detected these isotopes even at a resolving power of $1:1500$. It follows from this that the maxima found by Gettner and Bome have some other origin. Ashley and Jenkins$^{148}$, who studied the electronic bands of AgCl in the visible region, rule out the existence of $^{39}\mathrm{Cl}$ with still greater certainty: according to their measurements, $^{39}\mathrm{Cl}:^{35}\mathrm{Cl} < 1:4400$.

We have already mentioned that, according to the assertion of Kallmann and La-

Zareva[^56] succeeded in discovering \(^{39}\mathrm{Cl}\) (by means of a mass spectrograph), which has an even smaller abundance than that given above. However, even if this discovery is confirmed, it cannot serve as an argument in favor of the data of Gettner and Bome, since the abundance is too small for \(^{39}\mathrm{Cl}\) to be detected spectroscopically. For the recently discovered isotope \(^{2}\mathrm{H}\), as Hardy, Becker, and Dennison[^130] have shown, an isotopic shift can likewise be established. Since \(\rho\) for \(^{2}\mathrm{HCl}\) and \(^{1}\mathrm{HCl}\) is very large, the vibrational effect is very clearly expressed and the zero position of the band \((1,0)\) lies at \(4.8\,\mu\). Thus the band \((1,0)\) is sharply separated from the band \((1,0)\) of the molecule \(^{1}\mathrm{HCl}\), and each of its lines in turn consists of a doublet corresponding to the molecules \(^{2}\mathrm{H}^{35}\mathrm{Cl}\) and \(^{2}\mathrm{H}^{37}\mathrm{Cl}\). An exact measurement of \(\rho\), together with Aston’s values for the masses of \(^{1}\mathrm{H}\) and \(^{35}\mathrm{Cl}\), gives for the mass of \(^{2}\mathrm{H}\) the value \(2.01367 \pm 0.00010\), in excellent agreement with the value found (measured by means of a mass spectrograph) by Bainbridge[^71,^76]. Bramlley, on the basis of theoretical considerations, introduces a small correction into this value and obtains for the mass of \(^{2}\mathrm{H}\) the value \(2.01360 \pm 0.00010\).

In the electronic bands lying in the visible and ultraviolet regions, besides the new isotopes already mentioned, a whole series of isotopes first found on the mass spectrograph were also discovered. For \(^{2}\mathrm{H}\), the rotational and vibrational effects were observed by Ashlee[^132] in the spectrum of \(\mathrm{H}_{2}\) (an enriched sample), and by Johnston and Dawson[^133] in the spectrum of \(\mathrm{OH}\) (also an enriched sample). In unenriched samples (pure water), Chamberlain and Kutter[^134] found several weak lines of the \(\mathrm{OH}\) spectrum that should be attributed to \(^{2}\mathrm{H}\). In enriched samples the vibrational and rotational effect was also studied by Holst and Hulthen[^134a] in the spectrum of \(\mathrm{AlH}\), by Dieke and Blue[^134b] in the spectrum of \(^{1}\mathrm{H}^{2}\mathrm{H}\) and \(^{2}\mathrm{H}_{2}\), by Jeppson[^134c] in 12 bands of the \(^{1}\mathrm{H}^{2}\mathrm{H}\) system, and by Beutler and Mi[^134d] in the fluorescence spectra of \(^{1}\mathrm{H}^{2}\mathrm{H}\). Holst and Hulthen established, from precision measurements of \(\rho^{2}\), that the influence of the electronic system on the moment of inertia of the molecule cannot be neglected. For the isotopes \(^{6}\mathrm{Li}\) and \(^{7}\mathrm{Li}\), the vibrational and rotational isotope effect was found by Harvey and Jenkins[^135] in the spectrum of \(\mathrm{Li}_{2}\). For the abundance ratio \(^{7}\mathrm{Li}:{}^{6}\mathrm{Li}\), Nakamura[^136], on the basis of measurements of the \(\mathrm{LiH}\) spectrum, gives values lying within the limits from \(2:1\) to \(8:1\), depending on the excitation conditions. Wick and Covering[^137] find from measurements of line intensities in the spectrum of \(\mathrm{Li}_{2}\) the value \(^{7}\mathrm{Li}:{}^{6}\mathrm{Li}=7.2:1\). This result, however, is in contradiction with the best mass-spectrographic determination of the abundance, made by Bainbridge[^53].

A precision measurement of \(\rho\) from the vibrational isotope effect in \(\mathrm{Li}_{2}\), made by MacKellar and Jenkins[^138], gives

\[ ^{7}\mathrm{Li}:{}^{6}\mathrm{Li}=2\rho^{2}-1=1.1690\pm0.0003. \]

The same value of the abundance is also obtained in measuring \(\rho\) from the rotational isotope effect. These measurements still further strengthen the discrepancy with the data of the mass-spectroscopic investigations of Bainbridge[^53] and da Costa[^42]. For the isotopes

METHODS AND RESULTS OF ISOTOPE RESEARCH

For boron $^{10}\mathrm{B}$ and $^{11}\mathrm{B}$, the vibrational and rotational effect was found by Mulliken in the BO spectrum (the first observation of an isotopic shift in electronic bands). Mulliken’s observations were confirmed by Jenkins$^{140}$. Elliot, by carefully measuring the intensities in the BO spectrum, found for the relative abundance $^{11}\mathrm{B}:{}^{10}\mathrm{B}$ the value $3.63 \pm 0.02$. Paton and Elvey$^{141}$, from measurements in the BH spectrum, obtained for the relative abundance the value $4.86 \pm 0.15$. Since for hydrides the isotopic shift is relatively small, it may be thought that the effect of corrections in this case will also be less pronounced.

A precision measurement of $\rho$ for the BO spectrum enabled Jenkins and McKellar to determine with high accuracy the mass ratio $^{11}\mathrm{B}:{}^{10}\mathrm{B}$. It proved to be $1.09961 \pm 0.00006$. This value is in complete agreement with that calculated on the basis of Aston’s data. Thus, in the work of Jenkins and McKellar, the slope (at a definite point) of Aston’s curve of relative mass defects was measured for the first time independently of mass spectroscopy, with an accuracy considerably exceeding that of Aston’s measurements.

The isotope effect for the isotopes Mg 24, 25, and 26 was observed by Watson$^{143}$ in the MgH spectrum, by Pearse$^{144}$ in the MgH$+$ spectrum, and by Jenkins and Grinfeld in the MgF absorption spectrum. The isotopes Si: 28, 29, and 30 were observed by Mulliken$^{104}$, McKellar$^{104a}$, and Jenkins and de Laszlo$^{146}$. The chlorine isotopes 35 and 37 were observed by many investigators both in spectra of the infrared region and in spectra lying in the visible region. As already indicated, Elliot$^{102}$ made an accurate determination of the relative abundance in the absorption spectrum of $\mathrm{Cl}_2$ and found:
$^{35}\mathrm{Cl}\,{}^{35}\mathrm{Cl}:{}^{35}\mathrm{Cl}\,{}^{37}\mathrm{Cl} = 1.46:1$. This ratio should be increased somewhat (by a maximum of 20%), since for some of the lines superposition is possible. On the basis of data on the alternation of intensities in the $\mathrm{Cl}_2$ spectrum, the nuclear spin of $^{35}\mathrm{Cl}$ can also be determined. Cl isotopes were also observed by the following authors: Mahanti$^{146a}$ in the spectrum of $^{27}\mathrm{AlCl}$, Hedfeld$^{147}$ in the spectrum of $^{40}\mathrm{CaCl}$, Ashleigh and Jenkins in AgCl, Petrikaln and Hochberg$^{149}$ in $^{115}\mathrm{InCl}$, Jevons$^{150}$ and Ferguson$^{151}$ in $^{119}\mathrm{SnCl}$ (where Jevons also found very weak indications of an isotope effect corresponding to Sn, although the resolving power of the apparatus was too small for this effect to be established), Gibson$^{97}$, Wilson$^{182}$, Patkowski and Curtis$^{199}$, Derbidge$^{153}$, and Brown and Gibson$^{154}$—in the spectrum of $^{127}\mathrm{JCl}$, Ferguson$^{155}$ in $^{197}\mathrm{AuCl}$, Wieland$^{156}$ in $^{200,6}\mathrm{HgCl}$, and Butkov$^{157}$ in $^{204}\mathrm{TeCl}$. The potassium isotopes 39 and 41 were observed by Ritschl and Villars$^{158,159}$ in the $\mathrm{K}_2$ spectrum. For the isotopes Cu 63 and 65, Mulliken$^{160}$ identified the bands of the isotopic molecules $\mathrm{Cu}^{1}\mathrm{H}$. Ritschl$^{159}$ measured the isotopic shift (vibrational effect) in the spectra of copper halides—for the molecules $\mathrm{Cu}^{19}\mathrm{F}$, $\mathrm{Cu}^{35}\mathrm{Cl}$, $\mathrm{Cu}^{37}\mathrm{Cl}$, $\mathrm{Cu}^{79}\mathrm{Br}$, $\mathrm{Cu}^{81}\mathrm{Br}$, and $\mathrm{Cu}^{127}\mathrm{J}$. In the spectrum of $\mathrm{Cu}^{127}\mathrm{J}$ the isotope effect had already been discovered by Mulliken$^{160}$, who supposed that he had succeeded in finding a new

isotope \(^{61}\mathrm{Cu}\). Bromley \(^{161}\) finds from Ritchl’s measurements for the masses of the isotopes \(^{63}\mathrm{Cu}\) and \(^{65}\mathrm{Cu}\) the ratio
\(^{63}\mathrm{Cu}:{}^{65}\mathrm{Cu}=(63-x):(65-x)\). Thus the relative mass defect for both isotopes proves to be the same—to an accuracy of \(2:10^{-6}\). For Zn, Viland \(^{156}\) observed in the spectrum of Zn \(^{127}\mathrm{J}\) the two most abundant isotopes 64 and 66. For the Ga isotopes 69 and 71 the vibrational effect was measured in the spectrum of Ga \(^{35}\mathrm{Cl}\) by Petrikaln and Hochberg \(^{149}\). Shapiro, Gibbs, and Lautenbacher \(^{162}\) discovered in the spectrum of Ge \(^{32}\mathrm{S}\) an isotope effect corresponding to the germanium isotopes 70, 72, 74, and 76, and found for them the relative intensities \(61:68:100:11\). For the isotopes (71), (73), and (75) the effect was not detected. The Br isotopes 79 and 81 were detected by Brown \(^{163}\) in the absorption spectrum of \(\mathrm{Br}_2\), by Plemel \(^{163a}\) in the fluorescence spectrum of \(\mathrm{Br}_2\), by Viland \(^{156}\) in \(^{202}\mathrm{HgBr}\), and by Butkov \(^{157}\) in the absorption spectrum of \(^{204}\mathrm{TlBr}\). The isotopes Sr 88 and 86 were found by Harvey and Jenkins \(^{135}\) in the spectrum of Sr \(^{19}\mathrm{F}\) and by Hedfeld \(^{147}\) in the bands Sr \(^{35}\mathrm{Cl}\), Sr \(^{37}\mathrm{Cl}\), Sr \(^{73}\mathrm{Br}\), Sr \(^{81}\mathrm{Br}\). For the silver isotopes 107 and 109 an isotopic shift was observed by Bengtson and Losson \(^{164}\) in the spectrum of Ag \(^{1}\mathrm{H}\) and by Bryce \(^{165}\) in the spectra Ag \(^{35}\mathrm{Cl}\), Ag \(^{37}\mathrm{Cl}\), Ag \(^{79}\mathrm{Br}\), Ag \(^{81}\mathrm{Br}\), Ag \(^{127}\mathrm{J}\). For the Sb isotopes 121 and 123 the vibrational effect was found by Nöde \(^{166}\) in the absorption spectrum of \(\mathrm{Sb}_2\). The isotopes Ba 136 and 138 were detected by Hedfeld in the bands Ba \(^{35}\mathrm{Cl}\), Ba \(^{37}\mathrm{Cl}\), Ba \(^{39}\mathrm{Br}\), Ba \(^{81}\mathrm{Br}\). (The isotopes 135 and 137 were not found.) For Hg, Holten and Mrozowski \(^{168}\) found in the bands Hg \(^{1}\mathrm{H}\) an isotope effect corresponding to isotopes with mass numbers 198, 200, 202, and 204. Viland \(^{156}\) assigned some edges in the spectrum of Hg Br to the molecules \(^{198}\mathrm{Hg}\,^{79}\mathrm{Br}\), \(^{202}\mathrm{Hg}\,^{79}\mathrm{Br}\), \(^{198}\mathrm{Hg}\,^{81}\mathrm{Br}\), and \(^{202}\mathrm{Hg}\,^{81}\mathrm{Br}\).

The principal lead isotopes 206, 207, and 208, according to Blumenthal \(^{169}\), can be detected in the spectrum of Pb \(^{16}\mathrm{O}\). Lead was the first element for which a difference in the spectra of isotopes was discovered—by Grebe and Konen \(^{170}\).

These authors showed that the higher members of the spectral series they selected, corresponding to uranium lead, have wavelengths differing by \(0.055\,\text{\AA}\) from the wavelengths of the corresponding lines of ordinary lead.

c) Shifts caused by isotopy in the spectra of triatomic and polyatomic molecules

For this case there is as yet no developed theory. Attempts to construct such a theory were made by Salant and Rosenthal \(^{171}\) and by Adel \(^{171a}\). Henry and Gowell \(^{172}\) observed the isotope effect for Cl in the spectrum of \(\mathrm{COCl}_2\); Goudy and Stein \(^{173}\) found this effect in the spectrum of \(\mathrm{ClO}_2\). In addition, Bartholomew and Clusius \(^{173a}\) and Ellis and Sorge \(^{173b}\) observed isotopy of H in the infrared absorption bands of water vapor. Frank and Wood discovered isotopy in the ultraviolet spectrum of water vapor.

2. Shifts Caused by Isotopy in Line Spectra

To determine the difference in the line spectra of isotopes of one and the same element, in the general case it is necessary, besides the direct difference in mass, also to take other differences into account. Therefore, for line spectra the isotope effect proves to be much more complex and intricate, and its theoretical interpretation considerably less complete, than for band spectra.

a) Systems with One Electron

The relations prove to be simple only for systems with one electron (H, He\(^+\), Li\(^{++}\), Be\(^{+++}\) . . . .), for which the Bohr theory is fully valid. For a nucleus of mass \(M\) or \(M^i\), about which an electron of mass \(m_e\) revolves, the following well-known relation holds, when the motion of the nucleus is taken into account:

\[ \nu = \frac{2\pi^2 Z^2 e^2}{ch^3}\cdot \frac{Mm_e}{M+m_e}\cdot \left(\frac{1}{n''^2}-\frac{1}{n'^2}\right), \tag{45} \]

where \(Z\) is the charge of the nucleus, and \(n'\) and \(n''\) are the principal quantum numbers corresponding to the upper and lower levels. The shift caused by isotopy is therefore:

\[ \nu^i-\nu = \frac{2\pi^2 Z^2 e^2}{ch^3} \left( \frac{M^i m_e}{M^i+m_e} - \frac{Mm_e}{M+m_e} \right) \left(\frac{1}{n''^2}-\frac{1}{n'^2}\right) = \frac{m_e(M^i-M)\nu}{MM^i}. \tag{46} \]

Practically the only, but all the more expressive, example of the use of this relation is found for hydrogen. For the isotope H with mass 2, the calculated shift for the \(\alpha, \beta, \gamma\), and \(\delta\) lines of the Balmer series (\(n''=2\)) is respectively 4.16, 5.61, 6.29, 6.65 cm\(^{-1}\). Urey, Brickwedde, and Murphy\(^{179}\) were not only able to discover the existence of the isotope \({}^2\mathrm{H}\) in the study of weak satellites of the lines of the Balmer series, but also succeeded in enriching this rare isotope by fractional distillation of liquid hydrogen. The \({}^2\mathrm{H}\) lines must obviously have a doublet structure, like the lines of \({}^1\mathrm{H}\). For the \(\alpha\) line of \({}^2\mathrm{H}\), Urey and his collaborators actually succeeded in resolving the doublet; for the remaining lines it was possible only to establish their considerable broadening, which was to be expected for unresolved doublets. At the same time it turned out that the lines are less broad and diffuse than the corresponding lines of \({}^1\mathrm{H}\)—obviously because of the weaker Doppler effect. (Atoms of \({}^1\mathrm{H}\), owing to their smaller mass, at the same temperature move on average 1.4 times faster than atoms of \({}^2\mathrm{H}\).) The ratio of the content of \({}^2\mathrm{H}\) to \({}^1\mathrm{H}\) was estimated at approximately \(1/4000\). The estimate was made by determining the exposure time necessary for the corresponding lines of \({}^1\mathrm{H}\) or \({}^2\mathrm{H}\) to appear on the plate. This result agrees with the assumptions of Birge and Menzel\(^{227}\). These authors pointed out that, after the discovery of the isotopes of O, which possess comparatively

with great abundance, the previously found coincidence between the chemical atomic weight of hydrogen and the isotopic weight of \(^{1}\mathrm{H}\) measured by Aston becomes incomprehensible, and it can be explained only by admitting the existence of \(^{2}\mathrm{H}\) with a relative abundance equal to \(1:4500\).

However, the indicated coincidence was at first discredited by Bleakney’s measurements \(^{61}\), according to whose data the relative abundance of \(^{2}\mathrm{H}\) should be considerably smaller \((1:30\,000)\). Since Urey and his collaborators \(^{175}\) indicated that the value of the relative abundance obtained by them could have been distorted by the influence of absorption, and since Hardy, Barker, and Denni-

Son \(^{130}\), in studying band spectra, found a value close to Bleakney’s results (\(1:35000\)); for some time, therefore, the value given by Bleakney was considered the most correct. Rank \(^{176}\), in his measurements made with the H-line, estimated the relative abundance of the isotope \(^{2}\mathrm{H}\) still lower (\(1:80000\)).

After the new determination of the mass of the isotope \(^{1}\mathrm{H}\), carried out by Bainbridge \(^{72}\), who obtained a result practically coinciding with Aston’s, the question of the relative abundance of the heavy isotope of hydrogen became clearer. The latest measurements led to its complete solution. After Washburn and Urey \(^{225}\) discovered that water from old electrolytic baths is considerably richer in \(^{2}\mathrm{H}\) than ordinary water, Lewis and Macdonald \(^{226}\) developed a method for obtaining “heavy” water, consisting practically 100% of \(^{2}\mathrm{H}_{2}\mathrm{O}\). These authors were able, from their experiments on enriching water with the isotope \(^{2}\mathrm{H}\), to establish that the relative abundance of this isotope in ordinary water is about \(1:6500\) (the result of an approximate estimate). Bleakney and Gould \(^{62}\) showed, moreover, that the relative abundance of the isotope \(^{2}\mathrm{H}\) depends on the previous history of the water used, and that for hydrogen obtained from pure rainwater by means of a process in which enrichment in the light isotope has been eliminated, the relative abundance \(^{2}\mathrm{H}:^{1}\mathrm{H}\) is \(1:5000 \pm 10\%\).

Ballard and White \(^{177}\), working with Lewis’s heavy water, were able to detect the shift caused by isotopy also in the first six lines of the Lyman series (\(n''=1\)).

Lewis and Spedding \(^{178}\) succeeded, for hydrogen highly enriched in the heavy isotope, in resolving also the doublet \(^{2}\mathrm{H}\beta\) and in proving the smaller Doppler broadening of this line in comparison with the same line of the light isotope. They further showed that the hydrogen isotope of mass 3, which Urey and his collaborators \(^{174}\) had also tried to find, in any case cannot exist in an amount greater than \(1/1000\) relative to the amount of \(^{2}\mathrm{H}\). It follows from this that the relative abundance of this isotope in ordinary hydrogen is \(^{3}\mathrm{H}:^{1}\mathrm{H}<1:6\cdot 10^{6}\). However, as mentioned \(^{65a}\), the existence of \(^{3}\mathrm{H}\) has recently been firmly established, and for the ratio \(^{3}\mathrm{H}:^{1}\mathrm{H}\) the value \(1:10^{9}\) has been found.

Most recently Hertz \(^{179}\), with the aid of his special method, obtained pure \(^{2}\mathrm{H}\), in which the presence of \(^{1}\mathrm{H}\) cannot be detected either optically or by the methods of mass spectrography. This means that \(^{1}\mathrm{H}\) is present in an amount certainly less than 1%.

b) Systems with many electrons

The line shifts caused by isotopy have been established for many heavy elements. However, these shifts are, as a rule, considerably greater than would be expected on the basis of equation (46), in which the correction for nuclear mass is taken into account in the simplest way. Nevertheless, changes in intensity give

it possible to obtain very accurate values of the relative abundance. Thus, for example, for Li isotopes Schüler^180 finds, from measurements of the intensities of the components of the hyperfine structure—the line \(\lambda = 5485\ \text{Å}\), belonging to the spectrum of \(\mathrm{Li}^{+}\)—the relative abundance ratio \({}^{7}\mathrm{Li} : {}^{6}\mathrm{Li} = 10.5 \pm 10\% : 1\), whereas Ornstein, Brisuik and Wolfson^180a give, for the same ratio, the value \(8.1 \pm 0.4\).

For the boron isotopes, Ornstein and his collaborators obtained, in measurements with the line \(\lambda = 2497.7\ \text{Å}\), the ratio \({}^{11}\mathrm{B} : {}^{10}\mathrm{B} = 4.43 : 1\). The displacement caused by isotopy in the spectrum of Ne was observed and measured by Hansen^182, Thomas and Evans^183, and also by Nagaoka and Mishima^184. Ornstein and Brisuik^185 found, in measurements of the lines \(\lambda = 5852\ \text{Å}\) and \(\lambda = 6402\ \text{Å}\), the ratio \({}^{20}\mathrm{Ne} : {}^{22}\mathrm{Ne} = 10 : 1\).

The theory of the shift caused by isotopy for atoms with several electrons was given by Hughes and Eckart^186. In this theory, in addition to the elementary Bohr correction for the motion of the nucleus, which affects equally all terms of the atom and does not depend on whether the atom is in the normal or ionized state, a second correction for the mass of the nucleus is also taken into account, which does not apply to all states of the atom. It is equal to zero when all the electrons move completely independently of one another. Thus this additional correction depends essentially on the degree to which the rotation of the individual electrons about the nucleus is coordinated, i.e. on how strongly phase relationships are manifested in this motion. That phase relationships are indeed important for the mass correction can easily be seen: if all the electrons were to rotate about the nucleus in one direction, then in order to balance this motion the nucleus would have to move much more strongly than in the case when the motions of the individual electrons are not connected with one another, or in the case when the angular momenta corresponding to the motion of the individual electrons about the nucleus are compensated in pairs.

Hughes and Eckart^186 calculated this effect for two- and three-electron systems (\(\mathrm{Li}^{+}\) and Li). For the line \(\lambda = 5485\ \text{Å}\), belonging to the spectrum of \(\mathrm{Li}^{+}\) (\(2^{3}P \to 1^{3}S\)), elementary theory (equation 46) gives a displacement of the components corresponding to \({}^{6}\mathrm{Li}\) and \({}^{7}\mathrm{Li}\) equal to \((\nu^{i} - \nu)_{\mathrm{elem}} = -0.24\ \text{cm}^{-1}\). However, Schüler’s measurements^187 give for the displacement of the components the value \((\nu^{i} - \nu)_{\mathrm{obs}} = -1.41\ \text{cm}^{-1}\). The elementary theory therefore gives only one fifth of the observed value. Hence the main part of the isotope effect must be attributed to the influence of “phase relationships” regulating the motion of the electrons. The fact that in the present case the elementary correction plays a subordinate role is explained by the fact that it enters almost equally into the initial and final terms and therefore affects the line frequency very weakly; on the contrary, the correction connected with “phase relationships” affects only

to the initial term \(2^3P\), and therefore is fully manifested in the line frequency. The theory of Urey and Eckart, together with an elementary correction, gives for this case the value \((\nu^i-\nu)_{\mathrm{calc}}=-1.06\ \mathrm{cm}^{-1}\), in excellent agreement with the observational results. For Li, the shifts caused by isotopy were observed and measured by Urey\(^{189}\); moreover, the measurement results confirmed the theoretical calculations. Bartlett and Gibbons\(^{190}\) extended the theory of Urey and Eckart to systems with many electrons and found that, for Ne, the calculated values of the shifts agree in part with the values measured by Nagaoka and Mishima\(^{184}\).

Dickinson\(^{191}\) likewise finds that, for \(^{206}\mathrm{Pb}\) and \(^{208}\mathrm{Pb}\), the theoretically calculated shifts agree with those measured by Kopfermann\(^{210}\) and by Schüler and Jones\(^{201}\).

c) Investigations of hyperfine structure

\(\alpha\)) General considerations.

Here we encounter a question which, both from the experimental and from the theoretical side, is extremely complicated. The fact that the splitting of lines, called “hyperfine structure,” also exists in elements for which the absence of isotopes has been firmly established shows that isotopy is not the only cause of this splitting. By analogy with the coarse multiplet structure, which is determined by the laws of interaction of the orbital angular momentum and spin of the electron (quantum numbers \(L\) and \(S\)), in explaining hyperfine structure one must first of all take into account the interaction of the resultant rotational moment of the outer electron shell, determined by the quantum number \(J\), and the rotational moment (spin) of the atomic nucleus, determined by the quantum number \(I\). The resultant rotational moment of the atomic system \(F\), formed by coupling the vectors \(I\) and \(J\), can, for \(I \ge J\), assume \((2J+1)\) different values, and for \(J \ge I\), \((2I+1)\) values. Correspondingly, the terms split into \((2J+1)\) or \((2I+1)\) components. The splitting of terms (the distance between components) is proportional to the magnetic moment of the nucleus \(\mu\). The pattern of splitting of a spectral line, i.e. its hyperfine structure, can be determined by knowing the law of term splitting and by taking into account the selection rule for transitions between different states. In addition to the condition \(\Delta J=0,\pm1\), there is added here a completely analogous condition \(\Delta F=0,\pm1\), with the transition \(F=0\to0\) to be regarded as forbidden. From the number of terms and the relative intensities of the different components of the hyperfine structure one can calculate \(I\), while measurement of the absolute values of the intervals of this magnetic splitting gives the magnetic moment of the nucleus \(\mu\). Determination of both these quantities constitutes the main aim of investigations of hyperfine structure. If hyperfine structure is absent in the spectrum of a given element, this may be explained either by the smallness of the magnetic moment of the nucleus or by the fact that \(I=0\).

A zero nuclear spin can, therefore, be rigorously established only from the falling out of lines in band spectra.

The presence of isotopes complicates the hyperfine structure in two ways. First, in the case when the shift caused by isotopy is sufficiently large, the components corresponding to the individual isotopes are shifted relative to one another; second, \(I\), and with it the number of components of the hyperfine structure, may take different values for different isotopes.

Analysis of the hyperfine structure for a larger number of elements seems to indicate that the nuclear spin \(i\) for isotopes with even mass numbers is equal to 0 (or, perhaps, unity), whereas for isotopes with odd mass numbers it is equal to \(1/2\) or to an odd multiple of \(1/2\). Therefore the lines of even isotopes, as a rule, are not split, and only the shift described above, connected with the correction for the mass of the nucleus, is observed. Lines corresponding to odd isotopes break up into several components (the number of which is determined by the value of \(I\)). The centers of gravity of the lines corresponding to different odd isotopes are shifted relative to one another, and also with respect to the lines of even isotopes. In this case, however, it turns out that in many instances the shift caused by isotopy exceeds by one or two orders of magnitude the value of the correction given by equation (46). In some cases even the sign of the shift proves to be opposite to the sign of the elementary correction. From equation (46) it follows that the heavier isotope should correspond to a term of higher frequency. In a number of cases, however, experiment gives the opposite result. Thus, for example, the experiments of Schüler and Westmeyer \(^{204}\) show that, for Zn\(^+\), the lines of the heavy isotopes 68 and 66 are shifted with respect to the lines of isotope 64 toward lower frequencies, the magnitude of the shift for different lines varying from 0.08 to 0.10 cm\(^{-1}\). According to Schüler and Jones \(^{192}\), the sign of the shift caused by isotopy is the same for all terms of Hg, Hg\(^+\), and Tl\(^+\), but is opposite to the sign of the shift for the terms Tl, Pb\(^+\), and Pb. Further, although the distance between the centers of gravity of the split terms of two odd isotopes is equal to the distance between the terms of even isotopes, the centers of gravity of the terms of odd isotopes do not lie exactly in the middle of the interval between the terms of even isotopes. This apparently indicates that the theory of the correction for the mass of the nucleus, even after the substantial improvements made in it by Hughes and Eckart, is still not capable of giving a complete explanation of all the facts found in the investigation of the isotope effect in line spectra. It should be thought that, besides the difference in nuclear masses (and the difference in spin), other causes must also be taken into account to explain these phenomena. Bartlett pointed out the following possibility: if one admits that an electron belonging to the periphery of the atom and taking part in the formation of the spectrum can, in its motion, penetrate into the interior of the nucleus, then one must take into account the influence which it indicates on its interaction energy—

interaction with the intranuclear field. This interaction will be different for different isotopes of an element. The force field for \(r>r_0\) (\(r_0\)—the radius of the nucleus) may be regarded as Coulombic, and for \(r<r_0\)—as equal to \(-\dfrac{Ze}{r_0}\). Further, it may be assumed that the isotopes have different \(r_0\), with

\[ \frac{\Delta r_0}{r_0}\sim \frac{\Delta M}{M}. \]

Under these assumptions, as Bartlett showed, it is possible to explain the shifts caused by isotopy in the spectra of Tl, Pb, and Hg. Racah\(^{194}\) developed and supplemented Bartlett’s theory by taking into account the correction connected with the relativistic nature of the electron and the correction for shielding of the nucleus by the outer electrons, after which he calculated the shifts connected with isotopy; it turned out that the calculated values exceed the observed ones by more than a factor of one hundred. However, Rosenthal and Breit\(^{195}\) pointed out that in Racah’s calculation the probability for the electron to pass near the nucleus was determined incorrectly (overestimated). If this probability is determined more accurately, it turns out that, in order of magnitude, the results of theory and experiment agree (see, however, the relevant remarks by Schuler and Westmeyer\(^{196}\)).

β) New discoveries. Despite the fact that the influence of isotopy on the hyperfine structure of spectral lines still remains insufficiently clarified both experimentally and theoretically, the study of hyperfine structure already now provides valuable information concerning the existence of a number of isotopes and their relative abundances.

Schuler\(^{197}\), by measuring the intensities of the components of the lines, succeeded in showing that the relative abundance of the split isotopes of Hg lies between 25 and 32%, which is in complete agreement with Aston’s data (30%). Further, Schuler and Keyston\(^{198}\) measured the relative abundance of the splitting (odd) isotopes of Cd and found that it is equal to 23%. Since the shift caused by isotopy is absent in Cd, only the total relative abundance of the splitting and non-splitting isotopes was measured.

After Schuler and Brück\(^{199}\) found indications of the existence of isotopy in Tl, Schuler and Keyston\(^{200}\) showed (before Aston’s data relating to the same question were published) that the extra components appearing in the analysis of the hyperfine structure and not fitting into a simple scheme can be easily explained if it is assumed that Tl has two odd isotopes. The mass numbers of these isotopes can be determined by knowing the atomic weight and using Aston’s well-known rule for isotopes of elements with odd atomic number. These will be the numbers 203 and 205. Although the nuclear spin is the same for both isotopes, they can nevertheless be detected separately because: a) some terms give an isotopic shift, b) in some cases the splitting interval for 205 is 1–2% lower than for 203, i.e. \(\mu\) has a different value for the two isotopes. Intensity measurements give, for the relative abundances, \({}^{203}\mathrm{Tl}:{}^{205}\mathrm{Tl}=\)

\(=1:2.3\), which is in complete agreement with the international atomic weight \(204.39\). Analysis of the hyperfine structure also enabled Schuler and Jones \(^{201}\) to establish the existence of the lead isotope \(^{204}\mathrm{Pb}\), which Aston had at first classed among doubtful isotopes. This result was obtained by Schuler and Jones even before Aston had finally confirmed the existence of this isotope. The mass number was determined from the position of the line corresponding to \(^{204}\mathrm{Pb}\) relative to the lines of \(^{206}\mathrm{Pb}\), \(^{207}\mathrm{Pb}\), and \(^{208}\mathrm{Pb}\). The relative abundance was found by measuring the intensity of the lines and proved to be equal to \(1\%\).

The lead isotope of mass 209, which according to Aston should exist with such a relative abundance that it would be possible to detect its existence by spectroscopic methods, was not found by Schuler and Jones, despite the fact that at such an intensity it ought to have been observed.

The authors also point out that the isotopes found by Aston, \(^{203}\mathrm{Pb}\) and \(^{205}\mathrm{Pb}\), whose relative abundance, according to Aston’s data, is very small, are isobars of \(^{203}\mathrm{Tl}\) and \(^{205}\mathrm{Tl}\) and therefore may easily be confused with the latter, since the presence of Tl can be established spectroscopically even in pure (accumulator) lead.

\(\gamma\)) Measurements with known isotopes. In addition to the already mentioned investigations of Li and Ne, isotopic displacement was detected in the study of the hyperfine structure also in a whole series of elements whose isotopes are known. Tolansky \(^{202}\) explains by the influence of isotopy the broadening of lines observed in the spectrum of Cl. Schuler and Jones \(^{192}\) believe that the asymmetric broadening of the resonance lines of \(^{39}\mathrm{K}\) should be explained by the superposition of the \(^{41}\mathrm{K}\) line on the violet side of the \(^{39}\mathrm{K}\) line. Ritschl discovered a displacement caused by isotopy in the Cu spectrum. Schuler and Westmeyer \(^{204}\) studied the displacements in the Zn spectrum and confirmed the existence of the even isotopes 64, 66, and 68 and of the odd isotope 67. In In, Paschen and Campbell, in investigating the hyperfine structure, did not succeed in detecting any traces of a second isotope. This is in full agreement with Aston’s results \(^{30}\). In Ba, Kruger, Gibbs, and Williams \(^{205}\) found an isotopic displacement between the centers of gravity of the levels corresponding to the even isotopes 136, 138 and the odd isotopes 135, 137. Grace, White, and More \(^{206}\) consider that three components of the hyperfine structure in the spectrum of W, lying at equal distances from one another, correspond to the isotopes 182, 184, 186. Anomalies in the course of the intensity compel one to assume the existence of a fourth, odd isotope 183, which must have magnetic splitting. In mercury Schuler and Keyston \(^{207}\) found for a number of lines that the components of the unsplit even isotopes 198, 200, 202, and 204 are separated as a result of isotopic displacement and at the same time are arranged in the same order as the mass numbers. The centers of gravity of the lines of the magnetically split odd isotopes 199 and 201 are situated between the lines of the even isotopes in such a way that the sequence of mass numbers is preserved. Measure-

intensity measurements, carried out on a series of lines, fully confirm Aston’s values of the relative abundance. In lead, the difference in the spectrum of the isotopes was first established and measured by Aronberg and Merton209. By studying the hyperfine structure, Kopfermann210 firmly established the existence of the isotopes 206Pb, 207Pb, 208Pb and, comparing pure uranium lead (206) with pure thorium lead (208), found a strong displacement of the lines. The same result was obtained by Mac-Lennan, McLay, and Crawford211. Rose and Granath212 found that the relative abundance of the isotopes 207Pb and 208Pb in uranium lead is considerably greater than in ordinary lead. This observation agrees with Aston’s data. Further, it turns out that the displacements caused by isotopy for the three principal isotopes are considerably greater in the spectrum of Pb+ than in the spectrum of Pb. Murakawa213 maintains that analysis of the hyperfine structure not only makes it possible to detect the three principal isotopes of lead, but also permits confirmation of the existence of the isotopes 204 and 210. For the relative abundance of the lead isotopes Murakawa gives the following values (which are in sharp contradiction with Aston’s data): 204Pb : 210Pb = 8 : 1 and 208Pb : 207Pb : 206Pb = 47.7 : 25.9 : 26.4.

Kopfermann214 and Jackson215 established that in the Rb spectrum the centers of gravity of the levels corresponding to the two isotopes 85 and 87 coincide with one another, and therefore the displacement caused by isotopy cannot be measured. An analogous situation holds for Kr. Kopfermann and Wit-Knudsen216 found that the components of the even isotopes of Kr are not separated and coincide with the center of gravity of the odd isotope 83. At the same time, however, a small displacement of the order of 0.01 cm−1 is not excluded. In X, according to Kopfermann’s data, there is likewise coincidence of the centers of gravity of lines corresponding to the different isotopes. In this case, however, both odd isotopes—129X and 131X—have different magnetic and mechanical moments of the nuclei. For Ga, Campbell218 found that the centers of gravity of the components corresponding to the isotopes 69Ga and 71Ga almost coincide. Therefore, if an isotopic displacement exists, it must be very small. However, since the nuclei of both isotopes have different magnetic and mechanical moments, these isotopes can be detected separately. The same holds, according to Tolansky218a, for both isotopes of Sb. According to Grace and Morley206, in Mo, in contrast to the homologous W, if an isotopic displacement exists, then in any case it is of negligibly small magnitude. The observed splitting into two components may also be explained by the magnetic splitting of the odd isotopes. In Cr these authors find no hyperfine structure at all. A very small isotopic displacement of certain terms was established by Schuler and Westmeyer for Cd.

In general, as Schuler and Jones192 established, isotopic displacements for light and heavy elements are considerably greater than for elements of intermediate atomic weight, in which

it in many cases cannot be measured at all. The sign (or direction) of the displacement changes several times in passing from one element to the next following it. The magnitude of the displacement for different elements varies within very broad limits. For isotopes of one and the same element, as Grace and Moore[^206] have shown, approximately identical displacements correspond to identical mass differences.

IV. Results

1. Survey of methods

a) Method of canal rays. For finding new isotopes, for determining masses, and for measuring r.a., the methods of analysis of canal rays considered in Section II have the most general application. Of these, the parabola method gives not only a mass spectrum, but also, for each individual mass, an energy spectrum, thereby greatly facilitating the interpretation of the found values of \(\frac{m}{e}\). This method may be regarded as analogous to the method of crossed prisms in optics, if one considers one of the variables characterizing a canal ray, namely the velocity, as something corresponding to the wavelength of a light ray. With respect to the other variable, \(\frac{m}{e}\), all mass spectrographs act like prisms. Since the parabola method contains no possibilities for focusing, its refinement (an increase in the accuracy of mass determination) is possible only by strongly narrowing the beam, i.e., only with a considerable increase in its intensity. In the methods considered in subsections 2 and 3, focusing achieves a considerable increase in intensity and, consequently, in accuracy. Aston’s method brings to one point those parts of the parabolas bounded by the same parallels to the \(y\)-axis (focusing of velocities); owing to the diaphragming by two narrow collimator slits, the rays have negligible divergence. In this method the possibility of focusing directions is not used. Dempster brings to a point a beam of rays with comparatively large divergence in directions; owing to the fact that all masses pass through one and the same potential difference, they all possess approximately the same energy. Consequently, of the parabolas there remain only those points that lie on a definite parallel to the \(y\)-axis. An optical analogy of the instrument may be a lens, uncorrected for chromaticism, which gives a sharp image only for monochromatic rays. The methods described in subsections 6a and 6b (double focusing of directions and velocities) should give still greater sharpness of the line corresponding to a separate value of \(\frac{m}{e}\). In this respect these methods are equivalent to the use of achromatic lenses.

For all the methods indicated, a characteristic feature is the use of segments (or points) of parabolas corresponding to the same segments (or points) on the abscissa axis, i.e., to the same energy values. Therefore, in addition to mass measurement, by using these methods it is also possible to carry out measurement of the relative abundance of isotopes by comparing the intensities of individual lines. In doing so, however, for light isotopes it is necessary to take special precautions and introduce certain corrections.

The methods described in §§ 4a and 4b, in contrast to those just mentioned, use those points of the parabolas which lie on one and the same straight line passing through the origin, since in these methods particles with a definite linear velocity are selected from the primary beam. Method 4a enabled Bainbridge to make the most accurate determination of isotope masses. Since in method 4b (Smythe and Mattauch) it is easy to carry out an electrometric measurement of the energy distribution of all ions (the sum of the projections of the parabolas onto the \(x\)-axis), in certain cases it is possible to determine the relative abundance of isotopes. Moreover, this method should make possible a very accurate comparison of the masses of isotopes that are widely separated from one another, provided only that the mass numbers are related as the squares of integers. Somewhat apart from all these methods stands the method described in § 5, that of Lawrence and his collaborators, which, however, up to the present has not been applied to the study of isotopes.

b) Spectroscopy. All methods of canal-ray analysis have one common shortcoming, namely, they do not make it possible to establish with certainty the existence of a rare isotope in the case where the value of \(\frac{m}{e}\) belonging to it may coincide with one of the values of \(\frac{m}{e}\) corresponding to possible impurities or to the formation of hydrides. The methods of optical spectroscopy described in Section III are free from this shortcoming. Therefore they prove especially valuable for detecting weak isotopes. Since the isotope effect in band spectra is, in general, considerably stronger than in line spectra, the methods described in § 1 have made it possible to discover a large number of new isotopes of light and medium elements and to confirm the existence of a number of isotopes that had previously been in doubt. The detection of isotopy is also possible for heavy elements in the case where a band spectrum is known that corresponds to the element in its compound with a suitable partner. Moreover, under favorable circumstances this method makes it possible to determine with great accuracy the ratio of the masses of two isotopes of one and the same element. Measurement of the relative abundance of isotopes by this method is not so easy and is possible only when the corresponding corrections are taken into account.

The methods of analysis of line spectra considered in § 2 and the method of investigating hyperfine structure make it possible, precisely for the heaviest elements, to carry out very accurate measurement

i.e., isotopes. Owing to the insufficient elucidation of the isotope effect for line spectra, an exact determination of isotope masses by this method is not yet possible.

c) Other methods. In this review, the isotopes of radioactive elements and the methods of their detection have not been considered, nor have methods of splitting atoms. The latter not only make it possible to detect unstable isotopes, but also make it possible to measure accurately the masses of known isotopes from the energy balance of the reaction equation with the aid of Einstein’s equivalence relation. For this, however, on the one hand, knowledge of the remaining terms of the reaction equation is necessary (for which in many cases one has to use the values of isotope masses determined by other methods), and, on the other hand, it is necessary to be certain that the energy corresponding to a quantum of \(\gamma\)-rays does not drop out of the energy balance.

By this method, for example, Dziobak[^220] determined the mass of \(^{17}\mathrm{O}\) on the basis of data on atomic disintegration obtained by Curie, Chadwick, Constable, and Pollard.[^221] In exactly the same way, the new isotope \(^{3}\mathrm{H}\) was first found in the experiments of Oliphant, Harteck, and Rutherford[^221a] in the disintegration of deuterons by deuterons on the basis of the reaction \(^{2}\mathrm{H} + {}^{2}\mathrm{H} \to {}^{3}\mathrm{H} + {}^{1}\mathrm{H}\). For \(^{3}\mathrm{H}\), and also for the presumed \(^{3}\mathrm{He}\) \((^{2}\mathrm{H} + {}^{2}\mathrm{H} \to {}^{3}\mathrm{He} + n)\), there already exists a very large number of mass determinations.

Of the methods for separating isotopes, only those were mentioned which give 100-percent separation.[^18],[^19],[^43],[^66],[^179],[^225],[^226]

In addition, no mention was made of the so-called “magneto-optical method” of Allison,[^223] by means of which it is possible not only to detect elements and compounds in solutions if their concentration is only \(1:10^{11}\), but also to facilitate the discovery of rare isotopes. Allison asserts that, in the Faraday effect, there exists a very small, but nevertheless measurable, lag between the moment of application of the magnetic field to the solution in which the substance under investigation is located and the resulting rotation of the plane of polarization of light. This lag time must be characteristic of chemical compounds and of cation isotopes. However, even if one accepts the interpretation which Allison gives to his experiments (see the objections of Morey and Webb[^223]), it should nevertheless be noted that this method cannot give values of mass numbers and is capable of giving only very approximate indications of the relative abundance of the isotopes found. Although a certain number of “new” isotopes have been found by means of this method, nevertheless their correlation with definite mass numbers is based on special hypotheses.

2. Table of Isotopes

a) Isotopes and methods of their detection. The appended table of isotopes contains, in addition to the ordinal number \(Z\) of each

of the element, the mass numbers \(m\) of all presently known isotopes of this element, and also the differences \(m-Z\). If one assumes that the nucleus is built of protons and electrons, then the number of fundamental elements of the nuclear structure will be equal to \(m\) and \(m-Z\); if, however, one regards the nucleus as consisting of neutrons and protons, then the quantities determining the number of fundamental elements will be, respectively, \(Z\) and \(m-Z\). The values of \(m\) and \(Z\) obtained directly from experiment determine the number of fundamental elements for the case in which the nucleus consists of neutrons and positrons.

The 5th column gives brief information on the methods by which the given isotope was found and investigated, and, moreover, in historical sequence, so that the method by means of which the isotope was discovered is given first. Here \(m\) denotes investigations by the method of mass spectrography, \(b\)—by band spectra, \(l\)—by line spectra, and H.F.S.—by investigation of hyperfine structure. The letters placed in parentheses next to the designation of the method indicate the author, with \(A_1\) referring to those works of Aston in which he used his first apparatus, \(A_2\)—to Aston’s works carried out on the second apparatus, \(B\)—to Bainbridge, \(D\)—to Dempster, \(Th\)—to J. J. Thomson, \(Z\)—to Zeeman and de-Gier, and \(Bl\)—to Blackett and co-workers. If a letter is absent, this means that the corresponding isotope was observed with a mass spectrograph by a large number of investigators. For a whole series of elements, such as, for example, S, Cu, Ti, Cr, Sr, etc., Aston, working with the second apparatus, succeeded in finding new isotopes (in some cases having a rather large relative abundance) which he had been unable to detect with his first apparatus. Therefore, for those elements whose isotopes were studied by Aston only on the first apparatus and were not subsequently studied by anyone else (as, for example, Fe and Ni), one should expect the discovery of new isotopes absent from the table. For those elements (K, Cl) in which the investigation of H.F.S. can prove only the existence of a mixture, but does not permit the mass numbers to be determined, the H.F.S. mark refers to both isotopes at once.

b) Nonexistent or doubtful isotopes. Many isotopes that have been sought especially intensively may, with very great certainty, be regarded as nonexistent.

Thus, the relative abundance for \({}^{3}\mathrm{He}\) or \({}^{5}\mathrm{He}:{}^{4}\mathrm{He}<1.4\cdot10^{4}\) \(^{59b,61,63,64}\); for \({}^{23}\mathrm{Ne}:{}^{20}\mathrm{Ne}<1\cdot10^{4}\) \(^{57,75}\) (later Kalman and Lazarev \(^{56}\) reported that the intensity of \({}^{23}\mathrm{Ne}\) relative to \({}^{20}\mathrm{Ne}\) is \(1:2\cdot10^{3}\)); for \({}^{21}\mathrm{Na}\) or \({}^{25}\mathrm{Na}:{}^{23}\mathrm{Na}<1:3\cdot10^{3}\), and \({}^{22}\mathrm{Na}:{}^{23}\mathrm{Na}<1:8\cdot10^{2}\) \(^{53}\); for \({}^{39}\mathrm{Cl}:{}^{35}\mathrm{Cl}<1:1.6\cdot10^{3}\) \(^{129}\); for \({}^{39}\mathrm{Cl}:\mathrm{Cl}<1:4.2\cdot10^{3}\) \(^{148}\) (in contrast to these data, Kalman and Lazarev find that \({}^{39}\mathrm{Cl}\) exists with a relative abundance \({}^{39}\mathrm{Cl}:{}^{35}\mathrm{Cl}=1:6\cdot10^{3}\)); for \({}^{40}\mathrm{Cl}:{}^{37}\mathrm{Cl}:1:10^{4}\) \(^{56}\); for \({}^{43}\mathrm{K}:{}^{43}\mathrm{K}<1:1.5\cdot10^{3}\) \(^{53}\); for \({}^{42}\mathrm{K}:{}^{39}\mathrm{K}<1:6\cdot10^{2}\); for \({}^{40}\mathrm{K}:{}^{39}\mathrm{K}<1:3\cdot10^{2}\) \(^{53}\); further, the relative abundances for \({}^{132}\mathrm{Cs}\), \({}^{131}\mathrm{Cs}\), \({}^{130}\mathrm{Cs}\), and \({}^{129}\mathrm{Cs}\) must be less than \(1/10\) of the relative abundance which they would have had in order to give the international atomic weight of Cs,

TABLE

Symbol	Atomic number \(Z\)	Mass number \(m\)	\(m-Z\)	Methods of detection
H	1	1	0	\(m, l, b\)
H	1	2	1	\(l, m, b\)
H	1	3	2	\(m\) (B1)
He	2	4	2	\(m\)
Li	3	6	3	\(m, b, l,\) C. T. S.
Li	3	7	4	\(m, b, l,\) C. T. S.
Be	4	(8)	(4)	\((b)\)
Be	4	9	5	\(m,\) (Th, A\(_1\), B)
B	5	10	5	\(m\) (A\(_1\), A\(_2\), B) \(b, l\)
B	5	11	6	\(m\) (A\(_1\), A\(_2\), B) \(b, l\)
C	6	12	6	\(m, b\)
C	6	13	7	\(b, m\) (T)
N	7	14	7	\(m, b\)
N	7	15	8	\(b, m\) (T)
O	8	16	8	\(m, b\)
O	8	17	9	\(b, m\) (A\(_2\))
O	8	18	10	\(b, m\)
F	9	19	10	\(m\) (A\(_1\), A\(_2\))
Ne	10	20	10	\(m,\) C. T. S.
Ne	10	21	11	\(m\)
Ne	10	22	12	\(m,\) C. T. S., \(l\)
Na	11	23	12	\(m\)
Mg	12	24	12	\(m\) (D, A\(_1\)), \(b\)
Mg	12	25	13	\(m\) (D), \(b\)
Mg	12	26	14	\(m\) (D), \(b\)
Al	13	27	14	\(m\) (A\(_1\))
Si	14	28	14	\(m\) (A\(_1\), A\(_2\)), \(b\)
Si	14	29	15	\(m\) (A\(_1\)), \(b\)
Si	14	30	16	\(b, m\) (A\(_1\))
P	15	31	16	\(m\) (A\(_1\), A\(_2\))
S	16	32	16	\(m\) (A\(_1\), A\(_2\))
S	16	33	17	\(m\) (A\(_2\))
S	16	34	18	\(m\) (A\(_2\))
Cl	17	35	18	\(m, b\)
Cl	17	37	20	\(m, b\) } C. T. S.
Ar	18	36	18	\(m\)
Ar	18	38	20	\(m\) (\(Z\))
Ar	18	40	22	\(m\)
K	19	39	20	\(m, b\) } C. T. S.
K	19	41	22	\(m, b\)
Ca	20	40	20	\(m\) (D, A\(_1\), A\(_2\))
Ca	20	42	22	\(m\) (A\(_2\))
Ca	20	43	23	\(m\) (A\(_2\))
Ca	20	44	24	\(m\) (D, A\(_1\), A\(_2\))
Sc	21	45	24	\(m\) (A\(_1\), A\(_2\))
Ti	22	46	24	\(m\) (A\(_2\))
Ti	22	47	25	\(m\) (A\(_2\))
Ti	22	48	26	\(m\) (A\(_1\), A\(_2\))
Ti	22	49	27	\(m\) (A\(_2\))

Rel. abund. in %	Relative mass defect \(\pi \cdot 10^4\)	Isotopic weight \(M = m(1+\pi)\)
99,98	\([+\,77,75 \pm 0,35]\)	\(1,007775 \pm 0,000035\)
0,02	\([+\,68,15 \pm 0,4]\)	\(2,01363 \pm 0,00008\)
\(10^{-7}\)
100	\(+\,5,4 \pm 0,5\)	\(4,00216 \pm 0,0002\)
8,3	\([+\,24,2 \pm 0,5]\)	\(6,0145 \pm 0,0003\)
91,7	\([+\,20,9 \pm 0,9]\)	\(7,0146 \pm 0,0006\)
(0,05)
99,9	\([+\,17,2 \pm 0,7]\)	\(9,0155 \pm 0,0006\)
20	\(+\,13,5 \pm 0,5\)	\([10,0135 \pm 0,0005]\)
80	\(+\,10,0 \pm 0,5\)	\([11,0110 \pm 0,00055]\)
98,92	\(+\,3,0 \pm 0,3\)	\([12,0036 \pm 0,00036]\)
1,08	\([+\,3,0 \pm 1,1]\)	\([13,0039 \pm 0,0014]\)
99,62	\(+\,5,7 \pm 1\)	\([14,008 \pm 0,0014]\)
0,38	\([+\,1,8 \pm ?]\)	\([15,0027 \pm ?]\)
99,76		\(16\)
0,04
0,20	\([+\,3,6 \pm 0,1]\)	\([18,0065 \pm 0,00018]\)
100	\(0,0 \pm 0,3\)	\([19,0000 \pm 0,0006]\)
90,00	\([-\,1,65 \pm 0,45]\)	\(19,9967 \pm 0,0009\)
0,27
9,73	\([-\,2,4 \pm 0,4]\)	\(21,9917 \pm 0,0009\)
100
77,4
11,5
11,1
100
89,6	\(-\,6,5 \pm 1\)	\([27,9818 \pm 0,0028]\)
6,2
4,2
100	\(-\,5,6 \pm 0,5\)	\([30,9826 \pm 0,0016]\)
96
1
3
76	\([-\,5,8 \pm 0,3]\)	\(34,9796 \pm 0,0012\)
24	\([-\,6,0 \pm 0,5]\)	\(36,9777 \pm 0,0019\)
0,33	\(-\,6,6 \pm 0,5\)	\([35,9762 \pm 0,0018]\)
0,05
99,62	\(-\,7,3 \pm 0,3\)	\([39,9708 \pm 0,0012]\)
94,7
5,3
97
0,8
0,2
2,3
\(> 97\)
rare
”
very common
rare

Symbol	Atomic number \(Z\)	Mass number \(m\)	\(m - Z\)	Methods of detection
V	23	51	28	\(m\) \((A_2)\)
Cr	24	50	26	\(m\) \((A_1)\)
Cr	24	52	28	\(m\) \((A_2)\)
Cr	24	53	29	\(m\) \((A_1, A_2)\)
Cr	24	54	30	\(m\) \((A_2)\)
Mn	25	55	30	\(m\) \((A_1)\)
Fe	26	54	28	\(m\) \((A_1)\)
Fe	26	56	30	\(m\) \((A_1)\)
Co	27	59	32	\(m\) \((A_1)\)
Ni	28	58	30	\(m\) \((A_1, A_2)\)
Ni	28	60	32	\(m\) \((A_1)\)
Cu	29	63	34	\(m\) \((A_1)\), \(b\), C. T. C.
Cu	29	65	36	\(m\) \((A_1)\), \(b\), C. T. C.
Zn	30	64	34	\(m\) \((D, A_1, A_2, B)\), \(b\), C. T. C.
Zn	30	66	36	\(m\) \((D, A_1, A_2, B)\), \(b\), C. T. C.
Zn	30	67	37	\(m\) \((D, A_1, A_2, B)\), C. T. C.
Zn	30	68	38	\(m\) \((D, A_1, A_2, B)\), C. T. C.
Zn	30	70	40	\(m\) \((D, A_1, A_2, B)\)
Ga	31	69	38	\(m\) \((A_1)\), C. T. C., \(b\)
Ga	31	71	40	\(m\) \((A_1)\), C. T. C., \(b\)
Ge	32	70	38	\(m\) \((A_1, A_2, B)\), \(b\)
Ge	32	72	40	\(m\) \((A_1, A_2, B)\), \(b\)
Ge	32	73	41	\(m\) \((A_2, B)\)
Ge	32	74	42	\(m\) \((A_1, A_2, B)\), \(b\)
Ge	32	76	44	\(m\) \((A_2, B)\), \(b\)
As	33	75	42	\(m\) \((A_1, A_2)\)
Se	34	74	40	\(m\) \((A_1, A_2, B)\)
Se	34	76	42	\(m\) \((A_1, A_2, B)\)
Se	34	77	43	\(m\) \((A_1, A_2, B)\)
Se	34	78	44	\(m\) \((A_2, A_2, B)\)
Se	34	80	46	\(m\) \((A_1, A_2, B)\)
Se	34	82	48	\(m\) \((A_1, A_2, B)\)
Br	35	79	44	\(m\) \((A_1, A_2)\), \(b\)
Br	35	81	46	\(m\) \((A_1, A_2)\), \(b\)
Kr	36	78	42	\(m\) \((A_1, A_2, B)\)
Kr	36	80	44	\(m\) \((A_1, A_2, B)\)
Kr	36	82	46	\(m\) \((A_1, A_2, B)\)
Kr	36	83	47	\(m\) \((A_1, A_2, B)\)
Kr	36	84	48	\(m\) \((A_1, A_2, B)\)
Kr	36	86	50	\(m\) \((A_1, A_2, B)\)
Rb	37	85	48	\(m\) \((A_1, A_2)\)
Rb	37	87	50	\(m\) \((A_1, A_2)\)
Sr	38	86	48	\(m\) \((A_1, A_2)\), \(b\)
Sr	38	87	49	\(m\) \((A_2)\)
Sr	38	88	50	\(m\) \((A_1, A_2)\), \(b\)
Y	39	89	50	\(m\) \((A_1)\)
Zr	40	90	50	\(m\) \((A_1, A_2)\)
Zr	40	91	51	\(m\) \((A_2)\)

Continuation

Relative abundance in %	Relative mass defect \(\pi \cdot 10^4\)	Isotopic weight \(M = m(1+\pi)\)
rare
100
4,9
81,6	\(-10,0 \pm 3\)	\([51,948000 \pm 0,016]\)
10,4
3,1
100
95
5
100
95	\(-10,0 \pm 2\)	\([57,942 \pm 0,012]\)
5
68
32
50,4	\(-9,9 \pm 3\)	\([63,937 \pm 0,017]\)
27,2
4,2
17,8
0,4
60
40
21,2
27,3
7,9
37,1
6,5
100	\(-8,8 \pm 0,5\)	\([74,934 \pm 0,004]\)
0,9
9,5
8,3
24,0	\(-8 \pm 2\)	\([77,938 \pm 0,016]\)
48,0	\(-7,3 \pm 1\)	\([79,942 \pm 0,008]\)
9,3
50	\(-9,0 \pm 0,5\)	\([78,929 \pm 0,004]\)
50	\(-8,6 \pm 0,5\)	\([80,930 \pm 0,004]\)
0,42	\(-9,4 \pm 1\)	\([77,927 \pm 0,008]\)
2,45	\(-9,1 \pm 1\)	\([79,927 \pm 0,008]\)
11,79	\(-8,8 \pm 0,5\)	\([81,928 \pm 0,004]\)
11,79	\(-8,7 \pm 0,5\)	\([82,928 \pm 0,004]\)
56,85	\(-8,5 \pm 0,5\)	\([83,929 \pm 0,004]\)
16,70	\(-8,2 \pm 0,5\)	\([85,929 \pm 0,004]\)
75
25
10,0
6,6
83,4
100
59
add. abundance.

Symbol	Atomic number $Z$	Mass number $m$	$m - Z$	Methods of detection
Nb	41	92	52	$m$ ($A_1$, $A_2$)
Nb	41	94	54	$m$ ($A_1$, $A_2$)
Nb	41	96	56	$m$ ($A_1$, $A_2$)
Mo	42	93	52	$m$ ($A_2$)
Mo	42	92	50	$m$ ($A_2$)
Mo	42	94	52	$m$ ($A_2$)
Mo	42	95	53	$m$ ($A_2$)
Mo	42	96	54	$m$ ($A_2$)
Mo	42	97	55	$m$ ($A_2$)
Mo	42	98	56	$m$ ($A_2$)
Mo	42	100	58	$m$ ($A_2$)
Ru	44	96	52	$m$ ($A_2$)
Ru	44	(98)	(54)	$m$ ($A_2$)
Ru	44	99	55	$m$ ($A_2$)
Ru	44	100	56	$m$ ($A_2$)
Ru	44	101	57	$m$ ($A_2$)
Ru	44	102	58	$m$ ($A_2$)
Ru	44	104	60	$m$ ($A_2$)
Rh	45	103	58	$m$ ($A_2$)
Ag	47	107	60	$m$ ($A_1$), $b$
Ag	47	109	62	$m$ ($A_1$), $b$
Cd	48	(108)	(60)	$b$
Cd	48	110	62	$m$ ($A_1$, B), $b$
Cd	48	111	63	$m$ ($A_1$, B)
Cd	48	112	64	$m$ ($A_1$, B), $b$
Cd	48	113	65	$m$ ($A_1$, B)
Cd	48	114	66	$m$ ($A_1$, B), $b$
Cd	48	116	68	$m$ ($A_1$, B), $b$
Cd	48	(118)	(70)	$b$
In	49	115	66	$m$ ($A_1$)
Sn	50	112	62	$m$ ($A_2$)
Sn	50	114	64	$m$ ($A_2$)
Sn	50	115	65	$m$ ($A_2$)
Sn	50	116	66	$m$ ($A_1$, $A_2$)
Sn	50	117	67	$m$ ($A_1$, $A_2$)
Sn	50	118	68	$m$ ($A_1$, $A_2$)
Sn	50	119	69	$m$ ($A_1$, $A_2$)
Sn	50	120	70	$m$ ($A_1$, $A_2$)
Sn	50	121	71	$m$ ($A_2$)
Sn	50	122	72	$m$ ($A_1$, $A_2$)
Sn	50	124	74	$m$ ($A_1$, $A_2$)
Sb	51	121	70	$m$ ($A_1$, $A_2$), $b$, C. T. C.
Sb	51	123	72	$m$ ($A_1$, $A_2$), $b$, C. T. C.
Te	52	122	70	$m$ (B)
Te	52	123	71	$m$ (B)
Te	52	124	72	$m$ (B)
Te	52	125	73	$m$ ($A_2$, B)
Te	52	126	74	$m$ ($A_1$, $A_2$, B)
Te	52	(127)	(75)	$m$ (B)

Continuation

Rel. ab. in %	Relative mass defect $\pi \cdot 10^4$	Isotopic weight $M = m(1+\pi)$
12
24
5
(100)	$-\,8 \pm 5$	$[92{,}926000 \pm 0{,}047]$
14,2
10,0
15,5
17,8
9,6
23,0	$-\,5{,}5 \pm 5$	$[97{,}946000 \pm 0{,}049]$
9,8	$-\,5{,}5 \pm 5$	$[99{,}945000 \pm 0{,}050]$
5
?
12
14
22
30
17
100
51
49
?
14
12
24
10
35
5
?
(100)
1,07
0,74
0,44
14,19
9,81
21,48
11,02
27,04	$-\,7{,}3 \pm 1$	$[119{,}912 \pm 0{,}012]$
2,96
5,03
6,19
56
44
2,9
1,6
4,5
6
19,0	$-\,5 \pm 2$	$[125{,}937 \pm 0{,}025]$
?

Symbol	Atomic number $Z$	Mass number $m$	$m - Z$	Methods of detection
J	53	128	76	$m$ ($A_1$, $A_2$, B)
J	53	130	78	$m$ ($A_1$, $A_2$, B)
X	54	127	74	$m$ ($A_1$, $A_2$)
X	54	124	70	$m$ ($A_1$, $A_2$)
X	54	126	72	$m$ ($A_1$, $A_2$)
X	54	128	74	$m$ ($A_1$, $A_2$)
X	54	129	75	$m$ ($A_1$, $A_2$), C. T. C.
X	54	130	76	$m$ ($A_1$, $A_2$)
X	54	131	77	$m$ ($A_1$, $A_2$), C. T. C.
X	54	132	78	$m$ ($A_1$, $A_2$)
X	54	134	80	$m$ ($A_1$, $A_2$)
X	54	136	82	$m$ ($A_1$, $A_2$)
Cs	55	133	78	$m$ ($A_1$, $A_2$, B)
Ba	56	135	79	$m$ ($A_2$)
Ba	56	136	80	$m$ ($A_2$), $b$
Ba	56	137	81	$m$ ($A_2$)
Ba	56	138	82	$m$ ($A_1$, $A_2$), $b$
La	57	139	82	$m$ ($A_1$)
Ce	58	140	82	$m$ ($A_1$)
Ce	58	142	84	$m$ ($A_1$)
Pr	59	141	82	$m$ ($A_1$)
Nd	60	142	82	$m$ ($A_1$, $A_2$)
Nd	60	143	83	$m$ ($A_2$)
Nd	60	144	84	$m$ ($A_1$, $A_2$)
Nd	60	145	85	$m$ ($A_2$)
Nd	60	146	86	$m$ ($A_1$, $A_2$)
Sm	62	144	82	$m$ ($A_2$)
Sm	62	147	85	$m$ ($A_2$)
Sm	62	148	86	$m$ ($A_2$)
Sm	62	149	87	$m$ ($A_2$)
Sm	62	150	88	$m$ ($A_2$)
Sm	62	152	90	$m$ ($A_2$)
Sm	62	154	92	$m$ ($A_2$)
Eu	63	151	88	$m$ ($A_2$)
Eu	63	153	90	$m$ ($A_2$)
Gd	64	155	91	$m$ ($A_2$)
Gd	64	156	92	$m$ ($A_2$)
Gd	64	157	93	$m$ ($A_2$)
Gd	64	158	94	$m$ ($A_2$)
Gd	64	160	96	$m$ ($A_2$)
Tb	65	159	94	$m$ ($A_2$)
Dy	66	161	95	$m$ ($A_2$)
Dy	66	162	96	$m$ ($A_2$)
Dy	66	163	97	$m$ ($A_2$)
Dy	66	164	98	$m$ ($A_2$)
Ho	67	165	98	$m$ ($A_2$)
Er	68	166	98	$m$ ($A_2$)
Er	68	167	99	$m$ ($A_2$)
Er	68	168	100	$m$ ($A_2$)

Continuation

Rel. abund. in %	Relative mass defect $\pi \cdot 10^4$	Isotopic weight $M = m(1+\pi)$
32,8	— 5 ± 2	[127,936000 ± 0,026]
33,1
100	— 5,3 ± 1	[126,933 ± 0,013]
0,08
0,08
2,30
27,13
4,18
20,67
26,45
10,31	— 5,3 ± 1	[133,929 ± 0,013]
8,79
100	— 5 ± 2	[132,934 ± 0,027]
5,9
8,9
11,1
74,1	— 6,1 ± 2	[137,916 ± 0,028]
100
90
10
100
34
? rare
33
?
33
rare
"
"
"
"
very widespread
"
50 "
50
(100)
(100)
very widespread

Symbol	Atomic number \(Z\)	Mass number \(m\)	\(m - Z\)	Methods of detection
Tu	69	170	102	\(m\) (A\(_2\))
Yb	70	169	100	\(m\) (A\(_2\))
Yb	70	171	101	\(m\) (A\(_2\))
Yb	70	172	102	\(m\) (A\(_2\))
Yb	70	173	103	\(m\) (A\(_2\))
Yb	70	174	104	\(m\) (A\(_2\))
Yb	70	176	106	\(m\) (A\(_2\))
Cp	71	175	104	\(m\) (A\(_2\))
Hf	72	176	104	\(m\) (A\(_2\))
Hf	72	177	105	\(m\) (A\(_2\))
Hf	72	178	106	\(m\) (A\(_2\))
Hf	72	179	107	\(m\) (A\(_2\))
Hf	72	180	108	\(m\) (A\(_2\))
Ta	73	181	108	\(m\) (A\(_2\))
W	74	182	108	\(m\) (A\(_2\)), S. T. S.
W	74	183	109	\(m\) (A\(_2\)), S. T. S.
W	74	184	110	\(m\) (A\(_2\)), S. T. S.
W	74	186	112	\(m\) (A\(_2\)), S. T. S.
Re	75	185	110	\(m\) (A\(_2\))
Re	75	187	112	\(m\) (A\(_2\))
Os	76	186	110	\(m\) (A\(_2\))
Os	76	187	111	\(m\) (A\(_2\))
Os	76	188	112	\(m\) (A\(_2\))
Os	76	189	113	\(m\) (A\(_2\))
Os	76	190	114	\(m\) (A\(_2\))
Os	76	192	116	\(m\) (A\(_2\))
Hg	80	196	116	\(m\) (A\(_2\))
Hg	80	197	117	\(m\) (A\(_2\))
Hg	80	198	118	\(m\) (A\(_1\), A\(_2\), B), \(b\), S. T. S.
Hg	80	199	119	\(m\) (A\(_1\), A\(_2\), B), S. T. S.
Hg	80	200	120	\(m\) (A\(_1\), A\(_2\), B), \(b\), S. T. S.
Hg	80	201	121	\(m\) (A\(_2\), B), S. T. S.
Hg	80	202	122	\(m\) (A\(_1\), A\(_2\), B), \(b\), S. T. S.
Hg	80	203	123	\(m\) (A\(_2\))
Hg	80	204	124	\(m\) (A\(_1\), A\(_2\), B), \(b\), S. T. S.
Tl	81	203	122	S. T. S., \(m\) (A\(_2\))
Tl	81	205	124	S. T. S., \(m\) (A\(_2\))
Pb	82	203	121	\(m\) (A\(_2\))
Pb	82	204	122	S. T. S. \(m\) (A\(_2\))
Pb	82	205	123	\(m\) (A\(_2\))
Pb	82	206	124	\(m\) (A\(_2\)), \(b\), S. T. S.
Pb	82	207	125	\(m\) (A\(_2\)), \(b\), S. T. S.
Pb	82	208	126	\(m\) (A\(_2\)), \(b\), S. T. S.
Pb	82	209	127	\(m\) (A\(_2\))
Pb	82	210	126	\(m\) (A\(_2\)), S. T. S.
Bi	83	209	126	\(m\) (A\(_1\))
Th	90	232	142	\(m\) (A\(_2\))
U	92	238	146	\(m\) (A\(_2\))

METHODS AND RESULTS OF ISOTOPE RESEARCH

Continuation

R. a. in %	Relative mass defect $\pi \cdot 10^4$	Isotopic weight $M = m(1+\pi)$
rare (100)
very abundant
100
rare
very abundant
less abundant
very abundant
less abundant
$> 98$	$-\,4 \quad \pm 3$	$[180{,}928000 \pm 0{,}054]$
22,6
17,3
30,2	$0 \quad \pm 5$	$[184{,}00 \quad \pm 0{,}09]$
29,9
38,2
61,8	$-\,1 \quad \pm 2$	$[186{,}981 \quad \pm 0{,}037]$
1,0
0,6
13,4
17,4
25,1	$-\,1 \quad \pm 2$	$[189{,}981 \quad \pm 0{,}038]$
42,5	$-\,1 \quad \pm 2$	$[191{,}981 \quad \pm 0{,}038]$
0,10
0,01
9,89
16,45
23,77	$-\,0{,}8 \quad \pm 2$	$[200{,}016 \quad \pm 0{,}020]$
13,67
29,27
0,006
6,85
29,4	$-\,1{,}8 \quad \pm 2$	$[203{,}037 \quad \pm 0{,}041]$
70,6	$-\,1{,}8 \quad \pm 2$	$[205{,}037 \quad \pm 0{,}041]$
0,04
1,50
0,03
27,75
20,20
49,55
0,85
0,08
100
$> 97$
$> 97$

The most recent determination of the chemical atomic weight of Cs, carried out by Baxter and Thomas \(^{223}\), gives the value 132.91, which is in agreement with the fact that Cs is a simple element. The relative abundance with respect to Cs in pollucite is less than \(1:3.5 \cdot 10^{7}\), and in lepidolite less than \(1:7.3 \cdot 10^{8}\); however, according to a preliminary communication by Barnes and Gibbs \(^{58}\), an ion with mass 220 and a relative content of \(1:10^{4}\) (with respect to Cs) was found in a special sample. The elements Sc, Nb, In, Tb, Ta, and U (judging from their atomic weights) should have second isotopes. Aston, however, finds that the relative abundance of these isotopes is less than \(2\text{–}3\%\). Indium also proves to be a simple element on the basis of data from the investigation of hyperfine structure \(^{204a}\).

Isotopes whose existence has not yet been fully proved, and concerning which there are contradictory opinions among different investigators, such as, for example, \(^{98}\mathrm{Ru}\), \(^{108}\mathrm{Cd}\), \(^{118}\mathrm{Cd}\), \(^{128}\mathrm{Te}\), or \(^{8}\mathrm{Be}\), are enclosed in parentheses in the table. In connection with them one should mention the mass numbers 65 and 69 for Zn and 71, 75, and 77 for Ge, which were first found by Aston \(^{35,36}\) and assigned by him to isotopes of the corresponding elements (Zn and Ge); subsequently Bainbridge \(^{68,75}\) showed that they are connected with the formation of hydrides. For Pb, Aston was able to measure the extent to which the intensity of the lines corresponding to the various isotopes of lead is distorted by the presence of hydrides. A sample of pure thorium lead gave the lines 206, 207, 208, and 209 with relative intensities 4.9, 1.5, 100, and 2.3. Since \(^{209}\mathrm{Pb}\) is present in ordinary lead only in a very small amount, the 209 line could not have arisen from contamination by ordinary lead. Further, since pure thorium lead hardly contains an element with mass number 209, it turns out that the lead isotopes form monohydrides, and the relative content of the hydrides is about 2.3% of the total content of the given isotope. In calculating the relative abundances of the isotopes of ordinary lead, this circumstance must be taken into account. There still remains, however, a residue of 0.85%, which Aston attributes to a real isotope of lead. However, as Schuler and Jones \(^{201a}\) point out, at such a relative abundance the presence of this isotope should have affected the hyperfine structure, which is not in fact the case. These authors also draw attention to the fact that \(^{203}\mathrm{Pb}\) and \(^{205}\mathrm{Pb}\) are isobars of both Tl isotopes and point to the possibility of the presence of Tl in pure lead, since its presence is in fact detected spectroscopically. Thus the existence of the lead isotopes 203, 205, and 209 requires confirmation. The situation is analogous for its still weaker mercury lines 197 and 203, of which the first is probably an isobar of the principal isotope of Au (the isotopes of gold have not yet been studied).

Recently Kendall, Smythe, and Tate \(^{224}\) expressed the supposition that the somewhat higher atomic weight of Ca obtained from very old feldspar varieties in the geological sense,

containing approximately 0.3% CaO and 9% K₂O, is explained by the existence of the new isotope ⁴¹Ca, formed from ⁴¹K by β-decay. However, Aston asserts that in material purified from K there are no traces whatever of ⁴¹Ca, despite the fact that the presence of this isotope in an amount of 0.1% could still have been detected.

c) Relative abundances. The 6th column of the table contains the r. a. of the isotopes of the elements of the periodic system. These numbers, however, are not especially accurate, although they are given with two or three decimal places. The numbers printed in italics have not been measured, but have been estimated by Aston from the darkening or calculated from the atomic weight. The number 100 without parentheses means that the given element must be simple, if one also judges from its chemical atomic weight.

For light elements, for which the masses of the isotopes differ comparatively greatly from one another in percentage terms, the results of measurements of r. a. carried out by different investigators disagree very strongly among themselves. It cannot, however, be asserted with certainty that this discrepancy can be explained by errors of measurement; rather, one should think that the r. a. of isotopes for different samples actually varies within fairly wide limits owing to partial artificial or natural separation of the isotopes. Thus, for example, at first it was thought that the r. a. of the hydrogen isotopes ²H and ¹H lay between \(1:4\cdot 10^3\) and \(1:8\cdot 10^5\) 174,61,130,176; however, Bleakney and Gould 62 showed that in commercial electrolytic hydrogen there is appreciable separation of isotopes, which was first discovered by Urey and Washburn and used as a method for obtaining heavy hydrogen by Lewis and Macdonald. The number given in the table belongs to Bleakney and Gould 62 (\(1:5000 \pm 10\%\)). This value agrees with that calculated by Birge and Menzel 227 from the international atomic weight of H and the mass of the isotope ¹H determined by Aston. The same value of r. a. (\(1:6500\)) is given by Lewis and Macdonald on the basis of studies of enrichment in ²H. Initially the ratios \(^{3}\mathrm{H}:^{1}\mathrm{H}<1:6\cdot 10^6\) 178 or \(^{3}\mathrm{H}:^{1}\mathrm{H}<1:5\cdot 10^8\) 62 were accepted. Subsequently Bleakney et al. 65a found that this ratio is equal to \(1:10^9\).

The value of r. a. given in the table for Li (\(^{6}\mathrm{Li}:^{7}\mathrm{Li}=1:11.28 \pm 0.07\)) was measured by Bainbridge 53 and recalculated by Aston (\(1:12.187\)). Values obtained by other authors lie in the range from \(1:2\) to \(1:37\) 46,48,49,137,136,37,180,180a,55. For B Aston obtained the ratio \(^{10}\mathrm{B}:^{11}\mathrm{B}=1:3.85\). After correction for the higher velocity and greater depth of penetration of the light isotope, the ratio proves to be \(1:4.04\). Values for r. a. obtained by other authors lie in the range from \(1:3.63 \pm 0.02\)103 to \(1:4.85 \pm 0.15\)141. For C the value given is that obtained by Vohn, Williams, and Tate 596 — \(^{13}\mathrm{C}:^{12}\mathrm{C}=1:91.6 \pm 2.2\), which is in agreement with the number obtained in the latest investigations of band spectra (\(^{13}\mathrm{C}:^{12}\mathrm{C}=1:106\)117) and

with atomic weight C, which according to the most recent determinations by Wichard and Whytlaw-Gray \(^{228}\) is equal to 12.011. The old value, obtained from spectroscopic data and equal to \(1:400\), cannot claim any accuracy. The value for N is likewise taken from the data of Vaughan, Williams, and Tate \(^{596}\). According to these data
\[ {}^{15}\mathrm{N}:{}^{14}\mathrm{N}=1:265\pm 8. \]
The most reliable spectroscopic determination gives \(1:346^{120}\). The old value \(1:700\pm140^{118}\) was too small, for in its determination the value
\[ {}^{18}\mathrm{O}:{}^{16}\mathrm{O}, \]
obtained by Mecke and Childs and equal to \(1:409\), was used as the basis. For O the table gives a new value, obtained by Smyth \(^{83a}\), who found that
\[ {}^{16}\mathrm{O}:{}^{18}\mathrm{O}=503\pm10. \]
On comparison with the value
\[ {}^{18}\mathrm{O}:{}^{17}\mathrm{O}=1:0.2, \]
obtained by Mecke and Childs, we have
\[ {}^{17}\mathrm{O}:{}^{18}\mathrm{O}:{}^{16}\mathrm{O}=0.2:1:630\pm20. \]
The latter ratio is well confirmed by mass-spectrographic investigations, which give for
\[ {}^{18}\mathrm{O}:{}^{16}\mathrm{O} \]
the values \(1:600^{89}\), \(1:640\pm8^{56}\), and for
\[ {}^{17}\mathrm{O}:{}^{18}\mathrm{O}:{}^{16}\mathrm{O} \]
the value \(1/4.2:1:536^{39}\).

For the isotopes of Ne the latest measurements by Vaughan, Williams, and Tate \(^{596}\) were used, giving
\[ {}^{20}\mathrm{Ne}:{}^{21}\mathrm{Ne}=337\pm20 \]
and
\[ {}^{20}\mathrm{Ne}:{}^{22}\mathrm{Ne}=9.25\pm0.08. \]
Other authors obtained the following values:
\[ {}^{20}\mathrm{Ne}:{}^{21}\mathrm{Ne}:{}^{22}\mathrm{Ne}=100:0.28:8.25^{9} \]
(measurements on singly charged ions) and
\[ 100:0.30:9.25^{9} \]
(measurements on doubly charged ions, when the formation of hydrides is excluded);
\[ 88:2:10^{50};\quad 93.7:1:9.75^{56}; \]
\[ {}^{22}\mathrm{Ne}:{}^{20}\mathrm{Ne}=10:1^{185}. \]
For the isotopes of Mg and K only Dempster’s numbers \(^{46,47}\) are so far known; these are the ones given in the table. For the isotopes of Si new data have been used, obtained by MacKellar by the method of studying band spectra. In the row Cl there stands the ratio
\[ {}^{37}\mathrm{Cl}:{}^{35}\mathrm{Cl}=1:3.18, \]
obtained from the international atomic weight. A series of measurements gave values from \(1:2.92^{102}\) to \(1:3.25^{56}\). For A, the measurements of Vaughan, Williams, and Tate gave
\[ {}^{36}\mathrm{A}:{}^{40}\mathrm{A}=1:304\pm12. \]
These data are also given in the table together with data for the new isotope \({}^{38}\mathrm{A}\), for which, according to Zeeman and de Gier \(^{21a}\), the relative abundance with respect to \({}^{36}\mathrm{A}\) lies between \(1/5\) and \(1/10\).

The data on the relative abundances of isotopes for all the remaining elements, with the exception of the three lightest isotopes of Te, are taken from Aston’s works \(^{35,36,37,38}\). The results of investigations of hyperfine structure also agree well with these data (for those elements with which they were carried out). Thus, for Tl, measurements of the hyperfine structure gave
\[ {}^{203}\mathrm{Tl}:{}^{205}\mathrm{Tl}=1:2.3^{200} \]
(the corresponding value obtained by Aston is \(1:2.4\)). Aston’s results for the mercury isotopes 198, 199, 200, 201, 202, 204 were confirmed in exactly the same way. The same agreement between the results of hyperfine-structure investigations and Aston’s data is also found for the lead isotope \({}^{204}\mathrm{Pb}\), discovered by Schüler and Jones, whose relative abundance is \(1\%\).

The relative abundance of the odd isotopes of Cd, according to hyperfine-structure investigations \(^{198}\), is \(23\%\). According to Aston’s estimate it is \(22\%\). Measurements of the relative abundance of isotopes of Bi by the band-spectra method \(^{162}\) also give results close to Aston’s measurements.

\({}^{70}\mathrm{Ge} : {}^{72}\mathrm{Ge} : {}^{74}\mathrm{Ge} : {}^{76}\mathrm{Ge} = 81 : 68 : 100 : 11\), whereas according to Aston
\({}^{70}\mathrm{Ge} : {}^{72}\mathrm{Ge} : {}^{74}\mathrm{Ge} : {}^{76}\mathrm{Ge} = 56.23 : 72.44 : 100 : 17.37\).

d) Relative mass defects and isotope weights.
In the 7th and 8th columns of the table are given the relative mass defects and the weights of the isotopes (for those isotopes for which they are known). For each number the magnitude of the error indicated by the investigator is also given. Since Aston always gives first of all the values of the relative mass defects measured directly by him (and often only these values), in those cases where the data given in the table are taken from his measurements, the isotope weights are placed in square brackets. The relative mass defect is obtained from this by calculation. Therefore, in the table the values of the relative mass defect that are taken from Aston’s works are likewise placed in square brackets. Conversely, Bainbridge always gives the isotope weight together with the magnitude of the error. Where both numbers are enclosed in square brackets (as for \({}^{13}\mathrm{C}\), \({}^{15}\mathrm{N}\), \({}^{18}\mathrm{O}\)), they are taken from band-spectrum measurements made by Birge. These measurements always give only the mass ratios of different isotopes of one and the same element, and therefore only O gives directly the isotope weights and the relative mass defects. For all other elements, it is evidently necessary that the mass of at least one of the isotopes be determined with the aid of a mass spectrograph. The isotope weight (together with the error determining the accuracy of its measurement) can obviously be obtained by simple recalculation from the corresponding relative mass defect (and vice versa). In the table, however, both the weight and the relative mass defect are given for each isotope. This is of use if only because, in the corresponding table in Handbuch der Physik, Vol. XXIV, Part I, p. 794, 1933, the values of both errors are placed in one column, and moreover they are completely confused, and in some cases incorrect values are given. In addition, in a number of cases the origin of the data presented is incorrectly indicated. The same ambiguity—especially with respect to the results of investigations of band spectra—also exists in the isotope table included in Aston’s new book.

The values of the errors included in our table are based on Birge’s data \(^{112, 116}\), supplemented by Aston’s measurements.

In our table the value of the weight of the isotope \({}^{17}\mathrm{O}\) is entirely absent; in other tables its determination was erroneously attributed to spectroscopic methods. In reality this isotope was discovered in experiments on the disintegration of atomic nuclei by Chadwick, Pollard, and Constable \(^{221}\).

In Aston’s table the relative mass defects of the isotopes, calculated from Bainbridge’s measurement results, are, owing to a typographical error, mistakenly reduced by a factor of 10. The same incorrect values are also given in Aston’s book in Figs. 30 and 31.

For the Cl isotopes our table gives the values given by Bainbridge, since his measurement errors are somewhat smaller,

the values cited by Aston. According to Aston, the corresponding values of the relative mass defects will be, for \(^{35}\mathrm{Cl}\): \(-5.0 \pm 0.5\), and for \(^{37}\mathrm{Cl}\): \(-4.8 \pm 0.5\). For \({}^{1}\mathrm{H}\) the Bainbridge number has likewise been taken, since with regard to the value given by Aston it is known only that it lies between \(77.8\) and \(78.3 \pm 1\). For the Ne isotopes Aston himself gives preference to Bainbridge’s data. The study of band spectra gives, for the weight of the hydrogen isotope \({}^{2}\mathrm{H}\), the values \(2.01367 \pm 0.00010^{130}\) and \(2.01360 \pm 0.00010^{131}\), which, within the limits of the errors of measurement, agree with the value given in the table.

The first confirmation of Aston’s results by the method of band spectra occurred for the boron isotopes. The mass ratio \(^{11}\mathrm{B}:{}^{10}\mathrm{B}\), calculated from measurements of spectral bands, proves to be \(1.09961 \pm 0.00006\) (with Aston’s value for \(^{10}\mathrm{B}\)). This ratio practically coincides with the ratio of Aston’s numbers for \(^{11}\mathrm{B}\) and \(^{10}\mathrm{B}\).

A discrepancy between the data of band-spectrum analysis and the results obtained with the aid of the mass spectrograph occurs for Li. Spectroscopy gives in this case the precise value \(^{7}\mathrm{Li}:{}^{6}\mathrm{Li} = 1.1690 \pm 0.0003^{138}\), whereas according to Bainbridge’s data the ratio of the weights of the isotopes is \(1.1663 \pm 0.00016\), and according to da Costa’s old measurements it is \(1.1663 \pm 0.07\). Further, from measurements of band spectra \(^{158}\) it follows \(^{161}\) that the deviations from integral values (the mass defect) for both Cu isotopes must have one and the same magnitude to an accuracy of \(1 \cdot 10^{-4}\).

Figure 29 shows Aston’s curve of relative mass defects, constructed on the basis of the data that are given in our table. As usual, in the region of the light elements the curve splits into two branches, of which the lower contains isotopes with mass numbers divisible by 4: \({}^{4}\mathrm{He}\), \({}^{12}\mathrm{C}\), and \({}^{16}\mathrm{O}\). The value of the relative mass defect for \({}^{13}\mathrm{C}\), obtained from band-spectrum analysis, also lies rather close to the lower branch.

3. Attempts to Construct a Systematics of Isotopes

Similar attempts were undertaken from various sides by Beck \(^{229}\), Harkins \(^{230}\) (see there also the earlier literature), Barton \(^{231}\), Johnston \(^{232}\), Urey \(^{233}\), Bartlett \(^{234}\), and others. These attempts are, as a rule, characterized by the construction of a graph in which, for each isotope, the numbers of the presumed principal elements of nuclear structure are plotted along the abscissa and ordinate axes (\(m\) and \(m - Z\), or respectively \(Z\) and \(m - Z\)), or else the “isotopic number” \(m - 2Z\) and the atomic number \(Z\); and from the regularities appearing on the graph, conclusions are drawn about the existence of isotopes not yet discovered. Some authors, in addition, consider separately isotopes with weights \(4n\), \(4n + 1\), \(4n + 2\), \(4n + 3\). Although these inferences were in part invalidated by the discovery of new isotopes and the exclusion of former ones which proved to be hydrides, nevertheless they exerted, in կա

as a working hypothesis, an important service to Urey and his collaborators in the discovery of the heavy isotope of hydrogen. The general rules which can still be established with regard to the existence of isotopes are clearly set forth in Aston’s new book.

First of all, attention is drawn to the isotopes of elements with an odd atomic number. If one sets aside the isotope \({}^{1}\mathrm{H}\), which occupies a special place, it turns out that all elements without exception with odd \(Z\) have only one or two isotopes. For the interval \(Z\) from 1 to 7 (from H to N), all the numbers \(m-Z\) from 1 to 8 are occupied by isotopes once each (\(m-Z=0\) corresponds to \({}^{1}\mathrm{H}\)). For \(Z>9\) the mass numbers are always odd (and for complex elements are separated from one another by 2), and therefore the numbers \(m-Z\) are always even; moreover, with rare exceptions, all even numbers \(m-Z\) are filled only once each or are not occupied at all. The first double filling is at \(m-Z=20\) (\({}^{37}\mathrm{Cl}\) and \({}^{39}\mathrm{K}\)), the second at \(m-Z=50\) (\({}^{37}\mathrm{Rb}\) and \({}^{39}\mathrm{Y}\)), the third at \(m-Z=82\) (\({}^{57}\mathrm{La}\) and \({}^{59}\mathrm{Pr}\)).

For filling the elements with an even atomic number, we may try to establish the following selection rule: isotopes with an odd atomic number have no isobars. Isobars of elements with an even atomic number, on the contrary, can always exist and are often encountered, but only at even mass numbers. For the occupation of mass numbers there is the following general rule. All mass numbers which lie within the “occupation limits” defined below and can exist according to the selection rules given above are in fact occupied by isotopes. For the whole system of elements one can establish three intervals of occupation. If by \(Z\) we denote any element with an even atomic number, by \(\bar m\) the mass number of the heavier isotope of the element \(Z-1\), and by \(m\) the mass number of the lighter isotope of the element \(Z+1\), then for elements from \(Z=1\) to \(Z=16\) there exists only one interval of occupation, from \(\bar m+1\) to \(m-1\). For \(Z>16\) there therefore exists a second interval of occupation: from \(\bar m-1\) to \(m+1\). This interval overlaps many times from Se to Sm. Therefore, for \(32<Z<62\) we have a third interval of occupation—from \(\bar m-3\) to \(m+3\), with, according to the selection rule, the places \(\bar m-2\) and \(m+2\) in each case forbidden. For \(Z>64\) the second interval of occupation again does not overlap anywhere.

Places not filled up to now should be assigned to isotopes not yet discovered. These unfilled places, up to \(Z=76\), belong in the first interval of occupation to the isotopes \({}^{5}\mathrm{He}\), \({}^{57}\mathrm{Fe}\), \({}^{58}\mathrm{Fe}\), \({}^{61}\mathrm{Ni}\), \({}^{62}\mathrm{Ni}\), \({}^{104}\mathrm{Pd}\), \({}^{105}\mathrm{Pd}\), \({}^{106}\mathrm{Pd}\), \({}^{134}\mathrm{Ba}\), \({}^{146}\mathrm{Sm}\), \({}^{154}\mathrm{Gd}\), \({}^{166}\mathrm{Dy}\), \({}^{170}\mathrm{Yb}\). Beginning with \(Z>16\), to these are added isotopes corresponding to places unfilled in the second interval of occupation, namely: \(({}^{36}\mathrm{S})\), \({}^{46}\mathrm{Ca}\), \({}^{44}\mathrm{Ti}\), \({}^{52}\mathrm{Ti}\), \({}^{56}\mathrm{Cr}\), \({}^{60}\mathrm{Fe}\), \({}^{64}\mathrm{Ni}\), \({}^{84}\mathrm{Sr}\), \({}^{90}\mathrm{Sr}\), \({}^{88}\mathrm{Zr}\), \({}^{102}\mathrm{Pd}\), \({}^{132}\mathrm{Ba}\), \({}^{140}\mathrm{Ba}\), \({}^{138}\mathrm{Ce}\), \({}^{140}\mathrm{Nd}\), \({}^{152}\mathrm{Gd}\), \({}^{158}\mathrm{Dy}\), \({}^{166}\mathrm{Dy}\), \({}^{164}\mathrm{Er}\), \({}^{168}\mathrm{Yb}\), \({}^{174}\mathrm{Hf}\), \({}^{182}\mathrm{Hf}\).

Between \(Z = 32\) and \(Z = 62\) in the third substitution interval, the unfilled places should be assigned to the isotopes \((^{68}\mathrm{Ge})\), \((^{78}\mathrm{Ge})\), \(^{72}\mathrm{Se}\), \(^{88}\mathrm{Kr}\), \(^{82}\mathrm{Sr}\), \(^{92}\mathrm{Sr}\), \(^{86}\mathrm{Zr}\), \(^{90}\mathrm{Mo}\), \(^{106}\mathrm{Ru}\), \(^{100}\mathrm{Pd}\), \(^{110}\mathrm{Pd}\), \(^{106}\mathrm{Cd}\), \(^{120}\mathrm{Te}\), \(^{130}\mathrm{Ba}\), \((^{142}\mathrm{Ba})\), \(^{136}\mathrm{Ce}\), \(^{144}\mathrm{Ce}\) \((^{138}\mathrm{Nd})\). These unknown isotopes, whose existence is necessary for the completeness of the system, constitute about one fifth of all isotopes known at the present time.

Thus, elements with an even atomic number possess, at \(Z \leq 6\), two isotopes with mass numbers differing by one; at \(Z > 8\) the number of isotopes is more than two up to \(Z < 16\), i.e. for those elements for which only the first substitution interval exists, each element corresponds to 3 isotopes with mass numbers differing by one, since the odd elements lying in this interval are simple and fill every fourth odd mass number. Since for \(^{37}\mathrm{Cl}\) and \(^{39}\mathrm{K}\) there is a double substitution of the even number \(m - Z\), only isotopes with odd mass numbers are possible for argon, which lies between Cl and K. The same holds for Ce, which is situated between La and Pr. Since, beginning with Cl, odd elements may have more than one isotope and, at the same time, at \(Z > 16\) (or 20) the substitution interval for even elements widens, the existence of more than three isotopes per element is possible for this group of even elements.

With the exception of Ar and Ce, all the remaining elements have either 3 or 5 isotopes whose mass numbers differ by one. Five-isotope groups appear where, in the neighboring odd elements, the even numbers \(m - Z\) remain unsubstituted. These groups always begin and end with even mass numbers; in addition, in many cases one or two more even mass numbers lying above or below the given group are substituted. The value (suitability) of the above selection rule first became clear to the reporter after Bainbridge\(^{68,75}\) established that the previously considered true isotopes \(^{65}\mathrm{Zn}\), \(^{69}\mathrm{Zn}\), \(^{71}\mathrm{Ge}\), \(^{75}\mathrm{Ge}\), and \(^{77}\mathrm{Ge}\), which are isobars of isotopes with odd atomic number \(^{65}\mathrm{Cu}\), \(^{69}\mathrm{Ga}\), \(^{71}\mathrm{Ga}\), \(^{75}\mathrm{As}\), and \(^{75}\mathrm{As} + 2\), in reality are hybrids. Since the isotopes of the elements H, Ar, Ca, Ti, Rh, Hf, and the rare earths (40 isotopes in all), discovered in this year by Aston\(^{41,41a}\), Dempster and de Gier\(^{21a}\), as well as Loucher, Smith, and Bleakney\(^{65}\), all without exception obey the selection rule established above, the reporter considers it possible to state this rule here, although it still has some exceptions.

The first exception is \(^{87}\mathrm{Rb}\), which is an isobar of \(^{87}\mathrm{Sr}\). If Rb owes its \(\beta\)-radioactivity to the heavier isotope, then we should have excluded this isotope from consideration altogether, for the rule indicated above is valid only for stable isotopes.

The next exceptions are both tin isotopes \(^{115}\mathrm{Sn}\) and \(^{121}\mathrm{Sn}\), isobars of \(^{115}\mathrm{In}\) and \(^{121}\mathrm{Sb}\). Since for the homologous elements Ge and Pb, Bainbridge and Aston\(^{38}\) proved the formation

hydrides, an analogous suspicion is also possible concerning the nature of both isotopes of Sn. The suspicion is strengthened if one pays attention to the r. a. of the Sn isotopes. If in the present case one uses the rule established by Aston for Pb, it turns out that \(^{121}\mathrm{Sn}\) may be a hydride of the most abundant isotope \(^{120}\mathrm{Sn}\). The r. a. ratio \(^{115}\mathrm{Sn} : {}^{114}\mathrm{Sn}\), it is true, is considerably greater than \(^{121}\mathrm{Sn} : {}^{120}\mathrm{Sn}\), but this may be caused simply by inaccuracy in the r. a. measurements for such rare isotopes. At the same time it should be noted that the two indicated isotopes of Sn are the only ones with respect to which there is possible suspicion that they are hydrides.

The following are exceptions: the weak isotope \(^{123}\mathrm{Te}\)—the isobar \(^{123}\mathrm{Sb}\), the still doubtful isotope \(^{127}\mathrm{Te}\)—the isobar \(^{127}\mathrm{I}\), and the weakest isotope of osmium \(^{187}\mathrm{Os}\)—the isobar \(^{187}\mathrm{Re}\). If these isotopes prove to be real, then the above selection rule should be restricted by pointing to the rarity of those cases in which isotopes of elements with odd atomic number have isobars. Predictions concerning isotopes with even atomic number and following from the selection rule are based on the regularities of isotopes with odd atomic number. It has not yet been possible to establish a general rule which would indicate precisely which isotopes with odd atomic number can and must exist.

If we attempt to extend, with the aid of our rule, the substitution of elements up to \(Z = 83\) (the last element, apart from Th and U, studied with the aid of the mass spectrograph), then we find first of all that the isotope of gold 199, proposed by Aston on the basis of measurements of atomic weight, is impossible. It would be an isobar of firmly established \(^{199}\mathrm{Hg}\). The contradiction with the data on the atomic weight of gold arising from this is no more serious than for Cs, which, as has already been indicated, is a simple element.

If we now take as a hypothesis that Ir is also a simple element with mass number 193, then for platinum, lying between Au and Ir, the mass numbers 194, 195, 196 remain, or also 192 and 198—depending on the substitution interval. Osmium, according to the selection rule, must have two isotopes—191 and 194. In addition, on the basis of the selection rule the existence of the isotopes \(^{197}\mathrm{Hg}\), \(^{203}\mathrm{Hg}\), \(^{203}\mathrm{Pb}\), \(^{205}\mathrm{Pb}\), and \(^{209}\mathrm{Pb}\) should be considered excluded. Somewhat earlier we have already mentioned that there is also a number of experimental facts indicating that the existence of the isotopes listed above is doubtful or, at least, not sufficiently firmly established.

Thus, up to \(Z = 83\), all places which, according to the selection rule and the substitution rule, can be occupied are in fact occupied by isotopes.

It is interesting to note that for two elements whose atomic weight is still unknown, namely \(Z = 43\) (Ma) and \(Z = 61\) (for which

there are already many names, but for which, nevertheless, there still remains no place in the system. The only possible mass numbers for \(Z=43\) would be 95, 97, 99, or 101. However, all these numbers are already occupied by rather widespread isotopes of both neighboring elements, namely \(^{95}\mathrm{Mo}\), \(^{97}\mathrm{Mo}\), \(^{99}\mathrm{Ru}\), \(^{101}\mathrm{Ru}\). The same is also true for the mass numbers which, according to the selection rule, should have corresponded to isotopes of element 61. These mass numbers—143, 145, 147, and 149—are already occupied by isotopes. It is possible that this explains the extraordinary rarity of both of the indicated elements.

It should also be mentioned that recently Bartlett \(^{234}\), on the basis of quite different premises, predicted the existence of a number of new isotopes. His predictions partly coincide with those made by us, while some of them contradict the selection rule stated by us. Moreover, a number of isotopes whose existence is assumed by us are absent in Bartlett’s work.

LITERATURE

1. B. V. Boltwood. Amer. Journ. Sci., 22, 537, 1906; 24, 370, 1907.
2. B. Keetman, Jahrb. Radioakt., 6, 269, 1909.
3. C. Auer v. Welsbach, Wien. Ber., 11a, 119, 1011, 1910.
4. W. Marckwald, Ber., 40, 3420, 1910.
5. F. Soddy, Trans. Chem. Soc., 99, 72, 1911.
6. —, Chem. Soc. Ann. Rep. Soc., 285, 1910.
7. A. G. Russell u. R. Rossi, Proc. Roy. Soc., (A) 77, 478, 1912.
8. K. Fajans, Z. Physik, 14, 131, 1913.
9. E. Rutherford a. E. N. da C. Andrade, Phil. Mag., 27, 854, 1914.
10. J. J. Thomson, Rays of positive electricity, London, 1913.
11. F. W. Aston, Proc. Camb. Phil. Soc., 19, 317, 1920.
12. —, Proc. Roy. Soc., (A) 89, 440, 1914.
13. G. P. Thomson, Phil. Mag., 42, 857, 1921.
14. R. Conrad, Z. Physik, 31, 888, 1930.
15. O. Elsenhut u. R. Conrad, Z. Elektrochem., 36, 654, 1930.
16. R. Conrad, Z. Physik, 75, 504, 1932.
17. E. Rüchardt, Naturw., 18, 534, 1930; Handb. d. Phys., 22, 2, S. 96.
18. G. Hertz, Z. Physik, 79, 108, 1932; see Naturwiss., 20, 493, 1932.
19. H. Harmsen, Z. Physik, 82, 589, 1933.
20. H. Lukanow u. W. Schütze, Z. Physik, 82, 610, 1933.
21. P. Zeeman u. J. de Gier, Proc. Amst., 36, 609, 716, 1933; 37, 2, 1934.
    21a. P. Zeeman, Proc. Amst., 37, 127, 1934.
22. F. W. Aston a. R. H. Fowler, Phil. Mag., 43, 514, 1922.
23. F. W. Aston, Phil. Mag., 39, 449, 1920.
24. —, Phil. Mag., 39, 611, 1920.
25. —, Phil. Mag., 40, 628, 1920.
26. —, Phil. Mag., 42, 140, 1921.

1. F. W. Aston, Phil. Mag., 42, 436, 1921.
2. ” Phil. Mag., 45, 934, 1923.
3. ” Phil. Mag., 47, 385, 1924.
4. ” Phil. Mag., 49, 1191, 1925.
5. ” Proc. Roy. Soc., (A) 103, 462, 1923.
6. ” Proc. Camb. Phil. Soc., 22, 548, 1925.
7. ” Proc. Roy. Soc., (A) 115, 487, 1927; Nature, 116, 208, 1925; 117, 893, 1926; 119, 489, 1927; 120, 956, 1927.
8. F. W. Aston, Proc. Roy. Soc., (A) 126, 511, 1930.
9. ” Proc. Roy. Soc., (A) 130, 302, 1931; Nature, 122, 345, 1928; 126, 200, 348, 1930.
10. F. W. Aston, Proc. Roy. Soc., (A) 132, 487, 1931; Nature, 122, 167, 1928; 126, 913, 1930; 127, 233, 591, 1931.
11. F. W. Aston, Proc. Roy. Soc., (A) 134, 571, 1932; Nature, 127, 813, 1931; 128, 140, 221, 725, 1931.
12. F. W. Aston, Proc. Roy. Soc., (A) 140, 535, 1933; Nature 120, 224, 1927; 123, 313, 1929; 129, 649, 1932.
13. F. W. Aston, Nature, 130, 21, 1932; 123, 488, 1929; 126, 913, 1930.
14. ” Nature, 130, 130, 1932.
15. ” Nature, 132, 930, 1933; 133, 327, 1934.
    41a. ” Nature, 133, 613, 869, 1934.
16. J. L. da Costa, Ann. de Phys., 4, 425, 1925.
17. K. P. Jakowlew, Z. Physik, 64, 378, 1930.
18. G. Stetter, Z. Physik, 34, 158, 1925; 42, 741, 1927.
19. A. J. Dempster, Phys. Rev., 11, 316, 1918.
20. ” Phys. Rev., 18, 415, 1921; Phys. Rev., 17, 427, 1921; Proc. Nat. Acad. Amst., 7, 45, 1921.
21. A. J. Dempster, Phys. Rev., 20, 631, 1922; Science, 54, 516, 1921; Phys. Rev., 19, 271, 431, 1922; 21, 209, 1923.
22. M. Morand, Ann. Phys., 7, 164, 1927; G. R., 182, 460, 1926.
23. J. L. Hundley, Phys. Rev., 30, 864, 1927.
24. T. R. Hogness a. H. M. Kvalnes, Nature, 122, 441, 1928.
25. K. T. Bainbridge, Phys. Rev., 34, 752, 1929.
26. ” Phys. Rev., 36, 1668, 1930.
27. ” J. Frankl. Inst., 212, 317, 1931.
28. ” J. Frankl. Inst., 212, 489, 1931; Phys. Rev., 37, 1717, 1931.
29. G. P. Harnwell a. W. Bleakney, Phys. Rev., 45, 117, 1934.
30. H. Kallmann u. W. Lasareff, Z. Physik, 80, 237, 1933.
31. W. Bleakney, Phys. Rev., 43, 1056, 1933.
32. LeRoy, L. Barnes a. R. C. Gibbs, Phys. Rev., 40, 318, 1932.
33. H. Murawkin, Ann. Phys., 8, 203, 353, 974, 1931.
    59a. J. T. Tate, P. T. Smith a. A. L. Vaughan, Phys. Rev., 43, 1054, 1933.
    59b. A. L. Vaughan, J. H. Williams a. J. T. Tate, Bull. Am. Phys. Soc., 9, 3, 23, 1934.
34. W. Bleakney, Phys. Rev., 40, 496, 1932; 40, 130, 1932.
35. ” Phys. Rev., 41, 32, 1932; 39, 536, 1932.
36. ” a. A. J. Gould, Phys. Rev., 44, 265, 1933; 45, 281, 1934.
37. H. Kallmann u. W. Lasareff, Naturw., 20, 206, 472, 1932.
38. J. T. Tate a. P. T. Smith, Phys. Rev., 43, 672, 1933.
39. P. T. Smith, W. W. Lozier a. W. Bleakney, Phys. Rev., 45, 655, 1934.
    65a. W. W. Lozier, P. T. Smith a. W. Bleakney, Phys. Rev., 45, 655, 1934.
    65b. J. P. Harnwell, H. D. Smyth, S. N. van Voorhis a. J. B. H. Kuper, Phys. Rev., 45, 655, 769, 1934.

1. W. R. Smythe, L. H. Rumbaugh and S. S. West, Phys. Rev., 45, 220, 1934.
2. W. Bleakney, Phys. Rev., 34, 157, 1929.
3. K. T. Bainbridge, Phys. Rev., 39, 847, 1932.
4. ” Phys. Rev., 39, 1021, 1932.
5. ” Phys. Rev., 40, 130, 1932.
6. ” Phys. Rev., 42, 1, 1932; 41, 115, 396, 1932.
7. ” Phys. Rev., 43, 103, 1933.
8. ” Phys. Rev., 43, 367, 1933.
9. ” Phys. Rev., 43, 424, 1933.
10. ” Phys. Rev., 43, 378, 1056, 1060, 1933.
11. ” Phys. Rev., 44, 56, 57, 1933.
12. ” Phys. Rev., 44, 123, 1933.
13. ” J. Frankl. Inst., 215, 509, 1933.
14. W. R. Smythe, Phys. Rev., 28, 1275, 1926.
15. ” and J. Mattauch, Phys. Rev., 40, 429, 1932.
16. J. Mattauch, Phys. Z., 33, 899, 1932.
17. A. Hughes and V. Rojansky, Phys. Rev., 34, 284, 1929.
18. R. Herzog and J. Mattauch, Ann. Phys., 19, 345, 1934.
    83a. W. R. Smythe, Phys. Rev., 45, 299, 1934.
19. D. H. Sloan and E. O. Lawrence, Phys. Rev., 38, 2021, 1931.
20. E. O. Lawrence and M. S. Livingston, Phys. Rev., 40, 19, 1932; 45, 608, 1934.
21. F. G. Dunnington, Phys. Rev., 43, 404, 1933; 42, 734, 1933.
22. W. Bartky and A. J. Dempster, Phys. Rev., 33, 1019, 1929.
23. H. Bondy and K. Popper, Ann. Phys., 17, 28, 1933.
24. W. Henneberg, Ann. Phys., 19, 335, 1934.
25. B. Herzog, Z. Physik, 89, 447, 1934.
    90a. W. E. Stephens, Phys. Rev., 45, 513, 1934.
26. A. E. Shaw, Phys. Rev., 44, 1006, 1933.
27. J. Mattauch and R. Herzog, Z. Phys. (in press).
28. F. W. Loomis, Astrophys. J., 52, 248, 1920.
29. A. Kratzer, Z. Physik, 3, 460, 1920.
30. A. Haas, Z. Physik, 4, 68, 1921.
31. R. S. Mulliken, Phys. Rev., 25, 119, 259, 1925.
32. G. E. Gibson, Z. Physik, 50, 692, 1928.
33. R. T. Birge, Trans. Far. Soc., 25, 718, 1929.
34. J. Patkowski and W. E. Curtis, Trans. Farad. Soc., 25, 725, 1929.
35. J. L. Dunham, Phys. Rev., 36, 1553, 1930.
36. G. Stenwinkel, Nature, 126, 649, 1930.
37. A. Elliott, Diss., Utrecht 1930; Proc. Roy. Soc., (A) 123, 629, 1929; (A) 127, 638, 1930; Nature, 126, 133, 1930.
38. A. Elliott, Z. Physik, 67, 75, 1931; Nature, 126, 203, 845, 1930.
39. R. S. Mulliken, Phys. Rev., 26, 319, 1925; Science, 58, 164, 1923; Phys. Rev., 23, 554, 1924; Nature, 113, 423, 1924; 116, 14, 1925.
    104a. A. McKellar, Phys. Rev., 45, 761, 1934.
40. G. H. Dieke and H. D. Babcock, Proc. Nat. Acad. Am., 13, 670, 1927.
41. R. S. Mulliken, Phys. Rev., 32, 880, 1928.
42. W. F. Giauque and H. L. Johnston, J. Am. Chem. Soc., 51, 1436, 1929; Nature, 123, 318, 1929; Phys. Rev., 34, 540, 1929.
43. H. D. Babcock, Proc. Nat. Acad. Amer., 15, 471, 1929; Nature, 123, 761, 1929.
44. W. F. Giauque and H. L. Johnston, J. Amer. Chem. Soc., 51, 3528, 1929; Nature, 123, 831, 1929.
45. R. T. Birge, Nature, 124, 13, 1929.
46. R. Mecke and K. Wurm, Z. Physik, 61, 37, 1930.
47. H. D. Babcock, W. P. Hoge, R. T. Birge, Phys. Rev., 37, 227, 233, 1931; Abstracts, 6, 7, 29.

METHODS AND RESULTS OF ISOTOPE RESEARCH

1. R. Mecke and W. H. J. Childs, Z. Physik, 68, 362, 1931.
2. A. S. King and R. T. Birge, Phys. Rev., 34, 376, 1929; Nature, 124, 127, 1929.
3. R. T. Birge, Phys. Rev., 34, 379, 1929; Nature, 124, 182, 1929.
4. A. S. King and R. T. Birge, Astrophys. J., 72, 19, 1930.
5. F. A. Jenkins and L. S. Ornstein, Proc. Amst., 35, 1212, 1933.
6. S. M. Naudé, Phys. Rev., 36, 333, 1930; 34, 1498, 1929; 35, 130, 1930.
7. G. Herzberg, Z. physik. Chem., (B) 9, 43, 1930.
8. G. M. Murphy and H. C. Urey, Phys. Rev., 41, 141, 1932.
9. R. T. Birge, Phys. Rev., 37, 841, 1931.
10. W. W. Watson and A. E. Parker, Phys. Rev., 37, 167, 1931; W. W. Watson, Phys. Rev., 36, 1019, 1930.
11. E. Olsson, Z. Physik, 73, 732, 1932.
12. E. Svensson, Nature, 131, 28, 1933.
13. E. S. Imes, Astrophys. J., 50, 251, 1919.
14. Ch. F. Meyer and A. A. Levin, Phys. Rev., 34, 44, 1929.
15. H. Becker, Z. Physik, 59, 583, 601, 1930.
16. G. Hettner and J. Böhme, Z. Physik, 72, 95, 1931.
17. J. D. Hardy and G. B. B. M. Sutherland, Phys. Rev., 41, 471, 1932.
18. J. D. Hardy, E. F. Barker and D. M. Dennison, Phys. Rev., 42, 279, 1932.
19. A. Bramley, Phys. Rev., 44, 309, 1933.
20. M. F. Ashley, Phys. Rev., 43, 770, 1933.
21. H. L. Johnston, Phys. Rev., 45, 79, 1932; D. H. Dawson, Naturwiss., 21, 495, 1933.
22. K. Chamberlain and H. B. Cutter, Phys. Rev., 43, 771, 1933.
    134a. W. Holst and E. Hulthén, Nature, 133, 294, 796, 1934.
    134b. G. Dieke and R. W. Blue, Nature, 133, 611, 1934.
    134c. C. R. Jeppeson, Phys. Rev., 45, 480, 1934.
    134d. H. Beutler and K. Mie, Naturwiss., 22, 419, 1934.
23. A. Harvey and F. A. Jenkins, Phys. Rev., 35, 789, 1930; 36, 1413, 1930.
24. G. Nakamura, Nature, 128, 759, 1931.
25. W. R. van Wijk and A. J. van Koeveringe, Proc. Roy. Soc., (A) 132, 98, 1931.
26. A. McKellar, Phys. Rev., 44, 155, 1933; 43, 215, 1933; F. A. Jenkins and A. McKellar, Phys. Rev., 44, 325, 1933.
27. R. S. Mulliken, Phys. Rev., 25, 259, 1925; 23, 554, 1924; Nature, 113, 423, 1924.
28. F. A. Jenkins, Proc. Nat. Acad., Amer., 13, 496, 1927.
29. R. T. Paton and G. M. Almy, Phys. Rev., 37, 1710, 1931.
30. F. A. Jenkins and A. McKellar, Phys. Rev., 42, 464, 1932; 39, 549, 1932.
31. W. W. Watson, Phys. Rev., 27, 801, 1926.
32. W. B. Pearse, Proc. Roy. Soc., (A) 122, 442, 1929.
33. F. A. Jenkins and R. Grinfeld, Phys. Rev., 43, 943, 1933; 45, 229, 1934.
34. F. A. Jenkins and H. de Laszlo, Proc. Roy. Soc., (A) 122, 103, 1929.
    146a. P. C. Mahanti, Z. Physik, 88, 550, 1934.
35. K. Hedfeld, Z. Physik, 68, 610, 1931.
36. M. Ashley and F. A. Jenkins, Phys. Rev., 37, 1712, 1931; 42, 43, 1932.
37. A. Petrikain and J. Hochberg, Z. Physik, 86, 214, 1933.
38. W. Jevons, Proc. Roy. Soc., (A) 110, 365, 1926.
39. W. F. C. Ferguson, Phys. Rev., 32, 607, 1928.
40. E. D. Wilson, Phys. Rev., 32, 611, 1928.
41. O. Darbyshire, Phys. Rev., 40, 366, 1932.

1. W. G. Brown and G. F. Cibson, Phys. Rev., 40, 529, 1932.
2. W. F. C. Ferguson, Phys. Rev., 31, 969, 1928.
3. K. Wieland, Helv. Phys. Acta, 2, 46, 1929.
4. K. Butkow, Z. Physik, 58, 232, 1929.
5. R. Ritschl and D. Villars. Naturwiss., 16, 219, 1928.
6. R. Ritschl, Z. Physik, 42, 172, 1927.

7. R. S. Mulliken, Phys. Rev., 26, 1, 1925; 23, 767, 1924; Nature, 113, 489, 1924.

8. A. Bramley, Phys. Rev., 44, 309, 1933.

9. C. V. Shapiro, R. C. Gibbs and A. W. Laubengayer, Phys. Rev., 40, 354, 1932.
10. W. G. Brown, Phys. Rev., 38, 1179, 1931; 39, 777, 1932.
    163a. H. J. Plumley, Phys. Rev., 45, 678, 1934.
11. E. Bengtsson and E. Losson, Z. Physik, 72, 163, 1931.
12. B. A. Brice, Phys. Rev., 35, 960, 1930; 38, 658, 1931.
13. S. M. Naudé, Phys. Rev., 45, 280, 1934.
14. E. Hulthén, Nature, 129, 56, 1932.
15. S. Mrozowski, Z. Physik, 72, 776, 1931; Nature, 129, 399, 1932.
16. S. Bloomenthal, Phys. Rev., 35, 34, 1930.
17. L. Grebe and H. Konen, Phys. Z., 22, 546, 1921.
18. E. O. Salant and J. E. Rosenthal, Phys. Rev., 42, 812, 1932.
    171a. A. Adel, Phys. Rev., 45, 56, 1934.
19. V. Henri and O. R. Howell, Proc. Roy. Soc., (A) 128, 178, 1930.
20. C. F. Goodeve and C. P. Stein, Trans. Farad. Soc., 25, 738, 1929.
    173a. E. Bartholomé and K. Clusius, Naturwiss., 22, 420, 1934.
    173b. E. Bartholomé and B. W. Sorge, Phys. Rev., 45, 757, 1934.
    173c. J. Franck and R. W. Wood, Phys. Rev., 45, 667, 1934.
21. H. C. Urey, F. G. Brickwedde and G. M. Murphy, Phys. Rev., 40, 1, 1932; Phys. Rev., 39, 164, 864, 1932.
22. H. C. Urey, F. G. Brickwedde and G. M. Murphy, Phys. Rev., 40, 464, 1932.
23. D. H. Rank, Phys. Rev., 42, 446, 1932.
24. St. S. Ballard and H. E. White, Phys. Rev., 43, 941, 1933.
25. G. N. Lewis and F. H. Spedding, Phys. Rev., 43, 964, 1933.
26. G. Hertz, Naturwiss., 21, 884, 1933.
27. H. Schüler, Naturwiss., 19, 772, 1931.
    180a. L. S. Ornstein, J. A. Vreeswijk jr. and G. Wolfsohn, Physica, 1, 53, 1934.
28. L. S. Ornstein and J. A. Vreeswijk jr., Z. Physik, 80, 57, 1933.
29. G. Hansen, Naturwiss., 15, 163, 1927.
30. E. Thomas and E. S. Evans, Phil. Mag., (7) 10, 128, 1930.
31. H. Nagaoka and T. Mishima, Sci. Pap. Tokyo, 13, 293, 1930.
32. L. S. Ornstein and J. A. Vreeswijk jr., Z. Physik, 75, 109, 1932.
33. D. T. Hughes and C. Eckart, Phys. Rev., 36, 694, 1930.
34. H. Schüler, Z. Physik, 58, 741, 1929; 66, 431, 1930.
35. P. Güttinger and W. Pauli, Z. Physik, 67, 743, 1931.
36. D. S. Hughes, Phys. Rev., 38, 857, 1931; H. Schüler and E. Wurm, Naturwiss., 15, 971, 1927.
37. J. H. Bartlett and J. J. Gibbons, Phys. Rev., 44, 325, 538, 1933.
38. B. H. Dickinson, Phys. Rev., 44, 329, 1933.
39. H. Schüler and E. G. Jones, Z. Physik, 76, 14, 1932.
40. J. A. Bartlett, Nature, 128, 408, 1931.
41. G. Racah, Nature, 129, 723, 1932.

1. J. R. Rosenthal and G. Breit, Phys. Rev., 41, 459, 1932; G. Breit, Phys. Rev., 42, 348, 1932.
2. H. Schüler and H. Westmeyer, Z. Physik, 83, 270, 1933.
3. H. Schüler, Naturwiss, 18, 895, 1930.
4. ″ and J. E. Keyston, Z. Physik, 67, 433, 1931.
5. ″ and H. Brück, Z. Physik, 58, 735, 1929.
6. ″ and J. E. Keyston, Z. Physik, 70, 1, 1931; Naturwiss., 19, 320, 1931.
7. H. Schüler and E. G. Jones, Z. Physik, 75, 536, 1932; Naturwiss., 20, 171, 1932; Nature, 129, 833, 1932.
8. S. Tolansky, Z. Physik, 73, 470, 1931.
9. R. Ritschl, Z. Physik, 79, 1, 1932.
10. H. Schüler and H. Westmeyer, Z. Physik, 81, 565, 1933.
    204a. F. Paschen and J. S. Campbell, Naturwiss., 22, 136, 1934.
11. P. G. Kruger, R. C. Gibbs and R. C. Williams, Phys. Rev., 41, 322, 1932.
12. N. S. Grace and K. R. More, Phys. Rev., 45, 166, 1934; N. S. Grace and H. E. White, Phys. Rev., 43, 1039, 1933; N. S. Grace and K. R. More, Phys. Rev., 44, 128, 1933.
13. H. Schüler and J. E. Keyston, Z. Physik, 72, 423, 1931; Naturwiss., 19, 676, 1931.
14. L. Aronberg, Proc. Nat. Acad. Amer., 3, 710, 1917; Astrophys. J., 47, 96, 1918.
15. T. R. Merton, Proc. Roy. Soc., (A) 96, 388, 1920; 100, 84, 1921.
16. H. Kopfermann, Z. Physik, 75, 363, 1932; Naturwiss., 19, 400, 675, 1931.
17. T. C. McLennan, A. B. McLay and M. F. Crawford, Proc. Roy. Soc., (A) 133, 652, 1931.
18. J. L. Rose and L. P. Granath, Phys. Rev., 40, 760, 1932; 39, 1017, 1932; 40, 467, 1932.
19. K. Murakawa, Sci. Pap. Tokyo, 18, 191, 245, 1932.
20. H. Kopfermann, Z. Physik, 83, 417, 1933.
21. D. A. Jackson, Z. Physik, 86, 131, 1933; Proc. Roy. Soc., (A) 139, 673, 1933.
22. H. Kopfermann and N. Wieth-Knudsen, Z. Physik, 85, 353, 1933; Naturwiss., 21, 547, 1933.
23. H. Kopfermann, 21, 704, 1933; E. Rindal, Z. Physik, 87, 460, 1934.
24. J. S. Campbell, Nature, 131, 204, 1933.
    218a. S. Tolansky, Nature, 133, 531, 1934.
25. H. Schüler and H. Westmeyer, Z. Physik, 82, 685, 1933.
26. W. F. Giaugue, Nature, 124, 265, 1929.
27. J. Chadwick, J. E. R. Constable and E. C. Pollard, Proc. Roy. Soc., (A) 130, 463, 1931.
    221a. M. L. Oliphant, P. Harteck and Lord Rutherford, Nature, 133, 413, 1934.
28. F. Allison, J. Chem. Soc., 10, 70, 1933; Phys. Rev., 43, 38—50, 1933.
29. D. R. Morey and J. S. Webb, Phys. Rev., 44, 517, 1933.
    223a. G. P. Baxter and J. S. Thomas, J. Amer. Chem. Soc., 55, 858, 1933; 56, 1108, 1932.
30. J. Kendall, W. W. Smith and T. Tait, Nature, 131, 688, 1933.
31. E. W. Washburn and H. C. Urey, Proc. Nat. Acad. Amer., 18, 496, 1932.
32. G. N. Lewis and R. T. Macdonald, J. Chem. Phys., 1, 341, 1933.
33. R. T. Birge and D. H. Menzel, Phys. Rev., 37, 1660, 1931.
34. M. Woodhead and R. Whytlaw-Gray, J. Chem. Soc., 846, 1933.

*

1. G. Beck, Z. Physik, 48, 407, 1928.
2. W. D. Harkins, Chem. Rev. 5, 371, 1928.
3. H. A. Berton, Phys. Rev., 34, 1228, 1929; 35, 408, 1930; H. C. Urey and H. Johnston, Phys. Rev., 35, 869, 1930.
4. H. L. Johnston, J. Amer. Chem. Soc., 53, 2866, 1931; 54, 824, 1932.
5. H. C. Urey, J. Amer. Chem. Soc., 53, 2872, 1931.
6. J. H. Bartlett jr., Nature, 130, 165, 1932; Phys. Rev., 42, 145, 1932; H. C. Urey, Nature, 130, 403, 1932.
   234a. J. H. Bartlett, Phys. Rev., 45, 847, 1932.