ISOTOPE SPECTRAL ANALYSIS

A. R. Striganov

1. INTRODUCTION

Most chemical elements are mixtures of several isotopes that differ in atomic weights. To study isotopic composition, a number of methods are used that are based either directly on differences in the atomic weights of isotopes (the mass-spectrometric method), or on differences in their physical properties (the densitometric method, the refractometric method of water analysis, analysis of gases by their thermal conductivity, methods of analysis by the radioactive properties of isotopes, and spectral methods based on atomic and molecular spectra).

The most effective and universal is the mass-spectrometric method, by means of which all the basic data on the isotopic composition of natural elements have been obtained and which is successfully used for the analysis of separated and enriched products. All the other methods are less universal and in most cases have lower accuracy. An important advantage of these methods over the mass-spectrometric one, however, is the considerably greater availability and low cost of the apparatus required for their implementation, as well as the simplicity of the methods themselves, which in a number of cases do not require a high level of training on the part of the operator.

Almost all the methods of isotope analysis mentioned above are considered, to one degree or another, in the very valuable book by A. I. Brodskii.^1 However, very little is said in that book about isotope spectral analysis by atomic spectra, although this method has long been used to investigate the isotopic composition of natural elements and separated products, and has been used especially successfully in recent years for isotope analysis of a number of important objects. It therefore seems useful to us to elucidate the foundations of the method of spectral analysis for determining

isotopic composition from atomic spectra and, within the limits of a short article, to give a survey of the published works on isotopic spectral analysis.

2. FUNDAMENTALS OF THE METHOD OF ISOTOPIC SPECTRAL ANALYSIS BY ATOMIC EMISSION SPECTRA

The basis of isotopic spectral analysis is formed by the same facts that are characteristic of ordinary spectral analysis of elements. By about 1930 it had been established that each isotope entering into the composition of a given element corresponds to its own characteristic components, into which ordinary spectral lines are split. In addition, it was found that the intensity of the components belonging to a given isotope increases with an increase in the concentration of this isotope in the sample. On this basis it was obvious that, by means of the spectral method, both qualitative and quantitative analyses of isotopic composition could be carried out by essentially the same procedures that at that time were accepted for the spectral analysis of elements.

These possibilities were successfully used in investigations of the isotopic composition of a number of elements. As a result, new isotopes were discovered by spectroscopic means, and for some elements the relative abundance of isotopes was found from the ratio of the intensities of the corresponding components. In a number of cases the spectral method was also successfully used to determine the degree of enrichment in the separation of isotopes.

In order to give a more complete idea of the essence of the method of isotopic spectral analysis, it is necessary to dwell briefly on the structure of spectral lines, and also to consider the question of the dependence of the intensity ratio of two components, belonging to different isotopes, on the relative concentration of these isotopes.

3. STRUCTURE OF SPECTRAL LINES

The electron shells of isotopes of one and the same element are identical both in their structure and in the number of electrons. Hence, owing to identical electronic configurations, the optical spectra of such isotopes outwardly do not differ from one another and fit into a single serial scheme of multiplets. However, careful investigations carried out with the aid of spectral instruments of high resolving power show that some spectral lines of almost all elements are complex, consisting of a number of components situated close to one another.

As an example, Figs. 1 and 2 show the complex structures of the spectral lines of cerium and rubidium, obtained with high-resolution instruments by exciting ordinary cerium and rubidium in a hollow-cathode discharge tube. The structure of the Ce II 4628.2 Å line in each order of interference consists, as we see, of two components belonging to the principal even-even isotopes Ce\(^{140}\) and Ce\(^{142}\), whose abundances in natural cerium are respectively 88.48 and 11.07%. In this case each isotope of cerium corresponds to its own and only one component belonging to the structure of the given spectral line. A somewhat different picture is obtained in the spectrum of rubidium. The structure of the Rb I 7800.2 Å line is represented by four components, although natural rubidium contains only two isotopes, Rb\(^{85}\) and Rb\(^{87}\), in concentrations of respectively 72.8 and 27.2%. Of these four components, the two middle components \(A\) and \(B\) belong to Rb\(^{85}\), whereas the two outer components \(a\) and \(b\) belong to Rb\(^{87}\). Thus,

Fig. 1. Isotopic structure of the Ce II 4628.2 Å line.

Fig. 2. Hyperfine structure of the resonance line Rb I 7800.2 Å.

in the case of rubidium, each isotope corresponds to two components, which make up the structure of the given spectral line.

In the examples considered, the structure of the cerium line is a purely isotopic structure, caused by the isotopic shift of the lines, whereas in the rubidium line, in addition to the isotopic shift, there appears the so-called hyperfine structure, which is caused by the splitting of the lines of one and the same isotope into several components. From this it is evident that two effects play a role in the formation of the complex structure of lines: isotopic shift and hyperfine splitting of lines. Both these effects are the result of the interaction of the electronic shell of the atom with the nucleus, and usually in the structure of lines they appear together. Therefore, generally speaking, the complex structure of the lines of most elements is the result of the superposition of hyperfine structure on isotopic structure.

From what has been said it is clear that the structure of the spectral lines of many elements consisting of several isotopes will have a rather complex character, since, in addition to the isotopic structure arising as a result of the different isotopic shift of the given line for each isotope, the lines of some of these iso-

isotopes give a hyperfine structure in the form of an additional splitting into a number of components. In other words, the spectral lines of such elements consist of unsplit and split components of the individual isotopes.

In Fig. 3, as an example, the structure of the line Hg I 5460.7 Å is schematically represented. This line consists of 15 components belonging to the six principal isotopes of mercury (the content of the seventh stable isotope Hg$^{196}$ in a natural sample is insufficient for detecting its components).

Fig. 3. Isotopic and hyperfine structures of the line Hg I 5460.7 Å.

Four components of this complex structure, 1, 2, 3, 4, are the displaced lines of the isotopes Hg$^{198}$, Hg$^{200}$, Hg$^{202}$, and Hg$^{204}$; three components $A$, $B$, $C$ belong to the split line of the isotope Hg$^{199}$; the remaining eight $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ belong to the split line of the isotope Hg$^{201}$. Components $e$ and $c$ coincide with one another; all the others were observed separately. The height of each component of the structure corresponds approximately to the relative intensity of this component in comparison with the others.

Fig. 4. Isotopic structure of the line Hg I 5460.7 Å.

We see that the structure of the mercury line under consideration is a complex picture. It consists of isotopic structure and hyperfine structure; moreover, only the lines of two isotopes, Hg$^{199}$ and Hg$^{201}$, undergo hyperfine splitting, whereas isotopic displacement is inherent in the lines of all mercury isotopes. The latter follows directly from Fig. 4, where dashed lines show the positions of the centers of gravity*) of the individual components of the isotopes Hg$^{199}$ and Hg$^{201}$ along with the components of the other isotopes.

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

From the examples given it is clear that the degree of complexity of the line structure is determined by three conditions: 1) the number of isotopes that make up the given element; 2) the number of isotope lines which, in addition to the isotopic shift, undergo hyperfine splitting; 3) the number of components into which the lines of these isotopes are split. The first condition is connected, as we see, with the isotopic shift, while the second and third are connected with hyperfine splitting.

The isotopic structure in atomic spectra arises as a result of the displacement, relative to one another, of the energy levels of atoms of different isotopes of one and the same element*). As an example, Fig. 5 gives a scheme of transitions for the Ce II 4628.2 Å line in the case of two isotopes, Ce\(^{140}\) and Ce\(^{142}\). To the left, opposite the upper and lower levels, are given the term symbols; to the right are given the mass numbers of the isotopes to which the corresponding terms belong. As a result of the displacement of the lower level of the isotopes Ce\(^{140}\) and Ce\(^{142}\), there is obtained, as follows from experiment (Fig. 1), a doublet structure of the line, represented schematically at the bottom of Fig. 5. The distance between the components is equal to \(0.054\ \mathrm{cm}^{-1}\).

Fig. 5. Scheme of levels and structure of the Ce II 4628.2 Å line.

It is not difficult to see that, regardless of whether the lower or upper term undergoes isotopic displacement, or both terms are displaced, the number of components of the isotopic structure of the lines will be equal to the number of isotopes that make up the given element. Thus, if an isotopic shift appears, then the complexity of the isotopic structure is directly determined by the isotopic composition of the element.

The isotopic composition of natural elements is very diverse. Elements with an odd atomic number \(Z\) contain no more than two stable isotopes, and 22 of them are simple, consisting of a single isotope (F, Na, Al, P, Sc, etc.). The majority of elements with even \(Z\) consist of 3–7 stable isotopes. A simple element in this group is beryllium; the two-isotope elements include helium and carbon; the greatest number of stable isotopes is possessed by tin (10 isotopes), xenon (9 isotopes), cadmium and tellurium (8 isotopes each).

*) The isotopic effect in the atomic spectra of light, medium, and heavy elements was recently examined in detail by A. R. Striganov and Yu. P. Dontsov\(^2\).

Proceeding from this, one can understand that the simplest isotopic structure will occur in elements with odd atomic number \(Z\); a more complex isotopic structure will occur in elements with even \(Z\) (with the exception of Be, He, and C). The concentration of a given isotope in a given element is also of substantial importance. Experience shows that isotope components whose concentration is below 1%, as a rule, do not appear in the isotopic structure of a line obtained with high-resolution instruments. This means that the isotopic structure of the spectral lines of some elements with 3–4 isotopes will consist of 1–2 components (O, Ne, S, A, Ca, Ce). Likewise, the isotopic structure of the elements most complex in number of isotopes (Sn, Xe, Cd, Te) will be no more complex than the structure of certain seven-isotope elements, such as Mo, Nd, Sm.

The hyperfine structure in atomic spectra is the result of splitting of the energy levels of an atom due to the magnetic interaction between the nucleus and the electron shell*). Theory shows that if a given energy level is characterized by the quantum number \(J\), corresponding to the total angular momentum of the electron shell, and the atomic nucleus is characterized by the quantum number \(I\) of the nuclear angular momentum (nuclear spin)**), then the number of split sublevels is determined as follows:

\[ \left. \begin{array}{ll} \text{for } J \geqslant I & \text{there will be } 2I+1 \text{ sublevels,}\\ \text{for } J \leqslant I & \text{there will be } 2J+1 \text{ sublevels.} \end{array} \right\} \tag{1} \]

Knowing the quantum numbers \(J\) and \(I\), it is easy to determine the values of the quantum number \(F\) of the total angular momentum of the atom for each sublevel by the formula

\[ F=(J+I),\ (J+I-1),\ \ldots,\ |J-I|. \tag{2} \]

It is easy to see that the levels of all atoms for which the spin \(I=0\) will always be single. Likewise, all levels for which the quantum number \(J=0\) will be single. If the nuclear spin \(I=\tfrac{1}{2}\), then all levels with \(J>0\) will split into two sublevels. Similarly, for nuclei with \(I>0\), all levels with \(J=\tfrac{1}{2}\) will be doublets.

Knowing the structure of the levels in an atom, one can determine the structure of the given spectral line. For this it is necessary to take into account the known selection rules:

\[ \Delta F=0,\ \pm 1, \tag{3} \]

except for the cases

\[ F_1=0 \to F_2=0, \]

*) A detailed discussion of hyperfine structure in atomic spectra is given in S. E. Frisch’s book\(^3\).

**) For brevity, the quantum number \(I\), as well as the angular momentum itself, is customarily called the nuclear spin.

according to which transitions between sublevels are possible only when the quantum number \(F\) either changes by \(\pm 1\), or remains unchanged (except for transitions in which the upper and lower levels correspond to the number \(F=0\)).

Proceeding from the foregoing, the spectral lines of atoms whose nuclei have spins \(I=0\) will always be single, i.e., they will have no hyperfine structure. The lines of all other atoms, whose nuclei have spin \(I>0\), will have a complex structure consisting of two or more components, with the exception of a very few cases of “forbidden” lines, when the upper and lower levels have quantum numbers \(J=0\), or, respectively, \(J=0\) and \(J=2\). Thus, the hyperfine structure of spectral lines is uniquely determined by the two quantum numbers \(I\) and \(J\). The larger the spin \(I\), with at the same time a large quantum number \(J\) for both levels, the more complex the hyperfine structure of the spectral lines will be.

Fig. 6. Level scheme and structure of the line Rb I 7800.2 Å.

In Fig. 6, as an example, a scheme is given of transitions between sublevels in the case of the line RbI 7800.2 Å, a photograph of whose hyperfine structure was given by us in Fig. 2. In this scheme the transitions for two isotopes, \(\mathrm{Rb}^{85}\) and \(\mathrm{Rb}^{87}\), are shown separately. Since the nuclear spins of these isotopes are respectively \(5/2\) and \(3/2\), the upper levels with \(J=3/2\) for the atoms of both isotopes should have split into four sublevels with quantum numbers, respectively, \(F=4,3,2,1\) and \(F=3,2,1,0\). However, the magnitude of the splitting for these sublevels is so small that in practice they may be regarded as unresolved. The lower levels with \(J=1/2\) will split, in the case of both isotopes, into two sublevels with quantum numbers \(F=3\) and \(2\) for \(\mathrm{Rb}^{85}\) and \(F=2\) and \(1\) for \(\mathrm{Rb}^{87}\). The splitting of the lower term of \(\mathrm{Rb}^{85}\) (\(0.096\ \mathrm{cm}^{-1}\)) is approximately half the splitting of the same term of \(\mathrm{Rb}^{87}\) (\(0.220\ \mathrm{cm}^{-1}\)). This is explained by the fact that the magnetic moment of the \(\mathrm{Rb}^{87}\) nucleus (\(\mu=+1.35\) nuclear magnetons) is half the magnetic moment of the \(\mathrm{Rb}^{85}\) nucleus (\(\mu=+2.75\) nuclear magnetons). As a result of transitions, in accordance with the selection rule, from the upper unsplit levels to the lower doublet levels, the spectral lines of both isotopes of rubidium split into two components.

The general hyperfine structure of the rubidium line is shown schematically in Fig. 6 below. Thus, the hyperfine structure of the rubidium line has a complete theoretical explanation.

On the basis of experimental data it has now been established:

1. The spins of even-even nuclei, consisting of an even number of protons and an even number of neutrons, are equal to zero.

2. The spins of even-odd and odd-even nuclei, consisting respectively of an even number of protons and an odd number of neutrons, and conversely, take only half-integral values from \(^{1}/_{2}\) to \(^{9}/_{2}\).

3. The spins of odd-odd nuclei, consisting of odd numbers of protons and neutrons, take integral values from 1 to 7.

It follows from what has been said above that, for every even-even isotope, the structure of a given spectral line always corresponds to only one component. All other isotopes give a more complex structure consisting of several components. Hence it follows that the spectral lines of elements whose composition includes only even-even isotopes possess a purely isotopic structure. The lines of the remaining elements undergo both isotopic displacement and hyperfine splitting.

Thus, the complexity of the hyperfine structure of a spectral line (if the magnitudes of the splitting and the intensities of the components are not taken into account) for elements with even atomic number \(Z\) is determined by the number of even-odd isotopes that make up the given element in spectroscopically detectable concentrations. Similarly, for elements with odd \(Z\), the complexity of the hyperfine structure is determined by the number of odd-even and odd-odd isotopes present in the composition of this element.

Experience shows that the hyperfine structure of spectral lines rapidly increases in its width from element to element with increasing \(Z\). For light elements the hyperfine structure in spectral lines is small. In the spectrum of hydrogen, for example, it cannot be detected even with the most advanced spectroscopic technique of high resolving power. In the spectrum of lithium, on the spark line 5484.7 Å, it reaches approximately \(0.5\ \mathrm{cm}^{-1}\) and is already quite easily resolved with an interferometer. For medium and heavy elements, the hyperfine structure of many spectral lines becomes so broad that in the case of some elements it is resolved by means of a spectrograph with a large diffraction grating having a dispersion of \(1.0\)—\(2.0\ \text{Å}/\mathrm{mm}\). Here one may point to the hyperfine structure of the spark lines of indium, the width of which reaches \(4\)—\(5\ \mathrm{cm}^{-1}\) for a number of lines. The hyperfine structure of 300 InI lines was measured on a diffraction spectrograph with a dispersion of \(2\ \text{Å}/\mathrm{mm}\). In the spark spectrum of praseodymium it was meas—

resolved the hyperfine structure of 200 lines with the aid of a spectrograph with a concave grating and a dispersion of 1.5 Å/mm, the width of the structure reaching \(0.7—1.5\ \mathrm{cm}^{-1}\). Recently the hyperfine structure of americium lines was resolved with the aid of a spectrograph with a six-meter grating and a dispersion of 1.5 Å/mm.

The isotopic structure also varies in its width depending on \(Z\). For light and heavy elements it is so considerable that, on some spectral lines, it can be detected without difficulty with the aid of a comparatively small spectrograph with a dispersion of \(2—5\) Å/mm. An example is the isotopic shift in the spectra of hydrogen and uranium. For the intermediate elements, however, the isotopic structure is so small that it is observed with difficulty with the aid of interferometers of the highest resolving power.

The hyperfine and isotopic structures of the overwhelming majority of natural elements have at present already been studied to one degree or another. The experimental data obtained up to 1952 have been collected in special tables\(^4\). On the basis of these data, using the appropriate sources for exciting spectra and spectroscopic technique, in many cases it is possible to investigate the isotopic composition of various samples. This method, as we shall show below, has been successfully applied to the solution of a number of important scientific and practical problems.

4. DEPENDENCE OF THE INTENSITY RATIO OF TWO ISOTOPIC LINES ON THE RELATIVE CONCENTRATION OF ISOTOPES

The principal task in developing any method of quantitative spectral analysis is to find the dependence between the intensity ratio of two compared lines (of the impurity and of the base of the sample) and the concentration of the element being analyzed. In the case of spectral analysis of elements this dependence is usually found experimentally from standard samples. It is then represented in the form of a so-called calibration graph, which is used for the analysis of unknown samples. We have shown that, in the case of isotopic spectral analysis, the dependence of the intensity ratio of two isotopic lines (or two components of isotopic structure belonging to different isotopes of one and the same element) on the relative concentration of isotopes can be investigated theoretically\(^*\).

As is known, the intensity of a spectral line in the absence of self-absorption in the light source is expressed by the formula

\[ i = N_0 \frac{g_m}{g_0} e^{-\frac{E_m}{kT}} A_{mn} h\nu, \tag{4} \]

where \(N_0\) is the total number of unexcited atoms in the discharge region, \(g_m\) and \(g_0\) are the statistical weights for the upper and lower levels, \(E_m\) is the energy of the upper level, \(k\) is the Boltzmann constant, \(T\) is the absolute temperature of the gas, \(A_{mn}\) is the probability of transition from the upper level \(E_m\) to the lower level \(E_n\), \(h\) is Planck’s constant, and \(\nu\) is the frequency of the emitted line. It may be assumed that the total number of atoms \(N_0\) located in the discharge region is proportional to the concentration \(C\) of the given element in the sample:

\[ N_0=\alpha C. \tag{5} \]

Then we finally obtain:

\[ I=\alpha C\,\frac{g_m}{g_0}\,e^{-\frac{E_m}{kT}}A_{mn}h\nu. \tag{6} \]

This expression may be represented in the form of the formula

\[ I=\alpha\beta C, \tag{7} \]

where the product \(\alpha\beta\) is the coefficient of proportionality; moreover, the quantity \(\alpha\) is determined by the properties of the sample, while the quantity \(\beta\) depends on the nature of the given spectral line.

Let us now express the ratio of the intensities of two isotopic lines in a two-isotope sample. The statistical weights and transition probabilities for identical quantum states, to which the components of the isotopic structure of one and the same spectral line correspond, will be the same for both isotopes. It is also easy to show that the numerical values of the quantities \(e^{-\frac{E_{m_1}-E_{m_2}}{kT}}\) and \(\frac{\nu_1}{\nu_2}\) for any pair of isotopes may, without appreciable error, be taken as equal to unity. Thus, for example, in the case of uranium isotopes (\(\mathrm{U}^{235}\) and \(\mathrm{U}^{238}\)) the product of these quantities for the line \(4244.4\ \text{\AA}\) is \(0.99992\); in the case of hydrogen isotopes (H and D), for the line \(6562.8\ \text{\AA}\), it reaches \(0.99892\). Hence, for the ratio of the intensities of two isotopic lines, a very simple expression is obtained:

\[ \frac{I_1}{I_2}=\frac{\alpha_1 C_1}{\alpha_2 C_2}. \tag{8} \]

The coefficient \(\alpha\) does not depend on the nature of the spectral line. For isotopes of one and the same element, this coefficient will differ only in the case of light elements, for which, owing to the relatively large difference in isotope masses, an appreciable difference is observed in certain physical properties. The physical properties of isotopes of medium and heavy elements are practically identical. Proceeding from this, it may be considered that for them

$x_1 = a_2$, then we obtain:

$$ \frac{I_1}{I_2}=\frac{C_1}{C_2}. \tag{9} $$

Thus, in the case of medium and heavy elements, in the absence of self-absorption of the lines, there is a simple proportionality between the ratio of the intensities of two isotopic components and the relative concentration of the isotopes. The calibration graph, plotted in the commonly used logarithmic coordinates, will be a straight line with a slope of $45^\circ$, passing through the origin at $C_1=C_2=50\%$.

For isotopes of light elements $a_1\ne a_2$. Taking the logarithm of formula (8), we obtain the expression for the calibration graph

$$ \lg \frac{I_1}{I_2}=\lg \frac{C_1}{C_2}+\lg \frac{a_1}{a_2}. \tag{10} $$

This calibration graph will be rectilinear with a slope of $45^\circ$, but, unlike the previous case, it will be shifted along the intensity axis by a segment numerically equal to $\lg \frac{a_1}{a_2}$. The magnitude and direction of this shift are determined by the difference in the physical properties of the two isotopes. Thus, for example, in hydrogen the dissociation energy of the molecules is smaller than in deuterium; therefore $a_1>a_2$, whence $\lg \frac{a_1}{a_2}>0$. Consequently, the calibration graph will shift along the axis $\lg \frac{I_1}{I_2}$ toward positive values. The same should also be observed in the case of other light elements, since the boiling temperature for light isotopes is lower than for heavy ones.

If the Doppler width of a line is comparable with the width of the instrumental function, then Doppler broadening will affect the ratio of the intensities of the isotopic lines, since this effect depends on the atomic weight of the isotopes. It is easy to show that the ratio of the intensities of isotopic lines at the maximum of the distribution curve will in this case be expressed by the formula

$$ \frac{I_1}{I_2}=\frac{a_1C_1}{a_2C_2}\sqrt{\frac{A_1}{A_2}}, \tag{11} $$

where $\sqrt{\frac{A_1}{A_2}}$ is the correction for Doppler broadening, and $A_1$ and $A_2$ are the atomic weights of the isotopes. In the case of heavy elements (for isotopes with atomic weight $A>100$) this correction may be neglected. Therefore, the simple dependence in the form (9) for isotopes

of heavy elements is preserved. For isotopes of light elements the quantity \(\sqrt{\frac{A_1}{A_2}}\), associated with Doppler broadening, has a substantial significance, and therefore the final expression for the calibration graph will be:

\[ \lg \frac{I_1}{I_2}=\lg \frac{C_1}{C_2}+\lg \frac{a_1}{a_2}+\lg \sqrt{\frac{A_1}{A_2}} . \tag{12} \]

It is evident from this expression that Doppler broadening affects only the displacement of the calibration graph. Since \(A_1<A_2\), the quantity \(\lg \sqrt{\frac{A_1}{A_2}}<0\). Thus, as a result of Doppler broadening, the calibration graph will be displaced along the axis \(\lg \frac{I_1}{I_2}\) toward negative values. Since the displacement determined by the quantity \(\lg \frac{a_1}{a_2}\) is directed in the opposite direction, the total displacement will be a superposition of both effects. This displacement can be estimated theoretically and found experimentally by investigating the calibration graph with the aid of standards.

Fig. 7. Calibration graphs in the absence of self-absorption.

In Fig. 7 theoretical calibration graphs are given for isotopic lines in the absence of self-absorption. Line \(I\) is characteristic of isotopes of heavy elements. Line \(II\) refers to isotopes of light elements. The dashed straight lines show where the straight line for light elements would be displaced when the difference in the coefficients \(\alpha_1\) and \(\alpha_2\) is taken into account, and also as a result of the difference in the atomic weights of the isotopes.

In the presence of self-absorption, the calibration graph in the case of isotopic analysis can be expressed as follows:

\[ \lg \frac{I_1}{I_2}=\lg \frac{C_1^{b_1}}{C_2^{b_2}}, \tag{13} \]

ISOTOPIC SPECTRAL ANALYSIS

where \(b_1\) and \(b_2\) are the slope of the “growth curve”1. The quantities \(b_1\) and \(b_2\) characterize the self-absorption of radiation of the given wavelength in a luminous cloud. Their values change depending on the concentration: at low concentrations \(C_1\) (or \(C_2\)) the value \(b_1\) (or \(b_2\)) is equal to 1; at high concentrations \(C_1\) (or \(C_2\)) the value \(b_1\) (or \(b_2\)) reaches a certain minimum value \(b_0\). Since, when the concentration of one isotope increases, the concentration of the other isotope correspondingly decreases, the value of the exponent in formula (13) over the entire concentration interval for one isotope decreases from 1 to \(b_0\), while for the other isotope it increases from \(b_0\) to 1.

Our consideration2 shows, as is evident from Fig. 8, that as a result of self-absorption the calibration graph (curve II) in the case of two isotopic lines may deviate appreciably from the normal straight line which, in the absence of self-absorption, passes at an angle of \(45^\circ\) to the concentration axis (straight line I). In the middle concentration interval the calibration graph in the presence of self-absorption becomes curvilinear, and only small sections of such a graph may approximately be regarded as rectilinear. The angle of inclination of these sections to the concentration axis will be less than \(45^\circ\). At both ends, at high concentrations of one of the isotopes (\(C_1'\) or \(C_2'\), with \(C_1' = C_2'\)), at which self-absorption reaches a maximum \((b_1 = b_2 = b_0)\), the calibration graph is a straight line with a slope close to \(45^\circ\). Both these sections are displaced relative to the origin along the intensity axis by the same amount, equal to \(A_0 = \pm(1 - b_0)\lg C_1'\).

Fig. 8. Calibration graph in the presence of self-absorption.

The true course of calibration graphs may be strongly distorted by the presence of background near the lines being photometered and by the mutual superposition of the compared components of the isotopic structure. When an identical background is superposed on both lines, the calibration graph in the middle concentration interval (Fig. 9) is rotated by

toward the concentration axis near the point that corresponds to lines of equal intensity \(\left(\lg \frac{J_1}{J_2}=0\right)\). In the lower and upper intervals the calibration graph bends, approaching the concentration axis. When background is superimposed on one of the lines being compared, the calibration graph, in addition to a change in slope and curvature at both ends, is shifted to the right or to the left along the abscissa axis (Fig. 9). If the background is taken into account, the calibration graph

Fig. 9. Effect of background on the calibration graph: I — in the absence of background, II — when identical background is superimposed on both lines, III and IV — when background is superimposed on one line.

must straighten out and pass with respect to the concentration axis at an angle of \(45^\circ\). Mutual overlap manifests itself when the intensities of two isotopic lines differ greatly from one another and the bright line of one isotope is superimposed by its wing on the weak line of another isotope. In this case the calibration graph in the lower and upper portions of the concentration interval will bend toward the concentration axis, as shown in Fig. 10. The inflection points \(a_1\) and \(a_2\) are determined by the specific experimental conditions (the distance between the isotopic lines, their width, exposure time, etc.). It is quite clear that if the mutual overlap of partially overlapping spectral lines is taken into account, the calibration graphs should straighten, approxi—

cling at the ends to the normal graph in the absence of overlap.

Thus we see that, in the case of isotopic spectral analysis, in order to obtain reliable results it is necessary to calibrate the method with the aim of establishing the character of the dependence between the quantities \(\dfrac{I_1}{I_2}\) and \(\dfrac{C_1}{C_2}\). The true dependence between these quantities, even in the absence of reabsorption, may be strongly distorted by mutual overlap of the compared lines and by the background in the spectrum. Calibration is necessary in this case in order to establish this distorted empirical dependence, or to find the correct way of estimating the background and the overlap and taking them into account. In light elements there is also added to this the effect of displacement of calibration graphs as a result of the different Doppler broadening of the lines of different isotopes and the different dissociation energies of the molecules, or the different boiling points for these isotopes. Therefore, in the case of isotopic analysis one cannot postulate in advance a simple equality between the quantities \(\dfrac{I_1}{I_2}\) and \(\dfrac{C_1}{C_2}\) without investigating this dependence using standards.

Fig. 10. Effect of mutual overlaps on the calibration graph: \(I\) — in the absence of overlap; \(II\) — in the presence of overlap.

5. APPLICATION OF ISOTOPIC SPECTRAL ANALYSIS

For the study of the isotopic composition of natural elements by means of the spectral method, molecular spectra were first used; in these spectra the isotopic shift in some cases reaches considerable values and is readily detected with spectrographs of medium dispersion. By means of molecular spectra, in 1927–1929 the heavy isotopes of carbon\({}^{7}\) (\(\mathrm{C}^{13}\)), nitrogen\({}^{8}\) (\(\mathrm{N}^{15}\)), and oxygen\({}^{9}\) (\(\mathrm{O}^{17}\) and \(\mathrm{O}^{18}\)) were discovered, which at that time could not be detected by the mass-spectrometric method. On the basis of molecular spectra, the first data (although very approximate) were also obtained on the abundance

these isotopes under natural conditions. In particular, for oxygen the best results were obtained by V. N. Kondrat'ev and D. I. Eropkin^10.

Isotopic and hyperfine structures in atomic spectra were first used for investigating the isotopic composition of elements by Schüller and his collaborators. In 1931 Schüller and Keyston^11, studying the hyperfine structure of arc and spark lines of thallium, established that natural thallium consists of two isotopes, Tl^203 and Tl^205. A year later Schüller and Jones^12 thoroughly analyzed the structure of a number of arc and spark lines of lead and discovered a fourth isotope, Pb^204, which had not been detected by the first mass-spectroscopic investigations. From the intensity of the components they estimated the concentration of this isotope in natural lead at 1%, which agrees quite well with the presently accepted value of 1.37%. In 1932 Urey, Brickwedde, and Murphy^13, using the spectral method, discovered deuterium in enriched hydrogen. From the intensity ratio of the isotopic lines of three members of the Balmer series (Hβ, Hγ, Hδ), they estimated the abundance of heavy hydrogen under natural conditions. It should also be noted that the four principal isotopes of platinum (Pt^194, Pt^195, Pt^196, Pt^198) were discovered by the spectral method^14, and the spectroscopic investigations were carried out before investigations by other methods. On the basis of measurements of the integral intensity of the components of the hyperfine and isotopic structure of a series of lines, the relative abundance of these isotopes in natural platinum^15 was found, which was subsequently confirmed by mass-spectroscopic measurements. Good results were obtained for the abundance of the isotopes of lead. Indeed, the isotopic composition of ordinary lead, found as early as 1936 on the basis of an analysis of the structure of the line Pb II 5372.1 Å, agrees well with the mass-spectrometric data^17 accepted at the present time, as is seen from the following table:

Method	Isotope	208	207	206	204
Spectral . . . . . . . .		51.5	21.3	26.3	0.8%
Mass-spectrometric . . .		52.38	21.11	25.15	1.37%

Great interest was shown in the investigation of the isotopic composition of lead of various origins. By spectroscopic means it was established that the isotopic composition of uranium and thorium

lead differs substantially from one another^18. It turned out that, in the case of lead obtained from uranium ore, the brightest component of the isotopic structure of the lines is the component of the isotope Pb^206, which corresponds to the final product of the radioactive decay of uranium. In exactly the same way, the brightest component in the spectrum of lead obtained from thorium ore is the component of the isotope Pb^208, which is formed as a result of the radioactive transformations of the thorium series. Rose and Stranathan^19, using samples of lead from minerals with different uranium contents, clearly showed the natural enrichment in the isotope Pb^206 that occurs as a result of the decay of primary products during the existence of the Earth. From the relative amounts of the isotopes Pb^206: Pb^207 the geological age of the Earth was estimated, and the value obtained proved to be in good agreement with results found by other methods^20.

The method of spectral quantitative analysis of isotopic composition was successfully used in some cases for monitoring the enrichment process and for determining the degree of enrichment in isotope separation. As early as 1932, Urey, Brickwedde, and Murphy^13 applied this method to the isotopic analysis of enriched hydrogen obtained by fractional distillation of liquid hydrogen. The enriched hydrogen in gaseous form entered Wood’s discharge tube^21, about 1 cm in diameter, and in it was excited by means of a high-voltage transformer, at whose output terminals the voltage reached 3000–4000 V. A six-meter diffraction spectrograph with a second-order dispersion of 1.3 Å/mm was used as the spectral instrument. In the same year, Herz^22 applied spectral analysis to estimate the enrichment of neon with the isotope Ne^22 by the diffusion method. From the ratio of the intensities of the components of the isotopes Ne^22 and Ne^20, the relative concentration of Ne^22 in the enriched fractions was determined with the aid of a Geissler tube. To resolve the components of the isotopic structure, a Fabry–Perot interferometer crossed with a prism spectrograph was used. Exactly the same method was also used for the isotopic analysis of enriched argon^23. In the latter case, in order to obtain narrower lines, the Geissler tube was cooled with liquid air. Later the spectral method was applied to the isotopic analysis of rubidium^24 and thallium^25, separated by means of the electromagnetic method. To excite the spectra of rubidium and thallium, a discharge tube with a hollow cathode, cooled with liquid air, was used. To resolve the isotopic and hyperfine structure of the spectral lines, a Fabry–Perot etalon crossed in both cases with a three-prism spectrograph was employed. In the case of rubidium, a sample weighing 0.05 mg was placed in the discharge tube, and this quantity proved quite sufficient for obtaining the hyperfine structure of the resonance lines.

This, perhaps, was what limited the application of the spectral method of isotope analysis in the first period, which is briefly described in Kopfermann’s book^26. The application of spectral analysis at that time, although not industrial in character, was nevertheless of great importance for studying the isotopic composition of natural elements and of certain enriched products. After the Second World War the spectral method was used for the isotope analysis of such products as heavy water, gaseous mixtures of hydrogen with deuterium, and separated and enriched isotopes of lithium and uranium. Below we shall briefly give a review of the literature on published materials concerning these four objects, which are of great importance for the problem of the utilization of nuclear energy.

6. ISOTOPIC SPECTRAL ANALYSIS OF GASEOUS MIXTURES OF HYDROGEN AND DEUTERIUM

More or less successful attempts to apply the spectral method to the quantitative analysis of gaseous mixtures of hydrogen isotopes were undertaken long ago. Thus, for example, Van Tiggelen^27 analyzed mixtures of H$_2$ and D$_2$ in the concentration range from 15 to 85% deuterium with an accuracy of $\pm 2\%$ of the content of the component of lower concentration. However, only in 1952 were more detailed investigations on this question published. Broïda, Mayer, and Morgane^28 succeeded in developing a sufficiently reliable photoelectric method of spectral analysis of gaseous mixtures of hydrogen and deuterium.

Fig. 11. Optical scheme of the monochromator:
$D$ — light source, $Q$ — condenser,
$S_1$ — entrance slit, $S_2$ — exit slit,
$M$ — spherical mirror, $G$ — diffraction grating,
$P$ — photomultiplier.

Spectral setup. As the spectral instrument, an autocollimating monochromator with a concave mirror ($F = 75$ cm) and a plane diffraction grating 7.5 cm long with 12,000 lines per 1 cm was used. The optical scheme of this instrument is given in Fig. 11. The grating gives the maximum intensity of the spectrum in the first order. The linear dispersion of the instrument in this order reaches 10.4 Å/mm. The resolving power was selected by changing the widths of the entrance and exit slits so that the H and D lines were reliably resolved while remaining sufficiently bright, and so that the height of the peaks corresponded, with good approximation, to the integral intensity of the lines. The width

the entrance and exit slits varied from 20 to 40 μ, and, in order to reduce the recorder lag, the exit slit was taken somewhat wider than the entrance slit. Along with the principal lines of the Balmer series, “ghosts” of Rowland were observed in the spectrum; these were located at distances of 1.2 Å and 2.7 Å from the principal line and amounted to approximately \(1/200\) of its intensity.

The monochromator was equipped with a photoelectric recording device consisting of a photomultiplier, a direct-current amplifier, and a recorder. The amplifier, with a narrow pass band, gave amplification in the range from 2 to 800, as a result of which very small signals could be observed. The lag of the recorder did not exceed 1 sec.*). The duration of a double measurement was less than 1 minute.

Excitation source and vacuum system. As the excitation source of the spectrum, an electrodeless high-frequency discharge was used in a discharge tube connected to the vacuum system. The discharge tube consisted of a glass (Pyrex) cylinder 8 mm in diameter and 10 cm long, placed in a glass jacket for cooling with running water during the discharge. The very first experiments showed that it was impossible to work with sealed tubes, since in the course of the discharge the composition of the sample being analyzed is strongly distorted because of the release of occluded gases. Therefore an apparatus was constructed that made it possible to obtain a regulated gas flow through the discharge tube. The scheme of this vacuum apparatus is given in Fig. 12. It is intended for the preparation and analysis of standard samples, and also for determining the isotopic composition of ready-made \(H_2 + D_2\) mixtures. Standard samples are prepared in a glass three-liter mixing vessel, into which gaseous hydrogen and deuterium are admitted in the required proportions through heated palladium tubes. The pressure in the mixing vessel is measured by means of a mercury U-shaped manometer. The mixing vessel is also used for receiving the ready-made gaseous mixtures to be analyzed, which are drawn off in advance into special sample vessels. For analysis, the gaseous sample from the mixing vessel is admitted by means of tap \(S_1\) into the vacuum system with the discharge tube. The gas flow is regulated by taps \(S_1\) and \(S_2\) and by several capillaries arranged parallel to one another, and is monitored by means of a mercury manometer \(M_F\). The pressure in the discharge tube can be measured either with a small mercury manometer \(M_T\), or with a McLeod gauge \(G\), graduated up to 2 mm Hg. Ballast vessels serve to maintain constant pressure and flow in the discharge tube. Traps intended for capturing water and mercury vapors were cooled with a mixture of dry ice and acetone.

*) A detailed description of the monochromator is given in work \(^{29}\).

The use of liquid air is not recommended, since selective absorption of H\(_2\) on the walls of the traps was found. The system was evacuated by successively arranged diffusion (mercury) and fore-vacuum pumps.

To excite the glow in the tube, a high-frequency generator of 400 W power with a frequency of 150 MHz was used. The generator was connected to the discharge tube by means of a high-frequency cable \(1/4\) wavelength long with a variable-capacitance capacitor connected in parallel for tuning (Fig. 12). To reduce interference that might distort the readings of the electronic circuits and instruments, it is useful to place the supply conductors together with the capacitor in a metal box. The intensity of the glow can be varied over a considerable range by changing the capacitance of the capacitor. Finer changes of intensity are made by changing the cathode resistance. During the analyses the intensity of the tube glow must be kept as constant as possible, since the compared H and D lines are measured not simul—

Fig. 12. Diagram of the vacuum apparatus for the analysis of H\(_2\)+D\(_2\): \(A\)—mixing vessel, \(B\)—ballast vessels of 0.5 and 1 l capacity, \(C\)—cylinder for the analyzed samples, \(F\)—capillary tubes, \(G\)—McLeod manometer, \(K\)—capillary aperture for admitting air, \(L\)—traps; \(M_A\), \(M_F\), and \(M_T\)—mercury manometers, \(P_D\) and \(P_H\)—palladium tubes, \(S_1\) and \(S_2\)—cocks for regulating pressure and flow rate, \(T\)—discharge tube, \(\oplus\)—two-way cocks.

temporally. A check showed that the intensity of the lines can be maintained during the discharge process with a constancy to within 1% per hour, whereas a single measurement lasts 1 minute. The authors tested a generator with a frequency of 2450 MHz and a power of 125 W. With this generator the glow in the tube was 10 times more intense, but considerable self-absorption was observed. The tube cooled by water did not operate in this case, since water at such a frequency loses its dielectric properties.

Selection of spectral lines. For carrying out analyses in the concentration range from 85 to 99% \(D_2\) in \(H_2\), the lines \(H_{\alpha}\) (6562.8 Å) and \(D_{\alpha}\) (6561.0 Å) were used. In Fig. 13 a photoelectric recording of these lines is given at two different speeds

Fig. 13. Photoelectric curve of the spectrum \(H_2 + D_2\).

(2 and 10 Å/min), with the width of the entrance slit of the monochromator equal to 15 μ, and the width of the exit slit 25 μ.

Selection of optimal working conditions. As already noted, the dependence of the ratio of the intensities of two isotopic lines on the relative concentration of the isotopes in the sample under investigation is expressed by formula (8). Here the proportionality coefficient \(\frac{a_1}{a_2}\) in the case of hydrogen may vary with the pressure and flow rate in the discharge tube, with the character of the discharge, etc. In developing the procedure it is desirable to select such experimental conditions under which the proportionality coefficient in formula (8) would be close to unity, and the ratio of the intensities of the deuterium and hydrogen lines \(\frac{I_D}{I_H}\) would be a function of the relative concentration

of these isotopes \(\dfrac{C_D}{C_H}\) in the sample under study and would not depend on small fluctuations in the experimental conditions. In other words, from a practical point of view the stability conditions require that changes in the quantity \(\dfrac{I_D}{I_H}\), caused by fluctuations of pressure, discharge temperature, etc., be incomparably smaller than the changes due to fluctuations in the quantity \(\dfrac{C_D}{C_H}\). In this connection the authors, having set themselves the task of increasing the accuracy and sensitivity of the method, studied in detail the influence on the intensity ratio \(\dfrac{I_D}{I_H}\) of pressure, flow rate, glow intensity in the discharge tube, generator parameters, the dimensions of the tube, and its position in front of the monochromator slit.

Fig. 14. Dependence of the line intensity ratio on pressure.

The experimental dependence of the intensity ratio \(\dfrac{I_D}{I_H}\) in the case of a sample with a concentration of 93.0% \(D_2\) is shown in Fig. 14 for three lines of the Balmer series (\(\alpha\), \(\beta\), and \(\gamma\)). The curves show that the ratio \(\dfrac{I_D}{I_H}\) decreases with increasing pressure; moreover, at low pressures this decrease is approximately proportional to the reciprocal of the pressure, while at higher pressures it is linear in character. At high pressures the smallest value of \(\dfrac{I_D}{I_H}\) is given by the \(\alpha\) line, and at low pressures by the \(\gamma\) line. This means that some fraction of the effect is due to atomic excitation.

In Fig. 15, using the same experimental data, the intensity of the lines is presented as a function of pressure. The maximum intensity of each spectral line is taken here as unity, and the intensities of these same lines at other pressures are taken relative to the corresponding maximum (it should be remembered that in fact the deuterium line is 13 times brighter than the hydrogen line). It is easy to see that in all three cases the deuterium line reaches its maximum at a lower pressure than the hydrogen line. In addition, it should be noted that the \(\gamma\) line reaches maximum inten-

... at the lowest pressure; after it comes line β, and then, already, line α, which gives a maximum at the highest pressure.

There is as yet no complete theoretical explanation of the discovered phenomenon of the dependence of the intensity ratio of the D and H lines on pressure. However, from a practical point of view, investigation of this effect is of great importance. The data obtained indicate that, when carrying out analyses, it is necessary to adhere strictly to a definite pressure in the discharge tube, reliably monitoring it with a manometer.

Fig. 15. Dependence of line intensities on pressure.

Fig. 16. Dependence of the intensity ratio on pressure at different flow velocities.

In Fig. 16 is given the experimentally found dependence of the intensity ratio \(\frac{I_D}{I_H}\) on pressure at different flow velocities in the discharge tube \((0.3;\ 1.0;\ 3.8\ \mathrm{cm^3/min})\). The graph shows that an increase in the flow velocity leads to an increase in the value of \(\frac{I_D}{I_H}\). From this it is clear how important it is to keep the flow velocity constant during operation. In the authors’ apparatus, the pressure drop in the manometer \(M_p\) (Fig. 12) was maintained within the limits \(19.0—22.0\ \mathrm{mm}\) Hg, which corresponds to limiting flow values of \(0.90—1.20\ \mathrm{cm^3/min}\). The ratio \(\frac{I_D}{I_H}\) under these conditions remained practically constant.

In order to find the effect of a change in the brightness of the discharge glow on the ratio \(\frac{I_D}{I_H}\), the distance between the capacitor plates was gradually increased (Fig. 12), as a result of which the glow intensity decreased. These experiments showed that at intermediate pressures (approximately \(0.5\)—\(1.0\) mm Hg) the ratio \(\frac{I_D}{I_H}\) decreases somewhat, while at lower and higher pressures it increases as the intensity of the H and D lines decreases. Hence follows the necessity of keeping the electrical conditions of the discharge unchanged (generator tuning, current strength, etc.).

A decrease in glow intensity also occurs in the absence of a cooling mixture in the traps. This is explained by the fact that part of the energy is spent on exciting mercury vapor. However, the ratio \(\frac{I_D}{I_H}\) does not change in this case. Similarly, when the discharge tube is changed from water cooling to air cooling, the glow intensity decreases, but in this case as well the ratio \(\frac{I_D}{I_H}\) changes very little.

Specially conducted investigations, the results of which are presented in Table 1, show that the intensity of the discharge glow and the ratio \(\frac{I_D}{I_H}\) depend quite noticeably on the diameter of the discharge tube.

Table 1

Tube diameter in mm	Pressure in the tube in mm Hg	\(I_D\)	\(I_H\)	\(\dfrac{I_D}{I_D + I_H} \cdot 100\)
2	0.520	268.8	29.36	90.14
4	0.550	83.8	6.69	92.60
8	0.510	22.92	1.686	93.13

If the increase in \(I_H\) with decreasing tube diameter occurs in the ratio \(1:3.8:17\), then the corresponding decrease in the cross section of the tube is expressed by the ratio \(1:4:16\). This means that the energy is absorbed by the gas independently of the diameter of the tube. The decrease in the value \(\frac{I_D}{I_D + I_H}\) with decreasing tube diameter is apparently due to an increase in self-absorption of the brighter D line.

At a given concentration of the sample, the value of \(\dfrac{I_D}{I_H}\) undergoes noticeable changes depending on the position of the discharge tube relative to the monochromator slit. Thus, for example, at a pressure of \(0.09\) mm Hg the ratio \(\dfrac{I_D}{I_H}\) for the lines \(\alpha\), \(\beta\), and \(\gamma\) is approximately 5% smaller if not the middle of the tube but its edge is in front of the slit (the transverse position of the tube relative to the optical axis is meant; the axis of the tube is directed along the slit). In the longitudinal position of the tube (when observing from the end, when the tube is directed along the optical axis), the value of \(\dfrac{I_D}{I_H}\) for the lines \(\beta\) and \(\gamma\) remains the same as when observing the middle of the tube in its transverse position; however, for the line \(\alpha\) this value is 30% smaller. At a pressure of \(0.6\) mm Hg the ratio \(\dfrac{I_D}{I_H}\) for the line \(\alpha\), when observed from the end, also proved to be considerably smaller. It is evident that the decrease found in the case of the line \(\alpha\), for both pressures, in the value of \(\dfrac{I_D}{I_H}\) when observed from the end is due to more pronounced self-absorption in the thickness of the discharge.

The data presented show how important the correct choice of the dimensions of the discharge tube and of its position relative to the slit of the spectral instrument is. It is necessary, in subsequent work, to adhere strictly to the dimensions and position of the tube once chosen.

Calibration of the method and accuracy. In connection with the discovered dependence of the intensity ratio \(\dfrac{I_D}{I_H}\) on a number of factors, it became necessary to calibrate the method using standards. The task of calibration was to select such working conditions under which the measured intensity ratio would directly give the concentration value. Electrolytic hydrogen and deuterium containing 0.3% \(H_2\) were used for preparing the standard samples. The pressures were read from manometer \(M_A\) (Fig. 12) with an accuracy of up to \(\pm 0.2\) mm Hg. If the temperature in the mixer is controlled, the accuracy of preparing standard samples can be brought to \(\pm 0.1\%\).

The reliability of any method of analysis is characterized by the accuracy, reproducibility, and correctness of the results. Accuracy is determined by the error or deviation of a single measurement from the mean of a given series of measurements obtained under identical conditions. Reproducibility is the error that characterizes the variation of the results of analysis when they are carried out

over a considerable period of time. Correctness is understood as the deviation of the results of analysis from the true concentration. It is easy to see that reproducibility includes the error of the measurements, which characterizes precision; correctness is determined by the precision and reproducibility of the measurements, as well as by the error in the initial concentrations of the reference samples. Usually these three notions of error in the results of analysis are characterized by the arithmetic mean or root-mean-square error, which may refer both to individual determinations and to the mean result of the analysis.

Table II presents the results of 10 determinations of one and the same sample, carried out over the course of 15 minutes, which characterize

Table II

$I_D$	$I_H$	$\dfrac{I_D}{I_H}$	$C_D$ in %	$\Delta C_D$
28,04	3,104	9,033	90,03	−0,04
28,16	3,104	9,072	90,07	−0,01
28,28	3,112	9,087	90,09	+0,09
28,20	3,108	9,073	90,07	0,00
28,24	3,116	9,062	90,06	−0,02
28,36	3,108	9,124	90,12	+0,04
28,24	3,112	9,074	90,07	0,00
28,28	3,116	9,075	90,07	0,00
28,24	3,108	9,086	90,08	+0,01
28,28	3,116	9,075	90,07	0,00
Mean		9,076	90,075	±0,02

the precision of the method at constant pressure (0.365 mm Hg) and flow rate. It follows from the table that the mean absolute error of an individual determination is ±0.02%. Experiments to evaluate reproducibility gave a mean error of ±0.07%. A check of the correctness of the analyses on a single reference sample showed that the absolute discrepancy between the measured value of the concentration and the true value (mass-spectrometric analysis) reaches ±0.35% for a concentration of 90%. This shows that the authors did not calibrate the method sufficiently well against standards.

Specially conducted experiments to study the influence of air on the results of analysis showed that additions of air to the mixture

$\mathrm{D}_2 + \mathrm{H}_2$, reaching even 50%, introduce no changes into the intensity ratio $\dfrac{I_{\mathrm{D}}}{I_{\mathrm{H}}}$.

Conclusions. As a result of the investigations carried out, a method has been developed for the isotope analysis of gaseous mixtures $\mathrm{D}_2 + \mathrm{H}_2$ in the deuterium concentration range of 85–99%. The method can also be applied to lower concentrations. The mean absolute error for a 90% sample is $\pm 0.35\%$, and the relative error is $\pm 0.4\%$. With more thorough calibration, the accuracy of the analyses can be substantially improved. In the case of the other component of the mixture ($\mathrm{H}_2$), the relative error of the spectral method will be approximately $\pm 1$–$2\%$ for a 10% concentration.

On the basis of literature data, the mean relative error of the mass-spectrometric method of isotope analysis of hydrogen and deuterium is approximately $\pm 2$–$3\%$30. This means that, in accuracy, the optical spectral method is not inferior to the mass-spectrometric method. As regards the independence of the results from extraneous impurities in the sample under investigation, the speed of the analyses, and their low cost, all the advantages are undoubtedly on the side of the spectral method.

7. ISOTOPIC SPECTRAL ANALYSIS OF HEAVY WATER

In Kirshenbaum’s well-known monograph30, only modest space is allotted to the spectral method of analyzing the isotopic composition of heavy water. The author indicates that the spectroscopic method is not very accurate and can be used only when a spectrograph of high resolving power is available. No experimental results characterizing the accuracy and reproducibility of the spectral method are given in the monograph. It is therefore quite appropriate to dwell on the recently published work of Brody, Morowitz, and Selgin31, who carried out detailed studies on the spectrometric determination of hydrogen isotopes in enriched water. As a result, they developed a photoelectric method of spectral quantitative analysis of the isotopic composition of water in the concentration range from 0.015 to 99% $\mathrm{D}_2\mathrm{O}$ in $\mathrm{H}_2\mathrm{O}$.

Spectral apparatus, excitation source, and vacuum system. The spectral apparatus and excitation source used by the authors of the above-mentioned work are described in the preceding section. The discharge tube, cooled by water, was connected to a vacuum system by means of which a stream of water vapor was produced. The diagram of the vacuum apparatus is shown in Fig. 17. It consists of two independent branches. The left branch is intended for pumping water vapor by means of diffusion (oil) and fore-vacuum pumps through the discharge tube. The right

branch serves to evacuate, by means of a single fore-vacuum pump, the space above the sample under investigation before the vapors are admitted into the left-hand part of the system. The sample vessels are provided with standard conical glass ground joints, which ensures rapid replacement of samples. Admission of the vapors into the left-hand part of the vacuum system is carried out by means of stopcocks located above the vessels. To regulate the pressure

Fig. 17. Diagram of the vacuum apparatus for the analysis of heavy water: \(A\)—sample vessels, \(K\)—capillary, \(B\)—valve, \(T\)—discharge tube, \(M\)—ionization manometer, \(Л\)—trap, \(\oplus\)—two-way stopcock.

in the discharge tube, parallel capillaries and a metal, slowly opening valve were used. The pressure in both branches of the system is measured in turn by means of an ionization manometer calculated for the pressure range from \(10^{-4}\) to \(10\) mm Hg. The high-frequency discharge was excited by means of a generator with a power of 150 W and a frequency of 150 MHz.

Choice of spectral lines. For carrying out the analyses the lines \(H_{\beta}\) (4861.3 Å) and \(D_{\beta}\) (4860.0 Å) were used; in photoelectric recording these gave large peaks, which is explained by the maximum sensitivity of the photomultiplier in this region of the spectrum. Figure 18 presents the pattern of the arrangement of the peaks of the lines \(H_{\beta}\), \(D_{\beta}\), and the “ghost” \(G_{H}\), obtained in recording the spectrum of a sample with a \(D_{2}O\) concentration of about 1%. In the concentration interval \(1\)–\(99\%\) \(D_{2}O\) in \(H_{2}O\), the intensities of the lines \(D_{\beta}\) and \(H_{\beta}\) were measured for carrying out the analyses, whereas for the concentration interval \(0.015\)–\(1\%\) \(D_{2}O\) it was more convenient to compare the intensities of \(D_{\beta}\) and \(G_{H}\). The possibility of such a compar-

...does not raise objections, since the ratio of the intensity of the “ghost” to the intensity of the line itself remains constant for the given optical system. In order to eliminate any arbitrariness, the base line of the line (or of the “ghost”) and the background were photometered at definite distances from the peak of the effect. This does not introduce errors into the final results, since the spectra of the standards, when the calibration curves are established, are photometered in the same way.

Choice of optimal working conditions. It turned out that contamination from previous samples manifests itself most strongly when a metal valve is used to admit water vapor into the system with the discharge tube, although the latter is indispensable for fine pressure adjustment. Therefore, in the case of serial analyses, it is recommended that for admission one use a glass stopcock and a capillary (Fig. 17), which, even with large differences in the concentrations of the introduced vapors, ensure cleaning of the vacuum system to within 1% in the course of three minutes and complete cleaning in the course of one hour. An ionization manometer, which has many metal parts, slows the removal of traces from previous samples. Therefore, during the measurement process it is recommended to switch off the manometer and to monitor the pressure by the change in the length of the discharge in the tube.

Fig. 18. Photoelectric curve of the spectrum of an \( \mathrm{H_2O} + \mathrm{D_2O} \) mixture.

Fig. 19. Effect of pressure on the intensity ratio.

The experimental material on the effect of the pressure in the tube on the ratio of the intensities of the \(D_\beta\) and \(H_\beta\) lines is presented in Fig. 19. Here

the dependence of the ratio of intensities \(\frac{I_D}{I_H}\) on pressure variation is given for seven samples of different concentrations of \(\mathrm{D_2O}\) in \(\mathrm{H_2O}\) (0, 3, 5, 10, 29, 50, 76, and 99%). The data obtained show that at a pressure of \(0.4\) mm Hg the ratio \(\frac{I_D}{I_H}\) changes by no more than 0.2% when the pressure is changed by 0.01 mm Hg. This means that if the pressure in the discharge tube is maintained with an accuracy of up to 0.01 mm Hg, then the relative error depending on pressure does not exceed 0.2%. The dependence of the total intensity \(I_H + I_D\) on pressure is shown in Fig. 20. This dependence proved to be the same for different concentrations of \(\mathrm{D_2O}\). The curve reaches a maximum at \(0.4\) mm Hg. This pressure

Fig. 20. Dependence of the total intensity \(I_H + I_D\) on pressure.

was adopted for the analyses, since in this case it is possible to work at the greatest intensity on the flat portion of the curve.

It should be noted that when the pressure in the discharge tube is changed, the flow rate also changes; therefore the experimental data mentioned above (Figs. 19 and 20) characterize the combined effect. The flow rate is a function of the pressure in the tube, the geometry of the vacuum system, and also depends on the pump capacity. For concentrations below 50% \(\mathrm{D_2O}\), the authors established that, at constant pressure, the ratio \(\frac{I_D}{I_H}\) increases with increasing flow rate (the flow rate is determined by weighing the sample before and after the experiment). For analyses the flow rate is selected depending on the size of the sample and the rate of measurement.

Because of the considerable difference in the vapor pressures of saturated vapors of H₂O and D₂O at room temperature, an evaporating water sample H₂O + D₂O will become enriched in D₂O. In order to determine the error associated with this effect, a 0.5 ml sample was introduced into the apparatus and then evaporated completely; during the evaporation, measurements were made of the ratio \(\frac{I_D}{I_H}\). The results obtained are presented in Fig. 21. From this curve it is possible to find the size of the minimum sample, and also to estimate the magnitude of the error associated with fractional evaporation. The relative vapor pressure of H₂O, D₂O, and DHO varies with the temperature of the sample. Experimental data show that the ratio \(\frac{I_D}{I_H}\) changes by 0.15% when the temperature changes by one degree. For the analyses, samples of up to 1 ml in volume at room temperature were taken.

Fig. 21. Effect of fractional evaporation.

The character of the discharge may change with changes in the current intensity, the tuning of the supply circuit with a capacitance, the position of the electrodes relative to the discharge tube, and the temperature of the water in the jacket. Experience shows that a change in some of these parameters has a substantial effect on the magnitude of \(\frac{I_D}{I_H}\). Therefore, in order to avoid additional errors in the analyses, it is necessary to keep the current intensity and the water temperature constant and not to change the tuning of the circuit or the position of the electrodes. The analyses were carried out at a current intensity of 175 mA. The tuning of the generator corresponded to the maximum-length discharge in the tube. The temperature of the water with which the tube was cooled was room temperature.

Calibration graphs and accuracy. To construct the calibration curves, standard samples were prepared in the form of mixtures of distilled water and heavy water of 99.8% enrichment,

which were weighed with an accuracy of up to 0.1 mg. Figures 22 and 23 give calibration plots for two concentration ranges

Fig. 22. Calibration curve for the concentration range 0–1% \(D_2O\).

Fig. 23. Calibration curve for the concentration range 1–80% \(D_2O\).

of heavy water in \(H_2O\). The analyses carried out showed that the spectral method has high accuracy and good reproducibil-

ness. Table III gives the absolute and relative root-mean-square error of the analysis for the concentration interval 0.01–99.8% \(D_2O\) under optimal conditions.

Table III

\(D_2O\) concentration, %	Root-mean-square error, %	Root-mean-square error, %
	absolute	relative
99.8	0.002	0.002
99.0	0.001	0.001
90.9	0.01	0.01
50.0	0.25	0.05
9.09	0.01	0.01
0.99	0.001	0.1
0.10	0.0007	0.7
0.01	0.0002	2.0

Fig. 24 shows the reproducibility in determining the isotopic composition of a sample with a concentration of 21% \(D_2O\). Here are given the results

Fig. 24. Results of analyses of four identical samples.

of analyses obtained at different times. Each point represents the average of 10 determinations with limits of the root-mean-square

errors. The mean of all readings is represented by a solid horizontal line. The two dashed lines give the limits of the mean-square error for the entire series of determinations. These data confirm the very high accuracy of the spectral method. Experience shows that the duration of a single analysis does not exceed 1 hour.

Conclusions. The accuracy of the mass-spectrometric method usually used for analyzing the isotopic composition of water is estimated, as is known[^30], at $\pm 2—5\%$. Thus we see that the method of optical spectroscopy is not inferior in accuracy to the mass-spectrometric method, while possessing, in comparison with the latter, a number of unquestionable advantages. Indeed, in the case of optical spectral analysis, water vapor is analyzed directly, unlike in the mass-spectrometric method. Here it is not necessary to decompose the water and convert it into the gaseous state. Another important advantage is that the results of spectral analysis are not affected by extraneous contaminants in the sample, and therefore no preliminary purification is required. It follows from this that the method of spectral analysis of the isotopic composition of water is faster and cheaper in comparison with many other methods.

In addition to the analysis of water, the spectroscopic method is already being used for the study of aqueous hydrate compounds and water-containing crystals, and for the determination of labile hydrogen that is part of molecules[^31]. This method is also being successfully used in medicine and biology to determine the content of $\mathrm{D_2O}$ in blood, to study water exchange in arteries, capillaries, extremities, etc.[^32]

8. ISOTOPIC SPECTRAL ANALYSIS OF LITHIUM

Exhaustive data on the structure of the spectral lines of lithium were first obtained by Schüler and his collaborators[^33], who reliably resolved the two brightest lines, Li I 6707.8 Å and Li II 5484.7 Å. It turned out that the spark line 5484.7 Å consists of 13 components arising as a result of multiplet and hyperfine splitting, as well as isotopic shifts of the terms.

In the structure of the arc line 6707.8 Å, three components were found that are the result of multiplet splitting and isotopic shift. Detailed data on the isotopic shift of the resonance line Li I 6707.8 Å were obtained by Hughes[^34]. In 1934 this line was used by Ornstein and his collaborators[^35] to determine the relative abundance of isotopes in natural lithium. It was established that the ratio $\frac{\mathrm{Li}^7}{\mathrm{Li}^6}$

is equal to 8.1. This value differs rather strongly from the presently accepted value of 12.7, found by the mass-spectrometric method[^17]. However, later, carefully performed determinations[^38], carried out by the photoelectric method with allowance for instrumental errors and self-absorption of the Li I 6707.8 Å line, gave a closer value, 13.5.

Much attention in recent years has been devoted to developing a method of isotope spectral analysis of enriched lithium. In 1952 a paper was published by Stakenbroker, Smith, Werker, and MacNelly[^36], who applied a photographic method for these purposes. Later the same authors published a paper[^37] in which they described a photoelectric method developed by them in the search for a better solution. Detailed investigations devoted to the development of a photoelectric method for analyzing the isotopic composition of lithium were carried out at the same time by Brody, Fred, and Tomkins[^38]. Below, on the basis of the two above-mentioned papers[^37,^36], a brief description will be given and the accuracy of the photographic and photoelectric methods will be considered. After this, the most interesting data from the third paper[^38] will be briefly presented.

A. Photographic method

Experimental setup. If one has in mind the Li I 6707.8 Å line, then, on the basis of the Rayleigh criterion, in order to resolve two isotopic lines of Li⁶ and Li⁷ equal in intensity and separated from one another by 0.16 Å, a resolving power of the order of 40,000 is required. To separate lines that differ strongly in intensity, a considerably greater resolving power is required. Accordingly, for isotope analysis of lithium a six-inch diffraction grating was used, with a radius of curvature of 6.4 m and with 12,000 lines per centimeter, mounted according to the Paschen–Runge scheme. This spectral setup provided, in the first order, a dispersion of 1.2 Å/mm and a resolving power of 75,000.

In isotope analysis of lithium, a low-temperature source is required in order to reduce Doppler broadening, together with a low vapor density so as to reduce self-absorption. As such a source, a hollow-cathode discharge tube was used, allowing the sample to be replaced rapidly. An improved design of such a tube, recommended by the authors in another paper[^37], is shown in Fig. 27 (see below, in the description of the photoelectric method). During the discharge the cathode was cooled (by means of a special jacket) with running water. The discharge current was of the order of 35 mA. Helium was used as the working gas, which during the discharge circulated through the tube and a charcoal trap. The scheme of the circulating vacuum ...

of the system is shown in Fig. 25. During the discharge, the pressure in the tube was maintained at \(2\) mm Hg, and it was monitored by means of an ionization vacuum gauge with an accuracy of up to 2%.

The samples to be analyzed were taken in the form of the compound \(\mathrm{Li_2SO_4}\), weighing \(4\) mg, and were placed directly in the hollow of the cathode. After this, distilled water was poured in, which

Fig. 25. Circulating vacuum system: \(T\)—tube, \(L\)—trap cooled with liquid nitrogen, \(M\)—manometer, \(A\)—cylinder with helium, \(D\)—diffusion pump, \(B\)—charcoal trap cooled with liquid nitrogen, \(C\)—ionization manometer.

was then evaporated in a furnace. As a result, lithium sulfate was deposited on the walls of the cathode cavity in the form of a uniform layer.

In order to make it possible to work with lines differing greatly in intensity, a rotating sector was placed before the slit of the spectrograph, weakening the upper half of the spectrum by a factor of 10. For each sample, six spectra were photographed with an exposure of 60 sec. The measurements were made with a densitometer. The ratio of intensities was reproduced on a given photographic plate with an accuracy of up to 1%. To control the exposures, a photomultiplier was used, which was set to the \(\mathrm{Li}\) \(4602.9\) Å line and was connected to an electrometer.

Selection of spectral lines. The most suitable spectral line for carrying out the analyses proved to be the line

Li I 6707.8 Å. It is the most intense line in the lithium spectrum that, in comparison with other lines, undergoes the greatest isotopic shift. The Li II 5484.7 Å line, although it has twice as large an isotopic shift, is harder to excite and its structure is much more complicated than that of the Li 6707.8 Å line.

Figure 26 schematically shows, below, the structure of the resonance line 6707.8 Å, and above gives the scheme of transitions explaining this structure. From the figure it is clear that, in addition to the isotopic shift, each of the spectral lines of the isotopes Li\(^6\) and Li\(^7\) is split, as a result of multiplet splitting of the upper term, into two components \((a\) and \(b,\ c\) and \(d)\). However, owing to equality of the magnitude of the isotopic shift and the width of the splitting, the short-wavelength component of the isotope Li\(^6\) coincides with the long-wavelength component of the isotope Li\(^7\). Therefore in the structure of the 6707.8 Å line, instead of four components, three are observed, of which the two extreme ones \(a\) and \(d\) belong respectively to the isotopes Li\(^6\) and Li\(^7\), while the middle one is the combined component \(b+c\) of the same isotopes.

Fig. 26. Structure of the Li I 6707.8 Å line.

Since the theoretical intensity ratio of the doublets \(I_a:I_b\) and \(I_c:I_d\) is \(1:2\), in the absence of self-absorption the intensity ratio of the three observed components in natural lithium is expressed as follows:

\[ I_a:I_{b+c}:I_d = 2.4:35.5:62.1. \]

When the 6707.8 Å line is used for spectral analysis, the intensities of the extreme components are measured. If the theoretical intensity ratio of the lines of the doublet is taken into account, then the ratio of the isotopic lines by which the isotope concentration in the sample is determined will be expressed as follows:

\[ \frac{I_6}{I_7}=2\frac{I'_6}{I'_7}, \]

where \(I'_6\) is the intensity of the weak component of the Li\(^6\) line, and \(I''_7\) is the intensity of the strong component of the Li\(^7\) line.

Selection of optimal working conditions. To select the experimental conditions ensuring the best reproducibility of the intensity ratio \(\frac{I_6}{I_7}\), the geometry of the discharge tube, the effect of the pressure in the tube, the effect of the sample size, the current strength, and the spacing between the anode and cathode were investigated.

On the basis of these studies it was found that the line intensities and their ratio \(\frac{I_6}{I_7}\) vary with the change in helium pressure in the tube, reaching a maximum at approximately 2 mm Hg. This pressure was adopted as the working pressure. The limiting sample was found to be 10 mg; samples larger by weight lead to greater self-absorption. For carrying out analyses, samples weighing 2 mg were used. During continuous burning of the discharge for 10 hours, the intensity ratio \(\frac{I_6}{I_7}\) decreased by approximately 3%, while the duration of photographing the spectra did not exceed 20 minutes.

The experimentally found dependence between the ratio \(\frac{I_6}{I_7}\) and the current strength shows that the value of \(\frac{I_6}{I_7}\) increases with increasing current strength. This is explained by the greater self-absorption of the line of the isotope Li\(^7\), which is present in the samples at higher concentrations than the isotope Li\(^6\). It is therefore recommended to work at the minimally low current strength—35 mA.

Calibration graphs and accuracy. To carry out the analyses, a calibration graph was constructed in the concentration range from 2.5 to 10% Li\(^6\) in Li\(^7\). The standards were analyzed by the mass-spectrometric method. The slope angle of the graph was found to be approximately \(40^\circ\). The absolute root-mean-square error in the analysis of natural lithium is estimated by the authors as \(\pm 0.23\%\) Li\(^6\). Hence the relative error is approximately \(\pm 3\%\) of the content of this isotope.

B. Photoelectric method

Spectral apparatus. The same diffraction spectrograph was used as the spectral instrument as in the case of the photographic method. In order to convert this spectrograph into a two-channel photoelectric monochromator, it was necessary to separate the spectral lines Li\(^6\)

and \(\mathrm{Li}^7\), located at a distance of \(0.1\ \mathrm{mm}\) from one another, for their separate photometry. For this purpose, the following were specially manufactured and fitted to the instrument: 1) a device for shifting the slit, 2) a cylindrical mirror for magnifying the spectrum, 3) a device for separating the beam, 4) two exit slits, and 5) a set of lenses for focusing the lines on the photomultipliers.

The device for shifting the slit is a glass plate \(8\ \mathrm{mm}\) thick, which is placed in the path of the rays between the slit and the grating. The plate can rotate about a vertical axis by means of a lever connected to a micrometric screw. A displacement of the micrometer by \(0.13\ \mathrm{mm}\) (this corresponds to a rotation of the glass plate by \(1.5\) minutes) shifts the image of the spectrum by approximately \(0.002\ \text{\AA}\). The cylindrical mirror \((F = 152\ \mathrm{mm})\) serves to deflect to the side from the optical axis of the grating and to refocus somewhat lower the doublet line \(\mathrm{Li}\ 6707.8\ \text{\AA}\). The magnification of the mirror is approximately 10; this means that the dispersion in the final image of the spectrum will be \(0.1\ \text{\AA}/\mathrm{mm}\). Hence the distance between the extreme components of the isotopes \(\mathrm{Li}^6\) and \(\mathrm{Li}^7\) will reach \(3.2\ \mathrm{mm}\). The device for separate measurement of the lines consists of two mirrors, by means of which each line is imaged on an exit slit located in front of a photomultiplier. The width of the entrance slit is \(0.1\ \mathrm{mm}\) (in wavelengths, \(0.125\ \text{\AA}\)); the width of the exit slit is approximately \(0.4\ \mathrm{mm}\) \((0.04\ \text{\AA})\). The authors found that the error in setting the micrometer reaches \(0.13\ \mathrm{mm}\). This gives an error of \(0.3\%\) in the results of measurements of the intensity ratio.

For the photoelectric measurements two photomultipliers were used, each of which received one line. The photomultipliers were connected to two capacitors of \(0.1\) and \(0.5\ \mu\mathrm{F}\) capacity, which could be charged up to \(5\ \mathrm{V}\) in the course of \(20\ \mathrm{s}\). The intensities of the lines were measured by comparing the voltage of the capacitors with a standard voltage of \(1\ \mathrm{V}\). The photomultipliers are only weakly sensitive to the \(6707.8\ \text{\AA}\) line, and therefore it proved necessary to compensate the dark current as accurately as possible. This is accomplished by means of a special device, which makes it possible automatically to introduce the necessary corrections into the measurements. During the first second the capacitors are charged by the total current (including the dark current); during the second second the slit of the spectrograph is closed and, at the same time, the direction of the current is changed, as a result of which the capacitors are discharged by the dark current. Thus, when the full exposure is counted, during \(10\ \mathrm{s}\) the capacitors are charged under the action of light, and during the other \(10\ \mathrm{s}\) compensation of the dark current takes place.

Discharge tube. For excitation of the spectrum a discharge tube with a hollow cathode was used, the design of which in assembled form

and, in disassembled form, is shown in Fig. 27. This tube makes it possible to change the sample just as quickly as in the case of an ordinary arc. The aluminum cathode is a cup whose external dimensions are: length—30 mm, diameter—12 mm. The recess of the cathode, 6 mm in diameter, reaches 25 mm in length. The copper anode with two tubes for the inlet and outlet of helium can be reliably fixed on the optical axis of the spectrograph, thereby ensuring a reproducible setting of the tube for a series of analyses. The copper insert imparts a definite direction to the gas stream. The anode and the insert can be used without replacement for a number of analyses.

Fig. 27. Discharge tube with a hollow cathode: K—aluminum cathode, B—gasket, C—intermediate glass, A—copper anode, D—copper insert, F—window, a and b—openings for the inlet and outlet of gas.

For the window, ordinary microscope cover glasses are recommended; these must be replaced periodically. Ring-shaped sealing gaskets are easily cleaned and can be used for a long time without replacement. The intermediate glass for insulation (1 mm thick) can serve for carrying out a hundred analyses.

Cooling of the cathode with running water was carried out by means of a special clamp consisting of two halves joined together by a thick-walled rubber tube (Fig. 28). This clamp is placed on the cathode before the start of the analysis; when a new sample is inserted, it is quickly and easily removed. The lithium samples under investigation, weighing 0.4 mg, were used in the form of sulfate. As the working

gas helium was used. Instead of the circulation system, a vacuum pump, a cylinder of pure helium, a valve for admitting gas, and a manometer for measuring pressure were connected to the tube. The analyses were carried out in a stream of helium at a pressure of 4 mm Hg. The band spectrum caused by extraneous gaseous impurities, even in the presence of a small leak, was very weak and did not introduce noticeable interference.

A specially made generator was used to supply the discharge tube; it provided sufficient constancy of the excitation conditions. The maximum voltage of the generator was 760 V,

Fig. 28. Clamp for cooling the cathode.

the current could be varied within the range from 10 to 140 mA. The current during the discharge was set at approximately 100 mA and was maintained constant to an accuracy of up to ±0.5 mA.

Results and accuracy. Calibration graphs were constructed from standards analyzed by the mass-spectrometric method. The studies showed that, as a result of changes in the spectrograph adjustment over time, an additional error is introduced into the data of the analyses. If necessary, this error can be eliminated by comparing, one after another, the sample being analyzed and the standard sample. Table IV gives the results of spectral analysis of 12 samples with a 30% content of Li⁶, which were corrected in the course of the measurements with the aid of standards. On the basis of these data it is seen that the mean relative error for a 30% sample is ±0.5% of the concentration. It follows from this that the photoelectric-

chemical method surpasses the photographic method both in accuracy and in speed of analysis.

Table IV

Sample No.	Concentration in %	Deviation from the mean	Relative error in %
1	30.3	+0.1	+0.3
2	30.2	0.0	0.0
3	30.0	−0.2	−0.7
4	30.1	−0.1	−0.3
5	30.3	+0.1	+0.3
6	30.3	+0.1	+0.3
7	30.6	+0.4	+1.3
8	30.3	+0.1	+0.3
9	30.0	−0.2	−0.7
10	30.1	−0.1	−0.3
11	30.1	−0.1	−0.3
12	30.0	−0.2	−0.7
Mean	30.2	+0.14	+0.46

Determination of relative abundance. The photoelectric method was used to determine the relative abundance of isotopes in natural lithium. In this work an ordinary discharge tube with a hollow Schuler–Hallow cathode was used, connected to a circulation system for purifying the working gas. It is interesting to note that, in order to reduce self-absorption, the authors developed a source with low luminous intensity. For this purpose a graphite electrode in the form of a small cup was inserted into the cathode cavity, and potassium was added to the lithium sample as a “carrier.” Under these conditions it proved possible to reduce the lithium sample to 2 μg. The graphite cup was impregnated with Apiezon N dissolved in petroleum ether.

The technique for introducing the sample was as follows. First, 10 ml of an aqueous solution containing 1 mg of potassium in the form of chloride was poured into the cup. After evaporation, lithium sulfate (2 μg) was introduced into the cup in the form of an aqueous solution. For uniform deposition, the volume of this solution was brought up to 50 ml. After evaporation, a uniform layer was obtained on the walls of the cup. The working gas was a mixture of helium and argon, prepared in the ratio 1:1. During the dis-

a pressure of 1.5 mm Hg was maintained. The cathode was cooled with liquid air. In order to degas the graphite electrode, the tube was first operated at a current of 10–40 ma. Then the entire vacuum system was evacuated, a new portion of gas was introduced into the tube, and the current was increased to 70 ma. Under these conditions the normal glow intensity was maintained for approximately two hours.

The structure of the Li I 6707.8 Å line was resolved with the aid of a large diffraction spectrograph with a 9-meter grating, set up according to the Paschen–Runge scheme. The work was carried out in the fourth

Fig. 29. Calibration graph for lithium.

order, where the dispersion reached 0.35 Å/mm. As an internal standard in the photoelectric recording of the spectrum, the same lithium line in the first order was used. This line was received by a second photomultiplier. The signals from the photomultipliers were amplified and then recorded by a self-recording instrument. By measuring the ratios of the intensities of the analytical line and of the internal standard, it is possible automatically to take into account fluctuations in the intensity of the source.

In order to establish whether there is agreement between the ratios \(\frac{I_6}{I_7}\) and \(\frac{C_6}{C_7}\), standard samples of \(Li^6 + Li^7\) were prepared with an accuracy of up to 5%. In determining the intensity ratio, the background was taken into account and the corresponding corrections were made. The results obtained by the authors in the concentration interval from 7 to 70% are presented in Fig. 29 in the form of an ordinary calibration graph

in logarithmic coordinates. Since the slope of the calibration straight line does not change, self-absorption in the light source is practically absent. Along with this it is important to note that the angle of inclination is less than 45° and the straight line is shifted, relative to the point

\[ \frac{C_6}{C_7}=1, \]

to the right along the intensity axis. It follows from this that a simple equality between the quantities

\[ \frac{I_6}{I_7} \quad \text{and} \quad \frac{C_6}{C_7} \]

is absent. This circumstance is apparently explained by an insufficiently correct allowance for the background, and also by the different rates of diffusion of the atoms Li\(^6\) and Li\(^7\) in the discharge tube, and perhaps by other effects as well, which we discussed in considering the course of calibration graphs for isotopic analysis of light elements. From the detected shift of the calibration graph it is seen that the value of the relative abundance

\[ \frac{C_7}{C_6}=13.5, \]

found by the authors\({}^{38}\) for natural lithium, is too high. The latter is confirmed by the deviation of this value from the presently generally accepted value (12.7), found by the mass-spectrometric method\({}^{17}\).

9. ISOTOPIC SPECTRAL ANALYSIS OF URANIUM

In 1949, Bechert, Stukenbroeker, and Adams\({}^{39}\) discovered a large isotopic shift of a number of lines in the spectrum of uranium. At the same time they pointed out the possibility of spectral analysis of the isotopic composition of uranium. In 1952 Brody\({}^{40}\) used a photoelectric method for the isotopic spectral analysis of enriched uranium. Below the apparatus is described and the results obtained by him for the isotopic analysis of uranium are presented.

Spectral apparatus. To obtain the spectrum, a three-meter Baird diffraction spectrograph was used, whose resolving power in the first order is 60,000 and whose linear dispersion is 5.6 Å/mm. In front of the camera slit, in place of the cassette, a special photoelectric device for recording the spectrum was installed. The principal elements of this device were two photomultipliers, which were placed opposite the lines to be photometered in special holders. During recording, these holders with the photomultipliers were moved across the lines by means of a micrometer screw driven by a synchronous motor. Special switches changed the direction of travel as soon as the photomultipliers reached the limiting position. The recording speed could be varied within the range from 0.2 to 16 mm per minute. To obtain correct peaks, the width of the entrance slit was taken as 0.03 mm, the width of the exit slit as 0.01 mm, and the recording speed as 0.2 mm per minute.

During recording, the light from the source was interrupted 60 times per second by means of a rotating sector. As a result, in the photomultiplier-

lines a pulsating current was generated. This current was then amplified by an alternating-current amplifier with a frequency of 60 cycles, the dark current being thereby eliminated. The amplified alternating-current signal was rectified by a vibrator and after this was sent to a recorder, which made it possible to record directly the ratios of the signals from two photometered lines. If desired, the recorder could be used as a microammeter; in this case the signal from the photomultiplier was compared with a known current from a battery.

Since the current of the photomultiplier is proportional to the intensity of the light falling on the photocathode, and since the intensity of a spectral line is proportional to the concentration of the given isotope in the sample, then, in the absence of background and self-absorption, the current of the photomultiplier should be proportional to the concentration.

Source. As excitation sources, a hollow-cathode tube and a condensed spark were tested. Into the water-cooled hollow cathode a sample weighing several milligrams was placed. Argon served as the working gas; it circulated through the discharge tube at a pressure of 3 mm Hg. A voltage of 300 V from a special generator was applied to the tube, and the current was taken as 100 mA. Measurements were begun after the discharge tube had burned for 20 minutes, when the discharge had reached sufficient brightness.

Proceeding from the desire to reduce the duration of the analyses, the author used a spark, employing the very same spectral setup and the same spectral lines. Since in this case the lines proved to be broader, the work was carried out in the third-order spectrum. To excite the spark, a Baird generator of 0.5 kW power was used, with a voltage in the secondary circuit of the transformer of 25,000 V. The oscillatory circuit consisted of a capacitance of 0.12 μF, an inductance of 50 μH, and an auxiliary spark gap connected in series with respect to the main gap. The spark gave two discharges per half-period. The upper electrode was a copper rod 6 mm in diameter, and the lower one was the investigated sample of metallic uranium. The current in the primary circuit of the transformer was maintained during the discharge at 2.7 A.

Spectral lines. The analytical line was the uranium spark line 4244.37 Å, which is resolved into two components: U²³⁵ 4244.126 Å and U²³⁸ 4244.372 Å[^41]. As the comparison line (internal standard) the line U II 4090.13 Å was taken, which does not undergo isotopic shift. The photoelectric device recorded the ratios of the intensities of the lines \(\frac{4244.13}{4090.13}\) and \(\frac{4244.37}{4090.13}\). If, on the basis of the photometric curves, the background corrections are taken into account, then one can determine the ratio of the intensities of the two isotopic lines \(\frac{I_{235}}{I_{238}}\). In the case of the hollow cathode the photometered lines,

located in the second-order spectrum; when working with a spark, the third-order spectrum was used.

Results and accuracy. Since the isotope \(U^{234}\) is present in enriched-uranium samples and self-absorption in the excitation source is not excluded, it was first necessary to establish the nature of the dependence between the quantities \(\dfrac{I_{235}}{I_{238}}\) and \(\dfrac{C_{235}}{C_{238}}\). In the case of a hollow cathode it turned out that these two quantities are simply equal to one another. Apparently this was explained by the mutual compensation of two effects acting in opposite directions. The scatter in analyses of the same sample, carried out one after another with continuous burning of the tube, proved to be small, with the exception of the natural sample. The mean relative error of a single determination for a concentration of \(7.04\%\) reaches \(\pm 1.1\%\); for \(3.72\%\) it is \(\pm 1.8\%\); for the natural sample, \(\pm 12\%\). The mean results from 10 determinations give very good accuracy. For the concentrations mentioned, the relative error in this case is respectively \(\pm 0.3\), \(\pm 0.5\), and \(\pm 3.5\%\). The comparatively large error in determining the concentration in the natural sample is explained by errors in allowing for the background and for mutual overlap.

In the case of the spark, equality between the quantities \(\dfrac{I_{235}}{I_{238}}\) and \(\dfrac{C_{235}}{C_{238}}\) is absent. The author explains this by the presence of self-absorption and by insufficient resolution of the lines \(U^{235}\) and \(U^{238}\). By introducing corrections one can obtain correct values, which will correspond to mass-spectrometric data. The results obtained make it possible in this case to estimate the mean relative error as \(\pm 0.3\%\) for concentrations of \(93\%\) \(U^{235}\).

These data show that, despite a number of unsuccessful solutions of certain particular questions (comparison of different spectral lines, use of a spark with electrodes made of metallic uranium, insufficiently careful allowance for background and mutual line overlap), good accuracy was nevertheless obtained. It is therefore clear that more careful work on the development of the method of spectral analysis of the isotopic composition of enriched uranium should give still better results.

10. ACCURACY OF METHODS OF ISOTOPIC SPECTRAL ANALYSIS

In accordance with experimental data, the relative error in determining isotopic composition by the spectral method decreases as the concentration of one isotope in a two-isotope sample increases. Let us consider in more detail the nature of the change in the error with change in concentration. In the general case the calibration graph is expressed by the formula

\[ \lg \frac{I_1}{I_2}=b\lg \frac{C_1}{C_2}=b\lg \frac{C_1}{1-C_1}, \tag{14} \]

where \(I_1\) and \(I_2\) are the intensities of two isotopic lines, \(C_1\) and \(C_2\) are the relative concentrations of the isotopes in the sample, and \(b\) is the tangent of the angle of inclination of the calibration graph. For simplicity denote \(\lg \dfrac{I_1}{I_2}=y\). Then the analytical expression of the calibration graph takes the form

\[ y=b\lg\frac{C_1}{1-C_1}. \tag{15} \]

The value \(y\) is the measured quantity from which the unknown concentration \(C_1\) is found. It may be assumed that the absolute error in determining the value \(y\) remains constant, not changing with the change in the absolute value of \(y\), or, in other words, with the change in the concentration \(C_1\) within the limits from 0 to 100%. In fact, the absolute error is somewhat greater at the edges of this region, where the mutual superposition of isotopic lines has to be taken into account, but for the time being we shall not take this into consideration, since accounting for this circumstance would not in principle change our conclusions.

For the absolute error one may write the expression

\[ dy=\pm d\left(b\lg\frac{C_1}{1-C_1}\right). \]

Differentiating it, we find:

\[ dy=\pm \frac{b}{\ln 10}\frac{dC_1}{C_1(1-C_1)}, \]

whence the absolute error in determining the concentration will be:

\[ dC_1=\pm \frac{\ln 10}{b}\,dy\cdot C_1(1-C_1). \tag{16} \]

If \(dy=\mathrm{const}\), then expression (16) may be written in the form

\[ dC_1=\pm K C_1(1-C_1), \tag{17} \]

where \(K=\dfrac{2.3}{b}dy\). Studying this function for a minimum and maximum, it is easy to establish that at \(C_1=1/2\) (concentration 50%) the absolute error reaches a maximum, with \(dC_{\max}=\pm \dfrac{K}{4}\); for \(C_1>1/2\) and \(C_1<1/2\) the absolute error gradually decreases, approaching zero at \(C_1=0\) and \(C_2=1\).

The relative error will be expressed by the formula

\[ \frac{dC_1}{C_1}=\pm K(1-C_1), \tag{18} \]

whence it follows that the relative error will gradually decrease with increasing concentration: at \(C_1=0\) it will reach

of the greatest value, \(\dfrac{dC_1}{C_1}=\pm K\); at \(C_1=1/2\) it will be half as large, \(\dfrac{dC_1}{C_1}=\pm \dfrac{K}{2}\); at \(C_1=1\) it will approach zero.

In Fig. 30, the dashed lines give a graphical representation of the character of the change in the absolute and relative error with a change in the concentration of one isotope in another, in accordance with the conclusions set out above. The solid lines represent the experimental dependence of the absolute and relative error on concentration, taking into account the mutual superposition of isotopic lines and the conditions of their resolution.

Fig. 30. Dependence of error on concentration.

The high accuracy of the methods of isotopic spectral analysis, and also the independence of the measurement results from extraneous impurities in the sample, is explained by a number of favorable circumstances that are inherent in the spectral method. First of all it should be noted that the isotopic lines being compared belong to transitions between the same (slightly displaced) terms. As a result, the isotopic lines are, to a high degree, homologous pairs, the ratio of whose intensities practically does not depend on random changes in the discharge conditions during the photometry of the lines or the photographing of the spectrum. If, in ordinary spectral analysis, those lines are considered homologous for which the difference of excitation potentials does not exceed \(0.06\ \mathrm{eV}^{42}\), then in the case, for example, of isotopic analysis of uranium, the condition of homology is 4000 times higher. The second favorable factor is the use of the doubled effect of the change in the ratio of intensities of two isotopic lines as a function of the change in concentration. In the case of ordinary spectral quantitative analysis, the impurity line is compared with a line of the principal substance, whose intensity, under identical conditions of photometry or recording in different samples, remains unchanged. In isotopic analysis, if one of the lines becomes brighter with a change in concentration, the other line becomes weaker. Therefore the intensity ratio of such lines changes more strongly with a change in the concentration ratio of the two isotopes. And, finally, a third important circumstance, which applies only to photographic methods, is the close location of the lines relative to one another.

group of the isotopic lines being compared. In this case, the nonuniformity of the photographic properties of emulsions distorts the true ratio of line intensities to a much lesser degree. Thus, in the case of isotope analysis, errors caused by the excitation source and the photographic plate, which play the chief role in ordinary spectral analysis, are reduced to a minimum.

11. CONCLUSION

On the basis of the literature data considered, one may conclude that in the last four years substantial advances have been made in the development and application of isotopic spectral analysis. Using hydrogen, heavy water, lithium, and uranium as examples, the effectiveness of this method and its applicability for practical purposes have been demonstrated. It has proved possible to determine quantitatively the isotopic composition of the objects mentioned over a wide range of concentrations (approximately from 1 to 90%) and with higher accuracy (a relative error on the average of less than 1% of the content) than the accuracy of ordinary spectral analysis of impurities in elements and alloys. A very important advantage of the spectral method for determining isotopic composition is the independence of the analysis results from the presence of any amounts of foreign impurities in the sample. Therefore, unlike all other methods used to determine the isotopic composition of enriched products, spectral analysis requires no preliminary purification of the sample.

The experimental calibration graphs considered fully confirm the dependence we theoretically found between the relative intensity of isotopic lines and the relative concentration of isotopes for light and heavy elements; they also confirm the character of the change in this dependence in the presence of self-absorption, background, and mutual overlap of lines. This shows that, in developing spectroanalytical methods of isotope analysis, calibration of the method against standards is of great importance for establishing the true dependence of the relative intensity of isotopic lines on the relative concentration of isotopes.

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