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EXPERIMENTAL STUDY OF THE FINE STRUCTURE OF SINGLY IONIZED HELIUM
In our journal the results of studies of the fine structure of the spectra of hydrogen and singly ionized helium, carried out by Lamb and co-workers, have already been reported.¹˒² These studies, performed by means of radio-spectroscopic methods, showed that, contrary to Dirac’s theoretical prediction, states of these atoms with the same principal quantum numbers $n$ and the same quantum numbers of the total angular momentum $j$ are not degenerate. The observed energy difference between the levels $2^2S_{1/2}$ and $2^2P_{1/2}$ was, in the case of hydrogen, $1052 \pm$ MHz, and in the case of singly ionized helium, $14020 \pm 100$ MHz ($1$ MHz $= 4.1 \cdot 10^{-9}$ eV).
The explanation of this splitting of the terms of hydrogen-like atoms was found to require taking into account the interaction of the atomic electron with the field of radiation—“zero-point oscillations of the vacuum.” A very detailed treatment of the theoretical questions connected with allowing for this interaction is contained in the reviews cited, and we shall not touch upon it. We shall only point out that theoretical calculation gives, for the energy difference of the levels $2^2S_{1/2}$ and $2^2P_{1/2}$, in the case of hydrogen, $1051$ MHz, and in the case of singly ionized helium, $13820$ MHz—in good agreement with the results of the measurements. Our note is intended to describe the experimental method by which the above results were obtained; in the cited reviews no attention was given to it. We shall confine ourselves here only to the last paper of Lamb and Skinner,³ devoted to the study of the fine structure of singly ionized helium.
An indication of the anomalous (from the point of view of theory) fine structure of He$^+$ was obtained as early as 1916 by Paschen, who investigated the structure of the line with wavelength $\lambda = 4686$ Å. After Lamb and Retherford discovered the splitting of the hydrogen terms in 1947, a number of spectroscopists undertook the study of the fine structure of He$^+$ in the visible and ultraviolet regions of the spectrum. The investigation was carried out on ...
lines \(\lambda = 1640\ \text{\AA}\) (transition \(n=3 \to n=2\)) and \(\lambda = 4686\ \text{\AA}\) (transition \(n=4 \to n=3\)). In all cases a doublet splitting was found; when reduced to the transition \((n=3 \to n=2)\), its magnitude varied (according to different authors) from \((11.4 \pm 1.4)\cdot 10^3\) to \((13.9 \pm 1.5)\cdot 10^3\) Mc/sec and, in general, proved to be several percent smaller than the expected value. The accuracy of the indicated measurements was apparently considerably lower than the accuracy of the theoretical calculations, and therefore, in order to obtain greater accuracy, it was quite natural to turn to microwave radio spectroscopy. The possibility of applying it in the present case was due to the fact that the magnitude of the doublet splitting corresponds to wavelengths of approximately 2.2 cm, i.e., lies in a range readily accessible to modern microwave technology. The greater precision of the measurements was determined by the fact that, in addition to the possibility of direct measurements of the transition frequency (without analysis of complex components), the influence of the Doppler effect, which is substantial in spectroscopic measurements in the visible and ultraviolet regions of the spectrum, becomes negligible here.
As in the case of the investigation of hydrogen, the experimental method was based on the metastability of the state \(2^2S_{1/2}\). Whereas the lifetime of the state \(2^2P_{1/2}\) is \(10^{-10}\) sec (16 times less than in hydrogen), the estimate of the lifetime of the state \(2^2S_{1/2}\) leads to the value \(2.2\cdot 10^{-3}\) sec (64 times less than in hydrogen). The idea of the method was that, after producing (by electron bombardment of atomic helium) a sufficient number of \(\mathrm{He}^+\) ions, they were subjected to microwave radiation inducing the transition of helium ions from the state \(2^2S_{1/2}\) to the state \(2^2P_{1/2}\), and the change in the population of the state \(2^2S_{1/2}\) was investigated as a function of the microwave radiation frequency.
In the case of hydrogen, a beam of metastable atoms was directed at a metallic target, ejecting electrons from it. Owing to the sharp difference in the ejecting ability of metastable and unexcited atoms, the change in the population of the metastable state was readily detected from the change in the electron current from the target. In the case of \(\mathrm{He}^+\), such a method proves unsuitable because of the high ejecting ability of unexcited helium ions and the relatively small concentration of metastable ions (when atomic helium is bombarded by electrons, the number of metastable ions formed is about 1% of the number of nonmetastable ions). Therefore, in the work with \(\mathrm{He}^+\) a photoelectric detection method was used. If a transition of an ion is induced from the state \(2^2S_{1/2}\) to the state \(2^2P_{1/2}\), then within \(10^{-10}\) sec it passes to the ground state with emission of a photon of energy 41 ev. The photoelectric current caused by these photons can serve as a measure of the number of induced transitions. However, a serious difficulty arises along this path. When atomic helium is bombarded by electrons, along with the metastable ions there arise (in considerably greater number) ions in the states \(n^2P\), which then pass to the ground ionic state with emission of a photon of energy 41 ev. These photons, striking the photoelectric detector (a copper plate was used as the photoelectric detector), create a photoelectric current forming a background against which the effect under study may be lost. Further, during bombardment there arise helium atoms excited to the metastable atomic state \(2^3S_0\), which, on reaching the detector, knock electrons out of its surface, thereby increasing the background level. Finally, the greater part of the atoms excited during bombardment to the atomic state \(n^1P_1\) pass to the ground atomic state
with the emission of a photon with energy 20 eV; these photons, striking the detector, also produce a photoelectric current, increasing the background.
Both theoretical estimates and direct experiments led to a value of the ratio of the useful-signal level to the background level (under optimal conditions) of about 1.5%. However, in order to carry out high-quality measurements by this means it would be necessary to observe an excess of the signal level over the background level of only 0.1%, which in turn required stability of the background level with an accuracy up to 0.05%—quantities that are clearly unsatisfactory. To increase the ratio of the signal level to the background level, the authors placed between the detector and the region in which the metastable atoms were located a thin collodion film ($\sim 7\,\text{mg}/\text{cm}^2$). Such a film completely retained the metastable atoms and substantially reduced the number of 20-eV photons reaching the detector. Owing to this, the ratio of the signal level to the background level under optimal conditions increased to 13%, and it became possible to carry out measurements with a ratio of the signal level to the background level (at the signal maximum) of about 4–5%, which already ensured the obtaining of reliable results.
The next difficulty that the authors had to overcome was the necessity of varying the frequency of the microwave radiation over wide limits. The state $2^2P_{1/2}$ has a natural width of 1600 MHz. Thus, in order to obtain a resonance curve, it is necessary to vary the frequency of the microwave radiation within approximately 3000 MHz, while maintaining the power of the radiation strictly constant throughout this interval. The authors overcame this difficulty by using an additional constant magnetic field and studying the induced transitions between individual components of the Zeeman splitting as a function of the magnitude of this splitting (i.e., of the strength of the constant field) at an unchanged frequency of the microwave radiation. In other words, varying the radiation frequency at unchanged splitting of the levels was replaced by varying the splitting at an unchanged radiation frequency. The basic data were obtained for the transition
$$ 2^2S_{1/2}\left(m=+\frac{1}{2}\right)\to 2^2P_{1/2}\left(m=-\frac{1}{2}\right). $$
The probability of the induced transition
$$ \mu \sim \frac{1}{(\nu-\nu_p)^2+\left(\frac{A}{4\pi}\right)^2}, \tag{1} $$
where $\nu$ is the frequency of the microwave radiation, $\nu_p$ is the frequency corresponding to the transition $2^2S_{1/2}\to 2^2P_{1/2}$ (the resonance frequency), and $A=10^{10}\ \text{s}^{-1}$ is the radiative half-width of the state $2^2P_{1/2}$. For the indicated components of the Zeeman splitting in a field of strength $H$,
$$ \nu_p=\nu_0+aH+bH^2, \tag{2} $$
where $\nu_0$ is the resonance frequency in the absence of a magnetic field,
$$ a=1.866\ \text{MHz/gauss}\quad\text{and}\quad b=2.5\cdot 10^{-6}\ \text{MHz/gauss}^2. $$
Thus, for the Zeeman components under consideration,
$$ \mu \sim \frac{1}{(\nu-\nu_0-aH+bH^2)^2+\left(\frac{A}{4\pi}\right)^2}. \tag{3} $$
In the case of low power of the radio-frequency field, this expression gives the form of the resonance curve obtained when varying \(H\). If by \(H_{\mathrm{m}}\) we denote the value of \(H\) corresponding to the maximum of this curve, then
\[ \nu_0=\nu-aH_{\mathrm{m}}-bH_{\mathrm{m}}^2 . \tag{4} \]
The half-width of the resonance maximum is then approximately 855 gauss.
A transverse section of the apparatus is shown in Fig. 1; \(A\) and \(B\) are pole pieces of an electromagnet, which produced a uniform field with strength up to 4000 gauss. The field strength was calibrated by means of a rotating coil. Deviations from homogeneity in the working region
Fig. 1. Schematic cross-section of the apparatus:
\(A\) and \(B\) — pole pieces of the electromagnet; \(C\) — K-band waveguide (this waveguide could be removed from the apparatus, as a result of which an X-band waveguide was formed); \(D\) — detector plate; \(E\) — collecting electrode; \(F\) — collodion film; \(J\) — diaphragm defining the solid angle under which the detector plate is visible from the excitation region.
did not exceed 0.5%. The field strength itself was determined with the same accuracy. The source of radio-frequency radiation was a magnetron, radiating (in the principal experiments) in the wavelength region of about \(1.6\ \mathrm{cm}\).
The radiation propagated along waveguide \(C\) with a cross-section of \(4.5\times 9.16\ \mathrm{mm}^2\). Round holes of diameter \(3.81\ \mathrm{mm}\) were cut in the walls of the waveguide, opening access to the electron beam that pierced the waveguide and allowing photons to enter from the waveguide onto a
detector. Since the wavelength of the microwave radiation in the waveguide was about 33 mm, the presence of holes did not introduce appreciable perturbations into the wave of type \(TE_{01}\) traveling along the waveguide.
The wavelength was measured with a wavemeter to an accuracy of 0.1%. The radiation power was about 40 mW (which corresponded to the induction of transitions of approximately 66% of the ions in the state \(2^3S_{1/2}\left(m=-+\frac{1}{2}\right)\)) and was continuously monitored. The entire apparatus was evacuated to a pressure of \(2\cdot 10^{-6}\) mm Hg. Helium was introduced through a “leak,” which was obtained by drawing a short (0.25-m) tungsten wire of 0.76-mm diameter into a Pyrex capillary. The helium pressure in the instrument was measured with an ionization manometer with the necessary corrections introduced. In the experiments two batches of helium were used, the degree of purification of which was 99.95 and 99.99%.
Fig. 2. One of the experimentally obtained resonance curves.
Excitation was carried out by means of an electron beam produced by a heated filament \(O\), located under a negative potential. A beam of electrons with an energy of about 200 eV, crossing the waveguide, was directed toward the opposite pole shoe of the electromagnet and was collected on a metal plate \(H\). Since the number of ions formed is proportional to the emission from the filament, special measures were required to keep the filament temperature strictly unchanged.
In contrast to the experiments with hydrogen, here no measures were taken to form a beam of the helium ions being produced. Instead, these ions moved freely in the apparatus with thermal velocities.
The photons arising as a result of the induced transition \(2^3S_{1/2}\to 2^3P_{1/2}\) and of the subsequent transition to the ground ionic state entered the detector plate \(D\), separated from the excitation region by a cold nickel partition \(K\). The photoelectrons were collected by the electrode \(E\). The photoelectric current was measured with an electrometer tube connected according to Penick’s circuit, and with a sensitive galvanometer.
The current sensitivity of the circuit was \(1.2\cdot 10^{-15}\) A/mm and did not undergo noticeable changes over more than a year. The fluctuation instability was about 1 mm of the scale, increasing in individual cases to 2–3 mm.
To obtain the resonance curves, several fixed frequencies were used, as a result of which the resonance maxima were obtained at values of the magnetic-field strength varying over fairly wide limits. The dependence thereby obtained
of the resonance frequency from \(H_M\) showed that the resonance curves did indeed correspond to the transition
\[ 2^3S_{1/2}\left(m=+\frac{1}{2}\right)\to 2^2P_{1/2}\left(m=-\frac{1}{4}\right). \]
A typical resonance curve is shown in Fig. 2.
The value of \(H_M\) was determined graphically from the curve, and then formula (4) was used to determine \(\nu_0\), i.e., the frequency corresponding to the transition \(2^3S_{1/2}\to 2^2P_{1/2}\) in the absence of a magnetic field. An estimate of the errors showed that the largest possible error in determining \(\nu_0\) is \(\pm 100\) Mc/s. The statistical error is considerably smaller, \(\sim \pm 10\) Mc/s.
The mean value obtained by the authors, \(14\,020\) Mc/s, is 1.4% higher than the theoretical value, \(13\,820\) Mc/s, and both values are somewhat higher than the most accurate values obtained spectroscopically in the visible region of the spectrum.
The authors regard their measurements as preliminary, while pointing out that, in the case of singly ionized helium, an accuracy greater than in the case of hydrogen can be achieved, since here there is no hyperfine structure complicating the measurements.
G. R.
CITED LITERATURE
- Ya. A. Smorodinskii, UFN 39, 325 (1949).
- V. F. Weisskopf, UFN 41, 165 (1950).
- W. E. Lamb and M. Skinner, Phys. Rev. 78, 539 (1950).