FROM THE CURRENT LITERATURE

SPECTRUM OF RECOIL ATOMS IN $K$-CAPTURE

1. INTRODUCTION

To explain the difficulties associated with the apparent nonconservation of the conservation laws in beta transformations, Pauli (1931) proposed the hypothesis of the neutrino. According to this hypothesis, in beta transformation the nucleus emits a neutrino—a particle having no charge, possessing zero (or vanishingly small) rest mass, half-integral spin, and a very small magnetic moment. The assumption of the existence of the neutrino was the only hypothesis capable of reconciling, on the basis of the conservation laws, all the experimental facts then known in the field of beta transformations.

A natural generalization of the neutrino hypothesis and of the nucleon theory of nuclear structure[^1] was Fermi’s phenomenological theory of beta transformations,[^2] constructed on the basis of a formal analogy with the process of electromagnetic radiation. The theory of beta transformations, despite its incompleteness and unfinished character, proved capable of explaining the principal experimental data obtained in the study of the distribution of beta particles over energies and momenta, both for allowed and for forbidden beta transitions.[^3] The agreement between the conclusions of the theory of beta transformations and the experimental results is indirect evidence for the existence of the neutrino.

More substantial evidence for the existence of the neutrino is provided by quantitative results of experiments on the study of the spectra of recoil atoms in beta transformations.[^4][^5][^6] In those experiments in which, for one reason or another, quantitative conclusions about the spectrum of recoil atoms are impossible,[^7][^8][^9] a consistent explanation of the qualitative conclusions found regarding the spectrum can be given only on the basis of the assumption of the existence of the neutrino.

Direct proof of the correctness of the neutrino hypothesis, i.e. proof that neutrinos really exist and possess the properties postulated for them, could be furnished by experiments in which the interaction of the neutrino with matter would be demonstrated. Unfortunately, attempts[^10] to detect the existence of such processes have not been successful, despite the successive increase in the power of the neutrino flux. The failure of these experiments in no way contradicts the neutrino hypothesis or the theory of beta decay, since the cross section for such processes is exceedingly small.

At present the range of application of the working hypothesis of the neutrino has expanded considerably. In particular, it is necessary in order to

explain the known features of meson decay. For example, the decay of $\mu$-mesons can be represented in the form of the reaction

\[ \mu^{\pm} \to e^{\pm} + 2\nu \tag{1} \]

in accordance with the experimental fact that the decay electrons have a continuous distribution in energy. Conversely, the decay of $\pi$-mesons fits the scheme

\[ \pi^{\pm} \to \mu^{\pm} + , \tag{2} \]

since the $\mu$-mesons formed in this decay have one and the same energy.

In this connection, experiments on the study of recoil atoms produced when a nucleus captures orbital electrons ($E$-capture) are of interest.

One of the principal tasks of experiments of this kind is to determine the number of neutrinos emitted in an individual act of beta transformation. If one assumes, as is done in the theory of beta transformations$^{2}$, that in each act of beta decay the nucleus emits single neutrinos, then all recoil nuclei formed in $E$-capture must have identical momenta and energies (a line spectrum of recoil nuclei*). Conversely, if one assumes that in beta transformations the nucleus emits two or more neutrinos in different directions, then the spectra of recoil nuclei will be continuous, i.e., the recoil nuclei may have any value of momentum from 0 to $(P_R)_{\max}$ or energy from 0 to $(E_R)_{\max}$. If, furthermore, one neglects the energy received by the recoil atom as a result of transitions in the electron shell, then the foregoing statements are valid for recoil atoms.

2. EXPERIMENTS ON THE STUDY OF RECOIL IN $K$-CAPTURE

For experiments of the indicated type, the radioactive nuclei $\mathrm{Be}^7$ and $\mathrm{A}^{37}$ proved to be very convenient objects of study. The schemes of their decays are presented in Fig. 1. The principal radioactive constants of $\mathrm{Be}^7$ and $\mathrm{A}^{37}$ and the initial characteristics of their recoil atoms ($\mathrm{Li}^7$ and $\mathrm{Cl}^{37}$, respectively) are collected in Table 1. The advantages of using $\mathrm{Be}^7$ and $\mathrm{A}^{37}$ as sources for experiments on the study of recoil atoms may be summarized briefly as follows:

1. Simple and well-established decay schemes (Fig. 1).
2. Small masses of the recoil atoms, owing to which the initial energies and velocities of the recoil atoms are comparatively large (see Table 1). The initial kinetic energy and velocity of the recoil atom are determined by the formulas

\[ E_R = 536 \frac{(\Delta M)^2}{M_R} = 536 \frac{E_\nu^2}{M_R}, \tag{3} \]

and

\[ v_R = 1.41 \cdot 10^6 \left(\frac{E_R}{M_R}\right)^{1/2} = 32.65 \left(\frac{E_\nu}{M_R}\right)\cdot 10^6 . \tag{4} \]

* Naturally, if in a separate act of $E$-capture several neutrinos are emitted simultaneously in one and the same direction, then the spectrum of nuclei will also be a line spectrum. In this sense, the study of recoil spectra does not give a final answer to the question of how many neutrinos (one or several) are emitted in an individual act of beta transformation. Such a final conclusion could be obtained, for example, from experiments on the study of the cross section of inverse beta transformations$^{5}$.

Here \(\Delta M\) is the mass difference of the initial atom and the recoil atom in \(Mэв\), \(M_R\) is the mass of the recoil atom in atomic units, \(E_\nu\) is the kinetic energy of the neutrino in \(Mэв\), \(E_R\) is the kinetic energy of the recoil atom in \(эв\), \(v_R\) is the velocity of the recoil atom in \(\frac{см}{сек}\). It is assumed here that single neutrinos are emitted and that their rest mass is taken to be zero.

1. A sufficiently long lifetime (cf. Table I).

Table I

Characteristics of \(\mathrm{Be}^7\) and \(\mathrm{A}^{37}\) and of the recoil atoms formed in their decay

1. Almost all recoil atoms in the decay of \(\mathrm{Be}^7\) and \(\mathrm{A}^{37}\) are ionized owing to the high probability of conversion Auger transitions in the recoil atoms \(^{6,9}\).

Fig. 1. Decay schemes of \(\mathrm{Be}^7\) and \(\mathrm{A}^{37}\).

1. The energies of Auger electrons are comparatively small (see Table I). The effect of the recoil experienced by atoms upon the emission of conversion

Auger electrons is small:

\[ (\Delta E_R)_{\text{Auger}}=\frac{536}{M_R}\left(E_{\text{e}}^2+1.02E_{\text{e}}\right). \tag{5} \]

Here the energy of the Auger electrons is expressed in MeV, and the energy \((\Delta E_R)_{\text{Auger}}\) in eV.

The proposal to use Be\(^7\) in experiments on the study of recoil atoms in \(E\)-capture was made by Alikhanov and Alikhanyan (see \(^{11,12}\)). Unfortunately, the war prevented the completion of the experiments they had begun, which were being carried out at the Leningrad Physico-Technical Institute. In 1942 Allen \(^{8}\), using a thin layer of Be\(^7\) on platinum as the source and an electron multiplier as the recording device, obtained the integral energy spectrum of recoil atoms (the “\(R\)-spectrum”) by the retarding-potential method. The results of this experiment and possible imperfections of the method used were discussed in detail both by the author himself and by the authors of a number of reviews \(^{12,13}\).

In brief, the result of the experiments amounts to the following: recoil ions have a continuous energy distribution from zero to a certain maximum value \((E_R)_{\max}\), with the greater part of the recoil ions having an energy close to zero (curves were obtained similar, for example, to those shown in Fig. 3). The value \((E_R)_{\max}\) obtained by Allen from the experiments lay between 41 and 47 eV (depending on the thermal treatment of the source). If the indeterminacy connected with surface phenomena in the source and the absence of an absolute calibration of the system of grids by energy are taken into account, the agreement of the value \((E_R)_{\max}\) found experimentally with the value \(E_R\) predicted by the theory should be regarded as satisfactory (see Table I). The fact that the experimentally obtained “\(R\)-spectrum” was continuous, whereas the theory of beta transformations predicted a line “\(R\)-spectrum” in \(K\)-capture, prompted Allen to repeat the experiment under better experimental conditions.

Fig. 2. Schematic of Allen’s apparatus for studying the energy spectrum of recoil ions in the decay of Be\(^7\). \(I\)—source; \(C_1\), \(C_3\), \(C_4\)—protective grids; \(C_2\)—retarding-potential grid; E.M.—electron multiplier; \(C_{\text{sc}}\)—scintillation crystal counter.

The layout of the apparatus used by Allen in the second series of experiments \(^{9}\) is shown in Fig. 2 and basically repeats the layout of the apparatus used in the first series of experiments \(^{8}\). It is significant, however, to note the new features introduced by Allen into the experimental technique when repeating the work:

1) a more refined construction of the electron multiplier was used \(^{14}\);

2) the grid system was improved; protective grids \(C_1\) and \(C_3\) were introduced (Fig. 2), reducing the influence of external fields on the retarding-potential grid \(C_2\);

3) calibration of the grid system with monoenergetic lithium ions from a spodumene source was carried out;

4) separate vacuum evaporation\(^4\) was used for preparing the source (a fraction of a monolayer of active beryllium on tantalum); the “thickness” and efficiency of the source were more carefully monitored;

5) a scintillation crystal counter was used for registering \(\gamma\)-rays from the 11-percent branch of the decay (Fig. 1, a), in order to monitor the efficiency of the sources and to determine the influence of recoil from \(\gamma\)-rays on the form of the energy “\(R\)-spectrum.”

As regards the answer to the main question—how many neutrinos are emitted in a single act of \(K\)-capture—the results of Allen’s second work repeat the results of the first work. Namely: recoil ions “possess” a continuous spectrum, the form of which depends on the thickness of the source and the degree of its heating (Fig. 3). Such a distribution with a predominance of low-energy ions sharply contradicts the distribution predicted by theory. Although the other results of the experiment (the precise determination of the recoil energy corresponding to the maximum decay energy \((E_R)_{\max} = 56.6 \pm 1\) eV; information on Auger transitions in the recoil atom) are very valuable, the absence of a linear “\(R\)-spectrum” suggests a strong influence of surface phenomena (scattering) in the source, which leads to distortion of the true “\(R\)-spectrum.”

Fig. 3. Curves of retarding potential (integral “\(R\)-spectra”), obtained by Allen\(^9\) on the apparatus of Fig. 2. 1 — curve obtained with the source at a temperature of \(500^\circ\)C; 2 — curve obtained 5 min after the source was heated to room temperature; 3 — the same after 35 min; 4 — the same after 65 min.

The dotted line shows the curve predicted by theory for ideal experimental conditions.

The authors9 came to the conclusion that the results of works 8, 9 force one to accept one of the alternative proposals: “either the predictions of the theory of beta transformations are not fulfilled with respect to the process of K-capture, or the recoil spectrum is distorted by surface phenomena. Since at present there are no serious grounds for doubting the correctness of the theory of beta transformations, it was natural to continue experiments on the study of ‘R-spectra’ in K-capture, using, for example, gaseous A37. In this case the automatic device eliminates the influence of surface phenomena, and the results of the experiment should determine the choice between the two above-mentioned alternative conclusions.” Rodabaugh and Allen6 followed this path.

It was also possible to take another path, namely, using the experience of applying film (surface) sources4, 15, to try to reduce to a minimum the influence of surface phenomena. Davis5 followed this path.

3. SOURCES IN EXPERIMENTS ON THE STUDY OF RECOIL SPECTRA

Before proceeding to the presentation of the results of the latest works by Allen and Davis, it is necessary to discuss certain questions connected with the preparation and use of sources in experiments on the study of recoil atoms. This is all the more necessary because the preparation of sources and their skillful use constitute the most responsible part of the experiments. The interpretation of the quantitative results of these experiments is directly dependent on the quality of the source and the conditions of its application4,5.

In order to obtain reliable quantitative results in the study of recoil atoms in beta transformations, the source must satisfy a number of strict requirements.

These requirements may be formulated as follows:

1) the initial characteristics of the recoil atoms (charge, momentum) must not be distorted in the source;

2) the source must be sufficiently intense;

3) the stability (invariance of properties) of the source over the time required for the experiment;

4) the “pointlike character” of the source or, at least, sufficiently small geometrical dimensions of it.

To satisfy all these requirements simultaneously and equally well is a difficult, if not impossible, task. In practice it is reduced to obtaining sources whose properties differ as little as possible from the properties of the “ideal” source.

The second and third requirements are fulfilled by choosing an active material with suitable radioactive constants, by choosing a suitable “backing” for the source, and by creating special conditions under which the source is prepared and used. The second requirement is met to a considerable extent if, as the registering device for the recoil atoms, highly efficient electron multipliers14 are used.

The first requirement is best satisfied by volumetric gaseous sources. An important advantage of gaseous sources over film sources lies in the fact that they are free from the surface phenomena that necessarily occur in film sources. This advantage can be fully realized if a monoatomic noble gas is chosen and if it is used at such a pressure that the mean free path of the recoil atom is greater than the dimensions of the instrument.

However, the use of gaseous volumetric sources also has its drawbacks.

This is, above all, a difficulty of a “geometrical” character—the impossibility of creating a “point” source (requirement 4) or of defining with sufficient accuracy the “effective” volume in which recoil atoms arise. If to this are added the uncertainties associated with fluctuations in the density of the active gas in the volume, and also with possible scattering in the volume and on the walls, it becomes clear that obtaining high resolution in the momenta or energies of recoil atoms in experiments using volume sources is far from an easy task. This circumstance compels one not to reject film sources, since they make it possible to satisfy the condition of “point-likeness” in the best possible way and thereby to ensure the possibility of using analyzing devices of high resolving power.

Since, on the one hand, it is impossible to obtain a film source free from surface phenomena, and since, on the other hand, the detailed picture of surface phenomena changes from case to case, so that a theoretical analysis of these phenomena is hardly possible, obtaining surface sources suitable for experiments with recoil atoms is associated with great difficulties.

The performance of an experiment ultimately comes down to obtaining sufficiently “good” sources (with respect to scattering and absorption of recoil atoms in the surface layer) and, most importantly, to identifying those recoil atoms that have left the surface without substantial changes in their initial state. Here it is necessary to clarify the concept of “without substantial changes in the initial state.”

A recoil ion arising in beta decay possesses a definite initial charge \(ne^*\) and kinetic energy. One may say that a recoil atom has left the surface “without substantial changes in the initial state” if it has retained its initial charge and if the relative change in its initial momentum is small. This imposes certain requirements on the choice of the active material and of the “substrate” for it. The “substrate” must be sufficiently “smooth,” ensuring a minimum probability of neutralization of the recoil atoms and making it possible to reduce the influence of surface forces to a minimum. At the same time, the atoms of the active substance must be distributed on the “substrate” in the form of a fraction of a monolayer. Finally, the inclusion of foreign atoms in the active layer (including atoms or molecules of gases occluded by the surface) must be reduced to a minimum. The latter requirements are obvious, since even a single “unfortunate” collision of a recoil atom with its “neighbor” or with an atom of an occluded gas can change its state in such a way that it loses any resemblance to the initial state.

The production of highly effective surface layers satisfying the indicated requirements is achieved by special techniques for producing and using the sources. Without going into the details of the technique of preparing sources\(^{4,9,5}\), we shall note here the following important circumstances:

1. The material of the “substrate” must have a work function greater than the first ionization potential of the recoil atoms. The recoil atoms—

*) It is important to note that if the charge of an individual recoil atom in beta decay is a multiple of the electron charge, then the average charge of the recoil atoms \(\langle ne\rangle_{\mathrm{cp}}\) may, generally speaking, be a fractional number. For beta decay, as a rule, \(\langle n\rangle_{\mathrm{cp}}\) is close to 1. However, in \(E\)-capture the average initial charge of the recoil atom may be of the order of several units\(^{16}\). At present there are not sufficiently convincing experimental data concerning the detailed picture of the external ionization of recoil atoms.

sources (\(\mathrm{Li}^{7}\)), obtained as a result of the decay of \(\mathrm{Be}^{7}\), have \(V_1 = 5.36\ \mathrm{eV}\). Allen, in his experiments \(^{8,9}\), used thoroughly degassed platinum and oxidized tantalum as a “substrate”; Davis \(^{5}\), oxidized tungsten.

In addition, the material of the “substrate” must have a high melting point and low vapor pressure even at relatively high temperatures, so that the source can be heated during operation. Davis showed that the best results are obtained with a source in which degassed and oxidized tungsten is used as the “substrate”; LiF, for example, evaporated noticeably at a temperature of \(\sim 250\text{--}300^\circ\mathrm{C}\).

1. The application of the active material to the backing and the transfer of the source into the operating position should preferably be carried out under good vacuum conditions (the vacuum must be, at the very least, better than \(1\cdot 10^{-6}\ \mathrm{mm}\) Hg). For this purpose Davis \(^{5}\), for example, used an auxiliary vacuum chamber in which the source was prepared and “formed,” and this chamber could be connected to and disconnected from the working chamber without breaking the vacuum.

The deposition of thin layers of active material can be carried out with equal success both by the method of fractional distillation \(^{4,9,15}\) and by the method of evaporating contaminants from the source \(^{8,5}\). In both cases preliminary experiments are required to select the optimum evaporation or heat-treatment regime of the source, in order to obtain a thin active layer free from contamination.

To establish the optimum conditions under which sources with a minimal impurity content are obtained, the technique of weighing on vacuum microbalances \(^{5}\) is useful, for example. However, such weighing does not make it possible to judge the “thickness” and the quality of the active layer*.

The latter is achieved—and this is essential for all recoil experiments in which film sources are used—only by analyzing the results obtained with the given source \(^{4,5}\).

The “thickness” of the active layer and the efficiency of the source, however, can be determined by the counting method \(^{9}\). Yet this method also does not make it possible to judge the quality of the source and the presence of contamination in the surface layer.

1. It is desirable to heat the sources during operation. This sharply increases the yield of recoil ions and, most importantly, decreases distortions of the recoil spectrum caused by surface phenomena. This circumstance was noted by Allen in his first work \(^{8}\). Sherrin’s experimental results \(^{4}\) especially strongly emphasized the importance of heat treatment of the sources. Finally, Davis \(^{5}\) showed that the obtaining of “good,” undistorted spectra is in direct dependence on the conditions under which the source operates and, in particular, on the thermal regime of the source. Concrete examples will be given below.

The influence of heat treatment affects the properties of the source in at least three respects. First, upon heating, occluded gases are removed from the surface of the source; in the case of monolayer sources these constitute the main cause of scattering of recoil atoms and, consequently, of distortion of the spectrum*). Second, the influence of surface—

*) A monolayer of \(\mathrm{Be}^{7}\) atoms \(3\cdot 10^{-8}\ \mathrm{cm}\) thick and with an average density of \(\sim 3\ \mathrm{g/cm^2}\) has a weight of \(\sim 10^{-7}\ \mathrm{g}\), which is beyond the sensitivity of microbalances.

**) At a pressure of \(\sim 5\cdot 10^{-7}\ \mathrm{mm}\) Hg, the surface of the source is bombarded by residual-gas atoms so intensely that a monolayer of gas is formed in 15 sec. (if the accommodation coefficient is assumed equal to unity).

... surface adsorption forces, and, consequently, the energy losses upon the “stripping” of recoil atoms from the surface become relatively smaller. Finally, the probability of neutralization of the recoil atoms decreases and the efficiency of counting increases.

What has been said is illustrated by the integral distribution curves of recoil ions in the decay of Be\(^{7}\) (Fig. 3), obtained by Allen\(^{9}\) at a source temperature of 500° C and at room temperature at various times after cooling.

A good illustration of how the “quality” of the spectra obtained depends on the heat treatment of the source and on the temperature at which it “operates” is the difference between the differential “\(R\)-spectra” obtained by Davis\(^{5}\) under different operating conditions of the source (Be\(^{7}\) on oxidized tungsten). These curves are shown in Fig. 4. The necessary explanations are given in the captions to the figures.

Fig. 4. Energy spectra of recoil ions in the decay of Be\(^{7}\), obtained by Davis\(^{5}\) under different regimes of heat treatment of the source. \(a\)—the source, formed before use, was heated for 15 sec at a temperature of 1000° C. Then the temperature was lowered to 300° C. Measurements were made 3.5 hours after the temperature was set to 300°; \(b\)—the same, except that the curve was obtained in the time interval 5–180 sec after lowering the temperature to 300° C.

4. PROOFS OF THE LINEARITY OF RECOIL SPECTRA UNDER \(K\)-CAPTURE

Taking these remarks into account, let us consider the results of Davis’s work\(^5\) and try to assess the possible shortcomings of Allen’s work\(^{8,9}\).

A diagram of Davis’s experimental arrangement is shown in Fig. 5. Davis used as a source \(\mathrm{Be}^7\) (\(\sim 1/200\) monolayer) on oxidized tungsten. To obtain differential energy “\(R\)-spectra,” a brass cylindrical electrostatic analyzer with a resolving power of \(\sim 2\%\) was used. The recoil ions were registered by an electron multiplier.

Fig. 5. Diagram of the apparatus used by Davis\(^5\) for studying differential energy “\(R\)-spectra.” \(A\) — electrostatic analyzer, \(E.M.\) — electron multiplier, \(C_1C_2\) — grounded grids, \(I\) — source, \(H\) — heater.

The principal result of Davis’s experiments is represented by the curves shown in Figs. 4, \(b\) and 6, \(a\) and \(b\). The curve in Fig. 4, \(b\) was obtained at zero potential of the “tray” of the source relative to the grounded entrance grid of the electrostatic analyzer. In this case the source was heated for \(\sim 50\) sec. to \(1000^\circ\)C, and the recoil ions were counted for several minutes after the temperature of the source had reached a constant value of \(300^\circ\)C.

A characteristic feature of the resulting “\(R\)-spectrum,” which is typical of the results of Davis’s work with heated sources, is the presence of a clearly expressed peak in the region \(E_R \sim 50\) eV and a relatively small number of ions of low energy. The presence of a peak in the energy region close to that predicted theoretically is evidence that the recoil ions arising in \(K\)-capture are monoenergetic. The presence of a “tail” of low-energy ions indicates that surface effects must necessarily occur in this case as well, although their influence, owing to the heating of the source, is greatly reduced in comparison, for example, with what was observed in the experiments of Smith and Allen\(^9\) (cf. Fig. 3).

The curve of Fig. 6, \(a\) was obtained under approximately the same conditions, except that the source had a potential \(V_g=-146.4\) V relative to the grounded entrance grid \(C\). As a result, all recoil ions acquired the same energy \(eV_g\) in addition to their initial energy (after leaving the surface layer), and the spectrum was displaced by \(V_g\) volts along the energy axis.

In this case, the counting efficiency, both of low-energy ions and of high-energy analyzer ions, remains comparable. Owing to this, the error in measuring the low-energy end of the spectrum is reduced. The second advantage of this method is that the possible errors in measuring \((E_R)_{\max}\), introduced by the contact potential difference, are reduced.

In interpreting the results (Fig. 6, \(a\)) obtained by this method, it is necessary, first, to introduce corrections for the analyzer resolution (which is comparatively easy to do) and corrections for the binding energy of Li ions on the surface (which cannot be done, at least theoretically). Secondly, a correction must be introduced for the so-called “discrimination effect,” responsible for the presence of the second peak of recoil ions at low energies.

Fig. 6. Energy differential “\(R\)-spectra” in the decay of \(Be^7\), obtained by Lewis. The accelerating potential \(V_a = 46.4\) V. The conditions of heat treatment of the source are approximately the same as when obtaining the curve in Fig. 4, \(b\). \(a\)—curve obtained experimentally; \(b\)—curve obtained with corrections for the discrimination effect taken into account.

The “discrimination effect” consists in the fact that the accelerating field changes the initial direction of motion of low-energy ions, relative to the field lines, much more strongly than it changes the initial direction of ions of high energy. Therefore, in the presence of an accelerating field \(V_a\), the collection efficiency of low-energy ions increases sharply in comparison with the collection efficiency of high-energy ions. The influence of this effect on the recoil spectrum, for a given geometry of the apparatus and given accelerating potentials, is readily amenable to calculation, as was done by Lewis. The curve obtained with corrections for the discrimination effect taken into account is given in Fig. 6, \(b\).

The presence of a clearly expressed peak on the differential distribution curves (Figs. 4,b and 6,b) at an energy close to that predicted by theory may be regarded as experimental proof that the recoil ions in the decay of Be\(^7\) are monoenergetic. It follows from this that in each act of decay a single neutrino is emitted, in agreement with the theory of beta transformations; or, at least, if several neutrinos are emitted in an individual act of decay, then they fly out in one and the same direction. If several neutrinos with random directions of emission arose in an individual decay, then the recoil spectrum would have a bell-shaped form and (allowing for scattering in the source) an excess of recoil ions would be observed at the low-energy end of the spectrum. Since, however, experiment shows that recoil ions with an energy in the region close to \((E_R)_{\max}\) predominate, while the low-energy part of the spectrum (due to diffusely scattered ions) has a progressive decline toward low energies, there simply are no grounds for doubting the correctness of the conclusion drawn above.

Fig. 7. Diagram of the apparatus used by Rodebäck and Allen\(^7\) to study the spectrum of recoil ions in the decay of A\(^ {37}\). E.M.-1—electron multiplier for registering Auger electrons; E.M.-2—electron multiplier for registering recoil ions; \(C_1, C_2, C_3\)—grids; \(P_1, P_2, P_3\)—partitions limiting the effective volume of the source (hatched). The recoil ions arising in this volume travel a distance of \(\sim 6\) cm in a field-free space. An accelerating ion potential (45 kV) is applied between the grids \(C_2\) and \(C_3\).

In this connection one can understand the cause of the distortion of the recoil spectrum in Allen’s experiments. Surface effects in his experiments completely altered the form of the true recoil spectrum. Therefore, from the results of his experiments\(^{8,9}\) one cannot draw the conclusion that the predictions of the beta-transformation theory are not fulfilled in the process of \(K\)-capture.

The value of the maximum recoil energy determined by Davis, \((E_R)_{\max} = 55.9 \pm 1\) eV, agrees with the value obtained by Allen (\(55.6 \pm 1\) eV), although it is lower than that predicted theoretically (Table I). One of the reasons for this small discrepancy may be the failure to allow for the binding energy of the recoil ions on the surface.

Rodebäck and Allen\(^7\) studied the spectrum of recoil ions in the decay of A\(^ {37}\) (cf. Table I) in the apparatus shown in Fig. 7. Argon was introduced into the working chamber in a small amount, so that the total pressure in the chamber was about \(10^{-5}\) mm Hg. Collisions of recoil ions with molecules of the residual gas at such a pressure are improbable
\[ \left(\frac{\lambda}{d} \sim 80\right). \]
The effective volume in which the registered recoil ions arose was determined by the system of partitions in front of the entrance slits of the multipliers. The spectra of the recoil ions were studied by the method of measuring the time of flight of the recoil ions (Cl\(^ {37}\)) in a field-free space. The initial moment in this was fixed by the time of registration of the Auger electron corresponding to the given decay. To measure the distribution of the flight times of the recoil ions, the “recoil ion—Auger electron” coincidence method (\(R\)-\(e\)-coincid-

...decay), and 20 delayed-coincidence channels were used. The results obtained were recorded in the form of differential curves of the time distribution of recoil ions arising in the effective volume of the source. From these curves, using the known mean distance traversed by the recoil ions from the “effective” volume to the electron multiplier, and the resolving power of the apparatus, one can obtain a differential curve of the recoil-ion distribution by impulses (or energies).

From the fact that the experimentally obtained differential curve of the flight-time distribution (the solid curve in Fig. 8) agrees satisfactorily with the predicted one (the dashed curve in Fig. 8), the authors conclude that the recoil ions in the decay of \(A^{37}\) are monoenergetic. The calculation of the predicted shape of the main peak (in the region \(\Delta t \sim 6,\ 6\ \mu\mathrm{sec}\)) is based on taking into account the following factors:

Fig. 8. Curve of the flight-time distribution of recoil ions in the decay of \(A^{37}\). The dashed curve is the expected distribution of recoil ions.

1. Geometry of the apparatus.
2. Influence of the thermal velocities of \(A^{37}\) atoms in the gaseous medium.
3. Flight time of the ions in the accelerating field between grids \(C_2\) and \(C_3\) (Fig. 7).
4. Influence of the “sagging” of the accelerating field between the “effective” volume and grid \(C_2\).
5. Flight time of Auger electrons from the “effective” volume to \(E.U.\)-1 (Fig. 7).
6. Influence of the recoil received by the atom upon emission of Auger electrons (cf. formulae (5) and table I). It has been found that the influence of the last two factors is negligibly small. In the calculation it was assumed that the recoil ions (\(\mathrm{Cl}^{37}\)) are singly charged, although additional observations showed that there is some number of recoil ions with a charge different from unity. Quantitative results on the question of the external ionization of recoil atoms could not be obtained in the work*).

*) It has recently been shown \(^{16}\) that the mean charge of the recoil ion in the decay of \(A^{37}\) is \(+3.85 \pm 0.2\). Since in the experiments of Allen and Rodeback the method used measured the flight time in a field-free space, the influence of the charge manifested itself in a small region from the grid before the electron multiplier to the 1st dynode.

Since the experimental curve in Fig. 8 shows the presence of an additional peak in the energy region close to zero, and also the presence of some number of ions with a short flight time, it was necessary to show that they arise as a result of “false coincidences.” This was proved by a series of additional experiments, the results of which are presented in the curves of Fig. 9.

Here the number of \(R\)-\(e\) coincidences is plotted, obtained for various delay times (\(\Delta t = 0\) and \(\Delta t = 6.6\ \mu\text{sec}\), which corresponds to the peak in Fig. 8) and for various negative potentials of grid \(C_1\) and of the first dynode of E. U.-1 (Fig. 7), which registers electrons. Analysis of the conditions under which these curves were obtained, and of the course of the curves, leads to the conclusion that the peak in the curve of Fig. 8 (at \(\Delta t = 0\)) and the coincidences at short flight times are due in origin to the decay of \(A^{37}\) atoms in the space between grids \(C_2\) and \(C_3\) (Fig. 7). In this case recoil ions are registered by E. U.-2, while Auger electrons, accelerated by the field between grids \(C_2\) and \(C_3\), are scattered on the walls of the partitions limiting the “effective” volume and are registered by E. U.-1. The part of the coincidences that occurs at a delay \(\Delta t = 6.6\ \mu\text{sec}\) (Fig. 9) and at a retarding potential greater than 2800 V can be explained by x-ray–recoil-ion coincidences. Indeed (cf. Table I), approximately 8% of the recoil atoms pass to the normal state by emission of x-ray quanta and do not emit Auger electrons.

Fig. 9. Counting curves of \(R\)-\(e\) coincidences as a function of the retarding potential applied to grid \(C_1\) (Fig. 7), for different delay times.

5. CONCLUSION

The results of the work of Davis[^6] and Rodaback and Allen[^7] may be regarded as experimental proof that the spectrum of recoil ions in \(E\)-capture is linear, in accordance with the assertion of the theory on the emission of a single neutrino in the beta-transformation process.

A. R.

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