NUCLEAR DISINTEGRATIONS CAUSED BY COSMIC PARTICLES OF HIGH ENERGY*)

K. F. Powell, U. Camerini, P. Fowler, et al.

III. THE NATURE OF SHOWER PARTICLES**)

INTRODUCTION

In Part II of the present series of articles it was shown that, in favorable cases, the rest mass and energy of fast charged particles emitted in the explosive disintegration of nuclei can be determined by measuring the grain density in the tracks of these particles and the deviations in the direction of the tracks caused by multiple Coulomb scattering. The basic principles of the methods used for measuring scattering were described by Goldschmidt-Clermont et al.\(^1\) and by Davis et al.\(^2\).

In Part II it was indicated that the new method of particle identification is in principle similar to the method used in experiments with a Wilson chamber operating in a magnetic field. The mean deflection of a track per unit length \((\bar{\alpha})\) makes it possible to determine the quantity \(\frac{p\beta}{Z}\) for the corresponding particle, where \(p\) is the momentum of the particle, \(Z\) is its charge, and \(\beta\) is the ratio of the particle velocity \(v\) to the velocity of light \(c\). On the other hand, if the charge of the particle is known, the grain density in its track makes it possible to determine the velocity of the particle and, consequently, the corresponding value of \(\beta\).

These two measurements of one and the same track therefore make it possible to determine both the mass and the velocity of the particle, if its kinetic energy lies in a definite range of values, namely, if the kinetic energy is less than \(0.6\,mc^2\), where \(m\) is the rest mass of the particle, and greater than approximately \(15\) MeV. This method has the advantage over the Wilson-chamber method that it permits the specific ionization of a cosmic-ray particle to be determined, by measuring the grain density in its track in a photographic emulsion, with much greater accuracy than in counting droplets on photographs taken with a controlled—

*) Parts I and II were published in UFN, Vol. XL, No. 1, p. 76 (1950).
**) P. H. Fowler, Phyl. Mag. 41, 163 (1950).

by a Wilson cloud chamber. In addition, with the aid of scattering measurements in favorable cases it is apparently possible to determine the quantity \(\frac{p\beta}{Z}\) with an accuracy comparable to that usually attained in experiments measuring the deflections of charged cosmic particles in a magnetic field.

The importance of the new method lies in the fact that it very greatly extends the range of application of the photographic method. Until now, measurements of mass by the scattering method have been restricted to particles of low energy, whose tracks end in the emulsion. In a certain energy region the new method removes this limitation and enables us to determine the mass and energy of particles, independently of whether or not they come to rest in the emulsion.

Although the measurements described in Part II have already shown the possibilities of this method, the determination of the scattering parameter \(\alpha\) was nevertheless difficult, since it required measuring the positions of individual grains along the track from an image projected onto a screen with the aid of a projection microscope. This feature limited the range of application of the scattering method, since with the instruments at our disposal a detailed analysis, necessary for obtaining results with good statistical accuracy, would have required several years of work. Thus it was desirable to develop simpler methods for measuring scattering. It turned out that a method using microscopes of ordinary design has the advantage of greatly increasing the speed of measurements, without any noticeable deterioration in their accuracy, and of extending the range of energies of particles that can be identified.

DETERMINATION OF THE SCATTERING PARAMETER

In order to determine the scattering parameter \(\alpha\) for a given track, the plate is placed on the movable stage of a microscope so that the track is approximately parallel to one of the directions of motion of the stage, for example the \(X\) axis. The coordinate \(y_1\) of the point at which the image of the track intersects the \(Y\) axis in the eyepiece is determined by means of the eyepiece scale. The microscope stage is then moved along the \(X\) axis by a definite distance, for example by \(100\,\mu\), and the coordinate \(y_2\) of the point of intersection of the track image with the \(Y\) axis of the eyepiece scale is again read. This operation is repeated along the entire length of the track.

The first differences \(s_1 = y_1 - y_2\), \(s_2 = y_2 - y_3\), etc., make it possible to measure the inclinations \(i_1\), \(i_2\), etc., of successive chords in the track, while the second differences \(D_1 = s_1 - s_2\), \(D_2 = s_2 - s_3\), etc., determine changes in the directions of each two neighboring chords. The mean

the quantity \(|D|\)

\[ \bar D=\frac{\sum_{n}|D|}{n} \]

makes it possible to determine the scattering parameter \(\bar s\), from which one can estimate the value \(\dfrac{p\beta}{Z}\).

For successful application of this method it is necessary that any apparent changes in the direction of the track, associated with mechanical imperfections in the motion of the microscope stage, be smaller than the deviations caused by scattering. It would seem difficult to check the suitability of a given microscope stage for measurements of multiple scattering without a special optical device. However, observations of tracks of particles with very large momentum showed that, when working with a good microscope, inaccuracies in the motion of its stage lead to errors that do not exceed the error in measuring the coordinate of the point of intersection of the image of the track with the axis of the eyepiece scale.

The results of a characteristic series of measurements of a section of a track whose total length was \(8\) mm are given in Table IX. In this case the quantity \(y\) was determined for successive points every \(50\,\mu\). The analysis of the observations was performed with a segment length \(c\) equal to \(100\,\mu\), and it was found that the mean value \(\bar\alpha(100)\) of the angular deviation between the directions of adjacent chords of length \(100\,\mu\) is equal to \(0.105^\circ\). Although the individual values of \(D\) obtained by the above method are not independent, nevertheless the use of overlapping segments leads to a reduction in the errors of counting.

A similar analysis was also carried out for other segment lengths \(C\), equal to \(200\,\mu\), \(300\,\mu\), etc. The mean values \(\bar\alpha\) for different values of \(C\) are related by the relation \(\bar\alpha(C_1)/\bar\alpha(C_2)=\sqrt{C_1}/\sqrt{C_2}\). The values \(\bar\alpha(200)\), \(\bar\alpha(300)\), etc., can therefore be used to determine the quantity \(\bar\alpha(100\,\mu)\), to which they are equivalent. The corresponding quantities, given in Table IX, show that \(\bar\alpha(100\,\mu)\) decreases as the segment length increases, reaching an approximately constant value. This is due to the fact that, for too small a segment length, the apparent changes in the direction of the track associated with counting errors are greater than the true deviations caused by scattering, which leads to an overestimate of the value of \(\alpha\). On the other hand, if the segment length is unreasonably large, the number of statistically independent observations decreases, and the calculated values of \(\bar\alpha\) will accordingly be more strongly subject to statistical fluctuations. Consequently, it is desirable to choose the smallest

Table IX

Characteristic results of measurements of particle scattering, the track of which has a total length of 8 cm. Counts were made every 50 μ on the first millimeter of the track. The grain density in the track is 19.2 grains per 50 μ, \(C = 100\ \mu\)

\(x\), mm	\(y\), divisions	\(S\) \(y_n-y_{n+2}\)	\(D\) \(s_n-s_{n+2}\)	\(x\), mm	\(y\), divisions	\(S\) \(y_n-y_{n+2}\)	\(D\) \(s_n-s_{n+2}\)
12,8	5,6	0,5	0,3	12,25	7,0	0,4	0,3
12,75	5,7	0,6	0,6	12,2	7,3	0,2	0,2
12,7	6,1	0,2	0,1	12,15	7,4	0,1	0,1
12,65	6,3	0,0	−0,2	12,1	7,5	0,0	−0,3
12,6	6,3	0,1	0,0	12,05	7,5	0,0	−0,3
12,55	6,3	0,2	0,1	12,0	7,5	0,3	0,3
12,5	6,4	0,1	−0,2	11,95	7,5	0,3	0,3
12,45	6,5	0,1	−0,3	11,9	7,8	0,0	−0,1
12,4	6,5	0,3	−0,2	11,85	7,8	0,0	−0,1
12,35	6,6	0,4	0,0	11,8	7,8	0,1	−0,0
12,3	6,8	0,5	0,3

Values of \(\overline{\alpha}\) \((100\ \mu)\) for various values of \(C\)

\(C\mu\)	\(\overline{D}\)	\(\overline{\alpha}\ (100\ \mu)\), in degrees
50	0,141	0,191
100	0,22	0,105
150	0,285	0,074
200	0,35	0,060
300	0,54	0,050
400	0,78	0,047
600	1,56	0,051
1000	3,74	0,057
1400	6,28	0,058
2000	11,0	0,058

the possible length of the segment for which the uncertainty in the value of \(\overline{\alpha}\), caused by counting errors, is small¹.

It was found that, for small segment lengths \(C\), the values of \(\overline{D}\), owing to counting errors, approach a constant value and do not depend on \(C\). Following the preceding authors, we shall call this value the “noise level” \((\overline{D}_n)\). The quantity \(\overline{D}_n\) depends on the grain density in the track, on the quality and arrangement of the optical system of the microscope, and on the individual skill of the observer, but it is of the order of \(0.2\ \mu\). As the segment length \(C\) is increased, the quantity \(\overline{D}\) becomes greater than \(\overline{D}_n\). The corresponding value of \(\overline{\alpha}\ (100\ \mu)\) then rapidly approaches a constant

significance, if statistical fluctuations are not counted. These results may be generalized by saying that the measurement errors become negligible if \(\overline{D} > 4\overline{D}_n\).

When measuring the quantity \(\bar{\alpha}\) (100 \(\mu\)), the minimum value \(C\) is chosen for which this condition is satisfied. It is clear that if it is desired to measure the quantity \(\bar{\alpha}\) with a specified statistical accuracy, then the minimum length of track suitable for measurement will depend on the value of \(\bar{\alpha}\) (100 \(\mu\)); moreover, for measuring the scattering of particles of higher energy, longer tracks are needed. Thus, if \(L\) is the track length, then the number of statistically independent quantities \(D\) is equal to

\[ \left(\frac{L}{C}-1\right). \]

Table X

Minimum track lengths required for determining the scattering parameter \(\alpha\) (100 \(\mu\)) with an accuracy of up to 25% in the absence of emulsion distortions

Track length in mm	\(\alpha\) (100 \(\mu\)) in degrees
3	0.017
5	0.008
10	0.003
20	0.001

It can be shown that, for the track considered above, the root mean square deviation of the individual values of \(D\) from the true mean value corresponding to the particle momentum is equal to \(\sim 0.5D\). Approximate minimum track lengths required for determining \(\overline{D}\) with a probable error of less than 25% are given for various values of \(\alpha\) (100 \(\mu\)) in Table X.

EFFECTS OF EMULSION DISTORTION

During development of an exposed plate, the volume of its emulsion increases greatly. If there is good adhesion between the gelatin and the glass, then the swelling in the central regions of the plate corresponds approximately to a uniform linear transformation of the coordinate \(Z\) for any point, i.e. to a displacement of the point in a direction perpendicular to the glass. During drying of the plates after fixing and washing, the well-known shrinkage of the emulsion occurs, associated with the loss of water and the removal of silver bromide from the emulsion during fixing. Since we are interested only in scattering in the plane of projection of the track, such shrinkage of the emulsion will not, in principle, affect the measurements.

In practice, however, especially when working with thick emulsions and in regions near the edges of the plate, certain distortions of the emulsion are also observed in the plane of projection of the track, so that a strictly rectilinear track will usually appear curved.

Distortions of the emulsion arise, at least in part, as a result of the stresses formed when water is absorbed, and also because the upper surface and the sides of the emulsion are not fixed in any way. Thus, the cross section of the emulsion (rectangular in a dry emulsion) is distorted if the plate is placed in the developing solution. As the emulsion dries, the distortions may remain or increase.

The difficulties can be substantially reduced if large plates are used and observations are confined to the central region of the plates. But even with these precautions, deviations of tracks caused by distortions can lead to a serious error in determining the energy of the fastest particles \([\alpha(100\mu) \ll 0.01^\circ]\). In the worst cases, a correction is necessary even for tracks for which the true value of \(\alpha(100\mu)\) is equal to \(0.05^\circ\).

The presence of distortions can be established in the following two ways: 1) it is sometimes found that the values of \(D\), instead of being distributed approximately equally between positive and negative quantities, are all of one sign. In such cases, examination of more inclined tracks, whose projections are approximately parallel to the projection of the track under consideration, may indicate the presence of distortions of the emulsion.

Observations show that the distortions usually have such a character that the deviation which they introduce into a strictly rectilinear track can be written in the form

\[ \Delta = F\delta z, \tag{1} \]

where \(\Delta\) is the change in the direction of the particle as it passes from depth \(z\) to depth \(z+\delta z\) of the emulsion, and \(F\) is a constant for the given direction of projection of the track in the given region of the emulsion. From observations of more inclined tracks one can determine the constant \(F\) and then make the corresponding corrections to the track under consideration. If these corrections change the magnitude of the deviations attributed to scattering so that they are now symmetrically distributed about zero, apart from differences connected with fluctuations, then the corrected quantities are considered suitable for determining the true value of \(\bar D\).

2) A more refined method is usually used more often. The presence of emulsion distortions of the type described by equation (1) leads to the fact that the observed deviations of the track are caused by the superposition of the curvature \(D\), connected with scattering, and the deviation \(\Phi\), connected with distortion of the emulsion. The observed deviations \(D_a\) may therefore be written as \(D_a = D + \Phi\), where \(\Phi\) is a constant term*). If the emulsion distortions

*) With sufficient accuracy it may be assumed that the inclination of the track to the plane of the emulsion is a constant quantity.

Table XI

Observation of a strongly distorted track

$x$, mm	$y$, scale divisions	$S$ $y_n-y_{n+2}$	$D$ $s_n-s_{n+2}$	$D^*=D-\Phi$ $\Phi=4$
9,1	10,0	−0,5	5,0	1,0
8,9	9,5	1,0	7,0	3,0
8,7	9,5	4,5	6,0	2,0
8,5	10,5	8,0	4,5	0,5
8,3	14,0	10,25	5,0	1,0
8,1	18,5	12,5	5,25	1,25
7,9	24,25	15,25	4,0	0,0
7,7	31,0	17,75	2,0	−2,0
7,5	39,5	19,25	1,25	−2,75
7,3	48,75	19,25	2,75	−1,25
7,1	58,75	20,5	3,25	−0,75
6,9	68,5	22,5	2,0	−2,0
6,7	79,25	23,75
6,5	91,0	24,5
6,3	103,0
6,1	115,5

\[ \Phi=\frac{\Sigma D}{n}=4,\qquad \frac{\Sigma |D^*|}{n}=1,46. \]

Table XIa

Values of $a^*(100\ \mu)$ for various values of $\Phi$

$\Phi$	$\Sigma |D-\Phi|$	$a^*(100\ \mu)$
0	48,0	0,12
1	36,0	0,09
2	25,5	0,064
3	20,0	0,050
4	17,5	0,044
5	18,5	0,046
6	26,0	0,065
7	36,0	0,090
8	48,0	0,12

sharp, so that \(\Phi\) is large, then all the values \(D_a\) will have the same sign, despite the fact that positive and negative values of \(D\) are observed equally often, if statistical fluctuations are not taken into account. An approximate value of \(\Phi\) may be obtained from the relation \(\frac{\Sigma D_a}{n} \sim \Phi\), where \(\Sigma D_a\) is the algebraic sum of the individual values \(D_a\). Knowing the magnitude of \(\Phi\), one can determine the individual values \(D\) and, consequently, the approximate value \(|\overline{D}|\). The quantity \(|\overline{D}|\), determined in this way, depends only weakly on the value of \(\Phi\) used, provided that the error in \(\Phi\) is less than \(\frac{|\overline{D}|}{2}\).

Table XI gives the results of measurements of a strongly distorted track with the correction \(\Phi\), calculated by means of the method described above. Table XIa gives the values of \(\alpha\) (100 \(\mu\)) obtained from measurements of the very same track, but for various arbitrarily chosen values of \(\Phi\). It is seen from the table that the final results of the measurements depend only weakly on the degree of accuracy of the correction term \(\Phi\).

LIMITATIONS OF THE METHOD

The minimum value \(\overline{D}\) that can be regarded as caused by scattering, and, consequently, the maximum value of the particle energy that can be measured, depend on the length of the track being measured, the magnitude of the “noise level,” and the degree of distortion of the gelatin. The minimum value of \(\alpha\) (100 \(\mu\)) that can be measured for different track lengths, given in Table X, corresponds to conditions in which distortions of the emulsion are absent.

The minimum value of \(\overline{\alpha}\) (100 \(\mu\)) actually observed in the present experiments is \(0.001^\circ\). Whether the theoretical value \(\overline{\alpha}(100 \mu)=0.0005^\circ\) for a track \(2\ \mathrm{cm}\) long can be reached under our experimental conditions, we do not know. Such a deviation of the track can be attributed to scattering if, when the microscope stage is displaced by \(2\ \mathrm{cm}\), its deviation from a strictly rectilinear direction of motion does not exceed \(0.4\mu\). The possibility of attaining this order of accuracy for shorter displacements of the stage is confirmed by the observation of long tracks with a value \(\overline{\alpha}(100\mu)=0.001^\circ\).

If distortions of the emulsion occur, then the minimum value \(\overline{D}\) that can be attributed to scattering for a track of a given length will be greater than the values given in Table X. However, if one confines oneself to observations of tracks in the central region of the plates, under the condition that the correction for distortion \(\Phi\) is less than \(2D\), then for the majority of tracks Table X remains valid.

CALIBRATION

If the particle track is divided into segments of length \(C\), then, as is known\(^{3,4}\), the mean angular deflection \(\bar{\alpha}\) between successive chords is related to the particle velocity \(v\) by the relation

\[ \bar{\alpha}=\frac{kZ}{m_0v^3}(1-\beta^2)^{\frac12} \sqrt{\ln \frac{\theta_{\max}}{\theta_{\min}}}, \tag{2} \]

where \(Z\) is the charge of the particle, and \(m_0\) is its rest mass. The constant factor \(k\) in equation (2) depends on the composition of the emulsion and on the chosen segment length \(C\). Without introducing a substantial error, one may neglect the change in the magnitude of the logarithmic factor in equation (2); then

In the figure: vertical axis — “Tenth logarithm of the number of grains (number of grains per 50 µ)”; horizontal axis — “Proton range in mm”; legend — “+ For protons and deuterons”; “○ For \(\pi\)- and \(\mu\)-mesons.”

Fig. 24. Grain density in a proton track as a function of its residual range.

\[ \bar{\alpha}=\frac{k_1 Z}{m_0v^2}(1-\beta^2)^{\frac12} =\frac{k_2 Z}{pv}, \tag{3} \]

where \(p\) is the momentum of the particle.

In order to determine the relation between the grain density \(g\) in the track and the specific energy loss of the particle that produced this track, the tracks of mesons, protons, and deuterons of large range stopping in the emulsion were investigated. The results are shown in Fig. 24. This curve and the range–energy relation for the same particles make it possible to determine the grain density in the track of any particle of charge \(e\) as a function of its velocity.

Direct determinations of the energy–range dependence for protons in electron-sensitive plates have not yet been published. It is known, however, that in Ilford C-2 emulsions the range \(R\) (in microns) of protons with energy \(E\) MeV is given by the relation

\[ R = cE^n \]

and that

\[ \frac{dE}{dR} = k Z^2 \beta^{-n}. \tag{4} \]

Let us suppose that for the G-5 emulsion, too, relations of the same type are valid. The corresponding quantities \(k\) and \(n\)

Fig. 25. Relation between grain density and specific energy loss along the track. Ilford G-5 emulsion.

can be determined in the following way. The mean range in this emulsion of \(\mu\)-mesons emitted with kinetic energy \(4.1\) MeV in the decay of stopped \(\pi\)-mesons is \(595 \pm 10\,\mu\). Next, from Fig. 24 we determine the residual range of the proton whose initial ionization is equal to the initial ionization of “primary” \(\alpha\)-particles of very high energy (equal grain densities in the tracks). The specific ionization of these \(\alpha\)-particles is equal to four times the minimum ionization for particles with charge \(|e|\). From these data it was found that

\[ \frac{dE}{dR} = \frac{0.587 Z^2}{\beta^{1.46}} \ \text{keV}/\mu . \tag{5} \]

The value of \(n\) for the G-5 and C-2 emulsions turns out to be the same. Using this relation, as well as the curve for the change in grain density along the proton track given in Fig. 24, we can determine the grain density in the track as a function of the particle’s specific energy loss (Fig. 25). It is evident from the figure,

that the relation is almost linear for values of the specific energy loss up to four times the minimum value. From equation (5) and Fig. 25 one can derive a relation between the grain density in the track of a high-energy proton and the corresponding value of the scattering parameter $\bar{\alpha}$.

EXPERIMENTAL RESULTS

The methods described in the preceding paragraphs were used in the study of 350 tracks associated with stars, the length of the tracks in the emulsion exceeding $3000\,\mu$. The observations were carried out on Ilford G-5 plates with an emulsion thickness of $400\,\mu$, exposed at high altitudes, as described in Part II of the present series of papers. The results are presented in Fig. 26, where, on a logarithmic scale, the dependence of the grain density on the scattering parameter is shown for the tracks investigated.

The shape of the curves in Fig. 26, denoted by $\pi$, $P$, $D$, and $T$, is similar to the shape of the theoretical curves obtained by the methods of the preceding paragraphs for particles with charge $e$. However, these curves are displaced along the axis of the scattering parameter by approximately 7% relative to the theoretical curves for the corresponding particles. Their actual position was determined by the following method.

The calculated curve for protons was shifted until the best agreement was obtained with the points corresponding to the most numerous group of particles. These tracks were undoubtedly produced by protons. Then other curves of the same shape as the first were drawn so as to correspond to particles with mass $286m_e$ for $\pi$-mesons and to the mass values of deuterons and tritons. The corresponding curves for $\alpha$-particles, lithium nuclei, etc., can therefore be derived from the curve for protons, if the curve shown in Fig. 25 is used. The difference of only 7% between the calculated curve for protons and the observed distribution for these particles is not unexpected, in view of the uncertainty in some of the quantities on which the calculations were based.

When plotting the results shown in Fig. 26, primary and secondary particles were distinguished by means of the criteria described in Part I. In disintegrations in which many shower particles are emitted, such a particle was considered primary if its track is located in the upper hemisphere, and its direction is closest to the axis of the cone of shower particles.

In disintegrations with a smaller number of emitted particles, especially in observations at altitudes exceeding $4000\,m$, where the directions of the primary particles are distributed over a wider range of angles, the identification of primary particles is subject to

some uncertainty, and should be treated with caution. However, since the energies of the primary particles are large

Fig. 26. Relation between grain density and \(\alpha\) \((100\ \mu)\) for tracks of 350 particles. “Primary” particles are denoted by ●, “secondary” particles by ○. The curve \(P\) has the same form as the calculated curve for protons, but it is shifted to the left along the \(\alpha\)-axis by 7%. The lines \(\pi\), \(D\), and \(T\) are drawn in the correct relation to the line \(P\), so as to correspond to particles with masses \(286\,m_e\), \(3674\,m_e\), and \(5511\,m_e\). \(A\), \(B\), and \(C\) are lines of constant momentum and correspond to momenta of 500, 1000, and 2000 MeV/\(c\). The energy scales (in MeV) for \(\pi\)-mesons and protons are given in the upper part of the figure. It is evident that all high-energy \(\alpha\)-particles are “primary” particles. Two points corresponding to particles of larger mass belong to the “primary” nitrogen nucleus and the “secondary” lithium nucleus produced in the same event.

in comparison with the energies of the secondary particles (see Fig. 26), it may be thought that errors in identification are rare.

From consideration of Fig. 26 it is seen that the primary $\alpha$-particles produce disintegrations with the same frequency as the protons in the very same energy interval. On the other hand, $\alpha$-particles of high energy are emitted in nuclear disintegrations very rarely in comparison with protons and deuterons. Measurement of the distance from the proton curve in a direction parallel to the $\alpha$-axis to any observed point in Fig. 26 makes it possible to determine the mass of the corresponding particle. In Fig. 27 is shown the distribution of the mass values of the particles obtained in this way as a result of measurements of all the points (Fig. 26) with a grain density in the track in the interval from 15 to 98 grains per 50 $\mu$. It may be seen that, besides

Fig. 27. Distribution of masses of particles of charge $|e|$, corresponding to the measurements shown in Fig. 26.

the large maximum due to protons, there are groups of lower intensity which merge with the proton maximum and which may be attributed to deuterons and tritons, and, in addition, there is a distinct maximum due to mesons with an average mass of $283 \pm 14\,m_e$.

It may be emphasized that if the distance of the individual points in Fig. 26 from the theoretical curve for protons were measured, then the resulting distribution of particle mass values would be identical in form with the distribution shown in Fig. 27, but all the mass values would be reduced by 7%. However, it is more reasonable to obtain the mass of the mesons by direct comparison with the mass of the protons, which was done in constructing the graph in Fig. 27. Thus, the overwhelming majority of the mesons can be identified with $\pi$-mesons.

The observed distribution of the grain-density values in the tracks of particles of charge $e$, presented in Fig. 26, is shown

on Fig. 28. This curve may be compared with the results of similar measurements (Part I, Fig. 1), carried out on plates that were exposed on the Jungfrau. These two distributions are similar in form, but in the present observations there are relatively more particles with a low grain density. This corresponds to the higher mean energy of nuclear explosions at a greater altitude (see Part II).

Study of Fig. 26 shows that the majority of particles forming tracks with a grain density of less than 14 grains per 50 μ,

Fig. 28. Distribution of grain-density values in the tracks of particles with charge \(|e|\). In the lower figure—the distribution for all particles; in the upper—for particles identified as π-mesons.

are either mesons or “primary” particles. The nature of secondary particles for which \(a(100\mu)<0.02\) cannot be determined by the present methods. It is possible, however, to extrapolate the observed energy distribution of protons and π-mesons to the high-energy region and thus estimate the ratio of these two types of particles in the region of high energies. By making the appropriate corrections for particles emerging from the emulsion, one can estimate the relative frequency of occurrence of the various types of particles in stars. The results are given in Table XII.

Table XII

Average number of particles of various types with a range exceeding 3 mm in the “star” at an altitude of 23,000 m

π-mesons	Protons	Deuterons	Tritons	α-particles
0.6	1.4	0.5	0.2	0.02

Average number of tracks in a star, 7.4

CONCLUSIONS

From the present experiments the following conclusions may be drawn:

1) The overwhelming majority of mesons emitted with kinetic energy less than 150 MeV as a result of nuclear disintegrations produced by high-energy protons and α-particles are π-mesons. If μ-mesons are sometimes produced directly, their number amounts to less than 2% of the number of π-mesons.

2) If electrons or other charged particles with small rest mass and with energy less than 150 MeV are sometimes emitted in nuclear explosions, then their number is less than 2% of the number of π-mesons. Since the greater part of the electrons of the soft component of cosmic radiation has energies less than 200 MeV, the observations indicate that the soft component is of secondary origin.

3) The observations do not confirm the assumption that mesons of intermediate mass, τ-mesons with a lifetime greater than \(10^{-12}\) sec., are emitted with energy less than \(\sim 300\) MeV during nuclear explosions. If such particles with mass between \(350 m_e\) and \(1200 m_e\) do sometimes arise in these processes, their number is less than 2% of the number of π-mesons and less than 1% of the number of protons. This conclusion is confirmed by the work of Franzinetti \(^{5}\) on the determination of the masses of slow charged particles produced by cosmic rays by the method of deflection in a magnetic field.

4) A final conclusion as to the nature of particles with mass less than the proton mass, which form tracks with a minimum grain density, cannot be made. Taking into account observations of particles of lower energy, it is reasonable to assume that the faster particles are particles of the same type. In what follows they will therefore be regarded as π-mesons.

5) At least 90% of the “shower” particles emitted in nuclear explosions are π-mesons, the shower particle being defined as a particle with grain density less than \(1.5 g_{\min}\) (see Part I). Let us consider, however, all particles in the momentum interval between 200 and 700 MeV/\(c\) (the interval usually dealt with in experiments with a Wilson chamber operating in a magnetic field). Besides π-mesons, among the particles in this inter-

there is a large number of protons in the momentum interval. If positive and negative \(\pi\)-mesons occur equally often, then there should be approximately six times as many positive particles as negative ones, which agrees with the value observed in experiments with a Wilson chamber\(^6\).

b) The observed uniformity in the mass values of mesons emitted with energy less than 150 MeV allows us to exclude certain assumptions concerning the origin of the particles. From experiments on the artificial production of \(\pi\)-mesons it is known that the threshold of this process corresponds to such an energy of the bombarding protons or \(\alpha\)-particles that, from the standpoint of conservation laws, the emergence of a particle with mass \(\sim 280\,m_e\) is just possible. It follows from this that \(\pi\)-mesons must be formed directly, and that they cannot be regarded as products of the decay of mesons of substantially greater mass with a very short lifetime.

In the nuclear transformations considered by us, which take place in cosmic rays, there are no restrictions on the energy. In principle, therefore, the possibility cannot be excluded that the \(\pi\)-mesons observed in showers are formed in the decay of short-lived mesons of greater mass, which are the primary products of nucleon–nucleon collisions. The observed uniformity in the mass values of the emitted mesons indicates, however, that any presumed unstable “ancestor” of the \(\pi\)-mesons, if it exists, must have a very short lifetime, since, if it could exist on the average over a path length of 2 mm, the observed mass values would be intermediate between the masses of the primary and daughter particles. As a result we might expect a dispersed collection of mass values instead of the compact group actually found. The present observations, consequently, prove that any “parent” mesons of greater mass, if they exist, have lifetimes less than \(10^{-13}\) sec. It is, of course, also reasonable to suppose that \(\pi\)-mesons are formed directly in acts of nuclear collisions.

IV. ENERGY SPECTRUM AND SECONDARY INTERACTION OF PARTICLES EMITTED IN STARS\(^*\)

MEASUREMENTS OF PARTICLE MASS BY THE SCATTERING METHOD

New measurements were also carried out with plates coated with Ilford G-5 emulsion 400 \(\mu\) thick and exposed during balloon-sonde flights, the duration of which

\(^*\) U. Camerini, P. H. Fowler, W. O. Lock, H. Muirhead, Phyl. Mag. 41, 413 (1950).

Fig. 29. Dependence of the grain density on \(\bar{\alpha}\) (100 \(\mu\)) for tracks of 700 particles. \(\bullet\)—primary particles, \(\circ\)—secondary particles, \(-\circ-\)—particles joining double stars. The curves \(\mu, \pi, P, D, T\) mean the same as in Fig. 26. \(A, B, C\)—showers corresponding to a minimum pulse equal respectively to 500, 1000, and 2000 MeV/s. The kinetic energy of protons and \(\pi\)-mesons is given in the upper part of the figure.

Vertical axis: Grain density (number of grains per 50 \(\mu\))

Horizontal axis: Mean deflection \(\bar{\alpha}\), in degrees per 100 \(\mu\)

is represented by curve b in Fig. 17 (Part II). Plates with the same coating were also used in the measurements described in Part III.

The experimental data obtained for 700 tracks of charged particles with a grain density of less than 50 grains per 50 μ are presented in Fig. 29. The curves μ, π, P, D, and T are the same as the corresponding curves in Part III, and the value of \(k_2\) used [equation (3), Part III] is equal to 32.7. This value of \(k_2\) was chosen because it gives the best agreement with the experimental distribution for protons producing tracks with a grain density in the interval between 16 and 50 grains per 50 μ.

Fig. 30. Mass distribution of particles with charge \(|e|\), corresponding to the results presented in Fig. 29.

For large values of the grain density, a slight deviation of the experimental points for protons is observed, corresponding to a change in the value of \(k\) by approximately 7%. This, however, does not affect the determination of the meson mass, since a π-meson with a residual range of \(\sim 3\ \mu\text{m}\) produces a track with a grain density approximately equal to 45 grains per 50 μ.

The mass spectrum obtained by the method described in Part III is shown in Fig. 30. The number of meson tracks with a grain density greater than 15 grains per 50 μ, suitable for mass measurement, is 58. The mean mass of these particles is \((283 \pm 7)m_e\).

In order to confirm the correctness of the mass determination for these particles, analogous measurements were carried out on 22 tracks of π-mesons and 13 tracks of μ-mesons. The π-mesons were identified by the \(\pi \to \mu\) decay or by nuclear disintegration at the end of the track, and the μ-mesons by decay with emission of an electron. The results of these

measurements are shown in Fig. 31. It seems to us that these results leave little doubt that all or almost all mesons formed in stars with kinetic energy less than 150 MeV are in fact π-mesons.

The nature of the particles with kinetic energy between 150 and 1500 MeV, producing tracks with a grain density of less than 16 grains per 50 μ, cannot be determined by such measurements. Piccioni’s work\(^7\), however, has shown that the majority of them are also π-mesons. The possibility of the presence of a large number of electrons among the shower particles can also be excluded on the following grounds: 12 out of 300 shower particles have a range greater than 2 cm. Since the \(t\)-unit for the emulsion is equal to 2.3 cm, a noticeable decrease in energy should have been observed at least on some of these tracks, if a significant fraction of these tracks had been produced by electrons.

Fig. 31. Dependence between the grain density and \(\bar{\alpha}\) (100 μ). ○ — mesons emitted from a star, the same as in Fig. 29 for the corresponding interval of grain densities; + — π-mesons, either decaying into a μ-meson or giving a star at the end of the range; × — μ-mesons decaying into an electron at the end of the range.

ENERGY SPECTRUM OF PARTICLES EMITTED IN NUCLEAR DISINTEGRATIONS

The results shown in Fig. 29 enable us to construct the energy spectrum of secondary particles. Such a spectrum is presented in Fig. 32 for π-mesons, protons, and deuterons together with tritons. The results are plotted on a logarithmic scale. By π-mesons here are meant all particles lighter than

protons; however, the arguments given above allow us to consider, with greater justification, that most of the points in the energy interval 150–1500 MeV on or near the π-curve in Fig. 29 do indeed belong to π-mesons. The point in Fig. 32 corresponding to π-mesons with energy \(\sim 1.6\) MeV was obtained from the distribution of the residual ranges of emitted slow π-mesons that stop in the emulsion and form “stars.” To obtain

Fig. 32. Differential spectrum of kinetic energy for particles of various types. The point corresponding to a meson energy of 1.6 MeV was obtained only for \(\pi^-\)-mesons produced in stars and producing nuclear disintegration at the end of their range. To obtain the number of particles per star per energy interval of 1 MeV, the corresponding values must be multiplied approximately by \(2.5 \cdot 10^{-5}\).

this point it was assumed that only 72% of \(\pi^-\)-mesons form stars recorded in the photographic emulsion,\(^{8}\) and a geometrical correction for “losses” of particles was also introduced. The energy spectrum is cut off at energies corresponding to an average scattering angle of \(0.022^\circ\) per 100 \(\mu\), since the nature of faster particles could not be established.

Figure 33 gives the differential spectrum of mesons possessing energy \(E\) at the point of their production. In the interval

energies from 250 to 1400 MeV the differential spectrum can be represented in the form of a power function \(\dfrac{dE}{E^{1.5}}\).

No appreciable difference has been found between the energy spectra of mesons produced in stars with different numbers of strongly ionizing particles. This indicates that the energy spectrum does not depend strongly on the nature of the nucleus in which the mesons are generated. Consequently, the observed energy spectrum shown in Fig. 33 should be similar to the spectrum of mesons produced in the atmosphere at a depth of the order of

Fig. 33. Differential spectrum of the total energy for mesons emitted in stars.
The solid curve is the Sands spectrum.

50 g/cm\(^3\), i.e., the depth of that layer in which mesons are mainly produced in the atmosphere. It is therefore of interest to compare our data with the results of Sands\(^9\), who calculated the energy spectrum of mesons at the point of their generation from

a) measurements of the intensity of slow \(\mu\)-mesons at various altitudes in the atmosphere and

b) the spectrum of mesons at sea level.

\(\pi\)-mesons produced in a nuclear interaction occurring in the atmosphere must decay in flight with the formation of \(\mu\)-mesons. In the coordinate system associated with the flying \(\pi\)-meson,

μ-mesons are distributed isotropically. Since the velocity of μ-mesons in the center-of-mass system is small compared with the velocity of the π-meson, in the laboratory system the μ-mesons have essentially the same velocity as the π-mesons. Thus, in order to compare our data, shown in Fig. 33, with Sands’s curve, it is necessary only to make a correction for the difference in the masses of the π- and μ-mesons and then to add the rest mass of the π-meson. Sands’s data are represented by the solid curve in Fig. 33. It is easy to see that the two spectra agree with each other rather well.

In the experimental spectrum there is an indication of a point of inflection corresponding to the minimum at 450 MeV on Sands’s curve. However, the statistical errors of the experimental data obtained are too large for significance to be attached to such a coincidence.

For values of the kinetic energy greater than \(10mc^2\), where \(m\) is the rest mass of the meson, Sands’s spectrum follows a power law of the form \(\dfrac{dE}{E^{2.5}}\). The results shown in Fig. 32 indicate that the energy spectrum of the emitted protons also follows such a law at energies \(\sim 100\) MeV. To obtain the relative number of mesons and protons for which the mean scattering angle is less than \(0.22^\circ\) per 100 μ, we extrapolated the proton spectrum according to the law \(\dfrac{dE}{E^{2.5}}\) for high energies, and the meson spectrum according to Sands’s curve.

Good agreement was obtained between the total number of particles found by means of such an extrapolation and that observed experimentally.

The present results make it possible to obtain the relative number of particles of different nature formed in nuclear disintegration. Among particles with ionization less than fourfold, \((36 \pm 2.5)\%\) are π-mesons, \((51 \pm 2.9)\%\) are protons, and \((13 \pm 1.5)\%\) are deuterons and tritons. For shower particles with ionization less than 16 grains per 50 μ, the corresponding values are \((79 \pm 6)\%\), \((17 \pm 3)\%\), and \((4 \pm 1.3)\%\).

NUCLEAR INTERACTIONS OF SHOWER PARTICLES

In Part II it was noted that in some cases shower particles produced in the disintegration of one of the emulsion nuclei subsequently undergo nuclear collisions. We called such cases double stars. From the number of double stars (see microphotographs XII, XIII, XIV at the end of the issue) one can determine the mean free path for nuclear interactions. Since it is now known that 80% of shower particles are mesons, and predominantly π-mesons, such observations may provide important information on the character of the interaction of π-mesons with nucleons.

Method I. The simplest way of determining the mean nuclear path is a direct comparison of the total path in the emul-

sion \((\Sigma L)\) of all shower particles whose tracks have a length greater than \(3000\,\mu\), with the number of nuclear interaction events produced by them. In doing so one may restrict oneself only to those shower particles which, from their scattering, can be identified as mesons.

In the examination of \(48\ \mathrm{cm}^3\) of emulsion, 241 mesons were found, formed in “stars” with kinetic energy between 150 and 1500 MeV and with a range exceeding \(3000\,\mu\). The total range of such particles is \(200\ \mathrm{cm}\), and 6 of them form secondary “stars.” A meson must have a range in the emulsion of at least \(3000\,\mu\) in order for its nature to be established. If a nuclear collision occurred at a range less than \(3000\,\mu\), then such a meson is excluded both from the number of cases of secondary nuclear interaction and from the total number of tracks. Thus, the effective range of the particles that we must take into account is equal to \(200 - (241 \times 0.3) = 128\ \mathrm{cm}\), and the mean range per disintegration is \((21 \pm 9)\ \mathrm{cm}\), or \(82 \pm 35\ \mathrm{g}/\mathrm{cm}^2\). This value is comparable with the value obtained for the nuclear range under the assumption that the effective interaction cross section is equal to the geometrical cross section of the nucleus, i.e.

\[ L_{\mathrm{nuc}} = 90\ \mathrm{g}/\mathrm{cm}^2. \]

Fig. 34.
Legend: mesons; protons; deuterons, tritons.
Vertical axis: number of cases.
Horizontal axis: number of grains per \(50\,\mu\).

Method II. Similar calculations can be made for all shower particles with a range greater than \(400\,\mu\). Such a value of the minimum range makes it possible to determine the grain density for tracks connecting two “stars” with an accuracy of about 15%. Figures 34 and 35 give the distribution according to grain-density values for all connecting tracks. Tracks of a length sufficient for identification of the particles are marked by hatching. In some cases identification proved possible although the tracks were shorter than \(3000\,\mu\). The corresponding distribution for all observed tracks formed in “stars,” divided by 10, is given in the same figures. The grain-density distribution for particles of various types is given in Fig. 36.

The mean free path of shower particles can be calculated by the method described in the appendix. For the mean thickness of that layer of emulsion through which the particles pass, without ex-

NUCLEAR DISINTEGRATIONS

Fig. 35. Distribution by grain density of tracks of particles causing secondary disintegrations. The corresponding distribution, reduced by a factor of 10, for all tracks emerging from stars is shown by the hatched area.

Fig. 36. Distribution by grain density of tracks of particles of various types emerging from stars. To obtain the number of particles per one star, one must multiply approximately by \(1.6 \cdot 10^{-4}\).

... by undergoing a nuclear interaction, a value of \(26 \pm 7\) cm was obtained. This corresponds to \(102 \pm 27\ \mathrm{g/cm^2}\), i.e., it also agrees with the mean free path corresponding to the geometrical cross section.

We shall show in the next paragraph that the effective cross section for the production of stars by protons with kinetic energy between 100 and 600 MeV is also approximately equal to the geometrical one. It is reasonable to assume that protons of higher energies as well, such as those which we have called shower particles, possess the same effective cross section. Then it follows from the preceding result that the average effective cross section for \(\pi\)-mesons also corresponds to the geometrical cross section.

COMPARISON WITH OTHER WORKS

The values given above for the nuclear free path of shower particles, and consequently also for \(\pi\)-mesons, are smaller than the values obtained by other authors.

A summary of all the data on ranges is given in Table XIII. Since the interaction of shower particles was studied in various substances,

Table XIII

Author	Method	Mean free path	\(L_{\text{obs}}/L_{\text{geom}}\)	Energy interval (mesons)
Piccioni \(^{7}\)	Counters at high altitude	\(1200\ \mathrm{g/cm^2}\) Fe	14.0	\(>400\) MeV
Fretter \(^{10}\)	Wilson chamber with lead plates at high altitude	\(750\ \mathrm{g/cm^2}\) Pb (without correction)	4.7	\(>150\) MeV \(n_s \geq 2\)
Lovati, Mura, Salvini and Tagliaferri \(^{11}\)	Wilson chamber with lead plates at high altitude	\(300 \pm 100\ \mathrm{g/cm^2}\) Pb (with correction)	1.9	150 MeV \(n_s > 2\)
Brown and McKay \(^{12}\)	Wilson chamber with lead plates at high altitude	\(316 \pm 70\ \mathrm{g/cm^2}\) Pb (with correction)	2.0	150 MeV
Butler, Rochester and Barker \(^{13}\)	Wilson chamber with lead plates at sea level	\(400\ \mathrm{g/cm^2}\) Pb (without correction) \(200\ \mathrm{g/cm^2}\) Pb (with correction)	2.5 1.4	Mean energy of showers \(\sim 7000\) MeV
Harding, Perkins \(^{14}\)	Photographic plates exposed under ice	\(120\ \mathrm{g/cm^2}\) ice	2.0	\(>100\) MeV
Camerini, Fowler, Muirhead (see Part II)	Electron-sensitive plates exposed at high altitudes	\(100\ \mathrm{g/cm^3}\) emulsion	1.1	\(>150\) MeV

then in the table the ratio is given of the observed range to the range corresponding to the geometrical cross section for the given substance.

All experiments with counters and with the Wilson chamber have the disadvantage that in them the path of the particle cannot be followed continuously over its entire length. Work with the Wilson chamber is carried out using lead plates of various thicknesses inside the chamber. The value obtained in these experiments for the range must be corrected, taking into account those nuclear disintegrations whose products have a range ending inside the plate.

Table XIV gives the characteristics of those stars which are produced by shower particles in interaction with the nuclei of the emulsion.

Table XIV

Primary star $N_{\mathrm{H}} + N_{\mathrm{S}}$	Secondary star $N_{\mathrm{H}} + N_{\mathrm{S}}$	Length of the connecting track in $\mu$	Density of grains per $50\,\mu$	$\bar{\alpha}^{\circ}$ at $100\,\mu$	Nature of the connecting particle
$10 + 1_p$	$4 + 0_p$	3 800	14,9	0,191	Meson
$17 + 5_p$	$6 + 0_p$	700	13,3	—	Not identified
$14 + 1_p$	$9 + 0_p$	4 600	10,1	0,025	Meson
$16 + 1_p$	$4 + 0_p$	750	14,5	—	Not identified
$8 + 2_n$	$1 + 0_p$	3 000	15,0	0,21	Meson
$3 + 4_p$	$5 + 1_p$	10 600	12,2	0,0087	Not identified
$18 + 5_p$*)	$5 + 0_p$	10 600	10,4	0,049	Meson
$6 + 5_n$	$16 + 0_p$	5 800	12,4	0,0215	Not identified
$9 + 1_n$	$5 + 0_p$	13 000	11,7	0,032	Meson
$4 + 4_p$	$4 + 0_p$	450	13,4	—	Not identified
$0 + 7_p$**)	$8 + 5_p$	1 200	11,7	—	”
$6 + 6_p$	$14 + 3_p$	1 500	12,1	0,017	”
$18 + 15_p$	$17 + 1_p$	1 700	11,4	—	”
$9 + 1_p$	$5 + 0_p$	1 750	10,0	—	”
$8 + 1_p$	$1 + 0_p$	11 200	10,1	0,068	Meson

) See microphotograph XIII.
*) See microphotograph XIV.

From this table it is clear that, in the main, stars with a small number of rays are formed, and this is confirmed by the study of stars produced by fast $\pi$-mesons. We therefore suppose that the corrections introduced in processing some data obtained with the chamber are in fact considerably underestimated. Our results, however, do not differ greatly from Butler’s new data1.

NUCLEAR INTERACTIONS OF PROTONS, DEUTERONS, AND TRITONS

The same method as that applied above for determining the nuclear path length of shower particles can be used to obtain the corresponding values for protons, deuterons, and tritons. Figure 37 gives the mass spectrum of particles for whose tracks a grain density exceeding 50 grains per 50 μ is observed.

Fig. 37. Mass distribution of particles emitted in stars with a grain density greater than 50 grains per 50 μ.

It is clear from the figure that protons and deuterons can be distinguished, but not deuterons and tritons.

The results obtained for the mean free path are summarized in Table XV. The figures given in the sixth row were obtained in the following way: the ratio of the number of protons to the number of deuterons and tritons was determined in various grain-density intervals for the identified tracks giving secondary stars.

The unidentified tracks were then assigned to various particle types according to this ratio. The fact that the nuclear path length for deuterons and tritons differs strongly from the path length for protons means that the difference between protons and deuterons is in fact real.

NUCLEAR DISINTEGRATIONS

Table XV*)

Type of particles	Energy in MeV	Grain density (grains per 50 μ)	Effective path length in emulsion	Number of interacting particles	Mean range in cm	Volume of examined emulsion
Stopped mesons	150—1500	10—16	128 cm	6±2.5	21±9	48 cm³
Shower particles	150—1500 (mesons)	10—16	389 cm	15±4	26±7	48 cm³
Shower particles	600—5000 (protons)	10—16	389 cm	15±4	26±7	48 cm³
Protons	40—100	40—66	174 cm	5.5±2.7	32±16	87 cm³
Protons	100—600	10—16	291 cm	9.6±3.3	30±10	48 cm³
Deuterons + tritons	75—250	40—76	127 cm	10.7±3.9	12±4.5	87 cm³

*) Data for shower particles are given for comparison.

APPENDIX

Determination of the mean nuclear range of various particles in emulsion can be carried out in the following way. Studies of the angular distribution of shower particles have shown that in the lower hemisphere this distribution is close to isotropic\(^{15}\). Consequently, with sufficient accuracy it may be assumed that the distribution is truly isotropic and that this is valid for protons, deuterons, and tritons. The tracks may be considered straight, and then it is easy to show that the number of tracks \(N(L)\,dL\), for which the true length lies between \(L\) and \(L+dL\), is

\[ N(L)\,dL=N_0\,\frac{D}{2L^2}\,dL \quad (\text{for } L>D), \tag{A} \]

where \(N_0\) is the total number of particles emitted from the “star” in the energy interval under consideration, and \(D\) is the thickness of the emulsion. The mean length \(\overline{L}\) of tracks in the emulsion, for which \(L\) lies in the interval \(L_1—L_2\), is

\[ \overline{L}=\frac{\ln \dfrac{L_2}{L_1}}{\dfrac{1}{L_1}-\dfrac{1}{L_2}} . \]

For \(D=400\,\mu\), \(L_1=400\,\mu\), and \(L_2=20\,000\,\mu\) we obtain: \(\overline L=1596\). We took the upper limit equal to \(20\,000\,\mu\) for convenience of analysis.

In \(48\ \mathrm{cm}^3\) of emulsion the number of tracks considered, with grain density less than 16 grains per \(50\,\mu\) and with path length greater than \(3000\,\mu\), is 349. The calculated value of \(N_0\) is 5235, and the number of tracks longer than \(400\,\mu\) is equal to half this value, i.e. 2617.5.

From (A), for the number of tracks longer than \(20\,000\,\mu\), we obtain:

\[ \frac{N_0 D}{2\times 20\,000}=52.35. \]

Hence the number of tracks with length between 400 and \(20\,000\,\mu\) is equal to 2565, and the mean effective path is \(1596-400=1196\,\mu\), since the first \(400\,\mu\) of path are not included in the analysis.

To this we must add the contribution made by tracks of particles whose range is greater than \(20\,000\,\mu\), which is equal to \(52.35\times(2.00-0.04)\ \mathrm{cm}\). Further, we have so far assumed that the emulsion covers an infinite layer. Simple geometrical considerations show that for the plates used, the total path length must be reduced by 5%. Thus, the total effective length in the emulsion is

\[ \Sigma L=(2565\times0.1196+52.35\times1.96)\times0.95=389\ \mathrm{cm}. \]

The corresponding number of observed secondary “stars” is equal to 15. Consequently, the nuclear range of the shower particles is equal to \((26\pm7)\ \mathrm{cm}\) of emulsion.

V. NEUTRAL MESONS*)

INTRODUCTION

Recent experiments carried out by Bjorklund, Crandall, Moyer, and York\(^{16}\) lead to the idea that high-energy \(\gamma\)-rays arising when matter is irradiated by protons with energies above 200 MeV have a secondary origin and arise in the decay into two photons of neutral mesons with mass about \(300\,m_e\) and lifetime less than \(10^{-11}\ \mathrm{sec}\). These neutral mesons are the primary products of nuclear collisions. Further, Bradt, Kaplan, and Peters\(^{17}\) carried out a detailed investigation of the splitting produced in photographic emulsion by \(\alpha\)-particles with energies between \(10^{-12}\) and \(10^{-13}\) eV, as a result of which 56 shower particles were formed. In the “stem” of the shower particles, the majority of which are mesons\(^{7}\), these authors observed the formation of several electron pairs as well, and concluded that charged mesons are accompanied by \(\gamma\)-radiation. If this radiation arises in the decay of neutral mesons and if these

\[ \text{*) A. G. Carlson, J. E. Hooper, D. T. King, Phil. Mag. 41, 701 (1950).} \]

particles are produced in a nuclear explosion with kinetic energy close to the energy of charged mesons,—it can be shown that the proper lifetime of neutral mesons is less than \(3\cdot 10^{-13}\) sec.

Finally, apparently, the existence of neutral mesons with a mass of about \(280\,m_e\) and their mode of decay into two photons follow from experiments on the capture of \(\pi^{-}\)-mesons in liquid hydrogen\({}^{17a}\). This process leads to transformations which may be represented by the following equations:

\[ \begin{aligned} \mathrm{H}^{1}+\pi^{-1} &\to n^{0}+\pi^{0}; \qquad \pi^{0}\to 2h\nu\,(h\nu\sim 70\ \text{MeV}), &&(\mathrm{a})\\ \mathrm{H}^{1}+\pi^{-1} &\to n^{0}+h\nu \qquad\qquad\quad (h\nu\sim 140\ \text{MeV}), &&(\mathrm{b}) \end{aligned} \]

where \(\pi^{0}\) is the neutral meson.

The degree of monochromaticity of the observed \(\gamma\)-rays formed as a result of process (a), which is the more probable one, shows that the \(\pi^{0}\)-mesons are emitted with small kinetic energy. Since the masses of the other particles participating in the transformations are known, this fact shows that the mass of the neutral meson \((\pi^{0})\) is only a few electron masses less than the mass of the charged \(\pi\)-mesons; namely \(m_{\pi^0}\sim 280\,m_e\). Apparently, these experiments are of great importance for work carried out during the last several years with the Wilson chamber and counters\({}^{18-21}\). These works have convincingly shown that cascade showers of the soft component are usually associated with penetrating showers of charged mesons.

Section I. DETERMINATION OF THE MASS AND ENERGY SPECTRUM OF NEUTRAL MESONS

Spectrum of \(\gamma\)-rays at an altitude of 23 km

During the past year in our laboratory experiments have been carried out with photographic plates exposed at an altitude of 23 km (see Part II).

Several of these plates were examined under a microscope at relatively high magnification, in order to detect electron pairs and other events consisting only of particles with charge \(e\), moving with relativistic velocities. Such events are usually missed by the observer if the plates are examined at lower magnification. By determining the energies of electron pairs by the scattering method (in those favorable cases in which the tracks in the emulsion have sufficient length), it was possible to calculate the energy spectrum and the direction of motion of the photons of the “soft” component at an altitude of 23 km.

The method used to determine the electron energies is similar to the method described in Part III, and also by King\({}^{22}\). Measurements were made only of those pairs for which the track of each component was longer than 1 mm. With a track length of less than \(2000\,\mu\), it is impossible to determine the energy of the corresponding particle if it

greater than 500 MeV. However, satisfactory estimates of particle energies (up to 400 MeV) can be made with track lengths of only 1000 μ.

The characteristic values of the probable errors in measuring particle energies from tracks 1000 μ long are given in Table XVI.

Table XVI

Energy range (in 1 MeV)	Length of segment (in microns)	Probable error in energy (in %)
10—100	100	19
100—200	200	28
200—400	250	32

The energy of the pair is usually substantially greater than the energy of one of its components. We may therefore say that the total energy \(E\) of the pair can be measured with a probable error of less than 30% for values \(E < 600\) MeV if both tracks of the pair have lengths greater than 1000 μ, and for values \(E < 1500\) MeV if the tracks are longer than 2000 μ. Measurements of pairs with very low energy require a small correction taking into account the effect of scattering on the exit of particles from the emulsion. However, such corrections for \(E > 200\) MeV are very small, and in the present measurements they were not introduced. The results are shown in Fig. 38.

Fig. 38. Distribution of the energies of electron pairs observed at an altitude of 23 km. The smooth line shows the expected spectrum of photons arising in the decay of neutral mesons with a mass of \(280\,m_e\) and of the kinetic energy of charged \(\pi\)-mesons formed in “stars.”

In determining the orientation of electron pairs with respect to the vertical line drawn through the plate at the time

exposure, it was considered sufficient to determine only the angle \(\theta\) of inclination of the projection of the “bisector” of the pair. The “bisector” of a pair of electrons produced by a photon with an energy lying in the region under consideration makes it possible to determine the direction of the photon that produced the pair with a probable error of less than \(0.2^\circ\).

The distribution of the values \(\theta\), calculated from observations of 500 pairs, is shown in Fig. 39. From Fig. 39 it is evident that the distribution of the directions of motion of the photons is close to isotropic in the range of angles from \(0\) to \(90^\circ\) with respect to the vertical directed downward, and that the intensity of any radiation directed upward is relatively very small. Charged mesons emitted in nuclear disintegrations in the emulsion during this same exposure exhibit the same angular distribution\(^{15}\).

Fig. 39. Angular distribution of the directions of motion of photons at an altitude of 23 km.

Features of the radiation produced in the decay of neutral mesons

After the publication of the results of work\(^{16}\), it occurred to us that if \(\gamma\)-radiation in the atmosphere arises from the decay of neutral mesons, and if it is studied under conditions in which bremsstrahlung and other processes associated with the development of cascades of the soft component do not have a substantial influence, then its spectrum should have certain features of internal structure and should make it possible to determine the mass of the neutral mesons that have arrived.

The method of determining the mass is based on the following considerations.

Suppose that a neutral particle of mass \(m_0\), moving with velocity \(\beta c\), is transformed into two quanta with energies \(h\nu_1\) and \(h\nu_2\), emitted in directions forming angles \(\theta_1\) and \(\theta_2\) with the line of motion of the neutral particle (Fig. 40).

The equations of conservation of energy and momentum may be written in the form

\[ \frac{h\nu_1}{c}\sin\theta_1=\frac{h\nu_2}{c}\sin\theta_2, \]

\[ \frac{h\nu_1}{c}\cos\theta_1+\frac{h\nu_2}{c}\cos\theta_2 = \frac{m_0\beta c}{\sqrt{1-\beta^2}}, \]

\[ h\nu_1+h\nu_2= \frac{m_0c^2}{\sqrt{1-\beta^2}} = Bm_0c^2=\varepsilon. \]

Solving these equations, it is then easy to show that the energy of a quantum emitted at an angle \(\theta\) is given by the well-known formula of the relativistic Doppler effect

\[ h\nu=\frac{m_0c^2}{2B(1-\beta\cos\theta)}. \tag{1} \]

Fig. 40.

Curves showing the variation of \(h\nu\) with \(\theta\) for different values of \(B\) and for a specified value of \(m_0c^2\), equal to \(140\) MeV, are shown in Fig. 41. From equation (1) it follows that, for a given value of \(\beta\), the maximum and minimum values of \(h\nu\) are given by the following relations:

\[ (h\nu)_{\max}=\frac{Bm_0c^2}{2}(1+\beta), \]

\[ (h\nu)_{\min}=\frac{Bm_0c^2}{2}(1-\beta), \]

and that the greater the velocity of the monoenergetic beam of neutral mesons, the greater the width of the spectrum of the emitted \(\gamma\)-radiation.

Fig. 41. Polar diagram showing the dependence of photon energy on the direction of their emission, if the photons were formed in the decay of neutral mesons. The calculations were carried out for two values of the total energy of neutral mesons.

Assuming that the $\gamma$ radiation is emitted isotropically in a coordinate system moving with a velocity equal to the meson velocity, one can calculate the form of the $\gamma$-ray spectrum (see the Appendix), and it can be shown that it is given by the relation

\[ N(E)\,dE = K\,dE, \]

where $N(E)\,dE$ is the number of emitted quanta with energy between $E$ and $E + dE$, and $K$ is a quantity constant for values of $E$ lying between $(h\nu)_{\max}$ and $(h\nu)_{\min}$, and equal to zero for values of $E$ lying outside this interval. Characteristic spectra for various values

Fig. 42. Energy distribution of photons produced in the decay of monoenergetic neutral mesons.

of $B$, normalized so as to correspond to one and the same total number of quanta, are shown in Fig. 42.

In the case of an inhomogeneous beam of neutral mesons, it can be shown that the resulting $\gamma$-ray spectrum has the form shown in Fig. 43. The spectrum has a maximum if a limited number of neutral mesons is emitted with zero velocity.

For any value of the intensity below the maximum there are two corresponding values of the quantum energy, $E_1$ and $E_2$, and it can be shown that these values are related to the rest mass of the initial neutral particles by the relation

\[ \sqrt{(E_1E_2)} = \frac{m_0c^2}{2}. \tag{2} \]

The proof of this theorem is given in the Appendix. It follows from this result that, if the greater part of the $\gamma$ radiation at an altitude of 23 km is produced in the decay of neutral mesons

and if this radiation is not significantly altered by the processes that lead to the formation of cascade showers, then there must exist an internal consistency of the spectral form expressed by equation (2). It seems to us that observation of such consistency of the spectrum would be a very strong argument in favor of

Fig. 43. Spectrum of γ-radiation at an altitude of 23 km. The smooth line is the line giving the best agreement with the experimental results; the mass of the neutral meson is calculated from it.

the assumption that the γ-radiation actually arises as a result of the decay of neutral mesons, and would make it possible to estimate their mass.

Comparison with experiment. Mass of the neutral meson

Figure 43 gives the energy distribution of γ-radiation. It was calculated from the energy spectrum of pairs (see Fig. 38) under the assumption that the effective cross section for pair production by photons varies with energy according to the formulas of quantum electrodynamics[^23]. The smooth line is drawn in best agreement with the distribution, and the sections for determining \(E_1\) and \(E_2\) were made at various arbitrarily chosen intensities. The corresponding values of \(\sqrt{E_1E_2}\) are shown by circles. From Fig. 43 and Table XVII it is seen that these values are in sufficiently good agreement with one another and indicate that the γ-radiation arises in the decay of neutral mesons with mass \(295 \pm 20m_e\). The results may be subject to a small systematic error arising from the variation with energy of the scattering constant \(K\), determined by the equation \(E = \dfrac{K}{\bar{\alpha}}\), where \(\bar{\alpha}\) is the mean observed scattering angle.

Table XVII

Measurements of the mass of neutral mesons from the observed energy spectrum of photons

Weights (number of pairs)	\(E_1\)	\(E_2\)	\(\dfrac{\varepsilon_0}{2}=\sqrt{E_1E_2}\)	“Weighted” values *) \(\dfrac{\varepsilon_0}{2}\)
26	63	91	75.5	196
22	53	106	75.0	165
18	45	124	75.0	135
14	39	156	78.0	109
10	31	200	79.0	79
6	21	274	75.5	42

Total 96 pairs.
All energies are given in MeV.

Mean value

\[ \frac{\varepsilon_0}{2}=75.8\ \text{MeV}. \]

Mass of the neutral particle

\[ m_{\pi^0}=295\pm20\,m_e. \]

) By the “weighted” value of \(\dfrac{\varepsilon_0}{2}\) the authors mean the product of the experimentally determined value of \(\dfrac{\varepsilon_0}{2}\) by the corresponding number of pairs. (Translator’s note.)*

for a segment of length \(100\,\mu\). Such a change is the result of shielding effects.

The error given above in the determination of the mass of the neutral meson corresponds only to statistical fluctuations in the number of measured electron pairs. When this result was first obtained, it was not clear to what extent it was affected by the increased role of cascade showers. However, recent observations show that the “distortion” in the form of the spectrum is small. Apparently, this is connected with the following features of the experiments.

First, the plates were exposed near the principal meson-producing layer of the atmosphere. At this altitude the probability that a given quantum will produce a cascade, even for photons that originated at the boundary of the atmosphere, is small. Second, if the cascade process begins within the stack of plates itself, it usually forms relatively narrow showers consisting of particles moving almost parallel to one another and very close to one another. The development of such cascade showers was observed in the present experiments, but they are rare, and any pairs associated with them were excluded from consideration.

For these reasons we believe that our observations are very convincing evidence that the greater part of the γ-radiation in the atmosphere arises in the decay of neutral mesons with mass \(\sim 295 \pm 20\,m_e\), identical with the particles whose existence also follows from experiments\(^{16,17}\).

Energy spectrum of neutral mesons

From the observed spectrum of γ-rays one can calculate the energy distribution of the primary neutral mesons. The result of the calculation, obtained by the methods described in the Appendix, is shown in Fig. 44. The analogous distribution obtained in Part IV for charged mesons formed in nuclear disintegrations and recorded on the same plates is also shown in Fig. 44. It is seen from Fig. 44 that these two spectra are identical. On the other hand, it is easy to calculate the γ-ray spectrum that would be formed in the decay of neutral mesons with mass \(280m_e\) and with an energy distribution similar to the energy distribution of charged mesons. On the basis of the results of Part II, for charged mesons the γ-ray spectrum shown in Fig. 38 by the smooth line was obtained. It is seen from Fig. 38 that the calculated form of the spectrum is indistinguishable from the observed one. This gives additional weight to the basic supposition concerning the origin of the γ-rays and to the supposition that the primary neutral mesons arise with an energy distribution very close to the distribution of the charged particles.

Fig. 44. Energy spectrum of neutral mesons formed in nuclear disintegrations. The spectrum of neutral mesons is denoted by crosses; the spectrum of charged mesons according to observations (Part IV et al.) is denoted by circles.

Section II. Experiments to establish the independent existence of neutral mesons and to measure their lifetime and frequency of occurrence

Method for determining the lifetime

The evidence for the existence of neutral mesons described so far is very strong, but nevertheless indirect evidence. It was obtained in the study of the γ-radiation spectrum, the features of which agree with the supposition of the ob—

rupture of this radiation in the decay of neutral mesons. Although the conclusion as to the existence of neutral mesons can hardly be called into question, it is nevertheless essential to carry out an indisputable verification of the independent existence of these particles.

In addition, it is known^24 that the determination of the lifetime of a neutral meson is of great importance for choosing the theoretical interpretation of the nature of the particle. The discussion below shows that such proof can be obtained by means of the photographic method, if the particle lifetime is greater than \(2 \cdot 10^{-14}\) sec.

Let us suppose that a neutral meson is emitted with total energy \(\epsilon = Bm_0c^2\) in a nuclear disintegration in an emulsion and that it decays into two photons after traversing a distance \(l\). We have seen that the two photons are usually emitted at an angle to the direction of the original particle. Let us further suppose that one of the photons produces in the emulsion an electron pair at a distance of 1 or 2 mm from the point of its origin. The probability of such a process is, for each emitted neutral meson, 3%, since the mean free path of a photon in the emulsion before pair production is 46 mm*) and each neutral particle produces two photons; because of the finite thickness of the emulsion, a considerable fraction of all cases will escape observation.

We have seen that, for neutral mesons with kinetic energy in the region under consideration, the electrons of the pairs will be emitted making an angle of \(0.2^\circ\) with one another. If the momentum transferred to the nucleus participating in the formation of the pair^25 is neglected, then the bisector of the angle between the tracks of the two electrons will determine, consequently, the direction of motion of the incident photon with an accuracy up to an error of the order of \(0.2^\circ\) (Fig. 45). It is clear from Fig. 45 that if the mean line of the pair is extended in the opposite direction, it will pass not exactly through the disintegrated nucleus in which the neutral meson was formed, but at a distance \(r\) from it. The greater the value of the meson lifetime, the greater the mean distance \(l\) and the corresponding value of \(r\).

Fig. 45. Principle of the method for determining the lifetime of a neutral meson.

*) This value is calculated taking into account the observed spectrum of \(\gamma\)-radiation and the change of the radiation unit with the quantum energy.

The value \(r\) for any given case will depend on the distance traveled by the neutral meson before its decay and on the direction of photon emission relative to the line of motion of the meson. For a monoenergetic beam of neutral mesons, the expected distribution of the values \(r\) can be calculated by the methods mentioned in the Appendix. From the results obtained in this way one can then pass to the general case, in which the neutral mesons are distributed in energy. Fig. 46 presents the results of such calculations, carried out under the assumption that the energy distribution coincides with the energy distribution of charged mesons (Part IV), and that the lifetime of the neutral mesons is \(10^{-13}\) sec.

Fig. 46. 1 — calculated distribution of the values \(r\) for an assumed lifetime of the neutral meson of \(10^{-13}\) sec.; 2 — calculated distribution due to random pairs.

A characteristic feature of the method is that, for high-energy mesons (\(B \geqslant 2\)), the calculated distributions depend almost not at all on the value of \(B\) and on the energy distribution of the neutral particles. This is due to the fact that although the emitted radiation, with increasing \(B\), tends to concentrate near the direction of motion of the original mesons (a factor that leads to a decrease in the value of \(r\)), this effect, on the other hand, is compensated by the relativistic dilation of the time scale of the moving particles, which increases the free path length of the neutral mesons in the emulsion.

Experimental Results

In order to use the above-mentioned method at large magnification, an extensive study was undertaken of the region of the emulsion in the immediate vicinity of nuclear explosions in which three or more “shower” particles (\(n_s \geqslant 3\)) were formed. For all electron pairs found in the volume of the emulsion selected, as shown in Fig. 47, according to the direction of the projection of the mean line of the pair relative to the star and its “slope,” the value \(r\) was determined. These measurements and the possible errors are discussed in more detail in the following paragraph.

The observed distribution of the measured values \(r\) is given in Fig. 48. From the figure it is seen that the distribution contains a group of cases corresponding to small values of \(r\), along with a markedly pronounced general background. These features of the distribution may

be interpreted as follows. We saw that there is a random distribution of pairs of electrons in the plates, caused by the flux of γ-radiation, and that these pairs can sometimes fall into the volume of the emulsion around the star which is being carefully examined. The frequency of appearance of these “random” pairs can be calculated from the observed number of showers \((n_s \geqslant 3)\) and of electron pairs per unit area of the plates.

Fig. 47. Area around a “shower” that was examined in detail in the search for pairs.

Fig. 48. Observed distribution of the values \(r\) for \(r < 400\,\mu\). The group of “correlated” pairs observed at small values of \(r\) is clearly visible.

The calculated frequency of occurrence of these “random” pairs as a function of \(r\) is shown by the smooth line in Fig. 48; it can be seen that it is in good agreement with the observations. Indeed, in the interval of \(r\) from \(40\,\mu\) to \(200\,\mu\) the expected number of pairs is \(22 \pm 5\), while the observed number is 23. On the other hand, the majority of pairs for which \(r < 40\,\mu\) must be attributed to γ-radiation associated with the corresponding nuclear explosions, since the expected number of random pairs in this interval of \(r\) is only 0.2, while the observed number is 15. The observations, therefore, prove that γ-radiation is produced by some

mechanism as a result of nuclear interactions leading to the formation of three or more shower particles, \(n_s \geq 3\).

Observations of correlated pairs show that the lifetime of neutral mesons is less than \(5 \cdot 10^{-14}\) sec and, apparently, they lead to the value \(\tau_{\pi^0} = 3 \cdot 10^{-14}\) sec. It has already been emphasized above that such a result, apart from its interest for establishing the decay constant of neutral \(\pi\)-mesons, is also direct evidence of their independent existence. Taking into account the significance of the result obtained, it is essential to establish the reliability of our observations. It is essential to determine whether these observations can be explained by measurement errors, while the mean lifetime of neutral mesons is in fact considerably shorter than the time that can be measured by this method.

Experimental errors in determining the quantity \(r\)

The quantity \(r\), which was chosen as the criterion for associating a pair with a given nuclear explosion, is shown in Fig. 49. \(r\) is the distance between the center of the star and the bisector of the angle formed by the directions of motion of the two electrons of the pair. To determine \(r\), the distance \(x\) is first measured from the star to the vertical plane containing the bisector of the electron pair. By “vertical” is meant here the direction of the normal to the plane of the emulsion, which is regarded as “horizontal.”

Fig. 49. Measurement of the quantity \(r\).

The distance \(x\) is represented by the segment \(OC\) in Fig. 49 and is measured in the following way: the plate is rotated until the bisector of the pair becomes parallel to one of the directions of motion of the microscope stage, for example \(Y\), and coincides in the field of view with the line of the eyepiece crosshair. The crosshair is also provided with a scale perpendicular to the line. By moving the stage parallel to the \(Y\)-axis and refocusing the image, the star is finally brought into the field of view. Then the distance of the star from the bisector of the pair can be measured by means of the eyepiece scale.

This method of determining \(x\) is subject to errors which depend on the distance \(bC=d\) (Fig. 49) and on the accuracy with which the bisector of the pair can be determined. The study of the scattering of fast particles\(^3\) shows that the horizontal projection of a rectilinear track can be drawn with an error of less than \(0.1^\circ\). In the case of pairs formed by photons of very high energy, the two tracks are not scattered appreciably over the first \(100\,\mu\) of their trajectories, and the angle \(\delta\) between them is small; for \(100\) MeV \(\gamma\)-rays \(\delta \sim 0.2^\circ\); for \(400\) MeV \(\delta \sim 0.05^\circ\). For radiation of such energy, which constitutes the principal part of the spectrum (see Fig. 38), determination of the bisector can be made with an accuracy of the order of \(0.25^\circ\), and the corresponding error in \(x\) may be written as

\[ \delta x \sim \frac{d}{200} \]

(where \(\delta x\) and \(x\) are measured in microns).

The second quantity needed to determine \(r\) is found by measuring the angle of “inclination” of the bisector of the pair. In Fig. 49, \(AC'\) is the projection of the bisector onto a plane parallel to the surface of the emulsion and passing through the center of the star \(O\), and \(f\) is the depth of the point of origin of the pair \(F\) relative to this plane. If the length of the bisector to the point \(E\) is denoted by \(\Delta d\), then the depth of the point \(E\) is equal to \(f+\Delta f\). It is easy to show that

\[ z^2 = CD^2 = (d\Delta f - f\Delta d)^2 / [(\Delta d)^2 + (\Delta f)^2] \]

and that

\[ r^2 = x^2 + z^2. \]

To determine the probable errors arising from errors in measuring depth, numerous repeated measurements were made for each case, and the standard deviations were calculated empirically. Similar observations were made for high-energy protons producing nuclear disintegrations and undergoing, as the scattering measurements showed, an average change in direction of motion of less than \(0.01^\circ\) per \(100\,\mu\).

The quantities \(r\) can thus be calculated for tracks which certainly pass through the center of the star and whose deviation from straightness is negligible.

These observations show that the probable errors in determining the angle of inclination are about \(0.2^\circ\), and that the uncertainty in the true value of \(r\) is made up almost equally of the errors of the two measurements \(x\) and \(z\).

In accordance with the specific features of individual cases, each of these two measurements may make a greater or lesser contribution to the total error. Taking both measurements into account, it was estimated that the probable error in determining the direction of the bisector is \(0.3^\circ\).

Fig. 50. Distribution of the observed values of \(r\) for correlated pairs. Curves \(A\) and \(B\) show the expected distributions of the values of \(r\), due to probable errors of \(0.25^\circ\) and \(0.5^\circ\) in the measurements of the direction of motion of the photon. The lifetime of the meson is assumed to be infinitely small.

An important question remains the accuracy with which the bisector characterizes the direction of motion of the photon. Usually the lines of motion of the two electrons form unequal angles with the direction of the photon that produced them. Further, the nucleus that participated in the formation of the pair receives a momentum of the order of \(m_e c^{25,26}\). For the energies with which we are dealing, the probable errors arising from these effects can hardly exceed \(0.2^\circ\). Thus we obtain the final value of the probable error in determining the direction of motion of the photon equal to \(0.4^\circ\). It follows from this that the errors in measuring the distance from the star to this line may be expressed by the relation

\[ \delta r \approx \frac{d}{150}. \]

Measurements of the value of \(r\) for 15 correlated stars are given in Table XVIII. The error limits given in the last column are errors calculated from the standard deviations associated with the scatter of a large number of measurements of one and the same case. They therefore do not include errors due to the difference between the directions of the photon and the bisector of the pair,

NUCLEAR DISINTEGRATIONS

Figure 50 shows a diagram presenting the distribution of the quantities \(r\) on a larger scale.

Figure 50 gives the distributions of the quantities \(r\) that could be expected for standard deviations of \(0.25\) and \(0.5^\circ\) in the measurement errors arising from all causes, it being assumed that the lifetime of the neutral particles is small in comparison with \(10^{-14}\) sec. It can be seen that there are only four pairs for which the quantities \(r\) are substantially greater than the expected values of \(r\) for a standard deviation of \(0.5^\circ\), and Table XVIII

Table XVIII

Individual measurements of \(r\) and their errors

Pair	Type of star	\(x\)	\(d\)	Slope \(\dfrac{\Delta f}{\Delta d}\)	\(f\)	\(r\)
1	\(7+6\,N\)	0	0	—	—	0
2	\(6+4\,p\)	\(0.5\pm0.5\)	73	\(0.050\pm0.013\)	\(2.4\pm0.5\)	\(1.5\pm1.5\)
3	\(3+5\,p\)	\(0.0\pm1.0\)	123	\(0.096\pm0.010\)	\(9.5\pm0.7\)	\(2.0\pm2.0\)
4	\(12+3\,p\)	\(2.0\pm2.0\)	497	\(0.120\pm0.010\)	\(60.0\pm8.0\)	\(2.0\pm4.0\)
5	\(9+7\,p\)	\(1.0\pm1.0\)	722	\(0.122\pm0.004\)	\(91.5\pm0.8\)	\(3.5\pm3.5\)
6	\(6+7\,''\)	\(3.5\pm1.0\)	1035	\(1.011\pm0.020\)	\(132.9\pm0.4\)	\(3.5\pm2.5\)
7	\(7+12\,p\)	\(1.0\pm1.0\)	378	\(0.094\pm0.004\)	\(31.8\pm0.5\)	\(3.5\pm2.0\)
8	\(21+18\,p\)	\(3.5\pm2.0\)	660	\(0.172\pm0.003\)	\(113.6\pm0.4\)	\(4.5\pm2.0\)
9	\(3+26\,p\)	\(3.0\pm2.0\)	1027	\(0.158\pm0.002\)	\(157.5\pm0.3\)	\(5.0\pm3.5\)
10	\(7+6\,N\)	\(6.0\pm1.0\)	226	\(0.337\pm0.013\)	\(78.6\pm0.5\)	\(7.0\pm2.0\)
11	\(3+26\,p\)	\(2.0\pm2.0\)	536	\(0.042\pm0.016\)	\(27.3\pm0.4\)	\(7.5\pm5.5\)
12	\(13+5\,p\)	\(6.0\pm3.0\)	1993	\(0.0008\pm0.0008\)	\(17.4\pm1.2\)	\(17.0\pm3.0\)
13	\(17+6\,p\)	\(7.0\pm1.0\)	395	\(0.320\pm0.006\)	\(145.0\pm1.1\)	\(17.5\pm3.0\)
14	\(14+3\,N\)	\(10.0\pm3.0\)	1080	\(0.049\pm0.003\)	\(81.6\pm0.8\)	\(29.0\pm4.0\)
15	\(11+4\,N\)	\(23.0\pm5.0\)	745	\(0.170\pm0.004\)	\(94.6\pm0.7\)	\(38.0\pm4.5\)

All errors are calculated from standard deviations; \(x\), \(d\), \(f\), \(r\) are measured in microns.

shows that for these four cases the quantities \(r\) are many times greater than the measurement errors. It is very unlikely that four such cases would have been found among the selected fifteen electron pairs associated with “stars” if in

actually, the $\gamma$-rays were emitted directly as a result of nuclear explosions; on the other hand, we have seen that the number of expected random pairs for $r < 40\ \mu$ is only $0.2$. However, the number of observed cases is too small to serve as indisputable experimental proof that the lifetime of neutral mesons has a finite value. We note only that, if our observations are confirmed by further experiments, the value of the mean lifetime of the neutral meson can be measured by this method if it is greater than $2 \cdot 10^{-14}$ sec. The conclusion that the mean lifetime of neutral mesons is less than $5 \cdot 10^{-14}$ sec may be regarded as established.

Intensity of neutral mesons

In order to compare the frequency of occurrence of neutral mesons with the intensity of emission of charged mesons, the total length $L$ of the tracks of shower particles in the area of the emulsion examined around each star was determined. Recent experiments have shown that about 80% of the shower particles formed in “stars” are $\pi$-mesons. This makes it possible to determine the number of fast protons among the shower particles. Suppose that, on average, the emission of each charged $\pi$-meson is accompanied by $\Phi$ neutral mesons,

$$ \frac{N(\pi^0)}{N(\pi^\pm)}=\Phi. $$

Then the path length in the emulsion of photons formed in the decay of neutral mesons is given, without substantial error, by the expression

$$ l=\frac{4}{5}L2\Phi=1.6L\Phi. $$

Over this path length, under the conditions of our experiment, the photons formed 15 electron pairs. Bearing in mind that the value of the radiation unit for the emulsion, calculated with allowance for the photon spectrum, is equal to $4.6\ \mathrm{cm}^{23}$, we may therefore write

$$ \frac{1.6L\Phi}{15}=4.6 $$

and, for $L=95\ \mathrm{cm}$, we obtain

$$ \Phi=0.45\pm0.10. $$

This result leads to the idea that, in the formation of showers of charged particles, for every two charged mesons there arises one neutral meson. This conclusion is confirmed by a recently completed study in our laboratory^37 of the energy balance in nuclear processes. In that work the energies of the primary particles were compared with the energy of the mesons and nucleons emitted in the disintegration. The authors showed that there is an apparent disappearance of energy, which can be explained by the formation of neutral particles. A similar result was also obtained by Green et al.^28.

Conclusions

We regard our experiments as evidence that, in explosive disintegrations of nuclei, the number of neutral mesons produced is approximately equal to one half the number of charged \(\pi\)-mesons. The mass of these neutral mesons is \(295 \pm 20\,m_e\), and they decay into two photons with a lifetime of less than \(5\cdot 10^{-14}\) sec.; they may therefore be identified with the neutral \(\pi\)-mesons discovered at Berkeley^17a. It is well known^24,29 that decay into two photons indicates zero spin of the neutral meson.

The energy spectrum of the neutral mesons is very close to the spectrum of the charged mesons, and they arise in the same nuclear processes.

Consequently, neutral mesons may be regarded as particles possessing the properties of the neutral “heavy quanta” of Yukawa’s theory.

The observed frequency of occurrence of neutral mesons makes it possible to give a simple explanation of Rossi’s result, who established that the energy fluxes of the “hard” and “soft” components of cosmic radiation are almost identical—a result which in those years was accepted as an empirical fact, but which, as Rossi believed, might have very great significance. Our results confirm the assumption that both the “hard” and the “soft” components arise as a result of nuclear disintegrations produced by protons and heavy particles of high energy, which are accompanied by the emission of charged and neutral \(\pi\)-mesons. It seems unnecessary to attribute the soft component to “primary” electrons.

The lifetime of \(\pi^0\)-mesons is, of course, less than \(5\cdot 10^{-14}\) sec., and to determine it directly and thereby establish the independent existence of these particles will be difficult by the methods at our disposal. The lifetime of the \(\pi^0\)-meson can be determined by the present method only if it lies in the interval from \(2\cdot 10^{-14}\) to \(5\cdot 10^{-14}\) sec.

APPENDICES

For completeness, we shall present the main calculations in general outline.

We have seen that the energy of photons emitted at an angle \(\theta\) with respect to the line of motion of a neutral meson having velocity \(v=\beta c\) and total energy \(B m_0 c^2=\varepsilon\) is given by the formula

\[ h\nu=E=\frac{m_0 c^3}{2B(1-\beta\cos\theta)}. \tag{1} \]

a) Angular distribution of the radiation

It may be assumed that the photons formed in the spontaneous decay of a monoenergetic beam of neutral mesons are emitted isotropically in the coordinate system moving with a velocity equal to the velocity of the particles (the \(C\)-system).

We now pass to the laboratory coordinate system (the \(L\)-system). Let \(\alpha\) be the angle of emission of the photon relative to the direction of motion of the meson in the \(C\)-system, and let the components of the photon velocity \(c\) (parallel and perpendicular to the direction of motion of the particle) be, respectively, \(c\cos\alpha\) and \(c\sin\alpha\) (Fig. 51). Let \(u_x\)

Fig. 51.

and \(u_y\) be the corresponding components in the \(L\)-system. By Einstein’s theorem on the addition of velocities we obtain:

\[ u_x=\frac{c\cos\alpha+v}{\left(1-\dfrac{v\cos\alpha}{c}\right)},\qquad u_y=\frac{c\sin\alpha}{B\left(1-\dfrac{v\cos\alpha}{c}\right)}, \]

where

\[ B=\frac{1}{\sqrt{1-\beta^2}}. \]

It follows from this that

\[ \tg\theta=\frac{u_y}{u_x} =\frac{\sin\alpha}{B(\cos\alpha+\beta)}, \]

or, denoting

\[ \sin\xi=\sqrt{1-\beta^2}\quad\text{and}\quad \cos\xi=\beta, \]

we obtain

\[ \tg\theta=\frac{\sin\alpha\sin\xi}{\cos\alpha+\cos\xi}; \tag{2} \]

then, bearing in mind that \(\sec^2\theta=\tg^2\theta+1\), we finally obtain—

expressions

\[ \cos \theta=\frac{\cos \xi+\cos \alpha}{1+\cos \xi \cos \alpha} \tag{3} \]

and

\[ \sin \theta=\frac{\sin \xi \sin \alpha}{1+\cos \xi \cos \alpha}, \tag{4} \]

and from (2), (3), and (4):

\[ \frac{d\alpha}{d\theta}= \frac{(\cos \xi+\cos \alpha)^3 \sec^3 \theta} {\sin \xi(1+\cos \xi\cos \alpha)}. \tag{5} \]

Let \(I(\theta)\) be the intensity of the radiation in the \(L\)-system, expressed as the number of quanta per unit solid angle, and \(I(\alpha)\) the intensity in the \(C\)-system. Then we may write:

\[ I(\theta)\sin\theta\,d\theta = I(\alpha)\sin\alpha\,d\alpha = K\sin\alpha\,d\alpha. \]

Taking, for convenience, \(K=1\), we obtain the expression

\[ I(\theta)=\frac{\sin\alpha}{\sin\theta}\frac{d\alpha}{d\theta}, \]

which is transformed into

\[ I(\theta)=\frac{1}{B^2(1-\beta\cos\theta)^2}. \tag{6} \]

Fig. 52. Polar diagram showing the angular distribution of the intensity of \(\gamma\)-rays formed in the decay of monoenergetic beams of neutral mesons of various energies.

The form of the distribution for two values of \(B\) is shown as a polar diagram in Fig. 52.

b) Distribution of \(\gamma\)-rays with respect to energy

We may write:

\[ N(E)\,dE=\frac{1}{2}I(\theta)\sin\theta\,d\theta. \]

From (1) and (6) we obtain that

\[ \frac{d\theta}{dE} = -\frac{2(1-\beta\cos\theta)^2} {\beta\varepsilon(1-\beta^2)\sin\theta} \]

(\(\varepsilon\) is the total energy of the neutral meson), and hence

\[ N(E)=\frac{1}{\beta\varepsilon}=\text{const}, \tag{7} \]

where the distribution is bounded by the energies

\[ E_1=(1-\beta)\frac{\varepsilon}{2} \quad\text{and}\quad E_2=(1+\beta)\frac{\varepsilon}{2}. \tag{8} \]

c) Properties of the γ-ray spectrum

Let us now consider a nonuniform beam of neutral mesons, where \(F(\varepsilon)\,d\varepsilon\) is the number of mesons with energies from \(\varepsilon\) to \(\varepsilon+d\varepsilon\). The intensity of the γ-rays produced by neutral mesons in this energy interval is given, according to (7), by the expression

\[ \frac{F(\varepsilon)\,d\varepsilon}{\beta\varepsilon} = \frac{F(\varepsilon)\,d\varepsilon}{\sqrt{\varepsilon^2-\varepsilon_0^2}} \tag{9} \]

in the energy interval between

\[ E_1=\frac{1}{2}\left\{\varepsilon-\sqrt{\varepsilon^2-\varepsilon_0^2}\right\} \quad \text{and} \quad E_2=\frac{1}{2}\left\{\varepsilon+\sqrt{\varepsilon^2-\varepsilon_0^2}\right\}, \tag{10} \]

where \(\varepsilon_0\) is the rest energy of the neutral meson. For γ-rays with energy \(E\), the intensity is obtained by integrating expression (9) over the limits from \(\varepsilon=E+\dfrac{\varepsilon_0^2}{4E}\) to infinity; here each neutral meson with energy above \(\varepsilon\) contributes to the intensity of γ-rays with energy \(E\). This is true both for energies \(E_1 \leqslant \dfrac{\varepsilon_0}{2}\) and for \(E_2 \geqslant \dfrac{\varepsilon_0}{2}\), since, solving (10) for \(\varepsilon\), we obtain in both cases

\[ \varepsilon=E_{1,2}+\frac{\varepsilon_0^2}{4E_{1,2}}. \tag{11} \]

Thus, the γ-ray spectrum is given by the expression

\[ N(E)=\int_{\varepsilon}^{\infty}\frac{F(\varepsilon)\,d\varepsilon}{\sqrt{\varepsilon^2-\varepsilon_0^2}}. \tag{12} \]

Differentiating (12), we obtain the relation

\[ F\left(E+\frac{\varepsilon_0^2}{4E}\right) = E\left|\frac{dN}{dE}\right|, \tag{13} \]

which makes it possible to determine the energy distribution of mesons from the observed γ-radiation spectrum. At photon energy \(E=\dfrac{\varepsilon_0}{2}\), the γ-radiation intensity has a maximum value.

It is clear from (13) that the derivative \(\dfrac{dN}{dE}\) at this point is equal to zero only if \(F(\varepsilon_0)=0\), i.e., if the beam contains no mesons with velocity equal to zero.

d) Rest energy of the neutral meson

From expression (12), which is valid both for \(E_1 \leqslant \dfrac{\varepsilon_0}{2}\) and for energies \(E_2 \geqslant \dfrac{\varepsilon_0}{2}\), it follows that the intensity

the \(\gamma\)-radiation is the same for \(E_1\) and \(E_2\), since, according to (11),

\[ E_1+\frac{\varepsilon_0^2}{4E_1}=E_2+\frac{\varepsilon_0^2}{4E_2}. \]

This gives:

\[ \varepsilon_0^2=4E_1E_2 \]

or

\[ \varepsilon_0=2\sqrt{(E_1E_2)}. \tag{14} \]

Relation (14), which makes it possible to determine the rest energy \(\varepsilon_0\) from the observed spectrum of \(\gamma\)-radiation, is usually valid, since it is the result of taking the Doppler effect into account. Therefore it does not depend on the form of the spectrum of neutral mesons.

d) Determination of the distribution of the quantity \(r\)

Consider a neutral meson emitted from a star (Fig. 45) and traveling, before decay, a distance \(l\). Suppose that it then emits a photon in a direction making an angle \(\theta\) with the direction of motion of the meson, so that the distance from the star to the line of motion of the photon is equal to \(r\). Consequently,

\[ r=l\sin\theta, \]

and, since we have:

\[ N(\theta)\,d\theta=I(\theta)\sin\theta\,d\theta, \]

we can write, for one meson:

\[ N(r)=N(\theta)\frac{d\theta}{dr}=I(\theta)\frac{\operatorname{tg}\theta}{l}. \]

Next, \(l=c\beta Bt\), where \(t\) is the time interval between the moments of creation and decay of the neutral meson in the \(C\)-system. Suppose that the meson decays according to the usual laws of radioactive decay; then the number of mesons that decay in the time interval from \(t\) to \(t+dt\) will be:

\[ N(t)\,dt=\frac{N_0}{\tau_0}\exp\left(-\frac{t}{\tau_0}\right)dt, \]

where \(\tau_0\) is the meson’s proper mean lifetime. Therefore,

\[ N(l)\,dl=\frac{N_0}{c\beta B\tau_0}\exp\left(-\frac{l}{c\beta B\tau_0}\right)dl, \]

and since

\[ l=\frac{r}{\sin\theta}, \]

\[ N(l)\,dl=\frac{N_0}{c\beta B\tau_0}\frac{r}{\operatorname{tg}\theta\sin\theta}\exp\left(-\frac{r}{c\beta B\tau_0\sin\theta}\right)d\theta \]

and, finally,

\[ N(r)=\frac{N_0}{c\beta B\tau_0}\int_0^\pi I(\theta)\exp\left(-\frac{r}{c\beta\tau_0\sin\theta}\right)d\theta . \]

We carried out this integration graphically for several energy values and found that the calculated distributions of the quantities \(r\) are approximately exponential for values \(r>10\mu\). For values \(B\gg2\) the curves are almost identical. The final distribution was obtained by adding the curves for different values of \(B\), the distribution of which was taken in accordance with the energy spectrum of neutral mesons.

e) Random pairs

The probability of detecting random pairs whose bisector passes from the star at a distance in the interval between \(r\) and \(r+dr\) can be calculated from the known angular distribution and the frequency of occurrence of such pairs in the plate.

Our observations show that the \(\gamma\)-rays are distributed isotropically in the lower half of space (Fig. 39). As a consequence of the geometry of the scanned area (Fig. 47), two thirds of the random pairs are directed toward the star. If \(N(r)dr\) is the distribution of the quantity \(r\) associated with these pairs, then we may write that

\[ N(r)=\frac14\int_r^R I(\theta)\sin\theta\,\frac{d\theta}{dr}N(l)\,dl, \]

where \(l\) is the distance between the star and the point of formation of the pair, \(\theta\) is the angle between the bisector and the line joining the vertex of the pair to the star, and \(N(l)dl\) is the number of pairs whose vertices lie in the spherical shell bounded by spheres of radii \(l\) and \(l+dl\). Thus, \(N(l)dl=\rho\,4\pi l^2dl\), where \(\rho\) is the observed number of pairs per unit volume. The angular distribution \(I(\theta)\) is constant. \(R\) is the effective radius of the scanned area. Using the relation \(r=l\sin\theta\), we obtain:

\[ N(r)=\text{const}\int_r^R l\,\tg\theta\,d\theta=\text{const}\, r\sqrt{R^2-r^2}. \]

For \(r<600\mu\) we have \(N(r)\sim\text{const}\,Rr\), a relation that is valid to within 10%. The constant is then calculated from the observed density of “background” pairs and the known value of \(R\).

CITED LITERATURE

1. Goldschmidt-Clermont, King, Muirhead, Ritson, Proc. Phys. Soc. 61, 183 (1948).
2. Davies, Lock, Muirhead, Phil. Mag. 40, 1250 (1949).

1. Williams, Proc. Roy. Soc. A, 169, 531 (1939).
2. B. Rossi, K. Greisen, Interaction of Cosmic Rays with Matter, 1948.
3. Franzinetti, Phil. Mag. 41, 86 (1950).
4. Rochester, Report to Como Conference, 1949.
5. Piccioni, Phys. Rev. 77, 1, 6 (1950).
6. Adelman, Jones, Phys. Rev. 75, 1468 (1949).
7. Sands, Phys. Rev. 77, 180 (1950).
8. Fretter, Phys. Rev. 76, 511 (1949).
9. Lovati, Mura, Salvini, Tagliaferri, Nature, 163, 1003 (1949); Phys. Rev. 77, 284 (1950).
10. Brown, McKay, Phys. Rev. 77, 342 (1950).
11. Butler, Rosser, Barker, Proc. Phys. Soc. 63, 145 (1950).
12. Harding, Perkins, Nature 164, 285 (1949).
13. Davies, Franzinetti, Perkins, Phil. Mag. (in press).
14. Bjorklund, Crandall, Moyer, York, Phys. Rev. 77, 213 (1950).
15. Bradt, Kaplon, Peters, Helv. Phys. Acta 23, 24 (1950).
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16. Cocconi, Loverdo, Tongiorgi, Phys. Rev. 70, 852 (1946)*).
17. Chao, Phys. Rev. 75, 581 (1949)**).
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*) See also G. T. Zatsepin, DAN 67, 993 (1949).
**) See also N. G. Birger, V. I. Veksler et al., ZhETF 19, 826 (1949). (Translator’s note.)

To the article: “Nuclear Disintegrations Caused by High-Energy Particles.”

Microphotograph XII. A shower particle is observed, emitted in a star of type \(7 + 8_p\), which produces a secondary nuclear disintegration of type \(3 + 3_p\). The connecting track 5 has a minimal grain density and cannot be identified because of its short length. Track 2 of the primary star is produced by a meson (\(\bar{\alpha} = 0.15^\circ\) per \(100\,\mu\); grain density equal to 12.9 grains per \(50\,\mu\)); track \(a\) of the secondary star is also produced by a meson (\(\bar{\alpha} = 0.03^\circ/100\,\mu\); grain density equal to 12.1 grains/\(50\,\mu\)).

To the article: “Nuclear Disintegrations Caused by Particles of High Energy”

Microphotograph XIII. A shower particle is observed, emitted in a star of type \(18+5n\), which produces a secondary nuclear disintegration of type \(5+0p\). The connecting track has a length of \(10600\,\mu\) and is identified as a meson \((a = 0.049^\circ/100\,\mu;\) the grain density is \(10.4\) grains per \(50\,\mu)\).

To the article: “Nuclear Disintegrations Caused by High-Energy Particles.”

Microphotograph XIV. A shower particle, produced at point \(A\) as part of the shower, forms a secondary shower of type \(8 + 5_p\) at point \(B\). This case was found at the corner of the plate, so that the strong distortion of the emulsion did not allow a scattering measurement to be made.

1. 13. ↩