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SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS
V. I. Gol’danskii, A. L. Lyubimov, and B. V. Medvedev
IV. TOTAL NUCLEAR CROSS SECTIONS FOR HIGH-ENERGY NEUTRONS*)
Total nuclear cross sections for high-energy neutrons are composed of the cross sections of inelastic collisions and the cross sections of elastic (diffraction) scattering.
Here, by elastic scattering, as usual, is meant such an interaction of a neutron with a nucleus in which the neutron energy is distributed between the kinetic energy of the scattered neutron and the kinetic energy of the recoil nucleus as a whole, while the internal state of the nucleus remains the same as before the scattering.
By inelastic collisions are meant any collisions that lead to a change in the nature or state of the colliding particles. Any interaction between the bombarding neutron and some individual nucleon of the target nucleus must therefore be regarded as an inelastic collision of the neutron with the nucleus. Special cases of inelastic collisions are, for example, charge exchange on a complex nucleus and quasifree scattering: bombarding nucleon—nuclear nucleon.
It follows from theory³ that at such neutron energies, when the neutron wavelength is much smaller than the nuclear radius \((\lambda \ll R)\), but the mean free path of neutrons in nuclear matter is also still appreciably smaller than the nuclear radius, the cross sections of elastic scattering \((\sigma_d)\) and of inelastic collisions of neutrons with nuclei \((\sigma_a)\) are equal, and each of them is equal to the geometrical cross section of the nucleus \(\pi R^2\). The conditions stated correspond to a neutron-energy interval of approximately 14 to 40 MeV. In this energy region, the total nuclear cross sections must therefore be equal to \(2\pi R^2\).
*) For the beginning see UFN, vol. 48, issue 4, 1952. The bibliography for Nos. 1–86 is also given there.
At higher energy the “transparency” of nuclei should already begin to make itself felt—the path of neutrons in nuclear matter becomes comparable with the radii even of heavy nuclei, and the cross sections for elastic scattering and inelastic collisions, and consequently also the total nuclear cross sections, should decrease with increasing energy.
Qualitatively, just such a dependence was observed experimentally; however, the quantitative predictions of the theory proved to be in strong contradiction with the experiments. A comparison of the experimental and theoretical data both for the total cross section and for the components of the cross sections for the interaction of high-energy neutrons with nuclei is given below in Section VII. Let us proceed to a consideration of the methods and results of the experiments on measuring total cross sections.
Experiments to determine total neutron cross sections were carried out under conditions of the so-called “good” geometry, when the dimensions of the scatterer exceeded the diameter of the neutron (collimated) beam, and the detector was placed along the axis of the beam very far from the scatterer, so that any collision—elastic or inelastic—with the nuclei of the scatterer removed the neutron from the beam, since even at very small scattering angles (down to 20 minutes in some experiments) the neutrons, under “good” geometry, no longer reached the detector.
Under such conditions the attenuation of the neutron beam is explained both by inelastic collisions and by elastic scattering, and occurs according to the law \(e^{-n\sigma t}\), where \(n\) (atoms/\(\mathrm{cm}^{3}\)) is the thickness of the scatterer.
In experiments on determining total cross sections the choice of the neutron detector cannot influence the results so strongly as under conditions of “poor” geometry (see Section VI), when the detector must have as high a threshold as possible in order not to record comparatively slow secondary neutrons. However, in view of the broad energy distribution of the primary neutrons, detectors with different thresholds and different types of excitation functions correspond to different mean effective energies of the registered neutrons, and this can affect the magnitude of the total cross sections being determined.
If the neutron spectrum has the form \(f(E)\,dE\), the efficiency of the detector for neutrons of different energies is \(\sigma(E)\), and the detector threshold is \(E_{1}\), then the mean effective energy of the registered neutrons is equal to
\[ \overline{E}= \frac{ \displaystyle \int_{E_{1}}^{E_{\max}} E\sigma(E) f(E)\,dE }{ \displaystyle \int_{E_{1}}^{E_{\max}} \sigma(E) f(E)\,dE }. \]
Thus, for neutrons produced in the stripping reaction from deuterons with an energy of 190 MeV, the mean effective energy when registered by a carbon-
...with a hydrogen detector is equal to 84 MeV⁹¹, when registered by the fission of bismuth—to 95 MeV⁸⁷, and by the fission of gold—to 100–105 MeV⁸⁷.
To determine the total cross sections, in most cases three ratios are found: the counting rates or activities of the detector (placed behind the scatterer along the beam axis) and of the monitor (placed in front of the scatterer or to the side of the beam axis) in the absence of the scatterer \((r_0)\), in the presence of a scatterer of length \(l\) cm \((r)\), and when the scatterer is replaced by a very thick absorber, which practically completely absorbs the neutrons (background ratio \(r_\phi\)).
The range of neutrons in the substance of the scatterer
\[ \lambda=\frac{1}{N\sigma_t} \]
(where \(N\) atoms/cm³ is the density of the scatterer) is determined from the relation
\[ e^{-\frac{l}{\lambda}}=\frac{r-r_\phi}{r_0-r_\phi}. \]
Thus, total nuclear cross sections were determined directly for a number of elements, and for other elements by comparing the ranges in various substances containing these elements (for a mixture,
\[ \lambda=\frac{1}{\sum N_i\sigma_i} \]
). Thus, for example, the total cross section for carbon was determined directly; then from experiments with paraffin \((\mathrm{CH}_2)_n\) the cross section for hydrogen was obtained, from experiments with water—for oxygen, with melamine \((\mathrm{C}_3\mathrm{N}_6\mathrm{H}_6)\)—for nitrogen, and with heavy water—for deuterium.
The principal experiments on determining total nuclear cross sections for high-energy neutrons reduce to the following:
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Neutron energy 42 MeV, detector—carbon; data obtained for 33 elements⁹⁰.
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Neutron energy 84 MeV, detector—carbon; data for 15 elements⁹¹.
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Neutron energy 95 MeV, detector—bismuth fission chambers; data for 12 elements⁸⁷.
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Neutron energy 156 MeV, detector—a telescope of three proportional counters of recoil protons with a filter setting the registration threshold; data for 10 elements⁹².
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Neutron energy 280 MeV, detector—a telescope of proportional counters of recoil protons, with a filter setting the registration threshold; data for 12 elements¹⁸.
As an example of the arrangement of the experiments, let us consider the determination of cross sections at a neutron energy of 280 MeV¹⁸. The neutron beam was collimated: the first time by a tube 1.27 cm in diameter, placed in a concrete cube 214 cm thick, and the second time by a tube with a cross section of \(6.6 \times 3.5\) cm in a concrete wall 306 cm thick. The axis of the beam was determined with the aid of photographic film—by blackening. The area of the beam beyond the second collimator was 23.2 cm²; the scatterer samples were made in the form of cylindrical blocks 7.6 cm in diameter \((S = 45.5\ \text{cm}^2\), i.e. twice the area of the beam). A counter with \(\mathrm{BF}_3\), placed in the concrete wall near the second collimator, served as the beam monitor. The beam detector was a telescope of two anthracene counters, registering protons knocked out of a paraffin target at an angle of \(15^\circ\) to the direction of the neutron beam. Between the two counters there was...
a copper filter 5 cm thick, which stopped protons knocked out at a given angle by neutrons with energies below 250 MeV.
The distance from the scatterer to the paraffin target was 214–275 cm, and any deflection by an angle greater than 40′ removed neutrons from the beam (in other experiments[^22] the distance to the paraffin target was increased to 920 cm, which reduced the limiting angle to 20 minutes). Under the indicated geometrical conditions, the counting rate of the detector in the absence of the scatterer was 800 pulses/sec for the first counter, 150 pulses/sec for the second, and 3 pulses/sec for coincidences of the two counters. The background of random coincidences did not exceed 0.06 pulse/sec, i.e. 2% of the principal effect (the resolving time of the setup was \(\tau = 2.5 \cdot 10^{-7}\) sec.).
Since a number of differences enter into the calculation of the cross sections (both in determining ranges from the quantities \(r\), \(r_0\), and \(r_\phi\), and in recalculating neutron ranges in mixtures into cross sections of individual nuclei), the principal experiments were carried out in such a way that the total statistical errors were sufficiently small. The accuracy in determining total cross sections increases from light nuclei to heavy ones—with the increase in the absolute value of the cross sections—and in most of the works reviewed amounts from 1.5–2.5% to 4–6% (for hydrogen and deuterium).
A summary of all data on total nuclear cross sections for high-energy neutrons is given in Table VII and is partially illustrated by Fig. 34, as well as by Fig. 24 in Section III (for hydrogen and deuterium). Table VII also gives data on total cross sections for neutrons with energies of 14 and 25 MeV[^87],[^89]. At these energies it should be assumed that \(\sigma_t = 2\pi R^2\), and, proceeding from this, it was possible to determine nuclear radii well described by the relation \(R = (1.3 \div 1.37)\, A^{1/3}\,10^{-13}\) cm. It is precisely these values of the radii that were used in Table VII in determining the “geometrical” cross section of nuclei, equal to \(\pi R^2\). The table also gives, in addition, the neutron mean free paths in various elements calculated from the geometrical cross section \((\pi R^2)\) (in g/cm\(^2\)).
At energies above 40 MeV for light nuclei and above 70–90 MeV for heavy nuclei, the effects of nuclear “transparency” begin to appear, and the total cross sections fall rather rapidly with increasing energy up to 150–200 MeV. However, even in this energy interval the decrease of the cross sections is slower than according to the law \(\sigma \sim \frac{1}{E}\). At energies above 160–200 MeV, for the heaviest nuclei the cross sections still decrease, although only very slightly (for lead, in the interval from 240 to 406 MeV the reduction of the total cross section is about 7%); for light nuclei, however, the cross sections remain practically constant.
This fact contradicts all existing theoretical predictions and is of considerable interest. The constancy of the total cross sections of complex nuclei in the energy region above 200 MeV is undoubtedly a consequence of the energy dependence of the \(np\)- and \(pp\)-scattering cross sections, which was discussed in the preceding section.
Fig. 34. Dependence of the total effective cross sections of various nuclei on neutron energy.
V. I. GOLDANSKII, A. L. LYUBIMOV, AND B. V. MEDVEDEV
Total effective nuclear cross sections for neutrons
| Nucleus | \(2\pi R^2\) | \(k_{\mathrm{geom}}^{2}\), in g/cm\(^2\) | 14 [87, 88*] | 25 [87, 88] | 39 [89] | 42 [90] | 64.5 [89] | 84 [91] | 95 [37] | 97 [89] | 100—105 [37] |
|---|---|---|---|---|---|---|---|---|---|---|---|
| H* | 0.77 | 0.39 | 0.223 | 0.203 | 0.125 | 0.083 | 0.073 | 0.074 | |||
| D* | 0.80 | 0.289 | 0.117 | 0.104 | |||||||
| Li | 0.965 | 24 | 0.684 | 0.314 | |||||||
| Be | 1.08 | 0.853 | 0.431 | 0.396 | |||||||
| B | 1.18 | 1.16 | 0.985 | ||||||||
| C | 1.24 | 32 | 1.24 | 1.29 | 1.100 | 1.089 | 0.784 | 0.550 | 0.498 | 0.508 | 0.48 |
| N* | 1.33 | 1.220 | 0.570 | ||||||||
| O* | 1.42 | 1.60 | 1.358 | 0.765 | 0.663 | ||||||
| F* | 1.61 | 1.603 | |||||||||
| Na* | 1.69 | 1.67 | |||||||||
| Mg* | 1.75 | 1.8 | 1.723 | ||||||||
| Al | 1.84 | 49 | 1.85 | 1.782 | 1.12 | 0.993 | |||||
| S* | 2.01 | 1.974 | |||||||||
| Cl* | 2.12 | 1.88 | 2.11 | 1.28 | |||||||
| Ca | 2.25 | 2.21 | |||||||||
| Fe | 2.67 | 2.441 | |||||||||
| Ni | 2.75 | 2.510 | |||||||||
| Cu | 2.85 | 74 | 2.50 | 2.540 | 2.22 | 2.00 | 1.92 | ||||
| Zn | 2.92 | 2.618 | |||||||||
| Br* | 3.25 | 2.93 | |||||||||
| Sr* | 3.43 | 2.99 | |||||||||
| Mo | 3.60 | 3.11 | |||||||||
| Ag | 3.83 | 3.70 | 3.229 | ||||||||
| Sn | 4.04 | 98 | 3.251 | 3.28 | 3.18 | ||||||
| J* | 4.22 | 3.51 | |||||||||
| Ba* | 4.41 | 3.57 | |||||||||
| Ta | 5.14 | 4.20 | |||||||||
| W | 5.19 | 4.31 | |||||||||
| Hg | 5.44 | 5.64 | 5.25 | 4.51 | |||||||
| Pb | 5.56 | 125 | 5.0 | 4.44 | 4.53 | 4.48 | |||||
| Bi | 5.60 | 5.2 | 4.58 | ||||||||
| Th | 5.92 | 5.03 | |||||||||
| U | 6.02 | 130 | 5.12 | 5.03 | 4.92 |
*) The numbers in brackets denote literature references.
Table VII
high energies (in \(10^{-24}\ \mathrm{cm}^2\))
neutrons in MeV
| 110—120 [24] | 145 [24] | 156 [89, 90] | 160 [24] | 180 [24] | 190 [24] | 220 [24] | 240 [24] | 260 [20] | 270 [22] | 280 [12] | *406 [50] |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0,046± | 0,0512 | 0,041 | 0,035 | 0,038 | 0,033 | ||||||
| 0,0707 | 0,057 | 0,049 | |||||||||
| 0,164 | |||||||||||
| 0,258 | 0,229 | 0,225 | |||||||||
| 0,330 | 0,291 | 0,285 | 0,288 | 0,279 | 0,288 | ||||||
| 0,430 | 0,372 | 0,380 | |||||||||
| 0,733 | 0,677 | 0,575 | 0,540 | 0,576 | 0,576 | 0,555 | 0,566 | ||||
| 1,238 | |||||||||||
| 1,49 | 1,31 | 1,30 | 1,25 | 1,15 | 1,15 | 1,15 | 1,145 | 1,19 | 1,14 | ||
| 1,90 | 1,87 | 1,83 | |||||||||
| 2,476 | |||||||||||
| 2,61 | |||||||||||
| 2,80 | |||||||||||
| 3,71 | 3,499 | 3,03 | 2,85 | 2,99 | 2,88 | 2,84 | 2,89 | 2,69 | |||
| 3,28 | 3,29 | 3,14 |
In concluding the discussion of the experimental data on total cross sections, it should be noted that there is a certain doubtfulness in the data of the American work \({}^{24}\), devoted to the determination of total cross sections at energies between 90 and 270 MeV, connected with the insufficient reliability of the determination of the mean effective energy of the neutrons. In work \({}^{34}\) the neutrons were obtained by bombarding a beryllium target with protons of energies from 180 to 310 MeV. The authors consider that, over the entire indicated interval of proton energies, the “drop” upon charge exchange amounts to 70 MeV, so that the peak of the neutron spectrum lies from 110 to 240 MeV. Meanwhile, direct investigations of neutron spectra, considered in Section I, showed that at \(E_p = 170\) MeV the “drop” is equal to about 20 MeV, while at \(E_p = 245\) MeV it is about 30 MeV, and only at \(E_p = 340\) MeV does the “drop” reach 70 MeV. Therefore the values of the neutron energies given in \({}^{24}\) should apparently be increased by 20–40 MeV, which improves the agreement between the data of \({}^{24}\) and \({}^{93}\), decreases the rather large scatter of the points on the curves of Fig. 34 for Al, Cu, and Pb, and somewhat smooths the excessively sharp decrease of the cross sections of these nuclei from 90 to 156 MeV.
The indicated possible inaccuracies do not, however, refute the characteristic dependence of the total nuclear cross sections on energy that was discussed above.
Let us note that at high neutron energies, as is seen, for example, from the data of Table VII for 280 MeV, the total cross sections of light nuclei (approximately up to oxygen) are very close to the additive cross sections of the neutron—deuteron and neutron—neutron interactions (we take as the latter the \(pp\)-interaction cross section from Table VI). Thus, for the nucleus \(\mathrm{Be}^9\), \(\sigma_{\mathrm{exp}} = 0.225 \cdot 10^{-24}\ \mathrm{cm}^2\), while \(4\sigma_{nd} + \sigma_{nn} = 0.220 \cdot 10^{-24}\ \mathrm{cm}^2\); for the nucleus \(\mathrm{O}^{16}\), \(\sigma_{\mathrm{exp}} = 0.380 \cdot 10^{-24}\ \mathrm{cm}^2\), while \(8\sigma_{nd} = 0.392 \cdot 10^{-24}\ \mathrm{cm}^2\). For heavier nuclei this additivity is violated the more strongly, the heavier the nucleus; for example, for uranium the additive value of the cross section is 80% higher than the experimental one. Since, as will be seen from Section VII, the cross sections of inelastic collisions at high energies (\(\sim 280\) MeV) constitute, for all nuclei, an approximately equal fraction of the total cross sections, what has been said also applies to the cross sections of inelastic collisions. The presence of additivity of the cross sections for light nuclei testifies in favor of considering collisions of nucleons with such nuclei as sums of nucleon—nucleon collisions.
V. ELASTIC SCATTERING OF HIGH-ENERGY NUCLEONS
By elastic scattering is understood, as was already mentioned above, the interaction between a high-energy nucleon and the nucleus as a whole, without changing the internal properties of the nucleus.
At high energies, when the wavelength of the bombarding nucleon \(\lambda \ll R\) (\(R\) is the radius of the nucleus) and at the same time the radius of the nucleus
SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS
is close in magnitude to the range of nucleons in nuclear matter, elastic scattering may be regarded as diffraction by a semitransparent screen having the size and shape of the nucleus. Then the parameters determining the angular distribution of elastically scattered particles and the elastic-scattering cross section should be the absorption coefficient and the refractive index of the nucleon wave in the nucleus, i.e., ultimately, the nuclear radius, the depth of the nuclear potential well, and the elementary cross sections for the interaction of free nucleons.
The diffraction angular distribution should be characterized by a strong forward directionality; the angle of the first diffraction minimum is equal to several \(\frac{\lambda}{R}\), and under the condition \(\lambda \ll R\) several diffraction maxima may be observed.
In elastic scattering through a small angle \(\theta\), the energy of the particle scattered by the nucleus (of mass \(M\)) (of mass \(m\) and initial energy \(E_0\)) is approximately equal to
\[ E_\theta = E_0 - E_0 \frac{m^2 \sin^2 \theta}{(m+M)^2}, \]
and the average energy loss in elastic scattering is very small and, even for such light nuclei as carbon, does not exceed \(3\text{--}5\%\). Therefore it is desirable to register elastically scattered particles with a detector having as high a threshold as possible, in order to cut off particles traveling at the same angle after inelastic collisions and therefore having lower energy.
In the experiment, differential angular cross sections were determined for elastic scattering of neutrons with energy 84 MeV by nuclei of carbon \(^{37}\), aluminum, copper, and lead \(^{93}\). Spheres of the indicated materials with diameters \(2.5\text{--}3.8\ \mathrm{cm}\), placed in a collimated neutron beam, were used as scatterers. The flux of neutrons scattered by these spheres was measured at angles from \(2.5^\circ\) to \(40^\circ\) for \(C^{37}\) and up to \(24^\circ\) for Al, Cu, and Pb\(^{93}\). Elastic scattering on carbon was measured with the aid of a bismuth fission chamber, and scattering on Al, Cu, and Pb with the aid of carbon detectors. The intensity of the primary beam was determined with the aid of the same detectors moved into the position of the scatterers. The background, measured in the absence of scatterers, proved negligibly small. The experiments showed that a sharp forward directionality of the scattered neutrons is observed.
To check the data obtained with relatively low-threshold (\(\sim 20\) MeV) carbon detectors, experiments were also carried out in \(^{93}\) with another detector—a coincidence telescope of three proportional counters, registering recoil protons from neutrons with energy not less than 60 MeV. As is seen from Fig. 35, where the curves of the angular distribution of neutrons elastically scattered by Al, Cu, and Pb nuclei are given, changing the detector threshold from 20 to 60 MeV in this case did not change the results obtained.
Fig. 35. Diffraction scattering of neutrons with energy 84 MeV by aluminum, copper, and lead. The coarse dashed line corresponds to the calculation from the integral elastic-scattering cross sections obtained by integrating the differential cross sections. The fine dashed line corresponds to the calculation from the integral elastic-scattering cross sections obtained from the difference between the total cross sections and the cross sections for inelastic collisions.
The results of experiments^93 were compared with the theoretical angular distribution of elastically scattered neutrons obtained from the diffraction formulas for opaque and semi-transparent^94 screens. The best agreement with experiment—especially at small angles—was obtained for calculations of diffraction by opaque screens. In Fig. 35 the experimental data from ^93 and the calculated curves for diffraction by opaque screens are presented. The two types of curves in Fig. 35 (coarse and fine dashed) were obtained for different values of the integral elastic-scattering cross sections chosen for normalization.
At large angles, the calculation for diffraction by semi-transparent screens also agreed rather well with experiment.
Let us note the presence—in accordance with the theory—of a second diffraction maximum for lead at an angle of \(18^\circ\).
The integral elastic-scattering cross sections \(\sigma_d\) were determined in ^93 and ^37 in two ways: by integrating the differential cross sections and by comparing simultaneous measurements of beam attenuation in good and poor geometry (see Section VI)—from the difference between the total cross sections and the cross sections for inelastic collisions. Both methods gave similar results, namely (in \(10^{-24}\ \text{cm}^2\)):
by integration of the differential angular cross sections:
\[ \begin{aligned} \mathrm{Al}:&\ 0.71 \pm 0.04,\\ \mathrm{Cu}:&\ 1.37 \pm 0.07,\\ \mathrm{Pb}:&\ 2.79 \pm 0.14; \end{aligned} \]
by comparing data for good and poor geometry:
\[ \mathrm{C}: 0.275^{37}, \qquad \mathrm{Al}: 0.65 \pm 0.04^{93}, \]
\[ \mathrm{Cu}: 1.32 \pm 0.03^{93}, \qquad \mathrm{Pb}: 2.64 \pm 0.08^{93}. \]
If one introduces into the values of \(\sigma_d\), obtained by integrating the differential angular cross sections, a correction that takes into account the possibility of recording some of the inelastically scattered small-angle neutrons having sufficiently high energy, then lower limits are obtained for the values of \(\sigma_d\), for allowance for the differential scattering at large angles, not investigated experimentally, can only increase the elastic-scattering cross section. The authors\(^{93}\) introduced a correction for fast neutrons of inelastic origin on the basis of data on the production of fast protons in the interaction of 90-MeV neutrons with C, Cu, and Pb nuclei\(^{95}\), assuming that the number of fast secondary protons and neutrons is the same. Taking this correction into account, the integral cross sections of elastic scattering \(\sigma_d\) (in \(10^{-24}\ \mathrm{cm}^2\)) are equal to:
\[ \mathrm{Al}: 0.64, \qquad \mathrm{Cu}: 1.24 \quad \text{and} \quad \mathrm{Pb}: 2.62. \]
At higher energies, no similar investigations of the elastic scattering of neutrons were carried out. However, experiments were performed on the elastic scattering of protons with energy 340 MeV by carbon, copper, and lead nuclei\(^{96}\). The experiments were carried out with a collimated proton beam, the scattered protons being recorded by a coincidence telescope of three scintillation counters placed 120 cm from the scatterer. The filter in the telescope was chosen so as to record protons scattered by Cu and Pb nuclei with energy not less than 330 MeV, and by C nuclei—not less than 315 MeV. The background of accidental coincidences was extremely small—especially owing to the high resolving power of the coincidence circuit (\(2 \cdot 10^{-6}\) sec.).
Fig. 36. Diffraction scattering of 340-MeV protons on carbon (theory and experiment).
In Figs. 36–38 are given the curves of the angular distribution of protons elastically scattered by C, Cu, and Pb nuclei. For carbon the angular-distribution pattern does not yet have a sharply expressed diffraction form; for copper there already appears
the second diffraction maximum at \(\sim 13^\circ\), and for lead—the second and third maxima at \(9^\circ\) and \(15^\circ\).
The curves plotted in Figs. 36–38 on the basis of theoretical considerations\(^{94}\) agree rather poorly with experiment, as do all the predictions made on the basis of the theoretical work\(^{94}\) concerning cross sections at energies of the order of \(150\)—\(300\) MeV.
Fig. 37. Diffraction scattering of protons with an energy of 340 MeV by copper (theory and experiment).
Fig. 38. Diffraction scattering of protons with an energy of 340 MeV by lead (theory and experiment).
In a later work by the same authors\(^{91}\), values are given for the integral cross sections of elastic scattering, equal to \(0.098 \cdot 10^{-24}\ \text{cm}^2\) for C, \(0.515 \cdot 10^{-24}\ \text{cm}^2\) for Cu, and \(0.934 \cdot 10^{-24}\ \text{cm}^2\) for Pb. The difference between the total cross sections and the integral cross sections of elastic scattering gives upper limiting values for the cross sections of inelastic collisions: for C, \(0.190 \cdot 10^{-24}\ \text{cm}^2\); for Cu, \(0.62 \cdot 10^{-24}\ \text{cm}^2\); for Pb, \(1.76 \cdot 10^{-24}\ \text{cm}^2\) (from the total cross sections given for 280 and 406 MeV in Table VII).
As will be seen below, it is very important to limit the values of the cross sections of inelastic collisions by some upper bound, for the methods of determining these cross sections leave room for doubt as to whether the obtained values are not underestimated.
In this connection the question arises whether one can confidently regard the diffraction cross sections of nuclei for neutrons and protons as identical. Since, at the energies used, Coulomb scattering is appreciable only at the very smallest angles, while the nuclear interaction of the neutron and proton with nuclei is the same (on the basis of the charge symmetry of nuclear forces; see Section IIIb), it would seem that equality of the cross sections should be expected. However, it should be noted that the Coulomb interaction can affect the elastic-scattering cross section “indirectly.” If one proceeds from formula (23) in determining the refractive index of the neutron wave in the nucleus (see Section VII, p. 26), then the presence of an additional Coulomb interaction in pp scattering, as compared with np scattering, can substantially change the refractive index and, consequently, the magnitude of the elastic cross section, increasing it. Therefore, the use of the elastic-scattering cross section of protons as a lower estimate for the elastic-scattering cross sections of neutrons can be regarded only as approximate.
VI. INELASTIC COLLISIONS OF HIGH-ENERGY NEUTRONS
The first act of an inelastic collision of high-energy neutrons with nuclei may be regarded as the interaction of two free nucleons—the incident nucleon and a nuclear nucleon—possessing definite values of energy and momentum, the sum of which is distributed between the two nucleons. The simplest example of such an interaction is the case in which both nucleons leave the nucleus without further collisions—the so-called quasi-free nucleon–nucleon scattering. In the general case, however, one or both nucleons participating in the primary collision may subsequently collide with other nuclear nucleons, transferring to them part of their energy and momentum. In this process, some number of nucleons may be knocked out of the nucleus during a time of the order of the nuclear time,
\[ 10^{-22}—10^{-21}\ \text{sec}. \]
After the loss of particles the nucleus remains in an excited state, and additional emission of particles by the boiling nucleus may occur through the evaporation mechanism over a much longer time. It is clear that the energy and angular distributions of secondary particles arising as a result of inelastic collisions of high-energy neutrons with nuclei must differ substantially for knocked-out and evaporated particles.
The former must be characterized by higher energy and a predominant forward direction; the latter, by lower energy and isotropy in angle in the center-of-mass system. It is in this spirit that the processes of inelastic collisions are treated in the theoretical works that will be considered below, in Section VII.
and chambers measuring the total ionization (thresholdless detector)$^{98}$. In typical experiments the radius of the front cylindrical wall of the absorber was 20–30 cm, which ensured the condition of uniform illumination by neutrons$^{37,22}$.
The angular aperture of the absorber cone was $\pm 45^\circ{}^{37}$ or $\pm 21$ and $27^\circ{}^{93}$, which exceeds, as is seen from Section 5, the angles of elastic scattering. The thickness of the absorber in typical experiments did not exceed one and a half to two ranges for inelastic collisions (20–30 cm).
Only in individual cases, in order to verify the logarithmic law $\ln \dfrac{I_0}{I}=\dfrac{L}{\lambda_a}$, were large absorber thicknesses used—up to 3.5 $\lambda_a$ ranges in lead in experiments at 270 MeV$^{22}$. In this case the linearity of the logarithmic relation was well satisfied over the entire interval of absorber thicknesses.
For experiments with carbon and bismuth detectors at 84 and 95 MeV, the relation $\sigma_t=\sigma_a+\sigma_d$ was well satisfied and thus confirmed the correctness of the determination of inelastic-collision cross sections. Different results were obtained in experiments with thresholdless chamber detectors registering the total ionization$^{98}$ under the action of secondary particles produced in inelastic collisions of neutrons with an energy of 84 MeV.
For thin absorber layers in this case, transition effects were observed which apparently led to completely incorrect results in determining inelastic-collision cross sections for heavy nuclei. Thus, for lead the value of $\sigma_a$ determined in$^{98}$ amounted to only 40% of that obtained in$^{37,93}$ and consistent with the values of $\sigma_t$ and $\sigma_d$. The absorber thicknesses used in$^{98}$ were clearly insufficient, especially in the presence of transition effects—the attenuation of the neutron beam in copper and aluminum reached no more than a factor of 2, and in paraffin only 30%. Relatively good agreement with the data of other experiments was obtained only for carbon.
A summary of data on the determination of inelastic-collision cross sections is given in Table VIII.
In addition to the measurements mentioned above, the table includes data on inelastic collisions of neutrons with an energy of 100–105 MeV$^{37}$ (detector—a gold fission chamber), as well as information reported in the literature$^{99}$ on the relative inelastic-collision cross sections of neutrons with energies of 90 and 280 MeV with beryllium, copper, and tantalum nuclei.
It is seen from Table VIII that the inelastic-collision cross sections at the highest of the investigated energies (270 MeV) for light and heavy nuclei amount to about one half of the total cross section. The values of $\sigma_a$ vary relatively weakly within the range from 90 to 270 MeV, especially for heavy nuclei. Thus, the energy depen-
Table VIII
Ratio of cross sections for inelastic collisions to total cross sections
| Nucleus | 84 [93]*) | 90 [99] relative | 95 [37] | 270 [22] | 280 [99] relative |
|---|---|---|---|---|---|
| Be | 1 | 1 | |||
| C | 0,45±0,015 | 0,505±0,02 | |||
| Al | 0,43 | 0,42±0,015 | |||
| Cu | 0,396 | 4,3 | 0,39±0,005 | 0,50±0,02 | 5 |
| Ta | 8,7 | 11 | |||
| Pb | 0,41 | 0,40±0,01 | 0,51±0,01 0,49±0,02 |
*) The figures in square brackets denote literature references.
Absolute values of the cross sections of inelastic collisions
(in \(10^{-24}\ \mathrm{cm}^2\)) and ranges \(\lambda_a\) (\(\mathrm{g}/\mathrm{cm}^2\))
| Nucleus | \(\sigma_a\), 84 | \(\sigma_a\), 90 | \(\sigma_a\), 95 | \(\sigma_a\), 270 | \(\sigma_a\), 280 | \(\lambda_a\), 84 | \(\lambda_a\), 90 | \(\lambda_a\), 95 | \(\lambda_a\), 270 | \(\lambda_a\), 280 |
|---|---|---|---|---|---|---|---|---|---|---|
| Be | — | (0,205) | — | — | (0,114) | — | 73 | — | — | 131 |
| C | — | — | 0,224 | 0,144 | — | — | — | 89 | 138 | — |
| Al | 0,48 | — | 0,416 | — | — | 94 | — | 108 | — | — |
| Cu | 0,88 | (0,88) | 0,78 | 0,572 | (0,572) | 123 | (123) | 137 | 187 | (187) |
| Ta | — | (1,78) | — | 1,42 | (1,26) | — | 168 | — | — | 238 |
| Pb | 1,86 | — | 1,79 | — | — | 185 | — | 193 | 243 | — |
The dependence of \(\sigma_a\) is close to the analogous dependences for \(\sigma_t\) and for the cross sections of interaction of elementary particles. It is interesting to note that the ranges \(\lambda_a\) at energies of 270–280 MeV attain values characteristic of the absorption of the components of cosmic radiation that generate nuclear disintegrations of low and high energy. For the absorption of the component of cosmic radiation that generates high-energy nuclear disintegrations (showers), convincing experiments by Soviet authors have shown that the absorption ranges exceed the ranges between nuclear collisions, since the absorption of particles of relativistic energies is not a single-act process (see Section VIII).
In this connection, the question of whether the values of \(\sigma_a\) obtained by the method of “poor”
geometry,” are true or whether they are underestimated owing to the registration of secondary particles. In the experiments at 84 and 95 MeV, proof of the truth of \(\sigma_a\) is contained in the independent determination of \(\sigma_t\) and \(\sigma_d\). At an energy of 270 MeV no experiments to determine diffraction cross sections for neutrons were carried out. Moreover, whereas at an energy of 95 MeV, close to the threshold for fission of bismuth, one could with sufficient confidence assume that secondary particles either are not registered at all in bismuth fission, or are registered with considerably lower efficiency, such a point of view is no longer applicable at an energy of 270 MeV. In favor of the truth of the measured values of \(\sigma_a\) speaks the fact that the linearity of the relation \(\ln I=f(L)\) is preserved over a wide interval of absorber thicknesses—up to \(3.5\lambda_a\), i.e., the fact that there is no transition effect. But, given the closeness of the values of \(\sigma_a\) for primary and secondary particles, this argument is not confirmation of the applicability of the “poor geometry” method to the determination of \(\sigma_a\).
Very important, therefore, are the measurements, considered in Section V, of the integral cross sections of elastic scattering of protons with energy 340 MeV; from these the maximum values of \(\sigma_a\) at 270 MeV are estimated as: for C—\(0.190\cdot10^{-24}\ \text{cm}^2\), for Cu—\(0.62\cdot10^{-24}\ \text{cm}^2\), and for Pb—\(1.76\cdot10^{-24}\ \text{cm}^2\).
The small distortion of the cross sections of inelastic collisions, determined by the “poor geometry” method when neutrons are registered by a bismuth fission chamber at an energy of 270 MeV, is evidently connected with the following main causes:
1) The intranuclear momentum distribution of nucleons leads to a very broad angular distribution of the knocked-out particles. Therefore secondary neutrons, even when possessing energy sufficient for registration by the detector, often fly out at such large angles that they no longer strike the detector. The role of the momentum distribution was already evident in the description of the angular distribution of charge-exchange neutrons (Section I). The angular distribution of high-energy secondary neutrons in inelastic collisions should be still broader, since \(nn\)-scattering, like \(pp\)-scattering, evidently leads to a more isotropic (in the c.m. system) angular distribution than \(np\)-scattering. An additional decrease in the efficiency of registration of high-energy secondary neutrons is connected with the fact that, even when directed toward the detector after emission at large angles, they pass through greater absorber thicknesses and are more strongly absorbed as a result of new inelastic collisions.
2) Among the secondary particles, as the energy increases, charged particles begin to play a large role; for them the ionization ranges are still considerably smaller than the nuclear ranges. Thus, for protons with energy 250 MeV the ionization range is \(48\ \text{g}/\text{cm}^2\) of aluminum \({}^{100}\). Thus even the reverse charge exchange neutron—pro-
ton, not to mention other processes of formation of charged particles, corresponds to the knocking out of a neutron from the beam.
The secondary charged particles knocked out by neutrons with a mean energy of 90 MeV from nuclei of carbon, copper, and lead have been studied in the greatest detail.^95
With the aid of a coincidence telescope placed in a magnetic field, the type (p, d, t) and energy of these particles were distinguished by \(H\rho\) and specific ionization. The total knock-out cross section, spectrum, and angular distribution were determined for protons with energies above 20 MeV, deuterons with energies above 27 MeV, and tritons above 33 MeV (the latter only for carbon nuclei).
The data of work^95 are given in Table IX.
Table IX
Cross sections for the formation of secondary charged particles (in \(10^{-27}\ \text{cm}^2\))
| Nucleus | Particles | 0—25° | 25—45° | 45—180° | All angles |
|---|---|---|---|---|---|
| C | Sum | 48 | 39 | (30) | (117) |
| C | p \(>20\) MeV | 35 | 31 | (24) | (90) |
| C | p \(>35\) » | 24 | 19 | (9) | (52) |
| C | d \(>27\) » | 12 | 8 | (6) | (26) |
| Cu | Sum | 88 | 82 | (123) | (293) |
| Cu | p \(>20\) MeV | 70 | 68 | (103) | (241) |
| Cu | d \(>27\) » | 18 | 14 | (20) | (52) |
| Pb | Sum | 123 | 153 | (223) | (499) |
| Pb | p \(>20\) MeV | 100 | 132 | (192) | (424) |
| Pb | d \(>27\) MeV | 23 | 21 | (31) | (75) |
The data in parentheses were obtained by extrapolation to large angles according to the form of the angular distribution. The cross section for the formation of protons with energies above 35 MeV is regarded by the authors^95 as the cross section for the charge-exchange process \(n \to p\).
As a result it is found that per one inelastic collision in a carbon nucleus there are formed 0.53 fast secondary charged particles (0.41 p and 0.12 d), in a copper nucleus—0.38 (0.31 p and 0.067 d), and in a lead nucleus—0.28 (0.24 p and 0.042 d). Since the mean energy of the secondary charged particles is, of course, lower than the mean energy of the incident neutrons, it is clear that the fraction of the neutron energy transferred to charged particles is much smaller than 53% for C, 38% for Cu, and 28% for Pb. In Fig. 39 are given the spectra of particles knocked out from carbon nuclei at different angles.
With increasing neutron energy, the fraction of energy transferred to charged particles increases strongly. According to the data of work^11, in which the formation of stars by high-energy neutrons was investigated with the aid of thick-layer photographic emulsions, this fraction for neu-
trons with an energy of about 310 MeV (from charge exchange of protons with an energy of 385 MeV) is about 50%.
Thus, direct experiments confirm that a considerable part of the neutron energy in inelastic collisions (especially at high energies) is spent on the formation of secondary charged particles. This confirms the correctness of one of the explanations given above for the applicability of the “poor-geometry” method for determining \(\sigma_a\).
Fig. 39. Spectra of secondary charged particles knocked out by neutrons with an energy of about 90 MeV from carbon nuclei at various angles.
Another explanation amounted to a relatively broad angular distribution of high-energy secondary neutrons. There are no direct experimental data confirming this proposition. However, proceeding from the assumption of equality of the \(pp\)- and \(nn\)-interactions, we can consider data on secondary fast protons knocked out of nuclei by high-energy protons.
According to experiments with thick-layer photoemulsions\(^{102,107}\), in stars formed by protons with an energy of 350–400 MeV, there are observed, on average, 0.4 tracks with energy \(>100\) MeV and 0.35 tracks with energy 300–100 MeV. The mean angle of direction of these tracks is close to \(30^\circ\).
In studying the spectrum of protons knocked out of Al nuclei at angles of \(45\)–\(60^\circ\) to the primary beam of protons with an energy of 250 MeV\(^{103}\), it was found that this spectrum (allowing for the momentum distribution of nuclear nucleons) corresponds to simple proton–proton collisions. For heavier nuclei (Cu, Pb) the picture turned out to be more complicated, for multiple collisions are probable.
The spectra of protons scattered by deuterium and carbon nuclei at angles of 30–40° (at an initial energy of 340 MeV), when compared with simple pp scattering, gave good agreement with a theoretical calculation based on the assumption of collisions of free nucleons, taking into account the momentum distribution of nuclear nucleons[^104].
The energy distribution of protons knocked out of carbon nuclei at an angle of 90° by protons with an energy of 240 MeV was studied in detail[^105].
By the method of thick photographic emulsions, protons with energies from 60 to 190 MeV were investigated.
Table X gives the differential cross sections for the emission of such protons.
Table X
| \(E_p\), MeV | \(\dfrac{d^2\sigma}{dE\cdot d\Omega}\times 10^{29}\), \(\dfrac{\text{cm}^2}{\text{MeV}\cdot\text{steradian}}\) | \(E_p\), MeV | \(\dfrac{d^2\sigma}{dE\cdot d\Omega}\times 10^{29}\), \(\dfrac{\text{cm}^2}{\text{MeV}\cdot\text{steradian}}\) |
|---|---|---|---|
| 61.2 | \(4.69\pm0.36\) | 132.1 | \(0.81\pm0.10\) |
| 73.1 | \(4.89\pm0.36\) | 143.8 | \(0.59\pm0.07\) |
| 83.6 | \(3.17\pm0.25\) | 155.7 | \(0.52\pm0.06\) |
| 93.5 | \(2.17\pm0.16\) | 167.5 | \(0.54\pm0.07\) |
| 103.1 | \(2.01\pm0.19\) | 173.8 | \(0.51\pm0.07\) |
| 113.8 | \(1.45\pm0.12\) | 191.8 | \(0.50\pm0.08\) |
Integration over the indicated energy limits gives the value \(2.2\cdot10^{-27}\ \text{cm}^2/\text{steradian}\), and in all, assuming isotropic scattering, about \(28\cdot10^{-27}\ \text{cm}^2\), which amounts to about 20% of the cross section for inelastic neutron collisions with energy 270 MeV with carbon nuclei. This figure, obtained from a rough estimate, cannot, of course, give the true number of secondary fast protons directed predominantly forward (which we have not taken into account here), but it convincingly testifies to a sufficiently broad angular distribution of secondary fast protons in the bombardment of carbon nuclei by protons and, consequently, of secondary fast neutrons in the bombardment of nuclei by neutrons.
Thus, another explanation is also justified for the applicability of the “poor geometry” method for determining the cross sections of inelastic neutron collisions in the energy interval considered.
In conclusion of the present section, we note that a recent paper has appeared[^106], the data of which contradict those set forth above.
In this work the cross sections of inelastic collisions of protons with energy 240 MeV with nuclei contained in Ilford G-5 emulsion were determined. Over a total scanned track length of 9907 cm, 198 stars, 10 “disappearances” of protons (recharging without the appearance of other secondary particles except neutrons), 124 scatterings through angles greater than 4°, and 5 pp-scattering events were observed. After introducing corrections for Coulomb and elastic scattering, the path length in the emulsion corresponding to inelastic collisions was found to be \(36.1\,(+2.6\, -2.1)\) cm. Calculating from the theoretical formula[^24] the ratio
\[ \frac{\sigma_a}{\pi R^2} \]
(where \(R\) was taken equal to \(1.37A^{1/3}\cdot10^{-13}\) cm) for light (C, N, O) and heavy (Ag, Br) compo-
nent of the emulsion, the author\(^ {106}\) obtained the following values (in \(10^{-24}\ \text{cm}^2\)) of the cross sections for inelastic collisions of protons with energy \(240\ \text{MeV}\) with various nuclei: \(\mathrm{Ag} — 1.18 \pm 0.08,\ \mathrm{Br} — 0.97 \pm 0.07,\ \mathrm{O} — 0.25 \pm 0.03,\ \mathrm{N} — 0.23 \pm 0.03,\ \mathrm{C} — 0.21 \pm 0.03\). These quantities amount to \(65—72\%\) of the total cross sections of the nuclei for neutrons with such energy (see Fig. 34), instead of the \(50\%\) obtained by the poor-geometry method.
The data of work\(^ {106}\) contradict not only the experiments with neutrons under conditions of poor geometry, but also other experiments with photographic emulsions. Thus, from the distribution of star-formation events, for protons at \(240\ \text{MeV}\) a mean free path equal to \(50\ \text{cm}\) is obtained, whereas in a preceding paper by the same author\(^ {108}\) the value \(82\ \text{cm}\) was given, in exact agreement with the result of another author’s investigation\(^ {109}\), in which the mean free paths for star formation were determined over a broad interval of proton energies from 100 to \(350\ \text{MeV}\) (in this interval the mean free paths vary from 170 to \(70\ \text{cm}\)). Moreover, the accuracy of the calculations performed in\(^ {106}\) for separating the effect for different nuclei and accompanied by the use of the theoretical formula from\(^ {93}\), which agrees very poorly with experiment at an energy of \(240\ \text{MeV}\), is doubtful.
It should therefore be considered that the data of the neutron experiments under poor-geometry conditions are more reliable than the result of\(^ {106}\). Nevertheless, refinement of the data on the inelastic cross sections of the interaction of nucleons with energies of the order of hundreds of \(\text{MeV}\) with nuclei is of undoubted interest, in particular for comparison with data on the absorption of various components of cosmic radiation (see Section VIII).
VII. INTERACTION OF HIGH-ENERGY NUCLEONS WITH NUCLEI (THEORETICAL CONCEPTS)
The peculiarities of nuclear reactions at high energies are associated with concepts which, perhaps, are simpler, but are entirely different from those used at “small” (less than several tens of \(\text{MeV}\)) energies. The collision time of an incident particle with one of the nucleons of the nucleus is much shorter than the time between collisions of particles in the nucleus. Therefore the first stages of the process of interaction of an incident high-energy particle with a nucleus may be regarded as individual collisions with separate nucleons of the nucleus. The de Broglie wavelength of high-energy nucleons is much smaller than the dimensions even of light nuclei. Therefore the penetration of such a nucleon into the nucleus can be described quasiclassically: one may speak of which part of the nucleus the nucleon entered, where it was scattered, where it flew after scattering, etc. The wave properties of the nucleon must be taken into account only for such “fine” processes as, for example, diffraction of the nucleon wave by the nucleus; here it is natural to assume that, for the incident nucleon, the nucleus is a macroscopic body whose properties can be described by such “macroscopic” quantities as the absorption coefficient and the refractive index.
This immediately indicates two directions of investigation: a) calculation of elastic and total cross sections of nuclei, requiring allowance for the wave properties of incident nucleons, in the course of which one restricts oneself to establishing the interaction of the nucleon with the nucleus without considering
their subsequent fate, and b) a detailed investigation of the fate of a nucleon that has entered the nucleus and of the recoil nucleons arising in nucleon collisions (as long as the energy of the latter does not become sufficiently small), which is carried out with complete neglect of wave properties, by methods customary for calculating diffusion in macroscopic bodies. Such a separation is, of course, not too strict, and investigations in both regions mutually penetrate one another.
Let us begin with an exposition of the first group of investigations. If a plane wave, whose wave vector in vacuum is equal to \(k\) (for a particle \(k=\sqrt{2mE}/\hbar\)), passes through a layer of thickness \(s\) of a substance characterized by absorption coefficient \(K\) and refractive index \(n\), then its amplitude is multiplied by the (complex) factor
\[ a(s)=\exp\left\{-\frac{1}{2}K+ik(n-1)\right\}s. \]
If the area of the layer facing the beam is \(df\), then its contribution to the elastic cross section is
\[ d\sigma_d=df\,|1-a|^2, \tag{15} \]
and to the inelastic cross section:
\[ d\sigma_a=df(1-|a|^2). \tag{16} \]
The total cross sections are obtained from this by integration over the entire obstacle placed in the path of the wave:
\[ \sigma_d=\int df\,|1-a(s)|^2;\qquad \sigma_a=\int df(1-|a(s)|^2). \tag{17} \]
Here any process in which the incident nucleon interacts with one of the nucleons of the nucleus is regarded as inelastic, while the elastic process is the diffraction of the nucleon by the obstacle.
If the obstacle is a nucleus of radius \(R\), then \(df=2\pi\rho\,d\rho\), \(s=2\sqrt{R^2-\rho^2}\), and the integration in (17) is performed from \(0\) to \(R\). We obtain\(^{94}\):
\[ \sigma_a=\pi R^2\left\{1-\frac{1-(1+2KR)e^{-2KR}}{2K^2R^2}\right\} \tag{18} \]
and
\[ \sigma_d=\pi R^2\left[1-\frac{1-(1+2KR)e^{-2KR}}{2K^2R^2} -\frac{1}{\left(\frac{K^2}{4}+k_1^2\right)^2R^2}\times \right. \]
\[ \left. \times\left\{\left(\frac{K^2}{4}-k_1^2\right) +e^{-KR}\left[2k_1R\left(\frac{K^2}{4}+k_1^2\right)+k_1K\right]\sin 2k_1R-\right. \right. \]
\[ \left. \left. -e^{-KR}\left[\left(\frac{K^2}{4}-k_1^2\right)+KR\left(\frac{K^2}{4}+k_1^2\right)\right]\cos 2k_1R \right\}\right], \tag{19} \]
where \(k_1=k(n-1)\). In a completely analogous way, \({}^{94}\) also yields the formula for the angular distribution of elastically scattered nucleons:
\[ \left. \begin{aligned} d\sigma_a(\vartheta)&=|f(\vartheta)|^2\,d\Omega;\\ f(\vartheta)&=k\int_0^R(1-a)J_0(k\rho\sin\vartheta)\,\rho\,d\rho, \end{aligned} \right\} \tag{20} \]
in which the evaluation of the integral does not lead to simple results. The formulas given above were obtained in an approximation analogous to that usually used in solving optical diffraction problems. An exact numerical calculation carried out for the case of scattering of \(90\)-MeV neutrons by Al showed that this approximation may lead to errors of \(\sim 10\text{–}20\%\), which hardly exceed the errors of the quasi-macroscopic treatment.
The absorption coefficient \(K\) is determined in the classical treatment of the motion of a nucleon in a homogeneous medium of “nuclear matter”:
\[ K=\rho\,\frac{Z\sigma'_{np}+(A-Z)\sigma'_{nn}}{A}, \tag{21} \]
where \(\rho=\dfrac{3A}{4\pi R^3}\) is the density of nucleons in nuclear matter. The cross sections for nucleon—nuclear-nucleon scattering (\(\sigma'_{np}\) and \(\sigma'_{nn}\)) are reduced (for \(90\) MeV by \(15\text{–}30\%\)) in comparison with the scattering cross sections of free nucleons (\(\sigma_{np}\) and \(\sigma_{nn}\)) because of the influence of the Pauli principle, which restricts the possible states after scattering. The magnitude of this effect should decrease as the energy increases.
To find the index of refraction \(n\), in \({}^{94}\) a model of the nucleus was used in the form of a Fermi gas enclosed in a potential well of depth \(V=E_f+S\) MeV—the sum of the energy of the Fermi sphere and the binding energy. Under this assumption, for \(n\) the value
\[ n=\sqrt{1+\frac{V}{E}} \tag{22} \]
was obtained
(\(E\) is the energy of the incident nucleon).
It should be emphasized that the identification, contained in such a procedure, of the averaged potentials acting on a nucleon bound in the nucleus and on an incident high-energy nucleon is a far-reaching assumption, whose validity is by no means immediately obvious. This prompted some authors \({}^{93}\) to abandon its use and to regard \(V\) as a certain phenomenological parameter, which can be adjusted to achieve agreement with experiment. Recently the expression \({}^{114}\)
\[ n=1+\frac{\pi}{k^2}\frac{\rho}{A}\left(Zf_{np}(0)+(A-Z)f_{nn}(0)\right), \tag{23} \]
directly relating \(n\) to the amplitudes of np- and nn-scattering forward \(\bigl(f_{\mathrm{np}}(0)\) and \(f_{\mathrm{nn}}(0)\bigr)\), which is of undoubted interest.
The authors were able to use only one series of measurements of nuclear cross sections for neutrons with energy \(84\) MeV\(^{91}\); therefore they finally settled on the following values of the constants entering (18) and (19):
\(K=2.2\cdot 10^{12}\ \text{cm}^{-1}\), \(k_1=3.3\cdot 10^{12}\ \text{cm}^{-1}\), \(R=1.37\cdot 10^{-13} A^{1/3}\ \text{cm}^{10}\), which led to direct proportionality between \(R\) and \(A^{1/3}\). The chosen quantities agreed reasonably well with the values of the potential in the nucleus and of the absorption coefficient estimated from other considerations.
Further series of measurements at close neutron energies\(^{93,37}\) required that \(n\) be somewhat reduced and \(K\) increased; however, the new values still corresponded to their original physical meaning. Experiments at high energies served as an actual test of the theory developed in \(^{94}\). In accordance with the physical meaning of the theory, the potential \(V\) should not have depended on energy at all, while the dependence of \(K\) on energy should have been determined by the energy behavior of the np- and nn-cross sections.
Experiments carried out at an energy of \(156\) MeV\(^{92}\) required a decrease of the value of \(V\) at least to \(6\) MeV, which is in no way consistent with the values required at lower energies and grossly contradicts the usual notions. The situation turned out to be still worse at an energy of \(280\) MeV\(^{18,22}\), when it was necessary to set \(V=0\), and nevertheless the theoretical values of the cross sections, especially for heavy nuclei, proved to be larger than the observed ones. Thus, the theory proposed in \(^{94}\) turned out to be incapable of explaining the energy dependence of nuclear cross sections. However, it is still not clear whether this failure is a consequence of the defectiveness of the basic physical assumption about the possibility of a quasimagroscopic description of the nucleus, or of insufficient work on the less fundamental assumptions of the theory.
Let us now consider what happens to a fast nucleon and to a nucleus after an inelastic collision. The main idea of the treatment here consists in dividing all processes into two phases.
In the first phase, while the primary neutron and the recoil nucleons are still sufficiently fast, one may apply the classical description and assume that the interaction with the nucleus of a nucleon moving through it reduces to pair collisions with individual nucleons of the nucleus. As a result of one or several such collisions, the primary nucleon (or recoil nucleon) may either reach the surface of the nucleus and leave it, or expend all its energy and remain in the nucleus. The duration of this phase does not exceed \(10^{-22}\) sec. The theory must then determine: a) the number of nucleons leaving the nucleus after a small number of collisions (i.e., with a noticeable fraction of the energy), b) their energy distribution, c) their angular distribution, and d) the energy transferred to the nucleus as a whole in the form of kinetic energy of slow nucleons. We shall call
nucleons leaving the nucleus during the first phase, knocked out by the nucleons.
In the second, incomparably longer phase, which will not be considered in detail here, a compound nucleus is formed, strongly excited by the energy of the nucleons that have become stuck in the nucleus during the first phase, and emitting particles by the usual evaporation mechanism.
The properties of the knocked-out nucleons were considered in greatest detail in work 111, in which the case of 86.6 MeV neutrons incident on a lead nucleus was calculated.
The main results of this work are discussed in review 34, and we shall not dwell on them here.
Experience has not confirmed either the quantitative predictions or a number of qualitative predictions of these calculations. First, the predicted dip in the angular distribution of knock-out nucleons at angles close to zero was not found experimentally. This is apparently due to the fact that in 111 the refraction of the \(\psi\)-wave of the incident nucleon at the boundary of the nucleus was not taken into account, which leads to a concentration of particles in the region of small angles.
Second, it turned out that many deuterons and even tritons are emitted from nuclei, especially light ones, which was completely unexpected from the point of view of the ideas developed in 111. The physical essence of the theoretical explanation of this phenomenon proposed in 112, 113 is that if the incident neutron undergoes a collision with one of the protons present in the nucleus, and in such a way that after this collision both the neutron and the recoil proton fly out of the nucleus without further collisions, then they will have a certain probability (the greater, the smaller the momentum of their relative motion) of being in a bound state, i.e., of forming a deuteron. The role of the nucleus is then reduced only to carrying away the excess energy*), and to the fact that the protons captured are not at rest, but possess a certain momentum distribution.
The momentum distribution of protons in the target nucleus determines the form of the angular distribution of the deuterons. At present, the only method for theoretically finding this distribution is the application of the Fermi-gas model. However, even if one admits the possibility of applying to the nucleus a one-body model with a certain self-consistent potential, there are no grounds
*) We note that a deuteron cannot be formed in collisions of neutrons with free protons, since this would contradict the conservation laws. Indeed, upon formation of a deuteron the energy of the neutron–proton system must sharply decrease, while the momentum must remain constant. In the capture of free protons this excess energy can be carried away only by simultaneous emission of a quantum, which at high energies is extremely improbable because of the short collision time.
SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS
one should expect that the radial dependence of such a potential must necessarily have the form of a rectangular potential well, as is assumed in the Fermi-gas model. Motion in a rectangular potential well differs from motion in a potential field with any other radial dependence precisely in that states with a definite energy have in this case a very small (in the case of an infinitely deep well—zero) spread in momenta (the eigenfunctions in a rectangular well are standing waves). Therefore the authors\(^{113}\) preferred not to determine the distribution of nucleons in the nucleus over momenta from theoretical considerations, but to obtain it from the experimental angular distribution of deuterons. The results obtained from experiments\(^{95}\) for \(C^{13}\) are shown in Fig. 40; moreover, it is not excluded that the distribution obtained is somewhat overestimated in the region of high momenta.
Fig. 40. Distribution of protons in the \(C^{12}\) nucleus by momenta.
Comparing the energy distribution of secondary deuterons obtained in the experiment\(^{95}\) for \(C^{13}\) (see Fig. 39) with the theoretical one, the authors\(^{112}\) come to the conclusion that in all cases nearly the same amount of energy was transferred to the carbon nucleus. On the other hand, the energy transferred to the nucleus must correspond to the proton separation energy. Thus it turns out that the separation energy of any proton from \(C^{13}\) is approximately the same.
In the work\(^{107}\), calculations were carried out by the same method as in\(^{111}\) for the case of incidence of nucleons with energy 350–400 MeV on a nucleus with \(A=100\) (this value was chosen as an average between Ag and Br in order to make it possible to compare the results with experiments in photographic plates). For the calculation it was assumed that the nuclear radii are given by the formula \(R=1.4\cdot 10^{-13} A^{1/3}\) cm (for simplicity of calculation the authors used a two-dimensional model), the pp- and np-scattering cross sections were taken from experiment, and the nn-scattering cross section was taken equal to the pp-scattering cross section. As the nuclear model, a Fermi gas in a well of depth 35 MeV with maximum kinetic energy 22 MeV was used. (The unusually large binding energy—13 MeV—was chosen as an average between the neutron binding energy and the binding energy plus the Coulomb barrier for the proton.)
The interaction with the nucleus of 90 nucleons was considered, of which 60 produced inelastic processes. For comparison with the phenomena observed in photographic plates, all emitted nucleons were divided into three groups: those corresponding to gray tracks in the emulsion (\(E>100\) MeV), to semiblack tracks (\(30\ \text{MeV}<E<100\ \text{MeV}\))
and black \((E<30\ \text{MeV})\) tracks. The results of the calculations are summarized as follows.
One half of the particles emitted by the nucleus are protons, and one half are neutrons. The average excitation energy of the remaining compound nucleus is \(50\ \text{MeV}\), and the largest encountered is \(200\ \text{MeV}\). The distribution found for protons by energy and by the number of simultaneously emitted protons (the number of prongs in a star) is given in Table XI in comparison with the experimental data. In calculating the number of evaporation protons it was assumed that their average energy is \(35\ \text{MeV}\). Comparison with experiment indicates, apparently, a certain overestimate of the calculated energy of the emitted particles.
Table XI
Distribution of knocked-out protons by energy
| Gray tracks \((>100\ \text{MeV})\) | Semi-black tracks \((30—100\ \text{MeV})\) | Black tracks (knock-out) | Black tracks (evaporation) | All black tracks | |
|---|---|---|---|---|---|
| Calculated . . . | \(0,6 \pm 0,12\) | \(0,42 \pm 0,1\) | \(0,58 \pm 0,12\) | \(1,5 \pm 0,2\) | \(2,1 \pm 0,4\) |
| Measured . . . | \(0,42 \pm 0,4\) | \(0,35 \pm 0,4\) | — | — | \(2,5 \pm 0,2\) |
Distribution of proton stars by number of prongs
| Gray tracks \((>100\ \text{MeV})\) | Gray tracks \((>100\ \text{MeV})\) | Gray tracks \((>100\ \text{MeV})\) | Semi-black tracks \((30—100\ \text{MeV})\) | Semi-black tracks \((30—100\ \text{MeV})\) | Semi-black tracks \((30—100\ \text{MeV})\) |
|---|---|---|---|---|---|
| number of prongs | % of cases (calculated) | % of cases (observed) | number of prongs | % of cases (calculated) | % of cases (observed) |
| 0 | \(46 \pm 11\) | \(57 \pm 4\) | 0 | \(30 \pm 7\) | \(35 \pm 1\) |
| 1 | \(48 \pm 11\) | \(40 \pm 4\) | 1 | \(48 \pm 8\) | \(54 \pm 4\) |
| 2 | \(7 \pm 4\) | \(2,5 \pm 1\) | 2 | \(21 \pm 7\) | \(9 \pm 2\) |
| 3 | \(3 \pm 1,5\) | \(1,7 \pm 0,7\) |
It should be noted, however, that despite the generally satisfactory agreement of the experiments with the calculations of \({}^{107}\), there is still no theory that explains with any degree of reliability the phenomena of knock-out of secondary particles. Nor is there yet such a theory for explaining the absolute values and the energy dependence of the total and component \((\sigma_a\) and \(\sigma_d)\) cross sections for the interaction of high-energy nucleons with complex nuclei.
Any formulas for deriving such cross sections will inevitably have a semi-empirical character until a complete theoretical explanation is given of the picture of np-, pp-, and nn-scattering of high-energy nucleons.
VIII. INTERACTION WITH NUCLEI AND ABSORPTION OF NUCLEONS CONSTITUTING COSMIC RAYS
The study of cosmic rays makes it possible to obtain information about nuclear processes occurring at energies considerably exceeding those attainable by means of accelerators. In particular, effective cross sections for interaction with nuclei have been measured for particles with energies up to tens of Bev. Naturally, the character of the processes of interaction with nuclei of particles of such energies differs substantially from the nuclear processes occurring at energies of several hundred Mev.
Accordingly, the methods for recording nuclear interactions and measuring effective cross sections for various regions of nucleon energies are also different.
Thus, for example, an experiment on measuring the absorption of particles of relativistic energies, analogous to the experiment on absorption of artificially accelerated particles carried out in “poor geometry,” determines not the mean free path for interaction with nuclei, but a different quantity—the path length for absorption.
In the interaction with nuclei of cosmic-ray nucleons with energies of the order of hundreds of Mev, naturally, the same processes take place as for nucleons of equal energies obtained with accelerators, i.e. the formation of stars (which has been studied in greatest detail), as well as elastic scattering and charge exchange. With increasing nucleon energy the role of meson-inelastic collisions grows; these become significant for nucleon energies \(> 1\) Bev*).
At nucleon energies of several Bev and higher, interaction with nuclei in most cases leads to the formation of electron-nuclear showers.
Electron-nuclear showers, in the discovery and investigation of which the works of Soviet physicists played a very important role, are high-energy nuclear processes in which groups of mesons, both charged and neutral, are produced**).
The composition of electron-nuclear showers is very complex. Besides \(\pi^{+}\)-, \(\pi^{-}\)-, and \(\pi^{0}\)-mesons, they include knock-on nucleons (designated in the cosmic-ray literature as \(\delta\)-nucleons) and nucleons, as well as
*) On the generation of mesons by means of accelerators, see the review \(^{115}\).
**) In the foreign literature there is no established term for designating electron-nuclear showers. The names “penetrating shower,” “mixed shower,” etc., are used.
heavier particles formed as a result of evaporation from the nucleus excited by the collision.
$\pi^0$-mesons, in decaying, produce photons, which give rise to electron–photon cascades. Among the electron–nuclear showers, $V$-particles have been observed, and it should apparently be assumed that the formation of other unstable particles discovered recently is also connected with electron–nuclear showers.
Nucleons and $\pi$-mesons that arise in electron–nuclear showers can interact with nuclei (they are nuclear-active particles). At high energies they are in turn capable of again producing electron–nuclear showers. Thus there arises a nuclear-cascade process, discovered in the work of Soviet physicists. For more detail on electron–nuclear showers and the nuclear-cascade process, see $^{116}$, where a bibliography of numerous works by Soviet physicists on this question up to 1949 is given, and $^{117}$.
For the energies at which electron–nuclear showers are produced, neither charge exchange $^{118}$ nor elastic scattering is observed.
The absence of observed charge exchange is connected with the fact that, in the process of charge exchange between a proton and a neutron, a considerable amount of energy is released, leading to the formation of a shower, one of whose particles is the charge-exchange nucleon.
Elastic scattering of nucleons in this energy region cannot be detected by the methods employed, owing to the extremely small magnitude of the angles of such scattering.
The composition of the particles producing nuclear interactions in cosmic rays differs in different energy regions. Moreover, in many cases it proves impossible to separate out results pertaining only to nucleons from data obtained for the totality of nuclear-active particles, i.e. nucleons and $\pi$-mesons.
The energies of the nuclear-active particles of cosmic rays (or else the energy released in nuclear interactions) are estimated by various methods.
In thick photographic emulsions the energy of the charged particles constituting a star can be determined directly. It is usually assumed that the energy of the neutrons in a star is $1.25$ times the energy of the protons. In a Wilson chamber one can estimate the energies of the electron–photon cascades of electron–nuclear showers, and also, with the aid of a magnetic field, determine the momenta of penetrating particles. In work with counters and with ionization chambers, the latitude effect is used to estimate the energy, and also the determination of the number of shower particles (chiefly electrons).
The energy magnitude of nuclear-active particles was determined directly from the magnitude of multiple Coulomb scattering in emulsions in $^{119,120}$. In the work $^{119}$, from which we borrow Figs. 41—43, a systematic investigation was made of the change in the character of nuclear interactions with increasing nucleon energy.
Figure 41 shows the change in the probability of the formation by a proton of a star without “shower” particles as a function of the energy of the proton generating the star.
“Shower” particles are what the authors call particles producing a track with ionization \(<1.4 I_{\min}\) (grain density \(<16\) per \(50\mu\) of emulsion). These are \(\pi\)-mesons with energies \(>80\) MeV and, in smaller numbers, protons with energies \(>500\) MeV.
Along the abscissa axis (at the top) in the figure is plotted the total proton energy \(\gamma_p\), expressed in units of the proton \(Mc^2\). Along the ordinate axis is plotted the fraction of stars of type \(O_p^{*)}\) relative to the total number of stars. The number of “shower” particles in the first approximation corresponds to the total number of generated mesons, since the presence of protons among the “shower” particles to some extent compensates for the presence of mesons with energies below 80 MeV. Therefore the obtained experimental values of the fraction of stars of type \(O_p\) can be compared with the theoretical curve for the probability of inelastic collision of a proton without meson generation. In the figure, the corresponding curve, calculated on the basis of Fermi’s theory, is drawn as a solid line.
At a kinetic energy \(E_p=1.6\) Bev, one half of all collisions are meson-inelastic.
Fig. 41. Dependence of the probability of formation by a proton of stars without “shower” particles on the proton energy.
In Fig. 42 is shown the dependence of the average number of tracks of different ionization on the kinetic energy of the particles that formed the star. \(N_S\) is the number of “shower” particles, \(N_G\) is the number of “gray” tracks (grain density 16–80 per \(50\mu\) of emulsion). These are mainly protons with energies 25–500 MeV, and also \(\pi\)-mesons with energies below 80 MeV and, in part, deuterons and tritons. Since the energy of protons formed by “evaporation” rarely reaches 25 MeV, the “gray” tracks for the most part belong to \(\delta\)-protons. “Black” tracks (grain density \(>80\) per \(50\mu\)) are produced by protons with energies below 25 MeV, deuterons and protons with energies below, respectively, 50 MeV and 75 MeV, and \(\alpha\)-particles with energies below 800 MeV. \(N_H\) denotes the sum of “gray” and “black” tracks.
In the figure, the data relating to stars from protons, from \(\pi\)-mesons, and from \((\pi,p)\)-particles are indicated separately. The authors apply the last designation to charged particles of such high energy that it was impossible to distinguish protons and \(\pi\)-mesons (by the magnitude of Coulomb scattering and ionization).
\(*\) The number in the designation of the star type (in the present case 0) denotes the number of “shower” particles; the indices \(p\) or \(n\) indicate whether the star was produced by a charged or a neutral particle.
The quantities \(\overline{N}_G\) and \(\overline{N}_H\) fall, respectively, on a single curve for stars produced by protons and by \(\pi\)-mesons, i.e. the number of “gray” and “black” tracks is determined only by the kinetic energy of the particle interacting with the nucleus and does not depend on its nature. It is possible that this is also true with respect to the number of “shower” tracks, but, unfortunately, the data concerning the value \(\overline{N}_S\) in the region where it was possible to establish the nature of the generating particle have very low statistical accuracy.
The total energy of the particles of a star forming “gray” and “black” tracks (as well as of neutrons of the same energies) can be estimated with the aid of the empirical relation
\[ E_{N_H}=155\,N_H-100\ \text{Mev} \]
(\(N_H>1\)), valid for all types of stars except \(O_{\mathrm p}\). With the inclusion of stars of this type, for small \(N_h\) the relation
\[ E_{N_h}=37N_H+4N_H^2 \]
holds.
Fig. 42. Dependence of the mean number of tracks of different ionization on the kinetic energy of particles producing a star (\(\bullet\)—for stars from protons, \(\delta\)—for stars from \(\pi\)-mesons, \(\circ\)—for stars from \((\pi,p)\)-particles).
In \({}^{120}\) the energy of the particles generating the star (up to \(680\ \text{Mev}\)) and the energy of the particles composing the stars were measured independently. Taking into account the energy carried away by neutrons, both quantities practically coincide. This, in particular, shows that the formation of \(\pi\)-mesons at these energies is a rare process.
The mean number of shower particles increases with increasing energy; moreover, as is seen from Fig. 42, in the interval \(10^9\)—\(10^{10}\ \text{ev}\) the change in \(\overline{N}_S\) is approximately proportional to the change in energy, while at higher energies the growth of \(\overline{N}_S\) becomes slower.
In the region above \(10\ \text{Bev}\) the energy of individual nuclear-active particles could not be directly measured, and therefore dependences similar to those shown in Fig. 42 (and also in Fig. 43—see below) could not be studied.
One may note the results obtained with the Wilson chamber \({}^{121}\): at energies of electron-nuclear showers \(\sim 3\ \text{Bev}\) the mean number of penetrating particles is \(7 \pm 1\), and at an energy \(\sim 6\ \text{Bev}\) it is \(11 \pm 2\). In comparing these numbers with the data of Fig. 42, one should keep in mind a certain difference between the criteria used to determine shower particles in emulsion and penetrating particles in the Wilson chamber.
However, the number of “shower” particles cannot serve as a measure of the energy of the particle that caused the given interaction. Indeed, Fig. 43, where
given the distribution of \(N_S\) in stars formed by protons and \((\pi, p)\)-particles of known energy, shows that generating particles of a given energy can produce any number of “shower” particles from 0 to \(N_{S\max}\).
Thus, a star with a given \(N_S\) can be generated by a particle of any energy above some minimum, and only the average number of penetrating particles \(\overline{N}_S\) determines the energy of the generating particles.
Fig. 43. Distribution of the number of “shower” tracks in stars formed by protons and \((\pi, p)\)-particles of different energies.
Practically every apparatus that records nuclear interactions selects a very broad and, moreover, not always quite well-defined interval of energies of the generating particles. The lower boundary of this interval is determined by the presence of a certain sensitivity threshold of the apparatus (for example, the requirement that discharges coincide in three counters placed inside a lead block), and an exact determination of the energy corresponding to this threshold usually presents considerable difficulties. From above, the selected energy interval is limited by the steep decline of the spectrum of the recorded particles, or else is formed by excluding processes corresponding to a higher threshold.
Let us now proceed to the consideration of ranges for interaction and for absorption of nucleons.
The mean free path for interaction \(\lambda_{\mathrm{v}}\) is defined as the average distance traversed by a particle in a substance before colliding with one of the nuclei. In accordance with this, the ratio of the number of nuclear-active particles that have passed without interaction through a layer of substance of thickness \(d\ \mathrm{g/cm}^{2}\) to the number of particles incident on this layer is equal to
\[ \frac{N_1}{N_0}=e^{-\frac{d}{\lambda_{\mathrm{v}}}}. \]
The path length for absorption \(\lambda_{\mathrm{p}}\) determines the decrease in the number of nuclear-active particles when passing through matter. If \(\Phi(\varepsilon,x)\) is the number of nuclear-active particles at a depth \(x\ \mathrm{g/cm}^{2}\) with energy above \(\varepsilon\) (where \(\varepsilon\) may mean, in particular, the threshold energy of the apparatus), then
\[ d\Phi(\varepsilon,x)=-\frac{1}{\lambda_{\mathrm{p}}}\Phi(\varepsilon,x)\,dx. \]
Since the absorption of the nuclear-active component occurs practically exponentially, the path length for absorption may be defined as the thickness of the layer of matter in which the number of nuclear-active particles decreases by a factor of \(e\). If \(N_2\) is the number of nuclear-active particles beneath a layer of matter \(d\ \mathrm{g/cm}^{2}\), then
\[ \frac{N_2}{N_0}=e^{-\frac{d}{\lambda_{\mathrm{p}}}}. \]
If in each nuclear interaction a nuclear-active particle disappears and does not form other nuclear-active particles, then \(\lambda_{\mathrm{p}}=\lambda_{\mathrm{v}}\). If, however, the loss of energy of the nuclear-active particle in an interaction is incomplete (energy above the threshold energy necessary for registering the particle is retained), or new nuclear-active particles are formed (a nuclear-cascade process takes place), then \(\lambda_{\mathrm{p}}>\lambda_{\mathrm{v}}\). The magnitude of the path length for absorption depends on the magnitude of the path length for interaction, on the spectrum of nuclear-active particles, on the character of their interaction with nuclei, and on the threshold energy of the recording apparatus.
Between the quantities \(\lambda_{\mathrm{v}}\) and \(\lambda_{\mathrm{p}}\) there exists the following relation, valid under quite general assumptions\({}^{136}\):
\[ \overline{\lambda}_{\mathrm{v}}=\overline{\lambda}_{\mathrm{p}}(1-\overline{s}), \]
where \(\overline{s}\) is the average number of nuclear-active particles with energy above \(\varepsilon\) produced in one act of interaction. The averaging of the quantities \(\lambda_{\mathrm{v}}\), \(\lambda_{\mathrm{p}}\), and \(s\) must be carried out over the entire thickness of the absorber and over the energy spectrum of the nuclear-active particles.
In this relation \(s\) denotes the effective number of secondary nuclear-active particles, i.e. the number of particles produced multiplied by the ratio of the effective cross sections of secondary and primary particles for generating nuclear processes of the type to which the quantities \(\lambda_{\mathrm{p}}\) and \(\lambda_{\mathrm{v}}\) refer (for example, electron-nuclear showers with energy above a certain value).
Data on the magnitude of the mean path lengths may conveniently be divided into three groups, determined mainly by different methods of recording nuclear interactions, especially since each of these methods corresponds to its own region of effectively registered energies.
nuclear-active particles. Of course, such a division is to a considerable extent conventional, since the boundaries between the energy regions are not sharply defined and these regions partially overlap.
1. Formation of electron-nuclear showers
Registration is carried out by means of counters, controlled Wilson chambers, photoemulsions sensitive to relativistic particles, and by tracks in ionization chambers (by coincidences of tracks with counter discharges or by tracks in high-pressure chambers). The energy interval of nuclear-active particles is from several Bev and higher, in most of the setups used up to tens of Bev.
The flux of generating particles with energies up to several tens of Bev in air consists mainly of protons and neutrons, the number of protons being of the same order as the number of neutrons.
The measured ratio of the number of neutral generating particles to the number of charged ones in 122 is \(0.84 \pm 0.13\), and in 123 \(0.87 \pm 0.2\). At higher energies, at which the decay path of \(\pi\)-mesons becomes comparable with the interaction path in air, there should be a significant fraction of \(\pi\)-mesons among the nuclear-active particles in air. The role of \(\pi\)-mesons is also substantial in the formation of electron-nuclear showers under layers of dense substances whose thickness is not small compared with \(\lambda_{\mathrm{B}}\).
In 124, where events of higher energy were recorded (according to the authors’ estimate \(> 15\) Bev), ionizing generating particles constituted 83.5%, non-ionizing ones 4%, and in 12.5% of cases the character of the generating particles could not be established. This result is in agreement with the idea that the fraction of \(\pi\)-mesons increases as the energy of nuclear-active particles increases.
For particles generating electron-nuclear showers, both the values of the mean free paths for interaction and the values of the mean free paths for absorption were measured. Let us first consider the methods and results of measuring the mean free path for interaction.
By means of a Wilson chamber controlled by counters, the value \(\lambda_{\mathrm{B}}\) was determined from the distribution of the points of formation of electron-nuclear showers over the plates inside the chamber. With the aid of counter systems, \(\lambda_{\mathrm{B}}\) was measured by two methods.
The first method consists in measuring the number of particles generating electron-nuclear showers that pass through a layer of absorber without interaction. This method may be applied to single ionizing generating particles or to neutral particles.
The scheme of one of the simplest arrangements of this type 125, in which the value \(\lambda_{\mathrm{B}}\) was measured for single ionizing particles, is shown in Fig. 44.
For different thicknesses of absorbers \(\Sigma\) and \(\Sigma'\), the number of coincidences \(A + B + C + D + 1E\) was recorded, signifying the operation of no fewer
one of the counters in groups \(A, B, C\), and \(D\), and only one counter in row \(E\). Electron-nuclear showers registered by coincidences of counters \(B, C\), and \(D\) arise in the lead block \(P\), for if they are produced in the absorber, then necessarily more than one counter \(E\) will operate. If a particle generating an electron-nuclear shower in \(P\), registered by coincidences of counters \(B, C\), and \(D\), also interacted inside \(\Sigma\) or \(\Sigma'\), then particles produced in this interaction will emerge from the absorbers, and more than one counter will operate in row \(E\). Thus, coincidences \(A+B+C+D+1E\) correspond to the passage of single ionizing particles that generate, without interaction, through the absorbers \(\Sigma\) and \(\Sigma'\), and \(\lambda_{\mathrm{B}}\) can be determined from the relation
\[ \frac{N_1}{N_0}=e^{-\frac{d}{\lambda_{\mathrm{B}}}}, \]
With the use of this method for measuring \(\lambda_{\mathrm{B}}\) of neutral generating particles (neutrons), the condition is imposed that no discharge occur in any of the counters surrounding the substance block in which electron-nuclear showers are generated \(^{122,126,127}\). A similar method was applied in \(^{128}\).
Fig. 44. Diagram of one of the arrangements for measuring the mean free path of particles generating electron-nuclear showers.
Despite the apparent simplicity of the scheme of experiments for measuring the value \(\lambda_{\mathrm{B}}\), in reality, in order to obtain accurate values it is necessary to take into account a whole series of corrections and carefully analyze the experimental conditions and the directly obtained experimental data.
Second method—by the transition curve for the generation of electron-nuclear showers. Above an arrangement registering electron-nuclear showers, a block of material is placed, and the curve of the dependence of the number of registered showers on the thickness of the block is recorded.
The results of measurements of \(\lambda_{\mathrm{B}}\) are given in Table XII.
As is evident from the table (in particular from work \(^{133}\)), the measured values of \(\lambda_{\mathrm{B}}\) are the same both for neutral nuclear-active particles and for charged ones. Before turning to a discussion of the values presented in Table XII, it is necessary to note that in the methods indicated above for measuring \(\lambda_{\mathrm{B}}\) for particles generating electron-nuclear showers, there are a number of methodological shortcomings.
The most essential of them is connected with the following circumstance: in a certain fraction of collisions of nuclear-active particles of very high energy with nuclei, “weak” interactions may occur, as a result of which neither charged penetrating particles nor neutral \(\pi\)-mesons arise.
Apparently, the probability of such “weak” interactions in light nuclei is greater than in heavy ones.
SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS
Table XII
Values of the mean free paths for the interaction of particles generating electron-nuclear showers
| Method of measurement and work | Substance | \(\lambda_{\mathrm{v}}\) (in \(g\,cm^{-2}\)) | Note |
|---|---|---|---|
| A. Counters, according to the number of noninteracting particles | |||
| a) Ionizing particles | |||
| [125] | Pb | \(160\pm15\) | |
| [125] | Fe | \(135\pm15\) | |
| [125] | C | \(100\pm5\) | |
| [122] | Pb | \(157\pm12\) | |
| [122] | C | \(82\pm8\) | |
| [123] | Pb | \(147\pm10\) | Number of shower particles (counters triggered) \(>7\) |
| [123] | \(180\pm10\) | Number of shower particles (counters triggered) \(4–5\) | |
| [129] | Pb | \(162\pm10\) | Range of shower particles \(>200\ g\,cm^{-2}\) |
| [129] | \(196\pm13\) | Range of shower particles \(>100\ g\,cm^{-2}\) | |
| b) Nonionizing particles | |||
| [122] | Pb | \(164\pm15\) | |
| [122] | C | \(80\pm7\) | |
| [126] | Paraffin | \(61\pm6\) | |
| [128] | Pb | 130 | |
| [127] | Pb | 165–215 | Depending on the threshold for selecting showers (\(\lambda_{\mathrm{v}}\) decreases with more “severe” selection) |
| [127] | C | 85–105 | Depending on the threshold for selecting showers (\(\lambda_{\mathrm{v}}\) decreases with more “severe” selection) |
| B. Counters, by transition effect | |||
| [130] | Pb | \(180\pm40\) | |
| [130] | Al | \(85\pm15\) | |
| V. Wilson chamber | |||
| [124] | Au | \(145\pm3\) |
Only strongly absorbing products of nuclear disintegrations escape from the nucleus. For the registration of such weak interactions, however, counters are not very effective, since the particles formed in this case are for the most part absorbed inside the same absorber in which they arise. As a result, measurement of \(\lambda_{\mathrm{v}}\) always gives only an upper limit of the measured quantity, or a lower limit for the effective interaction cross section. This applies to all three measurement methods described.
For the first of the measurement methods considered, the determination of \(\lambda_{\mathrm{v}}\) by means of counters, which is the most commonly used, the influence of the indicated
the effect is aggravated by the event-selection system used in these measurements. Every detector of electron–nuclear showers (in Fig. 44, the block \(P\) with counters \(B, C\), and \(D\) immersed in it) has a certain sensitivity threshold. The presence of this threshold and the steep fall of the spectrum of the generating particles lead to the fact that, in interactions inside the absorber, the particles generating the electron–nuclear shower recorded in the detector usually release less energy than nuclear-active particles of the same energy should lose on average.
These very substantial circumstances (as well as some others) were not taken into account in the cited works, which led to a certain overestimate of the measured values of \(\lambda_{\mathrm{B}}\) over the true ones.
It may be said that most of the apparatuses with which the results presented in Table XII were obtained did not directly measure the free path for interaction, but rather a quantity close to the free path for meson-inelastic collisions or for the generation of electron–nuclear showers.
Naturally, the greater the energy of the electron–nuclear showers selected by the apparatus, the greater the average energy of the generating particles and the smaller the fraction of the weak interactions, produced by them, that are difficult to record. In particular, in the first of the methods considered above for determining \(\lambda_{\mathrm{B}}\) by means of counters, with a harder selection of the recorded showers there should be fewer cases in which an interaction that occurred inside the absorber layer is not recorded by the counters situated below this layer. As a result, the value of \(\lambda_{\mathrm{B}}\) measured with a more “rigid” selection of electron–nuclear showers will be smaller (and closer to the true one) than that measured with a “softer” selection of showers. It is by this essentially instrumental effect that one should explain the dependence of \(\lambda_{\mathrm{B}}\) on the energy of the selected electron–nuclear showers found in \({}^{133,137}\) and \({}^{139}\). The interpretation given by the authors of these works, namely of the indicated result as a dependence of the effective cross section on the energy of nuclear-active particles, is, in our opinion, unjustified.
The geometrical dimensions of the nucleus, determined by the range of nuclear forces, correspond to the radius \(R = r_0 A^{1/3}\), where \(r_0 = \dfrac{\hbar}{m_\pi c}\) (\(m_\pi\) is the mass of the \(\pi\)-meson). The geometrical cross section \(\sigma = \pi R^2\) is proportional to \(A^{2/3}\). The corresponding mean free path is, in lead, \(\sim 160\ \mathrm{g\,cm^{-2}}\), and in carbon \(\sim 60\ \mathrm{g\,cm^{-2}}\).
The measured values of \(\lambda_{\mathrm{B}}\) are close to the ranges corresponding to the geometrical dimensions of nuclei.
Let us compare the data of Table XII with measurements performed at accelerators (Tables VII and VIII).
As is seen, the cross section for inelastic collisions increases approximately twofold in passing from the energies obtained at accelerators to the energies of generation of electron–nuclear showers. At the same time
with increasing energy, the role of meson-inelastic collisions grows, and they become dominant in the processes of nuclear interaction at energies of several Bev.
It should be noted that the measured value of \(\lambda_{\mathrm{int}}\) for particles that generate electron–nuclear showers is very close to the ranges corresponding to the total effective cross sections at the highest energies reached in accelerators.
Let us now turn to consideration of the absorption of particles that generate electron–nuclear showers. For these particles the mean range for absorption \(\lambda_{\mathrm{a}}\) is substantially greater than the mean range for interaction \(\lambda_{\mathrm{int}}\), which is explained by the presence of a nuclear-cascade process and by the possibility for a single particle to undergo several successive interactions. Thus, the absorption of particles that generate electron–nuclear showers is a very complex process, fundamentally different from the absorption of particles obtained with accelerators. Since it is not possible, within the scope of the present article, to set forth all questions connected with this process, we shall give only a few results connected with the measurement of the quantity \(\lambda_{\mathrm{a}}\).
Numerous measurements of the growth in the number of electron–nuclear showers with increasing altitude of the observation site, carried out above all in the work of Soviet physicists (see \(^{116, 131, 132}\)), gave for the value \(\lambda_{\mathrm{a}}^{\mathrm{air}}\) the value \(120\text{—}125\ \mathrm{g\,cm^{-2}}\). Thus, the range for absorption in air is approximately twice as large as the range for interaction*).
When measuring the quantity \(\lambda_{\mathrm{a}}\) in dense substances, the result may be distorted by the presence of the so-called transition density effect, the discovery and detailed investigation of which belongs entirely to Soviet physicists \(^{116, 134, 135}\). Measurements of \(\lambda_{\mathrm{a}}\) under conditions in which the influence of the transition density effect was eliminated, carried out in the works of Soviet researchers \(^{136}\), as well as the few data on the absorption of electron–nuclear showers obtained by means of photographic emulsions, give the following basic results:
a) \(\lambda_{\mathrm{a}}\) in water and graphite is substantially greater than in air. This shows that in the composition of the component generating electron–nuclear showers there are decaying particles (apparently, \(\pi\)-mesons).
b) Absorption is approximately the same in layers of different substances equivalent in the number of values \(\lambda_{\mathrm{int}}\) contained in them. Thus, the absorption of the component generating electron–
*) The change in the number of electron–nuclear showers generated by charged particles in the very upper layers of the atmosphere corresponds to \(\lambda_{\mathrm{a}}^{\mathrm{air}} \sim 60\ \mathrm{g\,cm^{-2}}\) \(^{131,132}\). G. T. Zatsepin \(^{133}\) interpreted this result as a manifestation of the proton–neutron charge-exchange process.
cosmic-nuclear showers, i.e., the decrease in the number of nuclear-active particles, is determined not by the number of nucleons in the absorber layer, but, to a first approximation, by the number of interactions with nuclei. In this case the ratio $\dfrac{\lambda_n}{\lambda_{\mathrm{int}}}$ in all substances is $3$—$3.5$.
This means that the absorption of nuclear-active particles, together with the secondary nuclear-active particles produced by them, occurs on the average after approximately three interactions with nuclei.
And since the cross section for interaction with nuclei of the generating particles is close to the geometrical dimensions of the nucleus, then, when these particles pass through heavy nuclei, successive collisions with nucleons inside one and the same nucleus should occur. Nevertheless, the decrease in the number of nuclear-active particles in passing through a single nucleus is approximately the same for light and for heavy substances, despite the different number of collisions with nucleons inside light and heavy nuclei.
This very substantial experimental result may be explained either by the assumption that, at energies of the order of $10^{10}$ ev, the incident fast particle interacts with the nucleus as with a whole, and the results of the interaction depend little on the atomic number of the nucleus. Or else, if one proceeds from the idea that, in passing through a nucleus, nuclear-active particles may successively interact with various nucleons or groups of nucleons, the indicated phenomenon may be explained on the basis of the notion of the existence of a nuclear-cascade process inside the nucleus, proceeding with the participation of neutral $\pi$-mesons (and also, possibly, of other nuclear-active particles which decay with the formation of nuclear-passive particles over a path small or comparable with the mean free path for interaction).
2. Nuclear interactions of charged penetrating particles of electron-nuclear showers
Registration is carried out mainly by means of controlled Wilson chambers and photographic emulsions sensitive to relativistic particles.
$\lambda_{\mathrm{int}}$ is measured by determining the total length of the path traversed by the particles under study in plates inside a Wilson chamber or in emulsion, and the number of interactions produced by these particles. The energy interval of the particles covered in this way is very wide and is not in all cases sufficiently well defined. The “penetrating” particles of electron-nuclear showers observed in a Wilson chamber, or the “shower” particles in a photographic emulsion, consist of protons and $\pi$-mesons. At the same time the fraction of $\pi$-mesons, apparently, is the larger-
…than the high energies of electron–nuclear showers. In those cases where protons and \(\pi\)-mesons can be identified, i.e., for energies \(<10^9\) eV, the corresponding values of \(\lambda_{\mathrm{v}}\) agree within the limits of experimental errors.
The results of measurements of \(\lambda_{\mathrm{v}}\) are given in Table XIII.
Table XIII
Mean free path for the interaction of penetrating particles in electron–nuclear showers
| Work | Material | \(\lambda_{\mathrm{p}}\) | Note |
|---|---|---|---|
| A. In a Wilson chamber | |||
| [137] | Pb | \(200\pm50\ \mathrm{g\,cm^{-2}}\) | |
| [138] | » | \(200\pm80\) » | |
| [139] | » | \(300\pm100\) » | |
| [140] | » | \(316\pm70\) » | |
| [141] | » | \(172\pm30\) » | |
| Al | \(164\pm50\) » | ||
| [121] | Au | \(230\pm60\) » | |
| [142] | C | \(108\pm25\) » | |
| B. In photographic emulsion | |||
| [143] | Photographic emulsion | \(\dfrac{102\pm27\ \mathrm{g\,cm^{-2}}}{(26\pm7\ \mathrm{cm})}\) | Shower particles (protons \(>600\) MeV \(\pi\)-mesons \(>150\) MeV) |
| [143] | Photographic emulsion | \(\dfrac{82\pm35\ \mathrm{g\,cm^{-2}}}{(21\pm9\ \mathrm{cm})}\) | \(\pi\)-mesons \(150\)—\(1500\) MeV |
| [120] | Photographic emulsion | \(120\pm20\ \mathrm{g\,cm^{-2}}\) | Shower particles |
| [120] | Photographic emulsion | \(95\pm27\ \mathrm{g\,cm^{-2}}\) | Mesons \(80\)—\(1100\) MeV |
| [120] | Photographic emulsion | \(54\pm20\ \mathrm{cm}\) | Protons \(40\)—\(100\) MeV \((\bar E=60\ \mathrm{MeV})\ N_H>3\) |
| [120] | Photographic emulsion | a) \(51\pm20\ \mathrm{cm}\) | Protons \(100\)—\(500\) MeV \((\bar E=175\ \mathrm{MeV})\ N_H\geq3\) |
| [120] | Photographic emulsion | b) \(47\pm10\ \mathrm{cm}\) | Protons \(100\)—\(500\) MeV \((\bar E=195\ \mathrm{MeV})\ N_H\geq3\) |
| Data a) and b) refer to different methods of identifying protons. \(N_H\) is the number of tracks in secondary stars. |
Let us note that the mean free path corresponding to the geometrical cross section of a nucleus \(\sigma=\pi r_0^2 A^{2/3}\) is \(\sim 90\ \mathrm{g\,cm^{-2}}\), or \(23\ \mathrm{cm}\) of emulsion.
A considerable scatter of the data obtained by different authors with the aid of a Wilson chamber is explained by the different accounting for the incomplete efficiency of the apparatus in registering secondary nuclear interactions. As a result of these interactions, stars of low energies are often formed, the “rays” of which are absorbed inside the plates in which these stars arose; consequently, the observed number of interactions of penetrating particles proves to be smaller than the true one.
For comparison we note that in work \(^{144}\), where no correction for this circumstance was taken into account, the value \(\lambda_{\mathrm{B}} = 750\ \text{g cm}^{-2}\) Pb was obtained. From this point of view the nuclear-emulsion method is more reliable.
The data on the relative role of the processes of star formation and elastic scattering deserve attention.
Thus, the mean free path for anomalous scattering through an angle \(>5^\circ\) in graphite \(^{145}\) is \(800 \pm 220\ \text{g cm}^{-2}\) (the free path measured in this work for the formation of secondary disintegrations is \(237 \pm 29\ \text{g cm}^{-2}\), without correction). In \(^{124}\) it is indicated that nuclear scattering (through an angle \(>10^\circ\)) amounts to about \(1/4\) of the total number of secondary interactions.
In \(^{144}\), among 78 observed cases of the generation by penetrating particles of secondary showers and stars, 12 cases of anomalous scattering (through angles \(>15^\circ\)) were found.
Of 38 nuclear interactions of penetrating particles found in \(^{138}\), no fewer than 8 are cases of anomalous scattering (exceeding by no less than a factor of 5 the calculated value of Coulomb scattering).
The most detailed data on the ratio of the number of elastic scatterings and stars formed by shower particles are contained in \(^{137}\). In particular, it is stated that, out of 17 secondary interactions of shower particles in lead plates \(0.5\ \text{mm}\) thick, in one case scattering through an angle \(>10^\circ\) occurred, and in 16 cases stars were formed (including 14 stars with weakly ionizing particles).
It is necessary to point out a certain arbitrariness in the criterion adopted in the cited works for the process of elastic scattering.
In cases observed in a Wilson chamber as elastic scattering in a plate, there may apparently occur the formation of low-energy stars not emerging from the plates, which leads to an overestimation of the measured fraction of elastic-scattering processes relative to the process of star formation.
From comparison with the data of Tables VII and VIII it follows that, with increasing nucleon energy, the cross section for inelastic collisions increases, approaching the geometrical cross section of the nucleus. At what energies these quantities begin to coincide cannot yet be stated at present. In any case, at an energy \(\sim 1\ \text{Bev}\),
apparently, in practice this coincidence already takes place. At the same time, the fraction of recorded elastic-scattering processes falls, as should be expected, owing to the decrease in the magnitude of the scattering angles.
3. Star Formation
In conclusion, we shall briefly discuss the processes of star formation, recorded mainly by means of thick-layer photoemulsions, and also by tracks in ionization chambers and by neutrons detected with counters filled with BF$_3$.
The energy region covered is 100–1000 MeV. The largest number of cases, which chiefly determine the results, pertains to energies up to 400 MeV, i.e., to an interval that overlaps to a considerable extent with the interval of energies attainable in accelerators.
Stars without “shower” particles are formed for the most part by neutrons and, to a lesser extent, by protons, because the intensity of the proton flux is small in comparison with the neutron flux in the energy region for which the range of protons due to ionization stopping is less than the range for nuclear interaction. In the formation of stars with “shower” particles, protons and neutrons participate in approximately equal measure (their ratio depends on the altitude of the observation site and on the number of shower particles $N_S$).
The quantity $\lambda_{\mathrm{p}}$ was measured predominantly. Data on $\lambda_{\mathrm{n}}$ relating to this energy interval were given earlier, in Table XIII.
A considerable number of works is devoted to the measurement of $\lambda_{\mathrm{p}}$ in air, i.e., to the increase in the number of stars with altitude.
The measured values of $\lambda_{\mathrm{p}}$, despite considerable scatter, are grouped around the values $\lambda_{\mathrm{p}}^{\mathrm{air}} = 140$–$150\ \mathrm{g\,cm^{-2}}$.
Data on the value of $\lambda_{\mathrm{p}}$ measured in dense substances are given in Table XIV.
$\lambda_{\mathrm{p}}^{\mathrm{air}}$ is smaller than $\lambda_{\mathrm{p}}$ in dense substances of the same (even somewhat smaller) atomic weight (graphite and ice). This shows that the component generating stars in dense substances includes a certain number of unstable particles that decay in air. These particles are apparently $\pi$-mesons.
Nucleons with energies of several hundred MeV usually form only one star. Therefore the measured value of $\lambda_{\mathrm{p}}$ for the nucleons generating stars is close to the free path for star formation, i.e., to the free path for inelastic collisions $\lambda_a$.
Some difference may be caused only by nucleons of higher energies present in the spectrum of particles generating-
Table XIV
Mean absorption path in dense substances of the component generating stars (in g cm\(^{-2}\))
| Work | Pb | Al | C | Ice | Note |
|---|---|---|---|---|---|
| [146] | \(310 \pm 20\) | \(160 \pm 13\) | Stars \(\geqslant 3\) rays | ||
| [147] | \(300 \pm 20\) | 220 | Stars \(\geqslant 3\) rays | ||
| [148] | 320 | Stars \(\geqslant 3\) rays | |||
| [149] | \(166 \pm 7\) | Stars \(\geqslant 3\) rays | |||
| [150] | \(200 \pm 10\) | Stars \(\geqslant 3\) rays | |||
| [151] | \(170 \pm 10\) | Stars \(\geqslant 3\) rays | |||
| [152] | \(270 \pm 75\) | Stars \(> 3\) rays | |||
| [152] | \(320 \pm 60\) | Single strongly ionizing particles | |||
| [153] | \(305 \pm 7\) | All stars | |||
| [153] | \(405 \pm 31\) | Stars from ionizing particles | |||
| [153] | \(260 \pm 34\) | Stars from neutral particles |
…stars. Naturally, in the region of energies of cosmic-ray nucleons that overlaps with the energies of nucleons obtained by means of accelerators, the measured values \(\lambda_n \approx \lambda_a\) are in sufficiently good agreement with one another.
On the basis of the material presented in this section, the following conclusion may be drawn:
The magnitude of the effective cross section for the interaction of nucleons with nuclei, in going from the energies obtained by means of accelerators to the relativistic energies of cosmic-ray particles, increases approximately by a factor of two, reaching a value close to the geometrical dimensions of the nucleus. In the interval of relativistic energies of nuclear-active particles studied up to the present time (from \(\sim 1\) Bev to ten or several tens of Bev), no noticeable change occurs in the magnitude of the effective cross section for interaction with nuclei.
The mean absorption path in air of the component generating stars is somewhat greater than that of the component generating electron-nuclear showers. This is connected with the corresponding decrease in the mean free path for interaction as the particle energy increases.
In dense substances, however, when the energy of nuclear-active particles increases from several hundred Mev to several tens of Bev, the magnitude of the mean absorption path systematically increases.
The difference between the absorption path length in air and in dense substances of the same atomic weight, increasing with increasing energy, indicates the growing role of mesons in nuclear processes.
The authors of review^154 come to the conclusion that, with an increase in the energy of nuclear-active particles, beginning with the energies of the star-producing component, the quantity \(\lambda_{\mathrm{p}}\) systematically decreases, tending toward the value \(\lambda_{\mathrm{n}}\). This conclusion is erroneous.
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