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Inelastic interactions of cosmic ray particles with atomic nuclei at very high energies
I I
9.B
1
i
Nuclear Physics 87 (1966) 241--255; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
I N E L A S T I C I N T E R A C T I O N S OF C O S M I C RAY P A R T I C L E S W I T H A T O M I C N U C L E I AT VERY H I G H E N E R G I E S I. Z. A R ' I Y K O V , V. S. B A R A S H E N K O V
and S. M. E L I S E E V
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna Received 18 February 1966 Abstract: T h e nucleon-nucleus interactions in the energy range T :- 100-1000 GeV are analysed from the point o f view o f the m e c h a n i s m o f intranuclear cascades in its generally accepted form (as a series o f independent two-particle interactions). According to the presently available experimental data, it was taken into account that a m o n g secondary particles p r o d u c e d in high-energy inelastic nucleon-nucleon a n d pion-nucleon collisions there is a leading particle carrying away a b o u t 70 % o f the total energy. Consideration o f this fact leads to the result that the cascade calculation is strongly contradictory with the experimental data. To obtain a g r e e m e n t between theory and experiment it is necessary, in the calculation o f the intranuclear cascades, to take into consideration many-particle interactions o f fast particles inside the nucleus. A change in the usual form o f the cascade m e c h a n i s m o f nucleon-nucleus interactions occurs at an energy o f several dozens o f GeV.
1. Introduction It was convincingly shown in many papers that in the energy range from several dozens of MeV to several GeV the inelastic interactions of elementary particles with atomic nuclei proceed via the development of the intranuclear cascade which, with a good accuracy, may be considered as consisting of a series of successive twoparticle interactions (for a detailed discussion of the relevant problems see, e.g. refs. ~' 2) where an extensive list of references is given). Many-particle interactions are essential only in some particular cases, such as reactions involving the knock-out of deuterons, tritons and heavier fragments. The calculations 3,8) show also that such a cascade mechanism can be used to explain the experimental data on nucleon-nucleus and pion-nucleus interactions t at higher energies too, up to T = 20-30 GeV, although at present this conclusion does not seem to be so well proved as in the region of energies lower than several GeV. The fact is that at energies T = 10-30 GeV the results of theoretical calculations were compared with a comparatively small number of average experimental data which contained very large measurement errors. The overwhelming majority o f these data has been obtained in nuclear emulsions and is related to a mixture of light and heavy nuclei. Therefore for T = 10-30 GeV one can speak only in a crude way about the agreement between experiment and the usual model regarding t Here and in what follows T is the kinetic energy o f the primary particle in the lab system. 241
242
I.z. ARTYKOVet
al.
intranuclear cascades as a series of successive two-particle interactions. Though we do not know any serious disagreement between theory and experiment, this problem is still to be studied. In the superhigh-energy range T > 30 GeV, the situation is still less definite. First, this is due to inaccuracy of experimental data on the interactions of cosmic ray particles with atomic nuclei (in particular, in many cases when we are concerned with nuclear emulsion data, the energy of the primary particle is known within an accuracy of one order of magnitude only). The second cause is that theoretical calculations are very laborious since many dozens and, sometimes, hundreds of particlcs are to be taken into account. Besides, in the range of cosmic ray energies we know still little about the properties of the elementary particle interactions, therefore the information used in cascade calculations is very tentative and this affects the results of calculations, as well. Nevertheless use of modern high-speed electronic computers and the statistical analysis of available experimental data allow one to draw some important conclusions about the mechanism of nucleon-nucleus interactions at superhigh energies. Now it is known that among secondary particles produced in superhigh-energy, inelastic z - N or N - N collisions there is a particle which by its properties is essentially singled out as compared to the other particles. This leading particle carries away, on the average, 70 ~o of the total energy of the colliding particles. This fact turns out to be especially important for calculating the intranuclear cascades. A similar effect takes place in the region of accelerator energies, but here it is not so appreciable since the kinetic energy of the leading particle is not yet high compared to its rest mass, therefore its properties do not strongly differ from those of the other produced particles. In the region of superhigh cosmic ray energies the presence of such a "pivotal" particle, the energy of which slowly decreases and the kinetic energy by far exceeds the average one of other secondary particles, essentially affects the development of the intranuclear cascade. In papers 9, ~0) we have calculated the cascades under the assumption that for a fixed value of the total energy the characteristics of particles produced in an elementary inelastic act depend, in their c.m. system, neither on the kind of colliding particles nor on the kind of produced ones. At an energy higher than several dozens of GeV such an assumption is equivalent to an exclusion of the part of the intranuclear cascade produced by the leading particle, since this particle changes only slightly a distant "tail" of the total energy spectrum and the kinetic energies of the particles determined from this spectrum by the Monte-Carlo method are hardly ever equal to the energy of the leading particle. In this case the mean characteristics of the nucleonnucleus interactions turn out to be rather close to the experimentally observed ones. Of course, the theoretical characteristics of shower particles can be compared only with that part of the experimental data which does not include the leading particle. This is especially important for the mean particle energy which is naturally far lower than the mean total energy of shower particles measured cxperimentally.
INELASTIC INTERACTIONS
243
In the present paper we give some calculation results of the total intranuclear cascade including the contribution of the "pivotal" leading particle and the contributions of subsequent generations of all other produced particles. For the sake of concreteness, we take the nucleon as the leading particle. This is however a non-essential restriction since the energy of such a particle is very high as compared with its rest mass.
2. Experimental Data Used in Calculations As in all our previous papers we used the Fermi gas model to describe the atomic nucleus. The radius of a nucleus with atomic number A was assumed to be R = roA~fm, where r o = 1.4. The diffuseness of the nucleus boundary was neglected. It should be noted that the results of calculation are very sensitive to the choice of the coefficient r 0 ; in particular, due to a large multiplicity of particles produced per act of inelastic interaction, even a small increase of the mean free path of particles inside the nucleus 5..~ 1.3o leads to a noticeable decrease of the number of shower particles outgoing from the nucleus. According to the available experimental data on elementary particle interactions (see reviews i t - lz)), we assumed that at T ) ~ 1 GeV the average number of particles produced in inelastic collision is independent of the kind of colliding particles and is entirely determined by the magnitude of the kinetic energy T:
n(T) = 3T ~.
(1)
Further we assume that in inelastic pion-nucleon collision at T > 200 GeV the average number of heavy particles (K-mesons, baryons, antibaryons) amounts to 10 % of the total number of all secondary particles: nit = 0.1 n. At T < 200 GeV the average number of heavy particles is assumed to be equal to unity. The arguments underlying these assumptions consist in the ratio nH/n at accelerator energies being about half as much for the n - N interactions as for the N - N interactions. We assumed this relationship to hold for higher energies. The choice of the value of the boundary energy is very arbitrary and is made by using the formula n H = 0.1 n only in the region where 0.1 n > 1 (see refs. 11-13)). For inelastic collisions of heavy particles with intranuclear nucleons it was assumed that n H = 0.2 n provided T > 200 GeV and n H = 2 provided T < 200 GeV (refs. i -i
Relativistic transformations from the centre-of-mass system of colliding particles, where all characteristics describing an elementary act were given, to the lab system of coordinates were performed according to such mass distributions of the produced particles. In the energy range above several dozens of GeV only the total momentum distributions of produced particles averaged over heavy and light particles are known with sufficient reliability. Therefore at these energies the identical energy distributions
244
I.Z.
ARTYKOV e t
al.
to(z) were used for all kinds of produced particles, but for the leading nucleon. These distributions were taken from the experimental papers ~4-~6). It was assumed that they are independent of the kind of colliding particles. (Note that, unlike the energy distributions, the momentum distributions of secondary heavy and light particles are very different.) In the region of accelerator energies, T < 30 GeV, distinct distributions for the produced pion and nucleons were taken from refs. ~7, ~9). The mean energy of the leading nucleon eL = MN + ~'~L was determined in the c.m. system of the colliding particles as the difference M N + J ' L = e--
[n.(M.+~)+(nH-- 1)(MN+.Y)],
(2)
where e is the total energy in the c,m. system, MN and M. the nucleon and pion masses, nH and n. the average numbers of produced heavy particles and pions. The mean kinetic energy ~-- of secondary particles (excluding the leading nucleon) is determined from the condition
n,(M, +,~-) + (nil -- 1)(M N+ , ~ ) = K~.
(3)
The inelasticity coefficient K is chosen to be one third + according to the experimental data. The analytical dependence 3 - = ,~-'-(T) determined in such a way is in good agreement with the mean kinetic energy calculated for the corresponding values of T directly from the distributions og(Q. At very high. energies it is obvious that ,~-" zc T 1,
'~-"L OZ T ÷.
(4)
After singling out the leading nucleon the angular distributions of all remaining secondary particles for T >> 10 GeV as well as their energy distributions were assumed to be independent of the kind of particle (in the c.m. system). However, unlike the energy distributions, the angular distributions of secondary particles essentially depend on the kind of colliding particles; in N - N collisions secondary particles are on the average, scattered symmetrically with respect to the angle 0 = ½~, while in n - N collisions a noticeable asymmetry takes place t~). As to the angular distribution of the leading particle we know only that it is limited to a small angle around the direction of the primary particle. Accordingly all other particles in the c.m. system go out mainly in the opposite direction in order to compensate the large momentum of the leading particle. To estimate roughly the emission angles of the leading nucleon, use can be made of the relations PL COS 0L = --(nil-- 1)PH COS O--n,P, cos 0,
(5)
The calculations in which the values of the coefficient K were sampled according to the experimental histogram were performed separately. However, the results of such calculations do not practically differ from those obtained when K was usecl, as a mean value.
INELASTIC INTERACTIONS
245
which is an approximate consequence of momentum conservation in the c.m. system In this relation PL, P~t and P~ are the momenta of the leading nucleon, produced heavy particles and pions, respectively, cos 0L and cos 0 the mean cosines of the emission angles of the leading particle and of all the remaining particles. (We remind that the angular distributions w(O) = ~,(0).) Taking the values of cos 0 calculated on the basis of the experimental angular distributions w(0), we can estimate the quantity cos 0e and construct an approximate distribution COL(0) with such a value of the mean cosine. In the case of N - N collisions by virtue of the symmetry of the initial system, the leading nucleon goes out at an angle 0 c in half of the cases and in the other half at an angle 0r--0L). Therefore the resulting angular distribution is obtained by adding the two distributions
WL(O) =
'OL(O)+~',.(~--O).
The angular distributions of all other secondary particles are symmetrical, as well: W(O) = c,~(O)+ , o ( ~ - 0). In the casc of N - N collisions the distributions w(O) are taken from rcfs. i4-17). Since for 7r-N collisions at Ty,> l0 GcV the experimental angular distributions of secondary particlcs arc unknown, the function w(O) was chosen approximately the samc as for particles produccd in N - N collisions for 0 < ~2n (or 0 > ~zr). It should however be stressed that in spite of the rough qualitative character of the constructed distributions no noticeable errors are introduced in the results of cascade calculations because owing to a relativistic narrowing of angles a detailed form of the angular distributions in the c.m. system in the transformation to the lab. system turns out not to be very important. At T < 30 GeV, for nucleons and pions produced in inelastic collisions, the experimental histograms taken from refs. 18-2o) were used. To describe the angular distributions of elastic N - N and rr-N interactions at T > 30 GeV where there are no experimental data, the optical model approach was used which, as is well known agrees with experiment in the accelerator energy region. However, since at high energies the overwhelming majority of elastically scattered particles is concentrated in the region of very small angles a concrete choice of angular distributions is not very essential. In calculating the intranuclear cascades initiated by particles of very high energy, after the leading particle has been singled out, detailed knowledge of the angular and energy distribution of particles in elementary collisions is far less important than in calculations in the accelerator energy region. At very high energies the kinematic factors of the relativistic transformations are the determining ones. We assumed all the possible isospin states to be equally probable from the statistical point of view. At high energies such an assumption is quite justified since with increasing energy the isospin dependence of the interactions becomes weak. All the
246
i.z. ARTYKOVet
al.
used experimental characteristics of the elastic and inelastic interactions of elementary particles at low energies, when the isospin dependence is still important, were averaged over the isospin. In the conclusive results it was assumed that half of all the heavy particles and two thirds of the pions are charged ones. In the calculations the whole energy region was divided into eleven intervals < 0.5; 0.5-1; 1-3; 3-7; 7-10; 10-30; 30-175; 175-375; 375-625; 625-875; > 875 GeV. In each of these intervals the characteristics of particle interactions obtained by averaging the experimental data were considered to be constant. In considering the cross sections of the interaction of nucleons and pions with internuclear nucleons (O't, O'in , O'el), which determine the paths of particles inside the nucleus, the energy was divided into much smaller intervals. These cross sections were practically considered as continuous functions of energy. The charge-exchange cross section a¢x was included in the cross section ac~ since the signs of the electric charge of particles were not distinguished. In the region T > 20 GeV the interaction cross sections were assumed to be constant: at(NN ) = 39.5 mb,
trin(NN ) = 32.5 rob,
trel(NN) = 7 mb;
at(teN ) = 24 mb,
ai.(nN) = 20 mb,
tr¢l(nN ) = 4 mb.
3. M e t h o d
of Calculation
All the calculations were performed on the electronic computer M-20 by the Monte-Carlo method, taking into account the relativistic three-dimensional kinematics. The applied scheme was generally the same as in ref. 3). The energy .~7"and the angle of emission 0 were sampled for each of the heavy and light particles produced in the inelastic intranuclear collisions (with the exception of the leading nucleon). The energy of the leading nucleon was determined by eq. (2) and the angle of emission 0 L was sampled in just the same way as for all other particles. The average number of produced particles n and that of heavy particles n . were calculated according to the rules given in the previous section. It turned out that one or both of these numbers were fractional, i.e. n = ( n ) i . t , g , , + A,
n u = (n.)i.t~
r + AH.
In these cases it was sampled into nearest integers with weights equal to (l-A),
and
(1--AH),
AH.
The number of produced pions was put to be equal to n~ =
(n) ...... ,i.tes~
-- (nH) . . . . . . ti.tcger.
INELASTIC
247
INTERACTIONS
T o get sutticient statistics, a b o u t 400 intranuclear cascades were calculated for each value of the primary nucleon energy T. In all cases when the interaction with the nuclear emulsion was calculated it was preliminarily sampled according to the nuclear c o m p o n e n t of the emulsion with which the interaction proceeds. It should be however noted that the calculation results turn out to be very close to those o b t a i n e d in a direct treatment of the average nucleus 7°Ga. 4. Results o f Calculations. C o m p a r i s o n with Experimental Cross S e c t i o n s for N u c l e o n - N u c l e u s Interactions W i t h i n the limits of statistical errors the reaction cross sections ar a n d the cross sections a,d for elastic non-diffraction interactions calculated by the M o n t e - C a r l o method are i n d e p e n d e n t of the energy of an incident nucleon and agree well with the results of calculations according to the optical model. This is seen, e.g. from table 1, where the results of calculations are given for the average emulsion nucleus 7 ° G a at three different energies. The independence of the nucleon-nucleus interaction cross sections o n the energy is a direct consequence of the a s s u m p t i o n that at high energies the N - N and ~ - N interaction cross sections are constant. TABLE 1 Cross sections of interaction of nucleons with the nucleus calculated by the Monte-Carlo method and by the optical model T (GeV) 100 500 1000
a t (rob) 872+32 (860) 880__-.33 (860) 884 !_34 (860)
ona (rob)
84_-i=4 (80) 82±4 (80) 85.1-4 (80)
The values by the optical model are given in parentheses. TABLE 2 Cross sections for inelastic nucleon-nucleus interactions determined from the experimental value of the mean free path of protons in nuclear emulsion up to the interaction T (GeV)
Mean free path L (cm)
o"r (mb)
170 a)
41 :.:10
"170 490__'100
950! 550 b)
39 t=12
5~^+240 LU_ 130 --'360 940_'300
2.8 • 103 e, ,I) 3" 10se,d) 3.5 • 103e,e) 7" 103e,r) '~) Ref. ~). b) Ref. 24). c) Ref. a).
10 22 + --6 27 -t-32 --10 ~ 20 ~ 27 d) Ref. 2s). e) Ref. 26). t) Ref. ~v).
760 + 460 -- 420 1040 760
248
!. z. AR'rVKov et al.
The calculated values of a r agree with the estimates obtained from the experimental mean free paths of cosmic ray protons in nuclear emulsion. But the errors of measurement are in this case very large (see table 2). As in the region of accelerator energies, the cross section and amounts to about 40 % of the cross section for a purely diffractional scattering on the nucleus (the latter was calculated by means of the optical model). The total cross section of interaction of a cosmic ray nucleon with the average nucleus of the emulsion is about 1 b. Similar results were obtained for the light nuclei 12C and 27A1. All these results are determined by the cross sections for elastic and inelastic N - N collisions in the nucleus and they are independent of detailed assumptions on the mechanism of intranuclear cascade. 4.1. M U L T I P L I C I T Y O F P R O D U C T E D P A R T I C L E S
The theoretical and experimental values of the average multiplicity of produced shower and cascade particles are t given in tables 3 and 4. The theoretical values agree well with experiment in the region of accelerator energies (see refs. 7.8)) and at higher energies they increase much faster than the experinaental values. The energy dependence of the calculated multiplicity of produced particles at T > 100 GeV remains about the same as for accelerator energies. In particular, the number of shower particles ns ~: T m where m ~ 0.6; the experimental energy dependence n~(T) corresponds to the exponent m .~ 0.25 and is close to what is observed for the case of N - N collisions. At energies higher than several dozens of GeV the difference between the calculated and experimental values of n~ and n~- is so great that we are unable to eliminate it by any reasonable variation of the experimental data on high-energy N - N and n - N interactions. The cause of so strong a disagreement between theory and experiment is that the kinetic energy of the leading particle decreases rather slowly with the development of the intranuclear cascade. This results in the production of a large number of secondary particles not only in the first but in each subsequent inelastic interaction of the leading particle with the nucleon of the nucleus. l f t h e contribution of the leading particle is not taken into account the mean number of produced particles is rather close to the experimental one; however, even in this case theory gives somewhat overestimated values 9.1o). In the region of accelerator energies the properties of the leading particle differ little from those of other secondary particles. The intranuclear cascade taking into account the leading particle (within the limits of measurement errors and statistical errors of calculation) does not practically differ from the cascade calculated under Here and in what follows the indices s and g denote the quantities belonging to shower and cascade particles, respectively. The criteria for the choice of these particles are given in refs. 3,2~).
249
INELASTIC INTERACTIONS
the assumption of a division of the whole energy on an equal footing among all the particles produced in inelastic N - N and n - N collisions. Independently of the assumption on the role of the leading particle, g-particles almost completely consist of nucleons in accordance with experiment. TABLE 3 Average number o f particles produced in inelastic interactions o f a proton with nuclear emulsion T
ns
(GeV)
theory
75 I00 250 500 1000 3000 3500
I1 a ":"
16.2±0.9 24.3_:_ 1.1 46.2:2.4 71.7.-4.3 95.1 rz9.5
" theory 11.8±0.6 15.4_:_0.8 31.8:!:1.6 46.6=2.8 61.3:-6.1
Ilg
--experiment 8 :-.0.5 a) 12.9 =.1.8a) 18.8~4 b-°)
theory 11 _!:0.4 12.4:=0.6 25 ±1.2 33.5±2.0 42.6--4.2
IIg ~
theory 5.5--0.2 6.3--0.3 12.5.!-0.6 16.8 i-l.0 21.3.t:2.1
exper-iment 5 _:_1.6 b)
4-::0.8 t',e) 4.i 1.6 b)
2 2 . 5 : 3 a)
n is the total number o f produced particles. nt is the number o f produced charge particles. ~) Ref...3). o) Ref. ao). t,) Ref. 2s). e) Ref. 3x). e) Ref. 29).
TABLE 4 Average number o f particles produced in inelastic interactions o f protons with the nucleus ~2C at T = 100 GeV
n~
Theory
Experiment
16.6.:.0.9
8.81 ::.0.32 a) 7.35~ 1.0b)
10.5.!.0.3
5.6 ! 0 . 2 a ) 4.5 --1 ~) 4.0 -- 1 c)
n s'ng
ng=
5.3 =0.3 2.7 ± 0.2
All notations are the same as in table 3. a) Ref. a2). ~') Ref. ~3). c) Ref. 3~).
4.2. E N E R G Y S P E C T R A O F P R O D U C E D P A R T I C L E S
From table 5 where the calculated and the available cxperimental values of the mean kinetic energy of produced particles are given and from the histograms of fig. 1, it is seen that the calculated kinetic energy of shower particles increases rapidly with an increasing energy of the primary nucleon and considerably changes in the transition from light to heavy nuclei. On the contrary, the kinetic energy of g-par-
250
L Z. AR'rYKov et
al.
ticles r e m a i n s p r a c t i c a l l y c o n s t a n t in a w i d e r e g i o n f r o m l0 to 10 a G e V a n d v e r y w e a k l y d e p e n d s o n the a t o m i c n u m b e r o f the target nucleus. .
,
.~,
i
1_ _
__
L_ rm
Fig. 1. Distribution of shower particles produced in emulsion over their kinetic energy. The primary proton energy T = 100 GeV. The dashed line is the experimental histogram from ref. s0). TABLE 5 Average kinetic energy of particles produced in inelastic interactions of protons with nuclei 1~C and 7°Ga
~
T
.y-
\\
•~'-s (GeV) '~-'8 (GeV) •~L (GeV) .Y-g (GeV)
I00 GeV 12C 5.7___0.3 2.1 : - 0 . 1 (2.9-t-0.3) a) 63+3 0.16--0.01
TOGa 3.8 !0.2 1.53:z0.07(2.4±0.9) b) 58 :~1.5 0.147±0.007
500 GeV 7°Ga
1000 GeV t°Ga
6.5 +0.4 2.96---0.02
10.0 - 0 . 9 4.0 _: 0.4
260 _~14 0.14 ! 0 . 0 1
590 =60 0.14+-0.01
d~8 and .Y-g are the average energy of all shower and cascade particles. .Y-'~ is the average energy of shower particles without the leading particle "Y-Lis the average energy of the leading particle. The experimental values are given in brackets. '~) Ref. $1). l,) Ref. ~').
S i n c e t h e l e a d i n g p a r t i c l e e m i t t e d f r o m t h e nucleus carries a w a y a b o u t (54+_3) 9/oo o f t h e e n e r g y o f t h e p r i m a r y n u c l e o n , t h e m e a n v a l u e s o f J - s a n d J ' ~ essentially differ. B e t w e e n t h e e x p e r i m e n t a l a n d t h e t h e o r e t i c a l e n e r g i e s o f s h o w e r particles t h e r e is a n o t i c e a b l e d i s a g r e e m e n t , b u t it is n o t so striking as for the m e a n n u m b e r s ns. T h e t h e o r e t i c a l v a l u e o f J - g is v e r y close t o t h e e x p e r i m e n t a l ones J - g = 120+_ 12 M e V a n d J - ~ = 170 M e V o b t a i n e d in e x p e r i m e n t s 2 1 , 2 2 ) at T - - 9 a n d 16 G e V .
INELASTIC INTERACTIONS
251
4.3. DISTRIBUTIONS OVER THE TRANSVERSE M O M E N T U M T h e t r a n s v e r s e m o m e n t a o f p r o d u c e d s h o w e r a n d c a s c a d e particles are p r a c t i c a l l y i n d e p e n d e n t o f the p r i m a r y n u c l e o n e n e r g y a n d o f the a t o m i c n u m b e r o f t h e t a r g e t nucleus o v e r the e n t i r e e n e r g y i n t e r v a l T_>_ 1 G e V . T h e m e a n v a l u e o f p± for s h o w e r particles p r o d u c e d in n u c l e a r e m u l s i o n at T = 100, 500 a n d
1000 G e V is 0.41+_0.02, 0.44+_0.02 a n d 0.49_+0.05 G e V / c , respectively,
for c a s c a d e particles p± = 0.35+_0.02, 0.30_+0.02 a n d 0.32+_0.03 G e V / c . TABLE 6 Average transverse momentum of shower particles produced in inelastic interactions of cosmic ray particles at energy T = 10-10 5 GeV with nuclear emulsion T (GeV)
p± (GeV/c)
103-104
0.41 ]_0.13 a,b) 0.8 _LO.I 0.5 L0.1 a,~,) 0.3 ±0.1 e) 0 . 4 : 0 . 1 d)
~,b)
250 10-100
Theoretical value at T = I000 GeV, Pj. -- 0.49-i-0.05 GeV. a) Three values of P± are obtained by different experimental methods. b) Ref. as). e) Ref. 23). d) Ref. s.). r
~o.,
WtP,: "I
i r~ 20 t ~
0 F-
C~a T=~OO~eV
_+
~:~.'2":'~-
20il_J-~ ~,=5000e',i G'~° "
~ ,.,
"i
c-i_5 ~--F q
........ l., q
2o11-I
c-~,o
[
/
T.Js~ T'~c:~"1
0
0.5
4
P,.~ev/~
Fig. 2. Distribution of shower particles over their transversal momentum. The dashed lines are the experimental histograms from refs.SS,3s). The dotted lines are the experimental data from ref. 3=).
252
I . Z . ARTYKOV e t
aL
In the interaction of a 100 GeV proton with the carbon nucleus Pi~ = 0 . 4 3 + 0 . 0 3 GeV/c, p±g = 0 . 3 3 + 0 . 0 2 GeV/c. (See table 6.) The results of calculations agree well with experimental data (see fig. 2). 4.4. A N G U L A R
DISTRIBUTIONS
A part of the obtained results is given in table 7 and fig. 3. The theoretical distribution for shower particles presented in fig. 3 is rather close to the experimental one. Cascade particles are, with a great probability, emitted at large angles 0 > 90 ° too. However, there are yet no experimental data for cascade particles at an energy T > 30 GeV. TABLE 7
Average angle in the lab system into which half of the cascade or shower particles produced in the interaction of the proton with nuclei azC and 7°Ga is emitted "-..
01
T
\\
~ 0 `7 0°g,~
100 GeV ..............
1"C
\
1000 GeV T°Ga
:°Ga
9.3 • 0.5 52
500 GeV TOGa
11.5:0.6
---3
5.1-:-0.3
51.5 ! 2.0
53
_-t-3.2
4.0-0.3 53
•6
The indices s and g denote the angles belonging to shower a nd cascade particles, respectively.
...... ,
20L
i
..... ]
] mzq . . . . .
2
I 0
Oi
0.2
8 ra~.
Fig. 3. A n g u l a r distribution of shower particles produced in the interaction of 500 GeV p r o t o n w i t h
emulsion. The dashed line denotes the experimental histogram from ref. '-'~). In the case of N - N collisions, simple kinematic considerations based on the Lorentz transformations show that the angle into which half of all the produced particles is emitted in the lab system is 0~ .w~ T -'~. In nucleon-nucleus collisions the angle 0~. decreases with increasing energy more slowly (see table 7). This is due to the influence of subsequent collisions inside the nucleus which correspond to lower values of the energy T. Therefore use of the relation 7¢ = ctg 0~. ~ for the determination o f the primary particle energy, as sometimes has been done in experimental works, may lead to essential errors.
INELASTIC INTERACTIONS
253
In fig. 4 the angular distributions of shower particles are presented as a function of the variable log~ o tg 0. In these distributions two maxima are seen which are direct consequences of the two-maximum character of the angular distributions of particles in N - N collisions we have employed. The latter is well seen from fig. 5, where the w'~')~, f T=',gOgeV
0
7=EJgGeV
z,
.
S
-
-an _
T=f~,ogGev
Fig. 4. Angular distribution of shower particles expressed in terms of the variables x = log~o tg 0.
~ . % 1
'
. . . . . . . . . . . .
"
.....
,V=,~
e.,s =3'
N=2
~,~= 3 8~
/
8'
i
]
8"
-2.Y
-2
-'~5
-'1
-~5
0 X
Fig. 5. Angular distribution of shower particles after N intranuclcar interactions expressed in terms of the variables x = log~o tg 0 for the nucleus 7"Ga.
evolution of the angular distribution of shower particles with an increasing number of intranuclear collisions is given. As the intranuclear cascade develops, the secondary interactions of the produced particles smooth out the angular distribution, therefore in the final angular distribution of shower particlcs the two maxima are less pronounced than for the initial N - N interactions. It should be also stressed that the angular distributions of particles in N - N collisions at T > 30 GeV are poorly known and their variation essentially affects the shape of the curves of figs. 4 and 5. In particular, if for particles produced in N - N
254
i.z. ARTYKOVet
al.
collisions we choose the angular distributions mainly concentrated in the regions 0 ~ 0 and 0 ~ zt, then the two maxima of fig. 4 become even more pronounced but this affects very weakly other calculated characteristics. Inaccuracies of N - N angular distributions affect much more weakly the histograms expressed as a function of 0 or of cos 0 (see fig. 3) than the distributions W(logt0 tg 0). 5. Conclusion Thus, the model of nucleon-nucleus interactions which is based on the cascade mechanism considered as a series of two-particle interactions contradicts the available experimental data at high energies T ) 7 10 GeV. The largest disagreement takes place for the multiplicity of produced particles which in experiments at T > I00 GeV is several times smaller than the theory predicts. Other experimental and theoretical characteristics of cascade and shower particles do not differ so strongly. In principle, we might hope to bring these characteristics in agreement by varying in a corresponding manner the experimental data on N - N and n - N interactions (within the limits of their experimental errors). However in all cases the calculated and experimental values of the multiplicity of produced particles essentially differ. Since the overestimation of the multiplicity is mainly due to the leading particle and the angular and energy distributions are mainly determined by the secondary particle interactions, further improvements of the nucleon-nucleus interaction model must be first concentrated on the decrease of the number of particles produced in the intranuclear interactions of the leading particle. Such a decrease may be, in particular, due to a simultaneous absorption by one intranuclear nucleon of the leading particle and of some accompanying particles produced with it in the previous N - N collision. Since in the high-energy region, the produced particles are emitted within a narrow solid angle such many-particle interactions seem to be rather likely ~0). In the limiting case when all particles produced in the interaction with one nucleon are then absorbed also by one nucleon the characteristics of nucleon-nucleus and nucleon nucleon-interactions do not differ from one another. It should be expected that this would also lead to some increase of the two maxima of the angular distributions of created particles. The cascade calculations taking into account many-particle interactions inside the nucleus are now being performed. We are indebted to D. I. Blokhintsev for discussion of the obtained results. References 1) N. A. Pcrfilov, O. V. Lojkin and V. i. Ostrournov, ladernye reaktsii pod dcistvicm chastits vysokih energii (Izd. A N SSSR, Moskva, 1962) 2) V. S. Barashenkov, V. M. Maltscv and V. D. Toneev, Izv. AN SSSR (ser. liz.) in press
INELASTIC INTERAC'TIONS
255
3) V. S. Barashenkov, V. M. Maltsev and E. K. Mihul, Nuclear Physics 24 (1961) 642 4) V. S. Barashenkov, A. V. Boyadgiev, L. A. Kulyukina and V. M. Maltsev, Nuclear Physics 55 (1964) 79 5) N. M. Viryasov, kandidatskaya dissertatsia, OIYI (1964) 6) E. G. Boos e t al., Z h E T F 47 (1964) 2041 7) I. Z. Artykov, V. S. Barashenkov and S. M. Eliseev, ladernaya fizika, in press; preprint J I N R P-2279 (1965) 8) I. Z. Artykov, V. S. Barashenkov and S. M. Eliseev, Nuclear Physics, in press; preprint J I N R P-2373 (1965) 9) V. S. Barashcnkov and S. M. Eliseev, preprints J I N R P-1678, P-1796 (1964) 10) V. S. Barashenkov and S. M. Eliseev, Izv. AN SSSR (ser. fiz.) 29 (1965) 1631 11) V. S. Barashenkov, V. M. Maltsev and I. Patera, preprint JINR P-1577 (1964) 12) V. S. Barashenkov, V. M. Maltscv and V. D. Toneev, Forts. Phys., in press; preprint J I N R P-2392 (1965) 13) V. S. Barashcnkov and J. Patera, Forts. Phys. 11 (1963) 469, 479 14) M. Schcin, D. M. Haskin, E. Lohrmann and M. W. Teucher, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 7 15) S. Matsumoto, J. Phys. Soc. Japan 18 (1963) 1 16) D. H. Perkins, Int. Conf. on Theoretical Aspects of Very High Energy Phenomena, Geneva (1961) CERN 99 125, p. 99 17) N. A. Dobrotin et al., Nuclear Physics 35 (1962) 152 18) F. F. Abraham and R. M. Kalbach, Nuovo Cim. 26 (1962) 717 19) R. J. Piscrchio and R. M. Kalbach, Nuovo Cim. 26 (1962) 729 20) V. A. Beliakov et al., Z h l T F 39 (1960) 937 21) V. S. Barashenkov et al., Nuclear Physics 14 (1959) 522 22) U. Camerini et al., Phil. Mag. 42 (1951) 1241 23) E. Lohrmann, M. W. Toucher and M. Schein, Phys. Rev. 122 (1961) 672 24) M. G. Bowler et al., Proc. Int. Conf. on Cosmic Rays and the Earth Storm, Kyoto (1961)p. 423 25) G. Day, C. F. Gaould, C. B. A. McCusker and L. S. Peak, Nuovo Cim. 27 (1963) 977 26) A. G. Barkow et al., Phys. Rcv. 122 (1961) 617 27) R. R. Daniel, N. Purgaprasad, P. K. Malhotra and B. Vijayalakshmi, Suppl. Nuovo Cim. 1 (1963) 1163 28) I. 1. Gurevich, A. P. Mishakova, B. A. Nikolsky and L. V. Surkova, Z h E T F 34 (1958) 265 29) J. S. Takibaev, A. A. Loktionov, L. A. Sanko and C. I. Shakhova, Proc. Int. Conf. on Cosmic Rays, Moscow, Vol. 1 (1959) 51 30) A. H. Vinishky, I. G. Golyak, J. S. Takibaev and I. Y. Chasnikov, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 61 31) T. P. kazarcva and P. A. Usik, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 71 32) S. Lal, Y. Pal and R. Ragha~an, Nuclear Physics 31 (1962) 415 33) W. B. Fretter and L. F. Hansen, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 134 34) L. F. Hanscn and W. B. Frcttcr, Phys. Rcv. 118 (1960) 812 35) B. Edwards e t al., Phil. Mag. 3 (1958) 237 36) E. Fenyves, E. Gombosi and P. Suranyi, Nuovo Cim. 11 (1959) 21
9.B
1
i
Nuclear Physics 87 (1966) 241--255; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
I N E L A S T I C I N T E R A C T I O N S OF C O S M I C RAY P A R T I C L E S W I T H A T O M I C N U C L E I AT VERY H I G H E N E R G I E S I. Z. A R ' I Y K O V , V. S. B A R A S H E N K O V
and S. M. E L I S E E V
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna Received 18 February 1966 Abstract: T h e nucleon-nucleus interactions in the energy range T :- 100-1000 GeV are analysed from the point o f view o f the m e c h a n i s m o f intranuclear cascades in its generally accepted form (as a series o f independent two-particle interactions). According to the presently available experimental data, it was taken into account that a m o n g secondary particles p r o d u c e d in high-energy inelastic nucleon-nucleon a n d pion-nucleon collisions there is a leading particle carrying away a b o u t 70 % o f the total energy. Consideration o f this fact leads to the result that the cascade calculation is strongly contradictory with the experimental data. To obtain a g r e e m e n t between theory and experiment it is necessary, in the calculation o f the intranuclear cascades, to take into consideration many-particle interactions o f fast particles inside the nucleus. A change in the usual form o f the cascade m e c h a n i s m o f nucleon-nucleus interactions occurs at an energy o f several dozens o f GeV.
1. Introduction It was convincingly shown in many papers that in the energy range from several dozens of MeV to several GeV the inelastic interactions of elementary particles with atomic nuclei proceed via the development of the intranuclear cascade which, with a good accuracy, may be considered as consisting of a series of successive twoparticle interactions (for a detailed discussion of the relevant problems see, e.g. refs. ~' 2) where an extensive list of references is given). Many-particle interactions are essential only in some particular cases, such as reactions involving the knock-out of deuterons, tritons and heavier fragments. The calculations 3,8) show also that such a cascade mechanism can be used to explain the experimental data on nucleon-nucleus and pion-nucleus interactions t at higher energies too, up to T = 20-30 GeV, although at present this conclusion does not seem to be so well proved as in the region of energies lower than several GeV. The fact is that at energies T = 10-30 GeV the results of theoretical calculations were compared with a comparatively small number of average experimental data which contained very large measurement errors. The overwhelming majority o f these data has been obtained in nuclear emulsions and is related to a mixture of light and heavy nuclei. Therefore for T = 10-30 GeV one can speak only in a crude way about the agreement between experiment and the usual model regarding t Here and in what follows T is the kinetic energy o f the primary particle in the lab system. 241
242
I.z. ARTYKOVet
al.
intranuclear cascades as a series of successive two-particle interactions. Though we do not know any serious disagreement between theory and experiment, this problem is still to be studied. In the superhigh-energy range T > 30 GeV, the situation is still less definite. First, this is due to inaccuracy of experimental data on the interactions of cosmic ray particles with atomic nuclei (in particular, in many cases when we are concerned with nuclear emulsion data, the energy of the primary particle is known within an accuracy of one order of magnitude only). The second cause is that theoretical calculations are very laborious since many dozens and, sometimes, hundreds of particlcs are to be taken into account. Besides, in the range of cosmic ray energies we know still little about the properties of the elementary particle interactions, therefore the information used in cascade calculations is very tentative and this affects the results of calculations, as well. Nevertheless use of modern high-speed electronic computers and the statistical analysis of available experimental data allow one to draw some important conclusions about the mechanism of nucleon-nucleus interactions at superhigh energies. Now it is known that among secondary particles produced in superhigh-energy, inelastic z - N or N - N collisions there is a particle which by its properties is essentially singled out as compared to the other particles. This leading particle carries away, on the average, 70 ~o of the total energy of the colliding particles. This fact turns out to be especially important for calculating the intranuclear cascades. A similar effect takes place in the region of accelerator energies, but here it is not so appreciable since the kinetic energy of the leading particle is not yet high compared to its rest mass, therefore its properties do not strongly differ from those of the other produced particles. In the region of superhigh cosmic ray energies the presence of such a "pivotal" particle, the energy of which slowly decreases and the kinetic energy by far exceeds the average one of other secondary particles, essentially affects the development of the intranuclear cascade. In papers 9, ~0) we have calculated the cascades under the assumption that for a fixed value of the total energy the characteristics of particles produced in an elementary inelastic act depend, in their c.m. system, neither on the kind of colliding particles nor on the kind of produced ones. At an energy higher than several dozens of GeV such an assumption is equivalent to an exclusion of the part of the intranuclear cascade produced by the leading particle, since this particle changes only slightly a distant "tail" of the total energy spectrum and the kinetic energies of the particles determined from this spectrum by the Monte-Carlo method are hardly ever equal to the energy of the leading particle. In this case the mean characteristics of the nucleonnucleus interactions turn out to be rather close to the experimentally observed ones. Of course, the theoretical characteristics of shower particles can be compared only with that part of the experimental data which does not include the leading particle. This is especially important for the mean particle energy which is naturally far lower than the mean total energy of shower particles measured cxperimentally.
INELASTIC INTERACTIONS
243
In the present paper we give some calculation results of the total intranuclear cascade including the contribution of the "pivotal" leading particle and the contributions of subsequent generations of all other produced particles. For the sake of concreteness, we take the nucleon as the leading particle. This is however a non-essential restriction since the energy of such a particle is very high as compared with its rest mass.
2. Experimental Data Used in Calculations As in all our previous papers we used the Fermi gas model to describe the atomic nucleus. The radius of a nucleus with atomic number A was assumed to be R = roA~fm, where r o = 1.4. The diffuseness of the nucleus boundary was neglected. It should be noted that the results of calculation are very sensitive to the choice of the coefficient r 0 ; in particular, due to a large multiplicity of particles produced per act of inelastic interaction, even a small increase of the mean free path of particles inside the nucleus 5..~ 1.3o leads to a noticeable decrease of the number of shower particles outgoing from the nucleus. According to the available experimental data on elementary particle interactions (see reviews i t - lz)), we assumed that at T ) ~ 1 GeV the average number of particles produced in inelastic collision is independent of the kind of colliding particles and is entirely determined by the magnitude of the kinetic energy T:
n(T) = 3T ~.
(1)
Further we assume that in inelastic pion-nucleon collision at T > 200 GeV the average number of heavy particles (K-mesons, baryons, antibaryons) amounts to 10 % of the total number of all secondary particles: nit = 0.1 n. At T < 200 GeV the average number of heavy particles is assumed to be equal to unity. The arguments underlying these assumptions consist in the ratio nH/n at accelerator energies being about half as much for the n - N interactions as for the N - N interactions. We assumed this relationship to hold for higher energies. The choice of the value of the boundary energy is very arbitrary and is made by using the formula n H = 0.1 n only in the region where 0.1 n > 1 (see refs. 11-13)). For inelastic collisions of heavy particles with intranuclear nucleons it was assumed that n H = 0.2 n provided T > 200 GeV and n H = 2 provided T < 200 GeV (refs. i -i
Relativistic transformations from the centre-of-mass system of colliding particles, where all characteristics describing an elementary act were given, to the lab system of coordinates were performed according to such mass distributions of the produced particles. In the energy range above several dozens of GeV only the total momentum distributions of produced particles averaged over heavy and light particles are known with sufficient reliability. Therefore at these energies the identical energy distributions
244
I.Z.
ARTYKOV e t
al.
to(z) were used for all kinds of produced particles, but for the leading nucleon. These distributions were taken from the experimental papers ~4-~6). It was assumed that they are independent of the kind of colliding particles. (Note that, unlike the energy distributions, the momentum distributions of secondary heavy and light particles are very different.) In the region of accelerator energies, T < 30 GeV, distinct distributions for the produced pion and nucleons were taken from refs. ~7, ~9). The mean energy of the leading nucleon eL = MN + ~'~L was determined in the c.m. system of the colliding particles as the difference M N + J ' L = e--
[n.(M.+~)+(nH-- 1)(MN+.Y)],
(2)
where e is the total energy in the c,m. system, MN and M. the nucleon and pion masses, nH and n. the average numbers of produced heavy particles and pions. The mean kinetic energy ~-- of secondary particles (excluding the leading nucleon) is determined from the condition
n,(M, +,~-) + (nil -- 1)(M N+ , ~ ) = K~.
(3)
The inelasticity coefficient K is chosen to be one third + according to the experimental data. The analytical dependence 3 - = ,~-'-(T) determined in such a way is in good agreement with the mean kinetic energy calculated for the corresponding values of T directly from the distributions og(Q. At very high. energies it is obvious that ,~-" zc T 1,
'~-"L OZ T ÷.
(4)
After singling out the leading nucleon the angular distributions of all remaining secondary particles for T >> 10 GeV as well as their energy distributions were assumed to be independent of the kind of particle (in the c.m. system). However, unlike the energy distributions, the angular distributions of secondary particles essentially depend on the kind of colliding particles; in N - N collisions secondary particles are on the average, scattered symmetrically with respect to the angle 0 = ½~, while in n - N collisions a noticeable asymmetry takes place t~). As to the angular distribution of the leading particle we know only that it is limited to a small angle around the direction of the primary particle. Accordingly all other particles in the c.m. system go out mainly in the opposite direction in order to compensate the large momentum of the leading particle. To estimate roughly the emission angles of the leading nucleon, use can be made of the relations PL COS 0L = --(nil-- 1)PH COS O--n,P, cos 0,
(5)
The calculations in which the values of the coefficient K were sampled according to the experimental histogram were performed separately. However, the results of such calculations do not practically differ from those obtained when K was usecl, as a mean value.
INELASTIC INTERACTIONS
245
which is an approximate consequence of momentum conservation in the c.m. system In this relation PL, P~t and P~ are the momenta of the leading nucleon, produced heavy particles and pions, respectively, cos 0L and cos 0 the mean cosines of the emission angles of the leading particle and of all the remaining particles. (We remind that the angular distributions w(O) = ~,(0).) Taking the values of cos 0 calculated on the basis of the experimental angular distributions w(0), we can estimate the quantity cos 0e and construct an approximate distribution COL(0) with such a value of the mean cosine. In the case of N - N collisions by virtue of the symmetry of the initial system, the leading nucleon goes out at an angle 0 c in half of the cases and in the other half at an angle 0r--0L). Therefore the resulting angular distribution is obtained by adding the two distributions
WL(O) =
'OL(O)+~',.(~--O).
The angular distributions of all other secondary particles are symmetrical, as well: W(O) = c,~(O)+ , o ( ~ - 0). In the casc of N - N collisions the distributions w(O) are taken from rcfs. i4-17). Since for 7r-N collisions at Ty,> l0 GcV the experimental angular distributions of secondary particlcs arc unknown, the function w(O) was chosen approximately the samc as for particles produccd in N - N collisions for 0 < ~2n (or 0 > ~zr). It should however be stressed that in spite of the rough qualitative character of the constructed distributions no noticeable errors are introduced in the results of cascade calculations because owing to a relativistic narrowing of angles a detailed form of the angular distributions in the c.m. system in the transformation to the lab. system turns out not to be very important. At T < 30 GeV, for nucleons and pions produced in inelastic collisions, the experimental histograms taken from refs. 18-2o) were used. To describe the angular distributions of elastic N - N and rr-N interactions at T > 30 GeV where there are no experimental data, the optical model approach was used which, as is well known agrees with experiment in the accelerator energy region. However, since at high energies the overwhelming majority of elastically scattered particles is concentrated in the region of very small angles a concrete choice of angular distributions is not very essential. In calculating the intranuclear cascades initiated by particles of very high energy, after the leading particle has been singled out, detailed knowledge of the angular and energy distribution of particles in elementary collisions is far less important than in calculations in the accelerator energy region. At very high energies the kinematic factors of the relativistic transformations are the determining ones. We assumed all the possible isospin states to be equally probable from the statistical point of view. At high energies such an assumption is quite justified since with increasing energy the isospin dependence of the interactions becomes weak. All the
246
i.z. ARTYKOVet
al.
used experimental characteristics of the elastic and inelastic interactions of elementary particles at low energies, when the isospin dependence is still important, were averaged over the isospin. In the conclusive results it was assumed that half of all the heavy particles and two thirds of the pions are charged ones. In the calculations the whole energy region was divided into eleven intervals < 0.5; 0.5-1; 1-3; 3-7; 7-10; 10-30; 30-175; 175-375; 375-625; 625-875; > 875 GeV. In each of these intervals the characteristics of particle interactions obtained by averaging the experimental data were considered to be constant. In considering the cross sections of the interaction of nucleons and pions with internuclear nucleons (O't, O'in , O'el), which determine the paths of particles inside the nucleus, the energy was divided into much smaller intervals. These cross sections were practically considered as continuous functions of energy. The charge-exchange cross section a¢x was included in the cross section ac~ since the signs of the electric charge of particles were not distinguished. In the region T > 20 GeV the interaction cross sections were assumed to be constant: at(NN ) = 39.5 mb,
trin(NN ) = 32.5 rob,
trel(NN) = 7 mb;
at(teN ) = 24 mb,
ai.(nN) = 20 mb,
tr¢l(nN ) = 4 mb.
3. M e t h o d
of Calculation
All the calculations were performed on the electronic computer M-20 by the Monte-Carlo method, taking into account the relativistic three-dimensional kinematics. The applied scheme was generally the same as in ref. 3). The energy .~7"and the angle of emission 0 were sampled for each of the heavy and light particles produced in the inelastic intranuclear collisions (with the exception of the leading nucleon). The energy of the leading nucleon was determined by eq. (2) and the angle of emission 0 L was sampled in just the same way as for all other particles. The average number of produced particles n and that of heavy particles n . were calculated according to the rules given in the previous section. It turned out that one or both of these numbers were fractional, i.e. n = ( n ) i . t , g , , + A,
n u = (n.)i.t~
r + AH.
In these cases it was sampled into nearest integers with weights equal to (l-A),
and
(1--AH),
AH.
The number of produced pions was put to be equal to n~ =
(n) ...... ,i.tes~
-- (nH) . . . . . . ti.tcger.
INELASTIC
247
INTERACTIONS
T o get sutticient statistics, a b o u t 400 intranuclear cascades were calculated for each value of the primary nucleon energy T. In all cases when the interaction with the nuclear emulsion was calculated it was preliminarily sampled according to the nuclear c o m p o n e n t of the emulsion with which the interaction proceeds. It should be however noted that the calculation results turn out to be very close to those o b t a i n e d in a direct treatment of the average nucleus 7°Ga. 4. Results o f Calculations. C o m p a r i s o n with Experimental Cross S e c t i o n s for N u c l e o n - N u c l e u s Interactions W i t h i n the limits of statistical errors the reaction cross sections ar a n d the cross sections a,d for elastic non-diffraction interactions calculated by the M o n t e - C a r l o method are i n d e p e n d e n t of the energy of an incident nucleon and agree well with the results of calculations according to the optical model. This is seen, e.g. from table 1, where the results of calculations are given for the average emulsion nucleus 7 ° G a at three different energies. The independence of the nucleon-nucleus interaction cross sections o n the energy is a direct consequence of the a s s u m p t i o n that at high energies the N - N and ~ - N interaction cross sections are constant. TABLE 1 Cross sections of interaction of nucleons with the nucleus calculated by the Monte-Carlo method and by the optical model T (GeV) 100 500 1000
a t (rob) 872+32 (860) 880__-.33 (860) 884 !_34 (860)
ona (rob)
84_-i=4 (80) 82±4 (80) 85.1-4 (80)
The values by the optical model are given in parentheses. TABLE 2 Cross sections for inelastic nucleon-nucleus interactions determined from the experimental value of the mean free path of protons in nuclear emulsion up to the interaction T (GeV)
Mean free path L (cm)
o"r (mb)
170 a)
41 :.:10
"170 490__'100
950! 550 b)
39 t=12
5~^+240 LU_ 130 --'360 940_'300
2.8 • 103 e, ,I) 3" 10se,d) 3.5 • 103e,e) 7" 103e,r) '~) Ref. ~). b) Ref. 24). c) Ref. a).
10 22 + --6 27 -t-32 --10 ~ 20 ~ 27 d) Ref. 2s). e) Ref. 26). t) Ref. ~v).
760 + 460 -- 420 1040 760
248
!. z. AR'rVKov et al.
The calculated values of a r agree with the estimates obtained from the experimental mean free paths of cosmic ray protons in nuclear emulsion. But the errors of measurement are in this case very large (see table 2). As in the region of accelerator energies, the cross section and amounts to about 40 % of the cross section for a purely diffractional scattering on the nucleus (the latter was calculated by means of the optical model). The total cross section of interaction of a cosmic ray nucleon with the average nucleus of the emulsion is about 1 b. Similar results were obtained for the light nuclei 12C and 27A1. All these results are determined by the cross sections for elastic and inelastic N - N collisions in the nucleus and they are independent of detailed assumptions on the mechanism of intranuclear cascade. 4.1. M U L T I P L I C I T Y O F P R O D U C T E D P A R T I C L E S
The theoretical and experimental values of the average multiplicity of produced shower and cascade particles are t given in tables 3 and 4. The theoretical values agree well with experiment in the region of accelerator energies (see refs. 7.8)) and at higher energies they increase much faster than the experinaental values. The energy dependence of the calculated multiplicity of produced particles at T > 100 GeV remains about the same as for accelerator energies. In particular, the number of shower particles ns ~: T m where m ~ 0.6; the experimental energy dependence n~(T) corresponds to the exponent m .~ 0.25 and is close to what is observed for the case of N - N collisions. At energies higher than several dozens of GeV the difference between the calculated and experimental values of n~ and n~- is so great that we are unable to eliminate it by any reasonable variation of the experimental data on high-energy N - N and n - N interactions. The cause of so strong a disagreement between theory and experiment is that the kinetic energy of the leading particle decreases rather slowly with the development of the intranuclear cascade. This results in the production of a large number of secondary particles not only in the first but in each subsequent inelastic interaction of the leading particle with the nucleon of the nucleus. l f t h e contribution of the leading particle is not taken into account the mean number of produced particles is rather close to the experimental one; however, even in this case theory gives somewhat overestimated values 9.1o). In the region of accelerator energies the properties of the leading particle differ little from those of other secondary particles. The intranuclear cascade taking into account the leading particle (within the limits of measurement errors and statistical errors of calculation) does not practically differ from the cascade calculated under Here and in what follows the indices s and g denote the quantities belonging to shower and cascade particles, respectively. The criteria for the choice of these particles are given in refs. 3,2~).
249
INELASTIC INTERACTIONS
the assumption of a division of the whole energy on an equal footing among all the particles produced in inelastic N - N and n - N collisions. Independently of the assumption on the role of the leading particle, g-particles almost completely consist of nucleons in accordance with experiment. TABLE 3 Average number o f particles produced in inelastic interactions o f a proton with nuclear emulsion T
ns
(GeV)
theory
75 I00 250 500 1000 3000 3500
I1 a ":"
16.2±0.9 24.3_:_ 1.1 46.2:2.4 71.7.-4.3 95.1 rz9.5
" theory 11.8±0.6 15.4_:_0.8 31.8:!:1.6 46.6=2.8 61.3:-6.1
Ilg
--experiment 8 :-.0.5 a) 12.9 =.1.8a) 18.8~4 b-°)
theory 11 _!:0.4 12.4:=0.6 25 ±1.2 33.5±2.0 42.6--4.2
IIg ~
theory 5.5--0.2 6.3--0.3 12.5.!-0.6 16.8 i-l.0 21.3.t:2.1
exper-iment 5 _:_1.6 b)
4-::0.8 t',e) 4.i 1.6 b)
2 2 . 5 : 3 a)
n is the total number o f produced particles. nt is the number o f produced charge particles. ~) Ref...3). o) Ref. ao). t,) Ref. 2s). e) Ref. 3x). e) Ref. 29).
TABLE 4 Average number o f particles produced in inelastic interactions o f protons with the nucleus ~2C at T = 100 GeV
n~
Theory
Experiment
16.6.:.0.9
8.81 ::.0.32 a) 7.35~ 1.0b)
10.5.!.0.3
5.6 ! 0 . 2 a ) 4.5 --1 ~) 4.0 -- 1 c)
n s'ng
ng=
5.3 =0.3 2.7 ± 0.2
All notations are the same as in table 3. a) Ref. a2). ~') Ref. ~3). c) Ref. 3~).
4.2. E N E R G Y S P E C T R A O F P R O D U C E D P A R T I C L E S
From table 5 where the calculated and the available cxperimental values of the mean kinetic energy of produced particles are given and from the histograms of fig. 1, it is seen that the calculated kinetic energy of shower particles increases rapidly with an increasing energy of the primary nucleon and considerably changes in the transition from light to heavy nuclei. On the contrary, the kinetic energy of g-par-
250
L Z. AR'rYKov et
al.
ticles r e m a i n s p r a c t i c a l l y c o n s t a n t in a w i d e r e g i o n f r o m l0 to 10 a G e V a n d v e r y w e a k l y d e p e n d s o n the a t o m i c n u m b e r o f the target nucleus. .
,
.~,
i
1_ _
__
L_ rm
Fig. 1. Distribution of shower particles produced in emulsion over their kinetic energy. The primary proton energy T = 100 GeV. The dashed line is the experimental histogram from ref. s0). TABLE 5 Average kinetic energy of particles produced in inelastic interactions of protons with nuclei 1~C and 7°Ga
~
T
.y-
\\
•~'-s (GeV) '~-'8 (GeV) •~L (GeV) .Y-g (GeV)
I00 GeV 12C 5.7___0.3 2.1 : - 0 . 1 (2.9-t-0.3) a) 63+3 0.16--0.01
TOGa 3.8 !0.2 1.53:z0.07(2.4±0.9) b) 58 :~1.5 0.147±0.007
500 GeV 7°Ga
1000 GeV t°Ga
6.5 +0.4 2.96---0.02
10.0 - 0 . 9 4.0 _: 0.4
260 _~14 0.14 ! 0 . 0 1
590 =60 0.14+-0.01
d~8 and .Y-g are the average energy of all shower and cascade particles. .Y-'~ is the average energy of shower particles without the leading particle "Y-Lis the average energy of the leading particle. The experimental values are given in brackets. '~) Ref. $1). l,) Ref. ~').
S i n c e t h e l e a d i n g p a r t i c l e e m i t t e d f r o m t h e nucleus carries a w a y a b o u t (54+_3) 9/oo o f t h e e n e r g y o f t h e p r i m a r y n u c l e o n , t h e m e a n v a l u e s o f J - s a n d J ' ~ essentially differ. B e t w e e n t h e e x p e r i m e n t a l a n d t h e t h e o r e t i c a l e n e r g i e s o f s h o w e r particles t h e r e is a n o t i c e a b l e d i s a g r e e m e n t , b u t it is n o t so striking as for the m e a n n u m b e r s ns. T h e t h e o r e t i c a l v a l u e o f J - g is v e r y close t o t h e e x p e r i m e n t a l ones J - g = 120+_ 12 M e V a n d J - ~ = 170 M e V o b t a i n e d in e x p e r i m e n t s 2 1 , 2 2 ) at T - - 9 a n d 16 G e V .
INELASTIC INTERACTIONS
251
4.3. DISTRIBUTIONS OVER THE TRANSVERSE M O M E N T U M T h e t r a n s v e r s e m o m e n t a o f p r o d u c e d s h o w e r a n d c a s c a d e particles are p r a c t i c a l l y i n d e p e n d e n t o f the p r i m a r y n u c l e o n e n e r g y a n d o f the a t o m i c n u m b e r o f t h e t a r g e t nucleus o v e r the e n t i r e e n e r g y i n t e r v a l T_>_ 1 G e V . T h e m e a n v a l u e o f p± for s h o w e r particles p r o d u c e d in n u c l e a r e m u l s i o n at T = 100, 500 a n d
1000 G e V is 0.41+_0.02, 0.44+_0.02 a n d 0.49_+0.05 G e V / c , respectively,
for c a s c a d e particles p± = 0.35+_0.02, 0.30_+0.02 a n d 0.32+_0.03 G e V / c . TABLE 6 Average transverse momentum of shower particles produced in inelastic interactions of cosmic ray particles at energy T = 10-10 5 GeV with nuclear emulsion T (GeV)
p± (GeV/c)
103-104
0.41 ]_0.13 a,b) 0.8 _LO.I 0.5 L0.1 a,~,) 0.3 ±0.1 e) 0 . 4 : 0 . 1 d)
~,b)
250 10-100
Theoretical value at T = I000 GeV, Pj. -- 0.49-i-0.05 GeV. a) Three values of P± are obtained by different experimental methods. b) Ref. as). e) Ref. 23). d) Ref. s.). r
~o.,
WtP,: "I
i r~ 20 t ~
0 F-
C~a T=~OO~eV
_+
~:~.'2":'~-
20il_J-~ ~,=5000e',i G'~° "
~ ,.,
"i
c-i_5 ~--F q
........ l., q
2o11-I
c-~,o
[
/
T.Js~ T'~c:~"1
0
0.5
4
P,.~ev/~
Fig. 2. Distribution of shower particles over their transversal momentum. The dashed lines are the experimental histograms from refs.SS,3s). The dotted lines are the experimental data from ref. 3=).
252
I . Z . ARTYKOV e t
aL
In the interaction of a 100 GeV proton with the carbon nucleus Pi~ = 0 . 4 3 + 0 . 0 3 GeV/c, p±g = 0 . 3 3 + 0 . 0 2 GeV/c. (See table 6.) The results of calculations agree well with experimental data (see fig. 2). 4.4. A N G U L A R
DISTRIBUTIONS
A part of the obtained results is given in table 7 and fig. 3. The theoretical distribution for shower particles presented in fig. 3 is rather close to the experimental one. Cascade particles are, with a great probability, emitted at large angles 0 > 90 ° too. However, there are yet no experimental data for cascade particles at an energy T > 30 GeV. TABLE 7
Average angle in the lab system into which half of the cascade or shower particles produced in the interaction of the proton with nuclei azC and 7°Ga is emitted "-..
01
T
\\
~ 0 `7 0°g,~
100 GeV ..............
1"C
\
1000 GeV T°Ga
:°Ga
9.3 • 0.5 52
500 GeV TOGa
11.5:0.6
---3
5.1-:-0.3
51.5 ! 2.0
53
_-t-3.2
4.0-0.3 53
•6
The indices s and g denote the angles belonging to shower a nd cascade particles, respectively.
...... ,
20L
i
..... ]
] mzq . . . . .
2
I 0
Oi
0.2
8 ra~.
Fig. 3. A n g u l a r distribution of shower particles produced in the interaction of 500 GeV p r o t o n w i t h
emulsion. The dashed line denotes the experimental histogram from ref. '-'~). In the case of N - N collisions, simple kinematic considerations based on the Lorentz transformations show that the angle into which half of all the produced particles is emitted in the lab system is 0~ .w~ T -'~. In nucleon-nucleus collisions the angle 0~. decreases with increasing energy more slowly (see table 7). This is due to the influence of subsequent collisions inside the nucleus which correspond to lower values of the energy T. Therefore use of the relation 7¢ = ctg 0~. ~ for the determination o f the primary particle energy, as sometimes has been done in experimental works, may lead to essential errors.
INELASTIC INTERACTIONS
253
In fig. 4 the angular distributions of shower particles are presented as a function of the variable log~ o tg 0. In these distributions two maxima are seen which are direct consequences of the two-maximum character of the angular distributions of particles in N - N collisions we have employed. The latter is well seen from fig. 5, where the w'~')~, f T=',gOgeV
0
7=EJgGeV
z,
.
S
-
-an _
T=f~,ogGev
Fig. 4. Angular distribution of shower particles expressed in terms of the variables x = log~o tg 0.
~ . % 1
'
. . . . . . . . . . . .
"
.....
,V=,~
e.,s =3'
N=2
~,~= 3 8~
/
8'
i
]
8"
-2.Y
-2
-'~5
-'1
-~5
0 X
Fig. 5. Angular distribution of shower particles after N intranuclcar interactions expressed in terms of the variables x = log~o tg 0 for the nucleus 7"Ga.
evolution of the angular distribution of shower particles with an increasing number of intranuclear collisions is given. As the intranuclear cascade develops, the secondary interactions of the produced particles smooth out the angular distribution, therefore in the final angular distribution of shower particlcs the two maxima are less pronounced than for the initial N - N interactions. It should be also stressed that the angular distributions of particles in N - N collisions at T > 30 GeV are poorly known and their variation essentially affects the shape of the curves of figs. 4 and 5. In particular, if for particles produced in N - N
254
i.z. ARTYKOVet
al.
collisions we choose the angular distributions mainly concentrated in the regions 0 ~ 0 and 0 ~ zt, then the two maxima of fig. 4 become even more pronounced but this affects very weakly other calculated characteristics. Inaccuracies of N - N angular distributions affect much more weakly the histograms expressed as a function of 0 or of cos 0 (see fig. 3) than the distributions W(logt0 tg 0). 5. Conclusion Thus, the model of nucleon-nucleus interactions which is based on the cascade mechanism considered as a series of two-particle interactions contradicts the available experimental data at high energies T ) 7 10 GeV. The largest disagreement takes place for the multiplicity of produced particles which in experiments at T > I00 GeV is several times smaller than the theory predicts. Other experimental and theoretical characteristics of cascade and shower particles do not differ so strongly. In principle, we might hope to bring these characteristics in agreement by varying in a corresponding manner the experimental data on N - N and n - N interactions (within the limits of their experimental errors). However in all cases the calculated and experimental values of the multiplicity of produced particles essentially differ. Since the overestimation of the multiplicity is mainly due to the leading particle and the angular and energy distributions are mainly determined by the secondary particle interactions, further improvements of the nucleon-nucleus interaction model must be first concentrated on the decrease of the number of particles produced in the intranuclear interactions of the leading particle. Such a decrease may be, in particular, due to a simultaneous absorption by one intranuclear nucleon of the leading particle and of some accompanying particles produced with it in the previous N - N collision. Since in the high-energy region, the produced particles are emitted within a narrow solid angle such many-particle interactions seem to be rather likely ~0). In the limiting case when all particles produced in the interaction with one nucleon are then absorbed also by one nucleon the characteristics of nucleon-nucleus and nucleon nucleon-interactions do not differ from one another. It should be expected that this would also lead to some increase of the two maxima of the angular distributions of created particles. The cascade calculations taking into account many-particle interactions inside the nucleus are now being performed. We are indebted to D. I. Blokhintsev for discussion of the obtained results. References 1) N. A. Pcrfilov, O. V. Lojkin and V. i. Ostrournov, ladernye reaktsii pod dcistvicm chastits vysokih energii (Izd. A N SSSR, Moskva, 1962) 2) V. S. Barashenkov, V. M. Maltscv and V. D. Toneev, Izv. AN SSSR (ser. liz.) in press
INELASTIC INTERAC'TIONS
255
3) V. S. Barashenkov, V. M. Maltsev and E. K. Mihul, Nuclear Physics 24 (1961) 642 4) V. S. Barashenkov, A. V. Boyadgiev, L. A. Kulyukina and V. M. Maltsev, Nuclear Physics 55 (1964) 79 5) N. M. Viryasov, kandidatskaya dissertatsia, OIYI (1964) 6) E. G. Boos e t al., Z h E T F 47 (1964) 2041 7) I. Z. Artykov, V. S. Barashenkov and S. M. Eliseev, ladernaya fizika, in press; preprint J I N R P-2279 (1965) 8) I. Z. Artykov, V. S. Barashenkov and S. M. Eliseev, Nuclear Physics, in press; preprint J I N R P-2373 (1965) 9) V. S. Barashcnkov and S. M. Eliseev, preprints J I N R P-1678, P-1796 (1964) 10) V. S. Barashenkov and S. M. Eliseev, Izv. AN SSSR (ser. fiz.) 29 (1965) 1631 11) V. S. Barashenkov, V. M. Maltsev and I. Patera, preprint JINR P-1577 (1964) 12) V. S. Barashenkov, V. M. Maltscv and V. D. Toneev, Forts. Phys., in press; preprint J I N R P-2392 (1965) 13) V. S. Barashcnkov and J. Patera, Forts. Phys. 11 (1963) 469, 479 14) M. Schcin, D. M. Haskin, E. Lohrmann and M. W. Teucher, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 7 15) S. Matsumoto, J. Phys. Soc. Japan 18 (1963) 1 16) D. H. Perkins, Int. Conf. on Theoretical Aspects of Very High Energy Phenomena, Geneva (1961) CERN 99 125, p. 99 17) N. A. Dobrotin et al., Nuclear Physics 35 (1962) 152 18) F. F. Abraham and R. M. Kalbach, Nuovo Cim. 26 (1962) 717 19) R. J. Piscrchio and R. M. Kalbach, Nuovo Cim. 26 (1962) 729 20) V. A. Beliakov et al., Z h l T F 39 (1960) 937 21) V. S. Barashenkov et al., Nuclear Physics 14 (1959) 522 22) U. Camerini et al., Phil. Mag. 42 (1951) 1241 23) E. Lohrmann, M. W. Toucher and M. Schein, Phys. Rev. 122 (1961) 672 24) M. G. Bowler et al., Proc. Int. Conf. on Cosmic Rays and the Earth Storm, Kyoto (1961)p. 423 25) G. Day, C. F. Gaould, C. B. A. McCusker and L. S. Peak, Nuovo Cim. 27 (1963) 977 26) A. G. Barkow et al., Phys. Rcv. 122 (1961) 617 27) R. R. Daniel, N. Purgaprasad, P. K. Malhotra and B. Vijayalakshmi, Suppl. Nuovo Cim. 1 (1963) 1163 28) I. 1. Gurevich, A. P. Mishakova, B. A. Nikolsky and L. V. Surkova, Z h E T F 34 (1958) 265 29) J. S. Takibaev, A. A. Loktionov, L. A. Sanko and C. I. Shakhova, Proc. Int. Conf. on Cosmic Rays, Moscow, Vol. 1 (1959) 51 30) A. H. Vinishky, I. G. Golyak, J. S. Takibaev and I. Y. Chasnikov, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 61 31) T. P. kazarcva and P. A. Usik, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 71 32) S. Lal, Y. Pal and R. Ragha~an, Nuclear Physics 31 (1962) 415 33) W. B. Fretter and L. F. Hansen, Proc. Int. Conf. on Cosmic Rays, Moscow, vol. 1 (1959) p. 134 34) L. F. Hanscn and W. B. Frcttcr, Phys. Rcv. 118 (1960) 812 35) B. Edwards e t al., Phil. Mag. 3 (1958) 237 36) E. Fenyves, E. Gombosi and P. Suranyi, Nuovo Cim. 11 (1959) 21