Energy determination of electron-photon showers induced by heavy cosmic-ray primaries

317

Nuclear Instruments and Methods in Physics Research A276 (1989) 317-339 North-Holland, Amsterdam

ENERGY DETERMINATION OF ELECTRON-PHOTON SHOWERS INDUCED BY HEAVY COSMIC-RAY PRIMARIES Takashi FUJINAGA, Masakazu ICHIMURA, Yasuo NIIHORI and Toru SHIBATA Department of Physics, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo, Japan Received 1 September 1987 and in revised form 27 May 1988 On the basis of extensive simulation calculations, we establish a method of energy determination, in the region 10 1- -10 16 eV/nucleus, for electromagnetic cascade showers initiated by heavy cosmic-ray primaries as well as by protons and alphas . We study in detail the accuracy of the present method, focusing on the errors arising from the chamber configuration and from both fluctuations in the production spectrum of source -y-rays and subsequent cascade development . All errors are well within our allowance, comparable with, or less than, experimental error. 1 . Introduction Direct observation of primary cosmic rays in the energy region 10'4-1016 eV gives us, particularly nowadays, essential and fruitful information for both highenergy particle physics and high-energy astrophysics . In the former field, the detection of "exotic" fireballs with a high temperature and high energy density a new state of matter (so-called "quark-gluon plasma"), will make QCD theory indisputable . In the latter field, the direct confirmation of the chemical composition in the region 10 14-10' 6 eV gives us invaluable information of the origin of high-energy cosmic rays and the mechanisms of their acceleration . With this motivation, there are several balloon-borne emulsion- chamber projects, aiming to extend the present flight-duration time ( - 30 h/flight) to more than 100 h/flight . The JACEE (Japanese-American Cooperative Emulsion Experiment) group [1] has already started flight construction and will produce valuable data in the near future . There are several methods to determine the primary energy of cosmic rays in the region - 100 GeV/nucleon, of which two methods in particular are available: using the half angle of a log(tan 0) plot for charged secondaries (Castagnoli method [2]), or the opening angle of fragmented a [3]. However, these become less accurate as the energy increases, as the resolution of the secondaries becomes much more difficult. Electromagnetic cascade showers initiated by -y-rays, which are radiated from nucleus interactions (hereafter called heavy-initiated showers), are recorded on highsensitivity X-ray film (for instance Fuji #200-type) as dark spots. These spots are detectable by the naked eye for showers with energy larger than - 1 TeV, and their darkness is nearly proportional to the shower energy . So detection of the signal becomes much more efficient as the incident energy increases . 0168-9002/89/$03 .50 (0 Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)

Methods of energy determination for y (electron)initiated cascade showers have already been established through large-area emulsion chamber experiments at mountain level [4], and those for hadron-initiated showers in lead (called Pb-jets) have been developed by Dake [5], Jones et al . [6] and others [7]. However, the energy determination for heavy-initiated showers is still not accurately known. In the present paper, we describe a unified method for the energy determination for any type of chamber design, either with wide gaps or alternate mixed substances, and for any kind of initial conditions, i.e ., types of incident nucleus, position of fragment in target layer and so on. In section 2 we summarize various problems inherent in the analysis of heavy-initiated showers which make the energy estimation difficult. In section 3 we present the procedure of the simulation calculation, and give numerical results for a standard type of emulsion chamber (hereafter called EC). In section 4, on the basis of extensive simulation calculations, we show how the above-mentioned problems are solved, and in section 5 we describe the practical process of the energy determination, and discuss the accuracy of the energy thus obtained . Section 6 is reserved for discussion . 2. Problems inherent in the analysis of heavy-initiated showers 2 .1 . Fluctuations in the process of nucleus-nucleus collisions

The relation between the energy of the incident nucleus, Eo , and that released into secondary -y-rays (,To -> 2y), EEC is expressed by

Y E,, = rl,r Eo

( ,q, : conversion factor),

31 8

T. Fujinaga et al . / Energy determination of electron- photon showers

where q y is closely related to the model of nucleus interaction, particularly to the fraction of "wounded" nucleons in the projectile nucleus, the inelasticity of each jet and the proportion of y's in the secondaries . Of course, in the case of nucleon-nucleus interactions, ply is equal to k y , the inelasticity converted into -y-rays, with (k y ) -- 1/4 [8] . Since the conversion factor rl,, fluctuates greatly, it is not meaningful to start simulation calculations from a fixed energy of the projectile nucleus E o and investigate the average behavior of heavy-initiated showers . From the practical point of view, it is important for us to estimate the energy 2E Y radiated into y's as accurately as possible, regardless of the subprocess of nucleus-nucleus collisions, all of which are incorporated into q,, . The distribution form of rl,, can be obtained either by simulation calculation, or experimentally by using heavy-ion accelerators, which will be reported elsewhere . From the above-mentioned viewpoint, we set beforehand EEY instead of E o , and we prepare a large amount of jet events having various types of y-ray multiplicity n Y as well as production spectra (Ei , B, ; i = 1, 2, . . ., n y ), which are all stored on "DISK" . The simulation procedure to generate such events is given explicitly in section 3 . Combining these events with the simulation program of cascade showers initiated by y-rays, we can obtain the transition curve of electron number vs depth in calorimeter for events initiated by heavy-primary collisions . 2 .2 . Problems relevant to the chamber structure ECs designed for cosmic-ray observations at the stratospheric level represent heterogeneous media for shower development ; as a result the following problems arise : (a) a large dilution factor, defined by (L +,A)/L, where 4 is the spacing between successive Pb plates, and L the thickness of each plate, (typically larger than in mountain experiments), (b) the measurement of electron number occurs a significant distance (8 - 5 mm) below the Pb plates, (c) heavy primaries interact in the target layers, a significant distance H above the Pb calorimeter, resulting in angular spread of photons . In the present paper, we call the above three problems d-, 8- and H-effect, respectively . 3. Simulation calculations 3 .1 . Outline of simulation calculations for electron photon showers Recently Okamoto and one of the present authors (T .S .) [9] developed three-dimensional Monte Carlo

calculations for determining the energies of electronphoton showers detected in EC . The calculations provide better accuracy and higher precision by taking account of contributions from the Landau effect, the transition effects, etc ., which are hard to solve analytically . Their simulation is applicable, even in the extremely high-energy region (-- 1000 TeV), for any type of chamber design, either with wide gap or alternate mixed-substances . We can summarize the essence of their calculations in the following . (a) The Landau effect [10], which is particularly important for heavy media such as lead in the region > 10 TeV . (b) Effects of ionization loss and deviation from the complete screening cross section (Bethe-Heitler cross section), which are effective in the region < 1 GeV . (c) The transition effect, coming from the inhomogeneity due to the insertion of photosensitive materials (X-ray films, nuclear emulsion plates, and so on) . (d) The effect of the "immediate gap", i .e., the effect of the gap between the bottom surface of the overlying dense materials and the position of the X-ray film or nuclear emulsion plate where the number of electron tracks is actually counted . (e) The effect of alternate mixed-substances, i.e ., mixtures of dense absorbers, such as carbon, lead and so on . The authors of ref . [9] developed two powerful methods to save computing time without sacrificing the accuracy of the numerical results, taking into account the above-mentioned effects (a)-(e) . The first employs approximate formulae for the cross sections of bremsstrahlung and pair creation processes, which reproduce well those obtained from the analytical calculations by Bethe and Heitler [11] and by Migdal [12] over the wide energy range 1 MeV to 1000 TeV . The second uses analytical formulae of multiple Coulomb scattering applicable when including successive electromagnetic processes, bremsstrahlung, pair creation and ionization loss . Consistency of their simulation is confirmed by comparing their results with FNAL electron-beam data [131, both for average cascades and for fluctuations . 3 .2 . Generation of y-rays originated in nuclear interactions Although we need only the production spectra of y's (Ei , Bi ; i = 1, 2, . . ., n y ) for fixed energy flow EE ., we have to start the simulation calculation from the process of nucleus collision with primary energy Eo , which should be set so that the energy sum radiated into y's is as close as the above fixed one EE, . Recently, two of the authors (Y .N . and T .S .) and Martin et al . [14] performed simulations of nuclear

T. Fujinaga et al. / Energy determination of electron -photon showers

31 9

Table 1 Statistics of simulation events Mass

EE Y [TeV]

Proton (A =1) Alpha (A = 4) Carbon (A =12) Silicon (A = 24) Iron (A = 56)

300 300 300 300 300

1

2 200 200

5 150 150

10

100 100 100 100 100

interactions on the basis of SPS data (CERN) [15] for nucleon-nucleon interactions and JACEE data [16] for nucleus-nucleus interactions . For nucleon-nucleon interactions, their simulations reproduced satisfactorily the single-particle inclusive distribution and the dispersion of the multiplicity obtained at both ISR and SPS. They assumed that nucleus-nucleus interactions are described by the superposition of nucleon-nucleon interactions, and showed the consistency in the portion of "wounded" nucleons in the nucleus-nucleus interaction by comparing with JACEE data . We summarize the essence of the simulation process to generate secondary particles in appendix A, combined with the cascade shower program package given by ref. [9]. On the basis of the simulation calculation of ref. [14], we put (kY ) - 1/4 (inelasticity released into y's), and (pw) - 1/3 (portion of "wounded" nucleons in the projectile nucleus), though the latter is not yet clear, which leads to

20

50

100

200

500

100

50

50

50

50

100

50

50

50

50

magnitude of the energy flow EEY is the most essential for the cascade shower development, and the effect of the production spectrum of each source y-ray is a second-order contribution, while that of scaling violation is of third order, as will be discussed in section 5. We performed simulations for several kinds of projectile on carbon targets and stored them on "DISK" after the above-mentioned procedure, which is summarized in table 1 . 3.3 . Chamber structure and initial condition As illustrated in fig. 1, we must explicitly define the chamber configuration as well as the type of incident particle in order to perform the simulation . Let us call the three variables 1, S and H configuration parame-

(71Y)=(kY)(pw)= ;~ . Now, as the region of energy flow EE, of interest here is 1-500 TeV, we set the energies of the primary nucleus .,/(q,,)) to 12, 24, 60, 120, 240, 600, 1200, Eo (=EE 2400 and 6000 TeV/nucleus, each corresponding to the energy flow EE, listed in table 1. After performing simulations using the above energies, we get a large number of events with various kinds of y-ray multiplicity, n Y, and various types of ;, B,'; i = 1, 2, . - -, n Y ). The enproduction spectra (E ergy sum of y's (=EEY) thus obtained is of course different from the fixed one given in table 1, from which we start the simulation of the electron-photon cascade. In order to normalize EE,' to EE,, we perform the following replacements for individual y's : =Ei' Ei' -Ei (3a) (FEY/yEY), Bí ->Br =Br (~EYIY_E,) .

H

(3b)

One may worry that such replacements, assuming scale invariance, might yield significant effects on the energy estimation of the practical shower. This does not, however, result in any serious problem, since the

Fig. 1. Illustration of the emulsion-chamber structure and features of the primary cosmic-ray interaction.

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T. Fujinaga et al. / Energy determination of electron -photon showers

Table 2 Numerical values of configuration parameters set for simulation calculations 4 [mm] ô [mm] H [cm]

1 0 0

2 1 5

3 2 10

In the practical chamber, many sensitive materials, such as X-ray film, nuclear emulsion plate, CR39 and so on, are inserted in the air gap d, and also in the target . Though the simulation program developed by ref. [9] is applicable even in such heterogeneous media, we regard them simply as air gaps in the present calculation, and later discuss how the effect of materialization of electromagnetic components in such media is taken into account. In table 1, we show five kinds of projectile . The calculations in the cases of proton and iron are performed completely for all energies and for all combinations of configuration parameters . Only selected en-

4 3 20

ters . We set these configuration parameters to cover the dimensions of practical chambers which are given in table 2. We fix the thickness of each lead plate to L = 2.5 mm, the standard value in balloon-borne experiments.

10 4

10 3

10Z

0

10 -1

0

1

2

3

4

5

6

7

8

9

10 depth (cm Pb)

depth (cm Pb)

104

10 3

102

10 1

10 '

10-1

0

1

2

3

4

5

6

7

8

9

10

depth (cm Pb)

Fig. 2 . Transition curves of the proton- and the iron-initiated showers for the configuration parameters (a) ô= mm, (b) H= 21 = 0, 5 = 0, 1, 2, 3 mm and (c) 21 = 8 = 0, H = 0, 5, 10, 20 cm .

H= 0, d =1,

2, 3, 4,

Table 3 Numerical values of electron number within a radius r for the cases of proton-initiated (a) and iron-initiated (b) showers (a) EEC, [TeV]

r :5 [JIM]

Depth [cm Pb] 1 2

1

25 50 100 200 500 1000

21 .2 32 .3 43 .6 54.5 68 .1 77 .0

40 .6 83 .4 141 .2 214.4 329 .6 423 .1

39 .0 101 .2 209 .4 372.2 690.5 1001 .7

29 .1 86 .7 209 .1 425 .9 921 .4 1478 .8

18 .1 57 .3 156 .0 357 .4 895 .5 1590.9

9 .8 33 .9 100.7 253 .3 712.6 1372 .7

2

25 50 100 200 500 1000

33 .7 45 .6 58 .1 70.0 86 .0 95 .8

86 .2 150.0 227 .8 324.3 473 .2 594.3

104.8 221 .2 403 .7 661 .9 1143.4 1603.4

86.7 215 .4 451 .5 841 .7 1694 .7 2632.9

55 .4 159.3 377 .6 787.5 1816.3 3098.7

5

25 50 100 200 500 1000

51 .9 66.6 81 .9 96.9 115 .9 129.3

172.8 266.7 385 .3 525 .4 743 .9 914.3

274.1 488 .5 807.5 1250 .0 2053.9 2804.1

279.5 566 .7 1058 .8 1836 .1 3476 .9 5217.5

10

25 50 100 200 500 1000

67 .7 84.8 100.7 116.7 137 .9 151 .2

267 .5 391 .9 546.4 723 .9 997 .9 1214 .8

490 .3 817 .5 1295.7 1947 .6 3099 .1 4143 .1

20

25 50 100 200 500 1000

85 .0 104.6 125 .6 144.3 168.7 185 .2

402 .3 567.0 771 .1 1006.0 1356.8 1632.5

50

25 50 100 200 500 1000

106.0 127.6 151 .1 173 .7 200.4 218 .8

100

25 50 100 200 500 1000

200

500

8

9

10

5 .0 17.5 56 .5 154.2 482 .3 1005 .8

2 .1 8.2 27 .9 83 .7 290 .0 647 .0

1 .1 4 .0 14 .3 43 .1 159 .2 377 .5

0.5 1 .8 6.5 21 .0 82.9 206.3

31 .7 100 .0 261 .1 599.1 1553.1 2881 .1

17.2 57 .5 159 .1 394 .2 1131 .7 2256 .8

8 .3 29 .2 89 .1 236 .6 738 .0 1564 .6

3 .8 14.4 45 .8 129 .9 431 .2 974.7

2.0 6.9 23 .1 67.1 237.5 560.6

226.4 507 .9 1042.7 1984 .5 4240.5 6926.1

152.0 375 .2 837.8 1732 .6 4115 .4 7245.6

91 .2 242 .0 578 .9 1286 .7 3360 .8 6336 .1

48 .1 138 .3 357 .4 852 .9 2423 .2 4857 .8

23 .7 73 .6 202 .9 511 .6 1564.1 3318.1

11 .2 36 .4 106.3 286 .0 931 .9 2082.3

582 .9 1089 .2 1918 .3 3184 .6 5769 .3 8446 .6

520 .3 1077 .3 2089.9 3820 .3 7785 .4 12348 .0

390 .4 884 .9 1859 .7 3670 .0 8246 .5 14074 .8

246 .6 607 .1 1377 .6 2941 .5 7306 .4 13331 .5

141 .9 375 .1 911 .3 2072 .8 5609 .8 10904 .0

75 .6 212.6 549.3 1319.8 3841 .6 7889.6

38.2 111 .8 306.8 768 .4 2390 .3 5187 .4

854 .4 1360.3 2069 .4 3009 .8 4650 .6 6122 .0

1126 .4 2016 .2 3408 .0 5497 .4 9620 .0 13778 .1

1100 .3 2171 .8 4053 .0 7169 .2 14063 .5 21784 .9

885 .9 1901 .4 3851 .5 7350 .4 15997 .8 26676 .7

607 .0 1408 .7 3076 .2 6315 .9 15090 .7 26855 .0

379 .1 940 .2 2181 .9 4752 .8 12281 .7 23227 .5

218 .7 570.4 1402.3 3232.9 8961 .0 17868.8

118 .1 321 .1 826 .5 2011 .8 5958 .3 12454 .4

563 .4 779.6 1042 .1 1339.3 1777.1 2115 .2

1357 .8 2088 .4 3089 .8 4402 .8 6633 .9 8612 .5

2141 .4 3614.0 5869 .6 9118 .0 15367 .8 21498 .2

2652 .7 4698 .7 8237 .6 13794 .0 25643 .0 38434 .1

2481 .3 4874 .6 199 .1 16588 .1 33686 .1 53803 .5

2030 .9 4302 .4 8644 .3 16599 .8 36655 .4 62067 .1

1478 .4 3313 .7 7095 .7 14443 .5 34322 .4 61373 .2

963.1 2293.5 5220.6 11204.3 28541 .0 53582.3

577 .8 1453 .5 3473 .5 7840 .3 21335 .8 42001 .5

122.3 144.9 167 .8 188.7 216.1 233 .6

728.9 978.3 1273 .5 1608.8 2100 .1 2481 .1

2006 .6 3010.3 4344 .1 6040.7 8889 .4 11351 .2

3487 .5 5692 .4 9000 .5 13693 .1 22485 .1 30939 .5

4569 .3 8107.1 13807 .2 22578.8 40660 .8 59774 .3

4822 .3 9167 .7 16772 .0 29366 .6 57659 .1 90117 .1

4312 .9 8730 .0 17016 .0 31682 .9 67403 .7 111530 .5

3444 .6 7354.7 15134 .7 29746 .6 67883 .6 118204 .5

2493 .6 5572.5 12067 .8 24899.1 60591 .5 110543 .2

1696 .5 3947 .6 8863 .5 19061 .5 49020 .8 93227 .2

25 50 100 200 500 1000

134.7 157.6 181 .8 204.4 234.4 253.7

867.1 1157 .6 1499 .3 1882 .7 2447 .0 2871.2

2537 .2 3738 .7 5331 .8 7371 .3 10753 .1 13659 .4

4806 .6 7710 .3 11964 .8 17906 .1 28874 .3 39316 .1

6903 .3 11847 .5 19719 .4 31634 .6 55776 .2 80850 .1

8223 .1 14939 .0 26372 .8 44821 .6 85190 .6 130495 .6

8406 .4 16156 .4 30044 .4 53797 .2 109355 .5 175744.6

7557 .9 15297 .3 29916 .9 56189 .8 121642 .2 204733 .6

6189.3 13089.8 26750.4 52452.4 120351 .6 211134.0

4675 .8 10214 .2 21778 .3 44525 .2 107739 .3 196471 .9

25 50 100 200 500 1000

156.3 185 .5 215 .2 244 .5 281 .3 305 .3

1048 .4 1384 .1 1780 .0 2224 .7 2888 .7 3386 .9

3388 .4 4897 .6 6876 .7 9366 .4 13420 .7 16890 .7

7288 .3 11329 .7 17124 .6 25045 .7 39398 .2 52808 .3

11917 .0 19740 .7 31792 .1 49507 .0 84337 .3 119627 .9

15801 .4 27756 .0 47350 .8 77952 .0 142673 .0 213124 .3

18261 .8 33541 .9 59959 .6 103563 .4 202006 .1 315914.9

18646.1 35795 .6 66882.1 120622.4 248889 .6 405301 .4

17391 .2 34695.9 67342.8 126105 .6 273497 .0 462189 .1

15054 .8 30996 .3 62199 .1 120450 .6 273543 .6 478224 .6

3

4

5

6

7

Table 3 (continued) (b) EE, [TeV]

r< [gym]

6

7

8

9

1

25 50 100 200 500 1000

12 .6 30 .8 66 .1 120 .6 211 .9 281 .9

14 .3 41 .3 107 .4 240 .1 557.3 884.0

11 .9 37.3 107.6 273 .0 757 .5 1379.5

7 .9 27 .7 84 .6 229 .3 727 .5 1468 .5

4 .7 17 .1 55 .2 160 .0 553 .4 1219 .9

2 .6 9 .5 32 .8 100 .3 371 .7 864 .1

1 .3 4 .7 17 .2 55 .5 222 .0 547 .7

0 .7 2 .3 8 .3 28 .6 120 .2 313 .6

0 .2 0 .9 3 .5 12 .8 59 .0 163.1

0 .1 0 .4 1 .5 5 .7 27 .4 80 .5

2

25 50 100 200 500 1000

30.8 66 .9 131 .8 216 .0 330.5 411 .2

43 .8 109.8 249 .2 507 .8 1040.4 1528 .4

39 .6 113 .4 283 .3 647 .6 1586 .6 2656 .5

26 .6 85 .7 238 .7 591 .4 1650 .7 3079 .4

16 .3 55 .3 165 .0 437 .7 1358 .1 2763 .4

9 .1 31 .8 100 .2 281 .1 942.1 2057 .7

4 .5 16 .9 54 .7 162 .4 584 .5 1361 .2

2 .3 8 .7 28 .8 87.4 332.3 809 .3

0 .9 4 .0 14 .3 46 .0 180 .3 451 .3

0 .4 1 .9 7 .0 22 .1 91 .6 237 .5

5

25 50 100 200 500 1000

89 .2 165 .3 263 .5 375 .3 522 .6 619 .2

148 .3 338 .7 665 .0 1146 .3 2008 .1 2753 .0

148 .3 390 .9 885 .1 1753 .4 3638 .4 5615 .1

114 .3 332 .2 833 .5 1837 .9 4384.7 7481 .0

74 .8 231 .2 623 .1 1500 .7 4048 .9 7535 .4

42.2 138.8 399.2 1032 .3 3080.4 6181 .6

23 .1 76 .6 229.2 628 .3 2050.9 4407.3

12 .1 39.9 122 .3 349 .2 1231 .7 2809.3

5 .5 19.7 63 .1 185 .9 686 .3 1640.5

2 .6 8 .9 29 .6 91 .0 358.9 899 .9

10

25 50 100 200 500 1000

152 .7 252 .9 369 .1 490 .6 650 .8 756 .1

329 .0 642 .1 1130 .7 1786 .8 2875 .2 3795 .5

397 .0 877 .6 1746 .9 3144 .3 5945 .0 8746 .7

372.5 884 .4 1902.2 3754.1 8088 .1 13054.4

273 .6 695 .1 1630 .0 3481 .8 8387 .0 14689 .6

171 .4 474.3 1187 .1 2715 .7 7181 .9 13475 .7

96 .0 280 .1 753 .6 1836 .2 5270 .2 10581 .6

53 .4 158 .3 440 .8 1126 .8 3481 .8 7394 .4

28 .5 86 .1 245 .7 650.3 2120 .6 4708 .4

13 .5 43 .3 128 .1 355 .8 1226 .6 2830.1

20

25 50 100 200 500 1000

240 .2 353 .6 476 .5 602 .5 762 .3 870 .0

623 .5 1094 .8 1749 .3 2564 .3 3888 .1 4975 .4

891 .9 1754 .1 3156 .2 5227 .2 9187 .8 13015 .2

912 .5 1968 .7 3895 .3 7100.6 14130.6 21858 .7

748 .8 1743 .7 3732 .9 7383 .1 16358 .4 27321 .4

531 .9 1309.6 2988 .0 6339 .5 15399 .0 27596 .4

332 .7 870.2 2120 .1 4756 .0 12506 .2 23799 .7

187 .7 517 .5 1330 .3 3164 .4 9004 .2 18140 .6

99 .0 291 .0 777 .9 1938 .1 5897 .0 12492 .3

47 .1 146.3 417 .6 1100.4 3562 .4 7915 .9

50

25 50 100 200 500 1000

507 .8 699 .3 901 .5 1103 .6 1363 .1 1535 .5

1541 .6 2527 .7 3816 .1 5378 .7 7858 .8 9864 .0

2391 .9 4438.7 7579 .8 12064 .3 20374.6 28207.3

2505 .7 5197 .3 9906 .4 17420 .9 33423 .6 50639 .0

2068 .1 4686 .3 9747 .7 18763 .7 40354 .4 66291 .7

1477 .5 3573 .4 7997 .6 16563 .2 39283 .4 69306 .4

948 .3 2410 .0 5731 .5 12706 .5 32727 .0 61471 .5

556 .5 1480 .0 3726 .7 8733 .2 24265 .6 48098 .8

315 .7 859 .1 2239 .0 5453 .6 16228 .7 33858 .4

168 .6 475 .8 1275 .3 3245 .8 10129 .4 22040.6

100

25 50 100 200 500 1000

631 .6 816 .6 1014.2 1211 .6 1464.2 1630.4

2307 .9 3487 .2 4982.5 6767 .3 9530 .6 11768.2

4109.7 7019 .3 11292.5 17180.5 27767 .8 37544.3

4962 .7 9375 .6 16605 .4 27833 .2 50888 .9 74932 .2

4704 .3 9606 .7 18519 .0 33650 .4 68418 .8 108696 .3

3827 .7 8350 .0 17152 .3 33363 .7 74199 .5 125913 .6

2819 .0 6452 .3 14004 .1 28735 .2 68900 .4 123640 .4

1888 .0 4515 .3 10304 .3 22229 .1 56944 .9 107319 .9

1179 .3 2920 .6 6946 .1 15664 .7 42617 .4 84082 .7

723 .9 1826 .0 4472 .7 10415 .4 29719 .9 60866 .2

200

25 50 100 200 500 1000

739.2 919.3 1110 .6 1298 .4 1534.9 1689 .8

3240.3 4675 .4 6434.3 8487 .6 11620.6 14085 .9

6678 .0 10832.5 16690 .3 24580 .3 38387 .0 50825 .2

9140 .1 16297 .1 27619 .6 44617 .3 78300 .6 112494 .4

9706 .8 18721 .5 34371 .7 60016 .3 116569 .2 180012 .9

8686 .3 17892 .4 35143 .3 65490 .9 138881 .8 228343 .4

6969 .6 15058 .7 31163 .9 61439 .6 140296 .2 243804 .6

5039 .3 11429 .7 24849 .6 51389.2 125220.5 228363 .0

3401 .0 8020 .1 18169 .6 39212 .7 101136 .1 192586 .1

2139 .3 5255 .9 12373 .4 27754 .7 75377 .9 149268 .9

500

25 50 100 200 500 1000

1101 .2 1340 .8 1597 .4 1845 .2 2159 .0 2371 .7

5429 .5 7581 .5 10206 .9 13271 .1 17886 .3 21473 .4

12699 .6 19730 .1 29500 .4 42342 .7 64510 .4 84186 .1

19635 .3 33403 .6 54561 .6 85393 .6 144975 .5 204137.3

23163.9 42579.8 75103.4 126816.1 237275 .7 357554.5

22542 .0 44348.9 83802.9 151070.8 307903 .1 493442.2

19237 .6 40008 .1 79974.2 152674.1 335526 .6 567924.4

14810 .0 32329.1 67940.5 136192 .8 316949 .2 568090 .5

10664 .6 24102.1 52702.7 110311 .8 274190.7 508703 .4

7220 .6 16875 .1 38220 .2 82836 .4 216355 .6 416960.3

Depth [cm Ph] 1 2

3

4

5

10

T Fujinaga et al. / Energy determination of electron -photon showers

source -y-rays, and on the configuration parameters ( .A, 8, H), i .e . the electron number is expressed as Ne (A, EE Y , r, t; A, 8, H), where r and t are radius and material depth respectively. In the following discussion, however, we express it as Ne (4, 8, H) for the sake of simplicity . We wish to find a general relation between N,(1, 0, 0) and Ne (4, 8, H) . Firstly, we assume the following approximate relation ;

ergies are used for alpha, carbon and silicon calculations, since the purpose of the latter three is to check the mass number dependence of the energy estimation (see section 5 .2) .

3.4. Numerical results We call the chamber with configuration parameters A = 1 mm, 8 = H = 0 and L = 2 .5 mm the standard one . In tables 3a and 3b we present explicitly the electron number within radii of 25, 50, 100, 200, 500 and 1000 I.Lm for the standard chamber in two cases of the initial conditions : initiated by a proton jet and by an iron jet respectively. To examine the effect of changing the configuration parameters, we show three cases of transition curves in figs . 2a, 2b and 2c, corresponding to the following sets of configuration parameters respectively; (a) (8, H) = (0, 0), and A = 1, 2, 3, 4 mm; (b) (H, A) = (0, 1), and 8 = 0, 1, 2, 3 mm ; (c) (1, 8) = (1, 0), and H = 0, 5, 10, 20 cm . It is obvious that the configuration parameters are important for the energy estimation, particularly in the case of heavy primaries such as iron in an energy region as low as a few TeV .

N,(á, 8, H)

_

1 .0 F

(q,( A )g2(8)q3(HANe(1,0,0),

v

+

:

i ron

qi (x) cc x -R=

(i = 1, 2, 3) for large x,

1 .0 r- N e (L O,H)

N e (1 , b, 0)

0,5

0 .5

0 .2

0 .2

0

0.2

0,1

.+

5

10

0\

20

(4)

with 3=A-1 mm . Since the validity of the above approximation is limited to small values of 8 and H, we have to perform a higher-order correction, which will be presented later. Secondly, we demonstrate the features of the three quantities q, (i = 1, 2, 3) in eq. (4) for A, 8 >> 1 mm and H >> 1 cm at shower maximum in fig. 3, where we performed simulation calculations in the case of EEY = 1 TeV besides the numerical sets presented in table 2 . Fig. 3 indicates that all display a powerlike attenuation at large values of the configuration parameters, i.e.,

N e (1,0,0) proton

0,5

s .e

XNe (1, 0, 0)

Generally, the electron number Ne depends on initial conditions, mass number A and energy flow, EE-Y, of

N e (1,0,0)

N,(1, 8, H) N, (1, 0, H) ) H: fix

N~(A, 0, 0) Ne (1, 8, 0) Ne (1, 0, H) Ne (1, 0, 0) N, (1, 0, 0) Ne (1, 0, 0) )

4.1 . Correction for (A, 8, H) effects

N e (A,0,0)

N, (A, 8, H) Ne (1, 8, H)

) (I, N,(1, 0, X~ H) Ne 0, 0) N,(1,0,0)

4 . General relation of transition curves between standard chambers and practical chambers

1 .0 F

323

0,1

u1 10

20

5

10

A - 1 (mm) Fig. 3 . Attenuation of the electron number at large values of the configuration parameters .

20

324

T. Fujinaga et al. / Energy determination of electron -photon showers ö (mm)

1 .0

0

1

2

3

4

5

6

7

0

ó

s

8

9

Ó

tan 6 0 .0

0 .5

0 .5

X O ZE

X

á

:

O

0 .2

.

simulation data ~

(

tan 9 = 0 .0 )

experimental data ( tan 0 = 0 .0 - 0 .5 )

Fig. 4. Attenuation of the electron number within a radius of 100 O m after leaving the heavy media (iron plate here) at the shower maximum. Open circles are obtained by electron-track counting in a nuclear emulsion plate, and open-dot ones estimated from the spot darkness on X-ray film .

where ß, seems to depend on the type of projectile. Of course, we have the condition q ; (x) - 1 for x - 0,

(6)

so that, considering eqs. (5) and (6), it is reasonable to

0,90

0,95

1 .00

1 .05

1 .10

assume the empirical functional form of q,(x) to be of the type

Rr(x) -

0 .95

1,00

1 - e - U;(x) U (x)

1 .05

0 .95

Ncorrection 7 N true

Fig. 5. Histogram of Ncorrection/Nrue for proton jets and iron jets .

(7a)

1 .00

1 .05

325

T. Fujinaga et al . / Energy determination of electron - photon showers

with U, (x) =-,x ß ,

10 5

N e (r<_100

(7b)

The parameters a, and ß, are determined by fitting eq . (7) to simulation data using the least-squares method . The explicit parametrizations, which depend on A (the mass number of the projectile), EE, r and t, are summarized in appendix B. In order to check the validity of the functional form of eq. (7), we present the experimental data of q2(8) at shower maximum in fig. 4, obtained by the wide-gap-type EC with iron absorber exposed at Mt. Fuji [7], where shower energies are 5-10 TeV and zenith angles within 30 °. Also shown are simulation data calculated by one of the authors (T .S .) [7]. The attenuation curves from the present work, drawn as solid lines, are obtained by putting a = 1 .435 and ß = 0.631 in eq. (7), using the least-squares method . The simulation data are in good agreement with the experimental data, and the functional form of eq . (7) reproduces both surprisingly well . Thirdly, we further refine the approximate expression eq . (4) . On the basis of the simulation data with various sets of (a, 8, H) (see table 2) we found it was necessary to introduce two more parameters, is and p, in order to correct the deviation, and finally we obtain the following expression : N, (a, s, H) = C(3, 8, H) Ne (1, 0, 0),

(8a)

C (J, 8, H)=K[q,( 3 )q,(8)q,(H)1 ",

(8b)

where

micron)

materialization in target neglected O : materialization in target included " : Parallel shifting by 1 c,u, for O o94 9o0OZ 5ooi5o6tó

a~.oáQx, 10

o°z le o --a °-ä

10 2.

ooó~a

o--a x

o .RQM

oooó66

66

oöxx oot 1 TeV

A = 1 mm

0_4

a-á x

101~~ 0 1

a =

0 mn

4

5

6 %_

x

H=20 cm 2

TeV

3

6

7

8

9

10

depth (cm Pb) Fig. 6. Transition curves of the electron number within a radius of 100 Wm initiated by an iron jet, which occurred at the hypothetical target with a radiation length of 20 cm and critical energy 20 MeV. Interaction height H is assumed to be 20 cm above the calorimeter.

4 .2. Correction for heterogeneous effects

p = 1 + CO SH + CI H3 + C2 2'S + C3a6H,

(9a)

In ic = D08H + D 1 113 + D2 38 + D33SH .

(9b)

The parameters Ct , D l (l = 0, 1, 2, 3) depend on A, EE,, and t, and are independent of r . The explicit numerical values for these are summarized in appendix B. In order to check the accuracy of eq . (8), we calculate the electron number, Ncorrecdon, estimated using the correction factor C(,~, 8, H) and Ne(1, 0, 0), and compare it with the true one, N«e (= Ne(A, 6, H)) . In fig. 5, we show the ratio Ncorrection/Ntrue, indicating that the error due to the approximate relation eq . (8) is negligible (a few %) in comparison with that coming from fluctuations of the cascade shower (see section 5 .2). Finally, we can derive transition curves for an arbitrary structure of EC with configuration parameters (A, 8, H), if we have the standard transition curves Ne (1, 0, 0) and the correction parameters (a i , ß i , ic, w) . Practically, we store all the numerical information about Ne (1, 0, 0) and (a,, ß i , ic, p,) on "DISK", and calculate automatically transition curves for the chamber we need, using a microcomputer (see section 5.1).

First, let us consider the behavior of a cascade shower in a mixed substance as shown in fig. 1, where we assume a hypothetical target with Xo = 20 cm (radiation length) and e = 20 MeV (critical energy) on an emulsion calorimeter . Now, we start a full Monte Carlo calculation of the cascade shower, taking into account the materialization of electromagnetic components in the target section, from a height of 20 cm in the target, i.e. 1 c.u . above the calorimeter section, where we regard the space A in the calorimeter part as air gap. Here, we prepare jet events with energy flow EE,, = 1, 10 TeV, created by iron collisions . We demonstrate the transition curves of electron number within a 100 ltm radius in fig. 6, where we show two results together, one including materialization in the target (white circle), and the other neglecting it (cross mark). It is immediately apparent that the former grows significantly faster than the latter, as expected . Looking carefully, however, it is remarkable that the difference is just the parallel shifting by 1 c.u . along the horizontal axis, corresponding to the thickness of target through which the source y-rays pass .

T Fujinaga et al. / Energy determination of electron- photon showers

326

homogeneous lead absorber . For such heterogeneous media, we must apply full Monte Carlo calculations, taking into account the exact chamber configuration as well as material constants such as radiation length, critical energy, etc. Practically, however, the approximate method mentioned here is quite acceptable for common types of EC both in mountain and balloonborne experiments . Finally, we conclude that we can correct for the heterogeneous effects as follows : step 1 : treat all materials, except heavy absorber (lead), as air gap, step 2 : convert the thickness of each material into c.u ., and then add it to those of the lead plate sequentially .

( N e (r550 [Im) 100

10

depth (cm Pb) Fig. 7. Transition curves of the electron number within a radius of 50 ~tm induced by an electron-pair primary.

5. Practical method for energy determination 5.1 . Automatic system for energy determination

Using a similar method, we can correct the heterogeneous effect due to the insertions of sensitive materials, such as nuclear emulsion plate, X-ray film and so on, in the calorimeter section by shifting the integrated thickness of sensitive materials in units of c .u., as shown in fig. 7, where we compare two cases, one assuming a 500 pm air gap and the other a 500 ltm nuclear-emulsion gel filling . The above considerations are, of course, not applicable for thick heterogeneous media inserted between heavy elements (lead), since the critical energy effect becomes serious, and the lateral displacement of cascade showers will be significantly deformed from those in a

TRACK-COUNTING FOR ELECTRON NUMBER ATI EACH DEPTH tr,tz. . . . . t Ne(1;, r
2. CALCULATION OF TRANSITION CURVE Ne(A,IE,,r,T; n, d, 01 A = proton and iron LES = 1, 2, . , 500 TeV tang = 0, 0.2, . . ., 2.0

(write) 3. MOST LIKELIHOOD METHOD FOR FITTING Ne( calculation ) TO Ne( track-counting )

In this subsection, we summarize * a procedure for energy determination and demonstrate examples of fitting automatically theoretical transition curves to experimental ones . Here we focus our discussions only on electron-track-counting data obtained from a nuclear emulsion plate. The flow chart of both processes of microscopic measurement and energy determination is illustrated in fig. 8, and essential points are explained below. After registering the chamber structure, we calculate the transition curves using eq . (8), where we always calculate two kinds of curves, one initiating from a proton jet and the other from an iron jet. At this stage, we also calculate the relation between Nmax (maximum electron number) and EE .,, which we use to determine the final energy . In fig. 9, we show this relation in the case of 8 = 1 .39 and S = 480 ~Lm, corresponding to the configuration designed for a recent balloon-borne EC experiment [19] . In fig. 9, we draw curves in the case of electron-pair-initiated primaries (y - e +e- ) together, which is obtained from the simulation program package given in ref. [9]. It is remarkable that the difference among the three initial projectiles is rather small in spite of the fact that the multiplicities and the production energy spectra of source y-rays are extremely different. This is because the EM-cascade development reflects approximately the energy conservation, and the nature of source -y-rays is not so significant . Of course, we should take care that the difference among the three becomes significant as the radius becomes small, so that for heavy-initiated showers we should use a large slit

4. DETERMINATION OF £E-

Fig. 8. Block diagram of the automatic energy-determination system which makes use of a microcomputer .

* Each step of the software system on the energy determination is summarized in detail in ref. [25] . Practical application of this software system is given in ref. [26] .

327

T. Fujinaga et al. / Energy determination of electron- photon showers 1000

100

Nm 10

0

2

e

4

ZE, = ~.75 TeV 8

10

12

14

16

77 18

20

Fig. 10 . Example of the automatic curve fitting to experimental plots, where the electron-track counting is done within a radius of 75 (rm.

size, say at least 300 ~Lm, to measure the shower spot darkness on X-ray film using a photodensitometer.

Since experimentally the interaction positions differ from event to event, we set H = 0 in this step, and take this effect into account for each event in the final stage

(see the end of this subsection). 10 10 010 10 , 2

Now we fit the theoretical transition curves thus

3

obtained to those obtained by experiment, using the

least-squares method, details of which are summarized

IE, (TeV)

in ref. [191 . Let us demonstrate an example of curve

Fig. 9. Correlation between N,ax and EEY for three sets of initial conditions, electron pair (dotted curves), proton jet (solid one) and iron jet (broken ones).

fitting in

fig.

showers [191 .

10, originating from

proton-initiated

30

20

10

0.6

0 .8

1.0

1 .2

1.4

1 .6

0 .8

1 .0

1 .2

JE Y /FEY

Fig. 11 . Histogram of EE,, /EE Y for proton jets and iron jets .

0.8

1 .0

1 .2

328 0 .5

T Fujinaga et al. /

nergy determination of electron - photon showers of mass number A . Then we determine the energy EEC, using the following equation obtained by the propor-

Q

tional allotment method :

Proton jet

0 .4 +

.

iron iet

rEY =

0 .3-

(I -

p)rE ( ') + pl.E (2),

p = In A/In 56 .

0 .2

O

0 .1-

5.2. Accuracy of energy determination

+

O

O

0 .01

L

In this section we investigate the accuracy of the

n d`

+ O

energy determination presented in section 5 .1 .

There are two main sources of error in the present

method, one arising from the correction for the cham-

JE Y (TeV)

ber structure and the other from both fluctuations in the production spectrum of source y-rays and subse-

Fig. 12 . Correlation between a and EEC where a denotes the . standard dispersion of EEC /Y-E,

quent cascade development.

For the first error, due to (4, S, H) effects, we have

already shown in fig. 5 that it is of the magnitude of a

In order to get the final energy, we have to further

few %, which is negligible in comparison with the latter caused by the above-mentioned fluctuations . So we

take into account the two following effects. For the H effect, it is necessary to perform the

consider here the latter error only .

following replacement (see eq . (8)), Nmax - Nmax/ [ 93 ( H)] ~It=rm .~r

The transition curves we presented in sections 3 and 4 are the average ones, obtained by superposing a large

(10)

number of simulated data with statistics summarized in

replacement is made, we can obtain the corrected energy from fig. 9. For the mass-number effect, the following procedure

table 1 . Therefore, one worries naturally that those of

Once this

individual events deviate considerably from the average one.

is used . Since we have two transition curves, one originating from a proton jet and the other from an iron jet, we get two energies EE(" and EE,t(2' by fitting them

In fig. 11 we present the distribution of the energy

ratio Y_EY/Y,E ., for proton- and iron-initiated showers, in

the case of the standard chamber configuration, where EE., is the energy determined by the method of

to the experimental one originating from the projectile

30

ó

£E Y = 10 TeV

20

10

0 0 .8

1 .0

1 .2

0 .8

1 .0

1 .2

0.8

1 .0

1.2

EEY/£E Y

Fig. 13 . Histogram of EEy/EE C, for alpha, carbon and silicon jets, obtained by the proportional allotment method .

329

T Fujinaga et al. / Energy determination of electron- photon showers

section 5 .1, and LE Y the true one. We find that the spread is a widest in the lower energy region, particularly for iron jets . To see more quantitatively the reliability of the energy determination, let us show the correlation between a and EE,( in fig. 12, where a is the standard deviation of Y-EY/EEY. These results are well within our allowance for proton jets, comparable with or less than the experimental error (t1 Ne/N, - 15%), while somewhat worse for iron jets with energies as low as 1-2 TeV. However, we only detect iron-induced showers with EEY > 2-3 TeV, because of the detection limit of dark spots on X-ray film, so that the accuracy of the present method is sufficient for practical purposes . Here, we check further the reliability of the proportional allotment method to get the energy for projectiles other than proton and iron as presented in section 5 .1 . In fig. 13, we present the Y-EY/EEY distributions for alpha-, carbon- and silicon-initiated showers, obtained by the proportional allotment method, showing that they distribute around EE,'/EE, = 1, and that the spreads become wider with increasing projectile mass.

10 3

10 2

10°1 0

6. Discussion On the basis of simulation programs given by refs . [9] and [14], we have performed extensive calculations in order to establish a method of energy determination for cascade showers initiated by heavy cosmic-ray primaries as well as by light ones such as protons and alphas . We have succeeded in correcting various effects due to the chamber structure to within a few % . We have quantitatively investigated the error due to fluctuations coming from both the production spectrum of source Y-rays and subsequent cascade development . In this section we discuss further problems not treated sufficiently in the previous sections. The first problem is the thickness of the lead plate, which we fix at 2.5 mm in the present paper. Although the value L = 2.5 mm is a standard one in EC experiments at stratospheric levels, we sometimes use lead plates with a thickness of 1 mm or 5 mm, the latter used often for airplane experiments [20] . This may raise some doubts as to whether or not the present method is applicable for those values different from the standard value of 2.5 mm . In regard to this question, one of the present authors (T .S .) [7] has pointed out that the following relation holds, so long as A/L is not extremely large, say > 5 : Ne (A, L)=N,(a', L')

10 4

if o/L=a'/L',

(12)

i.e . the transition curve is not affected by the choice of (,~A, L) if the dilution factor (A+L)/L remains unchanged. In fig. 14, we show examples of transition curves for two different chamber configurations, one

1

2

3

4

5

6

7

8

9

10

t (cm Pb) Fig. 14 . Transition curves of the electron number within a radius of 100 wm for two chamber configurations, (8, L)=(1 mm, 2 .5 mm) and (d, L) _ (4 mm, 1 cm).

corresponding to the standard type of chamber 8 = H = 0, 4 = 1 mm and L = 2.5 mm, and the other to a configuration with 8 = H = 0, A = 4 mm and L = 1 cm, where one should note that both have the same dilution factor 1.4. One finds the deviation is - 5% at shower maximum even in the worst case, EEY = 1 TeV, and almost negligible in the region EEY > 10 TeV. Practically, therefore, we make the following replacement for a chamber with a configuration (4', L') (L' 0 2.5 mm), Al

-a-a'XL,

(L=2 .5mm),

(13)

and then we apply the numerical method summarized in sections 4 and 5 for the chamber we need . The second problem is the zenith-angle dependence of the projectile. Remembering the above-mentioned consideration, the correction factor defined by eq . (8) is now modified to be C(0, 8, H, cos 0)

=K[ q,(i)g2( 8/cos 0)g3(H/cos 0)1 ~`. (14) Here, recalling further that we count electron tracks within some radius r in a sensitive material set horizontally, we need to integrate electrons inside an ellipse

330

T Fujinaga et al. / Energy determination of electron -photon showers 100

104

directly the above integration as follows:

Tev

N,(r, cos 0)

y) = f f dzNe(x> 3x oy, dx dy

s S = x 2 + V 2/COs 29 < r2 .

10 2

10 0

t/cos9 (cm Pb)

Fig. 15 . Transition curves of the electron number within a radius of 100 hm for inclined projectiles (proton and iron).

(15)

In fig. 15, we illustrate examples of transition curves for several cases of tan 9, where the configuration parameters are the same as in fig. 9. The third problem is the effect of successive nuclear interactions induced by fragments of the first primary interaction during the development of the electromagnetic cascdde shower . The present transition curves, on the other hand, are obtained by the primary nuclear interaction only, so that if fragments (p, n, a, iT ±, - - - ) of the primary produce nuclear interactions subsequently in the calorimeter section, we must take these contributions into account. For this problem, we superpose two or more theoretical curves together, fitting them to the experimental plots using the least-squares method . An example of this is shown in fig. 16, obtained by a recent balloonborne experiment [19], where the second peak is induced by a residual proton . Here the experimental data denote the transition of spot darkness registered on X-ray film, obtained by a photodensitometer, and the theoretical curves are those converted from electron density to spot darkness D. Details of technical studies of cascade showers using X-ray film will be reported in the near future. Acknowledgements

along the incident direction, with major axis 2r and minor axis 2rcos 9. In the simulation process we save all electron numbers within the small area AS = 6 .25 X 6 .25 p m2 on magnetic tape, so that we can perform

1 .0m

0.10

0.02

wl .

0

2

4

S -

30 pm

H =

1.3 cm

~~_

6

8

10

L

12

fE 1 r

~~

14

= 4.24 TeV

1E

\

18

20

Fig. 16 . An example of a successive nuclear interaction. The vertical axis denotes the darkness of the cascade showers registered on X-ray film, which is measured by a photometer with 300 x 300 11m2 slit size .

We are greatly indebted to the JACEE members, particularly S. Dake, M. Fuki, T. Ogata, T. Tabuki and T. Tominaga, who have given us valuable discussions and comments since the beginning of the present work . Acknowledgement is also due to P. Edwards for a careful reading of the manuscript and valuable comments. We are much indebted to H. Nanjo and H. Matsutani for stimulating discussions and for allowing us the use of the Sanriku balloon data . Appreciation is also extended to K. Yokoi who gave us valuable and helpful comments . Numerical calculations were performed using of ACOS 950 (Aoyama Gakuin University), and the microcomputer used for practical energy determination is the NEC PC9801 VM system . Appendix A Summary of the simulation process of nuclear interactions Since we assume that nucleus-nucleus interactions are described by the superposition of nucleon-nucleon

331

T. Fujinaga et al. / Energy determination of electron -photon showers

interactions, we first summarize the assumptions of nucleon-nucleon interactions, and then summarize the process of the nucleus-nucleus one.

UA5

INELASTIC

A1 . Nucleon-nucleon interaction

3

(a) Average multiplicity [15] : (N)=A+B lnf +C ln2f,

(16)

with ~s the cms energy in GeV, A = 1.97, B = 0.42 and C = 0.592. (b) Multiplicity distribution [15] :

2

(N)/k P((N), N)_ ((N)+k-I N )(1 + (N)/k x

1

(1 + (N)lk)k

_ 1 =a+P In V-s , k with a = - 0 .098 and ß = 0.0564. (c) K/ar ratio [15] :

53

(17a) (17b) I- __L -1- __L

-6

K : ir=5 .2 :32.3 .

(d) Inelasticity distribution :

71(K) dK=dK

x : ax

=

(1 +PT/mó)

(20a)

3

Ea (20b) aa =BO (1- x) K , d p Pr : fix where the numerical values of AO , Bo, lc, v are summarized in ref. [21] . Practically, we must take into account the energy conservation, x1 + x2 + - - - + x = K. Details of the sampling method for such constraints are summarized in appendix A of ref. [14] . The pseudorapidity distribution of charged pions is shown in fig. 17, where Ybeam = 1nf /M N (M N is nucleon mass), and we plot the experimental data together [22] . The statistics of our simulated events are one thousand in all cases. We find that our calculations reproduce well the experimental data in the fragmentation region, though there exists a slight discrepancy in the central region in the higher energy region . Such a discrepancy is, however, not important for our calculations . A2 . Nucleus-nucleus interaction (a) Fragmentation probability : In the process A + target (carbon) - A' + anything,

I

-3

I

-2

I

-1

0

0

beam

(uniform) . A0

-4

Fig. 17 . Pseudorapidity distribution of the charged pions in the energy range f = 53-900 GeV.

(e) Single-particle inclusive distribution [21] : d ;v Es dp

-5

where A, A' are the projectile and the fragment nuclei respectively, it is quite important to obtain the so-called fragmentation probability P(carbon ; A, A'). In the present simulation calculation, we used the numerical table summarized by Tsao, Silberberg and Letaw [23] . (b) Production rate of a: Freier and Waddington [24] have investigated the production rate of a in the energy region - 1 GeV/nucleon, and gave the following average numbers of a, (N,), for several groups of projectile nuclei, 0.61 t 0.11 for L-nuclei projectiles (A = 3-5),

: uns 0 : ise



®

.

100

. ° x 10 1 1

(21)

Fe - C

.

® x

O ce

o

0

x

x

o

°

( 10

o x c®

x

o

x

x

o

$o

x ®o

x

c-c

o Q-P (simulatior)

i 100

1 ( 000

E o (TeV/nucleus) Fig. 18 . Average charged multiplicity in the energy range Eo =1-1000 TeV/nucleus for various projectiles.

332

T Fujinaga et al. / Energy determination of electron -photon showers - ( N)P(Z) Fe - Corbon ; p - p;

30 ---------

P - p

;

160 TeV/nucleus

160 TeV (VS - 540 GeV)

-

0 .48 TeV ( Vs -

GeV)

(our simulation)

0.72 ± 0.06 for M-nuclei projectiles (A = 6-9), 0.77 ± 0.10 for LH-nuclei projectiles (A = 10-15), 1 .17 ± 0.19 for MH-nuclei projectiles (A = 16-19), 1.71 ± 0.10 for VH-nuclei projectiles (A = 20-26), where the target is an air nucleus. The number of the produced a, Na , is sampled from a Poisson distribution with the mean value presented above, P(~Na) " NJ

=N N, a

e -< N°' .

(22)

(c) Number of wounded nucleons :

In the present simulation, we determine the number of evaporated nucleons, Nevap, from the following distribution [14] : f(~) d~=2i; d~ with

(23a)

~= N_P /(A -A' - 4 NJ,

Z

=

N/~N)

Fig. 19 . Normalized multiplicity distribution in the cases of p-p and Fe-C interactions at energies of f = 30 .4 and 540 GeV.

r

dN 2rrpTdpT Fe - Carbon ;

10

0 -

4

(our

1)

;

(23b)

which is determined so as to be consistent with JACEE data . We then obtain the number of wounded nucleons as follows : (24)

Nw,=A-A'-4Na- Napp ..

The individual interaction process for wounded nucleons is simulated by the nucleon-nucleon interaction model mentioned previously . Let us show several essential results obtained by the present simulation calculations in the following.

160 TeV/nucleus

160 TeV ( Ts - 540 GeV)

simulation)

\I \

10 3 OA1 540 GeV (1983)

\ v

pT

(GeV/a)

Fig. 20 . Transverse momentum distribution of the charged pions for p-p and Fe-C interactions at f = 540 GeV.

beam

Fig. 21 . Pseudorapidity distribution of the charged pions for various kinds of projectiles at 100 TeV/nucleus .

333

T Fujinaga et al. / Energy determination of electron-photon showers

Table 4 Numerical values of correction parameters (a ;, P,) (i =1, 2, 3) except /3, in the cases of proton-initiated (a) and iron-initiated (b) showers (a) r [ ~L m]

al

a2

ß2

1

25 50 100 200 500 1000

5 .485 4 .253 3 .876 3 .468 2.923 2.640

4 .700 4 .502 4 .608 4 .561 4 .764 4 .827

2

25 50 100 200 500 1000

4.351 3 .946 3 .402 3 .060 2 .786 2 .730

5

25 50 100 200 500 1000

10

0 .8170 0 .8056 0 .8330 0 .8924 1 .102 1 .422

03 4 .642 2 .175 9 .509(-1) 4 .485(-1) 1 .446(-1) 6 .002(-2)

83 1 .057 1 .093 1 .094 1 .115 1 .144 1 .183

4.740 4.586 4 .460 4 .531 4 .765 4 .836

0.8060 0.7964 0.8153 0.8876 1 .100 1 .421

2.338 9 .878(-1) 4 .667(-1) 2 .007(-1) 6 .080(-2) 2 .281(-2)

1 .075 1 .062 1 .121 1 .160 1 .167 1 .205

3 .498 3 .151 2.881 2.823 2 .683 2 .599

4 .732 4 .563 4 .453 4 .486 4.682 4 .794

0 .7899 0 .7978 0 .8191 0 .8879 1 .094 1 .417

6 .982(-1) 3 .201(-1) 1 .556(-1) 6 .700(-2) 2 .160(-2) 9 .285(-3)

1 .039 1 .063 1 .076 1 .113 1 .126 1 .095

25 50 100 200 500 1000

3 .303 3 .005 2 .888 2 .732 2 .706 2 .654

4.867 4.663 4 .581 4 .464 4 .725 4 .838

0 .7999 0 .7994 0 .8249 0.8818 1 .099 1 .424

3 .023(-1) 1 .416(-1) 5 .900(-2) 2 .618(-2) 8 .429(-3) 3 .160(-3)

1 .149 1 .115 1 .100 1 .127 1 .165 1 .184

20

25 50 100 200 500 1000

3 .280 3 .061 3 .022 2.909 2 .841 2 .740

4 .802 4 .568 4.516 4 .432 4.685 4 .799

0 .7946 0 .7947 0 .8217 0 .8819 1 .095 1 .422

1 .525(-1) 6 .653(-2) 2 .722(-2) 1 .191(-2) 3 .657(-3) 1 .418(-3)

1 .129 1 .132 1 .129 1 .149 1 .153 1 .172

50

25 50 100 200 500 1000

3 .105 2 .977 3 .079 2 .963 2 .892 2 .787

4.747 4.546 4 .529 4 .462 4 .694 4 .779

0.7900 0.7967 0 .8242 0.8829 1 .099 1 .421

5 .309(-2) 2 .731(-2) 1 .097(-2) 5 .013(-3) 1 .729(-3) 6 .739(-4)

1 .091 1 .052 1 .166 1 .152 0.8484 0.7303

100

25 50 100 200 500 1000

2.958 2.905 2 .812 2 .808 2 .748 2 .624

4 .758 4 .647 4 .479 4.472 4.677 4.773

0 .7927 0 .7997 0 .8219 0 .8858 1 .097 1 .422

2 .343(-2) 1 .037(-2) 5 .035(-3) 2.101(-3) 5 .392(-4) 1 .923(-4)

0 .9493 1 .221 1 .139 0 .7082 0 .3702 0 .3587

200

25 50 100 200 500 1000

2 .866 2 .779 2 .774 2 .715 2.753 2.626

4.783 4 .586 4 .550 4 .480 4 .783 4 .823

0 .8310 0 .8115 0 .8234 0 .8729 1 .095 1 .432

9 .238(-3) 4 .080(-3) 1 .735(-3) 8 .631(-4) 2 .491(-4) 8 .911(-5)

1 .145 1 .141 0 .8754 0 .5620 0 .3296 0 .3015

500

25 50 100 200 500 1000

2.492 2.497 2 .549 2 .592 2 .594 2 .491

4.648 4.576 4 .580 4.593 4 .884 4 .955

0 .8398 0 .8102 0 .8156 0 .8667 1 .096 1 .449

1 .761(-3) 1 .019(-3) 4.967(-4) 2 .001(-4) 7 .987(-5) 1 .985(-5)

1 .397 0 .7730 0 .4927 0 .3767 0 .2111 0.2026

Ee,, [TeV]

334

T. Fujinaga et al. / Energy determination

of electron-photon showers

Table 4 (continued) (b) EEY [TeV]

r [gm]

al

aZ

25 50 100 200 500 1000

7 .280 6 .400 5 .909 5 .002 4 .229 3 .603

4 .691 4 .647 4 .566 4.308 4 .552 4 .505

ßz 0 .8280 0 .8331 0 .8632 0 .9009 1 .105 1 .386

a3 7 .927 5 .171 3 .025 2 .311 1 .279 7 .337(-1)

t~

1

2

25 50 100 200 500 1000

6 .133 5 .891 5 .177 4 .511 3 .879 3 .203

4.617 4.742 4.585 4.534 4.766 4.751

0 .8216 0 .8324 0 .8599 0 .9253 1 .129 1 .434

5 .983 2 .987 2 .019 1 .373 7 .518(-1) 4.323(-1)

1 .176 0 .9483 0.9007 0 .8754 0 .9108 1 .009

5

25 50 100 200 500 1000

5 .999 4 .967 4 .334 3 .955 3 .361 2 .898

4.832 4.605 4.449 4.558 4 .744 4.720

0 .8264 0 .8194 0 .8292 0 .9003 1 .104 1 .409

3 .578 2 .305 1 .330 7 .653(-1) 3 .434(-1) 1 .797(-1)

0.9603 0 .9301 0.9266 0.9446 0 .9920 1 .034

10

25 50 100 200 500 1000

4 .860 4 .112 3 .673 3 .416 3 .053 2 .752

4 .903 4 .578 4 .483 4 .509 4 .756 4 .762

0.8081 0.8030 0 .8296 0.8932 1 .102 1 .414

1 .404 1 .036 6 .741(-1) 3 .723(-1) 1 .596(-1) 7 .942(-2)

0 .8819 0 .8792 0 .9018 0 .9353 1 .016 1 .086

20

25 50 100 200 500 1000

3 .956 3 .653 3 .286 3 .117 2 .902 2 .659

4 .700 4 .617 4 .447 4 .537 4 .738 4.768

0 .7976 0.8021 0 .8234 0 .8925 1 .098 1 .417

1 .000 5 .682(-1) 3 .418(-1) 1 .709(-1) 6 .419(-2) 3 .001(-2)

0 .8599 0 .9057 0 .9591 1 .018 1 .080 1 .110

50

25 50 100 200 500 1000

3 .900 3 .480 3 .245 2 .955 2 .813 2 .688

4 .744 4 .579 4 .526 4 .474 4 .727 4 .804

0 .7937 0 .7969 0.8245 0 .8867 1 .101 1 .421

7 .828(-1) 4 .392(-1) 2 .183(-1) 1 .105(-1) 3 .741(-2) 1 .512(-2)

0 .9048 0 .9609 1 .017 1 .072 1 .124 1 .136

100

25 50 100 200 500 1000

3 .255 2 .994 2 .873 2.759 2.707 2 .592

4 .833 4 .568 4 .487 4 .434 4 .732 4 .788

0 .8013 0.7961 0 .8216 0 .8835 1 .101 1 .420

3 .805(-1) 1 .929(-1) 8 .601(-2) 4 .179(-2) 1 .236(-2) 4 .929(-3)

1 .019 1 .027 1 .064 1 .086 1 .136 1 .156

200

25 50 100 200 500 1000

3 .210 3 .088 2.822 2 .816 2 .745 2.680

4 .813 4 .724 4 .541 4 .586 4 .806 4 .855

0 .8276 0 .8136 0 .8192 0 .8778 1 .095 1 .436

2 .438(-1) 1 .077(-1) 5 .111(-2) 2 .095(-2) 6 .095(-3) 2 .040(-3)

1 .024 1 .069 1 .119 1 .174 1 .183 1 .198

500

25 50 100 200 500 1000

2 .969 2 .835 2 .760 2 .671 2 .682 2 .652

4 .827 4 .579 4 .607 4 .582 4 .795 4.903

0 .8421 0 .8161 0 .8197 0 .8672 1 .086 1 .442

1 .283(-1) 5 .261(-2) 2 .111(-2) 8 .910(-3) 2 .125(-3) 6 .602(-4)

1 .079 1 .145 1 .191 1 .206 1 .267 1 .206

1 .185 1 .063 0 .9632 0.9136 0.9033 0 .9414

Co C, C2 C3

CO C,

1

2

CO C, Cz

10

CO C, Cz C3

CO C, C2 C3

CO C, C2 C3

Co C, C2 C3

Co Cl CZ C3

20

50

100

200

500

C3

Co C, Cz C3

5

C3

C2

C,

EE, [TeV]

-5 .463(-2) -1 .110 3.850(-1) 2 .862 -5 .393(-2) -1 .128 3 .607(-1) 2 .912

-5 .090(-2) -1 .108 4.814(-1) 2 .829 -5 .332(-2) -1 .085 2.240(-1) 2.883 -5.229(-2) -8.674(-1) -2 .550(-1) 2 .395

-4.046(-2) -8 .864(-1) -2 .985(-1) 2.484 -4 .288(-2) -7.573(-1) -7 .428(-1) 2 .278 -4 .011(-2) -9 .117(-1) -4 .289(-1) 2 .587 -4 .247(-2) -2 .633 6 .217 4.986 -3 .992(-2) -3.012 7.765 5.437

-6.082(-2) -8.974(-1) -1 .739(-1) 2.539

-2 .517(-2) -2 .535 6 .074 4 .846

-1 .036(-1) 2 .42 -1 .478(+1) -7 .668(-1)

-1 .291(-1) 2.916 -1 .786(+1) -5 .763(-1)

-4.218(-2) 2.680 -1 .233(+1) -3 .090

-8 .329(-2) -3 .439 8 .535 6 .824

-7 .704(-2) -3 .995 1 .012(+1) 7 .843

-1 .299(-1) -1 .691(+1) 5 .769(+1) 2 .711(+1)

-1 .068(-1) -2 .676 6 .820 5 .226

-8.044(-2) -1 .082 1.179 2 .746

-5 .408(-2) -1 .086 7 .357(-1) 2 .772

-6 .367(-2) -7 .200(-1) -6 .133(-1) 2 .237

-5 .977(-2) -1 .409 1 .522 3 .360

-5 .916(-2) -1 .047 4 .201(-1) 2 .729

-1 .893(-2) -3 .389 6 .217 7.461

-2 .831(-2) -3 .037 5 .727 6 .554

-2 .284(-2) -1 .520 1 .349 3 .574

-7 .330(-2) -1 .045 3 .646(-1) 2 .704

-6 .159(-2) -1 .096 4 .906(-1) 2 .846

-6.030(-2) -9.533(-1) 3 .493(-1) 2.553

-8 .560(-2) -1 .075 9.168(-1) 2 .845

-1 .944(-1) -7 .905(-1) -1 .401 2 .766 -1 .184(-1) -1 .522 1 .247 3 .830

-1 .717(-1) -1 .782 2 .647 4 .095 -1 .110(-1) -2.059 3 .334 4.490 -8 .463(-2) -1 .096 3.160(-1) 2.797

-1 .195(-1) -6 .809(-1) -1 .182 2.244 -8.543(-2) -1 .319 1 .021 3 .256 -7 .431(-2) -9 .223(-1) -3 .939(-1) 2 .604

10

-1 .072(-1) -1 .117 -5 .728(-2) 3 .132 -1 .075(-1) 4 .840(-1) -6 .943 1 .303 -1 .189(-1) 6.333(-1) -7 .978 1 .409

-6 .338(-2) -1 .655 1 .498 3 .973 -9 .627(-2) 7 .209(-1) -9 .637 1 .914 -1 .098(-1) 9.342(-1) -1 .194(+1) 2.471

-3 .339(-2) -2,004 2 .728 4 .450 -3 .798(-2) -1 .948 2 .545 4 .350

-2 .224(-2) -1 .881 2 .293 4 .237 -1 .842(-2) -1 .308 1 .684(-1) 3 .330 -1 .684(-3) -1 .450 3 .510(-1) 3 .605

-2 .614(-3) -2.478 4.259 5 .266 1 .718(-2) -2 .749 4 .739 5 .856

-2 .116(-2) -1 .469 1 .158 3 .495

-2 .436(-2) -2.182 3 .164 4.749

-8 .821(-2) -1 .359 7 .139(-1) 3 .431

-6 .851(-2) -1 .170 1 .941(-2) 3 .128 -6 .615(-2) -1 .150 4.581(-1) 3 .055

-7 .141(-2) -1 .241 3 .343(-1) 3.232 -5 .550(-2) -1 .297 5 .564(-1) 3 .281

-6 .486(-2) -8 .905(-1) -3 .811(-1) 2 .510

-1 .645(-1) 5 .546(-1) -6.264 9.275(-1)

-1 .462(-1) 4 .335(-1) -5 .770 9 .875(-1)

-1 .631(-1) -2 .160(-1) -2 .500 1 .608

-8 .774(-2) -2 .180 3 .022 4 .890 -7 .913(-2) -1 .293 9 .914(-2) 3 .405

-7 .583(-2) -1 .083 -8 .014(-2) 2 .920 . -7 .479(-2) -8 .895(-1) -4 .894(-1) 2 .538

-3 .874(-2) -1 .132 4 .612(-1) 2 .821

-1 .005(-1) -4.673(-1) -2.198 1 .999

-8.672(-2) -1 .194 4.358(-1) 3 .059

-6 .287(-2) -1 .045 3 .528(-1) 2 .656

-8 .244(-2) -1 .052 1 .968(-1) 2 .805 -8.068(-2) -1 .203 6 .288(-1) 3 .042

-1 .553(-1) -2 .423 3 .557 5 .517

-9 .982(-2) -1 .860 3 .372 3 .819

-1 .128(-1) -7 .685(-1) -1 .139 2.492

-1 .596(-1) -6 .863(-1) -2 .013 2 .598

9

8

7

-1 .262(-1) -1 .974 3 .363 4.239

-1 .209(-1) -1 .590 1 .777 3 .757

-8 .974(-2) -1 .862 2 .954 4 .061

-8 .009(-2) -1 .699 2 .699 3 .717

-7 .401(-2) -1 .376 1 .676 3.228

-7 .535(-2) -1 .135 1 .348 2.749

-8 .368(-2) -8 .384(-1) 2 .839(-1) 2 .438

-1 .480(-1) -1 .593 1 .325 4.031

-1 .418(-1) -1 .950 2 .552 4 .631

-8 .654(-2) -2 .483 4 .843 5 .299

-1 .174(-1) -1 .524 1 .554 3 .863

-1 .036(-1) -2 .199(-1) -1 .423 1 .340

-1 .175(-1) -2.015 3 .377 4 .451

4

6

3

2

5

Depth [cm Pb] 1

Table 5 Numerical values of the correction parameters on (a) C, and (b) D, (!= 0, 1, 2, 3) for proton-initiated showers (a)

w

y

ó

b

i s 0

n

;3

ril

A

wA ;o p

!y

D,

Do D, DZ D3

Do D, DZ D3

Do D, Dz D3

Do D, Dz D3

Do D, Dz D3

D D, Dz D3

Do Dl Dz D3

Do D, Dz D3

Do D, Dz D3

FE,. [TeV]

1

2

5

10

20

50

100

200

500

-2 .267(-3) -6 .721(-2) 9.861(-2) 1 .546(-1)

-3 .457(-2) 5 .504(-1) -3 .501 1 .320(-1)

-2 .627(-3) -1 .943(-1) 5 .577(-1) 2 .325(-1)

-3 .464(-3) -2 .115(-1) 4 .950(-1) 3 .320(-1)

-2 .480(-2) 5 .497(-1) -3 .114 -1 .511(-1)

-5 .758(-3) -1 .040(-1) 2 .345(-1) 2 .094(-1)

-4 .746(-3) 7 .650(-2) -7 .557(-1) 1 .408(-2)

-8 .316(-4) -3 .257(-1) 9 .282(-1) 5 .520(-1)

-1 .525(-4) 5 .303(-1) -2 .063 -7 .399(-1)

-1 .621(-3) 5 .352(-2) -6 .582(-1) 4 .250(-2)

-6.933(-3) 3 .919(-2) -5 .175(-1) 3 .745(-2)

-4.101(-3) 1 .539(-2) -5 .941(-1) 1 .336(-1)

-7 .027(-3) -1 .321(-1) -5 .764(-2) 3 .656(-1)

-8 .921(-3) 6.647(-2) -5 .896(-1) 2 .216(-2)

-7 .533(-3) -1 .895(-2) -2.494(-1) 1 .392(-1)

-7 .907(-3) -6 .244(-2) -3 .369(-1) 2 .584(-1)

-1 .105(-2) -1 .796(-1) 7 .405(-2) 4 .258(-1)

4 .779(-4) -6 .107(-1) 8 .845(-1) 1 .490

-1 .295(-3) -5 .703(-1) 9 .719(-1) 1 .295

-1 .905(-3) -8 .299(-2) -3 .642(-1) 2 .990(-1)

-2 .571(-3) -5 .699(-2) -4 .237(-1) 2 .399(-1)

-6 .026(-3) -6 .885(-2) -4 .871(-1) 3 .233(-1)

-4 .707(-3) -8 .451(-2) -3 .531(-1) 3 .118(-1)

-7 .329(-3) -1 .039(-1) -3 .373(-1) 3 .822(-1)

-1 .226(-2) -3 .957(-1) 6 .705(-1) 7 .701(-1)

4 -2 .246(-2) -5 .167(-1) 8 .275(-1) 1 .016

3 -1 .131(-2) -5 .025(-1) 8 .790(-1) 1 .040

-7 .230(-3) 2 .612(-2) -4 .041(-1) 6 .227(-2)

-1 .451(-2) -9 .142(-2) 1 .558(-2) 2 .589(-1)

-2 .729(-2) -1 .215(-1) -3 .483(-1) 4 .062(-1)

2

-2 .179(-2) -2 .424 8 .930 3 .570

-1 .562(-2) -2.338(-1) 7 .297(-1) 3 .809(-1)

-8 .649(-3) 3.033(-2) -2 .936(-1) 1 .087(-2)

-9 .628(-3) 5 .964(-2) -3 .758(-1) -3 .913(-2)

-1 .488(-2) 1 .272(-1) -6 .454(-1) -7 .967(-2)

-2 .991(-2) 2 .534(-1) -1 .086 -2 .801(-1)

Depth [cm Pb] 1

Table 5 (continued) (b)

-1 .086(-2) -1 .216(-1) -6 .559(-1) 4 .614(-1)

-8 .242(-3) -1 .778(-1) -2 .098(-1) 4 .785(-1) -9.729(-3) -1 .175(-1) -5 .549(-1) 4 .442(-1) -4 .849(-3) -2 .544(-1) -2 .617(-1) 6 .882(-1)

-6 .095(-3) -1 .208(-1) -3 .965(-1) 4 .168(-1) -6 .804(-3) -1 .130(-1) -4 .522(-1) 4 .082(-1) -1 .648(-3) -1 .860(-1) -1 .936(-1) 4 .966(-1)

9 .359(-3) -4 .896(-1) 4 .766(-1) 1 .170

4 .549(-3) -4 .754(-1) 5 .889(-1) 1 .074

-4 .041(-3) -1 .706(-1) -2 .771(-1) 5 .153(-1)

-1 .457(-2) -3 .442(-1) 9 .654(-2) 8 .276(-1)

-1 .44'7(-2) -1 .618(-1) -3 .906(-1) 5 .121(-1)

-1 .135(-2) -1 .472(-1) -2 .640(-1) 4 .387(-1)

-3 .342(-2) 5 .639(-1) -4 .828 4 .213(-1)

-1 .126(-2) -3 .946(-1) 3 .417(-2) 1 .021 -1 .071(-2) -4 .021(-1) -2 .330(-2) 1 .075

-3 .319(-3) -1 .265(-1) -7 .827(-1) 5 .223(-1) -9.859(-4) -1 .068(-1) -9 .475(-1) 5 .138(-1)

-4 .368(-2) 6 .605(-1) -5 .929 7 .514(-1)

-2 .030(-2) -4.887(-1) 2 .309(-1) 1 .216

-1 .145(-2) -5 .459(-1) 5 .814(-1) 1 .255 -7 .133(-3) -3 .710(-1) 1 .482(-1) 9.186(-1)

-9 .653(-3) -1 .438(-1) -6 .837(-1) 5 .259(-1)

-1 .257(-2) -2 .924(-1) -3 .756(-1) 8 .125(-1) -8 .663(-3) -2 .067(-1) -4 .560(-1) 6 .148(-1)

-1 .441(-2) -1 .941(-1) -4 .848(-1) 6 .091(-1)

-1 .367(-2) -3 .155(-1) -7 .298(-2) 7 .930(-1)

-1 .532(-2) -3 .186(-1) -9 .209(-2) 7 .605(-1)

10

-3 .982(-2) 7 .590(-1) -4 .848 -2 .101(-1)

-3 .242(-2) 6 .323(-1) -4 .188 -1 .982(-1)

-3 .086(-2) -3 .121(-1) -2 .727(-1) 9 .119(-1)

-3 .757(-2) 8 .371(-1) -4 .383 -7 .583(-1)

-3 .305(-2) 7 .008(-1) -3 .917 -5 .926(-1)

-3 .836(-2) 3 .749(-2) -1 .143 2 .275(-1)

-2 .057(-2) -2 .494(-1) -4 .236(-1) 7 .246(-1)

-1 .656(-2) -6.419(-1) 6 .055(-1) 1 .398

-1 .122(-2) -3 .257(-1) -5 .634(-1) 9 .023(-1) -1 .332(-2) -1 .598(-1) -7 .484(-1) -016(-1)

-2 .127(-2) -1 .104(-2) -1 .474 4 .038(-1)

-2 .778(-2) -9 .408(-1) 1 .913 1 .717

-3 .745(-2) -1 .102 1 .537 2 .256

-1 .211(-2) -1 .148(-1) 2 .619 1 .961

-1 .580(-2) -3 .465(-1) -1 .443(-1) 8 .632(-1)

-2 .481(-2) -1 .556(-1) -9.675(-1) 6 .156(-1)

-2 .272(-2) -5 .717(-1) 2 .855(-1) 1 .356

-2 .369(-2) -1 .215(-1) -7 .786(-1) 4 .355(-1)

-2.360(-2) -4 .431(-1) 3 .595(-1) 9 .298(-1)

-2 .029(-2) -8 .895(-1) 1 .669 1 .692

9 -4 .428(-2) -1 .670(-1) -1 .327 6 .285(-1)

8 -3 .848(-2) -8 .184(-1) 1 .481 1 .544

7 -3 .050(-2) -1 .869(-2) -1 .660 3 .604(-1)

-1 .384(-2) -4 .967(-1) 7 .977(-1) 9.633(-1)

6 -2 .998(-2) -4 .438(-1) 5 .713(-2) 1 .036

5 -3 .022(-2) -4 .883(-1) 3 .234(-1) 1 .112

0

°

b-

0

n

ó. o

0

m

c

c

s..

y

w w

CO Cl C2 C3

CO C1 C2

1

2

CO C,

10

-1 .191(-1) 2 .661 -1 .606(+1) -9.758(-1)

-1 .215(-1) 3.453 -2 .002(+1) -1 .579

CO

CO C, C2 C3

200

500

C, C2 C3

C3

-1 .013(-1) -1 .080 3.970(-1) 3.123

CO C, C2

100

-9.931(-2) -1 .421 1 .646 3 .622

CO Cl C2 C3

50

-1 .061(-1) -2 .204 4.048 4.980

CO C, C2 C3

-1 .087(-1) -2 .837 6.191 5.992

-1 .187(-1) -2.449 4.325 5.537

-1 .685(-1) -1 .370 -1 .225(-1) 4.236

5.355

-L333(-1) -1 .990 1.387

1

Depth [cm Pb]

20

C3

C2

CO C, C2 C3

5

C3

C,

EE., [TeV]

-8 .069(-2) -1 .362 1.211 3.377 -6 .721(-2) -1,140 5.175(-1) 2.942

-8.406(-2) -1 .708 2.301 4.022

-8 .081(-2) -1 .150 6.712(-1) 3.033

-4 .056(-2) 9.412(-2) -5 .062 1.764

-6.152(-2) -1 .886(-1) -3 .902 2.117 -1478(-2) -3.914(-1) -2.641 2.049

-4.836(-2) -5 .763(-1) -1 .987 2.338

-9.418(-2) -1 .419 1 .147 3.527

-9.858(-2) -1 .846 2.441 4.310

-1 .023(-1) -1 .981 1866 4.647

6.663

-1 .601(-1) 8.379 -4 .069(+1) -7 .410

-1 .346(-1) 7.064 -3 .378(+1) -6 .756

-2 .042(-2) -3 .788 1.068(+l) 6.479 1.468(-2) -4.110 1 .229(+l)

-6.053(-2) -1 .438 1 .222 3.446

-7 .751(-2) -1 .390 1 .186 3.369

-8 .942(-2) -1 .082 9.712(-2) 2.902

-5 .812(-2) -1 .201 6.038(-1) 3.028

-7 .814(-2) -1 .415 1.360 3.419

-1 .167(-1) -2 .005 2.961 4.559

-1 .191(-1) -2 .389 4.040 5.299

-1 .253(-1) -3 .458 7.636 7.145

-1.083(-1) -1 .543 1 .494 3.711

-1 .444(-1) -1 .967 2.259 4.683

-1 .282(-1) -2 .461 4.000 5.648

-1 .417(-1) -1 .990 2.393 4.750

-2 .096(-1) -1 .370 -6 .833(-1) 4 .157

-1 .246(-1) -2 .443 3.821 5 .547

-1 .637(-1) -1 .946 8.410(-1) 5.271

-1 .768(-1) -2 .556 3.163 6.301

-2.182(-1) -1 .787 1 .054(-1) 5 .156

5

-2 .150(-1) -2 .855 4.891 6.372

-1 .688(-1) -1 .564 -2 .712(-1) 4.504

-1 .760(-1) -1 .919 5.734(-1) 5 .338

4

-1 .880(-1) -2 .626 3.468 6 .252

3

2

-1 .281(-1) -9.386(-1) -5 .862(-1) 2.788 -1 .166(-1) -1 .467 9.255(-1) 1709

-1 .098(-1)

-1 .056(-1) -1 .467 1 .153 3.642 -6 .155(-2) -1 .334 8.634(-1) 3.257

-8 .605(-2) -1 .271 6.639(-1) 3.217 -6 .024(-2) -1 .351 8.938(-1) 3.300

-7 .860(-2) 1.159 -9 .208 1.763(-1) -6.847(-2) 1.399 -1 .049(+1) 7.945(-2)

-5 .681(-2) 2.594 -1 .484(+1) -1 .721

-1 .687 1 .971 3 .965

-6 .649(-2) 2.150 -1 .276(+1) -1 .348

-8 .479(-2) -1 .411 1 .303 3.372

1.990 4.049

-1 .090(-1) -2 .759 5.306 5.727

-1 .074(-1) 2.715 -1 .562(+1) -1,940 -1 .013(-1) 3.452 -1 .905(+1) -2 .727

-9 .615(-2) 1.634 -1 .108(+1) -5 .345(-1)

-1 .305(+1) -9 .111(-1)

-1 .124(-1) -2.243 3.657 4.882

-9 .559(-2) -1 .114 -1 .336(-1) 3.067 -7 .484(-2) -1 .114 -1 .031(-1) 3.019

-8 .919(-2) 2 .050

-1 .169(-1) -1 .622 1.380 4.005

-1 .313(-1) -8 .738(-1) -1 .089 2.836

-1 .185(-1) -1 .405 7.622(-1) 3 .566 -1 .183(-1) -9 .786(-1) -3 .558(-1) 2.785

-1 .046(-1) -1 .418 7.503(-1) 3.595

-1 .289(-1) -1 .201 8.025(-1) 3.056

-1 .129(-1) -1 .829 2.596 4.055

-1 .183(-l) -1 .725

-1 .122(-1) -1 .699 1 .756 4.044

-2.031(-1) -2 .575 4.881 5.510

-1 .343(-1) -8 .843(-1) -4.094(-1) 2.644

-2 .065(-1) -5 .020(-1) -3 .042 2.421

-2.377(-1) -1 .908 1.816 4.843

-1 .617(-1) -1 .563 7 .910(-1) 4 .079

-1 .989(-1) -1 .129 -9 .122(-2) 3.140

-1 .536(-1) -2 .008 2.611 4.647

-1 .797(-1) 1.390 -9.572 -5 .494(-1)

10

-2 .286(-1) -1 .518 4.024(-2) 4.398

-2.193(-1) -3 .284 6.465 6.999

-1 .901(-1) -1 .349 -6 .348(-1) 4.032

9

-4 .980(-1) -3 .789 8.932 7.745

8 -3 .117(-1) -9.480(-1) -3 .116 3.908

-3 .036(-1) -9 .546 2.728(+l) 1 .712(+1)

7 -2 .361(-1) -1 .470 1 .260 3.318

6 -2 .554(-1) -7 .269(-1) -1 .816 2.669

Table 6 Numerical values of the correction parameters on (a) C, and (b) D, (1= 0, 1, 2, 3) for iron-initiated showers (a)

w

F

s

ô

b s 0

ô

m^

a

-4.717(-2) -1 .188(-1) -6 .100(-1) 4.826(-1)

-2 .718(-2) -2 .790(-1) 3.683(-1) 6.043(-1)

Do Dl

Do Dl DZ

Do D, Dz D3

Do D1

5

10

20

50

9..154(-3) 8 .452(-1) -3 .405 -1 .199

Do Dl

500

D2 D3

Dz D3

-2 .781(-3) 6.585(-1) -2.797 -8 .597(-1)

Do D,

200

1.433(-1) -7 .712(-1) -9.936(-2)

-1 .679(-2)

Da D3 Dz D3

-1 .591(-2) 8.874(-2) -6 .521(-1) -6 .159(-3)

-2.188(-2) -4 .676(-2) -2 .553(-1) 2.280(-1)

100

D3

Dz

D3

Dz D3

-1 .352(-1) 4.106(-1) -3 .557 1.541(-1)

Do D, D2 D3

2

4.762(-1) -4 .353 -3 .350(-2)

Do D, D2 D3

-6 .577(-2)

1

1

Depth [cm Pb]

D,

EE, [TeV]

Table 6 (continued) (b)

2.203(-2) 3 .726(-1) -1 .482 -6.079(-1)

7.453(-3) 2.149(-1) -1 .101 -2 .712(-1)

1.546(-2) -4.422(-1) 1.090(-1)

-1 .182(-2)

-1 .389(-2) -1 .245(-1) -1 .059(-1) 3.732(-1)

-1 .711(-2) -1349(-1) -1 .945(-1) 4.196(-1)

-2 .815(-2) -6 .101(-1) 1.227 1 .196

-3 .367(-2) -3 .188(-1) -6 .804(-2) 8326(-1)

-9 .512(-2) -4 .063(-1) -9347(-1) 1.394

3.965(-1) -3 .923 -8 .149(-2)

2

-7 .705(-2)

-1 .262 -3 .519(-1)

1 .679(-2) 2.458(-1)

-4.637(-2)

5.579(-3) 9.407(-2) -8 .672(-1)

-8 .758(-3) -8 .281(-2) -3 .206(-1) 2.934(-1)

-1 .242(-2) -1 .964(-1) -5 .229(-2) 5.298(-1)

-1 .405(-2) -2 .056(-1) -5 .361(-2) 5.202(-1)

-1 .921(-2) -4 .349(-1) 4.824(-1) 9.186(-1)

-2 .104(-2) -4 .536(-1) 2.717(-1) 1.014

-5 .595(-2) -2 .990(-1) -1 .573 1.176

-3 .760(-2) 1.829(-1) -2 .886 4 .888(-2)

3

2.686(-2) -8 .812(-1) 2.984 1.126

1 .280(-2) -9.417(-1) 2.975 1 .362

-5 .766(-3) -1 .954(-1) -1 .752(-1) 5 .266(-1)

-1 .169(-2) -2 .980(-1) 1 .386(-1) 7 .086(-1)

-1 .323(-2) -1 .693(-1) -2 .616(-1) 4.770(-1)

-1 .678(-2) -4 .401(-1) 4.660(-1) 9.269(-1)

-2 .562(-2) -4 .755(-1) 2.202(-1) 1.050

2.166

-6 .380(-2) -9 .567(-1) 8 .391(-1)

-2 .200 8 .645(-1)

4 -6 .407(-2) -1 .532(-1)

5

-4.193(-2) 3 .175 -1 .525(+1) -2 .816

-2 .732

-3 .134(-2) 2.746 -1 .283(+l)

-6 .677(-3) -3 .063(-1) 3.655(-2) 7.356(-1)

-1 .040(-2) -3 .588(-1) 1 .670(-1) 8.298(-1)

-1 .292(-2) -1 .504(-1) -4 .679(-1) 4.731(-1)

-1 .437(-2) -3 .500(-1) 6.217(-2) 7.851(-1)

-2 .439(-2) -5 .748(-1) 3 .717(-1) 1.219

2.730

-7 .159(-2) -1 .408 2 .665

-4 .862(-2) -1 .316(-1) -1 .869 6 .266(-1)

6

-6 .687(-3) 1 .570 -7 .541 -1 .573

-8.541(-3) 1 .311 -6 .429 -1 .339

-6.803(-3) -3 .211(-1) -1 .951(-2) 7.845(-1)

-1 .227(-2) -2 .898(-1) -1 .741(-1) 7.316(-1)

-1 .332(-2) -4 .142(-1) 2.717(-1) 9.304(-1)

-1 .575(-2) -4 .707(-1) 2.397(-1) 1 .028

-2 .478(-2) -6 .770(-1) 7.543(-1) 1 .328

1 .504

-5 .648(-2) -5 .553(-1) -6 .727(-1)

-7 .716(-2) -3 .877(-2) -1 .484 9.784(-2)

-1 .407(-2) 1 .300 -6 .853 -1 .029

-1 .487(-2) 1.096 -5 .887 -9 .067(-1)

-7 .512(-3) -3 .812(-1) 1.021(-1) 8.987(-1)

-1 .302(-2) 1 .696 -8 .230 -1 .800

-1 .556(-2) 1.366 -6 .855 -1 .425

-1 .101(-2) -2 .725(-1) -4 .568(-1) 7.843(-1)

-2 .544(-1) 8.521(-1)

-2 .323(-2) -3 .359(-1)

-1 .767(-2) -3 .491(-1) -9 .372(-2) 8.567(-1)

5.043(-1)

-2 .318(-2) -1 .436(-1) -8 .084(-1)

-1 .964(-2) -7.111(-1) 1 .118 1 .354

-4.150(-2) -3 .840(-1) -2 .214(-1) 7 .795(-1)

-2 .108(-2) -5 .505(-1) 5 .904(-1) 1 .183

-1 .988(-2) -5.455(-1) 4.995(-1) 1 .160

-2 .853(-2) -5 .110(-1) -1 .031(-1) 1 .183

-5 .979(-2) -1 .656 3.832 2.979

-6 .679(-2) -1 .890 4.145 3.513

8 -9 .010(-2) -7 .411(-2) -3 .231 7.261(-1)

7 -5 .301(-2) -8 .102(-1) 1 .504 8.385(-1)

9

-5 .688(-3) 2.508 -1 .141(+1) -3 .103

-1 .281(-2) 1.964 -9.197 -2 .355

-1 .720(-2) -2 .904(-1) -4 .489(-1) 8.193(-1)

-1 .528 2.838(-1)

-3 .395(-2) 5.265(-2)

-1 .622(-2) -3 .455(-1) -3 .295(-1) 8.726(-1)

-2 .739(-2) -4 .503(-1) 3.359(-1) 9.913(-1)

-5 .585(-2) -8 .244(-1) 6.819(-1) 1.728

-6 .743(-2) -6 .866(-1) -3 .870(-1) 1.754

-7 .573(-2) -6 .856 2.103(+1) 1 .089(+1)

10

-4 .047(-2) -9 .132(-1) 1.761 1.880

-3 .577(-2) -6 .682(-1) 9.129(-1) 1 .457

-2 .365(-2) -4 .430(-1) 1 .221(-2) 1 .066

-1 .023 3.464(-1)

-3 .595(-2) -3 .650(-2)

-2 .326(-2) -3 .892(-1) -2 .007(-1) 9 .407(-1)

-3 .110(-2) -3 .255(-1) -3 .323(-1) 8 .940(-1)

-2 .569 3.074(-1)

-4 .384(-2) 5 .826(-2)

-4 .895(-2) 1 .026 -6 .186 -1 .033

-1 .646(-1) -2 .385 6.659 3.350

0

óg

0

m

0

c

w .. s 0 oa

y

00

w w

T Fujinaga et al. / Energy determination of electron- photon showers In fig. 18 we show the relation between the average charged multiplicity and the incident primary energy in the laboratory system for several kinds of projectiles. In fig. 19 we demonstrate the normalized multiplicity distribution in the cases of p-p and Fe-C interactions . In the latter we find a significant fraction near Z(= NI(N~) - 0, which arises from peripheral collisions . In fig. 20 we present the PT-distribution of charged pions with l11 I < 2.5 for p-p and Fe-C interactions . In the present model, the pT-distribution does not differ between these two. In fig. 21 we demonstrate the pseudorapidity distribution for several kinds of projectiles with a total energy of 100 TeV, where we set Yheam = In 2 Eo/AMN . Appendix B Summary of correction factors In the text, we introduced variables a;, ß; (i = 1, 2, 3), and ic, W, where the former two are the main correction parameters, while the latter two are the second-order ones . On the basis of the simulation data, we found ß1 is nearly unity, irrespective of EE C r and t, and so we put here /31 = 1. Now, we expand a;, ,ß; with respect to r in the following way: 5

In ai = A ;,0 In r + F_ A i_trt-t , t=1 5

In ßi = B;,o In r + Y_ B;, lrt-1, t=1

(25a) (25b)

;,, by fitting eq . and we determine the parameters Ai,,, B (7) to simulation data using the least-squares method . After determining these parameters, finally we can determine the parameters Ct and Dt in eq . (9) by fitting QA, 8, H) (see eq. (8b)) to Ne (A, 8, H)/Ne( 1 , 0, 0) obtained by simulation data with the least-squares method . In tables 4a and 4b, we summarize the numerical values a; and ßr (i = 1, 2, 3 except ß1) at shower maximum in the energy range 1-500 TeV for proton jets and iron jets respectively, where we set six cases of radii, 25, 50, 100, 200, 500 and 1000 [Lm, within which the electron number is actually counted. In tables 5 and 6 we present Ct and Dt (1= 0, 1, 2, 3) in the cases of proton jets and iron jets . Here, the units for each parameter are millimeters for r, centimeters for A and 8, and 10 centimeters for H. References [1] T. Ogata, private communication . [2] C. Castognoli, G. Gortini, D. Moreno, C. Franzinetti and A. Manfredini, Nuovo Cimento 10 (1953) 1539. [3] M.F . Kaplon, B. Peters and D.M. Ritson, Phys . Rev. 85 (1952) 295;

339

R. Kullberg and T. Otterlund, Z. Phys . 259 (1973) 245. [4] T. Shibata et al ., Proc. Int. Conf. on Cosmic Rays, Paris, 1981, vol. 5, pp . 214, 218 and 222; K. Kasahara, Phys . Rev. D31 (1985) 2737. [5] S. Dake, Proc . Int. Cosmic Ray Conf . on High Energy Phenomena (ICR, Univ. of Tokyo, 1974) p. 137 . [6] W.V . Jones, Phys . Rev. 187 (1969) 1868 ; ibid . DI (1970) 2201 . [7] T. Shibata, Talk at Int . Symp. on Cosmic Rays and Particle Physics, Tokyo (March, 1984). [8] For example, see T. Shibata, Phys . Rev. D22 (1980) 100; R.W . Ellsworth, G.B . Yodh and T.K . Gaisser, Proc . Bartol Conf. on Cosmic Rays and Particle Physics, 1978, ed . T.K . Gaisser (AIP, New York, 1979). [9] M. Okamoto and T. Shibata, Nucl . Instr. and Meth. A257 (1987) 155. [10] L.D . Landau and I.J . Pomeranchuk, Dokl . Akad . Nauk USSR 92 (1953) 535, 735. [11] H.A . Bethe and W. Heitier, Proc . Roy. Soc. London A146 (1934) 83 . [12] A.B. Migdal, Phys. Rev. 103 (1956) 1811. [13] N. Hotta, N. Munakata, M. Sakata, Y. Yamamoto, S. Dake, H. Ito, M. Miyanishi, K. Kasahara, T. Yuda, K. Mizutani and I. Ohta, Phys . Rev. D22 (1980) 1; Y. Sato and H. Sugimoto, Proc. Int. Conf . on Cosmic Rays, Kyoto (ICR, Univ. of Tokyo, 1979) vol. 7, p. 42. [14] Y. Niihori, T. Shibata, I.M . Martin, E.H . Shibuya and A. Turtelli Jr ., Phys. Rev . D36 (1987) 783. [15] UA5 collaboration, CERN Report No . EP/85-62 . [16] JACEE collaboration, in : Quark Matter '84, Proc . 4th Int. Conf. on Ultrarelativistic Nucleus-Nucleus Collisions, Helsinki, Finland, 1984, ed. K. Kajante (Lecture Notes in Physics, vol. 221) (Springer, New York, 1985). [17] M. Okamoto, Master Thesis, Aoyama Gakuin University (1981). [18] M. Niwa, M. Ichimura, Y. Eguchi, T. Fujinaga and Y. Horiguchi, ICR-report (ICR, Univ . of Tokyo) ICR-50-85-4 (1985). [19] H. Nanjo, H. Matsutani, K. Teraoka, K. Toda, S. Mase, M. Ichimura, T. Kobayashi, T. Shibata, Y. Niihori and Y. Yosizumi, Proc . Balloon Symp ., ISAS (Institute of Space and Aeronautical Science, Tokyo, Japan, Dec. 1987) pp . 115, 120. [20] J.N . Capdevielle, M. Ichimura, T. Fujinaga, T. Ogata and T. Shibata, Proc. Int. Conf. on Cosmic Rays, Moscow (1987) vol. 5, p. 182. [21] F.E. Taylor, D.C . Carrey, J.R. Johnson, R.K. Kammerud, D.J . Richie, A. Roberts, J.R. Sauer, R. Shafer, D. Theriot and J.D. Walker, Phys . Rev. D14 (1976) 1217 . [22] UA5 collaboration, Z. Phys . 33 (1986) 1. [23] C .H . Tsao, R. Silberberg and J.R . Letaw, Proc . Int. Conf. on Cosmic Rays, Bangalore (1983) vol. 2, p. 294. [24] P.S . Freier and C.J . Waddington, Astrophys. Space Sci. 38 (1975) 419. [25] T. Fujinaga, M. Ichimura, Y. Niihori and T. Shibata, ICR-Report 173-88-19 (Institute of Cosmic Ray Research, University of Tokyo, 1988). [26] Y. Kawamura, H. Matsutani, H. Nanjyo, K. Teraoka, K. Toda, M. Ichimura, K. Kirii, T. Kobayashi, Y. Niihori, T. Shibata, K. Shibuta and Y. Yoshizumi, ICR-Report 172 88-18 (Institute of Cosmic Ray Research, University of Tokyo, 1988) to be published in Phys . Rev . D."