Phenomenological analysis of the fragmentation cross section

Volume 92B, number 1,2

PHYSICS LETTERS

5 May 1980

PHENOMENOLOGICAL ANALYSIS OF THE FRAGMENTATION CROSS SECTION

V.V. AVDEICHIKOV E G. Khlopin Radium Institute, Leningrad 197022, USSR

Received 13 February 1979

The phenomenologicalsystematics of the cross sections for fragments produced in high-energyinteractions and deep inelastic processes of heavy ions with nuclei is discussed.

The description of the mechanism of emission of heavy charged particles (fragments) produced in highenergy interactions is based on the cascade-evaporation model [1]. However, this model, in which the reaction is assumed to proceed in two distinct steps (cascade + evaporation) with the fragments being emitted from the excited nuclei at the second slow step, cannot account for the data. The model is unable to describe the relative yields of various produced fragments. Analysis of the experimental data shows that emission of the fragments occurs on a time scale comparable to that of the cascade step itself, i.e. ~10 -21 s [1]. The excited cascade residual nucleus seems analogous to the systems created in deep inelastic collisions (DIC) of heavy ions (at energies below 20 MeV/ nucleon, A < 40) with nuclei which emit the fragments in a time of -~10 -21 s [2]. The analogy may be justified by the fact that the main characteristics of the fragments produced in DIC processes and in high-energy interactions (HEI) are similar: (a) The most probable kinetic energy of the fragments is independent of the type and bombarding energy of primary particles and is approximately determined by the exit Coulomb barrier (DIC [2], HEI [1]). An increase in the width of the fragment energy spectra is observed with increasing excitation energy of the decaying nuclei (DIC [3,4], HEI [1,18]). (I0) The angular distributions of the fragments are forward peaked, so the fragments retain some memory of the incident beam direction (DIC [2], HEI [1 ]). (c) There is a strong correlation between the direc74

tions of the emitted fragment and of the recoil nucleus, an angle of 180 ° being favoured (DIe [5], HEI [61). (d) In both processes, apart from the fragment and the residual nucleus in the final state, a few light particles are formed (DIC [5], HEI [1,6]). (e) Both fragmenting systems have a continuous excitation energy spectrum. The maximum value of the excitation energy is well defined only in DIC processes (DIC [1], HEI [2]). (f) Fragment energy spectra have high-energy "tails". In DIC processes these "tails" extend to the energy restricted by the kinematic limit (DIC [2,7], HEI [11). (g) Fragment production cross sections obey the same systematics of two-body decay. In this paper we discuss this systematics. We consider the emission of fragments by the excited cascade residue in HEI reactions as a two-body breakup. Then, to describe the yield of fragments, Z 1, A 1- NI by the excited residue Z, A, N we can use the Qgg systematics [8] of the fragment production cross section in DIC processes (Z 1 < Z/2, A 1 < A/2): O(Z l, A ) = c(Z 1, A ) exp(Qgg/T),

(I)

where -Qgg is the binding energy of the fragment Z 1 , A 1 in the nucleus Z, A; T is some parameter characterizing the excited nucleus Z, A. Fig. 1 shows the fragment yield in the reaction p(5.5 GeV) + 238U against the values of Q.. calculated for the average emitting nucleus 221Rn ]~1]. Eq. (1) does not describe the yield of fragments in this

Volume 92B, number 1,2 I

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PHYSICS LETTERS I

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5 May 1980

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Volume 92B, number 1,2

PHYSICS LETTERS

Qgg systematics has been proposed to describe the fragment yield in DIC and HEI processes:

reaction at T = const. This failure is not remedied by different choices of residual emitting nuclei. A similar picture is seen in DIC processes [2]. A great number of fragments deviate significantly from the simple Qgg systematics (1). This is the reason why more sophisticated regularities have been proposed ,2. One of them [2,12] is founded on the assumed mechanism of nucleon exchange between the incident ion and the target nucleus in the DIC process, and non-pairing corrections for Qgg have been introduced for neutron and proton pair transfer. In our paper [13] a phenomenological version of the

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5 May 1980

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:~1 Lukyanov and Titov [ 10] have proposed a Qgg model of projectile fragmentation reactions of relativistic nuclei 260 (2.1 GeV/nucleon). Assuming the excitation of the nucleus by peripherical interaction and its subsequent statistical decay after equilibrium is reached, they have obtained eq. (1). In their next paper [11] they have introduced the factor £i (relative weight of the decaying channel) into eq, (1).

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76

(2)

Volume 92B, number 1,2

PHYSICS LETTERS

where t 3 = (Z 1 - N1)/2 , and Z 1
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5 May 1980

from Monte Carlo calculations of the cascade step in these reactions give a number of knockout nucleons of 5 - 7 . Since the value of Qgg slightly changes with decreasing Z, A of the emitting nuclei, in fig. 2 the same value of Qgg is presented as for the primary emitting target nucleus. Figs. lc and 2 demonstrate that the experimental data with T = const. (at A = const.) are well described by eq. (2). Fig. 3a gives the values of T (MeV) extracted from fitting the data on the fragmentation cross sections by eq. (2). A clear correlation T = F(A) is seen. The values of T are practically not changed for A > 200. The characteristics of the emitting systems produced in DIC and HEI processes are listed in table 1. The T values slightly increase in the range of nuclear excitation energy U* = 50-1000 MeV. It should be noted that the calculated U* values [1 ] listed in table 1 are much larger than the experimental ones. In a range o f E = 1 - 8 GeV experiment gives P U* = 160-200 MeV [19]. The Qgg systematics in DIC processes is regarded as evidence for partial statistical equilibrium in the decaying system Z, A with effective "temperature" T [20,21]. For c(Z1,A ) one gets [20,21]

c(Z1, A) = c(A ) e x p ( - V/T) ,

(3)

. Ep=1.0 GeV

Table 1 The values of T extracted using two-body energetics [eq. (2)1 for decaying systemsA > 200. Interaction

Kinetic energy of the incident particles (MeV)

Excitation energy of the system (MeV)

T (MeV)

Reference

2°Ne+ 197Au 11B + 238 U a) 22Ne + 238 U a) 22Ne+232Thb) p + 238U d + 232Th d + 238U + 238U ~He + 238U

110 85 165 174 1000 3000 4200 5500 8400

53 67 92 105 200 600 800 900 1000

~2.9 ~3.5 3.6 +- 0.2 3.1-+ 0.2 3.6 -+ 0.1 3.6 +- 0.2 3.6 -+ 0.2 3.6 -+ 0.1 3.7 +- 0.3

[3] [16] [16] [4] [14] [17] [18] [9] [18]

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Fig. 3. (a) The parameter T as a function o f the atomic number of the fragmenting nucleus. The values of T were ex-

tracted from fitting by eq. (2) the data of the reactions: p(1.0 GeV) + nuclei [14,15] (e); 22Ne(8 MeV/nucleon) + 232Th [4] and 94Zr [23] (o); l l B ( 8 MeV/nucleon) + 238U [16] and 22Ne(8 MeV/nucleon) + 238U [16] (A); and in the projectile-fragmentation reactions: 12C(2.1 GeV/nucleon) [24] (=); 4°At(0.21 GeV/nucleon) [25] (v). (b) The experimental and calculated [eq. (4)] values ofc(Z1) in the reactions p(1.0 GeV) + U, Au, Ag. (c) The same as (B) in the reactions p(1.0 GeV) + U, p(5.5 GeV) + U and 22Ne(8 MeV/ nucleon) + Th.

c) c) c) c) c)

a) Measured values only for the yields of the isotopesRb and Cs. b) Measured values at 12° (lab. system). c) Calculated values, after the cascade step [1].

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Volume 92B, number 1,2

PHYSICS LETTERS

where V is the Coulomb barrier between the fragment Z 1 and the residual nucleus Z 2. We have used eq. (3) to describe experimental c(Z 1, A ) in HEI processes. The calculated c(Z1) are denoted by solid lines in fig. 3b, the distance between the particles Z 1 and Z 2 is R = 1.25 (A ~/3 + A 1/3) + 1.67 fm [22], and the values of T are taken from fig. 2. The calculated curves are normalized to the experimental data at Z 1 = 4. It is seen that T extracted using eq. (2) reasonably fits the data. Using eqs. (1) or (2), we obtain the same result for the relative yield of fragments Z1, A 1 in the reactions p + ll2Sn and p + 124Sn, Y ~ exp[(Aagg + AV)/T] .

(4)

A comparison of the experimental relative yields of fragments (from fig. 2b) with the calculated ones gives an average value of T = 4.4 MeV. This value agrees with those extracted in the description of the fragment yields by eq. (2) for each target nucleus. separately due to the factor exp(t3) (T = 4.3 and 4.5 MeV for 122Sn and 124Sn, respectively). It seems that the empirical factor exp(t3) helps to reestablish a "true" T. It is worth mentioning that the correction exp(t3) in the description of the isotropic yields 83Rb-97Rb (table 1) amounts from 1 up to 103. The temperature extracted using two-body energetics is in great discrepancy with those from quasimaxwellian evaporation particle spectra [9,14]. In the latter case the apparent T is very high ( ~ 1 0 - 1 3 MeV) and the apparent V is too low to be physically realistic. It seems that the evaporation formula which is used to fit the fragment spectra with its three [1,6] or five [9] free parameters (including T and V) must be regarded only as a reasonable functional form to systematize experimental data. The yield of newly discovered isotopes [9], which are denoted by asterisks in fig. lc, obey eq. (2), so this systematics may be useful to predict the yield of yet unknown isotopes.

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5 May 1980

References [1] V.S. Barashenkov and V.D. Toneev, High-energy interaction of atomic nuclei and particles with nuclei (Moscow, 1972). [2] V.V. Volkov, Phys. Rep. 44C (1978) 94. [3] A.G. Artukh at al., Yad. Fiz. 28 (1978) 1154. [4] A.G. Artukh et aL, Nucl. Phys. A283 (1977) 350. [5] R. Babinet et al., Nucl. Phys. A296 (1978) 160. [6] N.A. Perfilov, O.V. Lozhkin and V.P. Shamov, Usp. Fiz. Nauk 60 (1960) 3; E. Makowska et al., Nucl. Phys. 79 (1966) 449. [7] E. Gerlic, A.M. Kalinin, R. Kalpakchieva, Yu.Ts. Oganesyan and Yu.G. Kharitonov, preprint JINR, E 7-12922, Dubna (1979). [8] A.G. Artukh et al., Nucl. Phys. A160 (1971) 511. [9] A.M. Poskanzer et al., Phys. Rev. Lett. 17 (1966) 1271; Phys. Rev. C4 (1971) 1759; J.D. Bowman et al., Phys. Rev. C9 (1974) 836. [10] V.K. Lukyanov and A.1. Titov, Phys. Lett. 57B (1975) 10. [11 ] V.K. Lukyanov, Yu.A. Panebratsev and A.I. Titov, preprint JINR, E2-9089, Dubna (1975). [12] A.G. Artukh et al., Izv. Akad. Nauk SSSR Set. Fiz. 39 (1975) 2. [13] V.V. Avdeichikov, preprint JINR, 4-11262, Dubna (1978). [14] E.N. Volnin et al., preprint FTI-101, Leningrad (1974). [15] E.N. Volnin et al., Phys. Lett. 55B (1975) 409. [16] R. Klapish, preprint JINR, D7-9734, Dubna (1975) p. 155. [17] G.G. Beznogikh et al., Pis'ma Zh. Eksp. Teor. Fiz. 30 (1979) 349. [18] A.M. Zebelman et al., Phys. Rev. Cll (1975) 1280. [19] N.A. Perfilov, Physics and chemistry of fission, Vol. II (Vienna, 1965) p. 285. [20] J.P. Bondorf et al., J. de Phys. 32 (1971) C6-145. [21] A.Y. Abul-Magd et al., Phys. Lett. 39B (1972) 166. [22] G. Igo, L.F. Hansen and T.J. Gooding, Phys. Rev. 131 (1963) 337. [23] A.G. Artukh et al., Yad. Fiz. 25 (1977) 255. [24] D.E. Greiner et al., Phys. Rev. Lett. 35 (1975) 152. [25] Y.P. Voyogi et al., Phys. Rev. Lett. 42 (1979) 33.