Zeitschrift Physik A

Z. Physik A277, 3 7 9 - 384 (1976)

'IC by Springer-Verlag 1976

Momentum Distributions in Fragmentation Reactions with Relativistic Heavy Ions A. Abul-Magd Department of Physics, Cairo University, Giza, Egypt J. Hiifner Institut fiir Theoretische Physik, Universit~it Heidelberg and Max-Planck-Institut fiir Kernphysik, Heidelberg, Germany Received February 2, 1976 The momentum distribution (or angular distribution) in inelastic heavy ion reactions is calculated by using a two-step model ("abrasion" and "ablation"). First, all nucleons in the volume element where projectile and target overlap spatially during the collision are sheared away. The remaining prefragment (the projectile minus the sheared off nucleons) has a recoil momentum proportional to the Fermi momentum. The prefragment is left in an excited state and emits nucleons, the recoil momentum given to the final fragment is proportional to the nuclear temperature. This two-step model reproduces the overall trend and the isotopic dependence for the widths of the experimental momentum distributions. Contrary to previous theoretical studies we find that surface and friction phenomena lead to an anisotropy: The momentum distributions in transverse direction are always broader than in the longitudinal direction by about 5 to 10 %.

1. Introduction

The first experiments with heavy ions at relativistic energies already brought us a surprise: The momentum distributions after a fragmentation reaction were found to scale with the pion mass [1]: in a fragmentation reaction of the type a z + t a r g e t ~ a v z r + x , a projectile AZ with relativistic energy (typically 2 GeV/ nucleon) hits a target and a fast fragment AFZv (also about 2 GeV/nucleon) is recorded in the counter. The momentum of the fragment is measured. When transformed into the rest system of the projectile, the fragment momenta P are found to follow a Gaussian probability distribution

d3W d3p

= c-exp(-P2/2a2) - exp(-(Pir - (Prl))2/2a~), (1)

where c is independent of P. The components of P have been decomposed into a longitudinal component PII in beam direction and into two transversal components, PL. The first analysis [1] of the data yielded the Gaussian shape Equation (1), showed isotropy (all =a=) and, most surprising, all =m~c= 140MeV/c, practically independent of the type of final fragment A~Ze. The relation to the pion mass rn, gave rise

to many high-brow speculations but finally found a rather sobering solution (Feshbach et al. [2] and Goldhaber [3]): The relevant quantity which determines the widths all and a• of the momentum distribution is not the pion mass, but the Fermi-momentum Pv in the projectile, more precisely pv/l/5, which is numerically rather close* to m~c (for Pv = 300 MeV/c, pv/t/5= 135 MeV/c). Recently a more refined analysis of the data has been published by Greiner et al. [4]. The main features of the experiments can now be summarized as follows: (i) The momentum distributions are Gaussians, like Equation (1), to a good approximation. Emulsion studies (Jacobson et al. [5]) essentially confirm the Gaussian shape, but find a non-Gaussian tail for large momenta of light fragments. (ii) The distributions are isotropic, i.e. a l l = ~ l to within 10 %. (iii) The width all depends on the final fragment A~Zv. * The near equality pt/1/5~-m~c is not a mere numerical coincidence. The range of the nuclear forces (among them one-pion exchange) determines the nuclear density and therefore the Fermi momentum.

380

A. Abul-Magd and J. Hfifner: Momentum Distributions in Fragmentation Reactions

The overall trend of the experimental data seems compatible with a "parabolic" form

[ A-n'~ ~/2 a,t (n)= a o kn ~--1-1} '

(2)

where n = A - A v is the number of nucleons by which projectile and fragment differ. However, the form, Equation (2), does not account for the observed strong isotopic dependences. For instance, in the reaction 1 6 0 + 9 B e ~ AvZ F+ X, experimental values of all for the final fragments ~2C and 12B (n=4) are (120• MeV/c and (163 • 8)MeV/c, respectively. The features (i) to (iii) are roughly compatible with several, even mutually exclusive models for the fragmentation reaction. Goldhaber [-3] shows this most clearly by studying a model of direct reaction and a model of intermediate compound nucleus formation. Symmetry considerations and momentum conservation yield in each case o'11 =o'•

ff,l=6O ( n A - n ~ 1/2 A-l!

(3)

"

In the direct mechanism, n nucleons are suddenly sheared away from the projectile. The fragment carries the recoil momentum of the sheared off nucleons. One derives a o = p v / ] / / 5 for a Fermi gas model of the nucleus. In the compound nucleus picture, the projectile is excited during the collision with the target. The excitation energy thermalizes and finally nucleons are evaporated. The recoil momentum given to the fragment is then related to the nuclear temperature O by a o = ~ / 3 . Our point of view lies between the two extremes: Fragmentation proceeds in two steps. First, some projectile nucleons are sheared away ("abrasion"). These nucleons have been located in the overlap volume between projectile and target during the moment of collision. The remaining system (projectile minus sheared off nucleons) is called the prefragment. It is highly excited and loses nucleons ("ablation"), presumably by compound nucleus decay*. This model has been proposed by Bowman et al. [6] and has been successful in the calculation of fragmentation cross sections [7] and friction effects [8]. As shown in this paper, the model also explains quantitatively all features of the momentum distributions without the necessity of introducing a single adjustable parameter. * We only consider what happens to the projectile. Of course, since the situation is completely symmetric with respect to projectile and target, also the target undergoes an abrasion-ablation reaction which, however, is not observed in these experiments.

2. Abrasion

Abrasion is the first step in a peripheral encounter of two fast heavy ions. All nucleons in the volume element, where projectile and target overlap, are scraped away. Because of the relativistic velocity, the nucleons are removed "suddenly" (on the time scale of nucleon motion inside the nucleus). The prefragment receives as recoil the total momentum which the removed nucleons had just before abrasion: The validity of the sudden approximation has been assumed by Goldhaber [3] and Lepore et al. [9]. We show here that it is inherent in the Glauber approximation, which is so successful for reactions at high velocities. The following derivation starts from scratch. It only assumes the Glauber approximation to be valid. Some other simplifying assumptions are introduced to exhibit the physics more clearly, but can be removed [3, 9]. In the rest system of the projectile the target passes by and interacts with the nucleons of the projectile. We neglect all intrinsic degrees of freedom of the target and describe its interaction with a projectile nucleon i at position x/ by a phase shift function Zi(Xi). As shown in Reference 7, Xi(x~) is related to the optical potential for nucleon-target scattering. The scattering amplitude for the reaction ~ into 7'f, where the initial state ~ of the projectile is changed into 7~: during the collision, is written in Glauber approximation k

a

F,:(q)=~i~d2be-iqb(~fl[le'Z~"'-b)]~

) .

(4)

j=l

In standard notation, the impact parameter is denoted by b and q is the momentum transfer during the collision. According to our intuitive picture for abrasion, the final state ~: of the projectile consists of the pre-fragment with intrinsic wave function ~m and of the n nucleons which are scraped away. Their intrinsic wave function is denoted by 4/2L Thus, we decompose (neglecting antisymmetrisation) (17 (r ~e: = {a~,,~

..., G - , ) e iPRI}

9 {~(m2](?/1, " " , t / n ) e - i { q

P)R2},

(5)

where m1 and r?/2 denote quantum numbers, P and q - P describe the center-of-mass momenta of prefragment and of the removed nucleons respectively (R1 and R 2 a r e center-of-mass coordinates and ~i and r/~ intrinsic variables). While the factorization, Equation (5), seems appropriate for ~u:, a corresponding decomposition of the initial state ~/= @g1'(~1, ..., CA-n) ~g2)(!11, "'", T/n)r

R2)

(6)

may not always be very realistic. But Equation (6) will be assumed mainly for pedagogical reasons.

A. Abul-Magdand J. Hi.ifner:MomentumDistributionsin FragmentationReactions The target interacts only with those nucleons of the projectile which are in the overlap volume element, and therefore not with nucleons of the prefragment. This means formally Z j ( x j - b ) = z j ( ~ j + R l - b ) = O for all xj in the pre-fragment. This property together with the factorization assumptions Equations (5) and (6) permit the evaluation of the momentum dependence of the transition matrix element: F/f (q, P) = {S d 3 r ~o(r) e-iP'r}

f/f (q),

(7)

where f~y does not depend on P. As exprected on intuitive grounds, the momentum distribution after collision reflects the relative motion of prefragment and removed nucleons before the collision. According to experiment, the momentum distribution is Gaussian, Equation (1). Therefore (in our formalism) the Fourier transform of the wave function of relative motion, must be Gaussian. This property cannot be generally true, but must be related to a special feature of light nuclei. The harmonic oscillator potential approximates well the average shell-model potential. We use this fact to derive the parabolic dependence, Equation (2), on the number of removed nucleons. The intrinsic Hamiltonian of the projectile is assumed to describe A nucleons moving in a harmonic oscillator well: 1 ~z m A 1 ~ P, P~,.~ 3-~co 2(xi-R)2; H = 2 m i= , -A . i-1

P~m=Zpi (8)

R=~1 ~Xi, which can be transformed exactly into H=H{1)(~i)~_ H(21(71i)_~~

kt= n

A-n A

p2 + 2 co2(RI_ R2)2;

m.

(9)

Here H(~)(r is the intrinsic Hamiltonian of the prefragment, H(2)(qi) the one of the n nucleons to be sheared off. P and R refer to the relative motion, which is also harmonic. Therefore, the Fourier transform ~3(P) of ~o(R~- R2) is a harmonic oscillator function q/.o

I(~(p)I2 = q/no ( ~ ) e

(lO)

29

If the function q/HO is always the same quantum state, independent of the number n of removed nucleons, only the reduced mass kt depends on n (cf. Eq. (9)) and we find for the width of the momentum distribution after abrasion (indicated by superscript "abr"): A-n (a"br (n))2 = (P2)/3 = n ~

a"br(1).

(11)

381

For symmetry reasons a~br(n)=aaibr(n)=~r"br(n). The derivation, Equations (8) to (11), is simplified on purpose, to exhibit the physics. Goldhaber shows that Equation (11) is a consequence of momentum conservation alone. However, the Gaussian shape of the distribution is related to harmonic oscillator singleparticle wave functions (Lepore et al. [9]). Instead of relating the constant of proportionality o'abr(l) to the spring constant co of the harmonic oscillator (Ref. 9), we prefer to express it in terms of the Fermi momentum Pv (Ref. 3) since this quantity can be measured in other independent experiments. In a Fermi gas model for the projectile nucleus PF

[pF

30"abr(l)=0~ d 3 p p 2 / o ~ d 3 p = 3 p 2 / 5 ; (12)

~7abr(1)= pF/]//-5- .

Moniz et al. [10] deduce values ofpv from experiments of quasielastic scattering on light nuclei. Unfortunately 160 is not contained in their list. Therefore, we interpolate values from neighboring nuclei to obtain Pr = (225 _+5) MeV/c for Oxygen. The numbers of all shown in column 2 of Table 1 are calculated from Equations(ll) and (12) with this value of PF" Compared to experiment (column 7 in Table 1) the calculated values are systematically too large. The discrepancy increases with n and isotopic dependencies are not reproduced. Before we proceed to a discussion of ablation, which cures these deficiencies, we draw attention to a surface phenomenon. It reduces the values of all slightly, but, more important, it destroys the isotropy aji =a• The effect is the following: During abrasion only nucleons from the projectile surface are sheared off. These nucleons move on the edge of the potential well and therefore are slower than predicted by the Fermi gas model. We define the density of square momentum pZ(x), where i refers to the x, y or z directions, by p2(x) P(x)=

~=IZq/~(X) \ 0X2/ q/~(X)

A p(x)= ~, q/~*(x)q/=(x).

(13)

The sums run over all occupied single-particle orbits in the projectile. The abrasion cross section cr~t for the removal of n nucleons receives its intensity from a small band of impact parameters a,tot= Sd2 b crtot( b - b,) around a peak value at b,. The function 6,,tot( b - b.) is displayed in Reference 7. We define the mean square momentum of a nucleon in the overlap region by 2 j"d3xPi2(x) p(x) ~~ (P~),= ~-d~x~a~oot~)

b.) ;

x=(b,z).

(14)

382

A. Abul-Magd and J. Hiifner: M o m e n t u m Distributions in Fragmentation Reactions

Table 1. A comparison of calculated and experimental values for the width ali [MeV/c] of the m o m e n t u m distribution in the reaction ~60 +9Be--*a~Z v + X at 2 GeV/nucleon. Experimental values are from Reference 4. The contributions from the various effects during the process of abrasion-ablation are listed separately. Details in the text Fragment

all (Only abrasion)

Influence of ablation

Final value of a ,

Ablation coefficients

Isotropic Fermi gas

Surface correction

No recoil from decay

Calc.

co

c1

c2

c3

~50 15N

101+2 101+2

-3 -3

1 1

0 0

0 0

0 0

1~O 14N 14C

137__2 137-1-2 137_+2

-1 -1 -!

130 13N

163_+3 163-+3 163-+3 163 -+ 3

-1 -1 -1

AFZF

13C 13B 12N

12C LZB

180_+4 180-+4 180_+4

-

Recoil correction

0 0

0 0

24-? 9_+3 9_+8

0.5 1.5_+0.5 1.5_+0.5

Exp.

98+2 98_+2

94+3 95+3

135_+? 130+6 130_+10

99_+6 112_+3 125_+3

0.95 0.7 0.7

0.05 0.3 0.3

0 0 0

0 0 0

1

0 --20-+4 -39_+5 8 _+?

0 +4-+1 +5_+2 < 0.5

163_+3 147+8 128-+10 155 + ?

143_+14 1344-2 130_+3 166 -+ 10

1 0.4 0.2 0.8

0 0.4 0.4 0.1

0 0.2 0.4 0.1

0 0 0 0

0 0 0

--21_+12 -52-+35 -36_+12

+4_+1 +9_+3 +5_+2

163_+17 137-+42 149+18

153_+11 120_+4 163_+8

0 0 0

1 0.3 0.6

0 0.3 0.2

0 0.4 0.2

We have adjusted the spring constant in the harmonic oscillator functions ~b, in such a way that Equation (14) without at,~ yields ( p 2 ) = p 2 / 5 " The difference between this value and the one calculated from Equation (14) is called the surface correction and is shown in Table 1. As expected, the correction is largest for one-nucleon removal9 The correction is not very large, but differs for transverse and longitudinal components because the localization function on al~ depends only on b and not on z. The difference a N - a ~ deriving from the surface effect is significant, at least for n = 1, cf. Table 2.

3. Ablation The nucleus, which is formed from the projectile by the abrasion of n nucleons, is called a pre-fragment. It is a highly excited nucleus, in general, and decays by emitting nucleons or clusters until a particle-stable nucleus is formed. This final fragment is observed in the counter. The momentum distribution of the prefragment is calculated in the previous section. Here we compute the additional momentum due to the recoil of the decay nucleons. Of course, the magnitude of the recoil momentum P~or is simply related to the decay energy E a by P2r162 Until now the decay nucleons have not been observed and we do not know the magnitude of E a nor the angular distribution. Therefore we have to make a model. We assume that the excitation energy of the prefragment thermalizes until a compound nucleus is formed, which then decays. One consequence is immediate: the decay is isotropic and does not modify the difference a l l - a• Furthermore, the decay energy

is proportional to the nuclear temperature 6) in the daughter nucleus, Ea,,~eO, where ~ is a numerical constant of order one. The precise value of e depends on the energy dependence of the optical potential transmission coefficients and the level density. We find e,-~3/4 for light nuclei. The sequence of compound nucleus decays starting from the pre-fragment and ending at the final fragment is calculated from a realistic cascade program, Reference 11. Level densities, nuclear temperatures, branching ratios and recoil momenta are computed at each step. Typical values of O are found to be 2-3 MeV. Therefore, recoil momenta in direction i are of order arec~--(Pi2)1"2-1/2m~O/3"~-30-4OMeV/c. These values are rather small (note that they have to be added quadratically

to (~br)2). Consider a fragment A~'Zv with n = A - A r observed in the counter. It can be formed from different intermediate pre-fragments APZv with n' -- A - Av. The final value of the momentum distribution is given by

n'--O

9[(a~lb' (A"zp))2 + (a~ ~(A"Zp, a~Zv))2 ]

(15)

where c,_,,>0

and

~, C,_,,(a"Zv, A~ZF)=I.

(16)

n'=O

The ablation coefficients c,_,, indicate with which weight a given pre-ffagment contributes to a final fragment. The values of c,_,, are calculated in Reference 7 and are quoted in the last columns of Table 1. The abrasion values a~lb~ arm computed as discussed in Section 2. The main effect of ablation is to reduce

A. Abul-Magd and J. Hiifner: Momentum Distributions in Fragmentation Reactions

The word "friction" introduced in Reference 7 describes the following phenomenon: During the process of abrasion, a few nucleons of the projectile are knocked out because they collide with nucleons from the target. Microscopically, each projectile nucleon i to be knocked out receives an additional large m o m e n t u m q (we find ( q 2 ) t / 2 = 400 MeV/c from experimental angular distributions in proton-proton collisions) which is sufficient to overcome the potential well. The energy necessary to break the binding must be taken from the kinetic energy of relative motion between the two heavy ions. Breaking of bonds leads to a slow-down, which we call friction. The magnitude of the effect can be understood in the following, very simple model: Assume that initially a projectile nucleon rests at the bottom of the potential well U. It receives m o m e n t u m q by a nucleon-nucleon collision, starts to climb "'uphill" and slows down. Outside the potential well it moves with m o m e n t u m ( 1 - f ) q (we have neglected a possible change in the direction), where f is determined from energy conservation

the calculated values of the widths. This paradoxical result can be understood easily from Equation (16), when one neglects the small values of ( o ~ ) 2. Then the final value o fin is calculated from an average of widths aabr(n'), where n'
[.~,n,AF7 ~"12 ~'tl ~ "~vIa.o recoil

, 2 = ~,.., c ..... ,(oilabr (n)) <(aab~(n)) 2.

383

(17)

n" = 0

Recoil effects modify this result only slightly (Table 1, columns 4 and 5). The ablation coefficients e depend strongly on the binding energy of the final fragment. Therefore, isotopic effects in ~tt originate in the ablation mechanism. Error bars are given to the ablation corrections. They reflect the degree of agreement or disagreement which is obtained in the abrasion-ablation calculation for the fragmentation cross sections. Question marks indicate isotopes where computed and experimental cross sections differ by more than a factor of two. Within the error bars, experimental and theoretical values of the widths all agree for most fragments.

q2 ( 1 _ f)2q2 ---2m {UI= 2m

IUI therefore f ~-qz/2 m.

(18)

We call f the friction coefficient. It is of order 0.1 to 0.3. Because of m o m e n t u m conservation, the m o m e n t u m loss of the nucleon must be taken up by the pre-fragment, pF~=jq. After the removal of n nucleons from the projectile

4. F r i c t i o n

According to experiment [4], the m o m e n t u m distribution after a fragmentation reaction is isotropic within an accuracy of 10 ~o. Detailed numbers about a possible anisotropy are not published. However, Figure 16 in Reference 1 shows the transverse and longitudinal m o m e n t u m distributions for ~60 + 9Be--, ~ 3 C + X and displays a definite anisotropy: o• is larger than oil by about 7 ~o. Thus, isotropy does not seem to be a fundamental property of fragmentation reactions, but only an approximate one. The abrasionablation model supports this point of view. While most effects discussed so far lead to an isotropic m o m e n t u m distribution, surface phenomena for example act differently in longitudinal and transverse directions. Friction, discussed in this section, is another effect which increases o~ but leaves oil unchanged.

t~

w = E J;q,.

09)

i=1

The m o m e n t u m transfer in a nucleon-nucleon collision is symmetric in transverse direction; therefore, we have for the averaged transverse m o m e n t u m transfer ( q i ) = 0, but in longitudinal direction: (q It) = - (1 + r/) (qZ)/2m cf. Reference7 (q=0.3 at an energy of 2 GeV). Therefore ( p , , r ) = ~ ([iq, l l ) = _ n j ~ ( l + , / )

2m

'

(20)

i=t

Table 2. The anisotropy in the momentum distributions. The surface effectand the friction contribute (in opposite directions) to the difference era-nil. Details in the text

A~'ZI

Fragment

Fermi gas model Surfaceeffectin abrasion O1 ~ all EMeV/c] . . . . . . . . Act, [MeWc] Aoz [MeV/c]

Friction zl~r~r [MeV/c]

Anisotropy (o-• a ll)/89 I + ~rll) . . . . . . . . . Calc. Exp.

150, lSN t40, 14N, 1'~C

I01 137

-4 -2

- 15 4

+ 15 + 12

(4-t- 2) I~o (8_+2)3{;

1aN, 13C 130,13B ~2C 12N. t2B

163 163 180 180

-1 1 0

- 1 - 1 0

(7• o (15• (2+ 1)'%

0

0

+11 +23 + 4 ~-.13

(7 • 3)'!~,

(7 _+?)~%

384

A. Abul-Magd and J. Hiifner: Momentum Distributions in Fragmentation Reactions

where an average friction coefficient f is introduced. The discussion o f f and a comparison of experimental, Equation(l), and calculated values for (PII), Equation (20), are the subject of Reference 8. In this paper we take Equation (20) to derive f from experimental values of (PII) and use them to calculate friction contributions to the momentum distribution via tl

=(P~ ) = ~ f ~ 2 ( q ~ z ) = n f 2 ( q 2 ) .

(21)

i=1

If we assume .f2=,]:2 and eliminate f from Equations (19) and (20), we find (O.Fr) 2

1 (Pll)exp (2m)2 -~2 n (1 +q)2(q2) '

(22)

In longitudinal direction friction does not appreciably change the momentum distribution. Table 2 shows calculated values for the anisotropy arising from surface and friction effects. All computed values are compatible with the upper limit of 10% given in Reference4. The value for 13C compares very favourably with the number read from the figure of Reference 1. Unfortunately we are unable to assign an error to the experimental number. 5. Conclusion

The two step model of abrasion-ablation leads to a quantitative (and parameter free) understanding of cross sections [7], friction effects [8] and of the momentum distributions (this paper). The abrasion step is well founded by Glauber theory but ablation still remains the weak point. Not only does it introduce

large uncertainties, but even more important: The fundamental assumption of thermalization and compound nucleus decay is not yet proven in an unambiguous way. References 1. Heckman, H.H.: Proc. of the 5th Int. Conf. on High-Energy Physics and Nuclear Structure, Uppsala 1973, (G. Tibell ed.) p. 403. Amsterdam: North-Holland Publ. Comp. 1974 2. Feshbach, H., Huang, K.: Phys. Lett. 47B, 300 (1973) 3. Goldhaber, A.S.: Phys. Lett. 53B, 306 (1974) 4. Greiner, D. E., Lindstrom, P. J., Heckman, H. H., Cork, B., Bieser, E.S.: Phys. Rev. Lett. 35, 152 (1975) 5. Jakobson, B., Kullberg, R., Otterlund, I.: Presented at the 6th Int. Conf. on High-Energy Physics and Nuclear Structure, Santa Fe 1975 6. Bowman, J.D., Swiatecki, W.J., Tsang, C.F.: July 1973, unpublished 7. Hfifner, J., Sch~fer, K., Schtirmann, B.: Phys. Rev. C 12, 1888 (1975) 8. Abul-Magd, A., Htiflaer, J., Schiirmann, B.: Phys. Lett. 60B, 327 (1976) 9. Lepore, J.V., Ridell, R.J.: Proc. 2nd High Energy Heavy Ion Summer Study, Berkeley 1975, LBL-3675, p. 283 10. Moniz, E.J., Sick, I., Whitney, R.R., Ficenec, J.R., Kephart, R.D., Trower, W.P.: Phys. Rev. Lett. 26, 445 (1971) 11. Bondorf, J., NSrenberg, W. : Unpublished A. Abul-Magd Department of Physics Cairo University Giza, Egypt J. Hiifner Institut far Theoretische Physik der Universitiit Heidelberg Philosophcnweg 19 D-6900 Heidelberg 1 Federal Republic of Germany