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Proton inclusive cross sections from 720 MeV α-nucleus reactions
Nuclear Physics @ North-Holland
A362 (1981) 431-444 Publishing Company
PROTON INCLUSIVE CROSS SECTIONS FROM 720 MeV a-NUCLEUS REACTIONS K.R.
CORDELL,
S.T. THORNTON, L.C. DENNIS*, R.R. R.L. PARKS and T.C. SCHWEIZER***
DOERING**,
Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA Received 30 October (Revised 29 December
1980 1980)
Abstract: Proton inclusive cross sections have been measured from 30”-1.50” (lab) for outgoing energies up to 500 MeV from the bombardment of 720 MeV a-particles on Al and Ta. An exhaustive survey of applicable models indicates that the intranuclear cascade and knock-out models provide the best description of the data. Single scattering mechanisms seem to be important, and there is some indirect evidence that high nuclear density effects occur.
E
NTJCLEAR
REACTIONS Al, Ta(a, p), E = 720 MeV; measured tion mechanism. Intranuclear cascade, knockout
a(&,
0); deduced
reac-
models.
1. Introduction Nucleus-nucleus collisions at energies greater than 100 MeV/nucleon in which regions of high nuclear density may be formed could provide means to observe novel phenomena and to learn more about the nuclear equation of state. Most of our knowledge about nuclear behavior to date has been derived from nuclear matter at “normal” density, p. - 0.17 fme3. At higher densities, p/pa- 2-5, field theory calculations predict a secondary minimum in the equation of state which represents a new state, or density isomer ‘), of nuclear matter. Other phenomena such as pion condensation ‘) and nuclear shock waves 3, have also been predicted. Before exotic new phenomena can be observed, a more immediate goal of medium energy interacts
physics should be realized. during
the collision
The mechanisms
process
must be better
by which normal understood.
nuclear
matter
At energies
typical
of Van de Graaff accelerators (- 10 MeV/nucleon) mechanisms such as few-nucleon mass transfer and compound-nucleus formation with some degree of preequilibrium emission appear to give good descriptions of the predominant reaction mechanisms in nuclear collisions. At energies considered in this work (180 MeV/nucleon), quite different mechanisms may dominate the collision process. Unlike small-impactparameter collisions at lower energies, medium- or high;energy central collisions * Present address: Dept. of Physics, Florida State University, Tallahassee, Fl. 32306. ** Present address: Texas Instruments, Box 1443, Houston, TX. 77001. *** Present address: Science Applications, Inc., 1710 Goodridge Dr., McLean, Va. 22102 431
432
K.R.
Cordell et al. / Proton inclusive cross sections
may result in complete destruction of the target nucleus, spallation, or the production of several thermally decaying sources which are assumed in some models to travel at relativistic velocities along the beam direction. Peripheral collisions which occur at energies
much higher than the nuclear
lead to nucleons
being knocked
binding
energy per nucleon
out of the target
are more likely to
than in few-nucleon
transfer.
Despite the complex nature of these collisions, measurements of single-particle inclusive spectra can help us to understand the mechanisms by which they were produced. By measuring the double differential cross sections one obtains information on the average multiplicity of such a reaction and the average energy and momentum dissipation 4). Although it is unlikely that one reaction mechanism is responsible for the observed spectra, some kinematical regions of the data might display the gross features of the dominant mechanism. Because of the uniquely available 720 MeV a-beam of the Space Radiation Effects Laboratory (SREL) at Newport News, Virginia, we chose to study relativistic cY-particle-induced reactions and to measure high-momentum components of the inclusive spectra. We have measured proton-inclusive spectra for Al and Ta targets from 180 MeV/nucleon a-bombardment for outgoing proton kinetic energies up to 500 MeV at lab angles of 30”-150”. Some of these data have been published in earlier reports5’6). The experimental procedure description of several reaction results of calculations. Finally,
and results are presented in the next section, and a mechanism models is given in sect. 3 along with the the conclusions are presented in sect. 4.
2. Experimental We have previously
published
procedure
and results
inclusive
proton data at 60”, 90”, 120” and 150” up and up to energies of 150 MeV [refs. ‘*“)I f or b om b ar d ment by 720 MeV a-particles, to energies of 550 MeV [ref. ‘)I for bombardment by 600 MeV protons. The outgoing protons in excess of 150 MeV are not completely stopped by the NaI (Tl) detector used for proton detection. In the present experiment we have used time-of-flight (TOF) techniques to increase the measurable proton energies up to 500 MeV and have also measured data at 30” by both techniques [NaI(Tl) only and also in conjunction with TOF]. Since the experimental details have been discussed elsewhere 5-7), only a brief mention will be given here. A “stochastic tee” was used to detune the beam of the SREL synchrocyclotron from the prompt bursts which caused the detectors to be overloaded. Typical beam currents of 0.2 nA were used, and the time structure of the beam was monitored by the use of plastic scintillator monitor detectors. Natural targets of 2.57 g/cm2 Al and 2.52 g/cm2 Ta were used. The TOF system 7, consisted of plastic scintillators separated by flight paths of 2.5-5.0 m. A conically shaped 10.2 cm thick NaI(T1) detector preceeded by two 1000 wrn thick Si detectors (400 mm2 area) were used in conjunction with the TOF system for particle
433
K.R. Cordell et al. / Proton inclusive cross sections
identification. analyzed
The data were written
on the CYBER
reduction technique tered in the present
event
172 computer
by event
on magnetic
at the University
tape
of Virginia.
has been described elsewhere ‘*‘). No problems experiment due to pion production.
and later The data
were encoun-
The double differential cross sections (fig. 1) were normalized to previously measured absolute cross sections 5*6) and the lower energy 30” data measured in the present experiment. At 120” and 150” the cross sections measured by TOF do not extend the proton energies previously measured. For the higher proton energies the horizontal lines reflect the energy uncertainty from the TOF measurement. The absolute cross section uncertainties are estimated to be 20%.
‘.*’ 10-a -150"(xlo-~)**.
I 0
t
I
I
100
1
__
Cascade
---~~
Knock-on
1
200
I
300
I
I
LOO
_ I
II 500
E (MN)
on Fig. 1. The double differential cross sections for protons (lab energy) produced by 720 MeV a-particles Ta and Al are displayed for lab angles of 30”-150”. The lines are the results of calculations by the cascade model of Stevenson 16) and knock-out model of Hatch and Koonin lo).
3. Discussion The theoretical treatment of collisions involving many nucleons is clearly a very difficult problem without some simplifying approximations. At one end of the spectrum, the many-body problem can be treated microscopically by assuming that only a small number of nucleons interact and that the others are left undisturbed. This assumption is basic to the knock-out, quasi-two-body scaling and cluster
434
K.R. CordeN et al. / Proton inclusive
cross sections
models. At the other extreme one can assume that many nucleons that enough interactions occur so that a thermal equilibrium thermodynamic
models
scopic description
such as the fireball
of the collision.
In between
and firestreak
depend
discussion
and The
on this macro-
there are models which include
microscopic and macroscopic features. The following mod,els beginning with a simple two-body interaction.
3.1. KNOCK-OUT
are involved is attained.
will feature
various several
MODEL
One of the simplest and most transparent of the microscopic high-energy, heavyion reaction mechanisms is a single-scattering mechanism in which one projectile nucleon and one target nucleon interact in the early stages of the collision and are mutually scattered without disturbing other nucleons 9’10). The observation of knock-out nucleons may be masked in forward and backward directions by projectile and target evaporation but should be apparent in directions transverse to the nucleon-nucleon c.m. velocity. Hatch and Koonin lo) have formulated a parameter-free model which uses a hard-scattering mechanism governed by two-body kinematics to calculate absolute cross sections for proton and pion production. Experimental nucleon-nucleon cross sections are input using polynomial fits. A key feature of the model is that empirical momentum distributions for nucleons in nuclei are used rather than a sharp Fermi momentum distribution that leads to grossly underestimated cross sections in the high-momentum region of the data. There is evidence from backward-angle data that the nucleon
momentum
distribution
must fall off exponentially
11S12).Theoreti-
cal calculations 13) indicate that the momentum distribution should eventually fall off as a power at high momenta. Hatch and Koonin find their model is quite successful in describing
proton
800 MeV/nucleon
pion
production
from
bombarding
and
energies,
especially
3.2. QUASI-TWO-BODY
SCALING
Ne +NaF,
C+ C and
C+ Pb
at
near 90”.
AND HIGH-MOMENTUM
COMPONENTS
IN
THE NUCLEUS
As previously mentioned, there have been several experiments performed at backward angles which have led to various explanations of the data 6311-14). If an incident nucleon collides with a target nucleon which happens to be traveling in the opposite direction with a very high momentum, both nucleons are scattered and the target fragment is emitted at a backward angle in the lab frame 11-14). Frankel 11’12) has used his quasi-two-body scaling formalism to explain several high-energy backward-angle measurements. Since we have already compared our 150” data with Frankel’s and other models elsewhere 6), no further consideration of it will be made in the present paper. We concluded 6, in that work that although our data were
K.R. CordeN et al. / Proton inclusive cross sections
consistent
with the quasi-two-body
with multiple-scattering
3.3. INTRANUCLEAR
scaling formalism,
thermodynamic
CASCADE
435
the data were also consistent
models.
MODEL
The
treatment of the many-body problem has been attempted by several using Monte Carlo techniques. They devote most of their efforts to authors 15*i6) following the energy and momenta of many nucleons through the time evolution of the collision process, and most of the information about each two-body collision is taken from experimental nucleon-nucleon cross sections. Thus, these calculations yield no new information about the nucleon-nucleon interaction. They do, however, give us some insight into the special features of nuclear collisions, such as nuclear density, which single-scattering calculations do not provide. If we consider the possibility that each projectile nucleon may interact with several target nucleons, that each of those target nucleons may interact with other projectile and target nucleons, and that the projectile nucleons may then interact with each other, then we have reached the pinnacle of computational complexity. Such a model has been constructed by Smith and Danos is). Peripheral collisions result in most nucleons being emitted after only one collision, whereas central collisions generally lead to interactions among many nucleons and to some degree of thermalization. Calculations show that during the collision the local nuclear density can reach three to five times the normal nuclear density. Other features of this model include pion production and absorption, diffuse nuclear surfaces, the exclusion principle and binding effects. The complexity of the model necessitates very large and costly computer codes. Another classical many-body (cascade) calculation was formulated by Stevenson 16) using the assumption that relativistic nucleus-nucleus collisions may be treated as a succession of nucleon-nucleon collisions in a potential well. Before each collision,
all nucleons
are assigned
randomly
chosen
positions
and momenta
from a Fermi distribution. Nucleons follow straight lines and interact at the point of closest approach, provided that their separation d satisfies d2< cr(E,.,.) where cr((E,.,.) is the nucleon-nucleon total cross section at c.m. energy EC,,.. The angle of one of the scattered nucleons is randomly chosen from experimental elastic scattering angular distributions, and the momentum of each nucleon must satisfy the exclusion principle p > pF (265 MeV/c) in the laboratory frame. One of the interesting aspects of this calculation is the number of scatterings per emitted nucleon. Stevenson 16) calculates that for the 250 MeV/nucleon 20Ne on U reaction single scattering accounts for 13% of the emitted nucleons and the average number of scatterings is about five. The most probable number of scatterings is about two. If only the results of central collisions (impact parameter b <3 fm) are calculated, the average number of scatterings is six, with the most probable number
436
K.R.
Cordell et al. / Proton inclusive cross sections
being 2 or 3 collisions. However, for collisions with lighter particles such as alphas, single scattering is likely to account for a larger fraction of the events.
3.4. ROWS-ON-ROWS
MODEL
In an attempt to simplify the 3n-dimensional geometry of the intranuclear cascade problem, the “rows-on-rows model” 17)assumes straight-line geometry with a linear cascade occurring within each pair of colliding rows or tubes of the target and the projectile which lie along the beam direction. To further simplify the problem, interactions among target nucleons and among projectile nucleons are ignored. In the regime of “relativistic” reactions where the incident energy/nucleon, E,/A, is larger than the rest mass (E,/A > m,,c’), Glauber theory is used to calculate the angular distribution resulting from each nucleon-nucleon scattering. For lower energy “fast reactions”, where the incident energy is much greater than the Fermi energy but less than the rest mass (cr<< EB/A < mgc2), experimental p + n and p + p cross sections are used as input to determine the outcome of each collision. Unlike the full cascade calculations where the time evolution of each collision is followed and many collisions are required for good statistics, the rows-on-rows model reduces the problem to an equation for the probability distribution which may be solved analytically. The price paid for the appealing simplicity of this model is the obvious violations of its assumptions at energies around 2.50 MeV/nucleon. At these energies, nucleonnucleon scattering is nearly isotropic, not forward peaked as the assumption of non-interacting rows would require. Also, for these “low” energies the projectile may be stopped in the target and the assumption of no interactions among target and among projectile nucleons must be violated because all nucleons are then on an equal footing. A surprising result of this calculation is that, in spite of all the restrictions on the allowed interactions, after only a few collisions the momentum distribution within the nucleus appears to be thermalized. 3.5. THERMODYNAMIC
MODEL
Since the many-body calculations seem to indicate that some degree of thermalization occurs after only a few interactions, it seems appropriate to discuss a macroscopic thermodynamic model. Such a model might include one or more thermal sources, each traveling at different velocities along the beam direction and emitting nucleons isotropically in their respective rest frames. A Maxwell-Boltzmann distribution has been used to describe the energy distribution among the nucleons within a highly excited nucleus “). The observed energy distribution is D(E)
exp (-(E-/&)/T)
dk,
(1)
K.R. Cordell et al. I Proton inclusive cross sections
437
where 7 is the nuclear temperature, k is the effective Coulomb barrier fraction and A is the smearing parameter needed to reproduce the experimental peak widths. The Coulomb barrier energy is given by Z(Z,-Z)e2 ’ = 1.44[A”3+(AT-A)1’3]’
(2)
where Z(Z,) is the atomic number and A(AT) is the mass number of the emitted (target) nucleus. Typical values of k range from 0.4 to 1.0. The pre-exponential factor D(E) -E corresponds to a non-relativistic Maxwell-Boltzmann distribution of particles emitted from the surface of a hot object where the higher velocity particles are more likely to escape. The form D(E) - JE, on the other hand, corresponds to the energy distribution of all nucleons, including the lower velocity particles i9). The cross sections can be calculated as by Westfall et al. ‘O). Previous experiments 5,21) h ave shown that an evaporation peak was apparent with a source velocity less than or equal to the c.m. velocity of the projectile and target. However, at higher outgoing energies such as in the present experiment, projectile de-excitation is not likely to occur thermally, but perhaps the projectile might pick up enough nucleons as it plows through the target to become a “nuclear fireball”, One would then expect to have a thermally evaporating source traveling at a velocity much less than the beam velocity but greater than the c.m. velocity. Forward-backward symmetry in the c.m. frame of the invariant differential cross section (l/p) d*g/dE dL&where p is the momentum of a proton with total energy E, is indicative of a thermal source. The shape of the perpendicular (to the beam direction) versus parallel momentum distribution should reflect the assumed gaussian momentum distribution within the source. Transformation to the laboratory frame, however, will distort the shape of the distribution. If we plot the perpendicular momenta versus the rapidity along the beam (yll= tanh-’ &), the transformation from the moving frame to the laboratory frame results in a translation along the yli axis. The rapidity plots for the cy+ Al(Ta) + p + X reactions (fig. 2) show contours of constant invariant cross section which approximately center around /? = 0.2 in all cases. Another feature may be added to the thermodynamic model when one considers that, for beam energies > 100 MeV/nucleon, the collision velocity is greater than the speed of sound in nuclear matter 22). As the projectile sweeps out a part of the target, the resulting fireball is not only highly excited but also somewhat compressed 22).The compressional energy in this dense nuclear fireball forms a blast wave, or explosion, which continues as long as collisions are sufficiently frequent in the region of the source to direct the average momenta of the nucleons outward. The characteristic features of the explosion are the peaking of the observed velocity distribution about the mean radial velocity (in the c.m. of the fireball) and the reduction in the excitation due to the cooling accompanying the expansion. Siemens and Rasmussen 23)give an expression for the final momentum-space density of particles of momentum p and
438
K.R. Cord&l et al. / Proton inclusive cross sections 800.)
,
I
;; s -f600-
5 I
,
,
-
’
I' 0 ,' ;$
,
,
,
__
/
d' I
-_
\\
b. \ ,-x '. '\ ,10-5 ,j '\ '\
,>d
',,'wJ-~
6
f5200-
,Id‘ :
I(
6;
10-a
d --
o1o-2
,' ' d
d'
OJ""""""
-0.4
0
0.4
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_
d d'
Beam
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;--o., \\ I I \ \ I' ,p \ -6 ' I -a'\ '010 P, *'\ \\ bW5 -' ' /* D ,' ,d ,' lo-' -o--__ -- ,' I' p' /' *1o-3 -
P.,
p’-
,
,
0 -;
vi
E
,
/-hy
/+(yJ -
P
,
a+Ta
/'
5 t-
,
0.8 LAB
-0.4
0
0.4
I'
0.8
RAPIDITY
Fig. 2. The transverse momentum is plotted versus lab rapidity (see text) for 720 MeV a-particle bombardment of Ta and Al. The arrows represent the rapidity of the incident beam. The dashed lines are contours of constant invariant cross sections.
energy E of a spherically symmetric fireball expanding relativistically with a radial velocity p and a temperature r. We have used corrected equations ‘) of Siemens and Rasmussen 23) to calculate cross sections from this blast wave formalism. The simple thermodynamic model describes the energy spectra of nuclei emitted from a fireball-like entity which has a temperature r and velocity /3 without regard to the mechanism which produced it. A more complete model should include the kinematics and the geometry of the collision to determine r and p.
3.6. FIREBALL
AND
FIRESTREAK
MODELS
The fireball model of Westfall et al. 24) attempts to calculate geometrically the number of nucleons which are mutually swept out of the target and projectile as a function of impact parameter. The participant nucleons, those which are swept out, become a fireball, and the remainder of the nucleons are classified as spectators. The projectile spectators continue forward with the beam energy/nucleon while the target spectators remain at rest in the lab. The excitation and subsequent decay of spectator fragments is not considered in this model. The projectile participants are assumed to transfer all of their momentum to the c.m. of the fireball which moves forward at a velocity determined by conservation of energy. We have used the equations of refs. 24’25) to calculate cross sections for 720 MeV a-bombardment. The fireball model later evolved overlap region is treated as a series and participants during the collision “firestreak” by the velocity shear in
to include diffuse nuclear surfaces where the of tubes. Some interaction between spectators process results in the fireball being drawn into a the overlap region 26). The ratio of the particles
439
K.R. Cordell et al. / Proton inclusive cross sections
from the projectile
to the total number
of particles
in the firestreak
is given by the
dimensionless parameter 7, whose range is 0 to 1. We have also performed section calculations for our data for the firestreak model of ref. “).
3.7. REACTION
MODEL
cross
COMPARISON
The results of several of the model calculations discussed previously were compared with the data. The present data are unique in that we measured the reaction protons to quite high momenta relative to the beam momentum which may allow a more rigorous test of the various reaction mechanism models. Although some of the calculated cross sections completely fail to fit the data, several of the models yield surprisingly good agreement with a wide range of the data. The high-momentum components of the spectra reported here are not easily calculated in the intranuclear cascade calculations. A great many collisions must be followed in order to obtain good statistics for the nanobarn cross sections. Calculations using very complicated codes, such as the one by Smith and Danos i5), are simply not feasible because of the computer time involved. In fig. 1 cascade calculations from the method of Stevenson i6) are shown for Al and Ta targets to reasonably high energies. The over-all agreement with the data is satisfactory. Even a shoulder in the 30” data is predicted. The agreement with the Ta data is perhaps better than with that of the Al data for the forward angles and somewhat poorer for the backward angles. Smith 28) has observed that for central collisions after as few as 2-3 scatterings the thermalized. projectile and target nucleons of the “Ne + U system become Stevenson
i6) has found
peripheral
collisions
apparent
that
similar
results.
most nucleons
intranuclear
cascade
On the other
are ejected calculations
hand,
Smith
states
after only one collision. contain
features
that for It is quite
of both
single
scattering and complete thermalization. The computational difficulty of the cascade calculations prompted some authors to look for a simpler model which approximates the cascade results. The rows-on-rows model *‘) attempts to accomplish this. Unfortunately this parameter-free model reproduces neither the shape nor the normalization of the a-induced data. The results of the rows-on-rows calculations shown in fig. 3 for the (Y+ Al reaction are the predicted cross sections divided by 5.5. The assumption of non-interacting rows which requires very forward-peaked nucleon-nucleon cross sections is not appropriate for the reactions discussed here. According to Hiifner and Knoll I’), the “stopping length” of a 180 MeV nucleon in the target nucleus is on the order of 7 fm. In a near-central collision the 180 MeV/nucleon a-particle is stopped in the Al nucleus. The longitudinal momentum is partially deposited in the target matter and partially converted into transverse kinetic energy of the projectile. The target and projectile nucleons are indistinguishable; thus the assumption that target and projectile nucleons do not interact among themselves is clearly violated.
440
K.R. CordeN et al. / Proton inclusive cross sections
720
MeV
1
a + Al
-
I 0
I
I
100
i
I
F,restreak - Ftreball -Rows-on-rows 1
200
I
300
I
1
400
i
4
I
II 500
E (MeV)
Fig. 3. See fig. 1 for description of the data. The lines are results of calculations by the firestreak *‘), fireball *‘) and rows-on-rows I’) models. The calculations for the rows-on-rows model have been divided by 5.5 for the results shown.
Calculations for the knock-out model are displayed in fig. 1. The calculations were performed for the Al target and scaled as in eq. (3) of ref. lo) for Ta. The nucleon momentum distribution used for the calculation was a fit to the I60 theoretical calculation of Zabolitzky and Ey 29). The agreement with the data, especially for Ta, is surprisingly good. The hard, single-scattering assumption in the knock-out model is not expected to be particularly good at such a low bombarding energy of 180 MeV/nucleon. The calculation is also expected to be most applicable for the high momenta components, but the agreement is good over a wide energy and angular range. We point out that this calculation has no adjustable parameters and is primarily based on kinematics, NN cross sections, and the nucleon momentum distribution in the nucleus. Calculations for the nuclear fireball model *‘) are shown in fig. 3. The rapid fall-off of the calculations compared with the data, especially for Ta, indicate that the predicted average temperature of the fireball is too low. The firestreak calculations 27), on the other hand, give an average temperature which is too high to fit the data (fig. 3) for Al and Ta. Perhaps if the energy deposited in the spectator nucleons were considered in the calculation, the effective temperature of the firestreak would be lowered enough to fit the data. Gosset etal. 25)describe a method
441
K.R. Cordell et al. / Proton inclusive cross sections
of calculating
the contribution
to the spectator
excitation
energy
arising
from the
non-equilibrium shape of the (target) spectator nucleus after the collision. The difference in the surface area after the collision and the surface area of the equilibrium shape (assumed spherical) is multiplied by the nuclear surface-energy coefficient (-0.90 MeV/fm*) to obtain the surface energy in the spectator nucleus. The excess yield predicted by the firestreak model at low energies comes from an overestimate of the interaction between the nuclei at large impact parameters. If the transparency of the diffuse nuclear tails were included in the firestreak calculations, a better fit might be obtained 30). The transverse momentum versus lab rapidity plots (fig. 2) for the (Y+Ta and (Y+A1 reaction data show that contours of invariant cross sections are centered around p = 0.2 and are approximately symmetric except for the highest rapidity points (30” data). Calculations with a one-source thermodynamic model (fig. 4) indicate that the evaporation spectra of a source with a velocity PC.,. = 0.2 and a temperature r = 26 MeV give very good agreement with the data for 8 = 60”-150”, but underpredict the 30” data by as much as an order of magnitude. It appears that a fireball-like mechanism may be responsible for at least some of the features of the inclusive cross sections. It is possible to estimate the average number of nucleons that participate in the fireball. The available kinetic energy per nucleon in the fireball c.m.
I
I
0
100
I
1
200
I
1
300
400
II 500
E (MeV)
Fig. 4. See fig. 1 for description of the data. The lines are results of calculations with a one-source thermodynamic model, with PC.,. = 0.2 and temperature 7 = 26 MeV, and the blast wave model 23).
442
K.R. Cordell et al. / Proton inclusive cross sections
for a temperature of 26 MeV is 40 MeV/nucleon. The kinetic energy of the c.m. of a fireball with velocity of 0.2~ is 19 MeV/nucleon. If we neglect the recoil and excitation
of the spectator
nucleons
in the fireball:
possible
that
nucleons
we arrive
N = 720 MeV/(40
12 nucleons
can thermalize
at an upper
limit for the number
+ 19) MeV/nucleon their
energy
during
= 12 nucleons. the collision
of Is it time
(10pz2 s)? It is not necessary that the equilibration of a fireball occur outside the target nucleus. Cascade calculations, which predict that thermalization occurs after only a few collisions, also predict that a local “hot spot” of high nuclear density is formed as the projectile plows into the target nucleus at a supersonic velocity. Equilibration of the nucleons in the hot spot probably can occur during the collision time. Although most reaction data (60”-150”) can be fit using a simple thermodynamic model with one or several sources, the shoulder in the 30” spectra is in qualitative disagreement with a simple thermodynamic model. If we consider the possibility that the fireball expands rapidly from a compressed state, we obtain the blast wave model which also gives reasonable agreement with the data (fig. 4), especially for Ta. The c.m. velocity, PC.,. = 0.2, which was obtained from the rapidity plots, reproduces the anisotropy of the angular distribution very well. The temperature of 10 MeV was chosen to fit the slope, and the normalization cr = 1.5 mb was chosen to fit the magnitude of the 90” data. The blast velocity Pblast = 0.32 was chosen to reproduce the shoulder in the 30” data. Siemens and Rasmussen 23) fit only data for 90” in the c.m. frame, therefore they did not consider the source velocity. The extension of the model in this work to include the transformation from the fireball c.m. to the laboratory was necessary to fit all of the data. Siemens and Rasmussen give no justification for the value of the expansion velocity required to fit the data, except that the expansion degree of freedom contains the portion of the full initial kinetic energy which does not appear as pion mass or thermal energy. Calculations with a one-fluid hydrodynamic model “) predict enhancement in the differential cross sections for forward angles as a result of shock waves formed as the Experiments 3, with Ag and Cl targets projectile traverses the target nucleus. indicate peaks near 43” for 0.87 GeV/N I60 and c-u-beams and at 36” for a 0.25 GeV/N 12C beam. The velocity of a 0.18 GeV/N a-particle is high enough to cause shock waves at forward angles. Recall that most of the model predictions underestimate the present 30” data from the a + Al and cx+ Ta reactions. Perhaps the enhancement of the cross sections at 30” is due to shock waves. More detailed angular distributions would have to be measured in order to determine if shock waves occur in the reaction. 4. Conclusions Somewhat different reaction mechanisms may be responsible for the observed spectra at forward and backward angles. Forward-angle data (30”) scale approxi-
443
K.R. CordeN et al. / Proton inclusive cross sections
mately as Ag’3 for the Al and Ta targets, which possibly indicates a surface-type interaction. The more backward-angle data scale simply as AT, which indicates a volume-type
interaction.
The forward-angle
data
show some
structure
near
the
quasi-elastic-scattering energy. Intranuclear cascade and knock-out calculations reproduce both the normalization and the shape of the 720 MeV a-induced spectra. The other parameter-free models, rows-on-rows, fireball and firestreak, fail to fit the observed cross sections. A simple thermodynamic model can reproduce most of the data using the source velocity from the rapidity plots and treating the temperature and normalization as free parameters constrained only by conservation of energy, but cannot account for the shoulders in the 30” spectra. The blast-wave model with three free parameters reproduces the high-momentum components of the spectra and the shoulders at 30”. At least one additional source would be needed to explain the lower momentum data. The most surprising result of the present work is the agreement of the knock-out model calculation with the data considering the assumptions and applicability of the model. For many years it has been hoped that high-energy, light projectiles can probe the internal momentum distribution of nuclei. The nucleon momentum distribution of Zabolitzky and Ey 29) give good agreement with the present data and lead us to believe that probing the internal nucleon momentum distribution is possible with more sophisticated experiments. It is interesting to note that two of the model calculations, intranuclear cascade and blast wave, which give reasonable agreement with our data, also predict local regions of high density during the collision process. Perhaps the success of these models is indirect evidence that nuclear densities up to five times greater than normal nuclear matter are achieved through these reactions. Finally, it must be noted that two seemingly quite different models, intranuclear cascade and knock-out, give the best description of the data. One should remember, however, that many of the collisions described by the intranuclear cascade model are in fact single scattering, and the model is based on nucleon-nucleon cross sections. Therefore, there seems to be evidence that a description of the data must include a significant fraction of single scattering. The resolution
of these last two points,
possible
high nuclear
density
effects and
significant single scattering, will have to await further coincident and multiplicity experiments where more collision details are studied. The present and future experiments of nuclei.
show great promise
as tools to study the nuclear
momentum
distribution
The authors wish to acknowledge interesting discussions with G.D. Westfall, T. Fujita, R.K. Smith, W.D. Meyer and R.L. Hatch. We are especially indebted to J. Stevenson for performing the intranuclear cascade model calculation, to R.L. Hatch for performing the knock-out model calculation, and to G.D. Westfall for providing
444
K.R. Cordell et al. f Proton inclusive cross sections
the firestreak model and thermodynamic model computer programs. We also acknowledge the help of T.N. Taddeucci in obtaining the experimental data. This work was supported in part by the US National Science Foundation. We gratefully acknowledge the University of Virginia Academic Computer Center for financial support and the Max-Planck-Institute fiir Kernphysik, Heidelberg, W. Germany where the figures were prepared. References 1) T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291; Rev. Mod. Phys. 47 (1975) 267; E.N. Nyman and M. Rho, Nucl. Phys. A268 (1976) 408 2) G.E. Brown and W. Weise, Phys. Reports 276 (1976) 1 3) H.G. Baumgart, J.U. Schott, Y. Sakamoto, E. Schopper, H. StBdser, J. Hofmann, W. Scheid and W. Greiner, Z. Phys. A273 (1975) 359 4) H.H. Gutbrod, Relativistic heavy-ion physics Symp., GSI, Darmstadt, West Germany, 1978, LBL-7730 5) R.R. Doering, T.C. Schweizer, S.T. Thornton, L.C. Dennis, K.R. Cordell, K.O.H. Ziock and J.C. Comiso, Phys. Rev. Lett. 40 (1978) 1433 6) S.T. Thornton, K.R. Cordell, L.C. Dennis, R.R. Doering and T.C. Schweizer, Phys. Rev. Cl9 (1979) 913 7) K.R. Cordell, S.T. Thornton, L.C. Dennis, R.R. Doering, R.L. Parks and T.C. Schweizer, Nucl. Phys. A352 (1981) 485 8) K.R. Cordell, Ph.D. dissertation, University of Virginia (1979), available from University Microfilms Int., 300 N. Zeeb Rd., Ann Arbor, Michigan 48106 9) SE. Koonin, Phys. Rev. Lett. 39 (1977) 680 10) R.L. Hatch and S.E. Koonin, Phys. Lett. 81B (1979) 1 11) S. Frankel, Phys. Rev. Lett. 38 (1977) 1338; Phys. Rev. Cl7 (1978) 694 12) S. Frankel et al., Phys. Rev. Lett. 36 (1976) 642 13) R.D. Amado and R.M. Woloshyn, Phys. Rev. Lett. 36 (1976) 1435 14) R.D. Amado, Phys. Rev. Cl4 (1976) 1264 15) R.K. Smith and M. Danos, Proc. Conf. on heavy-ion collisions, Fall Creek Falls, Tenn., June 1977, p. 363 16) J.D. Stevenson, Phys. Rev. Lett. 41(1978) 1702 17) J. Hiifner and J. Knoll, Nucl. Phys. A290 (1977) 460 18) A.M. Poskanzer, G.W. Butler and E.K. Hyde, Phys. Rev. C3 (1973) 882 19) AS. Goldhaber;Phys. Rev. Cl7 (1978) 2243 20) G.D. Westfall, R.G. Sextro, A.M. Pozkanzer, A.M. Zebelman, G.W. Butler and E.K. Hyde, Phys. Rev. Cl7 (1978) 1368 21) T.C. Schweizer, R.R. Doering, S.T. Thornton, L.C. Dennis, K.R. Cordell and R.L. Parks, Phys. Rev. Cl9 (1979) 1408 22) M. Gyulassy, Int. Symp. on nuclear collisions and their microscopic descriptions, Bled, Yugoslavia, 1977, LBL-6594 23) P.J. Siemens and J.O. Rasmussen, Phys. Rev. Lett. 42 (1979) 880 24) G.D. Westfall, J. Gosset, P.J. Johansen, A.M. Poskanzer, W.G. Meyer, H.H. Gutbrod, A. Sandoval and R. Stock, Phys. Rev. Lett. 37 (1976) 1202 25) J. Gosset, H.H. Gutbrod, W.G. Meyer, A.M. Poskanzer, A. Sandoval, R. Stock and G.D. Westfall, Phys. Rev. Cl6 (1977) 629 26) W.D. Meyer, Nucl. Phys. A296 (1978) 177 27) J. Gosset, J.I. Kapusta and G.D. Westfall, Phys. Rev. Cl8 (1978) 844 28) R.K. Smith, pricate communication 29) J.G. Zabolitzky and W. Ey, Phys. Lett. 76B (1978) 527 30) W.D. Meyer, private communication
A362 (1981) 431-444 Publishing Company
PROTON INCLUSIVE CROSS SECTIONS FROM 720 MeV a-NUCLEUS REACTIONS K.R.
CORDELL,
S.T. THORNTON, L.C. DENNIS*, R.R. R.L. PARKS and T.C. SCHWEIZER***
DOERING**,
Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA Received 30 October (Revised 29 December
1980 1980)
Abstract: Proton inclusive cross sections have been measured from 30”-1.50” (lab) for outgoing energies up to 500 MeV from the bombardment of 720 MeV a-particles on Al and Ta. An exhaustive survey of applicable models indicates that the intranuclear cascade and knock-out models provide the best description of the data. Single scattering mechanisms seem to be important, and there is some indirect evidence that high nuclear density effects occur.
E
NTJCLEAR
REACTIONS Al, Ta(a, p), E = 720 MeV; measured tion mechanism. Intranuclear cascade, knockout
a(&,
0); deduced
reac-
models.
1. Introduction Nucleus-nucleus collisions at energies greater than 100 MeV/nucleon in which regions of high nuclear density may be formed could provide means to observe novel phenomena and to learn more about the nuclear equation of state. Most of our knowledge about nuclear behavior to date has been derived from nuclear matter at “normal” density, p. - 0.17 fme3. At higher densities, p/pa- 2-5, field theory calculations predict a secondary minimum in the equation of state which represents a new state, or density isomer ‘), of nuclear matter. Other phenomena such as pion condensation ‘) and nuclear shock waves 3, have also been predicted. Before exotic new phenomena can be observed, a more immediate goal of medium energy interacts
physics should be realized. during
the collision
The mechanisms
process
must be better
by which normal understood.
nuclear
matter
At energies
typical
of Van de Graaff accelerators (- 10 MeV/nucleon) mechanisms such as few-nucleon mass transfer and compound-nucleus formation with some degree of preequilibrium emission appear to give good descriptions of the predominant reaction mechanisms in nuclear collisions. At energies considered in this work (180 MeV/nucleon), quite different mechanisms may dominate the collision process. Unlike small-impactparameter collisions at lower energies, medium- or high;energy central collisions * Present address: Dept. of Physics, Florida State University, Tallahassee, Fl. 32306. ** Present address: Texas Instruments, Box 1443, Houston, TX. 77001. *** Present address: Science Applications, Inc., 1710 Goodridge Dr., McLean, Va. 22102 431
432
K.R.
Cordell et al. / Proton inclusive cross sections
may result in complete destruction of the target nucleus, spallation, or the production of several thermally decaying sources which are assumed in some models to travel at relativistic velocities along the beam direction. Peripheral collisions which occur at energies
much higher than the nuclear
lead to nucleons
being knocked
binding
energy per nucleon
out of the target
are more likely to
than in few-nucleon
transfer.
Despite the complex nature of these collisions, measurements of single-particle inclusive spectra can help us to understand the mechanisms by which they were produced. By measuring the double differential cross sections one obtains information on the average multiplicity of such a reaction and the average energy and momentum dissipation 4). Although it is unlikely that one reaction mechanism is responsible for the observed spectra, some kinematical regions of the data might display the gross features of the dominant mechanism. Because of the uniquely available 720 MeV a-beam of the Space Radiation Effects Laboratory (SREL) at Newport News, Virginia, we chose to study relativistic cY-particle-induced reactions and to measure high-momentum components of the inclusive spectra. We have measured proton-inclusive spectra for Al and Ta targets from 180 MeV/nucleon a-bombardment for outgoing proton kinetic energies up to 500 MeV at lab angles of 30”-150”. Some of these data have been published in earlier reports5’6). The experimental procedure description of several reaction results of calculations. Finally,
and results are presented in the next section, and a mechanism models is given in sect. 3 along with the the conclusions are presented in sect. 4.
2. Experimental We have previously
published
procedure
and results
inclusive
proton data at 60”, 90”, 120” and 150” up and up to energies of 150 MeV [refs. ‘*“)I f or b om b ar d ment by 720 MeV a-particles, to energies of 550 MeV [ref. ‘)I for bombardment by 600 MeV protons. The outgoing protons in excess of 150 MeV are not completely stopped by the NaI (Tl) detector used for proton detection. In the present experiment we have used time-of-flight (TOF) techniques to increase the measurable proton energies up to 500 MeV and have also measured data at 30” by both techniques [NaI(Tl) only and also in conjunction with TOF]. Since the experimental details have been discussed elsewhere 5-7), only a brief mention will be given here. A “stochastic tee” was used to detune the beam of the SREL synchrocyclotron from the prompt bursts which caused the detectors to be overloaded. Typical beam currents of 0.2 nA were used, and the time structure of the beam was monitored by the use of plastic scintillator monitor detectors. Natural targets of 2.57 g/cm2 Al and 2.52 g/cm2 Ta were used. The TOF system 7, consisted of plastic scintillators separated by flight paths of 2.5-5.0 m. A conically shaped 10.2 cm thick NaI(T1) detector preceeded by two 1000 wrn thick Si detectors (400 mm2 area) were used in conjunction with the TOF system for particle
433
K.R. Cordell et al. / Proton inclusive cross sections
identification. analyzed
The data were written
on the CYBER
reduction technique tered in the present
event
172 computer
by event
on magnetic
at the University
tape
of Virginia.
has been described elsewhere ‘*‘). No problems experiment due to pion production.
and later The data
were encoun-
The double differential cross sections (fig. 1) were normalized to previously measured absolute cross sections 5*6) and the lower energy 30” data measured in the present experiment. At 120” and 150” the cross sections measured by TOF do not extend the proton energies previously measured. For the higher proton energies the horizontal lines reflect the energy uncertainty from the TOF measurement. The absolute cross section uncertainties are estimated to be 20%.
‘.*’ 10-a -150"(xlo-~)**.
I 0
t
I
I
100
1
__
Cascade
---~~
Knock-on
1
200
I
300
I
I
LOO
_ I
II 500
E (MN)
on Fig. 1. The double differential cross sections for protons (lab energy) produced by 720 MeV a-particles Ta and Al are displayed for lab angles of 30”-150”. The lines are the results of calculations by the cascade model of Stevenson 16) and knock-out model of Hatch and Koonin lo).
3. Discussion The theoretical treatment of collisions involving many nucleons is clearly a very difficult problem without some simplifying approximations. At one end of the spectrum, the many-body problem can be treated microscopically by assuming that only a small number of nucleons interact and that the others are left undisturbed. This assumption is basic to the knock-out, quasi-two-body scaling and cluster
434
K.R. CordeN et al. / Proton inclusive
cross sections
models. At the other extreme one can assume that many nucleons that enough interactions occur so that a thermal equilibrium thermodynamic
models
scopic description
such as the fireball
of the collision.
In between
and firestreak
depend
discussion
and The
on this macro-
there are models which include
microscopic and macroscopic features. The following mod,els beginning with a simple two-body interaction.
3.1. KNOCK-OUT
are involved is attained.
will feature
various several
MODEL
One of the simplest and most transparent of the microscopic high-energy, heavyion reaction mechanisms is a single-scattering mechanism in which one projectile nucleon and one target nucleon interact in the early stages of the collision and are mutually scattered without disturbing other nucleons 9’10). The observation of knock-out nucleons may be masked in forward and backward directions by projectile and target evaporation but should be apparent in directions transverse to the nucleon-nucleon c.m. velocity. Hatch and Koonin lo) have formulated a parameter-free model which uses a hard-scattering mechanism governed by two-body kinematics to calculate absolute cross sections for proton and pion production. Experimental nucleon-nucleon cross sections are input using polynomial fits. A key feature of the model is that empirical momentum distributions for nucleons in nuclei are used rather than a sharp Fermi momentum distribution that leads to grossly underestimated cross sections in the high-momentum region of the data. There is evidence from backward-angle data that the nucleon
momentum
distribution
must fall off exponentially
11S12).Theoreti-
cal calculations 13) indicate that the momentum distribution should eventually fall off as a power at high momenta. Hatch and Koonin find their model is quite successful in describing
proton
800 MeV/nucleon
pion
production
from
bombarding
and
energies,
especially
3.2. QUASI-TWO-BODY
SCALING
Ne +NaF,
C+ C and
C+ Pb
at
near 90”.
AND HIGH-MOMENTUM
COMPONENTS
IN
THE NUCLEUS
As previously mentioned, there have been several experiments performed at backward angles which have led to various explanations of the data 6311-14). If an incident nucleon collides with a target nucleon which happens to be traveling in the opposite direction with a very high momentum, both nucleons are scattered and the target fragment is emitted at a backward angle in the lab frame 11-14). Frankel 11’12) has used his quasi-two-body scaling formalism to explain several high-energy backward-angle measurements. Since we have already compared our 150” data with Frankel’s and other models elsewhere 6), no further consideration of it will be made in the present paper. We concluded 6, in that work that although our data were
K.R. CordeN et al. / Proton inclusive cross sections
consistent
with the quasi-two-body
with multiple-scattering
3.3. INTRANUCLEAR
scaling formalism,
thermodynamic
CASCADE
435
the data were also consistent
models.
MODEL
The
treatment of the many-body problem has been attempted by several using Monte Carlo techniques. They devote most of their efforts to authors 15*i6) following the energy and momenta of many nucleons through the time evolution of the collision process, and most of the information about each two-body collision is taken from experimental nucleon-nucleon cross sections. Thus, these calculations yield no new information about the nucleon-nucleon interaction. They do, however, give us some insight into the special features of nuclear collisions, such as nuclear density, which single-scattering calculations do not provide. If we consider the possibility that each projectile nucleon may interact with several target nucleons, that each of those target nucleons may interact with other projectile and target nucleons, and that the projectile nucleons may then interact with each other, then we have reached the pinnacle of computational complexity. Such a model has been constructed by Smith and Danos is). Peripheral collisions result in most nucleons being emitted after only one collision, whereas central collisions generally lead to interactions among many nucleons and to some degree of thermalization. Calculations show that during the collision the local nuclear density can reach three to five times the normal nuclear density. Other features of this model include pion production and absorption, diffuse nuclear surfaces, the exclusion principle and binding effects. The complexity of the model necessitates very large and costly computer codes. Another classical many-body (cascade) calculation was formulated by Stevenson 16) using the assumption that relativistic nucleus-nucleus collisions may be treated as a succession of nucleon-nucleon collisions in a potential well. Before each collision,
all nucleons
are assigned
randomly
chosen
positions
and momenta
from a Fermi distribution. Nucleons follow straight lines and interact at the point of closest approach, provided that their separation d satisfies d2< cr(E,.,.) where cr((E,.,.) is the nucleon-nucleon total cross section at c.m. energy EC,,.. The angle of one of the scattered nucleons is randomly chosen from experimental elastic scattering angular distributions, and the momentum of each nucleon must satisfy the exclusion principle p > pF (265 MeV/c) in the laboratory frame. One of the interesting aspects of this calculation is the number of scatterings per emitted nucleon. Stevenson 16) calculates that for the 250 MeV/nucleon 20Ne on U reaction single scattering accounts for 13% of the emitted nucleons and the average number of scatterings is about five. The most probable number of scatterings is about two. If only the results of central collisions (impact parameter b <3 fm) are calculated, the average number of scatterings is six, with the most probable number
436
K.R.
Cordell et al. / Proton inclusive cross sections
being 2 or 3 collisions. However, for collisions with lighter particles such as alphas, single scattering is likely to account for a larger fraction of the events.
3.4. ROWS-ON-ROWS
MODEL
In an attempt to simplify the 3n-dimensional geometry of the intranuclear cascade problem, the “rows-on-rows model” 17)assumes straight-line geometry with a linear cascade occurring within each pair of colliding rows or tubes of the target and the projectile which lie along the beam direction. To further simplify the problem, interactions among target nucleons and among projectile nucleons are ignored. In the regime of “relativistic” reactions where the incident energy/nucleon, E,/A, is larger than the rest mass (E,/A > m,,c’), Glauber theory is used to calculate the angular distribution resulting from each nucleon-nucleon scattering. For lower energy “fast reactions”, where the incident energy is much greater than the Fermi energy but less than the rest mass (cr<< EB/A < mgc2), experimental p + n and p + p cross sections are used as input to determine the outcome of each collision. Unlike the full cascade calculations where the time evolution of each collision is followed and many collisions are required for good statistics, the rows-on-rows model reduces the problem to an equation for the probability distribution which may be solved analytically. The price paid for the appealing simplicity of this model is the obvious violations of its assumptions at energies around 2.50 MeV/nucleon. At these energies, nucleonnucleon scattering is nearly isotropic, not forward peaked as the assumption of non-interacting rows would require. Also, for these “low” energies the projectile may be stopped in the target and the assumption of no interactions among target and among projectile nucleons must be violated because all nucleons are then on an equal footing. A surprising result of this calculation is that, in spite of all the restrictions on the allowed interactions, after only a few collisions the momentum distribution within the nucleus appears to be thermalized. 3.5. THERMODYNAMIC
MODEL
Since the many-body calculations seem to indicate that some degree of thermalization occurs after only a few interactions, it seems appropriate to discuss a macroscopic thermodynamic model. Such a model might include one or more thermal sources, each traveling at different velocities along the beam direction and emitting nucleons isotropically in their respective rest frames. A Maxwell-Boltzmann distribution has been used to describe the energy distribution among the nucleons within a highly excited nucleus “). The observed energy distribution is D(E)
exp (-(E-/&)/T)
dk,
(1)
K.R. Cordell et al. I Proton inclusive cross sections
437
where 7 is the nuclear temperature, k is the effective Coulomb barrier fraction and A is the smearing parameter needed to reproduce the experimental peak widths. The Coulomb barrier energy is given by Z(Z,-Z)e2 ’ = 1.44[A”3+(AT-A)1’3]’
(2)
where Z(Z,) is the atomic number and A(AT) is the mass number of the emitted (target) nucleus. Typical values of k range from 0.4 to 1.0. The pre-exponential factor D(E) -E corresponds to a non-relativistic Maxwell-Boltzmann distribution of particles emitted from the surface of a hot object where the higher velocity particles are more likely to escape. The form D(E) - JE, on the other hand, corresponds to the energy distribution of all nucleons, including the lower velocity particles i9). The cross sections can be calculated as by Westfall et al. ‘O). Previous experiments 5,21) h ave shown that an evaporation peak was apparent with a source velocity less than or equal to the c.m. velocity of the projectile and target. However, at higher outgoing energies such as in the present experiment, projectile de-excitation is not likely to occur thermally, but perhaps the projectile might pick up enough nucleons as it plows through the target to become a “nuclear fireball”, One would then expect to have a thermally evaporating source traveling at a velocity much less than the beam velocity but greater than the c.m. velocity. Forward-backward symmetry in the c.m. frame of the invariant differential cross section (l/p) d*g/dE dL&where p is the momentum of a proton with total energy E, is indicative of a thermal source. The shape of the perpendicular (to the beam direction) versus parallel momentum distribution should reflect the assumed gaussian momentum distribution within the source. Transformation to the laboratory frame, however, will distort the shape of the distribution. If we plot the perpendicular momenta versus the rapidity along the beam (yll= tanh-’ &), the transformation from the moving frame to the laboratory frame results in a translation along the yli axis. The rapidity plots for the cy+ Al(Ta) + p + X reactions (fig. 2) show contours of constant invariant cross section which approximately center around /? = 0.2 in all cases. Another feature may be added to the thermodynamic model when one considers that, for beam energies > 100 MeV/nucleon, the collision velocity is greater than the speed of sound in nuclear matter 22). As the projectile sweeps out a part of the target, the resulting fireball is not only highly excited but also somewhat compressed 22).The compressional energy in this dense nuclear fireball forms a blast wave, or explosion, which continues as long as collisions are sufficiently frequent in the region of the source to direct the average momenta of the nucleons outward. The characteristic features of the explosion are the peaking of the observed velocity distribution about the mean radial velocity (in the c.m. of the fireball) and the reduction in the excitation due to the cooling accompanying the expansion. Siemens and Rasmussen 23)give an expression for the final momentum-space density of particles of momentum p and
438
K.R. Cord&l et al. / Proton inclusive cross sections 800.)
,
I
;; s -f600-
5 I
,
,
-
’
I' 0 ,' ;$
,
,
,
__
/
d' I
-_
\\
b. \ ,-x '. '\ ,10-5 ,j '\ '\
,>d
',,'wJ-~
6
f5200-
,Id‘ :
I(
6;
10-a
d --
o1o-2
,' ' d
d'
OJ""""""
-0.4
0
0.4
‘\\
,
a+Al
_
d d'
Beam
5
,
I’
:
6
b ”
2
-4.
,
;--o., \\ I I \ \ I' ,p \ -6 ' I -a'\ '010 P, *'\ \\ bW5 -' ' /* D ,' ,d ,' lo-' -o--__ -- ,' I' p' /' *1o-3 -
P.,
p’-
,
,
0 -;
vi
E
,
/-hy
/+(yJ -
P
,
a+Ta
/'
5 t-
,
0.8 LAB
-0.4
0
0.4
I'
0.8
RAPIDITY
Fig. 2. The transverse momentum is plotted versus lab rapidity (see text) for 720 MeV a-particle bombardment of Ta and Al. The arrows represent the rapidity of the incident beam. The dashed lines are contours of constant invariant cross sections.
energy E of a spherically symmetric fireball expanding relativistically with a radial velocity p and a temperature r. We have used corrected equations ‘) of Siemens and Rasmussen 23) to calculate cross sections from this blast wave formalism. The simple thermodynamic model describes the energy spectra of nuclei emitted from a fireball-like entity which has a temperature r and velocity /3 without regard to the mechanism which produced it. A more complete model should include the kinematics and the geometry of the collision to determine r and p.
3.6. FIREBALL
AND
FIRESTREAK
MODELS
The fireball model of Westfall et al. 24) attempts to calculate geometrically the number of nucleons which are mutually swept out of the target and projectile as a function of impact parameter. The participant nucleons, those which are swept out, become a fireball, and the remainder of the nucleons are classified as spectators. The projectile spectators continue forward with the beam energy/nucleon while the target spectators remain at rest in the lab. The excitation and subsequent decay of spectator fragments is not considered in this model. The projectile participants are assumed to transfer all of their momentum to the c.m. of the fireball which moves forward at a velocity determined by conservation of energy. We have used the equations of refs. 24’25) to calculate cross sections for 720 MeV a-bombardment. The fireball model later evolved overlap region is treated as a series and participants during the collision “firestreak” by the velocity shear in
to include diffuse nuclear surfaces where the of tubes. Some interaction between spectators process results in the fireball being drawn into a the overlap region 26). The ratio of the particles
439
K.R. Cordell et al. / Proton inclusive cross sections
from the projectile
to the total number
of particles
in the firestreak
is given by the
dimensionless parameter 7, whose range is 0 to 1. We have also performed section calculations for our data for the firestreak model of ref. “).
3.7. REACTION
MODEL
cross
COMPARISON
The results of several of the model calculations discussed previously were compared with the data. The present data are unique in that we measured the reaction protons to quite high momenta relative to the beam momentum which may allow a more rigorous test of the various reaction mechanism models. Although some of the calculated cross sections completely fail to fit the data, several of the models yield surprisingly good agreement with a wide range of the data. The high-momentum components of the spectra reported here are not easily calculated in the intranuclear cascade calculations. A great many collisions must be followed in order to obtain good statistics for the nanobarn cross sections. Calculations using very complicated codes, such as the one by Smith and Danos i5), are simply not feasible because of the computer time involved. In fig. 1 cascade calculations from the method of Stevenson i6) are shown for Al and Ta targets to reasonably high energies. The over-all agreement with the data is satisfactory. Even a shoulder in the 30” data is predicted. The agreement with the Ta data is perhaps better than with that of the Al data for the forward angles and somewhat poorer for the backward angles. Smith 28) has observed that for central collisions after as few as 2-3 scatterings the thermalized. projectile and target nucleons of the “Ne + U system become Stevenson
i6) has found
peripheral
collisions
apparent
that
similar
results.
most nucleons
intranuclear
cascade
On the other
are ejected calculations
hand,
Smith
states
after only one collision. contain
features
that for It is quite
of both
single
scattering and complete thermalization. The computational difficulty of the cascade calculations prompted some authors to look for a simpler model which approximates the cascade results. The rows-on-rows model *‘) attempts to accomplish this. Unfortunately this parameter-free model reproduces neither the shape nor the normalization of the a-induced data. The results of the rows-on-rows calculations shown in fig. 3 for the (Y+ Al reaction are the predicted cross sections divided by 5.5. The assumption of non-interacting rows which requires very forward-peaked nucleon-nucleon cross sections is not appropriate for the reactions discussed here. According to Hiifner and Knoll I’), the “stopping length” of a 180 MeV nucleon in the target nucleus is on the order of 7 fm. In a near-central collision the 180 MeV/nucleon a-particle is stopped in the Al nucleus. The longitudinal momentum is partially deposited in the target matter and partially converted into transverse kinetic energy of the projectile. The target and projectile nucleons are indistinguishable; thus the assumption that target and projectile nucleons do not interact among themselves is clearly violated.
440
K.R. CordeN et al. / Proton inclusive cross sections
720
MeV
1
a + Al
-
I 0
I
I
100
i
I
F,restreak - Ftreball -Rows-on-rows 1
200
I
300
I
1
400
i
4
I
II 500
E (MeV)
Fig. 3. See fig. 1 for description of the data. The lines are results of calculations by the firestreak *‘), fireball *‘) and rows-on-rows I’) models. The calculations for the rows-on-rows model have been divided by 5.5 for the results shown.
Calculations for the knock-out model are displayed in fig. 1. The calculations were performed for the Al target and scaled as in eq. (3) of ref. lo) for Ta. The nucleon momentum distribution used for the calculation was a fit to the I60 theoretical calculation of Zabolitzky and Ey 29). The agreement with the data, especially for Ta, is surprisingly good. The hard, single-scattering assumption in the knock-out model is not expected to be particularly good at such a low bombarding energy of 180 MeV/nucleon. The calculation is also expected to be most applicable for the high momenta components, but the agreement is good over a wide energy and angular range. We point out that this calculation has no adjustable parameters and is primarily based on kinematics, NN cross sections, and the nucleon momentum distribution in the nucleus. Calculations for the nuclear fireball model *‘) are shown in fig. 3. The rapid fall-off of the calculations compared with the data, especially for Ta, indicate that the predicted average temperature of the fireball is too low. The firestreak calculations 27), on the other hand, give an average temperature which is too high to fit the data (fig. 3) for Al and Ta. Perhaps if the energy deposited in the spectator nucleons were considered in the calculation, the effective temperature of the firestreak would be lowered enough to fit the data. Gosset etal. 25)describe a method
441
K.R. Cordell et al. / Proton inclusive cross sections
of calculating
the contribution
to the spectator
excitation
energy
arising
from the
non-equilibrium shape of the (target) spectator nucleus after the collision. The difference in the surface area after the collision and the surface area of the equilibrium shape (assumed spherical) is multiplied by the nuclear surface-energy coefficient (-0.90 MeV/fm*) to obtain the surface energy in the spectator nucleus. The excess yield predicted by the firestreak model at low energies comes from an overestimate of the interaction between the nuclei at large impact parameters. If the transparency of the diffuse nuclear tails were included in the firestreak calculations, a better fit might be obtained 30). The transverse momentum versus lab rapidity plots (fig. 2) for the (Y+Ta and (Y+A1 reaction data show that contours of invariant cross sections are centered around p = 0.2 and are approximately symmetric except for the highest rapidity points (30” data). Calculations with a one-source thermodynamic model (fig. 4) indicate that the evaporation spectra of a source with a velocity PC.,. = 0.2 and a temperature r = 26 MeV give very good agreement with the data for 8 = 60”-150”, but underpredict the 30” data by as much as an order of magnitude. It appears that a fireball-like mechanism may be responsible for at least some of the features of the inclusive cross sections. It is possible to estimate the average number of nucleons that participate in the fireball. The available kinetic energy per nucleon in the fireball c.m.
I
I
0
100
I
1
200
I
1
300
400
II 500
E (MeV)
Fig. 4. See fig. 1 for description of the data. The lines are results of calculations with a one-source thermodynamic model, with PC.,. = 0.2 and temperature 7 = 26 MeV, and the blast wave model 23).
442
K.R. Cordell et al. / Proton inclusive cross sections
for a temperature of 26 MeV is 40 MeV/nucleon. The kinetic energy of the c.m. of a fireball with velocity of 0.2~ is 19 MeV/nucleon. If we neglect the recoil and excitation
of the spectator
nucleons
in the fireball:
possible
that
nucleons
we arrive
N = 720 MeV/(40
12 nucleons
can thermalize
at an upper
limit for the number
+ 19) MeV/nucleon their
energy
during
= 12 nucleons. the collision
of Is it time
(10pz2 s)? It is not necessary that the equilibration of a fireball occur outside the target nucleus. Cascade calculations, which predict that thermalization occurs after only a few collisions, also predict that a local “hot spot” of high nuclear density is formed as the projectile plows into the target nucleus at a supersonic velocity. Equilibration of the nucleons in the hot spot probably can occur during the collision time. Although most reaction data (60”-150”) can be fit using a simple thermodynamic model with one or several sources, the shoulder in the 30” spectra is in qualitative disagreement with a simple thermodynamic model. If we consider the possibility that the fireball expands rapidly from a compressed state, we obtain the blast wave model which also gives reasonable agreement with the data (fig. 4), especially for Ta. The c.m. velocity, PC.,. = 0.2, which was obtained from the rapidity plots, reproduces the anisotropy of the angular distribution very well. The temperature of 10 MeV was chosen to fit the slope, and the normalization cr = 1.5 mb was chosen to fit the magnitude of the 90” data. The blast velocity Pblast = 0.32 was chosen to reproduce the shoulder in the 30” data. Siemens and Rasmussen 23) fit only data for 90” in the c.m. frame, therefore they did not consider the source velocity. The extension of the model in this work to include the transformation from the fireball c.m. to the laboratory was necessary to fit all of the data. Siemens and Rasmussen give no justification for the value of the expansion velocity required to fit the data, except that the expansion degree of freedom contains the portion of the full initial kinetic energy which does not appear as pion mass or thermal energy. Calculations with a one-fluid hydrodynamic model “) predict enhancement in the differential cross sections for forward angles as a result of shock waves formed as the Experiments 3, with Ag and Cl targets projectile traverses the target nucleus. indicate peaks near 43” for 0.87 GeV/N I60 and c-u-beams and at 36” for a 0.25 GeV/N 12C beam. The velocity of a 0.18 GeV/N a-particle is high enough to cause shock waves at forward angles. Recall that most of the model predictions underestimate the present 30” data from the a + Al and cx+ Ta reactions. Perhaps the enhancement of the cross sections at 30” is due to shock waves. More detailed angular distributions would have to be measured in order to determine if shock waves occur in the reaction. 4. Conclusions Somewhat different reaction mechanisms may be responsible for the observed spectra at forward and backward angles. Forward-angle data (30”) scale approxi-
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K.R. CordeN et al. / Proton inclusive cross sections
mately as Ag’3 for the Al and Ta targets, which possibly indicates a surface-type interaction. The more backward-angle data scale simply as AT, which indicates a volume-type
interaction.
The forward-angle
data
show some
structure
near
the
quasi-elastic-scattering energy. Intranuclear cascade and knock-out calculations reproduce both the normalization and the shape of the 720 MeV a-induced spectra. The other parameter-free models, rows-on-rows, fireball and firestreak, fail to fit the observed cross sections. A simple thermodynamic model can reproduce most of the data using the source velocity from the rapidity plots and treating the temperature and normalization as free parameters constrained only by conservation of energy, but cannot account for the shoulders in the 30” spectra. The blast-wave model with three free parameters reproduces the high-momentum components of the spectra and the shoulders at 30”. At least one additional source would be needed to explain the lower momentum data. The most surprising result of the present work is the agreement of the knock-out model calculation with the data considering the assumptions and applicability of the model. For many years it has been hoped that high-energy, light projectiles can probe the internal momentum distribution of nuclei. The nucleon momentum distribution of Zabolitzky and Ey 29) give good agreement with the present data and lead us to believe that probing the internal nucleon momentum distribution is possible with more sophisticated experiments. It is interesting to note that two of the model calculations, intranuclear cascade and blast wave, which give reasonable agreement with our data, also predict local regions of high density during the collision process. Perhaps the success of these models is indirect evidence that nuclear densities up to five times greater than normal nuclear matter are achieved through these reactions. Finally, it must be noted that two seemingly quite different models, intranuclear cascade and knock-out, give the best description of the data. One should remember, however, that many of the collisions described by the intranuclear cascade model are in fact single scattering, and the model is based on nucleon-nucleon cross sections. Therefore, there seems to be evidence that a description of the data must include a significant fraction of single scattering. The resolution
of these last two points,
possible
high nuclear
density
effects and
significant single scattering, will have to await further coincident and multiplicity experiments where more collision details are studied. The present and future experiments of nuclei.
show great promise
as tools to study the nuclear
momentum
distribution
The authors wish to acknowledge interesting discussions with G.D. Westfall, T. Fujita, R.K. Smith, W.D. Meyer and R.L. Hatch. We are especially indebted to J. Stevenson for performing the intranuclear cascade model calculation, to R.L. Hatch for performing the knock-out model calculation, and to G.D. Westfall for providing
444
K.R. Cordell et al. f Proton inclusive cross sections
the firestreak model and thermodynamic model computer programs. We also acknowledge the help of T.N. Taddeucci in obtaining the experimental data. This work was supported in part by the US National Science Foundation. We gratefully acknowledge the University of Virginia Academic Computer Center for financial support and the Max-Planck-Institute fiir Kernphysik, Heidelberg, W. Germany where the figures were prepared. References 1) T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291; Rev. Mod. Phys. 47 (1975) 267; E.N. Nyman and M. Rho, Nucl. Phys. A268 (1976) 408 2) G.E. Brown and W. Weise, Phys. Reports 276 (1976) 1 3) H.G. Baumgart, J.U. Schott, Y. Sakamoto, E. Schopper, H. StBdser, J. Hofmann, W. Scheid and W. Greiner, Z. Phys. A273 (1975) 359 4) H.H. Gutbrod, Relativistic heavy-ion physics Symp., GSI, Darmstadt, West Germany, 1978, LBL-7730 5) R.R. Doering, T.C. Schweizer, S.T. Thornton, L.C. Dennis, K.R. Cordell, K.O.H. Ziock and J.C. Comiso, Phys. Rev. Lett. 40 (1978) 1433 6) S.T. Thornton, K.R. Cordell, L.C. Dennis, R.R. Doering and T.C. Schweizer, Phys. Rev. Cl9 (1979) 913 7) K.R. Cordell, S.T. Thornton, L.C. Dennis, R.R. Doering, R.L. Parks and T.C. Schweizer, Nucl. Phys. A352 (1981) 485 8) K.R. Cordell, Ph.D. dissertation, University of Virginia (1979), available from University Microfilms Int., 300 N. Zeeb Rd., Ann Arbor, Michigan 48106 9) SE. Koonin, Phys. Rev. Lett. 39 (1977) 680 10) R.L. Hatch and S.E. Koonin, Phys. Lett. 81B (1979) 1 11) S. Frankel, Phys. Rev. Lett. 38 (1977) 1338; Phys. Rev. Cl7 (1978) 694 12) S. Frankel et al., Phys. Rev. Lett. 36 (1976) 642 13) R.D. Amado and R.M. Woloshyn, Phys. Rev. Lett. 36 (1976) 1435 14) R.D. Amado, Phys. Rev. Cl4 (1976) 1264 15) R.K. Smith and M. Danos, Proc. Conf. on heavy-ion collisions, Fall Creek Falls, Tenn., June 1977, p. 363 16) J.D. Stevenson, Phys. Rev. Lett. 41(1978) 1702 17) J. Hiifner and J. Knoll, Nucl. Phys. A290 (1977) 460 18) A.M. Poskanzer, G.W. Butler and E.K. Hyde, Phys. Rev. C3 (1973) 882 19) AS. Goldhaber;Phys. Rev. Cl7 (1978) 2243 20) G.D. Westfall, R.G. Sextro, A.M. Pozkanzer, A.M. Zebelman, G.W. Butler and E.K. Hyde, Phys. Rev. Cl7 (1978) 1368 21) T.C. Schweizer, R.R. Doering, S.T. Thornton, L.C. Dennis, K.R. Cordell and R.L. Parks, Phys. Rev. Cl9 (1979) 1408 22) M. Gyulassy, Int. Symp. on nuclear collisions and their microscopic descriptions, Bled, Yugoslavia, 1977, LBL-6594 23) P.J. Siemens and J.O. Rasmussen, Phys. Rev. Lett. 42 (1979) 880 24) G.D. Westfall, J. Gosset, P.J. Johansen, A.M. Poskanzer, W.G. Meyer, H.H. Gutbrod, A. Sandoval and R. Stock, Phys. Rev. Lett. 37 (1976) 1202 25) J. Gosset, H.H. Gutbrod, W.G. Meyer, A.M. Poskanzer, A. Sandoval, R. Stock and G.D. Westfall, Phys. Rev. Cl6 (1977) 629 26) W.D. Meyer, Nucl. Phys. A296 (1978) 177 27) J. Gosset, J.I. Kapusta and G.D. Westfall, Phys. Rev. Cl8 (1978) 844 28) R.K. Smith, pricate communication 29) J.G. Zabolitzky and W. Ey, Phys. Lett. 76B (1978) 527 30) W.D. Meyer, private communication