Angular Dependence in Proton-Proton Correlation Functions in Central 40 Ca + 40 Ca and 48 Ca + 48 Ca Reactions
V. Henzl, M. A. Kilburn, Z. Chajecki, D. Henzlova, W. G. Lynch, D. Brown, A. Chbihi, D. Coupland, P. Danielewicz, R. deSouza, M. Famiano, C. Herlitzius, S. Hudan, Jenny Lee, S. Lukyanov, A. M. Rogers, A. Sanetullaev, L. Sobotka, Z. Y. Sun, M. B. Tsang, A. Vander Molen, G. Verde, M. Wallace, M. Youngs
aa r X i v : . [ nu c l - e x ] A ug Angular Dependence in Proton-Proton Correlation Functions in Central Ca+ Caand Ca+ Ca Reactions
V. Henzl, ∗ M. A. Kilburn,
1, 2
Z. Chaj¸ecki, D. Henzlova, ∗ W. G. Lynch,
1, 2, † D. Brown,
1, 2
A. Chbihi, D. Coupland,
1, 2
P. Danielewicz,
1, 2
R. deSouza, M. Famiano, C. Herlitzius,
1, 6
S.Hudan, Jenny Lee,
1, 2
S. Lukyanov, A. M. Rogers,
1, 2, ‡ A. Sanetullaev,
1, 2
L. Sobotka, Z. Y.Sun,
1, 9
M. B. Tsang, A. Vander Molen, G. Verde, M. Wallace,
1, 2, ∗ and M. Youngs
1, 2 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48864, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48864, USA GANIL, CEA et IN2P3/CNRS, F-14076 Caen, France Department of Chemistry, Indiana University, Bloomington, IN 47405, USA Department of Physics, Western Michigan University, Kalamazoo, MI 49008, USA Joint Institute of Nuclear Astrophysics, Michigan State University, East Lansing, MI 48864, USA FLNR, JINR, 141980 Dubna, Moscow region, Russian Federation Department of Chemistry, Washington University, St. Louis, MO 63130, USA Institute of Modern Physics, CAS, Lanzhou 730000, Peoples Republic of China INFN, Laboratori Nazionali del Sud, Catania, Italy (Dated: October 10, 2018)The angular dependence of proton-proton correlation functions is studied in central Ca+ Caand Ca+ Ca nuclear reactions at E = 80 MeV/ A . Measurements were performed with the HiRAdetector complemented by the 4 π Array at NSCL. A striking angular dependence in the laboratoryframe is found within p - p correlation functions for both systems that greatly exceeds the measuredand expected isospin dependent difference between the neutron-rich and neutron-deficient systems.Sources measured at backward angles reflect the participant zone of the reaction, while much largersources observed at forward angles reflect the expanding, fragmenting and evaporating projectileremnants. The decrease of the size of the source with increasing momentum is observed at backwardangles while a weaker trend in the opposite direction is observed at forward angles. The results arecompared to the theoretical calculations using the BUU transport model. I. INTRODUCTION
The spectra of particles emitted in nuclear reactionscan include contributions from a variety of dynamicaland statistical mechanisms characterized by vastly differ-ent timescales. Dynamical emission typically occurs overtimescales as short as 10 − seconds. Statistical emis-sion can extend to much longer times. The descriptionsof dynamical and statistical emission mechanisms requirecompletely different theoretical formalisms. This compli-cates theoretical interpretations of measured spectra, asmost experimental observables do not allow a model inde-pendent distinction between earlier dynamical and laterstatistical emission.The correlation functions relevant to intensity interfer-ometry investigation [1–3], however, do not suffer thislimitation. Due to their ability to probe the space-timeextent of the sources of emission, two particle correla-tion functions allow a distinction between early dynami-cal and later statistical emission. This has been used toprobe the emission mechanisms of a variety of different ∗ Current address: Los Alamos National Laboratory, Los Alamos,NM 87545, USA † Email comments to: [email protected] ‡ Current address: Physics Division, Argonne National Laboratory,Argonne, Illinois, 60439 USA particle types for a number of different reactions studiedover a wide range of collision energies [4–7].The sensitivity of the two proton correlation functionsto the space-time extent of the source arises from themutual nuclear (attractive) and Coulomb (repulsive) in-teractions between the two protons and from the anti-symmetrized nature of their wave functions [3]. Gateson the proton pair velocity provide information aboutthe sources of these protons at different times during thereaction. Proton pairs with higher total momenta in therest frame of the source preferentially reflect the space-time extent of that source at earlier emission times whenthe source is smaller. Smaller sources typically displaylarger and broader correlation functions [4]. In contrast,proton pairs with lower total momenta tend to be emittedat later times after the source has expanded and cooled.Such sources typically display narrower, and weaker cor-relation functions. Thus, correlation functions can trackthe time evolution of a cooling, expanding source.Transport models [8, 9] have revealed the existence ofa sensitivity of two-nucleon ( p - p , n - p and n - n ) correla-tion functions to the density dependence of the symme-try energy and some sensitivity to isospin in two-particlecorrelation functions have been observed [10]. Physically,this sensitivity was shown to come from the effect of thesymmetry energy on proton and neutron potentials andtheir influence on the emission times of particles dur-ing the pre-equilibrium stages of the collision [8]. Thissuggests that investigations of isospin effects on reactiondynamics and their links to the density dependence ofthe symmetry energy [11], may profit from a more clearunderstanding of the time characteristics of different par-ticle emission processes and by the capability of isolatingemissions from the early pre-equilibrium stages of the re-action [8, 9].Stimulated by these ideas, we have measured p - p cor-relations over a wide angular and kinematic range withhigh statistics. In order to investigate the existence ofisospin effects, we have compared results from reactionsystems with different N/Z asymmetries, i.e. Ca+ Ca(N/Z =1) and Ca+ Ca (N/Z =1.4), at beam energiesE/A=80 MeV.Studies have been performed for the Ar+ Sc reac-tion at E/A=80 MeV, providing some guidance for thedependence of the source on the momenta of the outgoingprotons [12, 13]. The correlation functions measured in Ar+ Sc reactions show a strong decrease in the sourcesize with proton momentum, consistent with emissionfrom an expanding and cooling participant source. BUUtransport calculations generally reproduce these experi-mental trends.In our investigations of Ca + Ca and Ca + Cacollisions, we have measured the correlation functionsover a broader range of angles than previous measure-ments, and have studied in detail the momentum depen-dent two-proton correlation functions at different angles.The obtained results show that the applied momentumgates have strikingly different effects on the size of sourcescorresponding to particles emitted at forward angles ascompared to those detected at backward angles in thelaboratory frame. The measurements show a strong in-fluence of emission from the expanding, fragmenting andcooling spectator matter that was not evident in previousmeasurements. We also extract the fraction of protonsemitted over short timescales during the collisions fromthe height of the correlation function and the integral ofthe imaged source distribution. We see surprisingly littlesensitivity of these fractions to the angle or momentumof the measured protons. In order to better distinguishdynamical from statistical emission mechanisms, we alsocompared the extracted results to expectations of a BUUtransport model [14].
II. TWO PARTICLE CORRELATIONS
The two-proton correlation function probes the spa-cial and temporal information about the particle emittingsource because the magnitude of the final-state interac-tions and anti-symmetrization effect depend on both thespacial separation and the relative momentum of parti-cles [15].Theoretically, the correlation function is related to thespace-time extent of the source by the angle-averagedKoonin-Pratt equation [16, 17] C ( q ) = 1 + R ( q ) = 1 + 4 π Z K ( q, r ) S ( r ) r dr, (1) where the two-particle source function, S(r) , is the prob-ability of emitting two protons with a spatial separation r . In general protons are not emitted simultaneously.Then r in Eq. 1 refers to the separation at the time thesecond proton is emitted. The source function satisfiesthe following normalization condition4 π Z S ( r ) r dr = 1 (2)The angle-averaged kernel K ( q, r ) is given by K ( q, r ) = | φ q ( r ) | − . (3)where φ q ( r ) is the two proton wave function measured atthe separation distance between particles r and at pairrelative momentum q , defined in the center of mass ofthe pair by: q = | ~q | = 12 | ~p − ~p | (4)Within this approach, the shape of the proton-protoncorrelation function is affected by the nature of the final-state interactions. The anti-correlation at low q is a re-sult of Coulomb repulsion. More importantly, there is acharacteristic final S interaction peak at a relative mo-mentum of 20 MeV/ c . If there are two distinct emissiontimescales, a fast dynamical emission and a slow statis-tical emission, the height of this final interaction peakshould primarily reflect the fraction of protons emittedduring the fast pre-equilibrium stage of the reaction andthe size of the emitting source. The width of the peak at20 MeV/ c is solely affected by the space-time of the fastpre-equilibrium source [18]. Therefore a detailed studyof the overall shape of the correlation function allowsone to extract the space-time extent of the source andconstrain the relative contributions from fast and slowproton emitting components.Experimentally the correlation function can be writtenas C ( p , p ) = N A ( p , p ) B ( p , p ) . (5)Here, the numerator from Eq. 5 is the distribution of twoprotons with momentum p and p detected in the sameevent. The denominator describes the uncorrelated back-ground distribution and is constructed using so-called event-mixing method [19, 20] where each particle withina pair comes from a different event, taking into accountthe experimental two proton detection efficiency. N isa normalization factor, which typically results in corre-lation functions that are close to unity at large relativemomenta [21, 22].We used two different methods to extract the sizes ofthe sources presented in this paper. In one method, weemployed the imaging technique [23–25] to extract boththe size of the source and the source distribution pro-file S ( r ) from the measured correlation functions. In theother method, we obtained source sizes by fitting exper-imental correlation functions with the Koonin-Pratt for-mula (Eq. 1) assuming the Gaussian source distribution S ( r ) given by S ( r ) = λ G (2 √ πR G ) e − r R G (6)For the Gaussian source, there are three free parameters:the normalization of the correlation function, N , λ G andthe source size R G parameters of Eq. 6.According to Eq. 2, λ G = 1 if the emission of allprotons used to construct the correlation function is de-scribed by the source function. While some protons areemitted over a short time scale after the collision and arestrongly correlated, other protons can be emitted oververy long timescales due to evaporation processes andsecondary decays. Since the strength of the correlationsreflects the spatial separation between the two protons atthe time the second proton is emitted, early protons arenot correlated with protons emitted at later times andlate protons are only weakly correlated with each other.When both early and later emission occurs, thewidth of the peak in the p - p correlation function at20 MeV/ c primarily reflects the early emitted particles(fast source with smaller source sizes). Slowly emittedparticles, coming from long-lived and more extended sec-ondary decay sources primarily influence the correlationfunction at low q -values [18]. If one is not primarily con-cerned with low q -values, these late emissions of protonslargely reduce the magnitude of the correlation functionwhile not usually strongly modifying its shape [18]. Inthis case, Equation 2 has a more general form that re-flects the fact that not all protons are correlated witheach other, given to a good approximation by4 π Z S ( r ) r dr = λ. (7)The λ parameter represents the fraction of pairs whereboth protons are emitted by the fast source representedby S ( r ) over the range of r represented in the integral inEq. 7. The remainder 1 − λ contains the contributionsfrom pairs at large separation r outside of this range,where either one or both protons are emitted by the slowsource at the late secondary decay stage of the reaction.The relevant proton pair fraction comes from the fastsource; thus, λ , can be well approximated by [18] λ = f (8)where f and the remainder 1 − f are the fractions of thetotal protons yields produced by the fast source and theslow source, respectively.To minimize apriori assumptions about the sourcefunction we follow the imaging techniques described inRef. [18, 23–26], and describe the source function S ( r )by an expression involving three positive definite splinefunctions, which decreases monotonically with radius. We take the half-width half-maximum of the extractedsource profile, r / , as a measure of the spatial extent ofthe source. This provides a simple size parameter thatcan be easily calculated even in the case of non-Gaussiansource profiles where a R G parameter as in Eq. 6 cannotbe defined. In the specific case of a Gaussian source, therelation between r / and the size of the Gaussian sourcedistribution (Eq. 6) is given by r / = 2 √ ln R G . (9) III. EXPERIMENTAL DETAILS
We performed an experiment at the National Super-conducting Cyclotron Laboratory (NSCL), where pri-mary beams of Ca ( Ca) with an E/A=80 MeV im-pinged on Ca ( Ca) targets in a form of a thin mono-isotopic metallic foils of isotopic purity of about 97%(92.4%) by mass. We mounted the target near the centerof the 4 π Array, which housed 215 fast/slow phoswichescovering 85% of 4 π solid angle around the target in thelaboratory reference frame. The 4 π Array, with an in-side diameter of nearly 2 m, was instrumental in select-ing central events by requiring a high transverse energy, E t = P i E i sin ( θ i ) >
150 MeV [27]. Here, θ i and E i correspond to the angles and energies of charged par-ticles detected in the 4 π detector array. Assuming thetransverse energy to monotonically decrease with impactparameter, this gate on E t corresponds approximately toan impact parameter range of 0 < b ( f m ) < π Array with the High Resolution Array(HiRA) [28]. In our setup, HiRA consisted of 17 in-dividual telescopes in a hexagonal configuration, eachhousing a 65 µm thin single-sided silicon strip detectorfollowed by a 1.5 mm thick double-sided Si strip detec-tor with each face having 32 strips with a pitch of 2 mmand an active area of 62.3 x 62.3 mm . The orthogonalorientation of the front and back strips of the thick Sidetector, which was located 63 cm from the target, al-lowed for angular resolution of δθ ≈ . ◦ . In order toallow the high-precision angular determination of the po-sition, we measured the position of the target and siliconstrips in HiRA with the Laser Based Alignment System(LBAS) [29]. Additionally, the Si detectors were backedby a cluster of four 39 mm long CsI(TI) crystals whichserved as the calorimeters. For this paper, we only an-alyzed protons which stopped in the CsI(TI) crystals.This resulted in a proton momentum range of approxi-mately 200-500 MeV/ c . The angular coverage of HiRAwith respect to the beam was 18 < θ Lab ( deg ) <
58 inthe laboratory frame and 30 < θ CM ( deg ) <
110 in thecenter-of-mass frame. q [MeV/c]
Gaussian with time (cid:176) (cid:176) (cid:176) Ca Ca+ Ca Ca+ C ( q ) FIG. 1: Experimental correlation functions from Ca+ Ca(left) and Ca+ Ca (right). The upper panels include pro-tons with low total momentum of the pair (500-640 MeV/ c )while the lower panels represent proton pairs with a high totalmomentum (740-900 MeV/ c ). The dashed-dotted lines repre-sent the results of the fit assuming the Gaussian source distri-bution. The solid lines are reconstructed correlation functionsfrom imaging. The dashed lines represent the calculations as-suming the Gaussian source distribution with non-zero life-time (Eq. 10); see Sec. IV for more details. IV. EXPERIMENTAL RESULTS
The correlation functions measured in our experimentare shown in Fig. 1. The left panels present resultsfrom Ca+ Ca and the right panels from Ca+ Cacollisions. The upper and lower panels are for pro-tons with total momentum of the pair in the laboratoryframe of 500-640 MeV/ c and 740-900 MeV/ c , respectively.The correlation functions at the most backward angles(33 − ◦ ) in the laboratory frame are represented bysquares and at forward angles (18 − ◦ ) are shown ascircles. The results at intermediate angles (26 − ◦ ) areplotted as diamonds.In order to get quantitative information about the pro-ton emitting source we use the imaging technique to ex-tract the imaged source function. The fits to the correla-tion function are shown as the solid lines in Fig. 1. Thecorresponding extracted source functions are presentedas the light cross-hatched and dark solid bands in Fig. 2for 33 − ◦ and 26 − ◦ , respectively. In general, the cor-relation functions at backward angles have source func-tions that are larger and more localized around r = 0 f m .Imaging allowed us to reconstruct source distributionsonly at backward and intermediate angles. The imag-ing technique fails at forward angles when the peak at q = 20 MeV/ c is not well defined. If the source were r [fm] (cid:176) Imaging 33-58 (cid:176)
Imaging 26-33 (cid:176)
Gaussian 33-58 (cid:176)
Gaussian 26-33 (cid:176)
Gaussian 18-26 S ( r ) x Ca Ca+ Ca Ca+ FIG. 2: Comparison of imaging technique to Gaussian fit of p - p correlation functions for Ca+ Ca (left) and Ca+ Ca(right). The upper panels include proton pairs with low totalmomentum (500-640 MeV/ c ) while the lower panels representproton pairs with a high total momentum (740-900 MeV/ c ). Gaussian, the peak would become negligible for large val-ues of the R G parameter in Eq. 6, e.g. R G > − f m .Both the presence of sources with such large spatial ex-tensions, and large statistical errors in the correlationfunction make convergence of the imaging method diffi-cult to achieve at forward angles.The fit quality and the normalization of the recon-structed source distribution, λ I from Eq. 7, are givenin Table I. If no constraints are placed on the shapeof the source function, the imaging method can provideother solutions i.e. source functions S ( r ), with compara-bly small values of χ /dof , where dof ≈
30 is the numberof data points minus the number of fit parameters. How-ever, some of those solutions have unphysical properties,such as S ( r ) < λ I parameter.We also performed fits to the experimental correlationfunctions using Eq. 1 and assuming the Gaussian sourcefunction given by Eq. 6. The corresponding fits to thecorrelation functions are denoted by the dashed-dottedcurves in Fig. 1. For the angular ranges of θ = 26 − ◦ and θ = 33 − ◦ , these fits are nearly indistinguishablefrom the fits obtained via the imaging procedure, the lat-ter shown as thick lines in Fig. 1. In these fits there arethree fitting parameters: 1) the size of the source, R G ;2) the λ G parameter (from Eq. 6); and 3) the normal-ization of the correlation function, N (from Eq. 5). Thebest fit parameters are presented in Table I. The sourcedistributions obtained from the Gaussian fit are plottedas the solid, dashed-dotted and dashed lines for 18 − ◦ ,26 − ◦ and 33 − ◦ in Fig. 2, respectively. System P Angle Gaussian fit Imaging BUU[MeV/ c ] [ ◦ ] R G [fm] r / [fm] λ G f G χ /dof r / [fm] λ I f I χ /dof r / [fm] Ca+ Ca 500-640 33-58 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
48 5 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
18 6 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − − − − . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
06 4 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
58 4 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − − − − . ± . Ca+ Ca 500-640 33-58 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
41 5 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
17 6 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − − − − . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
51 4 . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . .
15 5 . ± . . +5 . − . . +8 . − . . +0 . − . . +0 . − . − − − − . ± . The correlation functions reconstructed from imagingand obtained from the Gaussian fit are very similar andmatch the data well at most angles, as it is shown inFig. 1. For the lowest momentum gate at θ = 33 − ◦ , thepeaks in correlation functions for the Gaussian sourcesare narrower and their tails lie consistently below thedata and the imaging results for q ≈
40 MeV/ c . Thisgives rise to the slightly wider widths of the correspond-ing Gaussian sources shown in Fig. 2 for these data. Forthe other gates, the results for Gaussian and imaginganalyses are very similar; in some cases, the source func-tions provided by imaging method are slightly more lo-calized at r = 0 f m than the corresponding fits withthe Gaussian source functions. At the most forward an-gles where the size of the source is large and the cor-relation effect is not as strong as in the experimentaldata collected at backward angles, it was not possible toconstrain the source function adequately via the imagingtechnique. There we used the more constrained Gaussiansource function in order to extract information about thespace-time extent of the source. Fortunately, the similar-ity between Gaussian and imaging analyzes at the otherangles provides support for us to use the Gaussian ap-proach and lends confidence to the information it pro-vides.To provide the simplified measure of the source, wecharacterize the extracted sources using r / (also usedin e.g. [18, 26, 30]) for each set of data and method usedto extract the source distribution or its size. Resultsare presented in Table I for both reaction systems, bothpair momentum ranges in the laboratory frame, and allthree angular selections. With the exception of the lowest momentum gate at θ = 33 − ◦ , the values for r / areconsistent between imaging and a Gaussian fit.The sources from the collisions with larger initial ge-ometry, Ca+ Ca (N/Z=1.4), are systematically largerthan those from Ca+ Ca collisions (N/Z=1). The av-erage increase in source size with A somewhat exceeds A / , which suggests that the average freezeout densityis somewhat lower for the Ca than for Ca. Due tothe large value of the neutron-proton cross section whichsignificantly exceeds the p - p cross section, the relevantdensity for proton freezeout may be the neutron den-sity rather than the total nuclear density. In this case,the additional neutrons in the Ca+ Ca may shift thefreezeout to lower overall density. Such a shift reflectsdetailed differences in the transport of neutrons and pro-tons that could be used to extract information about therelevant in-medium cross section. Calculations indicate,however, that such effects are subtle and dwarfed by thequalitative difference between the sizes at forward andbackward angles and we, therefore, defer such detailedmodel investigations to a latter publication.Clearly, the strong angular and momentum depen-dence of the extracted source size is a much more dra-matic trend. The observed large increase in the sourcesize occurs at forward angles at velocities comparable tothat of the beam. (The total momentum of two beamvelocity protons is approximately 800 MeV/c.) Cor-relation functions of similar magnitudes have been re-ported for protons evaporated from heavy residues pro-duced in
Xe+ Al reactions [31], and for protons emit-ted at energies comparable to the Coulomb barrier in Ar+
Au [32] and
Xe+
Au reactions [33]. In the
600 700 800 600 700 800
P [MeV/c]
Gaussian fit Imaging BUU Ca+ Ca Ca+ [f m ] / r (cid:176) (cid:176) (cid:176) FIG. 3: r / as a function of total momentum for Ca+ Caand Ca+ Ca collisions and all three angular ranges. Thesizes of source from data using the imaging technique are givenby red closed circles while those from the Gaussian techniqueare shown as blue closed triangles. Source sizes from BUUare shown as black open circles. latter case, however, a relatively small fraction of fastprotons (f=0.30) was reported. From the λ G and λ I parameters obtained from the Gaussian fit and imagingmethod we calculated the fraction f of short time scaleemitted protons [18], according to Eq. 8. The results aresummarized in Table I. We show that the values of this f parameter are consistent between both methods. Allproton fractions exceed 0.5 and there is very little mo-mentum or angular dependence of the f parameter. It isinteresting that the large sources at θ = 18 − ◦ and totalmomenta of 740 −
900 MeV/ c observed for both reactionsalso have relatively large fast fractions eg. f > .
7. Thisimplies that more than 70% of the two-proton emissionoccurs at relative separations of r <
15 fm. This appearsto exclude significant contributions ( > ̺ fre ≈ A spec / { π ( √ R G ) } .Assuming, the projectile contains A spec = 20 nucleonsprior to fragmentation and R G = 3 − ̺ fre of ̺ fre = 0 . − . ρ . This is somewhatbelow the density range assumed by statistical simultane-ous multi-fragmentation models [34, 35]. It is also belowthe density range, ρ = 0 . − . ρ , extracted from d − α correlations for the participant source in Xe+
Aucollisions [33]. Both comparisons suggest that the sourcefragments over a non-zero timescale.Alternatively, we assume that decay occurs from aspherical source with R G and vary the timescale of thedecay. Following Koonin [16] we assume a Gaussian emis-sion time distribution: i.e. emission rate ∝ exp ( − t /τ ).This leads to a source function of the form: S ( r ) = λ G (4 π ) / R G p R G + 0 . vτ ) e − r ⊥ R G − r || R G +0 . vτ )2) . (10)Here, v = |−→ v | , where −→ v = −→ V − −→ V is the magnitude ofthe velocity −→ V of the center of mass of the two protonsrelative to the velocity −→ V of the source, r ⊥ ( r || ) is thecomponent of ~r perpendicular (parallel) to ~v .The beam momentum per nucleon is roughly equalto the average proton momentum for the data with18 ◦ < θ < ◦ and 740 ≤ P ( M eV /c ) < v ≈ . c . In this scenario, thespace-time extent measured for those particles is a com-bination of the spatial dimension ( R G ) and the lifetimeof the source ( τ ). The dashed lines in the lower panels ofFig. 1 correspond to the correlation functions obtainedwith the source distribution from Eq. 10, where R G = 3fm and τ = 100 fm/c ( ≈ . × − s) for Ca+ Caand R G = 3 . τ = 135 fm/c ( ≈ . × − s) Ca+ Ca reactions.Those calculations show reasonable agreement with theexperimental correlation functions. This ”lifetime” is rel-atively short for a statistical evaporation process, butcomparable to the times for expansion and disassemblyduring a multifragmentation process [36–41].To illustrate the inconsistency of the large sourcesizes at forward angles with a straightforward dynami-cal origin, we simulated Ca+ Ca and Ca+ Ca col-lisions at E/A=80 MeV. We chose a parametrization ofBUU such that an energy dependent in-medium nucleon-nucleon cross section reduction was employed [42]. Wealso included momentum dependence in the mean fieldwith a soft equation of state [43]. We chose the densitydependence of the symmetry energy to be γ = 0 .
7, whichis in agreement with Ref. [44]. We also included the pro-duction of A ≤ r / by approximately 1 fm. From the information pro-vided by the transport model we constructed the sourcefunctions for the same momentum and particle emissionangle in the laboratory as used in the experimental anal-ysis. We included only the protons emitted at energiesand angles that could have been detected in the exper-iment. We calculated the quantity r / from the sourcedistribution. We show the comparison between theoreti-cal and experimental values for r / in Fig. 3.In Fig. 3, we can see that BUU can reproduce the ex-perimental data well at backward and intermediate an-gles for both pair momentum ranges measured in thelaboratory frame, but underpredicts the sizes at forwardangles, especially for protons in the high momentum gate.These high momentum particles move at close to thebeam velocity. We have calculated source radii for a widevariety of different mean fields and nucleon-nucleon crosssections, but have not been able to find a choice of trans-port parameters that result in significantly larger sourceradii at forward angles and beam velocity. Such largeradii indicate emission from a source that is much largeror longer-lived or both compared to the source that canbe predicted by a dynamical model such as the BUUapproach. A long lived source could explain the discrep-ancy with the BUU calculations, however a very long-lived source is inconsistent with the large fast fractions f > . V. CONCLUSIONS
We studied the angular and momentum dependence of p - p correlations for central Ca+ Ca and Ca+ Canuclear reactions at E/A=80 MeV. We found a strongangular dependence within p - p correlation functions re-flecting the different space-time extent of the source se-lected. Sources observed at backward angles, in the labo-ratory frame, reflect the participant zone of the reaction,while much larger sources are seen at forward angles aredominated by expanding, fragmenting and evaporatingprojectile-like residues. The obtained results show a de-creasing source size with increasing momentum of theproton pair emitted at backward and intermediate an-gles. In contrast we observe a weak trend in the op-posite direction at forward angles. At some level, thesetrends are consistent. In the rest frames of the respec- tive sources, higher velocity protons are more stronglycorrelated than their lower velocity counterparts, consis-tent with emission from expanding and cooling sources.The protons with small laboratory momenta at backwardangles move slowly in the rest frame of the participantsource. Protons with large laboratory momenta at back-ward angles move rapidly in the rest frame of the partici-pant source. In contrast, the highest momentum protonsat forward laboratory angles are nearly at rest relative tothe fragmenting projectile remnants, and the lower mo-mentum protons at forward angle are actually movingat a higher relative velocity to the fragmenting projectileremnants. In both angular domains, we therefore observesmaller sources for protons moving at higher velocities inthe frame of the source.Long evaporation times are not consistent with thefast fractions extracted from the correlation functionsforward angles. The time scales estimated from ourcorrelation functions are consistent with bulk multi-fragmentation time scales that have been extracted byfragment-fragment correlation functions.We find that BUU transport calculations reproduce thedata well at backward and intermediate angles, but un-derpredict to reproduce the source sizes at forward anglesat high momentum. There the data are consistent withexpansion, multi-fragmentation and subsequent evapora-tion. The failure of the BUU to reproduce the sourcefunctions for this case can be attributed to the suppres-sion of the fluctuations leading to multi-fragmentation inthis approach.In all cases, the Ca+ Ca reaction system resultsin larger sources than the Ca+ Ca reaction system,which can be partly attributed to a sensitivity of thesource distribution to the initial size of the projectile andtarget nuclei. However, the effect appears to be some-what larger than the A / scaling expected from suchgeometrical arguments. VI. ACKNOWLEDGMENTS
We would like to thank S. Pratt and D. A. Brownfor their help with the imaging process. We wish to ac-knowledge the support of Michigan State University, theJoin Institute for Nuclear Astrophysics, the National Sci-ence Foundation Grants No. PHY-0216783, No. PHY-0606007, No. PHY-0822648, and No. PHY-0855013,and the U.S. Department of Energy, Division of NuclearPhysics Grant No. DE-FG02-87ER-40316 and ContactNo. DE-AC02-06CH11357. [1] R. Hanbury Brown and R. Q. Twiss, Phil. Mag. , 663(1954).[2] G. Goldhaber, S. Goldhaber, W.-Y. Lee, and A. Pais, Phys. Rev. , 300 (1960).[3] D. H. Boal, C. K. Gelbke, and B. K. Jennings, Rev. Mod.Phys. , 553 (1990). [4] F. Zhu, W. G. Lynch, T. Murakami, C. K. Gelbke, Y. D.Kim, T. K. Nayak, R. Pelak, M. B. Tsang, H. M. Xu,W. G. Gong, et al., Phys. Rev. C , R582 (1991).[5] S. J. Gaff, C. K. Gelbke, W. Bauer, F. C. Daffin, T. Glas-macher, E. Gualtieri, K. Haglin, D. O. Handzy, S. Han-nuschke, M. J. Huang, et al., Phys. Rev. C , 2782(1995).[6] D. O. Handzy, M. A. Lisa, C. K. Gelbke, W. Bauer, F. C.Daffin, P. Decowski, W. G. Gong, E. Gualtieri, S. Han-nuschke, R. Lacey, et al., Phys. Rev. C , 858 (1994).[7] Z. Chen, C. K. Gelbke, W. G. Gong, Y. D. Kim, W. G.Lynch, M. R. Maier, J. Pochodzalla, M. B. Tsang,F. Saint-Laurent, D. Ardouin, et al., Phys. Rev. C ,2297 (1987).[8] L.-W. Chen, V. Greco, C. M. Ko, and B.-A. Li, Phys.Rev. C68 , 014605 (2003), nucl-th/0305036.[9] L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev.
C69 ,054606 (2004), nucl-th/0403049.[10] R. Ghetti et al., Phys. Rev.
C69 , 031605 (2004), nucl-ex/0310012.[11] B.-A. Li, L.-W. Chen, and C. M. Ko, Phys. Rept. ,113 (2008), 0804.3580.[12] M. A. Lisa et al., Phys. Rev. Lett. , 3709 (1993).[13] M. A. Lisa et al., Phys. Rev. Lett. , 2863 (1993).[14] P. Danielewicz and G. F. Bertsch, Nucl. Phys. A ,712 (1991).[15] G. Verde, A. Chibihi, R. Ghetti, and J. Helgesson, Eur.Phys. J. A , 81 (2006).[16] S. E. Koonin, Phys. Lett. B , 43 (1977).[17] S. Pratt, T. Cs¨org˝o, and J. Zim´anyi, Phys. Rev. C ,2646 (1990).[18] G. Verde, D. A. Brown, P. Danielewicz, C. K. Gelbke,W. G. Lynch, and M. B. Tsang, Phys. Rev. C , 054609(2002).[19] G. I. Kopylov, Phys. Lett. B , 472 (1974).[20] M. Lisa, W. Gong, C. Gelbke, and W. Lynch, Phys. Rev.C , 2865 (1991).[21] J. Adams et al. (STAR), Phys. Rev. C71 , 044906 (2005),nucl-ex/0411036.[22] M. M. Aggarwal et al. (STAR) (2010), 1004.0925.[23] D. A. Brown and P. Danielewicz, Phys. Lett. B , 252 (1997).[24] D. A. Brown and P. Danielewicz, Phys. Rev. C , 2474(1998).[25] D. Brown and P. Danielewicz, Phys. Rev. C , 014902(2001).[26] G. Verde, P. Danielewicz, W. G. Lynch, D. A. Brown,C. K. Gelbke, and M. B. Tsang, Phys. Rev. C , 034606(2003).[27] L. Phair, D. R. Bowman, C. K. Gelbke, W. G. Gong,Y. D. Kim, M. A. Lisa, W. G. Lynch, G. F. Peaslee,R. T. de Souza, M. B. Tsang, et al., Nucl. Phys. A ,489 (1992).[28] M. S. Wallace, M. A. Famiano, M. J. van Goethem, A. M.Rogers, W. G. Lynch, J. Clifford, F. Delaunay, J. Lee,S. Labostov, M. Mocko, et al., Nucl. Instr. and Meth. A , 302 (2007).[29] A. M. Rogers et al., in preparation (2011).[30] P. Chung, N. Ajitanand, J. Alexander, M. Anderson,D. Best, et al., Phys.Rev.Lett. , 162301 (2003), nucl-ex/0212028.[31] M. A. Lisa et al., Phys. Rev. C49 , 2788 (1994).[32] J. Pochodzalla et al., Phys. Rev.
C35 , 1695 (1987).[33] G. Verde et al., Phys. Lett.
B653 , 12 (2007), 0708.0081.[34] J. P. Bondorf, A. S. Botvina, A. S. Ilinov, I. N. Mishustin,and K. Sneppen, Phys. Rept. , 133 (1995).[35] D. H. E. Gross, Physics Reports ,119 (1997), ISSN 0370-1573, URL .[36] Y. D. Kim et al., Phys. Rev. Lett. , 14 (1991).[37] D. R. Bowman et al., Phys. Rev. Lett. , 3534 (1993).[38] B. Kampfer et al., Phys. Rev. C48 , 955 (1993).[39] D. Fox et al., Phys. Rev.
C47 , R421 (1993).[40] E. Bauge et al., Phys. Rev. Lett. , 3705 (1993).[41] E. Cornell et al., Phys. Rev. Lett. , 1475 (1995).[42] P. Danielewicz, Acta. Phys. Pol. B , 45 (2002).[43] P. Danielewicz, Nucl. Phys. A , 375 (2000).[44] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano,Z. Li, W. G. Lynch, and A. W. Steiner, Phys. Rev. Lett. , 122701 (2009).[45] P. Danielewicz, Nucl. Phys. A545