High Energy Gamma-Ray Production in Nuclear Reactions

High Energy Gamma-Ray Production in Nuclear Reactions H. NIFENECKER*,t and J. A. PINSTON* *Institutdes zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Sciences Nucltkires (INZP3, USTM G) 53, Avenue des M artyrs - F- 3086, Grenoble Cedex, France fand CEAIDRFISPhAN zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA INTRODUCTION High energy gamma-ray production in heavy ion collisions has been the object of a number of recent studies between 15 and 86 MeV/Nucleon incident energies (Grosse,1985; Grosse,1986; Hingmann,1987; Berthollet,l986, Stevenson,l986; Stevenson.1987; Alamanos,l986; Berthollet,l987; Kwato Njock,1986). Photons have attracted attention since they are not as seriously affected by absorption phenomena as pions, for example ; they can serve as unambiguous probes to study the reaction dynamics in the early stage of the collision. The main drawback of hard photon studies is the smallness of the -r-cross sections. Several models have been proposed to predict the photon production yields. Some of them suggest that incoherent nucleon-nucleon bremsstrahlung is the main source of the -r-emission. These collisions can take place either in the initial stage of the reaction (Cassing,1986; Nakayama.1986; 8auer,1986; Bauer,1987; Che min Ko,1987; Remington.1986; Randrup,l988) or within an equilibrated hot participant zone (Nifenecker,l985; Prakash,1987). Other models suppose that photons are produced by coherent bremsstrahlung where both nuclei or substantial parts of them act as a whole (Vasak,1985; Vasak,1986; Stah1,1987) In the following, after a short overview of the experimental techniques involved we first present the experimental results. Our main emphasis will be with the inclusive measurements of differential production cross-sections. These include the shape of the spectra, the angular distributions and the absolute cross-sections. We shall show, on qualitative grounds, that these inclusive experiments tend to favor a picture where the photons are produced in first nucleon-nucleon collisions. Results from exclusive reactions which display the impact parameter dependence of the multiplicity and spectra of the gamma-rays will be presented. After summarizing the experimental status, we give a first, semi-classical, approach of a theoretical account of the photon production in many nucleon systems. The main aim of this presentation is to show how, and in what limits. it mav be justified to schematized NucleusNucleus reactions into an incoherent sum of independent nucleon-nucleon contributions. It will clearly appear that the understandino of the Nucleus-Nucleus reactions reauires both knowledge and-understanding of the simple; nucleon-nucleon and nucleon-Nucleus cases. We shall, therefore, review the experimental and theoretical aspects of these simpler reactions insofar as they relate to our main subject. Doina so we are conscious that we mav omit important deveiopments, both experimental or theoretical, but it would have been out of scope of this review to give a full account of such an extended field. It will appear that the contribution of charged pion exchange currents to photon production is probably very important. Finally, we present an overview of the available theoretical approaches, and try to balance these different approaches with the experimental results, as well as with our knowledge of the more elementary processes. 271 272 H. Nifenecker and J. A. Pinston I. EXPERIMENTALTECHNIQUES. In this section we review the experimental techniques most commonlyused for the measurement of photons with energies between 20 and 200 MeV. The probability for producing such photons in nuclear encounters is approximately 10.3 per collision. These rare events have to be discriminated against a large neutrons and charged particles background. The requirement of both good energy resolution and good background rejection has lead to the use of multl-stage detecting systems. Such gamma-raytelescopes consist, schematically, of a converter element where photons are converted into pairs of electron and positron, followed by thin scintillators detecting the leptons, and, f i n a l l y , by a large volume total absorption detector. The classification adopted in the following is based on the nature of the final absorber which can be lead glass, plastic Cerenkov, or various inorganic scintillators. Lead Glasses. This material is expected, due to the Cerenkov mechanism, to have a very low sensitivity to protons with less t h a n 200 MeV. However, several authors (Edgington,1966; Lebrun,1979; Alamanos,1986) have found that lead glasses have a small but f i n i t e efficiency for detecting neutrons and protons below 200 MeV. Lebrun and colleagues(Ig79) found that i) the detection efficiency is similar for protons and neutrons of 65 MeV and i i ) that the probability that a 65 MeV nucleon produced a light signal above the detection threshold of 25 MeV is .3!H).7 10.4 These energy neutrons and protons are abundantly produced in medium Discrimination against those may be done by time of f l i g h t collisions. whenever possible. An exampleof time of f l i g h t spectrum is shown on Fig. l . I . I t is seen on this figure that the background is, indeed, important, even under the gammatime peak. Another limitation of lead glass counters is their poor resolution for photons under ~ucleus-Nucleus I00 MeV. Their energy resolution as measured in (Herrman,1986) FWHM 13.2% E ~ is larger than that obtained with BaF2 and Nal scintillators. Moreoverthe response function shows a pronounced high-energy t a i l (Edgington,Ig66;Herrmann,1986),as can be seen on Fig.l.2. This t a i l may cause significant errors, both on the slopes and yields of exponential like spectra (Herrmann,1986). Various set-ups, using lead glasses have been used. The simplest (Alamanos,1986) consists of S lead glasses detectors with a Cerenkov plastic veto paddle in front oF the central one. The veto is used to eliminate cosmic rays and fast electrons. A photon is characterized by a signal in the central counter and no signal above 42 MeV in the four surroundings blocks. I t is also required that the time of f l i g h t corresponds to the gammapeak. In other systems the lead glasses are associated to converters. In (Edgington,1966) and (Budiansky,1982) passive converters are used. The thickness of the converter results from a compromise between efficiency and the loss in energy resolution that i t produces. Thicknesses between1.2 and |.9 g/ cmz of lead have been used with conversion efflciencies between I I and 17%.The positive and negative electrons were detected in plastic scintillators, Cerenkov plastic detectors and lead glasses. Active converters give the possibility to add the energies deposited in the converter and in the absorber. This allows to use thick converters without loss of resolution. In the GSl detector (Michel,1986) a 4.35 cms. active lead glass converter corresponding to 1.8 radiation length gives a conversion efficiency of 60% for ISO MeV gamma-rays. The electrons escaping from the converter are detected in a multiwire proportional counter allowing the localization of the centroid of the electromagnetic shower. A schematic lay-out of the GSl detector is shown on Fig.1.3. This set-up has been extensively used to detect ~o and single photons. Note that the active converter, also, reduces considerably the background. Plastic Cerenkov d~tector@, The M.S.U. group has developed a high energy gammadetector (Stevenson,1986) which consists of an active BaF2converter, 0.625 cm thick, followed by a stack of eight plastic Cerenkov counters (Bicron-480 lucite with wavelength shifter additive). The gamma-rays are converted High Energy Gamma-Ray Production 2000 273 I. -\ I I 0 I iO ZO I 50 ~' [.11 Fig. 1.1. Time spectrum of a lead glass detector measured in coincidence with the beampulse (A1amanos,]986). ~t Io° IT . ,,;]l . i .'-lrr . , ."1m.'1 Fig. 1.2 Lead glass energy spectra for monoenergetic photons(Herrmann,lg86) lOe IT 0 . , I ct~ nat ,'/T/ Ig~ . , I00 . i,1 mm 274 H. Nifenecker and J. A. Pinston (b) F i g . l . 3 . The large solid angle detector of the GSI group (Hichel,lg86). The lower part shows a vertical cut CONYERTER VETI~ Fig. 1.4. Concept of the range telescope of Che M.S.U. group (Stevenson1986). High Energy Gamma-Ray Production 275 into an electron-positon pair in the BaF2. The pair is slowed down and stoppped in the Cerenkov stack without producing a shower, due to the low atomic number of the constituents of the plastic. Fig.1.4 is a schematic drawing of the system. The detector is used as a range telescope. Unlike the case of lead glasses counters the l i g h t output from the different elements are not summedup to determine the energy loss by the leptonic pair but is used to determinethe range of each memberof the pair. From this range, the original energy of each memberof the pair is determined and summed to give the energy of the original photon. Scintillator@. The energy resolution of inorganic scintilators is much better than that of lead glasses. At high gamma- ray energies the resolution is predominantly determined by the loss of electrons and, even more, photons, escaping the detector volume. The use of very large crystals allows to reach very good resolutions. For example, the detector used in (Kishimoto,1982) and (Shibata,lg84) is a large volume NaI(TI) crystal, severely collimated and operated in anticoincidence with plastic scintillators surrounding the crystal. A resolution of 2.6% is reported for 60 MeV gamma-rays. Such a good resolution is not necessary for the study of the continuous spectra produced in heavy-lon reactions, which allows the use of smaller and less expansive crystals. Resolutions of 6.5%/~=Ti~eV-~- and 8%/~E(GeV) have been obtained for a 20 cm. long Nal crystal and a 14 cm. long BaF2 crystal respectively. Such values are quite appropriate, especially in view of the fact that scintillators resolution functions display a low energy t a i l rather than a high energy one, as shown on Fig. I.5. Such t a i l s have small influence on the shape of exponential like spectra. Scintillators are very sensitive to charged particles and neutrons. Charged particles can be eliminated by anticoincidence shielding as well as pulse shape discrimination. I t is much more d i f f i c u l t to discriminate against neutrons, since neutrons are detected essentially by the photons they produce by nuclear interactions, lhls is usually achieved by time of f l i g h t techniques. BaF2 is probably the best s c i n t i l l a t o r for the study of high energy photons in intermediate energy physics. I t a l l i e s a good energy resolution to excellent timing properties (timing resolution bettter t h a n 400 psecs). I t also allows pulse shape discrimination against hadronic particles as shown in Fig.l.6. I t is possible to associate inorganic scintillators to active converters and plastic scintillators in a telescope arrangement such as that shown on Fig.].7 (Bertholet,1987). This system consists of a BaF2converter (6"4"I cm3) ,two plastic s c i n t i l l a t o r s used to identify the electrons and positons of the shower, and a large volume, 20 cms long and 15 cms in diameter, NaI(TI) total energy absorber. A veto plastic s c i n t i l l a t o r in front of the telescope allows to eliminate charged particles, lime of f l i g h t between the accelerator RF pulse and the BaF2 converter allows discrimination against neutrons. This system has very good background rejection capability and a moderately good energy resolution of 6.5% Jr[GeVT" Calibration Drocedure~, Whatever the experimental set-up, i t is necessary to determine its response to monoenergetic gamma-rays, over the range of interest. As an example, we consider the calibration procedure which was used in the case of the telescope described in the preceeding paragraph. The response function of the various elements of the telescope was determined from a measurement performed with tagged photons in the energy range 40 s ~ s ]50 MeV at the Saclay electron Linac(Veyssi~res,1983). Two different experiments werei)erformed. In the f i r s t one, the NaI(ll) detector was used alone (mode ]) and only the central part of the s c i n t i l l a t o r was illuminated with the photon beam. In the second one, the whole telescope was used and the size of the v-beamwas comparablewith the surface of the converter (mode 2). In this case the energies deposited respectively in NaI(ll) and BaF2detectors were added together in order to improve the energy resolution. Examplesof pulse height spectra for monoenergetic v-rays are shown in Fig.1.8 for the two different operatlng modes. The experimental llne shapes, in the two cases, are in good agreementwith the theoretical calculations using the Stanford EGS simulation code (Ford,]gTg).However the simulation cannot reproduce the shift observed in Fig.1.g between the energy calibration curves corresponding respectively to the H. Nifenecker and J. A. Pinston 276 i !~I i~E~,~~L........ v ! :iE- i/t ! . '5 c:) . , . , . , . . . , , . . . , . , ~ Fig. 1.5. Response functions of a BaF2 of 14cms length and lOcms to monoenergettc photons(Htngmann,1987) . E~=112 MeV i t U , 5 . 0 ; . ," 20 ~0 . 60 60 tO0 , 120 E i (IvteV) f.! - " 20, A v i (. I ' I i¼1ili~i..- ~. I I t (j') c- LL C> lO Fig. 1.6. Example of a)~-neutron and b)charged particles discrimination obtained with a single BaFI crystal. The discrimination between a and b is achieved by pulse shape analysis while the n-~ discrimination is obtained by time of flight(Hingmann,lg87). I-..- ' ""-.... l" I 1 1 20 ' 25 , 3~0 20. A v U) C: LL C) 10 I-- 5, 10 , 1~ E T (MeV) 35 High Energy Gamma-Ray Production 277 PILaf PLt Nil VETO Fig. 1.7. Concept of the gamma-ray telescope using a NaI(TI) absorber and a 8aF2 converter(Bertholet,]987). @ @ E (NO1) E (NCI1)'E ( e a F z ) ! Ftg. ].8. a)Pulse height of the Na](T1) s c i n t i l l a t o r alone, when the central part of the crystal was illuminated with monoenergettc photons. b) response of the telescope to monoenergettc photons.(Bertholet,1987) ill .,~J . . . . , . . . . ~o., i . . . . PULSE HEIGHT i . . . . 278 H. N i f e n e c k e r and J. A . Pinston 12001 11 O0 IOO0 9oo 2 8oo ~1 Too 5OO 500 ~,OO x I 40 1 50 1 60 1 I 70 Bo I 90 i 100 I = 110 E}f (MtV) Fig. 1.9. Energy calibration curves in mode I and mode 2(see text) FE.GS. 02 01 I , 50 . , , | A i A L I i iIOO 150 Ir~ (MeV) Fig. ].10. Comparison of efficiency calculated with the EGS code with the experimental data. High Energy Gamma-Ray Production 279 two different operating modes. This shift amounts to ~ 8 MeV and is nearly independent of the incident v-energy. Most probably, the main contribution to this effect, corresponds to the energy lost by the charged particles of the shower in the non-actlve part of the different scintillators (especially in the front part of the NaI(Tl) detector).This shift illustrates the necessity to control experimentally the v a l i d i t y of EGS type simulations for the specific arrangment of the detection system. The energy deposited in the detector by cosmic rays is often used as a secondary standard during the experiments.The measuredefficiencies are comparedto those obtained from the EGS code on Fig.I.IO. I t is clear that the experimental spectra have to be corrected for efficiency of detection. In principle i t would be desirable to unfold the experimental spectra with the resolution function. Howeverthe effect of the f i n i t e width of the resolution function is found to be small. This can be seen on F i g . l . l l , where i t is found that that a v-ray distribution with an exponential shape and a slope parameter of 14 MeV is not significantly modified by the response function of the above mentioned telescope. In most cases correction for detector resolution is not carried through. Larae Ar~& Mul~i~tector SYstems. The very good quality of the BaF2 scintillators(Laval,lg83) were first used in large arrays of detectors in the field of high-spin physic(Beck, lg85). It has become clear that they could lead to significant improvements, also, for the study of continuum gamma-rays. For example, the NBI-Unlversity of Milano collaboration uses half a dozen of large (~ - 15cms, h - 20cms) BaF2 scintillators for giant resonances studies. A very ambitious project, TAPS (for Two Arms Photon Spectrometer) is, now, under construction, by a collaboration between GSI, the Universities of Giessen and Gronlngen and GANIL. This multidetector system will incorporate more than 300 BaF2 hexagonal scintillators with ~ - 6cms and h - 2Scms. The small diameter of the detectors gives the possibility of good angular resolution. This is especially desirable for the detection of neutral mesons. A large fraction of the gamma ray energy may escape from the individual detectors, due to their small transverse dimension. Therefore the detectors will be grouped in clusters of Ig, and the pulse heights of neighbour detectors will be added, restoring a good energy resolution. Although the main aim of this system is to study neutral meson emission processes near threshold, it will also be used for giant resonances and bremsstrahlung studies. 280 H. Nifenecker and J. A. Pinston ' I 'I b I ' I I000 ' o x x x ~ ~ (. I - ' I ' t 5 H .V £.ICCHV) - 1 & .5 H..V 100 10 ~o X 0 X 0 X 0 X 0 X 0 X ¢' X 0 X I /.0 : I 60 , I OO , L_, 100 I i I i 120 lhO EII~V| Fig. l . I I Comparison of exponential spectra before(o) and after(x) folding by the telescope resolution function (Fig. 1.8) The unfolded spectrum has an inverse slope of IS MeV. The folded spectrum has a slope of 14.5 MeV. 2.HIGH-ENERGY PHOTONSPRODUCTIONIN NUCLEUS-NUCLEUSCOLLISIONS. Sp(¢tral @hap~. Above 20-30 MeV the gamma-ray spectra show a distinct exponential behaviour for all systems studied between 20 and 8SAMeVincident energies (Grosse,1985; Grosse,lg86; Kwato Njock,I986; Berthollet,1987; Kwato Njock,1988b; Stevenson,1986; Hingmann,1987). A typical spectrum is shown on Fig. 2.1. I t is also found that the slopes of the spectra depend on the angle of observation,larger angles corresponding to steeper slopes. Fig. 2.2 shows an example of such a behaviour. I t suggests emission by a moving source. Assuming a single source with velocity ~, the laboratory spectra are related to those observed in the source frame via: 1 Etab =Eem~(I'~c°sOe~) Eem -Etab ~r(l-lscose.~ ) (2.1) sin O~ =sin Ot~ d3a(Ot~, ,Et,b ) dEt, b dNtab i ~(1-~cos et~ ) d3a(O<~, E= ) ! dE~ d~, "~(1-1~cos Or,b ) High Energy Gamma-Ray Production I ' 1 ' I ' I ' I ' I ' I 281 ' I ' 1 ' , 1 , l , Ar+A1 (B5 I"h=V/n ) 10. O0 > Q) I.oo .£3 -I O. 10 Ill "13 0.0~ nO I 20 , 1 , /-.0 I 60 , 1 80 , ! , 1 , I "100 3.20 11,0 160 '180 E T [MeV] Fig. 2.1. Photon energy spectrum, at etmb - go', for the reaction ~Ar+Al at 85 AMev. Io IQ a°°ooo0tl 4 6LUa30' I 0.1 :~ 0o °°a @og Fig. 2.2 Photon energy spectra, at different lab. angles, for the reaction tL~=~* aaKr+wT Au at 44 AMeV o# °000 °°°tOltt b~o.! I0 °Ill I a ! 9WI, J'tS3.S" aO 0 0.1 . . . . 30 I &o . . . . I SO . . . . I 60 i i , i I 70 . . . . 1 ~ 8O 282 H. Nifenecker and J. A. Pinston Consider, for example, an exponential spectrum in the source frame: Es dSa(O,,E,) -E-'o" dE, dfl,, - f(stn B,) e (2.2) i t comes Et~-f(1-1)cos Ot.~ ) d'°(Otab 'Et,~ ) dEtab d~tab 1 fI' E° stn Otab ./ (2.3) (,~(l-13cos eL,b )J " -f(l-cos Or,b )e showing that , at each angle, corresponds an exponential spectrum with an inverse slope Eo Or,b ) ~(1-~cos On Fig. 2.3 we have plotted, for two cases, the variations of the slopes with cos St,b . I t is seen that these variations are almost l i n e a r , supporting the assumption of p-hoton emission by a single source. I t is seen that the source v e l o c i t i e s are close to h a l f the beam v e l o c i t y . As said in the f i r s t section, the spectra should be corrected for the f i n i t e resolution of the detection system. However, exponential spectra are not affected by Gaussian resolution functions for energies s i g n i f i c a n t l y larger than the resolution width. An example of this r e l a t i v e i n s e n s i t i v i t y is shown on Fig. 1.11. On the other hand, high energy t a i l s of the resolution function may have s i g n i f i c a n t consequences. An exponential t a i l provides a larger value of the inverse slope. Such t a i l i n g are most noticeable for lead-glass Cerenkov scintillators(Grosse) and might lead to s i g n i f i c a n t overestimates of the inverse slopes. Fig. 2.4 shows the systemattcs of the inverse slopes of the spectra as a function of the incident energy per nucleon, as measured at 90", in order to suppress the Doppler e f f e c t . I t is to be noticed that for, the same experimental group, the slopes appear to be weakly dependent of the target and p r o j e c t i l e masses. The scattering which appears on the figure is, therefore, p a r t l y , a consequence of d i f f e r e n t systematic errors for the d i f f e r e n t experimental and analysis techniques. However, a closer examination of the slope parameters observed by the same group, and at the same incident energy per nucleon, displays a systematic v a r i a t i o n of the slopes with the mass of the target. In general more massive targets lead to larger values of the inverse slope parameter Eo . Below 20 HeV, departures from the simple exponential behavtour are observed, which can be attributed to the presence of gama-rays produced in the s t a t i s t i c a l decay of hot nuclei produced in the reaction. The Eo systematic reported on Fig. 2.4 shows a smooth v a r i a t i o n with the beam energy, Ep and 1 ] a one finds Eo(HeV) ~ to ~Ep(ReV / n ) . Below 30 AHeV, the bremsstrahlung and s t a t i s t i c 1 components are d i f f i c u l t to separate experimentally and the Eo values reported in Fig. 2.4 are somewhat uncertain. This is especially true for symmetric or almost symmetric systems. I t is possible to specify the conditions under which the normal s t a t i s t i c a l component may compete successfully with the bremsstrahlung one. We consider a p r o j e c t i l e with mass Ap Ep incident on a target with mass At with incident energy per nucleon % = ~ - . We assume that a compound nucleus excitation of energy per temperature T = 8~('- y i e l d i n g %< 72 ApAr - - (Ap+AT)2 mass Ap+AT nucleon of ~pApAI Ap+ AT is formed. the fused This Neglecting system is will binding E* energies effects, the ApA; c ' - Ap+A'~" (p (Ap+Al)Z, y i e l d i n g a (p be the dominant contribution for T> - 3 . For symmetric systems, one obtains %< 18 MeV; for High Energy Gamma-RayProduction O. 100 I KR÷X & O. 0 8 0 4~ MEV/N ,5 %7 {3 ' I .... g , > 283 ) t A O. 0 6 0 3-0.040 0 LU "-' O. 0 2 0 0.000 l , , , 1 ~ ~ , -0.5 J I , I i J 0.5 0.0 I I i l COS ( 8 FIG 2.3 Variations of the slopes of the gamma spectra as a function of cos Ot~ for the reactions Kr+X at 44 AMev o Kr+Au, ~ Kr+Ag, V Kr+Au '''l''''l''''l''''l''''l' 'li'''l 30.0 ..... a 20.i I0. '' 1 i , " ~ • O. lllll,,,,J,,,,I,,,.|,,,.l, 0 10 20 30 40 50 ,I,,,,I,,, ,,, 60 70 80 BEAM ENERGY (MeV/~) FIG.2.4. Inverse slope En versus beam energy. The bars correspond to Eo variations for different p r o j e c t i l e - t a r g e t combinations. PPp--J 284 H. Nifeneckcr and J. A. Pinston AT-IOA , • < 5.9 MeV . These numbers should, estimates, ~ithin a factor of 2. of course, be considered as very rough Anqular Distributions. As an example the angular distribution analysis of the three Kr + (C,Ag,Au) reactions at 44 MeV/n are reported in Fig. 2.5. The data of the three systems studied are normalized to the Kr + Au reaction according to their relative ~-cross sections. The angular distribution is forward peaked (o(30')/a(153")-3.05) and the shape is almost identical for the three targets used. The data are then consistent with a v-emission from a recoiling source with a source velocity almost identical for the three reactions The source velocity can be extracted from a two dimensional plot of the invariant photon l d3~ 1 O+cosB~ cross section ~ - d E v ~ versus the rapidity y - ~ In U-cosO) and the transverse energy E± - Ez sinB . The data for the three targets are reported in Fig. 2.6 ; they are normalized accordlng to their relative ~-cross sections. The rapidity distribution is almost identical for the three targets and i t is nearly symmetrically distributed about a centroid with an average rapidity y - 0.16 ± 0.02 . Assuming ~ emission from a single moving source the source velocity is close to half the beamrapidity y - 0.153. Moreover we observe in Fig. 2.6 that the contour plots are close to those expected for an isotropic distribution. Least-squares analysis of the experimental spectra, at different laboratory angles, allow ,also, to extract values of the source velocities. Fig. 2.7 shows the results of such f i t s for a number of systems and show, once more, a clustering of the observed experimental values around the half beam velocity. However a slight tendancy seems to exist (Berthollet,1987; Tam,lgBs), which biases the source velocity towards that of the nucleus-nucleus center of mass. This may be the effect of secondary collisions where the proportion of participants originating from the heavier nuclear partner is increased, as compared to the case of the primary ones. Angular distributions may be computed in the half-beam velocity frame. Someof them are displayed on Fig. 2.8.1n this frame, the angular distribution is almost isotropic. However a small anisotropic component of El character is also evidenced. The relative amplitude, ~, of the El component increases with beamenergy and takes the values ~ = 0., 0.25 (Berthollet,1986) and 0.40 (Grosse,1986) respectively at 30, 44 and 60 MeV/n. At B4 MeV/n Grosse (Grosse,1986) has found a nearly isotroplc angular distribution with a possible minimumat 90". Such a behaviour is to be expected i f the origin of the radiation is attributed to incoherent p-n or n-p collisions.This can be seen simply within the frame of the classical theory of bremstrahlung (Jackson,1975). We consider a proton-neutron c o l l i s i o n in their center of mass frame. I t w i l l be shown in section 4 that, in the soft photon l i m i t , and for ~2sin26 isotropic scattering, the radiated intensity is proportional to a + b . In this (I-~ cose) 4 expression the isotropic part b comes from the post collision contributions, and, also, from any fluctuations in the initial direction of the incident nucleons. Such fluctuations may be due to the Fermi motion of the nucleons within the projectile or the target. Consider the sum of the contributions of two p-n collisions, one with the proton belonging to the target, the other with the proton belonging to the projectile. In this case the non-isotropic term writes: ~Zsin2B I I + l _ ~ , which gives, to the f i r s t lowest order in (I-~ cose)4 (I+~ cose)4) 2~ZsinZB(l + lO BZcosZe). This term displays a quadrupole contribution which grows in importance with energy. Note that the incoherent summation of the contributions of the two nucleon-nucleon collisions is only j u s t i f i e d " i f they are s u f f i c i e n t l y separated in space time. This point w i l l be discussed later. However one should be aware that, as such, the characteristics of the angular distributions do not rule out a collective origin of the radiation, where each nucleus would radiate as an entity. In this case, also, the natural frame of reference, for not too high energy gamma rays, has half the beam velocity, at least for systems having the same charge densities. Since this might appear surprising when deallng with projectile and target having very different masses, we give a short derivation of this fact, here again in the classical, soft photons, approximation. We consider a projectile with mass AI , charge ZI , and i n i t i a l velocity v . The target has mass A , charge Z , and i n i t i a l velocity -v . The veloclty High Energy Gamma-Ray Production ++ 285 Kr.C 300 o Kr + +1~ +00 , I 20 + I AO , l 60 , I 80 . l , 100 I , 120 I , 1&0 I , 160 l~ l~- eLOb Fig. 2.5. Plot of the laboratory angular distribution of high energy photons for the 86Kr + 197 Au reaction at 44 MeV/n.The angular d i s t r i b u t i o n for Kr + C and Kr + Ag reactions are also reported on the same plot ; they are normalized to thea6Kr + +9rAu reaction according to their total ~-cross sections. I . . . . 80 - I . . . . . I . . . . I A.. ':' ~. ' ' I ' ' ' ' I ' ' ' , , J ' 0.00~ I 60 ~0 20 I n i -1.5 , , I , -1.0 ~ t J I t -0.5 J J J I ,!, ~ J I ~ i 0.01 0.5 Y.. RAPIDITY i , I 1.0 ~ Y Fig. 2.6. Contours of constant Invarlant photon cross-sectlon~, in ~/HeVa-Sr, versus the rapidity y and the photonenergy E~, for the a6Kr + 19r Au reaction. The data for a6Kr + C and 86Kr + Ag reactions are reported on the same drawing ; they are normalized to the a6Kr + 197Au according to their relative Y-cross sections. 286 H. Nifenecker and J. A. Pinston changes are &v1 for the projectile and, for the target, &vz A1 m -- ~ JAkVl . The energy radiated is then proporttonnal to : [ Zi~vie '~ Zz~vze"'~ / z g " sinZe 1;(1-~--~°-~=) ~ z + (1-~ z cose)ZJ where R and -R are the centers of the projectile aqd t~rget £1 &v1 and &vz , and assuming that A1 L2 Az ZZ ' A , Az 'z / ~'k._~* W= stnZe AT ~ &vz ~(l-~lc°se )z respectively. Z ~we obtain: e"~ / z. (]_~zCOse)Zj Using the ,here average acceleration relationship between we have put Z = ZI+ Z2, A - AI+ Az, &v - &vl+ &v2. I t is clear that the angular distribution of W is independent of the r a t i o of the projectile and target masses. Let alone the t r i v i a l case Bl" ~2, i t is symetric with respect to 90" i f B~ - -Be. Note, also, that in this frame , in the lowest ~ order, i t is quadrupolar. This is true since we have assumed that the acceleration remained Rarallel to the velocity at all times, and since we have assumed central collisions with R II ~. Under different assumptions the angular d i s t r i b u t i o n may be more complex(Herrmann,]986), but the v a l i d i t y of the half beam velocity frame remains. The dependence of the angular distributions of the gamma-rays as function of the mass of the projectile and target has been systematically studied by the M.S.U. group (Tam,]g88). In particular, these authors have studied symmetric systems with dif f er ent total masses. Fig. 2.9 shows the angular distributions observed in the nucleon-nucleon center of mass frame. The energy of the beamswas 30 AHeV. For a ll cases a dipole component is apparent. Its intensity, r e l a t i v e to the isotroptc component ranges between 0.29 and 0.49, in agreement with the values found in (Berthollet,]987) and (Grosse,1986). I t seems to disagree with the result reported in (Kwato Njock,1986) for the system Ar+Au at 30 AHeV. However the dipolar character of the radiation observed in the Ar+Pb reaction (Tam,1988) is essentially caused by the smallest angle measurement. Excluding this point would lead to a very small intensity of the dipole component. In general, i t seems that the intensity of the dipole component is a decreasing function of the total mass of the system. As pointed out by the authors(Tam,1988), this might be a consequence of an increased influence of secondary collisions in the more massive systems. Svstematics of the Photon Production Cross-sections. I t would be tedious to review a l l the values of photon production cross-sections which have been reported solar. Rather, we shall try to find i f there exist trends in these cross-sections allowing some kind of phenomenological description. We have already stressed two important points, in this respect: the almost exponential shape of the spectra, above about 30 HeY gamma-energy. This allows a description of the production cross- section with only two constants, for example, the slope of the exponential and the value of the integrated cross-section above some specific energy. The second characteristic of importance is the relevance of the nucleon-nucleon center of mass for the angular distributions. The knowledge of the cross-section at 90" lab. allows the estimation of the angle integrated cross-section within an error of around 201~. Table 2.1 gives a compilation of a number of systems which have been studied. Part of this table is extracted from (Prakash,1987). We give both the total and d i f f e r e n t i a l cross-sections for producing gamma-rays above the minimum energy E. In general the total cross-section was obtained by multiplying the d i f f e r e n t i a l one ~ 90°'by 49. A commonly used presentation of the cross-section systematics is used in Fig. 2.10. The cross-sections for production of photons above some specific energy are divided by a scaltng factor which accounts for the p r o j e c t i l e and target masses dependences of the cross-sections. These normalized quantities are plotted as a function of the beam energy per nucleon above the Coulombbarrier. Such a representation would be perfectly legitimate i f a l l spectra measured at the same beam energy per nucleon had the same slope. This would insure that the scaling factor, once determined, would be valid, independently of the lower energy l i m i t . However, slope parameters show significant variations for different systems which are, in part, genuine, and, in part, due to the High Energy Gamma-Ray Production 0.30 .... I .... I .... 287 l''''l'J I .... Jl 0.28 0 0 0.20 Y+ 0.15 U 0 or) 0 -ca 0 i O.lO 0.05 0.00 0 0.1 0.3 0.2 0.4 0.5 be~rn velocity ~bearn Fig. 2.7. Variations of the source velocity with the incident energy of the beam. i 05 0 6 r"- ~ ,03 20 - '}0 Mev o~ 02 ~ ug o °, 04 E b~ 06 o~ 04 02 1: O8 06 04 02 0 50- 60MeV , 30 I . 510 . 70 . . . 90 . 1 1 1 0 '130 1 ' 1 ' 7 150 1 0 -ectn Fig. 2.8. Angular distributions as observed in the half velocity frame. (Berthollet,1987; Grosse,1986) 288 H . N i f e n e c k e r a n d J. A . P i n s t o n '' I ' ' '" I E/A=30 ' ' ' MeV x 4°Ar+Ca 4°Ar+Pb • VLi+Pb ~l~le+Mg • I0-I "E" x • vLi+Li ~ 10 - 2 b ,,l 10-3 .... ,,I l,,, 60 .... l,,, 120 80 120 8~=(deg) Fig. tO-I .... I ........ l ........ p h o t o n s f r o m in m e d i u m p-n collisions 0 '4 Angular distributions as observed in the half velocity frame.(lam,1988) 2.9. to-Z (E 7 > 30 ~eV) / / ./" o 10 - 3 / I ./ .- : /° I : ~m = 10_4 x >, X tO-5 ! as/ / (3 i /' ./ otons f r o m fre~ 0 / I0 - 6 / p-n [ ~". . . . I 5 collisions : (E~, > 30 MeV) ....... I0 ( ( E ,I ~0 -Vc I ........ 100 500 I000 ) / A )lab Fig. 2.10. Normallzed cross-sections for production of v-rays with energies above 50 MeV, as a function of the beam energy per nucleon, above the Coulomb barrier. From {Metag,1988) High Energy Gamma-Ray Production 289 different resolution functions of the d i ff e r e n t experimental set-ups. In the following, we present a semt-phenomenological method which is not subjected to this drawback. We, then, shall discuss some other proposed scaling laws. The main characteristics of the photon emission deduced from the experimental data : shapes of the angular distributions and of the spectra which are almost independent of the p r o j e c t i l e - t a r g e t combinations at a fixed bombarding energy. - source velocity close to the nucleon-nucleon c.m. velocity. shapes of the c . m . angular distributions close to that of a neutron-proton reaction. suggest that the f i r s t nucleon-nucleon c o l l i s i ons , which take place In the early stage of the reaction, are the main source of the htgh energy photons. In this hypothesis photons are produced when charged protons are accelerated or decelerated in the nucleon-nucleon Interaction.Only neutron-proton (n-p) collisions are e f f i c i e n t to produce photons and proton-proton collisions can be neglected.(N1fenecker,1985; Koehler, 1966; Edgington,1967) In this f i r s t c o l l i s i o n hypothesis the total v-cross section in nucleus-nucleus collisions follows the simple relation : ox - oR(Nr~ ) PT (2.4) where <N_> is the average number of f i r s t n-p collisions.An estimate of this number can be made from~the equal participant model described in (Nifenecker,1985). (A e ) (Nnp) - ~ (ZpNT + ZTNp) (2.5) with Ae,Zp,Np,AT,ZT,Ni the mass charge and neutron numbers of the p r o j e c t i l e and target respectively. Here, i t is assumed that Ap<AT.(Ae) ts the average mass of the participant zone. I t can be obtained by weighted integration of the expressions given in (Ntfenecker,1985) and equals (Ntfenecker,]g88a) (A,) -Ap 5 A~/]- A~/3 + )2 (2.6) Further o, - ~ x ].z2 (A~,3 +R,3 )2 (2.7) is the total reaction cross section. Below 30 MeV photon energy a s t a t i s t i c a l component is present which cannot be subtracted experimentally. However we have found i t possible to compare the d i f f e r e n t i a l cross sections, for the same value of the r a t i o (E~/tEo), where EotS the inverse slope parameter. For this purpose, the quantity : dzG(B-90") 1 (2.8) Ed~_o~d n (Nnp) G, is plotted versus (E~AEo) in Fig. 2.1], for a number of examples. The product (N,~)Ge is the scaling factor defihed in equation 2.4. Only small r e l a t i v e variations with beam energies are observed in Fig. 2.]] for d i ff e r e n t nucleus-nucleus reactions, measured at beam energies between 30 and 85 AMeV . The quantity defined in equation 2.8 is the d i f f e r e n t i a l probability to produce a photon in a stngle n-p c o l l i s i o n , in the nuclear medium, at a fixed beam energy. The equation of the universal curve obtained in Fig. 2.11 writes : dz P~ (e.g0") ET P, e - Eo (2.9) d ~'~o~ Assuming that the bremsstrahlung v-rays have an exponential shape down to zero v-energy, one 290 H. Nifenecker and J. A. Pinston can compute the collision : total d i f f e r e n t i a l P~ " probability PT, PT e Eo d for photon emission in = 10"4/sterad. a single n-p (2.10) This quantity is nearly constant in nucleus-nucleus c o l l i s i o n s at beam energies between 20 and 85 MeV/n. This can be seen on Table 2.1 which shows the values of P_ obtained from a 4 large number of experimental cross-sections. The values of the number of proton-neutron c o l l i s i o n s are also reported on the t a b l e , so that i t is possible to reconstruct the experimental cross-sections with i t s help. Systematic v a r i a t i o n s of P~ with the size of the system or the incident beam energy are d i f f i c u l t to e x t r a c t from - t h e t a b l e , because i t countatns data obtained by d i f f e r e n t groups using d i f f e r e n t techniques. To study such v a r i a t i o n s i t may be more convenient to look f o r them in coherent sets of data obtained by the same group. Fig. 2.12 shows the variations of Pr with <N_> for a number of systems measured at energies at 85 AMeV, as well as for reactions induce~by a Kr beamwith 44 AMeV. Pz seems to be only weakly dependent upon <Nr~, An increase of Pr as a function of the incident beamenergy is observed, above 80 AMeV. However the data were only grossly, i f at a l l , corrected for % contamination, and the increasing trend may be related to an increasing importance of this contamination. Furthermore, the experimental results were obtained by different groups, and, therefore some systematic differences between them might be possible. In any case i t is striking that formulae 2.9 and 2.10 allow a prediction of the photon production cross-sections within 50% over a very broad range of p r o j e c t i l e , target and beamenergy combinations. In table 2.2 , we compare some experimental data with the prediction of the f i r s t collision model and two other phenomenological laws (ApAT)z~ and Ai AI~ + Az A~/3 as a function of target mass. In order to do this comparison we normalize all calculations to the Izc or rLi experimental value and show the ratio of the so normalized values to the experimental value as a function of the target mass. From the table one can see that the f i r s t c o l l i s i o n model seems to do best for the 84-85 AMeV reactions. The situation is less clear for the 30 AMeV case where the (ApAT)z/3 scaling does as well, or as bad, as the f i r s t c o l l i s i o n one. The lack of a good scaling for symmetric systems is problematic. Note, however, that the energy at which the comparison was made (Tam,]g88) is rather low, and that s t a t i s t i c a l gamma-rays may be d i f f i c u l t to separate from the Bremsstrahlung component. Indeed, for symmetric systems, the temperature of a nucleus resulting from total or partial fusion would be close to 8 MeV, that is, very close to the "Bremsstrahlung slope". High Energy Gamma-Ray Production 291 Ir,,,l,,,,li,,,l,,,,l,,,,l,,,,l,rd,l,,, z- • + C • C I0 0 gr t40 fl~,/v 4 C 4 4 Dq~V/v .~ . 10.7 P~C • ~ w~ • I~V/u • e j J Y C • c IN ~ ¥ ~ lO-e ! 3 2 ~ 5 G 8 7 ErAE, d~(e-90") Fig. 2.11. Plot of the invariant quantity E as a function of x - ; - - f o r to ] , , . I , , ' I " ' ' | ' ' " dx 1 aa<Nm> d i f f e r e n t systems. I' "" I ' ' '1 ' 10.~ e. 0( % 6. OO t 1 M " L~ 0. K <N n-p> Fig. 2.12. Variations of P_ as a function of Nr~ for reactions observed between 84~VIeV and 86AMeV(n) and for the reactions Kr+X(~) at 44 AMeV. ppp--J° 292 H. Nifenecker and J. A. Pinston TABLE 2.1(a) Values of the inverse slope Eo and of the probability Pr of emitting a photon per proton neutron collision for a number of systems. (see text) SYSTEM E beam Eo Emin a(ET>E®In :90") G(ET>E=in ) Nnp Pr MeV/N ~barns/sterad ~barns x104 REFERENCES 160 +n,t W 15 5.2 19F +2r Al 19 6.8 2.24 1.7 Gossett,1988 19 F +6o Ni 19 6.5 3.03 1.6 Gossett,1988 19 F +1oo Mo 19 6.5 3.56 1.5 Gossett,19~ 4.16 1.0 Gossett,1988 19 F +181 Ta 19 6.8 3z S +zr A1 22 10.8 3z S +nat Ni 22 3z S +19r Au 22 25 2.5 50 2.78 10.0 50 9.1 50 31. 3.69 0.27 Breitbach,1986 35. 2.93 0.56 Stachel,1987 4.77 60. 4.23 0.73 Stachel,lg87 7.24 91. 6.3 0.76 Stachel,1987 TABLE 2.1(b) Values of the inverse slope Eo and of the probability PT of emitting a photon per proton neutron collision for a number of systems. (see text) SYSTEM E beam Eo E,in o(E~>E,in :90") MeV/N ~barns/sterad ~(E,>E.,. ) Nr~ PT ~barns xlO4 REFERENCES ZLi +Li 7 30 g.o 30 3.8 47.75 0.68 2.34 Tam,19~ ZLi +2oaPb 30 7.9 30 14.8 185.9 1.89 1.25 Tam,1988 2°Ne +24Mg 30 8.B 30 10.7 134.46 2.18 1.04 Tam,19~ 40Ar +40Ca 30 8.3 30 36.3 456.16 4.00 1.59 Tam,1988 4OAr +2oaPbl 30 7.4 30 74.6 937.45 7.34 1.48 Tam,1988 ~Ar +19rAu 30 7.5 50 3.97 SO. 7.23 1.11 Kwato Njock, lg86 14N +n,t Ni 35 13.5 50 3.97 SO. 2.47 0.36 Alamanos,1986 4OAr +15~ Gd 44 12.6 31.83 400. 6.73 0.70 Hingmann~1987 50 High Energy Gamma-Ray Production 293 TABLE 2 . I ( c ) Values of the inverse slope Eo and of the p r o b a b i l i t y Px of emitting a photon per proton neutron c o l l i s i o n for a number of systems. (see text) SYSTEM E beam E0 E,In G(Er>EmIn :90") MeV/N ~barns/sterad ~(E~>E,,. ) Nnp pbarns PT REFERENCES xI(P 44 11.7 50 4.37 55. 2.46 0.621 Bertholet,1987 12.5 50 35.80 450. 9.47 0.54 • s Kr +i~ Au 44 44 ]2.] 50 35.01 440. I I . g 0.381 Bertholet,1987 12C +Iz C 48 16.5 50 3.02 38. ].2 0.55 Grosse,1986 12 C +lz C 60 2].5 50 4.77 60. 1.2 0.42 Grosse,1986 12 C 4"12 C 74 23.0 50 7.95 100. 1.2 0.61 Grosse,1986 12 C 4-12 C 84 27.2 50 ]2.73 ]60. 1.2 0.70 Grosse,1986 85 25.6 64 21. 263. ].95 0.99 Kwato Njock,1988c Kr +Iz C • s Kr +nat Ag Ar +12 C Bertholet,1987 TABLE 2.1(dl Values of the inverse slope Eo and of the probability PT of emitting a photon per proton neutron collision for a number of systems. (see text) SYSTEM Ar +27A1 E beam MeV/N Eo 85 28.4 E,i n G(Er>E,t n :90") a(F--r>E.,n pbarns/sterad ~arns 64 6]. 766. )i Nm Pr REFERENCES x]O4 3.25 ].04 Kwato Njock,]988c Ar +~t Cu 85 29.9 64 114. 1432. 4.9 Ar +natAg 85 29.6 64 177. 2224. 6.0] 0.93 Kwato Njock,]988c Ar +n,t Tb 85 29.8 64 224. 2814. 7.0 0.86 Kwato Njock,]988c Ar +197Au 85 28.3 64 254. 3191. 7.2 0.94 Kwato Njock,]988c l~S Xe + N,t Sn 89 26.83 0 6735. 84634. 12.32 1.19 Stevenson,]988 l~s Xe + .at Sn 124 35.23 0 22000. 276460. 12.32 3.89 Stevenson,]988 0.88 Kwato NJock,]988c 294 H. Nifenecker and J. A. Pinston TABLE 2.2(a) (Nifenecker,1988) Comparison of three different scaling laws: ScI: f i r s t collision, equal participants scaling(see text) Sc2: A~/3A~/3 scaling SC3: ApA~13 +ATA~I3 scaling The beamenergy was 85 AMeV.Results are normalized on the reaction Ar+Al. For completeness the data of the C+C reaction(GR086) at 84 AMeVare given. SYSTEM a(ET>64 MeV Eo pbarns/sterad. Z = a x e ~/E 0 Z(X) ScI(AI) Z(X) Sc2(Al) Z(X) Sc3(A1) Z(AI) Sc1(X) Z(A1) Sc2(X) Z(AI) Sc3(X) Ar+IzC 25.6 22. 268. 0.97 0.79 0.9 ~Ar+ ~TAl 28.4 61. 580. 1. 1. 1. 36Ar+Cu 29.9 114. 969. 0.82 0.94 0.81 Ar+Ag 29.6 177. 1538. 0.87 1.03 0.81 Ar+Is9 Tb 29.8 224. 1918. 0.84 ] .02 0.72 ~Ar+ 197Au 28.3 254. 2437. 0.91 1.12 0.77 IsC+Izc 27.2 7.61 80.3 0.67 0.5 0.69 TABLE 2.2(b).(Stevenson,19B8) Same as above The beamenergy was 30 AMeV The results are normalized to the Ar+Ca reaction. SYSTEM a(E~>30 MeV Eo ~barns/sterad. 7Li+ZLi 9.0 7 Li+Pb 30/E Z - a x e olZ(x) sc](c)Jz(x) Sc2(C) z(x) sc3(c) Z(C) ScI(X) z(c) sc2(X) z(c) Sc3(X) 3.8 106. 1.47 0.82 1.44 7.9 14.8 659. 0.78 0.51 0.44 2o Ne+Z4Mg 8.8 10.7 323. 0.65 0.52 0.63 4o Ar+4O Ca 8.3 36.3 1347. 1. 1. 1. 4OAr+Pb 7.4 74.6 4299. 0.92 0.99 0.78 Impact Parameter Filtered Hiqh-Enerqv Gamma-RaysProduction. Only a few experimental r e s u l t s where the high-energy gamma-rays were observed in coincidence with s p e c i f i c reaction channels have been reported so far(Htngmann,1987; Gaardhoje,19B7; Kwato Njock, I g ~ b ) . In general, the gamma m u l t i p l i c i t y follows the violence of the reaction, while the inverse slopes of the spectra are somewhat smaller f o r peripheral c o l l i s i o n s than f o r central ones. The impact parameter selection has been done in a v a r i e t y of means. In (Hingmann,1987) two d i f f e r e n t types of detectors were positioned downstream from the t a r g e t . One d e t e c t o r was sensitive to slow, heavy fragments and i t s f i r i n g was considered as a signal f o r rather central c o l l i s i o n s . The other detector was sensitive to p a r t i c l e s having v e l o c i t i e s in the beam v e l o c t t y domain. I t s f i r i n g was considered to be a signal f o r r a t h e r peripheral c o l l i s i o n s . When no detector, except the gamma ones f i r e d , very High Energy Gamma-Ray Production 295 peripheral event was assumed. Fig. 2.13 shows the gamma-rayspectra observed for the three different c e n t r a l i t i e s of the reactions. In (Gaardhoje,]g87), fragmentswere detected at forward angles with a PPAC gas detector. Two main types of fragments were identified corresponding to partial fusion residues and p r o j e c t i l e l l k e fragments respectively. In the f i r s t case central reactions were infered while , in the second case, rather peripheral ones. The beams were Ar at 44 AMeV for (Hingmann,lg87) and Ar at 24 AMeV for (Gaardhoje,lg87). BUU (Cassing,lg86)and transport(Randrup,]g88) calculations have been able to reproduce both the magnitude of the partial cross-sectlons and the change of slope from peripheral to central reactions. Results of such calculations are displayed on Fig. 2.]3. In general i t can be seen that the inverse slope of the spectra tend to be larger for the more central collisions, both for the experiments and for the calculations. The experiments j u s t refered to (Hingmann,]g87; Gaardhoje,1987) are of a rather qualitative nature. The experiment described in (Kwato Njock,lg88b) is more quantitative and we shall spend sometime describing i t in the followlng.ln this experiment a set of 24 plastic s c i n t i l l a t o r s was used to detect charged fragments. The detectors were assembled on two concentric rings covering angles between 5" and 13". This m u l t i p l i c i t y dectector was used to study the ~Ar+ 27Al systemat 85 AMeV. Due to the inverse kinematic of this reaction, the m u l t i p l i c i t y array had a rather large efficiency, especially for complexparticles. The charge i d e n t i f i c a t i o n of the particles was achieved by a combined measurementof their time of f l i g h t and energy loss in the thin plastic s c i n t i l l a t o r s . Several telescopes, situated at different angles were used to detect the high energy gammarays. Both coincidences events and single events in the charged particles and gamma-raydetctors were recorded. Fig. 2.14 showsm u l t i p l i c i t y histograms obtained without and with coincidence requirement with the gamma detectors.The fragment m u l t i p l i c i t y detector allowed distinction between fragments with charges one, two and heavier ones only. Due to i t s f i n i t e efficiency i t was far from being exhaustive for charged particle detection. Therefore, a rather broad original m u l t i p l i c i t y d i s t r i b u t i o n corresponded to each measured m u l t i p l i c i t y . Furthermore, the tota] number of recorded coincidences (around ]06) did not give significant s t a t i s t i c s for the higher m u l t i p l i c i t y bins. Due to these circumstances coincident events were classified in a few categories corresponding to I - No counts 2 - I or 2 counts 3 - 3 or 4 counts 4 - 5 or 6 counts in the m u l t i p l i c i t y array. Cases where a l l detected particles in the array had a charge larger than one were also considered, expecting such events to correspond to multifragmentation and therefore to the more violent collisions. Two such cases were considered. 5 - ! or 2 counts 6 - 3 or 4 counts in the array. Therefore events were classified in supposedly increasing violent character of the reactions. The gamma spectra had the usual exponential character for a l l cases In line with the analysis presented in the inclusive case the gamma m u l t i p l i c i t y for each particle m u l t i p l i c i t y bin could be written. Mr(V, E~) GT(u) o(v) PT Eo N r~ (v)e - o E (2.11) The slope Eo was obtained from the ~ spectra corresponding to the different bins. Py was taken from the inclusive measurement to be approximately 10.4 . The m u l t i p l i c i t y M~(u, E~) was d i r e c t l y obtained for m u l t i p l i c i t y bins from 2 to 6, where at least one partlcle was detected in the m u l t i p l i c i t y array since the "singles" values of G(v) were obtained in the standard way from the number of counts per incident particle and from the target thickness. The value of ~_(v) was obtained in the same manner, taking into account the gammaray .8 detector efficlency. The cross sections a(u) so obtained are given in Table 2.3. For the case v - 1 of 0 count in the p a r t i c l e m u l t i p l i c i t y array the gammacoincidence cross section is obtained as said above. However the "singles" cross-section is obtained by the difference 296 H. Nifenecker and J. A. Pinston 1 44 ¢ i MeV/A '°Ar * ~Gd 4= cencrsl ~P~ = (14 ± 2) MeV i i i _.-- i0o perlpheral ~E Brazing if " 0 i • 1 20 40 E7 (MeV) , = 60 Fig. 2.13. Measured energy spectra for central,peripheral (Inelastlc),and grazing collisions from 40Ar+ISaGd at 44 /Q4eV, in comparison to calculatJons(Randrup,19~). Taken from (Metag,lg~) and (Randrup,3988). High Energy Gamma-Ray Production 297 4 o(~-1) - a,- Z (2.12) o(~) and depends upon the assumed value o f c . We have chosen ~e-w 1.2Z(A~/3 +Al/3) .z We also present the results obtained with t~e reaction cross section a. estimated from Kox's(]987)formalism.From Equation 2.1] one can derive an averagenumber'of n-p co111sions for each bin N~(v). These numbers are shown in table 2.3. I t is found that, indeed, the number of n-p coTlisions increases when the reaction becomesmore violent. Also shown in the table are the values of the reduced impact parameter x corresponding to the values of Nn,(U). The reduced impact parameter is given by the ratio of the impact parameter to the s~m of the nuclear radii. Also displayed in the table is the overlap distance R.+ R,- b corresponding to the reduced impact parameter x and the probability for this quantity ~o be larger than the posted quantlty which is simply equal to xz. From this i t can be seen that the probability for a collislon to be mere violent than those related to bin 6 is less than 10%. Fig.2.15. shows the variation of the inverse slope with the number of proton-neutron collisions. Fig. 2.16. shows the variation of the samequantity with the overlap distance. On the same figure we have added values obtained from the inclusive measurementswith the other targets. These values are also shown in table 2.3. Fig. 2.16. shows a clear correlation of the inverse slope with the overlap distance.lt is, of course, tempting to explain the trend observed in Fig. 2.16. by the spatial dependanceof the Fermi momentum distribution of the nucleons. The decrease of the nuclear density close to the nuclear surface would imply a softer momentumdistribution, and therefore a softer photon spectrum. Other interpretations are also possible. For example thermal models would predict larger temperatures for more central collisions, also corresponding to higher gamma-ray m u l t i p l i c i t i e s . At this point, i t does not seem, therefore, that exclusive experiments such as those just discussed can unambiguouslyhelp in choslng between competing models. An i l l u s t r a t i o n of a clear thermal origin of high-energy gamma-raysis provided by the experiment of Herrrmann and colleagues(Herrmann,lg87). These authors studied gamma production in the reaction 9ZMo+gZMo at 19.5 AMeV. The gamma-rays were detected in coincidence with the two heavy nuclei resulting from the deep inelastic scattering of the projectile and target nuclei. The gamma-raym u l t i p l i c i t y as well as the inverse slope of the spectra were increasing with the total kinetic energy loss of both nuclei. These observation were in agreementwith a statistical origin of the gamma-rays. In view of what we have said in section 2.1, this is not surprising. The expected inverse slope of the Bremstrahlung component for 20 AMeV incident energy is around 5 MeV. The range of temperature which could be reached in the deep inelastic process was between 0 and 6.5 MeV, corresponding to total kinetic energy losses between 0 and 900 MeV. A temperature of 5 MeV would correspond to a total excitation energy , and therefore kinetic energy loss of around 575 MeV, i.e. well inside the possible values. In the case of a thermal statistical emission of the gamma-rays, i t is possible to derive a relationship between the gammaintensities and the temperature. Consider a nucleus at excitation energy E" and temperature T. For simplification we assume that only gamma-rays and neutron can be emitted. At each evaporation step the probability E~- Bn FT ~ T~e 1 for gammaemission is approximately given by Fnn T . The total gammaintensity is then obtained by multiplying the emission probability at each step by the number of steps E" Tz u ~ < - - B n +2T> ~ ~ above T some definite Bn <Bn+2T~e T . A f t e r i n t e g r a t i o n over Er we obtain t h a t the t o t a l gamma i n t e n s i t y multiple of the temperature is approximately proportional to This expression is only valid well above the giant dipole resonance since any structure in the gamma strength function was ignored in its derivation. Note that for temperatures significantly higher than the neutron binding energy the fractional gamma intensity becomes almost independent of the temperature. This should be the case for the exclusive experiment at 85 AMeV described above, in contrast with the observed trend, which displays a fast variation of the gamma intensity with the slope of the spectra. 298 H. Nifenecker and J. A. Pinston 'l'''l'''l'''l'''l'''l' z Q 100.00 r 10. O0 F 1. O0 r 0.10 ! Ar~'AI 85 MtV/A 0 0 0 o 0 o o 0.01 r E~cluo~,t ,I,,,(,,,I,,,I,,,l,,,I, 2 ~ 6 8 I0 PARTICLE MULTIPLICITY 0 Fig. 2.14 a)Charged particles multiplicities without gamma coincidence. b)with gamma colncldence(Kiato Njock, lg88b,c) 'I'''I'''I'''I'''I'''I' 100. O0 r ~- Q ~ 1o. oo r z I. O0 Q 0.10 0.01 0 At+A1 85 MeVIA l 0 1 _= 0 0 ? o -! o Inelu¢~ , , , l,,,I,,,l,,,l,,,l,,,l~ 0 2 ~ 6 8 PART] CLE MULTIPLICITY 0 = 1 10 High Energy Gamma-Ray Production 1 2 Bin Multiplicity Number 1 0 2 3 4 3 4 5 o(mbarns) Singles E0 Nnp 6 x 299 7 Ri+~-b 8 P(~x)-x.x 550(*) 850(+) 22.5 1.1(*) 0.7(+) 0.77 0.82 1.7(*) 1.4(+) 0.6(*) 0.74(+) 1+2 All Z 908 26.S 3.1 0.61 2.9 0.37 3+4 All Z 292 30.0 5.1 0.48 3.9 0.23 5+6 46 30.4 6.2 0.42 4.4 0.17 All Z 5 I+2 Z>2 596 29.1 5.4 0.46 4.0 0.2] 6 3+4 Z>2 52 32.3 8.5 0.3] 5.2 0.09 13.5 0.0 7.2 O. Central collisions TABLE 2.3 Dependence on the charged particle m u l t i p l i c i t y of different quantities. Col.3 gives the value of the cross-section for each of the m u l t i p l i c i t y bins, measured without requirement of a gamma coincidence. Two estimates were made for the cross section associated with no charged particle detected. The f i r s t (*) corresponds to thebblack nucleus approximation, the second (+) to Kox formula(KOX87), x- Ri---~. Nnp is the number of p-n f i r s t c o l l i s i o n s as calculated from the equal participant model(Nlfenecker,1985). 300 H. Nifenecker and J. A. Pinston '"l'"'l'"'l'"'l'"'l'"'l'"'l'"'l'"' 34 - Exelu*|~* - At*A1 85 HeV/A 32 30 # " , +÷ { 28 26 24 ÷ 22 1 2 3 ~ 5 G 7 8 Nn-p Fig. 2.15 Variation of the inverse slope Eo with the number of f i r s t p-n collisions N~ '1 .... I''''1 .... I .... I''''1 3,5 .... I' { > 30 o ~'XCl.U$I Vl{ ,=, 2 5 UJ + • INCLUSIVI[ 20 0 1 2 3 4 5 6 RI+R2-b FIG.2.I6.Variation of the inverse slope Eowlth the overlap distance between projectile and target Rp+Rt-b. The impact parameter b was obtained from the number of participants ~AF,, using the formalism of Nifenecker(1985). High Energy Gamma-Ray Production 301 3.SEMI-CLASSICAL DESCRIPTIONSOF THE NUCLEUS-NUCLEUS-GAMMAREACTIONS. Most of the theoretical approaches to the high energy gamma-productionin heavy-ion reactions have used classical or semi-classlcal approximations, both to the nucleonic motion and to the electromagnetic f i e l d . I t is believed that the qualitative characteristics of the high energy radiation can be obtained through such a treatment. Only recently f u l l quantal treatment of the radiation in the simple nucleon-nucleon case has been incorporated in otherwise semi-classical treatments of the motion of ensembles of nucleons. In this section we shall review the different classical or seml-classical approaches to the problem of highenergy gamma-rays production in nuclear encounters. We shall start with the elementary semi-classlcal treatment of the electromagnetic radiation associated to the scattering of two point particles, say two schematized nuclei or nucleons. E1ementarv Treatment of Radiation in Point Particle ScatterinQ. In order t o discuss the relevance of the different theoretical approachesto the production of high energy gamma-rays, i t i t useful to recall the classical theory of radiation emitted by an assembly of accelerated particles. We follow, here, the book of Jackson(1975). The photon production cross-section is expressed in terms of the radiated intensity: da((,O) d(dfl aR dI(o,e) ,E f-ld(~d("~ (3.]) dl where ( - i ~ , G R is the scattering cross section, ~ is the energy radiated by the accelerated charges per unit frequency interval and unit so"~'rfd angle. The radiated intensity is expressed in term of the vector potential: dl d~dg~ ez 4~ c At ( o , e ) • (3.2) where A I(o,O) - Zl . ~ - - d t dt (]'¢1" (3.3a) n is the direction o f observation, - i , i the position and velocity of the particle i. In the simple scattering case i t is convenient to define the parallel and perpendicular components of these vectors with respect to the i n i t i a l direction of the relative velocity vector. In the soft photon approximation the details of the motion can be ignored, and a scattering event can be schematize~ into a double and simultaneous acceleration sequence where the velocity decreases from ~I to 0 in the i n i t i a l direction and increases from 0 to ~ i n the final direction. Again In the soft photon approximation, the radiation f i e l d is not sensitive to the details of the time dependenceof the acceleration. We, therefore, assume that i t takes place during a time T, at a constant rate. I t is then possible to write for any i: 302 H. Nifenecker and J. A. Pinston Z {'1 Pt X(~,0) = ~ _ Pl cOs O PF 1 - I%cos 0 +2"i~t c T_e 2 dt Finally we only consider moderately relativistic motions. Within these approximations, the parallel and perpendicular components potential may be written: Ai -Z ( i ~ t n 01 (l+2q31>cos 01+3q32>cos20! ) 1" T e - (3.3b) of the vector c )dt 2 (3.4) Ai -Z ~i in 0 t 1+2<0/ > c°se(+3q3(Z>c°sZ°( _ r e - c )dt 2 Here 711 and (t are polarization vectors corresponding to the parallel and perpendicular components of the acceleration, respectively, q3> and <B 2> are suitable averages of final and i n i t i a l velocities. Practically ~13- ~i-13r, <~>- T ' <~z>. 3 " el and 0( are the angles made between the observation direction ~ and the i n i t i a l I and final F directions of the scattered particle respectively. Because of parity conservation, and for non polarized particles, opposite directions of (~ are produced wlth equal probability, leading to a cancellation of the interference terms expected from 3.2,if the scattered particles are not detected and an average is taken over the final velocities directions.We concentrate on the PBrallel compone~)i~, which for two particles, is: i~{r 1-r2)cos e o 2c l¢(r i - r z}cos O -Z stn o +3<I>cos2o)+ (3.S) zc r I +r 2 . ~(~,?) sumartzes the where we have taken the origin of the axis at the position - ~ i ntegratlons in equation 3.4. Momentum conservation allows to write ~I AB2 A(Blh3z) ~13 . For small values of ¢ one can , then write: A,+A2 A,+Az A2 AI m e/ A I l ' ~ * l l ~ AAI+A 1 A zz ~A+ in e{l+2q~l>c°se +3q31Z>cosZ9~ { I - ic°[rl-rz]c°s 2c Z2 { i ' [ r , - r z ]c°s O_}] ~-(~,-) - - ~ s i n e<l-Zq3z>cos 0 +3q3zZ>cosZO) 1+ A2 2c (3.6) High Energy Gamma-Ray Of special interest are the Production 303 Z2 Z1 cases where ~'2" 0 and where AI Z2 ~-which, as we shall see correspond to the pure dipole and quadrupole patterns respectively. In the f i r s t case one obtains, in the f i r s t two orders in 0 (3.7) In nucleus-nucleus collisions one, usually, has to add incoherently the contributions of proton-neutron and neutron-proton collisions. One then gets: r~ 12 - 16-~ 2 sin20(2+4[<~1-~,>]cos 0 + [6<~12>+4<0,>' (3.8) +6<~ >+4<~2 >a ] cos2e)~ ( . , . ) 1 ,F • F Z Recall that <Bi,a>" T (~I,2 + Bi,2 ) with BI,2 "131,z cos ~,2 where 6~.z are the scattering angles. Since, in the nucleon-nucleon center of mass ~-4S2it is the only frame in which the angular distribution has the go" symmetry. Assuming-that, in this frame the particle scattering angular distribution is symmetric with respect to go', one has <~>-0.5~= and <~2>. 3 BZ(l+ cos ~ ) . Finally, the angular distribution becomes: where an average has to be taken over the scattering angle 8. As observed in the experimental data (Grosse,1986) i t is seen that the distribution evolves from pure dipole to a mixture of dipole and quadrupole when the projectile velocity increases. Zl Zz In the second case putting Pz" ~'1 " A2 one gets for moderately r e l a t i v i s t i c XII . ~.11z~t3A1Az AI+A2 pzsin o cos 0 I ~2{<131+t3z>} i~{r~ -r z} c r I I r 2~ + cos motions: (3.]0) o Here again, the go" symmetry can only be obtained for ~I" 02" As noted in section 2.3 this AI is true, irrespective of the ratio A2"2" However the cancellation of the cos 0 term may not be complete, due to the final velocities mismatch. In this "nucleon-nucleon" frame one gets: which is a pure quadrupole pattern. The two origins of this pattern are clearly apparent. The f i r s t is due to the r e l a t i v i s t i c Doppler effect acting on both charges, the second is due to the fact that the acceleration centers of the two charges are separated by a distance Irl-r~l . Note that only this term shows an additional J frequency dependence which cures the infrared divergence present otherwise. This term only gives rise to the simple sin2B cos2e pattern when the acceleration is parallel to the relative distance vector. In other cases the angular distributions are more complex and may even look like dipolar ones (Vasak,lg86; Herrmann,lg87; Kwato Njock,1988*). In the lowest orders one obtains a combination of spherical harmonics of order two. We give a derivation of the angular 304 H. Nifenecker and J. A. Pinston distribution (;o~,~-~.~ b.i-b.j-O, and the R2 --sin20 expected for a non-relativistic quadrupole. Let ~ rl-r z , I ~- ~, -t. -t . We~a~so~d~fine the angle such that cos x " ~.~, and thg axis ~,j such ~h~&t and i . J - J.n _~ O. ~ is the angle between the projection of won the plane ( i , j ) axis vector i . The non-relativistic quadrupole component has the form: cosZw. I t is easy to show that cos W " cos Xcos 0 + sin x cos ~o sin O. After C2 substitution and integration one obtains the radiation pattern: I stnaO (cosZe cosZx + ~- sinZe sinZx ) f(e) (3.12) For ~e-O one obtains the familiar pattern with vanishing values of the cross-section at 0 and ~/Z. For ;e.~/2, the distribution has a single maximum at e-~/2, much like a dipole pattern. When the charge densities of the projectile and target differ, i t can be shown that, in the colinear assumption, the dipole and quadrupole components add incoherently. For example, in the non r e l a t i v i s t i c cases and in the lowest orders in I r l - r 21it comes: IX"l " (A1Az ~z ((Zl Z.zT:'- Z1Zz sin oL( -A2) ,- cos o {ri-r2} 2 c~_/ ~(o,z) (3.13) we now consider the Peroendicular comoonentX~ of the vector potential. We concentrate, mere particularly, on the angular pattern of the radiation. Let ~be the ang1~ of the scattered particle as projected on the plane perpendicular to the beamdirection b .In ~his plane, the angle ~ Is measured with respect to the~projection of the observation vector n . W is the angle between the obse~vatio~ direction n and the projected scattering angle. As above, e is the angle between n and b . Then one has cos W" sin e cos ~ . For a given value of ~, the general form of the angular distribution is, f r o m what has been said in the preceeding, P(w:w)-asinaqs+bsinZ~cosZq~-a+(b-a)cosZ~t-bcos4w. After substitution of W and integration over ~ i t comes: P(e)-a-sin z e ( a~ - ~)+ sinZe cosZe 3b (3.14) The main effect of the perpendicular component is to decrease the order of the angular distribution pattern. A dipole distribution is modified by the addition of an isotropic component, while a quadrupolar distribution is modified by the addition of a dipolar component. We apply the elementary treatments sketched above to the classical treatment of nucleon-nucleon bremstrahlung. The f i r s t case is the Proton-neutron ~qatterinq, This corresponds to a dipole like radiation. The parallel component writes: IX" 12- Z~I311241-~inZO(l+4<13>cos O +{4<@>z+6<13z>)cosaO)3~ (o,T) (3.1s) ]+<COS I~> ]+<cos ~> + <cos2~c, Here z~311z-t3z <(I-cos e)z>, <t3>'13 2 , and <13z>-~2 3 with J~oc os"e T dZ~(e) d(! <cosne~J~OT dZ~( aen) Angular distributions will be considered in the nucleon-nucleon center of mass, and i t w i l l be assumed that they have, in this frame, a 90" symmetry. Then <cos e~-0. For fixed value of the scattering angle the perpendicular component writes: iX±l 2 . 41sinZw(i+4<t3±>cosW +(4<13~>z+6<13j~> )cosZw)~(~,T ) (3.16) High Energy Gamma-Ray 305 zyxwvutsr Production The odd term in <gl> vanishes after integration over the angle. p*<sin*6> 3 . Therefore, after integration and making use of (3.14) one 4+* - p*tsin*@, cpU>= gets: 1 - sin29 'G +<sin*& sin*0 cos*e e*<sin*@ *(w,T) I Finally one gets for the sum of the components: $lI*+ IXL12 = $P(w,~)<sin*o> <sin*@(l-0.5p2<sin2@) + $ P(~,~)sin*e~+<cos*O*- + $@(w,r)sin*e t $-*(~,~)sin*e 2 1 (3.17) cos 0 2p(lt<cos*o>) cos*e p* (l+~cos2~)(3t2~cos20>)+ ( i <sin*@* I A non relativistic has been approximation to this expression (Nifenecker,l985; Bauer,1986; Remington,l987). In this case one obtains: d* o(c,e) = -.!& % used 3ia(c,-r)$(tsin2Ea + sin*e{l+tcos*0>- +sin*@}] &dn Recall tiat @=pLab <sin*@=-, extensively (3.18) . ForIisotropic scattering one has and <COSTS)=-, and the angular part of 3.18 reduces to sin*e t This expression was obtiined by Nifen&ker(l985). A similar expression is obtained for t $r;n2;jre(Jackson,1975j Cassing,1986). For a l/sin zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM 8 distribution one would obtain The distribution 1s more anlsotropic than the preceeding one. An extreme case is obtained5 for pure forward backward scattering. The angular distribution becomes a pure dipole 2 sin*e. It is seen that the total gamma cross-section is not a very sensitive function of the scattering angles distribution. We now consider the classical treatment of the Proton-oroton scatterino. This is a quadrupolar process. for simplicity we give the expression for the parallel component. I$‘I*. $. sin*0 As compared to reduced by a the p-n cos*e +&$_J@(,,T) I4p2(1t~os2m) (3.19) case we see th;t, in the p-p system the radiative cross-section is factor of order p*+ AR* which for a gamma energy of 100 Mev and typical values of p amounts to Aside from quadrupole also consider magnetic dipole radiation which might be induced by the spin flips of the nucleons during the scattering process. It can be shown(Jackson,l975) that the magnetic dipole radiation is reduced with respect to the electric dipole by a factor of order 22= that is around 0.01. 306 H. Nifenecker and J. A. Pinston As w i l l be seen later, most theories of nucleus-nucleus Bremstrahlung consider nuclei as more collections of nuleons. Most add incoherently the contributions of the different nucleon-nucleon-~ processes. Howeverone expects that collisions between a projectile proton and a target neutron should interfere negatively with collisions between a projectile neutron and a target proton. In fact, for nuclei having the same charge density, one would naively expect a quadrupolar pattern. That this is not the case, due to the loss of correlation between the protons, after scattering, can be shown by examining the Deuteron-deuteron-~ammaprocess. For simplification we consider the non-relativistic case.lt is straight-forward to generalize the calculations to moderately r e l a t i v i s t i c situations, using formulas given at the beginning of this section. For reference we recall the cross-sections obtained for the incoherent sum of p-n-~ and n-p-~ processes: 132 OR .~_(<sin2~>+sinZO[]+<cosZ~{>_ .~ e2 4~t~c "~ ~ s i n 2e~}):~ (~,T) Similarly a quasi-elastic d-d-~ process where the deuteron would keep their identity in the final state would lead to: ~z d2 ~(~'e) dd d(dIl e2 ~, 4~i~c ~ 4 slnZe c°sZe l~z(]+<cos2e~) j__~zl~ 3 + c~ } (('T) In the d-d process the projectile proton(]) is scattered from O' t o ~ i while the target proton is scattered from ]80"to ~ . Becauseof the presence of the neutrons el needs not to be equal t o ( ~ . Therefore, in the dipole approximation the parallel componentof the vector potential writes: sin 2 B/ ix,,i _T_ <cos. >+<cosZ ,>-z<cos ,cos (3.zo) while the perpendicular componentwrites : iX±iz. Bz (l- sinZe~" z T Tit<sin a ~>+<sin ~>-2<sln 81sin ~>) (3.2]) As expected both components disappear in case of complete c o r r e l a t i o n between ~and E~. In case of complete decorrelation one gets the gamma production cross section: dz a(~e) d~dn . d,np ez ~, 4~T~C ~ ~z 2 (<sinZE~ + sinZe(<c°sZ~c~-O'5<stnZ(~ } ) ~ ( ~ , T ) As compared to the incoherent pn+np case we see that the lsotropic component is kept unmodified. The sinZe term is very s i g n i f i c a n t l y reduced. For tsotropic scattering i t even vanishes, while for ] / s i n ~ scattering i t is reduced by a factor 5. In the extreme backward forward scattering i t is only reduced by a factor 2. Another cause of decorrelation l i e s In the Fermi motion.A complete treatment of this effect is outside the scope of this elementary outlook. However, a feeling for the Fermi motion effect can be obtained assuming d i f f e r e n t v e l o c i t i e s for both p-n pairs. The f i r s t p-n pair is assumed to have v e l o c i t i e s ± ( ~ I + ~ , ) . The second n-p pair have v e l o c i t i e s ±(~l-~131).The p a r a l l e l component of the vector potential writes: ~fl ~11 A =~ sin e(2~13I - (~l+z~13i)cos ~ + (~i-z~131}cos E)z) which, a f t e r squaring and averaging gives: r slnZe 2<c°sZ 1+ ~------>~ ) (3.22) The <~13~>may contribute significantly to the gamma-rayproduction. Of course the Fermi motion w111, also, smooth the angular distributions, which w i l l look more isotropic. This simple treatment shows that, due to the effect of various fluctuations, i t might be justified to treat the f i r s t nucleon-nucleon collisions incoherently. Howeverthe elementary High Energy Gamma-Ray Production 307 cross-sections. They are related, in a rather complex way to the true cross-sections. In the preceeding, we have only discussed angular distributions. Only those can be obtained from classical considerations. The shapes of the spectra w i l l be discussed, in a quantum environment, in a later section. For the highest gamma-rayenergies, momentumand energy conservation might put constraints on the angular distributions and alter the conclusions reached here. The classical treatment given here deals exclusively with the external contributions to the gamma-production process. Internal contributions are expected from the chargedpions exchange betweennucleons. Thesewill be considered later. However a few remarks are in order. First internal contributions have no infra-red divergences(Jackson,1975). They should only be of importance in the proton-neutron scattering process, which is the only one mediated by charged pions. I t is expected that the angulardistributions induced by the internal contributions should be rather isotropic and would, therefore be more justified of an incoherent treatment. 3.2 Multinle Scatterina Effects. Equation 3.3 shows that contributions of scattering events occuring at different times have to be added coherently. Consider, for example, a charged particle suffering two successive collisions with neutral scattering centers. Three contributions to the vector potential become apparent. The f i r s t and third correspond to the ingoing and outgoing channels respectively, in the non-relativistic limit, the second t e ~ writes: ~zBzsin ez 1- e where T = . Therefore, the dominant contribution to the second tem writes: • L~r ~z~2sin e2 I]_ el ~ - ~ ~2 i ~ 2--L~r sin ez" Although dipolar with respect to i t s angular distribution, this contribution countains the t e ~ °~r which is analogous to that displayed by quadrupolar radiation. In particular, i t disappears in the soft photon (~ = O) l i m i t . This effect was not taken into account in previous multiple scattering treatments of the gamma emission process(Nifenecker,1985; Nakayama,1986; Bauer,1986; Remtngton,1987). In such kinetic models i t may lead to serious errors, especially at low gamma energies. On the other hand, Knoll (Kno11,1988) has shown that these models may violate electromagnetic sum rules. In the following we t r y to evaluate the connection between the quantum mechanical treatment of Knoll and the unproper treatment of t i ~ correlations in the classical calculations. Limitations due to the electromaqnetic sum-rules. We summarize the t r e a t ~ n t of Knoll and Guet (Knoll,1988). One starts with the expression of the transition probability : - ~ cz. 2 Wlf ( ~ ) where - 12~I~21I/z J c Z e |k T r I/xfI .. Ii}I and (E _E,_+ E,) 3.23 ;3 3 The absorption cross-section and production rates are obtained from: 1 W (ET) = ~-~f if (kr)e(Ef>Ei) 3.24 8~.~ ~Wif (k~)e(Ei>Ef) It is then possible to derive the following f i n i t e temperaturedipole sum-rules, for a 308 H. Nifenecker and J. A. Pinston thermally equilibrated system: 8w J~odEtr~-3 (eEr/' +(-l)n) Nr(Er,T) - (2,.~:) - 3 S.(T) 3.25 ~odF_~r~" (l+(-|)ne'Er/T)oaW (E~,T) . 1~ Sn(T) The S. sum rule is model independent. Simple expressions for the SO and S2 sum rules can be obtained i f two body correlations are neglected in their expression. Such ~implificatlon can be shownto have modest effects on the estimates of the sum-rules. Under these conditions one obtains, for isospin symmetric matter: So - 4 ~ A A Si e2 ~- c ~ -2~" ~ c e2 ~ <<i Iz 2 li>> i $2C2 3.26 mc2 e2 z <<i Sz -4'n2 ~A c tic mc2 m Ip:li>>, Several interesting remarks can be done from these sum-rules. Firtsly, they are independent upon the density of the emitting system. This implies that the pure incoherent summation of the radiative contributions of independent collisions cannot be correct. Indeed, in this case the radiated intensity should be proportional to the numberof collisions, and therefore, to the density. Furthermore, the finiteness of the sum-rule n-O implies that N(Ey) goes to 0 faster than E~+~ for E ~ 0 . This is in strong divergence with the E~I infFared divergence displayed -by the f r e e n-p cross-sectlon. At high energies, the -E / t production rate has to decrease faster than E~e r . Knoll and Guet(lg88) show that, as expected, kinetic models (Nlfenecker,1985; Neuhahser,lg87) based on the incoherent summation of individual contributions of the collisions violate the sum-rules in a very serious way. They trace the origin of this violation to the fact that nucleons are rescattered before the long wave-lengthphotons are produced(infrared region) and to specific quantum effects at the high energy end of the spectrum. I t should be stressed that the treatment of Knoll and Guet(Ig88) does not apply directly to nuclei. I t deals with the total number of photons emitted in thermally equilibrated nuclear matter. Someof these photons may be reabsorbed before they escape from the nuclear medium. Therefore the sum-rules obtained by Knoll and Guet are upper limits, when applied to real nuclei. In practice, this l i m i t is reached , since absorption is low. However, the sum-rules could be applied, in principle to massive systems, like stars or plasmas, in which the absorption can, no longer, be neglected. For large systems, the asymptotic behavior should be the black body l i m i t . This l i m i t , which predicts a E~ dependenceat low energies leads to an infrared convergence of the nmI,2 sum-rules, anda logarithmic divergence of the n-O sum-rule. This is related to the f i n i t e size of the radiating body which limits the validity of the black-body formula to wavelengths larger than its dimensions. At high energies the black-body formula gives an ultraviolet divergence for all sum-rules. Since the black-body emission rate, in this energy region, has to be smaller than that considered in the derivation of the sum-rule, because i t incorporates the effect of absorption, this divergence implies a failure of the sum-rule at the higher energies. This is, probably, due to the failure of the dipole assumption, at these energies. This assumption is only valid for kr > 1. Taking as a characteristic distance the internucleon distance this leads to a transition energy of around 100 MeV. The considerations of the beginning of this section are clearly related to the violation of the sum-rules In the soft photons region. We explore shortly the possibility to incorporate High Energy Gamma-Ray Production 309 these into a Modified kinetic model. We consider a gas of protons and neutrons. Only proton-neutron collisions are taken into account, for simplicity. This is j u s t i f i e d since the n-p cross-section is about three times larger than the p-p cross-section and since the p-p collisions give only small contributions to the radiation, because of their quadrupolar nature. In the non-relativistic approximation, the potential vector can be written: Z Z ~ I I(o'rl i, A - j~- ] ~ ; ^ ; ^ ('~j,k+"l 4,.lk" ) ei''°Tj.k = ~-1 ~ n^n^,e "" e .~rj,k+` )~j.k assuming t~at ~he velocities are uncorrelated before and after collisions, one may write that ~j,k "~t., ~ " B~.k 6j,t 6k,= and after squaring and averaging over angles i t comes: Z IXI2 " 2 j~l ~ sin2ej,k ~,k (]-COS ~ (lr j, k --q'j,k+l }) < 3.27 > 2 Further averagingconcerns the angles and gives sinZOjk - ~ . The cosine term has, then to be averaged over the free paths distribution. In doi'ng so we shall assume that the mean Xj,k free path is velocity independent. Then Tj, k -~rj,k+I - ClZj,kcan be averaged over the free path Xj,k . I t comes: Xj ,k <COS m CIZj,k > C2^2 Pj,k 2 2 + (~2X2 • Finally one obtains: C 13j,k IXI 2 " T161r Z v~oi.NcotL (v) E ~ 2 + ~2v2 dv 3• 28 where Ncott (v) is the collision rate for particles with velocity v. x is the mean free path I - --. p(7 For a Boltzmann gas: ~vz Ncott (v) - I ]/2 ---/^l~./ vZe 2T where rL is the A ~ITIJ( relative velocity of the two nucleons. The gamma production rate is the written: N(E~,T) 16 I Z c . __m 411 /~c213/2 reduced mass of the =.. z 2 2 p-n system, and v the EZe TdE (3.29) where the doubly bracketted expression is the additional term induced by the negative interferences at both ends of a segment of the trajectory. This term assures convergence, in the infrared region, of the sum-rules with n>O. The divergence of the n-O sum-rule is related to the fact that we have not constrained the hot gas into a f i n i t e volume• Therefore the right hand tertnof equation (3.26) is , in that case, unbounded. The bracketted term equals I/2 when E~- ~ '~.p . Typical values of this energy range between 10 and 30 MeV. This is to be compared to tee lower cut-off energies obtained by Knoll and Guet (1988). I t is also seen from (3.29) that, for low energies the gamma-productionrate is inversely proportional to the nuclear density. For high energies, one comes back to the more usual situation where the photon production rate Is proportional to the nuclear density. The dependenceof the total photon production with respect to the nuclear density is weak. I t is interesting to note that the f i r s t and last collisions are not included in 310 H. Nifenecker and J. A. Pinston considerations about the sum-rules. Therefore, one of the effect of the negative interferences which take place along the particles path, is to decrease the isotropic component found in the d-d elementary treatment, with respect to the anisotroptc component. The intensity of this anisotroptc component w i l l be governed by the fluctuations of the i n i t i a l velocities and directions, induced by the Fermi momentum. In any case, i t appears that, i f these fluctuatfons j u s t i f y calculations where the contributions of the nucleon-nucleon c o l l i s i o n s are added incoherently, the quantitative use of the classical n-p-~ cross-section is very questionable, even on the basis of purely classical considerations, due to the additive nature of the electromagnetic vector-potential. The treatment presented above is purely classical, and therefore can be considered as an approximation to the external contributions exclusively. I f internal contributions, such as those produced by the charged ptons exchange between protons and neutrons are important, the treatment looses i t s relevance and the sum-rules should be modified accordingly. This has not been done, to our knowledge. Note, however that the internal contributions do not suffer from the infrared divergence since, for low energies t h e i r intensity is proporttonnal to the n~u" ] t is timely to examine the v a l i d i t y of a semi-classical expression of the cleon-nucleon-~ cross-sections. 4.NUCLEON-NUCLEON-GAHHACROSS-SECTIONS. Our experimental knowledge of the nucleon-nucleon-gamma production is Irregular. The proton-proton system has been thoroughly studied (Stgnell,]967; Kitchin9,]987 ), since i t is the simplest, experimentally. In this case, a good measurement of the momenta and energies of the two protons, after scattering, is basically s u f f i c i e n t to describe the reaction. The neutron-neutron system is unknown, for obvious reasons. Our knowledge of the neutron-proton-gamma process is poor(Brady,]970; Edgington,]g74; Dupont,]g~). This is due to the small intensity and bad resolution of the neutron beams. Furthermore the neutron and photon detectors are more d i f f i c u l t and less performing than proton detectors. This is unfortunate since it is known, experimentally(Koehler,1967; Edgtngton,]966) and theoretically (Baier,]969; Ntfenecker, tg85) that the n-p-~ process is, by far, more e f f i c i e n t than the p-p-~ process to produce gamma-rays. Due to these circumstances, calculations of the nucleus-nucleus-gamma process have made use, almost exclusively of theoretical estimates of the neutron-proton-~ cross-sections. This is why we begin by a summary of the relevant theoretical calculations. Theoretical Descriotions of the Proton-Neutron-Gamma Process. The f i r s t nucleus-nucleus calculations h a v e used the classtcal(Nifenecker,1985; Nakayama,]986; Bauer, 1986; Remington,J987) approximation, with the inclusion of energy conservation, in the semi-classical fashion. This approximation has been amply described in Chapter 3. For completeness we recall a form of the employed semi-classical cross-section, most commonly used(Remtngton,]987; Nakayama,Ig86). For each n-p c o l l i s i o n the photon production rate is given by: j . II-n. -P-* j2 , e = ¢4.1) Eo where e is the step function, and ~ , x . -~-the maximum gamma energy. Eo Is the tncldent beam energy.The d l f f t c u l t i e s met by the calculations to account for experimental results at the highest incident energies (Grosse,]g86; Kwato NJock,I988b,c)led to the questioning of the semi-classical approximation (Neuhauser,]g87 ;Remington,1987). A f i r s t improvement would consist in treating the n-p-~ process within a quantum potential scattering model. The f t r s t treatment of this sort, to our knowledge is that of Ashkin and Harshak(1948). Since this calculation incorporates most of the significant ingredients of potential medels, we shall report on i t in the following. High Energy Gamma-Ray Production 311 Potential ouantumcal~u1atlons. In the classlcal theory, radiation only occurs when a charge Is accelerated. Similarly, in quantum electrodynamlcs, radiation emission is a second order process. This is obvious from energy and momentum conservation. In its own frame, an elementaryparticle cannot emit radiation, in its ground state. In order to display the basic features of the n-p bremsstrahlung calculation we start by accounting for the perturbative treatment of Ashkin and Marshak. In their Born approximation for b o t h the scattering and the radiation processes, the Hamiltonian writes: H-H.+ V.+ V whereHo is the hamlltonlan of the free particles, VWthe Nuclear potential. Ash~in "and'~4arshak chose a Yukawanuclear potential which reproduced the n-p scattering data, as they were known at that time. This potential has a merit to lead to analytical ]+P. e-Xr expressions for the photon spectra. VW(r) = = g l where P. Is the Majorana operator r and g~(i=1,3) refer to the singlet and t r i p l e t s~ates respectively. The electromagnetic part of the Hamiltonian writes, in the non-relativlstic l i m i t : v- - - ~ e (~.X+~.X)+ x2 - ~ ( + ; , + ..;.).curl ~ e' (,.2) wlth the following usual expression for the vector potential: x. ~,c + Z k,~ eiE.;/~ . ,..,--~-(ak+ a, ) (+.3) where f~ , ~ are the momentum and energy of the photons. ~ k are the two polarization vectors, w i c h are supposed to be perpendicular to ~ . a~and al are the creation and destruction operators. In the right hand side of equation 4.Z the f i r s t two terms correspond to electric transitions and the third to magnetic transitions. The second term is of second order, and furthermore, couples only states differing by an even number of photons. I t may be dropped out. The magnetic contributions requires a spin f l i p of the nucleons, and, therefore, do not interfere with the electric contribution. For simplicity we only consider the electric contribution. The perturbative expansion, up to second order writes, for the transition matrix T: T. - (vl,+ vr,)+ ~,m [vl+v~ ~ + ~I:.'v~/ El - E, J (4.4) The f i r s t term of the right hand side of equation 4.4 corresponds to the nucleon-nucleon elastic scattering and can be ignored. The second term writes: V~ ~ ~- r,c e ~, .~. 6 ~,- ~,- ~kV.~) (4.5) mc This matrix element conserves momentumbut not energy. As said above, i t has to be dropped out. The f i r s t second order term corresponds to a situation where the photon is emitted before the scattering, while the reverse is true for the second term. The energy is conserved betweenthe I n i t l a l state i and the final state f. Therefore, neglecting the recoil terms, Et=EI+ ~ and EfmE.+~. Flnally, taking the exchangeterm into account one gets: Where Vp is the Fourier transform of the nuclear potential. Squaring T gives: 312 H. Nifenecker and J. A. Pinston Z Z 2"z"z (V, ,(pi-pf)+ Ve~I+p,>) z /p, sinZeo+pfsin2Of-Zp, pfsi~asinOfcos[~%~,])(4.7) Ti2f = ~ ~here e~, ,~, e are the azimuthal and polar angles of the proton velocities with respect to the photon d|Yectton. After Integration over the angles and using the Fermi golden rule Ashkin and Narshak finally get the electric cross-section: 3cZ~Pz 'lg~ doe (4.8) J ('P') p' where : F_,(x) xZ(l+xZ) { 2 + l - [AZ+(l+x)Zl} I _ (l-x z) [AZ+(I-x) z ] [AZ+(I+x) z ] 2x[^Z.l+xZ]C°g[Az+(l-x)Z where A=~m .A11 quantltles are Pl measured In the Initlal (4.9) nucleon-nucleon center of mass Eo frame. The gamma energy Is f~oJ k= Emmx (l-x z) with Emx - -~-. As expected the cross-section goes to 0 when the gamma energy approaches its maximum value. This is the main difference of this result as compared to the semi-classlcal approximation. Similar expressions are obtained for the magnetic component. Z do.- 2(~-~) 2 ~gl Ifig'l gl) (4.1o) wlth: (l-xZ) z ~(x) - ~ E ( x ) (4.11) l+x 2 The Ashkln-Marshak treatment does not glve correct angular distribution since It does not include the Doppler shift of the gamma-rays. Their formalism has been modified by Kwato NJock(Ig88d) who has been able, again, to obtaln an analytic, formulation of the cross-section. This formalism has been used to calculate the proton-nucleus process(see Section 5). I For small gamma energies (p ~ Po) the spectrum goes llke ~-, while for high gamma energies (p ~ O) It goes llke - - . It Is interesting to note that the traditional modification of the seml-classlcal sharp cut-off has the same properties. It amounts(Heltler,1953) to replace the 6( of the classical formula (equation 4.1) by the geometrical mean 616f. Since ~I~ ~ , this leads to the modified semi-classlcal spectrum Lr e(~aX-Ev). A comparison between the Ashkln-Marshak expression and the modified as well as unmodified semi-classical ones Is shown on fig. 4.1. In fact, the gamma spectrum shape is not very sensitive to the detailed shape of the nuclear potential, as long as the photon wave length exceeds the nuclear potentlal range, or, equivalently, as long as the photon energy Is less than the plon mass. This Is also examplified by the fact that the elaborate calculations of BlrO and colleagues(BirO,Ig87) which use a a,u descrlptlon of the nuclear potential give results very similar to that of Kwato Njock(Ig88a). The only significant sensitivity to the High Energy Gamma-Ray Production 313 nuclear potential is through the scattering angles distribution.The Ashkin and Marshak approach considers only on-energy shell nuclear potential matrix elements. This is a consequence of their assumption that l p i l ~ I p f l . However, i f the photon emission occurs before scattering, the energy of the proton may be significantly affected. Such off-energy shell effects were investigated by Brown(Ig6g) and Brown and Franklin(1973). Neuhauser and Koonin(]g87) integrated the results of Brown and Franklin over the neutron and proton scattering angles, to obtain integral gamma-ray spectra. The same approach was used, again, by Nakayama(Ig88a). On Fig. 4.1, the gamma spectrum calculated with the inclusion of off-energy shell effects is displayed. It increases markedly the photon production close to the end of the spectrum. Neuhauser and Koonin suggest that this increase is due to the fast decrease of the nucleon-nucleon cross-section with incident energy. They account, partially for this effect by replacing the seml-classical cross-section at the center of mass incident l energy Eo, =J ~ dOE-~_Eo) by the geometrical mean;dUE-~Eo)dUE--~Eo-E~). More intuitively, i t is possible to consider that the electromagnetic part of the interaction is unchangedwhile the nuclear part, alone, is modified when the photon is emitted before scattering. Therefore one may write the modified off-energy shell cross-section: %n (E°)+%n (E°-Er) off " on (4.12) 2°nn(Eo) Note that the j u s t i f i c a t i o n of such an approach is related to the f i n a l energy of the proton, and not to the properties of the intermediate state where energy is not conserved. We use the following expression of the nucleon-nucleon cross-sectton(Bertsch,1977): Gnn= 62.5 - 50.06 22.36 +~ (4.13) The result of such a simple calculation is shown on Fig. 4.1, together with the more elaborate calculation of Nakayama(Ig88a). I t is seen that the off-energy shell effects put the unmodified semi-classical cross-section back into business. In the preceedlng, i t was assumed that the nucleons suffered only one nuclear scattering. I t is possible, however, that they might suffer several and that the photon might be emitted between two successive scattering. This corresponds to the so-called rescattertng terms. The contribution of these terms is small(Brown,Ig73; Nakayama,1988a). This can be understood since they correspond to quadrupolar type processes, and, therefore, to a reduction factor where ] / x is the range of the nuclear potential. We shall now see of order f]-cos % / / ~cx) that, even at high gamma energies, this contribution is small, as compared to pionic contributions. In p-n scattering, charged plons may be exchanged.These transient plonic currents may lead to photon emission, with characteristics similar to that due to the rescatterlng process. However, here, the mass of the pion is small and, therefore, the bremsstrahlung emission is expected to be enhanced. Considering, as in the Ashkln-Marshak potential, a one plon )xcha~e, the range of the potential is simply related to the plon mass by - = ~ 1.4 fermi. As well known this relation is obtained by use of the uncertainty x n~c2 principle which gives the off-energy characteristic time, and by the assumption that the pions propagate with the speed of l i g h t . Using the same assumptions, i t is possible to obtain a rough estimate of the bremsstrahlung of the v i r t u a l pions. I t is given, in comparison to the proton external contributions by: ~ad. which gives: o~ ] ] ¢~x _ ~ - ~°O/_ COs _.~/e XXdx (4.14} 314 H. Nifenecker and J. A. Pinston 10-3 T~b= 150 rn ::> MeV 8=30 ° 10-4 0.) v .,'1 10-5 '%,. C ,10 3 '0 % Q I "%e t-} % • 10-6 % b "O ! i a) CONVECTION 10 -7 0 20 40 60 80 (MeV) Fig. 4.1.Convection radiation calculated under diverse assumptions . . . . Nakayamacalculation neglecting off-energy shell effects • Ashkin and Marshak type of calculation from Kwato NJock ---Nakayama calculation with off-energy shell effects x I/ET semi classical = E ~ - E ~ , ~ modified semi-classical o modified semi-classical with off-energy shell effects High Energy Gamma-Ray Production 315 ~ET ~2 1 Lm c+) (4.1s) 1+ I ' I Lmc+) We recall that, here, B is the velocity of the proton in the center-of-mass frame. As an example, for 150 MeV incident energy, the maximum enhancementfactor is around 13, and around 7 at 300 MeV incident energy. I t is, therefore, quite a significant effect. Calculations incorporating e x p l i c i t l y the meson exchangeswere made by Baler and colleagues(Ig69) in a OBE type calculation. As w i l l be seen in the next section they obtained rather satisfactory agreementwith experiment. Brown and Franklin(lg73) used a non-local potential formalism which can be shown to incorporate pion exchangeeffects, via the f i n i t e range of the potential. As said, this analysis was used again by Neuhauserand Koonin(1987) and by Nakayama(Ig88a).Fig. 4.2 shows some results obtained by Nakayama.The convectlve(external), magneticand exchangecontributions are displayed. Also shown are the results of an Ashkin-Marshaktype calculatlon and the exchange contribution obtained by applying to i t the correction given by equation (4.15). The success of the s i d l e formula 4.15 is striking, especially at 150 MeV. Of course, such a success may be partly fortuitous, and does not prevent from doing more elaborate and exact calculatlons. This is why we think i t useful to give, as an i l l u s t r a t i o n , an overview of the treatment of Nakayama. At variance with Ashkin, the nuclear scattering is treated non-perturbatively. The distorted wave functions are denoted by the notation ~, while the unperturbed plane waves are denoted ~. The nuclear transition matrix is given by(Messiah,1963): Tt," (++ IT I++ ) " <% IV" l+i> (4.16) the distorted wave functions are, then given by(Messiah,1963): I ~ > - I~> + Go!'T!~l~) where + corresponds to outgoing and - to ingoing waves. G~ of the free equal to: M - p ~ t ~ c l e + Ha+il+onian. The t r a n s i t i o n (4.17) ] E - Ho±i~ is the Green function amplitude for emitting a photon ~,~ is Using equation(4.]7) one gets: ~k,,++;/V--/+~;. , - Iv" I+,) + I T.+ + Iv " F+'I+,) t I (4.18) ! ++.+;+,IT-+ + v - F T+I+,) The f i r s t term is the zero scattering and does not contribute to photon emission, due to momentum and energy conservation. The second and third terms are single scattering. The fourth term is the rescattering contribution. Nakayama only keeps its contribution corresponding to plon exchange. The electromagnetic potential is dlvided into three contributions: V,,- Vcon, + V,,,~ + Ve,~ . The single scattering of the convection potential Vconv is the electric externa~TcontFTbution. The same is true for the magnetic contribution V.,,~ . The two-bodyexchange contribution is purely internal. Except for off-energy shell effects, the external contributions are essentially the same as in (Ashkln,1948). We shall not treat, any further, the magnetic contribution and refer to (Nakayama,lgBSa)for a complete treatment. The convection and exchangeelectromagnetic potential, then, can be written: FI31PK- 316 H. Nifeneckcr and J. A. Pinston 10 -3 . . . . , . . . . , . . . . . . . . , 10 -s Tw,b-150 MeV ~ ::> ~ 8"30° 10 -4 10 -4 Q) ,.o • .." ... % ..." .... i0 -s ......... ~ o'~..o"tS" 10 - I ~ - _ ~i o .. % i 0 ~.. .''" "-..~ o • . 0 / c~ -o ,lO 0.'" ." 10-' b '0 ¢ XO -e ¢" ""~ ~. t ..." s" #' / t b) o) 10-7 . # ............... 0 20 40 60 80 10-7 . . . . . . . . 0 25 50 75 100 125 150 Fig. 4.2. Various contributions to the photon production(Nakayama,lg88a) for neutron-proton reactions at 150 and 300 HeV at 8-30" . .... Magnetic contribution .... Convection contrlbutlon ..... Exchange contribution --Total ca]cu]ated cross-sectlon(Nakayama) • Ashkin-Harshak type ca]cu]ation(Kwato Njock,1988d) o Approximate exchange c o n t r i b u t i o n oex¢ -o=onv ~ - I+(EJI~cZ~Z'/" %onv was the va]ue c a ] c u ] a t e d by Nakayama. High Energy Gamma-Ray Production 317 (4.]9) el,e2,e(,e~ are the electric charges of the interacting nucleons. Clearly, the exchange poteBtlal -acts only in the p-n channel. Nakayamauses the Bonn potential in his calculation of the nuclear potential(Machleidt,1987). We shall not enter Into the details of the calculation, but shall, rather make some physlcal interpretation of the terms involved. Consider, f i r s t the convectiveterm. The gammaenergy dependence of the cross-section, for small gamma energy will be determined by three factors: the relation between the cross-section and the transition matrix involves a multiplication by a level density, that is a factor proportional to E~. The electromagnetic potential matrix element has a E~ly2 dependence. The energy denomlnators El,f are equal to E~, neglecting the recoil energies. Finally, after squaring the transition matrix element, one obtains the familiar E~i dependence at low energy, the infrared divergence. Consider, now the exchange term. For a local potential: VW(~,,~,) - ~oei (P'-P')'~VW (r)dSr (4.20) and one gets: Vexch (~f,~,) . J ~ 1~-(e~- e(- e2+ e , ) ~0 ~,.~ ei(pf-p,).~VW (r)d3r (4.2l) It appears that a non-vanishlng value of this expression requires charge exchange e~- ei , e(One can choose, for example, e~- e( - O. Further, the integral Is proportional ~ the range of the nuclear potential: This leads to a dominance ofr l -the one plon exchangeas compared to other mesons as a source of photons. Except for the J6term, the exchangepotential contribution is, like the convection one f i n i t e when th~Kphoton energy goes to O. However, while the second and third terms of equation (4.18) diverge for vanishing gammaenergy, the last one does not, due to the integration over all intermediate energies. Thls is easily seen since for zero gammaenergy, this last term reduces to (~I~), which is f i n i t e since ~-~ is localized. Therefore, in the l i m i t of small gamma energies, the exchange contribution will behave like Er. Exoerimental Data on the Neutron-Proton-GammaProcess. Very few data h a v e b e e n reported on the neutron-proton-gamma process(Brady,1968; Edgtngton,1974; Dupont,Ig88). Genuine n-p-~ measurements require neutron beams of high intensities and well defined energies. These neutrons are usually produced tn p+Be or p+Lt~ as well as d+Be reactions. Beam intensities obtained wtth these techniques range between lOr (Brady, lgTO) and 6.10S(Edgtngton,lg74) neutrons per second. In the three cases mentioned, a l i q u i d hydrogen target was used, combining the advantage of high target thickness and low background, provided thin windows are used. Both scattered proton and neutron were detected in coincidence. The proton energies were determined from the pulse height delivered by s c i n t i l l a t o r s , whtle the neutron energies were obtained from t i m e - o f - f l i g h t measurement.The set up used by Brady(1968) ts shown on Fig. 4.3. The proton energies were measured, with an accuracy of about 1%, with a NaI crystal placed behind the three plastic s c i n t i l l a t o r s of the proton telescope. Neutrons were detected in a large volume Ne]02 plastic s c i n t i l l a t o r , in front of which was a chaged particles veto thin plasttc s c i n t i l l a t o r . The time difference between the time of f l i g h t s of the proton and neutron was recorded. In the experiments at 130 MeV(Brady,1968) and 208 MeV(Edgtngton,1974) almost coplanar geometries were used. The angle between the proton and the neutron is smaller than 90°, as soon as a gamma is emitted. In the n o n - r e l a t i v i s t i c l i m i t , the momentumof the photon can be neglected. Then, any couple of scattering angles of the proton and neutron corresponds to a 318 H. Nifenecker and J. A. Pinston .......... o Fig. 4.3 Experimental set-up used by Brady(1970) for the study of the n-p-Y process. 0.8- :L T ) z 0.6 T . 'i r ~-p ! 'T o . J~ o-. K,.eM*T,CS " l! o-0 K,.EM.T,¢S i~ I M,v] FET > ~ T" I , ~ 0.2 . I . . . UEOA 0 0~o . I , i E !:'HOTON m ANGLE i m tN C . M SYSTEM Fig. 4.4 Differential cross-section for p-p and p-d Bremsstrahlung (ET>40 MeV). From Edglngton and Rose (1966), High Energy Gamma-Ray Production 319 given gamma-energy. Neglecting the photon momentumand in the n o n - r e l a t i v i s t i c approximation the gammaenergy is then given by: where H t s the mass of the nucleon, P the i n i t i a l momentumof the p r o j e c t i l e , e,, e° the final angles of the neutron and proton.For moderately r e l a t i v i s t i c velocity of the-Incident particle, a narrow gamma-spectrum corresponds to each pair of scattering angles wtth respect to the beam. In the Harvard geometry, the neutron and proton are detected at symmetric angles. In this case, the relation between the detection angle and the gammaenergy is especially simple, and wrttes: I~cos 20 I 2 + ~ cos OTI Erm 88COSZO COS2O) where e Is the common angle of detection of the nucleons and er the angle of emission of the unobserved photon. The average of the gammaenergy is obtained for cos Or =0. The relative width of the gamma spectrum ts given by ~ - ± 2~ 1- where E~"x ts the maximumgamma d2o ~ f o r various neutron and proton exit , p angles, as obtained by Brady(1968) and Edgtngton(1974) are reported tn Table 4.].The data reported were compiled by Remington(1987) and Nakayama(lg88a). The experimental results are compared wtth predictions of various theoretical calculations(Brown,1973; Nakayama,lg88a; Baier,lg6g, Remtngton,1987). The Remington calculation(1987), ts, essentially, semi-classical, and, therefore, excludes internal charged mesons current contributions. The other calculations are quanta] and do include internal contributions. In the 76 HeV expertment(Dupont,1988) the highest energy gamma-ray region was selected by measuring both protons and neutrons at small angles. A total cross-section of 602_~33 nbarns was obtained when both nucleons were detected between 1 and 12", wtth respect to the beam direction. Agreement was found with a calculation obeying gauge tnvartance, and, therefore, including internal contributions. I t Is also possible to extract information on the elementary n-p-> process from the radiation emitted tn p-d-~ reactions. Such studies are, also, scarce. Koehler and colleagues (Koehler,1967; Rothe,1966) measured the p-p-~ and p-d-~ reactions at 148 HeY and 198 HeY. The comparison between the two reactions confirms that the p-p-> contribution is small. At Gn~ 198 MeV they deduce from t h e i r measurements a r a t i o between 30 and 70, for gamma Gpp~ energies larger than 40 HeY. The total cross-section measured at 148 MeV was 26 i~barns for gamma-energies larger than 25 HeV. Edgington and colleagues(Edgtngton,1966), measured the p-p-~ and p-d-~ reactions at 140 HeV. They also found that the p-p-> cross-section was negligible, as compared to the p-d-v, and therefore, n-p-> cross-sections. Their result ts shown on Ftg. 4.4. One should, however, consider the absolute n-p-~ cross-section obtained by Edgington(1966), wtth caution,as w i l l be seen later. The integrated cross-section they found was 11 ~barns for the p-d-~ reaction and gammaenergies larger than 25 HeV. Thts Is, therefore, a factor of more than two less than the value found by the Rochester group(Rothe,1966). The results considered here are of limited precision and do not allow a clear evaluation of the d i f f e r e n t theoretical approaches. The results of Edgtngton(1966) seem to favor calculations which do not incorporate internal contributions, while the reverse seems to be true for the Rochester results. A comparison between the p-d-~ results of Edgtngton(1966) and calculations ts made on Ftg. 4.5. This Figure is due to Remington and colleagues (1987). They compare the experimental results with semi-classical calculations, and with a quantal calculation which follows the approach of Neuhauser and Koonin(1987). The available direct n-p-~ results, especially those at 208 MeV, exceed the theoretical estimates, even those tnc]uding large internal contributions. In absence of more r e l i a b l e data on the elementary energy. The n-p-~ d i f f e r e n t i a l cross-sections 320 H. Nifenecker and J. A. Pinston i i i ; ~ ~,. 140~V i IO0 lO-Z \\ i ""-. I0-' l lO-e 10 "1 I I I I I I ETIM~I Fig. 4.5 Comparison of calculattons(Remtngton,lg87a),wtth data(Edgtngton,1966) At the top, are displayed the values corresponding to the d-p-~ reaction. The short-dashed l i n e corresponds to a semi-classical fomula for free nucleon-nucleon scattering. The long-dashed 11ne is the same, inc]uding Fermi morton of the neutron. The solid ltne is the same mu]ttplied by 2. The dot-dashed ltne corresponds to the use of the formalism of Neuhauser(1987) for the elementary n-p-~ process. The lower curves are the corresponding values for p+C and p+Pb. The ftgure was taken from(Remtngton,1987). High Energy Gamma-Ray Production 321 process, i t ts interesting to examinewhat can be learnt from the p-nucleus-~ reactions. TABLE 4.1 Double differential cross sections for measuring a neutron at angle e, and proton at ep after gammaemission, in ~arns/sterad. 2. En Bnte p Experimental 1208 30,30 " 35,35 " 38,38 " 40,30 " 45,30 130 23,20 " 26,20 " 29,20 " 38,20 " 23,32 " 26,32 " 29,32 " 38.3Z results 35±14 57±13 116~_20 114_+44 132±53 47±35 16±.29 35±28 64±24 17+_29 66±29 77±32 116+-21 Er 69. 53. 40. 51. 41. 55. 52. 50. 40. 46. 44. 42. HM86 OBE BS HJ Semi-Classical Remington,1987 Baier,1969 Brown,lg73 Brown,1973 Na kayama, 1988a 34.8 14. 46. 34.1 34.6 43.1 44.7 44.0 38. 60. 64.9 96. 71.4 69.8 92. 58.8 69.9 48. 72. 94.2 121.0 105. 128. 38.5 40.1 8. 36. 42.0 10. 40.4 34. 13. 44.4 42.8 32. 26. 53.9 56.2 32. 8. 23.3 23.5 11. 26.5 26.6 15. 30.6 30.6 55.2 55.6 44, 5.HIGH ENERGYGAN4A-RAYSPRODUCTIONIN PROTON-NUCLEUSREACTIONS. Informatlons on the pn~ elomentary process can also be deduced from proton-nucleus reactions. In t h i s case the phase space problem is much simpler to solve than f o r a heavy ion c o l l i s i o n and thus, d i f f e r e n t theories of the pn~ elementary process can be tested. Exoerlmental Data on Proton-Nucleus-G~rmaReactionl are also very poor. In the early experiments of Wilson(1952) and Cohen(1963) the order of magnitude of the photon cross section have been measured. More complete data have been reported by Edgington and Rose at 140 MeV(Edgington,1966) In this work the photon production cross sections were measured for a wide range of targets and angles. Someof the spectra are displayed on Fig. 4.5. Several important results were obtained. The shapes of the spectra were similar fo~ targets ranging betweendeuterium and lead. The scaling of the cross-sections followed a law. The angular distributions were peaked forward, and were AI~ similar for p-d-~ and p-C-~ or p-O-~. All these characteristics led the authors to conclude that the origin of the photons were individual n-p collisions, as suggested, previously by Beckham(Ig62). Very recently, new results have becomeavailable concerning gammaemission induced by 72,168 and 200 Mev protons colliding wlth several target nuclei(Kwato,lg~a; Pinston,lg~).In the following we report on these recent experiments. The 72 MeV experiment was performed at the SIN Philips cyclotron.The energy spectra for the p+Au reaction, measured at three different angles, are reported in Fig.5.1a. For this reaction we have also plotted in Fig. 5.2a the angular distribution in the frame of the nucleon-nucleon c.m.. In Fig.5.3 a comparison is madebetween the spectra of different targets measured at eL.~ - 90" and 150". The scaling factor, NT A~I/3 , used to normalize the differential cross se~ion of the photon production for different targets was that suggested by Edgington(1966) I t is apparent in Fig. 5.1a that the energy spectra for the p + Au reaction are "harder" for forward than for backward production. This difference is an indication of a photon emission 322 H. Nifenecker and J. A. Pinston from a moving source. The source v e l o c i t y deduced from the contour p l o t of the tnvartant photon cross section versus the transverse photon energy and the r a p i d i t y is close to the nucleon-nucleon c.m. v e l o c i t y . In t h i s midraptdtty frame the shape o f the energy spectra measured at d i f f e r e n t angles are almost i d e n t i c a l as observed in Ftg.5. l b . Table 5.1 - Source v e l o c i t y at 168 HeY bombarding energy. TARGET C A1 Cu Ag Tb Au mean value ET < 85 MeV Er > 85 MeV 0.28 0.30 0.24 0.20 0.27 0.22 0.28 0.20 0.18 0.17 0.19 0.14 ± ± ± ± ± ± 0.03 0.03 0.03 0.03 0.03 0.03 0.25 + ± ± ± ± ± 0.05 0.05 0.05 0.05 0.05 0.05 0.19 The experiments(Pinston,1988) at 168 and 200 HeV were performed at the Orsay Synchrocyclotron. Targets of C, A1, Cu, Ag, Tb and Au were used.Since one of the aim of the experiment was to compare absolute cross-sections, both with the previously measured ones(Edgtngton,1966) and with c a l c u l a t i o n s , the beam i n t e n s i t y had to be determined on an absolute basis. I t was monitored by a Faraday cup c a l i b r a t e d by two d i f f e r e n t techniques. In the f i r s t one an A1 t a r g e t was bombarded with protons f o r a fixed amount of time . The Z4Na v - a c t i v i t y was measured o f f - l i n e and the proton number impinging on the t a r g e t was deduced from the known cross section of the 27Al(p, 3pn) 24 Na reaction(Yule,1960) . In the second experiment the proton number was deduced from a proton-proton s c a t t e r i n g measurement performed at 30 ° in the laboratory frame. For t h i s purpose we have subtracted the normalized counting rates obtained f o r a CH2 and C t a r g e t respectively.The proton i n t e n s i t i e s deduced by these two methods agreed between themselves and with that deduced from the Faraday readings w i t h i n 20%. The energy spectra f o r the p + Tb reaction at 168 MeV and f o r three d i f f e r e n t angles are reported in Fig. 5.4. Again, the spectra are "harder" f o r fonvard than f o r backward production. The source v e l o c i t y at 168 HeV was extracte~ from a two dimensional p l o t of the tnvartant photon cross section versus the r a p i d i t y y - ~ Ln ( ( I + c o s e ) / ( I - c o s e ) ) and the transverse photon energy E1- Ey stnO, where e is the photon Observation angle in the laboratory frame. We have found ~ h a t t h e r a p i d i t y d i s t r i b u t i o n s , f o r the d i f f e r e n t targets measured, are nearly i d e n t i c a l . The source v e l o c i t y deduced from the centrotd of the r a p i d i t y d i s t r i b u t i o n , is close to the nucleon-nucleon c.m. v e l o c i t y ~ - 0.28 f o r photons below 85 MeV while i t is s u b s t a n t i a l l y smaller than t h i s value f o r photons above 85 HeY (see Table 5.1). This can be understood, in the frame of a nucleon-nucleon model of photon production, since high-energy gamma-ray requires a c o l l i s i o n of the beam proton with a t a r g e t neutron having a high v e l o c t t y opposite to the beam. This tends to slow down the nucleon-nucleon center of mass, in the laboratory frame. Table 5.2 shows the t o t a l cross sections f o r bremsstrahlung production with photon energies greater than 40 HeY and measured at 168 and 200 HeY r e s p e c t i v e l y . F o r proton-nucleus reactions at 168 and 200 HeV one must consider the c o n t r i b u t i o n due to photons coming from the decay of the neutral pions. At 168 MeV, t h i s c o n t r i b u t i o n was estimated to be less than 9% f o r C and 24% f o r Au r e s p e c t i v e l y , of the t o t a l cross section f o r E~ > 40 HeY. At 200 MeV bombarding energy the pton background represents a c o n t r i b u t i o n of 46¢ f o r C and 59% f o r Au r e s p e c t i v e l y , of the t o t a l cross section f o r E~ > 40 MeV. The cross sections reported in Table 5.2 are corrected f o r the neutral pton c o n t r | b u t i o n at 200 MeV. High Energy Gamma-Ray Production 100¢ .... i .... ILl Be r .... P*Av i .... i .... (T2 i,~v) I • o 30* "~" . . . . • e I t t i . . . . i ,o ° , "' ,o l''' t.4b. tt . . . . i . ,o . . . . . . ~ .... I ; EI(MeV) ,, - f P*Au r ? 2 f~v) c.~. .............. , ......... 1.... t 1ooo * *~ L tr e ~ . . . . J . . . . i 30 . . . . h ~0 . . . . ~0 E r ', L . , , I L , , , , GO ~0 (NeVl Fig. 5.1 (a) Energy spectra of high energy photons for the p+Au reaction at 72 HeV, measured at ot,h =30",g0" and 150". (b) The same spectra transformed in The nucleon-nucleon c.m. frame. The solid curve is the result of a theoretical calculation at Oc,b = 90". P*A. r 7 2 ~..v) ExP. .o c r,) Ioo0o" 3 0 - ~ 0 ~,v i o 20 ~o 60 80 ec Eooo ~oo0 1oo 12o 1~o i 1~o .m. P*A~ t?2 ~ w r~o. ~vl 3o-~o c 2i_. ~.1,1 2o o ~o .,.,.,., Ko io ec. Ioo 12o . ' ~,.,J o ,~6o m. Fig. 5.2. Experimental (a) and theoretical (b) angular d i s t r i b u t i o n in the nucleon-nucleon c.m. frame for the p + Au reaction at 72 NeV. PI~--K* 323 324 H. Nifenecker and J. A. Pinston Table 5.2 - Photon cross sections and P_ values at 168 and 200 HeY f o r F~ > 40 MeV. The p + d cross sectto~ comes from Koehler(]967). Ep- 168 MeV Ep- ]68 MeV Ep- 200 HeY Ep- 200 HeY TARGET oR <Nnp>a ~ (mb) (mb) C A1 Cu Ag Tb Au d 210 158 450 344 750 588 ]ogo 865 1 6 2 0 1316 1 7 6 0 1436 60 45 °T(~b) 90 221 36] 606 806 911 ± ± ± ± ± ± 9 22 36 61 81 91 ~xl~ Prxl04 5.7±0.6 6.4±0.6 6.1±0.6 7.0±0.7 6.1±0.6 6.3±0.6 155 ± 32 9.8±2.0 ]049 +_ 208 12.1±2.4 910 ± 269 23+_4 6.3±1.9 5.1±0.8 The characteristics of the angular distributions suggest, as in the case of nucleus-nucleus collisions, that the hard photons are mainly produced in p-n collisions. As mentioned above, this mechanism was already suggested by Beckham(1962) and Edglngton(]966). Under this assumption the ~ production cross sections can be written as : o~ " o, Pn P~ (5.1) where aB i s the t o t a l reaction cross section (Bauhoff,1986; Hess,1958) Pn is the p r o b a b i l i t y f o r the incident proton to c o l l i d e with a t a r g e t neutron and P~ is the p r o b a b i l i t y to produce a photon in a single n-p c o l l i s i o n . From experimental data (Hess,1958) %~3a~ and then : Pn " o ~ N /(Gnp N + on Z) - 3 1t/'(3 N + Z) (5.2) where N and Z are the neutron and proton number of the t a r g e t . As expected from the f i r s t c o l l i s i o n hypothesis the P_ values deduced from these measurements and reported in Table 5.2 are almost Independent o f ' t h e t a r g e t and p r o j e c t i l e combination. Of course the 200 HeV data are less precise than the 168 MeV data due to the neutral pion subtraction. In Table 5.2 we have also reported the previous p + d measurement of Koehler(1967) at 197 MeV . In t h i s work a quasi free nFY cross section o - 23 ~b, was deduced from an exclusive experiment. The comparison between the deuterium and the heavier targets we have measured at 200 MeV is not s t r a i g h t forward. For deuterium the maximum a v a i l a b l e energy in the c o l l i s i o n , f o r photon production is 137 HeV while i t is about 207 HeY f o r the heavier t a r g e t s , f o r which the r e c o i l is almost n e g l i g i b l e . A smaller Pr value is then expected f o r the former reaction. With these l i m i t a t i o n s in mind one can conclude that the agreement between the two experiments Is b e t t e r than ] . 8 ± 0.4. In Fig. 5.5 we have compared our data at 72 and 168 MeV with the previous measurements of Edgington and Rose(J966) at ]40 MeV . For thts purpose the photon production cross-sections are displayed as a function of the reduced v a r i a b l e E ~ b m , . Using t h i s v a r i a b l e we see that the curves corresponding to our 72 and 168 HeY measurements l i e almost on top of one another, except f o r the highest energies. In contrast the curve corresponding to the measurement of Edgtngton and Rose l i e s very much below our data. One can conclude, that the cross section measured f o r E~> 40 MeV f o r the p + Au reaction at 140 HeY is underestimated by a f a c t o r 3.3] ± 0.33. This f a c t o r is probably related to the difference between the cross sections measured f o r the r a d i a t i v e process d(p,~)3He, a - 0.5 ± 0.2 pb and o - 1.35 pb r e s p e c t i v e l y found by Edgington and Rose(Edgington,1966) and in the recent measurement of Hugi(1984). I t is i n s t r u c t i v e to compare the c h a r a c t e r i s t i c s of the p-nucleus-~ process to those of the nucleus-nucleus-~. While the spectra were very nearly exponential, in the nucleus-nucleus case, departure from the exponential shape are noticed at the higher energy end of the proton-nucleus spectra. This is clearly due to the lower energy available in the l a t t e r case. The departure of the exponential shape occurs r e l a t i v e l y e a r l i e r at higher proton incident energies, as can be seen on Fig. 5.5, where a comparison is made, in a reduced plot, between the 72 and 168 MeV data. This effect may be attributed to a decreased influence of the Fermi momentum at the higher energy. For ratios E~/E..x between 0.2 and High Energy Gamma-Ray Production i .... i .... i .... P * X t ? 2 MIV) 100,0¢ I0.0( i .... Leb. I : l | | i i l l l 325 1 .... 1.-, e-150" e t.o¢ • cv > o . 1o ~o. . . . ,~. . . . 20. . . . .,o. . . . ~ .... 4o' ~ E, ( H E Y ) -% Z ~o . . . . i i .... P*X ('?z Mev) . . . . i .... t.l,. • o e lO.OC 1.o¢ o A. • ¢v • ~ ' ' " ~ 1 .... i,.. ~90 ° ! 1 .... £ol . . . . 501. . . . E s ( GOI' ' ' "/o'/ . , MtV ) FLg. 5.3 Energy spectra for the p + (C, Cu, Ag, Au) r e a c t i o n s , measured at e L,b " go" and ]50" and normalized by the s c a l i n g f a c t o r N, Ar"l/] • At ~L~ - go" an increase of the photon y i e l d i s observed t o t p + C r e a c t i o n compared to the r e a c t i o n s on h e a v i e r t a r g e t s . A I ' I ' I ' I ' I ' I ' 1 ' 1 Q P+Tb > 1000.0 n Ep=168 M,V . C v Z o 100.0 u bJ (/) co o no ooo 10.0 I , I lttl ., , I , I L I , i . 1 , I 20 40 60 80 100 120 140160 E~ ( Me V ) Fig. 5.4 - Example of photon spectra : p + Tb r e a c t i o n at 168 HeY. 326 H. Nifenecker and J. A. Pinston 0.6, the spectrum shape seems to be independent of the incident energy and scales l i k e E... , or, equivalently, l i k e the incident energy. This is similar to the observed dependence-of the inverse slope of the spectra, tn the nucleus-nucleus reaction(Fig. 2.4). However, in the proton induced reactions, the cross-section increases almost l i n e a r l y with incident energy, while i t remains almost constant in the heavy ion case. I t is l i k e l y that this difference is related to the larger influence of Pault blocking, in the l a t t e r case. Furthermore, the comparison between the two cases is only partly meaningful, since the proton data considered range from 72 to 200 HeY incident energies, while the heavy-ton data correspond to energies between 30 and 89 AHeV. We have noted that there are evidences for an increase of the photon production cross-section between 89 and 124 MeV in the Xe+Sn reactton(Tam,1g88). Between 30 and 65 HeY gamma energies, and for 72 MeV incident proton energy, the photon spectrum is very nearly exponential, with an inverse slope of around 14 ReV. lhts is s i g n i f i c a n t l y less than the value expected from heavy-ton reactions at a similar incident energy per nucleon. Indeed, Fig. Z.4 gives a value of the inverse slope Eo around 22 HeY for 72 AMeV incident energy. This lower value, in the proton-nucleus case, may be explained by the absence of Fermi motion in the p r o j e c ti l e . I t is d i f f i c u l t to compare the absolute probabilities of photon emission in the proton and heavy-ion cases, since this quantity depends upon energy, in the f i r s t case. In order to do so we approximate the gamma spectrum observed at 72 MeV proton energy by an exponential with inverse slope 14 HeY and d i f f e r e n t i a l cross-section of 1 ~barn/HeV/sterad. at e-g0" and e~qm32MeV (See Fig. 5 . ] ) . The d i f f e r e n t i a l probability for photon emission is, then, using uatton (5.2): dSG 3NT+ AT e]Z/14 14 1.4 ]0 "4 . p, This value is to be compared to those displayed in Table 2.1 which are close to I. 10.4 . The closeness of the two figures should not be taken too seriously, however i t shows that nucleon-nucleon collisions may be the origin of high energy gammaray production in nucleus-nucleus reactions. The angular distributions, in the nucleon-nucleon center of mass frame, displayed in Fig 5.2 do not peak at 90", as do those displayed on Fig 2.8. Rather,the maximum is around 80". This can be explained as due to a Doppler s h i f t of the gamma-rays emitted by the proton, even In the n-n frame. From equation (3.17) i t is seen that the sin zecos e term induces a shift of the maximumof the angular distribution to smaller angles. Applying this equation to a 72 MeV proton one gets an angle, for the maximum, of 77", in qualit at iv e agreement with the experimental angular distribution. The r e l a t i v e amplitude of the antsotropic component is around 30%, again, tn q u a l i t a t i v e agreement with the values observed in nucleus-nucleus reactions. In summary, we have seen that proton-nucleus-~ and nucleus-nucleus-~ reactions share many characteristics. However, the absence of Fermi motion and Pauli blocking in the proton projectile makes a detailed comparison rather uncertain. This ts why the recent measurements of Ltoht ton-Nucleus-~ orocesses, by the MSU group,(Tam, lg88b) are interesting. Fig. 5.6 shows energy spectra observed at 90" for deuterons and alphas of 25 and 53 AMeV bombarding Carbon and Lead targets. I t is clear, from the figure, that the relevant quantity is the beam energy per nucleon rather than the total energy of the beam. This agrees with f i r s t c o l l i s i o n models, and not with thermal models. I t also agrees with the trends observed with heavier projectiles. I t is also seen that the shape of the spectra, above 20 MeV, are very nearly exponential for alpha projectiles. They diverge from the exponential, at the higher energy end, for the deuteron projectiles, very much ltke what was observed with protons. For He projectiles, the inverse slopes vary from around 6.5 MeV a~ 25 AMeV to around 12 MeV at -beam 53 AMeV. This is close to, although s l i g h t l y smallerthan the --i---systemattcs observed for heavier projectiles. The inverse slope of around 9.5 MeV for-the deuterons of 53 AMeV ts sig n i f i c a n t l y smaller than the heavy projectile systematics, and close to the value of 10 MeV, extrapolated from the 72 MeV proton data. The values of the probability of photon emtsston per p-n c o l l i s i o n P~ are close to 10"4 for alpha projectiles, at both incident energies and for both targetS. This is, again, very close to the values observed with heavier projectiles. On the other hand, the values of Pr are close to 2.10 -4 for deuterons. I t seems, therefore, possible, that for hydrogen isotopes, the gammaemission probability per p-n c o l l i s i o n is higher than for heavier projectile. Wether this might be due to a small or vanishing Fermi motion and Pauli blocking in the p r o je c t ile, or to interference effects is an open question. High Energy Gamma-Ray Production L " ' n ~ ' .... I .... , ......... I .... [ .... 327 , .... ' ~ E I000'01 + + i's*~l'~ "t~. + P'^" ¢ Z r-1 {-u ~ooo~ I ' ) +f+l i~ '°'°I u) u') 0r,(_} t+ ,.+i. '+++ " i .... i ..i .... i .... i .... [ .... :.... -I ,.. ; 0.2 0.3 o.~ o.~ O.S 0.7 o.e o.9 '~o Es/Ep Fig. 5.5. Comparison, in a reduced plot, of our results at 72 (squares) and 168 HeV (triangles) with those of Edglngton and Rose at 140 HeY (crosses). The system studied was p + Au at 90 ° . 100 r ! I i ~ II 10-1 : 10_2 r- 10-3 F X E/A=,53 MeV sH+Pb x10 • E/A,-53 U e V / H e + P b • •m ~ . • E/A=25 M~),m ' _ • li'a 10_4 .~ 10-5 • *0 r x IN mm = m_ • X xllelmm XX • E/.=53 ,..v ~+c" " . 10_8 -e • 10-7 0 E/A=53 -~li_ XT1 // ""."" ueV ~Te+C .:v • | . . . . 50 E7 xl0 MeV 4 H e + P b xl0 mm|o0nt°° o • I • ,J, J= T. 100 (MeV) Fig. 5.6 Energy spectra of high-energy gamma rays at 90" for the reactions zH+C,Pb at 53 AMeV, and He+C,Pb at 53 an~25 AHeV. From (Tam,1988b). 328 H. Nifenecker and J. A. Pinston Fig. 5.7 shows angular distributions as observed in the nucleon-nucleon center of mass frame. In a l l cases a dipolar component is present. For the heavier targets, i t is small and comparable to that observed with heavier projectiles. On the other hand, i t is very strong for the Carbon target. This is reminiscent of the trend observed in the Lt+Li reaction(Tam,]g88), which we have discussed in section 2. This decreased antsotropy for heavier targets may be due to secondary c o l l i s i o n s . I t ts, however, surprising that these collisions do not show up in the cross-section systemattcs. Another p o s s i b i l i t y would be that of a direct capture contribution for the ltghest sysytems. Note that, in the proton case, also, at 72 NeV, the angular d i s t r i b u t i o n observed with a carbon target seems peculiar. Thts is certainly a point which deserves further studies. In conclusion, we have seen that there is a general agreement between proton-gamma, l t g h t - and heavy-ton gamma reactions with a smooth transition from the lightest projectiles to the heavier ones. The most s i g n i f i c a n t differences between the Z - ] projectiles and the heavter ones, starting with alpha particles, can be attributed to smaller total avallable energies as well as Fermi and Pault effects. Theoretical Calculations of the Proton-Nucleus-Gamma Process. Following the approach f i r s t taken by Beckham(1962), the photon production rates in proton-nucleus c o l l i s i o n s are deduced from the elementary p-n-~ cross sections, under the assumption that the proton makes a c o l l i s i o n with one of the target neutrons. The neutrons are affected by the Fermi motion and the Pault exclusion principle is taken into account in the final state(Nakayama,lg86; Bauer,lg86; Remington,lg87; Kwato NJock, lg88a). The contribution to the radiation of the acceleration produced by the mean nuclear f i e l d has been esttmeted(Beckham,]g62; Nakayama,lg86) and found to be sma]l as compared to the nucleon-nucleon c o l l i s i o n s . Two approaches have been used, in order to take tnto account the Fermi motion and Pauli Blocking. The most commonly used(Beckham,1962; Nakayama,1986; Kwato Njock, lg88a) consider p a r t i c l e - p a r t i c l e collisions between the incident proton and the target neutrons, characterized by t h e i r Fermi momentum d i s t r i b u t i o n . The other approach(Remtngton,lg87) considers the tnctdent proton as an exctton inside the target nucleus, and follows tts decay vla a Boltzmann equation. In both cases gamma emission may occur when a nucleon-nucleon c o l l i s i o n takes place. One defines, therefore, a probability for gamma emission per c o l l i s i o n . Thts probability is not the free n-p-~ probability, due to Pauli blocking. Further, the incident energy of the proton is increased by an amount equal to the nuclear potential depth minus the Coulomb potential. The Pault blocking is most simply taken tnto account tn the ~qltzmann master eouatlon aooroach(BHE)o In this case PTm~ (cy,(p,Cn) i p~r~ ,, ( ~ ) ~ where B are Paull blocklng factors, c~,c~ are the final energies for an elastic scattering event. Thus ¢;+ c~- % + (n" ~; and c; are the final energies when a photon Is emitted. Therefore, (;+ c~- ~p+ On-( T. The blocking factors also include the level densltles. For example, B ~ o , % I " go(c~)g,(%)(]- n o [ ( o ] ) ( ] - n , [ ( , ] ) , where the g's are stngle particle level densfttes-and the'n's are occupational ~umbers. The evolution of the occupational numbers is given by the Boltzmann Master Equation: gxt ~ I f((p,n)- n xI.~-~. w x -r ~E~ o) ~x~y ~ x, y/ ~ xy i j k L (s.3) Y where x,y specify the nature of the particle, f l ( p , n ) is the injection term, which, for a single proton reduces to a delta function at t-O. ~ t is the transition rate between two states of the scattering nucleons pair. Remington and Blann(1g87) define ~kL . Gxy(Vv,I )Vv,I PW wtth vv,j " ~ ((j+ el)" Gxy IV) ts the free nucleon-nucleon cross-section for relattve velocity v. ml-~ is the escape rate to the continuum state u. I t is obtained by applying the detatled balance to the capture of a nucleon x with energy ( u t n the continuum. The photon production rate is now equal to: High Energy Gamma-Ray Production dZ° = dErd t °n ~ pT((v) ~ ,j~t g~gylB(( x j , k,(ly) ljkl 329 (5.4) Remington and B1ann(1987) have applied this formalism to the p-nucleus-~ data of Edgtngton and Rose(1966). The results of t h e i r calculation is compared to the experimental data in Fig. 4.5. I t ts seen that, provided the elementary cross-section given, tn this case, by the p-d-~ reaction ts we1] reproduced, the Raster equation approach allows a good reproduction of p-nucleus data. The s i m i l a r i t y of the spectra, as well as the v a l i d i t y of the N A"1/3 sca]tng rule, indicate that multiple scattering may not contribute very much to the photon yteld. The I~IE has several attractive features. I t is simple and easy to extend to the nucleus-nucleus case, as wt]l be seen later. I t takes, naturally, into account the successive c o l l i s i o n s , up to complete thermaltzatton. I t allows the study of the correlations between gamma and preequtltbrtum particle emission. In principle, t t conserves energy at each step. Note, however, that i t is not immune to spurious energy effects. Indeed, after a f i r s t scattering a situation may arise where, for examp]e, the bottom 1eve] has a small, but f i n i t e probabt]tty to be empty while the entrance energy level ts not e n t i r e l y depleted. I t is, then possible that a photon would be emitted in a process where the particle with the i n i t i a l energy ct+V has 0 ftnal energy, and therefore, the photon energy c_ could be ]arger than (~+V wht]e the i n i t i a l available energy was only ( , . Therefore" one should be cautious, using the ~E for very large gamma energies, close to the absolute threshold. This also appltes, of course, to pton production. This d i f f i c u l t y is encountered in a]l models where the information on the system, at a given time, is summarized by a set of non-integer occupation numbers. Such are BUU type calculations, thermal models, and Fermi gas calculations. The ~E equation does not consider the momenta of the particle. This is why, the entire incident proton energy can be released in the f i r s t scattering event. No energy needs to be stored tn the center of mass motion. In prtncip]e the ~E equation does not provide angu]ar distributions. However, Remington and B1ann(1987b) have reproduced angular distributions by adding an additional ansatz to the ortgtna] ~E model. They distinguish f i r s t and subsequent co]ltsions. First co]ltsions contributions to the angu]ar d i s t r i b u t i o n are obtained from a nucleon-nucleon semi-classical calcu]ation. Later c o l l i s i o n s are assumed to give tsotroptc distributions in the g]obal center of mass. This procedure amounts to take partly into account the Fermi oas model aooroach. In thts case, the target nucleus is schemattzed as a sphere, both in configuration and momentum sphere. In the simplest approach(Nakayama,1986; Kwato NJock,lg88a) only f i r s t c o l l i s i o n s are considered. The consideration of secondary c o l l i s i o n s requires more elaborate treatment of the momentum d i s t r i b u t i o n . T h i s has been done in the frame of the BUU theory(Bauer,lg86), which w i l ] be discussed later. The photon production cross-section is then given by: dZo - ~ (s.s) Note that the probabillty for photon emission depends, this time, not only on ~ but on the flnal and Inltlal proton momenta. This is necessary in order to obtaln angular distributions. Thls formallsm has been applied by Nakayama and Bertsch(1986), using the seml-classical expression of the photon production cross-section. Their result Is compared to the p+Be data of Edglngton and Rose(I966) In Fig. 5.8. Nakayama and Bertsch(I986) summarize the result of thelr first collislon calculatlon for the p-nucleus-~ process in a seml-emplrlcal expression: d2~ - 2.5 I0"~, ( ~ - E')2 dEvdn cot ~ Er (5.6) I t is interesting to note that the cross-section for a gamma-energy equal to a fraction of the incident beam energy is a constant. This Is in agreement with the energy scaling we used In Fig. 5.5. The contribution given by equation (5.6) can be compared with a simllar 330 H. Nifenecker and J. A. Pinston ''1 10-1 .... [ I ...... , , , , [ i , , m , E / x = 2 5 geV~I-Ie E/A-S3 UeV '~I x Pb xlO x Pb • Zn x5 • Zn • Cx5 i,C X ~ , [ , i , ,[11 E/A=53 ureV ¢FIe x Pb • Zn eC 10_2 b 10-3 60 120 120 60 60 1gO e==(deg) Fig. 5.7 Angular distribution of high-energy gammarays with energy>30 MeV in the nucleon-nucleon center of mass frame, for the reactions He+C at 53 and 25 AMeV and zH+Cat 53 AMeV. 100 - , , , , =0 100 i >. ¢ I~=,- 140 MeV e,,=0' 1=0 4O =0 "1=0 (. (MeV) Fig. 5.8 Comparison o f the p+Be data o f Edgington(1966) w i t h Fermi gas c a l c u l a t i o n s using a s e m i - c l a s s i c a l n - p - ~ cross-section(Nakayama,1986) High Energy Gamma-Ray Production expression obtained by Nakayama and Bertsch(1986) contribution. This writes, at 90": i . ,0,o./¼) 331 and concerning the nuclear potential I, zl' (5.7) BUU type calculations(Bauer,1986) have, also, been applied to the data of Edgtngton and Rose(1966). The results of the calculations are compared to the experimental data on Fig. 5.9 . The agreement is, again, satisfactory. From the preceeding i t appears, therefore, that, provided the elementary p-d-~ or p-n-~ are well reproduced, i t is possible to reproduce, also, the proton-nucleus-~ data, in the frame of nucleon-nucleon c o l l i s i o n s models. I t does not seem that m u l t i p l e c o l l i s i o n s play an important r o l e in determining the photon production rate. Since we have seen that new measurements of the proton-nucleus-~ process seem to contradict the older measurements of Edgtngton and Rose, i t was important to compare the new data to calculations s i m i l a r to those Just mentioned.We have, therefore, compared the p-nucleus-~ results at 72,168 and 200 MeV with a Fermi gas type calculatlon(PJnston,1988). Two d i f f e r e n t assumptions were made concerning the elementary n-p-~ cross-section. In the f i r s t we made use of model of AshkJn and Marshak(lg4g), described in section 4. We have used the same Yukawa potential as Ashkin and Marshak, which reproduces s a t l f a c t o r t l y the nucleon-nucleon e l a s t i c scattering t o t a l cross-section. I t has equal strength of ordinary and exchange forces : e-Xr 1 + P. V(r) 2 gl r with X"l - 1.18 . 10"13 cm, gl " 0.280 t¢ and g3 " 0.404 ~c, where P. is the Majorana operator and gl and g3 r e f e r to the single~ and t r i p l e t s t a t e s ' r e s p e c t i v e l y In thts n o n - r e l a t i v i s t i c model, radiation from charged meson exchange ts neglected. I t gives photon cross-sections which are very s i m i l a r to the semi-classical ones. The result of our calculation for 140 MeV p-nucleus (Edgtngton,lg66)colltsions shows a good agreement with the much more sophisticated c a l c u l a t i o n of Bit6 and colleagues (Bir~,I987). These authors used a neutral meson o and ~ model to evaluate the pn~ elementary process and the Boltzmann-Uehltng-Uhlenbeck equation to f o l l o w the p-n c o l l t s t o n a l history. These two models reproduce equally well the proton-nucleus data at 140 MeV. In the second approach we have used the formulation given recently by Nakayama(lg88a) . Here the contribution of the internal ptontc current is e x p l i c i t l y accounted for and is shown to be several times l a r g e r than the pure external contribution. For t h i s second calculation we have used the d i f f e r e n t i a l cross sections computed by Nakayema f~r n-p c.m. energies of 50 and 200 MeV and found that for the same value of the r a t i o "T ~--the cross sections are -p proportional to E~~ , where Ep is the beam energy. This simple r e l a t t o n was supposed to be v a l i d for the c.m. energies range 50 • Em• 200. This second computation has to be considered as a crude approximation. The results of the two calculations in the reaction p + Au at 72 and 168 MeV are shown together with experimental data on Fig. 5.10. Both calculations and measurements are d i f f e r e n t i a l cross sections at 90 ° in the laboratory frame. I t can be seen that the calculation which includes the internal piontc contributtons(Nakayama,lg88) gives a much better agreement with experiment than the most commonly used one. In Fig. 5.2, the experimental and calculated angular d i s t r i b u t i o n s , for the p+AU reaction at 72 MeV are displayed. The c a l c u l a t i o n , In t h i s case, used the Ashktn(]g4g) formalism, excluding internal contributions. I t is seen that the c a l c u l a t i o n overestimates the antsotropy of the angular d i s t r i b u t i o n . On the other hand, the internal contributions is expected to gtve an almost tsotroptc angular distribution(Brown,]g73), in agreement with the observational trends.Nakayama(lg88a) has, also, used his formalism for a comparison with the data of Edgington and Rose(1966), concerning the p+Be reaction at ]40 MeV. He makes use of the Fermt gas model of the nucleus, as has been described above. In order to account for the experimental results, Nakayama has to consider the Fermi momentum as a parameter. S i m i l a r l y he considers the presence or the absence of the nuclear mean p o t e n t i a l . Two Fermi momenta are considered, 1.36 and 0.65 fm"l corresponding to lO0"k and 10~ of normal nuclear matter density respectively. Fig. 5.11 shows the results of the c a l c u l a t i o n for four d i f f e r e n t combinations. When there is no nuclear p o t e n t i a l , the t o t a l ava|lable energy for gamma H. Nifenecker and J. A. Pinston 332 1 ' I m I ' I At (.100) ~> OJ 100 10 .Q . I,I ~ ' ~ ~ - -_-.- • "0 "0 0.1 I 0.01 i I 4.0 60 j I I I 80 10( [MeV] Fig. 5.9 Comparison of the data of Edgtngton(1966) for p+C,Al,d with a BUU ca]cu]atton using the semi-classical expression for the n-p-~ cross-section(Bauer,1986). A 'r .... i .... 1''''1''''1 .... fi'm P+Au E#=72 HeY I ' ] ' I " I ' T ~ = I ' [ ' HeY Ep=168 ~1000.0 c v z loo.o. p0 W (./3 \ lO.g O rY U 1 , I • I . ! . I i I i I i 20 E~(MeV) 40 gO 80 100 120 140 EB.(MeV) F i g . 5 . ] 0 . Comparison of the experimental spectra (squares) with two calculations. The f i r s t (dashed) does not include internal pton contributions. The second (continuous) does include them. Experiment and calculations refer to p+ Au reaction at 72 and 168 MeV. High Energy Gamma-Ray Production 10 -4 " " " " I . . . . I . . . '1 . . . . . i . . . . 1 333 . . . . Tlab= 140 MeV 7 0=90 ° 10 -5 10-6 %'... ~ C nO ~ • . % nO i0 -v "0 10 -8 0 20 40 60 80 100 120 (MeV) Fig. 5.11. The proton-Nucleus bremsstrahlung p r o b a b i l i t y rate in the l a b o r a t o r y system at the incident energy T -140 MeV and photon emission angle lab -1 e=90". The dot-dashed l i n e is the r e s u l t at k -1.36fm and no mean f i e l d F 1 p o t e n t i a l , while the dotted curve corresponds to k =0.65fM- and no mean F Field p o t e n t i a l . The s o l i d and dashed curves are the f u l l r e s u l t s f o r -1 k =1.36 and 0.65fm r e s p e c t i v e l y . From (Nakayama,1988a). F 334 H. Nifenecker and J. A. Pinston 100 . . . . ! . . . . I . . . . I . . . . ~ . . . . a . . . . I f~ p> Q} V jn 10 C "c/ 3 T1ab= 140 NeV b 1 l 0=90 ° 1 ...................... /, ] 0 20 40 60 80 100 120 ca (MeV) Fig. 5.12. Comparison between the p+Be results of Edgington(I966) and the Fermi gas calculation using a realistic description of the n-p-~ process as given by Nakayama(1988a). The Fermi gas calculation assumed -I k -I.36 fm and no mean field. F High Energy Gamma-Ray Production 335 emission is reduced by the Fermi energy, due to Pault blocking. Therefore the higher Fermi momentum gives smaller photon production. The Nuclear Potential V adds, to the incident proton, inside nuclear matter, an energy equal to V, and larger than (F. The total available energy for gammaenergy is increased by the proton binding energy, a small quantity. On the other hand, the coupling to the Fermi motion increases the photon production rate, and the higher Fermi momentumcorresponds to higher production rates. In Fig. 5.]2, the experimental data of Edgtngton and Rose(1966) are compared to the result of the calculation of Nakayama(lg88a). To obtain a reasonable f i t to the experiment Nakayamahad to make the unreasonable assumption that the Fermi mo~ntum corresponded to normal nuclear matter density(1.36 fm"1 ) while there was no nuclear mean f i e l d . This strange finding is due to the fact that the experimental data were erroneous. They are underestimated by a factor of about 3 as compared to the new data at 72,168 and 200 MeV.In conclusion, i t appears that the internal charged pion exchangecurrents contribution is Important, even dominant, in the n-p-~ process. The careful study of proton-nucleus, and, even more, of nucleus-nucleus reactions may, therefore, give the p o s s i b i l i t y to examine the in medtum modifications of the ptontc currents. We shall now proceed with a review of the existing theoretical approaches to the nucleus-nucleus-~ process. 6.THEORETICAL OESCRIPTIONSOF pHOTON[MISSION IN NUCLEUS-NUCLEUS COLLISIONS. Many d i f f e re n t theoretical approaches have been advocated, in the recent past, in order to explain the hard photon production in nucleus-nucleus c ollis io n s . We have shown, in the preceedtng chapters, that a rather extensive set of experimental results Is available to compare with theoretical predictions. I t is our feeling that any serious theoretical attempt to explain photon production in nuclear encounters should take into account the main trends displayed by the experimental data. I t is unfortunate that some recent theoretical attempts have used only those few experimental results which could lend some kind of support to them, and ignored the others. This w i l l make the comparison between theory and experiment less thorough than we might have liked to. Nevertheless, we hope that the following w i l l show that such a comparison makes possible to infer the dominant mechanisms responsible for photon production in nucleus-nucleus reactions. I t is possible to distinguish three main types of theoretical approaches . These approaches have also been applied to other types of problems, such as pion production or fast particles emission. We classify these models into collective, thermal and dynamical ones. The CQllective Mod~l~. In the extreme c o l l e c t i v e approach, the nuclei are considered as simple e n t i t i e s scattered in the f i e l d of one another(Vasak,1985; Vasak,lg86).Thts situation was already considered in Section 3. (equations 3.10 to 3.13). I t was shown )hat the angular distributions were sy~trtcal with respect to 90°, only in the half-beam velocity frame, irrespective of the mass r a t i o of the p r o j e c t i l e and target. The shape of the angular distributions depend upon the r e l a t i v e orientations of the position and velocity vectors.For coltnear velocity and position vectors, l i k e in central c o l l i s i o n s , the angular d i s t r i b u t i o n is of the sinZO cosZB type, while i t would be of the stn4e for orthogonal position and velocity vectors. Of course, integration over impact parameters mixes the d i f f e r e n t situations and leads to more complex angular distributions. Examples of laboratory angular dttrtbuttons, for different impact parameters are given in Fig. 6.1. Such characteristics are not in disagreement with the experimentally observed angular distributions, except for the absence of an tsotroptc component. The shape of the v-spectrum, following the classical theory(Jackson,1975), corresponds to the square of the Fourier transform of the acceleration. For the observed exponential shapes of the spectra, one, therefore~expects Brett and Wtgner shapes of the acceleration function ~(t) where T " is the characteristic time of the deceleration of the two (t-t°)z+ T ro nuclei. The acceleration function is characterized by long times which do not seem to be physically sound. The dgc~eration time can be related to a characteristic deceleration S. @TermlS distance d ~ 2 v~=, T where we have made use of the fact that the inverse slope Pw. E was approximately equal to one fourth of the incident beam energy per nucleon. Although 336 H. Nifenecker and J. A. Pinston the magnitude of the stopping distance is not unreasonable, i t s decrease with beam ene~ly is d i f f i c u l t to understand. The predicted scaltng law has a strong dependence upon the nuclear charges. For symmetric reactions, the collective model predicts a aeZ2~ Za~ dependence.Experiment points to a much slower Z5~ one, as shown by the v a l t d t t y of the scaltng law described in section 2. For thts reason, i t seems that the extreme col]ecttve model does not apply to nucleus-nucleus encounters at intermediate energies. In the extreme collective model, Just described, the c o l l e c t i v i t y is a property of the nuclei whtch behave l i k e e n t i t i e s , all nucleons feeling simultaneously the accelerating ftelds, due to a very low compressibility. I f one considers, at the opposite, that the nuclei are mere collections of nucleons interacting independently wtth one another, the electromagnetic f i e l d , i t s e l f , ts a possible source of c o l l e c t i v i t y , due to i t s additive properties. A simplified semf-classtcal treatment of the c o l l i s i o n of nucleonic ensembles was made by Ntfenecker and Bondorf(Ntfenecker,]g85).]t shows that the expected c o l ] e c t t v t t y is small since, as seen tn section 3. positive Interferences between c o l l i s i o n s involving protons from the same nucleus tend to be cancelled by negative Interferences with collisions Involving neutrons from this nucleus. Thts is because the vector potential ts an odd function of the charged particles velocities. As shown in section 3,the emitted radiation is the result of fluctuations in the Individual scattering processes, especially tn the distribution of f i n a l parallel velocities, after scattering of two nucleons. The collecttveness of the radiation, t f i t exists, decreases with gamma-energy. As an example, for central colltsfons, the collective gamma-cross section reads: d3~ dE.,.(~ " a, ~ (Fosln2e + FQslnZe cos28) NT Np 2 F0 - ~--/Zp ~,-t - ZT ~') e 2o~( (6.1) ~z (F~ ~z F° " T Cre - l- -J NTNP 2~ e o, where of is the characteristic stopping time of the projectile. ~R is the distance between the deceleration centers of the proJectt]e and target, respectively. In the absence of a compression zone this quantity vanishes. Typical values of ac are given, for a small projectile, with radius R femts, absorbed into a larger target (Ntfenecker,1985), by oe" R-~-ISt.b • This gives, for example, around 70 HeV for a 84 AHeV C projectile.Similar result~s were also obtatned recently (Heuer,]988) tn a quantum calculation based on the quantum molecular model for nucleus-nucleus c o l l i s i o n s . In this case a remaining collective component was obtained for the lowest photon energies, namely, below 30 Hey. Previous schmattc quantum calculations(Bauer,]g87; Nakayama,lg86) also concluded to the weakness of collective effects tn nucleonic ensembles encounters. Such results Justify to treat photon production in nucleus-nucleus reaction as an incoherent summation of Individual n-p--r processes. The Thermal Models use this assumption. These models can be classifled Into two categories, depending upon the treatment of the time evolutlon of the hot system. The first type of models has been used extenslvely In intermediate energy heavy-lon physics, in order to predict pion and composite particles production (Knoll,]g7g; Das Gupta,]g81; Bondorf, lg82, Gross,1982; Awes,lg8]; Bondorf,]g85; Shyam,]g84; Jacak, lg87). Those are essentlally phase space models. A freeze-out configuration Is selected out for .hlch thermodynamlcal equilibrium is assumed. The relatlve production of partlcles is then governed by the chemical potentlal.Thls High Energy Gamma-Ray Production approach has not 337 been used extensively for photon production(Shyam,Ig86; Grosse,1985).The photon production, then g e n e r a l l y , reads: dSa 2V dETd~l - o, (hc)s E~ Er (6.2) e T +1 Here V is the volume of the participant zone with temperature T, formed with cross-section oR . This expression was written in the emitter frame. For the IzC+12C reaction, at 84 AMeV(Grosse,lgB5), one gets T - 18 MeV, and a value of the product cmV - 37 fermi s. With an averageformation cross-section of 0.5 barn, one gets a particlpaBt average volume of 0.74 fermi 3 , a very small value corresponding to around 0.1 nucleon at normal matter density. Alternatively, i f one would use reasonable values of the number of participants, one would get cross-sections close to two order of magnitude too large. This is probably, a consequence of the small interaction of gamma-rayswith nuclear matter, which prevents reaching equilibrium. The same property makes i t problematic to define a freeze out configuration for photons. Note, also, that the shapesof the black body spectra are not pure exponentials and do not agreewell with the experimental ones.The photon production cross-sectlons predicted wlth this approach have the desirable feature to be proportional to the numberof participants, in qualitative agreementwith experiment. The other features of this type of model are commonto all thermal models and w i l l be discussed later. The other thermal models were inspired by the traditional evaporation theory, applied to a hot participant zone. The participant zone was defined in a variety of ways. Nifenecker and Bondorf(Ig85) assumedan equal number of projectile and target participants. This has the advantage to explain, by construct, the sources velocities and the weak dependence of the spectrum temperatures upon the relative masses of projectile and target. However, this ansatz Is arbitrary and d i f f i c u l t , i f not impossible to j u s t i f y for very asymmetric collisions(Gosset,Ig77). Prakash and colleagues(1987) use the geometrical, clear-cut, participant model(Gosset,1977). Bonasera and colleagues(Ig88) havemodified this model, in order to take into account the interaction between participants and spectators which is expected to play a significant role at intermediate energies. I t is, then, possible to obtain source velocities, which,ln principle, d i f f e r from the half beamvalue. The deviations from this value are not very large. For example, at 44 AMeVBm" 0.151, while the participant velocity Bp,rt " 0.184 for IzC+~Kr and ~ = r t - 0 . 1 3 3 for 197Au+~Kr. These values would correspond to ratios ~??:?, of 3.27, 4.27, and" 2.84 respectively. The experimental values vary, as seen on F i g . a ~ ~rom 3.0 to 3.2.The agreement is, therefore not very good. However, Bonasera and colleagues claim that the source velocities they obtain are very close to the half beamvalue. The models of Nifenecker(1985) and Prakash(1987), d i f f e r mostly in the way they treat the photon production. Nifenecker assumed that the photons are produced in individual nucleon-nucleon collisions, each collision being treated semi-classically, as described in section 3. Schematically, the p h o t o n production cross-section is given by a_a.p • K nip P(E_)T -7 pert where o.K is the hot zone formation cross-section, unip the numberof proton-neutron collisions per time unit, within the hot zone, P(E~) the probability for emission of a gamma-rayin a p-n collision and T=rt the l i f e time of the hot zone. This life-timo is determined by particle evaporation. The expressions obtained by Nifenecker do not comply to the sum-rules derived by Knoll and Guet(]9~), due to the infrared divergence which was not cured in their model. A possible modification of their model, in order to cure this defect, is sketched In section 3. Prakash uses the detailed balance theorem to obtain the photon production rate from the photon absorption cross-sections, estimated for the hot participant nucleus. The photon production cross-section number of evaporation is t h e n given schematically ~st~ by oT- ORFT ~ . v,tep~ is the steps. Since the particle escape width is F,- ~---, one can write that Te Pstepe - - - Tpert - - expression. . Both Only the Prakash and Nifenecker models, t h e r e f o r e lead photon production rates are d i f f e r e n t , to the same ktnd of but they are both p r o p o r t i o n a l 338 H. Nifenecker and J. A. Pinston to the total mass of the participant zone. The particle escape width is proportional to the surface of the emitting system, while, at fixed temperature, the total excitation energy is proportional to i t s volume. Therefore, the participant zone ] if e- t imo is proportional to the radius of the zone.For symmetric systems, the photon production cross-section w i l l be, for both models, proportional to Az. This is faster than observed experimentally, as can be seen in section 2 . In Table 6.1, we compare the experimental data with the predictions of both thermal models. In the table we have treated the theoretical results in the same way as the experimental data to obtain values of P.. As explained In Section 2, thts treatment has the advantage to be insenttttve to the slope of the spectra. Using the scaling described in section 2 i t is possible, from the data displayed on Table 6.1 (Equations 2.8 and 2.9) to reconstruct the theoretical cross-sections. TABLE 6.1 Comparison of the experimental values of the inverse slopes Eo and of the photon emission probability per i n i t i a l p-n c o l l i s i o n with the predictions of the thermal models(Prakash,1987; Ntfenecker,1985). In the Ntfenecker case the experimental value of EowaS used. In the Prakash case, i t was calculated (see text). SYSTEM • s Kr+IZ C E beam Eo Pvx104 P v x l 0 4 PTxI04 Eo MeV/N Experiment Prakash Experiment Prakash Nlfenecker 44 11.7 10.5 0.62 Kr+n't Ag 44 12.5 12.9 0.54 2.9 4.2 ~s Kr+i~ Au 44 12.1 12.7 0.38 3.43 4.47 Ar+TM C 85 25.6 18.5 0.99 7.9 2.3 Ar+27A1 3.8 2.3 85 28.4 19.0 1.04 10.9 2.9 36At+nat Cu 85 29.9 1g.I 0.88 11.9 3.33 ]6 Ar+~,t Ag 85 29.6 18.0 0.93 14.6 3.4 36Ar+Is9 Tb 85 29.8 17.8 0.86 13.g 3.6 Ar+~gzAu 85 28.3 17.5 0.94 14.1 3.6 In the model of Ntfenecker, i t was found that the temperatures, i f calculated from a Boltzmann or Fermi gas model applied to the participant zone, were systematically too low, by around 65%. However, Neuhauser and Koonln(1986), showed that a proper treatment of Fermi motion and Paul1 blocking, as well as a r e a l i s t i c treatment of the elementary n-p-~ cross-sections could improve the situation significantly. In the practical calculation shown on Table 6.1 the experimental tnverse slopes were used as effective temperatures. The inverse slopes (temperatures) of the spectra obtained from the model of Prakash are, a l s o , s i g n i f i c a n t l y smaller than the observed ones, especially at the htgher energies. The modified version of Bonasera(1988) improves the situation, in that respect. Both models overestimate s i g n i f i c a n t l y the photon production cross-section, as seen in Table 6.1. The overestimate is larger for larger systems, tn agreement with the mass number dependence discussed above. The origin of this overestimate is easy to understand in the frame of the model of Ntfenecker. I t reflects the average number of proton neutron collisions which a given proton suffers within the f i r e b a l l lif et ime . This number ranges between 5 and 8. Decreasing the l i f e t i m e of the hot system would decrease this number and i t is seen that, in order to agree with the experimental results i t would be necessary to reduce the number of p-n collisions per proton to about one. The thermal models should have, also, great d i f f i c u l t i e s in explaining two important experimental features:a) the s i m i l a r i t i e s of the p-nucleus and nucleus-nucleus reactions. I t is d i f f i c u l t to imagine an equilibrated hot zone made of only two or three nucleons.b) the existence of an anlsotropic High Energy Gamma-Ray Production 339 component in the angular d i s t r i b u t i o n . Resorting to angular momentumeffects would not be helpful, here, since the antsotropy appears to be larger for small systems(Tam,lg88). The breakdown of the thermal model approach to high energy photon production carries interesting information on the dynamics of the nucleus-nucleus reaction. In fact, i f any object l i k e a very hot participant nucleus existed, in an equilibrated state, for a f i n i t e amount of time, a thermal component should be present, in the photon spectra. The absence of this component t e l l s us that the participant zone has a very short l i f e - t i m e , governed by dynamical instabilities. Dynamical Calculations have been used extensively, in order to explain the main characteristics of high energy photon production in nucleus-nucleus collisions. One of the f i r s t attempt(Nakayama,1986) extended the Fermi aas model calculation described in Section 5 from proton-nucleus to nucleus-nucleus reactions. According to equation 5.5 the probability for photon emission per f i r s t proton c o l l i s i o n reads: I d'W --dEydN o~ j<k, %n+O~o d'pnl Pr(E~,pp,pn) B(p;,p;) ~ p + ( n - ( ; - c ; - ( r ) J dip; (6.3) I<k, d3Pnl B(p;,p~)~p+(n-(;-(;)d3p; Now, the Paull blocking factors B refer to a double momentum sphere geometry, one for the projectile, one for the target. The centers of the two spheres are separated by an amount equal to the relative momentum per nucleon between projectile and target. Due to the stronger Paull blocking of the two spheres geometry it is not possible to assume that all incident protons do interact with the target nucleons. Nakayama and Bertsch, therefore define an effective photon emission probability per incident proton: d2P where r is the inverse collision cross-section is given by: d2o d~ " ~Ap 1 dZW e"U" (6.41 rate of the particle. Finally the photon production 3 d]pp dZW(pp)( kf 4"n~) ~ - .~l-e (6.S) In this f i r s t calculation, the semi-classical n-p-~ cross-section was used, including the angular d i s t r i b u t i o n 3.18. The experimental angular distributions were r e l a t i v e l y well reproduced, as seen on Fig. 6.2. On figure 6.3 the calculated spectrum is also shown. The calculation underestimates the experiment by a factor of 3. In fact, the data displayed(Stevenson,]g86) h a v e b e e n subsequently renormaltzed (Benenson,lg87) by approximately, a factor 2. Therefore the true underestimation is by a factor 6. I t is, even, probably more. In effect, the authors assumed that all projectile nucleons interacted,with a cross-section smaller than the total cross-section since i t is taken as the proton-nucleus cross-section. For the 14N+Pb system, the product ~tt2A,- 3024 whtle o,<Ar>-2|35 reflects more exactly the number of participants. Finally, the unaeresttmation factor is close to an order of magnitude. I t seems that the origin of this underestimation comes, essentially from two effects. First, as pointed out in Section 4, the semi-classical cross-section underestimates the true n-p-~ cross-section. Nakayama and Bertsch(Nakayama,lg88b) have used the n-p-~ cross-section derived by Nakayama(lg88a) as an input for the two Fermi spheres calculations. The cross-section is, then, increased by a factor 3, as seen in Fig. 6.3. The other possible source of underestimation ltes in an oversimplified treatment of the 340 H. Nifeneckcr and J. A. Pinston ~il 5.0 <o> 1 \ @) =c+-c 4+0" ~ !kO" ~2.0I0" O0 o 3o 6o 90 0,= 30 60 90 120 150 180 120 e. Fig. 6.1 Angular distributions obtained from the collective model, for two nuclear systems a) 14N+~ Ni at 35 AMeV b) Iz C+TM C at 84 AMeV . The curves are labelled after the distance between the two nuclear centers.(from Kwato Njock,lg88c) .. 6 T 4 - - - i . . . . i i i J - i - - - l - - - : ~ + Pb . ll-30 Mo¥ 3 2 I b ¢%1 bl 0 0 • t - - - * * * e (deg) Fig. 6.2 Angular distribution, in the lab frame of 30 MeV photons emitted in the reaction N14+Pb at 40 AMeV - - c a l c u l a t i o n s (Nakayama,lg86) x Experimentaldata (Stevenson,1986) High Energy G a m m a - R a y Production 341 Paull blocking by the two Fermi spheres model.A simple remark shows that this is, indeed, the case. For not too high energies, the two spheres have an overlap region. I t corresponds to a region of small relative velocities of the projectile and target nucleons. Two nucleons which have small relative velocity and are separated in space w i l l , obviously, not be l i k e l y to overlap spatially later on. I t is, therefore, expected that the nucleons which overlap spatially have different velocities. The two overlapping Fermi spheres, while being a reasonable approximation when projectile and target matter do not overlap, should, therefore, break down in the spatial overlap region. This is an i l l u s t r a t i o n of the Liouvllle theorem which states that, in an energy conserving system, the density, in phase space, is conserved along the trajectories. I t is, therefore, not possible to increase, at the same time the density in configuration and in momentum space. This klnd of behavior is, indeed, demonstrated by the Phase soace distributions evolution. This evolution is derived from extensions of the Boltzmann equation to fermionlc ensembles. There are several denominations for, baslcally, the same equation which has been used to follow the evolution of the phase space distributions( Landau-Vlassov , V1assov-Uehling-Uhlenbeck, Nordheim, Boltzmann-Uehling-Uhlenbeck...). In the following, we shall retain the single denomination BUU, since i t has been the most used in the frame of photon production calculatlons. Considering a phase space distribution f ( r , p : t ) , the Liouvllle theorem, in presence of conservative forces reads: d f(r,p:t) dt I f the forces equation: af dr dp 8t + ~ "vrf + ~.vpf = 0 dp derive from a potential, then ~ - - VrU and the Llouville becomes the Vlassov 8f dr aT + ~ "vrf - VrU'Vpf = 0 (6.6) This equation can be shown to be the classical l i m i t of the TDHF equation, i f the d i s t r i b u t i o n functions are taken to be the Wigner transforms of the many-body wave functions. Practically, in the semi-classical approximation, used so-far, f is taken to be a probability d i s t r i b u t i o n . The only quantum features kept in the calculations are the Fermi motion, present in the i n i t i a l state, and the Pault blocking. In fact, the Landau equation assures that, i f the Hetsenberg and Pault principles are obeyed by the system at some i n i t i a l stage, they are, also, obeyed l a t e r on (Bertsch,1977). Fig. 6.4 shows the result of a computation of the evolution of a system, following the Vlassov equation. I t can be seen that, indeed, the Fermi distributions are modified in such a way that, in the spattal overlap region, a hole is created in the region of small relative velocities of the nucleons. The Vlassov equation assumes that the forces acting on the particles derive from a potential. Practically, two body c o l l i s i o n s , in nucleus-nucleus reactions, give rise to stochastic Forces which do not obey this condition. The effect of these forces is, usually, treated as a non-zero second member added to equation 6.6. This term is the c o l l i s i o n term. I t measures the change of f due to c o l l i s i o n s . I t must, therefore, include a depletion term corresponding to c o l l i s i o n s of the particle at r,p, and of a f i l l i n g term which corresponds to c o l l i s i o n s which lead, in t h e i r f i n a l state , to a particle with r,p. The modified equation, is the Boltzmann equation. For fermtontc systems, the c o l l i s i o n term has to take the Pault blocking into account. This specific form of the Boltzmann equation is the so-called BUU equation, which reads (Bauer,]986): af dr a"t" + ~ ' ' v r f - VrU'VPf = 4 (2~)~ ~ d3kzdSk3d(I vlz ~_~ 3(ki+kz_k3_k4) (6.7) x [ff2 (I-f3 ) (I-f4 )-f) f4 (l-f ) (I-fz )] The c o l l i s i o n term corresponds to the elastic scattering of nucleons tn states kl,I ~ to H. Nifenecker and J. A. Pinston zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON 342 10-l - ‘k + 20@F’b 2 * s > 10-a 10-a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA E - lo-* 2 3 q 1o-s 4 w zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO (MeV) Fig. 6.3 Comparison of an experimental spectrum with two calculations. x experimental data (Stevenson,l986) The data should be renormalized by a factor 2. semi classical calculation (Nakayama,l986) --_ quantum calculation (Nakayama,l988a) t=O fm/c *=7.5 fm/c t=l5 t=22.5fm/c *=3Ofm/c t=37.5 fmk fm/c Fig. 6.4 Evolution of the momentum distributions as a function of time following a BUU simulation of a 12C+12C collision at 40 AMeV. (Bauer,1986) abcsissa:position along the separation axis ordinate: relative momentum along the same axis Note that the window in position becomes a window in momentum. High Energy Gamma-Ray Production 343 do states k3 4,k with cross-section-Equation 6 " 7 gives the evolution of the phase space C~" density distribution during the nucleus-nucleus reaction. I t is then assumed that each proton-neutron collision may lead, with a small probability, to photon production. The probability for such emission is taken to be the probability observed in a free n-p collision modified by the Pauli blocking, in the final state. I t reads: dZa " ~ I c~ 4w E~~ 1 d2a I m ( ] - f , ) l l - f , ) dE~dnl~ (6.8) pn E~ om coll. This expression being, subsequently, integrated over impact parameter. This analysis has been extensively applied by the MSU,Glessen collaboration using either, the seml-classical n-p-Y cross-section (Bauer, lg86; Casslng,lg86; Casslng,lg87) or a relativistic a,, model (Blr~,Ig87). One of the striking results of these calculations is the time evolution of the gamma emission. It is found that, due to the two-body collisions, the hole in phase space, at small relative velocities, fills up. This filling up prevents further photon emission. Fig. 6.5 displays the time evolution of the photon production with time. It is found, that, for the C+C reaction at 84 AMeV, virtually all photons are emitted in a time smaller than 1.5 I0 "z2 sec. This time Is close to the flying by time of the projectlle.A close correlation between the photon yield and the number of first proton-neutron collisions is found. Fig. 6.6 shows the origin, in momentum space, of the emitted photon, as well as the final states of the nucleons, after photon emission. It is seen that photons are emitted by collisions between nucleons having as much as possible relative velocity, and that, due to the Paull blocking, the photon energy is close to the maximum allowed in the collision, leaving the nucleons with a small relative momentum. As soon as this posslbily becomes unavailable, the high energy photon production stops. A broad range of experimental results have been satisfactorily reproduced. A sampling of these is shown on Fig. 6.7 and 6.8. Fig. 6.7 shows examples of spectra and Fig. 6.8, of angular distributions. The calculated photon yields follow, approximately, a A~-~ A~ .91 law, close to the observed one. The angular distributions are reasonably well re)roduced. Similarly, the impact parameter dependence of the spectra is well reproduced. However, one can see, on Fig. 6.7, that the calculation is not able to reproduce the data at high incident projectile energies. It seems that a serious discrepancy with the shape of the experimental spectra appears above 60 AMeV. As will appear, in the following, the most probable origin of this discrepancy is a wrong representation of the elementary n-p-~ process. Results very similar to those just described have been obtained with two other approaches, the preequlllbrium model of Remington and Blann(I987), and the exchange model of Randrup and Vandenbosch(Ig88). In The exchanoe model, the dynamics of a dlnuclear system is followed in time. This dynamics is controled by the interplay of the potential and of the friction caused by the exchange of nucleons through the window which opens up between the two nuclei, as soon as the reaction begins (Randrup,1987 and references therein). This model has been very successful In explaining the characteristics of deep inelastic collisions, in the low energy regime of nucleus-nucleus collisions. It has been extended to the higher energy regime by allowing for the possibility of escape of the transfered nucleons into the continuum, leading to preequllibrium particle emission (Randrup,1987). If the transfer is Paull allowed, the nucleon is propagated through the receptor nucleus, with a temperature dependent mean-free path. Whenever a collision occurs, the probability that a photon may be emitted is considered, using the semi-classical expression of the n-p-Y cross-section, modified by Pauli blocking in the final state. This model is simpler and less computing time demanding than the BUU calculations. It is very helpful for examining the correlation between photon fast particle emission and for studying impact parameter selected gamma emission. An example of such impact parameter selected calculation is shown on Fig. 2.13. Fig. 6.9 shows the variations of the gamma multiplicity wlth impact parameter, for the system Ne+Ho . Also shown on the figure is the variation of the number of preequillbrlum neutrons, as calculated with the exchange model. A striking correlation is predicted between fast particle and photon emission. This is also displayed in Fig. 6.10, where the time evolution of photons and preequlllbrlum neutrons production is displayed. The same type of correlation have been found In the frame of BUU (Cassing,lg88) and BME calculations (Remlngton,lg87b). The recent measurement of Lampls and colleagues (Lampls,lg88) deals with such a correlation. dErd~l t= H. Nifenecker 344 and J. A. Pinston z;[yt2, 0 10 20 t Fig. 30 ,1 40 50 60 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK (fm/ c) 6.5 Time dependence of the photon production rate as obtained from a BUU calculation (Bauer,1986) The photon energy was 40 MeV and the reaction 1rC+12C at 40 AMeV. t,is the contact time and t, the time when maximum overlap is reached. -2 - -2 -2 tia/‘m) t -I 2 -2 k, t&n) -I ' k, Fig. 6.6 Initial (left) and final (right) momentum distributions of nucleons producing 100 MeV photons in a r2C+12C collision at 40 AMeV. From (Biro,1988) 2 High Energy Gamma-Ray Production ~ 100~ I (ll' | ' I I , ~ O.i 0.01 0.001 ' ~0 60 80 100 E 1 [MeV] Fig. 6.7 Comparison of BUU calculations with experiment. c a l c u l a t i o n including r e l a t i v i t y effects(Btro,1988) --- the same without r e l a t i v i t y effects 0,4,0 14 N+lZC at 20,30,40 AHeV (Stevenson,]g86) o 12 C+12C at 84 AHeV (Grosse,]986) • ]~Ar+2I'A1 at 85 AHeV renormaltzed to Iz C+lZ C following equ. 2.4 (Kwato Njock,1988c) ! I ! d) I ~e - 100 !0 (~ U.I ! I 0 i -I cosetab Fig. 6.8 comparison of BUU angular d t s t r i b u t i o n s ( B i r o , ] 9 8 8 ) with experiment (Stevenson,1986) for the reaction 14N+lZC at 40 AReV =,o,o 80,60,40 HeV photons, respectively. 345 346 H. Nifenecker and J. A. Pinston These authors measured p-~ and p-p coincidences. They found that the proton multiplicity measured in coincidence with a high energy photon was almost the same as that measured in coincidence with a proton. The true proton m u l t i p l i c i t y is larger than the measured one by one unit in the second case, since one has to take into account the coincident proton. Therefore, the measurement seems to indicate that whenever a photon is emitted, the intervening proton cannot escape. The last dynamical model which has been applied in the context of high energy photon emission is The oreeoutltbrtum medel (Remington,1987). The application of t h i s model to p-nucleus-~ reactions has been presented in section 5. Its extension to nucleus-nucleus reaction is straightforward. I t is assumedthat a compoundsystem is formed, with an excitation energy equal to the energy available in the center of mass. This excitation energy is partitioned between the projectile (tightest partner) nucleons, exclusively. The number of nucleons having energies between ( and ( + ~ is then given by: --~()~E AP[ (E-()%-I - ( E - ( - ~ )* P-~ ] - EAp-1 AP(AP-]) (E-()AP-' -~ EAp-1 ~ (6.9) dn I t is easy to see that integration of ~-L(c) from 0 to E, the total excitation energy, gives t~o " Remington and Blann modify the dis[Fibution given by equation 6.9 to prevent a particle take too large an energy. They constrain all i n i t i a l particle energies to be within bounds given by: where (f is the Fermi energy of the projectile and (w,. the beamenergy per nucleon. At low beam energy (,i,-O. Remington anBBlann obtain results qua]itatively similar to those obtained by the other dynamical models. They find that most photons are emitted at the beginning of the reaction, the more so for the most energetic ones. They also find a significant contribution of secondary collisions, although f i r s t collisions are predominant. This behaviour is shown on Fig. 6.11. One of the peculiarity of their work has been to study the influence of the chosen n-p-~ elementary cross-section on the photon production cross-sections. An exampleof this study is shownon Fig. 6.12. The calculation used either a semi-classical or a quantum expression of the elementary cross-sections. The quantum expression was taken from Neuhauser and Koonin(lg87) and included the contribution of charged pionic currents (see section 4.). I t is seen that, at the highest beamenergies, only the quantum calculations give agreement with the experimental data. At 84 AMeVthe difference between the two calculations reaches two order of magnitude, at the highest gammaenergies, and one order of magnitude for typical gammaenergies. The finding that exchange currents had a decisive influence on photon emission at photon energies higher than 50 MeV Is corroborated by the recent two Fermi gas calculation of Nakayamaand Bertsch (Nakayama,lg88b) as shown on Fig. 6.13 where i t appears that the exchangecontribution exceeds the convection one above 50 MeV. I t might seem surprising that three models as different as the BUU, BME and nucleon exchange give comparable results, concerning the high energy photon production. In fact, except, possibly, for the BUU calculations,it is a commonfeature of these models to have, f i r s t , been devised in order to take into account the fast particle preequilibrium emission. This was used to normalize the elementary nucleon-nucleon scattering cross-sectlons, both in the preequilibrium(BME) and in the nucleon exchange models. In particular, this may take care of the fact that, in the preequilibrium model, fusion is assumed for all impact parameters, although with a reduced cross-section. In other words, the study of fast partlcle emission defined the nucleon mean-free path. This defined the number of nucleon-nucleon collisions, and, therefore, the photon production cross-section, from the photon production rate per collision. This, in turn, is given by the elementary n-p-~ cross-section assumed, and, also, by Paull blocking effects. In most calculations, the semi-classical cross-section was assumedfor the n-p-~ process. The main difference might come from the treatment of the Pauli blocking. In this respect, i t is interesting that both the exchangeand the preequilibrium models deal with one Fermi sphere situations. In the exchange model, the donatednucleons are treated as isolated nucleons in the sea of the High Energy Gamma-Ray Production 1 4!10_4 I I I 1 347 I 400 MeV 2°Ne+ lesHo 2 0 total pre-equilibrium neutrons 2 I I 20 t I 40 I I 6 0 _ _ 80 I00 Z I 120 140 Fig. 6.9 Variations of photons and neutrons m u l t i p l i c i t y as a function of impact parameter(angular momentum) from (Randrup,1988) I 1 400 MeV ZONe+ I~SHo I =46 ~E <3 ~ ' ~ gommos . f3 ,-- aOneutrons <3 00 I t (sec) 2 3xld ~ Fig. 6.10 Time dependence for emission of preequilibrium neutrons and hard photons .From (Randrup,1988). PPP--L 348 H. Nifenecker and J. A. Pinston so loo 150 E, WV) zyxwvutsrqponmlkjihgfedcbaZYXWVU Fig. 6.11 Proportion of photons originating from first collisions as a function of photon energy and for a number of systems.(Remington,lg86) 0 40 11 II -Sharp Cutoff -.-.-Ericson Fig. 6.12 Comparison between experimental gamma spectra and 8ME calculations (Remington,lg86) The data (Grosse,1986) are from 12C+12C at 84 (upper) and 48 (lower) AMeV _ semi classical n-p-7 cross section with cut-off -.-.- the same without cut-off (see text) --- quantum n-p-7 cross-section. High Energy Gamma-Ray Production 349 receptor nucleons. In fact, except for the change of temperature and macroscop|c characteristics of the dt-nuclear system, the donated nucleons are treated Independently• The transfered nucleons correspond, mostly, to the endcap of the Fermi distribution of the donor nucleus, since they are the most energetic vis-a-vis the receptor. This is analogous to the situation displayed on Fig. 6.6. However, the exchange model does not provide a hole in phase space for the fi n a l state of the nucleons, after photon emission• In the preequtltbrium model, the few p r o j e c t i l e nucleons are disposed on concentric shells around the target Fermi sphere. Due to thetr small number and to the larger phase space associated to larger radii they only occupy a small fraction of the phase space, allowing easy transitions from shell to shell. Here again, no hole in phase space seems to appear. I t seems that, in both the exchange and the preequtltbrtum model, the early character of photon emission is more related to the energy available tn the transition than to a property of the phase space. In this respect, i t is instructive to consider the maximumenergy available in a c o l l i s i o n for the three kind of approaches. In the BUU model two nucleons from opposite endcaps may end up at 0 r e l a t i v e velocity. This leads to a maximumphoton energy E~ m IIUUw 2of + 2Jc~m c(+_---=--. (mx I t may seem strange that this formula gives a f i n i t e , and large z value, even when the beam energy vanishes. This is due to the fact that the formation of the hole in phase space requires energy, which has to be taken from the collective r e l a t i v e kinetic energy of the two nuclei, in th e i r approach phase. The smallest the r e l a t i v e energy, the highest the collective energy needed to create the hole for one nucleon, and therefore, the smallest the number of overlapping nucleons. In the l i m t t of vanishing beam velocity, no participant nucleons is l e f t . This is an i l l u s t r a t i o n of the subtle collective effects which can happen in BUU. In the preequtlibrtum case the maximumenergy available for a transition is E, the total excitation energy or ( ~ - ((+ 2 ~ + (bu. , whichever is smallest• In fact, the preequiltbrtum model assumes some kind bT-parttaTequtltbration between the i n i t i a l excttons, without specifying the mechanism underlying i t . We feel that its remarkable success for accounting a large body of data is a real puzzle. The maximume n e ~ available in the nucleon exchange model is less exotic and reads" ( ~ - ( + 2 , ~ . Note that the same expression is obtained from the two sharp Fermi sphere mo~e~.Cer~a~n~, closer analysis of the connection between the three dynamical approaches would be worthwhile. I t may be possible in the frame of the BUU model. I t remains that, even with the normalization procedure on the fast particles emission the close agreement between the three approaches remains surprising. Table 6.2 summarizes the considerations made above. As seen on the table, we are not aware of BUU or NE calculations which incorporate the charged pion exchange contribution to photon production. Certainly, as stated above, this contribution should improve the situation at the higher beam energies. However, I t may deteriorate the quality of the agreement with data at lower energies. 350 H. Nifenecker and J. A. Pinston lO-Z " " " - i . . . . I . . . . I . . . . I . . . . 1 " " " zZC + zzC 10 -3 > m v °....'~ 10 -4 .-" '"~ '. i 10 -5 X .~'.. x C ".~ 3 10 -6 Tlab=84 M e V / N 0=90 10-7 ........ 0 25 ° ,. . . . . . . . . 50 75 , ......... I00 125 150 (MeV) Fig. 6.13 Comparison of two Fermi spheres c a l c u l a t i o n (Nakayama,lg88b) with experiment (Grosse,1986) The r e a c t i o n was 12 C+lZ C at 84 AHeV x experimental data . . . . magnetization c u r r e n t c o n t r i b u t i o n convection current c o n t r i b u t i o n . . . . . exchange current c o n t r i b u t i o n total calculated rate High Energy Gamma-Ray Production 351 TABLE 6.2 Comparison of certain characteristics of the available dynamical models £be(m Real Ist Ic Dynamics Z Fermi Spheres NO m (b BUU ~E YES NO Impact NO Parameter NO YES Oependence Naxtmum -2% +2%~-~'f+ cb Energy per (b +2(b~'E'~';-@ T Cf+2~Cb( t, 41[b n-n c o l l i s i o n Semi-classical YES YES YES n-p-~ quantum n-p-~r Maximum beam energy appl ted AI4eV Nucleon Exchange YES Z matn bodies YES YES YES NO YES NO 84 84 84 44 7. SUMMARYAND OUTLOOK. In this s t r o l l i n g around in the realm of high energy photon production, we have found that, during the last few years, a vast amount of experimental work delineated the math characteristics of t h i s process. At the same time, several theortical attempts have been made to account for them, but only a few have been reasonably successful. This is, certainly, one of the few domains in intermediate energy heavy-ion physics where so l i t t l e time has elapsed between the discovery of a phenomenon, at the occasion of pton studies (No11,1984; Beard,1985; Julien,1985), the recognition of i t s interest, and i t s elucidation. At this point it is possible to say that we have acquired a firts order (in the sense of perturbation theory), knowledge of high energy gamma-rays production in nuclear reactions. Hard photon production, in nucleus-nucleus collisions, have been studied at beam energies between I0 and 125 AMeV. From these, mostly inclusive, measurements, the main characteristics of the photon emission have been deduced.lt was found thatthe photons seemed to be emitted in a frame having close to the half beam velocity, that is the velocity of the nucleon-nucleon center of mass. In this frame, the angular distributions were sywetrical with respect to 90"with, in most cases, a dipolar character built upon an isotroplc component. The spectra had almost exponentially decaying shapes, above ZO MeV, the inverse slope of which increased almost linearly with beam energy per nucleon. These characteristics are found for very light systems, like d+C or LI+LI as well a for very heavy ones like Kr+Au or Xe+Sn. The photon production yields follow a very simple scaling rule, which relates them to the number of first n-p collisions. These characteristics of the photon emission suggest strongly that the neutron-proton collisions in the early stageof the reaction are the main source of high energy Y-rays. Some hints of a second order of knowledge are already present in recent data. The slopes of the spectra, for the same beam energy per nucleon, seem to depend ,in a significant way, upon the target mass, as well as upon the impact parameter. This may be a consequence of changes in the Fermi motion of nucleons, frem system to system. The angular distributions appear to be more anlsotroplc for lighter systems than for heavier ones. This may be a signal of a persisting influence of multiple collisions, which becomes more important in heavy systems. Although the scaling law gives photon emission probability per n-p collision which appear remarkably, and surprisingly, constant for the large variety of systems studied so far, some 352 H. Nifenecker and J. A. Pinston significant variations are probably present. I t is, stl11 d l f f i c u l t to extract them, due to the different experimental and analysis techniquesusedby the different groups.There is, however, some evidence that this emission probability increases with beam energy. The only theoretical approacheswhich have been, so far, able to reproduce the main trends of the data are dynamical calculations which evaluate the numberof nucleon-nucleon collisions as a function of time. To each such co111slon is associated a small but finite probability for photon emission. These calculations show that, indeed, most photons are produced early in the collision. They, also, associate closely photon and fast particle emission. However, these calculations, when using semi-classical expressions of the elementary nuceon-nucleon-Y cross-section, are not able to reproduce the shape of the photon spectra, above, approximately, 60 AMeV incident energy. This failure is due to the important contribution of charged plon exchange currents to the photon production. This importance is demonstrated in recent proton -nucleus-Y experimental studies. Theoretical attempts to include these contributions in calculations of p-nucleus-Y and nucleus-nucleus-~ reactions show significant improvements in their account of the experimental data. However, it remains to verify that the agreement obtained by the previous calculations at lower energies is not deteriorated by this change in the elementary cross-sections. Future experimental work should aim at reducing the remaining systematic errors, so that a precise study of the effects of multiple collisions could be carried out. Exclusive experiments will certainly be realized, in order to investigate the change of spectral shapes, as well as angular distributions, with impact parameter. These studies may lead to the observation of unambiguous collective effects, which we have failed to identify, so far, although they have been predicted. The impact parameter dependence of the photon yields may, also, be used as a tool in studying heavy-ion reaction mechanisms. It will be interesting to disentangle, mere precisely, the respective contributions of statistical and bremsstrahlung gamma rays, at the lower incident energy range. At the higher energy range the contribution to the gamma spectra of the neutral plons decays has to be precisely estimated, so that we could gain knowledge of photon production at energies as high as possible. At those enegles, it is expected that Fermi and Pauli effects would be minimized, leading, therefore, to an easier analysis of the production mechanisms. The most needed experimental work is a careful and extensive study of the elementary n-p-Y process. The study of small systems, like p-d, d-d , ~-p is also important, since it should demonstrate the possible cooperative effects expected from electrodynamlcs. How such effects could affect the calculations of p-nucleus-~ and nucleus-nucleus-~ processes is important to elucidate. One of the most exciting perspective offered by the study of hard photons emission in nucleus-nucleus reactions is the possibility to examine,if and how much the nuclear medium modifies the elementary n-p-Y process. Such modifications are expected, for example, if the pion mass, in nuclear matter, is different from its vacuum value. In this context, it is important to note that it seems that, above SO MeV, most photons are produced by the exchange currents. Photon production is, probably, the most sensitive probe of charged pions exchange currents. ACKNOWLEDGEMENTS. I t is a pleasant duty to thank N.Alamanos, W.Bauer, W.Benenson, G.Bertsch, g.Casstng, M.Durand, A.Gobbl, E.Grosse, C.Guet, K.Knoll, M.Blann, V.Metag, U.Mosel, M.Prakash, J.Randrup, B.Remtngton, J.Stachel, J.Stevenson, R.Vandenbosch for many illuminating discussions and for providing us with unpublished data. N.Alamanos was kind enough to carry some thermal calculations concerning the Ar measurements at 85 AMeV. The part of the work reported here in which we had a personal contribution was done wtth the invaluable help of O.Barneoud, M.Maurel, C.Rtstort, F.Schussler. S.Drtsst, d.Kern, J.P.Vorlet, Y.Shutz, S.Bjornholm, were most effective in part of the experimental program in which we have been involved. We thank deeply the staffs of the following accelerators: GANIL, Orsay SC, SARA, SIN cyclotron where our experiments were carried through. One of us(H.N.) is deeply indebted to the Ntels Bohr Institute where his interest for photon production was aroused, owing to an illuminating collaboration with J.Bondorf. We aknowledge the support of the Instttut des Sciences Nucl~atres and of the D~partement de Recherche Fondamentale, Grenoble, where most of this work was carried on. More especially we High Energy Gamma-Ray Production 353 thank B.Vignon for the constant interest he has shown for our work. Last, but not the least we thank Molse Kwato Njock with whom we were lucky enough to be closely associated during these recent years, who was the working horse of most the analysis and whose thesis was of invaluable help for writing this review. H. Nifenecker 354 and J. A. Pinston zyxwvutsrqponmlkjihgfedcbaZYXWV REFERENCES. Alamanos,N. and colleagues(1986).Phys.Lett.,Z738 ,392-394. Ashkin,J. and R.E.Marshak(l949). Phys.Rev.,76, 58-61. Awes,T.C.(1981).Phys.Lett.,ZO38, 417-421. BaSer,R. and colleagues(1969).Nucl.Phys.,BZZ,675-691. Bauhoff,W.,(1986). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA At omic and Nucl. Dat a Tables 35,4 29-4 50. Bauer,W. and colleagues(1986). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC Phys./?ev.,C34,2127-2141. Bauer,W. and colleagues(1987). Pr oceedings of t he XXV Int er n. Mint er Meet ing on Nuclear Phy sics, Bor mio, It aly , Jan.1987. Beard,K.B. and colleagues(1985).Phys.Rev.,C32,1111-1113 Beck,F.A.(1985).In D.Shapira(Ed.),Proc.Conf. on Inst r ument at ion f or Heavy -Zon Nuclear Resear ch, Oak Ridge,October 22-25 1984,pp.129-134. Beckham,C.(1962).Thesis. Benenson,W.(1988).Pr ivat e communicat ion. Bertholet,R.and colleagues(1986).JournaI de Physique,C4, 201-209. Bertholet,R.and colleagues(l987).Nucl.Phys.,A474, 541-560. Biro,T.S.and colleagues(1987). Nucl. Phys.,A475, 579-593. Bonasera,A. and colleagues(l988).Nucl.Phys.,A483, 738-751. Bondorf,J.P.(1982).NucZ.Phys.,A387, 25c-36~. Bondorf,J.P. and colleagues(1985).Phys.Lett.,Z628, 30-34. Brady,F.P., and colleagues(1968).Phys.Rev.Lett.P0,750-755 Brady,F.P. and J.C.Young(1970). Phys. Rev .,CZ, 1579-1595. Brown,V.R.(1969).Phys.Rev.,Z70, 1498-1511. Brown,V.R.(1970).Phys.Lett.,328, 259-261. Brown,V.R. and J.Franklin(l973). Phys. Rev.,C8,1706-1735. Budiansky,M.P. and colleagues(1982). N.Z.M.,1299,453-457. Cassing,W. and colleagues(1986). Phy s.Let t .,ZBZB, 217-221. Che Ming Ko and J.Aichelin(l987). Phys. Rev .,C35, 1976-1991. Cohen,D. and colleagues(1963). Phys. Rev.,Z30, 1505-1523. Das Gupta,S. and A.Z.Mekjan(l981).Phys.Rep.,72,131-201. Dupont,C. and colleagues(1988).NucI.Pbys.A481,424-444. Edgington,J.A. and B.Rose(1966). Nucl. Phy s.,89, 523-54 7. Edgington,J.A.and colleagues(1974) .Nucl. Phy s.,A218, 151-167. Ford,R.L.and W.R.Nelson(l979). SLAC Repor t ,n’2ZB (June 1979). Gaardhoje,J.J and colleagues(1987).Phys. Rev.Let t .,56, 1783-1787. Gosset,J. and colleagues(1977).Phys.Rev.,CZ6,629-654. Gross,D.H. and colleagues(l982).Zeits.f& Phys.,A309,41-56. Grosse,E. and colleagues(1986), Eur ophys. Let t .,2, 9-14. Grosse,E.(1985). In H.Feldmaier(Ed.), Pr oc. Int er n. Wor k shop Pr oper t ies of nuclei and nuclear ex cit at ion, on Gr oss Hir shegg,pp.65-70. Herrmann,N. and colleagues(1986).GSZ scient if ic r epor t , 284 . Herrmann,N. and colleagues(1987).GSZ scient if ic r epor t , 4 8. and Herrmann and colleagues(l988).Phys.Rev.Lett.,60,1630-1635. Hess,W.N,(1958). Rev. of Rod. Phy s.,30,368-380. Heuer,R. and colleagues(l988). Z. f ur Phys. At omic Nuclei,330, 315-333. Hingmann,R.(1987). Thesis. Hingmann,R. and colleagues(1987). Phys.Rev.Let t ., 58 759-764 . Hugi,M.(1984). Can. Jour n. Phy s.,62, 1120-1139. Jackson,J.D.(1975). in John Wiley & sons (Ed.),CZassical Elect r odynamics. Jacak,B.V. and colleagues(1987).Phys.Rev.,C35, 1751-1776. Julien,J. and colleagues(1984 ).unpublished. Kishimoto,T. and colleagues( Kitching,P. and colleagues(1987).Nucl. Phy s.,A4 63,87c_94 c Knoll,J.(1979).Phys.Rev.,C20, 773-786. Knoll,C. and C.Guet(1988).GSZ Pr epr int 88-25. Koehler,P.F.M.and colleagues(1967). Phy s.Rev.Let t .,ZB, 933-938. Kox,S. and colleagues(1987). Phys. Rev .,C35, 1678-1701. Kwato Njock,M. and colleagues(1986). Phys. Let t .,BZ75, 125-130 t he Zst t opical Reet ing Kwato Njock,M. and colleagues(1987). Pr oceedingsof on Giant Resonance Ex cit at ion in Heavy -ion Collisions, Padova(1987) High Energy Gamma-Ray Production Collisions,St-Halo(June 1988). Kwato NJock,H. and colleagues(19~c), to be published. Lampis,A. and colleagues(IN88), g.S.U.Preprint(1988) Laval,M. and colleagues(1983).N.I.g.,206,169-181. Lebrun,M. and colleagues(1979).N.I.g.166,151-165 Machleidt,R. and colleagues(1987).Phys.Rep.149,1 Hetag.9. and SImon,R.S.(1987)GS! Report 87-19 Metag,V.(1988).to be published in Nucl. Phys., and Proceedings of the 3rd International Conference on Nucleus-NucTeus Collisions,St-Halo(dune 1988). Michel,C. and colleagues(IN86). N.I.N.,A243,453-467. Nakayama,K. and G.Bertsch(1986). Phys. Rev.,C34, 2190-2215. Nakayama,K.(1988a) Nakayama,K. and Bertsch,G.(1988b) Neuhauser,D. and S.£.Koonin(1987).Nucl. Phys.,A462, 163-175. Nifenecker,H. and d.P.Bondorf(1985). Nucl. Phys.,A442, 478-496. Nifenecker,H. and colleagues(1986) Proceedings of the 24th International Winter Meeting on Nuclear Physics, Bormio(1986). Nifenecker,H. and colleagues(1988a).Proceedings of the 20th summer school in Nuclear Physics,Mikolajki(1988). Nifenecker,H. and colleagues(1988b).Proceedings of the International Workshop on Nuclear dynamics at gedium and Nigh Energies,Bad Honef Ntita,K. and colleagues(lg87).Texas A & g Symp. on Not Nuclei(dec.]gB7) Noll,H. and colleagues(1984).Phys.Rev.Lett.,52,1284-1290. Prakash,M. and colleagues(1986).Phys.Rev.,C33, 937-951. Prakash,M. and colleagues(1987), to be published. Randrup,J. and R.Vadenbosch(1987).Nucl.Phys.,A474,219-245 Randrup,J. and R.Vadenbosch(1988).To be published and Proceedings of the International Workshop on Nuclear dynamics at gedium and High Energies,Bad Honef Remington,B.A. and colleagues(1987a). Phys.Rev.Lett.,57, 2909-2914, and Phys. Rev.,C35 1720-1743. Remington,B.A. and M.Blann(1987b).Phys.Rev.,C36,1387-1396. Rothe,K.W. and colleagues(1966). Contribution to the Williamsburg Conf. on intermediate energy physics,(1966) Shtbata,T. and colleagues(1984), in H.Ejiri and T.Fukuda(Ed.)Proceedings of the Intern. Symp. on Nucl. Spectroscopy and Nucl. Interactions, Osaka 1984,pp.517-529. Shyam,R. and J.Knoll(1984).Nucl.Phys.,A426, 606-631. Shyam,R. and J.Knoll(1986).Nucl.Phys.,A448, 322-344. Signell,P.(1967). Some theoretical Aspects of p-p Brehmsstrahlung. in G.Paic and I.Slaus(Ed.),Few Body problems, light Nuclei and Nuclear Interactions. Vol.1 pp.287-305 Stahl,T. and colleagues(1987). Zeit. for Phys.,A327,311-335. Stevenson, J. and colleagues(1982).N.I.N.,198,269-280. and K.B.Beard, Thesis. Stevenson,J. and colleagues(1986). Phys.Rev.Lett.,57, 555-560. Stevenson,J. and colleagues(1987). Nucl. Phys.,A47], 371c Tam,C.L., and colleagues(1988). NSU Preprint. Tam,C.L., and colleagues(lg88b). NSU Preprint. Vasak,D. and colleagues(1985), dourn, of Phys.,G]], 1309-1331. Vasak,D.(1986). Phys. Lett.,176B, 276-280. Veyssi~re,A. and colleagues(1983). N.I.N.,212,1-27. Wilson,R.(1952). Phys. Rev.,85, 563-587. Yule,H.P. and Turkevich,A.,(1960). Phys. Rev. 118,1591-1607. 355