dp ] d 8'p den dq 8'~~ d~ dq

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1 PHYSCAL REVE% C VOLUME 40, NUMBER 6 DECEMBER 989 Hard photons in proton-nuceus coisions K. Nakayama* nstitut fur Kernphysik, Kernforschungsanage Juich, D 5-70, Juiich, West Germany G. F. Bertsch Department of Physics and Astronomy and Cycotron Laboratory, Michigan State University, East Lansing, Michigan (Received 4 June 989) A cacuation of energetic photon production in proton-nuceus reactions at intermediate bombarding energies is performed assuming the first chance nuceon-nuceon coisiona mechanism. The eementary nuceon-nuceon bremsstrahung ampitude is cacuated within a meson exchange potentia mode which incudes the one-body as we as the two-body current contributions. The resuts are in reasonabe agreement with the recent data by the Grenobe group at the proton incident energy of T,b =68 MeV and indicate that these energetic photons are created in a reativey owdensity region in the nuceus (-40% of the nucear matter norma density). The present mode, however, cannot account for the strong anguar dependence observed at T, b =72 MeV.. NTRODUCTON. NUCLEON-NUCLEUS BREMSSTRAHLUNG The investigation of energetic products in nucear reactions is of great importance since these products carry information about nucear reaction dynamics. Among them, the photon is of particuar interest because the weakness of the eectromagnetic interaction makes it a cean probe. n fact, in the ast few years the observation of hard photons produced in intermediate-energy heavyion coisions has ed to considerabe efforts to understand the underying dynamics. To test the theoretica modes of photon production, it is important to study nuceon-nuceus coisions, since these are much simper to treat than heavy-ion coisions. ndeed, in Ref. 6, in connection to the nuceon-nuceon (NN) bremsstrahung, we have aso investigated nuceon-nuceus bremsstrahung. However, there we were abe to reproduce the data then avaiabe ony under certain extreme conditions. Very recenty, new data on the hard photon production in proton-nuceus coisions became avaiabe, which show that the earier data of Ref. 9 is off by a factor of -3. Therefore, in the present work we anayze these new data. The formaism we use to cacuate the nuceon-nuceus bremsstrahung is the same as in Ref. 6. We express the photon emission probabiity per unit soid ange and unit photon energy as dp ] d 8p den dq 8 d dq (2. ) where d 8 z&/den d Q denotes the nuceon-nuceus bremstrahung differentia transition rate and 8&& the tota NN coisiona rate irrespective of the bremsstrahung. We cacuate these quantities in the nucear matter approximation. The photon emission probabiity shoud be rather insensitive to certain kind of effects such as the distortion of the wave functions or the strength of the interaction since it is given as a ratio between the two quantities. The nuceon-nuceus bremsstrahung transition rate is obtained by foding the eementary NN bremsstrahung transition rate with the phase-space distribution function of the target nuceus d W (pi) dco dq d p2 "f (2)e, 27TCO t pz (2)e, (2)E, 4 g g V &i&2co(e, k;p, p, SMs. V, 0;p,p2S )+E,s, SMs SMs, X 5 (p, + p2 p, p2 k)5( E, + e2 e, E2 co). (2.2) n the above equation, the primed (unprimed) quantities refer to the fina (initia) two interacting nuceons energies E E2 (E e2), momenta p,, p2 (p,, p2), tota spin S (S), and its projection Ms., (Ms). The matrix eement in the arge parentheses denotes the photon emission ampitude for a photon with poarization e and momentum k; co denotes its The American Physica Society

2 GpipPMs 40 HARD PHOTONS N PROTON-NUCLEUS COLLSONS 252 energy. The target is characterized ony by its Fermi momentum k&. The Paui bocking operator Q excudes the target Fermi sphere; in the nucear matter rest frame Q = 8(p, kj )0(pz k&) Anaogousy, the tota XN coisiona rate is given by d p2 d pi d 2 p2 NN(p ) (k Q (ppzMs )&Eisz "y (2rr) Ez (2) E (2m) Ez sm S SMs X (2m. ) 6(p, +pz p, pz)5(e&+ Ez &i &z), (2.4) (2.3) where G denotes the NN G matrix associated with the bare X% potentia used. The singe-partice energies appearing in Eqs. (2.2) and (2.4) contain not ony the kinetic energy but aso the potentia energy due to the mean fied of the target nuceus. We assume the singe-partice energies to have the reativistic form E(p)=(p +m* )i where the effective mass m * is defined as (2.5) m*=m+u,. (2.6) Here u, is the scaar potentia; m stands for the nuceon mass. The actua singe-partice energy E(p) is reated to that of Eq. (2.5) by (in the nucear matter rest frame) E (p) =E(p)+ uo, (2.7) where uo denotes the timeike component of the vector potentia. n principe, both u, and uo depend on the momentum p of the partice. However, here we assume them to be momentum independent, which seems to be a very reasonabe approximation. " The bremsstrahung ampitude in Eq. (2.2) has been cacuated within a meson exchange potentia mode where the strongy interacting partices are treated to a orders in perturbation theory whie the couping to the photon is considered ony in first order. The mode incudes the convection, magnetization, as we as important exchange current contributions, so that the photon emission potentia V, is written as a sum of three terms Vem Vconv+ Vmagn+ exch (2.8) where V n V,, and V,h denote the convection, magnetization, and two-body current contributions, respectivey. The atter is dominanty the exchange currents. We have previousy parametrized the ampitudes in Eq. (2.8) by simpe functions of the nuceon and photon energies, and we use that parametrized form in the present cacuation. Here we consider contributions ony from proton-neutron (pn ) coisions because they are much more efficient to produce photons than protonproton (pp) coisions. The former processes originate a dipoe radiation in owest order whie the atter contribute to the quadrupoe radiation. ndeed, the pp bremsstrahung is roughy a factor of 4 smaer than the corresponding pn bremsstrahung in the reevant region of photon energy here. n principe, the bremsstrahung ampitude aso contains medium effects (in particuar, we shoud use the G matrix instead of the T matrix); however, we have shown in Ref. 6 that these effects are sma and therefore may be negected. For consistency, the T matrix is used aso in Eq. (2.4). A more fundamenta probem that arises in medium is reated to the photonnuceon couping vertex itsef. The eectromagnetic form factors may (and probaby wi) change in the nucear medium from those in the free space. This is as yet an unsoved probem and certainy beyond our scope here. We keep them unchanged from their vaues in free space. The cross section is obtained by mutipying the probabiity rate given by Eq. (2.) by the cross sectiona area of the target do. d co d 0 dp d co d A (2.9) with R =.252 fm and 3 being the mass number of the target nuceus. We now consider the parameters in the present cacuation. They are the Fermi momentum k&, the effective mass m * (or the scaar potentia u, ) and the vector potentia uo. Note that uo enters ony for the determination of the oca momentum of the incident nuceon. The resuts wi be rather sensitive to the Fermi momentum, so we wi estimate it taking nucear surface effects expicity into account. We first find the average density at the %2V coision point from an eikona mode, f d b f dz p(b, z)op(b, z)exp f dzop(b, z) f d b f dz op(b, z)exp f dzcrp(b, z) (2.)

3 2522 K. NAKAYAMA AND G. F. BERTSCH Here o. is the XN cross section and p is the nucear density. t is assumed to have the form p+tb p(b, z)= (2.) +exp[(r R )/a] with po=0. 7 fm, a =0. 65, and r =(b +z ) Given the average density from Eq. (2.), we find the Fermi momentum from the Thomas-Fermi reation (2. 2) Q Z MeV Next we extract the singe-partice energy on the Fermi surface. For a given target nuceus and proton incident energy T&,b, the maximum photon energy co is Nm = T)b +Sp (2.3) where S is the proton separation energy in the nuceus with one more proton than the target. Now, within our mode, energy conservation yieds for the proton energy on the Fermi surface C U 3 D b m m m» m m 0.8 m 0.9 m.0 m 70 0 E(kf )+be, = S. (2.4) u (Nev) Here, be, -e Z/R, stands for the Couomb energy of the proton. The preceding equation, thus, gives us the proton energy (the nucear part) E(p) at p =kf. This determines the vector potentia u 0 in terms of the effective mass m *, u =E(k ) (k +* ) f f (2.5) We treat m * as a free parameter to be adjusted to best reproduce the data. For ater convenience we define the mean-fied potentia energy as. RESULTS (2. 6) n this section we compare our resuts with the recent data by the Grenobe group. Figure shows our resuts for p+ Tb at T&, b =68 MeV and for a photon emission ange of 0=90 as a function of photon energy and for various vaue of the effective mass. At this incident energy the NX cross section inside nuceus is about o.= 4 mb. Equations (2. ) (2.2), then, give for the Fermi momentum the vaue of kf =.04 fm which corresponds to about 45% of the nucear matter norma density. Then, the meanfied potentia energy at the Fermi surface, extracted from Eq. (2.6), is U(kf =.04 fm )= 4.7 MeV. This is a quite reasonabe vaue as compared, for exampe, to those from phenomenoogica optica potentias. As can be seen from Fig., the inhuence of the effective mass is to enhance the cross sections. One of the effective mass dependences arises from the determination of the oca momentum of the incident nuceon. As m* T \ FG.. The bremsstrahung differentia cross section {in the aboratory system) for p+ Tb at the incident energy of T, b = 68 MeV as a function of photon energy and for different choices of the effective mass m {in units of nuceon mass m). The photon emission ange is 0=90. The Fermi momentum is fixed to be kf =.04 fm. The data are from Ref. 8. decreases the oca momentum increases which means that the effective nuceon energy increases. Now, the XN bremsstrahung transition rate depends very itte on the incident energy, whie the tota NX coisiona rate has a rather strong incident energy dependence and it decreases as the incident energy increases. Since what determines the photon cross section for nuceon-nuceus bremsstrahung is the ratio between the bremsstrahung rate and the tota coisiona rate (the atter is independent on the photon energy), we see that the cross section wi be shifted up if the effective mass is decreased. The effective mass has aso another effect on the photon cross sections which comes from energy conservation. n the NX center-ofmass (c.m. ) frame we have 2e(p) = «p+ &/2 )+s( p t/2 )+. where p(p) denotes the initia (fina} NN reative momenturn and co, the photon energy. This gives (3.2) From the preceding equation we see that if the effective mass is decreased the avaiabe fina-state phase space wi increase since the singe-partice energy s(p) as defined in Eq. (2.5} wi decrease. t is aso easy to see that this effect increases as the photon energy increases. This expains the arger enhancement of the photon cross sections for higher photon energies as m * decreases in Fig.. We observe that the conditions given by Eqs. (2. 4)

4 40 HARD PHOTONS N PROTON-NUCLEUS COLLSONS 2523 and (2.5) are aways satisfied for difterent vaues of m *. From Fig., we see that the effective mass of m*-0. 9m seems to best reproduce the data in the high photon energy region (co) 75 MeV). This vaue of the effective mass aso seems reasonabe. n the ow-energy region, the data prefer a much smaer vaue of m *, but for such a ow density this is probaby unreaistic. We wi discuss this probem ater and take for the moment the vaue of m *=0. 9m since we are primariy interested in the high-energy tais. Once the m* is fixed, a the parameters are fixed and we can now compare the present predictions with the data. n Fig. 2 the resuts for the same system as in Fig. are shown for different photon emission anges. As we can see, the agreement is reasonabe for high-energy photons. Here, it is worthwhie to mention that, in the aboratory frame, the avaiabe phase space for backward emissions is ess than for forward emissions. This is easy to see from Eq. (3.2) together with the fact that the photon energy transforms from the aboratory to the NN c.m. system according to co, = y( P cos8)co, (3.3) where co and 0 denote the photon energy and emission ange in the aboratory frame and 3 and y are the usua Lorentz frame transformation parameters. For a given photon energy in the aboratory frame, the cross section wi decrease if the photon emission ange is increased since co, gets arger eaving ess avaiabe phase space. We aso observe that the rather pronounced anguar distribution shown by the data in the high-energy region can be reproduced ony for a reativey sma vaue of the Fermi momentum as in the present case. n fact, the anguar distribution becomes more and more isotropic as the Fermi momentum increases (see Fig. 4). This indicates that these energetic photons are originated more from the surface region rather than from the interior of the nuceus. n Fig. 3 we show the resuts for p+ Ag with the same condition as in Fig. 2, but for very forward photon emission ange of L9=30. Since in our mode the dominant dependence on the target nuceus arises from the geometrica cross sectiona area [see Eq. (2.9)] we have just rescaed the resuts for p +Tb for 0=30, i.e., the parameters of the mode are the same as in Fig. 2. We see that the quaity of the agreement with the data is simiar to that of Fig. 2. n the ow-energy region (co (70 MeV) our resuts underestimate the data by a factor of -2. There are two ways to cure this probem within the present mode. One way is to decrease the effective mass, as has been seen in Fig., and at the same time aso to decrease the vaue of the Fermi momentum kf. The atter is necessary in order to keep the agreement with the data in the high-energy region. n fact, we can get a good agreement over the entire energy region if we choose m*-0. 7m and kf -0.8 fm. However, this parameter set is not reaistic because for such a ow density we do not expect a sma effective mass. The second possibiity is that the owenergy photons come from a higher-density region than T ab = 68 MeV T i = 68 MeV M S Z C: b =50 8 = 90 8 = (x&0) (x0) s (MeV) FG. 2. The bremsstrahung differentia cross section (in the aboratory system) for p +" Tb at the incident energy of T,b = 68 MeV as a function of photon energy and for different photon emission ange 0. The vaues of the Fermi momentum and the effective mass are kf =.04 fm and m*=0. 9m, respectivey. The data are from Ref. 8. M S R g Cm O 3 o CVb =30 (ii / f / (NeV) FG. 3. The bremsstrahung differentia cross section (in the aboratory system) for p + Ag at the incident energy of T»b = 68 mev as a function of photon energy and for photon emission ange of 0=30. The parameters are the same as in Fig. 2. The data are from Ref. 8.

5 2524 K. NAKAYAMA AND G. F. BERTSCH the high-energy photons. For higher densities we have much stronger mean-fied potentia and consequenty the cross section may increase. n order to iustrate this point, in Fig. 4 we show the resuts when we use kf =.36 fm and m*=0. 7m. Of course, we cannot answer the question here why ow-energy photons woud come from higher densities than high-energy photons. Perhaps some of the ower-energy photons come from mutistep coision processes. Certainy, these are more important for ow-energy photons than they are for high-energy photons. We now examine the Fermi momentum dependence of the photon cross sections in more detai. The kf dependence enters in two ways: (i) Negecting the Paui principe, the avaiabe phase space goes up when kf is increased. This is easiy seen from Eq. (3.2); the average p in this equation is arger when kf is increased. (ii) Second, the Paui bocking is more effective for forward photon emissions than for backward emissions. This can easiy be visuaized from Fig. 5, where the momenta of the nuceons before and after the photon emission are dispayed together with the corresponding photon momentum. Around the target nuceon we have the Fermi sphere (actuay an eipsoid due to the Lorentz boost) which bocks the fina-state phase space. Then, for high-energy photons, due to the recoi momentum ( k/2) of the scattered nuceons, it is easy to see that the Paui bocking is more effective for forward photon emission than for backward emission. From the above considerations it is cear that these two effects, (i) and (ii), of the Fermi momentum kf on the bremsstrahung cross section work against each other. m M QJ b 8 =50 8=90 e 8= ) p+th T = 68 MeV ab (xyo) {xioo) u (MeV) FG. 4. Same as in Fig. 2, but for kf =.36 fm and =0.7m. FG. 5. The reative momenta of the two interacting nuceons in the initia nuceon-nuceon center-of-mass frame before (p and p) and after (p and p) the emission of a photon with momentum k. The Lorentz boosted Fermi sphere with momentum kf bocks the fina-state phase space around the target nuceon with momentum p. y is the usua Lorentz transformation parameter, The situation in the figure corresponds to the case where the target nuceon in the aboratory frame has its momentum aong the z direction. The fact that the cross section is amost insensitive on kf for forward anges in the present case (compare Figs. 2 and 4, and aso Figs. 6 and 7) is that they compensate each other. For backward emission anges the effect (i) dominates. -4 QJ R g c:» -6 3 / ) $ p+au 72 MeV {x) {xioo): = 90 8 = 50 (x 000). 8 = (MeV) FG. 6. The bremsstrahung di6erentia cross section (in the aboratory system) for p+ Au at the incident energy of T&,b =72 MeV as a function of photon energy and for various photon emission ange 8. The vaues of the Fermi momentum and the effective mass are kf =0.93 fm and m =0. 9m, respectivey. The data are from Ref. 7.

6 40 HARD PHOTONS N PROTON-NUCLEUS COLLSONS 2525 S Z 5 p 3 D -6 p+au = 72 hfev (x) e e = ao e = 50 (x00) e = u (Mev) FG. 7. Same as in Fig. 6, but for kf =0.7 fm We next consider photon production at a ower incident energy, p + Au at T] b =72 MeV. Here we used a cross section of o. 86 mb, which is a weighted average of the pn and pp cross sections at this energy. The interaction density is then, po and the equivaent Fermi momentum is 0.93 fm. We have aso chosen m*=0. 9m as in the previous case. We see from Fig. 6 that, athough for 0=50 and 90 the agreement seems reasonabe, we cannot describe the data for very forward and backward anges. We might think that these highenergy photons come from a ower-density region than that corresponding to the vaue of kf estimated here. t may be that for these incident energies a simpe estimate of the density as in Eq. (2.) is not as good as for higher incident energies. To iustrate this, in Fig. 7 we show the resuts when the Fermi momentum is chosen to have the very sma vaue of kf =0.7 fm. The agreement now is much better for 0=50. However, we sti underestimate the data at 0=30 which are a factor of -3 5 arger than those at 0=50. Such a strong anguar dependence is not observed at higher incident energies (see Figs. 2 and 3). The reason for the strong sensitivity of high-energy photon cross sections on kf for backward photon emission anges has aready been mentioned. A comparison of Figs. 6 and 7 reveas that, in the region of Fermi momentum invoved here, the forward emission ange cross sections are insensitive to the vaue of kf even for high-energy photons. This means that the ony way to increase the cross section at these anges is to decrease the e6ective mass whie aso decreasing the Fermi momentum at the same time in order to satisfy Eq. (2. 4) (compare aso Figs. 2 and 4). The point is that we wi not be abe to describe this strong anguar dependence in the present mode within a reasonabe range of the Fermi momentum and the mean-fied potentia. We aso mention that the eementary pn bremsstrahung cross section does not show such a strong anguar dependence, athough, this fact doesnt impy that the nuceon-nuceus bremsstrahung cannot have a stronger anguar dependence. V. CONCLUSONS We have investigated the energetic photons produced in intermediate-energy nuceon-nuceus coisions considering the first chance NX coisiona mechanism. The phase-space distribution of the target nuceus have been simuated by a Fermi sphere. For the eementary NX bremsstrahung ampitude we use the parametrized version of Ref. 3 which reproduces the meson exchange potentia mode cacuation of Ref. 6. The present cacuation reproduces rather we the recent data by the Grenobe group at Tj,b =68 MeV and indicates that these hard photons are created in a rather ow-density region in the nuceus (-40% of the nucear matter norma density). t aso indicates that owerenergy photons are produced in a difterent way than the high-energy photons. t is very interesting to repeat the cacuation within a mode in which other processes, such as the contribution of secondary coisions, can be incuded. This wi hep a nore compete understanding about the origin of ow as we as high-energy photons. The present mode cannot describe the strong anguar dependence shown by the data at the incident energy of T&,b=72 MeV. However, it is not cear whether the discrepancy is due to the inadequacy of our mode. A more detaied cacuation seems to be needed. On the other hand, it is of crucia importance to have more compete anguar distribution data in order to carify this probem. Note added in proof. After submitting the present work for pubication, we have earned from Pinston that their data for p+ Au at 72 MeV may contain a pieup between photons and fast protons for forward photon emission anges (9-30 ). This coud expain the disagreement between the present prediction and their data observed in the high-energy part of the y spectrum. The authors wish to thank J. Pinston for providing us with his data prior to pubication. *Aso at Department of Physics and Astronomy, University of Georgia, Athens, GA T. S. Biro, K. Niita, A. A. DePaoi, W. Bauer, W. Cassing, and U. Mose, Nuc. Phys. A475, 579 (987). B. A. Remington, M. Bann, and Cz. F. Bertsch, Phys. Rev. C 35, 720 (987). 3R. Hingmann et a., Phys. Rev. Lett. 58, 759 (987). 4V. Metag, Nuc. Phys. A482, 483c (988). 5K. Kwato Njock, M. Maure, E. Monnand, H. Nifenecker, J. A. Pinston, F. Schusser, D. Barneoud, and Y. Schutz, Nuc.

7 2526 K. NAKAYAMA AND G. F. BERTSCH 40 Phys. A482, 489c (988). K. Nakayama, Phys. Rev. C 39, 475 (989). K. Kwato Njock, M. Maure, H. Nifenecker, J. A. Pinston, F. Schusser, D. Barneoud, S. Drissi, J. Kern, and J. P. Voret, Phys. Lett. B 207, 269 (988). J. A. Pinston, D. Barneoud, V. Beini, S. Drissi, J. Guiot, J. Juien, M. Kwato Njock, M. Maure, H. Nifenecker, F. Schusser, and J. P. Voret, Phys. Lett. B 28, 28 (989);J. A. Pinston, private communication. J. Edgington and B. Rose, Nuc. Phys. 89, 523 (966). K. Nakayama, S. Drozdz, S. Krewad, and J. Speth, Nuc. Phys. A470, 573 (987); A484, 685 (988). M. R. Anastasio, L. S. Ceenza, W. S. Pong, and C. M. Shakin, Phys. Rep. 0, 327 (983). V. R. Brown and J. Frankin, Phys. Rev. C 8, 706 (973). K. Nakayama and G. F. Bertsch, Phys. Rev. C 40, 685 (989). V. Herrmann, J. Speth, and K. Nakayama, Phys. Rev. C (submitted). H. W. L. Naus and J. H. Koch, Phys. Rev. C 39, 907 {989); J. H. Koch, private communication. J. Franz, H. P. Grotz, L. Lehmann, E. Rosse, H. Schmitt, and L. Schmitt, Nuc. Phys. A490, 667 (988). J. Pinston, private communication.