arXiv:0709.4155v1 [nucl-th] 26 Sep 2007 ∗ htatog h sincosscina h in dipo- giant is the is at reason section resonance The cross lar fission [1]. the beams although that -rich intense get to lwddw natnse W ovre rdrcl in directly or converter generating (U), (W) target tungsten the a in down slowed en- The ( beam deuterons/. the ergetic of that than significant / ≈ h alyo tblt,epcal ntevcnt of vicinity the from in away especially far stability, nuclei of valley exotic the very very explore new to neutron- with open perspectives highly will fission photofission of through study nuclei production radioactive to the rich Moreover, possible detec- avalanche yields. it low plate make parallel which and track tors state Re- using solid Table. by improved Periodic the are the techniques of experimental in- the half all cently upper limit the practically to reasons vestigations These elements. distinguishing light in particularly for difficulty processes, fragmentation the various also from in fission is fission are There high sections induce cross small. With the to quite elements possible lighter For element. is elements. it the most energy on each dependent by enough released is neutrons emission event of the number fission on with the incident along and is neutrons occur element of may above an fission of element, (photon) that energy gamma binding a nuclear When the interest. considerable r xie ypooscvrn h eko h giant the of peak the covering resonance. the nuclei dipolar by for production, excited method beam photofission are ion the radioactive In (neutron-rich) fission. induce may -al1 [email protected];Emi :[email protected] 2: E-mail [email protected]; 1: E-mail h s fteeegtceetosi rmsn mode promising a is electrons energetic the of use The nrcn er h td fpoosinhsattracted has photofission of study the years recent In htncerratoso ciieadpeatnd nucle pre- and actinide of reactions Photonuclear . anfr4 e etosidcdfiso)btthe but fission) induced neutrons MeV 40 for barn 1.6 γ pooscneso ffiinyi uhmore much is efficiency conversion -photons sinpoeso h opudncesi iuae naMont manner a systematic in a simulated in is analysed are nucleus sections compound consi cross the is reaction process of b fission process Thus competition fission nucleus. of excited process the the for fission by followed model toabsorption Keywords h the ASnmes 52.x 79.b 58.g 52.c 24. 25.20.Dc, 25.85.Jg, 27.90.+b, 25.20.-x, numbers: PACS td erdcsstsatrl elteaalbeexperi available the well satisfactorily reproduces study o h ciie n r-ciie titreit energi intermediate at pre-actinides and actinides the for htncerrato sdsrbdwt napoc ae o based approach an with described is reaction Photonuclear ∼ .INTRODUCTION I. 0MV)o nieteetoscnbe can electrons incident of ) MeV 50 aibeEeg ylto ete /FBda aa,Kolka Nagar, Bidhan 1/AF Centre, Cyclotron Energy Variable ∼ htncerratos htfiso;Ncerfissility; Nuclear Photofission; reactions; Photonuclear : 07 e n h nraigtedo ula siiywt t with fissility nuclear of trend increasing the and MeV 50-70 ≈ 232 Th, .6br for barn 0.16 233 U, 235 aa Mukhopadhyay Tapan U, 238 238 γ ry which -rays (against U and U Dtd uut9 2021) 9, August (Dated: 78 t.in Ni. 237 padtepeatnd nuclei pre-actinide the and Np 1 h ciie r eyrdooi eas hyas have are of also and Many they half-lives because less). biological radiotoxic or long very years are 30 actinides shorter- of years, the be half-lives of to with tend (most thousands lived products many fission of radioactive whereas half-lives with long-lived ina nemdaeeege sdsrbdb the by described evaluate to employed is is which energies [2] intermediate model quasideuteron at tion inadcmae ihteaalbeeprmna data experimental available energies the at with compared calcula- and evaporation-fission a tion the using for calculated are method nuclei Monte-Carlo various of sections Photofis- a cross sion by competition. followed evaporation-fission nucleus, by of inside absorbed mechanism pair be pro- to [n-p] step assumed two neutron- is a a photon as incoming described The be cess. can reaction a such MeV] evr ffcie but effective, very be poo)ry.The electromagnetic rays. with (photon) spent bombarding been by has proposed decay much radioactive a Accelerated with management one. it very shorter-term replaces waste a decay and a hazard eliminates will radioactive From actinides long-term activity of time. transmutation the short ra- viewpoint, of very a majority are within away products the fission actinides but The the dioactive, convert products. to fission is into intention the transmutation xiainpoes titreit nris[ de- energies subsequent intermediate absorp- the At by and quantum nucleus process. be high-energy a excitation also a of can of excitation Information tion the varies. about barrier obtained the especially mechanism, how fission pos- the about is into insight it gain reactions, to photonuclear sible the From section. cross actinide target the to product. tuned fission precisely or be to need may and I H HTASRTO MECHANISM PHOTOABSORPTION THE II. n ..Basu D.N. and stpso ltnu n te ciie edt be to tend actinides other and of h oiatmcaimfrncerphotoabsorp- nuclear for mechanism dominant The h i ftepeetwr st banphotofission obtain to is present the of aim The 10.Lx etldt fpoosincosscin at sections cross photofission of data mental s[ es ee sadcymd.Teevaporation- The mode. decay a as dered ∼ nteeeg range energy the in tenlgtpril vprto and evaporation particle light etween 010MeV]. 20-140 ∼ h usdueo ula pho- nuclear quasideuteron the n -al rmwr.Photofission framework. e-Carlo Monte-Carlo 07 MeV. 50-70 2 a7004 India 064, 700 ta efislt parameter fissility he titreit energies intermediate at i γ γ ry aeas ensgetdto suggested been also have -rays ry r eydffiutt produce to difficult very are -rays 208 band Pb ∼ 07 e for MeV 50-70 209 α ∗ mtesa el In well. as emitters i The Bi. Z 2 /A ∼ 20-140 2 the total photoabsorption cross section in heavy nuclei. III. THE NUCLEAR EXCITATION AND It is based on the assumption that the incident photon FISSION PROBABILITY is absorbed by a correlated n-p pair inside the nucleus, leaving the remaining as spectators. Such an After absorption of a photon (γ) with incident energy assumption is enforced when wavelength of the incident Eγ (as measured in the laboratory frame), the nucleus photon is small compared to the nuclear dimensions. The with rest m0 recoils with a velocity vr given by T total nuclear photoabsorption cross section σa is then proportional to the available number of n-p pairs inside nucleus and also to the free deuteron Eγ c vr = 2 (5) 0 cross section σd(Eγ ) and is given by : [Eγ + m c ]

which is also the velocity of the centre of mass vcm of the T L γ-nucleus system where c is the in . σa = NZσd(Eγ )fP (Eγ ) (1) A In the centre of mass frame, momenta of the γ and the where Eγ is the incident photon enegy, N, Z and A nucleus are the same and equal to pcm: are the neutron, proton and mass numbers respectively, L/A factor represents the fraction of correlated n-p pairs. Eγm0c Thus, Levinger’s constant [2] L is a factor which measures pcm = 2 2 (6) the relative probability of two nucleons being near each [m0c (2Eγ + m0c )] other in a complex nucleus compared with that in a free p and the Er of the recoiling nucleus in the deuteron. The function fP (Eγ ) accounts for the reduc- laboratory frame is tion of the n-p phase space due to the Pauli exclusion principle. A systematic study of the total nuclear pho- ′ 2 toabsorption cross section data [3] in the intermediate m0c ′ 2 Er = m0c energy range shows that 1 v2/c2 − − r p 2 2 2 1/2 = Eγ + m0c [m0c (2Eγ + m0c )] (7) −D/Eγ 0.81 fP (Eγ )= e where D =0.72A MeV. (2) − ′ where m0 and m0 are the rest of the nucleus be- For photon energies upto the pion threshold, the Eq. (2) fore and after the photon absorption respectively. along with the damping parameter D provided above The recoiling nucleus can be viewed as a compound nu- agrees reasonably well with the approach based upon cleus having the same composition as the target nucleus phase space considerations [4] using Fermi gas state den- but with the excitation energy sities that conserve linear momentum for the Pauli block- ing effects in the quasideuteron regime of hard photon ∗ ′ 2 2 2 2 1/2 absorption. E = m0c m0c = m0c [(1 + 2E /m0c ) 1] − γ − The free deuteron photodisintegration cross section [5] 2 2 1/2 2 = [m0c (2E + m0c )] m0c (8) is given by γ − and in this case E∗ is also equal to the kinetic energy 3/2 61.2 [Eγ B] in the centre of mass frame (which is sum of the kinetic σd(Eγ )= 3− [mb] (3) energies of the γ and the nucleus moving in the centre Eγ of mass frame). The kinetic energy Er of the recoiling where B=2.224 MeV is the of the nucleus in the laboratory frame can now be rewritten as deuteron. The quasideuteron model of nuclear photoab- the obvious result sorption is used together with the modern root-mean- square radius data to obtain Levinger’s constant ∗ E = E E . (9) r γ − −2/3 −4/3 L =6.8 11.2A +5.7A (4) This excited compound nucleus then undergoes succes- − sive evaporation of neutrons and other light particles or of nuclei throughout the and is in good fission. Thus, the fission is considered as a decay mode. agreement [6] with those obtained from the experimen- The photofission cross section σ is a product of the nu- tally measured σT values. At the quasideuteron energy f a clear photoabsorption cross section σT and the total fis- range, as a consequence of the primary photointerac- a sion probability (fissility) f and is, therefore, given by tion, γ+(n+p) n*+p*, in most of the cases excited com- pound nuclei are→ formed with the same composition as target nucleus where both neutron and proton are re- T σf = σ f. (10) tained inside the nucleus and the probabilities that either a neutron escapes or proton escapes or both neutron and The statistical approach for and light-particle proton escape from the nucleus are extremely low. evaporation and nuclear fission is an appropriate scheme 3

∗ ∗ ∗ ∗ for calculation of the relative probabilities of different where En = E Bn and Ef = E Bf are the nu- decay modes of the compound nucleus. Such statistical clear excitation energies− after the emission− of a neutron decay of the compound nucleus is the slow stage of the and after fission, resepctively, where Bn is the binding photonuclear reaction. According to the standard Weis- energy of the emitted neutron. Γn and Γf are the partial skopf evaporation scheme [7], the partial width Γj for the widths for the decay of the excited compound nucleus evaporation of a particle j = n, p, 2H, 3H, 3He or 4He is via and fission, resepctively, and the given by parameters an and af are the level density parameters for the neutron emission and the fission, respectively and 2 2 ∗ K0 =¯h /2mr0 where m and r0 are the neutron mass and E −Bj (2s + 1)µ ∗ Γ = j j σj (E)ρ (E B E)EdE radius parameter respectively. The emission probability j 2 2 ∗ Z inv j j π ¯h ρCN (E ) Vj − − of particle k relative to neutron emission is calculated (11) according to the Weisskopfs statistical model [7] where sj , µj , Vj and Bj are the spin, reduced mass, Coulomb barrier and the binding energy of the parti- ∗ Γ γ E a ∗ 1 ∗ 1 j k k k k 2 2 cle j, respectively. σ (E) is the cross section for the = exp 2[(akE ) (anE ) ] inv Γ γ E∗ a k − n inverse reaction which means the capture reaction cross n  n  n  n    (14) section of the particle j to create the compound nucleus. where ak is the level density parameter for the emis- ρCN and ρj are the nuclear level densities for the ini- sion of the particle k, γk/γn = 1 for k = p, 2 for tial and final (after the emission of the particle j) nuclei, 4 2 3 3 k = He, 1 for k = H, 3 for k = H and 2 for k = He. respectively. ∗ E = E (B + V ) is the nuclear excitation energy The Bohr-Wheeler statistical approach [8] is used to k k k after the emission− of particle k [13]. B are the binding calculate the fission width of the excited compound nu- k energies of the emitted particles and V are the Coulomb cleus. This width is proportional to the nuclear level k potentials. The evaporation-fission competition of the density ρ at the fission saddle point: f compound nucleus is then described in a Monte-Carlo framework. Any particular reaction channel r is then ∗ E −Bf 1 ∗ defined as the formation of the compound nucleus via Γf = ∗ ρf (E Bf E)dE (12) photoabsorption and its decay via particle emission or 2πρ (E ) Z0 − − CN fission. Thus, fission is considered as a decay mode. The where Bf is the fission barrier height. The diffuse surface photonuclear reaction cross sections σr are calculated us- T nucleus Sierk’s [9] fission barriers (Bf ) are used for these ing the equation σr = σa nr/N where nr is the number calculations. The decay of the excited compound nucleus of events in a particular reaction channel r and N is to- [10] is simulated using the Monte-Carlo method [11]. The tal number of events that is the number of the incident competition between the various decay channels at each photons. step of the evaporation chain is determined by the rela- tion between the partial widths for particle evaporation and fission, Eqs. (11) and (12), respectively. Finally, in V. COMPARISON OF THE CROSS SECTION ESTIMATES WITH order to calculate the fission probability, the total num- MEASURED DATA ber of fission events in a computer run is counted and divided by the total number of photoabsorption events. Evaporation from excited fission fragments is also taken Each calculation is performed with 40000 events using into account. a Monte-Carlo technique for the evaporation-fission cal- culation. This provides a reasonably good computational statistics. The photofission cross sections are calculated IV. COMPETITION BETWEEN LIGHT at different energies for various elements. Results of these PARTICLE EVAPORATION AND FISSION calculations corresponding to the available experimental data at 52 MeV, 69 MeV, 120 MeV and 145 MeV are The quantitative description of the process is based listed in Table-I. The statistical error in the theoretical on the liquid drop model (LDM) for nuclear fission by estimates for the photofission cross sections are calcu- T Bohr and Wheeler [8] and the statistical model of nuclear lated using the equation σf ∆σf = σa [nf √nf ]/N ± T ± evaporation developed by Weisskopf [7]. Accordingly, the which implies that ∆σf = σa σf /N. For the cases, probability of fission relative to neutron emission is calcu- where not a single fission eventp occured in N events, the lated using Vandenbosch-Huizenga’s equation [12] given upper limits of the cross sections are calculated using the T by equation σf = σa /N where N[= 40000] is the number of incident photons. ∗ 1 The results of the present calculations should be com- 2 Γf K0an[2(af Ef ) 1] ∗ 1 ∗ 1 − 2 2 pared with measured photofission cross sections with cer- = 2 ∗ exp 2[(af Ef ) (anEn) ] Γ 3 − n 4A af En   tain degree of caution. In this regard, it is pertinent to (13) mention here that the experimental data listed in Table-I 4

TABLE I: Comparison between the measured and the calcu- Eγ=52 MeV Eγ=69 MeV lated photofission reaction cross sections. Pb 10.0000 208Pb Target Eγ Expt. Calc. Calc. Pb Pb T nuclei σf σa σf Pb MeV mb [Ref.] mb mb 1.0000 237Np 52 20±2 [14] 17.6 17.61± 0.09 Pb Pb f 69 19±4 [14] 15.6 15.59 ±0.05 0.1000 Tl f 235U 52 16±4 [15] 17.6 17.58± 0.09 (mb) Pb R 0.0100 Hg 69 15±3 [15] 15.5 15.54 ±0.08 E Tl Tl Tl σ Tl 238U 52 14±2 [14] 17.5 17.46± 0.09 0.0010 Pb ± Hg Hg 16 2 [15] Hg 32±2 [16] 0.0001 25±3 [17] Bi 10.0000 Bi Bi 209Bi 69 15±1 [15] 15.5 15.52± 0.08 23±1 [15] Bi f 1.0000 ± f 13 2 [14] f 232 ± ± Pb Bi (mb) Bi f Th 52 8.6 0.6 [15] 17.5 12.41 0.07 0.1000

ER Pb Pb 69 9±1 [15] 15.4 13.82 ±0.07 Tl Pb σ 209 −3 ± × ± Tl Pb Bi 52 (16 1) 10 [16] 17.4 0.15 0.01 0.0100 Tl − (70±12)×10 3 [18] Tl Bi ± × −3 (24 3) 10 [19] 0.0010 −2 69 (8.0±0.6)×10 [16] 15.0 0.26 ±0.01 Tl Tl Pb ± × −2 (18 3) 10 [18] 0.0001 120 0.20±0.06 [20] 9.3 0.87 ±0.01 198 200 202 204 206 208 210 A 145 0.31±0.08 [20] 7.6 1.21 ± 0.02 R 208Pb 52 (1.9±0.3)×10−3 [16] 17.3 0.039 ±0.004 −3 FIG. 1: The plots of cross sections σ as a function of mass 3×10 [21] ER − number AR of the evaporation residues for the pre-actinides ± × 3 ± 208 209 69 (12 2) 10 [16] 14.9 0.097 0.006 Pb and Bi at Eγ = 52 MeV, 69 MeV. The fission cross −3 (18±3)×10 [21] sections of the fissioning nuclei are also shown (marked f). − 197Au 120 (59±38)×10 3 [20] 9.0 0.073 ±0.004 − 145 (11±4)×10 2 [20] 7.3 0.117±0.005 − 181Ta 120 (13±4)×10 3 [20] 8.5 0.0023±0.0007 145 (9.7±3.0)×10−3 [20] 6.9 0.0035±0.0008 measured photofission cross sections is reasonable. − − 174Yb 52 (3.2±0.5)×10 5 [16] 16.7 < 4.2×10 4 − − 69 (6±1)×10 5 [16] 13.9 < 3.5×10 4 − − VI. RESULTS AND DISCUSSION 154Sm 69 (1.8±4)×10 7 [16] 13.0 < 3.3×10 4 − 51V 120 (76±25)×10 3 [20] 3.0 0.0019±0.0004 −3 The photonuclear reaction cross section calulations at 145 (78±29)×10 [20] 2.3 0.0033 ±0.0004 intermediate energies within a Monte-Carlo framework for simulation of the evaporation-fission competition are performed assuming 40000 incident photons for each cal- for Eγ = 69 MeV, 120 MeV and 145 MeV correspond to culation which provide a reasonably good computational the effective mean energies of 69 MeV, 120 MeV and 145 statistics. The cross sections of fission and evapora- MeV respectively, for the incident quasi-monochromatic tion residues (σER) as a function of AR 208 beams of photons. However, those for Eγ = 52 MeV of the evaporation residues for the pre-actinides Pb actually correspond to the incident beam energy of 52 and 209Bi at the incident photon energies of 52 MeV and MeV monochromatic photons. The same holds for the 69 MeV for the present calculations are shown in Fig.1. cross sections for the evaporation residues calculated for This is to show the relative probabilities of other evapo- these photon induced reactions. But the theoretical cal- ration processes compared to fission. The results of the culations are performed exactly at photon energies of fission cross sections and the fissility at Eγ = 20, 40, 60, 52 MeV, 69 MeV, 120 MeV and 145 MeV. Neverthe- 80, 100, 120 and 140 MeV using 40000 events for the less, present calculations provide good estimates of the Monte-Carlo calculations are tabulated in Table-II. The photofission cross sections for the actinides. In the pre- increasing trend of the nuclear fissility with the fissility actinide to medium mass region the agreement with the parameter Z2/A for the actinides at intermediate ener- 5

TABLE II: Variation of the calculated photofission reaction cross sections of actinides with the incident photon energy (Eγ ) and the fissility parameter Z2/A. 2 Target Z /A Calculated Eγ [MeV] Eγ [MeV] Eγ [MeV] Eγ [MeV] Eγ [MeV] Eγ [MeV] Eγ [MeV] nuclei Quantity 20 40 60 80 100 120 140 239 T Pu 36.97 σa (mb) 10.19 10.04 16.78 14.24 11.97 10.13 8.68

σf (mb) 10.19 10.04 17.78 14.24 11.97 10.13 8.68 f (%) 99.98 100.0 100.0 100.0 100.0 100.0 99.99 237 T Np 36.49 σa (mb) 10.31 18.06 16.74 14.18 11.90 10.07 8.62

σf (mb) 10.24 18.05 16.73 14.18 11.90 10.07 8.62 f (%) 99.38 99.99 99.99 99.99 100.0 99.99 99.99 233 T U 36.33 σa (mb) 10.58 10.15 16.71 14.11 11.82 9.99 8.54

σf (mb) 10.40 10.14 16.70 14.11 11.81 9.98 8.53 f (%) 98.34 99.95 99.97 99.99 99.97 99.96 99.93 235 T U 36.02 σa (mb) 10.42 18.07 16.69 14.12 11.84 10.01 8.57

σf (mb) 9.98 18.05 16.68 14.11 11.84 10.00 8.56 f (%) 95.83 99.91 99.97 99.96 99.99 99.97 99.95 238 T U 35.56 σa (mb) 10.19 17.94 16.66 14.13 11.87 10.04 8.60

σf (mb) 8.94 17.76 16.64 14.12 11.86 10.04 8.59 f (%) 87.71 98.97 99.87 99.96 99.95 99.96 99.97 232 T Th 34.91 σa (mb) 10.57 18.05 16.59 13.99 11.72 9.89 8.46

σf (mb) 2.20 10.18 13.86 13.27 11.45 9.70 8.27 f (%) 20.87 56.38 83.53 94.79 97.76 98.06 97.83 gies [ 20-140 MeV] are observed. VII. SUMMARY AND CONCLUSION Present∼ calculations provide excellent estimates of the photofission cross sections for the actinides except for 232Th where it somewhat overestimates the fission cross sections. However, the general increasing trend of nu- In summary, the cross sections for the fission and the clear fissility with fissility parameter is retained. For the evaporation residues are calculated for photon induced pre-actinides, 208Pb and 209Bi, the photonuclear reaction reactions at intermediate energies. Monte-Carlo calcu- cross sections show peaks for evaporation residues around lations for the evaporation-fission competition are per- AR = 203 (Pb), 201 (Pb) and 204 (Bi), 203 (Bi) respec- formed assuming 40000 incident photons for each calcu- tively, for the incident photon energies Eγ = 52 MeV, 69 lation. These many number of events for each calculation MeV. The total number of events (that is the number of provide a reasonably good statistics for computationally incident photons) for each run limits the calculations of stable results. Present calculations provide excellent esti- too low fission cross-sections. The fissility for mates of the photofission cross sections for the actinides. [22] and several isotopes [23] was found to be In the pre-actinide to medium mass region the agreement lower than that for , showing that nuclear fis- with measured photofission cross sections is reasonable. sility does not saturate for those nuclei, remaining at a The process of photofission of heavy nuclei is considered value below hundred percent even at high incident pho- in terms of production of fission fragments and is pro- ton energies. The present theoretical study corroborates jected as a viable method for the production of neutron this behaviour. We have included 239Pu in our study, rich nuclei for radioactive ion beam (RIB). though it is not a naturally occuring element but is read- ily produced as spent fuel.

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