84 ZHANG Jing-Shang Vol. 39 of which would emit 5 He rather than 3 He. 5 He i untable and eparated into n + pontaneouly, which can alo be treated a if

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1 Commun. Theor. Phy. (Beijing, China) 39 (003) pp. 83{88 c International Academic Publiher Vol. 39, No. 1, January 15, 003 Theoretical Analyi of Neutron Double-Dierential Cro Section of n+ 11 B at 14. MeV ZHANG Jing-Shang China Intitute of Atomic Energy, P.O. Box 75(41), Beijing 10413, China (Received June 10, 00) Abtract A new reaction model for light nuclei i propoed to analyze the meaured data, epecially for the doubledierential cro ection. In thi paper the calculation with thi model i employed to analyze meaurement of the total outgoing neutron double-dierential cro ection for n+ 11 B reaction at En 14: MeV. The repreentation of the double-dierential cro ection of the econd emitted particle i given in detail. The calculation reult indicate that the recoil eect in light nuclear reaction i eentially important. The reaction channel are dicued in detail. PACS number: Key word: light nucleu reaction, double-dierential cro ection 1 Introduction Element boron ha long been elected a the hielding material in nuclear engineering becaue of light ma and large neutron aborption cro-ection. The reaction mechanim of n+ 11 B i very complex, many reaction channel are opened even below incident neutron energie of 0 MeV. A new approach for decription of neutron induced light nucleu reaction ha been propoed. [1] In thi model the angular momentum conervation in the emiion from a compound nucleu to the dicrete level of the reidual nuclei through pre-euilibrium emiion i involved properly. [] Meanwhile, the energy balance i taken into account by the accurate kinematic. Only one doubledierential meaurement of the total outgoing neutron for n+ 11 B wa meaured by M. Baba et al. in 1985, [3] which ha been analyzed with thi new approach in thi paper. In the cae of E n < 0 MeV, the total cro ection, elatic cattering, non-elatic cro ection, elatic cattering angular ditribution, and the energy-angular pectra of outgoing neutron have been calculated. Baed on the tting meaured data, the optical potential parameter both for neutron and charged particle are obtained. In the cae of neutron induced light nucleu reaction, the pre-euilibrium emiion dominate the reaction mechanim a ame a the other light nuclei, like lithium, [4 5] carbon, [1] a well a oxygen. [] In Sec. the reaction channel of n+ 11 B are lited in detail. The repreentation of the double-dierential cro ection of the outgoing particle and reidual nucleu from dicrete level to dicrete level are preented in Sec. 3, which are ued for the econd emitted particle in the (n n), (n np), (n n), (n nt), (n pn), (n ), (n nd), (n n), and (n dn) reaction channel of n+ 11 B reaction. Becaue of the level width and energy reolution in the meaurement, according to the Heienberg' uncertainty relation, the meaured data are alway in a broadening form. Therefore, in tting procedure the broadening eect mut be taken into account with the Gauian expanion form. Since all of the calculation are carried out in CMS, the tranformation from CMS to LS mut be performed in the tting procedure. The formulation of the broadening expanion and formulation of the motion ytem tranformation can be found in Sec. 5 and 6 of Ref. [1]. The calculated energy-angular ditribution of outgoing neutron at E n 14: MeV are hown in Fig. 1 3 in Sec 4. The calculated reult of the outgoing neutron agree fairly well with the experimental data. The remark are given in Sec. 5. Reaction Channel For n+ 11 B reaction below E n < 0 MeV, the opened reaction channel and the correponding reaction Q-value are lited a follow: B Q 3:370 MeV n B Q 0:000 MeV p + 11 Be Q ;10:74 MeV + 8 Li Q ;6:630 MeV 5 He + 7 Li Q ;9:559 MeV d + >< 10 Be Q ;9:004 MeV n + 11 t + 9 Be Q ;9:558 MeV B n + 10 B Q ;11:453 MeV n + p + 10 Be Q ;11:8 MeV n Li Q ;8:663 MeV n + d + 9 Be Q ;15:815 MeV n + t + Q ;11:3 MeV + >: 4 H Q ;16:413 MeV + t + 5 He Q ;1:05 MeV The reaction channel of (n 3 He) i not open due to the threhold energy over 0 MeV, while the (n 5 He) channel i opened with the threhold energy of MeV. Thi ituation alo happen in the neutron induced nucleu reaction with 9 Be, 10 B, 1 C, 14 N, a well a 16 O, all

2 84 ZHANG Jing-Shang Vol. 39 of which would emit 5 He rather than 3 He. 5 He i untable and eparated into n + pontaneouly, which can alo be treated a if the neutron i emitted from 5 He a the euential particle emiion, only the binding energy of the neutron in 5 He i negative. The reaction channel of (n t) i reached by emitting a triton to the ground tate of 9 Be, while the excited tate of 9 Be decay to one neutron and two particle, o they belong to the (n nt) reaction channel. The reaction mechanim of n+ 11 B leading to the decay into one triton, one neutron and two alpha-particle may proceed through dierent channel. The reaction to 11 B(n nt) reaction channel involved in the calculation are a follow: (a) n + 11 B! n + t + 8 Be, and 8 Be! +, (b) n + 11 B! t + n + 8 Be, and 8 Be! +, (c) n + 11 B! + n + 7 Li, and 7 Li! + t, (d) n + 11 B! n Li, and 7 Li! + t, (e) n + 11 B! H, and 4 H! n + t. In n+ 11 B (E n 0 MeV) reaction, there i no threebody breakup proce, all reaction channel are proceeded by the euential particle emiion. In the cae of n+ 11 B, the reaction ituation from the compound nucleu 1 Be to the dicrete level of the reidual nuclei up to 0 MeV i preented in Table 1. Table 1 The reaction ituation from the compound nucleu 1 B to open k level of the reidual nucleu via k 1 level for variety reaction up to 0 MeV. Channel k1 k (n n) 0 ; g. (n n) 3 ; 6 g. ; 1 (n n) 7 ; 9 g. ; 3 (n n) 30 ; 31 g. ; 4 5 (n n) 3 ; 33 g. ; 8 9 (n n) 34 g. ; 11 (n np) 18 ; 9 g. (n np) 30 ; 31 g. ; 1 (n np) 3 ; 33 g. ; 3 (n np) 34 g. ; 5 (n n) 11 ; 6 g. ; 1 (n n) 7 ; 30 g. ; (n n) 31 ; 34 g. ; 4 (n nt) 19 ; 9 g. (n nt) 30 ; 34 g. ; 1 (n pn) ; 4 g. (n pn) 5 ; 7 g. ; 1 (n pn) 8 g. ; 5 (n ) 10 g. ; 1 (n nd) 31 ; 34 g. (n t) 5 ; 9 g. (n t) 10 g. ; 1 (n n) g. (n n) 3 ; 9 g. ; 1 (n n) 10 g. ; 4 (n dn) 6 g. The letter k 1 and k refer to the order number of the excited level in the correponding reaction channel. The acronym g. tand for the ground tate (a ame a in Table 3 and 4). There are much more dicrete level lited in the table, all of which are needed in the model calculation. The dicrete cheme, including the individual level energy, pin and parity ued for every reaction channel, are taken from the Table of Iotope 8-th. [6] From Table 1 one can ee that the rt ten excited level of 11 B purely belong to the inelatic cattering reaction. 3 The Double-Dierential Cro Section of the Second Emitted Particle In thi ection the repreentation of the doubledierential cro ection of the econd emitted particle from dicrete level to dicrete level i given with the derivation procedure in detail. Two motion ytem are ued in the formulation, and the phyical uantitie are indicated by the upercript c and r for center of ma ytem (CMS) and recoil reidual nucleu ytem (RNS), repectively. Nomenclature B i binding energy of the emitted particle in the nucleu of the i-th particle emiion proce E n incident neutron energy in LS E ki level energy with the level order number k i v V velocitie of emitted particle and it reidual nucleu, repectively E energie of emitted particle and it reidual nucleu, repectively fl c(m 1) fl c() Legendre expanion coecient of the rt emitted particle and it reidual nucleu, repectively, in CMS m M mae of the emitted particle and it reidual nucleu, repectively. The ubcript i 1 appearing in the following euation refer to the rt and the econd emitted particle and their reidual nuclei, repectively. The normalized angular ditribution of the recoil reidual nucleu after the rt particle emiion i expreed in the tandard form a d X l f l c ( )P l (co c ) : (1) l Baed on the momentum conervation, the relation of angle and energy of the emitted econd particle in the CMS and RNS are obtained by and co c co r + co c M p co + () c r (1 + co + ) (3)

3 No. 1 Theoretical Analyi of Neutron Double-Dierential Cro Section of n+ 11 B at 14. MeV 85 where i dened by E c 1 m r : (4) r in E. (4) tand for the energy carried by the econd emitted particle in RNS, r M (E k1 ; B ; E k ) (5) and tand for the angle between the velocitie ~v r and ~V M c 1. The angle between ~v c and V ~ M c 1 i denoted by, which atie the relation co co c co c + in c in c co( c ; c ) : (6) Since r can alo be determined by c and c with the velocity compoition relation ~v r ~v c ; V ~ M c 1, and uing E. (3), in energy cale we nd co c r [co c co c + in c in c co( c ; c )] ; : (7) By uing the velocity compoition ~v c ~v r + ~ V c, the maximum and the minimum energie of the econd emitted particle in CMS can be obtained by c max r (1 + ) (8) c min r (1 ; ) : (9) Thu, the pectra of the emitted econdary particle from level to level are in ring-type form. At low incident energie, the double-dierential cro ection of m in RNS i in the iotropic ditribution with a contant energy, which read d d r dr 1 4 (c ; r (1 + co + )) : (10) Starting from the baic relation of the doubledierential cro ection in the CMS and RNS, d c d dc r d r (11) dr the double-dierential cro ection in CMS can be obtained by the double-dierential cro ection of m in RNS averaged by the angular ditribution of the reidual nucleu after the rt particle emiion in CMS. Hence, Z d d c d c d dc M r d r : (1) dr d c M Subtituting E. (10) into E. (1), the double-di- erential cro ection of m in CMS read Z d 1 c dc 4 r d c d d c ( c ; r (1 + co + )) : (13) By mean of the property of function Z F (x)(g(x))dx X i F (x 0 i) jdg(x)dxj xx0 i (14) where g(x 0 i) 0 for i 1 :::. Carrying out the integration over c, the denominator in E. (14) can be reduced by d(c ; r (1 + co + )) d c r d co d c By uing E. (3) and (7) we have where i dened by in c in c co( c ; c ) c r r in c in M c 1 1 ; co ( c ; c ) : (15) r c (co + ) ; co c co M c 1 ; co c co M c 1 Therefore, euation (15) become into the form c r r Denote r c c r ; 1 + in c in c M1 ; ( ; co c co c M1 ) c r r : (16) 1 + co c co c ; : (17) x co c (18) and c 1 ; co c ; : (19)

4 86 ZHANG Jing-Shang Vol. 39 The denominator i obtained by d(c ; r (1 + co + )) d c c r p r c + co c x ; x : (0) One hould emphaize that beide in c there i another zero point in( ; c ) in the integrated function, o the factor mut be put in the reult. Now, the double-dierential cro ection of m CMS i d dc Z x 1 d dx p 16 r x 1 dx c + x co c ; x (1) where the normalized angular ditribution of the reidual nucleu become d dx 1 X l in (l + 1)f c l ( )P l (x) : () The uare root term in E. (1) demand x ; x co r ; c 0 : (3) Thi i jut the exiting condition of the integration E. (14). Then, the integration limit over x in E. (1) can be obtained by x 1 co c ; (1 ; ) in c (4) and x co c + (1 ; ) in c : (5) Subtituting E. (8) and (9) into E. (1), the condition, which mut hold for any c within c min and c max, implie that the ineuality ;1 1 i held. Changing the integration variable from x to t with the following relation x (1 ; )in c co t + co c (6) we have dx p dt: (7) c + x co c ; x The integration variable change to t, from t 0 to t, and euation (1) i reduced to d 1 dc 16 r Z 0 X l (l + 1)f c l ( ) dt P l (1 ; in c co t + co c : (8) The integration in E. (8) can be carried out analytically by uing the compoite formula of the Legendre polynomial P l (co ) P l (co )P l (co 0 ) + P m l lx (l ; m)! (l + m)! m1 ; co )P m l (co 0 )com( ; 0 (9) where tand for the angle between and 0, which ha the relation co co co 0 + in in 0 co( ; 0 ) : (30) In E. (9), taking co c co, co 0, and t ; 0, and carrying out the integration over t, the integrated reult can be obtained, Z 0 ; dt P l (1 ; in c co t + co c P l ()P l (co c ) : (31) Finally, euation (1) can be integrated analytically. The double-dierential cro ection of m in CMS i obtained, d X dc l l f c l ( c )P l (co c m ) (3) the Legendre expanion coecient of the econd emitted particle m in E. (3) i given by fl c ( c ) (;1)l 4 r fl c (m 1 )P l () (33) where i dened by E. (16). It i eay to check that the energy pectrum f c 0 i normalized and independent of c. In fact, by uing E. (8) and (9) Z c max c min f0( c c )d c c 4 r r (1+) r (1;) 1 : (34) With the ame procedure the expreion of the doubledierential cro ection of the reidual nucleu M can be obtained by replacing m, M with M, m, repectively. 4 Calculated Reult and Dicuion The LUNF code of n+ 11 B reaction i developed for calculating the cro ection and the double-dierential cro ection of outgoing neutron of each reaction channel below 0 MeV. Fig. 1 The energy-angular pectra of 5, 30, 45 for n+ 11 B at En 14: MeV. The calculation of the double-dierential cro ection of the total outgoing neutron from each reaction

5 No. 1 Theoretical Analyi of Neutron Double-Dierential Cro Section of n+ 11 B at 14. MeV 87 channel have been performed. The comparion between the calculated reult and the meaured data are hown in Fig. 1 3 at the incident neutron energy E n 14: MeV for outgoing angle of 5, 30, 45, 60, 75, 85, 100, 10, 135, and 150, repectively. The entire tting agree fairly well with the meaurement. At low energy region, epecially at E n < MeV, the partial pectra mainly come from the econdary outgoing neutron. Becaue too many partial pectra occur in thi gure, they are too dene to be pecied with deignation, o only the rt three excited level of 11 B are labelled by 1 3 a the inelatic cattering pectra. The full line i the calculated total neutron energy-angular pectrum. The percentage of euilibrium and pre-euilibrium emiion procee from dierent J, channel and the correponding aborption cro ection at E n 14: MeV are given in Table. Fig. The energy-angular pectra of 60, 75, and 85 for n+ 11 B at En 14: MeV. Fig. 3 The energy-angular pectra of 100, 10, 135, and 150 for n+ 11 B at En 14: MeV. The meaured data are taken from Ref. [3]. Fig. 4 The partial pectra of the emitted neutron of 90 at En 14: MeV. The total energy-angular pectrum for each angle conit of many partial pectra from dierential reaction channel, which are hown in Fig. 4, a an example, for Q L 90 at E n 14: MeV. Table The euilibrium and pre-euilibrium percentage and the aborption cro ection at En 14: MeV. J e(%) pre-e(%) (+) a (b) (;) a (b) One can ee that the pre-euilibrium emiion are dominant reaction mechanim, epecially at the mall angular momentum region (J 0 6) with large value of aborption cro ection. Meanwhile, the euilibrium emiion mainly contribute to the high angular momentum part, which correpond to the multi-tep colliion procee. Averaged by the angular momentum the percentage of pre-euilibrium emiion i 74.69%, while percentage of euilibrium i only 5.31%. Thu, only the euilibrium theory, like Hauer{Fehbach model, could not work on the light nucleu reaction calculation. 5 Remark A new model for neutron induced light nucleu reaction ha been developed. In thi model, the particle emiion from the compound nucleu to dicrete level in preeuilibrium mechanim i taken into account, becaue the pre-euilibrium emiion mechanim dominate the whole reaction procee with the angular momentum conervation in the whole reaction procee. Beide the nuclear reaction mechanim, on the other hand, the kinematic play an important role in the light nucleu reaction, ince the recoil eect of the particle emiion i very trong. A an example, the calculated cro ection from dicrete level to dicrete level of the econd emitted particle related to the neutron emiion and the energy region of the ring-type pectra after the econd particle emiion are hown in Table 3 and 4, repectively, at E n 14: MeV.

6 88 ZHANG Jing-Shang Vol. 39 Table 3 The calculated cro ection of the inelatic cattering and the partial reaction channel of (n n 0 ), and (n n) to the k level of the reidual nucleu via the k 1 level at En 14: MeV. k1 k mb (min{max) (n n) 1 40: (n n) 56:56 (n n) 3 34:3 (n n) 4 33:71 (n n) 5 9:74 (n n) 6 4:75 (n n) 7 13:8 (n n) 8 10:6 (n n) 9 13:3 (n n) 10 :11 (n n) 0 g. :14 0:0687 ; 0:199 (n n) 1 g. 0:84 0:863 ; 0:5180 (n n) g. 3:48 0:3815 ; 0:684 (n n) 3 1 0:11 0:845 ; 0:435 The acronym (min{max) tand for the energy range of the ring-type pectra in unit of MeV (a ame a in Table 4). In general the econd nucleon emiion (neutron or proton) give the energy range with everal hundred kev of the ring-type pectra, while the econd alpha-particle emiion (ee Table 4) give the large energy range with the order of magnitude of MeV, due to the relative heavier ma. However, all of the ring-type pectra mut be treated a the continuum pectra to give the reaonable hape of the energy-angular pectra. In thi model calculation, we nd that the optical model work well to give the emiion branch from level to level in the cae of light nucleu reaction. From the calculated reult, one can ee that thi model work well in calculation for the double-dierential cro ection of outgoing neutron. Table 4 The calculated cro ection of the inelatic cattering and the partial reaction channel of (n n) (n pn), and (n n) to the k level of the reidual nucleu via the k 1 level at En 14: MeV. k1 k mb (min{max) (n n) 10 g. 13:84 0:0557 ; 0:8403 (n n) 11 g. 1:87 0:086 ; 0:918 (n n) 1 g. 3:35 0:995 ; 1:3663 (n n) 13 g. 6:44 0:355 ; 1:408 (n n) :8 0:1414 ; 0:9844 (n n) 14 g. 5:41 0:5177 ; 1:6815 (n n) 15 g. 6:56 0:5551 ; 1:789 (n n) :09 0:343 ; 1:3338 (n n) 16 g. 7:47 0:7039 ; 1:9037 (n n) :33 0:4793 ; 1:505 (n n) 17 g. 4:03 0:91 ; :165 (n n) :61 0:6836 ; 1:756 (n n) 18 g. 7:6 1:1169 ; :3005 (n n) 19 g. :90 1:376 ; :3968 (n n) 0 g. 1:86 1:3465 ; :4770 (n n) 1 g. 1:39 1:5563 ; :6139 (n n) 1 1 0:9 1:931 ; :69 (n n) g. 0:17 1:6441 ; :6643 (n n) 3 g. 0:11 :13 ; :8613 (n pn) g. 0:74 1:083 ; 1:958 (n n) g. 8:81 0:0007 ; 0:7313 (n n) 3 g. 1:49 0:444 ; 1:8993 (n n) 3 1 0:88 0:1755 ; 1:315 Reference [1] J.S. Zhang, Y.L. Han, and L.G. Cao, Nucl. Sci. Eng. 133 (1999) 18. [] J.S. Zhang, et al., Commun Theor. Phy. (Beijing, China) 35 (001) 579. [3] M. Baba, et al., \Scattering of 14.1 MeV Neutron from B-10, B-11, C, N, O, F and SI," C, 85 Santa, 1 (1985) 3. [4] J.S. Zhang and Y.L. Han, CommunTheor. Phy. (Beijing, China) 36 (001) 437. [5] J.S. Zhang and Y.L. Han., Commun Theor. Phy. (Beijing, China) 37 (001) 465. [6] R.B. Firetone and V.S. Shirley, Table of Iotope 8th., John Wiley & Son (1996).

Theoretical Analysis of Neutron Double-Differential Cross Section of n + 19 F at 14.2 MeV

Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 102 106 c International Academic Publishers Vol. 47, No. 1, January 15, 2007 Theoretical Analysis of Neutron Double-Differential Cross Section of n +