@1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Determining of Cross Sections for 16O(n,α)13C reaction from Cross Sections of 13C(α,n)16O for the ground state Sameera A. Ebrahiem Maher N. Sarsam Hermez M. Youhana Nebras T. Abd-Al-Hameed Dept. of Physics/College of Education For Pure Science(Ibn Al-Haitham) / University of Baghdad Received in : 11March 2008 , Accepted in: 16July 2008 Abstract In this study, light elements for 13C , 16O for (α,n) and (n,α) reactions as well as α-particle energy from 2.7 MeV to 3.08 MeV are used as far as the data of reaction cross sections are available. The more recent cross sections data of (α,n) and (n,α) reactions are reproduced in fine steps 0.02 MeV for 16O (n,α) 13C in the specified energy range, as well as cross section (α,n) values were derived from the published data of (n,α) as a function of α-energy in the same fine energy steps by using the principle inverse reactions. This calculation involves only the ground state of 13C , 16O in the reactions 13C (α,n) 16O and 16O (n,α) 13C. Keywords : Cross sections, Inverse reaction ,Statistical factor , Dirac constant 109 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Introduction Since interactions in a reaction take place with individual target nuclei independently of each other, it is useful to refer the probability of a nuclear reaction to one target nucleus. Assume that in a given experiment a thin slab of target material is struck by a mono energetic beam consisting of I particles per unit time distributed uniformly over an area A, as shown in this schematic [1]. Schematic of Cross-section [1]. If the nuclear reaction produces N light product particles per unit time. It can be pretended that with each target nucleus there is an associated area σ (perpendicular to the incident beam) such that if the center of a bombarding particle strikes inside of σ, there is a hit and a reaction is produced, and if the center of the bombarding particle misses σ , no reaction is produced. The quantity σ is called cross-section and gives a measure of the reaction probability per target nucleus. It is a fictitious area, which need not be related to the cross sectional area ( π R2) of the struck nucleus. The reaction probability can also be described by the ratio N / I , but this quantity depends on the target density as well as its thickness Δx, whereas σ is associated with an individual target nucleus .The probability that any one bombarding particle has a hit is equal to N/I and is also equal to the projected total cross- section of all target nuclei lying within the area A,as seen along the beam direction ,divided by A . If there are n target nuclei per unit volume in the target material, n A Δx, such nuclei are within reach of any bombarding particle in the beam [1]. Each target nucleus has an associated cross-section σ so that This relation can be used in two ways: It can serve as a definition of cross-section, by writing The unit of cross-section is cm2 or barn (1b = 10-24cm2). Theory If the cross-sections of the reaction A(α,n)B are measured as a functions of Tα (Kinetic energy of α-particle) the cross –sections of the inverse reaction B(n,α)A can be calculated as a function of Tn (Kinetic energy of neutron) using the reciprocity theorem [2] which states that : σ (α,n) σ (n,α) ──────── = ──────— ---------(2) g α,n  α2 g n,α  n2 Where σ(α,n) and σ(n,α) represent cross- sections of (α,n) and (n,α) reactions respectively , g is a statistical factor and  is the de–Broglie wave length divided by 2π . ħ Number of light product particles per unit time, per unit incident flux, and per target nucleus. )1.(.......... A xnA I N σ∆ = ( )( ) = ∆ = xnAAI N / σ 110 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013  = ───── -------(a) Mv Where ħ is Dirac constant (h /2π ) , h is plank constant , M and v are mass and velocity of α or n particle . From eq.(a) ,we have ħ² λ² = ───── ---------(b) 2 MT The statistical g-factors are givens by [2] 2Jc + 1 2Jc + 1 g α,n = ────────── , g n,α = ───────── ------(c) (2IA + 1)(2Iα + 1) (2IB +1)(2In +1) The reactions A(α,n)B and B(n, α)A can be represented with the compound nucleus. It is clear that there are some important and useful relations between the kinetic energies of the neutron and alpha particle . One can calculate the separation energies of α-particle (Sα) and neutron (Sn) using the following relations: Sα = 931.5 [ MA + Mα - Mc ] ----------(3) Sn = 931.5 [ MB + Mn - Mc ] ----------(4) Sα and Sn are separation energies of α and n from compound nucleus. Then MA E = Sα + ─────── Tα ----------(5a) MA+ Mα MB E = Sn + ─────── Tn ----------(5b) MB+ Mn Where E is reaction energy Combining (3) , (4) ,(5a) , (5b) and as the Q- value of the reaction A( α , n )B is given by : Q = 931.5 [ MA + Mα − MB – Mn ] ----------(6) Then MB MA Q = ─────── Tn − ─────── Tα ---------(7) MB+ Mn MA + Mα Or : MB + Mn MA Tn = ──────── [ ─────── Tα + Q ] --------(8) MB MA + Mα The threshold energy Eth is given by MA + Mα Eth = – Q ──────── --------(9a) MA Or MA Q = – ───────── Eth --------(9b) MA + Mα 111 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Then MB + Mn MA Tn = ──────── * ──────── (Tα - Eth ) --------(10) MB MA + Mα Thus eq . (2) can be written as follows [2] : g n,α Mα Tα σ (n,α) = ──────────── σ (α,n) --------(11) g α,n Mn Tn It is clear form this equation that the cross sections of reverse reaction are related by a variable parameters which can be calculated if the nuclear characteristics of the reactions are known. Results and Discussion The 16O(n, α )13C cross sections as plotted in fig.(2) and as shown in table (1) was calculated from cross sections for 13C(α,n)16O as shown in fig.(1) by using the principle of the inverse reaction [3] by using equ.(11) these calculated on the ground state with parity of 13C , 16O , 1/2- , 0 + respectively[4] and with threshold energy 2.354(MeV) .The cross sections for 16O(n, α )13C were measured from 2.7 ≤ En ≥ 3.18 (MeV) [5]. The atomic mass of these elements is used in the present work [6] . It is clear that our result , especially , when the energy between 2.8 (MeV) to 3.08 (MeV) is close to with author [5] as shown in fig.(3) . References 1- Meyerhof ,W. E. (1967) Elements Of Nuclear Physics, Mc Graw – Hill Book co. 2- Macklin , R.L. and Gibbons J.H.( 1968) phys. Rev. 165 ,1147. 3- Drotleff , H. W. ; Denker A. ; Knee H. ; Soine M.; Wolf, G. and Hammer, J.W. , A. (1993) J. Astrophysical Journal ,414 , 735 . 4- Firestons , R. B. and Shirley, V. S.( 1999 ) Table of Isotopes eighth edition, John Wiley and Sons , New York. 5- Hale ,G. M. and Young, P. G.( 2001) ENDF / B – VI . 6- Tuli , J. K.( 2005) Nuclear Wallet Cards for Radioactive Nuclides ,National Nuclear Data Center NNDC . 112 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Table (1) : The Cross Sections of the Reaction 16O(n,α)13C as a Function Of Neutron Energy with Thresholds Energy of 2.354 MeV Neutron Energy(MeV) Cross section(barn) P.Work Cross section(barn) by ENDF[5] 2.70 1.81e-10 2.09e-10 2.72 5.29e-10 5.20e-10 2.74 1.33e-9 1.46e-9 2.76 2.78e-9 3.31e-9 2.78 6.27e-9 7.37e-9 2.80 1.44e-8 1.48e-8 2.82 3.04e-8 3.00e-8 2.84 5.96e-8 6.03e-8 2.86 1.09e-7 1.19e-7 2.88 2.01e-7 2.05e-7 2.90 3.72e-7 3.65e-7 2.92 6.66e-7 6.72e-7 2.94 1.14e-6 1.21e-6 2.96 1.85e-6 2.10e-6 2.98 2.99e-6 3.47e-6 3.00 4.90e-6 5.05e-6 3.02 7.95e-6 7.81e-6 3.04 1.26e-5 1.40e-5 3.06 1.96e-5 2.19e-5 3.08 2.98e-5 3.36e-5 3.10 5.20e-5 4.47e-5 3.12 7.64e-5 6.61e-5 3.14 11.1e-5 9.62e-5 3.16 1.55e-4 1.38e-4 3.18 2.16e-4 1.96e-4 Fig.(1): The cross sections of the reaction 13C(α,n )16O 113 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Fig.(2): The cross sections of the reaction 16O(n,α)13C as given by P.Work Fig.(4):Cross sections of O16(n,α)C13 reaction Data-1 represent of p.workData-2 represent of ENDF-library [5] 114 | Physics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 من المقاطع العرضیة لتفاعل 16O(n,α)13C حساب المقاطع العرضیة لتفاعل 13C(α,n)16O االرضي ىباستعمال نظریة التعاكس في المستو سمیره احمد ابراھیم ماھر ناصر سرسم ھرمز موشي یوحنا تحسین عبد الحمیدنبراس جامعة بغداد ) /ابن الھیثمللعلوم الصرفة ( كلیة التربیھ/ الفیزیاءعلوم قسم 2008تموز 16قبل البحث في ، 2008اذار 11استلم البحث في الخالصة في ھذه الدراسة ، اعی�دت حس�ابات المق�اطع العرض�یة للتف�اعالت النووی�ة (الف�ا ، نی�وترون) و(نی�وترون ، الف�ا) للن�وى ) لجس�یمات الف�ا والنیوترون�ات . (10MeVفرة و للمدى الطاقي من طاقة العتبة ال�ى اللبیانات المتو ) , 13C ) 16Oالخفیفة 0.02بخطوات طاقیة ( استخدثت ر حداثة للتفاعالت (الفا ،نیوترون) و(نیوترون ،الفا) قد ان بیانات المقاطع العرضیة االكث MeV 16 ) لتفاعلO(n,α)13C لتفاع�ل ة ، وكذلك حسبت المقاطع العرض�یة لتفاع�ل (الف�ا،نیوترون) م�ن المق�اطع العرض�ی مبدأ التفاعل المعاكس.عمال باست نفسھا یةالخطوات الطاقب(نیوترون،الفا) المنشورة في االدبیات دالة لطاقة الفا و , 13C(α,n)16O ف��ي التف�اعلین ) , 13C ) 16Oتم�ت ھ��ذه الحس�ابات فق��ط المس�توى االرض��ي للن��وى 16O(n,α)13C. ثابت دیراك ,العامل االحصائي ، التفاعل المعاكس ، المقاطع العرضیھ الكلمات المفتاحیة : 115 | Physics