212 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Calculation the Cross Sections of 10B(n,p)10Be Reaction by Using the Reciprocity Theory for the First Excited State S. A. Ebrahiem, K. H. Mahdi ,A. K.Taki Department of Physics ,College of Education Ibn Al-Haytham , University of Baghdad Received in: 10 April 2011, Accepted in: 11February 2012 Abstract In this study light elements 10B , 10Be for 10B(n,p)10Be reaction as well as proton energy from 0.987 MeV to 2.028 MeV with threshold energy (1.04MeV) are used according to the available data of reaction cross sections. The more recent cross sections data of 10Be(p,n)10B reaction is reproduced in fin steps in the specified energy range , as well as cross section (p,n) values were derived from the published data of (n,p) as a function of energy in the same fine energy steps by using the reciprocity theory of principle inverse reaction . This calculation involves only the first excited state of 10B , 10Be in the reactions 10Be(p,n)10B and 10B(n,p)10Be. Key word: Cross Sections , Reverse Reaction , Stopping Power, Neutron Yield Introduction The interaction of particles with matter is described in terms of quantities known as cross sections which is defined in the following way [1]. Consider a thin target of area (a) and thickness (X) containing(N) atoms per unit volume, placed in a uniform mono-directional beam of incident particles (neutrons for example of intensity Io, which strikes the entire target normal to its surface as shown in fig.(1). It is found that the rate at which interactions occur within the target is proportional to the beam intensity and to the atom density, area and thickness of the target Summarizing this experimental result by an equation, we define the interaction rate (in the entire target) = σ I N a X ------ (1) Where the proportionality constant σ is known as the cross section , Thus σ = interaction rate / INaX ------ (2) As NaX is equal to the total number of atoms in the target, it follow s that σ is the interaction rate per atom in the target per unit intensity of the incident beam [2] . Reciprocity Theory If the cross-sections of the reaction A(p,n)B are measured as a functions of Tp (Tp = Kinetic energy of proton ) the cross –sections of the inverse reaction B(n,p)A can be calculated as a function of Tn (Tn = Kinetic energy of neutron) using the reciprocity theorem [3] which states that : σ (p,n) σ (n,p) ─────── = ────── ---------(3) g p,n D p2 g n,p D n2 213 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Where σ(p,n) and σ(n,p) represent cross- sections of (p,n) and (n,p) reactions respectively , g is a statistical factor and D is the de–Broglie wave length divided by 2π and is given by ħ D = ───── -----------(4) Mv Where ħ is Dirac constant (h /2π ) , h is plank constant , M and v are mass and velocity of p or n particle . From eq.(4),we have ħ² D 2 = ───── ---------(5) 2 MT The statistical g-factors are givens by [3] 2Jc + 1 g p,n = ────────────── ---------(6) (2IA + 1)(2Ip + 1) And 2Jc + 1 g n,p = ───────────── ----------(7) (2IB +1)(2In +1) The conservation low of the momentum implique that : IA + Ip = Jc = IB + In ----------(8) And πA . πp (-1)ℓp = πc = πB . πn (-1) ℓn ----------(9) Jc and πc are total angular momentum and parity of the compound nucleus . IA and πA are total angular momentum and parity of nucleus A. IB and πB are total angular momentum and parity of nucleus B. Ip and πp are total angular momentum and parity of proton. In and πn are total angular momentum and parity of neutron . πp = πn = +1 ----------(10) 214 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Ip = sp + ℓp ----------(11) Where Ip is the total angular momentum of proton Sp is spin of proton = 1\2 ℓp is the orbital angular momentum of proton And In = sn + ℓn ----------(12) Where In is the total angular momentum of the neutron sn is spin of neutron = 1/2 ℓn is the orbital angular momentum of neutron From eq.(1-8),we have : │ Jc - IA│≤ lp ≤ │ Jc + IA│----------(13) And │ Jc - IB │≤ ln ≤ │ Jc + IB │----------(14) The reactions A(p,n)B and B(n,p)A can be represented with the compound nucleus c as in the following schematic diagram. It is clear that there are some important and useful relations between the kinetic energies of the neutron and proton[4] . One can calculate the separation energies of proton (Sp) and neutron (Sn) using the following relations: Sp and Sn are separation energies of p and n from c as shown Fig(2)[4]. Then MA E = Sp + ─────── Tp ----------(15a) MA+ Mp MB E = Sn + ─────── Tn ----------(15b) MB+ Mn With Sp = 931.5 [ MA + Mp - Mc ] -------(16) Sn = 931.5 [ MB + Mn - Mc ] ------(17) Combining (15a) , (15b) , (16) and (17) and as the Q- value of the reaction A(p, n)B is given by : Q = 931.5 [ MA + Mp − MB – Mn ] ----------(18) Then MB MA Q = ─────── Tn − ─────── Tp ----------(19) MB+ Mn MA + Mp Or : MB + Mn MA Tn = ───────── ──────── Tp + Q --------(20) MB MA + Mp The threshold energy Et h is given by : MA + Mp Et h = – Q ──────── ------(21a) MA MA Q = – ──────── Et h ------(21b) 215 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 MA + Mp Then MB + Mn MA Tn = ──────── * ──────── (Tp - Et h ) --------(22) MB MA + Mp eq . (3) can be written as follows : 8 g n,p Mp Tp σ (n,p) = ──────────── σ (p,n) --------(23) g p,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 cross section of (p,n) reactions for the elements 10B , 10Be for 10B(n,p)10Be reaction available in the literature[5] , have been taken and re-plotted for a defined energy level as shown in Fig.(3).These plots were analyzed using the Matlab computer program to obtain the cross sections for the selected energies The atomic mass of elements and isotopes mentioned in this study have been taken from the latest nuclear wallet cards released by the National Nuclear Data Center(NNDC)[6] and the energy level, parity and spin scheme of isotopes from [7]. By using the reciprocity theory we derive the mathematical formula for 10B(n,p)10Be reaction for first excited state : np n p pn T T ,, 664.1 ss = The evaluated cross sections as a function of neutron energy from (0.0320) MeV to (0.9881) MeV of present work are listed in tables (1). These data plotted in Fig.(4) we get mathematical equation representing the cross sections distribution in the indicated range of energy and percentage error ( ± 0.3164) for every data : y = 3.3e+4*x^10 - 1.7e+5*x^9 + 3.9e+5*x^8 -4.9e+5*x^7 + 3.8e+5*x^6-1.9e+5*x^5+5.9e+4*x^4 - 1.1e+4*x^3 + 1.1e+3*x^2 - 43*x +0.52 We get the maximum cross section to produced the 10Be by neutron energy (0.357MeV) and (0.9881MeV) are (1.1601 mbarn) and (1.3061mbarn) respectively and 10Be very important in technology field. In Fig.(4) we observed that the high probability( high cross sections ) to produced 10Be in intermediate and fast neutrons. References 1. Alex, D.R. Green, E.S., (1955) , Nuclear Physics, Mcgraw-Hill Book Company , Inc.. 2. Huizenga, J.R. and Igo, G. (1962) , Nuclear Physics. 29 :462-473. 3. Macklin, R.L. and Gibbons, J.H. (1968) ,phys. Rev. 165:1147. 216 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 4. Ebrahiem, S.A. (2007) , Cross Sections of (n,α) reaction from Cross Sections of(α,n) reaction using the reciprocity theory for the ground state" , P (34-40) . 5. Eremin, N.V. Zeinalov SH.S. & et al. ,(1987) , Investigation of Resonance Structures in 10B(P,N) Reaction at Low Proton Energies , C,87JURMAL, P (300). 6. Audi, G. and Wapstra, A . (1995) , Nuclear physics , A 595 (4):409. 7. Firestons R.B and Shirley, V.S , (1999 ), Table of isotopes eighth edition , Newyork . Table (1):The cross sections of 10B(n,p)10Be Re action as a function of ne utron e ne rgy pre s e nt work ne utron - e ne rgy (Me V) X- s e ctions (mbarn) P.Work ne utron - e ne rgy (Me V) X- s e ctions (mbarn) P.Work ne utron - e ne rgy (Me V) X- s e ctions (mbarn) P.Work 0.0320 0.0517 0.2280 0.7492 0.5000 0.5801 0.0530 0.0164 0.2350 0.9339 0.5100 0.5274 0.0670 0.0387 0.2440 0.9996 0.5150 0.4834 0.0760 0.0691 0.2640 1.2036 0.5280 0.4041 0.0780 0.6758 0.2890 0.9068 0.5520 0.3558 0.0840 0.4118 0.2920 0.5685 0.5770 0.3379 0.1040 1.2568 0.3120 0.6081 0.5990 0.3292 0.1280 0.8024 0.3330 0.9112 0.6200 0.3243 0.1310 0.5743 0.3570 1.1601 0.6481 0.3591 0.1400 0.6356 0.3790 0.8667 0.7061 0.5129 0.1590 0.6400 0.3870 0.7787 0.7411 0.4771 0.1690 0.8377 0.4080 0.7260 0.7931 0.8604 0.1810 0.9605 0.4350 0.7081 0.8951 0.8938 0.1910 0.9208 0.4680 0.7120 0.9441 0.9329 0.2170 0.7318 0.4800 0.4969 0.9881 1.3061 217 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 n nB B T MM M + Io I x Fig. (1): A schematic diagram illustrating the definition of total cross section in terms of the reduction of intensity[1]. C E B + n A + p Sn Sp q p pA A T MM M + 218 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 C Fig. (2):Sce matic diagram of the re actions [4] 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 0.05 0.1 0.15 0.2 0.25 0.3 Proton Energy (MeV) C ro ss S ec ti on s( m ba rn ) N.V.EREMIN, SH.S.ZEINALOV, A.P.KABACHENKO shape-preserving(present work) Fig.( 3): Cros s s e ctions of 10Be(p,n)10B Re action [5] Fig.(4): Cros s s e ctions of 10B(n,p)10Be Re action P.Work 219 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 بأستعمال نظرية التعاكس 10B(n,p)10Beحساب المقاطع العرضية لتفاعل للمستو المتهيج االول سميرة احمد ابراهيم ¡ علي كاظم تقي ¡خالد هادي معة بغدادجا ¡ابن الهيثم –ه كلية التربي ¡قسم الفيزياء 2012كانون الثاني 11قبل البحث في 2011نيسان 10لبحث في Çاستلم الخالصة للبيانـات 10B(n,p)10Be) للتفاعـل 10B , 10Beفـي هـذه الدراسـة اعيـد حسـاب المقـاطع العرضـية للنـوى الخفيفـة ( وبطاقـة عتبــه مقــدارها MeV (2.028) الـى MeV (0.987)المتـوفرة فــي االدبيـات العالميــة وللمـدى الطــاقي مـن (1.04)MeV دالـة للمقـاطع العرضــية وبخطـوات طاقيــة معينـة . بأســتعمال نظريـة التعــاكس اشـتقت معادلــة لحسـاب المقــاطع 10Be(p,n)10Bوللمستو المتهيج االول وذلك باالعتماد علـى المقـاطع العرضـية لتفاعـل 10B(n,p)10Beالعرضية لتفاعل . تــم جدولــة ورســـم Matlab-6.5)ول علــى معادلـــة للرســم البيــاني مــن خــالل اســـتخدام بــرامج الحاســوب (ومــن ثــم الحصــ . 10Beالنتائج فضال عن مناقشة النتائج وتحديد نوع النيوترون ألنتاج الحصيلة النيوترونية ،قدرة االيقاف ،التفاعل المعاكس ¡المقاطع العرضية: الكلمات المفتاحية